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    Circuit Design Know It All (Newnes, 2008)

    Circuit Design

    The Newnes Know It All Series PIC Microcontrollers: Know It All Lucio Di Jasio, Tim Wilmshurst, Dogan Ibrahim, John Morton, Martin Bates, Jack Smith, D.W. Smith, and Chuck Hellebuyck ISBN: 978-0-7506-8615-0 Embedded Software: Know It All Jean Labrosse, Jack Ganssle, Tammy Noergaard, Robert Oshana, Colin Walls, Keith Curtis, Jason Andrews, David J. Katz, Rick Gentile, Kamal Hyder, and Bob Perrin ISBN: 978-0-7506-8583-2 Embedded Hardware: Know It All Jack Ganssle, Tammy Noergaard, Fred Eady, Lewin Edwards, David J. Katz, Rick Gentile, Ken Arnold, Kamal Hyder, and Bob Perrin ISBN: 978-0-7506-8584-9 Wireless Networking: Know It All Praphul Chandra, Daniel M. Dobkin, Alan Bensky, Ron Olexa, David Lide, and Farid Dowla ISBN: 978-0-7506-8582-5 RF & Wireless Technologies: Know It All Bruce Fette, Roberto Aiello, Praphul Chandra, Daniel Dobkin, Alan Bensky, Douglas Miron, David Lide, Farid Dowla, and Ron Olexa ISBN: 978-0-7506-8581-8 Electrical Engineering: Know It All Clive Maxfield, Alan Bensky, John Bird, W. Bolton, Izzat Darwazeh, Walt Kester, M. A. Laughton, Andrew Leven, Luis Moura, Ron Schmitt, Keith Sueker, Mike Tooley, DF Warne, Tim Williams ISBN: 978-1-85617-528-9 Audio Engineering: Know It All Douglas Self, Richard Brice, Don Davis, Ben Duncan, John Linsley Hood, Morgan Jones, Eugene Patronis, Ian Sinclair, Andrew Singmin, John Watkinson ISBN: 978-1-85617-526-5 Circuit Design: Know It All Darren Ashby, Bonnie Baker, Stuart Ball, John Crowe, Barrie Hayes-Gill, Ian Grout, Ian Hickman, Walt Kester, Ron Mancini, Robert A. Pease, Mike Tooley, Tim Williams, Peter Wilson, Bob Zeidman ISBN: 978-1-85617-527-2 Test and Measurement: Know It All Jon Wilson, Stuart Ball, GMS de Silva,Tony Fischer-Cripps, Dogan Ibrahim, Kevin James, Walt Kester, M. A. Laughton, Chris Nadovich, Alex Porter, Edward Ramsden, Stephen Scheiber, Mike Tooley, D. F. Warne, Tim Williams ISBN: 978-1-85617-530-2 Mobile Wireless Security: Know It All Praphul Chandra, Alan Bensky, Tony Bradley, Chris Hurley, Steve Rackley, John Rittinghouse, James Ransome, Timothy Stapko, George Stefanek, Frank Thornton, Chris Lanthem, Jon Wilson ISBN: 978-1-85617-529-6 For more information on these and other Newnes titles visit: www.newnespress.com

    Circuit Design Darren Ashby Bonnie Baker Stuart Ball J. Crowe Barrie Hayes-Gill Ian Hickman Walt Kester Ron Mancini Ian Grout Robert A. Pease Mike Tooley Tim Williams Peter Wilson Bob Zeidman

    AMSTERDAM • BOSTON • HEIDELBERG • LONDON NEW YORK • OXFORD • PARIS • SAN DIEGO SAN FRANCISCO • SINGAPORE • SYDNEY • TOKYO

    Newnes is an imprint of Elsevier

    Newnes is an imprint of Elsevier 30 Corporate Drive, Suite 400, Burlington, MA 01803, USA Linacre House, Jordan Hill, Oxford OX2 8DP, UK Copyright

    #

    2008, Elsevier Inc. All rights reserved.

    No part of this publication may be reproduced, stored in a retrieval system, or transmitted in any form or by any means, electronic, mechanical, photocopying, recording, or otherwise, without the prior written permission of the publisher. Permissions may be sought directly from Elsevier’s Science & Technology Rights Department in Oxford, UK: phone: (+44) 1865 843830, fax: (+44) 1865 853333, E-mail: [email protected]You may also complete your request online via the Elsevier homepage (http://elsevier.com), by selecting “Support & Contact” then “Copyright and Permission” and then “Obtaining Permissions.” Library of Congress Cataloging-in-Publication Data Application submitted British Library Cataloguing-in-Publication Data A catalogue record for this book is available from the British Library. ISBN: 978-1-85617-527-2 For information on all Newnes publications visit our Web site at www.elsevierdirect.com Printed in the United States of America 08 09 10 10 9 8 7 6 5 4 3

    2

    1

    Contents About the Authors............................................................................... xv Chapter 1: The Fundamentals ................................................................ 1 1.1 Electrical Fundamentals..................................................................................1 1.2 Passive Components ..................................................................................... 36 1.3 DC Circuits.................................................................................................. 87 1.4 Alternating Voltage and Current ................................................................. 124 1.5 Circuit Simulation ...................................................................................... 151 1.6 Intuitive Circuit Design .............................................................................. 166 1.7 Troubleshooting Basic ................................................................................ 177 References .......................................................................................................... 208

    Chapter 2: The Semiconductor Diode....................................................211 References .......................................................................................................... 215

    Chapter 3: Understanding Diodes and Their Problems .............................217 3.1 Speed Demons ........................................................................................... 219 3.2 Turn ’em off—turn ’em on. . . .................................................................... 220 3.3 Other Strange Things that Diodes Can Do to You. . . .................................. 222 3.4 Zener, Zener, Zener.................................................................................... 225 3.5 Diodes that Glow in the Dark, Efficiently................................................... 228 3.6 Optoisolators .............................................................................................. 228 3.7 Assault and Battery .................................................................................... 231 References .......................................................................................................... 232

    Chapter 4: Bipolar Transistors ............................................................235 References .......................................................................................................... 247

    Chapter 5: Field-Effect Transistors.......................................................249 References .......................................................................................................... 254

    Chapter 6: Identifying and Avoiding Transistor Problems..........................255 6.1 6.2

    More Beta—More Better? .......................................................................... 257 Field-Effect Transistors .............................................................................. 258

    www.newnespress.com

    vi

    Contents

    6.3 Power Transistors may Hog Current ........................................................... 261 6.4 Apply the 5-Second Rule ........................................................................... 264 6.5 Fabrication Structures make a Difference.................................................... 264 6.6 Power-Circuit Design Requires Expertise.................................................... 267 6.7 MOSFETs Avoid Secondary Breakdown .................................................... 269 References .......................................................................................................... 270

    Chapter 7: Digital Circuit Fundamentals ...............................................271 7.1 Digital Technology..................................................................................... 273 References .......................................................................................................... 278

    Chapter 8: Number Systems ...............................................................279 8.1 8.2 8.3 8.4 8.5 8.6 8.7

    Introduction................................................................................................ 279 Decimal–Unsigned Binary Conversion........................................................ 280 Signed Binary Numbers.............................................................................. 284 Gray Code ................................................................................................. 289 Binary Coded Decimal ............................................................................... 290 Octal-Binary Conversion ............................................................................ 291 Hexadecimal-Binary Conversion................................................................. 294

    Chapter 9: Binary Data Manipulation ..................................................301 9.1 Introduction................................................................................................ 301 9.2 Logical Operations ..................................................................................... 302 9.3 Boolean Algebra ........................................................................................ 303 9.4 Combinational Logic Gates ........................................................................ 306 9.5 Truth Tables............................................................................................... 308 References .......................................................................................................... 317

    Chapter 10: Combinational Logic Design...............................................319 10.1 Introduction.............................................................................................. 319 10.2 NAND and NOR Logic ............................................................................ 332 10.3 Karnaugh Maps ........................................................................................ 334 10.4 Don’t Care Conditions.............................................................................. 341 References .......................................................................................................... 341

    Chapter 11: Sequential Logic Design ....................................................343 11.1 11.2 11.3 11.4 11.5 11.6

    Introduction.............................................................................................. 343 Level Sensitive Latches and Edge-triggered Flip-flops .............................. 348 The D Latch and D-type Flip-Flop ........................................................... 348 Counter Design ........................................................................................ 354 State Machine Design............................................................................... 366 Moore Versus Mealy State Machines........................................................ 377

    www.newnespress.com

    Contents

    vii

    11.7 Shift Registers.......................................................................................... 377 11.8 Digital Scan Path ..................................................................................... 379 References .......................................................................................................... 382

    Chapter 12: Memory.........................................................................383 12.1 12.2 12.3

    Introduction ............................................................................................ 383 Random Access Memory ........................................................................ 385 Read-only Memory................................................................................. 386

    Chapter 13: Selecting a Design Route...................................................389 13.1 13.2 13.3 13.4 13.5 13.6

    Introduction ............................................................................................ 389 Discrete Implementation ......................................................................... 391 Mask Programmable ASICs .................................................................... 400 Field-Programmable Logic...................................................................... 414 VHDL .................................................................................................... 434 Choosing a Design Route ....................................................................... 436

    Chapter 14: Designing with Logic ICs...................................................441 14.1

    Logic ICs ............................................................................................... 441

    Chapter 15: Interfacing......................................................................455 15.1 15.2 15.3 15.4 15.5 15.6

    Mixing Analog and Digital ..................................................................... 455 Generating Digital Levels from Analog Inputs ........................................ 458 Protection Against Externally Applied Overvoltages ............................... 461 Isolation ................................................................................................. 462 Classic Data Interface Standards ............................................................. 465 High-Performance Data Interface Standards ............................................ 471

    Chapter 16: DSP and Digital Filters ....................................................477 16.1 Origins of Real-World Signals and Their Units of Measurement ............. 477 16.2 Reasons for Processing Real-World Signals ............................................ 478 16.3 Generation of Real-World Signals........................................................... 480 16.4 Methods and Technologies Available for Processing Real-World Signals 480 16.5 Analog Versus Digital Signal Processing ................................................ 481 16.6 A Practical Example ............................................................................... 482 16.7 Finite Impulse Response (FIR) Filters ..................................................... 489 16.8 FIR Filter Implementation In DSP Hardware Using Circular Buffering ... 494 16.9 Designing FIR Filters ............................................................................. 497 16.10 Infinite Impulse Response (IIR) Filters ................................................... 508 16.11 IIR Filter Design Techniques .................................................................. 511 16.12 Multirate Filters...................................................................................... 514 16.13 Adaptive Filters ...................................................................................... 519 References .......................................................................................................... 523

    www.newnespress.com

    viii

    Contents

    Chapter 17: Dealing with High-Speed Logic ...........................................525 References on Dealing with High-Speed Logic ........................................... 532 Chapter 18: Bridging the Gap between Analog and Digital .......................533 18.1 Try to Measure Temperature Digitally .................................................... 536 18.2 Road Blocks Abound.............................................................................. 540 18.3 The Ultimate Key to Analog Success...................................................... 548 18.4 How Analog and Digital Design Differ ................................................... 549 18.5 Time and Its Inversion............................................................................ 556 18.6 Organizing Your Toolbox ....................................................................... 556 18.7 Set Your Foundation and Move On, Out of The Box .............................. 557 References .......................................................................................................... 558

    Chapter 19: Op-Amps .......................................................................559 19.1 The Magical Mysterious Op-Amp ........................................................... 559 19.2 Understanding Op-Amp Parameters ........................................................ 572 19.3 Modeling Op-Amps ................................................................................ 599 19.4 Finding the Perfect Op-Amp ................................................................... 600 References .......................................................................................................... 618

    Chapter 20: Analog-to-Digital Converters..............................................619 20.1 20.2 20.3 20.4 20.5 20.6 20.7 20.8 20.9 20.10 20.11 20.12 20.13 20.14

    ADCs ..................................................................................................... 621 Types of ADCs ...................................................................................... 624 ADC Comparison ................................................................................... 633 Sample and Hold .................................................................................... 634 Real Parts............................................................................................... 636 Microprocessor Interfacing ..................................................................... 637 Clocked Interfaces .................................................................................. 643 Serial Interfaces...................................................................................... 644 Multichannel ADCs ................................................................................ 650 Internal Microcontroller ADCs................................................................ 650 Codecs ................................................................................................... 652 Interrupt Rates........................................................................................ 652 Dual-Function Pins On Microcontrollers ................................................. 653 Design Checklist .................................................................................... 655

    Chapter 21: Sensors..........................................................................657 21.1 21.2 21.3 21.4 21.5

    Instrumentation and Control Systems ...................................................... 657 Transducers ............................................................................................ 659 Sensors................................................................................................... 660 Switches................................................................................................. 667 Semiconductor Temperature Sensors ....................................................... 672

    www.newnespress.com

    Contents 21.6 21.7 21.8 21.9 21.10 21.11 21.12 21.13

    ix

    Thermocouples ....................................................................................... 672 Threshold Detection................................................................................ 674 Outputs .................................................................................................. 676 LED Indicators ....................................................................................... 676 Driving High-Current Loads ................................................................... 678 Audible Outputs ..................................................................................... 678 Motors.................................................................................................... 681 Driving Mains Connected Loads ............................................................. 682

    Chapter 22: Active Filters ..................................................................685 22.1 Introduction ............................................................................................ 685 22.2 Fundamentals of Low-Pass Filters........................................................... 686 22.3 Low-Pass Filter Design........................................................................... 697 22.4 High-Pass Filter Design .......................................................................... 707 22.5 Band-Pass Filter Design.......................................................................... 714 22.6 Band-Rejection Filter Design .................................................................. 724 22.7 All-Pass Filter Design............................................................................. 729 22.8 Practical Design Hints ............................................................................ 734 22.9 Filter Coefficient Tables ......................................................................... 744 References .......................................................................................................... 752

    Chapter 23: Radio-Frequency (RF) Circuits ...........................................753 23.1 Modulation of Radio Waves ................................................................... 753 23.2 Low-Power RF Amplifiers...................................................................... 759 23.3 Stability.................................................................................................. 762 23.4 Linearity................................................................................................. 767 23.5 Noise and Dynamic Range ..................................................................... 771 23.6 Impedances and Gain ............................................................................. 773 23.7 Mixers.................................................................................................... 778 23.8 Demodulators ......................................................................................... 783 23.9 Oscillators .............................................................................................. 787 References .......................................................................................................... 795

    Chapter 24: Signal Sources.................................................................797 24.1 Voltage References................................................................................. 797 24.2 NonsinusoidaI Waveform Generators ...................................................... 800 24.3 Sine Wave Generators ............................................................................ 808 24.4 Voltage-Controlled Oscillators And Phase Detectors ............................... 817 References .......................................................................................................... 828

    Chapter 25: EDA Design Tools for Analog and RF .................................829 25.1 25.2

    The Old Pencil and Paper Design Process............................................... 835 Is Your Simulation Fundamentally Valid?............................................... 838

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    Contents 25.3 Macromodels: What Can They Do? .......................................................... 843 25.4 VHDL-AMS............................................................................................. 849 References .......................................................................................................... 867

    Chapter 26: Useful Circuits ................................................................869 26.1 Introduction.............................................................................................. 869 26.2 Boundary Conditions ................................................................................ 873 26.3 Amplifiers ................................................................................................ 873 26.4 Computing Circuits .................................................................................. 891 26.5 Oscillators ................................................................................................ 902 26.6 Some Favorite Circuits ............................................................................. 910 References .......................................................................................................... 915

    Chapter 27: Programmable Logic to ASICs............................................917 27.1 Programmable Read-Only Memory (PROM)............................................. 918 27.2 Programmable Logic Arrays (PLAs) ......................................................... 922 27.3 Programmable Array Logic (PALs) .......................................................... 923 27.4 The Masked Gate Array ASIC.................................................................. 929 27.5 CPLDs and FPGAs................................................................................... 931 27.6 Summary.................................................................................................. 932 References .......................................................................................................... 932

    Chapter 28: Complex Programmable Logic Devices (CPLDs) ....................933 28.1 CPLD Architectures ................................................................................. 933 28.2 Function Blocks ....................................................................................... 934 28.3 I/O Blocks................................................................................................ 936 28.4 Clock Drivers ........................................................................................... 937 28.5 Interconnect ............................................................................................. 938 28.6 CPLD Technology and Programmable Elements ....................................... 940 28.7 Embedded Devices ................................................................................... 940 28.8 Summary: CPLD Selection Criteria .......................................................... 944 References .......................................................................................................... 946

    Chapter 29: Field Programmable Gate Arrays (FPGAs) ..........................947 29.1 29.2 29.3 29.4 29.5 29.6 29.7 29.8

    FPGA Architectures ................................................................................. 947 Configurable Logic Blocks ....................................................................... 948 Configurable I/O Blocks........................................................................... 951 Embedded Devices ................................................................................... 954 Programmable Interconnect ...................................................................... 955 Clock Circuitry......................................................................................... 957 SRAM vs. Antifuse Programming............................................................. 957 Emulating and Prototyping ASICs ............................................................ 961

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    29.9 Summary.................................................................................................. 964 References .......................................................................................................... 965

    Chapter 30: Design Automation and Testing for FPGAs ..........................967 30.1 Simulation .............................................................................................. 967 30.2 Libraries................................................................................................. 971 30.3 Synthesis ................................................................................................ 974 30.4 Physical Design Flow ............................................................................. 977 30.5 Place and Route ..................................................................................... 977 30.6 Timing Analysis ..................................................................................... 978 30.7 Design Pitfalls ........................................................................................ 978 30.8 VHDL Issues for FPGA Design .............................................................. 979 30.9 Summary................................................................................................ 979 References .......................................................................................................... 980

