Passive Macromodeling

Stefano Grivet-Talocia, PhD, is an Associate Professor of Circuit Theory at the Politecnico di Torino in Turin, Italy, and President of IdemWorks. Dr. Grivet-Talocia is author of over 150 technical papers published in international journals and conference proceedings. He invented several algorithms in the area of passive macromodeling, making them available through IdemWorks.Bjørn Gustavsen, PhD, is a Chief Research Scientist in Energy Systems at SINTEF Energy Research in Trondheim, Norway. More than ten years ago, Dr. Gustavsen developed the original version of the vector fitting method with Prof. Semlyen at the University of Toronto. The vector fitting method is one of the most widespread approaches for model extraction. Dr. Gustavsen is also an IEEE fellow.

• Preface xix1 Introduction 11.1 Why Macromodeling? 11.2 Scope 41.3 Macromodeling Flows 61.3.1 Macromodeling via Model Order Reduction 61.3.2 Macromodeling from Field Solver Data 71.3.3 Macromodeling from Measured Responses 81.4 Rational Macromodeling 91.5 Physical Consistency Requirements 111.6 Time-Domain Implementation 151.7 An Example 161.8 What Can Go Wrong? 172 Linear Time-Invariant Circuits and Systems 232.1 Basic Definitions 242.1.1 Linearity 242.1.2 Memory and Causality 262.1.3 Time Invariance 262.1.4 Stability 272.1.5 Passivity 282.2 Linear Time-Invariant Systems 282.2.1 Impulse Response 292.2.2 Properties of LTI Systems 322.3 Frequency-Domain Characterizations 332.4 Laplace and Fourier Transforms 342.4.1 Bilateral Laplace Transform and Transfer Matrices 342.4.2 Causal LTI Systems and the Unilateral Laplace Transform 362.4.3 Fourier Transform 362.5 Signal and System Norms∗ 372.5.1 Signal Norms 382.5.2 System Norms 412.6 Multiport Representations 442.6.1 Ports and Terminals 442.6.2 Immittance Representations 452.6.3 Scattering Representations 462.6.4 Reciprocity 482.7 Passivity 492.7.1 Power and Energy 502.7.2 Passivity and Causality 512.7.3 The Static Case 522.7.4 The Dynamic Case 532.7.5 Positive Realness Bounded Realness and Passivity 542.7.6 Some Examples 562.8 Stability and Causality 592.8.1 Laplace-Domain Conditions for Causality 612.8.2 Laplace-Domain Conditions for BIBO Stability 622.8.3 Causality and Stability 622.9 Boundary Values and Dispersion Relations∗ 642.9.1 Assumptions 642.9.2 Reconstruction of H(s) for s ∈ ℂ+ 652.9.3 Reconstruction of H(s) for s ∈ jℝ 652.9.4 Causality and Dispersion Relations 672.9.5 Generalizations 682.10 Passivity Conditions on the Imaginary Axis∗ 70Problems 713 Lumped LTI Systems 733.1 An Example from Circuit Theory 743.1.1 Variation on a Theme 763.1.2 Driving-Point Impedance 773.2 State-Space and Descriptor Forms 773.2.1 Singular Descriptor Forms 773.2.2 Internal Representations of Lumped LTI Systems 793.3 The Zero-Input Response 803.4 Internal Stability 813.4.1 Lyapunov Stability 813.4.2 Internal Stability of LTI Systems 833.5 The Lyapunov Equation 843.6 The Zero-State Response 873.6.1 Impulse Response 883.7 Operations on State-Space Systems 893.7.1 Interconnections 903.7.2 Inversion 913.7.3 Similarity Transformations 913.8 Gramians 913.8.1 Observability 923.8.2 Controllability 933.8.3 Minimal Realizations 953.9 