Capacitance of a plate capacitor with one band-limited fractal rough surface

Abstract

The problem of the capacitance between a band-limited, zero-mean, fractal shaped-rough surface and a plane electrode is investigated. Five parameters are required to define the rough surface: $\sigma $, the rms height, $D$ (1 $<$ $D$ $<$ 2), the fractal dimension of the roughness; $K_0 $, the fundamental spatial frequency; $b$ ($b$ $>$ 1), the spatial frequency scaling parameter; and $N$, the number of spatial frequency components in the surface structure. We find that the graph of inverse capacitance against nearest electrode separation depends on $\sigma $ and $D$, whereas it is nearly independent of $K_0 $, $b$, and $N$ for $N >$ 4. The numerical results also indicate that the surface roughness can be interpreted as an equivalent dielectric film with an effective dielectric constant and effective thickness for surprisingly small minimum electrode separations. Our findings in this paper can be used to complement established techniques for the experimental determination of the statistical parameters of the surface roughness of conducting surfaces.