2972 IEEE TRANSACTIONS ON INDUSTRY APPLICATIONS, VOL. 51, NO. 4, JULY/AUGUST Jian Li, Member, IEEE, and Thomas A. Lipo, Life Fellow, IEEE

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1 97 IEEE TRANSACTIONS ON INDUSTRY APPLICATIONS, VOL. 51, NO. 4, JULY/AUGUST 015 Design Procedure of Dual-Stator Spoke-Array Vernier Permanent-Magnet Machines Dawei Li, Student Member, IEEE, Ronghai Qu, Senior Member, IEEE, WeiXu,Senior Member, IEEE, Jian Li, Member, IEEE, and Thomas A. Lipo, Life Fellow, IEEE Abstract The dual-stator spoke-array vernier permanentmagnet (DSSA VPM) machines proposed in the previous papers have been proven to be with high torque density and high power factor. However, the design procedure on the DSSA VPM machines has not been well established, and there is little design experience to be followed, which makes the DSSA VPM machine design quite difficult. This paper presents the detailed DSSA VPM machine design procedure including decision of design parameter initial values, analytical sizing equation, geometric size relationship, and so on. In order to get reasonable design parameter initial values which can reduce the number of design iteration loop, the influence of the key parameters, such as rotor/stator pole combination, slot opening, magnet thickness, etc., on the performances is analyzed based on the finite-element algorithm (FEA) in this paper, and the analysis results can be regarded as design experience during the selection process of the initial values. After that, the analytical sizing equation and geometric relationship formulas are derived and can be used to obtain and optimize the size data of the DSSA VPM machines with little time consumption. The combination of the analytical and FEA methods makes the design procedure time-effective and reliable. Finally, the design procedure is validated by experiments on a DSSA VPM prototype with 000 N m. Index Terms Design procedure, dual-stator spoke-array vernier permanent-magnet (DSSA VPM) machines, finite-element algorithm (FEA), sizing equation. I. INTRODUCTION WITH ever-increasing concerns on various newly developing applications such as wind generation and electric vehicles, high-torque-density electrical machines, such as the so-called pseudo-pm machines [1], dual-rotor PM machines [], harmonic machines [3], and so on, are attracting more and more attention from academia and industry. Vernier permanent- Manuscript received November 1, 014; revised January 17, 015; accepted January 5, 015. Date of publication February 10, 015; date of current version July 15, 015. Paper 014-EMC-0766.R1, presented at the 014 IEEE Energy Conversion Congress and Exposition, Pittsburgh, PA, USA, September 0 4, and approved for publication in the IEEE TRANSACTIONS ON INDUSTRY APPLICATIONS by the Electric Machines Committee of the IEEE Industry Applications Society. This work was supported by the National Natural Science Foundation of China (NSFC) under Project Number (Corresponding author: Ronghai Qu.) D. Li, R. Qu, W. Xu, and J. Li 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 , China ( lidawei_zs@163.com; ronghaiqu@hust.edu.cn; weixuforhappy@qq.com; jianli@hust.edu.cn). T. A. Lipo is with the Department of Electrical and Computer Engineering, University of Wisconsin Madison, Madison, WI USA ( thomas.lipo1@gmail.com). Color versions of one or more of the figures in this paper are available online at Digital Object Identifier /TIA Fig. 1. DSSA VPM machine. magnet (VPM) machines have become popular over the recent years for several advantages including high torque density, smooth torque performance, etc. [4] [8]. A nonoverlapping winding VPM machine was presented in [9]. Dual-rotor and dual-stator VPM machines were proposed in [10], which are reported to further improve the torque density of VPM machines. The linear VPM machines were proposed [11], in which the features such as high thrust force density and low cogging force are reported. However, compared to a regular PM machine, VPM machines suffer from a low power factor [10], [1], which makes the VPM machines require a large-capability converter for a given output power, which results in higher cost and larger volume in the converter. In order to improve the power factor, dual-stator spoke-array VPM (DSSA VPM) machines as shown in Fig. 1 were proposed in [13]. Theoretical analysis by the finite-element algorithm (FEA) and prototype experiments have proved that the DSSA VPM machines have not only a higher torque density than that of regular VPM machines but also a comparable power factor, viz., 0.85, with traditional PM machines. These features attribute to their vernier pole-slot structures and special relative position of their inner and outer stators which significantly reduce magnet leakage and improve the main flux. So far, the study on DSSA VPM machines is limited to introducing the topology, operation principle, and its performance features. For the DSSA VPM machine, the dual-side structure makes its design work much more complex than that of single-side VPM machines, and this design work has not been done. This paper elaborates on the analysis of the detailed design procedure of the DSSA VPM machines and develops an effective and efficient design methodology, including design parameter initial value selection, analytical sizing equation, IEEE. Personal use is permitted, but republication/redistribution requires IEEE permission. See for more information.

