The permanent magnet toroidal motor (PMTM) is a special motor that combines a permanent magnet motor with a deceleration mechanism. Because of the special structure of the PMTM, its reluctance torque includes cogging torque and end torque. In order to reduce the reluctance torque ripple of the PMTM and improve its output stability, the cogging torque and end torque of the motor are optimized, respectively. The structure of eccentric permanent magnet teeth improves the sine of the air gap magnetic induction of the motor, thereby effectively reducing the fluctuation of the cogging torque. It is proposed to add compensation windings at both ends of the central worm stator to reduce the fluctuation of the end torque. The dynamic compensation effect and the change in air gap magnetic induction intensity with the current applied to the compensation windings are analyzed by finite element simulation. The relationship between the compensation current and the structure parameters of the motor is derived. Finally, it is verified that the output torque ripple can be effectively reduced by adding the magnetic isolation material between the compensation winding and the three-phase winding. The research results show that the reluctance torque fluctuation can be greatly reduced by using eccentric permanent magnet teeth and compensation windings. It provides an important reference basis for the optimization design of the electromagnetic structure of the PMTM.

Permanent magnet synchronous motor has the advantages of simple structure, reliable operation, and high torque density. It has been widely used in industrial production, electric vehicles, wind power generation, and other fields.1,2 However, the traditional permanent magnet motor needs a gearbox to achieve torque output, which results in large space occupation and low transmission accuracy.3 Therefore, a new space motor that integrates permanent magnet motor and planetary worm drive is proposed, which is called permanent magnet toroidal motor (PMTM). The motor has the advantages of stable operation and large reduction ratio, and it can transfer a large torque in a small space. The PMTM has broad application prospects in the fields of transportation, robot, and aerospace field.4 

The reluctance torque usually stems from the cogging effect and end effect,5 which will cause the output torque ripple of the PMTM and affect the control accuracy of the drive system. The cogging torque of the PMTM is due to the slotting of the central worm stator. The cogging torque can be optimized in many ways. The method of permanent magnet teeth reverse magnetization combined with rotor segmentation can effectively reduce the cogging torque and load torque fluctuation of the permanent magnet motor.6 The torque ripple can be reduced effectively by using fractional slot concentrated winding.7 The influence of the cogging torque can be reduced by adjusting the distance between magnetic poles, optimizing the shape of magnetic poles, and adopting the installation method of the inclined permanent magnet teeth.8,9 The end torque of the PMTM is mainly caused by the sudden change of magnetic induction at both ends of the central worm stator. In order to compare the magnitude of the cogging torque and the end torque, a new method of separating the cogging torque and the end torque is proposed in Ref. 10. It shows that the fluctuation amplitude of the end torque is much greater than that of the cogging torque. The end torque is mainly optimized by changing the mechanical structure. Aiming at the end torque fluctuation caused by the finite stator length of the permanent magnet synchronous linear motor, the optimization design method of adding auxiliary magnetic poles is adopted.11 For the arc-shaped permanent magnet synchronous motor, the stator length is optimized to adjust the phase difference between the left and right end-effect forces, so as to reduce the torque fluctuation.12 It can be seen from the above analysis that there are currently two main methods for optimizing the end torque. One is to optimize the length of the stator, but the end torque cannot be eliminated completely because of the high-order harmonic effect in the end torque. The other is to add the auxiliary magnetic pole, but the size and shape of the auxiliary magnetic pole need to be calculated by complicated calculation, and the magnetic pole with complex shape is difficult to manufacture.

This paper studies the reluctance torque of the PMTM, and the reluctance torque is separated into cogging torque and end torque for optimization analysis. First, the cogging torque is optimized by using the eccentric permanent magnet teeth. Second, it is proposed to add compensation windings at both ends of the central worm stator to optimize the end torque. The dynamic compensation effect of compensation winding and its influence on the air gap magnetic induction intensity are analyzed by finite element simulation. Finally, the influence of the combination of the two structural optimization methods on the reluctance torque and output torque of the motor is analyzed.

