Under what conditions does the inductance vary with frequency?

[Mike Sulzer]

There are some unsettled questions regarding the variation of inductance with frequency, and so here is the beginning of a new discussion on this topic. First consider a definition of inductance. This is a translation of of equation 5.157 of the third edition of Jackson into words:

The inductance L of a loop of wire (possibly a multiple turn coil) carrying a current I is found by making the sum of the amplitude squared of the magnetic field (divided by the permeability) in each small volume of space. This sum is then divided by the square of the current.

The sum must be taken over the space inside and outside of the wire. This means that if the distribution of current in the wire changes, the inductance changes. It is well known that current tends to flow near the surface of a wire, not the center, at high frequencies. This is characterized by the "skin depth", which decreases with increasing frequency. Thus it is reasonable to suppose that an inductor using heavy wire would have an inductance that changes with frequency as the distribution of the current is shifted more towards the outside of the wire.

The skin depth of copper is 8.47 mm at 60 Hz and .66 mm at 10 KHz. The diameter of wire used in pickups is much less than .66 mm. Therefore, the inductance of the coil does not change with frequency due to this effect. This does not exclude the possibility that the inductance might change as a function of frequency due to mutual impedance effects with some other conductor(s).

I believe that this is what Joe Gwinn intended when he wrote this in post 52 of this discussion: http://music-electronics-forum.com/t14110-2/:

This cannot be a new problem, so I did a little research, and found the answer in the classic textbook, "Classical Electrodynamics", 2nd edition, J.D. Jackson, Wiley 1975. On page 298, the last sentence of the paragraph just below equation (7.77) is the answer (taken from a discussion of eddy currents and skin depth):

"One simple consequence [of eddy currents] is that the high-frequency inductance of circuit elements is somewhat smaller than the low-frequency inductance because of the expulsion of flux from the interior of the conductors."

Note that [of eddy currents] was inserted by Joe. I have read that section of Jackson, and I think that Jackson is describing the effect that I just explained above, that is, the skin effect due to current in the inductor coil itself, not due to eddy currents in another conductor. That is a more complicated question. I believe that I have already answered the question for pickups, but others do not agree, and I think the issue is worth discussing.

[Joe Gwinn]

The part of Jackson that I quoted was as you suspect discussing eddy currents in the wire, but any nearby metal will do its bit in much the same way.

The page in Terman that you mentioned in turn referenced (in a footnote) an article by Legg that I found very informative.

There is a large literature on the effect of "eddy current shielding" and its effect on inductance coils near metal. The math may be best done in the literature on how electromagnetic shields work. I cited some good references in one of our threads, but I don't recall the details, but a search on electromagnetic should pull it up.

[Mike Sulzer]

Joe, I do not understand why you say "eddy currents in the wire". Jackson is discussing the distribution of the current in the inductor, that is, the coil of wire or in some cases a bar or rod, or whatever. I think of eddy currents as currents induced in a second conductor to to mutual inductance effects with the first conductor.

The effect of eddy currents in a conductor near an inductor is quite complicated. As you wrote, the literature is large. In this discussion, I hope we can cover a general model of how it works with some specific application to pickups.

The part of Jackson that I quoted was as you suspect discussing eddy currents in the wire, but any nearby metal will do its bit in much the same way.

The page in Terman that you mentioned in turn referenced (in a footnote) an article by Legg that I found very informative.

There is a large literature on the effect of "eddy current shielding" and its effect on inductance coils near metal. The math may be best done in the literature on how electromagnetic shields work. I cited some good references in one of our threads, but I don't recall the details, but a search on electromagnetic should pull it up.

[Joe Gwinn]

Joe, I do not understand why you say "eddy currents in the wire". Jackson is discussing the distribution of the current in the inductor, that is, the coil of wire or in some cases a bar or rod, or whatever.

Because there are eddy currents in the wire itself, independent of eddy currents in nearby masses of metal. The typical effect of eddy currents in the wire is to force AC currents to flow near the surface, and for a large enough wire carrying a high enough frequency, the current is confined to a thin surface layer on the wire.

Mitigation of this effect is the purpose of litz wire: Litz wire.

I think of eddy currents as currents induced in a second conductor to to mutual inductance effects with the first conductor.

This also happens, but one can have eddy current effects in a straight wire isolated from everything else.

More generally, every changing field induces currents in everything conductive, including the conductors carrying the varying currents that generate the original varying fields.

[Mike Sulzer]

Because there are eddy currents in the wire itself, independent of eddy currents in nearby masses of metal. The typical effect of eddy currents in the wire is to force AC currents to flow near the surface, and for a large enough wire carrying a high enough frequency, the current is confined to a thin surface layer on the wire.

That kind of makes makes sense, but I have never before heard the term used this way. For instance, that section of Jackson you have referred to, 7.7 in the second edition, derives the skin depth from a wave propagation point of view without mentioning eddy currents. Describing the absence of current near the center of the wire as the result of an eddy current seems physically cumbersome to me. We have a current in the wire flowing near the surface. Is this the eddy current? Are all ac currents eddy currents? If so, the term seems to have no particular significance.

