Complex vector model of the squirrel-cage induction machine including instantaneous rotor bar currents

In this paper, a new detailed mathematical derivation of the squirrel-cage induction machine d-q model is introduced. The model is based on coupled magnetic circuit theory and complex space-vector notation and takes into account the actual nonsinusoidal rotor bar distribution. It is shown for the first time that, given the structural symmetry of the induction machine, both stator and rotor circuits can be modeled by the simple set of only four coupled differential equations, i.e., the d-q model. More importantly, the number of equations does not depend on the number of rotor bars, and the model is valid even if the number of bars per pole is not an integer number. This enormous simplification is achieved without loss of generality nor loss of any information contained in the full set of equations, and it is valid for any operating condition. The actual n rotor bars and end-ring currents are fully included in the model, and they are obtained directly by using a simple vector transformation. In addition, the three-phase rotor equivalent parameters are obtained. Second-order effects, such as skin effect in the rotor bars, can be taken into account by simply modifying the bar and end-ring resistance values. An equivalent circuit based on the model is also derived.