Numerical Evaluation of the Frequency-Dependent Impedance of Hemispherical Ground Electrodes through Finite Element Analysis

Metallic electrodes are widely used in many applications, the analysis of their frequency-domain behavior is an important subject, particularly in applications related to earthing/grounding systems, from dc up into the MHz range. In this paper, a numerical evaluation of the frequency-dependent complex impedance of the hemispherical ground electrode is implemented. A closed-form solution for non-zero frequencies is still a difficult task to achieve as evidenced in a previous paper dedicated to the subject and, therefore, numerical approaches should be an alternative option. The aim of this article is to present a solution based on a numerical method using finite element analysis. In typical commercial FE tools, electric currents exhibit azimuthal orientation and, as such, the magnetic field has a null azimuthal component but non-null axial and radial components. On the contrary, a dual problem is considered in this work, with a purely azimuthal magnetic field. To overcome the difficulty of directly using a commercial FE tool, a novel formulation is developed. An innovative 2D formulation, the ι -form, is developed as a modification of the H-formulation applied to axisymmetric magnetic field problems. The results are validated using a classical 3D H-formulation; comparisons showed very good agreement. The electrode complex impedance is analyzed considering two different cases. Firstly, the grounding system is constituted by a hemispherical electrode surrounded by a remote concentric electrode; in the second case, the grounding system is constituted by two identical thin hemispherical electrodes. Computed results are presented and discussed, showing how the grounding impedance depends on the frequency and, also, on the radius of the remote concentric electrode (first case) or on the distance between the two hemispherical electrodes (second case).