3.2 Elliptic Integrals in Transmission-Line Analysis

The analysis, by use of conformal transformation techniques ^{[1] , [2]}, of transmission lines whose cross-sections include right-angled configurations involves, at one stage or another, the use of elliptic integrals. Since a complete treatment of these is obviously inappropriate for inclusion in the present text, the discussion will be limited to the most frequently encountered example (the interested reader will find a detailed and comprehensive description of elliptic integrals in ^[3]).

This is the "complete elliptic integral of the 1st kind," usually denoted by K( k), which is defined as follows:

k is known as the "modulus" of K.

The associated "complementary" function K ?( k) is defined by

where

The evaluation of K and K ? involves extremely tedious hand-calculation and/or computer programming, and in general it is preferable to make use of available tables, e.g. ^[4]. However, in transmission-line applications, these elliptic functions are most frequently encountered in the form of the ratio K/K ?, and there are in existence some simple, and very accurate, closed-form formulae for this ratio in terms of k. Which formula is to be used depends upon the accuracy required: Hilberg ^[5] has quoted three versions, as follows:

1. For 1 ? K/K ? ? ? and 0.5 ? k ² ? 1

   
   

   For 0 ? K/K ? ? 1 and 0 ? k ² ? 0.5

   
   

   Claimed accuracy is...