I would like to make a Tesla coil, so I bought a 10 kV 120 W 30 mA neon sign transformer from Aliexpress.
First potential problem, while playing with it, I noticed that when it stops arcing (because the probes are too far apart), if I put them closer again it doesn't arc anymore and I have to turn off and on the transformer to generate the arc again. From what I saw online, the arc comes back with a regular NST so I don't know if it will be a problem. On the other hand, if I "short circuit" the probes and then separate them everything works fine, the arc is still there.
Next I created a capacitor using wine bottles inside a bucket and salty water, I get around 4.5 nF with 3 bottles. I connected the capacitor in parallel with the NST and I added a spark gap also in parallel (that's all I did for the moment, no secondary or anything, I just wanted to check how it's going to react).
When I power everything on I get no sparks on the spark gap, the NST makes the same sound as if it was "short circuited" but nothing! If I put the probes of the spark gap really close together (like 1 mm or 0,04 inches) I do get some really tiny sparks but I don't get this "high frequency discharge" sound (I mean the tazer sound that we hear on teslas).
Do you know what's the problem? My guess is the way the NST is built, maybe it's not built the same way as a regular one. Also I tried using 2 bottles and 1 so about 2.25 nF and 1.5 nF, same results.
UDAPTE :
@DerStrom8 answered to my question but I'll still send you what I was writing, maybe it will help me for later on, first here is my experimental schematic 
Using an online calculator, I know that with my 10 000V and 30 mA, the optimum value for my capacitor is 9.5 nF (of course I will change the value if I find a heavy 50Hz transformer), my capacitor for the moment is less than that so I think it’s ok for experiment purposes. Then I know the formula F = 1/(2pi(L*C)^1/2), which can be used to sinc/tune the frequency or the “primary circuit” and “secondary circuit” (by that I mean the first capacitor/coil and the second “capacitor”/coil). I also know that the easiest way to tune the frequency is to chance the inductance of the “primary coil” as it’s pretty easy. Since most people tune their tesla coil with trial and error, I deduced that If I randomly size my “primary” coil for experiments, I shouldn’t be a problem There is one thing I don’t quite get, tell me if it works like the first option the second, or both are wrong
First : I can almost randomly size my primary coil, I then find my resonance frequency using the capacitance C1 of the wine capacitor and the inductance L1 of my primary that I randomly sized. Then I’ll have to to match this resonance frequency with the correctly sized L2 and C2 of the secondary. And since the calculation won’t be perfect, I then tune a tiny bit the primary coil.
Second : I can’t randomly size the primary coil, and in that case can you tell me how to size it ?
Also, I don’t think I need the secondary circuit to have the spark effect on the spark gap for the experiment. Also, is 9.5 nF the best value to have the largest sparks in the end ?
Best Answer
You have not mentioned what type of transformer it is. Solid-state transformers will not work for Tesla coils, as their outputs are a much higher frequency. They often look similar to the image below:
You really need the big, heavy, bulky type shown below.
If yours is a little, lightweight box then it will not work. From your description of the results I am willing to bet you have a solid-state one. With a higher frequency the capacitor will act more like a short circuit (as JRE mentioned in the comments) and it will not charge/discharge in the desired manner.
If you do have one of the big bulky ones, however, with a 50/60 Hz output, then my guess is that it has built-in fault protection. Transformers like this often shut off automatically if a fault occurs, which could happen if it detects an open output (example, if your spark gap is too far apart). It will need to be reset (unplugged and plugged back in) before it will run again. These transformers also do not work well for Tesla coils for this very reason.