Under what conditions does the inductance vary with frequency?

[Mike Sulzer]

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]

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]

...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]

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]

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]

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]

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]

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]

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.

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.

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]

I think those two should get a room!

J/K

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

[Joe Gwinn]

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]

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.