Analysis and Optimization of a Moving Magnet Permanent Magnet Synchronous Planar Motor with Split Halbach Arrays

Abstract

This paper investigates an improved permanent magnet synchronous planar motor (PMSPM) featuring a moving magnet configuration to enhance thrust density and positioning accuracy. A novel split Halbach permanent magnet (PM) array is introduced, and the optimization begins with adjusting the pole size ratio α, analyzing the flux density distribution, and calculating thrust using an electromagnetic force model. Results demonstrate that the optimized Halbach array reduces thrust fluctuations and improves the uniformity of the air gap magnetic field. Multi-objective optimization using the non-dominated sorting genetic algorithm-II (NSGA-II) fine-tunes auxiliary magnet width and magnetization angles, resulting in a segmented auxiliary permanent magnet structure that achieves a 9.1% improvement in thrust density over conventional designs. Additionally, the optimized Halbach array effectively reduces thrust fluctuations and improves the uniformity of the air gap magnetic field, addressing key challenges in planar motor design. Extensive simulations and experimental validation demonstrate the superior performance of the proposed structure in terms of thrust density and positioning precision. These enhancements make the PMSPM suitable for high-precision, high-dynamic industrial applications. A detailed comparison of motor parameters and thrust performance validates the effectiveness of the proposed structure.

1. Introduction

Entering the 21st century, with the rapid development of photolithography technology, IC integration, and other fields, the semiconductor industry requires wafer positioning systems with higher acceleration and position tracking accuracy. The permanent magnet synchronous planar motor (PMSPM) is widely used in precision processing equipment because of its high dynamic response and high positioning accuracy. Planar motors are divided into moving coil-type [1,2,3] and moving magnet-type [4,5,6] PMSPM according to their different actuating parts. Unlike the moving coil-type planar motor, the actuator of the moving magnet-type PMSPM has no direct cable connection to the outside, so its positioning accuracy will not be interfered with. However, the acceleration of the actuator cannot be improved by simply increasing the amounts of permanent magnets as in the case of the moving coil-type, because this way also increases the weight of the actuator. Researchers around the world have been studying planar motors more and more intensively. The current optimization for planar motors focuses on positioning accuracy, load capacity, dynamic response, and control strategy. To improve the positioning accuracy and load capacity of the motor, the key lies in how to reduce the thrust fluctuation as well as increase the output thrust.

J.W Jansen et al. proposed a comparison of several different topologies of moving magnet PMSPM in an attempt to improve the thrust fluctuation of the motor during levitation and acceleration [7]. In reference [8], Tim Hofmann et al. of Technical University of Munich designed segmented magnetized linear Halbach arrays, which make the sinusoidal of magnetic flux components better and facilitates the compact design. Yiming Zhang et al. proposed a moving magnet PMSPM with high-temperature superconducting coils to provide driving force and torque that cannot be provided by the traditional coils [9], and Haiyue Zhu proposed to use Halbach permanent magnet (PM) array to reduce the size and mass of the magnetic levitation planar position to improve the positioning accuracy of the motor as well as the thrust density [10]. Reference [11] effectively smoothed the axial magnetic field distribution by introducing gradient-varying magnetization directions or gradient-varying magnet block sizes. Reference [12] combines finite element analysis and genetic algorithms to optimize the geometrical parameters of Halbach arrays to suppress local magnetic field fluctuations. It can be found that in order to improve the acceleration and the thrust density of the planar motor, researchers start from the direction of improving the stator and the actuating parts’ topology construction. This paper proposes a new type of moving magnet PMSPM with split magnet steel structure, in the case of the same amount of magnet. The output thrust is larger, and the thrust fluctuation is smaller. This is conducive to improving the load capacity and positioning accuracy of the planar motor.

