LINEAR MOTORS produce a direct thrust force without

5436 IEEE TRANSACTIONS ON INDUSTRIAL ELECTRONICS, VOL. 60, NO. 12, DECEMBER 2013 Investigation and General Design Principle of a New Series of Complementary and Modular Linear FSPM Motors Ruiwu Cao, Student Member, IEEE, Ming Cheng, Senior Member, IEEE, andweihua,member, IEEE Abstract The conventional linear flux-switching permanentmagnet (FSPM) motors directly split from a rotary FSPM motor suffer from drawbacks such as unbalanced magnetic circuit of end coil, bigger cogging force, and force ripple. In this paper, to reduce cogging force and thrust force ripple, some complementary and modular linear FSPM (LFSPM) (MLFSPM) motors with mover/stator pole pitch ratio τ m /τ s of about one, namely, τ m /τ s =10/12, 11/12, 12/12, 13/12, 14/12, 15/12, will be investigated using finite-element method (FEM) and experimental method at first. Then, another new LFSPM motor with τ m /τ s = 3 is analyzed. Based on τ m /τ s =3, some new MLFSPM motors are designed, investigated, and compared using FEM. To fully investigate these motors, the optimal MLFSPM motor with τ m /τ s =3is quantitatively compared with the two optimal motors with τ m /τ s 1. Finally, the general design principle for this series of MLFSPM motors with different τ m /τ s values is concluded. Index Terms Complementary and modular, flux-switching permanent-magnet (PM) (FSPM) motor, linear motor, PM motor. I. INTRODUCTION LINEAR MOTORS produce a direct thrust force without the need of conversion from rotational torque to linear force. Therefore, they have been widely used in transportation systems and industrial application [1] [5]. In recent years, new kinds of linear permanent-magnet (PM) motor [6] [9], namely, the linear structure of stator-pm motors [10], doubly salient PM (DSPM) motors [11], flux reversal PM motors [12], and flux-switching PM (FSPM) motors [13], have attracted wide attentions, in which both the PMs and the armature windings are all located in the short primary mover, while the long secondary stator is only made of iron. Hence, this kind of linear motors incorporates the merits of simple structure of linear induction motors and linear switched reluctance motors and Manuscript received July 26, 2012; revised October 29, 2012; accepted November 16, 2012. Date of publication November 30, 2012; date of current version June 21, 2013. This work was supported in part by the 973 Program of China under Project 2013CB035603, in part by the Specialized Research Fund for the Doctoral Program of Higher Education of China under Project 20090092110034, in part by the National Natural Science Foundation of China under Project 50907031, in part by the Scientific Research Foundation of the Graduate School of Southeast University, and part by the Program for Postgraduate Research Innovation in General Universities of Jiangsu Province 2010 under Project X10B_066Z. The authors are with the School of Electrical Engineering, Southeast University, Nanjing 210096, China (e-mail: ruiwucao@gmail.com; mcheng@seu.edu.cn; huawei1978@seu.edu.cn). Color versions of one or more of the figures in this paper are available online at http://ieeexplore.ieee.org. Digital Object Identifier 10.1109/TIE.2012.2230605 Fig. 1. Cross section of conventional LFSPM motor with additional tooth and the E-shaped module. (a) Motor 4 ABC. (b) E-shaped module. high power density of linear synchronous PM motors, which are perfectly suited for long stator applications. It has been identified that the FSPM motor can offer high power density [14], sinusoidal back electromotive force (EMF), and fault-tolerance capacities [15], [16], as compared with the DSPM motors. The conventional linear FSPM (LFSPM) motors directly split from the rotary P m /P s =12/14-pole FSPM motors without additional teeth will suffer from drawbacks of unbalanced magnetic circuit in the end coil and bigger cogging force, where P m and P s are the stator and rotor pole numbers of a rotary FSPM motor, respectively. In order to balance the end effect for the end coil of the conventional LFSPM motors, two additional teeth can be added at each end of its mover, as shown in Fig. 1(a). This motor is named as 4 ABC in this paper, and its mover and stator pole pitch ratio τ m /τ s is equal to 14/12. Also, its cogging force can be reduced by adjusting the additional teeth position [17]. However, the two additional teeth cannot totally balance the unbalanced magnetic circuit of the end coils due to the fact that the flux linkage in the middle coils is excited by two PMs while the one of the end coils is just excited by one PM. To solve the unbalanced magnetic circuit problem, a modular LFSPM (MLFSPM) motor shown in Fig. 2(a) based on the motor shown in Fig. 1(a) has been investigated in [18]. So, the back EMFs of its middle coils, end coils, 0278-0046/$31.00 2012 IEEE

