# Computational Technologies for Low-Frequency Torque Ripple Suppression in Exterior-Rotor Permanent Magnet Motors

Authors: Ding Hui; Mao Jun
Journal: Security and Communication Networks
 (2022)

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## Abstract

Exterior-rotor permanent magnet direct-drive motors (ERPMDDMs), which have a simple structure and high energy efficiency, can achieve high torque output at low speed. Because of the superior performance, ERPMDDMs have been widely used in low-speed and high-torque (LSHT) mechanical equipment. The stability of the LSHT drive of ERPMDDMs directly affects the reliability of the equipment, which is thoroughly studied in this work by taking 250 kW ERPMDDMs as an example. The influence of different pole-to-slot ratios and pole-arc coefficients on the cogging torque is analyzed by the electromagnetic field calculation and analysis software Ansoft, and several parameters are optimized to reduce the torque ripple in energized motors and ensure the stability and reliability of the equipment.


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## Body

## 1. Introduction

A standard motor turns electrical energy into mechanical energy by generating a magnetic field by connecting a current in the motor winding; however, a permanent magnet motor (PMM) does not need an excitation coil or an excitation current to sustain the magnetic field. Motors' structures may be substantially simplified, their volume and bulk lowered even more, and their performance is superior to that of typical electric excitation motors. The stator is on the inside and the rotor is on the outside of the exterior-rotor permanent magnet direct-drive motor (ERPMDDM) [1]. We can reduce the transmission chain, simplify the equipment structure, and increase operation reliability by adopting drive systems with this topology in transmission systems. Figure 1 depicts the outline of an exterior-rotor permanent magnet motor [2].Figure 1
Outline drawing of exterior-rotor permanent magnet motor.ERPMDDMs can achieve high-torque output under low-speed conditions, with a simple structure and high energy efficiency, and have been widely used in low-speed and high-torque (LSHT) mechanical equipment because of their superior performance [3]. The stability of the LSHT drive of ERPMDDMs directly affects the reliability of the equipment, so it is particularly important to investigate the suppression of the low-frequency torque ripple in ERPMDDMs. Cogging torque is generated by the interaction between the air-gap permeance and the harmonic magnetic potential of the motor, which will cause vibration and noise in the motor, leading to unstable torque output. Many methods have been proposed by scholars around the world to reduce cogging torque. Different pole-to-slot ratios and pole-arc coefficients have an impact on cogging torque. This paper takes 250 kW ERPMDDMs as an example to thoroughly analyze this problem [4].


## 2. Influencing Factors of Motor Cogging Torque

### 2.1. Influence of Different Pole-to-Slot Ratios on Cogging Torque

The cogging torque is related to the Fourier decomposition of the air-gap magnetic flux density in the circumferential direction, but it is determined by only thenz/2p-order Fourier decomposition coefficients [5]. When the rotor spins a pitch from the initial static position, the period number of its cogging torque ripple depends on the combination of the number of magnetic poles and that of slots. It can be seen from the formula that the period number of the cogging torque ripple is the minimum value of n that makes nz/2p an integer, denoted as np; then,(1)np=2pGCDz,2p,where GCDz,2p is the greatest common divisor of the slot number z and the pole number 2p.The peak value of cogging torque primarily depends onBrnpz/2p, and the larger nz/2p the smaller Brnpz/2p, so that the larger the cogging torque period, the smaller the cogging torque amplitude [6, 7].Taking 250 kW ERPMDDM products as an example, the 40-pole 60-slot, 48-pole 60-slot, and 50-pole 60-slot motors were analyzed, respectively, and the analysis results are presented in Figures2–4.Figure 2
40-pole 60-slot cogging torque (peak value of 5302 N m).Figure 3
48-pole 60-slot cogging torque (peak value of 1361 N m).Figure 4
50-pole 60-slot cogging torque (peak value of 590 N m).The cogging torque of the 50-pole 60-slot motor was the least without pole-arc optimization, as seen in the figures. To optimize the motor characteristics, the 50-pole 60-slot combination was employed as the design approach in this research [8].


