Preface xix
1 Introduction 1
1.1 Why Macromodeling? 1
1.2 Scope 4
1.3 Macromodeling Flows 6
1.3.1 Macromodeling via Model Order Reduction 6
1.3.2 Macromodeling from Field Solver Data 7
1.3.3 Macromodeling from Measured Responses 8
1.4 Rational Macromodeling 9
1.5 Physical Consistency Requirements 11
1.6 Time-Domain Implementation 15
1.7 An Example 16
1.8 What Can Go Wrong? 17
2 Linear Time-Invariant Circuits and Systems 23
2.1 Basic Definitions 24
2.1.1 Linearity 24
2.1.2 Memory and Causality 26
2.1.3 Time Invariance 26
2.1.4 Stability 27
2.1.5 Passivity 28
2.2 Linear Time-Invariant Systems 28
2.2.1 Impulse Response 29
2.2.2 Properties of LTI Systems 32
2.3 Frequency-Domain Characterizations 33
2.4 Laplace and Fourier Transforms 34
2.4.1 Bilateral Laplace Transform and Transfer Matrices 34
2.4.2 Causal LTI Systems and the Unilateral Laplace Transform 36
2.4.3 Fourier Transform 36
2.5 Signal and System Norms? 37
2.5.1 Signal Norms 38
2.5.2 System Norms 41
2.6 Multiport Representations 44
2.6.1 Ports and Terminals 44
2.6.2 Immittance Representations 45
2.6.3 Scattering Representations 46
2.6.4 Reciprocity 48
2.7 Passivity 49
2.7.1 Power and Energy 50
2.7.2 Passivity and Causality 51
2.7.3 The Static Case 52
2.7.4 The Dynamic Case 53
2.7.5 Positive Realness, Bounded Realness, and Passivity 54
2.7.6 Some Examples 56
2.8 Stability and Causality 59
2.8.1 Laplace-Domain Conditions for Causality 61
2.8.2 Laplace-Domain Conditions for BIBO Stability 62
2.8.3 Causality and Stability 62
2.9 Boundary Values and Dispersion Relations? 64
2.9.1 Assumptions 64
2.9.2 Reconstruction of H(s) for s ∈ C+ 65
2.9.3 Reconstruction of H(s) for s ∈ jR 65
2.9.4 Causality and Dispersion Relations 67
2.9.5 Generalizations 68
2.10 Passivity Conditions on the Imaginary Axis? 70
Problems 71
3 Lumped LTI Systems 73
3.1 An Example from Circuit Theory 74
3.1.1 Variation on a Theme 76
3.1.2 Driving-Point Impedance 77
3.2 State-Space and Descriptor Forms 77
3.2.1 Singular Descriptor Forms 77
3.2.2 Internal Representations of Lumped LTI Systems 79
3.3 The Zero-Input Response 80
3.4 Internal Stability 81
3.4.1 Lyapunov Stability 81
3.4.2 Internal Stability of LTI Systems 83
3.5 The Lyapunov Equation 84
3.6 The Zero-State Response 87
3.6.1 Impulse Response 88
3.7 Operations on State-Space Systems 89
3.7.1 Interconnections 90
3.7.2 Inversion 91
3.7.3 Similarity Transformations 91
3.8 Gramians 91
3.8.1 Observability 92
3.8.2 Controllability 93
3.8.3 Minimal Realizations 95
3.9 Reciprocal State-Space Systems 95
3.10 Norms 97
3.10.1 L2 Norm 98
3.10.2 H? Norm 99
Problems 100
4 Distributed LTI Systems 103
4.1 One-Dimensional Distributed Circuits 104
4.1.1 The Discrete-Space Case 104
4.1.2 The Continuous-Space Case 106
4.1.3 Discussion 109
4.2 Two-Dimensional Distributed Circuits? 111
4.2.1 The Discrete-Space Case 112
4.2.2 The Continuous-Space Case 114
4.2.3 A Closed-Form Solution 116
4.2.4 Spatial Discretization 118
4.2.5 Discussion 120
4.3 General Electromagnetic Characterization 123
4.3.1 3D Electromagnetic Modeling 126
4.3.2 Summary and Outlook 130
Problems 131
5 Macromodeling Via Model Order Reduction 135
