Basic Electronic: Electrical and Electronic Systems

ELECTRONICSA SYSTEMS APPROACH

Neil Storey

Fourth Edition

Fourth Edition

ELECTRONICSA SYSTEMS APPROACHFourth Edition

Electronics plays a central role in our everyday lives, being at the heart of much of todays essential technology from mobile phones to computers and from cars to power stations. As such, all engineers, scientists and technologists need a basic understanding of thisarea, whilst many will require a far greater knowledge of the subject.

The fourth edition of Electronics: A Systems Approach is an outstanding introduction to this fast-moving, important field. Fully updated, it covers the latest changes and developments in the world of electronics. It continues to use Neil Storeys well-respected systems approach, firstly explaining the overall concepts to build students confidence and understanding before looking at the more detailed analysis that follows. This allows the student to contextualise what the system is designed to achieve, before tackling the intricacies of the individual components. The book also offers an integrated treatment of analogue and digital electronics, highlighting and exploring the common ground betweenthe two fields.

This fourth edition represents a significant update and a major expansion of previous material, and now provides a comprehensive introduction to basic electrical engineering circuits and components, in addition to a detailed treatment of electronic systems. This extended coverage allows the book to be used as a stand-alone text for introductory courses in both Electronics and Electrical Engineering.

This new edition includes:l A range of new chapters covering the basics of Electrical Circuits and Componentsl An introduction to Resistive, Capacitive and Inductive elements, Alternating Voltages and Currents, and AC Powerl New chapters on the Frequency Characteristics of AC circuits and on Transient Behaviourl A new, consolidated treatment of Noise and Electromagnetic Compatibility (EMC)l A new chapter on the Internal Circuitry of Operational Amplifiers.

Throughout the book learning is reinforced by chapter objectives, end-of-chapter summaries, worked examples and exercises.

Dr. Neil Storey is a member of the School of Engineering at the University of Warwick, where he has many years of experience in teaching electronics to undergraduate, post-graduate and professional engineers. He is also the author of Electrical and Electronic Systems and Safety-Critical Computer Systems, both published by Pearson Education.

www.pearson-books.com

ELECTRONICSA SYSTEM

S APPROACHNeil Storey

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E L E C T RO N I C SE L E C T RO N I C S

Visit the Electronics, fourth edition Companion Websiteat www.pearsoned.co.uk/Storey-elec to find valuablestudent learning material including:

Computer-marked self-assessment questions to checkyour understanding

Demonstration files that can be downloaded for usewith the Computer Simulation exercises.

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We work with leading authors to develop the strongest educational materials in electronics, bringing cutting-edge thinking and best learning practice to a global market.

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ELECTRONICSA SYSTEMS APPROACH

Fourth Edition

Neil StoreyUniversity of Warwick

ELECTRONICSELECTRONICS

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Pearson Education LimitedEdinburgh GateHarlowEssex CM20 2JEEngland

and Associated Companies throughout the world

Visit us on the World Wide Web at:www.pearsoned.co.uk

First published 1992Fourth edition published 2009

Addison-Wesley Publishers Limited 1992 Pearson Education Limited 1998, 2006, 2009

The right of Neil Storey to be identified as author of this work hasbeen asserted by him in accordance with the Copyright, Designs and Patents Act 1988.

All rights reserved. No part of this publication may be reproduced, stored in a retrievalsystem, or transmitted in any form or by any means, electronic, mechanical,photocopying, recording or otherwise, without either the prior written permission of thepublisher or a licence permitting restricted copying in the United Kingdom issued by theCopyright Licensing Agency Ltd, Saffron House, 610 Kirby Street, London EC1N 8TS.

All trademarks used herein are the property of their respective owners. The use of any trademark in this text does not vest in the author or publisher any trademark ownership rights in such trademarks, nor does the use of such trademarks imply any affiliation with or endorsement of this book by such owners.

ISBN: 978-0-273-71918-2

British Library Cataloguing-in-Publication DataA catalogue record for this book is available from the British Library

Library of Congress Cataloging-in-Publication DataStorey, Neil.

Electronics : a systems approach / Neil Storey. 4th ed.p. cm.

ISBN 978-0-273-71918-2 (pbk.)1. Electronic systems. 2. Electronic circuits. I. Title.

TK7870.S857 2009621.381dc22

2008052845

10 9 8 7 6 5 4 3 2 113 12 11 10 09

Typeset in 10/11pt Ehrhardt by 35Printed by Ashford Colour Press Ltd., Gosport

The publishers policy is to use paper manufactured from sustainable forests.

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15 Operational Amplifiers 27516 Semiconductors and Diodes 30517 Field-effect Transistors 33518 Bipolar Junction Transistors 38719 Power Electronics 45320 Internal Circuitry of Operational

Amplifiers 47921 Noise and Electromagnetic Compatibility 49322 Positive Feedback, Oscillators and

Stability 51923 Digital Systems 53324 Sequential Logic 59125 Digital Devices 62326 Implementing Digital Systems 67527 Data Acquisition and Conversion 73528 System Design 751

Appendix A 765Appendix B 768Appendix C 770Appendix D 775Appendix E 778

Index 783

Brief Contents

Preface xii

Part 1 Electrical Circuits and Components 1

1 Basic Electrical Circuits and Components 32 Measurement of Voltages and Currents 213 Resistance and DC Circuits 454 Capacitance and Electric Fields 715 Inductance and Magnetic Fields 876 Alternating Voltages and Currents 1077 Power in AC Circuits 1298 Frequency Characteristics of AC Circuits 1419 Transient Behaviour 175

Part 2 Electronic Systems 191

10 Electronic Systems 19311 Sensors 20312 Actuators 22513 Amplification 23714 Control and Feedback 257

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Contents

Preface xii

Part 1 Electrical Circuits and Components 1

1 Basic Electrical Circuits and Components 3

1.1 Introduction 31.2 Systme International units 41.3 Common prefixes 51.4 Electrical circuits 51.5 Direct current and alternating current 81.6 Resistors, capacitors and inductors 81.7 Ohms law 91.8 Kirchhoff s laws 101.9 Power dissipation in resistors 121.10 Resistors in series 121.11 Resistors in parallel 131.12 Resistive potential dividers 141.13 Sinusoidal quantities 161.14 Circuit symbols 17

Key points 18Exercises 18

2 Measurement of Voltages and Currents 21

2.1 Introduction 212.2 Sine waves 222.3 Square waves 292.4 Measuring voltages and currents 302.5 Analogue ammeters and voltmeters 322.6 Digital multimeters 362.7 Oscilloscopes 36


3 Resistance and DC Circuits 45

3.1 Introduction 453.2 Current and charge 453.3 Voltage sources 463.4 Current sources 473.5 Resistance and Ohms law 473.6 Resistors in series and parallel 483.7 Kirchhoff s laws 493.8 Thvenins theorem and Nortons

theorem 523.9 Superposition 563.10 Nodal analysis 593.11 Mesh analysis 623.12 Solving simultaneous circuit equations 653.13 Choice of techniques 66


4 Capacitance and Electric Fields 71

4.1 Introduction 714.2 Capacitors and capacitance 714.3 Capacitors and alternating voltages and

currents 734.4 The effect of a capacitors dimensions on its

capacitance 754.5 Electric field strength and electric flux

density 754.6 Capacitors in series and in parallel 774.7 Relationship between voltage and current in

a capacitor 794.8 Sinusoidal voltages and currents 814.9 Energy stored in a charged capacitor 824.10 Circuit symbols 83


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5 Inductance and Magnetic Fields 87

5.1 Introduction 875.2 Electromagnetism 875.3 Reluctance 915.4 Inductance 925.5 Self-inductance 925.6 Inductors 935.7 Inductors in series and in parallel 955.8 Relationship between voltage and current

in an inductor 965.9 Sinusoidal voltages and currents 985.10 Energy storage in an inductor 995.11 Mutual inductance 1005.12 Transformers 1015.13 Circuit symbols 1025.14 The use of inductance in sensors 103


6 Alternating Voltages and Currents 107

6.1 Introduction 1076.2 Relationship between voltage and current 1086.3 Reactance of inductors and capacitors 1096.4 Phasor diagrams 1126.5 Impedance 1186.6 Complex notation 120


7 Power in AC Circuits 129

7.1 Introduction 1297.2 Power dissipation in resistive components 1297.3 Power in capacitors 1307.4 Power in inductors 1317.5 Power in circuits with resistance and

reactance 1317.6 Active and reactive power 1337.7 Power factor correction 1357.8 Three-phase systems 1367.9 Power measurement 137


viii CONTENTS

8 Frequency Characteristics of AC Circuits 141

8.1 Introduction 1418.2 Two-port networks 1418.3 The decibel (dB) 1438.4 Frequency response 1468.5 A high-pass RC network 1468.6 A low-pass RC network 1518.7 A low-pass RL network 1548.8 A high-pass RL network 1568.9 A comparison of RC and RL networks 1568.10 Bode diagrams 1588.11 Combining the effects of several stages 1598.12 RLC circuits and resonance 1618.13 Filters 1678.14 Stray capacitance and inductance 171