2. Topology and Electromagnetic Principle of PMSPM

The structure of the moving magnet PMSPM optimized in this paper is shown in Figure 1, which consists of a substrate and stator coils as the primary, and Halbach PM array and yoke as the actuating parts. Each set of coils is separated into inner and outer coil units in the stator, which is a concentric combination of coils. The actuator is powered by a Halbach permanent magnet array that is two-dimensional. The array is characterized by a magnetic field that is more powerful on one side and less powerful on the other side. Unilateral enhancement occurs and has a significant impact on reducing the motor’s thrust fluctuations. It is ideal for planar motor applications and enhances its positioning accuracy [13,14,15,16,17,18,19].

2.1. Analysis of Halbach PM Arrays

In this paper, there are four auxiliary permanent magnets around each main magnet in the two-dimensional Halbach PM array. Two main permanent magnets and seven auxiliary permanent magnets are present in an overall Halbach array cell. Inward and perpendicular to the paper, the S-pole is magnetized in red, and outside and perpendicular to the paper is magnetized in green. Additionally, the auxiliary permanent magnets are magnetized parallel to the paper from the S-pole to the N-pole in Figure 2. The width of the main permanent magnets is ${\tau }_m$ and the pole pitch of the magnets is $\tau$.

By using this magnetization technique, the array of permanent magnets experiences a unilateral enhancement of the magnetic field, whereas the magnetic properties on the opposite side are weak. This array format assists in decreasing the thrust fluctuation of PMSPM with coreless concentric windings, which in turn improves their positioning accuracy.

In order to define the planar motor, three coordinate systems must be used, which are the global coordinate system and two local coordinate systems. The global coordinate system is fixed. The first local coordinate system has the center of the magnet array as the origin and is denoted by the superscript m. The second local coordinate system is at the center of the coil and is denoted by the superscript c. The analytical model of the 2D Halbach PM array mentioned in this paper is shown in Figure 3.

Based on Laplace’s equation and the corresponding boundary conditions, the components of the magnetic induction intensity in the x, y, and z directions generated by the two-dimensional Halbach PM array in the air gap can be solved for as Equation (1).

$$\begin{array}{l}{}^m\overrightarrow{B}\left({}^{m}x,{}^{m}y,{}^{m}z\right)\\ ={\displaystyle {\displaystyle \sum }_{k=1}^{\infty }}{\displaystyle {\displaystyle \sum }_{l=1}^{\infty }}{e}^{{(}^{m}z\lambda )m}{\overrightarrow{B}}_{xy}{(}^m\overrightarrow{x},k,l)\\ \left.={\displaystyle {\displaystyle \sum }_{k=1}^{\infty }}{\displaystyle {\displaystyle \sum }_{l=1}^{\infty }}{e}^{{(}^{m}z\lambda )}\left[\begin{array}{c}{B}_x\left(k,l\right)cos\left({\displaystyle \frac{k{\pi }^{m}x}{\tau }}\right)sin\left({\displaystyle \frac{l{\pi }^{m}y}{\tau }}\right)\\ B_y\left(k,l\right)sin\left({\displaystyle \frac{k{\pi }^{m}x}{\tau }}\right)cos\left({\displaystyle \frac{l{\pi }^{m}y}{\tau }}\right)\\ B_z\left(k,l\right)sin\left({\displaystyle \frac{k{\pi }^{m}x}{\tau }}\right)sin\left({\displaystyle \frac{l{\pi }^{m}y}{\tau }}\right)\end{array}\right.\right]\end{array}$$

(1)

where k and l are the harmonic numbers for the x-direction and y-direction, respectively, B_x (k, l), B_y (k, l), and B_z (k, l) are the effective amplitudes of the (k, l) harmonic of the flux density at ^mz = 0, and λ (k, l) can be calculated by Equation (2).

$$\lambda =\sqrt{{\left({\displaystyle \frac{k\pi }{\tau }}\right)}^2+{\left({\displaystyle \frac{l\pi }{\tau }}\right)}^2}$$

(2)

The Halbach magnetic homogeneous density resolution results are plotted as shown in Figure 4a–c. Through the simulation, the side length of the main permanent magnet is taken as 14 mm and the width of the secondary magnet is taken as 8 mm. In total, 22 points are taken around the center of the model in the x-direction [−2${\tau }_n$,2${\tau }_n$] range, and 88 points are taken in the y-direction [−2${\tau }_n$,2${\tau }_n$] range, which is a total of 1936 points. The 3D mapping of the air gap magnetic field of the 2D Halbach PM array is plotted as Figure 4d–f.

