## 1 Introduction  
Permanent magnet motors can mainly be divided into radial motors and axial motors according to the magnetic circuit direction. In recent years, with the improvement of rare earth permanent magnet material performance and industrial manufacturing level, axial flux permanent magnet motors (APMSM) have once again received widespread attention and attention in fields such as wind power generation, electric vehicles, aerospace, industrial robots, and flywheel energy storage due to their advantages of short axial size, high efficiency, and power density [1; 2; 3; 4; 5; 6; 7; 8]. Axial permanent magnet motors are mostly flat disks with a three-dimensional distribution of the air gap magnetic field. The topology of the stator and rotor of this type of motor is relatively rich, which provides greater space for motor design. According to its topological structure [6], it can be divided into single stator single rotor, double rotor intermediate stator, double stator intermediate rotor, and multi- stator multi-rotor structures. At the same time, according to the different magnetic circuit structures, they can also be divided into iron core structures and non-iron core structures. In disc motors, due to their circular ring structure and the space limitation between the inner diameter of the ring and the rotor shaft, distributed winding heat dissipation and processing will face challenges. Therefore, fractional slot concentrated winding is widely used in disc motors [9]. The calculation of electromagnetic fields inside a motor is the foundation of motor design, and its importance is self-evident. Since the establishment of [PERSON]'s equationsin 1865, after nearly a century of development, electromagnetic field calculation technology has rapidly developed and can now be roughly divided into two categories: numerical simulation and analytical calculation:  
(1) The finite element method is the most accurate method for calculating the electromagnetic field inside a motor. In recent years, commercial finite element software has developed rapidly, and the calculation speed has been greatly improved. However, due to the unique magnetic circuit structure of the disc-type permanent magnet motor, it has significant three-dimensional characteristics, and using three-dimensional finite element calculation would be time-consuming. Therefore, although the finite element method is a good calculation and verification tool for disc permanent magnet motors, it is not an initial fast design method suitable for disc motors [10]. (2) Analytical calculation methods include the magnetic network method and two-dimensional analytical calculation method. The lumped parameter magnetic network method based on the traditional magnetic circuit method is widely adopted by design workers [11]. The magnetic network method can effectively consider the end leakage and saturation effects, but its calculation accuracy is closely related to the number of nodes, and it is also difficult to partition the air gap reluctance in the dynamic magnetic field model. The theoretical basis of the motor electromagnetic analysis is Fourier series expansion, so it is also known as the Fourier series method. For example, the conformal transformation method and the Schwarzschild transform use complex magnetic permeability equations [12; 13] to calculate the tangential magnetic density distribution of the air gap. These magnetic field analysis methods first calculate the air gap magnetic field without slots and then use the air gap relative permeability equation to consider the slot effect, so there is a certain error in calculating the slot torque. Subdomain analysis technology is a development based on the Fourier series method, and its main idea is to divide the two-dimensional plane of the motor or electromagnetic device into finite subregions and combine relevant magnetic field boundary conditions in each subregion. By solving partial differential equations, an analytical solution of vector magnetic potential in Fourier series form is obtained.