Extrapolating VSWR of a simple series resonant antenna

An online expert helped recently helped his Small Transmitting Loop (STL) disciples with:

Also remember that the bandwidth given by the calculators is the half power point. That’s equivalent to an SWR of about 4.3 at the ends.

Whats that?

Most STL, and lots of other resonant antenna systems exhibit a classic VSWR curve being that of a approximatly constant resistance in series with an ideal capacitor and inductor.

Above is that classic VSWR curve.

The meaning of half power bandwidth

Industry, universities, colleges and reputable textbooks carry a shared meaning of the term half power bandwidth. (Terman 1955) gives a consistent definition.

Modern hams on the other hand are wont to talk the talk, but mean something different just to ensure that they are not really communication… it is after all evolving into a social hobby focussed on DX spotting.

The half power bandwidth of a single series RLC circuit is the difference between upper and lower frequencies where the response is half that at the centre frequency. This occurs where the current (with constant voltage drive of course) is reduced to 0.707 that at the centre frequency, and therefore the power is half that at the centre frequency.

VSWR at half power points

Like an ordinary RLC circuit, the half power points of a series resonant antenna where R changes much more slowly than X occur approximately where X=±R.

The VSWR wrt R0 where Z=R0±jR0 is 2.618, the ReturnLoss is 6.99dB.

VSWR at half power points is 2.618

Extrapolation of VSWR for simple series resonant antenna

A VSWR curve for a simple series resonant circuit with constant R is easy to derive and antennas where R changes much more slowly than X exhibit a VSWR curve that is well approximated by this classic VSWR curve.

This also allows us to estimate the half power bandwidth BW-3dB of an antenna that exhibits a classic VSWR curve from arbitrary VSWR bandwidth BWv:

BW-3dB=BWv*v^0.5/(v-1)

We could rearrange that expression to find that given the VSWR=4.3 bandwidth stated by the online expert, that the half power bandwidth is in fact 63% of that VSWR=4.3 bandwidth.

Much if not most of the ham discussion about STL is based on some pretty wooly thinking and dodgy formulas most of which are traceable back to one author, and it all prevails because of a lack of understanding of fundamentals and mindless regurgitation.

References

• Duffy, O, May 2014. Derivation of expression for X in terms of VSWR for a normalised load where R is constant and X changes with frequency. VSWRCurveDerivation.
• Terman, Frederick. 1955. Electronic and Radio Engineering – 4th ed. New York: McGraw-Hill.