The comparison reveals that the air gap magnetic field strength obtained by simulation is basically consistent with the air gap strength obtained analytically. The air gap magnetic field on the enhanced side of the two-dimensional Halbach PM array has good sinusoidal nature.

2.2. Thrust Analysis of PMSPM

The actuating part of the moving magnet PMSPM consists of an array of Halbach PM, and the stator part consists of two concentric coils.

The thrust of the planar motor is due to the interaction of the traveling wave magnetic field generated by the stator current with the magnetic field of the moving permanent magnets. In the previous section, the flux density of the moving son permanent magnets has been represented and the two-phase windings are shown in Figure 5, and the expression for the currents passed by them can be described as Equation (3).

$$\left\{\begin{array}{l}{i}_a=I_{xm}\sin \left({\omega }_{x}t\right)+I_{ym}\sin \left({\omega }_{y}t\right)\\ i_b=I_{xm}\sin \left({\omega }_{x}t\right)+I_{ym}\sin \left({\omega }_{y}t-\pi /2\right)\\ i_c=I_{xm}\sin \left({\omega }_{x}t-\pi /2\right)+I_{ym}\sin \left({\omega }_{y}t\right)\\ i_d=I_{xm}\sin \left({\omega }_{x}t-\pi /2\right)+I_{ym}\sin \left({\omega }_{y}t-\pi /2\right)\end{array}\right.$$

(3)

The synthetic magnetic field formed in space by the stator current can be expressed as Equation (4).

$$B_s\left(x,y,t\right)=B_{s}cos\left(k_{s}x-\omega t+{\theta }_s\right)$$

(4)

where $B_s$ is the amplitude of the stator magnetic field and $k_s$ is the number of waves of the traveling wave magnetic field generated by the stator current, which is related to the pole pitch of the windings. ω is the angular frequency of the current and ${\theta }_s$ is the initial phase of the traveling wave magnetic field.

According to the electromagnetic force formula, the electromagnetic force originates from the Laplace density between the magnetic field gradient and the magnetic field. $\mathit{J}$ is the current density vector.

$$\mathit{F}={{\displaystyle \int }}_V\left(\mathit{J}\times \mathit{B}\right)dV$$

(5)

For a permanent magnet synchronous motor, the thrust force is mainly Lorentzian and can be expressed by considering the thrust force in the x-direction. $J_y$ is the y-direction component of the current density vector.

$$F_x={{\displaystyle \int }}_V{J}_y{B}_{z}dV$$

(6)

3. Original Topology of the PMSPM

3.1. Topology and Parameters

Figure 1 shows the model of the motor in this paper, which consists of four parts: substrate, stator, magnet, and yoke. The parameters of the motor are shown in

Table 1. Original PMSPM size parameter.

Symbol	Quantity	Value
hm	magnetic thickness	10 mm
τm	main magnetic width	14 mm
τ	magnetic polar distance	22 mm
τn	magnetic polar distance rotates 45°	$11\sqrt{2}$ mm
hc	coil height	7.1 mm
wc	coil thickness	10.15 mm
lin	inner circle distance	3 τn
lout	outer circle distance	5 τn
gap	air gap height	1 mm

.