Reciprocal State-Space Systems 953.10 Norms 973.10.1 L2 Norm 983.10.2 H∞ Norm 99Problems 1004 Distributed LTI Systems 1034.1 One-Dimensional Distributed Circuits 1044.1.1 The Discrete-Space Case 1044.1.2 The Continuous-Space Case 1064.1.3 Discussion 1094.2 Two-Dimensional Distributed Circuits∗ 1114.2.1 The Discrete-Space Case 1124.2.2 The Continuous-Space Case 1144.2.3 A Closed-Form Solution 1164.2.4 Spatial Discretization 1184.2.5 Discussion 1204.3 General Electromagnetic Characterization 1234.3.1 3D Electromagnetic Modeling 1264.3.2 Summary and Outlook 130Problems 1315 Macromodeling Via Model Order Reduction 1355.1 Model Order Reduction 1355.2 Moment Matching 1365.2.1 Moments 1365.2.2 Padé Approximation and AWE 1385.2.3 Complex Frequency Hopping 1395.3 Reduction by Projection 1405.3.1 Krylov Subspaces 1415.3.2 Implicit Moment Matching: The Orthogonal Case 1425.3.3 The Arnoldi Process 1435.3.4 PRIMA 1455.3.5 Multipoint Moment Matching 1475.3.6 An Example 1485.3.7 Implicit Moment Matching: The Biorthogonal Case 1515.3.8 Padé Via Lanczos (PVL) 1545.4 Reduction by Truncation 1555.4.1 Balancing 1565.4.2 Balanced Truncation 1585.5 Advanced Model Order Reduction∗ 1595.5.1 Passivity-Preserving Balanced Truncation 1595.5.2 Balanced Truncation of Descriptor Systems 1605.5.3 Reducing Large-Scale Systems 161Problems 1666 Black-Box Macromodeling and Curve Fitting 1696.1 Basic Curve Fitting 1716.1.1 Linear Least Squares 1726.1.2 Maximum Likelihood Estimation 1746.1.3 Polynomial Fitting 1766.2 Direct Rational Fitting 1826.2.1 Polynomial Ratio Form 1836.2.2 Pole–Zero Form 1836.2.3 Partial Fraction Form 1846.2.4 Partial Fraction Form with Fixed Poles 1846.2.5 Nonlinear Least Squares 1856.3 Linearization via Weighting 1876.4 Asymptotic Pole–Zero Placement 1916.5 ARMA Modeling 1936.5.1 Modeling from Time-Domain Responses 1956.5.2 Modeling from Frequency Domain Responses 1976.5.3 Conversion of ARMA Models 2016.6 Prony’s Method 2036.7 Subspace-Based Identification∗ 2046.7.1 Discrete-Time State-Space Systems 2046.7.2 Macromodeling from Impulse Response Samples 2056.7.3 Macromodeling from Input–Output Samples 2076.7.4 From Discrete-Time to Continuous-Time State-Space Models 2106.7.5 Frequency-Domain Subspace Identification 2116.7.6 Generalized Pencil-of-Function Methods 2126.7.7 Examples 2146.8 Loewner Matrix Interpolation∗ 2156.8.1 The Scalar Case 2166.8.2 The Multiport Case 218Problems 2227 The Vector Fitting Algorithm 2257.1 The Sanathanan–Koerner Iteration 2267.1.1 The Steiglitz–McBride Iteration 2297.2 The Generalized Sanathanan–Koerner Iteration 2317.2.1 General Basis Functions 2317.2.2 The Partial Fraction Basis 2337.3 Frequency-Domain Vector Fitting 2347.3.1 A Simple Model Transformation 2347.3.2 Computing the New Poles 2367.3.3 The Vector Fitting Iteration 2377.3.4 From GSK to VF 2397.4 Consistency And Convergence 2417.4.1 Consistency 2417.4.2 Convergence 2427.4.3 Formal Convergence Analysis 2457.5 Practical VF Implementation 2477.5.1 Causality Stability and Realness 2477.5.2 Order Selection and Initialization 2537.5.3 Improving Numerical Robustness 2547.6 Relaxed Vector Fitting 2567.6.1 Weight Normalization Noise and Convergence 2567.6.2 Relaxed