Design Procedure of Dual-Stator Spoke-Array Vernier ...lipo.ece.wisc.edu/2015pubs/2015-22.pdf · LI...

2972 IEEE TRANSACTIONS ON INDUSTRY APPLICATIONS, VOL. 51, NO. 4, JULY/AUGUST 2015

Design Procedure of Dual-Stator Spoke-ArrayVernier Permanent-Magnet Machines

Dawei Li, Student Member, IEEE, Ronghai Qu, Senior Member, IEEE, Wei Xu, Senior Member, IEEE,Jian Li, Member, IEEE, and Thomas A. Lipo, Life Fellow, IEEE

Abstract—The dual-stator spoke-array vernier permanent-magnet (DSSA VPM) machines proposed in the previous papershave been proven to be with high torque density and high powerfactor. However, the design procedure on the DSSA VPM machineshas not been well established, and there is little design experienceto be followed, which makes the DSSA VPM machine design quitedifficult. This paper presents the detailed DSSA VPM machinedesign procedure including decision of design parameter initialvalues, analytical sizing equation, geometric size relationship, andso on. In order to get reasonable design parameter initial valueswhich can reduce the number of design iteration loop, the influ-ence of the key parameters, such as rotor/stator pole combination,slot opening, magnet thickness, etc., on the performances is ana-lyzed based on the finite-element algorithm (FEA) in this paper,and the analysis results can be regarded as design experienceduring the selection process of the initial values. After that, theanalytical sizing equation and geometric relationship formulasare derived and can be used to obtain and optimize the sizedata of the DSSA VPM machines with little time consumption.The combination of the analytical and FEA methods makes thedesign procedure time-effective and reliable. Finally, the designprocedure is validated by experiments on a DSSA VPM prototypewith 2000 N · m.

Index Terms—Design procedure, dual-stator spoke-arrayvernier permanent-magnet (DSSA VPM) machines, finite-elementalgorithm (FEA), sizing equation.

I. INTRODUCTION

W ITH ever-increasing concerns on various newly devel-oping applications such as wind generation and electric

vehicles, high-torque-density electrical machines, such as theso-called pseudo-PM machines [1], dual-rotor PM machines[2], harmonic machines [3], and so on, are attracting more andmore attention from academia and industry. Vernier permanent-

Manuscript received November 21, 2014; revised January 17, 2015; ac-cepted January 25, 2015. Date of publication February 10, 2015; date ofcurrent version July 15, 2015. Paper 2014-EMC-0766.R1, presented at the2014 IEEE Energy Conversion Congress and Exposition, Pittsburgh, PA, USA,September 20–24, and approved for publication in the IEEE TRANSACTIONS

ON INDUSTRY APPLICATIONS by the Electric Machines Committee of theIEEE Industry Applications Society. This work was supported by the NationalNatural Science Foundation of China (NSFC) under Project Number 51337004.(Corresponding author: Ronghai Qu.)

D. Li, R. Qu, W. Xu, and J. Li are with the State Key Laboratory ofAdvanced Electromagnetic Engineering and Technology, School of Electricaland Electronic Engineering, Huazhong University of Science and Technology,Wuhan 430074, China (e-mail: [email protected]; [email protected];[email protected]; [email protected]).

T. A. Lipo is with the Department of Electrical and Computer Engineer-ing, University of Wisconsin–Madison, Madison, WI 53706 USA (e-mail:[email protected]).

Color versions of one or more of the figures in this paper are available onlineat http://ieeexplore.ieee.org.

Digital Object Identifier 10.1109/TIA.2015.2402273

Fig. 1. DSSA VPM machine.

magnet (VPM) machines have become popular over the recentyears for several advantages including high torque density,smooth torque performance, etc. [4]–[8]. A nonoverlappingwinding VPM machine was presented in [9]. Dual-rotor anddual-stator VPM machines were proposed in [10], which are re-ported to further improve the torque density of VPM machines.The linear VPM machines were proposed [11], in which thefeatures such as high thrust force density and low cogging forceare reported. However, compared to a regular PM machine,VPM machines suffer from a low power factor [10], [12], whichmakes the VPM machines require a large-capability converterfor a given output power, which results in higher cost and largervolume in the converter.

In order to improve the power factor, dual-stator spoke-arrayVPM (DSSA VPM) machines as shown in Fig. 1 were proposedin [13]. Theoretical analysis by the finite-element algorithm(FEA) and prototype experiments have proved that the DSSAVPM machines have not only a higher torque density than thatof regular VPM machines but also a comparable power factor,viz., ∼0.85, with traditional PM machines. These featuresattribute to their vernier pole-slot structures and special relativeposition of their inner and outer stators which significantlyreduce magnet leakage and improve the main flux.

