I have done several experiments with an old capacitor of 10uf and a 20w .27a choke, on 120v. The first thing I did was using the capacitor after the ballast and then using the setup to run an f40t12. It worked, but the tube is overdriven and will only manual start. I also did it with an f32t8, but still overdriven. I have tried fs2 and fs25: they both just blink. Fs4 doesn't do squat. I tried using the cap in parallel with the f40and all it did was glow the first time but when I cut and reapplied power, it made a hellish squeal and flashed violently, but the tube, ballast and cap are ok. When I did the 20w lamp with cap in parallel, it started no problem but would be dim until I took the cap off when the lamp striked. I was thinking that I could use the very first setting to power a 50w merc lamp. Just food for thought, but I got a 100w merc to strike but didn't let it run too long for fear id kill the lamp or the ballast. Thoughts?
Edit: I have done further testing with interesting results.  I first used a  17 what fluorescent tube of T8 size and all it did was glow on the ends. I tried an F 14 T8 and 12 tube  and the results of that were puzzling. The F 14 tube of T 12 size started up fine but glowed dimmer than normal until I removed the capacitor. The F-14 tube of T8 size had a rough start up. Whenever I started it at first,it started in cold cathode mode and sputtered crazily in the ends glowed a vibrant blue. I heard some crackling but that was due to a loose connection. I have yet to try on an f15 tube of both t8 and 12 size, but I expect results similar to the f14 test, with fluorescent tubes of respective diameters. I believe I created some sort of resonant start circuit, perhaps ideal start up and brightness of an f20t12 tube could be achieved by tweaking the cap value.

Any configuration of coil and capacitor makes a circuit which would be resonant at some frequency, not necessarily 60 Hz. But even at off-resonance frequencies, it will have effects of its own....

What is more, when the lamp is involved, the striking and fade-in of the lamp changes the circuit, and its resonance frequency changes. It might as it changes approach 60 Hz as well



While at that, you dont have to just plug it in and hope for the best (and for no damage). This sort of stuff can be calculated... And in short, that is how it goes :

Coil, capacitor and resistor all resist to flow of current

Resistor's resistance is basic. The voltage and current in a resistor are in phase with each other, and they are related through Ohms law - E = IR. The wire from which the choke's coil is wound is a resistor, but to keep things simpler, we can disregard that at this stage

Coil and capacitor have reactance, which is some sort of resistance - sorta. But thing is, the reactance of coil XL and capacitor XC in seies does not add up, it subtracts. And the reactance does not just add up with resistance either.....

This leads us to what happens at resonence : Their reactance depends on frequency. At the resonance frequency they exactly cancel out each other XL - XC = 0, so the case of a resonance would be same as if you connect 120V straight to either end of the tube. (there is still the resistance from the wire, but we disregard it)

At other frequencies one is bigger than the other, so the cancelation is only partial - X = XL - XC, when XL is larger X is positive and the current lags behind the voltage, when XC is larger X is negativee and the current leads. Without resistance (so just the coil and/or capacitor), it will lag or lead by 1/4 cycle

Looking at a cycle like a complete circle (a complete turn of the armature in the generator supplying the AC if you wish - though this is only correct for synchronous generator), 1/4 cycle is 1/4 turn, so 90 deg angle between the current and voltage

In a series circuit (choke, capacitor, tube) the current is one for all components. What we get is, in the choke/capacitor the voltage drop will be at 90 deg angle to the current. In the tube the voltage and current are fairly in phase. So summing up the voltage on the tube and on the ballasting components is like summing the 2 right angle sides of triangle to find the 3rd side :

(E choke/cap)^2 + (E tube)^2 = (E mains)^2

The voltage drop on the tube is known, the voltage of the mains is known, so wit that, we can estimate the voltage on the ballasting components : For example F20T12 with 55V arc, on 120V power :

E choke = sqrt (120^2 - 55^2) = 106V

The reactance of a choke, capacitor, or what results from the 2 together, does satisfy Ohms law by itself - Just substitute the reactance for resistance. E = IX
X choke = 106V / 0.27A = 392 Ohm

Coils inductance is measured in Henry (Hy) and the reactance is :
XL = 2 pi F L where F = 60 Hz, L = inductance in Henry

Capacitor :
XC = 1/ (2 pi F C), C = capacitance in Farads (so 10uF is 0.00001 F)



With that you can allready estimate in quantities what you are doing....

The circuit where the choke and capacitor are not just in series but with capacitor parallel to tube etc are more complicated, so get okay with the basic things first (plain series). The concepts dont change, just more math to do