MATLAB introduction in circuit analysis , Exercises for Circuit Theory. University of Botswana
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tlotlo-oratilwe-seba

MATLAB introduction in circuit analysis , Exercises for Circuit Theory. University of Botswana

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Frontmatter

ELECTRONICS and CIRCUIT

ANALYSIS using MATLAB

JOHN O. ATTIA Department of Electrical Engineering

Prairie View A&M University

Boca Raton London New York Washington, D.C. CRC Press

© 1999 CRC Press LLC

This book contains information obtained from authentic and highly regarded sources. Reprinted material is quoted with permission, and sources are indicated. A wide variety of references are listed. Reasonable efforts have been made to publish reliable data and information, but the author and the publisher cannot assume responsibility for the validity of all materials or for the consequences of their use.

Neither this book nor any part may be reproduced or transmitted in any form or by any means, electronic or mechanical, including photocopying, microfilming, and recording, or by any information storage or retrieval system, without prior permission in writing from the publisher.

The consent of CRC Press LLC does not extend to copying for general distribution, for promotion, for creating new works, or for resale. Specific permission must be obtained in writing from CRC Press LLC for such copying.

Direct all inquiries to CRC Press LLC, 2000 Corporate Blvd. N.W. , Boca Raton, Florida 33431.

Trademark Notice:

Product or corporate names may be trademarks or registered trade- marks, and are used only for identification and explanation, without intent to infringe.

© 1999 by CRC Press LLC

No claim to original U.S. Government works International Standard Book Number 0-8493-1176-4

Library of Congress Card Number 98-46071 Printed in the United States of America 1 2 3 4 5 6 7 8 9 0

Printed on acid-free paper

Library of Congress Cataloging-in-Publication Data

Attia, John Okyere. Electronics and circuit analysis using MATLAB / John Okyere Attia p. cm. Includes bibliographical references and index. ISBN 0-8493-1176-4 (alk. paper) 1. Electronics--Data processing. 2. Electric circuit analysis-

-Data processing. 3. MATLAB (Computer file) I Title. TK7835.A88 1999 621.381’0285--dc21

CIP 98-46071

© 1999 CRC Press LLC

PREFACE MATLAB is a numeric computation software for engineering and scientific calculations. MATLAB is increasingly being used by students, researchers, practicing engineers and technicians. The causes of MATLAB popularity are legion. Among them are its iterative mode of operation, built-in functions, simple programming, rich set of graphing facilities, possibilities for writing additional functions, and its extensive toolboxes. The goals of writing this book are (1) to provide the reader with simple, easy, hands-on introduction to MATLAB; (2) to demonstrate the use of MATLAB for solving electronics problems; (3) to show the various ways MATLAB can be used to solve circuit analysis problems; and (4) to show the flexibility of MATLAB for solving general engineering and scientific problems. Audience The book can be used by students, professional engineers and technicians. The first part of the book can be used as a primer to MATLAB. It will be useful to all students and professionals who want a basic introduction to MATLAB. Parts 2 and 3 are for electrical and electrical engineering technology students and professionals who want to use MATLAB to explore the characteristics of semiconductor devices and the application of MATLAB for analysis and design of electrical and electronic circuits and systems. Organization The book is divided into three parts: Introduction to MATLAB, Circuit analysis applications using MATLAB, and electronics applications with MATLAB. It is recommended that the reader work through and experiment with the examples at a computer while reading Chapters 1, 2, and 3. The hands-on approach is one of the best ways of learning MATLAB. Part II consists of Chapters 4 to 8. This part covers the applications of MATLAB in circuit analysis. The topics covered in Part II are dc analysis, transient analysis, alternating current analysis, and Fourier analysis. In addition, two-port networks are covered. I have briefly covered the underlying theory and concepts, not with the aim of writing a textbook on circuit analysis and electronics. Selected problems in circuit analysis have been solved using MATLAB.

