This article explores a generic VSWR curve for simple series resonant matched antennas.
From Antennas and Q:
Some antennas, but not all, are resonant, and of those, some exhibit a simple series resonance near the desired operating frequency. Examples include a half wave dipole, a quarter wave monopole, shortened loaded monopoles, small transmitting loops.
A significant difference between such antennas and the simple RLC circuit discussed above is that R is not necessarily constant, it may vary with frequency. However, for many antennas of this type, R changes very slowly compared to X and the results are reasonably well approximated by considering R to be constant.
Constant R
Above is a chart of VSWR for a series resonant circuit with constant R=Zo. The frequency offset is relative to the half power points (|X|=R) of the circuit.
This curve is generic for this type of circuit, a single series resonator with constant R. It is also a good approximation of the case where R is not constant, but R varies very slowly with frequency compared to X (|dR/df|<<|dX/df|).
The relationship allows us to estimate the VSWR at one relative offset from VSWR known at another offset. For example, lets assume that we know that the VSWR=2 bandwidth of a certain antenna is 20kHz, and we want to know the bandwidth at VSWR=1.5. We can look VSWR=2 up on the chart and find it has an offset of about 0.7, and the offset for VSWR=1.5 is about 0.4, so the VSWR=1.5 bandwidth is about 0.4/0.7 times the VSWR=2 bandwidth, so about 0.4/0.7*20=11.4kHz.
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)
and since Qr=fr/BW:
Qr=fr/BWv*(v-1)/(v^0.5).
For enquiring minds, (Duffy 2014) derives the underlying relationship between |X| and VSWR for the scenario discussed. My thanks to David Ryeburn (VE7EZM) for reviewing the derivation and for his constructive comments.
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.