Torque Performance Enhancement of Flux-Switching Permanent Magnet Machines With Dual Sets of Magnet Arrangements

Torque performance, especially torque density, is a critical performance index for the flux-switching permanent magnet (FSPM) machine that is attractive for the propulsion system. In this article, a novel FSPM machine with dual sets of magnet arrangements is proposed. With the novel topology, the torque density of the proposed machine is significantly improved due to much increased working harmonic contents of magnetomotive force (MMF). Moreover, the cogging torque is also inherently reduced, which makes the proposed machine a promising candidate in the FSPM machine family. The operating principle of the proposed FSPM machine is revealed based on the MMF–permeance model and the numerical finite element analysis (FEA). The effect of geometric parameters, such as magnet thickness, auxiliary tooth width, and rotor tooth width on the average torque and cogging torque, is also investigated. Finally, a prototype has been manufactured to validate the analysis conclusion. With experimental test results, it is demonstrated that the proposed topology can achieve 30.8% higher torque density, 79.4% lower cogging torque, and 15.6% higher power factor than the conventional counterpart.


I. INTRODUCTION
I N RECENT decades, permanent magnet (PM) machines have been widely used in various industrial applications, such as ship propulsion, aerospace equipment, and electric vehicles. The flux-switching PM (FSPM) machine did not draw much attention when first proposed in [1] due to its relatively poor electromagnetic performance and immature design methods. With the extension of the electric machine theory and the development of high-performance magnetic materials, the FSPM machine has received much increased attention Manuscript  due to its simple rotor structure and fault-tolerance capability [2]- [8]. Besides, the stator PM topology benefits thermal management. FSPM machines have become attractive candidates for low-speed high-torque applications [9], [10], while their key performance index still needs to be further improved in terms of torque density and cogging torque level [11], [12], which has become research hotspots in recent years.
To improve torque density, many experts have developed and investigated various high-performance FSPM machines. Chen et al. [13] proposed a novel E-core FSPM machine that could exhibit 15% higher torque capability than a conventional 12/10 stator/rotor pole machine. Moreover, the C-core FSPM topology [14] was proposed, and it was demonstrated that 40% higher torque density could be realized compared with the conventional 12/10 stator/rotor pole FSPM machine. Shao et al. [15] investigated a novel FSPM machine with overlapping windings, and the machine could provide 17.6% higher torque density than conventional 12/10 stator/rotor pole nonoverlapping winding machine. To increase utilization of armature slot, Hua et al. [16] proposed an outer-rotor FSPM machine with wedge-shaped magnets, which could produce 40% higher torque than a rectangular-shaped magnet-based machine. Besides, consequent-pole FSPM machines with flux bridges in the stator core [17] and sandwiched FSPM machines using V-shaped magnets [18] were introduced to increase the PM utilization ratio. The multitooth FSPM machines were investigated for the first time in [19], and the optimal slot/pole combinations were discussed to exhibit high-torque capabilities in [20]. It is shown that the multitooth FSPM machine can exhibit higher torque density than the conventional 12/10 stator/rotor pole machine. The multitooth topology provides the feasibility of dual sets of magnet arrangements to further increase the torque density.
In this article, a novel topology of an FSPM machine with dual sets of magnet arrangements is introduced. With the novel topology, the torque density of the proposed machine is significantly improved due to much increased working harmonic contents of magnetomotive force (MMF). The operating principle of the proposed FSPM machine is revealed based on MMF-permeance model and numerical finite element analysis (FEA) in Section II. In Section III, the electromagnetic performance is compared with the conventional counterpart, and it is demonstrated that the proposed topology can achieve 30.8% higher torque density, 79.4% lower cogging torque, and 15.6% higher power factor. Then, Section IV will be devoted to the analysis of the influence of geometric parameters on torque performance. Furthermore, a prototype has been manufactured to verify the analysis results in Section V. Finally, some conclusions will be drawn in Section VI.

II. MACHINE TOPOLOGY AND OPERATING PRINCIPLE
In this section, the operating principle of the proposed FSPM machine with dual sets of magnet arrangements is analyzed, and some expressions are derived to investigate the contributions of the working harmonics. For simplicity, some assumptions are made as follows.
1) The symmetrical three-phase sinusoidal current sources are applied to drive the FSPM machines.
2) The axis of phase A, the initial position of the rotor tooth axis, and the origin of the coordinates are in alignment.