CAO et al.: INVESTIGATION AND GENERAL DESIGN PRINCIPLE OF A NEW SERIES OF LINEAR FSPM MOTORS 5437 these two types of motors, namely, one with τ m /τ s 1 and another with τ m /τ s =3, and disclose their essential principles. Based on this analysis, some new MLFSPM motors with different τ m /τ s values are proposed, investigated, and compared. Then, the general design principles of this series of MLFSPM motors with different τ m /τ s values are concluded. II. MLFSPM MOTORS WITH τ m /τ s NEAR ONE A. Structure and Operation Principle The basic E-shaped module and some key parameters of this motor with τ m /τ s =14/12 are defined as shown in Fig. 1(b). It can be seen that the relative displacement between the two E-shaped modules of motor ABCABC in Fig. 2(a)λ 1 is equal to 7τ s and the relative displacement between the two E-shaped modules of two adjacent phases λ 2 is equal to (2 + (1/3))τ s. So, this motor is a three-phase motor without the complementary performance as in the rotary FSPM motor [21]. In fact, it is not necessary to obtain a three-phase CMLFSPM motor with different τ m /τ s values from the rotary structure, which can be obtained directly using the following relationships. 1) To obtain a complementary structure, the relative displacement between the two E-shaped modules of one phase needs to satisfy the following relationships: λ 1 =(j ± 1/2)τ s. (1) Fig. 2. Cross section of LFSPM motors. (a) Motor ABCABC. (b) Motor AABBCC [19]. (c) Motor ABC [20]. (d) Motor ABC ABC. (e) Motor 2A2B2C. and three-phase coils are balanced. Also, the thrust force generated by per unit volume of PM of this MLFSPM motor is higher than that of the motor 4 ABC when the dimension of each PM of both motors is the same. However, this MLFSPM motor based on the 12/14-pole rotary FSPM motor suffers from drawbacks of nonsinusoidal back EMF particularly at higher air-gap flux density, bigger cogging force, and thrust force ripple. This is because the two coils of each phase have no complementary performance. So, this motor is named as ABCABC in this paper. To incorporate the merits and mitigate the deficiency of the motor ABCABC, a complementary MLFSPM (CMLFSPM) motor shown in Fig. 2(b) has been proposed and investigated in [19], in which the relative displacements between the two E-shaped modules of one phase are mutually 180 electrical degrees (180 ) apart. So, it can offer sinusoidal back EMF, smaller cogging force, and thrust force ripple. This motor is named as AABBCC in this paper. Currently, another LFSPM motor with τ m /τ s =3[20] first proposed by Lin and Heilig shown in Fig. 2(c) has attracted some attentions. However, the relationships and differences between these motors have not been fully investigated and compared. The key of this paper is to fully investigate and quantitatively compare 2) For a three-phase motor, the relative displacement between the two E-shaped modules of the adjacent two phases must satisfy λ 2 =(k ± 1/3)τ s or λ 2 =(k ± 1/6)τ s (2) where j and k are all positive integers. Also, there are two structures for these CMLFSPM motors satisfying the relationships (1) and (2). 1) First, the two E-shaped modules of one phase are set next to each other [19], as shown in Fig. 2(b), named as AABBCC. 2) Second, the first E-shaped modules of three phases are put together in sequence, as shown in Fig. 2(d). Because the two E-shaped modules of one phase are mutually 180 apart, this structure is named as ABC ABC. It should be mentioned that, if τ m /τ s =1, the negative and positive back-emf waveforms induced in coil A1, coil A2, and phase coil A are all symmetrical. Hence, the structure with λ 2 satisfying (2) and λ 1 satisfying the following relationship can also be adopted: λ 1 = jτ s. (3) Because the two E-shaped modules of one phase of this motor are mutually 360 apart, this structure is named as 2A2B2C, as shown in Fig. 2(e), or ABCABC. To clearly understand these structures, their acronym name and description are summarized and listed in Table I.