    Chapter 31: Integrating Processors onto FPGAs .....................................981 31.1 31.2 31.3 31.4

    Introduction ............................................................................................ 981 A Simple Embedded Processor ............................................................... 982 Soft Core Processors on an FPGA ........................................................ 1004 Summary.............................................................................................. 1004

    Chapter 32: Implementing Digital Filters in VHDL ...............................1005 32.1 32.2 32.3 32.4 32.5 32.6 32.7

    Introduction .......................................................................................... 1005 Converting S-Domain to Z-Domain ...................................................... 1006 Implementing Z-Domain Functions in VHDL ....................................... 1008 Basic Low-Pass Filter Model ................................................................ 1013 FIR Filters............................................................................................ 1017 IIR Filters............................................................................................. 1018 Summary.............................................................................................. 1018

    Chapter 33: Microprocessor and Microcontroller Overview .....................1019 33.1 33.2 33.3 33.4 33.5 33.6 33.7 33.8 33.9

    Microprocessor Systems ....................................................................... 1019 Single-Chip Microcomputers................................................................. 1020 Microcontrollers ................................................................................... 1020 Microprocessor Systems ....................................................................... 1020 Data Types ........................................................................................... 1024 Data Storage......................................................................................... 1024 The Microprocessor .............................................................................. 1025 Microprocessor Operation ..................................................................... 1032 A Microcontroller System..................................................................... 1038

    Chapter 34: Microcontroller Toolbox ..................................................1043 34.1

    Microcontroller Supply and Reference .................................................. 1043

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    Contents 34.2 34.3 34.4 34.5 34.6 34.7 34.8 34.9

    Resistor Networks ................................................................................ 1045 Multiple Input Control.......................................................................... 1046 AC Control........................................................................................... 1049 Voltage Monitors and Supervisory Circuits ........................................... 1050 Driving Bipolar Transistors................................................................... 1051 Driving MOSFETs ............................................................................... 1054 Reading Negative Voltages................................................................... 1057 Example Control System ...................................................................... 1059

    Chapter 35: Power Supply Overview and Specifications .........................1071 35.1 35.2 35.3

    Power Supplies..................................................................................... 1071 Specifications ....................................................................................... 1078 Off-the-Shelf or Roll Your Own ........................................................... 1078

    Chapter 36: Input and Output Parameters ..........................................1081 36.1 36.2 36.3 36.4 36.5 36.6 36.7 36.8 36.9 36.10 36.11 36.12 36.13

    Voltage ................................................................................................ 1081 Current ................................................................................................. 1082 Fuses.................................................................................................... 1082 Switch-On Surge, or Inrush Current ...................................................... 1084 Waveform Distortion and Interference .................................................. 1087 Frequency............................................................................................. 1090 Efficiency............................................................................................. 1090 Deriving the Input Voltage from the Output.......................................... 1092 Low-Load Condition............................................................................. 1094 Rectifier and Capacitor Selection .......................................................... 1095 Load and Line Regulation .................................................................... 1097 Ripple and Noise .................................................................................. 1099 Transient Response............................................................................... 1101

    Chapter 37: Batteries......................................................................1103 37.1 37.2 37.3 37.4

    Initial Considerations ............................................................................ 1103 Primary Cells ....................................................................................... 1108 Secondary Cells.................................................................................... 1110 Charging .............................................................................................. 1114

    Chapter 38: Layout and Grounding for Analog and Digital Circuits..........1117 38.1 38.2 38.3 38.4 38.5 38.6 38.7 38.8

    The Similarities of Analog and Digital Layout Practices ....................... 1117 Where the Domains Differ—Ground Planes Can Be a Problem............. 1121 Where the Board and Component Parasitics Can Do the Most Damage. 1123 Layout Techniques that Improve ADC Accuracy and Resolution........... 1131 The Art of Laying Out Two-Layer Boards ............................................ 1135 Current Return Paths With or Without a Ground Plane ......................... 1140 Layout Tricks for a 12-Bit Sensing System ........................................... 1142 General Layout Guidelines—Device Placement..................................... 1144

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    38.9 General Layout Guidelines—Ground and Power Supply Strategy .......... 1144 38.10 Signal Traces........................................................................................ 1147 38.11 Did I Say Bypass and Use an Anti-Aliasing Filter?............................... 1148 38.12 Bypass Capacitors ................................................................................ 1148 38.13 Anti-Aliasing Filters ............................................................................. 1149 38.14 PCB Design Checklist .......................................................................... 1150 References ........................................................................................................ 1151

    Chapter 39: Safety .........................................................................1153 39.1 39.2 39.3 39.4 39.5

    The Hazards of Electricity .................................................................... 1154 Safety Classes ...................................................................................... 1155 Insulation Types ................................................................................... 1156 Design Considerations for Safety Protection.......................................... 1156 Fire Hazard .......................................................................................... 1158

    Chapter 40: Design for Production.....................................................1161 40.1 40.2

    Checklist .............................................................................................. 1162 The Dangers of ESD ............................................................................ 1164

    Chapter 41: Testability ....................................................................1167 41.1 41.2 41.3 41.4

    In-Circuit Testing ................................................................................. 1167 Functional Testing ................................................................................ 1168 Boundary Scan and JTAG .................................................................... 1170 Design Techniques ............................................................................... 1174

    Chapter 42: Reliability ....................................................................1177 42.1 42.2 42.3 42.4 42.5

    Definitions............................................................................................ 1177 The Cost of Reliability ......................................................................... 1179 Design for Reliability ........................................................................... 1180 The Value of MTBF Figures ................................................................ 1184 Design Faults ....................................................................................... 1186

    Chapter 43: Thermal Management.....................................................1187 43.1 43.2 43.3 43.4

    Using Thermal Resistance .................................................................... 1187 Heatsinks.............................................................................................. 1193 Power Semiconductor Mounting ........................................................... 1197 Placement and Layout .......................................................................... 1201

    Appendix A Standards......................................................................1203 A.1 A.2

    British Standards .................................................................................. 1203 IEC Standards ...................................................................................... 1206

    Index.............................................................................................1207 www.newnespress.com

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    About the Authors

    Darren Ashby (Chapters 1, 19, 26, and 35) author of Electrical Engineering 101, is a self-described “techno geek with pointy hair.” He considers himself a Jack of all trades, master of none. He figures his common sense came from his dad and his book sense from his mother. Raised on a farm and graduating from Utah State University seemingly ages ago, he has nearly 20 years of experience in the real world as a technician, engineer and a manager. He has worked in diverse areas of compliance, production, testing and his personal favorite, R&D. He jumped at a chance some years back to teach a couple of semesters at his alma mater. For about two years, he wrote regularly for the online magazine “chipcenter. com.” He is currently the Director of electronics R&D at a billion dollar consumer product company. His passions are boats, snowmobiles, motorcycles and pretty much anything with a motor in it. When not at his day job, he spends most his time with his family and a promising R&D consulting/manufacturing firm he started a couple of years ago. He lives with his beautiful wife, four strapping boys and cute little daughter next to the mountains in Richmond, Utah. Bonnie Baker (Chapters 18, 19, 25 and 38) author of A Baker’s Dozen, also writes the monthly “Baker’s Best” for EDN magazine. She has been involved with analog and digital designs and systems for over 20 years. Bonnie started as a manufacturing product engineer supporting analog products at Burr-Brown. From there, Bonnie moved up to IC design, analog division strategic marketer, and then corporate applications engineering manager. In 1998, she joined Microchip Technology and served as their analog division analog/mixed-signal applications engineering manager and staff architect engineer for one of their PICmicro divisions. This expanded her background to

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    About the Authors

    not only include analog applications, but microcontroller solutions as well. At present, she has returned to the Precision Analog fold at Texas Instruments in Tucson, Arizona. Bonnie holds a Masters of Science in Electrical Engineering from the University of Arizona (Tucson, AZ) and a bachelor’s degree in music education from Northern Arizona University (Flagstaff, AZ). In addition to her fascination with analog design, Bonnie has a drive to share her knowledge and experience and has written almost 300 articles, design notes, and application notes and she is a frequent presenter at technical conferences and shows. Stuart Ball, P.E., (Chapters 20, 34) author of Analog Interfacing to Embedded Microprocessors, is an electrical engineer with over 20 years of experience in electronic and embedded systems. He is currently employed with Seagate Technologies, a manufacturer of computer hard disc drives. Bruce Carter (Chapter 19) a contributor to Electrical Engineering 101, is currently an Engineer for the Test and Measurement group of Texas Instruments. Carter earned a BS in Engineering Physics from Texas Tech University, and a BS in Electrical Engineering from the University of Texas. He authored several technical articles, including four chapters in Op-Amps for Everyone. New edition publishing soon. John Crowe (Chapter 13) co-author of Introduction to Digital Electronics, is Reader in Biomedical Informatics in the School of Electrical & Electronic Engineering, University of Nottingham, UK. His contribution to this book is based upon material used in a Digital Electronics module delivered to 1st and 2nd year undergraduate students. His research concerns the development of novel biomedical instrumentation such as fetal heart rate monitors and integrated optical and electronics ASICs for imaging skin blood flow. Ian Grout (Chapters 7, 8, 9, 10, 11, and 12) the author of Digital Systems Design received his B.Eng in Electronic Engineering (1991) and PhD (1994) from Lancaster University (UK). He has worked in both industry and the academic field in microelectronic circuit and electronics design and test. He currently works in the areas of mixed-signal integrated circuit (IC) design for testability (DfT) and digital electronic circuit design using programmable logic. The author is currently a lecturer within the Department of Electronic and Computer Engineering at the University of Limerick (Ireland). He currently teaches programmable logic and integrated circuit design and

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    About the Authors

    xvii

    test principles within the university and has worked in Limerick since 1998. Prior to this he was a lecturer in the Engineering Department at Lancaster University (UK). Barrie Hayes-Gill (Chapter 13) co-author of Introduction to Digital Electronics, is Associate Professor in Integrated Circuit Design and Electronic Instrumentation in the School of Electrical & Electronic Engineering, University of Nottingham, UK. He has lectured inintegrated circuit design both within the University of Nottingham and at international locations around the World. His research and industrial work concerns the development of compact and low noise instrumentation for medical devices and instrumentation where he deploys off-the-shelf electronic components and semi-custom and full custom integrated circuits for integrated optical sensors. He has published widely with over 150 publications and 10 patents on medical devices and VLSI systems. In addition to his University post he is also an Executive Directorat Monica Healthcare Ltd. Ian Hickman, Eur. Ing. D. I. H. May B.Sc.Hons, C.Eng., MIEE, MIEEE (Chapters 2, 4, 5, 23, 24) is the author of Analog Electronics. He has been interested in electronics since the late 1940s, and professionally involved in it since 1954. Starting with a crystal set, his interests over the years have covered every aspect of electronics, though mainly concentrating on analog. Now retired, Ian was a consultant to Electronics World for many years. He is a Member of the Institution of Engineering and Technology: and a Life Member of the Institute of Electrical & Electronics Engineers. He has also written several books including Practical RF Handbook, Hickman’s Analog and RF Circuits, and Analog Circuits Cookbook, to name just a few. Walt Kester (Chapters 16, 17) is the author of Mixed Signal and DSP Design Techniques. He is a corporate staff applications engineer at Analog Devices. For over 35 years at Analog Devices, he has designed, developed, and given applications support for high-speed ADCs, DACs, SHAs, op-amps, and analog multiplexers. Besides writing many papers and articles, he prepared and edited eleven major applications books, which form the basis for the Analog Devices world-wide technical seminar series including the topics of op-amps, data conversion, power management, sensor signal conditioning, mixed-signal, and practical analog design techniques. He also is the editor of The Data Conversion Handbook, a 900þ page comprehensive book on data conversion published in 2005 by Elsevier. Walt has a BSEE from NC State University and MSEE from Duke University.

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    About the Authors

    Thomas Kugelstadt (Chapter 22) was a contributor to Op Amps for Everyone. He is a senior application engineer at Texas Instruments. He is writing many technical articles on various subjects, often system related. He also provides freelance writing services if your company were ever interested in a technical subject but experienced difficulties finding a writer. Ron Mancini (Chapter 26) the editor of Op Amps for Everyone has spent nearly fifty years in electronics. Recently retired, he was a Staff Scientist at Texas Instruments for many years. He was also a regular columnist for EDN. Richard Palmer (Chapter 26) was a contributor to Op Amps for Everyone. Robert A. Pease (Chapters 1, 3, 6) author of Troubleshooting Analog Circuits, attended Mt. Hermon School, and graduated from MIT in 1961 with a BSEE. He worked at Philbrick Researches up to 1975 and designed many Op-Amps and Analog Computing Modules. Pease joined National Semiconductor in 1976. He has designed about 24 analog ICs including power regulators, voltage references, and temp sensors. He has written 65þ magazine articles and holds about 21 US patents. Pease is the self-declared Czar of Bandgaps since 1986. He enjoys hiking and trekking in Nepal, and ferroequinology. His position at NSC is Staff Scientist. He is a Senior Member of the IEEE. Pease is a columnist in Electronic Design magazine, with over 240 columns published. The column, PEASE PORRIDGE, covers a wide range of technical topics. Pease also has posted many technical and semi-technical items on his main website: http://www.national.com/rap Many of Pease’s recent columns are accessible there. Pease was inducted into the E.E. Hall Of Fame in 2002. Refer to: http://www.elecdesign.com/Articles/Index.cfm?ArticleID=17269&Extension=pdf See Pease’s other web site at http://www.transtronix.com

    Mike Tooley (Chapters 1, 21, and 33) author of Electronic Circuits, is a technical author and consultant. He was formerly Vice Principal at Brooklands College in Surrey, England, where he was responsible for the delivery of learning to over 10,000 Further and Higher Education students increasingly by flexible, open and online distance learning. Mike is the well-known author of several popular engineering and related text books, including widely adopted course texts for BTEC, GCE A-level and GCSE

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    About the Authors

    xix

    qualifications in Engineering. Mike’s hobbies include astronomy, amateur radio, aviation, computing and electronic circuit design and construction. Tim Williams (Chapters 14, 15, 19, 35, 36, 37, 39, 40,41, 42, 43, and Appendix A) is the author of The Circuit Designer’s Companion, 2nd Edition. He works at Elmac Services, which provides consultancy and training on all aspects of EMC, including design, testing and the application of standards, to companies manufacturing electronic products and concerned about the implications of the EMC Directive. Tim Williams gained a BSc in Electronic Engineering from Southampton University in 1976. He has worked in electronic product design in various industry sectors including process instrumentation and audio visual control. He was design group leader at Rosemount Ltd before leaving in 1990 to start Elmac Services. He is also the author of “EMC for Product Designers” (now in its fourth edition, Elsevier 2006), and has presented numerous conference papers and seminars. He is also author of “EMC for Systems & Installations” with Keith Armstrong. He is an EMC technical assessor for UKAS and SWEDAC. Peter Wilson (Chapters 25, 30, 31, 32) author of Design Recipes for FPGAs, is Senior Lecturer in Electronics at the University of Southampton. He holds degrees from Heriot-Watt University, an MBA from Edinburgh Business School and a PhD from the University of Southampton. He worked in the Avionics and Electronics Design Automation Industries for many years at Ferranti, GEC-Marconi and Analogy prior to rejoining academia . He has published widely in the areas of FPGA design, modeling and simulation, VHDL, VHDL-AMS, magnetics and power electronics. He is a Senior Member of the IEEE, member of the IET, and a Chartered Engineer. Bob Zeidman (Chapters 27, 28, 29) author of Designing with FPGAs and CPLDs, is the president of Zeidman Consulting (www.ZeidmanConsulting.com), a premiere contract research and development firm in Silicon Valley. He is also the president of Zeidman Technologies (www.zeidman.biz), a developer of tools for embedded systems hardware and software development, and president of Software Analysis and Forensic Engineering Corporation (www.SAFE-corp.biz), the leading provider of software intellectual property analysis tools. Bob has designed ASICs, FPGAs, and PC boards for RISC-based parallel processor systems, laser printers, network switches and routers, and other systems for clients including Apple Computer, Cisco Systems, Mentor Graphics, and Ricoh. He is the inventor of SynthOSTM, a tool for synthesizing software

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    About the Authors

    from a high-level description, and CodeSuiteW, a tool for measuring software source code correlation. His publications include papers on hardware and software design methods and three textbooks: Designing with FPGAs and CPLDs, Verilog Designer’s Library, and Introduction to Verilog. Bob has taught courses at conferences throughout the world. He holds several patents and earned bachelor’s degrees in physics and electrical engineering at Cornell University and a master’s degree in electrical engineering at Stanford University.

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    CHAPTER 1

    The Fundamentals Mike Tooley Darren Ashby Robert Pease

    1.1 Electrical Fundamentals This chapter has been designed to provide you with the background knowledge required to help you understand the concepts introduced in the later chapters. If you have studied electrical science, electrical principles, or electronics then you will already be familiar with many of these concepts. If, on the other hand, you are returning to study or are a newcomer to electronics or electrical technology this chapter will help you get up to speed.

    1.1.1

    Fundamental Units

    You will already know that the units that we now use to describe such things as length, mass and time are standardized within the International System of Units (SI). This SI system is based upon the seven fundamental units (see Table 1.1).

    1.1.2

    Derived Units

    All other units are derived from these seven fundamental units. These derived units generally have their own names and those commonly encountered in electrical circuits are summarized in Table 1.2, together with the corresponding physical quantities. (Note that 0K is equal to 273 C and an interval of 1K is the same as an interval of 1 C.) If you find the exponent notation shown in Table 1.2 a little confusing, just remember that V1 is simply 1/V, s1 is 1/s, m2 is 1/m2, and so on.

