Circuit analysis i with MATLAB applications

Orchard Publications, Fremont, California

www.orchardpublications.com

G=[35/50 −j*3/50; −1/5 1/10+j*1/10]; I=[1 0]'; V=G\I;

Ix=5*V(2,1)/4; % Multiply Vc by 5 and divide by 4 to get current Ix

magIx=abs(Ix); theta=angle(Ix)*180/pi; % Convert current Ix to polar form

fprintf(' \n'); disp(' Ix = ' ); disp(Ix);...

fprintf('magIx = %4.2f A \t', magIx); fprintf('theta = %4.2f deg \t', theta);...

fprintf(' \n'); fprintf(' \n');

Ix = 2.1176-1.7546i magIx = 2.75 A theta = -39.64 deg

Steven T. Karris

Circuit Analysis I

with MATLAB® Applications

Students and working professionals will find Circuit

Analysis I with MATLAB® Applications to be a con￾cise and easy-to-learn text. It provides complete,

clear, and detailed explanations of the principal elec￾trical engineering concepts, and these are illustrated

with numerous practical examples.

This text includes the following chapters and appendices:

• Basic Concepts and Definitions • Analysis of Simple Circuits • Nodal and Mesh Equations -

Circuit Theorems • Introduction to Operational Amplifiers • Inductance and Capacitance

• Sinusoidal Circuit Analysis • Phasor Circuit Analysis • Average and RMS Values, Complex

Power, and Instruments • Natural Response • Forced and Total Response in RL and RC

Circuits • Introduction to MATLAB • Review of Complex Numbers • Matrices and Determinants

Each chapter contains numerous practical applications supplemented with detailed instructions

for using MATLAB to obtain quick and accurate answers.

Steven T. Karris is the president and founder of Orchard Publications. He earned a bachelors

degree in electrical engineering at Christian Brothers University, Memphis, Tennessee, a mas￾ters degree in electrical engineering at Florida Institute of Technology, Melbourne, Florida, and

has done post-master work at the latter. He is a registered professional engineer in California

and Florida. He has over 30 years of professional engineering experience in industry. In addi￾tion, he has over 25 years of teaching experience that he acquired at several educational insti￾tutions as an adjunct professor. He is currently with UC Berkeley Extension.

Orchard Publications

Visit us on the Internet

www.orchardpublications.com

or email us: [email protected]

ISBN 0-9744239-3-9

$39.95

Circuit Analysis I

with MATLAB® Applications

Circuit Analysis I

with MATLAB® Applications

Steven T. Karris

Orchard Publications

www.orchardpublications.com

Circuit Analysis I with MATLAB® Applications

Copyright © 2004 Orchard Publications. All rights reserved. Printed in Canada. No part of this publication may be

reproduced or distributed in any form or by any means, or stored in a data base or retrieval system, without the prior

written permission of the publisher.

Direct all inquiries to Orchard Publications, 39510 Paseo Padre Parkway, Fremont, California 94538, U.S.A.

URL: http://www.orchardpublications.com

Product and corporate names are trademarks or registered trademarks of the MathWorks®, Inc., and Microsoft®

Corporation. They are used only for identification and explanation, without intent to infringe.

Library of Congress Cataloging-in-Publication Data

Library of Congress Control Number 2004093171

ISBN 0-9744239-3-9

Disclaimer

The author has made every effort to make this text as complete and accurate as possible, but no warranty is implied.

The author and publisher shall have neither liability nor responsibility to any person or entity with respect to any loss

or damages arising from the information contained in this text.

This book was created electronically using Adobe Framemaker®.

Circuit Analysis I with MATLAB Applications

Orchard Publications

Preface

This text is an introduction to the basic principles of electrical engineering. It is the outgrowth of

lecture notes prepared by this author while teaching for the electrical engineering and computer

engineering departments at San José State University, DeAnza college, and the College of San Mateo,

all in California. Many of the examples and problems are based on the author’s industrial experience.

It can be used as a primary text or supplementary text. It is also ideal for self-study.

This book is intended for students of college grade, both community colleges and universities. It

presumes knowledge of first year differential and integral calculus and physics. While some

knowledge of differential equations would be helpful, it is not absolutely necessary. Chapters 9 and 10

include step-by-step procedures for the solutions of simple differential equations used in the

derivation of the natural and forces responses. Appendices B and C provide a thorough review of

complex numbers and matrices respectively.