So far, the study on DSSA VPM machines is limited tointroducing the topology, operation principle, and its perfor-mance features. For the DSSA VPM machine, the dual-sidestructure makes its design work much more complex than thatof single-side VPM machines, and this design work has notbeen done. This paper elaborates on the analysis of the detaileddesign procedure of the DSSA VPM machines and developsan effective and efficient design methodology, including designparameter initial value selection, analytical sizing equation,

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LI et al.: DESIGN PROCEDURE OF DUAL-STATOR SPOKE-ARRAY VERNIER PERMANENT-MAGNET MACHINES 2973

Fig. 2. Exploded view of a DSSA VPM machine.

geometric size relationship, etc., to design the DSSA VPMmachine, and the analysis results are experimentally verifiedby a large-size natural-cooling prototype whose rated torque is2000 N · m.

II. STRUCTURE OF THE DSSA VPM MACHINE

This topology consists of dual stators and a sandwiched rotoras shown in Figs. 1 and 2, respectively. The inner and outerstators have half tooth pitch displacement; in other words, theinner/outer stator teeth face the outer/inner stator slots.

It should be noticed that the same phase axis of the inner andouter stators is not consistent, and the electrical phase shift forthe fundamental space harmonic caused by the angle betweenthe inner and outer stator phase axes as shown in Fig. 3(a) canbe expressed as

αio = Pr

(π

Pr− αmio

)= Pr

(π

Pr− π

Z

)=

Psπ

Pr + Ps(1)

where Z is the number of slots, Pr is the pole pairs of magnets,and Ps is the number of stator pole pairs.

In order to verify this deduction, the FEA model of one12/22 stator teeth/rotor pole DSSA VPM machine is built, andFig. 3(b) shows that the angle between the inner and outer statorphase axes is 15◦. By further investigation, it is found that theflux linkage amplitude per pole of the inner stator is 4% smallerthan that of the outer stator. In order to simplify the theoreticalanalysis latter, it is assumed in this paper that the outer statorflux Φm and inner stator flux Φ2 are equal to each other.

Since the flux linkage and induced voltage of the inner andouter stator coils have some phase shift, the same phase windingarrays of the inner and outer stators should be connected inseries or driven by two converters to avoid circulating current.In this paper, the same phase winding arrays of the inner andouter 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 outerstator, while A2 and O are the two terminals of phase A in theinner stator. The same phase winding arrays of the inner andouter stators are connected in series as shown in Fig. 4(c).

III. DESIGN FLOW

Machine design is the project, including works ranging fromsetting machine specification to finishing delivery as shown inFig. 5, and always includes more segments.

Fig. 3. Two stators of a 12/22 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, performancecalculation, and test verification, of the DSSA VPM machines.

Fig. 6 shows the electromagnetic design flow of the DSSAVPM machines. The first step in the machine design is notcalculating the machine size but analyzing the specificationswhich give much important information, such as the targetperformances of the machine and external condition limitations.

The next step is the “true” machine design process. First,since the inherent features including torque density, torqueripple, etc., are quite different for different configurations suchas stator/rotor pole combinations, stator/rotor configurations,etc., the configuration of the DSSA VPM machine should becarefully selected at first. Meanwhile, some design parameters,such as pole arc, slot opening, magnetic loading, etc., should begiven an initial value to start the machine size design, where thepole arc is defined as the ratio of the outer pole shoe arc lengthτp1 to the pole pitch, as shown in Fig. 1.

After that, the machine detailed sizes, such as the inner andouter diameters of the rotor and tooth width, can be obtainedfrom the sizing equation and geometrical relationship formulas,and then, the performance of this DSSA VPM machine canbe evaluated. If the predicted performances and specificationsdo not match very well, the fixed value of the parametersat the start of this design should be refined until they match


Fig. 4. Winding connection of the DSSA VPM machine. (a) Phase A windingconnection in the outer stator. (b) Phase A winding connection in the innerstator. (c) Winding connection of inner and outer stators in the DSSA VPMmachine.

Fig. 5. Simplified flowchart of the electrical machine design.

well with each other. This process may take several iterativeloops to get content results. FEA is employed to check theanalysis results of the magnetic circuit method. Then, severalparameter optimization loops using the design process are takento achieve different design purposes. Table I presents the majorspecifications 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 onthe DSSA VPM machines by FEA, and the major designparameters of the FEA model are summarized in Table II. Theanalysis results can help designers suitably select the majorelectromagnetic parameter initial values and further optimizethe machine dimensions.

Fig. 6. Design flowchart of the DSSA VPM machine design.

TABLE IMAJOR SPECIFICATIONS

TABLE IIMAJOR PARAMETERS

A. Stator and Rotor Pole Numbers

The stator and rotor pole numbers are unequal, and thisis quite different from that of a regular electrical machine.Hence, there is a special design parameter for the DSSA VPMmachines, i.e., the stator and rotor pole numbers, which hasa significant effect on the performances of the DSSA VPMmachine and optimal values of the other design parameters.Hence, the combination of stator and rotor pole numbers shouldbe analyzed and confirmed at the early stages of the design.

