Welcome here at EEVblog.
So what do you mean with "HID driver". I assume the transformer right?
I am not sure you need a bigger transformer, if all you want is some better arc's, this one seems reasonable, the weakness of the arc may come from other things. What are the specs on the transformer?
Also be careful with bigger transformers. You may think you are working isolated and safe, but even isolated voltage can hurt and even damage you if it has sufficient power behind it, and you accidentally touch both wires. Which is why I suggest you should improve build quality before you improve the energy used.
And your multimeter is not rated for anything above 1000V. In any case, please also understand your multimeter is also a source of leakage for the generator. In fact, your arcs may improve if you disconnect the multimeter. To simulate the multimeter in your simulator, you should add a 1M Ohm resistor. Most good multimeters are only 10M Ohm, but this (in)famous cheap one may be less at 1M
As to the construction of your schematic: I think a breadboard may not be the best. All these strips and wires probably has a lot of capacitance in the wrong places. Rebuild it using good sturdy solder connections. Though admittedly that may loose structural integrity quickly. Can you get hold of some
prototyping PCB?
As to the component values: Those blue capacitors you see are
Y class protection capacitors.
Purely theoratically, both the diode and the capacitor, each individual one, gets a maximum of
twice the input voltage.
So at 380V in, which is 380*root(2) or 535V peak, you would need 1070V rated elements.
(it also means you likely only have one stage working in your setup)
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It is all a bit high though. Maybe you should it first more focus on the theoretical part. See if you can get multiple stages working, with lower voltage. Yes, it is not so exciting, but it does build better experience and understanding:
Give it only 12V directly from your train variac, and see if you can get 5 stages working, pushing it to, I don't know, 40 or 50 volts
Model your simulator better (not those absurdly high 1uF values) , and see if each stage gives you the correct output in real life. (though remember, measuring it will affect its working, so how to measure is a bit of a challenge)
Build it all properly on experiment board rather then breadboard.
Once you understand this 12v to 50v, measure each component individually, and see what voltages each components get (should be double) and then build up.
Oh, and try and keep it high voltage, but low energy. Meaning low capacity in your capacitors. If you check those Y capacitors, you'll see they are in the pico and nano range, not the 1uF you have in your simulation
PS: if you source stuff, try and find a Romanian store rather then ALiExpress.
I could argue chinese quality, or supporting local shops, but also consider, every $1 piece you get from china comes with a 1 month wait time, only to then find you need yet another component, wait yet another month. To keep the speed and enthusiasm going, finding a good local supplier, even if 5x more expensive, is vital.