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Post by aquin43 on Feb 22, 2019 4:42:19 GMT -5
Modelling the stratocaster pickup as a variable reluctance device. I have tried a variable reluctance model of a stratocaster type pickup using the 2D finite element program FEMM to map the magnetic field. Because other motions and positions are rather difficult to calculate with a 2D system such as FEMM I have so far only modelled the case of a string passing directly over the centre of the magnet face and moving perpendicularly to that face. Fortunately, this is the string motion that is known to produce the major part of a pickup's output. The variable reluctance model used here assumes that the flux distribution within the coil is determined by the magnet and the geometry of the magnetic circuit. The magnetic reluctance of the pickup is high mainly because of the large air gap. In this case, the low reluctance string removes flux from the system, returning it via a remote path that couples to the coil only very weakly. Thus the string modulates the total flux in a pattern already set up by the magnet and the pickup geometry. It is also assumed that the magnet has a high enough coercivity that the string's influence on it is negligible. The actual reluctance of the string need not be known as long as it can be assumed to be small. In what follows the string is supposed to be moving bodily in a direction perpendicular to the magnet face, i.e. the pickup is not near the string end and the wavelength on the string is long compared to the pickup width. The problem was programmed in Octave, using the octavefemm interface to FEMM with FEMM running under wine on Linux - all free software. The first task is to model the magnet using FEMM in the axisymmetric mode. The magnet is a 16mm high cylinder of 5mm diameter made of Alnico 5. It is modelled as a half cylinder on the zero vertical (Z) axis. The properties of Alnico 5 from the FEMM library are used. Next, the integral of the flux density along lines extending from the magnet axis to 20mm away at various heights above the magnet pole is approximated as an average of a series of values along those lines. Due to symmetry, 20mm represents a 40mm length of string. There is some evidence that this is just outside the limit where reducing the string length begins to affect the pickup output. [/url] A plot of these values is made and a polynomial is fitted to the data giving a curve that the value of the flux intercepted by the moving string can now be assumed to follow. It turns out that the polynomial fit is very close. Assuming a sinusoidal movement of the string, the waveform of the intercepted flux is derived, in this example with a peak string movement of 1mm at 5mm above the pole. The flux waveform is harmonically analysed using an FFT, which is more convenient to program than expanding the curve algebraically about a bias point. The harmonic levels are normalised to the level of the fundamental and are multiplied by the harmonic number to account for the coil voltage being dependent on the rate of change of flux. Arthur
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Post by ms on Feb 23, 2019 14:45:57 GMT -5
When you consider a "bit" of string right over the pole piece, that is, right along the axis of the coil, the field from the string, as magnetized by the magnet, points along this axis. As your first figure shows, as the "bit" of string under consideration is further from the axis, it is illuminated by the magnet at an off vertical angle. It is the component of the field from the string pointed along the axis that contributes to the output. The field from the "bit" can be well approximated by that of a dipole, and the field of a dipole falls off significantly as its field is sampled away from its axis, and, as I said, it is the projection of this field along the axis of the coil that counts, already reduced in magnitude by the angle from the axis of the dipole. If I understand what you are doing, you are not allowing for this, but I could be wrong.
So I think the curves in your second plot do not fall off with distance from the axis nearly fast enough.
Also you do not seem to be considering the effect of the permeability of the magnets in amplifying the field through the coil from the vibrating string. But that is another matter.
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Post by aquin43 on Feb 23, 2019 19:29:56 GMT -5
When you consider a "bit" of string right over the pole piece, that is, right along the axis of the coil, the field from the string, as magnetized by the magnet, points along this axis. As your first figure shows, as the "bit" of string under consideration is further from the axis, it is illuminated by the magnet at an off vertical angle. It is the component of the field from the string pointed along the axis that contributes to the output. The field from the "bit" can be well approximated by that of a dipole, and the field of a dipole falls off significantly as its field is sampled away from its axis, and, as I said, it is the projection of this field along the axis of the coil that counts, already reduced in magnitude by the angle from the axis of the dipole. If I understand what you are doing, you are not allowing for this, but I could be wrong. So I think the curves in your second plot do not fall off with distance from the axis nearly fast enough. Also you do not seem to be considering the effect of the permeability of the magnets in amplifying the field through the coil from the vibrating string. But that is another matter. All of the evidence I have points to the magnetisation of the string being directed along its length with the exception of a very small region directly over the magnet. This region becomes smaller as the permeability of the string is increased. The string absorbs and redirects flux along its length regardless of the direction it arrives from. Flux that passes through the magnet is already diverging at the magnet pole, so it is not just the axial component of the flux near the string that passes through the coil.
