The current in the solenoid is exactly the same whether it moves or not,
but its slope is not, because L changes with the armature moving. I
suppose low L for unmoving and higher L *after* moving. If you look at the
energy of the decay pulse (when you release the Darlington), the high L
should give you higher energy. If the solenoid closes the magnetic circuit
fully when it strokes to the end, the increase will be large, otherwise
pretty small. You could measure this with a current sensor in the kickback
circuit driving a simple integrator. Quick diagram:

                        Coil drive +Vcc
                              o
                              |
Kickback diode Cathode o-----EBC-----R1--*--C1---GND
                                         |
                                         o Out (measure voltage vs. GND
                                               *after* turning off driver)

You will have to know something about the coil to choose R1 and C1. If
not, make trials. I suggest R1 = 1K and C1 = 0.01uF for a start. Measure
voltage on C1 with a peak-holding DVM.

Peter

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