The stator of this planar motor has concentric-type coils. Each set of coils has two concentric-type coils inside and outside. Individual coils are shown in Figure 6. The actuator part is a two-dimensional array of Halbach PM. The magnet pole pitch is 22 mm. The main magnets are 14 × 14 × 10 mm cubes and the secondary magnets are 8 × 14 × 10 mm cubes. Each main magnet is surrounded by four secondary magnets. The actuator part is a two-dimensional Halbach PM array after rotating by 45 degrees. The new array has a pole pitch of $11\sqrt{2}$ mm. The range of motion of this planar motor is 300 × 300 mm. Due to the symmetry of the performance of this PMSPM in the x and y directions, the model is a 3D model and the simulation is time consuming, so to save the simulation time while guaranteeing the accuracy, the subsequent consideration is only the effect of the performance in the x direction.

3.2. Parametric Modeling and Optimization

PMSPM requires high thrust density and high efficiency in industrial applications. Conflicting objectives in the design process can arise due to the thrust output being usually proportional to the motor losses. This study is designed to simplify the optimization problem by focusing on maximizing the motor thrust output as a single-objective optimization problem with the objective function defined as Equation (7).

$$f=\mathrm{Maximize} F_{X-\mathrm{thrust}}$$

(7)

The pole–slot fit can be used to determine the pole pitch of the magnets after determining the size of the actuating parts of the motor. This motor’s poles have a diameter of 22 mm. Before optimizing the motor, the first step is to pre-determine the pole size ratio α, which can be defined as the ratio of the width of the main permanent magnets ${\tau }_m$ and the pole pitch of the magnets $\tau$, with the Equation (8).

$$\alpha ={\tau }_m/\tau$$

(8)

As shown in Figure 7, the air gap magnetic field strength and the base wave duty cycle for different α values are shown. Normally, the larger the magnetic field amplitude is, the larger the motor thrust is according to Equation (5). However, the magnetic field amplitude consists of the fundamental wave and the harmonics together. A single magnetic field amplitude curve cannot directly reflect the motor thrust and positioning accuracy. So, the fundamental wave occupancy is introduced to measure together. The larger the fundamental percentage is, the lower the harmonic content is, and the positioning accuracy is relatively higher. From the results, it can be seen that when α = 0.571, when the width of the main permanent magnet is 14 mm and the width of the auxiliary permanent magnet is 8 mm, the air gap magnetic field strength is larger while the base wave duty cycle is also higher.

The simulation results in Figure 7 are the basis for the subsequent optimization. The thickness of the Halbach PM array is modified once the pole size ratio α is established. The increase in magnet thickness has been observed to gradually increase the output thrust of the planar motor. However, the amounts of permanent magnets are also increasing with the increase in the thickness of the magnet. It is evident that measuring the performance advantages or disadvantages of this planar motor cannot be accomplished by maximizing thrust output alone. This paper defines the thrust density of the motor by measuring the size of the motor thrust produced by a unit volume of permanent magnets. Equation (9) is how it can be expressed. The thrust density is now introduced to results. Figure 8 illustrates the simulation results.

$$F_{density}=F_{X-Thrust}/V$$

(9)

The results in Figure 8 indicate that the motor thrust gradually increases with the increase in the permanent magnet thickness, but the thrust density gradually decreases with the increase in the permanent magnet thickness. Additionally, its comprehensive thrust and thrust density considerations at hm = 10 mm is the best performance of a moving magnet PMSPM, with an output thrust of 264.21 N and a thrust density of 0.403 N/cm³. Since the previous part only considered the magnetic pole size ratio on the air gap magnetic field, now the force of the motor at hm = 10 mm is considered. The width of the main permanent magnet is changed, and the results are shown in Figure 9.

The motor thrust increases and then decreases with an increase in the width of the main permanent magnet, which is not hard to discover. As the width of the main permanent magnet increases, the thrust density decreases. When the main permanent magnet width is 14 mm, the motor performs better. At the moment, the motor has a thrust of 264 N and a thrust density of 0.403 N/cm³, which means that the main parameters of the motor have been determined.

