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: measuring inductance with a meat-and-potatoes DMM

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.

Originally Posted by 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.

[Joe Gwinn]

Originally Posted by 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.

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]

Originally Posted by Joe Gwinn

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]

Originally Posted by Mike Sulzer

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]

Originally Posted by Joe Gwinn

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]

Originally Posted by Mike Sulzer

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]

Originally Posted by Joe Gwinn

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.

Originally Posted by 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.

[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]

Originally Posted by David Schwab

...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.

[Joe Gwinn]

Originally Posted by 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.

The cause of skin depth is eddy currents. In this case the variation of wire AC resistance with frequency is eddy currents in the wire itself.

In electromagnetics, all things that are possible are required, and all will happen to some degree. In most practical situations , only a few of the many possible effects are large enough to matter, and it is these few effects that each practical design concentrates on.

In guitar pickups, eddy currents in the very thin wire used for winding are not important, but eddy currents in nearby pieces of metal (like magnets, slugs, baseplates, and covers) are quite important.

So the AC resistance of a pickup coil is due to energy losses caused by eddy currents in that nearby metal, and is not due to eddy currents in the wire (which does however contribute its DC resistance).

[David Schwab]

Originally Posted by Joe Gwinn

So the AC resistance of a pickup coil is due to energy losses caused by eddy currents in that nearby metal, and is not due to eddy currents in the wire (which does however contribute its DC resistance).

I know you were talking to Mike but... That's what I thought.

But does that decrease inductance? I had never heard about that effect.

[Joe Gwinn]

Originally Posted by David Schwab

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

Exactly. Yes to both.

Take an iron-cored coil. In the absence of eddy currents, the entire mass of iron would be available to increase the flux through the coil, yielding a proportionate increase in inductance. With eddy currents in that core, the AC magnetic field from the coil is to some degree pushed out of the core, so there is less field for a given coil current, and thus less inductance.

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

The AC resistance is a consequence of the finite resistance of the core material through which the eddy current flows. This is best explained by comparing some extreme cases:

Iron core with superconducting lead plating, in liquid helium. The iron will not increase the magnetic flux because the eddy currents in the superconducting surface layer will totally exclude magnetic flux from the iron interior. The inductance will be much reduced, but the AC resistance will not change because eddy currents in a superconductor are lossless.

Ferrite core. No eddy currents so no flux expulsion, so inductance goes up (compared to air), and the AC resistance (excess over DC resistance of the coil) will only go up a tad, due to the magnetic hysteresis of the ferrite material.

Plain old iron core. The higher the frequency the larger the eddy currents in the iron, the greater the expulsion of flux from the core, the lower the inductance, and the greater the excess AC resistance due to eddy current losses in the core.

[RedHouse]

Originally Posted by Joe Gwinn

...In guitar pickups, eddy currents in the very thin wire used for winding are not important, but eddy currents in nearby pieces of metal (like magnets, slugs, baseplates, and covers) are quite important....

The tone, and attack changes slightly, this we can work around because we know it's there. Sometimes we are making a full-meal-deal out of eddy currents when they really should be more of a side-note.

We can use (or not) metal covers and back plates, a discussion of the types of metal used for covers and back plates seems much more useful area to the field of pickup making IMHO.

Speaking of back plates, isn't it funny how if we add a backplate to a Tele bridge pickup (or Jaguar s-c) it gets more treble, but when we cover a pickup with a metal cover between the pickup and strings it tends to loose some treble.

(yes I know, focusing the field etc.)

[Joe Gwinn]

Originally Posted by RedHouse

Speaking of back plates, isn't it funny how if we add a backplate to a Tele bridge pickup (or Jaguar s-c) it gets more treble, but when we cover a pickup with a metal cover between the pickup and strings it tends to loose some treble.

(yes I know, focusing the field etc.)

It's actually not focusing, it's a form of electromagnetic shielding:

Consider a simple pickup without baseplate or cover. The motion of the strings generates an AC magnetic field, which then induces a AC voltage in the nearby pickup coil. Now, introduce a thin sheet of brass.

If we put the sheet between strings and coil, some part of the AC field will be reflected away from the coil, reducing the AC field at the coil and thus reducing the AC voltage from the coil.

If instead we put sheet behind coil, the reflection will increase the AC field at the coil.

