Technick.net

Guides

Sun, 01 Mar 1998 00:00:00 +0000

Search

Sun, 01 Mar 1998 00:00:00 +0000

Tools

Sun, 01 Mar 1998 00:00:00 +0000

Legal

Mon, 01 Jan 0001 00:00:00 +0000

Disclaimer

Please be aware that any information you find on this website may be inaccurate, misleading or out of date.

Some information on Technick.net may create a risk for readers who choose to apply or use it in their own activities, or who promote it for use by third parties. None of the authors or contributors connected with Technick.net can be held responsible, in any way whatsoever, for your use of the information contained in or linked from these pages.

JSON Editor

Tue, 09 May 2017 00:00:00 +0000

JSON Editor

Load JSON Schema

Validation

This will update whenever the form changes to show validation errors if there are any.

JSON Output

You can also make changes directly to the JSON here.

Load

AWG to Metric Converter

Sun, 01 Mar 1998 00:00:00 +0000

Reference and sources:

Wire Parameter Calculator by Brad Suppanz
Temperature Coefficients of Resistivity
Wikipedia: Resistivity
Wikipedia: American wire gauge
American Wire Gauge (AWG) & Metric Wire Gauge Wire Sizes
Van den Hul: AWG to Metric Conversion Chart

Electronics

Sun, 01 Mar 1998 00:00:00 +0000

Hardware

Sun, 01 Mar 1998 00:00:00 +0000

Inductance Calculator

Sun, 01 Mar 1998 00:00:00 +0000

Select the box with the geometry that you would like to calculate.

Round-Wire geometries (loops of wire with various shapes)
Circular Loop	Rectangular Loop	Square Loop	Equilateral Triangle Loop	Isosceles Triangle Loop

Transmission Line geometries (cross sections) the values calculated are the inductance per unit length
Twin Lead	Round Wire over a Ground Plane	Vertically Spaced Traces	Wide Trace over a Ground Plane	Coplanar Traces

Morse Code Converter

Sun, 01 Mar 1998 00:00:00 +0000

Alphanumeric	Morse Code


Spelling

PCB Impedance and Capacitance Calculator

Sun, 01 Mar 1998 00:00:00 +0000

Select the box with the geometry that you would like to calculate.
Microstrip	Embedded Microstrip	Stripline	Dual Stripline
Asymmetric Stripline	Differential Impedance of Microstrip	Differential Impedance of Stripline

THESE FORMULAS ARE APPROXIMATIONS!
They should not be used when a high degree of accuracy is required.
The approximate formulas are provided along with restrictions on the relative size of the various parameters.

All the units are in SI (Iternational System):

[m] = meter
[s] = second
[H] = Henry
[F] = Farad

Software

Sun, 01 Mar 1998 00:00:00 +0000

Theory

Sun, 01 Mar 1998 00:00:00 +0000

Software Metrics

Tue, 23 Dec 2014 00:00:00 +0000

Introduction

Quantitative measurements are essential in all sciences, including computer science, as we can’t optimise what we can’t measure.

In this context the term “metric” and “measure” will be often used as synonymous, even if by definition metrics are functions, while measurements are the numbers obtained by the application of metrics.

Software metrics are used to record events taking place in the execution of a software program in order to obtain objective, reproducible and quantifiable measurements. These measures provide an audit trail that have numerous valuable applications, including understanding the activity, quality assurance testing, software debugging, software performance optimisation, cost estimation, budget planning, etc.

Software Naming

Wed, 19 Nov 2014 00:00:00 +0000

Introduction

Choosing a good name for a software project is a critical task, as it will be referenced multiple times in building pipelines, configurations, logs, documentation, deployment packages, software dependencies, emails, discussions, and early morning support calls.

To avoid common issues related to software names it is important to follow a convention that minimise incompatibilities between technologies, platforms and usage case scenarios.

Here is proposed a convention that works well in the aforementioned multiple situations.

Simple API JSON Response Format

Fri, 17 Oct 2014 00:00:00 +0000

Introduction

The JavaScript Object Notation (JSON) format is a widely adopted standard to deliver HTTP RESTful API responses. This mostly because of the following properties:

It is a standard open lightweight data-interchange format;
Along with XML is the main format for data interchange used on the modern web;
It is easy for humans to read and write;
It is easy for machines to parse and generate;
Supports all the basic data types (numbers, strings, boolean, arrays and hashes);
It is developer-friendly, as it can be generated and parsed from almost any programming languages;
Popular databases (e.g. MongoDB, MySQL, PostgreSQL) can store the JSON format natively.

Following a shared convention promotes reuse and helps increasing the productivity by leaving more time to focus on the actual development task. However, while there are many common patterns for structuring JSON-based HTTP/REST API responses, there is no consistency in things like naming or types of responses.

Software Configuration

Sat, 13 Sep 2014 00:00:00 +0000

Introduction

In this context for “configuration” we consider the set of initial parameter settings that are read at run-time by computer programs.

Traditionally the configuration data has been provided mostly in plain-text format but in recent years structured standards have gained more traction. This mostly because in modern distributed environments the configurations are often provided via a Configuration Management System (CMS), transmitted via network and consumed by multiple applications, or multiple instances of the same application. Additionally, some of the configuration parameters are often automatically determined by the CMS instead of being manually entered by humans.

Software Logging Format

Mon, 11 Aug 2014 00:00:00 +0000

Introduction

Logs are used to record events taking place in the execution of a software program in order to provide an audit trail that can be used to understand the activity and diagnose problems.

In modern distributed environments the logs produced by multiple applications, or multiple instances of the same application, are often collected and processed in a single place. A typical stack involves the use of Elasticsearch, Logstash and Kibana (ELK). In this environments the first level consumer of the log messages is not anymore a human but a machine. This mostly because humans cannot process information in the massive flows that are created by concurrent and distributed systems. This poses new challenges when using traditional plain-text log formats designed for human consumption:

Build Software with Docker

Mon, 07 Jul 2014 00:00:00 +0000

Introduction

Two of the major requirements for a Continuous Integration/Delivery system (CI/CD) are:

Isolation: the ability to run a well isolated environment from other builds executed on the same infrastructure;
Reproducibility : the ability to reproduce the build environment from a pre-defined setup.

These requirements also apply to local development environments, ensuring developers the ability of consistently reproduce building and testing processes.

A great way to bootstrap reproducible and isolated environments is using Docker containers.

Software Automation

Thu, 05 Jun 2014 00:00:00 +0000

Introduction

The software development and deployment cycle usually involves numerous and tedious steps other than just “coding”. These steps often includes: download dependencies, build with different options, run various static analysis tools to check if the project meets the minimum predefined set of quality metrics, execute unit tests and check code coverage, execute functional and other types of tests, generate documentation, packaging, etc.

The ability to run all the development steps easily and frequently in a consistent and automatic way is important to minimize human errors, improve the software quality and speeds up the whole development cycle.

Software Structure

Sat, 03 May 2014 00:00:00 +0000

This document illustrates a portable directory structure for software projects to promote reusability and simplify the development cycle.

The software development and deployment cycle usually involves numerous and tedious steps other than just "coding". Some steps can be automated by Integration or Continuous Delivery systems (CI/CD), others needs to be performed frequently on the development machine before committing the code. Common development steps often includes: download dependencies, run various static analysis tools to check if the project meets the minimum predefined set of quality metrics, execute unit tests and check code coverage, execute functional and integration tests, check or generate documentation, etc.

Software Versioning

Wed, 02 Apr 2014 00:00:00 +0000

This document illustrates a strategy used for versioning software projects that is portable across different languages and technologies.

Having a universal way of versioning software development projects is a good thing to help us keep track of what's going on.

From Wikipedia (https://en.wikipedia.org/wiki/Software_versioning): "Software versioning is the process of assigning either unique version names or unique version numbers to unique states of computer software. [...] Modern computer software is often tracked using two different software versioning schemes: an internal version number that may be incremented many times in a single day, such as a revision control number, and a released version that typically changes far less often, such as semantic versioning or a project code name."

Formatting bytes in a spreadsheet cell

Sat, 01 Mar 2014 00:00:00 +0000

The following is a cell function that can be used with a spreadsheet application (e.g. MS Excel or LibreOffice Calc) to format a cell number using byte multiples.

The example below uses the content of the cell A1 as input.

For an explanation of IEC Prefixes for binary multiples please consult the guide: Multiples and Submultiples Prefixes Tables.

=IF((A1>=POWER(2,80)), CONCATENATE(TEXT((A1/POWER(2,80)),"##0.00"), " YiB"), IF((A1>=POWER(2,70)), CONCATENATE(TEXT((A1/POWER(2,70)),"##0.00"), " ZiB"), IF((A1>=POWER(2,60)), CONCATENATE(TEXT((A1/POWER(2,60)),"##0.00"), " EiB"), IF((A1>=POWER(2,50)), CONCATENATE(TEXT((A1/POWER(2,50)),"##0.00"), " PiB"), IF((A1>=POWER(2,40)), CONCATENATE(TEXT((A1/POWER(2,40)),"##0.00"), " TiB"), IF((A1>=POWER(2,30)), CONCATENATE(TEXT((A1/POWER(2,30)),"##0.00"), " GiB"), IF((A1>=POWER(2,20)), CONCATENATE(TEXT((A1/POWER(2,20)),"##0.00"), " MiB"), IF((A1>=POWER(2,10)), CONCATENATE(TEXT((A1/POWER(2,10)),"##0.00"), " KiB"), CONCATENATE(TEXT((A1),"##0.00"), " B") ) ) ) ) ) ) ) )

Ammonium Persulfate: Etchant for copper-plated PC Boards

Sun, 01 Mar 1998 00:00:00 +0000

AMMONIUM PERSULFATE – (NH₄)₂S₂O₈
(ammonium peroxydisulfate; peroxydisulfuric acid, diammonium salt)
Description:

Colorless to light straw crystals or powder. Odorless.
Strong oxidizer.
Hazard classes: 5.1 oxidizer; 6.3 acute health hazard.
CAS No.: 7727-54-0
UN-NA No.: 1444

NFPA 704 Placarding Information:

Health – 2
Flammability – 0
Reactivity – 1
Special Notice – OX

Uses:

Oxidizer and bleaching agent.
Used in photography; electroplating; preserving food; depolorizing batteries; and washing infected yeast.
Etchant for printed circuit boards and copper.
In the manufacture of other persulfates, deodorizing and bleaching oils, and aniline dyes.

Reactivity and Fire Risk:

ASCII Table

Sun, 01 Mar 1998 00:00:00 +0000

ASCII = American Standard Code for Information Interchange
Dec Hex Char Description Dec Hex Char Dec Hex Char Dec Hex Char
0 0 NUL null 32 20 64 40 @ 96 60 `
1 1 SOH start of heading 33 21 ! 65 41 A 97 61 a
2 2 STX start of text 34 22 " 66 42 B 98 62 b
3 3 ETX end of text 35 23 # 67 43 C 99 63 c
4 4 EOT end of transmission 36 24 $ 68 44 D 100 64 d
5 5 ENQ enquiry 37 25 % 69 45 E 101 65 e
6 6 ACK acknowledge 38 26 & 70 46 F 102 66 f
7 7 BEL bell 39 27 ' 71 47 G 103 67 g
8 8 BS backspace 40 28 ( 72 48 H 104 68 h
9 9 TAB horizontal tab 41 29 ) 73 49 I 105 69 i
10 A LF NL line feed, new line 42 2A * 74 4A J 106 6A j
11 B VT vertical tab 43 2B + 75 4B K 107 6B k
12 C FF NP form feed, new page 44 2C , 76 4C L 108 6C l
13 D CR carriage return 45 2D - 77 4D M 109 6D m
14 E SO shift out 46 2E . 78 4E N 110 6E n
15 F SI shift in 47 2F / 79 4F O 111 6F o
16 10 DLE data link escape 48 30 0 80 50 P 112 70 p
17 11 DC1 device control 1 49 31 1 81 51 Q 113 71 q
18 12 DC2 device control 2 50 32 2 82 52 R 114 72 r
19 13 DC3 device control 3 51 33 3 83 53 S 115 73 s
20 14 DC4 device control 4 52 34 4 84 54 T 116 74 t
21 15 NAK negative acknowledge 53 35 5 85 55 U 117 75 u
22 16 SYN synchronous idle 54 36 6 86 56 V 118 76 v
23 17 ETB end of transmission block 55 37 7 87 57 W 119 77 w
24 18 CAN cancel 56 38 8 88 58 X 120 78 x
25 19 EM end of medium 57 39 9 89 59 Y 121 79 y
26 1A SUB substitute 58 3A : 90 5A Z 122 7A z
27 1B ESC escape 59 3B ; 91 5B [ 123 7B {
28 1C FS file separator 60 3C < 92 5C \ 124 7C |
29 1D GS group separator 61 3D = 93 5D ] 125 7D }
30 1E RS record separator 62 3E > 94 5E ^ 126 7E ~
31 1F US unit separator 63 3F ? 95 5F _ 127 7F DEL

AWG to Metric Conversion Chart

Sun, 01 Mar 1998 00:00:00 +0000

Reference and Sources:

Wikipedia: American wire gauge

American Wire Gauge (AWG) & Metric Wire Gauge Wire Sizes

Van den Hul: AWG to Metric Conversion Chart

This table gives closest equivalent size cross references between metric and American wire sizes. In Europe, wire sizes are expressed in cross sectional area in mm² and also as the number of strands of wires of a diameter expressed in mm. For example 7/0.2 means 7 strands of wire each 0.2mm diameter. This example has a cross sectional area of 0.22mm². In America, the commonest system is the AWG numbering scheme, where the numbers are applied not only to individual strands but also to equivalent size bunches of smaller strands. For example 24AWG could be made of 1 strands of 24AWG wire (1/24) or 7 strands of 32 AWG wire (7/32).

See also: AWG to metric Parameter Calculator.

Basic Soldering Guide

Sun, 01 Mar 1998 00:00:00 +0000

Types of Iron | How to Solder | How to Desolder | Summary | Trouble Shooting Guide | First Aid
Photo Gallery | Contacting the Author | Copyright & Disclaimer |

This written guide will help beginners and novices to obtain effective results when soldering electronic components. If you have little or no experience of using a soldering iron, then we recommend that you practice your soldering technique on some fresh surplus components and clean stripboard (protoboard), before experimenting with a proper constructional project. This will help you to avoid the risk of disappointment when you start to assemble your first prototypes. If you've never soldered before, then read on!

Types of Iron

Topics in this section include: Voltage, Wattage, Temperature Control, Soldering Stations, Anti-Static Protection, Bits (Tips), Spare Parts, and Gas-Powered Irons.
The most fundamental skill needed to assemble any electronic project is that of soldering. It takes some practice to make the perfect joint, but, like riding a bicycle, once learned is never forgotten! The idea is simple: to join electrical parts together to form an electrical connection, using a molten mixture of lead and tin (solder) with a soldering iron. A large range of soldering irons is available -- which one is suitable for you depends on your budget and how serious your interest in electronics is.

Electronics catalogues often include a selection of well-known brands of soldering iron. Excellent British-made ones include the universally popular Antex, Adcola and Litesold makes. Other popular brands include those made by Weller and Ungar. A very basic mains electric soldering iron can cost from under 5 UK Pounds (8 US Dollars), but you can expect a reasonable model to be approximately 10 to 12 UKP (16 to 20 US Dollars), and it's quite possible to spend into three figures on a "soldering station" if you're really serious! You can check suppliers' catalogues for some typical types of iron. Certain factors you need to bear in mind include:

Voltage: Most irons run from the mains at 240V (110V in the US). However, low voltage types (e.g.
12V or 24V) generally form part of a "soldering station," and are designed to be used with a special controller made by the same manufacturer.

Wattage: Typically, soldering irons may have a power rating of between 15-25 watts or so, which is fine for most work. A higher wattage does not mean that the iron runs hotter -- it simply means that there is more power in reserve for coping with larger joints. This also depends partly on the design of the "bit" (the tip of the iron). Consider a higher wattage iron simply as being more "unstoppable" when it comes to heavier-duty work, because it won't cool down so quickly.

Temperature Control: The simplest and cheapest types don't have any form of temperature regulation. Simply plug them in and switch them on! Thermal regulation is "designed in" (by physics, not electronics!). These irons may be described as "thermally balanced" so that they have some degree of temperature "matching," but their output will otherwise not be controlled. Unregulated irons form an ideal general purpose iron for most users, and they generally cope well with printed circuit board soldering and general interwiring. Most of these "miniature" types of iron will be of little use when attempting to solder large joints (e.g. very large terminals or very thick wires) because the component being soldered will "sink" heat away from the tip of the iron, cooling it down too much. (This is where a higher wattage comes in useful.)

A proper temperature-controlled iron will be quite a lot more expensive -- retailing at say 40 UKP (60 USD) or more. This type of iron will have some form of built-in thermostatic control, to ensure that the temperature of the bit (the tip of the iron) is maintained at a fixed level (within limits). This is desirable, especially during more frequent use, since it helps to ensure that the temperature does not "overshoot" in between times, and also guarantees that the output will be relatively stable. Some irons have a bimetallic strip thermostat built into the handle which gives an audible "click" in use: other types use all-electronic controllers, and some may be adjustable using a screwdriver.

Yet more expensive still, soldering stations cost from 70 UKP (115 USD) upwards (the iron may
be sold separately, so you can pick the type you prefer). Soldering stations consist of a complete bench-top control unit into which a special low-voltage soldering iron is plugged. Some versions might have a built-in digital temperature readout, and will have a control knob to enable you to vary the setting. The temperature could be boosted for soldering larger joints, for example, or for using higher melting-point solders (e.g. silver solder). These are designed for the most discerning users, or for continuous production line and professional use. The best stations have irons which are well balanced, with comfort-grip handles which remain cool all day. A thermocouple will be built into the tip or shaft, which monitors temperature.

Anti-Static Protection: If you're interested in soldering a lot of static-sensitive parts (e.g. CMOS chips or MOSFET transistors), more advanced and expensive soldering iron stations use static-dissipative
materials in their construction to ensure that static does not build up on the iron itself. You may see these listed as "ESD safe" (electrostatic discharge proof). The cheapest irons won't necessarily be ESD-safe but never the less will still probably perform perfectly well in most hobby or educational applications if you take the usual anti-static precautions when handling the components. The tip would need to be well earthed (grounded) in these circumstances.

Bits: It's useful to have a small selection of manufacturer's bits (soldering iron tips) available with different diameters or shapes, which can be changed depending on the type of work in hand. You'll probably find that you become accustomed to, and work best with, a particular shape of tip. Often, tips are iron-coated to preserve their life, or they may be bright-plated instead. Copper tips are seldom seen these days.

Spare Parts: it's nice to know that spare parts may be available, so if the element blows, you don't need to replace the entire iron. This is especially so with expensive irons. Check through some of the larger mail-order catalogues.

Gas-Powered Irons: You will occasionally see gas-powered soldering irons which use butane rather than the mains electrical supply to operate. They have a catalytic element which, once warmed up, continues to glow hot when gas passes over them. Service engineers use them for working on repairs where there may be no power available, or where a joint is tricky to reach with a normal iron, so they are really for occasional "on the spot" use for quick repairs, rather than for mainstream construction or assembly work.

And Finally: A solder gun is a pistol-shaped iron, typically running at 100W or more, and is completely unsuitable for soldering modern electronic components: they're too hot, heavy and unwieldy for micro-electronics use. Plumbing, maybe..!

Soldering irons are best used along with a heat-resistant bench-type holder, so that the hot iron can be safely parked in between use. Soldering stations already have this feature, otherwise a separate soldering iron stand is essential, preferably one with a holder for tip-cleaning sponges. Now let's look at how to use soldering irons properly, and how to put things right when a joint goes wrong ......

How to Solder

Topics in this section include: Quick Summary Guide, Cleanliness of Components, Temperature, Time, and Amount.
Quick Summary Guide: Turning to the actual techniques of soldering, firstly it's best to secure the work somehow so that it doesn't move during soldering and affect your accuracy. In the case of a printed circuit board, various holding frames are fairly popular especially with densely populated boards. The idea with a holding frame is to insert all the parts on one side (this may be referred to as "stuffing" or "populating" the board), hold them in place with a special foam pad to prevent them from falling out, turn the board over, and then snip off the wires with cutters before making the joints. The frame saves an awful lot of turning the board over and back again, especially with large boards. Other parts could be held firm in a modeller's small vice, for example.

Solder joints may need to possess some degree of mechanical strength in some cases, especially with wires soldered to, say, potentiometer or switch tags, and this means that the wire should be looped through the tag and secured before solder is applied. The down side is that it is more difficult to desolder the joint (see later) and remove the wire afterwards, if required. Otherwise, in the case of an ordinary circuit board, components' wires are bent to fit through the board, inserted flush against the board's surface, splayed outwards a little so that the part grips the board, and then soldered.

In my view -- opinions vary -- it's generally better to snip the surplus wires leads off first, to make the joint more accessible and avoid applying a mechanical shock to the printed circuit board (PCB) joint. However, in the case of
semiconductors, I often tend to leave the snipping until after the joint has been made, since the excess wire will help to sink away some of the heat from the semiconductor junction. Integrated circuits can either be soldered directly into place if you are confident enough, or better, use a dual-in-line socket to prevent heat damage. The chip can then be swapped out if needed.

Parts which become hot in operation (e.g. some resistors), are raised above the board slightly to allow air to circulate. Some components, especially large electrolytic capacitors, may require a mounting clip to be screwed down to the board first, otherwise the part may eventually break off due to vibration.

The perfectly soldered joint will be nice and shiny looking, and will prove reliable in service. I would say that the key factors affecting the quality of the joint are:

o) Cleanliness
o) Temperature
o) Duration
o) Adequate solder coverage

A little effort spent now in soldering the perfect joint may save you -- or somebody else -- a considerable amount of time in troubleshooting a defective joint in the future. The basic principles are as follows.

Really Clean: Firstly, and without exception, all parts -- including the iron tip itself -- must be clean and free from contamination. Solder just will not "take" to dirty parts! Old components or copper board can be notoriously difficult to solder, because of the layer of oxidation which builds up on the surface of the leads. This repels the molten solder and this will soon be evident because the solder will "bead" into globules, going everywhere except where you need it. Dirt is the enemy of a good quality soldered joint!

Hence, it is an absolute necessity to ensure that parts are free from grease, oxidation and other contamination. In the case of old resistors or capacitors, for example, where the leads have started to oxidise, use a small hand-held file or perhaps scrape a knife blade or rub a fine emery cloth over them to reveal fresh metal underneath. Stripboard and copper printed circuit board will generally oxidise after a few months, especially if it has been fingerprinted, and the copper strips can be cleaned using an abrasive rubber block, like an aggressive eraser, to reveal fresh shiny copper underneath.

Also available is a fibre-glass filament brush, which is used propelling-pencil-like to remove any surface contamination. These tend to produce tiny particles which are highly irritating to skin, so avoid accidental contact with any debris. Afterwards, a wipe with a rag soaked in cleaning solvent will remove most grease marks and fingerprints. After preparing the surfaces, avoid touching the parts afterwards if at all possible.

Another side effect of having dirty surfaces is the tendency for people to want to apply more heat in an attempt to "force the solder to take." This will often do more harm than good because it may not be possible to burn off any contaminants anyway, and the component may be overheated. In the case of semiconductors, temperature is quite critical and they may be harmed by applying such excessive heat.

Before using the iron to make a joint, it should be "tinned" (coated with solder) by applying a few millimetres of solder, then wiped on a damp sponge preparing it for use: you should always do this
immediately with a new bit, anyway. Personally, I always re-apply a very small amount of solder again, mainly to improve the thermal contact between the iron and the joint, so that the solder will flow more quickly and easily. It's sometimes better to tin larger parts as well before making the joint itself, but it isn't generally necessary with PCB work. A worthwhile product is Weller's Tip Tinner & Cleaner, a small 15 gram tinlet of paste onto which you dab a hot iron -- the product cleans and
tins the iron ready for use. An equivalent is Adcola Tip-Save.

Normal electronics grade solder is usually 60% lead - 40% tin, and it contains cores of "flux," which helps the molten solder to flow more easily over the joint. Flux removes oxides which arise during
heating, and is seen as a brown fluid bubbling away on the joint. Acid fluxes (e.g. as used by plumbers) must never be applied. Other solders are available for specialist work, including aluminium and silver-solder. Different solder diameters are produced, too; 20-22 SWG (19-21 AWG) is 0.91-0.71mm diameter and is fine for most work. Choose 18 SWG (16 AWG) for larger joints requiring more solder.

Temperature: Another step to successful soldering is to ensure that the temperature of all the parts is raised to roughly the same level before applying solder. Imagine, for instance, trying to solder a resistor into place on a printed circuit board: it's far better to heat both the copper PCB and the resistor lead at the same time before applying solder, so that the solder will flow much more readily over the joint. Heating one part but not the other is far less satisfactory joint, so strive to ensure that the iron is in contact with all the components first, before touching the solder to it. The melting point of most solder is in the region of 188°C (370°F) and the iron tip temperature is typically 330°C to 350°C (626°F to 662°F).

Now is the time: Next, the joint should be heated with the bit for just the right amount of time -- during which a short length of solder is applied to the joint. Do not use the iron to carry molten solder over to the joint! Excessive time will damage the component and perhaps the circuit board copper foil too! Heat the joint with the tip of the iron, then continue heating whilst applying solder, then remove the iron and allow the joint to cool. This should take only a few seconds, with experience. The heating period depends on the temperature of your iron and size of the joint -- and larger parts need more heat than smaller ones -- but some parts (semiconductor diodes, transistors and integrated circuits), are sensitive to heat and should not be heated for more than a few seconds. Novices sometimes buy a small clip-on heat-shunt, which resembles a pair of aluminium tweezers. In the case of, say, a transistor, the shunt is attached to one of the leads near to the transistor's body. Any excess heat then diverts up the heat shunt instead of into the transistor junction, thereby saving the device
from over-heating. Beginners find them reassuring until they've gained more experience.

Solder Coverage: The final key to a successful solder joint is to apply an appropriate amount of solder. Too much solder is an unnecessary waste and may cause short circuits with adjacent joints. Too little and it may not support the component properly, or may not fully form a working joint. How much to apply, only really comes with practice. A few millimetres only, is enough for an "average" PCB joint, (if there is such a thing).

How to Desolder

A soldered joint which is improperly made will be electrically "noisy," unreliable, and is likely to get worse over time. It may even not have made any electrical connection at all, or could work initially and then cause the equipment to fail at a later date! It can be hard to judge the quality of a solder joint purely by appearances, because you cannot say how the joint actually formed on the inside, but by following the guidelines there is no reason why you should not obtain perfect results.

A joint which is poorly formed is often called a "dry joint." Such a joint usually results from dirt or grease preventing the solder from melting onto the parts properly, and is often noticeable because of the tendency of the solder not to "spread," but to form beads or globules instead, perhaps partially. Alternatively, if it seems to take an inordinately long time for the solder to spread, this is another sign of possible dirt and that the joint may potentially be a dry one.

There will undoubtedly come a time when you need to remove the solder from a joint: possibly to replace a faulty component or fix a dry joint. The usual way is to use a desoldering pump which works like a small spring-loaded bicycle pump, only in reverse! (More demanding users using CMOS devices might need a pump which is ESD safe.) A spring-loaded plunger is released at the push of a button and the molten solder is then drawn up into the pump. It may take one or two attempts to clean up a joint this way, but a small desoldering pump is an invaluable tool especially for PCB work.

Sometimes, it's effective to actually add more solder and then desolder the whole lot with a pump, if the solder is particularly awkward to remove. Care is needed, though, to ensure that the boards and parts are not damaged by excessive heat; the pumps themselves have a P.T.F.E. nozzle which is heat proof but may need replacing occasionally.

An excellent alternative to a pump is to use desoldering braid, including the famous American "Soder-Wick" (sic) or Adcola "TISA-Wick" which are packaged in small dispenser reels. This product is a specially treated fine copper braid which draws molten solder up into the braid where it solidifies. The best way is to use the tip of the hot iron to press a short length of braid down onto the joint to be desoldered. The iron will subsequently melt the solder, which will be drawn up into the braid. Take extreme care to ensure that you don't allow the solder to cool with the braid adhering to the work, or you run the risk of damaging PCB copper tracks when you attempt to pull the braid off the joint.

I recommend buying a small reel of de-soldering braid, especially for larger or difficult joints which would take several attempts with a pump. It is surprisingly effective, especially on difficult joints where a desoldering pump may prove a struggle.

Summary

Here's a summary of how to make the perfect solder joint:

All parts must be clean and free from dirt and grease.

Try to secure the work firmly.

"Tin" the iron tip with a small amount of solder. Do this immediately, with new tips being used for the first time.

Clean the tip of the hot soldering iron on a damp sponge (many people then add a tiny amount of fresh solder to the cleansed tip).

Heat all parts of the joint with the iron for under a second or so.

Continue heating, then apply sufficient solder only, to form an adequate joint.

Remove and return the iron safely to its stand.

It only takes two or three seconds at most, to solder the average PCB joint.

Do not move parts until the solder has cooled.

Troubleshooting Guide

The solder won't "take": If grease or dirt are present, desolder and clean up the parts. Or, the material may simply not be suitable for soldering with lead/tin solder.
The joint is crystalline or grainy-looking: The parts forming the joint may have been moved before being allowed to cool, or the joint was not heated adequately (too small an iron or too large a joint).

The solder joint forms a "spike": The joint was probably overheated, burning away the flux.

First Aid

If you are unlucky enough to receive burns which require treatment, here's what to do:

Immediately cool the affected area with cold running water, ice, or even frozen peas, for ten minutes.
Remove any rings etc. before swelling starts.
Apply a sterile dressing to protect against infection.
Do not apply lotions, ointments etc., nor prick any blisters which form later.
Seek professional medical advice where necessary.

Basics of Electrostatic Discharge (ESD)

Sun, 01 Mar 1998 00:00:00 +0000

Batteries in a Portable World 2nd Ed.

Sun, 01 Mar 1998 00:00:00 +0000

Controlling Electrical Hazards

Sun, 01 Mar 1998 00:00:00 +0000

This article provides an overview of basic electrical safety for individuals with little or limited training or familiarity with electrical hazards. The concepts and principles presented will help further an understanding of OSHA's electrical safety standards for general industry, Title 29 Code of Federal Regulations (CFR), Part 1910.302, Sub-part S-Design Safety Standards for Electrical Systems, and 1910.331 Electrical Safety-Related Work Practices Standard (1990).

Contents / Menu
Introduction
How Electricity Acts?
How Shocks Occur?
Severity of the Shock
Burns and Other Injuries
Preventing Electrical Hazards
( Insulation, Guarding, Grounding, Circuit Protection Devices, Safe Work Practices, Training, Overhead Lines )
Conclusion

Definitions of the SI base and derived units

Sun, 01 Mar 1998 00:00:00 +0000

(Definitions of the SI base units, SI derived units, Other not SI units)
The International System of Units, universally abbreviated SI (from the French Le Système International d'Unités), is the modern metric system of measurement. The SI was established in 1960 by the 11th General Conference on Weights and Measures (CGPM, Conférence Générale des Poids et Mesures). The CGPM is the international authority that ensures wide dissemination of the SI and modifies the SI as necessary to reflect the latest advances in science and technology.

Dielectric Constants of Material

Sun, 01 Mar 1998 00:00:00 +0000

Reference and Sources:
Zed Z. Chang
Material Min. Max.
Air 1 1
Amber 2.6 2.7
Asbestos fiber 3.1 4.8
Bakelite 5 22
Barium Titanate 100 1250
Beeswax 2.4 2.8
Cambric 4 4
Carbon Tetrachloride 2.17 2.17
Celluloid 4 4
Cellulose Acetate 2.9 4.5
Durite 4.7 5.1
Ebonite 2.7 2.7
Epoxy Resin 3.4 3.7
Ethyl Alcohol 6.5 25
Fiber 5 5
Formica 3.6 6
Glass 3.8 14.5
Glass Pyrex 4.6 5
Gutta Percha 2.4 2.6
Isolantite 6.1 6.1
Kevlar 3.5 4.5
Lucite 2.5 2.5
Mica 4 9
Micarta 3.2 5.5
Mycalex 7.3 9.3
Neoprene 4 6.7
Nylon 3.4 22.4
Paper 1.5 3
Paraffin 2 3
Plexiglass 2.6 3.5
Polycarbonate 2.9 3.2
Polyethylene 2.5 2.5
Polyimide 3.4 3.5
Polystyrene 2.4 3
Porcelain 5 6.5
Quartz 5 5
Rubber 2 4
Ruby Mica 5.4 5.4
Selenium 6 6
Shellac 2.9 3.9
Silicone 3.2 4.7
Slate 7 7
Soil dry 2.4 2.9
Steatite 5.2 6.3
Styrofoam 1.03 1.03
Teflon 2.1 2.1
Titanium Dioxide 100 100
Vaseline 2.16 2.16
Vinylite 2.7 7.5
Water distilled 34 78
Waxes, Mineral 2.2 2.3
Wood dry 1.4 2.9

Digital Audio Resampling

Sun, 01 Mar 1998 00:00:00 +0000

NOTE: THIS DOCUMENT IS OBSOLETE, PLEASE CHECK THE NEW VERSION: "The Digital Audio Resampling Home Page", by Julius O. Smith III, Copyright © 2016-05-17 by Julius O. Smith III - Center for Computer Research in Music and Acoustics (CCRMA), Stanford University

Elementary Digital Filter Theory

Sun, 01 Mar 1998 00:00:00 +0000

NOTE: THIS DOCUMENT IS OBSOLETE, PLEASE CHECK THE NEW VERSION: "Introduction to Digital Filters with Audio Applications", by Julius O. Smith III, Copyright © 2017-11-26 by Julius O. Smith III - Center for Computer Research in Music and Acoustics (CCRMA), Stanford University

Ferric Chloride Solution: Etchant for copper-plated PC Boards

Sun, 01 Mar 1998 00:00:00 +0000

Ferric Chloride CAS #7705-08-0 Green-black solution in water with a mild acid-like odor. May cause eye and skin burns. May be harmful if swallowed. Ingestion may cause rapid heartbeat, low blood pressure, shock, and possible coma. May cause kidney and liver damage. Inhalation of vapor may cause severe respiratory tract irritation. Substance has caused adverse reproductive effects in animals. Target Organs: Kidneys, liver, cardiovascular system. Adverse Health Effects If in Eyes: May cause eye burns. Contact produces irritation, tearing, and burning pain. If on Skin: May be absorbed through the skin. May cause severe irritation and possible burns. If Swallowed: May cause irritation of the digestive tract. May cause liver and kidney damage. Causes severe pain, nausea, vomiting, diarrhea, and shock. May cause severe irritation of the mouth and throat. May cause low blood pressure, rapid heartbeat, skin discoloration, and possible coma. If Inhaled: Vapor may cause severe respiratory tract irritation. Prolonged or repeated exposure may cause adverse reproductive effects. Repeated exposure may increase an increased body load of iron. FIRST AID MEASURES: Eyes: Immediately flush eyes with plenty of water for at least 15 minutes, occasionally lifting the upper and lower lids. Get medical aid immediately. Do NOT allow victim to rub or keep eyes closed. Skin: Immediately flush skin with plenty of soap and water for at least 15 minutes while removing contaminated clothing and shoes. Get medical aid if irritation develops or persists. Ingestion: If victim is conscious and alert, give 2-4 cupfuls of milk or water. Get medical aid immediately. Inhalation: Remove from exposure to fresh air immediately. If not breathing, give artificial respiration. If breathing is difficult, give oxygen. Get medical aid. Notes to Physician: Treat symptomatically and supportively. The use of an iron chelator should be determined only by qualified medical personnel. Handling: Wash thoroughly after handling. Remove contaminated clothing and wash before reuse. Use with adequate ventilation. Do not get in eyes, on skin, or on clothing. Do not ingest or inhale. Storage: Do not store in direct sunlight. Store in a cool, dry, well-ventilated area away from incompatible substances. OSHA Vacated PELs: Ferric Chloride: No OSHA Vacated PELs are listed for this chemical.
Physical State: Liquid Appearance: Green-black solution. Odor: Mild acid-like. pH: 2.0 (0.1M sol.) Vapor Pressure: Not available. Vapor Density: Not available. Evaporation Rate: Not available. Viscosity: Not available. Boiling Point: 212 degrees F Freezing/Melting Point: Not available. Decomposition Temperature: Not available. Solubility: Soluble in water. Specific Gravity/Density: 1.4 (water=1) Molecular Formula: FeCl3 Molecular Weight: 162.206 Chemical Stability: Stable under normal temperatures and pressures. Incompatible with Alkalis, allyl chloride, ethylene oxide, nylon, oxidizers, potassium, sodium. Corrosive to most metals. Hazardous Decomposition Products: Hydrogen chloride, chlorine. Hazardous Polymerization: None reported. LD50/LC50: CAS# 7732-18-5: Oral, rat: LD50 = >90 mL/kg. CAS# 7732-18-5: Oral, rat: LD50 = >90 mL/kg. Carcinogenicity: Not listed by ACGIH, IARC, NIOSH, NTP, or OSHA. Epidemiology: No information available. Teratogenicity: No information available. Reproductive Effects: Fertility: Pre-implantation mortality, intravaginal-rat TDLo=29mg/kg. Paternal Effects: Spermatogenesis and Testes/sperm duct/epididymis, intratesticular-rat TDLo=12976ug/kg. Neurotoxicity: No information available. Mutagenicity: No information available. Other Studies: None. Ecotoxicity: Daphnia (fresh water) TLm=15ppm/96H Mosquito fish TLm=74ppm/96H Minnow TLm=540ppm/1.5H Environmental Fate: No information available. Dispose of in a manner consistent with federal, state, and local regulations. RCRA D-Series Maximum Concentration of Contaminants: Not listed. RCRA D-Series Chronic Toxicity Reference Levels: Not listed. RCRA F-Series: Not listed. RCRA P-Series: Not listed. RCRA U-Series: Not listed. Not listed as a material banned from land disposal according to RCRA. Transportation Information: Shipping Name: FERRIC CHLORIDE SOLUTION Hazard Class: 8(9.2) UN Number: UN2582 TSCA: CAS# 7705-08-0 is listed on the TSCA inventory. Not listed on the Health & Safety Reporting List. Not listed under TSCA Section 12b. Does not have a SNUR under TSCA. CERCLA/SARA: Section 302 (RQ) NONE Section 302 (TPQ) NONE Section 313 Not reportable under Section 313. Clean Air Act: This material does not contain any hazardous air pollutants. This material does not contain any Class 1 Ozone depletors. This material does not contain any Class 2 Ozone depletors. Clean Water Act: CAS# 7705-08-0 is listed as a Hazardous Substance under the CWA. Not are listed as Priority Pollutants under the CWA. Not listed as Toxic Pollutants under the CWA. OSHA: Not are considered highly hazardous by OSHA. Ferric Chloride can be found on the following state right to know lists: New Jersey, Pennsylvania, Massachusetts. Not present on state lists from CA, PA, MN, MA, FL, or NJ. California No Significant Risk Level: Not listed. Canada CAS# 7705-08-0 is listed on Canada’s DSL/NDSL List. CAS# 7705-08-0 is not listed on Canada’s Ingredient Disclosure List. European Labeling in Accordance with EC Directives Hazard Symbols: Not available. Risk Phrases: Not available. Safety Phrases: Not available. Exposure Limits: OEL-DENMARK:TWA 1 mg(Fe)/m3 JANUARY 1993. OEL-FINLAND:TWA 1 mg(Fe)/m3 JANUARY 1993. OEL-THE NETHERLANDS:TWA 1 mg(Fe)/m3 JANUARY 1993. OEL-SWITZERLAND:TWA 1 mg(Fe)/m3 JANUARY 1993. OEL-UNITED KINGDOM:TWA 1 mg(Fe)/m3;STEL 2 mg(Fe)/m3 JANUARY 1993. OEL IN BULGARIA, COLOMBIA, JORDAN, KOREA check ACGIH TLV. OEL IN NEW ZEALAND, SINGAPORE, VIETNAM check ACGI TLV

Back to PCBs Fabrication Methods Guide

Inductance of Circular Loop

Sun, 01 Mar 1998 00:00:00 +0000

Loop of wire with circular shape
N [ ] number of turns
R [m] radius of the circle
a [m] wire radius
µ_r [ ] relative permeability of the medium
L [H] Inductance

NOTE: numbers are in scientific notation (e.g.: 1.427e-9 H = 1.1427·10^-9 H = 1.427 nH)

Inductance of Coplanar Traces

Sun, 01 Mar 1998 00:00:00 +0000

Cross section of Coplanar Traces transmission line
w [m] width of the traces
d [m] center-to-center distance
t [m] thickness of the traces
µ_r [ ] relative permeability of the medium
L/l [H/m] Inductance per unit length

NOTE: numbers are in scientific notation (e.g.: 1.427e-9 H = 1.1427·10^-9 H = 1.427 nH)

See also notes on Inductance Calculations page.

Inductance of Equilateral Triangle Loop

Sun, 01 Mar 1998 00:00:00 +0000

Loop of wire with Equilateral Triangle shape
N [ ] number of turns
s [m] length of one side
a [m] wire radius
µ_r [ ] relative permeability of the medium
L [H] Inductance

NOTE: numbers are in scientific notation (e.g.: 1.427e-9 H = 1.1427·10^-9 H = 1.427 nH)

See also notes on Inductance Calculations page.

Inductance of Isosceles Triangle Loop

Sun, 01 Mar 1998 00:00:00 +0000

Loop of wire with Isosceles Triangle shape
N [ ] number of turns
b [m] length of the base
c [m] length of the equal sides
a [m] wire radius
µ_r [ ] relative permeability of the medium
L [H] Inductance

NOTE: numbers are in scientific notation (e.g.: 1.427e-9 H = 1.1427·10^-9 H = 1.427 nH)

See also notes on Inductance Calculations page.

Inductance of Rectangular Loop

Sun, 01 Mar 1998 00:00:00 +0000

Loop of wire with rectangular shape
N [ ] number of turns
w [m] width of the rectangle
h [m] height of the rectangle
a [m] wire radius
µ_r [ ] relative permeability of the medium
L [H] Inductance

NOTE: numbers are in scientific notation (e.g.: 1.427e-9 H = 1.1427·10^-9 H = 1.427 nH)

See also notes on Inductance Calculations page.

Inductance of Round Wire over a Ground Plane

Sun, 01 Mar 1998 00:00:00 +0000

Cross section of Round Wire transmission line over a Ground Plane
h [m] height of center from ground plane
a [m] wire radius
µ_r [ ] relative permeability of the medium
L/l [H/m] Inductance per unit length

NOTE: numbers are in scientific notation (e.g.: 1.427e-9 H = 1.1427·10^-9 H = 1.427 nH)

See also notes on Inductance Calculations page.

Inductance of Square Loop

Sun, 01 Mar 1998 00:00:00 +0000

Loop of wire with square shape
N [ ] number of turns
w [m] width of the rectangle
a [m] wire radius
µ_r [ ] relative permeability of the medium
L [H] Inductance

NOTE: numbers are in scientific notation (e.g.: 1.427e-9 H = 1.1427·10^-9 H = 1.427 nH)

See also notes on Inductance Calculations page.

Inductance of Twin Lead

Sun, 01 Mar 1998 00:00:00 +0000

Cross section of Twin Lead transmission line
d [m] distance from center to center
a [m] wire radius
µ_r [ ] relative permeability of the medium
L/l [H/m] Inductance per unit length

NOTE: numbers are in scientific notation (e.g.: 1.427e-9 H = 1.1427·10^-9 H = 1.427 nH)

See also notes on Inductance Calculations page.

Inductance of Vertically Spaced Traces

Sun, 01 Mar 1998 00:00:00 +0000

Cross section of Vertically Spaced Traces transmission line
w [m] width of the traces
h [m] center-to-center spacing of traces
t [m] thickness of the traces
µ_r [ ] relative permeability of the medium
L/l [H/m] Inductance per unit length

NOTE: numbers are in scientific notation (e.g.: 1.427e-9 H = 1.1427·10^-9 H = 1.427 nH)

See also notes on Inductance Calculations page.

Inductance of Wide Trace over a Ground Plane

Sun, 01 Mar 1998 00:00:00 +0000

Cross section of Wide Trace transmission line over a Ground Plane
w [m] width of the trace
h [m] height of the trace above ground
t [m] thickness of the trace
µ_r [ ] relative permeability of the medium
L/l [H/m] Inductance per unit length

NOTE: numbers are in scientific notation (e.g.: 1.427e-9 H = 1.1427·10^-9 H = 1.427 nH)

See also notes on Inductance Calculations page.

Mathematics of the Discrete Fourier Transform (DFT)

Sun, 01 Mar 1998 00:00:00 +0000

NOTE: THIS DOCUMENT IS OBSOLETE, PLEASE CHECK THE NEW VERSION: "Mathematics of the Discrete Fourier Transform (DFT), with Audio Applications --- Second Edition", by Julius O. Smith III, W3K Publishing, 2007, ISBN 978-0-9745607-4-8. - Copyright © 2017-09-28 by Julius O. Smith III - Center for Computer Research in Music and Acoustics (CCRMA), Stanford University

Mouse Cleaning

Sun, 01 Mar 1998 00:00:00 +0000

When you run your mouse over the desktop it picks up all kinds of debris, gunk, and goo that can get drawn up into the insides of the mouse and cause it to stop working well.
If you find that the pointer on the screen doesn't move smoothly or that the trackball inside the mouse doesn't move freely, you should clean the mouse.

You could go to a computer store and buy one of those fancy mouse-cleaning kits, but, in my opinion, there really isn't any need. You're going to be able to clean it as well as anything using these inexpensive items:

Few clean cotton swabs.
Denatured alcohol. (I recommend denatured alcohol over rubbing alcohol, which has glycerine in it and can leave some slippery residue behind).
Water.
Mild soap.
A clean, soft, lint-free cloth (100% cotton works best).
An old toothbrush.
To clean your mouse, follow these steps:

1. Shut down and turn off your computer.

2. Disconnect the mouse cable from the computer.

3. Remove the trackball

3.1 Turn your mouse upside down.
3.2 Unlock the disk that's holding the trackball in place by either rotate counter-clockwise or slide in one direction.
3.3 Turn the mouse right side up with one hand and catch the ring and trackball with your other hand. Give it a shake if they don't dislodge themselves.

4. Wash the trackball and the retainer ring with warm soapy water using an old toothbrush. Rub the trackball thoroughly to remove any oily build-up.

5. Dry the trackball and the retainer ring thoroughly with a clean lint-free cloth.

6. Set the trackball and the retainer ring aside.

7. Moisten a lint-free cloth with denatured alcohol and gently rub the mouse back surface and trackball compartment. Blow into the trackball compartment to remove any dust or lint from inside.

8. Take a close look at the three rollers inside the mouse. You'll probably see lines of dirt that have accumulated over many months or even years of usage. These lines of dirt build up and cause jerky or erratic mouse movements. Clean the three small rollers inside the mouse with a cotton swab moistened with denatured alcohol. The cotton should be just a little damp. Rotate the rollers to clean all around them. Work the tip along each roller side to side until the line of dirt disappears. If there is a build-up of material on the rollers which the cotton swab does not remove, you can use your fingernail or a tweezer to gently remove the material.
WARNING: The rollers must remain smooth so be very careful not to gouge them with anything!

9. When it's clean, let it dry out for a moment or two.

10. Put the trackball back into the mouse and close the mouse trackball cover with a clockwise motion.

11. Reconnect the mouse cable to the computer.

12. Restart your computer and test mouse's movement.

13. Your mouse should be as good as new.

NOTE: Always use a clean mouse pad, free from dust, when moving your mouse.

Multiples and Submultiples Prefixes Tables

Sun, 01 Mar 1998 00:00:00 +0000

(SI Prefixes Table, IEC Prefixes for binary multiples)
SI Prefixes Table
The International System of Units, universally abbreviated SI (from the French Le Système International d'Unités), is the modern metric system of measurement.
The 20 SI prefixes used to form decimal multiples and submultiples of SI units are:
Symbol Name Factor Symbol Name Factor
Y yotta 10²⁴ y yokto 10^-24
Z zetta 10²¹ z zepto 10^-21
E exa 10¹⁸ a atto 10^-18
P peta 10¹⁵ f femto 10^-15
T tera 10¹² p pico 10^-12
G giga 10⁹ n nano 10^-9
M mega 10⁶ µ micro 10^-6
k kilo 10³ m milli 10^-3
h hecto 10² c centi 10^-2
da deka 10¹ d deci 10^-1
It is important to note that the kilogram is the only SI unit with a prefix as part of its name and symbol. Because multiple prefixes may not be used, in the case of the kilogram the prefix names of Table are used with the unit name "gram" and the prefix symbols are used with the unit symbol "g." With this exception, any SI prefix may be used with any SI unit, including the degree Celsius and its symbol °C.

Networking

Sun, 01 Mar 1998 00:00:00 +0000

On Cables and Connections

Sun, 01 Mar 1998 00:00:00 +0000

The original On Cables and Connections - A discussion by Dr. J. Kramer is Copyright © KRAMER ELECTRONICS LTD. (www.kramerelectronics.com). All rights reserved. No part of this publication may be reproduced, transmitted, transcribed, stored in a retrieval system or translated into any language or computer language in any form or by any means without written permission of KRAMER ELECTRONICS LTD. Sapir Center, Building 5, Jerusalem 95461, Israel - TEL:(972-2)-6524715, FAX:(972-2)-6535369 - Email: kramerel@netvision.net.il This is a personalized version reproduced by permission.
Introduction
Rules-of-thumb to prevent cable problems
Cable length and quality
Solving cable related problems
When longer distances are needed
Line Amplifiers
The twisted-pair solution
The fiber-optic solution
The RF solution
Problems related to audio wires
Dealing with high capacitance audio wires
Dealing with noise pickup
Signal loss and deterioration due to cable impedance
Dealing with amplifier output quality degradation due to cable resistance
A final word

Introduction
We are frequently asked - "what length of cable can I use for a specific application?" Seemingly a simple question, but the answer is quite complicated.

Basically, cables connecting different types of equipment should ideally behave as "shorts", meaning that the output from the receiving device should be as if it is connected directly to the transmitting device. Unfortunately, they don't. Connecting cables have their own characteristics, which they impose on the signals they carry. An ideal cable should behave as an ideal "transmission line", having no effect on the signal. Nevertheless, cables behave otherwise.

Two major factors play an important role in signal deterioration due to poor cables:
First, cables do not behave as ideal transmission lines with known impedance due to flaws in cable design and production. Sometimes the impedance of the cable is simply not as stated on the cable.
Second, due to imperfection in cable design, stray (parasite) capacitance and inductance appear along the cable and they have an adverse effect on signal quality.
The longer the cables used the more adverse effects they have on the signals being transmitted. Better cables alleviate this problem. A very popular cable used for video is "RG-59" type. For short distances - a meter or two - this cable is fine, and the signals, which emerge, look as if they were connected directly by an ideal short. At composite video frequencies, up to 10 MHz, the high frequency loss of this cable is about 3 dB after 100 meters. Therefore, deterioration is not apparent even at 20 meters. However, at 300 MHz, (the frequencies encountered in high-resolution or digital video) the loss at 100 meters is close to 20 dB, which is disastrous to the signal. When using SDI (Serial Digital) signals, at 270 Mbits/sec, a loss of 40 dB may occur at 300 meters. This represents a signal deterioration of 99% (down to 1% of the original signal!)

Rules-of-thumb to prevent cable problems:
Use the best cables you can afford.
Use the shortest cables possible.
Check that impedance of the cable really matches your application.
Check the connectors on the cables. Poor quality connectors may have worse effects than the cables themselves.
Use appropriate amplifiers if needed.
Keep cables away from sources of interference - electric motors, mains cables, transmitters, triac operated equipment (light dimmers) etc.

Cable length and quality:
For composite video, PAL or NTSC, RG-59 cable can be used for distances of up to 20 meters between the video source and the acceptor. No signal deterioration should show on the screen. For longer distances, up to 30 meters, use RG-11 type cable. For still longer distances, better cables should be used, such as Belden 8281 or Belden 1694A (which are also suitable for SDI signals). Those cables extend the range by 50% and more.
For s-Video (Y/C, Super-VHS and Hi-8), standard, inexpensive cables allow operating distances of 5-7 meters. Very little deterioration is apparent. At this distance lower quality cables would show color "smear" and blurred images. Better than standard quality cables will show no deterioration whatsoever. The quality of the s-Video connector, often overlooked, plays a very important role in final signal quality.
Analog component video signals - whether Y, R-Y and B-Y or RGB should be treated very carefully. If the origin of the component video signal is a VCR or a camera, in order to maintain good quality (especially for the "Y" signal), RG-59 cables are suitable for up to 10 meters. For longer distances, up to 20-25 meters, a better cable should be used - either RG-11 or an appropriate Belden cable (see above). Component signals such as RGB signals from computer graphics sources should be treated as computer sources (see below.).
Computer graphics cards generating VGA, Super-VGA or XGA signals need a different approach, depending on resolution. Standard, commonly available VGA cables are suitable for 640x480 resolution at distances of up to 1.5 meters. At 800x600, such cables are suitable for distances of not more than 80 centimeters. They are not suitable for use at higher resolutions. Special VGA/XGA cables should be used for resolutions of 1024x768 and above. The best quality cables of the packed-in-one-sleeve type are suitable for up to 10 meters, albeit with some apparent deterioration. When computer graphics or other high frequency component signals are used, the compound output should be transmitted over coax cables which are divided into individual, separately sleeved lines, of one of the types mentioned above. For VGA/XGA signals, 5 cables are needed for setups of Red, Green, Blue, Horizontal Sync and Vertical Sync, and four cables are needed for setups with Red, Green, Blue and Composite Sync. Three cables are needed for setups either with Red, Blue, Green+Sync or when the composite sync signal is superimposed on all signals (Red Green and Blue).

Solving cable related problems
The problems cable generate that can be completely or partially solved divide into several categories:
General signal loss and attenuation due to incorrect impedance characteristics. If you feed a long cable with 1-Volt composite video, you may find on the far end only about 0.9 Volts.
High frequency losses due to the wrong capacitance and inductance. Such high frequency loss in video shows up as loss of color, from color smears to up to full color loss, loss of picture resolution and loss of fine details.
Standing wave and reflection effects due to impedance mismatch between circuit and cable. The visible result of this problem is the "ringing" effect, where ghost lines appear on vertical image transitions, thus upsetting image integrity.

When longer distances are needed
As we have seen, cable lengths impose limitations in all formats. The available solutions are as follows, assuming that the best cables and connectors are used:
Use line amplifiers to extend the range.
Use twisted pair transmitters and receivers for longer ranges.
Use fiber-optic transceivers to extend the range even further.
Use RF transmission (microwave or HF) or satellite uplink for very long distances.

Line Amplifiers
Line amplifiers overcome the following shortcomings of long cable installations:
Loss of overall signal level.
Loss of high frequency signals due to stray capacitance and inductance.
Mismatch of cable impedance.
Line amplifiers should be installed as close as possible to the video source to achieve the best signal to noise ratio.
Line amplifiers are equipped with several controls or trimmer adjustments to control signal level. Some are for boosting and compensating for overall signal loss. Others control high frequency losses, and are called EQ (Cable Equalization) or HF (High Frequency) controls. EQ controls apply a variable, frequency dependent amplification to the signal, to "straighten" the tilted frequency response curve of the cable. In this way, the distortions of the cable can be eliminated.
Line amplifiers cure source impedance mismatch by driving the cable with an accurate impedance drive.

Line amplifiers differ in their functions, depending on their use:
Line amplifiers for composite video have simple controls, as mentioned above.
Line amplifiers for S-Video (Y/C) have separate controls for Y and C. Sometimes Y/C line amplifiers have additional controls such as Chroma phase control.
Line amplifiers for twisted-pair setups have simple controls similar to those for composite.
Line amplifiers for component video, RGB or VGA/XGA signals are more complicated. For example, in the case of VGA/XGA, one control must adjust three different signals.
Line amplifiers for SDI (Serial Digital) video have two-correction circuitry built-in: cable compensation and reclocking. Digital signals, besides being attenuated, become jittery when using long cables and the reclocking circuitry time-base corrects them.
With a reasonable quality line amplifier, effective operation range may be multiplied by a factor of from 2-10, depending on the format and cable used. It is not uncommon to find installations with video lines running 300 meter, using high quality cables and one or more line amplifiers inserted in series.
Line amplifiers may be cascaded. Inserting a line amplifier every 80 meters in a composite video setup is quite common. In component or Y/C setups, line amplifiers should be inserted closer together, 40 meters or so, though good cables must be used. To transmit high resolution computer graphics signals over long distances, the best way is to split the compound VGA/XGA cable to RGB coax cables, as explained above, and use coax line amplifiers. If this is not possible, bear in mind that a line amplifier should be inserted in the signal path every 10 meters maximum, and this only when very high quality cable is used.

When long distance line-amplification is used, ground-related problems such as hum and ground loops often arise. This is due to different ground potentials between the local and remote locations. Ground isolation and differential amplification is often needed to cure the ground-related problems.

The twisted-pair solution
Twisted pair may be the most appropriate solution to transmit composite video signals over long distances, of 100 meters up to 1km. Such a system includes a twisted pair video or AV transmitter which converts video into a twisted-pair, differential input, balanced format. On the receiving end, a twisted-pair receiver converts the twisted pair signal back to a standard video signal.

Any type of twisted pair or dual-wire cables can be used because, in most cases, the transmitter can force the impedance of the twisted-pair cable to match that of the transmitter. However, the twisted pair solution is dependent on wire quality. Low quality twisted pair wires, which have high capacitance or are over-shielded, limit the effective range of the system. Twisted-pair is also less prone to ground related problems.

Twisted pair setups linking distances of 500 meters between source and acceptor are quite common. Line amplifiers specially designed for twisted pair applications extend the range of operation. The rule of thumb is that when a system performs well at 500 meters, a line amplifier inserted at that point effectively doubles the operating distance.

Twisted pair works well for composite video where bandwidths approach 5 MHz. A good quality, color-rich signal is obtained on the far end. This system is not suitable for Y/C, Component video or VGA signals, due to the high-frequency nature of those signals.

The fiber-optic solution
For operating distances of over 1km, fiber optic is recommended. It converts electric signals to beams of light, and transmits them over an optic fiber. The beam of light is usually infrared and can have different wavelengths. A fiber optic system includes a transmitter and a receiver. The transmitter converts the electric signal to a light beam, and the receiver converts the light beam back into an electric, such as video, signal.

There are two types of optic fiber, multi-mode and single-mode. Using multi-mode systems with transmitter, optic fiber and receiver, the useful distance for analog composite video is up to 5km. Using single-mode, this range may be extended to up to 25km. For still longer distances, special transmitting elements create an intense laser beam which can be carried by the appropriate optic fiber for 50km or more. Line amplifiers may be used with fiber optics as well, but they are far more complicated.

The RF solution
Microwave transmitter / receivers, or satellite communication transceiving systems can extend the range of transmission to many kilometers, but this is different technology and will therefore not be discussed here.

Problems related to audio wires
As mentioned above, transmitting audio signals over long distances can also create a host of problems. The audio system may transmit low power signals (line, Mic or balanced) or high power signals, such as those used in standard loudspeakers. Audio cabling system and design is far less sensitive to impedance matching problems than it is in video.

The main difficulties in audio cabling are as follows:
High capacitance audio cables can overload the driving amplifier, and thereby cause instability and oscillations.
Audio cables may pick up noise and interference which will result in poor signal quality and bad signal to noise ratio.
Cable impedance problems cause signal loss and deterioration.
Cable resistance and the loss of the high dumping factor of the amplifier can degrade the amplifier's output quality.
All these problems can be solved satisfactorily.

Dealing with high capacitance audio wires
High capacitance audio wires may create signal deterioration and instability in the output amplifier stage. To avoid these problems, the output amplifier stages should use stable amplifier chips, with appropriate output resistors isolating the amplifier from the adverse effect of the cable.

The load on the cable should be high (low impedance) and the driving amplifier should be able to "pump" current into the cable. The printed circuit board design around the amplifier should eliminate PCB induced instability.

Dealing with noise pickup
Electric noise may be picked up by long wires, thereby causing audio signal quality to deteriorate. The problem is more severe when low-level signals such as microphone generated signals are involved.

There are two solutions to this problem:
One solution is to use good quality cables with proper shielding. The shielding substantially reduces noise pickup. This solution for line-level signals is appropriate for up to 10 meters, provided the cables are not adjacent to electric noise sources (see above).
The other solution, commonly used in broadcast applications, is to use a balanced-wire system. A typical system uses a three-wire assembly - two wires carry the signal in opposite polarities, and the third is the ground-reference and/or shield wire. This way a very clean signal is obtained at output, though at the cost of more complicated electronic circuitry. When using the balanced-wire system, the line level signal's range may be effectively extended by up to 100 meters, again, provided the cables are not adjacent to electric noise sources (see above).
Microphone level signals of a few milli-volts are more sensitive to external interference. The effective range of low level microphone signals in a balanced-wire system is 10-15 meters (ten times better than when using simple, unbalanced shielded wires).
A very common trick is used nowadays in order to improve signal to noise ratio. The signal level is elevated in order to elevate the signal to noise ratio. This trick is based on the assumption that interference is constant. In the past, audio/video signal levels of 1 Volt were normal. Today, signal levels of 6-30 Volts, travelling on balanced audio wire systems, are very common.

Signal loss and deterioration due to cable impedance
Cables impose specific impedance in a signal path. Sometimes, the resistance part of this compound impedance is too large and creates a voltage divider with the load impedance. The result is signal level deterioration. Active amplifying circuitry compensates for signal level changes where signal gain, either at the receiver end or, preferably, at the transmitter end, can be fine-tuned by the user, using internal or external controls. In some cases, some specific frequency bands within the audio signal spectrum are attenuated due to cable capacitance. It is usually the high frequency range which is effected. To overcome this problem, a tone-effecting amplifier or an audio equalizer should be inserted in the signal path.

Dealing with amplifier output quality degradation due to cable resistance
In an amplifier-loudspeaker system, the cables leading to the amplifier, if not heavy enough, create a voltage divider with the resistance of the loudspeaker reducing signal level. This problem can be dealt with by using the volume control of the amplifier. A much worse problem is the depreciation of the amplifier's dumping factor due to cable resistance. This results in lower quality and "muddiness" in the sound. Using heavy-conductor, low capacitance wires overcomes this problem.

A final word
Cables and long distance operation can be problematic, but understanding the nature of the problems, accurately identifying one's needs and using the tips above will help in getting a good job done, even from a remote location.

Passive Components Codes Guide

Sun, 01 Mar 1998 00:00:00 +0000

Codes for Resistors, Capacitors, Inductors, Termistors NTC and PTC, Varistors VDR.
Pictures
Color Code
Alpha-Numeric Code
One Letter Code
Pictures
Resistors

5 Band Ceramic Capacitors

Radial or Ayial Lead Ceramic Capacitors (6 Dot or Band System)

Ceramic Disk Capacitors

Ceramic Feed Through Capacitors

Dipped Tantalum Capacitors

Film Type Capacitors

5 Dot or Band Ceramic Capacitors (one wide band)

Postage Stamp Mica Capacitors

Standard Button Mica Capacitors

Polistirene Capacitors

Electrolitic Tantalum Capacitors (4 colors code)

Electrolitic Tantalum Capacitors (3 colors code)

Ceramic Capacitors Class II

Ceramic Capacitors Class I

Ceramic Capacitors Pin-up

PCB Impedance and Capacitance of Asymmetric Stripline

Sun, 01 Mar 1998 00:00:00 +0000

W [m] width of the trace
H [m] height of trace above return plane
H1 [m] height of trace above return plane
T [m] trace thickness
E_r [ ] relative permittivity of the dielectric

Are there distributed capacitive loads on this trace?
NoYes
L_a [m] average length of the traces attaching the loads
C_a [pF] average load capacitance

OUTPUT
Z₀ [Ohm] characteristic impedance
C₀ [F/m] capacitance per unit length
t_pd [s/m] propagation delay
L₀ [H/m] inductance per unit length
Z_c [Ohm] effective characteristic impedance
t'_pd [s/m] effective propagation delay

NOTE: numbers are in scientific notation (e.g.: 1.427e-9 s/m = 1.1427·10^-9 s/m = 1.427 ns/m)

CONDITIONS: (0.1 < W/H < 2.0); ( T/H < 0.25);
Formulas:

Capacitive Load Modify:

See also notes on PCB Impedance and Capacitance Calculator page.

PCB Impedance and Capacitance of Differential Microstrip

Sun, 01 Mar 1998 00:00:00 +0000

Z₀ [Ohm] characteristic impedance
H [m] height of trace above return plane
S [m] space between traces

OUTPUT
Z_d [Ohm] differential impedance

NOTE: numbers are in scientific notation (e.g.: 1.427e-9 s/m = 1.1427·10^-9 s/m = 1.427 ns/m)
Formulas:

See also notes on PCB Impedance and Capacitance Calculator page.

PCB Impedance and Capacitance of Differential Stripline

Sun, 01 Mar 1998 00:00:00 +0000

Z₀ [Ohm] characteristic impedance
H [m] height of trace above return plane
S [m] space between traces

OUTPUT
Z_d [Ohm] differential impedance

NOTE: numbers are in scientific notation (e.g.: 1.427e-9 s/m = 1.1427·10^-9 s/m = 1.427 ns/m)
Formulas:

See also notes on PCB Impedance and Capacitance Calculator page.

PCB Impedance and Capacitance of Dual Stripline

Sun, 01 Mar 1998 00:00:00 +0000

W [m] width of the trace
H [m] height of trace above return plane
T [m] trace thickness
C [m] distance between the differential stripline pair
E_r [ ] relative permittivity of the dielectric

Are there distributed capacitive loads on this trace?
NoYes
L_a [m] average length of the traces attaching the loads
C_a [pF] average load capacitance

OUTPUT
Z₀ [Ohm] characteristic impedance
C₀ [F/m] capacitance per unit length
t_pd [s/m] propagation delay
L₀ [H/m] inductance per unit length
Z_c [Ohm] effective characteristic impedance
t'_pd [s/m] effective propagation delay

NOTE: numbers are in scientific notation (e.g.: 1.427e-9 s/m = 1.1427·10^-9 s/m = 1.427 ns/m)

CONDITIONS: (0.1 < W/H < 2.0); ( T/H < 0.25);
Formulas:

Capacitive Load Modify:

See also notes on PCB Impedance and Capacitance Calculator page.

PCB Impedance and Capacitance of Embedded Microstrip

Sun, 01 Mar 1998 00:00:00 +0000

W [m] width of the trace
H [m] height of trace above return plane
H1 [m] height of dielectric above return plane
T [m] trace thickness
E_r [ ] relative permittivity of the dielectric

Are there distributed capacitive loads on this trace?
NoYes
L_a [m] average length of the traces attaching the loads
C_a [pF] average load capacitance

OUTPUT
Z₀ [Ohm] characteristic impedance
C₀ [F/m] capacitance per unit length
t_pd [s/m] propagation delay
L₀ [H/m] inductance per unit length
Z_c [Ohm] effective characteristic impedance
t'_pd [s/m] effective propagation delay

NOTE: numbers are in scientific notation (e.g.: 1.427e-9 s/m = 1.1427·10^-9 s/m = 1.427 ns/m)

CONDITIONS: ( H1 > 1.2 H);
Formulas:

Capacitive Load Modify:

See also notes on PCB Impedance and Capacitance Calculator page.

PCB Impedance and Capacitance of Microstrip

Sun, 01 Mar 1998 00:00:00 +0000

W [m] width of the trace
H [m] height of dielectric above return plane
T [m] trace thickness
E_r [ ] relative permittivity of the dielectric

Are there distributed capacitive loads on this trace?
NoYes
L_a [m] average length of the traces attaching the loads
C_a [pF] average load capacitance

OUTPUT
Z₀ [Ohm] characteristic impedance
C₀ [F/m] capacitance per unit length
t_pd [s/m] propagation delay
L₀ [H/m] inductance per unit length
Z_c [Ohm] effective characteristic impedance
t'_pd [s/m] effective propagation delay

NOTE: numbers are in scientific notation (e.g.: 1.427e-9 s/m = 1.1427·10^-9 s/m = 1.427 ns/m)

CONDITIONS: ( 0.1 < W/H < 3.0);
Formulas:

Capacitive Load Modify:

See also notes on PCB Impedance and Capacitance Calculator page.

PCB Impedance and Capacitance of Stripline

Sun, 01 Mar 1998 00:00:00 +0000

W [m] width of the trace
H [m] height of trace above return plane
T [m] trace thickness
E_r [ ] relative permittivity of the dielectric

Are there distributed capacitive loads on this trace?
NoYes
L_a [m] average length of the traces attaching the loads
C_a [pF] average load capacitance

OUTPUT
Z₀ [Ohm] characteristic impedance
C₀ [F/m] capacitance per unit length
t_pd [s/m] propagation delay
L₀ [H/m] inductance per unit length
Z_c [Ohm] effective characteristic impedance
t'_pd [s/m] effective propagation delay

NOTE: numbers are in scientific notation (e.g.: 1.427e-9 s/m = 1.1427·10^-9 s/m = 1.427 ns/m)

CONDITIONS: (0.1 < W/H < 2.0); ( T/H < 0.25);
Formulas:

Capacitive Load Modify:

See also notes on PCB Impedance and Capacitance Calculator page.

PCBs Fabrication Methods

Sun, 01 Mar 1998 00:00:00 +0000

Printed Circuit Board (PCB) is a mechanical assembly consisting of layers of fiberglass sheet laminated with etched copper patterns. It is used to mount electronic parts in a rigid manner suitable for packaging. Also known as a Printed Wiring Board (PWB).
[Project] [Different methods to make PCBs] [Final work] [Bibliography/Reference]

Project
[Electric Scheme] [Part List - Bill of Materials] [Choose components from Data Sheets] [Choose Board type and dimension] [Draw the PCB layout] [Draw Fabrication scheme] [Draw Assembly scheme]

POST Beep Codes

Sun, 01 Mar 1998 00:00:00 +0000

When you turn the computer on, it performs Power On System Test (POST), during which it checks and initializes the system's internal components.
If a serious error occurs, the computer does not display a message but emits a series of long and short beeps instead.
Beeps are your computer's way of letting you know what's going on when the video signal is not working.
These codes are built in to the BIOS of the PC.
There is no official standard for these codes due to the many brands of BIOS that are out there.

To decode the meaning of your computer POST beep codes you must consult the manual of your motherboard.
If you don't have a motherboard manual or if it's incomplete you must search on the site of your computer manufacturer.
Another way is take off the computer case and look for BIOS manufacturer (just see if it says "AMI" or "Phoenix" ...).
For more information about BIOS and how to identify it please check this site: Wim's BIOS Page - Everything you want to know about BIOS.
Once you have determined your BIOS make, consult the following tables to see what's wrong with your computer.

RAID Technology

Sun, 01 Mar 1998 00:00:00 +0000

Reference and Sources:
The most part of text in this guide has been taken from copyrighted document of Adaptec, Inc. on site (www.adaptec.com)
Perceptive Solutions, Inc.
RAID stands for Redundant Array of Inexpensive (or sometimes "Independent") Disks.

RAID is a method of combining several hard disk drives into one logical unit (two or more disks grouped together to appear as a single device to the host system). RAID technology was developed to address the fault-tolerance and performance limitations of conventional disk storage. It can offer fault tolerance and higher throughput levels than a single hard drive or group of independent hard drives. While arrays were once considered complex and relatively specialized storage solutions, today they are easy to use and essential for a broad spectrum of client/server applications.

SCSI Technology

Sun, 01 Mar 1998 00:00:00 +0000

SCSI stands for Small Computer System Interface.
A standard for high-speed connections to peripherals.
[Source and Reference] [History] [Usage] [SCSI Hard Drives] [APPLE SCSI] [Hardware] [Single Ended or Differential] [Cabling] [SCSI Termination][SCSI Connectors]

Source and Reference
comp.periphs.scsi FAQ
ANSI T10 Technical Commitee Archive (SCSI draft spec's)
SCSI Trade Association
Adaptec, Inc.
Gary Field's SCSI Info Central

History
SCSI has it's roots in the mainframe world, but it's first implementation in the PC world came soon after the first PC. Shugart Associates devised an interface that they designated the SASI, or "Shugart Associates Standard Interface" They proposed that SASI be adopted by ANSI for small computers, but during the work required for ratification, they discovered the process would take too much effort, and that the IPI groups were already well into their effort. (which had many features the same as SASI) A decision was made to take features of both interfaces, and put forth a new specification for a new interface, SCSI was born, and ratified in 1986 by ANSI. Since then, many have said that the original spec. was not tight enough, and that it allowed Manufacturers to make drives that met the ANSI spec., but would not talk to each other. Recently, the ANSI SCSI committee has proposed newer, tighter, more extended specs., for SCSI-2, and now SCSI-3.

SODIUM METASILICATE, PENTAHYDRATE

Sun, 01 Mar 1998 00:00:00 +0000

Reference and Sources:
Rhodia

1. PRODUCT DESCRIPTION
Chemical Name or Synonym: DISODIUM TRIOXOSILICATE, PENTAHYDRATE ; CRYSTAMET
Molecular Formula: Na₂SiO_3^.nH₂O n=5

2. COMPOSITION/INFORMATION ON INGREDIENTS

Component CAS Reg Number OSHA
Hazard Percentage

SODIUM METASILICATE 6834-92-0 Y 100

3. HAZARDS IDENTIFICATION
A. EMERGENCY OVERVIEW:
Physical Appearance and Odor:
white granules solid, odorless.
Warning Statements:
DANGER! CAUSES SEVERE BURNS.

B. POTENTIAL HEALTH EFFECTS:
Acute Eye:
Corrosive. Causes burns, irritation.
Acute Skin:
Corrosive. Causes burns.
Acute Inhalation:
May cause upper respiratory tract irritation, dizziness, headache, shortness of breath, chest pain, muscle weakness, nausea, vomiting, serious damage to lung tissue and respiratory tract.
Acute Ingestion:
Causes nausea, vomiting, abdominal pain, chest pain, burns to mouth and esophagus.
Chronic Effects:
This product does not contain any ingredient designated by IARC, NTP, ACGIH or OSHA as probable or suspected human carcinogens.

4. FIRST AID MEASURES
FIRST AID MEASURES FOR ACCIDENTAL:
Eye Exposure:
Hold eyelids open and flush with a steady, gentle stream of water for at least 15 minutes. Seek immediate medical attention, preferably with an ophthalmologist. If the physician is not immediately available, eye irrigation should be continued for an additional 15 minutes. If it is necessary to transport the patient to a physician and the eye needs to be bandaged, use a dry sterile cloth pad and cover both eyes.
Skin Exposure:
In case of contact, immediately wash with plenty of soap and water for at least 15 minutes. Seek medical attention. Remove contaminated clothing and shoes while washing. Clean contaminated clothing and shoes before re-use or discard if they cannot be thoroughly cleaned.
Inhalation:
Remove victim from immediate source of exposure and assure that the victim is breathing. If breathing is difficult, administer oxygen, if available. If victim is not breathing, administer CPR (cardio-pulmonary resuscitation). Seek medical attention.
Ingestion:
If victim is conscious and alert, give 2-3 glasses of water to drink and do not induce vomiting. Seek immediate medical attention. Do not leave victim unattended. To prevent aspiration of swallowed product, lay victim on side with head lower than waist. Vomiting may occur spontaneously. If vomiting occurs and the victim is conscious, give water to further dilute the chemical.

MEDICAL CONDITIONS POSSIBLY AGGRAVATED BY EXPOSURE:
Inhalation of product may aggravate existing chronic respiratory problems such as asthma, emphysema or bronchitis. Skin contact may aggravate existing skin disease.

NOTES TO PHYSICIAN:
All treatments should be based on observed signs and symptoms of distress in the patient. Consideration should be given to the possibility that overexposure to materials other than this product may have occurred.
Treat symptomatically. No specific antidote available.

5. FIRE FIGHTING MEASURES
FIRE HAZARD DATA:
Flash Point:
Not Applicable
Extinguishing Media:
Not combustible. Use extinguishing method suitable for surrounding fire.
Special Fire Fighting Procedures:
Firefighters should wear NIOSH/MSHA approved self-contained breathing apparatus and full protective clothing. Dike area to prevent runoff and contamination of water sources. Dispose of fire control water later.
Unusual Fire and Explosion Hazards:
Not combustible.
Hazardous Decomposition Materials (Under Fire Conditions):
none known

6. ACCIDENTAL RELEASE MEASURES
Evacuation Procedures and Safety:
Wear appropriate protective gear for the situation. See Personal Protection information in Section 8.
Containment of Spill:
Follow procedure described below under Cleanup and Disposal of Spill.
Cleanup and Disposal of Spill:
Shovel up into an appropriate closed container (see Section 7: Handling and Storage). Avoid creation of dusty conditions. Decontaminate tools and equipment following cleanup.
Environmental and Regulatory Reporting:
If spilled on the ground, the affected area should be scraped clean and placed in a appropriate container for disposal. Prevent material from entering public sewer system or any waterways. Large spills should be handled according to a predetermined plan. For assistance in developing a plan contact the Technical Service Department using the Product Information phone number in Section 1.

7. HANDLING AND STORAGE
Minimum/Maximum Storage Temperatures:
Not Available
Handling:
Do not get on skin or in eyes. Do not breathe dusts. Do not ingest.
Storage:
Store in tightly closed containers. Store in an area that is cool, dry, well-ventilated, away from foodstuffs or animal feed.

8. EXPOSURE CONTROLS/PERSONAL PROTECTION
Introductory Remarks:
These recommendations provide general guidance for handling this product. Because specific work environments and material handling practices vary, safety procedures should be developed for each intended application. While developing safe handling procedures, do not overlook the need to clean equipment and piping systems for maintenance and repairs. Waste resulting from these procedures should be handled in accordance with Section 13: Disposal Considerations.
Assistance with selection, use and maintenance of worker protection equipment is generally available from equipment manufacturers.
Exposure Guidelines:
Exposure limits represent regulated or recommended worker breathing zone concentrations measured by validated sampling and analytical methods, meeting the regulatory requirements. The following limits apply to this material, where, if indicated, S=skin and C=ceiling limit:

PARTICULATES NOT OTHERWISE REGULATED RESPIRABLE FRACTION

Notes TWA STEL

OSHA 5 mg/cu m

PARTICULATES NOT OTHERWISE REGULATED TOTAL DUST

Notes TWA STEL

OSHA 15 mg/cu m

Engineering Controls:
Where engineering controls are indicated by use conditions or a potential for excessive exposure exists, the following traditional exposure control techniques may be used to effectively minimize employee exposures: local exhaust ventilation at the point of generation.
Respiratory Protection:
When respirators are required, select NIOSH/MSHA approved equipment based on actual or potential airborne concentrations and in accordance with the appropriate regulatory standards and/or industrial recommendations.
Eye/Face Protection:
Eye and face protection requirements will vary dependent upon work environment conditions and material handling practices. Appropriate ANSI Z87 approved equipment should be selected for the particular use intended for this material.
Eye contact should be prevented through use of chemical safety glasses with side shields or splash proof goggles. An emergency eye wash must be readily accessible to the work area. Face contact should be prevented through use of a face shield.
Skin Protection:
Skin contact should be prevented through use of suitable protective clothing, gloves and footwear, selected with regard for use conditions and exposure potential. Consideration must be given both to durability as well as permeation resistance.
Work Practice Controls:
Personal hygiene is an important work practice exposure control measure and the following general measures should be taken when working with or handling this material:

(1) Do not store, use, and/or consume foods, beverages, tobacco products, or cosmetics in areas where this material is stored.

(2) Wash hands and face carefully before eating, drinking, using tobacco, applying cosmetics, or using the toilet.

(3) Wash exposed skin promptly to remove accidental splashes of contact with this material.

9. PHYSICAL AND CHEMICAL PROPERTIES
Physical and Chemical properties here represent typical properties of this product. Contact the business area using the Product Information phone number in Section 1 for its exact specifications.
Physical Appearance:
white granules solid.
Odor:
odorless.
pH:
12.4 at 1 wt/wt%.
Specific Gravity:
Not Available
Water Solubility:
soluble 61 wt/wt% at 30 C (86 F).
Melting Point Range:
72 C (162 F)
Boiling Point Range:
Not Available
Vapor Pressure:
Not Available
Vapor Density:
Not Available
Molecular Weight:
212.14

10. STABILITY AND REACTIVITY
Chemical Stability:
This material is stable under normal handling and storage conditions described in Section 7.
Conditions To Be Avoided:
moisture
Materials/Chemicals To Be Avoided:
aluminum
fluorine
zinc
acids
The Following Hazardous Decomposition Products Might Be Expected:
Decomposition Type: thermal
none known
Hazardous Polymerization Will Not Occur.
Avoid The Following To Inhibit Hazardous Polymerization: not applicable
Avoid The Following To Inhibit Hazardous Polymerization:
not applicable

11. TOXICOLOGICAL INFORMATION
Acute Eye Irritation:

Toxicological Information and Interpretation

eye - eye irritation, 0.1 ml, rabbit. Corrosive.

Acute Skin Irritation:

Toxicological Information and Interpretation

skin - skin irritation, rabbit.

Non-irritating. Dry skin. (At 4 hours.).

skin - skin irritation, rabbit. Corrosive. Moist skin. (At 4 hours.).

Acute Dermal Toxicity:
No test data found for product.
Acute Respiratory Irritation:
No test data found for product.
Acute Inhalation Toxicity:
No test data found for product.
Acute Oral Toxicity:

Toxicological Information and Interpretation

LD50 - lethal dose 50% of test species, 847 mg/kg, rat.

Chronic Toxicity:
This product does not contain any substances that are considered by OSHA, NTP, IARC or ACGIH to be "probable" or "suspected" human carcinogens.

12. ECOLOGICAL INFORMATION
Ecotoxicological Information:
No data found for product.
Chemical Fate Information:
No data found for product.

13. DISPOSAL CONSIDERATIONS
Chemical additions, processing or otherwise altering this material may make the waste management information presented in this MSDS incomplete, inaccurate or otherwise inappropriate. Please be advised that state and local requirements for waste disposal may be more restrictive or otherwise different from federal laws and regulations. Consult state and local regulations regarding the proper disposal of this material.
EPA Hazardous Waste - NO

14. TRANSPORTATION INFORMATION
Transportation Status: IMPORTANT! Statements below provide additional
data on listed DOT classification. The listed Transportation Classification does not address regulatory variations due to changes in package size, mode of shipment or other regulatory descriptors.
US Department of Transportation
Hazard Class..... 8
Shipping Name:
DISODIUM TRIOXOSILICATE
ID Number....... UN3253
Packing Group.... III
Labels........... CORROSIVE
Emergency Guide #.... 154

15. REGULATORY INFORMATION
Inventory Status

Inventory Status

UNITED STATES (TSCA) Y

CANADA (DSL) Y

EUROPE (EINECS/ELINCS) Y

AUSTRALIA (AICS) Y

JAPAN (MITI) Y

SOUTH KOREA (KECL) Y

Y = All ingredients are on the inventory.

E = All ingredients are on the inventory or exempt from listing.

P = One or more ingredients fall under the polymer exemption or are on the no longer polymer list. All other ingredients are on the inventory or exempt from listing.

N = Not determined or one or more ingredients are not on the inventory and are not exempt from listing.

FEDERAL REGULATIONS
Inventory Issues:
The hydrated components of this product are listed on the TSCA Inventory under their parent anhydrous compounds.
SARA Title III Hazard Classes:

Fire Hazard - NO

Reactive Hazard - NO

Release of Pressure - NO

Acute Health Hazard - YES

Chronic Health Hazard - NO

STATE REGULATIONS:
This product does not contain any components that are regulated under California Proposition 65.

16. OTHER INFORMATION
National Fire Protection Association Hazard Ratings--NFPA(R):

3 Health Hazard Rating--Serious

0 Flammability Rating--Minimal

0 Instability Rating--Minimal

National Paint & Coating Hazardous Materials Identification System--HMIS(R):

3 Health Hazard Rating--Serious

0 Flammability Rating--Minimal

0 Reactivity Rating--Minimal

Reason for Revisions:
Change and/or addition made to Section 11.
Key Legend Information:
ACGIH - American Conference of Governmental Industrial Hygienists
OSHA - Occupational Safety and Health Administration
TLV - Threshold Limit Value
PEL - Permissable Exposure Limit
TWA - Time Weighted Average
STEL - Short Term Exposure Limit
NTP - National Toxicology Program
IARC - International Agency for Research on Cancer
ND - Not determined
RPI - Rhone-Poulenc Established Exposure Limits
Disclaimer:
The information herein is given in good faith but no warranty, expressed or implied, is made.

Standard Certification Marks

Sun, 01 Mar 1998 00:00:00 +0000

[VDE Testing and Certification Institute] [Underwriters Laboratories Inc.] [CSA International] [NEMKO] [DEMKO] [FIMKO] [SEMKO] [VARIOUS]
VDE Testing and Certification Institute
Mark Description
VDE Mark for appliances as technical equipment according to the Appliance Safety Law (GSG), for Medical Device Law (MPG), components and installation materials.The VDE Mark indicates conformity with the VDE standards or European or internationally harmonized standards resp. and confirms compliance with protective requirements of the applicable EC Directive(s). The VDE Mark is a symbol for electrical, mechanical, thermal, toxic, radiological and other hazards.
For appliances as technical equipment according to the GSG.For ready-to-use equipment, the licence holder may chose to affix the VDE Mark or the VDE GS Mark.
For products certified on the basis of harmonized certification agreements.Testing is based on harmonized European standards listed in the ENEC Agreement. Products (at present luminaires and related components, energy saving lamps, IT equipment, transformers, switches for appliances, electrical controls, certain types of capacitors and EMI suppression components) tested to tested to the listed standards may be marked with the ENEC Mark of the VDE. The approval of any other body participating in the ENEC Agreement is not required.
For appliances in compliance with standards for electromagnetic compatibility.The VDE EMC Mark expresses the conformity of a product with applicable standards for electromagnetic compatibility. The reliable function of the product in its electromagnetic environment is also included. The requirements for granting this mark comprise automatically and without restriction the compliance with applicable standards.
For cables, insulated cords, installation conduits and ducts, the VDE Cable Mark is applicable.

For cables and cords, the VDE Identification Thread may be used.
VDE-HARmonization Marking

The VDE HARmonization Marking or VDE HARmonization Thread resp. for cables and insulated cords according to harmonized certification procedures.Testing is based on the Harmonization Documents (HD) listed in the HAR Agreement. Products (harmonized power cables) tested and found in compliance with with the requirements of the mentioned standards may be marked with the VDE HARmonization Marking. Further information is available from the Laboratory for Cables and Cords, Materials and Special Tests.
The VDE Component Mark may be used for electronic components.
The CECC Mark for electronic components according to CECC Specifications.For electronic components according to CECC Specifications (CECC: CENELEC Electronic Components Committee) the CECC Mark may be used.
VDE-Reg.-Nr. XXXXX
VDE-Reg.-Nr. (VDE Certificate of Conformity in conjunction with factory surveillance)This mark is used in two cases: firstly, for products in compliance with applicable clauses of VDE standards in the absence of a fully applicable VDE standard, and secondly, if a product, e.g. a sub-assembly, requires the fulfillment of additional conditions when incorporated into complete equipment. For cables and insulated cords, the VDE-Reg.-Nr. or the relevant mark resp. is applicable in absence of special regulations for products which were tested on the basis other standards. Special constructions and all variations of non-harmonized cables and insulated cords belong to this category of products.

Standard Resistors Values

Sun, 01 Mar 1998 00:00:00 +0000

Reference and Sources:
Manuale di elettronica e telecomunicazioni - Giuseppe Biondo, Enrico Sacchi. Vol.II - IX-40
http://users.eecs.ukans.edu/~kuku/Standard_Resistors.php
The technical specifications for a resistor are:
nominal resistance value [Ohm]
tolerance [%] (indicate the maximum difference (+/-) between nominal and real resistance value)
nominal dissipable power [W]
power-temperature diagrams
temperature coefficient
nominal maximum voltage [V]
noise tension
tension coefficient
resistance-frequency characteristic
the first three are always indicated.
The succession of nominal resistance values fit a geometric progression:
N = 10^((n-1)/k)
where: N is the nominal resistance value at n position;
k is a coefficient related to tolerance in this way:
Tolerance [%] k Series Name
20 6 E6
10 12 E12
5 24 E24
2 48 E48
1 96 E96
0.5 192 E192
0.25 192 E192
0.1 192 E192

Stripline Impedance

Sun, 01 Mar 1998 00:00:00 +0000

This guide was written by Nathan Towne (2009-09)

Introduction

There are quite a few calculators for microstrip and stripline, and commercial products that do the same. But I was looking for a method of constructing a balun of controlled and fairly low impedances, which requires broadside-coupled stripline, where the widths of the traces mostly control the even (common-mode) impedance, and the spacing the odd (differential) impedance. I had in mind using copper tape for the traces and ground planes, polystyrene sheets for dielectric between the traces and the ground plane, and polyethylene or polystyrene sheets setting the spacing between the traces. The problem I encountered is the apparent dearth of information on the impedances of broadside-coupled stripline.
I ended up calculating those impedances using the Superfish/Poisson finite-element codes available from the Accelerator Code group at Los Alamos National Laboratory. The codes are used to calculate the capacitances of the even and odd modes by applying the appropriate boundary condition of each in a Laplace boundary-value calculation. The formula

where where μ and ε are the permeability and permittivity of the medium, respectively, and C includes the dielectric constant, is then applied to each mode's capacitance in turn.

This note gives the results of those calculations.

First a note about the meaning of the impedances, since there can be a factor of two between definitions. The figure to the right showing the simultaneous matched termination of the even and odd modes of a twin-lead transmission line shows how the even- and odd-mode impedances Z_even and Z_odd are to be interpreted, where Z_odd^* in parallel with Z_even is Z_odd:

The Ultimate Memory Guide v.6

Sun, 01 Mar 1998 00:00:00 +0000

0. Introduction to Static Electricity

Sun, 01 Mar 1998 00:00:00 +0000

Next page >>
Introduction to Static Electricity
In ordinary circumstances, static electricity and ESD are little more than an annoyance. However, in an increasingly technological age, the familiar static shock we receive when walking across a carpet can be costly or dangerous.
This same static discharge can ignite flammable mixtures and damage electronic components. Static electricity can attract contaminants in clean environments or cause products to stick together.
The cost of ESD-damaged electronic devices alone ranges from only a few cents for a simple diode to several hundred dollars for complex hybrids. Loss of production time in web processing industries due to static attraction is significant. When associated costs of repair and rework, shipping, labor, and overhead are included, clearly the opportunities exist for significant improvements in reducing losses to ESD and static electricity.
Index

1. An Introduction to ESD
2. Principles of ESD Control
3. An Overview of ESD Control Procedures and Materials
4. Auditing and Training
5. Device Sensitivity and Testing
6. ESD Standards
7. Bibliography

Links to Other ESD Resources
Associations
Arizona ESD Association
Electronic Industries Alliance
Electrostatics Society of America
IDEMA
Institute of Electrostatics - Japan
Institute of Environmental Sciences and Technology
IPC - Association Connecting Electronic Industries
JEDEC
Midwest Chapter, ESD Association
National Association of Radio and Telecommunications Engineers (NARTE)
Northeast Chapter, ESD Association
Surface Mount Technology Association
Publications
Circuits Assembly
Compliance Engineering
ESD Journal
Evaluation Engineering
Test and Measurement World
Test and Measurement Online
Standards
American National Standards Institute
American Society for Testing and Materials (ASTM)
British Standards Institution
DIN (Deutsches Institut fur Normung)
Electronic Industries Alliance
IDEMA
IPC - Association Connecting Electronic Industries
IEEE
International Electrotechnical Commission (IEC)
JEDEC

0. Table of Contents

Sun, 01 Mar 1998 00:00:00 +0000

This is an original HTML version of THE ULTIMATE MEMORY GUIDE^TM reproduced by permission. ©2000 Kingston Technology Company, Inc. All rights reserved. All trademarks and registered trademarks are the property of their respective owners. PowerPC is a trademark of International Business Machines Corporation, used under license therefrom. Windows is a trademark of Microsoft Corporation. All other trademarks and registered trademarks are the property of their respective owners. This publication may contain typographical errors or technical inaccuracies. Any errors will be periodically corrected in future updates of this publication. Kingston Technology reserves the right to make changes to text and/or illustrations in this document at any time. This publication is the sole property of Kingston Technology and may not be copied or modified in part or in whole without the express permission of Kingston Technology.
TOC Next page >>

00. About the Author

Sun, 01 Mar 1998 00:00:00 +0000

INDEX
About the Author

A versatile inventor, researcher and writer, Mr. Isidor Buchmann is the president, founder and CEO of Cadex Electronics Inc., located in Richmond (Vancouver), Canada.

Fascinated by electronics during his high school years, Mr. Buchmann took to inventing at an early age, designing a fuel-powered engine that was based on continuous combustion. His drawings and theory of operation were reviewed by Felix Wankel, inventor of the Wankel Rotary Engine, who kindly replied that while the design was indeed unique and original, manufacturing would be too expensive to be commercially viable. Further to his credit, Mr. Buchmann invented a broadcast radio that ran on no power — it required only an antenna and a ground connection (it didn’t even use a battery). Mr. Buchmann sold several of these radio receivers to his family and colleagues and later set up a workshop in the attic where he restored and resold old radios. After high school, a four-year apprenticeship as a Radio Technician brought him practical experience in a workshop environment as well as academic theory. Finally, his experience with radio communications in the Swiss army led to his decision to make electronics his life's work.

Realizing that conservative Switzerland would not satisfy his entrepreneurial spirit, Mr. Buchmann emigrated to Canada in 1966, eventually finding employment in the radio communications department at General Electric. There he realized that a major problem with two-way radios was the battery's short life and, as part of his job, tested a wide variety.Chapter 18: Beginnings and Horizons of customer batteries that came in. In his spare time at home, he continued to research and develop electronic devices in his spare time, developing a battery analyzer that featured a ‘recondition’ program which restored weakened nickel cadmium batteries.

To prevent a conflict of interest with his employer, Mr. Buchmann quit his job with GE and started Cadex. The first battery analyzer, the Cadex 450, was introduced in 1981 but failed to achieve the anticipated market acceptance. Still in his spare time, Mr. Buchmann designed the modular Cadex 550 battery analyzer. This model sold reasonably well at first but it soon became evident that manufacturing methods needed to be improved to make it cost effective.

In 1984, Cadex moved from a small room in Mr. Buchmann's residence to rented facilities. With increased overhead costs, a staff to maintain and sluggish sales, cash flow became tight. Bank loans for start-up companies, especially high-tech firms, were almost non-existent at the time, so Mr. Buchmann worked from home during the day looking after his growing children and spent time in the office during the evening. Happily, the company survived the slump and managed to add a number of new products. Profitability returned and the staff grew.

Knowing that the wealth of an organization is in human resources, Isidor resolved to provide an environment that was conducive to attracting good people with skills that complemented his own strengths and weaknesses. Spacious new headquarters in a park-like setting overlooking the scenic Fraser River add to a pleasant working experience. Under Mr. Buchmann’s leadership, new and innovative products were developed that generated rapid growth and created global recognition of Cadex.

Today, Cadex is a world leader in the design and manufacture of battery analyzers and chargers.

00. Author's Note

Sun, 01 Mar 1998 00:00:00 +0000

<< Previous page INDEX Next page >>
Author's Note

Battery user groups have asked me to write an edited version of Batteries in a Portable World. The first edition was published in 1997. Much has changed since then.

My very first publication in book form was entitled Strengthening the Weakest Link. It was, in part, a collection of battery articles which I had written. These articles had been published in various trade magazines and gained the interest of many readers. This goes back to the late 1980s and the material covered topics such as the memory effect of NiCd batteries and how to restore them.

In the early 1990s, attention moved to the nickel-metal hydride (NiMH) and the articles compared the classic nickel cadmium (NiCd) with the NiMH, the new kid on the block. In terms of longevity and ruggedness, the NiMH did not perform so well when placed against the NiCd and I was rather blunt about it. Over the years, however, the NiMH improved and today this chemistry performs well for mobile phones and other applications.

Then came the lithium-ion (Liion), followed by the lithium-ion polymer (Liion polymer). Each of these new systems, as introduced, claimed better performance, freedom from the memory effect and longer runtimes than the dated NiCd. In many cases, the statements made by the manufacturers about improvements were true, but not all users were convinced.

The second edition of Batteries in a Portable World has grown to more than three times the size of the previous version. It describes the battery in a broader scope and includes the latest technologies, such as battery quick test.

Some new articles have also been woven in and some redundancies cannot be fully avoided. Much of this fresh material has been published in trade magazines, both in North America and abroad.

In the battery field, there is no black and white, but many shades of gray. In fact, the battery behaves much like a human being. It is mystical, unexplainable and can never be fully understood. For some users, the battery causes no problems at all, for others it is nothing but a problem. Perhaps a comparison can be made with the aspirin. For some, it works to remedy a headache, for others the headache gets worse. And no one knows exactly why.

Batteries in a Portable World is written for the non-engineer. It addresses the use of the battery in the hands of the general public, far removed from the protected test lab environment of the manufacturer. Some information contained in this book was obtained through tests performed in Cadex laboratories; other knowledge was gathered by simply talking to diverse groups of battery users. Not all views and opinions expressed in the book are based on scientific facts. Rather, they follow opinions of the general public, who use batteries. Some difference of opinion with the reader cannot be avoided. I will accept the blame for any discrepancies, if justified.

Readers of the previous edition have commented that I favor the NiCd over the NiMH. Perhaps this observation is valid and I have taken note. Having been active in the mobile radio industry for many years, much emphasis was placed on the longevity of a battery, a quality that is true of the NiCd. Today’s battery has almost become a disposable item. This is especially true in the vast mobile phone market where small size and high energy density take precedence over longevity.

Manufacturers are very much in tune with customers’ demands and deliver on maximum runtime and small size. These attributes are truly visible at the sales counter and catch the eye of the vigilant buyer. What is less evident is the shorter service life. However, with rapidly changing technology, portable equipment is often obsolete by the time the battery is worn out. No longer do we need to pamper a battery like a Stradivarius violin that is being handed down from generation to generation. With mobile phones, for example, upgrading to a new handset may be cheaper than purchasing a replacement battery. Small size and reasonable runtime are key issues that drive the consumer market today. Longevity often comes second or third.

In the industrial market such as public safety, biomedical, aviation and defense, requirements are different. Longevity is given preference over small size. To suit particular applications, battery manufacturers are able to adjust the amount of chemicals and active materials that go into a cell. This fine-tuning is done on nickel-based as well as lead and lithium-based batteries.

In a nutshell, the user is given the choice of long runtime, small size or high cycle count. No one single battery can possess all these attributes. Battery technology is truly a compromise.

00. Frequently Asked Questions about Batteries (FAQ)

Sun, 01 Mar 1998 00:00:00 +0000

INDEX
Frequently Asked Questions about Batteries (FAQ)

Which battery chemistry is best suited for my application?

Chapter 2: Battery Chemistries
Chapter 6: The Secrets of Battery Runtime, Increasing Internal Resistance
Chapter 8: Choosing the Right Battery
Chapter 9: Internal Battery Resistance
Chapter 15: Caring for Your Batteries from Birth to Retirement

Is the lithium polymer superior to other systems?

Chapter 2: Battery Chemistries, The Lithium Polymer Battery

What do I need to know about chargers?

Chapter 4: Proper Charge Methods

How deeply can I discharge a battery?

Chapter 5: Discharge Methods

What should I know about temperature extremes?

Chapter 4: Proper Charge Methods, Charging at High and Low Temperatures
Chapter 5: Discharge Methods

Is the ‘smart’ battery a help or deterrent?

Chapter 7: The ‘Smart’ Battery

What causes a battery to fail?

Chapter 6: The Secrets of Battery Runtime
Chapter 10: Getting the Most from your Batteries

Does a new battery always perform flawlessly?

Chapter 12: Battery Maintenance Equipment, Battery Analyzers
Chapter 15: Caring for Your Batteries from Birth to Retirement, Priming
Chapter 16: Practical Battery Tips, Battery Analyzers for Critical Missions

Do I need to discharge a battery before charging it?

Chapter 10: Getting the Most from your Batteries, How to Restore and Prolong Nickel-based Batteries

How does a battery analyzer improve battery reliability?

Chapter 11: Maintaining Fleet Batteries
Chapter 12: Battery Maintenance Equipment
Chapter 15: Caring for Your Batteries from Birth to Retirement, The Million Dollar Battery Problem

Can I restore weak batteries and how well do they perform?

Chapter 10: Getting the Most from your Batteries, Battery Recovery Rate

How accurate is battery quick testing?

Chapter 13: Making Battery Quick-Test Feasible

How do I store a battery?

Chapter 15: Caring for Your Batteries from Birth to Retirement, Storage

How do I recycle a battery?

Chapter 15: Caring for Your Batteries from Birth to Retirement, Battery Recycling

00. Introduction

Sun, 01 Mar 1998 00:00:00 +0000

<< Previous page INDEX Next page >>
Introduction

During the last few decades, rechargeable batteries have made only moderate improvements in terms of higher capacity and smaller size. Compared with the vast advancements in areas such as microelectronics, the lack of progress in battery technology is apparent. Consider a computer memory core of the sixties and compare it with a modern microchip of the same byte count. What once measured a cubic foot now sits in a tiny chip. A comparable size reduction would literally shrink a heavy-duty car battery to the size of a coin. Since batteries are still based on an electrochemical process, a car battery the size of a coin may not be possible using our current techniques.

Research has brought about a variety of battery chemistries, each offering distinct advantages but none providing a fully satisfactory solution. With today’s increased selection, however, better choices can be applied to suit a specific user application.

The consumer market, for example, demands high energy densities and small sizes. This is done to maintain adequate runtime on portable devices that are becoming increasingly more powerful and power hungry. Relentless downsizing of portable equipment has pressured manufacturers to invent smaller batteries. This, however, must be done without sacrificing runtimes. By packing more energy into a pack, other qualities are often compromised. One of these is longevity.

Long service life and predictable low internal resistance are found in the NiCd family. However, this chemistry is being replaced, where applicable, with systems that provide longer runtimes. In addition, negative publicity about the memory phenomenon and concerns of toxicity in disposal are causing equipment manufacturers to seek alternatives.

Once hailed as a superior battery system, the NiMH has also failed to provide the universal battery solution for the twenty-first century. Shorter than expected service life remains a major complaint.

The lithium-based battery may be the best choice, especially for the fast-moving commercial market. Maintenance-free and dependable, Li-ion is the preferred choice for many because it offers small size and long runtime. But this battery system is not without problems. A relatively rapid aging process, even if the battery is not in use, limits the life to between two and three years. In addition, a current-limiting safety circuit limits the discharge current, rendering the Li-ion unsuitable for applications requiring a heavy load. The Li-ion polymer exhibits similar characteristics to the Li-ion. No major breakthrough has been achieved with this system. It does offer a very slim form factor but this quality is attained in exchange for slightly less energy density.

With rapid developments in technology occurring today, battery systems that use neither nickel, lead nor lithium may soon become viable. Fuel cells, which enable uninterrupted operation by drawing on a continuous supply of fuel, may solve the portable energy needs in the future. Instead of a charger, the user carries a bottle of liquid energy. Such a battery would truly change the way we live and work.

This book addresses the most commonly used consumer and industrial batteries, which are NiCd, NiMH, Lead Acid, and Li-ion/polymer. It also includes the reusable alkaline for comparison. The absence of other rechargeable battery systems is done for reasons of clarity. Some weird and wonderful new battery inventions may only live in experimental labs. Others may be used for specialty applications, such as military and aerospace. Since this book addresses the non-engineer, it is the author’s wish to keep the matter as simple as possible.

00. Links

Sun, 01 Mar 1998 00:00:00 +0000

INDEX
Links

General Links

Elsevier Science, Journal of Power Sources

Energizer Battery Home Page

Electrochemistry Refresher

Online battery simulator

Canadian power technologies VRLA information

ThermoAnalystics Inc.

Battery Review

Huret Associates, Inc.

Manufacturers

www.cadex.com

www.polystor.com/technology.php

www.molienergy.com/products.php

www.sanyobatteries.net/NEWprodsIND.html

INDEX

00. Order Batteries in a Portable World

Sun, 01 Mar 1998 00:00:00 +0000

INDEX
Order Batteries in a Portable World

A paperback book of Batteries in a Portable World can be purchased on this site.

The 300-page book is easy and entertaining to read; it makes minimal use of technical jargon.

The well-organized layout helps you find information quickly. The book is
especially helpful to the busy professional who needs a crash course in
understanding this marvelous power source, the battery.

You can order the book in 3 ways:

Print the order form and mail or fax it to us. If faxed, payment must be made by credit card. If mailed, you can alternatively pay by check.

Coming soon: Fill out and send our secure on-line order form. Payment by credit card is required.

Order by phone: 604-231-7777

00. Part Four – Beyond Batteries

Sun, 01 Mar 1998 00:00:00 +0000

<< Previous page INDEX Next page >>
Part Four – Beyond Batteries
Chapter 18: Beginnings and Horizons
18.1 About the Author
18.2 About Cadex
18.3 Working with Natural Beauty
18.4 Customer Comments
18.5 Cadex Products
<< Previous page INDEX Next page >>

00. Part One - Battery Basics Everyone Should Know

Sun, 01 Mar 1998 00:00:00 +0000

<< Previous page INDEX Next page >>
Part One - Battery Basics Everyone Should Know
Author's Note

Introduction

Chapter 1: When was the battery invented?

Chapter 2: Battery Chemistries
2.1 Chemistry Comparison
2.2 The Nickel Cadmium (NiCd) Battery
2.3 The Nickel-Metal Hydride (NiMH) Battery
2.4 The Lead Acid Battery
2.5 The Lithium Ion Battery
2.6 The Lithium Polymer Battery
2.7 Reusable Alkaline Batteries
2.8 The Supercapacitor

Chapter 3: The Battery Pack
3.1 The Cylindrical Cell
3.2 The Button Cell
3.3 The prismatic Cell
3.4 The Pouch Cell
3.5 Series and Parallel Configurations
3.6 Protection Circuits

Chapter 4: Proper Charge Methods
4.1 All About Chargers
4.2 Charging the Nickel Cadmium Battery
4.3 Charging the Nickel-Metal Hydride Battery
4.4 Charging the Lead Acid Battery
4.5 Charging the Lithium Ion Battery
4.6 Charging the Lithium Polymer Battery
4.7 Charging at High and Low Temperatures
4.8 Ultra-fast Chargers
4.9 Charge IC Chips

Chapter 5: Discharge Methods
5.1 C-rate
5.2 Depth of Discharge
5.3 Pulse Discharge
5.4 Discharging at High and Low Temperature

00. Part Three – Knowing Your Battery

Sun, 01 Mar 1998 00:00:00 +0000

<< Previous page INDEX Next page >>
Part Three – Knowing Your Battery
Chapter 14: Non-Correctable Battery Problems
14.1 High Self-discharge
14.2 Low Capacity Cells
14.3 Cell Mismatch
14.4 Shorted Cells Loss of Electrolyte

Chapter 15: Caring for Your Batteries from Birth to Retirement
15.1 Storage
15.2 Priming
15.3 The Million Dollar Battery Problem
15.4 To the Service Counter, and No Further
15.5 The Quick Fix
15.6 Battery Warranty
15.7 Battery Recycling

Chapter 16: Practical Battery Tips
16.1 Personal Field Observations
16.2 The Correct Battery for the Job
16.3 Battery Analyzers for Critical Missions

Chapter 17: Did you know . . . ?
17.1 The Cost of Mobile Power
17.2 The Fuel Cell
17.3 The Electric Vehicle
17.4 Strengthening the Weakest Link

00. Part Two – You and the Battery

Sun, 01 Mar 1998 00:00:00 +0000

<< Previous page INDEX Next page >>
Part Two – You and the Battery
Chapter 6: The Secrets of Battery Runtime
6.1 Declining Capacity
6.2 Increasing Internal Resistance
6.3 Elevated Self-Discharge
6.4 Premature Voltage Cut-off

Chapter 7: The ‘Smart’ Battery
7.1 The Single Wire Bus
7.2 The SMBus
7.3 The State-of-Charge Indicator
7.4 The Tri-State Fuel Gauge
7.5 The Target Capacity Selector
7.6 Fuel Gauges for Large Batteries

Chapter 8: Choosing the Right Battery
8.1 What’s the best battery for mobile phones?
8.2 What’s the best battery for two-way radios?
8.3 What’s the best battery forlaptops?
8.4 Selecting a Lasting Battery

Chapter 9: Internal Battery Resistance
9.1 Why do seemingly good batteries fail on digital equipment?
9.2 How is the internal battery resistance measured?
9.3 What’s the difference between internal resistance and impedance?

Chapter 10: Getting the Most from your Batteries
10.1 Memory: myth or fact?
10.2 How to Restore and Prolong Nickel-based Batteries
10.3 The Effect of Zapping
10.4 How to Restore and Prolong Sealed Lead Acid Batteries
10.5 How to Prolong Lithium-based Batteries
10.6 Battery Recovery Rate

Chapter 11: Maintaining Fleet Batteries
11.1 The ‘Green Light’ Lies
11.2 Battery Maintenance, a Function of Quality Control
11.3 Battery Maintenance Made Simple
11.4 Battery Maintenance as a Business

Chapter 12: Battery Maintenance Equipment
12.1 Conditioning Chargers
12.2 Battery Analyzers
12.3 Battery Analyzers for Maintenance-Free Batteries
12.4 Battery throughput
12.5 Battery Maintenance Software

Chapter 13: Making Battery Quick-Test Feasible
13.1 Battery Specific Quick Testing
13.2 Three-Point Quick Test
13.3 The Evolving Battery
13.4 The Cadex Quicktest Method
13.5 How does the Cadex Quicktest work?
13.6 Battery Testing and the Internet
13.7 Electrochemical Impedance Spectroscopy

00. Table Of Content

Sun, 01 Mar 1998 00:00:00 +0000

NOTE: THIS DOCUMENT IS OBSOLETE, PLEASE CHECK THE NEW VERSION: "The Digital Audio Resampling Home Page", by Julius O. Smith III, Copyright © 2016-05-17 by Julius O. Smith III - Center for Computer Research in Music and Acoustics (CCRMA), Stanford University

Next page >>

Table Of Content
What is Bandlimited Interpolation?
Available Software
Theory of Operation
Abstract
Introduction
Theory of Ideal Bandlimited Interpolation
From Theory to Practice
Implementation
Quantization Issues
Choice of Table Size
Choice of Interpolation Resolution
Conclusions
Appendix A: Exact Sinc-Interpolation of Sampled Periodic Signals
Appendix B: Relation between Sinc and Lagrange Interpolation
Bibliography
About this document ...

00. Table Of Content

Sun, 01 Mar 1998 00:00:00 +0000

NOTE: THIS DOCUMENT IS OBSOLETE, PLEASE CHECK THE NEW VERSION: "Introduction to Digital Filters with Audio Applications", by Julius O. Smith III, Copyright © 2017-11-26 by Julius O. Smith III - Center for Computer Research in Music and Acoustics (CCRMA), Stanford University

INDEX Next page >>
Table Of Content
Positive Real Functions
Relation to Stochastic Processes
Relation to Schur Functions
Relation to functions positive real in the right-half plane
Special cases and examples
Minimum Phase (MP) polynomials in
Conjectured Properties
Introduction to Digital Filter Theory
Linearity and Time-Invariance
Difference Equation
Convolution Representation
Frequency Response
Phase Delay and Group Delay
Vector Space Concepts
Specific Norms
Concavity (Convexity)
Concave Norms
Gradient Descent
Taylor's Theorem
Newton's Method
Maxims of Signal Processing
Index
Bibliography
About this document ...
INDEX Next page >>

00. TOC

Sun, 01 Mar 1998 00:00:00 +0000

Table of Contents
Battery FAQ
New Articles
About the Author
Order Book
Links

Next page >> The battery has become our steady travel companion - it helps call a friend; it expands our workplace beyond four walls; and it supports critical missions for people in need.

Performance specifications for batteries and chargers are often based on ideal conditions.

Manufacturers carry out battery tests on brand new equipment within protected environments. In Batteries in a Portable World, Mr. Buchmann observes the battery as used by consumers in everyday life.

The battery handbook addresses the strengths and limitations of the modern battery. It illustrates which battery chemistry is most appropriate for each application. You will learn about charger technology, battery maintenance to prolong battery life and methods to restore weak batteries with a battery analyzer.

Table of Contents

01. Positive Real Functions

Sun, 01 Mar 1998 00:00:00 +0000

NOTE: THIS DOCUMENT IS OBSOLETE, PLEASE CHECK THE NEW VERSION: "Introduction to Digital Filters with Audio Applications", by Julius O. Smith III, Copyright © 2017-11-26 by Julius O. Smith III - Center for Computer Research in Music and Acoustics (CCRMA), Stanford University

<< Previous page TOC INDEX Next page >>
Positive Real Functions
Any passive driving-point impedance, such as the impedance of a violin bridge, is positive real. Positive real functions have been studied extensively in the continuous-time case in the context ofnetwork synthesis [Brune 1931,Van Valkenburg 1960]. Very little, however, seems to be available in the discrete time case. The purpose of this home page is to collect some facts about positive real transfer functions for discrete-time linear systems.
Definition. A complex valued function of a complex variable is said to bepositive real (PR) if
We now specialize to the subset of functions representable as a ratio of finite-order polynomials in . This class of ``rational'' functions is the set of all transfer functions of finite-order time-invariant linear systems, and we write to denote a member of this class. We use the convention that stable, minimum phase systems are analytic and nonzero in the strict outer disk.¹*The strict outer disk is defined as the region in the extended complex plane. Condition (1) implies that for to be PR, the polynomial coefficients must be real, and therefore complex poles and zeros must exist in conjugate pairs. We assume from this point on that satisfies (1). From (2) we derive the facts below.
Theorem. A real rational function is PR iff .
Proof. Expressing in polar form gives
(1)
(2)

since the zeros of are isolated.
Theorem. is PR iff is PR.
Proof. Assuming is PR, we have by Thm. (1),

Theorem. A PR function is analytic and nonzero in the strict outer disk.
Proof. (By contradiction)
Without loss of generality, we treat only order polynomials

which are nondegenerate in the sense that . Since facts about are readily deduced from facts about , we set at no great loss.
The general (normalized) causal, finite-order, linear, time-invariant transfer function may be written
(3)
(4)
(5)
(6)

where is the number of distinct poles, each of multiplicity ,and

Suppose there is a pole of multiplicity outside the unit circle. Without loss of generality, we may set , and with . Then for near , we have
(7)
(8)

Consider the circular neighborhood of radius described by . Since we may choose so that all points in this neighborhood lie outside the unit circle. If we write the residue of the factor in polar form as , then we have, for sufficiently small ,

Therefore, approaching the pole at an angle gives

which cannot be confined to satisfy Thm. (1) regardless of the value of the residue angle , or the pole angle ( cannot be zero by hypothesis). We thus conclude that a PR function can have no poles in the outer disk. By Thm. (1), we conclude that positive real functions must be minimum phase.
Corollary. In equation Eq. (6), .
Proof. If , then there are poles at infinity. As , , we must have .
Corollary. The log-magnitude of a PR function has zero mean on the unit circle.
This is a general property of stable, minimum-phase transfer functions which follows immediately from the argument principle [Markel and Gray 1976,Nehari 1952].
Corollary. A rational PR function has an equal number of poles and zeros all of which are in the unit disk.
This really a convention for numbering poles and zeros. In Eq. (6), we have , and all poles and zeros inside the unit disk. Now, if then we have extra poles at induced by the numerator. If , then zeros at the origin appear from the denominator.
Corollary. Every pole on the unit circle of a positive real function must be simple with a real and positive residue.
Proof. We repeat the previous argument using a semicircular neighborhood of radius about the point to obtain

In order to have near this pole, it is necessary that and .
Corollary. If is PR with a zero at , then

must satisfy

(9)
(10)

Proof. We may repeat the above for .
Theorem. Every PR function has a causal inverse -transform .
Proof. This follows immediately from analyticity in the outer disk [Papoulis 1977, pp. 30-36] However, we may give a more concrete proof as follows. Suppose is non-causal. Then there exists such that . We have,
(11)
(12)

Hence, has at least one pole at infinity and cannot be PR by Thm. (1). Note that this pole at infinity cannot be cancelled since otherwise
(13)
(14)
(15)

which contradicts the hypothesis that is non-causal.
Theorem. is PR iff it is analytic for , poles on the unit circle are simple with real and positive residues, and for .
Proof. If is positive real, the conditions stated hold by virtue of Thm. (1) and the definition of positive real.
To prove the converse, we first show nonnegativity on the upper semicircle implies nonnegativity over the entire circle.
(16)
(17)
(18)
(19)
(20)
(21)

Alternatively, we might simply state that real even in .
Next, since the function is analytic everywhere except at, it follows that is analytic wherever is finite. There are no poles of outside the unit circle due to the analyticity assumption, and poles on the unit circle have real and positive residues. Referring again to the limiting form Eq. (1) of near a pole on the unit circle at , we see that
(22)
(23)
(24)
(25)
(26)

since the residue is positive, and the net angle does not exceed . From Eq. (26) we can state that for points with modulus , we have For all , there exists such that . Thus is analytic in the strict outer disk, and continuous up to the unit circle which forms its boundary. By the maximum modulus theorem [Churchill 1960],

occurs on the unit circle. Consequently,

For example, if a transfer function is known to be asymptotically stable, then a frequency response with nonnegative real part implies that the transfer function is positive real.
Note that consideration of leads to analogous necessary and sufficient conditions for to be positive real in terms of its zeros instead of poles.

01. What is Bandlimited Interpolation?

Sun, 01 Mar 1998 00:00:00 +0000

NOTE: THIS DOCUMENT IS OBSOLETE, PLEASE CHECK THE NEW VERSION: "The Digital Audio Resampling Home Page", by Julius O. Smith III, Copyright © 2016-05-17 by Julius O. Smith III - Center for Computer Research in Music and Acoustics (CCRMA), Stanford University

<< Previous page TOC Next page >>
What is Bandlimited Interpolation?
Bandlimited interpolation of discrete-time signals is a basic tool having extensive application in digital signal processing. In general, the problem is to correctly compute signal values at arbitrary continuous times from a set of discrete-time samples of the signal amplitude. In other words, we must be able to interpolate the signal between samples. Since the original signal is always assumed to be bandlimited to half the sampling rate, (otherwisealiasing distortion would occur upon sampling), Shannon's sampling theorem tells us the signal can be exactly and uniquely reconstructed for all time from its samples by bandlimited interpolation.
There are many methods for interpolating discrete points. For example,Lagrange interpolation is the classical technique of finding an order polynomial which passes through given points.
The technique known as cubic splines fits a third-order polynomial through two points so as to achieve a certain slope at one of the points. (This allows for a smooth chain of third-order polynomial passing through a set of points.)
You may also have heard of Bezier splines which interpolate a set of points using smooth curves which don't necessarily pass through the points. (Bezier curves are commonly used in graphics and drawing programs, such as Adobe Illustrator.)
The above methods are suitable for graphics and other uses, but they are not ideal for digital audio. In digital audio, what matters is the audibility of interpolation error between samples. Since Shannon's sampling theorem says that it is possible to restore an audio signal exactly from its samples, it makes sense that the best digital audio interpolators would be based on that theory. Such ``ideal'' interpolation is called bandlimited interpolation.
A bandlimited interpolation algorithm designed along these lines is described in the theory of operation tutorial. There is also open-source softwareavailable in the C programming language.

