# Accurate Modeling of Conductor Rough Surfaces in Waveguide Devices
## 1 Introduction  
Waveguides are a basic component in radar and communication applications [1; 2; 3]. The accurate modeling of electromagnetic wave propagation is an important problem, especially as majority of present and future radar and communication systems operate at higher frequencies in the millimeter-wave bands. The roughness of real physical waveguide surfaces is usually caused by either fabrication-related irregularities (caused by the material used for fabrication or the fabrication process) or environmental effects (such as corrosion). At higher frequencies, for example at the Ka band, the conductor skin depth and the root mean square (RMS) heights of the inner surface roughness are both in the micrometer range. As the skin depth would extend and fall below the dimensions of the surface roughness, the propagation losses caused by this roughness are not negligible in the accurate modeling of waveguide performance. Thus, advances in accurately modeling practical or realistic waveguide structures with finite conductivity and inner surface roughness are needed in engineering problems; this is especially important for several applications, such as in microwave measurement devices, to improve the measurement accuracy and extend the measurement range, and in high power space communication components, to improve electromagnetic reliability and operation life. If the surface roughness must be taken into account using numerical methods, this process will involve co-simulation of multi-scale structures and will require considerable numerical effort. In 1949, [PERSON] [4] studied the power dissipation in a metallic plane with a rough surface at microwave frequencies using a finite difference approach. In this study, the surface roughness was modeledas periodic grooves of various shapes and sizes; the conductor was copper with a conductivity of \(5.8\times 10^{7}\)S/m. Using [PERSON]'s model, [PERSON] and [PERSON] [5] derived a widely used empirical formula that matches the numerical results produced by [PERSON]. This formula, given in (1), provides a correction factor describing the ratio of the power absorption of a rough surface \(P_{rough}\) to that of a corresponding smooth surface \(P_{smooth}\). \[K_{sr}=\frac{P_{rough}}{P_{smooth}}=1+\frac{2}{\pi}\tan^{-1}\left(1.4\left( \frac{h}{\delta}\right)^{2}\right), \tag{1}\]  
where \(h\) is the RMS height of the rough surface profile and \(\delta\) is the skin depth of the conductor. The formula given in (1) is currently the most common model that quantifies the impact of conductor surface roughness on Ohmic losses; this formula has been integrated in commercial field solvers, such as Ansys' High Frequency Structure Simulator (HFSS\({}^{\text{TM}}\)) ([Online]. Available: [[http://ansoft.com/maxwell/](http://ansoft.com/maxwell/)]([http://ansoft.com/maxwell/](http://ansoft.com/maxwell/))). However, the correction factor saturates at a value of two and therefore fails to model effects at high frequencies (small \(\delta\)) or high value of RMS \(h\), since it can be shown that the additional loss can easily exceed the value of two with measurement results [6]. The study of conductor surface roughness has attracted significant interest in planar transmission lines and interconnects on high-speed circuits. [PERSON] et al. [7] introduced a periodic model to calculate the equivalent roughened conductor surface impedance that can then be used to modify the per-unit-length transmission line parameters to evaluate signal attenuation and phase delay. [PERSON] et al. [8] proposed a model to substitute the conductor surface roughness of printed circuit boards by a layer with an effective material (lossy dielectric) for numerical simulations. A few parameters of the model, as statistical values, can be estimated from Scanning Electron Microscope (SEM) images of the roughened surfaces. [PERSON] et al. [9] characterized the rough surface by a stochastic random process for high-speed interconnects and microelectronic packages.