Introduction to Electric Circuits (9th Edition) PDF

# Courtney Keating/iStockphoto 9TH EDITION Introduction to Electric Circuits 9TH EDITION Introduction to Electric Circuits James A. Svoboda Clarkson University Richard C. Dorf University of California PUBLISHER Don Fowley EXECUTIVE EDITOR Dan Sayre CONTENT MANAGER Kevin Holm PRODUCTION EDITOR Tim Lindner EXECUTIVE MARKETING MANAGER Chris Ruel MARKETING ASSISTANT Marissa Carroll DESIGN DIRECTOR Harry Nolan PRODUCT DESIGNER Jenny Welter EDITORIAL OPERATIONS MANAGER Melissa Edwards EDITORIAL OPERATIONS ASSISTANT Courtney Welsh SENIOR DESIGNER Madelyn Lesure PHOTO EDITOR Sheena Goldstein SENIOR CONTENT EDITOR Wendy Ashenberg EDITORIAL PROGRAM ASSISTANT Jessica Knecht CONTENT ASSISTANT Helen Seachrist PRODUCTION MANAGEMENT SERVICES Bruce Hobart/Laserwords Maine Cover Photos: # Jivko Kazakov/iStockphoto.com; Alberto Pomares/Getty Images; # choicegraphx/iStockphoto.com; # mattjeacock/iStockphoto.com This book was set in 10/12 pt in Times New Roman by Laserwords Maine, and printed and bound by RRD Jefferson City. The cover was printed by RRD Jefferson City. 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Evaluation copies are provided to qualiﬁed academics and professionals for review purposes only, for use in their courses during the next academic year. These copies are licensed and may not be sold or transferred to a third party. Upon completion of the review period, please return the evaluation copy to Wiley. Return instructions and a free of charge return shipping label are available at www.wiley.com/go/returnlabel. Outside of the United States, please contact your local representative. ISBN-13: 978-1-118-47750-2 BRV ISBN: 978-1-118-52106-9 Printed in the United States of America 10 9 8 7 6 5 4 3 2 1 The scientiﬁc nature of the ordinary man Is to go on out and do the best he can. —John Prine But, Captain, I cannot change the laws of physics. —Lt. Cmdr. Montogomery Scott (Scotty), USS Enterprise Dedicated to our grandchildren: Ian Christopher Boilard, Kyle Everett Schafer, and Graham Henry Schafer and Heather Lynn Svoboda, James Hugh Svoboda, Jacob Arthur Leis, Maxwell Andrew Leis, and Jack Mandlin Svoboda About the Authors James A. Svoboda is an associate professor of electrical and computer engineer- ing at Clarkson University, where he teaches courses on topics such as circuits, electronics, and computer programming. He earned a PhD in electrical engineering from the University of Wisconsin at Madison, an MS from the University of Colorado, and a BS from General Motors Institute. Sophomore Circuits is one of Professor Svoboda’s favorite courses. He has taught this course to 6,500 undergraduates at Clarkson University over the past 35 years. In 1986, he received Clarkson University’s Distinguished Teaching Award. Professor Svoboda has written several research papers describing the advantages of using nullors to model electric circuits for computer analysis. He is interested in the way technology affects engineering education and has developed several software packages for use in Sophomore Circuits. Richard C. Dorf, professor of electrical and computer engineering at the University of California, Davis, teaches graduate and under- graduate courses in electrical engineering in the ﬁelds of circuits and control systems. He earned a PhD in electrical engineering from the U.S. Naval Postgraduate School, an MS from the University of Colorado, and a BS from Clarkson University. Highly concerned with the discipline of electrical engineering and its wide value to social and economic needs, he has written and lectured internationally on the contributions and advances in electrical engineering. Professor Dorf has extensive experience with education and industry and is professionally active in the ﬁelds of robotics, automa- tion, electric circuits, and communications. He has served as a visiting professor at the University of Edinburgh, Scotland, the Massachusetts Institute of Technology, Stanford University, and the University of California at Berkeley. A Fellow of the Institute of Electrical and Electronic Engineers and the American Society for Engineering Education, Dr. Dorf is widely known to the profession for his Modern Control Systems, twelfth edition (Pearson, 2011) and The International Encyclopedia of Robotics (Wiley, 1988). Dr. Dorf is also the coauthor of Circuits, Devices and Systems (with Ralph Smith), ﬁfth edition (Wiley, 1992). Dr. Dorf edited the widely used Electrical Engineering Handbook, third edition (CRC Press and IEEE press), published in 2011. His latest work is Technology Ventures, fourth edition (McGraw-Hill 2013). ix Preface The central theme of Introduction to Electric Circuits is the concept that electric circuits are part of the basic fabric of modern technology. Given this theme, we endeavor to show how the analysis and design of electric circuits are inseparably intertwined with the ability of the engineer to design complex electronic, communication, computer, and control systems as well as consumer products. Approach and Organization This book is designed for a one- to three-term course in electric circuits or linear circuit analysis and is structured for maximum ﬂexibility. The ﬂowchart in Figure 1 demonstrates alternative chapter organizations that can accommodate different course outlines without disrupting continuity. The presentation is geared to readers who are being exposed to the basic concepts of electric circuits for the ﬁrst time, and the scope of the work is broad. Students should come to the course with the basic knowledge of differential and integral calculus. This book endeavors to prepare the reader to solve realistic problems involving electric circuits. Thus, circuits are shown to be the results of real inventions and the answers to real needs in industry, the ofﬁce, and the home. Although the tools of electric circuit analysis may be partially abstract, electric circuits are the building blocks of modern society. The analysis and design of electric circuits are critical skills for all engineers. What’s New in the 9th Edition Revisions to Improve Clarity Chapter 10, covering AC circuits, has been largely rewritten to improve clarity of exposition. In addition, revisions have been made through the text to improve clarity. Sometimes these revisions are small, involving sentences or paragraphs. Other larger revisions involved pages or even entire sections. Often these revisions involve examples. Consequently, the 9th edition contains 36 new examples. More Problems The 9th edition contains 180 new problems, bringing the total number of problems to more than 1,400. This edition uses a variety of problem types and they range in difﬁculty from simple to challenging, including: Straightforward analysis problems. Analysis of complicated circuits. Simple design problems. (For example, given a circuit and the speciﬁed response, determine the required RLC values.) Compare and contrast, multipart problems that draw attention to similarities or differences between two situations. MATLAB and PSpice problems. Design problems. (Given some speciﬁcations, devise a circuit that satisﬁes those speciﬁcations.) How Can We Check... ? (Verify that a solution is indeed correct.) xi xii Preface Color Matrices, Code Determinants E A 1 2 3 4 ELECTRIC CIRCUIT RESISTIVE METHODS OF CIRCUIT ELEMENTS CIRCUITS ANALYSIS OF VARIABLES RESISTIVE CIRCUITS Complex Numbers B, C, D 9 10 11 12 THE COMPLETE SINUSOIDAL AC STEADY-STATE THREE-PHASE RESPONSE OF STEADY-STATE POWER CIRCUITS CIRCUITS WITH ANALYSIS TWO ENERGY STORAGE ELEMENTS FIGURE 1 Flow chart showing alternative paths through the topics in this textbook. Features Retained from Previous Editions Introduction Each chapter begins with an introduction that motivates consideration of the material of that chapter. Examples Because this book is oriented toward providing expertise in problem solving, we have included more than 260 illustrative examples. Also, each example has a title that directs the student to exactly what is being illustrated in that particular example. Various methods of solving problems are incorporated into select examples. These cases show students that multiple methods can be used to derive similar solutions or, in some cases, that multiple solutions can be correct. This helps students build the critical thinking skills necessary to discern the best choice between multiple outcomes. Much attention has been given to using PSpice and MATLAB to solve circuits problems. Two appendices, one introducing PSpice and the other introducing MATLAB, brieﬂy describe the capabilities of the programs and illustrate the steps needed to get started using them. Next, PSpice Preface xiii PSpice F, G 5 6 7 8 CIRCUIT THE ENERGY THE COMPLETE THEOREMS OPERATIONAL STORAGE RESPONSE OF AMPLIFIER ELEMENTS RL AND RC CIRCUITS 14 LAPLACE TRANSFORM 16 FILTER CIRCUITS 13 14 15 16 17 FREQUENCY THE FOURIER FILTER TWO-PORT RESPONSE LAPLACE SERIES CIRCUITS NETWORKS TRANSFORM AND FOURIER TRANSFORM 6 17 THE TWO-PORT OPERATIONAL NETWORKS AMPLIFIER Legend: Primary flow Chapter Appendix Optional flow and MATLAB are used throughout the text to solve various circuit analysis and design problems. For example, PSpice is used in Chapter 5 to ﬁnd a Thevenin equivalent circuit and in Chapter 15 to represent circuit inputs and outputs as Fourier series. MATLAB is frequently used to obtain plots of circuit inputs and outputs that help us to see what our equations