Pickup imp: apparent fall of L and rise of R with f

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The green line in the first plot below shows the real part of the pickup impedance of an SD SH1N, with the effect of the coil capacitance removed.  It rises with frequency as a result of eddy currents in metal parts, especially the cores.  The magenta curve shows the imaginary part; its downward curve indicates an apparent fall in coil inductance with increasing frequency.  Both of these effects can be the result of resistance in parallel with the coil, as we will see.  In a pickup this can come from the mutual coupling to metal parts, which act like secondaries of a very poor transformer.  The load from a transformer secondary appears in parallel with the primary.  The effect of eddy currents has dependence on frequency; for example, the resistance must be infinity at DC.




Look at the impedance of a very simple model, shown below, just an L with both a series and parallel resistor (frequency independent).  The colors match those in the plot above with values listed in the title.  It is remarkable how close this is to the measurement above. It reproduces all the major features, but the shapes of the curves are not right.  Next I will try to improve it, making the parallel resistance dependent on frequency (as we expect in the actual case).

[aquin43]

These frequency dependent or semi-inductors can be approximated as a chain of simple inductors, each with a parallel resistor. I find that two inductors are usually enough to model a pickup impedance up to the open circuit resonance, beyond which the simple model begins to fail anyway with humbuckers.

Here are the results for a De Armond type floating single coil, using three inductors. I plot the inductor Q, rather than just the resistance. The dashed lines are the model. The capacitance comes out as part of the impedance fit and is applied to the real pickup when calculating its inductance. I expect that you are doing something similar.

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I think the chain of simple inductors, each with a parallel resistor, works so well because the first link of the chain, one inductor and one parallel resistor (and the coil resistance in series), is the exact solution when k, the coupling constant from the pickup coil to the "metal", is one.
 So even when k is not so high it is a good first step.

Another way of looking at it uses a rearrangement of equation 7 from this:
guitarnuts2.proboards.com/thread/8455/derivation-coupling-effect

Equation 7:

Z_p = j\omega L_c + R_c + \frac{\omega^2k^2L_c^2}{j\omega L_c + R_{se}}

R_{se} is the resistance across the coil resulting from coupling to the "metal".

Adding and subtracting k^2j\omega L_c gives:

Z_p = k^2j\omega L_c + (1 - k^2)j\omega L_c + R_c + k^2\frac{\omega^2L_c^2}{j\omega L_c + R_{se}}


Z_p = R_c + k^2\frac{j\omega L_c R_{se}}{j\omega L_c + R_{se}} + (1 - k^2)j\omega L_c

So the pickup impedance is the linear combination of two simple cases.    The first simple case, when k = 1, is the coil resistance in series with the parallel combination of the coil inductance and the parallel resistance. (The fraction in the second term is of the form Z_AZ_B/(Z_A + Z_B).) The second simple case, when k = 0, is the coil resistance in series with the coil inductance.