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• 1. ale29559_IFC.qxd 07/11/2008 07:40 PM Page 2 PRACTICAL APPLICATIONS Each chapter devotes material to practical applications of the concepts covered in Fundamentals of Electric Circuits to help the reader apply the concepts to real-life situations. Here is a sampling of the practical applications found in the text: • • • • • • • • • • • • • • • • • • • • • • • • • • • • • Rechargeable ﬂashlight battery (Problem 1.11) Cost of operating toaster (Problem 1.25) Potentiometer (Section 2.8) Design a lighting system (Problem 2.61) Reading a voltmeter (Problem 2.66) Controlling speed of a motor (Problem 2.74) Electric pencil sharpener (Problem 2.78) Calculate voltage of transistor (Problem 3.86) Transducer modeling (Problem 4.87) Strain gauge (Problem 4.90) Wheatstone bridge (Problem 4.91) Design a six-bit DAC (Problem 5.83) Instrumentation ampliﬁer (Problem 5.88) Design an analog computer circuit (Example 6.15) Design an op amp circuit (Problem 6.71) Design analog computer to solve differential equation (Problem 6.79) Electric power plant substation—capacitor bank (Problem 6.83) Electronic photo ﬂash unit (Section 7.9) Automobile ignition circuit (Section 7.9) Welding machine (Problem 7.86) Airbag igniter (Problem 8.78) Electrical analog to bodily functions—study of convulsions (Problem 8.82) Electronic sensing device (Problem 9.87) Power transmission system (Problem 9.93) Design a Colpitts oscillator (Problem 10.94) Stereo ampliﬁer circuit (Problem 13.85) Gyrator circuit (Problem 16.69) Calculate number of stations allowable in AM broadcast band (Problem 18.63) Voice signal—Nyquist rate (Problem 18.65)
• 2. ale29559_IFC.qxd 07/11/2008 07:40 PM Page 3 COMPUTER TOOLS promote ﬂexibility and meet ABET requirements • PSpice is introduced in Chapter 3 and appears in special sections throughout the text. Appendix D serves as a tutorial on PSpice for Windows for readers not familiar with its use. The special sections contain examples and practice problems using PSpice. Additional homework problems at the end of each chapter also provide an opportunity to use PSpice. • MATLAB® is introduced through a tutorial in Appendix E to show its usage in circuit analysis. A number of examples and practice problems are presented throughout the book in a manner that will allow the student to develop a facility with this powerful tool. A number of end-of-chapter problems will aid in understanding how to effectively use MATLAB. • KCIDE for Circuits is a working software environment developed at Cleveland State University. It is designed to help the student work through circuit problems in an organized manner following the process on problem-solving discussed in Section 1.8. Appendix F contains a description of how to use the software. Additional examples can be found at the web site, http://kcide.fennresearch.org/. The actual software package can be downloaded for free from this site. One of the best beneﬁts from using this package is that it automatically generates a Word document and/or a PowerPoint presentation. CAREERS AND HISTORY of electrical engineering pioneers Since a course in circuit analysis may be a student’s ﬁrst exposure to electrical engineering, each chapter opens with discussions about how to enhance skills that contribute to successful problem-solving or career-oriented talks on a sub-discipline of electrical engineering. The chapter openers are intended to help students grasp the scope of electrical engineering and give thought to the various careers available to EE graduates. The opening boxes include information on careers in electronics, instrumentation, electromagnetics, control systems, engineering education, and the importance of good communication skills. Historicals throughout the text provide brief biological sketches of such engineering pioneers as Faraday, Ampere, Edison, Henry, Fourier, Volta, and Bell.
• 3. ale29559_IFC.qxd 07/11/2008 07:40 PM Page 4 OUR COMMITMENT TO ACCURACY You have a right to expect an accurate textbook, and McGraw-Hill Engineering invests considerable time and effort to ensure that we deliver one. Listed below are the many steps we take in this process. OUR ACCURACY VERIFICATION PROCESS First Round Step 1: Numerous college engineering instructors review the manuscript and report errors to the editorial team. The authors review their comments and make the necessary corrections in their manuscript. Second Round Step 2: An expert in the field works through every example and exercise in the final manuscript to verify the accuracy of the examples, exercises, and solutions. The authors review any resulting corrections and incorporate them into the final manuscript and solutions manual. Step 3: The manuscript goes to a copyeditor, who reviews the pages for grammatical and stylistic considerations. At the same time, the expert in the field begins a second accuracy check. All corrections are submitted simultaneously to the authors, who review and integrate the editing, and then submit the manuscript pages for typesetting. Third Round Step 4: The authors review their page proofs for a dual purpose: 1) to make certain that any previous corrections were properly made, and 2) to look for any errors they might have missed. Step 5: A proofreader is assigned to the project to examine the new page proofs, double check the authors' work, and add a fresh, critical eye to the book. Revisions are incorporated into a new batch of pages which the authors check again. Fourth Round Step 6: The author team submits the solutions manual to the expert in the field, who checks text pages against the solutions manual as a final review. Step 7: The project manager, editorial team, and author team review the pages for a final accuracy check. The resulting engineering textbook has gone through several layers of quality assurance and is verified to be as accurate and error-free as possible. Our authors and publishing staff are confident that through this process we deliver textbooks that are industry leaders in their correctness and technical integrity.
• 4. ale29559_fm.qxd 07/28/2008 11:54 AM Page i Fundamentals of Electric Circuits
• 5. ale29559_fm.qxd 07/28/2008 11:54 AM Page ii
• 6. ale29559_fm.qxd 07/28/2008 11:54 AM fourth Page iii edition Fundamentals of Electric Circuits Charles K. Alexander Department of Electrical and Computer Engineering Cleveland State University Matthew N. O. Sadiku Department of Electrical Engineering Prairie View A&M University
• 8. ale29559_fm.qxd 07/28/2008 11:54 AM Page v Dedicated to our wives, Kikelomo and Hannah, whose understanding and support have truly made this book possible. Matthew and Chuck
• 9. ale29559_fm.qxd 07/28/2008 11:54 AM Page vi
• 10. ale29559_fm.qxd 07/28/2008 11:54 AM Page vii Contents Preface xiii Acknowledgments xviii Guided Tour xx A Note to the Student xxv About the Authors xxvii Chapter 3 3.1 3.2 3.3 3.4 3.5 PART 1 DC Circuits 2 Chapter 1 Basic Concepts 3 3.6 1.1 1.2 1.3 1.4 1.5 1.6 1.7 Introduction 4 Systems of Units 4 Charge and Current 6 Voltage 9 Power and Energy 10 Circuit Elements 15 † Applications 17 3.7 3.8 3.9 3.10 Methods of Analysis 81 Introduction 82 Nodal Analysis 82 Nodal Analysis with Voltage Sources 88 Mesh Analysis 93 Mesh Analysis with Current Sources 98 † Nodal and Mesh Analyses by Inspection 100 Nodal Versus Mesh Analysis 104 Circuit Analysis with PSpice 105 † Applications: DC Transistor Circuits 107 Summary 112 Review Questions 113 Problems 114 Comprehensive Problem 126 1.7.1 TV Picture Tube 1.7.2 Electricity Bills 1.8 1.9 † Problem Solving 20 Summary 23 Chapter 4 Review Questions 24 Problems 24 Comprehensive Problems 27 4.1 4.2 4.3 4.4 4.5 4.6 4.7 Chapter 2 2.1 2.2 2.3 2.4 2.5 2.6 2.7 2.8 Basic Laws 29 Introduction 30 Ohm’s Law 30 † Nodes, Branches, and Loops 35 Kirchhoff’s Laws 37 Series Resistors and Voltage Division 43 Parallel Resistors and Current Division 45 † Wye-Delta Transformations 52 † Applications 58 4.8 4.9 4.10 Circuit Theorems 127 Introduction 128 Linearity Property 128 Superposition 130 Source Transformation 135 Thevenin’s Theorem 139 Norton’s Theorem 145 † Derivations of Thevenin’s and Norton’s Theorems 149 Maximum Power Transfer 150 Verifying Circuit Theorems with PSpice 152 † Applications 155 4.10.1 Source Modeling 4.10.2 Resistance Measurement 4.11 Summary 160 Review Questions 161 Problems 162 Comprehensive Problems 173 2.8.1 Lighting Systems 2.8.2 Design of DC Meters 2.9 Summary 64 Review Questions 66 Problems 67 Comprehensive Problems 78 Chapter 5 5.1 5.2 Operational Ampliﬁers 175 Introduction 176 Operational Ampliﬁers 176 vii
• 11. ale29559_fm.qxd 07/28/2008 11:54 AM Page viii Contents viii 5.3 5.4 5.5 5.6 5.7 5.8 5.9 5.10 Ideal Op Amp 179 Inverting Ampliﬁer 181 Noninverting Ampliﬁer 183 Summing Ampliﬁer 185 Difference Ampliﬁer 187 Cascaded Op Amp Circuits 191 Op Amp Circuit Analysis with PSpice 194 † Applications 196 5.10.1 Digital-to-Analog Converter 5.10.2 Instrumentation Ampliﬁers 5.11 Summary 199 Review Questions 201 Problems 202 Comprehensive Problems 213 Chapter 6 6.1 6.2 6.3 6.4 6.5 6.6 Introduction 216 Capacitors 216 Series and Parallel Capacitors 222 Inductors 226 Series and Parallel Inductors 230 † Applications 233 Summary 240 Review Questions 241 Problems 242 Comprehensive Problems 251 Chapter 7 7.1 7.2 7.3 7.4 7.5 7.6 7.7 7.8 7.9 7.10 First-Order Circuits 253 Introduction 254 The Source-Free RC Circuit 254 The Source-Free RL Circuit 259 Singularity Functions 265 Step Response of an RC Circuit 273 Step Response of an RL Circuit 280 † First-Order Op Amp Circuits 284 Transient Analysis with PSpice 289 † Applications 293 7.9.1 7.9.2 7.9.3 7.9.4 8.1 8.2 8.3 8.4 8.5 8.6 8.7 8.8 8.9 8.10 8.11 Capacitors and Inductors 215 6.6.1 Integrator 6.6.2 Differentiator 6.6.3 Analog Computer 6.7 Chapter 8 Delay Circuits Photoﬂash Unit Relay Circuits Automobile Ignition Circuit Summary Introduction 314 Finding Initial and Final Values 314 The Source-Free Series RLC Circuit 319 The Source-Free Parallel RLC Circuit 326 Step Response of a Series RLC Circuit 331 Step Response of a Parallel RLC Circuit 336 General Second-Order Circuits 339 Second-Order Op Amp Circuits 344 PSpice Analysis of RLC Circuits 346 † Duality 350 † Applications 353 8.11.1 Automobile Ignition System 8.11.2 Smoothing Circuits 8.12 Summary 356 Review Questions 357 Problems 358 Comprehensive Problems 367 PART 2 AC Circuits 368 Chapter 9 Sinusoids and Phasors 369 9.1 9.2 9.3 9.4 9.5 9.6 9.7 9.8 Introduction 370 Sinusoids 371 Phasors 376 Phasor Relationships for Circuit Elements 385 Impedance and Admittance 387 † Kirchhoff’s Laws in the Frequency Domain 389 Impedance Combinations 390 † Applications 396 9.8.1 Phase-Shifters 9.8.2 AC Bridges 9.9 Summary 402 Review Questions 403 Problems 403 Comprehensive Problems 411 Chapter 10 299 Review Questions 300 Problems 301 Comprehensive Problems 311 Second-Order Circuits 313 10.1 10.2 10.3 Sinusoidal Steady-State Analysis 413 Introduction 414 Nodal Analysis 414 Mesh Analysis 417
• 12. ale29559_fm.qxd 07/28/2008 11:54 AM Page ix Contents 10.4 10.5 10.6 10.7 10.8 10.9 Superposition Theorem 421 Source Transformation 424 Thevenin and Norton Equivalent Circuits 426 Op Amp AC Circuits 431 AC Analysis Using PSpice 433 † Applications 437 10.9.1 Capacitance Multiplier 10.9.2 Oscillators 10.10 Summary 441 Review Questions 441 Problems 443 Chapter 11 11.1 11.2 11.3 11.4 11.5 11.6 11.7 11.8 11.9 AC Power Analysis 457 Introduction 458 Instantaneous and Average Power 458 Maximum Average Power Transfer 464 Effective or RMS Value 467 Apparent Power and Power Factor 470 Complex Power 473 † Conservation of AC Power 477 Power Factor Correction 481 † Applications 483 11.9.1 Power Measurement 11.9.2 Electricity Consumption Cost 11.10 Summary 488 Review Questions 490 Problems 490 Comprehensive Problems 500 Chapter 12 12.11 Summary Introduction 504 Balanced Three-Phase Voltages 505 Balanced Wye-Wye Connection 509 Balanced Wye-Delta Connection 512 Balanced Delta-Delta Connection 514 12.6 Balanced Delta-Wye Connection 516 12.7 Power in a Balanced System 519 12.8 †Unbalanced Three-Phase Systems 525 12.9 PSpice for Three-Phase Circuits 529 12.10 †Applications 534 12.10.1 Three-Phase Power Measurement 12.10.2 Residential Wiring 543 Review Questions 543 Problems 544 Comprehensive Problems 553 Chapter 13 13.1 13.2 13.3 13.4 13.5 13.6 13.7 13.8 13.9 Magnetically Coupled Circuits 555 Introduction 556 Mutual Inductance 557 Energy in a Coupled Circuit 564 Linear Transformers 567 Ideal Transformers 573 Ideal Autotransformers 581 † Three-Phase Transformers 584 PSpice Analysis of Magnetically Coupled Circuits 586 † Applications 591 13.9.1 Transformer as an Isolation Device 13.9.2 Transformer as a Matching Device 13.9.3 Power Distribution 13.10 Summary 597 Review Questions 598 Problems 599 Comprehensive Problems 611 Chapter 14 14.1 14.2 14.3 14.4 14.5 14.6 14.7 14.8 Lowpass Filter Highpass Filter Bandpass Filter Bandstop Filter Active Filters 642 14.8.1 14.8.2 14.8.3 14.8.4 14.9 Frequency Response 613 Introduction 614 Transfer Function 614 † The Decibel Scale 617 Bode Plots 619 Series Resonance 629 Parallel Resonance 634 Passive Filters 637 14.7.1 14.7.2 14.7.3 14.7.4 Three-Phase Circuits 503 12.1 12.2 12.3 12.4 12.5 ix First-Order Lowpass Filter First-Order Highpass Filter Bandpass Filter Bandreject (or Notch) Filter Scaling 648 14.9.1 Magnitude Scaling 14.9.2 Frequency Scaling 14.9.3 Magnitude and Frequency Scaling 14.10 Frequency Response Using PSpice 652 14.11 Computation Using MATLAB 655
