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 (0.005,13);
- 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 (0.005,13);
- 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.

The key measurements were:

- centre frequency 7.014MHz, |Z|=51Ω, VSWR=1.1;
- VSWR=3 bandwidth 16.2kHz.

The step size of the analyser prevented measurement exactly at resonance, but R changes very closely with frequency near resonance so we can estimate it quite well. The above figures can be used to find R close to resonance.

Within the limits of measurement error, we can say that R at resonance should be very close to 51Ω, and VSWRmin close to 1.02.

We can infer the half power bandwidth from the quantities above, it is 14.2kHz.

A NEC-4.2 model of the antenna at 7MHz was built and calibrated to the implied half power bandwidth (14.2kHz). Model assumptions include:

- ‘average’ ground (0.005,13);
- Q of the tuning capacitor = 1000;
- 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 124Ω.

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 1.402% can be expressed as -18.5dB.

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 at the zenith and in this case it is -13dB. The Directivity show in an earlier screenshot (as RDF) is 5.84dB.

(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.

Adjusting Rr to the model (Rr/Rrfs=0.875), and Directivity to the model (5.84dB) we obtain an efficiency of -18.5dB and gain of -12.6dB which are both within 0.1dB of the model results.

The model applies to the scenarios described, and extension to other scenarios may not be valid. Note that different models of Alexloop may have significantly different behavior.

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

Now Alexloops seem to have undergone some evolution, and there does not seem to be a clear list of model names or numbers with features or specifications, so to some extent the antenna is a little non descript.

The article did not document the environment of the test antenna, but Findling explained in correspondence that it was relatively clear of conducting structures and about 1.2m above natural ground.

A NEC-4.2 model of the antenna at 7MHz was built and calibrated to their measured half power bandwidth (19kHz). Model assumptions include:

- ‘average’ ground (0.005,13);
- Q of the tuning capacitor = 1000;

Note that the model may depart from the actual test scenario in other ways, it is challenging to glean all the data that one would like from the article.

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

The NEC model reveals that the loop reactance is 124Ω.

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 1.073% can be expressed as -19.69dB.

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 at the zenith and in this case it is -14dB. The Directivity show in an earlier screenshot (as RDF) is 5.84dB.

The key experimental result was radiation efficiency of -15.77dB (2.65%) which is almost 4dB better than the model results.

There may be two main contributions to this, the use of term 2*Rrad in the efficiency calculation, and the value of radiation resistance Rr (Rrad) may not account for the effect of ground reflection on Rr.

(Findling, A & Siwiak 2012) mention “ideal loaded Q” in calculating Qrad which hints that they may be disciples of (Hart 1986) . They calculated Qtotal from the loop antenna alone, and for their given efficiency calculation, Qrad should be calculated for exactly the same scenario but with all of the dissipative losses extracted. Qrad should be of the order of 28,447 for their antenna, and their efficiency on that basis should be 10*log(376.29/28447)=-18.79dB (1.32%).

Adjusting Rr to the model (Rr/Rrfs=0.89), and Directivity to the model (5.84dB) we obtain an efficiency of -19.7dB and gain of -13.9dB which are both within 0.1dB of the model results.

The model applies to the scenarios described, and extension to other scenarios may not be valid. Note that different models of Alexloop may have significantly different behavior.

The radiation efficiency calculated by (Findling, A & Siwiak 2012) of their test antenna does not reconcile well with an NEC-4.2 model and appears to be optimistic, possibly due to a flawed concept of Qrad.

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

- Hart, Ted (W5QJR). 1986. Small, high efficiency loop antennas In QST June 1986.
- Findling, A & Siwiak, K. Summer 2012. How efficient is your QRP small loop antenna? In The QRP quarterly Summer 2012.

The STL used a main loop resonator and a separate small auxiliary loop for the 50Ω feed, a very common arrangement.

The main loop is a coaxial cable with, in this case, a tuning capacitor inserted between the inner conductors at each end. Above is a diagram of the main loop.

