Unified Formulation for Multiple Vias With or Without Circular Pads in High Speed Interconnects

A unified full-wave characterization of massive number of vias with or without circular pads is formulated analytically by means of equivalent magnetic frill array model and Galerkin´s procedure. The proposed method takes advantage of the parallel-plate structure and employs image theory and Fourier transform to simplify the problem from 3-D configuration into 2-D frame. Based on the cylindrical symmetry with Bessel´s functions and addition theorem, the final matrix equation is formulated analytically which can be used immediately for sensitivity analysis in both via dimensions and pad size. As a result, the new method is simple, efficient, and accurate. Numerical examples demonstrate good agreement between our analytical solution and the results obtained by commercial software (high frequency structure simulator) over a frequency range up to 20 GHz.