9 Transient Behaviour 175

9.1 Introduction 1759.2 Charging of capacitors and energising

of inductors 1759.3 Discharging of capacitors and de-energising

of inductors 1799.4 Generalised response of first-order systems 1819.5 Second-order systems 1879.6 Higher-order systems 189


Part 2 Electronic Systems 191

10 Electronic Systems 193

10.1 Introduction 19310.2 A systems approach to engineering 19410.3 Systems 19510.4 System inputs and outputs 19510.5 Physical quantities and electrical signals 19710.6 System block diagrams 198


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CONTENTS ix

11 Sensors 203

11.1 Introduction 20311.2 Describing sensor performance 20411.3 Temperature sensors 20611.4 Light sensors 20811.5 Force sensors 20911.6 Displacement sensors 21011.7 Motion sensors 21611.8 Sound sensors 21711.9 Sensor interfacing 21811.10 Sensors a summary 220


12 Actuators 225

12.1 Introduction 22512.2 Heat actuators 22512.3 Light actuators 22512.4 Force, displacement and motion actuators 22812.5 Sound actuators 23112.6 Actuator interfacing 23212.7 Actuators a summary 232


13 Amplification 237

13.1 Introduction 23713.2 Electronic amplifiers 23913.3 Sources and loads 24113.4 Equivalent circuit of an amplifier 24213.5 Output power 24613.6 Power gain 24913.7 Frequency response and bandwidth 25013.8 Differential amplifiers 25213.9 Simple amplifiers 254


14 Control and Feedback 257

14.1 Introduction 25714.2 Open-loop and closed-loop systems 25814.3 Automatic control systems 25914.4 Feedback systems 26114.5 Negative feedback 263

14.6 The effects of negative feedback 26814.7 Negative feedback a summary 271


15 Operational Amplifiers 275

15.1 Introduction 27515.2 An ideal operational amplifier 27715.3 Some basic operational amplifier circuits 27715.4 Some other useful circuits 28215.5 Real operational amplifiers 29115.6 Selecting component values for op-amp

circuits 29615.7 The effects of feedback on op-amp circuits 297


16 Semiconductors and Diodes 305

16.1 Introduction 30516.2 Electrical properties of solids 30516.3 Semiconductors 30616.4 pn junctions 30916.5 Diodes 31216.6 Semiconductor diodes 31316.7 Special-purpose diodes 32116.8 Diode circuits 325


17 Field-effect Transistors 335

17.1 Introduction 33517.2 An overview of field-effect transistors 33517.3 Insulated-gate field-effect transistors 33717.4 Junction-gate field-effect transistors 34017.5 FET characteristics 34117.6 FET amplifiers 34917.7 Other FET applications 37417.8 FET circuit examples 380


18 Bipolar Junction Transistors 387

18.1 Introduction 38718.2 An overview of bipolar transistors 387

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18.3 Bipolar transistor operation 38918.4 A simple amplifier 39218.5 Bipolar transistor characteristics 39318.6 Bipolar amplifier circuits 40318.7 Bipolar transistor applications 44318.8 Circuit examples 446


19 Power Electronics 453

19.1 Introduction 45319.2 Bipolar transistor power amplifiers 45419.3 Classes of amplifier 45819.4 Power amplifiers 46119.5 Four-layer devices 46719.6 Power supplies and voltage regulators 472


20 Internal Circuitry of Operational Amplifiers 479

20.1 Introduction 47920.2 Bipolar operational amplifiers 48020.3 CMOS operational amplifiers 48520.4 BiFET operational amplifiers 48920.5 BiMOS operational amplifiers 489


21 Noise and Electromagnetic Compatibility 493

21.1 Introduction 49321.2 Noise sources 49421.3 Representing noise sources within

equivalent circuits 49721.4 Noise in bipolar transistors 49821.5 Noise in FETs 49921.6 Signal-to-noise ratio 49921.7 Noise figure 50021.8 Designing for low-noise applications 50021.9 Electromagnetic compatibility 50221.10 Designing for EMC 507


22 Positive Feedback, Oscillators and Stability 519

22.1 Introduction 51922.2 Oscillators 51922.3 Stability 527


23 Digital Systems 533

23.1 Introduction 53323.2 Binary quantities and variables 53323.3 Logic gates 53723.4 Boolean algebra 54123.5 Combinational logic 54323.6 Boolean algebraic manipulation 54823.7 Algebraic simplification 55123.8 Karnaugh maps 55423.9 Automated methods of minimisation 56123.10 Propagation delay and hazards 56223.11 Number systems and binary arithmetic 56423.12 Numeric and alphabetic codes 57623.13 Examples of combinational logic design 581


24 Sequential Logic 591

24.1 Introduction 59124.2 Bistables 59224.3 Monostables or one-shots 60224.4 Astables 60424.5 Timers 60524.6 Memory registers 60624.7 Shift registers 60724.8 Counters 610


25 Digital Devices 623

25.1 Introduction 62325.2 Gate characteristics 62525.3 Logic families 63225.4 TTL 64325.5 CMOS 654

x CONTENTS

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27.5 Sample and hold gates 74527.6 Multiplexing 745


28 System Design 751

28.1 Introduction 75128.2 Design methodology 75128.3 Choice of technology 75428.4 Electronic design tools 758


Appendices

A Symbols 765B SI units and prefixes 768C Op-amp circuits 770D Complex numbers 775E Answers to selected exercises 778

Index 783

CONTENTS xi

25.6 Interfacing TTL and CMOS or logic usingdifferent supply voltages 662

25.7 Noise and EMC in digital systems 665Key points 672Exercises 673

26 Implementing Digital Systems 675

26.1 Introduction 67526.2 Array logic 67626.3 Microprocessors 69726.4 Programmable logic controllers (PLCs) 73026.5 Selecting an implementation method 730


27 Data Acquisition and Conversion 735

27.1 Introduction 73527.2 Sampling 73527.3 Signal reconstruction 73727.4 Data converters 737

Supporting resourcesVisit www.pearsoned.co.uk/storey-elec to find valuable online resources

Companion Website for students Computer-marked self-assessment questions to check your understanding Demonstration files that can be downloaded for use with the Computer Simulation exercises.

For instructors Full solutions manual PowerPoint slides with all figures from the book.

Also: The Companion Website provides the following features: Search tool to help locate specific items of content E-mail results and profile tools to send results of quizzes to instructors Online help and support to assist with website usage and troubleshooting.

For more information please contact your local Pearson Education sales representative or visit www.pearsoned.co.uk/storey-elec

ELEA_A01.qxd 2/10/09 2:17 PM Page xi

Preface

Electronics represents one of the most important, and rapidly changing, areasof engineering. It is used at the heart of a vast range of products that extendsfrom mobile phones to computers, and from cars to nuclear power stations.For this reason, all engineers, scientists and technologists need a basicunderstanding of such systems, while many will require a far more detailedknowledge of this area.

When the first edition of this book was published it represented a verynovel approach to the teaching of electronics. At that time most textsadopted a decidedly bottom-up approach to the subject, starting by lookingat semiconductor materials and working their way through diodes and tran-sistors before eventually, several chapters later, looking at the uses of the circuits being considered. Electronics: A Systems Approach pioneered a new,top-down approach to the teaching of electronics by explaining the uses andrequired characteristics of circuits, before embarking on detailed analysis.This aids comprehension and makes the process of learning much moreinteresting.

One of the great misconceptions concerning this approach is that it is insome way less rigorous in its treatment of the subject. A top-down approachdoes not define the depth to which a subject is studied but only the order andmanner in which the material is presented. Many students will need to lookin detail at the operation of electronic components and to understand thephysics of its materials; however, this will be more easily absorbed if thecharacteristics and uses of the components are understood first.

A great benefit of a top-down approach is that it makes the book moreaccessible for all its potential readers. For those who intend to specialise inelectronic engineering the material is presented in a way that makes it easy to absorb, providing an excellent grounding for further study. For thoseintending to specialise in other areas of engineering or science, the order ofpresentation allows them to gain a good grounding in the basics, and toprogress into the detail only as far as is appropriate for their needs.

This fourth edition represents a substantial revision and expansion of thetext. Previous editions have assumed that readers are already familiar withthe operation of basic electrical components and circuits, and this assump-tion has, perhaps, made the text inappropriate for some first-level courses.However, this new edition sees the text divided into two parts. Part 1 isalmost entirely new material and provides a thorough introduction to

New in this edition

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PREFACE xiii

Electrical Circuits and Components. This makes very few assumptionsabout prior knowledge, and gives a gentle and well-structured introductionto this interesting and vital area. Part 2 then provides a thorough introduc-tion to Electronic Systems, adopting the well-tried, top-down approachfor which this text is renowned. This part includes all the major materialfrom previous editions of the book, with revisions to account for develop-ments in this rapidly changing field and to take advantage of the preparatorymaterial now available within the new first part of the book. With the inclu-sion of the new material in Part 1, the book now offers a comprehensiveintroduction to both Electrical and Electronic Engineering, making it appro-priate for a wide range of first-level courses in areas such as ElectronicEngineering, Electrical Engineering and Electrical and ElectronicEngineering.

