Design Trade-Off Between Torque Density and Power Factor in Surface-Mounted PM Vernier Machines Through Closed-Form Per-Unit Equations

Benefited from flux modulation effects, surface-mounted permanent magnet vernier machines (SPMVMs) present the high-torque low-speed properties, which can become the potential candidates for direct-drive applications. However, their power factor is inferior to conventional PM machines due to the large inductance caused by parasitic no-working harmonics. This paper is devoted to propose some design considerations of SPMVMs to make a good compromise between torque density and power factor. Via constructing closed-from per-unit equations, the cross-coupling influences of slot/pole combinations and normalized geometric variables on machine performances are revealed. It is found that there is a tradeoff between torque density and power factor. The high magnet thickness to air-gap length ratio is beneficial to decrease inductance, while the flux modulation effect under the large equivalent air-gap length would be impaired. Moreover, the pole-pair ratio, i.e., the PM pole-pair to armature winding pole-pair, should be carefully selected, where the SPMVMs with high pole-pair ratio have the relatively high magnetic loading owing to the strong flux modulation effects, whereas they suffer from the low power factor due to serious harmonic leakage inductance. In particular, taking the end-winding length into consideration, the high torque advantages of high pole-pair ratio SPMVMs under the same copper losses can be well performed with high length-radius ratio. Combined with the analysis, the preferable selections of critical parameters to satisfy various technical indexes are determined. Then, the general design approach is proposed, and two design examples are built and compared. Meanwhile, finite element analysis (FEA) and prototype experiments are carried out to verify the feasibility.


I. INTRODUCTION
D IRECT-DRIVE motors have received increasing more attention for wind power generation, marine proportion, and in-wheel tractions. They have the advantages of high efficiency, small vibration noise, and high reliability due to the cancellation of mechanical gear boxes. However, there is still the challenges to overcome considering the high torque density requirements.
Benefited from flux modulation effects, permanent magnet vernier machines (PMVMs) present the low-speed high-torque properties, which can become the promising candidates for direct-drive applications [1], [2], [3]. However, PMVMs suffer from low power factor due to the parasitic harmonic inductances, which would increase the volume and cost of inverter [4], [5]. To promote the industrial applications of PMVMs, many researches have been done, which can be classified into two aspects.
1) Enhancing torque density: Based on the three elements of PMVMs, i.e., flux modulator, PM excitation, and armature winding, a lot of efforts have been presented to further improve the torque density of PMVMs. In [6] and [7], hybrid tooth surface-mounted permanent magnet vernier machines (SPMVM) has been proposed to enhance the flux modulation effects by introducing multiple working permeance harmonics. Further, interior PM and halbach PM have been applied to strengthen PM flux focusing effects [8], [9]. Alternate rotor flux bridges have been designed to resolve the flux barrier effects occurred in spoke-array PM rotor structure [10], [11]. Moreover, toroidal winding and split tooth concentrated winding structure have been used to reduce the end-winding length [12], [13]. Although the torque density of these topologies mentioned above has been enhanced, the low power factor issues are still existed. 2) Improving power factor: The dual-stator spoke-array PM vernier machine has been investigated in [14], where the power factor can be significantly improved by increasing the excitation capability. Further, the dual-side PM structure has been proposed in [15], [16] where the stator and rotor PMs work together to decrease flux leakage. Moreover, the hybrid excitation PMVM has been put forward to enhance the excitation magnetic fields and improve power factor [17]. Nevertheless, these topologies equipped with dual-stator, dual-PM, or dual-excitation, are relatively complex, which are not suitable for the direct-drive applications. Therefore, it is necessary to investigate more attractive PMVMs with simple structure and low cost for direct-drive applications. Especially, it is found in [18] and [19] that SPMVM with low pole-pair ratio (PR), i.e., the pole-pair ratio of PM to armature winding, performs a better power factor characteristic owing to the small parasitic harmonic inductances. Although the average torque of low PR SPMVMs are inferior to high PR counterparts, the low PR SPMVMs endowed with short end-windings are promising for small length-radius ratio conditions [20]. However, current researches are confined to specific size and power rating. There is a lack of the comprehensive analysis and design of SPMVMs to balance average torque and power factor, covering different length-radius ratios and power ratings.
This paper proposes some design considerations of SPMVMs for direct-drive applications, which is devoted to make a good compromise between torque density, power factor, and efficiency, covering different power ratings. According to construct the closed-form per-unit equations, the cross-coupling influences of critical parameters, including pole-pair ratio PR, magnet thickness to air-gap length ratio k g , stator slot opening width to slot pitch ratio k s , et al., on machine performances are investigated. And the preferable selection ranges of critical parameters are determined. Further, the general design approach of SPMVMs for direct-drive applications are presented, which can avoid the blind attempt and provide a good guidance for engineering designers.
The main contents are organized as follows: First, the closedfrom per-unit equations, including magnetic loading B, electric loading A, normalized inductance L, torque density σ, and power factor PF, are derived in Section II, where the negative correlations between torque and power factor are revealed. Second, Section III deeply investigates the influence of critical parameters, such as pole-pair ratio PR, magnet thickness to air-gap length ratio k g , stator slot opening width ratio k s , and length-radius ratio k shape , on machine performance. Moreover, the preferable selection ranges are determined. Then, Section IV propose the general design approach for SPMVMs. And the detailed design examples and Finite element analysis (FEA) are given in Section V. Section VI presents the conclusions.

