C
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Department of Electrical Engineering and Computer Sciences
University of California, Berkeley
New York St. Louis San Francisco
London Sydney Toronto
Mexico Panama

Preface,v
1. Lumped circuits, 2
2. Reference directions, 4
3. Kirchhoff's current law (KCL), 5
4. Kirchhoff's voltage law (KVL), 6
5. Wavelength and dimension of the circuit, 8
Summary, 10
Problems, 10
1. Resistors, 13
1.1 The linear time-invariant resistor, 15
1.2 The linear time-varying resistor, 16
1.3 The nonlinear resistor, 18
2. Independent sources, 23
2.1 Voltage source, 24
2.2 Current source, 26
2.3 Thevenin and Norton equivalent circuits, 27
2.4 Waveforms and their notation, 28
2.5 Some typical waveforms, 29
3. Capacitors, 34
3.1 The linear time-invariant capacitor, 35
3.2 The linear time-varying capacitor, 39
3.3 The nonlinear capacitor, 41
4. Inductors, 43
4.1 The linear time-invariant inductor, 44
4.2 The linear time-varying inductor, 47
4.3 The nonlinear inductor, 47
4.4 Hysteresis, 48
5. Summary of two-terminal elements, 51
6. Power and energy, 53
6.1 Power entering a resistor, passivity, 54

6.2 Energy stored in time-invariant capacitors, 55
6.3 Energy stored in time-invariant inductors, 57
7. Physical components versus circuit elements, 59
Summary, 61
Problems, 62
1. Series connection of resistors, 74
2. Parallel connection of resistors, 81
3. Series and parallel connection of resistors, 85
4. Small-signal analysis, 91
5. Circuits with capacitors or inductors, 96
5.1 Series connection of capacitors, 96
5.2 Parallel connection of capacitors, 97
5.3 Series connection of inductors, 99
5.4 Parallel connection of inductors, 100
Summary, 101
Problems, 101
1. Linear time-invariant first-order circuit, zero-input response, 110
1.1 The RC (resistor-capacitor) circuit, 110
1.2 The RL (resistor-inductor) circuit, 114
1.3 The zero-input response as a function of the initial state, 115
1.4 Mechanical example, 117
2. Zero-state response, 118
2.1 Constant current input, 118
2.2 Sinusoidal input, 121
3. Complete response: transient and steady-state, 124
3.1 Complete response, 124
3.2 Transient and steady state, 126
3.3 Circuits with two time constants, 128
4. The linearity of the zero-state response, 129
5. Linearity and time invariance, 133
5.1 Step response, 133
5.2 The time-invariance property, 134
5.3 The shift operator, 137
6. Impulse response, 141
7. Step and impulse response for simple circuits, 149
8. Time-varying circuits and nonlinear circuits, 154
Summary, 164
Problems, 165
Contents xii
1. Linear time-invariant RLC circuit, zero-input response, 177
2. Linear time-invariant RLC circuit, zero-state response, 185
2.1 Step response, 187

2.2 Impulse response, 190
3. The state-space approach, 196
3.1 State equations and trajectory, 197
3.2 Matrix representation, 201
3.3 Approximate method for the calculation of the trajectory, 202
3.4 State equations and complete response, 205
4. Oscillation, negative resistance, and stability, 207
5. Nonlinear and time-varying circuits, 211
6. Dual and analog circuits, 219
6.1 Duality, 219
6.2 Mechanical and electrical analog, 225
Summary, 227
Problems, 228
1. Some general definitions and properties, 235
2. Node and mesh analyses, 257
2.1 Node analysis, 238
2.2 Mesh analysis, 240
3. Input-output representation («th-order differential equation), 242
3.1 Zero-input response, 243
3.2 Zero-state response, 243
3.3 Impulse response, 245
4. Response to an arbitrary input, 247
4.1 Derivation of the convolution integral, 248
4.2 Example of a convolution integral in physics, 252
4.3 Comments on linear time-varying circuits, 253
4.4 The complete response, 254
5. Computation of convolution integrals, 254
Summary, 261
Problems, 262
1. Review of complex numbers, 269
1.1 Description of complex numbers, 269
1.2 Operations with complex numbers, 271
2. Phasors and ordinary differential equations, 272
2.1 The representation of a sinusoid by a phasor, 272
2.2 Application of the phasor method to differential equations, 278
3. Complete response and sinusoidal steady-state response, 281
3.1 Complete response, 281
3.2 Sinusoidal steady-state response, 285
3.3 Superposition in the steady state, 287
4. Concepts of impedance and admittance, 289
4.1 Phasor relations for circuit elements, 289
4.2 Definition of impedance and admittance, 292
5. Sinusoidal steady-state analysis of simple circuits, 295
5.1 Series-parallel connections, 296
5.2 Node and mesh analyses in the sinusoidal steady state, 299
Contents xiii

