If defects of one phase transformers can be improved by 3 phase transformer,then why not a 4 phase transformer used
Off the top of my head, we use 3-phase AC because you can make electric motors that go efficiently in one direction with no commutators or tricks with phasing. That would work with any number of phases greater than 2, but at increased complexity in wiring and rotor/stator construction, so there's no point.
What I see in the other answers look basically correct. There is a video on the General Electric website where a researcher tried transmissions of many types; all the way up to twelve phases. As already said; the advantage is quickly lost above three. Try the Edison Tech Center, Lionel Barthold, and also Steinmetz. For Reference.
If you use more than three the phase angle between phases will be different. Between phase 1 and 3 the voltage will be double where as between phase 1 and 2 cosine 45 of the phase voltage. So it will be very confusing to correctly connect phases.
Redudancy is just one small advantage of using three phase. The main advantage is that three is the minimum number of phases to generate a balanced magnetic field in transformers and a smooth rotating magnetic field in motors. The math for all these works out really nicely. I suppose the math could be worked out for four phases. But the above reasons mentioned, extra wires and complications in transformers and motors makes the four phase approach less attractive from an economic standpoint.