Vector Fitting 2597.7 Tuning VF 2647.7.1 Weighting and Error Control 2647.7.2 High-Frequency Behavior 2667.7.3 High-Frequency Constraints 2687.7.4 DC Point Enforcement 2697.7.5 Simultaneous Constraints 2717.8 Time-Domain Vector Fitting 2737.9 z-Domain Vector Fitting 2787.10 Orthonormal Vector Fitting 2817.10.1 Orthonormal Rational Basis Functions 2817.10.2 The OVF Iteration 2847.10.3 The OVF Pole Relocation Step 2857.10.4 Finding Residues 2867.11 Other Variants 2887.11.1 Magnitude Vector Fitting 2887.11.2 Vector Fitting with L1 Norm Minimization 2917.11.3 Dealing with Higher Pole Multiplicities 2937.11.4 Including Higher Order Derivatives 2947.11.5 Hard Relocation of Poles 2957.12 Notes on Overfitting and Ill-Conditioning 2967.12.1 Exact Model Identification 2967.12.2 Curve Fitting 2977.13 Application Examples 2997.13.1 Surface Acoustic Wave Filter 2997.13.2 Subnetwork Equivalent 3017.13.3 Transformer Modeling from Time-Domain Measurements 303Problems 3038 Advanced Vector Fitting for Multiport Problems 3078.1 Introduction 3078.2 Adapting VF to Multiple Responses 3088.2.1 Pole Identification 3088.2.2 Fast Vector Fitting 3108.2.3 Residue Identification 3118.3 Multiport Formulations 3128.3.1 Single-Element Modeling: Multi-SISO Structure 3148.3.2 Single-Column Modeling: Multi-SIMO Structure 3168.3.3 Matrix Modeling: MIMO Structure 3178.3.4 Matrix Modeling: Minimal Realizations 3188.3.5 Sparsity Considerations 3228.4 Enforcing Reciprocity 3228.4.1 External Reciprocity 3248.4.2 Internal Reciprocity∗ 3258.5 Compressed Macromodeling 3298.5.1 Data Compression 3298.5.2 Compressed Rational Approximation 3308.5.3 An Application Example 3318.6 Accuracy Considerations 3338.6.1 Noninteracting Models 3338.6.2 Interacting Models Scalar Case 3348.6.3 Error Magnification in Multiport Systems 3388.7 Overcoming Error Magnification 3408.7.1 Elementwise Inverse Weighting 3408.7.2 Diagonalization 3428.7.3 Mode-Revealing Transformations 3478.7.4 Modal Vector Fitting 3568.7.5 External and Internal Ports 358Problems 3639 Passivity Characterization of Lumped LTI Systems 3659.1 Internal Characterization of Passivity 3659.1.1 A First Order Example 3659.1.2 The Dissipation Inequality 3679.1.3 Lumped LTI Systems 3689.2 Passivity of Lumped Immittance Systems 3689.2.1 Rational Positive Real Matrices 3699.2.2 Extracting Purely Imaginary Poles 3729.2.3 The Positive Real Lemma 3769.2.4 Positive Real Functions Revisited 3789.2.5 Popov Functions and Spectral Factorizations 3799.2.6 Hamiltonian Matrices 3819.2.7 Passivity Characterization via Hamiltonian Matrices 3859.2.8 Determination of Local Passivity Violations 3879.2.9 Quantification of Passivity Violations via Bisection 3909.2.10 Quantification of Passivity Violations via Sampling 3939.2.11 Frequency Transformations 3949.2.12 Extended Hamiltonian Pencils 3969.2.13 Generalized Hamiltonian Pencils 3989.2.14 Positive Real Lemma for Descriptor Systems 3999.3 Passivity of Lumped Scattering Systems 4029.3.1 Rational Bounded Real Matrices 4029.3.2 The Bounded Real Lemma 4069.3.3 Bounded Real Functions Revisited 4089.3.4 Popov Functions Spectral Factorizations and Hamiltonian Matrices 4099.3.5 Passivity Characterization via Hamiltonian