Under what conditions does the inductance vary with frequency?

[Mike Sulzer]

Originally Posted by Joe Gwinn

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.

Yes, but does one effect matter more than the other? Measurements or calculations are necessary. Both in agreement would be best.

[Joe Gwinn]

Originally Posted by Mike Sulzer

Yes, but does one effect matter more than the other? Measurements or calculations are necessary. Both in agreement would be best.

Models need to be made to agree with experiment. Models can do anything, but the real world is what it is. So I'd be making measurements.

[Joe Gwinn]

Originally Posted by Mike Sulzer

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.

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.

Well, we've come full circle. Let me summarize it. I've been reading the relevant literature for thirty or forty years, and they all say than eddy current shielding reduces inductance, the Extech shows just that effect, and the pickup makers also report this effect. I also note that the national standards labs (like NBS and now NIST) for at least 100 years have modeled practical inductors as a series resistor with a parallel capacitor (for self-capacitance), and never add a parallel resistor, and yet the national standards labs are working to parts per million, far more precision than we will ever achieve with pickups.

So, I am presented with a choice: believe your model, or believe the industry and the standards labs and a century of experience.

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

I guess that that's true, but why does it matter?

[David Schwab]

Originally Posted by Joe Gwinn

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.

That's a very clear explanation! Thanks. I knew what it did, but not exactly why.

[Mike Sulzer]

Originally Posted by Joe Gwinn

So, I am presented with a choice: believe your model, or believe the industry and the standards labs and a century of experience.

That is a false choice.

In summary:
1. The complex impedance of a humbucker pickup versus frequency does not match that of a second order circuit (R, L, C). Additional element(s) are required. You cannot learn this with measurements at just a few frequencies. You actually need to look at impedance versus frequency to see this.
2. Unlike most iron cored coils, the humbucker uses solid, rather than laminated cores.
3. High permeability and moderate conductivity cause current to flow around the cores near the surface.
4. Each core thus behaves like a transformer secondary.
5. A transformer model is appropriate.
6. The secondary load reflects back on the primary in parallel. (This is why a transformer needs a large magnetizing inductance.)
7. The transformer model is poorly coupled, and so an inductance in series with the resistive load is required.
8. The frequency dependence of the skin effect means that the resistive load rises with frequency.

A model with these effects matches the measurements when the parameters are adjusted.

This does not disagree with "the industry and the standards labs and a century of experience". But you do have to measure the complex impedance as a function of frequency in order to relate model and measurements in a sensible way. A measurement of complex impedance at a couple of spot frequencies does not do it.

[Peter Naglitsch]

Sorry to interrupt this academic game of ping pong for something that I think is on topic although the complete other way around. It started with this statement from Possum:

Originally Posted by Possum

Actually a tele baseplate LOWERS the inductance readings. So do covers.

As I am curious and like to back anything up with real life experience and measurements I took two randomly chosen Strat pickups, one made by me and one that came out of a customers ESP that I made new pickups for. I measured the inductance with and without a steel plate under the pickup. I even tried different pieces of steel to be 100% sure. This is what I got:
SC1;
Inductance w/o steel = 2.44H
Inductance with steel = 2.55H
SC2;
Inductance w/o steel = 2.93H
Inductance with steel = 3.09H
(Measurements taken with my cheepo Voltcraft LCR-9063)

So adding a steel plate beneath the pickup doesn't lower the inductance. Actually the other way around. I did the same thing with two humbuckers and a couple of different covers + one Tele neck pickup ripped out of a squire:
HB1;
Inductance w/o covers = 4.75H
Inductance with covers = 4.68H
HB2;
Inductance w/o covers = 4.45H
Inductance with covers = 4.34H
SC3;
Inductance w/o covers = 2.26H
Inductance with covers = 2.17H