© 1999 CRC Press LLC

Part III includes Chapters 9, 10, 11 and 12. The topics discussed in this part are diodes, semiconductor physics, operational amplifiers and transistor circuits. Application of MATLAB for problem solving in electronics is discussed. Extensive examples showing the use of MATLAB for solving problems in electronics are presented. Each chapter has its own bibliography and exercises. Text Diskette Since the text contains a large number of examples that illustrate electronics and circuit analysis principles and applications with MATLAB, a diskette is included that contains all the examples in the book. The reader can run the examples without having to enter the commands. The examples can also be modified to suit the needs of the reader. Acknowledgments I appreciate the suggestions and comments from a number of reviewers including Dr. Murari Kejariwal, Dr. Reginald Perry, Dr. Richard Wilkins, Mr. Warsame Ali, Mr. Anowarul Huq and Mr. John Abbey. Their frank and positive criticisms led to considerable improvement of this work. I am grateful to Mr. Zhong You for typing and running some of the MATLAB programs in this book and I am also grateful to Mr. Carl Easton and Mr. Url Woods for drawing the circuit diagrams found in the text. I thank Ms. Debbie Hawkins and Cheryl Wright who typed several parts of this book. I am appreciative of Ms. Judith Hansen for her editing services. Special thanks go Ms. Nora Konopka, at CRC Press, who took an early interest in this book and offered me any assistance I needed to get it completed. I thank Ms. Mimi Williams, at CRC Press, for thoroughly proofreading the manuscript. The questions and comments from electrical engineering students at Prairie View A&M University led to rewriting some sections of this work. Special thanks go to the students who used various drafts of this book and provided useful comments. A final note of gratitude goes to my wife, Christine N. Okyere, who encouraged me to finish the book in record time. With equanimity and understanding, she stood by me during the endless hours I spent writing.

© 1999 CRC Press LLC

DEDICATION

Dedicated to my family members Christine, John II and Angela

for their unfailing love, support and encouragement

© 1999 CRC Press LLC

CONTENTS

CHAPTER ONE MATLAB FUNDAMENTALS

1.1 MATLAB BASIC OPERATIONS 1.2 MATRIX OPERATIONS 1.3 ARRAY OPERATIONS 1.4 COMPLEX NUMBERS 1.5 THE COLON SYMBOL ( : ) 1.6 M-FILES