There are several textbooks on the subject that have been used for years. The material of this book is

not new, and this author claims no originality of its content. This book was written to fit the needs of

the average student. Moreover, it is not restricted to computer oriented circuit analysis. While it is true

that there is a great demand for electrical and computer engineers, especially in the internet field, the

demand also exists for power engineers to work in electric utility companies, and facility engineers to

work in the industrial areas.

Circuit analysis is comprised of numerous topics. It would be impractical to include all related topics

in a single text. This book, Circuit Analysis I with MATLAB® Applications, contains the standard

subject matter of electrical engineering. Accordingly, it is intended as a first course in circuits and the

material can be covered in one semester or two quarters. A sequel, Circuit Analysis II with MATLAB®

Applications, is intended for use in a subsequent semester or two subsequent quarters.

It is not necessary that the reader has previous knowledge of MATLAB®. The material of this text

can be learned without MATLAB. However, this author highly recommends that the reader studies

this material in conjunction with the inexpensive MATLAB Student Version package that is available

at most college and university bookstores. Appendix A of this text provides a practical introduction

to MATLAB. As shown on the front cover, a system of equations with complex coefficients can be

solved with MATLAB very accurately and rapidly. MATLAB will be invaluable in later studies such as

the design of analog and digital filters.

In addition to several problems provided at the end of each chapter, this text includes multiple-choice

questions to test and enhance the reader’s knowledge of this subject. Moreover, answers to these

questions and detailed solutions of all problems are provided at the end of each chapter. The rationale

Preface

Circuit Analysis I with MATLAB Applications

Orchard Publications

is to encourage the reader to solve all problems and check his effort for correct solutions and

appropriate steps in obtaining the correct solution. And since this text was written to serve as a

self-study or supplementary textbook, it provides the reader with a resource to test his

knowledge.

The author has accumulated many additional problems for homework assignment and these are

available to those instructors who adopt this text either as primary or supplementary text, and

prefer to assign problems without the solutions. He also has accumulated many sample exams.

Like any other new book, this text may contain some grammar and typographical errors.

Accordingly, all feedback for errors, advice and comments will be most welcomed and greatly

appreciated.

Orchard Publications

Fremont, California

Circuit Analysis I with MATLAB Applications i

Orchard Publications

Contents

Chapter 1

Basic Concepts and Definitions

The Coulomb...................................................................................................................................................1-1

Electric Current and Ampere........................................................................................................................1-1

Two Terminal Devices...................................................................................................................................1-4

Voltage (Potential Difference) ......................................................................................................................1-5

Power and Energy...........................................................................................................................................1-8

Active and Passive Devices ........................................................................................................................ 1-12

Circuits and Networks................................................................................................................................. 1-12

Active and Passive Networks..................................................................................................................... 1-12

Necessary Conditions for Current Flow .................................................................................................. 1-12

International System of Units.................................................................................................................... 1-13

Sources of Energy........................................................................................................................................ 1-17

Summary........................................................................................................................................................ 1-18

Exercises........................................................................................................................................................ 1-21

Answers to Exercises................................................................................................................................... 1-25

Chapter 2

Analysis of Simple Circuits

Conventions.....................................................................................................................................................2-1

Ohm’s Law.......................................................................................................................................................2-1

Power Absorbed by a Resistor......................................................................................................................2-3

Energy Dissipated by a Resistor ...................................................................................................................2-4

Nodes, Branches, Loops and Meshes..........................................................................................................2-5

Kirchhoff’s Current Law (KCL)...................................................................................................................2-6

Kirchhoff’s Voltage Law (KVL)...................................................................................................................2-7

Analysis of Single Mesh (Loop) Series Circuits....................................................................................... 2-10

Analysis of Single Node-Pair Parallel Circuits......................................................................................... 2-14

Voltage and Current Source Combinations............................................................................................. 2-16

Resistance and Conductance Combinations............................................................................................ 2-18

Voltage Division Expressions.................................................................................................................... 2-22

Current Division Expressions.................................................................................................................... 2-24

Standards for Electrical and Electronic Devices..................................................................................... 2-26

Resistor Color Code .................................................................................................................................... 2-27

Power Rating of Resistors .......................................................................................................................... 2-28

