Circuit Analysis:A Systems Approach

Russell M. MersereauGeorgia Institute of Technology

Joel R. JacksonGeorgia Institute of Technology

PEARSON

PrenticeMall

Upper Saddle River, NJ 07458

Contents

FOREWORD xiii

PREFACE xv1 Circuit Elements and Models 1

1 -1 Introduction 11 -2 Network Elements 2

-2-1 Sign Conventions for Defining Element Variables 4-2-2 Resistors 5-2-3 Inductors 8-2-4 Capacitors 9-2-5 Short Circuits 10-2-6 Open Circuits 11-2-7 Switches 11

1-3 Independent Sources 111-4 Kirchhoff s Laws 12

1 -4-1 Kirchhoff's Current Law (KCL) 131-4-2 Kirchhoff's Voltage Law (KVL) 171-4-3 Solving for the Element Variables in a Circuit 21

A Confusing Issue: Selecting Reference Directions 221-4-4 Elements Connected in Series and Parallel 251-4-5 Circuits with Inductors and/or Capacitors 28

1-5 Dependent Sources 291-6 The Role of Models 331-7 Chapter Summary 34

1-7-1 Important Points Introduced 34

VI

1-7-2 New Abilities Acquired 351-7-3 Links 35

1-8 Problems 361-8-1 Drill Problems 361-8-2 Basic Problems 391-8-3 Advanced Problems 451-8-4 Design Problems 50

2 Writing Circuit Equations 522-1 The Exhaustive Method for Writing a Sufficient Set of Circuit Equations 53

2-1-1 Proof of Sufficiency of the Method* 532-1-2 Examples of the Method 55

2-2 Supernodes and Supermeshes 582-2-1 Supernodes 582-2-2 Supermeshes 592-2-3 Dependent Sources 62

2-3 Solving Circuit Equations 642-3-1 Matrices 642-3-2 Matrix Operations 642-3-3 Representing Linear Equations in MATLAB* 662-3-4 Matrix Descriptions of Resistive Circuits 672-3-5 Superposition of Independent Sources 69

2-4 The Node Method 722-5 The Mesh Method 782-6 Conservation of Power 83

A Confusing Issue: Proper Treatment of Sources 832-7 Chapter Summary 87

2-7-1 Important Points Introduced 872-7-2 New Abilities Acquired 87

2-8 Problems 882-8-1 Drill Problems 882-8-2 Basic Problems 922-8-3 Advanced Problems 1022-8-4 Design Problems 104

3 Subnetworks 1063-1 Resistor-Only Subnetworks 107

3-1-1 Resistors in Series 107

C o n t e n t s Contents

3-1-2 Resistors in Parallel 109A Confusing Issue: Identifying Series and Parallel Elements 110

3-2 The Vi Characteristics of Two-Terminal Networks 1163-3 Thevenin Equivalent Networks 120

3-3-1 Thevenin's Theorem for Resistive Networks 1203-3-2 Norton's Theorem for Resistive Networks 1253-3-3 Other Equivalence Relations 127

3-4 Selecting an Analysis Method 1283-5 Graphical Analysis* 1333-6 Chapter Summary 135

3-6-1 Important Points Introduced 1353-6-2 New Abilities Acquired 136

3-7 Problems 1363-7-1 Drill Problems 1363-7-2 Basic Problems 1383-7-3 Advanced Problems 1443-7-4 Design Problems 149

4 Operational Amplifiers 1504-1 The Ideal Opamp 1504-2 The Node Method for Opamp Circuits 154

A Confusing Issue: Writing KCL Equations at the Input Nodes 1554-3 Additional Examples of Operational-Amplifier Circuits 156

4-3-1 Noninverting Amplifier 1564-3-2 Difference Amplifier 1574-3-3 Summing Amplifier 158

A Confusing Issue: Opamp Output Current 1594-3-4 Buffer Amplifiers and Loading 1604-3-5 Voltage-Controlled Current Source 1614-3-6 A "Negative Resistor" 1624-3-7 Integrator 1634-3-8 Differentiator 164

4-4 Chapter Summary 1644-4-1 Important Points Introduced 1644-4-2 New Abilities Acquired 165

4-5 Problems 1654-5-1 Drill Problem 1654-5-2 Basic Problems 165

viii Contents

4-5-3 Advanced Problems 1704-5-4 Design Problems 175

5 Laplace Transforms 1765-1 Some Basic Signals 178

5-1-1 Sinusoids 1785-1-2 Exponentially Weighted Sinusoids 1805-1-3 Switched-Exponential Signals 182

5-2 Definition of the Laplace Transform 1835-3 Some Properties of the Laplace Transform 1855-4 Inverse Laplace Transforms 188

5-4-1 Case 1: More Poles than Zeros 1895-4-2 Case 2: Fewer Poles than Zeros (m > n) 192

A Confusing Issue: Using Limits to Compute Residues 1975-4-3 Case 3: Repeated Roots 198

5-5 Chapter Summary 1995-5-1 Important Points Introduced 1995-5-2 New Abilities Acquired 199

5-6 Problems 1995-6-1 Drill Problems 1995-6-2 Basic Problems 2005-6-3 Advanced Problems 202

6 Circuits in the Laplace Domain 2046-1 Circuits with One Reactive Element 205

6-1-1 Differential-Equation Descriptions 2056-1 -2 Solving Differential Equations with Exponential Inputs 2086-1-3 Solving Differential Equations by Using Laplace Transforms 211

6-2 Circuits in the Laplace Domain 2136-2-1 KCL in the Laplace Domain 2146-2-2 KVL in the Laplace Domain 2156-2-3 Element Relations in the Laplace Domain 215

A Confusing Issue: Modelling Initial Capacitor Voltages and Inductor Currents 2236-3 Inspection Methods for First-Order Circuits with Constant Inputs 225