2 LI et al.: DESIGN PROCEDURE OF DUAL-STATOR SPOKE-ARRAY VERNIER PERMANENT-MAGNET MACHINES 973 Fig.. Exploded view of a DSSA VPM machine. geometric size relationship, etc., to design the DSSA VPM machine, and the analysis results are experimentally verified by a large-size natural-cooling prototype whose rated torque is 000 N m. II. STRUCTURE OF THE DSSA VPM MACHINE This topology consists of dual stators and a sandwiched rotor as shown in Figs. 1 and, respectively. The inner and outer stators have half tooth pitch displacement; in other words, the inner/outer stator teeth face the outer/inner stator slots. It should be noticed that the same phase axis of the inner and outer stators is not consistent, and the electrical phase shift for the fundamental space harmonic caused by the angle between the inner and outer stator phase axes as shown in Fig. 3(a) can be expressed as ( ) ( π π α io = P r α mio = P r π ) = P sπ P r P r Z P r + Ps (1) where Z is the number of slots, P r is the pole pairs of magnets, and P s is the number of stator pole pairs. In order to verify this deduction, the FEA model of one 1/ stator teeth/rotor pole DSSA VPM machine is built, and Fig. 3(b) shows that the angle between the inner and outer stator phase axes is 15. By further investigation, it is found that the flux linkage amplitude per pole of the inner stator is 4% smaller than that of the outer stator. In order to simplify the theoretical analysis latter, it is assumed in this paper that the outer stator flux Φ m and inner stator flux Φ are equal to each other. Since the flux linkage and induced voltage of the inner and outer stator coils have some phase shift, the same phase winding arrays of the inner and outer stators should be connected in series or driven by two converters to avoid circulating current. In this paper, the same phase winding arrays of the inner and outer stators are connected in series. As shown in Fig. 4(a) and (b), A and A1 are the two terminals of phase A in the outer stator, while A and O are the two terminals of phase A in the inner stator. The same phase winding arrays of the inner and outer stators are connected in series as shown in Fig. 4(c). III. DESIGN FLOW Machine design is the project, including works ranging from setting machine specification to finishing delivery as shown in Fig. 5, and always includes more segments. Fig. 3. Two stators of a 1/ stator teeth/rotor pole DSSA VPM machine. (a) Relative position. (b) Phase back EMF waveforms predicted by FEA. This paper focuses on the detailed design procedure loop, including sizing equation, geometrical design, performance calculation, and test verification, of the DSSA VPM machines. Fig. 6 shows the electromagnetic design flow of the DSSA VPM machines. The first step in the machine design is not calculating the machine size but analyzing the specifications which give much important information, such as the target performances of the machine and external condition limitations. The next step is the true machine design process. First, since the inherent features including torque density, torque ripple, etc., are quite different for different configurations such as stator/rotor pole combinations, stator/rotor configurations, etc., the configuration of the DSSA VPM machine should be carefully selected at first. Meanwhile, some design parameters, such as pole arc, slot opening, magnetic loading, etc., should be given an initial value to start the machine size design, where the pole arc is defined as the ratio of the outer pole shoe arc length τ p1 to the pole pitch, as shown in Fig. 1. After that, the machine detailed sizes, such as the inner and outer diameters of the rotor and tooth width, can be obtained from the sizing equation and geometrical relationship formulas, and then, the performance of this DSSA VPM machine can be evaluated. If the predicted performances and specifications do not match very well, the fixed value of the parameters at the start of this design should be refined until they match

3 974 IEEE TRANSACTIONS ON INDUSTRY APPLICATIONS, VOL. 51, NO. 4, JULY/AUGUST 015 Fig. 6. Design flowchart of the DSSA VPM machine design. TABLE I MAJOR SPECIFICATIONS Fig. 4. Winding connection of the DSSA VPM machine. (a) Phase A winding connection in the outer stator. (b) Phase A winding connection in the inner stator. (c) Winding connection of inner and outer stators in the DSSA VPM machine. TABLE II MAJOR PARAMETERS Fig. 5. Simplified flowchart of the electrical machine design. well with each other. This process may take several iterative loops to get content results. FEA is employed to check the analysis results of the magnetic circuit method. Then, several parameter optimization loops using the design process are