(Purdum 2020) describes a small transmitting loop (STL) which is a little novel in that it uses an arrangement of four circular conductor loops, two in parallel in series with the other two in parallel.

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.

## 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.

## 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 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.

### 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.

Let us take efficiency

to mean radiation efficiency.

## Simple small loop models

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°).

## Basis of efficiency claims

The article gives a formula for efficiency \(Efficiency(\%)=\frac{R_{rad}}{R_{rad}+R_{loss}}\), a common enough formula (though the result needs multiplication by 100 to be in %) 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.

## Parallel loop conductors

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.

## Series loops

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.

## Extraordinary claims

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.

## Confirmation by measurement

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.

## References / links

- 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