I I

LINEAR

CIRCUIT ANALYSIS TIME DOMAIN, PHASOR, AND LAPLACE TRANSFORM APPROACHES

S E C O N D E D I T I O N

Raymond A. DeCarlo Purdue University

Pen-Min Lin Purdue University

New York Oxford OXFORD UNIVERSITY PRESS 2001

Contents

Preface xiii

Chapter 1

1. Role and Importance of Circuits in Engineering 3

2. Fields, Charge, and Current 5 Fields, 5 Charge, 5 Current, 6

3. Voltage 11 4. Energy Conversion in an Electric

Circuit 15 5. Relationships among Voltage, Current,

Power, and Energy 17 Power and Energy for Direct Voltages and Currents, 17 Non-DC Power and Energy Calculations, 20

6. Ideal Voltage and Current Sources 22

7. Resistance, Ohm's Law, and Power (a Reprise)

8. Additional Concepts: Characteristic, Memoryless, Model, and Lumped V-I Characteristic ofConstant Voltage and Current Sources, 31 Notion ofa Memoryless Device, 32 Notion of Model, 33 Frequency, Wavelength, and the Notion ofa Lumped Circuit Element, 33 Summary Terms and Concepts Problems

Chapter 2 KIRCHHOFF'S CURRENT AND VOLTAGE LAWS AND SERIES-PARALLEL

RESISTIVE CIRCUITS

Introduction 44 Terminology: Parallel, Series, Node, Branch, and so on 44 Kirchhoff's Current Law 46 Kirchhoff's Voltage Law 49 Equivalent Resistance, Series Resistances, and Voltage Division 53 Parallel Resistances and Current Division 55

10.

Series-Parallel Interconnections Dependent Sources Revisited Model for a Nonideal Battery and Battery Capacity Nonideal Sources Summary Terms and Concepts Problems

Chapter 3 NODAL AND LOOP ANALYSES

I • Introduction, Review, and Terminology 2. Concepts of Nodal and Loop Analysis 3. Nodal Analysis I: Grounded Voltage

Sources 4. Nodal Analysis II: Floating Voltage

Sources 5. Loop Analysis

*6. Modified Nodal Analysis

THE OPERATIONAL AMPLIFIER

88 *7. Some Theoretical Foundations 89 Planar and Nonplanar Circuit Graphs,

116 Meshes and Loops for Nonplanar 90 Circuits, 116 Number of Independent

KCL and KVL Equations, 118 98 Summary

102 Terms and Concepts 111 Problems

Chapter 4 1. Introduction 2. The Ideal Operational Amplifier 3. Design of General Summing

Amplifiers

134 Design Choicesfor the General Summing 135 Circuit, 142 Derivation

ofOp Amp Input-Output Characteristic, 142 145

viii CONTENTS

4. Saturation and the Active Region of the Op Amp 146

5. Op Amp Circuit for Digital-to-Analog Conversion 151 Elements ofA/D and D/A Conversion, 151

Binary-Weighted Summing Circuit, 152 Summary 154 Terms and Concepts 154 Problems 155

Chapter 5 LINEARITY, SUPERPOSITION, AND SOURCE TRANSFORMATIONS

1. Introduction 2. Linearity 3. Superposition and Proportionality 4. Source Transformations

167 *5. Modified Superposition Analysis 168 Summary 175 Terms and Concepts 181 Problems

166

188 192 193 193

Chapter 6 THEVENIN, NORTON, AND MAXIMUM POWER TRANSFER THEOREMS

1. Introduction 201 2. Thevenin and Norton Equivalent

Circuits for Passive Networks 202 3. Thevenin and Norton Equivalent

Circuits for Active Networks 208 4. Thevenin and Norton Equivalents for

Op Amp Circuits 214 Thevenin and Norton Equivalent

Circuits from Measured Data Theoretical Considerations: Pathological Cases and a Proof Maximum Power Transfer Theorem Summary Terms and Concepts Problems

