While you're at it, check one more thing! If the problem turns out to be
welded contacts on the thermostat switch, check that the MOV across the
secondary of the transformer is still good. I had one that repeatedly
welded the contacts till I replaced the MOV and has worked for 5 years
now without a problem.
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Not having such a Weller soldering iron myself ,nevertheless I am interested
in the MOV (firing) voltage against the normal operating voltage . Also , is
it DC or AC ?

Frank GM0CSZ / KN6WH