Torque Capacity Improvement of Flux-Switching PM Machines Based on Directional Stator Permeance Design

As one type of flux modulation machines, flux-switching permanent magnet (FSPM) machines present high sensitivity to airgap structures. Therefore, both stator/rotor teeth and slot/pole combinations have significant influences on machine performance. However, the relationships between the optimal stator structure and maximum torque capability of the FSPM machine are barely investigated. Therefore, in this article, we are devoted to proposing a directional stator permeance design approach to achieve the maximum torque of the FSPM machines under a given rotor and reveal the corresponding stator structure. First, the relations between torque and air-gap permeance are presented based on a constructed torque contribution equation, where amplitudes and phase angles of the stator permeance harmonics are determined. Then, main permeance harmonics are directionally optimized to enlarge positive torque while negative contributions are inversed to be positive. Especially, two FSPMs with 6-slot/19-pole and 6-slot/13-pole are chosen as design examples, and their optimal design processes and torque performances have been deeply analyzed, which verifies the effectiveness of the proposed design approach.

Abstract-As one type of flux modulation machines, fluxswitching permanent magnet (FSPM) machines present high sensitivity to airgap structures.Therefore, both stator/rotor teeth and slot/pole combinations have significant influences on machine performance.However, the relationships between the optimal stator structure and maximum torque capability of the FSPM machine are barely investigated.Therefore, in this article, we are devoted to proposing a directional stator permeance design approach to achieve the maximum torque of the FSPM machines under a given rotor and reveal the corresponding stator structure.First, the relations between torque and air-gap permeance are presented based on a constructed torque contribution equation, where amplitudes and phase angles of the stator permeance harmonics are determined.Then, main permeance harmonics are directionally optimized to enlarge positive torque while negative contributions are inversed to be positive.Especially, two FSPMs with 6-slot/19pole and 6-slot/13-pole are chosen as design examples, and their optimal design processes and torque performances have been deeply analyzed, which verifies the effectiveness of the proposed design approach.

I. INTRODUCTION
C ONVENTIONAL flux-switching permanent magnet (FSPM) machine is normally equipped with a segmented Hong Chen is with the School of Electrical Engineering and Automation, Shandong University of Science and Technology, Qingdao 266590, China (e-mail: hongchen@sdust.edu.cn).
Dawei Li and Ronghai Qu are with the State Key Laboratory of Advanced Electromagnetic Engineering and Technology, School of Electrical and Electronic Engineering, Huazhong University of Science and Technology, Wuhan 430074, China (e-mail: daweili@hust.edu.cn;ronghaiqu@hust.edu.cn).
Color versions of one or more figures in this article are available at https://doi.org/stator, fractional slot concentrated windings, stationary spoke-PMs, and a robust salient rotor.[1].Hence, the FSPM machines can achieve good thermal management and high-speed operation without magnet retention.In addition, the FSPM machines have an essentially sinusoidal phase back electromotive force (EMF), relatively high torque density, low torque ripple, and good fluxweakening capability [2].Therefore, the FSPM machines have become attractive candidates for electrical vehicle propulsion, servo system, and wind power generation applications [3], [4], [5].
According to the operation principle of the FSPM machines, there are lots of feasible stator/rotor pole combinations [8], [17], which have significant influences on torque capacity and performance.To achieve high torque capacity, the optimal stator/rotor pole combination of the FSPM machine must be determined.However, the FSPM machines with different stator structures have various optimal pole combinations.As investigated, for the U-core FSPM machines, the optimized rotor pole number should be close to the stator pole number, and the classic combinations are 12/10 and 12/13 [10].However, for the multitooth FSPM machine, the 6/19 combination is a common selection [11].Furthermore, for the E-core FSPM machines, it indicates in [12] that the optimal combinations should be that the rotor pole number is slightly smaller than twice the stator pole number, 0278-0046 © 2023 IEEE.Personal use is permitted, but republication/redistribution requires IEEE permission.
Authorized licensed use limited to the terms of the applicable license agreement with IEEE.Restrictions apply.such as the 6/11 combination.Moreover, for the C-core FSPM machines, the 6/13 combination is reflected to exhibit relatively higher torque density [15].However, most of the conclusions are based on finite-element analysis (FEA) comparisons, and difficult to use the results for practical design, especially for novel FSPM topologies.Since there are barely any pieces of literature to provide a general method to get the optimal stator structure for FSPM machines with random rotor pole numbers, it brings computation burden and heavy workloads for designers and researchers to try and compare separately [9], [15].To solve this problem, a mathematical method is needed to show the relationships between the torque capacity, the stator structure, and stator/rotor pole combinations.In previous research, although the traditional torque equations can be used to analyze the torque components conveniently from the perspective of the airgap field harmonics [18], [19], [20], it fails to show the relationships between torque capacity and stator structure clearly.
Therefore, this article proposes a directional stator permeance design approach to achieve the maximum torque capacity under a given rotor from a novel perspective of torque contributions by amplitudes and phase angles of the stator permeance harmonics.This article is organized as follows.The principle and flowchart of the directional stator design method are introduced in Section II.Then the torque contributions made by the stator permeance harmonics are calculated analytically, and the optimal strategy is presented in Section III.In Section IV, two FSPM machines with 6/19 and 6/13 combinations are taken as examples and optimized based on the proposed approach.Then, in Section V, the prototype of a 6/19 FSPM machine is manufactured and measured to validate the analytical and FEA.Finally, some conclusions are drawn in Section VI.

