A Generalized Multiple Scattering Method for Dense Vias With Axially Anisotropic Modes in an Arbitrarily Shaped Plate Pair

Numerical addition theorems of both axially anisotropic and axially isotropic parallel-plate modes are derived using a method based on boundary integral equations. This leads to a generalized multiple scattering (GMS) method for signal/power integrity analysis of dense vias in an arbitrarily shaped plate pair, which overcomes the limitation of the conventional multiple scattering method depending on the analytical addition theorems in an infinitely large or a finite circular plate pair. Both the numerical addition theorems and the GMS method have been validated by comparing results with either analytical expressions for special cases or full-wave simulations for more general cases. Several examples are provided to demonstrate the advantages of the generalized method over previous via models by taking into account the axially anisotropic modes due to the asymmetry caused by dense vias and/or arbitrarily shaped power/ground plate edges on via performance.