Contents

ii Circuit Analysis I with MATLAB Applications

Orchard Publications

Temperature Coefficient of Resistance .................................................................................................... 2-29

Ampere Capacity of Wires.......................................................................................................................... 2-30

Current Ratings for Electronic Equipment ............................................................................................. 2-30

Copper Conductor Sizes for Interior Wiring........................................................................................... 2-33

Summary........................................................................................................................................................ 2-38

Exercises........................................................................................................................................................ 2-41

Answers to Exercises................................................................................................................................... 2-50

Chapter 3

Nodal and Mesh Equations - Circuit Theorems

Nodal, Mesh, and Loop Equations .............................................................................................................3-1

Analysis with Nodal Equations....................................................................................................................3-1

Analysis with Mesh or Loop Equations .....................................................................................................3-8

Transformation between Voltage and Current Sources.........................................................................3-20

Thevenin’s Theorem....................................................................................................................................3-24

Norton’s Theorem.......................................................................................................................................3-35

Maximum Power Transfer Theorem ........................................................................................................3-38

Linearity.........................................................................................................................................................3-39

Superposition Principle ...............................................................................................................................3-41

Circuits with Non-Linear Devices.............................................................................................................3-45

Efficiency ......................................................................................................................................................3-47

Regulation .....................................................................................................................................................3-49

Summary........................................................................................................................................................3-49

Exercises........................................................................................................................................................3-52

Answers to Exercises...................................................................................................................................3-64

Chapter 4

Introduction to Operational Amplifiers

Signals .............................................................................................................................................................. 4-1

Amplifiers........................................................................................................................................................ 4-1

Decibels ........................................................................................................................................................... 4-2

Bandwidth and Frequency Response.......................................................................................................... 4-4

The Operational Amplifier........................................................................................................................... 4-5

An Overview of the Op Amp...................................................................................................................... 4-5

Active Filters................................................................................................................................................. 4-13

Analysis of Op Amp Circuits..................................................................................................................... 4-16

Input and Output Resistance ..................................................................................................................... 4-28

Summary........................................................................................................................................................ 4-32

Circuit Analysis I with MATLAB Applications iii

Orchard Publications

Contents

Exercises ........................................................................................................................................................4-34

Answers to Exercises...................................................................................................................................4-43

Chapter 5

Inductance and Capacitance

Energy Storage Devices................................................................................................................................. 5-1

Inductance ....................................................................................................................................................... 5-1

Power and Energy in an Inductor ............................................................................................................. 5-11

Combinations of Series and Parallel Inductors........................................................................................ 5-14

Capacitance.................................................................................................................................................... 5-17

Power and Energy in a Capacitor .............................................................................................................. 5-22

Capacitance Combinations ......................................................................................................................... 5-25

Nodal and Mesh Equations in General Terms........................................................................................ 5-28

Summary........................................................................................................................................................ 5-29

Exercises ........................................................................................................................................................ 5-31

Answers to Exercises................................................................................................................................... 5-36

Chapter 6

Sinusoidal Circuit Analysis

Excitation Functions...................................................................................................................................... 6-1

Circuit Response to Sinusoidal Inputs........................................................................................................ 6-1

The Complex Excitation Function.............................................................................................................. 6-3

Phasors in , , and Circuits ................................................................................................................. 6-8

Impedance ..................................................................................................................................................... 6-14

Admittance .................................................................................................................................................... 6-17

Summary........................................................................................................................................................ 6-21

Exercises........................................................................................................................................................ 6-25

Answers to Exercises................................................................................................................................... 6-30

Chapter 7

Phasor Circuit Analysis

Nodal Analysis................................................................................................................................................ 7-1

Mesh Analysis ................................................................................................................................................. 7-5

Application of Superposition Principle.......................................................................................................7-7

Thevenin’s and Norton’s Theorems ...........................................................................................................7-8

Phasor Analysis in Amplifier Circuits .......................................................................................................7-12

R L C

Contents

iv Circuit Analysis I with MATLAB Applications

Orchard Publications

Phasor Diagrams..........................................................................................................................................7-15

Electric Filters...............................................................................................................................................7-20

Basic Analog Filters .....................................................................................................................................7-21

Active Filter Analysis...................................................................................................................................7-26

Summary........................................................................................................................................................7-28

Exercises........................................................................................................................................................7-29