II. DIRECTIONAL STATOR PERMEANCE DESIGN METHOD
From the view of stator permeance, different stator structures, including the conventional U-core type, C-core type, E-core type, and multitooth topologies, can be expressed by a general analytical model.Thus, a C-core FSPM machine can be set as the primary topology, then other FSPM topologies with different stator types can be reflected by different values of permeance harmonics.In addition, a conventional rotor structure is applied in this general model.According to the simplified linear stator permeance model illustrated in Fig. 1, the stator permeance function Λ s (θ) considering winding slots and PM slots [20] can be expressed by Fourier series where Λ s0 and Λ sm are the amplitudes of the zeroth and mth stator permeance harmonics, respectively, Z s is the stator main tooth number, θ Λm is the phase angle of the mth stator permeance harmonic and the value is 0 or π for each specific harmonic.Moreover, the phase angle for the zeroth harmonic is set to θ Λ0 = 0. Then the procedure of the directional tooth design approach is shown in Fig. 2, which mainly consists of three steps.

A. Step I: Original Stator Permeance Investigation
According to the initial parameters set on the FSPM machine, the original amplitude Λ sk (k = 0, 1, 2, 3 …) and phase angle θ Λk of each stator permeance harmonic and corresponding torque contribution can be obtained, based on a proposed analytical model.

B. Step II: Directional Permeance Harmonic Optimization
To reveal the relationships between permeance harmonics and torque contributions, torque contribution coefficient λ k is Authorized licensed use limited to the terms of the applicable license agreement with IEEE.Restrictions apply.
proposed, which is calculated as where T avgk is the average torque produced by the kth order permeance harmonic.The amplitude of λ k reflects the weight given to the corresponding harmonic.Thus, permeance harmonics with low weight should be reduced, considering the interaction between the harmonic amplitudes.Moreover, the positive/negative value of λ k means that the produced torque increases/decreases the torque capacity.Based on the optimal strategy, optimized permeance harmonics with Λ sk and θ Λk can be obtained.

C. Step III: Integrated Permeance Waveform
The third step is to synthesize the stator permeance harmonics to get the integrated permeance waveform, which directly reveals the optimized stator tooth configuration with fuzzy dimensions.It should be noticed that the optimized amplitudes Λ sk and phase angles θ Λk for the maximum torque may synthesize an improper shape or small slot area as a stator, and the values should be revised and checked again.Then, optimize the detail dimensions of the stator based on FEA, including the tooth width, slot area, slot height, etc.
Finally, the stator structure and dimensions of the FSPM machine can be obtained.