are telling us. MALAB also helps us with some long and tedious arithmetic. For example, in Chapter 10, MATLAB helps us do the complex arithmetic that we must do in order to analyze ac circuits, and in Chapter 14, MATLAB helps with the partial fraction required to ﬁnd inverse Laplace transforms. xiv Preface Of course, there’s more to using PSpice and MATLAB than simply running the programs. We pay particular attention to interpreting the output of these computer programs and checking it to make sure that it is correct. Frequently, this is done in the section called “How Can We Check... ?” that is included in every chapter. For example, Section 8.9 shows how to interpret and check a PSpice “Transient Response,” and Section 13.7 shows how to interpret and check a frequency response produced using MATLAB or PSpice. Design Examples, a Problem-Solving Method, and “How Can We Check... ?” Sections Each chapter concludes with a design example that uses the methods of that chapter to solve a design problem. A formal ﬁve-step problem-solving method is introduced in Chapter 1 and then used in each of the design examples. An important step in the problem-solving method requires you to check your results to verify that they are correct. Each chapter includes a section entitled “How Can We Check... ? ” that illustrates how the kind of results obtained in that chapter can be checked to ensure correctness. Key Equations and Formulas You will ﬁnd that key equations, formulas, and important notes have been called out in a shaded box to help you pinpoint critical information. Summarizing Tables and Figures The procedures and methods developed in this text have been summarized in certain key tables and ﬁgures. Students will ﬁnd these to be an important problem-solving resource. Table 1.5-1. The passive convention. Figure 2.7-1 and Table 2.7-1. Dependent sources. Table 3.10-1. Series and parallel sources. Table 3.10-1. Series and parallel elements. Voltage and current division. Figure 4.2-3. Node voltages versus element currents and voltages. Figure 4.5-4. Mesh currents versus element currents and voltages. Figures 5.4-3 and 5.4-4. Thévenin equivalent circuits. Figure 6.3-1. The ideal op amp. Figure 6.5-1. A catalog of popular op amp circuits. Table 7.8-1. Capacitors and inductors. Table 7.13-2. Series and parallel capacitors and inductors. Table 8.11-1. First-order circuits. Tables 9.13-1, 2, and 3. Second-order circuits. Table 10.5-1. Voltage and current division for AC circuits. Table 10.16-1. AC circuits in the frequency domain (phasors and impedances). Table 11.5-1. Power formulas for AC circuits. Tables 11.13-1 and 11.13-2. Coupled inductors and ideal transformers. Table 13.4-1. Resonant circuits. Tables 14.2-1 and 14.2-2. Laplace transform tables. Preface xv Table 14.7-1. s-domain models of circuit elements. Table 15.4-1. Fourier series of selected periodic waveforms. Introduction to Signal Processing Signal processing is an important application of electric circuits. This book introduces signal processing in two ways. First, two sections (Sections 6.6 and 7.9) describe methods to design electric circuits that implement algebraic and differential equations. Second, numerous examples and problems throughout this book illustrate signal processing. The input and output signals of an electric circuit are explicitly identiﬁed in each of these examples and problems. These examples and problems investigate the relationship between the input and output signals that is imposed by the circuit. Interactive Examples and Exercises Numerous examples throughout this book are labeled as interactive examples. This label indicates that computerized versions of that example are available at the textbook’s companion site, www.wiley.com/ svoboda. Figure 2 illustrates the relationship between the textbook example and the computerized example available on the Web site. Figure 2a shows an example from Chapter 3. The problem presented by the interactive example shown in Figure 2b is similar to the textbook example but different in several ways: The values of the circuit parameters have been randomized. The independent and dependent sources may be reversed. The reference direction of the measured voltage may be reversed. A different question is asked. Here, the student is asked to work the textbook problem backward, using the measured voltage to determine the value of a circuit parameter. The interactive example poses a problem and then accepts and checks the user’s answer. Students are provided with immediate feedback regarding the correctness of their work. The interactive example chooses parameter values somewhat randomly, providing a seemingly endless supply of problems. This pairing of a solution to a particular problem with an endless supply of similar problems is an effective aid for learning about electric circuits. The interactive exercise shown in Figure 2c considers a similar, but different, circuit. Like the interactive example, the interactive exercise poses a problem and then accepts and checks the user’s answer. Student learning is further supported by extensive help in the form of worked example problems, available from within the interactive exercise, using the Worked Example button. Variations of this problem are obtained using the New Problem button. We can peek at the answer, using the Show Answer button. The interactive examples and exercises provide hundreds of additional practice problems with countless variations, all with answers that are checked immediately by the computer. Supplements and Web Site Material The almost ubiquitous use of computers and the Web have provided an exciting opportunity to rethink supplementary material. The supplements available have been greatly enhanced. Book Companion Site Additional student and instructor resources can be found on the John Wiley & Sons textbook companion site at www.wiley.com/college/svoboda. xvi Preface 4Ω 5Ω Voltmeter + + 12 V +– 3ia vm – ia – (a) Worked Examples 1.2 V R 27 Ω Voltmeter Calculator + – 12 V +– + 2ia vm New Problem ia – Show Answer The voltmeter measures a voltage in volts. What is the value of the resistance R in Ω? (b) Worked Examples 4Ω 2Ω Ammeter Calculator 12 V +– 3ia New Problem ia im Show Answer The ammeter measures a current in amps. What is the value of the current measured by the ammeter? (c) FIGURE 2 (a) The circuit considered Example 3.2-5. (b) A corresponding interactive example. (c) A corresponding interactive exercise. Student Interactive Examples The interactive examples and exercises are powerful support resources for students. They were created as tools to assist students in mastering skills and building their conﬁdence. The examples selected from the text and included on the Web give students options for navigating through the problem. They can immediately request to see the solution or select a more gradual approach to help. Then they can try their hand at a similar problem by simply electing to change the values in the problem. By the time students attempt the homework, they have built the conﬁdence and skills to complete their assignments successfully. It’s a virtual homework helper. Preface xvii PSpice for Linear Circuits, available for purchase. WileyPLUS option. Instructor Solutions manual. PowerPoint slides. WileyPLUS option. WileyPLUS Pspice for Linear Circuits is a student supplement available for purchase. The PSpice for Linear Circuits manual describes in careful detail how to incorporate this valuable tool in solving problems. This manual emphasizes the need to verify the correctness of computer output. No example is ﬁnished until the simulation results have been checked to ensure that they are correct. Acknowledgments and Commitment to Accuracy We are grateful to many people whose efforts have gone into the making of this textbook. We are especially grateful to our Executive Editor Daniel Sayre, Executive Marketing Manager Chris Ruel and Marketing Assistant Marissa Carroll for their support and enthusiasm. We are grateful to Tim Lindner and Kevin Holm of Wiley and Bruce Hobart of Laserwords Maine for their efforts in producing this textbook. We wish to thank Senior Product Designer Jenny Welter, Content Editor Wendy Ashenberg, and Editorial Assistant Jess Knecht for their signiﬁcant contributions to this project. We are particularly grateful to the team of reviewers who checked the problems and solutions to ensure their accuracy: Accuracy Checkers Khalid Al-Olimat, Ohio Northern William Robbins, University of Minnesota University James Rowland, University of Kansas Lisa Anneberg, Lawrence Mike Shen, Duke University Technological University Thyagarajan Srinivasan, Wilkes Horace Gordon, University of South University Florida Aaron Still, U.S. Naval Academy Lisimachos Kondi, SUNY, Buffalo Howard Weinert, Johns Hopkins University Michael Polis, Oakland University Xiao-Bang Xu, Clemson University Sannasi Ramanan, Rochester Institute Jiann Shiun Yuan, University of of Technology Central Florida xviii Preface Reviewers Rehab Abdel-Kader, Georgia Southern Seyed Mousavinezhad, Western University Michigan University Said Ahmed-Zaid, Boise State Philip Munro, Youngstown State University University Ahmad Naﬁsi, California Polytechnic State Farzan Aminian, Trinity University University Constantin Apostoaia, Purdue Arnost Neugroschel, University of Florida University Calumet Tokunbo Ogunfunmi, Santa Clara University Jonathon Bagby, Florida Atlantic University Gary