5438 IEEE TRANSACTIONS ON INDUSTRIAL ELECTRONICS, VOL. 60, NO. 12, DECEMBER 2013 TABLE I MOTOR STRUCTURE DESCRIPTION TABLE II DESIGN SPECIFICATIONS OF THE MOTOR AABBCC B. Comparison Rules and Design Specifications To optimize, analyze, and compare different MLFSPM motors, it is necessary to explain some key parameters and define some coefficients. 1) Different from rotary FSPM motors, the PMs are designed a little shorter than the mover tooth. The dimensions w mt, w ms, and w spm, as shown in Fig. 1(b) and Table II, satisfy the relationship w mt = w ms = w spm = τ m /4. 2) The dimensions h m, l m, g, h my, h st, h sy, τ m, h s, I max, and N coil and the total PM volume of all the MLFSPM motors with τ m /τ s that is about one are kept constant and designed the same as an optimal prototype motor as listed in Table II. 3) The stator teeth tip and stator teeth-yoke width coefficients are defined as k st = w st /w mt (4) k sty = w sty /w mt. (5) 4) The average thrust force and force ripple of all motors are calculated by means of finite-element method (FEM) based on i d =0control method, namely, the angle between back EMF and current that is equal to zero [19]. C. Electromagnetic Performance Comparison To obtain an optimal MLFSPM motor, some structures with different τ m /τ s values are designed and listed in Table III. Because coefficient k st is sensitive to the cogging force and force ripple, k st and k sty are optimized for different MLFSPM motors while other parameters are kept constant, as listed in Table II. Fig. 3 shows the cogging force F _cog, average thrust force F _avg, and thrust force ripple F _ripple at rated current of the two CMLFSPM motors with τ m /τ s =13/12 at different k st. It can be seen that k st =1.6 and k st =1.5 are chosen to be the optimal point for motor ABCABC with λ 2 =(2+(1/6))τ s and motor AABBCC with λ 2 =(5 (1/6))τ s, respectively. Based on the same methods, the detailed electromagnetic performances and dimensions of the optimal MLFSPM motors with different τ m /τ s values listed in Table III are chosen and listed directly in Table IV. By considering electromagnetic performance and mover length, ABCABC and AABBCC with τ m /τ s =13/12 are chosen to be the best two motors for the next step work. D. Experimental Results To validate the associated FEM results of the MLFSPM motors listed in Table IV, a three-phase motor prototype based on the optimized dimension of motor AABBCC with τ m /τ s = 14/12, λ 1 =(2+0.5)τ s, and λ 2 =(5+(1/3))τ s, as listed in Table II, has been built. The detailed structure including the mover, stator, U-shaped laminated segment, and the prototype motor is shown in Fig. 4. The simulation and measured opencircuit back-emf waveforms at a speed of 1.05 m/s are compared in Fig. 5. Also, Fig. 6(a) shows the FEM (I max =5, 7, and 9 A) and measured (I max =5A) locked thrust force waveforms versus electrical position β. It can be seen that the thrust force reaches the maximum value at β =90, where the back EMF of phase A reaches the maximum value. Fig. 6(b) shows the FEM and measured locked thrust force waveforms at different applied dc currents I max when the electrical position β =90. The applied dc currents in phases A, B, and C satisfy the following relationship: I a = I max and I b = I c = I max /2. It can be seen that the simulation results exhibit a good agreement with the experimental ones. The discrepancies between the experimental and simulation results are about 10%, which we believe are mainly caused by the end effects as in

CAO et al.: INVESTIGATION AND GENERAL DESIGN PRINCIPLE OF A NEW SERIES OF LINEAR FSPM MOTORS 5439 TABLE III MLFSPM MOTORS WITH τ m/τ s 1 Fig. 3. Force performance of the two CMLFSPM motors with τ m/τ s = 13/12 at different k st s. (a) Motor ABCABC. (b) Motor AABBCC. the stator-pm machines [22], manufacturing imperfection, and measurement error. III. MLFSPM MOTORS WITH τ m /τ s =3 A. Structure Analysis The structure and electromagnetic performances of different MLFSPM motors with τ m /τ s =14/12 have been investigated first from a rotary 12/14-pole FSPM motor. Hence, to clearly analyze the linear motor shown in Fig. 2(c), a rotary motor is also investigated first in this section. Fig. 7(a) shows a single-phase 4/6-pole DSPM motor with concentrated winding. Its operation principle has been investigated in [23]. In [24], a single-phase