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Preface v
Acknowledgments vi
1.7.1 TV Picture Tube
2.8.1 Lighting Systems
4.10.1 Source Modeling
5.10.1 Digital-to Analog Converter
C o n t e n t s
6.6.1 Integrator 6.6.2 Differentiator
7.9.1 Delay Circuits 7.9.2 Photoflash Unit 7.9.3 Relay Circuits
8.11.1 Automobile Ignition System
9.8.1 Phase-Shifters
10.9.1 Capacitance Multiplier
16.3.1 Even Symmetry
16.3.2 Odd Symmetry 16.7.1 Discrete Fourier Transform 16.8.1 Spectrum Analyzers 17.7.1 Amplitude Modulation 18.9.1 Transistor Circuits
- 1.1 Introduction A Note to the Student ix - 1.2 Systems of Units - 1.3 Charge and Current - 1.4 Voltage - 1.5 Power and Energy - 1.6 Circuit Elements
- †1.7 Applications
- †1.8 Problem Solving 1.7.2 Electricity Bills
- Review Questions
- Problems
- Comprehensive Problems - 2.1 Introduction - 2.2 Ohm’s Laws
- †2.3 Nodes, Branches, and Loops
- 2.4 Kirchhoff’s Laws
- 2.5 Series Resistors and Voltage Division
- 2.6 Parallel Resistors and Current Division
- †2.7 Wye-Delta Transformations
- †2.8 Applications
- 2.9 Summary 2.8.2 Design of DC Meters
- Review Questions
- Problems
- Comprehensive Problems - 3.1 Introduction - 3.2 Nodal Analysis - 3.3 Nodal Analysis with Voltage Sources - 3.4 Mesh Analysis - 3.5 Mesh Analysis with Current Sources
- †3.6 Nodal and Mesh Analyses by Inspection - 3.7 Nodal Versus Mesh Analysis - 3.8 Circuit Analysis with PSpice - †3.9 Applications: DC Transistor Circuits - 3.10 Summary - Review Questions - Problems - Comprehensive Problems - 4.1 Introduction - 4.2 Linearity Property - 4.3 Superposition - 4.4 Source Transformation - 4.5 Thevenin’s Theorem - 4.6 Norton’s Theorem - Theorems †4.7 Derivations of Thevenin’s and Norton’s - 4.8 Maximum Power Transfer - with PSpice 4.9 Verifying Circuit Theorems - †4.10 Applications - 4.11 Summary 4.10.2 Resistance Measurement - Review Questions - Problems - Comprehensive Problems - 5.1 Introduction - 5.2 Operational Amplifiers - 5.3 Ideal Op Amp - 5.4 Inverting Amplifier - 5.5 Noninverting Amplifier - 5.6 Summing Amplifier - 5.7 Difference Amplifier - 5.8 Cascaded Op Amp Circuits - with PSpice 5.9 Op Amp Circuit Analysis - †5.10 Applications - 5.11 Summary 5.10.2 Instrumentation Amplifiers - Review Questions - Problems - Comprehensive Problems - Chapter 2 Basic Laws xi - Chapter 3 Methods of Analysis
- PART 1 DC CIRCUITS - Chapter 1 Basic Concepts - Chapter 4 Circuit Theorems - Chapter 5 Operational Amplifiers - 6.1 Introduction - 6.2 Capacitors - 6.3 Series and Parallel Capacitors - 6.4 Inductors - 6.5 Series and Parallel Inductors
- †6.6 Applications
- 6.7 Summary 6.6.3 Analog Computer
- Review Questions
- Problems
- Comprehensive Problems - 7.1 Introduction - 7.2 The Source-free RC Circuit - 7.3 The Source-free RL Circuit - 7.4 Singularity Functions - 7.5 Step Response of an RC Circuit - 7.6 Step Response of an RL Circuit
- †7.7 First-order Op Amp Circuits
- 7.8 Transient Analysis with PSpice
- †7.9 Applications
- 7.10 Summary 7.9.4 Automobile Ignition Circuit
- Review Questions
- Problems
- Comprehensive Problems - 8.1 Introduction - 8.2 Finding Initial and Final Values - 8.3 The Source-Free Series RLC Circuit - 8.4 The Source-Free Parallel RLC Circuit - Circuit 8.5 Step Response of a Series RLC - Circuit 8.6 Step Response of a Parallel RLC - 8.7 General Second-Order Circuits - 8.8 Second-Order Op Amp Circuits - 8.9 PSpice Analysis of RLC Circuits
- †8.10 Duality