[Joe Gwinn]

That kind of makes makes sense, but I have never before heard the term used this way. For instance, that section of Jackson you have referred to, 7.7 in the second edition, derives the skin depth from a wave propagation point of view without mentioning eddy currents. Describing the absence of current near the center of the wire as the result of an eddy current seems physically cumbersome to me.

Well lots of people use the eddy-current analytical approach, but perhaps Jackson also thinks it clumsy.

There are many approaches, some equivalent, and some representing approximations to more complicated exact models.

We have a current in the wire flowing near the surface. Is this the eddy current?

I don't picture the situation, but the name eddy current came from the analogy with eddys in fluid flow. The math may be somewhat similar as well.

Are all ac currents eddy currents? If so, the term seems to have no particular significance.

All eddy currents are AC (perhaps of very low frequency), but not all AC currents are eddy currents.

[Mike Sulzer]

All eddy currents are AC (perhaps of very low frequency), but not all AC currents are eddy currents.

I prefer to use terminology directly related to solutions of Maxwell's equations.

[Joe Gwinn]

I prefer to use terminology directly related to solutions of Maxwell's equations.

One can certainly do that, but be aware that the literature is not uniform on this, especially where the theory is being applied in practice, where it is known which theoretical effects predominate and which theoretical effects may safely be neglected, and the resulting simplification is substantial.

[Mike Sulzer]

One can certainly do that, but be aware that the literature is not uniform on this, especially where the theory is being applied in practice, where it is known which theoretical effects predominate and which theoretical effects may safely be neglected, and the resulting simplification is substantial.

Yes, that is exactly how one approaches the solution.

[Joe Gwinn]

But to get back to the original question, while by Maxwell's Equations eddy currents are no different than any other current induced by a changing magnetic field, in the jargon of the industry not all induced currents are eddy currents. The basic distinction is topological:

Eddy currents flow in loops within a solid piece of metal, with paths unconfined except by the surfaces of the piece.

Currents that flow in a path determined by the topology of a conductor are not considered eddy currents. So, the currents in a shorted turn around a transformer core are not considered eddy currents.

Likewise, circular currents (eddies) in a body of water are considered eddy currents, while even circulating flow in pipes is not.

The distinction is practical as well, because the math of unconfined currents is more complex than that constrained to flow in wires or pipes.

[Mike Sulzer]

Yes, that is the way I understand it. Therefore, I do not classify the skin depth-limited current flow in the coil of a high frequency inductor as an eddy current. I guess I misunderstood you when I thought you wrote above that it is an eddy current.

But to get back to the original question, while by Maxwell's Equations eddy currents are no different than any other current induced by a changing magnetic field, in the jargon of the industry not all induced currents are eddy currents. The basic distinction is topological:

Eddy currents flow in loops within a solid piece of metal, with paths unconfined except by the surfaces of the piece.

Currents that flow in a path determined by the topology of a conductor are not considered eddy currents. So, the currents in a shorted turn around a transformer core are not considered eddy currents.

Likewise, circular currents (eddies) in a body of water are considered eddy currents, while even circulating flow in pipes is not.

The distinction is practical as well, because the math of unconfined currents is more complex than that constrained to flow in wires or pipes.

[David Schwab]

So... do eddy current lower the inductance? And if so, is that due to the opposing magnetic field produced by the eddies?

My understanding was that eddy currents effect the high frequencies more, which I assumed was due to AC resistance. I don't know how the magnetic field would tie into this.

What do you gentlemen think about this?

[RedHouse]

...What do you gentleman think about this?...

Well I'm no gentleman and being that as it is, one thing about eddy currents that gets ever tiring to me is the constant, ongoing argumentation about the minutiae involving eddy currents in pickups ...ad infinitum ...ad nausium.

It is what it is, we know it's there, let's move on.

(maybe we should create a sperate forum for eddy currents where Joe and Mike and anyone else can argue this out indefinately?)

...My understanding was that eddy currents effect the high frequencies more...

Yep, pretty much that simple for what we're doing here, Wheuuoow! glad that's solved now we can move on.

[David Schwab]

Yeah, but Brad, it's nice to know about in case you want to design around it. Some builders either use materials or methods that will resist eddy currents, or add eddy current producing parts or materials to the design.

Obviously if all someone does is copy Fender and Gibson pickups, then it doesn't matter. But some of us actually try new things!

[RedHouse]

...Yeah, but Brad, it's nice to know about in case you want to design around it. ...

Oh I so agree David, I wasn't knocking your post, I was pointing out that this eddy current thing seems to linger on ...and on ...like a bad fart.

The argueing about definitions of eddy currents, quotations of physics writings etc. We should be reducing this to it's practical applications in our field of interest (pickups) rather than expanding it out. To throw another well know quote: "everything tends toward entropy" and IMHO we're approaching it with this eddy current discussion being sprinkled around in every other thread.