5.1 Model Order Reduction 135
5.2 Moment Matching 136
5.2.1 Moments 136
5.2.2 Pade Approximation and AWE 138
5.2.3 Complex Frequency Hopping 139
5.3 Reduction by Projection 140
5.3.1 Krylov Subspaces 141
5.3.2 Implicit Moment Matching: The Orthogonal Case 142
5.3.3 The Arnoldi Process 143
5.3.4 PRIMA 145
5.3.5 Multipoint Moment Matching 147
5.3.6 An Example 148
5.3.7 Implicit Moment Matching: The Biorthogonal Case 151
5.3.8 Pade Via Lanczos (PVL) 154
5.4 Reduction by Truncation 155
5.4.1 Balancing 156
5.4.2 Balanced Truncation 158
5.5 Advanced Model Order Reduction? 159
5.5.1 Passivity-Preserving Balanced Truncation 159
5.5.2 Balanced Truncation of Descriptor Systems 160
5.5.3 Reducing Large-Scale Systems 161
Problems 166
6 Black-Box Macromodeling and Curve Fitting 169
6.1 Basic Curve Fitting 171
6.1.1 Linear Least Squares 172
6.1.2 Maximum Likelihood Estimation 174
6.1.3 Polynomial Fitting 176
6.2 Direct Rational Fitting 182
6.2.1 Polynomial Ratio Form 183
6.2.2 Pole-Zero Form 183
6.2.3 Partial Fraction Form 184
6.2.4 Partial Fraction Form with Fixed Poles 184
6.2.5 Nonlinear Least Squares 185
6.3 Linearization via Weighting 187
6.4 Asymptotic Pole-Zero Placement 191
6.5 ARMA Modeling 193
6.5.1 Modeling from Time-Domain Responses 195
6.5.2 Modeling from Frequency Domain Responses 197
6.5.3 Conversion of ARMA Models 201
6.6 Prony's Method 203
6.7 Subspace-Based Identification? 204
6.7.1 Discrete-Time State-Space Systems 204
6.7.2 Macromodeling from Impulse Response Samples 205
6.7.3 Macromodeling from Input-Output Samples 207
6.7.4 From Discrete-Time to Continuous-Time State-Space Models 210
6.7.5 Frequency-Domain Subspace Identification 211
6.7.6 Generalized Pencil-of-Function Methods 212
6.7.7 Examples 214
6.8 Loewner Matrix Interpolation? 215
6.8.1 The Scalar Case 216
6.8.2 The Multiport Case 218
Problems 222
7 The Vector Fitting Algorithm 225
7.1 The Sanathanan-Koerner Iteration 226
7.1.1 The Steiglitz-McBride Iteration 229
7.2 The Generalized Sanathanan-Koerner Iteration 231
7.2.1 General Basis Functions 231
7.2.2 The Partial Fraction Basis 233
7.3 Frequency-Domain Vector Fitting 234
7.3.1 A Simple Model Transformation 234
7.3.2 Computing the New Poles 236
7.3.3 The Vector Fitting Iteration 237
7.3.4 From GSK to VF 239
7.4 Consistency And Convergence 241
7.4.1 Consistency 241
7.4.2 Convergence 242
7.4.3 Formal Convergence Analysis 245
7.5 Practical VF Implementation 247
7.5.1 Causality, Stability, and Realness 247
7.5.2 Order Selection and Initialization 253
7.5.3 Improving Numerical Robustness 254
7.6 Relaxed Vector Fitting 256
7.6.1 Weight Normalization, Noise, and Convergence 256
7.6.2 Relaxed Vector Fitting 259
7.7 Tuning VF 264
7.7.1 Weighting and Error Control 264
7.7.2 High-Frequency Behavior 266
7.7.3 High-Frequency Constraints 268
7.7.4 DC Point Enforcement 269
7.7.5 Simultaneous Constraints 271
7.8 Time-Domain Vector Fitting 273
7.9 z-Domain Vector Fitting 278
7.10 Orthonormal Vector Fitting 281
7.10.1 Orthonormal Rational Basis Functions 281
7.10.2 The OVF Iteration 284
7.10.3 The OVF Pole Relocation Step 285
7.10.4 Finding Residues 286
7.11 Other Variants 288