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    Chapter 1 Table 1.1: SI units Quantity

    Unit

    Abbreviation

    Current

    ampere

    A

    Length

    meter

    m

    Luminous intensity

    candela

    cd

    Mass

    kilogram

    kg

    Temperature

    Kelvin

    K

    Time

    second

    s

    Matter

    mol

    mol

    Table 1.2: Electrical quantities Quantity

    Derived unit

    Capacitance

    farad

    F

    A s V1

    Charge

    coulomb

    C

    As

    Energy

    joule

    J

    Nm

    Force

    newton

    N

    kg m s1

    Frequency

    hertz

    Hz

    s1

    Illuminance

    lux

    lx

    lm m2

    Inductance

    henry

    H

    V s A1

    Luminous flux

    lumen

    lm

    cd sr

    Magnetic flux

    weber

    Wb

    Vs

    Potential

    volt

    V

    W A1

    Power

    watt

    W

    J s1

    Resistance

    ohm

    O

    V A1

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    Abbreviation

    Equivalent (in terms of fundamental units)

    The Fundamentals

    3

    Example 1.1 The unit of flux density (the tesla) is defined as the magnetic flux per unit area. Express this in terms of the fundamental units. Solution The SI unit of flux is the weber (Wb). Area is directly proportional to length squared and, expressed in terms of the fundamental SI units, this is square meters (m2). Dividing the flux (Wb) by the area (m2) gives Wb/m2 or Wb m2. Hence, in terms of the fundamental SI units, the tesla is expressed in Wb m2. Example 1.2 The unit of electrical potential, the volt (V), is defined as the difference in potential between two points in a conductor, which when carrying a current of one amp (A), dissipates a power of one watt (W). Express the volt (V) in terms of joules (J) and coulombs (C). Solution In terms of the derived units: Volts ¼ ¼

    Watts Joules=seconds ¼ Amperes Amperes Joules Joules ¼ Amperes  seconds Coulombs

    Note that: Watts ¼ Joules/seconds and also that Amperes  seconds ¼ Coulombs. Alternatively, in terms of the symbols used to denote the units: V¼

    W J=s J J ¼ ¼ ¼ ¼ JC1 A A As C

    One volt is equivalent to one joule per coulomb.

    1.1.3

    Measuring Angles

    You might think it strange to be concerned with angles in electrical circuits. The reason is simply that, in analog and AC circuits, signals are based on repetitive waves (often sinusoidal in shape). We can refer to a point on such a wave in one of two basic ways, either in terms of the time from the start of the cycle or in terms of the angle

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    Chapter 1

    FIGURE 1.1: One cycle of a sine wave voltage (a cycle starts at 0 and finishes as 360 —see Figure 1.1). In practice, it is often more convenient to use angles rather than time; however, the two methods of measurement are interchangeable and it’s important to be able to work in either of these units. In electrical circuits, angles are measured in either degrees or radians (both of which are strictly dimensionless units). You will doubtless already be familiar with angular measure in degrees where one complete circular revolution is equivalent to an angular change of 360 . The alternative method of measuring angles, the radian, is defined somewhat differently. It is the angle subtended at the center of a circle by an arc having length that is equal to the radius of the circle (see Figure 1.2). You may sometimes find that you need to convert from radians to degrees, and vice versa. A complete circular revolution is equivalent to a rotation of 360 or

    FIGURE 1.2: Definition of the radian

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    The Fundamentals

    5

    2p radians (note that p is approximately equal to 3.142). Thus, one radian is equivalent to 360/2p degrees (or approximately 57.3 ). Try to remember the following rules that will help you to convert angles expressed in degrees to radians and vice versa:  From degrees to radians, divide by 57.3.  From radians to degrees, multiply by 57.3. Example 1.3 Express a quarter of a cycle revolution in terms of: (a)

    degrees;

    (b)

    radians.

    Solution (a)

    There are 360 in one complete cycle (i.e., one full revolution. Hence, there are (360/4) or 90 in one quarter of a cycle).

    (b)

    There are 2p radians in one complete cycle. Thus, there are 2p/4 or p/2 radians in one quarter of a cycle.

    Example 1.4 Express an angle of 215 in radians. Solution To convert from degrees to radians, divide by 57.3. So, 215 is equivalent to 215/57.3 ¼ 3.75 radians. Example 1.5 Express an angle of 2.5 radians in degrees. Solution To convert from radians to degrees, multiply by 57.3. Hence, 2.5 radians is equivalent to 2.5  57.3 ¼ 143.25 .

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    Chapter 1

    1.1.4

    Electrical Units and Symbols

    Table 1.3 shows the units and symbols that are commonly encountered in electrical circuits. It is important to get to know these units and also be able to recognize their abbreviations and symbols. You will meet all of these units later in this chapter.

    Table 1.3: Electrical units Unit

    Abbrev.

    Symbol

    ampere

    A

    I

    Unit of electric current (a current of 1A flows when a charge of 1C is transported in a time interval of 1s)

    coulomb

    C

    Q

    Unit of electric charge or quantity of electricity

    farad

    F

    C

    Unit of capacitance (a capacitor has a capacitance of 1F when a potential of 1V across its plates produced a charge of 1C)

    henry

    H

    L

    Unit of inductance (an inductor has an inductance of 1H when an applied current changing at 1 A/s produces a potential difference of 1V across its terminals)

    hertz

    Hz

    f

    Unit of frequency (a signal has a frequency of 1 Hz if one cycle occurs in an interval of 1s)

    Joule

    J

    W

    Unit of energy

    Ohm

    O

    R

    Unit of resistance

    second

    s

    t

    Unit of time

    siemen

    S

    G

    Unit of conductance (the reciprocal of resistance)

    tesla

    T

    B

    Unit of magnetic flux density (a flux density of 1T is produced when a flux of 1 Wb is present over an area of 1 square meter)

    volt

    V

    V

    Unit of electric potential (e.m.f. or p.d.)

    watt

    W

    P

    Unit of power (equivalent to 1J of energy consumed in 1s)

    Weber

    Wb

    j

    Unit of magnetic flux

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    Notes

    The Fundamentals

    1.1.5

    7

    Multiples and Sub-Multiples

    Unfortunately, many of the derived units are either too large or too small for convenient everyday use, but we can make life a little easier by using a standard range of multiples and sub-multiples (see Table 1.4). Table 1.4: Multiples and sub-multiples Prefix

    Abbreviation

    Multiplier

    tera

    T

    1012 ( ¼ 1,000,000,000,000)

    giga

    G

    109 ( ¼ 1,000,000,000)

    mega

    M

    106 ( ¼ 1,000,000)

    kilo

    K

    103 ( ¼ 1,000)

    (none)

    (none)

    100 ( ¼ 1 )

    centi

    c

    102 ( ¼ 0.01)

    milli

    m

    103 ( ¼ 0.001)

    micro

    m

    106 ( ¼ 0.000001)

    nano

    n

    109 ( ¼ 0.000000001)

    pico

    p

    1012 ( ¼ 0.000000000001)

    Example 1.6 An indicator lamp requires a current of 0.075A. Express this in mA. Solution You can express the current in mA (rather than in A) by simply moving the decimal point three places to the right. Hence, 0.075A is the same as 75 mA. Example 1.7 A medium-wave radio transmitter operates on a frequency of 1,495 kHz. Express its frequency in MHz. Solution To express the frequency in MHz rather than kHz, we need to move the decimal point three places to the left. Hence, 1,495 kHz is equivalent to 1.495 MHz.

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    Chapter 1

    Example 1.8 Express the value of a 27,000 pF in mF. Solution To express the value in mF rather than pF we need to move the decimal point six places to the left. Hence, 27,000 pF is equivalent to 0.027 mF (note that we have had to introduce an extra zero before the 2 and after the decimal point).

    1.1.6

    Exponent Notation

    Exponent notation (or scientific notation) is useful when dealing with either very small or very large quantities. It’s well worth getting to grips with this notation as it will allow you to simplify quantities before using them in formulae. Exponents are based on powers of ten. To express a number in exponent notation the number is split into two parts. The first part is usually a number in the range 0.1 to 100 while the second part is a multiplier expressed as a power of ten. For example, 251.7 can be expressed as 2.517  100, i.e., 2.517  102. It can also be expressed as 0.2517  1,000, i.e., 0.2517  103. In both cases the exponent is the same as the number of noughts in the multiplier (i.e., 2 in the first case and 3 in the second case). To summarize: 251:7 ¼ 2:517  102 ¼ 0:2517  103 As a further example, 0.01825 can be expressed as 1.825/100; that is, 1.825  102. It can also be expressed as 18.25/1,000, i.e., 18.25  103. Again, the exponent is the same as the number of zeros but the minus sign is used to denote a fractional multiplier. To summarize: 0:01825 ¼ 1:825  102 ¼ 18:25  103 Example 1.9 A current of 7.25 mA flows in a circuit. Express this current in amperes using exponent notation. Solution 1 mA ¼ 1  103 A;

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    thus, 7.25 mA ¼ 7:25  103 A

    The Fundamentals

    9

    Example 1.10 A voltage of 3.75  106V appears at the input of an amplifier. Express this voltage in (a) V, and (b) mV, using exponent notation. Solution (a)

    1  106V ¼ 1 mV so 3.75  106V ¼ 3.75 mV

    (b)

    There are 1,000 mV in 1 mV so we must divide the previous result by 1,000 in order to express the voltage in mV. So 3.75 mV ¼ 0.00375 mV.

    1.1.7

    Multiplication and Division Using Exponents

    Exponent notation really comes into its own when values have to be multiplied or divided. When multiplying two values expressed using exponents, you simply need to add the exponents. Here’s an example: ð2  102 Þ  ð3  106 Þ ¼ ð2  3Þ  10ð2 þ 6Þ ¼ 6  108 Similarly, when dividing two values which are expressed using exponents, you only need to subtract the exponents. As an example: ð4  106 Þ  ð2  104 Þ ¼ 4=2  10ð64Þ ¼ 2  102 In either case it’s important to remember to specify the units, multiples and sub-multiples in which you are working (e.g., A, kO, mV, mF, etc.). Example 1.11 A current of 3 mA flows in a resistance of 33 kO. Determine the voltage dropped across the resistor. Solution Voltage is equal to current multiplied by resistance. Thus: V ¼ I  R ¼ 3 mA  33 kO Expressing this using exponent notation gives: V ¼ ð3  103 Þ  ð33  103 ÞV

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    10

    Chapter 1

    Separating the exponents gives: V ¼ 3  33  103  103 V Thus, V ¼ 99  10(3þ3) ¼ 99  100 ¼ 99  1 ¼ 99V Example 1.12 A current of 45 mA flows in a circuit. What charge is transferred in a time interval of 20 ms? Solution Charge is equal to current multiplied by time (see the definition of the ampere). Thus: Q ¼ It ¼ 45 mA  20 ms Expressing this in exponent notation gives: Q ¼ ð45  106 Þ  ð20  103 Þ coulomb Separating the exponents gives: Q ¼ 45  20  106  103 coulomb Thus, Q ¼ 900  10(63) ¼ 900  109 ¼ 900 nC Example 1.13 A power of 300 mW is dissipated in a circuit when a voltage of 1,500V is applied. Determine the current supplied to the circuit. Solution Current is equal to power divided by voltage. Thus: I ¼ P=V ¼ 300 mW=1; 500V amperes Expressing this in exponent notation gives: I ¼ ð300  103 Þ=ð1:5  103 ÞA

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    The Fundamentals

    11

    Separating the exponents gives: I ¼ ð300=1:5Þ  ð103 =103 ÞA I ¼ 300=1:5  103  103 A Thus, I ¼ 200  10(33) ¼ 200  106 ¼ 200 mA

    1.1.8

    Conductors and Insulators

    Electric current is the name given to the flow of electrons (or negative charge carriers). Electrons orbit around the nucleus of atoms just as the earth orbits around the sun (see Figure 1.3). Electrons are held in one or more shells, constrained to their orbital paths by virtue of a force of attraction toward the nucleus, which contains an equal number of protons (positive charge carriers). Since like charges repel and unlike charges attract, negatively charged electrons are attracted to the positively charged nucleus. A similar principle can be demonstrated by observing the attraction between two permanent magnets; the two North u.c. Poles of the magnets will repel each other, while a North and South u.c. Pole will attract. In the same way, the unlike charges of the negative electron and the positive proton experience a force of mutual attraction. The outer shell electrons of a conductor can be reasonably easily interchanged between adjacent atoms within the lattice of atoms of which the substance is composed. This makes it possible for the material to conduct electricity. Typical examples of conductors are metals such as copper, silver, iron and aluminum. By contrast, the outer shell

    FIGURE 1.3: A single atom of helium (He) showing its two electrons in orbit around its nucleus

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    12

    Chapter 1

    electrons of an insulator are firmly bound to their parent atoms and virtually no interchange of electrons is possible. Typical examples of insulators are plastics, rubber, and ceramic materials.

    1.1.9

    Voltage and Resistance

    The ability of an energy source (e.g., a battery) to produce a current within a conductor may be expressed in terms of electromotive force (e.m.f.). Whenever an e.m.f. is applied to a circuit a potential difference (p.d.), or voltage, exists. Both e.m.f. and p.d. are measured in volts (V). In many practical circuits there is only one e.m.f. present (the battery or supply), whereas a voltage will be developed across each component present in the circuit. The conventional flow of current in a circuit is from the point of more positive potential to the point of greatest negative potential (note that electrons move in the opposite direction!). Direct current results from the application of a direct e.m.f. (derived from batteries or a DC power supply). An essential characteristic of these supplies is that the applied e.m.f. does not change its polarity (even though its value might be subject to some fluctuation). For any conductor, the current flowing is directly proportional to the e.m.f. applied. The current flowing will also be dependent on the physical dimensions (length and cross-sectional area) and material of which the conductor is composed. The amount of current that will flow in a conductor when a given e.m.f. is applied is inversely proportional to its resistance. Therefore, resistance may be thought of as an opposition to current flow; the higher the resistance the lower the current that will flow (assuming that the applied e.m.f. remains constant).

    1.1.10

    Ohm’s Law

    Provided that temperature does not vary, the ratio of p.d. across the ends of a conductor to the current flowing in the conductor is a constant. This relationship is known as Ohm’s Law and it leads to the relationship: V=I ¼ a constant ¼ R

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    The Fundamentals

    13

    where V is the potential difference (or voltage drop) in volts (V), I is the current in amperes (A), and R is the resistance in ohms (O) (see Figure 1.4).

    FIGURE 1.4: Simple circuit to illustrate the relationship between voltage (V), current (I) and resistance (R). Note that the direction of conventional current flow is from positive to negative. The formula may be arranged to make V, I or R the subject, as follows: V ¼ I  R; I ¼ V=R

    and R ¼ V=I

    The triangle shown in Figure 1.5 should help you remember these three important relationships. However, it’s worth noting that, when performing calculations of currents, voltages and resistances in practical circuits it is seldom necessary to work with an accuracy of better than 1% simply because component tolerances are usually greater than this. Furthermore, in calculations involving Ohm’s Law, it can sometimes be convenient to work in units of kO and mA (or MO and mA) in which case potential differences will be expressed directly in V.

    FIGURE 1.5: Triangle showing the relationship between V, I and R

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    14

    Chapter 1

    Example 1.14 A 12O resistor is connected to a 6V battery. What current will flow in the resistor? Solution Here we must use I ¼ V/R (where V ¼ 6V and R ¼ 12O): I ¼ V=R ¼ 6V=12O ¼ 0:5Aðor 500 mAÞ Hence a current of 500 mA will flow in the resistor. Example 1.15 A current of 100 mA flows in a 56O resistor. What voltage drop (potential difference) will be developed across the resistor? Solution Here we must use V ¼ I  R and ensure that we work in units of volts (V), amperes (A), and ohms (O). V ¼ I  R ¼ 0:1A  56O ¼ 5:6V (Note that 100 mA is the same as 0.1A.) This calculation shows that a p.d. of 5.6V will be developed across the resistor. Example 1.16 A voltage drop of 15V appears across a resistor in which a current of 1 mA flows. What is the value of the resistance? Solution R ¼ V=I ¼ 15V=0:001A ¼ 15; 000O ¼ 15 kO Note that it is often more convenient to work in units of mA and V, which will produce an answer directly in kO, that is: R ¼ V=I ¼ 15V=l mA ¼ 15 kO

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    The Fundamentals

    15

    1.1.11 Resistance and Resistivity The resistance of a metallic conductor is directly proportional to its length and inversely proportional to its area. The resistance is also directly proportional to its resistivity (or specific resistance). Resistivity is defined as the resistance measured between the opposite faces of a cube having sides of 1 cm. The resistance, R, of a conductor is given by the formula: R ¼ r  l=A where R is the resistance (ft), r is the resistivity (Om), l is the length (m), and A is the area (m2). Table 1.5 shows the electrical properties of some common metals. Example 1.17 A coil consists of an 8m length of annealed copper wire having a cross-sectional area of 1 mm2. Determine the resistance of the coil. Solution We will use the formula, R ¼ r l/A. Table 1.5: Properties of some common metals Metal

    Resistivity (at 20˚C) (Vm)

    Silver

    1.626  108

    1.06

    0.0041

    Copper (annealed)

    1.724  108

    1.00

    0.0039

    Copper (hard drawn)

    1.777  108

    0.97

    0.0039

    Aluminum

    2.803  108

    0.61

    0.0040

    Mild steel

    1.38  107

    0.12

    0.0045

    0.08

    0.0040

    0.22

    0.0062

    7

    Lead

    2.14  10

    Nickel

    8.0  108

    Relative conductivity (copper = 1)

    Temperature coefficient of resistance (per ˚C)

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    16

    Chapter 1

    The value of r for annealed copper given in Table 1.5 is 1.724  108 Om. The length of the wire is 4m, while the area is 1 mm2 or 1  106 m2 (note that it is important to be consistent in using units of meters for length and square meters for area). Hence, the resistance of the coil will be given by: R¼

    1:724  108  8 ¼ 13:724  10ð8 þ 6Þ 1  106

    Thus, R ¼ 13.792  102 or 0.13792O Example 1.18 A wire having a resistivity of 1.724  108 Om, length 20m and cross-sectional area 1 mm2 carries a current of 5A. Determine the voltage drop between the ends of the wire. Solution First, we must find the resistance of the wire (as in Example 1.17): R¼

    rl 1:6  108  20 ¼ 32  102 ¼ 0:32O ¼ A 1  106

    The voltage drop can now be calculated using Ohm’s Law: V ¼ I  R ¼ 5A  0:32O ¼ 1:6V This calculation shows that a potential of 1.6V will be dropped between the ends of the wire.