- †8.11 Applications - 8.12 Summary 8.11.2 Smoothing Circuits - Review Questions - Problems - Comprehensive Problems - 9.1 Introduction - 9.2 Sinusoids - 9.3 Phasors - Elements 9.4 Phasor Relationships for Circuit - 9.5 Impedance and Admittance - Domain 9.6 Kirchhoff’s Laws in the Frequency - 9.7 Impedance Combinations - †9.8 Applications - 9.9 Summary 9.8.2 AC Bridges - Review Questions - Problems - Comprehensive Problems - 10.1 Introduction - 10.2 Nodal Analysis - 10.3 Mesh Analysis - 10.4 Superposition Theorem - 10.5 Source Transformation - Circuits 10.6 Thevenin and Norton Equivalent - 10.7 Op Amp AC Circuits - 10.8 AC Analysis Using PSpice - †10.9 Applications - 10.10 Summary 10.9.2 Oscillators - Review Questions - Problems - 11.1 Introduction - 11.2 Instantaneous and Average Power - 11.3 Maximum Average Power Transfer - 11.4 Effective or RMS Value - 11.5 Apparent Power and Power Factor - 11.6 Complex Power - †11.7 Conservation of AC Power - Chapter 8 Second-Order Circuits xii CONTENTS - Chapter 10 Sinusoidal Steady-State Analysis - Chapter 11 AC Power Analysis - Chapter 6 Capacitors and Inductors - Chapter 7 First-Order Circuits - PART 2 AC CIRCUITS - Chapter 9 Sinusoids and Phasors
- Review Questions
- Problems
- Comprehensive Problems - 16.1 Introduction - 16.2 Trigonometric Fourier Series - 16.3 Symmetry Considerations - 16.4 Circuit Applicatons 16.3.3 Half-Wave Symmetry - 16.5 Average Power and RMS Values - 16.6 Exponential Fourier Series - 16.7 Fourier Analysis with PSpice
- †16.8 Applications 16.7.2 Fast Fourier Transform
- 16.9 Summary 16.8.2 Filters
- Review Questions
- Problems
- Comprehensive Problems - 17.1 Introduction - 17.2 Definition of the Fourier Transform - 17.3 Properties of the Fourier Transform - 17.4 Circuit Applications - 17.5 Parseval’s Theorem - Transforms 17.6 Comparing the Fourier and Laplace
- †17.7 Applications - 17.8 Summary 17.7.2 Sampling - Review Questions - Problems - Comprehensive Problems - 18.1 Introduction - 18.2 Impedance Parameters - 18.3 Admittance Parameters - 18.4 Hybrid Parameters - 18.5 Transmission Parameters - †18.6 Relationships between Parameters - 18.7 Interconnection of Networks - PSpice 18.8 Computing Two-Port Parameters Using - †18.9 Applications - 18.10 Summary 18.9.2 Ladder Network Synthesis - Review Questions - Problems - Comprehensive Problems - Cramer’s Rule Appendix A Solution of Simultaneous Equations Using - Appendix B Complex Numbers - Appendix C Mathematical Formulas - Appendix D PSpice for Windows - Appendix E Answers to Odd-Numbered Problems - Selected Bibliography - Index - Chapter 16 The Fourier Series xiv CONTENTS - Chapter 17 Fourier Transform - Chapter 18 Two-Port Networks
Features
In spite of the numerous textbooks on circuit analysis available in the market, students often find the course difficult to learn. The main objective of this book is to present circuit analysis in a manner that is clearer, more interesting, and easier to understand than earlier texts. This objective is achieved in the following ways:
- A course in circuit analysis is perhaps the first exposure students have to electrical engineering. We have included several features to help stu- dents feel at home with the subject. Each chapter opens with either a historical profile of some electrical engineering pioneers to be mentioned in the chapter or a career discussion on a subdisci- pline of electrical engineering. An introduction links the chapter with the previous chapters and states the chapter’s objectives. The chapter ends with a summary of the key points and formulas.
