Passivity Enforcement Research Articles

Admittance-Based Modeling of Transmission Lines by a Folded Line Equivalent

This paper describes a new transmission-line model for frequency-dependent modeling of untransposed overhead lines and underground cables. The nodal admittance matrix is decomposed into two blocks that, respectively, represent the open- and short-circuit conditions of a half-length line obtained by ldquofoldingrdquo about the middle. By subjecting these matrices to rational fitting with inverse magnitude weighting, one obtains a model where the eigenvalues of the associated nodal admittance matrix are effectively fitted with high relative accuracy. This is shown to overcome the error magnification problem that occurs with direct fitting of the nodal admittance matrix. In addition, the modeling process (fitting and passivity enforcement) becomes faster. We show that this folded line equivalent (FLE) is particularly suitable as a companion form for phase-domain traveling-wave-type models, to be used when the time step is selected shorter than the line travel time. In this situation, the required model order is low and so the FLE gives highly efficient time-domain simulations.

Modal Vector Fitting: A Tool For Generating Rational Models of High Accuracy With Arbitrary Terminal Conditions

This paper introduces a new approach for rational macromodeling of multiport devices that ensures high accuracy with arbitrary terminal conditions. This is achieved by reformulating the vector fitting (VF) technique to focus on eigenpairs rather than matrix elements. By choosing the least squares (LS) weighting equal to the inverse of the eigenvalue magnitude, the modal components are fitted with a relative accuracy criterion. The resulting modal vector fitting (MVF) method is shown to give a major improvement in accuracy for cases with a high ratio between the largest and smallest eigenvalue, although it is computationally more costly than VF. It is also shown how to utilize the impedance characteristics of the adjacent network in the fitting process. The application of MVF is demonstrated for a two-conductor stripline, a coaxial cable, and a transformer measurement. We also show a simplified procedure which achieves similar results as MVF if the admittance matrix can be diagonalized by a constant transformation matrix. The extracted model is finally subjected to passivity enforcement by the modal perturbation method, which makes use of a similar LS formulation as MVF for the constrained optimization problem.

A Comparative Study of Passivity Enforcement Schemes for Linear Lumped Macromodels

This paper presents a comparative study of several passivity enforcement schemes for linear lumped macromodels. We consider three main classes of algorithms. First class is represented by those methods based on a direct enforcement of positive/bounded real lemma constraints via convex optimization. Second class includes those algorithms that enforce the passivity constraints at discrete frequency samples. These schemes are here formulated as second-order cone programs in order to optimize performance. Finally, we consider algorithms based on Hamiltonian eigenvalue perturbation. These three classes are applied to a significant set of benchmark examples, essentially various kinds of high-speed interconnects and packages, with the aim of comparing their performance in terms of accuracy, efficiency, applicability, and robustness. These examples are specifically selected in order to be critical for one or more algorithms, in terms of excessive accuracy degradation, computational complexity, or even lack of convergence. One important result is that carefully designed weighting schemes may dramatically improve performance for all considered algorithm classes.

Stability and Passivity Enforcement of Parametric Macromodels in Time and Frequency Domain

This paper presents a robust technique for the macromodeling of time-domain and frequency-domain responses, which are parameterized by one or more design variables. The representation of the multivariate macromodel ensures that stability of the transfer function poles is enforced by construction. Passivity of the parametric macromodel can be enforced in a post-processing step by perturbation of barycentric weights.

Fast Passivity Enforcement for Pole-Residue Models by Perturbation of Residue Matrix Eigenvalues

Rational models must be passive in order to avoid unstable time domain simulations. This paper introduces a fast approach for passivity enforcement of pole-residue models. This is achieved by perturbing the eigenvalues of the residue matrices, as opposed to the existing approach of perturbing matrix elements. This leads to large savings in computation time with only a small increase of the modeling error. This fast residue perturbation (FRP) approach is merged with the modal perturbation technique, leading to fast modal perturbation (FMP). Usage of FMP over FRP achieves to retain the relative accuracy of the admittance matrix eigenvalues. A complete approach is obtained by combining the passivity enforcement step with passivity assessment via the Hamiltonian matrix eigenvalues and a robust iteration scheme, giving a guaranteed passive model. Application of FMP to a six-port power transformer shows that the approach is able to remove large out-of band passivity violations without corrupting the in-band behavior. This is shown to mitigate an unstable simulation. The approach is also demonstrated for a high-speed interconnect and a transmission line.

