Mathematical Foundations for Linear Circuits and Systems in Engineering E-Book

John J. Shynk

• Herausgeber: John Wiley & Sons
• Kategorie: Wissenschaft und neue Technologien
• Sprache: Englisch

Seitenzahl: 346

Veröffentlichungsjahr: 2016

Beschreibung

Extensive coverage of mathematical techniques used in engineering with an emphasis on applications in linear circuits and systems

Mathematical Foundations for Linear Circuits and Systems in Engineering provides an integrated approach to learning the necessary mathematics specifically used to describe and analyze linear circuits and systems. The chapters develop and examine several mathematical models consisting of one or more equations used in engineering to represent various physical systems. The techniques are discussed in-depth so that the reader has a better understanding of how and why these methods work. Specific topics covered include complex variables, linear equations and matrices, various types of signals, solutions of differential equations, convolution, filter designs, and the widely used Laplace and Fourier transforms. The book also presents a discussion of some mechanical systems that mathematically exhibit the same dynamic properties as electrical circuits. Extensive summaries of important functions and their transforms, set theory, series expansions, various identities, and the Lambert W-function are provided in the appendices.

The book has the following features:

• Compares linear circuits and mechanical systems that are modeled by similar ordinary differential equations, in order to provide an intuitive understanding of different types of linear time-invariant systems.
• Introduces the theory of generalized functions, which are defined by their behavior under an integral, and describes several properties including derivatives and their Laplace and Fourier transforms.
• Contains numerous tables and figures that summarize useful mathematical expressions and example results for specific circuits and systems, which reinforce the material and illustrate subtle points.
• Provides access to a companion website that includes a solutions manual with MATLAB code for the end-of-chapter problems.

Mathematical Foundations for Linear Circuits and Systems in Engineering is written for upper undergraduate and first-year graduate students in the fields of electrical and mechanical engineering. This book is also a reference for electrical, mechanical, and computer engineers as well as applied mathematicians.

John J. Shynk, PhD, is Professor of Electrical and Computer Engineering at the University of California, Santa Barbara. He was a Member of Technical Staff at Bell Laboratories, and received degrees in systems engineering, electrical engineering, and statistics from Boston University and Stanford University.