• 13. ale29559_fm.qxd 07/28/2008 11:54 AM Page x Contents x 14.12 † Applications 657 17.3 14.12.1 Radio Receiver 14.12.2 Touch-Tone Telephone 14.12.3 Crossover Network 14.13 Summary 663 Review Questions 664 Problems 665 Comprehensive Problems 673 Symmetry Considerations 764 17.3.1 Even Symmetry 17.3.2 Odd Symmetry 17.3.3 Half-Wave Symmetry 17.4 17.5 17.6 17.7 Circuit Applications 774 Average Power and RMS Values 778 Exponential Fourier Series 781 Fourier Analysis with PSpice 787 17.7.1 Discrete Fourier Transform 17.7.2 Fast Fourier Transform 17.8 PART 3 Chapter 15 15.1 15.2 15.3 15.4 Advanced Circuit Analysis 674 † Applications 793 17.8.1 Spectrum Analyzers 17.8.2 Filters 17.9 Summary Introduction to the Laplace Transform 675 Chapter 18 18.1 18.2 18.3 The Convolution Integral 697 † Application to Integrodifferential Equations 705 Summary 708 18.4 18.5 18.6 Review Questions 708 Problems 709 15.7 Introduction 676 Deﬁnition of the Laplace Transform 677 Properties of the Laplace Transform 679 The Inverse Laplace Transform 690 15.4.1 Simple Poles 15.4.2 Repeated Poles 15.4.3 Complex Poles 15.5 15.6 796 Review Questions 798 Problems 798 Comprehensive Problems 807 18.7 Fourier Transform 809 Introduction 810 Deﬁnition of the Fourier Transform 810 Properties of the Fourier Transform 816 Circuit Applications 829 Parseval’s Theorem 832 Comparing the Fourier and Laplace Transforms 835 † Applications 836 18.7.1 Amplitude Modulation 18.7.2 Sampling Chapter 16 16.1 16.2 16.3 16.4 16.5 16.6 Applications of the Laplace Transform 715 Introduction 716 Circuit Element Models 716 Circuit Analysis 722 Transfer Functions 726 State Variables 730 † Applications 737 16.6.1 Network Stability 16.6.2 Network Synthesis 16.7 Summary 745 Review Questions 746 Problems 747 Comprehensive Problems 754 18.8 17.1 17.2 The Fourier Series 755 Introduction 756 Trigonometric Fourier Series 756 839 Review Questions 840 Problems 841 Comprehensive Problems 847 Chapter 19 19.1 19.2 19.3 19.4 19.5 19.6 19.7 19.8 Chapter 17 Summary 19.9 Two-Port Networks 849 Introduction 850 Impedance Parameters 850 Admittance Parameters 855 Hybrid Parameters 858 Transmission Parameters 863 † Relationships Between Parameters 868 Interconnection of Networks 871 Computing Two-Port Parameters Using PSpice 877 † Applications 880 19.9.1 Transistor Circuits 19.9.2 Ladder Network Synthesis
• 14. ale29559_fm.qxd 07/28/2008 11:54 AM Page xi Contents xi 889 Review Questions 890 Problems 890 Comprehensive Problems 901 Appendix A Appendix D PSpice for Windows A-21 Appendix E MATLAB A-46 Appendix F KCIDE for Circuits A-65 Appendix G 19.10 Summary Answers to Odd-Numbered Problems A-75 Simultaneous Equations and Matrix Inversion A Selected Bibliography B-1 Appendix B Complex Numbers A-9 Index I-1 Appendix C Mathematical Formulas A-16
• 15. ale29559_fm.qxd 07/28/2008 11:54 AM Page xii
• 16. ale29559_fm.qxd 08/08/2008 10:16 AM Page xiii Preface You may be wondering why we chose a photo of astronauts working in space on the Space Station for the cover. We actually chose it for several reasons. Obviously, it is very exciting; in fact, space represents the most exciting frontier for the entire world! In addition, much of the station itself consists of all kinds of circuits! One of the most signiﬁcant circuits within the station is its power distribution system. It is a complete and self contained, modern power generation and distribution system. That is why NASA (especially NASA-Glenn) continues to be at the forefront of both theoretical as well as applied power system research and development. The technology that has gone into the development of space exploration continues to ﬁnd itself impacting terrestrial technology in many important ways. For some of you, this will be an important career path. FEATURES New to This Edition A course in circuit analysis is perhaps the ﬁrst exposure students have to electrical engineering. This is also a place where we can enhance some of the skills that they will later need as they learn how to design. In the fourth edition, we have included a very signiﬁcant new feature to help students enhance skills that are an important part of the design process. We call this new feature, design a problem. We know it is not possible to fully develop a student’s design skills in a fundamental course like circuits. To fully develop design skills a student needs a design experience normally reserved for their senior year. This does not mean that some of those skills cannot be developed and exercised in a circuits course. The text already included openended questions that help students use creativity, which is an important part of learning how to design. We already have some questions that are open desired to add much more into our text in this important area and have developed an approach to do just that. When we develop problems for the student to solve our goal is that in solving the problem the student learn more about the theory and the problem solving process. Why not have the students design problems like we do? That is exactly what we will do in each chapter. Within the normal problem set, we have a set of problems where we ask the student to design a problem. This will have two very important results. The ﬁrst will be a better understanding of the basic theory and the second will be the enhancement of some of the student’s basic design skills. We are making effective use of the principle of learning by teaching. Essentially we all learn better when we teach a subject. Designing effective problems is a key part of the teaching process. Students xiii
• 17. ale29559_fm.qxd xiv 07/28/2008 11:54 AM Page xiv Preface should also be encouraged to develop problems, when appropriate, which have nice numbers and do not necessarily overemphasize complicated mathematical manipulations. Additionally we have changed almost 40% of the Practice Problems with the idea to better reﬂect more real component values and to help the student better understand the problem and have added 121 design a problem problems. We have also changed and added a total of 357 end-of-chapter problems (this number contains the new design a problem problems). This brings up a very important advantage to our textbook, we have a total of 2404 Examples, Practice Problems, Review Questions, and end-of-chapter problems! Retained from Previous Editions The main objective of the fourth edition of this book remains the same as the previous editions—to present circuit analysis in a manner that is clearer, more interesting, and easier to understand than other circuit text, and to assist the student in beginning to see the “fun” in engineering. This objective is achieved in the following ways: • Chapter Openers and Summaries Each chapter opens with a discussion about how to enhance skills which contribute to successful problem solving as well as successful careers or a career-oriented talk on a sub-discipline of electrical engineering. This is followed by an introduction that links the chapter with the previous chapters and states the chapter objectives. The chapter ends with a summary of key points and formulas. • Problem Solving Methodology Chapter 1 introduces a six-step method for solving circuit problems which is used consistently throughout the book and media supplements to promote best-practice problem-solving procedures. • Student Friendly Writing Style All principles are presented in a lucid, logical, step-by-step manner. As much as possible, we avoid wordiness and giving too much detail that could hide concepts and impede overall understanding of the material. • Boxed Formulas and Key Terms Important formulas are boxed as a means of helping students sort out what is essential from what is not. Also, to ensure that students clearly understand the key elements of the subject matter, key terms are deﬁned and highlighted. • Margin Notes Marginal notes are used as a pedagogical aid. They serve multiple uses such as hints, cross-references, more exposition, warnings, reminders not to make some particular common mistakes, and problem-solving insights. • Worked Examples Thoroughly worked examples are liberally given at the end of every section. The examples are regarded as a part of the text and are clearly explained without asking the reader to ﬁll in missing steps. Thoroughly worked examples give students a good understanding of the solution process and the conﬁdence to solve problems
• 18. ale29559_fm.qxd 07/28/2008 11:54 AM Page xv Preface • • • • • • • themselves. Some of the problems are solved in two or three different ways to facilitate a substantial comprehension of the subject material as well as a comparison of different approaches. Practice Problems To give students practice opportunity, each illustrative example is immediately followed by a practice problem with the answer. The student can follow the example step by step to aid in the solution of the practice problem without ﬂipping pages or looking at the end of the book for answers. The practice problem is also intended to test a student’s understanding of the preceding example. It will reinforce their grasp of the material before the student can move on to the next section. Complete solutions to the practice problems are available to students on ARIS. Application Sections The last section in each chapter is devoted to practical application aspects of the concepts covered in the chapter. The material covered in the chapter is applied to at least one or two practical problems or devices. This helps students see how the concepts are applied to real-life situations. Review Questions Ten review questions in the form of multiple-choice objective items are provided at the end of each chapter with answers. The review questions are intended to cover the little “tricks” that the examples and end-of-chapter problems may not cover. They serve as a selftest device and help students determine how well they have mastered the chapter. Computer Tools In recognition of the requirements by ABET® on integrating computer tools, the use of PSpice, MATLAB, KCIDE for Circuits, and developing design skills are encouraged in a student-friendly manner. PSpice is covered early on in the text so that students can become familiar and use it throughout the text. Appendix D serves as a tutorial on PSpice for Windows. MATLAB is also introduced early in the book with a tutorial available in Appendix E. KCIDE for Circuits is a brand new, state-of-the-art software system designed to help the students maximize their chance of success in problem solving. It is introduced in Appendix F. Finally, design a problem problems have been introduced, for the ﬁrst time. These are meant to help the student develop skills that will be needed in the design process. Historical Tidbits Historical sketches throughout the text provide proﬁles of important pioneers and events relevant to the study of electrical engineering. Early Op Amp Discussion The operational ampliﬁer (op amp) as a basic element is introduced early in the text. Fourier and Laplace Transforms Coverage To ease the transition between the circuit course and signals and systems courses, Fourier and Laplace transforms are covered lucidly and thoroughly. The chapters are developed in a manner that the interested instructor can go from solutions of ﬁrst-order xv
• 19. ale29559_fm.qxd xvi 07/28/2008 11:54 AM Page xvi Preface • • • • • • circuits to Chapter 15. This then allows a very natural progression from Laplace to Fourier to AC. Four Color Art Program A completely redesigned interior design and four color art program bring circuit drawings to life and enhance key pedagogical elements throughout the text. Extended Examples Examples worked in detail according to the six-step problem solving method provide a roadmap for students to solve problems in a consistent fashion. At least one example in each chapter is developed in this manner. EC 2000 Chapter Openers Based on ABET’s new skill-based CRITERION 3, these chapter openers are devoted to discussions as to how students can acquire the skills that will lead to a signiﬁcantly enhanced career as an engineer. Because these skills are so very important to the student while in college as well as in their career, we will use the heading, “Enhancing your Skills and your Career.” Homework Problems There are 358 new or changed end-of-chapter problems which will provide students with plenty of practice as well as reinforce key concepts. Homework Problem Icons Icons are used to highlight problems that relate to engineering design as well as problems that can be solved using PSpice or MATLAB. KCIDE for Circuits Appendix F A new Appendix F provides a tutorial on the Knowledge Capturing Integrated Design Environment (KCIDE for Circuits) software, available on ARIS. Organization This book was written for a two-semester or three-quarter course in linear circuit analysis. The book may also be used for a one-semester course by a proper selection of chapters and sections by the instructor. It is broadly divided into three parts. • Part 1, consisting of Chapters 1 to 8, is devoted to dc circuits. It covers the fundamental laws and theorems, circuits techniques, and passive and active elements. • Part 2, which contains Chapter 9 to 14, deals with ac circuits. It introduces phasors, sinusoidal steady-state analysis, ac power, rms values, three-phase systems, and frequency response. • Part 3, consisting of Chapters 15 to 19, is devoted to advanced techniques for network analysis. It provides students with a solid introduction to the Laplace transform, Fourier series, Fourier transform, and two-port network analysis. The material in three parts is more than sufﬁcient for a two-semester course, so the instructor must select which chapters or sections to cover. Sections marked with the dagger sign (†) may be skipped, explained brieﬂy, or assigned as homework. They can be omitted without loss of