01. When was the battery invented?

Sun, 01 Mar 1998 00:00:00 +0000

<< Previous page INDEX Next page >>
1. When was the battery invented?

One of the most remarkable and novel discoveries in the last 400 years has been electricity. One may ask, “Has electricity been around that long?” The answer is yes, and perhaps much longer. But the practical use of electricity has only been at our disposal since the mid-to late 1800s, and in a limited way at first. At the world exposition in Paris in 1900, for example, one of the main attractions was an electrically lit bridge over the river Seine.

The earliest method of generating electricity occurred by creating a static charge. In 1660, Otto von Guericke constructed the first electrical machine that consisted of a large sulphur globe which, when rubbed and turned, attracted feathers and small pieces of paper. Guericke was able to prove that the sparks generated were truly electrical.

The first suggested use of static electricity was the so-called “electric pistol”. Invented by Alessandro Volta (1745-1827), an electrical wire was placed in a jar filled with methane gas. By sending an electrical spark through the wire, the jar would explode.

Volta then thought of using this invention to provide long distance communications, albeit only addressing one Boolean bit. An iron wire supported by wooden poles was to be strung from Como to Milan, Italy. At the receiving end, the wire would terminate in a jar filled with methane gas. On command, an electrical spark is sent by wire that would detonate the electric pistol to signal a coded event. This communications link was never built.

Figure 1-1: Alessandro Volta, inventor of the electric battery.
Volta’s discovery of the decomposition of water by an electrical current laid the foundation of electrochemistry. ©Cadex Electronics Inc.

In 1791, while working at Bologna University, Luigi Galvani discovered that the muscle of a frog contracted when touched by a metallic object. This phenomenon became known as animal electricity — a misnomer, as the theory was later disproven. Prompted by these experiments, Volta initiated a series of experiments using zinc, lead, tin or iron as positive plates. Copper, silver, gold or graphite were used as negative plates.

Volta discovered in 1800 that a continuous flow of electrical force was generated when using certain fluids as conductors to promote a chemical reaction between the metals or electrodes. This led to the invention of the first voltaic cell, better know as the battery. Volta discovered further that the voltage would increase when voltaic cells were stacked on top of each other.

Figure 1-2: Four variations of Volta’s electric battery.
Silver and zinc disks are separated with moist paper. ©Cadex Electronics Inc.

01. When was the battery invented? 2

Sun, 01 Mar 1998 00:00:00 +0000

<< Previous page INDEX Next page >>

In the same year, Volta released his discovery of a continuous source of electricity to the Royal Society of London. No longer were experiments limited to a brief display of sparks that lasted a fraction of a second. A seemingly endless stream of electric current was now available.

France was one of the first nations to officially recognize Volta’s discoveries. At the time, France was approaching the height of scientific advancements and new ideas were welcomed with open arms to support the political agenda. By invitation, Volta addressed the Institute of France in a series of lectures at which Napoleon Bonaparte was present as a member of the Institute.

Figure 1-3: Volta’s experimentations at the French National Institute.
Volta’s discoveries so impressed the world that in November 1800, he was invited by the French National Institute to lectures in which Napoleon Bonaparte participated. Later, Napoleon himself helped with the experiments, drawing sparks from the battery, melting a steel wire, discharging an electric pistol and decomposing water into its elements. ©Cadex Electronics Inc.

New discoveries were made when Sir Humphry Davy, inventor of the miner’s safety lamp, installed the largest and most powerful electric battery in the vaults of the Royal Institution of London. He connected the battery to charcoal electrodes and produced the first electric light. As reported by witnesses, his voltaic arc lamp produced “the most brilliant ascending arch of light ever seen.”

Davy's most important investigations were devoted to electrochemistry. Following Galvani's experiments and the discovery of the voltaic cell, interest in galvanic electricity had become widespread. Davy began to test the chemical effects of electricity in 1800. He soon found that by passing electrical current through some substances, these substances decomposed, a process later called electrolysis. The generated voltage was directly related to the reactivity of the electrolyte with the metal. Evidently, Davy understood that the actions of electrolysis and the voltaic cell were the same.

In 1802, Dr. William Cruickshank designed the first electric battery capable of mass production. Cruickshank had arranged square sheets of copper, which he soldered at their ends, together with sheets of zinc of equal size. These sheets were placed into a long rectangular wooden box that was sealed with cement. Grooves in the box held the metal plates in position. The box was then filled with an electrolyte of brine, or watered down acid.

The third method of generating electricity was discovered relatively late — electricity through magnetism. In 1820, André-Marie Ampère (1775-1836) had noticed that wires carrying an electric current were at times attracted to one another while at other times they were repelled.

In 1831, Michael Faraday (1791-1867) demonstrated how a copper disc was able to provide a constant flow of electricity when revolved in a strong magnetic field. Faraday, assisting Davy and his research team, succeeded in generating an endless electrical force as long as the movement between a coil and magnet continued. The electric generator was invented. This process was then reversed and the electric motor was discovered. Shortly thereafter, transformers were developed that could convert electricity to a desired voltage. In 1833, Faraday established the foundation of electrochemistry with Faraday's Law, which describes the amount of reduction that occurs in an electrolytic cell.

In 1836, John F. Daniell, an English chemist, developed an improved battery which produced a steadier current than Volta's device. Until then, all batteries had been composed of primary cells, meaning that they could not be recharged. In 1859, the French physician Gaston Platé invented the first rechargeable battery. This secondary battery was based on lead acid chemistry, a system that is still used today.

Figure 1-4: Cruickshank and the first flooded battery.
William Cruickshank, an English chemist, built a battery of electric cells by joining zinc and copper plates in a wooden box filled with electrolyte. This flooded design had the advantage of not drying out with use and provided more energy than Volta’s disc arrangement. ©Cadex Electronics Inc.

01. When was the battery invented? 3

Sun, 01 Mar 1998 00:00:00 +0000

<< Previous page INDEX Next page >>

Toward the end of the 1800s, giant generators and transformers were built. Transmission lines were installed and electricity was made available to humanity to produce light, heat and movement. In the early twentieth century, the use of electricity was further refined. The invention of the vacuum tube enabled generating controlled signals, amplifications and sound. Soon thereafter, radio was invented, which made wireless communication possible.

In 1899, Waldmar Jungner from Sweden invented the nickel-cadmium battery, which used nickel for the positive electrode and cadmium for the negative. Two years later, Edison produced an alternative design by replacing cadmium with iron. Due to high material costs compared to dry cells or lead acid storage batteries, the practical applications of the nickel-cadmium and nickel-iron batteries were limited.

It was not until Shlecht and Ackermann invented the sintered pole plate in 1932 that large improvements were achieved. These advancements were reflected in higher load currents and improved longevity. The sealed nickel-cadmium battery, as we know it toady, became only available when Neumann succeeded in completely sealing the cell in 1947.

From the early days on, humanity became dependent on electricity, a product without which our technological advancements would not have been possible. With the increased need for mobility, people moved to portable power storage — first for wheeled applications, then for portable and finally wearable use. As awkward and unreliable as the early batteries may have been, our descendants may one day look at today’s technology in a similar way to how we view our predecessors’ clumsy experiments of 100 years ago.

History of Battery Development

1600 Gilbert (England) Establishment electrochemistry study

1791 Galvani (Italy) Discovery of ‘animal electricity’

1800 Volta (Italy) Invention of the voltaic cell

1802 Cruickshank (England) First electric battery capable of mass production

1820 Ampère (France) Electricity through magnetism

1833 Faraday (England) Announcement of Faraday’s Law

1836 Daniell (England) Invention of the Daniell cell

1859 Planté (France) Invention of the lead acid battery

1868 Leclanché (France) Invention of the Leclanché cell

1888 Gassner (USA) Completion of the dry cell

1899 Jungner (Sweden) Invention of the nickel-cadmium battery

1901 Edison (USA) Invention of the nickel-iron battery

1932 Shlecht & Ackermann (Germany) Invention of the sintered pole plate

1947 Neumann (France) Successfully sealing the nickel-cadmium battery

Mid 1960 Union Carbide (USA) Development of primary alkaline battery

Mid 1970 Development of valve regulated lead acid battery

1990 Commercialization nickel-metal hydride battery

1992 Kordesch (Canada) Commercialization reusable alkaline battery

1999 Commercialization lithium-ion polymer

2001 Anticipated volume production of proton exchange membrane fuel cell

Figure 1-5: History of battery development.
The battery may be much older. It is believed that the Parthians who ruled Baghdad (ca. 250 bc) used batteries to electroplate silver. The Egyptians are said to have electroplated antimony onto copper over 4300 years ago.

02. Available Software

Sun, 01 Mar 1998 00:00:00 +0000

NOTE: THIS DOCUMENT IS OBSOLETE, PLEASE CHECK THE NEW VERSION: "The Digital Audio Resampling Home Page", by Julius O. Smith III, Copyright © 2016-05-17 by Julius O. Smith III - Center for Computer Research in Music and Acoustics (CCRMA), Stanford University

<< Previous page TOC Next page >>
Available Software
resample-1.5.tar.gz(51 Kbytes) (v1.5 released on 10/07/99)
The resample software package contains free sampling-rate conversion and filter design utilities written in C, including a stand-alone command-line sampling-rate conversion utility called resample. The package compiles readily under Linux and most other UNIX operating systems.
README file for resample software package
sndlib-7 (required)
Older Version for NeXT Computers (49 Kbytes)
Open Source Audio Library Project (OSALP)
OSALP contains a C++ class (GNU LGPL) based on the above resample software.
To receive upgrade notifications, report bugs, or contribute enhancements, send email tojos@ccrma.stanford.edu.

02. Battery Chemistries

Sun, 01 Mar 1998 00:00:00 +0000

<< Previous page INDEX Next page >>
2. Battery Chemistries

Battery novices often argue that advanced battery systems are now available that offer very high energy densities, deliver 1000 charge/discharge cycles and are paper thin. These attributes are indeed achievable — unfortunately not in the same battery pack. A given battery may be designed for small size and long runtime, but this pack would have a limited cycle life. Another battery may be built for durability, but it would be big and bulky. A third pack may have high energy density and long durability, but would be too expensive for the commercial consumer.

Battery manufacturers are well aware of customer needs and have responded by offering battery packs that best suit the specific application. The mobile phone industry is an example of this clever adaptation. For this market, the emphasis is placed on small size and high energy density. Longevity comes in second.

The mention of NiMH on a battery pack does not automatically guarantee high energy density. A prismatic NiMH battery for a mobile phone, for example, is made for slim geometry and may only have an energy density of 60Wh/kg. The cycle count for this battery would be limited to around 300. In comparison, a cylindrical NiMH offers energy densities of 80Wh/kg and higher. Still, the cycle count of this battery will be moderate to low. High durability NiMH batteries, which are intended for industrial use and the electric vehicle enduring 1000 discharges to 80 percent depth-of discharge, are packaged in large cylindrical cells. The energy density on these cells is a modest 70Wh/kg.

Similarly, Li-ion batteries for defense applications are being produced that far exceed the energy density of the commercial equivalent. Unfortunately, these super-high capacity Li-ion batteries are deemed unsafe in the hands of the public. Neither would the general public be able to afford to buy them.

When energy densities and cycle life are mentioned, this book refers to a middle-of-the-road commercial battery that offers a reasonable compromise in size, energy density, cycle life and price. The book excludes miracle batteries that only live in controlled environments.

02. Battery Chemistries

Sun, 01 Mar 1998 00:00:00 +0000

<< Previous page INDEX Next page >>
2.8 The Supercapacitor

The supercapacitor resembles a regular capacitor with the exception that it offers very high capacitance in a small size. Energy storage is by means of static charge. Applying a voltage differential on the positive and negative plates charges the supercapacitor. This concept is similar to an electrical charge that builds up when walking on a carpet. Touching an object at ground potential releases the energy. The supercapacitor concept has been around for a number of years and has found many niche applications.

02. Battery Chemistries 2

Sun, 01 Mar 1998 00:00:00 +0000

<< Previous page INDEX Next page >>
2.1 Chemistry Comparison

Let's examine the advantages and limitations of today’s popular battery systems. Batteries are scrutinized not only in terms of energy density but service life, load characteristics, maintenance requirements, self-discharge and operational costs. Since NiCd remains a standard against which other batteries are compared, let’s evaluate alternative chemistries against this classic battery type.

Nickel Cadmium (NiCd) — mature and well understood but relatively low in energy density. The NiCd is used where long life, high discharge rate and economical price are important. Main applications are two-way radios, biomedical equipment, professional video cameras and power tools. The NiCd contains toxic metals and is not environmentally friendly.

Nickel-Metal Hydride (NiMH) — has a higher energy density compared to the NiCd at the expense of reduced cycle life. NiMH contains no toxic metals. Applications include mobile phones and laptop computers.

Lead Acid — most economical for larger power applications where weight is of little concern. The lead acid battery is the preferred choice for hospital equipment, wheelchairs, emergency lighting and UPS systems.

Lithium Ion (Li-ion) — fastest growing battery system. Li-ion is used where high-energy density and light weight is of prime importance. The Li-ion is more expensive than other systems and must follow strict guidelines to assure safety. Applications include notebook computers and cellular phones.

Lithium Ion Polymer (Li-ion polymer) — a potentially lower cost version of the Li-ion. This chemistry is similar to the Li-ion in terms of energy density. It enables very slim geometry and allows simplified packaging. Main applications are mobile phones.

Reusable Alkaline — replaces disposable household batteries; suitable for low-power applications. Its limited cycle life is compensated by low self-discharge, making this battery ideal for portable entertainment devices and flashlights.

Figure 2-1 compares the characteristics of the six most commonly used rechargeable battery systems in terms of energy density, cycle life, exercise requirements and cost. The figures are based on average ratings of commercially available batteries at the time of publication. Exotic batteries with above average ratings are not included.

NiCd NiMH Lead Acid Li-ion Li-ion polymer Reusable
Alkaline

Gravimetric Energy Density (Wh/kg) 45-80 60-120 30-50 110-160 100-130 80 (initial)

Internal Resistance
(includes peripheral circuits) in mW 100 to 200¹
6V pack 200 to 300¹
6V pack <100¹
12V pack 150 to 250¹
7.2V pack 200 to 300¹
7.2V pack 200 to 2000¹
6V pack

Cycle Life (to 80% of initial capacity) 1500² 300 to 500^2,3 200 to
300² 500 to 1000³ 300 to
500 50³
(to 50%)

Fast Charge Time 1h typical 2-4h 8-16h 2-4h 2-4h 2-3h

Overcharge Tolerance moderate low high very low low moderate

Self-discharge / Month (room temperature) 20%⁴ 30%⁴ 5% 10%⁵ ~10%⁵ 0.3%

Cell Voltage (nominal) 1.25V⁶ 1.25V⁶ 2V 3.6V 3.6V 1.5V

Load Current
- peak
- best result
20C
1C
5C
0.5C or lower
5C⁷0.2C
>2C
1C or lower
>2C
1C or lower
0.5C
0.2C or lower

Operating Temperature (discharge only) -40 to
60°C -20 to
60°C -20 to
60°C -20 to
60°C 0 to
60°C 0 to
65°C

Maintenance Requirement 30 to 60 days 60 to 90 days 3 to 6 months⁹ not req. not req. not req.

Typical Battery Cost
(US$, reference only) $50
(7.2V) $60
(7.2V) $25
(6V) $100
(7.2V) $100
(7.2V) $5
(9V)

Cost per Cycle (US$)¹¹ $0.04 $0.12 $0.10 $0.14 $0.29 $0.10-0.50

Commercial use since 1950 1990 1970 1991 1999 1992

Figure 2-1: Characteristics of commonly used rechargeable batteries.
The figures are based on average ratings of batteries available commercially at the time of publication; experimental batteries with above average ratings are not included.

Internal resistance of a battery pack varies with cell rating, type of protection circuit and number of cells. Protection circuit of Li-ion and Li-polymer adds about 100mW.

Cycle life is based on battery receiving regular maintenance. Failing to apply periodic full discharge cycles may reduce the cycle life by a factor of three.

Cycle life is based on the depth of discharge. Shallow discharges provide more cycles than deep discharges.

The discharge is highest immediately after charge, then tapers off. The NiCd capacity decreases 10% in the first 24h, then declines to about 10% every 30 days thereafter. Self-discharge increases with higher temperature.

Internal protection circuits typically consume 3% of the stored energy per month.

1.25V is the open cell voltage. 1.2V is the commonly used value. There is no difference between the cells; it is simply a method of rating.

Capable of high current pulses.

Applies to discharge only; charge temperature range is more confined.

Maintenance may be in the form of ‘equalizing’ or ‘topping’ charge.

Cost of battery for commercially available portable devices.

Derived from the battery price divided by cycle life. Does not include the cost of electricity and chargers.

Observation: It is interesting to note that NiCd has the shortest charge time, delivers the highest load current and offers the lowest overall cost-per-cycle, but has the most demanding maintenance requirements.

02. Battery Chemistries 3

Sun, 01 Mar 1998 00:00:00 +0000

<< Previous page INDEX Next page >>
2.2 The Nickel Cadmium (NiCd) Battery

Alkaline nickel battery technology originated in 1899, when Waldmar Jungner invented the NiCd battery. The materials were expensive compared to other battery types available at the time and its use was limited to special applications. In 1932, the active materials were deposited inside a porous nickel-plated electrode and in 1947, research began on a sealed NiCd battery, which recombined the internal gases generated during charge rather than venting them. These advances led to the modern sealed NiCd battery, which is in use today.

The NiCd prefers fast charge to slow charge and pulse charge to DC charge. All other chemistries prefer a shallow discharge and moderate load currents. The NiCd is a strong and silent worker; hard labor poses no problem. In fact, the NiCd is the only battery type that performs best under rigorous working conditions. It does not like to be pampered by sitting in chargers for days and being used only occasionally for brief periods. A periodic full discharge is so important that, if omitted, large crystals will form on the cell plates (also referred to as 'memory') and the NiCd will gradually lose its performance.

Among rechargeable batteries, NiCd remains a popular choice for applications such as two-way radios, emergency medical equipment, professional video cameras and power tools. Over 50 percent of all rechargeable batteries for portable equipment are NiCd. However, the introduction of batteries with higher energy densities and less toxic metals is causing a diversion from NiCd to newer technologies.

Advantages and Limitations of NiCd Batteries

Advantages

Fast and simple charge — even after prolonged storage.

High number of charge/discharge cycles — if properly maintained, the NiCd provides over 1000 charge/discharge cycles.

Good load performance — the NiCd allows recharging at low temperatures.

Long shelf life – in any state-of-charge.

Simple storage and transportation — most airfreight companies accept the NiCd without special conditions.

Good low temperature performance.

Forgiving if abused — the NiCd is one of the most rugged rechargeable batteries.

Economically priced — the NiCd is the lowest cost battery in terms of cost per cycle.

Available in a wide range of sizes and performance options — most NiCd cells are cylindrical.

Limitations

Relatively low energy density — compared with newer systems.

Memory effect — the NiCd must periodically be exercised to prevent memory.

Environmentally unfriendly — the NiCd contains toxic metals. Some countries are limiting the use of the NiCd battery.

Has relatively high self-discharge — needs recharging after storage.

Figure 2-2: Advantages and limitations of NiCd batteries.

02. Battery Chemistries 4

Sun, 01 Mar 1998 00:00:00 +0000

<< Previous page INDEX Next page >>
2.3 The Nickel-Metal Hydride (NiMH) Battery

Research of the NiMH system started in the 1970s as a means of discovering how to store hydrogen for the nickel hydrogen battery. Today, nickel hydrogen batteries are mainly used for satellite applications. They are bulky, contain high-pressure steel canisters and cost thousands of dollars each. In the early experimental days of the NiMH battery, the metal hydride alloys were unstable in the cell environment and the desired performance characteristics could not be achieved. As a result, the development of the NiMH slowed down. New hydride alloys were developed in the 1980s that were stable enough for use in a cell. Since the late 1980s, NiMH has steadily improved, mainly in terms of energy density. The success of the NiMH has been driven by its high energy density and the use of environmentally friendly metals. The modern NiMH offers up to 40 percent higher energy density compared to NiCd. There is potential for yet higher capacities, but not without some negative side effects. Both NiMH and NiCd are affected by high self-discharge. The NiCd loses about 10 percent of its capacity within the first 24 hours, after which the self-discharge settles to about 10 percent per month. The self-discharge of the NiMH is about one-and-a-half to two times greater compared to NiCd. Selection of hydride materials that improve hydrogen bonding and reduce corrosion of the alloy constituents reduces the rate of self-discharge, but at the cost of lower energy density. The NiMH has been replacing the NiCd in markets such as wireless communications and mobile computing. In many parts of the world, the buyer is encouraged to use NiMH rather than NiCd batteries. This is due to environmental concerns about careless disposal of the spent battery. The question is often asked, “Has NiMH improved over the last few years?” Experts agree that considerable improvements have been achieved, but the limitations remain. Most of the shortcomings are native to the nickel-based technology and are shared with the NiCd battery. It is widely accepted that NiMH is an interim step to lithium battery technology. Initially more expensive than the NiCd, the price of the NiMH has dropped and is now almost at par value. This was made possible with high volume production. With a lower demand for NiCd, there will be a tendency for the price to increase. <table border="0" cellspacing="0" cellpadding="3"> <tr> <td width="482" colspan="2" valign="top"> <table width="100%" border="0" cellspacing="0" cellpadding="0" bgcolor="#000000"> <tr> <td></td> </tr> </table> </td> </tr> <tr> <td width="482" colspan="2" valign="top" bgcolor="#E6E6E6"> Advantages and Limitations of NiMH Batteries </td> </tr> <tr> <td width="482" colspan="2" valign="top"> <table width="100%" border="0" cellspacing="0" cellpadding="0" bgcolor="#000000"> <tr> <td></td> </tr> </table> </td> </tr> <tr> <td width="65" valign="top"> Advantages </td> <td width="417" valign="top"> 30 – 40 percent higher capacity over a standard NiCd. The NiMH has potential for yet higher energy densities. Less prone to memory than the NiCd. Periodic exercise cycles are required less often. Simple storage and transportation — transportation conditions are not subject to regulatory control. Environmentally friendly — contains only mild toxins; profitable for recycling. </td> </tr> <tr> <td width="65" valign="top" bgcolor="#F3F3F3"> Limitations </td> <td width="417" valign="top" bgcolor="#F3F3F3"> Limited service life — if repeatedly deep cycled, especially at high load currents, the performance starts to deteriorate after 200 to 300 cycles. Shallow rather than deep discharge cycles are preferred. Limited discharge current — although a NiMH battery is capable of delivering high discharge currents, repeated discharges with high load currents reduces the battery’s cycle life. Best results are achieved with load currents of 0.2C to 0.5C (one-fifth to one-half of the rated capacity). More complex charge algorithm needed — the NiMH generates more heat during charge and requires a longer charge time than the NiCd. The trickle charge is critical and must be controlled carefully. High self-discharge — the NiMH has about 50 percent higher self-discharge compared to the NiCd. New chemical additives improve the self-discharge but at the expense of lower energy density. Performance degrades if stored at elevated temperatures — the NiMH should be stored in a cool place and at a state-of-charge of about 40 percent. High maintenance — battery requires regular full discharge to prevent crystalline formation. About 20 percent more expensive than NiCd — NiMH batteries designed for high current draw are more expensive than the regular version. </td> </tr> <tr valign="middle"> <td colspan="2" height="18"> <table width="100%" border="0" cellspacing="0" cellpadding="0" bgcolor="#000000"> <tr> <td></td> </tr> </table> </td> </tr> </table> <a name="_Toc507815236">Figure 2-</a>3:       Advantages and limitations of NiMH batteries</blockquote><a href="/guides/electronics/bpw/c02_03"><< Previous page</a>  <a href="/guides/electronics/bpw/00_toc">INDEX</a>  <a href="/guides/electronics/bpw/c02_05">Next page >></a>

02. Battery Chemistries 5

Sun, 01 Mar 1998 00:00:00 +0000

<< Previous page INDEX Next page >>
2.4 The Lead Acid Battery

Invented by the French physician Gaston Planté in 1859, lead acid was the first rechargeable battery for commercial use. Today, the flooded lead acid battery is used in automobiles, forklifts and large uninterruptible power supply (UPS) systems.

During the mid 1970s, researchers developed a maintenance-free lead acid battery, which could operate in any position. The liquid electrolyte was transformed into moistened separators and the enclosure was sealed. Safety valves were added to allow venting of gas during charge and discharge.

Driven by diverse applications, two designations of batteries emerged. They are the sealed lead acid (SLA), also known under the brand name of Gelcell, and the valve regulated lead acid (VRLA). Technically, both batteries are the same. No scientific definition exists as to when an SLA becomes a VRLA. (Engineers may argue that the word ‘sealed lead acid’ is a misnomer because no lead acid battery can be totally sealed. In essence, all are valve regulated.)

The SLA has a typical capacity range of 0.2Ah to 30Ah and powers portable and wheeled applications. Typical uses are personal UPS units for PC backup, small emergency lighting units, ventilators for health care patients and wheelchairs. Because of low cost, dependable service and minimal maintenance requirements, the SLA battery is the preferred choice for biomedical and health care instruments in hospitals and retirement homes.

The VRLA battery is generally used for stationary applications. Their capacities range from 30Ah to several thousand Ah and are found in larger UPS systems for power backup. Typical uses are mobile phone repeaters, cable distribution centers, Internet hubs and utilities, as well as power backup for banks, hospitals, airports and military installations.

Unlike the flooded lead acid battery, both the SLA and VRLA are designed with a low over-voltage potential to prohibit the battery from reaching its gas-generating potential during charge. Excess charging would cause gassing and water depletion. Consequently, the SLA and VRLA can never be charged to their full potential.

Among modern rechargeable batteries, the lead acid battery family has the lowest energy density. For the purpose of analysis, we use the term ‘sealed lead acid’ to describe the lead acid batteries for portable use and ‘valve regulated lead acid’ for stationary applications. Because of our focus on portable batteries, we focus mainly on the SLA.

The SLA is not subject to memory. Leaving the battery on float charge for a prolonged time does not cause damage. The battery’s charge retention is best among rechargeable batteries. Whereas the NiCd self-discharges approximately 40 percent of its stored energy in three months, the SLA self-discharges the same amount in one year. The SLA is relatively inexpensive to purchase but the operational costs can be more expensive than the NiCd if full cycles are required on a repetitive basis.
The SLA does not lend itself to fast charging — typical charge times are 8 to 16 hours. The SLA must always be stored in a charged state. Leaving the battery in a discharged condition causes sulfation, a condition that makes the battery difficult, if not impossible, to recharge.

Unlike the NiCd, the SLA does not like deep cycling. A full discharge causes extra strain and each discharge/charge cycle robs the battery of a small amount of capacity. This loss is very small while the battery is in good operating condition, but becomes more acute once the performance drops below 80 percent of its nominal capacity. This wear-down characteristic also applies to other battery chemistries in varying degrees. To prevent the battery from being stressed through repetitive deep discharge, a larger SLA battery is recommended.

Depending on the depth of discharge and operating temperature, the SLA provides 200 to 300 discharge/charge cycles. The primary reason for its relatively short cycle life is grid corrosion of the positive electrode, depletion of the active material and expansion of the positive plates. These changes are most prevalent at higher operating temperatures. Applying charge/discharge cycles does not prevent or reverse the trend.

There are some methods that improve the performance and prolong the life of the SLA. The optimum operating temperature for a VRLA battery is 25°C (77°F). As a rule of thumb, every 8°C (15°F) rise in temperature will cut the battery life in half. VRLA that would last for 10 years at 25°C would only be good for 5 years if operated at 33°C (95°F). The same battery would endure a little more than one year at a temperature of 42°C (107°F).

Advantages and Limitations of Lead Acid Batteries

Advantages

Inexpensive and simple to manufacture — in terms of cost per watt hours, the SLA is the least expensive.

Mature, reliable and well-understood technology — when used correctly, the SLA is durable and provides dependable service.

Low self-discharge —the self-discharge rate is among the lowest in rechargeable batterysystems.

Low maintenance requirements — no memory; no electrolyte to fill.

Capable of high discharge rates.

Limitations

Cannot be stored in a discharged condition.

Low energy density — poor weight-to-energy density limits use to stationary and wheeled applications.

Allows only a limited number of full discharge cycles — well suited for standby applications that require only occasional deep discharges.

Environmentally unfriendly — the electrolyte and the lead content can cause environmental damage.

Transportation restrictions on flooded lead acid — there are environmental concerns regarding spillage in case of an accident.

Thermal runaway can occur with improper charging.

Figure 2-4: Advantages and limitations of lead acid batteries.

The SLA has a relatively low energy density compared with other rechargeable batteries, making it unsuitable for handheld devices that demand compact size. In addition, performance at low temperatures is greatly reduced.

The SLA is rated at a 5-hour discharge or 0.2C. Some batteries are even rated at a slow 20 hour discharge. Longer discharge times produce higher capacity readings. The SLA performs well on high pulse currents. During these pulses, discharge rates well in excess of 1C can be drawn.

In terms of disposal, the SLA is less harmful than the NiCd battery but the high lead content makes the SLA environmentally unfriendly. Ninety percent of lead acid-based batteries are being recycled.

02. Battery Chemistries 6

Sun, 01 Mar 1998 00:00:00 +0000

<< Previous page INDEX Next page >>
2.5 The Lithium Ion Battery

Pioneer work with the lithium battery began in 1912 under G.N. Lewis but it was not until the early 1970s that the first non-rechargeable lithium batteries became commercially available. Attempts to develop rechargeable lithium batteries followed in the 1980s, but failed due to safety problems.

Lithium is the lightest of all metals, has the greatest electrochemical potential and provides the largest energy density per weight. Rechargeable batteries using lithium metal anodes (negative electrodes) are capable of providing both high voltage and excellent capacity, resulting in an extraordinary high energy density.

After much research on rechargeable lithium batteries during the 1980s, it was found that cycling causes changes on the lithium electrode. These transformations, which are part of normal wear and tear, reduce the thermal stability, causing potential thermal runaway conditions. When this occurs, the cell temperature quickly approaches the melting point of lithium, resulting in a violent reaction called ‘venting with flame’. A large quantity of rechargeable lithium batteries sent to Japan had to be recalled in 1991 after a battery in a mobile phone released flaming gases and inflicted burns to a person’s face.

Because of the inherent instability of lithium metal, especially during charging, research shifted to a non-metallic lithium battery using lithium ions. Although slightly lower in energy density than lithium metal, the Li-ion is safe, provided certain precautions are met when charging and discharging. In 1991, the Sony Corporation commercialized the first Li-ion battery. Other manufacturers followed suit. Today, the Li-ion is the fastest growing and most promising battery chemistry.

The energy density of the Li-ion is typically twice that of the standard NiCd. Improvements in electrode active materials have the potential of increasing the energy density close to three times that of the NiCd. In addition to high capacity, the load characteristics are reasonably good and behave similarly to the NiCd in terms of discharge characteristics (similar shape of discharge profile, but different voltage). The flat discharge curve offers effective utilization of the stored power in a desirable voltage spectrum.

The Li-ion is a low maintenance battery, an advantage that most other chemistries cannot claim. There is no memory and no scheduled cycling is required to prolong the battery’s life. In addition, the self-discharge is less than half compared to NiCd and NiMH, making the Li-ion well suited for modern fuel gauge applications.

The high cell voltage of Li-ion allows the manufacture of battery packs consisting of only one cell. Many of today’s mobile phones run on a single cell, an advantage that simplifies battery design. Supply voltages of electronic applications have been heading lower, which in turn requires fewer cells per battery pack. To maintain the same power, however, higher currents are needed. This emphasizes the importance of very low cell resistance to allow unrestricted flow of current.

Chemistry variations — During recent years, several types of Li-ion batteries have emerged with only one thing in common — the catchword 'lithium'. Although strikingly similar on the outside, lithium-based batteries can vary widely. This book addresses the lithium-based batteries that are predominantly used in commercial products.

Sony’s original version of the Li-ion used coke, a product of coal, as the negative electrode. Since 1997, most Li-ions (including Sony’s) have shifted to graphite. This electrode provides a flatter discharge voltage curve than coke and offers a sharp knee bend at the end of discharge (see Figure 2-5). As a result, the graphite system delivers the stored energy by only having to discharge to 3.0V/cell, whereas the coke version must be discharged to 2.5V to get similar runtime. In addition, the graphite version is capable of delivering a higher discharge current and remains cooler during charge and discharge than the coke version.

For the positive electrode, two distinct chemistries have emerged. They are cobalt and spinel (also known as manganese). Whereas cobalt has been in use longer, spinel is inherently safer and more forgiving if abused. Small prismatic spinel packs for mobile phones may only include a thermal fuse and temperature sensor. In addition to cost savings on a simplified protection circuit, the raw material cost for spinel is lower than that of cobalt.

Figure 2-5: Li-ion discharge characteristics.
The graphite Li-ion only needs to discharge to 3.0V/cell, whereas the coke version must be discharged to 2.5V/cell to achieve similar performance.

02. Battery Chemistries 7

Sun, 01 Mar 1998 00:00:00 +0000

<< Previous page INDEX Next page >>

As a trade-off, spinel offers a slightly lower energy density, suffers capacity loss at temperatures above 40°C and ages quicker than cobalt. Figure 2-6 compares the advantages and disadvantages of the two chemistries.

Cobalt Manganese (Spinel)

Energy density (Wh/kg) 140 ¹ 120 ¹

Safety On overcharge, the cobalt electrode provides extra lithium, which can form into metallic lithium, causing a potential safety risk if not protected by a safety circuit. On overcharge, the manganese electrode runs out of lithium causing the cell only to get warm. Safety circuits can be eliminated for small 1 and 2 cell packs.

Temperature Wide temperature range. Best suited for operation at elevated temperature. Capacity loss above +40°C. Not as durable at higher temperatures.

Aging Short-term storage possible. Impedance increases with age. Newer versions offer longer storage. Slightly less than cobalt. Impedance changes little over the life of the cell. Due to continuous improvements, storage time is difficult to predict.

Life Expectancy 300 cycles, 50% capacity at 500 cycles. May be shorter than cobalt.

Cost Raw material relatively high; protection circuit adds to costs. Raw material 30% lower than cobalt. Cost advantage on simplified protection circuit.

Figure 2-6: Comparison of cobalt and manganese as positive electrodes.
Manganese is inherently safer and more forgiving if abused but offers a slightly lower energy density. Manganese suffers capacity loss at temperature above 40°C and ages quicker than cobalt.

Based on present generation 18650 cells. The energy density tends to be lower for prismatic cells.

The choice of metals, chemicals and additives help balance the critical trade-off between high energy density, long storage time, extended cycle life and safety. High energy densities can be achieved with relative ease. For example, adding more nickel in lieu of cobalt increases the ampere/hours rating and lowers the manufacturing cost but makes the cell less safe. While a start-up company may focus on high energy density to gain quick market acceptance, safety, cycle life and storage capabilities may be compromised. Reputable manufacturers, such as Sony, Panasonic, Sanyo, Moli Energy and Polystor place high importance on safety. Regulatory authorities assure that only safe batteries are sold to the public.

Li-ion cells cause less harm when disposed of than lead or cadmium-based batteries. Among the Li-ion family, the spinel is the friendliest in terms of disposal.

Despite its overall advantages, Li-ion also has its drawbacks. It is fragile and requires a protection circuit to maintain safe operation. Built into each pack, the protection circuit limits the peak voltage of each cell during charge and prevents the cell voltage from dropping too low on discharge. In addition, the maximum charge and discharge current is limited and the cell temperature is monitored to prevent temperature extremes. With these precautions in place, the possibility of metallic lithium plating occurring due to overcharge is virtually eliminated.

Aging is a concern with most Li-ion batteries. For unknown reasons, battery manufacturers are silent about this issue. Some capacity deterioration is noticeable after one year, whether the battery is in use or not. Over two or perhaps three years, the battery frequently fails. It should be mentioned that other chemistries also have age-related degenerative effects. This is especially true for the NiMH if exposed to high ambient temperatures.

Storing the battery in a cool place slows down the aging process of the Li-ion (and other chemistries). Manufacturers recommend storage temperatures of 15°C (59°F). In addition, the battery should only be partially charged when in storage.

Extended storage is not recommended for Li-ion batteries. Instead, packs should be rotated. The buyer should be aware of the manufacturing date when purchasing a replacement Li-ion battery. Unfortunately, this information is often encoded in an encrypted serial number and is only available to the manufacturer.

Manufacturers are constantly improving the chemistry of the Li-ion battery. Every six months, a new and enhanced chemical combination is tried. With such rapid progress, it becomes difficult to assess how well the revised battery ages and how it performs after long-term storage.

Cost analysis — The most economical lithium-based battery in terms of cost-to-energy ratio is a pack using the cylindrical 18650 cell. This battery is somewhat bulky but suitable for portable applications such as mobile computing. If a slimmer pack is required (thinner than 18 mm), the prismatic Li-ion cell is the best choice. There is little or no gain in energy density per weight and size over the 18650, however the cost is more than double.

If an ultra-slim geometry is needed (less than 4 mm), the best choice is Li-ion polymer. This is the most expensive option in terms of energy cost. The Li-ion polymer does not offer appreciable energy gains over conventional Li-ion systems, nor does it match the durability of the 18560 cell.

Advantages and Limitations of Li-ion Batteries

Advantages

High energy density — potential for yet higher capacities.

Relatively low self-discharge — self-discharge is less than half that of NiCd and NiMH.

Low Maintenance — no periodic discharge is needed; no memory.

Limitations

Requires protection circuit — protection circuit limits voltage and current. Battery is safe if not provoked.

Subject to aging, even if not in use — storing the battery in a cool place and at 40 percent state-of-charge reduces the aging effect.

Moderate discharge current.

Subject to transportation regulations — shipment of larger quantities of Li-ion batteries may be subject to regulatory control. This restriction does not apply to personal carry-on batteries.

Expensive to manufacture — about 40 percent higher in cost than NiCd. Better manufacturing techniques and replacement of rare metals with lower cost alternatives will likely reduce the price.

Not fully mature — changes in metal and chemical combinations affect battery test results, especially with some quick test methods.

Figure 2-7: Advantages and limitations of Li-ion batteries.

Caution: Li-ion batteries have a high energy density. Exercise precaution when handling and testing. Do not short circuit, overcharge, crush, drop, mutilate, penetrate, apply reverse polarity, expose to high temperature or disassemble. Only use the Li-ion battery with the designated protection circuit. High case temperature resulting from abuse of the cell could cause physical injury. The electrolyte is highly flammable. Rupture may cause venting with flame.

02. Battery Chemistries 8

Sun, 01 Mar 1998 00:00:00 +0000

<< Previous page INDEX Next page >>
2.6 The Lithium Polymer Battery

The Li-polymer differentiates itself from other battery systems in the type of electrolyte used. The original design, dating back to the 1970s, uses a dry solid polymer electrolyte only. This electrolyte resembles a plastic-like film that does not conduct electricity but allows an exchange of ions (electrically charged atoms or groups of atoms). The polymer electrolyte replaces the traditional porous separator, which is soaked with electrolyte.

The dry polymer design offers simplifications with respect to fabrication, ruggedness, safety and thin-profile geometry. There is no danger of flammability because no liquid or gelled electrolyte is used.

With a cell thickness measuring as little as one millimeter (0.039 inches), equipment designers are left to their own imagination in terms of form, shape and size. It is possible to create designs which form part of a protective housing, are in the shape of a mat that can be rolled up, or are even embedded into a carrying case or piece of clothing. Such innovative batteries are still a few years away, especially for the commercial market.

Unfortunately, the dry Li-polymer suffers from poor conductivity. Internal resistance is too high and cannot deliver the current bursts needed for modern communication devices and spinning up the hard drives of mobile computing equipment. Although heating the cell to 60°C (140°F) and higher increases the conductivity to acceptable levels, this requirement is unsuitable in commercial applications.

Research is continuing to develop a dry solid Li-polymer battery that performs at room temperature. A dry solid Li-polymer version is expected to be commercially available by 2005. It is expected to be very stable; would run 1000 full cycles and would have higher energy densities than today’s Li-ion battery.

In the meantime, some Li-polymers are used as standby batteries in hot climates. One manufacturer has added heating elements that keeps the battery in the conductive temperature range at all times. Such a battery performs well for the application intended because high ambient temperatures do not affect the service life of this battery in the same way it does the VRLA, for example.

To make a small Li-polymer battery conductive, some gelled electrolyte has been added. Most of the commercial Li-polymer batteries used today for mobile phones are a hybrid and contain gelled electrolyte. The correct term for this system is ‘Lithium Ion Polymer’. For promotional reasons, most battery manufacturers mark the battery simply as Li-polymer. Since the hybrid lithium polymer is the only functioning polymer battery for portable use today, we will focus on this chemistry.

With gelled electrolyte added, what then is the difference between Li-ion and Li-ion polymer? Although the characteristics and performance of the two systems are very similar, the Li-ion polymer is unique in that it uses a solid electrolyte, replacing the porous separator. The gelled electrolyte is simply added to enhance ion conductivity.

Technical difficulties and delays in volume manufacturing have deferred the introduction of the Li-ion polymer battery. This postponement, as some critics argue, is due to ‘cashing in’ on the Li-ion battery. Manufacturers have invested heavily in research, development and equipment to mass-produce the Li-ion. Now businesses and shareholders want to see a return on their investment.

In addition, the promised superiority of the Li-ion polymer has not yet been realized. No improvements in capacity gains have been achieved — in fact, the capacity is slightly less than that of the standard Li-ion battery. For the present, there is no cost advantage in using the Li-ion polymer battery. The thin profile has, however, compelled mobile phone manufacturers to use this promising technology for their new generation handsets.

One of the advantages of the Li-ion polymer, however, is simpler packaging because the electrodes can easily be stacked. Foil packaging, similar to that used in the food industry, is being used. No defined norm in cell size has been established by the industry.

Advantages and Limitations of Li-ion Polymer Batteries

Advantages

Very low profile — batteries that resemble the profile of a credit card are feasible.

Flexible form factor — manufacturers are not bound by standard cell formats. With high volume, any reasonable size can be produced economically.

Light weight – gelled rather than liquid electrolytes enable simplified packaging, in some cases eliminating the metal shell.

Improved safety — more resistant to overcharge; less chance for electrolyte leakage.

Limitations

Lower energy density and decreased cycle count compared to Li-ion — potential for improvements exist.

Expensive to manufacture — once mass-produced, the Li-ion polymer has the potential for lower cost. Reduced control circuit offsets higher manufacturing costs.

Figure 2-8: Advantages and limitations of Li-ion polymer batteries.

02. Battery Chemistries 9

Sun, 01 Mar 1998 00:00:00 +0000

<< Previous page INDEX Next page >>
2.7 Reusable Alkaline Batteries

The idea of recharging alkaline batteries is not new. Although not endorsed by manufacturers, ordinary alkaline batteries have been recharged in households for many years. Recharging these batteries is only effective, however, if the cells have been discharged to less than 50 percent of their total capacity. The number of recharges depends solely on the depth of discharge and is limited to a few at best. With each recharge, less capacity can be reclaimed. There is a cautionary advisory, however: charging ordinary alkaline batteries may generate hydrogen gas, which can lead to explosion. It is therefore not prudent to charge ordinary alkaline unsupervised.

In comparison, the reusable alkaline is designed for repeated recharge. It too loses charge acceptance with each recharge. The longevity of the reusable alkaline is a direct function of the depth of discharge; the deeper the discharge, the fewer cycles the battery can endure.

Tests performed by Cadex on ‘AA’ reusable alkaline cells showed a very high capacity reading on the first discharge. In fact, the energy density was similar to that of a NiMH battery. When the battery was discharged, then later recharged using the manufacturer’s charger, the reusable alkaline settled at 60 percent, a capacity slightly below that of a NiCd. Repeat cycling in the same manner resulted in a fractional capacity loss with each cycle. In our tests, the discharge current was adjusted to 200mA (0.2 C-rate, or one fifth of the rated capacity); the end-of-discharge threshold was set to 1V/cell.

An additional limitation of the reusable alkaline system is its low load current capability of 400mA (lower than 400mA provides better results). Although adequate for portable AM/FM radios, CD players, tape players and flashlights, 400mA is insufficient to power most mobile phones and video cameras.

The reusable alkaline is inexpensive but the cost per cycle is high when compared to the nickel-based rechargeables. Whereas the NiCd checks in at $0.04 per cycle based on 1500 cycles, the reusable alkaline costs $0.50 based on 10 full discharge cycles. For many applications, this seemingly high cost is still economical when compared to the non-reusable alkaline that has a one-time use. If the reusable alkaline battery is only partially discharged before recharge, an improved cycle life is possible. At 50 percent depth of discharge, 50 cycles can be expected.

To compare the operating cost between the standard and reusable alkaline, a study was done on flashlight batteries for hospital use. The low-intensity care unit using the flashlights only occasionally achieved measurable savings by employing the reusable alkaline. The high-intensity unit that used the flashlights constantly, on the other hand, did not attain the same result. Deeper discharge and more frequent recharge reduced their service life and offset any cost advantage over the standard alkaline battery.

In summary, the standard alkaline offers maximum energy density whereas the reusable alkaline provides the benefit of allowing some recharging. The compromise of the reusable alkaline is loss of charge acceptance after the first recharge.

Advantages and Limitations of Reusable Alkaline Batteries

Advantages

Inexpensive and readily available — can be used as a direct replacement of non-rechargeable (primary) cells.

More economical than non-rechargeable – allows several recharges.

Low self-discharge — can be stored as a standby battery for up to 10 years.

Environmentally friendly — no toxic metals used, fewer batteries are discarded, reduces landfill.

Maintenance free — no need for cycling; no memory.

Limitations

Limited current handling — suited for light-duty applications like portable home entertainment, flashlights.

Limited cycle life — for best results, recharge before the battery gets too low.

Figure 2-9: Advantages and limitations of reusable alkaline batteries.

02. Relation to Stochastic Processes

Sun, 01 Mar 1998 00:00:00 +0000

NOTE: THIS DOCUMENT IS OBSOLETE, PLEASE CHECK THE NEW VERSION: "Introduction to Digital Filters with Audio Applications", by Julius O. Smith III, Copyright © 2017-11-26 by Julius O. Smith III - Center for Computer Research in Music and Acoustics (CCRMA), Stanford University

<< Previous page TOC INDEX Next page >>
Relation to Stochastic Processes
Theorem. If a stationary random process has a rational power spectral density corresponding to an autocorrelation function , then

03. Relation to Schur Functions

Sun, 01 Mar 1998 00:00:00 +0000

NOTE: THIS DOCUMENT IS OBSOLETE, PLEASE CHECK THE NEW VERSION: "Introduction to Digital Filters with Audio Applications", by Julius O. Smith III, Copyright © 2017-11-26 by Julius O. Smith III - Center for Computer Research in Music and Acoustics (CCRMA), Stanford University

<< Previous page TOC INDEX Next page >>
Relation to Schur Functions
Definition. A Schur function is defined as a complex function analytic and of modulus not exceeding unity in .
Theorem. The function

03. The Battery Pack

Sun, 01 Mar 1998 00:00:00 +0000

<< Previous page INDEX Next page >>
3. The Battery Pack

In the 1700 and 1800s, cells were encased in glass jars. Later, larger batteries were developed that used wooden containers. The inside was treated with a sealant to prevent electrolyte leakage. With the need for portability, the cylindrical cell appeared. After World War II, these cells became the standard format for smaller, rechargeable batteries.

Downsizing required smaller and more compact cell design. The button cell, which gained popularity in the 1980s, was a first attempt to achieve a reasonably flat geometry, or obtain higher voltages in a compact profile by stacking. The early 1990s brought the prismatic cell, which was followed by the modern pouch cell.

This chapter addresses the cell designs, pack configurations and intrinsic safety devices. In keeping with portability, this book addresses only the smaller cells used for portable batteries.
3.1 The Cylindrical Cell
The cylindrical cell continues to be the most widely used packaging style. The advantages are ease of manufacture and good mechanical stability. The cylinder has the ability to withstand high internal pressures. While charging, the cell pressure of a NiCd can reach 1379 kilopascals (kPa) or 200 pounds per square inch (psi). A venting system is added on one end of the cylinder. Venting occurs if the cell pressure reaches between 150 and 200 psi. Figure 3-1 illustrates the conventional cell of a NiCd battery.

Figure 3-1: Cross-section of a classic NiCd cell.
The negative and positive plates are rolled together in a metal cylinder. The positive plate is sintered and filled with nickel hydroxide. The negative plate is coated with cadmium active material. A separator moistened with electrolyte isolates the two plates. Design courtesy of Panasonic OEM Battery Sales Group, March 2001.

The cylindrical cell is moderately priced and offers high energy density. Typical applications are wireless communication, mobile computing, biomedical instruments, power tools and other uses that do not demand ultra-small size.

NiCd offers the largest selection of cylindrical cells. A good variety is also available in the NiMH family, especially in the smaller cell formats. In addition to cylindrical formats, NiMH also comes in the prismatic cell packaging.

The Li-ion batteries are only available in limited cells sizes, the most popular being the 18650. ‘Eighteen’ denotes the diameter in millimeters and ‘650’ describes the length in millimeters. The 18650 cell has a capacity of 1800 to 2000mAh. The larger 26650 cell has a diameter of 26 mm and delivers 3200mAh. Because of the flat geometry of the Li-ion polymer, this battery chemistry is not available in a cylindrical format.

Most SLA batteries are built in a prismatic format, thus creating a rectangle box that is commonly made of plastic materials. There are SLA batteries, however, that take advantage of the cylindrical design by using a winding technique that is similar to the conventional cell. The cylindrical Hawker Cyclone SLA is said to offer improved cell stability, provide higher discharge currents and have better temperature stability than the conventional prismatic design.

The drawback of the cylindrical cell is less than maximum use of space. When stacking the cells, air cavities are formed. Because of fixed cell size, the pack must be designed around the available cell size.

Almost all cylindrical cells are equipped with a venting mechanism to expel excess gases in an orderly manner. Whereas nickel-based batteries feature a resealable vent, many cylindrical Li-ion contain a membrane seal that ruptures if the pressure exceeds 3448 kPa (500 psi). There is usually some serious swelling of the cell before the seal breaks. Venting only occurs under extreme conditions.

03. The Battery Pack 2

Sun, 01 Mar 1998 00:00:00 +0000

<< Previous page INDEX Next page >>
3.2 The Button Cell

The button cell was developed to miniaturize battery packs and solve stacking problems. Today, this architecture is limited to a small niche market. Non-rechargeable versions of the button cell continue to be popular and can be found in watches, hearing aids and memory backup.

The main applications of the rechargeable button cell are (or were) older cordless telephones, biomedical devices and industrial instruments. Although small in design and inexpensive to manufacture, the main drawback is swelling if charged too rapidly. Button cells have no safety vent and can only be charged at a 10 to 16 hour charge rate. New designs claim rapid charge capability.

Figure 3-2: The button cell.
The button cell offers small size and ease of stacking but does not allow fast charging. Coin cells, which are similar in appearance, are normally lithium-based and are non-rechargeable. Photograph courtesy of Sanyo Corporation; design courtesy of Panasonic OEM Battery Sales Group, March 2001.
3.3 The Prismatic Cell

The prismatic cell was developed in response to consumer demand for thinner pack sizes. Introduced in the early 1990’s, the prismatic cell makes almost maximum use of space when stacking. Narrow and elegant battery styles are possible that suit today’s slim-style geometry. Prismatic cells are used predominantly for mobile phone applications. Figure 3-3 shows the prismatic cell.

Prismatic cells are most common in the lithium battery family. The Li-ion polymer is exclusively prismatic. No universally accepted cell size exists for Li-ion polymer batteries. One leading manufacturer may bring out one or more sizes that fit a certain portable device, such as a mobile phone. While these cells are produced at high volume, other cell manufacturers follow suit and offer an identical cell at a competitive price. Prismatic cells that have gained acceptance are the 340648 and the 340848. Measured in millimeters, ‘34’ denotes the width, ‘06’ or ‘08’ the thickness and ‘48’ the length of the cell.

Figure 3-3: Cross-section of a prismatic cell.
The prismatic cell improves space utilization and allows more flexibility in pack design. This cell construction is less cost effective than the cylindrical equivalent and provides a slightly lower energy density. Design courtesy of Polystor Corporation, March 2001.

Some prismatic cells are similar in size but are off by just a small fraction. Such is the case with the Panasonic cell that measures 34 mm by 50 mm and is 6.5 mm thick. If a few cubic millimeters can be added for a given application, the manufacturer will do so for the sake of higher capacities.

The disadvantage of the prismatic cell is slightly lower energy densities compared to the cylindrical equivalent. In addition, the prismatic cell is more expensive to manufacture and does not provide the same mechanical stability enjoyed by the cylindrical cell. To prevent bulging when pressure builds up, heavier gauge metal is used for the container. The manufacturer allows some degree of bulging when designing the battery pack.

The prismatic cell is offered in limited sizes and chemistries and runs from about 400mAh to 2000mAh and higher. Because of the very large quantities required for mobile phones, special prismatic cells are built to fit certain models. Most prismatic cells do not have a venting system. In case of pressure build-up, the cell starts to bulge. When correctly used and properly charged, no swelling should occur.

03. The Battery Pack 3

Sun, 01 Mar 1998 00:00:00 +0000

<< Previous page INDEX Next page >>
3.4 The Pouch Cell

Cell design made a profound advance in 1995 when the pouch cell concept was developed. Rather than using an expensive metallic cylinder and glass-to-metal electrical feed-through to insulate the opposite polarity, the positive and negative plates are enclosed in flexible, heat-sealable foils. The electrical contacts consist of conductive foil tabs that are welded to the electrode and sealed to the pouch material. Figure 3-4 illustrates the pouch cell.

The pouch cell concept allows tailoring to exact cell dimensions. It makes the most efficient use of available space and achieves a packaging efficiency of 90 to 95 percent, the highest among battery packs. Because of the absence of a metal can, the pouch pack has a lower weight. The main applications are mobile phones and military devices. No standardized pouch cells exist, but rather, each manufacturer builds to a special application.

The pouch cell is exclusively used for Li-ion and Li-ion polymer chemistries. At the present time, it costs more to produce this cell architecture and its reliability has not been fully proven. In addition, the energy density and load current are slightly lower than that of conventional cell designs. The cycle life in everyday applications is not well documented but is, at present, less than that of the Li-ion system with conventional cell design.

A critical issue with the pouch cell is the swelling that occurs when gas is generated during charging or discharging. Battery manufacturers insist that Li-ion or Polymer cells do not generate gas if properly formatted, are charged at the correct current and are kept within allotted voltage levels. When designing the protective housing for a pouch cell, some provision for swelling must be made. To alleviate the swelling issue when using multiple cells, it is best not to stack pouch cells, but lay them side by side.

Figure 3-4: The pouch cell.
The pouch cell offers a simple, flexible and lightweight solution to battery design. This new concept has not yet fully matured and the manufacturing costs are still high.
© Cadex Electronics Inc.

The pouch cell is highly sensitive to twisting. Point pressure must also be avoided. The protective housing must be designed to protect the cell from mechanical stress.

03. The Battery Pack 4

Sun, 01 Mar 1998 00:00:00 +0000

<< Previous page INDEX Next page >>
3.5 Series and Parallel Configurations

In most cases, a single cell does not provide a high enough voltage and a serial connection of several cells is needed. The metallic skin of the cell is insulated to prevent the ‘hot’ metal cylinders from creating an electrical short circuit against the neighboring cell.

Nickel-based cells provide a nominal cell voltage of 1.25V. A lead acid cell delivers 2V and most Li-ion cells are rated at 3.6V. The spinel (manganese) and Li-ion polymer systems sometimes use 3.7V as the designated cell voltage. This is the reason for the often unfamiliar voltages, such as 11.1V for a three cell pack of spinel chemistry.

Nickel-based cells are often marked 1.2V. There is no difference between a 1.2 and 1.25V cell; it is simply the preference of the manufacturer in marking. Whereas commercial batteries tend to be identified with 1.2V/cell, industrial, aviation and military batteries are still marked with the original designation of 1.25V/cell.

A five-cell nickel-based battery delivers 6V (6.25V with 1.25V/cell marking) and a six-cell pack has 7.2V (7.5V with 1.25V/cell marking). The portable lead acid comes in 3 cell (6V) and 6 cell (12V) formats. The Li-ion family has either 3.6V for a single cell pack, 7.2V for a two-cell pack or 10.8V for a three-cell pack. The 3.6V and 7.2V batteries are commonly used for mobile phones; laptops use the larger 10.8V packs.

There has been a trend towards lower voltage batteries for light portable devices, such as mobile phones. This was made possible through advancements in microelectronics. To achieve the same energy with lower voltages, higher currents are needed. With higher currents, a low internal battery resistance is critical. This presents a challenge if protection devices are used. Some losses through the solid-state switches of protection devices cannot be avoided.

Packs with fewer cells in series generally perform better than those with 12 cells or more. Similar to a chain, the more links that are used, the greater the odds of one breaking. On higher voltage batteries, precise cell matching becomes important, especially if high load currents are drawn or if the pack is operated in cold temperatures.

Parallel connections are used to obtain higher ampere-hour (Ah) ratings. When possible, pack designers prefer using larger cells. This may not always be practical because new battery chemistries come in limited sizes. Often, a parallel connection is the only option to increase the battery rating. Paralleling is also necessary if pack dimensions restrict the use of larger cells. Among the battery chemistries, Li-ion lends itself best to parallel connection.

03. The Battery Pack 5

Sun, 01 Mar 1998 00:00:00 +0000

<< Previous page INDEX Next page >>
3.6 Protection Circuits

Most battery packs include some type of protection to safeguard battery and equipment, should a malfunction occur. The most basic protection is a fuse that opens if excessively high current is drawn. Some fuses open permanently and render the battery useless once the filament is broken; other fuses are based on a Polyswitch™, which resembles a resettable fuse. On excess current, the Polyswitch™ creates a high resistance, inhibiting the current flow. When the condition normalizes, the resistance of the switch reverts to the low ON position, allowing normal operation to resume. Solid-state switches are also used to disrupt the current. Both solid-state switches and the Polyswitch™ have a residual resistance to the ON position during normal operation, causing a slight increase in internal battery resistance.

A more complex protection circuit is found in intrinsically safe batteries. These batteries are mandated for two-way radios, gas detectors and other electronic instruments that operate in a hazardous area such as oil refineries and grain elevators. Intrinsically safe batteries prevent explosion, should the electronic devices malfunction while operating in areas that contain explosive gases or high dust concentration. The protection circuit prevents excessive current, which could lead to high heat and electric spark.

There are several levels of intrinsic safety, each serving a specific hazard level. The requirement for intrinsic safety varies from country to country. The purchase cost of an intrinsically safe battery is two or three times that of a regular battery.

Commercial Li-ion packs contain one of the most exact protection circuits in the battery industry. These circuits assure safety under all circumstances when in the hands of the public. Typically, a Field Effect Transistor (FET) opens if the charge voltage of any cell reaches 4.30V and a fuse activates if the cell temperature approaches 90°C (194°F). In addition, a disconnect switch in each cell permanently interrupts the charge current if a safe pressure threshold of 1034 kPa (150 psi) is exceeded. To prevent the battery from over-discharging, the control circuit cuts off the current path at low voltage, which is typically 2.50V/cell.

The Li-ion is typically discharged to 3V/cell. The lowest ‘low-voltage’ power cut-off is 2.5V/cell. During prolonged storage, however, a discharge below that cut-off level is possible. Manufacturers recommend a ‘trickle’ charge to raise such a battery gradually back up into the acceptable voltage window.

Not all chargers are designed to apply a charge once a Li-ion battery has dipped below 2.5V/cell. A ‘wake-up’ boost will be needed to first engage the electronic circuit, after which a gentle charge is applied to re-energize the battery. Caution must be applied not to boost lithium-based batteries back to life, which have dwelled at a very low voltage for a prolonged time.

03. The Battery Pack 6

Sun, 01 Mar 1998 00:00:00 +0000

<< Previous page INDEX Next page >>

Each parallel string of cells of a Li-ion pack needs independent voltage monitoring. The more cells that are connected in series, the more complex the protection circuit becomes. Four cells in series is the practical limit for commercial applications.

The internal protection circuit of a mobile phone while in the ON position has a resistance of 50 to 100 mW. The circuit normally consists of two switches connected in series. One is responsible for high cut-off, the other for low cut-off. The combined resistance of these two devices virtually doubles the internal resistance of a battery pack, especially if only one cell is used. Battery packs powering mobile phones, for example, must be capable of delivering high current bursts. The internal protection does, in a certain way, interfere with the current delivery.

Some small Li-ion packs with spinel chemistry containing one or two cells may not include an electronic protection circuit. Instead, they use a single component fuse device. These cells are deemed safe because of small size and low capacity. In addition, spinel is more tolerant than other systems if abused. The absence of a protection circuit saves money, but a new problem arises. Here is what can happen:

Mobile phone users have access to chargers that may not be approved by the battery manufacturer. Available at low cost for car and travel, these chargers may rely on the battery’s protection circuit to terminate at full charge. Without the protection circuit, the battery cell voltage rises too high and overcharges the battery. Apparently still safe, irreversible battery damage often occurs. Heat buildup and bulging is common under these circumstances. Such situations must be avoided at all times. The manufacturers are often at a loss when it comes to replacing these batteries under warranty.

Li-ion batteries with cobalt electrodes, for example, require full safety protection. A major concern arises if static electricity or a faulty charger has destroyed the battery’s protection circuit. Such damage often causes the solid-state switches to fuse in a permanent ON position without the user’s knowledge. A battery with a faulty protection circuit may function normally but does not provide the required safety. If charged beyond safe voltage limits with a poorly designed accessory charger, the battery may heat up, then bulge and in some cases vent with flame. Shorting such a battery can also be hazardous.

Manufacturers of Li-ion batteries refrain from mentioning explosion. ‘Venting with flame’ is the accepted terminology. Although slower in reaction than an explosion, venting with flame can be very violent and inflicts injury to those in close proximity. It can also damage the equipment to which the battery is connected.

Most manufacturers do not sell the Li-ion cells by themselves but make them available in a battery pack, complete with protection circuit. This precaution is understandable when considering the danger of explosion and fire if the battery is charged and discharged beyond its safe limits. Most battery assembling houses must certify the pack assembly and protection circuit intended to be used with the manufacturer before these items are approved for sale.

03. Theory of Operation

Sun, 01 Mar 1998 00:00:00 +0000

NOTE: THIS DOCUMENT IS OBSOLETE, PLEASE CHECK THE NEW VERSION: "The Digital Audio Resampling Home Page", by Julius O. Smith III, Copyright © 2016-05-17 by Julius O. Smith III - Center for Computer Research in Music and Acoustics (CCRMA), Stanford University

<< Previous page TOC Next page >>
Theory of Operation

Subsections
Abstract
Introduction
Theory of Ideal Bandlimited Interpolation
From Theory to Practice
Implementation
Quantization Issues
Choice of Table Size
Choice of Interpolation Resolution
Conclusions
Appendix A: Exact Sinc-Interpolation of Sampled Periodic Signals
Appendix B: Relation between Sinc and Lagrange Interpolation

04. Abstract

Sun, 01 Mar 1998 00:00:00 +0000

NOTE: THIS DOCUMENT IS OBSOLETE, PLEASE CHECK THE NEW VERSION: "The Digital Audio Resampling Home Page", by Julius O. Smith III, Copyright © 2016-05-17 by Julius O. Smith III - Center for Computer Research in Music and Acoustics (CCRMA), Stanford University

<< Previous page TOC Next page >>
Abstract
This tutorial describes a technique for bandlimited interpolation of discrete-time signals which supports signal evaluation at an ``arbitrary'' time, and which performs well for smoothly changing sampling rates, such as needed for digital audio ``scrubbing.'' The method is based on interpolated look-up in a table of filter coefficients, so as to make the filter impulse response available effectively in continuous-time form. A single pre-computed filter table handles all interpolation times and sampling-rate conversion ratios. Formulas are given for determining the look-up table size needed for a given precision requirement. This tutorial is an expansion of the conference paper [Smith and Gossett 1984].

04. Proper Charge Methods

Sun, 01 Mar 1998 00:00:00 +0000

<< Previous page INDEX Next page >>
4. Proper Charge Methods

To a large extent, the performance and longevity of rechargeable batteries depends on the quality of the chargers. Battery chargers are commonly given low priority, especially on consumer products. Choosing a quality charger makes sense. This is especially true when considering the high cost of battery replacements and the frustration that poorly performing batteries create. In most cases, the extra money invested is returned because the batteries last longer and perform more efficiently.

4.1 All About Chargers

There are two distinct varieties of chargers: the personal chargers and the industrial chargers. The personal charger is sold in attractive packaging and is offered with such products as mobile phones, laptops and video cameras. These chargers are economically priced and perform well when used for the application intended. The personal charger offers moderate charge times.

In comparison, the industrial charger is designed for employee use and accommodates fleet batteries. These chargers are built for repetitive use. Available for single or multi-bay configurations, the industrial chargers are offered from the original equipment manufacturer (OEM). In many instances, the chargers can also be obtained from third party manufacturers. While the OEM chargers meet basic requirements, third party manufacturers often include special features, such as negative pulse charging, discharge function for battery conditioning, and state-of-charge (SoC) and state-of-health (SoH) indications. Many third party manufacturers are prepared to build low quantities of custom chargers. Other benefits third party suppliers can offer include creative pricing and superior performance.

Not all third party charger manufacturers meet the quality standards that the industry demands, The buyer should be aware of possible quality and performance compromises when purchasing these chargers at discount prices. Some units may not be rugged enough to withstand repetitive use; others may develop maintenance problems such as burned or broken battery contacts.

Uncontrolled over-charge is another problem of some chargers, especially those used to charge nickel-based batteries. High temperature during charge and standby kills batteries. Over-charging occurs when the charger keeps the battery at a temperature that is warm to touch (body temperature) while in ready condition.

Some temperature rise cannot be avoided when charging nickel-based batteries. A temperature peak is reached when the battery approaches full charge. The temperature must moderate when the ready light appears and the battery has switched to trickle charge. The battery should eventually cool to room temperature.

If the temperature does not drop and remains above room temperature, the charger is performing incorrectly. In such a case, the battery should be removed as soon as possible after the ready light appears. Any prolonged trickle charging will damage the battery. This caution applies especially to the NiMH because it cannot absorb overcharge well. In fact, a NiMH with high trickle charge could be cold to the touch and still be in a damaging overcharge condition. Such a battery would have a short service life.

A lithium-based battery should never get warm in a charger. If this happens, the battery is faulty or the charger is not functioning properly. Discontinue using this battery and/or charger.

It is best to store batteries on a shelf and apply a topping-charge before use rather than leaving the pack in the charger for days. Even at a seemingly correct trickle charge, nickel-based batteries produce a crystalline formation (also referred to as ‘memory’) when left in the charger. Because of relatively high self-discharge, a topping charge is needed before use. Most Li-ion chargers permit a battery to remain engaged without inflicting damage.

04. Proper Charge Methods 10

Sun, 01 Mar 1998 00:00:00 +0000

<< Previous page INDEX Next page >>
4.8 Ultra-fast Chargers

Some charger manufacturers claim amazingly short charge times of 30 minutes or less. With well-balanced cells and operating at moderate room temperatures, NiCd batteries designed for fast charging can indeed be charged in a very short time. This is done by simply dumping in a high charge current during the first 70 percent of the charge cycle. Some NiCd batteries can take as much a 10C, or ten times the rated current. Precise SoC detection and temperature monitoring are essential.

The high charge current must be reduced to lower levels in the second phase of the charge cycle because the efficiency to absorb charge is progressively reduced as the battery moves to a higher SoC. If the charge current remains too high in the later part of the charge cycle, the excess energy turns into heat and pressure. Eventually venting occurs, releasing hydrogen gas. Not only do the escaping gases deplete the electrolyte, they are also highly flammable!

Several manufacturers offer chargers that claim to fully charge NiCd batteries in half the time of conventional chargers. Based on pulse charge technology, these chargers intersperse one or several brief discharge pulses between each charge pulse. This promotes the recombination of oxygen and hydrogen gases, resulting in reduced pressure buildup and a lower cell temperature. Ultra-fast-chargers based on this principle can charge a nickel-based battery in a shorter time than regular chargers, but only to about a 90 percent SoC. A trickle charge is needed to top the charge to 100 percent.

Pulse chargers are known to reduce the crystalline formation (memory) of nickel-based batteries. By using these chargers, some improvement in battery performance can be realized, especially if the battery is affected by memory. The pulse charge method does not replace a periodic full discharge. For more severe crystalline formation on nickel-based batteries, a full discharge or recondition cycle is recommended to restore the battery.

Ultra-fast charging can only be applied to healthy batteries and those designed for fast charging. Some cells are simply not built to carry high current and the conductive path heats up. The battery contacts also take a beating if the current handling of the spring-loaded plunger contacts is underrated. Pressing against a flat metal surface, these contacts may work well at first, and then wear out prematurely. Often, a fine and almost invisible crater appears on the tip of the contact, which causes a high resistive path or forms an isolator. The heat generated by a bad contact can melt the plastic.

Another problem with ultra-fast charging is servicing aged batteries that commonly have high internal resistance. Poor conductivity turns into heat, which further deteriorates the cells. Battery packs with mismatched cells pose another challenge. The weak cells holding less capacity are charged before those with higher capacity and start to heat up. This process makes them vulnerable to further damage.

Many of today’s fast chargers are designed for the ideal battery. Charging less than perfect specimens can create such a heat buildup that the plastic housing starts to distort. Provisions must be made to accept special needs batteries, albeit at lower charging speeds. Temperature sensing is a prerequisite.

The ideal ultra-fast charger first checks the battery type, measures its SoH and then applies a tolerable charge current. Ultra-high capacity batteries and those that have aged are identified, and the charge time is prolonged because of higher internal resistance. Such a charger would provide due respect to those batteries that still perform satisfactorily but are no longer ‘spring chickens’.

The charger must prevent excessive temperature build-up. Sluggish heat detection, especially when charging takes place at a very rapid pace, makes it easy to overcharge a battery before the charge is terminated. This is especially true for chargers that control fast charge using temperature sensing alone. If the temperature rise is measured right on the skin of the cell, reasonably accurate SoC detection is possible. If done on the outside surface of the battery pack, further delays occur. Any prolonged exposure to a temperature of 45°C (113°F) harms the battery.

New charger concepts are being studied which regulate the charge current according to the battery's charge acceptance. On the initial charge of an empty battery when the charge acceptance is high and little gas is generated, a very high charge current can be applied. Towards the end of a charge, the current is tapered down.
4.9 Charge IC Chips

Newer battery systems demand more complex chargers than batteries with older chemistries. With today’s charge IC chips, designing a charger has been simplified. These chips apply proven charge algorithms and are capable of servicing all major battery chemistries. As the price of these chips decreases, design engineers make more use of this product. With the charge IC chip, an engineer can focus entirely on the portable equipment rather than devoting time to developing a charging circuit.

The charge IC chips have some limitations, however. The charge algorithm is fixed and does not allow fine-tuning. If a trickle charge is needed to raise a Li-ion that has dropped below 2.5V/cell to its normal operating voltage, the charge IC may not be able to perform this function. Similarly, if an ultra-fast charge is needed for nickel-based batteries, the charge IC applies a fixed charge current and does not take into account the SoH of the battery. Furthermore, a temperature compensated charge would be difficult to administer if the IC chips do not provide this feature.

Using a small micro controller is an alternative to selecting an off-the-shelf charge IC. The hardware cost is about the same. When opting for the micro controller, custom firmware will be needed. Some extra features can be added with little extra cost. They are fast charging based on the SoH of the battery. Ambient temperatures can also be taken into account. Whether an IC chip or micro controller is used, peripheral components are required consisting of solid-state switches and a power supply.

04. Proper Charge Methods 2

Sun, 01 Mar 1998 00:00:00 +0000

<< Previous page INDEX Next page >>

There are three types of chargers for nickel-based batteries. They are:

Slow Charger — Also known as ‘overnight charger’ or ‘normal charger’, the slow-charger applies a fixed charge rate of about 0.1C (one tenth of the rated capacity) for as long as the battery is connected. Typical charge time is 14 to 16 hours. In most cases, no full-charge detection occurs to switch the battery to a lower charge rate at the end of the charge cycle. The slow-charger is inexpensive and can be used for NiCd batteries only. With the need to service both NiCd and NiMH, these chargers are being replaced with more advanced units.

If the charge current is set correctly, a battery in a slow-charger remains lukewarm to the touch when fully charged. In this case, the battery does not need to be removed immediately when ready but should not stay in the charger for more than a day. The sooner the battery can be removed after being fully charged, the better it is.

A problem arises if a smaller battery (lower mAh) is charged with a charger designed to service larger packs. Although the charger will perform well in the initial charge phase, the battery starts to heat up past the 70 percent charge level. Because there is no provision to lower the charge current or to terminate the charge, heat-damaging over-charge will occur in the second phase of the charge cycle. If an alternative charger is not available, the user is advised to observe the temperature of the battery being charged and disconnect the battery when it is warm to the touch.

The opposite may also occur when a larger battery is charged on a charger designed for a smaller battery. In such a case, a full charge will never be reached. The battery remains cold during charge and will not perform as expected. A nickel-based battery that is continuously undercharged will eventually loose its ability to accept a full charge due to memory.

Quick Charger — The so-called quick-charger, or rapid charger, is one of the most popular. It is positioned between the slow-charger and the fast-charger, both in terms of charging time and price. Charging takes 3 to 6 hours and the charge rate is around 0.3C. Charge control is required to terminate the charge when the battery is ready. The well designed quick-charger provides better service to nickel-based batteries than the slow-charger. Batteries last longer if charged with higher currents, provided they remain cool and are not overcharged. The quick-chargers are made to accommodate either nickel-based or lithium-based batteries. These two chemistries can normally not be interchanged in the same charger.

Fast Charger — The fast-charger offers several advantages over the other chargers; the obvious one is shorter charge times. Because of the larger power supply and the more expensive control circuits needed, the fast-charger costs more than slower chargers, but the investment is returned in providing good performing batteries that live longer.

The charge time is based on the charge rate, the battery’s SoC, its rating and the chemistry. At a 1C charge rate, an empty NiCd typically charges in a little more than an hour. When a battery is fully charged, some chargers switch to a topping charge mode governed by a timer that completes the charge cycle at a reduced charge current. Once fully charged, the charger switches to trickle charge. This maintenance charge compensates for the self-discharge of the battery.

Modern fast-chargers commonly accommodate both NiCd and NiMH batteries. Because of the fast-charger’s higher charge current and the need to monitor the battery during charge, it is important to charge only batteries specified by the manufacturer. Some battery manufacturers encode the batteries electrically to identify their chemistry and rating. The charger then sets the correct charge current and algorithm for the battery intended. Lead Acid and Li-ion chemistries are charged with different algorithms and are not compatible with the charge methods used for nickel-based batteries.

It is best to fast charge nickel-based batteries. A slow charge is known to build up a crystalline formation on nickel-based batteries, a phenomenon that lowers battery performance and shortens service life. The battery temperature during charge should be moderate and the temperature peak kept as short as possible.

It is not recommended to leave a nickel-based battery in the charger for more than a few days, even with a correctly set trickle charge current. If a battery must remain in a charger for operational readiness, an exercise cycle should be applied once every month.

04. Proper Charge Methods 3

Sun, 01 Mar 1998 00:00:00 +0000

<< Previous page INDEX Next page >>

Simple Guidelines

A charger designed to service NiMH batteries can also accommodate NiCd’s, but not the other way around. A charger only made for the NiCd batteries could overcharge the NiMH battery.

While many charge methods exist for nickel-based batteries, chargers for lithium-based batteries are more defined in terms of charge method and charge time. This is, in part, due to the tight charge regime and voltage requirements demanded by these batteries. There is only one way to charge Li-ion/Polymer batteries and the so-called ‘miracle chargers’, which claim to restore and prolong battery life, do not exist for these chemistries. Neither does a super-fast charging solution apply.

The pulse charge method for Li-ion has no major advantages and the voltage peaks wreak havoc with the voltage limiting circuits. While charge times can be reduced, some manufacturers suggest that pulse charging may shorten the cycle life of Li-ion batteries.

Fast charge methods do not significantly decrease the charge time. A charge rate over 1C should be avoided because such high current can induce lithium plating. With most packs, a charge above 1C is not possible. The protection circuit limits the amount of current the battery can accept. The lithium-based battery has a slow metabolism and must take its time to absorb the energy.

Lead acid chargers serve industrial markets such as hospitals and health care units. Charge times are very long and cannot be shortened. Most lead acid chargers charge the battery in 14 hours. Because of its low energy density, this battery type is not used for small portable devices.

In the following sections various charging needs and charging methods are studied. The charging techniques of different chargers are examined to determine why some perform better than others. Since fast charging rather than slow charging is the norm today, we look at well-designed, closed loop systems, which communicate with the battery and terminate the fast charge when certain responses from the battery are received.

4.2 Charging the Nickel Cadmium Battery

Battery manufacturers recommend that new batteries be slow-charged for 24 hours before use. A slow charge helps to bring the cells within a battery pack to an equal charge level because each cell self-discharges to different capacity levels. During long storage, the electrolyte tends to gravitate to the bottom of the cell. The initial trickle charge helps redistribute the electrolyte to remedy dry spots on the separator that may have developed.

Some battery manufacturers do not fully form their batteries before shipment. These batteries reach their full potential only after the customer has primed them through several charge/discharge cycles, either with a battery analyzer or through normal use. In many cases, 50 to 100 discharge/charge cycles are needed to fully form a nickel-based battery. Quality cells, such as those made by Sanyo and Panasonic, are known to perform to full specification after as few as 5 to 7 discharge/charge cycles. Early readings may be inconsistent, but the capacity levels become very steady once fully primed. A slight capacity peak is observed between 100 and 300 cycles.

Most rechargeable cells are equipped with a safety vent to release excess pressure if incorrectly charged. The safety vent on a NiCd cell opens at 1034 to 1379 kPa (150 to 200 psi). In comparison, the pressure of a car tire is typically 240 kPa (35 psi). With a resealable vent, no damage occurs on venting but some electrolyte is lost and the seal may leak afterwards. When this happens, a white powder will accumulate over time at the vent opening.

Commercial fast-chargers are often not designed in the best interests of the battery. This is especially true of NiCd chargers that measure the battery’s charge state solely through temperature sensing. Although simple and inexpensive in design, charge termination by temperature sensing is not accurate. The thermistors used commonly exhibit broad tolerances; their positioning with respect to the cells are not consistent. Ambient temperatures and exposure to the sun while charging also affect the accuracy of full-charge detection. To prevent the risk of premature cut-off and assure full charge under most conditions, charger manufacturers use 50°C (122°F) as the recommended temperature cut-off. Although a prolonged temperature above 45°C (113°F) is harmful to the battery, a brief temperature peak above that level is often unavoidable.

04. Proper Charge Methods 4

Sun, 01 Mar 1998 00:00:00 +0000

<< Previous page INDEX Next page >>

More advanced NiCd chargers sense the rate of temperature increase, defined as dT/dt, or the change in temperature over charge time, rather than responding to an absolute temperature (dT/dt is defined as delta Temperature / delta time). This type of charger is kinder to the batteries than a fixed temperature cut-off, but the cells still need to generate heat to trigger detection. To terminate the charge, a temperature increase of 1°C (1.8°F) per minute with an absolute temperature cut-off of 60°C (140°F) works well. Because of the relatively large mass of a cell and the sluggish propagation of heat, the delta temperature, as this method is called, will also enter a brief overcharge condition before the full-charge is detected. The dT/dt method only works with fast chargers.

Harmful overcharge occurs if a fully charged battery is repeatedly inserted for topping charge. Vehicular or base station chargers that require the removal of two-way radios with each use are especially hard on the batteries because each reconnection initiates a fast-charge cycle. This also applies to laptops that are momentarily disconnected and reconnected to perform a service. Likewise, a technician may briefly plug the laptop into the power source to check a repeater station or service other installations. Problems with laptop batteries have also been reported in car manufacturing plants where the workers move the laptops from car to car, checking their functions, while momentarily plugging into the external power source. Repetitive connection to power affects mostly ‘dumb’ nickel-based batteries. A ‘dumb’ battery contains no electronic circuitry to communicate with the charger. Li-ion chargers detect the SoC by voltage only and multiple reconnections will not confuse the charging regime.

More precise full charge detection of nickel-based batteries can be achieved with the use of a micro controller that monitors the battery voltage and terminates the charge when a certain voltage signature occurs. A drop in voltage signifies that the battery has reached full charge. This is known as Negative Delta V (NDV).

NDV is the recommended full-charge detection method for ‘open-lead’ NiCd chargers because it offers a quick response time. The NDV charge detection also works well with a partially or fully charged battery. If a fully charged battery is inserted, the terminal voltage raises quickly, then drops sharply, triggering the ready state. Such a charge lasts only a few minutes and the cells remain cool. NiCd chargers based on the NDV full charge detection typically respond to a voltage drop of 10 to 30mV per cell. Chargers that respond to a very small voltage decrease are preferred over those that require a larger drop.

To obtain a sufficient voltage drop, the charge rate must be 0.5C and higher. Lower than 0.5C charge rates produce a very shallow voltage decrease that is often difficult to measure, especially if the cells are slightly mismatched. In a battery pack that has mismatched cells, each cell reaches the full charge at a different time and the curve gets distorted. Failing to achieve a sufficient negative slope allows the fast-charge to continue, causing excessive heat buildup due to overcharge. Chargers using the NDV must include other charge-termination methods to provide safe charging under all conditions. Most chargers also observe the battery temperature.

The charge efficiency factor of a standard NiCd is better on fast charge than slow charge. At a 1C charge rate, the typical charge efficiency is 1.1 or 91 percent. On an overnight slow charge (0.1C), the efficiency drops to 1.4 or 71 percent.

At a rate of 1C, the charge time of a NiCd is slightly longer than 60 minutes (66 minutes at an assumed charge efficiency of 1.1). The charge time on a battery that is partially discharged or cannot hold full capacity due to memory or other degradation is shorter accordingly. At a 0.1C charge rate, the charge time of an empty NiCd is about 14 hours, which relates to the charge efficiency of 1.4.

During the first 70 percent of the charge cycle, the charge efficiency of a NiCd battery is close to 100 percent. Almost all of the energy is absorbed and the battery remains cool. Currents of several times the C-rating can be applied to a NiCd battery designed for fast charging without causing heat build-up. Ultra-fast chargers use this unique phenomenon and charge a battery to the 70 percent charge level within a few minutes. The charge continues at a lower rate until the battery is fully charged.

Once the 70 percent charge threshold is passed, the battery gradually loses ability to accept charge. The cells start to generate gases, the pressure rises and the temperature increases. The charge acceptance drops further as the battery reaches 80 and 90 percent SoC. Once full charge is reached, the battery goes into overcharge. In an attempt to gain a few extra capacity points, some chargers allow a measured amount of overcharge. Figure 4-1 illustrates the relationship of cell voltage, pressure and temperature while a NiCd is being charged.

Ultra-high capacity NiCd batteries tend to heat up more than the standard NiCd if charged at 1C and higher. This is partly due to the higher internal resistance of the ultra-high capacity battery. Optimum charge performance can be achieved by applying higher current at the initial charge stage, then tapering it to a lower rate as the charge acceptance decreases. This avoids excess temperature rise and yet assures fully charged batteries.

Figure 4-1: Charge characteristics of a NiCd cell.
These cell voltage, pressure and temperature characteristics are similar in a NiMH cell.

Interspersing discharge pulses between charge pulses improves the charge acceptance of nickel-based batteries. Commonly referred to as ‘burp’ or ‘reverse load’ charge, this charge method promotes high surface area on the electrodes, resulting in enhanced performance and increased service life. Reverse load also improves fast charging because it helps to recombine the gases generated during charge. The result is a cooler and more effective charge than with conventional DC chargers.

Charging with the reverse load method minimizes crystalline formation. The US Army Electronics Command in Fort Monmouth, NJ, USA, had done extensive research in this field and has published the results. (See Figure 10-1, Crystalline formation on NiCd cell). Research conducted in Germany has shown that the reverse load method adds 15 percent to the life of the NiCd battery.

After full charge, the NiCd battery is maintained with a trickle charge to compensate for the self-discharge. The trickle charge for a NiCd battery ranges between 0.05C and 0.1C. In an effort to reduce the memory phenomenon, there is a trend towards lower trickle charge currents.

04. Proper Charge Methods 5

Sun, 01 Mar 1998 00:00:00 +0000

<< Previous page INDEX Next page >>
4.3 Charging the Nickel-Metal Hydride Battery

Chargers for NiMH batteries are very similar to those of the NiCd system but the electronics is generally more complex. To begin with, the NiMH produces a very small voltage drop at full charge. This NDV is almost non-existent at charge rates below 0.5C and elevated temperatures. Aging and cell mismatch works further against the already minute voltage delta. The cell mismatch gets worse with age and increased cycle count, which makes the use of the NDV increasingly more difficult.

The NDV of a NiMH charger must respond to a voltage drop of 16mV or less. Increasing the sensitivity of the charger to respond to the small voltage drop often terminates the fast charge by error halfway through the charge cycle. Voltage fluctuations and noise induced by the battery and charger can fool the NDV detection circuit if set too precisely.

The popularity of the NiMH battery has introduced many innovative charging techniques. Most of today’s NiMH fast chargers use a combination of NDV, voltage plateau, rate-of-temperature-increase (dT/dt), temperature threshold and timeout timers. The charger utilizes whatever comes first to terminate the fast-charge.

NiMH batteries which use the NDV method or the thermal cut-off control tend to deliver higher capacities than those charged by less aggressive methods. The gain is approximately 6 percent on a good battery. This capacity increase is due to the brief overcharge to which the battery is exposed. The negative aspect is a shorter cycle life. Rather than expecting 350 to 400 service cycles, this pack may be exhausted with 300 cycles.

Similar to NiCd charge methods, most NiMH fast-chargers work on the rate-of-temperature-increase (dT/dt). A temperature raise of 1°C (1.8°F) per minute is commonly used to terminate the charge. The absolute temperature cut-off is 60°C (140°F). A topping charge of 0.1C is added for about 30 minutes to maximize the charge. The continuous trickle charge that follows keeps the battery in full charge state.

Applying an initial fast charge of 1C works well. Cooling periods of a few minutes are added when certain voltage peaks are reached. The charge then continues at a lower current. When reaching the next charge threshold, the current steps down further. This process is repeated until the battery is fully charged.

Known as ‘step-differential charge’, this charge method works well with NiMH and NiCd batteries. The charge current adjusts to the SoC, allowing high current at the beginning and more moderate current towards the end of charge. This avoids excessive temperature build-up towards the end of the charge cycle when the battery is less capable of accepting charge.

NiMH batteries should be rapid charged rather than slow charged. The amount of trickle charge applied to maintain full charge is especially critical. Because NiMH does not absorb overcharge well, the trickle charge must be set lower than that of the NiCd. The recommended trickle charge for the NiMH battery is a low 0.05C. This is why the original NiCd charger cannot be used to charge NiMH batteries. The lower trickle charge rate is acceptable for the NiCd.

It is difficult, if not impossible, to slow-charge a NiMH battery. At a C-rate of 0.1C and 0.3C, the voltage and temperature profiles fail to exhibit defined characteristics to measure the full charge state accurately and the charger must depend on a timer. Harmful overcharge can occur if a partially or fully charged battery is charged on a charger with a fixed timer. The same occurs if the battery has lost charge acceptance due to age and can only hold 50 percent of charge. A fixed timer that delivers a 100 percent charge each time without regard to the battery condition would ultimately apply too much charge. Overcharge could occur even though the NiMH battery feels cool to the touch.

Some lower-priced chargers may not apply a fully saturated charge. On these economy chargers, the full-charge detection may occur immediately after a given voltage peak is reached or a temperature threshold is detected. These chargers are commonly promoted on the merit of short charge time and moderate price.

Figure 4-2 summarizes the characteristics of the slow charger, quick charger and fast charger. A higher charge current allows better full-charge detection.

Charge
C-rate Typical charge time Maximum permissible charge temperatures Charge termination method

Slow Charger 0.1C 14h 0°C to 45°C
(32°F to 113°F) Fixed timer. Subject to overcharge. Remove battery when charged.

Quick Charger 0.3-0.5C 4h 10°C to 45°C
(50°F to 113°F) NDV set to 10mV/cell, uses voltage plateau, absolute temperature and time-out-timer. (At 0.3C, dT/dt fails to raise the temperature sufficiently to terminate the charge.)

Fast Charger 1C 1h+ 10°C to 45°C
(50°F to 113°F) NDV responds to higher settings; uses dT/dt, voltage plateau absolute temperature and time-out-timer

Figure 4-2: Characteristics of various charger types.
These values also apply to NiMH and NiCd cells.

04. Proper Charge Methods 6

Sun, 01 Mar 1998 00:00:00 +0000

<< Previous page INDEX Next page >>
4.4 Charging the Lead Acid Battery

The charge algorithm for lead acid batteries differs from nickel-based chemistry in that voltage limiting rather than current limiting is used. Charge time of a sealed lead acid (SLA) is 12 to 16 hours. With higher charge currents and multi-stage charge methods, charge time can be reduced to 10 hours or less. SLAs cannot be fully charged as quickly as nickel-based systems.

A multi-stage charger applies constant-current charge, topping charge and float charge (see Figure 4-3). During the constant current charge, the battery charges to 70 percent in about five hours; the remaining 30 percent is completed by the slow topping charge. The topping charge lasts another five hours and is essential for the well-being of the battery. This can be compared to a little rest after a good meal before resuming work. If the battery is not completely saturated, the SLA will eventually lose its ability to accept a full charge and the performance of the battery is reduced. The third stage is the float charge, which compensates for the self-discharge after the battery has been fully charged.

Figure 4-3: Charge stages of a lead acid battery.
A multi-stage charger applies constant-current charge, topping charge and float charge.

Correctly setting the cell-voltage limit is critical. A typical voltage limit is from 2.30V to 2.45V. If a slow charge is acceptable, or the room temperature may exceed 30°C (86°F), the recommended voltage limit is 2.35V/cell. If a faster charge is required, and the room temperature will remain below 30°C, 2.40 to 2.45V/cell may be used. Figure 4-4 compares the advantages and disadvantages of the different voltage settings.

2.30V to 2.35V/cell

2.40V to 2.45V/cell

Advantage

Maximum service life; battery remains cool during charge; ambient charge temperature may exceed 30°C (86°F).

Faster charge times; higher and more consistent capacity readings; less subject to damage due to under-charge condition.

Disadvantage

Slow charge time; capacity readings may be low and inconsistent. If no periodic topping charge is applied, under-charge conditions (sulfation) may occur, which can lead to unrecoverable capacity loss.

Battery life may be reduced due to elevated battery temperature while charging. A hot battery may fail to reach the cell voltage limit, causing harmful over charge.

Figure 4-4: Effects of charge voltage on a plastic SLA battery.
Large VRLA and the cylindrical Hawker cell may have different requirements.

The charge voltage limit indicated in Figure 4-4 is a momentary voltage peak and the battery cannot dwell on that level. This voltage crest is only used when applying a full charge cycle to a battery that has been discharged. Once fully charged and at operational readiness, a float charge is applied, which is held constant at a lower voltage level. The recommended float charge voltage of most low-pressure lead acid batteries is between 2.25 to 2.30V/cell. A good compromise is 2.27V.

The optimal float charge voltage shifts with temperature. A higher temperature demands slightly lower voltages and a lower temperature demands higher voltages. Chargers that are exposed to large temperature fluctuations are equipped with temperature sensors to optimize the float voltage.

Regardless of how well the float voltage may be compensated, there is always a compromise. The author of a paper in a battery seminar explained that charging a sealed lead acid battery using the traditional float charge techniques is like 'dancing on the head of a pin'. The battery wants to be fully charged to avoid sulfation on the negative plate, but does not want to be over-saturated which causes grid corrosion on the positive plate. In addition to grid corrosion, too high a float charge contributes to loss of electrolyte.

Differences in the aging of the cells create another challenge in finding the optimum float charge voltage. With the development of air pockets within the cells over time, some batteries exhibit hydrogen evolution from overcharging. Others undergo oxygen recombination in an almost starved state. Since the cells are connected in series, controlling the individual cell voltages during charge is virtually impossible. If the applied cell voltage is too high or too low for a given cell, the weaker cell deteriorates further and its condition becomes more pronounced with time. Companies have developed cell-balancing devices that correct some of these problems but these devices can only be applied if access to individual cells is possible.

A ripple voltage imposed on the charge voltage also causes problems for lead acid batteries, especially the larger VRLA. The peak of the ripple voltage constitutes an overcharge, causing hydrogen evolution; the valleys induce a brief discharge causing a starved state. Electrolyte depletion may be the result.

Much has been said about pulse charging lead acid batteries. Although there are obvious benefits of reduced cell corrosion, manufacturers and service technicians are not in agreement regarding the benefit of such a charge method. Some advantages are apparent if pulse charging is applied correctly, but the results are non-conclusive.

Whereas the voltage settings in Figure 4-4 apply to low-pressure lead acid batteries with a pressure relief valve setting of about 34 kPa (5 psi), the cylindrical SLA by Hawker requires higher voltage settings. These voltage limits should be set according to the manufacturer’s specifications. Failing to apply the recommended voltage threshold for these batteries causes a gradual decrease in capacity due to sulfation. Typically, the Hawker cell has a pressure relief setting of 345 kPa (50 psi). This allows some recombination of the gases during charge.

An SLA must be stored in a charged state. A topping charge should be applied every six months to avoid the voltage from dropping below 2.10V/cell. The topping charge requirements may differ with cell manufacturers. Always follow the time intervals recommended by the manufacturer.

By measuring the open cell voltage while in storage, an approximate charge-level indication can be obtained. A voltage of 2.11V, if measured at room temperature, reveals that the cell has a charge of 50 percent and higher. If the voltage is at or above this threshold, the battery is in good condition and only needs a full charge cycle prior to use. If the voltage drops below 2.10V, several discharge/charge cycles may be required to bring the battery to full performance. When measuring the terminal voltage of any cell, the storage temperature should be observed. A cool battery raises the voltage slightly and a warm one lowers it.

04. Proper Charge Methods 7

Sun, 01 Mar 1998 00:00:00 +0000

<< Previous page INDEX Next page >>

Plastic SLA batteries arriving from vendors with less than 2.10V per cell are rejected by some buyers who inspect the battery during quality control. Low voltage suggests that the battery may have a soft short, a defect that cannot be corrected with cycling. Although cycling may increase the capacity of these batteries, the extra cycles compromise the service life of the battery. Furthermore, the time and equipment required to make the battery fully functional adds to operational costs.

The Hawker cell can be stored at voltages as low as 1.81V. However, when reactivating the cells, a higher than normal charge voltage may be required to convert the large sulfite crystals back to good active material.

Caution: When charging a lead acid battery with over-voltage, current limiting must be applied once the battery starts to draw full current. Always set the current limit to the lowest practical setting and observe the battery voltage and temperature during the procedure. If the battery does not accept a normal charge after 24 hours under elevated voltage, a return to normal condition is unlikely.

The price of the Hawker cell is slightly higher than that of the plastic equivalent, but lower than the NiCd. Also known as the ‘Cyclone’, this cell is wound similar to a cylindrical NiCd. This construction improves the cell’s stability and provides higher discharge currents when compared to the flat plate SLA. Because of its relatively low self-discharge, Hawker cells are well suited for defibrillators that are used on standby mode.

Lead acid batteries are preferred for UPS systems. During prolonged float charge, a periodic topping charge, also known as an ‘equalizing charge’, is recommended to fully charge the plates and prevent sulfation. An equalizing charge raises the battery voltage for several hours to a voltage level above that specified by the manufacturer. Loss of electrolyte through elevated temperature may occur if the equalizing charge is not administered correctly. Because no liquid can be added to the SLA and VRLA systems, a reduction of the electrolyte will cause irreversible damage. Manufacturers and service personnel are often divided on the benefit of the equalizing charge.

Some exercise, or brief periodic discharge, is believed to prolong battery life of lead acid systems. If applied once a month as part of an exercising program, the depth of discharge should only be about 10 percent of its total capacity. A full discharge as part of regular maintenance is not recommended because each deep discharge cycle robs service life from the battery.

More experiments are needed to verify the benefit of exercising lead acid batteries. Again, manufacturers and service technicians express different views on how preventive maintenance should be carried out. Some experts prefer a topping charge while others recommend scheduled discharges. No scientific data is available on the benefit of frequent shallow discharges as opposed to fewer deep discharges or discharge pulses.

Disconnecting the float charge while the VRLA is on standby is another method of prolonging battery life. From time-to-time, a topping charge is applied to replenish the energy lost through self-discharge. This is said to lower cell corrosion and prolong battery life. In essence, the battery is kept as if it was in storage. This only works for applications that do not draw a load current during standby. In many applications, the battery acts as an energy buffer and needs to be under continuous charge.

Important: In case of rupture, leaking electrolyte or any other cause of exposure to the electrolyte, flush with water immediately. If eye exposure occurs, flush with water for 15 minutes and consult a physician immediately.

04. Proper Charge Methods 8

Sun, 01 Mar 1998 00:00:00 +0000

<< Previous page INDEX Next page >>
4.5 Charging the Lithium Ion Battery

The Li-ion charger is a voltage-limiting device similar to the lead acid battery charger. The difference lies in a higher voltage per cell, tighter voltage tolerance and the absence of trickle or float charge when full charge is reached.

While the lead acid battery offers some flexibility in terms of voltage cut-off, manufacturers of Li-ion cells are very strict on setting the correct voltage. When the Li- ion was first introduced, the graphite system demanded a charge voltage limit of 4.10V/cell. Although higher voltages deliver increased energy densities, cell oxidation severely limited the service life in the early graphite cells that were charged above the 4.10V/cell threshold. This effect has been solved with chemical additives. Most commercial Li-ion cells can now be charged to 4.20V. The tolerance on all Li-ion batteries is a tight +/-0.05V/cell.

Industrial and military Li-ion batteries designed for maximum cycle life use an end-of-charge voltage threshold of about 3.90V/cell. These batteries are rated lower on the watt-hour-per-kilogram scale, but longevity takes precedence over high energy density and small size.

The charge time of all Li-ion batteries, when charged at a 1C initial current, is about 3 hours. The battery remains cool during charge. Full charge is attained after the voltage has reached the upper voltage threshold and the current has dropped and leveled off at about 3 percent of the nominal charge current.

Increasing the charge current on a Li-ion charger does not shorten the charge time by much. Although the voltage peak is reached quicker with higher current, the topping charge will take longer. Figure 4-5 shows the voltage and current signature of a charger as the Li-ion cell passes through stage one and two.

Some chargers claim to fast-charge a Li-ion battery in one hour or less. Such a charger eliminates stage 2 and goes directly to ‘ready’ once the voltage threshold is reached at the end of stage 1. The charge level at this point is about 70 percent. The topping charge typically takes twice as long as the initial charge.

No trickle charge is applied because the Li-ion is unable to absorb overcharge. Trickle charge could cause plating of metallic lithium, a condition that renders the cell unstable. Instead, a brief topping charge is applied to compensate for the small amount of self-discharge the battery and its protective circuit consume.

Depending on the charger and the self-discharge of the battery, a topping charge may be implemented once every 500 hours or 20 days. Typically, the charge kicks in when the open terminal voltage drops to 4.05V/cell and turns off when it reaches 4.20V/cell again.

Figure 4-5: Charge stages of a Li-ion battery.
Increasing the charge current on a Li-ion charger does not shorten the charge time by much. Although the voltage peak is reached quicker with higher current, the topping charge will take longer.

What if a battery is inadvertently overcharged? Li-ion batteries are designed to operate safely within their normal operating voltage but become increasingly unstable if charged to higher voltages. On a charge voltage above 4.30V, the cell causes lithium metal plating on the anode. In addition, the cathode material becomes an oxidizing agent, loses stability and releases oxygen. Overcharging causes the cell to heat up.

Much attention has been placed on the safety of the Li-ion battery. Commercial Li-ion battery packs contain a protection circuit that prevents the cell voltage from going too high while charging. The typical safety threshold is set to 4.30V/cell. In addition, temperature sensing disconnects the charge if the internal temperature approaches 90°C (194°F). Most cells feature a mechanical pressure switch that permanently interrupts the current path if a safe pressure threshold is exceeded. Internal voltage control circuits cut off the battery at low and high voltage points.

Exceptions are made on some spinel (manganese) packs containing one or two small cells. On overcharge, this chemistry produces minimal lithium plating on the anode because most metallic lithium has been removed from the cathode during normal charging. The cathode material remains stable and does not generate oxygen unless the cell gets extremely hot.

Important: In case of rupture, leaking electrolyte or any other cause of exposure to the electrolyte, flush with water immediately. If eye exposure occurs, flush with water for 15 minutes and consult a physician immediately.

04. Proper Charge Methods 9

Sun, 01 Mar 1998 00:00:00 +0000

<< Previous page INDEX Next page >>
4.6 Charging the Lithium Polymer Battery

The charge process of a Li-Polymer is similar to that of the Li-ion. Li-Polymer uses dry electrolyte and takes 3 to 5 hours to charge. Li-ion polymer with gelled electrolyte, on the other hand, is almost identical to that of Li-ion. In fact, the same charge algorithm can be applied. With most chargers, the user does not need to know whether the battery being charged is Li-ion or Li-ion polymer.

Almost all commercial batteries sold under the so-called ‘Polymer’ category are a variety of the Li-ion polymer using some sort of gelled electrolyte. A low-cost dry polymer battery operating at ambient temperatures is still some years away.
4.7 Charging at High and Low Temperatures

Rechargeable batteries can be used under a reasonably wide temperature range. This, however, does not automatically mean that the batteries can also be charged at these temperature conditions. While the use of batteries under hot or cold conditions cannot always be avoided, recharging time is controlled by the user. Efforts should be made to charge the batteries only at room temperatures.

In general, older battery technologies such as the NiCd are more tolerant to charging at low and high temperatures than the more advanced systems. Figure 4-6 indicates the permissible slow and fast charge temperatures of the NiCd, NiMH, SLA and Li-ion.

Slow Charge (0.1)

Fast Charge (0.5-1C)

Nickel Cadmium

0°C to 45°C (32°F to 113°F)

5°C to 45°C (41°F to 113°F)

Nickel-Metal Hydride

0°C to 45°C (32°F to 113°F)

10C° to 45°C (50°F to 113°F)

Lead Acid

0°C to 45°C (32°F to 113°F)

5C° to 45°C (41°F to 113°F)

Lithium Ion

0°C to 45°C (32°F to 113°F)

5C° to 45°C (41°F to 113°F)

Figure 4-6: Permissible temperature limits for various batteries.
Older battery technologies are more tolerant to charging at extreme temperatures than newer, more advanced systems.

NiCd batteries can be fast-charged in an hour or so, however, such a fast charge can only be applied within temperatures of 5°C and 45°C (41°F and 113°F). More moderate temperatures of 10°C to 30°C (50°F to 86°F) produce better results. When charging a NiCd below 5°C (41°F), the ability to recombine oxygen and hydrogen is greatly reduced and pressure build up occurs as a result. In some cases, the cells vent, releasing oxygen and hydrogen. Not only do the escaping gases deplete the electrolyte, hydrogen is highly flammable!

Chargers featuring NDV to terminate full-charge provide some level of protection when fast-charging at low temperatures. Because of the battery’s poor charge acceptance at low temperatures, the charge energy is turned into oxygen and to a lesser amount hydrogen. This reaction causes cell voltage drop, terminating the charge through NDV detection. When this occurs, the battery may not be fully charged, but venting is avoided or minimized.

To compensate for the slower reaction at temperatures below 5°C, a low charge rate of 0.1C must be applied. Special charge methods are available for charging at cold temperatures. Industrial batteries that need to be fast-charged at low temperatures include a thermal blanket that heats the battery to an acceptable temperature. Among commercial batteries, the NiCd is the only battery that can accept charge at extremely low temperatures.

Charging at high temperatures reduces the oxygen generation. This reduces the NDV effect and accurate full-charge detection using this method becomes difficult. To avoid overcharge, charge termination by temperature measurement becomes more practical.

The charge acceptance of a NiCd at higher temperatures is drastically reduced. A battery that provides a capacity of 100 percent if charged at moderate room temperature can only accept 70 percent if charged at 45°C (113°F), and 45 percent if charged at 60°C (140°F) (see Figure 4-7). Similar conditions apply to the NiMH battery. This demonstrates the typically poor summer performance of vehicular mounted chargers using nickel-based batteries.

Another reason for poor battery performance, especially if charged at high ambient temperatures, is premature charge cutoff. This is common with chargers that use absolute temperature to terminate the fast charge. These chargers read the SoC on battery temperature alone and are fooled when the room temperature is high. The battery may not be fully charged, but a timely charge cut-off protects the battery from damage due to excess heat.

The NiMH is less forgiving than the NiCd if charged under high and low temperatures. The NiMH cannot be fast charged below 10°C (45°F), neither can it be slow charged below 0°C (32°F). Some industrial chargers adjust the charge rate to prevailing temperatures. Price sensitivity on consumer chargers does not permit elaborate temperature control features.

Figure 4-7: Effects of temperature on NiCd charge acceptance.
Charge acceptance is much reduced at higher temperatures. NiMH cells follow a similar pattern.

The lead acid battery is reasonably forgiving when it comes to temperature extremes, as in the case of car batteries. Part of this tolerance is credited to the sluggishness of the lead acid battery. A full charge under ten hours is difficult, if not impossible. The recommended charge rate at low temperature is 0.3C.

Figure 4-8 indicates the optimal peak voltage at various temperatures when recharging and float charging an SLA battery. Implementing temperature compensation on the charger to adjust to temperature extremes prolongs the battery life by up to 15 percent. This is especially true when operating at higher temperatures.

An SLA battery should never be allowed to freeze. If this were to occur, the battery would be permanently damaged and would only provide a few cycles when it returned to normal temperature.

0°C (32°F) 25°C (77°F) 40°C (104°F)

Voltage limit on recharge 2.55V/cell 2.45V/cell 2.35V/cell

Continuous float voltage 2.35V/cell or lower 2.30V/cell or lower 2.25V/cell or lower

Figure 4-8: Recommended voltage limits on recharge and float charge of SLAs.
These voltage limits should be applied when operating at temperature extremes.

To improve charge acceptance of SLA batteries in colder temperatures, and avoid thermal runaway in warmer temperatures, the voltage limit of a charger should be compensated by approximately 3mV per cell per degree Celsius. The voltage adjustment has a negative coefficient, meaning that the voltage threshold drops as the temperature increases. For example, if the voltage limit is set to 2.40V/cell at 20°C, the setting should be lowered to 2.37V/cell at 30°C and raised to 2.43V/cell at 10°C. This represents a 30mV correction per cell per 10 degrees Celsius.

The Li-ion batteries offer good cold and hot temperature charging performance. Some cells allow charging at 1C from 0°C to 45°C (32°F to 113°F). Most Li-ion cells prefer a lower charge current when the temperature gets down to 5°C (41°F) or colder. Charging below freezing must be avoided because plating of lithium metal could occur.

04. Relation to functions positive real in the right-half plane

Sun, 01 Mar 1998 00:00:00 +0000

NOTE: THIS DOCUMENT IS OBSOLETE, PLEASE CHECK THE NEW VERSION: "Introduction to Digital Filters with Audio Applications", by Julius O. Smith III, Copyright © 2017-11-26 by Julius O. Smith III - Center for Computer Research in Music and Acoustics (CCRMA), Stanford University

<< Previous page TOC INDEX Next page >>
Relation to functions positive real in the right-half plane
Theorem. , where is any positive real number.
Proof. We shall show that the change of variable , provides a conformal map from the z-plane to the s-plane that takes the region to the region . The general formula for a bilinear conformal mapping of functions of a complex variable is given by

In general, a bilinear transformation maps circles and lines into circles and lines [Churchill 1960]. We see that the choice of three specific points and their images determines the mapping for all and . We must have that the imaginary axis in the s-plane maps to the unit circle in the z-plane. That is, we may determine the mapping by three points of the form and . If we predispose one such mapping by choosing the pairs and , then we are left with transformations of the form

or

Letting be some point on the imaginary axis, and be some point on the unit circle, we find that

which gives us that is real. To avoid degeneracy, we require , and this translates to finite and nonzero. Finally, to make the unit disk map to the left-half s-plane, and must have the same sign in which case .
There is a bonus associated with the restriction that be real which is that

We have therefore proven
Theorem. PR PR, where is any positive real number.
The class of mappings of the form Eq. (1.3) which take the exterior of the unit circle to the right-half plane is larger than the class Eq. (1.3). For example, we may precede the transformation Eq. (1.3) by any conformal map which takes the unit disk to the unit disk, and these mappings have the algebraic form of a first order complex allpass whose zero lies inside the unit circle.

where is the zero of the allpass and the image (also pre-image) of the origin, and is an angle of pure rotation. Note that Eq. (1.3) is equivalent to a pure rotation, followed by a realallpass substitution ( real), followed by a pure rotation. The general preservation of condition (2) in Def. 2 forces the real axis to map to the real axis. Thus rotations by other than are useless, except perhaps in some special cases. However, we may precede Eq. (1.3) by the first order real allpass substitution

which maps the real axis to the real axis. This leads only to the composite transformation,

which is of the form Eq. (1.3) up to a minus sign (rotation by). By inspection of Eq. (1.3), it is clear that sign negation corresponds to the swapping of points and , or and . Thus the only extension we have found by means of the general disk to disk pre-transform, is the ability to interchange two of the three points already tried. Consequently, we conclude that the largest class of bilinear transforms which convert functions positive real in the outer disk to functions positive real in the right-half plane is characterized by

Riemann's theorem may be used to show that Eq. (1.3) is also the largest such class of conformal mappings. It is not essential, however, to restrict attention solely to conformal maps. The pre-transform , for example, is not conformal and yet PR is preserved.
The bilinear transform is one which is used to map analog filters intodigital filters. Another such mapping is called the matched transform [Rabiner and Gold 1975]. It also preserves the positive real property.
Theorem. is PR if is positive real in the analog sense, where is interpreted as the sampling period.
Proof. The mapping takes the right-half -plane to the outer disk in the -plane. Also is real if is real. Hence PR implies PR. (Note, however, that rational functions do not in general map to rational functions.)
These transformations allow application of the large battery of tests which exist for functions positive real in the right-half plane [Van Valkenburg 1960].

05. Discharge Methods

Sun, 01 Mar 1998 00:00:00 +0000

<< Previous page INDEX Next page >>
5. Discharge Methods

The purpose of a battery is to store energy and release it at the appropriate time in a controlled manner. Being capable of storing a large amount of energy is one thing; the ability to satisfy the load demands is another. The third criterion is being able to deliver all available energy without leaving precious energy behind when the equipment cuts off.

In this chapter, we examine how different discharge methods can affect the deliverance of power. Further, we look at the load requirements of various portable devices and evaluate the performance of each battery chemistry in terms of discharge.
5.1 C-rate

The charge and discharge current of a battery is measured in C-rate. Most portable batteries, with the exception of the lead acid, are rated at 1C. A discharge of 1C draws a current equal to the rated capacity. For example, a battery rated at 1000mAh provides 1000mA for one hour if discharged at 1C rate. The same battery discharged at 0.5C provides 500mA for two hours. At 2C, the same battery delivers 2000mA for 30 minutes. 1C is often referred to as a one-hour discharge; a 0.5C would be a two-hour, and a 0.1C a 10 hour discharge.

The capacity of a battery is commonly measured with a battery analyzer. If the analyzer’s capacity readout is displayed in percentage of the nominal rating, 100 percent is shown if 1000mA can be drawn for one hour from a battery that is rated at 1000mAh. If the battery only lasts for 30 minutes before cut-off, 50 percent is indicated. A new battery sometimes provides more than 100 percent capacity. In such a case, the battery is conservatively rated and can endure a longer discharge time than specified by the manufacturer.

When discharging a battery with a battery analyzer that allows setting different discharge C-rates, a higher capacity reading is observed if the battery is discharged at a lower C-rate and vice versa. By discharging the 1000mAh battery at 2C, or 2000mA, the analyzer is scaled to derive the full capacity in 30 minutes. Theoretically, the capacity reading should be the same as a slower discharge, since the identical amount of energy is dispensed, only over a shorter time. Due to energy loss that occurs inside the battery and a drop in voltage that causes the battery to reach the low-end voltage cut-off sooner, the capacity reading is lower and may be 97 percent. Discharging the same battery at 0.5C, or 500mA over two hours would increase the capacity reading to about 103 percent.

The discrepancy in capacity readings with different C-rates largely depends on the internal resistance of the battery. On a new battery with a good load current characteristic or low internal resistance, the difference in the readings is only a few percentage points. On a battery exhibiting high internal resistance, the difference in capacity readings could swing plus/minus 10 percent or more.

One battery that does not perform well at a 1C discharge rate is the SLA. To obtain a practical capacity reading, manufacturers commonly rate these batteries at 0.05C or 20 hour discharge. Even at this slow discharge rate, it is often difficult to attain 100 percent capacity. By discharging the SLA at a more practical 5h discharge (0.2C), the capacity readings are correspondingly lower. To compensate for the different readings at various discharge currents, manufacturers offer a capacity offset.

Applying the capacity offset does not improve battery performance; it merely adjusts the capacity calculation if discharged at a higher or lower C-rate than specified. The battery manufacturer determines the amount of capacity offset recommended for a given battery type.

Li-ion/polymer batteries are electronically protected against high discharge currents. Depending on battery type, the discharge current is limited somewhere between 1C and 2C. This protection makes the Li-ion unsuitable for biomedical equipment, power tools and high-wattage transceivers. These applications are commonly reserved for the NiCd battery.

05. Discharge Methods 2

Sun, 01 Mar 1998 00:00:00 +0000

<< Previous page INDEX Next page >>
5.2 Depth of Discharge

The typical end-of-discharge voltage for nickel-based batteries is 1V/cell. At that voltage level, about 99 percent of the energy is spent and the voltage starts to drop rapidly if the discharge continues. Discharging beyond the cut-off voltage must be avoided, especially under heavy load.

Since the cells in a battery pack cannot be perfectly matched, a negative voltage potential (cell reversal) across a weaker cell occurs if the discharge is allowed to continue beyond the cut-off point. The larger the number of cells connected in series, the greater the likelihood of this occurring.

A NiCd battery can tolerate a limited amount of cell reversal, which is typically about 0.2V. During that time, the polarity of the positive electrode is reversed. Such a condition can only be sustained for a brief moment because hydrogen evolution occurs on the positive electrode. This leads to pressure build-up and cell venting.

If the cell is pushed further into voltage reversal, the polarity of both electrodes is being reversed, resulting in an electrical short. Such a fault cannot be corrected and the pack will need to be replaced.

On battery analyzers that apply a secondary discharge (recondition), the current is controlled to assure that the maximum allowable current, while in sub-discharge range, does not exceed a safe limit. Should a cell reversal develop, the current would be low enough as not to cause damage. A cell breakdown through recondition is possible on a weak or aged pack.

If the battery is discharged at a rate higher than 1C, the more common end-of-discharge point of a nickel-based battery is 0.9V/cell. This is done to compensate for the voltage drop induced by the internal resistance of the cell, the wiring, protection devices and contacts of the pack. A lower cut-off point also delivers better battery performance at cold temperatures.

The recommended end-of-discharge voltage for the SLA is 1.75V/cell. Unlike the preferred flat discharge curve of the NiCd, the SLA has a gradual voltage drop with a rapid drop towards the end of discharge (see Figure 5-1). Although this steady decrease in voltage is a disadvantage, it has a benefit because the voltage level can be utilized to display the state-of-charge (SoC) of a battery. However, the voltage readings fluctuate with load and the SoC readings are inaccurate.

Figure 5-1: Discharge characteristics of NiCd, NiMH and SLA batteries.
While voltage readings to measure the SoC are not practical on nickel-based batteries, the SLA enables some level of indication as to the SoC.

°C (77°F) with respect to the depth of discharge is:

150 – 200 cycles with 100 percent depth of discharge (full discharge)

400 – 500 cycles with 50 percent depth of discharge (partial discharge)

1000 and more cycles with 30 percent depth of discharge (shallow discharge)

05. Discharge Methods 3

Sun, 01 Mar 1998 00:00:00 +0000

<< Previous page INDEX Next page >>

The SLA should not be discharged beyond 1.75V per cell, nor can it be stored in a discharged state. The cells of a discharged SLA sulfate, a condition that renders the battery useless if left in that state for a few days.

The Li-ion typically discharges to 3.0V/cell. The spinel and coke versions can be discharged to 2.5V/cell. The lower end-of-discharge voltage gains a few extra percentage points. Since the equipment manufacturers cannot specify which battery type may be used, most equipment is designed for a three-volt cut-off.

Caution should be exercised not to discharge a lithium-based battery too low. Discharging a lithium-based battery below 2.5V may cut off the battery’s protection circuit. Not all chargers accommodate a recharge on batteries that have gone to sleep because of low voltage.

Some Li-ion batteries feature an ultra-low voltage cut-off that permanently disconnects the pack if a cell dips below 1.5V. This precaution prohibits recharge if a battery has dwelled in an illegal voltage state. A very deep discharge may cause the formation of copper shunt, which can lead to a partial or total electrical short. The same occurs if the cell is driven into negative polarity and is kept in that state for a while. A fully discharged battery should be charged at 0.1C. Charging a battery with a copper shunt at the 1C rate would cause excessive heat. Such a battery should be removed from service.

Discharging a battery too deeply is one problem; equipment that cuts off before the energy is consumed is another. Some portable devices are not properly tuned to harvest the optimal energy stored in a battery. Valuable energy may be left behind if the voltage cut-off-point is set too high.

Digital devices are especially demanding on a battery. Momentary pulsed loads cause a brief voltage drop, which may push the voltage into the cut-off region. Batteries with high internal resistance are particularly vulnerable to premature cut-off. If such a battery is removed from the equipment and discharged to the appropriate cut-off point with a battery analyzer on DC load, a high level of residual capacity can still be obtained.

Most rechargeable batteries prefer a partial rather than a full discharge. Repeated full discharge robs the battery of its capacity. The battery chemistry which is most affected by repeat deep discharge is lead acid. Additives to the deep-cycle version of the lead acid battery compensate for some of the cycling strain.

Similar to the lead acid battery, the Li-ion battery prefers shallow over repetitive deep discharge cycles. Up to 1000 cycles can be achieved if the battery is only partially discharged. Besides cycling, the performance of the Li-ion is also affected by aging. Capacity loss through aging is independent of use. However, in daily use, there is a combination of both.

The NiCd battery is least affected by repeated full discharge cycles. Several thousand charge/discharge cycles can be obtained with this battery system. This is the reason why the NiCd performs well on power tools and two-way radios that are in constant use. The NiMH is more delicate with respect to repeated deep cycling.

