At frequency 7.2MHz, the loop was octagonal with area of 1m^2 equivalent radius a=0.443m, ka=0.067rad, 3.15mm radius copper conductor, lossless tuning capacitor, and centre height above ground (σ=0.007 ε_{r}=17 ) was varied from 1.5 to 10m (0.036-0.240λ).

The model series was run in NEC-2, NEC-4.1, NEC-4.2 and NEC-5.0, and the results varied. NEC-4.1 showed serious problems, eg negative input resistance at some heights. The problem was discussed the Burke, and he explained that there was a known problem in NEC-4.1 for small loops near ground, and sent me an upgrade to NEC-4.2 to try with the GN 3 ground model, but that the better solution was in NEC-5 if it was ever released.

NEC-4.2 solved the negative resistance problem, but some issues remained.

With the recent release of NEC-5.0, opportunity arises to compare all four approaches.

(Burke 2019) p45 discusses loop antennas over ground and NEC-5.0.

The plot above of radiation efficiency gives an overall comparison of the different model techniques. (Burke 2019) states Since the mixed-potential solution ensures that the approximated integral of scalar potential around the loop is zero, whether the potential is accurate or not, it might be expected to do better than NEC-4.

Now, keep in mind that these plots assume a lossless capacitor. Hams tend to assume that the Q of a vacuum capacitor (and its connections) is around 10,000 (near lossless in this context).

Lets proceed on the assumption that NEC-5.0 provides the best result.

To be clear, the radiation efficiency is plotted above (recall that a lossless capacitor used). It is slightly better at the lowest heights, partly a result of the interesting interaction of radiation resistance Rr and equivalent ground loss resistance Rgnd.

Above is a plot of Antenna System Q vs height. The curve shows that Q is degraded at lowest heights, but not by much… a result of the interaction of the effect of near ground on Rr and Rgnd. A Q around 1000 implies half power bandwidth (VSWR=2.62) of 7.2kHz.

Addition of losses of a tuning capacitor as low as Q=1000 would roughly halve the Q plot, and double the half power bandwidth to 14.4kHz for this geometry.

The chart above plots the elements of feed point resistance – note the log scale. This provides part of the explanation for the earlier Radiation Efficiency and Q plots, Rgnd increases as low heights, more so below about 6m (0.14λ). There is a gradual increase in Rr as height is reduced, a consequence of ground reflection.

- Burke, G. Sep 2019. NEC-5 Validation Manual. Lawrence Livermore National Laboratory.
- Burke, G. Nov 2017. Numerical Electromagnetics Code – NEC-5 Method of Moments, User’s Manual. Lawrence Livermore National Laboratory.
- Burke, G. Sep 2019. NEC-5 Validation Manual. Lawrence Livermore National Laboratory.
- Burke, G. Jul 2011. Numerical Electromagnetics Code – NEC-4.2 Method of Moments Part I. User’s Manual. Lawrence Livermore National Laboratory.
- Burke and Poggio 1977b. Numerical Electromagnetic Code (NEC-1) Part II: NEC Program Description – Code: Lawrence Livermore Laboratory.
- Burke and Poggio 1977c. Numerical Electromagnetic Code (NEC-1) Part III: NEC User’s guide: Lawrence Livermore Laboratory.

The article goes on to claim some pretty extraordinary efficiency calculated from radiation resistance for a loop structure that is shown at a height of perhaps 2m above natural ground.

Unfortunately, the meaning of the terms efficiency and radiation resistance are often critical to understanding written work on antennas and it is best for authors to use accepted industry ‘standard’ meanings and to declare their interpretation for clarity.

Let us draw on the IEEE standard dictionary of electrical and electronic terms (IEEE 1988) for widely accepted meanings for some key terms.

A large sphere whose centre lies within the volume of the antenna and whose surface lies in the far field of the antenna, over which quantities characterising the radiation from the antenna are determined.

The definition of Radiation Sphere is important in that it defines **where** radiation is to be observed, it is to be observed in the far field.

The ratio of power radiated by an antenna to the square of the RMS antenna current referred to a specific point.

