Formulas for the Four-Terminal Network Parameters of Uniform Ladder Networks

Abstract

In the treatment of the over-all behavior of a smooth transmission line, or uniform ladder network, one frequently wishes to deal with the commonly termed A-B-C-D parameters rather than the propagation function, to which the former are related by hyperbolic functions, and the characteristic impedances. However, the latter are the functions which are simply related to the network components. The A-B-C-D parameters may be given as powerseries expansions of the variable α, where α <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</sup> = Z <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">T</sub> Y <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">T</sub> and Z <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">T</sub> and Y <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">T</sub> are the respective total series impedance and shunt admittance. For the smooth line these are infinite series; but they are terminating series for the ladder network, the number of terms and the coefficients of the terms being functions of the number (M) of elementary π or T sections of which the network is composed. Formulas for these coefficients are developed and they are given in several alternate forms. Also included are recurrence formulas and some relations among the coefficients of the different parameters. One form for the coefficients is particularly interesting because it shows that each coefficient for the ladder network is the corresponding smoothline coefficient multiplied by a polynomial consisting of unity plus a polynomial in nonzero powers of 1/M <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</sup> . This puts in evidence the manner in which the two coefficients approach each other as M increases; and also leads to easily computable upper and lower bounds of the ladder-network coefficients, which may provide useful estimates of their magnitudes.