DSPM motor with full-pitched winding was investigated, as shown in Fig. 7(b). Different from the motor in Fig. 7(a), its operation principle is the same as an FSPM motor. The linear structure of this motor shown in Fig. 7(c) can be obtained by splitting the motor in Fig. 7(b) along the radial direction and unrolling it. Fig. 7(d) shows an E-shaped module, which can be obtained from Fig. 7(c) by taking the end PMs away and adding slot height and PM length. It can be seen that this E-shaped module with τ m /τ s =3is the same as that in the linear motor in Fig. 2(c). Hence, the motor in Fig. 2(c) is a linear flux-switching motor. To compare this motor with the two CMLFSPM motors with τ m /τ s =13/12 listed in Table IV, it is necessary to explain some key parameters. 1) Parameters w m, τ m, τ u, h m, h my, h pm, w pm g, h st, h sy, h s, v, stack factor, and N coil of LFSPM motors with τ m /τ s =3 are designed the same as of the MLFSPM motor listed in Table II. 2) The same rated rms phase current as the motor with τ m /τ s =13/12 is applied in the MLFSPM motors with τ m /τ s =3. For the optimal motor with τ m /τ s =3,the thrust electromagnetic performance at the phase current value when the copper loss P cu (in watts) and current density J s (in amperes per square meter) are the same with the motor with τ m /τ s =13/12 will also be investigated. It can be seen that there are some differences between the two E-shaped modules shown in Fig. 1(b) and Fig. 7(d). 1) Because τ m /τ s =3in Fig. 7(d), if the mover speeds of both motors are the same, the electrical frequency of the motor in Fig. 7(d) is about three times that of the one with τ m /τ s that is about one. 2) To reduce flux leakage, its w mt is designed the same as w st. Hence, w mt in Fig. 1(b) is bigger than the one in Fig. 7(d), while w msm and w spm are smaller than the ones in Fig. 7(d). Therefore, if the N coil, filling ratio, and current density of both E-shaped modules are the same, the phase current of the MLFSPM motors based on Fig. 7(d) is bigger than that based on Fig. 1(b). Because τ m /τ s of the E-shaped module in Fig. 7(d) is equal to three, the positive and negative back-emf waveforms induced in the coil of one E-shaped module are symmetrical, which are the same as the ones with τ m /τ s =1, as discussed in Section II. So, it is not necessary to use two complementary E- shaped modules to constitute a three-phase motor with symmetrical back EMF. So, some three-phase LFSPM motors based on the E-shaped module with τ m /τ s =3can be obtained by using (2). Fig. 8(a) shows an LFSPM motor with λ 2 =(6 (1/6))τ s based on the ABC structure. Between the three-phase E- shaped modules, there are two pieces of flux barrier or iron. If the flux barrier is replaced by iron, the linear motor as shown in Fig. 8(a) is the same as the one in Fig. 2(c). The eight ABC motors with shorter mover can be expressed as λ 2 =(6 1/6)τ s, flux barrier or iron λ 2 =(6+1/6)τ s, flux barrier or iron ABC motors (6) λ 2 =(6 1/3)τ s, flux barrier or iron λ 2 =(6+1/3)τ s, flux barrier or iron.

5440 IEEE TRANSACTIONS ON INDUSTRIAL ELECTRONICS, VOL. 60, NO. 12, DECEMBER 2013 TABLE IV COMPARISON OF THE OPTIMAL MLFSPM MOTORS WITH τ m/τ s 1 Fig. 4. Prototype of motor AABBCC with τ m/τ s =14/12. Similarly, some MLFSPM motors with τ m /τ s =3 based on the structures ABCABC, ABC ABC, AABBCC, and 2A2B2C discussed in Section II can be obtained using (1) (3). Fig. 5. results. Back-EMF waveforms at 1.05 m/s. (a) FEM results. (b) Experimental B. Electromagnetic Performance Analysis It can be seen that motors ABC ABC and AABBCC are complementary, while motors ABCABC and 2A2B2C are not complementary. For the MLFSPM motors with τ m /τ s 1, the complementary performance can offer small cogging force and symmetrical back EMF. In this section, the electromagnetic performances of the MLFSPM motors with τ m /τ s =3based on the structures ABC ABC, AABBCC, ABCABC, and 2A2B2C all need to be analyzed. Before analyzing these MLFSPM motors with τ m /τ s =3,it is necessary to analyze the effect of flux barrier and optimize the stator teeth width w st to reduce the leakage flux. Hence, the effects of flux barrier and w st are first investigated based on motor ABC ABC with λ 1 =(17+0.5)τ s and λ 2 = (6 (1/6))τ s, as shown in Fig. 8(b). The coefficient k wst is defined as k wst = w st /τ s. (7) Fig. 9 shows the