3.3. Performance and Conclusions

According to the results above, the thrust density of the PMSPM with concentric coils remains small and the thrust fluctuation is relatively large. Mostly, the harmonic component of the air gap magnet density generates the thrust fluctuation. The thrust of the motor is partially disturbed, resulting in a decrease in the positioning accuracy of the planar motor. In this paper, we propose a magnet array structure with split magnets based on the Halbach PM array mentioned above. The auxiliary permanent magnets are split into several small pieces under the premise of no change in the dosage of permanent magnets to increase the air gap magnet density by modulating the magnetic line of force of the air gap magnetization by modulating the magnetic force lines, thereby increasing the thrust of the motor and reducing the thrust fluctuation of the motor.

4. PMSPM with Auxiliary Split Permanent Magnet

4.1. Novel Auxiliary Permanent Magnet Split Structure

Halbach arrays with double-layer segmentation proposed in reference [20] have a significant effect on the performance enhancement of permanent magnet brushless motors. The proposed moving magnet PMSPM with a new split magnet structure has the same stator part as the original motor. Additionally, the size and shape of the main permanent magnets of the Halbach PM array remain unchanged, while the auxiliary permanent magnets adopt a split structure. Considering the difficulty of assembly, the auxiliary permanent magnets are mainly bisected and trisected. The original auxiliary permanent magnets are divided into two and three small pieces, and the split magnets are shown in Figure 10a,c. The profile of the magnet array unit along the centerline is shown in Figure 10b,d. The effect of the width of each magnet of the split magnet and the direction of magnetization on the magnetic field of the air gap varies. Because of the low operating frequency of the motor, the effect of the motor temperature rise on the results is not considered in the subsequent optimization.

For the moving magnet PMSPM, only the auxiliary permanent magnets of the two-dimensional Halbach PM array are chunked when the stator coil array and actuating parts are determined. The mass of the actuating parts is maintained as unchanged, so the thrust magnitude is a direct response to the thrust density. However, the widths of the magnets after their chunking and the magnetization angles of the magnets in each chunking are various, and the differences in the widths of the chunks and the magnetization angles may make the result become better or worse. The output thrust and the thrust fluctuation of the motor are taken as the optimization objectives to optimize the parameters in the following.

4.2. Multi-Objective Optimization Based on NSGA-II

Single-objective optimization cannot be applied anymore due to the complexity of the motor thrust variation based on the width and magnetizing angle of the auxiliary permanent magnets. The new model of the Halbach PM array is now parametrically built. Multi-objective optimization is performed on the basis of this model.

The total width of the adopted auxiliary magnets is 8 mm. The parameter optimization of the split magnet length and magnetization angle is carried out. Through rough range screening in the previous stage, it is found that when the auxiliary permanent magnet is bisected, the thrust fluctuation is relatively small when the widths of the two permanent magnets are kept the same. In the trisection, the thrust fluctuation is relatively small compared to the asymmetric case when the widths of the first and the third blocks are kept the same, i.e., with symmetry, so as to carry out the subsequent optimization, which reduces the amount of simulation calculations. This model has relatively few parameterized variables, but it is a three-dimensional model. The amounts of operations are relatively large and the simulation takes a long time, so the non-dominated sorting genetic algorithm-II (NSGA-II) is used. First of all, a range search is carried out, and the range that meets the requirements of thrust and thrust fluctuations is narrowed down and then fine optimization is carried out on this range. The flow block diagram of this algorithm is shown in Figure 11.

At this time, the width b of the middle magnet of the auxiliary permanent magnet and the magnetization angle α of the magnets on both sides in Figure 10d are given. Equation (10) exhibits the initial range of parameters, while Equation (11) reveals the optimization objective.

$$Variables:\left\{\begin{array}{l}\mathrm{b}=\left[0.1\mathrm{mm},3.9\mathrm{mm}\right]\\ a_1=a_2=\left[2.05\mathrm{mm},3.95\mathrm{mm}\right]\\ {\alpha }_1={\alpha }_2=\left[5°,85°\right]\end{array}\right.$$

(10)

$$Objetives:\left\{\begin{array}{l}Maximum:(F_{X-Thrust})\\ Minimize:(RippleRate)\end{array}\right.$$

(11)

Table 2. Initial population parameters and results.