200

218

220 224 230 230 231

Chapter 7 INDUCTORS, CAPACITORS, AND DUALITY 242

1. Introduction 2. The Inductor

Some Physics, 244 Definition and Basic Examples, 246

3. The Capacitor Definitions and Properties, 254 Relationship of Charge to Capacitor Voltage and Current, 257 Principle of Conservation of Charge, 257 Energy Storage in a Capacitor, 259 Capacitance and Dielectrics, 261

4. Series and Parallel Inductors and Capacitors Inductors in Series, 261 Inductors in

244 244

254

261

Parallel, 263 Series-Parallel Inductor Combinations, 264 Capacitors in Series, 265 Capacitors in Parallel, 266 Series-Parallel Capacitor Combinations, 267 Smoothing Property of a Capacitor in a Power Supply The Duality Principle Basic Relationship ofDual Circuits, 269 Constructing the Dual N* of a Planar Circuit N, 271 Summary Terms and Concepts Problems

267 269

275 276 276

Chapter 8 FIRST-ORDER RL AND RC CIRCUITS

1. Introduction 288 2. Some Mathematical Preliminaries 289 3. Source-Free or Zero-Input Response 291 4. DC or Step Response of First-Order

Circuits 298 5. Superposition and Linearity 305

Response Classifications Further Points of Analysis and Theory First-Order RC Op Amp Circuits Summary Terms and Concepts Problems

286

308 309 313 317 318 318

CONTENTS

Chapter 9 SECOND-ORDER LINEAR CIRCUITS

1. Introduction 2. Discharging a Capacitor Through an

Inductor 3. Source-Free Second-Order Linear

Networks Development of Differential Equation Models for Series and Parallel RLC Circuits, 336 Solution ofthe General Second-Order Differential Equation Model, 337 Response Calculation of Parallel and Series RLC Circuits, 342 Application to Selected Second-Order Circuits, 345

:UITS

332

333

336

4.

*5.

6.

Second-Order Linear Networks with Constant Inputs Formulation of a Single Second-Order Differential Equation Writing the State Equations, 356 Reducing the State Equations to a Single Second-Order Differential Equation, 358 Oscillator Application Summary Terms and Concepts Problems

330

348

356

362 366 366 367

Chapter 10 SINUSOIDAL STEADY-STATE ANALYSIS BY PHASOR METHODS 378

1. Introduction 2. Brief Review of Comp lex Numbers 3. Naive Technique for Computing the

Sinusoidal Steady State 4. Complex Exponential Forcing Functions

in Sinusoidal Steady-State Computation 5. Phasor Representations of Sinusoidal

Signals 6. Elementary Impedance Concepts: Phasor

Relationships for Rs, Ls, and Cs 7. Phasor Impedance and Admittance

380 382

387

389

391

393 397

8.

9. 10.

11.

Steady-State Circuit Analysis Using Phasors Phasor Diagram Introduction to the Notion of Frequency Response Nodal Analysis of a Pressure-Sensing Device Summary Terms and Concepts Problems

402 405

408

414 417 418 418

Chapter 11 SINUSOIDAL STEADY-STATE POWER CALCULATIONS 432

1. Introduction 433 2. Instantaneous and Average Power 434 3. Effective Value of a Signal and Average

Power 437 4. Complex Power and its Components:

Average, Reactive, and Apparent Power 441 5. Conservation of Power in the Sinusoidal