F _cog, F _avg, and F _ripple at rated current (I rms =6A) of motor ABC ABC with and without flux barrier as shown in Fig. 8(b) when k wst is in the range of 0.4 0.6 and keeping k st =1. It can be seen that their F _cog and F _ripple reach the minimum value around k wst =0.475 and F _avg reaches the maximum value at k wst =0.45. Hence, k wst =0.45 is adopted in these motors in this paper. Also, it can be seen that F _avg, F _ripple, and F _cog of motor ABC ABC with flux barrier are about 104.3%, 79.5%, and 79.5% of the one without flux barrier at k wst =0.45, respectively. Hence, the flux barrier and k wst =0.45 will be adopted in all the MLFSPM motors with τ m /τ s =3. Fig. 10(a) shows the cogging force waveforms versus electrical position of one E-shaped module with different PM magnetized directions, namely, F _cog E_S denotes the PM magnetized direction that is from right to left while F _cog E_N denotes the PM magnetized direction that is from left to right. It can be seen that F _cog E_S and F _cog E_N are the same. Because some MLFSPM motors with τ m /τ s =3

CAO et al.: INVESTIGATION AND GENERAL DESIGN PRINCIPLE OF A NEW SERIES OF LINEAR FSPM MOTORS 5441 Fig. 6. FEM and measured locked thrust force. (a) Locked thrust force waveforms versus β at different currents. (b) Locked thrust force waveforms versus different applied currents at β =90. Fig. 8. Motor ABC, ABC ABC, and AABBCC with τ m/τ s =3.(a) Motor ABC with λ 2 =(6 (1/6))τ s. (b) Motor ABC ABC with λ 1 = 17.5τ s and λ 2 =(6 (1/6))τ s. (c) Motor AABBCC with λ 1 =6.5τ s and λ 2 =(13 (1/3))τ s. Fig. 7. Cross section of the rotary and linear structure of 4/6 DSPM motors and E-shaped module. (a) Concentrated coil. (b) Full-pitched coil. (c) Linear structure of full-pitched coil motor. (d) E-shaped module. based on ABC, ABCABC, ABC ABC, AABBCC, and 2A2B2C consist of three or six E-shaped modules, their cogging force can be analyzed first using the sum value of three or six cogging forces of one E-shaped module with different electrical degree shifts. For structure ABC, the cogging force of phase-b E-shaped module can be obtained by shifting the cogging force of phase Aby±60 or ±120. Similarly, the cogging force of phase-c Fig. 9. Force performance of motor ABC ABC with and without flux barrier versus k wst. (a)f _avg of both motors. (b) F _ripple and F _cog of both motors. E-shaped module can be obtained based on the same method. Fig. 10(b) shows the cogging force waveforms of three-phase E-shaped modules, namely, F _cog E, F _cog E +60, and F _cog E + 120, whose electrical positions are mutually 60 apart in sequence, namely, λ 2 =(k ± (1/6))τ s. F _sum1 is the sum of F _cog E, F _cog E +60, and F _cog E + 120. It can be seen that F _sum1 is a nearly sinusoidal waveform with 360 period. Hence, it can be predicted

5442 IEEE TRANSACTIONS ON INDUSTRIAL ELECTRONICS, VOL. 60, NO. 12, DECEMBER 2013 Fig. 10. Cogging force performance of one E-shaped module and the MLF- SPM motors. (a) One E-shaped module. (b) Motor ABC with λ 2 =(6 (1/6))τ s. (c) Motor ABC with λ 2 =(6 (1/3))τ s. (d) Module AA with λ 1 =(6+(1/2))τ s. (e) Motor AABBCC with λ 2 =(13 (1/3))τ s. that the cogging force of ABC ABC motors with λ 2 = (k ± (1/6))τ s is much smaller than F _sum1 while the cogging forces of motors ABCABC and 2A2B2C with λ 2 = (k ± (1/6))τ s are all twice that of F _sum1. Furthermore, the cogging force of motor ABC ABC ABC that consists of three ABC motors with λ 2 =(k ± (1/6))τ s, whose positions are mutually ±120 apart in sequence, will be much smaller than F _sum1, and their thrust force is three times that of motor ABC. Fig. 10(c) shows the cogging force waveforms of threephase E-shaped modules, namely, F _cog E, F _cog E + 120, and F _cog E 120, whose electrical positions are mutually 120 apart in sequence, namely, λ 2 =(k ± (1/6))τ s. F _sum2 with 60 period is the sum of F _cog E, F _cog E + 120, and F _cog E 120. It can be seen that F _sum2 is much smaller than F _cog E. This is because the cogging forces of three-phase E-shaped modules are reduced by each other. Hence, it can be predicted that the cogging forces of the MLFSPM motors with λ 2 =(k ± (1/3))τ s based on ABC ABC, ABCABC, and 2A2B2C structures are all twice that of F _sum2. Furthermore, the cogging force of motor ABC ABC ABC that consists of three ABC motors with λ 2 =(k ± (1/3))τ s, whose positions