Introduction to Electrical EngineeringMulukutla S. SarmaOXFORD UNIVERSITY PRESSI NTRODUCTI ON TOE L E CTRI CAL E NGI NE E RI NGthe oxford series in electrical and computer engineeringAdel S. Sedra, Series EditorAllen and Holberg, CMOS Analog Circuit DesignBobrow, Elementary Linear Circuit Analysis, 2nd EditionBobrow, Fundamentals of Electrical Engineering, 2nd EditionBurns and Roberts, Introduction to Mixed Signal IC Test and MeasurementCampbell, The Science and Engineering of Microelectronic FabricationChen, Analog & Digital Control System DesignChen, Digital Signal ProcessingChen, Linear System Theory and Design, 3rd EditionChen, System and Signal Analysis, 2nd EditionDeCarlo and Lin, Linear Circuit Analysis, 2nd EditionDimitrijev, Understanding Semiconductor DevicesFortney, Principles of Electronics: Analog & DigitalFranco, Electric Circuits FundamentalsGranzow, Digital Transmission LinesGuru and Hiziro glu, Electric Machinery and Transformers, 3rd EditionHoole and Hoole, A Modern Short Course in Engineering ElectromagneticsJones, Introduction to Optical Fiber Communication SystemsKrein, Elements of Power ElectronicsKuo, Digital Control Systems, 3rd EditionLathi, Modern Digital and Analog Communications Systems, 3rd EditionMartin, Digital Integrated Circuit DesignMcGillemand Cooper, Continuous and Discrete Signal and SystemAnalysis, 3rd EditionMiner, Lines and Electromagnetic Fields for EngineersRoberts and Sedra, SPICE, 2nd EditionRoulston, An Introduction to the Physics of Semiconductor DevicesSadiku, Elements of Electromagnetics, 3rd EditionSantina, Stubberud, and Hostetter, Digital Control System Design, 2nd EditionSarma, Introduction to Electrical EngineeringSchaumann and Van Valkenburg, Design of Analog FiltersSchwarz, Electromagnetics for EngineersSchwarz and Oldham, Electrical Engineering: An Introduction, 2nd EditionSedra and Smith, Microelectronic Circuits, 4th EditionStefani, Savant, Shahian, and Hostetter, Design of Feedback Control Systems, 3rd EditionVan Valkenburg, Analog Filter DesignWarner and Grung, Semiconductor Device ElectronicsWolovich, Automatic Control SystemsYariv, Optical Electronics in Modern Communications, 5th EditionINTRODUCTION TOELECTRICAL ENGINEERINGMulukutla S. SarmaNort heast ern Uni versi t yNew York OxfordOXFORD UNIVERSITY PRESS2001Oxford University PressOxford New YorkAthens Auckland Bangkok Bogot a Buenos Aires CalcuttaCape Town Chennai Dar es Salaam Delhi Florence Hong Kong IstanbulKarachi Kuala Lumpur Madrid Melbourne Mexico City MumbaiNairobi Paris S ao Paulo Shanghai Singapore Taipei Tokyo Toronto Warsawand associated companies inBerlin IbadanCopyright 2001 by Oxford University Press, Inc.Published by Oxford University Press, Inc.,198 Madison Avenue, New York, New York, 10016http://www.oup-usa.orgOxford is a registered trademark of Oxford University PressAll 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 permission of Oxford University Press.Library of Congress Cataloging-in-Publication DataSarma, Mulukutla S., 1938Introduction to electrical engineering / Mulukutla S. Sarmap. cm. (The Oxford series in electrical and computer engineering)ISBN 0-19-513604-7 (cloth)1. Electrical engineering. I. Title. II. Series.TK146.S18 2001621.3dc21 00-020033AcknowledgmentsTable 1.2.2 is adapted from Principles of Electrical Engineering (McGraw-Hill Series in Electrical Engineering), by Peyton Z.Peebles Jr. and Tayeb A. Giuma, reprinted with the permission of McGraw-Hill, 1991; gures 2.6.1, 2.6.2 are adapted from Getting Started withMATLAB 5: Quick Introduction, by Rudra Pratap, reprinted with the permission of Oxford University Press, 1998; gures 4.1.24.1.5, 4.2.14.2.3,4.3.14.3.2, are adapted from Electric Machines: Steady-State Theory and Dynamic Performance, Second Edition, by Mulukutla S. Sarma, reprintedwith the permission of Brooks/Cole Publishing, 1994; gure 4.6.1 is adapted fromMedical Instrumentation Application and Design, by John G. Webster,reprinted with the permission of John Wiley & Sons, Inc., 1978; table 4.6.1 is adapted from Electrical Safety in Industrial Plants, IEEE Spectrum, byRalph Lee, reprinted with the permission of IEEE, 1971; gure P5.3.1 is reprinted with the permission of Fairchild Semiconductor Corporation; gures5.6.1, 6.6.1, 9.5.1 are adapted fromElectrical Engineering: Principles and Applications, by Allen R. Hambley, reprinted with the permission of PrenticeHall, 1997; gure 10.5.1 is adapted from Power System Analysis and Design, Second Edition, by Duncan J. Glover and Mulukutla S. Sarma, reprintedwith the permission of Brooks/Cole Publishing, 1994; gures 11.1.2, 13.2.10 are adapted from Introduction to Electrical Engineering, Second Edition,by Clayton Paul, Syed A. Nasar, and Louis Unnewehr, reprinted with the permission of McGraw-Hill, 1992; gures E12.2.1(a,b), 12.2.212.2.5, 12.2.912.2.10, 12.3.112.3.3, 12.4.1, E12.4.1, P12.1.2, P12.4.3, P12.4.8, P12.4.12, 13.1.113.1.8, 13.2.113.2.9, 13.2.1113.2.16, 13.3.113.3.3, E13.3.2,13.3.4, E13.3.3, 13.3.513.3.6 are adapted from Electric Machines: Steady-State Theory and Dynamic Performance, Second Edition, by Mulukutla S.Sarma, reprinted with the permission of Brooks/Cole Publishing, 1994; gure 13.3.12 is adapted from Communication Systems Engineering, by John G.Proakis and Masoud Salehi, reprinted with the permission of Prentice Hall, 1994; gures 13.4.113.4.7, E13.4.1(b), 13.4.813.4.12, E13.4.3, 13.4.13,13.6.1 are adapted from Electric Machines: Steady-State Theory and Dynamic Performance, Second Edition, by Mulukutla S. Sarma Brooks/ColePublishing, 1994; gures 14.2.8, 14.2.9 are adapted from Electrical Engineering: Concepts and Applications, Second Edition, by A. Bruce Carlson andDavid Gisser, reprinted with the permission of Prentice Hall, 1990; gure 15.0.1 is adapted from Communication Systems, Third Edition, by A. BruceCarlson, reprinted with the permission of McGraw-Hill, 1986; gures 15.2.15, 15.2.31, 15.3.11 are adapted from Communication Systems Engineering,by John G. Proakis and Masoud Salehi, reprinted with the permission of Prentice Hall, 1994; gures 15.2.19, 15.2.27, 15.2.28, 15.2.30, 15.3.3, 15.3.4,15.3.9, 15.3.10, 15.3.20 are adapted from Principles of Electrical Engineering (McGraw-Hill Series in Electrical Engineering), by Peyton Z. PeeblesJr. and Tayeb A. Giuma, reprinted with the permission of McGraw-Hill, 1991; gures 16.1.116.1.3 are adapted from Electric Machines: Steady-StateTheory and Dynamic Performance, Second Edition, by Mulukutla S. Sarma, reprinted with the permission of Brooks/Cole Publishing, 1994; table16.1.3 is adapted from Electric Machines: Steady-State Theory and Dynamic Performance, Second Edition, by Mulukutla S. Sarma, reprinted with thepermission of Brooks/Cole Publishing, 1994; table 16.1.4 is adapted fromHandbook of Electric Machines, by S. A. Nasar, reprinted with the permissionof McGraw-Hill, 1987; and gures 16.1.413.1.9, E16.1.1, 16.1.1016.1.25 are adapted from Electric Machines: Steady-State Theory and DynamicPerformance, Second Edition, by Mulukutla S. Sarma, reprinted with the permission of Brooks/Cole Publishing, 1994.Printing (last digit): 10 9 8 7 6 5 4 3 2 1Printed in the United States of Americaon acid-free paperTo my grandchildrenPuja SreeSruthi LekhaPallavi Devi* * *and those to comeThis page intentionally left blank CONTENTSList of Case Studies and Computer-Aided Analysis xiiiPreface xvOverview xxiPART1 ELECTRICCIRCUITS1 Circuit Concepts 31.1 Electrical Quantities 41.2 Lumped-Circuit Elements 161.3 Kirchhoffs Laws 391.4 Meters and Measurements 471.5 Analogy between Electrical and Other Nonelectric Physical Systems 501.6 Learning Objectives 521.7 Practical Application: A Case StudyResistance Strain Gauge 52Problems 532 Circuit Analysis Techniques 662.1 Thvenin and Norton Equivalent Circuits 662.2 Node-Voltage and Mesh-Current Analyses 712.3 Superposition and Linearity 812.4 WyeDelta Transformation 832.5 Computer-Aided Circuit Analysis: SPICE 852.6 Computer-Aided Circuit Analysis: MATLAB 882.7 Learning Objectives 922.8 Practical Application: A Case StudyJump Starting a Car 92Problems 943 Time-Dependent Circuit Analysis 1023.1 Sinusoidal Steady-State Phasor Analysis 1033.2 Transients in Circuits 1253.3 Laplace Transform 1423.4 Frequency Response 154viiviii CONTENTS3.5 Computer-Aided Circuit Simulation for Transient Analysis, AC Analysis, andFrequency Response Using PSpice and PROBE 1683.6 Use of MATLAB in Computer-Aided Circuit Simulation 1733.7 Learning Objectives 1773.8 Practical Application: A Case StudyAutomotive Ignition System 178Problems 1794 Three-Phase Circuits and Residential Wiring 1984.1 Three-Phase Source Voltages and Phase Sequence 1984.2 Balanced Three-Phase Loads 2024.3 Measurement of Power 2084.4 Residential Wiring and Safety Considerations 2124.5 Learning Objectives 2154.6 Practical Application: A Case StudyPhysiological Effects of Current andElectrical Safety 216Problems 218PART2 ELECTRONICANALOGANDDIGITALSYSTEMS5 Analog Building Blocks and Operational Ampliers 2235.1 The Amplier Block 2245.2 Ideal Operational Amplier 2295.3 Practical Properties of Operational Ampliers 2355.4 Applications of Operational Ampliers 2445.5 Learning Objectives 2565.6 Practical Application: A Case StudyAutomotive Power-Assisted SteeringSystem 257Problems 2586 Digital Building Blocks and Computer Systems 2686.1 Digital Building Blocks 2716.2 Digital System Components 2956.3 Computer Systems 3166.4 Computer Networks 3206.5 Learning Objectives 3256.6 Practical Application: A Case StudyMicrocomputer-ControlledBreadmaking Machine 325Problems 3267 Semiconductor Devices 3397.1 Semiconductors 3397.2 Diodes 3407.3 Bipolar Junction Transistors 358CONTENTS ix7.4 Field-Effect Transistors 3677.5 Integrated Circuits 3787.6 Learning Objectives 3797.7 Practical Application: A Case StudyElectronic Photo Flash 380Problems 3808 Transistor Ampliers 3938.1 Biasing the BJT 3948.2 Biasing the FET 3958.3 BJT Ampliers 3998.4 FET Ampliers 4058.5 Frequency Response of Ampliers 4098.6 Learning Objectives 4148.7 Practical Application: A Case StudyMechatronics: Electronics Integratedwith Mechanical Systems 414Problems 4159 Digital Circuits 4229.1 Transistor Switches 4239.2 DTL and TTL Logic Circuits 4279.3 CMOS and Other Logic Families 4319.4 Learning Objectives 4379.5 Practical Application: A Case StudyCardiac Pacemaker, a BiomedicalEngineering Application 438Problems 439PART3 ENERGYSYSTEMS10 AC Power Systems 45110.1 Introduction to Power Systems 45210.2 Single- and Three-Phase Systems 45510.3 Power Transmission and Distribution 46010.4 Learning Objectives 46610.5 Practical Application: A Case StudyThe Great Blackout of 1965 466Problems 46811 Magnetic Circuits and Transformers 47111.1 Magnetic Materials 47211.2 Magnetic Circuits 47511.3 Transformer Equivalent Circuits 47911.4 Transformer Performance 48611.5 Three-Phase Transformers 49011.6 Autotransformers 492x CONTENTS11.7 Learning Objectives 49411.8 Practical Application: A Case StudyMagnetic Bearings for SpaceTechnology 494Problems 49512 Electromechanics 50512.1 Basic Principles of Electromechanical Energy Conversion 50512.2 EMF Produced by Windings 51412.3 Rotating Magnetic Fields 52212.4 Forces and Torques in Magnetic-Field Systems 52612.5 Basic Aspects of Electromechanical Energy Converters 53912.6 Learning Objectives 54012.7 Practical Application: A Case StudySensors or Transducers 541Problems 54213 Rotating Machines 55313.1 Elementary Concepts of Rotating Machines 55313.2 Induction Machines 56313.3 Synchronous Machines 58213.4 Direct-Current Machines 59413.5 Learning Objectives 61013.6 Practical Application: A Case StudyWind-Energy-ConversionSystems 610Problems 612PART4 INFORMATIONSYSTEMS14 Signal Processing 62514.1 Signals and Spectral Analysis 62614.2 Modulation, Sampling, and Multiplexing 64014.3 Interference and Noise 64914.4 Learning Objectives 65814.5 Practical Application: A Case StudyAntinoise Systems, NoiseCancellation 658Problems 65915 Communication Systems 66615.1 Waves, Transmission Lines, Waveguides, and Antenna Fundamentals 67015.2 Analog Communication Systems 68515.3 Digital Communication Systems 71015.4 Learning Objectives 73015.5 Practical Application: A Case StudyGlobal Positioning Systems 731Problems 732CONTENTS xiPART5 CONTROLSYSTEMS16 Basic Control Systems 74716.1 Power Semiconductor-Controlled Drives 74816.2 Feedback Control Systems 77916.3 Digital Control Systems 80516.4 Learning Objectives 81416.5 Practical Application: A Case StudyDigital Process Control 815Problems 816Appendix A: References 831Appendix B: Brief Review of Fundamentals of