The sketch at right represents a squirrel cage motor . Any two bars and the arcs that join them form a coil, as shown by the bold segments in the sketch. The problem with the motors I have sketched is that capacitors of high value and high voltage rating are expensive. One solution is the 'shaded pole' motor, but its rotating field has some directions where the torque is small, and it has a tendency to run backwards under some conditions. The neatest way to avoid this is to use multiple phase motors .
Single phase is used in domestic applications for low power applications but it has some drawbacks.
The second is that it makes it awkward to produce rotating magnetic fields. Industrial applications use three phase extensively, and the three phase induction motor is a standard workhorse for high power applications. Thus three stators give a smoothly rotating field. If one puts a permanent magnet in such a set of stators, it would be a synchronous three phase motor. If puts in a coil, or simply a conductor that will carry eddy currents, one has an induction motor capable of high efficiency, high power and high torques over a range of rotation rates.