Note that Radiation Sphere requires that radiated power must be measured / determined / summed in the far field.

The ratio of the total power radiated by an antenna to the net power accepted by the antenna from the connected transmitter.

Note that Radiation Sphere requires that radiated power must be measured / determined / summed in the far field.

Radiation fields decay inversely proportional to distance, other fields immediately around an antenna decay more quickly and are insignificant for the purpose of radio communications at great distances. Hence, Radiation is the usual objective of radio communications antennas.

Let us take efficiency

to mean radiation efficiency.

Simple small loop models rely upon a common formula for Rrad which assumes free space and uniform current around the loop, ie negligible change in amplitude and phase. The 6m perimeter double-double loop at 14.2MHz current phase delay around the loop of 100°, hardly negligible. In fact the formula falls down when the end to end length of the loop conductor exceeds λ/10 (36°).

The article gives a formula for efficiency \(Efficiency(%)=frac{R_{rad}}{R_{rad}+R_{loss}}\), a common enough formula and nothing too contentious about it if Rrad and Rloss are valid.

The article references two loop calculators, both of which use the common formula for calculation of Rrad of a STL in free space. As mentioned, the formula loses validity when the conductor length of the loop exceeds λ/10, and in any event does not capture the effects of ground reflection in real world antennas. See

Accuracy of estimation of radiation resistance of small transmitting loops for further discussion.

Neither calculator referenced captures all system losses, eg ground losses, and so they are overly optimistic / not relevant to real world antennas.

The article references AA8C as authority for their claim that paralleling two loop conductors raises Rrad by a two.

This is plainly a failure in thinking. The total current divides among the parallel conductors (though not necessarily equally) and the total current moment is the sum of the current moments of each conductor and for all intents and purposes is approximately that of a single conductor loop. In other words, the field strength at a great distance is simply proportional to the total current with either one conductor or two parallel conductors. For same distant field strength, the total feed current is the same whether there is one or two parallel conductors and hence Rrad is the same.

That is not to say that parallel conductors behave the same in all respects as a single conductor, conductor loss and inductance is clearly different and that will flow into antenna Q and bandwidth.

There may be advantage in two parallel loop conductors, but they do not include raising Rrad by a factor of two.

When n turns are arranged in series, if the loop is very small the current magnitude and phase is uniform throughout and the total current moment is due to the sum of the current moments of each turn, each of which carries the full input current. Since current moment is increased by a factor of n, then less input current is required to achieve a given distant field strength and so Rrad is increased… though it is overly simplistic to consider that it is increase by n^2 for other than the very smallest loops.

The article gives in Table 1 efficiency figures ranging to 89.9%, figures that would sound alarm bells to most readers.

The predicted efficiency is based on calculators which have two significant failings:

- the equation used for Rrad is inaccurate for all but the smallest loops, and it does not include the effect of ground reflection; and
- terms used in the efficiency equation are incomplete, they do not capture some losses.

On the back of these overly optimistic efficiency calculations, the authors misunderstand the effect of parallel conductors and wrongly apply a factor of 2 to Rrad due to use of two parallel conductors. The increase of Rrad by 2 due to the series conductors is naively simple as mentioned earlier.

Above Table 2 from the article gives predicted efficiency at 20m ranging to 90%.

An NEC-4.2 model of a 3m perimeter lossless loop was modelled at 2m height over “average” ground type (σ=0.005, ε_{r}=13).

Above, the results show no loss in the antenna structure, and radiation efficiency of 42%. Practical structure losses would further reduce radiation efficiency, the Table 1 predictions of up to 90% efficiency are extraordinary.

The performance of this type of structure is best assessed by direct measurement of field strength in the far field.

The article then sets out observations on air to support the calculated expected efficiency.

They mention under the heading Observations that the mag loop appears to perform at levels suggested by modelling

whilst observing that no account was made of ground effects etc.

The problem is that whilst measurements might question the theoretical workup, they do not rewrite sound principles. If the measurements support wrongly based theoretical figures, they do not prove the wrongly based methods to be now sound.

