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Post by aquin43 on Mar 6, 2019 17:35:16 GMT -5
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.
Given r_str = string radius, B_str = B at the string location, B_mag = axial B at the magnet due to the string, r, ry, rz as above.
We would have B_mag = K * B_str * r_str^2 * (1/r^3) * (rz/r)
B_mag / B_str = K * r_str^2 * (1/r^3) * (rz/r) ....... must be a dimensionless number.
This requires K to have dimension L because the rest of the rhs factors combine to have dimension L^-1.
What I don't understand is what is in K that has dimension L. It obviously includes the susceptibility, but that is dimensionless.
I note that Horton and Moore consider, not dipoles, but the magnetic "charge" induced on the bottom of the string by the total |B|, the complementary charge on the upper part presumably radiating into free space. They then only have a 1/r^2 initial dependency which makes K dimensionless and leads to their overall dependency z/(z^2+y^2)^3/2, or z/r^3. This, with the added dependency on the string cross sectional area (r_str^2) yields a dimensionless number. This seems to check out experimentally.
Arthur
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Post by ms on Mar 6, 2019 20:21:26 GMT -5
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.
Given r_str = string radius, B_str = B at the string location, B_mag = axial B at the magnet due to the string, r, ry, rz as above.
We would have B_mag = K * B_str * r_str^2 * (1/r^3) * (rz/r)
B_mag / B_str = K * r_str^2 * (1/r^3) * (rz/r) ....... must be a dimensionless number.
This requires K to have dimension L because the rest of the rhs factors combine to have dimension L^-1.
What I don't understand is what is in K that has dimension L. It obviously includes the susceptibility, but that is dimensionless.
I note that Horton and Moore consider, not dipoles, but the magnetic "charge" induced on the bottom of the string by the total |B|, the complementary charge on the upper part presumably radiating into free space. They then only have a 1/r^2 initial dependency which makes K dimensionless and leads to their overall dependency z/(z^2+y^2)^3/2, or z/r^3. This, with the added dependency on the string cross sectional area (r_str^2) yields a dimensionless number. This seems to check out experimentally.
Arthur
Take the last line in may attachment and divide it by the same thing with ry = 0. That is dimensionless, and is the axial field induced by a dipole at ry relative to the induced field from a dipole at ry = 0.
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Post by aquin43 on Mar 7, 2019 4:18:50 GMT -5
Given r_str = string radius, B_str = B at the string location, B_mag = axial B at the magnet due to the string, r, ry, rz as above.
We would have B_mag = K * B_str * r_str^2 * (1/r^3) * (rz/r)
B_mag / B_str = K * r_str^2 * (1/r^3) * (rz/r) ....... must be a dimensionless number.
This requires K to have dimension L because the rest of the rhs factors combine to have dimension L^-1.
What I don't understand is what is in K that has dimension L. It obviously includes the susceptibility, but that is dimensionless.
I note that Horton and Moore consider, not dipoles, but the magnetic "charge" induced on the bottom of the string by the total |B|, the complementary charge on the upper part presumably radiating into free space. They then only have a 1/r^2 initial dependency which makes K dimensionless and leads to their overall dependency z/(z^2+y^2)^3/2, or z/r^3. This, with the added dependency on the string cross sectional area (r_str^2) yields a dimensionless number. This seems to check out experimentally.
Arthur
Take the last line in may attachment and divide it by the same thing with ry = 0. That is dimensionless, and is the axial field induced by a dipole at ry relative to the induced field from a dipole at ry = 0. The ratio of two evaluations of a formula will always be dimensionless.
My point is that the formula itself is supposed to give the ratio of two B values so to be valid it must be dimensionless. This requires that the the factor I have called K must have the dimension of length, i.e. some linear factor in your formula must depend on a length of some sort in a way that is not made explicit in the formula as written. Otherwise the formula cannot describe a real physical system.
I can see that the formula describes the path from the dipole moment to the field returned to the magnet, so how do you get from |B| produced by the magnet at the string to the dipole moment?
