Modeling surface roughness with an array of hemispheres

This paper focuses on explaining the extra losses caused by the roughened surface of the signal conductor in a transmission line. At frequencies greater than a few GHz, the surface roughness of the conductor has a significant impact on the transmission line loss and could become a roadblock to high-speed platform design if not properly accounted for. Current surface roughness simulation methods utilize the empirically derived Hammerstad equation. In this paper we explain the fundamental physics of the surface roughness problem by deriving an analytical solution for a model of surface roughness based on hemispherical protrusions, dispersed on a flat surface. This model will provide insight to develop design alternatives to the Hammerstad equation. Using this model, we explain the physics of surface currents flowing on a roughened surface. We also calculate power absorption as a function of frequency and roughness density and compare the trends to both the Hammerstad equation and measured data.