    1.1.12

    Energy and Power

    At first you may be a little confused about the difference between energy and power. Simply put, energy is the ability to do work, while power is the rate at which work is done. In electrical circuits, energy is supplied by batteries or generators. It may also be stored in components such as capacitors and inductors. Electrical energy is converted into various other forms of energy by components such as resistors (producing heat), loudspeakers (producing sound energy), and light emitting diodes (producing light).

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    The Fundamentals

    17

    The unit of energy is the joule (J). Power is the rate of use of energy and it is measured in watts (W). A power of 1W results from energy being used at the rate of 1J per second. Thus: P ¼ W=t where P is the power in watts (W), W is the energy in joules (J), and t is the time in seconds (s). The power in a circuit is equivalent to the product of voltage and current. Hence: P¼IV where P is the power in watts (W), I is the current in amperes (A), and V is the voltage in volts (V). The formula may be arranged to make P, I or V the subject, as follows: P ¼ I  P; I ¼ P=V and

    V ¼ P=I

    The triangle shown in Figure 1.6 should help you remember these relationships. The relationship, P ¼ I  V, may be combined with that which results from Ohm’s Law (V ¼ I  R) to produce two further relationships. First, substituting for V gives: P ¼ I  ðI  RÞ ¼ I2 R Secondly, substituting for I gives: P ¼ ðV=RÞ  V ¼ V 2 =R Example 1.19 A current of 1.5A is drawn from a 3V battery. What power is supplied? Solution Here we must use P ¼ I  V (where I ¼ 1.5A and V ¼ 3V). P ¼ I  V ¼ 1:5A  3V ¼ 4:5W Hence, a power of 4.5W is supplied.

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    Chapter 1

    Example 1.20 A voltage drop of 4V appears across a resistor of 100O. What power is dissipated in the resistor? Solution Here we use P ¼ V2/R (where V ¼ 4V and R ¼ 100O). P ¼ V 2= R ¼ ð4V  4VÞ=100O ¼ 0:16W Hence, the resistor dissipates a power of 0.16W (or 160 mW). Example 1.21 A current of 20 mA flows in a 1 kO resistor. What power is dissipated in the resistor?

    FIGURE 1.6: Triangle showing the relationship between P, I and V Solution Here we use P ¼ I2  R but, to make life a little easier, we will work in mA and kO (in which case the answer will be in mW). P ¼ I2  R ¼ ð20 mA  20 mAÞ  1 kO ¼ 400 mW Thus, a power of 400 mW is dissipated in the 1 kO resistor.

    1.1.13

    Electrostatics

    If a conductor has a deficit of electrons, it will exhibit a net positive charge. If, on the other hand, it has a surplus of electrons, it will exhibit a net negative charge. An imbalance in charge can be produced by friction (removing or depositing electrons using materials such as silk and fur, respectively), or induction

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    The Fundamentals

    19

    (by attracting or repelling electrons using a second body, which is, respectively, positively or negatively charged).

    1.1.14 Force Between Charges Coulomb’s Law states that, if charged bodies exist at two points, the force of attraction (if the charges are of opposite polarity), or repulsion (if the charges have the same polarity), will be proportional to the product of the magnitude of the charges divided by the square of their distance apart. Thus: F¼

    kQ1 Q2 r2

    where Q1 and Q2 are the charges present at the two points (in coulombs), r the distance separating the two points (in meters), F is the force (in newtons), and k is a constant depending upon the medium in which the charges exist. In vacuum or “free space”: k¼

    1 4pe0

    where e0 is the permittivity of free space (8.854  1012 C/Nm2). Combining the two previous equations gives: F¼

    kQ1 Q2 Newtons 4  8:854  1012 r2

    1.1.15 Electric Fields The force exerted on a charged particle is a manifestation of the existence of an electric field. The electric field defines the direction and magnitude of a force on a charged object. The field itself is invisible to the human eye, but can be drawn by constructing lines, which indicate the motion of a free positive charge within the field; the number of field lines in a particular region being used to indicate the relative strength of the field at the point in question. Figures 1.7 and 1.8 show the electric fields between charges of the same and opposite polarity, while Figure 1.9 shows the field that exists between two charged parallel plates. You will see more of this particular arrangement when we introduce capacitors.

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    FIGURE 1.7: Electric field between two unlike electric charges

    FIGURE 1.8: Electric field between two like electric charges (in this case both positive)

    FIGURE 1.9: Electric field between two parallel plates

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    The Fundamentals

    21

    1.1.16 Electric Field Strength The strength of an electric field (E) is proportional to the applied potential difference and inversely proportional to the distance between the two conductors. The electric field strength is given by: E ¼ V=d where E is the electric field strength (V/m), V is the applied potential difference (V) and d is the distance (m). Example 1.22 Two parallel conductors are separated by a distance of 25 mm. Determine the electric field strength if they are fed from a 600V DC supply. Solution The electric field strength will be given by: E ¼ V=d ¼ 600=25  103 ¼ 24 kV=m

    1.1.17 Permittivity The amount of charge produced on the two plates shown in Figure 1.9 for a given applied voltage will depend not only on the physical dimensions, but also on the insulating dielectric material that appears between the plates. Such materials need to have a very high value of resistivity (they must not conduct charge) coupled with an ability to withstand high voltages without breaking down. A more practical arrangement is shown in Figure 1.10. In this arrangement the ratio of charge, Q, to potential difference, V, is given by the relationship: Q eA ¼ V d where A is the surface area of the plates (in m), d is the separation (in m), and e is a constant for the dielectric material known as the absolute permittivity of the material (sometimes also referred to as the dielectric constant). The absolute permittivity of a dielectric material is the product of the permittivity of free space (e0) and the relative permittivity (er) of the material. Thus:

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    Chapter 1

    FIGURE 1.10: Parallel plates with an insulating dielectric material e ¼ e0  e

    and

    Q e0 er A ¼ V d

    The dielectric strength of an insulating dielectric is the maximum electric field strength that can safely be applied to it before breakdown (conduction) occurs. Table 1.6 shows values of relative permittivity and dielectric strength for some common dielectric materials. Table 1.6: Properties of some common insulating dielectric materials Dielectric material

    Relative permittivity (free space = 1)

    Dielectric strength (kV/mm)

    Vacuum, or free space

    1

    1

    Air

    1

    3

    Polythene

    2.3

    50

    Paper

    2.5 to 3.5

    14

    Polystyrene

    2.5

    25

    Mica

    4 to 7

    160

    Pyrex glass

    4.5

    13

    Glass ceramic

    5.9

    40

    Polyester

    3.0 to 3.4

    18

    Porcelain

    6.5

    4

    Titanium dioxide

    100

    6

    Ceramics

    5 to 1,000

    2 to 10

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    The Fundamentals

    23

    1.1.18 Electromagnetism When a current flows through a conductor, a magnetic field is produced in the vicinity of the conductor. The magnetic field is invisible, but its presence can be detected using a compass needle (which will deflect from its normal North-South position). If two current-carrying conductors are placed in the vicinity of one another, the fields will interact with one another and the conductors will experience a force of attraction or repulsion (depending upon the relative direction of the two currents).

    1.1.19 Force Between Two Current-Carrying Conductors The mutual force that exists between two parallel current-carrying conductors will be proportional to the product of the currents in the two conductors and the length of the conductors but inversely proportional to their separation. Thus: F¼

    k I1 I2 l d

    where I1 and I2 are the currents in the two conductors (in amps), l is the parallel length of the conductors (in meters), d is the distance separating the two conductors (in meters), F is the force (in newtons), and k is a constant depending upon the medium in which the charges exist. In vacuum or “free space”, k¼

    m0 2p

    where m0 is a constant known as the permeability of free space (4p  107 or 12.57  107H/m). Combining the two previous equations gives: F¼

    m0 I1 I2 l 2pd

    or, F¼

    4p  107 I1 I2 l 2pd

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    Chapter 1

    or, F¼

    1.1.20

    2  107 I1 I2 l Newtons d

    Magnetic Fields

    The field surrounding a straight current-carrying conductor is shown in Figure 1.11. The magnetic field defines the direction of motion of a free North Pole within the field. In the case of Figure 1.11, the lines of flux are concentric and the direction of the field determined by the direction of current flow) is given by the right-hand rule.

    1.1.21

    Magnetic Field Strength

    The strength of a magnetic field is a measure of the density of the flux at any particular point. In the case of Figure 1.11, the field strength will be proportional to the applied current and inversely proportional to the perpendicular distance from the conductor. Thus: B¼

    kI d

    where B is the magnetic flux density (in tesla), I is the current (in amperes), d is the distance from the conductor (in meters), and k is a constant. Assuming that the medium is vacuum or ”free space,” the density of the magnetic flux will be given by: B¼

    m0 I 2p d

    where B is the flux density (in tesla), m0 is the permeability of free space (4p  107 or 12.57  107), I is the current (in amperes), and d is the distance from the center of the conductor (in meters). The flux density is also equal to the total flux divided by the area of the field. Thus: B ¼ F=A where F is the flux (in webers) and A is the area of the field (in square meters).

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    The Fundamentals

    25

    FIGURE 1.11: Magnetic field surrounding a straight conductor

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    26

    Chapter 1

    FIGURE 1.12: Forming a conductor into a loop increases the strength of the magnetic field in the center of the loop

    In order to increase the strength of the field, a conductor may be shaped into a loop (Figure 1.12) or coiled to form a solenoid (Figure 1.13). Note, in the latter case, how the field pattern is exactly the same as that which surrounds a bar magnet. Example 1.23 Determine the flux density produced at a distance of 50 mm from a straight wire carrying a current of 20A. Solution Applying the formula B ¼ m0I/2p d gives: B¼

    12:57  107  20 251:4  107 ¼ 314:2  103 2  3:142  50  103

    from which: B ¼ 0:8  104 tesla Thus, B ¼ 80  106 T or B ¼ 80 mT.

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    The Fundamentals

    27

    FIGURE 1.13: The magnetic field surrounding a solenoid coil resembles that of a permanent magnet

    Example 1.24 A flux density of 2.5 mT is developed in free space over an area of 20 cm2. Determine the total flux. Solution Rearranging the formula B ¼ F/A to make F the subject gives F ¼ B  A thus: F ¼ ð2:5  103 Þ  ð20  104 Þ ¼ 50  107 webers from which B ¼ 5 mWb

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    1.22

    Chapter 1

    Magnetic Circuits

    Materials such as iron and steel possess considerably enhanced magnetic properties. They are employed in applications where it is necessary to increase the flux density produced by an electric current. In effect, magnetic materials allow us to channel the electric flux into a “magnetic circuit,” as shown in Figure 1.14. In the circuit of Figure 1.14(B), the reluctance of the magnetic core is analogous to the resistance present in the electric circuit shown in Figure 1.14(A). We can make the following comparisons between the two types of circuit (see Table 1.7).

    FIGURE 1.14: Comparison of electric and magnetic circuits

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    The Fundamentals

    29

    Table 1.7: Comparison of electric and magnetic circuits Electric circuit Figure 1.14(A)

    Magnetic circuit Figure 1.14(A)

    Electromotive force, e.m.f. ¼ V

    Magnetomotive force, m.m.f. ¼ N  I

    Resistance ¼ R

    Reluctance ¼ S

    Current ¼ I

    Flux ¼ F

    e.m.f. ¼ current  resistance

    m.m.f. ¼ flux  reluctance

    V ¼ IR

    NI¼SF

    In practice, not all of the magnetic flux produced in a magnetic circuit will be concentrated within the core and some “leakage flux” will appear in the surrounding free space (as shown in Figure 1.15). Similarly, if a gap appears within the magnetic circuit, the flux will tend to spread out as shown in Figure 1.16. This effect is known as fringing.

    FIGURE 1.15: Leakage flux in a magnetic circuit

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    Chapter 1

    FIGURE 1.16: Fringing of the magnetic flux at an air gap in a magnetic circuit

    1.1.23

    Reluctance and Permeability

    The reluctance of a magnetic path is directly proportional to its length and inversely proportional to its area. The reluctance is also inversely proportional to the absolute permeability of the magnetic material. Thus: S¼

    l mA

    where S is the reluctance of the magnetic path, l is the length of the path (in meters), A is the cross-sectional area of the path (in square meters), and m is the absolute permeability of the magnetic material. The absolute permeability of a magnetic material is the product of the permeability of free space (m0) and the relative permeability of the magnetic medium (m0). Thus: m ¼ m0  m and S ¼

    l m0 mr A

    The permeability of a magnetic medium is a measure of its ability to support magnetic flux and it is equal to the ratio of flux density (B) to magnetizing force (H). Thus: m¼

    B H

    where B is the flux density (in tesla) and H is the magnetizing force (in ampere/meter). The magnetizing force (H) is proportional to the product of the number of turns and current but inversely proportional to the length of the magnetic path.

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    The Fundamentals H¼

    31

    NI l

    where H is the magnetizing force (in amperes/ meters), N is the number of turns, I is the current (in amperes), and l is the length of the magnetic path (in meters).

    1.1.24 B-H Curves Figure 1.17 shows four typical B-H (flux density plotted against permeability) curves for some common magnetic materials. If you look carefully at these curves you will notice that they flatten off due to magnetic saturation and that the slope of the curve (indicating the value of m corresponding to a particular value of H) falls

    FIGURE 1.17: B-H curves for three ferromagnetic materials

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    Chapter 1

    as the magnetizing force increases. This is important since it dictates the acceptable working range for a particular magnetic material when used in a magnetic circuit. Example 1.25 Estimate the relative permeability of cast steel (see Figure 1.18) at (a) a flux density of 0.6T, and (b) a flux density of 1.6T. Solution From Figure 1.18, the slope of the graph at any point gives the value of m at that point. We can easily find the slope by constructing a tangent at the point in question and then finding the ratio of vertical change to horizontal change.

    FIGURE 1.18: B-H curve for a sample of cast steel

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    The Fundamentals (a)

    33

    The slope of the graph at 0.6T is 0.6/800 ¼ 0.75  103

    Since m ¼ m0  mr, mr ¼ m/m0 ¼ 0.75  103/12.57  107, thus mr ¼ 597 at 0.6T. (b)

    The slope of the graph at 1.6T is 0.2/4,000 ¼ 0.05  103

    Since m ¼ m0  mr, mr ¼ m/m0 ¼ 0.05  103 / 12.57  107, thus mr ¼ 39.8 at 1.6T. (This example clearly shows the effect of saturation on the permeability of a magnetic material!) Example 1.26 A coil of 800 turns is wound on a closed mild steel core having a length 600 mm and cross-sectional area 500 mm2. Determine the current required to establish a flux of 0.8 mWb in the core. Solution Now B ¼ F/A ¼ (0.8  103)/(500  106) ¼ 1.6T From Figure 1.17, a flux density of 1.6T will occur in mild steel when H ¼ 3,500 A/m. The current can now be determined by re-arranging H ¼ N I/l as follows: I¼

    Hl 3; 500  0:6 ¼ ¼ 2:625A N 800

    1.1.25 Circuit Diagrams Finally, and just in case you haven’t seen them before, we will end this section with a brief word about circuit diagrams. We are introducing the topic here because it’s quite important to be able to read and understand simple electronic circuit diagrams before you can make sense of some of the components and circuits that you will meet later on. Circuit diagrams use standard symbols and conventions to represent the components and wiring used in an electronic circuit. Visually, they bear very little relationship to the physical layout of a circuit but, instead, they provide us with a “theoretical” view of the circuit. In this section we show you how to find your way around simple circuit diagrams.

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    To be able to understand a circuit diagram you first need to be familiar with the symbols that are used to represent the components and devices. It’s important to be aware that there are a few (thankfully quite small) differences between the symbols used in circuit diagrams of American and European origin. As a general rule, the input to a circuit should be shown on the left of a circuit diagram and the output shown on the right. The supply (usually the most positive voltage) is normally shown at the top of the diagram and the common, 0V, or ground connection is normally shown at the bottom. This rule is not always obeyed, particularly for complex diagrams where many signals and supply voltages may be present. Note also that, in order to simplify a circuit diagram (and avoid having too many lines connected to the same point) multiple connections to common, 0V, or ground may be shown using the appropriate symbol. The same applies to supply connections that may be repeated (appropriately labeled) at various points in the diagram. A very simple circuit diagram (a simple resistance tester) is shown in Figure 1.20. This circuit may be a little daunting if you haven’t met a circuit like it before but you can still glean a great deal of information from the diagram even if you don’t know what the individual components do.

    FIGURE 1.19: Various types of switches. From left to right: a mains rocker switch, an SPDT miniature toggle (changeover) switch, a DPDT side switch, an SPDT push-button (wired for use as an SPST push-button), a miniature PCB mounting DPDT push-button (with a latching action).