1.6.1 Script files 1.6.2 Function files

SELECTED BIBLIOGRAPHY EXERCISES

CHAPTER TWO PLOTTING COMMANDS

2.1 GRAPH FUNCTIONS 2.2 X-Y PLOTS AND ANNOTATIONS 2.3 LOGARITHMIC AND POLAR PLOTS 2.4 SCREEN CONTROL

SELECTED BIBLIOGRAPHY EXERCISES

CHAPTER THREE CONTROL STATEMENTS

3.1 FOR LOOPS 3.2 IF STATEMENTS 3.3 WHILE LOOP 3.4 INPUT/OUTPUT COMMANDS

SELECTED BIBLIOGRAPHY EXERCISES

© 1999 CRC Press LLC

CHAPTER FOUR DC ANALYSIS

4.1 NODAL ANALYSIS 4.2 LOOP ANALYSIS 4.3 MAXIMUM POWER TRANSFER

4.3.1 MATLAB diff and find Functions SELECTED BIBLIOGRAPHY EXERCISES

CHAPTER FIVE TRANSIENT ANALYSIS

5.1 RC NETWORK 5.2 RL NETWORK 5.3 RLC CIRCUIT 5.4 STATE VARIABLE APPROACH

5.4.1 MATLAB ode functions

SELECTED BIBLIOGRAPHY EXERCISES

CHAPTER SIX AC ANALYSIS AND NETWORK

FUNCTIONS

6.1 STEADY STATE AC POWER 6.1.1 MATLAB functions quad and quad8

6.2 SINGLE- AND THREE-PHASE AC CIRCUITS 6.3 NETWORK CHARACTERISTICS

6.3.1 MATLAB functions roots, residue and polyval

6.4 FREQUENCY RESPONSE 6.4.1 MATLAB Function freqs

SELECTED BIBLIOGRAPHY EXERCISES

© 1999 CRC Press LLC

CHAPTER SEVEN TWO-PORT NETWORKS

7.1 TWO-PORT NETWORK REPRESENTATIONS 7.1.1 z-parameters 7.1.2 y-parameters 7.1.3 h-parameters 7.1.4 Transmission parameters

7.2 INTERCONNECTION OF TWO-PORT NETWORKS 7.3 TERMINATED TWO-PORT NETWORKS

SELECTED BIBLIOGRAPHY EXERCISES

CHAPTER EIGHT FOURIER ANALYSIS

8.1 FOURIER SERIES 8.2 FOURIER TRANSFORMS

8.2.1 Properties of Fourier transform 8.3 DISCRETE AND FAST FOURIER TRANSFORMS

8.3.1 MATLAB function fft

SELECTED BIBLIOGRAPHY EXERCISES

CHAPTER NINE DIODES

9.1 DIODE CHARACTERISTICS

9.1.1 Forward-biased region 9.1.2 MATLAB function polyfit 9.1.3 Temperature effects

9.2 ANALYSIS OF DIODE CIRCUITS 9.3 HALF-WAVE RECTIFIER

9.3.1 MATLAB function fzero 9.4 FULL-WAVE RECTIFICATION 9.5 ZENER DIODE VOLTAGE REGULATOR CIRCUIT

SELECTED BIBLIOGRAPHY EXERCISES

© 1999 CRC Press LLC

10.1 INTRINSIC SEMICONDUCTOR

10.1.1 Energy bands 10.1.2 Mobile carriers

10.2 EXTRINSIC SEMICONDUCTOR 10.2.1 Electron and hole concentrations 10.2.2 Fermi level 10.2.3 Current density and mobility

10.3 PN JUNCTION: CONTACT POTENTIAL, JUNCTION CURRENT

10.3.1 Contact potential 10.3.2 Junction current

10.4 DEPLETION AND DIFFUSION CAPACITANCES 10.4.1 Depletion capacitance

10.4.2 Diffusion capacitance 10.5 BREAKDOWN VOLTAGES OF PN JUNCTIONS

REFERENCES

EXERCISES CHAPTER ELEVEN OPERATIONAL AMPLIFIERS

11.1 PROPERTIES OF THE OP AMP 11.2 INVERTING CONFIGURATION 11.3 NON-INVERTING CONFIGURATION 11.4 EFFECT OF FINITE OPEN-LOOP GAIN 11.5 FREQUENCY RESPONSE OF OP AMPS 11.6 SLEW RATE AND FULL-POWER BANDWIDTH 11.7 COMMON-MODE REJECTION

SELECTED BIBLIOGRAPHY EXERCISES

CHAPTER TWELVE TRANSISTOR CIRCUITS

12.1 BIPOLAR JUNCTION TRANSISTORS 12.2 BIASING OF BJT DISCRETE CIRCUITS

12.2.1 Self-bias circuit 12.2.2 Bias stability

12.3 INTEGRATED CIRCUIT BIASING 12.3.1 Simple current mirror

CHAPTER TEN SEMICONDUCTOR PHYSICS

© 1999 CRC Press LLC

12.3.2 Wilson current source 12.4 FREQUENCY RESPONSE OF COMMON EMITTER AMPLIFIER 12.5 MOSFET CHARACTERISTICS 12.6 BIASING OF MOSFET CIRCUITS 12.7 FREQUENCY RESPONSE OF

