The problem is, the SOX (and other cold starting low pressure lamps like slimlines or so) need not only to break down the gas, but also support quite significant current when the electrodes are still cold, so have high cathode drop voltage, in order to warm them up and so turn the discharge into a proper hot electrode arc. Better to say all lamps need this, but the high pressure ones (HPS, MH,...) have very low voltage drop over the rest of the discharge when cold, so the normal operating OCV is enough to support such current, so the ignitor is required only for the initial gas breakdown. The low pressure lamps (SOX, slimlines,...) operate at low temperatures, so the pressure so voltage drop across the rest of the arc is close to the normal operating drop. Wen the cold cathode drop adds on top of that normal voltage drop, it becomes too high for the lower OCV ballasts to support the current so the lamp does not start. To address this, in the past a ballast with just high enough OCV to overcome both the cathode drop and the anode column fall (with sufficient margin to allow reliable reignition after zero cross) was used. But high OCV means the ballast needs a lot of turns on the secondary, so a long wire, so yields high losses and very heavy and expensive ballast. But before the solid state electronic became usable, it was the only way to feed such lamps, so it was accepted. Now the SOX ignitors use a different approach: When they sense too high voltage drop (aka arc not there or with too high voltage to be supported by the ballast alone), it generates HF higher voltage, some 10kHz, superimposed on the ballast OCV. That rises the peak voltage to the rated 750V or so (sufficient for the gas breakdown), but mainly it is able to feed quite significant current at that voltage (it is an HF AC generator, not just pulser), so feed enough power at that higher voltage to allow the electrodes to warm up. With HPS/MH ignitors you may get higher peak voltage, but with very low power. And it is the power at the elevated (not necessarily high) voltage, what is missing for successful SOX or slimline fluorescent start. To be clear, we are talking about half second till few seconds after first gas breakdown, so a short time, just to allow the electrode filaments to warm up, but when such feed is not provided, the lamp can not pass the warmup state. And the only thing you will get is the HV ignitor causing cold electrode sputtering.
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