Harmonic Plane Decomposition: An Extension of the Vector-Space Decomposition - Part II

Abstract

In this paper the harmonic plane decomposition from part I is completed. Multiphase electrical machines with independently controlled stator coils benefit from a fixed amount of current controllers and unified models. In part I, the foundation for such models was laid. This paper shows that the choice of the base case has to fulfill certain criteria to properly represent vector-space quantities. More specifically, phase-pole configurations with an even number of pole pairs need adapted base cases with reduced rank as compared to the odd base case. Continuous modeling and control requires retention of the dimensionality of Clarke matrices. A different number of stator and rotor slots can lead to aliasing in the transformation, when keeping the dimension consistent. Therefore, a specific Clarke transformation for rotor quantities based on the developed theory is elaborated. Finally, it is demonstrated how the harmonic plane decomposition can be used to model each harmonic plane of an arbitrary phase-pole configuration with conventional machine equivalent circuits, e.g. the T-equivalent or inverse-Γ models.

Pole Transition Under Open Phase Fault Conditions in a Variable Pole Machine

“…The current space-vector īdq0,h=3 in harmonic plane h = 3 remains zero until the magnetic flux ψ R,h=3 has fully faded. Thus, harmonic planes h = [5,17] contain pulsating current space-vectors for minimizing the stator copper losses in the faulty case. Fig.…”

“…Fig. 7 shows the time domain currents īdq0,h>3 in harmonic planes h = [5,17]. In the first plot, it can be seen that before time instance 1, a pure sinusoidal current correlated to the frequency of harmonic plane h = p (1) = 3 exists due to the PFC.…”

“…For the modelling of VPPs, [3] further develops the previously in [4], [5] introduced the harmonic plane decomposition (HPD), an extension of the vector-space decomposition (VSD) which is independent of the number of electrical phases, but instead is formed based on the number of independent currents. This allows to have one unified model for any phasepole configuration (PPC) of the machine, enabling a fully controlled pole transition in a VPP.…”

Validation of FEM-Based Parameter Estimation for Variable Phase-Pole Induction Machines

“…In HPD, the lumped parameters computed in FEA are transformed into parameters of an equivalent circuit model based on the physical winding arrangement of the VPPIMs and following a similar transformation sequence to the VSD technique. In [13] and [20], the HPD transformation technique used for the different odd and even base-case scenarios of the pole-pair configurations is elaborated. 6) Parameter transformation in fundamental (123) reference frame: Compared to the conventional αβ and dq transform theory for three-phase machines, the VSD requires an additional intermediate reference frame, which is indicated as the 123 frame or fundamental reference frame [21].…”

“…7) Parameter transformation in αβ reference frame: Here a generalized Clarke transform that considers m-phase systems and a one-pole symmetry is used. The 123 fundamental reference frame parameters are transformed into the stationary αβ reference frame following the indications of [13], [20]. Additionally, the number of turns N s is used as a coupling factor between the stator and the rotor sides.…”