Effective length and soil surface voltage of grounding electrodes in causal-dispersive soils

In this paper, impact of causality of dispersive soils on the effective lengths of different grounding electrodes is investigated. These electrodes include single vertical/horizontal electrode, and two arrangements namely counterpoise wire and multiple electrodes. For computation of effective length, the computationally efficient multi-conductor transmission line method (MTL) under two typical lightning currents namely first and subsequent stroke currents is adopted. This paper consists of three parts. In the first part, the causality on the effective length of single vertical/horizontal electrode is applied which leads to increase or decrease of effective length with respect to that in non-causal counterpart depending on the low-frequency resistivity of lossy soil and current waveform. The difference between causal and non-causal values is more pronounced for subsequent stroke current with relative error less than 20%, whereas for first stroke current it is ignorable. This error is increased up to 33% when the ionization nonlinear phenomenon is occurred. In the case of the two mentioned arrangements, it is less pronounced with respect to single electrode. In the second part, lightning-induced voltage on the soil surface is computed considering causal and non-causal values of effective length of the mentioned grounding systems and accordingly touch and step voltages are computed from it. The simulation results show that for single vertical electrode and single horizontal electrode buried in dispersive soil, the non-causality error of the two voltages increased up to 15% and 50% respectively. In the case of the two arrangements, it is ignorable with error less than 4%. Finally, predicting formulae for effective length of the mentioned grounding electrodes based on the causality are presented. Integration of them with the equivalent resistivity in two-layer soils results in application of them in two-layer dispersive soils. Simulation results in two-layer dispersive soils show that the effective lengths of vertical and horizontal electrodes are different in despite of single-layer dispersive soils. Also, in comparison with the previously published studies in two-layer non-dispersive soils, dispersion in two-layer soil leads to reduction of effective length up to 50% and 25% respectively for the first and subsequent stroke currents.