• 20. ale29559_fm.qxd 07/28/2008 11:54 AM Page xvii Preface continuity. Each chapter has plenty of problems grouped according to the sections of the related material and diverse enough that the instructor can choose some as examples and assign some as homework. As stated earlier, we are using three icons with this edition. We are using to denote problems that either require PSpice in the solution process, where the circuit complexity is such that PSpice would make the solution process easier, and where PSpice makes a good check to see if the problem has been solved correctly. We are using to denote problems where MATLAB is required in the solution process, where MATLAB makes sense because of the problem makeup and its complexity, and where MATLAB makes a good check to see if the problem has been solved correctly. Finally, we use to identify problems that help the student develop skills that are needed for engineering design. More difﬁcult problems are marked with an asterisk (*). Comprehensive problems follow the end-of-chapter problems. They are mostly applications problems that require skills learned from that particular chapter. Prerequisites As with most introductory circuit courses, the main prerequisites, for a course using the text, are physics and calculus. Although familiarity with complex numbers is helpful in the later part of the book, it is not required. A very important asset of this text is that ALL the mathematical equations and fundamentals of physics needed by the student, are included in the text. Supplements McGraw-Hill’s ARIS—Assessment, Review, and Instruction System is a complete, online tutorial, electronic homework, and course management system, designed for greater ease of use than any other system available. Available on adoption, instructors can create and share course materials and assignments with other instructors, edit questions and algorithms, import their own content, and create announcements and due dates for assignments. ARIS has automatic grading and reporting of easy-to-assign algorithmically-generated homework, quizzing, and testing. Once a student is registered in the course, all student activity within McGraw-Hill’s ARIS is automatically recorded and available to the instructor through a fully integrated grade book that can be downloaded to Excel. Also included on ARIS are a solutions manual, text image ﬁles, transition guides to instructors, and Network Analysis Tutorials, software downloads, complete solutions to text practice problems, FE Exam questions, ﬂashcards, and web links to students. Visit www.mhhe.com/alexander. Knowledge Capturing Integrated Design Environment for Circuits (KCIDE for Circuits) This new software, developed at Cleveland State University and funded by NASA, is designed to help the student work through a circuits problem in an organized manner using the six-step problem-solving methodology in the text. KCIDE for Circuits allows students to work a circuit problem in PSpice and MATLAB, track the xvii
• 22. ale29559_fm.qxd 07/28/2008 11:54 AM Page xix Preface Dr. Gordon K. Lee, San Diego State University Gary Perks, Cal Poly State University, San Luis Obispo Dr. Raghu K. Settaluri, Oregon State University Ramakant Srivastava, University of Florida John Watkins, Wichita State University Yik-Chung Wu, The University of Hong Kong Xiao-Bang Xu, Clemson University Finally, we appreciate the feedback received from instructors and students who used the previous editions. We want this to continue, so please keep sending us emails or direct them to the publisher. We can be reached at c.alexander@ieee.org for Charles Alexander and sadiku@ieee.org for Matthew Sadiku. C. K. Alexander and M.N.O. Sadiku xix
• 24. ale29559_fm.qxd 07/28/2008 11:54 AM Page xxi Guided Tour Chapter 3 90 Example 3.3 2V v1 Solution: The supernode contains the 2-V source, nodes 1 and 2, and the 10-⍀ resistor. Applying KCL to the supernode as shown in Fig. 3.10(a) gives v2 +− 2Ω 2A Each illustrative example is immediately followed by a practice problem and answer to test understanding of the preceding example. Methods of Analysis For the circuit shown in Fig. 3.9, ﬁnd the node voltages. 10 Ω 2 ϭ i1 ϩ i2 ϩ 7 4Ω PSpice® for Windows is a student-friendly tool introduced to students early in the text and used throughout, with discussions and examples at the end of each appropriate chapter. Expressing i1 and i2 in terms of the node voltages 7A 2ϭ Figure 3.9 v1 Ϫ 0 v2 Ϫ 0 ϩ ϩ7 2 4 8 ϭ 2v1 ϩ v2 ϩ 28 1 or For Example 3.3. v2 ϭ Ϫ20 Ϫ 2v1 (3.3.1) To get the relationship between v1 and v2, we apply KVL to the circuit in Fig. 3.10(b). Going around the loop, we obtain Ϫv1 Ϫ 2 ϩ v2 ϭ 0 1 v2 ϭ v1 ϩ 2 xxi (3.3.2) From Eqs. (3.3.1) and (3.3.2), we write v2 ϭ v1 ϩ 2 ϭ Ϫ20 Ϫ 2v1 or 3v1 ϭ Ϫ22 1 v1 ϭ Ϫ7.333 V and v2 ϭ v1 ϩ 2 ϭ Ϫ5.333 V. Note that the 10-⍀ resistor does not make any difference because it is connected across the supernode. 2 v2 4Ω 2V 1 i2 7 A i1 2Ω + 7A +− 1 v1 2A 2A The last section in each chapter is devoted to applications of the concepts covered in the chapter to help students apply the concepts to real-life situations. 2 + v2 v1 − − (b) (a) Figure 3.10 Applying: (a) KCL to the supernode, (b) KVL to the loop. Practice Problem 3.3 3Ω +− 21 V + − Find v and i in the circuit of Fig. 3.11. 9V 4Ω + v − 2Ω Answer: Ϫ0.6 V, 4.2 A. i 6Ω Figure 3.11 For Practice Prob. 3.3. Chapter 3 106 Methods of Analysis 120.0000 1 3.9 81.2900 R1 2 20 + 120 V − R3 89.0320 3 10 IDC V1 R2 R4 30 40 3A I1 0 Figure 3.32 For Example 3.10; the schematic of the circuit in Fig. 3.31. are displayed on VIEWPOINTS and also saved in output ﬁle exam310.out. The output ﬁle includes the following: E NODE VOLTAGE NODE VOLTAGE NODE VOLTAGE (1) 120.0000 (2) 81.2900 (3) 89.0320 R1 1 100 Ω R2 + 24 V 2 3 − 2 R3 60 Ω 50 Ω + − 25 Ω 8 R4 4 V1 1.333E + 00 30 Ω E1 −+ R6 4 1 − + 2 For the circuit in Fig. 3.33, use PSpice to ﬁnd the node voltages. 2A 107 R5 indicating that V1 ϭ 120 V, V2 ϭ 81.29 V, V3 ϭ 89.032 V. Practice Problem 3.10 Applications: DC Transistor Circuits Solution: The schematic is shown in Fig. 3.35. (The schematic in Fig. 3.35 includes the output results, implying that it is the schematic displayed on the screen after the simulation.) Notice that the voltage-controlled voltage source E1 in Fig. 3.35 is connected so that its input is the voltage across the 4-⍀ resistor; its gain is set equal to 3. In order to display the required currents, we insert pseudocomponent IPROBES in the appropriate branches. The schematic is saved as exam311.sch and simulated by selecting Analysis/Simulate. The results are displayed on IPROBES as shown in Fig. 3.35 and saved in output ﬁle exam311.out. From the output ﬁle or the IPROBES, we obtain i1 ϭ i2 ϭ 1.333 A and i3 ϭ 2.667 A. 1.333E + 00 2.667E + 00 0 200 V Figure 3.35 The schematic of the circuit in Fig. 3.34. 0 Use PSpice to determine currents i1, i2, and i3 in the circuit of Fig. 3.36. Figure 3.33 Practice Problem 3.11 For Practice Prob. 3.10. i1 Answer: i1 ϭ Ϫ0.4286 A, i2 ϭ 2.286 A, i3 ϭ 2 A. Answer: V1 ϭ Ϫ40 V, V2 ϭ 57.14 V, V3 ϭ 200 V. 4Ω 2A Example 3.11 In the circuit of Fig. 3.34, determine the currents i1, i2, and i3. 3.9 p 2Ω Applications: DC Transistor Circuits 10 V 1Ω 2Ω 3vo +− 4Ω i2 i1 24 V + − 2Ω Figure 3.34 For Example 3.11. 8Ω i3 4Ω + vo − i2 Most of us deal with electronic products on a routine basis and have some experience with personal computers. A basic component for the integrated circuits found in these electronics and computers is the active, three-terminal device known as the transistor. Understanding the transistor is essential before an engineer can start an electronic circuit design. Figure 3.37 depicts various kinds of transistors commercially available. There are two basic types of transistors: bipolar junction transistors (BJTs) and ﬁeld-effect transistors (FETs). Here, we consider only the BJTs, which were the ﬁrst of the two and are still used today. Our objective is to present enough detail about the BJT to enable us to apply the techniques developed in this chapter to analyze dc transistor circuits. 1Ω i3 i1 2Ω + − Figure 3.36 For Practice Prob. 3.11. c h a p t e r 9 Sinusoids and Phasors Each chapter opens with a discussion about how to enhance skills that contribute to successful problem solving as well as successful careers or a careeroriented talk on a sub-discipline of electrical engineering to give students some real-world applications of what they are learning. He who knows not, and knows not that he knows not, is a fool— shun him. He who knows not, and knows that he knows not, is a child— teach him. He who knows, and knows not that he knows, is asleep—wake him up. He who knows, and knows that he knows, is wise—follow him. —Persian Proverb Enhancing Your Skills and Your Career ABET EC 2000 criteria (3.d), “an ability to function on multi-disciplinary teams.” The “ability to function on multidisciplinary teams” is inherently critical for the working engineer. Engineers rarely, if ever, work by themselves. Engineers will always be part of some team. One of the things I like to remind students is that you do not have to like everyone on a team; you just have to be a successful part of that team. Most frequently, these teams include individuals from of a variety of engineering disciplines, as well as individuals from nonengineering disciplines such as marketing and ﬁnance. Students can easily develop and enhance this skill by working in study groups in every course they take. Clearly, working in study groups in nonengineering courses as well as engineering courses outside your discipline will also give you experience with multidisciplinary teams. Photo by Charles Alexander 369 Icons next to the end-of-chapter homework problems let students know which problems relate to engineering design and which problems can be solved using PSpice or MATLAB. Appendices on these computer programs provide tutorials for their use.
• 25. ale29559_fm.qxd xxii 07/28/2008 11:54 AM Page xxii Guided Tour Supplements for Students and Instructors McGraw-Hill’s ARIS—Assessment, Review, and Instruction System is a complete, online tutorial, electronic homework, and course management system, designed for greater ease of use than any other system available. With ARIS, instructors can create and share course materials and assignments with other instructors, edit questions and algorithms, import their own content, and create announcements and due dates for assignments. ARIS has automatic grading and reporting of easy-to-assign algorithmically-generated homework, quizzing, and testing. Once a student is registered in the course, all student activity within McGraw-Hill’s ARIS is automatically recorded and available to the instructor through a fully integrated grade book that can be downloaded to Excel. www.mhhe.com/alexander Knowledge Capturing Integrated Design Environment for Circuits (KCIDE for Circuits) software, linked from ARIS, enhances student understanding of the six-step problem-solving methodology in the book. KCIDE for Circuits allows students to work a circuit problem in PSpice and MATLAB, track the evolution of their solution, and save a record of their process for future reference. Appendix F walks the user through this program.
• 26. ale29559_fm.qxd 07/28/2008 11:54 AM Page xxiii Guided Tour Other resources provided on ARIS. For Students: — Network Analysis Tutorials—a series of interactive quizzes to help students practice fundamental concepts in circuits. — FE Exam Interactive Review Quizzes—chapter based self-quizzes provide hints for solutions and correct solution methods, and help students prepare for the NCEES Fundamentals of Engineering Examination. — Problem Solving Made Almost Easy—a companion workbook to the text, featuring 150 additional problems with complete solutions. — Complete solutions to Practice Problems in the text — Flashcards of key terms — Web links For Instructors: — Image Sets—electronic ﬁles of text ﬁgures for easy integration into your course presentations, exams, and assignments. — Transition Guides—compare coverage of the third edition to other popular circuits books at the section level to aid transition to teaching from our text. xxiii
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• 29. ale29559_fm.qxd xxvi 07/28/2008 11:54 AM Page xxvi A Note to the Student When I was a student, I did not sell any of my engineering textbooks and was very glad I did not! I found that I needed most of them throughout my career. A short review on ﬁnding determinants is covered in Appendix A, complex numbers in Appendix B, and mathematical formulas in Appendix C. Answers to odd-numbered problems are given in Appendix G. Have fun! C. K. A. and M. N. O. S.