Engineering(FE) Examination 833Appendix C: Technical Terms, Units, Constants, and ConversionFactors for the SI System 835Appendix D: Mathematical Relations 838Appendix E: Solution of Simultaneous Equations 843Appendix F: Complex Numbers 846Appendix G: Fourier Series 847Appendix H: Laplace Transforms 851Index 855This page intentionally left blank LIST OF CASE STUDIES ANDCOMPUTER-AIDED ANALYSISCase Studies1.7 Practical Application: A Case StudyResistance Strain Gauge 522.8 Practical Application: A Case StudyJump Starting a Car 923.8 Practical Application: A Case StudyAutomotive Ignition System 1784.6 Practical Application: A Case StudyPhysiological Effects of Current and Electrical Safety2165.6 Practical Application: A Case StudyAutomotive Power-Assisted Steering System 2576.6 Practical Application: A Case StudyMicrocomputer-ControlledBreadmaking Machine 3257.7 Practical Application: A Case StudyElectronic Photo Flash 3808.7 Practical Application: A Case StudyMechatronics: Electronics Integrated with MechanicalSystems 4149.5 Practical Application: A Case StudyCardiac Pacemaker, a Biomedical EngineeringApplication 43810.5 Practical Application: A Case StudyThe Great Blackout of 1965 46611.8 Practical Application: A Case StudyMagnetic Bearings for Space Technology 49412.7 Practical Application: A Case StudySensors or Transducers 54113.6 Practical Application: A Case StudyWind-Energy-Conversion Systems 61014.5 Practical Application: A Case StudyAntinoise Systems, Noise Cancellation 65815.5 Practical Application: A Case StudyGlobal Positioning Systems 73116.5 Practical Application: A Case StudyDigital Process Control 815Computer-Aided Analysis2.5 Computer-Aided Circuit Analysis: SPICE 852.6 Computer-Aided Circuit Analysis: MATLAB 883.5 Computer-Aided Circuit Simulation for Transient Analysis, AC Analysis, and FrequencyResponse Using PSpice and PROBE 1683.6 Use of MATLAB in Computer-Aided Circuit Simulation 173xiiiThis page intentionally left blank PREFACEI. OBJECTIVESThe purpose of this text is to present a problem-oriented introductory survey text for the ex-traordinarily interesting electrical engineering discipline by arousing student enthusiasm whileaddressing the underlying concepts and methods behind various applications ranging from con-sumer gadgets and biomedical electronics to sophisticated instrumentation systems, computers,and multifarious electric machinery. The focus is on acquainting students majoring in all branchesof engineering and science, especially in courses for nonelectrical engineering majors, with thenature of the subject and the potentialities of its techniques, while emphasizing the principles.Since principles and concepts are most effectively taught by means of a problem-oriented course,judicially selected topics are treated in sufcient depth so as to permit the assignment of adequatelychallenging problems, which tend to implant the relevant principles in students minds.Inadditiontoanacademic-year (twosemestersor threequarters) introductorycoursetraditionally offered to non-EE majors, the text is also suitable for a sophomore survey coursegiven nowadays to electrical engineering majors in a number of universities. At a more rapid paceor through selectivity of topics, the introductory course could be offered in one semester to eitherelectrical and computer engineering (ECE) or non-EE undergraduate majors. Although this bookis written primarily for non-EE students, it is hoped that it will be of value to undergraduate ECEstudents (particularly for those who wish to take the Fundamentals of Engineering examination,whichisaprerequisiteforbecominglicensedasaProfessionalEngineer), tograduateECEstudentsfortheirreviewinpreparingforqualifyingexaminations, tomeet thecontinuing-education needs of various professionals, and to serve as a reference text even after graduation.II. MOTIVATIONThis text is but a modest attempt to provide an exciting survey of topics inherent to the electricaland computer engineering discipline. Modern technology demands a team approach in whichelectrical engineers and nonelectrical engineers have to work together sharing a common technicalvocabulary. Nonelectrical engineers must be introduced to the language of electrical engineers,just as the electrical engineers have to be sensitized to the relevance of nonelectrical topics.The dilemma of whether electrical engineering and computer engineering should be separatecourses of study, leading to distinctive degrees, seems to be happily resolving itself in the directionof togetherness. After all, computers are not only pervasive tools for engineers but also theirproduct; hence there is a pressing need to weave together the fundamentals of both the electricaland the computer engineering areas into the new curricula.Analmost total lackof contact betweenfreshmenandsophomore students andthe Departmentof Electrical and Computer Engineering, as well as little or no exposure to electrical and computerxvxvi PREFACEengineering, seems to drive even the academically gifted students away from the program. Aninitial spark that may have motivated them to pursue electrical and computer engineering has tobe nurtured in the early stages of their university education, thereby providing an inspiration tocontinue.This text is based on almost 40 years of experience teaching a wide variety of courses toelectrical as well as non-EE majors and, more particularly, on the need to answer many of thequestions raisedbysomanyof mystudents. I have always enjoyedengineering(teaching, research,and consultation); I earnestly hope that the readers will have as much fun and excitement in usingthis book as I have had in developing it.III. PREREQUISITES AND BACKGROUNDThestudent will beassumedtohavecompletedthebasiccollege-level coursesinalgebra,trigonometry, basicphysics, andelementarycalculus. Aknowledgeofdifferentialequationsis helpful, but not mandatory. For a quick reference, some useful topics are included in theappendixes.IV. ORGANIZATION AND FLEXIBILITYThe text is developed to be student-oriented, comprehensive, and up to date on the subject withnecessary and sufcient detailed explanation at the level for which it is intended. The key wordin the organization of the text is exibility.Thebookisdividedintovepartsinorder toprovideexibilityinmeetingdifferentcircumstances, needs, and desires. Aglance at the Table of Contents will showthat Part 1 concernsitself with basic electric circuits, in which circuit concepts, analysis techniques, time-dependentanalysisincludingtransients, aswell asthree-phasecircuitsarecovered. Part 2dealswithelectronic analog and digital systems, in which analog and digital building blocks are consideredalong with operational ampliers, semiconductor devices, integrated circuits, and digital circuits.Part3isdevotedtoenergysystems, inwhichacpowersystems, magneticcircuitsandtransformers, principles of electromechanics, and rotating machines causing electromechanicalenergy conversion are presented. Part 4 deals with information systems, including the underlyingprinciples of signal processing and communication systems. Finally, Part 5 presents control sys-tems, which include the concepts of feedback control, digital control, and power semiconductor-controlled drives.The text material is organized for optimum exibility, so that certain topics may be omittedwithout loss of continuity when lack of time or interest dictates.V. FEATURES1. The readability of the text and the level of presentation, from the students viewpoint,are given utmost priority. The quantity of subject matter, range of difculty, coverage of topics,numerous illustrations, a large number of comprehensive worked-out examples, and a variety ofend-of-chapter problems are given due consideration, to ensure that engineering is not a plug-inor cookbook profession, but one in which reasoning and creativity are of the highest importance.2. Fundamental physical concepts, which underlie creative engineering and become the mostvaluable and permanent part of a students background, have been highlighted while giving dueattention to mathematical techniques. So as to accomplish this in a relatively short time, muchthought has gone into rationalizing the theory and conveying in a concise manner the essentialdetails concerning the nature of electrical and computer engineering. With a good groundingPREFACE xviiin basic concepts, a very wide range of engineering systems can be understood, analyzed, anddevised.3. The theory has been developed fromsimple beginnings in such a manner that it can readilybe extended to new and more complicated situations. The art of reducing a practical device to anappropriate mathematical model and recognizing its limitations has been adequately presented.Sufcient motivation is provided for the student to develop interest in the analytical proceduresto be applied and to realize that all models, being approximate representations of reality, shouldbe no more complicated than necessary for the application at hand.4. Sincetheessenceofengineeringisthedesignofproductsusefultosociety, theendobjective of each phase of preparatory study should be to increase the students capability todesign practical devices and systems to meet the needs of society. Toward that end, the studentwill be motivated to go through the sequence of understanding physical principles, processes,modeling, using analytical techniques, and, nally, designing.5. Engineers habitually break systems up into their component blocks for ease of under-standing. The building-block approach has been emphasized, particularly in Part II concerninganalog and digital systems. For a designer using IC blocks in assembling the desired systems,the primary concern lies with their terminal characteristics while the internal construction of theblocks is of only secondary importance.6. Considering the world of electronics today, both analog and digital technologies are givenappropriate coverage. Since students are naturally interested in such things as op amps, integratedcircuits, and microprocessors, modern topics that can be of great use in their career are emphasizedin this text, thereby motivating the students further.7. The electrical engineering profession focuses on information and energy, which are thetwo critical commodities of any modern society. In order to bring the message to the forefront forthe students attention, Parts III, IV, and V are dedicated to energy systems, information systems,and control systems, respectively. However, some of the material in Parts I and II is critical to theunderstanding of the latter.Anunderstandingoftheprinciplesofenergyconversion, electricmachines, andenergysystems is important for all in order to solve the problems of energy, pollution, and poverty thatface humanity today. It can be well argued that todays non-EEs are more likely to encounterelectromechanical machines than some of the ECEs. Thus, it becomes essential to have sufcientbreadth and depth in the study of electric machines by the non-ECEs.Information systems have been responsible for the spectacular achievements in communica-tion in recent decades. Concepts of control systems, which are not limited to any particular branchof engineering, are very useful to every engineer involved in the understanding of the dynamicsof various types of systems.8. Consistent with modern practice, the international (SI) system of units has been usedthroughout the text. In addition, a review of units, constants and conversion factors for the SIsystem can be found in Appendix C.9. While solid-state electronics, automatic control, IC technology, and digital systems havebecome commonplace in the modern EE profession, some of the older, more traditional topics,such as electric machinery, power, and instrumentation, continue to form an integral part of thecurriculum, as well as of the profession in real life. Due attention is accorded in this text to suchtopics as three-phase circuits and energy systems.10. Appendixes provide useful information for quick reference on selected bibliography forsupplementary reading, the SI system, mathematical relations, as well as a brief review of theFundamentals of Engineering (FE) examination.xviii PREFACE11. Engineers whoacquire a basic knowledge of electric circuits, electronic analoganddigitalcircuits, energy systems, information systems, and control systems will have a well-roundedbackground and be better prepared to join a team effort in analyzing and designing systems.Therein lies the justication for the Table of Contents and the organization of this