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    FIGURE 1.20: A simple circuit diagram

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    The circuit uses two batteries, B1 (a 9V multi-cell battery) and B2 (a 1.5V single-cell battery). The two batteries are selected by means of a double-pole, double-throw (DPDT) switch. This allows the circuit to operate from either the 9V battery (B1) as shown in Figure 1.20(A) or from the 1.5V battery (B2) as shown in Figure 1.20(B) depending on the setting of S1. A variable resistor, VR1, is used to adjust the current supplied by whichever of the two batteries is currently selected. This current flows first through VR1, then through the milliammeter, and finally through the unknown resistor, RX. Notice how the meter terminals are labeled showing their polarity (the current flows into the positive terminal and out of the negative terminal). The circuit shown in Figure 1.20(C) uses a different type of switch but provides exactly the same function. In this circuit a single-pole, double-throw (SPDT) switch is used and the negative connections to the two batteries are “commoned” (i.e., connected directly together). Finally, Figure 1.20(D) shows how the circuit can be redrawn using a common “chassis” connection to provide the negative connection between RX and the two batteries. Electrically this circuit is identical to the one shown in Figure 1.20(C).

    1.2 Passive Components This section introduces several of the most common types of electronic component, including resistors, capacitors and inductors. These are often referred to as passive components as they cannot, by themselves, generate voltage or current. An understanding of the characteristics and application of passive components is an essential prerequisite to understanding the operation of the circuits used in amplifiers, oscillators, filters and power supplies.

    1.2.1

    Resistors

    The notion of resistance as opposition to current was discussed in the previous section. Conventional forms of resistor obey a straight line law when voltage is plotted against current (see Figure 1.21) and this allows us to use resistors as a means of converting current into a corresponding voltage drop, and vice versa (note that doubling the applied current will produce double the voltage drop, and so on). Therefore, resistors provide us with a means of controlling the currents and voltages present in electronic

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    FIGURE 1.21: Voltage plotted against current for three different values of resistor circuits. They can also act as loads to simulate the presence of a circuit during testing (e.g., a suitably rated resistor can be used to replace a loudspeaker when an audio amplifier is being tested). The specifications for a resistor usually include the value of resistance expressed in ohms (O), kilohms (kO) or megohms (MO), the accuracy or tolerance (quoted as the maximum permissible percentage deviation from the marked value), and the power rating (which must be equal to, or greater than, the maximum expected power dissipation). Other practical considerations when selecting resistors for use in a particular application include temperature coefficient, noise performance, stability and ambient temperature range. Table 1.8 summarizes the properties of five of the most common types of resistor. Figure 1.22 shows a typical selection of fixed resistors with values from 15O to 4.7 kO.

    1.2.2

    Preferred Values

    The value marked on the body of a resistor is not its exact resistance. Some minor variation in resistance value is inevitable due to production tolerance. For example, a resistor marked 100O and produced within a tolerance of 10% will have a value which falls within the range 90O to 110O. A similar component with a tolerance of 1% would have a value that falls within the range 99O to 101O. Thus, where accuracy is important it is essential to use close tolerance components.

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    Resistor type Carbon film

    Metal film

    Metal oxide

    Ceramic wirewound

    Vitreous wirewound

    Metal clad

    Resistance range (O)

    10 to 10M

    1 to 1M

    10 to 10 M

    0.47 to 22k

    0.1 to 22k

    0.05 to 10k

    Typical tolerance (%)

    5

    1

    2

    5

    5

    5

    Power rating (W)

    0.25 to 2

    0.125 to 0.5

    0.25 to 0.5

    4 to 17

    2 to 4

    10 to 300

    Temperature coefficient (ppm/ C)

    250

    þ50 to þ100

    þ250

    þ250

    þ75

    þ50

    Stability

    Fair

    Excellent

    Excellent

    Good

    Good

    Good

    Noise performance

    Fair

    Excellent

    Excellent

    n.a.

    n.a.

    n.a.

    Ambient temperature range ( C)

    45 to þ125

    45 to þ125

    45 to þ125

    45 to þ125

    45 to þ125

    55 to þ200

    Typical applications

    GeneralPurpose

    Amplifiers, test equipment, etc., requiring low-noise hightolerance components

    Property

    Power supplies, loads, medium and high-power applications

    Very high power applications

    Chapter 1

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    Table 1.8: Characteristics of common types of resistors

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    FIGURE 1.22: A selection of resistors including high-power metal clad, ceramic wirewound, carbon and metal film types with values ranging from 15V to 4.7 kV

    Resistors are available in several series of fixed-decade values, the number of values provided with each series being governed by the tolerance involved. In order to cover the full range of resistance values using resistors having a 20% tolerance it will be necessary to provide six basic values (known as the E6 series). More values will be required in the series, which offers a tolerance of 10%, and consequently, the E12 series provides twelve basic values. The E24 series for resistors of 5% tolerance provides no fewer than 24 basic values and, as with the E6 and E12 series, decade multiples (i.e., 1, 10, 100, 1 k, 10 k, 100k and 1M) of the basic series. Figure 1.23 shows the relationship between the E6, E12 and E24 series.

    1.2.3

    Power Ratings

    Resistor power ratings are related to operating temperatures and resistors should be derated at high temperatures. Where reliability is important resistors should be operated at well below their nominal maximum power dissipation. Example 1.27 A resistor has a marked value of 220O. Determine the tolerance of the resistor if it has a measured value of 207O.

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    FIGURE 1.23: The E6, E12, and E24 series Solution The difference between the marked and measured values of resistance (the error) is (220O  207O) ¼ 13O. The tolerance is given by: Tolerance ¼

    error  100% marked value

    The tolerance is thus, (13/220)  100 ¼ 5.9%.

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    Example 1.28 A 9V power supply is to be tested with a 39O load resistor. If the resistor has a tolerance of 10% find: (a)

    the nominal current taken from the supply;

    (b)

    the maximum and minimum values of supply current at either end of the tolerance range for the resistor.

    Solution (a)

    If a resistor of exactly 39O is used the current will be: I ¼ V=R ¼ 9V=39O ¼ 231 mA

    (b)

    The lowest value of resistance would be (39O  3.9O) ¼ 35.1O. In which case the current would be: I ¼ V=R ¼ 9V=35:1O ¼ 256:4 mA

    At the other extreme, the highest value would be (39O þ 3.9 O) ¼ 42.9O. In this case, the current would be: I ¼ V=R ¼ 9V=42:9O ¼ 209:8 mA The maximum and minimum values of supply current will thus be 256.4 mA and 209.8 mA, respectively. Example 1.29 A current of 100 mA (20%) is to be drawn from a 28V DC supply. What value and type of resistor should be used in this application? Solution The value of resistance required must first be calculated using Ohm’s Law: R ¼ V=I ¼ 28V=100 mA ¼ 280O The nearest preferred value from the E12 series is 270O (which will actually produce a current of 103.7 mA (i.e., within 4% > of the desired value). If a resistor

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    Chapter 1

    of 10% tolerance is used, current will be within the range 94 mA to 115 mA (well within the 20% accuracy specified). The power dissipated in the resistor (calculated using P ¼ IV) will be 2.9W and thus a component rated at 3W (or more) will be required. This would normally be a vitreous enamel coated wirewound resistor (see Table 1.8).

    1.2.4

    Resistor Markings

    Carbon and metal oxide resistors are normally marked with color codes which indicate their value and tolerance. Two methods of color-coding are in common use; one involves four colored bands (see Figure 1.24), while the other uses five color bands (see Figure 1.25).

    FIGURE 1.24: Four-band resistor color code

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    FIGURE 1.25: Five band resistor color code

    Example 1.30 A resistor is marked with the following colored stripes: brown, black, red, silver. What is its value and tolerance? Solution See Figure 1.26. Example 1.31 A resistor is marked with the following colored stripes: red, violet, orange, gold. What is its value and tolerance? Solution See Figure 1.27.

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    Chapter 1

    FIGURE 1.26: See Example 1.30

    FIGURE 1.27: See Example 1.31

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    Example 1.32 A resistor is marked with the following colored stripes: green, blue, black, gold. What is its value and tolerance? Solution See Figure 1.28. Example 1.33 A resistor is marked with the following colored stripes: red, green, black, black, brown. What is its value and tolerance? Solution See Figure 1.29.

    FIGURE 1.28: See Example 1.32

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    FIGURE 1.29: See Example 1.33

    Example 1.34 A 2.2 kO of 2% tolerance is required. What four-band color code does this correspond to? Solution Red (2), red (2), red (2 zeros), red (2% tolerance). Thus, all four bands should be red.

    1.2.5

    BS 1852 Coding

    Some types of resistor have markings based on a system of coding defined in BS 1852. This system involves marking the position of the decimal point with a letter to indicate the multiplier concerned as shown in Table 1.9. A further letter is then appended to indicate the tolerance as shown in Table 1.10.

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    Table 1.9: BS 1852 resistor multiplier markings Letter

    Multiplier

    R

    1

    K

    1,000

    M

    1,000,000

    Table 1.10: BS 1852 resistor tolerance markings Letter

    Multiplier

    F

    1%

    G

    2%

    J

    5%

    K

    10%

    M

    20%

    Example 1.35 A resistor is marked coded with the legend 4R7K. What is its value and tolerance? Solution 4.7O  10% Example 1.36 A resistor is marked coded with the legend 330RG. What is its value and tolerance? Solution 330O  2% Example 1.37 A resistor is marked coded with the legend R22M. What is its value and tolerance? Solution 0.22O  20%

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    1.2.6

    Chapter 1

    Series and Parallel Combinations of Resistors

    In order to obtain a particular value of resistance, fixed resistors may be arranged in either series or parallel as shown in Figures 1.30 and 1.31. The effective resistance of each of the series circuits shown in Figure 1.30 is simply equal to the sum of the individual resistances. So, for the circuit shown in Figure 1.30(A): R ¼ R1 þ R2 while for Figure 1.30(B): R ¼ R1 þ R2 þ R3 Turning to the parallel resistors shown in Figure 1.31, the reciprocal of the effective resistance of each circuit is equal to the sum of the reciprocals of the individual resistances. Hence, for Figure 1.31(A): 1 1 1 ¼ þ R R1 R2 while for Figure 1.32(B): 1 1 1 1 þ þ ¼ R R1 R2 R3 In the former case, the formula can be more conveniently rearranged as follows: R¼

    R1  R2 R1 þ R2

    You can remember this as the product of the two resistance values divided by the sum of the two resistance values.

    Example 1.38 Resistors of 22O, 47O, and 33O are connected (a) in series and (b) in parallel. Determine the effective resistance in each case.

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    FIGURE 1.30: Resistors in series

    FIGURE 1.31: Resistors in parallel Solution (a)

    In the series circuit R ¼ R1 þ R2 þ R3, thus R ¼ 22O þ 47O þ 33O ¼ 102O

    (b)

    In the parallel circuit: 1 1 1 1 þ þ ¼ R R1 R2 R3

    Thus, 1 1 1 1 ¼ þ þ R 22O 47O 33O Or, 1 ¼ 0:045 þ 0:021 þ 0:03 R

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    from which, 1 ¼ 0:096 ¼ 10:42O R Example 1.39 Determine the effective resistance of the circuit shown in Figure 1.32. Solution The circuit can be progressively simplified as shown in Figure 1.33. The stages in this simplification are: (a) R3 and R4 are in series and they can replaced by a single resistance (RA) of (12O þ 27O) ¼ 39O. (b)

    RA appears in parallel with R2. These two resistors can be replaced by a single resistance (RB) of (39O þ 47O)/(39O þ 47O) ¼ 21.3O.

    (c) RB appears in series with R1. These two resistors can be replaced by a single resistance (R) of (21.3O þ 4.7O) ¼ 26O.

    FIGURE 1.32: See Example 1.39

    FIGURE 1.33: See Example 1.39

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    Example 1.40 A resistance of 50O rated at 2W is required. What parallel combination of preferred value resistors will satisfy this requirement? What power rating should each resistor have? Solution Two 100O resistors may be wired in parallel to provide a resistance of 50O as shown below: R¼

    R1  R2 100  100 10; 000 ¼ ¼ 50O ¼ 100 þ 100 200 R1 þ R2

    Note, from this, that when two resistors of the same value are connected in parallel the resulting resistance will be half that of a single resistor. Having shown that two 100O resistors connected in parallel will provide us with a resistance of 50O we now need to consider the power rating. Since the resistors are identical, the applied power will be shared equally between them. Hence, each resistor should have a power rating of 1W.

    1.2.7

    Resistance and Temperature

    Figure 1.34 shows how the resistance of a metal conductor (e.g., copper) varies with temperature. Since the resistance of the material increases with temperature, this characteristic is said to exhibit a positive temperature coefficient (PTC). Not all materials have a PTC characteristic. The resistance of a carbon conductor falls with temperature and it is therefore said to exhibit a negative temperature coefficient (NTC). The resistance of a conductor at a temperature, t, is given by the equation: Rt ¼ R0 ð1 þ a t þ b t2 þ g t3 . . . Þ where a, b, g, etc. are constants and R0 is the resistance at 0 C. The coefficients, b, g, etc. are quite small and since we are normally only dealing with a relatively restricted temperature range (e.g., 0  C to 100  C), we can usually

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    Chapter 1

    approximate the characteristic shown in Figure 1.34 to the straight line law shown in Figure 1.35. In this case, the equation simplifies to: Rt ¼ R0 ð1 þ a tÞ where a is known as the temperature coefficient of resistance. Table 1.11 shows some typical values for a (note that a is expressed in O/O/ C or just / C). Example 1.41 A resistor has a temperature coefficient of 0.001/ C. If the resistor has a resistance of 1.5 kO at 0  C, determine its resistance at 80  C.

    FIGURE 1.34: Variation of resistance with temperature for a metal conductor

    FIGURE 1.35: Straight line approximation of Figure 1.34

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    Solution Now: Rt ¼ R0 ð1 þ a tÞ thus, Rt ¼ 1:5 kO  ð1 þ ð0:001  80ÞÞ Hence, Rt ¼ 1:5  1:08 ¼ 1:62 kO Example 1.42 A resistor has a temperature coefficient of 0.0005/ C. If the resistor has a resistance of 680O at 20  C, what will its resistance be at 80  C? Solution First we must find the resistance at 0 C. Rearranging the formula for Rt gives: R0 ¼

    Rt 680 680 ¼ ¼ 1 þ ð0:0005  20Þ 1 þ 0:01 1 þ at

    Hence, R0 ¼

    680 ¼ 673:3O 1 þ 0:01

    Now, Rt ¼ R0 ð1 þ a tÞ thus, R90 ¼ 673:3  ð1 þ ð0:0005  90ÞÞ Hence, R90 ¼ 673:3  1:045 ¼ 704O

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    Example 1.43 A resistor has a resistance of 40O at 0 C and 44O at 100 C. Determine the resistor’s temperature coefficient. Solution First we need to make a the subject of the formula: Rt ¼ R0 ð1 þ a tÞ Now,     1 Rt 1 44 a¼ 1 ¼ 1 t Ro 100 40 from which, a¼

    1 1 ð1:1  1Þ ¼  0:1 ¼ 0:001= C 100 100

    Table 1.11: Temperature coefficient of resistance

    1.2.8

    Material

    Temperature coefficient of resistance, a (/˚C)

    Platinum

    þ0.0034

    Silver

    þ0.0038

    Copper

    þ0.0043

    Iron

    þ0.0065

    Carbon

    0.0005

    Thermistors

    With conventional resistors we would normally require resistance to remain the same over a wide range of temperatures (i.e., a should be zero). On the other hand, there are applications in which we could use the effect of varying resistance to detect a temperature change. Components that allow us to do this are known as thermistors.

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    The resistance of a thermistor changes markedly with temperature and these components are widely used in temperature sensing and temperature compensating applications. Two basic types of thermistor are available, NTC and PTC (see Figure 1.36).

    FIGURE 1.36: Characteristics of (A) NTC and (B) PTC thermistors

    Typical NTC thermistors have resistances that vary from a few hundred (or thousand) ohms at 25  C to a few tens (or hundreds) of ohms at 100  C. PTC thermistors, on the other hand, usually have a resistance-temperature characteristic that remains substantially flat (typically at around 100O) over the range 0  C to around 75  C.

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    Above this, and at a critical temperature (usually in the range 80  C to 120  C) their resistance rises very rapidly to values of up to, and beyond, 10 kO (see Figure 1.36). A typical application of PTC thermistors is over-current protection. Provided the current passing through the thermistor remains below the threshold current, the effects of self-heating will remain negligible and the resistance of the thermistor will remain low (i.e., approximately the same as the resistance quoted at 25  C). Under fault conditions, the current exceeds the threshold value by a considerable margin and the thermistor starts to self-heat. The resistance then increases rapidly and, as a consequence, the current falls to the rest value. Typical values of threshold and rest currents are 200 mA and 8 mA, respectively, for a device which exhibits a nominal resistance of 25O at 25  C.

    1.2.9

    Light-Dependent Resistors

    Light-dependent resistors (LDR) use a semiconductor material (i.e., a material that is neither a conductor nor an insulator) whose electrical characteristics vary according to the amount of incident light. The two semiconductor materials used for the manufacture of LDRs are cadmium sulphide (CdS) and cadmium selenide (CdSe). These materials are most sensitive to light in the visible spectrum, peaking at about 0.6 mm for CdS and 0.75 mm for CdSe. A typical CdS LDR exhibits a resistance of around 1 MO in complete darkness and less than 1 kO when placed under a bright light source (see Figure 1.37).

