View source for Smith chart

{{Short description|Electrical engineers graphical calculator}}
{{For|other uses of the term Smith diagram|Smith diagram (disambiguation)}}
{{Redirect|Volpert diagram|the similar term Volpert graph|Volpert graph (disambiguation)}}
{{Use British English|date=August 2017}}
{{Use dmy dates|date=June 2023|cs1-dates=y}}
{{Use list-defined references|date=June 2023}}
The '''Smith chart''' (sometimes also called '''Smith diagram''', '''Mizuhashi chart''' ({{lang|ja|水橋チャート}}), '''Mizuhashi–Smith chart''' ({{lang|ja|水橋<!--・-->スミス<!--・-->チャート}}),<ref name="Okamura_1959"/><ref name="Kenichi_1999"/><ref name="Mori_2013"/> '''Volpert–Smith chart'''<!-- also "Volpert–Smith diagram" --> ({{lang|ru|Диаграмма Вольперта—Смита}})<ref name="Kurochkin_2009"/><ref name="Salov_2022"/> or '''Mizuhashi–Volpert–Smith chart''') is a graphical calculator or [[nomogram]] designed for [[electrical engineering|electrical and electronics engineers]] specializing in [[radio frequency]] (RF) engineering to assist in solving problems with [[transmission line]]s and [[impedance matching|matching]] circuits.<ref name="Ramo-Whinnery-Duzer_1965"/><ref name="Ramo-Whinnery-Duzer_1994"/><ref name="Smith_1969"/><ref name="Smith_1995"/><ref name="MAX_2012"/>

It was [[simultaneous invention|independently]]<ref name="Voltmer_2007"/><ref name="Kurochkin_2009"/><ref name="ETHW_2018"/><ref name="Salov_2022"/> proposed by [[Tōsaku Mizuhashi]]<!-- also transscribed as "Tōsaku Mizuhasi" --> ({{lang|ja|水橋東作}}) in 1937,<ref name="Mizuhashi_1937"/> and by {{ill|Amiel Rafailovich Volpert|ru|Вольперт, Амиэль Рафаилович|lt=Amiel R. Volpert}} ({{lang|ru|Амиэ́ль Р. Во́льперт}})<!-- many third-party sources state 1939 suggesting that there are earlier sources than 1940 --><ref name="Volpert_1940"/><ref name="Kurochkin_2009"/> and [[Phillip Hagar Smith|Phillip H. Smith]] in 1939.<ref name="Smith_1939"/><ref name="Smith_1944"/> Starting with a rectangular diagram<!-- according to <ref name="Inan_2005"/> "in 1931", but I could not find a more authorative source for this so far -->, 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 mapping]]s, was reworked into the final form in early 1937, which was eventually published in January 1939.<ref name="Smith_1939"/><ref name="Smith_1995"/><ref name="Inan_2005"/> While Smith had originally called it a "''transmission line chart''"<ref name="Smith_1939"/><ref name="Smith_1944"/> and other authors first used names like "''reflection chart''", "''circle diagram of impedance''", "''immittance chart''" or "''Z-plane chart''",<ref name="Smith_1995"/> early adopters at [[MIT]]'s [[MIT Radiation Laboratory|Radiation Laboratory]] started to refer to it simply as "''Smith chart''" in the 1940s,<ref name="Smith_1995"/><ref name="Inan_2005"/> a name generally accepted in the Western world by 1950<!-- possibly earlier? -->.<ref name="Linton_1950"/><ref name="GeneralRadio_1950"/>

The Smith chart can be used to simultaneously display multiple parameters including [[electrical impedance|impedances]], [[admittance]]s, [[reflection coefficient]]s, <math>S_{nn}\,</math> [[scattering parameters]], [[noise figure]] circles, constant gain contours and regions for [[stability theory|unconditional stability]].<ref name="Pozar_2005"/><ref name="Gonzalez_1997"/>{{rp|pages=93–103}} 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 [[electronic oscillator|oscillator]] design and [[stability theory|stability]] analysis.<ref name="Gonzalez_1997"/>{{rp|pages=98–101}} 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<ref name="Bevelacqua_2013"/> how RF parameters behave at one or more frequencies, an alternative to using [[table (information)|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.<ref name="Tektronix_2017"/>
[[File:Smith chart3.svg|thumb|290x290px|An impedance Smith chart (with no data plotted).]]

==Overview==
[[Image:NetworkAnalyzer.jpg|400px|thumb|A [[Network analyzer (electrical)|network analyzer]] ([[Hewlett-Packard|HP]] 8720A) showing a Smith chart.]]

The Smith chart is a mathematical transformation of the two-dimensional Cartesian complex plane. [[Complex numbers]] with positive real parts map inside the circle. Those with negative real parts map outside the circle.
If we are dealing only with impedances with non-negative resistive components, our interest is focused on the area inside the circle. The transformation, for an impedance Smith chart, is:

<math display="block">\Gamma = \frac{Z - Z_0}{Z + Z_0} = \frac{z - 1}{z + 1},</math>

where {{math|1= ''z'' = {{sfrac|''Z''|''Z''{{sub|0}}}}}}, i.e., the complex impedance, {{mvar|Z}}, normalized by the reference impedance, {{math|''Z''{{sub|0}}}}. The impedance Smith chart is then an [[Argand plot]] of impedances thus transformed. Impedances with non-negative resistive components will appear inside a circle with unit radius; the origin will correspond to the reference impedance, {{math|''Z''{{sub|0}}}}.

The Smith chart is plotted on the [[complex number|complex]] [[reflection coefficient]] plane in [[dimension|two dimensions]] and may be scaled in normalised [[Electrical impedance|impedance]] (the most common), normalised [[admittance]] or both, using different colours to distinguish between them. These are often known as the Z, Y and YZ Smith charts respectively.<ref name="Gonzalez_1997"/>{{rp|page=97}} Normalised scaling allows the Smith chart to be used for problems involving any [[characteristic impedance|characteristic]] or system impedance which is represented by the center point of the chart. The most commonly used normalization impedance is 50&nbsp;[[ohm]]s. Once an answer is obtained through the graphical constructions described below, it is straightforward to convert between normalised impedance (or normalised admittance) and the corresponding unnormalized value by multiplying by the characteristic impedance (admittance). Reflection coefficients can be read directly from the chart as they are unitless parameters.

The Smith chart has a scale around its [[circumference]] or periphery which is graduated in [[wavelengths]] and [[degree (angle)|degree]]s. The wavelengths scale is used in [[distributed-element model|distributed component]] problems and represents the distance measured along the transmission line connected between the [[signal generator|generator]] or source and the load to the point under consideration. The degrees scale represents the angle of the voltage reflection coefficient at that point. The Smith chart may also be used for [[lumped-element]] matching and analysis problems.

Use of the Smith chart and the interpretation of the results obtained using it requires a good understanding of [[alternating current|AC circuit theory]] and transmission-line theory, both of which are prerequisites for RF engineers.

As impedances and admittances change with frequency, problems using the Smith chart can only be solved manually using one [[frequency]] at a time, the result being represented by a [[point (geometry)|point]]. This is often adequate for [[frequency range|narrow band]] applications (typically up to about 5% to 10% [[Bandwidth (signal processing)|bandwidth]]) but for wider bandwidths it is usually necessary to apply Smith chart techniques at more than one frequency across the operating frequency band. Provided the frequencies are sufficiently close, the resulting Smith chart points may be joined by straight lines to create a [[locus (mathematics)|locus]].

A locus of points on a Smith chart covering a range of frequencies can be used to visually represent:
*how [[capacitance|capacitive]] or how [[inductance|inductive]] a load is across the frequency range
*how difficult matching is likely to be at various frequencies
*how well matched a particular component is.
The accuracy of the Smith chart is reduced for problems involving a large locus of impedances or admittances, although the scaling can be magnified for individual areas to accommodate these.

==Mathematical basis==
[[File:Smith chart explanation.svg|thumb|500px|right|Most basic use of an impedance Smith chart. A wave travels down a [[transmission line]] of [[characteristic impedance]] ''Z''<sub>0</sub>, terminated at a load with [[Electrical impedance|impedance]] ''Z''<sub>L</sub> and normalised impedance {{nowrap|1=''z''=''Z''<sub>L</sub>/''Z''<sub>0</sub>}}. There is a [[signal reflection]] with coefficient Γ. Each point on the Smith chart simultaneously represents both a value of ''z'' (bottom left), and the corresponding value of Γ (bottom right), related by {{nowrap|1=''z''=(1&nbsp;+&nbsp;Γ)/(1&nbsp;&minus;&nbsp;Γ).}}]]

===Actual and normalised impedance and admittance===
A transmission line with a characteristic impedance of <math>Z_0\,</math> may be universally considered to have a [[characteristic admittance]] of <math>Y_0\,</math> where
:<math>Y_0 = \frac{1}{Z_0}\,</math>
Any impedance, <math>Z_\text{T}\,</math> expressed in ohms, may be normalised by dividing it by the characteristic impedance, so the normalised impedance using the lower case ''z''<sub>T</sub> is given by 
:<math>z_\text{T} = \frac{Z_\text{T}}{Z_0}\,</math>
Similarly, for normalised admittance
:<math>y_\text{T} = \frac{Y_\text{T}}{Y_0}\,</math>
The [[International system of units|SI unit]] of [[Electrical impedance|impedance]] is the [[ohm]] with the symbol of the upper case [[Greek alphabet|Greek letter]] [[omega]] (Ω) and the [[SI unit]] for [[admittance]] is the [[Siemens (unit)|siemens]] with the symbol of an upper case letter S. Normalised impedance and normalised admittance are [[dimensionless]]. Actual impedances and admittances must be normalised before using them on a Smith chart. Once the result is obtained it may be de-normalised to obtain the actual result.

