Extension of the contour integral method to anisotropic modes on circular ports

In the analysis of power/ground planes in multilayer substrates, circular ports are often used for modeling of via transitions. The electric and magnetic fields on excited ports are usually assumed to be isotropic. This assumption may not hold in certain scenarios such as vias in very close proximity, where anisotropic modes can be excited. This paper first extends voltage and current definitions of circular ports to account for the non-uniform field distribution along the port perimeter and the anisotropic propagating modes. The effect of these modes on the parallel-plate impedance can be captured in the contour integral method (CIM) by discretizing the port perimeter with line segments. However, the computation time grows rapidly as the number of modeled ports increases. Therefore, the CIM is extended here to incorporate analytical modal expressions to improve the computational efficiency based on the new port definition. The derivation starts with solutions under the assumption of infinite planes, and then is expanded to take finite plane boundaries into consideration. Application examples using the extended CIM will be demonstrated and validated against the conventional CIM with ports modeled numerically. The significance of anisotropic propagating modes for dense via arrays will also be discussed.