Single phase motor design for two phase

Two phase motor works the same way as the single phase one. The only difference is in the voltage level applied. As with two phase motors voltage lever is higher the design must be adjusted so the capacitor voltage does not exceed its limit.

The question is why would anyone want to have two phase motor if all 3 phases are available and the far more reliable 3 phase motor can be used?

There is only one advantage of two phase (or call it single phase) motor. The direction of rotation is fixed regardless of supply. On the other hand 3 phase motors direction of rotation is affected by the order of supply phases connected to motor. This fixed direction of rotation is great feature with motors used in the refrigeration containers. These containers are being moved from ship to land and back to other ship and supply is being connected and disconnected all the time. With the motor fixed direction of rotation there is no chance of wrong connection and equipment malfunctioning because of that. I guess any other application where direction of rotation is critical and equipment is being moved from site to site could benefit from that feature.

With any electric motor the key is to have the force moving rotor consistently in the same direction.

Operation of universal motor with DC supply is very easy to understand. DC supply produces DC magnetic field in stator. Rotor winding is fed with DC supply and brushes/ commutator ensures that rotor current has direction that results in having produced force in the same direction.

In case of AC supply the dynamics are different but the essence the same. Stator magnetic field alternates with AC supply however that is in sync with the rotor current alteration resulting in produced force working again in the same direction.