II. RELATIONS BETWEEN TORQUE DENSITY AND POWER FACTOR
In this Section, the closed-from per-unit equations of torque density and power factor are constructed, where magnetic loading, electrical loading, and normalized inductance work as the bridges to reveal the negative correlations between torque density and power factor. To have a clear presentation, all the design parameters are summarized in Table I. And the dimension parameter model of SPMVM is depicted in Fig. 1.

A. Magnetic Loading
According to magnetomotive force (MMF)-permeance model, no-loading air-gap flux density in SPMVMs is defined as the product of air-gap MMF and permeance [1].
where k B is the factor that quantifies air-gap harmonics. Λ 0 and Λ 1 are constant and fundamental relative permeance, where the detailed expressions are given in Appendix. Both the Λ 0 and Λ 1 are the function of magnet thickness to air-gap length ratio k g , and stator slot opening width ratio k s . It can be noticed that SPMVMs exist multiple no-load airgap harmonics having different pole-pairs. And it has been confirmed in [12] that all the air-gap harmonics are working harmonics, which can contribute back-EMF and average torque. Combined with expression of back-EMF and winding function [14], the contribution of each harmonic can be determined.
Further, magnetic loading B of SPMVMs can be given as [7]: = k wP r B P r + P Rk wP r−Zs B P r−Zs where k w is winding factor of air-gap working harmonics, which has taken account of slot opening factor [2]. It can be found that magnetic loading B of SPMVMs is related to multiple parameters, including pole-pair ratio PR, magnet thickness to air-gap length ratio k g , stator slot opening width ratio k s , armature winding pole-pair P a , and winding factor k w .

B. Electric Loading
To make a fair comparison, end-winding length and the corresponding copper losses caused by which are taken into consideration. In particular, the copper loss factor k Copper , i.e., the copper losses per-unit volume, is set as a benchmark to reflect machine dissipation capability. Further, electric loading A can be defined as [21]: It can be noticed that electric loading A is related to multiple parameters. Among which, pole-pair ratio PR and length-radius ratio k shape have a significant influence on end-winding length factor k end , which are treated as the most critical parameters for electric loading A. Moreover, electric loading A is also related to stator slot opening width ratio k s , slot depth ratio k t , and split ratio k split .

