Circuits -2nd Edition by Fawwaz T. Ulaby and Michel M. Maharbiz

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CIRCUITSSecond Edition

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CIRCUITSSecond Edition

Fawwaz T. UlabyThe University of Michigan

Michel M. MaharbizThe University of California, Berkeley

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ISBN: 978-1-934891-19-3

10 9 8 7 6 5 4 3 2 1

Publisher: Tom RobbinsDevelopment Manager: Gretchen EdelmonProject Manager : Catherine PeacockCompositor: Paul Mailhot, PreTeX Inc.

Cover illustration: Used with permission of TechRepublic Copyright! 2012. All rights reserved.

!2013 National Technology and Science Press.

All rights reserved. Neither this book, nor any portion of it, may be copied or reproduced in any form or by any means withoutwritten permission of the publisher.

NTS Press respects the intellectual property of others, and we ask our readers to do the same. This book is protected by copyrightand other intellectual property laws. Where the software referred to in this book may be used to reproduce software or othermaterials belonging to others, you should use such software only to reproduce materials that you may reproduce in accordancewith the terms of any applicable license or other legal restriction.

LabVIEW, Multisim, and National Instruments are trademarks of National Instruments.

MATLAB" is a registered trademark of The MathWorks, Inc., 3 Apple Hill Drive, Natick, MA 01760-2098.

All other trademarks or product names are the property of their respective owners.

Library of Congress Control Number: 2012942576

Additional Disclaimers:

The reader assumes all risk of use of this book and of all information, theories, and programs contained or described in it.This book may contain technical inaccuracies, typographical errors, other errors and omissions, and out-of-date information.Neither the author nor the publisher assumes any responsibility or liability for any errors or omissions of any kind, to updateany information, or for any infringement of any patent or other intellectual property right.

Neither the author nor the publisher makes any warranties of any kind, including without limitation any warranty as tothe sufficiency of the book or of any information, theories, or programs contained or described in it, and any warrantythat use of any information, theories, or programs contained or described in the book will not infringe any patent or otherintellectual property right. THIS BOOK IS PROVIDED “AS IS.” ALL WARRANTIES, EITHER EXPRESS OR IMPLIED,INCLUDING, BUT NOT LIMITED TO, ANY AND ALL IMPLIED WARRANTIES OF MERCHANTABILITY, FITNESSFOR A PARTICULAR PURPOSE, AND NON-INFRINGEMENT OF INTELLECTUAL PROPERTY RIGHTS, AREDISCLAIMED.

No right or license is granted by publisher or author under any patent or other intellectual property right, expressly, or byimplication or estoppel.

IN NO EVENT SHALL THE PUBLISHER OR THE AUTHOR BE LIABLE FOR ANY DIRECT, INDIRECT, SPECIAL,INCIDENTAL, COVER, ECONOMIC, OR CONSEQUENTIAL DAMAGES ARISING OUT OF THIS BOOK OR ANYINFORMATION, THEORIES, OR PROGRAMS CONTAINED OR DESCRIBED IN IT, EVEN IF ADVISED OF THEPOSSIBILITY OF SUCH DAMAGES, AND EVEN IF CAUSED OR CONTRIBUTED TO BY THE NEGLIGENCE OFTHE PUBLISHER, THE AUTHOR, OR OTHERS. Applicable law may not allow the exclusion or limitation of incidental orconsequential damages, so the above limitation or exclusion may not apply to you.

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To an academic, writing a book is an endeavor of love.

We dedicate this book to Jean and Anissa.

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About NTS Press

National Technology & Science Press or NTS Pressis sponsored by engineers for engineering, science andmathematics students dedicated to the publication of scholarlymaterial of lasting value. Our products are for educatorswho desire a high degree of integration among classroom text,hardware and software to permit hands-on, visual learning. Oursuccess will be judged by the influence our publications havein inspiring you and others to pursue careers in engineering,science, and mathematics.

We believe the learning process is most rewarding if youbuild up an intuitive understanding of concepts by pausing totinker, explore and reflect. Thus, our publishing philosophyfollows the belief that learning takes place when you are activelyinvolved and we attempt to encourage this process in severalways. We build our textbooks with a plethora of worked-outexamples, with reinforcing and advanced problems, and withinteractive computer visuals. We address the computationalaspects of problem solving by integrating computer-basedlearning tools into these presentations to enhance the discussionand analysis of engineering and science applications. Inseveral cases our educational materials are developed withhardware experimentation platforms in mind, such as UniversalSerial Bus (USB) devices for acquiring, generating and

analyzing information. Owning your own portable laboratoryequipment permits you to perform experimentation and takemeasurements anywhere and at anytime, thereby reinforcingyour understanding of theoretical concepts. Having a little funis okay too.

Clearly, rising textbook costs have an enormous effect onthe way you view educational material. And there is littledoubt that the Internet has been the most influential agentof change in education in recent time. The prevalence ofsearch engine technology combined with a variety of open-source, open-content, and Wiki sites designed to deliver contenthas created a proliferation of freely available informationand has circulated it more widely. Authored, edited, printedmaterial is being overtaken by community libraries of digitizedcontent from which new content is constructed, remixed,reordered and reassembled, often absent of continuity, flow,and accountability. This free content seems to address concernsabout rising textbook prices, but it still has a cost. By contrastwe try carefully to design products that come together in onepiece with the author’s judgment, passion, and imaginationintact. These books are available with a reasonable price, andmay be counted on as “reputable islands of knowledge in thevast ocean of unscrutinized information.” www.ntspress.com

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Brief Contents

Chapter 1 Circuit Terminology 1

Chapter 2 Resistive Circuits 40

Chapter 3 Analysis Techniques 94

Chapter 4 Operational Amplifiers 150

Chapter 5 RC and RL First-OrderCircuits

203

Chapter 6 Circuit Analysis byLaplace Transform

227

Chapter 7 ac Analysis 346

Chapter 8 ac Power 412

Chapter 9 Frequency Response ofCircuits and Filters

453

Chapter 10 Three-Phase Circuits 512

Chapter 11 Magnetically CoupledCircuits

552

Chapter 12 Fourier AnalysisTechnique

578

Appendix A Symbols, Quantities,and Units

627

Appendix B Solving SimultaneousEquations

629

Appendix C Overview of Multisim 633

Appendix D Mathematical Formulas 636

Appendix E MATLAB and MathScript 639

Appendix F Answers to SelectedProblems

644

Index 645

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Contents

Preface xiii

Chapter 1 Circuit Terminology 1Cell-Phone Circuit Architecture 2

1-1 Historical Timeline 31-2 Units, Dimensions, and Notation 81-3 Circuit Representation 91-4 Electric Charge and Current 131-5 Voltage and Power 171-6 Circuit Elements 23TB1 Micro- and Nanotechnology 28

Chapter 1 Summary 32Problems 33

Chapter 2 Resistive Circuits 40Overview 41

2-1 Ohm’s Law 41TB2 Superconductivity 472-2 Kirchhoff’s Law 492-3 Equivalent Circuits 54TB3 Resistors as Sensors 632-4 Wye–Delta (Y–!) Transformation 652-5 The Wheatstone Bridge 682-6 Application Note: Linear versus

Nonlinear i–v Relationships69

2-7 Introducing Multisim 74Chapter 2 Summary 81Problems 82

Chapter 3 Analysis Techniques 82Overview 95

3-1 Node-Voltage Method 953-2 Mesh-Current Method 1003-3 By-Inspection Methods 105TB4 Light-Emitting Diodes 1093-4 Source Superposition 1113-5 Thevenin and Norton Equivalent Circuits 113TB5 Integrated Circuit Fabrication Process 1203-6 Maximum Power Transfer 1243-7 Application Note: Bipolar Junction

Transistor (BJT)127

3-8 Nodal Analysis with Multisim 130Chapter 3 Summary 133Problems 134

Chapter 4 Operational Amplifiers 150Overview 151

4-1 Op-Amp Characteristics 151TB6 Display Technologies 1554-2 Negative Feedback 1604-3 Ideal Op-Amp Model 161

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x

4-4 Inverting Amplifier 1634-5 Summing Amplifier 1654-6 Difference Amplifier 1674-7 Voltage Follower 1684-8 Op-Amp Signal-Processing Circuits 1694-9 Instrumentation Amplifier 1744-10 Digital-to-Analog Converters (DAC) 1764-11 The MOSFET as a Voltage-Controlled

Current Source179

TB7 Computer Memory Circuits 1844-12 Application Note: Neural Probes 1874-13 Multisim Analysis 188


Chapter 5 RC and RL First-OrderCircuits

203

Overview 2045-1 Nonperiodic Waveforms 2045-2 Capacitors 211TB8 Supercapacitors 2215-3 Inductors 2245-4 Response of the RC Circuit 2305-5 Response of the RL Circuit 2415-6 RC Op-Amp Circuits 2465-7 Application Note: Parasitic Capacitance

and Computer Processing Speed251

TB9 Hard Disk Drives 2575-8 Analyzing Circuit Response with

Multisim259


Chapter 6 Circuit Analysis byLaplace Transform

277

Overview 2786-1 Initial and Final Conditions 2786-2 The Series RLC Circuit 281TB10 Micromechanical Sensors and Actuators 2876-3 The Parallel RLC Circuit 2916-4 Unit Impulse Function 2936-5 Review of Complex Algebra 295

6-6 The Laplace Transform Technique 299TB11 Touchscreens and Active Digitizers 3036-7 Properties of the Laplace Transform 3066-8 Circuit Analysis Procedure 3086-9 Partial Fraction Expansion 3116-10 s-Domain Circuit Element Models 3116-11 s-Domain Circuit Analysis 3206-8 Multisim Analysis of Circuit Driven by

Non-Trivial Inputs326


Chapter 7 ac Analysis 346Overview 347

7-1 Sinusoidal Signals 3477-2 Phasor Domain 350TB12 Crystal Oscillators 3557-3 Phasor-Domain Analysis 3587-4 Impedance Transformations 3607-5 Equivalent Circuits 3677-6 Phasor Diagrams 371TB13 Electromagnetic Spectrum 3757-7 Phase-Shift Circuits 3777-8 Phasor-Domain Analysis Techniques 3807-9 Application Note: Power-Supply Circuits 3877-10 Multisim Analysis of ac Circuits 392


Chapter 8 ac Power 412Overview 413

8-1 Periodic Waveforms 4138-2 Average Power 417TB14 Noise-Cancellation Headphones 4198-3 Complex Power 4228-4 The Power Factor 4268-5 Maximum Power Transfer 431TB15 Night-Vision Imaging 3258-6 Measuring Power With Multisim 436


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xi

Chapter 9 Frequency Response ofCircuits and Filters

453

Overview 4549-1 The Transfer Function 4549-2 Scaling 460TB16 Smart Dust, Sensor Webs, and

Ubiquitous Computing462

9-3 Bode Plots 4649-4 Passive Filters 4749-5 Filter Order 4819-6 Active Filters 485TB17 Bandwidth, Data Rate, and

Communication487

9-7 Cascaded Active Filters 4899-8 Application Note: Modulation and the

Superheterodyne Receiver494

9-9 Spectral Response with Multisim 498Chapter 9 Summary 502Problems 503

Chapter 10 Three-Phase Circuits 512Overview 513

10-1 Balanced Three-Phase Generators 514TB18 Electrical Engineering and the

Audiophile518

10-2 Source-Load Configurations 52110-3 Y-Y Configuration 523TB19 Minaturized Energy Harvesting 52610-4 Balanced Networks 52910-5 Power in Balanced Three-Phase

Networks532

TB20 3-D TV 53710-6 Power-Factor Compensation 53910-7 Power Measurement in Three-Phase

Circuits542


Chapter 11 Magnetically CoupledCircuits

552

Overview 55311-1 Magnetic Coupling 55311-2 Transformers 558

TB21 Mapping the World in 3-D 56311-3 Energy Considerations 56511-4 Ideal Transformers 56611-5 Three-Phase Transformers 569


Chapter 12 Fourier AnalysisTechnique

578

Overview 57912-1 Fourier Series Analysis Technique 57912-2 Fourier Series Representation 52312-3 Circuit Applications 592TB22 Synthetic Biology 59612-4 Average Power 59812-5 Fourier Transform 59912-6 Fourier Transform Pairs 60512-7 Fourier versus Laplace 61112-8 Circuit Analysis with Fourier Transform 61212-9 Multisim: Mixed-Signal Circuits and the

Sigma-Delta Modulator613


Appendix A Symbols, Quantities,and Units

627

Appendix B Solving SimultaneousEquations

629

Appendix C Overview of Multisim 633

Appendix D Mathematical Formulas 636

Appendix E MATLAB and MathScript 639

Appendix F Answers to SelectedProblems

644

Index 645

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List of Technology Briefs

TB1 Micro- and Nanotechnology 28TB2 Superconductivity 47TB3 Resistors as Sensors 63TB4 Light-Emitting Diodes 109TB5 Integrated Circuit Fabrication

Process 120TB6 Display Technologies 155TB7 Computer Memory Circuits 184TB8 Supercapacitors 221TB9 Hard Disk Drives 257TB10 Micromechanical Sensors and

Actuators 287TB11 Touchscreens and Active

Digitizers 303

TB12 Crystal Oscillators 355TB13 Electromagnetic Spectrum 375TB14 Noise-Cancellation Headphones 419TB15 Night-Vision Imaging 325TB16 Smart Dust, Sensor Webs, and

Ubiquitous Computing 462TB17 Bandwidth, Data Rate, and

Communication 487TB18 Electrical Engineering and the

Audiophile 518TB19 Minaturized Energy Harvesting 526TB20 3-D TV 537TB21 Mapping the World in 3-D 563TB22 Synthetic Biology 596

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Preface

As the foundational course in the majority of electrical andcomputer engineering curricula, an Electric Circuits courseshould serve three vital objectives:

(1) It should introduce the fundamental principles of circuitanalysis and equip the student with the skills necessary toanalyze any planar, linear circuit, including those driven bydc or ac sources, or by more complicated waveforms such aspulses and exponentials.

