| It depends on how the mode control and mainly it's reset is implemented.
And that includes the supply of the controller part, mainly how big is the filtering capacitor there.
The PTC's need to really cool down to preheat the lamp again. It may be connected so, the PTC may cool down after the lamp starts (series VDR, connection of the PTC across only part of the series combination of the resonant capacitor, use of SIDAC's,...), but even that needs few menutes after the last turn ON. So if the lamp has been ON for only some minute or so, the PTC may have still not enough time to cool down (even when the circuit removes all power from it after start), so it won't preheat.
And when the fixture is really enclosed, it frequently happens the ballast and/or lamp heat keeps the PTC so warm, it effectively won't preheat the lamp unless the complete thing really cools down.
So the PTC will either start the lamp correctly, or need way longer time to cool down, so it does not correspond to your description.
With the electronic the preheat time is usually controlled by a capacitor (the timing itself) and a digital state machine. In order to work properly, all these things need a decent reset (discharge the timer capacitor, reset the flip flops) to be applied before the ballast starts. Normally, within the dedicated ballast control IC's this is done using a signal from the controller supply undervoltage lockout circuit (it keeps the reset active and the ballast OFF till the startup resistor charges the controller supply capacitor above certain voltage and only then lets the inverter stage to operate).
These designs count on the controller supply to get below the UV threshold once you shut the power down, so the next time they again start up through all the starting modes, same as when powered up after a long time.
But there could be a catch: To prevent false "fault" latch activation (due to hard switching) because of just a brief undervoltage on the ballast input, some ballasts use a main DC bus undervoltage comparator (monitoring the main power supply) detecting such event and trigger the restart once the power is restored. This works fine, but it has some unwanted consequences:
First any glitch on the supply line cause the ballast to go through complete restart, include the 2second preheat. So instead of just a brief glitch on the light, such installation shuts down the light for the preheat time. When these glitches are more frequent (heavy load switching,...), it becomes quite annoying.
Other consequence is, the controller IC needs an extra input and a few extra components for that.
And it means, the ballast power stage has to be designed so, it remains in the soft switching regime for all lamp, mains and all component tolerances. That restricts the design quite a lot.
Therefore many controller designs use other approach: They monitor the switching mode and during such undervoltage event by increasing the frequency they maintain the inverter in the soft switching region and the lamp lit, still preventing any harm from the eventual hard switching. And even when the lamp arc extinguish, after power restore the balalst then sweeps the frequency again over the resonance, so reignites the lamp. It is assumed, such events are short enough, so the cathodes have no time to actually cool down and so loose the emission. When the frequency reaches the maximum, it then shuts down - as that could happen only when the resonant circuit is broken, so either lamp filament failure, lamp removal and/or fault in the ballast power stage. This concept then does not need any voltage monitor on the DC bus, so may save some IC pins and mainly the breakdown problems with SMD resistors in the divider (no brownout detect needed, so no divider, so no resistors, so no PCB breakdown problems).
Usually after power OFF this concept boosts the voltage by the resonance effects to maintain the lamp powered, so by that discharges the main tank capacitor within a second or so. That means the controller looses it's supply, so the reset activates.
But if some crook things "larger is better" and put an excessive capacitor to the controller supply (normally a 1uF ceramic is more than enough, but still I've seen 220uF there - no idea why that), it may keep the controller powered for a few seconds before actually reaching the undervoltage and reset it. If you reapply power within this time, the ballast just restarts without any preheat (as the controller "thinks", there is just that brief power glitch), hence a possible cause of the behavior you observed.
Some discrete implementation may just count the timing/state control capacitor is just discharged at power ON, without providing any dedicated reset to it. These then may need the power to be really shut down for longer time to let all the capacitors to discharge and so may behave in the way you noticed.
What I do not understand is the motivation for that implementation - given the cost of the dedicated controller IC's, I doubt the discrete implementation could ever be cheaper (60..100 components) or any more reliable (200 solder joints or so, complex PCB), but I've seen such implementation used quite often. Really I do not know the motivation behind.
When someone implements the controller into a microcontroller, there the proper reset is the easiest - the rather high microcontroller active power consumption will discharge any of it's supply capacitors rather quickly after the ballast looses the main power input, so it should activate it's power-ON-reset quite easily. Plus giving the possibilities the microcontroller allows, it should not be that difficult to implement one extra time out for the mains glitches, so it may distinguish between just a short power glitch (there it reignites the lamp without preheat) and a real switch off, when the filaments have enough time to cool down, so need the preheat again. So I don't believe this would be the case of the ballasts you observe.
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