Indeed confirmation of wrongly based predictions raises questions about the experiment design, why did it support propositions that were not true.

Efficiency was prominent in prediction, yet nothing in the reported observations went close to directly measuring radiation efficiency.

- Purdum, J & Peter, A. Feb 2020. The double double magnetic loop. In Radcom Feb 2020.
- Accuracy of estimation of radiation resistance of small transmitting loops

]]>

It looks quite different to the expected behavior of the underlying loop, but it does contain an arc albeit rotated and offset. In fact it can be transformed in two simple steps.

An explanation of how the auxiliary loop feed operates is offered at Equivalent circuit of small transmitting loop with auxiliary loop feed.

The MFJ-1786 is a little more complicated in that there is a low pass L network (approximately 200nH and 100pF) between the coax socket and the auxiliary loop.

Let us look at a simulation where the main loop resistance is used for the load, the modelled inductance of the main loop and measured inductance of the auxiliary loop are used and the mutual inductance and tuning capacitor are adjusted to obtain a 50Ω match.

Above is the simulation circuit. The first line section is of negligible length and is there for later. The next line section simulates that used in the original measurements. The very small inductances are the mutual inductance components of the equivalent circuit of the coupled auxiliary and main loop inductors.

Above is the simulation result.

The first low pass L section acts a little like a transmission line section. Below the break point (40MHz in this case), attenuation is low and phase delay is roughly proportional to frequency… very much like a transmission line. With a 50Ω load at 10MHz, the phase delay is about 20°, which is roughly equivalent to about 1.1m of VF=0.66 50Ω line, or about 40° rotation on the Smith chart. There is also a phase delay in the network comprising the coupled coils. All of these contribute to the rotation of the ‘C’ curve.

Let us now simulate at 28.5MHz, replacing the load with a resistance of 0.8Ω obtained from the NEC simulation at 28.5MHz.

Above, the revised simulation circuit. The load resistance was changed to match the NEC simulation, and the tuning capacitor adjusted for best match… no other changes.

Above, the simulation result. It is not perfect, |s11|min is 0.08 which is equivalent to VSWR=1.17.

The broad tuning range of the auxiliary loop feed was the subject of (Dunlavey 1976), though the MFJ-1786’s cascaded L section may improve the tuning range (either in width of quality of match).

The referenced article stated there were two steps, the rotation and re-normalisation of impedance to remove the offset.

This simulation has dealt with the normalisation of impedance, and there is a residual rotation as could be seen in the following.

If we adjust the length of that first line section, we can demonstrate that the rotation is offset by a -8.3m length of 50Ω VF=0.66 lossless coax.

- Dunlavy, J. Oct 1967. Wide range tunable transmitting loop antenna, US Patent US3588905.

Some relveant theory: for a load where R is approximately constant and X varies, the half power points occur where R=|X|, and following on from that s11=0.2±j0.4, s11=0.4472∠63.43°, |s11|=-6.99dB, ReturnLoss=6.99dB (yes, the +ve sign is correct), VSWR=2.618 etc.

Finding the points where ReturnLoss is approximately 6.99dB with the cursor on the above diagram is quite easy.

With the VNWA client, Port Extensions is used to rotate the locus, and displaying a “Smith renormalised” plot allow specification of an alternate reference impedance (lower right in green).

Finding the half power points can be done by moving the markers searching for s11=0.2±j0.4.

There are lots of PC clients for analysing antenna impedance measurements, the two presented here are ones I use and are licenced by virtue of ownership of the associated Rigexpert and VNWA products. Not all tools can do these things.

Some of the new clients for the nanoVNA may be suitable… they have all sorts of bells and whisltes but I could not coax them to do these pretty fundamental transformations!

Of course can also write your own code in things like Octave, Matlab, Python etc.

]]>One of, if not the most popular loop calculator cited by hams is that by AA5TB. It is especially praised by ham loop enthusiasts.