===The normalised impedance Smith chart===
[[File:Transmission_line_pulse_reflections.gif|thumb|right|300px|Transmission lines terminated by an open circuit (top) and a short circuit (bottom). A pulse reflects perfectly off both these terminations, but the sign of the reflected voltage is opposite in the two cases. Black dots represent electrons, and arrows show the electric field.]]
Using transmission-line theory, if a transmission line is [[Electrical termination|terminated]] in an impedance (<math>Z_\text{T}\,</math>) which differs from its characteristic impedance (<math>Z_0\,</math>), a [[standing wave]] will be formed on the line comprising the [[resultant]] of both the incident or '''f'''orward (<math>V_\text{F}\,</math>) and the '''r'''eflected or reversed (<math>V_\text{R}\,</math>) waves. Using [[complex number|complex]] [[exponential function|exponential]] notation:
:<math>V_\text{F} = A \exp(j \omega t)\exp(+\gamma \ell)~\,</math> and
:<math>V_\text{R} = B \exp(j \omega t)\exp(-\gamma \ell)\,</math>
where

:<math>\exp(j \omega t)\,</math> is the [[time|temporal]] part of the wave
:<math>\exp(\pm\gamma \ell)\,</math> is the spatial part of the wave and
:<math>\omega = 2 \pi f\,</math> where
:<math>\omega\,</math> is the [[angular frequency]] in [[radian]]s per [[second]] (rad/s)
:<math>f\,</math> is the [[frequency]] in [[hertz]] (Hz)
:<math>t\,</math> is the time in seconds (s)
:<math>A\,</math> and <math>B\,</math> are [[Constant (mathematics)|constant]]s
:<math>\ell\,</math> is the distance measured along the transmission line from the load toward the generator in metres (m)
Also
:<math>\gamma = \alpha + j \beta\,</math> is the [[propagation constant]] which has [[SI units]] [[radians]]/[[meter]]
where
:<math>\alpha\,</math> is the [[attenuation constant]] in [[neper]]s per metre (Np/m)
:<math>\beta\,</math> is the [[phase constant]] in [[radian]]s per metre (rad/m)
The Smith chart is used with one frequency (<math>\omega</math>) at a time, and only for one moment (<math>t</math>) at a time, so the temporal part of the phase (<math>\exp(j \omega t)\,</math>) is fixed. All terms are actually multiplied by this to obtain the [[instantaneous phase]], but it is conventional and understood to omit it. Therefore,
:<math>V_\text{F} = A \exp(+\gamma \ell)\,</math> and
:<math>V_\text{R} = B \exp(-\gamma \ell)\,</math>

where <math>A\,</math> and <math>B\,</math> are respectively the forward and reverse voltage amplitudes at the load.

====The variation of complex reflection coefficient with position along the line====
[[File:SmithChartLineLength.svg|thumb|right|500px|Looking towards a load through a length <math>\ell\,</math> of lossless transmission line, the impedance changes as <math>\ell\,</math> increases, following the blue circle. (This impedance is characterized by its reflection coefficient <math>V_{\text{reflected}}/V_{\text{incident}}</math>.) The blue circle, centered within the impedance Smith chart, is sometimes called an ''SWR circle'' (short for ''constant [[standing wave ratio]]'').]]
The complex voltage reflection coefficient <math>\Gamma\,</math> is defined as the ratio of the reflected wave to the incident (or forward) wave. Therefore,
:<math>\Gamma = \frac{V_\text{R}}{V_\text{F}} = \frac{B \exp(-\gamma \ell)}{A \exp(+\gamma \ell)} = C \exp(-2 \gamma \ell)\,</math>
where {{math|''C''}} is also a constant.

For a uniform transmission line (in which <math>\gamma\,</math> is constant), the complex reflection coefficient of a standing wave varies according to the position on the line. If the line is [[attenuation (electronics)|lossy]] (<math>\alpha\,</math> is non-zero) this is represented on the Smith chart by a [[spiral]] path. In most Smith chart problems however, losses can be assumed negligible (<math>\alpha = 0\,</math>) and the task of solving them is greatly simplified. For the loss free case therefore, the expression for complex reflection coefficient becomes 
:<math>\Gamma = \Gamma_\text{L} \exp(-2 j \beta \ell)\,</math>
where <math>\Gamma_\text{L}\,</math> is the reflection coefficient at the load, and <math>\ell\,</math> is the line length from the load to the location where the reflection coefficient is measured. The phase constant <math>\beta\,</math> may also be written as 
:<math>\beta = \frac{2 \pi}{\lambda}\,</math>
where <math>\lambda\,</math> is the wavelength ''within the transmission line'' at the test frequency.

Therefore,
:<math>\Gamma = \Gamma_\text{L} \exp\left(\frac{-4 j \pi}{\lambda}\ell\right)\,</math>
This equation shows that, for a standing wave, the complex reflection coefficient and impedance repeats every half wavelength along the transmission line. The complex reflection coefficient is generally simply referred to as reflection coefficient. The outer circumferential scale of the Smith chart represents the distance from the generator to the load scaled in wavelengths and is therefore scaled from zero to 0.50.

====The variation of normalised impedance with position along the line====
If <math>\,V\,</math> and <math>\,I\,</math> are the voltage across and the current entering the termination at the end of the transmission line respectively, then
:<math>V_\mathsf{F} + V_\mathsf{R} = V \,</math>
and
:<math> V_\mathsf{F} - V_\mathsf{R} = Z_0\, I \,</math>.
By dividing these equations and substituting for both the voltage reflection coefficient
:<math> \Gamma = \frac{V_\mathsf{R}}{\, V_\mathsf{F} \,} \,</math>
and the normalised impedance of the termination represented by the lower case {{mvar|z}}, subscript T
:<math> z_\mathsf{T} = \frac{V}{\, Z_0\, I \,} \,</math>
gives the result:
:<math> z_\mathsf{T} = \frac{1 + \Gamma}{\, 1 - \Gamma \,} \,.</math>
Alternatively, in terms of the reflection coefficient
:<math> \Gamma = \frac{z_\mathsf{T} - 1}{\, z_\mathsf{T} + 1 \,} \,</math>
These are the equations which are used to construct the {{math|Z}} Smith chart. Mathematically speaking <math>\,\Gamma\,</math> and <math>\,z_\mathsf{T}\,</math> are related via a [[Möbius transformation]].

Both <math>\,\Gamma\,</math> and <math>\,z_\mathsf{T}\,</math> are expressed in [[complex number]]s without any units. They both change with frequency so for any particular measurement, the frequency at which it was performed must be stated together with the characteristic impedance.

<math>\,\Gamma\,</math> may be expressed in [[magnitude (mathematics)|magnitude]] and [[angle]] on a [[complex plane|polar diagram]]. Any actual reflection coefficient must have a magnitude of less than or equal to [[1 (number)|unity]] so, at the test frequency, this may be expressed by a point inside a circle of unity radius. The Smith chart is actually constructed on such a polar diagram. The Smith chart scaling is designed in such a way that reflection coefficient can be converted to normalised impedance or vice versa. Using the Smith chart, the normalised impedance may be obtained with appreciable accuracy by plotting the point representing the reflection coefficient ''treating the Smith chart as a polar diagram'' and then reading its value directly using the characteristic Smith chart scaling. This technique is a graphical alternative to substituting the values in the equations.

By substituting the expression for how reflection coefficient changes along an unmatched loss-free transmission line
:<math> \Gamma = \frac{B \exp(-\gamma \ell)}{A \exp(\gamma \ell)} = \frac{B \exp(-j \beta \ell)}{A \exp(j \beta \ell)} \,</math>
for the loss free case, into the equation for normalised impedance in terms of reflection coefficient
:<math> z_\mathsf{T} = \frac{1 + \Gamma}{\, 1 - \Gamma \,} \,.</math> 
and using [[Euler's formula]]
:<math> \exp(j\theta) = \text{cis}\, \theta = \cos \theta + j\, \sin \theta \,</math>
yields the impedance-version transmission-line equation for the loss free case:<ref name="Hayt_1981"/>
:<math>Z_\mathsf{in} = Z_0 \frac{\, Z_\mathsf{L} + j\, Z_0 \tan (\beta \ell) \,}{\, Z_0 + j\, Z_\mathsf{L} \tan (\beta \ell) \,} \,</math>
where
<math>\,Z_\mathsf{in}\,</math> is the impedance 'seen' at the input of a loss free transmission line of length <math>\,\ell\, ,</math> terminated with an impedance <math>\,Z_\mathsf{L}\,</math>

Versions of the transmission-line equation may be similarly derived for the admittance loss free case and for the impedance and admittance lossy cases.