A. Airgap Flux Density
The no-load airgap flux density B g (θ, t) can be calculated as [19] where F pm (θ) is the MMF produced by PMs, Λ r (θ, t) is the rotor permeance function with slotted rotor and slotless stator, g is the airgap length, and μ 0 is the vacuum permeability.The expressions of F pm (θ) and Λ r (θ, t) can be given by ( 4) and ( 5), respectively, as follows: where F pmi is the amplitude of the ith harmonic of the PM MMF, Λ r0 and Λ rn are the amplitudes of the zeroth and nth rotor permeance harmonics, respectively, Ω r is the mechanical angular velocity, and θ r0 is the initial angle.Therefore, the no-load airgap flux density ignoring stationary harmonics can be obtained as

B. Torque Contribution Equation
The electromagnetic torque T e of the FSPM machines can be calculated by where e q (q = A, B, C) is the back EMF of each phase, i q is the winding excitation, r g is the airgap radius, l st is the stack length, and N j (θ) is the winding function.Then the average torque with i d = 0 can be expressed as (8) where , N s is the turns in series per phase, I m is the amplitude of phase current, and k wν is the winding factor of the νth harmonic, i d is the d-axis current, and λ k is the contribution coefficients (k = 0, 1, 2, 3 …).
It can be observed that there are two components of the average torque, as listed in Table I.Under the condition of the same stator/rotor pole combination and volume limitation, it is reflected that Λ sk and θ Λk directly determine the torque capacity.Moreover, according to (8), since Λ sk is positive, only the positive torque contribution ecoefficiency λ k can result in positive and useful torque components, which is directly related to the phase angle θ Λk and the positive/negative of the winding factor coefficient K ski .In addition, the coefficients K s0i and K smi reflect the effects of the slot-pole combination and the winding configuration.

C. Directional Stator Permeance Optimization
Since the torque contribution made by Λ s0 accounts for a relatively large proportion of the total average torque to maximize the torque capacity, the mth stator permeance harmonic with the optimized θ Λm should generate the same torque polarity with that of the zeroth-order permeance harmonic.Hence, the optimal strategy of the phase angle θ Λm can be obtained by where T avg0 and T avgm are the original torque contributions of the zeroth-and mth-order harmonics.Therefore, when T avgm and T avg0 are both positive or negative, the optimal phase angle θ Λm is kept to be θ Λm .On the contrary, θ Λm should be inversed and become (θ Λm + π).
As for the optimal amplitude of the permeance harmonic Λ sk , it ranges from the minimum value Λ s.min to the maximum stator permeance Λ s.max .In addition, the amplitudes Λ sk are also restricted mutually considering integration.As a result, the amplitudes of the optimal permeance harmonic Λ sk are constricted and determined by where h s is the slot depth.
Based on the above analyses, the objective function of the optimal stator permeance parameters is defined as where T ripple % is the ratio of the peak-peak torque ripple T ripple to the average torque T eavg , and the torque ripple T ripple can be expressed as Then, by optimizing the amplitudes and phase angles of the stator permeance harmonics, an optimal stator structure can be obtained for the FSPM machine to achieve the maximum torque capacity.Importantly, it should be noted the proper slot area should be considered in the process of final result selection.

IV. OPTIMAL DESIGN AND PERFORMANCE ANALYSES
In this section, two FSPM machines with different rotor tooth numbers are designed and optimized based on the proposed directional stator optimization method.

