# Physically-Based Eddy Current Equivalent Circuit?

Discussion in 'Just Pickups' started by TeleTucson, Jul 8, 2018.

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Recent postings got me thinking about eddy currents and their sources. This posting is intended to introduce an equivalent circuit model which illustrates how eddy currents modify the LCR resonant circuit of the guitar by being driven by the dynamic magnetic field of the pickup coil. Consistent with experiment, this makes the effects of eddy currents measurable even in the absence of excitations by the dynamic magnetic field perturbation introduced by the string (the “string field”) - i.e., the effects of eddy currents are inherent to the circuit. This is consistent with the usual case where the resonant response of a circuit is fairly fundamental to the circuit and is less dependent on the specifics of the excitation.

The goal is to develop a physically-based equivalent circuit model for eddy currents. In looking on the web, I’ve seen the excellent work of Helmuth Lemme. In the case of eddy currents, Lemme adopted a model that splits the pickup coil inductance with a shunt resistor to ground. It’s a bit hard for me to see the physical justification for this model, so I tried to construct a model that seemed more physical to me.

Starting with the belief that the coil is coupling to the metals in which the eddy currents flow, it seemed reasonable to start thinking about the eddy currents essentially comprising another equivalent inductor (i.e., loops of current in a pickup cover, for example) and how those currents feed back into the pickup system.

This suggests to me a model of mutually coupled inductors, even though the inductance of the metal supporting the eddy currents will be very, very small. However, the resistance is very, very low so it can support significant currents – in effect the “eddy inductor” is nearly shorted by closed metal conductive paths with very low resistance.

This points to the following equivalent circuit using "transformer-like" mutually coupled inductors, but with adjustable coupling strength:

The pickup coil with it's inductance Lcoil is inductively coupled to an “eddy inductor” Leddy through coupling strength k (using the standard nomenclature for mutual inductance) and this eddy inductor is shunted with an “eddy resistance” Reddy.

The rest of the pickup is completely conventional apart from this simple, physically-based modification.

To pick reasonable values for these parameters, we can use something very down-to-earth. Let’s suppose a 14 gauge copper wire loop is placed at the top of a tele single coil, with a loop dimension of 15mm by 60mm to encompass the area of the pickup. For illustration purposes, we’ll use a 2.6H pickup with 7.5K resistance, although the latter doesn’t matter much.

The wire loop inductance can be calculated for 14 gauge wire to be Leddy = 79.2 nH, and its resistance for copper will be Reddy = 0.00124 ohms. We will assume that the loop placement right on top of the coil (i.e., like a cover) gives reasonably good coupling to the coil, and thus we will let k = 0.75.

This circuit is simulated with a 0 dBV signal, a combined winding and cable capacitance of 500pF, and a 250K ohm load (with load) or for no-load using 100pF and 10M ohm. To simulate "no eddy" we arbitrarily let Reddy change from 0.00124 ohms to 10K ohm to effectively take that effect out of the circuit (or you could put k=0 instead), and we get resonance peak heights of

Unloaded; no eddy current: 24.04 dBV

Unloaded; copper eddy current: 10.99 dBV (13.05 dB drop due to eddy current)

Loaded; no eddy current: 8.16 dBV

Loaded; copper eddy copper current: 3.04 dBV (5.12 dB drop due to eddy current)

These are quite large reductions using completely physically-based numbers.

If we use brass instead of copper for the loop, "no eddy” numbers obviously stay the same by definition. However the resistivity of the eddy current loop is increased by 3.5X, but still results in large resonant peak reductions:

Unloaded; brass eddy current: 6.99 dBV (17.05 dB drop due to eddy current)

Loaded; brass eddy current: 4.29 dBV (3.87 dB drop due to eddy current)

If we now go to nickel-silver, the resistivity is increased further still by 4.8X relative to brass, resulting in resonance peaks of:

Unloaded with nickel-silver eddy current: 14.40 dBV (9.64 dB drop due to eddy current)

Loaded with nickel-silver eddy current: 7.01dBV (only 1.15 dB drop due to eddy current).

To recapture this, the example shows that for the loaded guitar, which is most relevant, switching from a brass loop placed where the pickup cover would go to a nickel-silver loop of the same gauge would give you a 2.72 dB increase in resonant peak height. For the unloaded case the change is 7.41 dB, even more.

