A Time-Domain Approach for Extracting Broadband Macro-π Models of Differential Via Holes

A novel time-domain approach is proposed to synthesize the broadband macro-<formula formulatype="inline"><tex>π</tex></formula> model of the differential vias based on time-domain reflected/transmitted waveforms either measured by time-domain reflectometry (TDR) or simulated by finite-difference time-domain (FDTD) method. The step responses of the differential via are solved in terms of rational functions by the generalized pencil-of-matrix (GPOM) method. The macro-<formula formulatype="inline"><tex>π</tex></formula> model in terms of the rational functions pairs is obtained through an ABCD matrix transformation. The order of the macro-<formula formulatype="inline"><tex>π</tex> </formula> model can be reduced without losing the accuracy according to a residue criterion. The equivalent lumped circuits of the macro-<formula formulatype="inline"> <tex>π</tex></formula> model is synthesized by a lumped circuit extraction method (LCEM). The stability and passivity of the extracted models can be preserved based on this time-domain approach and the macro-<formula formulatype="inline"> <tex>π</tex></formula> topology. Two examples, one is an asymmentric via and the other is an differential via in four-layer PCB, are used to demonstrate the broadband accuracy of the proposed approach both in time-domain and frequency-domain.