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.

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. Index Terms-Bounded real lemma, Hamiltonian matrices, inverse weighting, linear macromodeling, passivity, positive real lemma, second-order cone programming.

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This paper presents a new technique for the elimination of passivity violations in linear lumped macromodels. The main algorithm is based on the perturbation of imaginary eigenvalues of suitably-defined Hamiltonian matrices, as documented in the existing literature. We introduce a modification aimed at the minimization of the relative error in the model responses during the passivity enforcement. This strategy allows the accurate modeling of structures characterized by a large dynamic range, as typically found in microwave filters or advanced packaging applications. Introduction and motivationsLinear macromodeling provides a flexible and effective solution for fast and accurate simulation of complex interconnects. However, macromodel identification from time-domain or frequency-domain tabulated data [1]-[8] often leads to nonpassive results. Though possibly accurate, non-passive models may lead to unstable behavior when used in a CAD tool for system design and verification. Therefore, passivity should be enforced in some way during the model identification process.Despite the intense research efforts that have been devoted to the subject, the passivity enforcement of linear macromodels still poses several challenges. Currently available solutions can be grouped into tree main classes. On one hand, methods based on convex optimization [9]-[12] are guaranteed to find the optimal solution. Unfortunately, these techniques are limited to small-scale models due to their large computational complexity. A second class is provided by a posteriori passivity correction techniques based on linear or quadratic programming [13,14]. These methods are based on inequality constraints imposed at discrete frequency samples. They are applicable to larger-size models but are not guaranteed to fully enforce passivity. Finally, other methods exploit the theory of Hamiltonian matrices [15]- [18]. They are applicable to larger-size models and do provide a global passivity characterization and enforcement. However, the convergence is not always guaranteed and the solution they offer is only sub-optimal.All passivity enforcement techniques apply some perturbation to the model until its passivity is achieved. This perturbation is performed using special constraints insuring that the model accuracy is preserved. These constraints have always been formulated so that the absolute error in the responses is minimized, except for the very recent results in [19]. In this work, we present a method allowing for the systematic preservation of the relative error during the passivity enforcement. We show that the proposed technique leads to superior performance with respect to standard schemes in all cases characterized by responses with large dynamic range. This scenario is typical, e.g., in packaging applications and RF component modeling.

This paper compares alternative methods for linear modeling of simulated time-domain responses by a rational approximation. The traditional approach based on autoregressive moving average (ARMA) is compared with two alternative approaches that have been introduced in recent years: Z-Domain Vector Fitting (ZD-VF) and Time-Domain Vector Fitting (TD-VF). Following a description of their implementation and fundamental properties, the methods are applied to the modeling of a frequency-dependent network equivalent. It is shown that TD-VF offers superior results in terms of accuracy and robustness, and the model has guaranteed stable poles. The ARMA approach requires higher orders than the others and the resulting model can be unstable. The ZD-VF approach is robust and gives a model with stable poles, but it produces incorrect simulation results when applied to truncated time-domain responses. The inaccuracy is caused mainly by the conversion of time-domain data into the -domain. The fitting process of ARMA is faster than with the other approaches since it does not involve iterative pole relocations. Index Terms-Autoregressive moving average (ARMA), frequency-dependent network equivalent (FDNE), macromodel, rational approximation, time-domain vector fitting, -domain vector fitting.