Any tips for reducing transformer Hum?

You have to distinguish between:-

1. Transformer 'rattle' - arises with loose laminations in E-core transformers
2. 'Growl' arises in toroids when DC is present on the mains
3. Hum - arises when the transformer (E-core or Toroid) radiated field intersects wiring loops internally or externally in an audio system

All three require different solutions. You best make a recording of your problems using your mobile and hen post it up - it should be possible to determine which it is.

🙂
 
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A proven solution consists of a rectifierbridge, with a bi-polar capacitor across it. It will stop the slight DC that can exist in power lines, that will tend to saturate transformers.
So every household is different.
These DC-buffers can also be bought ready dor. Use..
 
Sagen: like that but more often with a larger bipolar, say 1000 µF, that is two polarized 2.200F in series. It is to allow the AC a low resistance to overcome the 1.2V threshold. The 1µF silver cap you have would look better in its coupling capacity 🙂giggle🙂.
 
Coincidentally I had a technical problem last night partly caused by transformers. It was a very large US built device with power transformers for the electronics. These were totally encapsulated in a closed steel enclosure that offered no ventilation and functioned as shorted winding as well (unbelievable but true). When the device had a technical issue it shut itself down by software. Switching off power and then switching it one caused the primary fuses to blow immediately as the transformers were very low ohmic because of the still very high core/winding temperature. So it ended up with 2 issues 🙂