text.12. At the end of each chapter, the learning objectives of that chapter are listed so that thestudent can check whether he or she has accomplished each of the goals.13. At the very end of each chapter, Practical Application: A Case Study has been includedso that the reader can get motivated and excited about the subject matter and its relevance topractice.14. Basic material introduced in this book is totally independent of any software that mayaccompany the usage of this book, and/or the laboratory associated with the course. The commonsoftware in usage, as of writing this book, consists of Windows, Word Perfect, PSPICE, MathCAD, and MATLAB. There are also other popular specialized simulation programs such as SignalProcessing Workstation (SPW) in the area of analog and digital communications, Very HighLevel Description Language (VHDL) in the area of digital systems, Electromagnetic TransientsProgram(EMTP)intheeldofpower, andSIMULINKintheeldofcontrol. Inpractice,however, anycombinationof software that satises the needfor wordprocessing, graphics, editing,mathematical analysis, and analog as well as digital circuit analysis should be satisfactory.Inordertointegratecomputer-aidedcircuit analysis, twotypesofprogramshavebeenintroduced in this text: A circuit simulator PSpice and a math solver MATLAB. Our purposehere is not to teach students how to use specic software packages, but to help them developan analysis style that includes the intelligent use of computer tools. After all, these tools arean intrinsic part of the engineering environment, which can signicantly enhance the studentsunderstanding of circuit phenomena.15. The basics, to which the reader is exposed in this text, will help him or her to selectconsultantsexperts in specic areaseither in or out of house, who will provide the knowledgeto solve a confronted problem. After all, no one can be expected to be an expert in all areasdiscussed in this text!VI. PEDAGOGYA. OutlineBeyond the overview meant as an orientation, the text is basically divided into ve parts.Part 1: Electric Circuits This part provides the basic circuit-analysis concepts and tech-niques that will be used throughout the subsequent parts of the text. Three-phase circuits havebeen introduced to develop the background needed for analyzing ac power systems. Basic notionsof residential circuit wiring, including grounding and safety considerations, are presented.Part 2: Electronic Analog and Digital Systems With the background of Part I, the studentis then directed to analog and digital building blocks. Operational ampliers are discussed as anespeciallyimportant special case. After introducingdigital systemcomponents, computer systems,and networks to the students, semiconductor devices, integrated circuits, transistor ampliers, aswell as digital circuits are presented. The discussion of device physics is kept to the necessaryminimum, while emphasis is placed on obtaining powerful results from simple tools placed instudents hands and minds.Part 3: Energy Systems With the background built on three-phase circuits in Part I, acpower systems are considered. Magnetic circuits and transformers are then presented, before thestudent is introduced to the principles of electromechanics and practical rotating machines thatachieve electromechanical energy conversion.PREFACE xixPart 4: Information Systems Signal processing and communication systems (both analogand digital) are discussed using the block diagrams of systems engineering.Part 5: Control Systems By focusing on control aspects, this part brings together thetechniques and concepts of the previous parts in the design of systems to accomplish specictasks. A section on power semiconductor-controlled drives is included in view of their recentimportance. The basic concepts of feedback control systems are introduced, and nally the avorof digital control systems is added.Appendices The appendices provide ready-to-use information:Appendix A: Selected bibliography for supplementary readingAppendix B: Brief review of fundamentals of engineering (FE) examinationAppendix C: Technical terms, units, constants, and conversion factors for the SI systemAppendix D: Mathematical relations (used in the text)Appendix E: Solution of simultaneous equationsAppendix F: Complex numbersAppendix G: Fourier seriesAppendix H: Laplace transformsB. Chapter IntroductionsEach chapter is introduced to the student stating the objective clearly, giving a sense of whatto expect, and motivating the student with enough information to look forward to reading thechapter.C. Chapter EndingsAt the end of each chapter, the learning objectives of that chapter are listed so that the studentcan check whether he or she has accomplished each of the goals.In order to motivate and excite the student, practical applications using electrical engineeringprinciples are included. At the very end of each chapter, a relevant Practical Application: A CaseStudy is presented.D. IllustrationsA large number of illustrations support the subject matter with the intent to motivate the studentto pursue the topics further.E. ExamplesNumerous comprehensive examples are worked out in detail in the text, covering most of thetheoretical points raised. An appropriate difculty is chosen and sufcient stimulation is built into go on to more challenging situations.F. End-of-Chapter ProblemsA good number of problems (identied with each section of every chapter), with properly gradedlevelsofdifculty, areincludedat theendofeachchapter, therebyallowingtheinstructorconsiderable exibility. There are nearly a thousand problems in the book.G. Preparation for the FE ExamA brief review of the Fundamentals of Engineering (FE) examination is presented in AppendixB in order to aid the student who is preparing to take the FE examination in view of becoming aregistered Professional Engineer (PE).VII. SUPPLEMENTSA Solutions Manual to Accompany Introduction to Electrical Engineering, by M.S. Sarma(ISBN 019-514260-8), with complete detailed solutions (provided by the author) for all problemsin the book is available to adopters.xx PREFACEMicroSoft PowerPoint Overheads to Accompany Introduction to Electrical Engineering(ISBN 019-514472-1) are free to adopters. Over 300 text gures and captions are available forclassroom projection use.A web-site, MSSARMA.org, will include interesting web links and enhancement materials,errata, a forum to communicate with the author, and more.A CD-ROM Disk is packaged with each new book. The CD contains: Complete Solutions for Students to 20% of the problems. These solutions have beenprepared by the author and are resident on the disk in Adobe Acrobat (.pdf) format. Theproblems with solutions on disk are marked with an asterisk next to the problem in the text. ThedemonstrationversionofElectronicsWorkbenchMultisimVersion6, anin-novativeteachingandlearningsoftwareproduct that isusedtobuildcircuitsandtosimulateandanalyzetheirelectricalbehavior. Thisdemonstrationversionincludes20demo circuit les built from circuit examples from this textbook. The CD also includesanother 80 circuits from the text that can be opened with the full student or educationalversions of Multisim. These full versions can be obtained from Electronics Workbench atwww.electronicsworkbench.com.To extend the introduction to selected topics and provide additional practice, we recommendthe following additional items: Circuits: Allans Circuits Problems by Allan Kraus (ISBN 019-514248-9), which includesover 400 circuit analysis problems with complete solutions, many in MATLAB and SPICEform. Electronics: KCs Problems and Solutions to Accompany Microelectronic Circuits by K.C.Smith (019511771-9), which includes over 400 electronics problems and their completesolutions. SPICE: SPICE by Gordon Roberts and Adel Sedra (ISBN019-510842-6) features over 100examples and numerous exercises for computer-aided analysis of microelectronic circuits. MATLAB: Getting Started with MATLAB by Rudra Pratap (ISBN 019-512947-4) providesa quick introduction to using this powerful software tool.For more information or to order an examination copy of the above mentioned supplementscontact Oxford University Press at college@oup-usa.org.VIII. ACKNOWLEDGMENTSThe author would like to thank the many people who helped bring this project to fruition. Anumber of reviewers greatly improved this text through their thoughtful comments and usefulsuggestions.I am indebted to my editor, Peter C. Gordon, of Oxford University Press, who initiatedthisproject andcontinuedhissupport withskilledguidance, helpful suggestions, andgreatencouragement. The people at Oxford University Press, in particular, Senior Project Editor KarenShapiro, have been most helpful in this undertaking. My sincere thanks are also due to Mrs. SallyGupta, who did a superb job typing most of the manuscript.I would also like to thank my wife, Savitri, for her continued encouragement and support,without which this project could not have been completed. It is with great pleasure and joy that Idedicate this work to my grandchildren.Mulukutla S. SarmaNortheastern UniversityOVERVIEWWhat is electrical engineering? What is the scope of electrical engineering?Toanswertherst questioninasimpleway, electrical engineeringdealsmainlywithinformation systems and with power and energy systems. In the former, electrical means areused to transmit, store, and process information; while in the latter, bulk energy is transmittedfrom one place to another and power is converted from one form to another.The second question is best answered by taking a look at the variety of periodicals publishedby the Institute of Electrical and Electronics Engineers (IEEE), which is the largest technicalsociety in the world with over 320,000 members in more than 140 countries worldwide. Table Ilists 75 IEEE Society/Council periodicals along with three broad-scope publications.The transactions and journals of the IEEE may be classied into broad categories of devices,circuits, electronics, computers, systems, andinterdisciplinaryareas. All areasof electricalengineering require a working knowledge of physics and mathematics, as well as engineeringmethodologies and supporting skills in communications and human relations. A closely relatedeld is that of computer science.Obviously, one cannot deal with all aspects of all of these areas. Instead, the general conceptsand techniques will be emphasized in order to provide the reader with the necessary backgroundneeded to pursue specic topics in more detail. The purpose of this text is to present the basictheory and practice of electrical engineering to students with varied backgrounds and interests.After all, electrical engineering rests upon a few major principles and subprinciples.Some of the areas of major concern and activity in the present society, as of writing thisbook, are: Protecting the environment Energy conservation Alternative energy sources Development of new materials Biotechnology Improved communications Computer codes and networking Expert systemsThis text is but a modest introduction to the exciting eld of electrical engineering. However,it is the ardent hope and fervent desire of the author that the book will help inspire the readerto apply the basic principles presented here to many of the interdisciplinary challenges, some ofwhich are mentioned above.xxixxii OVERVIEWTABLE I IEEE PublicationsPublication Pub IDIEEE Society/Council PeriodicalsAerospace & Electronic Systems Magazine 3161Aerospace & Electronic Systems, Transactions on 1111Annals of the History of Computing 3211Antennas & Propagation, Transactions on 1041Applied Superconductivity, Transactions on 1521Automatic Control, Transactions on 1231Biomedical Engineering, Transactions on 1191Broadcasting, Transactions on 1011Circuits and Devices Magazine 3131Circuits & Systems, Part I, Transactions on 1561Circuits & Systems, Part II, Transactions on 1571Circuits & Systems for Video Technology, Transactions on 1531Communications, Transactions on 1201Communications Magazine 3021Components, Hybrids, & Manufacturing Technology, Transactions on 1221Computer Graphics & Applications Magazine 3061Computer Magazine 3001Computers, Transactions on 1161Computer-Aided Design of Integrated Circuits and Systems, Transactions on 1391Consumer Electronics, Transactions on 1021Design & Test of Computers Magazine 3111Education, Transactions on 1241Electrical Insulation, Transactions on 1301Electrical Insulation Magazine 3141Electromagnetic Compatibility, Transactions on 1261Electron Device Letters 3041Electron