Circuit Analysis I with MATLAB® Applications

Steven T. Karris

Orchard Publicationswww.orchardpublications.com

Circuit Analysis I with MATLAB® Applications

Copyright © 2004 Orchard Publications. All rights reserved. Printed in Canada. No part of this publication may bereproduced or distributed in any form or by any means, or stored in a data base or retrieval system, without the priorwritten 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 lossor damages arising from the information contained in this text.

This book was created electronically using Adobe Framemaker®.

Contents

Chapter 1

Basic Concepts and DefinitionsThe Coulomb...................................................................................................................................................1-1Electric Current and Ampere........................................................................................................................1-1Two Terminal Devices...................................................................................................................................1-4Voltage (Potential Difference) ......................................................................................................................1-5Power and Energy...........................................................................................................................................1-8Active and Passive Devices ........................................................................................................................ 1-12Circuits and Networks................................................................................................................................. 1-12Active and Passive Networks..................................................................................................................... 1-12Necessary Conditions for Current Flow .................................................................................................. 1-12International System of Units .................................................................................................................... 1-13Sources of Energy........................................................................................................................................ 1-17Summary........................................................................................................................................................ 1-18Exercises........................................................................................................................................................ 1-21Answers to Exercises................................................................................................................................... 1-25

Chapter 2

Analysis of Simple CircuitsConventions.....................................................................................................................................................2-1Ohm’s Law.......................................................................................................................................................2-1Power Absorbed by a Resistor......................................................................................................................2-3Energy Dissipated by a Resistor ...................................................................................................................2-4Nodes, Branches, Loops and Meshes..........................................................................................................2-5Kirchhoff’s Current Law (KCL)...................................................................................................................2-6Kirchhoff’s Voltage Law (KVL)...................................................................................................................2-7Analysis of Single Mesh (Loop) Series Circuits....................................................................................... 2-10Analysis of Single Node-Pair Parallel Circuits......................................................................................... 2-14Voltage and Current Source Combinations ............................................................................................. 2-16Resistance and Conductance Combinations............................................................................................ 2-18Voltage Division Expressions.................................................................................................................... 2-22Current Division Expressions.................................................................................................................... 2-24Standards for Electrical and Electronic Devices..................................................................................... 2-26Resistor Color Code .................................................................................................................................... 2-27Power Rating of Resistors .......................................................................................................................... 2-28

Circuit Analysis I with MATLAB Applications iOrchard Publications

Contents

Temperature Coefficient of Resistance .................................................................................................... 2-29Ampere Capacity of Wires.......................................................................................................................... 2-30Current Ratings for Electronic Equipment ............................................................................................. 2-30Copper Conductor Sizes for Interior Wiring........................................................................................... 2-33Summary........................................................................................................................................................ 2-38Exercises........................................................................................................................................................ 2-41Answers to Exercises................................................................................................................................... 2-50

Chapter 3

Nodal and Mesh Equations - Circuit TheoremsNodal, Mesh, and Loop Equations ............................................................................................................. 3-1Analysis with Nodal Equations.................................................................................................................... 3-1Analysis with Mesh or Loop Equations ..................................................................................................... 3-8Transformation between Voltage and Current Sources.........................................................................3-20Thevenin’s Theorem....................................................................................................................................3-24Norton’s Theorem.......................................................................................................................................3-35Maximum Power Transfer Theorem ........................................................................................................3-38Linearity .........................................................................................................................................................3-39Superposition Principle ...............................................................................................................................3-41Circuits with Non-Linear Devices.............................................................................................................3-45Efficiency ......................................................................................................................................................3-47Regulation .....................................................................................................................................................3-49Summary........................................................................................................................................................3-49Exercises........................................................................................................................................................3-52Answers to Exercises...................................................................................................................................3-64

Chapter 4

Introduction to Operational AmplifiersSignals .............................................................................................................................................................. 4-1Amplifiers........................................................................................................................................................ 4-1Decibels ........................................................................................................................................................... 4-2Bandwidth and Frequency Response.......................................................................................................... 4-4The Operational Amplifier ........................................................................................................................... 4-5An Overview of the Op Amp...................................................................................................................... 4-5Active Filters................................................................................................................................................. 4-13Analysis of Op Amp Circuits ..................................................................................................................... 4-16Input and Output Resistance ..................................................................................................................... 4-28Summary........................................................................................................................................................ 4-32

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Contents

Exercises ........................................................................................................................................................4-34Answers to Exercises ...................................................................................................................................4-43

