We have disassembled a transformer from a microvawe oven by cutting the core, put a secondary winding suitable for our purposes (so transformer outputs 16VAC rms) and then tig-welded the core back. Now the core is heating up while the transformer sits with no load on the secondary. By heating up i mean that core gets too hot to touch in about an hour. The primary and secondary do not heat by themselves, i.e. they are cooler than the core.
What might be causing this? Are there any voodoo to fix it?
Wait, you cut the core?
Well, congratulations, you have ruined/severely damaged it.
Transformers are made of lots of sheets of steel, with very thin insulating layers between them. This is to keep eddy-current losses from causing lots of heating, as you have discovered.
Microwave transformers are normally somewhat lossy, as they are not operated for a significant period of time. A stock microwave transformer will get noticeably warm if sitting unloaded for a while. You have just increased the losses by many times, by shorting out the laminations.
There is nothing you can do with the transformer you have. You need to get another transformer, and not cut the core to remove the secondary. You have to remove the secondary without damaging or dinging the core significantly, and then wind your new secondary in place. by threading the wire through the core.
For what it's worth, microwave transformers run pretty warm without any load. Have you compared this transformer to another, without the core damage?
I'd be interested in some measurements of no-load power draw on the hacked-up transformer vs a stock one. That would let you measure the increase in losses due to eddy currents.