Above is a screenshot of AA5TB’s calculator with the real antenna dimensions and “Added Loss Resistance” to calibrate the model to the measured 8kHz half power bandwidth. It predicts an efficiency of 30.6%, 2.9 times that of the NEC model. Perhaps it is popular because it provides overly generous estimates, IMHO it lacks credibility for many reasons.

The uncalibrated model uses Rr calculated for a small loop in free space and Directivity of a small loop in free space.

This loop is a little large for the simple formula for Rr to be very accurate.

Not only is the loop a little large for the simple formula for Rr to be accurate, but the proximity of ground modifies Rr, it also modifies the pattern so Directivity is different which flows into a different Gain figure.

Radiation efficiency of 10.188% reconciles well with the NEC-4.2 model’s 10.61%, and estimated gain of -4,4dBi reconciles well the the NEC-4.2 model’s -4.2dBi. The improved reconciliation demonstrates the need to factor in the effect of ground on radiation efficiency and gain.

]]>This article presents some modelling and analysis of the antenna principally to estimate its performance.

The loop was located at 2m above natural ground away from other conducting objects.

He tuned the loop for minimum VSWR at around 10.1MHz and took a sweep with a EU1KY antenna analyser looking through 0.5m of RG223 50Ω cable saving the results to a s1p file which was imported to Antscope.

Here is the impedance plot (excuse the |Z| plot as it is Rigexpert’s concession to hams who do not understand impedance and I cannot disable it).

Above, the impedance plot. The cursor is at point of minimum VSWR, and the associated R and X values at the measurement point are not very useful.

Above, the Smith chart presentation is much more revealing.

Let’s digress for a moment and consider the characteristics of a simple directly fed resonant loop with no coupling networks? Over a narrow frequency range about resonance, R will vary relatively slowly with frequency (in fact it will be approximately constant) and X will vary relatively quickly.

Above is an NEC4 model of such a loop, the impedance plot approximately coincides with the R’=1 circle about resonance.

But the measured impedance looks quite different, lets look at it again.

It looks quite different to the expected behavior of the underlying loop, but it does contain an arc albeit rotated and offset (the latter a consequency of not being properly matched). In fact it can be transformed in two simple steps.

First step: the plotted locus is of the ‘C’ shape that was expected, but it is rotated clockwise (ie towards the source) by some amount.

By eye, a line perpendicular to the locus at the point where it is closest to the Smith chart centre (min VSWR) through the centre is rotated clockwise by about 309° from the direction of the Gamma=1∠0° (or the X=0 axis).

That implies a transmission line like transformation of round trip delay 309°, or electrical length of 309/2=154.5°. Let us calculate the equivalent length of lossless line with VF=0.66, at 10.1MHz it is 8.4m.

Above is the plot with “cable subtracted” and in this case the length of cable subtracted was find tuned to 8.26m for best fit of the curve with a constant R circle, and in this case R=27.9Ω. The antenna is not unmatched, it is matched for a different impedance, and it is capable of analysis.

Step 2: the next step is to normalise the Smith chart to R=27.9Ω.

Above, the normalised plot now looks almost exactly as expected of the underlying loop, and here the cursor notes the lower 3dB bandwidth point (RL nominally 6.99dB). The upper point is 10.105MHz and without extrapolating, we can say the half power bandwidth is approximately 8kHz.

So let’s build and calibrate an NEC4 model for the same half power bandwidth as measured.

The model is based on measured dimensions and an assumption of “average” ground type (σ=0.005, ε_{r}=13) and loss of the tuning capacitor adjusted to calibrate the model to the same half power bandwidth as the measured antenna. The calculated efficiency, gain pattern etc should then be a good model of the real antenna.

Above is the calibrated model VSWR and |s11| response. The half power bandwidth of a matched antenna is the bandwidth between freqencies where |s11| is -6.99dB (or ReturnLoss is 6dB, or VSWR is 2.618), it is approximately 8kHz (to simulate the measured loop).

Above, the model summary gives us the radiation efficiency as 10.61% (-9.7dB).

