An analytical formulation of the electromagnetic field generated by lightning return strokes

The evaluation of the lightning performance of overhead lines requires, mainly for distribution lines, the calculation of the voltages induced by nearby lightning strokes hitting the ground. Generally, when calculating lightning-induced voltages using the time-domain approach and the TEM assumption, the most time-consuming stage is the calculation of the source terms of the coupling equations, which means, in turn, the electromagnetic field radiated by the lightning channel. Hence, simplified approaches are still desirable in order to save computational time. Analytical solution of the lighting-originated electric- and magnetic field is available in literature for the case of an ideal transmission-line (TL) return-stroke model with channel-base current having a stepwise waveshape. This paper describes an analytical approach that applies to the case of a general waveshape of the channel-base current, based on the piecewise straight-line approximation, in cylindrical coordinates, which is functional when using the TL return-stroke current model and the Agrawal et al. coupling model. This paper reports the exact solution for the case of trapezoidal waveshape of the current at the base of the channel and the analytical solution for the case of piecewise linear waveform.