A Method for the Evaluation of Cooray–Rubinstein Formula Based on Talbot's Method for Numerical Inverse Laplace Transform

The Cooray-Rubinstein formula (C-R formula) has been widely used to calculate the horizontal electric field generated by a lightning return stroke over the lossy ground. To evaluate the time-domain C-R formula by a recursive convolution, an alternative method is proposed in this paper, where the formulation in terms of the “step response” corresponding to the ground surface impedance, denoted as k
_ε
(t), is evaluated by the numerical inverse Laplace transform (NILT). Considering the good performance of Talbot's method for the NILT, Talbot's method is adopted to express k
_ε
(t) in terms of a series of exponential functions, which can avoid the tedious piecewise approximation used by the available method and make the recursive convolution easier to program. In addition, some key formulae for the recursive convolution with complex poles and residues are presented in this paper. Numerical examples are provided to validate the proposed method.