05. Discharge Methods 4

Sun, 01 Mar 1998 00:00:00 +0000

<< Previous page INDEX Next page >>
5.3 Pulse Discharge

Battery chemistries react differently to specific loading requirements. Discharge loads range from a low and steady current used in a flashlight, to intermittent high current bursts in a power tool, to sharp current pulses required for digital communications equipment, to a prolonged high current load for an electric vehicle traveling at highway speed. Because batteries are chemical devices that must convert higher-level active materials into an alternate state during discharge, the speed of such transaction determines the load characteristics of a battery. Also referred to as concentration polarization, the nickel and lithium-based batteries are superior to lead-based batteries in reaction speed. This reflects in good load characteristics.

The lead acid battery performs best at a slow 20-hour discharge. A pulse discharge also works well because the rest periods between the pulses help to disperse the depleted acid concentrations back into the electrode plate. In terms of capacity, these two discharge methods provide the highest efficiency for this battery chemistry.

A discharge at the rated capacity of 1C yields the poorest efficiency for the lead acid battery. The lower level of conversion, or increased polarization, manifests itself in a momentary higher internal resistance due to the depletion of active material in the reaction.

Different discharge methods, notably pulse discharging, also affect the longevity of some battery chemistries. While NiCd and Li-ion are robust and show minimal deterioration when pulse discharged, the NiMH exhibits a reduced cycle life when powering a digital load.

In a recent study, the longevity of NiMH was observed by discharging these batteries with analog and digital loads. In both tests, the battery discharged to 1.04V/cell. The analog discharge current was 500mA; the digital mode simulated the load requirements of the Global System for Mobile Communications (GSM) protocol and applied 1.65-ampere peak current for 12 ms every 100 ms. The current in between the peaks was 270mA. (Note that the GSM pulse for voice is about 550 ms every 4.5 ms).

With the analog discharge, the NiMH wore out gradually, providing an above average service life. At 700 cycles, the battery still provided 80 percent capacity. By contrast, the cells faded more rapidly with a digital discharge. The 80 percent capacity threshold was reached after only 300 cycles. This phenomenon indicates that the kinetic characteristics for the NiMH deteriorate more rapidly with a digital rather than an analog load.

05. Discharge Methods 5

Sun, 01 Mar 1998 00:00:00 +0000

<< Previous page INDEX Next page >>
5.4 Discharging at High and Low Temperature

Batteries function best at room temperature. Operating batteries at an elevated temperature dramatically shortens their life. Although a lead acid battery may deliver the highest capacity at temperatures above 30°C (86°F), prolonged use under such conditions decreases the life of the battery.

Similarly, a Li-ion performs better at high temperatures. Elevated temperatures temporarily counteracts the battery’s internal resistance, which is a result of aging. The energy gain is short-lived because elevated temperature promotes aging by further increasing the internal resistance.

There is one exception to running a battery at high temperature — it is the lithium polymer with dry solid polymer electrolyte, the true ‘plastic battery’. While the commercial Li-ion polymer uses some moist electrolyte to enhance conductivity, the dry solid polymer version depends on heat to enable ion flow. This requires that the battery core be kept at an operation temperature of 60°C to 100°C.

The dry solid polymer battery has found a niche market as backup power in warm climates. The battery is kept at the operating temperature with built-in heating elements. During normal operation, the core is kept warm with power derived from the utility grid. Only on a power outage would the battery need to provide power to maintain its own heat. To minimize heat loss, the battery is insulated.

The Li-ion polymer as standby battery is said to outperform VRLA batteries in terms of size and longevity, especially in shelters in which the temperature cannot be controlled. The high price of the Li-ion polymer battery remains an obstacle.

The NiMH chemistry degrades rapidly if cycled at higher ambient temperatures. Optimum battery life and cycle count are achieved at 20°C (68°F). Repeated charging and discharging at higher temperatures will cause irreversible capacity loss. For example, if operated at 30°C (86°F), the cycle life is reduced by 20 percent. At 40°C (104°F), the loss jumps to a whopping 40 percent. If charged and discharged at 45°C (113°F), the cycle life is only half of what can be expected if used at moderate room temperature. The NiCd is also affected by high temperature operation, but to a lesser degree.

At low temperatures, the performance of all battery chemistries drops drastically. While -20°C (-4°F) is threshold at which the NiMH, SLA and Li-ion battery stop functioning, the NiCd can go down to -40°C (-40°F). At that frigid temperature, the NiCd is limited to a discharge rate of 0.2C (5 hour rate). There are new types of Li-ion batteries that are said to operate down to -40°C.

It is important to remember that although a battery may be capable of operating at cold temperatures, this does not automatically mean it can also be charged under those conditions. The charge acceptance for most batteries at very low temperatures is extremely confined. Most batteries need to be brought up to temperatures above the freezing point for charging. The NiCd can be recharged at below freezing provided the charge rate is reduced to 0.1C.

05. Introduction

Sun, 01 Mar 1998 00:00:00 +0000

NOTE: THIS DOCUMENT IS OBSOLETE, PLEASE CHECK THE NEW VERSION: "The Digital Audio Resampling Home Page", by Julius O. Smith III, Copyright © 2016-05-17 by Julius O. Smith III - Center for Computer Research in Music and Acoustics (CCRMA), Stanford University

<< Previous page TOC Next page >>
Introduction
Bandlimited interpolation of discrete-time signals is a basic tool having extensive application in digital signal processing. In general, the problem is to correctly compute signal values at arbitrary continuous times from a set of discrete-time samples of the signal amplitude. In other words, we must be able to interpolate the signal between samples. Since the original signal is always assumed to be bandlimited to half the sampling rate, (otherwisealiasing distortion would occur upon sampling), Shannon's sampling theorem tells us the signal can be exactly and uniquely reconstructed for all time from its samples by bandlimited interpolation.
Considerable research has been devoted to the problem of interpolating discrete points. A comprehensive survey of ``fractional delay filter design'' is provided in [Laakso et al. 1996]. A comparison between classical (e.g., Lagrange) and bandlimited interpolation is given in [Schafer and Rabiner 1973]. The book Multirate Digital Signal Processing [Crochiere and Rabiner 1983] provides a comprehensive summary and review of classical signal processing techniques for sampling-rate conversion. In these techniques, the signal is first interpolated by an integer factor and then decimated by an integer factor. This provides sampling-rate conversion by any rational factor. The conversion requires a digital lowpass filter whose cutofffrequency depends on . While sufficiently general, this formulation is less convenient when it is desired to resample the signal at arbitrary times or change the sampling-rate conversion factor smoothly over time.
In this tutorial, a public-domain resampling algorithm is described which will evaluate a signal at any time specifiable by a fixed-pointnumber. In addition, one lowpass filter is used regardless of the sampling-rate conversion factor. The algorithm effectively implements the ``analog interpretation'' of rate conversion, as discussed in [Crochiere and Rabiner 1983], in which a certain lowpass-filter impulse response must be available as a continuous function. Continuity of the impulse response is simulated by linearly interpolating between samples of the impulse response stored in a table. Due to the relatively low cost of memory, the method is quite practical for hardware implementation.
In section 2, the basic theory is presented, section 3 addresses practical issues, and implementation details are discussed in section 4. Finally, section 5 discusses numerical requirements on the length, width, and interpolation accuracy of the filter coefficient table.

05. Special cases and examples

Sun, 01 Mar 1998 00:00:00 +0000

NOTE: THIS DOCUMENT IS OBSOLETE, PLEASE CHECK THE NEW VERSION: "Introduction to Digital Filters with Audio Applications", by Julius O. Smith III, Copyright © 2017-11-26 by Julius O. Smith III - Center for Computer Research in Music and Acoustics (CCRMA), Stanford University

<< Previous page TOC INDEX Next page >>
Special cases and examples
The sum of positive real functions is positive real.
The difference of positive real functions is conditionally positive real.
The product or division of positive real functions is conditionally PR.
PR not PR for .

06. Minimum Phase (MP) polynomials in Z

Sun, 01 Mar 1998 00:00:00 +0000

NOTE: THIS DOCUMENT IS OBSOLETE, PLEASE CHECK THE NEW VERSION: "Introduction to Digital Filters with Audio Applications", by Julius O. Smith III, Copyright © 2017-11-26 by Julius O. Smith III - Center for Computer Research in Music and Acoustics (CCRMA), Stanford University

<< Previous page TOC INDEX Next page >>
Minimum Phase (MP) polynomials in
All properties of MP polynomials apply without modification to marginallystable allpole transfer functions (cf. Thm. (1)).
Every first-order MP polynomial is positive real.
Every first-order MP polynomial is such that is positive real.
A PR second-order MP polynomial with complex-conjugate zeros,
(38)
(39)

satisfies

If , then has a double zero at

All polynomials of the form

are positive real. (These have zeros uniformly distributed on a circle of radius.)

06. The Secrets of Battery Runtime

Sun, 01 Mar 1998 00:00:00 +0000

<< Previous page INDEX Next page >>
6. The Secrets of Battery Runtime

Is the runtime of a portable device directly related to the size of the battery and the energy it can hold? In most cases, the answer is yes. But with digital equipment, the length of time a battery can operate is not necessarily linear to the amount of energy stored in the battery.

In this chapter we examine why the specified runtime of a portable device cannot always be achieved, especially after the battery has aged. We address the four renegades that are affecting the performance of the battery. They are: declining capacity, increasing internal resistance, elevated self-discharge, and premature voltage cut-off on discharge.
6.1 Declining Capacity

The amount of charge a battery can hold gradually decreases due to usage, aging and, with some chemistries, lack of maintenance. Specified to deliver about 100 percent capacity when new, the battery eventually requires replacement when the capacity drops to the 70 or 60 percent level. The warranty threshold is typically 80 percent.

The energy storage of a battery can be divided into three imaginary sections consisting of available energy, the empty zone that can be refilled and the rock content that has become unusable. Figure 6-1 illustrates these three sections of a battery.

In nickel-based batteries, the rock content may be in the form of crystalline formation, also known as memory. Deep cycling can often restore the capacity to full service. Also known as ‘exercise’, a typical cycle consists of one or several discharges to 1V/cell with subsequent discharges.

Figure 6-1: Battery charge capacity.
Three imaginary sections of a battery consisting of available energy, empty zone and rock content.

With usage and age, the rock content grows. Without regular maintenance, the user may end up carrying rocks instead of batteries.

The loss of charge acceptance of the Li-ion/polymer batteries is due to cell oxidation, which occurs naturally during use and as part of aging. Li-ion batteries cannot be restored with cycling or any other external means. The capacity loss is permanent because the metals used in the cells are designated to run for a specific time only and are being consumed during their service life.

Performance degradation of the lead acid battery is often caused by sulfation, a thin layer that forms on the negative cell plates, which inhibits current flow. In addition, there is grid corrosion that sets in on the positive plate. With sealed lead acid batteries, the issue of water permeation, or loss of electrolyte, also comes into play. Sulfation can be reversed to a certain point with cycling and/or topping charge but corrosion and permeation are permanent. Adding water to a sealed lead acid battery may help to restore operation but the long-term results are unpredictable.

06. The Secrets of Battery Runtime 2

Sun, 01 Mar 1998 00:00:00 +0000

<< Previous page INDEX Next page >>
Increasing Internal Resistance

To a large extent, the internal resistance, also known as impedance, determines the performance and runtime of a battery. If measured with an AC signal, the internal resistance of a battery is also referred to as impedance. High internal resistance curtails the flow of energy from the battery to the equipment.

A battery with simulated low and high internal resistance is illustrated below. While a battery with low internal resistance can deliver high current on demand, a battery with high resistance collapses with heavy current. Although the battery may hold sufficient capacity, the voltage drops to the cut-off line and the ‘low battery’ indicator is triggered. The equipment stops functioning and the remaining energy is undelivered.

Figure 6-2: Effects of impedance on battery load.
A battery with low impedance provides unrestricted current flow and delivers all available energy. A battery with high impedance cannot deliver high-energy bursts due to a restricted path, and equipment may cut off prematurely.

NiCd has the lowest internal resistance of all commercial battery systems, even after delivering 1000 cycles. In comparison, NiMH starts with a slightly higher resistance and the readings increase rapidly after 300 to 400 cycles.

Maintaining a battery at low internal resistance is important, especially with digital devices that require high surge current. Lack of maintenance on nickel-based batteries can increase the internal resistance. Readings of more than twice the normal resistance have been observed on neglected NiCd batteries. After applying a recondition cycle with the Cadex 7000 Series battery analyzer, the readings on the batteries returned to normal. Reconditioning clears the cell plates of unwanted crystalline formations, which restores proper current flow.

Li-ion offers internal resistance characteristics that are between those of NiMH and NiCd. Usage does not contribute much to the increase in resistance, but aging does. The typical life span of a Li-ion battery is two to three years, whether it is used or not. Cool storage and keeping the battery in a partially charged state when not in use retard the aging process.

The internal resistance of the Li-ion batteries cannot be improved with cycling. The cell oxidation, which causes high resistance, is non-reversible. The ultimate cause of failure is high internal resistance. Energy may still be present in the battery, but it can no longer be delivered due to poor conductivity.

With effort and patience, lead acid batteries can sometimes be improved by cycling or applying a topping and/or equalizing charge. This reduces the current-inhibiting sulfation layer but does not reverse grid corrosion.

Figure 6-3 compares the voltage signature and corresponding runtime of a battery with low, medium and high internal resistance when connected to a digital load. Similar to a soft ball that easily deforms when squeezed, the voltage of a battery with high internal resistance modulates the supply voltage and leaves the imprint of the load. The current pulses push the voltage towards the end-of-discharge line, resulting in a premature cut-off.

When measuring the battery with a voltmeter after the equipment has cut off and the load is removed, the terminal voltage commonly recovers and the voltage reading appears normal. This is especially true of nickel-based batteries. Measuring the open terminal voltage is an unreliable method to establish the state-of-charge (SoC) of the battery.

A battery with high impedance may perform well if loaded with a low DC current such as a flashlight, portable CD player or wall clock. With such a gentle load, virtually all of the stored energy can be retrieved and the deficiency of high impedance is masked.

Figure 6-3: Discharge curve.
This chart compares the runtime of batteries with similar capacities under low, medium and high impedance when connected to a pulsed load.

The internal resistance of a battery can be measured with dedicated impedance meters. Several methods are available, of which the most common are applying DC loads and AC signals. The AC method may be done with different frequencies. Depending on the level of capacity loss, each technique provides slightly different readings. On a good battery, the measurements are reasonably close; on a weak battery, the readings between the methods may disperse more drastically.

Modern battery analyzers offer internal resistance measurements as a battery quick-test. Such tests can identify batteries that would fail due to high internal resistance, even though the capacity may still be acceptable. Internal battery resistance measurements are available in the Cadex 7000 Series battery analyzers. (See Chapter 9: Internal Battery Resistance.)

06. The Secrets of Battery Runtime 3

Sun, 01 Mar 1998 00:00:00 +0000

<< Previous page INDEX Next page >>
6.3 Elevated Self-Discharge

All batteries exhibit a certain amount of self-discharge; the highest is visible on nickel-based batteries. As a rule, a nickel-based battery discharges 10 to 15 percent of its capacity in the first 24 hours after charge, followed by 10 to 15 percent every month thereafter.

The self-discharge on the Li-ion battery is lower compared to the nickel-based systems. The Li-ion self-discharges about five percent in the first 24 hours and one to two percent thereafter. Adding the protection circuit increases the self-discharge to ten percent per month.

One of the best batteries in terms of self-discharge is the lead acid system; it only self-discharges five percent per month. It should be noted, however, that the lead acid family has also the lowest energy density among current battery systems. This makes the system unsuitable for most hand-held applications.

At higher temperatures, the self-discharge on all battery chemistries increases. Typically, the rate doubles with every 10°C (18°F). Large energy losses occur through self-discharge if a battery is left in a hot vehicle. On some older batteries, stored energy may get lost during the course of the day through self-discharge rather than actual use.

The self-discharge of a battery increases with age and usage. For example, a NiMH battery is good for 300 to 400 cycles, whereas a NiCd adequately performs over 1000 cycles before high self-discharge affects the performance of the battery. Once a battery exhibits high self-discharge, no remedy is available to reverse the effect. Factors that accelerate self-discharge on nickel-based batteries are damaged separators (induced by excess crystalline formation, allowing the packs to cook while charging), and high cycle count, which promotes swelling in the cell.

Figure 6-4: Effects of high load impedance.
A battery may gradually self-discharge as a result of high temperature, high cycle count and age. In older batteries, stored energy may be lost during the course of the day through self-discharge rather than actual use.

At present, no simple quick-test is available to measure the self-discharge of a battery. A battery analyzer can be used by first reading the initial capacity after full charge, then measuring the capacity again after a rest period of 12 hours. The Cadex 7000 Series performs this task automatically. In the future, quick test methods may be available that are able to measure the self-discharge of a battery within a few seconds.
6.4 Premature Voltage Cut-off

Some portable equipment does not fully utilize the low-end voltage spectrum of a battery. The equipment cuts off before the designated end-of-discharge voltage is reached and some precious battery power remains unused.

A high cut-off voltage problem is more widespread than is commonly assumed. For example, a certain brand of mobile phone that is powered with a single-cell Li-ion battery cuts off at 3.3V. The Li-ion can be designed to be used to 3V and lower. With a discharge to 3.3V, only about 70 percent of the expected 100 percent capacity is utilized. Another mobile phone using NiMH and NiCd batteries cuts off at 5.7V. The four-cell nickel-based batteries are designed to discharge to 5V.

Figure 6-5: Illustration of equipment with high cut-off voltage.
Some portable devices do not utilize all available battery power and leave precious energy behind.

When discharging these batteries to their respective end-of-discharge threshold with a battery analyzer after the equipment has cut off, up to 60 percent residual capacity readings can be retrieved. High residual capacity is prevalent with batteries that have elevated internal resistance and are operated at warm ambient temperatures. Digital devices that load the battery with current bursts are more receptive to premature voltage cut-off than analog equipment.

A ’high cut-off voltage’ is mostly equipment related. In some cases the problem of premature cut-off is induced by a battery with low voltage. A low table voltage is often caused by a battery pack that contains a cell with an electrical short. Memory also causes a decrease in voltage; however, this is only present in nickel-based systems. In addition, elevated temperature lowers the voltage level on all battery systems. Voltage reduction due to high temperatures is temporary and normalizes once the battery cools down.

06. Theory of Ideal Bandlimited Interpolation

Sun, 01 Mar 1998 00:00:00 +0000

NOTE: THIS DOCUMENT IS OBSOLETE, PLEASE CHECK THE NEW VERSION: "The Digital Audio Resampling Home Page", by Julius O. Smith III, Copyright © 2016-05-17 by Julius O. Smith III - Center for Computer Research in Music and Acoustics (CCRMA), Stanford University

<< Previous page TOC Next page >>
Theory of Ideal Bandlimited Interpolation
We review briefly the ``analog interpretation'' of sampling rate conversion[Crochiere and Rabiner 1983] on which the present method is based. Suppose we have samples of a continuous absolutely integrable signal , where is time in seconds (real), ranges over the integers, and is the sampling period. We assume is bandlimited to, where is the sampling rate. If denotes the Fourier transform of , i.e., , then we assume for . Consequently, Shannon's sampling theorem gives us that can be uniquely reconstructed from the samples via
(1)

where

To resample at a new sampling rate , we need only evaluate Eq. (1) at integer multiples of .
When the new sampling rate is less than the original rate , the lowpass cutoff must be placed below half the new lower sampling rate. Thus, in the case of an ideal lowpass, , where the scale factor maintains unity gain in the passband.
A plot of the sinc function to the left and right of the origin is shown in Fig. 1. Note that peak is at amplitude , and zero-crossings occur at all nonzero integers. The sinc function can be seen as a hyperbolically weighted sine function with its zero at the origin canceled out.

Figure 1:The sinc function plotted for seven zero-crossings to the left and right.
If ``'' denotes the convolution operation for digital signals, then the summation in Eq. (1) can be written as .
Equation Eq. (1) can be interpreted as a superpositon of shifted and scaled sinc functions . A sinc function instance is translated to each signal sample and scaled by that sample, and the instances are all added together. Note that zero-crossings of occur at all integers except . That means at time, (i.e., on a sample instant), the only contribution to the sum is the single sample . All other samples contribute sinc functions which have a zero-crossing at time . Thus, the interpolation goes precisely through the existing samples, as it should.
A plot indicating how sinc functions sum together to reconstruct bandlimited signals is shown in Fig. 2. The figure shows a superposition of five sinc functions, each at unit amplitude, and displaced by one-sample intervals. These sinc functions would be used to reconstruct the bandlimited interpolation of the discrete-time signal . Note that at each sampling instant , the solid line passes exactly through the tip of the sinc function for that sample; this is just a restatement of the fact that the interpolation passes through the existing samples. Since the nonzero samples of the digital signal are all , we might expect the interpolated signal to be very close to over the nonzero interval; however, this is far from being the case. The deviation from unity between samples can be thought of as ``overshoot'' or ``ringing'' of the lowpass filter which cuts off at half the sampling rate, or it can be considered a ``Gibbs phenomenon'' associated with bandlimiting.

Figure 2:Bandlimited reconstruction of the signal from its samples . The dots show the signal samples, the dashed lines show the component sinc functions, and the solid line shows the unique bandlimited reconstruction from the samples obtained by summing the component sinc functions.
A second interpretation of Eq. (1) is as follows: to obtain the interpolation at time , shift the signal samples under one sinc function so that time in the signal is translated under the peak of the sinc function, then create the output as a linear combination of signal samples where the coefficient of each signal sample is given by the value of the sinc function at the location of each sample. That this interpretation is equivalent to the first can be seen as a result of the fact that convolution is commutative; in the first interpretation, all signal samples are used to form a linear combination of shifted sinc functions, while in the second interpretation, samples from one sinc function are used to form a linear combination of samples of the shifted input signal. The practical bandlimited interpolation algorithm presented below is based on the second interpretation.

07. Conjectured Properties

Sun, 01 Mar 1998 00:00:00 +0000

NOTE: THIS DOCUMENT IS OBSOLETE, PLEASE CHECK THE NEW VERSION: "Introduction to Digital Filters with Audio Applications", by Julius O. Smith III, Copyright © 2017-11-26 by Julius O. Smith III - Center for Computer Research in Music and Acoustics (CCRMA), Stanford University

<< Previous page TOC INDEX Next page >>
Conjectured Properties
The following conjectures are true for analog positive-real functions, but no rigorous attempt was made to establish them in the discrete-time case.
If all poles and zeros of a PR function are on the unit circle, then they alternate along the circle.
If is PR, then so is , where the prime denotes differentiation in .

07. From Theory to Practice

Sun, 01 Mar 1998 00:00:00 +0000

NOTE: THIS DOCUMENT IS OBSOLETE, PLEASE CHECK THE NEW VERSION: "The Digital Audio Resampling Home Page", by Julius O. Smith III, Copyright © 2016-05-17 by Julius O. Smith III - Center for Computer Research in Music and Acoustics (CCRMA), Stanford University

<< Previous page TOC Next page >>
From Theory to Practice
The summation in Eq. (1) cannot be implemented in practice because the ``ideal lowpass filter'' impulse response actually extends from minus infinity to infinity. It is necessary in practice to windowthe ideal impulse response so as to make it finite. This is the basis of the window method for digital filter design[Digital Signal Processing Committee 1979a,Rabiner and Gold 1975]. While many other filter design techniques exist, the window method is simple and robust, especially for very long impulse responses. In the case of the algorithm presented below, the filter impulse response is very long because it is heavily oversampled. Another approach is to design optimal decimated ``sub-phases'' of the filter impulse response, which are then interpolated to provide the ``continuous'' impulse response needed for the algorithm [Putnam and Smith 1997].
Figure 3 shows the frequency response of the ideal lowpass filter. This is just the Fourier transform of .

Figure 3:Frequency response of the ideal lowpass filter.
If we truncate at the fifth zero-crossing to the left and the right of the origin, we obtain the frequency response shown in Fig. 4. Note that the stopband exhibits only slightly more than 20 dB rejection.

Figure 4:Frequency response of the ideal lowpass filter after rectangularly windowing the ideal (sinc) impulse response at the fifth zero crossing to the left and right of the time origin. The vertical axis is in units of decibels(dB), and the horizontal axis is labeled in units of spectral samples between plus and minus half the sampling rate.
If we instead use the Kaiser window [Kaiser 1974,Smith 1992] to taper to zero by the fifth zero-crossing to the left and the right of the origin, we obtain the frequency response shown in Fig. 5. Note that now the stopband starts out close to dB. The Kaiser window has a single parameter which can be used to modify the stop-band attenuation, trading it against the transition width from pass-band to stop-band.

Figure 5:Frequency response of the ideal lowpass filter Kaiser windowed at the fifth zero crossing to the left and right.

07. The ‘Smart’ Battery

Sun, 01 Mar 1998 00:00:00 +0000

<< Previous page INDEX Next page >>
7. The ‘Smart’ Battery

Aspeaker at a battery seminar remarked that, “The battery is a wild animal and artificial intelligence domesticates it.” An ordinary or ‘dumb’ battery has the inherit problem of not being able to display the amount of reserve energy it holds. Neither weight, color, nor size provides any indication of the battery’s state-of-charge (SoC) and state-of-health (SoH). The user is at the mercy of the battery when pulling a freshly charged battery from the charger.

07. The ‘Smart’ Battery 2

Sun, 01 Mar 1998 00:00:00 +0000

<< Previous page INDEX Next page >>
The SMBus

The SMBus is the most complete of all systems. It represents a large effort from the portable electronic industry to standardize to one communications protocol and one set of data. The SMBus is a two-wire interface system through which simple power-related chips can communicate with the rest of the system. One wire handles the data; the second is the clock. It uses I²C as its backbone. Defined by Philips, the I²C is a synchronous multi-drop bi-directional communications system, which operates at a speed of up to 100 kilohertz (kHz).

The Duracell/Intel SBS, in use today, was standardized in 1993. In previous years, computer manufacturers had developed their own proprietary ‘smart’ batteries. With the new SBS specification, a broader interface standard was made possible. This reduces the hurdles of interfering with patents and intellectual properties.

In spite of an agreed standard, many large computer manufacturers, such as IBM, Compaq and Toshiba, have retained their proprietary batteries. The reason for going their own way is partly due to safety, performance and form factor. Manufacturers claim that they cannot guarantee safe and enduring performance if a non-brand battery is used. To make the equipment as compact as possible, the manufacturers explain that the common form factor battery does not optimally fit their available space. Perhaps the leading motive for using their proprietary batteries is pricing. In the absence of competition, these batteries can be sold for a premium price.

The early SMBus batteries had problems of poor accuracy. Electronic circuits did not provide the necessary resolution; neither was real time reporting of current, voltage and temperature adequate. On some batteries, the specified accuracy could only be achieved if the battery was new, operated at room temperature and was discharged at a steady rate of 1C. Operating in adverse temperatures or discharging at uneven loads reduced the accuracy dramatically. Most loads for portable equipment are uneven and fluctuate with power demand. There are power surges on a laptop at start up and refresh, high inrush currents on biomedical equipment during certain procedures and sharp pulse bursts on digital communications devices on transmit.

In the absence of a reliable reporting system on the older generation of ‘smart’ batteries, capacity estimation was inaccurate. This resulted in powering down the equipment before the battery was fully depleted, leaving precious energy behind. Most batteries introduced in the late 1990s have resolved some or all of these deficiencies. Further improvements will be necessary.

Design— The design philosophy behind the SMBus battery is to remove the charge control from the charger and assign it to the battery. With a true SMBus system, the battery becomes the master and the charger serves as a slave that must follow the dictates of the battery. This is done out of concerns over charger quality, compatibility with new and old battery chemistries, administration of the correct amount of charge currents and accurate full-charge detection. Simplifying the charging for the user is an issue that is important when considering that some battery packs share the same footprint but contain radically different chemistries.

The SMBus system allows new battery chemistries to be introduced without the charger becoming obsolete. Because the battery controls the charger, the battery manages the voltage and current levels, as well as cut-off thresholds. The user does not need to know which battery chemistry is being used.

The analogy of charging a ‘smart’ and ‘dumb’ battery can be made with the eating habits of an adult and a baby. Charging a ‘smart’ battery resembles the eating choices of a responsible adult who knows best what food to select how much to take. The baby, in on the other hand, has limited communications skills in expressing the type and amount of food desired. Putting this analogy in parallel with charging batteries, the charger servicing ‘dumb’ batteries can only observe the approximate SoC level and avoid overcharge conditions.

07. The ‘Smart’ Battery 3

Sun, 01 Mar 1998 00:00:00 +0000

<< Previous page INDEX Next page >>

Architecture — An SMBus battery contains permanent and temporary data. The permanent data is programmed into the battery at the time of manufacturing and include battery ID number, battery type, serial number, manufacturer’s name and date of manufacture. The temporary data is acquired during use and consists of cycle count, user pattern and maintenance requirements. Some of the temporary data is being replaced and renewed during the life of the battery.

The SMBus is divided into Level 1, 2 and 3. Level 1 has been eliminated because it does not provide chemistry independent charging. Level 2 is designed for in-circuit charging. A laptop that charges its battery within the unit is a typical example of Level 2. Another application of Level 2 is a battery that contains the charging circuit within the pack. Level 3 is reserved for full-featured external chargers.

Most external SMBus chargers are based on Level 3. Unfortunately, this level is complex and the chargers are costly to manufacture. Some lower cost chargers have emerged that accommodate SMBus batteries but are not fully SBS compliant. Manufacturers of SMBus batteries do not readily endorse this shortcut. Safety is always a concern, but customers buy these economy chargers because of the lower price.

Figure 7-2: Two-wire SMBus system.
The SMBus is based on a two-wire system using a standardized communications protocol. This system lends itself to standardized state-of-charge and state-of-health measurements.

Serious industrial battery users operating biomedical instruments, data collection devices and survey equipment use Level 3 chargers with full-fledged charge protocol. No shortcuts are applied. To assure compatibility, the charger and battery are matched and only approved packs are used. The need to test and approve the marriage between specific battery and charger types is unfortunate given that the ‘smart’ battery is intended to be universal.

Among the most popular SMBus batteries for portable computers are the 35 and 202 form-factors. Manufactured by Sony, Hitachi, GP Batteries, Moltech (formerly Energizer), Moli Energy and many others, this battery works (should work) in all portable equipment designed for this system.

Figure 7-3 illustrates the 35 and 202 series ‘smart' batteries. Although the ‘35’ has a smaller footprint compared to the ‘202’, most chargers are designed to accommodate all sizes, provided the common five-prong knife connector is used.

Figure 7-3: 35 and 202 series ‘smart’ batteries featuring SMBus.
Available in NiCd, NiMH and Li-ion chemistries, these batteries are used for mobile computing, biomedical instruments and high-end survey equipment. The same form factor also accommodates NiCd and NiMH chemistries but without SMBus (‘dumb’).

07. The ‘Smart’ Battery 4

Sun, 01 Mar 1998 00:00:00 +0000

<< Previous page INDEX Next page >>

Negatives of the SMBus — Like any good invention, the SMBus battery has some serious downsides that must be addressed. For starters, the ‘smart’ battery costs about 25 percent more than the ‘dumb’ equivalent. In addition, the ‘smart’ battery was intended to simplify the charger, but a full-fledged Level 3 charger costs substantially more than a regular dumb model.

A more serious issue is maintenance requirements, better known as capacity re-learning. This procedure is needed on a regular basis to calibrate the battery. The Engineering Manager of Moli Energy, a large Li-ion cell manufacturer commented, “With the Li-ion battery we have eliminated the memory effect, but are we introducing digital memory with the SMBus battery?”

Why is calibration needed? The answer is in correcting the tracking errors that occur between the battery and the digital sensing circuit during use. The most ideal battery use, as far as fuel-gauge accuracy is concerned, is a full charge followed by a full discharge at a constant 1C rate. This ensures that the tracking error is less than one percent per cycle. However, a battery may be discharged for only a few minutes at a time and commonly at a lower C-rate than 1C. Worst of all, the load may be uneven and vary drastically. Eventually, the true capacity of the battery no longer synchronizes with the fuel gauge and a full charge and discharge are needed to ‘re-learn’ or calibrate the battery.

How often is calibration needed? The answer lies in the type of battery application. For practical purposes, a calibration is recommended once every three months or after every 40 short cycles. Long storage also contributes to errors because the circuit cannot accurately compensate for self-discharge. After extensive storage, a calibration cycle is recommended prior to use.

Many applications apply a full discharge as part of regular use. If this occurs regularly, no additional calibration is needed. If a full discharge has not occurred for a few months and the user notices the fuel gauge losing accuracy, a deliberate full discharge on the equipment is recommended. Some intelligent equipment advises the user when a calibrating discharge is needed. This is done by measuring the tracking error and estimating the discrepancy between the fuel gauge reading and that of the chemical battery.

What happens if the battery is not calibrated regularly? Can such a battery be used in confidence? Most ‘smart’ battery chargers obey the dictates of the cells rather than the electronic circuit. In this case, the battery will be fully charged regardless of the fuel gauge setting. Such a battery is able to function normally, but the digital readout will be inaccurate. If not corrected, the fuel gauge information simply becomes a nuisance.

The level of non-compliance is another problem with the SMBus. Unlike other tightly regulated standards, such as the long play record introduced in the late 1950s, the audiocassette in the 1960s, the VCR in the 1970s, ISDN and GSM in the 1980s and the USB in the 1990s, some variations are permitted in the SMBus protocol. These are: adding a check bid to halt the service if the circuit crashes, counting the number of discharges to advise on calibration and disallowing a charge if a certain fault condition has occurred. Unfortunately, these variations cause problems with some existing chargers. As a result, a given SMBus battery should be checked for compatibility with the designated charger before use to assure reliable service. Ironically, the more features that are added to the SMBus charger and battery, the higher the likelihood of incompatibilities.

‘Smart’ battery technology has not received the widespread acceptance that battery manufacturers had hoped. Some engineers go so far as to suggest that the SMBus battery is a ‘misguided principal’. Design engineers may not have fully understood the complexity of charging batteries in the incubation period of the ‘smart’ battery. Manufacturers of SMBus chargers are left to clean up the mess.

The forecast in consumer acceptance of the ‘smart’ battery has been too optimistic. In the early 1990s when the SMBus battery was conceived, price many not have been as critical an issue as it is now. Then, the design engineer would include many wonderful options. Today, we look for scaled down products that are economically priced and perform the function intended. When looking at the wireless communications market, adding high-level intelligence to the battery is simply too expensive for most consumers. In the competitive mobile phone market, for example, the features offered by the SMBus would be considered overkill.

SMBus battery technology is mainly used by higher-level industrial applications and battery manufacturers are constantly searching for avenues to achieve a wider utilization of the ‘smart’ battery. According to a survey in Japan, about 30 percent of all mobile computing devices are equipped with a ‘smart’ battery.

Improvements in the ‘smart’ battery system, such as better compatibilities, improved error-checking functions and higher accuracies will likely increase the appeal of the ‘smart’ battery. Endorsement by large software manufacturers such as Microsoft will entice PC manufacturers to make full use of these powerful features.

07. The ‘Smart’ Battery 5

Sun, 01 Mar 1998 00:00:00 +0000

<< Previous page INDEX Next page >>
7.3 The State-of-Charge Indicator

Most SMBus batteries are equipped with a charge level indicator. When pressing a SoC button on a battery that is fully charged, all signal lights illuminate. On a partially discharged battery, half the lights illuminate, and on an empty battery, all lights remain dark. Figure 7-4 shows such a fuel gauge.

Figure 7-4: State-of-charge readout of a ‘smart’ battery.
Although the state-of-charge is displayed, the state-of-health and its predicted runtime are unknown.

While SoC information displayed on a battery or computer screen is helpful, the fuel gauge resets to 100 percent each time the battery is recharged, regardless of the battery’s SoH. A serious miscount occurs if an aged battery shows 100 percent after a full-charge, when in fact the charge acceptance has dropped to 50 percent or less. The question remains: “100 percent of what?” A user unfamiliar with this battery has little information about the runtime of the pack.

7.4 The Tri-State Fuel Gauge

The SoC information alone is incomplete without knowing the battery’s SoH. To fully evaluate the present state of a battery, three levels of information are needed. They are: SoC, SoH and the empty portion of the battery that can be replenished with a charge. (The empty portion is derived by deducting the SoC from the SoH.)

How can the three levels of a battery be measured and made visible to the user? While the SoC is relatively simple to produce, as discussed above, measuring the SoH is more complex. Here is how it works:

At time of manufacture, each SMBus battery is given its specified SoH status, which is 100 percent by default. This information is permanently programmed into the pack and does not change. With each charge, the battery resets to the full-charge status. During discharge, the energy units (coulombs) are counted and compared against the 100 percent setting. A perfect battery would indicate 100 percent on a calibrated fuel gauge. As the battery ages and the charge acceptance drops, the SoH begins to indicate lower readings. The discrepancy between the factory set 100 percent and the actual delivered coulombs is used to calculate the SoH.

Knowing the SoC and SoH, a simple linear display can be made. The SoC is indicated with green LED’s; the empty part remains dark; and the unusable part is shown with red LED’s. Figure 7-5 shows such a tri-state fuel gauge. As an alternative, the colored bar display may be replaced with a numeric display indicating SoH and SoC.

Figure 7-5: Tri-state fuel gauge.
The Battery Health Gauge reads the ‘learned’ battery information available on the SMBus and displays it on a multi-colored LED bar. This illustration shows a partially discharged battery of 50% SoC with a 20% empty portion and an unusable portion of 30%.

The most practical setting to place the tri-state-fuel gauge is on a charger. Only one display would be needed for a multi-bay charging unit. To view the readings of a battery, the user would simply press a button. The SoC and SoH information would be displayed within five seconds after inserting the battery into the charger bay. During charge, the gauge would reveal the charge level of each battery. This information would be handy when a functional battery is needed in a hurry. Cadex offers a series of SMBus chargers that feature the tri-state fuel gauge as an option.

07. The ‘Smart’ Battery 6

Sun, 01 Mar 1998 00:00:00 +0000

<< Previous page INDEX Next page >>
7.5 The Target Capacity Selector

For users that simply need a go/no go answer and do not want to bother about other battery information, chargers are available that feature a target capacity selector. Adjustable to 60, 70 or 80 percent, the target capacity selector acts as a performance check and flags batteries that do not meet set requirements.

If a battery falls below target, the charger triggers the condition light. The user is prompted to press the condition button to cycle the battery. Condition consists of charge/discharge/charge and performs calibration and conditioning functions. If the battery does not recover after the conditioning service, the fail light illuminates, indicating that the battery should be replaced. A green ready light at the completion of the program assures that the battery meets the required performance level.

An SMBus charger with the above described features acts as charger, conditioner and quality control system. Figure 76 illustrates a two-bay Cadex charger featuring the target capacity selector and discharge circuit. This unit is based on Level 3 and services both SMBus and ‘dumb’ batteries.

Some SMBus chargers can be fully automated to apply a conditioning cycle whenever the battery falls below the target setting. An override button cancels the discharge if a fast-charge is needed instead. Such a system maximizes the life of fleet batteries and assures that deadwood is identified and removed.

Figure 7-6:The Cadex SM2+ charger
This Level 3 charger serves as charger, conditioner and quality control system. It reads the battery’s true state-of-health and flags those that fall below the set target capacity. Each bay operates independently and charges NiCd, NiMH and Liion chemistries in approximately three hours. ‘Dumb’ batteries can also be charged.

By allowing the user to set the desired battery performance level, the question is raised as to what level to select. The answer is governed by the applications, reliability standards and cost policies.

It should be noted that the batteries are always charged to 100 percent, regardless of the target setting. The target capacity simply refers to the amount of charge the battery has delivered on the last discharge.

A practical target capacity setting for most applications is 80 percent. Decreasing the threshold to 70 percent will lower the performance standard but pass more batteries. A direct cost saving will result. The 60 percent level may suit those users who run a low budget operation, have ready access to replacement batteries and can live with shorter, less predictable runtimes.

Battery SoH readings are only available with Level 3 SMBus chargers servicing valid SMBus batteries. ‘Dumb’ batteries cannot provide SoH readings, even if they share the identical footprint of the ‘smart’ battery and can use the same charger.

07. The ‘Smart’ Battery 7

Sun, 01 Mar 1998 00:00:00 +0000

<< Previous page INDEX Next page >>
7.6 Fuel Gauges for Large Batteries

Most ‘smart’ battery applications today are limited to portable electronic equipment, and the electronic circuit is built right into the battery pack. In applications where larger ‘smart’ batteries are needed, such as electric wheelchairs, scooters, robots and forklifts, the electronic circuit may be placed in a box external to the battery.

The main benefit of adding intelligence to the battery is to enable the measurement of SoH and reserve energy. Most measuring devices used are based on voltage, which is known to be highly inaccurate.

Cadex is extending the ‘smart’ battery technology to wheeled applications. Called the Cadex Fuelcheck™, the device is based on the tri-state fuel gauge described earlier in this section. The system consists of a controller, current measuring device and a display unit. The Cadex Fuelcheck™ device can be added to new equipment when it is manufactured and can be installed in existing equipment as a retrofit. The installation is permanent and each wheeled appliance requires one system.

A tri-state linear bar graph consisting of colored LED’s would be the preferred display for a personal user, such as a wheelchair operator. By replacing the light display with a digital readout, additional information can be shown. For example, a display could indicate the remaining runtime based on the average power consumption logged. Corrections would be applied if the load factor changes during the course of the day. Figure 77 illustrates a digital fuel gauge using the LCD panel.

To initialize the Cadex Fuelcheck™ system, a one-time setup procedure is required. A PC will prompt the technician to enter information such as battery chemistry, desired end-of-discharge voltage, Ah rating of the battery, and designated service flags. The system is then calibrated by applying a full charge, followed by a full discharge. The initial discharge assesses the battery, or learns the performance. The SoH readings will be known after the first full discharge.

7-7: Cadex Fuelcheck™ digital fuel gauge.
The display indicates SoC, SoH, remaining power in hours, the voltage and current draw.

The simplest way to discharge the battery as part of calibration is to run it down through normal use in the equipment. The most accurate readings are obtained by using a load bank. Such a device may not be always available.

With a data logging option built into the system, the PC allows downloading service data that have been collected during use. Such information would benefit large equipment fleets and rental places that need to check the performance of the batteries on demand.

Similar to other ‘smart’ battery systems, calibration will be needed once every 6 to 12 months. More frequent calibrations may be required if the equipment is used for short durations between each recharge. The most practical way to calibrate the system is by occasionally allowing the battery to run down through continued use. Information relating to the date of the last calibration, or an early call for calibration if certain conditions occur, can be calculated and displayed.

Knowing the SoH of a battery at any time and scheduling timely service or replacement is a major benefit for industrial battery users. Such a system would be especially helpful for organizations in which different individuals use the equipment and no one is given maintenance responsibilities. Equipment rental places fall into this category.

08. Choosing the Right Battery

Sun, 01 Mar 1998 00:00:00 +0000

<< Previous page INDEX Next page >>
8. Choosing the Right Battery

What causes a battery to wear down — is it mechanical or chemical? The answer is ‘both’. A battery is a perishable product that starts deteriorating from the time it leaves the factory. Similar to a spring under tension, a battery seeks to revert to its lowest denominator. The rate of aging is subject to depth of discharge, environmental conditions, charge methods and maintenance procedures (or lack thereof). Each battery chemistry behaves differently in terms of aging and wear through normal use.
8.1 What’s the best battery for mobile phones?

When buying a replacement battery, the buyer often has the choice of different battery chemistries. Li-ion and Li-ion polymer batteries are used on newer phones, whereas the NiMH and NiCd are found in older models. If the buyer has a choice, the sales person may advise a customer to go for the highest capacity rating and to stay away from the NiCd because of the memory effect. The customer may settle for the slim-line NiMH because it offers relatively high capacity in a small package and is reasonably priced.

Seemingly a wise choice, an analysis in this chapter reveals that other chemistries may have served better. The NiMH offers good value for the price but falls short in expected cycle life. Although excellent when new, the performance trails off quickly after about 300 cycles due to decreased capacity and rising internal resistance. In comparison, the Li-ion can be used for about 500 cycles. The best cycle count is achieved with NiCd. Properly maintained, the NiCd delivers over 1000 cycles and the internal resistance remains low. However, the NiCd offers about 30 percent less capacity compared to the NiMH. In addition, the NiCd is being removed from the mobile phone market because of environmental concerns.

Switching to environmentally friendlier batteries is fitting, especially in the mobile phone market where the NiMH performs reasonably well and can be economical. The battery disposal issue is difficult to control, particularly in the hands of a diverse user group.

The NiMH and NiCd are considered high maintenance batteries, which require regular discharge cycles to prevent what is referred to as ‘memory’. Although the NiMH was originally advertised as memory-free, both NiCd and NiMH are affected by the phenomenon. The capacity loss is caused by crystalline formation that is generated by the positive nickel plate, a metal shared by both systems.

Nickel-based batteries, especially NiCd’s, should be fully discharged once per month. If such maintenance is omitted for four months or more, the capacity drops by as much as one third. A full restoration becomes more difficult the longer service is withheld.

It is not recommended to discharge a battery before each charge because this wears down the battery unnecessarily and shortens the life. Neither is it advisable to leave a battery in the charger for a long period of time. When not in use, the battery should be put on a shelf and charged before use. Always store the battery in a cool place.

Is the Li-ion a better choice? Yes, for many applications. The Li-ion is a low maintenance battery which offers high energy, is lightweight and does not require periodic full discharge. No trickle charge is applied once the battery reaches full charge. The Li-ion battery can stay in most chargers until used. The charging process of a Li-ion is, in many ways, simpler and cleaner than that of nickel-based systems, but requires tighter tolerances. Repeated insertion into the charger or cradle does not affect the battery by inducing overcharge.

On the negative side, the Li-ion gradually loses charge acceptance as part of aging, even if not used. For this reason, Li-ion batteries should not be stored for long periods of time but be rotated like perishable food. The buyer should be aware of the manufacturing date when purchasing a replacement battery.

The Li-ion is most economical for those who use a mobile phone daily. Up to 1000 charge/discharge cycles can be expected if used within the expected service life of about two to three years. Because of the aging effect, the Li-ion does not provide an economical solution for the occasional user. If the Li-ion is the only battery choice and the equipment is seldom used, the battery should be removed from the equipment and stored in a cool place, preferably only partially charged.

So far, little is known about the life expectancy of the Li-ion polymer. Because of the similarities with the Li-ion, the long-term performance of both systems is expected to be similar. Much effort is being made to prolong the service life of lithium-based systems. New chemical additives have been effective in retarding the aging process.

08. Choosing the Right Battery 2

Sun, 01 Mar 1998 00:00:00 +0000

<< Previous page INDEX Next page >>
8.2 What’s the best battery for two-way radios?

The two-way radio market uses mostly NiCd batteries. In the last few years, environmental agencies have been attempting to discourage the use of NiCd, especially in Europe. NiMH have been tried and tested in two-way radios for a number of years but the results are mixed. Shorter cycle life compared to NiCd is the major drawback.

The reasons for the relatively short life of NiMH are multi-fold. NiMH is less robust than NiCd and has a cycle life expectancy that is half or one third that of the standard NiCd. In addition, NiMH prefers a moderate discharge current of 0.5C or less. A two-way radio, on the other hand, draws a discharge current of about 1.5A when transmitting at 4W of power. High discharge loads shorten the life of the NiMH battery considerably.

NiCd has the advantage of maintaining a low and steady internal resistance throughout most of its service life. Although low when new, NiMH increases the resistance with advanced cycle count. A battery with high internal resistance causes the voltage to drop when a load is applied. Even though energy may still be present, the battery cannot deliver the high current flow required during transmit mode. This results in a drop in voltage, which triggers the ‘low battery’ condition and the radio cuts off. This happens mostly during transmission.

The Li-ion has been tested for use with two-way radios but has not been able to provide the ultimate answer. Higher replacement costs, restrictions posed by the safety circuit and aging pose limitations on this battery system.
8.3 What’s the best battery for laptops?

Batteries for laptops have a unique challenge because they must be small and lightweight. In fact, the laptop battery should be invisible to the user and deliver enough power to last for a five-hour flight from Toronto to Vancouver. In reality, a typical laptop battery provides only about 90 minutes of service.

Computer manufacturers are hesitant to add a larger battery because of increased size and weight. A recent survey indicated that, given the option of larger size and more weight to obtain longer runtimes, most users would settle for what is being offered today. For better or worse, we have learned to accept the short runtime of a laptop.

During the last few years, batteries have improved in terms of energy density. Any benefit in better battery performance, however, is being eaten up by the higher power requirements of the laptops. It is predicted that the even more power-hungry PC’s of the future will counteract any improvements in battery technology, as marginal as they might be. The net effect will result in the same runtimes but faster and more powerful computers.

The length of time the battery can be used will get shorter as the battery ages. A battery residing in a laptop ages more quickly than when used in other applications. After a warm-up, the official operating temperature inside a laptop computer is 45°C (113°F). Such a high ambient temperature drastically lowers the battery’s life expectation. At a temperature of 45°C, for example, the life expectancy of a NiMH battery is less than 50 percent as compared to running it at the ideal operating temperature of 20°C (68°F).

The Li-ion does not fare much better. At this high ambient temperature, the wear-down effect of the battery is primarily governed by temperature as opposed to cycle count. The situation is worsened by the fact that the battery resides in a high SoC most of the time. The combination of heat and high SoC promotes cell oxidation, a condition that cannot be reversed once afflicted.

A fully charged Li-ion battery that is stored at 45°C suffers a capacity loss from 100 percent to about 70 percent in as little as six months. If this condition persists, the capacity degrades further to 50 percent in twelve months. In reality, the battery in a laptop is exposed to elevated temperatures just during use and the battery is in a full charge state only part of the time. But leaving the laptop in a parked car under the hot sun will aggravate the situation.

Some Japanese computer manufacturers have introduced a number of sub-notebooks in which the battery is mounted externally, forming part of the hinge. This design improves battery life because the battery is kept at room temperature. Some models carry several size batteries to accommodate different user patterns.

What then is the best battery for a laptop? The choices are limited. The NiCd has virtually disappeared from the mobile computer scene and the NiMH is loosing steam, paving the way for the Li-ion. Eventually, very slim geometry will also demand thin batteries, and this is possible with the prismatic Li-ion polymer.

Besides providing reliable performance for general portable use, the Li-ion battery also offers superior service for laptop users who must continually switch from fixed power to battery use, as is the case for many sales people. Many biomedical and industrial applications follow this pattern also. Here is the reason why such use can be hard on some batteries:

On a nickel-based charging system, unless smart, the charger applies a full charge each time the portable device is connected to fixed power. In many cases, the battery is already fully charged and the cells go almost immediately into overcharge. The battery heats up, only to be detected by a sluggish thermal charge control, which finally terminates the fast charge. Permanent capacity loss caused by overcharge and elevated temperature is the result.

Among the nickel-based batteries, NiMH is least capable of tolerating a recharge on top of a charge. Adding elevated ambient temperatures to the charging irregularities, a NiMH battery can be made inoperable in as little as six months. In severe cases, the NiMH is known to last only 2 to 3 months.

For mixed battery and utility power use, the Li-ion system is a better choice. If a fully charged Li-ion is placed on charge, no charge current is applied. The battery only receives a recharge once the terminal voltage has dropped to a set threshold. Neither is there a concern if the device is connected to fixed power for long periods of time. No overcharge can occur and there is no memory to worry about.

NiMH is the preferred choice for a user who runs the laptop mostly on fixed power and removes the battery when not needed. This way, the battery is only engaged if the device is used in portable mode. The NiMH battery can thus be kept fresh while sitting on the shelf. NiMH ages well if kept cool and only partially charged.

08. Choosing the Right Battery 3

Sun, 01 Mar 1998 00:00:00 +0000

<< Previous page INDEX Next page >>
8.4 Selecting a Lasting Battery

As part of an ongoing research program to find the optimum battery system for selected applications, Cadex has performed life cycle tests on NiCd, NiMH and Li-ion batteries. All tests were carried out on the Cadex 7000 Series battery analyzers in the test labs of Cadex, Vancouver, Canada. The batteries tested received an initial full-charge, and then underwent a regime of continued discharge/charge cycles. The internal resistance was measured with Cadex’s Ohmtest™ method, and the self-discharge was obtained from time-to-time by reading the capacity loss incurred during a 48-hour rest period. The test program involved 53 commercial telecommunications batteries of different models and chemistries. One battery of each chemistry displaying typical behavior was chosen for the charts below.

When conducting battery tests in a laboratory, it should be noted that the performance in a protected environment is commonly superior to those in field use. Elements of stress and inconsistency that are present in everyday use cannot always be simulated accurately in the lab.

The NiCd Battery — In terms of life cycling, the standard NiCd is the most enduring battery. In Figure 8-1 we examine the capacity, internal resistance and self-discharge of a 7.2V, 900mA NiCd battery with standard cells. Due to time constraints, the test was terminated after 2300 cycles. During this period, the capacity remains steady, the internal resistance stays flat at 75mW and the self-discharge is stable. This battery receives a grade ‘A’ for almost perfect performance.

The readings on an ultra-high capacity NiCd are less favorable but still better than other chemistries in terms of endurance. Although up to 60 percent higher in energy density compared to the standard NiCd version, Figure 8-2 shows the ultra-high NiCd gradually losing capacity during the 2000 cycles delivered. At the same time, the internal resistance rises slightly. A more serious degradation is the increase of self-discharge after 1000 cycles. This deficiency manifests itself in shorter runtimes because the battery consumes some energy itself, even if not in use.

Figure 8-1: Characteristics of a standard cell NiCd battery.
This battery deserves an ‘A’ for almost perfect performance in terms of stable capacity, internal resistance and self-discharge over many cycles. This illustration shows results for a 7.2V, 900mA NiCd.

Figure 8-2: Characteristics of a NiCd battery with ultra-high capacity cells.
This battery is not as favorable as the standard NiCd but offers higher energy densities and performs better than other chemistries in terms of endurance. This illustrations shows results for a 6V, 700mA NiCd.

08. Choosing the Right Battery 4

Sun, 01 Mar 1998 00:00:00 +0000

<< Previous page INDEX Next page >>

The NiMH Battery — Figure 8-3 examines the NiMH, a battery that offers high energy density at reasonably low cost. We observe good performance at first but past the 300-cycle mark, the performance starts to drift downwards rapidly. One can detect a swift increase in internal resistance and self-discharge after cycle count 700.

Figure 8-3: Characteristics of a NiMH battery.
This battery offers good performance at first but past the 300-cycle mark, the capacity, internal resistance and self-discharge start to deteriorate rapidly. This illustrations shows results for a 6V, 950mA NiMH.

The Li-ion Battery — The Li-ion battery offers advantages that neither the NiCd nor NiMH can match. In Figure 8-4 we examine the capacity and internal resistance of a typical Li-ion. A gentle capacity drop is observed over 1000 cycles and the internal resistance increases only slightly. Because of low readings, self-discharge was omitted for this test.

The better than expected performance of this test battery may be due to the fact that the test did not include aging. The lab test was completed in about 200 days. A busy user may charge the battery once every 24 hours. With such a user pattern, 500 cycles would represent close to two years of normal use and the effects of aging would become apparent.

Manufacturers of commercial Li-ion batteries specify a cycle count of 500. At that stage, the battery capacity would drop from 100 to 80 percent. If operated at 40°C (104°F) rather than at room temperature, the same battery would only deliver about 300 cycles.

Figure 8-4: Characteristics of a Li-ion battery.
The above-average performance of this battery may be due to the fact that the test did not include aging. This illustration shows results for a 3.6V, 500mA Li-ion battery.

08. Implementation

Sun, 01 Mar 1998 00:00:00 +0000

NOTE: THIS DOCUMENT IS OBSOLETE, PLEASE CHECK THE NEW VERSION: "The Digital Audio Resampling Home Page", by Julius O. Smith III, Copyright © 2016-05-17 by Julius O. Smith III - Center for Computer Research in Music and Acoustics (CCRMA), Stanford University

<< Previous page TOC Next page >>
Implementation
Our implementation provides signal evaluation at an arbitrary time, where time is specified as an unsigned binary fixed-point number in units of the input sampling period (assumed constant). Figure 6 shows the time register , and Figure 7 shows an example configuration of the input signal and lowpass filter at a given time. The time register is divided into three fields: The leftmost field gives the number of samples into the input signal buffer, the middle field is an initial index into the filter coefficient table , and the rightmost field is interpreted as a number between and for doinglinear interpolation between samples and (initially) of the filter table. The concatenation of and are called which is interpreted as the position of the current time between samples and of the input signal.

08. Introduction to Digital Filter Theory

Sun, 01 Mar 1998 00:00:00 +0000

NOTE: THIS DOCUMENT IS OBSOLETE, PLEASE CHECK THE NEW VERSION: "Introduction to Digital Filters with Audio Applications", by Julius O. Smith III, Copyright © 2017-11-26 by Julius O. Smith III - Center for Computer Research in Music and Acoustics (CCRMA), Stanford University

<< Previous page TOC INDEX Next page >>
Introduction to Digital Filter Theory
In this section, linearity, time-invariance and four basic representations of digital filters are defined: thedifference equation coefficients, impulse response, transfer function, and frequency response. Next the concepts ofphase delay, group delay, poles and zeros, and filter stability are defined. This elementary material was taken from course notes for a class given at Stanford by the author in 1979.

09. Internal Battery Resistance

Sun, 01 Mar 1998 00:00:00 +0000

<< Previous page INDEX Next page >>
9. Internal Battery Resistance

With the move from analog to digital devices, new demands are being placed on the battery. Unlike analog equipment that draws a steady current, the digital mobile phone, for example, loads the battery with short, high current bursts.

Increasingly, mobile communication devices are moving from voice only to multimedia which allows sending and receiving data, still pictures and even video. Such transmissions add to the bandwidth, which require several times the battery power compared to voice only.

One of the urgent requirements of a battery for digital applications is low internal resistance. Measured in milliohms (mW), the internal resistance is the gatekeeper that, to a large extent, determines the runtime. The lower the resistance, the less restriction the battery encounters in delivering the needed power bursts. A high mW reading can trigger an early ‘low battery’ indication on a seemingly good battery because the available energy cannot be delivered in an appropriate manner.

Figure 9-1 examines the major global mobile phone systems and compares peak power and peak current requirements. The systems are the AMP, GSM, TDMA and CDMA.

AMP GSM TDMA¹ CDMA

Type Analog Digital Digital Digital

Used in USA, Canada Globally USA, Canada USA, Canada

Peak Power 0.6W 1-2W 0.6-1W 0.2W

Peak current² 0.3A DC 1-2.5A 0.8-1.5A 0.7A

In service since 1985 1986 1992 1995

Figure 9-1: Peak power requirements of popular global mobile phone systems.
Moving from voice to multi-media requires several times the battery power.

Some TDMA handsets feature dual mode (analog 800mA DC load; digital 1500mA pulsed load).

Current varies with battery voltage; a 3.6V battery requires higher current than a 7.2V battery.

9.1 Why do seemingly good batteries fail on digital equipment?

Service technicians have been puzzled by the seemingly unpredictable battery behavior when powering digital equipment. With the switch from analog to digital wireless communications devices, particularly mobile phones, a battery that performs well on an analog device may show irrational behavior when used on a digital device. Testing these batteries with a battery analyzer produces normal capacity readings. Why then do some batteries fail prematurely on digital devices but not on analog?

The overall energy requirement of a digital mobile phone is less than that of the analog equivalent, however, the battery must be capable of delivering high current pulses that are often several times that of the battery’s rating. Let’s look at the battery rating as expressed in C-rates.

A 1C discharge of a battery rated at 500mAh is 500mA. In comparison, a 2C discharge of the same battery is 1000mA. A GSM phone powered by a 500mA battery that draws 1.5A pulses loads the battery with a whopping 3C discharge.

A 3C rate discharge is fine for a battery with very low internal resistance. However, aging batteries, especially Li-ion and NiMH chemistries, pose a challenge because the mW readings of these batteries increase with use.

Improved performance can be achieved by using a larger battery, also known as an extended pack. Somewhat bulkier and heavier, an extended pack offers a typical rating of about 1000mAh or roughly double that of the slim-line. In terms of C-rate, the 3C discharge is reduced to 1.5C when using a 1000mAh instead of a 500mAh battery.

As part of ongoing research to find the best battery system for wireless devices, Cadex has performed life cycle tests on various battery systems. In Figure 9-2, Figure 9-3, and Figure 9-4, we examine NiCd, NiMH and Li-ion batteries, each of which generates a good capacity reading when tested with a battery analyzer but produce stunning differences on a pulsed discharge of 1C, 2C and 3C. These pulses simulate a GSM phone.

Figure 9-2: Talk-time of a NiCd battery under the GSM load schedule.
This battery has 113% capacity and 155m© internal resistance.

A closer look reveals vast discrepancies in the m© measurements of the test batteries. In fact, these readings are typical of batteries that have been in use for a while. The NiCd shows 155mW, the NiMH 778mW and the Li-ion 320mW, although the capacities checked in at 113, 107 and 94 percent respectively when tested with the DC load of a battery analyzer. It should be noted that the internal resistance was low when the batteries were new.

Figure 9-3: Talk-time of a NiMH battery under the GSM load schedule.
This battery has 107% capacity and 778m© internal resistance.

Figure 9-4: Talk-time of a Li-ion battery under the GSM load schedule.
This battery has 94% capacity and 320m© internal resistance.

From these charts we can see that the talk-time is in direct relationship with the battery’s internal resistance. The NiCd performs best and produces a talk time of 140 minutes at 1C and a long 120 minutes at 3C. In comparison, the NiMH is good for 140 minutes at 1C but fails at 3C. The Li-ion provides 105 minutes at 1C and 50 minutes at 3C discharge.

09. Internal Battery Resistance 2

Sun, 01 Mar 1998 00:00:00 +0000

<< Previous page INDEX Next page >>
9.2 How is the internal battery resistance measured?

A number of techniques are used to measure internal battery resistance. One common method is the DC load test, which applies a discharge current to the battery while measuring the voltage drop. Voltage over current provides the internal resistance (see Figure 9-5).

Figure 9-5: DC load test.
The DC load test measures the battery’s internal resistance by reading the voltage drop. A large drop indicates high resistance.

The AC method, also known as the conductivity test, measures the electrochemical characteristics of a battery. This technique applies an alternating current to the battery terminals. Depending on manufacturer and battery type, the frequency ranges from 10 to 1000Hz. The impedance level affects the phase shift between voltage and current, which reveals the condition of the battery. The AC method works best on single cells. Figure 9-6 demonstrates a typical phase shift between voltage and current when testing a battery.

Figure 9-6: AC load test.
The AC method measures the phase shift between voltage and current. The battery’s reactance is used to calculate the impedance.

Some AC resistance meters evaluate only the load factor and disregard the phase shift information. This technique is similar to the DC method. The AC voltage that is superimposed on the battery’s DC voltage acts as brief charge and discharge pulses. The amplitude of the ripple is utilized to calculate the internal battery resistance.

Cadex uses the discreet DC method to measure internal battery resistance. Added to the Cadex 7000 Series battery analyzers, a number of charge and discharge pulses are applied, which are scaled to the mAh rating of the battery tested. Based on the voltage deflections, the battery’s internal resistance is calculated. Known as Ohmtest™, the mW reading is obtained in five seconds. Figure 9-7 shows the technique used.

Figure 9-7: Cadex Ohmtest™.
Cadex’s pulse method measures the voltage deflections by applying charge and discharge pulses. Higher deflections indicate higher internal resistance.

09. Internal Battery Resistance 3

Sun, 01 Mar 1998 00:00:00 +0000

<< Previous page INDEX Next page >>

Figure 9-8 compares the three methods of measuring the internal resistance of a battery and observe the accuracy. In a good battery, the discrepancies between methods are minimal. The test results deviate to a larger degree on packs with poor SoH.

Impedance measurement alone does not provide a definite conclusion as to the battery performance. The mW readings may vary widely and are dependent on battery chemistry, cell size (mAh rating), type of cell, number of cells connected in series, wiring and contact type.

Figure 9-8: Comparison of the AC, DC and Cadex Ohmtest™ methods.
State-of-health readings were obtained using the Cadex 7000 Series battery analyzer by applying a full charge/discharge/charge cycle. The DC method on the 68% SoH battery exceeded 1000mW.

When using the impedance method, a battery with a known performance should be measured and its readings used as a reference. For best results, a reference reading should be on hand for each battery type. Figure 9-9kl; provides a guideline for digital mobile phone batteries based on impedance readings.

The milliohm readings are related to the battery voltage. Higher voltage batteries allow higher internal resistance because less current is required to deliver the same power. The ratio between voltage and milliohm is not totally linear. There are certain housekeeping components that are always present whether the battery has one or several cells. These are wiring, contacts and protection circuits.

Temperature also affects the internal resistance of a battery. The internal resistance of a naked Li-ion cell, for example, measures 50mW at 25°C (77°F). If the temperature increases, the internal resistance decreases. At 40°C (104°F), the internal resistance drops to about 43mW and at 60°C (140°F) to 40mW. While the battery performs better when exposed to heat, prolonged exposure to elevated temperatures is harmful. Most batteries deliver a momentary performance boost when heated.

Milli-Ohm Battery Voltage Ranking

75-150mOhm 3.6V Excellent

150-250mOhm 3.6V Good

250-350mOhm 3.6V Marginal

350-500mOhm 3.6V Poor

Above 500mOhm 3.6V Fail

Figure 9-9: Battery state-of-health based on internal resistance.
The milliohm readings relate to the battery voltage; higher voltage allows higher milliohm readings.

Cold temperatures have a drastic effect on all batteries. At 0°C (32°F), the internal resistance of the same Li-ion cell drops to 70mW. The resistance increases to 80mW at -10°C (50°F) and 100mW at -20°C (-4°F).

The impedance readings work best with Li-ion batteries because the performance degradation follows a linear pattern with cell oxidation. The performance of NiMH batteries can also be measured with the impedance method but the readings are less dependable. There are instances when a poorly performing NiMH battery can also exhibit a low mW reading.

Testing a NiCd on resistance alone is unpredictable. A low resistance reading does not automatically constitute a good battery. Elevated impedance readings are often caused by memory, a phenomenon that is reversible. Internal resistance values have been reduced by a factor of two and three after servicing the affected batteries with the recondition cycle of a Cadex 7000 Series battery analyzer. Of cause, high internal resistance can have sources other than memory alone.
9.3 What’s the difference between internal resistance and impedance?

The terms ‘internal resistance’ and ‘impedance’ are often intermixed when addressing the electrical conductivity of a battery. The differences are as follows: The internal resistance views the conductor from a purely resistive value, or ohmic resistance. A comparison can be made with a heating element that produces warmth by the friction of electric current passing through.

Most electrical loads are not purely resistive, rather, they have an element of reactance. If an alternating current (AC) is sent through a coil, for example, an inductance (magnetic field) is created, which opposes current flow. This AC impedance is always higher than the ohmic resistance of the copper wire. The higher the frequency, the higher the inductive resistance becomes. In comparison, sending a direct current (DC) through a coil constitutes an electrical short because there is only a very small ohmic resistance.

Similarly, a capacitor does not allow the flow of DC, but passes AC. In fact, a capacitor is an insulator for DC. The resistance that is present when sending an AC current flowing through a capacitor is called capacitance. The higher the frequency, the lower the capacitive resistance.

A battery as a power source combines ohmic, inductive and capacitive resistance. Figure 9-10 represents these resistive values on a schematic diagram. Each battery type exhibits slightly different resistive values.

Figure 9-10: Ohmic, inductive and capacitive resistance in batteries.

R_o = ohmic resistance

Q_c = constant phase loop (type of capacitance)

L = inductor

Z_w = Warburg impedance (particle movement within the electrolyte)

R_t = transfer resistance

09. Linearity and Time-Invariance

Sun, 01 Mar 1998 00:00:00 +0000

NOTE: THIS DOCUMENT IS OBSOLETE, PLEASE CHECK THE NEW VERSION: "Introduction to Digital Filters with Audio Applications", by Julius O. Smith III, Copyright © 2017-11-26 by Julius O. Smith III - Center for Computer Research in Music and Acoustics (CCRMA), Stanford University

<< Previous page TOC INDEX Next page >>
Linearity and Time-Invariance
In everyday terms, the fact that a filter islinear means simply that
the amplitude of the output is proportional to the amplitude of the input, and
when two signals are added together and fed to the filter, the filter output is the same as if one had put each signal through the filter separately and then added the outputs.
Definition. A filter is said to be linear if for any pair of signals and for all constant gains , we have
(40)
(41)

for all . These two conditions are a mathematical statement of the previous definition. For rational, property (2) implies (1).
Definition. A filter is said to be time-invariant if

where is understood to denote the waveform shifted right (or delayed) by samples.
From now on, all filters discussed will be linear and time-invariant. For brevity, these will be referred to as LTI filters.

09. Quantization Issues

Sun, 01 Mar 1998 00:00:00 +0000

NOTE: THIS DOCUMENT IS OBSOLETE, PLEASE CHECK THE NEW VERSION: "The Digital Audio Resampling Home Page", by Julius O. Smith III, Copyright © 2016-05-17 by Julius O. Smith III - Center for Computer Research in Music and Acoustics (CCRMA), Stanford University

<< Previous page TOC Next page >>
Quantization Issues
In this section, we investigate the requirements on the sampling density of the lowpass-filter impulse response, and the number of bits required in the interpolation factor . These quantities are determined by computing the worst-case error and comparing it to the filter coefficient quantization error.

1. An Introduction to ESD

Sun, 01 Mar 1998 00:00:00 +0000

<< Previous page INDEX Next page >>
1. An Introduction to ESD
History & Background
The decade of the 90's may be remembered as the Decade of Quality in the electronics industry. Increased competition, six-sigma quality, and ISO 9000 have forced a recommitment to quality even in those companies that might not have done so willingly. As we examine our environments for quality improvement areas, electrostatic discharge (ESD) remains a key target.
Static electricity has been an industrial problem for centuries. As early as the 1400’s, European and Caribbean forts were using static control procedures and devices to prevent electrostatic discharge ignition of black powder stores. By the 1860's, paper mills throughout the U.S. employed basic grounding, flame ionization techniques, and steam drums to dissipate static electricity from the paper web as it traveled through the drying process. The age of electronics brought with it new problems associated with static electricity and electrostatic discharge. And, as electronic devices became faster and smaller, their sensitivity to ESD increased.
Today, ESD impacts productivity and product reliability in virtually every aspect of today’s electronics environment. Many aspects of electrostatic control in the electronics industry also apply in other industries such as clean room applications and graphic arts.
Despite a great deal of effort during the past decade, ESD still affects production yields, manufacturing costs, product quality, product reliability, and profitability. Industry experts have estimated average product losses due to static to range from 8-33% (Table 1). Others estimate the actual cost of ESD damage to the electronics industry as running into the billions of dollars annually. The cost of damaged devices themselves range from only a few cents for a simple diode to several hundred dollars for complex hybrids. When associated costs of repair and rework, shipping, labor, and overhead are included, clearly the opportunities exist for significant improvements.

Table 1
Informal Summary of
Static Losses by Level

Static Losses Reported

Description

Min.
Loss

Max.
Loss

Est. Avg.
Loss

Component Manufacturers

4%

97%

16-22%

Subcontractors

3%

70%

9-15%

Contractors

2%

35%

8-14%

User

5%

70%

27-33%

Source: Stephen Halperin, "Guidelines for Static Control Management," Eurostat, 1990.

This first in a series of six articles on ESD focuses on how electrostatic charge and discharge occur, how various materials affect the level of charge, types of ESD damage, and how ESD events can damage electronic components. Future articles will cover various ways to control the problem.
Static Electricity: Creating Charge
Static electricity is defined as an electrical charge caused by an imbalance of electrons on the surface of a material. This imbalance of electrons produces an electric field that can be measured and that can influence other objects at a distance. Electrostatic discharge is defined as the transfer of charge between bodies at different electrical potentials.
Electrostatic discharge can change the electrical characteristics of a semiconductor device, degrading or destroying it. Electrostatic discharge also may upset the normal operation of an electronic system, causing equipment malfunction or failure. Another problem caused by static electricity occurs in clean rooms. Charged surfaces can attract and hold contaminants, making removal from the environment difficult. When attracted to the surface of a silicon wafer or a device's electrical circuitry, these particulates can cause random wafer defects and reduce product yields.
Controlling electrostatic discharge begins with understanding how electrostatic charge occurs in the first place. Electrostatic charge is most commonly created by the contact and separation of two similar or dissimilar materials. For example, a person walking across the floor generates static electricity as shoe soles contact and then separate from the floor surface. An electronic device sliding into or out of a bag, magazine or tube generates an electrostatic charge as the device's case and/or metal leads make multiple contacts and separations with the surface of the container. While the magnitude of electrostatic charge may be different in these examples, static electricity is indeed generated.

Figure 1: The Triboelectric Charge. Materials Make Intimate Contact

Figure 2: The Triboelectric Charge - Separation
Creating electrostatic charge by contact and separation of materials is known as "triboelectric charging." It involves the transfer of electrons between materials. The atoms of a material with no static charge have an equal number of positive (+) protons in their nucleus and negative (-) electrons orbiting the nucleus. In Figure 1, Material "A" consists of atoms with equal numbers of protons and electrons. Material B also consists of atoms with equal (though perhaps different) numbers of protons and electrons. Both materials are electrically neutral.
When the two materials are placed in contact and then separated, negatively charged electrons are transferred from the surface of one material to the surface of the other material. Which material loses electrons and which gains electrons will depend on the nature of the two materials. The material that loses electrons becomes positively charged, while the material that gains electrons is negatively charged. This is shown in Figure 2.
The actual level of charge is measured in coulombs. Commonly, however, we speak of the electrostatic potential on an object, which is expressed as voltage.
This process of material contact, electron transfer and separation is really a more complex mechanism than described here. The amount of charge created by triboelectric charging is affected by the area of contact, the speed of separation, relative humidity, and other factors. Once the charge is created on a material, it becomes an "electrostatic" charge (if it remains on the material). This charge may be transferred from the material, creating an electrostatic discharge, or ESD, event.

Table 2
Examples of Static Generation
Typical Voltage Levels

Means of Generation

10-25% RH

65-90% RH

Walking across carpet

35,000V

1,500V

Walking across vinyl tile

12,000V

250V

Worker at bench

6,000V

100V

Poly bag picked up from bench

20,000V

1,200V

Chair with urethane foam

18,000V

1,500V
An electrostatic charge also may be created on a material in other ways such as by induction, ion bombardment, or contact with another charged object. However, triboelectric charging is the most common.
Material Characteristics—How They Affect Static Charge
Virtually all materials, including water and dirt particles in the air, can be triboelectrically charged. How much charge is generated, where that charge goes, and how quickly, are functions of the materials' electrical characteristics.
Insulators
A material that prevents or limits the flow of electrons across its surface or through its volume is called an insulator. Insulators have an extremely high electrical resistance. A considerable amount of charge can be generated on the surface of an insulator. Because an insulative material does not readily allow the flow of electrons, both positive and negative charges can reside on insulative surface at the same time, although at different locations. The excess electrons at the negatively charged spot might be sufficient to satisfy the absence of electrons at the positively charged spot. However, electrons cannot easily flow across the insulative material's surface, and both charges may remain in place for a very long time.
Conductive Materials
A conductive material, because it has low electrical resistance, allows electrons to flow easily across its surface or through its volume. When a conductive material becomes charged, the charge (i.e., the deficiency or excess of electrons) will be uniformly distributed across the surface of the material. If the charged conductive material makes contact with another conductive material, the electrons will transfer between the materials quite easily. If the second conductor is attached to an earth grounding point, the electrons will flow to ground and the excess charge on the conductor will be "neutralized."
Electrostatic charge can be created triboelectrically on conductors the same way it is created on insulators. As long as the conductor is isolated from other conductors or ground, the static charge will remain on the conductor. If the conductor is grounded the charge will easily go to ground. Or, if the charged conductor contacts or nears another conductor, the charge will flow between the two conductors.
Static Dissipative Materials
Static dissipative materials have electrical resistance between insulative and conductive materials. There can be electron flow across or through the dissipative material, but it is controlled by the surface resistance or volume resistance of the material.
As with the other two types of materials, charge can be generated triboelectrically on a static dissipative material. However, like the conductive material, the static dissipative material will allow the transfer of charge to ground or other conductive objects. The transfer of charge from a static dissipative material will generally take longer than from a conductive material of equivalent size. Charge transfers from static dissipative materials are significantly faster than from insulators, and slower than from conductors.
Triboelectric Series
When two materials contact and separate, the polarity and magnitude of the charge are indicated by the materials’ positions in the triboelectric series. The triboelectric simply lists materials according to their relative triboelectric charging characteristics. When two materials contact and separate, the one nearer the top of the series takes on a positive charge, the other a negative charge. Materials further apart on the table typically generate a higher charge than ones closer together. See Table 3.

Table 3
Typical Triboelectric Series

+
Positive

Negative
-

Acetate
Glass
Nylon
Wool
Lead
Aluminum
Paper
COTTON
Wood
Steel
Nickel-Copper
Rubber
Polyester
PVC
Silicon
Teflon

ESD Damage—How Devices Fail
Electrostatic damage to electronic devices can occur at any point from manufacture to field service. Damage results from handling the devices in uncontrolled surroundings or when poor ESD control practices are used. Generally damage is classified as either a catastrophic failure or a latent defect.
Catastrophic Failure
When an electronic device is exposed to an ESD event it may no longer function. The ESD event may have caused a metal melt, junction breakdown, or oxide failure. The device's circuitry is permanently damaged causing the device fail. Such failures usually can be detected when the device is tested before shipment. If the ESD event occurs after test, the damage will go undetected until the device fails in operation.
Latent Defect
A latent defect, on the other hand, is more difficult to identify. A device that is exposed to an ESD event may be partially degraded, yet continue to perform its intended function. However, the operating life of the device may be reduced dramatically. A product or system incorporating devices with latent defects may experience premature failure after the user places them in service. Such failures are usually costly to repair and in some applications may create personnel hazards.
It is relatively easy with the proper equipment to confirm that a device has experienced catastrophic failure. Basic performance tests will substantiate device damage. However, latent defects are extremely difficult to prove or detect using current technology, especially after the device is assembled into a finished product.
Basic ESD Events--What Causes Electronic Devices to Fail?
ESD damage is usually caused by one of three events: direct electrostatic discharge to the device; electrostatic discharge from the device or field induced discharges.
Discharge to the Device
An ESD event can occur when any charged conductor (including the human body) discharges to an ESDS (electrostatic discharge sensitive) device. The most common cause of electrostatic damage is the direct transfer of electrostatic charge from the human body or a charged material to the electrostatic discharge sensitive (ESDS) device. When one walks across a floor, an electrostatic charge accumulates on the body. Simple contact of a finger to the leads of an ESDS device or assembly allows the body to discharge, possibly causing device damage. The model used to simulate this event is the Human Body Model (HBM).
A similar discharge can occur from a charged conductive object, such as a metallic tool or fixture. The model used to characterize this event is known as the Machine Model.
Discharge from the Device
The transfer of charge from an ESDS device is also an ESD event. The trend towards automated assembly would seem to solve the problems of HBM ESD events. However, it has been shown that components may be more sensitive to damage when assembled by automated equipment. A device may become charged, for example, from sliding down the feeder. If it then contacts the insertion head or another conductive surface, a rapid discharge occurs from the device to the metal object. This event is known as the Charged Device Model (CDM) event, and can be more destructive than the HBM for some devices. Although the duration of the discharge is very short--often less than one nanosecond--the peak current can reach several tens of amperes.
Field Induced Discharges
Another event that can directly or indirectly damage devices is termed Field Induction. As noted earlier, whenever any object becomes electrostatically charged, there is an electrostatic field associated with that charge. If an ESDS device is placed in that electrostatic field, a charge may be induced on the device. If the device is then momentarily grounded while within the electrostatic field, a transfer of charge from the device occurs.
Device Sensitivity: How Much Static Protection is Needed?
Damage to an ESDS device by the ESD event is determined by the device's ability to dissipate the energy of the discharge or withstand the voltage levels involved. This is known as device "ESD sensitivity". Test procedures based on the models of ESD events help define the sensitivity of components to ESD. Some devices may be more readily damaged by discharges occurring within automated equipment, while others may be more prone to damage from handling by personnel. Defining the ESD sensitivity of electronic components is the first step in determining the degree of ESD protection required.
Many electronic components are susceptible to ESD damage at relatively low voltage levels. Many are susceptible at less than 100 volts, and many disk drive components have sensitivities below 10 volts. Current trends in product design and development pack more circuitry onto these miniature devices, further increasing their sensitivity to ESD and making the potential problem even more acute. Table 4 indicates the ESD sensitivity of various types of components.

Table 4
Susceptibility of Electronic Components to ESD

Device Type

ESD Susceptibility
(Volts)

VMOS

30 - 1,200

Mosfet, GaAsfet, EPROM

100 - 300

JFET

150 - 7,000

OP-AMP

190 - 2,500

Schottky Diodes

300 - 2,500

Film Resistors

300 - 3,000

Schottky TTL

1,000 - 2,500

Summary
In this introductory article on electrostatic discharge, we have discussed the basics of electrostatic charge, discharge, types of failures, ESD events, and device sensitivity. We can summarize this discussion as follows:Virtually all materials, even conductors, can be triboelectrically charged.

The level of charge is affected by material type, speed of contact and separation, humidity, and several other factors.
Electrostatic discharge can create catastrophic or latent failures in electronic components.
Electrostatic discharge can occur throughout the manufacturing, test, shipping, handling, or operational processes.
Component damage can occur as the result of a discharge from the component as well as a direct discharge to the component.
Components vary significantly in their sensitivity to ESD.
Protecting your products from the effects of static damage begins by understanding these key concepts of ESD. Armed with this information, you can then begin to develop an effective ESD control program. In Part Two we will focus on some basic concepts of ESD control.

1. WHAT IS MEMORY?

Sun, 01 Mar 1998 00:00:00 +0000

<< Previous page TOC Next page >>
1. WHAT IS MEMORY?
1.1. INTRODUCTION
1.2. THE ROLE OF MEMORY IN THE COMPUTER
1.3. THE DIFFERENCE BETWEEN MEMORY AND STORAGE
1.4. MEMORY AND PERFORMANCE
1.5. HOW MUCH MEMORY DO YOU NEED?
<< Previous page TOC Next page >>

1.1. INTRODUCTION

Sun, 01 Mar 1998 00:00:00 +0000

<< Previous page TOC Next page >>
1.1. INTRODUCTION
These days, no matter how much memory your computer has, it never seems to be quite enough. Not long ago, it was unheard of for a PC (Personal Computer), to have more than 1 or 2 MB (Megabytes) of memory. Today, most systems require 64MB to run basic applications. And up to 256MB or more is needed for optimal performance when using graphical and multimedia programs.

As an indication of how much things have changed over the past two decades, consider this: in 1981, referring to computer memory, Bill Gates said, “640K (roughly 1/2 of a megabyte) ought to be enough for anybody.”

For some, the memory equation is simple: more is good; less is bad. However, for those who want to know more, this reference guide contains answers to the most common questions, plus much, much more.

1.2. THE ROLE OF MEMORY IN THE COMPUTER

Sun, 01 Mar 1998 00:00:00 +0000

<< Previous page TOC Next page >>
1.2. THE ROLE OF MEMORY IN THE COMPUTER
People in the computer industry commonly use the term “memory” to refer to RAM (Random Access Memory). A computer uses RAM to hold temporary instructions and data needed to complete tasks. This enables the computer’s CPU (Central Processing Unit), to access instructions and data stored in memory very quickly.

A good example of this is when the CPU loads an application program – such as a word processing or page layout program – into memory, thereby allowing the application program to work as quickly and efficiently as possible. In practical terms, having the program loaded into memory means you can get work done more quickly with less time spent waiting for the computer to perform tasks.

The process begins when you enter a command from your keyboard. The CPU interprets the command and instructs the hard drive to load the command or prgram into memory. Once the data is loaded into memory, the CPU is able to access it much more quickly than if it had to retrieve it from the hard drive.

This process of putting things the CPU needs in a place where it can get at them more quickly is similar to placing various electronic files and documents you’re using on the computer into a single file folder or directory. By doing so, you keep all the files you need handy and avoid searching in several places every time you need them.

1.3. THE DIFFERENCE BETWEEN MEMORY AND STORAGE

Sun, 01 Mar 1998 00:00:00 +0000

<< Previous page TOC Next page >>
1.3. THE DIFFERENCE BETWEEN MEMORY AND STORAGE
People often confuse the terms memory and storage, especially when describing the amount they have of each. The term memory refers to the amount of RAM installed in the computer, whereas the term storage refers to the capacity of the computer’s hard disk. To clarify this common mix-up, it helps to compare your computer to an office that contains a desk and a file cabinet.

The file cabinet represents the computer's hard disk, which provides storage for all the files and information you need in your office. When you come in to work, you take out the files you need from storage and put them on your desk for easy access while you work on them. The desk is like memory in the computer: it holds the information and data you need to have handy while you're working.

Consider the desk-and-file-cabinet metaphor for a moment. Imagine what it would be like if every time you wanted to look at a document or folder you had to retrieve it from the file drawer. It would slow you down tremendously, not to mention drive you crazy. With adequate desk space – our metaphor for memory – you can lay out the documents in use and retrieve information from them immediately, often with just a glance.

Here’s another important difference between memory and storage: the information stored on a hard disk remains intact even when the computer is turned off. However, any data held in memory is lost when the computer is turned off. In our desk space metaphor, it’s as though any files left on the desk at closing time will be thrown away.

1.4. MEMORY AND PERFORMANCE

Sun, 01 Mar 1998 00:00:00 +0000

<< Previous page TOC Next page >>
1.4. MEMORY AND PERFORMANCE
It’s been proven that adding more memory to a computer system increases its performance. If there isn’t enough room in memory for all the information the CPU needs, the computer has to set up what’s known as a virtual memory file. In so doing, the CPU reserves space on the hard disk to simulate additional RAM. This process, referred to as “swapping”, slows the system down. In an average computer, it takes the CPU approximately 200ns (nanoseconds) to access RAM compared to 12,000,000ns to access the hard drive. To put this into perspective, this is equivalent to what’s normally a 3½ minute task taking 4½ months to complete!

Access time comparision between RAM and a hard drive.

MEMORY UPGRADE ON A PC: LIFE IS GOOD
If you've ever had more memory added to your PC, you probably noticed a performance improvement right away. With a memory upgrade, applications respond more quickly, Web pages load faster, and you can have more programs running simultaneously. In short, additional memory can make using your computer a lot more enjoyable.

MEMORY UPGRADE ON A SERVER: LIFE IS EVEN BETTER
These days, more and more people are using computers in a workgroup and sharing information over a network. The computers that help distribute information to people on a network are called servers. And their performance has a huge impact on the performance of the network: if a server is performing poorly, everyone on the network "feels the pain." So, while a memory upgrade on an individual PC makes a big difference for the person who uses it, a memory upgrade in a server has even more far-reaching effects and benefits everyone who accesses the server.

To better understand the benefits of increasing memory on a server, take a look at these results from an independent study done on Windows NT-based servers.

Application servers are utilized to host a wide range of applications, such as word processing and spreadsheet programs. By increasing base memory from 64MB to 256MB, Windows NT Server was able to support five times as many clients before transactions per second dropped.

Web servers are employed to serve up Web pages in response to HTTP requests from users. Doubling memory can cut response time by more than 50%.

Directory servers are vital to corporate productivity, handling most email and messaging tasks. In this environment, more memory increases the speed with which a server can access information from linked databases. Doubling memory increased performance from 248 to 3000%.

1.5. HOW MUCH MEMORY DO YOU NEED?

Sun, 01 Mar 1998 00:00:00 +0000

<< Previous page TOC Next page >>
1.5. HOW MUCH MEMORY DO YOU NEED?
Perhaps you already know what it's like to work on a computer that doesn't have quite enough memory. You can hear the hard drive operating more frequently and the "hour glass" or "wrist watch" cursor symbol appears on the screen for longer periods of time. Things can run more slowly at times, memory errors can occur more frequently, and sometimes you can't launch an application or a file without first closing or quitting another.

So, how do you determine if you have enough memory, or if you would benefit from more? And if you do need more, how much more? The fact is, the right amount of memory depends on the type of system you have, the type of work you're doing, and the software applications you're using. Because the right amount of memory is likely to be different for a desktop computer than for a server, we've divided this section into two parts - one for each type of system.

MEMORY REQUIREMENTS FOR A DESKTOP COMPUTER
If you're using a desktop computer, memory requirements depend on the computer's operating system and the application software you're using. Today's word processing and spreadsheet applications require as little as 32MB of memory to run. However, software and operating system developers continue to extend the capabilities of their products, which usually means greater memory requirements. Today, developers typically assume a minimum memory configuration of 64MB. Systems used for graphic arts, publishing, and multimedia call for at least 128MB of memory and it's common for such systems to require 256MB or more for best performance.

The chart on the next page provides basic guidelines to help you decide how much memory is optimal for your desktop computer. The chart is divided by operating system and by different kinds of work. Find the operating system you're using on your computer, then look for the descriptions of work that most closely match the kind of work you do.

DESKTOP MEMORY MAP
WINDOWS^® 2000 PROFESSIONAL
Windows 2000 Professional runs software applications faster. Notebook- ready and designed with the future in mind, Windows 2000 Professional allows users to take advantage of a full- range of features today. Windows 2000 Professional is future- ready and promises to run today's and tomorrow's applications better.
Baseline: 64MB - 128MB
Optimal: 128MB - 512MB
Administrative & Service Light- Word processing, email, data- entry 64MB - 96MB
Medium- Fax /communications, database administration, spreadsheets; >2 applications open at a time 64MB - 128MB
Heavy- Complex documents, accounting, business graphics, presentation software, network connectivity 96MB - 256MB
Executives & Analysts Light- Proposals, reports, spreadsheets, business graphics, databases, scheduling, presentations 64MB - 96MB
Medium- Complex presentations, sales/ market analysis, project management, Internet access 96MB - 128MB
Heavy- Statistical applications, large databases, research/ technical analysis, complex presentations, video conferencing 128MB - 512MB
Engineers&Designers Light- Page layout, 2 - 4 color line drawings, simple image manipulation, simple graphics 96MB - 128MB
Medium- 2D CAD, rendering, multimedia presentations, simple photo- editing, Web development 128MB - 512MB
Heavy- Animation, complex photo- editing, real- time video, 3D CAD, solid modeling, finite element analysis 256MB - 1GB

WINDOWS^® 98
Windows 98 requires 16 - 32MB to run basic applications. Tests show 45 - 65% performance improvements at 64MB and beyond.
Baseline: 32MB - 64MB
Optimal: 128MB - 256MB
Students Light- Word processing, basic financial management, email and other light Internet use 32MB - 64MB
Medium- Home office applications, games, Internet surfing, downloading images, spreadsheets, presentations 64MB - 128MB
Heavy- Multimedia use such as video, graphics, music, voice recognition, design, complex images 128MB - 384MB
Home Users Light- Word processing, basic financial management, email and other light Internet use 32MB - 48MB
Medium- Home office applications, games, Internet surfing, downloading images, spreadsheets, presentations 48MB - 64MB
Heavy- Multimedia use such as video, graphics, music, voice recognition, design, complex images 64MB - 128MB

LINUX^®
The Linux operating system is quickly gaining popularity as an alternative to Microsoft Windows. It includes true multitasking, virtual memory, shared libraries, demand loading, proper memory management, TCP/ IP networking, and other features consistent with Unix- type systems.
Baseline: 48MB - 112MB
Optimal: 112MB - 512MB
Administrative&Service Light- Word processing, email, data- entry 48MB - 80MB
Medium- Fax /communications, database administration, spreadsheets; >2 applications open at a time 48MB - 112MB
Heavy- Complex documents, accounting, business graphics, presentation software, network connectivity 80MB - 240MB
Executives&Analysts Light- Proposals, reports, spreadsheets, business graphics, databases, scheduling, presentations 48MB - 80MB
Medium- Complex presentations, sales/ market analysis, project management, Internet access 80MB - 112MB
Heavy- Statistical applications, large databases, research/ technical analysis, complex presentations, video conferencing 112MB - 512MB
Engineers&Designers Light- Page layout, 2 - 4 color line drawings, simple image manipulation, simple graphics 80MB - 112MB
Medium- 2D CAD, rendering, multimedia presentations, simple photo- editing, Web development 112MB - 512MB
Heavy- Animation, complex photo- editing, real- time video, 3D CAD, solid modeling, finite element analysis 240MB - 1GB

MACINTOSH^™ OS
The Macintosh operating system manages memory in substantially different ways than other systems. Still, System 9.0 users will find that 48MB is a bare minimum. When using PowerMac ^® applications with Internet connectivity, plan on a range between 64 and 128MB as a minimum.
Baseline: 48MB - 64MB
Optimal: 128MB - 512MB
Administrative&Service Light- Word processing, email, data- entry 48MB - 64MB
Medium- Fax /communications, database administration, spreadsheets; >2 applications open at a time 64MB - 96MB
Heavy- Complex documents, accounting, business graphics, presentation software, network connectivity 96MB - 128MB
Executives&Analysts Light- Proposals, reports, spreadsheets, business graphics, databases, scheduling, presentations 64MB - 256MB
Medium- Complex presentations, sales/ market analysis, project management, Internet access 128MB - 1GB
Heavy- Statistical applications, large databases, research/ technical analysis, complex presentations, video conferencing 96MB - 128MB
Engineers&Designers Light- Page layout, 2 - 4 color line drawings, simple image manipulation, simple graphics 128MB - 512MB
Medium- 2D CAD, rendering, multimedia presentations, simple photo- editing, Web development 256MB - 1GB
Heavy- Animation, complex photo- editing, real- time video, 3D CAD, solid modeling, finite element analysis 512MB - 2GB

* Please Note: These figures reflect work done in a typical desktop environment. Higher-end workstation tasks may require up to 4GB. Naturally, a chart such as this evolves as memory needs and trends change. Over time, developers of software and operating systems will continue to add features and functionality to their products. This will continue to drive the demand for more memory. More complex character sets, like Kanji, may require more memory than the standard Roman based (English) character sets.

SERVER MEMORY REQUIREMENTS
How can you tell when a server requires more memory? Quite often, the users of the network are good indicators. If network-related activity such as email, shared appli-cations, or printing slows down, they'll probably let their Network Administrator know. Here are a few proactive strategies that can be used to gauge whether or not a server has sufficient memory:

Monitor server disk activity. If disk swapping is detected, it is usually a result of inadequate memory.

Most servers have a utility that monitors CPU, memory, and disk utilization. Review this at peak usage times to measure the highest spikes in demand.
Once it's determined that a server does need more memory, there are many factors to consider when deciding on how much is enough:

What functions does the server perform (application, communication, remote access, email, Web, file, multimedia, print, database)?

Some servers hold a large amount of information in memory at once, while oth-ers process information sequentially. For example, a typical large database server does a lot of data processing; with more memory, such a server would likely run much faster because more of the records it needs for searches and queries could be held in memory - that is, "at the ready." On the other hand, compared to a database server, a typical file server can perform efficiently with less memory because its primary job is simply to transfer information rather than to process it.
What operating system does the server use?

Each server operating system manages memory differently. For example, anetwork operating system (NOS) such as the Novell operating system handles information much differently than an application-oriented system such as Windows NT. Windows NT's richer interface requires more memory, while the traditional Novell functions of file and print serving require less memory.
How many users access the server at one time?

Most servers are designed and configured to support a certain number of users at one time. Recent tests show that this number is directly proportional to the amount of memory in the server. As soon as the number of users exceeds maximum capacity, the server resorts to using hard disk space as virtual memory, and performance drops sharply. In recent studies with Windows NT, additional memory allowed an application server to increase by several times the number of users supported while maintaining the same level of performance.
What kind and how many processors are installed on the server?

Memory and processors affect server performance differently, but they work hand in hand. Adding memory allows more information to be handled at one time, while adding processors allows the information to be processed faster. So, if you add processing power to a system, additional memory will enable the processors to perform at their full potential.
How critical is the server's response time?

In some servers, such as Web or e-commerce servers, response time directly affects the customer experience and hence revenue. In these cases, some IT Managers choose to install more memory than they think they would ever need in order to accommodate surprise surges in use. Because server configurations involve so many variables, it's difficult to make precise recommendations with regard to memory. The following chart shows two server upgrade scenarios.
SERVER MEMORY MAP
WINDOWS^® 2000 SERVER
Designed to help businesses of all sizes run better, Windows 2000 Server offers a manageable, reliable and internet- ready solution for today's growing enterprises. For optimal performance, consider adding more memory to take advantage of Windows 2000 Server's robust feature set. Windows 2000 Server is internet- ready and promises to run today's and tomorrow's applications better.
Baseline: 128MB
Optimal: 256MB - 1GB
Application Server Houses one or more applications to be accessed over a wide user base 256MB - 4GB
Directory Server Central Management of network resources 128MB - 1GB
Print Server Distributes print jobs to appropriate printers 128MB - 512MB
Communication Server Manages a variety of communications such as PBX, Voicemail, Email, and VPN 512MB - 2GB
Web Server Internet and intranet solutions 512MB - 2GB
Database Server Manages simple to complex databases of varying sizes 256MB - 4GB

LINUX^®
Linux is a reliable, cost- effective alternative to traditional UNIX servers. Depending on the distribution, the Linux server platform features a variety of utilities, applications, and services.
Baseline: 64MB - 128MB
Optimal: 256MB - 1GB
Application Server Houses one or more applications to be accessed over a wide user base 64MB - 4GB
Directory Server Central Management of network resources 128MB - 1GB
Print Server Distributes print jobs to appropriate printers 128MB - 512MB
Communication Server Manages a variety of communications such as PBX, Voicemail, Email, and VPN 512MB - 2GB
Web Server Internet and intranet solutions 512MB - 2GB
Database Server Manages simple to complex databases of varying sizes 256MB - 4GB

* Please Note: These figures reflect work done in a typical server environment. Higher- end workstation tasks may require up to 4GB. Naturally, a chart such as this evolves as memory needs and trends change. Over time, developers of software and operating systems will continue to add features and functionality to their products. This will continue to drive the demand for more memory. More complex character sets, like Kanji, may require more memory than the standard Roman based (English) character sets.

10. Choice of Table Size

Sun, 01 Mar 1998 00:00:00 +0000

NOTE: THIS DOCUMENT IS OBSOLETE, PLEASE CHECK THE NEW VERSION: "The Digital Audio Resampling Home Page", by Julius O. Smith III, Copyright © 2016-05-17 by Julius O. Smith III - Center for Computer Research in Music and Acoustics (CCRMA), Stanford University

<< Previous page TOC Next page >>
Choice of Table Size
It is desirable that the stored filter impulse response be sampled sufficiently densely so that interpolating linearly between samples does not introduce error greater than the quantization error. We will show that this condition is satisfied whenever the filter table contains at least entries per zero-crossing, where is the number of bits allocated to each table entry.

Linear Interpolation Error Bound.
Let denote the lowpass filter impulse response, and assume it is twice continuously differentiable for all . By Taylor's theorem[Goldstein 1967, p. 119], we have

for some , where denotes the time derivative of evaluated at , and is the second derivative at.
The linear interpolation error is defined as

where , , , and is the interpolated value given by

where and . Thus and are successive time instants for which samples of are available, and is the linear interpolation factor. (We ignore errors in the linear interpolation itself at this point.)
Expressing as

applying Eq. (2.6.1) to both terms on the right-hand side, and subtracting Eq. (2.6.1) gives

where both and are in . Defining

and noting that for some which implies

we obtain the upper bound

Application to the Ideal Lowpass Filter.
For the ideal lowpass filter, we have

where , and is the number of table entries per zero-crossing. Note that the rightmost form in Eq. (2.6.1) is simply the inverse Fourier transform of the ideal lowpass-filter frequency response. Twice differentiating with respect to , we obtain

from which it follows that the maximum magnitude is

Note that this bound is attained at . Substituting Eq. (2.6.1) into Eq. (2.6.1), we obtain the error bound

Thus for the ideal lowpass filter , the pointwise error in the interpolated lookup of is bounded by . This means that must be about half the coefficient word-length used for the filter coefficients. For example, if is quantized to bits, must be of the order of . In contrast, we will show that without linear interpolation, must increase proportional to for -bit samples of . In the -bit case, this gives . The use of linear interpolation of the filter coefficients reduces the memory requirements considerably.
The error bounds obtained for the ideal lowpass filter are typically accurate also for lowpass filters used in practice. This is because the error bound is a function of , the maximum curvature of the impulse response , and most lowpass designs will have a value of very close to that of the ideal case. The maximum curvature is determined primarily by the bandwidth of the filter since, generalizing equations Eq. (2.6.1) and Eq. (2.6.1),

which is just the second moment of the lowpass-filter frequency response (which is real for symmetric FIR filters obtained by symmetrically windowing the ideal sinc function [Rabiner and Gold 1975]). A lowpass-filter design will move the cut-off frequency slightly below that of the ideal lowpass filter in order to provide a ``transition band'' which allows the filter response to give sufficient rejection at the ideal cut-off frequency which is where aliasing begins. Therefore, in a well designed practical lowpass filter, the error bound should be lower than in the ideal case.

Relation of Interpolation Error to Quantization Error.
If is approximated by which is represented in two's complement fixed-pointarithmetic, then

where is the th bit, and the worst-case rounding error is

Letting , where , the interpolated look-up becomes

Thus the error in the interpolated lookup between quantized filter coefficients is bounded by

which, in the case of , can be written

If , then , and the interpolation error is less than the quantization error by more than a factor of .

Error in the Absence of Interpolation.
For comparison purposes, we derive the error incurred when no interpolation of the filter table is performed. In this case, assuming rounding to the nearest table entry, we have

(5)
(6)
(7)
(8)
(9)

where . For the ideal lowpass, we have

Note that and . Thus where . The no-interpolation error bound is then

10. Difference Equation

Sun, 01 Mar 1998 00:00:00 +0000

NOTE: THIS DOCUMENT IS OBSOLETE, PLEASE CHECK THE NEW VERSION: "Introduction to Digital Filters with Audio Applications", by Julius O. Smith III, Copyright © 2017-11-26 by Julius O. Smith III - Center for Computer Research in Music and Acoustics (CCRMA), Stanford University

<< Previous page TOC INDEX Next page >>
Difference Equation
Definition. The difference equation for a general linear time-invariant (LTI)digital filter is given by
(42)
(43)

where is the input signal, is the output signal, and the constants , , , are called difference equation coefficients, or more simply, filter coefficients. When the and coefficients are real numbers, then the filter is said to be real.
Definition. Equation Eq. (43) represents only causal LTI filters. A filter is said to be causal when the output does not depend on any ``future'' inputs. (In more colorful terms, a filter is causal if it does not ``laugh'' before it is ``tickled.'')
Definition. The maximum time span, in samples, used in creating each output sample is called the order of the filter. In Eq. (43), the order is the larger of and . Since and in Eq. (43) are assumed finite, Eq. (43) represents the class of finite order causal LTI filters.
In addition to difference equation coefficients, any LTI filter may be represented in the time domain by its response to a specific signal called the impulse.
Definition. The impulse is denoted as and is defined by

Definition. The impulse response of a filter is the response of the filter to and is most often denoted .
Definition. A filter is said to be stable if the impulse response approaches zero as goes to infinity.

10. Getting the Most from your Batteries

Sun, 01 Mar 1998 00:00:00 +0000

<< Previous page INDEX Next page >>
10. Getting the Most from your Batteries
A common difficulty with portable equipment is the gradual decline in battery performance after the first year of service. Although fully charged, the battery eventually regresses to a point where the available energy is less than half of its original capacity, resulting in unexpected downtime.
Downtime almost always occurs at critical moments. This is especially true in the public safety sector where portable equipment runs as part of a fleet operation and the battery is charged in a pool setting, often with minimal care and attention. Under normal conditions, the battery will hold enough power to last the day. During heavy activities and longer than expected duties, a marginal battery cannot provide the extra power needed and the equipment fails.

10. Getting the Most from your Batteries 2

Sun, 01 Mar 1998 00:00:00 +0000

<< Previous page INDEX Next page >>
10.2 How to Restore and Prolong Nickel-based Batteries

The effects of crystalline formation are most pronounced if a nickel-based battery is left in the charger for days, or if repeatedly recharged without a periodic full discharge. Since most applications do not use up all energy before recharge, a periodic discharge to 1V/cell (known as exercise) is essential to prevent the buildup of crystalline formation on the cell plates. This maintenance is most critical for the NiCd battery.

All NiCd batteries in regular use and on standby mode (sitting in a charger for operational readiness) should be exercised once per month. Between these monthly exercise cycles, no further service is needed. The battery can be used with any desired user pattern without the concern of memory.

The NiMH battery is affected by memory also, but to a lesser degree. No scientific research is available that compares NiMH with NiCd in terms of memory degradation. Neither is information on hand that suggests the optimal amount of maintenance required to obtain maximum battery life. Applying a full discharge once every three months appears right. Because of the NiMH battery’s shorter cycle life, over-exercising is not recommended.

A hand towel must be cleaned periodically. However, if it were washed after each use, its fabric would wear out very quickly. In the same way, it is neither necessary nor advisable to discharge a rechargeable battery before each charge — excessive cycling puts extra strain on the battery.

Exercise and Recondition — Research has shown that if no exercise is applied to a NiCd for three months or more, the crystals ingrain themselves, making them more difficult to break up. In such a case, exercise is no longer effective in restoring a battery and reconditioning is required. Recondition is a slow, deep discharge that removes the remaining battery energy by draining the cells to a voltage threshold below 1V/cell.

Figure 10-2: Exercising and reconditioning batteries on a Cadex battery analyzer.
This illustration shows the battery voltage during a normal discharge to 1V, followed by the secondary discharge (recondition). Recondition consists of a discharge to 1V/cell at a 1C load current, followed by a secondary discharge to 0.4V at a much reduced current. NiCd batteries affected by memory often restore themselves to full service.

Tests performed by the US Army have shown that a NiCd cell needs to be discharged to at least 0.6V to effectively break up the more resistant crystalline formation. During recondition, the current must be kept low to prevent cell reversal. Figure 10-2 illustrates the battery voltage during normal discharge to 1V/cell followed by the secondary discharge (recondition).

Figure 10-3 illustrates the effects of exercise and recondition. Four batteries afflicted with various degrees of memory are serviced. The batteries are first fully charged, then discharged to 1V/cell. The resulting capacities are plotted on a scale of 0 to 120 percent in the first column. Additional discharge/charge cycles are applied and the battery capacities are plotted in the subsequent columns. The solid black line represents exercise, (discharge to 1V/cell) and the dotted line recondition (secondary discharge at reduced current to 0.4V/cell). On this test, the exercise and recondition cycles are applied manually at the discretion of the research technician.

Figure 10-3: Effects of exercise and recondition.
Battery A improved capacity on exercise alone; batteries B and C required recondition. A new battery with excellent readings improved further with recondition.

Battery A responded well to exercise alone and no recondition was required. This result is typical of a battery that has been in service for only a few months or has received periodic exercise cycles. Batteries B and C, on the other hand, required recondition (dotted line) to restore their performance. Without the recondition function, these two batteries would need to be replaced.

10. Getting the Most from your Batteries 3

Sun, 01 Mar 1998 00:00:00 +0000

<< Previous page INDEX Next page >>

After service, the restored batteries were returned to full use. When examined after six months of field use, no noticeable degradation in the restored performance was visible. The regained capacity was permanent with no evidence of falling back to the previous state. Obviously, the batteries would need to be serviced on a regular basis to maintain the performance.

Applying the recondition cycle on a new battery (top line on chart) resulted in a slight capacity increase. This capacity gain is not fully understood, other than to assume that the battery improved by additional formatting. Another explanation is the presence of early memory. Since new batteries are stored with some charge, the self-discharge that occurs during storage contributes to a certain amount of crystalline formation. Exercising and reconditioning reverse this effect. This is why the manufacturers recommend storing rechargeable batteries at about 40 percent charge.

The importance of exercising and reconditioning NiCd batteries is emphasized further by a study carried out by GTE Government Systems in Virginia, USA, for the US Navy. To determine the percentage of batteries needing replacement within the first year of use, one group of batteries received charge only, another group was exercised and a third group received recondition. The batteries studied were used for two-way radios on the aircraft carriers USS Eisenhower with 1500 batteries and USS George Washington with 600 batteries, and the destroyer USS Ponce with 500 batteries.

With charge only (charge-and-use), the annual percentage of battery failure on the USS Eisenhower was 45 percent (see Figure 10-4). When applying exercise, the failure rate was reduced to 15 percent. By far the best results were achieved with recondition. The failure rate dropped to 5 percent. Identical results were attained from the USS George Washington and the USS Ponce.

Maintenance Method Annual Percentage of Batteries
Requiring Replacement

Charge only (charge-and-use) 45%

Exercise only (discharge to 1V/cell) 15%

Reconditioning (secondary deep discharge) 5%

Figure 10-4: Replacement rates of NiCd batteries.
The annual percentage of NiCd batteries requiring replacement when used without any maintenance decreases with exercise and recondition. These statistics were drawn from batteries used by the US Navy on the USS Eisenhower, USS George Washington and USS Ponce.

The GTE Government System report concluded that a battery analyzer featuring exercise and recondition functions costing $2,500US would pay for itself in less than one month on battery savings alone. The report did not address the benefits of increased system reliability, an issue that is of equal if not greater importance, especially when the safety of human lives is at stake.

Another study involving NiCd batteries for defense applications was performed by the Dutch Army. This involved battery packs that had been in service for 2 to 3 years during the Balkan War. The Dutch Army was aware that the batteries were used under the worst possible conditions. Rather than a good daily workout, the packs were used for patrol duties lasting 2 to 3 hours per day. The rest of the time the batteries remained in the chargers for operational readiness.

After the war, the batteries were sent to the Dutch Military Headquarters and were tested with Cadex 7000 Series battery analyzers. The test technician found that the capacity of some packs had dropped to as low as 30 percent. With the recondition function, 90 percent of the batteries restored themselves to full field use. The Dutch Army set the target capacity threshold for field acceptability to 80 percent. This setting is the pass/fail acceptance level for their batteries.

Based on the successful reconditioning results, the Dutch Army now assigns the battery maintenance duty to individual battalions. The program calls for a service once every two months. Under this regime, the Army reports reduced battery failure and prolonged service life. The performance of each battery is known at any time and any under-performing battery is removed before it causes a problem.

NiCd batteries remain the preferred chemistry for mobile communications, both in civil and defense applications. The main reason for its continued use is dependable and enduring service under difficult conditions. Other chemistries have been tested and found problematic in long-term use.

During the later part of the 1990s, the US Army switched from mainly non-rechargeable to the NiMH battery. The choice of chemistry was based on the benefit of higher energy densities as compared to NiCd. The army soon discovered that the NiMH did not live up to the expected cycle life. Their reasoning, however, is that the 100 cycles attained from a NiMH pack is still more economical than using a non-rechargeable equivalent. The army’s focus is now on the Li-ion Polymer, a system that is more predictable than NiMH and requires little or no maintenance. The aging issue will likely cause some logistic concerns, especially if long-term storage is required.

Simple Guidelines

Do not leave a nickel-based battery in a charger for more than a day after full charge is reached.

Apply a monthly full discharge cycle. Running the battery down in the equipment may do this also.

Do not discharge the battery before each recharge. This would put undue stress on the battery.

Avoid elevated temperature. A charger should only raise the battery temperature for a short time at full charge, and then the battery should cool off.

Use quality chargers to charge batteries.

10. Getting the Most from your Batteries 4

Sun, 01 Mar 1998 00:00:00 +0000

<< Previous page INDEX Next page >>
10.3 The Effect of Zapping

To maximize battery performance, remote control (RC) racing enthusiasts have experimented with all imaginable methods available. One technique that seems to work is zapping the cells with a very high pulse current. Zapping is said to increase the cell voltage slightly, generating more power.

Typically, the racecar motor draws 30A, delivered by a 7.2V battery. This calculates to over 200W of power. The battery must endure a race lasting about four minutes.

According to experts, zapping works best with NiCd cells. NiMH cells have been tried but they have shown inconsistent results.

Companies specializing in zapping NiCd for RC racing use a very high quality Japanese NiCd cell. The cells are normally sub-C in size and are handpicked at the factory for the application. Specially labeled, the cells are delivered in a discharged state. When measuring the cell in empty state-of-charge (SoC), the voltage typically reads between 1.11 to 1.12V. If the voltage drops lower than 1.06V, the cell is considered suspect and zapping does not seem to enhance the performance as well as on the others.

The zapping is done with a 47,000mF capacitor that is charged to 90V. Best results are achieved if the battery is cycled twice after treatment, then is zapped again. After the battery has been in service for a while, zapping no longer seems to improve the cell’s performance. Neither does zapping regenerate a cell that has become weak.

The voltage increase on a properly zapped battery is between 20 and 40mV. This improvement is measured under a load of 30A. According to experts, the voltage gain is permanent but there is a small drop with usage and age.

There are no apparent side effects in zapping, however, the battery manufacturers remain silent about this treatment. No scientific explanations are available why the method of zapping improves battery performance. There is little information available regarding the longevity of the cells after they have been zapped.

10. Getting the Most from your Batteries 5

Sun, 01 Mar 1998 00:00:00 +0000

<< Previous page INDEX Next page >>
10.4 How to Restore and Prolong Sealed Lead Acid Batteries

The sealed version of the lead acid battery is designed with a low over-voltage potential to prevent water depletion. Consequently, the SLA and VRLA systems never get fully charged and some sulfation will develop over time.

Finding the ideal charge voltage limit for the sealed lead acid system is critical. Any voltage level is a compromise. A high voltage limit produces good battery performance, but shortens the service life due to grid corrosion on the positive plate. The corrosion is permanent and cannot be reversed. A low voltage preserves the electrolyte and allows charging under a wide temperature range, but is subject to sulfation on the negative plate. (In keeping with portability, this book focuses on portable SLA batteries. Due to similarities between the SLA and VRLA systems, references to the VRLA are made where applicable).

Once the SLA battery has lost capacity due to sulfation, regaining its performance is often difficult and time consuming. The metabolism of the SLA battery is slow and cannot be hurried.

A subtle indication on whether an SLA battery can be recovered is reflected in the behavior of its discharge voltage. A fully charged SLA battery that starts its discharge with a high voltage and tapers off gradually can be reactivated more successfully than one on which the voltage drops rapidly when the load is applied.

Reasonably good results in regaining lost capacity are achieved by applying a charge on top of a charge. This is done by fully charging an SLA battery, then removing it for a 24 to 48 hour rest period and applying a charge again. This is repeated several times, then the capacity of the battery is checked with a full discharge. The SLA is able to accept some overcharge, however, too long an overcharge could harm the battery due to corrosion and loss of electrolyte.

The effect of sulfation of the plastic SLA can be reversed by applying an over-voltage charge of up to 2.50V/cell for one to two hours. During that time, the battery must be kept cool and careful observation is necessary. Extreme caution is required not to raise the cell pressure to venting point. Most plastic SLA batteries vent at 34 kPa (5 psi). Cell venting causes the membrane on some SLA to rupture permanently. Not only do the escaping gases deplete the electrolyte, they are also highly flammable!

The VRLA uses a cell self-regulating venting system that opens and closes the cells based on cell pressure. Changes in atmospheric pressure contribute to cell venting. Proper ventilation of the battery room is essential to prevent the accumulation of hydrogen gas.

Cylindrical SLA — The cylindrical SLA (made by Hawker) resembles an enlarged D sized cell. After long storage, the Hawker cell can be reactivated relatively easily. If affected by sulfation, the cell voltage under charge may initially raise up to 5V, absorbing only a small amount of current. Within about two hours, the small charging current converts the large sulfate crystals back into active material. The internal cell resistance decreases and the charge voltage eventually returns to normal. At a voltage between 2.10V and 2.40V, the cell is able to accept a normal charge. To prevent damage, caution must be exercised to limit the charge current.

The Hawker cells are known to regain full performance with the described voltage method, leaving few adverse effects. This, however, does not give credence to store this cell at a very low voltage. It is always best to follow the manufacturer’s recommended specifications.

Improving the capacity of an older SLA by cycling is mostly unsuccessful. Such a battery may simply be worn out. Cycling would just wear down the battery further. Unlike nickel-based batteries, the lead acid battery is not affected by memory.

SLA batteries are commonly rated at a 20-hour discharge. Even at such a slow rate, a capacity of 100 percent is difficult to obtain. For practical reasons, most battery analyzers use a 5-hour discharge when servicing SLA batteries. This typically produces 80 to 90 percent of the rated capacity. SLA batteries are normally overrated and manufacturers are aware of this.

Caution: When charging an SLA with over-voltage, current limiting must be applied to protect the battery. Always set the current limit to the lowest practical setting and observe the battery voltage and temperature during charge. Prevent cell venting.

Important: In case of rupture, leaking electrolyte or any other cause of exposure to the electrolyte, flush with water immediately. If eye exposure occurs, flush with water for 15 minutes and consult a physician immediately.

Simple Guidelines

Always keep the SLA charged. Never store below 2.10V/cell.

Avoid repeated deep discharges. Charge more often.

If repeated deep discharges cannot be avoided, use a larger battery to ease the strain.

Prevent sulfation and grid corrosion by choosing the correct charge and float voltages.

10. Getting the Most from your Batteries 6

Sun, 01 Mar 1998 00:00:00 +0000

<< Previous page INDEX Next page >>
10.5 How to Prolong Lithium-based Batteries

Today’s battery research is heavily focused on lithium chemistries, so much so that one could assume that all future batteries will be lithium systems. Lithium-based batteries offer many advantages over nickel and lead-based systems. Although maintenance free, no external service is known that can restore the battery’s performance once degraded.

In many respects, Li-ion provides a superior service to other chemistries, but its performance is limited to a defined lifespan. The Li-ion battery has a time clock that starts ticking as soon as the battery leaves the factory. The electrolyte slowly ‘eats up’ the positive plate and the electrolyte decays. This chemical change causes the internal resistance to increase. In time, the cell resistance raises to a point where the battery can no longer deliver the energy, although it may still be retained in the battery. Equipment requiring high current bursts is affected most by the increase of internal resistance.

Battery wear-down on lithium-based batteries is caused by two activities: actual usage or cycling, and aging. The wear-down effects by usage and aging apply to all batteries but this is more pronounced on lithium-based systems.

The Li-ion batteries prefer a shallow discharge. Partial discharges produce less wear than a full discharge and the capacity loss per cycle is reduced. A periodic full discharge is not required because the lithium-based battery has no memory. A full cycle constitutes a discharge to 3V/cell. When specifying the number of cycles a lithium-based battery can endure, manufacturers commonly use an 80 percent depth of discharge. This method resembles a reasonably accurate field simulation. It also achieves a higher cycle count than doing full discharges.

In addition to cycling, the battery ages even if not used. The amount of permanent capacity loss the battery suffers during storage is governed by the SoC and temperature. For best results, keep the battery cool. In addition, store the battery at a 40 percent charge level. Never fully charge or discharge the battery before storage. The 40 percent charge assures a stable condition even if self-discharge robs some of the battery’s energy. Most battery manufacturers store Li-ion batteries at 15°C (59°F) and at 40 percent charge.

Simple Guidelines

Charge the Li-ion often, except before a long storage. Avoid repeated deep discharges.

Keep the Li-ion battery cool. Prevent storage in a hot car. Never freeze a battery.

If your laptop is capable of running without a battery and fixed power is used most of the time, remove the battery and store it in a cool place.

Avoid purchasing spare Li-ion batteries for later use. Observe manufacturing date when purchasing. Do not buy old stock, even if sold at clearance prices.