COMMON-SOURCE AMPLIFIER SELECTED BIBLIOGRAPHY EXERCISES

© 1999 CRC Press LLC

LIST OF EXAMPLES IN TEXT

CHAPTER ONE MATLAB FUNDAMENTALS EXAMPLE

DESCRIPTION

1.1

Power Dissipation in a Resistor

1.2

Complex Number Representation

1.3

Equivalent Resistance

1.4

Quadratic Equation

CHAPTER TWO PLOTTING COMMANDS EXAMPLE

DESCRIPTION

2.1Voltage and Current of an RL Circuit 2.2

Gain versus Frequency of an RC Amplifier

2.3

Polar Plot of a Complex Number

CHAPTER THREE CONTROL STATEMENTS EXAMPLE

DESCRIPTION

3.1

Horizontal and Vertical Displacement

3.2

A 3-bit A/D Converter

3.3

Consecutive Integer Numbers

© 1999 CRC Press LLC

EXAMPLE

DESCRIPTION

4.1

Nodal Voltages of a Simple Circuit

4.2

Circuit with Dependent and Independent Sources

4.3

Loop Analysis of a Bridge Circuit

4.4Power Dissipation and Source Current

4.5Nodal Voltage Circuit with Dependent Sources4.6Maximum Power Dissipation

CHAPTER FIVE TRANSIENT ANALYSIS EXAMPLE

DESCRIPTION

5.1 Charging of a Capacitor with Different Time Constants

5.2

Charging and Discharging of a Capacitor

5.3Current Flowing through Inductor of RL Circuit

5.4

Current Flowing through a Series RLC Circuit

5.5

Voltage across a Parallel RLC Circuit

5.6

State Variable Approach to RC Circuit

5.7

State Variable Approach to an RLC Circuit Analysis

5.8State Variable Analysis of a Network

CHAPTER FOUR DC ANALYSIS

© 1999 CRC Press LLC

EXAMPLE

DESCRIPTION

6.1

Power Calculations of One-port Network

6.2AC Voltage of an RLC Circuit6.3

AC Current and Voltage of a Circuit with Two Sources

6.4

Unbalanced Wye-wye Connection

6.5

Network Function, Poles and Zeros of a Circuit

6.6

Inverse Laplace Transform

6.7

Magnitude and Phase Response of an RLC Circuit

CHAPTER SEVEN TWO-PORT NETWORKS EXAMPLE

DESCRIPTION

7.1 z-parameters of T-Network

7.2

y-parameters of Pi-Network

7.3

y-parameters of Field Effect Transistor

7.4h-parameters of Bipolar Junction Transistor

7.5

Transmission Parameters of a Simple Impedance Network

7.6

Transmission Parameters of a Simple Admittance Network

7.7

y-parameters of Bridge T-Network

7.8Transmission Parameters of a Simple Cascaded Network

7.9

Transmission Parameters of a Cascaded System

7.10z - parameters and Magnitude Responses of an Active Lowpass Filter

CHAPTER SIX AC ANALYSIS AND NETWORK FUNCTIONS

© 1999 CRC Press LLC

EXAMPLE

DESCRIPTION

8.1

Fourier Series Expansion of a Square Wave

8.2

Amplitude and Phase Spectrum of Full-wave Rectifier Waveform

8.3

Synthesis of a Periodic Exponential Signal

8.4

DFT and FFT of a Sequence

8.5

Fourier Transform and DFT of a Damped Exponential Sinusoid

8.6Power Spectral Density of a Noisy Signal

CHAPTER NINE DIODES EXAMPLE

DESCRIPTION

9.1

Determination of Diode Parameters from Data

9.2

I-V characteristic of a Diode at Different Temperatures

9.3

Operating Point of a Diode Using Graphical Techniques

9.4

Operating Point of a Diode using Iterative Technique

9.5

Battery Charging Circuit – Current, Conduction Angle and Peak Current

9.6

Capacitor Smoothing Circuit – Calculation of Critical Times

9.7

Full-wave Rectifier – Ripple Voltage, Dc Output Voltage, Discharge Time and Period of Ripple