7.11.1 Magnitude Vector Fitting 288
7.11.2 Vector Fitting with L1 Norm Minimization 291
7.11.3 Dealing with Higher Pole Multiplicities 293
7.11.4 Including Higher Order Derivatives 294
7.11.5 Hard Relocation of Poles 295
7.12 Notes on Overfitting and Ill-Conditioning 296
7.12.1 Exact Model Identification 296
7.12.2 Curve Fitting 297
7.13 Application Examples 299
7.13.1 Surface Acoustic Wave Filter 299
7.13.2 Subnetwork Equivalent 301
7.13.3 Transformer Modeling from Time-Domain Measurements 303
Problems 303
8 Advanced Vector Fitting for Multiport Problems 307
8.1 Introduction 307
8.2 Adapting VF to Multiple Responses 308
8.2.1 Pole Identification 308
8.2.2 Fast Vector Fitting 310
8.2.3 Residue Identification 311
8.3 Multiport Formulations 312
8.3.1 Single-Element Modeling: Multi-SISO Structure 314
8.3.2 Single-Column Modeling: Multi-SIMO Structure 316
8.3.3 Matrix Modeling: MIMO Structure 317
8.3.4 Matrix Modeling: Minimal Realizations 318
8.3.5 Sparsity Considerations 322
8.4 Enforcing Reciprocity 322
8.4.1 External Reciprocity 324
8.4.2 Internal Reciprocity? 325
8.5 Compressed Macromodeling 329
8.5.1 Data Compression 329
8.5.2 Compressed Rational Approximation 330
8.5.3 An Application Example 331
8.6 Accuracy Considerations 333
8.6.1 Noninteracting Models 333
8.6.2 Interacting Models, Scalar Case 334
8.6.3 Error Magnification in Multiport Systems 338
8.7 Overcoming Error Magnification 340
8.7.1 Elementwise Inverse Weighting 340
8.7.2 Diagonalization 342
8.7.3 Mode-Revealing Transformations 347
8.7.4 Modal Vector Fitting 356
8.7.5 External and Internal Ports 358
Problems 363
9 Passivity Characterization of Lumped LTI Systems 365
9.1 Internal Characterization of Passivity 365
9.1.1 A First Order Example 365
9.1.2 The Dissipation Inequality 367
9.1.3 Lumped LTI Systems 368
9.2 Passivity of Lumped Immittance Systems 368
9.2.1 Rational Positive Real Matrices 369
9.2.2 Extracting Purely Imaginary Poles 372
9.2.3 The Positive Real Lemma 376
9.2.4 Positive Real Functions Revisited 378
9.2.5 Popov Functions and Spectral Factorizations 379
9.2.6 Hamiltonian Matrices 381
9.2.7 Passivity Characterization via Hamiltonian Matrices 385
9.2.8 Determination of Local Passivity Violations 387
9.2.9 Quantification of Passivity Violations via Bisection 390
9.2.10 Quantification of Passivity Violations via Sampling 393
9.2.11 Frequency Transformations 394
9.2.12 Extended Hamiltonian Pencils 396
9.2.13 Generalized Hamiltonian Pencils 398
9.2.14 Positive Real Lemma for Descriptor Systems 399
9.3 Passivity of Lumped Scattering Systems 402
9.3.1 Rational Bounded Real Matrices 402
9.3.2 The Bounded Real Lemma 406
9.3.3 Bounded Real Functions Revisited 408
9.3.4 Popov Functions, Spectral Factorizations, and Hamiltonian Matrices 409
9.3.5 Passivity Characterization via Hamiltonian Matrices 410
9.3.6 Determination of Local Passivity Violations 413
9.3.7 Quantification of Passivity Violations via Bisection 416
9.3.8 Quantification of Passivity Violations via Sampling 420
9.3.9 Extended Hamiltonian Pencils 421
9.3.10 Generalized Hamiltonian Pencils 422
9.3.11 Bounded Real Lemma for Descriptor Systems 423
