[old_tv_nut]

I suspect the inductance is too small to make any difference at 60 Hz, and you should just measure the coil resistanceand compare it to 270. Obviously, when the coil is connected, it reduces the voltage across the 270 ohm to a safe level, so the impedance of the coil must be much lower than 270.

Measuring only the resistance of the coil will underestimate its total impedance, but I'd guess not by much.

It would be interesting to know the actual value, though.

I am trying to understand this circuit - it looks to me like maybe X1 charges up C1 and quickly reduces the current shortly after turn on (???). Do you have a good idea how it works?