• 30. ale29559_fm.qxd 07/28/2008 11:54 AM Page xxvii About the Authors Charles K. Alexander is professor of electrical and computer engineering of the Fenn College of Engineering at Cleveland State University, Cleveland, Ohio. He is also the Director of The Center for Research in Electronics and Aerospace Technology (CREATE), and is the Managing Director of the Wright Center for Sensor Systems (WCSSE). From 2002 until 2006 he was Dean of the Fenn College of Engineering. From 2004 until 2007, he was Director of Ohio ICE, a research center in instrumentation, controls, electronics, and sensors (a coalition of CSU, Case, the University of Akron, and a number of Ohio industries). From 1998 until 2002, he was interim director (2000 and 2001) of the Institute for Corrosion and Multiphase Technologies and Stocker Visiting Professor of electrical engineering and computer science at Ohio University. From 1994–1996 he was dean of engineering and computer science at California State University, Northridge. From 1989–1994 he was acting dean of the college of engineering at Temple University, and from 1986–1989 he was professor and chairman of the department of electrical engineering at Temple. From 1980–1986 he held the same positions at Tennessee Technological University. He was an associate professor and a professor of electrical engineering at Youngstown State University from 1972–1980, where he was named Distinguished Professor in 1977 in recognition of “outstanding teaching and research.” He was assistant professor of electrical engineering at Ohio University in 1971–1972. He received the Ph.D. (1971) and M.S.E.E. (1967) from Ohio University and the B.S.E.E. (1965) from Ohio Northern University. Dr. Alexander has been a consultant to 23 companies and governmental organizations, including the Air Force and Navy and several law firms. He has received over \$85 million in research and development funds for projects ranging from solar energy to software engineering. He has authored 40 publications, including a workbook and a videotape lecture series, and is coauthor of Fundamentals of Electric Circuits, Problem Solving Made Almost Easy, and the ﬁfth edition of the Standard Handbook of Electronic Engineering, with McGraw-Hill. He has made more than 500 paper, professional, and technical presentations. Dr. Alexander is a fellow of the IEEE and served as its president and CEO in 1997. In 1993 and 1994 he was IEEE vice president, professional activities, and chair of the United States Activities Board (USAB). In 1991–1992 he was region 2 director, serving on the Regional Activities Board (RAB) and USAB. He has also been a member of the Educational Activities Board. He served as chair of the USAB Member Activities Council and vice chair of the USAB Professional Activities Council for Engineers, and he chaired the RAB Student Activities Committee and the USAB Student Professional Awareness Committee. Charles K. Alexander xxvii
• 32. ale29559_fm.qxd 07/28/2008 11:54 AM Page 1 Fundamentals of Electric Circuits
• 33. ale29559_ch01.qxd 07/08/2008 10:37 AM Page 2 P A R T O N E DC Circuits OUTLINE 1 Basic Concepts 2 Basic Laws 3 Methods of Analysis 4 Circuit Theorems 5 Operational Ampliﬁers 6 Capacitors and Inductors 7 First-Order Circuits 8 Second-Order Circuits
• 35. ale29559_ch01.qxd 07/08/2008 10:37 AM Page 4 Chapter 1 4 1.1 Basic Concepts Introduction Electric circuit theory and electromagnetic theory are the two fundamental theories upon which all branches of electrical engineering are built. Many branches of electrical engineering, such as power, electric machines, control, electronics, communications, and instrumentation, are based on electric circuit theory. Therefore, the basic electric circuit theory course is the most important course for an electrical engineering student, and always an excellent starting point for a beginning student in electrical engineering education. Circuit theory is also valuable to students specializing in other branches of the physical sciences because circuits are a good model for the study of energy systems in general, and because of the applied mathematics, physics, and topology involved. In electrical engineering, we are often interested in communicating or transferring energy from one point to another. To do this requires an interconnection of electrical devices. Such interconnection is referred to as an electric circuit, and each component of the circuit is known as an element. An electric circuit is an interconnection of electrical elements. Current − + Battery Figure 1.1 A simple electric circuit. Lamp A simple electric circuit is shown in Fig. 1.1. It consists of three basic elements: a battery, a lamp, and connecting wires. Such a simple circuit can exist by itself; it has several applications, such as a ﬂashlight, a search light, and so forth. A complicated real circuit is displayed in Fig. 1.2, representing the schematic diagram for a radio receiver. Although it seems complicated, this circuit can be analyzed using the techniques we cover in this book. Our goal in this text is to learn various analytical techniques and computer software applications for describing the behavior of a circuit like this. Electric circuits are used in numerous electrical systems to accomplish different tasks. Our objective in this book is not the study of various uses and applications of circuits. Rather our major concern is the analysis of the circuits. By the analysis of a circuit, we mean a study of the behavior of the circuit: How does it respond to a given input? How do the interconnected elements and devices in the circuit interact? We commence our study by deﬁning some basic concepts. These concepts include charge, current, voltage, circuit elements, power, and energy. Before deﬁning these concepts, we must ﬁrst establish a system of units that we will use throughout the text. 1.2 Systems of Units As electrical engineers, we deal with measurable quantities. Our measurement, however, must be communicated in a standard language that virtually all professionals can understand, irrespective of the country where the measurement is conducted. Such an international measurement
• 36. ale29559_ch01.qxd 07/17/2008 11:15 AM Page 5 1.2 L1 0.445 ␮H Antenna C3 R1 47 8 7 U1 SBL-1 Mixer C1 2200 pF 3, 4 2, 5, 6 Oscillator R2 C 10 k B R3 EQ1 10 k 2N2222A R6 100 k R5 100 k U2B 1 2 TL072 C10 5 + 7 1.0 ␮F 16 V 6 − + + R7 1M C12 0.0033 R8 15 k C13 C11 100 ␮F 16 V C15 U2A 0.47 1 2 TL072 16 V + 3 8 C14 + 1 0.0022 − 4 2 R9 15 k C6 0.1 5 L2 22.7 ␮H (see text) C4 910 to U1, Pin 8 R11 47 C8 0.1 + C9 1.0 ␮F 16 V Y1 7 MHz C5 910 R4 220 L3 1 mH 5 0.1 1 C2 2200 pF Systems of Units C7 532 R10 10 k GAIN 3 2 + + C16 100 ␮F 16 V + 12-V dc Supply − 6 − 5 4 R12 10 U3 C18 LM386N Audio power amp 0.1 Audio + Output C17 100 ␮F 16 V Figure 1.2 Electric circuit of a radio receiver. Reproduced with permission from QST, August 1995, p. 23. language is the International System of Units (SI), adopted by the General Conference on Weights and Measures in 1960. In this system, there are six principal units from which the units of all other physical quantities can be derived. Table 1.1 shows the six units, their symbols, and the physical quantities they represent. The SI units are used throughout this text. One great advantage of the SI unit is that it uses preﬁxes based on the power of 10 to relate larger and smaller units to the basic unit. Table 1.2 shows the SI preﬁxes and their symbols. For example, the following are expressions of the same distance in meters (m): 600,000,000 mm 600,000 m 600 km TABLE 1.1 Six basic SI units and one derived unit relevant to this text. Quantity Basic unit Length Mass Time Electric current Thermodynamic temperature Luminous intensity Charge meter kilogram second ampere kelvin candela coulomb Symbol m kg s A K cd C TABLE 1.2 The SI preﬁxes. Multiplier 18 10 1015 1012 109 106 103 102 10 10Ϫ1 10Ϫ2 10Ϫ3 10Ϫ6 10Ϫ9 10Ϫ12 10Ϫ15 10Ϫ18 Preﬁx Symbol exa peta tera giga mega kilo hecto deka deci centi milli micro nano pico femto atto E P T G M k h da d c m m n p f a
• 37. ale29559_ch01.qxd 07/08/2008 10:38 AM Page 6 Chapter 1 6 1.3 Basic Concepts Charge and Current The concept of electric charge is the underlying principle for explaining all electrical phenomena. Also, the most basic quantity in an electric circuit is the electric charge. We all experience the effect of electric charge when we try to remove our wool sweater and have it stick to our body or walk across a carpet and receive a shock. Charge is an electrical property of the atomic particles of which matter consists, measured in coulombs (C). We know from elementary physics that all matter is made of fundamental building blocks known as atoms and that each atom consists of electrons, protons, and neutrons. We also know that the charge e on an electron is negative and equal in magnitude to 1.602 ϫ 10Ϫ19 C, while a proton carries a positive charge of the same magnitude as the electron. The presence of equal numbers of protons and electrons leaves an atom neutrally charged. The following points should be noted about electric charge: 1. The coulomb is a large unit for charges. In 1 C of charge, there are 1͞(1.602 ϫ 10Ϫ19) ϭ 6.24 ϫ 1018 electrons. Thus realistic or laboratory values of charges are on the order of pC, nC, or mC.1 2. According to experimental observations, the only charges that occur in nature are integral multiples of the electronic charge e ϭ Ϫ1.602 ϫ 10Ϫ19 C. 3. The law of conservation of charge states that charge can neither be created nor destroyed, only transferred. Thus the algebraic sum of the electric charges in a system does not change. − − I + − − − Battery Figure 1.3 Electric current due to ﬂow of electronic charge in a conductor. A convention is a standard way of describing something so that others in the profession can understand what we mean. We will be using IEEE conventions throughout this book. We now consider the ﬂow of electric charges. A unique feature of electric charge or electricity is the fact that it is mobile; that is, it can be transferred from one place to another, where it can be converted to another form of energy. When a conducting wire (consisting of several atoms) is connected to a battery (a source of electromotive force), the charges are compelled to move; positive charges move in one direction while negative charges move in the opposite direction. This motion of charges creates electric current. It is conventional to take the current ﬂow as the movement of positive charges. That is, opposite to the ﬂow of negative charges, as Fig. 1.3 illustrates. This convention was introduced by Benjamin Franklin (1706–1790), the American scientist and inventor. Although we now know that current in metallic conductors is due to negatively charged electrons, we will follow the universally accepted convention that current is the net ﬂow of positive charges. Thus, Electric current is the time rate of change of charge, measured in amperes (A). 1 However, a large power supply capacitor can store up to 0.5 C of charge.
• 38. ale29559_ch01.qxd 07/16/2008 12:22 PM Page 7 1.3 Charge and Current 7 Historical Andre-Marie Ampere (1775–1836), a French mathematician and physicist, laid the foundation of electrodynamics. He deﬁned the electric current and developed a way to measure it in the 1820s. Born in Lyons, France, Ampere at age 12 mastered Latin in a few weeks, as he was intensely interested in mathematics and many of the best mathematical works were in Latin. He was a brilliant scientist and a proliﬁc writer. He formulated the laws of electromagnetics. He invented the electromagnet and the ammeter. The unit of electric current, the ampere, was named after him. The Burndy Library Collection at The Huntington Library, San Marino, California. Mathematically, the relationship between current i, charge q, and time t is iϭ ¢ dq dt (1.1) where current is measured in amperes (A), and 1 ampere ϭ 1 coulomb/second The charge transferred between time t0 and t is obtained by integrating both sides of Eq. (1.1). We obtain I t Qϭ ¢ Ύ i dt (1.2) t0 The way we deﬁne current as i in Eq. (1.1) suggests that current need not be a constant-valued function. As many of the examples and problems in this chapter and subsequent chapters suggest, there can be several types of current; that is, charge can vary with time in several ways. If the current does not change with time, but remains constant, we call it a direct current (dc). 0 t (a) i A direct current (dc) is a current that remains constant with time. By convention the symbol I is used to represent such a constant current. A time-varying current is represented by the symbol i. A common form of time-varying current is the sinusoidal current or alternating current (ac). 0 An alternating current (ac) is a current that varies sinusoidally with time. Such current is used in your household, to run the air conditioner, refrigerator, washing machine, and other electric appliances. Figure 1.4 t (b) Figure 1.4 Two common types of current: (a) direct current (dc), (b) alternating current (ac).
• 39. ale29559_ch01.qxd 07/08/2008 10:38 AM Page 8 Chapter 1 8 −5 A 5A (a) (b) Figure 1.5 Conventional current ﬂow: (a) positive current ﬂow, (b) negative current ﬂow. Example 1.1 Basic Concepts shows direct current and alternating current; these are the two most common types of current. We will consider other types later in the book. Once we deﬁne current as the movement of charge, we expect current to have an associated direction of ﬂow. As mentioned earlier, the direction of current ﬂow is conventionally taken as the direction of positive charge movement. Based on this convention, a current of 5 A may be represented positively or negatively as shown in Fig. 1.5. In other words, a negative current of Ϫ5 A ﬂowing in one direction as shown in Fig. 1.5(b) is the same as a current of ϩ5 A ﬂowing in the opposite direction. How much charge is represented by 4,600 electrons? Solution: Each electron has Ϫ1.602 ϫ 10Ϫ19 C. Hence 4,600 electrons will have Ϫ1.602 ϫ 10Ϫ19 C/electron ϫ 4,600 electrons ϭ Ϫ7.369 ϫ 10Ϫ16 C Practice Problem 1.1 Calculate the amount of charge represented by four million protons. Answer: ϩ6.408 ϫ 10Ϫ13 C. Example 1.2 The total charge entering a terminal is given by q ϭ 5t sin 4 p t mC. Calculate the current at t ϭ 0.5 s. Solution: iϭ dq d ϭ (5t sin 4 p t) mC/s ϭ (5 sin 4 p t ϩ 20 p t cos 4 p t) mA dt dt At t ϭ 0.5, i ϭ 5 sin 2 p ϩ 10 p cos 2 p ϭ 0 ϩ 10 p ϭ 31.42 mA Practice Problem 1.2 If in Example 1.2, q ϭ (10 Ϫ 10eϪ2t ) mC, ﬁnd the current at t ϭ 0.5 s. Answer: 7.36 mA.
• 40. ale29559_ch01.qxd 07/08/2008 10:38 AM Page 9 1.4 Voltage 9 Example 1.3 Determine the total charge entering a terminal between t ϭ 1 s and t ϭ 2 s if the current passing the terminal is i ϭ (3t 2 Ϫ t) A. Solution: Qϭ Ύ 2 i dt ϭ tϭ1 ϭ at 3 Ϫ Ύ 2 (3t 2 Ϫ t) dt 1 2 2 t 1 b ` ϭ (8 Ϫ 2) Ϫ a1 Ϫ b ϭ 5.5 C 2 1 2 Practice Problem 1.3 The current ﬂowing through an element is iϭ e 2 A, 2t 2 A, 0 6 t 6 1 t 7 1 Calculate the charge entering the element from t ϭ 0 to t ϭ 2 s. Answer: 6.667 C. 1.4 Voltage As explained brieﬂy in the previous section, to move the electron in a conductor in a particular direction requires some work or energy transfer. This work is performed by an external electromotive force (emf), typically represented by the battery in Fig. 1.3. This emf is also known as voltage or potential difference. The voltage vab between two points a and b in an electric circuit is the energy (or work) needed to move a unit charge from a to b; mathematically, vab ϭ ¢ dw dq (1.3) where w is energy in joules (J) and q is charge in coulombs (C). The voltage vab or simply v is measured in volts (V), named in honor of the Italian physicist Alessandro Antonio Volta (1745–1827), who invented the ﬁrst voltaic battery. From Eq. (1.3), it is evident that 1 volt ϭ 1 joule/coulomb ϭ 1 newton-meter/coulomb Thus, Voltage (or potential difference) is the energy required to move a unit charge through an element, measured in volts (V). + a vab Figure 1.6 shows the voltage across an element (represented by a rectangular block) connected to points a and b. The plus (ϩ) and minus (Ϫ) signs are used to deﬁne reference direction or voltage polarity. The vab can be interpreted in two ways: (1) point a is at a potential of vab − Figure 1.6 Polarity of voltage vab. b
• 41. ale29559_ch01.qxd 07/16/2008 12:23 PM Page 10 Chapter 1 10 Basic Concepts Historical Alessandro Antonio Volta (1745–1827), an Italian physicist, invented the electric battery—which provided the ﬁrst continuous ﬂow of electricity—and the capacitor. Born into a noble family in Como, Italy, Volta was performing electrical experiments at age 18. His invention of the battery in 1796 revolutionized the use of electricity. The publication of his work in 1800 marked the beginning of electric circuit theory. Volta received many honors during his lifetime. The unit of voltage or potential difference, the volt, was named in his honor. The Burndy Library Collection at The Huntington Library, San Marino, California. volts higher than point b, or (2) the potential at point a with respect to point b is vab. It follows logically that in general + a (a) + b vab ϭ Ϫvba a −9 V 9V − − b (b) Figure 1.7 Two equivalent representations of the same voltage vab : (a) point a is 9 V above point b, (b) point b is Ϫ9 V above point a. Keep in mind that electric current is always through an element and that electric voltage is always across the element or between two points. (1.4) For example, in Fig. 1.7, we have two representations of the same voltage. In Fig. 1.7(a), point a is ϩ9 V above point b; in Fig. 1.7(b), point b is Ϫ9 V above point a. We may say that in Fig. 1.7(a), there is a 9-V voltage drop from a to b or equivalently a 9-V voltage rise from b to a. In other words, a voltage drop from a to b is equivalent to a voltage rise from b to a. Current and voltage are the two basic variables in electric circuits. The common term signal is used for an electric quantity such as a current or a voltage (or even electromagnetic wave) when it is used for conveying information. Engineers prefer to call such variables signals rather than mathematical functions of time because of their importance in communications and other disciplines. Like electric current, a constant voltage is called a dc voltage and is represented by V, whereas a sinusoidally time-varying voltage is called an ac voltage and is represented by v. A dc voltage is commonly produced by a battery; ac voltage is produced by an electric generator. 1.5 Power and Energy Although current and voltage are the two basic variables in an electric circuit, they are not sufﬁcient by themselves. For practical purposes, we need to know how much power an electric device can handle. We all know from experience that a 100-watt bulb gives more light than a 60-watt bulb. We also know that when we pay our bills to the electric utility companies, we are paying for the electric energy consumed over a certain period of time. Thus, power and energy calculations are important in circuit analysis.
