Optimization Design of Cogging Torque for Electric Power Steering Motors

Abstract

Excessive cogging torque can cause torque fluctuations, noise, and vibration in electric power steering (EPS) motors, which is a key factor in the high-precision and high-performance optimization design of EPS motors for electric vehicles. This article takes a 12-slot 10-pole electric power steering motor for a certain car as an example. By establishing the corresponding electromagnetic field model and theoretical analysis of the motor, the influence of the pole arc coefficient and eccentricity parameters of the permanent magnet on the cogging torque of the electric power steering motor is explored. A comprehensive optimization scheme for reducing the cogging torque of the motor structure is proposed. The effectiveness of the designed scheme was verified through finite-element simulation and experimental testing of motor electromagnetism. Compared with the original design, the optimized structure of the EPS motor resulted in an 86.62% reduction in cogging torque during experimental testing.

1. Introduction

With the increasing energy saving requirements of new energy electric vehicle systems, electric power steering (EPS) systems have become necessary. EPS motors require lower cogging torque to achieve comfortable vehicle steering [1,2]. Due to the fact that the EPS system does not require a hydraulic system, it can reduce the weight and volume of electric vehicles, and the cogging torque characteristics of EPS motors are very important. The surface-mounted permanent-magnet synchronous motor used in the vehicle EPS system is no exception [3]. EPS motors require lower cogging torque to make new energy electric vehicles turn more comfortably. Scholars at home and abroad have conducted extensive research on this topic, and reducing the cogging torque of permanent magnet motors is an important issue for low torque ripple applications [4,5]. An electromagnetic design adopts various optimization design methods to reduce cogging torque. By applying a tilt angle with a maximum cycle value of cogging torque, cogging torque can be eliminated. Notches or asymmetric shapes are used to reduce cogging torque [6,7,8].

Reference [9] has conducted theoretical analysis and simulation research to demonstrate that the tooth slot torque in permanent magnet motors can be eliminated by moving the magnet and optimizing the pole arc. However, this method placed high demands on the accuracy of manufacturing small motors at that time. Reference [10] weakens the cogging torque by changing the size and distribution of the magnetic flux density at each pole through magnetic pole offset. This method requires an accurate calculation of the offset angle of each magnetic steel, which brings inconvenience to motor manufacturing. Based on a large number of finite-element simulations, fast Fourier transform analysis, and experimental results, reference [11] concluded that the use of interlocking stator stack stacking leads to additional cogging torque components. Reference [12] reduces the cogging torque by using a combination of different pole arcs and magnetic materials. In terms of the structure, only the pole arc parameter is optimized, and the improvement effect is limited. Reference [13] proposed a design method based on orthogonal experiments to optimize the cogging torque of permanent-magnet motors. References [14,15,16,17] reduced the difference in spatial magnetic permeability by adjusting the inclination angle of the stator and rotor, the pole arc ratio of the permanent magnet, and the taper, thereby reducing the cogging torque. Currently, research is also being conducted to reduce cogging torque by optimizing slot length (which is the main cause of spatial magnetic permeability differences) or adding magnetic wedges [18,19,20]. In order to reduce the cogging torque, the main approach is to change the pole slot combination or optimize the generator shape design variables to reduce spatial magnetic permeability differences.

Based on relevant literature both domestically and internationally, current research on reducing motor vibration mainly focuses on radial electromagnetic forces, with relatively little research on reducing the cogging torque of EPS motors. This article takes the 10-pole 12-slot EPS motor used in electric vehicles as the research object. Through electromagnetic simulation and experimental measurement, the pole arc coefficient and eccentricity structural parameters of the permanent magnet are comprehensively optimized to reduce the cogging torque of the EPS motor, thereby reducing the vibration and noise of the EPS motor and improving the driver’s driving experience.