A. Machine Topology
To obtain high torque density, the stator auxiliary tooth-slot structure is adopted to improve the modulation effect [19], [21]. The conventional multitooth and proposed FSPM machines are depicted in Fig. 1(a) and (b), respectively. Different from the conventional counterpart, there are dual sets of magnet arrangements in the proposed topology, in which the yellow arrows represent the magnetization direction of the magnets. Besides, the proposed machine can be seen as the combination of machine I and machine II, as shown in Fig. 1(c) and (d), respectively. The sketch part in the red circle in Fig. 1(b) and the simplified equivalent MMF function neglecting the magnetic reluctance of the iron core are shown in Fig. 2, where t s is the spoke-type magnet thickness, w o is the outer auxiliary tooth width, w i is the inner auxiliary tooth width, d a is the auxiliary slot depth, and w r is the rotor tooth width. Due to the magnets in the auxiliary stator slots, there are differences between the simplified equivalent MMF functions of conventional multitooth and proposed FSPM machines. The different parts are marked as the red line. F 1 and F 2 are the amplitudes of the MMF drop in the air gap under the auxiliary teeth and slots, respectively.

B. Operating Principle
The no-load air-gap flux density in FSPM machines can be given as [22] where θ is the angle along the circumference in the air gap, g is the air-gap length, μ 0 is the permeability of air, F m (θ ) is the PM MMF, s (θ ) is the stator permeance with a slotted stator and a smoothed rotor, r (θ , t) is the rotor permeance with the slotted rotor and the smoothed stator, and F e (θ ) is the equivalent MMF after being modulated by the stator auxiliary teeth. The MMF, which refers to the equivalent MMF in this article, can be expressed as where F en is the amplitude of the nth-order harmonic and N s is the number of stator slots. The rotor permeance function can be written as where ω is the electrical angular velocity, N r is the number of the rotor slots, 0 is the constant permeance, and m is the amplitude of the mth-order harmonic.
Thus, the no-load air-gap flux density can be obtained as Considering the flux linkages interacting with phase A, the back EMF of phase A can be given as where r g is the radius of the stator bore and l s is the stack length. The winding function N(θ ) can be expressed as [23] where N c is the number of series turns per phase and k w j is the winding factor of the j -pole-pair flux density harmonic. From expression (5), the pole pair number of the no-load airgap flux density and armature field should be equal to produce back EMF. Thus, the following equation should be satisfied: In the concentrated-winding machine with six stator slots, the winding factors of the odd-pole-pair flux density harmonics are zero. Therefore, only the even-pole-pair flux density harmonics need to be considered when calculating the back EMF. The constant term of rotor permeance function can be ignored when analyzing the back EMF since only the rotating magnetic field can produce the back EMF. Based on the above analysis, the variable m can only be odd.
For symmetrical three-phase FSPM machines, the righthand side of (7) is a multiple of 3 when m is 3. However, the flux density harmonics, whose pole pair numbers are multiples of 3, are nonworking harmonics because of the zero-winding factors. When m is larger than or equal to 5, the corresponding amplitude of the permeance function is small enough to be neglected. Thus, only the fundamental permeance harmonic, i.e., m = 1, is taken into account when calculating the back EMF and electromagnetic torque.
Only considering main working flux density harmonics that contribute to back EMF, the expression (4) can be modified as Then, the back EMF of phase A can be expressed in (9), as shown at the bottom of the next page.
The characteristics of the main working harmonic are illustrated in Table I. The electromagnetic torque with i d = 0 can be given in (10), as shown at the bottom of the next page. where E A is the root-mean-square (rms) value of the phase back EMF and I A is the rms value of the phase current. Fig. 3 shows the comparison of FEA-predicted MMF distributions of conventional multitooth and proposed FSPM machines. This figure is shown based on the per-unit system, and the amplitude of the first-order MMF harmonic of conventional one is set as the base value. Fig. 4 shows FFT analysis results of MMF distributions of conventional multitooth and proposed machines. It is noted that the first-order MMF harmonic reduces significantly, while fifth-and seventh-order harmonics show a marked increase compared with the conventional one. To research the nature of the proposed topology, the comparison of FEA-predicted MMF distributions of machines I and II is shown in Fig. 5. Considering that the permeability of the iron core is nonlinear,   the frozen permeability method [24] has been taken to obtain accurate results. Fig. 6 shows the FFT analysis results of MMF distributions of machines I and II. The first-order MMF harmonic  of machine I is negative, while fifth-and seventh-order MMF harmonics are all positive. Thus, the reduction of first-order MMF harmonic is due to the negative contribution of machine I and decreased positive contribution of machine II. Besides, the different signs of third-and 11th-order MMF harmonics contribute to the decrease in the corresponding MMF harmonics.
The no-load air-gap flux density distributions at t = 0 are shown in Fig. 7, and the corresponding FFT analysis results are given in Fig. 8. It should be noted that the three/nine/15/ 21-pole-pair flux density harmonic originates from corresponding first-/third-/fifth-/seventh-order MMF harmonics, which is modulated by the constant term of rotor permeance. It is shown that the three-pole-pair flux density harmonic accounts for a large proportion and does not make contributions to back EMF due to the zero rotating speed. The amplitude of the three-pole-pair flux density harmonic decreases by 32.1%, and the 16/22-pole-pair flux density harmonic decreases at the same time. As shown in Table I, the two-polepair and three-pole-pair flux density harmonics are modulated from seventh-and fifth-order MMF harmonics, respectively. It is noted that the three-pole-pair flux density harmonic is modulated from the first-order MMF harmonic, i.e., the fundamental MMF harmonic. It can be seen from Fig. 2(b) that the red lines can weaken the fundamental MMF harmonic represented by the black lines. The enhancement of two-polepair and three-pole-pair flux density harmonics indicates the increase in seventh-and fifth-order MMF harmonics. Thus, the three-pole-pair flux density harmonic is reduced, while the 15-pole-pair and 21-pole-pair flux density harmonics increase. Then, the decreased saturation degree can be realized by the reduction of three-pole-pair flux density harmonic in the proposed FSPM machines, as shown in Fig. 9.
In conventional multitooth FSPM machines, nonworking harmonics account for a large proportion and cannot make contributions to the back EMF and torque. Due to the saturation limitation, the main working harmonics cannot be improved by further increasing the magnet usage. As a comparison, the novel FSPM machine with dual sets of magnet arrangements is proposed.
In order to validate the foregoing analysis, the back EMF and torque contributions of the main working harmonics are shown in Table II. It can be observed that the two-pole-pair and three-pole-pair flux density harmonics exhibit dominant back EMF and torque contributions. The total calculated and FEA-predicted back EMFs are 66.45 and 64.57 V, respectively, for the proposed FSPM machine.

