What do you mean by electromagnetic induction
Detection of a low pulse through electromagnetic induction
The anti-diluvian and inexpensive LM311 may not be ideal for this application, but should work fine with proper circuitry and pulses in the microsecond range or longer.
First, make sure that one end of the coil is firmly grounded so that common mode noise that may be capacitively coupled is connected directly to ground. Try connecting it directly to the inverting input.
Try a reference of around 30 mV with maybe 5 mV hysteresis (feedback from the output to the non-inverting input and a divider from the supply, i.e. three resistors - for example 100 ohms to ground, 16 K to +5 and 100 K to the output ( Output high) has a 4K7 pull-up, e.g. up to +5), assuming a +/- 5 V supply. Ground the common output. You can test this with a function generator (such as a 1 kHz-60 mV sine wave). It should deliver clean output pulses.
Use positive and negative supplies (e.g. +/- 5V to +/- 15V) to ensure you are in the common mode range of the LM311 (or it will not work properly at all). It can be customized for a single delivery later, but it works first.
If that doesn't give a clean pulse, send an oscilloscope recording of the voltage on the coil with one end grounded to your circuit. If your pulse isn't pretty clean at this point, you have another problem.
A coil cannon is little more than a special case of a linear motor that is just a special case of a brushless DC motor. Hall effect detectors are used in high power motor applications or those where you need to know where the magnetic field is in order to drive it most efficiently.
I suggest that while you can use the coils for both feedback and driving, you can save time and frustration by simply adding Hall effect sensors to the assembly.
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