C. Inductance
Different from regular PM machines, the constraints of equal pole-pair between armature winding and PM have been broken in SPMVMs. As a result, the definition of inductance components is also changed. From armature magnetic field perspective, only the armature harmonic which has the same pole-pair P r of PM, can interact with PM rotor and generate torque. As a result, the normalized main inductance L m of SPMVMs is defined as the flux linkage produced by the P r th armature harmonics under per-unit current. Whereas the normalized harmonic leakage inductance L σ is produced by other no-working armature harmonics, where P a th armature harmonics are the main components. Slot leakage inductances and end-winding leakage inductances are neglected due to the relatively small proportion.
Then, the inductance L of SPMVMs can be given as [2]: It can be observed that the inductance is mainly determined by pole-pair ratio PR, armature winding pole-pair P a , magnet thickness to air-gap length ratio k g , and winding factor k w .

D. Torque Density and Power Factor
Combined with magnetic loading B, electric loading A, and inductance L, both the torque density and power factor can be acquired. First, the total volume torque density of SPMVMs can be given as: Further, under I d = 0 control and neglecting winding resistance, as shown in Fig. 2, the power factor of SPMVMs can be written as: It can be seen that torque density σ is proportional to the product of k split 2 AB/k end , while power factor PF is inversely proportional to LA/B. That is, by selecting magnetic loading B, electric loading A, and inductance L as bridges, the negative correlation between torque density and power factor is revealed. Moreover, they are mutually restricted due to the cross-coupling influence of geometric parameters on which. Thus, there is a trade-off for the design of SPMVMs.

III. CROSS-COUPLING INFLUENCE OF PARAMETERS ON MACHINE PERFORMANCE
Based on the analysis mentioned above, torque density of SPMVMs is negatively correlated with power factor. To make a good compromise between which, the cross-coupling influence of critical parameters, including magnet thickness to air-gap length ratio k g , stator slot opening width ratio k s , pole-pair ratio PR, armature winding pole-pair P a , and length-radius ratio k shape , on machine performance are comprehensively investigated. The detailed analysis is given in what follows, where the variation of machine performances with design parameters are acquired by the analytical calculation equations.
To better reveal the influences of critical parameters on machine performances, six models with different slot/pole combinations are selected for comparisons, covering different polepair ratios PR and armature winding pole-pairs P a . Among which, the model with PR = 1, P a = 2 is the regular PM synchronous machine (PMSM) having the same pole-pair of armature winding and PM, which is worked as the benchmark. It can also be treated as the special SPMVM, where the flux modulation effect is weakest [7]. While other models have the different pole-pairs between armature windings and PMs, where the higher pole-pair ratio PR, the stronger flux modulation effects. It is worth noting that the models under different pole-pair ratios have the different coil pitches and end-winding lengths. As a result, under the same copper loss, the models with low PR and compact end-winding can acquire the high active phase current. However, the models with high PR suffer from the low active phase currents. Especially, the selected model with PR = 3.5, P a = 2 can have a better balance between flux modulation effects and end-winding lengths. Table II presents the original design parameters of the six models. To have a fair comparison, all the models have the same stator outer radiuses, air-gap lengths, stack lengths, series turns per phase, total copper losses, and so on. Moreover, the stator inner radiuses, split ratios and stator slot depths are flexibly adjusted to acquire the similar no-load yoke flux density. Although the models under different stator inner radiuses, split ratios and stator slot depths have the different electromagnetic performances, the variation tendencies of performances with targeted parameters are kept unchangeable. To avoid the interference of other parameters on machine performances, the variable control method is applied. That is, only the investigated single variable is changeable, while other parameters are kept invariable. Fig. 3 shows the influence of magnet thickness to air-gap length ratio k g on torque density σ and power factor PF, respectively. For regular PMSM, only B Pr and L m are the main components of air-gap flux density and inductance, while the modulated harmonics B Pr±Zs and harmonic leakage inductance L σ are barely existed. As a result, it can be seen that with the increase of k g , both the torque density σ and power factor PF of regular PMSM are enhanced due to the increase of magnetic loading B and decrease of inductance, as shown in Fig. 4.

