The article is devoted to developing the mathematical model of an induction machine in polar coordinates, a task that has not been studied to a sufficient detail for being used in engineering applications. The above-mentioned task was solved using the methods of automated control theory, electromechanics, and electrical engineering. The software implementations of the proposed mathematical models and numerical investigations of the processes simulated by these models were carried out in the MATLAB computation environment (the SIMULINK application software package). Three versions of induction machine mathematical models written in polar coordinates and taking into account the rotor current displacement effect are presented. The specific features of implementing such models by means of software are determined. It is found that these models satisfactorily describe the processes in an induction machine both in the working part of its mechanical characteristics and at high rotor slip values. It is shown that the mathematical models written in polar coordinates replicate the processes in an induction machine with the same accuracy as the widely used similar models written in the Cartesian system of coordinates.