Analytical investigation on the power factor of a flux-modulated permanent-magnet synchronous machine

In recent years, flux modulation is increasingly receiving attention since it is capable of adjusting the pole-pair number and rotational speed of the magnetic field. This principle has already been applied to magnetic gear (MG) to realize the low-speed high-torque operation [1]. Flux-modulated permanent magnet synchronous machine (FM-PMSM) is an integration of a coaxial MG with a conventional outer rotor PM machine. Compared with the coaxial MG, a stator with armature windings is employed in the FM-PMSM to replace the inner rotor of the coaxial MG [2-3], as shown in Fig. 1 (a). A modulation ring with a number of ferromagnetic pole-pieces is sandwiched between the outer rotor and stator, and two air-gaps are thereby formed to separate the above three components from each other. The modulation ring could act as a mediator to match the pole-pairs differential between the two magnetic fields generated by the PMs and armature windings, respectively. The advantage of the FM-PMSM is that it could provide high torque at low speed directly without employing mechanical gearbox, mechanical issues, such as friction loss and mechanical fatigue, are thus avoided and high reliability could be achieved. However, it is found that the power factor of the FM-PMSM is relatively low [4], and few details in the published literatures are provided to explain the intrinsic causes of this problem. Therefore, this paper will develop an analytical model to predict the power factor of the FM-PMSM and investigate the essence of the low power factor. Because of the existence of the flux modulation, many high frequency harmonics are introduced into the magnetic field of the FM-PMSM. It is important to have a good knowledge of the magnetic field distribution in the FM-PMSM in order to accurately predict its power factor. This paper gives an alternative to the commercial finite element method (FEM) to quickly obtain the electromagnetic solution. This approach could provide many physical insights into the electromagnetic solution. In this paper, partial differential equations are used to describe the magnet field behavior in terms of magnetic vector potentials. The whole calculation domain is partitioned into six regions, viz. the stator slot and slot opening (Region I_i and Region II_i), the inner and outer air-gaps (Region III and Region V), the slots between modulation pole-pieces (Region IV_j), and the PMs (Region VI), as shown in Fig. 1 (b). In Region II_i, III, IV_j and V, the magnetic vector potentials are governed by Laplace's equations. And in Region I and Region VI, magnetic vector potentials are of Poisson's equations due to the existence of current and PMs, respectively. By applying the constraints on the interfaces between these regions, the solution of the magnetic field could be derived. Fig. 2 (a) compares the analytically and numerically calculated distribution of the radial flux density in the inner air-gap due to the PMs on the outer rotor. As can be seen, the analytical results have a good agreement with the FEM calculation, which indicates the developed analytical model could be employed as a powerful tool for predicting and analyzing the power factor of the FM-PMSM. With the knowledge of the solution of the magnetic field, the flux linkage of each phase winding could be obtained through the surface integral of the magnetic vector potential in Region I_i. Then, the self and mutual inductances of each winding could be deducted, which are 4.06 mH and 2.02 mH, respectively. Similarly, the analytically and numerically calculated inductances have a good agreement. Substituting the inductances, currents and resistances into the voltage equation, the voltage of each phase winding could be obtained as well. Fig. 2 (b) presents the waveforms of the voltage and current of phase-A winding versus the time at the rated power. Finally, the active and reactive power could be calculated by expressing the flux linkage in terms of impedances, and the predicted power factor of the studied FM-PMSM is 0.6, which is quite low compared with conventional PM machines. And the low power factor will inevitably lead to the increase of the capacity, volume and loss of the inverter.