INTRODUCTION TO
NONLINEAR NETWORK
THEORY LEON 0. CHUA
Associate Professor of Electrical Engineering
Purdue Unl¥ersity
McGRAW-HILL BOOK COMPANY New York St, Louis San Francisco London
Sydney Toronto Mexico Panama
CONTENTS
Preface, vii
List of Reference Tables, xxv
Glossary of Symbols and Abbreviations, xxvi
PART 1 FOUNDATIONS OF NONLINEAR NETWORK THEORY
1 TWO-TERMINAL NETWORK ELEMENTS, 3
1-1 Review of Basic Physical Variables in Network Theory, 3 1-2 The Simultaneity Postulate in Lumped-network Theory, 5 1-3 Significance of the Reference Current Direction and the
Reference Voltage Polarity, 7 1-4 Independent Sources, 9 1-5 Characterization of a Two-terminal Black Box, 11
1-5-1 A Mechanical Black-box Analogy, 12 1-5-2 Static Characteristics of a Two-terminal Black Box, 15
1-6 Two-terminal Resistors, 17 1-6-1 Linear Resistors, 17 1-6-2 Nonlinear Resistors, 18 1-6-3 Classification o/v-i Curves, 19 1-6-4 v-i Curves of DC Sources and Ideal Diodes, 23 1-6-5 Some Practical Applications of Two-terminal
Nonlinear Resistors, 24 1-7 Two-terminal Capacitors, 30
1-7-1 Linear Capacitors, 30 1-7-2 Nonlinear Capacitors, 31 1-7-3 Some Practical Applications of Two-terminal
Nonlinear Capacitors, 32 1-8 Two-terminal Inductors, 35
1-8-1 Linear Inductors, 35 1-8-2 Nonlinear Inductors, 36 1-8-3 Some Practical Applications of Two-terminal
Nonlinear Inductors, 37 1-9 Energy and Power, 38 1-10 Time-varying Elements, 47
XV
1-11 Concepts of Modeling, 51 1-12 Summary, 53
CONTROLLED ELEMENTS, 62
2-1 Two-terminal Controlled Elements, 62 2-1-1 Two-terminal Controlled Resistors, 62 2-1-2 Two-terminal Controlled Capacitors, 67 2-1-3 Two-terminal Controlled Inductors, 67
2-2 Practical Applications of Controlled Elements, 68 2-3 Controlled Sources, 70
2-3-1 Transducers, 71 2-3-2 Sources Controlled by Electrical Variables, 72
2-4 A Basic Composition Technique, 75 2-5 Summary, 78
MULTITERMINAL ELEMENTS, 84
3-1 Characterization of a Multiterminal Black Box, 84 3-2 Three-terminal Resistors, 86
3-2-1 Forms of Representation, 89 3-2-2 Graphical Transformation of Representations, 95 3-2-3 Transformation to Another Common Terminal, 97 3-2-4 Some Practical Three-terminal Resistors, 98
3-3 Three-terminal Capacitors, 103 3-4 Three-terminal Inductors, 104 3-5 Multiterminal Elements, 106
3-5-1 Multiterminal Resistors Operating as Three-terminal Controlled Resistors, 107
3-5-2 Multiterminal Elements with Prescribed Constraints, 108
3-6 Some Practical Applications of Operational Amplifiers, 115 3-6-1 Operational Amplifier Used as a Nonlinear
Element, 115 3-6-2 Operational Amplifier Used as a Linear Element, 117
3-7 Scalers, Rotators, and Reflectors: A Class of Useful Two-port Resistors, 128 3-7-1 Scalers, 130 3-7-2 Rotators, 131 3-7-3 Reflectors, 134
3-8 Mutator: The Chameleon Black Box, 138 3-8-1 R-L Mutators, 138 3-8-2 R-C Mutators, 140 3-8-3 C-L Mutators, 142
3-9 Summary, 144
EQUATIONS OF MOTION, 152
4-1 Classification of Nonlinear Networks, 152 4-2 Laws of Elements and Laws of Interconnection, 153
