Something that has always bothered me about this- how do you get true three phase which is 120-120-120 degrees out of single phase which is 180-180 degrees unless you use a true motor generator? (Single phase motor coupled to a three phase generator.)
All rotary converters I have ever seen use the single phase hot legs and create a third phase from them. This works if a three phase motor loses a phase- it will generate the third. But the "input" is already two phases at 120-120 degrees.
There are three questions hidden here.
1) How do you get _any_ phase angle difference from the single phase machine?
The answer is that you have stored energy in the mechanically moving rotor, just as you would in a true motor-generator. The single phase supply attached to one of the phase bands causes the rotor to move. The moving magnetized rotor interacts with the other phase bands in turn, and since the other phase bands are mechanically displaced, the voltage on those phase bands is displaced in time.
2) How do you get _exaxtly_ 120-120-120 with balanced voltage?
Alas, you don't. You have three separate circuits which are _not_ the same. Current flows (and thus voltage drops) are different. The system will come close. If you use an ordinary three phase induction machine as your phase converter, the machine is designed with balanced coils, and _must_ produce unbalanced output. If you design the machine with unbalanced coils, then at a particular load condition it could produce balanced output.
3) If you have a three phase 120-120-120 supply, and one phase legs goes dead, what is the phase relation on the remaining two legs?
The phase to _neutral_ phase angle remains 120 degrees. However for a motor without neutral connection, losing _one_ of the phase legs means that you have lost _two_ of your three phase supply circuits. What is left is a _single_ phase circuit, with 180 degree phase relation between its two supply terminals. For _any_ load connected between _two_ supply terminals, be they two hots or a hot and a neutral, or the hot and a tap on an autotransformer, the current flowing in from one supply terminal must be balanced by the current flowing out on the other supply terminal.