Electronic – Mini slayer exciter (tesla coil) not working

The Slayer oscillator is actually a crude form of a resonant power supply. It runs at a resonant frequency. The resonant frequency is determined by L and C. Switching losses are reduced so that a BJT will run cool at sensible frequencies. You are stepping up so this makes L big and lots of turns make C significant. R has nothing to do with frequency. R is just for starting.

The transformer does have parasitic capacitance to ground. That is why C is drawn the way it is. If there was no parasitic capacitance to ground the oscillator would not work because its feedback path would be broken. Your diode does get reverse biased because AC current is flowing. I have used a red LED in place of D1 to indicate operation and give higher negative voltage on the transistor base to improve turnoff and possibly switching losses.

On your circuit, making R too small will actually make starting unreliable. You can ballpark R to give a small current that gives reasonable beta for Q. If your step-up ratio is too high, the oscillator won't start because the total gain would be less than unity. Make the step-up ratio N 10 or less so the oscillator won't need an exotic or nonexistent transistor.

The conductivity being only two thirds of that of copper wire of the same cross sectional area shouldn't be a problem, especially given a very low current application like a tesla coil secondary.

Enamelled aluminum wire will thus be just fine electrically, but mechanically it is problematic: It won't wet with normal solder, be it lead free or not, unless you actually have copper clad aluminum wire. There are some fancy solder/flux combinations that strip the oxide layer and bond with aluminum, but they aren't exactly common. Even crimp connections of AL with copper/brass fittings are problematic, as such joints tend to corrode, but they should still be your best bet in this case. Another option is to terminate the AL wire with conductive epoxy.