Uncertainty Assessment of Lossy and Dispersive Lines in SPICE-Type Environments

Abstract

Abstract-This contribution presents an alternative modeling strategy for the stochastic analysis of high-speed interconnects. The proposed approach takes advantage of the polynomial chaos framework and a fully SPICE-compatible formulation to avoid repeated circuit simulations, therefore alleviating the computational burden associated to traditional sampling-based methods like Monte Carlo. Nonetheless, the technique offers very good accuracy and the opportunity to easily simulate complex interconnect topologies which include lossy and dispersive transmission lines, thus overcoming the limitations of previous publications. Application examples involving the stochastic analysis of onchip and on-board interconnects validate the methodology and conclude the paper.

The relationship between Galerkin and collocation methods in statistical transmission line analysis

“…The substitution of the PCEs (2) and (3) into (1) and subsequent Galerkin projection yield the following coupled and augmented, yet deterministic equations in the unknown PCE coefficients [2]:…”

“…In particular, the polynomial chaos (PC) method [1] was widely adopted in the analysis of electrical circuits and interconnects [2]- [7]. The underlying idea of PC is to represent stochastic variables of interest (e.g., circuit voltages and currents) as expansions of suitable orthogonal polynomials.…”

“…The stochastic Galerkin method (SGM) requires the single solution of a modified (augmented) problem [2], [3]. On the other hand, stochastic collocation methods (SCM) only require to sample the solution at a limited set of points in the space of the random variables [4]- [7].…”

Review of Polynomial Chaos-Based Methods for Uncertainty Quantification in Modern Integrated Circuits

“…PC-based methods have been successfully adopted for different UQ problems with a relatively small number of random variables, see, for example, [13,22,24,38,47,48,70,74,92]. However, specific limitations and challenges arise in the application of the PC expansion when a high number of random variables is considered [93].…”

“…Using the Galerkin Projection method offers an elegant way of computing the PC coefficients, allowing its application to a wide variety of problems. For example, such an intrusive approach has been adopted to solve transmission lines [47,74] or stochastic full wave [75] problems. However, using the Galerkin method poses some challenges: Formulating and solving the obtained coupled system of equations for the determination of PC coefficients could be a difficult task.…”