Comparative Study of the Causality Property of Various Conductor Surface Roughness Models

Accurate modeling of conductor roughness is crucial for signal integrity analysis. Various models have been proposed in the literature, while many of the commonly-used ones are not causal, meaning that the time domain waveform will have nonphysical precursors before the actual arrival of the signal. The causality issue has been addressed in a few papers, while there is a lack of a thorough comparison between the available models. In this paper, through a 7-inch microstrip line example, we compare the causality of six models: Hammerstad, modified Hammerstad, causal Hammerstad, hemisphere, Huray, and causal Huray. It is found that the hemisphere and Huray models can have $a\gt 10$ mV precursor, the Hammerstad models have smaller precursor (<4 mV) due to its fast saturation, whereas all causal models have no precursors.