The Smith chart graphical equivalent of using the transmission-line equation is to normalise <math>\, Z_\mathsf{L} \, ,</math> to plot the resulting point on a {{math|Z}} Smith chart and to draw a circle through that point centred at the Smith chart centre. The path along the arc of the circle represents how the impedance changes whilst moving along the transmission line. In this case the circumferential (wavelength) scaling must be used, remembering that this is the wavelength within the transmission line and may differ from the free space wavelength.

====Regions of the {{math|Z}} Smith chart====
If a polar diagram is mapped on to a [[cartesian coordinate system]] it is conventional to measure angles relative to the positive {{mvar|x}}-axis using a [[counterclockwise]] direction for positive angles. The magnitude of a complex number is the length of a straight line drawn from the [[origin (mathematics)|origin]] to the point representing it. The Smith chart uses the same convention, noting that, in the normalised impedance plane, the positive {{mvar|x}}-axis extends from the center of the Smith chart at <math>\, z_\mathsf{T} = 1 \pm j 0 \,</math> to the point <math>\, z_\mathsf{T} = \infty \pm j \infty \,.</math> The region above the x-axis represents inductive impedances (positive imaginary parts) and the region below the {{mvar|x}}-axis represents capacitive impedances (negative imaginary parts).

If the termination is perfectly matched, the reflection coefficient will be zero, represented effectively by a circle of zero radius or in fact a point at the centre of the Smith chart. If the termination was a perfect open circuit or [[short circuit]] the magnitude of the reflection coefficient would be unity, all power would be reflected and the point would lie at some point on the unity circumference circle.

====Circles of constant normalised resistance and constant normalised reactance====
The normalised impedance Smith chart is composed of two families of circles: circles of constant normalised resistance and circles of constant normalised reactance. In the complex reflection coefficient plane the Smith chart occupies a circle of unity radius centred at the origin. In cartesian coordinates therefore the circle would pass through the points (+1,0) and (&minus;1,0) on the {{mvar|x}}-axis and the points (0,+1) and (0,&minus;1) on the {{mvar|y}}-axis.

Since both <math>\,\Gamma\,</math> and <math>\,z_\mathsf{T}\,</math> are complex numbers, in general they may be written as:
:<math>z_\mathsf{T} = a + j b \,</math>
:<math>~ \Gamma ~= c + j d \,</math>

with ''a'', ''b'', ''c'' and ''d'' real numbers.

Substituting these into the equation relating normalised impedance and complex reflection coefficient:
:<math>\Gamma = \frac{z_\mathsf{T} - 1}{\, z_\mathsf{T} + 1 \,} = \frac{(a - 1) + j\,b}{\, (a + 1) + j\,b \,} \,</math>
gives the following result:
:<math>\Gamma = c + j d = \left[\frac{a^2 + b^2 - 1}{\,(a + 1)^2 + b^2\,}\right] + j \left[\frac{2b}{\,(a + 1)^2 + b^2\,}\right]  = \left[ 1 + \frac{ -2(a + 1) }{\,(a + 1)^2 + b^2\,}\right] + j \left[\frac{+2b}{\,(a + 1)^2 + b^2\,}\right] \,.</math>
This is the equation which describes how the complex reflection coefficient changes with the normalised impedance and may be used to construct both families of circles.<ref name="Davidson_1989"/>

===The {{math|Y}} Smith chart===
The {{math|Y}} Smith chart is constructed in a similar way to the {{math|Z}} Smith chart case but by expressing values of voltage reflection coefficient in terms of normalised admittance instead of normalised impedance. The normalised admittance {{mvar|y}}{{sub|T}} is the reciprocal of the normalised impedance {{mvar|z}}{{sub|T}}, so
:<math> y_\mathsf{T} = \frac{1}{z_\mathsf{T}} \,</math>
Therefore:
:<math> y_\mathsf{T} = \frac{1-\Gamma}{\, 1 + \Gamma \,} \,</math>
and
:<math> \Gamma = \frac{ 1 - y_\mathsf{T} }{\, 1 + y_\mathsf{T} \,} \,</math>
The {{math|Y}} Smith chart appears like the normalised impedance, type but with the graphic nested circles rotated through 180°, but the numeric scale remaining in its same position (not rotated) as the {{math|Z}} chart.

Similarly taking
:<math> y_\mathsf{T} = \tilde{o} + j\,\tilde{p} \,</math>
for [[real number|real]] <math>\,\tilde{o}\,</math> and <math>\,\tilde{p}\,</math> gives an analogous result, although with more and different minus signs:
:<math>\Gamma = c + j d = \left[\frac{1 - \tilde{o}^2 - \tilde{p}^2}{\,(\tilde{o} + 1)^2 + \tilde{p}^2\,}\right] + j \left[\frac{ -2\tilde{p} }{\,(\tilde{o} + 1)^2 + \tilde{p}^2\,}\right]  = \left[ \frac{ 2(\tilde{o} + 1) }{\,(\tilde{o} + 1)^2 + \tilde{p}^2\,}- 1 \right] + j \left[\frac{ - 2 \tilde{p} }{\,(\tilde{o} + 1)^2 + \tilde{p}^2\,}\right] \,.</math>

The region above the {{mvar|x}}-axis represents capacitive admittances and the region below the {{mvar|x}}-axis represents inductive admittances. Capacitive admittances have positive [[Imaginary number|imaginary]] parts and inductive admittances have negative imaginary parts.

Again, if the termination is perfectly matched the reflection coefficient will be zero, represented by a 'circle' of zero radius or in fact a point at the centre of the Smith chart. If the termination was a perfect open or short circuit the magnitude of the voltage reflection coefficient would be unity, all power would be reflected and the point would lie at some point on the unity circumference circle of the Smith chart.

===Practical examples===
[[Image:SmithEx2.png|thumbnail|Example points plotted on the normalized impedance Smith chart.]]
A point with a reflection coefficient magnitude 0.63 and angle 60° represented in polar form as <math>0.63\angle60^\circ\,</math>, is shown as point P<sub>1</sub> on the Smith chart. To plot this, one may use the circumferential (reflection coefficient) angle scale to find the <math>\angle60^\circ\,</math> graduation and a ruler to draw a line passing through this and the centre of the Smith chart. The length of the line would then be scaled to P<sub>1</sub> assuming the Smith chart radius to be unity. For example, if the actual radius measured from the paper was 100&nbsp;mm, the length OP<sub>1</sub> would be 63&nbsp;mm.

The following table gives some similar examples of points which are plotted on the ''Z'' Smith chart. For each, the reflection coefficient is given in polar form together with the corresponding normalised impedance in rectangular form. The conversion may be read directly from the Smith chart or by substitution into the equation.

{| class="wikitable" style="margin-left: auto; margin-right: auto; border: none;"
|+Some examples of points plotted on the normalised impedance Smith chart
!width="200"|Point identity
!width="250"|Reflection coefficient (polar form)
!width="300"|Normalised impedance (rectangular form)
|-
|P<sub>1</sub> (Inductive)
|<math>0.63\angle60^\circ\,</math>
|<math>0.80 + j1.40\,</math>
|-
|P<sub>2</sub> (Inductive)
|<math>0.73\angle125^\circ\,</math>
|<math>0.20 + j0.50\,</math>
|-
|P<sub>3</sub> (Capacitive)
|<math>0.44\angle-116^\circ\,</math>
|<math>0.50 - j0.50\,</math>
|}

===Working with both the ''Z'' Smith chart and the ''Y'' Smith charts===
In RF circuit and matching problems sometimes it is more convenient to work with admittances (representing [[electrical conductance|conductance]]s and [[susceptance]]s) and sometimes it is more convenient to work with impedances (representing [[electrical resistance|resistance]]s and [[reactance (electronics)|reactance]]s). Solving a typical matching problem will often require several changes between both types of Smith chart, using normalised impedance for [[series and parallel circuits|series]] elements and normalised admittances for [[series and parallel circuits|parallel]] elements. For these a dual (normalised) impedance and admittance Smith chart may be used. Alternatively, one type may be used and the scaling converted to the other when required. In order to change from normalised impedance to normalised admittance or vice versa, the point representing the value of reflection coefficient under consideration is moved through exactly 180 degrees at the same radius. For example, the point P1 in the example representing a reflection coefficient of <math>0.63\angle60^\circ\,</math> has a normalised impedance of <math> z_P = 0.80 + j1.40\,</math>. To graphically change this to the equivalent normalised admittance point, say Q1, a line is drawn with a ruler from P1 through the Smith chart centre to Q1, an equal radius in the opposite direction. This is equivalent to moving the point through a circular path of exactly 180 degrees. Reading the value from the Smith chart for Q1, remembering that the scaling is now in normalised admittance,  gives <math>y_P = 0.30 - j0.54\,</math>. Performing the calculation
:<math>y_\text{T} = \frac{1}{ z_\text{T} }\,</math>
manually will confirm this.