    FIGURE 1.37: Characteristic of a light-dependent resistor (LDR)

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    1.2.10 Voltage Dependent Resistors The resistance of a voltage dependent resistor (VDR) falls very rapidly when the voltage across it exceeds a nominal value in either direction (see Figure 1.38). In normal operation, the current flowing in a VDR is negligible; however, when the resistance falls, the current will become appreciable and a significant amount of energy will be absorbed. VDRs are used as a means of “clamping” the voltage in a circuit to a predetermined level. When connected across the supply rails to a circuit (either AC or DC) they are able to offer a measure of protection against voltage surges.

    FIGURE 1.38: Characteristic of a voltage dependent resistor (VDR)

    1.2.11 Variable Resistors Variable resistors are available in several forms including those which use carbon tracks and those which use a wirewound resistance element. In either case, a moving slider makes contact with the resistance element. Most variable resistors have three (rather than two) terminals and as such are more correctly known as potentiometers. Carbon potentiometers are available with linear or semi-logarithmic law tracks (see Figure 1.39) and in rotary or slider formats. Ganged controls, in which several potentiometers are linked together by a common control shaft, are also available. Figure 1.40 shows a selection of variable resistors. You will also encounter various forms of preset resistors that are used to make occasional adjustments (e.g., for calibration). Various forms of preset resistor are commonly used including open carbon track skeleton presets and fully encapsulated carbon and multiturn cermet types, as shown in Figure 1.41.

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    FIGURE 1.39: Characteristics for linear and semi-logarithmic law variable resistors

    FIGURE 1.40: A selection of common types of carbon and wirewound variable resistors/potentiometers

    FIGURE 1.41: A selection of common types of standard and miniature preset resistors/potentiometers

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    1.2.12 Capacitors A capacitor is a device for storing electric charge. In effect, it is a reservoir into which charge can be deposited and then later extracted. Typical applications include reservoir and smoothing capacitors for use in power supplies, coupling AC signals between the stages of amplifiers, and decoupling supply rails (i.e., effectively grounding the supply rails as far as AC signals are concerned). A capacitor can consist of nothing more than two parallel metal plates as shown in Figure 1.10. To understand what happens when a capacitor is being charged and discharged take a look at Figure 1.42. If the switch is left open (position A), no charge will appear on the plates and in this condition there will be no electric field in the space between the plates nor will there be any charge stored in the capacitor. When the switch is moved to position B, electrons will be attracted from the positive plate to the positive terminal of the battery. At the same time, a similar number of electrons will move from the negative terminal of the battery to the negative plate. This sudden movement of electrons will manifest itself in a momentary surge of current (conventional current will flow from the positive terminal of the battery toward the positive terminal of the capacitor). Eventually, enough electrons will have moved to make the e.m.f. between the plates the same as that of the battery. In this state, the capacitor is said to be fully charged and an electric field will be present in the space between the two plates. If at some later time the switch is moved back to position A, the positive plate will be left with a deficiency of electrons while the negative plate will be left with a surplus of electrons. Furthermore, since there is no path for current to flow between the two plates the capacitor will remain charged and a potential difference will be maintained between the plates. Now assume that the switch is moved to position C. The excess electrons on the negative plate will flow through the resistor to the positive plate until a neutral state once again exists (i.e., until there is no excess charge on either plate). In this state the capacitor is said to be fully discharged and the electric field between the plates will rapidly collapse. The movement of electrons during the discharging of the capacitor will again result in a momentary surge of current (current will flow from the positive terminal of the capacitor and into the resistor). Figure 1.43 shows the direction of current flow in the circuit of Figure 1.42 during charging (switch in position B) and discharging (switch in position C). It should be

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    FIGURE 1.42: Capacitor charging and discharging

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    FIGURE 1.43: Current flow during charging and discharging

    noted that current flows momentarily in both circuits even though you may think that the circuit is broken by the gap between the capacitor plates!

    1.2.13 Capacitance The unit of capacitance is the farad (F). A capacitor is said to have a capacitance of 1F if a current of 1A flows in it when a voltage changing at the rate of 1 V/s is applied to it. The current flowing in a capacitor will thus be proportional to the product of the capacitance, C, and the rate of change of applied voltage. Hence: i ¼ C  ðrate of change of voltageÞ Note that we’ve used a small i to represent the current flowing in the capacitor. We’ve done this because the current is changing and doesn’t remain constant.

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    The rate of change of voltage is often represented by the expression dv/dt where dv represents a very small change in voltage and dt represents the corresponding small change in time. Expressing this mathematically gives: i¼C

    dV dt

    Example 1.44 A voltage is changing at a uniform rate from 10V to 50V in a period of 0.1s. If this voltage is applied to a capacitor of 22 mF, determine the current that will flow. Solution Now the current flowing will be given by: i ¼ C  ðrate of change of voltageÞ Thus,

        change in voltage 50  10 6 i¼C ¼ 22  10  change in time 0:1

    From which, i ¼ 22  106 

    

    40 0:1

    

    ¼ 22  106  400

    so, i ¼ 8:8  103 ¼ 8:8 mA

    1.2.14

    Charge, Capacitance and Voltage

    The charge or quantity of electricity that can be stored in the electric field between the capacitor plates is proportional to the applied voltage and the capacitance of the capacitor. Thus: Q ¼ CV where Q is the charge (in coulombs), C is the capacitance (in farads), and V is the potential difference (in volts).

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    Example 1.45 A 10 mF capacitor is charged to a potential of 250V. Determine the charge stored. Solution The charge stored will be given by: Q ¼ CV ¼ 10  106  250 ¼ 2:5 mC

    1.2.15 Energy storage The energy stored in a capacitor is proportional to the product of the capacitance and the square of the potential difference. Thus: W ¼ ½C V 2 where W is the energy (in joules), C is the capacitance (in farads), and V is the potential difference (in volts). Example 1.46 A capacitor of 47 mF is required to store 4J of energy. Determine the potential difference that must be applied to the capacitor. Solution The foregoing formula can be rearranged to make V the subject as follows: rffiffiffiffiffiffiffiffiffiffi rffiffiffiffiffiffi rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi E 2E 24 V¼ ¼ ¼ 0:5C C 47  106 from which, rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 8 ¼ 0:170  106 ¼ 0:412  103 ¼ 412V V¼ 47  106

    1.2.16 Capacitance and Physical Dimensions The capacitance of a capacitor depends upon the physical dimensions of the capacitor (i.e., the size of the plates and the separation between them) and the

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    Chapter 1

    dielectric material between the plates. The capacitance of a conventional parallel plate capacitor is given by: e0 er A C¼ d where C is the capacitance (in farads), e0 is the permittivity of free space, er is the relative permittivity of the dielectric medium between the plates), and d is the separation between the plates (in meters).

    Example 1.47 A capacitor of 1 nF is required. If a dielectric material of thickness 0.1 mm and relative permittivity 5.4 is available, determine the required plate area.

    Solution Rearranging the formula: C¼

    e0 er A d

    to make A the subject gives: A¼

    Cd 1  109  0:1  103 ¼ e0 er 8:854  1012  5:4

    from which: A¼

    0:1  1012 47:8116  1012

    thus, A ¼ 0:00209 m2

    or 20:9 cm2

    In order to increase the capacitance of a capacitor, many practical components employ multiple plates (see Figure 1.44). The capacitance is then given by:

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    FIGURE 1.44: A multi-plate capacitor



    e0 er ðn  1Þ A d

    where C is the capacitance (in farads), e0 is the permittivity of free space, er is the relative permittivity of the dielectric medium between the plates), and d is the separation between the plates (in meters) and n is the total number of plates. Example 1.48 A capacitor consists of six plates each of area 20 cm2 separated by a dielectric of relative permittivity 4.5 and thickness 0.2 mm. Determine the value of capacitance. Solution Using: C¼

    e0 er ðn  1Þ A d

    gives: C¼

    8:854  1012  4:5  ð6  1Þ  20  104 0:2  103

    from which, C¼

    3; 984:3  1016 ¼ 19:921  1013 ¼ 190  1012 0:2  103

    Thus, C ¼ 190  1012 F

    or 1:992 nF

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    1.2.17

    Chapter 1

    Capacitor Specifications

    The specifications for a capacitor usually include the value of capacitance (expressed in microfarads, nanofarads or picofarads), the voltage rating (i.e., the maximum voltage which can be continuously applied to the capacitor under a given set of conditions), and the accuracy or tolerance (quoted as the maximum permissible percentage deviation from the marked value). Other practical considerations when selecting capacitors for use in a particular application include temperature coefficient, leakage current, stability and ambient temperature range. Table 1.12 summarizes the properties of five of the most common types of capacitor. Note that electrolytic capacitors require the application of a polarizing voltage in order to the chemical action on which they depend for their operation.

    Table 1.12: Characteristics of common types of capacitor Capacitor type Property

    Ceramic

    Electrolytic

    Polyester

    Mica

    Polystyrene

    Capacitance range (F)

    2.2p to 100n

    100n to 10m

    10n to 2.2m

    0.47 to 22k

    10p to 22n

    Typical tolerance (%)

    10 and 20

    10 to þ50

    10

    1

    5

    Typical voltage rating (W)

    50V to 200V

    6.3V to 400V

    100V to 400V

    350V

    100V

    Temperature coefficient (ppm/ C)

    þ100 to 4700

    þ1000 typical

    þ100 to þ200

    þ50

    þ250

    Stability

    Fair

    Poor

    Good

    Excellent

    Good

    Ambient temperature range ( C)

    85 to þ85

    40 to þ80

    40 to þ100

    40 to þ125

    40 to þ100

    Typical applications

    High-frequency and low-cost

    Smoothing and decoupling

    GeneralPurpose

    Tuned circuits and oscillators

    GeneralPurpose

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    The polarizing voltages used for electrolytic capacitors can range from as little as 1V to several hundred volts depending upon the working voltage rating for the component in question. Figure 1.45 shows some typical nonelectrolytic capacitors (including polyester, polystyrene, ceramic and mica types), while Figure 1.46 shows a selection of electrolytic (polarized) capacitors. An air-spaced variable capacitor is shown later in Figure 1.54.

    FIGURE 1.45: A typical selection of nonelectrolytic capacitors (including polyester, polystyrene, ceramic and mica types) with values ranging from 10 pF to 470 nF and working voltages from 50V to 250V

    FIGURE 1.46: A typical selection of electrolytic (polarized) capacitors with values ranging from 1 mF to 470 mF and working voltages from 10V to 63V

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    1.2.18

    Capacitor Markings

    The vast majority of capacitors employ written markings which indicate their values, working voltages, and tolerance. The most usual method of marking resin dipped polyester (and other) types of capacitor involves quoting the value (mF, nF or pF), the tolerance (often either 10% or 20%), and the working voltage (often using _ and  to indicate DC and AC, respectively). Several manufacturers use two separate lines for their capacitor markings and these have the following meanings: First line:

    capacitance (pF or mF) and tolerance (K ¼ 10%, M ¼ 20%)

    Second line:

    rated DC voltage and code for the dielectric material

    A three-digit code is commonly used to mark monolithic ceramic capacitors. The first two digits of this code correspond to the first two digits of the value, while the third digit is a multiplier which gives the number of zeros to be added to give the value in picofarads. Other capacitors may use a color code similar to that used for marking resistor values (see Figure 1.48).

    FIGURE 1.47: Examples of capacitor markings

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    FIGURE 1.48: Capacitor color code

    Example 1.49 A monolithic ceramic capacitor is marked with the legend “103K”. What is its value? Solution The value (pF) will be given by the first two digits (10) followed by the number of zeros indicated by the third digit (3). The value of the capacitor is thus 10,000 pF or 10 nF. The final letter (K) indicates that the capacitor has a tolerance of 10%.

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    FIGURE 1.49: See Example 1.50 Example 1.50 A tubular capacitor is marked with the following colored stripes: brown, green, brown, red, brown. What is its value, tolerance, and working voltage? Solution See Figure 1.49.

    1.2.19

    Series and Parallel Combination of Capacitors

    In order to obtain a particular value of capacitance, fixed capacitors may be arranged in either series or parallel (Figures 1.50 and 1.51). The reciprocal of the effective capacitance of each of the series circuits shown in Figure 1.50 is equal to the sum of the reciprocals of the individual capacitances. Hence, for Figure 1.50(A): 1 1 1 þ ¼ C C1 C2 while for Figure 1.50(B): 1 1 1 1 þ þ ¼ C C1 C2 C3

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    FIGURE 1.50: Capacitors in series

    FIGURE 1.51: Capacitors in parallel

    In the former case, the formula can be more conveniently rearranged as follows: C¼

    C1  C2 C1 þ C2

    You can remember this as the product of the two capacitor values divided by the sum of the two values—just as you did for two resistors in parallel. For a parallel arrangement of capacitors, the effective capacitance of the circuit is simply equal to the sum of the individual capacitances. Hence, for Figure 1.51(A): C ¼ C1 þ C2

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    while for Figure 1.51(B): C ¼ C1 þ C2 þ C3 Example 1.51 Determine the effective capacitance of the circuit shown in Figure 1.52. Solution The circuit of Figure 1.52 can be progressively simplified as shown in Figure 1.53. The stages in this simplification are: (a) C1 and C2 are in parallel and they can be replaced by a single capacitor (CA) of (2 nF þ 4 nF) ¼ 6 nF. (b)

    CA appears in series with C3. These two resistors can be replaced by a single capacitor (CB) of (6 nF  2 nF)/(6 nF þ 2 nF) ¼ 1.5 nF.

    (c) CB appears in parallel with C4. These two capacitors can be replaced by a single capacitance (C) of (1.5 nF þ 4 nF) ¼ 5.5 nF.

    FIGURE 1.52: See Example 1.51

    Example 1.52 A capacitance of 50 mF (rated at 100V) is required. What series combination of preferred value capacitors will satisfy this requirement? What voltage rating should each capacitor have?

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    Solution Two 100 mF capacitors wired in series will provide a capacitance of 50 mF, as follows: C¼

    C1  C2 100  100 10; 000 ¼ ¼ ¼ 50 mF 100 þ 100 200 C1 þ C2

    FIGURE 1.53: See Example 1.51 Since the capacitors are of equal value, the applied DC potential will be shared equally between them. Thus each capacitor should be rated at 50V. Note that, in a practical circuit, we could take steps to ensure that the DC voltage was shared equally between the two capacitors by wiring equal, high-value (e.g., 100 kO) resistors across each capacitor.

    1.2.20 Variable Capacitors By moving one set of plates relative to the other, a capacitor can be made variable. The dielectric material used in a variable capacitor can be either air (see Figure 1.54) or plastic (the latter tend to be more compact). Typical values for variable capacitors tend to range from about 25 pF to 500 pF. These components are commonly used for tuning radio receivers.

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    FIGURE 1.54: An air-spaced variable capacitor. This component (used for tuning an AM radio) has two separate variable capacitors (each of 500 pF maximum) operated from a common control shaft.

    1.2.21

    Inductors

    Inductors provide us with a means of storing electrical energy in the form of a magnetic field. Typical applications include chokes, filters and (in conjunction with one or more capacitors) frequency selective circuits. The electrical characteristics of an inductor are determined by a number of factors including the material of the core (if any), the number of turns, and the physical dimensions of the coil. Figure 1.55 shows the construction of a typical toroidal inductor wound on a ferrite (high permeability) core. In practice every coil comprises both inductance (L) and a small resistance (R). The circuit of Figure 1.56 shows these as two discrete components. In reality the inductance and the resistance (we often refer to this as a loss resistance because it’s something that

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    we don’t actually want) are both distributed throughout the component but it is convenient to treat the inductance and resistance as separate components in the analysis of the circuit.

    FIGURE 1.55: A practical coil contains inductance and resistance

    FIGURE 1.56: A practical coil contains inductance and a small amount of series loss resistance To understand what happens when a changing current flows through an inductor, take a look at the circuit shown in Figure 1.57(A). If the switch is left open, no current will flow and no magnetic flux will be produced by the inductor. If the switch is closed, as shown in Figure 1.57(B), current will begin to flow as energy is taken from the supply in order to establish the magnetic field. However, the change in magnetic flux resulting from the appearance of current creates a voltage (an induced e.m.f.) across the coil which opposes the applied e.m.f. from the battery.

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    FIGURE 1.57: Flux and e.m.f. generated when a changing current is applied to an inductor

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    The induced e.m.f. results from the changing flux and it effectively prevents an instantaneous rise in current in the circuit. Instead, the current increases slowly to a maximum at a rate which depends upon the ratio of inductance (L) to resistance (R) present in the circuit. After a while, a steady-state condition will be reached in which the voltage across the inductor will have decayed to zero and the current will have reached a maximum value determined by the ratio of V to R (i.e., Ohm’s Law). This is shown in Figure 1.57(C). If, after this steady-state condition has been achieved, the switch is opened, as shown in Figure 1.57(D), the magnetic field will suddenly collapse and the energy will be returned to the circuit in the form of an induced back e.m.f., which will appear across the coil as the field collapses. For large values of magnetic flux and inductance this back e.m.f. can be extremely large!

    1.2.22 Inductance Inductance is the property of a coil which gives rise to the opposition to a change in the value of current flowing in it. Any change in the current applied to a coil/inductor will result in an induced voltage appearing across it. The unit of inductance is the henry (H) and a coil is said to have an inductance of 1H if a voltage of 1V is induced across it when a current changing at the rate of 1 A/s is flowing in it. The voltage induced across the terminals of an inductor will thus be proportional to the product of the inductance (L) and the rate of change of applied current. Hence: e ¼  L  ðrate of change of currentÞ Note that the minus sign indicates the polarity of the voltage, i.e., opposition to the change. The rate of change of current is often represented by the expression di/dt where di represents a very small change in current and dt represents the corresponding small change in time. Using mathematical notation to write this we arrive at: e ¼ L

    di dt

    You might like to compare this with the similar relationship that we obtained for the current flowing in a capacitor shown in Section 1.2.13.