Chapter 5

Inductance and CapacitanceEnergy Storage Devices................................................................................................................................. 5-1Inductance ....................................................................................................................................................... 5-1Power and Energy in an Inductor ............................................................................................................. 5-11Combinations of Series and Parallel Inductors........................................................................................ 5-14Capacitance.................................................................................................................................................... 5-17Power and Energy in a Capacitor .............................................................................................................. 5-22Capacitance Combinations ......................................................................................................................... 5-25Nodal and Mesh Equations in General Terms........................................................................................ 5-28Summary ........................................................................................................................................................ 5-29Exercises ........................................................................................................................................................ 5-31Answers to Exercises ................................................................................................................................... 5-36

Chapter 6

Sinusoidal Circuit AnalysisExcitation Functions...................................................................................................................................... 6-1Circuit Response to Sinusoidal Inputs ........................................................................................................ 6-1The Complex Excitation Function.............................................................................................................. 6-3Phasors in , , and Circuits ................................................................................................................. 6-8Impedance ..................................................................................................................................................... 6-14Admittance .................................................................................................................................................... 6-17Summary ........................................................................................................................................................ 6-21Exercises ........................................................................................................................................................ 6-25Answers to Exercises ................................................................................................................................... 6-30

Chapter 7

Phasor Circuit AnalysisNodal Analysis ................................................................................................................................................ 7-1Mesh Analysis ................................................................................................................................................. 7-5Application of Superposition Principle.......................................................................................................7-7Thevenin’s and Norton’s Theorems ...........................................................................................................7-8Phasor Analysis in Amplifier Circuits .......................................................................................................7-12

R L C

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Phasor Diagrams .......................................................................................................................................... 7-15Electric Filters............................................................................................................................................... 7-20Basic Analog Filters ..................................................................................................................................... 7-21Active Filter Analysis................................................................................................................................... 7-26Summary........................................................................................................................................................ 7-28Exercises........................................................................................................................................................ 7-29Answers to Exercises................................................................................................................................... 7-37

Chapter 8

Average and RMS Values, Complex Power, and InstrumentsPeriodic Time Functions............................................................................................................................... 8-1Average Values ............................................................................................................................................... 8-2Effective Values ............................................................................................................................................. 8-3Effective (RMS) Value of Sinusoids ........................................................................................................... 8-5RMS Values of Sinusoids with Different Frequencies............................................................................. 8-7Average Power and Power Factor............................................................................................................... 8-9Average Power in a Resistive Load ...........................................................................................................8-10Average Power in Inductive and Capacitive Loads ................................................................................8-11Average Power in Non-Sinusoidal Waveforms.......................................................................................8-14Lagging and Leading Power Factors.........................................................................................................8-15Complex Power - Power Triangle .............................................................................................................8-16Power Factor Correction ............................................................................................................................8-18Instruments ...................................................................................................................................................8-21Summary........................................................................................................................................................8-30Exercises........................................................................................................................................................8-33Answers to Exercises...................................................................................................................................8-39

Chapter 9

Natural ResponseThe Natural Response of a Series RL circuit............................................................................................. 9-1The Natural Response of a Series RC Circuit ......................................................................................... 9-10Summary........................................................................................................................................................ 9-17Exercises........................................................................................................................................................ 9-19Answers to Exercises................................................................................................................................... 9-25

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Contents

Chapter 10

Forced and Total Response in RL and RC CircuitsThe Unit Step Function...............................................................................................................................10-1The Unit Ramp Function............................................................................................................................10-6The Delta Function......................................................................................................................................10-8The Forced and Total Response in an RL Circuit ................................................................................10-14The Forced and Total Response in an RC Circuit ................................................................................10-21Summary ...................................................................................................................................................... 10-31Exercises ...................................................................................................................................................... 10-33Answers to Exercises ................................................................................................................................. 10-41

Appendix A

Introduction to MATLAB®MATLAB® and Simulink® ........................................................................................................................ A-1Command Window....................................................................................................................................... A-1Roots of Polynomials.................................................................................................................................... A-3Polynomial Construction from Known Roots ......................................................................................... A-4Evaluation of a Polynomial at Specified Values ....................................................................................... A-6Rational Polynomials .................................................................................................................................... A-8Using MATLAB to Make Plots ................................................................................................................ A-10Subplots ........................................................................................................................................................ A-19Multiplication, Division and Exponentiation.......................................................................................... A-20Script and Function Files ........................................................................................................................... A-26Display Formats .......................................................................................................................................... A-31

Appendix B

A Review of Complex NumbersDefinition of a Complex Number ...............................................................................................................B-1Addition and Subtraction of Complex Numbers......................................................................................B-2Multiplication of Complex Numbers ..........................................................................................................B-3Division of Complex Numbers....................................................................................................................B-4Exponential and Polar Forms of Complex Numbers ..............................................................................B-4