New material within Part 1 includes:

An introduction to Basic Electrical Circuits and Components. An outline of the Measurement of Voltages and Currents. Chapters dedicated to the basics of Resistance and DC Circuits,

Capacitance and Electric Fields and Inductance and MagneticFields.

An overview of Alternating Voltages and Currents, and the essentialelements of Power in AC Circuits.

The fundamentals of the Frequency Characteristics of AC Circuitsand a description of the Transient Behaviour of first-, second- andhigher-order systems.

New and revised material in Part 2 includes:

Dedicated and separate chapters on Sensors and Actuators. A new chapter on the Internal Circuitry of Operational Amplifiers

which provides useful insights into the design of sophisticated circuits,including Bipolar, CMOS, BiFET and BiMOS op-amps.

A new chapter on Noise and Electromagnetic Compatibility.

The new edition also benefits from:

A greatly increased number of worked examples within the text. Additional self-assessment exercises at the end of the chapters.

This text is intended for undergraduate students in all fields of engineeringand science. For students of electronics or electrical engineering it providesa first-level introduction to electronics that will give a sound basis for furtherstudy. For students of other disciplines it includes most of the electronicsmaterial that they will need throughout their course.

The book assumes very little by way of prior knowledge, except for anunderstanding of the basic principles of physics and mathematics.

Who should read this book

Assumed knowledge

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The text is supported by a comprehensive companion website that willgreatly increase both your understanding and your enjoyment of the book.The site contains a range of support material, including computer-markedself-assessment exercises for each chapter. These exercises not only give youinstant feedback on your understanding of the material, but also give usefulguidance on areas of difficulty. To visit the site, go to www.pearsoned.co.uk/storey-elec.

Circuit simulation offers a powerful and simple means of gaining an insightinto the operation of electronic circuits. Throughout the book there arenumerous Computer Simulation Exercises that support the material inthe text. These are marked by icons in the margin as shown on the left. Theexercises may be performed using any circuit simulation package, althoughperhaps the most widely used are those produced by Electronic Workbenchand by OrCAD. Both these packages are widely used within industry andwithin Universities and Colleges. Each comes as part of a suite of programsthat provides schematic capture of circuits and the graphical display of simulation results.

Many students will have access to simulation tools within their University orCollege, but you may also obtain the software for use on your own computerif you wish. For some packages demonstration versions may be downloadedfree of charge from the manufacturers website or obtained on a free CD.In other cases, low-cost student versions are available. Details of how toobtain your free or reduced-cost simulation software are given on the bookscompanion website.

xiv PREFACE

Circuit simulation

Circuit simulation using theElectronic Workbench Multisimpackage

Companion website

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PREFACE xv

To simplify the use of simulation as an aid to understanding the materialwithin the book, a series of demonstration files for the most widely used simulation packages can also be downloaded from the website. The name ofthe relevant demonstration file is given in the margin under the computericon for the associated computer simulation exercise. Computer icons are alsofound next to some circuit diagrams where simulation files have been pro-vided to aid understanding of the operation of the circuit. The demonstrationfiles come with full details of how to carry out the various simulation exercises.

Problems using simulation have also been included within the exercises atthe end of the various chapters. These exercises do not have demonstrationfiles and are set to develop and test the readers understanding of the use ofsimulation as well as the circuits concerned.

A comprehensive set of online support material is available for instructorsusing this book as a course text. This includes a set of editable PowerPointslides to aid in the preparation of lectures, plus an Instructors Manual thatgives fully worked solutions to all the numerical problems and sampleanswers for the various non-numerical exercises. Guidance is also given on course preparation and on the selection of topics to meet the needs of students with different backgrounds and interests. This material, togetherwith the various online study aids, simulation exercises and self-assessmenttests, should greatly assist both the instructor and the student to gain maxi-mum benefit from courses based on this text. Instructors adopting this bookshould visit the companion website at: www.pearsoned.co.uk/storey-elec fordetails of how to gain access to the secure website that holds the instructorssupport material.

Circuit simulation using theOrCAD Capture CIS package

To the instructor

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I would like to express my gratitude to the various people who have assistedin the preparation of this book. In particular I would like to thank my col-leagues at the University of Warwick who have provided useful feedback andsuggestions.

I would also like to thank the companies who have given permission toreproduce their material: RS Components for the photographs in figures2.14, 2.15, 11.2(a) and (b), 11.3(a) and (b), 11.4, 11.5(a) and (b), 11.6, 11.7,11.9(a) and (b), 11.12, 12.1, 12.2, 12.3, 12.4 and 12.6; Farnell ElectronicComponents Limited for the photographs in figures 2.12, and 2.13; TexasInstruments for figures 20.4(a) and (b), 20.6, 20.7(a) and (b), 25.22, 25.34;Lattice Semiconductor Corporation for figures 26.11, 26.12, 26.13; AnachipCorporation for figure 26.14; Microchip Technology for figures 26.40 and26.41. Certain materials herein are reprinted with the permission ofMicrochip Technology Incorporated. No further reprints or reproductionsmay be made of said materials without Microchip Technology Inc.s priorwritten consent. Cadence Design Systems Inc. for permission to use theOrCAD Capture CIS images in the Preface; and Electronic Workbench forpermission use the Multisim in the Preface.

Finally, I would like to give special thanks to my family for their help andsupport during the writing of this book. In particular I wish to thank my wifeJillian for her never-failing encouragement and understanding.

Tri-state is a trademark of the National Semiconductor Corporation.PAL is a trademark of Lattice Semiconductor Corporation.PEEL is a registered trademark of International CMOS Technology.PIC is a registered trademark of Microchip Technology Inc.SPI is a registered trademark of Motorola Inc.I2C is a registered trademark of Philips Semiconductors Corporation.PSpice, OrCAD and Capture CIS are trademarks of Cadence DesignSystems, Inc.Electronics Workbench and Multisim are registered trademarks ofInteractive Image Technologies, Ltd.Lin CMOS is a trademark of Texas Instruments Incorporated.

xvi PREFACE

Acknowledgements

Trademarks

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Part 1 Electrical Circuits and Components

1 Basic Electrical Circuits and Components

2 Measurement of Voltages and Currents

3 Resistance and DC Circuits

4 Capacitance and Electric Fields

5 Inductance and Magnetic Fields

6 Alternating Voltages and Currents

7 Power in AC Circuits

8 Frequency Characteristics of AC Circuits

9 Transient Behaviour

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Chapter one Basic Electrical Circuits and Components

Introduction

Objectives

When you have studied the material in this chapter, you should be able to: give the Systme International (SI) units for a range of electrical quantities use a range of common prefixes to represent multiples of these units describe the basic characteristics of resistors, capacitors and inductors apply Ohms law, and Kirchhoffs voltage and current laws, to simple electrical circuits calculate the effective resistance of resistors in series or in parallel, and analyse simple resistive

potential divider circuits define the terms frequency and period as they apply to sinusoidal quantities draw the circuit symbols for a range of common electrical components.

While the title of this book refers to electronic systems, this first part of the text refers to electrical circuits and components and it is perhaps appropriate to start by explaining what is meant by the terms electronic and electrical in this context. Both terms relate to the use of electricalenergy, but electrical is often used to refer to circuits that use only simplepassive components such as resistors, capacitors and inductors, while theterm electronic implies circuits that also use more sophisticated compon-ents such as transistors or integrated circuits. Therefore, before looking in detail at the operation of electronic systems, we need to have a basicunderstanding of the world of electrical engineering, since the componentsand circuits of this domain also form the basis of more sophisticated elec-tronic applications.

Unfortunately, while this use of the words electrical and electronic iscommon, it is not universal. Engineers sometimes use the term electricalwhen describing applications associated with the generation, transmission oruse of large amounts of electrical energy, and use electronic when describingapplications associated with smaller amounts of power, where the electricalenergy is used to convey information rather than as a source of power. Forthis reason, within this text we will be fairly liberal with our use of the twoterms, since much of the material covered is relevant to all forms of electricaland electronic systems.

Most readers of this book will have met the basic concepts of electricalcircuits long before embarking on study at this level, and later chapters ofthis book will assume that the reader is familiar with this elementary mater-ial. Later, we will look at these basic concepts in more detail and extend them

1.1


to give a greater understanding of the behaviour of the circuits and systemsthat we will be studying.

The list below gives an indication of the topics that you should be fam-iliar with before reading the following chapters:

The Systme International (SI) units for quantities such as energy,power, temperature, frequency, charge, potential, resistance, capacitanceand inductance. You should also know the symbols used for these units.

The prefixes used to represent common multiples of these units and theirsymbols (for example, 1 kilometre = 1 km = 1000 metres).

Electrical circuits and quantities such as charge, e.m.f. and potential difference.

