| The series reactor ballast has its OCV equal to the mains voltage. And to have stable arc, the OCV needs to be at least double the arc voltage over the whole lamp life. The reason is, each time the current crosses zero, the arc effectively extinguish and has to reignite at the opposite polarity again. And for that it needs somewhat higher voltage. And that has to be available from the ballast at that moment (with the series choke that is the exact instanteonus voltage at the mains). And for that you need either high enough amplitude of that voltage, or a very favorable phase shift, so the current zero cross moment is not too far from when the voltage is at its peak.
So when this is combined with the practical values (ballast losses so phase shifts, expected fast mains fluctuations, lamp parameters tolerances...), you end up with the ballpark relation mentioned above. Higher current density arcs may go a bit higher, as there is enough free charges left after zero cross to facilitate reignition at lower voltage, the MVs have still their probe active which does lower the reignition voltage, plus higher power ballasts tend to be designed with higher efficiency, so have a bit better phase shift, higher power lamps can be more expensive and are physically larger, so allow production with tighter parameter tolerances, so some lamp types, especially MV allowed the voltage to go a bit higher (135V arc on 220V mains).
What happens in your experiment: At cold, when the Hg is condensed, the arc is essentially low voltage, so the condition to sustain it is met, so it ignites. But as it warms up, the arc voltage is rising, so when reaching certain level either the reignition gets delayed so the power reduced, preventing it from warming up further.
Normally it as well may become unstable and extinguish so start to cycle. which way is pure coincidence of lamp parameters and the exact conditions, nothing guaranteed.
So if the lamp is supposed to operate at series choke at a given mains voltage, it must be designed so it meets that condition.
Why the lamps were designed with the arc voltages as they are in the first place?
There are other restrictions for selection of the lamp arc voltage for a given power level:
When too low, the current will be high, so the electrode drop dissipation would be high, so less power remain available to generate the light, so lower the efficacy (there is about 15V drop at each electrode pair in the circuit, which consumes power but does not generate any light). So to minimize this effect, the arc voltage must be as high as possible.
Then there is other aspect: Each chemistry needs certain arc power density to maximize output. Too low current means the arc would be very thin, with strong selfabsorbtion. So with given power, the arc should be of given length. But at low power that means the arc has to have high drop over short length, so operate at high pressure. High pressure then means high temperature and that means quite a lot of the power get spent on keeping the things warm enough, again less power available to generate light.
These two effect categories go obviously against each other, so there is for each power level an optimum sweet spot for the arc voltage to reach best performance.
But the lamp needs a ballast, which has losses too. With ballasts, each winding has its losses, which are given by the winding apparent power rating (maximum voltage at any stage of operation)x(current through that winding during normal operation), the winding losses use to be in the 5% of that apparent power rating for standard and about 2..3% for low loss construction. So to minimize that, you need to use the simplest ballast possible and use lamp voltage as close to the mains as possible (so to minimize the voltage drop).
So the real commercial lamp design (when the standard for a given power at given chemistry was formed) was a compromise among all the effects above.
The MV, together with the aim to suffice with a series reactor on 230V it led to 100..130V arc (it is very close to the bare lamp optimum). Lowering the arc voltage for 120V mains and series choke led to too much losses, so on 120V was better off using some transformer ballast. Although the transformer has about double losses, the result is still better than the eventual 60V arc would have. Because the technology started in the 240V UK, all standards were optimized for series ballast at 240V, so even higher power level stayed at the 130V arc.
A bit different story was with HPS: There the development started in the US, it was clear on 120V practically all lamps would need an autotransformer, so the exact ballast OCV did not make that much effect on ballast efficiency anymore, so standards were designed around the optimum of the lamp alone. The exceptions were the low power lamps, where the 55V arc was not that far from the optimum, so it made sense to tweak these so set that for the standard so to suffice with the less lossy and way cheaper series choke.
European 230V mains (so the 70..100V arc voltage limit for the HPS) was not that far from the optimum, so lamps for European market were standardized around the series choke operation on 230V, so become incompatible with the US specs.
The MH came when HPS and MV were quite established businesses, so there was very strong marketing push to make them as direct plugin replacement for either type or at least suffice with the ballast components already in common use (in Europe it is common to specify/order/install the ballast components separately, not as a complete kit as in the US).
The US probe start was designed to operate on then common MV CWA. But later was found, it need a bit higher OCV, so US ballasts were tweaked that way (OCV lifted to 270..300V from the previous ~220V).
In Europe that was a problem, because the mains could not be raised.
But it came at the time the HPS were getting a common thing. Because HPS needed the HV pulser ignitor anyway, so it allowed to get rid of the starting probe and optimize the lamp for the 230V OCV at the same time. Because HPS was already an established standard, most MHs were designed to directly replace them. The 400W MH tolerated the higher arc voltage, so some types were designed around MV choke (lower current, higher arc voltage than 400W HPS), but with added ignitor.
And because MHs were more of luxury, so less production volume than MV or HPS, there was the desire to have the same specs worldwide, plus the European specs were already close to the lamp optimum, so the European spec standards were reused in the US.
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