In both cases, the amount of reflection increases with the square root of frequency.

If one has a coil between a thin cover and a thick baseplate, the net effect will be some complicated kind of balance between the above two effects.

If one uses nickel silver or stainless steel, eddy currents are much reduced over those in brass, so the reflections are all proportionately reduced.

[Mike Sulzer]

Originally Posted by Joe Gwinn

The cause of skin depth is eddy currents. In this case the variation of wire AC resistance with frequency is eddy currents in the wire itself.

Electromagnetic induction causes eddy currents. Electromagnetic induction causes skin depth. I think it is somewhat misleading to say that eddy currents are the cause of skin depth. I prefer:

"The effect is caused by electromagnetic induction in the metal which opposes the currents set up by the wave E-field..."
--http://www.fas.harvard.edu/~scdiroff/lds/ElectricityMagnetism/SkinDepth/SkinDepth.html

[Mike Sulzer]

Originally Posted by Joe Gwinn

If we put the sheet between strings and coil, some part of the AC field will be reflected away from the coil, reducing the AC field at the coil and thus reducing the AC voltage from the coil.

The amount that is reflected at audio frequencies is very small. If the reflection at audio frequencies were large enough to matter, electromagnetic effects would be large enough so that we would not need coils with 5000 or more turns. (High permeability cores help, too.)

[Joe Gwinn]

Originally Posted by Mike Sulzer

The amount that is reflected at audio frequencies is very small. If the reflection at audio frequencies were large enough to matter, electromagnetic effects would be large enough so that we would not need coils with 5000 or more turns. (High permeability cores help, too.)

It has the effect of shaping the tone, and this is exactly what people hear.

The theory of shielding is laid out in some detail in the literature, and I gave some references in Someone having fun with magnets thread. From posting #19, bottom:

The theory is clearly explained in Edward F. Vance, “Coupling to Shielded Cables”, Wiley 1978, 183 pages. Reprinted by Krieger in 1987. The same theory applies to metal boxes.

If this isn't deep enough, the fundamental source is "Electromagnetic Waves”, S.A. Schelkunoff, Van Nostrand 1943. The foundations of the theory of electromagnetic shields are set forth in §8.18 “Shielding Theory”. This is the basic reference, appears in almost all bibliographies of articles on shielding, and can be heavy going for those not having had a course in E&M Field Theory.

[Mike Sulzer]

See here: http://www.naic.edu/~sulzer/coverEffects.png

It affects the pickup circuit. The effect shown in the figure is that it reduces the height of the peak. That is, losses from the eddy current induced in the cover from the current in the coil extract energy from the resonance, lowering the Q. Note that the peak frequency is also reduced a little bit. This is also consistent with lowering the Q.

The eddy currents do not lower the inductance. If they did, the resonant frequency would rise.

[Joe Gwinn]

Originally Posted by Mike Sulzer

See here: http://www.naic.edu/~sulzer/coverEffects.png

It affects the pickup circuit. The effect shown in the figure is that it reduces the height of the peak. That is, losses from the eddy current induced in the cover from the current in the coil extract energy from the resonance, lowering the Q. Note that the peak frequency is also reduced a little bit. This is also consistent with lowering the Q.

The eddy currents do not lower the inductance. If they did, the resonant frequency would rise.

How accurately can your setup measure inductance, and how do verify this?

What is the cover made of and how thick is it?

Try a thick piece of brass or copper.

[Mike Sulzer]

Originally Posted by Joe Gwinn

How accurately can your setup measure inductance, and how do verify this?

What is the cover made of and how thick is it?

Try a thick piece of brass or copper.

That was an old measurement before the I-V. Bit the effect of the loss dominates over the change in inductance. Otherwise the peak would rise.

We cannot directly measure the inductance at resonance anyway because of the capacitance. One must model the circuit. But the lack of rise in the resonant frequency is all we need to know.

It was thin; I do not remember the material.

I should do that; but it need not be much thicker than the skin depth.

[Mike Sulzer]

Originally Posted by Joe Gwinn

It has the effect of shaping the tone, and this is exactly what people hear.

You may be exaggerating how much conductors (as opposed to ferromagnetic materials) can shield out magnetic fields varying at audio frequencies. I think the tonal effects are adequately explained by losses at the high impedance resonance of the coil and cap.

