Under what conditions does the inductance vary with frequency?

[Mike Sulzer]

A toroidal transformer can be built with essentially zero leakage inductance, and yet shows eddy current variation if the frequency is too high for the thickness of the laminations. .

Now that is a very good point. But is that the dominant effect in a humbucker pickup, which has very large leakage reactance? To find out, we need to do some calculations using equation 31 in Legg. Note that the equation has two terms. The second one can be ignored for pickup work since the hysteresis loop is flat for the very small magnetic field deviations in a pickup.

The main problem is to determine what system of units he is using. We need to evaluate theta, which contains rho. He says that rho is in emu; what is the resistivity of steel in emu?

Another question is the value of the initial permeability of the material. He uses mu_0 for this. (Note that he does not use mu_0 as the permeability of free space as is standard today.) If he is working in emu, the permeability of free space should be unity (I think), and so mu_0 in his equations should be a number like 500 or 1000 for steel.

Then we need a value for t, the plate thickness in cm. We do not really have a plate thickness since the core has no laminations, but we could try using the radius.

If you have done some calculations, you might have the answers to these questions.

[Mike Sulzer]

The problem has two parts:

1. Equation 31 of Legg can be used to determine how much the effective permeability changes as a function of frequency.

2. Then it is necessary to compute how much a change in permeability affects the inductance of the pickup coil.

One might assume that the answer to 2 is that the inductance is proportional to the permeability. This would be nearly correct for a closed pole piece, especially a toroid. However, one would not expect this to be true for an open ferromagnetic pole piece since the field lines must complete through free space with unity relative permeability.

This question requires some careful thought and computation.

[Mike Sulzer]

It is possible to evaluate the effect of eddy current shielding on he inductance of a pickup coil without a lot of computation. Equation 31 of Legg gives the frequency variation of the permeability as a function of the variable theta, which varies as the square root of frequency. With hysteresis effects small, we have just the first term to deal with. A simple computation shows that this function has a value given by the dc value of the permeability up to a frequency where theta is about equal to one, and then it falls off with roughly 1/(theta) or the inverse of the square root of frequency. Thus over a frequency range of 100 (say 80 to 8,000 Hz), the permeability can fall by no more than about a factor of 10.

Now we need to look at the inductance of a steel core humbucker coil and see why it has a value much lower than you might expect from the high permeability of the cores. This plot, http://www.naic.edu/~sulzer/humCoilAirSteel.png, shows the amplitude and phase of the impedance of such a coil. The low frequency portion of the phase plot shows two lines, one with the cores in, the other with the cores out. The slopes of the lines differ by about a factor of two. The phase is given by tan(2*pi*f*L/R). The tan function is approximately linear for small angles. Therefore the inductance of the coil with the cores is approximately a factor of two greater than without the cores. The exact value is not important.

To understand this, we analyze this from the variable reluctance view point. The path around which the flux flow can be divided into two parts, that occupied by the core and the rest. With the core in place, the reluctance of the first part is very small. Now take the core out; the inductance drops by about a factor of two, and so the reluctance has doubled. This means that the reluctances of the two parts of the path with the cores out are approximately equal, and the most the inductance can increase no matter how high the permeability might be is about a factor of two.

In order to examine relative inductances, let the two parts of the paths have reluctance of .5. Then the total inductance is 1/(.5 + .5) = 1. If the permeability of the core is near infinite, then when it is inserted, the inductance is 1/(.5 + 0) = 2. When it is 1000 (reasonable for steel), it is 1/(.5 + .001) = .999. When it is 100 (what it could decrease to over the frequency range of the pickup), it is .99.

So the inductance of the pickup coil is buffered against changes in the permeability of the core. Eddy current shielding does not significantly affect the inductance of the pickup coil.

[Joe Gwinn]

Now that is a very good point. But is that the dominant effect in a humbucker pickup, which has very large leakage reactance? To find out, we need to do some calculations using equation 31 in Legg. Note that the equation has two terms. The second one can be ignored for pickup work since the hysteresis loop is flat for the very small magnetic field deviations in a pickup.

I'm not sure that equation 31 applies unmodified to a pickup core, as Legg assumed a zero-leakage toroid.

One effect of eddy current shielding pushing the flux out of the core is to increase leakage flux and thus leakage inductance. Because the degree of shielding varies with frequency, so will the leakage inductance.

This effect should be easy to demonstrate experimentally in a bifilar-wound singlecoil pickup: If one shorts one winding and measure the inductance of the other, the remaining inductance is that due to leakage flux. Hmm. While this is the standard approach in transformers, I'm not sure what it will mean in bifilar pickups. I'll think about it.

I agree that hysteresis may be neglected in the present analysis of pickups, but because the hysteresis is small, not because it's "flat". In pickups there will always be a small signal-induced Raleigh loop circulating around the static field value imposed by the permanent magnets. I'm sure that this will have some effect on tone, but don't know how important the effect will be in practice.

The main problem is to determine what system of units he is using. We need to evaluate theta, which contains rho. He says that rho is in emu; what is the resistivity of steel in emu?