Direct and alternating currents. The basic characteristics of resistors, capacitors and inductors. Ohms law, Kirchhoff s laws and power dissipation in resistors. The effective resistance of resistors in series and parallel. The operation of resistive potential dividers. The terms used to describe sinusoidal quantities. The circuit symbols used for resistors, capacitors, inductors, voltage

sources and other common components.

If, having read through the list above, you are confident that you are familiar with all these topics you can move on immediately to Chapter 2.However, just in case there are a few areas that might need some reinforce-ment, the remainder of this chapter provides what might be seen as a revisionsection on this material. This does not aim to give a detailed treatment ofthese topics (where appropriate this will be given in later chapters) but simply explains them in sufficient detail to allow an understanding of theearly parts of the book.

In this chapter, worked examples are used to illustrate several of the concepts involved. One way of assessing your understanding of the varioustopics is to look quickly through these examples to see if you can perform thecalculations involved, before looking at the worked solutions. Most readerswill find the early examples trivial, but experience shows that many will feel less confident on those related to potential dividers. This is a veryimportant topic, and a clear understanding of these circuits will make itmuch easier to understand the remainder of the book.

The exercises at the end of this chapter are included to allow you to testyour understanding of the assumed knowledge listed above. If you can perform these exercises easily you should have no problems with the tech-nical content of the next few chapters. If not, you would be well advised toinvest a little time in looking at the relevant sections of this chapter beforecontinuing.

The Systme International (SI) dUnits (International System of Units)defines units for a large number of physical quantities but, fortunately forour current studies, we need very few of them. These are shown in Table 1.1.In later chapters, we will introduce additional units as necessary, andAppendix B gives a more comprehensive list of units relevant to electricaland electronic engineering.

4 CHAPTER 1 BASIC ELECTRICAL CIRCUITS AND COMPONENTS

Systme International

units

1.2


1.4 ELECTRICAL CIRCUITS 5

Table 1.2 lists the most commonly used unit prefixes. These will suffice formost purposes although a more extensive list is given in Appendix B.

1.4.1 Electric charge

Charge is an amount of electrical energy and can be either positive or neg-ative. In atoms, protons have a positive charge and electrons have an equalnegative charge. While protons are fixed within the atomic nucleus, electronsare often weakly bound and may therefore be able to move. If a body orregion develops an excess of electrons it will have an overall negative charge,while a region with a deficit of electrons will have a positive charge.

1.4.2 Electric current

An electric current is a flow of electric charge, which in most cases is a flowof electrons. Conventional current is defined as a flow of electricity from a

Quantity Quantity Unit Unit symbol symbol

Capacitance C farad FCharge Q coulomb CCurrent I ampere AElectromotive force E volt VFrequency f hertz HzInductance (self ) L henry HPeriod T second sPotential difference V volt VPower P watt WResistance R ohm Temperature T kelvin KTime t second s

Table 1.1 Some importantunits.

Prefix Name Meaning(multiply by)

T tera 1012

G giga 109

M mega 106

k kilo 103

m milli 103

micro 106n nano 109

p pico 1012

Table 1.2 Common unitprefixes.

Common prefixes

1.3

Electrical circuits

1.4


positive to a negative region. This conventional current is in the oppositedirection to the flow of the negatively charged electrons. The unit of currentis the ampere or amp (A).

1.4.3 Current flow in a circuit

A sustained electric current requires a complete circuit for the recirculationof electrons. It also requires some stimulus to cause the electrons to flowaround this circuit.

1.4.4 Electromotive force and potential difference

The stimulus that causes an electric current to flow around a circuit istermed an electromotive force or e.m.f. The e.m.f. represents the energyintroduced into the circuit by a source such as a battery or a generator.

The energy transferred from the source to the load results in a change inthe electrical potential at each point in the load. Between any two points in the load there will exist a certain potential difference, which represents the energy associated with the passage of a unit of charge from one point tothe other.

Both e.m.f. and potential difference are expressed in units of volts, andclearly these two quantities are related. Figure 1.1 illustrates the relationshipbetween them: e.m.f. is the quantity that produces an electric current, whilea potential difference is the effect on the circuit of this passage of energy.

Some students have difficulty in visualising e.m.f., potential difference,resistance and current, and it is sometimes useful to use an analogy.Consider, for example, the arrangement shown in Figure 1.2. Here a waterpump forces water to flow around a series of pipes and through some formof restriction. While no analogy is perfect, this model illustrates the basic


Figure 1.1 Electromotive forceand potential difference.

Figure 1.2 A water-basedanalogy of an electrical circuit.


1.4 ELECTRICAL CIRCUITS 7

properties of the circuit of Figure 1.1. In the water-based diagram, the waterpump forces water around the arrangement and is equivalent to the voltagesource (or battery), which pushes electric charge around the correspondingelectrical circuit. The flow of water through the pipe corresponds to the flowof charge around the circuit and therefore the flow rate represents the currentin the circuit. The restriction within the pipe opposes the flow of water and isequivalent to the resistance of the electrical circuit. As water flows throughthe restriction the pressure will fall, creating a pressure difference across it.This is equivalent to the potential difference across the resistance within theelectrical circuit. The flow rate of the water will increase with the outputpressure of the pump and decrease with the level of restriction present. Thisis analogous to the behaviour of the electrical circuit, where the currentincreases with the e.m.f. of the voltage source and decreases with the magni-tude of the resistance.

1.4.5 Voltage reference points

Electromotive forces and potential differences in circuits produce differentpotentials (or voltages) at different points in the circuit. It is normal todescribe the voltages throughout a circuit by giving the potential at particu-lar points with respect to a single reference point. This reference is oftencalled the ground or earth of the circuit. Since voltages at points in the cir-cuit are measured with respect to ground, it follows that the voltage on theground itself is zero. Therefore, ground is also called the zero-volts line ofthe circuit.

In a circuit, a particular point or junction may be taken as the zero-voltreference and this may then be labelled as 0 V, as shown in Figure 1.3(a).Alternatively, the ground point of the circuit may be indicated using theground symbol, as shown in Figure 1.3(b).

1.4.6 Representing voltages in circuit diagrams

Conventions for representing voltages in circuit diagrams vary considerablybetween countries. In the UK, and in this text, it is common to indicate apotential difference by an arrow, which is taken to represent the voltage at the head of the arrow with respect to that at the tail. This is illustrated in Figure 1.4(a). In many cases, the tail of the arrow will correspond to the

Figure 1.3 Indicating voltagereference points.


zero-volt line of the circuit (as shown in VA in the figure). However, it canindicate a voltage difference between any two points in the circuit (as shownby VB).

In some cases, it is inconvenient to use arrows to indicate voltages in circuits and simple labels are used instead, as shown in Figure 1.4(b). Herethe labels VC and VD represent the voltage at the corresponding points withrespect to ground (that is, with respect to the zero-volt reference).

1.4.7 Representing currents in circuit diagrams

Currents in circuit diagrams are conventionally indicated by an arrow in thedirection of the conventional current flow (that is, in the opposite direction tothe flow of electrons). This was illustrated in Figure 1.1. This figure alsoshows that for positive voltages and currents the arrow for the currentflowing out of a voltage source is in the same direction as the arrow repres-enting its e.m.f. However, the arrow representing the current in a resistor isin the opposite direction to the arrow representing the potential differenceacross it.

The currents associated with electrical circuits may be constant or may varywith time. Where currents vary with time they may also be unidirectional oralternating.

When the current in a conductor always flows in the same direction thisis described as a direct current (DC). Such currents will often be associatedwith voltages of a single polarity. Where the direction of the current periodic-ally changes, this is referred to as alternating current (AC), and such currentswill often be associated with alternating voltages. One of the most commonforms of alternating waveform is the sine wave, as discussed in Section 1.13.

1.6.1 Resistors

Resistors are components whose main characteristic is that they provideresistance between their two electrical terminals. The resistance of a circuitrepresents its opposition to the flow of electric current. The unit of resistanceis the ohm (). One may also define the conductance of a circuit as its ability to allow the flow of electricity. The conductance of a circuit is equal


Figure 1.4 Indicating voltagesin circuit diagrams.

Direct current and alternating

current

1.5

Resistors, capacitors and

inductors

1.6


1.7 OHMS LAW 9

to the reciprocal of its resistance and has the units of siemens (S). We willlook at resistance in some detail in Chapter 3.

1.6.2 Capacitors

Capacitors are components whose main characteristic is that they exhibitcapacitance between their two terminals. Capacitance is a property of twoconductors that are electrically insulated from each other, whereby electricalenergy is stored when a potential difference exists between them. Thisenergy is stored in an electric field that is created between the two con-ductors. Capacitance is measured in farads (F), and we will return to lookat capacitance in more detail in Chapter 4.

1.6.3 Inductors

Inductors are components whose main characteristic is that they exhibitinductance between their two terminals. Inductance is the property of a coilthat results in an e.m.f. being induced in the coil as a result of a change in thecurrent in the coil. Like capacitors, inductors can store electrical energy andin this case it is stored in a magnetic field. The unit of inductance is thehenry (H), and we will look at inductance in Chapter 5.

