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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:
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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)
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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.
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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.
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Fundamentals of
Electric Circuits
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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
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Dedicated to our wives, Kikelomo and Hannah, whose understanding and
support have truly made this book possible.
Matthew
and
Chuck
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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
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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
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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
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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
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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
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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
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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
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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
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•
•
•
•
•
•
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
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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
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Preface
evolution of their solution, and save a record of their process for future
reference. In addition, the software automatically generates a Word
document and/or a PowerPoint presentation. Appendix F contains a
description of how to use the software. Additional examples can be
found at the web site, http://kcide.fennresearch.org/, which is linked
from ARIS. The software package can be downloaded for free.
Problem Solving Made Almost Easy, a companion workbook to Fundamentals of Electric Circuits, is available on ARIS for students who
wish to practice their problem-solving techniques. The workbook contains a discussion of problem-solving strategies and 150 additional
problems with complete solutions provided.
C.O.S.M.O.S This CD, available to instructors only, is a powerful solutions manual tool to help instructors streamline the creation of assignments, quizzes, and tests by using problems and solutions from the
textbook, as well as their own custom material. Instructors can edit
textbook end-of-chapter problems as well as track which problems have
been assigned.
Although the textbook is meant to be self-explanatory and act as
a tutor for the student, the personal contact in teaching is not forgotten. It is hoped that the book and supplemental materials supply the
instructor with all the pedagogical tools necessary to effectively present the material.
Acknowledgements
We would like to express our appreciation for the loving support we
have received from our wives (Hannah and Kikelomo), daughters
(Christina, Tamara, Jennifer, Motunrayo, Ann, and Joyce), son (Baixi),
and our extended family members.
At McGraw-Hill, we would like to thank the following editorial
and production staff: Raghu Srinivasan, publisher and senior sponsoring editor; Lora Kalb-Neyens, developmental editors; Joyce Watters,
project manager; Carrie Burger, photo researcher; and Brenda Rolwes,
designer. Also, we appreciate the hard work of Tom Hartley at the University of Akron for his very detailed evaluation of various elements
of the text and his many valued suggestions for continued improvement of this textbook.
We wish to thank Yongjian Fu and his outstanding team of students, Bramarambha Elka and Saravaran Chinniah, for their efforts in
the development of KCIDE for Circuits. Their efforts to help us continue to improve this software are also appreciated.
The fourth edition has beneﬁted greatly from the many outstanding reviewers and symposium attendees who contributed to the success
of the ﬁrst three editions! In addition, the following have made important contributions to the fourth edition (in alphabetical order):
Tom Brewer, Georgia Tech
Andy Chan, City University of Hong Kong
Alan Tan Wee Chiat, Multimedia University
Norman Cox, University of Missouri-Rolla
Walter L. Green, University of Tennessee
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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
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GUIDED TOUR
The main objective of this book is to present circuit analysis in a manner that is clearer, more interesting, and easier to understand than other
texts. For you, the student, here are some features to help you study
and be successful in this course.
The four color art program brings circuit drawings to life and enhances key concepts throughout
the text.
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
pϭ
¢
dw
dt
i
+
(1.5)
+
v
v
−
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 = +vi
pϭ
20
Chapter 1
1.8
Basic Concepts
p ϭ vi
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,
21
ϩPower absorbed ϭ ϪPower supplied
reduce effort and increase accuracy. Again, Now let us stress at this process for a student taking an electrical
we want to look that time
spent carefully deﬁning the problemand computer engineering foundations course. (The basic process also
and investigating alternative
approaches to its solution will pay big applies to later. Evaluatingengineering course.) 22
dividends almost every the
Keep in mind that
alternatives and determining which promises the the steps likelihood of
although greatest have been simpliﬁed to apply to academic types of
success may be difﬁcult but will be well worth the effort. Document always needs to be followed. We conproblems, the process as stated
this process well since you will want sider a simple example.the ﬁrst
to come back to it if
approach does not work.
4. Attempt a problem solution. Now is the time to actually begin
solving the problem. The process you follow must be well documented
Chapter 1
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.