Steady State 444 Basics and Examples, 444 Justification of

the Principle of Conservation of Power, 446

6. Power Factor and Power Factor Correction

7. Maximum Power Transfer in the Sinusoidal Steady State Summary Terms and Concepts Problems

448

455 458 459 459

Chapter 12 BALANCED THREE-PHASE CIRCUITS 466

1. Introduction 2. Ideal Three-Phase Voltage Sources and

Some Economical Aspects of Electric Power Transmission

3. Circuit Models for Practical Three-Phase Voltage Sources

467

468

473

4. Analysis of Balanced Three-Phase Circuits Summary Terms and Concepts Problems

478 485 486 486

CONTENTS

Chapter 13 LAPLACE TRANSFORM ANALYSIS, 1 : BASICS 492

1. Introduction 493 2. Review and Deficiencies of

"Second-Order" Time Domain Methods 494 3. Overview of Laplace Transform Analysis 498 4. Basic Signals 499 5. The One-Sided Laplace Transform 502 6. The Inverse Laplace Transform 508

Partial Fraction Expansions: Distinct Poles, 509 Partial Fraction Expansions:

Repeated Poles, 511 Partial Fraction Expansions: Distinct Complex Poles, 513 Elementary Properties and Examples 516 Solution of Integrodifferential Equations by the Laplace Transform 526 Summary 530 Terms and Concepts 530 Problems 531

Chapter 14 LAPLACE TRANSFORM ANALYSIS, 2: CIRCUIT APPLICATIONS

1. Introduction 542 2. Notions of Impedance and Admittance 542 3. Manipulation of Impedance and

Admittance 545 4. Notion of Transfer Function 550 5. Equivalent Circuits for Inductors and

Capacitors 554 6. Nodal and Loop Analyses in the

s-Domain 561 7. Switching in RLC Circuits 567

Switched Capacitor Circuits and Conservation of Charge Design of General Summing Integrators Design Choicesfor the General Summing Circuit of Figure 14.52, 578 Derivation of Op Amp Input-Output Characteristic, 580 Summary Terms and Concepts Problems

Chapter 15 LAPLACE TRANSFORM ANALYSIS, 3: TRANSFER FUNCTION APPLICATIONS

1. 2. 3. 4.

5. 6. 7.

Introduction Poles, Zeros, and the s -Plane Classification of Responses Computation of the Sinusoidal Steady-State Response for Stable Networks and Systems Frequency Response Impulse and Step Responses Initial- and Final-Value Theorems

598 598 604

611 616 620 623

8. BodePlots 9. Frequency Characteristics and Bode

Plots of Some Op Amp Circuits 10. Transfer Function Analysis of a DC

Motor Summary Terms and Concepts Problems

540

572 578

581 581 582

596

626

632

636 639 639 640

Chapter 16 TIME DOMAIN CIRCUIT RESPONSE COMPUTATIONS:

THE CONVOLUTION METHOD

1. Introduction 2. Definition, Basic Properties, and Simple

Examples 3. Convolution and Laplace Transforms 4. Time Domain Derivation of the

Convolution Integral for Linear Time-Invariant Circuits Rectangular Approximations to Signals, 662 Computation of Response for Linear Time-Invariant Systems, 663

654

655

657 661

662

5.

6. 7.

8.

Circuit Response Computations Using Convolution Convolution Properties Revisited Graphical Convolution and Circuit Response Computation Convolution Algebra Summary Terms and Concepts Problems

665 669

671 675 680 680 681

CONTENTS XI

Chapter 17 RESONANT AND BANDPASS CIRCUITS

1. Introduction 2. Resonant Frequency of Simple Circuits

with Applications 3. Frequency Response of a Parallel RLC

Circuit 4 General Structure of the Bandpass

Transfer Function with One Pair of Complex Poles

5. Bandpass Transfer Function with One Pair of Complex Poles and a Single Zero at the Origin

6. Bandpass Transfer Function with One Pair of Complex Poles and a Single Zero

CIRCUITS

692

693

699

707

708

7.

8.

