Editing Standing wave ratio (section)

==Relationship to the reflection coefficient==
[[File:Standing wave 2.gif|thumb|350px|right|Incident wave (blue) is fully reflected (red wave) out of phase at short-circuited end of transmission line, creating a net voltage (black) standing wave. Γ = −1, SWR = ∞.]]
[[File:StandingWaves-3.png|thumb|500px|right|Standing waves on transmission line, net voltage shown in different colors during one period of oscillation. Incoming wave from left (amplitude = 1) is partially reflected with (top to bottom) Γ = 0.6, −0.333, and 0.8 ∠60°. Resulting SWR = 4, 2, 9.]]
The voltage component of a standing wave in a uniform [[transmission line]] consists of the forward wave (with [[Phasor|complex]] amplitude <math>V_f</math>) superimposed on the reflected wave (with complex amplitude <math>V_r</math>).

A wave is partly reflected when a transmission line is terminated with an impedance unequal to its [[characteristic impedance]].  The [[reflection coefficient]] <math>\Gamma</math> can be defined as:

:<math>\Gamma = \frac{V_r}{V_f}.</math>

or

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

<math>\Gamma</math> is a [[complex number]] that describes both the magnitude and the phase shift of the reflection.  The simplest cases with <math>\Gamma</math> ''measured at the load'' are:
* <math>\Gamma = -1</math>: complete negative reflection, when the line is short-circuited,
* <math>\Gamma = 0</math>: no reflection, when the line is perfectly matched,
* <math>\Gamma = +1</math>: complete positive reflection, when the line is open-circuited.

The SWR directly corresponds to the [[magnitude (mathematics)#Complex numbers|magnitude]] of <math>\Gamma</math>.

At some points along the line the forward and reflected waves [[wave interference|interfere]] constructively, exactly in phase, with the resulting amplitude <math>V_\text{max}</math> given by the sum of those waves' amplitudes:

:<math>\begin{align}
  |V_\text{max}| &= |V_f| + |V_r| \\
                 &= |V_f| + |\Gamma V_f| \\
                 &= (1 + |\Gamma|) |V_f|.
\end{align}</math>

At other points, the waves interfere 180° out of phase with the amplitudes partially cancelling:

:<math>\begin{align}
  |V_\text{min}| &= |V_f| - |V_r| \\
                 &= |V_f| - |\Gamma V_f| \\
                 &= (1 - |\Gamma|) |V_f|.
\end{align}</math>

The voltage standing wave ratio is then

:<math>\text{VSWR} = \frac{|V_\text{max}|}{|V_\text{min}|} = \frac{1 + |\Gamma|}{1 - |\Gamma|}.</math>

Since the magnitude of <math>\Gamma</math> always falls in the range [0,1], the SWR is always greater than or equal to unity. Note that the ''phase'' of ''V''<sub>f</sub> and ''V''<sub>r</sub> vary along the transmission line in opposite directions to each other. Therefore, the complex-valued reflection coefficient <math>\Gamma</math> varies as well, but only in phase. With the SWR dependent ''only'' on the complex magnitude of <math>\Gamma</math>, it can be seen that the SWR measured at ''any'' point along the transmission line (neglecting transmission line losses) obtains an identical reading.

Since the power of the forward and reflected waves are proportional to the square of the voltage components due to each wave, SWR can be expressed in terms of forward and reflected power:

:<math>\text{SWR} = \frac{1 + \sqrt{P_r/P_f}}{1 - \sqrt{P_r/P_f}}.</math>

By sampling the complex voltage and current at the point of insertion, an SWR meter is able to compute the effective forward and reflected voltages on the transmission line for the characteristic impedance for which the SWR meter has been designed. Since the forward and reflected power is related to the square of the forward and reflected voltages, some SWR meters also display the forward and reflected power.

In the special case of a load {{mvar|R}}<sub>L</sub>, which is purely resistive but unequal to the characteristic impedance of the transmission line {{mvar|Z}}<sub>0</sub>, the SWR is given simply by their ratio:

:<math>\text{SWR} = \max \left\{  \frac{R_\text{L}}{\,Z_\text{0}\,} \, , \frac{\,Z_\text{0}\,}{R_\text{L}} \right\}</math>

with the ratio or its reciprocal is chosen to obtain a value greater than unity.