Mainly the HPS lamps for the US market are specifically designed to suppress (compare to the European spec lamps) the arc voltage response to the change in power delivered to the lamp. So even when the arc voltage rises over the lifetime (due to any reason) so does the power delivered by the ballast (because the current does not change that much), the further voltage rise as a result of this extra power is prevented. Most common way is to place the amalgam reservoir outside of the arctube, so there is limited heat transfer from the hot central arctube part and the amalgam reservoir has its own separate cooling surface. Because of the T^4 law of radiated power and the hear transfer from the arctube to the reservoir being just first order proportional, the reservoir temperature is stabilized by two 4-root functions (first is the arctube temperature as a response to the lamp power, then the power transfer to the reservoir is just linear to the arctube tempersture, then other 4-root is the reservoir temperature vs the transferred heat), the reservoir temperature variation so the pressure gets significantly supressed.
But the truth is, this suppression falls apart once the radiated heat from the arctube gets reflected and focused back to the reservoir. And another consequence is the lamp arc voltage tolerance being quite large, so even when the lamp seem to have the same electrical specs (like S56 vs European HPS150W), it may not work correctly on an European ballast (European ballast relies on tighter tolerance and the thermal feedback to correct it a bit).
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