Integral Equation Methodologies for the Signal Integrity Analysis of PCB and Interconnect Structures in Layered Media From DC to Multi-Ghz Frequencies

Abstract

Also introduced is a three-dimensional dynamic field solver for the analysis of interconnect structures with electrodynamic properties that cannot be quantified accurately using transmission line theory. To date, the development of electrodynamic integral equation solvers that can handle realistic interconnect and printed circuit board structures has been hindered by the large number of unknowns in the discrete model. While under quasi-static assumptions, fast multipole methods and precorrected FFT methods have been proven very successful in dealing with the computational complexity of such problems. Their extension to the electrodynamic, however, has been hindered by issues such as low frequency numerical instability, the development of electrodynamic Green's functions for lossy layered media, as well as the efficient modeling of the skin effect loss inside the metalization. The aforementioned difficulties are addressed effectively in this thesis through the following contributions. A fast iterative solver based on the low-storage CG-FFT algorithm is combined with the loop-tree decomposition of the unknown current densities to enable DC to multi-GHz numerically stable solutions with complexity that grows linearly with the number of unknowns. A new computational procedure for developing the three-dimensional layered media Green's function based on a Chebyshev polynomial approximation of the spectrum of the spatial Green's function is introduced capable of handling arbitrary, lossy, layered substrates. A model compatible with the CG-FFT methodology is proposed for efficiently handling the frequency dependent field penetration in thick conductors.