Circuits involve active electronics and a dummy coil.

[Helmholtz]

Hi cheezit,
That is how I interpreted it.
For my application, the circuit wasn't working with unity gain. I was using a dummy in the neck position to cancel out hum from an overwound bridge. So I modified it slightly as per my post.

I have tried it on 3 guitars now:

1) Neck P-90 cancelling out hum from bridge P-90: Worked great. So much noise reduction.
2) Strat pickup under metal pickguard cancelling out hum from Jazzmaster pickup: Works so-so. It is a marked improvement though.
3) Strat pickup under pickguard cancelling out hum from other strat pickups: Also works fairly well, removes enough hum for me, but not as quiet as positions 2 and 4 with RWRP.

It sometimes gets overlooked that magnetic (interference) fields vary with position and especially orientation. The dummy coil needs to have its magnetic axis oriented parallel to the PUs' and should be placed as close to the PUs as possible.

When you remove the ferromagnetic core parts from a PU coil, it will produce less hum than the original PU. Thus it is advisable to use an overwound dummy coil or a coil with larger area than the PUs (e.g. P-90 dummy coil in Strat) in conjunction with the unity gain buffer circuit.

Your modified Jfet source amplifier circuit is critical with such low battery voltage and will work properly only with specially selected Fets having a very low pinch-off voltage. Also, to use it as an amplifier means to take the output from the drain. But the internal impedance at the drain is considerably higher than at the source. Thus you sacrifice one of the benefits of the simple source follower, namely inserting only little series resistance with the PUs.

Placing the duimmy coil under a metal pickguard not only reduces the HC signal but also changes its frequency content, as the conductive metal preferably absorbs the higher frequency components of the magnetic interference fields. But for perfect HC you want the dummy coil's signal to replicate the PUs' interference content as close as possible.

[cheezit]

Your modified Jfet source amplifier circuit is critical with such low battery voltage and will work properly only with specially selected Fets having a very low pinch-off voltage. Also, to use it as an amplifier means to take the output from the drain. But the internal impedance at the drain is considerably higher than at the source. Thus you sacrifice one of the benefits of the simple source follower, namely inserting only little series resistance with the PUs.

BTW, in my searching around I consider this approach to be very similar in spirit to https://patents.google.com/patent/US5569872, which is the Dudley Gimpel patent for Ernie Ball. Take a look at Figure 5, which shows the same approach: a buffered noise-sensing coil inserted on the ground side of the pickup.

The circuits are much more complex, though, and I wondered why; your points about the sensitivity of the circuit to the characteristics of the JFET and the amplification problem may be it.

If I'm right, it's the same reason DIY fuzz pedals can be made out of 8 components but Boss equivalents have dozens and dozens; they have to design for consistency and low cost, and can't mess around with parts that are operating at (or beyond) the edge of their design parameters.

[cheezit]

Update here...I had an interesting experience with this yesterday. I have a dummy coil i'd harvested from an old P90, and I had it wired in to a small PCB board to try this circuit. Wasn't working, so I detached it and put a connector inline that allowed me to insert the dummy coil, unbuffered, on the ground side of the actual pickup, tucked away behind the volume pots.

So now I have a P90 with a P90 dummy coil in series, no buffering. Didn't know what to expect....I was assuming it would be mud. But the guitar itself is a little bright, and the existing P90s (GFS Vintage) are not high-output, at least not for P90s. The result was:

1) Some reduction of hum, but highly directional. If I stand in the right spot it's almost silent, if i turn 90 degrees it's almost as bad as if the dummy coil weren't there. This is likely due to it being down in the control cavity, and parallel to but not on the same plane as the pickup. Might be able to move it a bit.

2) Significant darkening of tone, but highs are still there. Brings the bridge EQ closer in similarity to the neck, which doesn't have this dummy coil.

3) Here's the surprise - big increase in low-end punch through high gain. This thing is now way "chunkier" and crunchier than before and just feels really great to play. It's not more output overall; it's that the lows are louder and more dynamic in a way that differs from what an EQ pedal might do.

The low-end behavior is (I'm guessing) related to the increased inductance of the additional coil, which seems to have lowered the resonant frequency of the pickup in a pleasing way.

[Helmholtz]

Update here...I had an interesting experience with this yesterday. I have a dummy coil i'd harvested from an old P90, and I had it wired in to a small PCB board to try this circuit. Wasn't working, so I detached it and put a connector inline that allowed me to insert the dummy coil, unbuffered, on the ground side of the actual pickup, tucked away behind the volume pots.

So now I have a P90 with a P90 dummy coil in series, no buffering. Didn't know what to expect....I was assuming it would be mud. But the guitar itself is a little bright, and the existing P90s (GFS Vintage) are not high-output, at least not for P90s. The result was:

1) Some reduction of hum, but highly directional. If I stand in the right spot it's almost silent, if i turn 90 degrees it's almost as bad as if the dummy coil weren't there. This is likely due to it being down in the control cavity, and parallel to but not on the same plane as the pickup. Might be able to move it a bit.

2) Significant darkening of tone, but highs are still there. Brings the bridge EQ closer in similarity to the neck, which doesn't have this dummy coil.

3) Here's the surprise - big increase in low-end punch through high gain. This thing is now way "chunkier" and crunchier than before and just feels really great to play. It's not more output overall; it's that the lows are louder and more dynamic in a way that differs from what an EQ pedal might do.

The low-end behavior is (I'm guessing) related to the increased inductance of the additional coil, which seems to have lowered the resonant frequency of the pickup in a pleasing way.

No real surprise, here.

You may achieve the same lowering of the PU's resonant frequency by wiring a capacitor of around 470pF (exact value depends on your PUs and guitar cable) in parallel with the PU - no noise reduction though.

The buffer avoids lowering of the resonant frequency.

Perfect noise cancelling via dummy coil would ideally require the dummy coil to be identical to the PU coil and being in the same place - or surround the PU as with low impedance frame shaped dummy coils. Results of real systems may not be completely satisfying if the noise source(s) are located in close proximity, as in these cases the magnetic interference fields vary strongly with position and orientation.

There is a problem with placing the dummy coil in the control cavity: The casings of the pots are typically made of ferromagnetic steel and thus change the noise fields close-by. This may prevent the dummy coil's signal to be identical to the PU's noise and consequently cancellation will suffer.

[Helmholtz]

Update: Mike Sulzer's Jfet buffer does not really have low output impedance. While typical textbook examples of the source follower give values of some hundred Ohms, a closer look at this circuit reveals that its output impedance increases noticeably with lower drain currents. In our case with Rs=27K, the output impedance will be around 3K (!) or even higher. (In can be measured directly with an LCR meter).

lnserting a series resistance of several k Ohms will damp the PU's resonance and might be audible.