In summary, this equivalent circuit model is fairly physically based and quite simple and appears to give numbers that are reasonably consistent with observed behavior. In addition to the examples illustrated here, the model also illustrates that the eddy current peak reductions are quite dependent on the mutual inductance coupling parameter k, which also illustrates why metals that are not directly over the pickup coil will have much less impact.

The model is easy to explore on your favorite circuit simulator, and seems to cover a range of observed behavior without invoking anything unreasonable.

If this model is well-known, and already recognized for its value and/or shortcomings, I apologize in advance . Maybe it's old news and discarded. But it seemed helpful to me base on what I've seen.

Edit: I changed the "no-load" numbers from the original posting to make for a more realistic "no load" value. Previous numbers kept the capacitance at 500pF for no load, but now the breakdown assumes the no-load capacitance is just 100pF, representing a more typical coil capacitance, while the loaded capacitance is 500pF. Also fixed a few typos ...

Last edited: Jul 8, 2018
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2. ### rigateleTele-Afflicted

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Did you observe the frequency dependence that you do in practice? The 6 component model of John H.'s GuitarFreak spreadsheet program has so far proved to be the one that provided the best fit for a large number of different pickups.

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Oh, maybe I wasn't clear - this isn't meant to replace any of that. I agree that GuitarFreak has almost all the important features! It's very versatile and well-written with a great user interface, and I'm a big fan. But it doesn't allow for the inclusion of eddy currents, unless he has an updated version I haven't seen. I hadn't even bothered to put in tone controls, etc., because I wasn't trying to displace that model at all.

The point of the post above would be to just add to his model in one simple way. I.e., it would just switch out the inductance of the pickup coil for one that has the inductance of the pickup plus the mutual inductance coupled by "k" to the "eddy inductor" and the "eddy resistor". I'm proposing a way to add in eddy currents if anyone cares - and while that has also been done by others and discussed online, the approach above seems more physically based to me and seems to produce impact that is about the right level without invoking anything particularly ad-hoc.

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4. ### rigateleTele-Afflicted

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Yes, it models eddy losses since version 6.4. It took me a while to find a diagram:

I tried a while back to make Spice models with a loosely coupled transformer, along the lines of what you are suggesting. I can't find them now, they might be on a different computer. Those never succeeded in tracking the frequency dependence.

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OK - I hadn't seen Hewitt's update to include an eddy current model. I only had 6.3 so I'll update that. I'm proposing something a little different that seems physically-based to me. Maybe there's a solid rationale to have elements shunted to ground as Hewitt has implemented, but it is hard for me to see it right away. I was just surprised that the mutual inductance version I posted provided a reasonable level of peak reduction with such miniscule, but realistic, additional inductances and resistances. As I tried to say, "If this model is well-known, and already recognized for its value and/or shortcomings, I apologize in advance. Maybe it's old news and discarded!"

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6. ### rigateleTele-Afflicted

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There was a long thread about this on another forum. I can't seem to find it. I also pleaded for a more physically based model, but I could not make it work. Of course, that doesn't mean that there isn't one that works.

There was so much stuff done and never indexed or catalogued that it is hard to dig it up sometimes. We've all been focused on moving ahead with new questions. I periodically complain that not enough people are doing and publishing research, so here I will do it again. Nobody has had time to make it into a unified book or giant website. Other than "the professor" - Manfred Zoller.

So, if I recall strictly from memory, we did this experiment where the cover was moveable towards and away from the pickup, so that it was either closer to the exciter, or the pickup, respectively. It was supposed to determine the answer to the very question that you are raising - how closely the loss is coupled to the coil circuit. It is my recollection that the values didn't change very much. But I have to admit that my memory is not clear enough to be sure. It could be repeated, but as I mentioned, I have no lab now. But thinking on it now, that one should be done as a 2 terminal measurement.

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I do like the model to be physically-based, otherwise modeling starts to lose it's purpose (to reliably predict how modifications will impact performance). So introducing circuit components or architectures that don't really correspond to physical phenomena and fitting those models to match observed behavior has limited value and can even be misleading in my experience.

