A new model the impedance of a humbucker, compared to measurements

[Joe Gwinn]

Joe,

I read Lemme's section on eddy currents. He claims that they add another 6 db/octave to the high frequency rolloff, and his equivalent circuit is intended to exhibit this property. This is not what I expect, but it is not something I can check with the pickup used here because its resonant frequency is too high. (I do have others, though.)

I also wonder about the 12 db/octave he discusses for pickups without eddy losses included, because the pickup capacitance should not really be independent of the series resistance of the coil. If you add in the cable capacitance, sure, and that might be what he is doing. But who knows, he does not show actual measurements in that article, just generic drawings.

I would not take Lemme as Gospel. Two inductors and a resistor is a pretty crude approximation. Note that the transformer guys use about ten inductors.

What Lemme's analyzer did (before eddy currents) is automate a standard test approach where one resonates the coil with a series of capacitors of known value, and mathematically solves for the added capacitance needed to make it come out right. This added capacitance was the self-capacitance of the coil, as the theory goes. The fly in the ointment was that the inductance varied with frequency due to eddy currents. I don't think he knew why his answers didn't quite add up, and just found some kind of workable average. Adding an adequate eddy current model would allow this same approach to give far better answers.

The apparent purpose of the article is to sell his pickup analyzer. While I don't doubt that it does exactly what is claimed, for that kind of money one can buy standard test equipment that can do more than one thing, including measure pickups.

I corresponded with him some years back. He was trying to convince me that a Maxwell-Wein Impedance Bridge could not be used to measure pickups, which is very interesting, as this is exactly what the various national standards labs used for at least a century, until digital techniques took over. Bridges were literally the gold standard.

[Mike Sulzer]

The transformer model

Magnetic circuits can be quite complicated, but we want to use something relatively simple to explain, approximately, the impedance of a pickup coil. A transformer model is potentially useful in modeling the eddy current losses in pickup cores because current flowing in the cores due to changing flux has the same basic physical cause as current flowing in a transformer winding. The cores in a pickup are short and open, and so one can expect a significant loss in coupling between the coil and the winding representing the current in the cores. [High coupling, or mutual inductance, results from a high high permeability closed core such as a toroid even if the windings are located on different parts of the core, or when the windings are very close, bifilar wound for example. We have neither in a pickup.]

Transformers for audio and power use closed cores to minimize flux leakage, but it is not difficult to include the effects of leakage flux in a transformer model. The model used here is from Jon Hagen's book "Radio Frequency Electronics". The basic idea is to find a circuit that models some of the effects in a real transformer by adding some components to an ideal transformer. An ideal transformer is described simply by its turns ratio.

This figure (http://www.naic.edu/~sulzer/transformerModel.png, similar to Figure 22.6 in Jon's book) shows the equivalency between a transformer with arbitrary coupling and an ideal transformer. Two components are added:
1. The magnetizing inductance. This has nothing to do with the coupling, but is needed in any model except "ideal". This magnetizing inductance represents the inductance of the pickup coil in the application to a pickup.
2. An inductor in series with the secondary which models the effect of the leakage inductance. One should read the first few pages of Chapter 22 of Jon's book to understand how this is derived. Note also that the turns ratio of the original transformer (N1/N2) is not the same as the ratio n1/n2 of the ideal transformer in the model. The square of the latter ratio is given by L1/(kL2). L1 and L2 are the inductances of the primary and secondary. k is a function of the coupling between the coils. k = 0 indicates no mutual inductance; k = 1 means no leakage. When k = 1, the leakage inductance is zero and the two ratios are the same.

We want to model the eddy current effect as simply as possible and see how well the model performs, and so we will use just one resistor to complete the circuit in the secondary. The total impedance in the secondary can be transformed by the square of the turns ratio to get the load on the primary. Then the model consists of the magnetizing inductance in parallel with the series combination of the leakage inductance and the eddy resistor. Now we can get an estimate of the size of the leakage inductor as a function of the leakage flux. As shown in the figure, one can solve a simple equation to show that k = .707 (that is, 1/sqrt(2)) when L1 and the leakage flux are equal. Thus the leakage inductor could be several Henries in size.

[Mike Sulzer]

The new model and the measurements

The pickup model is drawn here: http://www.naic.edu/~sulzer/pickupModel.png. It is as described above, but a resistor (Rc) has been added in series with the capacitor. This is a sort of "effective resistance"; charge cannot get into or out of the pickup without losing some energy in the wire, but we do not know what the value should be. On the other hand, Rp, the resistance of the pickup, can be measured, and so one has the other five variables (C, Rc, L1, LLp, and Re) that need to be varied until the computed impedance is close the the measured impedance (amplitude and phase). This is not easy to do, and one really needs an automatic process, non-linear least squares fitting, or some other similar method. In any case the results are here: http://www.naic.edu/~sulzer/newModelAmp.png and here: http://www.naic.edu/~sulzer/newModelPhase.png. This is much better than the simple model, especially at the frequencies that really matter. The values are L1 = 4.75H, Rp = 8500, LLp = 7 H Re = 240K, C = 45 pf, Rc = 35K.

Despite the pretty good agreement, I am not completely happy with this. The measured curves are relatively smooth, while the model is bumpier. It is possible that more inductor-resistor combinations are necessary in order to smooth the model. It is not unreasonable that the current in the pole pieces would require a better description.

[Joe Gwinn]

The pickup model is drawn here: http://www.naic.edu/~sulzer/pickupModel.png. It is as described above, but a resistor (Rc) has been added in series with the capacitor. This is a sort of "effective resistance"; charge cannot get into or out of the pickup without losing some energy in the wire, but we do not know what the value should be. ...

