Reduced-order macromodeling of complex multiport interconnects

This paper illustrates a technique for the generation of reduced-order lumped macromodels of linear distributed interconnect structures. The method is based on a robust state-space identification algorithm using as raw data discrete-time sequences of input/output waveforms at the accessible ports of the structure. Such sequences can be the result of a full-wave transient electromagnetic simulation using finite differences or finite elements. The poles describing the dominant dynamics of the structure are determined very accurately, allowing for the estimation of a state-space representation of the macromodel. The latter can be easily transformed into equivalent circuits for reduced-order system-level simulations. INTRODUCTION The high complexity of modern electronic systems calls for simplified modeling tools in order to perform systemlevel simulations for Signal Integrity and Electromagnetic Compatibility applications. Indeed, it is widely recognized that a system-level 3D electromagnetic simulation, including the effects of nonlinear drivers/receivers, is non-feasible even with today’s powerful computing tools. Macromodeling techniques tackle the modeling problem under a different perspective. Various subparts of the system are characterized separately at their accessible ports, either via numerical simulation or direct measurement. This procedure can be applied, e.g., to linear interconnects, junctions, packages, or connectors. Macromodeling techniques allow for the generation of simple equivalents starting from such characterizations. The macromodels mimic the port behavior of the structure and can be easily synthesized as SPICE-like subcircuits for system-level simulations, in order to include nonlinear effects of drivers/receivers. Several approaches have been presented in the very recent literature for the generation of macromodels. Such methods usually aim at the identification of a set of dominant poles of the Device Under Modeling, henceforth DUM, allowing for reduced-order approximations that are closely related to its actual dynamics. Some methods can be applied to process frequency-domain measurements/characterizations through rational functions approximations, usually performed via nonlinear least squares algorithms or by vector fitting [4]. Some other methods process the large matrices stemming from a full-wave discretization of Maxwell’s equations aiming at the construction of a smaller (reduced-order) system capable of preserving the DUT behavior over a prescribed bandwidth [2]. A third class of methods processes measured or simulated time-domain waveforms of input/output port responses. The subject of this paper is related to this latter class of techniques. Herewith we focus on the reduced-order macromodeling of complex interconnects and packaging structures from either measured or simulated transient port scattering waveforms. In particular, we address and compare several different algorithms trying to identify which are most suitable for the characterization of real-world structures having a very large number of ports. The two main algorithms that are considered in this work are the Block Complex Frequency Hopping (BCFH) algorithm [3] and the Subspace-based State-Space System Identification (4SID) techniques [6]. The BCFH method is a well-known technique, which has been extensively used for several applications. Therefore, we will use BCFH mainly for comparisons, pointing the reader to the vast literature on the subject, and we will focus on 4SID techniques The 4SID methods determine a state-space representation of the DUM via direct identification of the state matrices {A,B,C,D}. The poles distribution is a byproduct of this one-step procedure, since such state-space representation can be directly used to synthesize equivalent circuits. The identification algorithm makes use of highly reliable numerical tools (QR and SVD decompositions), and proves less sensitive to time-domain truncation of the port signals with respect to BCFH. Next section outlines the main 4SID-based algorithm that we propose for the specific macromodeling application. Some validations and numerical examples will follow.