Passivity Enforcement of Transfer Functions

Passivity of an identified model is crucial in simulation and prediction applications. This paper proposes a two-step procedure that generates guaranteed passive transfer function models from noisy data. It consists of an unconstrained optimal noise removal step, followed by passivity enforcement of the unconstrained model.

Passivity Enforcement for Transmission Line Models Based on the Method of Characteristics

The universal line model (ULM) has been implemented in several EMT programs for simulation of electromagnetic transients. In some cases, instability problems have been encountered. This paper shows that the current approach for rational function approximation adopted in ULM can lead to large out-of-band passivity violations, thereby causing an unstable simulation. An approach is introduced which prevents the occurrence of large passivity violations. Low-frequency violations are avoided by adding an artificial shunt conductance to the diagonal elements of the shunt admittance matrix while high-frequency violations are avoided by introducing artificial attenuation using a low-pass filter. In addition, high-frequency asymptotic passivity is enforced for the characteristic admittance. Any remaining violations are removed by adding a second-order correction term to the model ports. The approach is shown to mitigate instabilities from a cable system transient simulation, without impairing the quality of the model.

Passivity Enforcement Using an Infeasible-Interior-Point Primal-Dual Method

Application of the network equivalent concept for external system representation in electromagnetic transient studies is well known. However, the challenge in application of an equivalent model, approximated by rational functions, is to guarantee passivity of the corresponding model. Therefore, there is a need to enforce passivity of the equivalent model. In this paper, the passivity enforcement problem is first formulated as a <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">quadratic</i> <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">programming</i> (QP) problem, and then solved based on an efficient implementation, using an infeasible-interior-point primal-dual method. An application of this method for passivity enforcement of a six-port admittance model (with large passivity violations) demonstrates its effectiveness and computational efficiency.

Perturbation Schemes for Passivity Enforcement of Delay-Based Transmission Line Macromodels

This paper introduces two different algorithms for passivity enforcement of transmission-line macromodels based on the generalized method of characteristics (MoC). The first scheme corrects passivity violations via perturbation of purely imaginary solutions of a nonlinear Hamiltonian eigenvalue problem. The second scheme is based on a linearly constrained quadratic optimization formulated at carefully selected frequency samples. Both schemes can be viewed as extensions to the case of MoC-based macromodels of existing techniques that were only available for lumped macromodels. The resulting passive delay-based macromodels can be safely employed in system-level signal integrity simulations, due to their inability to produce any spurious energy gain. Several numerical examples illustrate the behavior of both schemes on transmission line structures of practical interest, highlighting their main differences and the open problems deserving further research work.

Radiation admittance of an open‐ended parallel plate metallic waveguide with exterior planar dielectric coating

A methodology is presented for the calculation of the radiation admittance of an open‐ended, homogeneously filled, parallel plate metallic waveguide with exterior planar dielectric coating, operated at the fundamental (transverse electromagnetic wave) mode. The methodology relies on the definition of an auxiliary electromagnetic boundary value problem that is simple to solve using an integral equation approach. In order to provide for broadband calculation of the radiation admittance of the open‐ended waveguide from DC to several tens of gigahertz, a loop‐tree decomposition scheme is used in the method of moments solution of the resulting integral equation. A rational function‐fitting process with passivity enforcement is used to generate a broadband, passive macromodel for the calculated radiation impedance. Such a macromodel is suitable for use as a local truncation boundary condition in the finite difference/finite element modeling of electromagnetic interactions in multilayered signal and power distribution networks encountered in printed circuit boards and packages.

