Design and Analysis of Novel Modular Stator Axial-Flux Permanent Magnet Vernier Machine With Improved Power Factor

Permanent magnet Vernier machines (PMVMs) have drawn increasing attention in recent years due to the advantage of high torque density. However, the inherent low power factor issue of PMVMs remains a key challenge. In this paper, a novel axial-flux permanent magnet vernier machine (AFPMVM) with modular stator is proposed. Benefiting from the unique separation slot structure, the proposed machine can obtain not only the improved power factor but also the higher electromagnetic torque, and this is explained theoretically by air-gap field modulation theory. A magnetomotive force (MMF)-permeance model is established to predict the air-gap field harmonics modulated by the stator slots. Moreover, by quantifying the contribution of air-gap field harmonics to the torque generation, the working and non-working harmonic components of the air-gap field in the proposed machine can be determined. Meanwhile, the relationship between electromagnetic torque and power factor of the proposed AFPMVM is revealed from the perspective of the air-gap field harmonics. Then, the power factor of the proposed machine is significantly improved due to the lower non-working harmonic contents of armature reaction field. The analysis results show that the proposed AFPMVM can achieve 4.1% higher electromagnetic torque and 6.9% higher power factor than the existing AFPMVM. Finally, the influences of some key design parameters on electromagnetic torque and power factor of the proposed AFPMVM are analyzed and verified by finite element analysis.


I. INTRODUCTION
Due to the inherent merit of high torque density and simple mechanical structure, permanent magnet (PM) vernier machines (PMVMs) have received increasing attention and have become promising candidate for direct-drive applications such as ship propulsion, wind power generation, and electric vehicle, etc. [1], [2], [3], [4].Significant works on the magnetic gearing effect of the PMVM have been presented in [5], [6], [7], and [8], which reveal that the PMVM exhibits higher output torque than that of the regular PM The associate editor coordinating the review of this manuscript and approving it for publication was Feifei Bu .
In order to select proper slot/pole combination to acquire relatively high torque density, the influence of different slot/pole combinations on the electromagnetic performance of PMVMs is evaluated in [13], which reveals that the PMVMs with high pole ratio (pole-pair number ratio of PM rotor to armature winding) have the merit of high torque density.However, PMVMs with high pole ratio generally suffer from relatively low power factor, resulting in increased requirement of inverter capacity and higher cost [14], [15], [16].This is mainly because the leakage flux and armature reaction reactance of PMVMs will increase with the increase of pole-pair number (PPN) of PMs [17].Recently, numerous research works have been conducted to improve the power factor by increasing PM flux linkage [18], [19].In [20], the spoke-type PMVM is investigated to enhance the PM flux linkage and power factor with the aid of the flux focusing effect, while the torque density is weakened by the unique flux barrier.To overcome the flux barrier issue and further improve the power factor, a double-stator spoke-type PMVM is proposed, which provides a flux bridge between two stators to improve the flux path and PM flux density.As a result, the power factor and torque density are both improved [21].However, this machine with two-layer air-gaps suffers from issues of mechanical complexity and reliability.Besides, due to the limitations caused by the property of PM material, it is actually far from to improve the power factor of the PMVM only by increasing PM flux linkage.Thus, the way to reduce armature reaction flux linkage has been identified as alternative method.In [22], a PMVM with modular stator and yokeless rotor is proposed to improve power factor by weakening the armature reaction flux linkage.A PMVM with coil pitch of two slot pitches is presented in [23].The power factor of the PMVM can be improved to a certain extent, whereas its torque density is uncompetitive.It means that the armature reaction field is closely related to the torque density and power factor.
Generally, integral slot distributed windings (ISDWs) featured by longer end-winding are mostly adopted in PMVMs, which may significantly increase the machine volume and offset the advantage in torque density.To shorten endwinding length and reduce copper usage, the PMVMs with fractional slot concentrated windings (FSCWs) have been investigated in [24], [25], [26], and [27].It should be noted that the FSCW structure typically generates abundant armature reaction magnetomotive force (MMF) harmonics.Based on the air-gap field modulation theory, the effective armature reaction field harmonics play a crucial role in the research of the PMSM torque generation mechanism [28], [29], [30].Meanwhile, there is a conflicting relationship between electromagnetic torque and power factor of PMVMs.Therefore, analyzing the relationship between torque density and power factor from the perspective of the armature reaction field harmonics is of great significance for PMVMs to improve power factor and maintain high torque density.
In this paper, a novel axial-flux PMVM (AFPMVM) with separation slots in the stator is proposed.The relationship between electromagnetic torque and power factor of the proposed AFPMVM is revealed from the perspective of air-gap field harmonics.The separation slots of the proposed AFP-MVM can not only work as magnetic barriers to separate magnetic circuits, but also provide non-uniform distributed field modulation poles (FMPs) to generate additional permeance harmonics.Thus, the armature reaction MMF can be modulated to produce a higher proportion of the armature reaction field working harmonic.With the help of the stator separation slot structure, the proposed AFPMVM can obtain improved power factor and higher torque density.
The rest of this paper is organized as follows: the topology of the proposed AFPMVM will be introduced in Section II.In Section III, an MMF-permeance model will be established to predict the air-gap field harmonics accounting for air-gap field modulation effect.In Section IV, in order to investigate the relationship between the electromagnetic torque and power factor of the proposed AFPMVM, the working and non-working harmonic components of the air-gap field will be further determined.Furthermore, the influences of key design parameters on the output torque and power factor of the proposed AFPMVM will be investigated in Section V. Finally, some conclusions will be drawn in Section VI.