The focussing effect of the magnet is a function of the distribution of the pickup reluctance. That is not modelled; the reluctance is just assumed to be high compared with the reluctance of the string and the distribution not to change as the string moves so that the total flux redirected by the string is directly reflected in the coil flux variations. This version of the model can only provide the waveform, not the voltage, although that could be done by computing the reluctance distribution.
In this model, the string is not a source of magnetism but a sink.
Arthur
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Post by antigua on Feb 24, 2019 0:59:10 GMT -5
I agree that the variable reluctance model makes sense. In fact, I don't think there is really another model to speak of. Even the well regarded Kirk T McDonald model is ultimately a variable reluctance model. I don't know enough about FEMM to understand what all is going on here, but I agree with the overall principle. In the case of the French model, I don't think the model was wrong so much as it contained some mistake in the math which was never found, because they never tested its predictions.
Even with the variable reluctance model, the string is still the central factor because the string is where the maximum amount of variability occurs, and the pickup's voltage output is ultimately the derivative of that "string variable" with respect to time.
I think the problem more generally is that when people intuitively think about magnetic circuits, they underestimate the reluctance of air, and the permeability of steel. People can feel with a magnet in hand that magnets start to attract more than a centimeter away, and so it seems as though magnetic fields travel through air just fine, not realizing that when it comes to crunching the numbers and determining an output voltage, air gap reduces the efficiency of a magnetic circuit in a very big way, and so they look at the bottom of the pickup, thinking the parts down there are so important, not realizing how the amount of free space that abounds undercuts the contribution of distant metal parts. For example, someone asked about the influence of the "claw" on the Jaguar pickup, and I had to explain that there is too much "air" between the claw, the string and coil for it to contribute much.
Similarly, people don't realize that AlNiCo has a much lower permeability than steel, so when they imagine the magnetic path, they're envisioning the AlNiCo and the steel doing the same thing, not realizing that the AlNiCo functions somewhere in between air and steel, depending on the grade. For example, the steel base plate on the bottom of the Tele bridge pickup, it doesn't really matter because, among other things, the reluctance of the AlNiCo pole pieces is so high. If the pickup where to have steel pole pieces, the steel base plate would have a stronger magnetic connection to the guitar string.
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Post by ms on Feb 24, 2019 6:15:40 GMT -5
When you consider a "bit" of string right over the pole piece, that is, right along the axis of the coil, the field from the string, as magnetized by the magnet, points along this axis. As your first figure shows, as the "bit" of string under consideration is further from the axis, it is illuminated by the magnet at an off vertical angle. It is the component of the field from the string pointed along the axis that contributes to the output. The field from the "bit" can be well approximated by that of a dipole, and the field of a dipole falls off significantly as its field is sampled away from its axis, and, as I said, it is the projection of this field along the axis of the coil that counts, already reduced in magnitude by the angle from the axis of the dipole. If I understand what you are doing, you are not allowing for this, but I could be wrong. So I think the curves in your second plot do not fall off with distance from the axis nearly fast enough. Also you do not seem to be considering the effect of the permeability of the magnets in amplifying the field through the coil from the vibrating string. But that is another matter. All of the evidence I have points to the magnetisation of the string being directed along its length with the exception of a very small region directly over the magnet. This region becomes smaller as the permeability of the string is increased. The string absorbs and redirects flux along its length regardless of the direction it arrives from. Flux that passes through the magnet is already diverging at the magnet pole, so it is not just the axial component of the flux near the string that passes through the coil.
The focussing effect of the magnet is a function of the distribution of the pickup reluctance. That is not modelled; the reluctance is just assumed to be high compared with the reluctance of the string and the distribution not to change as the string moves so that the total flux redirected by the string is directly reflected in the coil flux variations. This version of the model can only provide the waveform, not the voltage, although that could be done by computing the reluctance distribution.
In this model, the string is not a source of magnetism but a sink.
Arthur
If the field induced in the string pointes even closer along the string than the field of the magnet, then the portion of the string that contributes signal is even narrower than I said. If you want just the shape of the waveform, a good approximation is to calculate using just the part of the string over the magnet (or coil) axis.