This is a problem, one that has bedeviled me as well. Electromagnetic units were a real mess, with about four systems in wide use, and woe betide anyone who accidentally used units from more than one such system in a given calculation. If one was lucky, the results were obvious nonsense. If not, the results were merely wrong. This is one reason the the MKS and later SI systems of units were developed.

As for the resistivity of steel in emu, I have no idea, but old reference books should tell.

Oh. People are wondering what is an emu, maybe it's a beast like a llama. No, emu= electro magnetic units, a now obsolete system of units of measure

Another question is the value of the initial permeability of the material. He uses mu_0 for this. (Note that he does not use mu_0 as the permeability of free space as is standard today.) If he is working in emu, the permeability of free space should be unity (I think), and so mu_0 in his equations should be a number like 500 or 1000 for steel.

Legg doesn't say, but from the publication date alone most likely he is working in emu.

The 500-1000 is a good range for the incremental permeability of mild steel, with air having a value of unity. If one anneals the steel, the value can be higher than 1000.

Then we need a value for t, the plate thickness in cm. We do not really have a plate thickness since the core has no laminations, but we could try using the radius.

The theory has also been developed for rods of a specified diameter, as well as for spheres (used for iron-dust cores). If I recall, the equation for rods contains the diameter squared, in parallel with thickness squared, but with a different coefficient.

If you have done some calculations, you might have the answers to these questions.

Despite the temptation, I have not done these calculations, if for no other reason that computing such a thing for a complicated thing like a pickup is far too much work. Basically, one must use finite-element methods on a computer, reinventing at least FEMM.

But simpler and thus tractable models can give useful albeit qualitative information.

[Joe Gwinn]

It is possible to evaluate the effect of eddy current shielding on he inductance of a pickup coil without a lot of computation. Equation 31 of Legg gives the frequency variation of the permeability as a function of the variable theta, which varies as the square root of frequency. With hysteresis effects small, we have just the first term to deal with. A simple computation shows that this function has a value given by the dc value of the permeability up to a frequency where theta is about equal to one, and then it falls off with roughly 1/(theta) or the inverse of the square root of frequency. Thus over a frequency range of 100 (say 80 to 8,000 Hz), the permeability can fall by no more than about a factor of 10.

Now we need to look at the inductance of a steel core humbucker coil and see why it has a value much lower than you might expect from the high permeability of the cores. This plot, http://www.naic.edu/~sulzer/humCoilAirSteel.png, shows the amplitude and phase of the impedance of such a coil. The low frequency portion of the phase plot shows two lines, one with the cores in, the other with the cores out. The slopes of the lines differ by about a factor of two. The phase is given by tan(2*pi*f*L/R). The tan function is approximately linear for small angles. Therefore the inductance of the coil with the cores is approximately a factor of two greater than without the cores. The exact value is not important.

To understand this, we analyze this from the variable reluctance view point. The path around which the flux flow can be divided into two parts, that occupied by the core and the rest. With the core in place, the reluctance of the first part is very small. Now take the core out; the inductance drops by about a factor of two, and so the reluctance has doubled. This means that the reluctances of the two parts of the path with the cores out are approximately equal, and the most the inductance can increase no matter how high the permeability might be is about a factor of two.

I've done this in the lab, using the Extech at 1000Hz to measure inductance, and a factor two or three effect is reasonable with mild steel slug cores in high-Q solenoid coils wrapped on a large plastic straw. A brass slug will reduce the inductance quite noticeably, but I don't recall the numbers.

In order to examine relative inductances, let the two parts of the paths have reluctance of 0.5. Then the total inductance is 1/(.5 + .5) = 1. If the permeability of the core is near infinite, then when it is inserted, the inductance is 1/(.5 + 0) = 2. When it is 1000 (reasonable for steel), it is 1/(.5 + .001) = .999. When it is 100 (what it could decrease to over the frequency range of the pickup), it is 0.99.

So the inductance of the pickup coil is buffered against changes in the permeability of the core. Eddy current shielding does not significantly affect the inductance of the pickup coil.

This part does not agree with experience. A few on-point experiments seem to be in order.

[Joe Gwinn]

Legg equation 31 and the esu system

It occurs to me that there is an easy solution to the problem of getting around the obsolete system of units. In Legg, equation 31 depends on the parameter Theta, which is simply the thickness of the sheet expressed in skin depths at the test frequency f, and this we can figure out using modern SI units.

If Theta is two, the sheet thickness is twice the skin depth.

Some skin depths at 1000 Hz:

Titanium: 11.8 mm.
Brass: 4.02 mm.
Aluminum: 2.59 mm.
Copper: 2.06 mm.
Silicon steel: 0.39 mm.
Mumetal: 0.13 mm.

Skin Depth Calculator - Microwave Encyclopedia

Permeabilities of ferrous metals gleaned from other sources.

[Mike Sulzer]

It occurs to me that there is an easy solution to the problem of getting around the obsolete system of units. In Legg, equation 31 depends on the parameter Theta, which is simply the thickness of the sheet expressed in skin depths at the test frequency f, and this we can figure out using modern SI units.