4-2-1 Equations of Motion Pertaining to the Laws of Elements, 155
4-2-2 Equations of Motion Pertaining to the Laws of Interconnection, 155
Contents
4-3 Introduction to Network Topology, 157 V" 4-3-1 Criteria for Writing Independent KVL Equations, 160 4-3-2 Criteria for Writing Independent KCL Equations, 163
4-4 Equations of Motion for Resistive Networks, 166 4-4-1 Networks Containing Only Two-terminal Resistors
and Independent Sources, 166 4-4-2 Networks Containing Multiterminal Resistors and
Controlled Sources, 173 4-5 Practical Methods for Solving Nonlinear Functional
Equations, 179 4-5-1 The Fixed-point Concept, 180 4-5-2 The Newton-Raphson Method, 185 Y/" 4-5-3 Equations with Many Unknowns, 187
4-6 Equations of Motion for Dynamic Networks, 188 4-6-1 Selection of State Variables of Normal-form
Equations, 192 4-6-2 Autonomous and Nonautonomous Networks, 195
4-7 Practical Methods for Solving Nonlinear Differential Equations, 196 4-7-1 Euler Algorithm for One Differential Equation, 196 4-7-2 Euler Algorithm for Two or More Differential
Equations, 199 4-8 Principles of Duality, 201
4-8-1 Duality Relationships from the Laws of Elements, 203 4-8-2 Duality Relationships from the Laws of
Interconnection, 204 4-8-3 Algorithm for Drawing the Dual of a Planar
Network, 209 4-9 Summary, 213
PART 2 RESISTIVE NONLINEAR NETWORKS
5 THREE BASIC CONCEPTS OF RESISTIVE NONLINEAR NETWORKS, 225
5-1 The Operating-point Concept, 225 5-2 Concepts of Driving-point and Transfer-characteristic
Plots, 228 5-2-1 The Driving-point Characteristic Plot (DP Plot), 231 5-2-2 The Transfer-characteristic Plot (TC Plot), 232
5-3 Some Practical Examples, 236 5-3-1 Isolation Block for Integrated Circuits, 236 5-3-2 A Square-law Transfer-characteristic Plot, 238
5-4 The Power-Transfer Plot, 241 5-5 The Operating-point Paradox, 243 5-6 The Three Fundamental Theorems of Resistive Networks, 244 5-7 Summary, 246
6 GRAPHICAL ANALYSIS OF RESISTIVE NONLINEAR NETWORKS, 253
6-1 A Bird's-eye View, 253 6-2 Graphical Determination of the Operating Point, 254
XVIII Contents
6-2-1 Basic Network Configuration 1, Interconnection between 2 Two-terminal Resistors, 255
6-2-2 Basic Network Configuration 2, Interconnection between 2 Two-terminal Resistors and a Three-terminal Resistor, 260
6-3 Graphical Determination of DP Plots of Series-Parallel Networks, 266 6-3-1 The Series-combination Technique, 267 6-3-2 The Parallel-combination Technique, 271 6-3-3 Combination of Series-parallel Techniques, 272
6-4 Some Practical Applications of DP Plot and Operating-point Concepts, 273 6-4-1 Concave and Convex Resistors, 273 6-4-2 Elements with a Horizontal Segment, 276 6-4-3 Elements with a Vertical Segment, 277 6-4-4 Multithreshold Elements, 277 6-4-5 Automatic Sorting Circuit, 280 6-4-6 Automatic Comparison and Null-detection Circuit, 282
6-5 Graphical Determination of TC Plots of Series-parallel Networks, 286 6-5-1 Nonlinear Voltage Divider, 286 6-5-2 Nonlinear Current Divider, 290 6-5-3 Series-parallel Nonlinear Networks, 290
