A generalized transformation methodology for polyphase electric machines and networks

This paper introduces a methodology by which the dq electromagnetic model of an AC machine or network can be extended to a system of any number of phases. The methodology proposed here is based on pole-symmetry (symmetry with respect to π rather than 2π) and separates the electrical configuration from the magnetic configuration of the machine, leading to the concept of the fundamental winding configuration. It is shown that any possible winding configuration can be accounted for using the generic fundamental winding configuration together with a winding configuration matrix. Symmetry with respect to π rather than 2π suggests a change to the complex operator that has served as the basis of Fortescue's method of symmetrical components for nearly 100 years. Using the new complex operator, the authors derive a modified symmetrical component transformation for the fundamental winding configuration. It is shown that when the configuration matrix is applied, the modified transformation yields the same results as the original, but lends itself to a more systematic generalization methodology. Then, following the historical progression, the generalized Clarke and Park transformations are derived from the modified symmetrical component transformation. These transformations presented here enable the systematic generalization of the dq electromagnetic machine model, including the effects of saliency. The generalized dq model, along with laboratory results of a nine-phase permanent magnet synchronous machine, will be presented in a series of follow-up papers.