Arthur
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Post by ms on Mar 7, 2019 6:05:48 GMT -5
"My point is that the formula itself is supposed to give the ratio of two B values so to be valid it must be dimensionless."
I do not need that. All I need is the equation for the r dependence of a dipole, valid from rz to arbitrarily large r.
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Post by aquin43 on Mar 11, 2019 4:58:03 GMT -5
"My point is that the formula itself is supposed to give the ratio of two B values so to be valid it must be dimensionless." I do not need that. All I need is the equation for the r dependence of a dipole, valid from rz to arbitrarily large r. The problem stems from your hypothesis that, on the scale of the pickup dimensions, the string can be modelled as an array of dipoles. This appears not to be valid. If you look at the local pattern of flux in the string it is clear it is far from being dipolar. Flux entering the string over a point on the magnet travels down the string to a considerable extent before exiting, which means that the locally induced pole has no immediate complementary pole with which to approximate a dipole. In fact, at the sub centimetre scale, the string looks rather like an array of monopoles.* This, presumably, is why others model the axial flux in the coil as varying with z/r^3. That is 1/r^2 for the monopole times z/r for the cosine of the angular offset. Arthur * There are no observed true monopoles, but the pole at the end of a long thin magnet produces a good representation of a monopole field.
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Post by aquin43 on Mar 11, 2019 5:05:06 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. It seems that someone has actually made a reasonably meaningful measurement, slightly limited by the practicalities of making the aperture of the test equipment small enough:
Arthur
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Post by ms on Mar 11, 2019 6:50:46 GMT -5
"My point is that the formula itself is supposed to give the ratio of two B values so to be valid it must be dimensionless." I do not need that. All I need is the equation for the r dependence of a dipole, valid from rz to arbitrarily large r. The problem stems from your hypothesis that, on the scale of the pickup dimensions, the string can be modelled as an array of dipoles. This appears not to be valid. If you look at the local pattern of flux in the string it is clear it is far from being dipolar. Flux entering the string over a point on the magnet travels down the string to a considerable extent before exiting, which means that the locally induced pole has no immediate complementary pole with which to approximate a dipole. In fact, at the sub centimetre scale, the string looks rather like an array of monopoles.* This, presumably, is why others model the axial flux in the coil as varying with z/r^3. That is 1/r^2 for the monopole times z/r for the cosine of the angular offset. Arthur * There are no observed true monopoles, but the pole at the end of a long thin magnet produces a good representation of a monopole field.
Of course the magnetized string can be modeled as dipoles; it literally is made up of magnetic dipoles. The question is how simple can the representation be. Any magnetic field looks dipolar if you get far enough away, that is, view at a distance large compared to the physical size of the magnetized region. Also magnetic fields add linearly here. So divide the string into bits that are short compared to the distance to the viewing point. The assumption is that the string is thin, and the bits are short. I have made a further assumption about the orientation of the dipoles (all point to viewing point) and their degree of magnetization (all the same) to represent a limiting case. So all I have done is to use the 1/r^3 variation and take the component aligned with the coil axis. I find the variation with ry including rz where it is a necessary parameter in this ry variation; otherwise it need not appear because it is held fixed as ry varies. Have I done the math wrong? Consider a long thin magnet. For simplicity let it be uniformly magnetized, that is, all microscopic dipoles have the same intensity and direction, a situation that can be approached with a material such as neo. Far away it looks like a dipole; that is, it can be represented by two monopoles. Up close, the field is not dipolar since a whole range of microscopic dipoles contribute significantly. At no distance does it look monopolar, nor does either pole have any particular significance alone. The string does not look like an array of monopoles. When you say that it does , you are not looking at the string alone but as a part of the system where the monopoles are complemented by monopoles of the opposite sign elsewhere. However, because of linearity, it is possible to look at the string alone, and that is the simplest way to look at it.
