# 3.14: Standing Wave Ratio

- Page ID
- 6280

[m0081_Standing_Wave_Ratio]

Precise matching of transmission lines to terminations is often not practical or possible. Whenever a significant mismatch exists, a standing wave (Section [m0086_Standing_Waves]) is apparent. The quality of the match is commonly expressed in terms of the *standing wave ratio* (SWR) of this standing wave.

*Standing wave ratio* (SWR) is defined as the ratio of the maximum magnitude of the standing wave to minimum magnitude of the standing wave.

In terms of the potential: \[\boxed{ \mbox{SWR} \triangleq \frac{\mbox{maximum}~|\widetilde{V}|}{\mbox{minimum}~|\widetilde{V}|} }\]

SWR can be calculated using a simple expression, which we shall now derive. In Section [m0086_Standing_Waves], we found that: \[\left|\widetilde{V}(z)\right| = |V_0^+| \sqrt{ 1 + \left|\Gamma\right|^2 + 2\left|\Gamma\right| \cos\left( 2\beta z + \phi \right) }\] The maximum value occurs when the cosine factor is equal to \(+1\), yielding: \[\mbox{max}~\left|\widetilde{V}\right| = |V_0^+| \sqrt{ 1 + \left|\Gamma\right|^2 + 2\left|\Gamma\right| }\] Note that the argument of the square root operator is equal to \(\left( 1 + \left|\Gamma\right| \right)^2\); therefore: \[\mbox{max}~\left|\widetilde{V}\right| = |V_0^+| \left( 1 + \left|\Gamma\right| \right)\] Similarly, the minimum value is achieved when the cosine factor is equal to \(-1\), yielding: \[\mbox{min}~\left|\widetilde{V}\right| = |V_0^+| \sqrt{ 1 + \left|\Gamma\right|^2 - 2\left|\Gamma\right| }\] So: \[\mbox{min}~\left|\widetilde{V}\right| = |V_0^+| \left( 1 - \left|\Gamma\right| \right)\] Therefore: \[\boxed{ \mbox{SWR} = \frac{1 + \left|\Gamma\right|}{1 - \left|\Gamma\right|} } \label{m0081_eSWR}\]

This relationship is shown graphically in Figure [m0081_fSWRvGamma]. Note that SWR ranges from 1 for perfectly-matched terminations (\(\Gamma=0\)) to infinity for open- and short-circuit terminations (\(\left|\Gamma\right|=1\)).

It is sometimes of interest to find the magnitude of the reflection coefficient given SWR. Solving Equation [m0081_eSWR] for \(\left|\Gamma\right|\) we find: \[\left|\Gamma\right| = \frac{\mbox{SWR}-1}{\mbox{SWR}+1} \label{m0081_eGammaFromSWR}\]

SWR is often referred to as the *voltage standing wave ratio* (VSWR), although repeating the analysis above for the current reveals that the current SWR is equal to potential SWR, so the term “SWR” suffices.

SWR \(<2\) or so is usually considered a “good match,” although some applications require SWR \(<1.1\) or better, and other applications are tolerant to SWR of 3 or greater.

Reflection coefficient for various values of SWR.

What is the reflection coefficient for the above-cited values of SWR? Using Equation [m0081_eGammaFromSWR], we find:

SWR = 1.1 corresponds to \(\left|\Gamma\right|=0.0476\).

SWR = 2.0 corresponds to \(\left|\Gamma\right|=1/3\).

SWR = 3.0 corresponds to \(\left|\Gamma\right|=1/2\).