So inductance does really go down on HBs and SCs using covers. HOWEVER I noticed one peculiar thing that really surprised me. I make a Gretsch lookalike pickup using these covers:

and when putting them on the same pickups the inductance didn't change much at all:
HB1;
Inductance w/o covers = 4.75H
Inductance with covers = 4.74H
HB2;
Inductance w/o covers = 4.45H
Inductance with covers = 4.46H

Conclusion:
Adding steel to the bottom of SC style pickups add to the inductance, traditional covers on HBs lower it and open type covers on HBs doesn't change it much. And from personal experiences (call it empirical knowledge if that makes anyone more comfortable) that I think most of us here share and can testify to: Adding a piece of steel under a SC type pickup add "bite" or treble to the sound, adding a cover removes a bit of treble or "dulls" the sound a bit and finally adding a top-less (no naughty thoughts now guys) or open cover like the ones pictured above or like this one:

doesn't change the sound much at all. So what does that tell us about inductance and how changes in inductance changes the frequency response of the pickup? Or how we can use that knowledge and adapt it to pickup design? This is more or less the total opposite of the origin of this discussion. Not "how does the inductance wary with frequency" but rather "how can we alter the inductance to get the frequencies that we want?"

Sorry if this is considered OT. Please continue as before...

[Mike Sulzer]

Originally Posted by Peter Naglitsch

Sorry to interrupt this academic game of ping pong ...

Peter, how do you measure inductance?

Ping pong or not, it is real measurements and how to interpret them that we are discussing.

[David Schwab]

Originally Posted by Peter Naglitsch

So adding a steel plate beneath the pickup doesn't lower the inductance. Actually the other way around.

I don't have an inductance meter, but I based my comment on an added steel plate increasing inductance, based on my experiments with pickups with low inductance, such as wound on a ceramic magnet with no steel poles. Adding a steel plate increases the output, which I attributed to an increase in inductance.

I did the same thing with two humbuckers and a couple of different covers + one Tele neck pickup ripped out of a squire...

So inductance does really go down on HBs and SCs using covers. HOWEVER I noticed one peculiar thing that really surprised me. I make a Gretsch lookalike pickup using these covers...
and when putting them on the same pickups the inductance didn't change much at all

And that makes sense because of eddy currents. The Gretsch style covers were made to reduce eddy currents. Rowe had a similar patent on a cover with a slit in it and an open front.

[Mike Sulzer]

Sorry, Peter, I did not see that that you said how you measure it..

[Joe Gwinn]

Originally Posted by Mike Sulzer

That is a false choice.

In summary:

[... a repeat of the questioned model ...]

A model with these effects matches the measurements when the parameters are adjusted.

This does not disagree with "the industry and the standards labs and a century of experience". But you do have to measure the complex impedance as a function of frequency in order to relate model and measurements in a sensible way. A measurement of complex impedance at a couple of spot frequencies does not do it.

It should be clear that I have some difficulty believing this model, for the reasons already explained at some length. And this model does disagree with that century of experience. You might wish to make some Maxwell-Wein Bridge measurements at various frequencies.

[Mike Sulzer]

Originally Posted by Joe Gwinn

It should be clear that I have some difficulty believing this model, for the reasons already explained at some length. And this model does disagree with that century of experience. You might wish to make some Maxwell-Wein Bridge measurements at various frequencies.

Let's examine Joe's contention above that a series resistor model is adequate for a guitar humbucker pickup coil. Let's do this with a measurement and some analysis, but in such a way that questions about the accuracy of the measurement device do not arise. For this purpose, we compare two measurements of the impedance of a humbucker coil, one with the steel cores removed, the other with them in place. The amplitude and phase measurements are here:http://www.naic.edu/~sulzer/humCoilAirSteel.png. (Please note that the upper end of the frequency scale is 20 KHz; there should be a 0 after the 2.)

From the locations of the peak in the amplitude plot, and from the location of the zero crossing points in the phase plot, one can see that the steel cores lower the resonance point by increasing the inductance. One can also see from the widths of the peaks and slopes of the phase responses that the Q is lower when the cores are in. Let's examine how this loss happens: that is, should we attribute it to a resistor in series or parallel with the coil?

