First, I hope I'm not stepping on any toes here; I respect your input and I think you could wire a house. I used to wire them for a living and I know what to connect to where, believe me. As I said, I think the whole problem we're having is a difference in definitions of two phase rather than refering to this situation as single phase with two legs of the secondary of a single phase transformer.I think I see where some of the confusion is, looking at the batteries in the drawing they are facing the same way, but compared to ground they are reversed. If the same polarity was connected to ground then there would not be any voltage out.
As you know, the power company sends our localities 3 phase voltage over three wires. These voltages are 120 degrees apart. Industrial areas get all phases stepped down through a bank of three xformers so they can run 3 phase motors. But any particular area will get only one phase. Look at the xformer that feeds your house. Above it you will see one small diameter wire.
This feeds one lead of the primary of your xformer, the other is connected to a grounded wire. At the secondary we have a center-tapped coil. It is all wound in the same direction and between its outer legs is 240 volts. The center tap is connected to earth ground.
Now, with respect to this transformer it is polarized as per your battery drawing. Notice the batteries are in series, not in parallel. Also, the xformer diagram should be marked with the same polarizations as per your battery drawing. Your phase drawing is incorrect. From top to bottom it should be + - + -. These are in series and in phase with each other and therefore 120 volts adds to 240 when you connect them to an appliance. The grounded center tap -- neutral -- is not a part of the circuit, except short circuit protection for the appliance casing.
This whole discussion came about because I objected to Sgt W's calling this two phase (180 degrees apart) which it is not. It is single phase and the correction on your drawing so that the voltages are in phase, not 180 degrees apart is needed.
But you're looking at the case inside the house where the center tap is indeed grounded and you get two legs of 120 volts each. From the transformer's point of view you're mearly making a connection between its legs, not its centertap, and therefore getting the whole 240 volts across the entire secondary winding. This means it is still single phase, not 2-phase. To get two phase the power company would have to deliver two, not one, of its feeders.
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