2. Analysis of the Mechanism of Cogging Torque

2.1. Deduction of a Mathematical Model for Cogging Torque

The cogging torque, also known as the positioning torque, is the torque formed by the tangential component of the interaction force between the permanent magnet and the armature iron core when the EPS motor winding is not energized. Essentially, it is the no-load reluctance torque. Reducing the cogging torque can significantly reduce torque ripple under light load or deep weak magnetic conditions, thereby reducing EPS motor vibration and noise. The cogging torque is usually defined as the negative derivative of the magnetic field energy W when the motor is not powered on, with respect to the relative position angle α of the stator and rotor, as shown in Equation (1):

$$T_{cog}=-{\displaystyle \frac{\partial W}{\partial \alpha }}$$

(1)

where α is the relative position angle between the stator and rotor, referring to the angle between the centerline of a certain armature tooth and the centerline of a certain permanent magnet.

After expanding Equation (1) in Fourier series, the analytical formula for the cogging torque of the electric power steering motor can be obtained Equation (2):

$$T_{cog}(a)={\displaystyle \frac{\pi z{L}_a}{4{\mu }_0}}(R_2^2-R_1^2){\displaystyle \sum _{n=1}^{\infty }n{G}_n}{B}_{r{\displaystyle \frac{nz}{2p}}}\sin nza$$

(2)

where z is the number of slots; 2p is the number of motor poles; and L_a is the length of the iron core. The motor in this article has an inner rotor structure, with R₁ and R₂ representing the outer radius of the rotor yoke and the inner radius of the stator, respectively; N is an integer that makes nz/2p an integer.

2.2. Structural Parameters of the EPS

The structural parameters of the optimized electric power steering motor in this article are shown in

Table 1. Structural parameters of the EPS motor.

Structural Parameter	Value	Structural Parameter	Value
Stator out diameter/mm	85	Axial length/mm	77
Stator in diameter/mm	44.5	Permanent magnet thickness/mm	3.23
Rotor out diameter/mm	43	Permanent magnet	NdFeB
Rotor in diameter/mm	16.9	Polar arc coefficient	0.75

, and the simulation model of the EPS motor after finite-element optimization is shown in Figure 1.

3. Parameter Analysis and Optimization

3.1. Analysis and Optimization of the Polar Arc Coefficient

The pole arc coefficient α_p is the ratio of the axial length of the permanent magnet of the electric power steering motor to the pole distance, as shown in Equation (3):

$${\alpha }_p={\displaystyle \frac{L_p}{\tau }}$$

(3)

where L_p is the width of the permanent magnet.

For any relative position β, the radial component distribution of the motor air gap magnetic flux density along the armature surface can be expressed as shown in Equation (4):

$$B\left(\theta ,\beta \right)=B_r(\theta ){\displaystyle \frac{h_m(\theta )}{h_m}}{\displaystyle \frac{h_m}{h_m(\theta )+\delta (\theta ,\beta )}}$$

(4)

where h_m is the length of the magnetization direction at the center position of the permanent magnet pole; h_m (θ) is the length of the magnetization direction at an angle of θ with the line where h_m is located.

The magnetic field energy inside the EPS motor is as shown in Equation (5):

$$W={\displaystyle \frac{1}{2{\mu }_0}}{\displaystyle {\int }_V{B^{\prime }}_r^2}(\theta ){[{\displaystyle \frac{h_m(\theta )}{h_m(\theta )+\delta (\theta ,\beta )}}]}^{2}dV$$

(5)

Performing Fourier expansion, B′_r²(θ) can be expressed as shown in Equation (6):

$${B^{\prime }}_r^2(\theta )={\displaystyle \frac{B_r^2}{h_m^2}}[R_{m1}^2+R_{m2}^2\cos 2\theta -2R_{m1}{R}_{m2}\cos \theta \sqrt{1-{({\displaystyle \frac{R_{m2}}{R_{m1}}}\sin \theta )}^2}]$$

(6)

The magnitude of the cogging torque is not related to all B′_r²(θ) Fourier decomposition coefficients, but only to Fourier decomposition coefficients with nz/2p times. Therefore, optimizing the pole arc coefficient of permanent magnets can effectively reduce the cogging torque.