First, examine the phase response at the lower frequencies. Note that we have an inductor, and a resistor in series. The phase rises from near zero at very low frequencies. The reason is that the inductive reactance is very low at low frequencies, and so the resistance dominates. A resistor, of course, does not cause a phase shift. As the frequency rises, the phase shift increases; the inductive reactance, which increases with frequency, takes over and dominates the series combination. (The two reactances are equal at 45 degrees. Note that this happens at a lower frequency when the cores are in place because the inductance is larger.)

Next look at the phase response when the cores are in place between roughly 2 and 8 KHz. The phase response decreases, meaning that the impedance is becoming less inductive and more resistive. A resistor in parallel with the coil could cause this: as the inductive reactance rises with increasing frequency, the parallel resistance limits the combined impedance, and it becomes closer to resistive.

Can this be done with a series resistor instead? Yes, but it requires a resistance that increases very quickly with frequency. In order to maintain the phase at a constant value as the frequency increase, the resistance would have to increase at the same rate as the inductive reactance does. For the phase to decrease with frequency, it would have to increase even faster.

In order to explain the coil loss as a series resistor, one needs to find a physical process that could cause a resistance to rise so fast with increasing frequency. Even if one did that, it would then be necessary to check for consistency with the amplitude response. One would also need to find something wrong with the model using the parallel resistance. I do not think it is possible to do any of this.

[Joe Gwinn]

Exactly how were those curves generated?

It would be instructive to use a brass core. Unless laminated, iron cores yield a complex and confusing mixture of eddy-current and magnetic-permeability effects. Non-magnetic stainless steel and/or german silver would yield another useful data point.


Maxwell-Wein Bridges and LCR meters are intended for use well below resonance. There are other bridge types and instruments intended for use at resonance.


Some background:

The balance equations for both the Maxwell Bridge and the Maxwell-Wein Bridge are independent of frequency. This was necessary in Maxwell's day because although the mathematics of alternating current was well developed, there were no really practical sources of alternating current with which to drive a bridge. So, what people did was to use a telegraph key to pulse the DC from a battery, and adjust the bridge until the galvanometer no longer kicked when the key was operated. Because the random square-wave signal from a telegraph key contains all frequencies, a bridge requiring a specific frequency could not be balanced, so people developed frequency-independent bridges.

A Maxwell Bridge balances two inductors against one another, while a Maxwell-Wein Bridge balances an inductor against a capacitor. Both kinds of bridge balance resistances against resistances.

A lot of the original work on the behavior of ferromagnetic materials was done with the telegraph key source and galvanometer detectors as well, giving rise to the various "surge" parameters. When vacuum tubes were invented, it became practical to generate and detect AC signals, and surge quantities were gradually supplanted with the corresponding incremental quantities.


As for "Joe's contention" you might also wish to take the argument up with all the national standards lab. Perhaps they have been missing something for all these years. It is dead simple in a Maxwell-Wein Bridge to add a potentiometer to balance a resistance parallel to the inductor under test, and yet one never sees such a potentiometer.

I assume there is a learned article from the late 19th century explaining why for the Maxwell-Wein Bridge no parallel resistor is needed. Actually, I bet that the original analysis was by Max Wein himself. Legg references this article, which is in German. Perhaps my schoolboy German is up to it.

[Mike Sulzer]

Originally Posted by Joe Gwinn

Exactly how were those curves generated?

With the same I-V circuited and computer set up as described in the discussion "A new model for..." and in earlier discussions.

[RedHouse]

Originally Posted by Mike Sulzer

...Ping pong or not, it is real measurements and how to interpret them that we are discussing....

I don't think anyone calls these dialogs between you and Joe ..."discussions" we need a separate forum for you guys to argue in.

[Dave Kerr]

Originally Posted by RedHouse

we need a separate forum for you guys to argue in.