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Post by aquin43 on Mar 11, 2019 9:49:21 GMT -5
The problem stems from your hypothesis that, on the scale of the pickup dimensions, the string can be modelled as an array of dipoles. This appears not to be valid. If you look at the local pattern of flux in the string it is clear it is far from being dipolar. Flux entering the string over a point on the magnet travels down the string to a considerable extent before exiting, which means that the locally induced pole has no immediate complementary pole with which to approximate a dipole. In fact, at the sub centimetre scale, the string looks rather like an array of monopoles.* This, presumably, is why others model the axial flux in the coil as varying with z/r^3. That is 1/r^2 for the monopole times z/r for the cosine of the angular offset. Arthur * There are no observed true monopoles, but the pole at the end of a long thin magnet produces a good representation of a monopole field.
Of course the magnetized string can be modeled as dipoles; it literally is made up of magnetic dipoles. The question is how simple can the representation be. Any magnetic field looks dipolar if you get far enough away, that is, view at a distance large compared to the physical size of the magnetized region. Also magnetic fields add linearly here. So divide the string into bits that are short compared to the distance to the viewing point. The assumption is that the string is thin, and the bits are short. I have made a further assumption about the orientation of the dipoles (all point to viewing point) and their degree of magnetization (all the same) to represent a limiting case. So all I have done is to use the 1/r^3 variation and take the component aligned with the coil axis. I find the variation with ry including rz where it is a necessary parameter in this ry variation; otherwise it need not appear because it is held fixed as ry varies. Have I done the math wrong? Consider a long thin magnet. For simplicity let it be uniformly magnetized, that is, all microscopic dipoles have the same intensity and direction, a situation that can be approached with a material such as neo. Far away it looks like a dipole; that is, it can be represented by two monopoles. Up close, the field is not dipolar since a whole range of microscopic dipoles contribute significantly. At no distance does it look monopolar, nor does either pole have any particular significance alone. The string does not look like an array of monopoles. When you say that it does , you are not looking at the string alone but as a part of the system where the monopoles are complemented by monopoles of the opposite sign elsewhere. However, because of linearity, it is possible to look at the string alone, and that is the simplest way to look at it. You divide the string into slices, but then you assume that these slices behave like dipoles. Well so they would, if it were a true physical division. Each piece would develop a pole on its opposite side and the pole pair would behave as an approximate dipole. But the actual division is hypothetical. The string slices remain part of the original string and retain their part of the magnetisation pattern of the whole string. In particular, the flux inside the string is inaccessible and does not exit near enough to the original induced pole in the slice for the separation of the exit pole to be neglected. The net result is that the induced poles on the string surface appear independent to a degree. They do not simply combine in such a fashion that they can be represented as infinitesimal dipoles localised at the slices.
Also, near one pole of a long thin magnet, the behaviour of the field can be pretty close to that of a monopole. Such magnets (with ball ends) are used in school demonstrations of the inverse square law.
Arthur
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Post by ms on Mar 11, 2019 12:56:13 GMT -5
Of course the magnetized string can be modeled as dipoles; it literally is made up of magnetic dipoles. The question is how simple can the representation be. Any magnetic field looks dipolar if you get far enough away, that is, view at a distance large compared to the physical size of the magnetized region. Also magnetic fields add linearly here. So divide the string into bits that are short compared to the distance to the viewing point. The assumption is that the string is thin, and the bits are short. I have made a further assumption about the orientation of the dipoles (all point to viewing point) and their degree of magnetization (all the same) to represent a limiting case. So all I have done is to use the 1/r^3 variation and take the component aligned with the coil axis. I find the variation with ry including rz where it is a necessary parameter in this ry variation; otherwise it need not appear because it is held fixed as ry varies. Have I done the math wrong? Consider a long thin magnet. For simplicity let it be uniformly magnetized, that is, all microscopic dipoles have the same intensity and direction, a situation that can be approached with a material such as neo. Far away it looks like a dipole; that is, it can be represented by two monopoles. Up close, the field is not dipolar since a whole range of microscopic dipoles contribute significantly. At no distance does it look monopolar, nor does either pole have any particular significance alone. The string does not look like an array of monopoles. When you say that it does , you are not looking at the string alone but as a part of the system where the monopoles are complemented by monopoles of the opposite sign elsewhere. However, because of linearity, it is possible to look at the string alone, and that is the simplest way to look at it. You divide the string into slices, but then you assume that these slices behave like dipoles. Well so they would, if it were a true physical division. Each piece would develop a pole on its opposite side and the pole pair would behave as an approximate dipole. But the actual division is hypothetical. The string slices remain part of the original string and retain their part of the magnetisation pattern of the whole string. In particular, the flux inside the string is inaccessible and does not exit near enough to the original induced pole in the slice for the separation of the exit pole to be neglected. The net result is that the induced poles on the string surface appear independent to a degree. They do not simply combine in such a fashion that they can be represented as infinitesimal dipoles localised at the slices.