Ohms law states that the current I flowing in a conductor is directly propor-tional to the applied voltage V and inversely proportional to its resistance R.This determines the relationship between the units for current, voltage andresistance, and the ohm is defined as the resistance of a circuit in which acurrent of 1 amp produces a potential difference of 1 volt.

The relationship between voltage, current and resistance can be repres-ented in a number of ways, including:

V = IR (1.1)

(1.2)

(1.3)

A simple way of remembering these three equations is to use the virtual tri-angle of Figure 1.5. The triangle is referred to as virtual simply as a way ofremembering the order of the letters. Taking the first three letters of VIRtualand writing them in a triangle (starting at the top) gives the arrangementshown in the figure. If you place your finger on one of the letters, the remain-ing two show the expression for the selected quantity. For example, to findthe expression for V put your finger on the V and you see I next to R, so V = IR. Alternatively, to find the expression for I put your finger on the Iand you are left with V above R, so I = V/R. Similarly, covering R leavesV over I, so R = V/I.

RVI

=

IVR

=

Ohms law1.7

Figure 1.5 The relationshipbetween V, I and R.


Example 1.1 Voltage measurements (with respect to ground) on part of an elec-trical circuit give the values shown in the diagram below. If theresistance of R2 is 220 , what is the current I flowing through thisresistor?

From the two voltage measurements, it is clear that the voltage differenceacross the resistor is 15.8 12.3 = 3.5 V. Therefore using the relationship

we have

1.8.1 Current law

At any instant, the algebraic sum of all the currents flowing into any junctionin a circuit is zero:

I = 0 (1.4)A junction is any point where electrical paths meet. The law comes aboutfrom consideration of conservation of charge the charge flowing into apoint must equal that flowing out.

1.8.2 Voltage law

At any instant, the algebraic sum of all the voltages around any loop in a circuit is zero:

V = 0 (1.5)

I .

. = =

3 5220

15 9V

mA

IVR

=


Kirchhoffs laws1.8

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1.8 KIRCHHOFFS LAWS 11

The term loop refers to any continuous path around the circuit, and the lawcomes about from consideration of conservation of energy.

With both laws, it is important that the various quantities are assigned thecorrect sign. When summing currents, those flowing into a junction are giventhe opposite polarity to those flowing out from it. Similarly, when summingthe voltages around a loop, clockwise voltages will be assigned the oppositepolarity to anticlockwise ones.

Example 1.2 Use Kirchhoff s current law to determine the current I2 in the following circuit.

From Kirchhoff s current law

I2 = I1 I3= 10 3

= 7 A

Example 1.3 Use Kirchhoff s voltage law to determine the magnitude of V1 in the following circuit.

From Kirchhoff s voltage law (summing the voltages clockwise around theloop)

E V1 V2 = 0

or, rearranging,

V1 = E V2= 12 7

= 5 V

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The instantaneous power dissipation P of a resistor is given by the product ofthe voltage across the resistor and the current passing through it. Combiningthis result with Ohms law gives a range of expressions for P. These are

P = VI (1.6)

P = I 2R (1.7)

(1.8)

Example 1.4 Determine the power dissipation in the resistor R3 in the followingcircuit.

P

VR

=2

From Equation 1.7

P = I 2R

= 32 50

= 450 W

The effective resistance of a number of resistors in series is equal to the sumof their individual resistances:

R = R1 + R2 + R3 + + Rn (1.9)

For example, for the three resistors shown in Figure 1.6 the total resistanceR is given by

R = R1 + R2 + R3

Example 1.5 Determine the equivalent resistance of the following combination.


Power dissipation in resistors

1.9

Resistors in series

1.10

Figure 1.6 Three resistors inseries.

From above

R = R1 + R2 + R3 + R4= 10 + 20 + 15 + 25

= 70

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1.11 RESISTORS IN PARALLEL 13

The effective resistance of a number of resistors in parallel is given by thefollowing expression:

(1.10)

For example, for the three resistors shown in Figure 1.7 the total resistanceR is given by

1 1 1 1

1 2 3R R R R = + +

1 1 1 1 1

1 2 3R R R R Rn = + + + +

Example 1.6 Determine the equivalent resistance of the following combination.

From above

Note that the effective resistance of a number of resistors in parallel willalways be less than that of the lowest-value resistor.

= = . R203

6 67

=

320

= + 1

10120

1 1 1

1 2R R R = +

Resistors in parallel

1.11

Figure 1.7 Three resistors inparallel.

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When several resistors are connected in series the current flowing througheach resistor is the same. The magnitude of this current is given by the volt-age divided by the total resistance. For example, if we connect three resistorsin series, as shown in Figure 1.8, the current is given by

The voltage across each resistor is then given by this current multiplied byits resistance. For example, the voltage V1 across resistor R1 will be given by

V IRV

R R RR1 1

1 2 31

= =

+ +

IV

R R R

=

+ +1 2 3

To calculate the voltage at a point in a chain of resistors, one must deter-mine the voltage across the complete chain, calculate the voltage across those


Resistive potential dividers

1.12

Figure 1.8 A resistive potentialdivider.

Therefore, the fraction of the total voltage across each resistor is equal to itsfraction of the total resistance, as shown in Figure 1.9, where

or, rearranging,

V V

RR R R3

3

1 2 3

=

+ +

V V

RR R R

V VR

R R R11

1 2 32

2

1 2 3

=

+ +=

+ +

VV

RR R R

VV

RR R R

VV

RR R R

1 1

1 2 3

2 2

1 2 3

3 3

1 2 3

=

+ +=

+ +=

+ +

Figure 1.9 The division ofvoltages in a potential divider.

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1.12 RESISTIVE POTENTIAL DIVIDERS 15

Example 1.7 Determine the voltage V in the following circuit.

As described above, we first determine the voltage across the chain (by subtracting the voltages at either end of the chain). Then we calculate thevoltage across the relevant resistor and add this to the voltage at the appro-priate end of the chain.

In this case one end of the chain of resistors is at zero volts, so the calcu-lation is very straightforward. The voltage across the chain is 10 V, and V issimply the voltage across R2, which is given by

= 6 V

Note that a common mistake in such calculations is to calculate R1/(R1 +R2 ), rather than R2 /(R1 + R2). The value used as the numerator in thisexpression represents the resistor across which the voltage is to be calculated.

Potentiometer calculations are slightly more complicated where neitherend of the chain of resistors is at zero volts.

=

+

10

300200 300

V

RR R

=

+10 2

1 2

Figure 1.10 A simple potentialdivider.

resistors between that point and one end of the chain and add this to the volt-age at that end of the chain. For example, in Figure 1.10

(1.11) V V V V

RR R

( )

= + +

2 1 22

1 2

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Example 1.8 Determine the voltage V in the following circuit.

Again, we first determine the voltage across the chain (by subtracting thevoltages at either end of the chain). Then we calculate the voltage across the relevant resistor and add this to the voltage at the appropriate end of thechain. Therefore

= 3 + 4

= 7 V

In this case we pick one end of the chain of resistors as our reference point(we picked the lower end) and calculate the voltage on the output withrespect to this point. We then add to this calculated value the voltage at thereference point.

Sinusoidal quantities have a magnitude that varies with time in a mannerdescribed by the sine function. The variation of any quantity with time canbe described by drawing its waveform. The waveform of a sinusoidal quan-tity is shown in Figure 1.11. The length of time between correspondingpoints in successive cycles of the waveform is termed its period, which isgiven the symbol T. The number of cycles of the waveform within 1 secondis termed its frequency, which is usually given the symbol f.

= +

+

3 12

5001000 500

V

RR R

( )

= + +

3 15 3 21 2


Sinusoidal quantities

1.13

Figure 1.11 A sine wave.

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In later chapters we will meet a number of additional component symbols,but these are sufficient for our current needs.

1.14 CIRCUIT SYMBOLS 17

The frequency of a waveform is related to its period by the expression

(1.12)

Example 1.9 What is the period of a sinusoidal quantity with a frequency of 50 Hz?

From above we know that

and therefore the period is given by

The following are circuit symbols for a few basic electrical components.

T

f . = = = =

1 150

0 02 20s ms

f

T =

1

f

T =

1

Circuit symbols1.14

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Key points Since this chapter introduces no new material there are very few keypoints. However, the importance of a good understanding of thisassumed knowledge encourages me to emphasise the following:

Understanding the next few chapters of the book relies on under-standing the various topics covered in this chapter.

A clear concept of voltage and current is essential for all readers.

Ohms law and Kirchhoff s voltage and current laws are used exten-sively in later chapters.

Experience shows that students have most problems with potentialdividers a topic that is widely used in these early chapters. You aretherefore advised to make very sure that you are happy with thismaterial before continuing.

Exercises

1.1 Give the prefixes used to denote the following powers: 1012; 109; 106; 103; 103; 106; 109; 1012.

1.2 Explain the difference between 1 ms, 1 m/s and 1 mS.1.3 Explain the difference between 1 m and 1 M.1.4 If a resistor of 1 k has a voltage of 5 V across it,

what is the current flowing through it?1.5 A resistor has 9 V across it and a current of 1.5 mA

flowing through it. What is its resistance?1.6 A resistor of 25 has a voltage of 25 V across it.

