Analysis of direct current flowing inside a linear increasing conductivity half-space

The paper analyses the injection of a direct current through an above-ground circular plate earth electrode. The assumed medium in which the current flows, filling a half-space, has a nonuniform linear increasing conductivity. A parameter having the dimension of length is considered, in order to describe the in-depth linear increase of conductivity inside the half-space. Solving the steady-state electromagnetic field equations describing the phenomena, a homogenous second order partial differential equation (PDE) for the electric potential is obtained. The analytical solution of the PDE, based on the separation of variables method, allows an accurate computation of both the electric field strength and Poynting´s vector in the considered area. Thus, an exact analytical relationship of the earth electrode resistance is derived, whose calculated limit gives the already reported formula of the same quantity in the constant conductivity medium case, and hence validating the obtained results.