Matrices 4109.3.6 Determination of Local Passivity Violations 4139.3.7 Quantification of Passivity Violations via Bisection 4169.3.8 Quantification of Passivity Violations via Sampling 4209.3.9 Extended Hamiltonian Pencils 4219.3.10 Generalized Hamiltonian Pencils 4229.3.11 Bounded Real Lemma for Descriptor Systems 4239.4 Advanced Passivity Characterization 4269.4.1 On the Computation of Imaginary Hamiltonian Eigenvalues 4269.4.2 Large-Scale Hamiltonian Eigenvalue Problems∗ 4279.4.3 Half-Size Passivity Test Matrices 430Problems 43310 Passivity Enforcement of Lumped LTI Systems 43710.1 Passivity Constraints for Lumped LTI Systems 43710.1.1 Passive State-Space Immittance Systems 43810.1.2 Passive State-Space Scattering Systems 43910.2 State-Space Perturbation 44010.2.1 Asymptotic Perturbation 44110.2.2 Dynamic Perturbation 44110.2.3 Input-State Perturbation 44210.2.4 State-Output Perturbation 44310.2.5 A Perturbation Strategy for Passivity Enforcement 44410.3 Asymptotic Passivity Enforcement 44510.3.1 Immittance Systems 44510.3.2 Scattering Systems 44610.4 Imaginary Poles of Immittance Systems 44710.5 Local Passivity Enforcement 44810.5.1 Local Passivity Constraints 44910.5.2 Enforcing Local Passivity Constraints 45410.6 Passivity Enforcement Via Hamiltonian Perturbation 46010.6.1 Hamiltonian Perturbation of Immittance Systems 46210.6.2 Hamiltonian Perturbation of Scattering Systems 46410.6.3 Hamiltonian Perturbation Strategies 46510.6.4 Slopes 46810.6.5 Global Passivity Enforcement via Hamiltonian Perturbation 47110.7 Linear Matrix Inequalities 47410.7.1 Parameterizations 47610.8 Computational Cost 47710.9 Advanced Accuracy Control 47810.9.1 Frequency-Selective Norms 47810.9.2 Individual Response Weighting 48010.9.3 Bandlimited Norms 48110.9.4 Relative Norms 48410.9.5 Data-Based Cost Functions 48610.10 Least-Squares Residue Perturbation 48710.10.1 Basic Residue Perturbation (RP) 48710.10.2 Spectral Residue Perturbation (SRP) 49210.10.3 Mode-Revealing Transformations 49310.10.4 Modal Perturbation (MP) 49410.10.5 Robust Iterations 49510.11 Alternative Formulations 49610.11.1 Passivity Constraints Based on H∞ norm∗ 49610.11.2 Iterative Update by Fitting Passivity Violations 50310.11.3 Pole Perturbation Approaches 50510.11.4 Parameterization via Positive Fractions 50610.12 Descriptor Systems∗ 50810.12.1 Perturbation of Generalized Hamiltonian Pencils 50810.12.2 Handling Singular Direct Coupling Terms 50910.12.3 Proper Part Extraction 51010.12.4 Handling Impulsive Terms 51110.12.5 Accuracy Control 512Problems 51211 Time-Domain Simulation 51711.1 Discretization of ODE Systems 51811.2 Interconnection of Macromodels 52011.3 Direct Convolution 52211.3.1 Equivalent Circuit Implementations 52411.3.2 Discussion 52711.4 Interfacing State-Space Macromodels 52811.4.1 Equivalent Circuit Interfaces 53011.5 Interfacing Pole-Residue Macromodels 53311.5.1 Scalar Single-Pole System 53311.5.2 General Multiport High-Order Systems 53511.5.3 Discussion 53711.6 Equivalent Circuit Synthesis 53711.6.1 Direct Admittance Synthesis 53811.6.2 Direct State-Space Synthesis 54111.6.3 Sparse Synthesis 54311.6.4 Classical RLCT Synthesis∗ 545Problems 55912 Transmission Lines