- All principles are presented in a lucid, logical, step-by-step manner. We try to avoid wordiness and superfluous detail that could hide concepts and impede understanding the material.
- Important formulas are boxed as a means of helping students sort what is essential from what is not; and to ensure that students clearly get the gist of the matter, key terms are defined and highlighted.
- Marginal notes are used as a pedagogical aid. They serve multiple uses—hints, cross-references, more exposition, warnings, reminders, common mis- takes, and problem-solving insights.
- Thoroughly worked examples are liberally given at the end of every section. The examples are regard- ed as part of the text and are explained clearly, with- out asking the reader to fill in missing steps. Thoroughly worked examples give students a good understanding of the solution and the confidence to solve problems themselves. Some of the problems are solved in two or three ways to facilitate an understanding and comparison of different approaches.
- To give students practice opportunity, each illus- trative example is immediately followed by a practice problem with the answer. The students can follow the example step-by-step to solve the prac- tice problem without flipping pages or searching the end of the book for answers. The practice prob-
lem is also intended to test students’ understanding of the preceding example. It will reinforce their grasp of the material before moving to the next section.
- In recognition of ABET’s requirement on integrat- ing computer tools, the use of PSpice is encouraged in a student-friendly manner. Since the Windows version of PSpice is becoming popular, it is used instead of the MS-DOS version. PSpice is covered early so that students can use it throughout the text. Appendix D serves as a tutorial on PSpice for Windows.
- The operational amplifier (op amp) as a basic ele- ment is introduced early in the text.
- To ease the transition between the circuit course and signals/systems courses, Fourier and Laplace transforms are covered lucidly and thoroughly.
- The last section in each chapter is devoted to appli- cations of the concepts covered in the chapter. Each chapter has at least one or two practical problems or devices. This helps students apply the concepts to real-life situations.
- Ten multiple-choice review questions are provided at the end of each chapter, with answers. These are intended to cover the little “tricks” that the exam- ples and end-of-chapter problems may not cover. They serve as a self-test device and help students determine how well they have mastered the chapter.
Organization
This book was written for a two-semester or three-semes- ter course in linear circuit analysis. The book may also be used for a one-semester course by a proper selec- tion of chapters and sections. 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 the- orems, circuit techniques, passive and active ele- ments.
- Part 2, consisting of Chapters 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 18, is devoted to advanced techniques for network analysis. It provides a solid introduction to the Laplace transform, Fourier series, the Fourier transform, and two-port network analysis. The material in three parts is more than suffi- cient for a two-semester course, so that the instructor
PREFACE
v
Aniruddha Datta, Texas A&M University John Bay, Virginia Tech Wilhelm Eggimann, Worcester Polytechnic Institute A. B. Bonds, Vanderbilt University Tommy Williamson, University of Dayton Cynthia Finelli, Kettering University John A. Fleming, Texas A&M University Roger Conant, University of Illinois at Chicago Daniel J. Moore, Rose-Hulman Institute of Technology Ralph A. Kinney, Louisiana State University Cecilia Townsend, North Carolina State University Charles B. Smith, University of Mississippi H. Roland Zapp, Michigan State University Stephen M. Phillips, Case Western University Robin N. Strickland, University of Arizona David N. Cowling, Louisiana Tech University Jean-Pierre R. Bayard, California State University
Jack C. Lee, University of Texas at Austin E. L. Gerber, Drexel University
The first author wishes to express his apprecia- tion to his department chair, Dr. Dennis Irwin, for his outstanding support. In addition, he is extremely grate- ful to Suzanne Vazzano for her help with the solutions manual. The second author is indebted to Dr. Cynthia Hirtzel, the former dean of the college of engineering at Temple University, and Drs.. Brian Butz, Richard Klafter, and John Helferty, his departmental chairper- sons at different periods, for their encouragement while working on the manuscript. The secretarial support provided by Michelle Ayers and Carol Dahlberg is gratefully appreciated. Special thanks are due to Ann Sadiku, Mario Valenti, Raymond Garcia, Leke and Tolu Efuwape, and Ope Ola for helping in various ways. Finally, we owe the greatest debt to our wives, Paulette and Chris, without whose constant support and cooperation this project would have been impossible. Please address comments and corrections to the publisher.
