Step-Function Response of a Line with Ground Return - Empirical Vs. Asymptotic Approximations for Impedance

A first-order asymptotic approximation to the distributed series impedance of an overhead transmission line with ground return (frequency-dependent resistance) is insufficient for the prediction of the step-function response at distances of a mile (1. 6km) or more from the source. Series solutions based on a third-order asymptotic impedance approximation have useful convergence properties only for distances up to perhaps three miles. An empirical impedance function, designed to approximate the true function closely over a finite band of somewhat more than one decade, and utilizing corrective terms proportional to √s exp (-k
₁
,√s) and s exp (-k
₂
√s), yields a series solution which is usable for much longer distances. These series solutions predict a more rapid rise in voltage than does the first-order asymptotic approximation and they indicate that, at distances of more than a few miles, rise time tends to be directly proportional to the distance.