(2) It should guide the student into the seemingly magicalworld of domain transformations—such as the Laplace andFourier transforms, not only as circuit analysis tools, but alsoas mathematical languages that are “spoken” by many fields ofscience and engineering.

(3) It should expand the student’s technical horizon byintroducing him/her to some of the many allied fields of scienceand technology.

This book aims to accomplish exactly those objectives.Among its distinctive features are:

Technology Briefs The book contains 22 Technology Briefs,each providing an overview of a topic that every electricaland computer engineering professional should become familiarwith. Electronic displays, data storage media, sensors andactuators, supercapacitors, and 3-D imaging are typical of thetopics shared with the reader. The Briefs are presented at atechnical level intended to challenge the reader to pursue thesubject further on his/her own.

Application Notes Most chapters include a section focusedon how certain devices or circuits might be used in practical

applications. Examples include power supplies, CMOSinverters in computer processors, signal modulators, and severalothers.

Multisim SPICE circuit simulators have been part of teachingand learning how circuits respond to electrical stimuli for at leastthe past two decades. Multisim, a relatively recent SPICE-based software simulator, has the distinct advantage over itspredecessors that it offers a friendlier computer-use interface,thereby making it easier to use and manipulate. In addition tointroducing its functionality through examples throughout thebook, Multisim is highlighted through 43 modules containedon the DVD-ROM accompanying the book. The student isstrongly encouraged to take advantage of this rich resource.

DVD-ROM The two DVD-ROMs accompanying the bookcontain:

(1) All Figures and Tables, and many of the major equations.

(2) Solutions to all of the Exercises contained in the book. Theicon on the text pages indicates that related material

can be found on the enclosed DVD.

(3) 43 Multisim Modules (see Appendix C for details).

(4) NI Multisim and LabVIEW Student Edition software.

(5) MathScript software, which can perform matrix inversionand many other calculations, much like The MathWorks,Inc. MATLAB software.

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xiv PREFACE

Second Edition Additions

• Two new chapters: one on three-phase circuits and anotheron magnetically coupled circuits.

• myDAQ tutorials: In addition to solving a circuit problemanalytically and simulating its behavior with Multisim, acomplementary third approach is to physically build thecircuit on a circuit board and use myDAQ to interface thecircuit to a computer. The interface allows the studentto use the computer both as a function generator toexcite the circuit and as a multimeter or oscilloscope to

measure its response. A myDAQ tutorial developed byProfessor Ed Doering of the Rose-Hulman Institute ofTechnology is available on the DVD accompanying thebook. This is in addition to video tutorials on how touse computer-based instruments and 36 end-of-chaptermyDAQ problems. Vendor information is available athttp://www.ni.com/mydaq.

• About 200 additional end-of-chapter problems, raising thetotal to 834.

• Updated Technology Briefs.

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AcknowledgmentsA science or engineering textbook is the product of an integratedeffort by many professionals. Invariably, the authors receivefar more of the credit than they deserve, for if it were not forthe creative talents of so many others, the book would neverhave been possible, much less a success. We are indebtedto many students and colleagues, most notably the followingindividuals:

Richard Carnes: For his meticulous typing of themanuscript, careful drafting of its figures, and overallstewardship of the project. Richard imparted the samecombination of precision and passion to the manuscript as healways does when playing Chopin on the piano.

Dr. Adib Nashashibi: For his superb attention to detail as the“Quality Control Officer” of the project. He checked many ofthe derivations in the text, as well as the solutions of numerousend-of-chapter problems.

Joe Steinmeyer: For testing the Multisim problems con-tained in the text and single-handedly developing all of theMultisim modules on the DVD-ROMs. Shortly thereafter, Joewent to MIT to pursue a Ph.D. in electrical engineering.

Professor Ed Doering: For developing a comprehensivetutorial that includes 36 circuit problems, each of whichis solved analytically, with Multisim, and with myDAQ.In addition, he created instructive video tutorials on howto use a variety of computer-based instruments, includingthe multimeter, oscilloscope, waveform generator, and Bodeanalyzer.

For their reviews of the overall manuscript and for offeringmany constructive criticisms, we are grateful to ProfessorsFred Terry and Jamie Phillips of the University of Michigan,

Keith Holbert of Arizona State University, Ahmad Safaai-Jaziof Virginia Polytechnic Institute and State University, RobinStrickland of the University of Arizona, and Frank Merat ofCase Western Reserve University. The manuscript was alsoscrutinized by a highly discerning group of University ofMichigan graduate students: Mike Benson, Fikadu Dagefu,Scott Rudolph, and Jane Whitcomb. Multisim sections werereviewed by Peter Ledochowitsch.

Many of the approximately 200 new end-of-chapter problemswere developed and solved by students from the Universityof Michigan and the University of California at Berkeley.They include Holly Chiang, David Hiskins, Tonmoy Monsoor,Zachary Hargeaves, James Dunn, Christopher Lo, ChrisBuonocore, and Randolf Tjandra. We thank them for theircontributions.

Editing and compositioning the manuscript to generate anappealing look in a functional format is an art unto itself. Ourthanks go to Paul Mailhot of PreTeX, Inc.

NTS Press offers an innovative approach to publishingscience and engineering textbooks. With today’s computer-savvy student in mind, NTS’s goal is to publish textbooks thathelp the student understand how the fundamentals connect toreal-world applications, and to market its books at affordableprices. NTS Press is the brainchild of Tom Robbins, an old handin the textbook publishing business, who recently decided thatthe time is ripe for a different publishing paradigm. We supportTom’s endeavor and we are grateful for the opportunity topublish this book under NTS Press, which provides a dedicatedweb site for the book (www.ntspress.com).

We enjoyed writing this book, and we hope you enjoylearning from it.

Fawwaz Ulaby and Michel Maharbiz

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Photo Credits

Page 2 (Figure 1-1): Fawwaz UlabyPage 4 (left) Dorling Kindersley/Getty Images; (right) c!

Bettmann/CORBIS; Chuck EbyPage 5 (left) Chuck Eby; John Jenkins, sparkmuseum.com; IEEE

History Center; History San Jose; (right) LC-USZ62-39702, Library of Congress; History San Jose; c!Bettmann/ CORBIS

Page 6 (left) MIT Museum; c! Bettmann/ CORBIS;Emilio Segre Visual Archives/American Institute ofPhysics/Science Photo Library; (right) Emilio Segre VisualArchives/American Institute of Physics/Science PhotoLibrary;

Page 7 (left) Courtesy of Dr. Steve Reyer; Courtesy of TexasInstruments Incorporated; NASA; Digital EquipmentCorporation; (right) Used with permission of SRIInternational; Courtesy of Texas Instruments Incorporated

Page 8 (left) Courtesy of IBM; Courtesy of Palra Inc., US Robotics,Inc.

Page 28 Courtesy Office of Basic Energy Sciences, Office ofScience, U.S. Department of Energy

Page 31 From “When will computer hardware match the humanbrain?” by Hans Moravec, Journal of Transhumanism, Vol.1, 1998 Used with permission

Page 47 Pacific Northwest National LaboratoryPage 48 Courtesy of Central Japan Railway Company; Courtesy

General Electric Healthcare

Page 64 Courtesy of Khalil Najafi, University of MichiganPage 120 Courtesy of Veljko MilanovicPage 123 Courtesy of International Business Machines CorporationPage 184 Courtesy of Intel CorporationPage 187 Courtesy of Prof. Ken Wise, University of MichiganPage 221 (photo) AP Photo/Rob Widdis; (figure) from Science Au-

gust, 18 2006, Vol. 313 (#5789) Reprinted with permissionof AAAS

Page 225 Fawwaz UlabyPage 258 c! Steve Allen/Brand X/CorbisPage 289 (left to right) Analog Devices; Courtesy of Prof. Khalil

Najafi, University of MichiganPage 290 Analog DevicesPage 355 AltzonePage 357 NISTPage 434 (left to right) c! Reuters/CORBIS; c! Colin Ander-

son/Brand X/CorbisPage 462 Courtesy of Prof. Kristopher J. Pister of the University of

California at BerkeleyPage 520 Martin LoganPage 563 NASAPage 564 NASAPage 564 ROBYN BECK/AFP/Getty ImagesPage 597 Aaron Chevalier and Nature (Nov. 24, 2005)Page 614 Courtesy of Renaldi Winoto

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C H A P T E R 1Circuit Terminology

ContentsCell-Phone Circuit Architecture, 2

1-1 Historical Timeline, 31-2 Units, Dimensions, and Notation, 81-3 Circuit Representation, 91-4 Electric Charge and Current, 131-5 Voltage and Power, 171-6 Circuit Elements, 23TB1 Micro- and Nanotechnology, 28

Chapter 1 Summary, 32Problems, 33

ObjectivesLearn to:

! Differentiate between active and passive devices;analysis and synthesis; device, circuit, andsystem; and dc and ac.

! Point to important milestones in the history ofelectrical and computer engineering.

! Relate electric charge to current; voltage toenergy; power to current and voltage; and applythe passive sign convention.

! Describe the properties of dependent andindependent sources.

! Describe the operation of SPST and SPDTswitches.

The iPhone is a perfect example of an integrated electronicarchitecture composed of a large number of interconnectedcircuits. Learning a new language starts with the alphabet.This chapter introduces the terms and conventions used in thelanguage of electronics.

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2 CHAPTER 1 CIRCUIT TERMINOLOGY

Cell-Phone Circuit Architecture

Electronic circuits are contained in just about every gadget weuse in daily living. In fact, electronic sensors, computers, anddisplays are at the operational heart of most major industries,from agricultural production and transportation to healthcareand entertainment. The ubiquitous cell phone (Fig. 1-1), whichhas become practically indispensable, is a perfect exampleof an integrated electronic architecture made up of a largenumber of interconnected circuits. It includes amplifier circuits,oscillators, frequency up- and down-converters, and circuitswith many other types of functions (Fig. 1-2). Factors such ascompatibility among the various circuits and proper electricalconnections between them are critically important to the overalloperation and integrity of the cell phone.

Usually, we approach electronic analysis and design througha hierarchical arrangement where we refer to the overall entityas a system, its subsystems as circuits, and the individualcircuit elements as devices or components. Thus, we mayregard the cell phone as a system (which is part of a muchlarger communication system); its audio-frequency amplifier,for example, as a circuit, and the resistors, integrated circuits(ICs), and other constituents of the amplifier as devices. Inactuality, an IC is a fairly complex circuit in its own right,but its input/output functionality is such that usually it can berepresented by a relatively simple equivalent circuit, thereby

Figure 1-1: Cell phone.

allowing us to treat it like a device. Generally, we refer todevices that do not require an external power source in order tooperate as passive devices; these include resistors, capacitors,and inductors. In contrast, an active device (such as a transistoror an IC) cannot function without a power source.

Received Signal

Transmitted SignalD/A and A/DConverters

andFilters

Antenna andPropagation

RF Front-End IF Block Back-End Baseband

~ ~

Demodulator

Modulator

MicroprocessorControl

LOLO

Human Interface,Dialing, Memory

Battery Power Control

Antenna

Transmit Path

Receive Path

Diplexer/Filter

RF Filter

RF Power Amp

Mixer

RF LowNoise Amp

Mixer

IF Amp

IF Amp

IF Filter

(Speech, video, data)

In Out

RF = Radio FrequencyIF = Intermediate FrequencyLO = Local OscillatorMixer = Frequency Up- or Down-Converter

Figure 1-2: Cell-phone block diagram.

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1-1 HISTORICAL TIMELINE 3

This book is about electric circuits. A student once asked:“What is the difference between an electric circuit and anelectronic circuit? Are they the same or different?” Strictlyspeaking, both refer to the flow of electric charge carriedby electrons, but historically, the term “electric” preceded“electronic,” and over time the two terms have come to signifydifferent things:

! An electric circuit is one composed of passive devices,in addition to voltage and current sources, and possiblysome types of switches. In contrast, the term “electronic”has become synonymous with transistors and other activedevices. "

The study of electric circuits usually precedes and sets the stagefor the study of electronic circuits, and even though a course onelectric circuits usually does not deal with the internal operationof an active device, it does incorporate active devices in circuitexamples by representing them in terms of equivalent circuits.

An electric circuit, as defined by Webster’s EnglishDictionary, is a “complete or partial path over which currentmay flow.” The path may be confined to a physical structure(such as a metal wire connecting two components), or it maybe an unbounded channel carrying electrons through it. Anexample of the latter is when a lightning bolt strikes theground, creating an electric current between a highly chargedatmospheric cloud and the earth’s surface.

The study of electric circuits consists of two complementarytasks: analysis and synthesis (Fig. 1-3). Through analysis, wedevelop an understanding of “how” a given circuit works. Ifwe think of a circuit as having an input—a stimulus—and anoutput—a response, the tools we use in circuit analysis allowus to relate mathematically the output response to the inputstimulus, enabling us to analytically and graphically “observe”the behavior of the output as we vary the relevant parameters ofthe input. An example might be a specific amplifier circuit,in which case the objective of circuit analysis might be toestablish how the output voltage varies as a function of theinput voltage over the full operational range of the amplifierparameters. By analyzing the operation of each circuit in asystem containing multiple circuits, we can characterize theoperation of the overall system.

As a process, synthesis is the reverse of analysis. Inengineering, we tend to use the term design as a synonym forsynthesis. The design process usually starts by defining theoperational specifications that a gadget or system should meet,and then we work backwards (relative to the analysis process)to develop circuits that will satisfy those specifications. Inanalysis, we are dealing with a single circuit with a specific

Circuit

Circuit

Functionality

Specs

Analysis

Analysis vs. Synthesis

Synthesis

(Design)

Figure 1-3: The functionality of a circuit is discerned by applying thetools of circuit analysis. The reverse process, namely the realizationof a circuit whose functionality meets a set of specifications, is calledcircuit synthesis or design.

set of operational characteristics. We may employ differentanalysis tools and techniques, but the circuit is unique, andso are its operational characteristics. That is not necessarilythe case for synthesis; the design process may lead to multiplecircuit realizations—each one of which exhibits or satisfies thedesired specifications.