Devices, Transactions on 1151Electronic Materials, Journal of 4601Energy Conversion, Transactions on 1421Engineering in Medicine & Biology Magazine 3091Engineering Management, Transactions on 1141Engineering Management Review 3011Expert Magazine 3151Geoscience & Remote Sensing, Transactions on 1281Image Processing, Transactions on 1551Industrial Electronics, Transactions on 1131Industry Applications, Transactions on 1321Information Theory, Transactions on 1121Instrumentation & Measurement, Transactions on 1101Knowledge & Data Engineering, Transactions on 1471Lightwave Technology, Journal of 4301LTS (The Magazine of Lightwave Telecommunication Systems) 3191Magnetics, Transactions on 1311Medical Imaging, Transactions on 1381Micro Magazine 3071Microelectromechanical Systems, Journal of 4701Microwave and Guided Wave Letters 1511Microwave Theory & Techniques, Transactions on 1181Network Magazine 3171Neural Networks, Transactions on 1491Nuclear Science, Transactions on 1061Oceanic Engineering, Journal of 4201Parallel & Distributed Systems, Transactions on 1501Pattern Analysis & Machine Intelligence, Transactions on 1351Photonics Technology Letters 1481Plasma Science, Transactions on 1071Power Delivery, Transactions on 1431ContinuedOVERVIEW xxiiiTABLE I ContinuedPublication Pub IDPower Electronics, Transactions on 4501Power Engineering Review 3081Power Systems, Transactions on 1441Professional Communication, Transactions on 1251Quantum Electronics, Journal of 1341Reliability, Transactions on 1091Robotics & Automation, Transactions on 1461Selected Areas in Communication, Journal of 1411Semiconductor Manufacturing, Transactions on 1451Signal Processing, Transactions on 1001Signal Processing Magazine 3101Software Engineering, Transactions on 1171Software Magazine 3121Solid-State Circuits, Journal of 4101Systems, Man, & Cybernetics, Transactions on 1271Technology & Society Magazine 1401Ultrasonics, Ferroelectrics & Frequency Control, Transactions on 1211Vehicular Technology, Transactions on 1081Broad Scope PublicationsIEEE Spectrum 5001Proceedings of the IEEE 5011IEEE Potentials 5061Ahistorical perspective of electrical engineering, in chronological order, is furnished in TableII. A mere glance will thrill anyone, and give an idea of the ever-changing, fast-growing eld ofelectrical engineering.TABLE II Chronological Historical Perspective of Electrical Engineering17501850 Coulombs law (1785)Battery discovery by VoltaMathematical theories by Fourier and LaplaceAmperes law (1825)Ohms law (1827)Faradays law of induction (1831)18501900 Kirchhoffs circuit laws (1857)Telegraphy: rst transatlantic cables laidMaxwells equations (1864)Cathode rays: Hittorf and Crookes (1869)Telephony: rst telephone exchange in New Haven, ConnecticutEdison opens rst electric utility in New York City (1882): dc power systemsWaterwheel-driven dc generator installed in Appleton, Wisconsin (1882)First transmission lines installed in Germany (1882), 2400 V dc, 59kmDc motor by Sprague (1884)Commercially practical transformer by Stanley (1885)Steinmetzs ac circuit analysisTeslas papers on ac motors (1888)Radio waves: Hertz (1888)First single-phase ac transmission line in United States (1889): Ac power systems, Oregon City toPortland, 4 kV, 21 kmFirst three-phase ac transmission line in Germany (1891), 12 kV, 179 kmFirst three-phase ac transmission line in California (1893), 2.3 kV, 12 kmGenerators installed at Niagara Falls, New YorkHeavisides operational calculus methodsxxiv OVERVIEW19001920 Marconis wireless telegraph system: transatlantic communication (1901)Photoelectric effect: Einstein (1904)Vacuum-tube electronics: Fleming (1904), DeForest (1906)First AM broadcasting station in Pittsburgh, PennsylvaniaRegenerative amplier: Armstrong (1912)19201940 Television: Farnsworth, Zworykin (1924)Cathode-ray tubes by DuMont; experimental broadcastingNegative-feedback amplier by Black (1927)Boolean-algebra application to switching circuits by Shannon (1937)19401950 Major advances in electronics (World War II)Radar and microwave systems: Watson-Watts (1940)Operational ampliers in analog computersFM communication systems for military applicationsSystem theory papers by Bode, Shannon, and WienerENIAC vacuum-tube digital computer at the University of Pennsylvania (1946)Transistor electronics: Shockley, Bardeen, and Brattain of Bell Labs (1947)Long-playing microgroove records (1948)19501960 Transistor radios in mass productionSolar cell: Pearson (1954)Digital computers (UNIVAC I, IBM, Philco); Fortran programming languageFirst commercial nuclear power plant at Shippingport, Pennsylvania (1957)Integrated circuits by Kilby of Texas Instruments (1958)19601970 Microelectronics: Hoernis planar transistor from Fairchild SemiconductorsLaser demonstrations by Maiman (1960)First communications satellite Telstar I launched (1962)MOS transistor: Hofstein and Heiman (1963)Digital communications765 kV AC power lines constructed (1969)Microprocessor: Hoff (1969)19701980 Microcomputers; MOS technology; Hewlett-Packard calculatorINTELs 8080 microprocessor chip; semiconductor devices for memoryComputer-aided design and manufacturing (CAD/CAM)Interactive computer graphics; software engineeringPersonal computers; IBM PCArticial intelligence; roboticsFiber optics; biomedical electronic instruments; power electronics1980Present Digital electronics; superconductorsNeural networks; expert systemsHigh-density memory chips; digital networksI NTRODUCTI ON TOE L E CTRI CAL E NGI NE E RI NGThis page intentionally left blank PARTELECTRICAL CIRCUITSONEThis page intentionally left blank 1Circuit Concepts1.1 Electrical Quantities1.2 Lumped-Circuit Elements1.3 Kirchhoffs Laws1.4 Meters and Measurements1.5 Analogy between Electrical and Other Nonelectric Physical Systems1.6 Learning Objectives1.7 Practical Application: A Case StudyResistance Strain GaugeProblemsElectriccircuits, whicharecollectionsofcircuit elementsconnectedtogether, arethemostfundamental structures of electrical engineering. A circuit is an interconnection of simple elec-trical devices that have at least one closed path in which current may ow. However, we mayhave to clarify to some of our readers what is meant by current and electrical device, atask that we shall undertake shortly. Circuits are important in electrical engineering becausethey process electrical signals, which carry energy and information; a signal can be any time-varying electrical quantity. Engineering circuit analysis is a mathematical study of some usefulinterconnection of simple electrical devices. An electric circuit, as discussed in this book, isanidealizedmathematicalmodel ofsomephysicalcircuitorphenomenon. Theidealcircuitelements are the resistor, the inductor, the capacitor, and the voltage and current sources. Theideal circuit model helps us to predict, mathematically, the approximate behavior of the actualevent. The models also provide insights into how to design a physical electric circuit to perform adesired task. Electrical engineering is concerned with the analysis and design of electric circuits,systems, and devices. In Chapter 1 we shall deal with the fundamental concepts that underlieall circuits.Electricalquantitieswillbeintroducedrst. Thenthereaderisdirectedtothelumped-circuit elements. Then Ohms law and Kirchhoffs laws are presented. These laws are sufcient34 CIRCUIT CONCEPTSforanalyzinganddesigningsimplebutillustrativepracticalcircuits. Later, abriefintroduc-tion is given to meters and measurements. Finally, the analogy between electrical and othernonelectric physical systems is pointed out. The chapter ends with a case study of practicalapplication.1.1 ELECTRICAL QUANTITIESIndescribingthe operationof electric circuits, one shouldbe familiar withsuchelectrical quantitiesas charge, current, and voltage. The material of this section will serve as a review, since it willnot be entirely new to most readers.Charge and Electric ForceThe proton has a charge of 1.602 1019coulombs (C), while the electron has a charge of1.6021019C. The neutron has zero charge. Electric charge and, more so, its movementare the most basic items of interest in electrical engineering. When many charged particles arecollected together, larger charges and charge distributions occur. There may be point charges (C),line charges (C/m), surface charge distributions (C/m2), and volume charge distributions (C/m3).A charge is responsible for an electric eldand charges exert forces on each other. Likecharges repel, whereas unlike charges attract. Such an electric force can be controlled and utilizedfor some useful purpose. Coulombs law gives an expression to evaluate the electric force innewtons (N) exerted on one point charge by the other:Force on Q1 due to Q2=F21 =Q1Q240R2 a21(1.1.1a)Force on Q2 due to Q1 =F12 =Q2Q140R2 a12(1.1.1b)where Q1 and Q2 are the point charges (C); R is the separation in meters (m) between them; 0is the permittivity of the free-space medium with units of C2/N m or, more commonly, faradsper meter (F/m); and a21 and a12 are unit vectors along the line joining Q1 and Q2, as shown inFigure 1.1.1.Equation (1.1.1) shows the following:1. ForcesF21 andF12 are experienced byQ1 andQ2, due to the presence ofQ2 andQ1,respectively. They are equal in magnitude and opposite of each other in direction.2. The magnitude of the force is proportional to the product of the charge magnitudes.3. The magnitude of the force is inversely proportional to the square of the distance betweenthe charges.4. The magnitude of the force depends on the medium.5. The direction of the force is along the line joining the charges.Note that the SI system of units will be used throughout this text, and the student should beconversant with the conversion factors for the SI system.The force per unit charge experienced by a small test charge placed in an electric eld isknown as the electric eld intensity E, whose units are given by N/C or, more commonly, voltsper meter (V/m),E =limQ0FQ(1.1.2)1.1 ELECTRICAL QUANTITIES 5RQ1Q2a21a12F21F12Figure 1.1.1Illustration of Coulombs law.Equation (1.1.2) is the dening equation for the electric eld intensity (with units of N/C or V/m),irrespective of the source of the electric eld. One may then conclude:F21 = Q1 E2(1.1.3a)F12 = Q2 E1(1.1.3b)whereE2 is the electric eld due to Q2 at the location of Q1, andE1 is the electric eld due toQ1 at the location of Q2, given byE2 =Q240R2 a21(1.1.4a)E1 =Q140R2 a12(1.1.4b)Note that the electric eld intensity due to a positive point charge is directed everywhereradially away from the point charge, and its constant-magnitude surfaces are spherical surfacescentered at the point charge.EXAMPLE 1.1.1(a) A small region of an impure silicon crystal with dimensions 1.25106m 103m103m has only the ions (with charge 1.6 1019C) present with a volume density of1025/m3. The rest of the crystal volume contains equal densities of electrons (with charge1.6 1019C) and positive ions. Find the net total charge of the crystal.(b)Consider the charge of part (a) as a point charge Q1. Determine the force exerted by thison a charge Q2 = 3C when the charges are separated by a distance of 2 m in free space,as shown in Figure E1.1.1.Q3 = 2 106 CF12F2F32Q2 = 3 106 C2 m1 m76yxQ1 = 2 106 C+ +Figure E1.1.16 CIRCUIT CONCEPTS(c) If another charge Q3 = 2C is added to the system 1 m above Q2, as shown in FigureE1.1.1, calculate the force exerted on Q2.S ol ut i on(a) In the region where both ions and free electrons exist, their opposite charges cancel, andthe net charge density is zero. From the region containing ions only, the volume-chargedensity is given by = (1025)(1.6 1019) = 1.6 106C/m3The net total charge is then calculated as:Q = v = (1.6 106)(1.25 106103103) = 2 106C(b)The rectangular coordinate system shown denes the locations of the charges: Q1 =2106C; Q2 = 3106C. The force that Q1 exerts on Q2 is in the positive directionof x, given by Equation (1.1.1),F12 =(3 106)(2 106)4(109/36)22 ax = ax 13.5 103NThis is the force experienced by Q2 due to the effect of the electric eld of Q1. Note thevalue used for free-space permittivity, 0, as (8.8541012), or approximately 109/36F/m. ax is the unit vector in the positive x-direction.(c) When Q3 is added to the system, as shown in Figure E1.1.1, an additional force on Q2directed in the positive y-direction occurs (since Q3 and Q2 are of opposite sign),F32 =(3 106)(2 106)4(10936)12( ay) = ay 54 103NThe resultant forceF2 acting onQ2 is the superposition ofF12 andF32 due toQ1 andQ3, respectively.The vector combination of F12 and F32 is given by:F2 =