The basic structure of the permanent magnet toroidal motor is shown in Fig. 1. It is mainly composed of (1) central worm stator, (2) planet carrier rotor, (3) planet gears, and (4) outer stator. The central worm stator is provided with uniform spiral slots, which is used to place the three-phase winding to generate a rotating spiral magnetic field. The permanent magnet teeth are evenly distributed on the planet gear. The outer stator is composed of several special helical permanent magnet beams. The N pole and S pole are arranged alternately on the outer stator bracket to guide the planet gear. When the three-phase winding in the central worm stator is supplied with alternating current, a spiral magnetic field will be generated on its surface. The planet gear moves spirally around the central worm stator under the combined action of the spiral magnetic field generated by the central worm stator and the outer stator. The spiral motion of the planet gear can be regarded as rotation and revolution around the central worm stator axis. The planet gear drives the planet carrier rotor to rotate when it revolves, so as to realize the power output with deceleration.

FIG. 1.

The basic model of PMTM.

FIG. 1.

The basic model of PMTM.

Close modal

During the rotation of the planet gear, the cogging torque always exists. Only when the permanent magnet teeth of the planet gear enter and exit the wrap angle of the central worm stator, the air gap magnetic induction intensity at both ends of the central worm stator suddenly changes, which will cause the end torque. This torque fluctuation occurs during the rotation of the planet gear. Therefore, the following analysis of the reluctance torque mainly focuses on the rotation of the planet gear.

The central worm stator is disconnected at both ends, so there is a large ferromagnetic attraction between the stator core at the end position and the permanent magnet tooth, which causes the end torque fluctuation. The spiral groove of the central worm stator will cause the change of air gap permeance in the local area, which will lead to the fluctuation of the cogging torque. Therefore, the PMTM has a relatively large reluctance torque. The reluctance torque of the PMTM is analyzed, and the main structural parameters for finite element simulation are shown in Table I. The distribution of the magnetic field line between the planet gear and the central worm stator is shown in Fig. 2(a). The comparison result of the cogging torque and the end torque in the reluctance torque is shown in Fig. 2(b).

TABLE I.

Main parameters of PMTM simulation research.

SymbolParametersValuesUnit
R Radius of the planet gear 50 mm 
w Thickness of the planet gear 15 mm 
θ Wrap angle of the central worm stator 100 ° 
Br PM remanence 0.96 
δ Air gap length mm 
h Height of permanent magnet teeth mm 
a Center distance 120 mm 
αp Polar arc coefficient 0.68  
M Number of permanent magnet teeth  
Q Number of central worm stator slots 12  
SymbolParametersValuesUnit
R Radius of the planet gear 50 mm 
w Thickness of the planet gear 15 mm 
θ Wrap angle of the central worm stator 100 ° 
Br PM remanence 0.96 
δ Air gap length mm 
h Height of permanent magnet teeth mm 
a Center distance 120 mm 
αp Polar arc coefficient 0.68  
M Number of permanent magnet teeth  
Q Number of central worm stator slots 12  
FIG. 2.

Simulation model of the PMTM and analysis of reluctance torque: (a) distribution of the magnetic field line in the planet gear and the central worm stator and (b) distribution of the end torque and the cogging torque in the reluctance torque.

FIG. 2.

Simulation model of the PMTM and analysis of reluctance torque: (a) distribution of the magnetic field line in the planet gear and the central worm stator and (b) distribution of the end torque and the cogging torque in the reluctance torque.

Close modal

It can be seen from Fig. 2(a) that the magnetic field lines are dense within the wrap angle range of the central worm stator and the magnetic field lines outside the wrap angle range are sparse. It can be found from Fig. 2(b) that the end torque is the main part of the reluctance torque. The end torque is a sine wave, but the reluctance torque is a non-sine wave because of the influence of the cogging torque. The sag in the peak part of the reluctance torque is caused by the inconsistency of the fluctuation period between the end torque and the cogging torque.

The better the sinusoidal of the reluctance torque fluctuation and the larger the proportion of the fundamental wave, the simpler and more effective the corresponding optimization strategy. It can be seen from the analysis in Fig. 2(b) that the cogging torque is the main factor affecting the non-sinusoidal variation in the reluctance torque, so the optimization analysis of the cogging torque is carried out first. After reducing the influence of the cogging torque, the end torque is analyzed to achieve the maximum optimization effect on the reluctance torque.

The position relationship between the spiral groove on the central worm stator and the planet gear is shown in Fig. 3.

