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This paper discusses a reduced-order modeling and simulation approach for fast transient power integrity verification at full system level. The reference structure is a complete power distribution network (PDN) from platform voltage regulator module (VRM) to multiple cores, including board, package, decoupling capacitors, and per-core fully integrated voltage regulators (FIVR). All blocks are characterized and known through high-fidelity models derived from first-principle solvers (full-wave electromagnetic and circuit-level extractions). The complexity of such detailed characterization grows very large and becomes intractable, especially for power integrity verification of massive multicore platforms subjected to real workload scenarios. We approach this problem by exploiting a multi-stage macromodeling and compression process, leading to a compact representation of the system dynamics in terms of a linearized state-space structure with multiple feedback loops from the FIVR controllers. The PDN macromodel is obtained through a data-driven approach starting from reference smallsignal frequency responses, obtaining a sparse and structured representation specifically designed to match the behavior of the reference system. The resulting compact model is then solved in time-domain very efficiently. Results on mobile and enterprise server benchmarks demonstrate a speedup in runtime up to 50× with respect to HSPICE, with negligible loss of accuracy.

Behavioral models are effective tools used to relieve the computational burden of large-scale system-level simulations. In electrical and electronic applications, the Vector Fitting (VF) iteration often represents the algorithm of choice for generating low order equivalent circuits for complex multiport components in a data-driven setting. Although accurate and reliable in general, macromodels generated via VF are inherently represented in terms of a rational approximation of one specific input-output transfer function of the structure under modeling, e.g., its scattering matrix. However, accuracy in the scattering representation does not necessarily imply a good accuracy when solving the macromodel in a system-level setting, under different termination conditions. In fact, the sensitivity of the macromodel with respect to its loading conditions may be large and needs to be addressed and controlled. In this work, we present a modified VF scheme that overcomes this issue, by introducing in the rational approximation algorithm the requirement that the macromodel remains accurate when interconnected with a known class of admissible networks. The proposed formulation is based on an augmentation of the cost function minimized at each VF iteration; further, it does not require additional expensive data gathering steps when compared to standard approaches. The effectiveness of the scheme is tested over a set of relevant examples, in particular for Power Integrity applications. I. INTRODUCTIONVector Fitting (VF) [1] is currently the most common algorithm for the identification of behavioral models of Linear and Time-Invariant (LTI) systems for electrical and electronic applications. Similarly to other data-driven macromodeling approaches [2], [3], VF generates a reduced order network based on the availability of a collection of frequency [1] or time domain [4], [5] measurements, that characterize the behavior of a target system at its electrical ports. The modeling stage relies on solving a sequence of rational fitting problems, with the aim of generating a model that matches one specific transfer function of the underlying system, typically, its impedance, admittance or scattering matrix. The residual fitting error of a VF model is practically always extremely small, ensuring that the model accurately reproduces the prescribed target transfer function.Since its original formulation [1], several VF improvements have been documented, including efficient methods for handling large electrical systems [6] and related implementation to multicore computing hardware [7], [8], increasing the A.

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