F212F232 , tan1F32F12=

13 .52542103, tan15413 .5= 55.7 103,76 NConductors and InsulatorsIn order to put charge in motion so that it becomes an electric current, one must provide a paththrough which it can ow easily by the movement of electrons. Materials through which chargeows readily are called conductors. Examples include most metals, such as silver, gold, copper,and aluminum. Copper is used extensively for the conductive paths on electric circuit boards andfor the fabrication of electrical wires.1.1 ELECTRICAL QUANTITIES 7Insulators are materials that do not allow charge to move easily. Examples include glass,plastic, ceramics, and rubber. Electric current cannot be made to ow through an insulator, sincea charge has great difculty moving through it. One sees insulating (or dielectric) materials oftenwrapped around the center conducting core of a wire.Although the term resistance will be formally dened later, one can say qualitatively thata conductor has a very low resistance to the ow of charge, whereas an insulator has a veryhigh resistance to the ow of charge. Charge-conducting abilities of various materials vary ina wide range. Semiconductors fall in the middle between conductors and insulators, and havea moderate resistance to the ow of charge. Examples include silicon, germanium, and galliumarsenide.Current and Magnetic ForceThe rate of movement of net positive charge per unit of time through a cross section of a conductoris known as current,i(t ) =dqdt(1.1.5)TheSIunit ofcurrent istheampere(A), whichrepresents1coulombpersecond. Inmostmetallic conductors, such as copper wires, current is exclusively the movement of free electronsinthewire. Sinceelectronsarenegative, andsincethedirectiondesignatedforthecurrentis that of the net positive charge movement, the charges in the wire are thus moving in thedirection opposite to the direction of the current designation. The net charge transferred at aparticular time is the net area under the currenttime curve from the beginning of time to thepresent,q(t ) =t

i() d (1.1.6)WhileCoulombslawhastodowiththeelectricforceassociatedwithtwochargedbodies,Amperes lawof force is concerned with magnetic forces associated with two loops of wire carryingcurrents by virtue of the motion of charges in the loops. Note that isolated current elements donot exist without sources and sinks of charges at their ends; magnetic monopoles do not exist.Figure 1.1.2 shows two loops of wire in freespace carrying currents I1 and I2.Considering a differential element dl1 of loop 1 and a differential element dl2of loop 2,the differential magnetic forcesd F21 andd F12 experienced by the differential current elementsI1dl1, and I2dl2, due to I2 and I1, respectively, are given byd F21 = I1dl1

04I2dl2 a21R2

(1.1.7a)d F12 = I2dl2

04I1dl1 a12R2

(1.1.7b)where a21 and a12 are unit vectors along the line joining the two current elements, R is the distancebetween the centers of the elements,0 is the permeability of free space with units of N/A2orcommonly known as henrys per meter (H/m). Equation (1.1.7) reveals the following:1. The magnitude of the force is proportional to the product of the two currents and theproduct of the lengths of the two current elements.8 CIRCUIT CONCEPTSLoop 1Loop 2RI1I2a12dl1dl2a21Figure 1.1.2Illustration of Amperes law (offorce).2. The magnitude of the force is inversely proportional to the square of the distance betweenthe current elements.3. To determine the direction of, say, the force acting on the current element I1 dl1, the crossproductdl2 a21 must be found. Then crossingdl1 with the resulting vector will yieldthe direction of d F21.4. Each current element is acted upon by a magnetic eld due to the other current element,d F21 = I1dl1

B2(1.1.8a)d F12 = I2dl2

B1(1.1.8b)where B is known as the magnetic ux density vector with units of N/A m, commonlyknown as webers per square meter (Wb/m2) or tesla (T).Current distribution is the source of magnetic eld, just as charge distribution is the sourceof electric eld. As a consequence of Equations (1.1.7) and (1.1.8), it can be seen thatB2 =04I2dl2 a21(1.1.9a)B1 =04I1dl1 a12R2(1.1.9b)which depend on the medium parameter. Equation (1.1.9) is known as the BiotSavart law.Equation (1.1.8) can be expressed in terms of moving charge, since current is due to the owof charges. WithI =dq/dt anddl = vdt , where v is the velocity, Equation (1.1.8) can berewritten asd F =

dqdt

( vdt ) B = dq ( v B) (1.1.10)Thus it follows that the forceFexperienced by a test chargeq moving with a velocity v in amagnetic eld of ux density B is given byF = q ( v B) (1.1.11)The expression for the total force acting on a test chargeq moving with velocity v in a regioncharacterized by electric eld intensity E and a magnetic eld of ux density B isF =FE FM = q ( E v B) (1.1.12)which is known as the Lorentz force equation.1.1 ELECTRICAL QUANTITIES 9EXAMPLE 1.1.2Figure E1.1.2 (a) gives a plot of q(t ) as a function of time t .(a) Obtain the plot of i(t ).(b)Find the average value of the current over the time interval of 1 to 7 seconds.3t, seconds101 2 3 4 5 6 7 8 9 10q(t), coulombs(a)t, secondsi(t), amperes2.0(b)1.01.51 2 3 4 5 6 7 8 9 10FigureE1.1.2(a)Plot of q(t ).(b) Plot of i(t ).Sol ut i on(a) Applying Equation (1.1.5) and interpreting the rst derivative as the slope, one obtainsthe plot shown in Figure E1.1.2(b).(b) Iav = (1/T )

T0idt . Interpreting the integral as the area enclosed under the curve, onegets:Iav =1(7 1)[(1.5 2) (2.0 2) (0 1) (1 1)] = 0Note that the net charge transferred during the interval of 1 to 7 seconds is zero in this case.10 CIRCUIT CONCEPTSEXAMPLE 1.1.3Consider an innitesimal length of 106m of wire whose center is located at the point (1, 0, 0),carrying a current of 2 A in the positive direction of x.(a) Find the magnetic ux density due to the current element at the point (0, 2, 2).(b)Let another current element (of length 103m) be located at the point (0, 2, 2), carryinga current of 1 A in the direction of ( ay az). Evaluate the force on this current elementdue to the other element located at (1, 0, 0).Sol ut i on(a) I1dl1 = 2 106 ax. The unit vector a12 is given by a12 =(0 1) ax (2 0) ay (2 0) az122222=( ax 2 ay 2 az)3Using the BiotSavart law, Equation (1.1.9), one gets[B1](0,2,2) =04I1dl1 a12R2where0 is the free-space permeability constant given in SI units as 4107H/m,and R2in this case is {(0 1)2(2 0)2(2 0)2}, or 9. Hence,[B1](0,2,2) = 4 1074(2 106 ax) ( ax 2 ay 2 az)9 3= 107274 106( az ay) Wb/m2= 0.15 1013( az ay) T(b) I2dl2 = 103( ay az)d F12 = I2dl2

B1=103( ay az)