I don't understand the issue here - I skim these discussions, avoiding the stuff that's clearly over my head, and latching on to stuff like Peter's post upthread. If the arcane back and forth is so troublesome to skim over, can't you size up the gist of the thread in about 5 seconds and ignore it?

[Mike Sulzer]

Originally Posted by Joe Gwinn

As for "Joe's contention" you might also wish to take the argument up with all the national standards lab. Perhaps they have been missing something for all these years. It is dead simple in a Maxwell-Wein Bridge to add a potentiometer to balance a resistance parallel to the inductor under test, and yet one never sees such a potentiometer.

We are discussing a steel core coil. A transformer is a steel core coil with a secondary added. So think about a transformer model; leave the secondary open or take it off. This does not affect modeling the core losses in the circuit model.

We are discussing core losses here; the losses get bigger when the cores are put in. These losses appear in parallel. Look here: Transformer - Wikipedia, the free encyclopedia

[RedHouse]

Originally Posted by Dave Kerr

I don't understand the issue here - I skim these discussions, avoiding the stuff that's clearly over my head, and latching on to stuff like Peter's post upthread. If the arcane back and forth is so troublesome to skim over, can't you size up the gist of the thread in about 5 seconds and ignore it?

Yes Dave, I can and do, I just thought I'd try being as persistant with an annoying chime-in.

I guess it isn't working.

[Joe Gwinn]

Originally Posted by Mike Sulzer

We are discussing a steel core coil. A transformer is a steel core coil with a secondary added. So think about a transformer model; leave the secondary open or take it off. This does not affect modeling the core losses in the circuit model.

We are discussing core losses here; the losses get bigger when the cores are put in. These losses appear in parallel. Look here: Transformer - Wikipedia, the free encyclopedia

Yes, I know about transformers. Power transformer folk often model eddy currents as a shorted turn because all they care about is power loss. The power transformer folk are not worrying about changes in inductance due to eddy current shielding. The standards labs use a series resistor instead, and do worry about inductance changes.

If I recall, Legg implied that one can handle eddy current losses either way, which may be true or at least true enough, and then chose the series-resistor model. I bet the series approach yields smaller and more easily measured values than the parallel approach.

But the key issue is that a transformer model cannot replicate the key aspect of eddy currents in a core, the exclusion of flux. The currents in transformers are not eddy currents, by the topological definition given earlier, and so are inherently incapable of phenomena such as skin depth and eddy current shielding.

More generally, lumped-element models are usually inadequate to explain or model phenomena requiring partial differential equations to express.

I've ordered Max Wein's 1898 article. We'll see how long it takes to arrive. I also see that Legg cites some other articles by Wein.

[LtKojak]

Originally Posted by RedHouse

I don't think anyone calls these dialogs between you and Joe ..."discussions" we need a separate forum for you guys to argue in.

I think those two should get a room!

J/K

[Mike Sulzer]

Originally Posted by Joe Gwinn

Yes, I know about transformers. Power transformer folk often model eddy currents as a shorted turn because all they care about is power loss. The power transformer folk are not worrying about changes in inductance due to eddy current shielding. The standards labs use a series resistor instead, and do worry about inductance changes.

Suppose one shorts the secondary of the transformer in the model I referred to.
1. The resistance is very small. In this case the leakage inductance appears across the magnetizing inductance and the parallel combination has a lower inductance.
2. The resistance is large compared to the reactance of the leakage inductance. Then there is no significant change in inductance.

From the description of the model I referred to: "Since the core flux is proportional to the applied voltage, the iron loss can be represented by a resistance RC in parallel with the ideal transformer." This is not just a power model, it is a circuit model. Effects due to leakage inductance can affect the core loading as well. Of course you can represent the effect at a single frequency as a series resistor if you want, but this produces extreme variations in resistance if you want to do it across frequency. A measuring system must be able to account for the complexity of the system.