Leseprobe

Table of Contents

COVER

TITLE PAGE

COPYRIGHT

DEDICATION

PREFACE

NOTATION AND BIBLIOGRAPHY

ABOUT THE COMPANION WEBSITE

CHAPTER 1: OVERVIEW AND BACKGROUND

1.1 INTRODUCTION

1.2 MATHEMATICAL MODELS

1.3 FREQUENCY CONTENT

1.4 FUNCTIONS AND PROPERTIES

1.5 DERIVATIVES AND INTEGRALS

1.6 SINE, COSINE, AND π

1.7 NAPIER'S CONSTANT AND LOGARITHMS

PROBLEMS

PART I: CIRCUITS, MATRICES, AND COMPLEX NUMBERS

CHAPTER 2: CIRCUITS AND MECHANICAL SYSTEMS

2.1 INTRODUCTION

2.2 VOLTAGE, CURRENT, AND POWER

2.3 CIRCUIT ELEMENTS

2.4 BASIC CIRCUIT LAWS

2.5 MECHANICAL SYSTEMS

PROBLEMS

CHAPTER 3: LINEAR EQUATIONS AND MATRICES

3.1 INTRODUCTION

3.2 VECTOR SPACES

3.3 SYSTEM OF LINEAR EQUATIONS

3.4 MATRIX PROPERTIES AND SPECIAL MATRICES

3.5 DETERMINANT

3.6 MATRIX SUBSPACES

3.7 GAUSSIAN ELIMINATION

3.8 EIGENDECOMPOSITION

3.9 MATLAB FUNCTIONS

PROBLEMS

CHAPTER 4: COMPLEX NUMBERS AND FUNCTIONS

4.1 INTRODUCTION

4.2 IMAGINARY NUMBERS

4.3 COMPLEX NUMBERS

4.4 TWO COORDINATES

4.5 POLAR COORDINATES

4.6 EULER'S FORMULA

4.7 MATRIX REPRESENTATION

4.8 COMPLEX EXPONENTIAL ROTATION

4.9 CONSTANT ANGULAR VELOCITY

4.10 QUATERNIONS

PROBLEMS

PART II: SIGNALS, SYSTEMS, AND TRANSFORMS

CHAPTER 5: SIGNALS, GENERALIZED FUNCTIONS, AND FOURIER SERIES

5.1 INTRODUCTION

5.2 ENERGY AND POWER SIGNALS

5.3 STEP AND RAMP FUNCTIONS

5.4 RECTANGLE AND TRIANGLE FUNCTIONS

5.5 EXPONENTIAL FUNCTION

5.6 SINUSOIDAL FUNCTIONS

5.7 DIRAC DELTA FUNCTION

5.8 GENERALIZED FUNCTIONS

5.9 UNIT DOUBLET

5.10 COMPLEX FUNCTIONS AND SINGULARITIES

5.11 CAUCHY PRINCIPAL VALUE

5.12 EVEN AND ODD FUNCTIONS

5.13 CORRELATION FUNCTIONS

5.14 FOURIER SERIES

5.15 PHASOR REPRESENTATION

5.16 PHASORS AND LINEAR CIRCUITS

PROBLEMS

CHAPTER 6: DIFFERENTIAL EQUATION MODELS FOR LINEAR SYSTEMS

6.1 INTRODUCTION

6.2 DIFFERENTIAL EQUATIONS

6.3 GENERAL FORMS OF THE SOLUTION

6.4 FIRST-ORDER LINEAR ODE

6.5 SECOND-ORDER LINEAR ODE

6.6 SECOND-ORDER ODE RESPONSES

6.7 CONVOLUTION

6.8 SYSTEM OF ODE

PROBLEMS

CHAPTER 7: LAPLACE TRANSFORMS AND LINEAR SYSTEMS

7.1 INTRODUCTION

7.2 SOLVING ODE USING PHASORS

7.3 EIGENFUNCTIONS

7.4 LAPLACE TRANSFORM

7.5 LAPLACE TRANSFORMS AND GENERALIZED FUNCTIONS

7.6 LAPLACE TRANSFORM PROPERTIES

7.7 INITIAL AND FINAL VALUE THEOREMS

7.8 POLES AND ZEROS

7.9 LAPLACE TRANSFORM PAIRS

7.10 TRANSFORMS AND POLYNOMIALS

7.11 SOLVING LINEAR ODE

7.12 IMPULSE RESPONSE AND TRANSFER FUNCTION

7.13 PARTIAL FRACTION EXPANSION

7.14 LAPLACE TRANSFORMS AND LINEAR CIRCUITS

PROBLEMS

CHAPTER 8: FOURIER TRANSFORMS AND FREQUENCY RESPONSES

8.1 INTRODUCTION

8.2 FOURIER TRANSFORM

8.3 MAGNITUDE AND PHASE

8.4 FOURIER TRANSFORMS AND GENERALIZED FUNCTIONS

8.5 FOURIER TRANSFORM PROPERTIES

8.6 AMPLITUDE MODULATION

8.7 FREQUENCY RESPONSE

8.8 FREQUENCY RESPONSE OF SECOND-ORDER FILTERS

8.9 FREQUENCY RESPONSE OF SERIES RLC CIRCUIT

8.10 BUTTERWORTH FILTERS

PROBLEMS

CHAPTER: APPENDICES

INTRODUCTION TO APPENDICES

APPENDIX A: EXTENDED SUMMARIES OF FUNCTIONS AND TRANSFORMS

A.1 FUNCTIONS AND NOTATION

A.2 LAPLACE TRANSFORM

A.3 FOURIER TRANSFORM

A.4 MAGNITUDE AND PHASE

A.5 IMPULSIVE FUNCTIONS

A.6 PIECEWISE LINEAR FUNCTIONS