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Appendix C

Matrices and DeterminantsMatrix Definition ...........................................................................................................................................C-1Matrix Operations..........................................................................................................................................C-2Special Forms of Matrices ............................................................................................................................C-5Determinants ..................................................................................................................................................C-9Minors and Cofactors................................................................................................................................. C-12Cramer’s Rule .............................................................................................................................................. C-16Gaussian Elimination Method .................................................................................................................. C-19The Adjoint of a Matrix ............................................................................................................................. C-20Singular and Non-Singular Matrices ........................................................................................................ C-21The Inverse of a Matrix ............................................................................................................................. C-21Solution of Simultaneous Equations with Matrices............................................................................... C-23Exercises....................................................................................................................................................... C-30

vi Circuit Analysis I with MATLAB ApplicationsOrchard 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 coulombwhen 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 AmpereElectric 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 ofone coulomb per second. In other words,

(1.2)

T

10 7– c2

c velocity of light 3 108 m s⁄×≈=

Vacuum

q q1 m

F 10 7– c2 N=q=1 coulomb

C

1.6 10 19– C× 6.24 1018×q Q

i

i dqdt------ q∆

t∆------

t∆ 0→lim= =

A q

1 ampere 1 coulomb1 ondsec

-----------------------------=

Circuit Analysis I with MATLAB Applications 1-1Orchard Publications

Chapter 1 Basic Concepts and Definitions

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

q i qt0 t

i dqdt------=

q

q t0

t i tdt0

t

∫=

q t( ) q t0( )– i tdt0

t

∫ =

q t( ) i tdt0

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( )

1-2 Circuit Analysis I with MATLAB ApplicationsOrchard 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 mCFor 6 < t < 8 s, area = ½ × (2 × 30 mA) = 30 mCFor 8 < t < 9 s, we observe that the slope of the straight line for t > 6 s is −30 mA / 2 s, or −15mA / s. Then, for 8 < t < 9 s, area = ½ × {1×(−15)} = −7.5 mC. Therefore, for 0 < t < 9 s, totalcharge = 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 currentnext to the device in which the current flows. For example, the designation shown in Figure 1.3indicates either a current of is flowing from left to right, or that a current of is movingfrom 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 laterin Chapters 2 and 3 that in some circuits (to be defined shortly), the actual direction ofthe current cannot be determined by inspection. In such a case, we assume a directionwith an arrow for said current ; then, if the current with the assumed direction turns outto be negative, we conclude that the actual direction of the current flow is opposite to thedirection of the arrow. Obviously, reversing the direction reverses the algebraic sign ofthe current as shown in Figure 1.3.

In the case of time-varying currents which change direction from time-to-time, it is convenient tothink or consider the instantaneous current, that is, the direction of the current which flows at someparticular instant. As before, we assume a direction by placing an arrow next to the device in whichthe current flows, and if a negative value for the current i is obtained, we conclude that the actualdirection is opposite of that of the arrow.

q t 0=t i td

0

t

∫ Area 0t= =

i

2 A 2– A

2 A −2 A

Device

i

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Chapter 1 Basic Concepts and Definitions

1.3 Two Terminal Devices

In this text we will only consider two-terminal devices. In a two-terminal device the current enteringone terminal is the same as the current leaving the other terminal* as shown in Figure 1.4.

Figure 1.4. Current entering and leaving a two-terminal device

Let us assume that a constant value current (commonly known as Direct Current and abbreviated asDC) enters terminal and leaves the device through terminal in Figure 1.4. The passage of cur-rent (or charge) through the device requires some expenditure of energy, and thus we say that a poten-tial difference or voltage exists “across” the device. This voltage across the terminals of the device is ameasure of the work required to move the current (or charge) through the device.

Example 1.2

In a two-terminal device, a current enters the left (first) terminal.

a. What is the amount of current which enters that terminal in the time interval ?

b. What is the current at ?

c. What is the charge at given that ?

Solution:

a.

b.

c.

* We will see in Chapter 5 that a two terminal device known as capacitor is capable of storing energy.

Two terminal device

Terminal A Terminal B

7 A 7 A

A B

i t( ) 20 100πt mAcos=

10 t 20 ms≤ ≤–

t 40 ms=

q t 5 ms= q 0( ) 0=

i t0

t 20 100cos πt10– 10 3–

×

20 10 3–× 20 100cos π 20 10 3–×( ) 20 100cos π 10– 10 3–×( )–= =

20 2π 20 π–( )cos–cos 40 mA==

i t 0.4 ms=20 100cos πt t 0.4 ms=

20 40πcos 20 mA= = =

q t( ) i t q 0( )+d0

5 10 3–×

∫ 20 100cos πt td0

5 10 3–×

∫ 0+= =

0.2π

------- 100πt 05 10 3–×sin 0.2

π------- π

2--- 0–sin 0.2

π------- C= ==

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