# Accurate Modeling of Conductor Rough Surfaces in Waveguide Devices
## 3 Examples
From \(f_{\text{smooth\_HFSS}}\) and \(f_{\text{rough\_HFSS}}\), we can see that surface roughness have negligible effect on the resonator frequencies but have significant effects on quality factors in comparison \(Q_{\text{smooth\_HFSS}}\) with \(Q_{\text{rough\_HFSS}}\) for the three modes. This indicates that the loss increase due to the surface roughness can be accurately predicted with the effective conductivity. On the other hand, resonance frequencies of resonators are influenced by the inner inductance of metals which  
Figure 4: (**a**) Attenuation and (**b**) phase constants for the first four WR-28 rectangular modes computed with smooth surface (solid lines) and surface roughness walls (dashed lines, The lines in green color are for TE\({}_{11}\) mode and the lines in red color are for TM\({}_{11}\) mode). is related to the magnetic field response inside the conductor, the increased inner inductance can be modeled by an effective, frequency-dependent permeability \(\mu_{r\,\textit{eff}}\) obtained by comparing the magnetic field energies of a rough and a smooth surface [13]. In addition, compared with the TM\({}_{010}\) and TE\({}_{111}\) modes, the field distribution for the TE\({}_{011}\) mode is concentrated at the center (or axis) of the cylindrical resonator, and the fields nearly vanish around the surface roughness wall. Hence, the TE\({}_{011}\) has highest quality factor among the three modes and the surface roughness may have a smaller influence on TE\({}_{011}\) mode than the other two TM\({}_{010}\) and TE\({}_{111}\) modes. Finally, the surface roughness effects in a four-pole filter operating in the Ka band are studied [21]. The filter is made of aluminum alloy with a conductivity equal to \(1.2\times 10^{7}\)S/m. The filter is symmetrical (Figure 5) and the cavities have equal widths (\(a\)= 8.636 mm) and lengths (\(l_{1}\) = 4.696 mm and \(l_{2}\) = 5.555 mm). The cavities are made with the inductive coupling irises that have equal length (\(d\) = 2.5 mm) and widths (\(a_{1}\) = 4.939 mm, \(a_{2}\) = 3.799 mm, and \(a_{3}\) = 3.578 mm). The RMS value of the surface roughness wall is 1 \(\upmu\)m.  
The filter has been designed to have a band-pass response of 800 MHz bandwidth centered at 28 GHz. A detailed view of the filter insertion losses is shown in Figure 6a, which includes the simulated results from HFSS\({}^{\textsuperscript{TM}}\), considering the finite conductivity value of the employed aluminum alloy with inner surface roughness, as well as a comparison with the result from the inner smooth surface. It can be seen that the insertion loss is larger than that of the smooth surface, especially at the high frequencies of the pass band. Moreover, at the center frequency, the insertion loss increases by about 0.6 dB for the surface roughness. The complete reflection and transmission responses of this filter are shown in Figure 6b. It can be observed that there are slight differences between the two simulated results for insertion loss and bandwidth from the two models of the smooth and rough surfaces. However, for higher frequencies (microwave and millimeter waves), accurately modeling the effects of surface roughness on increasing the insertion loss and shifting the bandwidth are essential for engineering applications, such as microwave measurement and space communication. Considering manufacturing and fabrication factors, accurate modeling of surface roughness in microwave and millimeter components will bridge the gap between the simulation and measurement results.