10. Getting the Most from your Batteries 7

Sun, 01 Mar 1998 00:00:00 +0000

<< Previous page INDEX Next page >>
10.6 Battery Recovery Rate

The battery recovery rate by applying controlled discharge/charge cycles varies with chemistry type, cycle count, maintenance practices and age of the battery. The best results are achieved with NiCd. Typically 50 to 70 percent of discarded NiCd batteries can be restored when using the exercise and recondition methods of a Cadex battery analyzer or equivalent device.

Not all batteries respond equally well to exercise and recondition services. An older battery may show low and inconsistent capacity readings with each cycle. Another will get worse when additional cycles are applied. An analogy can be made to a very old man for whom exercise is harmful. Such conditions indicate instabilities caused by aging, suggesting that this pack should be replaced. In fact, some users of the Cadex analyzers use the recondition cycle as the acid test. If the battery gets worse, there is strong evidence that this battery would not perform well in the field. Applying the acid test exposes the weak packs, which can no longer hide behind their stronger peers.

Some older NiCd batteries recover to near original capacity when serviced. Caution should be applied when ‘rehiring’ these old-timers because they may exhibit high self-discharge. If in doubt, a self-discharge test should be carried out.

The recovery rate of the NiMH is about 40 percent. This lower yield is, in part, due to the NiMH’s reduced cycle count as compared to the NiCd. Some batteries may be afflicted by heat damage that occurs during incorrect charging. This deficiency cannot be corrected. Permanent loss of battery capacity is also caused by prolonged storage at elevated temperatures.

The recovery rate for lead acid batteries is a low 15 percent. Unlike nickel-based batteries, the restoration of the SLA is not based on reversing crystalline formation, but rather by reactivating the chemical process. The reasons for low capacity readings are prolonged storage at low terminal voltage, and poor charging methods. The battery also fails due to age and high cycle count.

Lithium-based batteries have a defined age limit. Once the anticipated cycles have been delivered, no method exists to improve the battery. The main reason for failure is high internal resistance caused by oxidation. Operating the battery at elevated temperatures will momentarily reduce this condition. When the temperature normalizes, the condition of high internal resistance returns.

The speed of oxidation depends on the storage temperature and the battery’s charge state. Keeping the battery in a cool place can prolong its life. The Li-ion battery should be stored at 40 percent rather than full-charge state.

An increasing number of modern batteries fall prey to the cut-off problem induced by a deep discharge. This is especially evident on Li-ion batteries for mobile phones. If discharged below 2.5V/cell, the internal protection circuit often opens. Many chargers cannot apply a recharge and the battery appears to be dead.

Some battery analyzers feature a boost, or wake-up function, to activate the protection circuit and enable a recharge if discharged too low. If the cell voltage has fallen too low (1.5V/cell and lower) and has remained in that state for a few days, a recharge should not be attempted because of safety concerns on the cell(s).

It is often asked whether a restored battery will work as good as a new one. The breakdown of the crystalline formation can be considered a full restoration. However, the crystalline formation will re-occur with time if the battery is denied the required maintenance.

When the defective component of a machine is replaced, only the replaced part is new; the rest of the machine remains in the same condition. If the separator of a nickel-based battery is damaged by excess heat or is marred by uncontrolled crystalline formation, that part of the battery will not improve.

Other methods, which claim to restore and prolong rechargeable batteries, have produced disappointing results. One method is attaching a strong magnet on the side of the battery; another is exposing the battery to ultrasound vibrations. No scientific evidence exists that such methods will improve battery performance, or restore an ailing battery.

10. THE GLOSSARY

Sun, 01 Mar 1998 00:00:00 +0000

<< Previous page TOC
10. THE GLOSSARY
#A B C D E F G H I J K L M N O P Q R S T U V W X Y Z
Accelerated Graphics Port: (AGP) – An Intel-developed interface that enables high-speed graphics. Graphics data move between the PC’s graphics controller and computer memory directly, instead of being cached in video memory.
Access Time: The average time (in nanoseconds) for RAM to complete one access. Access time is composed of address setup time and latency (the time it takes to initiate a request for data and prepare access).
ANSI: (American National Standards Institute) – The U.S. organization responsible for setting information technology standards.
ASCII: (American Standard Code for Information Interchange) – A method of encoding text as binary values. The ASCII coding system contains 256 combinations of 7-bit or 8-bit binary numbers to represent every possible keystroke.
Backside Bus: (BSB) – The data path that runs between the CPU and L2 cache. The amount of data moved on electronic lines, such as a bus, per second.
Bandwidth: Bandwidth is usually measured in bits per second, bytes per second, or cycles per second (Hertz).
Bank: See memory bank.
Bank Schema: A method of diagramming memory configurations. The bank schema consists of rows and/or columns that represent memory sockets on a computer board. Rows indicate independent sockets; columns represent banks.
Base Rambus: The first generation of Rambus technology, first shipped in 1995.
BGA: (Ball Grid Array) – A chip package having solder balls on the underside for mounting. BGA allows for a reduction in die package size, better heat dissipation, and greater module densities.
Binary: A numbering system that uses combinations of 0 and 1 to represent data. Also known as Base 2.
BIOS: (Basic Input-Output System) – Startup routines that prepare the computer for operation.
Bit: The smallest unit of information a computer processes. A bit is 1 or 0.
Buffer: A holding area for data shared by devices that operate at different speeds or have different priorities. A buffer allows a device to operate without the delays that other devices impose.
Buffered Memory: A memory module that contains buffers. Buffers re-drive the signals through the memory chips and allow the module to include more memory chips. Buffered and unbuffered memory cannot be mixed. The design of the computer memory controller dictates whether memory must be buffered or unbuffered.
Burst EDO RAM: (BEDO) – EDO memory that can process four memory addresses in one burst. Bus speeds range from 50MHz to 66MHz (compared to 33MHz for EDO and 25MHz for Fast Page Mode).
Burst Mode: High-speed transmission of a block of data (a series of consecutive addresses) when the processor requests a single address.
Bus: A data path in a computer, consisting of various parallel wires to which the CPU, memory, and all input/output devices are connected.
Bus Cycle: A single transaction between main memory and the CPU.
Byte: Eight bits of information. The byte is the fundamental unit of computer processing; almost all specifications and measures of computer performance are in bytes or multiples thereof. See kilobytes and megabytes.
Cache Memory: A small amount (normally less than 1MB) of high-speed memory residing on or close to the CPU. Cache memory supplies the processor with the most frequently requested data and instructions. Level 1 cache (primary cache) is the cache closest to the processor. Level 2 cache (secondary cache) is the cache second closest to the processor and is usually on the motherboard.
CAS: (Column Address Strobe) – A memory chip signal that latches the column address of a particular location in a row-column matrix.
CAS Latency: The ratio between column access time and clock cycle time. CAS Latency 2 (CL2) offers a slight performance increase over CAS Latency 3 (CL3).
ccNUMA: (Cache-Coherent, Non-uniform Memory Access) – A flexible architecture that uses modular, low-cost components and offers multidimensional scaling potential to high-end servers.
Chipset: Microchips that support the CPU. The chipset usually contains several controllers that govern how information travels between the processor and other components.
Chip-Scale Package: (CSP) – Thin chip packaging whereby electrical connections are typically through a ball grid array. Chip-scale packaging is used in RDRAM and flash memory.
CompactFlash: A small, lightweight form factor for removable storage cards. CompactFlash cards are durable, operate at low voltages, and retain data when power is off. Uses include digital cameras, cell phones, printers, handheld computers, pagers, and audio recorders.
Composite: An Apple Computer, Inc. term for a memory module that used an older technology and contained more but lower-density chips.
Concurrent Rambus: The second generation of Rambus technology. Concurrent Rambus has been used in graphics-based computers, digital TVs, and video game applications (such as Nintendo 64 since 1997).
Continuity RIMM: (C-RIMM) – A Direct Rambus memory module that does not contain memory chips. C-RIMM provides a continuous channel for the signal. In a Direct Rambus system, open connectors must be populated with C-RIMMs.
CPU: (Central Processing Unit) – The computer chip that has primary responsibility for interpreting commands and running programs. The CPU is also known as the processor or microprocessor.
Credit Card Memory: A type of memory typically in laptop and notebook computers. Credit card memory is the size of a credit card.
DDR SDRAM: (Double Data Rate Synchronous Dynamic Random-Access Memory) – The latest generation of SDRAM technology. Data is read on both the rising and the falling edge of the computer clock, thereby delivering twice the bandwidth of standard SDRAM. With DDR SDRAM, memory speed doubles without increasing the clock frequency.
DIMM: (Dual In-line Memory Module) – A printed circuit board with gold contacts and memory devices. A DIMM is similar to a SIMM, but with this primary difference: unlike the metal leads on either side of a SIMM, which are “tied together” electrically, the leads on either side of a DIMM are electrically independent.
Direct Rambus: Rambus technology’s third generation, which offers a completely new DRAM architecture for high-performance PCs. Data transfers at speeds up to 800MHz over a narrow 16-bit channel, compared to current SDRAM, which operates at 100MHz on a wide 64-bit bus.
DIP: (Dual In-line Package) – A DRAM component packaging. DIPs can be installed in sockets or permanently soldered into holes on the printed circuit board. The DIP package was extremely popular when memory was installed directly on the motherboard.
DRAM: (Dynamic Random-Access Memory) – The most common form of RAM. DRAM can hold data for only a short time. To retain data, DRAM must be refreshed periodically. If the cell is not refreshed, the data disappear.
Dual-Banked: A memory module having two banks.
Dual Independent Bus: (DIB) – An Intel-developed bus architecture that offers greater bandwidth by having two separate buses (frontside and backside) access the processor. Pentium II computers have DIBs.
ECC: (Error Correction Code) – A method of checking the integrity of data in DRAM. ECC provides more elaborate error detection than parity; ECC can detect multiple-bit errors and can locate and correct single-bit errors.
EDO: (Extended Data-Out) – A DRAM technology that shortens the read cycle between memory and CPU. On computers that support it, EDO memory allows a CPU to access memory 10 to 20 percent faster than comparable fast-page mode memory.
EDRAM: (Enhanced DRAM) – Enhanced Memory Systems, Inc. DRAM that contains a small amount of SRAM.
EEPROM: (Electrically Erasable Programmable Read-Only Memory) – A memory chip that retains data content after power has been removed. EEPROM can be erased and reprogrammed within the computer or externally.
EISA: (Extended ISA) – A bus architecture that extended the 16-bit ISA bus to 32 bits. EISA operates at 8MHz and has a peak data transfer rate of 33MB per second. EISA was introduced in 1988 as an open alternative to IBM’s proprietary Micro Channel bus.
EOS: (ECC on SIMM) – An IBM data-integrity-checking technology that features ECC data-integrity-checking on a SIMM.
EPROM: (Erasable Programmable Read-Only Memory) – A programmable and reusable chip that retains content until erasure under ultraviolet light. Special equipment erases and reprograms EPROMs.
ESDRAM: (Enhanced Synchronous DRAM) – A type of SDRAM developed by Enhanced Memory Systems, Inc. ESDRAM replaces expensive SRAM in embedded systems and offers comparable speed with less power consumption and lower cost.
Even Parity: A type of data integrity checking whereby the parity bit checks for an even number of 1s.
Fast-Cycle RAM: (FCRAM) – FCRAM is a memory technology currently being developed by Toshiba and Fujitsu. FCRAM is not intended for PC main memory but will be used in specialty applications such as high-end servers, printers, and telecommunications switching systems.
Fast-Page Mode: An early form of DRAM, fast-page mode’s advantage over previous page mode memory technologies was faster access to data in the same row.
Flash Memory: A solid-state, nonvolatile, rewritable memory that functions like a combination of RAM and hard disk. Flash memory is durable, operates at low voltages, and retains data when power is off. Flash memory cards are used in digital cameras, cell phones, printers, handheld computers, pagers, and audio recorders.
Form Factor: The size, configuration, and other specifications used to describe hardware. Examples of memory form factors are: SIMM, DIMM, RIMM, 30-pin, 72-pin, and 168-pin.
Frontside Bus: (FSB) – The data path that runs between the CPU and main memory (RAM).
Gigabit: Approximately 1 billion bits, or exactly 1 bit x 1,0243 (1,073,741,824) bits.
Gigabyte: Approximately 1 billion bytes, or exactly 1 byte x 1,0243 (1,073,741,824) bytes.
Heat Spreader: A sheath, usually aluminum, that covers an electronic device and dissipates heat.
Heat Sink: A component, typically zinc alloy, that dissipates heat. CPUs require heat sinks.
IC: (Integrated Circuit) – An electronic circuit on a semiconductor chip. The circuit includes components and connectors. A semiconductor chip is usually molded in a plastic or ceramic case and has external connector pins.
Interleaving: Techniques for increasing memory speed. For example, with separate memory banks for odd and even addresses, the next byte of memory can be accessed while the current byte refreshes.
JEDEC: (Joint Electron Device Engineering Council) – An Electronic Industries Alliance (EIA) body that sets semiconductor engineering standards.
Kilobit: Approximately one thousand bits, or exactly 1 bit x 210 (1,024) bits.
Kilobyte: Approximately one thousand bytes, or exactly 1 byte x 210 (1,024) bytes.
Level 1 Cache: (L1) – Also known as primary cache, L1 Cache is a small amount of high-speed memory that resides on or very close to the processor. L1 Cache supplies the processor with the most frequently requested data and instructions.
Level 2 Cache: (L2) – Also known as secondary cache, L2 Cache is a small amount of high-speed memory close to the CPU and usually on the motherboard. L2 Cache supplies the processor with the most frequently requested data and instructions. Depending on the motherboard, Level 2 cache may be upgraded.
Logic Board: See Motherboard.
Megabit: Approximately one million bits, or exactly 1 bit x 1,0242 (1,048,576) bits.
Megabyte: Approximately one million bytes, or exactly 1 byte x 1,0242 (1,048,576) bytes.
Memory: A computer’s random-access memory. Memory temporarily holds data and instructions for the CPU. See RAM.
Memory Bank: A logical unit of memory in a computer, the size of which the CPU determines. For example, a 32-bit CPU requires memory banks that provide 32 bits of information at a time. A bank can consist of one or more memory modules.
Memory Bus: The bus that runs from the CPU to the memory expansion slots.
Memory Controller Hub: (MCH) – The interface between the processor, Accelerated Graphics Port, and RDRAM on motherboards that use Intel’s 820 or 840 chipsets.
Memory Translator Hub: (MTH) – The interface that allows SDRAM memory to be supported on a Direct Rambus Channel for motherboards using Intel’s 820 chipset.
Micro BGA: (µBGA) – Tessera, Inc. BGA chip packaging technique allows for a reduction in die package size, improved heat dissipation, and greater module densities.
Motherboard: Also known as the logic board, main board, or computer board, the motherboard is the computer’s main board and in most cases holds all CPU, memory, and I/O functions or has expansion slots for them.
Nanosecond: (ns) – One billionth of a second. Memory data access times are in nanoseconds. For example, memory access times for typical 30- and 72-pin SIMM modules range from 60 to 100 nanoseconds.
Nibble: Half of an 8-bit byte, or 4 bits.
Non-Composite: An Apple Computer, Inc. term for a memory module that used a new technology and contained fewer but higher-density chips. Non-composite modules were more reliable and more expensive than composite modules.
Odd Parity: Data integrity checking in which the parity bit checks for an odd number of 1s.
Parity: Data integrity checking that adds a single bit to each byte of data. The parity bit is used to detect errors in the other 8 bits.
PCB: (Printed Circuit Board) – Generally flat, multi-layer boards made of fiberglass with electrical traces. The surface and sublayers use copper traces to provide electrical connections for chips and other components. Examples of PCBs include: motherboards, SIMMs, and credit card memory.
PC Card: (PCMCIA: Personal Computer Memory Card International Association) – A standard that allows interchangeability of various computing components on the same connector. The PCMCIA standard supports input-output devices, including memory, fax/modem, SCSI, and networking products.
PCI: (Peripheral Component Interconnect) – A peripheral bus that can send 32 or 64 bits of data simultaneously. PCI offers plug-and-play capability.
Pipeline Burst Cache: Cache that reduces wait states and accelerates memory access by using pipelining and bursting functions.
Pipelining: A technique in which memory loads the requested memory contents into a small cache composed of SRAM, then immediately begins fetching the next memory contents. This creates a two-stage pipeline, where data is read from or written to SRAM in one stage, and data is read from or written to memory in the other stage.
Proprietary Memory: Memory custom designed for a specific computer.
RAM: (Random-Access Memory) – A memory cell configuration that holds data for processing by a central processing unit (CPU). Random means the CPU can retrieve data from any address within RAM. See also Memory.
Rambus: (1) Rambus, Inc. develops and licenses high-performance memory logic and circuit design technology and provides licensees with product design, layout, and testing information. (2) Direct Rambus is a high-speed memory technology that uses a narrow 16-bit bus
Rambus Channel: (Rambus channel) to transmit data at speeds up to 800MHz. See Rambus Channel. The data path of Rambus systems. Because of the narrow data width (two bytes), Rambus modules transfer data at up to 800MHz.
RAS: A memory chip signal that latches the row address of a particular location in a row-column matrix.
Refresh: Refreshing maintains data stored in DRAM. The process of refreshing electrical cells on a DRAM component is similar to recharging batteries. Different DRAM components require different refresh methods.
Refresh Rate: The number of DRAM component rows that must be refreshed. Three common refresh rates are 2K, 4K and 8K.
Registered Memory: SDRAM memory that contains registers directly on the module. The registers re-drive the signals through the memory chips and allow the module to be built with more memory chips. Registered and unbuffered memory cannot be mixed. The design of the computer memory controller dictates which type of memory the computer requires.
RIMM^TM: The trademarked name for a Direct Rambus memory module. A RIMM™ conforms to the DIMM form factor and transfers data 16 bits at a time.
RIMM Connector: A Direct Rambus memory socket.
SDRAM: (Synchronous DRAM) – A DRAM technology that uses a clock to synchronize signal input and output on a memory chip. The clock is coordinated with the CPU clock so the timing of the memory chips and the timing of the CPU are in synch. Synchronous DRAM saves time in executing commands and transmitting data, thereby increasing the overall performance of the computer. SDRAM allows the CPU to access memory approximately 25 percent faster than EDO memory.
Self-Refresh: A memory technology that enables DRAM to refresh on its own and independent of the CPU or external refresh circuitry. Self-Refresh technology is built into the DRAM chip itself and reduces power consumption dramatically. Notebook and laptop computers use this technology.
Serial Presence Detect: An EEPROM chip that contains information about size and speed, as well as other specifications and manufacturer information of a memory module.
SGRAM: (Synchronous Graphics Random-Access Memory) – Video memory that includes graphicsspecific read/write features. SGRAM allows data to be retrieved and modified in blocks instead of individually. Blocking reduces the number of reads and writes the memory must perform and increases the performance of the graphics controller.
SIMM: (Single In-line Memory Module) – A printed circuit board having memory devices and gold or tin/lead contacts. A SIMM plugs into a computer memory expansion socket. SIMMs offer two main advantages: ease of installation and minimal consumption of board surface. A vertically mounted SIMM requires only a fraction of the space required by a horizontally mounted DRAM. A SIMM may have as few as 30 or as many as 200 pins. On a SIMM, the metal leads on either side of the board are electrically tied together.
SIMM Socket: A motherboard component that holds a single SIMM.
Single-Banked: A module that has only one bank or row.
SLDRAM: (Synclink) – Although obsolete today, SLDRAM was a main memory technology developed by a consortium of twelve DRAM manufacturers as an alternative to Direct Rambus technology.
SMART CARD: An electronic device, similar in size to a credit card, that can store data and programs while enhancing security. Applications include identification, mass transit, and banking.
SO DIMM: (Small-Outline Dual In-line Memory Module) – An enhanced version of a standard DIMM. A 72-pin small-outline DIMM is about half the length of a 72-pin SIMM. The trademarked name for a Direct Rambus memory module in notebook computers.
SO-RIMM^TM: SO-RIMM™s provide memory bandwidth comparable to desktop memory configurations.
SOJ: (Small-Outline J-lead) – A common form of surface-mounted DRAM packaging. An SOJ is a rectangular package with J-shaped leads on the two long sides.
Static RAM: (SRAM) – A memory chip that requires power to retain content. SRAM is faster than DRAM but more expensive and bulky. A typical use for SRAM is cache memory.
Storage: A data-holding device, such as a hard disk or CD-ROM.
Swapping: Using part of the hard drive as memory when RAM is full. See Virtual Memory.
System Board: See Motherboard.
Transmission Line Technology: A technology that supports the backside bus in Direct Rambus systems.Information is quickly pipelined in simultaneous packets. The memory controller reassembles the packets for frontside bus transfer and communication to the processor.
TSOP: (Thin Small-Outline Package) – A DRAM package that uses gull-wing leads on both sides. TSOP DRAM mounts directly on the surface of the printed circuit board. The TSOP package is one-third the thickness of an SOJ. TSOP components commonly occur in small-outline DIMMs and credit card memory.
Unbuffered Memory: Memory that does not contain buffers or registers located on the module. Instead, these devices are located on the motherboard.
VESA Local Bus: (VL-Bus) – A 32-bit local bus that runs between the CPU and peripheral devices at speeds up to 40MHz.
Virtual Channel Memory: (VCM) – VCM is a memory architecture developed by NEC. VCM allows different blocks of memory–each with its own buffer–to interface separately with the controller. This way, system tasks can be assigned their own virtual channels.
Virtual Memory: Information related to one function does not share buffer space with other tasks running simultaneously, thereby making overall operations much more efficient. Simulated memory. When RAM is full, the computer swaps data to the hard disk and back as needed. See Swapping.
VRAM: (Video Random-Access Memory) – Dual-ported (two separate data ports) memory typically on a video or graphics card. One port is dedicated to the CRT and refreshes and updates the image. The second port is for the CPU or graphics controller and changes the image data in memory.
Wait State: An inactive period for the processor. Wait states result from the different clock speeds of the processor and memory, the latter being typically slower.
Window Random Access Memory: (WRAM) – Samsung Electronics’ dual-ported (two separate data ports) memory typically on a video or graphics card. WRAM has a 25% higher bandwidth than VRAM but costs less.

11. Choice of Interpolation Resolution

Sun, 01 Mar 1998 00:00:00 +0000

NOTE: THIS DOCUMENT IS OBSOLETE, PLEASE CHECK THE NEW VERSION: "The Digital Audio Resampling Home Page", by Julius O. Smith III, Copyright © 2016-05-17 by Julius O. Smith III - Center for Computer Research in Music and Acoustics (CCRMA), Stanford University

<< Previous page TOC Next page >>
Choice of Interpolation Resolution
We now consider the error due to finite precision in the linear interpolation between stored filter coefficients. We will find that the number of bits in the interpolation factor should be about half the filter coefficient word-length .

Quantized Interpolation Error Bound.
The quantized interpolation factor and its complement are representable as

(10)
(11)

where, since are unsigned, . The interpolated coefficient look-up then gives

(12)
(13)

where second-order errors and are dropped. Since , we obtain the error bound

The three terms in Eq. (2.6.2) are caused by coefficient quantization, interpolation quantization, and linear-approximation error, respectively.

Ideal Lowpass Filter.
For the ideal lowpass, the error bound is

Let and require that the added error is at most . Then we arrive at the requirement

11. Convolution Representation

Sun, 01 Mar 1998 00:00:00 +0000

NOTE: THIS DOCUMENT IS OBSOLETE, PLEASE CHECK THE NEW VERSION: "Introduction to Digital Filters with Audio Applications", by Julius O. Smith III, Copyright © 2017-11-26 by Julius O. Smith III - Center for Computer Research in Music and Acoustics (CCRMA), Stanford University

<< Previous page TOC INDEX Next page >>
Convolution Representation
If is the output of an LTI filter with input and impulseresponse , then is the convolution of with ,

Since convolution is commutative (), we have also

Definition. The -transform of the discrete-time signal is defined to be

That and are transform pairs is expressed by writing or .
Theorem. The convolution theorem (Papoulis [Papoulis 1977]) states that

In words, convolution in the time domain is multiplication in the frequency domain.
Taking the -transform of both sides of Eq. (2.2.1) and applying the convolution theorem gives

where is the -transform of the filter impulse response. Thus the-transform of the filter output is the -transform of the input times the -transform of the impulse response.
Definition. The transfer function of a linear time-invariant discrete-time filter is defined to be the -transform of the impulse response .
Theorem. The shift theorem [Papoulis 1977] for -transforms states that

The general difference equation for an LTI filter appears as
(44)
(45)

Taking the -transform of both sides, denoting the transform by gives
(46)
(47)

using linearity and the shift theorem. Replacing by , by , and solving for , which equals the transfer function , yields

11. Maintaining Fleet Batteries

Sun, 01 Mar 1998 00:00:00 +0000

<< Previous page INDEX Next page >>
11. Maintaining Fleet Batteries

Unlike individual battery users, who come to know their batteries like a good friend, fleet users must share the batteries from a pool of unknown packs. While an individual user can detect even a slight reduction in runtime, fleet operators have no way of knowing the behavior or condition of the battery when pulling it from the charger. They are at the mercy of the battery. It’s almost like playing roulette.

It is recommended that fleet battery users set up a battery maintenance program. Such a plan exercises all batteries on a regular basis, reconditions those that fall below a set target capacity and ‘weeds out’ the deadwood. Usually, batteries get serviced only when they no longer hold a charge or when the equipment is sent in for repair. As a result, battery-operated equipment becomes unreliable and battery-related failures often occur. The loss of adequate battery power is as detrimental as any other malfunction in the system.

Implementing a battery maintenance plan requires an effort by management to schedule the required service for the battery packs. This should become an integral component of an organization’s overall equipment maintenance and repair activities. A properly managed program improves battery performance, enhances reliability and cuts replacement costs.

The maintenance plan should include all rechargeable batteries in use. Large organizations often employ a variety of batteries ranging from wireless communications, to mobile computing, to emergency medical equipment, to video cameras, portable lighting and power tools. The performance of these batteries is critical and there is little room for failure.

Whether the batteries are serviced in-house with their own battery analyzers or sent to an independent firm specializing in that service, sufficient spare batteries are required to replace those packs that have been temporarily removed. When the service is done on location and the batteries can be reinstated within 24 hours, only five spares in a fleet of 100 batteries are required. This calculation is based on servicing five batteries per day in a 20 workday month, which equals100 batteries per month. If the batteries are sent away, five spares are needed for each day the batteries are away. If 100 batteries are absent for one week, for example, 35 spare batteries are needed.

Manufacturers of portable equipment support battery maintenance programs. Not only does such a plan reduce unexpected downtime, a well-performing battery fleet makes the equipment work better. If the recurring problems relating to the battery can be eliminated, less equipment is sent to the service centers. A well-managed battery maintenance program also prolongs battery life, a benefit that looks good for the vendor.

11. Maintaining Fleet Batteries 2

Sun, 01 Mar 1998 00:00:00 +0000

<< Previous page INDEX Next page >>
11.1 The ‘Green Light’ Lies

When charging a battery, the ready light will eventually illuminate, indicating that the battery is fully charged. The user assumes that the battery has reached its full potential and the battery is taken in confidence.

In no way does the ‘green light’ guarantee sufficient battery capacity or assure good state-of-health (SoH). Similar to a toaster that pops up the bread when brown (or black), the charger fills the battery with energy and ‘pops’ it to ready when full (or warm).

The rechargeable battery is a corrosive device that gradually loses its ability to hold a charge. Many users in an organization are unaware that their fleet batteries barely last a day with no reserve energy to spare. In fact, weak batteries can hide comfortably because little demand is placed on them in a routine day. The situation changes when full performance is required during an emergency. Total collapse of portable systems is common and such breakdowns are frequently related to poor battery performance. Figure 11-1 shows five batteries in various states of degradation.

Figure 11-1: Progressive loss of charge acceptance.
The rechargeable battery is a corrosive device that gradually loses its ability to hold charge as part of natural aging, incorrect use and/or lack of maintenance. The unusable part of the battery that creeps in is referred to as ‘rock content’.

Carrying larger packs or switching to higher energy-dense chemistries does not assure better reliability if the weak batteries are not ‘weeded’ out at the appropriate time. Likewise, the benefit of using ultra-advanced battery systems offers little advantage if packs are allowed to remain in the fleet once their performance has dropped below an acceptable performance level.

Figure 11-2 illustrates four batteries with different ratings and SoH conditions. Batteries B, C and D show reduced performance because of memory problems and other deficiencies. The worst pack is Battery D. Because of its low charge acceptance, this battery might switch to ready after only 14 minutes of charge (assumed time). Ironically, this battery is a likely candidate to be picked when a fresh battery is required in a hurry. Unfortunately, it will last only for a brief moment. Battery A, on the other hand, has the highest capacity and takes the longest to charge. Because the ready light is not yet lit, this battery is least likely picked.

Figure 11-2: Comparison of charge and discharge times.
This illustration shows typical charge and discharge times for batteries with different ratings and SoH conditions. Carrying larger batteries or switching to high energy-dense chemistries does not necessarily assure longer runtime if deadwood is allowed to remain in the battery fleet.

The weak batteries are charged quicker and remain on ‘ready’ longer than the strong ones. The bad batteries tend to gravitate to the top. They become a target for the unsuspecting user. In an emergency situation that demands quick charge action, the batteries that show ready may simply be those that are deadwood.

A weak battery can be compared to a fuel tank with an indentation. Refueling this tank is quicker than a normal tank because it holds less fuel. Similar to the ‘green light’ on a charger, the fuel gauge in the vehicle will show full when filled to the brim, but the distance traveled before refueling will be short.

11. Maintaining Fleet Batteries 3

Sun, 01 Mar 1998 00:00:00 +0000

<< Previous page INDEX Next page >>
11.2 Battery Maintenance, a Function of Quality Control

The reliability of portable equipment relies almost entirely on the performance of the battery. A dependable battery fleet can only be assured if batteries are maintained on a periodic basis.

Battery maintenance also needs proper documentation. One simple method is attaching a color dot, each color indicating the month of service. A different color dot is applied when the battery is re-serviced the following month. A numbering system indicating the month of service also works well.

A better system is attaching a full battery label containing service date and capacity. Like the pending service on a car, the label shows the user when maintenance is due. For critical missions, the user will pick a battery with the highest capacity and the most recent service date. The label ensures a properly serviced replacement pack.

Battery analyzers are available that print a label revealing the organization, group, service date, expiry date (time to service the battery), battery capacity and ID number. The label is generated automatically when the battery is removed from the analyzer. Figure 11-3 illustrates such a label.

">

">11-3: Sample battery label.
The battery label keeps track of the battery in the same way a service sticker on a car reminds the owner of pending service.

The battery labeling system is simple to manage. It is self-governing in the sense that the users would only pick a battery that is properly labeled and has recently been serviced. The system does not permit batteries to fall though the cracks and be forgotten. It is in the interest of the user to ensure continued reliability by bringing in batteries with dated labels for service.
11.3 Battery Maintenance Made Simple

Several methods are available to maintain a fleet of batteries. A simple, self-governing system is illustrated in Figure 11-4 to Figure 11-6. Only 30 minutes per day should be required for a technician to maintain the system. One or several battery analyzers are needed that are capable of producing battery labels.

Figure 11-4: Sorting batteries for servicing.
Each time a battery is taken from the charger, the user checks the service date on the label attached to the battery. If the date has expired, the battery is placed in a box marked ‘To be serviced’.

Figure 11-5: Servicing expired batteries.
Batteries with expired dates are exercised; those that do not recover to the preset target capacity are reconditioned. Batteries that pass are re-certified by attaching a new label with dates and capacity reading.

Figure 11-6: Returning batteries to circulation.
After servicing, the restored batteries are returned to the charger; those that failed are replaced with new ones. Battery maintenance assures that all packs perform at the expected capacity level.

When taking a battery from the charger, the user checks the service date on the battery label. If expired, the battery is placed into the box marked: ‘To be serviced’. Periodically, the box is removed and the batteries are serviced and re-certified with a battery analyzer.

After service, the batteries are re-labeled and returned to the charger. Those batteries that fail to meet the target capacity are replaced with new packs. All batteries in the charger are now certified to meet a required performance standard.

Battery maintenance has been simplified with the introduction of battery analyzers that offer a target capacity selection. All batteries must meet a user-defined performance test or target capacity to pass. Nickel-based batteries that fall short of the required capacity are automatically restored with the analyzer’s recondition cycle. Those packs that fail to recover are subsequently replaced with new packs.

Recondition is only effective for nickel-based batteries. It is worth noting that batteries with high self-discharge and/or shorted cells cannot be corrected with recondition; neither can a battery be restored that is worn out or has been damaged through abuse.

Another group of batteries that cannot be deep discharged by recondition are ‘smart’ batteries. This includes any pack that contains a microchip that must be maintained by a continuous voltage supply. Discharging these batteries below a certain voltage point will put the battery to sleep. A recharge often fails to wake up these batteries.
11.4 Battery Maintenance as a Business

Some entrepreneurs have come up with the novel idea of providing a service to test and restore rechargeable batteries. They operate in convenient locations such as downtown cores, shopping malls and transportation hubs. Customers bring in their batteries to have them serviced. The packs are tested and reconditioned with automated battery analyzers. A full performance report is issued with each battery serviced, showing service date, performance status and the date for the next service. The suggested fee per battery is $10.00US. Higher prices can be requested on specialty batteries which are expensive to replace.

For organizations using a large number of batteries, a special pick-up and delivery service can be organized to provide scheduled maintenance. This ensures that fleet batteries used by organizations are regularly maintained. Such a service would benefit firms that do not want to bother with battery maintenance or do not have the expertise or resources to perform the task in-house.

Increasingly, dealers who sell mobile phones, laptops and camcorders also provide battery service. This activity increases traffic and helps foster good customer relations. A new battery is sold if the old one does not recover when serviced. By knowing that a battery can be checked and possibly restored, customers may try to salvage their weak batteries before investing in new ones. Some dealers may be reluctant to restore used batteries for fear of reduced battery sales.

12. Battery Maintenance Equipment

Sun, 01 Mar 1998 00:00:00 +0000

<< Previous page INDEX Next page >>
12. Battery Maintenance Equipment
With the increasing volume of batteries in circulation, battery manufacturing is outpacing the supply of suitable equipment to test these packs. This void is especially apparent in the mobile phone market where large quantities of batteries are being replaced under warranty without checking or attempting to restore them. The dealers are simply not equipped to handle the influx of returned batteries, neither is the staff trained to perform this task on a customer service level. Testing and conditioning these batteries is a complex procedure that lies outside the capabilities of most customer service clerks.
With the move to maintenance-free batteries and the need to test larger numbers of batteries, the function of battery test equipment is changing. Lengthy cycling is giving way to quick testing, improved battery preparation and better customer service. This shift in priority is especially apparent in the rapidly growing consumer market. In this chapter we examine modern battery analyzers and how they adapt to the changing needs of battery service.
12.1 Conditioning Chargers

Charging batteries is often not enough, especially when it comes to nickel-based chemistries. Periodic maintenance is needed to optimize battery life. Some innovative manufacturers offer chargers with conditioning features. The most basic charger models feature one or several bays with discharge opportunity. More advanced chargers include a display to reveal the capacity.

Some chargers offer pulse charge methods. This is done to improve charge efficiency and reduce the memory phenomenon on nickel-based batteries. Optimal charge performance is achieved by using a pulse charge that intersperses discharge pulses between charge pulses. Commonly referred to as ‘burp’ or ‘reverse load’ charge, this charge method promotes high surface area on the electrodes and helps in the recombination of the gases generated during charge.

Some manufacturers claim that the pulse charge method conditions and restores NiCd batteries and makes the periodic discharges redundant. Research carried out by the US Army has revealed that pulse charging does reduce the crystalline formation on the NiCd battery. If properly administered, batteries charged with these pulse chargers prolong service life. For batteries with advanced memory, however, the pulse charge method alone is not sufficient and a full discharge or recondition cycle is needed to break down the more stubborn crystalline formation.

12. Battery Maintenance Equipment 2

Sun, 01 Mar 1998 00:00:00 +0000

<< Previous page INDEX Next page >>
12.2 Battery Analyzers

There are two types of battery analyzers: the fixed current units and the programmable devices. While fixed current units are less expensive and generally simpler to operate, programmable analyzers are more accurate and faster. Programmable units can better adapt to different battery needs and are more effective in restoring weak batteries. One of the main advantages of the programmable battery analyzer is the ability to test the batteries against preset parameters.

Fixed current analyzers perform well in organizations that use medium size batteries ranging from 600mAh to 1500mAh. If smaller or larger batteries are serviced, the charge and discharge currents are compromised and the program time is prolonged. Here is the reason why.

A fixed current battery analyzer with a current of 600mA, for example, services a 600mAh battery in about three hours, roughly one hour for each cycle starting with charge, followed by discharge and a final charge. Servicing an 1800mAh battery would take three times as long. On the low end of the scale, a problem may arise if a 400mAh battery is serviced. This battery may not be capable of accepting a charge rate higher than 1C and the battery could be damaged.

When purchasing a battery analyzer, there is a tendency to buy according to price. With the need to service a larger volume of batteries of a wider variety, second-generation buyers find the advanced features on upscale models worth the extra cost. These features manifest themselves in reduced operator time, increased, throughput, simpler operation and the use of less trained staff. Adaptation to new battery systems is also made easier. Figure 12-1 illustrates an advanced battery analyzer.

Figure 12-1: Cadex 7400 battery analyzer
The Cadex 7400 services NiCd, NiMH, SLA and Li-ion/polymer batteries and is programmable to a wide range of voltage and current settings. Custom battery adapters simplify the interface with different battery types. A quick test program measures battery state-of-health in three minutes, independent of charge. Nickel-based batteries are automatically restored if the capacity falls below the user-defined target capacity.

An advanced battery analyzer evaluates the condition of a battery and implements the appropriate service to restore the battery’s performance. On nickel-based systems, a recondition cycle is applied automatically if a user-selected capacity level cannot be reached.

Battery chemistry, voltage and current ratings are user-programmable. These parameters are stored in interchangeable battery adapters and configure the analyzer to the correct function when the adapter is installed. In the Cadex 7000 Series battery analyzers, for example, each adapter is preprogrammed with up to ten distinct configuration codes (C-codes) to enable service for all batteries with the same footprint.

12. Battery Maintenance Equipment 3

Sun, 01 Mar 1998 00:00:00 +0000

<< Previous page INDEX Next page >>

Battery-specific adapters are available for all major batteries; user-programmable cables with alligator clips accommodate batteries for which no adapter is on hand. Batteries with shorted, mismatched or soft cells are identified in minutes and their deficiencies are displayed on the LCD panel.

User-selectable programs address different battery needs. The Cadex 7000 Series features ‘Prime’ to prepare a new battery for field use and ‘Auto’ to test and recondition weak batteries from the field. ‘Custom’ allows the setting of unique cycle sequences composed of charge, discharge, recondition, trickle charge or any combination, including rest periods and repeats.

More and more battery analyzers now measure the internal battery resistance, a feature that enables one to test a battery in a few seconds. The resistance check works best with lithium-based batteries because the level of internal cell resistance is in direct reflection to the performance. The resistance measurements can also be used for NiMH batteries but the readings do not fully disclose the battery’s condition.

One of the most powerful features offered in modern battery analyzers is battery quick testing. Within two to five minutes, reasonably accurate state-of-health (SoH) readings are available. The test is independent of the state-of-charge (SoC). Some charge is needed, however, to facilitate the test.

New requirements of battery analyzers are the ultra-fast charge and quick prime features. When a battery is inserted, the analyzer evaluates the battery, applies an ultra-fast charge if needed, and prepares the battery for service within minutes. Such a feature helps the mobile phone industry, which receives a large number of batteries under warranty. With the proper equipment, many of these presumably faulty batteries can be jump-started instead of replaced.

To accurately test batteries that power digital equipment, a modern battery analyzer is capable of discharging a battery under a simulated digital load. The GSM waveform, for example, transmits voice data in 567 ms bursts with currents of 1.5A and higher. By simulating these pulses, the performance of a battery can be tested under these field conditions. Not all analyzers are capable of simulating such short current bursts. Instead, medium-priced battery analyzers use a slower motion to accommodate the load signals. Pulse duration of 5 ms, or ten times slower than the true GSM, is commonly used.

Another application involving uneven load demand is the so-called 5-5-90 program used to simulate the runtime of analog two-way radios. The battery is loaded 5 percent of the time on transmit, 5 percent on receive and 90 percent on standby. Other combinations are 10-10-80. Each stage can be programmed to the appropriate discharge current. Because of the different load conditions, calculating the predicted runtime in the absence of a battery analyzer would be difficult.

Easy operation is an important feature of any battery analyzer. This quality is appreciated because the user is confronted with an ever-increasing number of battery types. Displaying the battery capacity in percentage of the nominal capacity rather than in milliampere-hours (mAh) is preferred by many users. With the percentage readout, the user does not need to memorize the ratings of each battery tested because this battery information is stored in the system. The percentage readout allows an added level of automation by implementing a recondition cycle if the set target capacity level cannot be reached.

Some analyzers are capable of setting the appropriate battery parameters automatically when a battery is inserted. An intelligent battery adapter reads a passive code that is imbedded in most batteries. The code may consist of a jumper, resistor or specified thermistor value. Some battery packs contain a memory chip that holds a digital code. On recognition of the battery, the adapter assigns the correct service parameters. Automatic battery identification minimizes training and allows battery service by untrained staff.

Most analyzers are capable of printing service reports and battery labels. This feature simplifies the task of keeping track of batteries. Marking batteries with the service date reminds the user when a battery is due for service. Labeling works well because the basic service history is attached right to the battery.

A battery analyzer should be automated and require minimal operator time. The task of the operator should be limited to scheduling incoming batteries for testing, marking the batteries after service, and replacing those that did not meet the performance criteria. Occasional selection of the correct current rating and chemistry may also be necessary. Properly used, a battery analyzer generates major cost savings in terms of longer battery life and more dependable service.

12. Battery Maintenance Equipment 4

Sun, 01 Mar 1998 00:00:00 +0000

<< Previous page INDEX Next page >>
12.3 Battery Analyzers for Maintenance-Free Batteries

In the past, the purpose of battery analyzers was to restore NiCd batteries affected by ‘memory’. With today’s nickel-free batteries, memory is no longer a problem and the modern battery analyzer assumes duties other than conditioning weak batteries. In an environment with nickel-free batteries, the purpose of an analyzer is shifting to performance verification, quality control, quick testing and quick priming.

12. Battery Maintenance Equipment 5

Sun, 01 Mar 1998 00:00:00 +0000

<< Previous page INDEX Next page >>
12.5 Battery Maintenance Software

Organizations servicing portable equipment need simplified battery testing. The difficulty of testing batteries is brought on by the proliferation of batteries, both in volume and diversity of models. With most standalone battery test equipment, servicing batteries with conventional methods is complex and time consuming. This task will only get more difficult as new battery models are added, almost weekly. New chemistries are being introduced which have different service requirements.

Manufacturers of battery test equipment are responding by introducing software packages that run on a PC. Many new systems enable operating the battery analyzers through a PC. Such products bring battery maintenance within reach of the untrained operator.

Cadex Batteryshop™ is a system that integrates with the Cadex 7000 Series battery analyzers. Although the analyzers are stand-alone units that can think on their own, the software overrides the analyzer to adjust the settings, and stores the test results obtained from the batteries. Figure 12-2 illustrates such a battery maintenance system.

Figure 12-2: Components of a battery maintenance system.
Cadex Batteryshop™ stores the battery test results on the database. Point and click technology programs the analyzer by selecting the battery from a listing of over 2000 commercial batteries. The system accommodates up to 120 analyzers for simultaneous service of 480 batteries.

Here are examples of how a computer-assisted battery testing system can simplify operation. To service a battery with Cadex Batteryshop™, for example, the user selects the battery model from the database, clicks the mouse, and the analyzer is automatically configured to the correct battery parameters. Programming the analyzer by scanning the bar code identifying the battery’s model number is also possible.

In the near future, the operator will be able to view a picture of the battery on a PC monitor. Clicking on the image will reveal the various models available in that battery family. Clicking on the correct model will program the analyzer.

For battery fleet operators, keeping track of a large battery fleet can be difficult, especially when observing the periodic maintenance requirements. With systems such as Cadex Batteryshop™, the battery test results can be stored in the database. This feature enables the operator to retain battery records from birth to retirement. Here is how this is done:

Each battery is marked with a permanent bar code label containing a unique battery ID number. When servicing the battery, the user scans the battery ID and the analyzer is automatically configured through the PC. All battery test results are stored and updated in the database under the assigned battery ID number. Any reference to this battery in terms of performance, maintenance history and even vendor information is available with a click of a mouse.

Delivering batteries with consistent high quality is a concern for all battery manufacturers and distributors. With advanced battery maintenance systems, battery batches can be tested and documented to satisfy quality control standards. Voltage, current and temperature information can be displayed in real-time graphics.

Cadex Batteryshop™, includes specialty programs that may not be available on other software products. For example, the program allows discharging a battery under a given pulsed current to simulate digital load requirements. Other programs include life cycling to evaluate the battery’s longevity, self-discharge tests, quick formatting and priming. The Internet allows updating the battery database to include new entries, fetching battery matrix settings for quick testing, sending battery test results to a central location, and downloading of new firmware for the Cadex 7000 Series battery analyzers.

12. Conclusions

Sun, 01 Mar 1998 00:00:00 +0000

NOTE: THIS DOCUMENT IS OBSOLETE, PLEASE CHECK THE NEW VERSION: "The Digital Audio Resampling Home Page", by Julius O. Smith III, Copyright © 2016-05-17 by Julius O. Smith III - Center for Computer Research in Music and Acoustics (CCRMA), Stanford University

<< Previous page TOC Next page >>
Conclusions
A digital resampling method has been described which is convenient forbandlimited interpolation at arbitrary times and for smoothly varyingsampling rates, and which is attractive for hardware implementation. We have presented the case which assumes uniform sampling of the input signal; however, extensions to variable sampling rates and isolated-point evaluation are straightforward.
A quantization error analysis led to the conclusion that for -bitfilter coefficients, the number of impulse-response samples stored in the filter lookup table should be on the order of times the number of ``zero-crossings'' in the impulse response, and the number of bits in the interpolation between impulse-response samples should be about . With these choices, the linear interpolation error and the error due to quantized interpolation factors are each about equal to the coefficient quantization error. A signal resampler designed according to these rules will typically be limited primarily by the lowpass filter design, rather than by quantization effects.
We note that the error analysis presented here is pessimistic in the sense that it assumes worst-case input signal conditions (e.g., a sinusoid at half the sampling rate or white noise). A different type of error analysis is possible by treating the filter coefficients as exact but subject to time jitter. In this approach, the error can be expressed in terms of the input signal Taylor series expansion, and consequently in terms of the input signal bandwidth (or maximum slope). Such an analysis reveals that for most practical signals, the quantization error is considerably less than the levels derived here.

12. Frequency Response

Sun, 01 Mar 1998 00:00:00 +0000

NOTE: THIS DOCUMENT IS OBSOLETE, PLEASE CHECK THE NEW VERSION: "Introduction to Digital Filters with Audio Applications", by Julius O. Smith III, Copyright © 2017-11-26 by Julius O. Smith III - Center for Computer Research in Music and Acoustics (CCRMA), Stanford University

<< Previous page TOC INDEX Next page >>
Frequency Response
Beginning with Eq. (2.2.1), we have

where is the -transform of the filter input, is the -transform of the output signal, and is the filter transfer function.
Definition. The frequency response of a linear time-invariant digital filter is defined to be the transfer function, , evaluated on the unit circle, that is, .
The frequency response is a complex-valued function of a real variable. The response at frequency Hz, for example, is , where is the sampling period in seconds.
Since every complex number can be represented as a magnitude and angle, the frequency response may be decomposed into two real-valued functions, the amplitude response and the phase response. Formally, we may define them as follows:
(48)
(49)

so that

Thus is the magnitude (or complex modulus) of , and is the phase (or complex angle) of .
Definition. The real valued function in Eq. (49) is called the filter amplitude response and it specifies the amplitude gain that the filter provides at each frequency.
Definition. The function is called the power response and it specifies the power gain at each frequency.
Definition. The real function in Eq. (49) is thephase response and it gives the phase shift in radians that each input component sinusoid will undergo.
If the filter input and output signals are and respectively, then
(50)
(51)

13. Exact Sinc-Interpolation of Sampled Periodic Signals

Sun, 01 Mar 1998 00:00:00 +0000

NOTE: THIS DOCUMENT IS OBSOLETE, PLEASE CHECK THE NEW VERSION: "The Digital Audio Resampling Home Page", by Julius O. Smith III, Copyright © 2016-05-17 by Julius O. Smith III - Center for Computer Research in Music and Acoustics (CCRMA), Stanford University

<< Previous page TOC Next page >>
Appendix A: Exact Sinc-Interpolation of Sampled Periodic Signals
It turns out all periodic sampled signals can be sinc-interpolatedexactly using the following formula [Schanze 1995]:

where the sampling rate is normalized to be , and the period is samples.
The first step in the derivation is the exact general formula
(14)
(15)

which follows immediately from the identity . This form can be used to develop a table-based sinc interpolation algorithm in which the function is sampled, windowed, and stored in a table over a small range of . (Reverting to the weighted sinc table is advisable near an argument of zero where there is a pole-zero cancellation in the definition of , i.e., when .) Note that when crosses , the table can be implemented as . In other words, the table between and can be computed from the table between and using a simple one-bit right-shift on the table address and the table output. If this trick is used, the table window must be applied separately, but there ways to synthesize simple windows (e.g., the Hanning or Hamming windows which consist of a single sinusoidal component) using waveform synthesis techniques, avoiding a separate table for the interpolated window function.

13. Making Battery Quick-Test Feasible

Sun, 01 Mar 1998 00:00:00 +0000

<< Previous page INDEX Next page >>
13. Making Battery Quick-Test Feasible
When Sanyo, one of the largest battery manufacturers in the world, was asked, “Is it feasible to quick test batteries?” the engineer replied decisively, “No”. He based his conclusion on the difficulty of using a universal test formula that applies to all battery applications, — from wireless communications to mobile computing, and from power tools to forklifts and electric vehicles.
Several universities, research organizations and private companies, including Cadex, are striving to find a workable solution to battery quick testing. Many methods have been tried, and an equal number have failed because they were inaccurate, inconsistent and impractical.

When studying the characteristics relating to battery state-of-health and state-of-charge (SoH and SoC, respectively) some interesting effects can be observed. Unfortunately, these properties are cumbersome and non-linear, and worst of all, the parameters are unique for every battery type. This inherent complexity makes it difficult, if not impossible, to create a formula that works for all batteries.

In spite of these seemingly insurmountable odds, battery quick testing is possible. But the question is asked, “how accurate will it be, and how well will it adapt to continuously changing battery chemistries?” The cost of a commercial quick tester and the ease-of-use are other issues of concern.
13.1 Battery Specific Quick Testing

The secret of battery quick testing lies, to a large extent, in understanding how the battery is being loaded. Battery loads vary from short current bursts for a mobile phone using the GSM protocol, to long and fluctuating loads on laptops, and to intermittent heavy loads for power tools.

Because of these differences in loads, a battery for a digital mobile phone should be tested primarily for low impedance to assure a clean delivery of the current bursts, whereas a battery for a notebook should be examined mainly for the bulk in energy reserve. Ultra-low impedance is of less importance here. A battery for a power tool, on the other hand, needs both — low impedance and good power reserve.

Some quick testers simulate the equipment load and observe the voltage signature of the battery under these conditions. The readings are compared with the reference settings, which are stored in the tester. The resulting discrepancies are calculated against the anticipated or ideal settings and displayed as the SoH readings.

The first step in obtaining quick test readings is measuring the battery’s internal resistance, often referred to as impedance. Internal resistance measurements take only a few seconds to complete and provide a reasonably accurate indication of the battery’s condition, especially if a reference reading from a good battery is available for comparison.

Unfortunately, the impedance measurement alone provides only a rough sketch of the battery’s performance. The readings are affected by various battery conditions, which cannot always be controlled. For example, a fully charged battery that has just been removed from the charger shows a higher impedance reading than one that has rested for a few hours after charge. The elevated impedance is due to the increased interfacial resistance present after charging. Allowing the battery to rest for an hour or two will normalize the battery. Temperature also affects the readings. In addition, the chemistry, the number of cells connected in series and the rating of a battery influence the results. Many batteries also contain a protection circuit that further distorts the readings.

13. Making Battery Quick-Test Feasible 2

Sun, 01 Mar 1998 00:00:00 +0000

<< Previous page INDEX Next page >>
13.2 Three-Point Quick Test

The three-point quick test uses internal battery impedance as a basis and adds the battery voltage under charge and discharge to the equation. The readings are evaluated and compared with reference settings stored in the tester. Let’s explore each of these fundamentals closer to see what it entails:

Internal resistance — To measure the impedance, a battery must be at least 50 percent charged. An empty or nearly empty battery exhibits a high internal resistance. As the battery reaches 50 percent SoH, the resistance drops, then increases again towards full discharge or full charge. Figure 13-1 shows the typical internal resistance curve of a NiMH as a function of charge. Note the decrease of impedance after the battery has rested for a while. To obtain accurate results, allow the battery to rest after discharge and charge.

Figure 13-1: Internal resistance in a NiMH battery.
Note the higher readings immediately after a full discharge and full charge. To obtain accurate results, allow the battery to rest after discharge and charge.

Charge Voltage — During charge, the voltage of a battery must follow a narrow predetermined path relating to time. Anomalies such as too high and too low voltages are identified. For example, a fast initial rise reveals that the battery may be fully charged. If the voltage overshoots, the battery may be ‘soft’. This condition often arises when one or more cells have developed dry spots. A frozen battery exhibits a similar effect. If, on the other hand, the voltage does not increase in the allotted time and remains constant, an electrical short is suspected.

Discharge Voltage — When applying a discharge, the voltage drops slightly, and then stabilizes for most of the period in which the energy is drawn. As the battery reaches the end-of-discharge point, the voltage drops rapidly. Observing the initial voltage drop and measuring the voltage delta during the flat part of the discharge curve provides some information as to the SoC. However, each battery type behaves differently and an accurate prediction is not easy. NiCd batteries that have a long flat voltage during most of the discharge period are more difficult to predict using this method than chemistries which exhibit a steady voltage drop under load.

Unfortunately, the battery’s SoC affects the three-point quick test. Even within a charge range of 50 to 90 percent, fluctuations in the test results cannot be avoided. Internal resistance readings further influence the final outcome. If used as a linear correlation with capacity, internal resistance measurements can be highly unreliable, especially with NiCd and NiMH batteries. Figure 13-2 compares the accuracy of six batteries when tested with the three-point quick test. To establish the true capacity, each battery was analyzed by applying a full charge/discharge/charge cycle.

Often referred to as the ‘Feel Good Battery Tester’ because of overly optimistic readings, the three-point quick test method fails to provide the accuracy and repeatability that serious battery users demand.

Figure 13-2: Comparison of battery quick test methods.
Six batteries with different state-of-health conditions were quick tested. The dark gray bars reflect the true state-of-health obtained with the Cadex 7000 Series battery analyzer by applying a full charge/discharge/ charge cycle; the light gray bars are readings derived using the Three-Point Quick Test.

The impression of casual battery users that this method is “better than nothing” will not stand up to the requirements of critical industries such as biomedical, law enforcement, emergency response, aviation and defense. Because of relatively low cost, the three-point tester finds a strong niche in the consumer market where a wrong reading is simply a nuisance and does not threaten human safety. Satisfactory readings are achieved in the mobile phone market where batteries are similar in format. It should be noted that the three-point quick test method provides better results than merely measuring the battery’s internal resistance or voltage.

13. Making Battery Quick-Test Feasible 3

Sun, 01 Mar 1998 00:00:00 +0000

<< Previous page INDEX Next page >>
13.3 The Evolving Battery

The Li-ion battery has not yet matured. Chemical compositions change as often as once every six months. According to Moli Energy, a large manufacturer of Li-ion batteries, the chemical composition of Li-based batteries changes every six months. New chemicals are discovered that provide better load characteristics, higher capacities and longer storage life. Although beneficial to consumers, these improvements wreak havoc with battery testing equipment that base quick test algorithms on fixed parameters. Why do these changes in battery composition affect the results of a quick tester?

The early Li-ion batteries, notably the coke-based variety, exhibited a gradual drop of voltage during discharge. With newer graphite-based Li-ion batteries, flatter voltage signatures are achieved. Such batteries provide a more stable voltage during most of the discharge cycle. The rapid voltage drop only occurs towards the end of discharge.

A ‘hardwired’ tester looks for an anticipated voltage drop and estimates the SoH according to fixed references. If the voltage-drop changes due to improved battery technology, erroneous readings will result.

Diverse metals used in the positive electrode also alter the open terminal voltage. Manganese, also referred to as spinel, has a slightly higher terminal voltage compared to the more traditional cobalt. In addition, spinel ages differently from cobalt. Although both cobalt and spinel systems belong to the Li-ion family, differences in readings can be expected when the batteries are quick tested side-by-side.

The Li-ion polymer has a dissimilar composition to the Li-ion and responds in a different way when tested. Instruments capable of checking Li-ion batteries may not provide reliable readings when quick testing Li-ion polymer batteries.
13.4 The Cadex Quicktest™ Method

A battery quick text must be capable of adapting to new chemical combinations as introduced from time to time. Cadex solves this by using a self-learning fuzzy logic algorithm. Used to measure analog variances in an assortment of applications, fuzzy logic is known to the industry as a universal approximator. Along with unique learning capabilities, this system can adapt to new trends. Similar to a student adapting to the complexity of a curriculum, the system learns with each battery tested. The more batteries that are serviced, the higher the accuracy becomes.

Cadex Quicktest™ is built on the new Cadex 7000 Series battery analyzer platform. This system features interchangeable battery adapters that contain the battery configuration codes (C-codes). When installed, the adapter sets the analyzer to the correct battery parameters (chemistry, voltage rating, etc.).

To enable quick testing, the battery adapters must also contain the matrix settings for the serviced battery. While matrices for the most common batteries are included when acquiring the adapter, the user is asked to enter the information on those adapters that have not yet been prepared for quick testing. This can be done in the field by ‘scanning’ the working battery.

The ‘Learn’ program of the Cadex 7000 Series battery analyzer performs this task by applying charge-discharge-charge activities on the test battery. Similar to downloading a program into a PC, the information derived from the battery sets the matrices and prepares the Cadex Quicktest™ function. The ‘Learn’ program completes its cycle within approximately four hours. One learning cycle is the minimal requirement to enable the Cadex Quicktest™ function.

With only one battery learned or scanned, the confidence level is ‘marginal’. Running additional batteries through the learning program will fill the matrix registers and the confidence level will increase to ‘good’ or ‘excellent’. Like a bridge that needs several pillars for proper support, the most accurate quick test results are achieved by scanning individual batteries that have SoH readings of around 100, 80 and 60 percent. The confidence level attained for a given battery adapter is indicated on the LCD panel of the analyzer.

The Cadex Quicktest™ can be performed with charge levels between 20 and 90 percent. Within this range, different charge levels do not affect the readings. If the battery is insufficiently charged, or has too high a charge, a message appears and the analyzer automatically applies the appropriate charge or discharge to bring the battery within testing range. Charging or discharging a battery immediately prior to taking the reading does not affect the Cadex Quicktest™ results.

The reader may ask whether the Cadex Quicktest™ system can also learn incorrectly. No — once the learning cycles have been completed for a given battery, the matrix settings are firm and resilient. Testing bad batteries will not affect the setting.

Spoilage is only possible if a number of bad batteries are purposely put through the ‘Learn’ program in an attempt to alter the existing matrix. Such would be the case when scanning a batch of batteries that have not been properly formatted, have been in prolonged storage, or are of poor quality. To protect the existing matrix from spoilage when adding learning cycles, the system checks each new vector reading for its integrity before accepting the information as a valid reference. Learned readings obtained from defective batteries are rejected.

If a battery adapter has lost its integrity as part of ‘bad learning’, the matrix setting can be erased and re-taught. As an alternative, Cadex will make recommended matrices available on the Internet. Users may also want to exchange learned matrix information with each other. Copying battery adapters by inserting a recognized adapter into the analyzer will achieve this. Another method is ‘Webcasting’ the matrices over the Internet.

13. Making Battery Quick-Test Feasible 4

Sun, 01 Mar 1998 00:00:00 +0000

<< Previous page INDEX Next page >>
13.5 How does the Cadex Quicktest work?

The first stage of the Cadex Quicktest™analysis uses a waveform to gather battery information under certain stresses, establishing probability levels for the given battery. Since there are many battery types with several interacting variables, a set of rules is applied to further evaluate the data. The results are averaged and an estimated battery capacity is predicted. The initial inference to categorize the batteries is computed from a set of specialized shapes called membership functions. These membership functions are unique to every battery model and are developed using a specialized trend-learning algorithm.

The raw data, consisting of three or more items, flows through the input layer. Vectors leading from the input layer are weighted and the derived values are passed through a function in the hidden layer. Another vector set channels the information to the output.

Figure 13-3: Flowchart of a neuro-network based on fuzzy logic.
The first three circles on the left are the inputs. The data entering is ‘fuzzified’ according to a set of curves called membership functions. A set of rules that depend on fixed knowledge is evaluated. The results of the rules are combined and distilled, or ‘defuzzified’. The result is a crisp, non-fuzzy number.

The weights are highly significant and function as the learning facility of the network. A run would proceed with a certain set of weights. If the result is off by a certain range, the weights are changed and the process is repeated until a certain number of iterations have passed or the algorithm produces the correct output.

The Cadex Quicktest™ requires less time than most other methods. While current quick test systems, such as those used in defense applications, need hundreds of learning cycles and run on large computers, the Cadex method requires minimal experience and can be performed on relatively simple hardware. Typically less than five learning cycles are necessary to achieve robust, model-specific solution sets, also known as matrices. This massive reduction in time is the result of a new self-learning algorithm that acquires numerous measures of the battery’s characteristics. The algorithm uses a unique decision-making formula that determines the best solution set for each battery model.

Of course, artificial intelligence is a complicated subject, and is beyond the scope of this book. With respect to complexity, Dr. Lofti Zadeh spoke these famous words: “As complexity rises, precise statements lose meaning and meaningful statements lose precision.”

Battery quick testing has raised the interest of manufacturers and users alike. The race is on to provide a product that is accurate, easy to use and cost effective. The true winner may not be an individual or organization that amasses the largest number of patents, but a company that can offer a product that is cost effective and truly works.
13.6 Battery Testing and the Internet

Increasingly, the Internet plays a pivotal role in battery testing. The ability to send all battery test results to a central global database is an exciting prospect. With this information on hand, battery manufacturers would be able to perform battery analysis based on battery type, geographic area and user pattern. Field failures could be identified quickly and appropriate corrections implemented.

Another application for the Internet is establishing a global database for all major battery types, complete with matrix settings. With compatible systems, users would be able to select and download battery information from a central database. Batteryshop™, a software product offered by Cadex, provides such a service. The database lists all common batteries, complete with battery specifications and matrix information. Point and click technology programs the battery analyzer to the correct battery parameters.

Collaborating with battery manufacturers enables Cadex to create the most accurate vector settings. Manufacturers welcome such a system because it reduces beta testing and puts the manufacturer in closer contact with the battery user. The aim is to reduce warranty returns and increase customer satisfaction.

Another powerful feature of the Internet is downloading new software for hardware upgrades. Since battery quick testing is still in its infancy, improved software will be made available in the future that allows upgrading existing equipment with the latest developments.

13. Making Battery Quick-Test Feasible 5

Sun, 01 Mar 1998 00:00:00 +0000

<< Previous page INDEX Next page >>
13.7 Electrochemical Impedance Spectroscopy

Electrochemical Impedance Spectroscopy (EIS) has been used for a number of years to test the SoH and SoC of industrial batteries. EIS is well suited for observing reactions in the kinetics of electrodes and batteries. Changes in impedance readings hint at minute intrusion of corrosion, which can be evaluated with the EIS methods. Impedance studies using the EIS technology have been carried out on lead acid, NiCd, NiMH, Li-ion and other chemistries. EIS test methods are also used to examine the cells on larger stationary batteries.

In its simplest manifestation, measurements of internal battery resistance can be taken by applying a load to a battery and observing the current-voltage characteristics. A secondary load of higher current is applied, again noting the voltage and current. The current and voltage relationship of the two loads can be utilized to provide the internal resistance using Ohm’s Law.

Rather than applying two load levels, an AC signal is injected. This AC voltage floats as a ripple on top of the battery DC voltage and charges and discharges the battery alternatively. The AC frequency varies from a low 100mHz to about 5kHz. 100mHz is a very low frequency that takes 10 seconds to complete a full cycle. In comparison, 5kHz completes 5000 cycles in one second. At about 1000Hz, the load behaves more like a DC resistance because the chemistry cannot follow the rapid changes between charge and discharge pulses. The information about electrolyte mass transport is ascertained at lower frequencies.

Additional information regarding the battery’s condition can be obtained by applying various frequencies. One can envision going through different layers of the battery and examining each level. Similar to tuning the dial on a broadcast radio, in which individual stations offer different types of music, so too does the battery provide different information of the internal processes. The EIS is an effective technique to analyze the mechanisms of interfacial structure and to observe the change in the formation when cycling the battery as part of everyday use.

When applying a sine wave to a battery, a phase shift between voltage and current occurs. The reactive load of the battery causes this phenomenon. The overall battery resistance consists of three resistance types: pure resistance, inductance and capacitance. Capacitance is responsible for the capacitor effect; and the inductance is accountable for the so-called magnetic field, or coil effect. The voltage on a capacitor lags behind the current. This process is reversed on a coil and the current lags behind the voltage. The level of phase shift that occurs when applying a current through a reactive load is used to provide information as to the battery’s condition.

One of the difficulties with the EIS method is interpreting the information. It is one thing to amass a large amount of data, and another to make practical use of it. Although the derived information reflects aging and other deficiencies, the readings are not universal and do not apply in the same way to all battery makes and types. Rather, each battery type generates its own set of signatures. Without a library of well-defined reference readings with which to compare, the EIS method has little meaning.

Modern technology can help. The vector settings of a given battery type can be stored in the test instrument and translated into meaningful readings by software. The readings can further be analyzed by coupling impedance spectroscopy with a fuzzy neuro-adaptive algorithm.

Electrochemical Impedance Spectroscopy is commonly used to research batteries in a lab environment. Best results are obtained on a single cell. EIS is also used in aviation and in-flight analysis of satellite batteries. Closer to earth, the EIS method examines stationary batteries for grid corrosion and water loss. Further refined, the EIS technology has the potential for wider applications, such as testing portable batteries. EIS may one day test batteries in a matter of seconds and achieve higher accuracy than current methods.

13. Phase Delay and Group Delay

Sun, 01 Mar 1998 00:00:00 +0000

NOTE: THIS DOCUMENT IS OBSOLETE, PLEASE CHECK THE NEW VERSION: "Introduction to Digital Filters with Audio Applications", by Julius O. Smith III, Copyright © 2017-11-26 by Julius O. Smith III - Center for Computer Research in Music and Acoustics (CCRMA), Stanford University

<< Previous page TOC INDEX Next page >>
Phase Delay and Group Delay
The phase response of a filter gives the radian phase shift experienced by each sinusoidal component of the input signal. Sometimes it is more meaningful to consider phase delay[Papoulis 1977].
Definition. The phase delay of an LTI filter with phase response is defined by

The phase delay gives the time delay in seconds experienced by each sinusoidal component of the input signal. For example, in filter , the phase response is which corresponds to a phase delay which is one-half sample.
More generally, if the input to a filter with frequency response is

then the output is

and it can be seen that the phase delay expresses phase response as time delay.
In working with phase delay, care must be taken to ensure all appropriate multiples of have been included in . We defined simply as the complex angle of the frequencyresponse , and this is not sufficient for obtaining a phase response which can be converted to true time delay. By discarding multiples of , as is done in the definition of complex angle, the phase delay is modified by multiples of the sinusoidal period. Since LTI filter analysis is based on sinusoids without beginning or end, one cannot in principle distinguish between ``true'' phase delay and a phase delay with discarded sinusoidal periods. Nevertheless, it is convenient to define the filter phase response as a continuous function of frequency with the property that (for real filters). This specifies a means of ``unwrapping'' the phase response to get a consistent phase delay curve.
Definition. A more commonly encountered representation of filter phase response is called the group delay, and it is defined by

For linear phase responses, the group delay and the phase delay are identical, and each may be interpreted as time delay.
For any phase function, the group delay may be interpreted as the time delay of the amplitude envelope of a sinusoid at frequency [Papoulis 1977]. The bandwidth of the amplitude envelope in this interpretation must be restricted to a frequency interval over which the phase response is approximately linear. While the proof will not be given here, it should seem reasonable when the process of amplitude envelope detection is considered. The narrow ``bundle'' of frequencies centered at the carrier frequency is translated to Hz. At this point, it is evident that the group delay at the carrier frequency gives the slope of the linear phase of the translated spectrum. But this is a constant phase delay, and therefore it has the interpretation of true time delay for the amplitude envelope.

14. Appendix B: Relation between Sinc and Lagrange Interpolation

Sun, 01 Mar 1998 00:00:00 +0000

NOTE: THIS DOCUMENT IS OBSOLETE, PLEASE CHECK THE NEW VERSION: "The Digital Audio Resampling Home Page", by Julius O. Smith III, Copyright © 2016-05-17 by Julius O. Smith III - Center for Computer Research in Music and Acoustics (CCRMA), Stanford University

<< Previous page TOC Next page >>
Appendix B: Relation between Sinc and Lagrange Interpolation
Lagrange interpolation is a well known, classical technique for interpolation [Hildebrand 1974]. Given a set of known samples, , the problem is to find the unique order polynomial which interpolates the samples. The solution can be expressed as a linear combination of elementary th order polynomials:

where

From the numerator of the above definition, we see that is an order polynomial having zeros at all of the samples except the th. The denominator is simply the constant which normalizes its value to at. Thus, we have

In other words, the polynomial is the th basis polynomial for constructing a polynomial interpolation of order over the sample points .
In the case of an infinite number of equally spaced samples, with spacing , the Lagrangian basis polynomials converge to shifts of the sinc function, i.e.,

where

The simplest argument is based on the fact that any analytic function is determined by its zeros and its value at one point. Since is zero on all the integers except , and since , it must coincide with the infinite-order Lagrangian basis polynomial for the sample at which also has its zeros on the nonzero integers and equals at.
A direct proof can be based on the equivalance between Lagrange interpolation and windowed-sinc interpolation using a ``binomial window'' [Kootsookos and Williamson 1996,Välimäki 1995]. That is, for a fractional sample delay of samples, multiply the shifted-by-, sampled, sinc function

by a binomial window

and normalize by [Välimäki 1995]

which normalizes the interpolating filter to have a unit norm, to obtain the th-order Lagrange interpolating filter

Since the binomial window converges to the Gaussian window as , and since the window gets wider and wider, approaching a unit constant in the limit, the convergence of Lagrange to sinc interpolation can be seen.

14. Non-Correctable Battery Problems

Sun, 01 Mar 1998 00:00:00 +0000

<< Previous page INDEX Next page >>
14. Non-Correctable Battery Problems
Non-correctable battery problems are those that cannot be improved through external means such as giving the battery a full charge or by applying repeated charge/discharge cycles. Deficiencies that denote the non-correctable status are high internal resistance, elevated self-discharge, electrical short of one or several cells, lack of electrolyte, oxidation, corrosion and general chemical breakdown. These degenerative effects are not only caused by normal usage and aging, but they include less than ideal field conditions and an element of neglect. The user may have poor charging equipment, may operate and store the battery in adverse temperatures and, in the case of nickel-based batteries, may not maintain the battery properly.
New battery packs are not exempt from deficiency syndromes and early failure. Some batteries may be kept in storage too long and sustain age-related damage, others are returned by the customer because of incorrect user preparation.

In this section we examine the cause of non-correctable battery problems and explore why they occur. We also look at ways to minimize premature failure.
14.1 High Self-discharge

Self-discharge is a natural phenomenon of any battery. It is not a manufacturing defect per se, although poor manufacturing practices and improper maintenance and storage by the consumer enhance the problem.

The level of self-discharge differs with each chemistry and cell design. High-performance nickel-based batteries with enhanced electrode surface area and super conductive electrolyte are subject to higher self-discharge than the standard version cell with lower energy densities. Self-discharge is non linear and is highest right after charge when the battery holds full capacity.

NiCd and NiMH battery chemistries exhibit a high level of self-discharge. If left on the shelf, a new NiCd loses about 10 percent of its capacity in the first 24 hours after being removed from the charger. The rate of self-discharge settles to about 10 percent per month afterwards. At a higher temperature, the self-discharge rate increases substantially. As a rule, the rate of self-discharge doubles with every 10°C (18°F) increase in temperature. The self-discharge of the NiMH is about 30 percent higher than that of the NiCd.

A major contributor to high self-discharge on nickel and lead-based batteries is a high cycle count and/or old age. With increased cycles, the battery plates tend to swell. Once enlarged, the plates press more firmly against the delicate separator, resulting in increased self-discharge. This is common in aging NiCd and NiMH batteries but can also be seen in lead acid systems.

Loading less active materials on the plates can reduce the plate swelling on nickel-based batteries. This improves expansion and contraction while charging and discharging. In addition, the load characteristic is enhanced and the cycle life prolonged. The downside is lower capacity.

Metallic dendrites penetrating into the separator are another cause of high self-discharge. The dendrites are the result of crystalline formation, also known as memory. Once marred, the damage is permanent. Poorly designed chargers that ‘cook’ the batteries also increase the self-discharge. High cell temperature causes irreversible damage to the separator.

While the nickel-based systems can withstand some abuse and tolerate innovative or crude charge methods, the Li-ion demands tight charging and discharging regimes. Keeping the voltage and current within firm boundaries prevents the growth of dendrites. The presence of dendrites in lithium-based batteries has more serious implications than just an increase in self-discharge — dendrites can cause an electrical short, which could lead to venting with flame.

The self-discharge of the Li-ion battery is five percent in the first 24 hours after charge and averages 1 to 2 percent per month thereafter. In addition to the natural self-discharge through the chemical cell, the safety circuit draws as much as 3 percent per month. High cycle count and aging has little effect on self-discharge on lithium-based batteries.

An SLA self-discharges at a rate of only five percent per month or 50 percent per year. Repeated deep cycling increases the self-discharge. When deep cycling, the electrolyte is drawn into the separator, resulting in a crystalline formation similar to that of a NiCd battery.

The self-discharge of a battery is best measured with a battery analyzer. The procedure starts by charging the battery. The capacity is read by applying a controlled discharge. The battery is then recharged and put on a shelf for 24 hours, after which the capacity is measured again. The discrepancy between the capacity readings reveals the level of self-discharge.

More accurate self-discharge measurements can be obtained by allowing the battery to rest for at least 72 hours before taking the reading. The longer rest period compensates for the relatively high self-discharge immediately after charge. At 72 hours, the self-discharge should be between 15 and 20 percent. The most uniform self-discharge readings are obtained after seven days. On some battery analyzers, the user may choose to adjust the desired rest periods in which the self-discharge is measured.

Research is being conducted to find a way to measure the self-discharge of a battery in minutes, if not seconds. The accuracy and repeatability of such technology is still unknown. The challenge is finding a formula that applies to all major batteries and includes the common chemistries.

14. Non-Correctable Battery Problems 2

Sun, 01 Mar 1998 00:00:00 +0000

<< Previous page INDEX Next page >>
14.2 Low Capacity Cells

Even with modern manufacturing techniques, the capacity of a cell cannot be accurately predicted. As part of the manufacturing process, each cell is measured and segregated into categories according to their inherent capacity levels. The high capacity A cells are commonly sold for special applications at premium prices; the large mid-range B cells are used for commercial and industrial applications such as mobile communications; and the low-end C cells are mostly sold in supermarkets at bargain prices. Cycling will not significantly improve the capacity of the low-end cell. When purchasing rechargeable batteries at a reduced price, the buyer should be aware of the different capacity and quality levels offered.

As part of quality control, the battery assembler should spot-check each batch of cells to examine cell uniformity in terms of voltage, capacity and internal resistance. Failing to observe these simple rules will often result in premature battery failures. When buying quality cells from a well-known manufacturer, battery assemblers are able to relax the matching requirements somewhat.
14.3 Cell Mismatch

Cell mismatch can be found in brand-new as well as aged battery packs. Poor quality control at the cell manufacturing level and inadequate cell matching when assembling the batteries cause unevenly matched cells. If only slightly off, the cells in a new pack adapt to each other after a few charge/discharge cycles, like players in a winning sports team.

A weak cell holds less capacity and is discharged more quickly than the strong one. This imbalance causes cell reversal on the weak cell if the battery is discharged below 1V/cell. The weak cell reaches full charge first and goes into heat-generating overcharge while the stronger cell still accepts charge and remains cool. In both situations, the weak cell is at a disadvantage, making it weaker and contributing to a more acute cell mismatch condition. An analogy can be made with a high school bully who picks on the weaker kid.

High quality cells are more consistent in capacity than lower quality counterparts. During their life span, high quality cells degrade at about the same rate, helping to maintain the matching. Manufacturers of power tools choose high quality cells because of their durability under heavy load conditions and temperature extremes. Lower-cost cells have been tried, but early failure and consequent replacement is costlier than the initial investment.

The capacity matching between the cells in a battery pack should be within +/- 2.5 percent. Tighter tolerances are required on batteries with high cell counts that also must generate high load currents and are operating under adverse temperatures. There is a strong correlation between well-balanced cells and the longevity of a battery.

Lithium-based cells have tighter matching tolerances than their nickel-based cousins. Tight matching of all cells in a pack is especially important on lithium-based chemistries. All cells must reach the end-of-discharge voltage threshold at the same time. The full-charge point must be attained in unison by all cells. If the cells are allowed to get out of match, the weaker cell will be discharged to a lower voltage point before the cut-off occurs. On charge, this weak cell will attain the full-charge status before the others, causing the voltage to go higher than on the stronger cells. This larger voltage swing will put undue strain on the weak cell.

Each cell in a lithium-based pack is electronically monitored to assure proper cell matching during the battery’s life. An electronic circuit is added to some packs that compensate the differences in cell voltages. This is done by connecting a shunt across each cell string to consume the excess energy of the cells which are more energetic. The low-voltage cut-off occurs when the weakest cell reaches the end-of-discharge point.

The Li-ion battery is controlled down to the cell level to assure safety at all times. Because this chemistry is still relatively new and unpredictable under extreme conditions, manufacturers do not want to take undue risks. There have been a few failures but such irregularities are often kept a secret. This chemistry is considered very safe, considering the large number of Li-ion batteries that are in use.

14. Non-Correctable Battery Problems 3

Sun, 01 Mar 1998 00:00:00 +0000

<< Previous page INDEX Next page >>
14.4 Shorted Cells

Manufacturers are often unable to explain why some cells develop high electrical leakage or an electrical short while the batteries are still relatively new. There are a number of possible reasons that contribute to this irreversible form of cell failure.

The suspected culprit is foreign particles that contaminate the cells during manufacture. Another possible cause is rough spots on the plates that damage the separator. Better quality control at the raw material level and minimal human interface during the manufacturing process has greatly reduced the ‘infant mortality’ rate of the modern rechargeable cells.

Cell reversal caused by deep discharging also contributes to shorted cells. This commonly occurs if a nickel-based battery is being fully depleted under a heavy load. A NiCd battery is designed with some reverse voltage protection and a small reverse current in the magnitude of milliamperes can be tolerated. A high current, however, causes the reversed-polarized cell to develop a permanent electrical short. Another cause of a short circuit is marring the separator through uncontrolled crystalline formation.

Applying momentary high-current bursts in an attempt to repair shorted cells has had limited success. The short may temporarily evaporate but the damage to the separator material remains. The repaired cell often exhibits a high self-discharge and the short frequently returns.

Replacing a shorted cell in an aging pack is not recommended unless the new cell is matched with the others in terms of voltage and capacity. Otherwise, an imbalance may occur. One may remember the biblical verse “No one puts a patch of unshrunken cloth on an old garment. . .” or “No man would put new wine into old wineskins. . .” (Mt 9.16-17). Attempts to replace faulty cells have commonly lead to battery failures after about six months of use. It is best not to disturb the cells in a battery pack but allow them to age naturally. Maintaining the batteries while they are still in good working condition will help to prevent premature failure.

Shorts in a Li-ion cell are uncommon. Protection circuits monitor an ailing Li-ion cell and render the pack unusable if serious voltage irregularities are detected. Charging such a pack would (protection circuit permitting) generate excess heat. The battery’s temperature control circuits are designed to terminate the charge.
14.5 Loss of Electrolyte

Although sealed, battery cells may lose some electrolyte during their life. Typical loss of moisture occurs if the seal opens due to excessive pressure. This occurs if the battery is charged at very low or very high temperatures. Once vented, the spring-loaded seal of nickel-based cells may never properly close again, resulting in a deposit of white powder around the seal opening. Losses may also occur if the cell cap is not correctly sealed in the manufacturing process. The loss of electrolyte results in a decrease of capacity, a defect that cannot be corrected.

Permeation, or loss of electrolyte in sealed lead acid batteries, is a recurring problem. Overcharge is the main cause. Careful adjustment of charging and float voltages reduces loss of electrolyte. In addition, the battery should operate at moderate temperatures. Air-conditioning is a prerequisite for VRLA batteries, especially in warmer climates.

Replenishing lost liquid in VRLA batteries by adding water has had limited success. Although lost capacity can often be regained with a catalyst, the performance of the stack is short-lived. After tampering with the cells, it was observed that the battery stack turned into high maintenance mode and needed to be closely supervised.

A properly designed, correctly charged Li-ion cell should never generate gases. As a result, the Li-ion battery does not lose electrolyte through venting.

But in spite of what is being said, the lithium-based cells can build up an internal pressure under certain conditions. Provisions are made to maintain safety of the battery and equipment should this occur. Some cells include an electrical switch that opens if the cell pressure reaches a critical level. Other cells feature a membrane that safely releases the gases if need be. Controlled release of the pressure prevents bulging of the cell during pressure buildup.

Most of the safety features of lithium-based batteries are one-way; meaning that once activated, the cells are inoperable thereafter. This is done for safety reasons.

14. Vector Space Concepts

Sun, 01 Mar 1998 00:00:00 +0000

NOTE: THIS DOCUMENT IS OBSOLETE, PLEASE CHECK THE NEW VERSION: "Introduction to Digital Filters with Audio Applications", by Julius O. Smith III, Copyright © 2017-11-26 by Julius O. Smith III - Center for Computer Research in Music and Acoustics (CCRMA), Stanford University

<< Previous page TOC INDEX Next page >>
Vector Space Concepts
Definition. A set of objects is called a metric spaceif with any two points and of there is associated a real number, called the distance from to , such that (a) if ; , (b) , (c) , for any [Rudin 1964].
Definition. A linear space is a set of ``vectors'' together with a field of ``scalars'' with an addition operation , and a multiplication opration taking , with the following properties: If , , and are in , and are in , then
.
.
There exists in such that for all in .
.
.
.
.
The element is written as thus coinciding with the notation for the real number zero. A linear space is sometimes be called a linear vector space, or a vector space.
Definition. A normed linear space is a linear space on which there is defined a real-valued function of called a norm, denoted , satisfying the following three properties:
, and .
, a scalar.
.
The functional serves as a distance function on , so a normed linear space is also a metric space.
Note that when is the space of continuous complex functions on the unit circle in the complex plane, the norm of a function is not changed when multiplied by a function of modulus on the unit circle. In signal processing terms, the norm is insensitive to multiplication by a unity-gain allpass filter (also known as a Blaschke product).
Definition. A pseudo-norm is a real-valued function of satisfying the following three properties:
, and .
, a scalar.
.
A pseudo-norm differs from a norm in that the pseudo-norm can be zero for nonzero vectors (functions).
Definition. A Banach Space is a complete normed linear space, that is, a normed linear space in which every Cauchy sequence²*A sequence is said to be a Cauchy sequence if for each there is an such that for all and larger than . converges to an element of the space.
Definition. A function is said to belong to the space if

Definition. A function is said to belong to the space if it is in and if its analytic continuation is analytic for. is said to be in if .
Theorem. (Riesz-Fischer) The spaces are complete.Proof. See Royden [Royden 1968], p. 117.
Definition. A Hilbert space is a Banach space with a symmetric bilinearinner product defined such that the inner product of a vector with itself is the square of its norm .

15. Bibliography

Sun, 01 Mar 1998 00:00:00 +0000

NOTE: THIS DOCUMENT IS OBSOLETE, PLEASE CHECK THE NEW VERSION: "The Digital Audio Resampling Home Page", by Julius O. Smith III, Copyright © 2016-05-17 by Julius O. Smith III - Center for Computer Research in Music and Acoustics (CCRMA), Stanford University

<< Previous page TOC Next page >>
Bibliography
Crochiere and Rabiner 1983
Crochiere, R., and L. R. Rabiner. 1983.
Multirate Digital Signal Processing.
Englewood Cliffs, NJ: Prentice-Hall, Inc.
Digital Signal Processing Committee 1979a
Digital Signal Processing Committee (ed). 1979a.
Programs for Digital Signal Processing.
New York: IEEE Press.
Digital Signal Processing Committee 1979b
Digital Signal Processing Committee (ed). 1979b.
Programs for Digital Signal Processing II.
New York: IEEE Press.
Goldstein 1967
Goldstein, A. A. 1967.
Constructive Real Analysis.
New York: Harper and Row.
Hildebrand 1974
Hildebrand, F. B. 1974.
Introduction to Numerical Analysis.
New York: Dover. second edition.
Kaiser 1974
Kaiser, J. F. 1974.
``Using the Window Function.''
IEEE Transactions on Circuits and Systems--I: Fundamental Theory and Applications, April 22-25:20-23. reprinted in [Digital Signal Processing Committee 1979b], pp. 123-126.
Kootsookos and Williamson 1996
Kootsookos, P. J., and R. C. Williamson. 1996.
``FIR Approximation of Fractional Sample Delay Systems.''
IEEE Transactions on Circuits and Systems--II: Analog and Digital Signal Processing, 43(2):40-48.
Laakso et al. 1996
Laakso, T. I., V. Välimäki, M. Karjalainen, and U. K. Laine. 1996.
``Splitting the Unit Delay--Tools for Fractional Delay Filter Design.''
IEEE Signal Processing Magazine, 13(1):30-60.
Putnam and Smith 1997
Putnam, W., and J. O. Smith. 1997.
``Design of Fractional Delay filters Using Convex Optimization.''
In: Proceedings of the IEEE Workshop on Applications of Signal Processing to Audio and Acoustics, New Paltz, NY.
New York: IEEE Press.
Rabiner and Gold 1975
Rabiner, L. R., and B. Gold. 1975.
Theory and Application of Digital Signal Processing.
Englewood Cliffs, NJ: Prentice-Hall, Inc.
Rossum 1993
Rossum, D. 1993.
``Constraint Based Audio Interpolators.''
Proceedings of the IEEE Workshop on Applications of Signal Processing to Audio and Acoustics, New Paltz, NY.
Schafer and Rabiner 1973
Schafer, R. W., and L. R. Rabiner. 1973.
``A Digital Signal Processing Approach to Interpolation.''
Proceedings of the IEEE, 61(June):692-702.
Schanze 1995
Schanze, T. 1995.
``Sinc Interpolation of Discrete Periodic Signals.''
IEEE Transactions on Signal Processing, 43(6):1502-1503.
Smith 1992
Smith, J. O. 1992 (Winter).
The Window Method for Digital Filter Design. mathematica notebook for Music 420 (EE 367A), available online as ftp://ccrma-ftp.stanford.edu/pub/DSP/Tutorials/Kaiser.ma.Z.
Smith and Gossett 1984
Smith, J. O., and P. Gossett. 1984.
``A Flexible Sampling-Rate Conversion Method.''
Pages 19.4.1-19.4.2 of: Proceedings of the International Conference on Acoustics, Speech, and Signal Processing, San Diego, vol. 2.
New York: IEEE Press. an expanded tutorial based on this paper and associated free software are available online at The Digital Audio Resampling Home Page: https://ccrma.stanford.edu/~jos//resample/.
Välimäki 1995
Välimäki, V. 1995 (Dec.).
Discrete-Time Modeling of Acoustic Tubes Using Fractional Delay Filters.
Ph.D. thesis, Report no. 37, Helsinki University of Technology, Faculty of Electrical Engineering, Laboratory of Acoustic and Audio Signal Processing, Espoo, Finland.

15. Caring for Your Batteries from Birth to Retirement

Sun, 01 Mar 1998 00:00:00 +0000

<< Previous page INDEX Next page >>
15. Caring for Your Batteries from Birth to Retirement
It is interesting to observe that batteries cared for by a single user generally last longer than those that operate in an open fleet system where everyone has access to, but no one is accountable for them. There are two distinct groups of battery users — the personal user and the fleet operator.
A personal user is one who operates a mobile phone, a laptop computer or a video camera for business or pleasure. He or she will most likely follow the recommended guidelines in caring for the battery. The user will get to know the irregularities of the battery. When the runtime gets low, the battery often gets serviced or replaced. Critical failures are rare because the owner adjusts to the performance of the battery and lowers expectations as the battery ages.

The fleet user, on the other hand, has little personal interest in the battery and is unlikely to tolerate a pack that is less than perfect. The fleet user simply grabs a battery from the charger and expects it to last through the shift. The battery is returned to the charger at the end of the day, ready for the next person. Little or no care is given to these batteries. Perhaps due to neglect, fleet batteries generally have a shorter service life than those in personal use.

How can fleet batteries be made to last longer? An interesting contrast in the handling of fleet batteries can be noted by comparing the practices of the US Army and the Dutch Army, both of which use fleet batteries. The US Army issues batteries with no maintenance program in place. If the battery fails, another pack is issued. Little or no care is given and the failure rate is high.

The Dutch Army, on the other hand, has moved away from the open fleet system by making the soldiers responsible for their batteries. This change was made in an attempt to reduce battery waste and improve reliability. The batteries are issued in the soldier’s name and the packs become part of their personal belongings. The results are startling. Since the Dutch Army adapted this new regime, the failure rate has dropped considerably and, at the same time, battery performance has increased. Unexpected down time has almost been eliminated.

It should be noted that the Dutch Army uses exclusively NiCd batteries. Each pack receives periodic maintenance to prolong service life. Weak batteries are systematically replaced. The US Army, on the other hand, uses NiMH batteries. They are evaluating the Li-ion polymer for the next generation battery.

Because of the high failure rate of fleet batteries and the uncertain situations such failures create, some organizations assign a person to maintain batteries. This person checks all batteries on a scheduled basis, exercises them for optimum service life, and replaces those that fall below an accepted capacity level and do not recover with maintenance programs. Batteries perform an important function; giving them the care they deserve is appropriate.

15. Caring for Your Batteries from Birth to Retirement 2

Sun, 01 Mar 1998 00:00:00 +0000

<< Previous page INDEX Next page >>
15.1 Storage

Batteries are a perishable product and start deteriorating right from the time they leave the manufacturing plant. For this reason, it is not advisable to stock up on batteries for future use. This is especially true with lithium-based batteries. The buyer should also be aware of the manufacturing date. Avoid acquiring old stock.

Keep batteries in a cool and dry storage area. Refrigerators are recommended, but freezers must be avoided because most battery chemistries are not suited for storage in sub-freezing temperatures. When refrigerated, the battery should be placed in a plastic bag to protect it against condensation.

The NiCd battery can be stored unattended for five years and longer. For best results, a NiCd should be fully charged, then discharged to zero volts. If this procedure is impractical, a discharge to 1V/cell is acceptable. A fully charged NiCd that is allowed to self-discharge during storage is subject to crystalline formation (memory).

Most batteries are shipped with a state-of-charge (SoC) of 40 percent. After six months storage or longer, a nickel-based battery needs to be primed before use. A slow charge, followed by one or several discharge/charge cycles, will do. Depending on the duration of storage and temperature, the battery may require two or more cycles to regain full performance. The warmer the storage temperature, the more cycles will be needed.

The Li-ion does not like prolonged storage. Irreversible capacity loss occurs after 6 to 12 months, especially if the battery is stored at full charge and at warm temperatures. It is often necessary to keep a battery fully charged as in the case of emergency response, public safety and defense. Running a laptop (or other portable device) continuously on an external power source with the battery engaged will have the same effect. Figure 15-1 illustrates the recoverable capacity after storage at different charge levels and temperatures.

The combination of a full charge condition and high temperature cannot always be avoided. Such is the case when keeping a spare battery in the car for a mobile phone. The NiMH and Li-ion chemistries are most severely affected by hot storage and operation. Among the Li-ion family, the cobalt has an advantage over the manganese (spinel) in terms of storage at elevated temperatures.

Temperature 40% charge level
(recommended storage charge level) 100% charge level
(typical user charge level)

0°C 98% after 1 year 94% after 1 year

25°C 96% after 1 year 80% after 1 year

40°C 85% after 1 year 65% after 1 year

60°C 75% after 1 year 60% after 3 months

Figure 15-1: Non-recoverable capacity loss on Li-ion batteries after storage.
High charge levels and elevated temperatures hasten the capacity loss. Improvements in chemistry have increased the storage performance of some Li-ion batteries.

The recommended storage temperature of a lithium-based battery is 15°C (59°F) or less. A charge level of 40 percent allows for some self-discharge that naturally occurs; and 15°C is a practical and economical storage temperature that can be achieved without expensive climate control systems.

While most rechargeable batteries cannot be stored at freezing temperatures, some newer commercial Li-ion batteries can be kept at temperatures of -40°C without apparent side effects. Such temperature tolerances enable long and cost-effective storage in the arctic.

The SLA battery can be stored for up to two years but must be kept in a charged condition. A periodic topping charge, also referred to as ‘refreshing charge’, is required to prevent the open cell voltage from dropping below 2.10V. (Depending on the manufacturer, some lead acid batteries may be allowed to drop to lower voltage levels). When self-discharged below a critical voltage threshold, sulfation occurs on most lead acid batteries. Sulfation is an oxidation layer on the negative plate that alters the charge and discharge characteristics. Although cycling can often restore the capacity loss, the battery should be recharged before the open cell voltage drops below 2.10V.

The SLA cannot be stored below freezing temperatures. Once a pack has been frozen, it is permanently damaged and its service life is drastically reduced. A previously frozen battery will only be able to deliver a limited number of cycles.

15. Caring for Your Batteries from Birth to Retirement 3

Sun, 01 Mar 1998 00:00:00 +0000

<< Previous page INDEX Next page >>
15.2 Priming

Some nickel-based batteries do not perform well when new. This deficiency is often caused by lack of formatting at the time of manufacturing. Batteries that are not sufficiently formatted are destined to fail because the initial capacity is low. The full potential is only reached after the battery has been cycled a few times. In many cases, the user does not have the patience to wait until the expected performance is reached. Instead, the customer exercises the warranty return option.

The most critical time in a battery’s life is the so-called priming stage. An analogy can be drawn with breaking in a new car engine. The performance and fuel efficiency may not be best at first, but with care and attention, the engine will improve over time. If overstressed when new, the engine may never provide the economical and dependable service that is expected.

Some poorly formatted batteries are known to produce less than 10 percent of capacity at the initial priming stage. By cycling, the capacity increases, and the battery will become usable after three to five cycles. Maximum performance on a NiCd, for example, is reached after 50 to 100 full charge/discharge cycles. This priming function occurs while the battery is being used. The gradual capacity increase during the early life of a battery is normally hidden to the user.

Quality cells from major Japanese manufacturers do not need extended priming and can be used almost immediately. After five full cycles, the performance is predictable and fully repeatable.

The manufacturer’s recommended priming procedure should be followed. In many cases, a 24-hour trickle charge is needed. Verifying the performance with a battery analyzer is advisable, especially if the batteries are used for critical applications.

Some nickel-based batteries are known to form a passivation layer if kept in prolonged storage. Little scientific knowledge is available on this subject and the battery manufacturers may deny the existence of such a layer. A full charge/discharge, followed by a complete recharge corrects the problem.

Li-ion cells need less priming than the nickel-based equivalent. Manufacturers of Li-ion cells insist that priming is not a requirement. The priming function on the Li-ion may be used to verify that the battery is fully functional and produces the capacity required.

In an earlier chapter, the question “Why are excessive quantities of batteries being returned under warranty?” was raised. This question has not been fully answered. It appears that all battery chemistries are represented among the packs being returned. It is unclear whether these batteries are inoperable as claimed. Perhaps the liberal warranty return offered by dealers provides an opportunity to acquire a new, and seemingly better, battery without charge. Some misuse of the warranty policy cannot be fully dismissed.

The internal protection circuit of lithium-based batteries may be the cause of some problems. For safety reasons, many of these batteries do not allow a recharge if the battery has been discharged below 2.5V/cell. If discharged close to 2.5V and the battery is not recharged for a while, self-discharge further discharges the pack below the 2.5V level. If, at this time, the battery is put into the charger, nothing may happen. The battery appears to have an open circuit and the user consequently demands a replacement.

Cadex has received a large number of supposedly dead Li-ion polymer batteries from various manufacturers. When measured, these batteries had no voltage at the terminals and appeared to be dead. Charging the packs in their respective chargers was unsuccessful. But after waking up the battery’s control circuit with the ‘Boost’ function of the Cadex 7000 Series battery analyzer, most of these batteries accepted normal charge. After a full charge, the performance was checked. Almost all packs reached capacities of 80 percent and higher and the batteries were returned to service.

15. Caring for Your Batteries from Birth to Retirement 4

Sun, 01 Mar 1998 00:00:00 +0000

<< Previous page INDEX Next page >>
15.3 The Million Dollar Battery Problem

In today’s surging mobile phone market, many batteries are returned to mobile phone carriers before the ink on the invoice has dried. The most common consumer complaint is ‘less than expected’ runtime.

The reasons for this failure are multi-fold. The battery may not have been properly formatted at the factory. Perhaps the packs remained on the shelf too long or have been discharged too low. Incorrect customer preparation is also to blame. The true reason for such failure may never be known.

Dealers are not equipped to handle the influx of returned batteries. To fulfill the warranty obligations and satisfy the customer, the dealer hands out a new battery and sends the faulty pack to the manufacturer. Truckloads of ‘worthless’ batteries are transported, only to be stockpiled in warehouses for eventual testing or recycling at the manufacturer’s expense. The cost of exchange, time lost by retail staff, shipping, warehousing and eventual disposal amounts to a million dollar problem.

On a recent visit to Europe, a Cadex staff member learned that a large phone manufacturer had received 17 tons of failed handset batteries in one year alone. The batteries were stockpiled in large barrels for recycling. He also discovered that 15,000 NiMH batteries were returned to the manufacturer within weeks after the release of a new phone. When spot-checking the failed batteries with a Cadex 7000 Series battery analyzer, most packs appeared to be operational.

On another occasion, a total of 14,000 Li-ion batteries were returned to a North American mobile phone provider. Of these, only 700 (or 5 percent), were faulty. Of these, ten random batteries were sent to Cadex for further testing. The Cadex lab reported that each of these failed packs indeed had genuine faults.

A European service center sent 40 Li-ion polymer batteries to Cadex for evaluation. These packs had failed in the field and were returned to the service center by customers. When servicing the batteries on a Cadex 7000 Series battery analyzer, 37 units were found to be fully functional with capacities of above 80 percent and impedances below 180mW.

Phone manufacturers report that 80 to 90 percent of returned batteries have no faults or can easily be repaired with battery analyzing equipment. The remaining 10 to 20 percent, which do not easily recover with basic service, can often be restored with extended programs. Only a small percentage of batteries returned under warranty exhibit non-correctable faults.

Not all batteries and portable equipment under warranty fail due to manufacturer’s defects. A service manager for a major mobile phone manufacturer hinted that submersion into a cup of coffee or soft drink is a sizable contributor to equipment and battery failures. Apparently, the acids in the beverages manage to corrode the electrical conductors. Submersion into coffee occurs when the user mistakes the coffee cup for the phone cradle.

In an effort to salvage returned batteries, a leading mobile phone manufacturer segregates battery packs according to purchase date. Packs returned within the thirty-day warranty period are marked as type B. The batteries are then sent to a regional service center where they are serviced with battery analyzers. If the batteries are clean, (have no coffee residue) and regain a capacity of 80 percent or higher, the packs are relabeled and sold as a B class product. Over 90 percent of their returned batteries have been reclaimed with this program.

On the strength of this success, some battery-refurbishing houses have extended the service to include batteries of up to one year old. The service center experiences a 40 to 70 percent restoration yield in repairing these older batteries. The battery-refurbishing centers are said to make a profit. Equally important, such programs reduce the environmental impact of battery disposal.

15. Caring for Your Batteries from Birth to Retirement 5

Sun, 01 Mar 1998 00:00:00 +0000

<< Previous page INDEX Next page >>
15.4 To the Service Counter, and No Further

Not all manufacturers and dealers offer battery-refurbishing centers. If not available, a program is gaining popularity in which the battery is serviced at the store level. When a customer returns a faulty battery, the pack goes no further than the store that sold the equipment.

The customer service clerk checks the battery on site with approved test equipment. An attempt is made to restore the battery. If not successful and a warranty replacement is needed, a service report is issued, which is sent to the manufacturer by fax or e-mail. After verifying the report, the manufacturer offers replacement batteries as part of the warranty replacement policy.

Warranty replacement can be further streamlined by using the Internet and compatible battery analyzers. Such a process will operate with a minimum of human resources and run independent of office hours and time zones. Here’s how it works:

The manufacturer first sends each participating store an appropriate number of replacement batteries. When a customer returns a faulty battery, service personnel test the pack with the in-store analyzer. If restoration is unsuccessful, the analyzer e-mails a report to the manufacturer, stating the nature of the deficiency. Other information, such as the date of purchase, battery type and customer name are also included. The computer at the manufacturer’s headquarters verifies the claim and, if valid, issues an inventory adjustment against the spare batteries allocated to the store. When the stock gets low, a re-stocking order is generated and additional batteries are sent out automatically.

Besides lowering overhead costs, a fully integrated warranty replacement system provides the manufacturer with accurate information regarding the nature of battery failures. User patterns leading to battery failure can be evaluated by geographic region. For example, a temperature related failure might be more likely to occur in warm climates than in cool ones. Batteries with higher temperature resiliency can be allocated for these regions. Recurring problems can be identified quickly and corrective measures implemented within months rather than years. Such measures can be as simple as providing the customer with better operating instructions in preparing a new battery before use.

One of the most difficult problems in servicing batteries at store-level is a lack of technical know-how by the customer service personnel. With the ever-increasing number of battery models, the task of identifying a battery type and setting the correct parameters is becoming increasingly more complex. Technology is not keeping pace in supplying the battery market with suitable test equipment that is both cost effective and easy to use.

To bring battery testing within reach of the untrained user, battery analyzers must be simple to operate and allow easy interface with all major battery types. Setting the correct battery parameters should be clear and concise. Uncertainties that can lead to errors must be minimized. The manufacturer of the battery test equipment should be aware that the task of operating a battery analyzer is not part of the clerk’s job description.

The Batteryshop™ software by Cadex has been developed for the purpose of simplifying battery maintenance. When installed in a PC, the operator simply selects the desired battery from the database of over 2000 battery listings. With the Cadex 7000 Series connected to the PC, the analyzer programs itself to the correct parameters with the click of the mouse. The user only needs to insert the battery into the appropriate battery adapter and everything else is done automatically.

Some batteries, such as those manufactured by Motorola, are equipped with bar code labels. If bar coded, the user can simply scan the bar code label and insert the battery into the analyzer. Here is how it works:

The scanned battery model number is matched with the battery listing in the database. Cadex Batteryshop™ then assigns the appropriate battery configuration code (C-code) to the battery and downloads it to the Cadex 7000 Series. The analyzer is now programmed to the correct parameters, ready to service the battery.

Not all battery packs come with bar code identification. If not available, a label printer connected to the PC can generate the missing bar code. These labels can be attached to a separate sheet on the service counter. The bar code labels may also be placed next to an illustration of the battery. The clerk simply refers to the correct battery and scans the bar code label associated with the battery. The system is now set to service the battery.

In the near future it will be possible to view the picture of the battery on the PC monitor. Clicking the mouse on the image will reveal all model numbers associated with this battery. A click on the correct model will program the analyzer.

When training global staff, simplification and automation make common sense. With tools now available that do the thinking, employees no longer need to be battery experts. Similar to a checkout clerk in a supermarket who, in the pre-computer days, required full product knowledge can now rely on the embedded bar code information. The price of all items purchased is flashed on the screen and an up-to-the-second inventory status is available. Such simplifications are also possible in servicing commercial batteries.

15. Caring for Your Batteries from Birth to Retirement 6

Sun, 01 Mar 1998 00:00:00 +0000

<< Previous page INDEX Next page >>
15.5 The Quick Fix

Checking a battery and assessing its status within a few minutes is one thing — finding a solution and actually fixing the problem is another. Increasingly, customers and dealers alike are seeking an alternative solution to replacing the batteries under warranty. They want a quick fix.

Fully automated test procedures are being developed which check the battery and apply a quick-prime program to wake up a sleeping battery. The program will last from a few minutes for an easy wake-up call, to an hour or longer for the deep-sleepers.

Batteries with minor deficiencies will be serviced while the customer enjoys a cup of coffee or browses through the store. If the battery has an electrical short or does not accept a charge, the likelihood of revitalizing the battery is slim. This pack is eliminated within seconds to clear the test equipment for other batteries. If a pack requires extensive priming, which will take a few hours to complete, the customer is asked to come back later.

Some battery analyzers indicate the estimated service time after the battery has passed through the early assessment stages. The customer can decide to wait, buy a spare battery, or come back for the repaired battery the next day.

A complete battery cycle offers the best service. Such a service makes optimal use of the restorative abilities of a battery analyzer. A full cycle may take five to eight hours and can be applied overnight. Multi-bay analyzers that service several batteries at the same time increase the throughput. Such analyzers operate 24 hours without user intervention.

A customer may not have time to wait for the outcome of a battery test. The prospect of having to buy a new battery is even less appealing. In such a case, a class B or replacement battery may be the answer. This pack can be drawn from a pool of refurbished batteries, which the store has built up from previous returns. This could become a lucrative side business as customers begin to realize the cost saving potential, especially if the battery is accompanied by a performance report.

Some battery analyzers offer ultra-fast charge functions. The maximum permissible charge current that can be applied to a battery is dictated by the battery’s ability to absorb charge. A fit battery, or one that has a partial charge, would charge to the 70 percent level in 30 minutes or less. A 70 percent charge level is often sufficient to complete a performance test or quick-fill the battery for a hurried customer. The topping charge from there to full charge is what demands the long charge time.

Some late model battery analyzers also offer a quick priming program that services a battery in a little more than an hour. This program applies an ultra-fast charge and ultra-fast discharge to check the integrity of the battery. By virtue of cycling, some priming and conditioning activities occur.

Customers demand a quick turnaround when a mobile phone fails. Manufacturers and service providers realize that better methods are needed to handle customer returns. The expensive and wasteful battery exchange policies practiced today may no longer be acceptable in the future. Fierce competition and tight product margins are part of the reason. Returned batteries account for a considerable after-sale burden. With modern technology, these costs can be reduced while improving customer service and enhancing satisfaction.

15. Caring for Your Batteries from Birth to Retirement 7

Sun, 01 Mar 1998 00:00:00 +0000

<< Previous page INDEX Next page >>
15.6 Battery Warranty

Some manufacturers of industrial batteries provide warranties of up to 18 months. A free exchange is offered if the battery fails to meet 80 percent of the rated capacity throughout the warranty period. (I hasten to mention that these warranty policies apply to markets other than mobile phones.)

But what happens if such a battery is returned for warranty? Will the dealer replace the pack without hesitation? Rarely.

With lack of battery standards, manufacturers are free to challenge warranty claims, even if a genuine problem exists. Many batteries reveal only the chemistry and voltage on the label and do not make reference to the milliampere-hour rating (mAh). How does the user know what capacity rating to use when testing the battery? What performance standards can be applied?

On battery packs that show the mAh rating, some battery manufacturers may have used the peak capacity rating. This is done for promotional reasons to make their packs look better than the competitor’s. Peak capacity is based on a lower discharge rate because a battery produces higher readings if discharged slowly. For warranty purposes, a discharge of 1C should be used.

Regulatory authorities stress the importance of marking all batteries with the average capacity rating. Portable batteries with a capacity of up to about 2A should be rated at a 1C discharge. Batteries above that capacity may be rated at 0.5C. No true standard exists in term of capacity rating.

With the increased popularity of battery analyzers, battery manufacturers and dealers are urged to follow industry-accepted standards regarding battery ratings. In an attempt to lower warranty claims, some battery manufacturers have moderated the published ratings of some batteries to be more consistent with reality.

Manufacturers are concerned about the high cost of providing free replacement batteries and disposing of returned units. If a battery analyzer is used, failures due to fading capacity can mostly be corrected. Warranty claims are exercised only on those packs that develop a genuine failure. If fewer batteries returned, the vendor can offer better pricing.

15. Caring for Your Batteries from Birth to Retirement 8

Sun, 01 Mar 1998 00:00:00 +0000

<< Previous page INDEX Next page >>
15.7 Battery Recycling

Even though the emphasis in battery research has shifted away from NiCd to newer technologies, the NiCd battery continues to be one of the most used rechargeable batteries. Over 75 million NiCd batteries were sold in the US during the year 2000. Market reports indicate that the demand of NiCd batteries is expected to rise six percent per year until 2003. The demand for other chemistries, such as the NiMH and Li-ion family, is rising at a more rapid pace. Where will the mountains of batteries go when spent? The answer is recycling.

15. Specific Norms

Sun, 01 Mar 1998 00:00:00 +0000

NOTE: THIS DOCUMENT IS OBSOLETE, PLEASE CHECK THE NEW VERSION: "Introduction to Digital Filters with Audio Applications", by Julius O. Smith III, Copyright © 2017-11-26 by Julius O. Smith III - Center for Computer Research in Music and Acoustics (CCRMA), Stanford University

<< Previous page TOC INDEX Next page >>
Specific Norms
The norms are defined on the space by

16. About this document

Sun, 01 Mar 1998 00:00:00 +0000

NOTE: THIS DOCUMENT IS OBSOLETE, PLEASE CHECK THE NEW VERSION: "The Digital Audio Resampling Home Page", by Julius O. Smith III, Copyright © 2016-05-17 by Julius O. Smith III - Center for Computer Research in Music and Acoustics (CCRMA), Stanford University

<< Previous page TOC
About this document
This document is a new HTML formatted version published by permission of:

The Digital Audio Resampling Home Page by Julius O. Smith III, Last updated January 17, 2000.
Julius O. Smith III, Ph.D. Dissertation, Center for Computer Research in Music and Acoustics (CCRMA), Department of Electrical Engineering, Stanford University, June 1983...

Copyright © 2000-09-19 by Julius O. Smith III

This document is also available in other formats:
Download PDF version (resample.pdf)
Download compressed PostScript version (resample.ps.gz)

Notes from original HTML document (https://ccrma.stanford.edu/~jos//resample):
This document was generated using theLaTeX2HTML translator Version 99.2beta6 (1.42)(patched)
Copyright © 1993, 1994, 1995, 1996,Nikos Drakos, Computer Based Learning Unit, University of Leeds.
Copyright © 1997, 1998, 1999, 2000Ross Moore, Mathematics Department, Macquarie University, Sydney.
The Makefiles and .latex2html-init file used to generate this website are available athttps://ccrma.stanford.edu/~jos//webpub/.
Hypertext links installed from a master glossary prepared using the Open Dictionary
.

16. Concavity (Convexity)

Sun, 01 Mar 1998 00:00:00 +0000

NOTE: THIS DOCUMENT IS OBSOLETE, PLEASE CHECK THE NEW VERSION: "Introduction to Digital Filters with Audio Applications", by Julius O. Smith III, Copyright © 2017-11-26 by Julius O. Smith III - Center for Computer Research in Music and Acoustics (CCRMA), Stanford University

<< Previous page TOC INDEX Next page >>
Concavity (Convexity)
Definition. A set is said to be concave if for every vector and in , is in for all . In other words, all points on the line between two points of lie in .
Definition. A functional is a mapping from a vector space to the real numbers .
Thus, for example, every norm is a functional.
Definition. A linear functional is a functional such that for each and in the linear space , and for all scalars and , we have .
Definition. The norm of a linear functional is defined on the normed linear space by

Definition. A functional defined on a concave subset of a vector space is said to be concave on if for every vector and in ,

A concave functional has the property that its values along a line segment lie below or on the line between its values at the end points. The functional is strictly concave on if strict inequality holds above for . Finally, isuniformly concave on if there exists such that for all ,

We have

Definition. A local minimizer of a real-valued function is any such that in some neighborhood of .
Definition. A global minimizer of a real-valued function on a set is any such that for all .
Definition. A cluster point of a sequence is any point such that every neighborhood of contains at least one .
Definition. The concave hull of a set in a metric space is the smallest concave set containing .

16. Practical Battery Tips

Sun, 01 Mar 1998 00:00:00 +0000

<< Previous page INDEX Next page >>
16. Practical Battery Tips
Batteries seem to have a mind of their own. Their stubborn and unpredictable behavior has left many battery users in awkward situations. In fact, the British Army could have lost the Falkland War in 1982 because of uncooperative batteries. The army assumed that a battery would always follow rigid military specifications. Not so. When the order was given to launch the portable missiles, nothing happened and the missiles did not fly that day. Such battery-induced letdowns happen on a daily basis. Some are simply a nuisance, others have serious consequences.
In this section we examine what the user can reasonably expect from a battery. We learn how to cope with the many moods of a battery and how to come to terms with its limitations.
16.1 Personal Field Observations

While working with General Electric, I had the opportunity to examine the behavior of many NiCd batteries for two-way radios. I noticed a trend with these batteries that was unique to NiCd. These particularities repeated themselves in various other applications.

A certain organization continually experienced NiCd battery failure after a relatively short service time. Although the batteries performed at 100 percent when new, their capacity dropped to 20 percent and below within one year. We discovered that their two-way radios were under-utilized; yet the batteries received a full recharge after each short field use.

After replacing the batteries, we advised the organization to exercise the new batteries once per month by discharging them to one-volt-per cell with a subsequent recharge. The first exercise took place after the batteries had been in service for four months. At that stage, we were anxious to find out how much the batteries had deteriorated. Here is what we found:

On half of the batteries tested, the capacity loss was between 25 to 30 percent; on the other half, the losses were around 10 to 20 percent. With exercise — and some needed recondition cycles — all batteries were fully restored. Had maintenance been omitted for much longer, the probability of a full recovery would have been jeopardized.

On another occasion, I noticed that two-way radios used by construction workers experienced fewer NiCd battery problems than those used by security guards. The construction workers often did not turn off the radios when they put down their hammers. As a result, the batteries got their exercise and kept performing well until they fell apart from old age. In many cases the batteries were held together with electrician’s tape.

In comparison, the security guards pampered their batteries to death by giving them light duty and plenty of recharge. These batteries still looked new when they had to be discarded after only 12 months of service. Because of the advanced state of memory, recondition was no longer effective to restore these batteries.

On a further application, I studied the performance of a two-way radio that was available with batteries of different capacities. It soon was apparent that the smaller battery lasted much longer, whereas the larger packs needed replacing more often. The small battery had to work harder and received more exercise during a daily routine.

Equipment manufacturers are aware of the weak link — the battery. For a more reliable energy source, higher capacity batteries are recommended. Not only are oversized batteries bulky, heavy and expensive, they hold more residual charge prior to recharge than smaller units. If the residual energy is never fully consumed before a recharge, and no exercise is applied, the nickel-based battery will eventually lose its ability to hold charge due to memory.

On the lithium and lead-base systems, a slightly oversized battery offers an advantage because the pack is less stressed on deep discharges. The battery does not need to be discharged as low for the given application. A high residual charge before recharge is a benefit rather than a disadvantage for these chemistries.

16. Practical Battery Tips 2

Sun, 01 Mar 1998 00:00:00 +0000

<< Previous page INDEX Next page >>
16.2 The Correct Battery for the Job

What is the best battery choice? The requirements differ between personal users and fleet operators. The personal user can choose batteries in various sizes and chemistries. Cost is a factor for many. If a smaller and less energy-dense battery is chosen, a spare battery may be carried to assure continued service.

The energy requirements are quite different with fleet operators. The equipment is matched with a battery designed to run for a specified number of hours per shift. A degradation factor to compensate for battery aging is taken into account. A reserve capacity is added to allow for unforeseen activities. Allowing an aging degradation factor of 20 percent and providing a reserve capacity of 20 percent will reduce the usable battery capacity from 100 percent to 60 percent in a worst-case scenario. Such a large percentage of reserve capacity may not always be practical but the equipment manufacturers should consider these safety factors when fitting the portable devices with a battery.

The best choice is not necessarily an oversized battery, but one that has sufficient safety margin and is well maintained. This is especially true of NiCd batteries. When adding large safety margins, the reserve capacity should be depleted once per month, if this is not done already through normal use.

The NiMH also needs exercising but less often. Cycling lithium-based batteries is only recommended for the purpose of measuring the performance.

Many battery users have a choice of switching from NiCd to NiMH to obtain longer runtimes and/or reduce weight. Regulatory bodies advise using less toxic alternatives because of the environment. But will the NiMH battery perform as well as the NiCd in industries that require repetitive deep discharges?

The NiMH will not match the cycle count of the NiCd chemistry. This lower life expectancy has serious consequences on applications that need one or several recharges per day. However, in a recent study on battery choice for heart defibrillators for emergency applications, it was observed that a battery may cycle far less than anticipated. Instead of the expected 200-cycle count after two years of use, less than 60 cycles had been delivered. Such service information is now available with the use of ‘smart’ batteries. With fewer cycles needed, the switch to lighter and higher energy-dense batteries becomes practical for these applications.

In most cases, NiMH can be used as a direct replacement for NiCd. When doing so, the charger must be checked. A NiMH charger can charge NiCd batteries, but a charger designed only for the NiCd battery should not be used to charge NiMH. Battery damage may result due to inaccurate full-charge detection and excessive trickle charge while in ready mode. If no alternative exists, the battery should be removed as soon as the green ready light appears. Battery temperature during charge should also be observed.

Remote control racecar enthusiasts rely heavily on high current capabilities and quick charging. NiMH batteries are now available that can handle very high discharge currents. This makes the battery ideally suited for competitions, because the weight and size of the battery can be reduced.

For most hobbyists, the NiCd remains the preferred choice. The reasons are: more consistent performance, longer cycle life and lower cost. NiCd needs replacement less often than NiMH. RC racing experts claim that NiMH is fragile, temperamental, and can be hurt easily. The storage of the NiMH battery is also erratic. Some cells are flat after a few weeks of storage; others still retain a charge.

High load currents have been problematic for NiMH. Discharge currents of 0.5C and higher rob the battery of cycle life. In comparison, NiCd delivers repetitive high load currents with minimal side effects.

The ultra-high capacity NiCd does not perform as well compared to the standard version in terms of load characteristics and endurance. Packing more active material makes the NiCd behave more like a NiMH battery.

The Li-ion battery has limited current handling capabilities. In many cases, it cannot be used as a replacement for such applications as defibrillators and power tools, not to mention RC racing. In addition, Li-ion requires a different charging system than the nickel-based battery chemistries.