### 2.2. Influence of Pole-Arc Coefficients on Cogging Torque

For different pole-arc coefficients, the Fourier decomposition coefficientsBrn of Br2θ are significantly different. If Brn=0, it must satisfy nαp= integer. For αp=2/3, when n is a multiple of 3, Brn=0; for αp=0.7, when n is a multiple of 10, Brn=0, and for αp=0.8, when n is a multiple of 5, Brn=0. Therefore, as long as the pole number and slot number of the motor are given, the orders of Fourier decomposition of Br2θ affecting the cogging torque are also determined. For different pole-arc coefficients, there are always some specific orders of Br2θ with very small Fourier decomposition coefficients. Therefore, by selecting the pole-arc coefficient reasonably, we can make Brn with small values have impact on the cogging torque, while Brn with large values have no effect on the cogging torque, thereby weakening the cogging torque.Taking 250 kW ERPMDDMs as an example, the electromagnetic field calculation software Ansoft was used to analyze a pole-slot combination of 50-pole 60-slot, and the calculation results are shown in Figure5.Figure 5
Cogging torque versus pole-arc coefficient.When the pole-arc coefficient of a 50-pole 60-slot motor was modified from 0.65 to 0.9, the cogging torque displayed periodic variations and three lowest values at pole-arc coefficients of 0.67 and 0.83, respectively, as shown in Figure5. The pole-arc coefficient should be used while constructing the motor such that the cogging torque value is close to the minimal value. The pole-arc coefficient in this design was 0.85, based on the motor's performance [9–11].


## 2.1. Influence of Different Pole-to-Slot Ratios on Cogging Torque

The cogging torque is related to the Fourier decomposition of the air-gap magnetic flux density in the circumferential direction, but it is determined by only thenz/2p-order Fourier decomposition coefficients [5]. When the rotor spins a pitch from the initial static position, the period number of its cogging torque ripple depends on the combination of the number of magnetic poles and that of slots. It can be seen from the formula that the period number of the cogging torque ripple is the minimum value of n that makes nz/2p an integer, denoted as np; then,(1)np=2pGCDz,2p,where GCDz,2p is the greatest common divisor of the slot number z and the pole number 2p.The peak value of cogging torque primarily depends onBrnpz/2p, and the larger nz/2p the smaller Brnpz/2p, so that the larger the cogging torque period, the smaller the cogging torque amplitude [6, 7].Taking 250 kW ERPMDDM products as an example, the 40-pole 60-slot, 48-pole 60-slot, and 50-pole 60-slot motors were analyzed, respectively, and the analysis results are presented in Figures2–4.Figure 2
40-pole 60-slot cogging torque (peak value of 5302 N m).Figure 3
48-pole 60-slot cogging torque (peak value of 1361 N m).Figure 4
50-pole 60-slot cogging torque (peak value of 590 N m).The cogging torque of the 50-pole 60-slot motor was the least without pole-arc optimization, as seen in the figures. To optimize the motor characteristics, the 50-pole 60-slot combination was employed as the design approach in this research [8].


## 2.2. Influence of Pole-Arc Coefficients on Cogging Torque

For different pole-arc coefficients, the Fourier decomposition coefficientsBrn of Br2θ are significantly different. If Brn=0, it must satisfy nαp= integer. For αp=2/3, when n is a multiple of 3, Brn=0; for αp=0.7, when n is a multiple of 10, Brn=0, and for αp=0.8, when n is a multiple of 5, Brn=0. Therefore, as long as the pole number and slot number of the motor are given, the orders of Fourier decomposition of Br2θ affecting the cogging torque are also determined. For different pole-arc coefficients, there are always some specific orders of Br2θ with very small Fourier decomposition coefficients. Therefore, by selecting the pole-arc coefficient reasonably, we can make Brn with small values have impact on the cogging torque, while Brn with large values have no effect on the cogging torque, thereby weakening the cogging torque.Taking 250 kW ERPMDDMs as an example, the electromagnetic field calculation software Ansoft was used to analyze a pole-slot combination of 50-pole 60-slot, and the calculation results are shown in Figure5.Figure 5
Cogging torque versus pole-arc coefficient.When the pole-arc coefficient of a 50-pole 60-slot motor was modified from 0.65 to 0.9, the cogging torque displayed periodic variations and three lowest values at pole-arc coefficients of 0.67 and 0.83, respectively, as shown in Figure5. The pole-arc coefficient should be used while constructing the motor such that the cogging torque value is close to the minimal value. The pole-arc coefficient in this design was 0.85, based on the motor's performance [9–11].