0.15 1013( az ay)= 0Note that the force is zero since the current element I2dl2 and the eld B1 due to I1dl1at (0, 2, 2) are in the same direction.The BiotSavart law can be extended to nd the magnetic ux density due to a current-carryinglamentarywireof anylengthandshapebydividingthewireintoanumber ofinnitesimal elements and using superposition. The net force experienced by a current loop canbe similarly evaluated by superposition.Electric Potential and VoltageWhen electrical forces act on a particle, it will possess potential energy. In order to describe thepotential energy that a particle will have at a point x, the electric potential at point x is dened as1.1 ELECTRICAL QUANTITIES 11v(x) =dw(x)dq(1.1.13)wherew(x) is the potential energy that a particle with chargeqhas when it is located at theposition x. The zero point of potential energy can be chosen arbitrarily since only differences inenergy have practical meaning. The point where electric potential is zero is known as the referencepoint or ground point, with respect to which potentials at other points are then described. Thepotential difference is known as the voltage expressed in volts (V) or joules per coulomb (J/C).If the potential at B is higher than that at A,vBA = vB vA(1.1.14)whichis positive. Obviouslyvoltages canbe either positive or negative numbers, andit follows thatvBA = vAB(1.1.15)The voltage at point A, designated asvA, is then the potential at point A with respect to theground.Energy and PowerIf a chargedq gives up energydw when going from pointa to pointb, then the voltage acrossthose points is dened asv =dwdq(1.1.16)Ifdw/dq is positive, point a is at the higher potential. The voltage between two points is thework per unit positive charge required to move that charge between the two points. If dw and dqhave the same sign, then energy is delivered by a positive charge going from a to b (or a negativecharge going the other way). Conversely, charged particles gain energy inside a source where dwand dq have opposite polarities.TheloadandsourceconventionsareshowninFigure1.1.3, inwhichpoint aisat ahigher potential than point b. The load receives or absorbs energy because a positive chargegoesinthedirectionof thecurrent arrowfromhigher tolower potential. Thesourcehasacapacitytosupplyenergy. Thevoltagesourceis sometimes knownas anelectromotiveforce, or emf, toconveythenotationthat it isaforcethat drivesthecurrent throughthecircuit.The instantaneous power p is dened as the rate of doing work or the rate of change ofenergy dw/dt,p =dwdt=

dwdq dqdt

= vi (1.1.17)The electric power consumed or produced by a circuit element is given by its voltagecurrentproduct, expressed in volt-amperes (VA) or watts (W). The energy over a time interval is foundby integrating power,w =T

0pdt (1.1.18)which is expressed in watt-seconds or joules (J), or commonly in electric utility bills in kilowatt-hours (kWh). Note that 1 kWh equals 3.6 106J.12 CIRCUIT CONCEPTSLoadiabvabab+Source ibavabaib+Figure 1.1.3Load and source conventions.EXAMPLE 1.1.4A typical 12-V automobile battery, storing about 5 megajoules (MJ) of energy, is connected to a4-A headlight system.(a) Find the power delivered to the headlight system.(b)Calculate the energy consumed in 1 hour of operation.(c) Expresstheauto-batterycapacityinampere-hours(Ah)andcomputehowlongtheheadlight system can be operated before the battery is completely discharged.Sol ut i on(a) Power delivered: P = VI = 124 = 48W.(b)Assuming V and I remain constant, the energy consumed in 1 hour will equalW = 48(60 60) = 172.8 103J = 172.8kJ(c) 1Ah =(1 C/s)(3600 s) =3600C. Forthebatteryinquestion, 5106J/12 V=0.417106C. Thus the auto-battery capacity is 0.417106/3600 = 116 Ah. Withoutcompletely discharging the battery, the headlight system can be operated for 116/4 =29hours.Sources and LoadsA sourceload combination is represented in Figure 1.1.4. A node is a point at which two ormore components or devices are connected together. A part of a circuit containing only onecomponent, source, or device between two nodes is known as a branch. A voltage rise indicatesan electric source, with the charge being raised to a higher potential, whereas a voltage dropindicates a load, with a charge going to a lower potential. The voltage across the source is thesame as the voltage across the load in Figure 1.1.4. The current delivered by the source goesthrough the load. Ideally, with no losses, the power (p = vi) delivered by the source is consumedby the load.When current ows out of the positive terminal of an electric source, it implies that non-electric energy has been transformed into electric energy. Examples include mechanical energytransformed into electric energy as in the case of a generator source, chemical energy changed1.1 ELECTRICAL QUANTITIES 13into electric energy as in the case of a battery source, and solar energy converted into electricenergy as in the case of a solar-cell source. On the other hand, when current ows in the directionof voltage drop, it implies that electric energy is transformed into nonelectric energy. Examplesinclude electric energy converted into thermal energy as in the case of an electric heater, electricenergy transformed into mechanical energy as in the case of motor load, and electric energychanged into chemical energy as in the case of a charging battery.Batteries and ac outlets are the familiar electric sources. These are voltage sources. An idealvoltage source is one whose terminal voltage v is a specied function of time, regardless of thecurrent i through the source. An ideal battery has a constant voltageVwith respect to time, asshown in Figure 1.1.5(a). It is known as a dc source, becausei =Iis a direct current. Figure1.1.5(b) shows the symbol and time variation for a sinusoidal voltage source with v = Vm cost .The positive sign on the source symbol indicates instantaneous polarity of the terminal at thehigher potential whenever cos t is positive. A sinusoidal source is generally termed an ac sourcebecause such a voltage source tends to produce an alternating current.The concept of an ideal current source, although less familiar but useful as we shall see later,is dened as one whose current i is a specied function of time, regardless of the voltage across itsterminals. The circuit symbols and the corresponding iv curves for the ideal voltage and currentsources are shown in Figure 1.1.6.Even though ideal sources could theoretically produce innite energy, one should recognizethat innite values are physically impossible. Various circuit laws and device representations ormodels are approximations of physical reality, and signicant limitations of the idealized conceptsor models need to be recognized. Simplied representations or models for physical devices arethe most powerful tools in electrical engineering. As for ideal sources, the concept of constant Vor constant I for dc sources and the general idea of v or i being a specied function of time shouldbe understood.When the source voltage or current is independent of all other voltages and currents, suchsources are known as independent sources. There are dependent or controlled sources, whoseSourceNode bNode aGroundi++LoadFigure 1.1.4Sourceload combination.+0vtV(a)Vi++0vt(b)v = Vm cos tVmVm2/iFigure 1.1.5Voltage sources. (a) Ideal dc source (battery). (b) Ideal sinusoidal ac source.14 CIRCUIT CONCEPTSvsis++0ivvs vsisis(a)i+0iv(b)vFigure 1.1.6Circuit symbols and iv curves. (a) Ideal voltage source. (b) Ideal current source.voltage or current does depend on the value of some other voltage or current. As an example, avoltage amplier producing an output voltage vout = Avin, where vin is the input voltage and Aisthe constant-voltage amplication factor, is shown in Figure 1.1.7, along with its controlled-sourcemodel using the diamond-shaped symbol. Current sources controlled by a current or voltage willalso be considered eventually.WaveformsWe are often interested in waveforms, which may not be constant in time. Of particular interestis a periodic waveform, which is a time-varying waveform repeating itself over intervals of timeT> 0.f (t ) = f (t nT ) n = 1,2,3, (1.1.19)The repetition time Tof the waveform is called the period of the waveform. For a waveformto be periodic, it must continue indenitely in time. The dc waveform of Figure 1.1.5(a) can beconsidered to be periodic with an innite period. The frequency of a periodic waveform is thereciprocal of its period,f =1THertz (Hz) (1.1.20)A sinusoidal or cosinusoidal waveform is typically described byf (t ) = A sin(t ) (1.1.21)where A is the amplitude, is the phase offset, and = 2f = 2/Tis the radian frequencyof the wave. When = 0, a sinusoidal wave results, and when = 90, a cosinusoidal waveresults. The average value of a periodic waveform is the net positive area under the curve for oneperiod, divided by the period,Fav =1TT

0f (t ) dt (1.1.22)++vinvout(a)Voltageamplifier++Avinvout(b)Figure 1.1.7Voltage amplier and itscontrolled-source model.1.1 ELECTRICAL QUANTITIES 15The effective, or root-mean square (rms), value is the square root of the average of f2(t ),Frms =

1TT

0f2(t ) dt (1.1.23)Determining the square of the functionf (t ), then nding the mean (average) value, and nallytaking the square root yields the rms value, known as effective value. This concept will be seento be useful in comparing the effectiveness of different sources in delivering power to a resistor.The effective value of a periodic current, for example, is a constant, or dc value, which deliversthe same average power to a resistor, as will be seen later.For the special case of a dc waveform, the following holds:f (t ) = F: Fav = Frms = F (1.1.24)For the sinusoid or cosinusoid, it can be seen thatf (t ) = A sin(t ): Fav = 0: Frms = A/2= 0.707 A (1.1.25)The student is encouraged to show the preceding results using graphical and analytical means.Other common types of waveforms are exponential in nature,f (t ) = Aet /(1.1.26a)f (t ) = A(1 et /) (1.1.26b)where is known as the time constant. After a time of one time constant has elapsed, looking atEquation (1.1.26a), the value of the waveformwill be reduced to 37%of its initial value; Equation(1.1.26b) shows that the value will rise to 63% of its nal value. The student is encouraged tostudy the functions graphically and deduce the results.EXAMPLE 1.1.5A periodic current waveform in a rectier is shown in Figure E1.1.5. The wave is sinusoidal for/3 t , and is zero for the rest of the cycle. Calculate the rms and average values of thecurrent.i103 2tFigure E1.1.5Sol ut i onIrms =

12/3

0i2d(t )

/3i2d(t ) 2

i2d(t )

16 CIRCUIT CONCEPTSNotice that t rather than t is chosen as the variable for convenience; =2f =2/T;and integration is performed over three discrete intervals because of the discontinuous currentfunction. Since i = 0 for 0 t < /3 and t 2,Irms =

12

/3102sin2td(t ) = 4.49 AIav =12

/310 sintd(t ) = 2.39 ANote that the base is the entire period 2, even though the current is zero for a substantial part ofthe period.1.2 LUMPED-CIRCUIT ELEMENTSElectric circuits or networks are formed by interconnecting various devices, sources, and com-ponents. Although the effects of each element (such as heating effects, electric-eld effects,or magnetic-eld effects) are distributed throughout space, one often lumps them together aslumped elements. The passive components are the resistance R representing the heating effect,the capacitance C representing the electric-eld effect, and the inductance L representing themagnetic-eld effect. Their characteristics will be presented in this section. The capacitor modelsthe relation between voltage and current due to changes in the accumulation of electric charge, andthe inductor models the relation due to changes in magnetic ux linkages, as will be seen later.While these phenomena are generally distributed throughout an electric circuit, under certainconditionstheycanbeconsideredtobeconcentratedatcertainpointsandcanthereforeberepresented by lumped parameters.ResistanceAn ideal resistoris a circuit element with the property that the current through it is linearlyproportional to the potential difference across its terminals,i = v/R = Gv,or v = iR (1.2.1)which is known as Ohms law, published in 1827. R is known as the resistance of the resistorwith the SI unit of ohms (), and G is the reciprocal of resistance called conductance, with the SIunit of siemens (S). The circuit symbols of xed and variable resistors are shown in Figure 1.2.1,along with an illustration of Ohms law. Most resistors used in practice are good approximationsto linear resistors for large ranges of current, and their iv characteristic (current versus voltageplot) is a straight line.The value of resistance is determined mainly by the physical dimensions and the resistivity of the material of which the resistor is composed. For a bar of resistive material of length l andcross-sectional area A the resistance is given byR =lA=lA(1.2.2)where is the resistivity of the material in ohm-meters ( m), and is the conductivity of thematerial in S/m, which is the reciprocal of the resistivity. Metal wires are often considered as ideal1.2 LUMPED-CIRCUIT ELEMENTS 17iabvabRa b(Fixed)+ iabiab = vab/R = GvabvabRa b(Variable)+ Figure 1.2.1Circuit symbols of xedand variable resistors and illustration ofOhms law.TABLE 1.2.1Resistivity of Some MaterialsType Material ( m)Conductors Silver 16 109(at 20C) Copper 17 109Gold 24 109Aluminum 28 109Tungsten 55 109Brass 67 109Sodium 0.04 106Stainless steel 0.91 106Iron 0.1 106Nichrome 1 106Carbon 35 106Seawater 0.25Semiconductors Germanium 0.46(at 27C or 300 K) Silicon 2.3 103Insulators Rubber 1 1012Polystyrene 1 1015conductors with zero resistance as a good approximation. Table 1.2.1 lists values of for somematerials.Theresistivityof conductor metalsvarieslinearlyover normal operatingtemperaturesaccording toT 2 = T 1

