Over recent weeks, I have run literally hundreds of thousands of NEC models of small transmitting loops (STL) over real ground. The objective was to try to discover some simple methods for initial design of a STL, particularly an estimate of ground loss of STL mounted near natural ground.

## Input resistance models

NEC models were constructed of a 0.1m diameter loop of thin lossless conductor at 7.2MHz. In fact the loop is octagonal and of the same area as a 0.1m diameter circle. The very small loop is chosen as a probe to explore ground loss effects, though of course it would not be practical for a transmitting loop at 7MHz.

### NEC2

Above is a plot of the total input resistance (Rin) from NEC2 of the small loop at varying height above three ground types.

There are obviously some glitches at the low end.

### NEC4.1

Above, the same model series run with NEC4.1.

There are obvious signs of numerical instability, and the results are of little value. At least one of the curves dips into negative Rin, a physical impossibility.

### NEC4.2

Above, the same model series run with NEC4.2.

There are some small glitches at the low end, but nowhere near the problems with NEC4.1, and trending more realistically than NEC2 at the low end.

### Summary of Rin models

Though all versions of NEC produce some defects in results, NEC4.2 is probably better than the others and might provide some useful indication of an underlying trend.

The NEC4.2 results are broadly consistent with (Vogler & Noble 1964) (see NEC-4 vs NEC-2 on a low small transmitting loop for more discussion).

## Components of Rin

Rin can be decomposed into three components, radiation resistance Rr, structure and network loss resistances Rlloss, and ground loss Rgnd.

Rr can be calculated from the NEC results as Average Power Gain (APG) over the sphere times Rin, Rlloss is 1-Efficiency/100 times Rin, and Rgnd is the balance of Rin. (Note that the models used for this study are lossless structures, so Rlloss is zero.)

### Radiation resistance

#### NEC2

Above is a plot or Rin from NEC2 models.

#### NEC4.1

Above is a plot or Rin from NEC4.1 models.

Above is a plot or Rin from NEC4.2 models.

#### Summary of Rin plots

The three plots are near identical and seem to behave sensibly. Note that for greater heights, Rr oscillates about and converges on the classic free space value.

None of the versions tested cast doubt on the calculations of the others, so although the total input resistance reported in the previous section is somewhat doubtful, it is entirely possible, plausible even, that the radiation resistance calculations are valid.

## Rgnd/Rr

If we accept that the calculated radiation resistance is probably correct, and total input resistance calculated by NEC4.2 shows an underlying trend that is consistent at the low end with (Vogler & Noble 1964), we can derive a likely characteristic for Rgnd/Rr from these models.

Rgnd/Rr varies with height, and depends on ground parameters. It is likely that Rgnd/Rr is a function of electrical height above ground, and varies with ground parameters, fairly independently of loop diameter for small loops.

This idea was floated at Small transmitting loop – ground loss relationship to radiation resistance which used NEC2 models (NEC-4.1 having shown instability).

Above is Rgnd/Rr vs electrical height using NEC4.2 at 7.2MHz for three ground types that bracket the most common ground types experienced. The curves probably apply more generally to ground types with the same magnitude of complex permittivity (Vogler's curves are for |N| where |N|=√|ε'+jε”|, the three ground cases studied are |N|=2.9, 4.2 and 4.9).

Though there are some glitches, there appears an underlying smooth curve which suggests that ground losses increase quite quickly for height below 0.07λ.

The application of this suggestion is that for a 1m diameter STL at say 1.5m height at 7.2MHz (0.035λ), we might expect that Rgnd is around 4.2 times Rr. The earlier plots suggest that Rr at that height is probably very close to the free space value Rrfs, so we can do some back-of-the-envelope calculations of a loss budget:

- Rr=0.0064Ω;
- Rgnd=4.2*0.0064=0.027Ω;
- Rloss (assuming 25mm diameter copper loop and 20mΩ capacitor ESR) 0.055Ω.

The pie chart shows the breakup of total input resistance, and shows that the Rgnd component is a significant element that the earlier curves show can be reduced dramatically by doubling height to 3m.

The model above shows that this loop has half power bandwidth well less than most constructors report for similar sized loops, so although calculated efficiency in this case is just 4%, it is probably two to three times that indicated by many constructor's reports.

The STL is unlikely to deliver high efficiency when mounted less than about 0.1λ above natural ground, and most loop designers ignore ground loss.

These results apply to STL at 7.2MHz with perimeter less than 0.1λ. Above this size, the ‘dipole' mode becomes increasingly significant and will result in some departure from pure STL behavior. Some departure can also be expected where the height is small wrt loop diameter.

## Acknowledgements

I wish to thank Brian Austin (G0GSF) for his invaluable input and discussion of the subject matter.

## References / links

- B Austin, A Boswell, M Perks. Aug 2014. Loss mechanisms in the electrically small loop antenna. Antennas and Propagation Magazine, IEEE (Volume:56 , Issue: 4 ) http://www.awasa.org.za/index.php/downloads/file/91-loss-mechanisms-in-the-electrically-small-loop-antenna (accessed 05/02/2015)
- J Belrose. Jun 2007. Performance analysis of small transmitting loop. In IEEE antennas and propagation magazine Vol 49, No 3, June 2007.
- A Boswell, A Tyler, A White. Apr 2005. Performance of a small loop antenna in the 3-10 MHz band. In IEEE antennas and propagation magazine Vol 47, No 2, April 2005.
- LE Vogler & JL Noble. Jan 1964. Curves of input impedance change due to ground for dipole antennas”. NBS72.