Editing Smith chart (section)

====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.