Passivity Enforcement of Rational Models via Modal Perturbation

Most methods for passivity enforcement of rational models are based on perturbing the model parameters in a postprocessing step. In some situations, even a small perturbation can result in a significant change to model behavior, depending on the model terminal conditions and the admittance matrix eigenvalues. This paper presents a new perturbation method that is significantly less prone to model corruption by focusing on the admittance matrix eigenvalues (modes), instead of elements. This is achieved by enforcing that the relative change to the eigenvalues is minimized in the least-squares sense. By combining this approach with precise passivity checking via the Hamiltonian matrix, a guaranteed passive model is obtained. The new approach (modal perturbation) is compared to previously developed methods in terms of perturbation size, model behavior, and computational efficiency.

Passivity Enforcement With Relative Error Control

This paper introduces a new error control strategy in passivity enforcement schemes for linear lumped macromodels. We consider the general class of a posteriori passivity enforcement algorithms based on Hamiltonian matrix perturbation. Standard available formulations preserve the accuracy during passivity enforcement using special matrix norms associated to the controllability Gramian of the macromodel. This procedure leads to absolute error control. On the other hand, it is well known that relative error control in the macromodel is sometimes preferable, especially for structures that are characterized by small coupling coefficients or high dynamic range in their responses. Here, we present a frequency-weighting scheme leading to the definition of a modified Gramian that, when employed during passivity enforcement, effectively leads to relative error control. Several examples illustrate the reliability of the proposed technique.

TermMerg: An Efficient Terminal-Reduction Method for Interconnect Circuits

In this paper, a novel method to efficiently reduce the terminal number of general linear-interconnect circuits with a large number of input or output terminals considering delay uncertainty is proposed. Our new algorithm is motivated by the fact that terminal reduction can lead to a more compact order-reduced model and the observation that very large-scale integration interconnect circuits have many similar terminals in terms of their timing and delay metrics due to their closeness in structure or due to the mathematical discretization using meshing in finite-difference or finite-element scheme during the extraction process. The new method, called TermMerg ( Proc. ICCAD, p. 821, 2005), is based on the moments of the circuits as the metrics for the timing or delay. It then employs a singular-value-decomposition (SVD) method to determine the best number of clusters based on the low-rank approximation. After this, the -means clustering algorithm is used to cluster the moments of the terminals into the different clusters. The proposed method can work with any passive-model order reduction and ensure the passive models. In contrast, we show that singular value decomposition model order reduction (SVDMOR) does not generate passive models in general. Passivity enforcement in SVDMOR will significantly hamper the terminal-reduction effectiveness. Experimental results on a number of real industry interconnect circuits demonstrate the effectiveness of the proposed method and show also that the proposed method is more accurate than SVDMOR when the used moment matrix does not give good terminal correlations.

A Passivity Enforcement Scheme for Delay-Based Transmission Line Macromodels

This letter presents an algorithm for the enforcement of passivity in transmission-line macromodels based on the generalized method of characteristics. The algorithm is based on first-order perturbations of the solutions to a frequency-dependent eigenvalue problem. The theoretical formulation provides an extension of techniques that were available only for lumped models to the more general class of delay-based models.

Computer Code for Passivity Enforcement of Rational Macromodels by Residue Perturbation

Passivity of rational macromodels is a necessary requirement for guaranteeing a stable time-domain simulation. This paper describes a freely available routine for passivity enforcement of multiport models (y parameters) using residue perturbation. The procedure calculates a correction to the original model by adjusting its residues and constant term as a postprocessing step while minimizing the model perturbation in the least squares (LS) sense. Control parameters are available for limiting the number of free variables and for specifying different LS weighting schemes. Two different constraint types can be specified, which helps to ensure reliable convergence when applied in an iterative scheme. Usage of a sparse solver is shown to greatly reduce the computation time. The software is demonstrated by several numerical examples

Identification of broadband passive macromodels for electromagnetic structures

PurposeThe paper aims to present an overview of techniques for the identification in the frequency domain of reduced order models for distributed passive electromagnetic structures.Design/methodology/approachMost known approaches proposed in different application contexts are described within a unified framework.FindingsA passive reduced order model of an unshielded twisted pair is fully developed with the combination of vector fitting algorithm and the passivity enforcement via Hamiltonian perturbation.Originality/valueA state‐of‐the‐art picture of the frequency domain identification and passivity enforcement techniques is given, and a test case of actual interest fully analysed.