Answers to Exercises...................................................................................................................................7-37

Chapter 8

Average and RMS Values, Complex Power, and Instruments

Periodic Time Functions...............................................................................................................................8-1

Average Values...............................................................................................................................................8-2

Effective Values ............................................................................................................................................. 8-3

Effective (RMS) Value of Sinusoids ...........................................................................................................8-5

RMS Values of Sinusoids with Different Frequencies.............................................................................8-7

Average Power and Power Factor...............................................................................................................8-9

Average Power in a Resistive Load...........................................................................................................8-10

Average Power in Inductive and Capacitive Loads ................................................................................8-11

Average Power in Non-Sinusoidal Waveforms.......................................................................................8-14

Lagging and Leading Power Factors.........................................................................................................8-15

Complex Power - Power Triangle.............................................................................................................8-16

Power Factor Correction ............................................................................................................................8-18

Instruments...................................................................................................................................................8-21

Summary........................................................................................................................................................8-30

Exercises........................................................................................................................................................8-33

Answers to Exercises...................................................................................................................................8-39

Chapter 9

Natural Response

The Natural Response of a Series RL circuit............................................................................................. 9-1

The Natural Response of a Series RC Circuit ......................................................................................... 9-10

Summary........................................................................................................................................................ 9-17

Exercises........................................................................................................................................................ 9-19

Answers to Exercises................................................................................................................................... 9-25

Circuit Analysis I with MATLAB Applications v

Orchard Publications

Contents

Chapter 10

Forced and Total Response in RL and RC Circuits

The Unit Step Function...............................................................................................................................10-1

The Unit Ramp Function............................................................................................................................10-6

The Delta Function......................................................................................................................................10-8

The Forced and Total Response in an RL Circuit ................................................................................10-14

The Forced and Total Response in an RC Circuit................................................................................10-21

Summary...................................................................................................................................................... 10-31

Exercises ...................................................................................................................................................... 10-33

Answers to Exercises................................................................................................................................. 10-41

Appendix A

Introduction to MATLAB®

MATLAB® and Simulink® ........................................................................................................................ A-1

Command Window....................................................................................................................................... A-1

Roots of Polynomials.................................................................................................................................... A-3

Polynomial Construction from Known Roots ......................................................................................... A-4

Evaluation of a Polynomial at Specified Values ....................................................................................... A-6

Rational Polynomials .................................................................................................................................... A-8

Using MATLAB to Make Plots ................................................................................................................ A-10

Subplots........................................................................................................................................................ A-19

Multiplication, Division and Exponentiation.......................................................................................... A-20

Script and Function Files........................................................................................................................... A-26

Display Formats .......................................................................................................................................... A-31

Appendix B

A Review of Complex Numbers

Definition of a Complex Number...............................................................................................................B-1

Addition and Subtraction of Complex Numbers......................................................................................B-2

Multiplication of Complex Numbers..........................................................................................................B-3

Division of Complex Numbers....................................................................................................................B-4

Exponential and Polar Forms of Complex Numbers ..............................................................................B-4

Contents

vi Circuit Analysis I with MATLAB Applications

Orchard Publications

Appendix C

Matrices and Determinants

Matrix Definition ...........................................................................................................................................C-1

Matrix Operations..........................................................................................................................................C-2

Special Forms of Matrices ............................................................................................................................C-5

Determinants ..................................................................................................................................................C-9

Minors and Cofactors................................................................................................................................. C-12

Cramer’s Rule .............................................................................................................................................. C-16

Gaussian Elimination Method .................................................................................................................. C-19

The Adjoint of a Matrix ............................................................................................................................. C-20

Singular and Non-Singular Matrices ........................................................................................................ C-21

The Inverse of a Matrix ............................................................................................................................. C-21

Solution of Simultaneous Equations with Matrices............................................................................... C-23

Exercises....................................................................................................................................................... C-30

Circuit Analysis I with MATLAB Applications 1-1

Orchard Publications

Chapter 1

Basic Concepts and Definitions

his chapter begins with the basic definitions in electric circuit analysis. It introduces the con￾cepts and conventions used in introductory circuit analysis, the unit and quantities used in cir￾cuit analysis, and includes several practical examples to illustrate these concepts.

