On simplified fast modal analysis for through silicon vias in layered media based upon full-wave solutions

Based on equivalent magnetic frill array model and Galerkin's procedure, we present a simplified full-wave algorithm to characterize the propagation behavior and signal integrity of massive number of through silicon vias (TSV) for the 3-D system-in-package (SIP) and system-on-chip (SOC) applications. The proposed method employs the Fourier transform and takes advantage of circular cylindrical shapes with Bessel's functions and the addition theorem to solve the Helmholtz equations without resorting numerical discretization. As a result, it provides closed form solutions with high precision. Since the algorithm does not rely on numerically generated meshes, it gains one to two orders of magnitude in speed, compared to popular commercial software packages. Numerical examples demonstrate that the new method provides good agreement with the HFSS results. As the number of vias increases the new method gains more in both speed and accuracy.