A. Initial Design
According to the design procedure in Fig. 2, the original stator structures of these two FSPM machines are set to be the C-core type, and the stator slot opening b so = 18 mm is initially built, as illustrated in Fig. 3(a) and (b).The main parameters are listed in Table II.
The nonoverlapping windings are adopted, and the pole pair number is set to 2. Then the amplitudes and phase angles of the main stator permeance harmonics can be calculated and listed in Table III.Under the initial condition, the 0th-3rd stator Authorized licensed use limited to the terms of the applicable license agreement with IEEE.Restrictions apply.permeance harmonics feature relatively large amplitudes, while the 2nd, 3rd, and 6th harmonics have opposite phase angles compared to the other harmonics.Furthermore, the airgap flux density distributions are calculated and presented in Fig. 4. It can be observed that the analytical results match well with the FEA results.
Moreover, the phase back EMF and the electromagnetic torque waveforms are calculated based on (7), as presented in Figs. 5 and 6(a).It can be observed that the back EMF and average torque of the 6/19 model are much lower than those of the 6/13 model.Thereinto, the amplitudes of the fundamental back EMF are 12.6 V and 62.8 V, respectively, and the average torques are 2 N•m and 9.8 N•m, respectively.The rated electric loadings are set to 253.3 A/cm for these two FSPM machines.
It can be explained by the different contributions made by the stator permeance harmonics, as investigated in Fig. 6(b).For the 6/19 model, the zeroth permeance harmonic generates the largest share of the average torque.Moreover, the zeroth, 1st, and 6th harmonics produce positive torque (7.1 N•m), while the torque contributions made by the 2nd-5th harmonics are negative (−5.0 N•m).For the 6/13 model, the torque contribution by the 1st permeance harmonic is the maximum and up to 8.8 N•m.In addition, only the 2nd harmonic contributes negative and low torque (−0.7 N•m), and all the other harmonics produce positive contributions.
Thus, although the gear ratio (ratio of the rotor pole pair number to the pole pair number of armature winding) of the 6/19 model is larger than that of the 6/13 model, the torque density of the 6/19 model is much lower because of the negative torque contributions made by several permeance harmonics.What is more, it can be extensively applied to explain the same phenomenon in other flux modulation machines.

B. Stator Permeance Optimization
To obtain the maximum average torque and minimized torque ripple, the multiobjective genetic algorithm is adopted to optimize the main stator permeance parameters based on the directional permeance design method.According to the torque contribution investigation of the 6/19 model above, the contribution coefficients λ k can be calculated and summarized in Table IV.Based on the directional permeance design method, the optimal phase angles θ Λk of the permeance harmonics with negative λ k (2nd-5th orders) should be inversed to produce positive torque.Furthermore, the amplitude of each permeance harmonic should be optimized to maximize the torque capacity.As listed in Table IV, it can be observed that λ 1 is only 0.09, which means that the 1st harmonic with a large amplitude produces rather low torque.Therefore, the optimal Λ s1 should be decreased.Meanwhile, the harmonics with relatively higher λ k , including the 2nd, 3rd, 4th, and 6th orders, should be enlarged.Since both  the original amplitudes Λ s5 and λ 5 are relatively low, it can be ignored to simplify the optimal process.As a result, the optimal amplitude ranges of the main permeance harmonics are set as listed in Table IV.
With the constrictions and optimal function equations, the average torque (larger than 14.8 N•m) and torque ripple of the cases with different Λ sk are scattered in Fig. 7(a).The red and green dots indicate the models with the maximum torque T max and minimum torque T min in the selected set.It can be seen that the maximum torque can reach to 18.1 N•m.In addition, the torque ripple is much less compared to that of the original stator due to the reduced Λ s1 .Then the integrated waveforms of  the main stator permeance harmonics are illustrated in Fig. 7(b).The directionally optimized permeance waveforms of both T max and T min models reveal that the optimal stator structure for the 6/19 FSPM machine is the multitooth type.Accordingly, the amplitudes and phase angles of the T max case and the original model are compared in Fig. 7(c).The sign (+/−) means that each optimized permeance harmonic increases/decreases the total average torque.It can be seen that all the phase angles of the permeance harmonics contributing to negative torques are inversed.Therefore, most of the permeance harmonics will produce positive torque contributions and the torque capacity of the FSPM machine with the optimized stator can be significantly enlarged.
Likewise, the 6/13 model is optimized based on the directional permeance optimization, and the optimal ranges of the permeance parameters are set in Table V.Since the 1st order of the permeance harmonic produces the most torque, the contribution coefficient λ k is set as the base.Although the zeroth harmonic generates much less torque, the amplitude Λ s0 should not be reduced a lot due to the constant airgap length.Consequently, the optimal ranges of the other harmonics are smaller than that of the 6/19 model.Then Fig. 8(a) shows the comparison of the optimized and the original permeance waveforms.It indicates that the variations of the optimal and original permeance are rather small, as compared to Fig. 8(b).Therefore, it can be expected that the torque improvement is less than that of the 6/19 model.