What power is being dissipated by the resistor?1.7 If a 400 resistor has a current of 5 A flowing

through it, what power is being dissipated by theresistor?

1.8 What is the effective resistance of a 20 resistor inseries with a 30 resistor?

1.9 What is the effective resistance of a 20 resistor inparallel with a 30 resistor?

1.10 What is the effective resistance of a series combinationof a 1 k resistor, a 2.2 k resistor and a 4.7 k resistor?

1.11 What is the effective resistance of a parallel combina-tion of a 1 k resistor, a 2.2 k resistor and a 4.7 kresistor?

1.12 Calculate the effective resistance between the ter-minals A and B in the following arrangements.

1.13 Calculate the effective resistance between the term-inals A and B in the following arrangements.

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EXERCISES 19

Exercises continued

1.14 Calculate the voltages V1, V2 and V3 in the followingarrangements.

1.15 Calculate the voltages V1, V2 and V3 in the followingarrangements.

1.16 A sinusoidal quantity has a frequency of 1 kHz.What is its period?

1.17 A sinusoidal quantity has a period of 20 s. What isits frequency?

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Chapter two Measurement of Voltages and Currents

Objectives

When you have studied the material in this chapter, you should be able to: describe several forms of alternating waveform, such as sine waves, square waves and triangular waves define terms such as peak value, peak-to-peak value, average value and r.m.s. value as they apply to

alternating waveforms convert between these various values for both sine waves and square waves write equations for sine waves to represent their amplitude, frequency and phase angle configure moving-coil meters to measure currents or voltages within a given range describe the problems associated with measuring non-sinusoidal alternating quantities using analogue

meters and explain how to overcome these problems explain the operation of digital multimeters and describe their basic characteristics discuss the use of oscilloscopes in displaying waveforms and measuring parameters such as phase shift.

In the previous chapter we looked at a range of electrical components andnoted their properties and characteristics. An understanding of the operationof these components will assist you in later chapters as we move on to ana-lyse the behaviour of electronic circuits in more detail. In order to do this,first we need to look at the measurement of voltages and currents in electricalcircuits, and in particular at the measurement of alternating quantities.

Alternating currents and voltages vary with time and periodically changetheir direction. Figure 2.1 shows examples of some alternating waveforms.

Introduction2.1

Figure 2.1 Examples ofalternating waveforms.

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Of these, by far the most important is the sinusoidal waveform or sinewave. Indeed, in many cases, when engineers use the terms alternating current or alternating voltage they are referring to a sinusoidal quantity.Since sine waves are so widely used, it is important that we understand thenature of these waveforms and the ways in which their properties aredefined.

In Chapter 1, we noted that the length of time between corresponding pointsin successive cycles of a sinusoidal waveform is termed its period T and thatthe number of cycles of the waveform within 1 s is termed its frequency f.The frequency of a waveform is related to its period by the expression

The maximum amplitude of the waveform is termed its peak value, and thedifference between the maximum positive and maximum negative values istermed its peak-to-peak value. Because of the waveforms symmetricalnature, the peak-to-peak value is twice the peak value.

Figure 2.2 shows an example of a sinusoidal voltage signal. This illus-trates that the period T can be measured between any convenient corres-ponding points in successive cycles of the waveform. It also shows the peakvoltage Vp and the peak-to-peak voltage Vpkpk. A similar waveform could beplotted for a sinusoidal current waveform indicating its peak current Ip andpeak-to-peak current Ipkpk.

f

T =

1

From the diagram the period is 20 ms or 0.02 s, so the frequency is 1/.02 =50 Hz. The peak voltage is 7 V and the peak-to-peak voltage is therefore 14 V.

22 CHAPTER 2 MEASUREMENT OF VOLTAGES AND CURRENTS

Sine waves2.2

Figure 2.2 A sinusoidal voltagesignal.

Example 2.1 Determine the period, frequency, peak voltage and peak-to-peakvoltage of the following waveform.

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2.2.2 Angular frequency

The frequency f of a waveform (in hertz) is a measure of the number of cyclesof that waveform that pass within 1 s. Each cycle corresponds to 2 radians,and it follows that there will be 2 f radians per second. The number of radians per second is termed the angular frequency of the waveform andis given the symbol . Therefore

= 2 f rad/s (2.1)

2.2.3 Equation of a sine wave

The angular frequency can be thought of as the rate at which the angle of thesine wave changes. Therefore, the phase angle at a particular point in thewaveform, , is given by

= t radThus our earlier expression for a sine wave becomes

y = A sin = A sin t

and the equation of a sinusoidal voltage waveform becomes

2.2 SINE WAVES 23

2.2.1 Instantaneous value

The shape of a sine wave is defined by the sine mathematical function. Thuswe can describe such a waveform by the expression

y = A sin where y is the value of the waveform at a particular point on the curve, A isthe peak value of the waveform and is the angle corresponding to thatpoint. It is conventional to use lower-case letters for time-varying quantities(such as y in the above equation) and upper-case letters for fixed quantities(such as A).

In the voltage waveform of Figure 2.2, the peak value of the waveform isVp, so this waveform could be represented by the expression

v = Vp sin One complete cycle of the waveform corresponds to the angle goingthrough one complete cycle. This corresponds to changing by 360, or 2radians. Figure 2.3 illustrates the relationship between angle and magnitudefor a sine wave.

Figure 2.3 Relationshipbetween instantaneous value andangle for a sine wave.

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v = Vp sin t (2.2)

or

v = Vp sin 2 ft (2.3)

A sinusoidal current waveform might be described by the equation

i = Ip sin t (2.4)

or

i = Ip sin 2 ft (2.5)

Example 2.2 Determine the equation of the following voltage signal.

From the diagram the period is 50 ms or 0.05 s, so the frequency is 1/0.05 =20 Hz. The peak voltage is 10 V. Therefore, from Equation 2.3

v = Vp sin 2 ft

= 10 sin 220t

= 10 sin 126t

2.2.4 Phase angles

The expressions of Equations 2.2 to 2.5 assume that the angle of the sinewave is zero at the origin of the time measurement (t = 0) as in the waveformof Figure 2.2. If this is not the case, then the equation is modified by addingthe angle at t = 0. This gives an equation of the form

y = A sin( t + ) (2.6)where is the phase angle of the waveform at t = 0. It should be noted thatat t = 0 the term t is zero, so y = A sin . This is illustrated in Figure 2.4.


Figure 2.4 The effects of phaseangles.

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In this example, the period is 100 ms or 0.1 s, so the frequency is 1/0.1 =10 Hz. The peak voltage is 10 V. Here the point corresponding to zerodegrees of the sine wave occurs at t = 25 ms, so at t = 0 the phase angle ()is given by 25/100 360 = 90 (or /2 rad). Therefore

v = Vp sin(2 ft + )= 10 sin(210t + )= 10 sin(63t /2)

2.2.5 Phase differences

Two waveforms of the same frequency may have a constant phase differ-ence between them, as shown in Figure 2.5. In this case, we will often saythat one waveform is phase shifted with respect to the other. To describethe phase relationship between the two, we often take one of the waveformsas our reference and describe the way in which the other leads or lags thiswaveform. In Figure 2.5, waveform A has been taken as the reference in eachcase. In Figure 2.5(a), waveform B reaches its maximum value some timeafter waveform A. We therefore say that B lags A. In this example B lags Aby 90. In Figure 2.5(b), waveform B reaches its maximum value beforewaveform A. Here B leads A by 90. In the figure the phase angles are shownin degrees, but they could equally well be expressed in radians.

It should be noted that the way in which the phase relationship isexpressed is a matter of choice. For example, if A leads B by 90, then clearlyB lags A by 90. These two statements are equivalent, and the one used willdepend on the situation and personal preference.

Figure 2.5 illustrates phase difference using two waveforms of the samemagnitude, but this is not a requirement. Phase difference can be measured

2.2 SINE WAVES 25

Example 2.3 Determine the equation of the following voltage signal.

Figure 2.5 Phase differencebetween two sine waves.

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between any two waveforms of the same frequency, regardless of their rel-ative size. We will consider methods of measuring phase difference later inthis chapter.

2.2.6 Average value of a sine wave

Clearly, if one measures the average value of a sine wave over one (or more)complete cycles, this average will be zero. However, in some situations weare interested in the average magnitude of the waveform independent of itspolarity (we will see an example of this later in this chapter). For a symmetr-ical waveform such as a sine wave, we can visualise this calculation as takingthe average of just the positive half-cycle of the waveform. In this case, theaverage is the area within this half-cycle divided by half the period. This pro-cess is illustrated in Figure 2.6(a). Alternatively, one can view the calculationas taking the average of a rectified sine wave (that is, a sine wave where thepolarity of the negative half-cycles has been reversed). This is shown inFigure 2.6(b).