Once a [[transformation (mathematics)|transformation]] from impedance to admittance has been performed, the scaling changes to normalised admittance until a later transformation back to normalised impedance is performed.

The table below shows examples of normalised impedances and their equivalent normalised admittances obtained by rotation of the point through 180°. Again, these may be obtained either by calculation or using a Smith chart as shown, converting between the normalised impedance and normalised admittances planes.

{| border="1" cellpadding="2"
|+Values of reflection coefficient as normalised impedances and the equivalent normalised admittances
!width="200"|Normalised impedance plane
!width="200"|Normalised admittance plane
|-
|P<sub>1</sub> (<math>z = 0.80 + j1.40\,</math>)
|Q<sub>1</sub> (<math>y = 0.30 - j0.54\,</math>)
|-
|P<sub>10</sub> (<math>z = 0.10 + j0.22\,</math>)
|Q<sub>10</sub> (<math>y = 1.80 - j3.90\,</math>)
|}
[[Image:SmithEx3.png|thumbnail|Values of complex reflection coefficient plotted on the normalized impedance Smith chart and their equivalents on the normalized admittance Smith chart.]]

===Choice of Smith chart type and component type===
The choice of whether to use the ''Z'' Smith chart or the ''Y'' Smith chart for any particular calculation depends on which is more convenient. Impedances in series and admittances in parallel add while impedances in parallel and admittances in series are related by a reciprocal equation. If <math>Z_\text{TS} </math> is the equivalent impedance of series impedances and <math>Z_\text{TP} </math> is the equivalent impedance of parallel impedances, then
:<math>Z_\text{TS} = Z_1 + Z_2 + Z_3 + ... \,</math>
:<math>\frac{1}{Z_\text{TP}} = \frac{1}{Z_1} + \frac{1}{Z_2} + \frac{1}{Z_3} + ... \,</math>
For admittances the reverse is true, that is
:<math>Y_\text{TP} = Y_1 + Y_2 + Y_3 + ... \,</math>
:<math>\frac{1}{Y_\text{TS}} = \frac{1}{Y_1} + \frac{1}{Y_2} + \frac{1}{Y_3} + ... \,</math>
Dealing with the [[multiplicative inverse|reciprocal]]s, especially in complex numbers, is more time-consuming and error-prone than using linear addition. In general therefore, most [[radio frequency|RF]] engineers work in the plane where the circuit topography supports linear addition. The following table gives the complex expressions for impedance (real and normalised) and admittance (real and normalised) for each of the three basic [[passive components|passive circuit elements]]: resistance, inductance and capacitance. Using just the characteristic impedance (or characteristic admittance) and test frequency an [[equivalent circuit]] can be found and vice versa.

{| border="1" cellpadding="1"
|+ <big>'''Expressions for impedance and admittance'''</big> <br/>normalised by impedance {{mvar|Z}}{{sub|0}} or admittance {{mvar|Y}}{{sub|0}} 
!rowspan="2" width="100" style="text-align:center;"| '''Element type'''
|colspan="2" width="180" style="text-align:center;"| '''Impedance''' ({{mvar|Z}} or {{mvar|z}}) or '''Reactance''' ({{mvar|X}} or {{mvar|x}})
|colspan="2" width="180" style="text-align:center;"| '''Admittance''' ({{mvar|Y}} or {{mvar|y}}) or '''Susceptance''' ({{mvar|B}} or {{mvar|b}})
|-
|style="text-align:center;"| '''Actual''' <br/>([[Ohm (unit)|&Omega;]])
|style="text-align:center;"| '''Normalised''' <br/>(no units)
|style="text-align:center;"| '''Actual''' <br/>([[Siemens (unit)|S]])
|style="text-align:center;"| '''Normalised''' <br/>(no units)
|-
|style="text-align:center;"| '''Resistance''' ({{mvar|R}})
| <math>\; Z = R \;</math>
| <math>\; z = \frac{R}{Z_0} = R Y_0 \;</math>
| <math>\; Y = G = \frac{1}{R} \;</math>
| <math>\; y = g = \frac{1}{R Y_0} = \frac{Z_0}{R} \;</math>
|-
|style="text-align:center;"| '''Inductance''' ({{mvar|L}})
| <math>\; Z = j X_\text{L} = j \omega L \;</math>
| <math>\; z = j x_\text{L} = j \frac{\omega L}{Z_0} = j \omega L Y_0 \;</math>
| <math>\; Y = -jB_\text{L} = \frac{-j}{\omega L} \;</math>
| <math>\; y = -jb_\text{L} = \frac{-j}{\omega L Y_0} = \frac{-j Z_0}{\omega L} \;</math>
|-
|style="text-align:center;"| '''Capacitance''' ({{mvar|C}})
| <math>\; Z = -j X_\text{C} = \frac{-j}{\omega C} \;</math>
| <math>\; z = -j x_\text{C} = \frac{-j}{\omega C Z_0} = \frac{-j Y_0}{\omega C} \;</math>
| <math>\; Y = j B_\text{C} = j \omega C \;</math>
| <math>\; y = j b_\text{C} = j\frac{\omega C}{Y_0} = j \omega C Z_0 \;</math>
|}

==Using the Smith chart to solve conjugate matching problems with distributed components==
Distributed matching becomes feasible and is sometimes required when the physical size of the matching components is more than about 5% of a wavelength at the operating frequency. Here the electrical behaviour of many lumped components becomes rather unpredictable. This occurs in microwave circuits and when high power requires large components in shortwave, FM and TV broadcasting.

For distributed components the effects on reflection coefficient and impedance of moving along the transmission line must be allowed for using the outer circumferential scale of the Smith chart which is calibrated in wavelengths.

The following example shows how a transmission line, terminated with an arbitrary load, may be matched at one frequency either with a series or parallel reactive component in each case connected at precise positions.

[[Image:SmithEx4.png|thumbnail|Smith chart construction for some distributed transmission-line matching.]]
Supposing a loss-free air-spaced transmission line of characteristic impedance <math>Z_0 = 50 \ \Omega</math>, operating at a frequency of 800&nbsp;MHz, is terminated with a circuit comprising a 17.5 <math>\Omega</math> resistor in series with a 6.5 nanohenry (6.5 nH) inductor. How may the line be matched?

From the table above, the reactance of the inductor forming part of the termination at 800&nbsp;MHz is
:<math>Z_L = j \omega L = j 2 \pi f L = j32.7 \ \Omega\,</math>
so the impedance of the combination (<math>Z_T</math>) is given by
:<math>Z_T = 17.5 + j32.7 \ \Omega\,</math>
and the normalised impedance (<math>z_T</math>) is
:<math>z_T = \frac{Z_T}{Z_0} = 0.35 + j0.65\,</math>
This is plotted on the Z Smith chart at point P<sub>20</sub>. The line OP<sub>20</sub> is extended through to the wavelength scale where it intersects at the point <math>L_1 = 0.098 \lambda\,</math>. As the transmission line is loss free, a circle centred at the centre of the Smith chart is drawn through the point P<sub>20</sub> to represent the path of the constant magnitude reflection coefficient due to the termination. At point P<sub>21</sub> the circle intersects with the unity circle of constant normalised resistance at 
:<math>z_{P21} = 1.00 + j1.52\,</math>.
The extension of the line OP<sub>21</sub> intersects the wavelength scale at <math>L_2 = 0.177 \lambda\,</math>, therefore the distance from the termination to this point on the line is given by
:<math>L_2 - L_1  = 0.177\lambda - 0.098\lambda = 0.079\lambda\,</math>
Since the transmission line is air-spaced, the wavelength at 800&nbsp;MHz in the line is the same as that in free space and is given by
:<math>\lambda = \frac{c}{f}\,</math>
where <math>c\,</math> is the velocity of [[speed of light|electromagnetic radiation in free space]] and <math>f\,</math> is the frequency in hertz. The result gives <math>\lambda = 375 \ \mathrm{mm}\,</math>, making the position of the matching component 29.6&nbsp;mm from the load.

The conjugate match for the impedance at P<sub>21</sub> (<math>z_{match}\,</math>) is 
:<math>z_{match} = - j (1.52),\!</math>
As the Smith chart is still in the normalised impedance plane, from the table above a series capacitor <math>C_m\,</math> is required where
:<math>z_{match} = - j 1.52 = \frac{-j}{\omega C_m Z_0} = \frac{-j}{2 \pi f C_m Z_0}\,</math>
Rearranging, we obtain
:<math>C_m=\frac{1}{(1.52) \omega Z_0} = \frac{1}{(1.52)(2 \pi f) Z_0}</math>.
Substitution of known values gives
:<math>C_m = 2.6 \ \mathrm{pF}\,</math>
To match the termination at 800&nbsp;MHz, a series capacitor of 2.6 pF must be placed in series with the transmission line at a distance of 29.6&nbsp;mm from the termination.

