Lumped approximation of the characteristic impedance of the RLGC transmission line with error analysis

A family of lumped circuits whose input impedance approximates the characteristic impedance function of an RLGC transmission line is presented. The input impedance matches the characteristic impedance exactly at both zero and infinity in the frequency domain so that the initial value and the final value of its step response are exact. Each circuit is a cascade connection of N sections of a symmetric ladder network. A tight upper bound on the maximum relative error is derived and is explicitly given in terms of the line parameters and the number of sections N. It is demonstrated that for most practical cases, four sections, or a fourth order lumped circuit, are sufficient to yield a relative error not greater than 10^-7