General relation between bandwidth and Q

I've seen the following relation between bandwidth, Q-factor and SWR:

BW = (SWR-1)/(Q√SWR).

See for example Pozar's article, Microstrip antennas: the analysis ... - Google Books

Then for the -3 dB bandwidth (SWR = 5.85) the expression reduces to BW = 2/Q, instead of 1/Q, as I was hoping for. Is that expression incorrect or what's going on here?

This thread got little attention. Does anyone know of a more general expression between the impedance bandwidth and Q, i.e. I'm not looking for the -3 dB bandwidth relation BW = 1/Q (or BW = f/Q for the absolute bandwidth).

I noticed the post earlier but did not have anything to add.

Last night I tripped over an article IEEE Antennas and Propagation Magazine; vol. 51 no. 5 October 2009; R. C. Hansen and R. E. Collin, A New Formula for Q.

Maybe this will help. At least there is a long list of references.

I have a question, how did you calculate SWR ? i.e, For -3dB bandwidth : SWR = 5.85 ?

And when designing antennas, you need to make sure that SWR ≤ 2.
And the same is given in the book Microstrip Antennas by Pozar.
" Impedance bandwidth is usually specified interms of return loss or maximum SWR typically less than 2 or 1.5 over a frequency range"

sv1437 ....


The appropriate VSWR is a moving target. There is no well defined standard (or at least a mutually agreed upon level). In some areas 2.0:1 is good. In others, 1.25:1 is a minimum acceptable level. Applications with extremely wide bandwidths generally accept higher reflections than narrow band antennas. Another way to think of the situation is that low vswr and high cost/price are usually directly related.

It really comes down to system requirements. A cynic would say that the specification is always a little more than what is routinely achievable in production for any given antenna.

Another point is that VSWR is often not the important parameter in an antenna. What makes it attractive is that it us generally relatively easy to measure. It is a good go/no-go test. If the match is poor the antenna is probably broken. The converse is not true. With VSWR alone a good antenna and an attenuator both have great matches.

Hi Jone...How did you calculate SWR based on 3dB bandwidth?

For the half-power bandwidth we have

Γ² = (V_refl/V_trans)² = P_refl/P_trans = 0.5
SWR = (1+|Γ|)/(1-|Γ|) = 5.828 => BW = 2/Q

For series and parallel RLC circuits, one mostly talkes about the half-power bandwidth, whereas for antennas, it is maybe more common to look at S-parameters. That's why I started wondering about this in the first place.