FIG. 3.

Position relationship between the helical groove of the central worm stator and the planet gear.

FIG. 3.

Position relationship between the helical groove of the central worm stator and the planet gear.

Close modal

In Fig. 3, M is a point on the spiral slot, β is the angle between the horizontal plane and the normal plane passing through the M point, φ is the permanent magnet tooth width, and τ is the polar distance of the planet gear. In the process of the planet gear rotation, the spiral slot corresponding to the permanent magnetic tooth can be regarded as the skewed slot to a certain extent. Then, the cogging torque of the planet gear can be expressed as

(1)

where

(2)

where α is the angle between the centerline of the permanent magnet teeth and the throat position of the central worm stator, b0 is the slot width of the central worm stator, and p is the number of pole pairs of the PMTM.

It can be seen from Eq. (1) that the cogging torque can be weakened by reducing Brn. The method of changing the shape of permanent magnet teeth is adopted to reduce Brn, so as to restrain the torque ripple. After the pole arc coefficient and height of the permanent magnet teeth are determined, the change in the reluctance torque is explored by changing the cutting thickness of the permanent magnet teeth. The structure model of the eccentric permanent magnet tooth is shown in Fig. 4.

FIG. 4.

The structure of the eccentric permanent magnet tooth.

FIG. 4.

The structure of the eccentric permanent magnet tooth.

Close modal

In Fig. 4, h is the cutting thickness, O1 is the center of the planet gear and O2 is the center of the outer circle of the permanent magnet tooth, R1 is the radius of the planet gear and R2 is the radius of the outer circle of the permanent magnet tooth, respectively, and γ is the angle between O1C and the centerline of the permanent magnet tooth. Thus, there is

(3)

The expression of the air gap magnetic induction intensity with pole cutting of the permanent magnet teeth is obtained as follows:

(4)

where the value range of γ is [−παp/2p, παp/2p] and g is the distance from the central worm stator core to the planet gear core.

According to the analytical model established by Eq. (4), the comparison of the air gap magnetic induction intensity of the permanent magnet teeth under different eccentricity is shown in Fig. 5.

FIG. 5.

Air gap magnetic induction intensity of the permanent magnet teeth with different eccentricities.

FIG. 5.

Air gap magnetic induction intensity of the permanent magnet teeth with different eccentricities.

Close modal

It can be seen from Fig. 5 that the air gap magnetic induction intensity becomes more and more close to the sinusoidal distribution with the increase in the eccentricity. However, when the eccentricity of the permanent magnet teeth is too large, the structural strength of the permanent magnet teeth and other electromagnetic characteristics of the motor will be affected. Therefore, the changes in the reluctance torque with different eccentricity permanent magnet teeth are presented to compare in Fig. 6.

FIG. 6.

Comparison of the reluctance torque of the permanent magnet teeth with different eccentricities.

FIG. 6.

Comparison of the reluctance torque of the permanent magnet teeth with different eccentricities.

Close modal

It can be seen from Fig. 6 that the structure of the eccentric permanent magnetic teeth has a certain inhibition effect on the reluctance torque. At hs = 10 mm and hs = 20 mm, the reluctance torque is concave in the peak part. The waveform of the reluctance torque is a sine wave at hs = 30 mm and hs = 35 mm. It shows that the cogging torque in the reluctance torque can be well optimized at hs = 30 mm and hs = 35 mm. When the eccentricity of the permanent magnet teeth is too large, both ends of the permanent magnet teeth will be too thin. Therefore, the best eccentricity hs = 30 mm is selected.

After the permanent magnet teeth are selected with better pole arc coefficient, height, and eccentricity, the three-dimensional magnetic field model of the planet gear is shown in Fig. 7(a). The change in the end torque and cogging torque in the reluctance torque after the optimization of the permanent magnet teeth is shown in Fig. 7(b).

FIG. 7.

Optimization model of the permanent magnet teeth and analysis of the reluctance torque: (a) optimized model of the permanent magnet teeth and (b) comparison of the reluctance torque after optimization of the permanent magnet teeth.

FIG. 7.

Optimization model of the permanent magnet teeth and analysis of the reluctance torque: (a) optimized model of the permanent magnet teeth and (b) comparison of the reluctance torque after optimization of the permanent magnet teeth.