Before optimization, the pole arc coefficient of the permanent magnet of the EPS motor was 0.85. With the other structural parameters of the EPS motor unchanged, the pole arc coefficients of the permanent magnet were set to 0.70, 0.75, 0.80, and 0.85 for the finite-element calculation. The resulting cogging torque is shown in Figure 2.

At this point, the cogging torque of the permanent magnet of the electric power steering motor is minimized when the pole arc coefficient is 0.75. In order to achieve comprehensive optimization with the eccentricity parameter and avoid affecting the performance of the electric power steering motor, the range of the comprehensive optimization pole arc coefficient is set to (0.75~0.80).

3.2. Analysis and Optimization of Eccentricity

The air gap magnetic field of a permanent magnet varies with the thickness of the magnetic pole, which in turn can change the cogging torque. O′ is the inner center of the permanent magnet, O is the outer center of the permanent magnet, and h is the eccentricity, as shown in Figure 3. The minimum air gap length, hm, generally remains unchanged.

When the permanent magnet is eccentric, the radial component of the air gap magnetic flux density is expressed as shown in Equation (7):

$${B^{\prime }}_r^{(\theta )}={B^{\prime }}_r(\theta ){\displaystyle \frac{h_m}{h_m+\delta (\theta )}}=B_r(\theta ){\displaystyle \frac{h_m(\theta )}{{h^{\prime }}_m}}{\displaystyle \frac{h_m}{h_m+\delta (\theta )}}$$

(7)

where h′(θ) is the changing thickness of the permanent magnet.

The relationship between the cogging torque and the eccentricity of the permanent magnet is shown in Equation (8), and the cogging torque can be optimized by changing the thickness of the permanent magnet.

$$T_{cog}(\beta )={\displaystyle \frac{\pi z{L}_a}{4{\mu }_0}}(R_2^2-R_1^2){\displaystyle \sum }_{n=1}^{\infty }n{G}_n{B}_r(\theta ){\displaystyle \frac{{h^{\prime }}_m(\theta )}{h_m(\theta )+\delta (\theta )}}\sin nz\beta$$

(8)

The eccentricity of the permanent magnet designed before EPS motor optimization was 0 mm, which means that the permanent magnet is an equally thick permanent magnet, and the generated cogging torque is relatively large. Under the condition that other structural parameters of the motor remain unchanged, the eccentricity of the permanent magnet is taken as 0 mm, 3 mm, 6 mm, 9 mm, 12 mm, etc. for the finite-element simulation calculation. The results are shown in Figure 4.

The larger the eccentricity of the permanent magnet in EPS motors, the smaller the cogging torque. However, excessive eccentricity can lead to difficulty and increase in the cost of permanent magnet processing, and it is difficult to ensure the electromagnetic performance of EPS motors. Therefore, this study selected a pole arc coefficient of (0.70~0.80) and an eccentricity of (0~12) mm for comprehensive optimization design analysis.

3.3. Double Parameter Optimization of the Polar Arc Coefficient and Eccentricity

A comprehensive optimization finite-element simulation analysis was conducted on a pole arc coefficient of (0.70~0.80) and an eccentricity of (0~12) mm. The variation curve of the cogging torque with the pole arc coefficient and permanent eccentricity was obtained as shown in Figure 5. A pole arc coefficient of 0.75 and a permanent magnet eccentricity of 12 mm were the optimal combination for comprehensive optimization, and the simulated cogging torque was 1.68 mNm.

4. EPS Motor Experimental Testing

4.1. Comparison of EPS Motor Performance before and after Optimization

The volume of the permanent magnet of the optimized EPS motor will change, leading to changes in magnetic flux, which in turn will alter parameters such as motor speed, locked rotor torque, and locked rotor current. Therefore, performance testing of the optimized EPS motor is required. The simulated and measured performance parameters of the EPS motor are shown in

Table 2. Comparison of EPS motor simulation and measured data.

