Impact of multiple scattering on passivity of equivalent-circuit via models

For fast analyses of multilayer printed circuit boards and packages, the parallel-plate impedance (Z
_pp
) is often used in equivalent-circuit via models to describe the current return path for vias traversing a power/ground plane pair. Analytical algorithms for the Z
_pp
calculation, such as the radial waveguide method, usually assumes solid planes and neglect the effect of multiple scattering among open via ports. However, this scattering can become significant when other vias lie in close proximity. Disregarding this fact can lead to potential passivity violations, especially for dense via arrays. We propose to use a new formulation for the Z
_pp
calculation based on the contour integral method that takes the multiple scattering effect into account to resolve the passivity problem. A via array example is demonstrated and, in particular, the passivity issue is discussed.