FWIW, the book
Antenna Theory, 3rd edition, by Constantine A. Balanis, givens in equation 5.14-a for the radiation resistance Rr of a loop as
Rr = 20 π
2 (C/λ)
4N
2 Ω
where λ is the wavelength, C is the circumference, N the number of turns. Sticking that in, for λ=2.08 m, diameter=225 mm, N=1 turn, one gets 2.52 Ω. So I have a nice set of conflicting results.
- Balanis 2.52 Ω
- HFSS 50 Ω
- VNA 119 Ω
The Balanis formula is being used outside the range of applicability, of C/λ < 0.1, where a constant current can be assumed But an example in the Balanis book gives the theoretical results for a loop of radius λ/25 with 1 and 8 turns, and gets 0.788 Ω for one turn and 50.43 Ω for 8 turns. In the book, λ/25 =0.04 λ. A loop radius of 0.04 λ means a circumference of 0.25 λ, so C/λ=0.25. My own antenna has C/λ=0.34.
I will make the loop smaller in HFSS, such that C/λ < 0.1 and see if the HFSS results agree with Balanis.
One thing Balanis does say is the loss resistance can be higher than the radiation resistance for a single turn, which is why multiple turns are preferable, as the radiation resistance goes up with the square of the number of turns, and the loss resistance roughly as the number of turns.
I assumed a perfect conductor in HFSS, but I'm using copper. I don't know if that could be the problem.
On thing I did notice on the VNA is that for low frequencies, below about 80 MHz, the resistance was very close to 0. That would be indicated by the Balanis formula, as at lower frequencies, C/λ would be less than 0.1.
Oh well, something to ponder over the weekend. Good job its only an amateur radio project, and not earning my living doing it !!
Dave