C. K. Alexander and M. N. O. Sadiku
PREFACE vii
DC CIRCUITS
P A R T 1
C h a p t e r 1 Basic Concepts
C h a p t e r 2 Basic Laws
C h a p t e r 3 Methods of Analysis
C h a p t e r 4 Circuit Theorems
C h a p t e r 5 Operational Amplifier
C h a p t e r 6 Capacitors and Inductors
C h a p t e r 7 First-Order Circuits
C h a p t e r 8 Second-Order Circuits
4 PART 1 DC Circuits
1.1 INTRODUCTION
Electric circuit theory and electromagnetic theory are the two fundamen- tal theories upon which all branches of electrical engineering are built. Many branches of electrical engineering, such as power, electric ma- chines, control, electronics, communications, and instrumentation, are based on electric circuit theory. Therefore, the basic electric circuit the- ory 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.
A simple electric circuit is shown in Fig. 1.1. It consists of three basic components: a battery, a lamp, and connecting wires. Such a simple circuit can exist by itself; it has several applications, such as a torch light, a search light, and so forth.
−
Current
Battery Lamp
Figure 1.1 A simple electric circuit.
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 accom- plish different tasks. Our objective in this book is not the study of various uses and applications of circuits. Rather our major concern is the anal- ysis 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 mea- surement, 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 measure- ment language is the International System of Units (SI), adopted by the General Conference on Weights and Measures in 1960. In this system,
CHAPTER 1 Basic Concepts 5
2, 5, 6
C
Oscillator
E
B
R 10 k
R 10 k
R1 47
Y 7 MHz C6 5
L 22.7 m H (see text)
to U1, Pin 8 R 10 k GAIN (^) +
C 100 m F 16 V
C 100 m F 16 V
C 1.0 m F 16 V
C 1.0 m F 16 V
C
16 V
C 100 m F 16 V
−
12-V dc Supply
Audio
C
R 10
1
2 4
3 C
C13 0.
U2A 1 ⁄2 TL
U2B 1 ⁄2 TL R 15 k
R 100 k
R 15 k
R 100 k
5
6 R 1 M C12 (^) 0.
L 1 mH
R 47 C
Q 2N2222A
7
C3 0. L 0.445 m H
Antenna (^) C 2200 pF C 2200 pF
1
8 U1^7 SBL- Mixer 3, 4
C 532
C 910 C 910 R 220
U LM386N Audio power amp
5 4
6 3
2
−
−
−
8
Figure 1.2 Electric circuit of a radio receiver.
(Reproduced with permission from QST, August 1995, p. 23.)
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 through- out 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 1.2 The SI prefixes.
Multiplier Prefix Symbol
10 18 exa E 10 15 peta P 10 12 tera T 10 9 giga G 10 6 mega M 10 3 kilo k 10 2 hecto h 10 deka da 10 −^1 deci d 10 −^2 centi c 10 −^3 milli m 10 −^6 micro μ 10 −^9 nano n 10 −^12 pico p 10 −^15 femto f 10 −^18 atto a
TABLE 1.1 The six basic SI units.
Quantity Basic unit Symbol
Length meter m Mass kilogram kg Time second s Electric current ampere A Thermodynamic temperature kelvin K Luminous intensity candela cd
CHAPTER 1 Basic Concepts 7
Electric current is the time rate of change of charge, measured in amperes (A).
Mathematically, the relationship between current i, charge q, and time t is
i =
dq dt
(1.1)
where current is measured in amperes (A), and
1 ampere = 1 coulomb/second
The charge transferred between time t 0 and t is obtained by integrating both sides of Eq. (1.1). We obtain
q =
∫ (^) t
t 0
i dt (1.2)
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 that may be represented by different kinds of mathematical functions. If the current does not change with time, but remains constant, we call it a direct current (dc).
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 com- mon form of time-varying current is the sinusoidal current or alternating current (ac).
An alternating current (ac) is a current that varies sinusoidally with time.
Such current is used in your household, to run the air conditioner, refrig- erator, washing machine, and other electric appliances. Figure 1.4 shows direct current and alternating current; these are the two most common types of current. We will consider other types later in the book.