Above is the gain pattern. Maximum gain is at the zenith, and whilst it is good at high elevations angles, it falls away quite quickly below 20° elevation. It is a good NVIS antenna at 10.1MHz, but there isn’t often paths for NVIS at that frequency.

]]>This article describes an NEC-4.2 model at 14MHz of an antenna similar to a commercial example.

The graphic shows the geometry. In this case the source is at the bottom of the lower loop, and the blue square is the tuning capacitor. The loop conductor is 22mm copper tube, the loop diameters are 1m, and the capacitor connection is 100mm wide. Commonly these are fed by a low loss auxiliary loop at the bottom of the lower loop, but the direct feed is quite fine for modelling the loop performance.

The antenna is quite affected by proximity and type of ground, the model locates the centre of the structure at 2m height (1m from ground to the bottom of the structure) above “average” ground. Capacitor Q is taken to be 1000.

Maximum gain (-1dBi) is at high elevation (52°), a result of proximity to ground, and it is nearly omni directional at very high elevation angles.

In fact, even at 30° elevation, there is not a deep pattern null, front to side is about 5dB.

Radiation efficiency in this configuration is 23%.

The antenna does not quite fit the criteria for a Small Transmitting Loop. Although the perimeter is 6.3m, the current amplitude and phase is more like a pair of 3.15m circumference loops in parallel in phase, a result of the shared connection to the tuning capacitor. So in some ways, it acts like a pair of co-phased parallel 1m diameter loops with their center displaced by the loop diameter.

It is compact in footprint, and a simple single loop using the same length of copper tube would performs fairly similarly, the simple loop performs significantly better than this figure 8 on 40m.

I might note that many of the construction pics I have seen show a vacuum cap but compromised by relatively high resistance connections, and it might not achieve the Q assumed by this model, and therefore have somewhat lower radiation efficiency.

]]>This article explores his 7MHz observations.

Assuming the measurements were made with the antenna clear of disturbing conductors etc, in good condition.

Above is his VSWR scan.

The key measurements were:

- centre frequency 7.175MHz, VSWRmin=1.1;
- VSWR=3 bandwidth 36kHz.

Based on that, we can estimate the half power bandwidth to be 30kHz if R is less than Ro, more like 33kHz in the other case, but we will be optimists.

A NEC-4.2 model of the antenna at 14MHz was built and calibrated to the implied half power bandwidth (30kHz). Model assumptions include:

- ‘average’ ground (σ=0.005, εr=13);
- Q of the tuning capacitor = 2000;
- conductivity of the loop conductor adjusted to calibrate the model half power bandwidth to measurement.

Note that the model may depart from the actual test scenario in other ways.

Above is the VSWR scan of the calibrated model, the load is matched at centre frequency and half power bandwidth is taken as the range between ReturnLoss=6.99dB points.

The NEC model reveals that the loop reactance is 121Ω.

Above is a result screen from the NEC model showing some key quantities that can be used to dissect the feed point resistance into important components. Radiation Efficiency given here as 0.914% can be expressed as -20.4dB.

Above, decomposition of the total feed point resistance into components Rr (radiation resistance), Rg (ground loss resistance). and Rstructure (structure loss resistance).

Also of interest is the gain calculated by the model.

Above is the radiation pattern. As expected for an STL near ground, the maximum gain is towards the zenith and in this case it is -14.8dB. The Directivity show in an earlier screenshot (as RDF) is 5.62dB.

(Duffy 2014) is an online calculator for finding STL gain from bandwidth. The basic calculator assumes free space conditions, but provision is made to tweak Rr and Directivity for ground effects.

Above is the uncalibrated model, uncalibrated to mean using the bandwidth measured near ground, but Rr and Directivity for free space conditions.

Note that the perimeter is 0.0725λ, within the stated accurate range of the underlying free space model.

More importantly, NEC’s calculation of Rr of the loop in the presence of ground being 0.0048Ω is significantly less than predicted by traditional formulas.

Adjusting Rr to the model (Rr/Rrfs=0.9), and Directivity to the model (5.6dB) we obtain an efficiency of -20.1dB and gain of -14.5dB which are both within tenths of a dB of the NEC model results.