• 42. ale29559_ch01.qxd 07/08/2008 10:38 AM Page 11 1.5 Power and Energy 11 To relate power and energy to voltage and current, we recall from physics that: Power is the time rate of expending or absorbing energy, measured in watts (W). We write this relationship as i dw dt i ¢ + (1.5) where p is power in watts (W), w is energy in joules (J), and t is time in seconds (s). From Eqs. (1.1), (1.3), and (1.5), it follows that + v pϭ v − − p = +vi pϭ dw dq dw ϭ ؒ ϭ vi dt dq dt p = −vi (a) (b) (1.6) Figure 1.8 Reference polarities for power using the passive sign convention: (a) absorbing power, (b) supplying power. or p ϭ vi (1.7) The power p in Eq. (1.7) is a time-varying quantity and is called the instantaneous power. Thus, the power absorbed or supplied by an element is the product of the voltage across the element and the current through it. If the power has a ϩ sign, power is being delivered to or absorbed by the element. If, on the other hand, the power has a Ϫ sign, power is being supplied by the element. But how do we know when the power has a negative or a positive sign? Current direction and voltage polarity play a major role in determining the sign of power. It is therefore important that we pay attention to the relationship between current i and voltage v in Fig. 1.8(a). The voltage polarity and current direction must conform with those shown in Fig. 1.8(a) in order for the power to have a positive sign. This is known as the passive sign convention. By the passive sign convention, current enters through the positive polarity of the voltage. In this case, p ϭ ϩvi or vi 7 0 implies that the element is absorbing power. However, if p ϭ Ϫvi or vi 6 0, as in Fig. 1.8(b), the element is releasing or supplying power. Passive sign convention is satisﬁed when the current enters through the positive terminal of an element and p ϭ ϩvi. If the current enters through the negative terminal, p ϭ Ϫvi. Unless otherwise stated, we will follow the passive sign convention throughout this text. For example, the element in both circuits of Fig. 1.9 has an absorbing power of ϩ12 W because a positive current enters the positive terminal in both cases. In Fig. 1.10, however, the element is supplying power of ϩ12 W because a positive current enters the negative terminal. Of course, an absorbing power of Ϫ12 W is equivalent to a supplying power of ϩ12 W. In general, ϩPower absorbed ϭ ϪPower supplied When the voltage and current directions conform to Fig. 1.8 (b), we have the active sign convention and p ϭ ϩvi. 3A 3A + − 4V 4V − + (a) (b) Figure 1.9 Two cases of an element with an absorbing power of 12 W: (a) p ϭ 4 ϫ 3 ϭ 12 W, (b) p ϭ 4 ϫ 3 ϭ 12 W. 3A 3A + − 4V 4V − + (a) (b) Figure 1.10 Two cases of an element with a supplying power of 12 W: (a) p ϭ Ϫ4 ϫ 3 ϭ Ϫ12W, (b) p ϭ Ϫ4 ϫ 3 ϭ Ϫ12 W.
• 43. ale29559_ch01.qxd 07/08/2008 10:38 AM Page 12 Chapter 1 12 Basic Concepts In fact, the law of conservation of energy must be obeyed in any electric circuit. For this reason, the algebraic sum of power in a circuit, at any instant of time, must be zero: apϭ0 (1.8) This again conﬁrms the fact that the total power supplied to the circuit must balance the total power absorbed. From Eq. (1.6), the energy absorbed or supplied by an element from time t0 to time t is t t t0 wϭ t0 Ύ p dt ϭ Ύ vi dt (1.9) Energy is the capacity to do work, measured in joules (J). The electric power utility companies measure energy in watt-hours (Wh), where 1 Wh ϭ 3,600 J Example 1.4 An energy source forces a constant current of 2 A for 10 s to ﬂow through a lightbulb. If 2.3 kJ is given off in the form of light and heat energy, calculate the voltage drop across the bulb. Solution: The total charge is ¢q ϭ i ¢t ϭ 2 ϫ 10 ϭ 20 C The voltage drop is vϭ Practice Problem 1.4 ¢w 2.3 ϫ 103 ϭ ϭ 115 V ¢q 20 To move charge q from point a to point b requires Ϫ30 J. Find the voltage drop vab if: (a) q ϭ 2 C, (b) q ϭ Ϫ6 C. Answer: (a) Ϫ15 V, (b) 5 V. Example 1.5 Find the power delivered to an element at t ϭ 3 ms if the current entering its positive terminal is i ϭ 5 cos 60 p t A and the voltage is: (a) v ϭ 3i, (b) v ϭ 3 di͞dt.
• 44. ale29559_ch01.qxd 07/08/2008 10:38 AM Page 13 1.5 Power and Energy 13 Solution: (a) The voltage is v ϭ 3i ϭ 15 cos 60 p t; hence, the power is p ϭ vi ϭ 75 cos2 60 p t W At t ϭ 3 ms, p ϭ 75 cos2 (60 p ϫ 3 ϫ 10Ϫ3) ϭ 75 cos2 0.18 p ϭ 53.48 W (b) We ﬁnd the voltage and the power as vϭ3 di ϭ 3(Ϫ60 p)5 sin 60 p t ϭ Ϫ900 p sin 60 p t V dt p ϭ vi ϭ Ϫ4500 p sin 60 p t cos 60 p t W At t ϭ 3 ms, p ϭ Ϫ4500 p sin 0.18 p cos 0.18 p W ϭ Ϫ14137.167 sin 32.4Њ cos 32.4Њ ϭ Ϫ6.396 kW Find the power delivered to the element in Example 1.5 at t ϭ 5 ms if the current remains the same but the voltage is: (a) v ϭ 2i V, (b) v ϭ a10 ϩ 5 Practice Problem 1.5 t Ύ i dtb V. 0 Answer: (a) 17.27 W, (b) 29.7 W. How much energy does a 100-W electric bulb consume in two hours? Example 1.6 Solution: w ϭ pt ϭ 100 (W) ϫ 2 (h) ϫ 60 (min/h) ϫ 60 (s/min) ϭ 720,000 J ϭ 720 kJ This is the same as w ϭ pt ϭ 100 W ϫ 2 h ϭ 200 Wh A stove element draws 15 A when connected to a 240-V line. How long does it take to consume 60 kJ? Answer: 16.667 s. Practice Problem 1.6
• 45. ale29559_ch01.qxd 07/08/2008 14 10:38 AM Page 14 Chapter 1 Basic Concepts Historical 1884 Exhibition In the United States, nothing promoted the future of electricity like the 1884 International Electrical Exhibition. Just imagine a world without electricity, a world illuminated by candles and gaslights, a world where the most common transportation was by walking and riding on horseback or by horse-drawn carriage. Into this world an exhibition was created that highlighted Thomas Edison and reﬂected his highly developed ability to promote his inventions and products. His exhibit featured spectacular lighting displays powered by an impressive 100-kW “Jumbo” generator. Edward Weston’s dynamos and lamps were featured in the United States Electric Lighting Company’s display. Weston’s well known collection of scientiﬁc instruments was also shown. Other prominent exhibitors included Frank Sprague, Elihu Thompson, and the Brush Electric Company of Cleveland. The American Institute of Electrical Engineers (AIEE) held its ﬁrst technical meeting on October 7–8 at the Franklin Institute during the exhibit. AIEE merged with the Institute of Radio Engineers (IRE) in 1964 to form the Institute of Electrical and Electronics Engineers (IEEE). Smithsonian Institution.
• 46. ale29559_ch01.qxd 07/08/2008 10:38 AM Page 15 1.6 1.6 Circuit Elements 15 Circuit Elements As we discussed in Section 1.1, an element is the basic building block of a circuit. An electric circuit is simply an interconnection of the elements. Circuit analysis is the process of determining voltages across (or the currents through) the elements of the circuit. There are two types of elements found in electric circuits: passive elements and active elements. An active element is capable of generating energy while a passive element is not. Examples of passive elements are resistors, capacitors, and inductors. Typical active elements include generators, batteries, and operational ampliﬁers. Our aim in this section is to gain familiarity with some important active elements. The most important active elements are voltage or current sources that generally deliver power to the circuit connected to them. There are two kinds of sources: independent and dependent sources. An ideal independent source is an active element that provides a speciﬁed voltage or current that is completely independent of other circuit elements. v In other words, an ideal independent voltage source delivers to the circuit whatever current is necessary to maintain its terminal voltage. Physical sources such as batteries and generators may be regarded as approximations to ideal voltage sources. Figure 1.11 shows the symbols for independent voltage sources. Notice that both symbols in Fig. 1.11(a) and (b) can be used to represent a dc voltage source, but only the symbol in Fig. 1.11(a) can be used for a time-varying voltage source. Similarly, an ideal independent current source is an active element that provides a speciﬁed current completely independent of the voltage across the source. That is, the current source delivers to the circuit whatever voltage is necessary to maintain the designated current. The symbol for an independent current source is displayed in Fig. 1.12, where the arrow indicates the direction of current i. An ideal dependent (or controlled) source is an active element in which the source quantity is controlled by another voltage or current. Dependent sources are usually designated by diamond-shaped symbols, as shown in Fig. 1.13. Since the control of the dependent source is achieved by a voltage or current of some other element in the circuit, and the source can be voltage or current, it follows that there are four possible types of dependent sources, namely: 1. 2. 3. 4. A voltage-controlled voltage source (VCVS). A current-controlled voltage source (CCVS). A voltage-controlled current source (VCCS). A current-controlled current source (CCCS). + V − + − (b) (a) Figure 1.11 Symbols for independent voltage sources: (a) used for constant or time-varying voltage, (b) used for constant voltage (dc). i Figure 1.12 Symbol for independent current source. v + − i (a) (b) Figure 1.13 Symbols for: (a) dependent voltage source, (b) dependent current source.
• 47. ale29559_ch01.qxd 07/08/2008 10:38 AM Page 16 Chapter 1 16 B A i + 5V − + − C 10i Figure 1.14 The source on the right-hand side is a current-controlled voltage source. Example 1.7 − + 12 V 20 V + − p1 Figure 1.15 For Example 1.7. p3 Dependent sources are useful in modeling elements such as transistors, operational ampliﬁers, and integrated circuits. An example of a current-controlled voltage source is shown on the right-hand side of Fig. 1.14, where the voltage 10i of the voltage source depends on the current i through element C. Students might be surprised that the value of the dependent voltage source is 10i V (and not 10i A) because it is a voltage source. The key idea to keep in mind is that a voltage source comes with polarities (ϩ Ϫ) in its symbol, while a current source comes with an arrow, irrespective of what it depends on. It should be noted that an ideal voltage source (dependent or independent) will produce any current required to ensure that the terminal voltage is as stated, whereas an ideal current source will produce the necessary voltage to ensure the stated current ﬂow. Thus, an ideal source could in theory supply an inﬁnite amount of energy. It should also be noted that not only do sources supply power to a circuit, they can absorb power from a circuit too. For a voltage source, we know the voltage but not the current supplied or drawn by it. By the same token, we know the current supplied by a current source but not the voltage across it. Calculate the power supplied or absorbed by each element in Fig. 1.15. Solution: We apply the sign convention for power shown in Figs. 1.8 and 1.9. For p1, the 5-A current is out of the positive terminal (or into the negative terminal); hence, p2 I=5A Basic Concepts 6A + 8V − p4 0.2 I p1 ϭ 20(Ϫ5) ϭ Ϫ100 W Supplied power For p2 and p3, the current ﬂows into the positive terminal of the element in each case. p2 ϭ 12(5) ϭ 60 W p3 ϭ 8(6) ϭ 48 W Absorbed power Absorbed power For p4, we should note that the voltage is 8 V (positive at the top), the same as the voltage for p3, since both the passive element and the dependent source are connected to the same terminals. (Remember that voltage is always measured across an element in a circuit.) Since the current ﬂows out of the positive terminal, p4 ϭ 8(Ϫ0.2I ) ϭ 8(Ϫ0.2 ϫ 5) ϭ Ϫ8 W Supplied power We should observe that the 20-V independent voltage source and 0.2I dependent current source are supplying power to the rest of the network, while the two passive elements are absorbing power. Also, p1 ϩ p2 ϩ p3 ϩ p4 ϭ Ϫ100 ϩ 60 ϩ 48 Ϫ 8 ϭ 0 In agreement with Eq. (1.8), the total power supplied equals the total power absorbed.