Given the complementary natures of analysis and synthesis,it stands to reason that developing proficiency with the toolsof circuit analysis is a necessary prerequisite to becoming asuccessful design engineer. This textbook is intended to providethe student with a solid foundation of the primary set of toolsand mathematical techniques commonly used to analyze bothdirect current (dc) and alternating current (ac) circuits, aswell as circuits driven by pulses and other types of waveforms.A dc circuit is one in which voltage and current sources areconstant as a function of time, whereas in ac circuits, sourcesvary sinusoidally with time. Even though this is not a bookon circuit design, design problems occasionally are introducedinto the discussion as a way to illustrate how the analysis andsynthesis processes complement each other.

Concept Question 1-1: What are the differences betweena device, a circuit, and a system?

Concept Question 1-2: What is the difference betweenanalysis and synthesis?

1-1 Historical Timeline

We live today in the age of electronics. No field of scienceor technology has had as profound an influence in shaping theoperational infrastructure of modern society as has the field ofelectronics. Our computers and communication systems are atthe nexus of every major industry, from food production andtransportation to health care and entertainment. Even though

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no single event marks the beginning of a discipline, electricalengineering became a recognized profession sometime inthe late 1800s (see chronology). Alexander Graham Bellinvented the telephone (1876); Thomas Edison perfectedhis incandescent light bulb (1880) and built an electricaldistribution system in a small area in New York City;Heinrich Hertz generated radio waves (1887); and GuglielmoMarconi demonstrated radio telegraphy (1901). The next50 years witnessed numerous developments, including radiocommunication, TV broadcasting, and radar for civilian andmilitary applications—all supported by electronic circuitry thatrelied entirely on vacuum tubes. The invention of the transistorin 1947 and the development of the integrated circuit (IC)shortly thereafter (1958) transformed the field of electronics bysetting it on an exponentially changing course towards “smaller,faster, and cheaper.”

Computer engineering is a relatively young discipline.The first all-electronic computer, the ENIAC, was built anddemonstrated in 1945, but computers did not become availablefor business applications until the late 1960s and for personaluse until the introduction of Apple I in 1976. Over the past 20years, not only have computer and communication technologiesexpanded at a truly impressive rate (see Technology Brief 1 onpage 28), but more importantly, it is the seamless integrationof the two technologies that has made so many business andpersonal applications possible.

Generating a comprehensive chronology of the events anddiscoveries that have led to today’s technologies is beyond thescope of this book, but ignoring the subject altogether wouldbe a disservice to both the reader and the subject of electriccircuits. The abbreviated chronology presented on the next fewpages represents our compromise solution.

Chronology: Major Discoveries, Inventions, andDevelopments in Electrical and ComputerEngineeringca. 1100 BC Abacus is the earliest known calculating device.

ca. 900 BC According to legend, a shepherd in northern Greece, Magnus,experiences a pull on the iron nails in his sandals by the black rock hewas standing on. The rock later became known as magnetite [a form ofiron with permanent magnetism].

ca. 600 BC Greek philosopher Thales describes how amber, after being rubbedwith cat fur, can pick up feathers [static electricity].

1600 William Gilbert (English) coins the term electric after the Greek word foramber (elektron) and observes that a compass needle points north tosouth because the Earth acts as a bar magnet.

1614 John Napier (Scottish) develops the logarithm system.

1642 Blaise Pascal (French) builds the first adding machine using multipledials.

1733 Charles Francois du Fay (French) discovers that electric charges areof two forms and that like charges repel and unlike charges attract.

1745 Pieter van Musschenbroek (Dutch) invents the Leyden jar, the firstelectrical capacitor.

1800 Alessandro Volta (Italian) develops the first electric battery.

1827 Georg Simon Ohm (German) formulates Ohm’s law relating electricpotential to current and resistance.

1827 Joseph Henry (American) introduces the concept of inductance andbuilds one of the earliest electric motors. He also assisted Samuel Morsein the development of the telegraph.

1837 Samuel Morse (American) patents the electromagnetic telegraph usinga code of dots and dashes to represent letters and numbers.

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1876 Alexander Graham Bell (Scottish-American) invents the telephone: therotary dial becomes available in 1890, and by 1900, telephone systemsare installed in many communities.

1879 Thomas Edison (American) demonstrates the operation of theincandescent light bulb, and in 1880, his power distribution systemprovided dc power to 59 customers in New York City.

1887 Heinrich Hertz (German) builds a system that can generate electromag-netic waves (at radio frequencies) and detect them.

Courtesy of John Jenkins (sparkmuseum.com)

1888 Nikola Tesla (Croatian-American) invents the ac motor.

1893 Valdemar Poulsen (Danish) invents the first magnetic sound recorderusing steel wire as recording medium.

1895 Wilhelm Rontgen (German) discovers X-rays. One of his first X-rayimages was of the bones in his wife’s hands. [1901 Nobel prize inphysics.]

1896 Guglielmo Marconi (Italian) files his first of many patents on wirelesstransmission by radio. In 1901, he demonstrates radio telegraphy acrossthe Atlantic Ocean. [1909 Nobel prize in physics, shared with Karl Braun(German).]

1897 Karl Braun (German) invents the cathode ray tube (CRT). [1909 Nobelprize, shared with Marconi.]

1897 Joseph John Thomson (English) discovers the electron and measuresits charge-to-mass ratio. [1906 Nobel prize in physics.]

1902 Reginald Fessenden (American) invents amplitude modulation fortelephone transmission. In 1906, he introduces AM radio broadcastingof speech and music on Christmas Eve.

1904 John Fleming (British) patents the diode vacuum tube.

1907 Lee De Forest (American) develops the triode tube amplifier for wirelesstelegraphy, setting the stage for long-distance phone service, radio, andtelevision.

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1917 Edwin Howard Armstrong (American) invents the superheterodyneradio receiver, dramatically improving signal reception. In 1933, hedevelops frequency modulation (FM), providing superior sound qualityof radio transmissions over AM radio.

1920 Birth of commercial radio broadcasting; Westinghouse Corporationestablishes radio station KDKA in Pittsburgh, Pennsylvania.

1923 Vladimir Zworykin (Russian-American) invents television. In 1926,John Baird (Scottish) transmits TV images over telephone wires fromLondon to Glasgow. Regular TV broadcasting began in Germany (1935),England (1936), and the United States (1939).

1926 Transatlantic telephone service established between London and NewYork.

1930 Vannevar Bush (American) develops the differential analyzer, an analogcomputer for solving differential equations.

1935 Robert Watson-Watt (Scottish) invents radar.

1945 John Mauchly and J. Presper Eckert (both American) develop theENIAC, the first all-electronic computer.

1947 William Shockley, Walter Brattain, and John Bardeen (all Americans)invent the junction transistor at Bell Labs. [1956 Nobel prize in physics.]

1948 Claude Shannon (American) publishes his Mathematical Theory ofCommunication, which formed the foundation of information theory,coding, cryptography, and other related fields.

1950 Yoshiro Nakama (Japanese) patents the floppy disk as a magneticmedium for storing data.

1954 Texas Instruments introduces the first AM transistor radio.

Courtesy of Dr. Steve Reyer

1955 The pager is introduced as a radio communication product in hospitalsand factories.

1955 Narinder Kapany (Indian-American) demonstrates optical fiber as a low-loss, light-transmission medium.

1956 John Backus (American) develops FORTRAN, the first major program-ming language.

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1958 Charles Townes and Arthur Schawlow (both Americans) develop theconceptual framework for the laser. [Townes shared 1964 Nobel prize inphysics with Aleksandr Prokhorov and Nicolay Bazov (both Soviets).] In1960 Theodore Maiman (American) builds the first working model of alaser.

1958 Bell Labs develops the modem.

1958 Jack Kilby (American) builds the first integrated circuit (IC) ongermanium, and independently, Robert Noyce (American) builds thefirst IC on silicon.

1959 Ian Donald (Scottish) develops an ultrasound diagnostic system.

1960 Echo, the first passive communication satellite is launched andsuccessfully reflects radio signals back to Earth. In 1962, the firstcommunication satellite, Telstar, is placed in geosynchronous orbit.

1960 Digital Equipment Corporation introduces the first minicomputer, thePDP-1, which was followed with the PDP-8 in 1965.

1962 Steven Hofstein and Frederic Heiman (both American) invent theMOSFET, which became the workhorse of computer microprocessors.

1964 IBM’s 360 mainframe becomes the standard computer for majorbusinesses.

1965 John Kemeny and Thomas Kurtz (both American) develop the BASICcomputer language.

1965 Konrad Zuse (German) develops the first programmable digitalcomputer using binary arithmetic and electric relays.

1968 Douglas Engelbart (American) demonstrates a word-processor system,the mouse pointing device, and the use of a Windows-like operatingsystem.

1969 ARPANET is established by the U.S. Department of Defense, which is toevolve later into the Internet.

1970 James Russell (American) patents the CD-ROM, as the first systemcapable of digital-to-optical recording and playback.

1971 Texas Instruments introduces the pocket calculator.

Courtesy of Texas Instruments

1971 Intel introduces the 4004 four-bit microprocessor, which is capable ofexecuting 60,000 operations per second.

1972 Godfrey Hounsfield (British) and Alan Cormack (South African–American) develop the computerized axial tomography scanner (CATscan) as a diagnostic tool. [1979 Nobel Prize in physiology or medicine.]

1976 IBM introduces the laser printer.

1976 Apple Computer sells Apple I in kit form, followed by the fully assembledApple II in 1977, and the Macintosh in 1984.

1979 Japan builds the first cellular telephone network:

• 1983 cellular phone networks start in the United States.

• 1990 electronic beepers become common.

• 1995 cell phones become widely available.

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1980 Microsoft introduces the MS-DOS computer disk operating system.Microsoft Windows is marketed in 1985.

1981 IBM introduces the PC.

1984 Worldwide Internet becomes operational.

1988 First transatlantic optical fiber cable between the U.S. and Europe isoperational.

1989 Tim Berners-Lee (British) invents the World Wide Web by introducing anetworking hypertext system.

1996 Sabeer Bhatia (Indian-American) and Jack Smith (American) launchHotmail as the first webmail service.

1997 Palm Pilot becomes widely available.

1997 The 17,500-mile fiber-optic cable extending from England to Japan isoperational.

2002 Cell phones support video and the Internet.

2007 The power-efficient White LED invented by Shuji Nakamura (Japanese)in the 1990s promises to replace Edison’s lightbulb in most lightingapplications.

2007 Apple iPhone released.

2009 Cloud computing goes mainstream. Companies begin to providecomputation as a scalable service (via the Internet) rather than asa product (i.e., selling physical computers to users). This allowsfor scalable on-demand computing from anywhere where an Internetconnection exists!

2011 IBM’s Watson supercomputer beats the top two human contestants ofJeopardy! for a $1M prize.

2011 8 ! 1012 (8 trillion) text messages sent worldwide.

2012 Approximately 85% of the world population is a mobile phone subscriber(5.9 billion people).

Table 1-1: Fundamental SI units.

Dimension Unit Symbol

Length meter mMass kilogram kgTime second sElectric Current ampere ATemperature kelvin KAmount of substance mole mol

1-2 Units, Dimensions, and NotationThe standard system used in today’s scientific literature toexpress the units of physical quantities is the InternationalSystem of Units (SI), abbreviated after its French name SystemeInternationale. Time is a fundamental dimension, and thesecond is the unit by which it is expressed relative to a specificreference standard. The SI configuration is based on the sixfundamental dimensions listed in Table 1-1, and their units arecalled Fundamental SI units. All other dimensions, such asvelocity, force, and energy, are regarded as secondary becausetheir units are based on and can be expressed in terms of the sixfundamental units. Appendix A provides a list of the quantitiesused in this book, together with their symbols and units.

In science and engineering, a set of prefixes commonly areused to denote multiples and submultiples of units. Theseprefixes, ranging in value between 10"18 and 1018, are listed inTable 1-2. An electric current of 3 ! 10"6 A, for example, maybe written as 3 µA.

Table 1-2: Multiple and submultiple prefixes.

Prefix Symbol Magnitude

exa E 1018

peta P 1015

tera T 1012

giga G 109

mega M 106

kilo k 103

milli m 10"3

micro µ 10"6

nano n 10"9

pico p 10"12

femto f 10"15

atto a 10"18

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1-3 CIRCUIT REPRESENTATION 9

The physical quantities we will discuss in this book (such asvoltage and current) may be constant in time or may vary withtime.

As a general rule, we shall use:

• A lowercase letter, such as i for current, to representthe general case:

i may or may not be time varying

• A lowercase letter followed with (t) to emphasizetime:

i(t) is a time-varying quantity

• An uppercase letter if the quantity is not timevarying; thus:

I is of constant value (dc quantity)

• A letter printed in boldface to denote that:

I has a specific meaning, such as a vector, amatrix, the phasor counterpart of i(t), or the Laplaceor Fourier transform of i(t)

Exercise 1-1: Convert the following quantities toscientific notation: (a) 52 mV, (b) 0.3 MV, (c) 136 nA,and (d) 0.05 Gbits/s.

Answer: (a) 5.2 ! 10"2 V, (b) 3 ! 105 V,(c) 1.36 ! 10"7 A, and (d) 5 ! 107 bits/s. (See )

Exercise 1-2: Convert the following quantities to a prefixformat such that the number preceding the prefix isbetween 1 and 999: (a) 8.32!107 Hz, (b) 1.67!10"8 m,(c) 9.79 ! 10"16 g, (d) 4.48 ! 1013 V, and (e) 762 bits/s.