taken to achieve different design purposes. Table I presents the major specifications of the DSSA VPM machine. IV. STATOR/ROTOR POLE COMBINATIONS AND INITIAL VALUE DECISION ON DESIGN PARAMETERS This section will analyze the design parameter effect on the DSSA VPM machines by FEA, and the major design parameters of the FEA model are summarized in Table II. The analysis results can help designers suitably select the major electromagnetic parameter initial values and further optimize the machine dimensions. A. Stator and Rotor Pole Numbers The stator and rotor pole numbers are unequal, and this is quite different from that of a regular electrical machine. Hence, there is a special design parameter for the DSSA VPM machines, i.e., the stator and rotor pole numbers, which has a significant effect on the performances of the DSSA VPM machine and optimal values of the other design parameters. Hence, the combination of stator and rotor pole numbers should be analyzed and confirmed at the early stages of the design. The relationship of the number of slots, rotor, and stator pole pairs of the DSSA VPM machines satisfies Z = P r ± P s ()

4 LI et al.: DESIGN PROCEDURE OF DUAL-STATOR SPOKE-ARRAY VERNIER PERMANENT-MAGNET MACHINES 975 TABLE III COMBINATION OF STATOR AND ROTOR POLE NUMBERS OF THREE-PHASE DSSA VPM MACHINES (PR/SPSP) Fig. 8. Optimized stator pole pair for the maximum back EMF amplitude versus outer diameter in the DSSA VPM machines with different pole ratios. Fig. 7. Fundamental back EMF versus pole ratio in the DSSA VPM machine. where Z is the number of slots, P r is the rotor pole pairs, and P s is the stator pole pairs The balanced three-phase winding configurations make the stator and rotor combinations satisfy the following equation: Z = mk, k =1,, 3,... (3) G.C.D(Z, P s ) where G.C.D is the shorthand word of the greatest common divisor, Z is the number of slots, P s is the number of stator pole pairs, and m is the number of phases. Table III gives several available combinations of rotor, stator pole pair number, pole ratio PR, and slots per phase per stator pole (SPSP). In order to analyze performance sensitivity to stator and rotor pole numbers, the other design parameters, such as pole arc, electrical and magnetic loading, etc., are fixed at first, and several FEA models with different pole combinations are built. The major parameters of these models are given in Table II. Fig. 7 gives the variation of the fundamental back EMF with pole ratio. It can be seen that the optimal pole ratio for the maximum fundamental back EMF varies with the stator pole number, and the optimized pole ratios are 17, 17 and 11 for 1,, 3-stator pole pair models in these cases, respectively. Fig. 8 shows the variation of the optimized stator pole pair number for the maximum back EMF amplitude with different outer diameters. The optimized stator pole pairs increase as the machine outer diameter augments, and this attributes to that large stator pole pair number can increase the airgap diameter by reducing the stator yoke thickness, but a too large stator pole number means a large rotor pole number which introduces a significant magnet leakage. Hence, it is very necessary to make careful optimization to get the reasonable number of stator/rotor pole pairs for the excellent performance. Fig. 9 summarizes the variation of the torque and power factor with the rotor and stator pole combination. The 17-pole ratio -stator pole pair DSSA VPM machine has the largest torque density but almost the lowest power factor among these DSSA VPM machines, while it can be seen that the candidate with higher torque always suffers from a lower power factor. Hence, it is almost impossible for models to obtain the largest power factor and torque density at the same time, several compromises should be carefully done during the design procedure, and the 4/ stator tooth/rotor pole pair combination is selected for further work in this paper. B. Magnetic Loading and Pole Arc The magnetic loading and pole arc are the key parameters in the machine design as illustrated in the foregoing sections and are fixed at the early design stage. The sensitivity of the two parameters on the performance of the DSSA VPM machines is analyzed in this section to help machine designers select the optimal magnetic loading and pole arc for a particular design work. As illustrated in [4], the stator teeth of the VPM machines work as not only part of the flux path but also of the flux modulator; hence, the slot opening ratio, i.e., the ratio of the slot opening to the slot pitch, is the key parameter in the DSSA VPM machine design. The slot opening ratio is fixed to remove its influence on performances during optimizing magnetic loading and pole arc.

