A compression strategy for rational macromodeling of large interconnect structures

Abstract

Rational macromodeling via Vector Fitting algorithms is a standard practice in Signal and Power Integrity analysis and design flows. However, despite the robustness and reliability of the Vector Fitting scheme, some challenges remain for those applications requiring models with a very large port count. Fully coupled signal and/or power distribution networks may require concurrent modeling of hundreds of simultaneously coupled ports over extended frequency bands. Direct rational fitting is impractical for such structures due to a large computational cost. In this work, we present a compression strategy aimed at representing the dynamic behavior of the structure through few carefully selected "basis functions". We show that model accuracy can be traded for complexity, with full control over approximation errors. Application of standard Vector Fitting to the obtained low-dimensional compressed system leads to the construction of a global state-space macromodel with significantly reduced runtime and memory consumption. Several benchmarks demonstrate the effectiveness of the approach.

Passivity verification and enforcement—A review paper

“…In their later work, 24 author resolves the issue by modifying the scaling function and introduced the VF as Relaxed VF. Afterwards, in the novel works of previous studies, [16][17][18][19][20][21][23][24][25] notable extension with necessary modifications have been proposed for improving the performance and integration for different area of applications. Although passivity of VF algorithm has been ensured in Gustavsen and Semlyen 26,27 but in Olivadese and Grivet-Talocia, 28 it has been cited that due to the band limited nature of the practical data, beyond the measurement bandwidth where there is no fitting restriction to minimal errors in the frequency domain data, may end up in generating a nonpassive model.…”

Convex passivity enforcement of linear macromodels via alternate subgradient iterations

“…where vector w ∈ R n collects the internal state variables and A ∈ R n,n , B ∈ R n,p , C ∈ R p,n , D ∈ R p,p are obtained as a result of a fitting process [1]- [4]. The input and output vectors u and y collect the p scattering incident and reflected waves at the device ports.…”

“…Starting from tabulated frequency samples of the scattering matrix coming from measurement or full-wave analysis, the VF algorithm produces accurate and guaranteed stable rational approximations of the system transfer function, which can in turn be synthesized into equivalent circuits or equation-based state-space macromodels for fast system-level simulation and design optimization. Recent developments essentially resolved the complexity issues due to large dynamic order and/or port counts required by modern designs, through reformulation [2], parallelization [3] or compression [4].…”

“…Despite the reliability and efficiency of recent VF implementations [2]- [4], the obtained macromodels are generally not passive. Since passivity is a fundamental requirement that guarantees numerically stable system-level simulations [5], any macromodel obtained by VF should be subject to some postprocessing in order to enforce its passivity.…”

Compressed Passive Macromodeling

“…Our main approach is based on the fundamental idea that there is often a lot of redundancy in the frequency responses of coupled multiport structures. Following the approach preliminary documented in [29], we show in Sec. II that a simple projection based on a truncated Singular Value Decomposition (SVD) [30], [31] leads to drastic compression of scattering responses, which can be cast as a linear combination of few carefully selected "basis functions".…”