An alternative shunt match could be calculated after performing a Smith chart transformation from normalised impedance to normalised admittance. Point Q<sub>20</sub> is the equivalent of P<sub>20</sub> but expressed as a normalised admittance. Reading from the Smith chart scaling, remembering that this is now a normalised admittance gives
:<math>y_{Q20} = 0.65 - j1.20\,</math>
(In fact this value is not actually used). However, the extension of the line OQ<sub>20</sub> through to the wavelength scale gives <math>L_3 = 0.152 \lambda\,</math>. The earliest point at which a shunt conjugate match could be introduced, moving towards the generator, would be at Q<sub>21</sub>, the same position as the previous P<sub>21</sub>, but this time representing a normalised admittance given by 
:<math>y_{Q21} = 1.00 + j1.52\,</math>.
The distance along the transmission line is in this case 
:<math>L_2 + L_3  = 0.177\lambda + 0.152\lambda = 0.329\lambda\,</math>
which converts to 123&nbsp;mm.

The conjugate matching component is required to have a normalised admittance (<math>y_{match}</math>) of
:<math>y_{match} = - j1.52\,</math>.
From the table it can be seen that a negative admittance would require an inductor, connected in parallel with the transmission line. If its value is <math>L_m\,</math>, then
:<math>-j1.52 = \frac{-j}{\omega L_m Y_0}= \frac{-jZ_0}{2\pi f L_m}\,</math>
This gives the result 
:<math>L_m = 6.5 \ \mathrm{nH}\,</math>
A suitable inductive shunt matching would therefore be a 6.5 nH inductor in parallel with the line positioned at 123&nbsp;mm from the load.

==Using the Smith chart to analyze lumped-element circuits==
The analysis of [[lumped-element]] components assumes that the wavelength at the frequency of operation is much greater than the dimensions of the components themselves. The Smith chart may be used to analyze such circuits in which case the movements around the chart are generated by the (normalized) impedances and admittances of the components at the frequency of operation. In this case the wavelength scaling on the Smith chart circumference is not used. The following circuit will be analyzed using a Smith chart at an operating frequency of 100&nbsp;MHz. At this frequency the free space wavelength is 3 m. The component dimensions themselves will be in the order of millimetres so the assumption of lumped components will be valid. Despite there being no transmission line as such, a system impedance must still be defined to enable normalization and de-normalization calculations and <math>Z_0 = 50 \ \Omega\,</math> is a good choice here as <math>R_1 = 50 \ \Omega\,</math>. If there were very different values of resistance present a value closer to these might be a better choice.

[[Image:SmithCctEx1.png|thumbnail|A lumped-element circuit which may be analyzed using a Smith chart.|alt=|325x325px]]
[[Image:SmithEx5.png|thumbnail|Smith chart with graphical construction for analysis of a lumped circuit.|alt=|325x325px]]
The analysis starts with a Z Smith chart looking into R<sub>1</sub> only with no other components present. As <math>R_1 = 50 \ \Omega\,</math> is the same as the system impedance, this is represented by a point at the centre of the Smith chart. The first transformation is OP<sub>1</sub> along the line of constant normalized resistance in this case the addition of a normalized reactance of -''j''0.80, corresponding to a series capacitor of 40 pF. Points with suffix P are in the ''Z'' plane and points with suffix Q are in the ''Y'' plane. Therefore, transformations ''P''<sub>1</sub> to ''Q''<sub>1</sub> and ''P''<sub>3</sub> to ''Q''<sub>3</sub> are from the Z Smith chart to the Y Smith chart and transformation ''Q''<sub>2</sub> to ''P''<sub>2</sub> is from the Y Smith chart to the Z Smith chart. The following table shows the steps taken to work through the remaining components and transformations, returning eventually back to the centre of the Smith chart and a perfect 50 ohm match. 
{| border="1" cellpadding="2"
|+Smith chart steps for analysing a lumped-element circuit
!width="100"|Transformation
!width="100"|Plane
!width="100"|''x'' or ''b'' Normalized value
!width="100"|Capacitance/Inductance
!width="150"|Formula to Solve
!width="100"|Result
|-
|<math> O \rightarrow P_1\,</math>
|<math>Z\,</math>
|<math>-j0.80\,</math>
|Capacitance (Series)
|<math>-j0.80 = \frac{-j}{\omega C_1 Z_0}\,</math>
|<math>C_1 = 40 \ \mathrm{pF}\,</math>
|-
|<math> Q_1 \rightarrow Q_2\,</math>
|<math>Y\,</math>
|<math> -j1.49\,</math>
|Inductance (Shunt)
|<math>-j1.49 = \frac{-j}{\omega L_1 Y_0}\,</math>
|<math>L_1 = 53 \ \mathrm{nH}\,</math>
|-
|<math> P_2 \rightarrow P_3\,</math>
|Z
|<math>-j0.23\,</math>
|Capacitance (Series)
|<math>-j0.23 = \frac{-j}{\omega C_2 Z_0}\,</math>
|<math>C_2 = 138 \ \mathrm{pF}\,</math>
|-
|<math> Q_3 \rightarrow O\,</math>
|Y
|<math>+j1.14\,</math>
|Capacitance (Shunt)
|<math>+j1.14 = \frac{j \omega C_3}{Y_0}\,</math>
|<math>C_3 = 36 \ \mathrm{pF}\,</math>
|}

==3D Smith chart==
[[File:3D Smith Chart representation.jpg|thumb|100px|alt=3D Smith chart representation.|3D Smith chart representation.]]
A generalization of the Smith chart to a three dimensional sphere, based on the extended complex plane ([[Riemann sphere]]) and [[inversive geometry]], was proposed by Muller, ''et al'' in 2011.<ref name="Muller-Soto-Dascalu-Neculoiu-Boria_2011"/>

The chart unifies the passive and active circuit design on little and big circles on the surface of a unit sphere, using a stereographic [[conformal map]] of the reflection coefficient's generalized plane. Considering the point at infinity, the space of the new chart includes all possible loads: The north pole is the perfectly matched point, while the south pole is the completely mismatched point.<ref name="Muller-Soto-Dascalu-Neculoiu-Boria_2011"/>

The 3D Smith chart has been further extended outside of the spherical surface, for plotting various scalar parameters, such as group delay, quality factors, or frequency orientation. The visual frequency orientation (clockwise vs. counter-clockwise) enables one to differentiate between a negative / capacitance and positive / inductive whose reflection coefficients are the same when plotted on a 2D Smith chart, but whose orientations diverge as frequency increases.<ref name="Muller-Asavei-Moldoveanu-Sanabria-Codesal-Khadar-Popescu-Dascalu-Ionescu_2020"/>

== See also ==
* [[Binary tiling]]
* [[Bode plot]]
* [[Nyquist plot]]
* [[Heyland–Ossanna circle diagram]]
* [[cis (mathematics)]]
* [[Transversal (instrument making)]]