A. Magnet Thickness to Air-Gap Length Ratio k g
For SPMVMs, the power factor can also be improved by the increase of k g , due to the reduction of inductance. While the influences of k g on torque density σ are more complicated due to there are multiple air-gap working harmonics. On the one  hand, increasing k g , that is, increasing magnet consumption, is beneficial to enhance harmonic B Pr , as shown in Fig. 4(a). On the other hand, the flux modulation effects are deteriorated due to the relatively large equivalent air-gap length. It can be noticed in Fig. 4(b) that the modulated harmonics B Pr±Zs are increased first and then dropped due to the decrease of air-gap relative fundamental permeance Λ 1 . Thus, with the increase of k g , the torque density σ of SPMVMs is increased first and then reduced. Under the optimal k g , the torque density σ of SPMVMs with high pole-pair ratio PR, can be up to several times of regular PMSMs. Therefore, the selection of k g can be concluded that: 1) In terms of torque density σ, the optimal k g of SPMVMs with the same PR is similar. With the increase of PR, the optimal k g is decreased to strengthen flux modulation effects, as listed in Tables I and II. 2) In terms of power factor PF, the relatively large k g is preferable due to the weakened inductance.

B. Stator Slot Opening Width to Slot Pitch Ratio k s
Further, the cross-coupling influences of stator slot opening width to slot pitch ratio k s on torque density σ and power factor PF are investigated, as shown in Fig. 5.
With the increase of k s , the harmonics B Pr is gradually decreased due to the impaired air-gap relative constant permeance Λ 0 , as shown in Fig. 6(a). Whereas there is the optimal k s about 0.6 to acquire the maximum modulated harmonics B Pr±Zs , as shown in Fig. 6(b). As a result, it can be noticed in Fig. 6(c) that magnetic loading B increases first and then reduces. Moreover, Fig. 6(d) and (e) show that with the increase of k s , the electric loading A under the same copper losses can be improved due to the enlarged slot area. While inductance L is decreased, where the harmonics leakage L σ is the main components.
On the whole, the effects of k s on torque density σ is in accordance with magnetic loading B, which climbs up and then declines. However, the effects of k s on PF are relatively weak, due to the variation of electric loading A and inductance L with k s are inverse.
The selection of k s can be concluded that: 1) For torque density σ, the optimal k s of SPMVMs with the same PR is nearly kept identical, which is around 0.4∼0.7.
With the increase of PR, the optimal k s is slightly enlarged to strengthen the contribution of modulated harmonics, as shown in Tables I and II. 2) In terms of power factor PF, the influence of k s on which is relatively weak, which can be neglected.

C. Pole-Pair Ratio PR and Length-Radius Ratio k shape
Combined with the analysis mentioned above, both the selections of k g and k s are strongly associated with the selection of pole-pair ratio PR. Thus, for the design of SPMVMs, it is necessary to determine PR at first. The effects of PR on torque density σ and power factor PF are explored under the same copper loss factor k Copper = 76 kW/m 3 , covering different length-radius ratio k shape .
It can be observed in Fig. 7(a) that with the increase of k shape , torque density σ is increased because of the negative influence of end-winding is weakened, while the electric loading A can be enhanced. The end-winding length factor k end of SPMVMs with high PR is significantly higher than low PR counterparts, especially under small k shape condition. As a result, compared to high PR SPMVMs, conventional PMSMs with fractional slot concentrated winding and low PR SPMVMs, can be up to the upper limit of torque density σ quickly. Moreover, the contributions of regular air-gap flux density harmonics B Pr and modulated harmonics B Pr±Zs are compared, as depicted in Fig. 7(c). With the increase of PR, the effects of modulated harmonics B Pr±Zs on torque density σ are enhanced. Further, Fig. 7(b) shows that high PR SPMVMs suffer from low power factor, due to large inductance.
Thus, the selection of PR is strongly related to k shape , which can be summarized that: 1) For torque density σ, the high PR SPMVMs are competitive when length-radius ratio k shape > 4, where the negative effects of long end-winding can be neglected. However, with k shape ≤ 4, the regular fractional slot  Table II. 2) For power factor PF, the high PR SPMVMs suffer from low PF due to the abundant harmonic leakage inductances.

