Thrust Fluctuation Optimization of Permanent Magnet Linear Motor Based on Composite Slot Wedge

In order to reduce the thrust fluctuation of the linear motor, combined with its structural characteristics, the method of embedding a magnetic slot wedge is adopted. Based on the design of a common slot wedge structure, the genetic algorithm is used to optimize the slot wedge size, select the best size of the slot wedge structure, and then carry out finite element analysis. The results show that the thrust fluctuation of the single-sided linear motor is reduced by 66.42%, and that of the double-sided linear motor is reduced by 61.23%, which effectively reduces the thrust fluctuation. It has certain practical application value.


Introduction
Compared with the rotating motor, the performance of the permanent magnet synchronous linear motor (PMLSM) is different, but its use of permanent magnets makes its thrust very large, so it is widely used in the linear field.Due to its ability to provide large thrust, high speed, and high precision, PMLSM has become a hot spot in the field of motors [1].Although linear motors have many advantages, thrust fluctuations due to slot forces can lead to instability in the application.Therefore, reducing thrust fluctuation as much as possible is the focus of PMLSM research [2].The existing solutions include control strategies to suppress thrust pulsation [3][4][5], such as repetitive control strategies, iterative sliding mode control, deep neural networks, etc.At the same time, the body structure of PMLSM can also be optimized to reduce thrust fluctuation.In [6], the working principle of PMLSM was described and some structural optimization methods were summarized based on the analysis of thrust and thrust fluctuation.Because the groove force is the main cause of the thrust pulsation, it can be reduced by optimizing the motor body structure.The structural optimization of thrust fluctuation is mainly divided into two aspects: stator and actuator.Stator structure optimization [7] includes the method of inclined slots, the matching of the number of slot poles, changing the width of slots, the method of auxiliary teeth, the method of unequal slot widths, etc.Among them, the method of inclined slots is widely used and can effectively suppress thrust fluctuations, but it will reduce the output torque, and lead to a more complex motor structure.In addition, reducing the opening width of the slot can also reduce the variation of the air gap permeability, improve the air gap permeability, and thus reduce the thrust fluctuation.In [8], the effect of fractional slots per pole on the thrust of linear motors with unilateral structure and bilateral structure was studied, and the optimal slot pole number coordination was obtained.In [9], the optimization of the permanent magnet shape was also studied.
Because the methods of control and body optimization are relatively complicated, this paper adopts the method of inserting slot wedges, which costs low and is simple.Based on the commonly used slot wedges, the composite slot wedge is designed and the algorithm is used to optimize the size, and the slot wedge structure that meets the actual requirements and has the least thrust fluctuation is obtained.

Effect of slot wedge on air gap
The effective length of the motor air gap depends on the air gap coefficient [10], which affects the performance of the motor such as the tooth slot force.When linear motors are studied, ideally, when the primary notch is embedded in the slot wedge, the structure is similar to that without the slot wedge.When analyzing the two-dimensional electromagnetic field of the motor, the following assumptions are made: (1) the influence of the leakage magnetic field generated by the end of the motor is ignored; (2) the influence of the leakage magnetic field outside the stator is ignored.The calculation formula [11] of the air gap coefficient is as follows: > @ 10 10 In the formula, ‫ݐ‬ represents the distance between the stator teeth, while the stator width is expressed by ܾ ௧ ; ‫ܭ‬ ఋ and ‫ܭ‬ ఋఓ represent the relative permeability of the slot wedge and the non-magnetic slot wedge; ߜ is the length of the air gap between the stator and the actuator, and ߝ is the correction factor.Formulas (1) -( 3) are taken to calculate the air gap coefficient, and it is obtained that ‫ܭ‬ ఋ is 2.612, and ‫ܭ‬ ఋఓ is 1.207.It can be seen that the magnetic slot wedge is 46.21% lower than the non-magnetic slot wedge.Due to the reduction of the effective slot length of the motor and the unchanged slot spacing, the equivalent slot width is reduced, so that the motor has the effect of the half slot.

