# A Novel Design of Permanent Magnets for the Air Gap Magnetic Field of Hollow-Cup Motor
## 1 Introduction  
As a spacecraft attitude control system, the inertial actuator mainly realises the attitude stabilisation and attitude manoeuvres of the spacecraft. The main principle is to change the angular momentum of the rotor of the high-speed motor and output the control torque. Inertial actuators are mainly divided into magnetically suspended flywheels and magnetically suspended control momentum gyroscopes [1]. As an important part of the space inertial actuator, the drive motor mainly plays the role of controlling or adjusting the rotor speed and then outputting the control torque required to control, adjust, and stabilise the aircraft attitude. For the magnetically suspended flywheel, the attitude control by acceleration and deceleration requires the motor to have higher-speed stabilisation accuracy, smoothness, and lower power consumption. In order to eliminate the iron loss and cogging torque pulsation caused by the presence of the rotor iron core of the conventional DC brushless motor, and to improve the speed accuracy and speed tracking accuracy of the inertia actuator, most inertia actuators use HCM. The main feature of the HCM structure is the use of a rotor without iron core, which overcomes all the problems caused by the iron core, and which greatly improves the main performance of the motor, thus having a wide application field [2]. Due to its superior performance, the HCM has been a hot topic of research in recent years and is used in various fields such as the military, aerospace, automation, industrial robotics, and medical instrument industries [3; 4; 5; 6]. In particular, for a magnetically suspended flywheel, it requires the motor to have less torque ripple, thereby improving control accuracy and stability. However, the HCM has a difference between the AMF and the ITW, which causes the back electromotive force (BEMF) waveform to be inconsistent with the ITW, resulting in increased torque ripple and affecting output torque accuracy [7]. At the same time, due to the existence of inter-pole leakage, the pole arc widthof the static permanent magnet field is often smaller than the actual pole arc width, causing the stator current to interact with the rotor field, which in turn generates torque fluctuations and causes motor vibration. It is, therefore, necessary to design the structure of the HCM so that the flat section width of the AMF is increased and brought closer to the ITW, further suppressing torque fluctuations and increasing the accuracy of motor torque control. At the same time, due to the extreme temperature difference in the operation of high-speed motors in spacecrafts, the PMs in the motors are susceptible to high-temperature demagnetisation, requiring higher-performance permanent magnet materials. The torque ripple suppression problem of HCMs is mainly solved by control methods [8; 9; 10] or structural design methods [11; 12; 13; 14]. [PERSON] et al. [8] proposed an automatic control method based on the measurement of BEMF parameters to adjust the control current with real-time feedback, thus improving the control accuracy and reducing torque pulsation. [PERSON] et al. [9] proposed a torque prediction analysis based on a neural network fitting analysis incorporating the effects of temperature, with a hybrid braking torque control structure for precise torque control of a small inductance brushless DC motor. In 2017, they [10] adopted the singular perturbation technique to improve the torque tracking performance. For HCMs with a large air gap, it is difficult to reduce the torque pulsation directly through precise control of the winding coil. Regarding the torque ripple caused by nonideal BEMF, from the structural design perspective, the optimisation is mainly carried out by designing the structural design of the PMs or the rotor [11; 12; 13; 14]. [PERSON] et al. [11; 12] used the ESCM of PMs to solve the analytical mathematical model of the radial magnetic density generated by the surface-mounted PMs on the stator surface for the brushless DC motor.