That is good insight; so it should not be too difficult to compute.

"This part does not agree with experience. A few on-point experiments seem to be in order. "

The inductance rises a factor of two or three with a relative permeability increase of 500 or a thousand. How could the sensitivity of the inductance to permeability changes be significant?

If you disagree, I think it is up to you to show that it is.

[Joe Gwinn]

"This part does not agree with experience. A few on-point experiments seem to be in order. "

The inductance rises a factor of two or three with a relative permeability increase of 500 or a thousand. How could the sensitivity of the inductance to permeability changes be significant?

If you disagree, I think it is up to you to show that it is.

I am not proposing a new model, you are. The usual journal article shows the theoretical curve and the experimental dots on the same plot, and assesses how well they match.

[Mike Sulzer]

I am not proposing a new model, you are. The usual journal article shows the theoretical curve and the experimental dots on the same plot, and assesses how well they match.

Perhaps you do not understand what I am responding to?

You have proposed that eddy currents in the core reduce the inductance of a wide range of coils, including pickup coils. You have used the results of the Legg paper as evidence of this.

It is not correct to do so. The short open cores are a very different case from the closed loop cores that Legg deals with. Since the inductance only rises a factor of two or three with a core with a perm. of 500 or so over air, the inductance of the coil cannot be very sensitive to the perm. of the core.

If 500 gives a factor of two, how much does 250 give? Any reasonable functional form gives a similar answer: it does not change much. The method of variable reluctance shows that most of the increase comes as the perm. increases from near unity while increasing from a couple hundred to 500 or 1000 or 10000 does very little. This is analogous to two resistors in series. If you start with equal values and decrease the value of one, most of the changes in the resistance of the series combination happens when the resistor is close to one, not when it is very small.

You just cannot change the inductance much once the perm. is large.

[Joe Gwinn]

Sure I understand, but I don't believe, and so proposed that the quickest way to sort through the conflicting claims and models is an experiment or two.

Legg and other authors established that eddy current shielding is real. While Legg derived equations for a toroid, one would expect to see the effect regardless of the core shape, even though Legg's equations are not exact for non-toroids.

Now I do agree that in magnetic circuits having more air than steel the changes in inductance due to eddy current shielding will be diluted, but I don't think it's 0.1% either, based on experiments where a sheet of aluminum is brought near to an air coil. I posted the results a while ago, but don't recall where. The change in inductance reading was quite apparent, in the percents as I recall. I guess that your implicit claim is that this must be due to something other than inductance change, but the debate won't be settled until we know what this other thing is.

Even if were to turn out that the effect of eddy current shielding in the steel components is negligible (I don't think so, but we can assume it for argument), we still have to deal with the non-ferrous metal components.

Legg and the authors of his era spent a lot of energy teasing apart the various physical effects that affect magnetic materials and components, and the key to progress was experiment after experiment, each more precisely focused than the last as the physics clarified.


More generally, all models must be verified in the lab before one can depend on them. It's very interesting to read the papers of the scientists who figured electricity and magnetism out, yielding the theories (which are models by another name) that Maxwell later summarized into his four equations. These scientists had many debates on which mathematical model nature really followed, and these debates were most often settled only by a clever but dead simple experiment, impossible to refute. Only then was the issue settled.

[Mike Sulzer]

Now I do agree that in magnetic circuits having more air than steel the changes in inductance due to eddy current shielding will be diluted, but I don't think it's 0.1% either, based on experiments where a sheet of aluminum is brought near to an air coil.

How is the stability of the inductance of a coil with an open short steel core related to that of an air core with a piece of non-ferrous metal nearby? It is not, by my thinking.

So if you do not accept the functional form for the reduction of inductdance with frequency of the pickup coil that I derived, then let us see how much it might be by some really simple assumptions. If adding a core of permeability 500 causes a factor of two increase in inductance, then assume that the change is linear with permeability change rather than the saturating function I derived. The slope of the line is a part in 250. If we had a meter that worked at 1000 and 100 Hz, this is a factor of ten and so we could get a factor of the square root of ten (about three) decrease in the permeability using the relationship in Legg. At a part in 250 this gives a 1.2% change in the inductance, much less than the 20% you have stated as roughly typical of a measurement.

So even using a functional relationship giving a much larger change than the a physics-based estimate, the change is still not enough. The problem with your measurement must be the combination of the series and parallel loss.

[Joe Gwinn]

I neither accept nor deny the claim.

The parallel-versus-series issue did occur to me, and perhaps that's what I saw, and perhaps not (or not entirely). It need not be one or the other.

I say only that more theory won't settle the question, that it's time for some lab work.

Just as did Legg and colleagues when figuring out how magnetic materials and devices work, and finding the correct set of models. Only the reasonably successful models were published; this will be a fraction of those tried.

[Mike Sulzer]

I say only that more theory won't settle the question, that it's time for some lab work.

OK, there is nothing wrong with more measurements. On the other hand, the reason for those early measurements that you mentioned above was to help construct a theory. Then one can use the theory instead of having to make a measurement to settle each question that arises.