Close modal

It can be seen from Fig. 7(b) that the amplitude of the optimized reluctance torque fluctuation is about 0.35 N·m. The amplitude of the reluctance torque fluctuation before the optimization of the permanent magnet teeth is about 0.6 N·m. The fluctuation amplitude of the reluctance torque is reduced by about 40%. By optimizing the shape of the permanent magnet teeth structure, the fluctuation amplitude of the cogging torque basically tends to zero. The fluctuation range of the end torque and the reluctance torque are similar.

When the permanent magnet teeth of the planet gear enter or leave the wrap angle of the central worm stator, the sudden change in magnetic induction intensity occurs at both ends of the central worm stator, which leads to the end torque. The fluctuation period of the end torque is a pole pitch of the planet gear, which has nothing to do with the polarity of the permanent magnet teeth. This means that a dynamic compensation is needed to optimize the end torque. Therefore, the optimization method of adding compensation windings at both ends of the central worm stator is proposed. The compensation winding is supplied with alternating current to induce the corresponding N pole and S pole on the iron cores at both ends of the central worm stator and form magnetic coupling with the permanent magnetic teeth of the planet gear. This can reduce the sudden change in the magnetic induction intensity at the two ends of the central worm stator during the operation of the PMTM and achieve the optimization effect of the end torque. The distribution of magnetic field lines between the planet gear and the central worm stator when the compensation winding is added to both ends of the central worm stator is shown in Fig. 8(a). The three-dimensional magnetic field model of the compensation winding and three-phase winding is shown in Fig. 8(b).

FIG. 8.

Finite element model of PMTM with compensation winding: (a) distribution of magnetic field lines with compensation winding and (b) magnetic field model of compensation winding and three phase winding.

FIG. 8.

Finite element model of PMTM with compensation winding: (a) distribution of magnetic field lines with compensation winding and (b) magnetic field model of compensation winding and three phase winding.

Close modal

It can be seen from Fig. 8(a) that the magnetic field is induced by the compensating windings at both ends of the central worm stator and forms magnetic coupling with the permanent magnet teeth of the planet gear at the end position. The magnetic permeability of the central worm stator is large. Therefore, a part of the magnetic field generated by the compensation windings at both ends forms magnetic coupling with the permanent magnet teeth in the wrap angle range through the central worm stator. It can be seen from Fig. 8(b) that the three-phase winding is spirally distributed in the middle position and the compensation windings is evenly distributed at the left and right ends. The magnetic coupling mode of the spiral magnetic field generated by three-phase winding and the planet gear is different in different positions, so the planet gear in different positions corresponds to different compensation windings. The 1–4 in Fig. 8(b) show different compensation windings.

The dynamic compensation effect of the compensation winding can be expressed by the vector flux density. The vector magnetic density distribution of the planet gear and the compensation winding at different times is shown in Fig. 9, where T represents the time taken for the planet gear to rotate one cycle.

FIG. 9.

Vector magnetic density distribution of the planet gear and the compensation windings at different times: (a) t = 0, (b) t = T/32, (c) t = T/16, and (d) t = 3T/32.

FIG. 9.

Vector magnetic density distribution of the planet gear and the compensation windings at different times: (a) t = 0, (b) t = T/32, (c) t = T/16, and (d) t = 3T/32.

Close modal

In Fig. 9(a), the N-pole permanent magnet teeth at both ends of the central worm stator form magnetic coupling with the S-pole induced by the compensation windings. In Fig. 9(b), the left end of the central worm stator is between two permanent magnet teeth. At this position, the magnetic leakage of the planet gear at the left end is small, so the amplitude of the compensation current at the left end is 0. The N-pole permanent magnet tooth at the right end of the central worm stator forms magnetic coupling with the S-pole induced by the compensation windings. In Fig. 9(c), the S-pole permanent magnet tooth at the left end of the central worm stator is about to enter the wrap angle, and the N-pole permanent magnet tooth at the right end will be completely separated from the central worm stator. The corresponding N-pole and S-pole are induced by the compensation windings at both ends to form magnetic coupling with them. In Fig. 9(d), the S-pole permanent magnet tooth at the left end of the central worm stator moves into the wrap angle, and the N-pole induced by the compensation windings forms a magnetic coupling with it. The right end of the central worm stator is between the two permanent magnet teeth of the planet gear, so the current amplitude of the compensation winding is 0. The current variation law of the compensation winding is consistent with the motion state of the planet gear, which plays an auxiliary role in planet gear movement and reduces the influence of the end torque.