Groove wedge structure design
The existing slot wedge structures mainly include structures A, B, and C as shown in Figure 1.Structure A is to place ferrite on top, the length is 1 mm, and the silicon steel sheet is stuck under the ferrite.Structure B is the opposite.Structure C is a single ferrite.In this paper, the composite notch is further optimized to obtain a double composite slot wedge (DCMSW), which is a horizontal embedding of two materials.No wedge structure (NSW) means no wedge embedded.A single composite slot wedge (SCMSW) is a single composite of ferrite and silicon steel sheet, and the ferrite material is embedded into the silicon steel sheet, with the same length and thickness as the silicon steel sheet.The models for NCMSW and DCMSW are shown in Figure 2. In this paper, the DCMSW slot wedge is optimized by the genetic algorithm.The genetic algorithm was used to optimize the composite size of the wedge, and the optimization target was set as thrust and slot force to determine the best size of the silicon steel sheet and ferrite.The first priority was thrust fluctuation, and the second priority was slot force.The optimization idea is shown in Figure 3. Due to the setting of the groove height and the groove wedge material, the height y2 has been determined and cannot be changed, but the width of the groove wedge material can be changed.By setting the minimum output thrust fluctuation and groove force, the material width of the groove wedge is optimized.The optimization data is shown in Table 1.Since the optimization process is linear, the optimal value is approximately rounded according to the actual production needs, and the minimum value is obtained around ‫2ݔ‬ equal to 2.4 mm and ‫3ݔ‬ near 2mm-3mm.Table 1 mainly shows the optimal slot wedge size of the single-sided linear motor.For the double-sided linear motor, the same algorithm is used to obtain the optimal size similar to that of the single-sided linear motor.

Motor Model
In order to better know the effect of slot wedge on electromotor generation, unilateral linear motor and bilateral linear motor are analyzed at the same time without considering the end effect and the uneven distribution of the axial magnetic field.The model is shown in Figure 4 and Figure 5.The permanent magnet synchronous linear motor is designed in this paper to distinguish the effect of slot wedge, the coil is used as the actuator and the stator is a permanent magnet.The permanent magnet of the double-sided linear motor is the moving part, the stator is a coil, and the stator structure is a fully open slot.In order to ensure the convenience of system installation and maintenance, both kinds of permanent magnet synchronous motors use centralized winding.Some design parameters of PMLSM are listed below.Some design parameters of PMLSM are shown in Table 2.

Performance Comparison
The fewer the harmonics of the air gap flux density is, the smoother the flux density curve is, and the more smoothly the motor runs.According to Formula (3), the embedded slot wedge material can effectively change the unevenness of the notch, and the verification of the analysis is shown in Figure 6 and Figure 7.The slot force will affect the thrust of the motor, resulting in thrust fluctuation, and the air gap permeability changes due to the slotting of the motor.The existence of the slot wedge will make the notch and the tooth have a connection, and reduce the cogging force to a certain extent.Figure 8 shows the cogging force fluctuations of unilateral and bilateral linear motors.The cogging force fluctuations of the slot wedge structure B of the unilateral linear motor and the SCMSW structure of the bilateral linear motor are small and reduced by 91.14% and 66.85% respectively.Due to the existence of the normal force of the unilateral structure, the tooth groove force is greatly affected, while the normal force of the bilateral structure is offset a lot, so the tooth groove force is very small.Therefore, the tooth groove force of the unilateral motor in the figure is very large, but the embedded groove wedge structure reduces the tooth groove force to different degrees.The thrust of a permanent magnet motor is generated by the coupling between the primary winding coil and the secondary permanent magnet.The existence of the slot wedge can alleviate the peak value of the electromagnetic force wave, thereby reducing the thrust fluctuation.Figure 9 is a comparison chart of thrust fluctuation.For the single-sided linear motor structure A, the thrust fluctuation is reduced by 47.43%, and DCMSW reduces the thrust fluctuation by 42.43%.The most effective slot-wedge structure of the bilateral linear motor is DCMSW, which reduces the fluctuation by 43.42%.

Conclusion
In this paper, based on the common slot wedges, the composite mode of slot wedges is changed, and the structure of slot wedges is optimized by the genetic algorithm.The optimum size of the composite slot wedge is obtained, and then the optimal analysis is carried out on the single-side short primary motor and double-side long primary motor.The results show that the magnetic slot wedge can reduce the air gap coefficient by 46.21%.For the single-side linear motor slot-wedge structure, the thrust fluctuation is reduced by 66.42%.For the double-sided linear motor, the thrust is increased by 16.79% and the thrust fluctuation is reduced by 43.42%.It shows that the structure can improve the performance of linear motors.

Figure 3 .
Figure 3. Process of optimizing slot wedge by genetic algorithm

Table 1 .
Unilateral Motor Thrust Fluctuation Under Different Slot Wedge Widths

Table 2 .
The Main Parameters of the Motor