This technique appears to be employed in some commercial devices and some DIY STL designs that may have been inspired by pics of the commercial ones.

For discussion purposes, lets consider that it is a circular loop of perimeter 3m of RG213 coax, the inductance between shield ends is 2.9µH.

The first important concept is that of skin effect in coaxial cables, and its influence on setting TEM mode propagation inside the coax. Readers might benefit from the article Small single turn un-tuned shielded loop which explains some relevant concepts.

This antenna is intended for HF, and at those frequencies with good coaxial cable, skin effect is sufficiently well developed that we can consider it to be fully effective.

In the presence of a HF electromagnetic field, a voltage will be induced in the conductor formed by the outer surface of the outer conductor, and some voltage V will appear between the shield ends at the gap.

That voltage V appears across three components in series, the left hand differential mode impedance of the coax, the tuning capacitor, and the right hand differential mode impedance of the coax.

Note that the polarity of the differential mode voltages is opposite. For that reason, the differential mode voltage half way around the coax (ie opposite to the gap) is zero even though current flows at that point. The impedance looking into each differential mode pair of terminals is that of a S/C stub, and of half of 3m in length.

At 7MHz, the impedance of a 1.5m S/C stub of RG213 is 0.266+j17.32Ω, and that of the 2.9µH loop is j128Ω plus some resistance comprising the radiation resistance Rr, ground loss resistance Rg, capacitor loss resistance Rc, plus some loss resistance Rco of the external surface of the coax.

Voltage V is applied to a total impedance of 2*(0.266+j17.32)+j128+Rr+Rg+Rc+Rco Ω.

A sobering though is that in this instance where Rr is of the order of 0.005Ω, the total stub resistance of 0.53Ω consumes a hundred times as much power as is actually radiated.

So this arrangement has increased the inductive reactance of the loop by 27% in this case so it tunes lower in frequency with a given capacitor, but it has been at a cost of an additional 0.53Ω of loss resistance. The effect on radiation efficiency depends on the other loss elements and is best evaluated by measurement or inference from a bandwidth measurement (using the corrected total inductance).

Buyers of these things are not usually concerned about radiation efficiency, indeed they prefer wider bandwidth which is easily achieved by compromising efficiency.

The calcs above were for a given length of RG213 coax at 7MHz. Changing any of these parameters changes the scenario and the results are different. For example, the same length of a ‘foam RG213’ like LMR400 has the same outside diameter, but the impedance looking into each 1.5m S/C stub is 0.19+j13.24Ω.

Beware that some implementations may achieve connection of inner to outer conductor in both of the connectors on the loop cable, and making it a ‘normal’ loop with tuning C connected between shield ends.

Don’t assume that STL using coax cable for the main loop directly connect the tuning capacitor between ends of the outer conductor, some use the connection described above and it needs different analysis.

The structure will work. It increases inductive reactance in the loop circuit and increases loss resistance which will in most cases contribute to reduced efficiency. If you have one of these you might explore shorting the inner to outer at each coax jack for improved efficiency, though it will change the tuning, and you may need a slightly longer loop.

The outside surface of the outer conductor of the loop coax is the antenna per se, claims that the inner conductor is the antenna and is shielded from electric field by the outer conductor are unsound, naive, and based on archetypal ham myths.

Behaviour is explained by traditional basic linear circuit analysis and transmission line concepts.

The model is drawn from AA5TB’s calculator’s initial values.

The model is in NEC-4.2, and is a 20 segment helix in free space, and tuned for resonance at 7.000MHz. (If you repeat this using NEC-2, you may need fewer segments to avoid violating NEC-2’s segment limits.)

Above, the model was resonated at approximately 7.000MHz, and the feedpoint R noted, it is 0.02976Ω.

Above, this is entered directly into the modified 4NEC2 dialogue (the change does not ‘stick’ in the unmodified version, it quietly ignores Zo<0.1Ω).

