The meaning of the terms efficiency and radiation resistance are often critical to understanding written work on antennas, yet different authors use them differently, often without declaring their intended meaning.

Mike Underhill (G3LHZ) is an enthusiastic proponent of Small Transmitting Loops and in his slide presentation (Underhill 2006) challenges the proposition that their efficiency is low.

The line taken broadly is to introduce his own interpretation of efficiency and to challenge by experimental evidence other views on expected efficiency.

## Shared language

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.

Readers must keep the author’s meaning in mind when interpreting their work, and comparing their findings with other works.

## Underhill’s meaning of efficiency

(Underhill 2006) right up front gives some insight into his use of the term efficiency:

The Scientific Definition , based on the First Law of Thermodynamics (– the law of energy conservation) is that:

- Antenna ‘intrinsic’ efficiency η is the total RF power Prad radiated away from all the surfaces of the antenna divided by the input power Pin to the antenna η=Prad/Pin
- The power not radiated is Pdiss = Pin – Prad. This is dissipated as heat in the conducting surface of the antenna and makes it ‘get hot’. We therefore have:

η=Prad/Pin=(Pin–Pdiss)/Pin=1-Pdiss/Pin

- We can also measure the dissipated power Pdiss directly by the ‘heat balance method’ to get the efficiency. Measurements take about 10 to 20 mins at each frequency.
- We can also estimate the dissipated power Pdiss by calculating the loss resistance to get the efficiency indirectly, by the new ‘Rho-Q’method.
Measurements (of Q) take about one to two minutes

You might be impressed by his calling up the First Law of Thermodynamics as the basis for his definition of efficiency and evidence that it is soundly based, but we need to look carefully at Underhill’s accounting for energy, does the detail honour the principle.

A key statement is The power not radiated is Pdiss = Pin – Prad. This is dissipated as heat in the conducting surface of the antenna and makes it ‘get hot’

. In this statement, the only source of loss in an ‘antenna’ that he gives is power lost in the conductors of the antenna.

There is a clear denial of all things that reduce radiation other than conductor loss. This is important because often, it is the things that are unidentified or that we casually ignore as irrelevant that ruin solution accuracy rather than the things we estimated incorrectly.

To illustrate his method, he gives Rho-Q Loop Efficiency Spreadsheet, a table of measured and calculated values for a 1m diameter loop.

The values given for the first seven columns seem reasonable, and the value calculated for columns 8 and 9 are quite consistent with his definition of efficiency quoted above.

However, his definition restricts an ‘antenna’ to exclude all things that convert RF energy to heat other than the loop conductor. Importantly, his ‘antenna’ excludes:

- loss in the loop tuning capacitor or equivalent;
- loss in any matching network;
- loss in nearby media due to electric and magnetic fields, especially soil.

Little wonder that Underhill argues higher antenna efficiency than many other authors, he has:

- excluded some really important loss elements from his definition of ‘antenna’; and
- by default, allocated their equivalent resistance to radiation resistance, implying that the power consumed by them is converted to Electro Magnetic Radiation.

Underhill is dismissive of the conventional theoretical estimate for radiation resistance of a small loop (which he labels ‘Kraus’s formula’):

In his table from (Underhill 2006) mentioned above, Underhill shows his value for radiation resistance as around 4000 times the above formula for the 1m loop at 1.8MHz.

Likewise he report’s Chu’s radiation resistance also thousandths of his own value for that case.

Similar disparity with conventional methods exists in a later work, the table at page 13 of (Underhill 2013).

## Some widely accepted definitions for the remainder of this article

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

### Radiation Sphere (for a given antenna)

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 characteristing 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.

### Radiation Resistance

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.

### Radiation Efficiency

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.

## Example – John Huggins’ STL

John Huggins described his STL (Huggins 2015) and gave a graph of measured Return Loss around the matched frequency. We are working here with the information he published and have no clarification on my interpretation of the data, and I make some reasonable assumptions for the purpose of demonstration.