Answer: (a) 83.2 MHz, (b) 16.7 nm, (c) 979 ag,(d) 44.8 TV, and (e) 762 bits/s. (See )

Exercise 1-3: Simplify the following operationsinto a single number, expressed in prefix format:(a) A = 10 µV + 2.3 mV, (b) B = 4THz " 230 GHz,(c) C = 3 mm/60 µm.

Answer: (a) A = 2.31 mV, (b) B = 3.77 THz,(c) C = 50. (See )

1-3 Circuit Representation

Figure 1-4 contains three parts: (a) a photograph of acircuit, designed as a capacitor-touch-sensor, (b) the circuit’sprinted-circuit-board (PCB) layout, and (c) a circuit diagramrepresenting the circuit’s electrical configuration. The circuit

(a) Actual circuit

(b) PC board

(c) Circuit diagram

Metal plate Capacitor IC Resistor5-V power supply to be connected here

Output voltage

Figure 1-4: (a) Photograph of a touch-sensor circuit, (b) printed-circuit-board (PCB) layout, and (c) circuit diagram.

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includes a flat conducting plate, two ICs, one diode, and severalresistors and capacitors. When the plate is touched by a finger,the capacitance introduced by the finger causes the outputvoltage to rise above a preset threshold, signifying the fact thatthe plate has been touched. The voltage rise can then be usedto trigger a follow-up circuit.

The PCB layout shown in part (b) of Fig. 1-4 displaysthe intended locations of the circuit elements and the printedconducting lines needed to connect the elements to each other.These lines are used in lieu of wires. The diagram in part (c)is the symbolic representation of the physical circuit. In thisparticular representation the resistors are drawn as rectangularboxes instead of the more familiar symbol . Designingthe PCB layout and the circuit’s physical architecture is animportant step in the production process, but it is outside thescope of this book. Our prime interest is to help the readerunderstand how circuits work, and to use that understanding todesign circuits to perform functions of interest. Accordingly,circuit diagrams will be regarded as true representations ofthe many circuits and systems we will discuss in this and thefollowing chapters.

1-3.1 Circuit Elements

Table 1-3 provides a partial list of the symbols used in this bookto represent circuit elements in circuit diagrams.

By way of an example, the diagram in Fig. 1-5 contains thefollowing elements:

• A 12-V ac source, denoted by the symbol +!~

• A 6-V dc source, denoted by the symbol+_

• Six resistors, all denoted by the symbol

• One capacitor, denoted by the symbol

• One inductor, denoted by the symbol

• An important integrated circuit known as an operationalamplifier (or op amp for short), denoted by a triangularsymbol (the wiring internal to the op amp is not shown).

1-3.2 Circuit Architecture

The vocabulary commonly used to describe the architecture ofan electric circuit includes a number of important terms. Short,but precise, definitions follow.

Table 1-3: Symbols for common circuit elements.

Conductor Two conductorselectrically joined

at node A

orA A

Two conductorsnot joinedelectrically

Fixed-valueresistor

Variable resistor Capacitor

Inductor 10-V dcbattery

10 V+_

12-V acsource

12 V +!~

6-A currentsource

6 A

Switch Operationalamplifier

+_

Transistor Voltmeter

Volts

Ammeter

AmpsI

• Ordinary node: an electrical connection point thatconnects to only two elements, such as the point betweenthe ac source and R1 in Fig. 1-5.

• Extraordinary node: node connected to three or moreelements. Figure 1-5 contains four extraordinary nodes,denoted N1 through N4, of which N4 has been selected as

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1-3 CIRCUIT REPRESENTATION 11

Ordinary nodeExtraordinarynode

Op amp

Same nodeGround

Conductingwire

Loop 1Loop 2

Capacitor

Inductor

Branch containing R1

N2

N3

N1

N4

R1 R3

R2

R6R5L

C

R4

+_

+_

+!~!1 = 12 cos (377t) V

!2 = 6 Vdc source

ac source

Figure 1-5: Diagram representing a circuit that contains dc and ac sources, passive elements (six resistors, one capacitor, and one inductor), andone active element (operational amplifier).

a reference voltage node, often referred to as the groundnode. When two points with no element between themare connected by a conducting wire, they are regarded asthe same node. Hence, the four connection points locatedat the bottom of the circuit in Fig. 1-5 represent the samenode, which in this case is the ground node.

• Branch: the trace between two consecutive nodescontaining one and only one element between them.

• Path: any continuous sequence of branches, provided thatno one node is encountered more than once. The pathbetween nodes N1 and N2 consists of two branches, onecontaining R3 and another containing C.

• Loop: a closed path in which the start and end node is oneand the same. Figure 1-5 contains several loops, of whichtwo are shown explicitly.

• Mesh: a loop that encloses no other loop. In Fig. 1-5,Loop 1 is a mesh, but Loop 2 is not.

• In-series: elements that share the same current,requiring that all nodes along the path containing the in-series elements be ordinary nodes. In Fig. 1-5, the twosources and R1 are all in series, as are R2 and L, and R3and C.

• In-parallel: elements that share the same voltage, whichmeans they share the same pair of nodes. In Fig. 1-5 the

series combination (!2 ! !1 ! R1) is in parallel with theseries combination (R2 ! L).

A summary of circuit terminology is given in Table 1-4.

Table 1-4: Circuit terminology.

Ordinary node: An electrical connection point thatconnects to only two elements.

Extraordinary node: An electrical connection point thatconnects to three or more elements.

Branch: Trace between two consecutive nodes with onlyone element between them.

Path: Continuous sequence of branches with no nodeencountered more than once.

Extraordinary path: Path between two adjacentextraordinary nodes.

Loop: Closed path with the same start and end node.

Independent loop: Loop containing one or morebranches not contained in any other independent loop.

Mesh: Loop that encloses no other loops.

In-series: Elements that share the same current.

In-parallel: Elements that share the same voltage.

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Example 1-1: In-Series and In-Parallel

For the circuits in Fig. 1-6, which elements, or combination ofelements, are connected in series and which are connected inparallel?

(a)

(b)

12 V 5 V6 !4 !

1 ! 8 !

1 V

+_+_

+ _

+_

+_

+ _

2 ! 6 V4 ! 2 ! 4 !+_

Figure 1-6: Circuits for Example 1-1.

Solution: Two or more elements are connected electrically inseries if the same current flows through all of them, and theyare connected in parallel if they share the same nodes.

(a) Circuit in Fig. 1-6(a):

In-series: 8-! resistor and 5-V voltage source (call itcombination 1).

In-series: 1-! resistor and 12-V voltage source (call itcombination 2).

In-parallel: 6-! resistor and combination 1.

In-parallel: 4-! resistor and combination 2.

(b) Circuit in Fig. 1-6(b):

In-series: none.

In-parallel: all five elements.

1-3.3 Planar Circuits

! A circuit is planar if it is possible to draw it on a two-dimensional plane without having any two of its branchescross over or under one another (Fig. 1-7). "

If such a crossing is unavoidable, then the circuit is nonplanar.To clarify what we mean, we start by examining the circuitin Fig. 1-7(a). An initial examination of the circuit topologymight suggest that the circuit is nonplanar because the branchescontaining resistors R3 and R4 appear to cross one anotherwithout having physical contact between them (absence of asolid dot at crossover point). However, if we redraw the branchcontaining R4 on the outside, as shown in configuration (b) ofFig. 1-7, we would then conclude that the circuit is planar after

Redrawn(b)

Original circuit(a)

R1

R3

!0

R5

R4

R2+-+_

not a nodeR1

R3

R4

!0

R5

R2+-+_

Figure 1-7: The branches containing R3 and R4 in (a) appear tocross over one another, but redrawing the circuit as in (b) avoids thecrossover, thereby demonstrating that the circuit is planar.

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1-4 ELECTRIC CHARGE AND CURRENT 13

R1

R3

R4!0

R5

R2

R7R9R8R6

+-+_

Figure 1-8: Nonplanar circuit.

all, and that is so because it is possible to draw it in a singleplane without crossovers. In contrast, the circuit in Fig. 1-8 isindeed nonplanar because no matter how we might try to redrawit, it will always include at least one crossover of branches.

! The planar-circuit condition shall be presumed to betrue throughout the material covered in this book. "

Concept Question 1-3: What is the difference betweenthe symbol for a dc voltage source and that for an acsource?

Concept Question 1-4: What differentiates anextraordinary node from an ordinary node? A loop froma mesh?

1-4 Electric Charge and Current

1-4.1 Charge

At the atomic scale, all matter contains a mixture of neutrons,positively charged protons, and negatively charged electrons.The nature of the force induced by electric charge wasestablished by the French scientist Charles Augustin deCoulomb (1736–1806) during the latter part of the 18th century.This was followed by a series of experiments on electricityand magnetism over the next 100 years, culminating in J. J.Thompson’s discovery of the electron in 1897. Through theseand more recent investigations, we can ascribe to electric chargethe following fundamental properties.

!

1. Charge can be either positive or negative.

2. The fundamental quantity of charge is that of a singleelectron or proton. Its magnitude usually is denotedby the letter e.

3. According to the law of conservation of charge, the(net) charge in a closed region can neither be creatednor destroyed.

4. Two like charges repel one another, whereas twocharges of opposite polarity attract. "

The unit for charge is the coulomb (C) and the magnitude of e

is

e = 1.6 ! 10"19 (C). (1.1)

The symbol commonly used to represent charge is q. Thecharge of a single proton is qp = e, and that of an electron,which is equal in magnitude but opposite in polarity, is qe = "e.It is important to note that the term charge implies “net charge,”which is equal to the combined charge of all protons present inany given region of space minus the combined charge of allelectrons in that region. Hence, charge is always an integralmultiple of e.

The last of the preceding properties is responsible for themovement of charge from one location to another, therebyconstituting an electric current. Consider the simple circuit

RVI

e-

e- e-

Atom

Expanded view of wire

Electron

+_

Figure 1-9: The current flowing in the wire is due to electron transportthrough a drift process, as illustrated by the magnified structure of thewire.

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in Fig. 1-9 depicting a battery of voltage V connected across aresistor R using metal wires. The arrangement gives rise to anelectric current given by Ohm’s law (which will be discussedin more detail in Chapter 2):

I = V

R. (1.2)

As shown in Fig. 1-9:

! The current flows from the positive (+) terminal ofthe battery to its negative (!) terminal, along the pathexternal to the battery. "

Through chemical or other means, the battery generates a supplyof electrons at its negatively labeled terminal by ionizing someof the molecules of its constituent material. A convenient modelfor characterizing the functionality of a battery is to regard theinternal path between its terminals as unavailable for the flowof charge, forcing the electrons to flow from the (!) terminal,through the external path, and towards the (+) terminal toachieve neutrality. It is important to note that:

! The direction of electric current is defined to be thesame as the direction of flow that positive charges wouldfollow, which is opposite to the direction of flow ofelectrons. "

Even though we talk about electrons flowing through the wiresand the resistor, in reality the process is a drift movementrather than free-flow. The wire material consists of atomswith loosely attached electrons. The positive polarity of the(+) terminal exerts an attractive force on the electrons of thehitherto neutral atoms adjacent to that terminal, causing someof the loosely attached electrons to detach and jump to the (+)terminal. The atoms that have lost those electrons now becomepositively charged (ionized), thereby attracting electrons fromtheir neighbors and compelling them to detach from their hostsand to attach themselves to the ionized atoms instead. Thisprocess continues throughout the wire segment (between

the (+) battery terminal and the resistor), into the longitudinalpath of the resistor, and finally through the wire segmentbetween the resistor and the (!) terminal. The net result isthat the (!) terminal loses an electron and the (+) terminalgains one, making it appear as if the very same electron thatleft the (!) terminal actually flowed through the wires and theresistor and finally appeared at the (+) terminal. It is as if the

i

60 m

t = 0

Switch

Wire

100 !8 V+_

Figure 1-10: After closing the switch, it takes only 0.2 µs to observea current in the resistor.

path itself were not involved in the electron transfer, which isnot the case.

The process of sequential migration of electrons from oneatom to the next is called electron drift, and it is this processthat gives rise to the flow of conduction current through acircuit. To illustrate how important this process is in terms ofthe electronic transmission of information, let us examine theelementary transmission experiment represented by the circuitshown in Fig. 1-10. The circuit consists of an 8-V battery and aswitch on one end, a resistor on the other end, and a 60-m-longtwo-wire transmission line in between. The wires are madeof copper, and they have a circular cross section with a 2-mmdiameter. After closing the switch, a current will start to flowthrough the circuit. It is instructive to compare two velocitiesassociated with the consequence of closing the switch, namelythe actual (physical) drift velocity of the electrons inside thecopper wires and the transmission velocity (of the informationannouncing that the switch has been closed) between the batteryand the resistor. For the specified parameters of the circuitshown in Fig. 1-10, the electron drift velocity—which is theactual physical velocity of the electrons along the wire—canbe calculated readily and shown to be on the order of only10!4 m/s. Hence, it would take about 1 million seconds (" 10days) for an electron to physically travel over a distance of120 m. In contrast, the time delay between closing the switchat the sending end and observing a response at the receiving end(in the form of current flow through the resistor) is extremelyshort (# 0.2 µs). This is because the transmission velocity ison the order of the velocity of light c = 3 $ 108 m/s. Thus:

! The rate at which information can be transmittedelectronically using conducting wires is about 12 ordersof magnitude faster than the actual transport velocity ofthe electrons flowing through those wires! "

This fact is at the heart of what makes electronic communicationsystems possible.

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1-4 ELECTRIC CHARGE AND CURRENT 15

!

!

!

!

!

!

!

!

iCurrent direction

Direction of electron flow

Electron

Cross sectionWire

Figure 1-11: Direction of (positive) current flow through a conductoris opposite that of electrons.

1-4.2 Current

Moving charge gives rise to current.

! Electric current is defined as the time rate of transferof electric charge across a specified boundary. "

For the wire segment depicted in Fig. 1-11, the current i flowingthrough it is equal to the amount of charge dq that crosses thewire’s cross section over an infinitesimal time duration dt , givenas

i = dq

dt(A), (1.3)

and the unit for current is the ampere (A). In general, bothpositive and negative charges may flow across the hypotheticalinterface, and the flow may occur in both directions.