6-3-1 Circuit Behavior at / = oo 2256-3-2 Circuit Behavior at t = 0 2266-3-3 Calculating the Time Constant x 227

6-4 Impedances and Admittances 229

Contents

6-4-1 Impedances Connected in Series and Parallel 2306-4-2 Thevenin and Norton Equivalent Circuits* 232

6-5 Systematic Analysis Methods 2346-5-1 A Set of Sufficient Equations for Analyzing a Circuit* 2346-5-2 Node and Mesh Methods in the Laplace Domain 2376-5-3 Operational Amplifiers in the Laplace Domain 239

6-6 Chapter Summary 2406-6-1 Important Points Introduced 2406-6-2 New Abilities Acquired 241

6-7 Problems 2416-7-1 Drill Problems 2416-7-2 Basic Problems 2446-7-3 Advanced Problems 2496-7-4 Design Problems 252

7 System Functions 2537-1 Circuits as Systems 254

7-1-1 The Input-Output Point of View 2547-1-2 The Complete Solution of a Circuit 2557-1-3 Circuits at Initial Rest 2597-1-4 Impulse Responses, Poles, and Zeros 2607-1-5 The Unit-Step Response 264

7-2 Linearity and Time Invariance 2667-2-1 Circuits at Initial Rest as Linear Systems 2667-2-2 Circuits at Initial Rest as Time-Invariant Systems 2677-2-3 Exploiting Linearity and Time Invariance 268

7-3 Responses to Switched-Exponential Inputs 2727-3-1 The General Case 2737-3-2 Impulse Responses of First- and Second-Order Systems 274

7-4 Two Additional Circuit Descriptions* 2777-4_ i Differential-Equation Characterization at Initial Rest 2777-4-2 Impulse-Response Characterization 281

7-5 Chapter Summary 2847-5-1 Important Points Introduced 2847-5-2 New Abilities Acquired 285

7-6 Problems 2857-6-1 Drill Problems 2857-6-2 Basic Problems 286

Contents

7-6-3 Advanced Problems 2917-6-4 Design Problems 294

8 Sinusoidal Input Signals 2978-1 The Sinusoidal Steady State 298

8-1-1 Sinusoidal Signals 2988-1-2 Physical Circuits and Mathematical Models 2988-1-3 Responses to Real Sinusoidal Inputs 2998-1-4 Responses to Unswitched Signals 303

8-2 Analyzing Circuits by Using Phasors 3048-2-1 A Simple Circuit with a Complex Exponential Input 3048-2-2 The General Case 3088-2-3 Networks with Multiple Sources 316

8-3 Frequency-Domain Circuit Simplifications 3198-3-1 Series and Parallel Connections 3198-3-2 Thevenin and Norton Equivalent Circuits 3218-3-3 The Node and Mesh Methods in the Phasor Domain 3238-3-4 Operational-Amplifier Circuits 329

8-4 Power in the Phasor Domain 3308-4-1 Average Power 3308-4-2 RMS Values 3318-4-3 Power in Circuits with Sinusoidal Excitations 332

8-5 Maximum Power Transfer 3368-5-1 Resistive Circuits 3368-5-2 General Circuits 337

8-6 Chapter Summary 3398-6-1 Important Points Introduced 3398-6-2 New Abilities Acquired 340

8-7 Problems 3408-7-1 Drill Problems 3408-7-2 Basic Problems 3428-7-3 Advanced Problems 3478-7-4 Design Problems 349

9 Frequency Responses of Circuits 3519-1 Spectrum Representations of Periodic Signals 352

9-1-1 Spectrum of a Sum of Sinusoids 3529-1-2 Periodic Waveforms 354

Contents xj

9-1-3 Fourier-Series Representations 3559-1-4 Response of a Circuit to a Periodic Input 358

9-2 Spectrum Representations for Aperiodic Signals 3629-2-1 The Fourier Transform 3629-2-2 Circuit Analysis by Using Fourier Transforms 363

9-3 The System Function and the Frequency Response 3669-3-1 Graphical Interpretation of Frequency Response 3669-3-2 Resonance 371

9-4 Bode Plots 3759-4-1 First-Order Systems 3769-4-2 Second-Order Systems 3789-4-3 Multifactor Bode Plots 379

A Confusing Issue: Normalized Forms for System Functions 3829-5 Chapter Summary 385

9-5-1 Important Points Introduced 3859-5-2 New Abilities Acquired 386

9-6 Problems 3869-6-1 Drill Problems 3869-6-2 Basic Problems 3879-6-3 Advanced Problems 3939-6-4 Design Problems 396

10 Filter Circuits 39810-1 Lowpass Filters 399

10-1-1 Ideal vs. Buildable Filters 40010-1-2 Butterworth and Chebyshev Lowpass Filters 40110-1-3 Realizations of Lowpass Second-Order Sections That Use Operational Amplifiers 406

10-2 Transforming Basic Filters 40810-2-1 Frequency Scaling 40910-2-2 Lowpass-to-Highpass Transformations 41010-2-3 Lowpass-to-Bandpass Transformations 41210-2-4 Impedance Scaling 414

10-3 Chapter Summary 41510-3-1 Important Points Introduced 41510-3-2 New Abilities Acquired 415

10-4 Problems 41510-4-1 Drill Problems 41510-4-2 Basic Problems 416

xii Contents

10-4-3 Advanced Problems 41910-4-4 Design Problems 421

A Review of Complex Numbers 423A-l Definitions 423A-2 Graphical Interpretation of Complex Numbers 424A-3 Complex Exponentials 426A-4 The Complex Conjugate 429A-5 Drill Problems 429A-6 Basic Problems 431

B Answers to Selected Basic Problems 433INDEX 439