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I'm aware that neon sign transformers (NSTs?) produce high voltage - up to 15kv. But I cannot picture what's inside of one. Wikipedia seems to indicate that they may be a form of resonant transformer but provides no diagram. Googling "resonant transformer" shows images that may be laminated core or toroidal based, close proximity series transformers, but it's hard to tell.

What would a typical NST look like if it was cut open, and would it be possible (not that I want to try) for someone to home-build a HV transformer like a NST?

Note: I'd never trust myself to build one, so that may answer the second part of the question - if I wanted a tesla coil, I'd buy a working NST

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Ordinary power transformers are designed to have good voltage regulation (the ratio of input to output voltage is fairly constant regardless of load)- the flux is mostly linked between primary and secondary and the leakage inductance is minimized.

Neon sign transformers are made to have high leakage inductance- ferromagnetic "shunts" are inserted (magnetically) in parallel with the secondary, which is similar electrically to having a large inductor in series with the output. That makes the transformer more of an AC constant current source than a constant voltage source. It allows the voltage across the neon tube to rise in order to start the ionization, and prevents excessive current from being drawn once the discharge starts. Below is a photo of a disassembled Neon Sign Transformer from this website (arrows and caption added).

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It's possible to fool with the shunts in order to change this behavior (removing them increases the short-circuit current).

Modern neon sign "transformers" are typically switching power supplies designed to have similar characteristics to the older style of mains-frequency transformer. The old style ones are practically indestructible, and the new ones are not going to last nearly as long.

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As far as building one yourself, I think it would be far cheaper/easier to buy a used one than to source the core and wire, but there is nothing mysterious in them-- just a lot of turns of wire with some vacuum impregnated coils and some laminations. –  Spehro Pefhany May 9 at 20:47
    
I am glad I read that wrong the first time. "Lamentations" sounded very mysterious. –  David Wilkins May 12 at 13:48
    
@DavidWilkins It was right this time, but auto-correct has done worse to me. –  Spehro Pefhany May 12 at 13:59
    
Having hand wound some small coils, I think "lamentations" is what would result from setting out to wind that yourself ;-) On the other hand, my grandfather used to fix small appliances as a hobby by rewinding their motors himself, so it certainly can be done. –  RBerteig Jul 18 at 22:42
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Mostly a "neon sign" transformer is a transformer with a high step-up ratio. The amount the voltage is stepped up from input to output is the turns ratio of turns of the secondary winding (the output) to the turns of the primary winding (the input).

The secondary also has to have a significant impedance, because neon signs really run on current. It takes a high voltage to get the gas ionized, but after that you keep it going with a fixed current at a much lower voltage. High impedance largely comes along with many turns. Put another way, ideally you want to drive a neon sign with a current source, not a voltage source. A transformer with significant output impedance approximates this well enough.

You mention resonance, so apparently some types have a capacitor in there to form a tank circuit at the line frequency. A friend in college had a neon sign transformer that we used to power up the CRT from a old TV with, but I don't remember seeing a capacitor in there. As I remember, there were just two wires coming out of one end of a sealed block with a transformer in it.

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