Editing Reflection coefficient (section)

=== Relation to load impedance ===
The reflection coefficient is determined by the load impedance at the end of the transmission line, as well as the [[characteristic impedance]] of the line. A load impedance of <math>Z_L</math> terminating a line with a characteristic impedance of <math>Z_0\,</math> will have a reflection coefficient of

:<math>  \Gamma ={Z_L-Z_0  \over Z_L+Z_0} .</math>

This is the coefficient at the load.  The reflection coefficient can also be measured at other points on the line.  The ''magnitude'' of the reflection coefficient in a lossless transmission line is constant along the line (as are the powers in the forward and reflected waves). However its ''phase'' will be shifted by an amount dependent on the [[electrical length|electrical distance]] <math>\phi</math> from the load.  If the coefficient is measured at a point <math>L</math> meters from the load, so the [[electrical length|electrical distance]] from the load is <math>\phi = 2\pi L/\lambda</math> radians, the coefficient <math>\Gamma'</math> at that point will be 

: <math>\Gamma' =\Gamma e^{-i \, 2 \phi}  </math>

Note that the phase of the reflection coefficient is changed by ''twice'' the phase length of the attached transmission line. That is to take into account not only the phase delay of the reflected wave, but the phase shift that had first been applied to the forward wave, with the reflection coefficient being the quotient of these. The reflection coefficient so measured,  <math>\Gamma'</math>, corresponds to an impedance which is generally dissimilar to <math>Z_L</math> present at the far side of the transmission line.

The complex reflection coefficient (in the region <math>| \Gamma| \le 1</math>, corresponding to passive loads) may be displayed graphically using a [[Smith chart]]. The Smith chart is a polar plot of <math>\Gamma</math>, therefore the magnitude of <math>\Gamma</math> is given directly by the distance of a point to the center (with the edge of the Smith chart corresponding to <math>|\Gamma|=1</math>). Its evolution along a transmission line is likewise described by a rotation of <math>2\phi</math> around the chart's center. Using the scales on a Smith chart, the resulting impedance (normalized to <math>Z_0</math>) can directly be read. Before the advent of modern electronic computers, the Smith chart was of particular use as a sort of  [[Nomogram|analog computer]] for this purpose.