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Mathematical Foundations for Linear Circuits and Systems in Engineering

ISBN: 978-1-119-07343-7
656 pages
March 2016
Mathematical Foundations for Linear Circuits and Systems in Engineering (111907343X) cover image

Description

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.

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Table of Contents

Preface xiii

Notation and Bibliography xvii

About the Companion Website xix

1 Overview and Background 1

1.1 Introduction 1

1.2 Mathematical Models 3

1.3 Frequency Content 12

1.4 Functions and Properties 16

1.5 Derivatives and Integrals 22

1.6 Sine, Cosine, and 𝜋 33

1.7 Napier’s Constant e and Logarithms 38

PART I CIRCUITS, MATRICES, AND COMPLEX NUMBERS 51

2 Circuits and Mechanical Systems 53

2.1 Introduction 53

2.2 Voltage, Current, and Power 54

2.3 Circuit Elements 60

2.4 Basic Circuit Laws 67

2.4.1 Mesh-Current and Node-Voltage Analysis 69

2.4.2 Equivalent Resistive Circuits 71

2.4.3 RC and RL Circuits 75

2.4.4 Series RLC Circuit 78

2.4.5 Diode Circuits 82

2.5 Mechanical Systems 85

2.5.1 Simple Pendulum 86

2.5.2 Mass on a Spring 92

2.5.3 Electrical and Mechanical Analogs 95

3 Linear Equations and Matrices 105

3.1 Introduction 105

3.2 Vector Spaces 106

3.3 System of Linear Equations 108

3.4 Matrix Properties and Special Matrices 113

3.5 Determinant 122

3.6 Matrix Subspaces 128

3.7 Gaussian Elimination 135

3.7.1 LU and LDU Decompositions 146

3.7.2 Basis Vectors 148

3.7.3 General Solution of 𝐀𝐲 = 𝐱 151

3.8 Eigendecomposition 152

3.9 MATLAB Functions 156

4 Complex Numbers and Functions 163

4.1 Introduction 163

4.2 Imaginary Numbers 165

4.3 Complex Numbers 167

4.4 Two Coordinates 169

4.5 Polar Coordinates 171

4.6 Euler’s Formula 175

4.7 Matrix Representation 182

4.8 Complex Exponential Rotation 183

4.9 Constant Angular Velocity 189

4.10 Quaternions 192

PART II SIGNALS, SYSTEMS, AND TRANSFORMS 203

5 Signals, Generalized Functions, and Fourier Series 205

5.1 Introduction 205

5.2 Energy and Power Signals 206

5.3 Step and Ramp Functions 208

5.4 Rectangle and Triangle Functions 211

5.5 Exponential Function 214

5.6 Sinusoidal Functions 217

5.7 Dirac Delta Function 220

5.8 Generalized Functions 223

5.9 Unit Doublet 233

5.10 Complex Functions and Singularities 240

5.11 Cauchy Principal Value 242

5.12 Even and Odd Functions 245

5.13 Correlation Functions 248

5.14 Fourier Series 251

5.15 Phasor Representation 261

5.16 Phasors and Linear Circuits 265

6 Differential Equation Models for Linear Systems 275

6.1 Introduction 275

6.2 Differential Equations 276

6.3 General Forms of The Solution 278

6.4 First-Order Linear ODE 280

6.4.1 Homogeneous Solution 283

6.4.2 Nonhomogeneous Solution 285

6.4.3 Step Response 287

6.4.4 Exponential Input 287

6.4.5 Sinusoidal Input 289

6.4.6 Impulse Response 290

6.5 Second-Order Linear ODE 294

6.5.1 Homogeneous Solution 296

6.5.2 Damping Ratio 304

6.5.3 Initial Conditions 306

6.5.4 Nonhomogeneous Solution 307

6.6 Second-Order ODE Responses 311

6.6.1 Step Response 311

6.6.2 Step Response (Alternative Method) 313

6.6.3 Impulse Response 319

6.7 Convolution 319

6.8 System of ODEs 323

7 Laplace Transforms and Linear Systems 335

7.1 Introduction 335

7.2 Solving ODEs Using Phasors 336

7.3 Eigenfunctions 339

