The Smith chart is a circular nomogram used in radio frequency (RF) engineering to solve transmission line and impedance-matching problems. It plots a complex reflection coefficient ( Γ {\displaystyle \Gamma } ) on a grid of normalized electrical impedance ( z {\displaystyle z} ). Depending on the region, it is also known as the Smith diagram, Volpert–Smith chart, or Mizuhashi chart. Because normalized impedance is complex, the chart maps two families of curves: circles of constant resistance ( Re ( z ) {\displaystyle \operatorname {Re} (z)} ) and arcs of constant reactance ( Im ( z ) {\displaystyle \operatorname {Im} (z)} ). Standard charts focus on passive circuits where resistance is non-negative ( Re ( z ) ≥ 0 {\displaystyle \operatorname {Re} (z)\geq 0} ). Regions outside the unit circle ( Re ( z ) < 0 {\displaystyle \operatorname {Re} (z)<0} ) correspond to negative resistance, which is used for oscillator design and stability analysis. The display allows engineers to evaluate multiple RF parameters at once, including admittance, S n n {\displaystyle S_{nn}} scattering parameters, noise figures, and stability boundaries. Although paper charts have mostly been replaced by software for calculations, the display format remains standard across RF simulation software and vector network analyzers to visualize how parameters change with frequency.
History The Smith chart emerged in the late 1930s through the independent work of Tōsaku Mizuhashi, Amiel R. Volpert, and Phillip H. Smith. It was initially known by several names before Smith chart became the standard name in the Western world by 1950. Tōsaku Mizuhashi (水橋東作) independently proposed the chart in 1937, while Amiel R. Volpert (Амиэ́ль Р. Во́льперт) and Phillip H. Smith independently proposed it in 1939. Smith initially developed a rectangular diagram, followed by a polar coordinate chart by 1936. With input from colleagues Enoch B. Ferrell and James W. McRae, who were familiar with conformal mapping, he refined it into its final form in early 1937. The chart was published in January 1939. Smith originally called it the "transmission line chart". Early authors also used names such as "reflection chart", "circle diagram of impedance", "immittance chart", and "Z-plane chart". During the 1940s, researchers at MIT's Radiation Laboratory began referring to it simply as the "Smith chart". By 1950, the name had become the generally accepted term in the Western world.
Overview The Smith chart converts normalized impedance into the complex reflection coefficient using a Möbius transformation. Impedances with positive real parts (passive loads) plot inside the unit circle, while those with negative real parts fall outside it.
For an impedance chart, the transformation is:
Γ = Z − Z 0 Z + Z 0 = z − 1 z + 1 , {\displaystyle \Gamma ={\frac {Z-Z_{0}}{Z+Z_{0}}}={\frac {z-1}{z+1}},}
where z = Z / Z 0 {\displaystyle z=Z/Z_{0}} is the complex impedance Z {\displaystyle Z} normalized by the reference impedance Z 0 {\displaystyle Z_{0}} . Charts can show normalized impedance (Z chart), normalized admittance (Y chart), or both (YZ chart). On combined YZ charts, different colors distinguish the two grids. The outer border has scales marked in wavelengths and degrees. The wavelength scale measures distance along a transmission line in distributed-element circuits, while the degree scale shows the phase angle of the reflection coefficient. Because the chart uses normalized values, it applies to any system impedance. Its center represents the reference impedance (typically 50 Ω). To find the actual impedance or admittance, multiply the chart value by Z 0 {\displaystyle Z_{0}} or Y 0 {\displaystyle Y_{0}} . Reflection coefficients are unitless and read directly off the chart. Engineers use the chart for both distributed and lumped-element circuit analysis. For manual calculations, plotting a single point per frequency works well for narrowband applications (typically 5–10% bandwidth). Across wider bandwidths, connecting points from multiple frequencies forms a locus. This path shows how capacitive or inductive a load is, how difficult it is to match, and how well the component performs across the frequency range. If a locus covers a wide impedance range, details can become hard to read, though specific regions can be enlarged for accuracy.
Mathematical basis
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