16. Practical Battery Tips 3

Sun, 01 Mar 1998 00:00:00 +0000

<< Previous page INDEX Next page >>
16.3 Battery Analyzers for Critical Missions

Occasionally, a customer will call Cadex because their battery analyzer appears faulty. The complaint: the battery no longer indicates correct capacity readings. In most cases, the customer has just purchased new batteries. When testing these new packs, the capacities read 50 to 70 percent. The customer assumes that, “Naturally, if two or more of these brand new batteries show low readings, it can only be the analyzer’s fault.”

Battery analyzers play a critical role in identifying non-performing batteries, new or old. Conventional wisdom says that a new battery always performs flawlessly. Yet many users realize that a fresh battery may not always meet the manufacturer's specifications. Weak batteries can be identified and primed. If the capacity does not improve, the packs can often be returned to the vendor for warranty replacement. Whole batches of new batteries have been sent back because of unacceptable performance. Had these batteries been released without prior inspection, the whole system would have been jeopardized, resulting in unpredictable performance and frequent down time.

In addition to getting new batteries field-ready, battery analyzers perform the important function of weeding out the deadwood in a battery fleet. Weak batteries can often hide among their peers. However, when the system is put to the test in an emergency, these non-performers become a real nuisance.

Organizations tend to postpone battery maintenance until a crisis situation develops. One fire brigade using two-way radios experienced chronic communication problems, especially during emergency calls which lasted longer than two hours. Although their radios functioned in the receive mode, they were not able to transmit and firefighters were left unaware that their calls did not get through.

The fire brigade acquired a Cadex battery analyzer and all batteries were serviced through exercise and recondition methods. Those batteries that did not recover to a preset target capacity were replaced.

Shortly thereafter, the firefighters were summoned to a ten-hour call that demanded heavy radio traffic. To their astonishment, none of the two-way radios failed. The success of this flawless operation was credited to the excellent performance of their batteries. The following day, the Captain of the fire brigade personally contacted the manufacturer of the battery analyzer and enthusiastically endorsed the use of the device.

Batteries placed on prolonged standby commonly fail. Such was the case when a Cadex representative was allowed to view the State Emergency Management Facility of a large US city. In the fortified underground bunker, over one thousand batteries were kept in chargers. The green lights glowed, indicating that the batteries where ready at a moment’s notice. The officer in charge stood tall and confidently said, “We are prepared for any emergency”.

The representative then asked the officer to hand over a battery from the charger to check the state-of-health (SoH). Within seconds, the battery analyzer detected a fail condition. In an effort to make good, the officer grabbed another battery from the charger bank but, it failed too. Subsequent batteries tested also failed.

Scenarios such as these are common but such flaws do not get rectified quickly. Political hurdles and lack of funding are often to blame. In the meantime, all the officer can do is pray that no emergency occurs.

Eventually, a new set of batteries is installed and the system returns to full operational readiness. However, the same scenario will reoccur, unless a program is implemented to exercise the batteries on a regular basis. Advanced battery analyzers, such as the Cadex 7000 Series, apply a conditioning discharge every 30 days to prevent the memory phenomenon on nickel-based batteries.

Figure 16-1: Results of neglecting your battery’s state-of-health.
Maintenance helps keep deadwood out of your arsenal.

The military also relies heavily on batteries. Defense organizations take great pride in employing the highest quality and best performing equipment. When it comes to rechargeable batteries, however, there are exceptions. The battery often escapes the scrutiny of a full military inspection and only its visual appearance is checked. Maintenance requirements are frequently ignored. Little effort is made in keeping track of the battery’s state of health, cycle count and age. Eventually, weak batteries get mixed with new ones and the system becomes unreliable. This results in soldiers carrying rocks instead of batteries. A battery analyzer, when used correctly, keeps deadwood out of the arsenal.

The task of keeping a battery fleet at an acceptable capacity level has been simplified with battery analyzers that offer target capacity selection. This novel feature works on the basis that all batteries must pass a user-defined performance test. Batteries that fall short are restored with the recondition cycle. If they fail to recover, the packs are replaced.

The target capacity setting of a battery analyzer can be compared to a student entry-exam for college. Assuming that the passing mark is 80 percent, the students who do not obtain that level are given the opportunity to take a refresher course and are thereafter permitted to rewrite the exam. In our analogy, the refresher course is the recondition cycle that is applied to nickel-based batteries. If the passing mark is set to 90 percent, for example, fewer but higher qualified students are admitted.

A practical target capacity setting for batteries in public safety is 80 percent. Increasing the capacity requirement to 90 percent will provide an extra 10 percentage points of available energy. However, higher settings will yield fewer batteries since more packs will fail as they age.

Many organizations allocate the top performing batteries for critical applications and assign the lower performers for lighter duties. This makes full use of the available resources without affecting reliability.

Some battery analyzers display both the reserve capacity (motor fuel left in the tank before refill) and the full-charge capacity (full tank) of the batteries serviced. The user is then able to calculate how much energy was consumed during the day by subtracting the reserve from the full-charge capacity. To ensure a reasonable safety margin after a routine day, the reserve capacity should be about 20 percent. If less reserve capacity is available, the target capacity should be set higher. By allowing reasonable reserve capacity, unexpected downtime in an emergency or on extra-strenuous field activities can be eliminated.

17. Concave Norms

Sun, 01 Mar 1998 00:00:00 +0000

NOTE: THIS DOCUMENT IS OBSOLETE, PLEASE CHECK THE NEW VERSION: "Introduction to Digital Filters with Audio Applications", by Julius O. Smith III, Copyright © 2017-11-26 by Julius O. Smith III - Center for Computer Research in Music and Acoustics (CCRMA), Stanford University

<< Previous page TOC INDEX Next page >>
Concave Norms
A desirable property of the error norm minimized by a filter-designtechnique is concavity of the error norm with respect to the filter coefficients. When this holds, the error surface ``looks like a bowl,'' and the global minimumcan be found by iteratively moving the parameters in the ``downhill'' (negative gradient) direction. The advantages of concavity are evident from the following classical results.
Theorem. If is a vector space, a concave subset of , and a concave functional on , then any local minimizer of is a global minimizer of in .
Theorem. If is a normed linear space, a concave subset of , and a strictly concave functional on , then hasat most one minimizer in .
Theorem. Let be a closed and bounded subset of . If is continuous on , then hasat least one minimizer in .
Theorem (4.1) bears directly on the existence of a solution to the general filter design problem in the frequency domain. Replacing ``closed and bounded'' with ``compact'', it becomes true for a functional on an arbitrary metric space (Rudin [Rudin 1964], Thm. 14). (In , ``compact'' is equivalent to ``closed and bounded'' [Royden 1968].) Theorem (4.1) implies only compactness of and continuity of the error norm on need to be shown to prove existence of a solution to the general frequency-domain filter design problem.

17. Did you know . . . ?

Sun, 01 Mar 1998 00:00:00 +0000

<< Previous page INDEX Next page >>
17. Did you know . . . ?
Technological advancements usually take off shortly after a major breakthrough has occurred. Electricity was discovered circa 1600 AD (or earlier). At that time, electric power had few other applications than creating sparks and experimenting with twitching frog legs. Once the relationship with magnetism was discovered in the mid 1800s, generators were invented that produced a steady flow of electricity. Motors followed that enabled mechanical movement and the Edison light bulb was invented to conquer the dark.
In the early 1900s, the electronic vacuum tube was invented, which enabled generating and amplifying signals. Soon thereafter broadcasting through the air by radio waves became possible. The discovery of the transistor in 1947 led to the development of the integrated circuit ten years later. Finally, the microprocessor ushered in the Information Age and revolutionized the way we live.

How much has the battery improved during the last 150 years when compared to other advancements? The progress has been moderate. A battery holds relatively little power, is bulky, heavy, and has a short life span. Battery power is also very expensive.

Yet humanity depends on the battery as a power source. In the year 2000, the total battery energy consumed globally by laptops and mobile phones alone is estimated to be 2,500MW. This equals 25,000 cars powered by a 100kW engine (134hp) driving at freeway speed.

Many travelers have experienced the exhilaration of take-off in a jumbo jet. At a full weight of over 396 tons, the Boeing 747 requires 90MW of energy to get airborne. The global battery power consumed by mobile phones and laptops could simultaneously lift off 28 jumbo jets. The energy consumption while cruising at high altitude is reduced to about half, or 45MW. The batteries that power our mobile phones and laptops could keep 56 Boeing 747s in the air.

The mighty Queen Mary, an 81,000 ton cruise ship measuring over 300 m (1000 ft) in length, was propelled by four steam turbine engines producing a total of 160,000hp. The energy consumed globally by mobile phones and laptops could power 20 Queen Mary ships, with 3000 passengers and crew aboard, traveling at a speed of 28.5 knots (52 km/hr). The Queen Mary was launched in 1934 and is now retired in Long Beach, California.

In this concluding chapter, we compare the cost of battery power against energy created by the combustion engine and the emerging fuel cell. We also examine the cost of electricity delivered through the electric utility system.

17. Did you know . . . ? 2

Sun, 01 Mar 1998 00:00:00 +0000

<< Previous page INDEX Next page >>
17.1 The Cost of Mobile Power

Among the common power sources, energy from non-rechargeable batteries is the most expensive. Figure 17-1 reflects the cost per kWh using non-rechargeable batteries, also referred to as primary batteries. In addition, non-rechargeable batteries have a high internal cell resistance, which limits their use to light loads with low discharge currents.

In the last few decades, there has been a shift from non-rechargeable to rechargeable batteries, also known as secondary batteries. The convenience of recharging, low cost and reliable operation have contributed to this. Another reason for the increased popularity of the secondary battery is the larger energy densities available. Some of the newer rechargeable lithium systems now approach or exceed the energy density of a primary battery.

AAA Cell AA Cell C Cell D Cell 9 Volt

Capacity (alkaline) 1100mAh 2500mAh 7100mAh 14,300mAh 600mAh

Energy (single cell) 1.4Wh 3Wh 9Wh 18Wh 4.2Wh

Cost per Cell (US$) $1.25 $1.00 $1.60 $1.60 $3.10

Cost per KWh (US$) $890 $330 $180 $90 $730

Figure 17-1: Energy and cost comparison of primary alkaline cells.
Energy from primary batteries is most expensive. The cost increases with smaller battery sizes.

Figure 17-2 compares the cost of power when using rechargeable batteries. The analysis is based on the purchase cost of the battery and the number of discharge-charge cycles it can endure before replacement is necessary. The cost does not include the electricity needed for charging, nor does it account for the cost of purchasing and maintaining the charging equipment.

NiCd
AA Cell NiMH
AA Cell Lead Acid
(typical pack) Li-ion
18650 Cell Reusable Alkaline AA Cell

Capacity 600mAh 1000mAh 2000mAh 1200mAh 1400mAh ¹

Battery Voltage 7.5V 7.5V 12V 7.2V 7.5V

Energy per cycle 4.5Wh 7.5Wh 24Wh 8.6Wh 6.3Wh

Cycle life 1500 500 250 500 10

Cost per battery (ref. only) $50 $70 $50 $100 $6.00

Cost per kWh ($US) $7.50 $18.50 $8.50 $24.00 $95.00

Figure 17-2: Energy and cost comparison using rechargeable cells.
Older battery technologies offer lower energy costs compared to new systems. In addition, larger cells are more cost-effective than small cells. The battery packs taken for comparison are for commercial applications at over-the-counter prices.

For this calculation, 840mA is used since subsequent capacities are rated at 840mA (60% of initial capacity). If the battery is discharged partially, a higher cycle life can be obtained.

Figure 17-3 evaluates the cost to generate 1kW of energy. We take into account the initial investment, add the fuel consumption and include the eventual replacement of each system.

Power obtained through the electrical utility grid is most cost effective. Consumers in industrialized countries pay between $0.05 and 0.15US per kWh. The typical daily energy consumption of a household is 25kWh.

Investment
of equipment to
generate 1kW Lifespan
of equipment before major overhaul or replacement Cost of fuel
per kWh Total Cost
per kWh, incl. fuel, maintenance and equipment replacement

NiCd
for portable use $7,000, based on 7.2V, 1000mAh at $50/pack 1500 h, based on 1C discharge $0.15 for electricity $7.50

Gasoline Engine for mobile use $30, based on $3,000/100kW (134hp) 4000 h $0.10 $0.14

Diesel Engine
for stationary use $40, based on $4,000/100kW (134hp) 5000 h $0.07 $0.10

Fuel Cell $3,000 – 7,500 $0.35

- for portable use 2000 h --> $1.85 – 4.10

- for mobile use 4000 h --> $1.10 – 2.25

- for stationary use 40,000 h --> $0.45 – 0.55

Electricity
from electric grid All inclusive All inclusive $0.10 $0.10

Figure 17-3: Cost of generating 1kW of energy.
This takes into account the initial investment, fuel consumption, maintenance and eventual replacement of the equipment. The most economical power source is by far the utility; the most expensive is portable batteries.

The fuel cell offers the most effective means of generating electricity, but is expensive in terms of cost per kWh. This high cost is made economical when comparing with portable rechargeable batteries. For mobile and stationary applications, the fuel cell is considerably more expensive than conventional methods.

Note: The costing information obtained on the fuel cell is based on current estimates and assumptions. It is anticipated that improvements and wider use of this technology will eventually lower the cost to be competitive with conventional methods.

17. Did you know . . . ? 3

Sun, 01 Mar 1998 00:00:00 +0000

<< Previous page INDEX Next page >>
17.2 The Fuel Cell

A fuel cell is an electrochemical device which combines hydrogen fuel with oxygen to produce electric power, heat and water. In many ways, the fuel cell resembles a battery. Rather than applying a periodic recharge, a continuous supply of oxygen and hydrogen is supplied from the outside. Oxygen is drawn from the air and hydrogen is carried as a fuel in a pressurized container. As alternative fuel, methanol, propane, butane and natural gas can be used.

The fuel cell does not generate energy through burning; rather, it is based on an electrochemical process. There are little or no harmful emissions. The only release is clean water. In fact, the water is so pure that visitors to Vancouver’s Ballard Power Systems, the leader in the development of the proton exchange membrane fuel cell (PEMFC), drank clear water emitted from the tailpipes of buses powered by a Ballard fuel cell.

The fuel cell is twice as efficient in converting fuel to energy through a chemical process than combustion. Hydrogen, the simplest element consisting of one proton and one electron, is plentiful and is exceptionally clean as a fuel. Hydrogen makes up 90 percent of the composition of the universe and is the third most abundant element on the earth’s surface. Such a wealth of fuel would provide an almost unlimited pool of energy at relatively low cost. But there is a price to pay. The fuel cell core (or ‘stack’), which converts oxygen and hydrogen to electricity, is expensive to build.

Hydrogen must be carried in a pressurized bottle. If propane, natural gas or diesel are used, a reformer is needed to convert the fuel to hydrogen. Reformers for PEMFCs are bulky and expensive. They start slowly and purification is required. Often the hydrogen is delivered at low pressure and additional compression is required. Some fuel efficiency is lost and a certain amount of pollution is produced. However, these pollutants are typically 90 percent less than what comes from the tailpipe of a car.

The fuel cell concept was developed in 1839 by Sir William Grove, a Welsh judge and gentleman scientist. The invention never took off, partly because of the success of the internal combustion engine. It was not until the second half of the 20^th century when scientists learned how to better utilize materials such as platinum and TeflonÔ, that the fuel cell could be put to practical use.

A fuel cell is electrolysis in reverse, using two electrodes separated by an electrolyte. Hydrogen is presented to the negative electrode (anode) and oxygen to the positive electrode (cathode). A catalyst at the anode separates the hydrogen into positively charged hydrogen ions and electrons. On the PEMFC system, the oxygen is ionized and migrates across the electrolyte to the anodic compartment where it combines with hydrogen. The byproduct is electricity, some heat and water. A single fuel cell produces 0.6 to 0.8V under load. Several cells are connected in series to obtain higher voltages.

The first practical application of the fuel cell system was made in the 1960s during the Gemini space program, when this power source was favored over nuclear or solar power. The fuel cell, based on the alkaline system, generated electricity and produced the astronauts’ drinking water. Commercial application of this power source was prohibitive because of the high cost of materials. In the early 1990s, improvements were made in stack design, which led to increased power densities and reduced platinum loadings at the electrodes.

High cost did not hinder Dr. Karl Kordesch, the co-inventor of the alkaline battery, from converting his car to an alkaline fuel cell in the early 1970s. Dr. Kordesch drove the car for many years in Ohio, USA. The hydrogen tank was placed on the roof and the trunk was utilized to store the fuel cell and back-up batteries. According to Dr. Kordesch, there was “enough room for four people and a dog”.

17. Did you know . . . ? 4

Sun, 01 Mar 1998 00:00:00 +0000

<< Previous page INDEX Next page >>

Types of fuel cells — Several variations of fuel cell systems have emerged. The most common are the previously mentioned and most widely developed PEMFC systems using a polymer electrolyte. This system is aimed at vehicles and portable electronics. Several developers are also targeting stationary applications. The alkaline system, which uses a liquid electrolyte, is the preferred fuel cell for aerospace applications, including the space shuttle. Molten carbonate, phosphoric acid and solid oxide fuel cells are reserved for stationary applications, such as power generating plants for electric utilities. Among these stationary systems, the solid oxide fuel cell system is the least developed but has received renewed attention due to breakthroughs in cell material and stack designs.

The PEMFC system allows compact designs and achieves a high energy to weight ratio. Another advantage is a quick start-up when hydrogen is applied. The stack runs at a low temperature of about 80°C (176°F). The efficiency is about 50 percent (in comparison, the internal compaction motor has an efficiency of about 15 percent).

The limitations of the PEMFC system are high manufacturing costs and complex water management issues. The stack contains hydrogen, oxygen and water. If dry, the input resistance is high and water must be added to get the system going. Too much water causes flooding.

The PEMFC has a limited temperature range. Freezing water can damage the stack. Heating elements are needed to keep the fuel cell within an acceptable temperature range. The warm-up is slow and the performance is poor when cold. Heat is also a concern if the temperature rises too high.

The PEMFC requires heavy accessories. Operating compressors, pumps and other apparatus consumes 30 percent of the energy generated. The PEMFC stack has an estimated service life of 4000 hours if operated in a vehicle. The relatively short life span is caused by intermittent operation. Start and stop conditions induce drying and wetting, which contribute to membrane stress. If run continuously, the stationary stack is good for about 40,000 hours. The replacement of the stack is a major expense.

Type of Fuel Cell Applications Advantages Limitations Status

Proton Exchange Membrane
(PEMFC) Mobile (buses, cars), portable power, medium to large-scale stationary power generation (homes, industry). Compact design; relatively long operating life; adapted by major automakers; offers quick start-up, low temperature operation, operates at 50% efficiency. High manufacturing costs, needs heavy auxiliary equipment and pure hydrogen, no tolerance for contaminates; complex heat and water management.p> Most widely developed; limited production; offers promising technology.

Alkaline
(AFC) Space (NASA), terrestrial transport (German submarines). Low manufacturing and operation costs; does not need heavy compressor, fast cathode kinetics. Large size; needs pure hydrogen and oxygen; use of corrosive liquid electrolyte.p> First generation technology, has renewed interest due to low operating costs.

Molten Carbonate
(MCFC) Large-scale power generation. Highly efficient; utilizes heat to run turbines for co-generation. Electrolyte instability; limited service life. Well developed; semi-commercial.

Phosphoric Acid
(PAFC) Medium to large-scale power generation.p> Commercially available; lenient to fuels; utilizes heat for co-generation.p> Low efficiency, limited service life, expensive catalyst. Mature but faces competition from PEMFC. p>

Solid Oxide (SOFC) Medium to large-scale power generation. High efficiency, lenient to fuels, takes natural gas directly, no reformer needed. Operates at 60% efficiency; utilizes heat for co-generation. High operating temperature; requires exotic metals, high manufacturing costs, oxidation issues; low specific power. Least developed. Breakthroughs in cell material and stack design sets off new research.

Direct Methanol
(DMFC) Suitable for portable, mobile and stationary applications. Compact design, no compressor or humidification needed; feeds directly off methanol in liquid form. p> Complex stack structure, slow load response times; operates at 20% efficiency. Laboratory prototypes.

Figure 17-4: Advantages and disadvantages of various fuel cell systems.
The PEMFC is the most widely developed system.

Figure 17-5: 1kW portable fuel cell generator.
The unit is a fully automated power system, which converts hydrogen fuel and oxygen from air directly into DC electricity. Water is the only by-product of the reaction. This fuel cell generator, which operates at low pressures, provides reliable, clean, quiet and efficient power. It is small enough to be carried to wherever power is needed. Illustration courtesy of Ballard Power Systems Inc., February 2001.

The SOFC is best suited for stationary applications. The system requires high operating temperatures (about 1000°C). Newer systems are being developed which can run at about 700°C.

A significant advantage of the SOFC is its leniency on fuel. Due to the high operating temperature, hydrogen is produced through a catalytic reforming process. This eliminates the need for an external reformer to generate hydrogen. Carbon monoxide, a contaminant in the PEMFC system, is a fuel for the SOFC. In addition, the SOFC system offers a fuel efficiency of 60 percent, one of the highest among fuel cells. The 60 percent efficiency is achieved with co-generation, meaning that the heat is utilized.

Higher stack temperatures add to the manufacturing cost because they require specialized and exotic materials. Heat also presents a challenge for longevity and reliability because of increased material oxidation and stress. High temperatures, however, can be utilized for co-generation by running steam generators. This improves the overall efficiency of this fuel cell system.

The AFC has received renewed interest because of low operating costs. Although larger in physical size than the PEMFC system, the alkaline fuel cell has the potential of lower manufacturing and operating costs. The water management is simpler, no compressor is usually needed, and the hardware is cheaper. Whereas the separator for the PEMFC stack costs between $800 and $1,100US per square meter; the equivalent of the alkaline system is almost negligible. (In comparison, the separator of a lead acid battery is $5 per square meter.) As a negative, the alkaline fuel cell needs pure oxygen and hydrogen to operate. The amount of carbon dioxide in the air can poison the alkaline fuel cell.

17. Did you know . . . ? 5

Sun, 01 Mar 1998 00:00:00 +0000

<< Previous page INDEX Next page >>

Applications — The fuel cell is being considered as an eventual replacement for the internal combustion engine for cars, trucks and buses. Major car manufacturers have teamed up with fuel cell research centers or are doing their own development. There are plans for mass-producing cars running on fuel cells. However, because of the low operating cost of the combustion engine, and some unresolved technical challenges of the fuel cell, experts predict that a large scale implementation of the fuel cell to power cars will not occur before 2015, or even 2020.

Large power plants running in the 40,000kW range will likely out-pace the automotive industry. Such systems could provide electricity to remote locations within 10 years. Many of these regions have an abundance of fossil fuel that could be utilized. The stack on these large power plants would last longer than in mobile applications because of steady use, even operating temperatures and absence of shock and vibration.

Residential power supplies are also being tested. Such a unit would sit in the basement or outside the house, similar to an air-conditioning unit of a typical middle class North American home. The fuel would be natural gas or propane, a commodity that is available in many urban settings.

Fuel cells may soon compete with batteries for portable applications, such as laptop computers and mobile phones. However, today’s technologies have limitations in meeting the cost and size criteria for small portable devices. In addition, the cost per watt-hour is less favorable for small systems than large installations.

Let’s examine once more the cost to produce 1kW of power. In Figure 17-5 we learned that the investment to provide 1kW of power using rechargeable batteries is around $7,000. This calculation is based on 7.2V; 1000mAh NiCd packs costing $50 each. High energy-dense batteries that deliver fewer cycles and are more expensive than the NiCd will double the cost.

The high cost of portable power opens vast opportunities for the portable fuel cell. At an investment of $3,000 to $7,500 to produce one kilowatt of power, however, the energy generated by the fuel cell is only marginally less expensive than that produced by conventional batteries.

The DMFC, the fuel cell designed for portable applications would not necessarily replace the battery in the equipment but serve as a charger that is carried separately. The feasibility to build a mass-produced fuel cell that fits into the form factor of a battery is still a few years away.

The advantages of the portable fuel cell are: relatively high energy density (up to five times that of a Li-ion battery), liquefied fuel as energy supply, environmentally clean, fast recharge and long runtimes. In fact, continuous operation is feasible. Miniature fuel cells have been demonstrated that operate at an efficiency of 20 percent and run for 3000 hours before a stack replacement is necessary. There is, however, some degradation during the service life of the fuel cell. Portable fuels cells are still in experimental stages.

Advantages and limitations of the fuel cell — A less known limitation of the fuel cell is the marginal loading characteristic. On a high current load, mass transport limitations come into effect. Supplying air instead of pure oxygen aggregates this phenomenon.

The issue of mass transport limitation is why the fuel cell operates best at a 30 percent load factor. Higher loads reduce the efficiency considerably. In terms of loading characteristics, the fuel cell does not match the performance of a NiCd battery or a diesel engine, which perform well at a 100 percent load factor.

Ironically, the fuel cell does not eliminate the chemical battery — it promotes it. Similar to the argument that the computer would make paper redundant, the fuel cell needs batteries as a buffer. For many applications, a battery bank will provide momentary high current loads and the fuel cell will serve to keep the battery fully charged. For portable applications, a supercapacitor will improve the loading characteristics and enable high current pulses.

Most fuel cells are still handmade and are used for experimental purposes. Fuel cell promoters remind the public that the cost will come down once the cells are mass-produced. While an internal combustion engine requires an investment of $35 to $50 to produce one kilowatt of power, the equivalent cost in a fuel cells is still a whopping $3,000 to $7,500. The goal is a fuel cell that would cost the same or less than diesel engines.

The fuel cell will find applications that lie beyond the reach of the internal combustion engine. Once low cost manufacturing is feasible, this power source will transform the world and bring great wealth potential to those who invest in this technology.

It is said that the fuel cell is as revolutionary in transforming our technology as the microprocessor has been. Once fuel cell technology has matured and is in common use, our quality of life will improve and the environmental degradation caused by burning fossil fuels will be reversed. However, the maturing process of the fuel cell may not be as rapid as that of microelectronics.

17. Did you know . . . ? 6

Sun, 01 Mar 1998 00:00:00 +0000

<< Previous page INDEX Next page >>
17.3 The Electric Vehicle

In a bid to lower air pollution in big cities, much emphasis has been placed on the electric car. The notion of driving a clean, quiet and light vehicle appeals to many city dwellers. Being able to charge the car at home for only a dollar a day and escape heavy fuel taxes (at least for the time being) makes the electric car even more attractive.

The battery is still the main challenge in the development of the electric car. Distance traveled between recharge, charge time and the limited cycle count of the battery continue to pose major concerns. Unless the cycle life of the battery can be increased significantly, the cost per mile will be substantially higher than that of a fuel-powered vehicle. The added expense is the need to replace the battery after a given number of recharges. This could offset any advantage of lower energy costs or the absence of fuel taxes. Disposing the spent batteries also adds to the expenditure.

Another challenge associated with the electric vehicle is the high power demand that would be placed on the electric grid if too many cars were charged at a certain time. Each recharge consumes between 15 to 20kW of power, an amount that is almost as much as the daily power requirement of a smaller household. By adding one electric car per family, the amount of electric power a residence requires would almost double. Delayed charging could ease this problem by only drawing power during the night when the consumption is low.

A rapid shift to the electric car could create shortages of electric power. With the move to reduce the generation of electricity due environmental concerns, electricity would need to be imported at high costs. This would make the electric car less attractive.

If the electricity was generated with renewable energy such as hydroelectric generators and windmills, the electric vehicle would truly clear the air in big cities. The generation of electricity by means of nuclear power or fossil fuels simply shifts the pollution problem elsewhere. However, a central source of pollution is easier to contain than many polluting objects in a metropolitan area.

A hybrid car is an alternative to vehicles running solely on battery power. Here, a small combustion engine works in unison with an electric motor. During acceleration, both the electric and combustion engines are engaged. Because of superior torque, the electric motor takes precedence during acceleration. Once cruising, the combustion engine maintains the speed and keeps the batteries charged. Hybrid cars achieve fuel savings of 30 percent or better compared to the combustion engine alone.

A hybrid car is less strenuous on a battery than a conventional electric car because the battery is not being deeply discharged during regular use. A deep discharge only occurs on a long mountain climb where the small combustion engine could not sustain the load and would need assistance from the electric motor and its battery bank. Driving habits would, to a large extent, determine the service life of the battery. A light foot on the pedal will help the pocket book also with the hybrid car.

Another alternative to powering cars is the fuel cell. Although much cleaner running than the combustion engine, the fuel cell must solve a number of critical problems before the product can be offered to the consumer as an economical alternative. The major challenge is cost reduction. If fossil fuel remains as low-priced is it is today, many drivers owning high-powered cars, SUVs and trucks would be reluctant to switch to a new technology. Concerns over pollution only persuade a limited number of drivers to switch to a cleaner-running vehicle. With the slow and gradual progress in the fuel cell, it will be some time before this technology renders the combustion engine obsolete.

Europe is talking about the three-liter motor, an internal combustion engine running on gasoline or diesel fuel. Remarkably, ‘three’ does not denote the engine displacement but stands for liters of fuel consumed per 100 km traveled. There is talk about the one-liter engine also. Major car manufacturers are divided on the fuel that will power our cars in the future. Within one large auto manufacturer in Europe, opinions regarding the fuel cell and an economical three-liter engine are divided fifty-fifty.
17.4 Strengthening the Weakest Link

The speed at which mobility can advance hinges much on the battery. So important is this portable energy that engineers design handheld devices around the battery, rather than the other way around. With each incremental improvement of the battery, the doors swing open for new products and applications. It is the virtue of the battery that provides us the freedom to move around and stay in touch. The better the battery, the greater the freedom we can enjoy.

The longer runtime of newer portable devices is not only credited to higher energy-dense batteries. Much improvement has been made in reducing the power consumption of portable equipment. These advancements are, however, counteracted with the demand for more features and faster processing time. In mobile computing, for example, high speed CPUs, large screens and wireless interface are a prerequisite. These features eat up the reserve energy that the more efficient circuits save and the improved battery provides. The result is similar runtime to an older system, but with increased performance. It is predicted that the improvements in battery technology will keep par with better performance.

Wide-band mobile phones, dubbed G3 for third generation, are being offered as replacements for the digital voice phone. There is public demand for Internet access in a tiny handset that connects to the world by the push of a few buttons, twenty-four hours a day. But these devices require many times the power compared to voice only when operating on wideband. Higher capacity batteries are needed, preferably without added size and weight. In fact, the success of the G3 system could hinge on the future performance of the battery. G3 technology may be ready but the battery lags behind.

The battery has not leap-frogged at the same speed as microelectronics. Only 5 to 10 percent gains in capacity per year have been achieved during the last decades and the ultimate miracle battery is still nowhere in sight. As long as the battery is based on an electro-chemical process, limitations of power density and life expectancy must be taken into account.

The battery remains the ‘weak link’ for the foreseeable future. A radical turn will be needed to satisfy the unquenchable thirst for mobile power. What people want is an inexhaustible pool of energy in a small package. It is anyone’s guess whether the electro-chemical battery of the future, the fuel cell or some groundbreaking new energy storage device will fulfill this dream.

17. Footnotes

Sun, 01 Mar 1998 00:00:00 +0000

NOTE: THIS DOCUMENT IS OBSOLETE, PLEASE CHECK THE NEW VERSION: "The Digital Audio Resampling Home Page", by Julius O. Smith III, Copyright © 2016-05-17 by Julius O. Smith III - Center for Computer Research in Music and Acoustics (CCRMA), Stanford University

<< Previous page TOC
Footnotes
... kHz.¹
We arbitrarily define the % guard band as a percentage of half the sampling rate actually used, not as % of the desired kHz bandwidth which would call for a kHz sampling rate.
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
<< Previous page TOC

18. Beginnings and Horizons

Sun, 01 Mar 1998 00:00:00 +0000

<< Previous page INDEX Next page >>
18. Beginnings and Horizons
18.1 About the Author

A versatile inventor, researcher and writer, Mr. Isidor Buchmann is the president, founder and CEO of Cadex Electronics Inc., located in Richmond (Vancouver), Canada.

Fascinated by electronics during his high school years, Mr. Buchmann took to inventing at an early age, designing a fuel-powered engine that was based on continuous combustion. His drawings and theory of operation were reviewed by Felix Wankel, inventor of the Wankel Rotary Engine, who kindly replied that while the design was indeed unique and original, manufacturing would be too expensive to be commercially viable. Further to his credit, Mr. Buchmann invented a broadcast radio that ran on no power — it required only an antenna and a ground connection (it didn’t even use a battery). Mr. Buchmann sold several of these radio receivers to his family and colleagues and later set up a workshop in the attic where he restored and resold old radios. After high school, a four-year apprenticeship as a Radio Technician brought him practical experience in a workshop environment as well as academic theory. Finally, his experience with radio communications in the Swiss army led to his decision to make electronics his life's work.

Realizing that conservative Switzerland would not satisfy his entrepreneurial spirit, Mr. Buchmann emigrated to Canada in 1966, eventually finding employment in the radio communications department at General Electric. There he realized that a major problem with two-way radios was the battery's short life and, as part of his job, tested a wide variety of customer batteries that came in. In his spare time at home, he continued to research and develop electronic devices in his spare time, developing a battery analyzer that featured a ‘recondition’ program which restored weakened nickel cadmium batteries.

To prevent a conflict of interest with his employer, Mr. Buchmann quit his job with GE and started Cadex. The first battery analyzer, the Cadex 450, was introduced in 1981 but failed to achieve the anticipated market acceptance. Still in his spare time, Mr. Buchmann designed the modular Cadex 550 battery analyzer. This model sold reasonably well at first but it soon became evident that manufacturing methods needed to be improved to make it cost effective.

In 1984, Cadex moved from a small room in Mr. Buchmann's residence to rented facilities. With increased overhead costs, a staff to maintain and sluggish sales, cash flow became tight. Bank loans for start-up companies, especially high-tech firms, were almost non-existent at the time, so Mr. Buchmann worked from home during the day looking after his growing children and spent time in the office during the evening. Happily, the company survived the slump and managed to add a number of new products. Profitability returned and the staff grew.

Knowing that the wealth of an organization is in human resources, Isidor resolved to provide an environment that was conducive to attracting good people with skills that complemented his own strengths and weaknesses. Spacious new headquarters in a park-like setting overlooking the scenic Fraser River add to a pleasant working experience. Under Mr. Buchmann’s leadership, new and innovative products were developed that generated rapid growth and created global recognition of Cadex.

Today, Cadex is a world leader in the design and manufacture of battery analyzers and chargers.

18. Beginnings and Horizons 2

Sun, 01 Mar 1998 00:00:00 +0000

<< Previous page INDEX Next page >>
18.2 About Cadex

Cadex Electronics Inc. was established in 1980 in Vancouver, Canada. Isidor Buchmann, founder, president and CEO recognized that the full potential of nickel cadmium batteries was not being achieved and developed a battery analyzer to exercise and rejuvenate them.

In its early days, the company operated under the name Buchmann Enterprises Inc. Until 1983, all activities were conducted in a small room of the founder’s residence. In 1985, after the registered trademark for the name ‘Cadex’ was granted, the company changed the corporate name to Cadex Electronics Inc. Cadex is derived from ‘CADmium-EXerciser.’

The first product, the Cadex 450, entered the market in 1981. Only a few units sold. Mr. Buchmann then designed a modular battery analyzer that was able to service three batteries simultaneously, with expansion to ten. Called the Cadex 550, this unit sold reasonably well and became the workhorse for many two-way radio users, such as railways, public safety and oil companies.

The Government of Canada awarded Cadex funds to develop a new generation of battery analyzers and in 1988 the Cadex 6000 was launched. This new product was capable of servicing up to 64 batteries unattended. A private company then commissioned Cadex to develop and manufacture an intelligent fast-charger for the End-of-Train Unit, a device that replaced the caboose on a freight train. This charger was later expanded into a four-station battery analyzer called Cadex 2000. Packaged into a compact desktop housing, the Cadex 2000 provided a low cost alternative to the modular Cadex 6000 system.

Towards the end of the 1980s, batteries began to diversify and it became evident that a battery analyzer needed to adapt to a large pool of different battery models. With the help of the Science Council of British Columbia and the National Research Council of Canada, Cadex designed an open platform battery analyzer that was software driven similar to a PC. In 1991, the first user-programmable battery analyzer was introduced. Called the Cadex 4000 for its four independent stations, this instrument immediately gained the interest of many battery users, both in North America and overseas. Some of the main features were the interchangeable battery adapters that contained a memory chip holding the unique battery configuration code, ‘C-code’ in short. With a few key stokes, the user was able to program and reprogram the analyzer to fit virtually any battery type.

In 1992, as part of a five-year program for a US defense project, Racal Communications commissioned Cadex to build intelligent fast-chargers and battery reconditioners. During this partnership, Cadex was the recipient of several Racal awards. The Supplier Recognition Award for Outstanding Performance, which Cadex received on two separate occasions, was granted to only six of over one thousand suppliers.

In 1995, Cadex introduced the Cadex 7000 Series, an upgraded version of the Cadex 4000 battery analyzer. The Cadex SnapLock adapters allowed quick and convenient changes from one battery type to another. The Cadex 7000 Series analyzer became the company’s flagship and established a new global standard to which competitive products were compared.

In 1996, an agreement was reached with Medtronics, (then Physio Control Corporation) to design and supply intelligent battery chargers/ conditioners as part of a five-year contract. In the same year, Cadex received the British Columbia Export Award for outstanding achievements in export.

In 1997, Cadex published the book Batteries in a Portable World — A Handbook on Rechargeable Batteries for Non-Engineers. Many of Mr. Buchmann’s articles on battery technology also gained recognition by appearing in leading trade magazines. By then, Cadex had achieved international market recognition with a customer base of over 100 countries.

In 1998, Motorola and Cadex engaged in a partnership to manufacture battery analyzers for distribution through Motorola’s global network. In the same year, Mr. Buchmann was selected as a finalist in the Entrepreneur of the Year program.

In 1999, Cadex received ISO 9001 certification. The company moved to the custom built headquarters in Fraserwood Industrial Park, Richmond, British Columbia, a suburb of Vancouver. In the same year, Cadex Batteryshop™ was released. This high level software integrates the Cadex 7000 Series battery analyzers with a PC to bring battery testing and maintenance within the reach of the untrained user. The task of entering battery parameters was reduced to either scanning a bar code label or the ‘point and click’ of the mouse.

In the same year, Cadex released the SM1 and SM2+ intelligent battery chargers. The Cadex SM2+ charger features a target capacity selector that passes or fails a battery based on state-of-health (SoH). If low, the user is prompted to restore the battery by pressing the condition key. Today, these chargers service batteries for mobile computing, medical instrumentation and survey equipment.

In the year 2000, Cadex developed Quicktest™, a technique that checks the SoH of a battery in three minutes. The system works on a neuro-logic network based on fuzzy logic, is self-learning and adapts to new chemical combinations as introduced from time to time.

During the year 2001, Cadex will introduce a new generation of 7000 Series battery analyzers. A two-station Cadex 7200 has been added to serve smaller battery users. Retaining the powerful priming and reconditioning features of the previous models, the emphasis is moving towards quick testing, boosting and ultra-fast charging of batteries. These services take only minutes and can restore a battery on the run.

18. Beginnings and Horizons 3

Sun, 01 Mar 1998 00:00:00 +0000

<< Previous page INDEX Next page >>
18.3 Working with Natural Beauty

The new Cadex headquarters is nestled in natural surroundings. The scenic Fraser River lies to the south, a public park to the east and the Coastal Mountains to the north. For joggers and cyclists, there is a nature path between the building and the river.

The interior of the building is designed with employee comfort in mind. It includes a snooker table, a gym, shower rooms, and several televisions. Balconies with a river view overlook the outdoor patio, which is used for summer lunches.

With the wonders of nature at its door, Cadex offers its staff a tranquil alternative to the noise and hustle of crowded city streets. And yet, the location is within easy reach of Vancouver’s downtown, is central to the surrounding suburban areas and is close to Vancouver International Airport.

Figure 18-1: Cadex headquarters in Richmond, Canada.
With the wonders of nature at its door, Cadex offers its staff a tranquil alternative to the noise and hustle of crowded city streets.
18.4 Customer Comments

A manager who is in charge of manufacturing and warranties for a large mobile phone provider writes: “I feel that by having the ability to analyze and condition our customer’s batteries, we have also increased our battery sales. We now recommend that all customers stop by and allow us to condition their batteries at least once every six months. Many customers will buy another battery to use while their pack is being conditioned. The customers feel that the maintenance program enables them to achieve longer battery life. Providing battery conditioning service has helped us increase customer loyalty and satisfaction, which in turn helps our bottom line in this competitive industry.”

When setting up a battery refurbishing service, the quality of battery analyzers is of importance. A manager in charge of battery warranty with a leading mobile phone manufacturer commented: “With the Cadex 7000 Series analyzers, our team was able to achieve a 90 percent recovery rate for all warranty batteries, compared to just 60 percent before the Cadex analyzers were brought on stream. The savings were absolutely immense.”

Another Regional Service Manager for a mobile phone provider wrote: “The recovery rate of batteries has been far better than expected. We used to run them on our old equipment and if they failed we would then run them on the Cadex. Over 50 percent of the batteries would pass our target capacity of 80 percent. That in itself saved us a tremendous amount of expense. It enables us to return the batteries to the customer for further use.”

These above mentioned reports are mostly based on servicing nickel-based batteries. The recovery rate is expected to be lower with the Li-ion batteries, which are considered maintenance free. But to everyone’s amazement, lithium-based batteries enjoy a similar recovery rate, especially in the mobile phone market. Here, the cause of failure is not memory-based, especially during the warranty period. Lack of customer preparation and failure to understand the behavior of a battery as a portable energy source may be to blame. The true reason for the failures may never be known.

18. Beginnings and Horizons 4

Sun, 01 Mar 1998 00:00:00 +0000

<< Previous page INDEX Next page >>
18.5 Cadex Products

Cadex products are built with one goal in mind — to make batteries run longer. Cadex has realized the importance of battery care and is offering equipment to charge, test, monitor, and restore batteries.

Cadex’s core competence is engineering. Over 25 percent of the Cadex staff is active in the Engineering Department. Existing products are improved on a continual basis, and new and creative products are added to adjust to the changing demands of battery users. Key products include:

Figure 18-2: Cadex 7200 battery analyzer.
This compact two-station battery analyzer brings battery maintenance within reach of all battery users.

">18-3: Cadex 7400 battery analyzer.
Provision to service four batteries simultaneously increases the service throughput. The Cadex 7400 offers parallel printer port and USP for easy interface to a PC.

Cadex 7000 Series battery analyzers solve the common battery problems of uncertain service and short life. Pre-configured ‘Snap Lock’ adapters enable quick interface with all major batteries for wireless communications devices, laptops, biomedical equipment, video cameras and other portable devices. Irregular batteries connect by universal cables that can be programmed with the analyzer’s menu function. The analyzer supports Li-ion/Polymer, NiMH, NiCd and Sealed Lead Acid (SLA) batteries.

The Cadex 7000 Series features the self-learning Cadex Quicktest™ program that performs an in-depth battery diagnosis in three minutes. Other programs include: ‘Boost’ to wake up low voltage batteries; ‘Auto’ to recondition weak batteries and ‘Prime’ to format new batteries. In addition, ‘Self-Discharge’ verifies charge retention; ‘CycleLife’ tests longevity and ‘Custom’ enables user-defined programs. The Cadex 7200 services two batteries simultaneously; the Cadex 7400 accommodates four.

The battery voltage is programmable from 1.2 to 15V with a current range of 100mA to 24A. If set high, the analyzer automatically reduces the current to remain within the 4A per station handling capabilities. With a printer, service reports and battery labels can be generated. The unit operates as stand-alone or with a PC.

Figure 18-4: Cadex Batteryshop™.
This Windows-based software allows untrained users to perform accurate and expedient battery tests. With the same system, a design engineer can collect valuable battery information running customized test programs.

18. Beginnings and Horizons 5

Sun, 01 Mar 1998 00:00:00 +0000

<< Previous page INDEX

Cadex Batteryshop™ software provides a simple, yet powerful PC interface to control and monitor the Cadex 7000 Series battery analyzers. Running on Windows 95, 98 and NT, the software enables untrained staff to test batteries as part of customer service. In addition, Cadex Batteryshop™ schedules periodic maintenance for fleet owners and assists battery manufacturers with quality control.

Cadex Batteryshop™ includes a database of over 2000 common battery models. Each battery listing contains the configuration code (C-code), the data that sets the analyzer to the correct parameters. A growing number of the battery listings also include matrices to perform Cadex Quicktest™.

Point and click technology selects the battery and programs the Cadex 7000 Series analyzer. Scanning the battery’s model number, if a bar code label is available, also programs the analyzer. Cadex Batteryshop™ supports up to 120 Cadex 7000 Series battery analyzers. The test results can be displayed on screen in real time graphs and printed in customized reports.

Figure 18-5: The Cadex SM1 battery charger.
This charger accommodates the widely used 202 format. Other batteries that fit the bay are the 2020, 1030, 1020, 210, 201, 36, 35, 30, 17 and 15. The Cadex SM1 charger supports ‘smart’ and ‘dumb’ batteries.

Cadex Smart Series battery chargers offer the consumer a continuous supply of freshly charged batteries. Conforming to the SMBus Level 3, the Cadex Smart Series chargers accommodate Li-ion, NiMH and NiCd batteries. The charge time is 2 to 3 hours. If faulty batteries are identified, the charge is halted. On compatible footprint, the chargers also accommodate ‘dumb’ batteries. Typical uses are mobile computing, biomedical and survey devices.

The Cadex SM1 charger is compact and charges one battery at a time. The Cadex SM2+ charger services two packs simultaneously and doubles as conditioner and quality control system. The charger reads the data stored in the SMBus battery, calculates the previous power delivered and compares the results with the target capacity setting. Adjustable to 60, 70 and 80 percent, the charger flags batteries that fall below the set capacity reading.

Figure 18-6: The Cadex SM2+ battery charger.
In addition to the features offered on the Cadex SM1 charger, this unit serves as charger, quality control system and battery conditioner. SMBus batteries with low state-of-health are identified. Conditioning and calibration occurs by pressing a button.

Figure 18-7: Cadex UCC1, MCC2 and UCC6.
The Cadex UCC Series chargers feature interchangeable battery adapters. The Cadex MCC2 serves both as desktop and mobile charger.

Cadex Universal Conditioning Chargers (UCC) offer battery users an alternate source of chargers to those provided by the original equipment manufacturer (OEM). Available in one, two and six bay configurations, the chargers feature intelligent battery adapters. This concept allows easy adaptation to a variety of battery types without compromising charge performance.

The adapters allow service of different battery types in one unit. Reconfiguration to other battery types can be done in the field; the one and six-bay chargers are desktop and wall-mountable. The two-bay unit also serves as a vehicular charger built to military shock and vibration specifications.

Custom Battery Chargers — Cadex designs and manufactures a wide variety of custom chargers to serve public safety, law enforcement, emergency response, healthcare, mobile computing, broadcast and defence applications. Cadex covers all aspects of product development, from circuit design to power supply, from plastic housing to mechanical battery interface, to testing and manufacturing.

Custom Battery Packs — Cadex completes the line of portable power source by offering specialty battery packs. To provide added safety, Cadex has the capability of designing specialty protection circuits for lithium ion chemistries and other battery systems.

18. Gradient Descent

Sun, 01 Mar 1998 00:00:00 +0000

NOTE: THIS DOCUMENT IS OBSOLETE, PLEASE CHECK THE NEW VERSION: "Introduction to Digital Filters with Audio Applications", by Julius O. Smith III, Copyright © 2017-11-26 by Julius O. Smith III - Center for Computer Research in Music and Acoustics (CCRMA), Stanford University

<< Previous page TOC INDEX Next page >>
Gradient Descent
Concavity is valuable in connection with the Gradient Method of minimizing with respect to .
Definition. The gradient of the error measure is defined as the column vector

Definition. The Gradient Method (Cauchy) is defined as follows.
Given , compute

where is the gradient of at , and is chosen as the smallest nonnegative local minimizer of

Cauchy originally proposed to find the value of which gave a global minimum of . This, however, is not always feasible in practice.
Some general results regarding the Gradient Method are given below.
Theorem. If is a local minimizer of , and exists, then .
Theorem. The gradient method is a descent method, i.e., .
Definition. , , is said to be in the class if all th order partial derivatives of with respect to the components of are continuous on .
Definition. The Hessian of at is defined as the matrixof second-order partial derivatives,

where denotes the th component of , , and denotes the matrix entry at the th row and th column.
The Hessian of every element of is a symmetric matrix [Williamson et al. 1972]. This is because continuous second-order partials satisfy

Theorem. If , then any cluster point of the gradient sequence is necessarily astationary point, i.e., .
Theorem. Let denote the concave hull of . If , and there exist positive constants and such that

for all and for all , then the gradient method beginning with any point in converges to a point . Moreover, is the unique global minimizer of in .
By the norm equivalence theorem [Ortega 1972], Eq. (5) is satisfied whenever is a norm on for each . Since belongs to , it is a symmetric matrix. It is also bounded since it is continuous over a compact set. Thus a sufficient requirement is that be positive definite on . Positive definiteness of can be viewed as ``positive curvature'' of at each point of which corresponds to strict concavity of on.

19. Taylor's Theorem

Sun, 01 Mar 1998 00:00:00 +0000

NOTE: THIS DOCUMENT IS OBSOLETE, PLEASE CHECK THE NEW VERSION: "Introduction to Digital Filters with Audio Applications", by Julius O. Smith III, Copyright © 2017-11-26 by Julius O. Smith III - Center for Computer Research in Music and Acoustics (CCRMA), Stanford University

<< Previous page TOC INDEX Next page >>
Taylor's Theorem
Theorem. (Taylor) Every functional in has the representation

for some between and , where is the gradientvector evaluated at , and is the Hessian matrix of at , i.e.,
(52)
(53)

Proof. See Goldstein [Goldstein 1967] p. 119. The Taylor infinite series is treated in Williamson and Crowell [Williamson et al. 1972]. The present form is typically more useful for computing bounds on the error incurred by neglecting higher order terms in the Taylor expansion.

2. A CLOSER LOOK

Sun, 01 Mar 1998 00:00:00 +0000

<< Previous page TOC Next page >>
2. A CLOSER LOOK
2.1. WHAT MEMORY LOOKS LIKE
2.2. WHERE MEMORY COMES FROM
2.3. WHERE MEMORY GOES IN THE COMPUTER
<< Previous page TOC Next page >>

2. Principles of ESD Control

Sun, 01 Mar 1998 00:00:00 +0000

<< Previous page INDEX Next page >>
2. Principles of ESD Control
In Part One, we discussed the basics of electrostatic charge, discharge, types of failures, ESD events, and device sensitivity. We summarized this discussion as follows:

Virtually all materials, even conductors, can be triboelectrically charged.
The level of charge is affected by material type, speed of contact and separation, humidity, and several other factors.
Electrostatic discharge can create catastrophic or latent failures in electronic components.
Electrostatic discharge can occur throughout the manufacturing, test, shipping, handling, or operational processes.
Component damage can occur as the result of a discharge from the component as well as a direct discharge to the component.
Components vary significantly in their sensitivity to ESD.
With this basic understanding of ESD and its impact on your environment, you can then begin to develop an effective ESD control program. In this column and the next, we will focus on basic Principles of ESD control.
Basic Principles of Static Control
Sometimes, controlling electrostatic discharge (ESD) in the electronics environment seems to be a formidable challenge. However, the task of designing and implementing ESD control programs becomes less complex if we focus on just four basic Principles of control. In doing so, we also need to keep in mind the ESD corollary to Murphy’s law, "no matter what we do, static charge will try to find a way to discharge."
1. Design In Immunity
The first Principle is to design products and assemblies to be as immune as reasonable from the effects of ESD. This involves such steps as using less static sensitive devices or providing appropriate input protection on devices, boards, assemblies, and equipment. For engineers and designers, the paradox is that advancing product technology requires smaller and more complex geometries that often are more susceptible to ESD.
2. Eliminate and Reduce Generation
Obviously, product design isn’t the whole answer. The second Principle of control is to eliminate or reduce the generation and accumulation of electrostatic charge in the first place. It’s fairly basic: no charge -- no discharge. We begin by reducing as many static generating processes or materials, such as the contact and separation of dissimilar materials and common plastics, as possible from the work environment. We keep other processes and materials at the same electrostatic potential. Electrostatic discharge does not occur between materials kept at the same potential or at zero potential. We provide ground paths, such as wrist straps, flooring and worksurfaces, to reduce charge generation and accumulation.
3. Dissipate and Neutralize
Because we simply can’t eliminate all generation of static in the environment, our third Principle is to safely dissipate or neutralize those electrostatic charges that do occur. Proper grounding and the use of conductive or dissipative materials play major roles. For example, workers who "carry" a charge into the work environment can rid themselves of that charge when they attach a wrist strap or when they step on an ESD floor mat while wearing ESD control footwear. The charge goes to ground rather than being discharged into a sensitive part. To prevent damaging a charged device, the rate of discharge can be controlled with static dissipative materials.
For some objects, such as common plastics and other insulators, grounding does not remove an electrostatic charge because there is no conductive pathway. Typically, ionization is used to neutralize charges on these insulating materials. The ionization process generates negative and positive ions that are attracted to the surface of a charged object, thereby effectively neutralizing the charge.
4. Protect Products
Our final ESD control Principle is to prevent discharges that do occur from reaching susceptible parts and assemblies. One way is to provide our parts and assemblies with proper grounding or shunting that will "dissipate" any discharge away from the product. A second method is to package and transport susceptible devices in proper packaging and materials handling products. These materials may effectively shield the product from charge, as well as reduce the generation of charge caused by any movement of product within the container.
Elements of an Effective ESD Control Program
While these four principles may seem rather basic, they can guide us in the selection of appropriate materials and procedures to use in effectively controlling ESD. In most circumstances, effective programs will involve all of these principles. No single procedure or product will do the whole job, rather effective static control requires a full ESD control program.
How to we develop and maintain a program that puts these basic principles into practice? How do we start? What is the process? What do we do first? Ask a dozen experts and you may get a dozen different answers. But, if you dig a little deeper, you will find that most of the answers center around similar key elements. You will also find that starting and maintaining an ESD control program is similar to many other business activities and projects. Although each company is unique in terms of its ESD control needs, there are at least 9 critical elements to successfully developing and implementing an effective ESD control program.
1. Establish an ESD Coordinator and ESD Teams.
As the problem-solving style of the decade, the team approach particularly applies to ESD because the problems and the solutions cross various functions, departments, divisions and even suppliers in most companies. Team composition includes line employees as well as department heads or other management personnel. ESD teams or committees help assure a variety of viewpoints, the availability of the needed expertise, and commitment to success. An active ESD committee helps unify the effort and brings additional expertise to the project. Committee or team membership should include representation from areas such as engineering, manufacturing, field service, training, and quality.
Heading this team effort is an ESD Program Coordinator. Ideally this responsibility should be a full-time job. However, we seldom operate in an ideal environment and you may have to settle for the function to be a major responsibility of an individual. The ESD coordinator is responsible for developing, budgeting, and administering the program. The coordinator also serves as the company’s internal ESD consultant to all areas.
2. Identify Your Losses.
Before seeking solutions to your problems, you will need to determine the extent of your losses to ESD. These losses may be reflected in receiving reports, QA and QC records, customer returns, in-plant yields, failure analysis reports, and other data that you may already have or that you need to gather. This information not only identifies the magnitude of the problem, but helps to pinpoint and prioritize areas that need attention.
Document your actual and potential ESD losses in terms of DOA components, rework, customer returns, and failures during final test and inspection. Use data from outside sources or the results of your pilot program for additional support. Develop estimates of the savings to be realized from implementing an ESD control program.
3. Evaluate Your Facility, Processes and Needs
Your next step is to gain a thorough understanding of your environment and its impact on ESD. Armed with your loss and sensitivity data, you can evaluate your facility, looking for areas and procedures that may be contributing to your defined ESD problems. Be on the lookout for things such as static generating materials and personnel handling procedures for ESD-sensitive items.
Document your processes. Observe the movement of people and materials through the areas. Note those areas that would appear to have the greatest potential for ESD problems. Remember, that ESD can occur in the warehouse just as it can in the assembly areas. Then conduct a thorough facility survey or audit. Measure personnel, equipment, and materials to identify the presence of electrostatic fields in your environment.
4. Identify ESD Sensitive Items
You will also want to identify those items (components, assemblies, and finished products) that are sensitive to ESD and the level of their sensitivity. You can test these items yourself, use data from suppliers, or rely on published data for similar items.
5. Build Justification for the Program
Once you understand your environment and the impact that ESD has on your products and services, you then build justification for your ESD control program. You may even need to conduct a pilot program if the experience of other companies is not sufficient to help prove your point.
6. Get the Support of Top Management
To be successful, an ESD program requires the support of your top management, at the highest level possible. What level of commitment is required? Recently, the CEO of a major electronics manufacturer was asked to leave the production area when he failed to wear the ESD control smock required by the company’s ESD procedures. He did so willingly and returned later wearing the proper apparel.
That’s the type of commitment you would like to achieve. To obtain it, your program justification will need to emphasize quality and reliability, the costs of ESD damage, the impact of ESD on customer service and product performance. Prepare a short corporate policy statement on ESD control. Have top management co-sign it with the ESD coordinator. Periodically, reaffirm the policy statement and management’s commitment to it.
7. Establish and Implement Procedures and Specifications
Now you can develop and implement the appropriate procedures that will help solve the ESD problems you have identified in your company. Prepare and distribute written procedures and specifications so that everyone has a clear understanding of what is to be done. Fully documented procedures will help you with ISO 9000 certification as well.
8. Train Personnel
Train and retrain your personnel in ESD and your company’s ESD control program and procedures. Proper training for line personnel is especially important. They are often the ones who have to live with the procedures on a day to day basis. A thorough understanding of what is expected and why will have a significant positive impact on proper long-term implementation.
9. Review, Audit, Analyze, Report, Feedback, Improve
Developing and implementing the program itself is obvious. What might not be so obvious is the need to continually review, audit, analyze, feedback and improve. You will be asked to continually identify the return on investment of the program and to justify the savings realized. Technological changes will dictate improvements and modifications. Feedback to employees and top management is essential. Management commitment will need reinforcement.
Remember to provide both reporting and feedback to management, the ESD team, and other employees. Management will want to know that their investment in time and money is yielding a return in terms of quality, reliability and profits. Team members need a pat on the back for a job well done and need to know where to put their next efforts. Other employees will want to know that the procedures you have asked them to follow are indeed worthwhile. Conduct periodic evaluations of your program and audits of your facility. You will find out if your program is successful and is giving you the expected return. You will spot weaknesses in the program and shore them up. You will discover whether the procedures are being followed.
As you find areas that need work, be sure to make the necessary adjustments to keep the program on track.
Four principles of static control and nine key elements to program development and implementation: your guideposts for effective ESD control programs.
In Part Three, we’ll take a close look at specific procedures and materials that become part of your program.
For Additional Information
Dangelmayer, Theodore, ESD Program Management: A Realistic Approach to Continuous, Measurable Improvement in Static Control, Van Nostrand Reinhold, 1999.
ESD ADV-2.0-1994, ESD Control Handbook, ESD Association, Rome, NY.
Halperin, Stephen A., "Facility Evaluation: Isolating Environmental ESD Issues," EOS/ESD Symposium Proceedings, 1980, ESD Association, Rome, NY

2.1. WHAT MEMORY LOOKS LIKE

Sun, 01 Mar 1998 00:00:00 +0000

<< Previous page TOC Next page >>
2.1. WHAT MEMORY LOOKS LIKE
Memory comes in a variety of sizes and shapes. In general, it looks like a flat green stick with little black cubes on it. Obviously, there's a lot more to memory than that. The illustration below shows a typical memory module and points out some of its most important features.

A closer look at a 168-pin SDRAM DIMM.

PCB(PRINTED CIRCUIT BOARD)
The green board that all the memory chips sit on is actually made up of several layers. Each layer contains traces and circuitry, which facilitate the movement of data. In general, higher quality memory modules use PCBs with more layers. The more layers a PCB has, the more space there is between traces. The more space there is between traces, the lesser the chance of noise interference. This makes the module much more reliable.
DRAM (DYNAMIC RANDOM ACCESS MEMORY)
DRAM is the most common form of RAM. It's called "dynamic" RAM because it can only hold data for a short period of time and must be refreshed periodically. Most memory chips have black or chrome coating, or packaging, to protect their circuitry. The following section titled "Chip Packaging" shows pictures of chips housed in different types of chip packages.
CONTACT FINGERS
The contact fingers, sometimes referred to as "connectors" or "leads," plug into the memory socket on the system board, enabling information to travel from the system board to the memory module and back. On some memory modules, these leads are plated with tin while on others, the leads are made of gold. To learn more about the type of metal on the contacts, refer to the section titled, "Tin versus Gold" on page 67.
INTERNAL TRACE LAYER
The magnifying glass shows a layer of the PCB stripped away to reveal the traces etched in the board. Traces are like roads the data travels on. The width and curvature of these traces as well as the distance between them affect both the speed and the reliability of the overall module. Experienced designers arrange, or "lay out," the traces to maximize speed and reliability and minimize interference.
CHIP PACKAGING
The term "chip packaging" refers to the material coating around the actual silicon. Today's most common packaging is called TSOP (Thin Small Outline Package). Some earlier chip designs used DIP (Dual In-line Package) packaging and SOJ (Small Outline J-lead). Newer chips, such as RDRAM use CSP (Chip Scale Package). Take a look at the different chip packages below, so you can see how they differ.
DIP (DUAL IN-LINE PACKAGE)
When it was common for memory to be installed directly on the computer's system board, the DIP-style DRAM package was extremely popular. DIPs are through-hole components, which means they install in holes extending into the surface of the PCB. They can be soldered in place or installed in sockets.
SOJ (SMALL OUTLINE J-LEAD)
SOJ packages got their name because the pins coming out of the chip are shaped like the letter "J". SOJs are surface-mount components - that is, they mount directly onto the surface of the PCB.
TSOP (THIN SMALL OUTLINE PACKAGE)
TSOP packaging, another surface-mount design, got its name because the package was much thinner than the SOJ design. TSOPs were first used to make thin credit card modules for notebook computers.
CSP (CHIP SCALE PACKAGE)
Unlike DIP, SOJ, and TSOP packaging, CSP packaging doesn't use pins to connect the chip to the board. Instead, electrical connections to the board are through a BGA (Ball Grid Array) on the underside of the package. RDRAM (Rambus DRAM) chips utilize this type of packaging.
CHIP STACKING
For some higher capacity modules, it is necessary to stack chips on top of one another to fit them all on the PCB. Chips can be "stacked" either internally or externally. "Externally" stacked chip arrangements are visible, whereas "internally" stacked chip arrangements are not.

Example of externally stacked chips.

2.2. WHERE MEMORY COMES FROM

Sun, 01 Mar 1998 00:00:00 +0000

<< Previous page TOC Next page >>
2.2. WHERE MEMORY COMES FROM
MAKING THE CHIP
Amazing but true: memory starts out as common beach sand. Sand contains silicon, which is the primary component in the manufacture of semiconductors, or "chips." Silicon is extracted from sand, melted, pulled, cut, ground, and polished into silicon wafers. During the chip-making process, intricate circuit patterns are imprinted on the chips through a variety of techniques. Once this is complete, the chips are tested and die-cut. The good chips are separated out and proceed through a stage called "bonding": this process establishes connections between the chip and the gold or tin leads, or pins. Once the chips are bonded, they're packaged in hermetically sealed plastic or ceramic casings. After inspection, they're ready for sale.
MAKING THE MEMORY MODULE
This is where memory module manufacturers enter the picture. There are three major components that make up a memory module: the memory chips, PCB, and other "on-board" elements such as resistors and capacitors. Design engineers use CAD (computer aided design) programs to design the PCB. Building a high-quality board requires careful consideration of the placement and the trace length of every signal line. The basic process of PCB manufacture is very similar to that of the memory chips. Masking, layering, and etching techniques create copper traces on the surface of the board. After the PCB is produced, the module is ready for assembly. Automated systems perform surface-mount and through-hole assembly of the components onto the PCB. The attachment is made with solder paste, which is then heated and cooled to form a permanent bond. Modules that pass inspection are packaged and shipped for installation into a computer.

2.3. WHERE MEMORY GOES IN THE COMPUTER

Sun, 01 Mar 1998 00:00:00 +0000

<< Previous page TOC Next page >>
2.3. WHERE MEMORY GOES IN THE COMPUTER
Originally, memory chips were connected directly to the computer's motherboard or system board. But then space on the board became an issue. The solution was to solder memory chips to a small modular circuit board - that is, a removable module that inserts into a socket on the motherboard. This module design was called a SIMM (single in-line memory module), and it saved a lot of space on the motherboard. For example, a set of four SIMMs might contain a total of 80 memory chips and take up about 9 square inches of surface area on the motherboard. Those same 80 chips installed flat on the motherboard would take up more than 21 square inches on the motherboard.

These days, almost all memory comes in the form of memory modules and is installed in sockets located on the system motherboard. Memory sockets are easy to spot because they are normally the only sockets of their size on the board. Because it's critical to a computer's performance for information to travel quickly between memory and the processor(s), the sockets for memory are typically located near the CPU.

20. Newton's Method

Sun, 01 Mar 1998 00:00:00 +0000

NOTE: THIS DOCUMENT IS OBSOLETE, PLEASE CHECK THE NEW VERSION: "Introduction to Digital Filters with Audio Applications", by Julius O. Smith III, Copyright © 2017-11-26 by Julius O. Smith III - Center for Computer Research in Music and Acoustics (CCRMA), Stanford University

<< Previous page TOC INDEX Next page >>
Newton's Method
The gradient method is based on the first-order term in the Taylor expansion for . By taking a second-order term as well and solving the quadratic minimization problem iteratively, Newton's method for functional minimization is obtained. Essentially,Newton's method requires the error surface to be close toquadratic, and its effectiveness is directly tied to the accuracy of this assumption. For most problems, the error surface can be well approximated by a quadratic form near the solution. For this reason, Newton's method tends to give very rapid (``quadratic'') convergence in the last stages of iteration.
Newton's method is derived as follows: The Taylor expansion of about gives

for some , where . It is now necessary to assume that . Differentiating with respect to , where is presumed to be minimum, this becomes

Solving for yields

Applying Eq. (7) iteratively, we obtain the following.
Definition. Newton's method is defined by

where is given as an initial condition.
When , the answer is obtained after the first iteration. In particular, when the error surface is a quadratic form in , Newton's method produces in one iteration, i.e., for every .
For Newton's method, there is the following general result on the existence of a critical point (i.e. a point at which the gradient vanishes) within a sphere of a Banach space.
Theorem. (Kantorovich) Let be a point in for which exists, and set

Let denote the sphere . Set . If there exists a number such that

for in , and such that , then for some in , and the Newton sequence Eq. (7) converges to it. Furthermore, the rate of convergence is quadratic, satisfying

Proof. See Goldstein [Goldstein 1967], p. 143.

21. Maxims of Signal Processing

Sun, 01 Mar 1998 00:00:00 +0000

NOTE: THIS DOCUMENT IS OBSOLETE, PLEASE CHECK THE NEW VERSION: "Introduction to Digital Filters with Audio Applications", by Julius O. Smith III, Copyright © 2017-11-26 by Julius O. Smith III - Center for Computer Research in Music and Acoustics (CCRMA), Stanford University

<< Previous page TOC INDEX Next page >>
Maxims of Signal Processing
Every technique is equivalent to the same operation, once you really understand it.
If one technique is superior to another, it is due to more averaging.
With sufficiently sophisticated processing, it is no longer necessary to have any input data.
Almost all unusual or interesting results are ultimately found to be artifacts of the processing.

22. Index

Sun, 01 Mar 1998 00:00:00 +0000

NOTE: THIS DOCUMENT IS OBSOLETE, PLEASE CHECK THE NEW VERSION: "Introduction to Digital Filters with Audio Applications", by Julius O. Smith III, Copyright © 2017-11-26 by Julius O. Smith III - Center for Computer Research in Music and Acoustics (CCRMA), Stanford University

<< Previous page TOC Next page >>
Index
-transform
Convolution Representation
norms
Specific Norms
norms
Specific Norms
amplitude response
Frequency Response | Frequency Response
Banach Space
Vector Space Concepts
Cauchy sequence
Vector Space Concepts
causal
Difference Equation | Specific Norms
Chebyshev norm
Specific Norms
cluster point
Concavity (Convexity)
concave
Concavity (Convexity)
concave error surface
Concave Norms
concave functional
Concavity (Convexity)
concave hull
Concavity (Convexity)
convergence, existence of minimizer
Concave Norms
convergence, global
Concave Norms
difference equation
Difference Equation
difference equation coefficients
Difference Equation
filter
Introduction to Digital Filter
filter coefficients
Difference Equation
filter delay for amplitude envelopes
Phase Delay and Group
filter order
Difference Equation
filter power response
Frequency Response
filter time delay
Phase Delay and Group
frequency response
Frequency Response
Frobenious norm
Specific Norms
functional
Concavity (Convexity)
global convergence, conditions for
Concave Norms
global minimizer
Concavity (Convexity)
gradient
Gradient Descent
Gradient Method
Gradient Descent
group delay
Phase Delay and Group
Hankel matrix
Specific Norms
Hankel norm
Specific Norms
Hessian
Gradient Descent
impulse
Difference Equation
impulse response
Difference Equation
induced norm
Specific Norms
linear
Linearity and Time-Invariance
linear filter
Linearity and Time-Invariance
linear functional
Concavity (Convexity)
linear space
Vector Space Concepts
local minimizer
Concavity (Convexity)
LTI filters
Linearity and Time-Invariance
metric space
Vector Space Concepts
Newton's method
Newton's Method
norm
Vector Space Concepts
normed linear space
Vector Space Concepts
outer disk
Positive Real Functions
phase delay
Phase Delay and Group
phase response
Frequency Response | Frequency Response
phase response at zero frequency
Phase Delay and Group
positive real
Positive Real Functions
pseudo-norm
Vector Space Concepts
real filter
Difference Equation
root mean square norm
Specific Norms
Schur function
Relation to Schur Functions
signal
Introduction to Digital Filter
spectral norm
Specific Norms
stable
Difference Equation
stationary point
Gradient Descent
strictly concave functional
Concavity (Convexity)
time-invariant
Linearity and Time-Invariance
transfer function
Convolution Representation
uniform norm
Specific Norms
uniformly concave
Concavity (Convexity)
weighted norms
Specific Norms
<< Previous page TOC Next page >>

23. Bibliography

Sun, 01 Mar 1998 00:00:00 +0000

NOTE: THIS DOCUMENT IS OBSOLETE, PLEASE CHECK THE NEW VERSION: "Introduction to Digital Filters with Audio Applications", by Julius O. Smith III, Copyright © 2017-11-26 by Julius O. Smith III - Center for Computer Research in Music and Acoustics (CCRMA), Stanford University

<< Previous page TOC INDEX Next page >>
Bibliography
Astrom 1970
Astrom, K. J. 1970.
Introduction to Stochastic Control Theory.
New York: Academic Press.
Brune 1931
Brune, O. 1931.
``Synthesis of a finite two terminal network whose driving point impedance is a prescribed function of frequency.''
Pages 191-236 of: J. Math. and Phys., vol. 10.
Churchill 1960
Churchill, R. V. 1960.
Complex Variables and Applications.
New York: McGraw-Hill.
Genin 1981
Genin, Y. 1981 (Aug).
``An Introduction to the Model Reduction Problem with Hankel Norm Criterion.''
In: Proc. European Conf. Circuit Theory and Design, The Hague.
Goldstein 1967
Goldstein, A. A. 1967.
Constructive Real Analysis.
New York: Harper and Row.
Golub and Van Loan 1989
Golub, G. H., and C. F. Van Loan. 1989.
Matrix Computations, 2nd Edition.
Baltimore: The Johns Hopkins University Press.
Markel and Gray 1976
Markel, J. D., and A. H. Gray. 1976.
Linear Prediction of Speech.
New York: Springer Verlag.
Nehari 1952
Nehari, Z. 1952.
Conformal Mapping.
New York: Dover.
Ortega 1972
Ortega, J. M. 1972.
Numerical Analysis.
New York: Academic Press.
Papoulis 1977
Papoulis, A. 1977.
Signal Analysis.
New York: McGraw-Hill.
Rabiner and Gold 1975
Rabiner, L. R., and B. Gold. 1975.
Theory and Application of Digital Signal Processing.
Englewood Cliffs, NJ: Prentice-Hall, Inc.
Roads 1989
Roads, C. (ed). 1989.
The Music Machine.
Cambridge, MA: MIT Press.
Royden 1968
Royden, H. L. 1968.
Real Analysis.
New York: Macmillan.
Rudin 1964
Rudin, W. 1964.
Principles of Mathematical Analysis.
New York: McGraw-Hill.
Smith 1985
Smith, J. O. 1985.
``Introduction to Digital Filter Theory.''.
In: Strawn, J. (ed), Digital Audio Signal Processing: An Anthology.
Los Altos, California: William Kaufmann, Inc. a shortened version appears in [Roads 1989]. Also available as Stanford University Department of Music Technical Report STAN-M-20, April 1985.
Smith 1987
Smith, J. O. 1987.
Music Applications of Digital Waveguides.
Tech. rept. STAN-M-39. CCRMA, Music Department, Stanford University. a compendium containing four related papers and presentation overheads on digital waveguide reverberation, synthesis, and filtering. CCRMA technical reports can be ordered by calling (415)723-4971 or by sending an email request to info@ccrma.stanford.edu.
Van Valkenburg 1960
Van Valkenburg, M. E. 1960.
Introduction to Modern Network Synthesis.
New York: John Wiley and Sons, Inc.
Williamson et al. 1972
Williamson, R. E., R. H. Crowell, and H. F. Trotter. 1972.
Calculus of Vector Functions.
Englewood Cliffs, NJ: Prentice-Hall.

24. About this document

Sun, 01 Mar 1998 00:00:00 +0000

NOTE: THIS DOCUMENT IS OBSOLETE, PLEASE CHECK THE NEW VERSION: "Introduction to Digital Filters with Audio Applications", by Julius O. Smith III, Copyright © 2017-11-26 by Julius O. Smith III - Center for Computer Research in Music and Acoustics (CCRMA), Stanford University

<< Previous page TOC INDEX
About this document...
This document is a new HTML formatted version published by permission of:

"Elementary Digital Filter Theory" by Julius O. Smith III,
Adpated from "Techniques for Digital FilterDesign and System Identification, with Application to the Violin",
Julius O. Smith III, Ph.D. Dissertation, Center for Computer Research in Music and Acoustics (CCRMA), Department of Electrical Engineering, Stanford University, June 1983...

Copyright © 2000-09-19 by Julius O. Smith III

This document is also available in other formats:
Download PDF version (filters.pdf)
Download compressed PostScript version (filters.ps.gz)
Order the whole thesis containing this material.

Notes from original HTML document (https://ccrma.stanford.edu/~jos//filters):
This document was generated using theLaTeX2HTML translator Version 99.2beta6 (1.42)(patched)
Copyright © 1993, 1994, 1995, 1996,Nikos Drakos, Computer Based Learning Unit, University of Leeds.
Copyright © 1997, 1998, 1999, 2000Ross Moore, Mathematics Department, Macquarie University, Sydney.
The Makefiles and .latex2html-init file used to generate this website are available athttps://ccrma.stanford.edu/~jos//webpub/.
Hypertext links installed from a master glossary prepared using the Open Dictionary
.

25. Footnotes

Sun, 01 Mar 1998 00:00:00 +0000

NOTE: THIS DOCUMENT IS OBSOLETE, PLEASE CHECK THE NEW VERSION: "Introduction to Digital Filters with Audio Applications", by Julius O. Smith III, Copyright © 2017-11-26 by Julius O. Smith III - Center for Computer Research in Music and Acoustics (CCRMA), Stanford University

<< Previous page TOC INDEX
Footnotes
... disk.
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
... sequence
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
<< Previous page TOC INDEX

3. An Overview of ESD Control Procedures and Materials

Sun, 01 Mar 1998 00:00:00 +0000

<< Previous page INDEX Next page >>
3. An Overview of ESD Control Procedures and Materials
In Part Two---Principles of ESD Control we introduced four principles of static control and nine key elements of ESD program development and implementation. In Part Three, we will cover some of the primary specific static control procedures and materials that become part of your program. First, a quick review.
Basic Principles of Static Control
We suggested that static control programs become more effective and less complex if we focus on just four basic principles of static control as follows:

Design In Immunity by designing products and assemblies to be as immune as reasonable from the effects of ESD.
Eliminate and Reduce Generation by reducing and eliminating static generating processes, keeping processes and materials at the same electrostatic potential, and by providing appropriate ground paths to reduce charge generation and accumulation.
Dissipate and Neutralize by grounding, ionization, and the use of conductive and dissipative static control materials.
Protect Products from ESD with proper grounding or shunting and the use of static control packaging and materials handling products.
In most facilities, our static control efforts center around the last three principles. In this column we will concentrate on the primary materials and procedures that eliminate and reduce generation, dissipate and neutralize charges, or protect sensitive products from ESD.
Identifying the Problem Areas
You may also recall from our previous article that we suggested that there were at least nine critical elements to successfully developing and implementing an effective ESD control program. Selecting appropriate static control materials and developing and implementing effective procedures begins with two of these critical elements:

Identifying and classifying ESD sensitive items used in your facility.
Evaluating your facility and processes to determine the areas that need to be protected.
One of the first questions answered with this information is "Which areas of our facility need ESD protection?" Often you will find that there are more areas that require protection than you originally thought, usually wherever ESDS devices are handled. Typical areas requiring ESD protection are shown in Table 1.

Table 1
Typical Facility Areas Requiring ESD Protection

Receiving

Inspection

Stores and warehouses

Assembly

Test and inspection

Research and development

Packaging

Field service repair

Offices and laboratories

Clean rooms
Personnel and Moving Equipment

In many facilities, people are one of the prime generators of static electricity. The simple act of walking around or repairing a board can generate several thousand volts on the human body. If not properly controlled, this static charge can easily discharge into a static sensitive device—a human body model (HBM) discharge.

Even in highly automated assembly and test processes, people still handle static sensitive devices…in the warehouse, in repair, in the lab, in transport. For this reason, static control programs place considerable emphasis on controlling personnel generated electrostatic discharge. Similarly, the movement of carts and other wheeled equipment through the facility also can generate static charges that can transfer to the products being transported on this equipment.

Wrist Straps
Typically, the primary means of controlling static charge on personnel is with a wrist strap. When properly worn and connected to ground, a wrist strap keeps the person wearing it near ground potential. Because the person and other grounded objects in the work area are at or near the same potential, there can be no hazardous discharge between them. In addition, static charges are safely dissipated from the person to ground and do not accumulate.
Wrist straps have two major components, the cuff that goes around the person’s wrist and the ground cord that connects the cuff to the common point ground. Most wrist straps have a current limiting resistor molded into the ground cord head on the end that connects to the cuff. The resistor most commonly used is a one megohm, 1/4 watt with a working voltage rating of 250 volts.
Wrist straps should be tested on a regular basis. Daily testing or continuous monitoring is recommended.
Floors, Floor Mats, Floor Finishes
A second method of controlling electrostatic charge on personnel is with the use of ESD protective floors in conjunction with ESD control footwear or footstraps. The combination of floor materials and footwear provides a ground path for the dissipation of electrostatic charge, thus reducing the charge accumulation on personnel and other objects to safe levels. In addition to dissipating charge, some floor materials (and floor finishes) also reduce triboelectric charging. The use of floor materials is especially appropriate in those areas where increased personnel mobility is necessary. In addition, floor materials can minimize charge accumulation on chairs, carts, walking stackers, lift trucks and other objects that move across the floor. However, those items require dissipative or conductive casters or wheels to make electrical contact with the floor.
Shoes, Grounders, Casters
Used in combination with ESD protective floor materials, static control shoes, grounders, casters and wheels provide the necessary electrical contact between the person or object and the floor material. Insulative footwear, casters, or wheels prevent static charges from flowing from the body to the floor to ground.
Clothing
Clothing is a consideration in some ESD protective areas, especially in clean rooms and very dry environments. Clothing materials can generate electrostatic charges when they contact and separate from other objects and the clothing itself. These charges may discharge into sensitive components or create electrostatic fields that may induce charges on the human body. Although a person may be grounded, that does not mean that insulative clothing fabrics can dissipate a charge to that person's skin and then to ground. Clothing usually is electrically insulated or isolated from the body. Grounded static control garments are intended to minimize the effects of electrostatic fields or charges that may be present on a person’s clothing.

Figure 1--Typical ESD Workstation
Workstations and Worksurfaces
Worksurfaces and workstations are important parts of an ESD protective program. Many ESDS devices and assemblies are handled, assembled or repaired at workstations.
An ESD protective workstation refers to the work area of a single individual that is constructed and equipped with materials and equipment to limit damage to ESD sensitive items. It may be a stand-alone station in a stockroom, warehouse, or assembly area, or in a field location such as a computer bay in commercial aircraft. A workstation also may be located in a controlled area such as a clean room.
The workstation provides a means for connecting all worksurfaces, fixtures, handling equipment, and grounding devices to a common point ground. In addition, there may be provision for connecting additional personal grounding devices, equipment, and accessories such as constant ground monitors and ionizers. The key ESD control elements comprising most workstations are a static dissipative worksurface, a means of grounding personnel (usually a wrist strap), a common grounding connection, and appropriate signage and labeling. A typical workstation is shown in Figure 1.
Static protective worksurfaces with a resistance to ground of 106 to 109 provide a surface that is at the same electrical potential as other ESD protective items in the workstation. They also provide an electrical path to ground for the controlled dissipation of any static potentials on materials that contact the surface. The worksurface also helps define a specific work area in which ESD sensitive devices may be safely handled. The worksurface is connected to the common point ground.
Production Equipment and Production Aids
Although personnel generated static is typically the primary ESD culprit in many environments, automated manufacturing and test equipment can also pose an ESD problem. For example, a device may become charged from sliding down the feeder. If the device then contacts the insertion head or another conductive surface, a rapid discharge occurs from the device to the metal object--a Charged Device Model (CDM) event.
In addition, various production aids may also pose an ESD problem. Production aids are those materials, tools and fixtures that help to produce finished products but do not become part of the finished product. Some examples are hand tools, soldering irons, tapes, solvents, and so forth.
Grounding is the primary means of controlling static charge on many production aids and equipment. The metal chassis or conductive enclosure of equipment that uses utility power is required by the National Electrical Code to be connected to the equipment ground (the green wire) in order to carry fault currents. This ground connection also will function for ESD purposes. All electrical tools and equipment used to process ESD sensitive hardware require the 3 prong grounded type AC plug. Hand tools that are not electrically powered, i.e., pliers, wire cutters, and tweezers, are usually grounded through the ESD worksurface and the (grounded) person using the conductive tools. Holding fixtures should be made of conductive or static dissipative materials when possible. If a conductive fixture is not sitting on a ESD worksurface or handled by a grounded person, a separate ground wire may be required. For those items that are composed of insulative materials, the use of ionization or application of topical antistats may be required to control generation and accumulation of static charges.
Packaging and Materials Handling
Direct protection of ESDS devices from electrostatic discharge is provided by packaging materials such as bags, corrugated, and rigid or semi-rigid packages. The primary use of these items is to protect the product when it leaves the facility, usually when shipped to a customer. In addition, materials handling products such as tote boxes and other containers primarily provide protection during inter or intra facility transport.
The main ESD function of these packaging and materials handling products is to limit the possible impact of ESD from triboelectric charge generation, direct discharge, and electrostatic fields. The initial consideration is to have low charging materials (antistatic) in contact with ESD sensitive items. For example, the antistatic property would control triboelectric charge resulting from sliding a board or component into the package or container. A second requirement is that the material provide protection from direct electrostatic discharge as well as shield from electrostatic fields.
Many materials are available that provide all three benefits: antistatic, discharge protection, and electric field suppression. The inside of these packaging materials have an antistatic layer, but also have an outer layer with a surface resistance generally in the dissipative range.
A material’s antistatic properties are not necessarily predicted by its resistance or resistivity. However, resistance or resistivity measurements help define the material’s ability to provide electrostatic shielding or charge dissipation. Electrostatic shielding attenuates electrostatic fields on the surface of a package in order to prevent a difference in electrical potential from existing inside the package. Electrostatic shielding is provided by materials that have a surface resistance equal to or less than 1.0 x 10E3 when tested according to EOS/ESD-S11.11 or a volume resistivity of equal to or less than 1.0 x 10E3 ohm-cm when tested according to the methods of EIA 541. In addition, shielding may be provided by packaging materials that provide an air gap between the package and the product.
Dissipative materials provide charge dissipation characteristics. These materials have a surface resistance greater than 1.0 x 10E4 but less than or equal to 1.0 x 10E11 when tested according to EOS/ESD-S11.11 or a volume resistivity greater than 1.0 x 10E5 ohm-cm but less than or equal to 1.0 x 10E12 ohm-cm when tested according to the methods of EIA 541. Be aware that the very wide range of resistance and resistivity results in a wide range of performance.
Grounding
In our discussion to this point, we have seen how important grounding is to effective ESD control. Consequently, effective ESD grounds are of critical importance in any operation, and ESD grounding should be clearly defined and regularly evaluated.
ESD Association Standard ANSI EOS/ESD 6.1-Grounding recommends a two-step procedure for grounding ESD protective equipment.
The first step is to ground all components of the work area (worksurfaces, people, equipment, etc.) to the same electrical ground point called the "common point ground." This common point ground is defined as a "system or method for connecting two or more grounding conductors to the same electrical potential."
Figure 2--Common Point Ground Symbol
This ESD common point ground should be properly identified. ESD Association standard EOS/ESD S8.1-1993 recommends the use of the symbol in Figure 2 to identify the common point ground.
The second step is to connect the common point ground to the equipment ground or the third wire (green) electrical ground connection. This is the preferred ground connection because all electrical equipment at the workstation is already connected to this ground. Connecting the ESD control materials or equipment to the equipment ground brings all components of the workstation to the same electrical potential. If a soldering iron used to repair an ESDS item were connected to the electrical ground and the surface containing the ESDS item were connected to an auxiliary ground, a difference in electrical potential could exist between the iron and the ESDS item. This difference in potential could cause damage to the item.
Any auxiliary grounds (water pipe, building frame, ground stake) present and used at the workstation must be bonded to the equipment ground to minimize differences in potential between the two grounds.
Ionization
As we have seen, the primary method of static charge control is direct connection to ground for conductors, static dissipative materials, and personnel. However, a complete static control program must also deal with isolated conductors that cannot be grounded, insulating materials (e.g., most common plastics). Topical antistats often are used to dissipate static charges from these items under some circumstances
More frequently, however, air ionization can neutralize the static charge on insulated and isolated objects by charging the molecules of the gases of the surrounding air. Whatever static charge is present on objects in the work environment will be neutralized by attracting opposite polarity charges from the air. Because it uses only the air that is already present in the work environment, air ionization may be employed even in clean rooms where chemical sprays and some static dissipative materials are not usable.
Air ionization is not a replacement for grounding methods. It is one component of a complete static control program. Ionizers are used when it is not possible to properly ground everything and as backup to other static control methods. In clean rooms, air ionization may be one of the few methods of static control available.
Clean Room Requirements
While the basic methods of static control discussed here are applicable in most environments, there are characteristics of the semiconductor manufacturing process that require special considerations.
Many objects integral to the semiconductor manufacturing process (quartz, glass, plastic, and ceramic) are inherently charge generating. Because these materials are insulators, this charge cannot be removed easily by grounding. Many static control materials contain carbon particles or surfactant additives that sometimes restrict their use in clean rooms. The need for personnel mobility and the use of clean room garments often make the use of wrist straps difficult. In these circumstances, ionization and flooring/footwear systems become key weapons against static charge.
Identification
A final element in our static control program is the use of appropriate symbols to identify static sensitive devices and assemblies, as well as products intended to control ESD. The traditional symbols traditionally used to identify ESDS parts or ESD control materials have been replaced with newer, more appropriate symbols. ESD Association Standard ANSI ESD S8.1-1993 — ESD Awareness Symbols provides two symbols for ESD identification.
Figure 3--ESD Susceptibility Symbol
The ESD Susceptibility Symbol (Figure 3), consists of a triangle, a reaching hand, and a slash through the reaching hand. The triangle means "caution" and the slash through the reaching hand means "Don’t touch." Because of its broad usage, the hand in the triangle has become associated with ESD and the symbol literally translates to "ESD sensitive stuff, don’t touch."
The ESD Susceptibility Symbol is applied directly to integrated circuits, boards, and assemblies that are static sensitive. It indicates that handling or use of this item may result in damage from ESD if proper precautions are not taken. If desired, the sensitivity level of the item may be added to the label.
Figure 4-- ESD Protective Symbol
The ESD Protective Symbol (Figure 4), consists of the reaching hand in the triangle. An arc around the triangle replaces the slash. This "umbrella" means protection. The symbol indicates ESD protective material. It is applied to mats, chairs, wrist straps, garments, packaging, and other items that provide ESD protection. It also may be used on equipment such as hand tools, conveyor belts, or automated handlers that is especially designed or modified to provide ESD control.
Neither symbol is applied on ESD test equipment, footwear checkers, wrist strap testers, resistance or resistivity meters or similar items that are used for ESD purposes, but which do not provide actual protection.
Click here to download copies of these two symbols or the common point ground symbol.
Summary
Effective static control programs require a variety of procedures and materials. In this column, we have provided a brief overview of the most commonly used elements of a program. Additional in-depth discussion of individual materials and procedures can be found in publications such as the ESD Handbook published by the ESD Association.
Your program is up and running. How do you determine whether it is effective? How do you make sure your employees follow it? In Part 4, we will cover the topics of Auditing and Training.
For Additional Information
ESD Association Standards

ESD-STM1.1—Evaluation, Acceptance, and Functional Testing of Wrist Straps.
ANSI EOS/ESD S3.1—Ionization
ESD STM 4.1 (Revised)--ESD Protective Worksurfaces—Resistance Measursements
ESD STM4.2—Worksurfaces - Charge Dissipation Characteristics
ANSI EOS/ESD S6.1—Grounding -- Recommended Practice
ANSI ESD S7.1—Floor Materials — Resistive Characterization of Materials
ANSI ESD S8.1—ESD Awareness Symbols
ESD S9.1—Resistive Characterization of Footwear
ANSI ESD S11.11—Surface Resistance Measurement of Static Dissipative Planar Materials
ANSI ESD S11.31—Evaluating the Performance of Electrostatic Discharge Shielding Bags
ANSI/ESD 20.20—Electrostatic Discharge Control Program
ESD STM2.1—Garments
ESD STM12.1—Seating-Resistive Characterization
ESD STM97.1—Floor Materials and Footwear – Resistance Measurement in Combination with a Person.
ESD STM97.2-1999—Floor Materials and Footwear – Voltage Measurement in Combination with a Person.
ESD DSP 3.3—Periodic Verification of Air Ionizers
ESD ADV53.1— ESD Protective Workstations
ESD ADV2.0--ESD Handbook
Other Standards

EIA-541, Packaging of Electronic Products for Shipment, Electronic Industries Alliance

3. HOW MEMORY WORKS

Sun, 01 Mar 1998 00:00:00 +0000

<< Previous page TOC Next page >>
3. HOW MEMORY WORKS
Earlier, we talked about how memory holds information in a place where the CPU can get to it quickly. Let's look at that process in more detail.

3.1. HOW MEMORYWORKS WITH THE PROCESSOR
3.2. MAXIMIZING PERFORMANCE
<< Previous page TOC Next page >>

3.1. HOW MEMORYWORKS WITH THE PROCESSOR

Sun, 01 Mar 1998 00:00:00 +0000

<< Previous page TOC Next page >>
3.1. HOW MEMORYWORKS WITH THE PROCESSOR

Main components of a computer system.
The CPU is often referred to as the brain of the computer. This is where all the actual computing is done.
The chipset supports the CPU. It usually contains several "controllers" which govern how information travels between the processor and other components in the system. Some systems have more than one chipset.

3.2. MAXIMIZING PERFORMANCE

Sun, 01 Mar 1998 00:00:00 +0000

<< Previous page TOC Next page >>
3.2. MAXIMIZING PERFORMANCE
Computer processor speeds have been increasing rapidly over the past several years. Increasing the speed of the processor increases the overall performance of the computer. However, the processor is only one part of the computer, and it still relies on other components in a system to complete functions. Because all the information the CPU will process must be written to or read from memory, the overall performance of a system is dramatically affected by how fast information can travel between the CPU and main memory.

So, faster memory technologies contribute a great deal to overall system performance. But increasing the speed of the memory itself is only part of the solution. The time it takes for information to travel between memory and the processor is typically longer than the time it takes for the processor to perform its functions. The technologies and innovations described in this section are designed to speed up the communication process between memory and the processor.

CACHE MEMORY
Cache memory is a relatively small amount (normally less than 1MB) of high speed memory that resides very close to the CPU. Cache memory is designed to supply the CPU with the most frequently requested data and instructions. Because retriev-ing data from cache takes a fraction of the time that it takes to access it from main memory, having cache memory can save a lot of time. If the information is not in cache, it still has to be retrieved from main memory, but checking cache memory takes so little time, it's worth it. This is analogous to checking your refrigerator for the food you need before running to the store to get it: it's likely that what you need is there; if not, it only took a moment to check.

The concept behind caching is the "80/20" rule, which states that of all the programs, information, and data on your computer, about 20% of it is used about 80% of the time. (This 20% data might include the code required for sending or deleting email, saving a file onto your hard drive, or simply recognizing which keys you've touched on your keyboard.) Conversely, the remaining 80% of the data in your system gets used about 20% of the time. Cache memory makes sense because there's a good chance that the data and instructions the CPU is using now will be needed again.
HOW CACHE MEMORY WORKS
Cache memory is like a "hot list" of instructions needed by the CPU. The memory controller saves in cache each instruction the CPU requests; each time the CPU gets an instruction it needs from cache - called a "cache hit" - that instruction moves to the top of the "hot list." When cache is full and the CPU calls for a new instruction, the system overwrites the data in cache that hasn't been used for the longest period of time. This way, the high priority information that's used continuously stays in cache, while the less frequently used information drops out.
LEVELS OF CACHE
Today, most cache memory is incorporated into the processor chip itself; however, other configurations are possible. In some cases, a system may have cache located inside the processor, just outside the processor on the motherboard, and/or it may have a memory cache socket near the CPU, which can contain a cache memory module. Whatever the configuration, any cache memory component is assigned a "level" according to its proximity to the processor. For example, the cache that is closest to the processor is called Level 1 (L1) Cache, the next level of cache is numbered L2, then L3, and so on. Computers often have other types of caching in addition to cache memory. For example, sometimes the system uses main memory as a cache for the hard drive. While we won't discuss these scenarios here, it's important to note that the term cache can refer specifically to memory and to other storage technologies as well.

You might wonder: if having cache memory near the processor is so beneficial, why isn't cache memory used for all of main memory? For one thing, cache memory typically uses a type of memory chip called SRAM (Static RAM), which is more expensive and requires more space per megabyte than the DRAM typically used for main memory. Also, while cache memory does improve overall system performance, it does so up to a point. The real benefit of cache memory is in storing the most frequently-used instructions. A larger cache would hold more data, but if that data isn't needed frequently, there's little benefit to having it near the processor.

It can take as long as 195ns for main memory to satisfy a memory request from the CPU. External cache can satisfy a memory request from the CPU in as little as 45ns.
SYSTEM BOARD LAYOUT
As you've probably figured out, the placement of memory modules on the system board has a direct effect on system performance. Because local memory must hold all the information the CPU needs to process, the speed at which the data can travel between memory and the CPU is critical to the overall performance of the system. And because exchanges of information between the CPU and memory are so intricately timed, the distance between the processor and the memory becomes another critical factor in performance.
INTERLEAVING
The term interleaving refers to a process in which the CPU alternates communication between two or more memory banks. Interleaving technology is typically used in larger systems such as servers and workstations. Here's how it works: every time the CPU addresses a memory bank, the bank needs about one clock cycle to "reset" itself. The CPU can save processing time by addressing a second bank while the first bank is resetting. Interleaving can also function within the memory chips themselves to improve performance. For example, the memory cells inside SDRAM chip are divided into two independent cell banks, which can be activated simultaneously. Interleaving between the two cell banks produces a continuous flow of data. This cuts down the length of the memory cycle and results in faster transfer rates.
BURSTING
Bursting is another time-saving technology. The purpose of bursting is to provide the CPU with additional data from memory based on the likelihood that it will be needed. So, instead of the CPU retrieving information from memory one piece of at a time, it grabs a block of information from several consecutive addresses in memory. This saves time because there's a statistical likelihood that the next data address the processor will request will be sequential to the previous one. This way, the CPU gets the instructions it needs without having to send an individual request for each one. Bursting can work with many different types of memory and can function when reading or writing data.

Both bursting and pipelining became popular at about the same time that EDO technology became available. EDO chips that featured these functions were called "Burst EDO" or "Pipeline Burst EDO" chips.
PIPELINING
Pipelining is a computer processing technique where a task is divided into a series of stages with some of the work completed at each stage. Through the division of a larger task into smaller, overlapping tasks, pipelining is used to improve performance beyond what is possible with non-pipelined processing. Once the flow through a pipeline is started, execution rate of the instructions is high, in spite of the number of stages through which they progress.

4. Auditing and Training

Sun, 01 Mar 1998 00:00:00 +0000

<< Previous page INDEX Next page >>
4. Auditing and Training
Your static control program is up and running. How do you determine whether it is effective? How do you make sure your employees follow it? Previously, we suggested that there were at least nine critical elements to successfully developing and implementing an effective ESD control program. In Part Four, we will focus on two more of these elements: training and auditing.
Personnel Training
The procedures are in place. The materials are in use. But, your ESD control program just does not seem to yield the expected results. Failures declined initially, but they have begun reversing direction. Or perhaps there was little improvement at all. The solutions might not be apparent in inspection reports of incoming ESD materials. Nor in the wrist strap log-in sheets. Sometimes simply turning the pages of the calendar will provide significant clues. When was the last time your employees attended an ESD training program? In fact, have your employees been trained at all?
In large companies or small, it is hard to underestimate the role of training in an ESD control program. There is significant evidence to support the contribution of training to the success of the program (References 1, 5, 9, 10, 14, 15, 16). We would not send employees to the factory floor without the proper soldering skills or the knowledge to operate the automated insertion equipment. We should provide them with the same skill level regarding ESD control procedures.
Elements of Effective Training Programs
Although individual requirements cause training programs to vary from company to company, there are several common threads that run through the successful programs.
1 -- Successful training programs cover all affected employees.
Obviously we train the line employees who test their wrist straps or place finished products in static protective packaging. But we also include department heads, upper management, and executive personnel in the process. Typically they are responsible for the day-to-day supervision and administration of the program or they provide leadership and support. The training will vary for these different groups, but awareness and understanding at all levels is key.
2 -- Effective training is comprehensive and consistent.
Training not only covers specific procedures, but also the physics of the problem and the benefits of the program as well. Consistent content across various groups, plants, and even countries (adjusted for cultural differences, of course) reduces confusion and helps assure conformance.
3 -- Use a variety of training tools and techniques.
Combine live instruction with training videos or interactive CD-ROM programs. Effective training involves employees in the process. Reinforce the message with demonstrations of ESD events and their impact. Bulletin boards, newsletters, and posters provide additional reminders and reinforcement.
4 -- Test, certify and retrain.
Your training should assure material retention and emphasize the importance of the effort. If properly implemented, testing and certification motivates and builds employee pride. Retraining is an ongoing process that reinforces, reminds, and provides opportunities for implementing new or improved procedures.
5 -- Feedback, auditing, and measurement.
Motivate and provide the mechanism for program improvement. Sharing yield or productivity data with employees demonstrates the effectiveness of the program and of their efforts. Tracking these same numbers can indicate that it’s time for retraining or whether modifications are required in the training program.
Design and delivery of an effective ESD training program can be just as important as the procedures and materials used in your ESD control program. A training program that is built on identifiable and measurable performance goals helps assure employee understanding, implementation and success.
Auditing
Developing and implementing an ESD control program itself is obvious. What might not be so obvious is the need to continually review, audit, analyze, feedback and improve. You will be asked to continually identify the program’s return on investment and to justify the savings realized. Technological changes will dictate improvements and modifications. Feedback to employees and top management is essential. Management commitment will need reinforcement.
Like training, regular auditing becomes a key ingredient in the successful management of ESD control programs. The mere presence of the auditing process spurs compliance with program procedures. It helps strengthen management’s commitment. Audit reports trigger corrective action and help foster continuous improvement.
The benefits to be gained from regular auditing of our ESD control procedures are numerous. They allow us to prevent problems before they occur rather than always fighting fires. They allow us to readily identify problems and take corrective action. They identify areas in which our programs may be weak and provide us with information required for continuous improvement. They allow us to leverage limited resources effectively. They help us determine when our employees need to be retrained. They help us improve yields, productivity, and capacity. And they help us bind our ESD program together into a successful effort.
An ESD audit measures performance to the defined standards and procedures of the ESD control program. Typically, we think of an ESD audit as a periodic review and inspection of the ESD work area covering use of the correct packaging materials, wearing of wrist straps, following defined procedures, and similar items. However, a good audit program is a bit more complicated.
Types of Audits
There are several types of ESD audits: the work place audit, program management audits, and quality process checking. Each type is distinctively different and each is vitally important to the success of the ESD program
Program management audits measure how well a program is managed and how strong management commitment is. The program management audit emphasizes factors such as the existence of an effective implementation plan, realistic program requirements, ESD training programs, regular audits, and other critical factors of program management. The program management audit typically is conducted by a survey specifically tailored to the factors being reviewed. Because it’s a survey, the audit can be conducted without actually visiting the site. The results of this audit indirectly measure work place compliance and are particularly effective as a means of self-assessment for small companies as well as large global corporations.
Quality process checking applies classical statistical quality control procedures to the ESD process and is performed by operations personnel. This is not a periodic audit, but rather daily maintenance of the program. Visual and electrical checks of the procedures and materials, wrist strap testing for example, are used to monitor the quality of the ESD control process. Checking is done on a daily, weekly or monthly basis.
Trend charts and detailed records trigger process adjustments and corrective action. They help assure that specified procedures are followed on a regular basis. The records are essential for quality control purposes, corrective action and compliance with ISO-9000.
Work area audits verify that program procedures are followed and that ESD control materials and equipment are within specification or are functioning properly. Audits are performed on a regular basis, often monthly, and utilize sampling techniques and statistical analysis of the results. The use of detailed checklists and a single auditor assures that all items are covered and that the audits are performed consistently over time.
Basic Auditing Instrumentation
Special instrumentation will be required to conduct work area audits. The specific instrumentation will depend on what you are trying to measure, the precision you require and the sophistication of your static control and material evaluation program. However, as a minimum, you will need an electrostatic field meter, a wide range resistance meter, a ground/circuit tester, and appropriate electrodes and accessories. Additional instrumentation might include a charge plate monitor, footwear and wrist strap testers, chart recorders and timing devices, discharge simulators, and ESD event detectors.
Although this equipment must be accurate, it need not be as sophisticated as laboratory instruments. The audit is intended to verify basic functions and not as a full qualification of ESD control equipment or materials. Remember, you want the right tool for job. Just as you would not buy a hammer if you are were planning to saw wood, you would not purchase an electrometer to measure static voltages on a production line. If you are making measurements according to specific standards, be sure the instrumentation meets the specifications of these standards.
With a hand held electrostatic field meter, you can measure the presence of electrostatic charge in your environment allowing you to identify problem areas and monitor your ESD control program. These instruments measure the electrostatic field associated with a charged object. Many field meters simply measure the gross level of the electrostatic field and should be used as general indicators of the presence of a charge and the approximate level of this charge. Others will provide more precise measurement for material evaluation and comparison.
For greater precision in facility measurements or for laboratory evaluation, a charge plate monitor can be attached to some field meters or connected to a voltmeter in the laboratory. With these additional tools you can evaluate the performance of flooring materials or balance ionizing equipment, for example.
Because resistance is one of the key factors in evaluating ESD control materials, a wide range resistance meter becomes a crucial instrument. Most resistance measurements are made at 100 volts, and some at 10 volts. The equipment you choose should be capable of applying these voltages to the materials being tested. In addition, the meter should be capable of measuring resistance ranges of 10⁵ to 10¹¹ ohms. With the proper electrodes and cables, you will be able to measure the resistance of flooring materials, worksurfaces, equipment, furniture, garments, and some packaging materials.
The final instrument is a ground/circuit tester. With this device you can measure the continuity of your ESD grounds and also check the impedance as well as neutral to ground shorts.
Areas, Processes, and Materials to be Audited
In our last column we stated that ESD protection was required wherever "wherever ESDS devices are handled." Obviously, our audits need to include these same areas. Table 1 indicates some of the physical areas that require ESD protection and auditing of the program.

Table 1
Typical Facility Areas To Be Audited

Receiving

Inspection

Stores and Warehouses

Assembly

Test and Inspection

Research and Development

Packaging

Field Service Repair

Offices and Laboratories

Clean Rooms
Similarly, we need to conduct work area and program management audits all of the various processes, materials, and procedures that are used in our ESD control programs. Some of these are shown in Table 2.

Table 2
Typical Processes, Materials and Procedures
To Be Audited

Personnel

Moving Equipment (Carts, lift trucks)

Wrist Straps

Floors, Floor Mats, Floor Finishes

Shoes, Grounders, Casters

Clothing

Workstations

Worksurfaces

Packaging and Materials Handling

Ionization

Grounding

Production Equipment

Production Aids (Soldering irons, fixtures, etc.)

Labeling and Identification

Purchasing Specifications and Requisitions

ESD Control Program Procedures and Specifications

ESD Measurement and Test Equipment

Personnel Training

Engineering Specifications and Drawings
Check Lists
Check lists can be helpful tools for conducting work place and program audits. However, it is important that ESD control program requirements are well documented and accessible to avoid a tendency for checklists becoming de facto lists of requirements. Table 3 indicates the type of questions and information that might be included in an auditing check list. Your own check lists, of course, will be based on your specific needs and program requirements. They should conform to your actual ESD control procedures and specifications and they should be consistent with any requirements you may have for ISO 9000.
In addition to check lists, you will use various forms for recording the measurements you make: resistance, voltage generation, etc. Part of your audit will also include the daily logs used on the factory floor such as those used for wrist strap checking.

Table 3
Sample Audit Check List
ESD Control Program

Function/Area Audited: Personnel

Date:

Auditor:

Audit Questions

Y

N

Comments

1. Where ESD protective flooring is used for personnel grounding, are foot grounding devices or conductive footwear worn?

2. Where conductive floors and footwear are used for personnel grounding, do personnel check continuity to ground upon entering the area?

3. Are personnel wearing grounded wrist straps at the ESD protective workstations?

4. Are personnel checking wrist straps for continuity or using a continuous ground monitor?

5. Where continuous ground monitors are not used, are wrist straps checked and logged routinely and at frequent intervals?

6. Are wriststrap checkers and continuous ground monitors checked and maintained periodically?

7. Do wrist straps and foot grounders fit correctly?

8. Are wrist straps and foot grounders working correctly?

9. Are wrist strap cords checked, on the person, at the workstation?

10. Are disposable foot grounders limited to one time use?

11. Are test records for wrist straps and foot grounders kept and maintained?

12. When required, are ESD protective garments correctly worn?

13. Are nonessential personal items kept out of ESD controlled areas?

14. Are finger cots and gloves being used on ESD sensitive hardware made of ESD protective material, or treated to be ESD protective?

15. Are personnel working in the ESD controlled area currently certified or escorted?

16. Are all personnel with access to the ESD controlled area trained?

17. Are ESD Control requirements imposed on visitors?

Reporting, Corrective Action
Upon completion of the auditing process, Reports should be prepared and distributed in a timely manner. Details of the audits need to be fully documented for ISO-9000. As with all audits, it is essential to implement corrective action if deficiencies are discovered. Trends need to be tracked and analyzed to help establish corrective action, which may include retraining of personnel, revision of requirement documents or processes, or modification of the existing facility.

Conclusion
Auditing and training are key elements in maintaining an effective ESD control program. They help assure that procedures are properly implemented and can provide a management tool to gauge program effectiveness and make continuous improvement.
For Further Reference

"A Successful ESD Training Program," L. Snow and G. T. Dangelmayer, EOS/ESD Symposium Proceedings, 1994, ESD Association, Rome, NY
"A Systematic ESD Program Revisited," G. T. Dangelmayer, EOS/ESD Symposium Proceedings, 1992, ESD Association, Rome, NY
"Auditing—The Essential Binding Force," G.T. Dangelmayer, EOS/ESD Symposium Tutorial, 1996
"Developing and Maintaining an Effective ESD Training Program," F. Dinger, EOS/ESD Symposium Proceedings, 1988, ESD Association, Rome, NY
"Effectiveness of ESD Training Using Multimedia," G. Smalanskas, J. Mason, EOS/ESD Symposium Proceedings, 1995, ESD Association, Rome, NY
"Employee Training for Successful ESD Control," G. T. Dangelmayer, E. S. Jesby, EOS/ESD Symposium Proceedings, 1985, ESD Association, Rome, NY
"ESD Demonstrations to Increase Engineering and Manufacturing Awareness," G. Baumgartner, EOS/ESD Symposium Proceedings, 1996, ESD Association, Rome, NY
Fundamentals of ESD Evaluation and Auditing, S. Halperin, Regional ESD Tutorial, 1998ESD Program Management: A Realistic Approach to Continuous Improvement in Static Control, G. T. Dangelmayer, Van Nostrand Reinhold, 1999
"Implementation of Computer-Based ESD Training: A Case Study Comparing the Computer Approach with Traditional Classroom Techniques," J. Woodward-Jack, H. Sommerfeld, EOS/ESD Symposium Proceedings, 1991, ESD Association, Rome, NY
"Implementing an ESD Program in a Multi-National Company: A Cross-Cultural Experience," W. H. Tan, EOS/ESD Symposium Proceedings, 1994, ESD Association, Rome, NY
"Internal Quality Auditing and ESD Control," D. H. Smith and C.D. Rier, EOS/ESD Symposium Proceedings, 1986, ESD Association, Rome, NY
"Modular ESD Certification Training Program," M. Berkowitz, B. Hamel, EOS/ESD Symposium Proceedings, 1989, ESD Association, Rome, NY
"The Production Operator: Weak Link or Warrior in the ESD Battle?", G. E. Hansel, EOS/ESD Symposium Proceedings, 1983, ESD Association, Rome, NY
"Tracking Results of an ESD Control Program Within a Telecommunications Service Company," R. J. Zezulka, EOS/ESD Symposium Proceedings, 1989, ESD Association, Rome, NY
"You’ve Implemented An ESD Program -- What’s Next?", W. Y. McFarland, R. A. Brin, EOS/ESD Symposium Proceedings, 1993, ESD Association, Rome, NY
ESD ADV2.0--ESD Handbook, ESD Association, Rome, NY
ANSI/ESD 20.20—Electrostatic Discharge Control Program, ESD Association, Rome, NY

4. HOW MUCH MEMORY IS ON A MODULE?

Sun, 01 Mar 1998 00:00:00 +0000

<< Previous page TOC Next page >>
4. HOW MUCH MEMORY IS ON A MODULE?
Up to now, we've discussed some of the technical attributes of memory and how memory functions in a system. What's left are the technical details - the "bits and bytes," as they say. This section covers the binary numbering system, which forms the basis of computing, and calculation of a memory module's capacity.

4.1. BITS AND BYTES
4.2. CPU AND MEMORY REQUIREMENTS
4.3. CALCULATING THE CAPACITY OF A MODULE
4.4. STACKING

4.1. BITS AND BYTES

Sun, 01 Mar 1998 00:00:00 +0000

<< Previous page TOC Next page >>
4.1. BITS AND BYTES
Computers speak in a "code" called machine language, which uses only two numerals: 0 and 1. Different combinations of 0s and 1s form what are called binary numbers. These binary numbers form instructions for the chips and microprocessors that drive computing devices - such as computers, printers, hard disk drives, and so on. You may have heard the terms "bit" and "byte." Both of these are units of information that are important to computing. The term bit is short for "binary digit." As the name suggests, a bit represents a single digit in a binary number; a bit is the smallest unit of information used in computing and can have a value of either 1 or a 0. A byte consists of 8 bits. Almost all specifications of your computer's capabilities are represented in bytes. For example, memory capacity, data-transfer rates, and data-storage capacity are all measured in bytes or multiples thereof (such as kilobytes, megabytes, or gigabytes).

This discussion of bits and bytes becomes very relevant when it comes to computing devices and components working together. Here, we'll address specifically how bits and bytes form the basis of measuring memory component performance and interaction with other devices like the CPU.

4.2. CPU AND MEMORY REQUIREMENTS

Sun, 01 Mar 1998 00:00:00 +0000

<< Previous page TOC Next page >>
4.2. CPU AND MEMORY REQUIREMENTS
A computer's CPU (central processing unit) processes data in 8-bit chunks. Those chunks, as we learned in the previous section, are commonly referred to as bytes. Because a byte is the fundamental unit of processing, the CPU's processing power is often described in terms of the maximum number of bytes it can process at any given time. For example, Pentium and PowerPC microprocessors currently are 64-bit CPUs, which means they can simultaneously process 64 bits, or 8 bytes, at a time.

Each transaction between the CPU and memory is called a bus cycle. The number of data bits a CPU can transfer during a single bus cycle affects a computer's performance and dictates what type of memory the computer requires. Most desktop computers today use 168-pin DIMMs, which support 64-bit data paths. Earlier 72-pin SIMMs supported 32-bit data paths, and were originally used with 32-bit CPUs. When 32-bit SIMMs were used with 64-bit processors, they had to be installed in pairs, with each pair of modules making up a memory bank. The CPU communicated with the bank of memory as one logical unit.

Interestingly, RIMM modules, which are newer than DIMMs, use smaller 16-bit data paths; however they transmit information very rapidly, sending several packets of data at a time. RIMM modules use pipelining technology to send four 16-bit packets at a time to a 64-bit CPU, so information still gets processed in 64-bit chunks.

4.3. CALCULATING THE CAPACITY OF A MODULE

Sun, 01 Mar 1998 00:00:00 +0000

<< Previous page TOC Next page >>
4.3. CALCULATING THE CAPACITY OF A MODULE
Memory holds the information that the CPU needs to process. The capacity of memory chips and modules are described in megabits (millions of bits) andmegabytes (millions of bytes). When trying to figure out how much memory you have on a module, there are two important things to remember:

A module consists of a group of chips. If you add together the capacities of all the chips on the module, you get the total capacity of the module. Exceptions to this rule are:

If some of the capacity is being used for another function, such as error checking.

If some of the capacity is not being used, for example some chips may have extra rows to be used as back-ups. (This isn't common.)
While chip capacity is usually expressed in megabits, module capacity is expressed in megabytes. This can get confusing, especially since many people unknowingly use the word "bit" when they mean "byte" and vice versa. To help make it clear, we'll adopt the following standards in this book:

When we talk about the amount of memory on a module, we'll use the term "module capacity"; when we are referring to chips, we'll use the term "chip density". Module capacity will be measured in megabytes (MB) with both letters capital, and chip density will be measured in megabits (Mbit), and we'll spell out the word "bit" in small letters.
COMPONENT CAPACITY EXPRESSION CAPACITY UNITS EXAMPLE
Chips Chip Density Mbit (megabits) 64Mbit
Memory Modules Module Capacity MB (megabytes) 64MB
CHIP DENSITY
Each memory chip is a matrix of tiny cells. Each cell holds one bit of information. Memory chips are often described by how much information they can hold. We call this chip density. You may have encountered examples of chip densities, such as "64Mbit SDRAM" or "8M by 8". A 64Mbit chip has 64 million cells and is capable of holding 64 million bits of data. The expression "8M by 8" describes one kind of 64Mbit chip in more detail.

In the memory industry, DRAM chip densities are often described by their cell organi-zation. The first number in the expression indicates the depth of the chip (in locations) and the second number indicates the width of the chip (in bits). If you multiply the depth by the width, you get the density of the chip. Here are some examples:
CURRENT AVAILABLE CHIP TECHNOLOGY
CHIP DEPTH
IN MILLIONS
OF LOCATIONS CHIP WIDTH
IN BITS CHIP DENSITY =
DEPTH x WIDTH
16Mbit Chips
4Mx4
1Mx16
2Mx8
16MxI

4
1
2
16

4
16
8
1

16
16
16
16
64Mbit Chips
4Mx16
8Mx8
16Mx4

4
8
16

16
8
4

64
64
64
128Mbit Chips
8Mx16
16Mx8
32Mx4

8
16
32

16
8
4

128
128
128
256Mbit Chips
32Mx8

32

8

256
MODULE CAPACITY

It's easy to calculate the capacity of a memory module if you know the capacities of the chips on it. If there are eight 64Mbit chips, it's a 512Mbit module. However, because the capacity of a module is described in megabytes, not megabits, you have to convert bits to bytes. To do this, divide the number of bits by 8. In the case of the 512Mbit module:
512Mbits

—————– = 64MB 8 bits per byte
You may hear standard memory modules in the industry being described as: “4Mx32” (that is, “4 Meg by 32”), or “16Mx64” (“16 Meg by 64”). In these cases, you can calculate the capacity of the module exactly as if it were a chip:

4Mx32 is 128Mbits. 16Mx64 is 1024Mbits.

128Mbits 1024Mbits

—————– = 16MB module —————– = 128MB module 8 bits per byte 8 bits per byte
Here are some additional examples:
STANDARD MODULE TYPES
STANDARD MODULE DEPTH
IN LOCATIONS MODULE WIDTH
IN DATA BITS CAPACITY IN
MBITS = DEPTH X WIDTH CAPACITY IN
MB = MBITS/8
72-Pin 1Mx32
2Mx32
4Mx32
8Mx32
16Mx32
32Mx32 1
2
4
8
16
32 32
32
32
32
32
32 32
64
128
256
512
1024 4
8
16
32
64
128
168-Pin 2Mx64
4Mx64
8Mx64
16Mx64
32Mx64 2
4
8
16
32 64
64
64
64
64 128
256
512
1024
2048 16
32
64
128
256
As we mentioned earlier, there’s only room for a certain number of chips on a PCB. Based on an industry standard 168-pin DIMM, the largest capacity module manufacturers can make using 64Mbit chips is 128MB; with 128Mbit chips, the largest module possible is 256MB; and with 256Mbit chips, the largest module possible is 512MB.

4.4. STACKING

Sun, 01 Mar 1998 00:00:00 +0000

<< Previous page TOC Next page >>
4.4. STACKING

Many large servers and workstations require higher capacity modules in order to reach system memory capacities of several gigabytes or more. There are two ways to increase the capacity of a module. Manufacturers can stack chips on top of one another, or they can stack boards.
CHIP STACKING

With chip stacking, two chips are stacked together and occupy the space that one chip would normally take up. In some cases, the stacking is done internally at the chip manufacturing plant and can actually appear to be one chip. In other cases the chips are stacked externally. The example below shows two externally stacked chips.

Example of externally stacked chips.
BOARD STACKING

As you might expect, board stacking involves putting two memory module printed circuit boards (PCBs) together. With board stacking, a secondary board mounts onto the primary board, which fits into the memory socket on the system motherboard.

