Smith Chart

A Smith chart is a circular coordinate diagram used in radio-frequency engineering to represent complex load impedances and admittances together with their complex voltage reflection coefficients. Consider a voltage wave on a transmission line with real positive characteristic impedance . If is the complex amplitude of the wave incident on a load and is the complex amplitude of the wave reflected by the load, then the voltage reflection coefficient at the load is . Its complex modulus is the reflected-to-incident voltage-amplitude ratio, its complex argument is the phase shift on reflection, and, for a lossless line, is the reflected-to-incident power ratio (Pozar 2012).

Define the normalized load impedance . The Möbius transformation

maps to its complex reflection coefficient , with inverse

Because passive loads satisfy , this conformal mapping sends their half-plane to the closed unit disk . The matched load maps to the center , a short circuit maps to , and an open circuit corresponds to .

Writing , curves of constant resistance and constant reactance become two families of circles in the -plane. These circles form the characteristic grid of the Smith chart and permit graphical calculations for impedance matching and transmission lines (Smith 1939).

A Smith chart and a Nyquist plot can both display a complex-valued frequency locus, but they answer different questions. The Smith chart applies a fixed Möbius transformation to impedance and emphasizes matching and reflection, whereas the Nyquist plot uses the contour winding number about to determine feedback stability.