How to prove that the number of poles in a squirrel-cage induction motor is the same as that of the stator?

In a squirrel-cage induction motor, because the rotor windings are assembled from conductor bars short-circuited by end rings, the number of poles is not an intrinsic characteristic of the windings, as it occurs in the case of wound rotor windings, but it is a consequence of the distribution of the currents induced in the bars. However, since the motor works, the number of stator and rotor poles must necessarily be the same, and the explanation of why this occurs is usually left to physical considerations. The paper proposes the analytical proof that in a squirrel-cage induction motor, the number of rotor poles is the same as that of the stator. The proposed demonstration is based on the determination of the air gap flux density distribution generated by any set of currents circulating in the rotor bars, without any hypothesis about their waveform. The distribution is expressed as sum of infinite space harmonics, for each of which a current space vector can be defined. First, the special case of sinusoidal stator currents is addressed, and then the general one of stator currents having any waveform. In this way, it is possible to show that the only nonzero rotor current space vector is the one corresponding to the space harmonic having a number of poles as that of the stator.