Basic Concepts
2Ω
Example 1.10
2Ω
Solution:
4Ω
5V
5V
+
−
8Ω
i1
v1
+ v −
2Ω
+
−
Loop 1
3V
i3
i2
+
v8Ω
−
8Ω
4Ω
+ v4Ω −
−
+
Loop 2
3V
1.9
Figure 1.19
2Ω
5V
+
−
Figure 1.20
Problem deﬁntion.
Summary
23
Figure 1.21
Illustrative example.
Using nodal analysis.
Therefore, we will solve for i8⍀ using nodal analysis.
4Ω
4. Attempt a problem solution. We ﬁrst write down all of the equationsi8Ω will need in order to ﬁnd i8⍀.
we
8Ω
−
+
3V
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
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.
Now we can solve for v1.
v1 Ϫ 5
v1 Ϫ 0
v1 ϩ 3
8c
ϩ
ϩ
d ϭ0
2
8
4
leads to (4v1 Ϫ 20) ϩ (v1) ϩ (2v1 ϩ 6) ϭ 0
v1
2
7v1 ϭ ϩ14,
v1 ϭ ϩ2 V,
i8⍀ ϭ
ϭ ϭ 0.25 A
8
8
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 (Ϫ(Ϫ1.5)2) ϩ (0.25 ϫ 8)
ϭ Ϫ5 ϩ 3 ϩ 2 ϭ 0
(Checks.)
Applying KVL to loop 2,
A six-step problem-solving methodology is introduced in Chapter 1 and
incorporated into worked examples
throughout the text to promote
sound, step-by-step problem-solving
practices.
When the voltage and current directions
conform to Fig. 1.8 (b), we have the active sign convention and p ϭ ϩvi.
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.
Solve for the current ﬂowing through the 8-⍀ resistor in Fig. 1.19.
1. Carefully Deﬁne the problem. This is only a simple example, but
we can already see that we do not know the polarity on the 3-V source.
We have the following options. We can ask the professor what the
polarity should be. If we cannot ask, then we need to make a decision
on what to do next. If we have time to work the problem both ways,
we can solve for the current when the 3-V source is plus on top and
then plus on the bottom. If we do not have the time to work it both
ways, assume a polarity and then carefully document your decision.
Let us assume that the professor tells us that the source is plus on the
bottom as shown in Fig. 1.20.
2. Present everything you know about the problem. Presenting all that
we know about the problem involves labeling the circuit clearly so that
we deﬁne what we seek.
Given the circuit shown in Fig. 1.20, solve for i8⍀.
We now check with the professor, if reasonable, to see if the problem is properly deﬁned.
3. Establish a set of Alternative solutions and determine the one that
promises the greatest likelihood of success. There are essentially three
techniques that can be used to solve this problem. Later in the text you
will see that you can use circuit analysis (using Kirchhoff’s laws and
Ohm’s law), nodal analysis, and mesh analysis.
To solve for i8⍀ using circuit analysis will eventually lead to a
solution, but it will likely take more work than either nodal or mesh
(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.
1. Carefully Deﬁne the problem. This may be the most important part
of the process, because it becomes the foundation for all the rest of the
steps. In general, the presentation of engineering problems is somewhat
incomplete. You must do all you can to make sure you understand the
problem as thoroughly as the presenter of the problem understands it.
Time spent at this point clearly identifying the problem will save you
1.8 Problem Solving
considerable time and frustration later. As a student, you can clarify a
problem statement in a textbook by asking your professor. A problem
in order to present a detailed solution if successful, and to evaluate the
presented to you in industry may require that you consult several individuals. At this step, it is important to process ifquestions not successful. This detailed evaluation may lead to
develop you are that need to
corrections that you have such
be addressed before continuing the solution process. If can then lead to a successful solution. It can also lead
questions, you need to consult with to new alternatives to try. Many times, it is wise to fully set up a soluthe appropriate individuals or
resources to obtain the answers to thosetion before With those answers, equations. This will help in checking
questions. putting numbers into
your results.
you can now reﬁne the problem, and use that reﬁnement as the problem statement for the rest of the solution5. Evaluate the solution and check for accuracy. You now thoroughly
process.
2. Present everything you know about evaluate whatYou are now ready
the problem. you have accomplished. Decide if you have an acceptable
solution, one that you possible
to write down everything you know about the problem and itswant to present to your team, boss, or professor.