Perks, California Polytechnic State Carlotta Berry, Tennessee State University University, San Luis Obispo Kiron Bordoloi, University of Louisville Owe Petersen, Milwaukee School of Engineering Mauro Caputi, Hofstra University Ron Pieper, University of Texas, Tyler Edward Collins, Clemson University Teodoro Robles, Milwaukee School of Glen Dudevoir, U.S. Military Academy Engineering Malik Elbuluk, University of Akron Pedda Sannuti, Rutgers University Prasad Enjeti, Texas A&M University Marcelo Simoes, Colorado School of Mines Ali Eydgahi, University of Maryland Ralph Tanner, Western Michigan University Eastern Shore Tristan Tayag, Texas Christian University Carlos Figueroa, Cabrillo College Jean-Claude Thomassian, Central Walid Hubbi, New Jersey Institute of Technology Michigan University Brian Huggins, Bradley University John Ventura, Christian Brothers University Chris Ianello, University of Central Florida Annette von Jouanne, Simone Jarzabek, ITT Technical Institute Oregon State University James Kawamoto, Mission College Ravi Warrier, Kettering University Rasool Kenarangui, University Gerald Woelﬂ, Milwaukee School of of Texas Arlington Engineering Jumoke Ladeji-Osias, Morgan State University Hewlon Zimmer, U.S. Merchant Mark Lau, Universidad del Turabo Marine Academy Contents CHAPTER 1 Electric Circuit Variables....................................................................................................................................... 1 1.1 Introduction............................................................................................................................. 1 1.2 Electric Circuits and Current................................................................................................... 1 1.3 Systems of Units...................................................................................................................... 5 1.4 Voltage.................................................................................................................................... 7 1.5 Power and Energy.................................................................................................................... 7 1.6 Circuit Analysis and Design.................................................................................................. 11 1.7 How Can We Check... ?................................................................................................... 13 1.8 Design Example—Jet Valve Controller................................................................................. 14 1.9 Summary............................................................................................................................... 15 Problems................................................................................................................................ 15 Design Problems................................................................................................................... 19 CHAPTER 2 Circuit Elements..................................................................................................................................................... 20 2.1 Introduction........................................................................................................................... 20 2.2 Engineering and Linear Models............................................................................................. 20 2.3 Active and Passive Circuit Elements..................................................................................... 23 2.4 Resistors................................................................................................................................ 25 2.5 Independent Sources.............................................................................................................. 28 2.6 Voltmeters and Ammeters..................................................................................................... 30 2.7 Dependent Sources................................................................................................................ 33 2.8 Transducers............................................................................................................................ 37 2.9 Switches................................................................................................................................. 39 2.10 How Can We Check... ?................................................................................................... 40 2.11 Design Example—Temperature Sensor................................................................................. 42 2.12 Summary............................................................................................................................... 44 Problems................................................................................................................................ 44 Design Problems................................................................................................................... 52 CHAPTER 3 Resistive Circuits................................................................................................................................................... 53 3.1 Introduction........................................................................................................................... 53 3.2 Kirchhoff's Laws................................................................................................................... 54 3.3 Series Resistors and Voltage Division................................................................................... 63 3.4 Parallel Resistors and Current Division................................................................................. 68 3.5 Series Voltage Sources and Parallel Current Sources............................................................ 74 3.6 Circuit Analysis..................................................................................................................... 77 3.7 Analyzing Resistive Circuits Using MATLAB..................................................................... 82 3.8 How Can We Check... ?................................................................................................... 86 3.9 Design Example—Adjustable Voltage Source...................................................................... 88 3.10 Summary............................................................................................................................... 91 Problems................................................................................................................................ 92 Design Problems................................................................................................................. 112 xix xx Contents CHAPTER 4 Methods of Analysis of Resistive Circuits...................................................................................................... 114 4.1 Introduction......................................................................................................................... 114 4.2 Node Voltage Analysis of Circuits with Current Sources.................................................... 115 4.3 Node Voltage Analysis of Circuits with Current and Voltage Sources............................... 121 4.4 Node Voltage Analysis with Dependent Sources................................................................ 126 4.5 Mesh Current Analysis with Independent Voltage Sources................................................. 128 4.6 Mesh Current Analysis with Current and Voltage Sources................................................. 133 4.7 Mesh Current Analysis with Dependent Sources................................................................. 137 4.8 The Node Voltage Method and Mesh Current Method Compared...................................... 139 4.9 Circuit Analysis Using MATLAB....................................................................................... 142 4.10 Using PSpice to Determine Node Voltages and Mesh Currents.......................................... 144 4.11 How Can We Check... ?................................................................................................. 146 4.12 Design Example—Potentiometer Angle Display................................................................ 149 4.13 Summary............................................................................................................................. 152 Problems.............................................................................................................................. 153 PSpice Problems.................................................................................................................. 167 Design Problems................................................................................................................. 167 CHAPTER 5 Circuit Theorems.................................................................................................................................................. 169 5.1 Introduction......................................................................................................................... 169 5.2 Source Transformations....................................................................................................... 169 5.3 Superposition....................................................................................................................... 