T2TT1T

(1.2.3)where T 2 and T 1 are resistivities at temperatures T2 and T1, respectively, and T is a temperatureconstant that depends on the conductor material. All temperatures are in degrees Celsius. Theconductor resistance also depends on other factors, such as spiraling, frequency (the skin effectwhich causes the ac resistance to be slightly higher than the dc resistance), and current magnitudein the case of magnetic conductors (e.g., steel conductors used for shield wires).Practical resistors are manufactured in standard values, various resistance tolerances, severalpower ratings (as will be explained shortly), and in a number of different forms of construction.The three basic construction techniques are composition type, which uses carbon or graphite andis molded into a cylindrical shape, wire-wound type, in which a length of enamel-coated wireis wrapped around an insulating cylinder, and metal-lm type, in which a thin layer of metal isvacuum deposited. Table 1.2.2 illustrates the standard color-coded bands used for evaluatingresistance and their interpretation for the common carbon composition type. Sometimes a fthband is also present to indicate reliability. Black is the least reliable color and orange is 1000times more reliable than black.For resistors ranging from 1 to 9.1 , the standard resistance values are listed in Table 1.2.3.Other available values can be obtained by multiplying the values shown in Table 1.2.3 by factors18 CIRCUIT CONCEPTSTABLE 1.2.2Standard Color-Coded Bands for Evaluating Resistance and Their InterpretationColor bands 14 b1b2b3b4Color of Band Digit of Band Multiplier % Tolerance in Actual ValueBlack 0 100Brown 1 101Red 2 102Orange 3 103Yellow 4 104Green 5 105Blue 6 106Violet 7 107Grey 8 108White 9 Gold 101 5%Silver 102 10%Black or no color 20%Resistance value = (10b1b2) 10b3.of 10 ranging from 10to about 22106. For example, 8.2 , 82 , 820 , . . . , 820 kare standard available values.The maximum allowable power dissipation or power rating is typically specied for com-mercial resistors. A common power rating for resistors used in electronic circuits is 14 W; otherratings such as18,12, 1, and 2 W are available with composition-type resistors, whereas largerratings are also available with other types. Variable resistors, known as potentiometers, with amovable contact are commonly found in rotary or linear form. Wire-wound potentiometers mayhave higher power ratings up to 1000 W.The advent of integrated circuits has given rise to packaged resistance arrays fabricated byusing lm technology. These packages are better suited for automated manufacturing and areusually less costly than discrete resistors in large production runs.An important property of the resistor is its ability to convert energy from electrical form intoheat. The manufacturer generally states the maximum power dissipation of the resistor in watts.If more power than this is converted to heat by the resistor, the resistor will be damaged due tooverheating. The instantaneous power absorbed by the resistor is given byp(t ) = v(t )i(t ) = i2R = v2/R = v2G (1.2.4)where v is the voltage drop across the resistance and i is the current through the resistance. It canbe shown (see Problem 1.2.13) that the average value of Equation (1.2.4) is given byPav = VrmsIrms = I2rmsR = V2rms/R = V2rmsG (1.2.5)for periodically varying current and voltage as a function of time. Equation (1.2.5) gives theexpression for the power converted to heat by the resistor.1.2 LUMPED-CIRCUIT ELEMENTS 19TABLE 1.2.3Standard Available Values of Resistors1.0 1.5 2.2 3.3 4.7 6.81.1 1.6 2.4 3.6 5.1 7.51.2 1.8 2.7 3.9 5.6 8.21.3 2.0 3.0 4.3 6.2 9.1Series and parallel combinations of resistors occur very often. Figure 1.2.2 illustrates thesecombinations.Figure1.2.2(a)showstworesistors R1andR2inseriessharingthevoltagevindirectproportion to their values, while the same current i ows through both of them,v = vAC = vABvBC = iR1iR2 = i(R1R2) = iReqor, when R1 and R2 are in series,Req = R1R2(1.2.6)Figure 1.2.2(b) shows two resistors in parallel sharing the current i in inverse proportion totheir values, while the same voltage v is applied across each of them. At node B,i = i1i2 =vR1vR2= v

1R11R2

= v/

R1R2R1R2

=vReqor, when R1 and R2 are in parallel,Req =R1R2R1R2or1Req=1R11R2or Geq = G1G2(1.2.7)Notice the voltage division shown in Figure 1.2.2(a), and the current division in Figure1.2.2(b).iAC(a) (b)B+v = vAC = vAB + vBCvAD = vBC = vvAB = iR1==vR1R1 + R2vBC = iR2R1R2ADBC+i = i1 + i2i2= v/R2= vG2i1= v/R1= vG1R1R2=iG2G1 + G2=iG1G1 + G2vR2R1 + R2Figure 1.2.2Resistances in series and in parallel. (a) R1 and R2 in series. (b) R1 and R2 in parallel.EXAMPLE 1.2.1A no. 14 gauge copper wire, commonly used in extension cords, has a circular wire diameter of64.1 mils, where 1 mil = 0.001 inch.(a) Determine the resistance of a 100-ft-long wire at 20C.20 CIRCUIT CONCEPTS(b)If such a 2-wire system is connected to a 110-V (rms) residential source outlet in orderto power a household appliance drawing a current of 1 A (rms), nd the rms voltage atthe load terminals.(c) Compute the power dissipated due to the extension cord.(d)Repeat part (a) at 50C, given that the temperature constant for copper is 241.5C.S ol ut i on(a) d = 64.1 mils =64.1103in = 64.1103 2.54cm/1in1m/100cm =1.628 103m. From Table 1.2.1, of copper at 20C is 17 109m,l = 100 ft = 100 ft 12 in1 ft

2.54 cm1 in

1 m100 cm = 30.48 mA =d24=(1.628 103)24= 2.08 106m2Per Equation (1.2.2),R20C = 17 10930.482.08 106= 0.25 (b)Rms voltage at load terminals, V =110 (0.25)2 =109.5 V (rms). Note that two100-ft-long wires are needed for the power to be supplied.(c) Power dissipated, per Equation (1.2.5), P = (1)2(0.25)(2) = 0.5 W.(d)Per Equation (1.2.3),50C = 20C

50 241.520 241.5

= 17 109291.5261.5= 18.95 109 mHence,R50C = 18.95 10930.482.08 106= 0.28 EXAMPLE 1.2.2(a) Consider a seriesparallel combination of resistors as shown in Figure E1.2.2(a). Findthe equivalent resistance as seen from terminals AB.(b)Determine the current I and power P delivered by a 10-V dc voltage source applied atterminals AB, with A being at higher potential than B.(c) Replace the voltage source by an equivalent current source at terminals AB.(d)Show the current and voltage distribution clearly in all branches of the original circuitconguration.1.2 LUMPED-CIRCUIT ELEMENTS 21S ol ut i on(a) The circuit is reduced as illustrated in Figure E1.2.2(b).(b) I = 5 A; P = VI = I2R = V2/R = 50 W [see Figure E1.2.2(c)].(c) See Figure E1.2.2(d).(d)See Figure E1.2.2(e).(a)ABB2 1 1 2 2 DCFigure E1.2.2ABD1 2 || 2 = 1 2 || 2 = 1 1 + 1 (In series)= 2 1 + 1 (In series)= 2 1 2 CBAB1 2 CBAB1 CBAB(b)Req = 2 22 CIRCUIT CONCEPTS10 VBAI = 10/2 = 5 A2 +(c)BAV = 5 2 = 10 V5 A2 +(d)(e)AB B2 1 1 5 A5 A2.5 A2.5 A2.5 V1.25 A 1.25 A5 V5 V10 V2 2 DC+++2.5 V+2.5 V++Maximum Power TransferIn order to investigate the power transfer between a practical source and a load connected to it,let us consider Figure 1.2.3, in which a constant voltage source v with a known internal resistanceRS is connected to a variable load resistance RL. Note that when RL is equal to zero, it is calleda short circuit, in which casevL becomes zero andiL is equal tov/RS. WhenRL approachesinnity, it is called an open circuit, in which caseiL becomes zero andvL is equal to v. One isgenerally interested to nd the value of the load resistance that will absorb maximum power fromthe source.The power PL absorbed by the load is given byPL = i2LRL(1.2.8)where the load current iL is given byiL =v2RS RL(1.2.9)Substituting Equation (1.2.9) in Equation (1.2.8), one getsPL =v2(RS RL)2RL(1.2.10)For given xed values of v andRS, in order to nd the value ofRL that maximizes the powerabsorbed by the load, one sets the rst derivative dPL/dRL equal to zero,dPLdRL=v2(RLRS)22v2RL(RLRS)(RLRS)4= 0 (1.2.11)1.2 LUMPED-CIRCUIT ELEMENTS 23vBARLRSvLiLiL+Source +LoadFigure 1.2.3Power transfer between source and load. Note:RL =0implies short circuit;vL = 0 andiL =vRSandRL implies opencircuit; iL = 0 and vL = v.which leads to the following equation:(RLRS)22RL(RLRS) = 0 (1.2.12)The solution of Equation (1.2.12) is given byRL = RS(1.2.13)That is to say, in order to transfer maximum power to a load, the load resistance must be matchedto the source resistance or, in other words, they should be equal to each other.A problem related to power transfer is that of source loading. Figure 1.2.4(a) illustrates apractical voltage source (i.e., an ideal voltage source along with a series internal source resistance)connected to a load resistance; Figure 1.2.4(b) shows a practical current source (i.e., an idealcurrent source along with a parallel or shunt internal source resistance) connected to a loadresistance. It follows from Figure 1.2.4(a) thatvL = v vint = v iLRS(1.2.14)where vint is the internal voltage drop within the source, which depends on the amount of currentdrawn by the load. As seen from Equation (1.2.14), the voltage actually seen by the loadvLis somewhat lower than the open-circuit voltage of the source. When the load resistanceRL isinnitely large, the load current iLgoes to zero, and the load voltagevLis then equal to theopen-circuit voltage of the source v. Hence, it is desirable to have as small an internal resistanceas possible in a practical voltage source.From Figure 1.2.4(b) it follows thatiL = i iint = i vLRS(1.2.15)where iint is the internal current drawn away from the load because of the presence of the internalsource resistance. Thus the load will receive only part of the short-circuit current available fromthe source. When the load resistanceRL is zero, the load voltagevL goes to zero, and the loadv(a)RLRSvLvintiL++ Voltage source +Loadi(b)RLRSvLiintiLCurrent source +LoadFigure 1.2.4Source-loadingeffects.24 CIRCUIT CONCEPTScurrent iL is then equal to the short-circuit current of the source i. Hence, it is desirable to haveas large an internal resistance as possible in a practical current source.CapacitanceAn ideal capacitor is an energy-storage circuit element (with no loss associated with it) repre-senting the electric-eld effect. The capacitance in farads (F) is dened byC = q/v (1.2.16)whereqisthechargeoneachconductor, andvisthepotentialdifferencebetweenthetwoperfect conductors. With v being proportional to q, C is a constant determined by the geometricconguration of the two conductors. Figure 1.2.5(a) illustrates a two-conductor system carryingq and q charges, respectively, forming a capacitor.The general circuit symbol for a capacitor is shown in Figure 1.2.5(b), where the currententering one terminal of the capacitor is equal to the rate of buildup of charge on the plateattached to that terminal,i(t ) =dqdt= Cdvdt(1.2.17)in which C is assumed to be a constant and not a function of time (which it could be, if theseparation distance between the plates changed with time).The terminal vi relationship of a capacitor can be obtained by integrating both sides ofEquation (1.2.17),v(t ) =1Ct