9.4 Advanced Passivity Characterization 426
9.4.1 On the Computation of Imaginary Hamiltonian Eigenvalues 426
9.4.2 Large-Scale Hamiltonian Eigenvalue Problems? 427
9.4.3 Half-Size Passivity Test Matrices 430
Problems 433
10 Passivity Enforcement of Lumped LTI Systems 437
10.1 Passivity Constraints for Lumped LTI Systems 437
10.1.1 Passive State-Space Immittance Systems 438
10.1.2 Passive State-Space Scattering Systems 439
10.2 State-Space Perturbation 440
10.2.1 Asymptotic Perturbation 441
10.2.2 Dynamic Perturbation 441
10.2.3 Input-State Perturbation 442
10.2.4 State-Output Perturbation 443
10.2.5 A Perturbation Strategy for Passivity Enforcement 444
10.3 Asymptotic Passivity Enforcement 445
10.3.1 Immittance Systems 445
10.3.2 Scattering Systems 446
10.4 Imaginary Poles of Immittance Systems 447
10.5 Local Passivity Enforcement 448
10.5.1 Local Passivity Constraints 449
10.5.2 Enforcing Local Passivity Constraints 454
10.6 Passivity Enforcement Via Hamiltonian Perturbation 460
10.6.1 Hamiltonian Perturbation of Immittance Systems 462
10.6.2 Hamiltonian Perturbation of Scattering Systems 464
10.6.3 Hamiltonian Perturbation Strategies 465
10.6.4 Slopes 468
10.6.5 Global Passivity Enforcement via Hamiltonian Perturbation 471
10.7 Linear Matrix Inequalities 474
10.7.1 Parameterizations 476
10.8 Computational Cost 477
10.9 Advanced Accuracy Control 478
10.9.1 Frequency-Selective Norms 478
10.9.2 Individual Response Weighting 480
10.9.3 Bandlimited Norms 481
10.9.4 Relative Norms 484
10.9.5 Data-Based Cost Functions 486
10.10 Least-Squares Residue Perturbation 487
10.10.1 Basic Residue Perturbation (RP) 487
10.10.2 Spectral Residue Perturbation (SRP) 492
10.10.3 Mode-Revealing Transformations 493
10.10.4 Modal Perturbation (MP) 494
10.10.5 Robust Iterations 495
10.11 Alternative Formulations 496
10.11.1 Passivity Constraints Based on H? norm? 496
10.11.2 Iterative Update by Fitting Passivity Violations 503
10.11.3 Pole Perturbation Approaches 505
10.11.4 Parameterization via Positive Fractions 506
10.12 Descriptor Systems? 508
10.12.1 Perturbation of Generalized Hamiltonian Pencils 508
10.12.2 Handling Singular Direct Coupling Terms 509
10.12.3 Proper Part Extraction 510
10.12.4 Handling Impulsive Terms 511
10.12.5 Accuracy Control 512
Problems 512
11 Time-Domain Simulation 517
11.1 Discretization of ODE Systems 518
11.2 Interconnection of Macromodels 520
11.3 Direct Convolution 522
11.3.1 Equivalent Circuit Implementations 524
11.3.2 Discussion 527
11.4 Interfacing State-Space Macromodels 528
11.4.1 Equivalent Circuit Interfaces 530
11.5 Interfacing Pole-Residue Macromodels 533
11.5.1 Scalar Single-Pole System 533
11.5.2 General Multiport High-Order Systems 535
11.5.3 Discussion 537
11.6 Equivalent Circuit Synthesis 537
11.6.1 Direct Admittance Synthesis 538
11.6.2 Direct State-Space Synthesis 541
11.6.3 Sparse Synthesis 543
11.6.4 Classical RLCT Synthesis? 545
Problems 559
12 Transmission Lines and Distributed Systems 563
12.1 Introduction 563
12.2 Multiconductor Transmission Lines 564
12.2.1 Per-Unit-Length Matrices 564