• 48. Page 17 1.7 Applications Compute the power absorbed or supplied by each component of the circuit in Fig. 1.16. 17 Practice Problem 1.7 8A 1.7.1 TV Picture Tube One important application of the motion of electrons is found in both the transmission and reception of TV signals. At the transmission end, a TV camera reduces a scene from an optical image to an electrical signal. Scanning is accomplished with a thin beam of electrons in an iconoscope camera tube. At the receiving end, the image is reconstructed by using a cathoderay tube (CRT) located in the TV receiver.3 The CRT is depicted in Fig. 1.17. Unlike the iconoscope tube, which produces an electron beam of constant intensity, the CRT beam varies in intensity according to the incoming signal. The electron gun, maintained at a high potential, ﬁres the electron beam. The beam passes through two sets of plates for vertical and horizontal deﬂections so that the spot on the screen where the beam strikes can move right and left and up and down. When the electron beam strikes the ﬂuorescent screen, it gives off light at that spot. Thus, the beam can be made to “paint” a picture on the TV screen. Horizontal deflection plates Bright spot on fluorescent screen Vertical deflection plates Electron trajectory Figure 1.17 Cathode-ray tube. D. E. Tilley, Contemporary College Physics Menlo Park, CA: Benjamin/ Cummings, 1979, p. 319. 2 p1 p3 + − Figure 1.16 In this section, we will consider two practical applications of the concepts developed in this chapter. The ﬁrst one deals with the TV picture tube and the other with how electric utilities determine your electric bill. Electron gun 3A p2 + 5V − Applications2 I=5A +− Answer: p1 ϭ Ϫ40 W, p2 ϭ 16 W, p3 ϭ 9 W, p4 ϭ 15 W. 1.7 2V The dagger sign preceding a section heading indicates the section that may be skipped, explained brieﬂy, or assigned as homework. 3 Modern TV tubes use a different technology. For Practice Prob. 1.7. 0.6I p4 3V − 10:38 AM + − 07/08/2008 + ale29559_ch01.qxd
• 49. ale29559_ch01.qxd 07/08/2008 10:38 AM Page 18 Chapter 1 18 Basic Concepts Historical Karl Ferdinand Braun and Vladimir K. Zworykin Zworykin with an iconoscope. © Bettmann/Corbis. Example 1.8 Karl Ferdinand Braun (1850–1918), of the University of Strasbourg, invented the Braun cathode-ray tube in 1879. This then became the basis for the picture tube used for so many years for televisions. It is still the most economical device today, although the price of ﬂat-screen systems is rapidly becoming competitive. Before the Braun tube could be used in television, it took the inventiveness of Vladimir K. Zworykin (1889–1982) to develop the iconoscope so that the modern television would become a reality. The iconoscope developed into the orthicon and the image orthicon, which allowed images to be captured and converted into signals that could be sent to the television receiver. Thus, the television camera was born. The electron beam in a TV picture tube carries 1015 electrons per second. As a design engineer, determine the voltage Vo needed to accelerate the electron beam to achieve 4 W. Solution: The charge on an electron is e ϭ Ϫ1.6 ϫ 10Ϫ19 C If the number of electrons is n, then q ϭ ne and i q Vo Figure 1.18 A simpliﬁed diagram of the cathode-ray tube; for Example 1.8. iϭ dq dn ϭe ϭ (Ϫ1.6 ϫ 10Ϫ19)(1015) ϭ Ϫ1.6 ϫ 10Ϫ4 A dt dt The negative sign indicates that the current flows in a direction opposite to electron ﬂow as shown in Fig. 1.18, which is a simpliﬁed diagram of the CRT for the case when the vertical deﬂection plates carry no charge. The beam power is p ϭ Voi or Vo ϭ p 4 ϭ 25,000 V ϭ i 1.6 ϫ 10Ϫ4 Thus, the required voltage is 25 kV. Practice Problem 1.8 If an electron beam in a TV picture tube carries 1013 electrons/second and is passing through plates maintained at a potential difference of 30 kV, calculate the power in the beam. Answer: 48 mW.
• 50. ale29559_ch01.qxd 07/17/2008 11:15 AM Page 19 1.7 Applications 19 TABLE 1.3 Typical average monthly consumption of household appliances. Appliance kWh consumed Water heater Freezer Lighting Dishwasher Electric iron TV Toaster 500 100 100 35 15 10 4 Appliance Washing machine Stove Dryer Microwave oven Personal computer Radio Clock kWh consumed 120 100 80 25 12 8 2 1.7.2 Electricity Bills The second application deals with how an electric utility company charges their customers. The cost of electricity depends upon the amount of energy consumed in kilowatt-hours (kWh). (Other factors that affect the cost include demand and power factors; we will ignore these for now.) However, even if a consumer uses no energy at all, there is a minimum service charge the customer must pay because it costs money to stay connected to the power line. As energy consumption increases, the cost per kWh drops. It is interesting to note the average monthly consumption of household appliances for a family of ﬁve, shown in Table 1.3. A homeowner consumes 700 kWh in January. Determine the electricity bill for the month using the following residential rate schedule: Example 1.9 Base monthly charge of \$12.00. First 100 kWh per month at 16 cents/kWh. Next 200 kWh per month at 10 cents/kWh. Over 300 kWh per month at 6 cents/kWh. Solution: We calculate the electricity bill as follows. Base monthly charge ϭ \$12.00 First 100 kWh @ \$0.16/k Wh ϭ \$16.00 Next 200 kWh @ \$0.10/k Wh ϭ \$20.00 Remaining 400 kWh @ \$0.06/k Wh ϭ \$24.00 Total charge ϭ \$72.00 Average cost ϭ \$72 ϭ 10.2 cents/kWh 100 ϩ 200 ϩ 400 Referring to the residential rate schedule in Example 1.9, calculate the average cost per kWh if only 400 kWh are consumed in July when the family is on vacation most of the time. Answer: 13.5 cents/kWh. Practice Problem 1.9
• 53. ale29559_ch01.qxd 22 07/08/2008 10:38 AM Page 22 Chapter 1 Basic Concepts analysis. To solve for i8⍀ using mesh analysis will require writing two simultaneous equations to ﬁnd the two loop currents indicated in Fig. 1.21. Using nodal analysis requires solving for only one unknown. This is the easiest approach. 2Ω i1 v1 + v − 2Ω 5V + − Loop 1 i3 i2 + v8Ω − 8Ω 4Ω + v4Ω − − + Loop 2 3V Figure 1.21 Using nodal analysis. Therefore, we will solve for i8⍀ using nodal analysis. 4. Attempt a problem solution. We ﬁrst write down all of the equations we will need in order to ﬁnd i8⍀. i8⍀ ϭ i2, i2 ϭ v1 , 8 i8⍀ ϭ v1 8 v1 Ϫ 5 v1 Ϫ 0 v1 ϩ 3 ϩ ϩ ϭ0 2 8 4 Now we can solve for v1. v1 Ϫ 5 v1 Ϫ 0 v1 ϩ 3 ϩ ϩ d ϭ0 2 8 4 leads to (4v1 Ϫ 20) ϩ (v1) ϩ (2v1 ϩ 6) ϭ 0 v1 2 v1 ϭ ϩ2 V, i8⍀ ϭ ϭ ϭ 0.25 A 7v1 ϭ ϩ14, 8 8 8c 5. Evaluate the solution and check for accuracy. We can now use Kirchhoff’s voltage law (KVL) to check the results. v1 Ϫ 5 2Ϫ5 3 ϭ ϭ Ϫ ϭ Ϫ1.5 A 2 2 2 i2 ϭ i8⍀ ϭ 0.25 A v1 ϩ 3 2ϩ3 5 i3 ϭ ϭ ϭ ϭ 1.25 A 4 4 4 i1 ϩ i2 ϩ i3 ϭ Ϫ1.5 ϩ 0.25 ϩ 1.25 ϭ 0 (Checks.) i1 ϭ Applying KVL to loop 1, Ϫ5 ϩ v2⍀ ϩ v8⍀ ϭ Ϫ5 ϩ (Ϫi1 ϫ 2) ϩ (i2 ϫ 8) ϭ Ϫ5 ϩ 3Ϫ(Ϫ1.5)2 4 ϩ (0.25 ϫ 8) ϭ Ϫ5 ϩ 3 ϩ 2 ϭ 0 (Checks.) Applying KVL to loop 2, Ϫv8⍀ ϩ v4⍀ Ϫ 3 ϭ Ϫ(i2 ϫ 8) ϩ (i3 ϫ 4) Ϫ 3 ϭ Ϫ(0.25 ϫ 8) ϩ (1.25 ϫ 4) Ϫ 3 ϭ Ϫ2 ϩ 5 Ϫ 3 ϭ 0 (Checks.)
• 54. ale29559_ch01.qxd 07/08/2008 10:38 AM Page 23 1.9 Summary 23 So we now have a very high degree of conﬁdence in the accuracy of our answer. 6. Has the problem been solved satisfactorily? If so, present the solution; if not, then return to step 3 and continue through the process again. This problem has been solved satisfactorily. The current through the 8-⍀ resistor is 0.25 A ﬂowing down through the 8-⍀ resistor. Try applying this process to some of the more difﬁcult problems at the end of the chapter. 1.9 Summary 1. An electric circuit consists of electrical elements connected together. 2. The International System of Units (SI) is the international measurement language, which enables engineers to communicate their results. From the six principal units, the units of other physical quantities can be derived. 3. Current is the rate of charge ﬂow. iϭ dq dt 4. Voltage is the energy required to move 1 C of charge through an element. vϭ dw dq 5. Power is the energy supplied or absorbed per unit time. It is also the product of voltage and current. pϭ dw ϭ vi dt 6. According to the passive sign convention, power assumes a positive sign when the current enters the positive polarity of the voltage across an element. 7. An ideal voltage source produces a speciﬁc potential difference across its terminals regardless of what is connected to it. An ideal current source produces a speciﬁc current through its terminals regardless of what is connected to it. 8. Voltage and current sources can be dependent or independent. A dependent source is one whose value depends on some other circuit variable. 9. Two areas of application of the concepts covered in this chapter are the TV picture tube and electricity billing procedure. Practice Problem 1.10
• 55. ale29559_ch01.qxd 07/08/2008 10:38 AM Page 24 Chapter 1 24 Basic Concepts Review Questions 1.1 One millivolt is one millionth of a volt. (a) True 1.2 (b) False (c) 10Ϫ3 (b) 103 (d) 10Ϫ6 1.9 (b) 2 kV (c) 2 MV (d) 2 GV (b) 1100 V (c) 110 V (b) time (d) current (c) voltage (e) power 1.10 The dependent source in Fig. 1.22 is: 1.5 A charge of 2 C ﬂowing past a given point each second is a current of 2 A. (a) voltage-controlled current source (a) True 1.4 (c) current-controlled voltage source (d) current-controlled current source The unit of current is: (b) ampere (c) volt (d) joule io vs Voltage is measured in: (a) watts (d) joules per second + − 6io (b) amperes (c) volts 1.7 (b) voltage-controlled voltage source (b) False (a) coulomb 1.6 (d) 11 V Which of these is not an electrical quantity? (a) charge The voltage 2,000,000 V can be expressed in powers of 10 as: (a) 2 mV The voltage across a 1.1-kW toaster that produces a current of 10 A is: (a) 11 kV The preﬁx micro stands for: (a) 106 1.3 1.8 A 4-A current charging a dielectric material will accumulate a charge of 24 C after 6 s. (a) True (b) False Figure 1.22 For Review Question 1.10. Answers: 1.1b, 1.2d, 1.3c, 1.4a, 1.5b, 1.6c, 1.7a, 1.8c, 1.9b, 1.10d. Problems Section 1.3 Charge and Current 1.1 How many coulombs are represented by these amounts of electrons? (a) 6.482 ϫ 1017 (c) 2.46 ϫ 1019 1.2 (b) 1.24 ϫ 1018 (d) 1.628 ϫ 10 20 Determine the current ﬂowing through an element if the charge ﬂow is given by (a) q(t) ϭ (3t ϩ 8) mC (b) q(t) ϭ (8t2 ϩ 4t Ϫ 2) C 1.4 A current of 3.2 A ﬂows through a conductor. Calculate how much charge passes through any cross-section of the conductor in 20 s. 1.5 Determine the total charge transferred over the time interval of 0 Յ t Յ 10 s when i(t) ϭ 1 t A. 2 1.6 The charge entering a certain element is shown in Fig. 1.23. Find the current at: (a) t ϭ 1 ms (b) t ϭ 6 ms (c) t ϭ 10 ms q(t) (mC) (c) q(t) ϭ (3eϪt Ϫ 5eϪ2t ) nC 80 (d) q(t) ϭ 10 sin 120 p t pC (e) q(t) ϭ 20eϪ4t cos 50t m C 1.3 Find the charge q(t) ﬂowing through a device if the current is: (a) i(t) ϭ 3 A, q(0) ϭ 1 C (b) i(t) ϭ (2t ϩ 5) mA, q(0) ϭ 0 (c) i(t) ϭ 20 cos(10t ϩ p͞6) mA, q(0) ϭ 2 mC (d) i(t) ϭ 10eϪ30t sin 40t A, q(0) ϭ 0 0 Figure 1.23 For Prob. 1.6. 2 4 6 8 10 12 t (ms)
• 56. ale29559_ch01.qxd 07/08/2008 10:38 AM Page 25 Problems 1.7 The charge ﬂowing in a wire is plotted in Fig. 1.24. Sketch the corresponding current. 25 1.13 The charge entering the positive terminal of an element is q ϭ 10 sin 4 p t mC q (C) while the voltage across the element (plus to minus) is 50 v ϭ 2 cos 4 p t V 0 2 4 8 6 t (s) −50 (b) Calculate the energy delivered to the element between 0 and 0.6 s. Figure 1.24 1.14 The voltage v across a device and the current i through it are For Prob. 1.7. 1.8 (a) Find the power delivered to the element at t ϭ 0.3 s. The current ﬂowing past a point in a device is shown in Fig. 1.25. Calculate the total charge through the point. i(t) ϭ 10 (1 Ϫ e Ϫ0.5t ) A v(t) ϭ 5 cos 2t V, Calculate: i (mA) (a) the total charge in the device at t ϭ 1 s (b) the power consumed by the device at t ϭ 1 s. 10 0 2 1 t (ms) Figure 1.25 (a) Find the charge delivered to the device between t ϭ 0 and t ϭ 2 s. For Prob. 1.8. 1.9 1.15 The current entering the positive terminal of a device is i(t) ϭ 3eϪ2t A and the voltage across the device is v(t) ϭ 5di͞dt V. The current through an element is shown in Fig. 1.26. Determine the total charge that passed through the element at: (a) t ϭ 1 s (b) t ϭ 3 s (c) t ϭ 5 s (b) Calculate the power absorbed. (c) Determine the energy absorbed in 3 s. Section 1.6 Circuit Elements 1.16 Find the power absorbed by each element in Fig. 1.27. i (A) 10 5 4A −3 A 2A + 0 1 2 3 4 Figure 1.26 − 10 V − 5 t (s) + 12 V − 5V + For Prob. 1.9. Sections 1.4 and 1.5 Voltage, Power, and Energy 1.10 A lightning bolt with 8 kA strikes an object for 15 ms. How much charge is deposited on the object? 1.11 A rechargeable ﬂashlight battery is capable of delivering 85 mA for about 12 h. How much charge can it release at that rate? If its terminal voltage is 1.2 V, how much energy can the battery deliver? Figure 1.27 For Prob. 1.16. 1.17 Figure 1.28 shows a circuit with ﬁve elements. If p1 ϭ Ϫ205 W, p2 ϭ 60 W, p4 ϭ 45 W, p5 ϭ 30 W, calculate the power p3 received or delivered by element 3. 1.12 If the current ﬂowing through an element is given by 3tA, 0 18A, 6 i(t) ϭ µ Ϫ12A, 10 0, Ϲt Ϲt Ϲt t 6 6 6 м 6s 10 s 15 s 15 s Plot the charge stored in the element over 0 6 t 6 20 s. 2 1 Figure 1.28 For Prob. 1.17. 4 3 5