III. ELECTROMAGNETIC PERFORMANCE
In this section, the electromagnetic performance of the proposed FSPM machine, including the back EMF, the cogging torque, the electromagnetic torque, and the power factor, will be compared with the conventional multitooth FSPM machine. To realize a fair comparison, the stator outer/inner diameter, the air-gap length, the stack length, and the current density  Table III. Fig. 10 shows the FEA-predicated phase back EMF waveforms and FFT analysis results. The amplitude of the phase  back EMF fundamental harmonic increases by 25.7% compared with the conventional multitooth FSPM machine. The calculated THD values in conventional multitooth and proposed machines are 1.5% and 1.0%, respectively. Therefore, the harmonic contents of the proposed FSPM machine are lower. The back EMF is less affected by rotor permeance higher harmonics. Hence, it is reasonable to neglect the rotor permeance harmonics in the previous analysis. Fig. 11 shows the FEA-predicted cogging torque waveforms of the conventional multitooth and proposed FSPM machines. The amplitudes of the cogging torque are 0.68 and 0.14 Nm, respectively. The cogging torque of the new topology decreases by 79.4% compared with the conventional one.

C. Electromagnetic Torque
The current density at the rated load is selected as 5 A/mm 2 , so the corresponding phase current is 7.9 A. Fig. 12 shows the FEA-predicted torque waveforms. The average torques of the conventional multitooth and proposed FSPM machines are 17.55 and 22.96 Nm, respectively; thus, the average torque increases by 30.8%. Thus, the torques per magnet volume values are 477 and 544 Nm/L, respectively. The torque ripples are 9.7% and 3.2%, respectively; therefore, the torque ripple decreases by 67.0%. The lower torque ripple is mainly because of the lower cogging torque. Fig. 13 shows the FEA-predicted average torque versus current density curves. It can be seen from the figure that the curve of the proposed FSPM machine has better linearity   than the conventional multitooth FSPM machine. It is mainly because the saturation degree of the iron core decreases by adopting the novel topology, as shown in Fig. 14.

