cutting posts?

[Joe Gwinn]

... the idea is to divide the pickup physics into two parts:
1. The static field magnetizes the string.
2. Then replace the magnetized string with a small magnet, remove the strong static field, and look at the induction resulting from the small "string magnet"

An alternative occurs to me. The motion of the string is fairly small, and the magnetic circuit is dominated by air, so the main effect is to cause the total magnetic flux to vary slightly. If FEMM won't handle moving the string, we can nonetheless model the variation in flux by slightly varying the strength of the magnet. In FEMM, permanent magnets are modeled as cleverly placed current sheets (basically representing a permanent magnet as if there were a solenoidal coil wound on unmagnetized magnet material). We can add our own little single-layer coil on the magnet, and feed this coil with AC, which will cause the net magnetic field to vary by the amplitude of the AC field induced by that little coil.

We can then go directly to the original question, the voltage induced in coils wound around the two blades, as we vary the dimensions and shapes of the blades.

[Mike Sulzer]

It looks the same to me, the first order, when you put the black tape over the two poles. I guess we need to discuss what "significant" means. More to the point, we need to hang a number on it.

Joe, what is the same and what is different as you see it?

[Mike Sulzer]

OK, but why does it matter?

Does it not seem remarkable that we make a large change in the magnetic circuit, increasing the field at the bottom, the field stays the same at the top? More than once, I have seen here the idea that the number of flux lines emerging from the magnet is fixed, and so diverting some to the bottom robs the top. It does not. As a significant quantity of additional steel becomes magnetized, this has an effect on the magnet, which is permeable. So its magnetization increases, as it can until it saturates.

[Mike Sulzer]

An alternative occurs to me. The motion of the string is fairly small, and the magnetic circuit is dominated by air, so the main effect is to cause the total magnetic flux to vary slightly. If FEMM won't handle moving the string, we can nonetheless model the variation in flux by slightly varying the strength of the magnet. In FEMM, permanent magnets are modeled as cleverly placed current sheets (basically representing a permanent magnet as if there were a solenoidal coil wound on unmagnetized magnet material). We can add our own little single-layer coil on the magnet, and feed this coil with AC, which will cause the net magnetic field to vary by the amplitude of the AC field induced by that little coil.

We can then go directly to the original question, the voltage induced in coils wound around the two blades, as we vary the dimensions and shapes of the blades.

If I understand the idea, we put a coil around the pickup magnet and vary the strength of the magnet by adjusting the current through the coil. This scales the flux pattern on both sides in the same way. This is not necessarily what happens a string vibrates over the pole pieces.

In the actual case we have two possibly important variations:
1. The vertical height gradient of the static field is generally different over the two different poles, and so the magnetization induced in the string is different.
2. The flux through the core and coil from the magnetized string decreases with distance from the string. (We know this from the fact that the original stacked humbucker works, this is also seen in modeling results.) The details of this could depend upon the distance of the string from the pole, and the size and shape of the pole, so we have to look at this carefully.

[David Schwab]

Does it not seem remarkable that we make a large change in the magnetic circuit, increasing the field at the bottom, the field stays the same at the top? More than once, I have seen here the idea that the number of flux lines emerging from the magnet is fixed, and so diverting some to the bottom robs the top. It does not. As a significant quantity of additional steel becomes magnetized, this has an effect on the magnet, which is permeable. So its magnetization increases, as it can until it saturates.

Is that based on the FEMM simulations, or real life? Because in real life you will get less pull on the top poles with the screws extending out he back. Cutting the screws flush increases the pull. It's very noticeable.

I've said here for a few years now that I feel the FEMM simulations look to be exaggerating the field in many cases. You haven't proven anything from the simulations IMO, but I'd like to see tests on real pickups. It's easy to set up a pull test.

But it just takes sticking a screw diver to the poles to see the increase in pull after you cut the backs off.

[Joe Gwinn]

Joe, what is the same and what is different as you see it?

I'm looking at the density and geometry of the flux lines in the air slightly away from the metal, but not their exact location.

[Joe Gwinn]

Does it not seem remarkable that we make a large change in the magnetic circuit, increasing the field at the bottom, the field stays the same at the top? More than once, I have seen here the idea that the number of flux lines emerging from the magnet is fixed, and so diverting some to the bottom robs the top. It does not. As a significant quantity of additional steel becomes magnetized, this has an effect on the magnet, which is permeable. So its magnetization increases, as it can until it saturates.

