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

Abstract

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 viaplate 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.Index Terms-Power integrity, signal-integrity (SI), via-barrel capacitance, via-pad capacitance, via-plate capacitance.

Improved Expression for the Via-Plate Capacitance Based on the Magnetic-Frill Model

“…2 as a function of the relative pad radius r p /r 0 . As can be seen, the former result [6] shows the unphysical initial decrease of the capacitance, as reported in [8] and [12]. The suggested alternative solution (15) shows a far better agreement with the numerical reference data.…”

“…As can be seen, this expression is quite different from the result obtained in [6]. To demonstrate the improved accuracy, the example structure from [6], [9], and [12] is analyzed. It consists of a two-layer board with plates shorted, in order to obtain a single total capacitance.…”

“…However, due to the 3-D structure, the magnetic field distribution within the antipad is more complex, so that the application of (5) leads to the discrepancies observed for the via-plate capacitance [12]. In the following, the magnetic field in the antipad is calculated using the exact Green's function of an infinite parallel-plate pair.…”

“…1. To keep the resulting expression as simple as possible, an infinite parallel-plane pair is investigated analogously to [9] and [12]. The Green's function for a magnetic-frill current within such a structure is given in [4] as an extension of the results in [7] …”

“…In [12], some significant deviations between the magneticfrill model [4]- [6] and a rigorous solution of the corresponding electrostatic boundary-value problem were observed. In particular, for vias with pads, the results obtained by the magnetic-frill approach show an unphysical behavior, as was also confirmed in [8] by numerical finite-difference method simulations.…”