II. MACHINE TOPOLOGIES
The configuration of the existing AFPMVM is shown in Fig. 1(a), which has dual identical surface-mounted PM rotors and one armature stator sandwiched in between [31].The axially magnetized PMs on the two back irons are circumferential symmetric and have opposite polarity.The armature winding adopts double-layer FSCWs for reducing end-winding length.Fig. 1(b) and (c) present the configuration of the proposed AFPMVM.It can be seen that the difference between these machines lies in the structure of the stator core.For the proposed AFPMVM, the FSCWs are embedded into the slots of adjacent H-type modular stator pieces.Meanwhile, the non-uniform distribution of FMPs caused by the separation slot structure in the proposed machine is beneficial for improving the air-gap field modulation effect.Fig. 2 shows the parametric model of the proposed AFP-MVM.To reduce the manufacture difficulty, the parallel armature winding slot and parallel separation slot stator is applied to the proposed AFPMVM as shown in Fig. 2(b).However, due to the parallel stator slot structure, the ratio of  armature winding slot width w sa and separation slot width w ss to slot pitch of the proposed machine decrease with the decrease of air-gap radius.Hence, it is necessary to introduce the coefficients θ ss , θ t , and θ sa at the average radius of the air-gap to investigate the air-gap field modulation effect of the proposed machine, in which θ ss , θ t , and θ sa represent the separation slot arc, stator tooth arc and armature winding slot arc, respectively.The main design parameters are tabulated in Table 1.
It should be noted that the separation slot structure only causes non-uniform distribution of FMPs and does not change the PPN of FMPs.In other words, the effective harmonic order of the armature reaction field of the proposed AFP-MVM is the same as that of the existing AFPMVM.Thus, the proposed AFPMVM with separation slot shares the same operation principle of the air-gap field modulation as that of the existing AFPMVM and the relationships of armature winding slot number N a , armature winding pole-pair P a , and PM rotor pole-pair P r can be expressed as (1)

III. AIR-GAP FIELD MODULATION PRINCIPLE
In this section, the harmonic generation and variation mechanisms in the air-gap field of the proposed AFPMVM are revealed by the air-gap field modulation theory, which lays the foundation for studying the relationship between the air-gap field harmonics and electromagnetic performances.
In order to obtain a simplified analytical model for air-gap flux density, some assumptions are made as follows.
1) The permeances of the H-type modular stator pieces and rotor back irons are considered infinite, so as to the saturation effect of the iron core can be ignored.
2) The relative permeability of PMs is the same as that of vacuum.
3) The leakage flux is ignored for simplicity.

