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LINEAR
CIRCUITS
TIME DOMAIN, PHASOR, AND LAPLACE
TRANSrORM APPROACHES
T H I R D E D I T I O N
Raymond A. DeCarlo
Purdue University
Pen-Min Lin
Purdue University
Kendall Hunt
p u b l i s h i n g c o m p a n y
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Copyright ©2001, 2009 Raymond A. DeCarlo and Pen-Min Lin
Copyright ©1995 Prentice-Hall, Inc.
ISBN 978-0-7575-6499-4
All rights reserved. No part of this publication may be reproduced,
stored in a retrieval system, or transmitted, in any form or by any means,
electronic, mechanical, photocopying, recording, or otherwise,
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without the prior written permission of the copyright owner.
Printed in the United States of America
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TABLE OF CONTENTS
Preface......................................................................................................................................................................vii
Chapter 1 • Charge, Current, Voltage and Ohm’s Law ............................................................................1
Chapter 2 • Kirchhoff’s Current & Voltage Laws and Series-Parallel Resistive Circuits..............51
Chapter 3 • Nodal and Loop Analyses.......................................................................................................107
Chapter 4 • The Operational Amplifier.....................................................................................................155
Chapter 5 * Linearity, Superposition, and Source Transformation...................................................191
Chapter 6 • Thevenin, Norton, and Maximum Power Transfer Theorems....................................227
Chapter 7 • Inductors and Capacitors.......................................................................................................269
Chapter 8 • First Order RL and RC Circuits...........................................................................................321
Chapter 9 • Second Order Linear Circuits................................................................................................379
Chapter 10 • Sinusoidal Steady State Analysis by Phasor Methods .................................................431
Chapter 11 • Sinusoidal State State Power Calculations.......................................................................499
Chapter 12 • Laplace Transform Analysis L Basics.................................................................................543
Chapter 13 • Laplace Transform Analysis II: Circuit Applications...................................................603
Chapter 14 • Laplace Transform Analysis III; Transfer Function Applications.............................683
Chapter 15 * Time Domain Circuit Response Computations: The Convolution Method......763
Chapter 16 • Band-Pass Circuits and Resonance....................................................................................811
Chapter 17 * Magnetically Coupled Circuits and Transformers........................................................883
Chapter 18 • Two-Ports...................................................................................................................................959
Chapter 19 • Principles of Basic Filtering .............................................................................................1031
Chapter 20 • Brief Introduction to Fourier Series ..............................................................................1085
Index...................................................................................................................................................................1119
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PREFACE
For the last several decades, EE/ECE departments of US universities have typically required two
semesters of linear circuits during the sophomore year for EE majors and one semester for other
engineering majors. Over the same time period discrete time system concepts and computer engi
neering principles have become required fare for EE undergraduates. Thus we continue to use
Laplace transforms as a vehicle for understanding basic concepts such as impedance, admittance,
fdtering, and magnetic circuits. Further, software programs such as PSpice, MATLAB and its tool
boxes, Mathematica, Maple, and a host of other tools have streamlined the computational drudg
ery of engineering analysis and design. MATLAB remains a working tool in this 3'''^ edition of
Linear Circuits.
In addition to a continuing extensive use of MATLAB, we have removed much of the more com
plex material from the book and rewritten much of the remaining book in an attempt to make the
text and the examples more illustrative and accessible. More importantly, many of the more diffi
cult homework exercises have been replaced with more routine problems often with numerical
answers or checks.
Our hope is that we have made the text more readable and understandable by today’s engineering
undergraduates.
C H A P T E R
Charge, Current, Voltage
and Ohm’s Law
CHAPTER OUTLINE
1. Role and Importance of Circuits in Engineering
2. Charge and Current
3. Voltage
4. Circuit Elements
5. Voltage, Current, Power, Energy, Relationships
6. Ideal Voltage and Current Sources
7. Resistance, Ohm’s Law, and Power (a Reprise)
8. V-I Characteristics of Ideal Resistors, Constant Voltage, and
Constant Current Sources
Summary
Terms and Concepts
Problems
CHAPTER OBjECTIVES
1. Introduce and investigate three basic electrical quantities: charge, current, and voltage,
and the conventions for their reference directions.
2. Define a two-terminal circuit element.
3. Define and investigate power and energy conversion in electric circuits, and demonstrate
that these quantities are conserved.
4. Define independent and dependent voltage and current sources that act as energy or sig
nal generators in a circuit.
5. Define Ohm’s law, v{t) = R i{t), for a resistor with resistance R.
6. Investigate power dissipation in a resistor.
7. Classify memoryless circuit elements by dieir terminal voltage-current relationships.
8. Explain the difference between a device and its circuit model.
chapter 1 • Charge, Current, Voltage and Ohm’s Law
1. ROLE AND IMPORTANCE OF CIRCUITS IN ENGINEERING
Are you curious about how fuses blow? About the meaning of different wattages on Hght bulbs?
About the heating elements in an oven? And how is the presence of your car sensed at a stoplight?
Circuit theory, the focus of this text, provides answers to all these questions.
When you learn basic circuit theory, you learn how to harness the power of electricity, as is done,
for example, in
• an electric motor that runs the compressor in an air conditioner or the pump in a dish
washer;
• a microwave oven;
• a radio, TV, or stereo;
• an iPod;
• a car heater.