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Post by ms on Feb 24, 2019 6:33:03 GMT -5
I agree that the variable reluctance model makes sense. In fact, I don't think there is really another model to speak of. Even the well regarded Kirk T McDonald model is ultimately a variable reluctance model. I don't know enough about FEMM to understand what all is going on here, but I agree with the overall principle. In the case of the French model, I don't think the model was wrong so much as it contained some mistake in the math which was never found, because they never tested its predictions. Even with the variable reluctance model, the string is still the central factor because the string is where the maximum amount of variability occurs, and the pickup's voltage output is ultimately the derivative of that "string variable" with respect to time. I think the problem more generally is that when people intuitively think about magnetic circuits, they underestimate the reluctance of air, and the permeability of steel. People can feel with a magnet in hand that magnets start to attract more than a centimeter away, and so it seems as though magnetic fields travel through air just fine, not realizing that when it comes to crunching the numbers and determining an output voltage, air gap reduces the efficiency of a magnetic circuit in a very big way, and so they look at the bottom of the pickup, thinking the parts down there are so important, not realizing how the amount of free space that abounds undercuts the contribution of distant metal parts. For example, someone asked about the influence of the "claw" on the Jaguar pickup, and I had to explain that there is too much "air" between the claw, the string and coil for it to contribute much. Similarly, people don't realize that AlNiCo has a much lower permeability than steel, so when they imagine the magnetic path, they're envisioning the AlNiCo and the steel doing the same thing, not realizing that the AlNiCo functions somewhere in between air and steel, depending on the grade. For example, the steel base plate on the bottom of the Tele bridge pickup, it doesn't really matter because, among other things, the reluctance of the AlNiCo pole pieces is so high. If the pickup where to have steel pole pieces, the steel base plate would have a stronger magnetic connection to the guitar string. Reluctance is analogous to resistance where the storage of magnetic energy is like the dissipation of electric energy. Both are circuit concepts, applying to situations where a three dimensional problem can be reduced to considering one or more "paths". For example, the consideration of electric fields in wires connecting small "components" that have some voltage/current relationship is a useful case. The equivalent magnetic case involves flux that is almost entirely contained within regions of high permeability. That does not describe a guitar pickup. So let's consider an electric problem analogous to the pickup magnetic problem. The electrical problem would have a source off electric field and various regions of high and low permittivity. Would you solve this problem with circuit theory? Of course not, you would have to solve the partial differential equations. So the same applies to the magnetic pickup problem: you need to solve the partial differential equations, that is, the applicable equations from the set of Maxwell's equations. Does it not mean anything that the most "professional" attempt to use reluctance got a completely wrong answer?
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Post by aquin43 on Feb 24, 2019 8:11:12 GMT -5
If the field induced in the string pointes even closer along the string than the field of the magnet, then the portion of the string that contributes signal is even narrower than I said. If you want just the shape of the waveform, a good approximation is to calculate using just the part of the string over the magnet (or coil) axis. Yes, of course, particularly in a symmetrical arrangement such as this the radial flux variation cannot influence the coil. So, reverting to using only the axial flux, the result becomes as follows. Black or green is total flux, red is axial flux only.
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Post by ms on Feb 25, 2019 6:12:18 GMT -5
Yes that looks good.
So you have modeled the non-linearity resulting from the spatial change in the strength of the permanent magnetic field. Then there is the other source of non-linearity, the one that MacDonald modeled. It is the change in the flux through the coil resulting from the change in distance of the string when illuminated with a spatially invariant field. I cannot tell if that is in your calculations or not. One way to proceed would be to extend MacDonald's method using your calculations of the permanent field. I think this would require numerical calculations as opposed to MacDonald's analytical calculation, but that is not so hard these days.
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Post by antigua on Feb 25, 2019 17:17:24 GMT -5
I agree that the variable reluctance model makes sense. In fact, I don't think there is really another model to speak of. Even the well regarded Kirk T McDonald model is ultimately a variable reluctance model. I don't know enough about FEMM to understand what all is going on here, but I agree with the overall principle. In the case of the French model, I don't think the model was wrong so much as it contained some mistake in the math which was never found, because they never tested its predictions. Even with the variable reluctance model, the string is still the central factor because the string is where the maximum amount of variability occurs, and the pickup's voltage output is ultimately the derivative of that "string variable" with respect to time. I think the problem more generally is that when people intuitively think about magnetic circuits, they underestimate the reluctance of air, and the permeability of steel. People can feel with a magnet in hand that magnets start to attract more than a centimeter away, and so it seems as though magnetic fields travel through air just fine, not realizing that when