	Simulation Data before Optimization	Actual Measurement Data before Optimization	Optimized Simulation Data	Optimized Measured Data
No-load speed/(r/min)	2082.58	1967.41	2237.19	2103.06
Locked rotor current/A	204.93	200.38	205.97	203.29
Locked rotor torque/Nm	44.17	42.62	41.85	40.82
Motor efficiency/%	78.95	77.67	81.63	80.74

. The magnetic density cloud map and magnetic field line map of the optimized EPS motor are shown in Figure 6.

Compared with the EPS motor before and after optimization, the optimized EPS motor has relatively small changes in parameters such as motor speed, locked rotor torque, and locked rotor current, which will not have a significant impact on the use of EPS motors. The finite-element simulation data of the optimized no-load speed is 2237.19 r/min. The torque curve of the EPS motor slot calculated by the finite-element simulation is shown in Figure 7. However, in reality, due to the friction of the EPS motor shaft and assembly process errors, the actual speed will be lower than the finite element simulation speed (100~150) r/min, so the finite element simulation results are reasonable and effective.

Based on the above analysis, finite-element simulation was conducted to verify the cogging torque of the EPS motor. The results showed that the maximum cogging torque of the optimized EPS motor was 1.68 mNm, a decrease of 96.09% compared to the 42.93 mNm before optimization. This indicates that the comprehensive optimization design of the pole arc coefficient and eccentricity can effectively reduce the cogging torque of the EPS motor.

4.2. Experimental Testing of EPS Motor Cogging Torque

The cogging torque of EPS motors is generated by the relative motion between the rotor permanent magnet and the stator cogging, which is related to the relative angle α of the stator and rotor, but not to the stator current. Therefore, during the test, the tested EPS motor can be powered off and driven by a device (stepper motor) that can precisely control the rotation angle, as shown in Figure 8.

During testing, the stepper motor, torque sensor, and test motor are rigidly connected to the same axis and powered on to rotate the stepper motor at a speed of 1 r/min for one revolution (the angle resolution can reach 0.01°). The test EPS motor rotor also rotates synchronously for one revolution (0~360°). During this period, the stator position of the tested motor remained unchanged, and the relative angle α between the stator and rotor varied from 0 to 360°. The trend of α variation will generate an oscillating torque (cogging torque) in the motor, which acts on the output shaft of the tested motor. The value is directly read out by the torque sensor and a real-time curve is drawn by data visualization software. Due to the small value of the cogging torque of the EPS motor, a lower speed should be used for the experiment, and the collection rate of the sensor should be as high as possible. The experimental platform for testing the cogging torque of the EPS motor is shown in Figure 9.

The specific values for the experimental testing of EPS motor cogging torque are shown in

Table 3. Comparison of the simulation and measured data of the EPS motor cogging torque.

	Before Optimization		After Optimization
	Simulation Data	Actual Measurement Data	Simulation Data	Actual Measurement Data
Cogging torque/(mNm)	42.93	50.74	1.68	6.79

. It should be noted that there is a certain error between the measured and simulated values of cogging torque. On the one hand, it is related to the angular resolution of the testing device, and on the other hand, the measured cogging torque also includes the friction torque during rotor rotation. However, the error is within a reasonable range. The maximum cogging torque of the optimized EPS motor measured is 6.79 mNm, which is a decrease of 86.62% compared to the 50.74 mNm tested before optimization.

5. Conclusions

This article first establishes the corresponding electromagnetic field model of an EPS motor through theoretical analysis, and studies the effects of the pole arc coefficient and eccentricity on cogging torque through finite-element simulation. Based on the results, the two objective parameters are comprehensively optimized, and the optimization design of the EPS motor cogging torque is achieved, while ensuring that the performance of the EPS motor does not decrease. In the study, the performance and magnetic field of the optimized EPS motor were simulated and verified. The performance and magnetic field distribution of the optimized EPS motor met the design requirements, and the cogging torque decreased from 42.93 mNm before optimization to 1.68 mNm, with a decrease of 96.09%. Finally, through experimental testing and comparison, the maximum value of the tooth slot torque test decreased from 50.74 mNm before optimization to 6.79 mNm, a decrease of 86.62%.