I realize I'm late to the thread, but I love this: "[T]he dimensionless quantity of mojo".
I also think that this is a pretty compelling analysis of the role of the electric guitar over time, and a nice explanation of how we ended up where we seem to be right now. Though I'd add that there's a distressing element of anti-science attitude, an attitude of "you have your truth and I have mine", which came out of both philosophy of science (e.g. Kuhn) and postmodernist literary theory (e.g. Derrida), and has ironically been embraced by the left and the right, at least in the US.
I don't understand it, I don't know why guitarists aren't clamoring to know the inductance or functional resonant peaks, or the gauss strength of pickups. When people talk about cars, they love to talk horsepower, torque, quarter mile times. With guitar amps, people talk about wattage, whether is Class A, solid state, etc. , and with speakers it's power efficiency, cone type, and all sorts of things. But when it comes to pickups, people seem to be happier knowing less. There are so many unanswerable questions about "tone wood" that you'd think people would be happy to know there's at least this one aspect of the electric guitar that can be easily quantified with a $100 LCR meter. The few of us who care about this stuff are a tiny minority, and I have no idea why that is.
Last Edit: Feb 24, 2021 22:48:05 GMT -5 by antigua
I tried an LTSpice simulation of a pair of 10 turn layers
The total currents through the capacitors are I(VSENSE_1) and I(VSENSE_2). They prove to be equal.
The output voltages out and outfb differ slightly at higher frequencies and outfb seems to be the most affected.
C11 is shorted out by VSENSE_1 but is included anyway. The inductors all have resistance, the voltage sources represent the volts per turn and everything is parameterised. The values are arbitrary but chosen so that the capacitive loading on each node is negligible.
The simulation file, to be copied into a text editor by anyone interested and saved as pu-turns.asc is:
The next exercise would be to model the two voltage distributions in FEMM to check whether the stored charges would be equal.
So this models the capacitance of two wires that are side by side, but it seems that the lumped capacitance is gradient that extends from the first wind to the very last, so the capacitance individual turns as depicted is but a small part of the lump. Maybe it's 0.1% or 10%, I have no idea.
I think in the transformer designs where they use the fold back technique, there might have been a less layers, and they might have been used for RF, so the improvement would have been more meaningful.
This is a naive model of the effective inter layer capacitance. It is a gross simplification of the actual layout. For example, the real turns are not face to face but nestled together. I thought it worthwhile, however, because the voltage gradient is actually similar to what would be produced by a current through the coil and it seemed reasonable to assume that the capacitances between turns would be localised to a marked degree, because capacitance falls with distance and the tightly interleaved turns shield each other . The physical layout of the turns is obviously the same for both winding methods. That leaves only the voltage distribution to make a difference.
If the two winding methods were to produce different capacitive currents, one would expect to see some sign of it in even this simplified model. In the audio range, there is none. I suspect that you are right and the effects only appear at RF, where the impedance of the turns becomes significant.
I think there is a limit to the localization. The current change due to the capacitance between turns a and b induces some voltages in all other turns because they are all magnetically coupled. It might be interesting to model a simple case, starting with low mutual inductance and increasing it until an effect is seen.
A single layer rf coil can be treated as a transmission line. The inductance and capacitance per unit distance determine the impedance. Its length is short compared to a quarter wavelength, and so it is capacitive, with the value computed from a very simple equation. I have read that this method works very well in many cases. I think that this works because the magnetic coupling is largest between turns that are close together, making this simple model possible.
But it does not work for a guitar pickup, a multi-layer coil with a huge number of turns. Perhaps a different approximation is possible taking advantage of a limit involving the large number of turns and the fact all the voltages resulting from magnetic coupling between turns appear in series.
There is, of course, also Yogi B's analysis which is based on the stored energy. While the capacitive currents may be the same, the distributions of voltages are not, so the way that the current is shared between the capacitors is different. Because the stored energy is dependent on the square of the voltage across the capacitor, the stored energy, and hence the capacitances, will differ in the bulk coil even though the current appears to be the same. This approach indicates that the foldback winding method has a lower effective capacitance than the zig zag method.
I think that I am following Yogi B: in this subsequent calculation.
We use the squares of the voltage differences to represent the stored energy,
assuming a total of 2V applied, 1V per winding and a winding length of 1.
delta_v = 2(1-x), delta_v^2 = 4(1 - 2x + x^2)
integral 4(1 - 2x + x^2) = [ 4(x - x^2 + x^3/3) ]
delta_v = 1, so delta_v^2 and its integral from 0 to 1 also evaluates to 1
Therefore, the foldback method stores 3/4 of the energy for a given voltage applied
Having been convinced by Yogi B's analysis, I thought that it might be interesting to try to model a pair of coil layers in such a way that the actual capacitance could be inferred.
I decided to use two resistor chains to represent the wires with ten windings per layer. The idea is to drive these resistor chains and their cross linking capacitors with a current. The voltage across the driving point would represent the impedance. Obviously, the different internal loading arrangements would upset the equal division of voltage at higher frequencies but perhaps at low frequency a similarity of input impedance could be found that would indicate the relative capacitance.
The simulation contains two chains and a simple parallel RC pair. The capacitors in the foldback array are scaled by the theoretical 4/3 necessary make its low frequency behaviour the same as that of the zig zag array. The capacitor in the RC pair is chosen experimentally to match the frequency response as far as possible.