Also, near one pole of a long thin magnet, the behaviour of the field can be pretty close to that of a monopole. Such magnets (with ball ends) are used in school demonstrations of the inverse square law.
Arthur
The small pieces behave like dipoles (at least to a useful level of approximation) because the distance to the observation point is significantly larger than the size of the source region. It has nothing to do with poles developing on physically divided pieces. Rather it has to do with representing the field from each small source region as a multipole expansion (https://en.wikipedia.org/wiki/Multipole_expansion). The higher order terms fall off faster than the lower order terms, and so as distance increases, only the lowest matters. The lowest order term in the magnetic case is the dipole term. Therefore a good approximation is to use a string of dipoles. My approximate analysis overestimates the width of the sampling window a bit because the dipoles further away point more along the string than I have assumed, and also because they are weaker than I assume. However, the dipoles further away do not contribute that much because of the 1/r^3 variation, and so the actual sampling window should not be very much narrower than I compute.
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Post by aquin43 on Mar 11, 2019 13:37:48 GMT -5
You divide the string into slices, but then you assume that these slices behave like dipoles. Well so they would, if it were a true physical division. Each piece would develop a pole on its opposite side and the pole pair would behave as an approximate dipole. But the actual division is hypothetical. The string slices remain part of the original string and retain their part of the magnetisation pattern of the whole string. In particular, the flux inside the string is inaccessible and does not exit near enough to the original induced pole in the slice for the separation of the exit pole to be neglected. The net result is that the induced poles on the string surface appear independent to a degree. They do not simply combine in such a fashion that they can be represented as infinitesimal dipoles localised at the slices.
Also, near one pole of a long thin magnet, the behaviour of the field can be pretty close to that of a monopole. Such magnets (with ball ends) are used in school demonstrations of the inverse square law.
Arthur
The small pieces behave like dipoles (at least to a useful level of approximation) because the distance to the observation point is significantly larger than the size of the source region. It has nothing to do with poles developing on physically divided pieces. Rather it has to do with representing the field from each small source region as a multipole expansion (https://en.wikipedia.org/wiki/Multipole_expansion). The higher order terms fall off faster than the lower order terms, and so as distance increases, only the lowest matters. The lowest order term in the magnetic case is the dipole term. Therefore a good approximation is to use a string of dipoles. My approximate analysis overestimates the width of the sampling window a bit because the dipoles further away point more along the string than I have assumed, and also because they are weaker than I assume. However, the dipoles further away do not contribute that much because of the 1/r^3 variation, and so the actual sampling window should not be very much narrower than I compute. So, are you saying that the flux distribution in the source doesn't matter? How would a point on the surface of the magnet pole appear as viewed from the string?
Arthur
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Post by ms on Mar 11, 2019 14:09:53 GMT -5
The source is the bit of string, that is, the magnetic material that is creating the magnetic field represented by the multipole expansion. It is magnetized by the permanent magnet, and this permanent magnetism has no further role to play. (The permeability of the material of the permanent magnet does, of course, but that is another matter.)
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Post by aquin43 on Mar 12, 2019 6:52:03 GMT -5
The source is the bit of string, that is, the magnetic material that is creating the magnetic field represented by the multipole expansion. It is magnetized by the permanent magnet, and this permanent magnetism has no further role to play. (The permeability of the material of the permanent magnet does, of course, but that is another matter.)