it comes to crunching the numbers and determining an output voltage, air gap reduces the efficiency of a magnetic circuit in a very big way, and so they look at the bottom of the pickup, thinking the parts down there are so important, not realizing how the amount of free space that abounds undercuts the contribution of distant metal parts. For example, someone asked about the influence of the "claw" on the Jaguar pickup, and I had to explain that there is too much "air" between the claw, the string and coil for it to contribute much. Similarly, people don't realize that AlNiCo has a much lower permeability than steel, so when they imagine the magnetic path, they're envisioning the AlNiCo and the steel doing the same thing, not realizing that the AlNiCo functions somewhere in between air and steel, depending on the grade. For example, the steel base plate on the bottom of the Tele bridge pickup, it doesn't really matter because, among other things, the reluctance of the AlNiCo pole pieces is so high. If the pickup where to have steel pole pieces, the steel base plate would have a stronger magnetic connection to the guitar string. Reluctance is analogous to resistance where the storage of magnetic energy is like the dissipation of electric energy. Both are circuit concepts, applying to situations where a three dimensional problem can be reduced to considering one or more "paths". For example, the consideration of electric fields in wires connecting small "components" that have some voltage/current relationship is a useful case. The equivalent magnetic case involves flux that is almost entirely contained within regions of high permeability. That does not describe a guitar pickup. So let's consider an electric problem analogous to the pickup magnetic problem. The electrical problem would have a source off electric field and various regions of high and low permittivity. Would you solve this problem with circuit theory? Of course not, you would have to solve the partial differential equations. So the same applies to the magnetic pickup problem: you need to solve the partial differential equations, that is, the applicable equations from the set of Maxwell's equations. Does it not mean anything that the most "professional" attempt to use reluctance got a completely wrong answer? It doesn't seem to me that the problem is with thinking of a pickup as a variable reluctance sensor, for example here is a diagram of a for-purpose variable reluctance sensor that detects the rotation speed of a toothed wheel, it looks a lot like a guitar pickup:
logansmith.org/tag/variable-reluctance/
I don't know how or why the "professionals" got it wrong, but my guess is the came from a background where this "flux tube" abstraction provided usable results, and then attempted to apply it to a real world situation where the results are less usable, or rather completely wrong. As an aside, it could be pointed out that if a steel guitar string has a remnant flux, the pickup will work, even if the pickup is nothing more than a coil of wire, which might break a model that presumes there is a permanent magnet which the string is relating to, but the thing is it would become a different model at that point, because the magnetism of the string would be a fixed value, where as in the variable reluctance model, the magnetism of the string is a function of it's location in space relative to the permanent magnet. So even though it would work either way, it would still be two distinct models.
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Post by ms on Feb 25, 2019 18:49:39 GMT -5
A field line goes from the closest tooth through the pole piece and returns to a nearby tooth through the steel housing. The gap is much smaller than that of a guitar ;pickup, and so a simple circuit model works much better.
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Post by aquin43 on Feb 26, 2019 9:41:35 GMT -5
Yes that looks good. So you have modeled the non-linearity resulting from the spatial change in the strength of the permanent magnetic field. Then there is the other source of non-linearity, the one that MacDonald modeled. It is the change in the flux through the coil resulting from the change in distance of the string when illuminated with a spatially invariant field. I cannot tell if that is in your calculations or not. One way to proceed would be to extend MacDonald's method using your calculations of the permanent field. I think this would require numerical calculations as opposed to MacDonald's analytical calculation, but that is not so hard these days. The McDonald model is two dimensional across the string. Whether it can simply be extended to fit in with two dimensions along the string I am not sure. Probably it can and would add another roughly square law attenuation with distance, narrowing the effective window. Before trying that, the test of a model is whether it agrees with experiment. The only experimental results I know of come from a 2017 AES paper by Leo Guadagnin and others in which a model based on the McDonald paper is compared with measurements. Unfortunately it is not freely available. www.aes.org/e-lib/browse.cfm?elib=19201 for those with library access (or $33 to spare). Their model gives values for the 2nd and 3rd harmonics that are 3db below the measured results. The experimental method was to construct a simplified version of a strat type pickup with a cylindrical coil and a Neo magnet and to excite it with piece of guitar string kept rigid and moved by a shaker at 80 Hz. The length of the string could be altered, as could the length of the coil along the magnet. Motion along and across the magnet axis could be measured. The magnet diameter was 5mm and its length 32mm. The string motion was sinusoidal. The Neo magnet was chosen because its permeability of 1.05 would make it effectively transparent to the flux from the string in the McDonald model. Strangely enough the simple variable reluctance model adapted to the same magnet parameters agrees pretty well with the H2 measurements, except up very close but less well with H3. -dB at 1mm pk axial vibration, distance d, M = measured. VR uses axial flux only.