176 5.4 Thevenin’s Theorem............................................................................................................ 180 5.5 Norton’s Equivalent Circuit................................................................................................. 187 5.6 Maximum Power Transfer................................................................................................... 191 5.7 Using MATLAB to Determine the Thevenin Equivalent Circuit........................................ 194 5.8 Using PSpice to Determine the Thevenin Equivalent Circuit.............................................. 197 5.9 How Can We Check... ?................................................................................................. 200 5.10 Design Example—Strain Gauge Bridge.............................................................................. 201 5.11 Summary............................................................................................................................. 203 Problems.............................................................................................................................. 204 PSpice Problems.................................................................................................................. 216 Design Problems................................................................................................................. 217 CHAPTER 6 The Operational Ampliﬁer.................................................................................................................................. 219 6.1 Introduction......................................................................................................................... 219 6.2 The Operational Ampliﬁer................................................................................................... 219 6.3 The Ideal Operational Ampliﬁer.......................................................................................... 221 6.4 Nodal Analysis of Circuits Containing Ideal Operational Ampliﬁers.................................. 223 6.5 Design Using Operational Ampliﬁers.................................................................................. 228 6.6 Operational Ampliﬁer Circuits and Linear Algebraic Equations......................................... 233 6.7 Characteristics of Practical Operational Ampliﬁers............................................................. 238 6.8 Analysis of Op Amp Circuits Using MATLAB.................................................................. 245 6.9 Using PSpice to Analyze Op Amp Circuits......................................................................... 247 6.10 How Can We Check... ?................................................................................................. 248 6.11 Design Example—Transducer Interface Circuit.................................................................. 250 Contents xxi 6.12 Summary............................................................................................................................. 252 Problems.............................................................................................................................. 253 PSpice Problems.................................................................................................................. 265 Design Problems................................................................................................................. 267 CHAPTER 7 Energy Storage Elements................................................................................................................................... 268 7.1 Introduction......................................................................................................................... 268 7.2 Capacitors............................................................................................................................ 269 7.3 Energy Storage in a Capacitor............................................................................................. 275 7.4 Series and Parallel Capacitors.............................................................................................. 278 7.5 Inductors.............................................................................................................................. 280 7.6 Energy Storage in an Inductor............................................................................................. 285 7.7 Series and Parallel Inductors................................................................................................ 287 7.8 Initial Conditions of Switched Circuits................................................................................ 288 7.9 Operational Ampliﬁer Circuits and Linear Differential Equations...................................... 292 7.10 Using MATLAB to Plot Capacitor or Inductor Voltage and Current.................................. 298 7.11 How Can We Check... ?................................................................................................. 300 7.12 Design Example—Integrator and Switch............................................................................ 301 7.13 Summary............................................................................................................................. 304 Problems.............................................................................................................................. 305 Design Problems................................................................................................................. 321 CHAPTER 8 The Complete Response of RL and RC Circuits............................................................................................. 322 8.1 Introduction......................................................................................................................... 322 8.2 First-Order Circuits.............................................................................................................. 322 8.3 The Response of a First-Order Circuit to a Constant Input.................................................. 325 8.4 Sequential Switching........................................................................................................... 338 8.5 Stability of First-Order Circuits........................................................................................... 340 8.6 The Unit Step Source........................................................................................................... 342 8.7 The Response of a First-Order Circuit to a Nonconstant Source......................................... 346 8.8 Differential Operators.......................................................................................................... 351 8.9 Using PSpice to Analyze First-Order Circuits..................................................................... 352 8.10 How Can We Check... ?................................................................................................. 355 8.11 Design Example—A Computer and Printer........................................................................ 359 8.12 Summary............................................................................................................................. 362 Problems.............................................................................................................................. 363 PSpice Problems.................................................................................................................. 374 Design Problems................................................................................................................. 375 CHAPTER 9 The Complete Response of Circuits with Two Energy Storage Elements................................................................................................................................................. 378 9.1 Introduction......................................................................................................................... 378 9.2 Differential Equation for Circuits with Two Energy Storage Elements............................... 379 9.3 Solution of the Second-Order Differential Equation—The Natural Response.................... 383 xxii Contents 9.4 Natural Response of the Unforced Parallel RLC Circuit...................................................... 386 9.5 Natural Response of the Critically Damped Unforced Parallel RLC Circuit........................ 389 9.6 Natural Response of an Underdamped Unforced Parallel RLC Circuit............................... 390 9.7 Forced Response of an RLC Circuit..................................................................................... 392 9.8 Complete Response of an RLC Circuit................................................................................ 396 9.9 State Variable Approach to Circuit Analysis....................................................................... 399 9.10 Roots in the Complex Plane................................................................................................ 403 9.11 How Can We Check... ?................................................................................................. 404 9.12 Design Example—Auto Airbag Igniter............................................................................... 407 9.13 Summary............................................................................................................................. 