6. time and frustration solved
solutions. This important step will save youHas the problem beenlater. Satisfactorily? If so, present the solution; if determine the one step
3. Establish a set of Alternative solutions andnot, then return tothat 3 and continue through the process
again. Now you problem will
promises the greatest likelihood of success. Almost everyneed to present your solution or try another alternative. At a solution. It is highly
have a number of possible paths that can lead tothis point, presenting your solution may bring closure to the
process. Often, however, presentation of a solution leads to further
desirable to identify as many of those paths as possible. At this point,
reﬁnement of to problem deﬁnition, and the process continues. Folyou also need to determine what tools are available the you, such as
lowing this process will eventually lead to a satisfactory conclusion.
PSpice and MATLAB and other software packages that can greatly
(b)
Figure 1.8
Reference polarities for power using the
passive sign convention: (a) absorbing
power, (b) supplying power.
Problem Solving
1. Carefully Deﬁne the problem.
2. Present everything you know about the problem.
3. Establish a set of Alternative solutions and determine the one that
promises the greatest likelihood of success.
4. Attempt a problem solution.
5. Evaluate the solution and check for accuracy.
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.
(a)
(1.6)
or
Although the problems to be solved during one’s career will vary in
complexity and magnitude, the basic principles to be followed remain
the same. The process outlined here is the one developed by the
authors over many years of problem solving with students, for the
solution of engineering problems in industry, and for problem solving
in research.
We will list the steps simply and then elaborate on them.
xx
dw dq
dw
ϭ
ؒ
ϭ vi
dt
dq dt
p = −vi
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.)
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
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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.
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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.
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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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Page xxv
A Note to the Student
This may be your ﬁrst course in electrical engineering. Although electrical engineering is an exciting and challenging discipline, the course
may intimidate you. This book was written to prevent that. A good textbook and a good professor are an advantage—but you are the one who
does the learning. If you keep the following ideas in mind, you will do
very well in this course.
• This course is the foundation on which most other courses in the
electrical engineering curriculum rest. For this reason, put in as
much effort as you can. Study the course regularly.
• Problem solving is an essential part of the learning process. Solve as
many problems as you can. Begin by solving the practice problem
following each example, and then proceed to the end-of-chapter
problems. The best way to learn is to solve a lot of problems. An
asterisk in front of a problem indicates a challenging problem.
• Spice, a computer circuit analysis program, is used throughout the
textbook. PSpice, the personal computer version of Spice, is the
popular standard circuit analysis program at most universities.
PSpice for Windows is described in Appendix D. Make an effort
to learn PSpice, because you can check any circuit problem with
PSpice and be sure you are handing in a correct problem solution.
• MATLAB is another software that is very useful in circuit analysis
and other courses you will be taking. A brief tutorial on MATLAB
is given in Appendix E to get you started. The best way to learn
MATLAB is to start working with it once you know a few commands.
• Each chapter ends with a section on how the material covered in
the chapter can be applied to real-life situations. The concepts in
this section may be new and advanced to you. No doubt, you will
learn more of the details in other courses. We are mainly interested
in gaining a general familiarity with these ideas.
• Attempt the review questions at the end of each chapter. They
will help you discover some “tricks” not revealed in class or in the
textbook.
• Clearly a lot of effort has gone into making the technical details in
this book easy to understand. It also contains all the mathematics
and physics necessary to understand the theory and will be very
useful in your other engineering courses. However, we have also
focused on creating a reference for you to use both in school as
well as when working in industry or seeking a graduate degree.
• It is very tempting to sell your book after you have completed your
classroom experience; however, our advice to you is DO NOT SELL
YOUR ENGINEERING BOOKS! Books have always been expensive, however, the cost of this book is virtually the same as I paid
for my circuits text back in the early 60s in terms of real dollars.
In fact, it is actually cheaper. In addition, engineering books of
the past are no where near as complete as what is available now.
xxv
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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.