D. Power Factor
The power factor of the FSPM machines can be given as where L s is the synchronous inductance. The phase synchronous inductances of the conventional multitooth and proposed FSPM machines are calculated as 11.9 and 12.2 mH, respectively, by adopting the frozen permeability method. Fig. 15 shows the FEA-predicted power factor versus current density curves. The power factor of the proposed FSPM machines is higher than the conventional one and increases by 15.6% at the rated current. The reason is that the phase back  EMF of the proposed FSPM machine is much greater than the conventional multitooth FSPM machine, while the inductances of the two topologies are very close.

E. Demagnetization Examination
In order to confirm the feasibility of PM thickness of the proposed FSPM machine with dual sets of magnet arrangements, demagnetization withstanding capability is examined. Fig. 16 shows the contour plots under two times overload operation, and the PM material and the working temperature are set as N40EH and 120 • C, respectively. It is observed that only a very small part of the spoke-type magnet is demagnetized since the reference knee flux density is −0.5 T.

F. Loss and Efficiency
The loss and efficiency of the conventional multitooth and proposed FSPM machines are listed in Table IV. It can be observed that the total losses of the conventional multitooth and proposed FSPM machines are comparable. However, the proposed topology has higher efficiency due to the higher output torque.

IV. DESIGN OPTIMIZATION AND INFLUENCE OF GEOMETRIC PARAMETERS
In this section, the proposed FSPM machine with dual sets of magnet arrangements will be optimized by adopting a multiobjective genetic algorithm. Based on the global optimization design, the influence of geometric parameters is investigated to reveal some design guidelines for the proposed FSPM topology.

A. Design Optimization
As shown in Fig. 17, a multiparameter optimization adopting the multiobjective genetic algorithm has been added to explain the parameter selection. It can be seen that the parameters are optimal. The current density at the rated load is 5 A/mm 2 , and the slot fill factor is 0.48. The main design parameters and the corresponding scopes are listed in Table V. The global optimization goal is to maximize the average torque and minimize the torque ripple.
It can be seen that the optimal case is selected when the average torque and torque ripple can be well balanced. The corresponding geometric parameters are selected to design the proposed FSPM machine.

B. Influences of Magnet Thickness
The magnet thickness, which is critical to the cost, has a significant influence on the performance of the proposed FSPM machines with dual sets of magnet arrangements. Figs. 18(a) and 19(a) show the influence of spoke-type and surface-mounted magnet thickness on average torque.  The influence of magnet thickness on average torque is analyzed separately with the magnet of another type fixed. The surface-mounted magnet thickness mainly influences the saturation degree of the stator core. It is found that the average torque is very sensitive to the position of the auxiliary tooth axis due to the modulation effect. Therefore, the auxiliary tooth axis position remains unchanged to possibly only research the impact of magnet thickness.
With the increase in the spoke-type magnet thickness, average torque first increases mainly due to the increase in flux density and enhancement of modulation effect. The average torque reaches the maximum value when the spoke-type magnet thickness is 3 mm. Then, the average torque decreases despite the increase in the magnet thickness, and this is mainly due to the saturated stator core and the weakened modulation effect.
Figs. 18(b) and 19(b) present the cogging torque versus the magnet thickness curves. Compared with the surface-mounted magnet, the cogging torque is more sensitive to the change of the spoke-type magnet thickness. It is mainly because the position of the inner auxiliary tooth changes with the increase in the spoke-type magnet thickness.
The cogging torque is very sensitive to the position of the auxiliary tooth axis due to the modulation effect. The influence of spoke-type magnet thickness on the cogging torque is analyzed with the surface-mounted magnet width changed to keep the axis of the auxiliary tooth still. When the spoke-type magnet thickness reaches 3.6 mm and continues to increase, the width of the inner auxiliary tooth decreases by twice the increase in the magnet thickness. When the spoke-type magnet thickness is small, the cogging torque is low due to the small MMF.   To obtain the high torque density and low cogging torque, the thicknesses of the spoke-type and surface-mounted magnet are 3 and 2 mm.

