I need a high voltage power supply for the homemade CO2 laser I am building. I understand that a 15kV 60mA transformer is a thing, but its very big and heavy and its out of phase meaning each lug is only 7,500V and the difference is 15,000V. This would work for most applications but not one where i need 15kV positive and GROUND. I have seen simple arc lighter things that generate a supposed 15kV from a small battery, but the amps must be low? How can i design a simple 15kV high amperage transformer that can be smaller than a neon sign transformer?
Unfortunately, if you have the high-voltage coils, they must be isolated thickly so that you don't get a short-circuit between loops of the coil. Add a rather thick wire to conduct the current needed without excessive losses, and add plenty of coils, because with fewer coils you generate much less of magnetic field in the transformer (in relation to what goes out through supply wires and the rest of circuitry) - and you have a massive device.
The arc lighters are massively lossy devices. They take a relatively high current on the relatively short primary coiling, and can produce massive voltage on the secondary (which still needs to be isolated well, but can be now much shorter; fewer loops of coil) - but most energy goes 'poof' as heat, induction of wires, and losses in general; the energy efficiency is very poor - but since to create the spark you need very little energy, you can afford these losses, at benefit of small, simple, handy device.
If you want to get reasonably good efficiency, you must retain the primary:secondary loops count ratio, but you need much more loops on the primary, and thicker wire on secondary. And you can't really afford low efficiency, because the puny 3.7V battery of a spark lighter will barely heat up your circuit, while massive losses with much higher current will melt the wires and set things on fire.
Voltage and Current multiply to give power. 15kV times 60mA is the same as 60Volt times 15 Amps (=900 Watts) You would not expect that to be a tiny thing, would you? And that power is without transformation losses.
Keep in mind that 900 watts a common size for an electric space heater. This is not trivial power you are dealing with.