Example of a stacked module.

5. Device Sensitivity and Testing

Sun, 01 Mar 1998 00:00:00 +0000

<< Previous page INDEX Next page >>
5. Device Sensitivity and Testing
In Part Two, we indicated that one of the key elements in a successful static control program was the identification of those items (components, assemblies, and finished products) that are sensitive to ESD and the level of their sensitivity. Damage to an ESDS device by the ESD event is determined by the device's ability to dissipate the energy of the discharge or withstand the current levels involved. This is known as device "ESD sensitivity" or "ESD susceptibility".
Some devices may be more readily damaged by discharges occurring within automated equipment, while others may be more prone to damage from handling by personnel. In this article we will cover the models and test procedures used to characterize, determine, and classify the sensitivity of components to ESD. These test procedures are based on the three primary models of ESD events: Human Body Model (HBM), Machine Model (MM), and Charged Device Model (CDM). The models used to perform component testing cannot replicate the full spectrum of all possible ESD events. Nevertheless, these models have been proven to be successful in reproducing over 95% of all ESD field failure signatures. With the use of standardized test procedures, the industry can

Develop and measure suitable on-chip protection.
Enable comparisons to be made between devices.
Provide a system of ESD sensitivity classification to assist in the ESD design and monitoring requirements of the manufacturing and assembly environments.
Have documented test procedures to ensure reliable and repeatable results.
Human Body Model
One of the most common causes of electrostatic damage is the direct transfer of electrostatic charge through a significant series resistor from the human body or from a charged material to the electrostatic discharge sensitive (ESDS) device. When one walks across a floor, an electrostatic charge accumulates on the body. Simple contact of a finger to the leads of an ESDS device or assembly allows the body to discharge, possibly causing device damage. The model used to simulate this event is the Human Body Model (HBM).
The Human Body Model is the oldest and most commonly used model for classifying device sensitivity to ESD. The HBM testing model represents the discharge from the fingertip of a standing individual delivered to the device. It is modeled by a 100 pF capacitor discharged through a switching component and a 1.5kW series resistor into the component. This model, which dates from the nineteenth century, was developed for investigating explosive gas mixtures in mines. It was adopted by the military in MIL-STD-883 Method 3015, and is also used in ESD Association standard ESD STM5.1-1998 Revised: Electrostatic Discharge Sensitivity Testing -- Human Body Model. The Human Body Model circuit is presented in Figure 1.

Figure 1: Typical Human Body Model Circuit
Testing for HBM sensitivity is typically performed using automated test systems. The device is placed in the test system and contacted through a relay matrix. ESD zaps are applied and the post stress I-V current traces are reviewed to see if the devices fail. The ESD Association HBM test standard was recently revised to include several technical changes. First, the number of zaps per stress level and polarity has been reduced from 3 to 1. Also, the minimum time interval between zaps has been reduced from 1 second to 300 milliseconds. The changes reduce the HBM qualification test time.
The second technical change is a revision in the HBM tester specifications. The maximum rise time for an HBM wave form measured through a 500 ohm load was increased from 20 to 25 nanoseconds. This will allow HBM test equipment manufacturers to build high pin count testers that typically have a higher parasitic test board capacitance that slows down the 500 ohm wave form.
Machine Model
A discharge similar to the HBM event also can occur from a charged conductive object, such as a metallic tool or fixture. Originating in Japan as the result of trying to create a worst-case HBM event, the model is known as the Machine Model. This ESD model consists of a 200 pF capacitor discharged directly into a component with no series resistor.
As a worst-case human body model, the Machine Model may be over severe. However, there are real-world situations that this model represents, for example the rapid discharge from a charged board assembly or from the charged cables of an automatic tester.
Testing of devices for MM sensitivity using ESD Association standard ESD STM5.2-1999 (Revised): Electrostatic Discharge Sensitivity Testing -- Machine Model is similar to HBM testing. The test equipment is the same, but the test head is slightly different. The MM version does not have a 1,500 ohm resistor, but otherwise the test board and the socket are the same as for HBM testing.

Figure 2: Typical Machine Model Circuit
Charged Device Model Testing
The transfer of charge from an ESDS device is also an ESD event. A device may become charged, for example, from sliding down the feeder in an automated assembler. If it then contacts the insertion head or another conductive surface, a rapid discharge may occur from the device to the metal object. This event is known as the Charged Device Model (CDM) event, and can be more destructive than the HBM for some devices. Although the duration of the discharge is very short--often less than one nanosecond--the peak current can reach several tens of amperes.
Several test methods have been explored to duplicate the real-world CDM event and provide a suitable test method that duplicates the types of failure that have been observed in CDM caused field failures. Current work in the area is concentrating on two separate CDM test methods. One is termed CDM and best replicates the real world charged device event. The other addresses devices that are inserted in a socket and then charged and discharged in the socket. It is termed the socketed discharge model (SDM).
A draft standard for CDM (ESD DS5.3.1-1996—Device Testing: Charged Device Model) was published in 1996. It is anticipated that the standard will be published as a full standard early in 2000. The test procedure involves placing the device on a field plate with its leads pointing up, then charging it and discharging the device. Figure 3 illustrates a typical CDM test circuit.

Figure 3: Typical Charged Device Model Test
SDM testing is similar to testing for HBM and MM sensitivity. The device is placed in a socket, charged from a high-voltage source and then discharged. This procedure is still a work in process and still has a number of limitations including too great a dependency on the specific design of the SDM tester. A technical paper on SDM testing was presented by the ESD Association SDM Device Testing standards working group at the October 1998 EOS/ESD Symposium.
Device Sensitivity Classification
Each of the device testing methods includes a classification system for defining the component sensitivity to the specified model (See Tables 1, 2, and 3). These classification systems have a number of advantages. They allow easy grouping and comparing of components according to their ESD sensitivity and the classification gives you an indication of the level of ESD protection that is required for the component.
Table 1
ESDS Component Sensitivity Classification - Human Body Model
(Per ESD STM5.1-1998)

Class

Voltage Range

Class 0

<250 volts

Class 1A

250 volts to <500 volts

Class 1B

500 volts to < 1,000 volts

Class 1C

1000 volts to < 2,000 volts

Class 2

2000 volts to < 4,000 volts

Class 3A

4000 volts to < 8000 volts

Class 3B

>= 8000 volts
Table 2
ESDS Component Sensitivity Classification - Machine Model
(Per ANSI/ESD-S5.2-1999)

Class

Voltage Range

Class M1

<100 volts

Class M2

100 volts to <200 volts

Class M3

200 volts to <400 volts

Class M4

> or = 400 volts
Table 3
ESDS Component Sensitivity Classification - Charged Device Model
(Per EOS/ESD-DS5.3-1996)

Class

Voltage Range

Class C1

<125 volts

Class C2

125 volts to <250 volts

Class C3

250 volts to <500 volts

Class C4

500 volts to <1,000 volts

Class C5

1,000 volts to <1,500 volts

Class C6

1,500 volts to <2,000 volts

Class C7

=>2,000 volts
A fully characterized component should be classified using all three models: Human Body Model, Machine Model, and Charged Device Model. For example, a fully characterized component may have the following: Class 1B (500 volts to <1000 volts HBM), Class M1 (<100 volts MM), and Class C3 (500 volts to <1000 volts CDM). This would alert a potential user of the component to the need for a controlled environment, whether assembly and manufacturing operations are performed by human beings or machines.
A word of caution, however. These classification systems and component sensitivity test results function as guides, not necessarily as absolutes. The events defined by the test data produce narrowly restrictive data that must be carefully considered and judiciously used. The three ESD models represent discrete points used in an attempt to characterize ESD vulnerability. The data points are informative and useful, but to arbitrarily extrapolate the data into a real world scenario can be misleading. The true utility of the data is in comparing one device with another and to provide a starting point for developing your ESD control programs.
Summary
Device failure models and device test methods define the sensitivity of the electronic devices and assemblies to be protected from the effects of ESD. With this key information, you can design more effective ESD control programs.
For Further Reference

Avery, L.R., "Beyond MIL HBM Testing - How to Evaluate the Real Capability of Protection Structures, EOS/ESD Symposium Proceedings, 1991, ESD Association, Rome, NY.
Avery, L.R., "Charged Device Model Testing: Trying to Duplicate Reality," EOS/ESD Symposium Proceedings, 1987, ESD Association, Rome, NY.
Chaine, M., Verhaege, K., Avery, L., Kelly, M., Gieser, H., Bock, K., Henry, L.G., Meuse, T., Brodbeck, T., Barth, J., "Investigation into Socketed CEM (SDM) Tester Parasitics," EOS/ESD Symposium Proceedings, 1998, ESD Association, Rome, NY.
EOS/ESD DS5.3-1996: Electrostatic Discharge Sensitivity Testing - Charged Device Model. ESD Association, Rome, NY.
ESD STM5.2-1999 (Revised): Electrostatic Discharge Sensitivity Testing -- Machine Model. ESD Association, Rome, NY.
ESD STM5.1-1998 Revised: Electrostatic Discharge Sensitivity Testing -- Human Body Model. ESD Association, Rome, NY.
Geiser, H., and Haunschild, M., "Very Fast Transmission Line Pulsing of Integrated Structures and the Charged Device Model," EOS/ESD Symposium Proceedings, 1996, ESD Association, Rome, NY.
Hyatt, Hugh, Calvin, Hugh, and Mellberg, Hans, "A Closer Look at the Human ESD Event," EOS/ESD Symposium Proceedings, 1981, ESD Association, Rome, NY.
Kelly, M., et al, "A Comparison of Electrostatic Discharge Models and Failure Signatures for CMOS Integrated Circuit Devices," EOS/ESD Symposium Proceedings, 1995, ESD Association, Rome, NY.
Pierce, Donald C., "Critical Issues Regarding ESD Sensitivity Classification Testing," EOS/ESD Symposium Proceedings, 1987, ESD Association, Rome, NY.
Renninger, Robert G., "Mechanisms of Charged-Device Electrostatic Discharges," EOS/ESD Symposium Proceedings, 1991. ESD Association, Rome, NY.
Russ, Christian., et al, "A Compact Model for the Grounded-Gate nMOS Behavior Under CDM ESD Stress," EOS/ESD Symposium Proceedings, 1996, ESD Association, Rome, NY.
Verhaege, Koen, "Component Level ESD Testing," Review Paper, Microelectronics Reliability Journal, 1998.
Verhaege, Koen., et al " Analysis of HBM ESD Testers and Specifications Using a 4^th Order Lumped Element Model," EOS/ESD Symposium Proceedings, 1993, ESD Association, Rome, NY.
Verhaege, Koen., et al, "Recommendations to Further Improvements of HBM ESD Component Level Test Specifications," EOS/ESD Symposium Proceedings, 1996, ESD Association, Rome, NY.
Wada, Tetsuaki. "Study of ESD Evaluation Methods for Charged Device Model," EOS/ESD Symposium Proceedings, 1995, ESD Association, Rome, NY.

5. DIFFERENT KINDS OF MEMORY

Sun, 01 Mar 1998 00:00:00 +0000

<< Previous page TOC Next page >>
5. DIFFERENT KINDS OF MEMORY
Some people like to know a lot about the computer systems they own - or are considering buying - just because. They're like that. It's what makes them tick. Some people never find out about their systems and like it that way. Still other people - most of us, in fact - find out about their systems when they have to - when something goes wrong, or when they want to upgrade it. It's important to note that making a choice about a computer system - and its memory features - will affect the experience and satisfaction you derive from the system. This chapter is here to make you smarter about memory so that you can get more out of the system you're purchasing or upgrading.

5.1. MODULE FORM FACTORS
5.2. MAJOR CHIP TECHNOLOGIES
5.3. MEMORY TECHNOLOGIES FOR VIDEO OR GRAPHICS PROCESSING
5.4. OTHER MEMORY TECHNOLOGIES YOU MAY HAVE HEARD ABOUT
5.5. ERROR CHECKING
5.6. OTHER SPECIFICATIONS

5.1. MODULE FORM FACTORS

Sun, 01 Mar 1998 00:00:00 +0000

<< Previous page TOC Next page >>
5.1. MODULE FORM FACTORS
The easiest way to categorize memory is by form factor. The form factor of any memory module describes its size and pin configuration. Most computer systems have memory sockets that can accept only one form factor. Some computer systems are designed with more than one type of memory socket, allowing a choice between two or more form factors. Such designs are usually a result of transitional periods in the industry when it's not clear which form factors will gain predominance or be more available.
SIMMS

As previously mentioned, the term SIMM stands for Single In-Line Memory Module. With SIMMs, memory chips are soldered onto a modular printed circuit board (PCB), which inserts into a socket on the system board.

The first SIMMs transferred 8 bits of data at a time. Later, as CPUs began to read data in 32-bit chunks, a wider SIMM was developed, which could supply 32 bits of data at a time. The easiest way to differentiate between these two different kinds of SIMMs was by the number of pins, or connectors. The earlier modules had 30 pins and the later modules had 72 pins. Thus, they became commonly referred to as 30-pin SIMMs and 72-pin SIMMs.

Another important difference between 30-pin and 72-pin SIMMs is that 72-pin SIMMs are 3/4 of an inch (about 1.9 centimeters) longer than the 30-pin SIMMs and have a notch in the lower middle of the PCB. The graphic below compares the two types of SIMMs and indicates their data widths.

Comparison of a 30-pin and a 72-pin SIMM
DIMMS

Dual In-line Memory Modules, or DIMMs, closely resemble SIMMs. Like SIMMs, most DIMMs install vertically into expansion sockets. The principal difference between the two is that on a SIMM, pins on opposite sides of the board are "tied together" to form one electrical contact; on a DIMM, opposing pins remain electrically isolated to form two separate contacts.

168-pin DIMMs transfer 64 bits of data at a time and are typically used in computer configurations that support a 64-bit or wider memory bus. Some of the physical differences between 168-pin DIMMs and 72-pin SIMMs include: the length of module, the number of notches on the module, and the way the module installs in the socket. Another difference is that many 72-pin SIMMs install at a slight angle, whereas 168-pin DIMMs install straight into the memory socket and remain completely vertical in relation to the system motherboard. The illustration below compares a 168-pin DIMM to a 72-pin SIMM.

Comparison of a 72-pin SIMM and a 168-pin DIMM.
SO DIMMS

A type of memory commonly used in notebook computers is called SO DIMM or Small Outline DIMM. The principal difference between a SO DIMM and a DIMM is that the SO DIMM, because it is intended for use in notebook computers, is significantly smaller than the standard DIMM. The 72-pin SO DIMM is 32 bits wide and the 144-pin SO DIMM is 64 bits wide.

Comparison of a 72-pin SO DIMM and a 144-pin SO DIMM.
RIMMS AND SO-RIMMS

RIMM is the trademarked name for a Direct Rambus memory module. RIMMs look similar to DIMMs, but have a different pin count. RIMMs transfer data in 16-bit chunks. The faster access and transfer speed generates more heat. An aluminum sheath, called a heat spreader, covers the module to protect the chips from overheating.

A 184-pin Direct Rambus RIMM shown with heat spreaders pulled away.

An SO-RIMM looks similar to an SO DIMM, but it uses Rambus technology.

A 160-pin SO-RIMM module.

FLASH MEMORY
Flash memory is a solid-state, non-volatile, rewritable memory that functions like RAM and a hard disk drive combined. Flash memory stores bits of electronic data in memory cells, just like DRAM, but it also works like a hard-disk drive in that when the power is turned off, the data remains in memory. Because of its high speed, durability, and low voltage requirements, flash memory is ideal for use in many applications - such as digital cameras, cell phones, printers, handheld computers, pagers, and audio recorders.

PC CARD AND CREDIT CARD MEMORY

Before SO DIMMs became popular, most notebook memory was developed using proprietary designs. It is always more cost-effective for a system manufacturer to use standard components, and at one point, it became popular to use the same "credit card" like packaging for memory that is used on PC Cards today. Because the modules looked like PC Cards, many people thought the memory cards were the same as PC Cards, and could fit into PC Card slots. At the time, this memory was described as "Credit Card Memory" because the form factor was the approximate size of a credit card. Because of its compact form factor, credit card memory was ideal for notebook applications where space is limited.

PC Cards use an input/output protocol that used to be referred to as PCMCIA (Personal Computer Memory Card International Association). This standard is designed for attaching input/output devices such as network adapters, fax/modems, or hard drives to notebook computers. Because PC Card memory resembles the types of cards designed for use in a notebook computer's PC Card slot, some people have mistakenly thought that the memory modules could be used in the PC Card slot. To date, RAM has not been packaged on a PCMCIA card because the technology doesn't allow the processor to communicate quickly enough with memory. Currently, the most common type of memory on PC Card modules is Flash memory.

On the surface, credit card memory does not resemble a typical memory module configuration. However, on the inside you will find standard TSOP memory chips.

This section presents the most common memory technologies used for main memory: This road map offers an overview of the evolution of memory.

YEAR INTRODUCED TECHNOLOGY SPEED LIMIT
1987 FPM 50ns
1995 EDO 50ns
1997 PC66 SDRAM 66MHz
1998 PC100 SDRAM 100MHz
1999 RDRAM 800MHz
1999/2000 PC133 SRAM 133MHz (VCM option)
2000 DDR SDRAM 266MHz

5.2. MAJOR CHIP TECHNOLOGIES

Sun, 01 Mar 1998 00:00:00 +0000

<< Previous page TOC Next page >>
5.2. MAJOR CHIP TECHNOLOGIES
It's usually pretty easy to tell memory module form factors apart because of physical differences. Most module form factors can support various memory technologies so, it's possible for two modules to appear to be the same when, in fact, they're not. For example, a 168-pin DIMM can be used for EDO, Synchronous DRAM, or some other type of memory. The only way to tell precisely what kind of memory a module contains is to interpret the marking on the chips. Each DRAM chip manufacturer has different markings and part numbers to identify the chip technology.
FAST PAGE MODE (FPM)
At one time, FPM was the most common form of DRAM found in computers. In fact, it was so common that people simply called it "DRAM," leaving off the "FPM". FPM offered an advantage over earlier memory technologies because it enabled faster access to data located within the same row.
EXTENDED DATA OUT (EDO)
In 1995, EDO became the next memory innovation. It was similar to FPM, but with a slight modification that allowed consecutive memory accesses to occur much faster. This meant the memory controller could save time by cutting out a few steps in the addressing process. EDO enabled the CPU to access memory 10 to 15% faster than with FPM.
SYNCHRONOUS DRAM (SDRAM)
In late 1996, SDRAM began to appear in systems. Unlike previous technologies, SDRAM is designed to synchronize itself with the timing of the CPU. This enables the memory controller to know the exact clock cycle when the requested data will be ready, so the CPU no longer has to wait between memory accesses. SDRAM chips also take advantage of interleaving and burst mode functions, which make memory retrieval even faster. SDRAM modules come in several different speeds so as to synchronize to the clock speeds of the systems they'll be used in. For example, PC66 SDRAM runs at 66MHz, PC100 SDRAM runs at 100MHz, PC133 SDRAM runs at 133MHz, and so on. Faster SDRAM speeds such as 200MHz and 266MHz are currently in development.
DOUBLE DATA RATE SYNCHRONOUS DRAM (DDR SDRAM)
DDR SDRAM, is a next-generation SDRAM technology. It allows the memory chip to perform transactions on both the rising and falling edges of the clock cycle. For example, with DDR SDRAM, a 100 or 133MHz memory bus clock rate yields an effective data rate of 200MHz or 266MHz. Systems using DDR SDRAM are expected to ship at the end of the year 2000.
DIRECT RAMBUS
Direct Rambus is a new DRAM architecture and interface standard that challenges traditional main memory designs. Direct Rambus technology is extraordinarily fast compared to older memory technologies. It transfers data at speeds up to 800MHz over a narrow 16-bit bus called a Direct Rambus Channel. This high-speed clock rate is possible due to a feature called "double clocked," which allows operations to occur on both the rising and falling edges of the clock cycle. Also, each memory device on an RDRAM module provides up to 1.6 gigabytes per second of bandwidth - twice the bandwidth available with current 100MHz SDRAM.

In addition to chip technologies designed for use in main memory, there are also specialty memory technologies that have been developed for video applications.

5.3. MEMORY TECHNOLOGIES FOR VIDEO OR GRAPHICS PROCESSING

Sun, 01 Mar 1998 00:00:00 +0000

<< Previous page TOC Next page >>
5.3. MEMORY TECHNOLOGIES FOR VIDEO OR GRAPHICS PROCESSING
VIDEO RAM (VRAM)
VRAM is a video version of FPM technology. VRAM typically has two ports instead of one, which allows the memory to allocate one channel to refreshing the screen while the other is focused on changing the images on the screen. This works much more efficiently than regular DRAM when it comes to video applications. However, since video memory chips are used in much lower quantities than main memory chips, they tend to be more expensive. So, a system designer may choose to use regular DRAM in a video subsystem, depending on whether cost or performance is the design objective.
WINDOW RAM (WRAM)
WRAM is another type of dual-ported memory also used in graphics-intensive systems. It differs slightly from VRAM in that its dedicated display port is smaller and it supports EDO features.
SYNCHRONOUS GRAPHICS RAM (SGRAM)
SGRAM is a video-specific extension of SDRAM that includes graphics-specific read/write features. SGRAM also allows data to be retrieved and modified in blocks, instead of individually. This reduces the number of reads and writes that memory must perform and increases the performance of the graphics controller by making the process more efficient.
BASE RAMBUS AND CONCURRENT RAMBUS
Before it even became a contender for main memory, Rambus technology was actually used in video memory. The current Rambus main memory technology is called Direct Rambus. Two earlier forms of Rambus are Base Rambus and Concurrent Rambus. These forms of Rambus have been used in specialty video applications in some workstations and video game systems like Nintendo 64 for several years.

5.4. OTHER MEMORY TECHNOLOGIES YOU MAY HAVE HEARD ABOUT

Sun, 01 Mar 1998 00:00:00 +0000

<< Previous page TOC Next page >>
5.4. OTHER MEMORY TECHNOLOGIES YOU MAY HAVE HEARD ABOUT
ENHANCED SDRAM (ESDRAM)
In order to increase the speed and efficiency of standard memory modules, some manufacturers have incorporated a small amount of SRAM directly into the chip, effectively creating an on-chip cache. ESDRAM is essentially SDRAM, plus a small amount of SRAM cache, which allows for burst operations of up to 200MHz. Just as with external cache memory, the goal of cache DRAM is to hold the most frequently used data in the SRAM cache to minimize accesses to the slower DRAM. One advantage of on-chip SRAM is that it enables a wider bus between the SRAM and DRAM, effectively increasing the bandwidth and speed of the DRAM.
FAST CYCLE RAM (FCRAM)
FCRAM, co-developed by Toshiba and Fujitsu, is intended for specialty applications such as high-end servers, printers, and telecommunications switching systems. It includes memory array segmentation and internal pipelining that speed random access and reduce power consumption.
SYNCLINK DRAM (SLDRAM)
Though considered obsolete today, SLDRAM was developed by a consortium of DRAM manufacturers as an alternative to Rambus technology in the late 1990s.
VIRTUAL CHANNEL MEMORY (VCM)
Developed by NEC, VCM allows different "blocks" of memory to interface inde-pendently with the memory controller, each with its own buffer. This way, different system tasks can be assigned their own "virtual channels," and information related to one function does not share buffer space with other tasks occurring at the same time, making operations more efficient.

5.5. ERROR CHECKING

Sun, 01 Mar 1998 00:00:00 +0000

<< Previous page TOC Next page >>
5.5. ERROR CHECKING
Ensuring the integrity of data stored in memory is an important aspect of memory design. Two primary means of accomplishing this are parity and error correction code (ECC).

Historically, parity has been the most commonly used data integrity method. Parity can detect - but not correct - single-bit errors. Error Correction Code (ECC) is a more comprehensive method of data integrity checking that can detect and correct single-bit errors.

Fewer and fewer PC manufacturers are supporting data integrity checking in their designs. This is due to a couple of factors. First, by eliminating support for parity memory, which is more expensive than standard memory, manufacturers can lower the price of their computers. Fortunately, this trend is complemented by the second factor: that is, the increased quality of memory components available from certain manufacturers and, as a result, the relative infrequency of memory errors.

The type of data integrity checking depends on how a given computer system will be used. If the computer is to play a critical role - as a server, for example - then a computer that supports data integrity checking is an ideal choice. In general:

Most computers designed for use as high-end servers support ECC memory.
Most low-cost computers designed for use at home or for small businesses support non-parity memory.

5.6. OTHER SPECIFICATIONS

Sun, 01 Mar 1998 00:00:00 +0000

<< Previous page TOC Next page >>
5.6. OTHER SPECIFICATIONS
In addition to form factors, memory technologies, and error checking methods, there are several other specifications important to understanding and selecting memory products.
SPEED
The speed of memory components and modules is one of the most important factors in optimizing a memory configuration. In fact, all computer systems specify a memory component speed. Ensuring memory compatibility requires conforming to this specification. This section covers three measurements of memory component and module speed: access time, megahertz, and bytes per second.

6. ESD Standards

Sun, 01 Mar 1998 00:00:00 +0000

<< Previous page INDEX Next page >>
6. ESD Standards
In a world that is characterized by change, the ESD industry seems to have jumped into the swirling eddy head-first. Control programs have mushroomed. Black has been replaced by green, blue and gold. Shielding bags dominate the warehouse. Ionizers exist along side wrist straps and ground cords. An early history of "smoke and mirrors," magic and lofty claims of performance is rapidly and safely being relegated to the past.
Today, meeting the complex challenge of reducing ESD losses requires more than reliance on faith alone. Users require a way to legitimately evaluate and compare competing brands and types of products. They need objective confirmation that their ESD control program provides effective solutions to their unique ESD problems.
That’s where standards come into play. They help define the sensitivity of the products manufactured and used. They help define the performance requirements for various ESD control instruments and tools. Standards are playing an ever increasing role in reducing marketplace confusion in the manufacture, evaluation, and selection of ESD control products and programs.
The Who and Why of Standards
Who uses ESD standards? Manufacturers of ESD control products; users of ESD control products; and third party testers of ESD control products.
Why use ESD standards? They help assure consistency of ESD sensitive products and consistency of ESD control products and services. They provide a means of objective evaluation and comparison among competitive ESD control products. They help reduce conflicts between users and suppliers of ESD control products. They help in auditing ESD control programs. And, they help reduce confusion in the marketplace.
In the United States, the use of standards is voluntary, although their use can be written into contracts or purchasing agreements between buyer and seller. In most of the rest of the world, the use of standards, where they exist, is compulsory.
Key Standards and Organizations
Just 20 years ago, there were relatively few reliable ESD standards and few ESD standards development organizations. Today’s ESD standards landscape is not only witnessing an increase in the number of standards, but also increasing cooperation among the organizations that develop them.
Today’s standards fall into three main groups. First, there are those that provide ESD program guidance or requirements. These include documents such as ANSI ESD S20.20-1999--Standard for the Development of an ESD Control Program, ANSI/ESD S8.1-ESD Awareness Symbols; ESD ADV2.0-ESD Handbook; EIA 625-Requirements for Handling Electrostatic Discharge Sensitive Devices; or EN100015-Protection of Electrostatic Sensitive Devices.
A second group covers requirements for specific products or procedures such as packaging or grounding. Typical standards in this group are ANSI/ESD S6.1-Grounding; or EIA 541-Packaging of Electronic Products for Shipment.
A third group of documents covers the standardized test methods used to evaluate products and materials. Historically, the electronics industry has relied largely on test methods established for other industries or even other materials (ASTM-257-DC Resistance or Conductance of Insulating Materials). Today, however, specific test method standards focus on ESD in the electronics environment, largely as a result of the ESD Association’s activity. These include standards such as ESD S5.1-Device Testing, Human Body Model and ANSI/ESD S7.1: Floor Materials -- Resistive Characterization to cite just a few.
Who Develops Standards?
Standards development and usage is a cooperative effort among all organizations and individuals affected by standards. There are several key ESD standards development organizations.
Military Standards
Traditionally the U.S. military developed and issued specific standards and specifications with regard to ESD control. Once a primary developer of ESD standards, US military agencies are taking a less proactive approach, relying on commercially developed standards rather than developing standards themselves. For example, the ESD Association has received the assignment from the Department of Defense to convert MIL-STD-1686 into a commercial standard. Among current military standards, MIL-STD-1686 - Electrostatic Discharge Control Program and MIL-HDBK-263 - Electrostatic Discharge Control Handbook are the predominant documents. The 1686 standard requires vendors to classify the ESD susceptibility of their components and have an ESD control program in place. The accompanying 263 Handbook provides guidelines and suggestions for ESD control programs and procedures.
Electronic Industries Alliance/JEDEC
The Electronic Industry Alliance (EIA) and EIA’s Joint Electronics Development Engineering Council (JEDEC) are also active in standards development. EIA 541 -- Packaging Material Standards for ESD Sensitive Items specifies test methods for the evaluation of packaging materials for protection of ESDS parts during shipment. JEDEC’s EIA 625 Requirements for Handling Electrostatic-Discharge-Sensitive (ESDS) Devices establishes methods for semiconductor manufacturers, distributors, processing and testing facilities and users to use in protecting components that have human body model sensitivities of 200 volts or greater.
ESD Association
The ESD Association has been a focal point for the development of ESD standards in recent years. An ANSI-Accredited Standards Development Organization, the Association is charged with the development of ESD standards and test methods. It also represents the US on the International Electrotechnical Commission Technical Committee 101-Electrostatics.
The ESD Association has published 27 standards documents and 4 technical reports. These voluntary standards cover the areas of material requirements, electrostatic sensitivity, and test methodology for evaluating ESD control materials and products. In addition to standards documents, the Association also publishes a number of informational advisories.
ESD Association Standards Classifications and Definitions
There are four types of ESD Association standards documents with specific clarity of definition. The four document categories are consistent with other standards development organizations. These four categories are defined below.
Standard: A precise statement of a set of requirements to be satisfied by a material, product, system or process that also specifies the procedures for determining whether each of the requirements is satisfied.
Standard Test Method: A definitive procedure for the identification, measurement and evaluation of one or more qualities, characteristics or properties of a material, product, system or process that yields a reproducible test result.
Standard Practice: A procedure for performing one or more operations or functions that may or may not yield a test result. Note: If a test result is obtained, it may not be reproducible between labs.
Technical Report: A collection of technical data or test results published as an informational reference on a specific material, product, system, or process.
As new documents are approved and issued, they will be designated into one of these four new categories. Existing documents are being reviewed and will be reclassified as appropriate.
International Standards
The international community, led by the European-based International Electrotechnical Commission, has also climbed on board the standards express. Europe’s CENELEC has issued a European electrostatic standard EN100015 - Protection of Electrostatic Sensitive Devices that was adopted as a European Norm. Additional work by the IEC to will result in a comprehensive series of standards that may someday be the successor to EN100015.
Japan also has released its proposed version of a national electrostatic Standard, which also shares many aspects of the European and U.S. documents.
Organizational Cooperation
Perhaps one of the more intriguing changes in ESD standards has been the organizational cooperation developing between various groups. In revising EIA541, the EIA and the ESD Association have implemented a cooperative effort to update the standard.
Another cooperative effort is underway between the ESD Association and the U.S. Department of Defense. The ESD Association prepared a commercial version of MIL-STD-1686. It is anticipated that defense agencies will use that commercial version as a reference.
Internationally, European standards development organizations and the ESD Association have developed working relationships that result in an expanded review of proposed documents, greater input, and closer harmonization of standards that impact the international electronics community.
For users of ESD standards, this increased cooperation will have a significant impact. First, we should see standards that are technically improved due to broader input. Second, we should see fewer conflicts between different standards. Finally, we should see less duplication of effort.
Summary
For the electronics community, the rapid propagation of ESD standards and continuing change in the standards environment mean greater availability of the technical references that will help improve ESD control programs. There will be recommendations to help set up effective programs. There will be test methods and specifications to help users of ESD control materials evaluate and select products that are applicable to their specific needs. And there will be guidelines for vendors of ESD products and materials to help them develop products that meet the real needs of their customers.
Standards will continue to fuel change in the international ESD community.
Principle ESD Standards
U.S. Military/Department of Defense
MIL-STD-1686C : Electrostatic Discharge Control Program for Protection of Electrical and Electronic Parts, Assemblies and Equipment (Excluding Electrically Initiated Explosive Devices)
This military standard establishes requirements for ESD Control Programs. It applies to U.S. military agencies, contractors, subcontractors, suppliers and vendors. It requires the establishment, implementation and documentation of ESD control programs for static sensitive devices, but does NOT mandate or preclude the use of any specific ESD control materials, products, or procedures. It is being updated and converted to a commercial standard by the ESD Association.
MIL-HBDK-263B: Electrostatic Discharge Control Handbook for Protection of Electrical and Electronic Parts, Assemblies and Equipment (Excluding Electrically Initiated Explosive Devices)
This document provides guidance, but NOT mandatory requirements, for the establishment and implementation of an electrostatic discharge control program in accordance with the requirements of MIL-STD-1686.
MIL-PRF 87893—Workstation, Electrostatic Discharge (ESD) Control
This document defines the requirements for ESD protective workstations.
MIL-B-81705—Barrier Materials, Flexible, Electrostatic Protective, Heat Sealable
This documents defines requirements for ESD protective flexible packaging materials.
MIL-STD-129—Marking for Shipment and Storage
Covers procedures for marketing and labeling ESD sensitive items.
EIA/JEDEC
EIA541—Packaging Material Standards for ESD Sensitive Items
This document covers packaging materials and definitions of material types for ESD sensitive items.
EIA-625: Requirements for Handling Electrostatic-Discharge-Sensitive (ESDS) Devices
This voluntary standard establishes minimum requirements for ESD control methods and materials used to protect electronic devices that have human body model sensitivities of 200 volts or greater. It is applicable to semiconductor manufacturers, semiconductor distributors, semiconductor processing and testing facilities, and semiconductor users. It is currently being revised and updated with a reduced scope.
International/European
EN100015: Protection of Electrostatic Sensitive Devices
Adopted in 1992 and 1993, this European Norm covers ESD handling practices for electronic devices.
ESD Association
Standards Documents
Standards, Standard Test Methods and Standard Practices have completed the industry comment and review process, and the document has been approved by the Standards Committee as a Final Standard Document.
ESD S1.1—1998: Evaluation, Acceptance, and Functional Testing of Wrist Straps.
A successor to the EOS/ESD S1.0, this document establishes test methods for evaluating the electrical and mechanical characteristics of wrist straps. It includes improved test methods and performance limits for evaluation, acceptance, and functional testing of wrist straps.
ESD STM2.1—1997: Resistance Test Method for Electrostatic Discharge Protective Garments
This Standard Test Method provides test methods for measuring the electrical resistance of garments used to control electrostatic discharge. It covers procedures for measuring sleeve-to-sleeve and point-to-point resistance.
ANSI EOS/ESD S3.1-1991: Ionization
Test methods and procedures for evaluating and selecting air ionization equipment and systems are covered in this standard. The document establishes measurement techniques to determine ion balance and charge neutralization time for ionizers.
ESD S4.1- 1997(Revised): Worksurfaces – Resistance Measurements
This Standard establishes test methods for measuring the electrical resistance of worksurface materials used at workstations for protection of ESD susceptible items. It includes methods for evaluating and selecting materials, and testing new worksurface installations and previously installed worksurfaces.
ESD STM4.2-1998: Worksurfaces - Charge Dissipation Characteristics
This Standard Test Method provides a test method to measure the electrostatic charge dissipation characteristics of worksurfaces used for ESD control. The procedure is designed for use in a laboratory environment for qualification, evaluation or acceptance of worksurfaces.
ESD STM5.1-1998 Revised: Electrostatic Discharge Sensitivity Testing -- Human Body Model
This Standard Test Method updates and revises an existing Standard. It establishes a procedure for testing, evaluating and classifying the ESD sensitivity of components to the defined Human Body Model (HBM).
ESD STM5.2-1999 (Revised): Electrostatic Discharge Sensitivity Testing -- Machine Model
This Standard establishes a test procedure for evaluating the ESD sensitivity of components to a defined Machine Model. It also provides a system of classifying the sensitivity of these components. The component damage caused by the Machine Model is often similar to that caused by the Human Body Model, but it occurs at a significantly lower voltage.
ESD S6.1-1999: Grounding -- Recommended Practice
This Standard recommends the parameters, procedures, and types of materials needed to establish an ESD grounding system for the protection of electronic hardware from ESD damage. This system is used for personnel grounding devices, worksurfaces, chairs, carts, floors, and other related equipment.
ANSI ESD S7.1-1994: Floor Materials -- Resistive Characterization of Materials
Measurement of the electrical resistance of various floor materials such as floor coverings, mats, and floor finishes is covered in this document. It provides test methods for qualifying floor materials before installation or application and for evaluating and monitoring materials after installation or application.
ANSI ESD S8.1-1993: ESD Awareness Symbols
Three types of ESD awareness symbols are established by this document. The first one is to be used on a device or assembly to indicate that it is susceptible to electrostatic charge. The second is to be used on items and materials intended to provide electrostatic protection. The third symbol indicates the common point ground
ESD S9.1-1995: Resistive Characterization of Footwear
This Standard defines a test method for measuring the electrical resistance of shoes used for ESD control in the electronics environment.
ANSI ESD S11.11-1993: Surface Resistance Measurement of Static Dissipative Planar Materials
This Standard defines a direct current test method for measuring electrical resistance. The Standard is designed specifically for static dissipative planar materials used in packaging of ESD sensitive devices and components.
ANSI ESD S11.31-1994: Evaluating the Performance of Electrostatic Discharge Shielding Bags
This Standard provides a method for testing and determining the shielding capabilities of electrostatic shielding bags.
ESD STM12.1-1997: Seating-Resistive Characterization
This Standard provides test methods for measuring the electrical resistance of seating used to control ESD. The test methods can be used for qualification testing as well as for evaluating and monitoring seating after installation. It covers all types of seating, including chairs and stools.
ANSI ESD S20.20-1999: Standard for the development of an ESD Control Program
This Standard provides administrative, technical requirements and guidance for establishing, implementing and maintaining an ESD Control Program.
ESD STM97.1-1999: Floor Materials and Footwear – Resistance Measurement in Combination with a Person.
This Standard Test Method provides for measuring the electrical resistance of floor materials, footwear and personnel together, as a system.
ESD STM97.2-1999: Floor Materials and Footwear – Voltage Measurement in Combination with a Person.
This Standard Test Method provides for measuring the electrostatic voltage on a person in combination with floor materials and footwear, as a system.
Draft Documents
Draft documents have not yet completed an industry comment and review stage necessary before being issued as full standards documents. Draft documents are subject to revision before being approved as full standards documents.
ESD DSP3.3—1997: Periodic Verification of Air Ionizers.
This Draft Standard practice provides a measurement technique to periodically determine ion balance and charge neutralization time for ionizers in actual use locations.
EOS/ESD DS5.3-1996: Electrostatic Discharge Sensitivity Testing - Charged Device Model
This Draft Standard establishes a test method for evaluating the ESD sensitivity of active and passive components to a defined charged device model.
ESD DSP10.1-1999: Automated Handling Equipment
This draft Standard Practice provides test methods for evaluating the ESD ground integrity; charge generation and charge accumulation on devices in automated handling equipment
ESD DS11.12-1996: Volume Resistance Measurement of Static Dissipative Planar Materials
This Draft Standard provides test methods for measuring the volume resistance of static dissipative, planar materials.
ESD DS13.1-1999: Measuring Electrical Potential from Electrical Soldering & Desoldering Hand Tools
This Draft Standard provides Test Methods for measuring the electrical leakage and tip to ground reference point resistance of electric soldering/desoldering hand tolls. It also provides parameters for EOS safe soldering operations.
Advisory Documents
Advisory Documents are not Standards, but provide general information for the industry or additional information to aid in better understanding of Association Standards.
ESD ADV1.0-1994: Glossary of Terms
Definitions and explanations of various terms used in Association Standards and documents are covered in this Advisory. It also includes other terms commonly used in the ESD industry.
ESD ADV2.0-1994: ESD Handbook
The ESD Handbook is a complete guide to static control in the work place. Nineteen chapters cover ESD basics, control procedures, auditing, symbols, device testing, standards.
ESD ADV3.2-1995: Selection and Acceptance of Air Ionizers
This Advisory document provides end users with guidelines for creating a performance specification for selecting air ionization systems. It reviews four types of air ionizers and discusses applications, test method references, and general design, performance and safety requirements.
ESD ADV11.2-1995: Triboelectric Charge Accumulation Testing
The complex phenomenon of triboelectric charging is discussed in this Advisory. It covers the theory and effects of tribocharging. It reviews procedures and problems associated with various test methods that are often used to evaluate triboelectrification characteristics. The test methods reviewed indicate gross levels of charge and polarity, but are unrepeatable if used to give more exact values in real world situations.
ESD ADV53.1-1995: ESD Protective Workstations
This Advisory document defines the minimum requirements for a basic ESD protective workstation used in ESD sensitive areas. It provides a test method for evaluating and monitoring workstations. It defines workstations as having the following components: support structure, static dissipative worksurface, a means of grounding personnel, and any attached shelving or drawers.
Sources of Standards

ESD Association, 7900 Turin Road, Building 3, Suite 2, Rome, NY 13440. Phone: 315-339-6937. Fax: +1 315-339-6793. Web Site: http://www.esda.org/
Electronic Industries Alliance/JEDEC, 2001 Pennsylvania Ave., NW, Washington, DC 20006. Phone: 202-457-4900. Fax: 202-457-4985. Web Site: http://www.eia.org/
HIS Global Engineering Documents, 15 Inverness Way East, Englewood, CO 80112. Phone: 800-854-7179. Fax: 303-397-2740. Web Site: http://global.ihs.com/
International Electrotechnical Commission, 3, rue de Varembe, Case postale 131, 1211 Geneva 20, Switzerland. Fax: 41-22-919-0300. Web Site: http://www.iec.ch/
Military Standards, Naval Publications and Forms Center, 5801 Tabor Avenue, Philadelphia, PA 19120.

6. WHAT TO CONSIDER WHEN BUYING MEMORY

Sun, 01 Mar 1998 00:00:00 +0000

<< Previous page TOC Next page >>
6. WHAT TO CONSIDER WHEN BUYING MEMORY
One of the quickest and easiest ways to identify the right memory modules and expansion options for your computer is to use a memory configurator. Like some of the other major memory manufacturers, Kingston makes a memory configurator available. You can access Kingston's configurator through their home page at www.kingston.com.

The most important thing to ensure when buying memory is compatibility with your system. In addition, you'll need to decide how much memory you need and beyond that lie considerations of price, quality, availability, service, and warranty. This section helps you address these important decision factors and helps you answer questions like these:

How much memory do I need?
How much memory will my system recognize?
What kind of memory is compatible with my system?
How many sockets are open and how should I fill them?
How do I determine the quality of memory?
What should I know about memory prices?
What other issues should I consider?

6.1. COMPATIBILITY
6.2. HOW TO READ A BANK SCHEMA
6.3. QUALITY
6.4. PRICING AND AVAILABILITY

6.1. COMPATIBILITY

Sun, 01 Mar 1998 00:00:00 +0000

<< Previous page TOC Next page >>
6.1. COMPATIBILITY
As mentioned earlier, compatibility of memory components with your computer system is arguably the most important factor to consider when upgrading memory. This section can get you started; it also makes frequent mention of the advantages of using a memory configurator.
WHAT KIND OF MEMORY IS COMPATIBLE WITH MY SYSTEM?
The easiest way to determine what type of memory goes with your system is to consult with your system documentation. If you need further assistance, consult a memory configurator available from many sources, including Kingston. Kingston and other brand-name memory companies offer such a tool to help you find the right memory configuration for your system.

With Kingston's configurator, you can search by five different criteria:

System manufacturer/model
Computer model name
Memory module part number (Kingston, distributor, manufacturer)
Specification
Generic memory
To access Kingston's Memory Configurator, click here
WHAT IF I CAN'T FIND MY SYSTEM IN A MEMORY CONFIGURATOR?
If you can't find your system in the memory configuration programs, you can still find out what kind of memory you need by consulting the manual that came with your system. In most cases, the manual will provide basic specifications such as the speed and technology of the memory you need. This information is usually enough to choose a module by specification. If you don't feel you have enough information, you can call your system manufacturer or Kingston's toll-free technical support number for assistance.
HOW MANY SOCKETS DO I HAVE OPEN?
You may or may not have an idea what the inside of your computer looks like and how memory is configured. You may have opened up your computer when you bought it to see the configuration inside, or you may have looked at a configuration diagram in your user's manual. Even if you have no idea of the memory configuration of your system, you can use Kingston's memory configuration tools to find out. For each system, the configuration includes a diagram, called a bank schema, which indicates how the memory sockets are arranged in your system and what the basic configuration rules are. The simple tutorial on the next page outlines how to use a bank schema diagram to determine the number of sockets in your system and how to fill them.

6.2. HOW TO READ A BANK SCHEMA

Sun, 01 Mar 1998 00:00:00 +0000

<< Previous page TOC Next page >>
6.2. HOW TO READ A BANK SCHEMA
A bank schema is a diagram of rows and columns that shows the number of memory sockets in your system. This diagram is a theoretical bank layout and not an actual system board layout; it is designed to help you quickly determine configuration requirements when adding memory modules.

In a bank schema, each represents a memory socket:

Example: = 4 memory sockets

Each column in the diagram represents a memory bank. The number of " " symbols in a column represents the number of memory sockets in a bank. Upgrading is performed one bank at a time. For example, if there are four columns with two in each column, upgrading is done two modules at a time. However, if there is just a single row of , upgrading may be performed one module at a time.

Examples:

8 sockets =
(Modules may be installed one at a time in any combination)

8 sockets (4 banks of 2) =

(Modules must be installed two at a time)

4 sockets (1 bank of 4) =

(Modules must be installed four
at a time)

The standard memory (base amount that the system was shipped with) appears in the diagram as either removable or non-removable.

Removable memory comes in the form of modules that fit into memory sockets, and, if desired, can be removed and replaced with modules of higher capacity. Removable memory is represented by a "" symbol with a number next to it: 4 that a 4MB module is in the first socket and that the second socket is empty.

Non-removable memory usually comes in the form of memory chips soldered directly onto the system board. It is represented in the bank schema in brackets: [_4MB_] indicates 4MB of non-removable memory soldered onto the board and two available memory sockets.

If your system is not included in the configurator, you may be able to find out how many sockets are in the system and how many are filled by pressing the F1 key during system startup. If your system supports this, a screen will appear that indicates how many memory sockets are in the system, which ones are filled and which are open, and what capacity modules are in each socket. If pressing the F1 key during startup doesn't produce this result, check your computer's system manual for more information.

As a last resort, you can open your computer and take a look at the sockets. (Important Note: Before removing the cover of your computer, refer to the computer's system manual and warranty information for instructions and other relevant information.) If you do open the computer, you may be able to identify "bank labels" that indicate whether memory are installed in pairs. Bank numbering typically begins with 0 instead of 1. So, if you have two banks, the first bank will be labeled "bank 0", and the second bank will be labeled "bank 1."

6.3. QUALITY

Sun, 01 Mar 1998 00:00:00 +0000

<< Previous page TOC Next page >>
6.3. QUALITY
As with any type of product, memory can vary in the quality from one manufacturer to another. In general, the larger, more established brand name companies are more consistent in adhering to tight design specifications, using high-quality components, and establishing certified quality control processes for manufacturing and thorough testing. That's not to say that lower-quality modules don't work fine - they may be the right solution, depending on how hard you work your system. In deciding the level of quality you require, consider the following:

If the memory you buy doesn't perform well, will you be comfortable returning it for a replacement? Would you have the time to deal with removing the memory and waiting a couple of days to a week to get the situation resolved?

When memory is of low quality, you often experience intermittent problems, such as the computer "freezing" unexpectedly, or having frequent errors. How often do you save your work, and if you were to lose your work, how much would that cost you? If you use your computer to play games, read email, and surf the Internet, such interruptions and losses may not be a big problem. But if you're running a business, losing a few hours of work could be a serious matter.

The biggest risk with unreliable memory is data corruption: that is, some bits of data may change or be read incorrectly. The result of data corruption could be as harmless as a syntax error in a document, or as potentially serious as a miscalculation in a spreadsheet. How important is the accuracy of the work you do on your computer? Again, if you use your computer for gaming, writing letters, and the Internet, it may not be a problem. But if you're managing your finances, you may want to do all you can to assure the reliability of your data.

Just like all products, the quality and durability you require depends on how you use it. Computer applications that require a lot of memory usually work the memory very hard. These applications often work better with memory that exceeds the system's speed and reliability specifications. If you're working in multimedia or using heavy number-crunching programs, the chance of a lower-quality memory module failing is greater than if you're only doing light work, such as simple word processing.

6.4. PRICING AND AVAILABILITY

Sun, 01 Mar 1998 00:00:00 +0000

<< Previous page TOC Next page >>
6.4. PRICING AND AVAILABILITY
This section contains information that helps make sense of the fluctuations that can occur in the memory market.

THE DRAM CHIP MARKET
Memory modules are made with DRAM chips, which are manufactured in mass quantities in enormous fabrication plants (often referred to as "fabs"). Fabs can take up to two years to build and require substantial capital investment: approximately $3 billion per plant. These time and cost factors directly affect on the ability of the memory market to adjust quickly to fluctuations in supply and demand. When demand for memory chips increases, chip manufacturers typically do not respond immediately because the investment required to add more capacity is substantial and may not pay off, especially if all the competitors are doing so at the same time. Therefore, the immediate effect is that prices rise as manufacturers assess whether the increase on demand is temporary or substantial enough to warrant investment. By the same token, when there is an oversupply situation in the market, chip manufacturers are willing to sustain losses for a long time while prices fall to below breakeven levels. This is because in many cases it costs more money to shut a plant down than to continue to produce and sell chips at below cost. Also, the longer a manufacturer can hold on, the greater the chance of "being there" to reap the rewards when competitors reduce capacity and the market turns around again.
WHY MEMORY PRICES FLUCTUATE
There are several factors that can affect memory prices. A few of these include: demand, DRAM manufacturing levels, inventory in the marketplace, time of the year, new operating system releases, and computer sales. All these things can affect memory prices at different times, either separately or simultaneously.

The most important thing to keep in mind when buying memory is that the price of 256MB today will most likely be different than the price of 256MB next quarter. The best rule of thumb is to compare memory prices close to your time of purchase. When doing price comparisons, it's more important to make sure you are making equivalent comparisons on module types than how the actual price per MB varies over time. If there are shortages in the market it's most important to be sure that what appears to be a "great deal" isn't a "short-cut" module built from off-spec components. In an oversupply market, you're much more likely to get a great price, but keep in mind that many manufacturers are losing money and may take shortcuts on testing and other expensive production quality measures to compensate. Refer to the quality section above for more details on this.

7. Bibliography

Sun, 01 Mar 1998 00:00:00 +0000

<< Previous page INDEX
7. Bibliography
Key Device and Design Papers EOS/ESD Symposium Proceedings
The following list contains some of the most significant papers on silicon device ESD phenomena and protection design from past proceedings of the EOS/ESD Symposium. After a careful review, the papers were selected based on their original contribution to the field and their effectiveness in advancing the understanding of ESD device phenomena. These papers provide further understanding of the concepts as well as implementation and practical use. This collection of articles will allow new researchers interested in the field to quickly gather pertinent information. The papers are categorized according to the topic of interest. A brief statement on the significance of each paper also is provided.
Copies of the papers may be purchased from the ESD Association
Compiled by: Charvaka Duvvury, May 2000

Section A: Device Phenomena

Year

Pages

Authors

Title

Significance

1981

120-131

Pierce & Durgin

An Overview of Electrical Overstress on Semiconductor Devices

Failure physics under ESD and EOS

1983

181-184

Duvvury et al.

A Summary of Effective Electrostatic Discharge Protection Circuits for MOS Memories and Their Observed Failure Modes

Study of npn turn- on of NMOS during ESD and basic ESD design issues

1985

67-76

Pierce

Electro-thermomigration as an Electrical Overstress Mechanism

Study of junction spiking for longer pulses

1988

201-205

Rountree et al.

A Process-Tolerant Input Protection Circuit for Advanced CMOS Processes

SCR design and analysis

1989

167-174

Polgreen & Chatterjee

Improving the ESD Failure Threshold of Silicided nMOS Transistors by Ensuring Uniform Current Flow

Fundamentals of multifinger NMOS behavior under ESD

1989

136-144

Krieger

The Dynamics of Electrostatic Discharge Prior to Bipolar Action Related Snap-back

Theoretical analysis of MOS bipolar turn-on during ESD

1995

162-174

Duvvury & Amerasekera

Advanced CMOS Device Trigger Mechanisms During CDM

Device behavior and simulations analysis under fast CDM

1999

1-10

Mergens

Analysis and Compact Modeling of Lateral DMOS Power Devices Under ESD Stress

Significant work in power device ESD phenomenon

Section B: Process & Technology Effects

Year

Pages

Authors

Title

Significance

1987

252-257

Fong & Hu

The Effects of High Electric Field Transients on Thin Gate Oxide MOSFETs

Oxide breakdown behavior under ESD pulses

1988

212-219

Chen

Effects of Interconnect Process and Snapback Voltage on the ESD Failure Threshold of NMOS Transistors

The first detailed study of NMOS ESD behavior with process changes

1990

143-150

Amerasekera et al.

An Analysis of Low Voltage ESD Damage in Advanced CMOS Processes

Study on the impact drain engineering on ESD damage

1993

117-122

Kuper et al.

Suppression of Soft ESD Failures in a Submicron CMOS Process

Analysis of the effects of LDD on ESD performance

1994

237-245

Amerasekera & Duvvury

The Impact of Technology Scaling on ESD Robustness and Protection Circuit Design

Analysis to establish ESD trends with scaling

1997

308-315

Bock et al.

Influence of Well Profile and Gate Length on the ESD Performance of a Fully Silicided 0.25 µm CMOS Technology

Study of substrate process on ESD

1997

316-329

Voldman et al.

ESD Robustness and Scaling Implications of Aluminum and Copper Interconnects in Advanced Semiconductor Technology

Current carrying capability of metal interconnects under ESD conditions

1998

161-169

Gupta et al.

ESD-Related Process Effects in Mixed Voltage Sub-0.50 µm Technologies

Study of process parameters on npn characteristics

1999

95-104

Bock

Influence of Gate Length on ESD Performance for Deep Submicron CMOS Technology

A new phenomenon for future technologies

Section C: Design Methods

Year

Pages

Authors

Title

Significance

1980

73-80

Keller

Protection of MOS Integrated Circuits from Destruction by Electrostatic Discharge

Fundamentals of ESD protection design

1981

90-96

Hulett

On-Chip Protection of High Density NMOS Devices

Layout effects of ESD protection design

1988

220-227

Maloney

Designing MOS Inputs and Outputs to Avoid Oxide Failure in the Charged Device Model

Failure phenomena & design techniques for CDM protection

1991

88-97

Duvvury & Rountree

A Synthesis of Input Protection Scheme

Analysis and design using discrete ESD device elements

1995

1-12

Maloney & Dabral

Novel Clamp Circuits for IC Power Supply Protection

Protection scheme designs using diode elements in complex circuit approach

1995

43-61

Voldman et al.

Analysis of Snubber-Clamped Diode-String Mixed Voltage Interface ESD Protection Network for Advanced Microprocessors

Detailed work on the operation and modeling of diodes

1997

230-239

Chen et al.

Design Methodology for Optimizing Gate-Driven ESD Protection Circuits In Submicron CMOS Processes

Issues related to gate modulated MOS protection

1999

62-69

Smith

An Anti-snapback Circuit Technique for Inhibiting Parasitic Bipolar Conduction During EOS/ESD Events

A new design method without using npn snapback in NMOS

Section D: Simulations/Modeling

Year

Pages

Authors

Title

Significance

1982

56-61

Pierce

Modeling Metalization Burnout of Integrated Circuits

A model for pulse width dependence of failure current density

1986

41-47

Scott et al.

A Lumped Element Model for Simulation of ESD Failures in Silicon Devices

Theoretical model that studied the effect of contact resistance for ESD

1988

137-146

Pierce

Electrical Overstress Testing of a 256K UVEPROM to Rectangular and Double Exponential Pulse

Modeling work that allows conversion to different types of waveforms

1993

73-80

Lin

ESD Sensitivity and VLSI Technology Trends: Thermal Breakdown and Dielectric Breakdown

Theoretical model to predict thermal and oxide breakdown sensitivities

1996

265-275

Beebe

Methodology for Layout Design and Optimization of ESD Protection Transistors

Analytical study of NMOS layout for ESD performance

Section E: Measurement/Characterization

Year

Pages

Authors

Title

Significance

1985

49-54

Maloney & Khurana

Transmission Line Pulsing Techniques for Circuit Modeling of ESD Phenomena

A pulsed method to measure ESD data

1989

59-71

Renninger et al.

A Field-Induced Charged Device Model Simulator

Description and operation of the CDM simulator

1990

119-130

Roozendaal et al.

Standard ESD Testing of Integrated Circuits

A comprehensive study of testers and parasitics

1992

129-134

Maloney

Integrated Circuit Metal in the Charged Device Model: Bootstrap Heating, Melt Damage and Scaling Laws

Metal heating under CDM events

1993

129-137

Verhaege et al.

Analysis of HBM ESD Testers and Specifications Using 4th Order Lumped Element Model

Analysis of HBM with parasitics

1998

170-176

Notermans et al.

Pitfalls When Correlating TLP, HBM, and MM Testing

Gives proper method to interpret data from pulse testing

1999

28-37

Wolf

Analyzing the Switching Behavior of ESD Protection Transistors by Very Fast Transmission Line Pulsing

A new test method to understand ESD behavior

1999

212-224

Voldman

A Strategy for Characterization and Evaluation of ESD Robustness of CMOS Semiconductor Technologies

The first paper to present ESD layout methods

7. HOW TO INSTALL MEMORY

Sun, 01 Mar 1998 00:00:00 +0000

<< Previous page TOC Next page >>
7. HOW TO INSTALL MEMORY
Congratulations! You have new memory for your computer. Now you have to install it. This chapter will guide you through the basics of memory module installation and refer you to resources that can help with problems.

7.1. BEFORE YOU INSTALL
7.2. IMPORTANT THINGS TO KEEP IN MIND
7.3. INSTALLING THE MEMORY
<< Previous page TOC Next page >>

7.1. BEFORE YOU INSTALL

Sun, 01 Mar 1998 00:00:00 +0000

<< Previous page TOC Next page >>
7.1. BEFORE YOU INSTALL
Before you start, make sure you have the following:
Your computer manual. To install memory, you must open the computer box (chassis) and locate the memory sockets. You may need to unplug cables and peripherals, and re-install them afterward. The manual will most likely provide instructions specific to your computer.

A small screwdriver. Most computer chassis assemble with screws. The screwdriver also comes in handy if the notches on memory sockets are too tiny for your fingers.

7.2. IMPORTANT THINGS TO KEEP IN MIND

Sun, 01 Mar 1998 00:00:00 +0000

<< Previous page TOC Next page >>
7.2. IMPORTANT THINGS TO KEEP IN MIND
ESD DAMAGE
Electro-Static Discharge (ESD) is a frequent causes of damage to the memory module. ESD is the result of handling the module without first properly grounding yourself and thereby dissipating static electricity from your body or clothing. If you have a grounded wrist strap, wear it. If you don't, before touching electronic components - especially your new memory module - make sure you first touch an unpainted, grounded metal object. Most convenient is the metal frame inside the computer. In addition, always handle the module by the edges. If ESD damages memory, problems may not show up immediately and may be difficult to diagnose.

Wearing a grounded wrist strap can prevent ESD damage.
SWITCHING OFF THE POWER
Before opening the chassis, always power-off your computer and all attached peripherals. Leaving power on can cause permanent electrical damage to your computer and its components.

7.3. INSTALLING THE MEMORY

Sun, 01 Mar 1998 00:00:00 +0000

<< Previous page TOC Next page >>
7.3. INSTALLING THE MEMORY
The vast majority of computers today have memory sockets that accept the following industry-standard memory modules:

Desktops, Workstations and Servers

72-pin SIMM
168-pin DIMM
184-pin RIMM
Notebooks and Mobile Computers

144-pin SO DIMM
Although sockets may be in different places on different computers, installation is the same. Consult the computer owner's manual to find out whether the memory sits on an expansion card or on the motherboard, and whether internal computer components must be moved to gain access.

In the section below are installation instructions for the standard modules listed above, followed by installation instructions for some of the more popular proprietary memory modules. If the computer requires proprietary memory, or the instructions below don't seem to apply to your situation, phone Kingston Technology's Technical Support Group at (800) 435-0640.
INSTALLING A 72-PIN SIMM

Installation of a 72-pin SIMM.
Place your computer's power switch in the off position and disconnect the AC power cord.

Follow the instructions in your owner's manual that describe how to locate your computer's memory expansion sockets.

Before touching any electronic components or opening the package containing your new module(s), make sure you first touch an unpainted, grounded metal object to discharge any static electricity you may have stored on your body or clothing.

Handle your new module(s) carefully; do not flex or bend the module(s). Always grasp the module by its edges.

As shown in the illustration, the module and the expansion socket are keyed. A small plastic bridge in the socket must align with the curved notch in the module. The bridge ensures the module can only be plugged into the socket one way.

Insert the module into the socket at a slight angle. Make sure the module is completely seated in the socket. If you're having problems inserting the module into the socket, stop and examine both the module and the socket; make sure the notch in the module is properly aligned with the keyed plastic bridge in the socket. Do not force the module into the socket. If too much force is used, both the socket and module could be damaged.

Once you are satisfied the module is seated properly in the socket, rotate the module upward until the clips at each end of the expansion socket click into place.

After all modules have been installed, close the computer, plug in the AC power cord, and reinstall any other cables that may have been disconnected during the installation process.
INSTALLING A 168-PIN DIMM

Installation of a 168-pin DIMM.
Locate the memory expansion sockets on the computer's motherboard. If all the sockets are full, you will need to remove smaller capacity modules to allow room for higher capacity modules.

For some installations, DIMM memory can be installed in any available expansion slot. Other installations may require the memory to be installed in a particular sequence based on the module's capacity. Check your owner's manual to determine the correct installation sequence for your configuration.

Insert the module into an available expansion socket as shown in the illustration. Note how the module is keyed to the socket. This ensures the module can be plugged into the socket one way only. Firmly press the module into position, making certain the module is completely seated in the socket. Repeat this procedure for any additional modules you are installing.

Most 168-pin DIMM modules have ejector tabs similar to those shown in the illustration. The ejector tabs are used only when you need to remove a module. By pressing down on the ejector tabs, the module will pop up from the socket and it can be removed.
INSTALLING A 184-PIN RIMM

Installation of a 184-pin RIMM. RIMMs install into RIMM connectors. Currently, there are two connectors on the computer board, and each must contain a RIMM or a C-RIMM (continuity RIMM). C-RIMMs do not contain memory devices. They are inexpensive pass-through modules that provide a continuous channel for the signal.
Turn off the computer and disconnect the AC power cord.

Locate your computer's memory expansion sockets by following the instruc-tions in your owner's manual.

Before touching any electronic components, make sure you first touch an unpainted, grounded metal object to discharge any static electricity stored on your clothing or body.

If all the sockets are full, you will need to remove smaller capacity modules to allow room for higher capacity modules.

The ejector tabs shown in the illustration are used to remove a module. By pushing outward on the ejector tabs, the module will pop up from the socket and it can then be removed.

For most installations, Rambus modules can be installed in any available expansion socket, but any empty sockets must contain a continuity module as shown in the illustration. Note that some modes may use a specific installation sequence for Rambus modules (e.g. Rambus dual-channel configurations); see your owner's manual for more details.

Insert the module into an available expansion socket as shown in the illustration. Note how the module is keyed to the socket. This ensures the module can be plugged into the socket one way only. Firmly press the module into position, making certain the module is completely seated in the socket. The ejector tabs at each end of the socket will automatically snap into the locked position. Repeat this procedure for any additional modules are installing.

Once the module or modules have been installed, close the computer.
EXAMPLES OF NOTEBOOK MEMORY INSTALLATION

Although more and more notebooks accept standard SO DIMM memory modules, many still require proprietary modules having unique form factors. No standard dictates where the memory should be installed in the notebook. Because notebooks differ greatly in how memory is installed, you should consult your computer manual for specific instructions.This example shows a typical 144-pin SO DIM installation.
Always turn off your computer and remove the rechargeable battery pack before installing the memory.

Insert the module into the socket at a slight angle (approximately 30 degrees). Note that the socket and module are both keyed, which means the module can be installed one way only.

To seat the module into the socket, apply firm, even pressure to each end of the module (see the arrows) until you feel it slip down into the socket. If you are having problems getting the module to seat properly, try rocking the module up and down slightly, while continuing to apply pressure. When properly seated, the contact fingers on the edge of the module will almost completely disappear inside the socket.

With the module properly seated in the socket, rotate the module downward, as indicated in the illustration. Continue pressing downward until the clips at each end of the socket lock into position. With most sockets, you will hear a distinctive click, indicating the module is correctly locked into position.

8. TROUBLESHOOTING MEMORY PROBLEMS

Sun, 01 Mar 1998 00:00:00 +0000

<< Previous page TOC Next page >>
8. TROUBLESHOOTING MEMORY PROBLEMS
8.1. COMMON MEMORY PROBLEMS
8.2. BASIC TROUBLESHOOTING
8.3. WHEN THE PROBLEM OCCURS
8.4. HANDLING SPECIFIC PROBLEMS
8.5. STILL NEED HELP?
<< Previous page TOC Next page >>

8.1. COMMON MEMORY PROBLEMS

Sun, 01 Mar 1998 00:00:00 +0000

<< Previous page TOC Next page >>
8.1. COMMON MEMORY PROBLEMS
When you have a problem with memory, the cause is usually one of three things:

Improper Configuration: You have the wrong part for your computer or did not follow the configuration rules.

Improper Installation: The memory may not be seated correctly, a socket is bad, or the socket may need cleaning.

Defective Hardware: The memory module itself is defective.

The fact that many computer problems manifest themselves as memory problems makes troubleshooting difficult. For example, a problem with the motherboard or software may produce a memory error message.

This chapter is designed to help you figure out if you have a memory problem, and if so, what kind of problem it is, so you can get to a solution as quickly as possible.

8.2. BASIC TROUBLESHOOTING

Sun, 01 Mar 1998 00:00:00 +0000

<< Previous page TOC Next page >>
8.2. BASIC TROUBLESHOOTING
The following basic steps apply to almost all situations:

Make sure you have the right memory part for your computer.At the manufacturer's Web site you can look up the part number. Many memory manufacturers have configurators, which indicate the compatibilities of your module. If not, phone the memory manufacturer, consult your computer manual, or phone the computer manufacturer.

Confirm that you configured the memory correctly.Many computers require module installation in banks of equal-capacity modules. Some computers require the highest capacity module to be in the lowest labeled bank. Other computers require that all sockets be filled; still others require single-banked memory. These are only a few examples of special configuration requirements. If you have a name-brand computer, visit Kingston's Web site (www.kingston.com) or use our upgrade manual to look up configuration rules specific to your computer. You can also contact technical support for your memory or computer manufacturer.

Re-install the module.Push the module firmly into the socket. In most cases you hear a click when the module is in position. To make sure you have a module all the way in the socket, compare the height of the module to the height of other modules in neighboring sockets.

Swap modules.Remove the new memory and see whether the problem disappears. Remove the old memory, reinstall the new, and see whether the problem persists. Try the memory in different sockets. Swapping reveals whether the problem is a particular memory module or socket, or whether two types of memory aren't compatible.

Clean the socket and pins on the memory module.Use a soft cloth to wipe the pins on the module. Use a PC vacuum or compressed air to blow dust off the socket. Do NOT use solvent, which may corrode the metal or prevent the leads from making full contact. Flux Off is a cleaner used specifically for contacts. You can purchase it at electronics or computer equipment stores.

Update the BIOS.Computer manufacturers update BIOS information frequently and post revisions on their Web sites. Make sure you have the most recent BIOS for your computer. This applies especially when you have recently installed new software or you are significantly upgrading memory.

8.3. WHEN THE PROBLEM OCCURS

Sun, 01 Mar 1998 00:00:00 +0000

<< Previous page TOC Next page >>
8.3. WHEN THE PROBLEM OCCURS
When the problem occurs is a clue as to the cause.

For example, your response to a memory error message depends on whether:

You have just bought a new computer.
You have just installed new memory.
You have just installed new software or a new operating system.
You have just installed or removed hardware.
Your computer has been running fine and you've made no other recent changes.

Here are rules to get started:

YOU'VE JUST BOUGHT A NEW COMPUTER

If you have just purchased a new computer and it is producing memory errors, the problem could be related to anything, including a bad computer board. In this case, you need to troubleshoot the entire computer, including memory. Because the computer dealer will have configured memory and run system tests before shipping, they can best help.

YOU'VE JUST INSTALLED NEW MEMORY

If you have just installed new memory, the first possibility is that you installed incorrect parts. Double-check the part numbers. Confirm that you have configured and installed the memory correctly.

YOU'VE INSTALLED NEW SOFTWARE OR OPERATING SYSTEM

Newer software or operating systems tend to push memory harder than older operating systems. Sometimes memory that worked fine prior to a software installation begins producing errors once it runs memory-intensive software. New software also has bugs, and beta versions are notorious for producing memory errors. In these cases, your first step should be to ensure you have the latest BIOS and service patches for your software. Otherwise contact the memory vendor. A technical support representative may have experience with other software incidents and can walk you through more-detailed troubleshooting.

YOU'VE INSTALLED OR REMOVED HARDWARE

If you have just installed or removed hardware and suddenly receive memory error messages, the first place to look is in the computer itself. A connection may have come loose during the installation or the new hardware may be defective; in either case the errors are manifesting themselves as memory problems. Make sure you have the latest drivers and firmware. Most hardware manufacturers will post updates on their Web sites.

UNEXPECTED PROBLEMS

If your system has been running fine, but suddenly starts to produce memory errors, and crash or lock up frequently, the chance of a hardware failure is more likely, since configuration and installation problems show up as soon as the computer turns on. Sometimes you can get memory problems if your computer is overheating, if you are having a problem with your power supply, or if corrosion has developed between the memory module and the socket, weakening the connection.

8.4. HANDLING SPECIFIC PROBLEMS

Sun, 01 Mar 1998 00:00:00 +0000

<< Previous page TOC Next page >>
8.4. HANDLING SPECIFIC PROBLEMS
Here is a list of the most common ways the computer informs you of a memory problem.

The computer won't boot, merely beeps.

The computer boots but the screen is blank.

The computer boots but the screen is blank.

The computer reports a memory error.
Memory mismatch error
Memory parity interrupt at xxxxx
Memory address error at xxxxx
Memory failure at xxxxx, read xxxxx, expecting xxxxx
Memory verify error at xxxxx

The computer has other problems caused by memory.
The computer intermittently reports errors, crashes frequently, or spontaneously reboots.
Registry Errors
General-protection faults, page faults, and exception errors

The server system manager reports a memory error.

The following translations help you understand what the computer means when it gives you one of these signals.

Computer won't boot, merely beeps.
Every time the computer starts, it takes inventory of hardware. Inventory consists of the computer BIOS recognizing, acknowledging, and in some cases, assigning addresses to, the components in the computer. If the computer won't boot, the CPU is unable to communicate with hardware. The cause can be improper installation or failure of the BIOS to recognize hardware. Follow basic troubleshooting, paying special attention to whether the memory module is completely installed and that you have the latest version of the BIOS.

Computer boots but doesn't recognize all the installed memory.When the computer boots, a part of the process is counting memory. On some machines the count appears on the screen and on others is masked. If the count is masked, from the computer set-up menu see how much memory the computer thinks it has. If the computer counts to or lists a number less than the memory you installed, the computer hasn't recognized all the memory.

Sometimes the computer will recognize only part of a module. This is almost always due to using the wrong kind of memory. For example, if your computer accepts only single-banked memory and you have installed dual-banked, the computer will read only half the memory on the module. Sometimes the computer will accept only modules containing memory chips with specific organizations. For example, the VX chipset doesn't work well with 64 Mbit chips.

In many computers the maximum amount of memory the computer can recognize is lower than the maximum amount you can physically install. For example, your computer may have three sockets, each of which can hold a 128MB module. If you filled every socket with 128MB, you would have 384MB of memory. However, your computer may recognize a maximum of 256MB. In most cases you can avoid this problem by consulting your computer manual or a memory configuration Web site before purchasing memory. Or visit the Kingston Web site.

The computer boots but the screen is blank.The most common reason for a blank screen is a dislodged card, memory not fully seated, or memory the computer doesn't support. Confirm that the memory is installed properly and that other components in the computer were not accidentally disconnected or dislodged while you installed memory.

Double-check that you have the right part number for the computer. If you have nonparity memory in a computer that requires error-checking memory, or SDRAM memory in a computer that supports only EDO, the screen may be blank at boot up.

The computer reports a memory error.
Memory mismatch error: This is not actually an error. Some computers require you to tell them that it's OK to have a new amount of memory. Use the set-up menu to tell the computer. Follow the prompts, enter the new amount, select Save, and exit.

Computer memory or address errors: All of the following errors, and those similar to them, indicate that the computer has a problem with memory:

Memory parity interrupt at xxxxx
Memory address error at xxxxx
Memory failure at xxxxx, read xxxxx, expecting xxxxx
Memory verification error at xxxxx

Typically the computer will perform a simple memory test as it boots. The computer will write information to memory and read it back. If the computer doesn't get what it was expecting, then it will report an error and sometimes give the address where the error occurred.

Such errors normally indicate a problem with a memory module but can sometimes indicate a defective motherboard or incompatibility between old and new memory. To verify that the new memory is causing the problem, remove the new memory and see whether the problem goes away. Then remove the old memory and install only the new memory. If the error persists, phone the memory manufacturer and ask for a replacement.

The computer has other problems caused by memory.
The Computer Intermittently Reports Errors, Crashes Frequently, or Spontaneously Reboots: Because of the large number of causes, these problems are difficult to diagnose. Possible causes are ESD (Electro-static Discharge), overheating, corrosion, or a faulty power supply. If you suspect ESD damage, contact the memory manufacturer and ask for a replacement. Before you install new memory, see page 85 for information on how to prevent ESD. If you suspect corrosion, clean the memory contacts and the sockets as explained on page 96. If you suspect the power supply, you will have to do overall computer troubleshooting with a focus on the power supply.

Registry Errors: Windows writes a large portion of the registry to RAM. Sometimes defective memory will cause registry errors. Windows reports a registry error and prompts you to restart and restore. If the prompts repeat, remove your newly installed memory and restart the computer. If the errors disappear, ask the memory manufacturer for replacement modules.

General-Protection Faults, Page Faults, and Exception Errors: The most common cause is software. For example, one application may not have released the memory after quitting or occupies the same memory addresses as another. In these cases, rebooting should solve the problem. If the computer suddenly displays general-protection faults, exception errors, or page faults after you have installed new memory, remove the new memory and see whether the errors stop. If they occur only when the new memory is installed, contact the memory manufacturer for assistance.

The server system manager reports a memory error.Most servers ship with system managers that monitor component utilization and test for abnormalities. Some of these system managers count soft errors in memory. Soft errors have been corrected by ECC memory. If the rate of soft errors is higher than specifications, however, the system manager issues a pre-failure warning. This warning enables the network administrator to replace the memory and prevent system downtime.

If the system manager on your server issues a pre-failure warning or other memory error, ask your memory manufacturer for a replacement. If the system manager continues to issue errors after memory replacement, make sure you have the latest BIOS, software service patches, and firmware. The chance of receiving two bad memory modules in a row is low. Contact the memory manufacturer for compatibility troubleshooting. Sometimes the server does not work well with certain types of memory chips or certain memory designs.

8.5. STILL NEED HELP?

Sun, 01 Mar 1998 00:00:00 +0000

<< Previous page TOC Next page >>
8.5. STILL NEED HELP?
Most memory manufacturers have FAQ or Q-and-A sections on their Web sites. There are also troubleshooting areas on the Web site of the computer manufacturer. If you can't find anything online, phone technical support for the computer or memory manufacturer. The next section gives information about Kingston and how to reach us.
<< Previous page TOC Next page >>

9. MORE ABOUT KINGSTON

Sun, 01 Mar 1998 00:00:00 +0000

<< Previous page TOC Next page >>
9. MORE ABOUT KINGSTON
9.1. COMPANY OVERVIEW
9.2. HOW TO REACH KINGSTON
9.3. WHY KINGSTON?
<< Previous page TOC Next page >>

9.1. COMPANY OVERVIEW

Sun, 01 Mar 1998 00:00:00 +0000

<< Previous page TOC Next page >>
9.1. COMPANY OVERVIEW
Founded in 1987, Kingston Technology is the world's largest independent manu-facturer of memory products. At the close of 1998, Kingston had sales in excess of $1.5 billion. Worldwide, the company presently staffs four manufacturing sites on four continents and six international sales and marketing facilities.

Kingston manufactures memory products that consistently meet or exceed the industry's stringent standards, possessing both the skill and equipment necessary to create the best modules in the marketplace.

The personal philosophy of company founders John Tu and David Sun, has been permanently imprinted into the corporate culture. From the company founders to the employees and ultimately, to the consumer, a special "magic" is infused into every product. At Kingston, it is the people that make the difference.

The Kingston spirit reflects an individual commitment...

CORE VALUES

Respect for one another in our culturally diverse environment

Loyalty to our long-term partnerships

Flexibility and adaptability in responding to our customers' needs

Investing in our employees to continuously improve our most valuable resource

Having fun working in the company of friends

9.2. HOW TO REACH KINGSTON

Sun, 01 Mar 1998 00:00:00 +0000

<< Previous page TOC Next page >>
9.2. HOW TO REACH KINGSTON
HOW TO REACH KINGSTON

U.S. HEADQUARTERS

Kingston Technology Company Toll Free: +1 (877) KINGSTON
17600 Newhope Street Tel: +1 (714) 435-2600
Fountain Valley, CA 92708 Fax: +1 (714) 435-2699
USA Internet: www.kingston.com

MEMORY-U.S.
PHONE/FAX NUMBER EMAIL ADDRESS
Memory Sales +1(877)KINGSTON
+1(714)435-2699 sales@kingston.com
High-end Memory
Sales +1(888)435-5446
+1(714)435-2618 specialtygroup-hem@kingston.com
ValueRam Sales +1(877)435-8726
+1(714)445-3496 valueram@kingston.com
Flash Memory Sales +1(888)822-6315
+1(714)438-1879 customer_sales_support@kingston.com
Technical Support +1(800)435-0640
+1(714)424-3939 tech_support@kingston.com
RMA +1(800)337-3719
+1(714)435-2643 rma@kingston.com
MEMORY-INTL.
(Supporting All
Regions Except
Europe) PHONE/FAX NUMBER EMAIL ADDRESS
Memory Sales
(in English) +1(714)437-3334
+1(714)438-1820 intlsales@kingston.com
Technical Support
(in English) +1(800)435-0640
+1(714)424-3939 tech_support@kingston.com
Technical Support
(in Spanish) +1(888)822-6316
+1(714)424-3939 soporte_technico@kingston.com
RMA (in Spanish) +1(714)437-3334
+1(714)438-1820 export_coordinator@kingston.com
STORCASE TECHNOLOGY, INC. PHONE/FAX NUMBER EMAIL ADDRESS
StorCase Sales +1(800)435-0642
+1(714)438-1847 info@storcase.com
StorCase Technical
Support +1(714)438-1858
+1(714)438-1847 tech_support@storcase.com

EUROPEAN HEADQUARTERS

Kingston Technology Europe, Inc. Tel: +44 (0) 1932 738 888
Kingston Court Fax: +44 (0) 1932 738 880
Sunbury-on-Thames
Middlesex TW16 7EP
England

EUROPEAN BRANCH OFFICES

Kingston Technology GmbH
Hoferstrasse 1
81737 Munich
Tel: +49 89 627 1560
Telefax: +49 49 89 627 15660

Kingston Technology France
171A, Avenue Charles De Gaulle
Tel: +33 1 46 43 9530
Tel: 0800 90 57 01 (Toll-free Customer Service within France)
Telefax: +33 1 46 43 9535
Telecopie: 0800 90 09 10 (Toll-free within France)

9.3. WHY KINGSTON?

Sun, 01 Mar 1998 00:00:00 +0000

<< Previous page TOC Next page >>
9.3. WHY KINGSTON?
ISO 9001 CERTIFICATION

Kingston is ISO 9001 certified as part of a continuing global effort to assure customers the highest quality service and products.

EXCEPTIONAL QUALITY

In addition to our ISO 9001 certification, we have established an extensive DCAT (Design, Components, Assembly, and Test) quality-control process to ensure total product reliability.

PROVEN INDUSTRY LEADER

Kingston has been making memory modules since 1987.

100% PRODUCT TESTING

Kingston tests every memory module before shipping. Our customized test equipment is so comprehensive that it can test every cell on every chip on every module-on a 64MB module, that's 512 million cells!

PREMIUM COMPONENTS

Kingston buys 100% of its memory chips from top-tier DRAM suppliers.

FREE TECHNICAL SUPPORT

Technical support professionals are available to answer your call 24 hours a day, 7 days a week at (800) 435-0640. Average wait time is less than 1 minute.

KINGSTON'S LIFETIME WARRANTY

With a lifetime warranty on all Kingston memory upgrades, network adapters, and processor upgrades, you have the most comprehensive warranty in the industry.

FREE EVALUATION UNITS

Kingston offers a free 30-day product evaluation program to qualified corporate customers.

100% COMPATIBILITY GUARANTEE

Kingston guarantees all memory products to be compatible with the system or family of systems for which they're designed.

OVERNIGHT PRODUCT REPLACEMENT

In the unlikely event your Kingston product needs repair or replacement, we will cross-ship a replacement product to you overnight (in most cases).

FREE LITERATURE

Kingston is committed to educating customers on memory and related technology. White papers, memory bits, and other informative tools are available free of charge in electronic and printed formats. Send requests to webmaster@kingston.com.

CORPORATE ALLIANCE PROGRAM (CAP)

As an extension of Kingston's service commitment, we offer Corporate Alliance membership to qualified IT professionals. Benefits include a dedicated Kingston representative, use of the Partner Purchase Program, on site tech support training, and special pricing.

FIELD SUPPORT

Kingston's field sales team is available for regional and on-site customer support. To find a representative in your area, go to www.kingston.com/contact/field.asp.

USER-FRIENDLY WEB SITE

Kingston's Web site www.kingston.com has the latest information on industry trends, promotions, product releases, and more.

Device Servers: Network-Enabling Nearly Any Device

Sun, 01 Mar 1998 00:00:00 +0000

Terminal and Print Servers -- Traditional Applications

Despite many of the advancements of the computer industry, there still exist a great many applications where serial I/O devices are the best or only solution. Terminal/print/serial servers have long been the best method for networking simple terminals, bar code readers, scanners or printers. Input devices attached to a serial port on a server can reach any host supporting the same protocols as that server. Printers can be shared over the network in the same manner, with a job from one protocol following a job from another. In the case of the print server that has multiple ports, jobs coming from hosts supporting different protocols can even print simultaneously.

Ethernet Tutorial Part I: Networking Basics

Sun, 01 Mar 1998 00:00:00 +0000

As companies rely on applications like electronic mail and database management for core business operations, computer networking becomes increasingly more important. This tutorial helps to explain Ethernet and Fast Ethernet, which are two of the most popular technologies used in networking.

LANs (Local Area Networks)

A network is any collection of independent computers that communicate with one another over a shared network medium. LANs are networks usually confined to a geographic area, such as a single building or a college campus. LANs can be small, linking as few as three computers, but often link hundreds of computers used by thousands of people. The development of standard networking protocols and media has resulted in worldwide proliferation of LANs throughout business and educational organizations.

Ethernet Tutorial Part II: Adding Speed

Sun, 01 Mar 1998 00:00:00 +0000

While repeaters allow LANs to extend beyond normal distance limitations, they still limit the number of nodes that can be supported. Bridges and switches, however, allow LANs to grow significantly larger by virtue of their ability to support full Ethernet segments on each port. Additionally, bridges and switches selectively filter network traffic to only those packets needed on each segment - this significantly increases throughput on each segment and on the overall network. By providing better performance and more flexibility for network topologies, bridges and switches will continue to gain popularity among network managers.

Ethernet Tutorial Part III: Sharing Devices

Sun, 01 Mar 1998 00:00:00 +0000

Device Servers

A device server is defined as a specialized, network-based hardware device designed to perform a single or specialized set of server functions. It is characterized by a minimal operating architecture that requires no per seat network operating system license, and client access that is independent of any operating system or proprietary protocol. In addition the device server is a “closed box,” delivering extreme ease of installation, minimal maintenance, and can be managed by the client remotely via a Web browser.

Guide to Using Fast Ethernet

Sun, 01 Mar 1998 00:00:00 +0000

The demand for higher transmission speeds for near-term network growth has been realized by the new Fast Ethernet specification (IEEE 802.3u) known as 100BASE-T. This new LAN standard has raised the Ethernet speed limit from 10 Megabits per second to 100 Megabits per second with only minimal changes to the existing cable structure. The building blocks of today’s networks call out for a mixture of legacy 10BASE-T Ethernet networks and Fast Ethernet.

Music 320 Background Reader Mathematics of the Discrete Fourier Transform (DFT)

Sun, 01 Mar 1998 00:00:00 +0000

NOTE: THIS DOCUMENT IS OBSOLETE, PLEASE CHECK THE NEW VERSION: "Mathematics of the Discrete Fourier Transform (DFT), with Audio Applications --- Second Edition", by Julius O. Smith III, W3K Publishing, 2007, ISBN 978-0-9745607-4-8. - Copyright © 2017-09-28 by Julius O. Smith III - Center for Computer Research in Music and Acoustics (CCRMA), Stanford University

<< Previous page TOC INDEX Next page >>
Music 320 Background Reader
Mathematics of the Discrete Fourier Transform (DFT)
Julius O. Smith III(jos@ccrma.stanford.edu)
Center for Computer Research in Music
and Acoustics (CCRMA)
Department of Music,Stanford University
Stanford, California 94305

Preface
Outline
Introduction to the DFT
The DFT
Mathematics of the DFT
DFT Math Outline
Introduction to Complex Numbers
Factoring a Polynomial with Real Roots
The Quadratic Formula
Complex Roots
The Fundamental Theorem of Algebra
Complex Numbers
The Complex Plane
More Notation and Terminology
Elementary Relationships
Euler’s Formula
De Moivre’s Theorem
Numerical Tools in Matlab
Numerical Tools in Mathematica
Proof of Euler’s Identity
Euler’s Theorem
Positive Integer Exponents
Properties of Exponents
The Exponent Zero
Negative Exponents
Rational Exponents
Special Case: The Mth Roots of Unity
Real Exponents
A First Look at Taylor Series
Imaginary Exponents
Derivatives of f(x)=a^x
Back to e
Sidebar on Mathematica
Back to e^(j theta)
Informal Derivation of Taylor Series Expansion
Derivation of Taylor Series Expansion with Remainder
Formal Statement of Taylor’s Theorem
The Weierstrass (Polynomial) Approximation Theorem
Further Notes on Differentiability of Audio Signals
Logarithms, Decibels, and Number Systems
Logarithms
Changing the Base
Logarithms of Negative and Imaginary Numbers
Decibels
Properties of DB Scales
Specific DB Scales
DBm Scale
DBV Scale
DB SPL
DB for Display
Dynamic Range
Linear Number Systems for Digital Audio
Pulse Code Modulation (PCM)
Binary Integer Fixed-Point Numbers
One’s Complement Fixed-Point Format
Two’s Complement Fixed-Point Format
General Formula for Two’s-Complement, Integer Fixed-Point Numbers
‘‘Little Endian’’ Formula for Two’s-Complement, Integer Fixed-Point Numbers
Fractional Binary Fixed-Point Numbers
How Many Bits are Enough for Digital Audio?
When Do We Have to Swap Bytes When Changing Computers?
Logarithmic Number Systems for Digital Audio
Floating-Point Numbers
Logarithmic Fixed-Point Numbers
-Law Companding
Appendix A: Round-Off Error Variance
Appendix B: Electrical Engineering 101
Sinusoids and Exponentials
Sinusoids
Example Sinusoids
Why Sinusoids are Important
In-Phase and Quadrature Sinusoidal Components
Sinusoids at the Same Frequency
Constructive and Destructive Interference of Sinusoids
Exponentials
Why Exponentials are Important
Audio Decay Time ()
Complex Sinusoids
Circular Motion
Projection of Circular Motion
Positive and Negative Frequencies
The Analytic Signal and Hilbert Transform Filters
Generalized Complex Sinusoids
Sampled Sinusoids
Powers of
Why (Generalized) Complex Sinusoids are Important
Comparing Analog and Digital Complex Planes
Mathematica for Selected Plots
Acknowledgement
Geometric Signal Theory
The DFT
Signals as Vectors
An Example Vector View:
Vector Addition
Vector Subtraction
Signal Metrics
Other Norms
The Inner Product
Linearity of the Inner Product
Norm Induced by the Inner Product
Cauchy-Schwarz Inequality
Triangle Inequality
Triangle Difference Inequality
Vector Cosine
Orthogonality
The Pythagorean Theorem in N-Space
Projection
Signal Reconstruction from Projections
An Example of Changing Coordinates in 2D
General Conditions
Gram-Schmidt Orthogonalization
Appendix: Matlab Examples
The Discrete Fourier Transform (DFT) Derived
The DFT Derived
Geometric Series
Orthogonality of Sinusoids
Orthogonality of the DFT Sinusoids
Norm of the DFT Sinusoids
An Orthonormal Sinusoidal Set
The Discrete Fourier Transform (DFT)
Frequencies in the ‘‘Cracks’’
Normalized DFT
The Length 2 DFT
Matrix Formulation of the DFT
Matrices
Matrix Multiplication
The DFT Matrix
Matlab Examples
Figure 7.2
Figure 7.3
DFT Matrix
Fourier Theorems for the DFT
The DFT and its Inverse
Notation and Terminology
Modulo Indexing, Periodic Extension
Signal Operators
Flip Operator
Shift Operator
Convolution
Graphical Convolution
Polynomial Multiplication
Multiplication of Decimal Numbers
Correlation
Stretch Operator
Zero Padding
Repeat Operator
Decimation Operator
Alias Operator
Even and Odd Functions
The Fourier Theorems
Linearity
Conjugation and Reversal
Symmetry
Shift Theorem
Linear Phase Terms
Application of the Shift Theorem to FFT Windows
Convolution Theorem
Dual of the Convolution Theorem
Correlation Theorem
Power Theorem
Rayleigh Energy Theorem (Parseval’s Theorem)
Stretch Theorem (Repeat Theorem)
Decimation Theorem (Aliasing Theorem)
Zero Padding Theorem
Bandlimited Interpolation in Time
Conclusions
Acknowledgement
Appendix A: Linear Time-Invariant Filters and Convolution
LTI Filters and the Convolution Theorem
Appendix B: Introductory Statistical Signal Processing
Cross-Correlation
Applications of Cross-Correlation
Autocorrelation
Coherence
Appendix C: Mathematica/Matlab Examples
Appendix D: The Similarity Theorem
Example Applications of the DFT
Spectrum Analysis of a Sinusoid: Windowing, Zero-Padding, and the FFT
Example 1: FFT of a Simple Sinusoid
Example 2: FFT of a Not-So-Simple Sinusoid
Example 3: FFT of a Zero-Padded Sinusoid
Example 4: Blackman Window
Example 5: Use of the Blackman Window
Example 6: Hanning-Windowed Complex Sinusoid
Spectral Phase
A Basic Tutorial on Sampling Theory
Introduction
Reconstruction from Samples–Pictorial Version
Reconstruction from Samples–The Math
Aliasing of Sampled Continuous-Time Signals
Shannon’s Sampling Theorem
Figuring Out Sampling Theory by Playing Around with Complex Sinusoids
What frequencies are representable by a geometric sequence?
Recovering a Continuous-Time Signal from its Samples
Method 1: Additive Synthesis
Does it Work?
Introduction to Digital Filter Analysis
Motivating Example and Overview
FIR Filters
Convolution Representation
Finiteness
Causal FIR Filters
Transfer Function
Order
The BiQuad Section
Digital Filter Theory Summary
Linearity and Time-Invariance
Difference Equation
Convolution Representation of LTI Filters
Frequency Response
Phase Delay and Group Delay
Bibliography
Index
About this document …

<< Previous page TOC INDEX Next page >>

Network Switching Tutorial

Sun, 01 Mar 1998 00:00:00 +0000

Switches are great, they add network capacity and speed, but they are not a cure-all. How can you tell if your network will benefit from switching? And how do you add switches to your network design for the most benefit? This tutorial is written to answer these questions. Along the way we’ll describe how switches work, and how they both harm and benefit your networking strategy. And we¹ll discuss different network types, so you can profile your network and gauge the potential benefit of switching for your environment.

00. Table Of Content

Sun, 01 Mar 1998 00:00:00 +0000

NOTE: THIS DOCUMENT IS OBSOLETE, PLEASE CHECK THE NEW VERSION: "Mathematics of the Discrete Fourier Transform (DFT), with Audio Applications --- Second Edition", by Julius O. Smith III, W3K Publishing, 2007, ISBN 978-0-9745607-4-8. - Copyright © 2017-09-28 by Julius O. Smith III - Center for Computer Research in Music and Acoustics (CCRMA), Stanford University

INDEX Next page >>
Table Of Content

Preface

Outline

Introduction to the DFT

The DFT

Mathematics of the DFT

DFT Math Outline

Introduction to Complex Numbers

Factoring a Polynomial with Real Roots

The Quadratic Formula

Complex Roots

The Fundamental Theorem of Algebra

Complex Numbers

The Complex Plane

More Notation and Terminology

Elementary Relationships

Euler's Formula

De Moivre's Theorem

Numerical Tools in Matlab

Numerical Tools in Mathematica

Proof of Euler's Identity

Euler's Theorem

Positive Integer Exponents

Properties of Exponents

The Exponent Zero

Negative Exponents

Rational Exponents

Special Case: The Mth Roots of Unity

Real Exponents

A First Look at Taylor Series

Imaginary Exponents

Derivatives of f(x)=a^x

Back to e

Sidebar on Mathematica

Back to e^(j theta)

Informal Derivation of Taylor Series Expansion

Derivation of Taylor Series Expansion with Remainder

Formal Statement of Taylor's Theorem

The Weierstrass (Polynomial) Approximation Theorem

Further Notes on Differentiability of Audio Signals

Logarithms, Decibels, and Number Systems

Logarithms

Changing the Base

Logarithms of Negative and Imaginary Numbers

Decibels

Properties of DB Scales

Specific DB Scales

DBm Scale

DBV Scale

DB SPL

DB for Display

Dynamic Range

Linear Number Systems for Digital Audio

Pulse Code Modulation (PCM)

Binary Integer Fixed-Point Numbers

One's Complement Fixed-Point Format

Two's Complement Fixed-Point Format

General Formula for Two's-Complement, Integer Fixed-Point Numbers

''Little Endian'' Formula for Two's-Complement, Integer Fixed-Point Numbers

Fractional Binary Fixed-Point Numbers

How Many Bits are Enough for Digital Audio?

When Do We Have to Swap Bytes When Changing Computers?

Logarithmic Number Systems for Digital Audio

Floating-Point Numbers

Logarithmic Fixed-Point Numbers

-Law Companding

Appendix A: Round-Off Error Variance

Appendix B: Electrical Engineering 101

Sinusoids and Exponentials

Sinusoids

Example Sinusoids

Why Sinusoids are Important

In-Phase and Quadrature Sinusoidal Components

Sinusoids at the Same Frequency

Constructive and Destructive Interference of Sinusoids

Exponentials

Why Exponentials are Important

Audio Decay Time ()

Complex Sinusoids

Circular Motion

Projection of Circular Motion

Positive and Negative Frequencies

The Analytic Signal and Hilbert Transform Filters

Generalized Complex Sinusoids

Sampled Sinusoids

Powers of

Why (Generalized) Complex Sinusoids are Important

Comparing Analog and Digital Complex Planes

Mathematica for Selected Plots

Acknowledgement

Geometric Signal Theory

The DFT

Signals as Vectors

An Example Vector View:

Vector Addition

Vector Subtraction

Signal Metrics

Other Norms

The Inner Product

Linearity of the Inner Product

Norm Induced by the Inner Product

Cauchy-Schwarz Inequality

Triangle Inequality

Triangle Difference Inequality

Vector Cosine

Orthogonality

The Pythagorean Theorem in N-Space

Projection

Signal Reconstruction from Projections

An Example of Changing Coordinates in 2D

General Conditions

Gram-Schmidt Orthogonalization

Appendix: Matlab Examples

The Discrete Fourier Transform (DFT) Derived

The DFT Derived

Geometric Series

Orthogonality of Sinusoids

Orthogonality of the DFT Sinusoids

Norm of the DFT Sinusoids

An Orthonormal Sinusoidal Set

The Discrete Fourier Transform (DFT)

Frequencies in the ''Cracks''

Normalized DFT

The Length 2 DFT

Matrix Formulation of the DFT

Matrices

Matrix Multiplication

The DFT Matrix

Matlab Examples

Figure 7.2

Figure 7.3

DFT Matrix

Fourier Theorems for the DFT

The DFT and its Inverse

Notation and Terminology

Modulo Indexing, Periodic Extension

Signal Operators

Flip Operator

Shift Operator

Convolution

Graphical Convolution

Polynomial Multiplication

Multiplication of Decimal Numbers

Correlation

Stretch Operator

Zero Padding

Repeat Operator

Decimation Operator

Alias Operator

Even and Odd Functions

The Fourier Theorems

Linearity

Conjugation and Reversal

Symmetry

Shift Theorem

Linear Phase Terms

Application of the Shift Theorem to FFT Windows

Convolution Theorem

Dual of the Convolution Theorem

Correlation Theorem

Power Theorem

Rayleigh Energy Theorem (Parseval's Theorem)

Stretch Theorem (Repeat Theorem)

Decimation Theorem (Aliasing Theorem)

Zero Padding Theorem

Bandlimited Interpolation in Time

Conclusions

Acknowledgement

Appendix A: Linear Time-Invariant Filters and Convolution

LTI Filters and the Convolution Theorem

Appendix B: Introductory Statistical Signal Processing

Cross-Correlation

Applications of Cross-Correlation

Autocorrelation

Coherence

Appendix C: Mathematica/Matlab Examples

Appendix D: The Similarity Theorem

Example Applications of the DFT

Spectrum Analysis of a Sinusoid: Windowing, Zero-Padding, and the FFT

Example 1: FFT of a Simple Sinusoid

Example 2: FFT of a Not-So-Simple Sinusoid

Example 3: FFT of a Zero-Padded Sinusoid

Example 4: Blackman Window

Example 5: Use of the Blackman Window

Example 6: Hanning-Windowed Complex Sinusoid

Spectral Phase

A Basic Tutorial on Sampling Theory

Introduction

Reconstruction from Samples--Pictorial Version

Reconstruction from Samples--The Math

Aliasing of Sampled Continuous-Time Signals

Shannon's Sampling Theorem

Figuring Out Sampling Theory by Playing Around with Complex Sinusoids

What frequencies are representable by a geometric sequence?

Recovering a Continuous-Time Signal from its Samples

Method 1: Additive Synthesis

Does it Work?

Introduction to Digital Filter Analysis

Motivating Example and Overview

FIR Filters

Convolution Representation

Finiteness

Causal FIR Filters

Transfer Function

Order

The BiQuad Section

Digital Filter Theory Summary

Linearity and Time-Invariance

Difference Equation

Convolution Representation of LTI Filters

Frequency Response

Phase Delay and Group Delay

Bibliography

Index

About this document ...

INDEX Next page >>

Index

Sun, 01 Mar 1998 00:00:00 +0000

NOTE: THIS DOCUMENT IS OBSOLETE, PLEASE CHECK THE NEW VERSION: "Mathematics of the Discrete Fourier Transform (DFT), with Audio Applications --- Second Edition", by Julius O. Smith III, W3K Publishing, 2007, ISBN 978-0-9745607-4-8. - Copyright © 2017-09-28 by Julius O. Smith III - Center for Computer Research in Music and Acoustics (CCRMA), Stanford University

<< Previous page TOC INDEX Next page >>
Index
20 dB boost
Properties of DB Scales
3 dB boost
Properties of DB Scales
convolution
Convolution Representation of LTI
Aliasing
Alias Operator
aliasing operator
Alias Operator
amplitude response
LTI Filters and the | Frequency Response | Frequency Response
anti-aliasing lowpass filter
Alias Operator
antilog
Logarithms
antilogarithm
Logarithms
antisymmetric functions
Even and Odd Functions
average power
Appendix A: Round-Off Error
base
Logarithms
base logarithm
Logarithms
base logarithm
Logarithms
bel
Decibels
bin number
Frequencies in the ''Cracks''
binary
Binary Integer Fixed-Point Numbers
binary digits
Binary Integer Fixed-Point Numbers
bits
Binary Integer Fixed-Point Numbers
canonical with respect to delay
Order
causal
Causal FIR Filters | Difference Equation
characteristic of the logarithm
Logarithms
coefficient of projection
The Discrete Fourier Transform
column-vector
Matrices
comb-filter
Constructive and Destructive Interference
common logarithm
Logarithms
commutativity of convolution
Convolution
companding
Dynamic Range
complex matrix
Matrices
complex matrix transpose
Matrices
convolution
Convolution
convolution operator ''''
Convolution Representation
convolution representation
Appendix A: Linear Time-Invariant
correlation operator
Correlation
cross-talk
Frequencies in the ''Cracks''
cyclic convolution
Convolution
dB scale
Decibels
decibel
Decibels
decimal numbers
Binary Integer Fixed-Point Numbers
DFT as a digital filter
Frequencies in the ''Cracks''
DFT matrix
The DFT Matrix
difference equation
Difference Equation
difference equation coefficients
Difference Equation
digit
Binary Integer Fixed-Point Numbers
digital filter
Appendix A: Linear Time-Invariant
Discrete Fourier Transform
The DFT and its
Discrete Fourier Transform (DFT)
The DFT
dynamic range
Dynamic Range
dynamic range of magnetic tape
Dynamic Range
energy
Decibels
Euler's Formula
Euler's Formula
Euler's Theorem
Euler's Theorem
even functions
Even and Odd Functions
expected value
Appendix A: Round-Off Error
fast convolution
Convolution Theorem
filter
Digital Filter Theory Summary
filter coefficients
Difference Equation
filter delay for amplitude envelopes
Phase Delay and Group
filter order
Difference Equation
filter power response
Frequency Response
filter time delay
Phase Delay and Group
finite-impulse-response filter
FIR Filters
FIR filter
FIR Filters
flip operator
Flip Operator
Fourier Dual
Dual of the Convolution
frequency bin
Frequencies in the ''Cracks''
frequency response
LTI Filters and the | Frequency Response
frequency-domain aliasing
Alias Operator | Alias Operator
geometric sequence
Geometric Series
geometric series
Geometric Series
group delay
Phase Delay and Group
Hermitian spectra
Symmetry
Hermitian symmetry
Conjugation and Reversal
Hermitian transpose
Matrices
hex
Binary Integer Fixed-Point Numbers
hexadecimal
Binary Integer Fixed-Point Numbers
ideal bandlimited interpolation
Zero Padding Theorem | Bandlimited Interpolation in Time
ideal lowpass filtering operation in the frequency domain
Bandlimited Interpolation in Time
identity matrix
Matrix Multiplication
IDFT
The DFT and its
impulse response
Appendix A: Linear Time-Invariant | FIR Filters | Difference Equation
impulse signal
Appendix A: Linear Time-Invariant | FIR Filters | Difference Equation
indicator function
Linear Phase Terms
Intensity
Decibels
intensity level
DB SPL
interpolation operator
Zero Padding Theorem
inverse DFT
The DFT | The DFT and its
inverse DFT matrix
The DFT Matrix
irrational number
Real Exponents
JND
Decibels
just-noticeable difference
Decibels
lag
Correlation
lagged product
Correlation
length even rectangular windowing operation
Bandlimited Interpolation in Time
linear
Linearity and Time-Invariance
linear filter
Linearity and Time-Invariance
linear phase FFT windows
Application of the Shift
linear phase signal
Linear Phase Terms
linear phase term
Shift Theorem | Linear Phase Terms
linear transformation
Matrix Multiplication
logarithm
Logarithms
loudness
DB SPL
LTI filters
Linearity and Time-Invariance
main lobe
Frequencies in the ''Cracks''
mantissa
Logarithms
matrix
Matrices
matrix columns
Matrices
matrix multiplication
Matrix Multiplication
matrix rows
Matrices
matrix transpose
Matrices
mean
Appendix A: Round-Off Error
mean of a random variable
Appendix A: Round-Off Error
mean of a signal
Appendix A: Round-Off Error
mean square
Appendix A: Round-Off Error
modulo
Modulo Indexing, Periodic Extension
moments
Appendix A: Round-Off Error
multiplication of large numbers
Logarithms
multiplying two numbers convolves their digits
Multiplication of Decimal Numbers
natural logarithm
Logarithms
non-commutativity of matrix multiplication
Matrix Multiplication
normalized inverse DFT matrix
The DFT Matrix
normalized DFT matrix
The DFT Matrix
normalized DFT sinusoids
An Orthonormal Sinusoidal Set | Normalized DFT
normalized frequency
The DFT and its
octal
Binary Integer Fixed-Point Numbers
odd functions
Even and Odd Functions
order
Order
orthogonal
The DFT Matrix
orthogonality of sinusoids
Orthogonality of Sinusoids
orthonormal
The DFT Matrix
PCM
Pulse Code Modulation (PCM)
periodic
Modulo Indexing, Periodic Extension
periodic extension
Frequencies in the ''Cracks'' | Modulo Indexing, Periodic Extension
phase delay
Phase Delay and Group
phase response
LTI Filters and the | Frequency Response | Frequency Response
phase response at zero frequency
Phase Delay and Group
phon amplitude scale
DB SPL
polar form
Rational Exponents
polynomial approximation
Derivation of Taylor Series
power
Decibels
pressure
Decibels
primitive th root of unity
Orthogonality of Sinusoids
rational
Rational Exponents
real filter
Difference Equation
rectangular window
Frequencies in the ''Cracks''
rms level
Appendix A: Round-Off Error
roots of unity
Orthogonality of Sinusoids
row-vector
Matrices
sample mean
Appendix A: Round-Off Error
sample variance
Appendix A: Round-Off Error
second central moment
Appendix A: Round-Off Error
sensation level
DB SPL
shift operator
Shift Operator
sidelobes
Frequencies in the ''Cracks''
signal
Digital Filter Theory Summary
signal dynamic range
Dynamic Range
similarity theorem
Appendix D: The Similarity
sinc function
Frequencies in the ''Cracks''
skew-Hermitian
Conjugation and Reversal
sone amplitude scale
DB SPL
Sound Pressure Level
DB SPL
spectral leakage
Frequencies in the ''Cracks''
spectrum
The Discrete Fourier Transform | The DFT and its
SPL
DB SPL
square matrix
Matrices
stable
Difference Equation
standard deviation
Appendix A: Round-Off Error
Stretch Operator
no title | Stretch Operator
symmetric functions
Even and Odd Functions
Taylor Series Expansion
no title | Informal Derivation of Taylor
time-domain aliasing
Alias Operator
time-invariant
Linearity and Time-Invariance
Toeplitz
Matrix Multiplication
transfer function
Transfer Function | Convolution Representation of LTI
transform pair
Notation and Terminology
transpose of a matrix product
Matrix Multiplication
unilateral transform
Convolution Representation of LTI
unit pulse signal
Appendix A: Linear Time-Invariant
unitary
The DFT Matrix
variance
Appendix A: Round-Off Error
window
Frequencies in the ''Cracks''
Zero padding
Zero Padding
zero phase signal
Symmetry
<< Previous page TOC INDEX Next page >>

Preface

Sun, 01 Mar 1998 00:00:00 +0000

NOTE: THIS DOCUMENT IS OBSOLETE, PLEASE CHECK THE NEW VERSION: "Mathematics of the Discrete Fourier Transform (DFT), with Audio Applications --- Second Edition", by Julius O. Smith III, W3K Publishing, 2007, ISBN 978-0-9745607-4-8. - Copyright © 2017-09-28 by Julius O. Smith III - Center for Computer Research in Music and Acoustics (CCRMA), Stanford University

<< Previous page TOC INDEX Next page >>
Preface
This chapter is an outgrowth of my course entitled''Introduction to Digital Signal Processing and the Discrete Fourier Transform (DFT)which I have given at the Center for Computer Research in Music and Acoustics (CCRMA) every year for the past 16 years. The course was created primarily as a first course in digital signal processing for entering Music Ph.D. students. As a result, the only prerequisite is a good high-school math background. Calculus exposure is desirable, but not required.

Subsections
Outline

<< Previous page TOC INDEX Next page >>

Outline

Sun, 01 Mar 1998 00:00:00 +0000

NOTE: THIS DOCUMENT IS OBSOLETE, PLEASE CHECK THE NEW VERSION: "Mathematics of the Discrete Fourier Transform (DFT), with Audio Applications --- Second Edition", by Julius O. Smith III, W3K Publishing, 2007, ISBN 978-0-9745607-4-8. - Copyright © 2017-09-28 by Julius O. Smith III - Center for Computer Research in Music and Acoustics (CCRMA), Stanford University

<< Previous page TOC INDEX Next page >>
Outline
Below is an overview of the chapters.
Introduction to the DFT
This chapter introduces the Discrete Fourier Transform (DFT) and points out the elements which will be discussed in this reader.
Introduction to Complex Numbers
This chapter provides an introduction to complex numbers, factoring polynomials, the quadratic formula, the complex plane, Euler's formula, and an overview of numerical facilities for complex numbers in Matlab and Mathematica.
Proof of Euler's Identity
This chapter for Music 320 outlines the proof of Euler's Identity which is an important tool for working with complex numbers. It is one of the critical elements of the DFT definition that we need to understand.
Logarithms, Decibels, and Number Systems
This chapter provides an introduction to logarithms (real and complex),decibels, and number systems such as binary integer fixed-point, fractional fixed-point, one's complement, two's complement, logarithmic fixed-point,-law, and floating-point number formats.
Sinusoids and Exponentials
This chapter provides an introduction to sinusoids, exponentials, complex sinusoids, , in-phase and quadrature sinusoidal components, theanalytic signal, positive and negative frequencies, constructive anddestructive interference, invariance of sinusoidal frequency in linear time-invariant systems, circular motion as the vector sum of in-phase and quadrature sinusoidal motions, sampled sinusoids, generating sampled sinusoids from powers of , and plot examples using Mathematica.
The Discrete Fourier Transform (DFT) Derived
This chapter derives the Discrete Fourier Transform (DFT) as a projection of a length signal onto the set of sampled complex sinusoids generated by the roots of unity.
Fourier Theorems for the DFT
This chapter derives various Fourier theorems for the case of the DFT. Included are symmetry relations, the shift theorem, convolution theorem,correlation theorem, power theorem, and theorems pertaining to interpolation and downsampling. Applications related to certain theorems are outlined, including linear time-invariant filtering, sampling rate conversion, andstatistical signal processing.
Example Applications of the DFT
This chapter goes through some practical examples of FFT analysis in Matlab. The various Fourier theorems provide a ''thinking vocabulary'' for understanding elements of spectral analysis.
A Basic Tutorial on Sampling Theory
This appendix provides a basic tutorial on sampling theory. Aliasing due to sampling of continuous-time signals is characterized mathematically. Shannon's sampling theorem is proved. A pictorial representation of continuous-time signal reconstruction from discrete-time samples is given.
Introduction to Digital Filter Theory
This appendix provides an introduction to the most basic elements ofdigital filter theory. Definitions are given for linearity,time-invariance and four basic representations of digital filters: the difference equation coefficients, impulse response, transfer function, and frequency response. Next the concepts of phase delay and group delay are defined. This material is a subset of that in [1].
<< Previous page TOC INDEX Next page >>

Introduction to the DFT

Sun, 01 Mar 1998 00:00:00 +0000

NOTE: THIS DOCUMENT IS OBSOLETE, PLEASE CHECK THE NEW VERSION: "Mathematics of the Discrete Fourier Transform (DFT), with Audio Applications --- Second Edition", by Julius O. Smith III, W3K Publishing, 2007, ISBN 978-0-9745607-4-8. - Copyright © 2017-09-28 by Julius O. Smith III - Center for Computer Research in Music and Acoustics (CCRMA), Stanford University

<< Previous page TOC INDEX Next page >>
Introduction to the DFT
This chapter introduces the Discrete Fourier Transform (DFT) and points out the elements which will be discussed in the Music 320 Background Reader.

Subsections
The DFT
Mathematics of the DFT
DFT Math Outline

<< Previous page TOC INDEX Next page >>

The DFT

Sun, 01 Mar 1998 00:00:00 +0000

NOTE: THIS DOCUMENT IS OBSOLETE, PLEASE CHECK THE NEW VERSION: "Mathematics of the Discrete Fourier Transform (DFT), with Audio Applications --- Second Edition", by Julius O. Smith III, W3K Publishing, 2007, ISBN 978-0-9745607-4-8. - Copyright © 2017-09-28 by Julius O. Smith III - Center for Computer Research in Music and Acoustics (CCRMA), Stanford University

<< Previous page TOC INDEX Next page >>
The DFT
For a length complex sequence , , thediscrete Fourier transform (DFT) is defined by

We are now in a position to have a full understanding of the transform kernel:

The kernel consists of samples of a complex sinusoid at discrete frequencies uniformly spaced between and the sampling rate. All that remains is to understand the purpose and function of the summation over of the pointwise product of times each complex sinusoid. We will learn that this can be interpreted as an inner product operation which computes the coefficient of projection of the signal onto the complex sinusoid . As such, , the DFT at frequency, is a measure of the amplitude and phase of the complex sinusoid at that frequency which is present in the input signal . This is the basic function of all transform summations (in discrete time) and integrals (in continuous time) and their kernels.
<< Previous page TOC INDEX Next page >>

Mathematics of the DFT

Sun, 01 Mar 1998 00:00:00 +0000

NOTE: THIS DOCUMENT IS OBSOLETE, PLEASE CHECK THE NEW VERSION: "Mathematics of the Discrete Fourier Transform (DFT), with Audio Applications --- Second Edition", by Julius O. Smith III, W3K Publishing, 2007, ISBN 978-0-9745607-4-8. - Copyright © 2017-09-28 by Julius O. Smith III - Center for Computer Research in Music and Acoustics (CCRMA), Stanford University

<< Previous page TOC INDEX Next page >>
Mathematics of the DFT
In the signal processing literature, it is common to write the DFT in the more pure form obtained by setting in the previous definition:

where denotes the input signal at time (sample) , and denotes the th spectral sample.^1.1 This form is the simplest mathematically while the previous form is the easier to think about physically.
There are two remaining symbols in the DFT that we have not yet defined:

The first, , is the basis for complex numbers. As a result, complex numbers will be the first topic we cover in this course (but only to the extent needed to understand the DFT).
The second, , is a transcendental number defined by the above limit. In this course we will derive and talk about why it comes up.
Note that not only do we have complex numbers to contend with, but we have them appearing in exponents, as in

We will systematically develop what we mean by imaginary exponents in order that such mathematical expressions are well defined.
With , , and imaginary exponents understood, we can go on to prove Euler's Identity:

Euler's Identity is the key to understanding the meaning of expressions like

We'll see that such an expression defines a sampled complexsinusoid, and we'll talk about sinusoids in some detail, from an audio perspective.
Finally, we need to understand what the summation over is doing in the definition of the DFT. We'll learn that it should be seen as the computation of the inner product of the signals and , so that we may write the DFT using inner-product notation as

where

is the sampled complex sinusoid at (normalized) radian frequency , and the inner product operation is defined by

We will show that the inner product of with the th ''basis sinusoid'' is a measure of ''how much'' of is present in and at ''what phase'' (since it is a complex number).
After the foregoing, the inverse DFT can be understood as theweighted sum of projections of onto , i.e.,

where

is the (actual) coefficient of projection of onto . Referring to the whole signal as a whole, the IDFT can be written as

Note that both the basis sinusoids and their coefficients of projection are complex.
Having completely understood the DFT and its inverse mathematically, we go on to proving various Fourier Theorems, such as the ''shift theorem,'' the ''convolution theorem,'' and ''Parsevals' theorem.'' The Fourier theorems provide a basic thinking vocabulary for working with signals in the time and frequency domains. They can be used to answer questions like
What happens in the frequency domain if I do this in the time domain?

DFT Math Outline

Sun, 01 Mar 1998 00:00:00 +0000

NOTE: THIS DOCUMENT IS OBSOLETE, PLEASE CHECK THE NEW VERSION: "Mathematics of the Discrete Fourier Transform (DFT), with Audio Applications --- Second Edition", by Julius O. Smith III, W3K Publishing, 2007, ISBN 978-0-9745607-4-8. - Copyright © 2017-09-28 by Julius O. Smith III - Center for Computer Research in Music and Acoustics (CCRMA), Stanford University

<< Previous page TOC INDEX Next page >>
DFT Math Outline
In summary, understanding the DFT takes us through the following topics:
Complex numbers
Complex exponents
Why ?
Euler's formula
Projecting signals onto signals via the inner product
The DFT as the coefficient of projection of a signal onto sinusoids
The IDFT as a weighted sum of the projections
Various Fourier theorems
Elementary time-frequency pairs
Practical spectrum analysis in Matlab
We will additionally discuss practical aspects of working with sinusoids, such as decibels (dB) and display techniques.
<< Previous page TOC INDEX Next page >>

Introduction to Complex Numbers

Sun, 01 Mar 1998 00:00:00 +0000

NOTE: THIS DOCUMENT IS OBSOLETE, PLEASE CHECK THE NEW VERSION: "Mathematics of the Discrete Fourier Transform (DFT), with Audio Applications --- Second Edition", by Julius O. Smith III, W3K Publishing, 2007, ISBN 978-0-9745607-4-8. - Copyright © 2017-09-28 by Julius O. Smith III - Center for Computer Research in Music and Acoustics (CCRMA), Stanford University

<< Previous page TOC INDEX Next page >>
Introduction to Complex Numbers
This chapter provides an introduction to complex numbers, factoring polynomials, the quadratic formula, the complex plane, Euler's formula, and an overview of numerical facilities for complex numbersin Matlab and Mathematica.