409 Problems.............................................................................................................................. 411 PSpice Problems.................................................................................................................. 422 Design Problems................................................................................................................. 423 CHAPTER 10 Sinusoidal Steady-State Analysis.................................................................................................................... 425 10.1 Introduction......................................................................................................................... 425 10.2 Sinusoidal Sources............................................................................................................... 426 10.3 Phasors and Sinusoids......................................................................................................... 430 10.4 Impedances.......................................................................................................................... 435 10.5 Series and Parallel Impedances............................................................................................ 440 10.6 Mesh and Node Equations................................................................................................... 447 10.7 Thevenin and Norton Equivalent Circuits........................................................................... 454 10.8 Superposition....................................................................................................................... 459 10.9 Phasor Diagrams.................................................................................................................. 461 10.10 Op Amps in AC Circuits...................................................................................................... 463 10.11 The Complete Response...................................................................................................... 465 10.12 Using MATLAB to Analyze AC Circuits........................................................................... 472 10.13 Using PSpice to Analyze AC Circuits................................................................................. 474 10.14 How Can We Check... ?.................................................................................................. 476 10.15 Design Example—An Op Amp Circuit............................................................................... 479 10.16 Summary............................................................................................................................. 481 Problems.............................................................................................................................. 482 PSpice Problems.................................................................................................................. 502 Design Problems................................................................................................................. 503 CHAPTER 11 AC Steady-State Power...................................................................................................................................... 504 11.1 Introduction......................................................................................................................... 504 11.2 Electric Power...................................................................................................................... 504 11.3 Instantaneous Power and Average Power............................................................................ 505 11.4 Effective Value of a Periodic Waveform............................................................................. 509 11.5 Complex Power................................................................................................................... 512 11.6 Power Factor........................................................................................................................ 519 11.7 The Power Superposition Principle..................................................................................... 527 11.8 The Maximum Power Transfer Theorem............................................................................. 530 11.9 Coupled Inductors............................................................................................................... 531 11.10 The Ideal Transformer......................................................................................................... 539 Contents xxiii 11.11 How Can We Check... ?................................................................................................. 546 11.12 Design Example—Maximum Power Transfer..................................................................... 547 11.13 Summary............................................................................................................................. 549 Problems.............................................................................................................................. 551 PSpice Problems.................................................................................................................. 566 Design Problems................................................................................................................. 567 CHAPTER 12 Three-Phase Circuits........................................................................................................................................... 568 12.1 Introduction......................................................................................................................... 568 12.2 Three-Phase Voltages.......................................................................................................... 569 12.3 The Y-to-Y Circuit.............................................................................................................. 572 12.4 The D-Connected Source and Load..................................................................................... 581 12.5 The Y-to-D Circuit............................................................................................................... 583 12.6 Balanced Three-Phase Circuits............................................................................................ 586 12.7 Instantaneous and Average Power in a Balanced Three-Phase Load................................... 588 12.8 Two-Wattmeter Power Measurement.................................................................................. 591 12.9 How Can We Check... ?................................................................................................. 594 12.10 Design Example—Power Factor Correction........................................................................ 597 12.11 Summary............................................................................................................................. 598 Problems.............................................................................................................................. 599 PSpice Problems.................................................................................................................. 602 Design Problems................................................................................................................. 603 CHAPTER 13 Frequency Response........................................................................................................................................... 604 13.1 Introduction......................................................................................................................... 604 13.2 Gain, Phase Shift, and the Network Function...................................................................... 604 13.3 Bode Plots............................................................................................................................ 616 13.4 Resonant Circuits................................................................................................................. 633 13.5 Frequency Response of Op Amp Circuits........................................................................... 640 13.6 Plotting Bode Plots Using MATLAB.................................................................................. 642 13.7 Using PSpice to Plot a Frequency Response....................................................................... 644 13.8 How Can We Check... ?................................................................................................. 646 13.9 Design Example—Radio Tuner........................................................................................... 650 13.10 Summary............................................................................................................................. 652 Problems.............................................................................................................................. 653 PSpice Problems.................................................................................................................. 666 Design Problems................................................................................................................. 668 CHAPTER 14 The Laplace Transform....................................................................................................................................... 