Editing Smith chart (section)

===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"/>