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    Example 1.53 A current increases at a uniform rate from 2A to 6A in a period of 250 ms. If this current is applied to an inductor of 600 mH, determine the voltage induced. Solution Now the induced voltage will be given by: e ¼  L  ðrate of change of currentÞ Thus,

        change in current 62 3 e ¼ L ¼  60  10  change in time 250  103

    From which, 3

    e ¼  600  10

    

     4  ¼  0:6  103  16 0:25

    so, e ¼ 9:6V

    1.2.23 Energy Storage The energy stored in an inductor is proportional to the product of the inductance and the square of the current flowing in it. Thus: W ¼ ½ L I2 where W is the energy (in joules), L is the capacitance (in henries), and I is the current flowing in the inductor (in amps). Example 1.54 An inductor of 20 mH is required to store 2.5J of energy. Determine the current that must be applied.

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    Solution The foregoing formula can be rearranged to make I the subject as follows: rffiffiffiffiffiffiffiffiffi rffiffiffiffiffiffi rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi E 2E 2  2:5 ¼ ¼ I¼ 0:5L L 20  103 From which

    rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffi 5 3 ¼ 250 ¼ 15:81A I¼ ¼ 0:25  10 20  103

    1.2.24 Inductance and Physical Dimensions The inductance of an inductor depends upon the physical dimensions of the inductor (e.g., the length and diameter of the winding), the number of turns, and the permeability of the material of the core. The inductance of an inductor is given by: L¼

    m0 mr n2 A l

    where L is the inductance (in henries), m0 is the permeability of free space, mr is the relative permeability of the magnetic core, l is the mean length of the core (in meters), and A is the cross-sectional area of the core (in square meters). Example 1.55 An inductor of 100 mH is required. If a closed magnetic core of length 20 cm, cross-sectional area 15 cm2 and relative permeability 500 is available, determine the number of turns required. Solution First we must rearrange the formula: L¼

    m0 mr n2 A l

    in order to make n the subject: sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Ll 100  103  20  102 ¼ n¼ m0 mr n2 A 12:57  107  500  15  104

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    Chapter 1

    From which: sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2  102 n¼ ¼ 21; 215 ¼ 146 11 94; 275  10 Hence, the inductor requires 146 turns of wire.

    1.2.25

    Inductor Specifications

    Inductor specifications normally include the value of inductance (expressed in henries, millihenries or microhenries), the current rating (i.e., the maximum current which can be continuously applied to the inductor under a given set of conditions), and the accuracy or tolerance (quoted as the maximum permissible percentage deviation from the marked value). Other considerations may include the temperature coefficient of the inductance (usually expressed in parts per million, p.p.m., per unit temperature change), the stability of the inductor, the DC resistance of the coil windings (ideally zero), the Q-factor (quality factor) of the coil, and the recommended working frequency range. Table 1.13 Table 1.13: Characteristics of common types of inductor Inductor type Property

    Air cored

    Ferrite cored

    Ferrite pot cored

    Iron cored

    Core material

    Air

    Ferrite rod

    Ferrite pot

    Laminated steel

    Inductance range (H)

    50n to 100m

    10m to 1m

    1m to 100m

    20m to 20

    Typical DC resistance (O)

    0.05 to 5

    0.1 to 10

    5 to 100

    10 to 200

    Typical tolerance (%)

    5

    10

    10

    20

    Typical Q-factor

    60

    80

    40

    20

    Typical frequency range (Hz)

    1M to 500M

    100k to 100M

    1k to 10M

    50 to 10k

    Typical applications

    Tuned circuits and filters

    Filters and HF transformers

    LF and MF filters and transformers

    Smoothing chokes and filters

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    summarizes the properties of four common types of inductor. Some typical small inductors are shown in Figure 1.58. These have values of inductance ranging from 15 mH to 1 mH.

    FIGURE 1.58: A selection of small inductors with values ranging from 15 mH to 1 mH

    1.2.26 Inductor Markings As with capacitors, the vast majority of inductors use written markings to indicate values, working current, and tolerance. Some small inductors are marked with colored stripes to indicate their value and tolerance (in which case the standard color values are used and inductance is normally expressed in microhenries).

    1.2.27 Series and Parallel Combinations of Inductors In order to obtain a particular value of inductance, fixed inductors may be arranged in either series or parallel as shown in Figs 1.59 and 1.60. The effective inductance of each of the series circuits shown in Figure 1.59 is simply equal to the sum of the individual inductances. So, for the circuit shown in Figure 1.59(A): L ¼ L1 þ L2 while for Figure 1.59(B): L ¼ L1 þ L2 þ L3

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    FIGURE 1.59: Inductors in series Turning to the parallel inductors shown in Figure 1.60, the reciprocal of the effective inductance of each circuit is equal to the sum of the reciprocals of the individual inductances. Hence, for Figure 1.60(A): 1 1 1 þ ¼ L L1 L2 while for Figure 1.60(B): 1 1 1 1 þ þ ¼ L L1 L2 L3 In the former case, the formula can be more conveniently re-arranged as follows: L¼

    L1  L2 L1 þ L2

    You can remember this as the product of the two inductance values divided by the sum of the two inductance values.

    FIGURE 1.60: Inductors in parallel

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    Example 1.56 An inductance of 5 mH (rated at 2A) is required. What parallel combination of preferred value inductors will satisfy this requirement? Solution Two 10 mH inductors may be wired in parallel to provide an inductance of 5 mH as shown below: L¼

    L1  L2 10  10 100 ¼ ¼ ¼ 5 mH 10 þ 10 20 L1 þ L2

    Since the inductors are identical, the applied current will be shared equally between them. Hence, each inductor should have a current rating of 1A. Example 1.57 Determine the effective inductance of the circuit shown in Figure 1.61. Solution The circuit can be progressively simplified as shown in Figure 1.62. The stages in this simplification are as follows: (a)

    L1 and L2 are in series and they can be replaced by a single inductance (LA) of (60 þ 60) ¼ 120 mH.

    (b)

    LA appears in parallel with L2. These two inductors can be replaced by a single inductor (LB) of (120  120)/(120 þ 120) ¼ 60 mH.

    (c)

    LB appears in series with L4. These two inductors can be replaced by a single inductance (L) of (60 þ 50) ¼ 110 mH.

    FIGURE 1.61: See Example 1.57

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    FIGURE 1.62: See Example 1.57

    1.2.28

    Variable Inductors

    A ferrite cored inductor can be made variable by moving its core in or out of the former onto which the coil is wound. Many small inductors have threaded ferrite cores to make this possible (see Figure 1.63). Such inductors are often used in radio and highfrequency applications where precise tuning is required.

    FIGURE 1.63: An adjustable ferrite cored inductor

    1.2.29

    Surface Mounted Components (SMC)

    Surface-mount technology (SMT) is now widely used in the manufacture of printed circuit boards for electronic equipment. SMT allows circuits to be assembled in a much smaller space than would be possible using components with conventional wire leads and pins that are mounted using through-hole techniques. It is also possible to mix the

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    two technologies, i.e., some through-hole mounting of components and some surface mounted components present on the same circuit board. The following combinations are possible:  Surface mounted components (SMC) on both sides of a printed circuit board.  SMC on one side of the board and conventional through-hole components (THC) on the other.  A mixture of SMC and THC on both sides of the printed circuit board. Surface mounted components are supplied in packages that are designed for mounting directly on the surface of a PCB. To provide electrical contact with the PCB, some SMC have contact pads on their surface. Other devices have contacts which extend beyond the outline of the package itself but which terminate on the surface of the PCB rather than making contact through a hole (as is the case with a conventional THC). In general, passive components (such as resistors, capacitors and inductors) are configured leadless for surface mounting, while active devices (such as transistors and integrated circuits) are available in both surface mountable types as well as lead as well as in leadless terminations suitable for making direct contact to the pads on the surface of a PCB. Most surface mounted components have a flat rectangular shape rather than the cylindrical shape that we associate with conventional wire leaded components. During manufacture of a PCB, the various SMC are attached using re-flow soldering paste (and in some cases adhesives) which consists of particles of solder and flux together with binder, solvents and additives. They need to have good “tack” in order to hold the components in place and remove oxides without leaving obstinate residues. The component attachment (i.e., soldering!) process is completed using one of several techniques including convection ovens in which the PCB is passed, using a conveyor belt, through a convection oven which has separate zones for preheating, flowing and cooling, and infra-red reflow in which infrared lamps are used to provide the source of heat. Surface mounted components are generally too small to be marked with color codes. Instead, values may be marked using three digits. For example, the first two digits marked on a resistor normally specify the first two digits of the value while the third digit gives the number of zeros that should be added.

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    Example 1.58 In Figure 1.65, R88 is marked “102”. What is its value? Solution R88 will have a value of 1,000O (i.e., 10 followed by two zeros).

    FIGURE 1.64: Conventional components mounted on a printed circuit board. Note that components such as C38, R46, etc. have leads that pass through holes in the printed circuit boards

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    FIGURE 1.65: Surface mounted components (note the appearance of capacitors C35, C52, and C53, and resistors, R87, R88, R91, etc.)

    1.3 DC Circuits In many cases, Ohm’s Law alone is insufficient to determine the magnitude of the voltages and currents present in a circuit. This section introduces several techniques that simplify the task of solving complex circuits. It also introduces the concept of exponential growth and decay of voltage and current in circuits containing capacitance and resistance and inductance and resistance. It concludes by showing how humble C-R

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    circuits can be used for shaping the waveforms found in electronic circuits. We start by introducing two of the most useful laws of electronics.

    1.3.1

    Kirchhoff’s Laws

    Kirchhoff’s Laws relate to the algebraic sum of currents at a junction (or node) or voltages in a network (or mesh). The term “algebraic” simply indicates that the polarity of each current or voltage drop must be taken into account by giving it an appropriate sign, either positive (þ) or negative (). Kirchhoff’s Current Law states that the algebraic sum of the currents present at a junction (node) in a circuit is zero (see Figure 1.66).

    FIGURE 1.66: Kirchhoff’s Current Law Example 1.59 In Figure 1.67, use Kirchhoff’s Current Law to determine: (a) the value of current flowing between A and B, and (b)

    the value of I3.

    Solution (a) I1 and I2 both flow toward Node A so, applying our polarity convention, they must both be positive. Now, assuming that a current I5 flows between A and B and that this

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    FIGURE 1.67: See Example 1.59 current flows away from the junction (obvious because I1 and I2 both flow toward the junction), we arrive at the following Kirchhoff’s Current Law equation: þI1 þ I2  I5 ¼ 0 From which: I5 ¼ I1 þ I2 ¼ 1:5 þ 2:7 ¼ 4:2A (b) Moving to Node B, let’s assume that I3 flows outward, so we can say that: þI4 þ I5  I3 ¼ 0 From which: I3 ¼ I4 þ I5 ¼ 3:3 þ 4:2 ¼ 7:5A Kirchhoff’s Voltage Law states that the algebraic sum of the potential drops in a closed network (or “mesh”) is zero (see Figure 1.68). Example 1.60 In Figure 1.69, use Kirchhoff’s Voltage Law to determine: (a)

    the value of V2, and

    (b)

    the value of E3.

    Solution (a)

    In Loop A, and using the conventions shown in Figure 1.68, we can write down the Kirchhoff’s Voltage Law equations: E1  V2  E2 ¼ 0

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    FIGURE 1.68: Kirchhoff’s Voltage Law

    FIGURE 1.69: See Example 1.60 From which: V2 ¼ E1  E2 ¼ 6  3 ¼ 3V (b)

    Similarly, in Loop B, we can say that: E2  V2 þ E3 ¼ 0

    From which: E3 ¼ V2  E2 ¼ 4:5  3 ¼ 1:5V

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    Example 1.61 Determine the currents and voltages in the circuit of Figure 1.70.

    FIGURE 1.70: See Example 1.61 Solution In order to solve the circuit shown in Figure 1.70, it is first necessary to mark the currents and voltages on the circuit, as shown in Figures 1.71 and 1.72. By applying Kirchhoff’s Current Law at Node A that we’ve identified in Figure 1.70: þI1 þ I2  I3 ¼ 0 Therefore: I1 ¼ I3  I2

    ðiÞ

    By applying Kirchhoff’s Voltage Law in Loop A we obtain: 12  V1  V3 ¼ 0 From which: V1 ¼ 12  V3

    ðiiÞ

    By applying Kirchhoff’s Voltage Law in Loop B we obtain: 9  V2  V3 ¼ 0 From which: V2 ¼ 9  V3

    ðiiiÞ

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    FIGURE 1.71: See Example 1.61

    FIGURE 1.72: See Example 1.61 Next we can generate three further relationships by applying Ohm’s Law: V1 ¼ I1 R1 V2 ¼ I2 R2

    V1 R1 V2 from which I2 ¼ R2 from which I1 ¼

    and, V3 ¼ I3 R3

    from which I3 ¼

    V3 R3

    Combining these three relationships with the Current Law equation (i) gives: V1 V3 V2 ¼  R1 R3 R2

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    from which: V1 V3 V2 ¼  110 22 33 Combining (ii) and (iii) with (iv) gives: ð12  V3 Þ V3 ð9  V3 Þ ¼  110 33 22 Multiplying both sides of the expression by 330 gives: 330ð12  V3 Þ 330V3 330ð9  V3 Þ ¼  110 33 22 3ð12  V3 Þ ¼ 15 V3  10ð9  V3 Þ From which: 36  3 V3 ¼ 15 V3  90 þ V3 36 þ 90 ¼ 15 V3 þ 10 V3 þ 3 V3 and: 126 ¼ 28V3

    so V3 ¼ 126=28 ¼ 4:5V

    From (ii): V1 ¼ 12  V3

    so

    V1 ¼ 12  4:5 ¼ 7:5V

    V2 ¼ 9  V3

    so

    V2 ¼ 9  4:5 ¼ 4:5V

    From (iii):

    Using the Ohm’s Law equations that we met earlier gives: V1 7:5 ¼ ¼ 0:068A ¼ 68 mA 110 R1 V2 4:5 I2 ¼ ¼ ¼ 0:136A ¼ 136 mA 33 R2 V3 4:5 I3 ¼ ¼ ¼ 0:204A ¼ 204 mA 22 R3 I1 ¼

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    94

    Chapter 1

    Finally, it’s worth checking these results with the Current Law equation (i): þI1 þ I2  I3 ¼ 0 Inserting our values for I1, I2 and I3 gives: þ0:068 þ 0:136  204 ¼ 0 Since the left and right hand sides of the equation are equal we can be reasonably confident that our results are correct.

    1.3.2

    The Potential Divider

    The potential divider circuit (see Figure 1.73) is commonly used to reduce voltages in a circuit. The output voltage produced by the circuit is given by: Vout ¼ Vin

    R2 R1 þ R2

    FIGURE 1.73: Potential divider circuit It is, however, important to note that the output voltage (Vout) will fall when current is drawn from the arrangement. Figure 1.74 shows the effect of loading the potential divider circuit. In the loaded potential divider (Figure 1.74) the output voltage is given by: Vout ¼ Vin

    Rp R1 þ Rp

    where: Rp ¼

    www.newnespress.com

    R2  RL R2 þ RL

    The Fundamentals

    95

    FIGURE 1.74: Loaded potential divider circuit

    Example 1.62 The potential divider shown in Figure 1.75 is used as a simple voltage calibrator. Determine the output voltage produced by the circuit: (a)

    when the output terminals are left open-circuit (i.e., when no load is connected); and

    (b)

    when the output is loaded by a resistance of 10 kO.

    FIGURE 1.75: See Example 1.62

    Solution (a)

    In the first case we can simply apply the formula: Vout ¼ Vin

    R2 R1 þ R2

    where Vin ¼ 5V, R1 ¼ 4 kO and R2 ¼ 1 kO.

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    96

    Chapter 1

    Hence: Vout ¼ 5  (b)

    1 ¼ 1V 4þ1

    In the second case we need to take into account the effect of the 10 kO resistor connected to the output terminals of the potential divider.

    First we need to find the equivalent resistance of the parallel combination of R2 and RL: Rp ¼

    R2  RL 1  10 10 ¼ ¼ 0:909 kO ¼ 1 þ 10 11 R2 þ RL

    Then we can determine the output voltage from: Vout ¼ Vin

    1.3.3

    Rp 0:909 ¼5 ¼ 0:925V 4 þ 0:909 R1 þ Rp

    The Current Divider

    The current divider circuit (see Figure 1.76) is used to divert a known proportion of the current flowing in a circuit. The output current produced by the circuit is given by: Iout ¼ Iin

    R1 R1 þ R2

    It is, however, important to note that the output current (Iout) will fall when the load connected to the output terminals has any appreciable resistance.

    FIGURE 1.76: Current divider circuit

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    The Fundamentals

    97

    Example 1.63 A moving coil meter requires a current of 1 mA to provide full-scale deflection. If the meter coil has a resistance of 100O and is to be used as a milliammeter reading 5 mA full-scale, determine the value of parallel shunt resistor required. Solution This problem may sound a little complicated so it is worth taking a look at the equivalent circuit of the meter (Figure 1.77) and comparing it with the current divider shown in Figure 1.76. We can apply the current divider formula, replacing Iout with Im (the meter full-scale deflection current) and R2 with Rm (the meter resistance). R1 is the required value of shunt resistor, Rs, Hence: Iout ¼ Iin

    Rs Rs þ Rm

    FIGURE 1.77: See Example 1.63

    Rearranging the formula gives: Im  ðRs þ Rm Þ ¼ Iin  Rs thus, Im Rs þ Im Rm ¼ Iin Rs

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    98

    Chapter 1

    or, Iin Rs  Im Rs ¼ Im Rm from which, Rs ðIin  Im Þ ¼ Im Rm So, Rs ¼

    Im Rm Iin  Im

    Now Iin ¼ 1 mA, Rm ¼ 100O and Iin ¼ 5 mA, thus: Rs ¼

    1.3.4

    1  100 100 ¼ ¼ 25O 51 4

    The Wheatstone Bridge

    The Wheatstone bridge forms the basis of a number of useful electronic circuits including several that are used in instrumentation and measurement. The basic form of Wheatstone bridge is shown in Figure 1.78. The voltage developed between A and B will be zero when the voltage between A and Y is the same as that between B and Y. In effect, R1 and R2 constitute a potential divider as do R3 and R4.

