# Accurate Modeling of Conductor Rough Surfaces in Waveguide Devices
## 2 Theories
### Frequency Dependent Effective Conductivity for Conductor Surface Roughness  
From the analysis presented earlier, field solutions have been obtained for conductor surface roughness in space using a gradient conductivity model. However, the gradient conductivity model cannot be directly employed in 3D commercial field solvers for the simulation of complex metal-based waveguide components, because the conductivity has an inhomogeneous distribution perpendicular to the mean plane of the surface roughness. In Figure 1, the conductivity increases gradually in the \(y\)-direction and exhibits bulk conductivity where the space is fully occupied by the conductor. As the field components around the air-metal rough interface space have been solved in Section 2.2, the time averaged conductor loss power density can be calculated easily with formula (4) as follows [18]:  
\[P_{rough}(y)=\frac{1}{2}\frac{\left|\overline{J}_{rough}\right|^{2}}{\sigma(y )}=\frac{1}{2}\frac{\left|\
abla\times\overline{H}\right|^{2}}{\sigma(y)}. \tag{11}\]  
Figure 2 shows the conductivity profile of copper (\(\sigma_{bulk}=5.8\times 10^{7}\)S/m) and the loss power density distribution in an air-metal plane with surface roughness. This can then be compared with the performance of an ideal or perfectly smooth surface in Figure 2. It should be noted that both cases are normalized to the loss power density for a smooth surface at \(y=0\). It can be seen from Figure 2 that the conductor loss density first increases when the signal enters the surface roughness conductor domain. However, as the depth that the signal enters increases, the signal attenuates quickly and the conductor loss density decreases accordingly afterwards. In order to transform the surface roughness to a smooth one for simulation with commercial field solvers, the frequency-dependent effective conductivity of a conductor with an equivalent smooth surface is defined. This can be obtained in terms of the equivalent relations of the conductor loss for the roughness surface and equivalent smooth surface models as follows  
\[\int P_{rough}(y,\omega)dy=\int P_{e,smooth}(y,\omega)dy \tag{12}\]  
where \(\int P_{rough}(y,\omega)dy\) is the conductor loss from the surface roughness model, which can be directly calculated with (11) using numerical integration. The conductor loss for the equivalent smooth surface model can then be expressed as [18]  
\[\int P_{e,smooth}(y,\omega)dy=\int\frac{1}{2}\frac{\left|\overline{J}_{e,smooth }\right|^{2}}{\sigma_{eff}(\omega)}dy \tag{13}\]where \(\overline{J}_{esmooth}=\sigma_{eff}\overline{E}_{eff}=\
abla\times\overline{H}_{ eff}\propto e^{-(1+j)y/\delta_{eff}}\). Thus, in summary, the calculation of the effective conductivity for an equivalent smooth surface can be obtained using (11), (12) and (13). This effective conductivity is frequency dependent and can be used to characterize the effects from surface roughness up to mm-wave region. The effective conductivity will be smaller than the actual bulk conductivity since the conductor loss for a surface roughness will be larger than that of the smooth surface. For conductors with a surface roughness of different RMS heights (\(R_{q}=0.5\upmu\)m, \(0.75\upmu\)m, \(1\upmu\)m), the frequency dependent effective conductivity is plotted in Figure 3, which has a good agreement with the results given in [14]. When the RMS height and frequency increases, the conductor loss increases dramatically, while the effective conductivity decreases with frequency. As shown in Figure 3, the effective conductivity is half of the DC bulk conductivity at 20 GHz with an RMS height \(R_{q}\) of 0.5 \(\upmu\)m for the conductor with surface roughness, and the effective conductivity decreases even more for larger RMS height values. Figure 3: Frequency dependent effective conductivity for a rough surface with different values of \(R_{q}\). Figure 2: Profiles of the conductivity and the time averaged conductor loss power density for a surface roughness (with root mean square (RMS) value of \(1\upmu\)m) and a smooth surface at 1 GHz.