and Distributed Systems 56312.1 Introduction 56312.2 Multiconductor Transmission Lines 56412.2.1 Per-Unit-Length Matrices 56412.2.2 Frequency-Domain Solution via Modal Decomposition 56612.2.3 Frequency-Domain Solution in the Physical Domain 57012.3 Direct Macromodeling Approaches 57312.3.1 Folded Line Equivalent Models 57312.4 Lumped Segmentation Approaches 57712.4.1 Segmenting 57712.4.2 Topology-Based Methods 57812.5 Matrix Rational Approximations 58212.5.1 Padé Matrix Rational Approximations 58312.5.2 Series Expansion into Eigenfunctions 58612.6 Traveling Wave Formulations 59012.6.1 Voltage Waves 59112.6.2 Current Waves 59212.6.3 Thévenin and Norton Equivalents 59312.6.4 Terminal Admittance from Traveling Wave Model 59312.6.5 Modal Traveling Waves 59412.7 Lossless Traveling Wave Modeling 59512.7.1 Delay Extraction for Lossless MTL 59712.8 Traveling Wave Modeling of Scalar Lossy Transmission Lines 59912.9 Representations Based on Multiple Reflections 60112.9.1 The Delayed Vector Fitting Scheme 60412.10 Basic Delay Extraction for Lossy MTL 60612.11 Frequency-Dependent Traveling Wave Modeling 60712.11.1 Modal Domain 60812.11.2 Physical Domain 61312.11.3 Delay Extraction and Optimization∗ 62512.12 General Delayed-Rational Macromodeling 62612.12.1 Delay Estimation 62912.12.2 Passivity Enforcement 63112.12.3 Equivalent Circuit Synthesis 63712.13 Passivity of Traveling Wave Models∗ 63812.14 Time-Domain Implementation for Traveling Wave Models 64112.14.1 The Scalar Lossless Line 64112.14.2 The Scalar Lossy Line 64312.14.3 Lossy Multiconductor Transmission Lines 64812.14.4 Examples 65212.15 Discussion 657Problems 65813 Applications 66313.1 Modeling for Signal and Power Integrity 66313.1.1 Prelayout Analysis of Backplane Interconnects 66413.1.2 Full Package Analysis 66713.1.3 Full Board Analysis and Simulation 67213.1.4 High-Speed Channel Modeling and Simulation 68113.1.5 Model Extraction from Measurements 68713.2 Computational Electromagnetics 69113.2.1 Dynamic Subcell Models in Time-Domain Solvers 69113.2.2 Automatic Stopping Criteria for Time-Domain Solvers 69513.2.3 VF-Based Adaptive Frequency Sampling 69813.3 Small-Signal Macromodels for RF and AMS Applications 70113.4 Modeling for High-Voltage Power Systems 70413.4.1 Subnetwork Equivalencing 70513.4.2 Power Transformer Modeling from Frequency Sweep Measurements 70813.4.3 Power Transformer Modeling from Manufacturer’s White-Box Model 71513.5 Fluid Transmission Lines 72013.6 Mechanical Systems 72613.7 Ship Motion in Irregular Seas 72813.8 Summary 73314 Summary and Outlook 73514.1 Parameterized Macromodels 73514.1.1 Parameterized Macromodels with Fixed Poles 73614.1.2 Fully Parameterized Macromodels 73814.1.3 Higher Dimensional Parameter Spaces 74214.2 Open Issues 74314.2.1 Optimal Passivity Enforcement 74314.2.2 Systems with Many Ports 74414.2.3 White-Box Model Identification and Tuning 74414.2.4 Transmission Line Models 74514.2.5 Delay Systems 74614.2.6 Extension to NL Systems 74914.2.7 Integration with other solvers 749Appendix A Notation 751Appendix B Acronyms 757Appendix C Linear Algebra 761Appendix D Optimization Templates 781Appendix E Signals and Transforms 805Bibliography 839Index 863