C. Influences of Auxiliary/Rotor Tooth Width
Figs. 20(a)-22(a) show the average torque versus the tooth width curves. Since auxiliary teeth and magnets are in contact, it is impossible to analyze the impact of a single parameter. To provide the main parameters, which affects the electromagnetic performance most, the influence of inner/outer auxiliary tooth width is analyzed with the surface-mounted magnet width changed and other parameters unchanged. The average torques all increase first, then reach the maximum values, and finally decrease.
Figs. 20(b)-22(b) present the cogging torque versus the tooth width curves. It is well known that the tooth width has a direct influence on the cogging torque. Compared with the curves in Figs. 18(b) and 19(b), the cogging torque is much more sensitive to the tooth width. The sensitivity of cogging torque mainly depends on the width and axis of auxiliary teeth.  Fig. 18, the width and axis of auxiliary teeth are both changed because the auxiliary teeth and magnets are in contact. Hence, cogging torque is more sensitive to the changes of inner and outer auxiliary tooth width. Thus, the high torque density and low cogging torque can be realized by choosing the appropriate tooth width.
To achieve the best electromagnetic performance of the proposed FSPM machine with dual sets of magnet arrangements, the widths of the inner auxiliary tooth, outer auxiliary tooth, and rotor tooth are 4.8, 5.3, and 4.4 mm.

D. Investigation of Different Stator/Rotor Combinations
The variation of torque performance of main stator/rotor combinations is shown in Table VI.
The ratio of rotor pole-pair number and the winding polepair number is defined as the pole ratio of FSPM machines. In general, the higher the pole ratio, the higher the torque density. For instance, the torque capability of 6/24/19 is greater than the counterpart of 6/24/17. However, it is noted that the torque density of 6/36/31 is lower than the counterpart of 6/24/19. The reason is that the spoke-type magnet is thinner with the increase in the auxiliary tooth number to maintain the modulation capability of the stator teeth. More basically, the PM excitation capability and modulation capability of the stator teeth should be balanced in some stator/rotor combinations. It can be seen that the average torque of 12/72/59 is much smaller than the counterpart of 6/24/19. Thus, the stator/rotor combinations of large auxiliary tooth numbers are not suitable for machines of small size.

V. PROTOTYPE AND EXPERIMENTS
In order to verify the foregoing analysis, a prototype of the proposed FSPM machine has been manufactured. The main parts of the prototype are shown in Fig. 23. The on-load experimental test bench for the prototype is illustrated in Fig. 24. Fig. 25 shows the cogging torque test platform for the prototype. Fig. 26 shows the FEA-predicted and tested line back EMF waveforms. The errors between the FEA-predicted and measured back EMF are mainly due to the end-effects and manufacturing tolerances. The FEA-predicted and measured   torque waveforms are shown in Fig. 27. The period of the torque ripple is influenced by the cogging torque, as shown in Fig. 31. Fig. 28 presents the average torque versus the current density curves predicted by FEA and measured by experiments. It is shown that the errors increase with the increase in the current, which causes the temperature to rise. It is noted that the measurements are slightly lower than the FEA predictions, which is mainly because the manufacture tolerance and end-effect are not considered in the simulation. Fig. 29 shows the power factor versus current density curves measured by experiments and predicted by FEA. Furthermore, the experimental waveforms are presented in Fig. 30,  Fig. 26. FEA-predicted and tested line back EMF waveforms.   including i d and i q . The measured cogging torque waveform is shown in Fig. 31. It is noted that the period of the tested cogging torque is very close to 18.95 mechanical  degrees, which is the period of rotor permeance. The additional cogging torque components are caused by stator tolerances. The manufacturing tolerances can be tolerance of PMs and tolerance of stator core. To verify the possible manufacturing tolerance, 4% is used for the remanence tolerance [25], i.e., the remanence of magnets under one stator main tooth is higher than other magnets. The cogging torque waveforms predicted by FEA and measured by experiments match well. Thus, the period of the cogging torque of the prototype may change due to the manufacturing tolerance. Nevertheless, the amplitude of the tested cogging torque is only 0.32 Nm.

VI. CONCLUSION
In this article, a novel FSPM machine with dual sets of magnet arrangements is proposed. Compared with the conventional multitooth FSPM machine, the torque density of the proposed machine is significantly improved due to much increased working harmonic contents of MMF and significantly reduced nonworking harmonics to maintain comparable saturation degrees. Furthermore, the cogging torque is also inherently reduced. The operating principle of the proposed topology is revealed based on MMF-permeance model and numerical FEA. The effect of geometric parameters, such as magnet thickness, auxiliary tooth width, and rotor tooth width on the average torque and cogging torque, is also investigated. A prototype has been manufactured to validate the analysis conclusion. With only a 14.7% increase in the magnet usage, it is demonstrated that the proposed topology can achieve 30.8% higher torque density, 79.4% lower cogging torque, and 15.6% higher power factor than the conventional counterpart.