Fast and Broadband Modeling Method for Multiple Vias With Irregular Antipad in Arbitrarily Shaped Power/Ground Planes in 3-D IC and Packaging Based on Generalized Foldy–Lax Equations

In this paper, we model multiple vias with irregular antipad in arbitrarily shaped 3-D integrated circuit and packaging system based on generalized Foldy–Lax equations method, boundary integral equation method, and generalized T matrix. We first obtain the impedance matrix for finite cavity, which includes the reflection features of the cavity boundaries. Then, the scattered field from a single via and a generalized T matrix, including the wall effects, are derived. The Foldy–Lax multiple scattering equations are generalized to include the wall effects using impedance matrix and the generalized T matrix. To obtain the incident field for the case of vias in the arbitrarily shaped antipad, we calculate the exciting and scattering field coefficients based on the transformation that converts surface integration of magnetic surface currents in antipad into 1-D line integration of surface charges on the vias and ground plane. The coupling among vertical vias are solved by applying Foldy–Lax multiple scattering equations. The scattering matrix of coupling among vias is calculated to make corresponding signal/power integrity analysis. Numerical results for the method are in a good agreement with a commercial full-wave numerical tool up to 50 GHz.