When 5V is driven from the PMOS to the inductive device under test, the current will ramp. After the 555 output goes low the inductance will free wheel at a nearly constant voltage. These ramps are amplified and given to the 555 timing inputs.
The ramp in this case goes from about 38mA to 80mA over about 200nSec time with 5V, and V = L (di/dt). So L = 5* (200nSec/42mA) = 24uH.
We can do the same calculation for the freewheeling current but at about 0.83V. The time can be better measured with an MCU's timer and/or capture hardware.
The Discharge output is an open collector and can pull down a pin on a a micro-controller (MCU) with a weak pull up e.g. 3V3 or 5V over a fair distance with good nose immunity.
UPDATE:
Mod to op-amp, now two stages each with less gain. Also added 400mV to see what a Schottky diode may look like. I think this could help the MCU measure a small inductance.
