# Single turn coaxial loop resonator analysis

Recent discussion online of a purported commercial HF small transmitting loop (STL) was challenged in analysing the structure, questioning whether such a connection was ‘correct’.

The STL used a main loop resonator and a separate small auxiliary loop for the 50Ω feed, a very common arrangement.

The main loop is a coaxial cable with, in this case, a tuning capacitor inserted between the inner conductors at each end. Above is a diagram of the main loop.

This technique appears to be employed in some commercial devices and some DIY STL designs that may have been inspired by pics of the commercial ones.

For discussion purposes, lets consider that it is a circular loop of perimeter 3m of RG213 coax, the inductance between shield ends is 2.9µH.

## How does it work?

The first important concept is that of skin effect in coaxial cables, and its influence on setting TEM mode propagation inside the coax. Readers might benefit from the article Small single turn un-tuned shielded loop which explains some relevant concepts.

This antenna is intended for HF, and at those frequencies with good coaxial cable, skin effect is sufficiently well developed that we can consider it to be fully effective.

In the presence of a HF electromagnetic field, a voltage will be induced in the conductor formed by the outer surface of the outer conductor, and some voltage V will appear between the shield ends at the gap.

That voltage V appears across three components in series, the left hand differential mode impedance of the coax, the tuning capacitor, and the right hand differential mode impedance of the coax.

Note that the polarity of the differential mode voltages is opposite. For that reason, the differential mode voltage half way around the coax (ie opposite to the gap) is zero even though current flows at that point. The impedance looking into each differential mode pair of terminals is that of a S/C stub, and of half of 3m in length.

At 7MHz, the impedance of a 1.5m S/C stub of RG213 is 0.266+j17.32Ω, and that of the 2.9µH loop is j128Ω plus some resistance comprising the radiation resistance Rr, ground loss resistance Rg, capacitor loss resistance Rc, plus some loss resistance Rco of the external surface of the coax.

Voltage V is applied to a total impedance of 2*(0.266+j17.32)+j128+Rr+Rg+Rc+Rco Ω.

A sobering though is that in this instance where Rr is of the order of 0.005Ω, the total stub resistance of 0.53Ω consumes a hundred times as much power as is actually radiated.

So this arrangement has increased the inductive reactance of the loop by 27% in this case so it tunes lower in frequency with a given capacitor, but it has been at a cost of an additional 0.53Ω of loss resistance. The effect on radiation efficiency depends on the other loss elements and is best evaluated by measurement or inference from a bandwidth measurement (using the corrected total inductance).

Buyers of these things are not usually concerned about radiation efficiency, indeed they prefer wider bandwidth which is easily achieved by compromising efficiency.

## Different coax

The calcs above were for a given length of RG213 coax at 7MHz. Changing any of these parameters changes the scenario and the results are different. For example, the same length of a ‘foam RG213’ like LMR400 has the same outside diameter, but the impedance looking into each 1.5m S/C stub is 0.19+j13.24Ω.

## Hidden shorts

Beware that some implementations may achieve connection of inner to outer conductor in both of the connectors on the loop cable, and making it a ‘normal’ loop with tuning C connected between shield ends.

## Conclusions

Don’t assume that STL using coax cable for the main loop directly connect the tuning capacitor between ends of the outer conductor, some use the connection described above and it needs different analysis.

The structure will work. It increases inductive reactance in the loop circuit and increases loss resistance which will in most cases contribute to reduced efficiency. If you have one of these you might explore shorting the inner to outer at each coax jack for improved efficiency, though it will change the tuning, and you may need a slightly longer loop.

The outside surface of the outer conductor of the loop coax is the antenna per se, claims that the inner conductor is the antenna and is shielded from electric field by the outer conductor are unsound, naive, and based on archetypal ham myths.

Behaviour is explained by traditional basic linear circuit analysis and transmission line concepts.

## References

Small single turn un-tuned shielded loop