Despite the pretty good agreement, I am not completely happy with this. The measured curves are relatively smooth, while the model is bumpier. It is possible that more inductor-resistor combinations are necessary in order to smooth the model. It is not unreasonable that the current in the pole pieces would require a better description.

I assume you are using Spice. Does the version you are using have the ability to handle user-defined black-box components of any kind? Controlled sources can often be pressed into service. If one can get the parameters of something to vary with the square root of frequency, the rest may become easy.

[Mike Sulzer]

I assume you are using Spice. Does the version you are using have the ability to handle user-defined black-box components of any kind? Controlled sources can often be pressed into service. If one can get the parameters of something to vary with the square root of frequency, the rest may become easy.

No, I am doing the calculation from scratch in a vector interpreter language I wrote some years ago. You can build these impedances out of just two complex functions, one for putting two complex impedances in series (trivial, just addition), and the other for putting them in parallel (just a few lines of code). Any impedance could vary arbitrarily in frequency without too much trouble. Ideas, Joe?

[Joe Gwinn]

No, I am doing the calculation from scratch in a vector interpreter language I wrote some years ago. You can build these impedances out of just two complex functions, one for putting two complex impedances in series (trivial, just addition), and the other for putting them in parallel (just a few lines of code). Any impedance could vary arbitrarily in frequency without too much trouble. Ideas, Joe?

Maybe. If it's your own private language, you can use any bit of math you like then, and there is no reason not to implement the true physics model.

There is a large literature on the effects of eddy currents in nearby metallic objects, the objective being non-destructive testing using eddy current tests to look for invisible cracks in highly stressed but critical components. Like turbine blades in aircraft jet engines.

Some of the mathematical analyses adopt a time-domain approach, and some a phase and frequency approach, depending on the kinds of tests (pulse induction versus impedance or impedance bridge). For pickup modeling we are interested in the phase and frequency approach, although the impulse response approach is useful for figuring out which component of the pickup did what.

Googling on "eddy current testing theory" yields far too many hits. Here is a very theoretical article with useful-looking references: http://class.ee.iastate.edu/jbowler/...JAP_EC_Inv.pdf.

Here is a whole list: http://www.pdf-search-engine.com/the...rrent-pdf.html.

Enough. The problem will not be too few good sources.

[Mike Sulzer]

Here is a very theoretical article with useful-looking references: http://class.ee.iastate.edu/jbowler/...JAP_EC_Inv.pdf.

Here is a whole list: http://www.pdf-search-engine.com/the...rrent-pdf.html.

I looked over that article (cannot say I "read" it; would not be finished yet!), but not at any of the references. We should be able to avoid the complexity of solving Maxwell's equations. Perhaps the most general we need to get is to think a pole piece as made up up several concentric cylinders, each of which is a secondary with its own resistance and leakage inductance. We should be able to determine the relative currents in each cylinder as a function of frequency by using using a well-known skin depth analysis, or so I would hope!

[Joe Gwinn]

I looked over that article (cannot say I "read" it; would not be finished yet!), but not at any of the references. We should be able to avoid the complexity of solving Maxwell's equations. Perhaps the most general we need to get is to think a pole piece as made up up several concentric cylinders, each of which is a secondary with its own resistance and leakage inductance. We should be able to determine the relative currents in each cylinder as a function of frequency by using using a well-known skin depth analysis, or so I would hope!

After all the fancy math, they should come to a simple answer. It's there; keep looking. I've seen it in pulse induction, but was not looking for phase and amplitude answers at the time.

The transformers approach has the problem that if the transformer is perfect, it all reduces to one resistance.

[Mike Sulzer]

After all the fancy math, they should come to a simple answer. It's there; keep looking. I've seen it in pulse induction, but was not looking for phase and amplitude answers at the time.

The transformers approach has the problem that if the transformer is perfect, it all reduces to one resistance.

The transformer is leaky. Each secondary has its own leakage inductor depending in value on its inductance (L_2x) and each has its own resistance (R_x). Thus the RL time constants are in general different.

[Mike Sulzer]

This is unexpectedly easy.

The skin depth of steel is surprisingly small because of the large mu (about 1000). This means that a single "secondary" is a pretty good approximation at low frequencies, and very good at high frequencies; that is, nearly all the current flows close to the outside of the pole piece. Since the depth varies with the square root of frequency, the resistance rises with frequency. When Re increases with the square root of frequency, the agreement gets much better. I will finish up some plots and post them in the next couple of days.

[Mike Sulzer]

Model with frequency variable resistance

The purpose of this post is to describe an improvement to the model in #18, as mentioned in #25. The problem is to find a physical effect that can do this, using Joe's suggestion that varying one of the components with the square root of frequency might make the necessary improvement. Skin depth (http://en.wikipedia.org/wiki/Skin_depth) varies inversely with the square root of frequency; so this seems like a possibility. If the skin depth is small (that is, most of the current flows close to the surface of the pole piece), then the analysis is simplified because it is not necessary to consider multiple layers of current with different values. Given the high permeability of the steel used in pole pieces (about 1000), and the fact that the skin depth varies with the inverse of the square root of permeability, the skin depth is quite a bit smaller than one might expect if used to that of non-magnetic conductors. It is small enough so that it is reasonable to proceed with a single "secondary" with the frequency dependent resistor (Re) in series.

So the constant Re used in #18 is replaced with a frequency varying value. The values of the components are varied to get good agreement with the measurements, and the results are here: http://www.naic.edu/~sulzer/hbAmpSD.png, and here: http://www.naic.edu/~sulzer/hbPhaseSD.png. The values are: L1 = 4.65, Rp = 8500, Llp = 7, Re = 130K at 1000 Hz, varies with inverse of frequency, Zc = 40.5 pf, Rc = 24K.

This is now looking pretty good. It might be possible to do better with an automatic method for searching for the best component values, and so it is time to implement that.