Just for fun, a few minutes ago I actually soldered up a Romex 14 gauge strand insulated loop like in the simple example, and put it on and off the tele neck pickup just sitting on the pickguard (a tele neck pickup with a nickel-silver cover). The difference in brightness of the sound was apparent, relatively easy to hear - and I would expect the "Romex loop/no Romex loop" comparison to be a more dramatic comparison if this pickup had no nickel-silver cover. The point is that this insulated Romex strand loop obviously has no connection to ground - only inductively coupled to the pickup down in the plane of the pickguard. To model that, you should use a physically-based model that corresponds to the reality of what was done - why should the model have non-inductive connections to the circuit that don't exist? In this simple case, it's actually easy to model the real deal pretty accurately. The more distributed eddy currents you'd find in a cover, etc., will flow in a more distributed way than the wire loop, but would still tend to concentrate at the perimeter and I'm guessing can be modeled fairly well by an effective Leddy and Reddy and a coupling strength k to a reasonable approximation.

For the record, I'm aware that this model alone won't produce an 18dB/octave roll-off at very high frequencies that is apparently reported in some high eddy current loss circumstances, so I'd still like to implement a circuit modification to see how to capture that but really want it to be physically based so it is readily understandable (including the insights that would come with that about specifically how to reduce it, etc.). For now I'm wanting to build upon this model because it makes sense to me.

Last edited: Jul 8, 2018
8. ### Old Tele manFriend of Leo's

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You are aware that those "eddy currents" are occurring INSIDE the individual core lamination sheets and NOT just within the pri-to-sec magnetic coupling field?

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There are NO core laminate sheets - a pickup is not a laminate core transformer. Also eddy currents are not within a "pri-to-sec" coupling field. The eddy currents relevant to a guitar pickup are genuine current loops in conductors exposed to the dynamically changing magnetic field of the pickup system. The transformer analogy in the model is just to capture the mutual inductive coupling, not a "real" transformer that might use core laminate sheets, etc. The eddy current for this model is flowing in the upper loop that is shunted with a very low eddy current resistance.

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10. ### Old Tele manFriend of Leo's

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Then maybe you should consider "skin effect" within the coil windings themselves, but at audio frequencies there ain't gonna be much to measure.

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The skin depth in copper at 1kHz is 2 mm. So I'll let you consider it.

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My point...

13. ### rigateleTele-Afflicted

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Dude, that means that there is no skin effect. Not unless your wire is much more than 4mm thick.

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14. ### Old Tele manFriend of Leo's

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Exactly my point...skin effect is a "high-frequency" (which audio ain't) effect noticeable in RF ranges...not audio!

15. ### Antigua TeleFriend of Leo's

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I suspect this model might be valid, and I have a suspicion that the dramatic frequency dependent aspect of eddy currents might have something to do with the use of an external exciter coil for testing.

For example, here's a Fender Fidel'tron, renown for it's impressive eddy currents, measured with an external excitation coil:

It shows a sharp drop ahead of the resonance, which agrees with what Helmuth Lemme said should occur in conjunction with eddy currents:

But Helmuth Lemme used an external exciter coil also:

So for all I know, his conclusions might be based on the use of an exciter coil, and an effect that is potentially related to its use.

Unfortunately, I can't really test this because I get an sort of inconclusive, non-linear result when I use the integrator circuit directly (black line), and without the integrator (blue line), there is a slope which is harder to make sense of:

The only way to know for sure would be to have a finely integrated direct measurement, so that it can be seen whether the drop off that precedes the resonances is present in both cases, or just the case where an external exciter is used.

And regarding the Guitar Freak eddy current model, I'm still convinced that it's essentially achieving a curve match by forcing a second resonance which doesn't occur in real life. There has to be something else going on, maybe the exciter coil is that something else. It makes sense to me that eddy currents should model like a transformer and a resistor, I can't conceive of why that model should not be valid. That being said, I would be a little surprised in the exciter coils are ever to blame, given the poor magnetic coupling involved.

Last edited: Jul 8, 2018
16. ### Old Tele manFriend of Leo's

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Was CEMF caused by the driven coil field accounted for?

