Design Space Exploration For Printed Circuit Board Vias Using Polynomial Chaos Expansion

This work discusses how the Williamson equivalent circuit for the modeling of via transitions is augmented using the polynomial chaos expansion (PCE) method to take into account stochastic variations of model parameters. In addition, it is shown how the PCE can be applied in combination with Monte-Carlo (MC) sampling to conduct an efficient design space analysis. The physics-based via (PBV) model is used to efficiently model multi-layer via interconnects by concatenating equivalent circuit models representing certain geometric sections of the interconnect. These equivalent circuits are expanded with respect to the stochastic model parameters by means of PCE and concatenated to derive an augmented model of the whole interconnect. From this representation, stochastic scattering parameters are extracted. Numerical examples verify the method and the computational advantages over MC sampling. The method provides a fast and comprehensive approach to design space exploration and variability assessment in the design stage of via interconnects. The presented approach is not limited to via models and can be applied to various microwave structures in a similar way.