A. AIR-GAP PERMEANCE MODULATION FUNCTION
Different from the existing AFPMVM with uniform distributed armature winding slot, in the proposed AFPMVM, not only do the armature winding slots serve as FMPs to generate air-gap permeance harmonics, but the separation slots also work as FMPs to further adjust the distribution of air-gap permeance harmonics.Based on the air-gap field modulation theory, the air-gap permeance modulation behavior of the proposed AFPMVM can be divided into two parts, namely, armature winding slot modulation function a (θ ) and separation slot modulation function s (θ ).Therefore, the simplified linear air-gap permeance model of the proposed machine can be obtained by establishing a coordinate system with the center of the armature winding slot as the origin, as shown in Fig. 3.
According to the armature winding slot permeance model shown in Fig. 3, the armature winding slot modulation function can be deduced by where θ is the air-gap circumferential position, a0 and av are the dc component and amplitude of the vth harmonic of armature winding slot permeance, respectively.a0 and av can be expressed as where sa and t are the amplitudes of air-gap permeance at the circumferential position of the armature winding slot and stator tooth, respectively.It should be noted that the armature winding slot modulation function can be regarded as the same as the air-gap permeance modulation function of the existing AFPMVM.By using a stator without armature winding slots, the separation slot modulation function can be obtained and expressed as where N s is the number of the separation slot, s0 and sv are the dc component and amplitude of the v th harmonic of separation slot permeance, respectively.These coefficients can be determined by where ss is the amplitude of air-gap permeance at the circumferential position of the separation slot.
The air-gap permeance modulation function of the proposed AFPMVM can be derived by the product of the modulation functions of the armature winding slot and separation slot.Nevertheless, since a (θ)and s (θ ) are even functions with the half period phase difference and same period, the air-gap permeance modulation function can be expressed as where 0 and v are the dc component and amplitude of the v th harmonic of air-gap permeance modulation function, respectively.Fig. 4 shows the air-gap permeance harmonics of the two AFPMVMs.It can be noticed that both AFPMVMs have the same main air-gap permeance harmonics, which are 0 th , 12 th , and 24 th , respectively.The normalized values of the main air-gap permeance harmonics of the existing and proposed AFPMVMs are summarized in Table 2.The 0 th air-gap permeance harmonic can be reduced from 1 to 0.957 with the proposed separation slot design.Compared with the existing AFPMVM, the proposed AFPMVM can achieve a 5.9% increase in the amplitude of the 12 th air-gap permeance harmonic, which can contribute to enhancing the air-gap field modulation effect.Meanwhile, the 24 th air-gap permeance is reduced from 0.272 to 0.201.