Originally Posted by Joe Gwinn

If I recall, Legg implied that one can handle eddy current losses either way, which may be true or at least true enough, and then chose the series-resistor model. I bet the series approach yields smaller and more easily measured values than the parallel approach.

One measures complex impedances and then interprets them according to a model. I do not see what you mean.

Originally Posted by Joe Gwinn

But the key issue is that a transformer model cannot replicate the key aspect of eddy currents in a core, the exclusion of flux. The currents in transformers are not eddy currents, by the topological definition given earlier, and so are inherently incapable of phenomena such as skin depth and eddy current shielding.

Currents in transformers certainly are affected by the skin effect, both in the conductors and the core. These effects can be handled by frequency dependent resistors as necessary.

[RedHouse]

Originally Posted by LtKojak

I think those two should get a room!

J/K

Maybe we need a group-buy ....to get them a room!

[Joe Gwinn]

Originally Posted by Mike Sulzer

Suppose one shorts the secondary of the transformer in the model I referred to.
1. The resistance is very small. In this case the leakage inductance appears across the magnetizing inductance and the parallel combination has a lower inductance.
2. The resistance is large compared to the reactance of the leakage inductance. Then there is no significant change in inductance.

From the description of the model I referred to: "Since the core flux is proportional to the applied voltage, the iron loss can be represented by a resistance RC in parallel with the ideal transformer." This is not just a power model, it is a circuit model. Effects due to leakage inductance can affect the core loading as well. Of course you can represent the effect at a single frequency as a series resistor if you want, but this produces extreme variations in resistance if you want to do it across frequency. A measuring system must be able to account for the complexity of the system.


One measures complex impedances and then interprets them according to a model. I do not see what you mean.

Currents in transformers certainly are affected by the skin effect, both in the conductors and the core. These effects can be handled by frequency dependent resistors as necessary.

Please read the Legg article.

[Mike Sulzer]

Originally Posted by Joe Gwinn

Please read the Legg article.

Is it available on line? I have not found it, and finding a paper copy in Puerto Rico seems unlikely.

But in any case is it really necessary? IS a resistor in series with the leakage inductance really a poor model?

Since you have a Wien bridge, you could make measurements at a series of frequencies on a humbucker coil (cores in and out) and see if they disagree with the ones I presented above.

[Steve Conner]

In transformer work, I've only ever seen core losses represented by a shunt resistor. The series resistor in the transformer model is there to account for copper losses.

If you have an LC circuit, you can transform between series resistance and parallel resistance: Rseries = (Z0^2)/Rshunt and vice versa. (where Z0 = sqrt(L/C)) But this transformation is only valid at the circuit's resonant frequency. (You can actually derive the transformation by considering what values of series and shunt resistances are needed to give the same Q factor.) For other frequencies, a circuit with a series resistor behaves differently to one with a shunt resistor: the differences being most pronounced for low-Q circuits such as pickups.

Lumped models are considered adequate for systems where everything is small compared with a wavelength. A guitar pickup doubtless has other resonant modes, but the lumped model only predicts the first one. The others are probably ultrasonic, though.

Resistances that vary with frequency, due to skin effect or whatever, can be modelled by a whole bunch of R-L or R-C networks with different time constants. This is still considered a lumped model.

[Mike Sulzer]

Steve,

Yes. I would add that the current into the shunt resistor that models the cores losses can be limited by an inductance due to flux leakage.

For a humbucker pickup, where we have both a high series coil resistance and large iron losses due to the solid cores, both types of losses are significant. The dominance of one or the other is a function of frequency. Both may be simultaneously important over part of the frequency range.

[Steve Conner]

So something like this:

Transformer - Wikipedia, the free encyclopedia

with the secondary terminals shorted?

[Mike Sulzer]

Originally Posted by Steve Conner

So something like this:

Transformer - Wikipedia, the free encyclopedia

with the secondary terminals shorted?

Yes, although I think that model might be a bit more complicated than necessary.