5 976 IEEE TRANSACTIONS ON INDUSTRY APPLICATIONS, VOL. 51, NO. 4, JULY/AUGUST 015 Fig. 11. Optimal torque and magnetic loading versus pole arc in the DSSA VPM machines (B y1 =1.4T, s o1 =0.6, ands o =0.6). Fig. 9. Influence of pole ratio on the performance of the DSSA VPM machine with different stator pole pair numbers. (a) Torque. (b) Power factor. Fig. 1. Power factor versus magnetic loading B g1m in the DSSA VPM machines with different pole arcs (B y1 =1.4T, s o1 =0.6, ands o =0.6). Fig. 10. Torque versus magnetic loading in the DSSA VPM machines with different pole arcs (the outer yoke flux density B y1 is 1.4 T, the outer slot opening ratio s o1 is 0.6, and the outer slot opening ratio so is 0.6). Fig. 10 gives the variation of the torque with magnetic loading in the DSSA VPM machines with different pole arcs. The optimal magnetic loading for the maximum torque reduces as the pole arc increases, as shown in Fig. 11. The optimal magnetic loading and pole arc for the maximum torque are around 1.6 T and 0.6, respectively. The variation of the power factor with magnetic loading and pole arc is presented in Fig. 1. It can be seen that the power factor of the models with smaller pole arc, which means larger magnet thickness, is always larger and increases as the magnetic loading goes up. Fig. 13. Torque versus s o1 and s o (B g1m =1.43 T, pole arc =0.586,and A = 04 A/cm). It can be seen that the maximum torque and power factor could not be satisfied at the same time. Therefore, a compromise between torque, power factor, and magnet usage has to be made. In this paper, magnetic loading and pole arc are selected as 1.43 T and for balancing these performances, such as torque density, power factor, etc. Figs. 13 and 14 present the variation of the torque and power factor with outer stator opening ratio s o1 and inner stator opening ratio s o. It can be seen that the slot opening ratio has a significant influence on the torque and power factor of the DSSA VPM machines, and in this case, the optimal

6 LI et al.: DESIGN PROCEDURE OF DUAL-STATOR SPOKE-ARRAY VERNIER PERMANENT-MAGNET MACHINES 977 Fig. 14. Power factor versus s o1 and s o (B g1m =1.43 T, pole arc = 0.586, anda = 04 A/cm). Fig. 16. Power factor versus ratio of outer stator to whole turns in series per phase (B g1m =1.43 T, pole arc =0.586, Ns = 440 turns, J =1.A/mm, based I rms =11.8 A, A conductor =0.78 mm, and five conductors in one turn). the shorter slot depth always means weaker field modulation effect, especially when the stator teeth are saturated. Therefore, there are optimal values of k for the maximum torque and power factor. Fig. 15. Torque versus ratio of outer stator to whole turns in series per phase (B g1m =1.43 T, pole arc =0.586, Ns = 440 turns, I rms =11.8A, A conductor =0.78 mm, and five conductors in one turn). combination of s o1 and s o for maximum torque are 0.65 and At the range of the inner and outer stator slot opening ratio from 0.5 to 0.7, the power factor increases as the s o1 and s o go up, as shown in Fig. 14. C. Electrical Loading The higher limit on electrical loading can be selected by the rules of thumb at the early design stage, and a more accurate higher limit of electrical loading should be estimated by the thermal computation. Adjusting electrical loading in the further stage may be an inevitable work to guarantee that the temperature rise is under the acceptable range. In other words, the whole electrical loading is limited by the thermal condition and cooling capability. For the DSSA VPM machines, there is another special design parameter, i.e., the ratio of the outer stator electrical loading A outer to whole electrical loading A, which is a relatively flexible parameter under the same whole electrical loading. k is defined as the ratio of the outer stator electrical loading A outer to the whole electrical loading A. The variations of the torque and power factor with k in the DSSA VPM machines with different phase currents, viz., different copper loss, are shown in Figs. 15 and 16. The smaller k leads to shorter outer stator slot depth and larger outer airgap diameter. Meanwhile, V. S IZING EQUATION OF THE DSSA VPM MACHINES After selecting the configurations and several design parameter initial values, the detailed design on DSSA VPM machine sizes can be done. Sizing equation is widely used to obtain the main sizes of the electrical machines, and the sizing equations of various topologies are different due to their specific structures and operation principle. There are numerous papers researching on the specific sizing equations for novel topologies such as stator-mounted permanent-magnet machines [14], transverse flux circumferential current machines [15], etc. However, since these novel and DSSA VPM machines have different principles and structures, the existing sizing equations are not available for the DSSA VPM machines. Hence, the sizing equation for the DSSA VPM machines is required. Generally, the total electromagnetic power of the surfacemounted PM machine is P e = m T T 0 e(t)i(t)dt (4) where m is the number of phases, T is the period of the back EMF, e(t) is the phase back EMF, and i(t) is the phase current. As shown in Figs. 17 and 18, the back EMF and flux linkage waveforms of the DSSA VPM machines are really sinusoidal even in single coil, and then, (4) can be rewritten as P e = 1 me mi m cos γ (5) where γ is the angle shift between the phase EMF and current, and E m and I m are the peak values of the fundamental phase back EMF and current, respectively. As illustrated in [8], the reluctance torque of the DSSA VPM prototype accounts for about 4% of the electromagnetic