d (mm) 3 4 5 6 7 M VR M VR M VR M VR M VR H2 14.3 12.7 14 14 15.5 15.2 16.3 16.2 17 17.7
H3 35.5 29.2 36 31.2 37 33.8 37 35.5 37.3 36.4
Arthur
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Post by antigua on Feb 26, 2019 10:59:21 GMT -5
Yes that looks good. So you have modeled the non-linearity resulting from the spatial change in the strength of the permanent magnetic field. Then there is the other source of non-linearity, the one that MacDonald modeled. It is the change in the flux through the coil resulting from the change in distance of the string when illuminated with a spatially invariant field. I cannot tell if that is in your calculations or not. One way to proceed would be to extend MacDonald's method using your calculations of the permanent field. I think this would require numerical calculations as opposed to MacDonald's analytical calculation, but that is not so hard these days. The McDonald model is two dimensional across the string. Whether it can simply be extended to fit in with two dimensions along the string I am not sure. Probably it can and would add another roughly square law attenuation with distance, narrowing the effective window. Before trying that, the test of a model is whether it agrees with experiment. The only experimental results I know of come from a 2017 AES paper by Leo Guadagnin and others in which a model based on the McDonald paper is compared with measurements. Unfortunately it is not freely available. www.aes.org/e-lib/browse.cfm?elib=19201 for those with library access (or $33 to spare). Their model gives values for the 2nd and 3rd harmonics that are 3db below the measured results. The experimental method was to construct a simplified version of a strat type pickup with a cylindrical coil and a Neo magnet and to excite it with piece of guitar string kept rigid and moved by a shaker at 80 Hz. The length of the string could be altered, as could the length of the coil along the magnet. Motion along and across the magnet axis could be measured. The magnet diameter was 5mm and its length 32mm. The string motion was sinusoidal. The Neo magnet was chosen because its permeability of 1.05 would make it effectively transparent to the flux from the string in the McDonald model. Strangely enough the simple variable reluctance model adapted to the same magnet parameters agrees pretty well with the H2 measurements, except up very close but less well with H3. -dB at 1mm pk axial vibration, distance d, M = measured. VR uses axial flux only.
d (mm) 3 4 5 6 7 M VR M VR M VR M VR M VR H2 14.3 12.7 14 14 15.5 15.2 16.3 16.2 17 17.7
H3 35.5 29.2 36 31.2 37 33.8 37 35.5 37.3 36.4
Arthur I'm not sure how to interpret the experiment you're describing, but it sounds like you/they attribute significance to the string's length. Would yo mind recapping for me why the string's length, or why the axis along the strings length, is a significant consideration? The Tillman demo www.till.com/articles/PickupResponseDemo/ shows how comb filtering can vary based on the width of the pickup, but the default width of 1 inch seems overly generous, and he admits in the text that "the sensitivity gradient of a pickup is a substantial topic on its own". If you have a loose pickup and some alligator clips to connect the pickup to a guitar cable and then to an amp, you can witness the aperture by moving the pickup close to moving guitar strings and observe that the sensitivity gradient of a pickup is concentrated around the immediate vicinity of the pole pieces. Therefore intuitively I consider the cross section of string movement to be the most relevant because (single coil) pickups in general don't seem to be sensitive enough over such a width to induce audible comb filtering on their own.
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Post by aquin43 on Feb 26, 2019 12:32:18 GMT -5
String Length
If you are going to compute the intercepted flux, you must decide on some length of string. The question is how long it should be. In the experiments, they tried reducing the length of the string to see what happened and depending on the distance they found that once the string was over about 40mm long, the string length ceased to have any effect (within the uncertainty of measurement). That is why I chose 20mm for the axially symmetrical model, to ensure that I was reading enough of the string to get the correct flux curve. If you drop the string length to 10mm in the model, there is a reduction in the total intercepted flux of over 10%, depending on the height clearance. The string cross sectional area would appear directly in the model if we were trying to make an absolute level prediction, but it is only implicit since I am only looking at the shape of the curves.
These figures still imply a narrow effective pickup window; not as narrow as just over the pole but narrow enough to be of no consequence.
The Stratocaster scale length is about 647mm, which would be the wavelength when fretted at the 12th fret. So, even with a pickup window of 15mm, say, in which cancellation could occur completely, first cancellation would be at the 43rd harmonic at the string octave. Does a stiff string even have a 43rd harmonic?
Arthur
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Post by ms on Feb 26, 2019 12:32:51 GMT -5
I wonder if one could predict the waveform (with its harmonics, of course), by a combination of modeling and measurements. I think your FEMM model of the field of a cylindrical Alnico magnet is probably more accurate than any measurements that would be reasonable to do. Then suppose you make measurements with a very small exciter coil, varying the distance between the coil and the magnet. Probably you hold the coil constant and vary the height of the pickup using its own adjustment mechanism. You need some way of measuring the distance between the coil and the face of the magnet. Maybe a set of feeler gauges would be adequate for a start. One could measure over a distance significantly more than that over which a string vibrates and then fit a function to the results, giving good resolution over the distance of string vibration..
Then you can multiply in the variation of the permanent field with height, and then compute the response and Fourier analyze it. (You have to make sure that you remember that it is the string velocity as a function of height that counts.)
I suppose that you could do horizontal vibration as well, but that would be harder.
