Power system earth return impedance and the complex ground return plane approximation

One of the most exotic concepts in the field of grounding is the ability of the earth to conduct electricity. Its sheer size means that the impedances produced by the passage of current through the soil are very low. Large earthing systems in low soil resistivity areas can produce very small earthing system impedances. What is not as well appreciated is that an overhead conductor using an earth return path to complete a circuit, such as is the case during a phase to earth fault, incurs an additional impedance due to the fields associated with the current flow through the earth. This paper presents a derivation of the closed-form solution to both the earth return self and mutual impedance expressions developed by Carson. The expressions derived agree with the closed-form expression first attained by Wait, who's contribution to here appears to have gone widely unrecognised. With the utilisation of this relatively simple close-form expression the applicability of Dubanton's approximation, that utilises the concept known as the complex ground return plane, is called into question. Its accuracy in comparison to the closed-form solution is explored with a focus on the implications to earthing system designs, third party asset LFI assessments, and easement restrictions.