9.

off the Origin Bandpass Transfer Function with One Pair of Complex Poles and No Finite Zero or Two Finite Zeros Magnitude Scaling and Frequency Scaling Practical Considerations in Tuned Circuits Quality Factor of Components, 729 Summary Terms and Concepts Problems

690

714

716

720

729

738 738 739

Chapter 18 MAGNETICALLY COUPLED CIRCUITS AND TRANSFORMERS 750

1. Introduction 2. Mutual Inductance and the Dot

Convention 3. Differential Equation, Laplace

Transform, and Phasor Models of Coupled Inductors

4. Applications: Automobile Ignition and RF Amplifier

5. Coefficient of Coupling and Energy Cakulation Justification that Mn = M%\ = M, 766

752

752

756

761

766

Cakulation of Stored Energy, 767 Upper Boundfor M and the Coefficient of Coupling, 768

6. Ideal Transformer as a Circuit Element and Applications

7. Coupled Inductors Modeled with an Ideal Transformer

*8. Models for Practical Transformers Summary Terms and Concepts Problems

770

778 782 784 785 785

Chapter 19 Two-PORTS 798

1. Introduction 2. One-Port Networks

Basic Impedance Calculations, 801 Thevenin and Norton Equivalent Circuits, 803 General One-Port Analysis, 806

3. Two-Port Admittance Parameters Two-Dependent Source Equivalent Circuit, 810

4. Admittance Parameter Analysis of Terminated Two-Ports Input and Output Admittance Calculations, 811 Gain Calculations, 812

5. Two-Port Impedance Parameters Relationship to y'-Parameters, 815 Two-Dependent Source Equivalent Circuit, 816

800 801

807

810

813

6. Impedance and Gain Calculations of Terminated Two-Ports Modeled by z-Parameters Input and Output Impedance Calculations, 817 Gain Calculations, 817

7. Hybrid Parameters Computation ofh-Parameters, 822 General Relations to z- and y-Parameters, 824 Impedance and Gain Calculations, 826

8. Generalized Two-Port Parameters 9. Transmission Parameters

10. Reciprocity Summary Terms and Concepts Problems

817

820

827 827 831 836 836 837

xii CONTENTS

Chapter20 ANALYSIS OF INTERCONNECTED TWO-PORTS

851 1. Introduction 2. Parallel, Series, and Cascade Connections

of Two-Ports 3. Indefinite Admittance Matrix of a

852

Chapter21 PRINCIPLES OF BASIC FILTERING

1. Introduction and Basic Terminology 880 Types of Filtering, 880 Basic Terminology, 881

2. Low-Pass Filter Basics 881 3. Butterworth Low-Pass Transfer

Characteristic 885 Phase 1: Development ofthe Butterworth Magnitude Response, 886 Phase 2: Development ofthe Butterworth Transfer Function, 888 Properties of the Butterworth Loss Function, 890

4. Computation of Butterworth Loss Functions from Brickwall Specifications 891

5. Basic Passive Realization of Butterworth Transfer Functions 894

10.

Three-Terminal Network Summary Terms and Concepts Problems

Basic Active Realization of Butterworth Transfer Functions Sallen and Key Active Low-Pass Filter, 897 Input Attenuation and Gain Enhancement for Active Circuit Design Input Attenuation, 899 Gain Enhancement, 900 Basic High-Pass Filter Design with Passive Realization Pole-Zero Movement under the LP to HP Transformation Active Realization of High-Pass Filters Summary Terms and Concepts Problems

Chapter22 FOURIER SERIES WITH APPLICATIONS TO ELECTRONIC CIRCUITS

1. Introduction 916 2. Fourier Series: Trigonometrie and

Exponential Forms 918 Two Properties ofthe Fourier Series, 921 *Convergence ofthe Fourier Series, 925

3. Additional Properties and Computational Shortcuts for the Fourier

APPENDICES

Appendix AI Matrices 954 Appendix A2 Solving Circuit Problems with MATLAB: Chapters 1 to 12 961

Series Representation 4. Harmonie Distortion in an Amplifier

*5. Ripple Factor in DC Power Supplies Summary Terms and Concepts Problems

Appendix A3 Circuits

Use of SPICE in Linear

850

863 868 869 869

878

896

899

901

904 905 907 907 908

914

927 935 940 949 949 950

976

Index 1001