6-6 Some Practical Applications of TC Plots, 294 6-6-1 Half-wave Rectifier, 294 6-6-2 Voltage Limiter, 296 6-6-3 Pulse Compressor, 297 6-6-4 Vertical-pulse Generator, 298
6-7 DP Plot and TC Plot of Networks Containing Three-terminal Resistors, 301 6-7-1 The Template-Double Load-line Method, 303 6-7-2 Some Practical Circuits, 306
6-8 Composite Characteristics of Three-terminal Resistive Black Boxes, 311
. 6-8-1 Four Basic Three-terminal Black Boxes, 311 6-8-2 Cascade Connection of Basic Three-terminal
Black Boxes, 313 6-9 Summary, 316
7 PRINCIPLES OF EQUIVALENCE AND SYMMETRY, 325
7-1 Some Shortcuts in Nonlinear Network Analysis, 325 7-2 Definition of Equivalent Networks, 325 7-3 Equivalence Based on Identical DP Plots, 327 7-4 Equivalence Based on Identical Operating Points, 330 7-5 The Shifting Theorems, 332 7-6 Graphical Analysis of Non-series-parallel Networks, 334 7-7 Principles and Applications of Symmetry, 342
7-7-1 Some Applications of Symmetry to the Operating-point Problem, 343
7-7-2 Some Applications of Symmetry to the Determination of DP Plots and TC Plots, 348
7-8 Complementary Networks, 353
»
Contents
7-9 Some Practical Circuits, 355 7-9-1 DC Amplifier, 355 7-9-2 Complementary Symmetric Amplifier, 356 7-9-3 Push-pull Amplifier, 357
7-10 Summary, 361
8 GRAPHICAL SYNTHESIS OF RESISTIVE NONLINEAR NETWORKS, 368
8-1 What Is Network Synthesis? 368 8-2 Synthesis of Operating Points, 369
8-2-1 Biasing a Two-terminal Nonlinear Resistor, 369 8-2-2 Biasing a Three-terminal Nonlinear Resistor, 374 8-2-3 Sensitivity Consideration in Three-terminal Biasing
Circuits, 382 8-3 Synthesis of DP Plots, 386
8-3-1 Synthesis of Monotonie DP Plots, 388 8-3-2 Synthesis of Nonmonotonic DP Plots, 396
8-4 Synthesis of TC Plots, 401 8-4-1 TC Plot Synthesis by a Nonlinear Voltage- or
Current-divider Network, 402 8-4-2 TC Plot Synthesis by Operational Amplifier
Circuits, 405 8-5 Synthesis of Jointly Prescribed DP Plot, TC Plot, and Load
v-i Curve, 408 8-6 Summary, 413
9 SYNTHESIS OF DC-RESISTIVE NONLINEAR FUNCTIONAL NETWORKS: THE BLACK-BOX APPROACH, 422
9-1 Basic Philosophy, 422 9-2 Functional Black Box for Waveform Generation, 424
9-2-1 Generation of Periodic Waveforms, 425 9-2-2 Generation of Aperiodic Waveforms, 426
9-3 Functional Black Box for Waveshaping, 427 9-3-1 Clipping Networks, 42.7 9-3-2 Analog-to-digital Converters, 431
9-4 Functional Black Box for Compensation, 433 9-5 Functional Black Box for Regulation, 437 9-6 Summary, 442
10 SYNTHESIS OF AC-RESISTIVE NONLINEAR FUNCTIONAL NETWORKS: THE BLACK-BOX APPROACH, 449
10-1 Characteristics of Electronically Controlled Switches, 449 10-2 Synthesis of Single-controlled Undirectional Switches, 452
10-2-1 A High-switching-sensitivity Transistor Switch, 454 10-2-2 A High-switching-sensitivity SCR Latching
Switch, 456 10-3 Synthesis of Single-controlled Bidirectional Switches, 458 10-4 Synthesis of Choppers and Amplitude Modulators, 461
10-4-1 Practical Choppers, 462 10-4-2 Practical Amplitude Modulators, 463
Contents