Do you have calculations or measurements that unambiguously show that induced currents in thin conductors such as pickup covers can shield at audio frequencies?

[David Schwab]

Originally Posted by RedHouse

The tone, and attack changes slightly, this we can work around because we know it's there. Sometimes we are making a full-meal-deal out of eddy currents when they really should be more of a side-note.

Where you really hear it is in the design of bass pickups. Eddy currents can kill the high end quickly. Modern bass pickups often have more high end than most guitar players would want.

We can use (or not) metal covers and back plates, a discussion of the types of metal used for covers and back plates seems much more useful area to the field of pickup making IMHO.

The less conductive the material, the better if you want to retain high end.

Speaking of back plates, isn't it funny how if we add a backplate to a Tele bridge pickup (or Jaguar s-c) it gets more treble, but when we cover a pickup with a metal cover between the pickup and strings it tends to loose some treble.

Tele and Jaguar pickups use steel back plates, which increase the inductance. It becomes part of the magnet circuit. You will hear the pickup get louder when you stick a piece of steel behind the magnets. This is especially true with non metallic magnets, like ceramic or neo.

Pickup covers are a different story, they aren't magnetic, and its the eddy currents in the cover that change the tone. Once again, the less conductive the material, the better if you want to retain high end.

[RedHouse]

Originally Posted by David Schwab

....Modern bass pickups often have more high end than most guitar players would want....

This I agree with. I've been learning this lesson winding (and rewinding) pickups for my own bass. What works for guitar doesn't work so well for bass.

[Joe Gwinn]

Originally Posted by Mike Sulzer

That was an old measurement before the I-V. Bit the effect of the loss dominates over the change in inductance. Otherwise the peak would rise.

Resonant frequency varies as the square root of inductance, and the voltage peak of a low-Q resonator is quite broad, so it's likely that the apparatus was simply not sufficiently precise to detect the change. A Maxwell-Wein bridge can easily detect such changes, as can an Extech.

On 25 October 2005, having just wound a drive coil (575 turns of #32 wire on a StuMac pickup bobbin), I made a few Extech measurements:

Rdc= 46.36 ohms (measured with a TX3 DMM, not the Extech)

L1KHz(air)= 16.199 mH
R1KHz(air)= 46.18 ohms

The above in air, with no metal nearby. Then, I set the coil on a sheet of 0.090" thick aluminum (6061 alloy T6 temper) plate:

L1KHz(metal)= 14.254 mH
R1KHz(metal)= 55.12 ohms

So, the inductance change is 16.199/14.252= 1.1365:1, or 14%.

I've made other measurements, but have not found my notes yet. Copper is a better conductor than 6061T6 aluminum, and has a somewhat larger effect.

We cannot directly measure the inductance at resonance anyway because of the capacitance. One must model the circuit. But the lack of rise in the resonant frequency is all we need to know.

One certainly can measure the inductance at resonance. At zero phase, the capacitive reactance is equal to the inductive reactance. One measures the unknown self-capacitance of the coil by measuring zero-phase resonant frequency with a series of capacitors of known value in parallel with the coil. The method is in Terman.

It was thin; I do not remember the material.

Then it's probably nickel plated brass, maybe 0.020" thick.

I should do that; but it need not be much thicker than the skin depth.

At audio frequency, the skin depth is large, and the thickness of such things as covers is a fraction of the skin depth. If the covers were not thinner than the skin depth, no music would get through to the coil.

The skin depth is where the signal voltage is 1/e of the incident voltage. This is a 8.7 dB loss, which is easily audible.

[Joe Gwinn]

Originally Posted by Mike Sulzer

You may be exaggerating how much conductors (as opposed to ferromagnetic materials) can shield out magnetic fields varying at audio frequencies. I think the tonal effects are adequately explained by losses at the high impedance resonance of the coil and cap.

At audio frequencies, the shielding effect of non-ferromagnetic metals is partial, giving rise to tone shaping versus total exclusion. The human ear is very sensitive to such shaping. Resonance also affects tone, but this is independent, and both effects act at once.

One can vary the resonant peak without affecting eddy currents by putting capacitors in parallel with the coil. For measuring eddy current effects independent of resonance effects, it's helpful to do eddy current tests at one tenth of the resonant frequency, which is naturally the case if one measures a pickup that resonates at 10 KHz using a 1 KHz test signal.