! By convention, the direction of i is defined to be thedirection of the net flow of (net) charge (positive minusnegative). "

The circuit segment denoted with an arrow in Fig. 1-12(a)signifies that a current of 5 A is flowing through that wire

Circuit 5 A Circuit !5 A

(a) (b)

Figure 1-12: A current of 5 A flowing “downward” is the same as!5 A flowing “upward” through the wire.

segment in the direction of the arrow. The same informationabout the current magnitude and direction may be displayed asin Fig. 1-12(b), where the arrow points in the opposite directionand the current is expressed as !5 A.

When a battery is connected to a circuit, the resultant currentthat flows through it usually is constant in time [Fig. 1-13(a)]—at least over the time duration of interest—in which case werefer to it as a direct current or dc for short. In contrast,the currents flowing in household systems (as well as in manyelectrical systems) are called alternating currents or simply ac,because they vary sinusoidally with time [Fig. 1-13(b)]. Othertime variations also may occur in circuits, such as exponentialdecays and rises [Fig. 1-13(c) and (d)], exponentially dampedoscillations [Fig. 1-13(e)], and many others.

Even though in the overwhelming majority of cases thecurrent flowing through a material is dominated by themovement of electrons (as opposed to positively charged ions),it is advisable to start thinking of the current in terms of positivecharge, primarily to avoid having to keep track of the fact thatcurrent direction is defined to be in opposition to the directionof flow of negative charges.

(b)(a)

dc

t

I

(d)(c)

Decaying

t

i(t)

Rising

t

i(t)

(e)

Damped oscillatoryi(t)

t

ac

t

i(t)

Figure 1-13: Graphical illustrations of various types of currentvariations with time.

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Example 1-2: Charge Transfer

In terms of the current i(t) flowing past a reference cross sectionin a wire:

(a) Develop an expression for the cumulative chargeq(t) thathas been transferred past that cross section up to time t . Applythe result to the exponential current displayed in Fig. 1-14(a),which is given by

i(t) =!

0 for t < 0,

6e!0.2t A for t " 0.(1.4)

(b) Develop an expression for the net charge !Q(t1, t2) thatflowed through the cross section between times t1 and t2, andthen compute !Q for t1 = 1 s and t2 = 2 s.

(a)

Current

Charge

(b)

t

i(t)

6 A

t

q(t)

30 C

Figure 1-14: The current i(t) displayed in (a) generates the cumulativecharge q(t) displayed in (b).

Solution:(a) We start by rewriting Eq. (1.3) in the form:

dq = i dt.

Then by integrating both sides over the limits !# to t , we have

t"

!#dq =

t"

!#i dt,

which yields

q(t) ! q(!#) =t"

!#i dt, (1.5)

where q(!#) represents the charge that was transferredthrough the wire “at the beginning of time.” We choose !#as a reference limit in the integration, because it allows us toset q(!#) = 0, implying that no charge had been transferredprior to that point in time. Hence, Eq. (1.5) becomes

q(t) =t"

!#i dt (C). (1.6)

For i(t) as given by Eq. (1.4), i(t) = 0 for t < 0. Uponchanging the lower integration limit to zero and inserting theexpression for i(t) in Eq. (1.6), the integration leads to

q(t) =t"

0

6e!0.2t dt = !60.2

e!0.2t###t

0= 30[1 ! e!0.2t ] C.

A plot of q(t) versus t is displayed in Fig. 1-14(b). Thecumulative charge that would transfer after a long periodof time is obtained by setting t = +#, which would yieldq(+#) = 30 C.

(b) The cumulative charge that has flowed through the crosssection up to time t1 is q(t1), and a similar definition appliesto q(t2). Hence, the net charge that flowed through the crosssection over the time interval between t1 and t2 is

!Q(t1, t2) = q(t2) ! q(t1)

=t2"

!#i dt !

t1"

!#i dt =

t2"

t1

i dt.

For t1 = 1 s, t2 = 2 s, and i(t) as given by Eq. (1.4),

!Q(1, 2) =2"

1

6e!0.2t dt = 6e!0.2t

!0.2

####2

1

= !30(e!0.4 ! e!0.2)

= 4.45 C.

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1-5 VOLTAGE AND POWER 17

Example 1-3: Current

The charge flowing past a certain location in a wire is given by

q(t) =!

0 for t < 0,

5te!0.1t C for t " 0.

Determine (a) the current at t = 0 and (b) the instant at whichq(t) is a maximum and the corresponding value of q.

Solution:(a) Application of Eq. (1.3) yields

i = dq

dt

= d

dt(5te!0.1t )

= 5e!0.1t ! 0.5te!0.1t

= (5 ! 0.5t)e!0.1t A.

Setting t = 0 in the expression gives i(0) = 5 A.Note that i #= 0, even though q(t) = 0 at t = 0.

(b) To determine the value of t at which q(t) is a maximum,we find dq/dt and then set it equal to zero:

dq

dt= (5 ! 0.5t)e!0.1t

= 0,

which is satisfied when

5 ! 0.5t = 0 or t = 10 s,

as well as when

e!0.1t = 0 or t = $.

The first value (t = 10 s) corresponds to a maximum and thesecond value (t = $) corresponds to a minimum (which canbe verified either by graphing q(t) or by taking the secondderivative of q(t) and evaluating it at t = 10 s and t = $).

At t = 10 s,

q(10) = 5 % 10e!0.1%10 = 50e!1 = 18.4 C.

Concept Question 1-5: What are the four fundamentalproperties of electric charge?

Concept Question 1-6: Is the direction of electric currentin a wire defined to be the same as or opposite to thedirection of flow of electrons?

Concept Question 1-7: How does electron drift lead tothe conduction of electric current?

Exercise 1-4: If the current flowing through a givenresistor in a circuit is given by i(t) = 5[1 ! e!2t ] A fort " 0, determine the total amount of charge that passedthrough the resistor between t = 0 and t = 0.2 s.

Answer: !Q(0, 0.2) = 0.18 C. (See )

Exercise 1-5: If q(t) has the waveform shown inFig. E1-5, determine the corresponding current wave-form.

q(t)

t (s)4 6 7 8

2 C

51 32

Figure E1-5

Answer:

i(t)

t (s)6

2 A

!2 A51 2 4 7 83

(See )

1-5 Voltage and Power

1-5.1 Voltage

The two primary quantities used in circuit analysis are currentand voltage. Current is associated with the movement ofelectric charge and voltage is associated with the polarity ofcharge. Before we offer a formal definition for voltage, let usexamine the energy implications of polarizing a hitherto neutral

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!ab

a

b

e_

e_

e_

Any material

Figure 1-15: The voltage !ab is equal to the amount of energy requiredto move one unit of negative charge from a to b through the material.

material, thereby establishing opposite electrical polarities onits two ends. Suppose we have a piece of material (such as aresistor) to which we connect two short wires and label theirend points a and b, as shown in Fig. 1-15. Starting out with anelectrically neutral structure, assume that we are able to detachan electron from one of the atoms at point a and move it topoint b. Moving a negative charge from the positively chargedatom against the attraction force between them requires theexpenditure of a certain amount of energy. Voltage is a measureof this expenditure of energy relative to the amount of chargeinvolved, and it always involves two spatial locations:

! Voltage often is denoted !ab to emphasize the fact thatit is the voltage difference between points a and b. "

The two points may be two locations in a circuit or any twopoints in space.

Against this background, we now offer the following formaldefinition for voltage:

! The voltage between location a and location b is theratio of dw to dq, where dw is the energy in joules (J)required to move (positive) charge dq from b to a (ornegative charge from a to b). "

That is,

!ab = dw

dq, (1.7)

and the unit for voltage is the volt (V), named after the inventorof the first battery, Alessandro Volta (1745–1827). Voltage alsois called potential difference. In terms of that terminology, if!ab has a positive value, it means that point a is at a potential

Circuit 12 V

a

b

Circuit !12 V

a

b

(a) (b)

Figure 1-16: In (a), with the (+) designation at node a, Vab = 12 V.In (b), with the (+) designation at node b, Vba = !12 V, which isequivalent to Vab = 12 V. [That is, Vab = !Vba .]

higher than that of point b. Accordingly, points a and b inFig. 1-15 are denoted with (+) and (!) signs, respectively. If!ab = 5 V, we often use the terminology: “The voltage risefrom b to a is 5 V”, or “The voltage drop from a to b is 5 V”.

Just as 5 A of current flowing from a to b in a circuitconveys the same information as !5 A flowing in the oppositedirection, a similar analogy applies to voltage. Thus, the tworepresentations in Fig. 1-16 convey the same information withregard to the voltage between terminals a and b. Also, the termsdc and ac defined earlier for current apply to voltage as well.A constant voltage is called a dc voltage and a sinusoidallytime-varying voltage is called an ac voltage.

Ground

Since by definition voltage is not an absolute quantity but ratherthe difference in electric potential between two locations, it issometimes convenient to select a reference point in the circuit,label it ground, and then define the voltage at any point in thecircuit with respect to that ground point. Thus, when we say thatthe voltage V1 at node 1 in Fig. 1-17(a) is 6 V, we mean that thepotential difference between node 1 and the ground referencepoint (node 4) is 6 V, which is equivalent to having assignedthe ground node a voltage of zero. Also, since V1 = 6 V andV2 = 4 V, it follows that

V12 = V1 ! V2 = 6 ! 4 = 2 V.

The voltage at node 3 is V3 = 12 V, relative to node 4. This isbecause nodes 3 and 4 are separated by a 12-V voltage sourcewith its (+) terminal next to node 3 and (!) terminal next tonode 4.

Had we chosen a node other than node 4 as our ground node,node voltages V1 to V4 would have had entirely different values(see Example 1-4). The takeaway message is:

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(a) Ground = Node 4

Voltage reference (ground)

R1 V1 = 6 V V2 = 4 VR2

R312 V

V3 = 12 V

V4 = 0

R4

Node 1Node 2Node 3

Node 4

+_

(b) Ground = Node 1

R1

V1 = 0V2 = !2 VR2

R312 V

V3 = 6 V

V4 = !6 V

R4

Node 1Node 2Node 3

Node 4

+_

Figure 1-17: Ground is any point in the circuit selected to serve as areference point for all points in the circuit.

! Node voltages are defined relative to a specificreference (ground) node whose voltage is assigned avoltage of zero. If a different node is selected as ground,the values of the node voltages will change to reflect thefact that the reference node has changed. "

Example 1-4: Node Voltages

In Fig. 1-17(a), node 4 was selected as the ground node.Suppose node 1 is selected as the ground node instead, as shownin Fig. 1-17(b). Use the information in Fig. 1-17(a) to determinenode voltages V2 to V4 when defined relative to V1 at node 1.

Solution: In the circuit of Fig. 1-17(a), V2 is 2V lower in levelthan V1 (4 V compared to 6 V). Hence, in the new configurationin Fig. 1-17(b), V2 will still be 2 V lower than V1, and sinceV1 = 0, it follows that V2 = !2 V. Similarly, V3 = 6 V andV4 = !6 V.

To summarize:

node 4 = ground node 1 = groundV1 = 6 V 0V2 = 4 V !2 VV3 = 12 V 6 VV4 = 0 V !6 V

When a circuit is constructed in a laboratory, the chassisoften is used as the common ground point—in which case itis called chassis ground. As discussed later in Section 10-1,in a household electrical network, outlets are connected tothree wires—one of which is called Earth ground because it isconnected to the physical ground next to the house.

Voltmeter and Ammeter

The voltmeter is the standard instrument used to measurethe voltage difference between two points in a circuit. Tomeasure V12 in the circuit of Fig. 1-18, we connect the (+)terminal of the voltmeter to terminal 1 and the (!) terminalto terminal 2. Connecting the voltmeter to the circuit doesnot require any changes to the circuit, and in the ideal case,the voltmeter will have no effect on any of the voltages andcurrents associated with the circuit. In reality, the voltmeterhas to extract some current from the circuit in order to performthe voltage measurement, but the voltmeter is designed suchthat the amount of extracted current is so small as to have anegligible effect on the circuit.

To measure the current flowing through a wire, it is necessaryto insert an ammeter in that path, as illustrated by Fig. 1-18.The voltage drop across an ideal ammeter is zero.

!

V12

Volts

AmpsI

RV

+

Voltmeter

1 2Ammeter

Figure 1-18: An ideal voltmeter measures the voltage differencebetween two points (such as nodes 1 and 2) without interfering with thecircuit. Similarly, an ideal ammeter measures the current magnitudeand direction without extracting a voltage drop across itself.

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R1 R2V

Open circuit Short circuit

1 2 3 4

+_

Figure 1-19: Open circuit between terminals 1 and 2, and short circuitbetween terminals 3 and 4.

Open and Short Circuits

! An open circuit refers to the condition of pathdiscontinuity (infinite resistance) between two points. Nocurrent can flow through an open circuit, regardless of thevoltage across it. "

The path between terminals 1 and 2 in Fig. 1-19 is an opencircuit. In contrast, a short circuit constitutes the conditionof complete path continuity (with zero electrical resistance)between two points, such as between terminals 3 and 4 inFig. 1-19.

! No voltage drop occurs across a short circuit, regardlessof the magnitude of the current flowing through it. "

Switches come in many varieties, depending on the intendedfunction. The simple ON/OFF switch depicted in Fig. 1-20(a)is known as a single-pole single-throw (SPST) switch. TheON (closed) position acts like a short circuit, allowing currentto flow while extracting no voltage drop across the switch’sterminals; the OFF (open) position acts like an open circuit.The specific time t = t0 denoted below or above the switch[Fig. 1-20(a)] refers to the time t0 at which it opens or closes.