are mutually ±120 apart in sequence, will be much smaller than F _sum2, and their thrust force is three times that of motor ABC. Fig. 10(d) shows the cogging force waveforms of the two E- shaped modules, whose relative position is mutually λ 1 =(j ± (1/2))τ s apart, namely, F _cog E and F _cog E + 180. F _AA is the sum of F _cog E and F _cog E + 180.It can be seen that the peak value of F _AA with 180 period is nearly the same as that of F _cog E. Hence, it can be predicted that the cogging forces of MLFSPM motors with λ 2 = (k ± (1/6))τ s or λ 2 =(k ± (1/3))τ s based on AABBCC are all the same as F _sum3, as shown in Fig. 10(e). Similarly, the cogging force of motor AABBCC AABBCC AABBCC consisting of three AABBCC motors, whose positions are mutually ±120 apart in sequence, will be much smaller than F _sum3, and their thrust force is three times that of motor AABBCC. Therefore, it can be concluded that motors ABC, ABC ABC, AABBCC, or 2A2B2C can also be used as one module to constitute a new motor with smaller cogging force and bigger thrust force. To validate the aforementioned analysis, the cogging forces of motor ABC with λ 2 =(6 (1/6))τ s and λ 2 =(6 (1/3))τ s and motor AABBCC with λ 2 =(13 (1/3))τ s shown in Fig. 8(c), namely, F _sum1, F _sum2 and F _sum3, are calculated directly by means of FEM and compared with F _sum1, F _sum2, and F _sum3, as shown in Fig. 11. It can be seen that F _sum1, F _sum2, and F _sum3 match well with F _sum1, F _sum2, and F _sum3, respectively. The small discrepancies between them are just because the end effects of both methods are different. Fig. 12 shows the back-emf waveforms at rated speed of motors ABC and ABC ABC with λ 2 =(k (1/6))τ s in Fig. 8(a) and (b), respectively. It can be seen that the peak value of the back EMF of motor ABC ABC is about twice that of motor ABC.

CAO et al.: INVESTIGATION AND GENERAL DESIGN PRINCIPLE OF A NEW SERIES OF LINEAR FSPM MOTORS 5443 TABLE V MLFSPM MOTORS WITH τ m/τ s =3 Fig. 11. Cogging force performance of MLFSPM motors by means of FEM and theoretical analysis. (a) Motor ABC with λ 2 =(6 (1/6))τ s and λ 2 = (6 (1/3))τ s. (b) Motor AABBCC with λ 2 =(13 (1/3))τ s. Fig. 13. Force performance of motor ABC ABC with λ 2 =(6+ (1/6))τ s and τ m/τ s =3at different k sty s. TABLE VI COMPARISON OF MLFSPM MOTORS WITH ABC ABC STRUCTURE Fig. 12. Back-EMF waveforms versus electrical positions of motors ABC and ABC ABC with λ 2 =(6 (1/6))τ s. Based on the aforementioned analysis, some MLFSPM motors based on the structures ABC ABC, AABBCC, AB- CABC, and 2A2B2C having small cogging force can be designed using (1) (3) and are listed in Table V. Fig. 13 shows the F _cog, F _avg, and F _ripple at the rated current of motor ABC ABC with τ m /τ s =3and λ 2 =(6+(1/6))τ s when k st =1and k sty is in the range of one to two. It can be seen that k sty =1.9 is chosen to be the optimal point. Based on the same methods, the detailed electromagnetic performances of these motors are calculated by means of FEM and listed in Tables VI VIII, respectively. By considering F _avg, F _cog, F _ripple, and mover length, motor ABC ABC with τ m /τ s =3, λ 1 =(18+(1/2))τ s, and λ 2 =(6+(1/6))τ s is chosen to be the best motor for the next step work. C. Electromagnetic Performance Comparison For MLFSPM motors with τ m /τ s =3, the optimal coefficient k wst is equal to 0.45, and w st is designed the same as w mt. TABLE VII COMPARISON OF MLFSPM MOTORS WITH AABBCC STRUCTURE Hence, the slot area of motor ABC ABC with τ m /τ s =3is 1.4 times that of the CMLFSPM motor with τ m /τ s =13/12.If the current densities J s of both motors are the same, the applied current of the optimal CMLFSPM motor with τ m /τ s =3 is

5444 IEEE TRANSACTIONS ON INDUSTRIAL ELECTRONICS, VOL. 60, NO. 12, DECEMBER 2013 TABLE VIII COMPARISON OF MLFSPM MOTORS WITH ABCABC AND 2A2B2C STRUCTURES TABLE IX COMPARISON OF CMLFSPM MOTORS WITH τ m/τ s =13/12 AND τ m/τ s =3 suitable for high-speed application than the one with τ m /τ s =3. 2) The phase resistance of the two CMLFSPM motors with τ m /τ s =13/12 is 1.4 times that of motor ABC ABC with τ m /τ s =3. 3) F _avg values of motor ABC ABC with τ m /τ s =3at I rms =6, 7.098, and 8.4 A are about 72.6%, 83%, and 94.1% of those of motors ABCABC and AABBCC with τ m /τ s =13/12 at I rms =6A, respectively. 