6. Resonant circuits, 304
6.1 Impedance, admittance, and phasors, 304
6.2 Network function, frequency response, 310
7. Power in sinusoidal steady state, 317
7.1 Instantaneous, average, and complex power, 318
7.2 Additive property of average power, 320
7.3 Effective or root-mean-square values, 321
7.4 Theorem on the maximum power transfer, 322
7.5 Q of a resonant circuit, 325
8. Impedance and frequency normalization, 326
Summary, 329
Problems, 331
1. Coupled inductors, 341
1.1 Characterization of linear time-invariant coupled inductors, 342
1.2 Coefficient of coupling, 346
1.3 Multiwinding inductors and their inductance matrix, 347
1.4 Series and parallel connections of coupled inductors, 349
1.5 Double-tuned circuit, 353
2. Ideal transformers, 356
2.1 Two-winding ideal transformer, 357
2.2 Impedance-changing properties, 361
3. Controlled sources, 362
3.1 Characterization of four kinds of controlled source, 362
3.2 Examples of circuit analysis, 365
3.3 Other properties of controlled sources, 368
Summary, 371
Problems, 372
1. The concept of a graph, 381
2. Cut sets and Kirchhoff's current law, 386
3. Loops and Kirchhoff's voltage law, 390
4. Tellegen's theorem, 392
5. Applications, 396
5.1 Conservation of energy, 396
5.2 Conservation of complex power, 397
5.3 The real part and phase of driving-point impedances, 398
5.4 Driving-point impedance, power dissipated, and energy stored, 401
Summary, 402
Problems, 403
1. Source transformations, 409
2. Two basic facts of node analysis, 414
Contents xiv

2.1 Implications of KCL, 414
2.2 Implications of KVL, 418
2.3 Tellegen's theorem revisited, 422
3. Node analysis of linear time-invariant networks, 423
3.1 Analysis of resistive networks, 424
3.2 Writing node equations by inspection, 429
3.3 Sinusoidal steady-state analysis, 431
3.4 Integrodifferential equations, 436
3.5 Shortcut method, 441
4. Duality, 444
4.1 Planar graphs, meshes, outer meshes, 444
4.2 Dual graphs, 448
4.3 Dual networks, 453
5. Two basic facts of mesh analysis, 457
5.1 Implications of KVL, 457
5.2 Implications of KCL, 459
6. Mesh analysis of linear time-invariant networks, 461
6.1 Sinusoidal steady-state analysis, 461
6.2 Integrodifferential equations, 464
Summary, 466
Problems, 469
Chapter 11 Loop and Cut-set Analysis
1. Fundamental theorem of graph theory, 477
2. Loop analysis, 480
2.1 Two basic facts of loop analysis, 480
2.2 Loop analysis for linear time-invariant networks, 483
2.3 Properties of the loop impedance matrix, 485
3. Cut-set analysis, 486
3.1 Two basic facts of cut-set analysis, 486
3.2 Cut-set analysis for linear time-invariant networks, 489
3.3 Properties of the cut-set admittance matrix, 490
4. Comments on loop and cut-set analysis, 491
5. Relation between В and Q, 493
Summary, 495
Problems, 496
1. Linear time-invariant networks, 501
2. The concept of state, 508
3. Nonlinear and time-varying networks, 510
3.1 Linear time-varying case, 510
3.2 Nonlinear case, 512
4. State equations for linear time-invariant networks, 516
Summary, 521
Problems, 522

1. Definition of the Laplace transform, 528
2. Basic properties of the Laplace transform, 532
2.1 Uniqueness, 532
2.2 Linearity, 533
2.3 Differentiation rule, 534
2.4 Integration rule, 539
3. Solutions of simple circuits, 542
3.1 Calculation of an impulse response, 542
3.2 Partial-fraction expansion, 544
3.3 Zero-state response, 551
3.4 The convolution theorem, 552
3.5 The complete response, 553
4. Solution of general networks, 555
4.1 Formulation of linear algebraic equations, 556
4.2 The cofactor method, 557
4.3 Network functions and sinusoidal steady state, 559
5. Fundamental properties of linear time-invariant networks, 562
6. State equations, 565
7. Degenerate networks, 565
8. Sufficient conditions for uniqueness, 571
Summary, 573
Problems, 574
1. Natural frequency of a network variable, 583
2. The elimination method, 588
2.1 General remarks, 588
2.2 Equivalent systems, 591
2.3 The elimination algorithm, 597
3. Natural frequencies of a network, 600
4. Natural frequencies and state equations, 603
Summary, 605
Problems, 606
1. Definition, examples, and general property, 609
2. Poles, zeros, and frequency response, 615
3. Poles, zeros, and impulse response, 625
4. Physical interpretation of poles and zeros, 628
4.1 Poles, 628
4.2 Natural frequencies of a network, 633
4.3 Zeros, 6J5
5. Application to oscillator design, 638
6. Symmetry properties, 641
Summary, 642
Problems, 643
Contents xvi