• 57. ale29559_ch01.qxd 07/08/2008 10:38 AM Page 26 Chapter 1 26 1.18 Calculate the power absorbed or supplied by each element in Fig. 1.29. 4A 4A 9V + − 1.22 A lightning bolt strikes an airplane with 30 kA for 2 ms. How many coulombs of charge are deposited on the plane? + 3V − 2 Section 1.7 Applications 1.21 A 60-W incandescent bulb operates at 120 V. How many electrons and coulombs ﬂow through the bulb in one day? + 6V − 1 Basic Concepts 1.23 A 1.8-kW electric heater takes 15 min to boil a quantity of water. If this is done once a day and power costs 10 cents/kWh, what is the cost of its operation for 30 days? (a) Io = 3 A + 10 V − 1.24 A utility company charges 8.5 cents/kWh. If a consumer operates a 40-W light bulb continuously for one day, how much is the consumer charged? 1 24 V + − 3Io + − 1.25 A 1.2-kW toaster takes roughly 4 minutes to heat four slices of bread. Find the cost of operating the toaster once per day for 1 month (30 days). Assume energy costs 9 cents/kWh. 2 − 5V + 3A 1.26 A 12-V car battery supported a current of 150 mA to a bulb. Calculate: (b) Figure 1.29 (a) the power absorbed by the bulb, For Prob. 1.18. (b) the energy absorbed by the bulb over an interval of 20 minutes. 1.19 Find I in the network of Fig. 1.30. I 1A + 9V − 4A + 3V − + 9V − + − 6V For Prob. 1.19. 1.20 Find Vo in the circuit of Fig. 1.31. 12 V + − 1A 3A + − 6A Figure 1.31 For Prob. 1.20. (b) how much energy is expended? (c) how much does the charging cost? Assume electricity costs 9 cents/kWh. (a) the current through the lamp. Io = 2 A 30 V (a) how much charge is transported as a result of the charging? 1.28 A 30-W incandescent lamp is connected to a 120-V source and is left burning continuously in an otherwise dark staircase. Determine: Figure 1.30 6A 1.27 A constant current of 3 A for 4 hours is required to charge an automotive battery. If the terminal voltage is 10 ϩ t͞2 V, where t is in hours, + Vo − (b) the cost of operating the light for one non-leap year if electricity costs 12 cents per kWh. + − 28 V 1.29 An electric stove with four burners and an oven is used in preparing a meal as follows. + − 28 V Burner 1: 20 minutes – + 5Io 3A Burner 2: 40 minutes Burner 3: 15 minutes Burner 4: 45 minutes Oven: 30 minutes If each burner is rated at 1.2 kW and the oven at 1.8 kW, and electricity costs 12 cents per kWh, calculate the cost of electricity used in preparing the meal.
• 58. ale29559_ch01.qxd 07/08/2008 10:38 AM Page 27 Comprehensive Problems 1.30 Reliant Energy (the electric company in Houston, Texas) charges customers as follows: Monthly charge \$6 First 250 kWh @ \$0.02/kWh All additional kWh @ \$0.07/kWh 27 1.31 In a household, a 120-W personal computer (PC) is run for 4 h/day, while a 60-W bulb runs for 8 h/day. If the utility company charges \$0.12/kWh, calculate how much the household pays per year on the PC and the bulb. If a customer uses 1,218 kWh in one month, how much will Reliant Energy charge? Comprehensive Problems 1.32 A telephone wire has a current of 20 mA ﬂowing through it. How long does it take for a charge of 15 C to pass through the wire? 1.33 A lightning bolt carried a current of 2 kA and lasted for 3 ms. How many coulombs of charge were contained in the lightning bolt? p (MW) 8 5 4 3 8.00 1.34 Figure 1.32 shows the power consumption of a certain household in 1 day. Calculate: 8.05 8.10 8.15 8.20 8.25 8.30 t Figure 1.33 For Prob. 1.35. (a) the total energy consumed in kWh, 1.36 A battery may be rated in ampere-hours (Ah). A lead-acid battery is rated at 160 Ah. (b) the average power per hour. 1200 W (a) What is the maximum current it can supply for 40 h? p 800 W (b) How many days will it last if it is discharged at 1 mA? 200 W t (h) 12 2 4 6 8 10 12 2 noon 4 6 8 10 12 Figure 1.32 For Prob. 1.34. 1.35 The graph in Fig. 1.33 represents the power drawn by an industrial plant between 8:00 and 8:30 A.M. Calculate the total energy in MWh consumed by the plant. 1.37 A unit of power often used for electric motors is the horsepower (hp), which equals 746 W. A small electric car is equipped with a 40-hp electric motor. How much energy does the motor deliver in one hour, assuming the motor is operating at maximum power for the whole time? 1.38 How much energy does a 10-hp motor deliver in 30 minutes? Assume that 1 horsepower ϭ 746 W. 1.39 A 600-W TV receiver is turned on for 4 h with nobody watching it. If electricity costs 10 cents/kWh, how much money is wasted?
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• 60. ale29559_ch02.qxd 07/21/2008 10:49 AM Page 29 c h a p t e r 2 Basic Laws There are too many people praying for mountains of difﬁculty to be removed, when what they really need is the courage to climb them! —Unknown Enhancing Your Skills and Your Career ABET EC 2000 criteria (3.b), “an ability to design and conduct experiments, as well as to analyze and interpret data. Engineers must be able to design and conduct experiments, as well as analyze and interpret data. Most students have spent many hours performing experiments in high school and in college. During this time, you have been asked to analyze the data and to interpret the data. Therefore, you should already be skilled in these two activities. My recommendation is that, in the process of performing experiments in the future, you spend more time in analyzing and interpreting the data in the context of the experiment. What does this mean? If you are looking at a plot of voltage versus resistance or current versus resistance or power versus resistance, what do you actually see? Does the curve make sense? Does it agree with what the theory tells you? Does it differ from expectation, and, if so, why? Clearly, practice with analyzing and interpreting data will enhance this skill. Since most, if not all, the experiments you are required to do as a student involve little or no practice in designing the experiment, how can you develop and enhance this skill? Actually, developing this skill under this constraint is not as difﬁcult as it seems. What you need to do is to take the experiment and analyze it. Just break it down into its simplest parts, reconstruct it trying to understand why each element is there, and ﬁnally, determine what the author of the experiment is trying to teach you. Even though it may not always seem so, every experiment you do was designed by someone who was sincerely motivated to teach you something. Photo by Charles Alexander 29
• 61. ale29559_ch02.qxd 07/17/2008 12:29 PM Page 30 Chapter 2 30 Basic Laws 2.1 Introduction Chapter 1 introduced basic concepts such as current, voltage, and power in an electric circuit. To actually determine the values of these variables in a given circuit requires that we understand some fundamental laws that govern electric circuits. These laws, known as Ohm’s law and Kirchhoff’s laws, form the foundation upon which electric circuit analysis is built. In this chapter, in addition to these laws, we shall discuss some techniques commonly applied in circuit design and analysis. These techniques include combining resistors in series or parallel, voltage division, current division, and delta-to-wye and wye-to-delta transformations. The application of these laws and techniques will be restricted to resistive circuits in this chapter. We will ﬁnally apply the laws and techniques to real-life problems of electrical lighting and the design of dc meters. 2.2 l i Material with resistivity ␳ + v − Cross-sectional area A (a) Figure 2.1 (a) Resistor, (b) Circuit symbol for resistance. Ohm’s Law Materials in general have a characteristic behavior of resisting the ﬂow of electric charge. This physical property, or ability to resist current, is known as resistance and is represented by the symbol R. The resistance of any material with a uniform cross-sectional area A depends on A and its length /, as shown in Fig. 2.1(a). We can represent resistance (as measured in the laboratory), in mathematical form, R Rϭr (b) / A (2.1) where r is known as the resistivity of the material in ohm-meters. Good conductors, such as copper and aluminum, have low resistivities, while insulators, such as mica and paper, have high resistivities. Table 2.1 presents the values of r for some common materials and shows which materials are used for conductors, insulators, and semiconductors. The circuit element used to model the current-resisting behavior of a material is the resistor. For the purpose of constructing circuits, resistors are usually made from metallic alloys and carbon compounds. The circuit TABLE 2.1 Resistivities of common materials. Material Silver Copper Aluminum Gold Carbon Germanium Silicon Paper Mica Glass Teﬂon Resistivity (⍀ؒm) Ϫ8 1.64 ϫ 10 1.72 ϫ 10Ϫ8 2.8 ϫ 10Ϫ8 2.45 ϫ 10Ϫ8 4 ϫ 10Ϫ5 47 ϫ 10Ϫ2 6.4 ϫ 102 1010 5 ϫ 1011 1012 3 ϫ 1012 Usage Conductor Conductor Conductor Conductor Semiconductor Semiconductor Semiconductor Insulator Insulator Insulator Insulator
• 62. ale29559_ch02.qxd 07/21/2008 10:49 AM Page 31 2.2 Ohm’s Law symbol for the resistor is shown in Fig. 2.1(b), where R stands for the resistance of the resistor. The resistor is the simplest passive element. Georg Simon Ohm (1787–1854), a German physicist, is credited with ﬁnding the relationship between current and voltage for a resistor. This relationship is known as Ohm’s law. Ohm’s law states that the voltage v across a resistor is directly proportional to the current i ﬂowing through the resistor. That is, v r i (2.2) Ohm deﬁned the constant of proportionality for a resistor to be the resistance, R. (The resistance is a material property which can change if the internal or external conditions of the element are altered, e.g., if there are changes in the temperature.) Thus, Eq. (2.2) becomes v ϭ iR (2.3) which is the mathematical form of Ohm’s law. R in Eq. (2.3) is measured in the unit of ohms, designated ⍀. Thus, The resistance R of an element denotes its ability to resist the ﬂow of electric current; it is measured in ohms (⍀). We may deduce from Eq. (2.3) that Rϭ v i (2.4) so that 1 ⍀ ϭ 1 V/A To apply Ohm’s law as stated in Eq. (2.3), we must pay careful attention to the current direction and voltage polarity. The direction of current i and the polarity of voltage v must conform with the passive Historical Georg Simon Ohm (1787–1854), a German physicist, in 1826 experimentally determined the most basic law relating voltage and current for a resistor. Ohm’s work was initially denied by critics. Born of humble beginnings in Erlangen, Bavaria, Ohm threw himself into electrical research. His efforts resulted in his famous law. He was awarded the Copley Medal in 1841 by the Royal Society of London. In 1849, he was given the Professor of Physics chair by the University of Munich. To honor him, the unit of resistance was named the ohm. 31
• 63. ale29559_ch02.qxd 07/21/2008 10:49 AM Page 32 Chapter 2 32 + i v=0 R=0 − Basic Laws sign convention, as shown in Fig. 2.1(b). This implies that current ﬂows from a higher potential to a lower potential in order for v ϭ i R. If current ﬂows from a lower potential to a higher potential, v ϭ Ϫi R. Since the value of R can range from zero to inﬁnity, it is important that we consider the two extreme possible values of R. An element with R ϭ 0 is called a short circuit, as shown in Fig. 2.2(a). For a short circuit, v ϭ iR ϭ 0 (a) + v i=0 R=∞ − showing that the voltage is zero but the current could be anything. In practice, a short circuit is usually a connecting wire assumed to be a perfect conductor. Thus, A short circuit is a circuit element with resistance approaching zero. Similarly, an element with R ϭ ϱ is known as an open circuit, as shown in Fig. 2.2(b). For an open circuit, i ϭ lim (b) RSϱ Figure 2.2 (a) Short circuit (R ϭ 0), (b) Open circuit (R ϭ ϱ). (2.5) v ϭ0 R (2.6) indicating that the current is zero though the voltage could be anything. Thus, An open circuit is a circuit element with resistance approaching inﬁnity. A resistor is either ﬁxed or variable. Most resistors are of the ﬁxed type, meaning their resistance remains constant. The two common types of ﬁxed resistors (wirewound and composition) are shown in Fig. 2.3. The composition resistors are used when large resistance is needed. The circuit symbol in Fig. 2.1(b) is for a ﬁxed resistor. Variable resistors have adjustable resistance. The symbol for a variable resistor is shown in Fig. 2.4(a). A common variable resistor is known as a potentiometer or pot for short, with the symbol shown in Fig. 2.4(b). The pot is a three-terminal element with a sliding contact or wiper. By sliding the wiper, the resistances between the wiper terminal and the ﬁxed terminals vary. Like ﬁxed resistors, variable resistors can be of either wirewound or composition type, as shown in Fig. 2.5. Although resistors like those in Figs. 2.3 and 2.5 are used in circuit designs, today most (a) (b) Figure 2.3 Fixed resistors: (a) wirewound type, (b) carbon ﬁlm type. Courtesy of Tech America. (a) (a) (b) (b) Figure 2.4 Figure 2.5 Circuit symbol for: (a) a variable resistor in general, (b) a potentiometer. Variable resistors: (a) composition type, (b) slider pot. Courtesy of Tech America.