Case	b	a	X-Thrust	Ripple Rate
1	3.47	13.4	230.84	10.9%
2	3.73	10.8	229.41	15.3%
3	3.01	18.2	239.04	11.5%
4	2.22	14.9	257.39	18.7%
5	2.06	5.8	270.42	17.8%
6	2.69	5.2	263.31	11.1%
7	3.30	15.9	231.67	12.8%
8	2.72	1.6	288.83	10.8%

shows an orthogonal array of methods for obtaining the initial population. The formation of the initial population is shown as well.

The NSGA-II algorithm’s Pareto front picture is illustrated in Figure 12. As the motor thrust increases, the inference fluctuation increases gradually. The final motor size parameters in

Table 3. PMSPM size parameter.

Symbol	Quantity	Value
a1, a2	auxiliary PM split block first, third width	2.7 mm
b	middle width of auxiliary PM split block	2.6 mm
α1, α2	auxiliary PM block magnetization angle	48°

require us to choose an appropriate value to meet the requirements.

4.3. Final Optimization Results

The motor’s optimized magnetic density cloud diagram can be seen in Figure 13. There is no apparent red saturated area present in the stator coil and actuating part.

Figure 14 illustrates the counter potential of the PMSPM after optimization completion. According to the phase relationship of the two-phase windings, phase A exceeds phase B by 90°. The waveform possesses a good sinusoidal nature. The amplitude of phase A of counter potential is 54.6 V, and phase B has an amplitude of 55.4 V.

Processing the auxiliary permanent magnet in two parts leads to an output thrust of 277 N in Figure 15. In Figure 16, the output thrust values for both the initial moving magnet PMSPM and the moving magnet PMSPM after bisection of the auxiliary permanent magnets are displayed. It is known from Section 3 of this paper that the output thrust of the original motor is 264 N, with the auxiliary permanent magnets not bisected for processing. Once the temporary magnet is divided into three components, the NSGA-II algorithm is used to improve the width of the partitions and the magnetization angle. The motor III has an output thrust of 288 N, and the ripple rate of thrust dropped to 8.8%.

Motor comparison results are shown in

Table 4. Comparison of three motors.

Motor	Thrust	Thrust Density	Ripple Rate
I	264	13.4	19.4%
II	277	10.8	15.3%
III	288	18.2	8.8%

. Compared to the original motor I, which has not been processed in two parts, the auxiliary permanent magnet is processed in three parts to make the direction of the motor’s output thrust more concentrated. The motor thrust has undergone a significant improvement. After the second chunking, the thrust of motor II increased by 4.9% compared to that of motor I. After the third chunking, the thrust of motor III increased by 9.1% compared to that of motor I. Under the condition that the mass of the actuating parts remains unchanged, the increase in the output thrust of the motor is a result of the increase in the thrust density of the motor. The maximum acceleration that the motor can give under the same working conditions is also increased.

5. Conclusions

This study presents an optimized moving magnet PMSPM featuring a split Halbach PM array, demonstrating significant improvements in thrust density and positioning accuracy. The novel design achieves a 9.1% thrust density enhancement using NSGA-II optimization, highlighting its potential for industrial applications requiring high precision and dynamic performance. The motor is able to increase the acceleration of the actuator by 9.1% with the same motor mass, which can enhance device processing efficiency. Considering the difficulty of processing the auxiliary permanent magnets, we will incorporate the processing cost into our future research to improve the reliability of the data. Future research will explore further refinements in auxiliary magnet configurations and adaptive control strategies to extend the motor’s applicability.