Someone having fun with magnets

[Joe Gwinn]

Yes, adequately modeled. Do you have any idea how hard it is to adjust four partially interacting parameters simultaneously by hand for a good fit? This really needs to be done by the usual minimization procedure one uses with non-linear least squares fitting by computer. But that is a lot of work to implement.

If one will write the optimizer engine from scratch, it is a lot of work. However, such engines are available in Mathematica, MATLAB, and probably SciLab as well.

And there are bound to be some additional small physical effects that are not accounted for.

But how much do the two effects found in this work matter? When the pickup is loaded by the cable capacitance, the peak comes way down in frequency. How bad is the simple model in this case? It would be interesting if part of the difference in the sound between pickups using steel cores and those using alnico magnets as cores is the differences in the shapes of the frequency response curves due to differences in these subtle eddy current effects. But this is certainly not obvious, and it would not be so easy to show that it is true.

Well I was teasing you about being so certain that this model was adequate, which is really in the eye of the beholder. On this point, a Golden Ear may have a different opinion than a Tin Ear. Nor is the amount of effort to achieve the model relevant to its adequacy.

To avoid the subjectivity of human hearing, a standard approach is to show that the model errors are less than the scatter in the properties of pickups in this case, so there is no point in doing a better job.

You are making such an argument above, but the classic rejoinder to the last sentence is to assert that in fact people can hear such differences, so the model isn't yet good enough. This quickly becomes circular. Which leads us into double-blind testing.

[Mike Sulzer]

Yes, I was talking about the changing magnetic field from the vibrating string. I know that the word electromagnetic makes one think only of radio waves, but there is more to it than that. The referenced theory is complete in that it handles all three limiting cases (magnetic, electric, and radio waves) with a single mathematical framework, and everything in between.

But for ease of understanding and discussion, it is good that we are in the low frequency limit where the magnetic and electric cases appear so separate and simple.

[Mike Sulzer]

If one will write the optimizer engine from scratch, it is a lot of work. However, such engines are available in Mathematica, MATLAB, and probably SciLab as well.

I have code for doing this kind of thing. It is part of what I do for a living. But to do it efficiently, you want the analytic partial derivatives of the function. This is what I was referring to. Of course in this case there probably is no reason to be efficient. I am just used to doing many thousands of such fits.

Well I was teasing you about being so certain that this model was adequate, which is really in the eye of the beholder. On this point, a Golden Ear may have a different opinion than a Tin Ear. Nor is the amount of effort to achieve the model relevant to its adequacy.

But is relevant to what you do with the results. At this point it is probably better to investigate how much these effects matter, rather than look for even more subtle effects in the model.

To avoid the subjectivity of human hearing, a standard approach is to show that the model errors are less than the scatter in the properties of pickups in this case, so there is no point in doing a better job.

Maybe there is. If the scatter means that each pickup has a different sound, one wants to understand even how that happens.

You are making such an argument above, but the classic rejoinder to the last sentence is to assert that in fact people can hear such differences, so the model isn't yet good enough. This quickly becomes circular. Which leads us into double-blind testing.

I am not convinced people are hearing the differences if the double blind testing has not been done.

[David Schwab]

Yes, adequately modeled. Do you have any idea how hard it is to adjust four patitally interacting parameters simultaneousy by hand for a good fit? This really needs to be done by the usual minimization procedure one uses with non-linear least squares fitting by computer. But that is a lot of work to implement.

And there are bound to be some additional small physical effects that are not accounted for.

Nice work Mike.

It would be interesting if part of the difference in the sound between pickups using steel cores and those using alnico magnets as cores is the differences in the shapes of the frequeny response curves due to differences in these subtle eddy current effects. But this is certainly not obvious, and it would not be so easy to show that it is true.

I'm sure it does have an effect, and also the steel poles would increase the inductance, wouldn't you think?

Probably everything that can effect the pickup, does, in some small way. Then you add those changes up and you can hear it.

[David Schwab]

You are making such an argument above, but the classic rejoinder to the last sentence is to assert that in fact people can hear such differences, so the model isn't yet good enough. This quickly becomes circular. Which leads us into double-blind testing.

At times you can hear differences in things, but the difference is so small, you can't even describe it.

I remembering listening to a pickup connected to a phase (polarity reversal) switch, and could hear a small difference when I switched it, even though I was listening to the pickup solo.

I couldn't even tell you what I heard, but it was different.

Probably wouldn't hear that anymore at my current age!

[Mike Sulzer]

Nice work Mike.



I'm sure it does have an effect, and also the steel poles would increase the inductance, wouldn't you think?

Probably everything that can effect the pickup, does, in some small way. Then you add those changes up and you can hear it.

That is exactly the problem. There are multiple effects from changing a material, and it is really hard to sort them all out.

[Joe Gwinn]

But for ease of understanding and discussion, it is good that we are in the low frequency limit where the magnetic and electric cases appear so separate and simple.