==References==
{{reflist|25em |refs=
<ref name="Mizuhashi_1937">{{cite journal |script-title=ja:四端子回路のインピーダンス変成と整合回路の理論 |title=Sì duānzǐ huílù no inpīdansu hensei to seigō kairo no riron |language=ja |trans-title=Theory of Four-Terminals Impedance Transformation Circuit and Matching Circuit |author-last=Mizuhashi<!-- also transscribed as: Mizuhasi --> [水橋] |author-first=Tōsaku [東作] |journal=The Journal of the Institute of Electrical Communication Engineers of Japan [電気通信学会雑誌] |issn=0914-5273 |publisher=[[Institute of Electrical Communication Engineers of Japan]] [電気通信学会] |volume=1937<!-- showa 12 --> |number=12 |date=December 1937 |orig-date=1937-11-19<!-- showa date: 12-11-19 --> |pages=1053–1058<!-- absolute pages --> (29–34<!-- relative pages -->) |url=http://www.linkclub.or.jp/~morikuni/mizuhashi-smithchart/Mizuhashi,J.IECEJ(1937).pdf |url-status=dead |archive-url=https://web.archive.org/web/20171116005011/http://www.linkclub.or.jp:80/~morikuni/mizuhashi-smithchart/Mizuhashi,J.IECEJ(1937).pdf |archive-date=2017-11-16}} (6 pages)</ref>
<ref name="Okamura_1959">{{cite journal |script-title=ja:スミスチャートは日本人の独創ではないか |title=Sumisuchāto wa nihonjin no dokusōde wanai ka |language=ja |trans-title="Smith Chart" May Have Origin in Japan |author-last=Okamura [岡村] |author-first=Fumiyoshi [史良]<!-- This given name mentioned in the article's Japanese header --> / Shirō [獅郎]<!-- This given name mentioned in the article's English footnote --> |journal=The Journal of the Institute of Electrical Communication Engineers of Japan [電気通信学会雑誌] |issn=0914-5273 |publisher=[[Institute of Electrical Communication Engineers of Japan]] [電気通信学会] |location=Tokyo, Japan |volume=1959<!-- showa 34 --> |number=8 |date=August 1959 |orig-date=1959-04-04<!-- showa date: 34-04-04 --> |pages=768–769<!-- absolute pages --> (44–45<!-- relative pages -->) |url=http://www.linkclub.or.jp/~morikuni/mizuhashi-smithchart/Okamura,J.IECEJ(1959).pdf |url-status=dead |archive-url=https://web.archive.org/web/20171116005338/http://www.linkclub.or.jp/~morikuni/mizuhashi-smithchart/Okamura,J.IECEJ(1959).pdf |archive-date=2017-11-16}} (2 pages) (NB. The article lists the author's [[given name]] as {{lang|ja|史良}} in Japanese, which would translate as "Fumiyoshi", whereas the English footnote in the same article transscribed it as "Shirō", which would be associated with {{lang|ja|獅郎}} in Japanese.)</ref>
<ref name="Kenichi_1999">{{cite book |script-title=ja:インピーダンスのはなし |title=Inpīdansu no hanashi |language=ja |trans-title=The story of impedance |author-last=Kenichi [伊藤健] |author-first=Ito [一著] |edition=1 |series=Science and Technology |publisher=[[Nikkan Kogyo Shimbun]] [日刊工業新聞社] |date=1999-11-01 |isbn=4-526-04463-6 |id={{EAN|978-4-526-04463-2}}. 1923054018007 |page=26}} (4+xi+1+207+3+4 pages)</ref>
<ref name="Mori_2013">{{cite web |title=The Mizuhashi-Smith Chart |author-last=Mori [森] |author-first=Kunihiko [邦彦] |date=2013 |work=morikuni_net |url=http://www.linkclub.or.jp/~morikuni/mizuhashi-smithchart/ |access-date=2023-06-24 |url-status=dead |archive-url=https://web.archive.org/web/20130303005317/http://www.linkclub.or.jp/~morikuni/mizuhashi-smithchart/ |archive-date=2013-03-03}}</ref>
<ref name="Smith_1939">{{cite magazine |title=<!-- A -->Transmission Line Calculator - A "cut-out" calculator for determining impedance and attenuation, in terms of the length of open-wire transmission lines |author-last=Smith |author-first=Phillip Hagar |author-link=Phillip Hagar Smith |magazine=[[electronics (magazine)|electronics]] - radio, communication, industrial applications of electron tubes ... engineering and manufacture |issn=0013-5070 |publisher=[[McGraw-Hill Publishing Company, Inc.]] |publication-place=New York, USA |volume=12 |issue=1 |pages=29–31 |date=January 1939 |url=https://worldradiohistory.com/Archive-Electronics/30s/Electronics-1939-01.pdf |access-date=2023-07-09 |url-status=live |archive-url=https://web.archive.org/web/20230708221149/https://worldradiohistory.com/Archive-Electronics/30s/Electronics-1939-01.pdf |archive-date=2023-07-08}} (3 pages)</ref>
<ref name="Smith_1944">{{cite magazine |title=An Improved Transmission Line Calculator - An extension of the "calculator" originally published in ELECTRONICS in January 1939. New parameters have been added and accuracy has been improved. |author-last=Smith |author-first=Phillip Hagar |author-link=Phillip Hagar Smith |magazine=[[electronics (magazine)|electronics]] |issn=0013-5070 |publisher=[[McGraw-Hill Publishing Company, Inc.]] |publication-place=New York, USA |volume=17 |issue=1 |pages=130–133, 318, 320, 322, 324–325 |date=January 1944 |url=https://worldradiohistory.com/Archive-Electronics/40s/Electronics-1944-01.pdf |access-date=2023-07-09 |url-status=live |archive-url=https://web.archive.org/web/20230131151634/https://worldradiohistory.com/Archive-Electronics/40s/Electronics-1944-01.pdf |archive-date=2023-01-31}} (4+5 pages)</ref>
<ref name="Smith_1969">{{cite book |title=Electronic Applications of the Smith Chart: In Waveguide, Circuit and Component Analysis |author-first=Phillip Hagar |author-last=Smith |author-link=Phillip Hagar Smith |date=June 1969 |edition=1 |publisher=[[McGraw-Hill Book Company]] / Kay Electric Company |publication-place=New York, USA |location=Pine Brook, New Jersey, USA |lccn=69-12411 |isbn=0-07058930-5 |id={{ISBN|978-0-07058930-8}}}} (xxvii+1+222 pages + envelope with 4 translucent plastic overlays + Kay Electric Company postcard) (NB. There is a 1983 reprint of the first edition by Robert E. Krieger Publishing Company with {{ISBN|978-0-89874-552-8|0-89874-552-7}}, and a {{citeref|Smith|2000|second edition|style=plain}} by Noble Publishing Corporation.)</ref>
<ref name="Smith_1995">{{cite book |title=Electronic Applications of the Smith Chart: In Waveguide, Circuit and Component Analysis |author-first=Phillip Hagar |author-last=Smith |author-link=Phillip Hagar Smith |date=October 2000<!-- reprint of 2nd edition still under Noble --> |orig-date=1995<!-- first printing of 2nd edition by Noble --> |edition=2 |publisher=[[Noble Publishing Corporation]] |publication-place=Atlanta, Georgia, USA |lccn=00-045239 |isbn=1-884932-39-8 |id={{ISBN|978-1-884932-39-7}} |quote-page=xiv |quote=[…] From [[John Ambrose Fleming|Fleming]]'s equation,{{citeref|Fleming|1912|A}} and in an effort to simplify the solution of the transmission line problem, he developed his first graphical solution in the form of a rectangular plot. […] the diagram gradually evolved through a series of steps. The first rectangular chart was limited by the range of data it could accommodate. He was aware of the limitations and kept working on the problem until some time in 1936, when he developed a new diagram that eliminated most of the difficulties. The new chart was a special [[polar coordinate]] form in which all values of impedance components could be accommodated. The data for this diagram was scaled from the earlier rectangular diagram. The impedance coordinates in this case were not orthogonal and were not true circles, but, in the form chosen, the standing wave ratio was linear. The chart closely resembled what ultimately became the final result. [[Phillip Hagar Smith|Phil]], however, suspected that a grid made up of a system of orthogonal circles might be more practical. He felt it would have distinct advantages, particularly as regards reproducibility. With this in mind, he spoke to two of his co-workers, [[Enoch B. Ferrell|E.B. Ferrell]] and [[James W. McRae|J.W. McRae]]. Because they were familiar with the principles of [[conformal mapping]], they were able to develop the transformation whereby all data from zero to infinity could be accommodated. Fortunately, curves of constant standing wave ratio, constant attenuation and constant reflection coefficient were all circles coaxial with the center of the diagram. The scales for these values, while not linear, were entirely satisfactory. A diagram designed along these lines was constructed in early 1937. It was essentially the form still being used today. Smith approached a number of technical magazines with regard to publication of the Chart, but acceptance was slow. There were not many technical magazines at the time, and none in the microwave area. However, in January of 1939, after a delay of two years, the {{citeref|Smith|1939|article was printed|style=plain}} in [[Electronics (magazine)|Electronics]] magazine. […]}} (xxvi+237+1 pages + envelope with 4 translucent plastic overlays) (NB. There is a 2006 reprint of the second edition by [[SciTech Publishing, Inc.]] under the same ISBN and LCCN.)</ref>
<ref name="ETHW_2018">{{cite web |title=Smith Chart |website=ETHW.org |date=2018-02-26 |url=https://ethw.org/Smith_Chart |access-date=2021-03-30 |url-status=live |archive-url=https://web.archive.org/web/20230708231337/https://ethw.org/Smith_Chart |archive-date=2023-07-08}}</ref>
<ref name="Ramo-Whinnery-Duzer_1965">{{cite book |title=Fields and Waves in Communications Electronics |author-last1=Ramo |author-first1=Simon "Si" |author-link1=Simon Ramo |author-last2=Whinnery |author-first2=John Roy |author-link2=John Roy Whinnery |author-last3=Van Duzer |author-first3=Theodore |date=1965 |edition=1 |publisher=[[John Wiley & Sons]] |pages=35–39}}</ref>
<ref name="Ramo-Whinnery-Duzer_1994">{{cite book |title=Fields and Waves in Communications Electronics |chapter=5.9. The Smith Transmission-Line Chart / 5.10. Some Uses of the Smith Chart |author-last1=Ramo |author-first1=Simon "Si" |author-link1=Simon Ramo |author-last2=Whinnery |author-first2=John Roy |author-link2=John Roy Whinnery |author-last3=Van Duzer |author-first3=Theodore |date=1994 |edition=3 |publisher=[[John Wiley & Sons, Inc.]] |isbn=978-0-471-58551-0 |pages=236–245}}</ref>
<ref name="Pozar_2005">{{cite book |title=Microwave Engineering |author-last=Pozar |author-first=David Michael |author-link=David Michael Pozar |date=2005 |edition=3 |publisher=[[John Wiley & Sons, Inc.]] |pages=64–71 |isbn=0-471-44878-8}}</ref>
<ref name="Gonzalez_1997">{{cite book |title=Microwave Transistor Amplifiers Analysis and Design |author-last=Gonzalez |author-first=Guillermo |author-link= |date=1997 |edition=2 |publisher=[[Prentice Hall]] |place=New Jersey, USA |isbn=0-13-254335-4 |pages=93–103}}</ref>
<ref name="Bevelacqua_2013">{{cite web |title=The Smith Chart |author-first=Peter Joseph |author-last=Bevelacqua |date=2013-01-11 |orig-date=2010 |website=www.antenna-theory.com |url=http://www.antenna-theory.com/tutorial/smith/chart.php |access-date=2023-07-09 |url-status=live |archive-url=https://web.archive.org/web/20230708231713/https://www.antenna-theory.com/tutorial/smith/chart.php |archive-date=2023-07-08}}</ref>
<ref name="Tektronix_2017">{{cite web |title=Antenna Matching with a Vector Network Analyzer |date=2017-10-06 |work=Tek |publisher=[[Tektronix, Inc.]] |url=https://www.tek.com/blog/antenna-matching-vector-network-analyzer |access-date=2023-07-09 |url-status=live |archive-url=https://web.archive.org/web/20230708231720/https://www.tek.com/en/blog/antenna-matching-vector-network-analyzer |archive-date=2023-07-08}}</ref>
<ref name="Hayt_1981">{{cite book |title=Engineering Electromagnetics |author-last=Hayt, Jr. |author-first=William Hart |author-link=:d:Q59626592 |date=1981 |edition=4 |publisher=[[McGraw-Hill, Inc.]] |pages=428–433 |isbn=0-07-027395-2}} (527 pages)</ref>
<ref name="Davidson_1989">{{cite book |title=Transmission Lines for Communications with CAD Programs |author-last=Davidson |author-first=Colin William |edition=2 |publisher=[[Macmillan Education Ltd]] |publication-place=Basingstoke, Hampshire, UK |isbn=0-333-47398-1 |date=1989 |id=ark:/13960/s2dzmfrhg24 |pages=80–85 |url=https://archive.org/details/transmissionline0000davi/page/n7/mode/1up= |access-date=2023-07-09}} (viii+244 pages)</ref>
<ref name="Muller-Soto-Dascalu-Neculoiu-Boria_2011">{{cite journal |title=A 3D Smith chart based on the Riemann sphere for active and passive microwave circuits |author-first1=Andrei A. |author-last1=Muller |author-first2=Pablo |author-last2=Soto Pacheco |author-first3=Dan |author-last3=Dascălu |author-link3=:ro:Dan Dascălu |author-first4=Dan |author-last4=Neculoiu |author-first5=Vicente<!-- also written as: Vincente --> E. |author-last5=Boria<!-- also written as: Boria-Esbert --> |date=2011 |journal=[[IEEE Microwave and Wireless Components Letters]] |issn=1531-1309 |publisher=[[IEEE Microwave Theory and Techniques Society]] |volume=21 |issue=6 |pages=286–288 |doi=10.1109/LMWC.2011.2132697 |hdl=10251/55107 |s2cid=38953650 |hdl-access=free}}</ref>
<ref name="Muller-Asavei-Moldoveanu-Sanabria-Codesal-Khadar-Popescu-Dascalu-Ionescu_2020">{{cite journal |title=The 3D Smith Chart: From Theory to Experimental Reality |author-last1=Muller |author-first1=Andrei A. |author-last2=Asavei |author-first2=Victor |author-last3=Moldoveanu |author-first3=Alin |author-last4=Sanabria-Codesal |author-first4=Esther |author-last5=Khadar |author-first5=Riyaz A. |author-last6=Popescu |author-first6=Cornel |author-last7=Dascălu |author-first7=Dan |author-link7=:ro:Dan Dascălu |author-last8=Ionescu |author-first8=Adrian M. |date=November 2020 |journal=[[IEEE Microwave Magazine]] |issn=1527-3342 |publisher=[[IEEE Microwave Theory and Techniques Society]] |volume=21 |issue=11 |pages=22–35 |doi=10.1109/MMM.2020.3014984 |s2cid=222296721 |url=http://infoscience.epfl.ch/record/280816}}</ref>
<ref name="Volpert_1940">{{cite news |script-title=ru:Номограмма для расчета длинных линий |title=Nomogramma dlya rascheta dlinnykh liniy |language=ru |trans-title=Nomogram for calculating long lines |author-first=Amiel Rafailovich [Амиэ́ль Рафаи́лович] |author-last=Volpert [Во́льперт] |author-link=:ru:Вольперт, Амиэль Рафаилович |newspaper=Производственно-технический бюллетень (Proizvodstvenno-tekhnicheskiy byulleten') [Industrial and technical bulletin] |location=[[Leningrad]]<!-- Ленинград -->, СССР |date=February 1940 |volume=1940 |number=2 |publisher={{ill|People's Commissariat of the Electrical Industry of the USSR|ru|Народный комиссариат электропромышленности СССР|lt=НКЭП}} |pages=14–18}} (5 pages)</ref>
<ref name="Kurochkin_2009">{{cite web |script-title=ru:Диаграмма Вольперта – Смита. Расчет и анализ характеристик усилителей радиосигналов<!--: метод. пособие по дисциплине «Радиоприем. устройства» для студентов специальностей «Радиотехника», «Радиоэлектр. системы», «Радиоинформ.», «Радиоэлектр. защита информ. --> |title=Diagramma Vol'perta – Smita. Raschet i analiz kharakteristik usiliteley radiosignalov |language=ru |trans-title=The Volpert–Smith diagram. Calculation and analysis of the characteristics of amplifiers of radio signals<!-- : Method. manual on the subject of "Radio reception devices" for students of the specialties "Radio engineering", "Radio electrical systems", "Radioinform.", "Radioelectro. information protection." --> |author-first=Alexander Evdokimovich [Александр Евдокимович] |volume=2009 |number=87 |author-last=Kurochkin [Курочкин] |date=2009 <!-- |orig-date=2004-04-30 --> |publisher=Department of Radio Engineering Devices, Belarusian State University of Informatics and Radio Electronics, Ministry of Education of the Republic of Belarus educational institution<!-- БГУИР [BGUIR] --> |location=Minsk, Belarus |isbn=978-9-85-488-422-6 <!-- |hdl=123456789/310 --> |url=https://libeldoc.bsuir.by/bitstream/123456789/310/2/Kurochkin_Diagr.pdf |access-date=2023-07-09 |url-status=live |archive-url=https://web.archive.org/web/20230709120141/https://libeldoc.bsuir.by/bitstream/123456789/310/2/Kurochkin_Diagr.pdf |archive-date=2023-07-09 |quote-page=4 |script-quote=ru:Диаграмма Смита остается одним из наиболее полезных графических инструментов для разработки высокочастотных усилительных каскадов. В нашей стране аналогичная диаграмма известна как круговая номограмма А. Р. Вольперта, который в 1939 г. независимо от Смита разработал и применил ее для пересчёта проводимостей и сопротивлений в отрезках линий передачи. |quote=Diagramma Smita ostayetsya odnim iz naiboleye poleznykh graficheskikh instrumentov dlya razrabotki vysokochastotnykh usilitel'nykh kaskadov. V nashey strane analogichnaya diagramma izvestna kak krugovaya nomogramma A. R. Vol'perta, kotoryy v 1939 g. nezavisimo ot Smita razrabotal i primenil yeye dlya pereschota provodimostey i soprotivleniy v otrezkakh liniy peredachi. |trans-quote=In our country, a similar diagram is known as a circular nomogram of A. R. Volpert, who in 1939, independently of Smith developed and applied it to recalculate conductances and resistances in segments of transmission lines.}} [https://web.archive.org/web/20230625131917/https://vk.com/wall-185886861_452?lang=en][https://vk.com/doc558553917_527458558?hash=ojEAr3vwhjjd3f2Fh1zr4iCnLRO1oIseQx1QPzHLxgP][https://web.archive.org/web/20230709120115/https://libeldoc.bsuir.by/handle/123456789/310] (40+1 pages)</ref>
<ref name="Voltmer_2007">{{cite book |title=Fundamentals Of Electromagnetics 2. Quasistatics And Waves |chapter=8.2. The Smith Chart |author-first=David |author-last=Voltmer |series=Synthesis Lectures On Computational Electromagnetics |volume=2 |date=2007-08-17 |edition=1 |number=Lecture 15 |editor-first=Constantine A. |editor-last=Balanis |editor-link=Constantine A. Balanis |issn=1932-1716<!-- series --> |publisher=[[Morgan & Claypool]] |doi=10.2200/S00078ED1V01Y200612CEM015 |isbn=978-1-59829172-8 |id={{ISBN|1-59829172-6}}. <!-- Arizona State University Library of Congress Cataloging-in-Publication Data Series -->LoC 1932-1252. MOBK081-FM |pages=135–141 [135] |s2cid=9045052 |url=https://vdoc.pub/documents/fundamentals-of-electromagnetics-2-quasistatics-and-waves-3qq83vm9q64g |access-date=2023-06-25 |quote-page=135 |quote=Though Volpert of the Soviet Union and Mizuhashi of Japan proposed essentially the same chart during the same year, Smith received the recognition.}}</ref>
<ref name="Salov_2022">{{cite book |title=Antenna Impedance Measurement and Matching |chapter=4. Volpert–Smith Chart |author-first=Mikhail |author-last=Salov |type=Application note |date=March 2022 |publisher=[[Texas Instruments]] |id=SWRA726 |page=11 |url=https://www.ti.com/lit/an/swra726/swra726.pdf?ts=1687781229182&ref_url=https%253A%252F%252Fwww.ti.com%252Fproduct%252FCC1352R |access-date=2023-06-26 |url-status=live |archive-url=https://web.archive.org/web/20230626154445/https://www.ti.com/lit/an/swra726/swra726.pdf?ts=1687781229182&ref_url=https%25253A%25252F%25252Fwww.ti.com%25252Fproduct%25252FCC1352R |archive-date=2023-06-26}} (51 pages)</ref>
<ref name="MAX_2012">{{cite web |title=Impedance Matching and Smith Chart Impedance |author= |date=2012 |orig-date=2002-07-22 |id=Tutorial 742 |type=Application note |publisher=[[Maxim Integrated Products, Inc.]] |url=https://www.analog.com/en/technical-articles/impedance-matching-and-smith-chart-impedance-maxim-integrated.html |access-date=2023-07-09 |url-status=live |archive-url=https://web.archive.org/web/20230709113110/https://www.analog.com/en/technical-articles/impedance-matching-and-smith-chart-impedance-maxim-integrated.html |archive-date=2023-07-09}}[https://web.archive.org/web/20230709113219/https://www.analog.com/media/en/technical-documentation/tech-articles/impedance-matching-and-smith-chart-impedance-maxim-integrated.pdf] (18 pages) (NB. A previous version of this article appeared in the July 2000 issue of ''[[RF Design (magazine)|RF Design]]''.)</ref>
<ref name="Linton_1950">{{cite magazine |author-last=Linton, Jr. |author-first=R. L. |location=Antenna Laboratory, University of California, Berkeley, California, USA |title=Graph for Smith Chart |magazine=[[electronics (magazine)|electronics]] |issn=0013-5070 |publisher=[[McGraw-Hill Publishing Company, Inc.]] |publication-place=New York, USA |volume=23 |issue=1 |date=January 1950 |pages=123, 158 |url=https://worldradiohistory.com/Archive-Electronics/50s/Electronics-1950-01.pdf |access-date=2023-07-22 |url-status=live |archive-url=https://web.archive.org/web/20230131151823/https://worldradiohistory.com/Archive-Electronics/50s/Electronics-1950-01.pdf |archive-date=2023-01-31}}</ref>
<ref name="GeneralRadio_1950">{{cite magazine |title=Impedance Measurement-Smith Chart with Example Plotted |magazine=[[General Radio Experimenter]] |publisher=[[General Radio Company]] |publication-place=Cambridge, Massachusetts, USA |volume=XXV |issue=6 |date=November 1950 |pages=5–7 |url=https://worldradiohistory.com/Archive-Company-Publications/Archive-General-Radio/GR%20Exp%201950_11.pdf |access-date=2023-07-22 |url-status=live |archive-url=https://web.archive.org/web/20230722174932/https://worldradiohistory.com/Archive-Company-Publications/Archive-General-Radio/GR%20Exp%201950_11.pdf |archive-date=2023-07-22}}</ref>
<ref name="Inan_2005">{{cite conference |title=Remembering Phillip H. Smith on his 100th Birthday |author-first=Aziz S. |author-last=Inan |date=2005-07-03<!-- /08 --> |conference=2005 [[IEEE Antennas and Propagation Society International Symposium Digest]] |publisher=[[IEEE Antennas and Propagation Society]] |publication-place=Washington, D.C., USA |location=School of Engineering, University of Portland, Portland, Oregon, USA |volume=3B |isbn=0-7803-8883-6 |issn=1522-3965 |eissn=1947-1491 |doi=10.1109/APS.2005.1552450 |pages=129–132 |url=https://faculty.up.edu/ainan/aps2005inanMarch05.pdf |access-date=2023-07-02 |url-status=live |archive-url=https://web.archive.org/web/20230702163157/https://faculty.up.edu/ainan/aps2005inanMarch05.pdf |archive-date=2023-07-02}} (NB. This is a corrected version of the originally published paper.)</ref>
}}