Equivalent SPICE Circuits With Guaranteed Passivity From Nonpassive Models

In this paper, a new fast technique of passivity enforcement of a nonpassive rational model is introduced. The technique disturbs poles of the model to restore passivity in such way that the frequency response of a device being modeled is preserved. The passivity enforcement procedure is defined as an optimization routine with the gradients of the cost function evaluated using the theory of matrix perturbation. The rational model can be based either on passive (electromagnetic simulations, measurements) or nonpassive (surrogate models) data. In the second case, the proposed technique can lead to a parameterized SPICE networks. Some advanced examples are given to show the application of proposed approach in interconnect, packaging, and signal integrity analysis

On the Generation of Large Passive Macromodels for Complex Interconnect Structures

This paper addresses some issues related to the passivity of interconnect macromodels computed from measured or simulated port responses. The generation of such macromodels is usually performed via suitable least squares fitting algorithms. When the number of ports and the dynamic order of the macromodel is large, the inclusion of passivity constraints in the fitting process is cumbersome and results in excessive computational and storage requirements. Therefore, we consider in this work a post-processing approach for passivity enforcement, aimed at the detection and compensation of passivity violations without compromising the model accuracy. Two complementary issues are addressed. First, we consider the enforcement of asymptotic passivity at high frequencies based on the perturbation of the direct coupling term in the transfer matrix. We show how potential problems may arise when off-band poles are present in the model. Second, the enforcement of uniform passivity throughout the entire frequency axis is performed via an iterative perturbation scheme on the purely imaginary eigenvalues of associated Hamiltonian matrices. A special formulation of this spectral perturbation using possibly large but sparse matrices allows the passivity compensation to be performed at a cost which scales only linearly with the order of the system. This formulation involves a restarted Arnoldi iteration combined with a complex frequency hopping algorithm for the selective computation of the imaginary eigenvalues to be perturbed. Some examples of interconnect models are used to illustrate the performance of the proposed techniques.

Time Domain Fluid Transmission Line Modelling using a Passivity Preserving Rational Approximation of the Frequency Dependent Transfer Matrix

Flow and pressure transients in fluid transmission lines can be analysed starting from a modal approximation of the frequency domain irrational transfer matrix, relating pressure and flow rate at the line ends in Laplace transform. The obtained rational approximation can be converted in a state space representation and used in variable time step simulators to evaluate the influence of the line on fluid servosystems dynamics. Particular attention must be given to the causality, to the stability and to the energy passivity of the resulting line model.In this paper the application of a numerical approximation technique (Vector Fitting) to the frequency dependent transfer matrix describing the pipeline dynamics is proposed. The admittance matrix formulation is chosen, introducing an effective passivity enforcing technique, to ensure the energy passivity of the approximated matrix, thus preserving in the model the physical meaning of the real system.The rational approximation of the transfer matrix, combined with the passivity enforcement methodology, is applied to the study of the transient response of a single uniform line and of compound hydraulic line systems, showing the agreement between the simulation and the solution obtained with inverse fast Fourier transform.

Global passivity enforcement algorithm for macromodels of interconnect subnetworks characterized by tabulated data

With the continually increasing operating frequencies, complex high-speed interconnect and package modules require characterization based on measured/simulated data. Several algorithms were recently suggested for macromodeling such types of data to enable unified transient analysis in the presence of external network elements. One of the critical issues involved here is the passivity violations associated with the computed macromodel. To address this issue, a new passivity enforcement algorithm is presented in this paper. The proposed method adopts a global approach for passivity enforcement by ensuring that the passivity correction at a certain region does not introduce new passivity violations at other parts of the frequency spectrum. It also provides an error estimate for the response of the passivity corrected macromodel.