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Forced plasma segregation
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Here's what happens where you drive a high-pressure sodium lamp with a (full-wave) rectified current. The mobile positive sodium ions are pulled towards the cathode (top), where they recombine into neutral sodium atoms that diffuse back. However, in HPS lamps the drift velocity of sodium ions is much greater than the natural diffusion of neutral sodium vapor, so, if you wait long enough you can see that all the sodium has been pulled on the cathode side of the arc tube, causing a visible segregation in the discharge. This results in an orange sodium-rich part and a blue sodium-depleted part. The latter discharge section burns primarily in mercury vapor.
Interestingly, because of differences in ionization and radiative properties between the metal vapors, the mercury discharge runs at a much higher temperature (~6000 K) than the sodium-dominated discharge (~4000 K). This large difference in temperature (~2000 K) also results in a thermophoretic barrier at the mercury-sodium discharge interface that impedes on the natural diffusion of neutral sodium atoms into the mercury-dominated section of the burner, thus further confining the sodium vapor near the cathode. These phenomena are why standard HPS and MH lamps are (almost) never run with a DC current.
The lamp featured here is a Penning-start sodium retrofit lamp designed to run on 125 W mercury lamp ballasts. The burner is provided with a capacitive antenna designed to enhance the electric field between the electrodes, which enables a reliable discharge ignition at 220 V mains voltage with the lamp's neon-argon fill. I chose this particular lamp for this experiment as it is easy to start, thus simplifying the design of the rectified-current circuit which consisted only of a series choke ballast and a full-bridge rectifier. The latter was placed between the ballast and the lamp to ensure that the series choke would still work properly (an AC current is required for that).
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Update: Just tried this with a 1kw SON on 400W ballast and it's worked very well! The temperature difference across the lamp is also noticeable. I first detected that by the smell of it running hotter than usual.