7.4 Laplace Transform 340

7.5 Laplace Transforms and Generalized Functions 347

7.6 Laplace Transform Properties 352

7.7 Initial and Final Value Theorems 364

7.8 Poles and Zeros 367

7.9 Laplace Transform Pairs 372

7.9.1 Constant Function 372

7.9.2 Rectangle Function 373

7.9.3 Triangle Function 374

7.9.4 Ramped Exponential Function 376

7.9.5 Sinusoidal Functions 376

7.10 Transforms and Polynomials 377

7.11 Solving Linear ODEs 380

7.12 Impulse Response and Transfer Function 382

7.13 Partial Fraction Expansion 387

7.13.1 Distinct Real Poles 388

7.13.2 Distinct Complex Poles 391

7.13.3 Repeated Real Poles 396

7.13.4 Repeated Complex Poles 402

7.14 Laplace Transforms and Linear Circuits 409

8 Fourier Transforms and Frequency Responses 423

8.1 Introduction 423

8.2 Fourier Transform 425

8.3 Magnitude and Phase 435

8.4 Fourier Transforms and Generalized Functions 437

8.5 Fourier Transform Properties 442

8.6 Amplitude Modulation 449

8.7 Frequency Response 453

8.7.1 First-Order Low-Pass Filter 455

8.7.2 First-Order High-Pass Filter 459

8.7.3 Second-Order Band-Pass Filter 460

8.7.4 Second-Order Band-Reject Filter 463

8.8 Frequency Response of Second-Order Filters 466

8.9 Frequency Response of Series RLC Circuit 475

8.10 Butterworth Filters 478

8.10.1 Low-Pass Filter 481

8.10.2 High-Pass Filter 484

8.10.3 Band-Pass Filter 487

8.10.4 Band-Reject Filter 490

APPENDICES 499

Introduction to Appendices 500

A Extended Summaries of Functions and Transforms 501

A.1 Functions and Notation 501

A.2 Laplace Transform 502

A.3 Fourier Transform 504

A.4 Magnitude and Phase 506

A.5 Impulsive Functions 511

A.5.1 Dirac Delta Function (Shifted) 511

A.5.2 Unit Doublet (Shifted) 514

A.6 Piecewise Linear Functions 514

A.6.1 Unit Step Function 514

A.6.2 Signum Function 517

A.6.3 Constant Function (Two-Sided) 517

A.6.4 Ramp Function 521

A.6.5 Absolute Value Function (Two-Sided Ramp) 523

A.6.6 Rectangle Function 524

A.6.7 Triangle Function 528

A.7 Exponential Functions 529

A.7.1 Exponential Function (Right-Sided) 529

A.7.2 Exponential Function (Ramped) 531

A.7.3 Exponential Function (Two-Sided) 533

A.7.4 Gaussian Function 537

A.8 Sinusoidal Functions 539

A.8.1 Cosine Function (Two-Sided) 539

A.8.2 Cosine Function (Right-Sided) 541

A.8.3 Cosine Function (ExponentiallyWeighted) 544

A.8.4 Cosine Function (ExponentiallyWeighted and Ramped) 544

A.8.5 Sine Function (Two-Sided) 549

A.8.6 Sine Function (Right-Sided) 550

A.8.7 Sine Function (ExponentiallyWeighted) 553

A.8.8 Sine Function (ExponentiallyWeighted and Ramped) 554

B Inverse Laplace Transforms 559

B.1 Improper Rational Function 559

B.2 Unbounded System 562

B.3 Double Integrator and Feedback 563

C Identities, Derivatives, and Integrals 565

C.1 Trigonometric Identities 565

C.2 Summations 566

C.3 Miscellaneous 567

C.4 Completing the Square 567

C.5 Quadratic and Cubic Formulas 568

C.6 Derivatives 571

C.7 Indefinite Integrals 573

C.8 Definite Integrals 574

D Set Theory 577

D.1 Sets and Subsets 577

D.2 Set Operations 579

E Series Expansions 583

E.1 Taylor Series 583

E.2 Maclaurin Series 585

E.3 Laurent Series 588

F Lambert W-Function 593

F.1 LambertW-Function 593

F.2 Nonlinear Diode Circuit 597

F.3 System of Nonlinear Equations 598

Glossary 601

Bibliography 609

Index 615

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