## 1 Introduction  
Planar motion is required for various applications in high-precision industry, for example, pick and place machines, semiconductor lithography, and transport devices [1; 2]. The planar movement can be realized through super-accurate 2-D positioning stages. Traditionally, these stages are often constructed by stacking several one-degree-of-freedom linear drives supported by mechanical bearings [3]. Such constructions inherently have a high moving mass and a limited stiffness, restricting the dynamic performance. In recent years, the 2-D positioning stage has been driven directly by the planar motor [4; 5; 6], which makes it an integrated structure featuring many advantages, such as direct driving, fast response, high sensitivity, good dynamic nature, and simple structure. Therefore, the traditional method of stacking several one-degree-of-freedom linear drives has been replaced by the planar motor. Among the planar motors, the permanent magnet synchronous planar motor has many advantages, such as simple structure, high power conversion ratio, low cost, and excellent control performance. Thus, it has been widely used in accurate and super-accurate manufacturing equipment,like photolithography machines [7; 8; 9]. Mechanical bearings [10], air bearings [11], and magnetic bearings [12; 13] are the three major bearings used in the planar motor. Compared with the other two methods, the adoption of magnetic bearings in the planar motor system makes the whole system enjoy fast response, high accuracy, and simple structure, etc. At the same time, the new-generation photolithography machine will adopt ultraviolet rays to achieve higher accuracy, which requires the machine to work in a vacuum environment to avoid the influence of the air [14; 15]. Therefore, the maglev permanent magnet synchronous planar motor has caught wide attention of researchers in relevant research fields [16; 17; 18; 19]. In the semiconductor industry, positioning accuracy, and acceleration are critical parameters [20]. To achieve the goal of high acceleration, high thrust density is a very important factor to be taken into account in the design process of the planar motor. At the same time, the winding structure should be simple for easy manufacturing. Besides, fewer drive units and better control performance are needed to be considered for a better control system. Therefore, a maglev ironless planar motor adopting the concentric winding structure and moving coils is proposed in this paper. This motor has many advantages such as high winding utilization rate, good dynamic performance, and no cogging force [21], thus the motor of this structure is very suitable for application in high-accuracy two-dimensional movement. Meanwhile, the effective winding length of each phase is increased in the concentric winding structure. Thus, on the condition that the whole system is controllable, fewer phases are needed for the same mover area, which makes fewer drive units needed accordingly. It is another big advantage of the proposed concentric winding structure. Concentric winding adopts the square structure instead of the traditional rectangle one [22]. The square structure leads to more complex electromagnetic model, the accuracy of which is essential for system control accuracy. To realize high-speed real-time control, a real-time computation model is needed, which reflects the relation between the current in each winding and the relevant electromagnetic force or torque. Therefore, an accurate real-time computation model contributes to the high-speed high-accuracy movement control. [PERSON] [23] in Eindhoven university points out in its research report that three different analytic methods can be used to calculate the electromagnetic force, namely virtual work method, Maxwell stress method, and the Lorentz force equation method. The virtual work method is normally used in non-carrier fluid electromagnetic force calculation, and the calculation process is relatively complex.