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the input of a guitar amp, a low noise current device driving the primary of a step-up transformer
Screw the theory.
Leo Fender sat there and wound pickups for "best performance".
If you are cheap, going direct into a vacuum tube, FET, or low-current BJT, you will not do any better than to stick with the classic pickup designs.
If you are rich, you can wind to a very low impedance, like 150 ohms, and go into a standard microphone input. Les Paul did this, and "recording" pickups like his were available. The low impedance balanced line makes line length uncritical, and reduces hum pickup in the line.
But line capacitance is PART of classic guitar sound. It rolls-out the super-highs which tend to grate. It resonates with the self-inductance to produce a boost.... where?
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credible document I've seen
Silly EEs. Well, OK, they divide and conquer; they measured an unplayable pickup, one with no cable on it. Their "Hot Strat" measurements imply 75pFd capacitance. Any in-use pickup has 100-1,000pFd hanging on it, in the cable.
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what does the pickup look like, in terms of impedance?
From the EE's data, a pickup is a ~5H inductor with 5K winding resistance, ~100pFd winding capacitance. In real life it will have 10pFd to 1,000pFd more capacitance in the cable, and 470K-5Meg+100pFd in the amplifier.
Taking 5H, 5K, 100,300,1000pFd, and 2Meg, I get these impedance curves.
The log-frequency scale is often useful. But for noise estimation in a flat amp, the linear frequency scale lets us eyeball the noise contribution. Though Z is 5K-50K over most of the musical band, it rises to ~1Meg at a resonance that depends on the pickup and the cable length. But the resonance is narrow. I'd expect the overall noise to be somewhat like a 50K-200K resistor.
Using a SPICE method I do not understand, I got 7 microVolts noise summed 80Hz-8KHz. (Guitar extends to 80Hz and no big cone has much total output at 8KHz.)
Using a rule of thumb I do understand, I get ~6uV noise. Improbably good agreement.
A 12AX7 can have noise below 3uV, if you are lucky; over 20uV if you get a hissy one. Splitting the difference, a large number of 12AX7 have noise below 8uV.
Leo was no fool. He got the pickup self-noise output above the noise of most 12AX7s. If wound with fewer turns, music output would be lower but total noise would not change much, so worse S/N. Winding with more turns would not improve S/N much, would lower the winding-cable resonance frequency, and most important the winding is already annoyingly fine and many-turn.
There is a matching too. Gain, bandwidth, and noise are ultimately limited by capacitance. The coil has ~100pFd, a minimum cable has 100pFd, a high-gain triode has 100pFd. Changing any one part would not change the total capacitance much. The player may add cable (100pFd is only 3 feet), get resonance at a lower frequency, clobber the extreme highs. Player's Choice; Kit does seem to be wanting less highs.
There is a fairly wide range of pickup values, but not incredibly wide. Throwing-out the extreme cases, 2.2H to 7H of inductance. Coil resistance matters hardly at all for reasonable values of grid resistor; they quote it only because it seems obvious, and is routinely measured to cast-out open coils. The other major variable is cable length. Grid resistor has a minor influence at resonance, unless it is very small (under 470K) in which case the peak is knocked-down.