7 978 IEEE TRANSACTIONS ON INDUSTRY APPLICATIONS, VOL. 51, NO. 4, JULY/AUGUST 015 machines, and its value is from 0.5 to k δ = for the DSSA VPM machines is obtained from several FEA results used in the primary design. The DSSA VPM machine has two airgaps, and the electrical loading A of the DSSA VPM machines is defined as A = 6N si rms = 6I rms (N sinner + N souter ) πd g1 πd g1 = A inner + A outer (10) Fig. 17. Coil and phase flux linkage waveforms of the DSSA VPM machine. where A inner is the inner stator electrical loading, A outer is the outer stator electrical loading, and N sinner and N souter are the number of turns in series per phase of the inner and outer stators, respectively. The distribution factor of the DSSA VPM machines should consider not only the phase shift among different coils in the outer or inner stator as regular PM machines do but also the influence of the angle α mio between the same phase axis of the inner and outer stators as shown in Fig. 3(a). Therefore, the distribution factor of the DSSA VPM machine is quite different from that of regular PM machines and is expressed as and the pitch factor is k dv = sin ( vq α ) s q sin ( v α ) sin s k pv =sin ( v α io sin ( v α io ( vp s α w ) ) ) (11) (1) Fig. 18. Phase back EMF waveform of the DSSA VPM machine. torque, and the reluctance torque can be neglected. Then, the electromagnetic torque expression can be given by T e = P e Ω = m P rψ pm I m cos γ (6) where Ψ pm is the magnet flux linkage per phase. As illustrated in Section II, the magnet flux per stator pole of two stators is assumed to be the same. Therefore, the flux linkage can be obtained from ψ pm = k w N s φ m (7) where Φ m is the flux per stator pole, N s is the turns in series per phase, and k w is the winding factor. Combining (6) and (7), it yields T e = m P rk w N s φ m I m. (8) For the DSSA VPM machines, the flux per stator pole can be expressed as πd g1 φ m = k δ α p1 B g1m L stk (9) 4P s where k δ is the leakage factor, α p1 is the rotor pole arc, P s is the number of stator pole pairs, L stk is the stack length, and B g1m is the peak flux density in the outer airgap excited by magnets. In [9], a leakage factor is defined for the flux reversal PM machines which have similar operation principle with VPM where v is the order of harmonics and α s and α w are the phase shift angles between the two adjacent EMF vectors in one phase and electrical angle of coil span for the fundamental space harmonic, respectively. The winding factor for the fundamental harmonics is k w = k d1 k p1. (13) Combining (5), (8), (9), and (10), the sizing equation is obtained f P e = 8 π3 k w k δ α p1 AB g1m D P g1l stk cos γ. (14) s Meanwhile, the electromagnetic torque can be calculated by P r T e = 16 π k w k δ α p1 AB g1m D P g1l stk cos γ. (15) s The rotor volume V r is given by V r = πd g1 L stk = 4 T e 4 π P r P s k w k δ α p AB g1m cos γ (16) and then the rotor outer diameter and lamination length can be obtained as ) 1 3 D g1 = ( 4k L V r π ( ) 1 (17) L stk = 4Vr 3 πk L where k L is the ratio of the outer airgap diameter D g1 to the stack length L stk.

8 LI et al.: DESIGN PROCEDURE OF DUAL-STATOR SPOKE-ARRAY VERNIER PERMANENT-MAGNET MACHINES 979 Fig. 19. Geometry of the outer stator. VI. GEOMETRICAL DESIGN A. Outer Stator Design Based on the sizing equation illustrated in Section IV, the outer rotor diameter D g1 can be obtained, which may be adjusted in the following process to make sure that the outer diameter and machine axial length meet the size limitations. The stator tooth width t w1 shown in Fig. 19 is given by Fig. 0. Flux contour line of a DSSA VPM machine. t w1 = πd g1α 1 B g1m (18) P r B t1 k stk where α 1 is the pole arc of the outer airgap, B t1 is the outer stator tooth flux density, and k stk is the stack factor. Given the required outer stator current density J 1 and linear loading A outer of the outer stator, the outer stator slot area viable for the conductor is expressed as A slot = πd g1a outer J 1 k cu = [ π ( Dg1 + h 1 + h 1 + h 13 ( Dg1 π + h 1 + h 13 ) ) zt w1h 1 ] (19) and then the outer stator tooth depth is obtained as h 1 =0.5 zt w1 π D g1 h 1 h 13 ( + Dg1 +h1 +h13 zt ) w1 + 4D g1a 1. (0) π J 1 k cu Since the stator yoke is required to support half of the flux per stator pole, the outer stator yoke depth h y1 can be obtained by the main magnet flux per stator pole Φ m and the designed stator yoke flux density B y1. h y1 is given by φ m h y1 =. (1) k stk B y1 L stk After obtaining the outer stator yoke depth h y1, stator slot depth h 1, and outer rotor diameter D g1, the outer stator diameter can be obtained as D o = D g1 +g +h y1 +h 1 +h 13 +h 1. () B. Rotor Design As