I mentioned in Findling & Siwiak 2012 measurements of an Alexloop issues with their efficiency calculation.

Above is an extract from (Findling & Siwiak 2012).

(Siwiak & Quick 2018) give an equivalent circuit of lossless loop structure in free space.

When tuned to resonance, the response is simply that of a series RLC circuit where R=Rr (the radiation resistance) which is dependent on frequency, but varies very slowly with frequency compared to the net reactance X.

Above is a NEC simulation of such a loop.

Making the assumption that the R component is approximately constant, we can determine that the power absorbed from a constant voltage source falls to half the maximum at frequencies where |X|=R, and we can determine the bandwidth BW between those points and calculate the Q of the antenna.

By eye from the graph above, the half power bandwidth is about 0.23kHz and Q is 7015.9/0.23=30,500 (by eye scaled from the graph).

For such a network, we can also find the half power points by exploiting the property that ReturnLoss=6.99dB or VSWR=2.618 when R=Ro and |X|=Ro.

Above is the calculated ReturnLoss plot for the same antenna, and again by eye, we can estimate half power bandwidth at 0.23kHz and Q is 7015.9/0.23=30,500 (by eye scaled from the graph).

This last method can be a very convenient way of determining half power bandwidth of an antenna matched to 50Ω using an antenna analyser. In typical practical antennas, almost all of the energy storage and almost all of the loss is due to the main loop, so the half power bandwidth measured looking into the antennas 50Ω port is approximately equal to that of the main loop.

The inductance of the loop can be estimated by formula to be 2.77µH (Grover 1945), and its reactance Xl calculated as 124Ω, or it could be measured using an antenna analyser. The Q of this antenna can also be calculated from Xl/R=124/0.0039=31,800.

This latter relationship allows finding R from Q and Xl, rearranging the terms R=Xl/Q.

This is all really straight forward conventional linear circuit theory, and this type of antenna is well represented by that model.

## A back of the envelope reality check

We can estimate Rr for the loop in free space at about 0.004Ω and from Grover, Xl=124, so Q of the lossless loop in free space should be about 124/0.004=31000.

If we accept the 19.2kHz measured half power bandwidth and calculated Qmeas value of 376.29, and ignore the small change in Rr due to proximity to ground, using (Siwiak & Quick 2018)'s formula we would calculate efficiency=376.29/31000=1.2% or -19.2dB.

## Did Findling & Siwiak go wrong?

So how do they come up with an efficiency figure of -15.77dB in both articles?

They explain how they determined their measured Q.

A point to remember is that the VSWR is independent of the source impedance, so no matter what kind of source is used here, matched or otherwise, VSWR is determined entirely by the impedance presented at the analyser terminals and its reference impedance. It is a ham myth engendered by Walt Maxwell's re-re-re-reflection model that VSWR depends on the Thevenin source impedance and can only be measured with a nominal source.

Their result for 40m was Qmeas=376.29, and if measured as explained above, Qmeas=Xl/Rtotal, and so Rtotal=Xl/Qmeas=124/376.29=0.330Ω. That is all credible.

They give a formula Qrad=Xl/(2Rr) referring to it as the ideal loaded Q

and calculate efficiency=Qmeas/Qrad (though they do not give the expression explicitly), so substituting we get efficiency=(Xl/Rtotal)/(Xl/2Rr)=2Rr/Rtotal when efficiency is actually Rr/Rtotal… they obtain a value 3dB too high.

The question is, what is the meaning of loaded? Is this in honor of (Hart 1986) and most things since?

This might explain why their measured efficiency is 3dB higher than a model calibrated for the same bandwidth, or G3CWI's measurements of the same type of antenna.

## On radiation resistance

Radiation resistance Rr (Rrad) is taken to mean that quantity that relates the total power radiated in the far far field to the feed point current, Rr=Pr/I^2.

From the article

Small transmitting loop – ground loss relationship to radiation resistance.

Above is a plot of Rr (Rrad) vs height for the three ground types and perfect ground. All curves oscillate at increasing height but converge on the free space radiation resistance Rrfs which is 6.4mΩ for that particular loop.

The methods used in (Findling & Siwiak 2012) and (Siwiak & Quick 2018) rely upon the expression efficiency=Ql/Qrad (given as Eq 5 in the second), and there is an implicit assumption that the radiation resistance component in each of the Q calcs are equal, but in fact they differ a little so the method hides a significant difference.

## References

- Duffy, O. 2014. Calculate small transmitting loop gain from bandwidth https://www.owenduffy.net/calc/SmallTransmittingLoopBw2Gain.htm.
- Findling, A & Siwiak, K. Summer 2012. How efficient is your QRP small loop antenna? In The QRP quarterly Summer 2012.
- Grover, F. 1945. Inductance calculations.
- Hart, Ted (W5QJR). 1986. Small, high efficiency loop antennas In QST June 1986.
- Siwiak, K & Quick, R. Sep 2018. Small gap resonated HF loop antennas In QST Sep 2018.
- Straw, Dean ed. 2007. The ARRL Antenna Book. 21st ed. Newington: ARRL. Ch5.