D. Armature Winding Pole-Pair P a
It can be found that except the pole-pair PR, the effects of armature winding pole-pair P a on torque density σ and power factor PF should also be taken into consideration. It can be found in Fig. 7 that under the same PR, the larger P a , the lower magnetic loading B. The reason is that the air-gap fundamental relative permeance and flux modulation effects are weakened under the large P a . However, the larger P a is conductive to improve end-winding, electric loading A, and inductance L.
Therefore, the selection of P a can be summarized as: 1) For torque density σ, the SPMVMs with large P a are promising when length-radius ratio k shape > 4, where the decrease of magnetic loading B can be compensated by the increase of electric loading A. However, with k shape ≤ 4, the SPMVMs with small P a are attractive owing to the stronger flux modulation effects. 2) For power factor PF, the SPMVMs with large P a is preferable due to decrease of inductance. In a word, combined with the analysis mentioned above, how the parameters of SPMVMs affect torque density σ and power factor PF are summarized in Table III. Among which, the higher k g , the higher PF due to the weakened inductance L under the large equivalent air-gap length. While with the increase of k g , σ is increased first owing to the improvement of regular air-gap flux density harmonics, and then decreased due to the impairment of modulated air-gap flux density harmonics. For k s , it is mainly related to σ, while the influence on PF can be neglected. The design of k s should balance both the flux modulation effects and slot area, that is, achieve the relatively high magnetic loading B and electric loading A simultaneously. In terms of PR, the SPMVM with high PR can obtain the high B due to the strong flux modulation effect. Whereas it suffers from the low PF due to the large harmonic leakage inductance. Moreover, the model with low armature winding pole-pair P a and high PR suffer from the long end-winding length, which lead to the low electric loading A. As a result, the high torque density advantages of SPMVM with high PR can only be exploited under the large length-radius ratio k shape . Further, the preferable selection ranges of critical parameters are determined, as listed in Table IV. It can be observed that with the increase of PR, the preferable k g is decreased to enhance the flux modulation effects by decreasing equivalent air-gap length. While the preferable k s is slightly increased to improve air-gap fundamental permeance, so as to have a better use of modulated air-gap flux density harmonics. Moreover, the high torque density advantages of SPMVMs with high PR can be performed under the relatively high k shape .

IV. GENERAL DESIGN APPROACH OF SPMVMS
Based on the relations between parameters and machine performances, as well as the preferable selection ranges of critical parameters, the general design approach of SPMVMs is proposed to balance torque density, power factor, and efficiency, covering different power ratings. And the detailed design flowchart is organized in Fig. 8.

A. Step 1: Determination of Initial Parameters
1) Based on the size of space, both the stator outer radius r so , and the upper limit of length-radius ratio k shape can be set. 2) According to power ratings and process craft, air-gap length g can be determined. Then, the air-gap length to stator outer radius ratio k P can be calculated. 3) Combined with dissipation capability and machine efficiency requirements, the upper limit of copper loss factor k Copper can be defined. 4) By materials characteristic, copper resistivity ρ and PM remanence B r can also be acquired. 5) The upper limit of PM consumption can be determined based on cost.

B. Step 2: Design of Key Parameters
1) Based on the indexes of torque and power factor, the initial pole-pair ratio PR can be set. 2) Combined with k shape and efficiency requirements, the armature winding pole-pair P a can be selected. 3) According to the selected PR, the preferable stator slot opening width ratio k s and magnet thickness to air-gap length ratio k g can be determined.

C. Step 3: Design of Detailed Size
1) Based on the selected P a and overloading capability requirements, the initial split ratio k split and slot depth ratio k t can be set.

D. Step 4: Evaluation of Performance
1) Based on the acquired parameters, initial model can be constructed, and the performance can be evaluated by FEA. 2) If all the performance are satisfied, flowchart can be completed. Otherwise, some iterations are needed. 3) If torque T is unsatisfied, P a can be decreased to enhance flux modulation effects and magnetic loading B. Moreover, k split can be increased to enhance air-gap radius. 4) If power factor PF is unsatisfied, inductance can be decreased by applying lower PR or larger P a . 5) If efficiency η is unsatisfied, copper losses can be reduced by increasing P a or decreasing PR to reduce end-winding. 6) If all the performance are unsatisfied, PR should be redefined.

