Exact Expressions for Lightning Electromagnetic Fields: Application to the Rusck Field-to-Transmission Line Coupling Model

An exact analytical expression for the electric field of the return stroke as excited by a propagating step current source is derived in this paper. This expression could be advantageously used to evaluate the disturbances caused by lightning on overhead lines. There are three equivalent procedures to evaluate the voltages induced by lightning on power lines, namely, the Agrawal-Price-Gurbaxani model, the Taylor-Satterwhite-Harrison model, and the Rachidi model. In the case of a vertical return stroke channel, the coupling model developed by Rusck becomes identical to these three coupling models. Due to its simplicity, the Rusck model is frequently used by engineers to study the effects of lightning on power distribution and transmission lines. In order to reduce the time involved in the electromagnetic field calculation, the Rusck model is incorporated with an analytical expression for the electromagnetic fields of the return stroke excited by a propagating step current pulse. Our research work shows that the Rusck expression can be used to calculate the peak values of lightning induced voltages to an accuracy of about 10%. However, the use of this analytical expression to calculate the time derivatives of lightning induced voltages may result in errors as large as 50%. The derived expression in this paper can be used to correct for this inaccuracy. We also provide an exact expression for the electric field at any given point in space when the propagating current is an impulse function. This expression can be combined with the convolution integral to obtain the electric field corresponding to waveforms similar to measured return stroke currents.