There are some unsettled questions regarding the variation of inductance with frequency, and so here is the beginning of a new discussion on this topic. First consider a definition of inductance. This is a translation of of equation 5.157 of the third edition of Jackson into words:

The inductance L of a loop of wire (possibly a multiple turn coil) carrying a current I is found by making the sum of the amplitude squared of the magnetic field (divided by the permeability) in each small volume of space. This sum is then divided by the square of the current.

The sum must be taken over the space inside and outside of the wire. This means that if the distribution of current in the wire changes, the inductance changes. It is well known that current tends to flow near the surface of a wire, not the center, at high frequencies. This is characterized by the "skin depth", which decreases with increasing frequency. Thus it is reasonable to suppose that an inductor using heavy wire would have an inductance that changes with frequency as the distribution of the current is shifted more towards the outside of the wire.

The skin depth of copper is 8.47 mm at 60 Hz and .66 mm at 10 KHz. The diameter of wire used in pickups is much less than .66 mm. Therefore, the inductance of the coil does not change with frequency due to this effect. This does not exclude the possibility that the inductance might change as a function of frequency due to mutual impedance effects with some other conductor(s).

I believe that this is what Joe Gwinn intended when he wrote this in post 52 of this discussion: http://music-electronics-forum.com/t14110-2/:

Note that [of eddy currents] was inserted by Joe. I have read that section of Jackson, and I think that Jackson is describing the effect that I just explained above, that is, the skin effect due to current in the inductor coil itself, not due to eddy currents in another conductor. That is a more complicated question. I believe that I have already answered the question for pickups, but others do not agree, and I think the issue is worth discussing.

The inductance L of a loop of wire (possibly a multiple turn coil) carrying a current I is found by making the sum of the amplitude squared of the magnetic field (divided by the permeability) in each small volume of space. This sum is then divided by the square of the current.

The sum must be taken over the space inside and outside of the wire. This means that if the distribution of current in the wire changes, the inductance changes. It is well known that current tends to flow near the surface of a wire, not the center, at high frequencies. This is characterized by the "skin depth", which decreases with increasing frequency. Thus it is reasonable to suppose that an inductor using heavy wire would have an inductance that changes with frequency as the distribution of the current is shifted more towards the outside of the wire.

The skin depth of copper is 8.47 mm at 60 Hz and .66 mm at 10 KHz. The diameter of wire used in pickups is much less than .66 mm. Therefore, the inductance of the coil does not change with frequency due to this effect. This does not exclude the possibility that the inductance might change as a function of frequency due to mutual impedance effects with some other conductor(s).

I believe that this is what Joe Gwinn intended when he wrote this in post 52 of this discussion: http://music-electronics-forum.com/t14110-2/:

This cannot be a new problem, so I did a little research, and found the answer in the classic textbook, "Classical Electrodynamics", 2nd edition, J.D. Jackson, Wiley 1975. On page 298, the last sentence of the paragraph just below equation (7.77) is the answer (taken from a discussion of eddy currents and skin depth):

"One simple consequence [of eddy currents] is that the high-frequency inductance of circuit elements is somewhat smaller than the low-frequency inductance because of the expulsion of flux from the interior of the conductors."

"One simple consequence [of eddy currents] is that the high-frequency inductance of circuit elements is somewhat smaller than the low-frequency inductance because of the expulsion of flux from the interior of the conductors."

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