## 1 Introduction  
In the past ten years, the development of modern industry has put forward the requirements of higher speed and higher power for machines. However, high-speed machines equipped with mechanical bearings have friction and wear, which limits further improvements in speed and power [1; 2]. Although air-bearing and liquid-bearing can greatly reduce bearing friction, theoretically, higher power and speed can be obtained. However, they need to be equipped with a special flow supply devices, which leads to the large volume and complex structure of the high-speed motor system and the inevitable problems of air and liquid leakage. Thanks to magnetic bearings (MBs), in which electromagnetic force is generated to suspend the rotor. Therefore, compared with mechanical bearing, air-bearing, and liquid-bearing, MBs have the advantages of no friction and wear, no lubrication and sealing, long life and maintenance free [3; 4; 5]. The high-speed rotor supported by MBs instead of other bearings can easily obtain higher speed and higher power. In addition, compared with bearings motors such as bearingless permanent magnet motors [6] and bearingless flux-switching permanent motors [7], the combined structure has obvious advantages in terms of simple control and small coupling [8]. They have received extensive attention in the past few decades throughout the world [9; 10; 11]. Scholars at home and abroad have developed many kinds of MBs. According to the mechanism of the suspension force producing, MBs can be divided into passive magnetic bearing (PMB) [12], active magnetic bearing (AMB) [13], and hybrid magnetic bearing (HMB) [14]. In particular, the HMB adopts high-performance permanent magnets to balance the rotor. If the rotor deviates from the balance position, the suspension windings need to be energized to pull the rotor back. Thus, the HMB has a large suspension force density, small volume, and low power consumption. Therefore, HMBs and their applications have become an international hotspot. In [15], an HMB was used in a ring-type flywheel and provided several advantages, such as low energy consumption, good heat dissipation, and linear magnetic force. Reference [16] proposed an AC radial-axial HMB and applied it in a wind energy generation system with running stability and excellent performance. In addition, according to the realized functions, the HMBs are also divided into an axial single degree of freedom (DOF) HMB [17], radial 2-DOF HMB [18], and 3-DOF HMB [19]. Especially, the radial 2-DOF HMB can generate the suspension forces to suspend the rotor in radial 2-DOF, which is often adopted as part of the magnetic suspension motor. At present, radial 2-DOF HMBs mainly include radial three-pole AC HMB, six-pole AC HMB, and four-pole DC HMB [20]. Compared with DC for four-pole HMB and AC three-pole HMB, the AC six-pole HMB not only has the advantages of a symmetrical structure but also can be driven by the three-phase inverter with small volume, low cost and low power consumption [21; 22; 23; 24; 25]. Furthermore, according to the bias magnetic circuit, radial HMBs can be divided into homopolar HMBs and heteropolar HMBs [26; 27]. In [28], a novel homopolar HMB was put forward to reduce the magnetic field coupling, and the magnetic fields of X and Y channels decoupled well from each other. However, the homopolar HMB has a long axial length because the permanent magnets are usually located in the axial space, which limits the further increase of critical speed. The permanent magnets of the heteropolar HMB are usually installed in the radial space, which has the advantages of short axial length, high critical speed, and high suspension force density [29; 30]. Reference [31] proposed one novel heteropolar radial HMB with low rotor core loss. The rotor core loss is just 22.53 W, which is only 41.6% of the conventional one. However, the permanent magnets occupy the stator space, and the area of the magnetic pole decreases, further reducing the levitation force.