==Further reading==
* {{cite journal |title=Cisoidal oscillations |author-link=George Ashley Campbell |author-first=George Ashley |author-last=Campbell |journal=[[Proceedings of the American Institute of Electrical Engineers]] |publisher=[[American Institute of Electrical Engineers]] |volume=XXX |issue=1–6 |date=April 1911 |doi=10.1109/PAIEE.1911.6659711 |s2cid=51647814 |pages=789–824 [Fig. 13 on p. 810] |url=https://ia800708.us.archive.org/view_archive.php?archive=/28/items/crossref-pre-1923-scholarly-works/10.1109%252Fpaiee.1910.6660428.zip&file=10.1109%252Fpaiee.1911.6659711.pdf |access-date=2023-06-24}} (37 pages) (NB. Includes an early graphical depiction of something similar to a Smith chart decades before Mizuhashi, Volpert and Smith published their works.)
* {{cite book |title=The Propagation of Electric Currents in Telephone and Telegraph Conductors: A Course of Post-graduate Lectures Delivered Before the University of London |author-first=John Ambrose |author-last=Fleming |author-link=John Ambrose Fleming |publisher=[[Constable & Company Ltd]] |location=University College, London, UK |date=January 1912 |orig-date=May 1911<!-- first edition --> |edition=revised 2nd |id=ark:/13960/t3bz6211d |url=https://archive.org/details/propagationofele030887mbp/page/n11/mode/1up |access-date=2023-07-23}} (xiv+316<!-- +4 blank --> pages)
* {{cite magazine |title=Transmission Line Charts |author-first=Herbert L. |author-last=Krauss |magazine=[[Electrical Engineering (magazine)|Electrical Engineering]] |issn=0095-9197 |eissn=2376-7804 |publisher= |volume=68 |issue=9 |date=September 1949 |doi=10.1109/EE.1949.6444963 |pages=766–774 [767]}}
* {{cite web |title=Smith Chart Tuning, Part I |author-first=Donald |author-last=Lee |publisher=SOC Business Unit, Advantest Test Cell Innovations, [[Advantest Corporation]] |date=2013-01-30 |url=https://www3.advantest.com/documents/11348/e453dfca-1fc8-4b53-a7be-3419f431b3f0 |access-date=2023-07-09 |url-status=live |archive-url=https://web.archive.org/web/20230709184336/https://www3.advantest.com/documents/11348/e453dfca-1fc8-4b53-a7be-3419f431b3f0 |archive-date=2023-07-09}} (29 pages)
* {{cite web |title=Mathematical Construction and Properties of the Smith Chart |author-first=Trevor |author-last=Gamblin |date=2015-07-23 |website=allaboutcircuits.com |url=http://www.allaboutcircuits.com/technical-articles/mathematical-construction-and-properties-of-the-smith-chart |access-date=2023-07-09 |url-status=live |archive-url=https://web.archive.org/web/20230709184018/https://www.allaboutcircuits.com/technical-articles/mathematical-construction-and-properties-of-the-smith-chart |archive-date=2023-07-09}}
* {{cite web |title=Impedance Matching and Smith Charts |author-first=John |author-last=Staples |date=2015 |publisher=U.S. Particle Accelerator School (USPAS) |url=https://uspas.fnal.gov/materials/08UCSC/mml13_matching+smith_chart.pdf |access-date=2023-07-09 |url-status=live |archive-url=https://web.archive.org/web/20230709114329/https://uspas.fnal.gov/materials/08UCSC/mml13_matching+smith_chart.pdf |archive-date=2023-07-09}} (27 pages)

