The Complex Impedance of the Hemispherical Ground Electrode: An Open Analytical Problem

Abstract

The hemispherical ground electrode is a basic electrode whose analysis appears in many textbooks on electromagnetics in chapters dedicated to steady currents. Considering a soil with a given resistivity and an electrode with a given perimeter, the electrode DC resistance is simply calculated from the ratio resistivity/perimeter. Strangely, the generalization of this result to AC regimes is missing. The issue of the frequency-domain impedance of the hemispherical ground electrode has been avoided in the literature despite its trivial geometry. But the problem is indeed not easy; electromagnetic field calculation involves Legendre and Bessel functions; the application of boundary conditions involves an infinite set of points, and some integrals involved need to be calculated recursively. We analyzed the math and physics of the problem but failed to find a closed-form solution. This article with “negative results” can, however, be useful; on one hand it may prevent researchers from wasting their time following the same steps, and, on the other hand, it may attract the interest of new researchers to the subject, ultimately, accelerating its analytical solution (if the solution exists).

Keywords

• analytical methods for electromagnetic field calculation
• boundary conditions
• grounding impedance
• hemispherical electrode