The large inductance values ...
... in the mHz range.
For very big inductances, maybe a multiplier (with gyrators) to emulate a big coil.
Never tried, but it should work. Same idea might be advantageous for capacitors, too, using a small capacitor multiplied instead of a leaky electrolytic.
Even more, either inductors or capacitors, they are not used here as energy storage, but to form filters. They can do the d/dt operation for voltage or current, and thus they can form filters. However, nowadays number crunching and ADC/DAC are cheap, might make sense to go digital and filter there.
Putting aside how practical will be to implement any of that, either analog or digital, for active noise cancellation, two things are needed:
- 1. to decide what to subtract from the input mix of Vref+noise (usually using RC filters)
- 2. to correct the output (usually an adder)
However, point 1. takes time, so we will also need
- 3. a delay line for the original signal, so the extracted Vnoise and the Vref+noise are in sync in the cancellation adder from point 2.
Speaking while writing this, once we have a delay line "component" we can use that to implement filters, and get rid entirely of big capacitors or big inductors.
I have zero hands-on experience with voltage references, and to implement such low frequency filters I would be tempted to go digital to process the noise (before adding it back to the analog circuit for cancelling).
Heaving an analog delay line would help filtering out the shot noise, too.
We can detect when a jump in voltage occurs (by comparing the current value with the most expected averaged value), and when a jump in voltage occurs, we disregard the current (wrong) voltage, and instead, copy to the analog output a previously known good voltage (we take the previously known good voltage from a tapped delay line, where each delay tap is connected to the input of an analog multiplexer, and we decide which input is considered as correct, or as heaving no pop-noise in it at that moment in the tapped delay line).