Transformation Groups of the Doubly-Fed Induction Machine

Abstract

Three-phase, doubly-fed induction (DFI) machines are key constituents in energy conversion processes. An ideal DFI machine is modeled by inductance matrices that relate electric and magnetic quantities. This work focuses on the algebraic properties of the mutual (rotor-to-stator) inductance matrix Lsr: its kernel, range, and left zero divisors are determined. A formula for the differentiation of Lsr with respect to the rotor angle θr is obtained. Under suitable hypotheses Lsr and its derivative are shown to admit an exponential representation. A recurrent formula for the powers of the corresponding infinitesimal generator A0 is provided. Historically, magnetic decoupling and other requirements led to the Blondel-Park transformation which, by mapping electric quantities to a suitable reference frame, simplifies the DGI machine equations. Herewith the transformation in exponential form is axiomatically derived and the infinitesimal generator is related to A0. Accordingly, a formula for the product of matrices is derived which simplifies the proof of the Electric Torque Theorem. The latter is framed in a Legendre transform context. Finally, a simple, “realistic” machine model is outlined, where the three-fold rotor symmetry is broken: a few properties of the resulting mutual inductance matrix are derived.