B. PM AND ARMATURE REACTION FIELD HARMONICS ANALYSIS
The air-gap flux density is derived from the interaction between primitive MMFs and air-gap permeance modulation function.The simplified PM MMF model of the AFPMVMs is shown in Fig. 5.It can be seen that the PM MMF rotates relative to the armature stator in the stationary coordinate system.Thus, the Fourier series expansion of the PM MMF can be expressed as where ω r represents the rotor mechanical angular speed, θ 0 is the initial rotor position, F PMn represents the amplitudes of the n th MMF harmonic generated by PMs, B r is the remanence of PMs, h m is the thickness of PMs, µ 0 is vacuum permeability, µ r is the relative permeability of PMs, θ PM is the PM pole arc coefficient.Based on the MMF-permeance method [1], the PM air-gap flux density in the proposed AFPMVM can be expressed as From (10), apart from the primitive PM MMF harmonic orders, i.e., nP r (n = 1, 3, 5, . . .), various PM field harmonics with PPN of |nP r ± vN a | (n = 1, 3, 5, . . ., v = 1, 2, 3, . . .), named as modulation harmonics, are modulated out by the armature stator as shown in Fig. 6.Theoretically, the order of PM field harmonics of the proposed AFPMVM is infinite because the combination of n and v is infinite.However, when n > 3 and v > 1, the corresponding amplitudes of the air-gap permeance harmonic and PM MMF harmonic are small enough to be neglected.Therefore, only n = 1 or 3 and v = 1 are taken into account when calculating the electromagnetic torque of the proposed AFPMVM.
To calculate the armature reaction MMF distribution of the proposed AFPMVM, the Fourier series of the winding  function for each phase can be expressed as where N c is coil turns number.Ignoring the harmonic components of armature current, the symmetrical three-phase currents can be expressed as where I is the amplitude of phase current, ϕ 0 is the initial phase angle, ω e = P r ω r .ω e denotes electrical angular speed.
According to the winding function and three-phase current, the resultant armature reaction MMF can be calculated as 108584 VOLUME 11, 2023 Authorized licensed use limited to the terms of the applicable license agreement with IEEE.Restrictions apply.Similarly, the armature reaction air-gap flux density of the proposed AFPMVM can be expressed as From ( 13)-( 14), it can be seen that the value of i not only determines the order of armature reaction field harmonics, but also determines the rotational directions of armature reaction field harmonics.The sources and rotational directions of the armature reaction field harmonic components are listed in Table 3.It can be observed that each harmonic in armature reaction field can be modulated out by different combinations of armature reaction MMF harmonics and air-gap permeance harmonics.
In order to confirm the analyses above, Fig. 7 shows the FFT spectrum of armature reaction field distributions for two AFPMVMs, in which the analysis results match well with finite-element analysis (FEA).Among which, 2 nd , 10 th , 14 th , 22 th , and 26 th armature reaction field harmonics have high amplitudes due to high winding factor.Besides, the separation slot configuration can effectively weaken the 2 nd harmonic of the armature reaction field, and has relatively little effect on the 10 th harmonic of the armature reaction field.As a result, the 10 th armature reaction field harmonic of the proposed AFPMVM has the highest amplitude.

IV. ELECTROMAGNETIC PERFORMANCE ANALYSIS A. FLUX-LINKAGE AND BACK-EMF
Based on the MMF-permeance model established in Section II, the PM flux linkage of phase A can be expressed as where r is the air-gap radius, R o and R i are the stator outer radius, stator inner radius, respectively.Similar to the PM flux linkage, the armature reaction flux linkage of phase A can be expressed as According to (15) and Faraday's law of induction, the back-EMF of phase A can be calculated by The open-circuit PM flux linkage waveforms and FFT spectrum for both AFPMVMs are calculated and shown in Fig. 8.The fundamental component of the flux linkage of the AFPMVM is about 3.3% large than that of the existing AFPMVM.Fig. 9 shows the open-circuit back EMFs of both AFPMVMs at the speed 300 r/min.It can be seen that the proposed AFPMVM has higher fundamental component of back EMF than that of the existing AFPMVM.Besides, there are more odd order harmonic components in the back EMF of the two AFPMVMs, and the main harmonic component is the 3 rd harmonic.

