Application of the Smith Chart to General Impedance Transformations

The Smith Chart is not generally used for making impedance calculations when the characteristic impedance of the two terminal-pair network involved is anything but a real number because normalization of some terminating impedances to the complex characteristic impedance may result in a normalized impedance whose real part is negative, and a "propagation constant" whose magnitude is greater than one, even though no active elements are involved in the network. This paper shows how impedances may be calculated using the Smith Chart and associated techniques in much the same manner, and almost as conveniently, as for lossless transmission lines, even though the network may be unsymmetrical, and have complex image impedances and propagation constant. The calculations are mathematically rigorous and simple using the Smith Chart only for converting back and forth between the complex numbers Г in polar form and Z in rectangular form where Г and Z are related by the mathematical expression Z- Z+ 1 No physical interpretation of these numbers is made during the computation as no such interpretation is necessary or useful as far as the method is concerned and may even confuse the main issue, that of a simple solution to a heretofore complex problem.