No, I was just asking how the dipole array methodology would be
applied to a point on the surface of the magnet as viewed from a
position near the string.
Consider McDonald's paper. At the origin of a system which has, I think,
sufficient similarities to the one we are considering, McDonald takes a
cross section and calculates analytically the field due to the string.
Retaining his cylindrical coordinates and combining his equations
(8) and (9) gives B outside the string
( 2 ) ( a (mu - 1) ) B = B0 ( 1 + ----------- ) ( 2 ) ( r (mu + 1) )
where a is the radius of the string. The variable part, which is
the flux due to the string viewed as a source, is
( 2 ) ( a (mu - 1) ) B = B0 ( ----------- ) ( 2 ) ( r (mu + 1) )
This would be at x = y = 0 in our system so the cosine y-offset term
is unity. McDonald's y is our z. The variation is with r^2, so the
overall expression is dimensionally correct.
The r^-2 formulation still makes the pickup window narrow,
as it is found to be experimentally.
Arthur
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Post by ms on Mar 12, 2019 7:17:04 GMT -5
The source is the bit of string, that is, the magnetic material that is creating the magnetic field represented by the multipole expansion. It is magnetized by the permanent magnet, and this permanent magnetism has no further role to play. (The permeability of the material of the permanent magnet does, of course, but that is another matter.)
No, I was just asking how the dipole array methodology would be
applied to a point on the surface of the magnet as viewed from a
position near the string.
Consider McDonald's paper. At the origin of a system which has, I think,
sufficient similarities to the one we are considering, McDonald takes a
cross section and calculates analytically the field due to the string.
Retaining his cylindrical coordinates and combining his equations
(8) and (9) gives B outside the string
( 2 ) ( a (mu - 1) ) B = B0 ( 1 + ----------- ) ( 2 ) ( r (mu + 1) )
where a is the radius of the string. The variable part, which is
the flux due to the string viewed as a source, is
( 2 ) ( a (mu - 1) ) B = B0 ( ----------- ) ( 2 ) ( r (mu + 1) )
This would be at x = y = 0 in our system so the cosine y-offset term
is unity. McDonald's y is our z. The variation is with r^2, so the
overall expression is dimensionally correct.
The r^-2 formulation still makes the pickup window narrow,
as it is found to be experimentally.
Arthur
Thank you for the clarification. I would expect integrating along a string of dipoles, each falling off as 1/r^3, to result in 1/r^2 if they all point towards the observation point. This would be for an infinitely long string. If the string has finite length, then far away it would be like a single dipole and fall off as 1/r^3.
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Post by aquin43 on Mar 16, 2019 11:54:59 GMT -5
Yes, I can see that it is the correlation between the dipoles in line along the string that reduces the active dimensions from three to two. From symmetry in the case of the strat type pickup the integral along the string in the y direction will be zero (at least at the origin). There still remains an overall problem with the transfer function from the string to the coil, though. The curve of total flux at the string position against distance from the pole is sufficient to explain the non-linearity of the pickup all by itself. There is no room for any further strong non-linearities. In the book by Manfred Zollner on the Gitec forum www.gitec-forum-eng.de/wp-content/uploads/2019/03/poteg-5-8-non-linear-pu-distortion.pdfhe reports from measurements that the strat pickup transfer curve fits neatly to an expression of the form K1 + K2 / (delta + x). The variation of total flux at the string position, considering just the magnet and the string, fits very closely that form and can match the reported H2 distortion figures. This suggests, perhaps, that the reluctance of the magnet/coil to string path is high enough that the flux modulation at the string is reflected largely unchanged in form in the coil/magnet combination. It would not be the raw magnetisation pattern of the pickup that was being modulated, rather it would be the alteration to that pattern caused by the introduction of the string. Zollner also points out that the field strength above the magnet is near to saturating the string and also that different results are obtained if the string is allowed to touch the magnet while installing it. Arthur
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