I didn't ask to be honest.
The lower the voltage the higher current required for a set volume. And as the ANR works best with max volume on the headset controls.... Hence the bat will get drained even quicker and won't give the same max output.
And I also presume that below a certain voltage the electrons stop jumping in the semiconductors which stops the op-amps and inverters working so the ANR stops working.
I suspect coodem and keef know far more about the workings of these things than I do. It would be interesting to know if the output of the Internal powered ANR connection is 12V and if there is a noticable difference between them and bat powered models.