We need to keep the original question in mind: How do the voltages generated by the two coils around the two poles compare if the size and shape of the poles differ?

Said another way, there is no place for field strength (in Gauss or Teslas) in Faraday's Law of Induction, which instead depends on changes in total flux (in Lines or Webers).

[Joe Gwinn]

If I understand the idea, we put a coil around the pickup magnet and vary the strength of the magnet by adjusting the current through the coil. This scales the flux pattern on both sides in the same way.

Exactly. But I would drive the coil with AC. Then FEMM can compute eddy currents and induced voltages for us.

This is not necessarily what happens a string vibrates over the pole pieces.

True enough. The claim is that it is nonetheless close enough to be instructive and useful, despite the limitations of FEMM.

In the actual case we have two possibly important variations:
1. The vertical height gradient of the static field is generally different over the two different poles, and so the magnetization induced in the string is different.

I'm not sure that the gradients are all that different, but a few FEMM plots with a long rectangle representing the string at various heights above the poles will tell us.

What is the actual magnitude of string motion above the pickup, and how much does this change the total reluctance of the magnetic path? No matter where the string is, most of the path is in air (with < 1/1000 the permittivity of steel), so the change in total flux is going to be fractional.

One rough way to estimate the relative effect of plucking is the following experiment, which requires an oscilloscope: First, slide a steel keeper between the humbucker and the strings, magnetically shorting the poles together. Abruptly pull the keeper out and measure the voltage generated. Second, pluck the string, and again measure the generated voltage. The ratio of voltages will tell the story.

If one plucks lightly, the guitar still works, and one can use the approximation that the shape and gradient of fields is unchanged, while the total flux changes slightly around the average. One way to change the flux but not the shape is to modulate the strength of the magnet.

2. The flux through the core and coil from the magnetized string decreases with distance from the string. (We know this from the fact that the original stacked humbucker works, this is also seen in modeling results.) The details of this could depend upon the distance of the string from the pole, and the size and shape of the pole, so we have to look at this carefully.

I'm not quite sure what this means, but I assume that the mental model is that of the magnetized string inducing flux in the nearby humbucker. One can certainly use this approach, one of many mathematically equivalent approaches, and the result will be that as the magnetized string becomes closer the reluctance of the magnetic circuit is reduced and the flux increases, and so on. The problem is that the strength of the induced magnet increases as well, complicating the analysis.

A traditional and simpler way to view this problem is as a variable-reluctance transducer. In this view, the only effect of changing the distance between the string and the poles is to cause the total magnetic flux through the two coils to vary. The sole source of flux is the magnet, and the string is treated as just another carrier of flux. The induced voltage is proportional to the time rate of change of the total magnetic flux.

[Mike Sulzer]

We need to keep the original question in mind: How do the voltages generated by the two coils around the two poles compare if the size and shape of the poles differ?

Said another way, there is no place for field strength (in Gauss or Teslas) in Faraday's Law of Induction, which instead depends on changes in total flux (in Lines or Webers).

I do understand how magnetic induction works; my reasons for looking at the static field are:
1. It is the first step in the two step method.
2. In analyzing a problem in science, it usually pays to disassemble it, and look at the pieces.

In this case, reason 2. has payed off. The fields of the flush and extended cases are the same, and so one possible contribution to the difference is removed. Thus, it is only necessary to perform the second step in the two step method:compute the differences in the flux changes through the cores (restricted to where the coil would be located) in the flush and extended cases. One can do this approximately by looking at the relative differences in the flux itself. One linearizes the magnetic circuit by computing the relative permeabilities (what FEMM wants for linear models) of the alnico and steel at approximately .5 Tesla (from the static field calculation. Thus one examines the B-H tables in FEMM, computing delta B and delta H at the operating point. One gets 150 for the alnico and 964 for the steel. One also turns off the static field of the alnico. Now one puts identical small magnets above the poles and looks at the flux through the poles in the two cases. I will try to do this final step in the next couple of days. One might also want to a magnet over just one pole at a time to get a measure of the coupling between the two poles.

[David Schwab]

The problem is that the strength of the induced magnet increases as well, complicating the analysis.

Thank you... I was trying to get that point across, but wasn't getting it into words correctly.

The moving string changes the reluctance of the circuit, and the closer the string is, the more sensitive it is to smaller movements, but the magnetic field is not getting any stronger than what the permanent magnet generates.

So moving a magnet over the pickup to simulate a string is not exactly right.