If the purpose of the switch is to combine two switchingfunctions so as to connect a common terminal to either oftwo other terminals, then we need to use the single-poledouble-throw (SPDT) switch illustrated in Fig. 1-20(b). Beforet = t0, the common terminal is connected to terminal 1; then att = t0, that connection ceases (becomes open), and it is replacedwith a connection between the common terminal and terminal 2.

1-5.2 Power

The circuit shown in Fig. 1-21(a) consists of a battery and alight bulb connected by an SPST switch in the open position.No current flows through the open circuit, but the battery has

(b) Switch initially connected to terminal 1, then moved to terminal 2 at t = t0

(a)

SPST switches

SPDT switcht = t0

1

2

Switch initially open,then closes at t = t0

Switch initially closed,then opens at t = t0

t = t0 t = t0

Figure 1-20: (a) Single-pole single-throw (SPST) and (b) single-poledouble-throw (SPDT) switches.

a voltage Vbat across it, due to the excess positive and negativecharges it has at its two terminals. After the switch is closedat t = 5 s, as indicated in Fig. 1-21(b), a current I will flowthrough the circuit along the indicated direction. The battery’sexcess positive charges will flow from its positive terminaldownward through the light bulb towards the battery’s negative

(a)

(b)

Vbat Vbulb

Switch closes at t = 5 s

+!

+

!

I

+!

Vbat

Switch open

Figure 1-21: Current flow through a resistor (light-bulb filament) afterclosing the switch.

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Passive Sign Convention

p > 0 power delivered to devicep < 0 power supplied by device

*Note: The i direction is defined as entering (+) side of !.

i Devicep = !i

!

Figure 1-22: Passive sign convention.

terminal, and (since current direction is defined to coincide withthe direction of flow of positive charge) the current direction willbe as indicated in the figure.

The consequences of current flow through the circuit are: (1)The battery acts as a supplier of power and (2) The light bulbacts as a recipient of power, which gets absorbed by its filament,causing it to heat up and glow, resulting in the conversion ofelectrical power into light and heat.

! A power supply, such as a battery, offers a voltage riseacross it as we follow the current from the terminal atwhich it enters (denoted with a (!) sign) to the terminalfrom which it leaves (denoted with a (+) sign). Incontrast, a power recipient (such as a light bulb) exhibitsa voltage drop across its corresponding terminals. Thisset of assignments of voltage polarities relative to thedirection of current flow for devices generating powerversus those consuming power is known as the passivesign convention (Fig. 1-22). We will adhere to itthroughout the book. "

Our next task is to establish an expression for the power p

delivered to or received by an electrical device. By definition,power is the time rate of change of energy,

p = dw

dt(W), (1.8)

and its unit is the watt (W), named after the Scottish engineerand inventor James Watt (1736–1819), who is credited with thedevelopment of the steam engine from an embryonic stage into

a viable and efficient source of power. Using Eqs. (1.3) and(1.7), we can rewrite Eq. (1.8) as

p = dw

dt= dw

dq· dq

dt

or simply

p = !i (W). (1.9)

Consistent with the passive sign convention:

! The power delivered to a device is equal to the voltageacross it multiplied by the current entering through its (+)voltage terminal. "

If the algebraic value of p is negative, then the device is asupplier of energy. For an isolated electric circuit composed ofmultiple elements, the law of conservation of power requiresthat the algebraic sum of power for the entire circuit be alwayszero. That is, for a circuit with n elements,

n!

k=1

pk = 0, (1.10)

which means that the total power supplied by the circuit alwaysmust equal the total power absorbed by it.

Power supplies are sometimes assigned ratings to describetheir capacities to deliver energy. A battery may be rated ashaving an output capacity of 200 ampere-hours (Ah) at 9 volts,which means that it can deliver a current i over a period oftime "t (measured in hours) such that i "t = 200 Ah, and itcan do so while maintaining a voltage of 9 V. Alternatively, itsoutput capacity may be expressed as 1.8 kilowatt-hours (kWh),which represents the total amount of energy it can supply,namely, W = !i "t (with "t in hours).

Example 1-5: Conservation of Power

For each of the two circuits shown in Fig. 1-23, determine howmuch power is being delivered to each device and whether it isa power supplier or recipient.

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(a)

(b)

12 V 12 V

0.2 A

+_

12 V 6 V

18 V

3 A

Device 2

Device

Device 1

+_

Figure 1-23: Circuits for Example 1-5.

Solution:(a) For the circuit in Fig. 1-23(a), the current entering the

(+) terminal of the device is 0.2 A. Hence, the power P (wherewe use an uppercase letter because both the current and voltageare dc) is:

P = V I = 12 ! 0.2 = 2.4 W,

and since P > 0, the device is a recipient of power. As weknow, the law of conservation of power requires that if thedevice receives 2.4 W of power, then the battery has to deliverexactly that same amount of power. For the battery, the currententering its (+) terminal is "0.2 A (because 0.2 A of current isshown leaving that terminal), so according to the passive signconvention, the power that would be absorbed by the battery(had it been a passive device) is

Pbat = 12("0.2) = "2.4 W.

The fact that Pbat is negative is confirmation that the batteryis indeed a supplier of power.

(b) For device 1 in Fig. 1-23(b), the current entering its (+)terminal is 3 A. Hence,

P1 = V1I1 = 18 ! 3 = 54 W,

and the device is a power recipient.For device 2,

P2 = V2I2 = ("6) ! 3 = "18 W,

and the device is a supplier of power (because P2 is negative).By way of confirmation, the power associated with the battery

is

Pbat = 12("3) = "36 W,

thereby satisfying the law of conservation of power, whichrequires the net power of the overall circuit to be exactly zero.

Example 1-6: Energy Consumption

A resistor connected to a 100-V dc power supply was consuming20 W of power until the switch was turned off, after which thevoltage decayed exponentially to zero. If t = 0 is defined asthe time at which the switch was turned to the off position andif the subsequent voltage variation was given by

!(t) = 100e"2t V for t # 0

(where t is in seconds), determine the total amount of energyconsumed by the resistor after the switch was turned off.

Solution: Before t = 0, the current flowing through theresistor was I = P/V = 20/100 = 0.2 A. Hence, theresistance R of the resistor is

R = V

I= 100

0.2= 500 ",

and the current variation after the switch was turned off is

i(t) = 0.2e"2t A for t # 0.

The instantaneous power is

p(t) = !(t) · i(t) = (100e"2t )(0.2e"2t ) = 20e"4t W.

We note that the power decays at a rate (e"4t ) much fasterthan the rate for current and voltage (e"2t ). The total energy

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1-6 CIRCUIT ELEMENTS 23

dissipated in the resistor after engaging the switch is obtainedby integrating p(t) from t = 0 to infinity, namely

W =!!

0

p(t) dt =!!

0

20e"4t dt = "204

e"4t"""!

0= 5 J.

Exercise 1-6: If a positive current is flowing through aresistor from its terminal a to its terminal b, is !ab positiveor negative?

Answer: !ab > 0. (See )

Exercise 1-7: A certain device has a voltage differenceof 5 V across it. If 2 A of current is flowing through itfrom its (") voltage terminal to its (+) terminal, is thedevice a power supplier or a power recipient, and howmuch energy does it supply or receive in 1 hour?

Answer: P = V I = 5("2) = "10 W. Hence, thedevice is a power supplier. |W | = |P | "t = 36 kJ.(See )

Exercise 1-8: A car radio draws 0.5 A of dc current whenconnected to a 12-V battery. How long does it take forthe radio to consume 1.44 kJ?

Answer: 4 minutes. (See )

1-6 Circuit Elements

Electronic circuits used in functional systems employ a widerange of circuit elements, including transistors and integratedcircuits. The operation of most electronic circuits and devices—no matter how complex—can be modeled (represented) interms of an equivalent circuit composed of basic elementswith idealized characteristics. The equivalent circuit offers acircuit behavior that closely resembles the behavior of the actualelectronic circuit or device over a certain range of specifiedconditions, such as the range of input signal level or output

load resistance. The set of basic elements commonly used incircuit analysis include voltage and current sources; passiveelements (which include resistors, capacitors, and inductors);and various types of switches. The basic attributes of switcheswere covered in Section 1-5.1. The nomenclature and current–voltage relationships associated with the other two groups arethe subject of this section.

1-6.1 i–! Relationship

The relationship between the current flowing through a deviceand the voltage across it defines the fundamental operation ofthat device. As was stated earlier, Ohm’s law states that thecurrent i entering into the (+) terminal of the voltage ! acrossa resistor is given by

i = !

R.

This is called the i–! relationship for the resistor. We notethat the resistor exhibits a linear i–! relationship, meaningthat i and ! always vary in a proportional manner, as shownin Fig. 1-24(a), so long as R remains constant. A circuitcomposed exclusively of elements with linear i–! responsesis called a linear circuit. The linearity property of a circuitis an underlying requirement for the various circuit analysistechniques presented in this and future chapters. Diodes andtransistors exhibit nonlinear i–! relationships, but we stillcan apply the analysis techniques specific to linear circuitsto circuits containing nonlinear devices by representing thosedevices in terms of linear subcircuits that contain dependentsources. The concept of a dependent source and how it is usedis introduced in Section 1-6.3.

1-6.2 Independent Sources

An ideal, independent voltage source provides a specifiedvoltage across its terminals, regardless of the type of load orcircuit connected to it. Hence, for a voltage source with aspecified voltage Vs, its i–! relationship is given by

! = Vs for any i,

so long as it is not connected to a short circuit. Similarly,an ideal, independent current source provides a specifiedcurrent flowing through it, regardless of the voltage across it

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(a)

Resistor

Slope =

!R

1R

i =i

!

(c)

VCVS

(b)

Slope = "

!s = "!x

!x

!s

Ideal voltage source

Ideal current sourcei = Is

! = Vs

Vs

Is

i

!

Figure 1-24: i–! relationships for (a) an ideal resistor, (b) ideal,independent current and voltage sources, and (c) a dependent,voltage-controlled voltage source (VCVS).

(but it cannot do so if connected to an open circuit). Its i–!

relationship is

i = Is for any !.

The i–! profile of an ideal voltage source is a vertical line,as illustrated in Fig. 1-24(b), whereas the profile for the idealcurrent source is a horizontal line.

The circuit symbol used for independent sources is acircle, as shown in Table 1-5, although for dc voltagesources the traditional “battery” symbol is used as well. Ahousehold electrical outlet connected through an electricalpower-distribution network to a hydroelectric- or nuclear-power generating station provides continuous power at an

approximately constant voltage level. Hence, it may beclassified appropriately as an independent voltage source. Ona shorter time scale, a flashlight’s 9-V battery may be regardedas a voltage source, but only until its stored charge has beenused up by the light bulb. Thus, strictly speaking, a battery isa storage device (not a generator), but we tend to treat it as agenerator so long as it acts like a constant voltage source.

In reality, no sources can provide the performancespecifications ascribed to ideal sources. If a 5-V voltage sourceis connected across a short circuit, for example, we run intoa serious problem of ambiguity. From the standpoint of thesource, the voltage is 5 V, but by definition, the voltage iszero across the short circuit. How can it be both zero and5 V simultaneously? The answer resides in the fact that ourdescription of the ideal voltage source breaks down in thissituation.

! More realistic models for voltage and current sourcesinclude a series resistor in the case of the voltage source,and a shunt (parallel) resistor in the case of the currentsource, as shown in Table 1-5. "

The real voltage source (which may have an elaboratecircuit configuration) behaves like a combination of anequivalent, ideal voltage source !s in series with an equivalentresistance Rs. Usually, Rs has a very small value for the voltagesource and a very large value for the current source.

1-6.3 Dependent Sources

As alluded to in the opening paragraph of Section 1-6, we oftenuse equivalent circuits to model the behavior of transistors andother electronic devices. The ability to represent complicateddevices by equivalent circuits composed of basic elementsgreatly facilitates not only the circuit analysis process butthe design process as well. Such circuit models incorporatethe relationships between various parts of the device throughthe use of a set of artificial sources known as dependentsources. The voltage level of a dependent voltage source isdefined in terms of a specific voltage or current elsewhere inthe circuit. An example of circuit equivalence is illustratedin Fig. 1-25. In part (a) of the figure, we have a Model741 operational amplifier (op amp), denoted by the triangularcircuit symbol, used in a simple amplifier circuit intended toprovide a voltage amplification factor of !2; that is, the outputvoltage !0 = !2!s, where !s is the input signal voltage. The opamp, which we will examine later in Chapter 4, is an electronic

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Table 1-5: Voltage and current sources.

Independent SourcesIdeal Voltage Source Realistic Voltage Source

or

dc source Any source*Battery

Vs+-

!s +!+_+

!Vs+_

Any source

!s

Rs

+!+_

Ideal Current Source Realistic Current Source

dc source Any source

Is is

Any source

is Rs

Dependent SourcesVoltage-Controlled Voltage Source (VCVS) Voltage-Controlled Current Source (VCCS)

!s = "!x+! is = g!x

Current-Controlled Voltage Source (CCVS) Current-Controlled Current Source (CCCS)

!s = rix+! is = !ix

Note: !, g, r , and " are constants; #x and ix are a specific voltage and a specific current elsewhere inthe circuit.!Lowercase # and i represent voltage and current sources that may or may not be time varying, whereasuppercase V and I denote dc sources.

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(a) Op-amp circuit (b) Equivalent circuit with dependent source

!s !o!i

15 k!

30 k!

Ro = 75 !

Ri = 3 M! + !2 = A!i+_ _!s+

_741

15 k!

30 k!

