High Frequency Numerical Solution to Evaluate the Impedance of the Vertical Grounding Electrode Using an Accurate Mathematical Approach

Considering the air/earth interface to compute the electromagnetic field at the desired point is the most important problem in transient analysis of grounding system buried to lossy media like earth. In order to consider this issue in the available proposed method and to obtain a solution for the problem, one needs to account an integral solution of the so-called Semmerfeld integral in the cylindrical coordinate system. Analytical solution for such integral is almost impossible which is due to the presence of the oscillating the zeroth-order Bessel function of the first kind, and singularities and branch-cuts in its integrand function.  In this paper, we investigate the behavior of the vertical grounding electrode in a high-frequency electromagnetic transient state based on the near field theory using the method of moments (MoM). To compute the electromagnetic field at the desired point, the main problem is calculating the well-known Sommerfeld integral in the cylindrical coordinate system, which its integral kernel includes the zeroth-order Bessel function of the first kind along with some singularities and branch-cuts. Since the analytical solution of this integral is not available in literature, we propose a numerical method, as well as a strategy based on the exact solution for far and near filed calculations. A detailed analysis of the obtained results compared with other techniques are provided to confirm the accuracy and validity of the proposed method.