The model applies to the scenarios described, and extension to other scenarios may not be valid.

Calculate small transmitting loop gain from bandwidth reconciles well with the NEC-4.2 model.

NEC-4.2 is a more complete model of the scenario and when calibrated to the measured half power bandwidth, it probably our best analytical estimator of radiation efficiency.

- Duffy, O. 2014. Calculate small transmitting loop gain from bandwidth https://www.owenduffy.net/calc/SmallTransmittingLoopBw2Gain.htm.

- https://www.hamradioandvision.com/measuring-loop-efficiency/
- https://dog-asparagus-b2eb.squarespace.com/100-watt-loop-antenna/

AE7PD gives the radiation efficiency on 20m as 30.5% or -5.2dB.

I present here an alternative analysis of the antenna as measured on 20m.

Assuming the measurements were made with the antenna clear of disturbing conductors etc, and that 5/8″ tube means 16mm OD.

The key measurements were:

- centre frequency 14.165MHz, VSWRmin=1.0;
- VSWR=2.62 bandwidth 22kHz.

A NEC-4.2 model of the antenna at 14MHz was built and calibrated to the measured half power bandwidth (22kHz). Model assumptions include:

- ‘average’ ground (σ=0.005, εr=13);
- Q of the tuning capacitor = 2000;
- conductivity of the loop conductor adjusted to calibrate the model half power bandwidth to measurement.

Note that the model may depart from the actual test scenario in other ways.

Above is the VSWR scan of the calibrated model, the load is matched at centre frequency and half power bandwidth is taken as the range between ReturnLoss=6.99dB points.

The NEC model reveals that the loop reactance is 256Ω.

Above is a result screen from the NEC model showing some key quantities that can be used to dissect the feed point resistance into important components. Radiation Efficiency given here as 16.6% can be expressed as -7.8dB.

Above, decomposition of the total feed point resistance into components Rr (radiation resistance), Rg (ground loss resistance). and Rstructure (structure loss resistance).

Also of interest is the gain calculated by the model.

Above is the radiation pattern. As expected for an STL near ground, the maximum gain is towards the zenith and in this case it is -2.24dB. The Directivity show in an earlier screenshot (as RDF) is 5.59dB.

(Duffy 2014) is an online calculator for finding STL gain from bandwidth. The basic calculator assumes free space conditions, but provision is made to tweak Rr and Directivity for ground effects.

Above is the uncalibrated model, uncalibrated to mean using the bandwidth measured near ground, but Rr and Directivity for free space conditions.

Note that the perimeter is 0.149λ, well above the stated accurate range of the underlying free space model, as will be seen by comparison with the NEC model which for instance gives Rrfs=0.093Ω, the classic formula gives Rrfs=0.098Ω.

Note that the loop is large enough that current is not sufficiently uniform for the classic formula to be accurate, so little surprise that there is a discrepancy.

More importantly, NEC’s calculation of Rr of the loop in the presence of ground being 0.0058Ω is substantially less than predicted by traditional formulas and the main contribution to the NEC’s efficiency being near half of AE7PD’s estimate.

Adjusting Rr to the model (Rr/Rrfs=0.594), and Directivity to the model (5.59dB) we obtain an efficiency of -8.0dB and gain of -2.4dB which are both within tenths of a dB of the model results.

The model applies to the scenarios described, and extension to other scenarios may not be valid.

Calculate small transmitting loop gain from bandwidth reconciles well with the NEC-4.2 model.

AE7PD’s estimate of efficiency is almost 3dB higher than indicated by the calibrated NEC-4.2 model, a result of its being based on some traditional formulas that have issues that I have discussed elsewhere.

NEC-4.2 is a more complete model of the scenario and when calibrated to the measured half power bandwidth, it probably our best analytical estimator of radiation efficiency.

- Duffy, O. 2014. Calculate small transmitting loop gain from bandwidth https://www.owenduffy.net/calc/SmallTransmittingLoopBw2Gain.htm.

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.

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.

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.

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.

- 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.