# Accurate Modeling of Conductor Rough Surfaces in Waveguide Devices
## 3 Examples  
We proceed to study the effects of surface roughness for waveguide components with the proposed effective conductivity technique through several examples. For comparative study, the simulation results for waveguide components with a smooth surface are also provided. The first example is a WR-28 rectangular waveguide (\(a=7.112\) mm and \(b=3.556\) mm) commonly used in space communications, where the waveguide walls are copper with a conductivity of \(\sigma_{bulk}=5.8\times 10^{7}\mathrm{S/m}\) and have an RMS value of surface roughness of \(1\)\(\upmu\)m. Figure 4a shows in a semi-logarithmic scale the attenuation constant of the first four waveguide modes as a function of the frequency. In this figure, the results for the smooth surface and the rough surface model are both implemented with the commercial software HFSS(tm). For waveguide, considering finite conductivity with a smooth surface, we have achieved the simulation with HFSS software, and the results have good agreement with the perturbed boundary condition method [20]. As discussed earlier, for the surface roughness model, the effective conductivity is frequency dependent, and, consequently, a conductivity datasheet file needs to be imported into the HFSS software. It should be noted that for propagation modes above the cutoff frequencies, the conductor losses of the four modes are all noticeably increased when considering the surface roughness model. However, the cutoff frequencies for the four modes are almost not affected by the surface roughness. It should be noted that the \(\mathrm{TE_{11}}\) and \(\mathrm{TM_{11}}\) modes are degenerate and have the same cutoff frequency, while the field distributions inside the waveguide are different and accordingly have different conductor losses. Furthermore, the conductor loss for the \(\mathrm{TE_{11}}\) mode is larger than the \(\mathrm{TM_{11}}\) mode, since the field strength near the conductor boundary for the \(\mathrm{TE_{11}}\) mode is larger than that of the \(\mathrm{TM_{11}}\) mode. To decrease the conductor loss from the surface roughness, a suitable way is to reduce the field distribution of the propagating signal near the surface roughness of the waveguide. In Figure 4b, the phase constants for the first four modes are presented. On thecontrary, we can see that below the cutoff frequencies the phase constants for the surface roughness waveguide are larger than that of the smooth surface waveguide. For the propagation modes above the cutoff frequencies, the effects of surface roughness on the phase constants can be assumed to be negligible for engineering applications. As the second example, a cylindrical resonator that has a rough inner wall surface with a diameter of 35 mm and a height of 23.7 mm is studied. The TM\({}_{010}\), TE\({}_{111}\), and TE\({}_{011}\) modes of the cylindrical resonator are analyzed with HFSS\({}^{\text{mt}}\) software simulation, and the surface roughness is characterized with the effective conductivity. The conductor wall is made of silver with a conductivity given by \(\sigma_{bulk}=6.2\times 10^{7}\text{S/m}\) for the resonator, while the RMS value of the surface roughness is 0.6 \(\upmu\)m. The resonance frequencies \(f_{\text{rough\_HFSS}}\) and quality factors \(Q_{\text{rough\_HFSS}}\) are listed in Table 1 for the three modes. For a comparative study, the results for a model with smooth surfaces from the analytical perturbation method (see (6.57) in [18]) and HFSS\({}^{\text{mt}}\) software simulation are also listed. From Table 1, we can see that resonant frequencies (\(f_{\text{smooth\_analy. }}\) and \(f_{\text{smooth\_HFSS}}\)) and quality factors (\(Q_{\text{smooth\_analy. }}\) and \(Q_{\text{smooth\_HFSS}}\)) have good agreement for smooth surfaces from the analytical perturbation method and HFSS simulation approach.