Above is an estimate of key loop parameters from bandwidth derived from the Return Loss chart. The model ignores self inductance of the tuning capacitor connection, and all impedances are expressed in terms of the main loop, but losses in the impedance transformation network, tuning capacitor and ground are captured in the measured bandwidth. Note that the Gain estimate is based on Directivity in free space and is likely to be more like -8.6dBi mounted near ground.

Antenna centre is taken from pics to be 2m above, ground and I assume ‘average ground’ (0.005/13).

Rr is taken to be very close to Rrfs based on the graphs in A method for initial ground loss estimates for an STL. Uncertainty in Rr plays into Efficiency and Gain estimates.

### Underhill’s efficiency calculation of Huggins’ loop

First, lets calculate total input power and power dissipated according to Underhill:

- Total input power is I^2*Rtotal=0.106*I^2.
- Power dissipated in the conductor is I^2*Rcond=0.029*I^2, this is the quantity Pdiss discussed by Underhill.

And from those, efficiency using Underhill’s formula η=Prad/Pin=(Pin–Pdiss)/Pin=1-Pdiss/Pin=1-0.029/0.106=0.726pu=72.6% or -1.4dB.

### IEEE based efficiency calculation of Huggins’ loop

The calculator screenshot above gives the key data for an estimate of efficiency, it is simply Rrad/Rtotal=3.8% or -14.2dB.

### Further dissection

It is informative to attempt allocation of the losses to various contributors, most of which are denied by Underhill’s model.

The tuned loop feed point resistance can be decomposed into three main components, radiation resistance, structure loss resistances (including conductors, capacitors, and matching network), and an equivalent ground loss resistance. This is discussed in a number of the Duffy references given below.

The calculator screenshot above provides most of the key data for allocation of those losses and calculation of radiation efficiency, and further:

- the RF resistance of the main loop conductor is calculated as 0.029Ω ignoring any additional resistance at joints;
- Qcap is taken to be 5000;
- Rgnd/Rr is estimated to be 3 using the graph in A method for initial ground loss estimates for an STL.

Above is a pie chart showing allocation of Rtotal to the various components based on the stated assumptions. dissection into these components attempts to explain the contribution to Rtotal by various mechanisms.

There is still a very large Runknown. The allocation leaves about 40% in the ‘unknown’ category, suggesting that there are significant losses in elements not identified (eg the ferrite matching transformer), and/or that allowances for some of the other elements are possibly low, and that is quite likely.

Rrad is estimated at 3.8% of Rtotal, so radiation efficiency is 3.8% or -14.2dB. It is unlikely that error in the estimate of Rrad would account for much of Runknown, and whilst the uncertainty of Rrad might well be +/-3dB, that still gives a confidence interval of efficiency as -11.2 to -17.2dB (7.6 to 1.9%) efficiency.

## Conclusions

Underhill’s definition and method of calculation of efficiency is ‘different’ and yield values that are quite high and might seem very attractive, but they are not comparable with ‘industry standard’ meanings of the terms and care should be exercised in interpretation.

If you wish to compare Underhill’s efficiency with conventional radiation efficiency, you must adjust it for unaccounted structure losses, matching loss and ground loss.

## References / links

- Duffy, O. 2015. A method for initial ground loss estimates for an STL
- ———. 2015. Accuracy of estimation of radiation resistance of small transmitting loops
- ———. 2015. Small transmitting loop – ground loss relationship to radiation resistance
- ———. 2015. Enhancement of Calculate small transmitting loop gain from bandwidth measurement
- ———. 2015. Analysis of a series of NEC-4 models of a low loss small transmitting loop at 7MHz at varying height
- ———. 2015. Antennas and Q
- Huggins, J. 2015. Small Loop HF Antenna. http://www.hamradio.me/antennas/small-loop-hf-antenna.html (accessed 05/10/2015).
- IEEE. 1988. IEEE standard dictionary of electrical and electronic terms, IEEE Press, 4th Edition, 1988.
- Underhill, M. May 2006. Loop efficiency.
- ———. M. Sep 2013. Impossible Antennas and Impossible Propagation.