10-5 Synthesis of Multicontrolled Electronic Switches, 465 10-5-1 Synthesis of a Coincidence Gate, 466 10-5-2 Synthesis of Matrix Transmission Gates, 468
10-6 Synthesis of Logic Circuit Building Blocks, 469 10-6-1 Synthesis of AND Gates, 472 10-6-2 Synthesis of OR Gates, 474 10-6-3 Synthesis of NAND, NOR, and NOT Gates, 476
10-7 Summary, 479
11 SYNTHESIS OF RESISTIVE NONLINEAR NETWORK MODELS, 489
11-1 What Is a Network Model? 489 11-2 Principles of Model Making, 492 11-3 Synthesis of Global Models for Nonlinear Controlled
Resistors, 494 11-4 Synthesis of Global Models for Nonlinear Three-terminal
Resistors, 497 11-4-1 A Basic T-network Model, 498 11-4-2 A Basic m-network Model, 501
11-5 Some Useful Techniques for Modifying Models, 503 11-5-1 Constraining v-i Curves to Half-planes, 503 11-5-2 Constraining v-i Curves to Quadrants, 505 11-5-3 Constraining v-i Curves to the First and Third
Quadrants, 506 11-6 Models of Practical Three-terminal Resistors, 508
11-6-1 Vacuum Triode Models, 508 11-6-2 Vacuum Pentode Models, 510 11-6-3 n-p-n Transistor Models, 512 11-6-4 p-n-p Transistor Models, 517 11-6-5 n-channel FET Models, 517 11-6-6 p'-channel FET Models, 517
11-7 Summary, 520
12 ITERATIVE PIECEWISE-LINEAR ANALYSIS AND SYNTHESIS OF RESISTIVE NONLINEAR NETWORKS, 526
12-1 Basic Philosophy of the Iterative Piecewise-linear Method, 526
12-2 Determination of Operating Points, 530 12-3 Determination of DP Plots, 539 12-4 Determination of TC Plots, 553 12-5 Computer Analysis of Resistive Nonlinear Networks, 558
12-5-1 Semiautomatic General Analysis Program, 558 12-5-2 Completely Automatic Meca Program, 559
12-6 TC Plot Synthesis by the Piecewise-linear Method, 562 12-6-1 TC Plot Synthesis by Nonlinear Voltage
Dividers, 562 12-6-2 Jointly Prescribed DP Plot, TC Plot, and Load v-i
Curve Synthesis, 565 12-7 Summary, 573
Contents
PART 3 DYNAMIC NONLINEAR NETWORKS
13 BASIC CONCEPTS OF DYNAMIC NONLINEAR у / NETWORKS, 583
13-1 Classification of Dynamic Nonlinear Networks, 583 13-1-1 Basis of Classification, 583 13-1-2 Definition of Order of Complexity, 585
13-2 Order of Complexity of Dynamic Networks, 586 13-2-1 Significance of the Initial Condition, 586 13-2-2 Determination of Order of Complexity by
Inspection, 589 13-2-3 Some Subtle Points Concerning Initial
Conditions, 592 13-3 Basic Principles for Analyzing Dynamic Networks, 593 13-4 Deriving Normal-form Equations from the Characteristics
of the 'H-port Subnetwork, 595 13-4-1 Normal-form Equations for First-order
Networks, 595 13-4-2 Normal-form Equations for Second-order
Networks, 599 13-4-3 Normal-form Equations for nth-order Networks, 604
13-5 Trajectory of Solutions in the State Space, 606 13-6 Equilibrium States of Autonomous Networks, 613 v - У
13-6-1 Equilibrium States of a Switching Circuit: A Numerical Example, 617
13-6-2 Relationship between Trajectories and Equilibrium States, 620
13-7 Stability of Equilibrium States, 622 13-7-1 Stability Definition for Second-order Networks, 624 13-7-2 Stability Definition for nth-order Networks, 626
13-8 Summary, 626
