Nonlinear and Distributed Circuits

Table of Contents......Page 8
1.2 Resistive Circuits......Page 9
Number of Solutions of a Resistive Circuit......Page 12
Bounds on Voltages and Currents......Page 18
Monotonic Dependence......Page 19
Introduction......Page 20
Convergence to DC-Operating Points......Page 21
Boundedness of the Solutions......Page 26
Unique Asymptotic Behavior......Page 27
References......Page 28
2.1 Introduction......Page 29
Unidimensional Functions......Page 31
Piecewise-Linear and Piecewise-Polynomial Approximants......Page 32
Gaussian and Bell-Shaped Basis Functions......Page 35
Radial Basis Functions......Page 36
Neuro-Fuzzy Interpolation......Page 37
2.3 Aggregation, Scaling, and Transformation Circuits......Page 38
Voltage-to-Current Transformation......Page 39
Voltage/Charge Domain Transformations for Sampled-Data Circuits......Page 40
Transformation among Transfer Characteristics......Page 41
From Driving-Point to Transfer and Vice Versa......Page 42
Scaling Operation......Page 43
2.4 Piecewise-Linear Circuitry......Page 44
Current Transfer Piecewise-Linear Circuitry......Page 45
Transresistance Piecewise-Linear Circuitry......Page 46
Piecewise-Linear Shaping of Voltage-to-Charge Transfer Characteristics......Page 47
Concepts and Techniques for Polynomic and Rational Functions......Page 48
Multiplication Circuitry......Page 49
Log-Antilog Multipliers......Page 51
Transconductance Multipliers......Page 52
Sigmoidal Characteristics......Page 54
Bell-Like Shapes......Page 55
Collective Computation Circuitry......Page 56
T-Norm Operator......Page 57
2.7 Extension to Dynamic Systems......Page 58
2.8 Appendix A: Catalog of Primitives......Page 59
2.9 Appendix B: Value and Slope Hermite Basis Functions......Page 60
References......Page 61
3.1 Introduction......Page 65
3.2 Representation......Page 66
Differential Equation and State-Space Representations......Page 67
Input-Output Representation......Page 70
Volterra Series Representation......Page 72
Least-Squares Approximation and Projections......Page 74
Min-Max (Uniform) Approximation......Page 77
Best (Uniform) Approximation of Signals (Functions)......Page 79
Polynomial and Rational Approximations......Page 80
Approximation via Splines and Radial Functions......Page 82
Approximation by Means of Interpolation......Page 83
Best Approximation of Linear Functionals......Page 85
Artificial Neural Network for Approximation......Page 86
3.4 Identification......Page 88
Linear Systems Identification......Page 89
Nonlinear Systems Identification......Page 90
References......Page 92
4.1 General Equivalence Theorems for Nonlinear Circuits......Page 97
4.2 Normal Forms......Page 100
4.3 Dimensionless Form......Page 104
4.4 Equivalence between Nonlinear Resistive Circuits......Page 110
4.5 Equivalence of Lumped n-Port Networks......Page 111
4.6 Equivalence between Nonlinear Dynamic Circuits......Page 112
References......Page 116
5.1 Introduction and Motivation......Page 118
5.2 Hierarchy of Piecewise-Linear Models and Their Representations......Page 119
5.3 Piecewise-Linear Models for Electronic Components......Page 125
5.4 Structural Properties of Piecewise-Linear Resistive Circuits......Page 130
Theorem Canonical PWL (Tableau Analysis)......Page 132
Algorithm......Page 134
5.6 Piecewise-Linear Dynamic Circuits......Page 135
References......Page 137
6: Simulation......Page 141
6.1 Numerical Solution of Nonlinear Algebraic Equations......Page 142
6.2 Numerical Integration of Nonlinear Differential Equations......Page 143
SPICE......Page 144
APLAC......Page 146
NAP2......Page 149
ESACAP......Page 154
DYNAST......Page 157
References......Page 158
7.1 Introduction: Definition and Classification......Page 160
Typical CNN Models......Page 161
7.2 The Simple CNN Circuit Structure......Page 162
7.3 The Stored Program CNN Universal Machine and the Analogic Supercomputer Chip......Page 165
Image Processing — Form, Motion, Color, and Depth......Page 167
Partial Differential Equations......Page 168
Relation to Biology......Page 169
7.5 Template Library: Analogical CNN Algorithms......Page 170
7.6 Recent Advances......Page 175
References......Page 176
Electrical and Electronic Circuits as Dynamical Systems......Page 178
Autonomous Continuous-Time Dynamical Systems......Page 179
Nonautonomous Dynamical Systems......Page 180
Discrete-Time Dynamical Systems......Page 181
Classification and Uniqueness of Steady-State Behaviors......Page 183
Periodic Steady-State......Page 184
Chaotic Steady-State......Page 186