shown in Fig. 0, the rotor is sandwiched between two stators, and the magnet flux is driven by magnets through the two airgaps to combine the two stators together. Fig. 1. Flux contour line of the spoke rotor configuration. (a) Traditional spoke-type PM machine. (b) DSSA VPM machine. The magnet flux circuit of the DSSA VPM machines is similar with that of traditional spoke-array PM machines as shown in Fig. 1; thus, the magnet flux density in the airgap can be calculated by gh g1m + H m L m =0; Ampere Law πd B g1m g1m 4P r = B m W m ; Gauss Law (3) B m = B r + u 0 u r H m ; B g1m = B r πd g1. (4) 4P r W m + μ rg L m In order to take the iron bridge effect into account, the remanence B r in (1) should be replaced by the new value B rn [16], i.e., B rn = B r N b B sat (5) W m where N b is the number of bridge of one magnet. From the view point of the electromagnetic structure design, the smaller the bridge width b r is, the smaller is the magnet leakage. The lower limit of the bridge width b r is determined by the mechanical machining accuracy. The magnet width W m shown in Fig. is obtained from (4) and (5) W m = α p1 πd g1 4P r b r + N bdb sat b r B g1m B r B g1m μ. (6) rg L m Finally, the inner rotor diameter D g can be given as D g = D g1 W m 4b r. (7)

9 980 IEEE TRANSACTIONS ON INDUSTRY APPLICATIONS, VOL. 51, NO. 4, JULY/AUGUST 015 TABLE IV COMPARISON OF ANALYTICAL METHOD AND FEA Fig.. Fig. 3. Geometry of the rotor. Geometry of the inner stator. C. Inner Stator Design First, the inner diameter of the rotor is obtained from (7), and the inner rotor pole arc α is expressed as α = πd g P r L m πd g. (8) Then, the maximum flux density excited by the magnets in the inner airgap is given by B gm = φ m α D g = B g1mα 1 D g1 α D g. (9) Meanwhile, the inner stator teeth width is t w = πd gα B g1m. (30) P r B t k stk The inner stator yoke thickness shown in Fig. 3 is given by h y = φ m k stk B y L stk. (31) Based on the same method as the outer stator does, the inner stator teeth depth can be obtained h =0.5 D g h h 3 zt w π ( Dg h h3 zt ) w 4 D ga. (3) π J k cu Then, the inner stator inner diameter is given by D i = D g h y h 1 h 13 h g. (33) So far, the main sizes and detailed dimensions of each part of the DSSA VPM machines can be obtained. In order to validate the analytical equations derived in the foregoing section, a FEA model of a DSSA VPM machine is built, and its main sizes are given in Table II. Table IV summarizes the results predicted by the FEA and analytical method, and it can be seen that the two result arrays match well. After obtaining the sizing equation and geometrical relationship, the detailed design procedure can be given as shown in Fig. 4 and summarized as follows. 1) The stator/rotor pole number and several design parameters of the DSSA VPM machines are selected as referring to the analysis results in Section IV. ) Then, the rotor outer diameter D g1 and lamination length L stk can be estimated by the sizing equation built in Section V. 3) The outer diameter of the outer stator can be obtained by the geometrical relationship formulas presented in Section VI. In order to make sure that both the calculated D o and stack length L stk satisfy the size limitations, adjusting the electrical and magnetic loading may be required. 4) In order to further make sure that the accuracy of the design results satisfies the specification demands, FEA verification on flux density, back EMF, and torque is employed. 5) The initial values are selected based on experience or the analysis results in Section IV and may not be optimal. Hence, the optimization of the design parameters is required in the whole design procedure. A. Prototype VII. EXPERIMENTS In order to verify the aforementioned analysis results, a DSSA VPM prototype has been built, and its major parameters are summarized in Table V. The exploded view of the DSSA VPM prototype is shown in Fig.. It can be seen that the prototype structure consists of several parts including the stator and rotor lamination stack, and auxiliary support part including rotor support, frame, etc. The active part of the prototype has two stators and one rotor sandwiched by the stators. Since the stator teeth work as not only a part of the flux path but also of the flux modulator, opened slots are used in the DSSA VPM machines to enhance the slot effect. Both rotor and stators are stacked by 0.5-mm-thickness laminations, and 44 pieces of magnets (14 mm thickness) are inserted in the rotor lamination. The magnets use N35UH material and are tangentially magnetized. In order to simplify the stack process, all rotor pole shoes are connected together with the 1-mm-thickness bridges as shown in Fig. 5. The