9.8

A Zener Diode Voltage Regulator

CHAPTER EIGHT FOURIER ANALYSIS

© 1999 CRC Press LLC

EXAMPLE

DESCRIPTION

10.1 Electron Concentration versus Temperature 10.2

Minority Carriers in Doped Semiconductor

10.3

Electron and Hole Mobilities versus Doping Concentration

10.4Resistivity versus Doping

10.5

Junction Potential versus Voltage

10.6

Effects of Temperature on Reverse Saturation Current

10.7

Depletion Capacitance of a PN Junction

10.8

Diffusion and Depletion Capacitance as a Function of Voltage

10.9Effect of Doping Concentration on the Breakdown Voltage of a PN Junction

CHAPTER ELEVEN OPERATIONAL AMPLIFIERS EXAMPLE

DESCRIPTION

11.1 Frequency Response of Miller Integrator

11.2

Transfer function, Poles, and Zeros of a Non- inverting Op Amp Circuit

11.3

Effect of Finite Open Loop Gain

11.4

Open Loop Gain Characteristics of an Op Amp

11.5

Effect of Closed Loop Gain on the Frequency Response of an Op Amp

11.6

Output Voltage versus Full-power Bandwidth

CHAPTER TEN SEMICONDUCTOR PHYSICS

11.7

Effect of CMRR on the Closed Loop Gain

© 1999 CRC Press LLC

12.2 Output Characteristics of an NPN Transistor

12.3

Self-Bias Circuit – Stability Factors and Collector Current as a Function of Temperature

12.4

Comparison of Simple Current Mirror and Wilson Current Source

12.5

Frequency Response of a Common Emitter Amplifier

12.6

I-V Characteristics of NMOS

12.7

Operating Point Calculation of NMOS Biasing Circuit

12.8

Voltage and Current Calculations for a MOSFET Current mirror

12.9Common-source Amplifier Gain, Cut-off Frequencies and Bandwidth

CHAPTER TWELVE TRANSISTOR CIRCUITS EXAMPLE

DESCRIPTION

12.1 Input Characteristics of a BJT

© 1999 CRC Press LLC

CHAPTER ONE

MATLAB FUNDAMENTALS MATLAB is a numeric computation software for engineering and scientific calculations. The name MATLAB stands for MATRIX LABORATORY. MATLAB is primarily a tool for matrix computations. It was developed by John Little and Cleve Moler of MathWorks, Inc. MATLAB was originally written to provide easy access to the matrix computation software packages LINPACK and EISPACK. MATLAB is a high-level language whose basic data type is a matrix that does not require dimensioning. There is no compilation and linking as is done in high-level languages, such as C or FORTRAN. Computer solutions in MATLAB seem to be much quicker than those of a high-level language such as C or FORTRAN. All computations are performed in complex-valued dou- ble precision arithmetic to guarantee high accuracy. MATLAB has a rich set of plotting capabilities. The graphics are integrated in MATLAB. Since MATLAB is also a programming environment, a user can extend the functional capabilities of MATLAB by writing new modules. MATLAB has a large collection of toolboxes in a variety of domains. Some examples of MATLAB toolboxes are control system, signal processing, neural network, image processing, and system identification. The toolboxes consist of functions that can be used to perform computations in a specific domain.

