A Hierarchical Approach to Dimensionality Reduction and Nonparametric Problems in the Polynomial Chaos Simulation of Transmission Lines

Polynomial chaos (PC)-based techniques recently became popular alternatives for the stochastic analysis of electrical circuits, especially in the context of signal integrity and electromagnetic compatibility investigations. Among the challenging issues of PC, there are two longstanding limitations: curse of dimensionality (i.e., efficiency decrease as the number of random parameters is increased) and the inability to handle nonparametric variations (i.e., random variables (RVs) that cannot be unambiguously parametrized upfront). This paper aims at covering this gap by putting forward a suitable hierarchical approach, with specific emphasis on transmission-line analysis. First, the problem is characterized by estimating the distribution of the transmission line per-unit-length parameters. Second, the obtained distribution is fitted using a multivariate mixture of Gaussians (MoGs). The MoGs is a flexible model that is capable of taking into account the existing dependence between inductance and capacitance entries in multiconductor lines. Next, appropriate orthogonal basis functions are generated based on a recent framework for non-Gaussian correlated RVs. Finally, the technique is combined with a stochastic Galerkin method to generate a deterministic and SPICE-compatible model that can be simulated in time domain. The proposed approach enables both dimensionality (hence, model order) reduction and handling of nonparametric variations. Application examples concerning cables with random cross-section illustrate and validate the methodology.