The existence of the compensation windings will generate an additional magnetic field to compensate for the sudden change in the magnetic field of the planet gear at both ends of the central worm stator. The three-dimensional model of air gap magnetic induction distribution with and without compensation windings is shown in Figs. 10(a) and 10(b).

FIG. 10.

Comparison of three-dimensional air gap magnetic field with and without compensation winding: (a) three-dimensional distribution of the magnetic induction intensity of uncompensated windings and (b) three-dimensional distribution of the magnetic induction intensity with compensation windings.

FIG. 10.

Comparison of three-dimensional air gap magnetic field with and without compensation winding: (a) three-dimensional distribution of the magnetic induction intensity of uncompensated windings and (b) three-dimensional distribution of the magnetic induction intensity with compensation windings.

Close modal

Figure 10(a) shows the three-dimensional distribution of the magnetic induction intensity of the outer air gap at 0.5 mm of the planet gear when there are no compensation windings. The permanent magnet teeth are completely magnetically coupled with the central worm stator core in the wrap angle range, so the air gap magnetic induction intensity is the highest, corresponding to the area A in Fig. 10(a). One part of the permanent magnet teeth at both ends of the central worm stator is within the wrap angle range, and the other part is outside the wrap angle range. Therefore, a sudden change in the magnetic induction intensity will occur at the end of the central worm stator, which corresponds to the cliff-like descending area B in Fig. 10(a). This is also the reason for the end torque. The air gap magnetic induction intensity of the permanent magnet teeth outside the wrap angle range is smaller, corresponding to the relatively flat C area on both sides of Fig. 10(a). Figure 10(b) shows the three-dimensional distribution of the magnetic induction intensity of the outer air gap at 0.5 mm of the planet gear with compensation windings. In Fig. 10(b), the magnetic induction intensity is not a cliff-like decline, but there are many semi-conical gentle slopes. These semi-conical gentle slopes are shown in area D in Fig. 10(b). It is the existence of these gentle slopes that reduces the sudden change in the magnetic induction intensity of the planet gear at the end of the central worm stator, which can fundamentally reduce the fluctuation of the end torque.

The distribution of the air gap magnetic induction intensity at different times with and without compensation windings is shown in Figs. 11(a) and 11(b).

FIG. 11.

Comparison of the air gap magnetic field with and without compensation windings: (a) distribution of the magnetic induction intensity at different times without compensation windings and (b) distribution of the magnetic induction intensity at different times with compensation windings.

FIG. 11.

Comparison of the air gap magnetic field with and without compensation windings: (a) distribution of the magnetic induction intensity at different times without compensation windings and (b) distribution of the magnetic induction intensity at different times with compensation windings.

Close modal

It can be seen from Fig. 11(a) that the magnetic induction intensity at any time is linearly decreased at both ends when there is no compensation winding. It can be found from Fig. 11(b) that there is a buffer when the magnetic induction intensity at any time decreases at both ends when there are compensation windings. The compensation effect of the compensation winding at different times can be obtained by comparing Fig. 11(b) with Fig. 9. When t = 0, the compensation winding has a compensation effect at both ends of the central worm stator, corresponding to the blue solid line in Fig. 11(b). It can be seen that the magnetic induction intensity at both ends of the central worm stator has been compensated and the magnitude is equal. When t = T/32, the compensation windings only have the compensation effect at the right end of the central worm stator, corresponding to the green dotted line in Fig. 11(b). It can be seen that the magnetic induction intensity of the right end of the central worm stator is higher than that of the left end. When t = T/16, the compensation windings have a compensation effect at both ends of the central worm stator. The permanent magnet teeth of the planet gear are at a certain distance from the both ends of the central worm stator, so the magnetic induction intensity on both ends is small, corresponding to the black double dashed line in Fig. 11(b). When t = 3T/32, the compensation windings only have the compensation effect at the left end of the central worm stator, so the magnetic induction intensity at the left end of the central worm stator is higher, corresponding to the red chain line in Fig. 11(b).