I
(^0) t
(a)
(b)
i
(^0) t
Figure 1.4 Two common types of
current: (a) direct current (dc), (b) alternating current (ac).
Once we define current as the movement of charge, we expect cur- rent 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 −5 A flowing in one direction as shown in Fig. 1.5(b) is the same as a current of +5 A flowing in the opposite direction.
5 A
(a)
−5 A
(b)
Figure 1.5 Conventional current flow:
(a) positive current flow, (b) negative current flow.
8 PART 1 DC Circuits
E X A M P L E 1. 1
How much charge is represented by 4,600 electrons? Solution: Each electron has − 1. 602 × 10 −^19 C. Hence 4,600 electrons will have − 1. 602 × 10 −^19 C/electron × 4 ,600 electrons = − 7. 369 × 10 −^16 C
P R A C T I C E P R O B L E M 1. 1
Calculate the amount of charge represented by two million protons. Answer: + 3. 204 × 10 −^13 C.
E X A M P L E 1. 2
The total charge entering a terminal is given by q = 5 t sin 4πt mC. Cal- culate the current at t = 0 .5 s. Solution:
i =
dq dt
d dt
( 5 t sin 4πt) mC/s = (5 sin 4πt + 20 πt cos 4πt) mA
At t = 0. 5 , i = 5 sin 2π + 10 π cos 2π = 0 + 10 π = 31 .42 mA
P R A C T I C E P R O B L E M 1. 2
If in Example 1.2, q = ( 10 − 10 e−^2 t^ ) mC, find the current at t = 0 .5 s. Answer: 7.36 mA.
E X A M P L E 1. 3
Determine the total charge entering a terminal between t = 1 s and t = 2 s if the current passing the terminal is i = ( 3 t^2 − t) A. Solution:
q =
t= 1
i dt =
1
( 3 t^2 − t) dt
t^3 −
t^2 2
2
1
= 5 .5 C
P R A C T I C E P R O B L E M 1. 3
The current flowing through an element is
i =
2 A, 0 < t < 1 2 t^2 A, t > 1 Calculate the charge entering the element from t = 0 to t = 2 s. Answer: 6.667 C.
10 PART 1 DC Circuits
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. 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
p =
dw dt
(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
p =
dw dt
dw dq
dq dt
= vi (1.6) or
p = 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 + sign, power is being delivered to or absorbed by the element. If, on the other hand, the power has a − 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 deter- mining 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 vol- tage 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 en- ters through the positive polarity of the voltage. In this case, p = +vi or vi > 0 implies that the element is absorbing power. However, if p = −vi or vi < 0, as in Fig. 1.8(b), the element is releasing or supplying power.
p = + vi (a)
v
− p = − vi (b)
v
−
i i
Figure 1.8 Reference
polarities for power using the passive sign conven- tion: (a) absorbing power, (b) supplying power.
Passive sign convention is satisfied when the current enters through
the positive terminal of an element and p = + vi. If the current
enters through the negative terminal, p = − vi.
CHAPTER 1 Basic Concepts 11
When the voltage and current directions con- form to Fig. 1.8(b), we have the active sign con- vention and p = + vi.
Unless otherwise stated, we will follow the passive sign convention throughout this text. For example, the element in both circuits of Fig. 1. has an absorbing power of +12 W because a positive current enters the positive terminal in both cases. In Fig. 1.10, however, the element is supplying power of −12 W because a positive current enters the negative terminal. Of course, an absorbing power of +12 W is equivalent to a supplying power of −12 W. In general,
Power absorbed = −Power supplied
(a)
4 V
3 A
(a)
−
3 A
4 V
3 A
(b)
−
Figure 1.9 Two cases of an
element with an absorbing power of 12 W: (a) p = 4 × 3 = 12 W, (b) p = 4 × 3 = 12 W.
3 A
(a)
4 V
3 A
(a)
−
3 A
4 V
3 A
(b)
−
Figure 1.10 Two cases of
an element with a supplying power of 12 W: (a) p = 4 × (− 3 ) = −12 W, (b) p = 4 × (− 3 ) = −12 W.
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:
∑ p = 0 (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 t 0 to time t is
w =
∫ (^) t
t 0
p dt =
∫ (^) t
t 0
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 = 3 ,600 J
E X A M P L E 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.
