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Bobhook149:
First, someone mentioned low voltage to a transformer might damage the transformer. That is total nonsense. The only way it could be a problem is with some peculiar load that required an increasing current as the load voltage was lowered was the load. A motor on the output of a transformer might fall in this category, but that is not the case here and there would be other factors to consider.
A tungsten filament lamp as a load falls some where a constant current load and a constant resistance load. Following are current vs voltage values for a 100 W 120 V bulb in volts and amps.
20 0.297
40 0.440
60 0.556
80 0.664
100 0.755
120 0.833
140 0.904
At 120 V the resistance is 144 ohms. If a constant resistance of 144 ohms was the load, then at 20 V the current would be 0.138 A instead of 0.297 .
The voltage drop line to line for the large 240 V loads that you mentioned is only 100*6/243 = 2.5%. That is reasonable and maybe quite good.
A well balanced lighting load on your center tapped transformer and a poor neutral from the transformer should have caused come lights to brighten, and others to dim when a large 120 load was applied.
From your description of the termination of neutrals in the main panel it is likely that a 120 V circuit with the light that was burning out shared a neutral with a 120 load from the opposite phase, and the opposite phase had a 120 V load that was large relative to the lamp or transformer, and that caused excessive voltage on the lamp or transformer.
Transformers do not care about under voltage, but not very much over-voltage can burn them out. When you get transformer input voltage above the nominal rating the magnetizing current grows rapidly.
Hopefully getting rid of the neutral terminations where the screw was on the insulation has solved the problem.
Also, note that a standard tungsten filament lamp's life increases as the applied voltage is reduced.
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