• 64. ale29559_ch02.qxd 07/09/2008 11:19 AM Page 33 2.2 Ohm’s Law circuit components including resistors are either surface mounted or integrated, as typically shown in Fig. 2.6. It should be pointed out that not all resistors obey Ohm’s law. A resistor that obeys Ohm’s law is known as a linear resistor. It has a constant resistance and thus its current-voltage characteristic is as illustrated in Fig. 2.7(a): its i-v graph is a straight line passing through the origin. A nonlinear resistor does not obey Ohm’s law. Its resistance varies with current and its i-v characteristic is typically shown in Fig. 2.7(b). Examples of devices with nonlinear resistance are the lightbulb and the diode. Although all practical resistors may exhibit nonlinear behavior under certain conditions, we will assume in this book that all elements actually designated as resistors are linear. A useful quantity in circuit analysis is the reciprocal of resistance R, known as conductance and denoted by G: 33 Figure 2.6 Resistors in a thick-ﬁlm circuit. 1 i Gϭ ϭ v R (2.7) The conductance is a measure of how well an element will conduct electric current. The unit of conductance is the mho (ohm spelled backward) or reciprocal ohm, with symbol , the inverted omega. Although engineers often use the mho, in this book we prefer to use the siemens (S), the SI unit of conductance: ⍀ ⍀ 1Sϭ1 ϭ 1 A/ V G. Daryanani, Principles of Active Network Synthesis and Design (New York: John Wiley, 1976), p. 461c. v (2.8) Slope = R Thus, i Conductance is the ability of an element to conduct electric current; it is measured in mhos ( ) or siemens (S). (a) ⍀ v The same resistance can be expressed in ohms or siemens. For example, 10 ⍀ is the same as 0.1 S. From Eq. (2.7), we may write i ϭ Gv (2.9) Slope = R The power dissipated by a resistor can be expressed in terms of R. Using Eqs. (1.7) and (2.3), i 2 p ϭ vi ϭ i 2R ϭ v R (b) (2.10) The power dissipated by a resistor may also be expressed in terms of G as p ϭ vi ϭ v2G ϭ i2 G (2.11) We should note two things from Eqs. (2.10) and (2.11): 1. The power dissipated in a resistor is a nonlinear function of either current or voltage. 2. Since R and G are positive quantities, the power dissipated in a resistor is always positive. Thus, a resistor always absorbs power from the circuit. This conﬁrms the idea that a resistor is a passive element, incapable of generating energy. Figure 2.7 The i-v characteristic of: (a) a linear resistor, (b) a nonlinear resistor.
• 65. ale29559_ch02.qxd 07/09/2008 11:19 AM Page 34 Chapter 2 34 Example 2.1 Basic Laws An electric iron draws 2 A at 120 V. Find its resistance. Solution: From Ohm’s law, Rϭ Practice Problem 2.1 v 120 ϭ ϭ 60 ⍀ i 2 The essential component of a toaster is an electrical element (a resistor) that converts electrical energy to heat energy. How much current is drawn by a toaster with resistance 10 ⍀ at 110 V? Answer: 11 A. Example 2.2 In the circuit shown in Fig. 2.8, calculate the current i, the conductance G, and the power p. i 30 V + − 5 kΩ Solution: The voltage across the resistor is the same as the source voltage (30 V) because the resistor and the voltage source are connected to the same pair of terminals. Hence, the current is + v − Figure 2.8 iϭ v 30 ϭ ϭ 6 mA R 5 ϫ 103 Gϭ For Example 2.2. 1 1 ϭ ϭ 0.2 mS R 5 ϫ 103 The conductance is We can calculate the power in various ways using either Eqs. (1.7), (2.10), or (2.11). p ϭ vi ϭ 30(6 ϫ 10Ϫ3) ϭ 180 mW or p ϭ i 2R ϭ (6 ϫ 10Ϫ3)25 ϫ 103 ϭ 180 mW or p ϭ v2G ϭ (30)20.2 ϫ 10Ϫ3 ϭ 180 mW Practice Problem 2.2 For the circuit shown in Fig. 2.9, calculate the voltage v, the conductance G, and the power p. i 2 mA Figure 2.9 For Practice Prob. 2.2 10 kΩ + v − Answer: 20 V, 100 mS, 40 mW.
• 66. ale29559_ch02.qxd 07/09/2008 11:19 AM Page 35 2.3 Nodes, Branches, and Loops 35 Example 2.3 A voltage source of 20 sin p t V is connected across a 5-k⍀ resistor. Find the current through the resistor and the power dissipated. Solution: iϭ v 20 sin p t ϭ ϭ 4 sin p t mA R 5 ϫ 103 Hence, p ϭ vi ϭ 80 sin2 p t mW Practice Problem 2.3 A resistor absorbs an instantaneous power of 20 cos2 t mW when connected to a voltage source v ϭ 10 cos t V. Find i and R. Answer: 2 cos t mA, 5 k⍀. 2.3 Nodes, Branches, and Loops Since the elements of an electric circuit can be interconnected in several ways, we need to understand some basic concepts of network topology. To differentiate between a circuit and a network, we may regard a network as an interconnection of elements or devices, whereas a circuit is a network providing one or more closed paths. The convention, when addressing network topology, is to use the word network rather than circuit. We do this even though the words network and circuit mean the same thing when used in this context. In network topology, we study the properties relating to the placement of elements in the network and the geometric conﬁguration of the network. Such elements include branches, nodes, and loops. A branch represents a single element such as a voltage source or a resistor. 5Ω a 10 V + − b 2Ω 3Ω 2A c In other words, a branch represents any two-terminal element. The circuit in Fig. 2.10 has ﬁve branches, namely, the 10-V voltage source, the 2-A current source, and the three resistors. Figure 2.10 Nodes, branches, and loops. A node is the point of connection between two or more branches. b A node is usually indicated by a dot in a circuit. If a short circuit (a connecting wire) connects two nodes, the two nodes constitute a single node. The circuit in Fig. 2.10 has three nodes a, b, and c. Notice that the three points that form node b are connected by perfectly conducting wires and therefore constitute a single point. The same is true of the four points forming node c. We demonstrate that the circuit in Fig. 2.10 has only three nodes by redrawing the circuit in Fig. 2.11. The two circuits in Figs. 2.10 and 2.11 are identical. However, for the sake of clarity, nodes b and c are spread out with perfect conductors as in Fig. 2.10. 5Ω 2Ω a 3Ω + − 10 V c Figure 2.11 The three-node circuit of Fig. 2.10 is redrawn. 2A
• 67. ale29559_ch02.qxd 07/09/2008 11:19 AM Page 36 Chapter 2 36 Basic Laws A loop is any closed path in a circuit. A loop is a closed path formed by starting at a node, passing through a set of nodes, and returning to the starting node without passing through any node more than once. A loop is said to be independent if it contains at least one branch which is not a part of any other independent loop. Independent loops or paths result in independent sets of equations. It is possible to form an independent set of loops where one of the loops does not contain such a branch. In Fig. 2.11, abca with the 2⍀ resistor is independent. A second loop with the 3⍀ resistor and the current source is independent. The third loop could be the one with the 2⍀ resistor in parallel with the 3⍀ resistor. This does form an independent set of loops. A network with b branches, n nodes, and l independent loops will satisfy the fundamental theorem of network topology: bϭlϩnϪ1 (2.12) As the next two deﬁnitions show, circuit topology is of great value to the study of voltages and currents in an electric circuit. Two or more elements are in series if they exclusively share a single node and consequently carry the same current. Two or more elements are in parallel if they are connected to the same two nodes and consequently have the same voltage across them. Elements are in series when they are chain-connected or connected sequentially, end to end. For example, two elements are in series if they share one common node and no other element is connected to that common node. Elements in parallel are connected to the same pair of terminals. Elements may be connected in a way that they are neither in series nor in parallel. In the circuit shown in Fig. 2.10, the voltage source and the 5-⍀ resistor are in series because the same current will ﬂow through them. The 2-⍀ resistor, the 3-⍀ resistor, and the current source are in parallel because they are connected to the same two nodes b and c and consequently have the same voltage across them. The 5-⍀ and 2-⍀ resistors are neither in series nor in parallel with each other. Example 2.4 Determine the number of branches and nodes in the circuit shown in Fig. 2.12. Identify which elements are in series and which are in parallel. Solution: Since there are four elements in the circuit, the circuit has four branches: 10 V, 5 ⍀, 6 ⍀, and 2 A. The circuit has three nodes as identiﬁed in Fig. 2.13. The 5-⍀ resistor is in series with the 10-V voltage source because the same current would ﬂow in both. The 6-⍀ resistor is in parallel with the 2-A current source because both are connected to the same nodes 2 and 3.
• 68. ale29559_ch02.qxd 07/09/2008 11:19 AM Page 37 2.4 5Ω 10 V 1 + − 6Ω 10 V 2A 5Ω 37 2 + − Figure 2.12 Kirchhoff’s Laws 6Ω 2A 3 For Example 2.4. Figure 2.13 The three nodes in the circuit of Fig. 2.12. Practice Problem 2.4 How many branches and nodes does the circuit in Fig. 2.14 have? Identify the elements that are in series and in parallel. Answer: Five branches and three nodes are identiﬁed in Fig. 2.15. The 1-⍀ and 2-⍀ resistors are in parallel. The 4-⍀ resistor and 10-V source are also in parallel. 5Ω 1Ω 2Ω 3Ω 1 + 10 V − 4Ω 1Ω + 10 V − 2Ω Figure 2.14 2 3 For Practice Prob. 2.4. Figure 2.15 Answer for Practice Prob. 2.4. 2.4 Kirchhoff’s Laws Ohm’s law by itself is not sufﬁcient to analyze circuits. However, when it is coupled with Kirchhoff’s two laws, we have a sufﬁcient, powerful set of tools for analyzing a large variety of electric circuits. Kirchhoff’s laws were ﬁrst introduced in 1847 by the German physicist Gustav Robert Kirchhoff (1824–1887). These laws are formally known as Kirchhoff’s current law (KCL) and Kirchhoff’s voltage law (KVL). Kirchhoff’s ﬁrst law is based on the law of conservation of charge, which requires that the algebraic sum of charges within a system cannot change. Kirchhoff’s current law (KCL) states that the algebraic sum of currents entering a node (or a closed boundary) is zero. Mathematically, KCL implies that N a in ϭ 0 (2.13) nϭ1 where N is the number of branches connected to the node and in is the nth current entering (or leaving) the node. By this law, currents 4Ω
• 69. ale29559_ch02.qxd 07/21/2008 10:49 AM Page 38 Chapter 2 38 Basic Laws Historical Gustav Robert Kirchhoff (1824–1887), a German physicist, stated two basic laws in 1847 concerning the relationship between the currents and voltages in an electrical network. Kirchhoff’s laws, along with Ohm’s law, form the basis of circuit theory. Born the son of a lawyer in Konigsberg, East Prussia, Kirchhoff entered the University of Konigsberg at age 18 and later became a lecturer in Berlin. His collaborative work in spectroscopy with German chemist Robert Bunsen led to the discovery of cesium in 1860 and rubidium in 1861. Kirchhoff was also credited with the Kirchhoff law of radiation. Thus Kirchhoff is famous among engineers, chemists, and physicists. entering a node may be regarded as positive, while currents leaving the node may be taken as negative or vice versa. To prove KCL, assume a set of currents ik (t), k ϭ 1, 2, p , ﬂow into a node. The algebraic sum of currents at the node is iT (t) ϭ i1(t) ϩ i2(t) ϩ i3(t) ϩ p i1 Integrating both sides of Eq. (2.14) gives i5 qT (t) ϭ q1(t) ϩ q2(t) ϩ q3(t) ϩ p i4 i2 (2.14) i3 Figure 2.16 Currents at a node illustrating KCL. where qk (t) ϭ ͐ ik (t) d