I used to know all this many years ago... how to work out the flux density, and choose a material which doesn't saturate, but I have long forgotten!
It needs to be good with a 13V peak square wave, 500Hz.
So we have 13V across 8mH for 1ms, which from v = L (di/dt) yields
1.625A. A quick hack in LTspice confirms this, for the first 5 cycles. I had a lot of trouble generating a square wave from -13V to +13V :) so I am not sure what the current waveform will look like when you switch to -13V when the current flowing is still 1.625A; I suspect it will not ever exceed 1.625A later though. With a 13V peak sinewave (a predefined function in LTspice) it looked to be just under 1A, but all positive which is obviously BS.The bit I have forgotten is how to calculate the flux density in the core. I would prefer the whole thing to be something the size of an RM10 core
A toroid would have less stray flux, and again there is a vast choice. I have a bag of these from many years ago
What I don't get is how much current this will carry. 83 turns at 1.6A is 132 AT which sounds an awful lot. I thus suspect I will need a bigger core, probably iron.
Looking on Ebay for ready made stuff, 10mH, I see e.g.
Another option is some toroidal transformer, and ignore the existing winding and put more turns on it. Ebay is full of vintage transformers but most are pretty big.
I was going to wind the TN16 with some turns and see what it does. I have an HP 3314 pulse generator and a power amp which can output 9V peak.
The 8mH needs to be +/- 0.3mH. It was determined experimentally using this amazing thing
I also have an LCR meter.
I'd be grateful for any suggestions.