A.7 EXPONENTIAL FUNCTIONS

A.8 SINUSOIDAL FUNCTIONS

APPENDIX B: INVERSE LAPLACE TRANSFORMS

B.1 IMPROPER RATIONAL FUNCTION

B.2 UNBOUNDED SYSTEM

B.3 DOUBLE INTEGRATOR AND FEEDBACK

APPENDIX C: IDENTITIES, DERIVATIVES, AND INTEGRALS

C.1 TRIGONOMETRIC IDENTITIES

C.2 SUMMATIONS

C.3 MISCELLANEOUS

C.4 COMPLETING THE SQUARE

C.5 QUADRATIC AND CUBIC FORMULAS

C.6 DERIVATIVES

C.7 INDEFINITE INTEGRALS

C.8 DEFINITE INTEGRALS

APPENDIX D: SET THEORY

D.1 SETS AND SUBSETS

D.2 SET OPERATIONS

APPENDIX E: SERIES EXPANSIONS

E.1 TAYLOR SERIES

E.2 MACLAURIN SERIES

E.3 LAURENT SERIES

APPENDIX F: LAMBERT W-FUNCTION

F.1 LAMBERT W-FUNCTION

F.2 NONLINEAR DIODE CIRCUIT

F.3 SYSTEM OF NONLINEAR EQUATIONS

GLOSSARY

BIBLIOGRAPHY

INDEX

END USER LICENSE AGREEMENT

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Guide

Cover

Table of Contents

Preface

Begin Reading

List of Illustrations

CHAPTER 1: OVERVIEW AND BACKGROUND

Figure 1.1 Systems with a single input and a single output (SISO). (a) General system with input and output . (b) Linear system with sinusoidal input and output.

Figure 1.2 Input/output characteristic for the nonlinear system in Example 1.1.

Figure 1.3 Output for the transfer characteristic in (1.1) in Example 1.1 with input for .

Figure 1.4 Output for the transfer characteristics in (1.2) and (1.4) in Example 1.2 with input for .

Figure 1.5 Multiple-input and multiple-output (MIMO) system.

Figure 1.6 Systems of equations. (a) Nonlinear system in (1.13) and (1.14) in Example 1.4 with . (b) Linear system in (1.5) and (1.6) in Example 1.3.

Figure 1.7 Integrator implementation of a second-order linear ODE.

Figure 1.8 Solutions for the second-order ODE in Example 1.5 with constant coefficients. The input is and the initial conditions are nonzero: .

Figure 1.9 Periodic rectangular waveform. (a) Time-domain representation. (b) Magnitude of frequency spectrum: Fourier series with harmonics and Hz.

Figure 1.10 Aperiodic waveforms. (a) Time-domain representation. (b) Magnitude of frequency spectrum: Fourier transform.

Figure 1.11 Two-dimensional image and spectrum. (a) Spatial representation. (b) Magnitude of frequency spectrum in two dimensions. White denotes a greater magnitude. (The vertical and horizontal white lines are the frequency axes where and . A log scale is used to better visualize variations in the spectrum.)

Figure 1.12 Device models used in Example 1.9. (a) Linear model for resistor . (b) Nonlinear model for diode with resistance .

Figure 1.13 Example of a function with a discontinuity at .

Figure 1.14 (a) Function with two pole singularities at . (b) Function with a removable pole singularity at .

Figure 1.15 Function with an essential singularity at .

Figure 1.16 Finite approximation of the derivative of at .

Figure 1.17 Vehicle position, velocity, and acceleration waveforms used in Example 1.14.

Figure 1.18 Cubic function in Example 1.15 and its derivatives.

Figure 1.19 Three functions in Example 1.17 with singularities at .

Figure 1.20 Lower and upper Riemann sums approximating the integral of on .

Figure 1.21 Unit circle with radius and circumference .