C. Sensitivity of Stator Tooth Dimensions
To gain the final stator structure from the stator permeance waveform, the relationships between the stator design parameters and the permeance harmonics should further be investigated.The main stator tooth parameters are depicted and labeled in Fig. 9, including b o , w t1 , w t2 , w t3 , w t4 , h s1 , and h s2 .Considering the symmetry and simplicity of the stator, the assisted tooth number is set to six.With the variations of the stator design parameters, the stator structure can be turned into the corresponding type.
Then a sensitivity analysis is proceeded to evaluate the influence of the stator design parameters on each permeance harmonic.Since the saturation effect of the FSPM machines with different stator types and slot-pole combinations is quite different, then the saturation impact on the stator permeance is ignored in the general analysis.Therefore, the sensitivity of the permeance harmonics can be calculated based on the DOE method and the Pearson correlation coefficient [21], [22] ρ where N is the sample number, x i is the ith design and Y i is the optimization objective.Fig. 10 shows the sensitivity of the stator tooth parameters with respect to permeance harmonics.The phase angle is considered in the value of each permeance as positive (0) or negative (π).The value of the zeroth permeance harmonic increases as b o increases and assisted tooth widths decrease.Besides, the influences of the stator design parameters on the 2nd and 4th harmonics are rather similar but contrary to that of the zeroth harmonic.Likewise, the sensitivities of the design parameters concerning the 1st and 3rd harmonics are contrary.Except for the influence of b o , the sensitivity to the 5th harmonic is similar to that of the 1st harmonic.Furthermore, since saturation is not considered in this section, the sensitivities of h and h s2 concerning each permeance harmonic are near zero and

D. Performance Analysis
Fig. 12 shows the back EMF and torque comparisons of the optimized and initial FSPM machines with the 6/19 model.With the optimized stator permeance, the back EMF of the optimized 6/19 model is largely increased compared to that of the initial waveforms predicted by FEA, as shown in Fig. 12(a).The RMS values predicted by FEA are 64.5 and 8.8 V, respectively.Moreover, as calculated in Fig. 12(b), the average torque of the optimized 6/19 model based on the analytical method can be improved to 18.1 N•m, which matches well with the FEA results without considering core saturation.However, according to the flux density distribution of the 6/19 model under the rated load shown in Fig. 13(a), there is severe saturation in the stator.Therefore, the permeance harmonics are influenced by the saturation effect, as presented in Fig. 12(c).As a result, the average torque is adjusted to 12.3 N•m based on FEA.
For the 6/13 FSPM machine, the back EMF improvement is less significant than that of the 6/19 model, and the RMS values of the optimized and initial models are 49.4V and 44.5 V, respectively, as shown in Fig. 14(a).Moreover, Fig. 13(b) reveals that the core saturation is less severe than that of the 6/19 model,  and then the torque reduction because the saturation would be As compared in Fig. 14(b), the analytical result of the average torque with the optimized permeance can be improved to be 11.8 N•m.Moreover, the average torque is adjusted to 10.9 N•m based on FEA, which is close to the analytical results.
Furthermore, the self-and mutual inductances of 6/19 and 6/13 models are analyzed and compared in Fig. 15, predicted by FEA.It can be observed that both the self-and mutual inductances of the optimized 6/19 model are enlarged, compared to that of the initial stator structure.Since the stator permeance harmonics have been optimized to maximize the average torque,  the 2nd and 4th airgap flux density harmonics are increased, which result in inductance enlargement [23].While for the 6/13 model, the inductances of the optimized model are slightly lower than that of the initial model due to the smaller Λ s0 .
In addition, Fig. 16 compares the FE-predicted power factor versus phase current curves of the 6/19 and 6/13 models.The power factor can be calculated by where ω is the electrical angular velocity, and L s is the synchronous inductance.Although the inductance of the optimized 6/19 model is enlarged, the back EMF increases much more.
Authorized licensed use limited to the terms of the applicable license agreement with IEEE.Restrictions apply.Therefore, the power factor gets higher than that of the initial model.For the 6/13 model, the power factor of the optimized model is slightly higher than that of the initial model.
Moreover, the detailed electromagnetic performances of the initial and optimized FSPM machines with the same electrical loading are summarized in Table VI.It can be observed that the torque capacities of the FSPM machines with directionally optimized stator structures are improved effectively.

