Analytical Expressions for Zero-Crossing Times in Lightning Return-Stroke Engineering Models

One of the characteristic features of the electromagnetic fields radiated by lightning is the far-field inversion of polarity with a zero-crossing occurring in the tens of microseconds range. This feature has been used in several studies to test the ability of return-stroke models to reproduce the observed electromagnetic fields. In this paper, we derive closed-form analytical expressions for the zero-crossing times associated with six engineering models for the lightning return strokes, namely the Bruce-Golde (BG) model, the transmission-line (TL) model, the traveling current source (TCS) model, the modified transmission-line linear (MTLL) model, the modified transmission-line exponential (MTLE) model, and the Diendorfer-Uman (DU) model. For the derivation, the late-time behavior of the current is expressed in terms of a single exponential function. It is shown from the derived expressions that except for the TL model, according to which the zero-crossing time is independent of the current decay constant tau, all the models exhibit greater zero-crossing time as the value of tau increases. This increase is found to be linear for the BG and TCS models, nearly linear for the DU model, and quasi-logarithmic for the MTLL and MTLE models. It is also shown that all the models exhibit a decrease of the zero-crossing time with increasing return-stroke speed, except for the BG model for which the zero-crossing time appears to be independent of v . The DU and TCS models predict almost the same zero-crossing times that vary quasi-linearly with the corresponding running variables. The zero-crossing times predicted by the MTLL and MTLE models are less sensitive to the current decay time and more sensitive to the return-stroke speed, compared to the predictions of the BG, TCS, and DU models. The BG, DU, and TCS models predict, in general, larger values for the zero-crossing times than those predicted by MTLL and MTLE models. The derived expressions are then used to discuss the conditions required for every engineering return-stroke model to reproduce the expected far-field zero-crossing times. The used procedure is to compute the zero-crossing times for each model, starting from channel-base current waveforms typical of the first and subsequent return strokes, and to examine how well the predicted values are in agreement with typical, experimentally observed zero-crossing times. Other adjustable parameters are varied within their typical range of variation. It is shown