How frequency-dependent is an air inductance?

Question

For a winding on a magnetic material, the value of the inductance is greatly impacted and is also frequency-dependent given the behavior of the magnetic material according to frequency. My question is, for a winding with no magnetic material, does the inductance vary with frequency? If yes(*), how can I explain it?

(*) I am more inclined to say that the permeability of the air does not vary with frequency. I am not sure though..

well, I don't have a single electronic device even close to the wavelength of that, so in terms of waves,it is! — Marcus Müller, Jun 24 '21 at 14:03
@MarcusMüller Oh for electronic devices, ok. What about the winding of an electric machine when no magnetic material is inserted? — Wallflower, Jun 24 '21 at 14:09
@Wallflower I've never seen an electric machine that operates anywhere close to that high of a frequency! Things won't start changing significantly until you're looking at coils comparable in size to a wavelength of your frequency of interest, a situation you're likely to encounter when looking at antennas and microwave electronics and basically nowhere else. — Hearth, Jun 24 '21 at 14:29

LorenzoDonati4Ukraine-OnStrike · Accepted Answer · 2021-06-24T14:27:43.550

When frequency rises both the inductance and the resistance of any length of wire changes, even if there's no magnetic core (dry air at normal pressure has almost the same permeability of vacuum).

This is due mainly to skin effect and proximity effect.

There is a math-heavy explanation of the phaenomena here: The Influence of Frequency upon the Self-Inductance of Coils - by J.G.Coffin, Clark University, Worcester, Mass. (this is a scan of an old document, but references basic EM theory).

Excerpts:

When currents of low frequency pass through the wires of a coil, the current distributes itself uniformly over the cross sections of the wires. With increasing frequency,this uniform current density no longer prevails, but, as is well known, at least for straight wires, the current density becomes greater at the surface of the wire at the expense of that of the interior. The corresponding lines of magnetic force become differently distributed, and in consequence the self-inductance suffers a change. A short calculation will show the direction and amount of the change for circuits in which the curvature of the wire may be assumed negligible, and the theory derived for straight wires used. The theory of this distribution of the current density in straight wires, which has been thoroughly worked out by Lord Rayleigh and by Stefan, is not applicable without modification to the distribution of current density in coils of wire. The following argument shows that the effect of increasing frequency is to diminish self-inductance.

Here is another relevant article.

Excerpts:

Whenever you alter the path of current, you alter the inductance. Because the skin effect modifies the distribution of current within the conductor, it must also change the inductance of that conductor. You can observe this in very careful measurements of transmission-line inductance at high and low frequencies.

[...]

POINTS TO REMEMBER

The distribution of current at high frequencies minimizes inductance.

At DC, the path of least DC resistance creates a slightly higher inductance.

Good models for skin effect take into account changes in both resistance and inductance with frequency.

Thank you for your response. So basically, in this case the inductance variation takes into account skin and proximity effects, meaning there is no need to use a resistance to model them right? — Wallflower, Jun 24 '21 at 13:58
@Wallflower The answer is not related only to the absolute frequency, but it depends also on the diameter of the wire (and how it is wound, for proximity effect). Resistance variation may or may not be relevant, depending on the application. Even at 50Hz resistance increase due to skin effect can impact efficiency of mains power distribution. — LorenzoDonati4Ukraine-OnStrike, Jun 24 '21 at 14:05
Oh I got it, so you basically are not negating the fact that inductance of an air winding varies with frequency, relatively, depending of the geometry of the winding as well as the inter-turn spacing, right? — Wallflower, Jun 24 '21 at 14:07
@Wallflower. Exactly. Resistance and self- and mutual- inductances depend on the specific paths of currents in the inductor. Since these paths change depending on frequency (skin and proximity effects), these parameters (R,L,M) also change with frequency, and the variation also depends on the geometry of the inductor, since skin and proximity effects depend also depend on that geometry. — LorenzoDonati4Ukraine-OnStrike, Jun 24 '21 at 14:13

score 2 · Answer 2 · answered Jun 24 '21 at 13:43

2

To first order, the inductance will not vary significantly with frequency, as the magnetic properties of air are essentially frequency independent.

If you connect the inductor to an inductance measuring device and sweep the test frequency, then the measured inductance (the imaginary part of its susceptance) will rise as the effect of the coil's self capacitance cancels some of the inductor's susceptance. Eventually, at the self resonant frequency (SRF), the measured impedance will be very large.

There are other minor contributions to inductance change with frequency, as the skin effect and the proximity effect change the effective geometry of the current flowing in the coil.

answered Jun 24 '21 at 13:43

Neil_UK

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Suppose I have a geometry of a winding on air and I am using FEM to compute the values of the inductance and resistance of each turn using magneto-dynamic formulation. For the inductance, each turn is set to 1A and the inductance is deduced from the following expression: E_m(f)= 1/2.L(f).i^2 are skin and proximity effect taken into account in the inductance? Because the results of the simulation yield a frequency-dependent inductance and I cannot justify this... – Wallflower Jun 24 '21 at 13:49
@Wallflower - "Because the results of the simulation yield a frequency-dependent inductance and I cannot justify this..." Don't forget parasitic capacitance. – Antonio51 Jun 25 '21 at 09:26
It is only a simulation. It allows the computation of the value of the inductance using finite element method by resolving the magneto-dynamic formulation, so no parasitic capacitance are involved, or are they? I would have been more keen to agreeing with you had it been the result of a measurement... – Wallflower Jun 25 '21 at 11:34
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Ok. (google translate). The question is: "To what extent is an air inductor dependent on the frequency?". But how often, until what frequency ? In fact, the value of the choke decreases when the frequency increases for many reasons (skin effect, proximity of the turns, proximity of metal, parasitic capacitance, "radiation", perhaps humidity ?, or in general the gaseous composition of the air?, etc ...). This component can even become a capacitor too ... The parasitic capacitance is generally evaluated by a "grid dip meter" or a measuring bridge of "quality coefficient Q" of the choke. – Antonio51 Jun 26 '21 at 07:54
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I think "capacity" can also be calculated by "finite elements" as inductance is. Short antenna equivalent capacitance is calculated as "static" by "simple integral" old electric method definition. – Antonio51 Jun 26 '21 at 07:58
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Ok. I see in a paper that "displacement current term in Maxwell equation" are generally neglected . – Antonio51 Jun 26 '21 at 08:11

How frequency-dependent is an air inductance?

2 Answers2