V. EXPERIMENTAL VALIDATION
To verify the foregoing analyses and prediction, a prototyped FSPM machine with a multitooth stator and 6/19 stator/rotor combination is manufactured and tested.The main parameters and materials are listed in Table VII.Fig. 17(a) presents photos of the main parts of the prototype, and the output torque is measured by the torque transducer, as shown in Fig. 17(b).
Fig. 18 shows the measured and FE-predicted back EMF waveforms and harmonic spectra at 600 r/min.It can be seen that the simulated and measured results match well, as shown in Fig. 18  than the 2-D-FEA due the relatively small ratio of axial length to stator outer diameter (40 mm/122.5 mm), which results in a significant end-effect.However, the measured result is slightly lower than the 3-D-FEA results because of fabrication tolerances.
The measured torque waveform is shown in Fig. 19.Fig. 20 presents the curves of the steady-state average torque versus phase current predicted by FEA and measured results.As can be seen, the difference gets larger with increasing loading due to the core saturation and increased magnet temperature.Furthermore, Fig. 21 compares the curves of the efficiency versus different phase currents at the rated speed.Due to the effect of mechanical friction and simulation errors of core loss prediction, the measured efficiencies are lower than the FE-predicted results.Overall, the tested and FE-predicted results are in good Authorized licensed use limited to the terms of the applicable license agreement with IEEE.Restrictions apply.agreement, which can verify the correctness of the theoretical and simulated analyses.

VI. CONCLUSION
In this article, a directional stator permeance design approach was proposed to maximize the torque capacity of the FSPM machines under a given rotor.To investigate relationships between the average torque and stator structures, the function of the torque and amplitudes and phase angles of the stator permeance harmonics T eavg = f (Λ sk , θ Λk ) was established.The main principle of the directional stator permeance design method was to make all the permeance harmonics with high torque contribution coefficients produce positive torque, and restrict the negative effects of other permeance harmonics by comprehensively designing the amplitudes and phase angles of the stator permeance harmonics.Based on the proposed design approach, two examples with different slot/pole combinations were designed and analyzed.Both the FEA and experimental results showed that the proposed method could derive the optimal stator structures to the torque capacities of FSPM machines with the original rotor and winding configuration.

Manuscript received 27
October 2022; revised 22 February 2023 and 3 April 2023; accepted 20 May 2023.Date of publication 5 June 2023; date of current version 18 December 2023.This work was supported in part by Shandong Provincial Natural Science Foundation under Grant ZR2020QE218 and in part by the National Natural Science Foundation of China under Grant 52122705.(Corresponding author: Hong Chen.)

Fig. 10 .
Fig. 10.Sensitivity of main stator design parameters with respect to permeance harmonic.
(a).In addition, as analyzed in Fig.18(b), the amplitudes of the fundamental components based on 3-D-FEA are lower

TABLE II MAIN
PARAMETERS OF THE INITIAL FSPM MACHINES

TABLE III PARAMETERS
OF THE ORIGINAL STATOR PERMEANCE HARMONICS

TABLE V OPTIMAL
PERMEANCE PARAMETER OF THE 6/13 MODEL

TABLE VI PERFORMANCE
COMPARISON OF THE FSPM MACHINES (I rms = 5 A, SPEED = 600 R/MIN)

TABLE VII KEY
PARAMETERS OF THE PROTOTYPE