V. DESIGN EXAMPLES AND FEA VALIDATIONS
In order to validate the analysis, two SPMVM examples named as Model 1 and Model 2 have been built based on the proposed design approach. Technical indexes and initial parameters are listed in Table V.
First, according to size upper limit, the initial k shape can be determined. Further, based on rated power and speed, the targeted torque density σ can be predicted. Then, combined with torque density and power factor indexes, as well as the relation between pole-pair ratio PR and machine performance, the initial  PR can be selected. In the design examples, the PR of Model 1 is selected as 3.5 to acquire high power factor PF, while the PR Model 2 is set as 14 to achieve high efficiency.
Further, based on the preferable selection range, the critical parameters, covering magnet thickness to air-gap length ratio k g and stator slot opening width ratio k s can be derived. In particular, the selection of k g should also meet magnet consumption limit. Model 2 with high PR has the smaller k g and larger k s to strengthen the flux modulation effects. Next, based on the overloading requirements, slot depth ratio k t and split ratio k split can be determined, where Model 2 with high PR and low armature winding pole-pair P a is easier to get saturated. Thus, the split ratio of which is smaller than Model 1. Fig. 9 shows machine structure and no-load flux density of two design examples. Fig. 10 presents the no-load air-gap flux density waveforms and spectrums of the two models. It can be seen that except for the regular air-gap flux density harmonics with the same pole-pair of PMs, the modulated air-gap flux density harmonics having the same pole-pair of armature winding are also generated.
Moreover, the torque waveforms of the two models are compared in Fig. 11. It can be observed in Fig. 11(a) that the cogging torques of the two models are low, where the peak-to-peak values  are lower than 50 mNm. In particular, the Model 2 with high PR presents the lower cogging torque due to the strong flux modulation effects [22]. Fig. 11(b) shows the on-load toque waveforms, where the torque ripples of the two models are also low.
The detailed parameters and performances of two models are concluded in Table VI , where the analysis results match well with finite element analysis (FEA), which verify the feasibility of the proposed design and analysis. Both Model 1 and Model 2 can achieve targeted power and torque within the limit of machine size, power factor, efficiency and PM consumption. It can be noticed that benefited from high PR, Model 2 has higher magnetic loading B, which is 2.7 times to Model 1. As a result, Model 2 can achieve targeted power with lower electric loading A and higher efficiency η, although it suffers from the longer end-winding. Moreover, the power factor of Model 2 with high PR is inferior to Model 1, due to the larger inductance. It is worth noting that for the proposed Model 1 and 2, only the length-radius ratio k shape = 0.6 is selected, which is based on the size upper limitation. For the conditions with k shape higher than 0.6, the copper loss proportion of end-winding can be decreased, which is conducive to enhance the active electric loading A under the same total copper losses. In other words, to achieve the targeted torque, the copper losses caused by end-winding can be decreased and machine efficiency can be improved. As a result, the efficiency advantage of Model 2 with high PR = 14 can be more significant.

