# 3.14: Standing Wave Ratio

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Precise matching of transmission lines to terminations is often not practical or possible. Whenever a significant mismatch exists, a standing wave (Section 3.13) 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 3.13, 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: \[\boxed{ \mbox{SWR} = \frac{1 + \left|\Gamma\right|}{1 - \left|\Gamma\right|} } \label{m0081_eSWR}\]

Figure \(\PageIndex{1}\): Relationship between SWR and \(|\Gamma|\).This relationship is shown graphically in Figure \(\PageIndex{1}\). 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 \ref{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.

Example \(\PageIndex{1}\): Reflection Coefficient for Various Values of SWR

What is the reflection coefficient for the above-cited values of SWR? Using Equation \ref{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 \(\left|\Gamma\right|=1/2\).

## Contributors and Attributions

Ellingson, Steven W. (2018) Electromagnetics, Vol. 1. Blacksburg, VA: VT Publishing. https://doi.org/10.21061/electromagnetics-vol-1 Licensed with CC BY-SA 4.0 https://creativecommons.org/licenses/by-sa/4.0. Report adoption of this book here. If you are a professor reviewing, adopting, or adapting this textbook please help us understand a little more about your use by filling out this form.