Smith chart

(a)

(b)

A smith chart is a graphical overlay that allows plotting a complex reflection coefficient, ${\displaystyle \Gamma }$, on top of grid lines of constant normalized impedance, ${\displaystyle z}$. Since the normalized impedance is also a complex quantity, the Smith Chart shows both lines of constant ${\displaystyle \Re e[z]}$ and lines of constant ${\displaystyle \Im m[z]}$. The Smith chart is limited to values of normalized resistance, ${\displaystyle z}$, for which ${\displaystyle \Re e[z]\geq 0}$, since Smith charts are mainly used for passive circuits.
(a) A sample Smith chart in which the lines of constant ${\displaystyle \Im m[z]}$ are depicted as blue arcs and lines of constant ${\displaystyle \Re e[z]}$ are depicted as red circles.
(b) an animated transformation of lines of constant ${\displaystyle \Re e[z]}$ and lines of constant ${\displaystyle \Im m[z]}$ from the ${\displaystyle z-}$space (where the lines appear straight vertical and horizontal) to the ${\displaystyle \Gamma -}$space (where the lines appear as circles). The transformation is a conformal mapping. Pink lines are used to denote ${\displaystyle \Re e[z]<0}$ and black lines are used to denote ${\displaystyle \Re e[z]\geq 0}$.

The Smith chart (sometimes also called Smith diagram, Mizuhashi chart (水橋チャート), Mizuhashi–Smith chart (水橋スミスチャート), Volpert–Smith chart (Диаграмма Вольперта—Смита) or Mizuhashi–Volpert–Smith chart) is a graphical calculator or nomogram designed for electrical and electronics engineers specializing in radio frequency (RF) engineering to assist in solving problems with transmission lines and matching circuits.

It was independently proposed by Tōsaku Mizuhashi (水橋東作) in 1937, and by Amiel R. Volpert (Амиэ́ль Р. Во́льперт) and Phillip H. Smith in 1939. Starting with a rectangular diagram, Smith had developed a special polar coordinate chart by 1936, which, with the input of his colleagues Enoch B. Ferrell and James W. McRae, who were familiar with conformal mappings, was reworked into the final form in early 1937, which was eventually published in January 1939. While Smith had originally called it a "transmission line chart" and other authors first used names like "reflection chart", "circle diagram of impedance", "immittance chart" or "Z-plane chart", early adopters at MIT's Radiation Laboratory started to refer to it simply as "Smith chart" in the 1940s, a name generally accepted in the Western world by 1950.

The Smith chart can be used to simultaneously display multiple parameters including impedances, admittances, reflection coefficients, ${\displaystyle S_{nn}\,}$ scattering parameters, noise figure circles, constant gain contours and regions for unconditional stability. The Smith chart is most frequently used at or within the unity radius region. However, the remainder is still mathematically relevant, being used, for example, in oscillator design and stability analysis. While the use of paper Smith charts for solving the complex mathematics involved in matching problems has been largely replaced by software based methods, the Smith chart is still a very useful method of showing how RF parameters behave at one or more frequencies, an alternative to using tabular information. Thus most RF circuit analysis software includes a Smith chart option for the display of results and all but the simplest impedance measuring instruments can plot measured results on a Smith chart display.