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the question may seem simple but I haven't found any fitting formula yet. The problem is the following:

consider a single-turn, circular coil made of reasonably thin wire (diameter of the wire much smaller than diameter of the coil). What is the inductance of this coil, based on the coil radius, and the wire radius?

The answer could be either an analytical solution, a good approximation, or a formula with elliptic integrals (for instance)

I found the analytical formula for the magnetic vector potential of an infinitely thin coil on wikipedia (here), but I do not have a good software to integrate the elliptic integrals properly; and moreover, this formula does not cover finite wire radius.

To have some first trends, I checked the value of the magnetic flux of such a single-turn coil on a physics modeling software, and the value of the integral (with sufficiently resolved mesh) seems to depend strongly (enough) on the wire diameter (going from 4cm to 2cm diameter wire increased the magnetic flux by 20% for a 2m diameter loop, although we have 4cm<<2m!). And the other thing, that actually surprised me, is that, with a constant current value (1 amp), a bigger loop shows a smaller magnetic flux, although the area is much bigger. I think this last effect is linked to the fact that I used 130kHz frequency to calculate the flux integral, so this needs to be clarified further, I will update on that.

EDIT 8 May: this is an error because the program handled badly the unit conversion from 4 cm to 1 m. As one might expect, the flux, and the inductance, is actually proportional to the radius.

That's all I can tell. Does anyone have more insight on this?

Thank you very much!

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Here is a good start:

ON THE SELF-INDUCTANCE OF CIRCLES.

A quote from the text:

Various formulae have been given by different authors for the self-inductance of circles; that is, for closed rings of circular cross section. Some of these formulae are at once convenient and accurate, while others are both inconvenient and unreliable, and should be avoided in numerical calculations. We therefore propose in this paper to critically examine and test these various formulae, and to show which of them are trustworthy and which are wrong. This seems the more necessary inasmuch as some of the latter have been given by writers of reputation and they have been quoted and used in the belief that they were correct.

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  • $\begingroup$ Great, this is exactly the kind of paper I was looking for. As expected by the canonical nature of the problem, the paper is quite old...but it looks correct! I will update if I can estimate the discrepancies between these formulas and the calculations with a good enough mesh! Thank you for your answer :-) $\endgroup$ – MrBrody May 8 '13 at 18:45
  • $\begingroup$ This does not actually provide an answer, only a link to an answer. The quotation by itself is not useful. $\endgroup$ – sammy gerbil Jul 28 '18 at 11:51
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You can find a standard coil inductance calculator here: http://deepfriedneon.com/tesla_f_calchelix.html You can specify a single turn of any diameter wire and coil diameter and the result is fairly accurate (I have tested it in practice and found it to be within 2% of the results obtained with an inductance meter). The formula for calculation is also shown.

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