1.1 MATLAB BASIC OPERATIONS When MATLAB is invoked, the command window will display the prompt >>. MATLAB is then ready for entering data or executing commands. To quit MATLAB, type the command

exit or quit MATLAB has on-line help. To see the list of MATLAB’s help facility, type

help The help command followed by a function name is used to obtain informa- tion on a specific MATLAB function. For example, to obtain information on the use of fast Fourier transform function, fft, one can type the command

© 1999 CRC Press LLC

help fft

The basic data object in MATLAB is a rectangular numerical matrix with real or complex elements. Scalars are thought of as a 1-by-1 matrix. Vectors are considered as matrices with a row or column. MATLAB has no dimension statement or type declarations. Storage of data and variables is allocated automatically once the data and variables are used. MATLAB statements are normally of the form:

variable = expression Expressions typed by the user are interpreted and immediately evaluated by the MATLAB system. If a MATLAB statement ends with a semicolon, MATLAB evaluates the statement but suppresses the display of the results. MATLAB is also capable of executing a number of commands that are stored in a file. This will be discussed in Section 1.6. A matrix

A =

1 2 3 2 3 4 3 4 5

  

  

may be entered as follows:

A = [1 2 3; 2 3 4; 3 4 5]; Note that the matrix entries must be surrounded by brackets [ ] with row elements separated by blanks or by commas. The end of each row, with the exception of the last row, is indicated by a semicolon. A matrix A can also be entered across three input lines as

A = [ 1 2 3 2 3 4 3 4 5];

In this case, the carriage returns replace the semicolons. A row vector B with four elements

B = [ 6 9 12 15 18 ] can be entered in MATLAB as

© 1999 CRC Press LLC

B = [6 9 12 15 18]; or

B = [6 , 9,12,15,18] For readability, it is better to use spaces rather than commas between the ele- ments. The row vector B can be turned into a column vector by transposition, which is obtained by typing

C = B’ The above results in

C = 6 9 12 15 18

Other ways of entering the column vector C are

C = [6 9 12 15 18]

or

C = [6; 9; 12; 15; 18] MATLAB is case sensitive in naming variables, commands and functions. Thus b and B are not the same variable. If you do not want MATLAB to be case sensitive, you can use the command

casesen off To obtain the size of a specific variable, type size ( ). For example, to find the size of matrix A, you can execute the following command:

size(A)

© 1999 CRC Press LLC

The result will be a row vector with two entries. The first is the number of rows in A, the second the number of columns in A. To find the list of variables that have been used in a MATLAB session, type the command

whos There will be a display of variable names and dimensions. Table 1.1 shows the display of the variables that have been used so far in this book:

Table 1.1 Display of an output of whos command

Name

Size Elements Byte Density Complex

A 3 by 3 9 72 Full No B 1 by 5 5 40 Full No C 5 by 1 5 40 Full No ans 1 by 2 2 16 Full No

The grand total is 21 elements using 168 bytes. Table 1.2 shows additional MATLAB commands to get one started on MATLAB. Detailed descriptions and usages of the commands can be obtained from the MATLAB help facility or from MATLAB manuals.

Table 1.2 Some Basic MATLAB Commands

Command Description % Comments. Everything appearing after % com-

mand is not executed. demo Access on-line demo programs length Length of a matrix clear Clears the variables or functions from workspace clc Clears the command window during a work session clg Clears graphic window diary Saves a session in a disk, possibly for printing at a

later date

© 1999 CRC Press LLC

1.2 MATRIX OPERATIONS The basic matrix operations are addition(+), subtraction(-), multiplication (*), and conjugate transpose(‘) of matrices. In addition to the above basic opera- tions, MATLAB has two forms of matrix division: the left inverse operator \ or the right inverse operator /. Matrices of the same dimension may be subtracted or added. Thus if E and F are entered in MATLAB as

E = [7 2 3; 4 3 6; 8 1 5]; F = [1 4 2; 6 7 5; 1 9 1];

and G = E - F H = E + F

then, matrices G and H will appear on the screen as

G = 6 -2 1 -2 -4 1 7 -8 4 H = 8 6 5 10 10 11 9 10 6

A scalar (1-by-1 matrix) may be added to or subtracted from a matrix. In this particular case, the scalar is added to or subtracted from all the elements of an- other matrix. For example,

J = H + 1 gives

J = 9 7 6 11 11 12 10 11 7

Matrix multiplication is defined provided the inner dimensions of the two op- erands are the same. Thus, if X is an n-by-m matrix and Y is i-by-j matrix,