VI. EXPERIMENTAL VALIDATIONS
In this section, two SPMVM prototypes of 18/14/4 (slot/PMpole-pair/winding-pole-pair) and 15/14/1 with pole-pair ratio PR 3.5 and 14 are manufactured and tested to verify the analysis mentioned above. Fig. 12 performs the pictures of stator, surface-mounted PM rotor, winding, and test platform, respectively.
The main parameters and electromagnetic performances are listed in Tables VI and VIII, where the two prototypes have the same stator outer diameter, stack length, and series turn per phase. And the same PM rotor has been used to save the manufacture cost.
At first, the no-load performances of proposed SPMVMs are tested. Fig. 13 depicts the phase back-EMF waveforms at rotation speed 300 r/min, where the analysis, FEA and measured results have the good agreement. It can be seen that prototype 15/14/1 with high pole-pair ratio PR of 14 has the higher back-EMF due to the strong flux modulation effects and high magnetic loading, where the fundamental amplitudes are 218.3 V and 88.1 V, respectively. And the waveforms are sinusoidal.
Then, the on-load performances of proposed SPMVMs are further investigated. Considering winding inductance has small variation with the rotation of rotor, the conventional I d = 0 Further, the power factor has been compared in Fig. 15, where prototype 18/14/4 can acquire the high power factor due to  the small inductance. Whereas with the increase of current, the power factor of prototype 15/14/1 is quickly declined due to the large inductance. At rated condition, the power factor of prototype 18/14/4 and prototype 15/14/1 are 0.95 and 0.86, respectively. Fig. 16 shows the measured torque, and phase current/voltage waveforms at the rated conditions. It can be seen that the torque ripples for the two prototypes are low. Moreover, it can be noticed that the phase angle difference between phase voltage and phase current of the prototype 18/14/4 is smaller than the prototype 15/14/1 counterpart due to the low inductance, which reflects the high power factor of SPMVMs with low PR.
Therefore, both the two prototypes can meet rated power, where the prototype 18/14/4 has the higher power factor, while the prototype 15/14/1 has the higher efficiency. Moreover, all the analysis, FEA, and measured results match well with each other, which confirmed the feasibility of the proposed analysis and design approach for SPMVM.

VII. CONCLUSION
This paper presents design considerations of SPMVMs to make a good compromise between machine performances. By means of closed-from per-unit equations, the negative correlation between torque density and power factor is revealed. Further, the cross-coupling influence of critical parameters, including magnet thickness to air-gap length ratio k g , stator slot opening width to slot pitch ratio k s , pole-pair ratio PR, armature winding pole-pair P a , and length-radius ratio k shape , on machine performances are clarified. Then, the preferable selection ranges are determined. In particular, the general design flowchart of SPMVMs is proposed, which can avoid the blind attempt and provide a good guidance for engineering designers.
The following conclusions are drawn: 1) When length-radius ratio k shape larger than 4, SPMVMs with high pole-pair ratio PR are preferable due to the stronger flux modulation effects. However, benefited compact end-winding lengths, regular fractional slot concentrated winding PMSMs and low PR SPMVMs are attractive when k shape smaller than 4. 2) Under the same PR and copper losses, the larger armature winding P a is conducive to decrease end-winding length and inductance, so as to improve electric loading A and power factor PF. However, with k shape smaller than 4, SPMVMs with smaller P a are attractive due to the higher magnetic loading B.
3) The preferable magnet thickness to air-gap length ratio k g of SPMVMs is lower than 6. With the increase of PR, the optimal k g is decreased to strengthen flux modulation effects. However, the relatively large k g is beneficial to weaken inductance. 4) The preferable stator slot opening width ratio k s of SP-MVMs is around 0.4∼0.7. With the increase of PR, the optimal k s is slightly enlarged to strengthen flux modulation effects.

APPENDIX
The coefficients of the air-gap relative permeance function can be figured out by conformal mapping method proposed in [2]. k s = w s t s , k P = g r so , k split = r g r so (18) Z s = P a + P r = (1 + P R)P a (19) Fig . 17 presents the variation of air-gap relative permeance with k s , under different stator slot number Z s and air-gap length to stator outer radius ratio k P . It can be seen that with the increase of k s , the air-gap relative constant permeance Λ 0 is gradually decreased, while the air-gap relative fundamental permeance Λ 1 increases first and then decrease. The relatively large Λ 1 can be acquired with k s around 0.4∼0.7. Moreover, with the increase of stator slot number Z s , Λ 0 can be increased, while Λ 1 is decreased, which means that under the same pole-pair ratio PR, the flux modulation effects would be impaired with large armature winding pole-pair P a . Similarly, with the increase of air-gap length g and the ratio k P , the flux modulation effects is deteriorated. In this paper, the air-gap length to stator outer radius ratio k P is set as 0.1% for direct-drive application [23].