Editing Smith chart (section)

==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>
|}