I do realize the idea is trying to come up with "tricks" to get the software to work in this situation.

[Joe Gwinn]

The moving string changes the reluctance of the circuit, and the closer the string is, the more sensitive it is to smaller movements, but the magnetic field is not getting any stronger than what the permanent magnet generates.

Almost true, but mixes a few things up. It's true that moving the string closer reduces the reluctance and thus increases the flux. The field strength as measured in Gauss at the poles will also increase, but it's the variation in total flux flowing through the coils that generates the voltage. It is also true that the flux change for a given string motion is larger the closer the string is to poles. However, the percentage change in all of these things isn't going to be large, and a possible approximation is to assume (slightly contrary to fact) that the average DC magnetic flux does not change, instead imposing an AC wiggle riding on top of this DC flux.

So moving a magnet over the pickup to simulate a string is not exactly right.

True, but it isn't exactly wrong either.

I do realize the idea is trying to come up with "tricks" to get the software to work in this situation.

No, no, no ... it's not a "trick", it's a valid approximation. Of course there is the small matter of deciding what "valid" means.

So, to summarize, we seek a set of approximations that allow us to use FEMM to answer some general questions about how humbuckers work, it being understood that the answers will not be exact.

[Mike Sulzer]

I'm looking at the density and geometry of the flux lines in the air slightly away from the metal, but not their exact location.

Plotting the vertical field along vertical contours through the middles of the narrow and wide poles does a better job of showing the differences. (The sign of the wide pole results has been reversed for a better comparison.)

http://www.naic.edu/~sulzer/narrowWideComp.png

[Mike Sulzer]

Exactly. But I would drive the coil with AC. Then FEMM can compute eddy currents and induced voltages for us.

I think this is an excellent way to determine the ac losses (ac resistance).

A traditional and simpler way to view this problem is as a variable-reluctance transducer. In this view, the only effect of changing the distance between the string and the poles is to cause the total magnetic flux through the two coils to vary. The sole source of flux is the magnet, and the string is treated as just another carrier of flux. The induced voltage is proportional to the time rate of change of the total magnetic flux.

A fine traditional view, and one that leads to simple solutions under certain conditions, especially when the flux path is closed, except for one or more small air gaps. But a guitar pickup is nothing like that. The low reluctance pole pieces connect to high reluctance air. If one thinks of a gap as something that drops most of the MMF and contains approximately straight flux lines, what we have is not a gap. The flux lines are curved; how does one solve for their paths in a simple way? The flux peels off along the poles, and is mostly gone before the ends. The other tiny bit of low reluctance material, the string, is separated from the poles by low reluctance air. One does not have a simple circuit, but rather a situation that requires solving a differential equation.

I doubt that a solution using this method will be any simpler than what I propose, but please prove me wrong.

[Joe Gwinn]

Plotting the vertical field along vertical contours through the middles of the narrow and wide poles does a better job of showing the differences. (The sign of the wide pole results has been reversed for a better comparison.)

http://www.naic.edu/~sulzer/narrowWideComp.png

Interestingly, the field is the same at the magnet and at the string. If the field is the same at the string, won't the induced magnet in the string be the same, leading to identical generated voltage?

The higher field in and above the thinner blade pole is as expected, as the same amount of flux is crowding into a smaller area in the thinner blade.

[Joe Gwinn]

A fine traditional view, and one that leads to simple solutions under certain conditions, especially when the flux path is closed, except for one or more small air gaps. But a guitar pickup is nothing like that. The low reluctance pole pieces connect to high reluctance air. If one thinks of a gap as something that drops most of the MMF and contains approximately straight flux lines, what we have is not a gap. The flux lines are curved; how does one solve for their paths in a simple way? The flux peels off along the poles, and is mostly gone before the ends. The other tiny bit of low reluctance material, the string, is separated from the poles by low reluctance air. One does not have a simple circuit, but rather a situation that requires solving a differential equation.

Umm. Most variable reluctance transducers resemble a guitar pickup in that the "gap" is quite large, and dominates the magnetic circuit. Often the transducer is in a iron cup, to reduce the air gap. Just like some guitar pickups with side shields made of steel.

And so on. For example, the transducers used to sense the passing of the teeth of a steel gear in a machine.

It's true that magnetic lines curve, but people manage somehow, and did so long before computers were invented.

I doubt that a solution using this method will be any simpler than what I propose, but please prove me wrong.

Well, there may be a reason why one method became the traditional approach. Nor are guitar pickups all so special.