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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
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Page xxviii
About the Authors
In 1998 he received the Distinguished Engineering Education
Achievement Award from the Engineering Council, and in 1996 he
received the Distinguished Engineering Education Leadership Award
from the same group. When he became a fellow of the IEEE in 1994,
the citation read “for leadership in the ﬁeld of engineering education
and the professional development of engineering students.” In 1984 he
received the IEEE Centennial Medal, and in 1983 he received the
IEEE/RAB Innovation Award, given to the IEEE member who best
contributes to RAB’s goals and objectives.
Matthew N. O. Sadiku
Matthew N. O. Sadiku is presently a professor at Prairie View A&M
University. Prior to joining Prairie View, he taught at Florida Atlantic
University, Boca Raton, and Temple University, Philadelphia. He has
also worked for Lucent/Avaya and Boeing Satellite Systems.
Dr. Sadiku is the author of over 170 professional papers and almost
30 books including Elements of Electromagnetics (Oxford University
Press, 3rd ed., 2001), Numerical Techniques in Electromagnetics (2nd
ed., CRC Press, 2000), Simulation of Local Area Networks (with M.
IIyas, CRC Press, 1994), Metropolitan Area Networks (CRC Press,
1994), and Fundamentals of Electric Circuits (with C. K. Alexander,
McGraw-Hill, 3rd ed., 2007). His books are used worldwide, and some
of them have been translated into Korean, Chinese, Italian, and Spanish.
He was the recipient of the 2000 McGraw-Hill/Jacob Millman Award for
outstanding contributions in the ﬁeld of electrical engineering. He was
the IEEE region 2 Student Activities Committee chairman and is an associate editor for IEEE “Transactions on Education.” He received his Ph.D.
at Tennessee Technological University, Cookeville.
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Page 1
Fundamentals of
Electric Circuits
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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
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c h a p t e r
1
Basic Concepts
Some books are to be tasted, others to be swallowed, and some few to
be chewed and digested.
—Francis Bacon
Enhancing Your Skills and Your Career
ABET EC 2000 criteria (3.a), “an ability to apply knowledge
of mathematics, science, and engineering.”
As students, you are required to study mathematics, science, and engineering with the purpose of being able to apply that knowledge to the
solution of engineering problems. The skill here is the ability to apply
the fundamentals of these areas in the solution of a problem. So, how
do you develop and enhance this skill?
The best approach is to work as many problems as possible in all
of your courses. However, if you are really going to be successful with
this, you must spend time analyzing where and when and why you have
difﬁculty in easily arriving at successful solutions. You may be surprised to learn that most of your problem-solving problems are with
mathematics rather than your understanding of theory. You may also
learn that you start working the problem too soon. Taking time to think
about the problem and how you should solve it will always save you
time and frustration in the end.
What I have found that works best for me is to apply our sixstep problem-solving technique. Then I carefully identify the areas
where I have difﬁculty solving the problem. Many times, my actual
deﬁciencies are in my understanding and ability to use correctly certain mathematical principles. I then return to my fundamental math
texts and carefully review the appropriate sections and in some cases
work some example problems in that text. This brings me to another
important thing you should always do: Keep nearby all your basic
mathematics, science, and engineering textbooks.
This process of continually looking up material you thought you
had acquired in earlier courses may seem very tedious at ﬁrst; however, as your skills develop and your knowledge increases, this process
will become easier and easier. On a personal note, it is this very process
that led me from being a much less than average student to someone
who could earn a Ph.D. and become a successful researcher.
Photo by Charles Alexander
3
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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
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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
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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.
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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).
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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.
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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
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Chapter 1
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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.
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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.
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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.
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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
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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.
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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.
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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.
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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
−
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+
−
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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
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Chapter 1
1.8
Basic Concepts
Problem Solving
Although the problems to be solved during one’s career will vary in
complexity and magnitude, the basic principles to be followed remain
the same. The process outlined here is the one developed by the
authors over many years of problem solving with students, for the
solution of engineering problems in industry, and for problem solving
in research.
We will list the steps simply and then elaborate on them.
1. Carefully deﬁne the problem.
2. Present everything you know about the problem.
3. Establish a set of alternative solutions and determine the one that
promises the greatest likelihood of success.
4. Attempt a problem solution.
5. Evaluate the solution and check for accuracy.
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.
1. Carefully deﬁne the problem. This may be the most important part
of the process, because it becomes the foundation for all the rest of the
steps. In general, the presentation of engineering problems is somewhat
incomplete. You must do all you can to make sure you understand the
problem as thoroughly as the presenter of the problem understands it.
