■' f-. ' # 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 o n o n o Cover image (^^J^ikiaui ^ Used under license from Shutterstock, Inc. Kendall Hunft publishing company www.kendallhunt.cpm Send all inquiries to: 4050 Westmark Drive Dubuque, lA 52004-1840 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, r^ without the prior written permission of the copyright owner. Printed in the United States of America 10 9 8 7 6 5 4 3 O 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 o n p , 0 o n O o 0 o n n o- o 0 0 0 ■ 0 0 n O n n ^ o 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.