14 ANALYSIS OF AUTONOMOUS FIRST-ORDER NONLINEAR NETWORKS, 636
14-1 Basic Philosophy and Approach, 636 14-2 Equilibrium States and Stability Criteria, 637 14-3 Stability Criteria for Two Common Circuit
Configurations, 640 14-4 Time Scaling the Trajectory of Solutions, 644 14-5 Realistic and Incomplete Models, 649 14-6 Jump Phenomenon and Oscillatory Solution in First-order
Networks, 652 14-7 Incomplete Models Requiring Additional Parasitic
Elements, 656 14-8 The Piecewise-linear Approach, 659
14-8-1 Networks Containing a Linear Energy-storage Element across a Nonlinear Resistive Network Characterized by Piecewise-linear Segments with Finite Nonzero Slopes, 660
Contents
14-8-2 Networks Containing a Linear Energy-storage Element across a Nonlinear Resistive Network Characterized by Piecewise-linear Segments with Arbitrary Slopes, 682
14-8-3 Networks Containing a Piecewise-linear Energy-storage Element across a Nonlinear Resistive Network Characterized by Piecewise-linear Segments with Arbitrary Slopes, 686
14-9 Summary, 689
15 ANALYSIS OF FIRST-ORDER NONLINEAR SWITCHING NETWORKS, 697
15-1 What Is a First-order Switching Network? 697 15-2 Analysis of First-order Linear Switching Networks, 698 15-3 Analysis of First-order Nonlinear Switching Networks, 708 15-4 Analysis of Autonomous First-order Networks Containing
One Nonlinear Inductor or Capacitor, 711 15-5 Summary, 718
16 SYNTHESIS OF FIRST-ORDER MULTIVIBRATORS: THE BLACK-BOX APPROACH, 725
16-1 What Is a First-order Multivibrator? 725 16-2 Astable (Free-running) Multivibrators, 726 16-3 Monostable (Single-shot) Multivibrators, 730
16-3-1 Some Applications of Monostable Multivibrators, 733
16-3-2 Minimum Triggering-pulse-amplitude Condition, 735
16-3-3 Minimum Triggering-pulse-width Condition, 736 16-4 Bistable (Double-shot, Flip-flop) Multivibrators, 737 16-5 Summary, 746
17 SYNTHESIS OF FIRST-ORDER TIME-BASE GENERATORS: THE BLACK-BOX APPROACH, 753
17-1 What Is a Time-base Generator? 753 17-2 Synthesis of Free-running Time-base Generators, 759 17-3 Synchronization of Free-running Time-base Generators, 763 17-4 Synthesis of Triggered Time-base Generators, 768 17-5 Summary, 71A
18 ANALYSIS OF NONAUTONOMOUS FIRST-ORDER NONLINEAR NETWORKS, 778
18-1 Motivation and Strategies, 778 18-2» Nonautonomous First-order Linear Network Analysis, 779
18-2-1 Zero-state Response, 779 18-2-2 Complete Response, 781 18-2-3 Very Large and Very Small Time-constant
Networks, 783 18-3 Nonautonomous First-order Piecewise-linear Network
Analysis, 786
Contents
18-3-1 One Nonlinear Resistor Case, 787 18-3-2 M Nonlinear Resistor Case, 790
18-4 The Constant-slope Network Approach, 793 18-4-1 Methods for Drawing the Vector Field, 794 18-4-2 The Concept of Constant-slope Networks, 797 18-4-3 Some Illustrative Examples, 800
18-5 Some Practical Applications of Nonautonomous First-order Nonlinear Networks, 807 18-5-1 The Monotonie Charging Property and Its
Applications, 807 18-5-2 The Pulse-smoothing Property, 811 18-5-3 The Pulse-sharpening Property, 817
18-6 Summary, 819