Above, the VSWR curve is now based on reference Z=0.02976Ω.

I have used a screen ruler to measure the Half Power Bandwidth (bandwidth between points where ReturnLoss is 6.99dB (or VSWR=2.618, see Antennas and Q).

In this case, Half Power Bandwidth is 1.96kHz. (No, it does not reconcile with the calculator, a defect of the calculator in this case.)

Of course, modelling the loop in the real world, you would include tuning capacitor loss, and add real ground to the model. All of these things influence the results.

If you then build such an STL, the model of the loop should reconcile with real world Half Power Bandwidth measurement… if your model was complete and valid. If you measure wider bandwidth, you might consider that a bonus but it is more likely a warning of unaccounted loss.

]]>I asked the developer to consider a change, but I gathered that he regarded 4NEC2 to be at End Of Life.

It appears that 4NEC2 enforces a requirement that Zo>=0.1, so having discovered that by trial and error, one wondered if it was possible to change that threshold by hacking the exe file.

The IEEE754 Double representation of 0.1 is 0x3FB999999999999A, and of course it would be stored backwords in the exe file. Searching for 0x9A9999999999B93F found only one occurrence, offset 0x1490. That was changed to 0xfca0f1d24d62503f (the backwords representation of 0.001) and the exe tested. (It might be tempting to set it so zero, but that would permit entering zero which may cause run time errors).

To my delight, it now permits directly changing Zo down to 0.001Ω.

There is a side effect of this change, it appears that the literal constant is used by (at least) one other function, and current graphs on the geometry window need to be scaled up with the page up key to be useful.

If you want to use this you can:

- edit 4NEC2.exe directly with your favourite hex editor (don’t forget to take a backup), or
- an xdelta3 patch file is available for 4NEC2 v5.8.16: 4NEC2-5.8.16-ZoPatch, the xdelta3 command is : xdelta3 -d -s 4nec2.exe patch.xd3 .

]]>

The response of a simple series resonant RLC circuit is well established, when driven by a constant voltage source the current is maximum where Xl=Xc (known as resonance) and falls away above and below that frequency. In fact the normalised shape of that response was known as the Universal Resonance Curve and used widely before more modern computational tools made it redundant.

Above is a chart of the Universal Resonance Curve from (Terman 1955). The chart refers to “cycles”, the unit for frequency before Hertz was adopted, and yes, these fundamental concepts are very old.

(Terman 1955) gives a general definition of Q that is widely accepted:

circuit Q is 2π(energy stored in the circuit)/(energy dissipated in circuit in one cycle) .

Q is easily assessed for a single reactor:

- inductor: energy stored is L*Ipk^2/2 and energy lost per cycle is I^2R/2f so Q=2πfL/R=Xl/Rs;
- capacitor: energy stored is C*Vpk^2/2 and energy lost per cycle is V^2/(2fRp) so Q= 2πfCRp=Rp/Xc.

A RLC series resonant circuit is nearly as easy as at the instance when the inductor current is maximum, capacitor voltage is zero and hence the energy stored is as per the simple inductor case and Q=Xl/Rs.

Likewise for a parallel RLC resonant circuit, at the instant of maximum capacitor voltage, inductor current is zero and hence the energy stored is as per the simple capacitor case and Q=Rp/Xc.

Q and bandwidth are related, high Q circuits have a narrow response, narrow bandwidth. Q is a factor used in normalising the Universal Resonance Curve (URC) above, and appears in several of the equations on the chart.

Lets focus in particular at the point on the URC corresponding to where |X|=R. When |X|=R, |Z| is 2^0.5 times that at resonance, and the current response is 2^-0.5 (or 0.707) times that at resonance. Since power=I^2*R, power at that point is half the maximum response. These are known at the half power points.

If you look at the URC, you will seen that the current response is 0.7 of maximum when a=0.5. Taking the case for a=0.5 (the half power points) and noting that Bandwidth (BW) is measured between the upper and lower points, BW=2*CyclesOffResonance, we can substitute into a=Q*(CyclesOffResonance/ResonantFrequency) to obtain BW=ResonantFrequency/Q. This is a well known relationship.