i() d (1.2.18)which may be rewritten asv(t ) =1Ct

0i() d 1C0

i() d =1Ct

0i() d v(0) (1.2.19)where v(0) is the initial capacitor voltage at t = 0.The instantaneous power delivered to the capacitor is given byp(t ) = v(t )i(t ) = C v(t )dv(t )dt(1.2.20)whose average value can be shown (see Problem1.2.13) to be zero for sinusoidally varying currentand voltage as a function of time. The energy stored in a capacitor at a particular time is foundby integrating,ABA B+q chargePotential vAq chargePotential vBPotential difference = v = vA vB;C = q/vCi(t)v(t)(a) (b)i(t) = = C ;dqdt C1v(t) =dvdti() d tC1=+ i() d + v(0)0tFigure 1.2.5 Capacitor. (a) Two perfect conductors carrying q and q charges. (b) Circuit symbol.1.2 LUMPED-CIRCUIT ELEMENTS 25w(t ) =t

p() d = Ct

v()dv()d=12Cv2(t ) 12Cv2() (1.2.21)Assuming the capacitor voltage to be zero at t = , the stored energy in the capacitor at sometime t is given byw(t ) =12Cv2(t ) (1.2.22)which depends only on the voltage of the capacitor at that time, and represents the stored energyin the electric eld between the plates due to the separation of charges.If the voltage across the capacitor does not change with time, no current ows, as seen fromEquation (1.2.17). Thus the capacitor acts like an open circuit, and the following relations hold:C =QV: I = 0, W =12CV2(1.2.23)An ideal capacitor, once charged and disconnected, the current being zero, will retain a potentialdifference for an indenite length of time. Also, the voltage across a capacitor cannot changevalue instantaneously, while an instantaneous change in the capacitor current is quite possible.The student is encouraged to reason through and justify the statement made here by recallingEquation (1.2.17).Series and parallel combinations of capacitors are often encountered. Figure 1.2.6 illustratesthese.It follows from Figure 1.2.6(a),v = vAC = vABvBCdvdt=dvABdtdvBCdt=iC1iC2= i

C1C2C1C2

=iCeqor, when C1 and C2 are in series,Ceq =C1C2C1C2or1Ceq=1C11C2(1.2.24)Referring to Figure 1.2.6(b), one getsi = i1i2 = C1dvdtC2dvdt= (C1C2)dvdt=Ceqdvdtor, when C1 and C2 are in parallel,Ceq = C1C2(1.2.25)Note that capacitors in parallel combine as resistors in series, and capacitors in series combine asresistors in parallel.BC1i1i2C2CDABC1C2ACv vi i++(a) (b)Figure 1.2.6Capacitors in series and in parallel. (a) C1and C2 in series. (b) C1 and C2 in parallel.26 CIRCUIT CONCEPTSThe working voltage for a capacitor is generally specied by the manufacturer, thereby givingthe maximum voltage that can safely be applied between the capacitor terminals. Exceeding thislimit mayresult inthe breakdownof the insulationandthenthe formationof anelectric arc betweenthe capacitor plates. Unintentional or parasitic capacitances that occur due to the proximity ofcircuit elements may have serious effects on the circuit behavior.Physical capacitors are often made of tightly rolled sheets of metal lm, with a dielectric(paper or nylon) sandwiched in between, in order to increase their capacitance values (or abilitytostore energy) for a givensize. Table 1.2.4lists the range of general-purpose capacitances togetherwith the maximum voltages and frequencies for different types of dielectric materials. Practicalcapacitors come in a wide range of values, shapes, sizes, voltage ratings, and constructions. Bothxed and adjustable devices are available. Larger capacitors are of the electrolytic type, usingaluminum oxide as the dielectric.TABLE 1.2.4Characteristics of General-Purpose CapacitorsCapacitance Maximum Voltage Frequency RangeMaterial Range Range (V) (Hz)Mica 1 pF to 0.1 F 50600 1031010Ceramic 10 pF to 1 F 501600 1031010Mylar 0.001 F to 10 F 50600 102108Paper 10 pF to 50 F 50400 102108Electrolytic 0.1 F to 0.2 F 3600 10104Note: 1 pF = 1012F; 1 F = 106F.EXAMPLE 1.2.3(a) Consider a 5-F capacitor to which a voltage v(t) is applied, shown in Figure E1.2.3(a),top. Sketch the capacitor current and stored energy as a function of time.(b)Let a current source i(t) be attached to the 5-F capacitor instead of the voltage source ofpart (a), shown in Figure E1.2.3(b), top. Sketch the capacitor voltage and energy storedas a function of time.(c) If three identical 5-F capacitors with an initial voltage of 1 mV are connected (i) inseries and (ii) in parallel, nd the equivalent capacitances for both cases.Sol ut i on(a) From Figure E1.2.3(a) it follows thatv(t ) = 0, t 1 s= 5(t 1) mV, 1 t 1 s= 10 mV, 1 t 3 s= 10(t 4) mV, 3 t 4s= 0, 4 t s1.2 LUMPED-CIRCUIT ELEMENTS 272 1v(t), mVv(t)t, s5 Ft, si(t), mA2 3 6 4 5 12 12550250102 3 6 4 5 1t, sw(t), pJ2 1(a)2 3 6 4 5 1v(t)i(t)++5090100125122.540102 1i(t), mAt, s2 3 6 4 5 110t, sv(t), mV2 176543212 3 6 4 5 1t, sw(t), pJ1(b)2 3 6 4 5 1v(t) 5 F i(t)i(t)+C3CeqC2C1+v1(0) +v2(0) +v3(0) v(0) = v1(0) + v2(0) + v3(0)A(c)i B AvB+ + + v + ABvii1i2i3+(d)Ceqv(0)ABv vi++C2v(0) +C1v(0) +C3v(0) +Figure E1.2.328 CIRCUIT CONCEPTSSincei(t ) = Cdvdt= (5 106)dvdtit follows thati(t ) = 0, t 1 s= 25 mA, 1 t 1 s= 0, 1 t 3 s= 50 mA, 3 t 4s= 0, 4 t swhich is sketched in the center of Figure E1.2.3(a).Since the energy stored at any instant isw(t ) =12Cv2(t ) =12(5 106)v2(t )it follows that:w(t ) = 0, t 1 s= 62.5 (t22t 1) pJ, 1 t 1 s= 250 pJ, 1 t 3 s= 250 (t28t 16) pJ, 3 t 4s= 0, 4 t swhich is sketched at the bottom of Figure E1.2.3(a).(b)From Figure E1.2.3(b) it follows thati(t ) = 0, t 1 s= 5 (t 1) mA, 1 t 1 s= 10 mA, 1 t 3 s= 10 (t 4) mA, 3 t 4s= 0, 4 t sSincev(t ) =1Ct

i() d =15 106t

i() dit follows thatv(t ) = 0, t 1 s=

t22 t 12

mV, 1 t 1 s1.2 LUMPED-CIRCUIT ELEMENTS 29= 2t mV, 1 t 3 s= t28t 9 mV, 3 t 4s= 7 mV, 4 t swhich is sketched in the center of Figure E1.2.3(b).Since the energy stored at any instant isw(t ) =12Cv2(t ) =12(5 106)v2(t )it follows thatw(t ) = 0, t 1 s= 2.5 (t22 t 12)2pJ, 1 t 1 s= 10t2pJ, 1 t 3 s= 2.5 (t28t 9)2pJ, 3 t 4s= 122.5pJ, 4 t swhich is sketched at the bottom of Figure E1.2.3(b).(c) (i)1Ceq=1C11C21C3=35 106, or Ceq = 53 106F = 53F,with an initial voltage v(0) = 3 mV [Figure E1.2.3(c)].(ii)Ceq = C1C2C3 = 3 5 106F = 15 Fwith an initial voltage v(0) = 1 mV [Figure E1.2.3(d)].InductanceAn ideal inductor is also an energy-storage circuit element (with no loss associated with it) like acapacitor, but representing the magnetic-eld effect. The inductance in henrys (H) is dened byL =i=Ni(1.2.26)where is the magnetic-ux linkage in weber-turns (Wbt), N is the number of turns of the coil,andN is the magnetic ux in webers (Wb) produced by the current i in amperes (A). Figure1.2.7(a) illustrates a single inductive coil or an inductor of N turns carrying a current i that islinked by its own ux.The general circuit symbol for an inductor is shown in Figure 1.2.7(b). According to Faradayslaw of induction, one can writev(t ) =ddt=d(N)dt= Nddt=d(Li)dt= Ldidt(1.2.27)30 CIRCUIT CONCEPTSi(t)i(t)(a)L++v(t) v(t) N turnsL = /i = N/i = N(b)v(t) = L ;didt L1i(t) = v() dtL1= v() d + i(0)0tFigure 1.2.7An inductor. (a) A single inductive coil of N turns. (b) Circuit symbol.where L is assumed to be a constant and not a function of time (which it could be if the physicalshapeofthecoilchangedwithtime). Mathematically, bylookingatEquations(1.2.17)and(1.2.27), the inductor is the dual of the capacitor. That is to say, the terminal relationship forone circuit element can be obtained from that of the other by interchanging v and i, and also byinterchanging L and C.The terminal iv relationship of an inductor can be obtained by integrating both sides ofEquation (1.2.27),i(t ) =1L

tv() d (1.2.28)which may be rewritten as