19 ANALYSIS OF AUTONOMOUS SECOND-ORDER NONLINEAR NETWORKS, 823
19-1 The Phase-plane Technique, 823 19-1-1 Isoclines and Phase Portraits, 823 19-1-2 Time Scaling the Trajectories, 828
19-2 The Isocline Network Method, 833 19-2-1 The Isocline Network, 833 19-2-2 Some Practical Networks, 836
19-3 From Qualitative to Quantitative Analysis, 842 19-4 Analytical Solutions of Autonomous Second-order Linear
Networks, 843 19-5 Piecewise-linear Analysis of Autonomous Second-order
Nonlinear Networks, 847 19-6 Behavior of Trajectories near an Equilibrium State, 859
19-6-1 Local Behavior of Trajectories in the Aperiodic Case, 861
19-6-2 Local Behavior of Trajectories in the Spiral Case, 867
19-6-3 Summary of Singular-point Classifications, 868 19-6-4 The Phase Portrait, 871
19-7 Summary, 873
20 ANALYSIS OF nTH-ORDER NONLINEAR NETWORKS, 880
20-1 Beyond the Second-order Autonomous Network Barrier, 880 20-2 The Numerical-integration Approach, 881 20-3 The Piecewise-linear Approach, 884 20-4 The Heuristic Approach, 885 20-5 The Qualitative Approach, 890 20-6 The Approximate Analytical Approach, 895 20-7 Summary, 895
APPENDIX A MATHEMATICAL REPRESENTATION OF NONLINEAR NETWORK ELEMENTS, 898
A-l Relations and Their Inverses, 898 A-2 Functions and Their Inverses, 901 A-3 Mathematical Representation of Functions, 904
XXIV Contents
A-4 Mathematical Representation of Multivalued Functions, 909
A-5 Parametric Representation of Nonlinear Network Elements, 913
APPENDIX В GRAPHICAL TECHNIQUES FOR BASIC MATHEMATICAL OPERATIONS, 919 B-1 Graphical Addition and Subtraction, 920 B-2 Graphical Multiplication and Division, 920 B-3 Graphical Composition, 921 B-4 Graphical Elimination, 923 B-5 Graphical Differentiation, 924 B-6 Graphical Integration, 926
APPENDIX С SCHEMATIC DIAGRAMS OF SOME BASIC CIRCUITS, 929
C-l Circuits for Tracing v-i, v-q, and i-<p Curves, 929 C-2 Circuits for Simulating Controlled Sources, 932 C-3 Circuits for Simulating Negative-impedance
Converters, 937 C-4 Circuits for Simulating Scalers, 938 C-5 Circuits for Simulating Rotators, 939 C-6 Circuits for Simulating Reflectors, 939 C-7 Circuits for Simulating Mutators, 940
APPENDIX D CHARACTERISTIC CURVES OF TYPICAL DEVICES, 943
APPENDIX E SELECTED LIST OF REFERENCES, 958
E-l References on Basic Circuit Theory, 958 E-2 References on Basic Electronic Circuits and
Devices, 959 E-3 References on Linear Active Networks, 959 E-4 References on Nonlinear Electronic Circuits, 960 E-5 References on Nonlinear Network Elements, 960 E-6 References on Integrated Circuits, 963 E-7 References on Device Modeling, 964 E-8 References on the Formulation of Equations of
Motion of Nonlinear Networks, 964 E-9 References on Numerical Techniques, 965 E-10 References on Computer-aided Analysis, 965 E-11 References on General Nonlinear Network
Properties and Theorems, 966 E-12 References on Approximation Techniques, 968 E-13 References on Qualitative Methods of Analysis, 968 E-14 References on Analytical Techniques, 969
Index of Theorems, 973
Subject Index, 975