In this text, we define and analyze common circuit elements and describe their interaction. Our
aim is to create a modular framework for analyzing circuit behavior, while simultaneously devel
oping a set of tools essential for circuit design. These skills are, of course, crucial to every electri
cal engineer. But they also have broad applicability in other fields. For instance, disciplines such
as bioengineering and mechanical engineering have similar patterns of analysis and often utilize
circuit analogies.
WHAT IS A CIRCUIT?
A circuit is an energy or signal/information processor. Each circuit consists of interconnections of
“simple” circuit elements, or devices. Each circuit element can, in turn, be thought of as an ener
gy or signal/information processor. For example, a circuit element called a “source” produces a
voltage or a current signal. This signal may serve as a power source for the circuit, or it may rep
resent information. Information in the form of voltage or current signals can be processed by the
circuit to produce new signals or new/different information. In a radio transmitter, electricity
powers the circuits that convert pictures, voices, or music (that is, information) into electromag
netic energy. This energy then radi
ates into the atmosphere or into
space from a transmitting antenna.
A satellite in space can pick up this
electromagnetic energy and trans
mit it to locations all over the
world. Similarly, a TV reception
antenna or a satellite dish can pick
up and direct this energy to a TV
set. The TV contains circuits
(Figure 1.1) that reconvert the
information within the received
signal back into pictures with
sound. FIGURE 1.1 Cathode ray tube with surrounding circuitry for
converting electrical signals into pictures.
Chapter 1 • Charge, Current, Voltage and Ohm’s Law
2. CHARGE AND CURRENT
CHARGE
Charge is an electrical property of matter. Matter consists of atoms. Roughly speaking, an atom
contains a nucleus that is made up of positively charged protons and neutrons (which have no
charge). The nucleus is surrounded by a cloud of negatively charged electrons. The accumulated
charge on 6.2415 x 10’^ electrons equals -1 coulomb (C). Thus, the charge on an electron is
-1.602176 X 10-19 C.
Particles with opposite charges attract each other, whereas those with similar charges repel. The
force of attraction or repulsion between two charged bodies is inversely proportional to the square
of the distance between them, assuming the dimensions of the bodies are very small compared
with the distance of separation. Two equally charged particles 1 meter (m) apart in free space have
charges of 1 C each if they repel each other with a force of 10“^ c^ Newtons (N), where c = 3 x
10^ m/s is the speed of light, by definition. The force is attractive if the particles have opposite
charges. Notationally, Q will denote a fixed charge, and q or q{t), a time-varying charge.
Exercise. How many electrons have a combined charge of-53.406 x 10 C?
ANSWER; 333,391,597
Exercise. Sketch the time-dependent charge profile q{t) = 3(l-^^0 C, ? > 0, present on a metal
plate. MATLAB is a good tool for such sketches.
A conductor refers to a material in which electrons can move to neighboring atoms with relative
ease. Metals, carbon, and acids are common conductors. Copper wire is probably the most com
mon conductor. An ideal conductor offers zero resistance to electron movement. Wires are
assumed to be ideal conductors, unless otherwise indicated.
Insulators oppose electron movement. Common insulators include dry air, dry wood, ceramic,
glass, and plastic. An ideal insulator offers infinite opposition to electron movement.
CURRENT
Current refers to the net flow of charge across any cross section of a conductor. The net move
ment of 1 coulomb (1 C) of charge through a cross section of a conductor in 1 second (1 sec)
produces an electric current of 1 ampere (1 A). The ampere is the basic unit of electric current
and equals 1 C/s.
The direction of current flow is taken by convention as opposite to the direction of electron flow,
as illustrated in Figure 1.2. This is because early in the history of electricity, scientists erroneously
believed that current was the movement of only positive charges, as illustrated in Figure 1.3. In
metallic conductors, current consists solely of the movement of electrons. However, as our under
standing of device physics advanced, scientists learned that in ionized gases, in electrolytic solu
chapter 1 • Charge, Current, Voltage and Ohm’s Law
tions, and in some semiconductor materials, movement of positive charges constitutes part or all
of the total current flow.
One Ampere
of Current "
One ; ; Cloud of\
second^.......|---- 6.24x10’®J 1
later i ; k electrons
Boundary
FIGURE 1.2 A cloud of negative charge moves past a cross section of an ideal conductor from right
to left. By convention, the positive current direction is taken as left to right.
One Ampere
of Current
One
One
Coulomb
'second
of positive
later
charge
Boundary
FIGURE 1.3 In the late nineteenth cenmry, current was thought to be the movement of a positive charge
past a cross section of a conduaor, giving rise to the conventional reference “direction of positive current flow.”
Both Figures 1.2 and 1.3 depict a current of 1 A flowing from left to right. In circuit analysis, we
do not distinguish between these two cases: each is represented symbolically, as in Figure 1.4(a).
The arrowhead serves as a reference for determining the true direction of the current. A positive
value of current means the current flows in the same direction as the arrow. A current of negative
value implies flow is in the opposite direction of the arrow. For example, in both Figures 1.4a and
b, a current of 1 A flows from left to right.
1A
-1A
> <
(a) (b)
FIGURE 1.4 1 A of current flows from left to right through a general circuit element.