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Post by ms on Feb 26, 2019 12:45:54 GMT -5
String Length
If you are going to compute the intercepted flux, you must decide on some length of string. The question is how long it should be. In the experiments, they tried reducing the length of the string to see what happened and depending on the distance they found that once the string was over about 40mm long, the string length ceased to have any effect (within the uncertainty of measurement). That is why I chose 20mm for the axially symmetrical model, to ensure that I was reading enough of the string to get the correct flux curve. If you drop the string length to 10mm in the model, there is a reduction in the total intercepted flux of over 10%, depending on the height clearance. The string cross sectional area would appear directly in the model if we were trying to make an absolute level prediction, but it is only implicit since I am only looking at the shape of the curves.
These figures still imply a narrow effective pickup window; not as narrow as just over the pole but narrow enough to be of no consequence.
The Stratocaster scale length is about 647mm, which would be the wavelength when fretted at the 12th fret. So, even with a pickup window of 15mm, say, in which cancellation could occur completely, first cancellation would be at the 43rd harmonic at the string octave. Does a stiff string even have a 43rd harmonic?
Arthur
I think the right questions are "how long does the 43rd harmonic last?" and, more important, "for a particular string fretted at a particular fret, is the 43rd harmonic in the frequency bandpass of the system?"
For example, the spacing between the two coils of a standard humbucker has an audible effect, especially on the open low E, but the effect decreases for the smaller strings, and is not significant for the plain strings. My conclusion is that a sampling window somewhat larger than a pole piece diameter is not significant for any of the six strings.
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Post by aquin43 on Feb 26, 2019 13:13:13 GMT -5
<abbr title="Feb 26, 2019 17:45:54 GMT" class="o-timestamp time" data-timestamp="1551203154000">Feb 26, 2019 17:45:54 GMT</abbr> ms said:I think the right questions are "how long does the 43rd harmonic last?" and, more important, "for a particular string fretted at a particular fret, is the 43rd harmonic in the frequency bandpass of the system?" For example, the spacing between the two coils of a standard humbucker has an audible effect, especially on the open low E, but the effect decreases for the smaller strings, and is not significant for the plain strings. My conclusion is that a sampling window somewhat larger than a pole piece diameter is not significant for any of the six strings.
I was thinking more in terms of the stiff bottom E string and whether it will in practice adopt the curvature required to express the highest harmonics to any degree - like the 83rd harmonic of the open string. But lots of guitar playing is done with extreme high pass filtering.
Arthur
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Post by antigua on Feb 26, 2019 20:51:21 GMT -5
String Length
If you are going to compute the intercepted flux, you must decide on some length of string. The question is how long it should be.
Is some length really necessary? AFAIK the McDonald model omits it www.physics.princeton.edu/~mcdonald/examples/guitar.pdf As far as anyone knows, the the length of the string merely effect the output amplitude, or is it implied / proven that it impacts the harmonic makeup as well? I think the harmonic composition as it relates the pickup placement in relation to the string is outside the purview of pickup analysis, only for the fact that a pickup can be placed anywhere and used with any length of string, so the more generalized the string is, the better. If there is some way to avoid having to care about whether the string is 20mm or 647mm in length, I'd opt for avoidance.
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Post by aquin43 on Feb 27, 2019 4:20:23 GMT -5
String Length
If you are going to compute the intercepted flux, you must decide on some length of string. The question is how long it should be.
Is some length really necessary? AFAIK the McDonald model omits it www.physics.princeton.edu/~mcdonald/examples/guitar.pdf As far as anyone knows, the the length of the string merely effect the output amplitude, or is it implied / proven that it impacts the harmonic makeup as well? I think the harmonic composition as it relates the pickup placement in relation to the string is outside the purview of pickup analysis, only for the fact that a pickup can be placed anywhere and used with any length of string, so the more generalized the string is, the better. If there is some way to avoid having to care about whether the string is 20mm or 647mm in length, I'd opt for avoidance. The McDonald paper considers a 2D cross section of a string in a uniform magnetic field. The length of McDonald's string is "infinite" i.e. he implicitly assumes that the string ends are far enough away to ignore. A model of a real pickup has to consider the string length because experiment has shown that it matters and also we know that conditions within a pickup vary from point to point.
An unfocussed pickup such as the stratocaster type seems to be influenced by string length up to 40mm, but only to a very minor degree at that distance so that the actual pickup "window" is probably under 10mm wide, with soft edges.
So, in practical terms, the actual string lengths found on a guitar are "infinite" as far as the model is concerned.
Arthur
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Post by JohnH on Feb 27, 2019 14:12:14 GMT -5
the last few posts are getting into areas which I thought about for GuitarFreak. I built in excitation with harmonics based on where you pluck and where the pickup is, with Tillman theory including a sensing window. The sensing window is clearly a rather loose concept, but I find that a 10mm window for Strat singles gives a result that is closet to a real recorded response. For humbuckers I add the distance between coils, and clearly there is a lot more complexity to it than that. I make the assumption that all strings are perfect, with the same tension (most sets are around 7kg tension per string, varying a bit), and that all give the same output if you pluck them with the same force.
