The electrical performance of Power Distribution Networks (PDNs) is usually assessed by computing frequency responses through quasi-static or full-wave electromagnetic solvers. Such responses, often available in the scattering form, are then fed to suitable macromodeling algorithms for the extraction of compact reduced-order behavioral models that can be seemlessly simulated in the time domain by standard circuit solvers. Such algorithms perform a rational fitting of the raw scattering responses, followed by a passivity check and enforcement step. The resulting macromodel is typically very accurate when compared to the raw scattering responses. It may however happen that the responses of the PDN macromodel exhibit significant deviation from the true system responses under realistic loading conditions, which include appropriate models for active device blocks, decoupling capacitors, voltage regulators, etc. We highlight the source of this accuracy loss, and we propose a sensitivity-based weighting strategy that is able to optimize and tune the macromodel accuracy based on its specific nominal termination network. The particular focus of this paper is the definition and the inclusion of optimal weigths in the passivity enforcement loop, which is recognized as the most challenging step. The result is a reliable macromodeling flow, which is able to produce passive, accurate and efficient reduced-order models of general PDN structures for power integrity analysis and verification.

This paper describes a novel approach for passivity enforcement of compact dynamical models of electrical interconnects. The proposed approach is based on a parameterization of general state-space scattering models with fixed poles. We formulate the passivity constraints as a unitary boundedness condition on the H∞norm of the system transfer function. When this condition is not verified, we use it as an explicit constraint within an iterative perturbation loop of the system state-space matrices. Since the resulting optimization framework is convex but nonsmooth, we solve it via localization based algorithms, such as the ellipsoid and the cutting plane methods. The proposed technique solves two critical bottleneck issues of the existing approaches for passivity enforcement of linear macromodels. Compared to quasi-optimal schemes based on singular value or Hamiltonian eigenvalue perturbation, we are able to guarantee convergence to the optimal solution. Compared to convex formulations based on direct Bounded Real Lemma constraints, we are able to reduce both memory and time requirements by orders of magnitude. We demonstrate the effectiveness of our approach on a number of cases for which existing algorithms either fail or exhibit very slow convergence.

The passivity characterization and enforcement of linear interconnect macromodels has received much attention in the recent literature. It is now widely recognized that the Hamiltonian eigensolution is a very reliable technique for such characterization. However, most available algorithms for the determination of the required Hamiltonian eigenvalues still require excessive comoputational resources for large-size macromodels with thousands of states. This work intends to break this complexity by introducing the first parallel implementation of a specialized Hamiltonian eigensolver, designed and optimized for shared memory multicore architectures. Our starting point is a multi-shift restarted and deflated Arnoldi process. Excellent parallel efficiency is obtained by running different Arnoldi iterations concurrently on different threads. The numerical results show that macromodels with several thousands states are characterized in few seconds on a 16-core machine, with close to ideal speedup factors.

This paper describes a robust and accurate black-box macromodeling technique, in which the constitutive equations combine both closed-form delay operators and low-order rational coefficients. These models describe efficiently electrically long interconnect links. The algorithm is based on an iterative weighted least-squares process and can be interpreted as a generalization of the well-known Vector Fitting. The paper is focused, in particular, on the passivity enforcement of these models. We present two perturbation methods and we show how the accuracy of the models is well-preserved during the passivity enforcement process.