12.2.2 Frequency-Domain Solution via Modal Decomposition 566
12.2.3 Frequency-Domain Solution in the Physical Domain 570
12.3 Direct Macromodeling Approaches 573
12.3.1 Folded Line Equivalent Models 573
12.4 Lumped Segmentation Approaches 577
12.4.1 Segmenting 577
12.4.2 Topology-Based Methods 578
12.5 Matrix Rational Approximations 582
12.5.1 Pade Matrix Rational Approximations 583
12.5.2 Series Expansion into Eigenfunctions 586
12.6 Traveling Wave Formulations 590
12.6.1 Voltage Waves 591
12.6.2 Current Waves 592
12.6.3 Thevenin and Norton Equivalents 593
12.6.4 Terminal Admittance from Traveling Wave Model 593
12.6.5 Modal Traveling Waves 594
12.7 Lossless Traveling Wave Modeling 595
12.7.1 Delay Extraction for Lossless MTL 597
12.8 Traveling Wave Modeling of Scalar Lossy Transmission Lines 599
12.9 Representations Based on Multiple Reflections 601
12.9.1 The Delayed Vector Fitting Scheme 604
12.10 Basic Delay Extraction for Lossy MTL 606
12.11 Frequency-Dependent Traveling Wave Modeling 607
12.11.1 Modal Domain 608
12.11.2 Physical Domain 613
12.11.3 Delay Extraction and Optimization? 625
12.12 General Delayed-Rational Macromodeling 626
12.12.1 Delay Estimation 629
12.12.2 Passivity Enforcement 631
12.12.3 Equivalent Circuit Synthesis 637
12.13 Passivity of Traveling Wave Models? 638
12.14 Time-Domain Implementation for Traveling Wave Models 641
12.14.1 The Scalar Lossless Line 641
12.14.2 The Scalar Lossy Line 643
12.14.3 Lossy Multiconductor Transmission Lines 648
12.14.4 Examples 652
12.15 Discussion 657
Problems 658
13 Applications 663
13.1 Modeling for Signal and Power Integrity 663
13.1.1 Prelayout Analysis of Backplane Interconnects 664
13.1.2 Full Package Analysis 667
13.1.3 Full Board Analysis and Simulation 672
13.1.4 High-Speed Channel Modeling and Simulation 681
13.1.5 Model Extraction from Measurements 687
13.2 Computational Electromagnetics 691
13.2.1 Dynamic Subcell Models in Time-Domain Solvers 691
13.2.2 Automatic Stopping Criteria for Time-Domain Solvers 695
13.2.3 VF-Based Adaptive Frequency Sampling 698
13.3 Small-Signal Macromodels for RF and AMS Applications 701
13.4 Modeling for High-Voltage Power Systems 704
13.4.1 Subnetwork Equivalencing 705
13.4.2 Power Transformer Modeling from Frequency Sweep Measurements 708
13.4.3 Power Transformer Modeling from Manufacturer's White-Box Model 715
13.5 Fluid Transmission Lines 720
13.6 Mechanical Systems 726
13.7 Ship Motion in Irregular Seas 728
13.8 Summary 733
14 Summary and Outlook 735
14.1 Parameterized Macromodels 735
14.1.1 Parameterized Macromodels with Fixed Poles 736
14.1.2 Fully Parameterized Macromodels 738
14.1.3 Higher Dimensional Parameter Spaces 742
14.2 Open Issues 743
14.2.1 Optimal Passivity Enforcement 743
14.2.2 Systems with Many Ports 744
14.2.3 White-Box Model Identification and Tuning 744
14.2.4 Transmission Line Models 745
14.2.5 Delay Systems 746
14.2.6 Extension to NL Systems 749
14.2.7 Integration with other solvers 749
Appendix A Notation 751
Appendix B Acronyms 757
Appendix C Linear Algebra 761
Appendix D Optimization Templates 781
Appendix E Signals and Transforms 805
Bibliography 839
Index 863
