But what if there is a small DC bias on, say, the mains? Shouldn't that, after a while, run the core into saturation as well? Ruining the efficiency? As far as I can tell transformers run on mains directly without any coupling caps or anything.
The mains is unlikely to have a significant DC bias, because your transformer is run through a coupling
transformer - i.e., the one outside your house (or down the street, depending on where you live).
Also, you seem to have confused a bit of the function of a transformer with the
memristor. The transformer doesn't store more and more flux as charge moves, and
keep it when the current is gone. The flux through a transformer winding is
phi =
Li - directly proportional to the instantaneous current. The flux is higher during the half of the wave that is biased up, but still drops to zero* at the zero points, it doesn't just build and build.
*
Real transformer cores have hysteresis due to the ferromagnetic core, but the basic idea still stands. The hysteresis will cause a
bit of this type of behavior, but unless the DC bias is large, it will be insignificant - the core will not become heavily magnetized.