670 14.1 Introduction......................................................................................................................... 670 14.2 Laplace Transform............................................................................................................... 671 14.3 Pulse Inputs......................................................................................................................... 677 14.4 Inverse Laplace Transform.................................................................................................. 680 14.5 Initial and Final Value Theorems........................................................................................ 687 14.6 Solution of Differential Equations Describing a Circuit...................................................... 689 xxiv Contents 14.7 Circuit Analysis Using Impedance and Initial Conditions................................................... 690 14.8 Transfer Function and Impedance....................................................................................... 700 14.9 Convolution......................................................................................................................... 706 14.10 Stability............................................................................................................................... 710 14.11 Partial Fraction Expansion Using MATLAB....................................................................... 713 14.12 How Can We Check... ?................................................................................................. 718 14.13 Design Example—Space Shuttle Cargo Door..................................................................... 720 14.14 Summary............................................................................................................................. 723 Problems.............................................................................................................................. 724 PSpice Problems.................................................................................................................. 738 Design Problems................................................................................................................. 739 CHAPTER 15 Fourier Series and Fourier Transform.............................................................................................................. 741 15.1 Introduction......................................................................................................................... 741 15.2 The Fourier Series................................................................................................................ 741 15.3 Symmetry of the Function f (t)............................................................................................. 750 15.4 Fourier Series of Selected Waveforms................................................................................. 755 15.5 Exponential Form of the Fourier Series............................................................................... 757 15.6 The Fourier Spectrum.......................................................................................................... 765 15.7 Circuits and Fourier Series.................................................................................................. 769 15.8 Using PSpice to Determine the Fourier Series..................................................................... 772 15.9 The Fourier Transform........................................................................................................ 777 15.10 Fourier Transform Properties............................................................................................... 780 15.11 The Spectrum of Signals...................................................................................................... 784 15.12 Convolution and Circuit Response...................................................................................... 785 15.13 The Fourier Transform and the Laplace Transform............................................................. 788 15.14 How Can We Check... ?................................................................................................. 790 15.15 Design Example—DC Power Supply.................................................................................. 792 15.16 Summary............................................................................................................................. 795 Problems.............................................................................................................................. 796 PSpice Problems.................................................................................................................. 802 Design Problems................................................................................................................. 802 CHAPTER 16 Filter Circuits......................................................................................................................................................... 804 16.1 Introduction......................................................................................................................... 804 16.2 The Electric Filter................................................................................................................ 804 16.3 Filters................................................................................................................................... 805 16.4 Second-Order Filters............................................................................................................ 808 16.5 High-Order Filters............................................................................................................... 816 16.6 Simulating Filter Circuits Using PSpice.............................................................................. 822 16.7 How Can We Check... ?................................................................................................. 826 16.8 Design Example—Anti-Aliasing Filter............................................................................... 828 16.9 Summary............................................................................................................................. 831 Problems.............................................................................................................................. 831 PSpice Problems.................................................................................................................. 836 Design Problems................................................................................................................. 839 Contents xxv CHAPTER 17 Two-Port and Three-Port Networks................................................................................................................. 840 17.1 Introduction......................................................................................................................... 840 17.2 T-to-P Transformation and Two-Port Three-Terminal Networks....................................... 841 17.3 Equations of Two-Port Networks........................................................................................ 843 17.4 Z and Y Parameters for a Circuit with Dependent Sources................................................... 846 17.5 Hybrid and Transmission Parameters.................................................................................. 848 17.6 Relationships Between Two-Port Parameters...................................................................... 850 17.7 Interconnection of Two-Port Networks............................................................................... 852 17.8 How Can We Check... ?................................................................................................. 855 17.9 Design Example—Transistor Ampliﬁer.............................................................................. 857 17.10 Summary............................................................................................................................. 859 Problems.............................................................................................................................. 859 Design Problems................................................................................................................. 863 APPENDIX A Getting Started with PSpice.............................................................................................................................. 865 APPENDIX B MATLAB, Matrices, and Complex Arithmetic................................................................................................ 873 APPENDIX C Mathematical Formulas...................................................................................................................................... 885 APPENDIX D Standard Resistor Color Code........................................................................................................................... 889 References............................................................................................................................................................. 891 Index....................................................................................................................................................................... 