4) The rms value of the back EMF of motor ABC ABC with τ m /τ s =3is only about 77.4% of that of motors ABCABC and AABBCC with τ m /τ s =13/12. 5) The mover iron volume V m_iron and copper volume V copper of motors ABCABC and AABBCC with τ m /τ s =13/12 are about 113.8% and 71.43% of those of motor ABC ABC with τ m /τ s =3, respectively. 6) F _cog of motor AABBCC with τ m /τ s =13/12 is very close to that of motor ABC ABC with τ m /τ s =3. However, its F _ripple is about twice of the one of motor ABC ABC. Its mover length is 107.8% of that of motor ABC ABC with τ m /τ s =3. 7) It can be seen that the flux density in the mover teeth, yoke, and stator yoke of the two motors is about in the range of 1.5 1.7 T. However, the flux density in the mover teeth tip of motor ABC ABC with τ m /τ s =3is nearly to 2.0 T. Also, for the two motors, there are some flux leakages at the mover end, mover top, and adjacent stator teeth in the inner part of the E-shaped module. This case is much more serious in motor ABC ABC with τ m /τ s =3, leading to poor utilization of the PM flux. 8) It can be seen that the CMLFSPM motor with τ m /τ s = 13/12 is better than the one with τ m /τ s =3. IV. GENERAL DESIGN PRINCIPLE 1.4 times that of the CMLFSPM motor with τ m /τ s =13/12, namely, I rms =8.4 A. If the copper losses of both motors are the same, the applied current of the optimal CMLFSPM motor with τ m /τ s =3is about 1.183 times that of the one with τ m /τ s =13/12, namely, I rms =7.098 A. Hence, I rms =6, 7.098, and 8.4 A are applied in the optimal CMLFSPM motor with τ m /τ s =3. For comparison, the detailed electromagnetic performances and dimensions of the optimal CMLFSPM motor with τ m /τ s = 3, λ 1 =(18+(1/2))τ s, and λ 2 =(6+(1/6))τ s and the two optimal CMLFSPM motors with τ m /τ s =13/12 are listed in Table IX. Also, the partial open-circuit magnetic flux density and flux line distribution of motor AABBCC with τ m /τ s = 13/12 and motor ABC ABC with τ m /τ s =3 are shown in Fig. 14. From Table IX and Fig. 14, the electromagnetic performances of these motors can be summarized as follows. 1) The electrical frequency f (in hertz) of motor ABC ABC with τ m /τ s =3is about three times those of the two CMLFSPM motors with τ m /τ s =13/12. Hence, the CMLFSPM motor with τ m /τ s =13/12 is much more In Sections II and III, some MLFSPM motors with and without complementary structure when τ m /τ s 1 and τ m /τ s =3 have been investigated. For these motors, the displacement between the two iron teeth on both sides of PM τ u1 is designed the same as the U-shaped teeth pitch τ u, namely, τ u1 = τ u = τ m /2. In fact, some modular flux-switching motors in this series can be obtained by using (1) (3) based on structures ABCABC, ABC ABC, AABBCC, and 2A2B2C, provided that the key parameters of one E-shaped module satisfy τ m = or kτ s, k =1, 2, 3, 4,... τ u1 = or (j +0.5)τ s, j =0, 1, 2, 3, 4,... (8) τ u = τ m τ u1 w mt w st. To understand (8) clearly, some expanded structures based the E-shaped module shown in Fig. 15 will be explained. For the E-shaped module shown in Fig. 15(a), τ m /τ s =1, and τ u1 = τ u = τ m /2, namely, k =1and j =0.First,ifτ s and τ u1 are kept constant, changing k from one to two, then a new E-shaped module as shown in Fig. 15(b) can be obtained, which also satisfies the flux-switching operation principle. Therefore, some MLFSPM motors using this E-shaped module can be obtained. Second, if k is changed from one to three and j is changed from

CAO et al.: INVESTIGATION AND GENERAL DESIGN PRINCIPLE OF A NEW SERIES OF LINEAR FSPM MOTORS 5445 Fig. 15. Two E-shaped modules with τ m/τ s =1 and 3. (a) τ m/τ s =1. (b) τ m/τ s =2.(c)τ m/τ s =3. Fig. 14. Partial open-circuit magnetic flux density and flux line distribution of the two optical CMLFSPM motors. (a) Flux density of motor AABBCC with τ m/τ s =13/12. (b) Flux line of motor AABBCC with τ m/τ s =13/12. (c) Flux density of motor ABC ABC with τ m/τ s =3. (d) Flux line of motor ABC ABC with τ m/τ s =3. zero to one, then τ m /τ s =3, and τ u = τ u1 =1.5τ s = τ m /2, namely, the E-shaped module in Fig. 15(a) is changed to the one in Fig. 15(c). It should be mentioned that, if τ m kτ s and τ u1 (j + 0.5)τ s, only the MLFSPM motors based on the two complementary structures ABC ABC and AABBCC can offer symmetrical back EMF and smaller cogging force. V. C ONCLUSION In this paper, the design principle, structure, dimensions, and electromagnetic