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Post by aquin43 on Feb 27, 2019 14:21:58 GMT -5
What is the shape of the window?
Arthur
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Post by JohnH on Feb 27, 2019 15:10:55 GMT -5
I think Tillman assumed a rectangular window. Any part inside it is 100% and anything outside is zero. He concluded that this resulted in a 6db roll-off above a cut-off frequency depending on width and fundamental. But I think you are going for a much deeper understanding and refinement, so that's all just background info.
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Post by aquin43 on Feb 28, 2019 8:46:27 GMT -5
I think Tillman assumed a rectangular window. Any part inside it is 100% and anything outside is zero. He concluded that this resulted in a 6db roll-off above a cut-off frequency depending on width and fundamental. But I think you are going for a much deeper understanding and refinement, so that's all just background info. Here are some graphs of the flux in the string for an Alnico stratocaster pickup that should help with the window problem: I have fitted a polynomial to the flux density curve so that it can easily be integrated. According to the variable reluctance model the total flux intercepted by the string is an indicator of the output voltage. I would read the last graph, for example as saying that at 7mm above the pole the first 5.6mm of the string contributes as much to the output as all of the rest, so by that criterion the window would be 12.2mm.
It is obviously very height dependent.
Arthur
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Post by antigua on Mar 1, 2019 13:54:06 GMT -5
I have fitted a polynomial to the flux density curve so that it can easily be integrated. According to the variable reluctance model the total flux intercepted by the string is an indicator of the output voltage. I would read the last graph, for example as saying that at 7mm above the pole the first 5.6mm of the string contributes as much to the output as all of the rest, so by that criterion the window would be 12.2mm.
It is obviously very height dependent.
Arthur
I don't know if you've given this thought when thinking about the window shape and size, but what effect the size and shape of the coil play. I would guess that the closer the coil's magnetic field is to that of the pole piece, the better, which is generally the case, so that the permanent field and the coil's field have significant overlap, but then you have pickups such as the Jazzmaster or P-90 where the coil's magnetic field is several times wider than the field of the pole piece, so obviously the overlap is not perfect, and it seems to me further out turns of wire would encompass both the primary, and much of the return path, of the pole piece, and therefore the string, I would assume, which would cause some degree of self cancellation. It should even make the S/N ratio worse, because while they would pick up voltage from the string less efficiently, they would pick up EMI just fine. You mentioned that you believe the string takes on a magnetic field whose polarity runs the parallel to the string, which seems reasonable, since the length of the string has the lowest reluctance away from the magnetic field that is induced by the pole piece. So you see this issue relating to the coil shape and size differently?
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Post by aquin43 on Mar 2, 2019 6:31:18 GMT -5
Well, those curves show only the string magnetisation on the assumption that this in itself is sufficient as an indicator of the pickup output. The model makes no mention of the coil shape or position, which is a weakness - possibly a fatal one. The assumption is that the string takes axial flux from the pickup, redirecting it radially where it will no longer interact with the coil. But introducing the string actually increases the flux through the coil.
Another approach is to consider the string as a moving magnet and integrate its effect within the coil, relying on this being so small that it is not necessary to consider the effect of the altered flux at the coil upon the string itself.
Or, again, one could treat the string as a "short circuit" of flux and integrate the susceptance of each part of the coil times the intercepted flux. I think that the equations would be similar.
We know that core of the coil is very important in concentrating the flux within the turns, even if it has quite a low permeability as with Alnico. Every study that I have seen has used a Neo magnet with its nearly unity permeability to avoid the necessity of considering the core, thereby limiting their usefulness. Wide coils are inefficient as are deep coils on Alnico because of the large leakage of flux from the pole pieces.
Arthur
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Post by antigua on Mar 4, 2019 23:16:03 GMT -5
Another approach is to consider the string as a moving magnet and integrate its effect within the coil, relying on this being so small that it is not necessary to consider the effect of the altered flux at the coil upon the string itself.
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We know that core of the coil is very important in concentrating the flux within the turns, even if it has quite a low permeability as with Alnico. Every study that I have seen has used a Neo magnet with its nearly unity permeability to avoid the necessity of considering the core, thereby limiting their usefulness. Wide coils are inefficient as are deep coils on Alnico because of the large leakage of flux from the pole pieces.
Arthur
I'm not sure that axially oriented magnetization along the string means it doesn't produce output. There was at least one pickup that worked by rubbing the strings with a magnet in order to give the strings a remnant magnetization, and in theory that magnetic field would be completely axial, but then again, maybe it isn't, that might be something I can look in to. Aside from the permanently magnetized guitar string, even ordinary guitar strings have some remnant magnetization, under normal circumstances. Unless I've forgetting something though, I don't think the string having a fixed residual flux introduces new non-linearities / harmonics, or impacts the aperture profile, so it just falls under the category of efficiency, how well it converts the physical input into a voltage output. THe permeability of the pole piece should be a similar issue, effecting overall output, but not the harmonic makeup or the aperture profile.
