Exact Analytical Solution for the Via-Plate Capacitance in Multiple-Layer Structures

An exact analytical solution for the via-plate capacitance in multilayer printed-circuit board (PCB) structures is presented. The formulation is based on Laplace's equation for the static electric potential within the via-plate region, which is electrically small for a large frequency range. The potential distribution is determined by the separation of Laplace's equation in cylindrical coordinates for the three subdomains: via barrel and pad, antipad, and the plate region. It is shown that the unknown via-plate capacitance is determined by only one single coefficient of the resulting linear equation system. The influence of additional via pads is readily included. The results are compared with previously published approaches and validated by 2-D static, as well as by 3-D full-wave numerical simulations.