893 CHAPTER 1 Electric Circuit Variables IN THIS CHAPTER 1.1 Introduction 1.5 Power and 1.8 DESIGN 1.2 Electric Circuits Energy EXAMPLE—Jet and Current 1.6 Circuit Analysis Valve Controller 1.3 Systems of and Design 1.9 Summary Units 1.7 How Can We Problems 1.4 Voltage Check... ? Design Problems 1.1 Introduction A circuit consists of electrical elements connected together. Engineers use electric circuits to solve problems that are important to modern society. In particular: 1. Electric circuits are used in the generation, transmission, and consumption of electric power and energy. 2. Electric circuits are used in the encoding, decoding, storage, retrieval, transmission, and processing of information. In this chapter, we will do the following: Represent the current and voltage of an electric circuit element, paying particular attention to the reference direction of the current and to the reference direction or polarity of the voltage. Calculate the power and energy supplied or received by a circuit element. Use the passive convention to determine whether the product of the current and voltage of a circuit element is the power supplied by that element or the power received by the element. Use scientiﬁc notation to represent electrical quantities with a wide range of magnitudes. 1.2 Electric Circuits and Current The outstanding characteristics of electricity when compared with other power sources are its mobility and ﬂexibility. Electrical energy can be moved to any point along a couple of wires and, depending on the user’s requirements, converted to light, heat, or motion. An electric circuit or electric network is an interconnection of electrical elements linked together in a closed path so that an electric current may ﬂow continuously. 1 2 1. Electric Circuit Variables Consider a simple circuit consisting of two well-known electrical elements, a battery and a resistor, as shown in Figure 1.2-1. Each element is represented by the two-terminal element shown in Figure 1.2-2. Elements are sometimes called devices, and terminals are sometimes called nodes. Wire Battery Resistor a b Wire FIGURE 1.2-2 A general two-terminal electrical element FIGURE 1.2-1 A simple circuit. with terminals a and b. Charge may ﬂow in an electric circuit. Current is the time rate of change of charge past a given point. Charge is the intrinsic property of matter responsible for electric phenomena. The quantity of charge q can be expressed in terms of the charge on one electron, which is 1.602 1019 coulombs. Thus, 1 coulomb is the charge on 6.24 1018 electrons. The current through a speciﬁed area is deﬁned by the electric charge passing through the area per unit of time. Thus, q is deﬁned as the charge expressed in coulombs (C). Charge is the quantity of electricity responsible for electric phenomena. Then we can express current as dq i¼ ð1:2-1Þ dt The unit of current is the ampere (A); an ampere is 1 coulomb per second. Current is the time rate of ﬂow of electric charge past a given point. Note that throughout this chapter we use a lowercase letter, such as q, to denote a variable that is a function of time, q(t). We use an uppercase letter, such as Q, to represent a constant. The ﬂow of current is conventionally represented as a ﬂow of positive charges. This convention was initiated by Benjamin Franklin, the ﬁrst great American electrical scientist. Of course, we now know that charge ﬂow in metal conductors results from electrons with a negative charge. Nevertheless, we will conceive of current as the ﬂow of positive charge, according to accepted convention. i1 Figure 1.2-3 shows the notation that we use to describe a current. There are two parts to a b this notation: a value (perhaps represented by a variable name) and an assigned direction. As a i2 matter of vocabulary, we say that a current exists in or through an element. Figure 1.2-3 shows that there are two ways to assign the direction of the current through an element. The current i1 FIGURE 1.2-3 Current is the rate of ﬂow of electric charge from terminal a to terminal b. On the other hand, the in a circuit element. current i2 is the ﬂow of electric charge from terminal b to terminal a. The currents i1 and i2 are Electric Circuits and Current 3 i I 0 t FIGURE 1.2-4 A direct current of magnitude I. similar but different. They are the same size but have different directions. Therefore, i2 is the negative of i1 and i1 ¼ i2 We always associate an arrow with a current to denote its direction. A complete description of current requires both a value (which can be positive or negative) and a direction (indicated by an arrow). If the current ﬂowing through an element is constant, we represent it by the constant I, as shown in Figure 1.2-4. A constant current is called a direct current (dc). A direct current (dc) is a current of constant magnitude. A time-varying current i(t) can take many forms, such as a ramp, a sinusoid, or an exponential, as shown in Figure 1.2-5. The sinusoidal current is called an alternating current (ac). i i = Mt, t 0 i i = I sin ω t, t 0 (A) (A) i (A) i = Ie–bt, t 0 M I I 1 0 0 t (s) t (s) 0 t (s) –I (a) (b) (c) FIGURE 1.2-5 (a) A ramp with a slope M. (b) A sinusoid. (c) An exponential. I is a constant. The current i is zero for t < 0. If the charge q is known, the current i is readily found using Eq. 1.2-1. Alternatively, if the current i is known, the charge q is readily calculated. Note that from Eq. 1.2-1, we obtain Z t Z t q¼ i dt ¼ i dt þ qð0Þ ð1:2-2Þ 1 0 where q(0) is the charge at t ¼ 0. EXAMPLE 1.2-1 Current from Charge Find the current in an element when the charge entering the element is q ¼ 12t C where t is the time in seconds. 4 1. Electric Circuit Variables Solution Recall that the unit of charge is coulombs, C. Then the current, from Eq. 1.2-1, is dq i¼ ¼ 12 A dt where the unit of current is amperes, A. Try it yourself E X A M P L E 1. 2 - 2 Charge from Current in WileyPLUS Find the charge that has entered the terminal of an element from t ¼ 0 s to t ¼ 3 s when the current entering the element is as shown in Figure 1.2-6. i (A) 4 3 2 1 –1 0 1 2 3 t (s) FIGURE 1.2-6 Current waveform for Example 1.2-2. Solution From Figure 1.2-6, we can describe i(t) as 8 > 0 t 0 t 0 s. The ideal switch changes state instantaneously. The switch in Figure 2.9-1b is initially closed. This switch changes state, becoming open, at time t ¼ 0 s. a a c c t=0 t=0 t=0 b t=0 b Initially open Initially closed Break before make Make before break (a) (b) (a) (b) FIGURE 2.9-1 SPST switches. (a) Initially open and (b) FIGURE 2.9-2 SPDT switches. (a) Break before make initially closed. and (b) make before break. Next, consider the single-pole, double-throw (SPDT) switch shown in Figure 2.9-1a. This SPDT switch acts like two SPST switches, one between terminals c and a, another between terminals c and b. Before t ¼ 0 s, the switch between c and a is closed and the switch between c and b is open. At t ¼ 0 s, both switches change state; that is, the switch between a and c opens, and the switch between c and b closes. Once again, the ideal switches are modeled as open circuits when they are open and as short circuits when they are closed. In some applications, it makes a difference whether the switch between c and b closes before, or after, the switch between c and a opens. Different symbols are used to represent these two types of single-pole, double-throw switch. The break-before-make switch is manufactured so that the switch between c and b closes after the switch between c and a opens. The symbol for the break- before-make switch is shown in Figure 2.9-2a. The make-before-break switch is manufactured so that the switch between c and b closes before the switch between c and a opens. The symbol for the make-before-break switch is shown in Figure 2.9-2b. Remember: the switch transition from terminal a to terminal b is assumed to take place instantaneously. This instantaneous transition is an accurate model when the actual make-before-break transition is very fast compared to the circuit time response. E X A M P L E 2. 9 - 1 Switches Figure 2.9-3 illustrates the use of open and short circuits for modeling ideal switches. In Figure 2.9-3a, a circuit containing three switches is shown. In Figure 2.9-3b, the circuit is shown as it would be modeled before t ¼ 0 s. The two single-pole, single-throw switches change state at time t ¼ 0 s. Figure 2.9-3c shows the circuit as it would be modeled when the time is between 0 s and 2 s. The single-pole, double-throw switch changes state at time t ¼ 2 s. Figure 2.9-3d shows the circuit as it would be modeled after 2 s. 40 2. Circuit Elements 5 kΩ 4 kΩ 5 kΩ 4 kΩ t=2s 12 kΩ 10 kΩ 12 kΩ 10 kΩ t=0s t=0s 12 V + + 12 V + 8 kΩ + – 8 kΩ 6V – – 6V – (a) (c) FIGURE 2.9-3 (a) A circuit containing 5 kΩ 4 kΩ 5 kΩ 4 kΩ several switches. (b) The equivalent 12 kΩ 10 kΩ 12 kΩ 10 kΩ circuit for t 0 s. (c) The equivalent 12 V + 8 kΩ 6 V +– 12 V + 8 kΩ 6 V +– circuit for 0 < t < 2 – – s. (d ) The equivalent circuit (b) (d) for t > 2 s. Try it yourself EXERCISE 2.9-1 What is the value of the current i in Figure E 2.9-1 at time t ¼ 4 s? in WileyPLUS Answer: i ¼ 0 amperes at t ¼ 4 s (both switches are open). EXERCISE 2.9-2 What is the value of the voltage v in Figure E 2.9-2 at time t ¼ 4 s? At t ¼ 6 s? Answer: v ¼ 6 volts at t ¼ 4 s, and v ¼ 0 volts at t ¼ 6 s. t=5s t=3s t=5s + 3 kΩ v + + 2 mA – 12 V 3 kΩ 6V – – i i FIGURE E 2.9-2 A circuit with a make-before-break FIGURE E 2.9-1 A circuit with two SPST switches. SPDT switch. 