!o+_

Figure 1-25: An operational amplifier is a complex device, but its circuit behavior can be represented in terms of a simple equivalent circuit thatincludes a dependent voltage source.

device with a complex architecture composed of transistors,resistors, capacitors, and diodes, but in practice, its circuitbehavior can be represented by a rather simple circuit consistingof two resistors (input resistor Ri and output resistor Ro) and adependent voltage source, as shown in Fig. 1-25(b). The voltage!2 on the right-hand side of the circuit in Fig. 1-25(b) is givenby !2 = A!i, where A is a constant and !i is the voltage acrossthe resistor Ri located on the left-hand side of the equivalentcircuit. In this case, the magnitude of !2 always depends onthe magnitude of !i, which depends in turn on the input signalvoltage !s and on the values chosen for some of the resistorsin the circuit. Since the controlling quantity !i is a voltage,!2 is called a voltage-controlled voltage source (VCVS). Hadthe controlling quantity been a current source, the dependentsource would have been called a current-controlled voltagesource (CCVS) instead. A parallel analogy exists for voltage-controlled and current-controlled current sources.

! The characteristic symbol for a dependent source is thediamond (Table 1-5). "

Proportionality constant " in Table 1-5 relates voltage tovoltage. Hence, it is dimensionless, as is #, since it relatescurrent to current. Constants g and r have units of (A/V)and (V/A), respectively. Because dependent sources arecharacterized by linear relationships, so are their i–! profiles.An example is shown in Fig. 1-24(c) for the VCVS.

Example 1-7: Dependent Source

Find the magnitude of the voltage V1 of the dependent sourcein Fig. 1-26. What type of source is it?

V1 = 4I110 V

5 "

2 "

I1+_

+_

Figure 1-26: Circuit for Example 1-7.

Solution: Since V1 depends on current I1, it is a current-controlled voltage source with a coefficient of 4 V/A.

The 10-V dc voltage is connected across the 2-$ resistor.Hence, the current I along the designated direction is

I1 = 102

= 5 A.

Consequently,

V1 = 4I1 = 4 ! 5 = 20 V.

Example 1-8: Switches

The circuit in Fig. 1-27 contains one SPDT switch that changesposition at t = 0, one SPST switch that opens at t = 0, and oneSPST switch that closes at t = 5 s. Generate circuit diagramsthat include only those elements that have current flowingthrough them for (a) t < 0, (b) 0 " t < 5 s, and (c) t # 5 s.

Solution: See Fig. 1-28.

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+

!V0

R1

R6

R7

R2

R5

R3

R4

t = 0

t = 0

t = 5 s

SPDT

SPST

SPST

Figure 1-27: Circuit for Example 1-8.

(a) t < 0

(b) 0 < t < 5 s

(c) t > 5 s

R1

R6

R7

R2V0

+!

V0

R1

R7

R3

R4

+!

V0

R1

R6

R7

R5

R3

R4

+!

Figure 1-28: Solutions for circuit in Fig. 1-27.

Concept Question 1-8: What is the difference betweenan SPST switch and an SPDT switch?

Concept Question 1-9: What is the difference betweenan independent voltage source and a dependent voltagesource? Is a dependent voltage source a real source ofpower?

Concept Question 1-10: What is an “equivalent-circuit”model? How is it used?

Exercise 1-9: Find Ix from the diagram in Fig. E1-9.

5 !

2 !

5 A Ix =V1

V1

4

+ _

Figure E1-9

Answer: Ix = 2.5 A. (See )

Exercise 1-10: In the circuit of Fig. E1-10, find I at (a)t < 0 and (b) t > 0.

3 ! 4 !12 V

t = 0

SPDTI

+_

Figure E1-10

Answer: (a) I = 4 A, (b) I = 3 A. (See )

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28 TECHNOLOGY BRIEF 1: MICRO- AND NANOTECHNOLOGY

Technology Brief 1Micro- and Nanotechnology

Scale of Things

Our ability as humans to shape and control theenvironment around us has improved steadily over time,most dramatically in the past 100 years. The degree ofcontrol is reflected in the scale (size) at which objects canbe constructed, which is governed by the tools availablefor constructing them. This refers to the construction ofboth very large and very small objects. Early tools—such as flint, stone, and metal hunting gear—were onthe order of tens of centimeters. Over time, we were able

Red blood cells(~7-8 µm)

Things NaturalThings Natural Things ManmadeThings Manmade

Fly ash~ 10-20 µm

Head of a pin1-2 mm

Quantum corral of 48 iron atoms on copper surfacepositioned one at a time with an STM tip

Corral diameter 14 nm

Human hair~ 60-120 µm wide

Ant~ 5 mm

Dust mite

200 µm

ATP synthase

~10 nm diameterNanotube electrode

Carbon nanotube~1.3 nm diameter

O O

O

OO

O O OOO O OO

O

S

O

S

O

S

O

S

O

S

O

S

O

S

O

S

POO

The Challenge

Fabricate and combine nanoscale building blocks to make useful devices, e.g., a photosynthetic reaction center with integral semiconductor storage.

Microworld

0.1 nm

1 nanometer (nm)

0.01 µm10 nm

0.1 µm100 nm

1 micrometer (µm)

0.01 mm10 µm

0.1 mm100 µm

1 millimeter (mm)

1 cm10 mm10-2 m

10-3 m

10-4 m

10-5 m

10-6 m

10-7 m

10-8 m

10-9 m

10-10 m

Visib

le

Nanoworld

1,000 nanometers =

Infra

red

Ultra

violet

Micr

owav

eSo

ft x-r

ay

1,000,000 nanometers =

Zone plate x-ray “lens”Outer ring spacing ~35 nm

Office of Basic Energy SciencesOffice of Science, U.S. DOE

Version 05-26-06, pmd

The Scale of Things The Scale of Things –– Nanometers and MoreNanometers and More

MicroElectroMechanical(MEMS) devices10 -100 µm wide

Red blood cellsPollen grain

Carbon buckyball

~1 nm diameter

Self-assembled,Nature-inspired structureMany 10s of nm

Atoms of siliconspacing 0.078 nm

DNA~2-1/2 nm diameter

Figure TF1-1: The scale of natural and man-made objects, sized from nanometers to centimeters. (Courtesy of U.S. Departmentof Energy.)

to build ever-smaller and ever-larger tools. The pyramidsof Giza (ca. 2600 BCE) are 100 m tall; the largest modernconstruction crane is the K10,000 Kroll Giant crane at100 m in length and 82 m in height; and the tallestbuildng in the world is the 828-m–tall Burj Khalifa in Dubai.Miniaturization also proceeded apace: the first hydraulicvalves, for example, were a few meters in length (ca. 400BCE); the first toilet valve was tens of centimeters in size(ca. 1596); and by comparison, the largest dimension ina modern microfluidic valve used in biomedical analysis-chips is less than 100 µm!

The chart in Fig. TF1-1 displays examples of man-made and natural things whose dimensions fall inthe range between 10!10 m and 1 cm, which en-compasses (1 µm = 10!6 m) and nanometer both the

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TECHNOLOGY BRIEF 1: MICRO- AND NANOTECHNOLOGY 29

micrometer (1 nm = 10!9 m) ranges. Microtechnology,which refers to our ability to manipulate matter at aprecision of 1 µm or better, became possible in the1960s, ushering in an electronics revolution that led tothe realization of the laptop computer and the ubiquitouscell phone. It then took another 30 years to improvethe control precision down to the nanometer scale,promising the development of new materials and deviceswith applications in electronics, medicine, energy, andconstruction.

Moore’s Law

With the invention of the semiconductor transistor in1947 and the subsequent development of the integratedcircuit in 1959, it became possible to build thousands(now trillions) of electronic components onto a singlesubstrate or chip. The 4004 microprocessor chip(ca. 1971) had 2250 transistors and could execute 60,000instructions per second; each transistor had a “gate”on the order of 10 µm (10!5 m). In comparison, the2006 Intel Core had 151 million transistors with eachtransistor gate measuring 65 nm (6.5 " 10!8 m); it couldperform 27 billion instructions per second. The 2011Intel Core i7 “Gulftown” processors have 1.170 billiontransistors and can perform # 150 billion instructions persecond. In recent years, the extreme miniaturization oftransistors (the smallest transistor gate in an i7 Core is# 32 nanometers wide!) has led to a number of design

innovations and trade-offs at the processor level, asdevices begin to approach the physical limits of classicsemiconductor devices. Among these, the difficulty ofdissipating the heat generated by a billion transistorshas led to the emergence of multi-core processors;these devices contain more than one processor operatingsimultaneously on the same chip (2 processors on thesame chip are called a dual core, 4 processors are calleda quad core, etc.). This type of architecture requiresadditional components to manage computation betweenprocessors and has led to the development of newsoftware paradigms to deal with the parallelism inherentin such devices.

Scaling of Integrated Circuits

In 1965, Gordon Moore, co-founder of Intel, predicted thatthe number of transistors in the minimum-cost processorwould double every two years (initially, he had guessedthey would double every year). Amazingly, this predictionhas proven true of semiconductor processors for 40 years,as demonstrated by Fig. TF1-2.

In order to understand Moore’s Law, we have to under-stand the basics about how transistors work. The basicswitching element in semiconductor microprocessors isthe transistor: All of the complex components in themicroprocessor (including logic gates, arithmetic logicunits, and counters) are constructed from combinations oftransistors. Within a processor, transistors have different

Transistors/Chip

Intel CPUs

10,000,000,000

1,000,000,0002,300,000,000Dual-Core Itanium 2

8-Core Xeon

Itanium 2Itanium

Pentium 4Pentium III

Pentium IIPentium II

386

2868086

6000

80088000

4004

100,000,000

10,000,000

1,000,000

100,000

10,000

1,0001970 1975 1980 1985 1990 1995 2000 2005 2011

Figure TF1-2: Moore’s Law predicts that the number of transistors per processor doubles every two years.

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30 TECHNOLOGY BRIEF 1: MICRO- AND NANOTECHNOLOGY

dimensions depending on the component’s function;larger transistors can handle more current, so the sub-circuit in the processor that distributes power may bebuilt from larger transistors than, say, the sub-circuitthat adds two bits together. In general, the smaller thetransistor, the less power it consumes and the faster itcan switch between binary states (0 and 1). Hence,an important goal of a circuit designer is to use thesmallest transistors possible in a given circuit. We canquantify transistor size according to the smallest drawndimension of the transistor, sometimes called the featuresize. In the Intel 4004, for example, the feature sizewas approximately 10 µm, which means that it wasnot possible to make transistors reliably with less than10-µm features drawn in the CAD program. As of 2012,the current production technology for processors (calledthe 22 nm node) produces transistors with gate lengthsapproximately 25 nm wide.

The following questions then arise: How small can wego? What is the fundamental limit to shrinking down thesize of a transistor? As we ponder this, we immediatelyobserve that we likely cannot make a transistor smallerthan the diameter of one silicon or metal atom (i.e., ! 0.2to 0.8 nm). But is there a limit prior to this? Well, as weshrink transistors down to the point that they are madeof just one or a few atomic layers (! 1 to 5 nm), we runinto issues related to the stochastic nature of quantumphysics. At these scales, the random motion of electronsbetween both physical space and energy levels becomessignificant with respect to the size of the transistor, and westart to get spurious or random signals in the circuit. Thereare even more subtle problems related to the statisticsof yield. If a certain piece of a transistor contained only

10 atoms, a deviation of just one atom in the device (to a9-atom or an 11-atom transistor) represents a hugechange in the device properties! This would makeit increasingly difficult to economically fabricate chipswith hundreds of millions of transistors. Additionally,there is an interesting issue of heat generation: Likeany dissipative device, each transistor gives off a smallamount of heat. But when you add up the heat producedby more than 1 billion transistors, you get a very largenumber! Figure TF1-3 compares the power density(due to heat) produced by different processors over time.The heat generated by single core processors increasedexponentially until the mid-2000s when power densitiesbegan approaching 100 W/cm2 (in comparison, a nuclearreactor produces about 200 W/cm2!). The inability topractically dissipate that much heat led, in part, to thedevelopment of multicore processors and a leveling off ofheat generation for consumer processors.

None of these issues are insurmountable. Challengessimply spur driven people to come up with innovativesolutions. Many of these problems will be solved, andin the process, provide engineers (like you) with jobsand opportunities. But, more importantly, the minimumfeature size of a processor is not the end goal ofinnovation: It is the means to it. Innovation seeks simplyto make increasingly powerful processors, not smallerfeature sizes. In recent years, processor companies havelessened their attempts at smaller, faster processors andstarted lumping more of them together to distribute thework among them. This is the idea behind the dualand quad processor cores that power the computers ofthe last few years. By sharing the workload amongvarious processors (called distributed computing) we

Light Bulb Integrated Circuit

Power dissipation

Surface area

Heat flux 0.9 W/cm2 33.3 W/cm2

100 W 50 W

106 cm2(bulb surface area) 1.5 cm2 (die area)

100

10

11990 1994 1998 2002 2006 2010

Year

CPU

pow

er d

ensit

y (W

/cm

2 ) AMD

Single cores

MulticoresIntelPower PC

FigureTF1-3: (a) Plot of heat power density generated by consumer processors over time, (b) comparison of heat power generationof a light bulb with that of a typical processor.

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TECHNOLOGY BRIEF 1: MICRO- AND NANOTECHNOLOGY 31

increase processor performance while using less energy,generating less heat, and without needing to run atwarp speed. So it seems, as we approach ever-smallerfeatures, we simply will transition into new physicaltechnologies and also new computational techniques. AsGordon Moore himself said, “It will not be like we hit abrick wall and stop.”

Scaling Trends and Nanotechnology

It is an observable fact that each generation of toolsenables the construction of a new, smaller, more powerfulgeneration of tools. This is true not just of mechanicaldevices, but electronic ones as well. Today’s high-power processors could not have been designed, muchless tested, without the use of previous processorsthat were employed to draw and simulate the nextgeneration. Two observations can be made in this regard.

First, we now have the technology to build tools tomanipulate the environment at atomic resolution. At leastone generation of micro-scale techniques (ranging frommicroelectromechanical systems—or MEMS—to micro-chemical methods) has been developed that, while usefulin themselves, are also enabling the construction ofnewer, nano-scale devices. These newer devices rangefrom 5-nm transistors to femtoliter (10!15) microfluidicdevices that can manipulate single protein molecules. Atthese scales, the lines between mechanics, electronics,and chemistry begin to blur! It is to these ever-increasing interdisciplinary innovations that the termnanotechnology rightfully belongs. Second, the rateat which these innovations are occurring seems to beincreasing exponentially! Consider Fig. TF1-4 and notethat the y-axis is logarithmic and the plots are very closeto straight lines.

