Small untuned loop for receiving set out a model for calculating the S/N degradation of an active untuned small loop antenna system.

The calculations in Small untuned loop for receiving – Trask noise and gain analysis might prompt the question of what is the optimal resistive load for an untuned small loop.

This article explores the topic for a simple model where the equivalent noise temperature of the amplifier is independent of source impedance.

## A simple model for a small loop

We can construct a simple model where the loop behaves as a fixed pure inductance, and its load is a fixed pure resistance.

This is a reasonably good model for a small loop, perimeter < wl/10, not too bad for perimeter up to wl/3.

The source impedance becomes the loop’s inductive reactance Xl which is proportional to frequency, and the load is Rl.

Above is a plot of the relative power developed in the load vs the ratio of Rl/Xl.

There is a maximum where Rl=Xl, and the power captured falls away either side.

Lets consider a 7MHz, untuned 1m square loop of 20mm tube. It has a source impedance Xl of 126Ω, calculating Antenna Factor, Antenna Gain and S/N degradation for ambient noise figure of 45dB.

### Rl=50Ω

- AF(50)=25.85dB
- Gain(50)=-38.7dB

Calculated S/N degradation is 1.7dB.

### Rl=100Ω

A round number close to Xl was chosen as 100Ω, it is often not easy to arbitrarily choose input resistance of the amplifier and the feedback schemes often used for these amplifiers make integral ratios easier to achieve. 100Ω is a little less than the optimal 126Ω.

- AF(100)=21.17dB
- Gain(100)=-37.0dB

Calculated S/N degradation is 1.2dB, 0.5dB better than the nominal 50Ω case.

### Rl=2Ω

- AF(2)=53.25dB
- Gain(2)=-52.14dB

Gain has been increased 10dB, but no amount of gain will get the S/N degradation lower than the calculated S/N degradation of 10.1dB. It is considerably worse than the nominal 50Ω case.

## Broadband behaviour

The cases above considered different load resistances on the loop at one frequency. At other frequencies, the loop reactance is different and the optimal load resistance is accordingly different.

This same antenna on 3.5MHz would be optimised with Rl=63Ω.

The loop inductance is sensitive to the loop dimensions, and whilst the open circuit voltage induced by a given field strength is not sensitive to conductor diameter, the loop inductance is very sensitive to conductor diameter. A loop with smaller conductor than that used above will have higher Antenna Factor, and lower Gain, and would be optimised with a higher resistance load.

The simple optimal load resistance for a given loop might just be the one that gives the best S/N degradation on the worst band, or the band of greatest interest to the user.

## Conclusions

If a small untuned loop has inductive reactance well removed from 50Ω at a frequency of interest, then an amplifier that loads to loop with Rl closer to Xl will give some improvement in S/N degradation.

Minimising inductance for given loop area may be a better measure for S/N improvement. It may be that a smaller loop of thicker conductor performs better.

## References / links

- ITU-R. Jul 2015. Recommendation ITU-R P.372-12 (7/2015) Radio noise.
- Lankford, D. May 2007. Common base transformer feedback Norton amplifiers rev 21 V0 7.
- Norton, D. Jun 1975. Transistor amplifier with impedance matching transformer – US Patent 3,891,934.
- Smith, J. Apr 2010. Z10040B broadband Norton amplifier.
- Trask, C. Aug 2010. Wideband loop antenna amplifier.
- owenduffy.net/files/NoiseAndReceivers.pdf
- Receiver sensitivity metric converter
- Calculate small loop Antenna Factor
- Convert Antenna Factor and Gain
- RxActiveNoise spreadsheet (zipped)