Use of the Smith Chart with Complex Characteristic Impedance

The Smith chart has long been recognized as a useful classroom aid in the study of transmission lines and waveguides. It could also be used for more general n-port network calculations, except that normalizing with respect to a complex characteristic impedance can lead to left half-plane normalized impedances and, consequently, points outside the standard Smith chart even for passive terminations. A simple transformation is presented which permits the direct use of the standard Smith chart in such cases. Detailed examples of a filter and low-frequency transmission line are presented.