Model of a five-phase induction machine allowing for harmonics in the air-gap field part II : transformation of co-ordinates and d-q models

The first part of the paper presented and discussed the calculation of the main parameters of a five-phase induction machine. In this second part, co-ordinate transformations are introduced and applied to the general equations, resulting in a simpler set which has proved more appropriate for machine simulation and control. The transformations contain complex values and the new set of equations is described in terms of generalized symmetrical components. It is shown that each symmetrical component relates to a set of air-gap field harmonics and produces a corresponding torque. Mathematical expressions relating the harmonic order and the symmetrical components are derived and discussed. It is also shown that for each harmonic wave considered in the model, there exists one independent equivalent circuit and one set of independent voltage equations. From the d-q model obtained, it is possible to evaluate the influence of each space harmonic on the machine behavior and establish the base for the vector control of high-phase machines. Finally, the transformed equations allow the introduction of an independent d-q reference system for each harmonic considered.