B. ELECTROMAGNETIC TORQUE
Based on the traditional electromagnetic theory, PM field harmonic and armature reaction field harmonic shared with the same PPN at the same rotational speed can be coupled with each other to generate the electromagnetic torque.The synthesized axial component B z and tangential component B t of the air-gap field harmonics can be expanded as where B zk and θ zk are the k th Fourier coefficient and corresponding phase angle of B z , respectively, B tk and θ tk are the k th Fourier coefficient and corresponding phase angle of B t , respectively.
To investigate the relationship between air-gap field harmonics and electromagnetic torque, the Maxwell stress tensor method is employed to obtain the contribution of each air-gap field harmonic to the electromagnetic torque.Hence, the electromagnetic torque of the proposed AFPMVM can be calculated based on the obtained axial component and tangential component of the air-gap field  where µ 0 is the vacuum permeability, T k denotes the electromagnetic torque produced by the kth air-gap field harmonic, which can be deduced as Under rated load condition, Fig. 10 gives the distribution of axial component and tangential component of the air-gap field harmonics of two AFPMVMs.The amplitude of the 10 th axial air-gap field harmonic is much larger than that of other axial air-gap field harmonics due to the higher amplitude of the 10th PM field harmonic.As shown in Fig. 7, it can be observed that both AFPMVMs have the same dominant armature reaction field harmonic orders, i.e., 2 rd , 10 th , 14 th , 22 th , and 26 th .Thus, these harmonic orders are taken as an example to investigate the variations of the air-gap field harmonics with respect to the rotor position.The phase angle of axial component and tangential component of the dominant air-gap field harmonic in the proposed AFPMVM versus   rotor position is presented in Fig. 11.It can be seen that the rotational direction of the 10 th and 22 th air-gap field harmonics is opposite to that of the 2 nd , 14 th , and 26 th harmonics, which is consistent with the rotational direction of the armature reaction field harmonics derived from Table 3.
The contribution of each air-gap field harmonic to the electromagnetic torque is presented in Fig. 12.It can be found that only the 10 th , 14 th , and 22 th air-gap field harmonics are responsible for electromagnetic torque generation.Among which, the 10 th air-gap field harmonic exhibits the highest torque proportion in the both AFPMVMs, which can be defined as the air-gap field working harmonics.In addition, due to the high amplitude of the 10 th tangential air-gap field harmonic of the proposed AFPMVM, the 10 th air-gap field harmonic in the proposed AFPMVM contributes more for electromagnetic torque generation than that of the existing AFPMVM.It should be noted that the 22 th harmonic provides a positive torque component in the existing AFPMVM while a negative torque component in the proposed AFPMVM, 108586 VOLUME 11, 2023 Authorized licensed use limited to the terms of the applicable license agreement with IEEE.Restrictions apply.
which is mainly due to the change in phase different between axial component and tangential component when the separation slot structure is adopted.
The comparison of electromagnetic torque between the two AFPMVMs is shown in Fig. 13.The average torques of the existing AFPMVM and proposed AFPMVM are 2.95 Nm and 3.07 Nm, respectively.Thus, the average torque increases by 4.1%.The torque ripples of the existing AFPMVM and proposed AFPMVM are 28.7% and 36.3%, respectively, which is mainly because the introduction of the separation slot increases the cogging torque.Fig. 14 shows the variation of average torque versus phase current of the existing and proposed AFPMVMs.It can be seen that the average torque of the proposed AFPMVM is larger than that of the existing AFPMVM under the same phase current.

