On the sum of the reflection and transmission coefficient on the Smith chart and 3D Smith chart

Abstract

The paper presents in premiere a simple mapping property of the sum of the reflection and transmission parameters of reciprocal two port networks. It is proved that although the reflection and transmission parameter may have very complicated paths on the Smith chart, their sum will be always moving on the unit circle if the circuit is symmetric and lossless. Further once symmetrical losses at the ports occur their sum path will switch on a family of circles through one point. Using inversive geometry we construct a new function which maps this family of circles in lines on the extended Smith chart. The proposed method for checking the symmetry uses just two parameters and avoids testing the phase of the corresponding input and output parameters. By means of the 3D Smith chart we propose in the end an alternative approach to visualize the parameters.