Actually, in the theory the distinction is between impedances. A magnetic field is low impedance (much less than 377 ohms), a radio wave is 377 ohms, and an electric field is high impedance (much larger than 377 ohms). Frequency is orthogonal to impedance, but both matter.

[Joe Gwinn]

I have code for doing this kind of thing. It is part of what I do for a living. But to do it efficiently, you want the analytic partial derivatives of the function. This is what I was referring to. Of course in this case there probably is no reason to be efficient. I am just used to doing many thousands of such fits.

Brute force, running overnight if needed. One can do without analytic partials, instead using a numerical approximation.

But is relevant to what you do with the results. At this point it is probably better to investigate how much these effects matter, rather than look for even more subtle effects in the model.

In the lab is where one gets some idea which effects are important and which can be neglected. Without this kind of practical information, modelling physical systems quickly becomes intractable.

Maybe there is. If the scatter means that each pickup has a different sound, one wants to understand even how that happens.

Yes, if it is shown that a particular kind of scatter leads to audible differences of a kind that people care about.

I am not convinced people are hearing the differences if the double blind testing has not been done.

For sure. We have had some threads on just that. And when there is a difference requiring a double-blind approach, most often the purported effect vanishes when subjectivity is excluded.

[Mike Sulzer]

In the lab is where one gets some idea which effects are important and which can be neglected. Without this kind of practical information, modelling physical systems quickly becomes intractable.

Yes, and this brings up another aspect of modeling, beyond just finding the impedance of the pickup: modeling the frequency response. The obvious way is to use the model found from the impedance and put a voltage source in series with the coil inductance.

There is something missing from this model, and it might be important in the situation discussed concerning eddy currents. The changing flux from the vibrating string induces a voltage around the pole piece. Suppose the pole piece were super-conducting, and let's neglect the leakage flux for the moment. Then we have a resistor of zero ohms appearing across the coil, and the voltage source would make an infinite current. This is not possible. The current in the pole piece induces a magnetic field that cancels the component from the vibrating string, so that there is no voltage around the pole piece.

With a finite load there should be partial cancellation. So the current in the eddy resistor (which is in parallel with the coil) should lower the pickup voltage. With the leakage inductor in series, the effect becomes frequency dependent, and so is a potential issue for determining the pickup frequency response. How does one compute the size of this effect?

[Joe Gwinn]

Yes, and this brings up another aspect of modeling, beyond just finding the impedance of the pickup: modeling the frequency response. The obvious way is to use the model found from the impedance and put a voltage source in series with the coil inductance.

OK.

There is something missing from this model, and it might be important in the situation discussed concerning eddy currents. The changing flux from the vibrating string induces a voltage around the pole piece. Suppose the pole piece were super-conducting, and let's neglect the leakage flux for the moment. Then we have a resistor of zero ohms appearing across the coil, and the voltage source would make an infinite current. This is not possible. The current in the pole piece induces a magnetic field that cancels the component from the vibrating string, so that there is no voltage around the pole piece.

That doesn't sound quite right. A superconducting slug will exclude magnetic fields. The resulting surface current is not infinite, it's exactly enough to cancel what would have been the internal field, by Lenz's Law. The effect of the zero resistance is not to yield infinite current, it's to ensure that the eddy current never decays.

A good way to keep out of trouble when dealing with actual or modelled zeros and/or infinities is to consider energy, because conservation of energy always applies. An infinite current in a loop (the surface of a slug) would generate an infinite magnetic field containing infinite energy. This may prove hard to arrange. If you succeed in getting this to work, a trip to Stockholm is assured.

By modelled I mean where one has simplified a model by setting some parameter to zero or infinity. If the simplification requires or implies that some kind of energy be infinite, look out. Nonsense results are likely.

With a finite load there should be partial cancellation. So the current in the eddy resistor (which is in parallel with the coil) should lower the pickup voltage. With the leakage inductor in series, the effect becomes frequency dependent, and so is a potential issue for determining the pickup frequency response. How does one compute the size of this effect?

I don't quite picture what you are proposing here, but I will say that one always tries lumped-element circuit models first, because the math is so much simpler. One resorts to distributed-element models only if cornered.

[Mike Sulzer]

OK.

That doesn't sound quite right. A superconducting slug will exclude magnetic fields. The resulting surface current is not infinite, it's exactly enough to cancel what would have been the internal field, by Lenz's Law. The effect of the zero resistance is not to yield infinite current, it's to ensure that the eddy current never decays.

Exactly. If one considers the model proposed in the preceding paragraph, the voltage source should cause infinite current to flow. If we consider the physics, we see that this cannot be true. Therefore, something is missing from the model.

I don't quite picture what you are proposing here, but I will say that one always tries lumped-element circuit models first, because the math is so much simpler. One resorts to distributed-element models only if cornered.

I am not proposing that we need distributed elements, but we do need something!