- Electromagnetics I
- Ch 3
- Loc 3.14

# 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 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:

SWR can be calculated using a simple expression, which we shall now derive. In Section 3.13, we found that:

The maximum value occurs when the cosine factor is equal to

, yielding:

Note that the argument of the square root operator is equal to

; therefore:

Similarly, the minimum value is achieved when the cosine factor is equal to

, yielding:

So:

Therefore:

This relationship is shown graphically in Figure 3.14.1. Note that SWR ranges from 1 for perfectly-matched terminations (

) to infinity for open- and short-circuit terminations (

).

It is sometimes of interest to find the magnitude of the reflection coefficient given SWR. Solving Equation 3.14.7 for

we find:

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.

### Exercise

What is the reflection coefficient for the above-cited values of SWR?

Using Equation 3.14.6, we find:

- SWR = 1.1 corresponds to = 0.0476.
- SWR = 2.0 corresponds to = 1/3.
- SWR = 3.0 corresponds to = 1/2.

Ellingson, Steven W. (2018) Electromagnetics, Vol. 1. Blacksburg, VA: VT Publishing. https://doi.org/10.21061/electromagnetics-vol-1 CC BY-SA 4.0

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