The decision I've come to is that if we (the people with knowledge) don't help him, he's going to try to do it anyway, quite possibly with bad information. I think that would be far more dangerous than helping him do it the RIGHT way.
Damn Straight !
I see the full spectrum of questions as well. Some people it is clear do not have a clue and will not progress past rubbing two sticks together to create fire.
There are some who of course will become arsonists when you do whether you help them or not.
Anyhow, there are a whole range of high voltage generators that are safe to build and operate even for the most complete novice.
One of my favourites is a Kelvin generator.
With tesla coils there is a lot to learn. Starting small and simple is the way to go.
Obviously anybody playing with a tesla coil should be aware that you can get seriously hurt and some points in the circuit contact with them can be lethal.
Just be aware that as a newbie, you're in the deep end without floaties so your own safety is your own concern and it should be priority.
The rest of us will give you enough information to make the next step as you learn what you need to get to each step.
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Tesla coils are more than just a transformer. Like a radio they are a tuned circuit.
There are many ways to run a tesla coil, but like the other guys that have suggested the simple methods are often the best.
A high voltage circuit which generates a spark is the simplest method, and it is not much different from a circuit one would use to run a jacobs ladder.
This is where a lot of your experimenting should begin. You want to be able to generate a spark which you can control and adjust the striking rate.
The simplest is to just have a spark gap which you can adjust with a screw. Another is to have a spark wheel which the spin rate can be adjusted.
I've seen many different ways to generate the voltage required for the spark. Ordinary power transformers used as isolation transformers and run above their primary voltage rating. There's obviously a little bit of an overhead, but I have seen that combination flash over.
Two microwave oven HV transformers used back to back. Damn scary but one of the best generators I've seen operated.
Walton-cockcroft multiplier is another I've seen used and of course TV flyback transformers, neon transformers, ignition coils and my personal favourite, the photocopier HV transformer.
The primary and secondary coils of the tesla coil look ok to start with. You'll refine them yourself later.
I'm not sure what that wire is coming down from the top of the secondary coil, you want to keep that end of the coil as much as an open circuit as possible.
Which then I have to ask, where the hell is the (supposedly) grounded end of the coil going ?
Think of a what I said about a tuned radio circuit. Open coils and sloppy wiring are not good RF practice, even if it is 500KHz.
While you might not have a nice dome for the top of your coil, you can still scrunch up some aluminium foil and sit it on top of the coil former with the wire tucked in under it. At least a fine high tension wire won't be flicking around trying to grab anything at ground potential.
Another thing to consider is the HV capacitor. A lot of coils I've seen running use a fixed value capacitor. I prefer to build mine so it is tunable.
My first coil was not much different from yours, but smaller. About 30cm tall and I used a TV flyback transformer. I learnt a lot on how to get a tesla coil to work. It used a fixed capacitor and an adjustable spark gap made from some tin cans nailed to the wood base. It would throw sparks about 5cm.
Tesla coils are addictive. If you build a small one, you always want to build something bigger !
My second coil was about double the size of the first. 60cm and before I tuned it the sparks were about 10cm long. I was really unimpressed and I did some careful reading (in the days before internet) down the local library. Used a small variable capacitor from an old valve radio. I had to make some modifications to it removing some of the plates so it wouldn't flash over. I also added more fixed capacitance in parallel to get a value close to what I wanted.
I tuned the coil with it running, and that moment of clarity came when it was sitting on the bench and a spark about 40cm long jumped off the top of the coil and stuck me fair and square on the forehead. I had that moment of clarity with the bright white flash which I saw not with my eyes !
It was something like god saying, "Congratulations idiot ! you got it right !" Later I used a photocopier transformer and there was no way I was going to be struck by it with that thing running it.
Lucky it was only a small coil. My third coil was not built well and was a failure. It worked, and I learnt a lot from it, but it wasn't any better than the second.
The fourth was over a metre tall. It was a brute. It wasn't so much about the coil, but more about what I could do with it. That kind of vandalism I'll leave for somebody else to discover on their own.
My last coil was also a failure. I have not attempted to build one since. It was 2.3 metres tall. I was using a cardboard former which I had wrapped in duct tape. (I was trying to avoid PVC). The coil only worked for a short time before it set fire to the cardboard. I didn't try to get it running after that. I gave the coils to a friend who also set fire to it. He later rewound the coil on a new plastic former.
I can think of reasons why I would not use the 555 circuit you're using. But then at a later stage after I had worked out some simple basics I would consider going back to it with what you have learned from ye olde tesla coil circuit.