Equivalent Distributed RC Networks of Transmission Lines

Abstract

There are any number of pairs of resistance and capacitance density functions, i.e., <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">r(x)</tex> and <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">c(x)</tex> , which can provide the same electrical performance as a given distributed RC laddernetwork. In general these equivalent networks will have different lengths. A criterion for equivalence of density functions is expressed in the following: Networks with the same <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">r</tex> to <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">c</tex> ratio, expressed as a function of <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">z</tex> , are equivalent at network-lengths which correspond to the same value of <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">z</tex> . The variable <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">z</tex> is related to the distance <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">x</tex> by <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">z = \int_0^x \sqrt{rc} dx</tex> . The equivalency criteria can be used to find alternate networks to a given network. Such alternates may be more suitable for fabrication. Examples of networks equivalent to the exponential, proportional and Bessel networks are given. The results apply to tapered transmission lines as well.