© 1999 CRC Press LLC

X*Y is defined provided m is equal to i. Since E and F are 3-by-3 matrices, the product

Q = E*F results as

Q = 22 69 27 28 91 29 19 84 26

Any matrix can be multiplied by a scalar. For example,

2*Q gives

ans = 44 138 54 56 182 58 38 168 52

Note that if a variable name and the “=” sign are omitted, a variable name ans is automatically created. Matrix division can either be the left division operator \ or the right division operator /. The right division a/b, for instance, is algebraically equivalent to a b

while the left division a\b is algebraically equivalent to b a

.

If Z I V* = and Z is non-singular, the left division, Z\V is equivalent to MATLAB expression I inv Z V= ( ) * where inv is the MATLAB function for obtaining the inverse of a matrix. The right division denoted by V/Z is equivalent to the MATLAB expression

I V inv Z= * ( ) There are MATLAB functions that can be used to produce special matrices. Examples are given in Table 1.3.

© 1999 CRC Press LLC

Table 1.3 Some Utility Matrices

FunctionDescriptionones(n,m) Produces n-by-m matrix with all the elements being

unity eye(n) gives n-by-n identity matrix zeros(n,m) Produces n-by-m matrix of zeros diag(A) Produce a vector consisting of diagonal of a square

matrix A

1.3 ARRAY OPERATIONS Array operations refer to element-by-element arithmetic operations. Preceding the linear algebraic matrix operations, * / \ ‘ , by a period (.) indicates an array or element-by-element operation. Thus, the operators .* , .\ , ./, .^ , represent element-by-element multiplication, left division, right division, and raising to the power, respectively. For addition and subtraction, the array and matrix op- erations are the same. Thus, + and .+ can be regarded as an array or matrix addition. If A1 and B1 are matrices of the same dimensions, then A1.*B1 denotes an ar- ray whose elements are products of the corresponding elements of A1 and B1. Thus, if

A1 = [2 7 6 8 9 10]; B1 = [6 4 3 2 3 4];

then C1 = A1.*B1 results in

C1 = 12 28 18 16 27 40

© 1999 CRC Press LLC

An array operation for left and right division also involves element-by-element operation. The expressions A1./B1 and A1.\B1 give the quotient of element- by-element division of matrices A1 and B1. The statement

D1 = A1./B1 gives the result

D1 = 0.3333 1.7500 2.0000 4.0000 3.0000 2.5000

and the statement

E1 = A1.\B1 gives

E1 = 3.0000 0.5714 0.5000 0.2500 0.3333 0.4000

The array operation of raising to the power is denoted by .^. The general statement will be of the form:

q = r1.^s1 If r1 and s1 are matrices of the same dimensions, then the result q is also a ma- trix of the same dimensions. For example, if

r1 = [ 7 3 5]; s1 = [ 2 4 3];

then

q1 = r1.^s1 gives the result

q1 = 49 81 125

© 1999 CRC Press LLC

One of the operands can be scalar. For example,

q2 = r1.^2 q3 = (2).^s1

will give

q2 = 49 9 25

and

q3 = 4 16 8

Note that when one of the operands is scalar, the resulting matrix will have the same dimensions as the matrix operand.

1.4 COMPLEX NUMBERS MATLAB allows operations involving complex numbers. Complex numbers are entered using function i or j. For example, a number z j= +2 2 may be entered in MATLAB as

z = 2+2*i or

z = 2+2*j Also, a complex number za

za j= 2 2 4exp[( / ) ]π can be entered in MATLAB as

za = 2*sqrt(2)*exp((pi/4)*j) It should be noted that when complex numbers are entered as matrix elements within brackets, one should avoid any blank spaces. For example, y j= +3 4 is represented in MATLAB as

© 1999 CRC Press LLC

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