10 LI et al.: DESIGN PROCEDURE OF DUAL-STATOR SPOKE-ARRAY VERNIER PERMANENT-MAGNET MACHINES 981 Fig. 4. Design procedure of the DSSA VPM machines. TABLE V MAJOR PARAMETERS OF THE PROTOTYPE Fig. 5. Photographs of the prototype. (a) Rotor lamination. (b) Rotor assembly. (c) Stator lamination. bridge can also be considered to be removed to reduce magnet leakage, and all rotor pole shoes are connected to the rotor support by screws. B. Test The no-load braking torque of the prototype is about 100 N m, and it is found that this value slightly varies with speed. Hence, this high no-load braking torque mainly attributes to the immature processing technology. Fig. 6 shows the test bed of the prototype. The measured line voltage and current waveforms at rated load are presented in Fig. 7. Figs. 8 and 9 show the comparison of the amplitude of the fundamental phase back EMF and output torque measured by the prototype experiments and predicted by FEA. The results evaluated by the FEA and experiments match well. The comparison of the simulation and measured performance indexes at rated conditions has been done and summarized in Table VI. Fig. 6. Test bed of the prototype. VIII. CONCLUSION This paper has presented the design procedure of the DSSA VPM machines, which includes the design parameter initial value setting, analytical sizing equation, key geometrical relationship formulas, and design parameter optimization. In addition, the feasible combinations of stator/rotor pole numbers of the DSSA VPM machines are given in this paper,

11 98 IEEE TRANSACTIONS ON INDUSTRY APPLICATIONS, VOL. 51, NO. 4, JULY/AUGUST 015 pole pairs for maximum torque increase as the outer diameter increases, and for a given diameter, the larger the pole ratio, the lower the optimal stator pole pairs are. The variation of optimal stator pole pairs to outer diameter is summarized in this paper. In addition, both the optimal inner and outer stator slot opening ratios are around 0.65, which is almost independent from other design parameters. The sizing equation and geometric relationship formulas of the DSSA VPM machines have been built based on the magnetic circuit method and validated by FEA. Fig. 7. Measured line voltage and current waveform (curve 1 line voltage; curve line current). Fig. 8. Fig A. Fundamental phase back EMF versus rotor speed. Output torque versus phase current. The rated phase current is TABLE VI COMPARISON OF SIMULATION AND MEASURED PERFORMANCE INDEXES REFERENCES [1] K. Atallah, J. Rens, S. Mezani, and D. Howe, A novel pseudo directdrive brushless permanent magnet machine, IEEE Trans. Magn.,vol.44, no. 11, pp , Nov [] R. Qu and T. Lipo, Dual-rotor, radial-flux, toroidally wound, permanentmagnet machines, IEEE Trans. Ind. Appl.,vol.39,no.6,pp , Nov./Dec [3] L. Jian, G. Xu, C. Mi, K. Chau, and C. Chan, Analytical method for magnetic field calculation in a low-speed permanent magnet harmonic machine, IEEE Trans. Energy Convers., vol. 6, no. 3, pp , Sep [4] A. Toba and T. Lipo, Generic torque-maximizing design methodology of surface permanent-magnet vernier machine, IEEE Trans. Ind. Appl., vol. 36, no. 6, pp , Nov./Dec [5] D. Li and R. Qu, Sinusoidal back-emf of vernier permanent magnet machines, in Proc. ICEMS, Oct. 01, pp [6] R. Qu, D. Li, and J. Wang, Relationship between magnetic gears and vernier PM machines, in Proc. IEEE Int. Conf. Elect. Mach. Syst., Aug. 011, pp [7] S. Niu, S. Ho, W. Fu, and L. Wang, Quantitative comparison of novel vernier permanent magnet machines, IEEE Trans. Magn., vol. 46, no. 6, pp , Jun [8] B. Kim and T. Lipo, Operation and design principles of a PM vernier motor, IEEE Trans. Ind. Appl., vol. 50, no. 6, pp , Nov./Dec [9] J. Li, K. Chau, J. Jiang, C. Liu, and W. Li, A new efficient permanentmagnet vernier machine for wind power generation, IEEE Trans. Magn., vol. 46, no. 6, pp , Jun [10] D. Li, R. Qu, and Z. Zhu, Comparison of Halhach and dual-side vernier permanent magnet machines, IEEE Trans. Magn., vol. 50, no., Feb. 014, Art. ID [11] Y. Du, K. Chau, M. Cheng et al., Design and analysis of linear stator permanent magnet vernier machines, IEEE Trans. Magn.,vol.47,no.10, pp , Oct [1] E. Spooner and L. Hardock, Vernier hybrid machines, Proc. Inst. Elect. Eng. Elect. Power Appl., vol. 150, no. 6, pp , Nov [13] D. Li, R. Qu, and T. Lipo, High power factor vernier permanent magnet machines, IEEE Trans. Ind. Appl., vol. 50, no. 6, pp , Nov./Dec [14] J. Zhang, M. Cheng, Z. Chen, and W. Hua, Comparison of stator mounted permanent magnet machines based on a general power equation, IEEE Trans. Energy Convers., vol. 4, no. 4, pp , Dec [15] S. Huang, J. Luo, and T. Lipo, Evaluation of the transverse flux circumferential current machine by the use of sizing equations, in Proc. Int. Elect. Mach. Drives Conf., May 1997, pp. WB/15.1 WB/15.3. [16] N. Bianchi and T. Jahns, Design, analysis, control of interior PM synchronous machines Tutorial course notes, presented at the IEEE IAS Annu. Meeting, Seattle, WA, USA, 004, pp and the performance sensitivity of the DSSA VPM machines to several key parameters including stator/rotor pole combination, pole arc, slot opening ratio, electrical and magnetic loading, etc., has been analyzed. It is found that the optimal stator Dawei Li (S 1) was born in China. He received the B.Eng. degree in electrical engineering from Harbin Institute of Technology, Harbin, China, in 010. He is currently working toward the Ph.D. degree in the School of Electrical and Electronic Engineering, Huazhong University of Science and Technology, Wuhan, China. His research interests include design and analysis of novel permanent-magnet brushless machines.