Dimension......Page 187
Stability of Steady-State Trajectories......Page 189
Eigenvalues......Page 190
Eigenvectors, Eigenspaces, Stable and Unstable Manifolds......Page 191
Poincaré Sections......Page 192
Horseshoes and Chaos......Page 193
Chaos in the Sense of Shil’nikov......Page 194
Lyapunov Exponents......Page 196
Structural Stability and Bifurcations......Page 198
Routes to Chaos......Page 199
Bifurcation Diagrams and Parameter Space Diagrams......Page 201
8.2 Chua’s Circuit: A Paradigm for Chaos......Page 203
Piecewise-Linear Dynamics......Page 204
Dynamics of D–1 and D1......Page 206
Equilibrium Point and Hopf Bifurcation......Page 207
Double-Scroll Chua’s Attractor......Page 208
Boundary Crisis......Page 209
Sensitive Dependence on Initial Conditions......Page 210
Practical Realization of Chua’s Circuit......Page 211
Simulation of Chua’s Circuit......Page 213
Dimensionless Coordinates and the Alpha-Beta Parameter-Space Diagram......Page 215
8.3 Chua’s Oscillator......Page 216
State Equations......Page 217
Topological Conjugacy......Page 219
Equivalent Eigenvalue Parameters......Page 220
Eigenvalues-to-Parameters Mapping Algorithm for Chua’s Oscillator......Page 221
8.4 Van der Pol Neon Bulb Oscillator......Page 222
Winding Numbers......Page 225
The Circle Map......Page 226
Experimental Observations of Mode-Locking and Chaos in van der Pol’s Neon Bulb Circuit......Page 227
C Bifurcation Sequence......Page 229
Circuit Model......Page 230
8.5 Synchronization of Chaotic Circuits......Page 231
Linear Mutual Coupling......Page 232
Pecora–Carroll Drive-Response Concept......Page 233
Cascaded Drive-Response Systems......Page 236
Pseudorandom Sequence Generation......Page 239
Chaotic Switching......Page 240
Chaotic Masking......Page 241
Vector Field Modulation......Page 242
Example: Communication via Vector Field Modulation Using Chua’s Circuits......Page 244
References......Page 245
Transmission Line Equations......Page 248
Characteristic Impedance......Page 249
Voltage Reflection Coefficient at the Input......Page 250
Input Admittance......Page 251
Transmission Line Parameters......Page 252
Wavelength and Phase Velocity......Page 253
Transmission Line Parameters......Page 254
Wavelength and Phase Velocity......Page 255
Modes of Operation......Page 256
Waveguide Parameters......Page 257
Modes of Operation......Page 258
Waveguide Parameters......Page 259
Transmission Line Parameters......Page 260
Geometric Structure......Page 262
Transmission Line Parameters......Page 263
References......Page 264
10.1 Introduction: Frequency vs. Time Domain Analysis......Page 265
Generalities......Page 266
Low-Frequency or Quasi-Transverse Electromagnetic Description......Page 268
Symmetric Stripline......Page 269
Coupled Striplines......Page 270
References......Page 271
One-Port Time Domain Reflectometry......Page 273
Time Domain Reflectometric Characterization of an Interconnection Structure......Page 274
Introduction......Page 276
Network Analyzer: Block Diagram......Page 278
One-Port Calibration: Error Correction Network......Page 279
Error Checking and Verification of the Calibration......Page 280
References......Page 281
12.1 Uniform Distributed RC Lines......Page 282
Superposition......Page 283
Finite DRC Line......Page 285
Solution in the Frequency Domain......Page 287
Uniform, Lossy DRC Lines......Page 288
12.2 Nonuniform Distributed RC Lines......Page 289
Approximation with Concatenated Uniform Sections......Page 291
Lumped Element Approximation......Page 292
12.3 Infinite-Length RC Lines......Page 293
Generalization of the Time Constant Representation......Page 294
Generalization of the Pole-Zero Representation......Page 296
Relations among R(zeta), Id (sigma) and the Impedance Function......Page 297
Practical Calculation of the R(ζ) Function......Page 298
12.4 Inverse Problem for Distributed RC Circuits......Page 299
Network Identification by Deconvolution......Page 300
References......Page 302
13.1 Generic Relations......Page 304
13.2 Synthesis of a Capacitance......Page 306
13.3 Synthesis of an Inductance......Page 307
13.4 Synthesis of a Resistance......Page 308
13.5 Synthesis of Transformers......Page 310
Series L-C Circuit......Page 312
Series L-C-R Circuit......Page 313
Parallel L-C-R Circuit......Page 315
Low-Pass Filters......Page 316
High-Pass Filters......Page 317
Bandpass Filters......Page 318
Bandstop Filters......Page 319
Proximity Couplers......Page 321
Quarter-Wavelength Couplers......Page 323
Lange Couplers......Page 324
References......Page 325