17. ### rigateleTele-Afflicted

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To get a direct readout of the pickup impedance Z, you can use the I-V method. The usual way is to apply a constant voltage sweep V and connect the pickup to a low valued resistor R, measure I and then use Z=V/I to graph the impedance 1/Z. Since the 2 terminal model of a pickup is a parallel resonant circuit, the parallel impedance is at a maximum at resonance hence the current is at a minimum. This makes the frequency plot "upside down" compared with the RLC low pass model, with a valley instead of a peak. The D. Carson lab implementation uses a 1.5M resistor in series with the applied test generator voltage to provide an almost constant current supply. The low valued resistor in that circuit is only used to record phase information. The voltage is measured across the pickup to obtain a plot of impedance which is not inverted, but is closely proportional to V since Z=V/I and I is almost constant.

The integrator could be connected across the pickup, as it usually is, but with the addition of a 1.5M resistor to the signal generator output instead of using the external exciter coil. Hope that the signal level is high enough.

Last edited: Jul 9, 2018

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Thanks for the comments. I was driven by related thoughts about getting to the root of roll-off issues, even to the point of putting an inductive exciter and integrator into Spice to explore this aspect a bit. But that'll have to wait because Monday beckons. Regardless of the roll-off issue, I do like starting with something physically justifiable and then asking what's missing if needed. I agree that there's probably something else additional going on if it is in fact a substantial and repeatable issue.

The model above is quite interesting (at least to me) because the response is also somewhat sensitive in quite a few ways to the values of the effective Leddy and Reddy , especially in circumstances where the usual resonance frequency fresonance = 1/(2 pi sqrt(LcoilC) ) happens to lie broadly in the vicinity of the frequency given by Reddy/(2pi Leddy). That's the frequency where the magnitude of the reactive and resistive components of the "eddy loop" match. These values of Leddy and Reddy are not only where it seems to generally damp the most, but also even seem to result in significant shifts in the resonance frequency of the whole circuit to my surprise. Maybe these resonance peak frequency shifts are just the usual damping-induced reductions, however. Having a lot less resistance Reddy has less effect, and a lot more resistance eventually makes the effect go away. I find it remarkable that reasonable, normal materials and design parameters (honestly not conjured up to hype an effect) result in values where it is having such a significant effect. Maybe it's Murphy's law. But if the model ends up seeming reasonable, it may also point to a challenge in anticipating which pickups will have strong eddy current effects because they are more sensitive to design choices than one might imagine. Usually in circumstances like that, the answer for a designer is to steer well-clear of the "bad lands", which at least some of industry seems to have tried to do over the years.

Last edited: Jul 9, 2018
19. ### rigateleTele-Afflicted

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To add: Those currents must complete a circuit. Hence the currents that form a route in the axes of the plane, are much much greater than those that form a route across it in the perpendicular direction. The significance of this is that the orientation of the laminations with respect to the changing magnetic field is important.

Last edited: Jul 9, 2018

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A bit more interesting findings about the model above. When the resistance Reddy gets very, very low (usually unrealistically low) such that it is much below 2pi Leddy* fresonance , where fresonance = 1/(2 pi sqrt(Lcoil*C) ), then the "eddy loop" starts to really look like it is truly shorted at the resonance frequency because that loop impedance is then just dominated by Leddy, and the damping effect goes away, and the shorted but mutually coupled Leddy actually subtracts away a substantial amount from the coil inductance Lcoil, in an amount that depends on the coupling factor k. In fact, you can show analytically for very, very low resistance the coil resistance Lcoil gets reduced to a value of (1-k^2)*Lcoil. So this can even result in increases in the resonance frequency of the whole circuit, and for k near unity and vanishingly small resistance Reddy this could even go so far, in principle, as to essentially remove the effect of Lcoil in the circuit entirely and turn it into an RC circuit with no resonance! It is of course highly unlikely that this condition gets realized in any practical situation, unless perhaps you also have an unusually large Leddy for some reason where maybe you could start to see increases in resonance frequency with little eddy current contribution to damping. But it's nonetheless a pretty interesting consequence to think about.

In the more practical situations for guitar pickup configurations, the model predicts damping with the damping decreasing with increasing resistance of the eddy current metal, consistent with most situations that are seen.

Last edited: Jul 10, 2018
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