# Small untuned loop for receiving – simple model with transformer

I have written several articles on untuned loops for receiving, as have others. A diversity of opinions abounds over several aspects, probably none more than the idea of an optimal load impedance for the loop.

This article analyses a simple untuned / unmatched loop in the context of a linear receive system (ie no IMD) of known Noise Figure.

The receiver has an idealised input impedance of 50+j0Ω, and known internal noise implied by its Noise Figure (NF). For the purpose of this analysis we will assume the NF is 5dB and from that we can derive an equivalent noise temperature Tr of 627K.

## External noise

After mentioning receiver internal noise, lets consider external noise.

For the purpose of estimating external noise, we can look to ITU-R P.372-14 for guidance, it gives us a median ambient noise figure from which we can calculate an external equivalent noise temperature. We will use the Residential precinct figures from P.372.

We can define S/N degradation (SND) to mean the reduction in external S/N  $$\frac{ExternalSignal}{ExternalNoise}$$  by the addition of internal noise

$$\frac{\left(\frac{ExternalSignal}{ExternalNoise}\right)}{\left(\frac{ExternalSignal}{ExternalNoise+InternalNoise}\right)}=\frac{ExternalNoise+InternalNoise}{ExternalNoise}$$ which we can convert to dB

$$SND=10log\frac{ExternalNoise+InternalNoise}{ExternalNoise}$$

## An example loop for discussion

Let’s consider a simple single turn untuned loop with an ideal broadband transformer. The example loop is 2.1m perimeter and 4mm diameter conductor situated in free space. The loop has perimeter less than λ/10 up to 15MHz, so we can regard that loop current is uniform in magnitude and phase. This simplifies analysis greatly. Above is a schematic diagram of the example loop.

A simple analysis is to consider the loop to have some fixed inductance (2.2µH in this case) and in series some resistance (radiation resistance Rr, and loss resistance which we will ignore). For simplicity, we are using an ideal transformer of some known turns ratio and a receiver.

We can consider the loop to have a Thevenin equivalent circuit of a voltage source with series equivalent impedance being Rr+j2πfL.

For this scenario, the loop is loaded with some impedance being the receiver input impedance transformed by the ideal n:1 turns ratio transformer. There is a large impedance mismatch, and the antenna system gain in this scenario is entirely due to mismatch loss, Gain=-MismatchLoss.

## Performance objective

One metric that could be used to assess performance is to calculate the SND of a receive system in a given noise environment.

To calculate SND, we need to refer internal and external noise to the same place to perform the calculation. In this instance, we will refer them to the receiver input terminals. We already have Tr=627, and we can calculate Tamr=Tam/Gain.

### Scenario: 0.5:1 transformer Above is the calculated SND from 0.3 to 15MHz using a 0.5:1 ratio transformer.

It is a small untuned / unmatched loop and the naked truth is that the SND is significant.

### Scenario: 1:1 transformer Above is the calculated SND from 0.3 to 15MHz using a 1:1 ratio transformer.

### Scenario: 0.2:1 transformer Above is the calculated SND from 0.3 to 15MHz using a 0.2:1 ratio transformer.

## Conclusions

The results apply to the scenario described. The results are sensitive to most parameters so they cannot be blindly applied to another scenario.