1.1 The Coulomb

Two identically charged (both positive or both negative) particles possess a charge of one coulomb

when being separated by one meter in a vacuum, repel each other with a force of newton

where . The definition of coulomb is illustrated in Figure 1.1.

Figure 1.1. Definition of the coulomb

The coulomb, abbreviated as , is the fundamental unit of charge. In terms of this unit, the charge

of an electron is and one negative coulomb is equal to electrons. Charge,

positive or negative, is denoted by the letter or .

1.2 Electric Current and Ampere

Electric current at a specified point and flowing in a specified direction is defined as the instanta￾neous rate at which net positive charge is moving past this point in that specified direction, that is,

(1.1)

The unit of current is the ampere abbreviated as and corresponds to charge moving at the rate of

one coulomb per second. In other words,

(1.2)

T

10–7

c

2

c velocity of light 3 108 = ≈ × m s⁄

Vacuum

q 1 m q

F 10–7

c2 = N

q=1 coulomb

C

1.6 10–19

× C 6.24 1018

×

q Q

i

i dq

dt

----- ∆q

∆t

------ ∆t → 0 = = lim

A q

1 ampere

1 coulomb

1 second = -----------------------------

Chapter 1 Basic Concepts and Definitions

1-2 Circuit Analysis I with MATLAB Applications

Orchard Publications

Note: Although it is known that current flow results from electron motion, it is customary to think

of current as the motion of positive charge; this is known as conventional current flow.

To find an expression of the charge in terms of the current , let us consider the charge trans￾ferred from some reference time to some future time . Then, since

the charge is

or

or

(1.3)

Example 1.1

For the waveform of current i shown in Figure 1.2, compute the total charge transferred between

a. and

b. and

Figure 1.2. Waveform for Example 1.1

qi q

t0 t

i dq

dt = -----

q

q t0

t i t d

t0

t

∫ =

q t( ) q t0 – ( ) i t d

t0

t

∫ =

q t( ) i t d

t0

t

∫ + q t0 = ( )

q

t 0 = t 3 = s

t 0 = t 9 = s

1 2 3 4 5 6 7 8

−30

−20

−10

0

30

20

10

9

i ( ) mA

t ( )s

Circuit Analysis I with MATLAB Applications 1-3

Orchard Publications

Electric Current and Ampere

Solution:

We know that

Then, by calculating the areas, we find that:

a. For 0 < t < 2 s, area = ½ × (2 × 30 mA) = 30 mC

For 2 < t < 3 s, area = 1 × 30 = 30 mC

Therefore, for 0 < t < 3 s, total charge = total area = 30 mC + 30 mC = 60 mC.

b. For 0 < t < 2 s, area = ½ × (2 × 30 mA) = 30 mC

For 2 < t < 6 s, area = 4 × 30 = 120 mC

For 6 < t < 8 s, area = ½ × (2 × 30 mA) = 30 mC

For 8 < t < 9 s, we observe that the slope of the straight line for t > 6 s is −30 mA / 2 s, or −15

mA / s. Then, for 8 < t < 9 s, area = ½ × {1×(−15)} = −7.5 mC. Therefore, for 0 < t < 9 s, total

charge = total area = 30 + 120 + 30 −7.5 = 172.5 mC.

Convention: We denote the current by placing an arrow with the numerical value of the current

next to the device in which the current flows. For example, the designation shown in Figure 1.3

indicates either a current of is flowing from left to right, or that a current of is moving

from right to left.

Figure 1.3. Direction of conventional current flow

Caution: The arrow may or may not indicate the actual conventional current flow. We will see later

in Chapters 2 and 3 that in some circuits (to be defined shortly), the actual direction of

the current cannot be determined by inspection. In such a case, we assume a direction

with an arrow for said current ; then, if the current with the assumed direction turns out

to be negative, we conclude that the actual direction of the current flow is opposite to the

direction of the arrow. Obviously, reversing the direction reverses the algebraic sign of

the current as shown in Figure 1.3.

In the case of time-varying currents which change direction from time-to-time, it is convenient to

think or consider the instantaneous current, that is, the direction of the current which flows at some

particular instant. As before, we assume a direction by placing an arrow next to the device in which

the current flows, and if a negative value for the current i is obtained, we conclude that the actual

direction is opposite of that of the arrow.

q t 0 =

t i t d

0

t

∫ Area 0

t = =

i

2 A 2– A

2 A −2 A

Device

i