The current of the compensation windings at the left and right ends of the central worm stator can be expressed as

(5)

where Im is the current amplitude, ω is the angular velocity of the planet gear rotation, z is a positive integer and satisfies θ > 0,(z − 1)τθ < 0, and k is the correction coefficient.

The variation in reluctance torque is shown in Fig. 12 when the current with different amplitudes is applied into the compensation windings.

FIG. 12.

Variation in reluctance torque with the compensation current.

FIG. 12.

Variation in reluctance torque with the compensation current.

Close modal

It can be seen from Fig. 12 that the amplitude of the reluctance torque is about 0.33 N m when there is no current in the compensation windings. The amplitude of the reluctance torque is obviously reduced as the current is applied to the compensation windings. From the initial moment, the reluctance torque first fluctuates downward and then upward when Im = 0 ∼ 6 A. The reluctance torque first fluctuates upward and then downward when Im = 9 A. This shows that the compensation is oversaturated when Im = 9 A, and the fluctuation at this time is the torque fluctuation caused by the compensation windings. The amplitude of the reluctance torque is about 0.02 N m when Im = 6 A. It can be seen that the amplitude of the reluctance torque fluctuation decreases by 94% when the appropriate current is applied into the compensation windings. This result shows that the compensation windings have an obvious optimization effect on the fluctuation of the reluctance torque.

Based on the above analysis, it can be found that the ripple of the cogging torque can be reduced by optimizing the structure of permanent magnet teeth, and the influence of the end torque can be reduced by adding compensation windings at both ends of the central worm stator. These methods have a good optimization effect on the fluctuation of the reluctance torque. In order to verify whether the compensation winding and the three-phase winding affect each other when they are energized, the output torque under different states needs to be compared.

The finite element simulation model and three-dimensional structure model with adding compensation windings and magnetic isolation materials at both ends of the central worm stator are shown in Fig. 13(a). It can be seen that the magnetic field distribution between the compensation winding and the three-phase winding is separated by the magnetic isolation material, which avoids the mutual influence. When there are compensation windings and magnetic isolation materials at both ends of the central worm stator, the influence on the output torque is shown in Fig. 13(b). It can be seen from Fig. 13(b) that the existence of the compensation winding has no effect on the output torque when there is no magnetic isolation materials. In both cases, the output torque fluctuates between 8.5 N m and 11.8 N m. When both the compensation winding and the magnetic isolation material exist, the fluctuation amplitude of the output torque is significantly reduced. In this case, the output torque fluctuates between 8.5 N m and 10.6 N m, and the fluctuation amplitude is reduced by about 36%. This shows that there is interaction between the compensation winding and the three-phase winding, so the magnetic isolation material must be added.

FIG. 13.

Analysis of the influence of compensation winding on three-phase winding: (a) the model of PMTM with the magnetic isolation material and (b) influence of compensation winding and magnetic isolation material on the output torque.

FIG. 13.

Analysis of the influence of compensation winding on three-phase winding: (a) the model of PMTM with the magnetic isolation material and (b) influence of compensation winding and magnetic isolation material on the output torque.

Close modal

The cogging torque and the end torque of the permanent magnet toroidal motor are studied in this paper. The influence of the cogging torque can be greatly reduced by using the eccentric permanent magnet teeth structure. The end torque is the main part of the reluctance torque after the cogging torque is optimized. It is proposed to add compensation windings at both ends of the central worm stator to solve the problem of torque fluctuation at the end. The dynamic compensation effect, the magnetic field distribution, and the optimization degree of the reluctance torque are analyzed through the finite element simulation after the compensation winding is added to the central worm stator. The relationship between the current in the compensation winding and the structural parameters of the PMTM is derived. Finally, it is verified that the compensation winding can significantly reduce the output torque fluctuation of the PMTM after adding the magnetic isolation material. The results show that the reluctance torque fluctuation is reduced by 94% and the output torque fluctuation is reduced by 36% after optimizing the shape of the permanent magnet teeth and adding compensation windings at both ends of the central worm stator.

This work was supported by the National Natural Science Foundation of China (Grant No. 51875408).

The data that support the findings of this study are available from the corresponding author upon reasonable request.

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