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Post by ms on Mar 6, 2019 7:24:31 GMT -5
Another approach is to consider the string as a moving magnet and integrate its effect within the coil, relying on this being so small that it is not necessary to consider the effect of the altered flux at the coil upon the string itself.
...
We know that core of the coil is very important in concentrating the flux within the turns, even if it has quite a low permeability as with Alnico. Every study that I have seen has used a Neo magnet with its nearly unity permeability to avoid the necessity of considering the core, thereby limiting their usefulness. Wide coils are inefficient as are deep coils on Alnico because of the large leakage of flux from the pole pieces.
Arthur
I'm not sure that axially oriented magnetization along the string means it doesn't produce output. There was at least one pickup that worked by rubbing the strings with a magnet in order to give the strings a remnant magnetization, and in theory that magnetic field would be completely axial, but then again, maybe it isn't, that might be something I can look in to. Aside from the permanently magnetized guitar string, even ordinary guitar strings have some remnant magnetization, under normal circumstances. Unless I've forgetting something though, I don't think the string having a fixed residual flux introduces new non-linearities / harmonics, or impacts the aperture profile, so it just falls under the category of efficiency, how well it converts the physical input into a voltage output. THe permeability of the pole piece should be a similar issue, effecting overall output, but not the harmonic makeup or the aperture profile. The contribution to the output of different bits of the string depends primarily on two things: 1. The contribution of a bit falls off as the cube of the distance between the bit and the observation point (says the top of the coil). 2. The contribution to the signal is the component along the axis of the coil. The diameter of the string is small compared to the distance from the coil. Therefore we can model the string as uniformly spaced magnetic dipoles, and we assume that they all have the same strength and point directly at the observation point. If r is the distance between a dipole and the observation point, then the field at this point is proportional to 1/r^3, that is, it falls off as the distance cubed. Let rz be the distance between the observation point and the point on the string right above it. Let ry be the distance from this point to a particular dipole. The distance between the dipole and the observation point is sqrt(ry^2 + rz^2). Thus, the contribution of a dipole is 1./ sqrt(ry^2 + rz^2)^3. Taking the component along the axis of the coil gives a total variation of 1/(ry^2 + rz^2)^2. A little computation shows that if rz is 3mm, then the contribution of a dipole with ry about 2mm is about half that of the dipole with ry = 0. Relaxing the assumption about how the dipoles point will decrease the 2mm number a bit since dipoles have weaker fields off axis. Thus, the sampling window along the string is narrow.
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Post by aquin43 on Mar 6, 2019 11:29:12 GMT -5
If we are looking for the ratio between B at the string and axially directed B returned to a point on the magnet or within the coil then his ratio, B/B must be dimensionless. We know that the cross sectional area of the string is a factor, giving L^2 (where L represents "length"), therefore the rest of the ratio must amount to L^-2. Horton and Moore use z/(z^2+y^2)^3/2, or z/r^3 which is dimensionally correct.
Arthur
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Post by ms on Mar 6, 2019 12:01:44 GMT -5
If we are looking for the ratio between B at the string and axially directed B returned to a point on the magnet or within the coil then his ratio, B/B must be dimensionless. We know that the cross sectional area of the string is a factor, giving L^2 (where L represents "length"), therefore the rest of the ratio must amount to L^-2. Horton and Moore use z/(z^2+y^2)^3/2, or z/r^3 which is dimensionally correct.
Arthur
Is this in response to what I wrote just above?
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Post by aquin43 on Mar 6, 2019 13:22:14 GMT -5
If we are looking for the ratio between B at the string and axially directed B returned to a point on the magnet or within the coil then his ratio, B/B must be dimensionless. We know that the cross sectional area of the string is a factor, giving L^2 (where L represents "length"), therefore the rest of the ratio must amount to L^-2. Horton and Moore use z/(z^2+y^2)^3/2, or z/r^3 which is dimensionally correct.
Arthur
Is this in response to what I wrote just above? Yes, sorry, quoting would have made it clearer. Your final formula 1/(ry^2 + rz^2)^2 is L^-4.
Arthur
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Post by ms on Mar 6, 2019 14:00:11 GMT -5
Is this in response to what I wrote just above? Yes, sorry, quoting would have made it clearer. Your final formula 1/(ry^2 + rz^2)^2 is L^-4.
Arthur
But I do not need to make the units correct to show the variation with ry. In any case, I think what I did is not clear enough to everyone without a drawing, and so one is attached.
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