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Next I'm waiting for one of you guys to go through the math of WHY when you have 208 VAC when you have each leg starting with 120VAC (and a 120 degree phase shift).

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Well, Steve...Lets turn that around and let you be the teacher.

Rotating Vectors are not my bag.....

Everyone knows that the AC voltages we usually speak of are RMS average, right? That is, the voltage is varying continuously as a sine wave so we describe it as the average of the absolute value (a straight average would be zero since it's negative half the time). For a sine wave the relationship between the RMS average and the peak to peak voltage is the square root of 2 divided by 2 which is about .707. So your 120V USA household voltage is 120/.707 or about 170 Vpp.

With single phase, the two legs from opposite ends of a transformer can be thought of as being 180° apart. The yellow line represents the difference between the two legs and would be the voltage you get connecting across them. I'm too lazy to show a scale but if the red and the blue are 170Vpp then the yellow is about 339Vpp, (times .707 = 240Vrms).

Here is three phase. Red and blue represent any pair of phases, and are 120° apart. Once again yellow is the difference between the two legs--you'll notice it is crossing the zero line when red and blue intersect. The lesser phase difference means this voltage isn't going to have as high a peak--only about 294Vpp. Times .707 gives you about 208Vrms.

Southern Japan is better off. They have the same strange voltages but at least they are on a 60 Hz system. (less damaging heat)

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Where A = the amplitude, Theda is the degree (in radians) of interest, and phi is the phase shift. If there is no phase shift, phi goes to 0 and it is a simple equation.

When I have the time, I'll go through the whole process.

Steve