C. POWER FACTOR
Based on the above analysis, the P r (10 th ) air-gap field harmonic is the working harmonic and provides the major torque component of AFPMVMs.However, the torque component generated by the P a (2 nd ) air-gap field harmonic is small enough to neglect and it can be defined as air-gap field nonworking harmonic.Besides, the air-gap field harmonic is formed by the interaction of the PM field and armature reaction field harmonic with the same order.Hence, the armature reaction field can be rewritten as where B aw is the armature reaction field working harmonic, B an is the armature reaction field non-working harmonic.
According to (16), the armature reaction flux linkage can be divided into armature reaction working flux linkage ψ aw and armature reaction non-working flux linkage ψ an , the armature reaction flux linkage of phase A can be rewritten as   It is noted that, from (24), both B aw and B an generate the fundamental armature reaction flux linkage.Therefore, the ψ aw component and ψ an component in the armature reaction flux linkage have the same order.Fig. 15 shows the phasor diagram of the proposed AFP-MVM.Under the control method of I d = 0, the phasor is in phase with the back-EMF phasor and the power factor angle is the same as the load angle.Therefore, the power factor can be expressed as  reduced to maintain the high torque and improve the power factor.In a large degree, this is mainly attributed to that the armature reaction field non-working harmonics have almost no contribution to electromagnetic torque.In PMVMs, the high-order harmonic of the armature reaction i.e., equal P r , is typically used to generate reaction working flux linkage.As a the field nonworking e.g., equal to P a , account for a large proportion of the armature reaction field due to the low-order of harmonics.This can explain why PMVMs exhibit relatively low power factor.As shown in Fig. 7, the 2 nd armature reaction field harmonic of the existing AFPMVM exhibits the highest amplitude, which has a negative impact on the power factor and has almost no contribution to electromagnetic torque.In order to the armature reaction field non-working harmonic that having long magnetic circuit, the separation slots are introduced in the armature stator of the proposed AFPMVM.It is evident from Fig. 7 that the 2 nd armature reaction field harmonic is reduced from 0.114 T to 0.036T.
The comparisons of average torque, power factor, and armature reaction flux linkage are presented in Fig. 16 and listed in Table 4.It can be seen that the armature reaction flux linkage of the proposed AFPMVM is decreased 22.4% owing to decline of the armature reaction field non-working harmonic.Benefiting from the unique separation slot structure, the proposed AFPMVM exhibits 6.9% higher power factor than the existing AFPMVM.

V. INFLUENCES OF DESIGN PARAMETERS
Compared with the other design parameters, the PM thickness, PM pole-arc coefficient, and separation slot width exhibit more influence on the air-gap field harmonic distributions.Therefore, the influences of these design parameters the electromagnetic torque and power factor are evaluated in order to reveal some design guidelines for the proposed AFPMVM.
A. PM THICKNESS Fig. 17 analyzes the influences of PM thickness on the electromagnetic torque and power factor of both AFPMVMs.It can be seen that the average torques of both AFPMVMs slightly decrease with the increase of PM thickness.This phenomenon is due to the fact that the increase of PM thickness can lead to the increase the length of effective air-gap and the serious magnetic saturation of stator teeth.Meanwhile, it can be found that the power factors of both AFPMVMs increase as the PM thickness rises, as shown in Fig. 17(b), which is mainly due the rising PM flux linkage.Compared with the existing AFPMVM, the electromagnetic performances of the proposed AFPMVM are much less sensitive to the PM thickness.

B. PM POLE-ARC COEFFICIENT
The PM pole-arc coefficient also has significant effect on electromagnetic performance of both AFPMVMs.The influences of PM pole-arc coefficient on electromagnetic torque and power factor are depicted in Fig. 18.It can be seen that the average torques rise with the increase of PM pole-arc coefficient, whereas the power factors are inverse to the PM  pole-arc coefficient due to the increase in the leakage flux.In addition, it can be observed that there are larger average torque and higher power factor in the proposed AFPMVM compared to the existing AFPMVM under the same design parameters.

C. SEPARATION SLOT WIDTH
For the proposed AFPMVM, the air-gap field modulation effect is not only determined by the separation slots but also determined by the armature winding slots.Therefore, before analyzing the influence of separation slot width on electromagnetic performance, the armature winding slot width ratio need to be defined The variations of the main axial armature reaction field harmonics with the separation slot width under various β are plotted in Fig. 19.It can be seen that the amplitude of the 2 nd harmonic is sharply decreased with increasing the separation slot width, which is due to the enhancement of the low-order harmonic weakening effect in the separation slot.Meanwhile, the amplitude of 10 th harmonic is also slightly decreased due to the decrease of dc component of air-gap permeance.The influences of the separation slot width on the electromagnetic torque and power factor are illustrated in Fig. 20(a) and (b), respectively.It can be observed that the average torques rise with the increase of separation slot first then drop.This is mainly attributed to the fact that the air-gap field modulation effect is enhanced with the increase of the separation slot width.However, as the separation slot width is higher than the optimal value, the magnetic saturation effect on the stator teeth results in the torque decrease.And, the power factor rises with the increase of the separation slot.This is mainly contributed by the fact that the non-working harmonic content of the armature reaction field is decreased.

