Editing Smith chart (section)

====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.