performance of some CMLFSPM motors based on AABBCC and ABC ABC with τ m /τ s 1 (τ m /τ s =10/12, 11/12, 12/12, 13/12, 14/12, and 15/12) and the MLFSPM 2A2B2C motor with τ m /τ s =1 have been investigated, optimized, and compared by means of FEM. To verify the FEM results of these LFSPM motors, a prototype motor has been built and tested. The experimental results agree well with the predicted results from FEM. Then, the structure and electromagnetic performance of another LFSPM motor with τ m /τ s =3have been investigated using FEM. Furthermore, some new MLFSPM motors with τ m /τ s =3based on ABCABC, ABC ABC, AABBCC, and 2A2B2C have been designed, investigated, and compared. Moreover, the electromagnetic performances of the optimal MLFSPM motors with τ m /τ s =3and τ m /τ s =13/12 have been quantitatively compared. The comparison results show that the MLFSPM motors with τ m /τ s =13/12 can offer higher thrust force and lower electrical frequency than the one with τ m /τ s =3.Finally, the general design principle of this series of MLFSPM motors with different τ m /τ s values has been concluded, based on which new MLFSPM motors with different τ m /τ s values can be easily developed. REFERENCES [1] R. Hellinger and P. Mnich, Linear motor-powered transportation: History, present status, future outlook, Proc. IEEE, vol. 97, no. 11, pp. 1892 1900, Nov. 2009. [2] R. J. Wai, J. D. Lee, and K. L. Chuang, Real-time PID control strategy for Maglev transportation system via particle swarm optimization, IEEE Trans. Ind. Electron., vol. 58, no. 2, pp. 629 646, Feb. 2011.

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Lipo, A novel doubly salient single phase permanent magnet generator, in Conf. Rec IEEE IAS Annu. Meeting, Oct. 1994, pp. 9 15. [24] J. Zhang, M. Cheng, and Y. Q. Zhang, Single phase doubly salient permanent magnet generator with full-pitched winding, in Proc. IEEE IEMDC, Miami, FL, May 2009, pp. 311 316. Ruiwu Cao (S 10) received the B.S. degree in electrical engineering from Yancheng Institute of Technology, Yancheng, China, in 2004 and the M.S. degree in electrical engineering from Southeast University, Nanjing, China, in 2007, where he has been working toward the Ph.D. degree in electrical engineering since 2009. From 2007 to 2009, he was a Hardware Electrical Engineer with the Electronic Drive System R&D Center, BSH Electrical Appliances (Jiangsu) Company, Ltd. From August 2010 to November 2011, he was a joint Ph.D. student funded by the China Scholarship Council with the College of Electrical and Computer Science, University of Michigan, Dearborn, where he worked on permanent-magnet (PM) motors for electric vehicle (EV), hybrid EV (HEV), and plug-in HEV applications. His areas of interest include design, analysis, and control of PM machines. Ming Cheng (M 01 SM 02) received the B.Sc. and M.Sc. degrees from the Department of Electrical Engineering, Southeast University, Nanjing, China, in 1982 and 1987, respectively, and the Ph.D. degree from the Department of Electrical and Electronic Engineering, The University of Hong Kong, Hong Kong, in 2001. Since 1987, he has been with Southeast University, where he is currently a Professor in the School of Electrical Engineering and the Director of the Research Center for Wind Power Generation. From January to April 2011, he was a Visiting Professor with the Wisconsin Electric Machine and Power Electronics Consortium, University of Wisconsin, Madison. His teaching and research interests include electrical machines, motor drives for electric vehicles, and renewable energy generation. He has authored or coauthored over 250 technical papers and four books and is the holder of 45 patents in these areas. Prof. Cheng is a fellow of the Institution of Engineering and Technology. He has served as chair and organizing committee member for many international conferences. Wei Hua (M 07) was born in Jiangsu, China, in 1978. He received the B.S. and Ph.D. degrees in electrical engineering from Southeast University, Nanjing, China, in 2001 and 2007, respectively. He is with Southeast University, where he is currently a Professor and the Deputy Dean of the School of Electrical Engineering. His areas of interest include design, analysis, and control of novel permanent-magnet machines and switched reluctance machines. He has authored more than 100 published papers on these topics.