So your 220V - 127V autotransformer I understand to effectively have 3 connections, which I'll call 0, 1, and sqrt(3) (which equals 220 / 127). I.e. if it has 127V and 220V two prong receptacles, then one prong on each side would just be connected straight through and I'm calling that 0; the other 127V prong is 1, and the other 220V prong is sqrt(3).
You can get ~277V between two wires with just one such autotransformer, but it requires exposing the autotransformer to 660V. Your solution I believe is the simplest way to get ~277V without exposing any autotransformer to more than 220V. I would explain it like this:
277V / sqrt(3) = 160V, so you'd like to get 277Y/160V from your 380Y/277V. I.e. it's enough to solve the single phase problem of getting 160V starting with 220V. That you can do in two stages: for Stage 1 connect neutral to sqrt(3), and 220V to 0; that puts the 1 terminal at 220V * (sqrt(3)-1)/sqrt(3)= 93V to neutral. Now for Stage 2 put the neutral to 0 and 93V to 1; that puts the sqrt(3) terminal at 161V. Doesn't seem to me that any capacitor is required.
As for VA, if each autotransformer is rated for 750VA, that means 750 VA / 220V = 3.4A may run through the coil between 1 and sqrt(3); and 750VA / 127V = 5.9A (= 3.4A * sqrt(3)) may run through the coil between 0 and 1.
Then if you put 1A across your 277V L1'-L2', you'll get 1A flowing L1'-N and N-L2'. That means 1A flowing through the Stage 2 coil 1 to sqrt(3), and sqrt(3) A flowing through the coil 0 to 1, as well as on the 93V conductor. Then Stage 1 sees sqrt(3) A * 93V = 161VA, and the current on the 220V supply is 161VA / 220V = (sqrt(3) - 1) A. That means the Stage1 transformer sees (sqrt(3) -1) A on its coil from 0 to 1, and 1A on its coil 1 to sqrt(3).
Comparing those coil currents with the maximums for the autotransformer rating in the previous paragraph, we see that you can draw at most 3.4A at 277V, or 940 VA.
Cheers, Wayne