Do you have calculations or measurements that unambiguously show that induced currents in thin conductors such as pickup covers can shield at audio frequencies?

The math was worked out during and before WW2, and the results were widely published after WW2. I would suggest reading Vance, which is short and fairly well written.

Actually, I may have some shielding experiment data as well. I recall doing this a few years ago. I'll look.

It's pretty simple to make such measurements with a signal generator, two coils, an oscilloscope, and a collection of metal sheets. Brass shimstock assortments are quite useful because they make varying the thickness easy.

[Possum]

Actually a tele baseplate LOWERS the inductance readings. So do covers. This is why I don't pay a huge amount of attention to Henry readings, AC resistance is a more logical thing to look at what is happening to the audio frequencies.

[RedHouse]

{whups, double posts}

[RedHouse]

Originally Posted by Possum

...AC resistance is a more logical thing to look at what is happening to the audio frequencies....

What frequency(s) are you looking at?
(Extech freq's?)

[Mike Sulzer]

Originally Posted by Joe Gwinn

Resonant frequency varies as the square root of inductance, and the voltage peak of a low-Q resonator is quite broad, so it's likely that the apparatus was simply not sufficiently precise to detect the change. A Maxwell-Wein bridge can easily detect such changes, as can an Extech.

The peak frequency went down; that is visible on my plot. If both types of effects, change in Q, and change in inductance are there, then the change in Q was the dominant effect.

Originally Posted by Joe Gwinn

On 25 October 2005, having just wound a drive coil (575 turns of #32 wire on a StuMac pickup bobbin), I made a few Extech measurements:

Rdc= 46.36 ohms (measured with a TX3 DMM, not the Extech)

L1KHz(air)= 16.199 mH
R1KHz(air)= 46.18 ohms

The above in air, with no metal nearby. Then, I set the coil on a sheet of 0.090" thick aluminum (6061 alloy T6 temper) plate:

L1KHz(metal)= 14.254 mH
R1KHz(metal)= 55.12 ohms

So, the inductance change is 16.199/14.252= 1.1365:1, or 14%.

I've made other measurements, but have not found my notes yet. Copper is a better conductor than 6061T6 aluminum, and has a somewhat larger effect.

Let's look at this measurement with some skepticism. The Extech measures the complex impedance. (It interpreted that measurement as an inductance and resistance in series because you told it to.) There are an infinite number of circuits that have the complex impedance that it measured. A simplified model of the situation you have measured is a poorly coupled transformer, which can be represented as a perfectly coupled transformer with a different turns ratio with a leakage inductance in series with the secondary. A resistor is connected across the secondary. The series inductance and resistance can be transformed back to the primary so that we have a series resistance and inductance across the coil. If that resistance is very small and the inductance very large, then the parallel combination of the two inductors would be what you measured. This might be correct situation or maybe not.

In the case of the humbucker pickup that I modeled, this interpretation is certainly not correct. The value of the series resistance is much greater the the inductive reactance at lower frequencies, allowing the resistance to load the pickup. At higher frequencies, the inductance reactance increases and the coil is partially unloaded, allowing the resonant peak to occur.

Please think about this carefully; I believe you are missing the point of the measurements I have made and interpreted. Remember, you are doing what I am doing: assuming a model. That is what happens when you put the Extech in a particular mode. In the case of the pickup, I think you are assuming the wrong model.

Originally Posted by Joe Gwinn

One certainly can measure the inductance at resonance. At zero phase, the capacitive reactance is equal to the inductive reactance. One measures the unknown self-capacitance of the coil by measuring zero-phase resonant frequency with a series of capacitors of known value in parallel with the coil. The method is in Terman.

Yes, that is such an indirect method.

Originally Posted by Joe Gwinn

Then it's probably nickel plated brass, maybe 0.020" thick.

At audio frequency, the skin depth is large, and the thickness of such things as covers is a fraction of the skin depth. If the covers were not thinner than the skin depth, no music would get through to the coil.

The skin depth is where the signal voltage is 1/e of the incident voltage. This is a 8.7 dB loss, which is easily audible.

But it is hard to separate the effects of coil loading from transmission loss, since both are frequency dependent.