FigureTF1-4: Time plot of computer processing power in MIPS per $1000. (From “When will computer hardware match the humanbrain?” by Hans Moravec, Journal of Transhumanism, Vol. 1, 1998.)

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Chapter 1 Summary

Concepts

• Active devices (such as transistors and ICs) require anexternal power source to operate; in contrast, passivedevices (resistors, capacitors, and inductors) do not.

• Analysis and synthesis (design) are complementaryprocesses.

• Current is related to charge by i = dq/dt ; voltagebetween locations a and b is !ab = dw/dq, where dw

is the work (energy) required to move dq from b to a;and power p = !i.

• Passive sign convention assigns i direction as enteringthe (+) side of !; if p > 0, the device is recipient(consumer) of power, and if p < 0, it is a supplier ofpower.

• Independent voltage and current sources are realsources of energy; dependent sources are artificialrepresentations used in modeling the nonlinear behaviorof a device in terms of an equivalent linear circuit.

Mathematical and Physical Models

Ohm’s law i = !/R

Current i = dq/dt

Direction of i = direction of flow of (+) charge

Charge transfer q(t) =t!

!"i dt

"Q = q(t2) ! q(t1) =t2!

t1

i dt

Voltage = potential energy difference per unit charge

Passive sign convention Direction of i is into +!

terminal of device

Power p = !i

If p > 0 device absorbs powerIf p < 0 device delivers power

Glossary of Important Terms Provide definitions or explain the meaning of the following terms:

acactive deviceampere-hoursanalysisconduction currentcumulative chargedcdependent source

designelectric chargeelectric circuitelectric currentelectron driftequivalent circuitindependent sourcei–! characteristic

kilowatt-hourslinear responseopen circuitpassive devicepassive sign conventionpolarizationpotential differencepower

prefixshort circuitSI unitsSPSTSPDTsynthesisvoltage

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PROBLEMS 33

PROBLEMS

Sections 1-2 to 1-4: Dimensions, Charge, and Current

1.1 Use appropriate multiple and submultiple prefixes toexpress the following quantities:

(a) 3,620 watts (W)*(b) 0.000004 amps (A)

(c) 5.2 ! 10"6 ohms (!)

(d) 3.9 ! 1011 volts (V)

(e) 0.02 meters (m)

(f) 32 ! 105 volts (V)

1.2 Use appropriate multiple and submultiple prefixes toexpress the following quantities:

(a) 4.71 ! 10"8 seconds (s)

(b) 10.3 ! 108 watts (W)

(c) 0.00000000321 amps (A)

(d) 0.1 meters (m)

(e) 8,760,000 volts (V)

(f) 3.16 ! 10"16 hertz (Hz)

1.3 Convert:

(a) 16.3 m to mm

(b) 16.3 m to km*(c) 4 ! 10"6 µF (microfarad) to pF (picofarad)

(d) 2.3 ns to µs

(e) 3.6 ! 107 V to MV

(f) 0.03 mA (milliamp) to µA

1.4 Convert:

(a) 4.2 m to µm

(b) 3 hours to µseconds

(c) 4.2 m to km

(d) 173 nm to m

(e) 173 nm to µm

(f) 12 pF (picofarad) to F (farad)

1.5 For the circuit in Fig. P1.5:

(a) Identify and label all distinct nodes.

(b) Which of those nodes are extraordinary nodes?

*Answer(s) in Appendix F.

(c) Identify all combinations of 2 or more circuit elements thatare connected in series.

(d) Identify pairs of circuit elements that are connected inparallel.

4 ! 5 !2 !

1 ! 3 !

16 V+_

Figure P1.5: Circuit for Problem 1.5.

1.6 For the circuit in Fig. P1.6:(a) Identify and label all distinct nodes.(b) Which of those nodes are extraordinary nodes?(c) Identify all combinations of 2 or more circuit elements that

are connected in series.(d) Identify pairs of circuit elements that are connected in

parallel.

2 !

4 ! 4 !

8 V12 V+_

+_

+_

+_


1.7 For the circuit in Fig. P1.7:(a) Identify and label all distinct nodes.(b) Which of those nodes are extraordinary nodes?(c) Identify all combinations of 2 or more circuit elements that

are connected in series.(d) Identify pairs of circuit elements that are connected in

parallel.

1 !0.1 !

0.2 !

0.3 !

0.4 !

1 !4 V

+_


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25 !40 !

30 !

10 !

20 !

60 !5 !

12 V

15 !

+_







3 ! 6 !

2 !2 ! 4 !

4 A

48 V+_






6 "16 "32 " 8 "

4 "12 V+_








6 !

2 !

4 !

3 !

1 !

5 !20 V

+_ +_


1.12 The total charge contained in a certain region of spaceis !1 C. If that region contains only electrons, how many doesit contain?

*1.13 A certain cross section lies in the x–y plane. If 3 " 1020

electrons go through the cross section in the z-direction in 4seconds, and simultaneously 1.5 " 1020 protons go throughthe same cross section in the negative z-direction, what is themagnitude and direction of the current flowing through the crosssection?

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PROBLEMS 35

1.14 Determine the current i(t) flowing through a resistor ifthe cumulative charge that has flowed through it up to time t isgiven by

(a) q(t) = 3.6t mC

(b) q(t) = 5 sin(377t) µC*(c) q(t) = 0.3[1 ! e!0.4t ] pC

(d) q(t) = 0.2t sin(120! t) nC

1.15 Determine the current i(t) flowing through a certaindevice if the cumulative charge that has flowed through it up totime t is given by

(a) q(t) = !0.45t3 µC

(b) q(t) = 12 sin2(800! t) mC

(c) q(t) = !3.2 sin(377t) cos(377t) pC*(d) q(t) = 1.7t[1 ! e!1.2t ] nC

1.16 Determine the net charge "Q that flowed through aresistor over the specified time interval for each of the followingcurrents:

(a) i(t) = 0.36 A, from t = 0 to t = 3 s*(b) i(t) = [40t + 8] mA, from t = 1 s to t = 12 s

(c) i(t) = 5 sin(4! t) nA, from t = 0 to t = 0.05 s

(d) i(t) = 12e!0.3t mA, from t = 0 to t = "1.17 Determine the net charge "Q that flowed through acertain device over the specified time intervals for each of thefollowing currents:

(a) i(t) = [3t + 6t3] mA, from t = 0 to t = 4 s*(b) i(t) = 4 sin(40! t) cos(40! t) µA, from t = 0 to

t = 0.05 s

(c) i(t) = [4e!t ! 3e!2t ] A, from t = 0 to t = "(d) i(t) = 12e!3t cos(40! t) nA, from t = 0 to t = 0.05 s

1.18 If the current flowing through a wire is given byi(t) = 3e!0.1t mA, how many electrons pass through the wire’scross section over the time interval from t = 0 to t = 0.3 ms?

1.19 The cumulative charge in mC that entered a certaindevice is given by

q(t) =

!"#

"$

0 for t < 0,

5t for 0 # t # 10 s,60 ! t for 10 s # t # 60 s

(a) Plot q(t) versus t from t = 0 to t = 60 s.

(b) Plot the corresponding current i(t) entering the device.

*1.20 A steady flow resulted in 3 $ 1015 electrons entering adevice in 0.1 ms. What is the current?

1.21 Given that the current in (mA) flowing through a wire isgiven by

i(t) =

!"#

"$

0 for t < 06t for 0 # t # 5 s30e!0.6(t!5) for t % 5 s

(a) Sketch i(t) versus t .(b) Sketch q(t) versus t .

1.22 The plot in Fig. P1.22 displays the cumulative amountof charge q(t) that has entered a certain device up to time t .What is the current at(a) t = 1 s

*(b) t = 3 s(c) t = 6 s

q(t)

t2 s 4 s

!4 C

4 C

06 s 8 s

Figure P1.22: q(t) for Problem 1.22.

1.23 The plot in Fig. P1.23 displays the cumulative amount ofcharge q(t) that has exited a certain device up to time t . Whatis the current at(a) t = 2 s(b) t = 6 s(c) t = 12 s

q(t)

t4 s 8 s

2 C

0

4 C4e!0.2(t!8)


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1.24 The plot in Fig. P1.24 displays the cumulative chargeq(t) that has entered a certain device up to time t . Sketch a plotof the corresponding current i(t).

q

t (s)1 2

20 C

!20 C

03 54


Sections 1-5 and 1-6: Voltage, Power, and Circuit Elements

1.25 In the circuit of Fig. P1.25, node V1 was selected as theground node.(a) What is the voltage at node V2?(b) What is the voltage difference V32 = V3 ! V2?(c) What are the voltages at nodes 1, 3, 4, and 5 if node 2 is

selected as the ground node instead of node 1?

R1

R3

R4

V2

V1 = 0

V3 = 32 VV5 = 20 VV4 = 10 V

48 V

R5

R2

+_


1.26 In the circuit of Fig. P1.26, node V1 was selected as theground node.(a) What is the voltage difference across R6?(b) What are the voltages at nodes 1, 3, and 4 if node 2 is

selected as the ground node instead of node 1?

1.27 For each of the eight devices in the circuit of Fig. P1.27,determine whether the device is a supplier or a recipient ofpower and how much power it is supplying or receiving.

R2

R3

R4R1

R6

10 V

R5

+_

20 V

V1 = 0 V2 = 4 V

V3 = 6 V

V4 = 12 V

+_


+ _+ _

6 V

4 A 3 A

16 V

1 A

1 A

2 A

+ _12 V

+

+__ 10 V

+_ 9 V

+

_

7 V

4 V+ _8 V

1

2

3

4 5

6

7

8


1.28 For each of the seven devices in the circuit of Fig. P1.28,determine whether the device is a supplier or a recipient ofpower and how much power it is supplying or receiving.

+ _6 V

+ _4 V

5 A

4 A

2 A3 A

1 A

2 A

24 V+

_

10 V 6 V

12 V

+_

+_

+_

+_

8 V

1

2

3 4

5

6 7


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PROBLEMS 37

*1.29 An electric oven operates at 120 V. If its power ratingis 0.6 kW, what amount of current does it draw, and how muchenergy does it consume in 12 minutes of operation?

1.30 A 9-V flashlight battery has a rating of 1.8 kWh. Ifthe bulb draws a current of 100 mA when lit; determine thefollowing:(a) For how long will the flashlight provide illumination?(b) How much energy in joules is contained in the battery?(c) What is the battery’s rating in ampere-hours?

1.31 The voltage across and current through a certain deviceare given by

!(t) = 5 cos(4" t) V, i(t) = 0.1 cos(4" t) A.

Determine:*(a) The instantaneous power p(t) at t = 0 and t = 0.25 s.(b) The average power pav, defined as the average value of

p(t) over a full time period of the cosine function (0 to0.5 s).

1.32 The voltage across and current through a certain deviceare given by

!(t) = 100(1 ! e!0.2t ) V, i(t) = 30e!0.2t mA.

Determine:(a) The instantaneous power p(t) at t = 0 and t = 3 s.(b) The cumulative energy delivered to the device from t = 0

to t = ".

1.33 The voltage across a device and the current through itare shown graphically in Fig. P1.33. Sketch the correspondingpower delivered to the device and calculate the energy absorbedby it.

10 A

5 A

1 s 2 s

i(t)

t

5 V

0

01 s 2 s

!(t)

t

Figure P1.33: i(t) and !(t) of the device in Problem 1.33.


10 A

1 s 2 s

i(t)

t

5 V

1 s 2 s

!(t)

t

0

0



10 A

4 s

i(t)

t1 s 3 s

4 s

1 s

3 s

0

2 s

!(t)

t

!5 V

0

5 V


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*1.36 After t = 0, the current entering the positive terminal ofa flashlight bulb is given by

i(t) = 2(1 ! e!10t ) (A),

and the voltage across the bulb is !(t) = 12e!10t (V).Determine the maximum power level delivered to the flashlight.

1.37 Apply the law of conservation of power to determinethe amount of power delivered to device 4 in the circuit ofFig. P1.37, given that that the amounts of power delivered tothe other devices are: p1 = !100 W, p2 = 30 W, p3 = 22 W,p5 = 67 W, p6 = !201 W, and p7 = 120 W.

P4 = ?2

6

4

5

1 3 7

Figure P1.37: Circuit of Problem 1.37.

1.38 Determine Vy in the circuit of Fig. P1.38.

10 ! 2 !

5 !

Vy12 V

1.2 A+

_

VxI = 0.1Vx+ _

+_


1.39 Determine V , the voltage of the dependent voltagesource in the circuit of Fig. P1.39.

1.40 For the circuit in Fig. P1.40, generate circuit diagramsthat include only those elements that have current flowingthrough them for(a) t < 0(b) 0 < t < 2 s(c) t > 2 s

20 !

10 V

Ix

30 !

5 !

10 ! V = 2Ix

+ _

+_ 15 V

+_


+_V0

R1R2 R3

R4

R6R5

t = 2 s

t = 0


1.41 For the circuit in Fig. P1.41, generate circuit diagramsthat include only those elements that have current flowingthrough them for

(a) t < 0

(b) 0 < t < 2 s

(c) t > 2 s

V1

V2

R1 SPST

SPST

SPDT R2R3R4

R6

R5+_

+_

t = 0

t = 0

t = 2 s12


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“book” — 2012/7/4 — 11:40 — page 39 — #39

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PROBLEMS 39

1.42 The switch in the circuit of Fig. P1.42 closes at t = 0.Which elements are in-series and which are in-parallel at (a)t < 0 and (b) t > 0?

+_!s R4

R6R5

R3R1

R2t = 0

2

1

3 4


Education

Circuits -2nd Edition by Fawwaz T. Ulaby and Michel M. Maharbiz