Regularized and Compressed Large-Scale Rational Macromodeling: Theory and Application to Energy-Selective Shielding Enclosures

Abstract

In this article, we introduce a robust procedure for the extraction of passive rational macromodels of low-loss electromagnetic structures with massive port counts. Such structures pose inherent challenges that make standard macromodeling tools and approaches inadequate, mainly due to complexity and sensitivity at low frequency. The proposed approach involves a preprocessing stage in which port response data from a full-wave electromagnetic solver are regularized and extrapolated to dc using an asymptotic modal representation. The resulting data samples are then processed by a dedicated compression algorithm that represents the full set of input-output responses in terms of a few basis functions, which are constructed by enforcing an exact low-frequency modal asymptotic behavior, possibly including higher order dc zeros. These zeros are preserved in any stage of rational fitting and passivity enforcement, resulting in dc and low-frequency compliant compressed passive macromodels. Numerical results with up to 400 ports demonstrate the superior performance and accuracy of the computed models with respect to state-of-the-art approaches. In particular, the resulting models preserve their accuracy irrespective of the loading conditions, including the limit cases of short and open terminations.

A Compressed Multivariate Macromodeling Framework for Fast Transient Verification of System-Level Power Delivery Networks

“…This is exactly the situation in which the complete set of responses can be represented by a reduced set of "basis" functions. This fact is common We exploit this redundancy by applying the compressed macromodeling framework originally presented in [15] and further elaborated/extended in [16]. In the present setting, this framework becomes a procedure by which we can represent the impedance matrix samples of a large dataset in a compressed form by identifying a low-dimensional subspace that is sufficient to describe the frequency dependence of all transfer matrix entries in (3) for virtually any value of d in the parametric domain.…”

A Waveform Relaxation Solver for Transient Simulation of Large-Scale Nonlinearly Loaded Shielding Structures

“…Well-known MoM or FEM field solver limitations may provide incorrect, inaccurate, or incomplete characterizations. For this reason, a fundamental step in the overall proposed flow is the data conditioning phase presented in [27], which involves enriching the original dataset H(jω ) with a set of self-consistent low-frequency samples obtained through an extrapolation (and regularization) procedure.…”

“…For this reason, we exploit the framework introduced in [27] and [30], which computes a low-complexity macromodel by rational fitting a compressed dataset obtained by a modified (causality-preserving) singular value decomposition (SVD). This entire procedure is well-documented in [27] and [30], including application to the very same shielding enclosures considered in this work, and is not repeated here. This optimized macromodel generation flow includes of course a dedicated passivity verification and enforcement, based on an iterative perturbation algorithm [31] of passivity violations identified through the adaptive sampling process [32].…”

“…10(a) shows the accuracy of a passive model with P = 1024 ports built following the proposed macromodeling procedure. The complete model extraction flow, fully documented in [27] involves a number of steps in order to properly handle all challenges posed by this structure: a large-scale system with more than 10 6 responses to be fitted concurrently, bandlimited data from the MoM solver, and especially very low losses making the port transfer function only marginally passive. We recall that in order to guarantee numerically stable transient simulations, model passivity is a necessary condition, requiring that σ max {H(jω)} ≤ 1 must hold ∀ω ∈ [0, ∞).…”

“…A passive model for the 400-port enclosure with 79 poles (31 600 states) has been obtained after a total of ≈4.3 h, considering all procedure steps (data compression, model fitting, and passivity enforcement), while a 100-port passive model with 79 poles has been generated in ≈ 10 min. A discussion on large-scale macromodel generation and related challenges is available in [27].…”