Ohh Yes you are right.
LOL
It would have been a lot easier if I'd have been wrong.
"But I see many variations in that yellow part"The way it recovers will depend a quite bit on how it saturated, and what's left of the real decay tail.
You might be able to do it perfectly well with an AC amp, but the best I could come up with is the DC coupled +5V and -12V version, I'll post a slightly updated version right here, It looks simple-ish too me, you might be able to pick something useful from it.
Possible advantages.
The coil is permanently connected to 0V, rather than 12V.
The decay curve is at 0V and positive going.
Uses the same n mosfet, now on the -12V supply.
The DC coupled non inverting op amp needs no coupling, or bias decoupling caps.
Can use very low value gain resistors and a low input impedance op amp.
Possible disadvantages.
The odd +5V and -12V supplies.
Switching the mosfet when it's on the -12V supply, but at these switching speeds an opto or 1 transistor would do, - probably.
The high DC gain of the op amp can cause a DC offset on the output, not too difficult to fix.
Probably quite a few more that I haven't thought of.
The gain of the simulation shown using the LT 28MHz op amp is about 1000 @ 10kHz, and about 650 @ 40kHz.
V9 is nothing, just a 1mV sine to measure the gain in the simulation.
SW S6 gets replaced by the n mosfet.
The
green V(out) is the output.
The
red V(out2) is a version with an analogue switch on the input, so that it doesn't saturate too much on the first large pulses.
Both versions are in the .asc, it's a bit messy but hopefully someone will run it and suggest improvements!
Phew.