In the discussion "A new model for...", the approach was a bit different. A simpler model (Hagen, RF Electronics) consisting of an ideal transformer with a leakage inductor in series with the secondary (no core loss) was used to derive the core loss model. The currents circulating around the cores were represented as a shorted secondary with some resistance. This was shown to provide the necessary loss in the low KHz range while partially unloading the coil at the higher frequencies so that the resonance occurs. This was not sufficient; it was necessary to make the "secondary" resistance frequency variable (from the skin effect) in order to get a reasonable match with the measurements.

[RedHouse]

Originally Posted by RedHouse

Maybe we need a group-buy ....to get them a room!

Appears it might need to be a ménage à trois!

[Joe Gwinn]

Originally Posted by Mike Sulzer

Is it available on line? I have not found it, and finding a paper copy in Puerto Rico seems unlikely.

I assumed that the observatory library is a university library and so could get anything. I don't think that it's online anywhere.

Anyway, here it is: Well, no it isn't. It exceeds the Forum's ~2 MByte file size limit by at least a megabyte, being a scan. So I need a direct email address. I will be offline from ~10 AM tomorrow (Friday) until Sunday the 30th, but I will send copies to whoever gets an email to me in time before I drop off the screen.

But in any case is it really necessary? Is a resistor in series with the leakage inductance really a poor model?

Yes and no. Did you mean in parallel?

Since you have a Wien bridge, you could make measurements at a series of frequencies on a humbucker coil (cores in and out) and see if they disagree with the ones I presented above.

I built the bridge from small parts and made such measurements many years ago, when I was qualifying the Extech LCR meter. The bridge and Extech agreed to a fraction of one percent. I can dig up the data in a week.

The trick with the bridge is to use ordinary 5% and 10% components but measure the resistance and capacitance values (which will be fairly pure) to within 1% using a digital multimeter, and use a calculator to solve for inductance and series resistance.

The only other thing that must be good is the oscillator - it must have low harmonic distortion. If it does not, the harmonics will not be nulled, and will obscure the null.

[LtKojak]

Originally Posted by RedHouse

Appears it might need to be a ménage à trois!

Yeah, the more, the merrier.

[Mike Sulzer]

Originally Posted by Joe Gwinn

I built the bridge from small parts and made such measurements many years ago, when I was qualifying the Extech LCR meter. The bridge and Extech agreed to a fraction of one percent. I can dig up the data in a week.

I am not disputing the accuracy of the Wien bridge and the Extech as regards to their ability to measure complex impedance (and in the case of the Extech, to interpret the measurement in terms of two simple models). My concern is with the interpretation of the measurements of complex impedance when the model is not so simple.

[Mike Sulzer]

I have read the Legg paper, which you kindly supplied.

The Legg paper represents the various losses in an inductor as a series resistor that is the sum of a resistor accounting for the the copper loss in the coil and and one accounting for the losses in the the core (equation 7). Let us consider eddy current loss. This resistance is a function of frequency squared (equations 11 through 19).

The frequency variation of this resistor suggests that it is an equivalent resistance, and that the physical process is more simply represented by a resistance in parallel. The question that arises then is whether or not this use of equivalence resistance affects the value of the inductance that is interpreted from the Wien bridge measurements.

Consider the equivalence relationship between an inductor with a series resistor (Rs) and a parallel resistance (Rp). That is, suppose we have a parallel resistor but we interpret the measurements of complex impedance as if it is in series (so that we can, for example, subtract off the copper loss resistor). How must we transform the values?

The impedance of the series combination is 2*pi*f*Ls*j + Rs. The parallel network consisting of Rp and Lp can be put in the same form (imaginary part plus real part) with some algebra. We equate the two expressions to look for the transformation relationships.