Time spent at this point clearly identifying the problem will save you
considerable time and frustration later. As a student, you can clarify a
problem statement in a textbook by asking your professor. A problem
presented to you in industry may require that you consult several individuals. At this step, it is important to develop questions that need to
be addressed before continuing the solution process. If you have such
questions, you need to consult with the appropriate individuals or
resources to obtain the answers to those questions. With those answers,
you can now reﬁne the problem, and use that reﬁnement as the problem statement for the rest of the solution process.
2. Present everything you know about the problem. You are now ready
to write down everything you know about the problem and its possible
solutions. This important step will save you time and frustration later.
3. Establish a set of alternative solutions and determine the one that
promises the greatest likelihood of success. Almost every problem will
have a number of possible paths that can lead to a solution. It is highly
desirable to identify as many of those paths as possible. At this point,
you also need to determine what tools are available to you, such as
PSpice and MATLAB and other software packages that can greatly
reduce effort and increase accuracy. Again, we want to stress that time
spent carefully deﬁning the problem and investigating alternative
approaches to its solution will pay big dividends later. Evaluating the
alternatives and determining which promises the greatest likelihood of
success may be difﬁcult but will be well worth the effort. Document
this process well since you will want to come back to it if the ﬁrst
approach does not work.
4. Attempt a problem solution. Now is the time to actually begin
solving the problem. The process you follow must be well documented
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Problem Solving
21
in order to present a detailed solution if successful, and to evaluate the
process if you are not successful. This detailed evaluation may lead to
corrections that can then lead to a successful solution. It can also lead
to new alternatives to try. Many times, it is wise to fully set up a solution before putting numbers into equations. This will help in checking
your results.
5. Evaluate the solution and check for accuracy. You now thoroughly
evaluate what you have accomplished. Decide if you have an acceptable
solution, one that you want to present to your team, boss, or professor.
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. Now you need to present your solution or try another alternative. At this point, presenting your solution may bring closure to the
process. Often, however, presentation of a solution leads to further
reﬁnement of the problem deﬁnition, and the process continues. Following this process will eventually lead to a satisfactory conclusion.
Now let us look at this process for a student taking an electrical
and computer engineering foundations course. (The basic process also
applies to almost every engineering course.) Keep in mind that
although the steps have been simpliﬁed to apply to academic types of
problems, the process as stated always needs to be followed. We consider a simple example.
Example 1.10
Solve for the current ﬂowing through the 8-⍀ resistor in Fig. 1.19.
2Ω
Solution:
1. Carefully deﬁne the problem. This is only a simple example, but
we can already see that we do not know the polarity on the 3-V source.
We have the following options. We can ask the professor what the
polarity should be. If we cannot ask, then we need to make a decision
on what to do next. If we have time to work the problem both ways,
we can solve for the current when the 3-V source is plus on top and
then plus on the bottom. If we do not have the time to work it both
ways, assume a polarity and then carefully document your decision.
Let us assume that the professor tells us that the source is plus on the
bottom as shown in Fig. 1.20.
2. Present everything you know about the problem. Presenting all that
we know about the problem involves labeling the circuit clearly so that
we deﬁne what we seek.
Given the circuit shown in Fig. 1.20, solve for i8⍀.
We now check with the professor, if reasonable, to see if the problem is properly deﬁned.
3. Establish a set of alternative solutions and determine the one that
promises the greatest likelihood of success. There are essentially three
techniques that can be used to solve this problem. Later in the text you
will see that you can use circuit analysis (using Kirchhoff’s laws and
Ohm’s law), nodal analysis, and mesh analysis.
To solve for i8⍀ using circuit analysis will eventually lead to a
solution, but it will likely take more work than either nodal or mesh
5V
+
−
4Ω
8Ω
3V
Figure 1.19
Illustrative example.
2Ω
4Ω
i8Ω
5V
+
−
Figure 1.20
Problem deﬁntion.
8Ω
−
+
3V
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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.)
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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
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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)
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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
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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.
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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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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
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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
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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
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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.
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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.
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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.
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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
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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.
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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
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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
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