Subsections
Factoring a Polynomial with Real Roots
The Quadratic Formula
Complex Roots
The Fundamental Theorem of Algebra
Complex Numbers
The Complex Plane
More Notation and Terminology
Elementary Relationships
Euler's Formula
De Moivre's Theorem
Numerical Tools in Matlab
Numerical Tools in Mathematica

<< Previous page TOC INDEX Next page >>

Factoring a Polynomial with Real Roots

Sun, 01 Mar 1998 00:00:00 +0000

NOTE: THIS DOCUMENT IS OBSOLETE, PLEASE CHECK THE NEW VERSION: "Mathematics of the Discrete Fourier Transform (DFT), with Audio Applications --- Second Edition", by Julius O. Smith III, W3K Publishing, 2007, ISBN 978-0-9745607-4-8. - Copyright © 2017-09-28 by Julius O. Smith III - Center for Computer Research in Music and Acoustics (CCRMA), Stanford University

<< Previous page TOC INDEX Next page >>
Factoring a Polynomial with Real Roots
Remember ''factoring polynomials''? Consider the second-order polynomial

It is second-order because the highest power of is (only non-negative integer powers are allowed). The polynomial is also monic because its leading coefficient, the coefficient of , is . Since it is second order, there are at most two real roots (or zeros) of the polynomial. Suppose they are denoted and . Then we have and , and we can write

This is the factored form of the monic polynomial . (For a non-monic polynomial, we may simply divide all coefficients by the first to make it monic, and this doesn't affect the zeros.) Multiplying out the symbolic factored form gives

Comparing with the original polynomial, we find we must have

This is a system of two equations in two unknowns. Unfortunately, it is anonlinear system of two equations in two unknowns.^2.1Nevertheless, because it is so small, the equations are easily solved. In beginning algebra, we did them by hand. However, nowadays we can use a computer program such as Mathematica:
In[]:= Solve[{x1+x2==5, x1 x2 == 6}, {x1,x2}] Out[]: {{x1 -> 2, x2 -> 3}, {x1 -> 3, x2 -> 2}}
Note that the two lists of substitutions points out that it doesn't matter which root is 2 and which is 3. In summary, the factored form of this simple example is

Note that polynomial factorization rewrites a monic th-order polynomial as the product of first-order monic polynomials, each of which contributes one zero (root) to the product. This factoring business is often used when working with digital filters.
<< Previous page TOC INDEX Next page >>

The Quadratic Formula

Sun, 01 Mar 1998 00:00:00 +0000

NOTE: THIS DOCUMENT IS OBSOLETE, PLEASE CHECK THE NEW VERSION: "Mathematics of the Discrete Fourier Transform (DFT), with Audio Applications --- Second Edition", by Julius O. Smith III, W3K Publishing, 2007, ISBN 978-0-9745607-4-8. - Copyright © 2017-09-28 by Julius O. Smith III - Center for Computer Research in Music and Acoustics (CCRMA), Stanford University

<< Previous page TOC INDEX Next page >>
The Quadratic Formula
The general second-order polynomial is

where the coefficients are any real numbers, and we assume since otherwise it would not be second order. Some experiments plotting for different values of the coefficients leads one to guess that the curve is always a scaled and translated parabola. The canonical parabola centered at is given by

where determines the width and provides an arbitrary vertical offset. If we can find in terms of for any quadratic polynomial, then we can easily factor the polynomial. This is called ''completing the square.'' Multiplying out , we get

Equating coefficients of like powers of gives

Using these answers, any second-order polynomial can be rewritten as a scaled, translated parabola

In this form, the roots are easily found by solving to get

This is the general quadratic formula. It was obtained by simple algebraic manipulation of the original polynomial. There is only one ''catch.'' What happens when is negative? This introduces the square root of a negative number which we could insist ''does not exist.'' Alternatively, we could invent complex numbers to accommodate it.
<< Previous page TOC INDEX Next page >>

Complex Roots

Sun, 01 Mar 1998 00:00:00 +0000

NOTE: THIS DOCUMENT IS OBSOLETE, PLEASE CHECK THE NEW VERSION: "Mathematics of the Discrete Fourier Transform (DFT), with Audio Applications --- Second Edition", by Julius O. Smith III, W3K Publishing, 2007, ISBN 978-0-9745607-4-8. - Copyright © 2017-09-28 by Julius O. Smith III - Center for Computer Research in Music and Acoustics (CCRMA), Stanford University

<< Previous page TOC INDEX Next page >>
Complex Roots

Figure 2.1:An example parabola defined by .
As a simple example, let , , and , i.e.,

As shown in Fig. 2.1, this is a parabola centered at (where) and reaching upward to positive infinity, never going below . It has no zeros. On the other hand, the quadratic formula says that the ''roots'' are given formally by . The square root of any negative number can be expressed as , so the only new algebraic object is . Let's give it a name:

Then, formally, the roots of of are , and we can formally express the polynomial in terms of its roots as

We can think of these as ''imaginary roots'' in the sense that square roots of negative numbers don't really exist, or we can extend the concept of ''roots'' to allow for complex numbers, that is, numbers of the form

where and are real numbers, and .
It can be checked that all algebraic operations for real numbers^2.2 apply equally well to complex numers. Both real numbers and complex numbers are examples of a mathematical field. Fields areclosed with respect to multiplication and addition, and all the rules of algebra we use in manipulating polynomials with real coefficients (and roots) carry over unchanged to polynomials with complex coefficients and roots. In fact, the rules of algebra become simpler for complex numbers because, as discussed in the next section, we can always factor polynomials completely over the field of complex numbers while we cannot do this over the reals (as we saw in the example ).
<< Previous page TOC INDEX Next page >>

The Fundamental Theorem of Algebra

Sun, 01 Mar 1998 00:00:00 +0000

NOTE: THIS DOCUMENT IS OBSOLETE, PLEASE CHECK THE NEW VERSION: "Mathematics of the Discrete Fourier Transform (DFT), with Audio Applications --- Second Edition", by Julius O. Smith III, W3K Publishing, 2007, ISBN 978-0-9745607-4-8. - Copyright © 2017-09-28 by Julius O. Smith III - Center for Computer Research in Music and Acoustics (CCRMA), Stanford University

<< Previous page TOC INDEX Next page >>
The Fundamental Theorem of Algebra
This is a very powerful algebraic tool. It says that given any polynomial

Complex Numbers

Sun, 01 Mar 1998 00:00:00 +0000

NOTE: THIS DOCUMENT IS OBSOLETE, PLEASE CHECK THE NEW VERSION: "Mathematics of the Discrete Fourier Transform (DFT), with Audio Applications --- Second Edition", by Julius O. Smith III, W3K Publishing, 2007, ISBN 978-0-9745607-4-8. - Copyright © 2017-09-28 by Julius O. Smith III - Center for Computer Research in Music and Acoustics (CCRMA), Stanford University

<< Previous page TOC INDEX Next page >>
Complex Numbers
This section introduces various notation and terms associated with complex numbers. As discussed above, complex numbers are devised by introducing the square-root of as a primitive new algebraic object among real numbers and manipulating it symbolically as if it were a real number itself:

Mathemeticians and physicists often use instead of as . The use of is common in engineering where is more often used for electrical current.
As mentioned above, for any negative number , we have , where denotes the absolute value of . Thus, every square root of a negative number can be expressed as times the square root of a positive number.
By definition, we have

and so on. Thus, the sequence , is a periodic sequence with period , since . (We'll learn later that the sequence is a sampled complex sinusoid havingfrequency equal to one fourth the sampling rate.)
Every complex number can be written as

where and are real numbers. We call the real part and the imaginary part. We may also use the notation

Note that the real numbers are the subset of the complex numbers having a zero imaginary part ().
The rule for complex multiplication follows directly from the definition of the imaginary unit :

In some mathematics texts, complex numbers are defined as ordered pairs of real numbers , and algebraic operations such as multiplication are defined more formally as operations on ordered pairs, e.g., . However, such formality tends to obscure the underlying simplicity of complex numbers as a straightforward extension of real numbers to include .
It is important to realize that complex numbers can be treated algebraically just like real numbers. That is, they can be added, subtracted, multiplied, divided, etc., using exactly the same rules of algebra (since both real and complex numbers are mathematicalfields). It is often preferable to think of complex numbers as being the true and proper setting for algebraic operations, with real numbers being the limited subset for which .
To explore further the magical world of complex variables, see any textbook such as [4,5].

Subsections
The Complex Plane
More Notation and Terminology
Elementary Relationships
Euler's Formula
De Moivre's Theorem

<< Previous page TOC INDEX Next page >>

The Complex Plane

Sun, 01 Mar 1998 00:00:00 +0000

NOTE: THIS DOCUMENT IS OBSOLETE, PLEASE CHECK THE NEW VERSION: "Mathematics of the Discrete Fourier Transform (DFT), with Audio Applications --- Second Edition", by Julius O. Smith III, W3K Publishing, 2007, ISBN 978-0-9745607-4-8. - Copyright © 2017-09-28 by Julius O. Smith III - Center for Computer Research in Music and Acoustics (CCRMA), Stanford University

<< Previous page TOC INDEX Next page >>
The Complex Plane

Figure 2.2:Plotting a complex number as a point in thecomplex plane.
We can plot complex numbers in a plane as ordered pairs , as shown in Fig. 2.2. The complex plane is any 2D graph in which the horizontal axis is the real part and the vertical axis is the imaginary part of a complex number or function. As an example, the number has coordinates in the complex plane while the number has coordinates .
Plotting as the point in the complex plane can be viewed as a plot in Cartesian or rectilinear coordinates. We can also express complex numbers in terms of polar coordinates as an ordered pair , where is the distance from the origin to the number being plotted, and is the angle of the number relative to the positive real coordinate axis (the line defined by and ). (See Fig. 2.2.)
Using elementary geometry, it is quick to show that conversion from rectangular to polar coordinates is accomplished by the formulas
The first equation follows immediately from the Pythagorean theorem, while the second follows immediately from the definition of the tangent function. Similarly, conversion from polar to rectangular coordinates is simply
These follow immediately from the definitions of cosine and sine, respectively,
<< Previous page TOC INDEX Next page >>

More Notation and Terminology

Sun, 01 Mar 1998 00:00:00 +0000

NOTE: THIS DOCUMENT IS OBSOLETE, PLEASE CHECK THE NEW VERSION: "Mathematics of the Discrete Fourier Transform (DFT), with Audio Applications --- Second Edition", by Julius O. Smith III, W3K Publishing, 2007, ISBN 978-0-9745607-4-8. - Copyright © 2017-09-28 by Julius O. Smith III - Center for Computer Research in Music and Acoustics (CCRMA), Stanford University

<< Previous page TOC INDEX Next page >>
More Notation and Terminology
It's already been mentioned that the rectilinear coordinates of a complex number in the complex plane are called the real part andimaginary part, respectively.
We also have special notation and various names for the radius and angle of a complex number expressed in polar coordinates :

The complex conjugate of is denoted and is defined by

where, of course, . Sometimes you'll see the notation in place of , but we won't use that here.
In general, you can always obtain the complex conjugate of any expression by simply replacing with . In the complex plane, this is a vertical flip about the real axis; in other words, complex conjugation replaces each point in the complex plane by its mirror image on the other side of the axis.
<< Previous page TOC INDEX Next page >>

Elementary Relationships

Sun, 01 Mar 1998 00:00:00 +0000

NOTE: THIS DOCUMENT IS OBSOLETE, PLEASE CHECK THE NEW VERSION: "Mathematics of the Discrete Fourier Transform (DFT), with Audio Applications --- Second Edition", by Julius O. Smith III, W3K Publishing, 2007, ISBN 978-0-9745607-4-8. - Copyright © 2017-09-28 by Julius O. Smith III - Center for Computer Research in Music and Acoustics (CCRMA), Stanford University

<< Previous page TOC INDEX Next page >>
Elementary Relationships
From the above definitions, one can quickly verify

Let's verify the third relationship which states that a complex numbermultiplied by its conjugate is equal to its magnitude squared:

<< Previous page TOC INDEX Next page >>

Euler's Formula

Sun, 01 Mar 1998 00:00:00 +0000

NOTE: THIS DOCUMENT IS OBSOLETE, PLEASE CHECK THE NEW VERSION: "Mathematics of the Discrete Fourier Transform (DFT), with Audio Applications --- Second Edition", by Julius O. Smith III, W3K Publishing, 2007, ISBN 978-0-9745607-4-8. - Copyright © 2017-09-28 by Julius O. Smith III - Center for Computer Research in Music and Acoustics (CCRMA), Stanford University

<< Previous page TOC INDEX Next page >>
Euler's Formula
Since is the algebraic expression of in terms of its rectangular coordinates, the corresponding expression in terms of its polar coordinates is

There is another, more powerful representation of in terms of its polar coordinates. In order to define it, we must introduce Euler's Formula:
(2.1)

A proof of Euler's identity is given in the next chapter. Just note for the moment that for , we have , as expected. Before, the only algebraic representation of a complex number we had was, which fundamentally uses Cartesian (rectilinear) coordinates in the complex plane. Euler's identity gives us an alternative algebraic representation in terms of polar coordinates in the complex plane:

This representation often simplifies manipulations of complex numbers, especially when they are multiplied together. Simple rules of exponents can often be used in place of more difficult trigonometric identities. In the simplest case,

A corollary of Euler's identity is obtained by setting to get

This has been called the ''most beautiful formula in mathematics'' due to the extremely simple form in which the fundamental constants , and all appear.
For another example of manipulating the polar form of a complex number, let's again verify , as we did above, but this time using polar form:

We can now add a fourth line to that set of examples:

Euler's identity can be used to derive formulas for sine and cosine in terms of :

Similarly, , and we have
<< Previous page TOC INDEX Next page >>

De Moivre's Theorem

Sun, 01 Mar 1998 00:00:00 +0000

NOTE: THIS DOCUMENT IS OBSOLETE, PLEASE CHECK THE NEW VERSION: "Mathematics of the Discrete Fourier Transform (DFT), with Audio Applications --- Second Edition", by Julius O. Smith III, W3K Publishing, 2007, ISBN 978-0-9745607-4-8. - Copyright © 2017-09-28 by Julius O. Smith III - Center for Computer Research in Music and Acoustics (CCRMA), Stanford University

<< Previous page TOC INDEX Next page >>
De Moivre's Theorem
As a more complicated example of the value of the polar form, we'll proveDe Moivre's theorem:

Working this out using sum-of-angle identities from trigonometry is laborious. However, using Euler's identity, De Moivre's theorem simply ''falls out'':

Moreover, by the power of the method used to show the result, can be any real number, not just an integer.
<< Previous page TOC INDEX Next page >>

Numerical Tools in Matlab

Sun, 01 Mar 1998 00:00:00 +0000

NOTE: THIS DOCUMENT IS OBSOLETE, PLEASE CHECK THE NEW VERSION: "Mathematics of the Discrete Fourier Transform (DFT), with Audio Applications --- Second Edition", by Julius O. Smith III, W3K Publishing, 2007, ISBN 978-0-9745607-4-8. - Copyright © 2017-09-28 by Julius O. Smith III - Center for Computer Research in Music and Acoustics (CCRMA), Stanford University

<< Previous page TOC INDEX Next page >>
Numerical Tools in Matlab
In Matlab, root-finding is always numerical:^2.3
>> % polynomial = array of coefficients in Matlab >> p = [1 0 0 0 5 7]; % p(x) = x^5 + 5*x + 7 >> format long; % print double-precision >> roots(p) % print out the roots of p(x)
ans = 1.30051917307206 + 1.10944723819596i 1.30051917307206 - 1.10944723819596i -0.75504792501755 + 1.27501061923774i -0.75504792501755 - 1.27501061923774i -1.09094249610903
Matlab has the following primitives for complex numbers:
>> help j

J Imaginary unit. The variables i and j both initially have the value sqrt(-1) for use in forming complex quantities. For example, the expressions 3+2i, 3+2i, 3+2i, 3+2j and 3+2*sqrt(-1). all have the same value. However, both i and j may be assigned other values, often in FOR loops and as subscripts.

See also I. Built-in function. Copyright (c) 1984-92 by The MathWorks, Inc.

>> sqrt(-1)

ans = 0 + 1.0000i

>> help real

REAL Complex real part. REAL(X) is the real part of X.

See also IMAG, CONJ, ANGLE, ABS.

>> help imag

IMAG Complex imaginary part. IMAG(X) is the imaginary part of X. See I or J to enter complex numbers.

See also REAL, CONJ, ANGLE, ABS.

>> help conj

CONJ Complex conjugate. CONJ(X) is the complex conjugate of X.

>> help abs

ABS Absolute value and string to numeric conversion. ABS(X) is the absolute value of the elements of X. When X is complex, ABS(X) is the complex modulus (magnitude) of the elements of X.

See also ANGLE, UNWRAP. ABS(S), where S is a MATLAB string variable, returns the numeric values of the ASCII characters in the string. It does not change the internal representation, only the way it prints. See also SETSTR.

>> help angle

ANGLE Phase angle. ANGLE(H) returns the phase angles, in radians, of a matrix with complex elements.

See also ABS, UNWRAP.</pre>Note how helpful the ''See also'' information is in Matlab.Let's run through a few elementary manipulations of complex

numbers in Matlab:
>> x = 1; % Every symbol must have a value in Matlab >> y = 2; >> z = x + j * y

z = 1.0000 + 2.0000i

>> 1/z

ans = 0.2000 - 0.4000i

>> z^2

ans = -3.0000 + 4.0000i

>> conj(z)

ans = 1.0000 - 2.0000i

>> z*conj(z)

ans = 5

>> abs(z)^2

ans = 5.0000

>> norm(z)^2

ans = 5.0000

>> angle(z)

ans = 1.1071
Now let’s do polar form:
>> r = abs(z)

r = 2.2361

>> theta = angle(z)

theta = 1.1071

>> r * exp(j * theta)

ans = 1.0000 + 2.0000i

>> z

z = 1.0000 + 2.0000i

>> z/abs(z)

ans = 0.4472 + 0.8944i

>> exp(j*theta)

ans = 0.4472 + 0.8944i

>> z/conj(z)

ans = -0.6000 + 0.8000i

>> exp(2jtheta)

ans = -0.6000 + 0.8000i

>> imag(log(z/abs(z)))

ans = 1.1071

>> theta

theta = 1.1071

>>
Some manipulations involving two complex numbers:
>> x1 = 1; >> x2 = 2; >> y1 = 3; >> y2 = 4; >> z1 = x1 + j * y1; >> z2 = x2 + j * y2; >> z1

z1 = 1.0000 + 3.0000i

>> z2

z2 = 2.0000 + 4.0000i

>> z1*z2

ans = -10.0000 +10.0000i

>> z1/z2

ans = 0.7000 + 0.1000i
Another thing to note about Matlab is that the transpose operator ' (for vectors and matrices) conjugates as well as transposes. Use .' to transpose without conjugation:
>>x = [1:4]*j

x = 0 + 1.0000i 0 + 2.0000i 0 + 3.0000i 0 + 4.0000i

>> x'

ans = 0 - 1.0000i 0 - 2.0000i 0 - 3.0000i 0 - 4.0000i

>> x.'

ans = 0 + 1.0000i 0 + 2.0000i 0 + 3.0000i 0 + 4.0000i

>>
<< Previous page TOC INDEX Next page >>

Numerical Tools in Mathematica

Sun, 01 Mar 1998 00:00:00 +0000

NOTE: THIS DOCUMENT IS OBSOLETE, PLEASE CHECK THE NEW VERSION: "Mathematics of the Discrete Fourier Transform (DFT), with Audio Applications --- Second Edition", by Julius O. Smith III, W3K Publishing, 2007, ISBN 978-0-9745607-4-8. - Copyright © 2017-09-28 by Julius O. Smith III - Center for Computer Research in Music and Acoustics (CCRMA), Stanford University

<< Previous page TOC INDEX Next page >>
Numerical Tools in Mathematica
In Mathematica, we can find the roots of simple polynomials in closed form, while larger polynomials can be factored numerically in eitherMatlab or Mathematica. Look to Mathematica to provide the most sophisticated symbolic mathematical manipulation, and look for Matlab to provide the best numerical algorithms, as a general rule.
One way to implicitly find the roots of a polynomial is to factor it in Mathematica:
In[1]: p[x_] := x^2 + 5 x + 6 In[2]: Factor[p[x]] Out[2]: (2 + x)*(3 + x)
Factor[] works only with exact Integers or Rational coefficients, not with Real numbers.
Alternatively, we can explicitly solve for the roots of low-order polynomials in Mathematica:
In[1]: p[x_] := a x^2 + b x + c In[2]: Solve[p[x]==0,x] Out[2]: {{x -> (-(b/a) + (b^2/a^2 - (4*c)/a)^(1/2))/2}, {x -> (-(b/a) - (b^2/a^2 - (4*c)/a)^(1/2))/2}}
Closed-form solutions work for polynomials of order one through four. Higher orders, in general, must be dealt with numerically, as shown below:
In[1]: p[x_] := x^5 + 5 x + 7 In[2]: Solve[p[x]==0,x] Out[2]: {ToRules[Roots[5*x + x^5 == -7, x]]} In[3]: N[Solve[p[x]==0,x]] Out[3]: {{x -> -1.090942496109028}, {x -> -0.7550479250175501 - 1.275010619237742*I}, {x -> -0.7550479250175501 + 1.275010619237742*I}, {x -> 1.300519173072064 - 1.109447238195959*I}, {x -> 1.300519173072064 + 1.109447238195959*I}}
Mathematica has the following primitives for dealing with complex numbers(The ''?'' operator returns a short description of the symbol to its right):
In[1]: ?I Out[1]: I represents the imaginary unit Sqrt[-1]. In[2]: ?Re Out[2]: Re[z] gives the real part of the complex number z. In[3]: ?Im Out[3]: Im[z] gives the imaginary part of the complex number z. In[4]: ?Conj* Out[4]: Conjugate[z] gives the complex conjugate of the complex number z. In[5]: ?Abs Out[5]: Abs[z] gives the absolute value of the real or complex number z. In[6]: ?Arg Out[6]: Arg[z] gives the argument of z.
<< Previous page TOC INDEX Next page >>

Proof of Euler's Identity

Sun, 01 Mar 1998 00:00:00 +0000

NOTE: THIS DOCUMENT IS OBSOLETE, PLEASE CHECK THE NEW VERSION: "Mathematics of the Discrete Fourier Transform (DFT), with Audio Applications --- Second Edition", by Julius O. Smith III, W3K Publishing, 2007, ISBN 978-0-9745607-4-8. - Copyright © 2017-09-28 by Julius O. Smith III - Center for Computer Research in Music and Acoustics (CCRMA), Stanford University

<< Previous page TOC INDEX Next page >>
Proof of Euler's Identity
This chapter outlines the proof of Euler's Identity, which is an important tool for working with complex numbers. It is one of the critical elements of the DFT definition that we need to understand.

Subsections
Euler's Theorem
Positive Integer Exponents
Properties of Exponents
The Exponent Zero
Negative Exponents
Rational Exponents
Special Case: The Mth Roots of Unity
Real Exponents
A First Look at Taylor Series
Imaginary Exponents
Derivatives of f(x)=a^x
Back to e
Sidebar on Mathematica
Back to e^(j theta)
Informal Derivation of Taylor Series Expansion
Derivation of Taylor Series Expansion with Remainder
Formal Statement of Taylor's Theorem
The Weierstrass (Polynomial) Approximation Theorem
Further Notes on Differentiability of Audio Signals

<< Previous page TOC INDEX Next page >>

Euler's Theorem

Sun, 01 Mar 1998 00:00:00 +0000

NOTE: THIS DOCUMENT IS OBSOLETE, PLEASE CHECK THE NEW VERSION: "Mathematics of the Discrete Fourier Transform (DFT), with Audio Applications --- Second Edition", by Julius O. Smith III, W3K Publishing, 2007, ISBN 978-0-9745607-4-8. - Copyright © 2017-09-28 by Julius O. Smith III - Center for Computer Research in Music and Acoustics (CCRMA), Stanford University

<< Previous page TOC INDEX Next page >>
Euler's Theorem
Euler's Theorem (or ''identity'' or ''formula'') is

To ''prove'' this, we must first define what we mean by ''.'' (The right-hand side is assumed to be understood.) Since is just a particular number, we only really have to explain what we mean by imaginary exponents. (We'll also see where comes from in the process.) Imaginary exponents will be obtained as a generalization of real exponents. Therefore, our first task is to define exactly what we mean by , where is any real number, and is any positive real number.

Subsections
Positive Integer Exponents
Properties of Exponents
The Exponent Zero
Negative Exponents
Rational Exponents
Special Case: The Mth Roots of Unity
Real Exponents
A First Look at Taylor Series
Imaginary Exponents
Derivatives of f(x)=a^x
Back to e
Sidebar on Mathematica
Back to e^(j theta)

<< Previous page TOC INDEX Next page >>

Positive Integer Exponents

Sun, 01 Mar 1998 00:00:00 +0000

NOTE: THIS DOCUMENT IS OBSOLETE, PLEASE CHECK THE NEW VERSION: "Mathematics of the Discrete Fourier Transform (DFT), with Audio Applications --- Second Edition", by Julius O. Smith III, W3K Publishing, 2007, ISBN 978-0-9745607-4-8. - Copyright © 2017-09-28 by Julius O. Smith III - Center for Computer Research in Music and Acoustics (CCRMA), Stanford University

<< Previous page TOC INDEX Next page >>
Positive Integer Exponents
The ''original'' definition of exponents which ''actually makes sense'' applies only to positive integer exponents:

where is real.
Generalizing this definition involves first noting its abstract mathematical properties, and then making sure these properties are preserved in the generalization.
<< Previous page TOC INDEX Next page >>

Properties of Exponents

Sun, 01 Mar 1998 00:00:00 +0000

NOTE: THIS DOCUMENT IS OBSOLETE, PLEASE CHECK THE NEW VERSION: "Mathematics of the Discrete Fourier Transform (DFT), with Audio Applications --- Second Edition", by Julius O. Smith III, W3K Publishing, 2007, ISBN 978-0-9745607-4-8. - Copyright © 2017-09-28 by Julius O. Smith III - Center for Computer Research in Music and Acoustics (CCRMA), Stanford University

<< Previous page TOC INDEX Next page >>
Properties of Exponents
From the basic definition of positive integer exponents, we have
(1)
(2)
Note that property (2) implies property (1). We list it explicitly for convenience below.
<< Previous page TOC INDEX Next page >>

The Exponent Zero

Sun, 01 Mar 1998 00:00:00 +0000

NOTE: THIS DOCUMENT IS OBSOLETE, PLEASE CHECK THE NEW VERSION: "Mathematics of the Discrete Fourier Transform (DFT), with Audio Applications --- Second Edition", by Julius O. Smith III, W3K Publishing, 2007, ISBN 978-0-9745607-4-8. - Copyright © 2017-09-28 by Julius O. Smith III - Center for Computer Research in Music and Acoustics (CCRMA), Stanford University

<< Previous page TOC INDEX Next page >>
The Exponent Zero
What should be? Multiplying it by gives

by property (1) of exponents. Solving for then gives

Again, we must have for this to make sense.
<< Previous page TOC INDEX Next page >>

Negative Exponents

Sun, 01 Mar 1998 00:00:00 +0000

NOTE: THIS DOCUMENT IS OBSOLETE, PLEASE CHECK THE NEW VERSION: "Mathematics of the Discrete Fourier Transform (DFT), with Audio Applications --- Second Edition", by Julius O. Smith III, W3K Publishing, 2007, ISBN 978-0-9745607-4-8. - Copyright © 2017-09-28 by Julius O. Smith III - Center for Computer Research in Music and Acoustics (CCRMA), Stanford University

<< Previous page TOC INDEX Next page >>
Negative Exponents
What should be? Multiplying it by gives

Solving for then gives

Similarly, we obtain

for all integer values of , i.e., .
<< Previous page TOC INDEX Next page >>

Rational Exponents

Sun, 01 Mar 1998 00:00:00 +0000

NOTE: THIS DOCUMENT IS OBSOLETE, PLEASE CHECK THE NEW VERSION: "Mathematics of the Discrete Fourier Transform (DFT), with Audio Applications --- Second Edition", by Julius O. Smith III, W3K Publishing, 2007, ISBN 978-0-9745607-4-8. - Copyright © 2017-09-28 by Julius O. Smith III - Center for Computer Research in Music and Acoustics (CCRMA), Stanford University

<< Previous page TOC INDEX Next page >>
Rational Exponents
A rational number is a real number that can be expressed as a ratio of two integers:

Applying property (2) of exponents, we have

Thus, the only thing new is . Since

we see that is the th root of . This is sometimes written

The th root of a real (or complex) number is not unique. As we all know, square roots give two values (e.g., ). In the general case of th roots, there are distinct values, in general.
How do we come up with different numbers which when raised to the th power will yield ? The answer is to consider complexnumbers in polar form. By Euler's Identity, the real number can be expressed, for any integer , as . Using this form for , the number can be written as

We can now see that we get a different complex number for each . When , we get the same thing as when . When , we get the same thing as when , and so on, so there are only cases using this construct. The th root for can be called the ''primitive th root of '', since integer powers of it give all of the others.

Subsections
Special Case: The Mth Roots of Unity

<< Previous page TOC INDEX Next page >>

Special Case: The Mth Roots of Unity

Sun, 01 Mar 1998 00:00:00 +0000

NOTE: THIS DOCUMENT IS OBSOLETE, PLEASE CHECK THE NEW VERSION: "Mathematics of the Discrete Fourier Transform (DFT), with Audio Applications --- Second Edition", by Julius O. Smith III, W3K Publishing, 2007, ISBN 978-0-9745607-4-8. - Copyright © 2017-09-28 by Julius O. Smith III - Center for Computer Research in Music and Acoustics (CCRMA), Stanford University

<< Previous page TOC INDEX Next page >>
Special Case: The Mth Roots of Unity
If , we have

The th roots of unity play an important role in the Discrete Fourier Transform. They are so important that they often have a special notation in the signal processing literature:

In this notation, the roots of unity can be expressed simply as, for .
We may also call the generator of the mathematicalgroup consisting of the th roots of unity and their products.
We will learn later that the th roots of unity are used to generate all the sinusoids used by the DFT and its inverse. The th sinusoid is given by

where , , and is the sampling interval in seconds.
<< Previous page TOC INDEX Next page >>

Real Exponents

Sun, 01 Mar 1998 00:00:00 +0000

NOTE: THIS DOCUMENT IS OBSOLETE, PLEASE CHECK THE NEW VERSION: "Mathematics of the Discrete Fourier Transform (DFT), with Audio Applications --- Second Edition", by Julius O. Smith III, W3K Publishing, 2007, ISBN 978-0-9745607-4-8. - Copyright © 2017-09-28 by Julius O. Smith III - Center for Computer Research in Music and Acoustics (CCRMA), Stanford University

<< Previous page TOC INDEX Next page >>
Real Exponents
The closest we can actually get to most real numbers is to compute arational number that is as close as we need. It can be shown that rational numbers are dense in the real numbers; that is, between every two real numbers there is a rational number, and between every two rational numbers is a real number. Anirrational number can be defined as any real number having a non-repeating decimal expansion. For example, is an irrational real number whose decimal expansion starts out as

(computed via N[Sqrt[2],80] in Mathematica). Every truncated, rounded, or repeating expansion is a rational number. That is, it can be rewritten as an integer divided by another integer. For example,

and, using to denote the repeating part of a decimal expansion,

Other examples of irrational numbers include

Let denote the -digit decimal expansion of an arbitrary real number . Then is a rational number (some integer over ). We can say

Since is defined for all , it is straightforward to define

<< Previous page TOC INDEX Next page >>

A First Look at Taylor Series

Sun, 01 Mar 1998 00:00:00 +0000

NOTE: THIS DOCUMENT IS OBSOLETE, PLEASE CHECK THE NEW VERSION: "Mathematics of the Discrete Fourier Transform (DFT), with Audio Applications --- Second Edition", by Julius O. Smith III, W3K Publishing, 2007, ISBN 978-0-9745607-4-8. - Copyright © 2017-09-28 by Julius O. Smith III - Center for Computer Research in Music and Acoustics (CCRMA), Stanford University

<< Previous page TOC INDEX Next page >>
A First Look at Taylor Series
Any ''smooth'' function can be expanded in the form of a Taylor series:

This can be written more compactly as

An informal derivation of this formula for is given in §3.2 and §3.3. Clearly, since many derivatives are involved, a Taylor series expansion is only possible when the function is so smooth that it can be differentiated again and again. Fortunately for us, all audio signals can be defined so as to be in that category. This is because hearing is bandlimited to kHz, and any sum of sinusoids up to some maximum frequency, i.e., any audible signal, is infinitely differentiable. (Recall that and , etc.). See §3.6 for more on this topic.
<< Previous page TOC INDEX Next page >>

Imaginary Exponents

Sun, 01 Mar 1998 00:00:00 +0000

NOTE: THIS DOCUMENT IS OBSOLETE, PLEASE CHECK THE NEW VERSION: "Mathematics of the Discrete Fourier Transform (DFT), with Audio Applications --- Second Edition", by Julius O. Smith III, W3K Publishing, 2007, ISBN 978-0-9745607-4-8. - Copyright © 2017-09-28 by Julius O. Smith III - Center for Computer Research in Music and Acoustics (CCRMA), Stanford University

<< Previous page TOC INDEX Next page >>
Imaginary Exponents
We may define imaginary exponents the same way that all sufficiently smooth real-valued functions of a real variable are generalized to the complex case--using Taylor series. A Taylor series expansion is just a polynomial (possibly of infinitely high order), and polynomials involve only addition, multiplication, and division. Since these elementary operations are also defined for complex numbers, any smooth function of a real variable may be generalized to a function of a complex variable by simply substituting the complex variable for the real variable in the Taylor series expansion.
Let , where is any positive real number. The Taylor series expansion expansion about (''Maclaurin series''), generalized to the complex case is then

which is well defined (although we should make sure the series converges for every finite ). We have , so the first term is no problem. But what is ? In other words, what is the derivative of at ? Once we find the successive derivatives of at , we will be done with the definition of for any complex .
<< Previous page TOC INDEX Next page >>

Derivatives of f(x)=a^x

Sun, 01 Mar 1998 00:00:00 +0000

NOTE: THIS DOCUMENT IS OBSOLETE, PLEASE CHECK THE NEW VERSION: "Mathematics of the Discrete Fourier Transform (DFT), with Audio Applications --- Second Edition", by Julius O. Smith III, W3K Publishing, 2007, ISBN 978-0-9745607-4-8. - Copyright © 2017-09-28 by Julius O. Smith III - Center for Computer Research in Music and Acoustics (CCRMA), Stanford University

<< Previous page TOC INDEX Next page >>
Derivatives of f(x)=a^x
Let's apply the definition of differentiation and see what happens:

Back to e

Sun, 01 Mar 1998 00:00:00 +0000

NOTE: THIS DOCUMENT IS OBSOLETE, PLEASE CHECK THE NEW VERSION: "Mathematics of the Discrete Fourier Transform (DFT), with Audio Applications --- Second Edition", by Julius O. Smith III, W3K Publishing, 2007, ISBN 978-0-9745607-4-8. - Copyright © 2017-09-28 by Julius O. Smith III - Center for Computer Research in Music and Acoustics (CCRMA), Stanford University

<< Previous page TOC INDEX Next page >>
Back to e
Above, we defined as the particular real number satisfying

which gave us when . From this expression, we have, as ,

or,

This is one way to define . Another way to arrive at the same definition is to ask what logarithmic base gives that the derivative of is . We denote by .
<< Previous page TOC INDEX Next page >>

Sidebar on Mathematica

Sun, 01 Mar 1998 00:00:00 +0000

NOTE: THIS DOCUMENT IS OBSOLETE, PLEASE CHECK THE NEW VERSION: "Mathematics of the Discrete Fourier Transform (DFT), with Audio Applications --- Second Edition", by Julius O. Smith III, W3K Publishing, 2007, ISBN 978-0-9745607-4-8. - Copyright © 2017-09-28 by Julius O. Smith III - Center for Computer Research in Music and Acoustics (CCRMA), Stanford University

<< Previous page TOC INDEX Next page >>
Sidebar on Mathematica
Mathematica is a handy tool for cranking out any number of digits intranscendental numbers such as :
In[]: N[E,50] Out[]: 2.7182818284590452353602874713526624977572470937
Alternatively, we can compute for small :
In[]: (1+delta)^(1/delta) /. delta->0.001 Out[]: 2.716923932235594 In[]: (1+delta)^(1/delta) /. delta->0.0001 Out[]: 2.718145926824926 In[]: (1+delta)^(1/delta) /. delta->0.00001 Out[]: 2.718268237192297
What happens if we just go for it and set delta to zero?
In[]: (1+delta)^(1/delta) /. delta->0 1 Power::infy: Infinite expression - encountered. 0 Infinity::indt: ComplexInfinity Indeterminate expression 1 encountered. Indeterminate
<< Previous page TOC INDEX Next page >>

Back to e^(j theta)

Sun, 01 Mar 1998 00:00:00 +0000

NOTE: THIS DOCUMENT IS OBSOLETE, PLEASE CHECK THE NEW VERSION: "Mathematics of the Discrete Fourier Transform (DFT), with Audio Applications --- Second Edition", by Julius O. Smith III, W3K Publishing, 2007, ISBN 978-0-9745607-4-8. - Copyright © 2017-09-28 by Julius O. Smith III - Center for Computer Research in Music and Acoustics (CCRMA), Stanford University

<< Previous page TOC INDEX Next page >>
Back to e^(j theta)
We've now defined for any positive real number and any complex number . Setting and gives us the special case we need for Euler's identity. Since is its own derivative, the Taylor series expansion for for is the simplest series there could be:

The simplicity comes about because for all and because we chose to expand about the point . We of course define

Note that all even order terms are real while all odd order terms are imaginary. Separating out the real and imaginary parts gives

Comparing the Maclaurin expansion for with that of and proves Euler's identity. Recall that

so that

Plugging into the general Maclaurin series gives

Separating the Maclaurin expansion for into its even and odd terms (real and imaginary parts) gives

thus proving Euler's identity.
<< Previous page TOC INDEX Next page >>

Informal Derivation of Taylor Series Expansion

Sun, 01 Mar 1998 00:00:00 +0000

NOTE: THIS DOCUMENT IS OBSOLETE, PLEASE CHECK THE NEW VERSION: "Mathematics of the Discrete Fourier Transform (DFT), with Audio Applications --- Second Edition", by Julius O. Smith III, W3K Publishing, 2007, ISBN 978-0-9745607-4-8. - Copyright © 2017-09-28 by Julius O. Smith III - Center for Computer Research in Music and Acoustics (CCRMA), Stanford University

<< Previous page TOC INDEX Next page >>

Informal Derivation of Taylor Series Expansion
We have a function and we want to approximate it using anth-order polynomial:

Derivation of Taylor Series Expansion with Remainder

Sun, 01 Mar 1998 00:00:00 +0000

NOTE: THIS DOCUMENT IS OBSOLETE, PLEASE CHECK THE NEW VERSION: "Mathematics of the Discrete Fourier Transform (DFT), with Audio Applications --- Second Edition", by Julius O. Smith III, W3K Publishing, 2007, ISBN 978-0-9745607-4-8. - Copyright © 2017-09-28 by Julius O. Smith III - Center for Computer Research in Music and Acoustics (CCRMA), Stanford University

<< Previous page TOC INDEX Next page >>

Derivation of Taylor Series Expansion with Remainder
We repeat the derivation of the preceding section, but this time we treat the error term more carefully.
Again we want to approximate with an th-order polynomial:

is the ''remainder term'' which we will no longer assume is zero.
Our problem is to find so as to minimize over some interval containing . There are many ''optimality criteria'' we could choose. The one that falls out naturally here is called ''Padé'' approximation. Padé approximation sets the error value and its first derivatives to zero at a single chosen point, which we take to be . Since all ''degrees of freedom'' in the polynomial coefficients are used to set derivatives to zero at one point, the approximation is termed ''maximally flat'' at that point. Padé approximation comes up often in signal processing. For example, it is the sense in which Butterworth lowpass filters are optimal. (Their frequencyreponses are maximally flat at dc.) Also, Lagrange interpolation filters can be shown to maximally flat at dc in the frequency domain.
Setting in the above polynomial approximation produces

where we have used the fact that the error is to be exactly zero at .
Differentiating the polynomial approximation and setting gives

where we have used the fact that we also want the slopeof the error to be exactly zero at .
In the same way, we find

for , and the first derivatives of the remainder term are all zero. Solving these relations for the desired constants yields the th-order Taylor series expansion of about the point

as before, but now we better understand the remainder term.
From this derivation, it is clear that the approximation error (remainder term) is smallest in the vicinity of . All degrees of freedomin the polynomial coefficients were devoted to minimizing the approximation error and its derivatives at . As you might expect, the approximation error generally worsens as gets farther away from .
To obtain a more uniform approximation over some interval in, other kinds of error criteria may be employed. This is classically called ''economization of series,'' but nowadays we may simply call itpolynomial approximation under different error criteria. In Matlab, the function polyfit(x,y,n) will find the coefficients of a polynomial of degree n that fits the data y over the points x in a least-squares sense. That is, it minimizes

where .
<< Previous page TOC INDEX Next page >>

Formal Statement of Taylor's Theorem

Sun, 01 Mar 1998 00:00:00 +0000

NOTE: THIS DOCUMENT IS OBSOLETE, PLEASE CHECK THE NEW VERSION: "Mathematics of the Discrete Fourier Transform (DFT), with Audio Applications --- Second Edition", by Julius O. Smith III, W3K Publishing, 2007, ISBN 978-0-9745607-4-8. - Copyright © 2017-09-28 by Julius O. Smith III - Center for Computer Research in Music and Acoustics (CCRMA), Stanford University

<< Previous page TOC INDEX Next page >>

Formal Statement of Taylor's Theorem
Let be continuous on a real interval containing (and ), and let exist at and be continuous for all . Then we have the following Taylor series expansion:

where is called the remainder term. There exists between and such that

In particular, if in , then

which is normally small when is close to .
When , the Taylor series reduces to what is called a Maclaurin series.
<< Previous page TOC INDEX Next page >>

The Weierstrass (Polynomial) Approximation Theorem

Sun, 01 Mar 1998 00:00:00 +0000

NOTE: THIS DOCUMENT IS OBSOLETE, PLEASE CHECK THE NEW VERSION: "Mathematics of the Discrete Fourier Transform (DFT), with Audio Applications --- Second Edition", by Julius O. Smith III, W3K Publishing, 2007, ISBN 978-0-9745607-4-8. - Copyright © 2017-09-28 by Julius O. Smith III - Center for Computer Research in Music and Acoustics (CCRMA), Stanford University

<< Previous page TOC INDEX Next page >>

The Weierstrass (Polynomial) Approximation Theorem
Let be continuous on a real interval . Then for any , there exists an th-order polynomial , where depends on, such that

for all .
Thus, any continuous function can be approximated arbitrarily well by means of a polynomial. Furthermore, an infinite-order polynomial can yield an error-free approximation. Of course, to compute the polynomial coefficients using a Taylor series expansion, the function must also be differentiable of all orders throughout .
<< Previous page TOC INDEX Next page >>

Further Notes on Differentiability of Audio Signals

Sun, 01 Mar 1998 00:00:00 +0000

NOTE: THIS DOCUMENT IS OBSOLETE, PLEASE CHECK THE NEW VERSION: "Mathematics of the Discrete Fourier Transform (DFT), with Audio Applications --- Second Edition", by Julius O. Smith III, W3K Publishing, 2007, ISBN 978-0-9745607-4-8. - Copyright © 2017-09-28 by Julius O. Smith III - Center for Computer Research in Music and Acoustics (CCRMA), Stanford University

<< Previous page TOC INDEX Next page >>

Further Notes on Differentiability of Audio Signals
As mentioned earlier, every audio signal can be regarded as infinitely differentiable due to the finite bandwidth of human hearing. One of the Fourier properties we will learn later in this course is thata signal cannot be both time limited and frequency limited.Therefore, by conceptually ''lowpass filtering'' every audio signal to reject all frequencies above kHz, we implicitly make every audio signal last forever! Another way of saying this is that the ''ideal lowpass filter `rings' forever''. Such fine points do not concern us in practice, but they are important for fully understanding the underlying theory. Since, in reality, signals can be said to have a true beginning and end, we must admit in practice that all signals we work with have infinite-bandwidth at turn-on and turn-off transients.^3.1
<< Previous page TOC INDEX Next page >>

Logarithms, Decibels, and Number Systems

Sun, 01 Mar 1998 00:00:00 +0000

NOTE: THIS DOCUMENT IS OBSOLETE, PLEASE CHECK THE NEW VERSION: "Mathematics of the Discrete Fourier Transform (DFT), with Audio Applications --- Second Edition", by Julius O. Smith III, W3K Publishing, 2007, ISBN 978-0-9745607-4-8. - Copyright © 2017-09-28 by Julius O. Smith III - Center for Computer Research in Music and Acoustics (CCRMA), Stanford University

<< Previous page TOC INDEX Next page >>
Logarithms, Decibels, and Number Systems
This chapter provides an introduction to logarithms (real and complex),decibels, and number systems such as binary integer fixed-point, fractional fixed-point, one's complement, two's complement, logarithmic fixed-point,-law, and floating-point number formats.
FIXME: HERE HERE HERE

Subsections
Logarithms
Changing the Base
Logarithms of Negative and Imaginary Numbers
Decibels
Properties of DB Scales
Specific DB Scales
DBm Scale
DBV Scale
DB SPL
DB for Display
Dynamic Range
Linear Number Systems for Digital Audio
Pulse Code Modulation (PCM)
Binary Integer Fixed-Point Numbers
One's Complement Fixed-Point Format
Two's Complement Fixed-Point Format
General Formula for Two's-Complement, Integer Fixed-Point Numbers
''Little Endian'' Formula for Two's-Complement, Integer Fixed-Point Numbers
Fractional Binary Fixed-Point Numbers
How Many Bits are Enough for Digital Audio?
When Do We Have to Swap Bytes When Changing Computers?
Logarithmic Number Systems for Digital Audio
Floating-Point Numbers
Logarithmic Fixed-Point Numbers
-Law Companding
Appendix A: Round-Off Error Variance
Appendix B: Electrical Engineering 101

<< Previous page TOC INDEX Next page >>

Logarithms

Sun, 01 Mar 1998 00:00:00 +0000

NOTE: THIS DOCUMENT IS OBSOLETE, PLEASE CHECK THE NEW VERSION: "Mathematics of the Discrete Fourier Transform (DFT), with Audio Applications --- Second Edition", by Julius O. Smith III, W3K Publishing, 2007, ISBN 978-0-9745607-4-8. - Copyright © 2017-09-28 by Julius O. Smith III - Center for Computer Research in Music and Acoustics (CCRMA), Stanford University

<< Previous page TOC INDEX Next page >>
Logarithms
A logarithm is fundamentally an exponent applied to a specific base . That is, . The term ''logarithm'' can be abbreviated as ''log''. The base is chosen to be a positive real number, and we normally only take logs of positive real numbers (although it is ok to say that the log of is ). The inverse of a logarithm is called anantilogarithm or antilog.
For any positive number , we have

for any valid base . This is just an identity arising from the definition of the logarithm, but it is sometimes useful in manipulating formulas.
When the base is not specified, it is normally assumed to be , i.e., . This is the common logarithm.
Base 2 and base logarithms have their own special notation:

(The use of for base logarithms is common in computer science. In mathematics, it may denote a base logarithm.) By far the most common bases are , , and . Logs base are called natural logarithms. They are ''natural'' in the sense that

while the derivatives of logarithms to other bases are not quite so simple:

(Prove this as an exercise.) The inverse of the natural logarithm is of course the exponential function , and is its own derivative.
In general, a logarithm has an integer part and a fractional part. The integer part is called the characteristic of the logarithm, and the fractional part is called the mantissa. These terms were suggested by Henry Briggs in 1624. ''Mantissa'' is a Latin word meaning ''addition'' or ''make weight''--something added to make up the weight [6].
The following Matlab code illustrates splitting a natural logarithm into its characteristic and mantissa:
>> x = log(3) x = 1.0986 >> characteristic = floor(x) characteristic = 1 >> mantissa = x - characteristic mantissa = 0.0986
>> % Now do a negative-log example >> x = log(0.05) x = -2.9957 >> characteristic = floor(x) characteristic = -3 >> mantissa = x - characteristic mantissa = 0.0043
Logarithms were used in the days before computers to performmultiplication of large numbers. Since , one can look up the logs of and in tables of logarithms, add them together (which is easier than multiplying), and look up the antilog of the result to obtain the product . Log tables are still used in modern computing environments to replace expensive multiplies with less-expensive table lookups and additions. This is a classic tradeoff between memory (for the log tables) and computation. Nowadays, large numbers are multiplied using FFT fast-convolution techniques.

Subsections
Changing the Base
Logarithms of Negative and Imaginary Numbers

<< Previous page TOC INDEX Next page >>

Changing the Base

Sun, 01 Mar 1998 00:00:00 +0000

NOTE: THIS DOCUMENT IS OBSOLETE, PLEASE CHECK THE NEW VERSION: "Mathematics of the Discrete Fourier Transform (DFT), with Audio Applications --- Second Edition", by Julius O. Smith III, W3K Publishing, 2007, ISBN 978-0-9745607-4-8. - Copyright © 2017-09-28 by Julius O. Smith III - Center for Computer Research in Music and Acoustics (CCRMA), Stanford University

<< Previous page TOC INDEX Next page >>
Changing the Base
By definition, . Taking the log base of both sides gives

which tells how to convert the base from to , that is, how to convert the log base of to the log base of . (Just multiply by the log base of .)
<< Previous page TOC INDEX Next page >>

Logarithms of Negative and Imaginary Numbers

Sun, 01 Mar 1998 00:00:00 +0000

NOTE: THIS DOCUMENT IS OBSOLETE, PLEASE CHECK THE NEW VERSION: "Mathematics of the Discrete Fourier Transform (DFT), with Audio Applications --- Second Edition", by Julius O. Smith III, W3K Publishing, 2007, ISBN 978-0-9745607-4-8. - Copyright © 2017-09-28 by Julius O. Smith III - Center for Computer Research in Music and Acoustics (CCRMA), Stanford University

<< Previous page TOC INDEX Next page >>
Logarithms of Negative and Imaginary Numbers
By Euler's formula, , so that

from which it follows that for any , .
Similarly, , so that

and for any imaginary number , , where is real.
Finally, from the polar representation forcomplex numbers,

where and are real. Thus, the log of the magnitude of a complex number behaves like the log of any positive real number, while the log of its phase term extracts its phase (times ).
As an example of the use of logarithms in signal processing, note that the negative imaginary part of the derivative of the log of aspectrum is defined as the group delay^4.1of the signal :

Another usage is in Homomorphic Signal Processing [8, Chapter 10] in which the multiplicative formants in vocal spectra are converted to additive low-frequency variations in the spectrum (with the harmonics being the high-frequency variation in the spectrum). Thus, the lowpass-filtered log spectrum contains only the formants, and the complementarily highpass-filtered log spectrum contains only the fine structure associated with the pitch.
Exercise: Work out the definition of logarithms using a complex base .
<< Previous page TOC INDEX Next page >>

Decibels

Sun, 01 Mar 1998 00:00:00 +0000

NOTE: THIS DOCUMENT IS OBSOLETE, PLEASE CHECK THE NEW VERSION: "Mathematics of the Discrete Fourier Transform (DFT), with Audio Applications --- Second Edition", by Julius O. Smith III, W3K Publishing, 2007, ISBN 978-0-9745607-4-8. - Copyright © 2017-09-28 by Julius O. Smith III - Center for Computer Research in Music and Acoustics (CCRMA), Stanford University

<< Previous page TOC INDEX Next page >>
Decibels
A decibel (abbreviated dB) is defined as one tenth of abel. The bel^4.2 is an amplitude unit defined for sound as the log (base 10) of the intensityrelative to some reference intensity,^4.3 i.e.,

The choice of reference intensity (or power) defines the particular choice of dB scale. Signal intensity, power, and energy are always proportional to the square of the signalamplitude. Thus, we can always translate these energy-related measures into squared amplitude:

Since there are 10 decibels to a bel, we also have

A just-noticeable difference (JND) in amplitude level is on the order of a quarter dB. In the early days of telephony, one dB was considered a reasonable ''smallest step'' in amplitude, but in reality, a series of half-dB amplitude steps does not sound very smooth, while quarter-dB steps do sound pretty smooth. A typical professional audio filter-design specification for ''ripple in the passband'' is 0.1 dB.
Exercise: Try synthesizing a sawtooth waveform which increases by 1/2 dB a few times per second, and again using 1/4 dB increments. See if you agree that quarter-dB increments are ''smooth'' enough for you.

Subsections
Properties of DB Scales
Specific DB Scales
DBm Scale
DBV Scale
DB SPL
DB for Display
Dynamic Range

<< Previous page TOC INDEX Next page >>

Properties of DB Scales

Sun, 01 Mar 1998 00:00:00 +0000

NOTE: THIS DOCUMENT IS OBSOLETE, PLEASE CHECK THE NEW VERSION: "Mathematics of the Discrete Fourier Transform (DFT), with Audio Applications --- Second Edition", by Julius O. Smith III, W3K Publishing, 2007, ISBN 978-0-9745607-4-8. - Copyright © 2017-09-28 by Julius O. Smith III - Center for Computer Research in Music and Acoustics (CCRMA), Stanford University

<< Previous page TOC INDEX Next page >>
Properties of DB Scales
In every kind of dB, a factor of 10 in amplitude gain corresponds to a 20 dB boost (increase by 20 dB):

and , of course.
Similarly, a factor of 2 in amplitude gain corresponds to a 6 dB boost:

and

A doubling of power corresponds to a 3 dB boost:

and

Finally, note that the choice of reference merely determines a vertical offset in the dB scale:

<< Previous page TOC INDEX Next page >>

Specific DB Scales

Sun, 01 Mar 1998 00:00:00 +0000

NOTE: THIS DOCUMENT IS OBSOLETE, PLEASE CHECK THE NEW VERSION: "Mathematics of the Discrete Fourier Transform (DFT), with Audio Applications --- Second Edition", by Julius O. Smith III, W3K Publishing, 2007, ISBN 978-0-9745607-4-8. - Copyright © 2017-09-28 by Julius O. Smith III - Center for Computer Research in Music and Acoustics (CCRMA), Stanford University

<< Previous page TOC INDEX Next page >>
Specific DB Scales
Since we so often rescale our signals to suit various needs (avoiding overflow, reducing quantization noise, making a nicer plot, etc.), there seems to be little point in worrying about what the dB reference is--we simply choose it implicitly when we rescale to obtain signal values in the range we want to see. Nevertheless, a few specific dB scales are worth knowing about.

Subsections
DBm Scale
DBV Scale
DB SPL
DB for Display

<< Previous page TOC INDEX Next page >>

DBm Scale

Sun, 01 Mar 1998 00:00:00 +0000

NOTE: THIS DOCUMENT IS OBSOLETE, PLEASE CHECK THE NEW VERSION: "Mathematics of the Discrete Fourier Transform (DFT), with Audio Applications --- Second Edition", by Julius O. Smith III, W3K Publishing, 2007, ISBN 978-0-9745607-4-8. - Copyright © 2017-09-28 by Julius O. Smith III - Center for Computer Research in Music and Acoustics (CCRMA), Stanford University

<< Previous page TOC INDEX Next page >>
DBm Scale
One common dB scale in audio recording is the dBm scale in which the reference power is taken to be a milliwatt (1 mW) dissipated by a 600Ohm resistor. (See Appendix B for a primer on resistors, voltage, current, and power.)
<< Previous page TOC INDEX Next page >>

DBV Scale

Sun, 01 Mar 1998 00:00:00 +0000

NOTE: THIS DOCUMENT IS OBSOLETE, PLEASE CHECK THE NEW VERSION: "Mathematics of the Discrete Fourier Transform (DFT), with Audio Applications --- Second Edition", by Julius O. Smith III, W3K Publishing, 2007, ISBN 978-0-9745607-4-8. - Copyright © 2017-09-28 by Julius O. Smith III - Center for Computer Research in Music and Acoustics (CCRMA), Stanford University

<< Previous page TOC INDEX Next page >>
DBV Scale
Another dB scale is the dBV scale which sets 0 dBV to 1 volt. Thus, a 100-volt signal is

and a 1000-volt signal is

Note that the dBV scale is undefined for current or power, unless the voltage is assumed to be across a standard resistor value, such as 600Ohms.
<< Previous page TOC INDEX Next page >>

DB SPL

Sun, 01 Mar 1998 00:00:00 +0000

NOTE: THIS DOCUMENT IS OBSOLETE, PLEASE CHECK THE NEW VERSION: "Mathematics of the Discrete Fourier Transform (DFT), with Audio Applications --- Second Edition", by Julius O. Smith III, W3K Publishing, 2007, ISBN 978-0-9745607-4-8. - Copyright © 2017-09-28 by Julius O. Smith III - Center for Computer Research in Music and Acoustics (CCRMA), Stanford University

<< Previous page TOC INDEX Next page >>
DB SPL
Sound Pressure Level (SPL) is defined using a reference which is approximately the intensity of 1000 Hz sinusoid that is just barely audible (0 ''phons''). In pressure units:

In intensity units:

which corresponds to a root-mean-square (rms) pressure amplitude of Pa, or about Pa, as listed above. The wave impedanceof air plays the role of ''resistor'' in relating the pressure- and intensity-based references exactly analogous to the dBm case discussed above.
Since sound is created by a time-varying pressure, we compute sound levels in dB-SPL by using the average intensity (averaged over at least one period of the lowest frequency contained in the sound).
Table 4.1 gives a list list of common sound levels and their dBequivalents^4.5 [9]:
Table 4.1:Ballpark figures for the dB-SPL level of common sounds.
Sound dB-SPL
Jet engine at 3m 140
Threshold of pain 130
Rock concert 120
Accelerating motorcycle at 5m 110
Pneumatic hammer at 2m 100
Noisy factory 90
Vacuum cleaner 80
Busy traffic 70
Quiet restaurant 50
Residential area at night 40
Empty movie house 30
Rustling of leaves 20
Human breathing (at 3m) 10
Threshold of hearing (good ears) 0

In my experience, the ''threshold of pain'' is most often defined as 120 dB.
The relationship between sound amplitude and actual loudness is complex [10]. Loudness is a perceptual dimension while sound amplitude is physical. Since loudness sensitivity is closer to logarithmic than linear in amplitude (especially at moderate to high loudnesses), we typically use decibels to represent sound amplitude, especially in spectral displays.
The sone amplitude scale is defined in terms of actual loudness perception experiments [10]. At 1kHz and above, loudness perception is approximately logarithmic above 50 dB SPL or so. Below that, it tends toward being more linear.
The phon amplitude scale is simply the dB scale at 1kHz [10, p. 111]. At other frequencies, the amplitude in phons is defined by following the equal-loudness curve over to 1 kHz and reading off the level there in dB SPL. In other words, all pure tones have the same loudness at the same phon level, and 1 kHz is used to set the reference in dB SPL. Just remember that one phon is one dB-SPL at 1 kHz. Looking at the Fletcher-Munson equal-loudness curves [10, p. 124], loudness in phons can be read off along the vertical line at 1 kHz.
Classically, the intensity level of a sound wave is its dB SPL level, measuring the peak time-domain pressure-wave amplitude relative to watts per centimeter squared (i.e., there is no consideration of the frequency domain here at all).
Another classical term still encountered is the sensation level of pure tones: The sensation level is the number of dB SPL above thehearing threshold at that frequency [10, p. 110].
For further information on ''doing it right,'' see, for example,
http://www.measure.demon.co.uk/Acoustics_Software/loudness.php.
<< Previous page TOC INDEX Next page >>

DB for Display

Sun, 01 Mar 1998 00:00:00 +0000

NOTE: THIS DOCUMENT IS OBSOLETE, PLEASE CHECK THE NEW VERSION: "Mathematics of the Discrete Fourier Transform (DFT), with Audio Applications --- Second Edition", by Julius O. Smith III, W3K Publishing, 2007, ISBN 978-0-9745607-4-8. - Copyright © 2017-09-28 by Julius O. Smith III - Center for Computer Research in Music and Acoustics (CCRMA), Stanford University

<< Previous page TOC INDEX Next page >>
DB for Display
In practical signal processing, it is common to choose the maximum signal magnitude as the reference amplitude. That is, we normalize the signal so that the maximum amplitude is defined as 1, or 0 dB. This convention is also used by ''sound level meters'' in audio recording. When displaying magnitude spectra, the highest spectral peak is often normalized to 0 dB. We can then easily read off lower peaks as so many dB below the highest peak.
Figure 4.1b shows a plot of the Fast Fourier Transform (FFT) of ten periods of a ''Kaiser-windowed'' sinusoid at Hz. (FFT windowswill be discussed later in this course. For now, just think of the window as selecting and tapering a finite-duration section of the signal.) Note that the peak dB magnitude has been normalized to zero, and that the plot has been clipped at -100 dB.

Figure 4.1:a) Ten Periods of a Sinusoid and b) its FFT magnitude.
Below is the Matlab code for producing Fig. 4.1. Note that it contains several elements (windows, zero padding, spectral interpolation) that we will not cover until later. They are included here as ''forward references'' in order to keep the example realistic and practical, and to give you an idea of ''how far we have to go'' before we know how to do practical spectrum analysis. Otherwise, the example just illustrates plotting spectra on an arbitrary dB scale between convenient limits.
% Example practical display of the FFT of a synthesized sinusoid
fs = 44100; % Sampling rate f = 440; % Sinusoidal frequency = A-440 nper = 10; % Number of periods to synthesize dur = nper/f; % Duration in seconds T = 1/fs; % Sampling period t = 0:T:dur; % Discrete-time axis in seconds L = length(t) % Number of samples to synthesize ZP = 5; % Zero padding factor (for spectral interpolation) N = 2^(nextpow2(L*ZP)) % FFT size (power of 2)

x = cos(2pift); % A sinusoid at A-440 (a “row vector”) w = kaiser(L,8); % We’ll learn a bit about “FFT windows” later xw = x . w’; % Need to transpose w to get a row vector sound(xw,fs); % Might as well listen to it xzp = [xw,zeros(1,N-L)];% Zero-padded FFT input buffer X = fft(xzp); % Spectrum of xw, interpolated by factor ZP

Xmag = abs(X); % Spectral magnitude Xdb = 20*log10(Xmag); % Spectral magnitude in dB

XdbMax = max(Xdb); % Peak dB magnitude Xdbn = Xdb - XdbMax; % Normalize to 0dB peak

dBmin = -100; % Don’t show anything lower than this Xdbp = max(Xdbn,dBmin); % Normalized, clipped, dB magnitude spectrum fmaxp = 2f; % Upper frequency limit of plot, in Hz kmaxp = fmaxpN/fs; % Upper frequency limit of plot, in bins fp = fs*[0:kmaxp]/N; % Frequency axis in Hz

% Ok, plot it already!

subplot(2,1,1); plot(1000*t,xw); xlabel(‘Time (ms)’); ylabel(‘Amplitude’); title(sprintf(‘a) %d Periods of a %3.0f Hz Sinusoid, Kaiser Windowed’,nper,f));

subplot(2,1,2); plot(fp,Xdbp(1:kmaxp+1)); grid; % Plot a dashed line where the peak should be: hold on; plot([440 440],[dBmin,0],’–’); hold off; xlabel(‘Frequency (Hz)’); ylabel(‘Magnitude (dB)’); title(sprintf(‘b) Interpolated FFT of %d Periods of %3.0f Hz Sinusoid’,nper,f));
The following more compact Matlab produces essentially the same plot, but without the nice physical units on the horizontal axes:
x = cos([0:2pi/20:102pi]); % 10 periods of a sinusoid, 20 samples/cycle L = length(x); xw = x’ . kaiser(L,8); N = 2^nextpow2(L*5); X = fft([xw’,zeros(1,N-L)]);

subplot(2,1,1); plot(xw); xlabel(‘Time (samples)’); ylabel(‘Amplitude’); title(‘a) 10 Periods of a Kaiser-Windowed Sinusoid’);

subplot(2,1,2); kmaxp = 2105; Xl = 20log10(abs(X(1:kmaxp+1))); plot([105+1,10*5+1],[-100,0],[0:kmaxp],max(Xl - max(Xl),-100)); grid; xlabel(‘Frequency (Bins)’); ylabel(‘Magnitude (dB)’); title(‘b) Interpolated FFT of 10 Periods of Sinusoid’);
<< Previous page TOC INDEX Next page >>

Dynamic Range

Sun, 01 Mar 1998 00:00:00 +0000

NOTE: THIS DOCUMENT IS OBSOLETE, PLEASE CHECK THE NEW VERSION: "Mathematics of the Discrete Fourier Transform (DFT), with Audio Applications --- Second Edition", by Julius O. Smith III, W3K Publishing, 2007, ISBN 978-0-9745607-4-8. - Copyright © 2017-09-28 by Julius O. Smith III - Center for Computer Research in Music and Acoustics (CCRMA), Stanford University

<< Previous page TOC INDEX Next page >>
Dynamic Range
The dynamic range of a signal processing system can be defined as the maximum dB level sustainable without overflow (or other distortion) minus the dB level of the ''noise floor''.
Similarly, the dynamic range of a signal can be defined as its maximum decibel level minus its average ''noise level'' in dB. For digital signals, the limiting noise is ideally quantization noise.
Quantization noise is generally modeled as a uniform random variable between plus and minus half the least significant bit (since rounding to the nearest representable sample value is normally used). If denotes the quantization interval, then the maximum quantization-error magnitude is, and its variance (''noise power'') is (see Appendix A for a derivation of this value).
The rms level of the quantization noise is therefore , or about 60% of the maximum error.
The number system and number of bits chosen to represent signal samples determines their available dynamic range. Signal processing operations such as digital filtering may use the same number system as the input signal, or they may use extra bits in the computations, yielding an increased ''internal dynamic range''.
Since the threshold of hearing is near 0 dB SPL, and since the ''threshold of pain'' is often defined as 120 dB SPL, we may say that the dynamic range of human hearing is approximately 120 dB.
The dynamic range of magnetic tape is approximately 55 dB. To increase the dynamic range available for analog recording on magnetic tape, companding is often used. ''Dolby A'' adds approximately 10 dB to the dynamic range that will fit on magnetic tape (by compressing the signal dynamic range by 10 dB), while DBX adds 30 dB (at the cost of more ''transient distortion'').^4.6 In general, any dynamic range can be mapped to any other dynamic range, subject only to noise limitations.
<< Previous page TOC INDEX Next page >>

Linear Number Systems for Digital Audio

Sun, 01 Mar 1998 00:00:00 +0000

NOTE: THIS DOCUMENT IS OBSOLETE, PLEASE CHECK THE NEW VERSION: "Mathematics of the Discrete Fourier Transform (DFT), with Audio Applications --- Second Edition", by Julius O. Smith III, W3K Publishing, 2007, ISBN 978-0-9745607-4-8. - Copyright © 2017-09-28 by Julius O. Smith III - Center for Computer Research in Music and Acoustics (CCRMA), Stanford University

<< Previous page TOC INDEX Next page >>
Linear Number Systems for Digital Audio
This section discusses the most commonly used number formats for digital audio.

Subsections
Pulse Code Modulation (PCM)
Binary Integer Fixed-Point Numbers
One's Complement Fixed-Point Format
Two's Complement Fixed-Point Format
General Formula for Two's-Complement, Integer Fixed-Point Numbers
''Little Endian'' Formula for Two's-Complement, Integer Fixed-Point Numbers
Fractional Binary Fixed-Point Numbers
How Many Bits are Enough for Digital Audio?
When Do We Have to Swap Bytes When Changing Computers?

<< Previous page TOC INDEX Next page >>

Pulse Code Modulation (PCM)

Sun, 01 Mar 1998 00:00:00 +0000

NOTE: THIS DOCUMENT IS OBSOLETE, PLEASE CHECK THE NEW VERSION: "Mathematics of the Discrete Fourier Transform (DFT), with Audio Applications --- Second Edition", by Julius O. Smith III, W3K Publishing, 2007, ISBN 978-0-9745607-4-8. - Copyright © 2017-09-28 by Julius O. Smith III - Center for Computer Research in Music and Acoustics (CCRMA), Stanford University

<< Previous page TOC INDEX Next page >>
Pulse Code Modulation (PCM)
The ''standard'' number format for sampled audio signals is officially called Pulse Code Modulation (PCM). This term simply means that each signal sample is interpreted as a ''pulse'' (e.g., a voltage or current pulse) at a particular amplitude which is binary encoded, typically in two's complement binary fixed-point format (discussed below). When someons says they are giving you a soundfile in ''raw binary format'', they pretty much always mean (nowadays) 16-bit, two's-complement PCM data. Most mainstream computer soundfile formats consist of a ''header'' (containing the length, etc.) followed by 16-bit two's-complement PCM.
You can normally convert a soundfile from one computer's format to another by stripping off its header and prepending the header for the new machine (or simply treating it as raw binary format on the destination computer). The UNIX ''cat'' command can be used for this, as can the Emacs text editor (which handles binary data just fine). The only issue usually is whether the bytes have to be swapped (an issue discussed further below).
<< Previous page TOC INDEX Next page >>

Binary Integer Fixed-Point Numbers

Sun, 01 Mar 1998 00:00:00 +0000

NOTE: THIS DOCUMENT IS OBSOLETE, PLEASE CHECK THE NEW VERSION: "Mathematics of the Discrete Fourier Transform (DFT), with Audio Applications --- Second Edition", by Julius O. Smith III, W3K Publishing, 2007, ISBN 978-0-9745607-4-8. - Copyright © 2017-09-28 by Julius O. Smith III - Center for Computer Research in Music and Acoustics (CCRMA), Stanford University

<< Previous page TOC INDEX Next page >>
Binary Integer Fixed-Point Numbers
Most prevalent computer languages only offer two kinds of numbers,floating-point and integer fixed-point. On present-day computers, all numbers are encoded using binary digits (called ''bits'') which are either 1 or 0.^4.7 In C, C++, and Java, floating-point variables are declared as float (32 bits) or double (64 bits), while integer fixed-point variables are declared as short int(typically 16 bits and never less), long int (typically 32 bits and never less), or simply int (typically the same as a long int, but sometimes between short and long). For an 8-bit integer, one can use the char datatype (8 bits).
Since C was designed to accommodate a wide range of hardware, including old mini-computers, some lattitude was historically allowed in the choice of these bit-lengths. The sizeof operator is officially the ''right way'' for a C program to determine the number of bytes in various data types at run-time, e.g. sizeof(long). (The word int can be omitted after short or long.) Nowadays, however, shorts are always 16 bits (at least on all the major platforms), ints are 32 bits, and longs are typically 32 bits on 32-bit computers and 64 bits on 64-bit computers (although some C/C++ compilers use long long int to declare 64-bit ints). Table 4.2 gives the lengths currently used by GNUC/C++ compilers (usually called ''gcc'' or ''cc'') on 64-bit processors.^4.8

Table 4.2:Byte sizes of GNU C/C++ data types for 64-bit architectures.
Type Bytes Notes
char 1
short 2
int 4
long 8 (4 bytes on 32-bit machines)
long long 8 (may become 16 bytes)
type * 8 (any pointer)
float 4
double 8
long double 8 (may become 10 bytes)
size_t 8 (type of sizeof())
T* - T* 8 (pointer arithmetic)

Java, which is designed to be platform independent, defines a long int as equivalent in precision to 64 bits, an intas 32 bits, a short int as 16 bits, and additionally a byte int as an 8-bit int. Similarly, the ''Structured Audio Orchestra Language'' (SAOL) (pronounced ''sail'')--the sound-synthesis component of the newMPEG-4audio compression standard--requires only that the underlying number system be at least as accurate as 32-bit floats. All ints discussed thus far are signed integer formats. C and C++ also support unsigned versions of all int types, and they range from to instead of to , where is the number of bits. Finally, an unsigned char is often used for integers that only range between 0 and 255.

Subsections
One's Complement Fixed-Point Format
Two's Complement Fixed-Point Format
General Formula for Two's-Complement, Integer Fixed-Point Numbers
''Little Endian'' Formula for Two's-Complement, Integer Fixed-Point Numbers

<< Previous page TOC INDEX Next page >>

One's Complement Fixed-Point Format

Sun, 01 Mar 1998 00:00:00 +0000

NOTE: THIS DOCUMENT IS OBSOLETE, PLEASE CHECK THE NEW VERSION: "Mathematics of the Discrete Fourier Transform (DFT), with Audio Applications --- Second Edition", by Julius O. Smith III, W3K Publishing, 2007, ISBN 978-0-9745607-4-8. - Copyright © 2017-09-28 by Julius O. Smith III - Center for Computer Research in Music and Acoustics (CCRMA), Stanford University

<< Previous page TOC INDEX Next page >>
One's Complement Fixed-Point Format
One's Complement is a particular assignment of bit patterns to numbers. For example, in the case of 3-bit binary numbers, we have the assignments shown in Table 4.3.

Table 4.3:Three-bit one's-complement binary fixed-point numbers.
Binary Decimal
000 0
001 1
010 2
011 3
100 -3
101 -2
110 -1
111 -0

In general, -bit numbers are assigned to binary counter values in the ''obvious way'' as integers from 0 to , and then the negative numbers are assigned in reverse order, as shown in the example.
The term ''one's complement'' refers to the fact that negating a number in this format is accomplished by simply complementing the bit pattern (inverting each bit).
Note that there are two representations for zero (all 0s and all 1s). This is inconvenient when testing if a number is equal to zero. For this reason, one's complement is generally not used.
<< Previous page TOC INDEX Next page >>

Two's Complement Fixed-Point Format

Sun, 01 Mar 1998 00:00:00 +0000

NOTE: THIS DOCUMENT IS OBSOLETE, PLEASE CHECK THE NEW VERSION: "Mathematics of the Discrete Fourier Transform (DFT), with Audio Applications --- Second Edition", by Julius O. Smith III, W3K Publishing, 2007, ISBN 978-0-9745607-4-8. - Copyright © 2017-09-28 by Julius O. Smith III - Center for Computer Research in Music and Acoustics (CCRMA), Stanford University

<< Previous page TOC INDEX Next page >>
Two's Complement Fixed-Point Format
In two's complement, numbers are negated by complementing the bit pattern and adding 1, with overflow ignored. From 0 to, positive numbers are assigned to binary values exactly as in one's complement. The remaining assignments (for the negative numbers) can be carried out using the two's complement negation rule. Regenerating the example in this way gives Table 4.4.

General Formula for Two's-Complement, Integer Fixed-Point Numbers

Sun, 01 Mar 1998 00:00:00 +0000

NOTE: THIS DOCUMENT IS OBSOLETE, PLEASE CHECK THE NEW VERSION: "Mathematics of the Discrete Fourier Transform (DFT), with Audio Applications --- Second Edition", by Julius O. Smith III, W3K Publishing, 2007, ISBN 978-0-9745607-4-8. - Copyright © 2017-09-28 by Julius O. Smith III - Center for Computer Research in Music and Acoustics (CCRMA), Stanford University

<< Previous page TOC INDEX Next page >>
General Formula for Two's-Complement, Integer Fixed-Point Numbers
Let denote the (even) number of bits. Then the value of a two's complement integer fixed-point number can be expressed in terms of its bits as

Little Endian Formula

Sun, 01 Mar 1998 00:00:00 +0000

NOTE: THIS DOCUMENT IS OBSOLETE, PLEASE CHECK THE NEW VERSION: "Mathematics of the Discrete Fourier Transform (DFT), with Audio Applications --- Second Edition", by Julius O. Smith III, W3K Publishing, 2007, ISBN 978-0-9745607-4-8. - Copyright © 2017-09-28 by Julius O. Smith III - Center for Computer Research in Music and Acoustics (CCRMA), Stanford University

<< Previous page TOC INDEX Next page >>
''Little Endian'' Formula for Two's-Complement, Integer Fixed-Point Numbers
The formula of the preceding section can be considered ''big endian'' because the bits are labeled ''left to right'' in the word as it is normally visualized. That is, is the most significant bit instead of the least significant bit. The ''little endian'' convention numbers bits in order of their significance instead.
For the ''little-endian'' formula, we simply reverse the bits in the previous formula:

The binary word is now written

The actual bits in the word are visualized the same in both big and little endian labelings. For example, in the three-bit case, a decimal 3 is still a binary and a decimal -4 is still a binary . Only the bit-symbol subscripts are different: vs. .
Note that this representation extends to any ''base'' (currently 2). For example, positive decimal integers can be expanded as

and an -digit integer is written as . This type of ''positional notation'' was a major ''breakthrough invention'' in ancient times that greatly facilitated the handling of large numbers. (Have you ever tried working extensively with Roman numerals?)
<< Previous page TOC INDEX Next page >>

Fractional Binary Fixed-Point Numbers

Sun, 01 Mar 1998 00:00:00 +0000

NOTE: THIS DOCUMENT IS OBSOLETE, PLEASE CHECK THE NEW VERSION: "Mathematics of the Discrete Fourier Transform (DFT), with Audio Applications --- Second Edition", by Julius O. Smith III, W3K Publishing, 2007, ISBN 978-0-9745607-4-8. - Copyright © 2017-09-28 by Julius O. Smith III - Center for Computer Research in Music and Acoustics (CCRMA), Stanford University

<< Previous page TOC INDEX Next page >>
Fractional Binary Fixed-Point Numbers
In ''DSP chips'' (microprocessors explicitly designed for digital signal processing applications), the most commonly used fixed-point format isfractional fixed point, also in two's complement.
Quite simply, fractional fixed-point numbers are obtained from integer fixed-point numbers by dividing them by . Thus, the only difference is a scaling of the assigned numbers. In the case, we have the correspondences shown in Table 4.6.

Table 4.6:Three-bit fractional fixed-point numbers.
Binary Decimal
000 0 (0/4)
001 0.25 (1/4)
010 0.5 (2/4)
011 0.75 (3/4)
100 -1 (-4/4)
101 -0.75 (-3/4)
110 -0.5 (-2/4)
111 -0.25 (-1/4)

<< Previous page TOC INDEX Next page >>

How Many Bits are Enough for Digital Audio?

Sun, 01 Mar 1998 00:00:00 +0000

NOTE: THIS DOCUMENT IS OBSOLETE, PLEASE CHECK THE NEW VERSION: "Mathematics of the Discrete Fourier Transform (DFT), with Audio Applications --- Second Edition", by Julius O. Smith III, W3K Publishing, 2007, ISBN 978-0-9745607-4-8. - Copyright © 2017-09-28 by Julius O. Smith III - Center for Computer Research in Music and Acoustics (CCRMA), Stanford University

<< Previous page TOC INDEX Next page >>
How Many Bits are Enough for Digital Audio?
Armed with the above knowledge, we can visit the question ''how many bits are enough'' for digital audio. Since the threshold of hearing is around 0db SPL, the ''threshold of pain'' is around 120 dB SPL, and each bit in a linear PCM format is worth about 6 dB of dynamic range, we find that we need bits to represent the full dynamic range of audio in a linear fixed-point format. This is a simplestic analysis because it is not quite right to equate the least-significant bit with the threshold of hearing; instead, we would like to adjust the quantization noisefloor to just below the threshold of hearing. Since the threshold of hearing is non-uniform, we would also prefer a shaped quantization noise floor (a feat that can be accomplished using filtered error feedback^4.9 Nevertheless, the simplistic result gives an answer similar to the more careful analysis, and 20 bits is a good number. However, this still does not provide forheadroom needed in a digital recording scenario. We also need both headroom and guard bits on the lower end when we plan to carry out a lot of signal processing operations, especially digital filtering. As an example, a 1024-point FFT (Fast Fourier Transform) can give amplitudes 1024 times the input amplitude (such as in the case of a constant ''dc'' input signal), thus requiring 10 headroom bits. In general, 24 fixed-point bits are pretty reasonable to work with, although you still have to scale very carefully, and 32 bits are preferable.
<< Previous page TOC INDEX Next page >>

When Do We Have to Swap Bytes When Changing Computers?

Sun, 01 Mar 1998 00:00:00 +0000

NOTE: THIS DOCUMENT IS OBSOLETE, PLEASE CHECK THE NEW VERSION: "Mathematics of the Discrete Fourier Transform (DFT), with Audio Applications --- Second Edition", by Julius O. Smith III, W3K Publishing, 2007, ISBN 978-0-9745607-4-8. - Copyright © 2017-09-28 by Julius O. Smith III - Center for Computer Research in Music and Acoustics (CCRMA), Stanford University

<< Previous page TOC INDEX Next page >>
When Do We Have to Swap Bytes When Changing Computers?
When moving a soundfile from one computer to another, such as from a ''PC'' to a ''Mac'' (Intel processor to Motorola processor), the bytes in each sound sample have to be swapped. This is because Motorola processors are big endian (bytes are numbered from ''left to right'' in a multi-byte word) while Intel processors are little endian. Any Mac program that supports a soundfile format native to PCs (such as .wav files) will swap the bytes for you. You only have to worry about swapping the bytes yourself when reading raw binary soundfiles from a foreign computer, or when digging the sound samples out an ''unsupported'' soundfile format yourself.
Since soundfiles typically contain 16 bit samples (not for any good reason, as we now know), there are only two bytes in each audio sample. Let L denote the least-significant byte, and M the most-significant byte. Then a 16-bit word is most naturally written , i.e., the most-significant byte is most naturally written to the left of the least-significant byte, analogous to the way we write binary or decimal integers. This ''most natural'' ordering is used in big-endian processors:

Little-endian machines, on the other hand, store bytes as

These orderings are preserved when the sound data are written to a disk file.
Since a byte (eight bits) is the small addressable unit in all modern day processor families, we don't have to additionally reverse the bitsin each byte. In other words, bits are not given explicit ''addresses'' in memory. They are extracted by means other than simple addressing (such as masking and shifting operations, table look-up, or by means of specialized processor instructions).
Table 4.7 lists popular present-day processors and their ''endianness'':^4.10
Table 4.7:Byte ordering in the major computing platforms.
Processor Family Endian
Pentium (Intel) Little
Alpha (DEC/Compaq) Little
680x0 (Motorola) Big
PowerPC (Motorola & IBM) Big
SPARC (Sun) Big
MIPS (SGI) Big

When compiling C or C++ programs under UNIX, there may be a BYTE_ORDER macro in endian.h or bytesex.h. In other cases, there may be a defined macro __INTEL__, __LITTLE_ENDIAN__, __BIG_ENDIAN__, or the like.
<< Previous page TOC INDEX Next page >>

Logarithmic Number Systems for Digital Audio

Sun, 01 Mar 1998 00:00:00 +0000

NOTE: THIS DOCUMENT IS OBSOLETE, PLEASE CHECK THE NEW VERSION: "Mathematics of the Discrete Fourier Transform (DFT), with Audio Applications --- Second Edition", by Julius O. Smith III, W3K Publishing, 2007, ISBN 978-0-9745607-4-8. - Copyright © 2017-09-28 by Julius O. Smith III - Center for Computer Research in Music and Acoustics (CCRMA), Stanford University

<< Previous page TOC INDEX Next page >>
Logarithmic Number Systems for Digital Audio
Since hearing is approximately logarithmic, it makes sense to represent sound samples in a logarithmic or semi-logarithmic number format.Floating-point numbers in a computer are partially logarithmic (the exponent part), and one can even use an entirely logarithmic fixed-pointnumber system. The -law amplitude-encoding format is linear at small amplitudes and becomes logarithmic at large amplitudes. This section discusses these formats.

Subsections
Floating-Point Numbers
Logarithmic Fixed-Point Numbers
-Law Companding

<< Previous page TOC INDEX Next page >>

Floating-Point Numbers

Sun, 01 Mar 1998 00:00:00 +0000

NOTE: THIS DOCUMENT IS OBSOLETE, PLEASE CHECK THE NEW VERSION: "Mathematics of the Discrete Fourier Transform (DFT), with Audio Applications --- Second Edition", by Julius O. Smith III, W3K Publishing, 2007, ISBN 978-0-9745607-4-8. - Copyright © 2017-09-28 by Julius O. Smith III - Center for Computer Research in Music and Acoustics (CCRMA), Stanford University

<< Previous page TOC INDEX Next page >>
Floating-Point Numbers
Floating-point numbers consist of an ''exponent,'' ''significand'', and ''sign bit''. For a negative number, we may set the sign bit of the floating-point word and negate the number to be encoded, leaving only nonnegative numbers to be considered. Zero is represented by all zeros, so now we need only consider positive numbers.
The basic idea of floating point encoding of a binary number is tonormalize the number by shifting the bits either left or right until the shifted result lies between 1/2 and 1. (A left-shift by one place in a binary word corresponds to multiplying by 2, while a right-shift one place corresponds to dividing by 2.) The number of bit-positions shifted to normalize the number can be recorded as a signed integer. The negative of this integer (i.e., the shift required to recover the original number) is defined as the exponent of the floating-point encoding. The normalized number between 1/2 and 1 is called the significand, so called because it holds all the ''significant bits'' of the number.
Floating point notation is exactly analogous to ''scientific notation'' for decimal numbers, e.g., ; the number of significant digits, 5 in this example, is determined by counting digits in the ''significand'' , while the ''order of magnitude'' is determined by the power of 10 (-9 in this case). In floating-point numbers, the significand is stored in fractional two's-complement binary format, and the exponent is stored as a binary integer.
Since the significand lies in the interval ,^4.11its most significant bit is always a 1, so it is not actually stored in the computer word, giving one more significant bit of precision.
Let's now restate the above a little more precisely. Let denote a number to be encoded in floating-point, and let denote the normalized value obtained by shifting either bits to the right (if ), or bits to the left (if ). Then we have , and . The significand of the floating-point representation for is defined as the binary encoding of .^4.12 It is often the case that requires more bits than are available for exact encoding. Therefore, the significand is typically rounded (or truncated) to the value closest to . Given bits for the significand, the encoding of can be computed by multiplying it by (left-shifting it bits), rounding to the nearest integer (or truncating toward minus infinity--the as implemented by the floor() function), and encoding the -bit result as a binary (signed) integer.
As a final practical note, exponents in floating-point formats may have abias. That is, instead of storing as a binary integer, you may find a binary encoding of where is the bias.^4.13
These days, floating-point formats generally follow the IEEE standards set out for them. A single-precision floating point word is bits (four bytes) long, consisting of sign bit, exponent bits, and significand bits, normally laid out as

where S denotes the sign bit, E an exponent bit, and M a significand bit. Note that in this layout, ordinary integer comparison can be used in the hardware.
A double-precision floating point word is bits (eight bytes) long, consisting of sign bit, exponent bits, and significand bits. In the Intel Pentium processor, there is also an extended precisionformat, used for intermediate results, which is bits (ten bytes) containing sign bit, exponent bits, and significand bits. In Intel processors, the exponent bias is for single-precision floating-point, for double-precision, and for extended-precision. The single and double precision formats have a ''hidden'' significand bit, while the extended precision format does not. Thus, the most significant significand bit is always set in extended precision.
The MPEG-4 audio compression standard (which supports compression using music synthesis algorithms) specifies that the numerical calculations in any MPEG-4 audio decoder should be at least as accurate as 32-bit single-precision floating point.
<< Previous page TOC INDEX Next page >>

Logarithmic Fixed-Point Numbers

Sun, 01 Mar 1998 00:00:00 +0000

NOTE: THIS DOCUMENT IS OBSOLETE, PLEASE CHECK THE NEW VERSION: "Mathematics of the Discrete Fourier Transform (DFT), with Audio Applications --- Second Edition", by Julius O. Smith III, W3K Publishing, 2007, ISBN 978-0-9745607-4-8. - Copyright © 2017-09-28 by Julius O. Smith III - Center for Computer Research in Music and Acoustics (CCRMA), Stanford University

<< Previous page TOC INDEX Next page >>
Logarithmic Fixed-Point Numbers
In some situations it makes sense to use logarithmic fixed-point. This number format can be regarded as a floating-point format consisting of an exponent and no explicit significand. However, the exponent is not interpreted as an integer as it is in floating point. Instead, it has a fractional part which is a true mantissa. (The integer part is then of course the ''characteristic'' of the logarithm.) In other words, a logarithmic fixed-point number is a binary encoding of the log-base-2 of the signal-sample magnitude. The sign bit is of course separate.
An example 16-bit logarithmic fixed-point number format suitable for digital audio consists of one sign bit, a 5-bit characteristic, and a 10-bit mantissa:

The 5-bit characteristic gives a dynamic range of about 6 dB = 192 dB. This is an excellent dynamic range for digital audio. (While 120 dB would seem to be enough for audio, consider that when digitally modeling a brass musical instrument, say, the internal air pressure near the ''virtual mouthpiece'' can be far higher than what actually reaches the ears in the audience.)
A nice property of logarithmic fixed-point numbers is that multiplies simply become additions and divisions become subtractions. The hard elementary operation are now addition and subtraction, and these are normally done using table lookups to keep them simple.
One ''catch'' when working with logarithmic fixed-point numbers is that you can't let ''dc'' build up. A wandering dc component will cause the quantization to be coarse even for low-level ''ac'' signals. It's a good idea to make sure dc is always filtered out in logarithmic fixed-point.
<< Previous page TOC INDEX Next page >>

Law Companding

Sun, 01 Mar 1998 00:00:00 +0000

NOTE: THIS DOCUMENT IS OBSOLETE, PLEASE CHECK THE NEW VERSION: "Mathematics of the Discrete Fourier Transform (DFT), with Audio Applications --- Second Edition", by Julius O. Smith III, W3K Publishing, 2007, ISBN 978-0-9745607-4-8. - Copyright © 2017-09-28 by Julius O. Smith III - Center for Computer Research in Music and Acoustics (CCRMA), Stanford University

<< Previous page TOC INDEX Next page >>
-Law Companding
A companding operation compresses dynamic range on encode andexpands dynamic range on decode. In digital telephone networks and voice modems (currently in use everywhere), standard CODEC^4.14 chips are used in which audio is digitized in a simple 8-bit -law format (or simply ''mu-law'').
Given an input sample represented in some internal format, such as ashort, it is converted to 8-bit mu-law format by the formula [12]

where is a quantizer which produces a kind of logarithmic fixed-point number with a 3-bit characteristic and a 4-bit mantissa, using a small table lookup for the mantissa.
As we all know from talking on the telephone, mu-law sounds really quite good for voice, at least as far as intelligibility is concerned. However, because the telephone bandwidth is only around 3 kHz (nominally 200-3200 Hz), there is very little ''bass'' and no ''highs'' in the spectrum above 4 kHz. This works out fine for intelligibility of voice because the first three formants (envelope peaks) in typical speech spectra occur in this range, and also because the difference in spectral shape (particularly at high frequencies) between consonants such as ''sss'', ''shshsh'', ''fff'', ''ththth'', etc., are sufficiently preserved in this range. As a result of the narrow bandwidth provided for speech, it is sampled at only 8 kHz in standard CODEC chips.
For ''wideband audio'', we like to see sampling rates at least as high as 44.1 kHz, and the latest systems are moving to 96 kHz (mainly becauseoversampling simplifies signal processing requirements in various areas, not because we can actually hear anything above 20 kHz). In addition, we like the low end to extend at least down to 20 Hz or so. (The lowest note on a normally tuned bass guitar is E1 = 41.2 Hz. The lowest note on a grand piano is A0 = 27.5 Hz.)
<< Previous page TOC INDEX Next page >>

Appendix A: Round-Off Error Variance

Sun, 01 Mar 1998 00:00:00 +0000

NOTE: THIS DOCUMENT IS OBSOLETE, PLEASE CHECK THE NEW VERSION: "Mathematics of the Discrete Fourier Transform (DFT), with Audio Applications --- Second Edition", by Julius O. Smith III, W3K Publishing, 2007, ISBN 978-0-9745607-4-8. - Copyright © 2017-09-28 by Julius O. Smith III - Center for Computer Research in Music and Acoustics (CCRMA), Stanford University

<< Previous page TOC INDEX Next page >>
Appendix A: Round-Off Error Variance
This section shows how to derive that the noise power of quantization error is , where is the quantization step size.
Each round-off error in quantization noise is modeled as a uniform random variable between and . It therefore has the probability density function (pdf)

Thus, the probability that a given roundoff error lies in the interval is given by

assuming of course that and lie in the allowed range . We might loosely refer to as a probability distribution, but technically it is a probability density function, and to obtain probabilities, we have to integrate over one or more intervals, as above. We use probability distributions for variables which take on discrete values (such as dice), and we use probabilitydensities for variables which take on continuous values (such as round-off errors).
The mean of a random variable is defined as

In our case, the mean is zero because we are assuming the use ofrounding (as opposed to truncation, etc.).
The mean of a signal is the same thing as theexpected value of , which we write as . In general, the expected value of any function of a random variable is given by

Since the quantization-noise signal is modeled as a series of independent, identically distributed (iid) random variables, we canestimate the mean by averaging the signal over time. Such an estimate is called a sample mean.
Probability distributions are often be characterized by theirmoments. The th moment of the pdf is defined as

Thus, the mean is the first moment of the pdf. The second moment is simply the expected value of the random variable squared, i.e., .
The variance of a random variable is defined as thesecond central moment of the pdf:

''Central'' just means that the moment is evaluated after subtracting out the mean, that is, looking at instead of . In the case of round-off errors, the mean is zero, so subtracting out the mean has no effect. Plugging in the constant pdf for our random variable which we assume is uniformly distributed on , we obtain the variance

Note that the variance of can be estimated by averaging over time, that is, by computing the mean square. Such an estimate is called the sample variance. For sampled physical processes, the sample variance is proportional to the average power in the signal. Finally, the square root of the sample variance (the rms level) is sometimes called the standard deviation of the signal, but this term is only precise when the random variable has a Gaussian pdf.
EE 278 is the starting course on statistical signal processing at Stanford if you are interested in this and related topics. A good text book on the subject is [13].
<< Previous page TOC INDEX Next page >>

Appendix B: Electrical Engineering 101

Sun, 01 Mar 1998 00:00:00 +0000

NOTE: THIS DOCUMENT IS OBSOLETE, PLEASE CHECK THE NEW VERSION: "Mathematics of the Discrete Fourier Transform (DFT), with Audio Applications --- Second Edition", by Julius O. Smith III, W3K Publishing, 2007, ISBN 978-0-9745607-4-8. - Copyright © 2017-09-28 by Julius O. Smith III - Center for Computer Research in Music and Acoustics (CCRMA), Stanford University

<< Previous page TOC INDEX Next page >>
Appendix B: Electrical Engineering 101
The state of an ideal resistor is completely specified by the voltage across it (call it volts) and the current passing through it ( Amperes, or simply ''amps''). The ratio of voltage to current gives the value of the resistor ( resistance in Ohms). The fundamental relation between voltage and current in a resistor is calledOhm's Law:

where we have indicated also that the voltage and current may vary with time (while the resistor value normally does not).
The electrical power in watts dissipated by a resistor R is given by

where is the voltage and is the current. Thus, volts times amps gives watts. Also, volts squared over ohms equals watts, and so on.
<< Previous page TOC INDEX Next page >>

Sinusoids and Exponentials

Sun, 01 Mar 1998 00:00:00 +0000

NOTE: THIS DOCUMENT IS OBSOLETE, PLEASE CHECK THE NEW VERSION: "Mathematics of the Discrete Fourier Transform (DFT), with Audio Applications --- Second Edition", by Julius O. Smith III, W3K Publishing, 2007, ISBN 978-0-9745607-4-8. - Copyright © 2017-09-28 by Julius O. Smith III - Center for Computer Research in Music and Acoustics (CCRMA), Stanford University

<< Previous page TOC INDEX Next page >>
Sinusoids and Exponentials
This chapter provides an introduction to sinusoids, exponentials, complex sinusoids, , in-phase and quadrature sinusoidal components, theanalytic signal, positive and negative frequencies, constructive anddestructive interference, invariance of sinusoidal frequency in linear time-invariant systems, circular motion as the vector sum of in-phase and quadrature sinusoidal motions, sampled sinusoids, generating sampled sinusoids from powers of , and plot examples using Mathematica.

Subsections
Sinusoids
Example Sinusoids
Why Sinusoids are Important
In-Phase and Quadrature Sinusoidal Components
Sinusoids at the Same Frequency
Constructive and Destructive Interference of Sinusoids
Exponentials
Why Exponentials are Important
Audio Decay Time ()
Complex Sinusoids
Circular Motion
Projection of Circular Motion
Positive and Negative Frequencies
The Analytic Signal and Hilbert Transform Filters
Generalized Complex Sinusoids
Sampled Sinusoids
Powers of
Why (Generalized) Complex Sinusoids are Important
Comparing Analog and Digital Complex Planes
Mathematica for Selected Plots
Acknowledgement

<< Previous page TOC INDEX Next page >>

Sinusoids

Sun, 01 Mar 1998 00:00:00 +0000

NOTE: THIS DOCUMENT IS OBSOLETE, PLEASE CHECK THE NEW VERSION: "Mathematics of the Discrete Fourier Transform (DFT), with Audio Applications --- Second Edition", by Julius O. Smith III, W3K Publishing, 2007, ISBN 978-0-9745607-4-8. - Copyright © 2017-09-28 by Julius O. Smith III - Center for Computer Research in Music and Acoustics (CCRMA), Stanford University

<< Previous page TOC INDEX Next page >>
Sinusoids
A sinusoid is any function of time having the following form:

where all variables are real numbers, and

The term ''peak amplitude'' is often shortened to ''amplitude,'' e.g., ''the amplitude of the sound was measured to be 5 Pascals.'' Strictly speaking, however, the ''amplitude'' of a signal is its instantaneous value at any time . The peak amplitude satisfies . The ''instantaneous magnitude'' or simply ''magnitude'' of a signal is given by , and the peak magnitude is the same thing as the peak amplitude.
Note that Hz is an abbreviation for Hertz which physically means ''cycles per second.'' You might also encounter the older (and clearer) notation ''c.p.s.'' for cycles per second.
Since is periodic with period , the phase is indistinguishable from the phase . As a result, we may restrict the range of to any length interval. When needed, we will choose

i.e., . You may also encounter the convention .

Subsections
Example Sinusoids
Why Sinusoids are Important
In-Phase and Quadrature Sinusoidal Components
Sinusoids at the Same Frequency
Constructive and Destructive Interference of Sinusoids

<< Previous page TOC INDEX Next page >>

Example Sinusoids

Sun, 01 Mar 1998 00:00:00 +0000

NOTE: THIS DOCUMENT IS OBSOLETE, PLEASE CHECK THE NEW VERSION: "Mathematics of the Discrete Fourier Transform (DFT), with Audio Applications --- Second Edition", by Julius O. Smith III, W3K Publishing, 2007, ISBN 978-0-9745607-4-8. - Copyright © 2017-09-28 by Julius O. Smith III - Center for Computer Research in Music and Acoustics (CCRMA), Stanford University

<< Previous page TOC INDEX Next page >>
Example Sinusoids
Figure 5.1 plots the sinusoid , for , , , and. Study the plot to make sure you understand the effect of changing each parameter (amplitude, frequency, phase), and also note the definitions of ''peak-to-peak amplitude'' and ''zero crossings.''
Figure 5.1:An example sinusoid.

The Mathematica code for generating this figure is listed in §5.4.
A ''tuning fork'' vibrates approximately sinusoidally. An ''A-440'' tuning fork oscillates at cycles per second. As a result, a tone recorded from an ideal A-440 tuning fork is a sinusoid at Hz. The amplitude determines how loud it is and depends on how hard we strike the tuning fork. The phase is set by exactly when we strike the tuning fork (and on our choice of when time 0 is). If we record an A-440 tuning fork on an analog tape recorder, the electrical signal recorded on tape is of the form

As another example, the sinusoid at amplitude and phase (90 degrees) is simply

Thus, is a sinusoid at phase 90-degrees, while is a sinusoid at zero phase. Note, however, that we could just as well have defined to be the zero-phase sinusoid rather than . It really doesn't matter, except to be consistent in any given usage. The concept of a ''sinusoidal signal'' is simply that it is equal to a sine or cosine function at some amplitude, frequency, and phase. It does not matter whether we choose or in the ''official'' definition of a sinusoid. You may encounter both definitions. Using is nice since ''sinusoid'' in a sense generalizes . However, using is nicer when defining a sinusoid to be the real part of a complex sinusoid (which we'll talk about later).
<< Previous page TOC INDEX Next page >>

Why Sinusoids are Important

Sun, 01 Mar 1998 00:00:00 +0000

NOTE: THIS DOCUMENT IS OBSOLETE, PLEASE CHECK THE NEW VERSION: "Mathematics of the Discrete Fourier Transform (DFT), with Audio Applications --- Second Edition", by Julius O. Smith III, W3K Publishing, 2007, ISBN 978-0-9745607-4-8. - Copyright © 2017-09-28 by Julius O. Smith III - Center for Computer Research in Music and Acoustics (CCRMA), Stanford University

<< Previous page TOC INDEX Next page >>
Why Sinusoids are Important
Sinusoids are fundamental in a variety of ways.
One reason for the importance of sinusoids is that they are fundamental in physics. Anything that resonates or oscillates produces quasi-sinusoidal motion. See simple harmonic motion in any freshman physics text for an introduction to this topic.
Another reason sinusoids are important is that they are eigenfunctionsof linear systems (which we'll say more about later). This means that they are important for the analysis of filters such as reverberators, equalizers, certain (but not all) ''effects'', etc.
Perhaps most importantly, from the point of view of computer music research, is that the human ear is a kind of spectrum analyzer. That is, the chochlea of the inner ear physically splits sound into its (near) sinusoidal components. This is accomplished by the basilar membrane in the inner ear: a sound wave injected at the oval window(which is connected via the bones of the middle ear to the ear drum), travels along the basilar membrane inside the coiled cochlea. The membrane starts out thick and stiff, and gradually becomes thinner and more compliant toward its apex (the helicotrema). A stiff membrane has a high resonance frequency while a thin, compliant membrane has a low resonance frequency (assuming comparable mass density, or at least less of a difference in mass than in compliance). Thus, as the sound wave travels, each frequency in the sound resonates at a particular place along the basilar membrane. The highest frequencies resonate right at the entrance, while the lowest frequencies travel the farthest and resonate near the helicotrema. The membrane resonance effectively ''shorts out'' the signal energy at that frequency, and it travels no further. Along the basilar membrane there are hair cells which ''feel'' the resonant vibration and transmit an increased firing rate along the auditory nerve to the brain. Thus, the ear is very literally a Fourier analyzer for sound, albeit nonlinear and using ''analysis'' parameters that are difficult to match exactly. Nevertheless, by looking at spectra (which display the amount of each sinusoidal frequency present in a sound), we are looking at a representation much more like what the brain receives when we hear.
<< Previous page TOC INDEX Next page >>

In-Phase and Quadrature Sinusoidal Components

Sun, 01 Mar 1998 00:00:00 +0000

NOTE: THIS DOCUMENT IS OBSOLETE, PLEASE CHECK THE NEW VERSION: "Mathematics of the Discrete Fourier Transform (DFT), with Audio Applications --- Second Edition", by Julius O. Smith III, W3K Publishing, 2007, ISBN 978-0-9745607-4-8. - Copyright © 2017-09-28 by Julius O. Smith III - Center for Computer Research in Music and Acoustics (CCRMA), Stanford University

<< Previous page TOC INDEX Next page >>
In-Phase and Quadrature Sinusoidal Components
From the trig identity , we have

From this we may conclude that every sinusoid can be expressed as the sum of a sine function (phase zero) and a cosine function (phase ). If the sine part is called the ''in-phase'' component, the cosine part can be called the ''phase-quadrature'' component. In general, ''phase quadrature'' means ''90 degrees out of phase,'' i.e., a relative phase shift of .
It is also the case that every sum of an in-phase and quadrature component can be expressed as a single sinusoid at some amplitude and phase. The proof is obtained by working the previous derivation backwards.
Figure 5.2 illustrates in-phase and quadrature components overlaid. Note that they only differ by a relative degree phase shift. (See §5.4 for the Mathematica code for this figure.)
Figure 5.2:In-phase and quadrature sinusoidal components.

<< Previous page TOC INDEX Next page >>

Sinusoids at the Same Frequency

Sun, 01 Mar 1998 00:00:00 +0000

NOTE: THIS DOCUMENT IS OBSOLETE, PLEASE CHECK THE NEW VERSION: "Mathematics of the Discrete Fourier Transform (DFT), with Audio Applications --- Second Edition", by Julius O. Smith III, W3K Publishing, 2007, ISBN 978-0-9745607-4-8. - Copyright © 2017-09-28 by Julius O. Smith III - Center for Computer Research in Music and Acoustics (CCRMA), Stanford University

<< Previous page TOC INDEX Next page >>
Sinusoids at the Same Frequency
An important property of sinusoids at a particular frequency is that they are closed with respect to addition. In other words, if you take a sinsusoid, make many copies of it, scale them all by different gains, delay them all by different amounts, and add them up, you always get a sinusoid at the same original frequency. This is a nontrivial property. It obviously holds for any constant signal (which we may regard as a sinusoid at frequency ), but it is not obvious for (see Fig. 5.2 and think about the sum of the two waveforms shown being precisely a sinusoid).
Since every linear, time-invariant (LTI^5.1) system (filter) operates by copying, scaling, delaying, and summing its input signal(s) to create its output signal(s), it follows that when a sinusoid at a particular frequency is input to an LTI system, a sinusoid at that same frequency always appears at the output. Only the amplitude and phase can be changed by the system. We say that sinusoids are eigenfunctionsof LTI systems. Conversely, if the system is nonlinear or time-varying, new frequencies are created at the system output.
To prove the important invariance property of sinusoids, we may simply express all scaled and delayed sinusoids in the ''mix'' in terms of their in-phase and quadrature components and then add them up. For example, consider the case of two sinusoids arbitrarily scaled by gains and arbitrarily delayed by time-delays :

Focusing on the first term, we have

We similarly compute

and add to obtain

This result, consisting of one in-phase and one quadrature signal component, can now be converted to a single sinusoid at some amplitude and phase (and frequency ), as discussed above.
<< Previous page TOC INDEX Next page >>

Constructive and Destructive Interference of Sinusoids

Sun, 01 Mar 1998 00:00:00 +0000

NOTE: THIS DOCUMENT IS OBSOLETE, PLEASE CHECK THE NEW VERSION: "Mathematics of the Discrete Fourier Transform (DFT), with Audio Applications --- Second Edition", by Julius O. Smith III, W3K Publishing, 2007, ISBN 978-0-9745607-4-8. - Copyright © 2017-09-28 by Julius O. Smith III - Center for Computer Research in Music and Acoustics (CCRMA), Stanford University

<< Previous page TOC INDEX Next page >>
Constructive and Destructive Interference of Sinusoids
Sinusoidal signals are analogous to monochromatic laser light. You might have seen ''speckle'' associated with laser light, caused by destructive inteference of multiple reflections of the light beam. In a room, the same thing happens with sinusoidal sound. For example, play a simple sinusoidal tone (e.g., ''A-440'' which is a sinusoid at frequency Hz) and walk around the room with one ear plugged. If the room is reverberant you should be able find places where the sound goes completely away due to destructive interference. In between such places (which we call ''nodes'' in the soundfield), there are ''antinodes'' at which the sound is louder by 6 dB (amplitude doubled) due to constructive interference. In a diffuse reverberant soundfield, the distance between nodes is on the order of a wavelength (the ''correlation distance'' within the random soundfield).
The way reverberation produces nodes and antinodes for sinusoids in a room is illustrated by the simple comb filter.^5.2 There is also shown in Fig. 5.3. A unit-amplitude sinusoid is present at the input, and the output must also be sinusoidal, since the comb filter is linear and time-invariant. The feedforward path of the comb filter has a gain of , and the delay is one period in one case and half a period in the other. With the delay set to one period, the unit amplitude sinusoid coming out of the delay line constructively interferes with the amplitude sinusoid from the feed-forward path, and the output amplitude is therefore . In the other case, with the delay set to half period, the unit amplitude sinusoid coming out of the delay line destructively interferes with the amplitude sinusoid from the feed-forward path, and the output amplitude therefore drops to .

Figure 5.3:A comb filter with a sinusoidal input.
comb-filter
Consider a fixed delay of seconds for the delay line. Constructive interference happens at all frequencies for which an exact integernumber of periods fits in the delay line, i.e., , or, for . On the other hand, destructive interference happens at all frequencies for which number of periods in the delay line is an integer plus a half, i.e., etc., or, , for . It is quick to verify that frequencies of constructive interference alternate with frequencies of destructive interference, and therefore the amplitude response of the comb filter (a plot of gain versus frequency) looks as shown in Fig. 5.4.

Figure 5.4:Comb filter amplitude response when the delay is sec.
The ampitude response of a comb filter has a ''comb'' like shape, hence the name.^5.3 Note that if the feedforward gain is increased from to , the comb-filter gain ranges between (complete cancellation) and . Negating the feedforward gain inverts the gain curve, placing a minumum at dc^5.4 instead of a peak.
<< Previous page TOC INDEX Next page >>

Exponentials

Sun, 01 Mar 1998 00:00:00 +0000

NOTE: THIS DOCUMENT IS OBSOLETE, PLEASE CHECK THE NEW VERSION: "Mathematics of the Discrete Fourier Transform (DFT), with Audio Applications --- Second Edition", by Julius O. Smith III, W3K Publishing, 2007, ISBN 978-0-9745607-4-8. - Copyright © 2017-09-28 by Julius O. Smith III - Center for Computer Research in Music and Acoustics (CCRMA), Stanford University

<< Previous page TOC INDEX Next page >>
Exponentials
The canonical form of an exponential function, as typically used in signal processing, is

where is called the time constant of the exponential. is the peak amplitude, as before. The time constant is the time it takes to decay by , i.e.,

A normalized exponential decay is depicted in Fig. 5.5.

Figure 5.5:The decaying exponential.

Subsections
Why Exponentials are Important
Audio Decay Time ()

<< Previous page TOC INDEX Next page >>

Why Exponentials are Important

Sun, 01 Mar 1998 00:00:00 +0000

NOTE: THIS DOCUMENT IS OBSOLETE, PLEASE CHECK THE NEW VERSION: "Mathematics of the Discrete Fourier Transform (DFT), with Audio Applications --- Second Edition", by Julius O. Smith III, W3K Publishing, 2007, ISBN 978-0-9745607-4-8. - Copyright © 2017-09-28 by Julius O. Smith III - Center for Computer Research in Music and Acoustics (CCRMA), Stanford University

<< Previous page TOC INDEX Next page >>
Why Exponentials are Important
Exponential decay occurs naturally when a quantity is decaying at a rate which is proportional to how much is left. In nature, all linear resonators, such as musical instrument strings and woodwind bores, exhibit exponential decay in their response to a momentary excitation. As another example, reverberant energy in a room decays exponentially after the direct sound stops. Essentially all undriven oscillations decay exponentially (provided they are linear and time-invariant). Undriven means there is no ongoing source of driving energy. Examples of undriven oscillations include the vibrations of a tuning fork, struck or plucked strings, a marimba or xylophone bar, and so on. Examples of driven oscillations include horns, woodwinds, bowed strings, and voice. Driven oscillations must be periodic while undriven oscillations normally are not, except in idealized cases.

Audio Decay Time

Sun, 01 Mar 1998 00:00:00 +0000

NOTE: THIS DOCUMENT IS OBSOLETE, PLEASE CHECK THE NEW VERSION: "Mathematics of the Discrete Fourier Transform (DFT), with Audio Applications --- Second Edition", by Julius O. Smith III, W3K Publishing, 2007, ISBN 978-0-9745607-4-8. - Copyright © 2017-09-28 by Julius O. Smith III - Center for Computer Research in Music and Acoustics (CCRMA), Stanford University

<< Previous page TOC INDEX Next page >>
Audio Decay Time ()
In audio, a decay by is too small to be considered a practical ''decay time.'' In architectural acoustics (which includes the design of concert halls), a more commonly used measure of decay is '''', which is defined as the time to decay by dB.^5.5 That is, is obtained by solving the equation

Using the definition of the exponential , we find

Thus, is about seven time constants. See where is marked on Fig. 5.5 compared with .
<< Previous page TOC INDEX Next page >>

Complex Sinusoids

Sun, 01 Mar 1998 00:00:00 +0000

NOTE: THIS DOCUMENT IS OBSOLETE, PLEASE CHECK THE NEW VERSION: "Mathematics of the Discrete Fourier Transform (DFT), with Audio Applications --- Second Edition", by Julius O. Smith III, W3K Publishing, 2007, ISBN 978-0-9745607-4-8. - Copyright © 2017-09-28 by Julius O. Smith III - Center for Computer Research in Music and Acoustics (CCRMA), Stanford University

<< Previous page TOC INDEX Next page >>
Complex Sinusoids
Recall Euler's Identity,

Multiplying this equation by and setting , we obtain the definition of the complex sinusoid:

Thus, a complex sinusoid consists of an in-phase component for its real part, and a phase-quadrature component for its imaginary part. Since , we have

That is, the complex sinusoid is constant modulus. (The symbol '''' means ''identically equal to,'' i.e., for all .) The phase of the complex sinusoid is

The derivative of the phase of the complex sinusoid gives itsfrequency

Subsections
Circular Motion
Projection of Circular Motion
Positive and Negative Frequencies
The Analytic Signal and Hilbert Transform Filters
Generalized Complex Sinusoids
Sampled Sinusoids
Powers of
Why (Generalized) Complex Sinusoids are Important
Comparing Analog and Digital Complex Planes

<< Previous page TOC INDEX Next page >>

Circular Motion

Sun, 01 Mar 1998 00:00:00 +0000

NOTE: THIS DOCUMENT IS OBSOLETE, PLEASE CHECK THE NEW VERSION: "Mathematics of the Discrete Fourier Transform (DFT), with Audio Applications --- Second Edition", by Julius O. Smith III, W3K Publishing, 2007, ISBN 978-0-9745607-4-8. - Copyright © 2017-09-28 by Julius O. Smith III - Center for Computer Research in Music and Acoustics (CCRMA), Stanford University

<< Previous page TOC INDEX Next page >>
Circular Motion
Since the modulus of the complex sinusoid is constant, it must lie on acircle in the complex plane. For example,

traces out counter-clockwise circular motion along the unit circle in the complex plane, while

is clockwise circular motion.
We call a complex sinusoid of the form , where , apositive-frequency sinusoid. Similarly, we define a complex sinusoid of the form , with , to be a negative-frequency sinusoid. Note that a positive- or negative-frequency sinusoid is necessarily complex.
REMARK: Add figure: circular motion (animation for web version)
<< Previous page TOC INDEX Next page >>

Projection of Circular Motion

Sun, 01 Mar 1998 00:00:00 +0000

NOTE: THIS DOCUMENT IS OBSOLETE, PLEASE CHECK THE NEW VERSION: "Mathematics of the Discrete Fourier Transform (DFT), with Audio Applications --- Second Edition", by Julius O. Smith III, W3K Publishing, 2007, ISBN 978-0-9745607-4-8. - Copyright © 2017-09-28 by Julius O. Smith III - Center for Computer Research in Music and Acoustics (CCRMA), Stanford University

<< Previous page TOC INDEX Next page >>
Projection of Circular Motion
We have

Interpreting this in the complex plane tells us that sinusoidal motion is the projection of circular motion onto any straight line. Thus, the sinusoidal motion is the projection of the circular motion onto the (real-part) axis, while is the projection of onto the (imaginary-part) axis.
Figure 5.7 shows a plot of a complex sinusoid versus time, along with its projections onto coordinate planes. This is a 3D plot showing the-plane versus time. The axes are the real part, imaginary part, and time. (Or we could have used magnitude and phase versus time.)

Figure 5.7:A complex sinusoid and its projections.
Note that the left projection (onto the plane) is a circle, the lower projection (real-part vs. time) is a cosine, and the upper projection (imaginary-part vs. time) is a sine. A point traversing the plot projects to uniform circular motion in the plane, and sinusoidal motion on the two other planes.
<< Previous page TOC INDEX Next page >>

Positive and Negative Frequencies

Sun, 01 Mar 1998 00:00:00 +0000

NOTE: THIS DOCUMENT IS OBSOLETE, PLEASE CHECK THE NEW VERSION: "Mathematics of the Discrete Fourier Transform (DFT), with Audio Applications --- Second Edition", by Julius O. Smith III, W3K Publishing, 2007, ISBN 978-0-9745607-4-8. - Copyright © 2017-09-28 by Julius O. Smith III - Center for Computer Research in Music and Acoustics (CCRMA), Stanford University

<< Previous page TOC INDEX Next page >>
Positive and Negative Frequencies
Earlier, we used Euler's Identity to show

Setting , we see that both sine and cosine (and hence all real sinusoids) consist of a sum of equal and opposite circular motion. Phrased differently, every real sinusoid consists of an equal contribution of positive and negative frequency components. This is true of all real signals. When we get to spectrum analysis, we will find that every real signal contains equal amounts of positive and negative frequencies, i.e., if denotes the spectrum of the real signal, we will always have .
Note that, mathematically, the complex sinusoid is really simpler and more basic than the real sinusoid because consists of one frequency while really consists of two frequencies and . We may think of a real sinusoid as being the sum of a positive-frequency and a negative-frequency complex sinusoid, so in that sense real sinusoids are ''twice as complicated'' as complex sinusoids. Complex sinusoids are also nicer because they have a constant modulus. ''Amplitude envelope detectors'' for complex sinusoids are trivial: just compute the square root of the sum of the squares of the real and imaginary parts to obtain the instantaneous peak amplitude at any time. Frequency demodulators are similarly trivial: just differentiate the phase of the complex sinusoid to obtain its instantaneous frequency. It should therefore come as no surprise that signal processing engineers often prefer to convert real sinusoids into complex sinusoids before processing them further.
<< Previous page TOC INDEX Next page >>

The Analytic Signal and Hilbert Transform Filters

Sun, 01 Mar 1998 00:00:00 +0000

NOTE: THIS DOCUMENT IS OBSOLETE, PLEASE CHECK THE NEW VERSION: "Mathematics of the Discrete Fourier Transform (DFT), with Audio Applications --- Second Edition", by Julius O. Smith III, W3K Publishing, 2007, ISBN 978-0-9745607-4-8. - Copyright © 2017-09-28 by Julius O. Smith III - Center for Computer Research in Music and Acoustics (CCRMA), Stanford University

<< Previous page TOC INDEX Next page >>
The Analytic Signal and Hilbert Transform Filters
A signal which has no negative-frequency components is called ananalytic signal.^5.6 Therefore, in continuous time, every analytic signal can be represented as

where is the complex coefficient (setting the amplitude and phase) of the positive-freqency complex sinusoid atfrequency .
Any sinusoid in real life may be converted to a positive-frequency complex sinusoid by simply generating a phase-quadrature component to serve as the ''imaginary part'':

The phase-quadrature component can be generated from the in-phase componentby a simple quarter-cycle time shift.^5.7
For more complicated signals which are expressible as a sum of many sinusoids, a filter can be constructed which shifts each sinusoidal component by a quarter cycle. This is called a Hilbert transform filter. Let denote the output at time of theHilbert-transform filter applied to the signal . Ideally, this filter has magnitude at all frequencies and introduces a phase shift of at each positive frequency and at each negative frequency. When a real signal and its Hilbert transform are used to form a new complex signal , the signal is the (complex) analytic signal corresponding to the real signal . In other words, for any real signal , the corresponding analytic signal has the property that all ''negative frequencies'' of have been ''filtered out.''
To see how this works, recall that these phase shifts can be impressed on a complex sinusoid by multiplying it by . Consider the positive and negative frequency components at the particular frequency:

Now let's apply a degrees phase shift to the positive-frequency component, and a degrees phase shift to the negative-frequency component:

Adding them together gives

and sure enough, the negative frequency component is filtered out. (There is also a gain of 2 at positive frequencies which we can remove by defining the Hilbert transform filter to have magnitude 1/2 at all frequencies.)
For a concrete example, let's start with the real sinusoid

Applying the ideal phase shifts, the Hilbert transform is

The analytic signal is then

by Euler's identity. Thus, in the sum , the negative-frequency components of and cancel out in the sum, leaving only the positive-frequency component. This happens for any real signal , not just for sinusoids as in our example.

