# Accurate Modeling of Conductor Rough Surfaces in Waveguide Devices
## 2 Theories
### Field Solutions for Conductivity with Inhomogeneous Distribution in Space  
In Section 2.1, the conductivity distribution for the surface roughness metal model was obtained. Based on this result, the field solutions from [PERSON]'s equations for conductivity with an inhomogeneous distribution can be derived. Here, the time dependence of the form \(\exp(j\omega t)\) is assumed for the time-harmonic fields and is omitted for brevity. The two Maxwell curl equations are as follows:  
\[\
abla\times\overline{H}=j\omega\epsilon_{0}\overline{E}+\overline{J}, \tag{4}\]  
\[\
abla\times\overline{E}=-j\omega\mu_{0}\overline{H}. \tag{5}\]  
For the surface roughness conductor model shown in Figure 1, \(\overline{J}=\sigma\overline{E}\). By solving [PERSON]'s equation, it should be possible to obtain the magnetic field distribution \(\overline{H}\). Taking the curl of (4), inserting [PERSON]'s law (5) to eliminate the electric field \(\overline{E}\), and imposing the condition \(\
abla\cdot\overline{H}=0\), we have  
\[\
abla^{2}\overline{H}+k_{0}^{2}\overline{H}+\
abla\times(\sigma\overline{E})=0, \tag{6}\]  
where \(k_{0}=\omega\sqrt{\mu_{0}\epsilon_{0}}\). Since \(\sigma\) varies with \(y\) and \(\overline{E}=\frac{\
abla\times\overline{H}}{j\omega\epsilon_{0}+\sigma}\) from (4), the curl of the conduction current density can be calculated as  
\[\
abla\times(\sigma\overline{E})=\sigma\
abla\times\overline{E}-\overline{E} \times\
abla\sigma=-j\omega\mu_{0}\sigma\overline{H}+\frac{1}{j\omega\epsilon _{0}+\sigma}\
abla\sigma\times\
abla\times\overline{H}. \tag{7}\]  
Substituting (7) into (6), the unknown magnetic field \(\overline{H}\) is left, and (6) can be modified as:  
\[\
abla^{2}\overline{H}+(k_{0}^{2}-j\omega\mu_{0}\sigma)\overline{H}+\frac{1}{ j\omega\epsilon_{0}+\sigma}\
abla\sigma\times(\
abla\times\overline{H})=0. \tag{8}\]  
Despite a location dependent \(\sigma(y)\) around the air-conductor interface, the displacement current density is much smaller than the conduction current density with \(\omega\epsilon_{0}<<\sigma\). Hence, the displacement current can be neglected and (8) can be approximated as  
\[\
abla^{2}\overline{H}-j\omega\mu_{0}\sigma\overline{H}+\frac{\
abla\sigma}{ \sigma}\times(\
abla\times\overline{H})=0. \tag{9}\]  
Figure 1: Profile of an air-metal plane interface with conductor surface roughness.  
It should be noted that \(\sigma(y)\) will gradually transition to bulk conductivity in the direction perpendicular to the mean surface of the air-conductor rough interface. In addition, the magnetic field strength will decrease exponentially in the \(+y\) direction and becomes a one dimension problem. Considering the conductor boundary shown in Figure 1, the magnetic field \(H_{x}\) decreases quickly with the increase of \(\sigma\) in the \(y\)- direction; while in the \(x\)- and \(z\)- directions, this change is less and the partial derivative in these two directions vanish in the Laplacian and curl terms of the equation (9). The \(H_{x}\) component can be solved from (9) as:  
\[\frac{\partial^{2}H_{x}}{\partial y^{2}}-j\omega\mu_{0}\sigma H_{x}-\frac{1}{ \sigma}\frac{\partial\sigma}{\partial y}\frac{\partial H_{x}}{\partial y}=0. \tag{10}\]  
In (10), the conductivity is location dependent as \(\sigma(y)\) derived from the CDF function. This equation can be efficiently solved using numerically techniques [17]. This process will yield the numerical magnetic field solution around the surface roughness and inside the conductor, and then the current density and conductor loss can be obtained directly.