This is not simple. But if we make the assumption Rp is much larger than 2*pi*f*Lp* over the frequency range of interest (high Q), then we see that Lp = Ls and Rp = ((2*pi*f*Ls)^2)/Rs. The latter shows that the parallel resistance has a simple frequency variation, given the squared dependence for Rs from the paper, as the physics suggests. The former says that the error in inductance goes to zero at very high Q because Ls and Lp are the same. (Measuring one is the same as measuring the other.)

When the Q is not high, the assumption that Ls = Lp fails, and there is error in the inductance measurement. (Rs and Rp are functions of both Rp and Lp.) Thus there is an assumption in Legg's work that the Q of the inductance is high. (I do not see where he has stated this, but I might have missed it.)

The same thing applies to measurements with the Extech. If the Q is high, then even if we have both a series and a parallel resistor, the measurement of the inductance is accurate if we assume just a series resistance.

But if the Q is low (humbucker pickup), the conditions are violated; the additional terms that were ignored for high Q are large in this case and there is error in the inductance measurement. Note that this is true even if the Extech measures the complex impedance with perfect accuracy. The problem is in the interpretation, not in the measurement itself.

[Joe Gwinn]

Originally Posted by Mike Sulzer

I have read the Legg paper, which you kindly supplied.

Quite welcome. Made me reread it, too. I think I'll put a copy up on my website too.

The Legg paper represents the various losses in an inductor as a series resistor that is the sum of a resistor accounting for the the copper loss in the coil and and one accounting for the losses in the the core (equation 7). Let us consider eddy current loss. This resistance is a function of frequency squared (equations 11 through 19).

The frequency variation of this resistor suggests that it is an equivalent resistance, and that the physical process is more simply represented by a resistance in parallel. The question that arises then is whether or not this use of equivalence resistance affects the value of the inductance that is interpreted from the Wien bridge measurements.

While it is clear that this frequency-dependent resistor is representing some deeper physical phenomena, largely eddy currents in the present case, how does it follow that the effect is better represented by a shunt resistor? While this has been your assertion all along, Legg says no such thing. Legg (on page 51) uses shunt capacitance and conductance only to model coil self-capacitance and DC leakage, and uses the series-resistor model for everything else.

Consider the equivalence relationship between an inductor with a series resistor (Rs) and a parallel resistance (Rp). That is, suppose we have a parallel resistor but we interpret the measurements of complex impedance as if it is in series (so that we can, for example, subtract off the copper loss resistor). How must we transform the values?

The impedance of the series combination is 2*pi*f*Ls*j + Rs. The parallel network consisting of Rp and Lp can be put in the same form (imaginary part plus real part) with some algebra. We equate the two expressions to look for the transformation relationships.

As discussed above, Legg does no such thing.

This is not simple. But if we make the assumption Rp is much larger than 2*pi*f*Lp* over the frequency range of interest (high Q), then we see that Lp = Ls and Rp = ((2*pi*f*Ls)^2)/Rs. The latter shows that the parallel resistance has a simple frequency variation, given the squared dependence for Rs from the paper, as the physics suggests. The former says that the error in inductance goes to zero at very high Q because Ls and Lp are the same. (Measuring one is the same as measuring the other.)

This assumes the shunt resistor model, which Legg does not use to model eddy current effects.

However, returning to the original question, Legg says in multiple places that eddy current shielding causes the observed inductance to decrease with increasing frequency. A clear statement appears above equation 30 on page 45:

"Also, eddy currents set up magnetizing forces within the magnetic material which more or less neutralize that applied by the coil winding, and thus effectively shield the inner parts of magnetic laminations of wires. Such eddy current shielding reduces the total flux in the core, thus decreasing the inductance and loss resistance observed at higher frequencies."

Equation 30 is a partial differential equation (PDE). As discussed in previous postings, PDEs are required to capture phenomena such as eddy currents, although for equation 30 the object is magnetic hysteresis. Probably Steinmetz (footnote 4 on page 41) is where this is done for eddy currents.

A supporting statement appears on page 46, near the bottom:

"The apparent permeability (mu sub fm), which is calculated from the measured inductance, decreases as the measuring frequency is increased."

