Toroid inductor correction factor

Question

Various equations and online calculators for the inductance of a wound toroid only include a parameter for the number of turns, not how the turns are wound (example). But the answer to this question, and page 4 of this Micrometals document, suggests that the spacing of the turns, and/or the angular distance around the toroid over which the turns occupy, can affect inductance by a significant amount (from a few percent to over 1.5x(!), without changing the number of turns.)

What is the physics behind this variance in inductance depending on turn spacing on a toroid? Why is this correction factor not including in (most?) canonical toroid inductance equations? Is there a formula that can be used to correct for this variance (e.g. an added parameter for turn spacing, or angular spread of all the turns, etc. in the inductance calculation)?

Or could it just be that the variation in parasitic capacitance due to turn spacing has affected the measurement of inductance?

Answer 1

What is the physics behind this variance in inductance depending on turn spacing?

If you had an air-core then turn-spacing is extremely significant. This is because, one turn might couple fairly well to the turns either side of it but, for sure, the turn at one end of a coil wound on a circular former (a la crystal set inductor) hardly couples at all to the turn at the other end. You reach the point where coil inductance tends to be linear with turns.

On the other hand, if you have a high-mu core then inductance is proportional with turns squared because, every single turn couples the field it produces to every other turn with very little loss.

So, at one end of the scale you have virtually no coupling between any of turns and you might as well regard this arrangement as a bunch of separate inductors in series.

At the other end of the scale you have virtually 100% coupling between all the turns and inductance is proportional to turns squared.

For a low-mu core, inductance might be proportional to \$N^{1.5}\$

Or could it just be that the variation in parasitic capacitance due to turn spacing has affected the measurement of inductance?

The theory is pretty-much established for what the inductance is; the problem is that it's hard to accurately predict inductance when permeability is moderate to low (as in air).

See also Coil Area vs Core Area - it discusses how inductance might be affected if the ferrite core cross sectional area is much smaller than the area taken-up by the turns (applies to HV transformers because we might need several mm gap between core and coil and between turns and other turns).

Answer 2

Something that might help, at least conceptually, is you can divide the inductor into the superposition of the air-core coil, plus the contribution due to the permeable core. Thus, roughly speaking, the spread-out coil is similar to a solenoid of much longer aspect ratio but the same turns (but not a solenoid, rather, one bent around into a toroidal helix), while the scrunched-up one resembles a short solenoid (with almost no bend, as it happens; well, perhaps, anyway!).

Then, when you add in the core's contribution, clearly it'll be more, but in a (1 + μx) sort of way, with x being some geometry factor, and μ the relative permeability. Clearly, that factor will be close to 1 for the evenly-distributed case, though how much less than 1 in the scrunched case, who knows.

This doesn't actually help analytically, as the coil itself is now massively harder to solve (it doesn't even have the axial symmetry of a helix to help out*), and the core at low μ isn't following any nice rules either (flux is hardly contained within the ring, leakage is comparable). This is basically the domain of, either shove it into FEM, build and measure it yourself, or pray to Thoth that someone's already done all the hard work for you and you can find their results in some obscure journal.

*The most accurate solenoid model starts with a helical sheath waveguide model, and goes from there, loading up with semi-theoretical and empirical correction factors to get an accurate result. See: http://hamwaves.com/antennas/inductance.html

Note also that, winding the coil in place, tightly, with the desired geometry, gives a slightly different result from scrunching a broader winding: the latter will be slightly larger diameter as the extra skew length is reduced.

Finally, mind what assumptions underlie the most common formulas. A uniform current sheet is usually used, or something similar to that. Real wires are not uniformly distributed over the winding area, nor are they perfectly transparent to magnetic fields. This will give a large divergence from the assumed ideal case, as only a tiny fraction of that winding area is actually occupied by wires, that are tilted significantly as they go.

Also, one more point of interest: note that, for the "100% width" case, even if the radius of the helix were shrunk to zero (ignore collision with the core), the winding makes a one-turn loop along the path of the core. This creates field lines perpendicular to the toroid plane (poloidal). This exists whether you have a core or not, turns or not, and arises simply from the fact that the winding goes around a loop path. It can be canceled out by doubling back on the winding (if you don't mind the added capacitance / transmission line effects), or starting with the wire tucked in along the core, then doubling back on that. (Why cancel it out? External fields: this can be relevant for EMI purposes, or say for reducing coupling between RF coils.)