    FIGURE 1.78: Basic Wheatstone bridge circuit

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    The Fundamentals

    99

    The bridge will be balanced (and VAB ¼ 0) when the ratio of R1:R2 is the same as the ratio R3:R4. Hence, at balance: R1 R3 ¼ R2 R4 A practical form of Wheatstone bridge that can be used for measuring unknown resistances is shown in Figure 1.79.

    FIGURE 1.79: See Example 1.64 In this practical form of Wheatstone bridge, R1 and R2 are called the ratio arms while one arm (that occupied by R3 in Figure 1.78) is replaced by a calibrated variable resistor. The unknown resistor, Rx, is connected in the fourth arm. At balance: R1 Rv ¼ R2 Rx

    thus Rx ¼

    R2  Rv R1

    Example 1.64 A Wheatstone bridge is based on the circuit shown in Figure 1.79. If R1 and R2 can each be switched so that they have values of either 100O or 1 kO and RV is variable between 10O and 10 kO, determine the range of resistance values that can be measured.

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    100

    Chapter 1

    Solution The maximum value of resistance that can be measured will correspond to the largest ratio of R2:R1 (i.e., when R2 is 1 kO and R1 is 100O) and the highest value of RV (i.e., 10 kO). In this case: Rx ¼

    1; 000  10; 000 ¼ 100; 000 ¼ 100 kO 100

    The minimum value of resistance that can be measured will correspond to the smallest ratio of R2:R1 (i.e., when R1 is 100O and R1 is 1 kO) and the smallest value of RV (i.e., 10O). In this case: Rx ¼

    100  10 ¼ 0:1  10 ¼ 1O 1; 000

    Hence the range of values that can be measured extends from 1O to 100 kO.

    1.3.5

    The´venin’s Theorem

    The´venin’s Theorem allows us to replace a complicated network of resistances and voltage sources with a simple equivalent circuit comprising a single voltage source connected in series with a single resistance (see Figure 1.70). The single voltage source in the The´venin equivalent circuit, Voc, is simply the voltage that appears between the terminals when nothing is connected to it. In other words, it is the open-circuit voltage that would appear between A and B. The single resistance that appears in the The´venin equivalent circuit, R, is the resistance that would be seen looking into the network between A and B when all of the voltage sources (assumed perfect) are replaced by short-circuit connections. Note that if the voltage sources are not perfect (i.e., if they have some internal resistance) the equivalent circuit must be constructed on the basis that each voltage source is replaced by its own internal resistance. Once we have values for Voc and R, we can determine how the network will behave when it is connected to a load (i.e., when a resistor is connected across the terminals A and B).

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    The Fundamentals

    101

    Example 1.65 Figure 1.81 shows a Wheatstone bridge. Determine the current that will flow in a 100O load connected between terminals A and B. Solution First we need to find the The´venin equivalent of the circuit. To find Voc we can treat the bridge arrangement as two potential dividers. The voltage across R2 will be given by: V ¼ 10 

    R2 600 ¼ 10  ¼ 5:454V 500 þ 600 R1 þ R2

    Hence, the voltage at A relative to Y, VAY, will be 5.454V. The voltage across R4 will be given by: V ¼ 10 

    R4 400 ¼ 10  ¼ 4:444V 500 þ 400 R3 þ R4

    Hence, the voltage at B relative to Y, VBY, will be 4.444V. The voltage VAB will be the difference between VAY and VBY. This, the open-circuit output voltage, VAB, will be given by: VAB ¼ VAY  VBY ¼ 5:454  4:444 ¼ 1:01V Next we need to find the The´venin equivalent resistance looking in at A and B. To do this, we can redraw the circuit, replacing the battery (connected between X and Y) with a short circuit, as shown in Figure 1.82. The The´venin equivalent resistance is given by the relationship: R¼

    R1  R2 R3  R4 500  600 500  400 þ þ ¼ 500 þ 600 500 þ 400 R1 þ R2 R3 þ R4

    From which: R¼

    300; 000 200; 000 þ ¼ 272:7 þ 222:2 ¼ 494:9O 1; 100 900

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    102

    Chapter 1

    FIGURE 1.80: The´venin equivalent circuit

    FIGURE 1.81: See Example 1.65

    FIGURE 1.82: See Example 1.65 The The´venin equivalent circuit is shown in Figure 1.83. To determine the current in a 100O load connected between A and B, we can simply add a 100O load to the The´venin equivalent circuit, as shown in Figure 1.84. By applying Ohm’s Law in Figure 1.84 we get: I¼

    Voc 1:01 1:01 ¼ ¼ 1:698 mA ¼ 494:9 þ 100 594:9 R þ 100

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    The Fundamentals

    103

    FIGURE 1.83: The´venin equivalent of Figure 1.81

    FIGURE 1.84: Determining the current when the The´venin equivalent circuit is loaded

    1.3.6

    Norton’s Theorem

    Norton’s Theorem provides an alternative method of reducing a complex network to a simple equivalent circuit. Unlike The´venin’s Theorem, Norton’s Theorem makes use of a current source rather than a voltage source. The Norton equivalent circuit allows us to replace a complicated network of resistances and voltage sources with a simple equivalent circuit comprising a single constant current source connected in parallel with a single resistance (see Figure 1.85). The constant current source in the Norton equivalent circuit, Isc, is simply the shortcircuit current that would flow if A and B were to be linked directly together. The resistance that appears in the Norton equivalent circuit, R, is the resistance that would be seen looking into the network between A and B when all of the voltage sources are replaced by short-circuit connections. Once again, it is worth noting that, if the voltage sources have any appreciable internal resistance, the equivalent circuit must be constructed on the basis that each voltage source is replaced by its own internal resistance.

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    104

    Chapter 1

    FIGURE 1.85: Norton equivalent circuit As with the The´venin equivalent, we can determine how a network will behave by obtaining values for Isc and R. Example 1.66 Three temperature sensors having the following characteristics shown in Table 1.14 are connected in parallel as shown in Figure 1.86: Determine the voltage produced when the arrangement is connected to a moving-coil meter having a resistance of 1 kO. Table 1.14: Temperature sensor characteristics Sensor

    A

    B

    C

    Output voltage (open circuit)

    20 mV

    30 mV

    10 mV

    Internal resistance

    5 kO

    3 kO

    2 kO

    FIGURE 1.86: See Example 1.66 Solution First we need to find the Norton equivalent of the circuit. To find Isc we can determine the short-circuit current from each sensor and add them together.

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    The Fundamentals

    105

    For sensor A: I¼

    V 20 mV ¼ ¼ 4 mA R 5 kO



    V 30 mV ¼ ¼ 10 mA R 3 kO



    V 10 mV ¼ ¼ 5 mA R 2 kO

    For sensor B:

    For sensor C:

    The total current, Isc, will be given by: Isc ¼ 4 mA þ 10 mA þ 5 mA ¼ 19 mA Next we need to find the Norton equivalent resistance. To do this, we can redraw the circuit showing each sensor replaced by its internal resistance, as shown in Figure 1.87.

    FIGURE 1.87: Determining the equivalent resistance in Figure 1.86 The equivalent resistance of this arrangement (think of this as the resistance seen looking into the circuit in the direction of the arrow shown in Figure 1.87) is given by: 1 1 1 1 1 1 1 þ þ ¼ ¼ þ þ R R1 R2 R3 5; 000 3; 000 2; 000 where R1 ¼ 5kO, R2 ¼ 3kO, R3 ¼ 2kO, hence: 1 1 1 1 1 1 1 ¼ þ þ þ þ ¼ R R1 R2 R3 5; 000 3; 000 2; 000

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    106

    Chapter 1

    or, 1 ¼ 0:0002 þ 0:00033 þ 0:0005 ¼ 0:00103 R from which: R ¼ 968O The Norton equivalent circuit is shown in Figure 1.88. To determine the voltage in a 1 kO moving coil meter connected between A and B, we can make use of the Norton equivalent circuit by simply adding a 1 kO resistor to the circuit and applying Ohm’s Law, as shown in Figure 1.89. The voltage appearing across the moving coil meter in Figure 1.90 will be given by: V ¼ Isc 

    R  Rm 1; 000  968 ¼ 19 mA  1; 000 þ 968 R þ Rm

    hence: V ¼ 19 mA  492O ¼ 9:35 mV

    FIGURE 1.88: Norton equivalent of the circuit in Figure 1.86

    FIGURE 1.89: Determining the output voltage when the Norton equivalent circuit is loaded with 1 kV

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    The Fundamentals

    107

    FIGURE 1.90: The voltage drop across the meter is found to be 9.35 mV

    1.3.7

    C-R Circuits

    Networks of capacitors and resistors (known as C-R circuits) form the basis of many timing and pulse shaping circuits and are thus often found in practical electronic circuits.

    1.3.8

    Charging

    A simple C-R circuit is shown in Figure 1.91. In this circuit C is charged through R from the constant voltage source, Vs. The voltage, nc, across the (initially uncharged) capacitor voltage will rise exponentially as shown in Figure 1.92. At the same time, the current in the circuit, i, will fall, as shown in Figure 1.93. The rate of growth of voltage with time (and decay of current with time) will be dependent upon the product of capacitance and resistance. This value is known as the time constant of the circuit. Hence: Time constant, t ¼ C  R where C is the value of capacitance (F), R is the resistance (F), and t is the time constant (s).

    FIGURE 1.91: A C-R circuit in which C is charged through R

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    108

    Chapter 1

    FIGURE 1.92: Exponential growth of capacitor voltage, nc, in Figure 1.92

    FIGURE 1.93: Exponential decay of current, i, in Figure 1.91

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    The Fundamentals

    109

    The voltage developed across the charging capacitor, nc, varies with time, t, according to the relationship:   t nc ¼ Vs 1  eCR where nc is the capacitor voltage, Vs is the DC supply voltage, t is the time, and CR is the time constant of the circuit (equal to the product of capacitance, C, and resistance, R). The capacitor voltage will rise to approximately 63% of the supply voltage, Vs, in a time interval equal to the time constant. At the end of the next interval of time equal to the time constant (i.e., after an elapsed time equal to 2CR) the voltage will have risen by 63% of the remainder, and so on. In theory, the capacitor will never become fully charged. However, after a period of time equal to 5CR, the capacitor voltage will to all intents and purposes be equal to the supply voltage. At this point, the capacitor voltage will have risen to 99.3% of its final value and we can consider it to be fully charged. During charging, the current in the capacitor, i, varies with time, t, according to the relationship: i¼

    Vs  t e CR R

    where Vs is the DC supply voltage, t is the time, R is the series resistance and C is the value of capacitance. The current will fall to approximately 37% of the initial current in a time equal to the time constant. At the end of the next interval of time equal to the time constant (i.e., after a total time of 2CR has elapsed) the current will have fallen by a further 37% of the remainder, and so on. Example 1.67 An initially uncharged 1 mF capacitor is charged from a 9V DC supply via a 3.3 MO resistor. Determine the capacitor voltage 1s after connecting the supply.

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    110

    Chapter 1

    Solution The formula for exponential growth of voltage in the capacitor is:   t nc ¼ Vs 1  eCR Here we need to find the capacitor voltage, nc, when Vs ¼ 9V, t ¼ 1s, C ¼ 1 mF and R ¼ 3.3 MO. The time constant, CR, will be given by: CR ¼ 1  106  3:3  106 ¼ 3:3s Thus:   t nc ¼ 9 1  e3:3 and, nc 9ð1  0:738Þ ¼ 9  0:262 ¼ 2:358V Example 1.68 A 100 mF capacitor is charged from a 350V DC supply through a series resistance of 1 kO. Determine the initial charging current and the current that will flow 50 ms and 100 ms after connecting the supply. After what time is the capacitor considered to be fully charged? Solution At t ¼ 0 the capacitor will be uncharged (nc ¼ 0) and all of the supply voltage will appear across the series resistance. Thus, at t ¼ 0: i¼

    Vs 350 ¼ 0:35A ¼ 1; 000 R

    When t ¼ 50 ms, the current will be given by: i¼

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    Vs  t e CR R

    The Fundamentals

    111

    Where Vs ¼ 350V, t ¼ 50 ms, C ¼ 100 mF, R ¼ 1 kO. Hence: i¼

    350 0:05 e 0:1 ¼ 0:35 e0:5 ¼ 0:35  0:607 ¼ 0:21A 1; 000

    When t ¼ 100 ms (using the same equation but with t ¼ 0.1s) the current is given by: i¼

    350 0:1 e0:1 ¼ 0:35 e1 ¼ 0:35  0:368 ¼ 0:129A 1; 000

    The capacitor can be considered to be fully charged when t ¼ 5CR ¼ 5  100  106  1  103 ¼ 0.5s. Note that, at this point the capacitor voltage will have reached 99% of its final value. Discharge Having considered the situation when a capacitor is being charged, let’s consider what happens when an already charged capacitor is discharged.

    FIGURE 1.94: C-R circuits are widely used in electronics. In this oscilloscope, for example, a rotary switch is used to select different C-R combinations in order to provide the various timebase ranges (adjustable from 500 ms/cm to 1 ms/cm). Each C-R time constant corresponds to a different timebase range.

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    112

    Chapter 1

    When the fully charged capacitor from Figure 1.89 is connected as shown in Figure 1.95, the capacitor will discharge through the resistor, and the capacitor voltage, nC, will fall exponentially with time, as shown in Figure 1.96. The current in the circuit, i, will also fall, as shown in Figure 1.97. The rate of discharge (i.e., the rate of decay of voltage with time) will once again be governed by the time constant of the circuit, C  R.

    FIGURE 1.95: A C-R circuit in which C is initially charged and then discharges through R

    FIGURE 1.96: Exponential decay of capacitor voltage, nc, in Figure 1.95

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    The Fundamentals

    113

    FIGURE 1.97: Exponential decay of current, i, in Figure 1.95

    The voltage developed across the discharging capacitor, nC, varies with time, t, according to the relationship: nc ¼ Vs eCR t

    where Vs, is the supply voltage, t is the time, C is the capacitance, and R is the resistance. The capacitor voltage will fall to approximately 37% of the initial voltage in a time equal to the time constant. At the end of the next interval of time equal to the time constant (i.e., after an elapsed time equal to 2CR) the voltage will have fallen by 37% of the remainder, and so on. In theory, the capacitor will never become fully discharged. However, after a period of time equal to 5CR, the capacitor voltage will to all intents and purposes be zero. At this point the capacitor voltage will have fallen below 1% of its initial value. At this point we can consider it to be fully discharged.

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    114

    Chapter 1

    As with charging, the current in the capacitor, i, varies with time, t, according to the relationship: i¼

    Vs  t e CR R

    where Vs, is the supply voltage, t is the time, C is the capacitance, and R is the resistance.The current will fall to approximately 37% of the initial value of current, Vs/R, in a time equal to the time constant. At the end of the next interval of time equal to the time constant (i.e., after a total time of 2CR has elapsed) the voltage will have fallen by a further 37% of the remainder, and so on. Example 1.69 A 10 mF capacitor is charged to a potential of 20V and then discharged through a 47 kO resistor. Determine the time taken for the capacitor voltage to fall below 10V. Solution The formula for exponential decay of voltage in the capacitor is: nc ¼ Vs eCR t

    where Vs ¼ 20V and CR ¼ 10 mF  47 kO ¼ 0.47s. We need to find t when nC ¼ 10V. Rearranging the formula to make t the subject gives:   nC t ¼ CR  ln Vs thus,

      10 t ¼ 0:47  ln ¼ 0:47  lnð0:5Þ 20

    or, t ¼  0:47 693 ¼ 0:325s In order to simplify the mathematics of exponential growth and decay, Table 1.15 provides an alternative tabular method that may be used to determine the voltage and current in a C-R circuit.

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    The Fundamentals

    115

    Table 1.15: Exponential growth and decay t/CR or t /(L/R)

    k (growth)

    k (decay)

    0.0

    0.0000

    1.0000

    0.1

    0.0951

    0.9048

    0.2

    0.1812

    0.8187 (1)

    0.3

    0.2591

    0.7408

    0.4

    0.3296

    0.6703

    0.5

    0.3935

    0.6065

    0.6

    0.4511

    0.5488

    0.7

    0.5034

    0.4965

    0.8

    0.5506

    0.4493

    0.9

    0.5934

    0.4065

    1.0

    0.6321

    0.3679

    1.5

    0.7769

    0.2231

    2.0

    0.8647 (2)

    0.1353

    2.5

    0.9179

    0.0821

    3.0

    0.9502

    0.0498

    3.5

    0.9698

    0.0302

    4.0

    0.9817

    0.0183

    4.5

    0.9889

    0.0111

    5.0

    0.9933

    0.0067

    Notes: (1) See Example 1.70 (2) See Example 1.74 k is the ratio of the value at time, t, to the final value (e.g., nc/Vs)

    Example 1.70 A 150 mF capacitor is charged to a potential of 150V. The capacitor is then removed from the charging source and connected to a 2 MO resistor. Determine the capacitor voltage 1 minute later.

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    116

    Chapter 1

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