Editing Reflection coefficient (section)

== Transmission lines ==
{{See also|Reflections of signals on conducting lines|Signal reflection}}

In [[telecommunications]] and [[transmission line]] theory, the reflection coefficient is the [[ratio]] of the [[complex amplitude]] of the reflected wave to that of the incident wave.  The voltage and current at any point along a transmission line can always be resolved into forward and reflected traveling waves given a specified reference impedance ''Z<sub>0</sub>''. The reference impedance used is typically the [[characteristic impedance]] of a transmission line that's involved, but one can speak of reflection coefficient without  any actual transmission line being present. In terms of the forward and reflected waves determined by the voltage and current, the reflection coefficient is defined as the [[complex number|complex]] ratio of the voltage of the reflected wave (<math>V^-</math>) to that of the incident wave (<math>V^+</math>). This is typically represented with a <math>\Gamma</math> (capital [[gamma]]) and can be written as:

:<math>\Gamma = \frac{V^-}{V^+} </math>

It can also be defined using the ''currents'' associated with the reflected and forward waves, but introducing a minus sign to account for the opposite orientations of the two currents:

:<math>\Gamma = -\frac{I^-}{I^+}  = \frac{V^-}{V^+}</math>

The reflection coefficient may also be established using other field or [[Electronic circuit|circuit]] pairs of quantities whose product defines power resolvable into a forward and reverse wave. With electromagnetic plane waves, one uses the ratio of the electric fields of the reflected to that of the incident wave (or magnetic fields, again with a minus sign); the ratio of each wave's electric field ''E'' to its magnetic field ''H'' is the medium's characteristic impedance, <math>Z_0</math>, (equal to the [[impedance of free space]] if the medium is a vacuum).<ref>Pozar, David M. (2012); p. 29.</ref>

[[Image:Reflection Coefficient Circuit.svg|thumb|right|Simple circuit configuration showing measurement location of reflection coefficient.]]
In the accompanying figure, a signal source with internal impedance <math>Z_S</math> possibly followed by a transmission line of characteristic impedance <math>Z_S</math> is represented by its [[Thévenin equivalent]], driving the load <math>Z_L</math>. For a real (resistive) source impedance <math>Z_S</math>, if we define <math>\Gamma</math> using the reference impedance <math>Z_0 = Z_S</math> then the source's [[Impedance matching#Reflection-less matching|maximum power is delivered]] to a load <math>Z_L = Z_0</math>, in which case <math>\Gamma=0</math> implying no reflected power. More generally, the squared-magnitude of the reflection coefficient <math>|\Gamma|^2</math> denotes the proportion of that power that is reflected back to the source, with the power actually delivered toward the load being <math>1-|\Gamma|^2</math>.

Anywhere along an intervening (lossless) transmission line of characteristic impedance <math>Z_0</math>, the magnitude of the reflection coefficient <math>|\Gamma|</math> will remain the same (the powers of the forward and reflected waves stay the same) but with a different phase. In the case of a short circuited load (<math>Z_L=0</math>), one finds <math>\Gamma=-1</math> at the load. This implies the reflected wave having a 180° phase shift (phase reversal) with the voltages of the two waves being opposite at that point and adding to zero (as a short circuit demands).

=== 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.

=== Standing  wave ratio ===
{{Main|Standing  wave ratio}}
The [[standing wave ratio]] (SWR) is determined solely by the ''magnitude'' of the reflection coefficient:

:<math> SWR =  {1+| \Gamma |   \over 1- | \Gamma | } .</math>

Along a lossless transmission line of characteristic impedance ''Z''<sub>0</sub>, the SWR signifies the ratio of the voltage (or current) maxima to minima (or what it would be if the transmission line were long enough to produce them). The above calculation assumes that  <math>\Gamma</math> has been calculated using ''Z''<sub>0</sub> as the reference impedance. Since it uses only the ''magnitude'' of <math>\Gamma</math>, the SWR intentionally ignores the specific value of the load impedance ''Z<sub>L</sub>'' responsible for it, but only the magnitude of the resulting [[impedance mismatch]]. That SWR remains the same wherever measured along a transmission line (looking towards the load) since the addition of a transmission line length to a load <math>Z_L</math> only changes the phase, not magnitude of <math>\Gamma</math>. While having a one-to-one correspondence with reflection coefficient, SWR is the most commonly used figure of merit in describing the mismatch affecting a [[radio antenna]] or antenna system. It is most often [[SWR meter|measured]] at the transmitter side of a transmission line, but having, as explained, the same value as would be measured at the antenna (load) itself.