## 3. Rotor Pole-Arc Shape Optimization

To improve the torque density, reduce the cogging torque, and improve the operation reliability, ERPMDDMs adopt a surface-mounted structure with double fixation protection measures of Loctite adhesive and stainless steel strip fastening, and its structure is shown in Figure6.Figure 6
Schematic diagram of rotor punch.Because the rotor magnetic pole structure is a surface-mounted magnetic circuit structure, we proposed optimizing the rotor pole-arc shape to improve the sine of the air-gap magnetic flux density waveform, thereby reducing the harmonic components of the air-gap magnetic flux density and thus mitigating torque ripple, noise, and vibrations. Figure7 shows a schematic model for optimizing rotor pole structure.Figure 7
Schematic diagram of pole-arc shape optimization.The optimal design of the rotor pole-arc shape was carried out by changing theR value as shown in Figure 7. The optimized surface of the pole shoe is an arc. Based on finite element analysis (FEA), the surface was affixed with 50-pole 60-slot, the pole-arc coefficient was 0.85, the diameter of the outer circle of the magnetic steel was 950 mm, and the diameter of the corresponding circle at the innermost point of the magnetic steel was 930 mm [12]. The ratio k of the maximum air gap to the minimum air gap can be changed by the radius R corresponding to the inner arc of the magnetic steel, thereby optimizing the pole-arc shape of the motor. The distributions of the magnetic lines of force at k of 1.35 and 2.5 are shown in Figures 8(a) and 8(b), respectively. It can be seen from these two figures that when the ratio k was different, the distribution densities of magnetic lines of force on the permanent magnet arc surface were different. The larger the ratio, the sparser the distribution of magnetic lines of force on both sides of the permanent magnet arc surface [13–15].The back electromotive force (Figure8) (EMF) waveform (Figure 9) and the radial air-gap magnetic flux density waveform (Figure 10) of the half period of the motor with different k values were obtained using FEA simulation to better understand the impact of the ratio k on the performance parameters of the motor. The effective value of the back EMF, the maximum value of the cogging torque, and the variation of the harmonic distortion rate of the no-load back EMF waveform of the motor as a function of k were also computed, and the results are shown in Figures 11–13, respectively.Figure 8
Schematic diagram of magnetic lines of force of optimized pole-arc shape. (a)k = 1.35. (b) k = 2.5.
(a)(b)Figure 9
Back EMF waveforms (V) at differentk.Figure 10
Radial air-gap magnetic density waveforms at differentk.Figure 11
Back EMF versusk.Figure 12
Cogging torque versusk.Figure 13
Harmonic distortion rate versusk.Through the simulation of the no-load performance parameters of the motor, it can be seen that with the increase ofk, the effective value of the no-load back EMF of the motor showed a downward trend, the cogging torque of the motor generally presented a downward trend, with rising phenomenon at several k values, and the harmonic distortion rate of the back EMF of the motor displayed a downward trend, with a rising phenomenon at k of 2. Obviously, the change of k will affect the running performance of the motor. To more comprehensively analyze the impact of uneven air gap on motor performance, the influence of k on the motor load performance was studied. For a given current source excitation, when the motor outputted rated torque, the amplitude of the motor torque ripple as a function of k was obtained, as shown in Figure 14. When the other parameters of the motor remain unchanged and the motor outputted rated torque, the effective current value of the motor changed with k, and the relationship curve is shown in Figure 15.Figure 14
Load torque ripple versusk.Figure 15
Rated current versusk.The modelling of the motor load performance characteristics revealed that whenk rose, the load torque ripple of the motor reduced and the motor's rated current increased. The load torque ripple of the motor may be minimised to some degree by optimizing the uneven air gap, but this would result in an increase in the motor's rated current, limiting its overload capacity. As a result, a reasonable choice of k is required to guarantee that the motor's overall performance indicators are ideal. In this work, the optimal k was 1.5.


## 4. Conclusion

(1)
Taking 250 kW ERPMDDMs as an example, the 40-pole 60-slot, 48-pole 60-slot, and 50-pole 60-slot combinations were analyzed, respectively. The results showed that the 50-pole 60-slot cogging torque was the smallest.(2)
Using the electromagnetic field calculation software Ansoft, for a 50-pole 60-slot motor, when the pole-arc coefficient of the motor changed from 0.65 to 0.9, the cogging torque exhibited periodic fluctuations and three minimum values at 0.67 and 0.83, respectively. Considering the overall performance of the motor, the pole-arc coefficient is 0.85 in our design.(3)
The simulation of the no-load and load performance parameters of motors was conducted to reveal the variation of the ratiok of the maximum air gap to the minimum air gap and the cogging torque. The optimization of various comprehensive performance indicators of the exterior-rotor permanent magnet motor was achieved at k of 1.5 in our work.



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*Source: 3973022-2022-05-04.xml*