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 that only two models, namely MTLL and MTLE, are able to reproduce the typical zero-crossing times.