Honestly, MadHatter's answer is the closest to being correct, although Tesla coils ARE still a type of transformer. Nobody else here seems to understand, however, that Tesla coils are resonant transformers, so they do not operate in the same way as common iron-core transformers. The most important factor required for a Tesla coil to run correctly is that the secondary coil and topload LC circuit have the same resonant frequency as the primary coil/capacitor LC circuit. This is how you get efficient energy transfer from the primary circuit to the secondary circuit. Putting too many turns on the secondary would add too much inductance (and self-capacitance) that the secondary circuit will be significantly out-of-tune with the primary circuit. You will get very little energy transfer between the two resonant circuits, causing there to be little to no output. You will also run into the issues MadHatter suggested (the waveform will be discontinuous due to current being induced in the wrong portions of the coil). Remove all but one layer of wire and just leave it as-is. Then make sure that the secondary resonates at the same frequency as the primary. You can use the following formula to calculate resonant frequency:
where 'f' is the resonant frequency, 'L' is the inductance of the coil, and 'C' is the capacitance of the system (the tank capacitor in the primary or the topload on the secondary plus the coil's self-capacitance).
Do the calculation for both the primary LC circuit and then the secondary LC circuit and make sure they match. Otherwise your Tesla coil won't work at all.
If they don't match, you can "tune" the Tesla coil using different methods:
If the primary resonant frequency is too low, do one or both of the following:
- "Tap" the primary coil at different points to decrease inductance of the primary coil (shorten the primary)
- Decrease the capacitance of the primary tank capacitor
If the primary resonant frequency is too high, do the opposite.
If the secondary resonant frequency is too low, do one or both of the following:
- Decrease the length of the secondary coil to reduce the inductance of the secondary coil
- Reduce the size of the topload to decrease its capacitance
If the secondary resonant frequency is too high, do the opposite.
You have to use math to determine which of the above to use, and how much to adjust each one.
Perhaps the simplest way is to wire the transformer up and test the whole unit, with a current limiting load in series with a primary. A mains filament light bulb would be suitable (make sure it's filament, not LED or CCFL). Or a mid power resistive load like curling tongs. Whatever load you choose, it should not draw more current than your transformer input is fused for. This does mean you'd have to connect things mains side. If you're not happy doing that safely, then you shouldn't be inside with the transformer mains connections anyway!
Wired correctly, the transformer will produce a significant output, and the bulb will be dim or off.
Wired incorrectly, the transformer will present a short circuit to the input. It will produce no output, and the bulb will light at more or less full brightness.
Best Answer
You mentioned replacing the secondary with 10-gauge wire - I'm guessing that you have some understanding of the thicker wire's greater current-carrying capacity, which you would need for an arc welder.
The problem here is that the magnetic core - the steel laminations - also have a maximum magnetic flux carrying capacity, just like the wire has a maximum current-carrying capacity. The bug-zapper manufacturer would have used the lowest-cost (smallest) laminated core that met the power needs of the zapper - which are MUCH less than what you'll need for an arc welder. You would find that the secondary current would max out at a too-low value.
One caution - if you do get hold of a microwave transformer, DO NOT power it up on the bench top. Microwave transformers are one of the few devices that supply both high voltage and relatively high current - and are pretty much guaranteed to kill you if you get across the HV side. Not sure about bug zappers, but I'd be very cautious with that as well.