VI. CONCLUSION
In this paper, a novel AFPMVM with modular stator is proposed to achieve higher power factor and larger torque density than the existing AFPMVM.An MMF-permeance model is established to reveal the modulation process of the air-gap field harmonics of the proposed machine.Then, the relationship between the electromagnetic torque and power factor of the proposed machine is investigated from the perspective of the air-gap field harmonics.It is indicated that the 10 th air-gap field harmonic is the air-gap field working harmonic and provides the major electromagnetic torque of both AFPMVMs, whereas in the existing AFPMVM, the 2 nd harmonic in the armature reaction field has the highest amplitude, which has a negative impact on the power factor and has almost no contribution to electromagnetic torque.Due to the weakening effect of the separation slots on low-order armature reaction field non-working harmonic, the proposed AFPMVM can achieve higher power factor and maintain high torque density.Compared to the existing AFPMVM, the electromagnetic torque and power factor of the proposed AFPMVM can be improved by 4.1% and 6.9%, respectively.Finally, the influences of some key design parameters on the electromagnetic performance of the proposed AFPMVM are analyzed and verified by finite element analysis.

FIGURE 2 .
FIGURE 2. Parametric model of the proposed AFPMVM.(a) Cross-sectional view.(b) Top view of the stator.(c) Top view of the rotor.

FIGURE 3 .
FIGURE 3. The simplified linearized air-gap permeance model of the proposed AFPMVM.

FIGURE 5 .
FIGURE 5.The simplified PM MMF model of the proposed AFPMVM.

FIGURE 6 .
FIGURE 6. FFT spectrum of PM field distributions for two AFPMVMs.

FIGURE 10 .
FIGURE 10.FFT spectrum of air-gap field distributions for two AFPMVMs.(a) Axial air-gap field harmonics.(b) Tangential air-gap field harmonics.

FIGURE 11 .
FIGURE 11.Phase angle of dominant air-gap field harmonics of the proposed AFPMVM versus rotor position.(a) Axial air-gap field harmonics.(b) Tangential air-gap field harmonics.

FIGURE 12 .
FIGURE 12. Torque contributions of the air-gap field harmonics.

FIGURE 14 .
FIGURE 14. Variation of average torque with phase current.

FIGURE 16 .
FIGURE 16.Comparisons of average torque, power factor, and armature reaction flux linkage between the two AFPMVM.

= 1 1 + 2 ( 25 )
(ψ aw + ψ an + ψ σ ) ψ PM where ψ σ denotes the leakage flux, E is the open-circuit back-EMF fundamental magnitude, U is the terminal voltage.It can be observed that the power factor is mainly determined by PM flux linkage, armature reaction flux linkage, and leakage flux.Since the PM flux linkage is limited by the property of PM material, the power factor can be effectively improved by reducing armature reaction flux linkage.Considering that the armature reaction working flux linkage is the component contributing to electromagnetic torque, the armature reaction non-working flux linkage can be

FIGURE 17 .
FIGURE 17. Influences of the PM thickness on the torque capability and power factor.(a) Average torque.(b) Power factor.

FIGURE 18 .
FIGURE 18. Influences of the PM pole-arc coefficient on the torque capability and power factor.(a) Average torque.(b) Power factor.

FIGURE 19 .
FIGURE 19.Influences of the separation slot width on the dominant axial armature reaction field harmonics under various β.

FIGURE 20 .
FIGURE 20.Influences of the separation slot width on the torque capability and power factor under various β.(a) Average torque.(b) Power factor.

TABLE 1 .
Main parameters of existing machine and proposed machine.

TABLE 3 .
Main parameters of existing machine and proposed machine.

TABLE 4 .
Electromagnetic performance comparisons under rated conditions.