1. The Substitution theorem, 653
1.1 Theorem, examples, and application, 655
1.2 Proof of the substitution theorem, 657
2. The superposition theorem, 658
2.1 Theorem, remarks, examples, and corollaries, 658
2.2 Proof of the superposition theorem, 664
3. Thevenin-Norton equivalent network theorem, 667
3.1 Theorem, examples, remarks, and corollary, 668
3.2 Special cases, 671
3.3 Proof of Thevenin theorem, 675
3.4 An application of the Thevenin equivalent network theorem, 678
4. The reciprocity theorem, 681
4.1 Theorem, examples, and remarks, 682
4.2 Proof of the reciprocity theorem, 694
Summary, 697
Problems, 699
1. Review of one-ports, 712
2. Resistive two-ports, 715
2.1 Various two-port descriptions, 718
2.2 Terminated nonlinear two-ports, 719
2.3 Incremental model and small-signal analysis, 720
3. Transistor examples, 724
3.1 Common-base configuration, 724
3.2 Common-emitter configuration, 728
4. Coupled inductors, 731
5. Impedance and admittance matrices of two-ports, 734
5.1 The (open-circuit) impedance matrix, 735
5.2 The (short-circuit) admittance matrix, 738
5.3 A terminated two-port, 741
Other two-port parameter matrices, 744
6.1 The hybrid matrices, 744
6.2 The transmission matrices, 746
Summary, 750
Problems, 751
1. Physical networks and network models, 761
2. Analysis of resistive networks from a power point of view, 765
2.1 Linear networks made of passive resistors, 765
2.2 Minimum property of the dissipated power, 770
2.3 Minimizing appropriate networks, 772
2.4 Nonlinear resistive networks, 775
3. The voltage gain and the current gain of a resistive network, 777
3.1 Voltage gain, 777
3.2 Current gain, 779
Summary, 781
Problems, 782
Contents xvii

Energy and Passivity
1. Linear time-varying capacitor, 788
1.1 Description of the circuit, 788
1.2 Pumping energy into the circuit, 790
1.3 State-space interpretation, 792
1.4 Energy balance, 793
2. Energy stored in nonlinear time-varying elements, 796
2.1 Energy stored in a nonlinear time-varying inductor, 797
2.2 Energy balance in a nonlinear time-varying inductor, 799
3. Passive one-ports, 802
3.1 Resistors, 802
3.2 Inductors and capacitors, 804
3.3 Passive one-ports, 806
4. Exponential input and exponential response, 807
5. One-ports made of passive linear time-invariant elements, 812
6. Stability of passive networks, 816
6.1 Passive networks and stable networks, 816
6.2 Passivity and stability, 817
6.3 Passivity and network functions, 821
7. Parametric amplifier, 822
Summary, 825
Problems, 827
Appendix A Functions and Linearity
1. Functions, 831
1.1 Introduction to the concept of function, 831
1.2 Formal definition, 833
2. Linear functions, 834
2.1 Scalars, 834
2.2 Linear spaces, 835
2.3 Linear functions, 837
Appendix В Matrices and Determinants
1. Matrices, 843
1.1 Definitions, 843
1.2 Operations, 844
1.3 More definitions, 844
1.4 The algebra o f « x « matrices, 845
2. Determinants, 846
2.1 Definitions, 846
2.2 Properties of determinants, 847
2.3 Cramer's rule, 848
2.4 Determinant inequalities, 850
3. Linear dependence and rank, 851
3.1 Linear independent vectors, 851
3.2 Rank of a matrix, 851
3.3 Linear independent equations, 853
4. Positive definite matrices, 853

Appendix C Differential Equations
1. The linear equation of order n, 857
1.1 Definitions, 857
1.2 Properties based on linearity, 858
1.3 Existence and uniqueness, 859
2. The homogeneous linear equation with constant coefficients, 860
2.1 Distinct characteristic roots, 861
2.2 Multiple characteristic roots, 861
3. Particular solutions of UD)y(t) = b(t), 862
4. Nonlinear differential equations, 863
4.1 Interpretation of the equation, 864
4.2 Existence and uniqueness, 865
Index, 869
List of Tables
2.1 Classification of Two-terminal Elements, 52
3.1 Series and Parallel Connection, 101
4.1 Step and Impulse Responses, 152-153
5.1 Zero-input Responses of a Second-order Circuit, 222-223
5.2 Classification of Parallel RLC Circuits, 227
7.1 Sinusoidal Steady-state Properties of Resonant Circuits, 316
10.1 Dual Terms, 456
10.2 Summary of Node and Mesh Analysis, 468
13.1 Laplace Transforms of Elementary Functions, 541
13.2 Basic Properties of the Laplace Transforms, 573
17.1 Conversion Chart of Two-port Matrices, 747
19.1 Summary of Energy Relations for Inductors and Capacitors, 801