And, at the top of page 52:

"Thus the observed inductance tends to increase at higher frequencies on account of distributed capacitance, in contrast to its tendency to decrease on account of magnetic shielding in the core according to equation (33)."

Equation 33 is one of the results of the PDE equation 30.

When the Q is not high, the assumption that Ls = Lp fails, and there is error in the inductance measurement. (Rs and Rp are functions of both Rp and Lp.) Thus there is an assumption in Legg's work that the Q of the inductance is high. (I do not see where he has stated this, but I might have missed it.)

Legg does not mention any limitation to high Q, and in fact his equations give the full story, including magnetic hysteresis and viscosity effects in addition to eddy current effects.

The closest approach to a Q limit in Legg is on page 55, where it is observed that for accurate measurement of magnetic material properties, it's helpful if the Q is high, and the two test coil examples Legg includes have inductances in the low millihenries and Q=20.

Note that while magnetic hysteresis and viscosity effects may possibly have some effect on tone in guitar pickups, they have limited effect on inductance and AC resistance. Hysteresis and viscosity are bulk properties of the magnetic material used for slugs, screws, magnets, and perhaps baseplates.

The same thing applies to measurements with the Extech. If the Q is high, then even if we have both a series and a parallel resistor, the measurement of the inductance is accurate if we assume just a series resistance.

But if the Q is low (humbucker pickup), the conditions are violated; the additional terms that were ignored for high Q are large in this case and there is error in the inductance measurement. Note that this is true even if the Extech measures the complex impedance with perfect accuracy. The problem is in the interpretation, not in the measurement itself.

Even if one accepts the necessity of a shunt-resistor model, all this depends on the definition of low and high Q.

The Extech users manual says that full accuracy (~1%) is achieved if the Q is not less than two, and my experience is that the Extech will work accurately well below that. One can test this by putting a potentiometer in series with an inductor and observing the effect on measured inductance as the series resistance is increased.

With the Maxwell-Wien Bridge, one can work down to very low Q values, but bridge balancing using a digital voltmeter as the detector becomes very slow because the null is so shallow. If one instead uses a phase-sensitive detector, one can easily balance the bridge despite the very shallow null.

To put some numbers on it, at 1000 Hz, a 2 H inductance has a reactance of 12,566 ohms, so if the AC resistance is 10,000 ohms, the Q is ~1.3. I recall making bridge and Extech measurements down to Q=0.5 or so.

[Joe Gwinn]

I just updated my website to include the Legg article:

V.E. Legg, "Magnetic measurements at low flux densities using the alternating current bridge", BSTJ, v.15, page 39, January 1939.

This is the real deal. Some math and physics background is advised, although it's possible to glean much of the gist by skimming the text while ignoring the equations.

Legg was interested in the design of RF and audio frequency inductors and transformers for use in telephone systems. Vacuum-tube amplifiers were well established and widely used in 1939, while the invention of the transistor (also at Bell Labs) was nine or ten years in the future.

[Mike Sulzer]

Originally Posted by Joe Gwinn

As discussed above, Legg does no such thing.

(Concerning the analysis that I presented) I never wrote that he did. It is my analysis, not Legg's, that shows that he needs the high Q approximation in order to get an accurate measurement of L. I think it is right. I can fill in the details if you have doubts.

Originally Posted by Joe Gwinn

The Extech users manual says that full accuracy (~1%) is achieved if the Q is not less than two, and my experience is that the Extech will work accurately well below that. One can test this by putting a potentiometer in series with an inductor and observing the effect on measured inductance as the series resistance is increased.

I have never disputed the accuracy with a series resistor. Suppose you have both a series and a parallel resistor. That is, with a series resistor of large enough value to make the Q low, and the parallel resistor of small enough value to also make the Q low. That is the issue. The coil core loads in parallel; that is the natural physical model. Using an equivalent series model requires checking with the analysis I presented above.