==External links==
{{commons category|Smith charts}}
* {{cite web |url=http://www.excelhero.com/blog/2010/08/excel-high-precision-engineering-chart-1.html |title=Excel Smith chart |website=excelhero.com |date=August 2010}} Non-commercial, interactive Smith Chart that looks best in Excel 2007+.
* {{cite web |url=http://www.ae6ty.com/Smith_Charts.html |title=SimSmith |website=ae6ty.com}} Non-commercial, available for Windows, Mac, and Linux. Many Smith chart tutorial videos. No circuit size restrictions. Not limited to ladder circuits.  
* {{cite web |url=http://fritz.dellsperger.net/smith.html |archive-url=https://web.archive.org/web/20150304202453/http://fritz.dellsperger.net/smith.html |url-status=dead |archive-date=2015-03-04 |title=Smith v3 |website=fritz.dellsperger.net}} Commercial and free Smith chart for Windows
* {{cite web |url=https://github.com/niyeradori/QuickSmith-Web |title=QuickSmith |website=github.com/niyeradori |date=2021-11-02}} Free web based Smith Chart Educational tool available on [[GitHub]].
* {{cite web |url=http://www.3dsmithchart.com/ |title=3D Smith chart tool |website=3dsmithchart.com}} 2D and 3D Smith chart generalized tool for active and passive circuits (free for academia/education).
{{Authority control}}

[[Category:Electrical engineering]]
[[Category:Charts]]