For a circuit where R is approximately constant with frequency, we can find the approximate BW by finding the frequencies at which |X|=R.

For loads, including antennas, that can be represented near their series resonance as a series RLC circuit, and where R changes very slowly with frequency compared to X, we can can approximate the bandwidth with an antenna analyser or VNA as the bandwidth between points where VSWR referenced to R is 2.62, or equivalently ReturnLoss referenced to R is 6.99dB

A series resonant circuit was constructed from an air cored solenoid inductor and 100pF silvered mica capacitor.

The following charts estimate the inductance and resistance of the inductor at 3.5MHz.

Above, the physical parameters.

Above, the electrical parameters estimates. The key ones are an inductance of 19.8µH and resistance of 3.7Ω.

The resistance of the inductor will dominate, silvered mica capacitors are very low loss, so we might expect the circuit Q to be just a little worse than that of the inductor, so Q somewhere towards 119.

Calculated resonant frequency is 3.58MHz, but real world tolerances will lead to a small error in that estimate.

Expectation is that half power bandwidth will be somewhere a little less than 3580/119=30kHz.

The series circuit was measured using a VNWA3.

Above is a plot of R, X and |Z| about resonance. Marker 1 is at resonance, Markers 2 and 3 are at the lower and upper points where |X|≅R.

Note that R is very slightly dependent on frequency, a natural consequence of skin effect in the coil conductor.

BW=29kHz, Q=121, very close to expectation.

One of the shortcomings of the VNWA3 software is that it does not calculate / display VSWR or Return Loss in terms of an arbitrary reference impedance, so we must calculate the VSWR and Return Loss at Markers 2 and 3 in terms of matched impedance using another calculator.

Above, at the lower |X|≅R point, VSWR=2.62 and ReturnLoss=6.97dB.

Above, at the upper |X|≅R point, VSWR=2.62 and ReturnLoss=6.98dB.

Hams tend to happily give different meaning to well known terms and concepts that have been part of the knowledge of a hundred years, indeed foundation knowledge on which bigger concepts are built.

There is a following in ham radio (where popularity defeats science) that the half power bandwidth for an antenna is the bandwidth between points where ReturnLoss=3dB (VSWR=5.85).

This is not the same thing as the half power bandwidth of a simple series L-R-C circuit (for one thing it includes an extra impedance being that of the generator, and possibly transmission line), and as a consequence:

- the bandwidth obtained does not give the correct Q for the antenna itself when substituted into the relationship Q=Fo/BW; and
- the condition does not imply |X|=R (in fact it implies |X|=2*R, a natural consequence of the unconsidered inclusion of the source R in the circuit).

Since VSWR=5.85 implies |X|=2*R in this scenario, |Z| is (1+2^2)^0.5 times that at resonance, and the current response under constant voltage drive is (1+2^2)^-0.5 (or 0.447) times that at resonance. Since power=I^2*R, power at that point is one fifth the maximum response.

Above, plotting the current response of |X|=2*R (I=0.447) on the URC, we get a=±1, twice the bandwidth of the half power points.

The half power bandwidth of the RLC network is in fact half the difference between VSWR=5.85 frequencies.

The article presents a theoretical analysis of a series RLC circuit using classical circuit analysis techniques, and theory based on a recognised text. The well known meaning of “half power points” of as series RLC circuit was derived in terms of R and X, and VSWR(R) and ReturnLoss(R).

The experiment designed and made an inductor, and with it a series RLC circuit which was measured to demonstrate the theoretical analysis. The experimental results reconciled very well with theoretical prediction, Fo was very close to expectation, measured half power bandwidth was in line with expectation, inferring a Q in line with expectation.

- Terman, Frederick. 1955. Electronic and Radio Engineering – 4th ed. New York: McGraw-Hill.