We can calculate this average value by integrating a sinusoidal quantityover half a cycle and dividing by half the period. For example, if we considera sinusoidal voltage v = Vp sin , the period is equal to 2, so

Therefore

(2.7)

and similarly, for a sinusoidal current waveform,

(2.8)I I Iav p p= = . 2

0 637

V V Vav p p= = . 2

0 637

= 2Vp

= [ cos ]Vp

0

V Vav p= sin

1

0

d


Figure 2.6 Calculation of theaverage value of a sine wave.

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2.2 SINE WAVES 27

2.2.7 The r.m.s. value of a sine wave

Often of more interest than the average value is the root mean square orr.m.s. value of the waveform. This is true not only for sine waves but alsofor other alternating waveforms.

In Chapter 1, we noted that when a voltage V is applied across a resistorR this will produce a current I (determined by Ohms law), and that thepower dissipated in the resistor will be given by three equivalent expressions:

P = VI P = I 2R

If the voltage has a varying magnitude, then the instantaneous power will berelated to the instantaneous voltage and instantaneous current in a similarmanner. As before, we use lower-case characters to represent varying quan-tities, so the instantaneous power p is related to the instantaneous voltage vand instantaneous current i by the expressions

p = vi p = i 2R

The average power will be given by the average (or mean) values of theseexpressions. Since the resistance is constant, we could say that the averagepower is given by

or

Pav = [average (or mean) of i2 ]R = [R

Placing a line (a bar) above an expression is a common notation for the meanof that expression. The term ] is referred to as the mean-square voltageand [ as the mean-square current.

While the mean-square voltage and current are useful quantities, we moreoften use the square root of each quantity. These are termed the root-mean-square voltage (Vrms) and the root-mean-square current (Irms) where

and

We can evaluate each of these expressions by integrating a correspond-ing sinusoidal quantity over a complete cycle and dividing by the period. For example, if we consider a sinusoidal voltage v = Vp sin t, we can see that

= Vp2=

( cos )

/

VT

t tpT2

0

1 2

12

1 2 d

VT

V t trmsT

p=

sin

/

1

0

2 2

1 2

d

Irms = [

Vrms = ]

Pv

R Rav= =

[ ]

average (or mean) of 2 ]

p

vR

=2

P

VR

=2

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Therefore

(2.9)

and similarly

(2.10)

Combining these results with the earlier expressions gives

and

Pav = [R = I 2rms R

If we compare these expressions with those for the power produced by a constant voltage or current, we can see that the r.m.s. value of an altern-ating quantity produces the same power as a constant quantity of the samemagnitude. Thus for alternating quantities

Pav = VrmsIrms (2.11)

(2.12)

Pav = I2rms R (2.13)

This is illustrated in the following example.

Example 2.4 Calculate the power dissipated in a 10 resistor if the applied volt-age is:

(a) a constant 5 V;(b) a sine wave of 5 V r.m.s.;(c) a sine wave of 5 V peak.

(a)

(b)

(c)

2.2.8 Form factor and peak factor

The form factor of any waveform is defined as

(2.14)form factorr.m.s. value

average value =

P

VR

VR

VRav

rms p p= = = = =

( / )

/

/ .

2 2 2 22 2 5 210

1 25

W

P

VRavrms

= = = . 2 25

102 5 W

P

VR

. = = =2 25

102 5 W

P

VRavrms

= 2

PR

VRavrms

= = ] 2

I I Irms p p= = .

12

0 707

V V Vrms p p= = .

12

0 707


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2.3.2 Phase angle

We can if we wish divide the period of a square wave into 360 or 2 radians,as in a sine wave. This might be useful if we were discussing the phase difference between two square waveforms, as shown in Figure 2.8. Here twosquare waves have the same frequency but have a phase difference of 90 (or/2 radians). In this case B lags A by 90. An alternative way of describingthe relationship between the two waveforms is to give the time delay of onewith respect to the other.

2.3 SQUARE WAVES 29

For a sine wave

(2.15)

The significance of the form factor will become apparent in Section 2.5.The peak factor (also known as the crest factor) for a waveform is

defined as

(2.16)

For a sine wave

= 1.414 (2.17)

Although we have introduced the concepts of average value, r.m.s. value,form factor and peak factor for sinusoidal waveforms, it is important toremember that these measures may be applied to any repetitive waveform. Ineach case, the meanings of the terms are unchanged, although the numericalrelationships between these values will vary. To illustrate this, we will nowturn our attention to square waves.

2.3.1 Period, frequency and magnitude

Frequency and period have the same meaning for all repetitive waveforms,as do the peak and peak-to-peak values. Figure 2.7 shows an example of asquare-wave voltage signal and illustrates these various parameters.

peak factor .

=

VV

p

p0 707

peak factor

peak valuer.m.s. value

=

form factor .

. .= =

0 707

0 6371 11

VV

p

p

Square waves2.3

Figure 2.7 A square wavevoltage signal.

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2.3.3 Average and r.m.s. values

Since the average value of a symmetrical alternating waveform is its averagevalue over the positive half-cycle, the average value of a symmetrical squarewave (as in Figure 2.8) is equal to its peak value. Thus for a voltage wave-form the average value is Vp and for a current waveform it is Ip.

Since the instantaneous value of a symmetrical square wave is alwaysequal to either its positive or its negative peak value, the square of this valueis constant. For example, for a voltage waveform the instantaneous value willalways be either +Vp or Vp and in either case the square of this value will be constant at V 2p. Thus the mean of the voltage squared will be V

2p, and

the square root of this will be Vp. Therefore, the r.m.s. value of a square waveis simply equal to its peak value.

2.3.4 Form factor and peak factor

Using the definitions given in Section 2.2.8, we can now determine the formfactor and peak factor for a square wave. Since the average and r.m.s. valuesof a square wave are both equal to the peak value, it follows that

= 1.0

= 1.0

The relationship between the peak, average and r.m.s. values depends on theshape of a waveform. We have seen that this relationship is very different for a square wave and a sine wave, and further analysis would show similardifferences for other waveforms, such as triangular waves.

A wide range of instruments is available for measuring voltages and currentsin electrical circuits. These include analogue ammeters and voltmeters, dig-ital multimeters, and oscilloscopes. While each of these devices has its own

peak factor

peak valuer.m.s. value

=

form factorr.m.s. value

average value =


Figure 2.8 Phase-shiftedsquare waves.

Measuring voltages and

currents

2.4

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2.4.2 Measuring current in a circuit

To measure the current flowing through a conductor or a component, weconnect an ammeter in series with the element, as shown in Figure 2.9(b).Note that the ammeter is connected so that conventional current flows fromthe positive to the negative terminal.

2.4.3 Loading effects

Unfortunately, connecting additional components to a circuit can change thebehaviour of that circuit. These loading effects can also occur when a volt-meter or an ammeter is connected to a circuit. The result is that the processof measurement actually changes the quantity being measured.

These loading effects are illustrated in Figures 2.9(c) and 2.9(d), whichshow equivalent circuits for the measurement processes of Figures 2.9(a) and

2.4 MEASURING VOLTAGES AND CURRENTS 31

characteristics, there are some issues that are common to the use of all theseinstruments.

2.4.1 Measuring voltage in a circuit

To measure the voltage between two points in a circuit, we place a voltmeter(or other measuring instrument) between the two points. For example, tomeasure the voltage drop across a component we connect the voltmeteracross the part as shown in Figure 2.9(a).

Figure 2.9 Measuring voltageand current.

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2.9(b). In each case, the measuring instrument is replaced by its equivalentresistance RM, and it is clear that the presence of these additional resistanceswill affect the operation of the circuits. When measuring voltages (as inFigure 2.9(c)), the presence of the voltmeter reduces the effective resistanceof the circuit and therefore tends to lower the voltage between these twopoints in the circuit. To minimise this effect, the resistance of the voltmetershould be as high as possible to reduce the current that it passes. When measuring currents (as in Figure 2.9(d)), the ammeter tends to increase theresistance in the circuit and therefore tends to reduce the current flowing. Tominimise this effect, the ammeter should have as low a resistance as possibleto reduce the voltage drop across it.

When using analogue voltmeters and ammeters (as described in the next section), loading effects should always be considered. Instruments willnormally indicate their effective resistance (which will usually be differ-ent for each range of the instrument), and this information can be used toquantify any loading errors. If these are appreciable it may be necessary tomake corrections to the measured values. When using digital voltmeters oroscilloscopes, loading effects are usually less of a problem but should still be considered.

Most modern analogue ammeters and voltmeters are based on moving-coilmeters and we will look at the characteristics of such meters in Chapter 12.These produce an output in the form of movement of a pointer, where thedisplacement is directly proportional to the current through the meter.Meters are characterised by the current required to produce full-scaledeflection (f.s.d.) of the meter and their effective resistance RM. Typicalmeters produce a full-scale deflection for a current of between 50 A and 1 mA and have a resistance of between a few ohms and a few kilohms.

2.5.1 Measuring direct currents

Since the deflection of the meters pointer is directly proportional to the current through the meter, currents up to the f.s.d. value can be measureddirectly. For larger currents, shunt resistors are used to scale the meterseffective sensitivity. This is illustrated in Figure 2.10, where a meter with anf.s.d. current of 1 mA is used to measure a range of currents.