12 LI et al.: DESIGN PROCEDURE OF DUAL-STATOR SPOKE-ARRAY VERNIER PERMANENT-MAGNET MACHINES 983 Ronghai Qu (S 01 M 0 SM 05) was born in China. He received the B.E.E. and M.S.E.E. degrees from Tsinghua University, Beijing, China, in 1993 and 1996, respectively, and the Ph.D. degree in electrical engineering from the University of Wisconsin Madison, Madison, WI, USA, in 00. In 1998, he joined the Wisconsin Electric Machines and Power Electronics Consortiums as a Research Assistant. He became a Senior Electrical Engineer with Northland, a Scott Fetzer Company, in 00. In 003, he joined the General Electric (GE) Global Research Center, Niskayuna, NY, USA, as a Senior Electrical Engineer with the Electrical Machines and Drives Laboratory. Since 010, he has been a Professor with Huazhong University of Science and Technology, Wuhan, China. He is the author of over 50 published technical papers and is the holder of over 40 patents/patent applications. Prof. Qu is a full member of Sigma Xi. He has been the recipient of several awards from the GE Global Research Center since 003, including the Technical Achievement and Management Awards. He was also the recipient of the 003 and 005 Best Paper Awards, Third Prize, from the Electric Machines Committee of the IEEE Industry Applications Society (IAS) at the 00 and 004 IAS Annual Meeting, respectively. Wei Xu (M 09 SM 13) received double B.E. and M.E. degrees in electrical engineering from Tianjin University, Tianjin, China, in 00 and 005, respectively, and the Ph.D. degree in electrical engineering from the Institute of Electrical Engineering, Chinese Academy of Sciences, Beijing, China, in 008. From 008 to 01, he held several academic positions with Australian and Japanese universities and companies. Since 013, he has been a Full Professor with the School of Electrical and Electronic Engineering, Huazhong University of Science and Technology, Wuhan, China. His research topics mainly cover electromagnetic design and control algorithms of linear/rotary machines, including induction, permanent-magnet, switched reluctance, and other emerging novel structure machines. Jian Li (M 10) received the B.E.E. degree from Dalian University of Technology, Dalian, China, in 005 and the M.S.E.E and Ph.D. degrees from Dong-A University, Busan, Korea, in 007 and 011, respectively. He is currently an Associate Research Professor with the School of Electrical and Electronic Engineering, Huazhong University of Science and Technology, Wuhan, China. His research interests include the design and analysis of PM and switched reluctance machines and electric drives. Thomas A. Lipo (M 64 SM 71 F 87 LF 00) is a native of Milwaukee, WI, USA. He received the B.E.E. and M.S.E.E. degrees from Marquette University, Milwaukee, WI, in 196 and 1964, respectively, and the Ph.D. degree in electrical engineering from the University of Wisconsin, Madison, WI, in From 1969 to 1979, he was an Electrical Engineer with the Power Electronics Laboratory, Corporate Research and Development, General Electric Company, Schenectady, NY, USA. He became a Professor of electrical engineering with Purdue University, West Lafayette, IN, USA, in 1979, and in 1981, he joined the University of Wisconsin Madison, Madison, WI, USA, where he served for 8 years as the W. W. Grainger Professor for Power Electronics and Electrical Machines. He is currently an Emeritus Professor with the University of Wisconsin. Dr. Lipo received the Outstanding Achievement Award from the IEEE Industry Applications Society, the William E. Newell Award from the IEEE Power Electronics Society, and the 1995 Nicola Tesla IEEE Field Award from the IEEE Power Engineering Society for his work. He was elected as a member of the Royal Academy of Engineering (U.K.) in 00, a member of the National Academy of Engineering (U.S.) in 008, and a member of the National Academy of Inventors (U.S.) in 013. In 014, he was selected to receive the IEEE Medal for Power Engineering. For the past 40 years, he has served the IEEE in numerous capacities, including President of the IEEE Industry Applications Society in 1994.