2.10 How Can We Check... ? Engineers are frequently called upon to check that a solution to a problem is indeed correct. For example, proposed solutions to design problems must be checked to conﬁrm that all of the How Can We Check... ? 41 speciﬁcations have been satisﬁed. In addition, computer output must be reviewed to guard against data- entry errors, and claims made by vendors must be examined critically. Engineering students are also asked to check the correctness of their work. For example, occasionally just a little time remains at the end of an exam. It is useful to be able quickly to identify those solutions that need more work. The following example illustrates techniques useful for checking the solutions of the sort of problem discussed in this chapter. EXAMPLE 2.10-1 How Can We Check Voltage and Current Values? The meters in the circuit of Figure 2.10-1 indicate that v1 ¼ 4 V, v2 ¼ 8 V and that i ¼ 1 A. How can we check that the values of v1, v2, and i have been measured correctly? Let’s check the values of v1, v2, and i in two ways: (a) Verify that the given values satisfy Ohm’s law for both resistors. (b) Verify that the power supplied by the voltage source is equal to the power absorbed by the resistors. – 4. 0 Voltmeter 1. 0 0 Ammeter – v1 + 8. 0 0 Voltmeter 4Ω + i + 12 V – 8Ω v2 – FIGURE 2.10-1 A circuit with meters. Solution (a) Consider the 8-V resistor. The current i ﬂows through this resistor from top to bottom. Thus, the current i and the voltage v2 adhere to the passive convention. Therefore, Ohm’s law requires that v2 ¼ 8i. The values v2 ¼ 8 V and i ¼ 1 A satisfy this equation. Next, consider the 4-V resistor. The current i ﬂows through this resistor from left to right. Thus, the current i and the voltage v1 do not adhere to the passive convention. Therefore, Ohm’s law requires that v1 ¼ 4(i). The values v1 ¼ 4 V and i ¼ 1 A satisfy this equation. Thus, Ohm’s law is satisﬁed. (b) The current i ﬂows through the voltage source from bottom to top. Thus the current i and the voltage 12 V do not adhere to the passive convention. Therefore, 12i ¼ 12(1) ¼ 12 W is the power supplied by the voltage source. The power absorbed by the 4-V resistor is 4i2 ¼ 4(12) ¼ 4 W, and the power absorbed by the 8-V resistor is 8i2 ¼ 8(12) ¼ 8 W. The power supplied by the voltage source is indeed equal to the power absorbed by the resistors. 42 2. Circuit Elements 2. 1 1 D E S I G N E X A M P L E Temperature Sensor Currents can be measured easily, using ammeters. A temperature sensor, such as Analog Devices’ AD590, can be used to measure temperature by converting temperature to current. Figure 2.11-1 shows a symbol used to represent a temperature sensor. For this sensor to operate properly, the voltage v must satisfy the condition 4 volts v 30 volts i(t) + v(t) AD590 – FIGURE 2.11-1 A temperature sensor. When this condition is satisﬁed, the current i, in mA, is numerically equal to the temperature T, in K. The phrase numerically equal indicates that the two variables have the same value but different units. mA i¼kT where k¼1 K The goal is to design a circuit using the AD590 to measure the temperature of a container of water. In addition to the AD590 and an ammeter, several power supplies and an assortment of standard 2 percent resistors are available. The power supplies are voltage sources. Power supplies having voltages of 10, 12, 15, 18, or 24 volts are available. Describe the Situation and the Assumptions For the temperature transducer to operate properly, its element voltage must be between 4 volts and 30 volts. The power supplies and resistors will be used to establish this voltage. An ammeter will be used to measure the current in the temperature transducer. The circuit must be able to measure temperatures in the range from 0 C to 100 C because water is a liquid at these temperatures. Recall that the temperature in C is equal to the temperature in K minus 273. State the Goal Use the power supplies and resistors to cause the voltage v of the temperature transducer to be between 4 volts and 30 volts. Use an ammeter to measure the current, i, in the temperature transducer. Generate a Plan Model the power supply as an ideal voltage source and the temperature transducer as an ideal current source. The circuit shown in Figure 2.11-2a causes the voltage across the temperature transducer to be equal to the power supply voltage. Because all of the available power supplies have voltages between 4 volts and 30 volts, any one of the power supplies can be used. Notice that the resistors are not needed. In Figure 2.11-2b, a short circuit has been added in a way that does not disturb the network. In Figure 2.11-2c, this short circuit has been replaced with an (ideal) ammeter. Because the ammeter will measure the current in the temperature transducer, the ammeter reading will be numerically equal to the temperature in K. Design Example 43 + + + Ammeter v(t) v(t) v(t) + + + – – – – – – i(t) Short i(t) circuit i(t) (a) (b) (c) FIGURE 2.11-2 (a) Measuring temperature with a temperature sensor. (b) Adding a short circuit. (c) Replacing the short circuit by an ammeter. Although any of the available power supplies is adequate to meet the speciﬁcations, there may still be an advantage to choosing a particular power supply. For example, it is reasonable to choose the power supply that causes the transducer to absorb as little power as possible. Act on the Plan The power absorbed by the transducer is p¼vi where v is the power supply voltage. Choosing v as small as possible, 10 volts in this case, makes the power absorbed by the temperature transducer as small as possible. Figure 2.11-3a shows the ﬁnal design. Figure 2.11-3b shows a graph that can be used to ﬁnd the temperature corresponding to any ammeter current. Verify the Proposed Solution Let’s try an example. Suppose the temperature of the water is 80.6 F. This temperature is equal to 27 C or 300 K. The current in the temperature sensor will be mA i¼ 1 300 K ¼ 300 mA K Next, suppose that the ammeter in Figure 2.11-3a reads 300 mA. A sensor current of 300 mA corresponds to a temperature of 300 mA T¼ ¼ 300 K ¼ 27 C ¼ 80:6 F mA 1 K The graph in Figure 2.11-3b indicates that a sensor current of 300 mA does correspond to a temperature of 27 C. This example shows that the circuit is working properly. Temperature, °C 100 Ammeter + 10 V – 0 i(t) 273 373 Ammeter reading, μ A (a) (b) FIGURE 2.11-3 (a) Final design of a circuit that measures temperature with a temperature sensor. (b) Graph of temperature versus ammeter current. 44 2. Circuit Elements 2.12 SUMMARY The engineer uses models, called circuit elements, to repre- A dependent source provides a current (or a voltage) that is sent the devices that make up a circuit. In this book, we dependent on another variable elsewhere in the circuit. The consider only linear elements or linear models of devices. A constitutive equations of dependent sources are summarized device is linear if it satisﬁes the properties of both superpo- in Table 2.7-1. sition and homogeneity. The short circuit and open circuit are special cases of inde- The relationship between the reference directions of the pendent sources. A short circuit is an ideal voltage source current and voltage of a circuit element is important. The having v(t) ¼ 0. The current in a short circuit is determined by voltage polarity marks one terminal þ and the other . The the rest of the circuit. An open circuit is an ideal current source element voltage and current adhere to the passive convention having i(t) ¼ 0. The voltage across an open circuit is determined if the current is directed from the terminal marked þ to the by the rest of the circuit. Open circuits and short circuits can also terminal marked . be described as special cases of resistors. A resistor with Resistors are widely used as circuit elements. When the resistance R ¼ 0 (G ¼ 1) is a short circuit. A resistor with resistor voltage and current adhere to the passive convention, conductance G ¼ 0 (R ¼ 1) is an open circuit. resistors obey Ohm’s law; the voltage across the terminals of An ideal ammeter measures the current ﬂowing through its the resistor is related to the current into the positive terminal terminals and has zero voltage across its terminals. An ideal as v ¼ Ri. The power delivered to a resistance is p ¼ i2R ¼ voltmeter measures the voltage across its terminals and has v2=R watts. terminal current equal to zero. Ideal voltmeters act like open An independent source provides a current or a voltage circuits, and ideal ammeters act like short circuits. independent of other circuit variables. The voltage of an Transducers are devices that convert physical quantities, independent voltage source is speciﬁed, but the current is such as rotational position, to an electrical quantity such not. Conversely, the current of an independent current source as voltage. In this chapter, we describe two transducers: is speciﬁed whereas the voltage is not. The voltages of potentiometers and temperature sensors. independent voltage sources and currents of independent Switches are widely used in circuits to connect and disconnect current sources are frequently used as the inputs to electric elements and circuits. An open switch is modeled as an open circuits. circuit and a closed switch is modeled as a short circuit. PROBLEMS Problem available in Wi

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