In Figure 2.10(a), the meter is being used to measure currents in the range01 mA, that is currents up to its f.s.d. value. In Figure 2.10(b), a shuntresistor RSH of RM /9 has been placed in parallel with the meter. Since thesame voltage is applied across the meter and the resistor, the current throughthe resistor will be nine times greater than that through the meter. To putthis another way, only one-tenth of the input current I will pass through themeter. Therefore, this arrangement has one-tenth the sensitivity of the meteralone and will produce an f.s.d. for a current of 10 mA. Figure 2.10(c) showsa similar arrangement for measuring currents up to 100 mA, and clearly thistechnique can be extended to measure very large currents. Figure 2.10(d)shows a switched-range ammeter arrangement, which can be used tomeasure a wide range of currents. It can be seen that the effective resistanceof the meter is different for each range.


Analogue ammeters and

voltmeters

2.5

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2.5 ANALOGUE AMMETERS AND VOLTMETERS 33

Example 2.5 A moving-coil meter produces an f.s.d. for a current 1 mA and has aresistance of 25 . Select a shunt resistor to turn this device into anammeter with an f.s.d. of 50 mA.

We need to reduce the sensitivity of the meter by a factor of

Therefore, we want 1/50 of the current to pass through the meter. There-fore, RSH must be equal to RM 49 = 510 m.

2.5.2 Measuring direct voltages

To measure direct voltages, we place a resistor in series with the meter andmeasure the resultant current, as shown in Figure 2.11. In Figure 2.11(a),the meter has an f.s.d. current of 1 mA and the series resistor RSE has beenchosen such that RSE + RM = 1 k. The voltage V required to produce a cur-rent of 1 mA is given by Ohms law and is 1 mA 1 k = 1 V. Therefore,an f.s.d. of the meter corresponds to an input voltage of 1 V.

In Figure 2.11(b), the series resistor has been chosen such that the totalresistance is 10 k, and this will give an f.s.d. for an input voltage of 10 V.In this way, we can tailor the sensitivity of the arrangement to suit our needs.Figure 2.11(c) shows a switched-range voltmeter that can be used to measurea wide range of voltages. As with the ammeter, the effective resistance of themeter changes as the ranges are switched.

501

50

mAmA

=

Figure 2.10 Use of a meter asan ammeter.

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Example 2.6 A moving-coil meter produces an f.s.d. for a current of 1 mA and hasa resistance of 25 . Select a series resistor to turn this device into avoltmeter with an f.s.d. of 50 V.

The required total resistance of the arrangement is given by the f.s.d. current divided by the full-scale input voltage. Hence

Therefore

RSE = 50 k RM= 49.975 k 50 k

2.5.3 Measuring alternating quantities

Moving-coil meters respond to currents of either polarity, each producingdeflections in opposite directions. Because of the mechanical inertia of themeter, it cannot respond to rapid changes in current and so will average thereadings over time. Consequently, a symmetrical alternating waveform willcause the meter to display zero.

R RSE M+ = =

501

50V

mAk


Figure 2.11 Use of a meter as avoltmeter.

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2.5 ANALOGUE AMMETERS AND VOLTMETERS 35

In order to measure an alternating current, we can use a rectifier to con-vert it into a unidirectional current that can be measured by the meter. Thisprocess was illustrated for a sine wave in Figure 2.6(b). The meter respondsby producing a deflection corresponding to the average value of the rectifiedwaveform.

We noted in Section 2.2 that when measuring sinusoidal quantities we are normally more interested in the r.m.s. value than in the average value.Therefore, it is common to calibrate AC meters so that they effectively multiply their readings by 1.11, this being the form factor of a sine wave.The result is that the meter (which responds to the average value of thewaveform) gives a direct reading of the r.m.s. value of a sine wave. However,a problem with this arrangement is that it gives an incorrect reading for non-sinusoidal waveforms. For example, we noted in Section 2.3 that the formfactor for a square wave is 1.0. Consequently, if we measure the r.m.s. valueof a square wave using a meter designed for use with sine waves, it will produce a reading that is about 11 per cent too high. This problem can beovercome by adjusting our readings to take account of the form factor of the waveform we are measuring.

Like all measuring devices, meters are only accurate over a certain rangeof frequencies determined by their frequency response. Most devices willwork well at the frequencies used for AC power distribution (50 or 60 Hz),but all will have a maximum frequency at which they can be used.

2.5.4 Analogue multimeters

General-purpose instruments use a combination of switches and resistors toachieve a large number of voltage and current ranges within a single unit.Such units are often referred to as analogue multimeters. A rectifier is alsoused to permit both unidirectional and alternating quantities to be measured,and additional circuitry is used to allow resistance measurement. While suchdevices are very versatile, they often have a relatively low input resistanceand therefore can have considerable loading effects on the circuits to whichthey are connected. A typical analogue multimeter is shown in Figure 2.12.

Figure 2.12 An analoguemultimeter.

ELEA_C02.qxd 2/10/09 2:18 PM Page 35

A standard measuring instrument in any electronics laboratory is a digitalmultimeter (DMM). This combines high accuracy and stability in a devicethat is very easy to use. It also normally has a very high input resistance whenused as a voltmeter and a very low input resistance when measuring currents,so minimising loading effects. While these instruments are capable of meas-uring voltage, current and resistance, they are often (inaccurately) referredto as digital voltmeters or simply DVMs. At the heart of the meter is an analogue-to-digital converter (ADC), which takes as its input a voltage signal and produces as its output a digital measurement that is usedto drive a numeric display. We will look at the operation of such ADCs inChapter 27.

Measurements of voltage, current and resistance are achieved by usingappropriate circuits to generate a voltage proportional to the quantity to bemeasured. When measuring voltages, the input signal is connected to anattenuator, which can be switched to vary the input range. When measuringcurrents, the input signal is connected across an appropriate shunt resistor,which generates a voltage proportional to the input current. The value of the shunt resistance is switched to select different input ranges. In order tomeasure resistance the inputs are connected to an ohms converter, whichpasses a small current between the two input connections. The resultantvoltage is a measure of the resistance between these terminals.

In simple DMMs, an alternating voltage is rectified, as in an analoguemultimeter, to give its average value. This is then multiplied by 1.11 (theform factor of a sine wave) to display the corresponding r.m.s. value. As dis-cussed earlier, this approach gives inaccurate readings when the alternatinginput signal is not sinusoidal. For this reason, more sophisticated DMMs use a true r.m.s. converter, which accurately produces a voltage propor-tional to the r.m.s. value of an input waveform. Such instruments can beused to make measurements of alternating quantities even when they are not sinusoidal. However, all DMMs are accurate over only a limited range offrequencies.

Figure 2.13(a) shows a typical hand-held DMM and Figure 2.13(b) is asimplified block diagram of a such a device.

An oscilloscope is an instrument that allows voltages to be measured by displaying the corresponding voltage waveform on a cathode ray tube(CRT) or LCD display. The oscilloscope effectively acts as an automatedgraph plotter that plots the input voltage against time.

2.7.1 Analogue oscilloscopes

In an analogue CRT oscilloscope a timebase circuit is used to scan a spot repeatedly from left to right across the screen at a constant speed byapplying a sawtooth waveform to the horizontal deflection circuitry. Aninput signal is then used to generate a vertical deflection proportional to themagnitude of the input voltage. Most oscilloscopes can display two inputquantities by switching the vertical deflection circuitry between two inputsignals. This can be done by displaying one complete trace of one waveform,


Digital multimeters

2.6

Oscilloscopes2.7

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2.7 OSCILLOSCOPES 37

then displaying one trace of the other (ALT mode) or by rapidly switchingbetween the two waveforms during each trace (CHOP mode). The choicebetween these modes is governed by the timebase frequency, but in eithercase the goal is to switch between the two waveforms so quickly that both aredisplayed steadily and with no noticeable flicker or distortion. In order toproduce a stable trace, the timebase circuitry includes a trigger circuit thatattempts to synchronise the beginning of the timebase sweep so that it alwaysstarts at the same point in a repetitive waveform thus producing a station-ary trace. Figure 2.14(a) shows a typical analogue laboratory oscilloscope andFigure 2.14(b) shows a simplified block diagram of such an instrument.

2.7.2 Digital oscilloscopes

In recent years, analogue oscilloscopes have largely been replaced by moreadvanced digital designs. These duplicate many of the basic features of ana-logue devices but use an ADC to change the input waveforms into a form

Figure 2.13 A digitalmultimeter (DMM).

ELEA_C02.qxd 2/10/09 2:18 PM Page 37

than can be more easily stored and manipulated than is possible with the analogue type. Digital oscilloscopes are particularly useful when looking atvery slow waveforms or short transients as their ability to store informationenables them to display a steady trace. Many instuments also provide meas-urement facilities to enable fast and accurate measurements to be made, aswell as mathematical functions for displaying information such as the fre-quency content of a signal. Figure 2.15(a) shows a typical digital laborato

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