Figure 5.8:Creation of the analytic signal from the real sinusoid and the derived phase-quadrature sinusoid , viewed in the frequency domain. a) Spectrum of . b) Spectrum of . c) Spectrum of . d) Spectrum of .
Figure 5.8 illustrates what is going on in the frequency domain. While we haven't ''had'' Fourier analysis yet, it should come as no surprise that the spectrum of a complex sinusoid will consist of a single ''spike'' at the frequency and zero at all other frequencies. (Just follow things intuitively for now, and revisit Fig. 5.8 after we've developed the Fourier theorems.) From the identity , we see that the spectrum contains unit-amplitude ''spikes'' at and . Similarly, the identity says that we have an amplitude spike at and an amplitude spike at . Multiplying by results in which is a unit-amplitude ''up spike'' at and a unit ''down spike'' at . Finally, adding together the first and third plots, corresponding to , we see that the two up-spikesadd in phase to give an amplitude 2 up-spike (which is ), and the negative-frequency up-spike in the cosine iscanceled by the down-spike in times sine at frequency. This sequence of operations illustrates how the negative-frequency component gets filtered outby the addition of and .
As a final example (and application), let , where is a slowly varying amplitude envelope (slow compared with). This is an example of amplitude modulation applied to a sinusoid at ''carrier frequency'' (which is where you tune your AM radio). The Hilbert transform is almost exactly ^5.8, and the analytic signal is . Note that AM demodulation^5.9 is now nothing more than the absolute value. I.e., . Due to this simplicity, Hilbert transforms are sometimes used in makingamplitude envelope followers for narrowband signals (i.e., signals with all energy centered about a single ''carrier'' frequency). AM demodulation is one application of a narrowband envelope follower.
<< Previous page TOC INDEX Next page >>

Generalized Complex Sinusoids

Sun, 01 Mar 1998 00:00:00 +0000

NOTE: THIS DOCUMENT IS OBSOLETE, PLEASE CHECK THE NEW VERSION: "Mathematics of the Discrete Fourier Transform (DFT), with Audio Applications --- Second Edition", by Julius O. Smith III, W3K Publishing, 2007, ISBN 978-0-9745607-4-8. - Copyright © 2017-09-28 by Julius O. Smith III - Center for Computer Research in Music and Acoustics (CCRMA), Stanford University

<< Previous page TOC INDEX Next page >>
Generalized Complex Sinusoids
We have defined sinusoids and extended the definition to include complex sinusoids. We now extend one more step by allowing for exponential amplitude envelopes:

where and are complex, and further defined as

When , we obtain

which is the complex sinusoid at amplitude , radian frequency , and phase .
More generally, we have

Defining , we see that the generalized complex sinusoid is just the complex sinusoid we had before with an exponential envelope:

<< Previous page TOC INDEX Next page >>

Sampled Sinusoids

Sun, 01 Mar 1998 00:00:00 +0000

NOTE: THIS DOCUMENT IS OBSOLETE, PLEASE CHECK THE NEW VERSION: "Mathematics of the Discrete Fourier Transform (DFT), with Audio Applications --- Second Edition", by Julius O. Smith III, W3K Publishing, 2007, ISBN 978-0-9745607-4-8. - Copyright © 2017-09-28 by Julius O. Smith III - Center for Computer Research in Music and Acoustics (CCRMA), Stanford University

<< Previous page TOC INDEX Next page >>
Sampled Sinusoids
In discrete-time audio processing, such as we must do on a computer, we work with samples of continuous-time signals. Let denote thesampling rate in Hz. For audio, we typically have kHz, since the audio band nominally extends to kHz. For compact discs (CDs), kHz (or very close to that--I once saw Sony device using a sampling rate of Hz), while for digital audio tape (DAT), kHz.
Let denote the sampling period in seconds. Then to convert from continuous to discrete time, we replace by , where is an integer interpreted as the sample number.
The sampled generalized complex sinusoid (which includes all other cases) is then

<< Previous page TOC INDEX Next page >>

Powers of

Sun, 01 Mar 1998 00:00:00 +0000

NOTE: THIS DOCUMENT IS OBSOLETE, PLEASE CHECK THE NEW VERSION: "Mathematics of the Discrete Fourier Transform (DFT), with Audio Applications --- Second Edition", by Julius O. Smith III, W3K Publishing, 2007, ISBN 978-0-9745607-4-8. - Copyright © 2017-09-28 by Julius O. Smith III - Center for Computer Research in Music and Acoustics (CCRMA), Stanford University

<< Previous page TOC INDEX Next page >>
Powers of
Choose any two complex numbers and , and form the sequence

What are the properties of this signal? Expressing the two complex numbers as

we see that the signal is always a discrete-time generalized complexsinusoid, i.e., an exponentially enveloped complex sinusoid.
Figure 5.9 shows a plot of a generalized (exponentially decaying) complex sinusoid versus time.

Figure 5.9:Exponentially decaying complex sinusoid and its projections.
Note that the left projection (onto the plane) is a decaying spiral, the lower projection (real-part vs. time) is an exponentially decaying cosine, and the upper projection (imaginary-part vs. time) is an exponentially enveloped sine wave.
<< Previous page TOC INDEX Next page >>

Why (Generalized) Complex Sinusoids are Important

Sun, 01 Mar 1998 00:00:00 +0000

NOTE: THIS DOCUMENT IS OBSOLETE, PLEASE CHECK THE NEW VERSION: "Mathematics of the Discrete Fourier Transform (DFT), with Audio Applications --- Second Edition", by Julius O. Smith III, W3K Publishing, 2007, ISBN 978-0-9745607-4-8. - Copyright © 2017-09-28 by Julius O. Smith III - Center for Computer Research in Music and Acoustics (CCRMA), Stanford University

<< Previous page TOC INDEX Next page >>
Why (Generalized) Complex Sinusoids are Important
As a preview of things to come, note that one signal^5.10 is projected onto another signal using an inner product. The inner product computes the coefficient of projection^5.11 of onto . If (a sampled, unit-amplitude, zero-phase, complex sinusoid), then the inner product computes the Discrete Fourier Transform (DFT), provided the frequencies are chosen to be . For the DFT, the inner product is specifically

Comparing Analog and Digital Complex Planes

Sun, 01 Mar 1998 00:00:00 +0000

NOTE: THIS DOCUMENT IS OBSOLETE, PLEASE CHECK THE NEW VERSION: "Mathematics of the Discrete Fourier Transform (DFT), with Audio Applications --- Second Edition", by Julius O. Smith III, W3K Publishing, 2007, ISBN 978-0-9745607-4-8. - Copyright © 2017-09-28 by Julius O. Smith III - Center for Computer Research in Music and Acoustics (CCRMA), Stanford University

<< Previous page TOC INDEX Next page >>
Comparing Analog and Digital Complex Planes
In signal processing, it is customary to use as the Laplace transform variable for continuous-time analysis, and as the -transform variable for discrete-time analysis. In other words, for continuous-time systems, the frequency domain is the '' plane'', while for discrete-time systems, the frequency domain is the '' plane.'' However, both are simply complex planes.

Figure 5.10:Generalized complex sinusoidsrepresented by points in the plane.

Figure 5.11:Generalized complex sinusoids represented by points in the plane.
Figure 5.10 illustrates the various sinusoids represented by points in the plane. The frequency axis is , called the '' axis,'' and points along it correspond to complex sinusoids, with dc at (). The upper-half plane corresponds to positive frequencies (counterclockwise circular or corkscrew motion) while the lower-half plane corresponds to negative frequencies (clockwise motion). In the left-half plane we have decaying (stable) exponential envelopes, while in the right-half plane we have growing (unstable) exponential envelopes. Along the real axis (), we have pure exponentials. Every point in the plane can be said to correspond to some generalized complex sinusoids, with special cases being complex sinusoids , exponentials , and the constant function (dc).
Figure 5.11 shows examples of various sinusoids represented by points in the plane. The frequency axis is the ''unit circle'' , and points along it correspond to sampledcomplex sinusoids, with dc at (). As in the plane, the upper-half plane corresponds to positive frequencies while the lower-half plane corresponds to negative frequencies. Inside the unit circle, we have decaying (stable) exponential envelopes, while outside the unit circle, we have growing (unstable) exponential envelopes. Along the positive real axis (), we have pure exponentials, but along the negative real axis (), we have exponentially enveloped sampled sinusoids at frequency (exponentially enveloped alternating sequences). The negative real axis in the plane is normally a place where all signal transforms should be zero, and all system responses should be highly attenuated, since there should never be any energy at exactly half the sampling rate (where amplitude and phase are ambiguously linked). Every point in the plane can be said to correspond to some sampled generalized complex sinusoids , with special cases being sampled complex sinusoids , exponentials , and the constant function (dc).
In summary, the exponentially enveloped (''generalized'') complex sinusoid is the fundamental signal upon which other signals are ''projected'' in order to compute a Laplace transform in the continuous-time case, or a transform in the discrete-time case. As a special case, if the exponential envelope is eliminated (set to ), leaving only a complex sinusoid, then the projection reduces to the Fourier transform in the continuous-time case, and either the DFT (finite length) or DTFT (infinite length) in the discrete-time case. Finally, there are still other variations, such as short-time Fourier transforms (STFT) and wavelet transforms, which utilize further modifications such as projecting onto windowed complex sinusoids. Music 420delves into these topics.
<< Previous page TOC INDEX Next page >>

Mathematica for Selected Plots

Sun, 01 Mar 1998 00:00:00 +0000

NOTE: THIS DOCUMENT IS OBSOLETE, PLEASE CHECK THE NEW VERSION: "Mathematics of the Discrete Fourier Transform (DFT), with Audio Applications --- Second Edition", by Julius O. Smith III, W3K Publishing, 2007, ISBN 978-0-9745607-4-8. - Copyright © 2017-09-28 by Julius O. Smith III - Center for Computer Research in Music and Acoustics (CCRMA), Stanford University

<< Previous page TOC INDEX Next page >>

Mathematica for Selected Plots
The Mathematica code for producing Fig. 5.1 (minus the annotations which were done using NeXT Draw and EquationBuilder from Lighthouse Design) is
Plot[10 Sin[2 Pi 2.5 t + Pi/4],{t,0,1}, PlotLabel->"10 Sin[2 Pi 2.5 t + Pi/4]", PlotPoints->500, AxesLabel->{" Sec", "Amp."}];
The Mathematica code for Fig. 5.2 is
Show[ Plot[Sin[2 Pi 2.5 t],{t,-0.1,1.1}, PlotPoints->500, AxesLabel->{'' Time (Sec)'', ''Amplitude''}], Plot[Cos[2 Pi 2.5 t],{t,-0.1,1.1}, PlotPoints->500, PlotStyle->Dashing[{0.01,0.01}] ];
For the complex sinusoid plots (Fig. 5.7 and Fig. 5.9), see the Mathematica notebookComplexSinusoid.nbon Craig Sapp's web pageof Mathematica notebooks. (The packageSCMTheory.mis required.)
<< Previous page TOC INDEX Next page >>

Acknowledgement

Sun, 01 Mar 1998 00:00:00 +0000

NOTE: THIS DOCUMENT IS OBSOLETE, PLEASE CHECK THE NEW VERSION: "Mathematics of the Discrete Fourier Transform (DFT), with Audio Applications --- Second Edition", by Julius O. Smith III, W3K Publishing, 2007, ISBN 978-0-9745607-4-8. - Copyright © 2017-09-28 by Julius O. Smith III - Center for Computer Research in Music and Acoustics (CCRMA), Stanford University

<< Previous page TOC INDEX Next page >>
Acknowledgement
Thanks to Craig Stuart Sapp (craig@ccrma.stanford.edu) for contributing Figures8.2,8.3,8.4,8.6,8.7,8.8,8.9, and8.10.
<< Previous page TOC INDEX Next page >>

Geometric Signal Theory

Sun, 01 Mar 1998 00:00:00 +0000

NOTE: THIS DOCUMENT IS OBSOLETE, PLEASE CHECK THE NEW VERSION: "Mathematics of the Discrete Fourier Transform (DFT), with Audio Applications --- Second Edition", by Julius O. Smith III, W3K Publishing, 2007, ISBN 978-0-9745607-4-8. - Copyright © 2017-09-28 by Julius O. Smith III - Center for Computer Research in Music and Acoustics (CCRMA), Stanford University

<< Previous page TOC INDEX Next page >>
Geometric Signal Theory
This chapter provides an introduction to the elements of geometric signal theory, including vector spaces, norms, inner products,orthogonality, projection of one signal onto another, and elementary vector space operations.

Subsections
The DFT
Signals as Vectors
An Example Vector View:
Vector Addition
Vector Subtraction
Signal Metrics
Other Norms
The Inner Product
Linearity of the Inner Product
Norm Induced by the Inner Product
Cauchy-Schwarz Inequality
Triangle Inequality
Triangle Difference Inequality
Vector Cosine
Orthogonality
The Pythagorean Theorem in N-Space
Projection
Signal Reconstruction from Projections
An Example of Changing Coordinates in 2D
General Conditions
Gram-Schmidt Orthogonalization
Appendix: Matlab Examples

<< Previous page TOC INDEX Next page >>

Signals as Vectors

Sun, 01 Mar 1998 00:00:00 +0000

NOTE: THIS DOCUMENT IS OBSOLETE, PLEASE CHECK THE NEW VERSION: "Mathematics of the Discrete Fourier Transform (DFT), with Audio Applications --- Second Edition", by Julius O. Smith III, W3K Publishing, 2007, ISBN 978-0-9745607-4-8. - Copyright © 2017-09-28 by Julius O. Smith III - Center for Computer Research in Music and Acoustics (CCRMA), Stanford University

<< Previous page TOC INDEX Next page >>
Signals as Vectors
For the DFT, all signals and spectra are length . A length sequence can be denoted by , , where may be real () or complex (). We now wish to regard as avector ^6.1 in an dimensional vector space. That is, each sample is regarded as a coordinate in that space. A vector is mathematically a single point in -space represented by a list of coordinates called an -tuple. (The notation means the same thing as .) It can be interpreted geometrically as an arrow in -space from the origin to the point .
We define the following as equivalent:

where is the th sample of the signal (vector) . From now on, unless specifically mentioned otherwise, all signals are length .

Subsections
An Example Vector View:

<< Previous page TOC INDEX Next page >>

An Example Vector View

Sun, 01 Mar 1998 00:00:00 +0000

NOTE: THIS DOCUMENT IS OBSOLETE, PLEASE CHECK THE NEW VERSION: "Mathematics of the Discrete Fourier Transform (DFT), with Audio Applications --- Second Edition", by Julius O. Smith III, W3K Publishing, 2007, ISBN 978-0-9745607-4-8. - Copyright © 2017-09-28 by Julius O. Smith III - Center for Computer Research in Music and Acoustics (CCRMA), Stanford University

<< Previous page TOC INDEX Next page >>
An Example Vector View:
Consider the example two-sample signal graphed in Fig. 6.1.

Figure 6.1:A length 2 signal plotted as a vector in 2D space.
Under the geometric interpretation of a length signal, each sample is acoordinate in the dimensional space. Signals which are only two samples long are not terribly interesting to hear, but they are easy to plot geometrically.
<< Previous page TOC INDEX Next page >>

Vector Addition

Sun, 01 Mar 1998 00:00:00 +0000

NOTE: THIS DOCUMENT IS OBSOLETE, PLEASE CHECK THE NEW VERSION: "Mathematics of the Discrete Fourier Transform (DFT), with Audio Applications --- Second Edition", by Julius O. Smith III, W3K Publishing, 2007, ISBN 978-0-9745607-4-8. - Copyright © 2017-09-28 by Julius O. Smith III - Center for Computer Research in Music and Acoustics (CCRMA), Stanford University

<< Previous page TOC INDEX Next page >>
Vector Addition
Given two vectors in , say and , the vector sum is defined byelementwise addition. If we denote the sum by , then we have for .
The vector diagram for the sum of two vectors can be found using the parallelogram rule, as shown in Fig. 6.2 for , , and.

Figure 6.2:Geometric interpretation of a length 2 vector sum.
Also shown are the lighter construction lines which complete the parallelogram started by and , indicating where the endpoint of the sum lies. Since it is a parallelogram, the two construction lines are congruent to the vectors and . As a result, the vector sum is often expressed as a triangle by translating the origin of one member of the sum to the tip of the other, as shown in Fig. 6.3.

Figure 6.3:Vector sum with translation of one vector to the tip of the other.
In the figure, was translated to the tip of . It is equally valid to translate to the tip of , because vector addition is commutative, i.e., = .
<< Previous page TOC INDEX Next page >>

Vector Subtraction

Sun, 01 Mar 1998 00:00:00 +0000

NOTE: THIS DOCUMENT IS OBSOLETE, PLEASE CHECK THE NEW VERSION: "Mathematics of the Discrete Fourier Transform (DFT), with Audio Applications --- Second Edition", by Julius O. Smith III, W3K Publishing, 2007, ISBN 978-0-9745607-4-8. - Copyright © 2017-09-28 by Julius O. Smith III - Center for Computer Research in Music and Acoustics (CCRMA), Stanford University

<< Previous page TOC INDEX Next page >>
Vector Subtraction
Figure 6.7 illustrates the vector difference between and . From the coordinates, we compute .

Figure 6.4:Geometric interpretation a difference vector.
Note that the difference vector may be drawn from the tip of to the tip of rather than from the origin to the point ; this is a customary practice which emphasizes relationships among vectors, but the translation in the plot has no effect on the mathematical definition or properties of the vector. Subtraction, however, is not commutative.
To ascertain the proper orientation of the difference vector , rewrite its definition as , and then it is clear that the vector should be the sum of vectors and , hence the arrowhead is on the correct endpoint.
<< Previous page TOC INDEX Next page >>

Signal Metrics

Sun, 01 Mar 1998 00:00:00 +0000

NOTE: THIS DOCUMENT IS OBSOLETE, PLEASE CHECK THE NEW VERSION: "Mathematics of the Discrete Fourier Transform (DFT), with Audio Applications --- Second Edition", by Julius O. Smith III, W3K Publishing, 2007, ISBN 978-0-9745607-4-8. - Copyright © 2017-09-28 by Julius O. Smith III - Center for Computer Research in Music and Acoustics (CCRMA), Stanford University

<< Previous page TOC INDEX Next page >>
Signal Metrics
This section defines some useful functions of signals.
The mean of a signal (more precisely the ''sample mean'') is defined as its average value:

The total energy of a signal is defined the sum of squaredmoduli:

Energy is the ''ability to do work.'' In physics, energy and work are in units of ''force times distance,'' ''mass times velocity squared,'' or other equivalent combinations of units. The energy of a pressure wave is the integral over time of the squared pressure divided by the wave impedance the wave is traveling in. The energy of a velocity wave is the integral over time of the squared velocity times the wave impedance. In audio work, a signal is typically a list of pressure samplesderived from a microphone signal, or it might be samples of forcefrom a piezoelectric transducer, velocity from a magnetic guitarpickup, and so on. In all of these cases, the total physical energy associated with the signal is proportional to the sum of squared signal samples. (Physical connections in signal processing are explored more deeply in Music 421.)
The average power of a signal is defined the energy per sample:

Another common description when is real is the ''mean square.'' When is a complex sinusoid, i.e., , then ; in other words, for complex sinusoids, the average power equals the instantaneous power which is the amplitude squared.
Power is always in physical units of energy per unit time. It therefore makes sense to define the average signal power as the total signal energydivided by its length. We normally work with signals which are functions of time. However, if the signal happens instead to be a function of distance (e.g., samples of displacement along a vibrating string), then the ''power'' as defined here still has the interpretation of a spatial energy density. Power, in contrast, is a temporal energy density.
The root mean square (RMS) level of a signal is simply . However, note that in practice (especially in audio work) an RMS level may be computed after subtracting out the mean value. Here, we call that the variance.
The variance (more precisely the sample variance) of the signal is defined as the power of the signal with its sample mean removed:

It is quick to show that, for real signals, we have

which is the ''mean square minus the mean squared.'' We think of the variance as the power of the non-constant signal components (i.e., everything but dc). The terms ''sample mean'' and ''sample variance'' come from the field of statistics, particularly the theory of stochastic processes. The field of statistical signal processing[16] is firmly rooted in statistical topics such as ''probability,'' ''random variables,'' ''stochastic processes,'' and ''time series analysis.'' In this course, we will only touch lightly on a few elements of statistical signal processing in a self-contained way.
The norm of a signal is defined as the square root of its total energy:

We think of as the length of in -space. Furthermore, is regarded as the distance between and . The norm can also be thought of as the ''absolute value'' or ''radius'' of a vector.^6.2
Example: Going back to our simple 2D example , we can compute its norm as . The physical interpretation of the norm as a distance measure is shown in Fig. 6.5.

Figure 6.5:Geometric interpretation of a signal norm in 2D.
Example: Let's also look again at the vector-sum example, redrawn in Fig. 6.6.

Figure 6.6:Length of vectors in sum.
The norm of the vector sum is

while the norms of and are and , respectively. We find that which is an example of the triangle inequality. (Equality occurs only when and are colinear, as can be seen geometrically from studying Fig. 6.6.)
Example: Consider the vector-difference example diagrammed in Fig. 6.7.

Figure 6.7:Length of a difference vector.
The norm of the difference vector is

Subsections
Other Norms

<< Previous page TOC INDEX Next page >>

Other Norms

Sun, 01 Mar 1998 00:00:00 +0000

NOTE: THIS DOCUMENT IS OBSOLETE, PLEASE CHECK THE NEW VERSION: "Mathematics of the Discrete Fourier Transform (DFT), with Audio Applications --- Second Edition", by Julius O. Smith III, W3K Publishing, 2007, ISBN 978-0-9745607-4-8. - Copyright © 2017-09-28 by Julius O. Smith III - Center for Computer Research in Music and Acoustics (CCRMA), Stanford University

<< Previous page TOC INDEX Next page >>
Other Norms
Since our main norm is the square root of a sum of squares, we are using what is called an norm and we may write to emphasize this fact.
We could equally well have chosen a normalized norm:

which is simply the ''RMS level'' of .
More generally, the norm of is defined

The most interesting norms are
: The , ''absolute value,'' or ''city block'' norm.
: The , ''Euclidean,'' ''root energy,'' or ''least squares'' norm.
: The , ''Chebyshev,'' ''supremum,'' ''minimax,'' or ''uniform'' norm.
Note that the case is a limiting case which becomes

There are many other possible choices of norm. To qualify as a norm on, a real-valued signal function must satisfy the following three properties:
,
The first property, ''positivity,'' says only the zero vector has norm zero. The second property is ''subadditivity'' and is sometimes called the ''triangle inequality'' for reasons which can be seen by studying Fig. 6.3. The third property says the norm is ''absolutely homogeneous'' with respect to scalar multiplication (which can be complex, in which case the phase of the scalar has no effect).
Mathematically, what we are working with so far is called a Banach space which a normed linear vector space. To summarize, we defined our vectors as any list of real or complex numbers which we interpret as coordinates in the -dimensional vector space. We also defined vector addition in the obvious way. It turns out we have to also define scalar multiplication, that is, multiplication of a vector by a scalar which we also take to be an element of the field of real or complex numbers. This is also done in the obvious way which is to multiply each coordinate of the vector by the scalar. To have a linear vector space, it must be closed under vector addition and scalar multiplication. That means given any two vectors and from the vector space, and given any two scalars and from the field of scalars, then any linear combination must also be in the space. Since we have used the field of complex numbers (or real numbers ) to define both our scalars and our vector components, we have the necessary closure properties so that any linear combination of vectors from lies in . Finally, the definition of a norm (any norm) elevates a vector space to a Banach space.
<< Previous page TOC INDEX Next page >>

The Inner Product

Sun, 01 Mar 1998 00:00:00 +0000

NOTE: THIS DOCUMENT IS OBSOLETE, PLEASE CHECK THE NEW VERSION: "Mathematics of the Discrete Fourier Transform (DFT), with Audio Applications --- Second Edition", by Julius O. Smith III, W3K Publishing, 2007, ISBN 978-0-9745607-4-8. - Copyright © 2017-09-28 by Julius O. Smith III - Center for Computer Research in Music and Acoustics (CCRMA), Stanford University

<< Previous page TOC INDEX Next page >>
The Inner Product
The inner product (or ''dot product'') is an operation on two vectors which produces a scalar. Adding an inner product to a Banach space produces a Hilbert space (or ''inner product space''). There are many examples of Hilbert spaces, but we will only need for this course (complex length vectors with complex scalars).
The inner product between two (complex) -vectors and is defined by

The complex conjugation of the second vector is done in order that a norm will be induced by the inner product:

As a result, the inner product is conjugate symmetric:

Note that the inner product takes to . That is, two length complex vectors are mapped to a complex scalar.
Example:For we have, in general,

Let

Then

Subsections
Linearity of the Inner Product
Norm Induced by the Inner Product
Cauchy-Schwarz Inequality
Triangle Inequality
Triangle Difference Inequality
Vector Cosine
Orthogonality
The Pythagorean Theorem in N-Space
Projection

<< Previous page TOC INDEX Next page >>

Linearity of the Inner Product

Sun, 01 Mar 1998 00:00:00 +0000

NOTE: THIS DOCUMENT IS OBSOLETE, PLEASE CHECK THE NEW VERSION: "Mathematics of the Discrete Fourier Transform (DFT), with Audio Applications --- Second Edition", by Julius O. Smith III, W3K Publishing, 2007, ISBN 978-0-9745607-4-8. - Copyright © 2017-09-28 by Julius O. Smith III - Center for Computer Research in Music and Acoustics (CCRMA), Stanford University

<< Previous page TOC INDEX Next page >>
Linearity of the Inner Product
Any function of a vector (which we may call an operator on ) is said to be linear if for all and , and for all scalars and in , we have

A linear operator thus ''commutes with mixing.''
Linearity consists of two component properties,
additivity: , and
homogeneity: .
The inner product is linear in its first argument, i.e.

This is easy to show from the definition:

The inner product is also additive in its second argument, i.e.,

but it is only conjugate homogeneous in its second argument, since

The inner product is strictly linear in its second argument with respect to real scalars:

Since the inner product is linear in both of its arguments for real scalars, it is often called a bilinear operator in that context.
<< Previous page TOC INDEX Next page >>

Norm Induced by the Inner Product

Sun, 01 Mar 1998 00:00:00 +0000

NOTE: THIS DOCUMENT IS OBSOLETE, PLEASE CHECK THE NEW VERSION: "Mathematics of the Discrete Fourier Transform (DFT), with Audio Applications --- Second Edition", by Julius O. Smith III, W3K Publishing, 2007, ISBN 978-0-9745607-4-8. - Copyright © 2017-09-28 by Julius O. Smith III - Center for Computer Research in Music and Acoustics (CCRMA), Stanford University

<< Previous page TOC INDEX Next page >>
Norm Induced by the Inner Product
We may define a norm on using the inner product:

It is straightforward to show that properties 1 and 3 of a norm hold. Property 2 follows easily from the Schwarz Inequality which is derived in the following subsection. Alternatively, we can simply observe that the inner product induces the well known norm on .
<< Previous page TOC INDEX Next page >>

Cauchy-Schwarz Inequality

Sun, 01 Mar 1998 00:00:00 +0000

NOTE: THIS DOCUMENT IS OBSOLETE, PLEASE CHECK THE NEW VERSION: "Mathematics of the Discrete Fourier Transform (DFT), with Audio Applications --- Second Edition", by Julius O. Smith III, W3K Publishing, 2007, ISBN 978-0-9745607-4-8. - Copyright © 2017-09-28 by Julius O. Smith III - Center for Computer Research in Music and Acoustics (CCRMA), Stanford University

<< Previous page TOC INDEX Next page >>
Cauchy-Schwarz Inequality
The Cauchy-Schwarz Inequality (or ''Schwarz Inequality'') states that for all and , we have

Triangle Inequality

Sun, 01 Mar 1998 00:00:00 +0000

NOTE: THIS DOCUMENT IS OBSOLETE, PLEASE CHECK THE NEW VERSION: "Mathematics of the Discrete Fourier Transform (DFT), with Audio Applications --- Second Edition", by Julius O. Smith III, W3K Publishing, 2007, ISBN 978-0-9745607-4-8. - Copyright © 2017-09-28 by Julius O. Smith III - Center for Computer Research in Music and Acoustics (CCRMA), Stanford University

<< Previous page TOC INDEX Next page >>
Triangle Inequality
The triangle inequality states that the length of any side of a triangle is less than or equal to the sum of the lengths of the other two sides, with equality occurring only when the triangle degenerates to a line. In , this becomes

We can show this quickly using the Schwarz Inequality.

<< Previous page TOC INDEX Next page >>

Triangle Difference Inequality

Sun, 01 Mar 1998 00:00:00 +0000

NOTE: THIS DOCUMENT IS OBSOLETE, PLEASE CHECK THE NEW VERSION: "Mathematics of the Discrete Fourier Transform (DFT), with Audio Applications --- Second Edition", by Julius O. Smith III, W3K Publishing, 2007, ISBN 978-0-9745607-4-8. - Copyright © 2017-09-28 by Julius O. Smith III - Center for Computer Research in Music and Acoustics (CCRMA), Stanford University

<< Previous page TOC INDEX Next page >>
Triangle Difference Inequality
A useful variation on the triangle inequality is that the length of any side of a triangle is greater than the absolute difference of the lengths of the other two sides:

Proof:

<< Previous page TOC INDEX Next page >>

Vector Cosine

Sun, 01 Mar 1998 00:00:00 +0000

NOTE: THIS DOCUMENT IS OBSOLETE, PLEASE CHECK THE NEW VERSION: "Mathematics of the Discrete Fourier Transform (DFT), with Audio Applications --- Second Edition", by Julius O. Smith III, W3K Publishing, 2007, ISBN 978-0-9745607-4-8. - Copyright © 2017-09-28 by Julius O. Smith III - Center for Computer Research in Music and Acoustics (CCRMA), Stanford University

<< Previous page TOC INDEX Next page >>
Vector Cosine
The Cauchy-Schwarz Inequality can be written

In the case of real vectors , we can always find a real number which satisfies

We thus interpret as the angle between two vectors in . In we can similarly define .
<< Previous page TOC INDEX Next page >>

Orthogonality

Sun, 01 Mar 1998 00:00:00 +0000

NOTE: THIS DOCUMENT IS OBSOLETE, PLEASE CHECK THE NEW VERSION: "Mathematics of the Discrete Fourier Transform (DFT), with Audio Applications --- Second Edition", by Julius O. Smith III, W3K Publishing, 2007, ISBN 978-0-9745607-4-8. - Copyright © 2017-09-28 by Julius O. Smith III - Center for Computer Research in Music and Acoustics (CCRMA), Stanford University

<< Previous page TOC INDEX Next page >>
Orthogonality
The vectors (signals) and are said to be orthogonal if , denoted . That is to say

Note that if and are real and orthogonal, the cosine of the angle between them is zero. In plane geometry (), the angle between twoperpendicular lines is , and , as expected. More generally, orthogonality corresponds to the fact that two vectors in-space intersect at a right angle and are thus perpendiculargeometrically.
Example ():
Let and , as shown in Fig. 6.8.

Figure 6.8:Example of two orthogonal vectors for .
The inner product is . This shows that the vectors are orthogonal. As marked in the figure, the lines intersect at a right angle and are therefore perpendicular.
<< Previous page TOC INDEX Next page >>

The Pythagorean Theorem in N-Space

Sun, 01 Mar 1998 00:00:00 +0000

NOTE: THIS DOCUMENT IS OBSOLETE, PLEASE CHECK THE NEW VERSION: "Mathematics of the Discrete Fourier Transform (DFT), with Audio Applications --- Second Edition", by Julius O. Smith III, W3K Publishing, 2007, ISBN 978-0-9745607-4-8. - Copyright © 2017-09-28 by Julius O. Smith III - Center for Computer Research in Music and Acoustics (CCRMA), Stanford University

<< Previous page TOC INDEX Next page >>
The Pythagorean Theorem in N-Space
In 2D, the Pythagorean Theorem says that when and are orthogonal, as in Fig. 6.8, (i.e., when the triangle formed by , , and , with translated to the tip of , is a right triangle), then we have

This relationship generalizes to dimensions, as we can easily show:

If , then and the Pythagorean Theorem holds in dimensions. If, on the other hand, we assume the Pythagorean Theorem holds, then since all norms are positive unless or is zero, we must have . Finally, if or is zero, the result holds trivially.
Note that we also have an alternate version of the Pythagorean theorem:

<< Previous page TOC INDEX Next page >>

Projection

Sun, 01 Mar 1998 00:00:00 +0000

NOTE: THIS DOCUMENT IS OBSOLETE, PLEASE CHECK THE NEW VERSION: "Mathematics of the Discrete Fourier Transform (DFT), with Audio Applications --- Second Edition", by Julius O. Smith III, W3K Publishing, 2007, ISBN 978-0-9745607-4-8. - Copyright © 2017-09-28 by Julius O. Smith III - Center for Computer Research in Music and Acoustics (CCRMA), Stanford University

<< Previous page TOC INDEX Next page >>
Projection
The orthogonal projection (or simply ''projection'') of onto is defined by

The complex scalar is called thecoefficient of projection. When projecting onto a unit length vector , the coefficient of projection is simply the inner product of with .
Motivation: The basic idea of orthogonal projection of onto is to ''drop a perpendicular'' from onto to define a new vector along which we call the ''projection'' of onto . This is illustrated for in Fig. 6.9 for and , in which case

Figure 6.9:Projection of onto in 2D space.
Derivation: (1) Since any projection onto must lie along the line colinear with , write the projection as . (2) Since by definition the projection is orthogonal to , we must have

<< Previous page TOC INDEX Next page >>

Signal Reconstruction from Projections

Sun, 01 Mar 1998 00:00:00 +0000

NOTE: THIS DOCUMENT IS OBSOLETE, PLEASE CHECK THE NEW VERSION: "Mathematics of the Discrete Fourier Transform (DFT), with Audio Applications --- Second Edition", by Julius O. Smith III, W3K Publishing, 2007, ISBN 978-0-9745607-4-8. - Copyright © 2017-09-28 by Julius O. Smith III - Center for Computer Research in Music and Acoustics (CCRMA), Stanford University

<< Previous page TOC INDEX Next page >>
Signal Reconstruction from Projections
We now know how to project a signal onto other signals. We now need to learn how to reconstruct a signal from its projections onto different vectors , . This will give us theinverse DFT operation (or the inverse of whatever transform we are working with).
As a simple example, consider the projection of a signal onto the rectilinear coordinate axes of . The coordinates of the projection onto the th coordinate axis are simply . The projection along coordinate axis has coordinates , and so on. The original signal is then clearly the vector sum of its projections onto the coordinate axes:

To make sure the previous paragraph is understood, let's look at the details for the case . We want to project an arbitrary two-sample signal onto the coordinate axes in 2D. A coordinate axis can be represented by any nonzero vector along its length. The horizontal axis can be represented by any vector of the form while the vertical axis can be represented by any vector of the form . For maximum simplicity, let's choose the positive unit-length representatives:

The projection of onto is by definition

Similarly, the projection of onto is

The reconstruction of from its projections onto the coordinate axes is then the vector sum of the projections:

The projection of a vector onto its coordinate axes is in some sense trivial because the very meaning of the coordinates is that they are scalars to be applied to the coordinate vectors in order to form an arbitrary vector as a linear combinationof the coordinate vectors:

Note that the coordinate vectors are orthogonal. Since they are also unit length, , we say that the coordinate vectors are orthonormal.
What's more interesting is when we project a signal onto a set of vectors other than the coordinate set. This can be viewed as a change of coordinates in . In the case of the DFT, the new vectors will be chosen to be sampled complex sinusoids.

Subsections
An Example of Changing Coordinates in 2D
General Conditions
Gram-Schmidt Orthogonalization

<< Previous page TOC INDEX Next page >>

An Example of Changing Coordinates in 2D

Sun, 01 Mar 1998 00:00:00 +0000

NOTE: THIS DOCUMENT IS OBSOLETE, PLEASE CHECK THE NEW VERSION: "Mathematics of the Discrete Fourier Transform (DFT), with Audio Applications --- Second Edition", by Julius O. Smith III, W3K Publishing, 2007, ISBN 978-0-9745607-4-8. - Copyright © 2017-09-28 by Julius O. Smith III - Center for Computer Research in Music and Acoustics (CCRMA), Stanford University

<< Previous page TOC INDEX Next page >>
An Example of Changing Coordinates in 2D
As a simple example, let's pick the following pair of new coordinate vectors in 2D

General Conditions

Sun, 01 Mar 1998 00:00:00 +0000

NOTE: THIS DOCUMENT IS OBSOLETE, PLEASE CHECK THE NEW VERSION: "Mathematics of the Discrete Fourier Transform (DFT), with Audio Applications --- Second Edition", by Julius O. Smith III, W3K Publishing, 2007, ISBN 978-0-9745607-4-8. - Copyright © 2017-09-28 by Julius O. Smith III - Center for Computer Research in Music and Acoustics (CCRMA), Stanford University

<< Previous page TOC INDEX Next page >>
General Conditions
This section summarizes and extends the above derivations in a somewhat formal manner (following portions of chapter 4 of ).

Definition: A set of vectors is said to form a vector space if given any two members and from the set, the vectors and are also in the set, where is any scalar.
Vectors defined as a list of complex numbers, using elementwise addition and multiplication by complex scalars , form a vector space. Similarly, real vectors and real scalars form a vector space.

Theorem: The set of all linear combinations of any set of vectors from or forms a vector space.
Proof: Let the original set of vectors be denoted , where can be any integer greater than zero. Then any member of the vector space is by definition a linear combination of them:

From this we see that is also a linear combination of the original vectors, and hence is in the vector space. Also, given any second vector from the space , the sum is

which is just another linear combination of the original vectors, and hence is in the vector space. It is clear here how the closure of vector addition and scalar multiplication are ''inherited'' from the closure of real or complex numbers under addition and multiplication.

Definition: If a vector space consists of the set of all linear combinations of a finite set of vectors , then those vectors are said to span the space.

Example: The coordinate vectors in span since every vector can be expressed as a linear combination of the coordinate vectors as

where , and , , , and so on up to .

Theorem: (i) If span a vector space, and if one of them, say , is linearly dependent on the others, then the vector space is spanned by the set obtained by omitting from the original set. (ii) If span a vector space, we can always select from these a linearly independent set that spans the same space.
Proof: Any in the space can be represented as a linear combination of the vectors . By expressing as a linear combination of the other vectors in the set, the linear combination for becomes a linear combination of vectors other than . Thus, can be eliminated from the set, proving (i). To prove (ii), we can define a procedure for forming the require subset of the original vectors: First, assign to the set. Next, check to see if and are linearly dependent. If so (i.e., is a scalar times ), then discard ; otherwise assign it also to the new set. Next, check to see if is linearly dependent on the vectors in the new set. If it is (i.e., is some linear combination of and ) then discard it; otherwise assign it also to the new set. When this procedure terminates after processing , the new set will contain only linearly independent vectors which span the original space.

Definition: A set of linearly independent vectors which spans a vector space is called a basis for that vector space.

Definition: The set of coordinate vectors in is called the natural basis for , where the th basis vectoris

Theorem: The linear combination expressing a vector in terms of basis vectors for a vector space is unique.
Proof: Suppose a vector can be expressed in two different ways as a linear combination of basis vectors :

where for at least one value of . Subtracting the two representations gives

Since the vectors are linearly independent, it is not possible to cancel the nonzero vector using some linear combination of the other vectors in the sum. Hence, for all .
Note that while the linear combination relative to a particular basis is unique, the choice of basis vectors is not. For example, given any basis set in , a new basis can be formed by rotating all vectors in by the same angle. In this way, an infinite number of basis sets can be generated.
As we will soon show, the DFT can be viewed as a change of coordinates from coordinates relative to the natural basis in , , to coordinates relative to the sinusoidal basis for , , where . The sinusoidal basis set for consists of length sampled complex sinusoids at frequencies . Any scaling of these vectors in by complex scale factors could also be chosen as the sinusoidal basis (i.e., any nonzero amplitude and any phase will do). However, for simplicity, we will only use unit-amplitude, zero-phase complex sinusoids as the Fourier ''frequency-domain'' basis set. To summarize this paragraph, the time-domain samples of a signal are its coordinates relative to the natural basis for , while its spectral coefficients are the coordinates of the signal relative to the sinusoidal basis for .

Theorem: Any two bases of a vector space contain the same number of vectors.
Proof: Left as an exercise (or see [3]).

Definition: The number of vectors in a basis for a particular space is called the dimension of the space. If the dimension is , the space is said to be an dimensional space, or -space.
In this course, we will only consider finite-dimensional vector spaces in any detail. However, the discrete-time Fourier transform (DTFT) and Fourier transform both require infinite-dimensional basis sets, because there is an infinite number of points in both the time and frequency domains.

Theorem: The projections of any vector onto any orthogonal basis set for can be summed to reconstruct exactly.
Proof: Let denote any orthogonal basis set for . Then since is in the space spanned by these vectors, we have

for some (unique) scalars . The projection of onto is equal to

(using the linearity of the projection operator which follows from linearity of the inner product in its first argument). Since the basis vectors are orthogonal, the projection of onto is zero for:

We therefore obtain

Therefore, the sum of projections onto the vectors is just the linear combination of the which forms .
<< Previous page TOC INDEX Next page >>

Gram-Schmidt Orthogonalization

Sun, 01 Mar 1998 00:00:00 +0000

NOTE: THIS DOCUMENT IS OBSOLETE, PLEASE CHECK THE NEW VERSION: "Mathematics of the Discrete Fourier Transform (DFT), with Audio Applications --- Second Edition", by Julius O. Smith III, W3K Publishing, 2007, ISBN 978-0-9745607-4-8. - Copyright © 2017-09-28 by Julius O. Smith III - Center for Computer Research in Music and Acoustics (CCRMA), Stanford University

<< Previous page TOC INDEX Next page >>
Gram-Schmidt Orthogonalization

Theorem: Given a set of linearly independent vectors from , we can construct an orthonormalset which are linear combinations of the original set and which span the same space.

Appendix: Matlab Examples

Sun, 01 Mar 1998 00:00:00 +0000

NOTE: THIS DOCUMENT IS OBSOLETE, PLEASE CHECK THE NEW VERSION: "Mathematics of the Discrete Fourier Transform (DFT), with Audio Applications --- Second Edition", by Julius O. Smith III, W3K Publishing, 2007, ISBN 978-0-9745607-4-8. - Copyright © 2017-09-28 by Julius O. Smith III - Center for Computer Research in Music and Acoustics (CCRMA), Stanford University

<< Previous page TOC INDEX Next page >>
Appendix: Matlab Examples
Here's how Fig. 6.1 was generated in Matlab:
>> x = [2 3]; % coordinates of x >> origin = [0 0]; % coordinates of the origin >> xcoords = [origin(1) x(1)]; % plot() expects coordinate lists, not endpoints >> ycoords = [origin(2) x(2)]; >> plot(xcoords,ycoords); % Draw a line from origin to x
Mathematica can plot a list of ordered pairs:
In[1]: ListPlot[{{0,0},{2,3}},PlotJoined->True]; (* Draw a line from (0,0) to (2,3) *)
In Matlab, the mean of the row-vector can be computed as

or by using the built-in function mean().
In Matlab, if x = [x1 ... xN] is a row vector, we can compute thetotal energy as

Matlab has a function orth() which will compute an orthonormal basis for a space given any set of vectors which span the space.
>> help orth
ORTH Orthogonalization. Q = orth(A) is an orthonormal basis for the range of A. Q’*Q = I, the columns of Q span the same space as the columns of A and the number of columns of Q is the rank of A.

See also QR, NULL.</pre>Below is an example of using <tt>orth()</tt> to orthonormalize a linearly

independent basis set for :
% Demonstration of the Matlab function orth() for % taking a set of vectors and returning an orthonormal set % which span the same space. v1 = [1; 2; 3]; % our first basis vector (a column vector) v2 = [1; -2; 3]; % a second, linearly independent column vector v1’ * v2 % show that v1 is not orthogonal to v2

ans =

6

V = [v1,v2] % Each column of V is one of our vectors

V =

1 1 2 -2 3 3

W = orth(V) % Find an orthonormal basis for the same space

W =

0.2673 0.1690 0.5345 -0.8452 0.8018 0.5071

w1 = W(:,1) % Break out the returned vectors

w1 =

0.2673 0.5345 0.8018

w2 = W(:,2)

w2 =

0.1690

-0.8452 0.5071

w1’ * w2 % Check that w1 is orthogonal to w2 (to working precision)

ans =

2.5723e-17

w1’ * w1 % Also check that the new vectors are unit length in 3D

ans =

1

w2’ * w2

ans =

1

W’ * W % faster way to do the above checks (matrix multiplication)

ans =

1.0000 0.0000 0.0000 1.0000

% Construct some vector x in the space spanned by v1 and v2: x = 2 * v1 - 3 * v2

x =

-1 10 -3

% Show that x is also some linear combination of w1 and w2: c1 = x’ * w1 % Coefficient of projection of x onto w1

c1 =

2.6726

c2 = x’ * w2 % Coefficient of projection of x onto w2

c2 =

-10.1419

xw = c1 * w1 + c2 * w2 % Can we make x using w1 and w2?

xw =

-1.0000 10.0000 -3.0000

error = x - xw

error =

1.0e-14 *

0.1332 0 0norm(error) % typical way to summarize a vector error

ans =

1.3323e-15

% It works!

% Now, construct some vector x NOT in the space spanned by v1 and v2: y = [1; 0; 0]; % Almost anything we guess in 3D will work

% Try to express y as a linear combination of w1 and w2: c1 = y’ * w1; % Coefficient of projection of y onto w1 c2 = y’ * w2; % Coefficient of projection of y onto w2 yw = c1 * w1 + c2 * w2 % Can we make y using w1 and w2?

yw =

0.1000 0.0000 0.3000

yerror = y - yw

yerror =

0.9000 0.0000

-0.3000

norm(yerror)

ans =

0.9487

% While the error is not zero, it is the smallest possible % error in the least squares sense. % That is, yw is the optimal least-squares approximation % to y in the space spanned by v1 and v2 (w1 and w2). % In other words, norm(yerror) <= norm(y-yw2) for any other vector yw2 made % using a linear combination of v1 and v2.
% In yet other words, we obtain the optimal least squares approximation % of y (which lives in 3D) in some subspace W (a 2D subspace of 3D) % by projecting y orthogonally onto the subspace W to get yw as above. % % An important property of the optimal least-squares approximation % is that the approximation error is orthogonal to the the subspace % in which the approximation lies. Let’s show this:

W’ * yerror % must be zero to working precision

ans =

1.0e-16 *

-0.2574 -0.0119
<< Previous page TOC INDEX Next page >>

The Discrete Fourier Transform (DFT) Derived

Sun, 01 Mar 1998 00:00:00 +0000

NOTE: THIS DOCUMENT IS OBSOLETE, PLEASE CHECK THE NEW VERSION: "Mathematics of the Discrete Fourier Transform (DFT), with Audio Applications --- Second Edition", by Julius O. Smith III, W3K Publishing, 2007, ISBN 978-0-9745607-4-8. - Copyright © 2017-09-28 by Julius O. Smith III - Center for Computer Research in Music and Acoustics (CCRMA), Stanford University

<< Previous page TOC INDEX Next page >>
The Discrete Fourier Transform (DFT)
Given a signal , the spectrum is defined by

or, as is most often written

That is, the th sample of the spectrum of is defined as the inner product of with the th DFT sinusoid . This definition is times the coefficient of projection of onto , i.e.,

The projection of onto itself is

The inverse DFT is simply the sum of the projections:

or, as we normally write,

In summary, the DFT is proportional to the set of coefficients of projection onto the sinusoidal basis set, and the inverse DFT is the reconstruction of the original signal as a superposition of its sinusoidal projections.

Subsections
The DFT Derived
Geometric Series
Orthogonality of Sinusoids
Orthogonality of the DFT Sinusoids
Norm of the DFT Sinusoids
An Orthonormal Sinusoidal Set
The Discrete Fourier Transform (DFT)
Frequencies in the ''Cracks''
Normalized DFT
The Length 2 DFT
Matrix Formulation of the DFT
Matrices
Matrix Multiplication
The DFT Matrix
Matlab Examples
Figure 7.2
Figure 7.3
DFT Matrix

<< Previous page TOC INDEX Next page >>

The DFT Derived

Sun, 01 Mar 1998 00:00:00 +0000

NOTE: THIS DOCUMENT IS OBSOLETE, PLEASE CHECK THE NEW VERSION: "Mathematics of the Discrete Fourier Transform (DFT), with Audio Applications --- Second Edition", by Julius O. Smith III, W3K Publishing, 2007, ISBN 978-0-9745607-4-8. - Copyright © 2017-09-28 by Julius O. Smith III - Center for Computer Research in Music and Acoustics (CCRMA), Stanford University

<< Previous page TOC INDEX Next page >>
The DFT Derived
In this section, the Discrete Fourier Transform (DFT) will be derived.

Subsections
Geometric Series
Orthogonality of Sinusoids
Orthogonality of the DFT Sinusoids
Norm of the DFT Sinusoids
An Orthonormal Sinusoidal Set
The Discrete Fourier Transform (DFT)
Frequencies in the ''Cracks''
Normalized DFT

<< Previous page TOC INDEX Next page >>

Geometric Series

Sun, 01 Mar 1998 00:00:00 +0000

NOTE: THIS DOCUMENT IS OBSOLETE, PLEASE CHECK THE NEW VERSION: "Mathematics of the Discrete Fourier Transform (DFT), with Audio Applications --- Second Edition", by Julius O. Smith III, W3K Publishing, 2007, ISBN 978-0-9745607-4-8. - Copyright © 2017-09-28 by Julius O. Smith III - Center for Computer Research in Music and Acoustics (CCRMA), Stanford University

<< Previous page TOC INDEX Next page >>
Geometric Series
Recall that for any compex number , the signal , , defines a geometric sequence, i.e., each term is obtained by multiplying the previous term by a (complex) constant. A geometric series is defined as the sum of a geometric sequence:

If , the sum can be expressed in closed form as

Proof: We have

<< Previous page TOC INDEX Next page >>

Orthogonality of Sinusoids

Sun, 01 Mar 1998 00:00:00 +0000

NOTE: THIS DOCUMENT IS OBSOLETE, PLEASE CHECK THE NEW VERSION: "Mathematics of the Discrete Fourier Transform (DFT), with Audio Applications --- Second Edition", by Julius O. Smith III, W3K Publishing, 2007, ISBN 978-0-9745607-4-8. - Copyright © 2017-09-28 by Julius O. Smith III - Center for Computer Research in Music and Acoustics (CCRMA), Stanford University

<< Previous page TOC INDEX Next page >>
Orthogonality of Sinusoids
A key property of sinusoids is that they are orthogonal at different frequencies. That is,

This is true whether they are complex or real, and whatever amplitude and phase they may have. All that matters is that the frequencies be different. Note, however, that the sinusoidal durations must be infinity.
For length sampled sinuoidal signal segments, such as used by the DFT, exact orthogonality holds only for the harmonics of the sampling rate divided by , i.e., only over the frequencies . These are the only frequencies that have an exact integer number of periods in samples (depicted in Fig. 7.2 for ).
The complex sinusoids corresponding to the frequencies are

These sinusoids are generated by the roots of unity in the complex plane:

These are called the roots of unity because each of them satisfies

The roots of unity are plotted in the complex plane in Fig. 7.1for . In general, for any , there will always be a point at , and the points equally subdivide the unit circle. When is even, there is a point at (corresponding to a sinusoid at exactly half the sampling rate), while if is odd, there is no point at .

Figure 7.1:The roots of unity for .
The sampled sinusoids corresponding to the roots of unity are plotted in Fig. 7.2. These are the sampled sinusoids used by the DFT. Note that taking successively higher integer powers of the point on the unit circlegenerates samples of the th DFT sinusoid, giving , . The th sinusoid generator is in turn theth power of the primitive th root of unity . The notation , , and are common in the digital signal processing literature.

Figure 7.2:The complex sinusoids used by the DFT for .
Note that in Fig. 7.2 the range of is taken to be instead of . This is the most ''physical'' choice since it corresponds with our notion of ''negative frequencies.'' However, we may add any integer multiple of to without changing the sinusoid indexed by . In other words, refers to the same sinusoid for all integer .
<< Previous page TOC INDEX Next page >>

Orthogonality of the DFT Sinusoids

Sun, 01 Mar 1998 00:00:00 +0000

NOTE: THIS DOCUMENT IS OBSOLETE, PLEASE CHECK THE NEW VERSION: "Mathematics of the Discrete Fourier Transform (DFT), with Audio Applications --- Second Edition", by Julius O. Smith III, W3K Publishing, 2007, ISBN 978-0-9745607-4-8. - Copyright © 2017-09-28 by Julius O. Smith III - Center for Computer Research in Music and Acoustics (CCRMA), Stanford University

<< Previous page TOC INDEX Next page >>
Orthogonality of the DFT Sinusoids
We now show mathematically that the DFT sinusoids are exactly orthogonal. Let

denote the th complex DFT sinusoid. Then

where the last step made use of the closed-form expression for the sum of ageometric series. If , the denominator is nonzero while the numerator is zero. This proves

While we only looked at unit amplitude, zero phase complex sinusoids, as used by the DFT, it is readily verified that the (nonzero) amplitude and phase have no effect on orthogonality.
<< Previous page TOC INDEX Next page >>

Norm of the DFT Sinusoids

Sun, 01 Mar 1998 00:00:00 +0000

NOTE: THIS DOCUMENT IS OBSOLETE, PLEASE CHECK THE NEW VERSION: "Mathematics of the Discrete Fourier Transform (DFT), with Audio Applications --- Second Edition", by Julius O. Smith III, W3K Publishing, 2007, ISBN 978-0-9745607-4-8. - Copyright © 2017-09-28 by Julius O. Smith III - Center for Computer Research in Music and Acoustics (CCRMA), Stanford University

<< Previous page TOC INDEX Next page >>
Norm of the DFT Sinusoids
For , we follow the previous derivation to the next-to-last step to get

which proves

<< Previous page TOC INDEX Next page >>

An Orthonormal Sinusoidal Set

Sun, 01 Mar 1998 00:00:00 +0000

NOTE: THIS DOCUMENT IS OBSOLETE, PLEASE CHECK THE NEW VERSION: "Mathematics of the Discrete Fourier Transform (DFT), with Audio Applications --- Second Edition", by Julius O. Smith III, W3K Publishing, 2007, ISBN 978-0-9745607-4-8. - Copyright © 2017-09-28 by Julius O. Smith III - Center for Computer Research in Music and Acoustics (CCRMA), Stanford University

<< Previous page TOC INDEX Next page >>
An Orthonormal Sinusoidal Set
We can normalize the DFT sinusoids to obtain an orthonormal set:

The orthonormal sinusoidal basis signals satisfy

We call these the normalized DFT sinusoids.
<< Previous page TOC INDEX Next page >>

Frequencies in the \

Sun, 01 Mar 1998 00:00:00 +0000

NOTE: THIS DOCUMENT IS OBSOLETE, PLEASE CHECK THE NEW VERSION: "Mathematics of the Discrete Fourier Transform (DFT), with Audio Applications --- Second Edition", by Julius O. Smith III, W3K Publishing, 2007, ISBN 978-0-9745607-4-8. - Copyright © 2017-09-28 by Julius O. Smith III - Center for Computer Research in Music and Acoustics (CCRMA), Stanford University

<< Previous page TOC INDEX Next page >>
Frequencies in the ''Cracks''
The DFT is defined only for frequencies . If we are analyzing one or more periods of an exactly periodic signal, where the period is exactly samples (or some integer divisor of ), then these really are the only frequencies present in the signal, and the spectrum is actually zero everywhere but at . However, we use the DFT to analyze arbitrary signals from nature. What happens when afrequency is present in a signal that is not one of the DFT-sinusoid freqencies ?
To find out, let's project a length segment of a sinusoid at an arbitrary freqency onto the th DFT sinusoid:

The coefficient of projection is proportional to

using the closed-form expression for a geometric series sum. As previously shown, the sum is at and zero at , for. However, the sum is nonzero at all other frequencies.
Since we are only looking at samples, any sinusoidal segment can be projected onto the DFT sinusoids and be reconstructed exactly by a linear combination of them. Another way to say this is that the DFT sinusoids form a basis for , so that any length signal whatsoever can be expressed as linear combination of them. Therefore, when analyzing segments of recorded signals, we must interpret what we see accordingly.
The typical way to think about this in practice is to consider the DFT operation as a digital filter.^7.1 The frequency response of this filter is what we just computed,^7.2 and its magnitude is

(shown in Fig. 7.3a for ). At all other integer values of, the response is the same but shifted (circularly) left or right so that the peak is centered on . The secondary peaks away from are called sidelobes of the DFT response, while the main peak may be called the main lobe of the response. Since we are normally most interested in spectra from an audio perspective, the same plot is repeated using a decibel vertical scale in Fig. 7.3b (clipped at dB). We see that the sidelobes are really quite high from an audio perspective. Sinusoids with frequencies near , for example, are only attenuated approximately dB in the DFT output .

Figure 7.3:Frequency response magnitude of a single DFT output sample.
We see that is sensitive to all frequencies between dc and the sampling rate except the other DFT-sinusoid frequencies for . This is sometimes called spectral leakageor cross-talk in the spectrum analysis. Again, there is no error when the signal being analyzed is truly periodic and we can choose to be exactly a period, or some multiple of a period. Normally, however, this cannot be easily arranged, and spectral leakage can really become a problem.
Note that spectral leakage is not reduced by increasing . It can be thought of as being caused by abruptly truncating a sinusoid at the beginning and/or end of the -sample time window. Only the DFT sinusoids are not cut off at the window boundaries. All other frequencies will suffer some truncation distortion, and the spectral content of the abrupt cut-off or turn-on transient can be viewed as the source of the sidelobes. Remember that, as far as the DFT is concerned, the input signal is the same as its periodic extension. If we repeat samples of a sinusoid at frequency , there will be a ''glitch'' every samples since the signal is not periodic in samples. This glitch can be considered a source of new energy over the entire spectrum.
To reduce spectral leakage (cross-talk from far-away frequencies), we typically use a window function, such as a ''raised cosine'' window, to taper the data record gracefully to zero at both endpoints of the window. As a result of the smooth tapering, the main lobe widens and the sidelobes decrease in the DFT response. Using no window is better viewed as using a rectangular window of length , unless the signal is exactly periodic in samples. These topics are considered further in Music 420 and in the ''Examples using the DFT'' chapter.
Since the th spectral sample is properly regarded as primarily a measure of spectral amplitude over the range to , this range is sometimes called a frequency bin (as in a ''storage bin'' for spectral energy). The frequency index is called the bin number, and can be regarded as the total energy in the th bin (see Parseval's Theorem in the ''Fourier Theorems'' chapter). Similar remarks apply to samples of any continuous bandlimited function; however, the term ''bin'' is only used in the frequency domain, even though it could be assigned exactly the same meaning mathematically in the time domain.
<< Previous page TOC INDEX Next page >>

Normalized DFT

Sun, 01 Mar 1998 00:00:00 +0000

NOTE: THIS DOCUMENT IS OBSOLETE, PLEASE CHECK THE NEW VERSION: "Mathematics of the Discrete Fourier Transform (DFT), with Audio Applications --- Second Edition", by Julius O. Smith III, W3K Publishing, 2007, ISBN 978-0-9745607-4-8. - Copyright © 2017-09-28 by Julius O. Smith III - Center for Computer Research in Music and Acoustics (CCRMA), Stanford University

<< Previous page TOC INDEX Next page >>
Normalized DFT
A more ''theoretically clean'' DFT is obtained by projecting onto thenormalized DFT sinusoids

In this case, the normalized DFT of is

which is also precisely the coefficient of projection of onto . The inverse normalized DFT is then more simply

While this definition is much cleaner from a ''geometric signal theory'' point of view, it is rarely used in practice since it requires more computation than the typical definition. However, note that the only difference between the forward and inverse transforms in this case is the sign of the exponent in the kernel.
<< Previous page TOC INDEX Next page >>

The Length 2 DFT

Sun, 01 Mar 1998 00:00:00 +0000

NOTE: THIS DOCUMENT IS OBSOLETE, PLEASE CHECK THE NEW VERSION: "Mathematics of the Discrete Fourier Transform (DFT), with Audio Applications --- Second Edition", by Julius O. Smith III, W3K Publishing, 2007, ISBN 978-0-9745607-4-8. - Copyright © 2017-09-28 by Julius O. Smith III - Center for Computer Research in Music and Acoustics (CCRMA), Stanford University

<< Previous page TOC INDEX Next page >>
The Length 2 DFT
The length DFT is particularly simple, since the basissinusoids are real:

The DFT sinusoid is a sampled constant signal, while is a sampled sinusoid at half the sampling rate.
Figure 7.4 illustrates the graphical relationships for the length DFT of the signal .

Figure 7.4:Graphical interpretation of the length 2 DFT.
Analytically, we compute the DFT to be

Note the lines of orthogonal projection illustrated in the figure. The ''time domain'' basis consists of the vectors , and theorthogonal projections onto them are simply the coordinate projections and . The ''frequency domain'' basis vectors are , and they provide an orthogonal basis set which is rotated degrees relative to the time-domain basis vectors. Projecting orthogonally onto them gives and , respectively. The original signal can be expressed as the vector sum of its coordinate projections (a time-domain representation), or as the vector sum of its projections onto the DFT sinusoids (a frequency-domain representation). Computing the coefficients of projection is essentially ''taking the DFT'' and constructing as the vector sum of its projections onto the DFT sinusoids amounts to ''taking the inverse DFT.''
<< Previous page TOC INDEX Next page >>

Matrix Formulation of the DFT

Sun, 01 Mar 1998 00:00:00 +0000

NOTE: THIS DOCUMENT IS OBSOLETE, PLEASE CHECK THE NEW VERSION: "Mathematics of the Discrete Fourier Transform (DFT), with Audio Applications --- Second Edition", by Julius O. Smith III, W3K Publishing, 2007, ISBN 978-0-9745607-4-8. - Copyright © 2017-09-28 by Julius O. Smith III - Center for Computer Research in Music and Acoustics (CCRMA), Stanford University

<< Previous page TOC INDEX Next page >>
Matrix Formulation of the DFT
The DFT can be formulated as a complex matrix multiply.

Subsections
Matrices
Matrix Multiplication
The DFT Matrix

<< Previous page TOC INDEX Next page >>

Matrices

Sun, 01 Mar 1998 00:00:00 +0000

NOTE: THIS DOCUMENT IS OBSOLETE, PLEASE CHECK THE NEW VERSION: "Mathematics of the Discrete Fourier Transform (DFT), with Audio Applications --- Second Edition", by Julius O. Smith III, W3K Publishing, 2007, ISBN 978-0-9745607-4-8. - Copyright © 2017-09-28 by Julius O. Smith III - Center for Computer Research in Music and Acoustics (CCRMA), Stanford University

<< Previous page TOC INDEX Next page >>
Matrices
A matrix is defined as a rectangular array of numbers, e.g.,

which is a (''two by two'') matrix. A general matrix may be , where is the number of rows, and is the number of columns. For example, the general matrix is

(Either square brackets or large parentheses may be used.) Theth element^7.3 of a matrix may be denoted by or . The rows and columns of matrices are normally numbered from instead of from; thus, and . When , the matrix is said to be square.
The transpose of a real matrix is denoted by and is defined by

Note that while is , its transpose is .
A complex matrix , is simply a matrix containing complex numbers. The transpose of a complex matrix is normally defined to includeconjugation. The conjugating transpose operation is called theHermitian transpose. To avoid confusion, in this tutorial, and the word ''transpose'' will always denote transpositionwithout conjugation, while conjugating transposition will be denoted by and be called the ''Hermitian transpose'' or the ''conjugate transpose.'' Thus,

Example: The transpose of the general matrix is

while the conjugate transpose of the general matrix is

A column-vector

is the special case of an matrix, and a row-vector

(as we have been using) is a matrix. In contexts where matrices are being used (only this section for this course), it is best to define all vectors as column vectors and to indicate row vectors using the transpose notation, as was done in the equation above.
<< Previous page TOC INDEX Next page >>

Matrix Multiplication

Sun, 01 Mar 1998 00:00:00 +0000

NOTE: THIS DOCUMENT IS OBSOLETE, PLEASE CHECK THE NEW VERSION: "Mathematics of the Discrete Fourier Transform (DFT), with Audio Applications --- Second Edition", by Julius O. Smith III, W3K Publishing, 2007, ISBN 978-0-9745607-4-8. - Copyright © 2017-09-28 by Julius O. Smith III - Center for Computer Research in Music and Acoustics (CCRMA), Stanford University

<< Previous page TOC INDEX Next page >>
Matrix Multiplication
Let be a general matrix and let denote a general matrix. Denote the matrix product by or . Then matrix multiplication is carried out by computing the inner product of every row of with every column of . Let theth row of be denoted by , , and theth column of by , . Then the matrix product is defined as

This definition can be extended to complex matrices by using a definition of inner product which does not conjugate its second argument.^7.4
Examples:

An matrix can only be multiplied on the right by an matrix, where is any positive integer. An matrix can only be multiplied on the left by a matrix, where is any positive integer. Thus, the number of columns in the matrix on the left must equal the number of rows in the matrix on the right.
Matrix multiplication is non-commutative, in general. That is, normally even when both products are defined (such as when the matrices are square.)
The transpose of a matrix product is the product of the transposes in reverse order:

The identity matrix is denoted by and is defined as

Identity matrices are always square. The identity matrix , sometimes denoted as , satisfies for every matrix . Similarly, , for every matrix .
As a special case, a matrix times a vector produces a new vector which consists of the inner product of every row of with

A matrix times a vector defines a linear transformationof . In fact, every linear function of a vector can be expressed as a matrix multiply. In particular, every linear filtering operation can be expressed as a matrix multiply applied to the input signal. As a special case, every linear, time-invariant (LTI) filtering operation can be expressed as a matrix multiply in which the matrix is Toeplitz, i.e., (constant along alldiagonals).
As a further special case, a row vector on the left may be multiplied by a column vector on the right to form a single inner product:

where the alternate transpose notation '''' is defined to include complex conjugation so that the above result holds also for complex vectors. Using this result, we may rewrite the general matrix multiply as

<< Previous page TOC INDEX Next page >>

The DFT Matrix

Sun, 01 Mar 1998 00:00:00 +0000

NOTE: THIS DOCUMENT IS OBSOLETE, PLEASE CHECK THE NEW VERSION: "Mathematics of the Discrete Fourier Transform (DFT), with Audio Applications --- Second Edition", by Julius O. Smith III, W3K Publishing, 2007, ISBN 978-0-9745607-4-8. - Copyright © 2017-09-28 by Julius O. Smith III - Center for Computer Research in Music and Acoustics (CCRMA), Stanford University

<< Previous page TOC INDEX Next page >>
DFT Matrix
The following example reinforces the discussion of the DFT matrix. We can simply create the DFT matrix in Matlab by taking the FFT of the identity matrix. Then we show that multiplying by the DFT matrix is equivalent to the FFT:
>> eye(4) ans = 1 0 0 0 0 1 0 0 0 0 1 0 0 0 0 1
>> S4 = fft(eye(4)) ans = 1.0000 1.0000 1.0000 1.0000
1.0000 0.0000 - 1.0000i -1.0000 0.0000 + 1.0000i 1.0000 -1.0000 1.0000 -1.0000
1.0000 0.0000 + 1.0000i -1.0000 0.0000 - 1.0000i

>> S4’ * S4 % Show that S4’ = inverse DFT (times N=4) ans = 4.0000 0.0000 0 0.0000 0.0000 4.0000 0.0000 0.0000 0 0.0000 4.0000 0.0000 0.0000 0.0000 0.0000 4.0000

>> x = [1; 2; 3; 4] x = 1 2 3 4

>> fft(x) ans = 10.0000
-2.0000 + 2.0000i -2.0000
-2.0000 - 2.0000i >> S4 * x ans = 10.0000
-2.0000 + 2.0000i -2.0000
-2.0000 - 2.0000i

<< Previous page TOC INDEX Next page >>

Matlab Examples

Sun, 01 Mar 1998 00:00:00 +0000

NOTE: THIS DOCUMENT IS OBSOLETE, PLEASE CHECK THE NEW VERSION: "Mathematics of the Discrete Fourier Transform (DFT), with Audio Applications --- Second Edition", by Julius O. Smith III, W3K Publishing, 2007, ISBN 978-0-9745607-4-8. - Copyright © 2017-09-28 by Julius O. Smith III - Center for Computer Research in Music and Acoustics (CCRMA), Stanford University

<< Previous page TOC INDEX Next page >>
Matlab Examples

Subsections
Figure 7.2
Figure 7.3
DFT Matrix

<< Previous page TOC INDEX Next page >>

Figure 7.2

Sun, 01 Mar 1998 00:00:00 +0000

NOTE: THIS DOCUMENT IS OBSOLETE, PLEASE CHECK THE NEW VERSION: "Mathematics of the Discrete Fourier Transform (DFT), with Audio Applications --- Second Edition", by Julius O. Smith III, W3K Publishing, 2007, ISBN 978-0-9745607-4-8. - Copyright © 2017-09-28 by Julius O. Smith III - Center for Computer Research in Music and Acoustics (CCRMA), Stanford University

<< Previous page TOC INDEX Next page >>
Figure 7.2
Below is the Matlab for Fig. 7.2:
N=8; fs=1;
n = [0:N-1]; % row t = [0:0.01:N]; % interpolated k=fliplr(n)’ - N/2; fk = kfs/N; wk = 2pifk; clf; for i=1:N subplot(N,2,2i-1); plot(t,cos(wk(i)t)) axis([0,8,-1,1]); hold on; plot(n,cos(wk(i)n),’’) if i==1 title(‘Real Part’); end; ylabel(sprintf(‘k=%d’,k(i))); if i==N xlabel(‘Time (samples)’); end; subplot(N,2,2i); plot(t,sin(wk(i)*t)) axis([0,8,-1,1]); hold on; plot(n,sin(wk(i)n),’’) ylabel(sprintf(‘k=%d’,k(i))); if i==1 title(‘Imaginary Part’); end; if i==N xlabel(‘Time (samples)’); end; end
<< Previous page TOC INDEX Next page >>

Figure 7.3

Sun, 01 Mar 1998 00:00:00 +0000

NOTE: THIS DOCUMENT IS OBSOLETE, PLEASE CHECK THE NEW VERSION: "Mathematics of the Discrete Fourier Transform (DFT), with Audio Applications --- Second Edition", by Julius O. Smith III, W3K Publishing, 2007, ISBN 978-0-9745607-4-8. - Copyright © 2017-09-28 by Julius O. Smith III - Center for Computer Research in Music and Acoustics (CCRMA), Stanford University

<< Previous page TOC INDEX Next page >>
Figure 7.3
Below is the Matlab for Fig. 7.3:
% Parameters (sampling rate = 1) N = 16; % DFT length k = N/4; % bin where DFT filter is centered wk = 2*pi*k/N; % normalized radian center-frequency for DFT_k() wStep = 2*pi/N; w = [0:wStep:2*pi - wStep]; % DFT frequency grid
interp = 10; N2 = interpN; % Denser grid showing “arbitrary” frequencies w2Step = 2pi/N2; w2 = [0:w2Step:2pi - w2Step]; % Extra dense frequency grid X = (1 - exp(j(w2-wk)N)) ./ (1 - exp(j(w2-wk))); % slightly offset to avoid divide by zero at wk X(1+k*interp) = N; % Fix divide-by-zero point (overwrite “NaN”)

% Plot spectral magnitude clf; magX = abs(X); magXd = magX(1:interp:N2); % DFT frequencies only subplot(2,1,1); plot(w2,magX,’-’); hold on; grid; plot(w,magXd,’*’); % Show DFT sample points title(‘DFT Amplitude Response at k=N/4’); xlabel(‘Normalized Radian Frequency (radians per sample)’); ylabel(‘Magnitude (Linear)’); text(-1,20,‘a)’);

% Same thing on a dB scale magXdb = 20log10(magX); % Spectral magnitude in dB % Since the zeros go to minus infinity, clip at -60 dB: magXdb = max(magXdb,-60ones(1,N2)); magXddb = magXdb(1:interp:N2); % DFT frequencies only subplot(2,1,2); hold off; plot(w2,magXdb,’-’); hold on; plot(w,magXddb,’*’); grid; xlabel(‘Normalized Radian Frequency (radians per sample)’); ylabel(‘Magnitude (dB)’); text(-1,40,‘b)’); print -deps ‘https://ccrma.stanford.edu/~jos//eps/dftfilter.eps'; hold off;
<< Previous page TOC INDEX Next page >>

Fourier Theorems for the DFT

Sun, 01 Mar 1998 00:00:00 +0000

NOTE: THIS DOCUMENT IS OBSOLETE, PLEASE CHECK THE NEW VERSION: "Mathematics of the Discrete Fourier Transform (DFT), with Audio Applications --- Second Edition", by Julius O. Smith III, W3K Publishing, 2007, ISBN 978-0-9745607-4-8. - Copyright © 2017-09-28 by Julius O. Smith III - Center for Computer Research in Music and Acoustics (CCRMA), Stanford University

<< Previous page TOC INDEX Next page >>
Fourier Theorems for the DFT
This chapter derives various Fourier theorems for the case of the DFT. Included are symmetry relations, the shift theorem, convolution theorem,correlation theorem, power theorem, and theorems pertaining to interpolation and downsampling. Applications related to certain theorems are outlined, including linear time-invariant filtering, sampling rate conversion, andstatistical signal processing.

Subsections
The DFT and its Inverse
Notation and Terminology
Modulo Indexing, Periodic Extension
Signal Operators
Flip Operator
Shift Operator
Convolution
Graphical Convolution
Polynomial Multiplication
Multiplication of Decimal Numbers
Correlation
Stretch Operator
Zero Padding
Repeat Operator
Decimation Operator
Alias Operator
Even and Odd Functions
The Fourier Theorems
Linearity
Conjugation and Reversal
Symmetry
Shift Theorem
Linear Phase Terms
Application of the Shift Theorem to FFT Windows
Convolution Theorem
Dual of the Convolution Theorem
Correlation Theorem
Power Theorem
Rayleigh Energy Theorem (Parseval's Theorem)
Stretch Theorem (Repeat Theorem)
Decimation Theorem (Aliasing Theorem)
Zero Padding Theorem
Bandlimited Interpolation in Time
Conclusions
Acknowledgement
Appendix A: Linear Time-Invariant Filters and Convolution
LTI Filters and the Convolution Theorem
Appendix B: Introductory Statistical Signal Processing
Cross-Correlation
Applications of Cross-Correlation
Autocorrelation
Coherence
Appendix C: Mathematica/Matlab Examples
Appendix D: The Similarity Theorem

<< Previous page TOC INDEX Next page >>

The DFT and its Inverse

Sun, 01 Mar 1998 00:00:00 +0000

NOTE: THIS DOCUMENT IS OBSOLETE, PLEASE CHECK THE NEW VERSION: "Mathematics of the Discrete Fourier Transform (DFT), with Audio Applications --- Second Edition", by Julius O. Smith III, W3K Publishing, 2007, ISBN 978-0-9745607-4-8. - Copyright © 2017-09-28 by Julius O. Smith III - Center for Computer Research in Music and Acoustics (CCRMA), Stanford University

<< Previous page TOC INDEX Next page >>
The DFT and its Inverse
Let denote an -sample complex sequence, i.e., . Then the spectrum of is defined by the Discrete Fourier Transform (DFT):

The inverse DFT (IDFT) is defined by

Note that for the first time we are not carrying along the sampling interval in our notation. This is actually the most typical treatment in the digital signal processing literature. It is often said that the sampling frequency is . However, it can be set to any desired value using the substitution

However, for the remainder of this reader, we will adopt the more common (and more mathematical) convention . In particular, we'll use the definition for this chapter only. In this case, a radian frequency is in units of ''radians per sample.'' Elsewhere in this course, always means ''radians persecond.'' (Of course, there's no difference when the sampling rateis really .) Another term we use in connection with the convention is normalized frequency: All normalized radian frequencies lie in the range , and all normalized frequencies in Hz lie in the range .

Subsections
Notation and Terminology
Modulo Indexing, Periodic Extension

<< Previous page TOC INDEX Next page >>

Notation and Terminology

Sun, 01 Mar 1998 00:00:00 +0000

NOTE: THIS DOCUMENT IS OBSOLETE, PLEASE CHECK THE NEW VERSION: "Mathematics of the Discrete Fourier Transform (DFT), with Audio Applications --- Second Edition", by Julius O. Smith III, W3K Publishing, 2007, ISBN 978-0-9745607-4-8. - Copyright © 2017-09-28 by Julius O. Smith III - Center for Computer Research in Music and Acoustics (CCRMA), Stanford University

<< Previous page TOC INDEX Next page >>
Notation and Terminology
If is the DFT of , we say that and form a transform pair and write

Another notation we'll use is

As we've already seen, time-domain signals are consistently denoted using lowercase symbols such as '','' while frequency-domainsignals (spectra), are denoted in uppercase ('''').
<< Previous page TOC INDEX Next page >>

Modulo Indexing, Periodic Extension

Sun, 01 Mar 1998 00:00:00 +0000

NOTE: THIS DOCUMENT IS OBSOLETE, PLEASE CHECK THE NEW VERSION: "Mathematics of the Discrete Fourier Transform (DFT), with Audio Applications --- Second Edition", by Julius O. Smith III, W3K Publishing, 2007, ISBN 978-0-9745607-4-8. - Copyright © 2017-09-28 by Julius O. Smith III - Center for Computer Research in Music and Acoustics (CCRMA), Stanford University

<< Previous page TOC INDEX Next page >>
Modulo Indexing, Periodic Extension
The DFT sinusoids are all periodichaving periods which divide . That is, for any integer . Since a length signal can be expressed as a linear combination of the DFT sinusoids in the time domain,

it follows that the ''automatic'' definition of beyond the range is periodic extension, i.e., for every integer .
Moreover, the DFT also repeats naturally every samples, since

because . (The DFT sinusoids behave identically as functions of and.) Accordingly, for purposes of DFT studies, we may define allsignals in as being single periods from an infinitely long periodic signal with period samples:

Definition: For any signal , we define

for every integer .
As a result of this convention, all indexing of signals and spectra^8.1 can be interpreted modulo , and we may write to emphasize this. Formally, '''' is defined as with chosen to give in the range .
As an example, when indexing a spectrum , we have that which can be interpreted physically as saying that the sampling rate is the samefrequency as dc for discrete time signals. In the time domain, we have what is sometimes called the ''periodic extension'' of . This means that the input to the DFT is mathematically treated as samples of a periodic signal with period seconds ( samples). The corresponding assumption in the frequency domain is that the spectrum is zero between frequency samples .
It is also possible to adopt the point of view that the time-domain signal consists of samples preceded and followed by zeros. In this case, the spectrum is nonzero between spectral samples, and the spectrum between samples can be reconstructed by means of bandlimited interpolation. This ''time-limited'' interpretation of the DFT input is considered in detail in Music 420 and is beyond the scope of Music 320 (except in the discussion of ''zero padding interpolation'' below).
<< Previous page TOC INDEX Next page >>

Signal Operators

Sun, 01 Mar 1998 00:00:00 +0000

NOTE: THIS DOCUMENT IS OBSOLETE, PLEASE CHECK THE NEW VERSION: "Mathematics of the Discrete Fourier Transform (DFT), with Audio Applications --- Second Edition", by Julius O. Smith III, W3K Publishing, 2007, ISBN 978-0-9745607-4-8. - Copyright © 2017-09-28 by Julius O. Smith III - Center for Computer Research in Music and Acoustics (CCRMA), Stanford University

<< Previous page TOC INDEX Next page >>
Signal Operators
It will be convenient in the Fourier theorems below to make use of the following signal operator definitions.

Subsections
Flip Operator
Shift Operator
Convolution
Graphical Convolution
Polynomial Multiplication
Multiplication of Decimal Numbers
Correlation
Stretch Operator
Zero Padding
Repeat Operator
Decimation Operator
Alias Operator

<< Previous page TOC INDEX Next page >>

Flip Operator

Sun, 01 Mar 1998 00:00:00 +0000

NOTE: THIS DOCUMENT IS OBSOLETE, PLEASE CHECK THE NEW VERSION: "Mathematics of the Discrete Fourier Transform (DFT), with Audio Applications --- Second Edition", by Julius O. Smith III, W3K Publishing, 2007, ISBN 978-0-9745607-4-8. - Copyright © 2017-09-28 by Julius O. Smith III - Center for Computer Research in Music and Acoustics (CCRMA), Stanford University

<< Previous page TOC INDEX Next page >>
Flip Operator

Definition: We define the flip operator by

which, by modulo indexing, is . The operator reverses the order of samples through of a sequence, leaving sample alone, as shown in Fig. 8.1a. Thanks to modulo indexing, it can also be viewed as ''flipping'' the sequence about the vertical axis, as shown in Fig. 8.1b. The interpretation of Fig. 8.1b is usually the one we want, and the operator is usually thought of as ''time reversal'' when applied to a signal or ''frequency reversal'' when applied to a spectrum .

Figure 8.1:Illustration of and for and two different domain interpretations: a) . b) .
<< Previous page TOC INDEX Next page >>

Shift Operator

Sun, 01 Mar 1998 00:00:00 +0000

NOTE: THIS DOCUMENT IS OBSOLETE, PLEASE CHECK THE NEW VERSION: "Mathematics of the Discrete Fourier Transform (DFT), with Audio Applications --- Second Edition", by Julius O. Smith III, W3K Publishing, 2007, ISBN 978-0-9745607-4-8. - Copyright © 2017-09-28 by Julius O. Smith III - Center for Computer Research in Music and Acoustics (CCRMA), Stanford University

<< Previous page TOC INDEX Next page >>
Shift Operator

Definition: The shift operator is defined by

and denotes the entire shifted signal. Note that since indexing is modulo , the shift is circular. However, we normally use it to represent time delay by samples. We often use the shift operator in conjunction with zero padding(appending zeros to the signal ) in order to avoid the ''wrap-around'' associated with a circular shift.

Figure 8.2:Successive one-sample shifts of a sampled periodic sawtooth waveform having first period .
Figure 8.2 illustrates successive one-sample delays of a periodic signal having first period given by .

Example: (an impulse delayed one sample).

Example: (a circular shift example).

Example: (another circular shift example).
<< Previous page TOC INDEX Next page >>

Convolution

Sun, 01 Mar 1998 00:00:00 +0000

NOTE: THIS DOCUMENT IS OBSOLETE, PLEASE CHECK THE NEW VERSION: "Mathematics of the Discrete Fourier Transform (DFT), with Audio Applications --- Second Edition", by Julius O. Smith III, W3K Publishing, 2007, ISBN 978-0-9745607-4-8. - Copyright © 2017-09-28 by Julius O. Smith III - Center for Computer Research in Music and Acoustics (CCRMA), Stanford University

<< Previous page TOC INDEX Next page >>
Convolution

Definition: The convolution of two signals and in is denoted '''' and defined by

Note that this is cyclic or ''circular'' convolution.^8.2 The importance of convolution in linear systems theory is discussed in §8.7
Convolution is commutative, i.e.,

Proof:

where in the first step we made the change of summation variable , and in the second step, we made use of the fact that any sum over all terms is equivalent to a sum from to .

Subsections
Graphical Convolution
Polynomial Multiplication
Multiplication of Decimal Numbers

<< Previous page TOC INDEX Next page >>

Graphical Convolution

Sun, 01 Mar 1998 00:00:00 +0000

NOTE: THIS DOCUMENT IS OBSOLETE, PLEASE CHECK THE NEW VERSION: "Mathematics of the Discrete Fourier Transform (DFT), with Audio Applications --- Second Edition", by Julius O. Smith III, W3K Publishing, 2007, ISBN 978-0-9745607-4-8. - Copyright © 2017-09-28 by Julius O. Smith III - Center for Computer Research in Music and Acoustics (CCRMA), Stanford University

<< Previous page TOC INDEX Next page >>
Graphical Convolution
Note that the cyclic convolution operation can be expressed in terms of previously defined operators as

where and . It is instructive to interpret the last expression above graphically.

Figure 8.3:Illustration of convolution of and its ''matched filter'' h=[1,0,0,0,0,1,1,1] ().
Figure 8.3 illustrates convolution of

For example, could be a ''rectangularly windowed signal, zero-padded by a factor of 2,'' where the signal happened to be dc (all s). For the convolution, we need

which is the same as . When , we say that ismatched filter for .^8.3 In this case, is matched to look for a ''dc component,'' and also zero-padded by a factor of . The zero-paddingserves to simulate acyclic convolution using circular convolution. The figure illustrates the computation of the convolution of and :

Note that a large peak is obtained in the convolution output at time . This large peak (the largest possible if all signals are limited to in magnitude), indicates the matched filter has ''found'' the dc signal starting at time . This peak would persist even if varioussinusoids at other frequencies and/or noise were added in.
<< Previous page TOC INDEX Next page >>

Polynomial Multiplication

Sun, 01 Mar 1998 00:00:00 +0000

NOTE: THIS DOCUMENT IS OBSOLETE, PLEASE CHECK THE NEW VERSION: "Mathematics of the Discrete Fourier Transform (DFT), with Audio Applications --- Second Edition", by Julius O. Smith III, W3K Publishing, 2007, ISBN 978-0-9745607-4-8. - Copyright © 2017-09-28 by Julius O. Smith III - Center for Computer Research in Music and Acoustics (CCRMA), Stanford University

<< Previous page TOC INDEX Next page >>
Polynomial Multiplication
Note that when you multiply two polynomials together, their coefficients are convolved. To see this, let denote the th-order polynomial

with coefficients , and let denote the th-order polynomial

with coefficients . Then we have [17]

<< Previous page TOC INDEX Next page >>

Multiplication of Decimal Numbers

Sun, 01 Mar 1998 00:00:00 +0000

NOTE: THIS DOCUMENT IS OBSOLETE, PLEASE CHECK THE NEW VERSION: "Mathematics of the Discrete Fourier Transform (DFT), with Audio Applications --- Second Edition", by Julius O. Smith III, W3K Publishing, 2007, ISBN 978-0-9745607-4-8. - Copyright © 2017-09-28 by Julius O. Smith III - Center for Computer Research in Music and Acoustics (CCRMA), Stanford University

<< Previous page TOC INDEX Next page >>
Multiplication of Decimal Numbers
Since decimal numbers are implicitly just polynomials in the powers of 10, e.g.,

it follows that multiplying two numbers convolves their digits. The only twist is that, unlike normal polynomial multiplication, we havecarries. That is, when a convolution result exceeds 10, we subtract 10 from the result and add 1 to the digit in the next higher place.
<< Previous page TOC INDEX Next page >>

Correlation

Sun, 01 Mar 1998 00:00:00 +0000

NOTE: THIS DOCUMENT IS OBSOLETE, PLEASE CHECK THE NEW VERSION: "Mathematics of the Discrete Fourier Transform (DFT), with Audio Applications --- Second Edition", by Julius O. Smith III, W3K Publishing, 2007, ISBN 978-0-9745607-4-8. - Copyright © 2017-09-28 by Julius O. Smith III - Center for Computer Research in Music and Acoustics (CCRMA), Stanford University

<< Previous page TOC INDEX Next page >>
Correlation

Definition: The correlation operator for two signals and in is defined as

We may interpret the correlation operator as

which is the coefficient of projection onto of advanced by samples (shifted circularly to the left by samples). The time shift is called the correlation lag, and is called a lagged product. Applications of correlation are discussed in §8.8.
<< Previous page TOC INDEX Next page >>

Stretch Operator

Sun, 01 Mar 1998 00:00:00 +0000

NOTE: THIS DOCUMENT IS OBSOLETE, PLEASE CHECK THE NEW VERSION: "Mathematics of the Discrete Fourier Transform (DFT), with Audio Applications --- Second Edition", by Julius O. Smith III, W3K Publishing, 2007, ISBN 978-0-9745607-4-8. - Copyright © 2017-09-28 by Julius O. Smith III - Center for Computer Research in Music and Acoustics (CCRMA), Stanford University

<< Previous page TOC INDEX Next page >>

Stretch Operator
Unlike all previous operators, the operator maps a length signal to a length signal, where and are integers. We use '''' instead of '''' as the time index to underscore this fact.

Figure:Illustration of .

Definition: A stretch by factor is defined by

Thus, to stretch a signal by the factor , insert zeros between each pair of samples. An example of a stretch by factor three is shown in Fig. 8.4. The example is

The stretch operator is used to describe and analyze upsampling, i.e., increasing the sampling rate by an integer factor.
<< Previous page TOC INDEX Next page >>

Zero Padding

Sun, 01 Mar 1998 00:00:00 +0000

NOTE: THIS DOCUMENT IS OBSOLETE, PLEASE CHECK THE NEW VERSION: "Mathematics of the Discrete Fourier Transform (DFT), with Audio Applications --- Second Edition", by Julius O. Smith III, W3K Publishing, 2007, ISBN 978-0-9745607-4-8. - Copyright © 2017-09-28 by Julius O. Smith III - Center for Computer Research in Music and Acoustics (CCRMA), Stanford University

<< Previous page TOC INDEX Next page >>
Zero Padding

Definition: Zero padding consists of appending zeros to a signal. It maps a length signal to a length signal, but need not be an integer multiple of :

For example,

The above definition is natural when represents a signal starting at time and extending for samples. If, on the other hand, we are zero-padding a spectrum, or we have a time-domain signal which has nonzero samples for negative time indices, then the zero padding is normally inserted between samples and for odd (note that ), and similarly for even . I.e., for spectra, zero padding is inserted at the point (). Figure 8.5 illustrates this second form of zero padding. It is also used in conjunction with zero-phase FFT windows(discussed a bit further below).

Figure 8.5:Illustration of frequency-domain zero padding: a) Original spectrum plotted over the domain where (i.e., as the spectral array would normally exist in a computer array). b) . c) The same signal plotted over the domain which is more natural for interpreting negative frequencies. d) .
Using Fourier theorems, we will be able to show that zero padding in the time domain gives bandlimited interpolation in the frequency domain. Similarly, zero padding in the frequency domain gives bandlimited interpolation in the time domain. This is how ideal sampling rate conversion is accomplished.
It is important to note that bandlimited interpolation is idealinterpolation in digital signal processing.
<< Previous page TOC INDEX Next page >>

Repeat Operator

Sun, 01 Mar 1998 00:00:00 +0000

NOTE: THIS DOCUMENT IS OBSOLETE, PLEASE CHECK THE NEW VERSION: "Mathematics of the Discrete Fourier Transform (DFT), with Audio Applications --- Second Edition", by Julius O. Smith III, W3K Publishing, 2007, ISBN 978-0-9745607-4-8. - Copyright © 2017-09-28 by Julius O. Smith III - Center for Computer Research in Music and Acoustics (CCRMA), Stanford University

<< Previous page TOC INDEX Next page >>
Repeat Operator
Like the operator, the operator maps a length signal to a length signal:

Definition: The repeat times operator is defined by

where . Thus, the operator simply repeats its input signal times.^8.4An example of is shown in Fig. 8.6. The example is

Figure:Illustration of .
A frequency-domain example is shown in Fig. 8.7. Figure 8.7a shows the original spectrum , Fig. 8.7b shows the same spectrum plotted over the unit circle in the plane, and Fig. 8.7c shows . The point (dc) is on the right-rear face of the enclosing box. Note that when viewed as centered about , is a somewhat ''triangularly shaped'' spectrum. The repeating block can be considered to extend from the point at to the point far to the left, or it can be considered the triangularly shaped ''baseband'' spectrum centered about .

Figure:Illustration of . a) Conventional plot of . b) Plot of over the unit circle in the plane. c) .
The repeat operator is used to state the Fourier theorem

That is, when you stretch a signal by the factor , its spectrum is repeated times around the unit circle.
<< Previous page TOC INDEX Next page >>

Decimation Operator

Sun, 01 Mar 1998 00:00:00 +0000

NOTE: THIS DOCUMENT IS OBSOLETE, PLEASE CHECK THE NEW VERSION: "Mathematics of the Discrete Fourier Transform (DFT), with Audio Applications --- Second Edition", by Julius O. Smith III, W3K Publishing, 2007, ISBN 978-0-9745607-4-8. - Copyright © 2017-09-28 by Julius O. Smith III - Center for Computer Research in Music and Acoustics (CCRMA), Stanford University

<< Previous page TOC INDEX Next page >>
Decimation Operator

Definition: Decimation by is defined as taking every th sample, starting with sample :

The operator maps a length signal down to a length signal. It is the inverse of the operator (but not vice versa), i.e.,

The stretch and decimation operations do not commute because they are linear time-varying operators. They can be modeled using time-varying switches controlled by the sample index .

Figure:Illustration of . The white-filled circles indicate the retained samples while the black-filled circles indicate the discarded samples.
An example of is shown in Fig. 8.8. The example is

<< Previous page TOC INDEX Next page >>

Alias Operator

Sun, 01 Mar 1998 00:00:00 +0000

NOTE: THIS DOCUMENT IS OBSOLETE, PLEASE CHECK THE NEW VERSION: "Mathematics of the Discrete Fourier Transform (DFT), with Audio Applications --- Second Edition", by Julius O. Smith III, W3K Publishing, 2007, ISBN 978-0-9745607-4-8. - Copyright © 2017-09-28 by Julius O. Smith III - Center for Computer Research in Music and Acoustics (CCRMA), Stanford University

<< Previous page TOC INDEX Next page >>
Alias Operator
Aliasing occurs when a signal is undersampled. If the signalsampling rate is too low, we get frequency-domain aliasing.
The topic of aliasing normally arises in the context of sampling a continuous-time signal. Shannon's Sampling Theorem says that we will have no aliasing due to sampling as long as the sampling rate is higher than twice the highest frequency present in the signal being sampled.
In this chapter, we are considering only discrete-time signals, in order to keep the math as simple as possible. Aliasing in this context occurs when a discrete-time signal is decimated to reduce its sampling rate. You can think of continuous-time sampling as the limiting case for which the starting sampling rate is infinity.
An example of aliasing is shown in Fig. 8.9. In the figure, the high-frequency sinusoid is indistinguishable from the lower frequency sinusoid due to aliasing. We say the higher frequencyaliases to the lower frequency. Undersampling in the frequency domain gives rise totime-domain aliasing. If time or frequency is not specified, the term ''aliasing'' normally means frequency-domain aliasing.

Figure 8.9:Example of aliasing due to undersampling in time.
The aliasing operator is defined by

Like the operator, the operator maps a length signal down to a length signal. The way to think of it is to partition the original samples in to blocks of length , with the first block extending from sample to sample , the second block from to , etc. Then just add up the blocks. This process is called aliasing. If the original signal is a time signal, it is called time-domain aliasing; if it is a spectrum, we call it frequency-domain aliasing, or just aliasing. Note that aliasing is not invertible. Once the blocks are added together, it is not possible to recover the original blocks, in general.
For example,

The alias operator is used to state the Fourier theorem

That is, when you decimate a signal by the factor , its spectrum is aliased by the factor .

Figure:Illustration of aliasing in the frequency domain. a) from Fig. 8.7c. b) First half of the original unit circle ( to ) wrapped around the new, smaller unit circle (which is magnified to the original size). c) Second half ( to ), also wrapped around the new unit circle. d) Overlay of components to be summed. e) Sum of components (the aliased spectrum). f) Both sum and overlay.
Figure 8.10 shows the result of applied to from Fig. 8.7c. Imagine the spectrum of Fig. 8.10a as being plotted on a piece of paper rolled to form a cylinder, with the edges of the paper meeting at (upper right corner of Fig. 8.10a). Then the operation can be simulated by rerolling the cylinder of paper to cut its circumference in half. That is, reroll it so that at every point, two sheets of paper are in contact at all points on the new, narrower cylinder. Now, simply add the values on the two overlapping sheets together, and you have the of the original spectrum on the unit circle. To alias by , we would shrink the cylinder further until the paper edges again line up, giving three layers of paper in the cylinder, and so on.
Figure 8.10b shows what is plotted on the first circular wrap of the cylinder of paper, and Fig. 8.10c shows what is on the second wrap. These are overlaid in Fig. 8.10d and added together in Fig. 8.10e. Finally, Fig. 8.10f shows both the addition and the overlay of the two components. We say that the second component (Fig. 8.10c) ''aliases'' to new frequency components, while the first component (Fig. 8.10b) is considered to be at its original frequencies. If the unit circle of Fig. 8.10a covers frequencies to , all other unit circles (Fig. 8.10b-c) cover frequencies to .
In general, aliasing by the factor corresponds to a sampling-rate reduction by the factor . To prevent aliasing when reducing the sampling rate, an anti-aliasing lowpass filter is generally used. The lowpass filter attenuates all signal components at frequencies outside the interval . If they are not filtered out, they will alias, and aliasing is generally non-invertible.
Conceptually, the unit circle is reduced by to a unit circlehalf the original size, where the two halves are summed. The inverse of aliasing is then ''repeating'' which should be understood as increasing the unit circle circumference using ''periodic extension'' to generate ''more spectrum'' for the larger unit circle. In the time domain,decimation is the inverse of the stretch operator. All of these relationships are precise only for integerstretch/decimation/aliasing/repeat factors; in continuous time, the restriction to integer factors is removed, and we obtain the (simpler) similarity theorem (proved in §8.10).
<< Previous page TOC INDEX Next page >>

Even and Odd Functions

Sun, 01 Mar 1998 00:00:00 +0000

NOTE: THIS DOCUMENT IS OBSOLETE, PLEASE CHECK THE NEW VERSION: "Mathematics of the Discrete Fourier Transform (DFT), with Audio Applications --- Second Edition", by Julius O. Smith III, W3K Publishing, 2007, ISBN 978-0-9745607-4-8. - Copyright © 2017-09-28 by Julius O. Smith III - Center for Computer Research in Music and Acoustics (CCRMA), Stanford University

<< Previous page TOC INDEX Next page >>
Even and Odd Functions
Some of the Fourier theorems can be succinctly expressed in terms of even and odd symmetries.

Definition: A function is said to be even if . An even function is alsosymmetric, but the term symmetric applies also to functions symmetric about a point other than .

Definition: A function is said to be oddif . An odd function is also calledantisymmetric.
Note that every odd function must satisfy . Moreover, for any with even, we also have since , i.e., and index the same point.

Theorem: Every function can be decomposed into a sum of its even part and odd part , where

Proof: In the above definitions, is even and is odd by construction. Summing, we have

Theorem: The product of even functions is even, the product of odd functions is even, and the product of an even times an odd function is odd.
Proof: Readily shown.
Since even times even is even, odd times odd is even, and even times odd is odd, we can think of even as and odd as : , , and .

Example: is an even signal since .

Example: is an odd signal since .

Example: is odd (even times odd).

Example: is even (odd times odd).

Theorem: The sum of all the samples of an odd signal in is zero.
Proof: This is readily shown by writing the sum as , where the last term only occurs when is even. Each term so written is zero for an odd signal .

Example: For all DFT sinusoidal frequencies ,

More generally,

for any even signal and odd signal in .
<< Previous page TOC INDEX Next page >>

AWG
Temperature	[ºC]
Material
	OUTPUT
Copper Diameter	[mm]
Copper Area	[mm²]
Resistance	[Ohm/m]

Dec	Hex	Char	Description	Dec	Hex	Char	Dec	Hex	Char	Dec	Hex	Char
0	0	NUL	null	32	20		64	40	@	96	60	`
1	1	SOH	start of heading	33	21	!	65	41	A	97	61	a
2	2	STX	start of text	34	22	"	66	42	B	98	62	b
3	3	ETX	end of text	35	23	#	67	43	C	99	63	c
4	4	EOT	end of transmission	36	24	$	68	44	D	100	64	d
5	5	ENQ	enquiry	37	25	%	69	45	E	101	65	e
6	6	ACK	acknowledge	38	26	&	70	46	F	102	66	f
7	7	BEL	bell	39	27	'	71	47	G	103	67	g
8	8	BS	backspace	40	28	(	72	48	H	104	68	h
9	9	TAB	horizontal tab	41	29	)	73	49	I	105	69	i
10	A	LF	NL line feed, new line	42	2A	*	74	4A	J	106	6A	j
11	B	VT	vertical tab	43	2B	+	75	4B	K	107	6B	k
12	C	FF	NP form feed, new page	44	2C	,	76	4C	L	108	6C	l
13	D	CR	carriage return	45	2D	-	77	4D	M	109	6D	m
14	E	SO	shift out	46	2E	.	78	4E	N	110	6E	n
15	F	SI	shift in	47	2F	/	79	4F	O	111	6F	o
16	10	DLE	data link escape	48	30	0	80	50	P	112	70	p
17	11	DC1	device control 1	49	31	1	81	51	Q	113	71	q
18	12	DC2	device control 2	50	32	2	82	52	R	114	72	r
19	13	DC3	device control 3	51	33	3	83	53	S	115	73	s
20	14	DC4	device control 4	52	34	4	84	54	T	116	74	t
21	15	NAK	negative acknowledge	53	35	5	85	55	U	117	75	u
22	16	SYN	synchronous idle	54	36	6	86	56	V	118	76	v
23	17	ETB	end of transmission block	55	37	7	87	57	W	119	77	w
24	18	CAN	cancel	56	38	8	88	58	X	120	78	x
25	19	EM	end of medium	57	39	9	89	59	Y	121	79	y
26	1A	SUB	substitute	58	3A	:	90	5A	Z	122	7A	z
27	1B	ESC	escape	59	3B	;	91	5B	[	123	7B	{
28	1C	FS	file separator	60	3C	<	92	5C	\	124	7C	\|
29	1D	GS	group separator	61	3D	=	93	5D	]	125	7D	}
30	1E	RS	record separator	62	3E	>	94	5E	^	126	7E	~
31	1F	US	unit separator	63	3F	?	95	5F	_	127	7F	DEL

Material	Min.	Max.
Air	1	1
Amber	2.6	2.7
Asbestos fiber	3.1	4.8
Bakelite	5	22
Barium Titanate	100	1250
Beeswax	2.4	2.8
Cambric	4	4
Carbon Tetrachloride	2.17	2.17
Celluloid	4	4
Cellulose Acetate	2.9	4.5
Durite	4.7	5.1
Ebonite	2.7	2.7
Epoxy Resin	3.4	3.7
Ethyl Alcohol	6.5	25
Fiber	5	5
Formica	3.6	6
Glass	3.8	14.5
Glass Pyrex	4.6	5
Gutta Percha	2.4	2.6
Isolantite	6.1	6.1
Kevlar	3.5	4.5
Lucite	2.5	2.5
Mica	4	9
Micarta	3.2	5.5
Mycalex	7.3	9.3
Neoprene	4	6.7
Nylon	3.4	22.4
Paper	1.5	3
Paraffin	2	3
Plexiglass	2.6	3.5
Polycarbonate	2.9	3.2
Polyethylene	2.5	2.5
Polyimide	3.4	3.5
Polystyrene	2.4	3
Porcelain	5	6.5
Quartz	5	5
Rubber	2	4
Ruby Mica	5.4	5.4
Selenium	6	6
Shellac	2.9	3.9
Silicone	3.2	4.7
Slate	7	7
Soil dry	2.4	2.9
Steatite	5.2	6.3
Styrofoam	1.03	1.03
Teflon	2.1	2.1
Titanium Dioxide	100	100
Vaseline	2.16	2.16
Vinylite	2.7	7.5
Water distilled	34	78
Waxes, Mineral	2.2	2.3
Wood dry	1.4	2.9

Symbol	Name	Factor	Symbol	Name	Factor
Y	yotta	10²⁴	y	yokto	10^-24
Z	zetta	10²¹	z	zepto	10^-21
E	exa	10¹⁸	a	atto	10^-18
P	peta	10¹⁵	f	femto	10^-15
T	tera	10¹²	p	pico	10^-12
G	giga	10⁹	n	nano	10^-9
M	mega	10⁶	µ	micro	10^-6
k	kilo	10³	m	milli	10^-3
h	hecto	10²	c	centi	10^-2
da	deka	10¹	d	deci	10^-1

W		[m]	width of the trace
H		[m]	height of trace above return plane
H1		[m]	height of trace above return plane
T		[m]	trace thickness
E_r		[ ]	relative permittivity of the dielectric
	Are there distributed capacitive loads on this trace? NoYes
L_a		[m]	average length of the traces attaching the loads
C_a		[pF]	average load capacitance
	OUTPUT
Z₀		[Ohm]	characteristic impedance
C₀		[F/m]	capacitance per unit length
t_pd		[s/m]	propagation delay
L₀		[H/m]	inductance per unit length
Z_c		[Ohm]	effective characteristic impedance
t'_pd		[s/m]	effective propagation delay

Component	CAS Reg Number	OSHA Hazard	Percentage
SODIUM METASILICATE	6834-92-0	Y	100

PARTICULATES NOT OTHERWISE REGULATED RESPIRABLE FRACTION
	Notes	TWA	STEL
OSHA		5 mg/cu m

PARTICULATES NOT OTHERWISE REGULATED TOTAL DUST
	Notes	TWA	STEL
OSHA		15 mg/cu m

(1)	Do not store, use, and/or consume foods, beverages, tobacco products, or cosmetics in areas where this material is stored.
(2)	Wash hands and face carefully before eating, drinking, using tobacco, applying cosmetics, or using the toilet.
(3)	Wash exposed skin promptly to remove accidental splashes of contact with this material.

	Toxicological Information and Interpretation
	eye - eye irritation, 0.1 ml, rabbit. Corrosive.

	Toxicological Information and Interpretation
	skin - skin irritation, rabbit.
	Non-irritating. Dry skin. (At 4 hours.).
	skin - skin irritation, rabbit. Corrosive. Moist skin. (At 4 hours.).

	Toxicological Information and Interpretation
	LD50 - lethal dose 50% of test species, 847 mg/kg, rat.

Inventory	Status
UNITED STATES (TSCA)	Y
CANADA (DSL)	Y
EUROPE (EINECS/ELINCS)	Y
AUSTRALIA (AICS)	Y
JAPAN (MITI)	Y
SOUTH KOREA (KECL)	Y

Y = All ingredients are on the inventory.
E = All ingredients are on the inventory or exempt from listing.
P = One or more ingredients fall under the polymer exemption or are on the no longer polymer list. All other ingredients are on the inventory or exempt from listing.
N = Not determined or one or more ingredients are not on the inventory and are not exempt from listing.

Fire Hazard	- NO
Reactive Hazard	- NO
Release of Pressure	- NO
Acute Health Hazard	- YES
Chronic Health Hazard	- NO

3	Health Hazard Rating--Serious
0	Flammability Rating--Minimal
0	Instability Rating--Minimal

3	Health Hazard Rating--Serious
0	Flammability Rating--Minimal
0	Reactivity Rating--Minimal

VDE Testing and Certification Institute
Mark	Description
	VDE Mark for appliances as technical equipment according to the Appliance Safety Law (GSG), for Medical Device Law (MPG), components and installation materials.The VDE Mark indicates conformity with the VDE standards or European or internationally harmonized standards resp. and confirms compliance with protective requirements of the applicable EC Directive(s). The VDE Mark is a symbol for electrical, mechanical, thermal, toxic, radiological and other hazards.
	For appliances as technical equipment according to the GSG.For ready-to-use equipment, the licence holder may chose to affix the VDE Mark or the VDE GS Mark.
	For products certified on the basis of harmonized certification agreements.Testing is based on harmonized European standards listed in the ENEC Agreement. Products (at present luminaires and related components, energy saving lamps, IT equipment, transformers, switches for appliances, electrical controls, certain types of capacitors and EMI suppression components) tested to tested to the listed standards may be marked with the ENEC Mark of the VDE. The approval of any other body participating in the ENEC Agreement is not required.
	For appliances in compliance with standards for electromagnetic compatibility.The VDE EMC Mark expresses the conformity of a product with applicable standards for electromagnetic compatibility. The reliable function of the product in its electromagnetic environment is also included. The requirements for granting this mark comprise automatically and without restriction the compliance with applicable standards.
	For cables, insulated cords, installation conduits and ducts, the VDE Cable Mark is applicable. For cables and cords, the VDE Identification Thread may be used.
	VDE-HARmonization Marking The VDE HARmonization Marking or VDE HARmonization Thread resp. for cables and insulated cords according to harmonized certification procedures.Testing is based on the Harmonization Documents (HD) listed in the HAR Agreement. Products (harmonized power cables) tested and found in compliance with with the requirements of the mentioned standards may be marked with the VDE HARmonization Marking. Further information is available from the Laboratory for Cables and Cords, Materials and Special Tests.
	The VDE Component Mark may be used for electronic components.
	The CECC Mark for electronic components according to CECC Specifications.For electronic components according to CECC Specifications (CECC: CENELEC Electronic Components Committee) the CECC Mark may be used.
	VDE-Reg.-Nr. XXXXX VDE-Reg.-Nr. (VDE Certificate of Conformity in conjunction with factory surveillance)This mark is used in two cases: firstly, for products in compliance with applicable clauses of VDE standards in the absence of a fully applicable VDE standard, and secondly, if a product, e.g. a sub-assembly, requires the fulfillment of additional conditions when incorporated into complete equipment. For cables and insulated cords, the VDE-Reg.-Nr. or the relevant mark resp. is applicable in absence of special regulations for products which were tested on the basis other standards. Special constructions and all variations of non-harmonized cables and insulated cords belong to this category of products.

Tolerance [%]	k	Series Name
20	6	E6
10	12	E12
5	24	E24
2	48	E48
1	96	E96
0.5	192	E192
0.25	192	E192
0.1	192	E192

	Slow Charge (0.1)	Fast Charge (0.5-1C)
Nickel Cadmium	0°C to 45°C (32°F to 113°F)	5°C to 45°C (41°F to 113°F)
Nickel-Metal Hydride	0°C to 45°C (32°F to 113°F)	10C° to 45°C (50°F to 113°F)
Lead Acid	0°C to 45°C (32°F to 113°F)	5C° to 45°C (41°F to 113°F)
Lithium Ion	0°C to 45°C (32°F to 113°F)	5C° to 45°C (41°F to 113°F)

Table 1 Informal Summary of Static Losses by Level
Static Losses Reported
Description	Min. Loss	Max. Loss	Est. Avg. Loss
Component Manufacturers	4%	97%	16-22%
Subcontractors	3%	70%	9-15%
Contractors	2%	35%	8-14%
User	5%	70%	27-33%
Source: Stephen Halperin, "Guidelines for Static Control Management," Eurostat, 1990.

Table 2 Examples of Static Generation Typical Voltage Levels
Means of Generation	10-25% RH	65-90% RH
Walking across carpet	35,000V	1,500V
Walking across vinyl tile	12,000V	250V
Worker at bench	6,000V	100V
Poly bag picked up from bench	20,000V	1,200V
Chair with urethane foam	18,000V	1,500V

Table 4 Susceptibility of Electronic Components to ESD
Device Type	ESD Susceptibility (Volts)
VMOS	30 - 1,200
Mosfet, GaAsfet, EPROM	100 - 300
JFET	150 - 7,000
OP-AMP	190 - 2,500
Schottky Diodes	300 - 2,500
Film Resistors	300 - 3,000
Schottky TTL	1,000 - 2,500

WINDOWS^® 2000 PROFESSIONAL Windows 2000 Professional runs software applications faster. Notebook- ready and designed with the future in mind, Windows 2000 Professional allows users to take advantage of a full- range of features today. Windows 2000 Professional is future- ready and promises to run today's and tomorrow's applications better. Baseline: 64MB - 128MB Optimal: 128MB - 512MB
Administrative & Service	Light- Word processing, email, data- entry	64MB - 96MB
	Medium- Fax /communications, database administration, spreadsheets; >2 applications open at a time	64MB - 128MB
	Heavy- Complex documents, accounting, business graphics, presentation software, network connectivity	96MB - 256MB
Executives & Analysts	Light- Proposals, reports, spreadsheets, business graphics, databases, scheduling, presentations	64MB - 96MB
	Medium- Complex presentations, sales/ market analysis, project management, Internet access	96MB - 128MB
	Heavy- Statistical applications, large databases, research/ technical analysis, complex presentations, video conferencing	128MB - 512MB
Engineers&Designers	Light- Page layout, 2 - 4 color line drawings, simple image manipulation, simple graphics	96MB - 128MB
	Medium- 2D CAD, rendering, multimedia presentations, simple photo- editing, Web development	128MB - 512MB
	Heavy- Animation, complex photo- editing, real- time video, 3D CAD, solid modeling, finite element analysis	256MB - 1GB

WINDOWS^® 98 Windows 98 requires 16 - 32MB to run basic applications. Tests show 45 - 65% performance improvements at 64MB and beyond. Baseline: 32MB - 64MB Optimal: 128MB - 256MB
Students	Light- Word processing, basic financial management, email and other light Internet use	32MB - 64MB
	Medium- Home office applications, games, Internet surfing, downloading images, spreadsheets, presentations	64MB - 128MB
	Heavy- Multimedia use such as video, graphics, music, voice recognition, design, complex images	128MB - 384MB
Home Users	Light- Word processing, basic financial management, email and other light Internet use	32MB - 48MB
	Medium- Home office applications, games, Internet surfing, downloading images, spreadsheets, presentations	48MB - 64MB
	Heavy- Multimedia use such as video, graphics, music, voice recognition, design, complex images	64MB - 128MB

LINUX^® The Linux operating system is quickly gaining popularity as an alternative to Microsoft Windows. It includes true multitasking, virtual memory, shared libraries, demand loading, proper memory management, TCP/ IP networking, and other features consistent with Unix- type systems. Baseline: 48MB - 112MB Optimal: 112MB - 512MB
Administrative&Service	Light- Word processing, email, data- entry	48MB - 80MB
	Medium- Fax /communications, database administration, spreadsheets; >2 applications open at a time	48MB - 112MB
	Heavy- Complex documents, accounting, business graphics, presentation software, network connectivity	80MB - 240MB
Executives&Analysts	Light- Proposals, reports, spreadsheets, business graphics, databases, scheduling, presentations	48MB - 80MB
	Medium- Complex presentations, sales/ market analysis, project management, Internet access	80MB - 112MB
	Heavy- Statistical applications, large databases, research/ technical analysis, complex presentations, video conferencing	112MB - 512MB
Engineers&Designers	Light- Page layout, 2 - 4 color line drawings, simple image manipulation, simple graphics	80MB - 112MB
	Medium- 2D CAD, rendering, multimedia presentations, simple photo- editing, Web development	112MB - 512MB
	Heavy- Animation, complex photo- editing, real- time video, 3D CAD, solid modeling, finite element analysis	240MB - 1GB

MACINTOSH^™ OS The Macintosh operating system manages memory in substantially different ways than other systems. Still, System 9.0 users will find that 48MB is a bare minimum. When using PowerMac ^® applications with Internet connectivity, plan on a range between 64 and 128MB as a minimum. Baseline: 48MB - 64MB Optimal: 128MB - 512MB
Administrative&Service	Light- Word processing, email, data- entry	48MB - 64MB
	Medium- Fax /communications, database administration, spreadsheets; >2 applications open at a time	64MB - 96MB
	Heavy- Complex documents, accounting, business graphics, presentation software, network connectivity	96MB - 128MB
Executives&Analysts	Light- Proposals, reports, spreadsheets, business graphics, databases, scheduling, presentations	64MB - 256MB
	Medium- Complex presentations, sales/ market analysis, project management, Internet access	128MB - 1GB
	Heavy- Statistical applications, large databases, research/ technical analysis, complex presentations, video conferencing	96MB - 128MB
Engineers&Designers	Light- Page layout, 2 - 4 color line drawings, simple image manipulation, simple graphics	128MB - 512MB
	Medium- 2D CAD, rendering, multimedia presentations, simple photo- editing, Web development	256MB - 1GB
	Heavy- Animation, complex photo- editing, real- time video, 3D CAD, solid modeling, finite element analysis	512MB - 2GB

WINDOWS^® 2000 SERVER Designed to help businesses of all sizes run better, Windows 2000 Server offers a manageable, reliable and internet- ready solution for today's growing enterprises. For optimal performance, consider adding more memory to take advantage of Windows 2000 Server's robust feature set. Windows 2000 Server is internet- ready and promises to run today's and tomorrow's applications better. Baseline: 128MB Optimal: 256MB - 1GB
Application Server	Houses one or more applications to be accessed over a wide user base	256MB - 4GB
Directory Server	Central Management of network resources	128MB - 1GB
Print Server	Distributes print jobs to appropriate printers	128MB - 512MB
Communication Server	Manages a variety of communications such as PBX, Voicemail, Email, and VPN	512MB - 2GB
Web Server	Internet and intranet solutions	512MB - 2GB
Database Server	Manages simple to complex databases of varying sizes	256MB - 4GB

LINUX^® Linux is a reliable, cost- effective alternative to traditional UNIX servers. Depending on the distribution, the Linux server platform features a variety of utilities, applications, and services. Baseline: 64MB - 128MB Optimal: 256MB - 1GB
Application Server	Houses one or more applications to be accessed over a wide user base	64MB - 4GB
Directory Server	Central Management of network resources	128MB - 1GB
Print Server	Distributes print jobs to appropriate printers	128MB - 512MB
Communication Server	Manages a variety of communications such as PBX, Voicemail, Email, and VPN	512MB - 2GB
Web Server	Internet and intranet solutions	512MB - 2GB
Database Server	Manages simple to complex databases of varying sizes	256MB - 4GB

Table 3 Sample Audit Check List ESD Control Program
Function/Area Audited: Personnel
Date: Auditor:
Audit Questions	Y	N	Comments
1. Where ESD protective flooring is used for personnel grounding, are foot grounding devices or conductive footwear worn?
2. Where conductive floors and footwear are used for personnel grounding, do personnel check continuity to ground upon entering the area?
3. Are personnel wearing grounded wrist straps at the ESD protective workstations?
4. Are personnel checking wrist straps for continuity or using a continuous ground monitor?
5. Where continuous ground monitors are not used, are wrist straps checked and logged routinely and at frequent intervals?
6. Are wriststrap checkers and continuous ground monitors checked and maintained periodically?
7. Do wrist straps and foot grounders fit correctly?
8. Are wrist straps and foot grounders working correctly?
9. Are wrist strap cords checked, on the person, at the workstation?
10. Are disposable foot grounders limited to one time use?
11. Are test records for wrist straps and foot grounders kept and maintained?
12. When required, are ESD protective garments correctly worn?
13. Are nonessential personal items kept out of ESD controlled areas?
14. Are finger cots and gloves being used on ESD sensitive hardware made of ESD protective material, or treated to be ESD protective?
15. Are personnel working in the ESD controlled area currently certified or escorted?
16. Are all personnel with access to the ESD controlled area trained?
17. Are ESD Control requirements imposed on visitors?

COMPONENT	CAPACITY EXPRESSION	CAPACITY UNITS	EXAMPLE
Chips	Chip Density	Mbit (megabits)	64Mbit
Memory Modules	Module Capacity	MB (megabytes)	64MB

	CHIP DEPTH IN MILLIONS OF LOCATIONS	CHIP WIDTH IN BITS	CHIP DENSITY = DEPTH x WIDTH
16Mbit Chips 4Mx4 1Mx16 2Mx8 16MxI	4 1 2 16	4 16 8 1	16 16 16 16
64Mbit Chips 4Mx16 8Mx8 16Mx4	4 8 16	16 8 4	64 64 64
128Mbit Chips 8Mx16 16Mx8 32Mx4	8 16 32	16 8 4	128 128 128
256Mbit Chips 32Mx8	32	8	256