The relationship of the number of slots, rotor, and stator polepairs of the DSSA VPM machines satisfies

Z = Pr ± Ps (2)


TABLE IIICOMBINATION OF STATOR AND ROTOR POLE NUMBERS OF

THREE-PHASE DSSA VPM MACHINES (PR/SPSP)

Fig. 7. Fundamental back EMF versus pole ratio in the DSSA VPM machine.

where Z is the number of slots, Pr is the rotor pole pairs, andPs is the stator pole pairs

The balanced three-phase winding configurations make thestator and rotor combinations satisfy the following equation:

Z

G.C.D(Z,Ps)= mk, k = 1, 2, 3, . . . (3)

where G.C.D is the shorthand word of the greatest commondivisor, Z is the number of slots, Ps is the number of stator polepairs, and m is the number of phases. Table III gives severalavailable combinations of rotor, stator pole pair number, poleratio PR, and slots per phase per stator pole (SPSP).

In order to analyze performance sensitivity to stator androtor pole numbers, the other design parameters, such as polearc, electrical and magnetic loading, etc., are fixed at first, andseveral 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 withpole ratio. It can be seen that the optimal pole ratio for the

Fig. 8. Optimized stator pole pair for the maximum back EMF amplitudeversus outer diameter in the DSSA VPM machines with different pole ratios.

maximum fundamental back EMF varies with the stator polenumber, and the optimized pole ratios are 17, 17 and 11 for1, 2, 3-stator pole pair models in these cases, respectively.Fig. 8 shows the variation of the optimized stator pole pairnumber for the maximum back EMF amplitude with differentouter diameters. The optimized stator pole pairs increase as themachine outer diameter augments, and this attributes to thatlarge stator pole pair number can increase the airgap diameterby reducing the stator yoke thickness, but a too large stator polenumber means a large rotor pole number which introduces asignificant magnet leakage. Hence, it is very necessary to makecareful optimization to get the reasonable number of stator/rotorpole pairs for the excellent performance.

Fig. 9 summarizes the variation of the torque and power fac-tor with the rotor and stator pole combination. The 17-pole ratio2-stator pole pair DSSA VPM machine has the largest torquedensity but almost the lowest power factor among these DSSAVPM machines, while it can be seen that the candidate withhigher torque always suffers from a lower power factor. Hence,it is almost impossible for models to obtain the largest powerfactor and torque density at the same time, several compromisesshould be carefully done during the design procedure, and the24/22 stator tooth/rotor pole pair combination is selected forfurther work in this paper.

B. Magnetic Loading and Pole Arc

The magnetic loading and pole arc are the key parameters inthe machine design as illustrated in the foregoing sections andare fixed at the early design stage. The sensitivity of the twoparameters on the performance of the DSSA VPM machinesis analyzed in this section to help machine designers select theoptimal magnetic loading and pole arc for a particular designwork.

As illustrated in [4], the stator teeth of the VPM machineswork as not only part of the flux path but also of the flux modul-ator; hence, the slot opening ratio, i.e., the ratio of the slot open-ing to the slot pitch, is the key parameter in the DSSA VPMmachine design. The slot opening ratio is fixed to remove itsinfluence on performances during optimizing magnetic loadingand pole arc.


Fig. 9. Influence of pole ratio on the performance of the DSSA VPM machinewith different stator pole pair numbers. (a) Torque. (b) Power factor.

Fig. 10. Torque versus magnetic loading in the DSSA VPM machines withdifferent pole arcs (the outer yoke flux density By1 is 1.4 T, the outer slotopening ratio so1 is 0.6, and the outer slot opening ratio so2 is 0.6).

Fig. 10 gives the variation of the torque with magneticloading in the DSSA VPM machines with different pole arcs.The optimal magnetic loading for the maximum torque reducesas the pole arc increases, as shown in Fig. 11. The optimalmagnetic loading and pole arc for the maximum torque arearound 1.6 T and 0.62, respectively.

The variation of the power factor with magnetic loading andpole arc is presented in Fig. 12. It can be seen that the powerfactor of the models with smaller pole arc, which means largermagnet thickness, is always larger and increases as the magneticloading goes up.

Fig. 11. Optimal torque and magnetic loading versus pole arc in the DSSAVPM machines (By1 = 1.4 T, so1 = 0.6, and so2 = 0.6).

Fig. 12. Power factor versus magnetic loading Bg1m in the DSSA VPMmachines with different pole arcs (By1 = 1.4 T, so1 = 0.6, and so2 = 0.6).

Fig. 13. Torque versus so1 and so2 (Bg1m = 1.43 T, pole arc = 0.586, andA = 204 A/cm).

It can be seen that the maximum torque and power factorcould not be satisfied at the same time. Therefore, a compro-mise between torque, power factor, and magnet usage has to bemade. In this paper, magnetic loading and pole arc are selectedas 1.43 T and 0.586 for balancing these performances, such astorque density, power factor, etc.

Figs. 13 and 14 present the variation of the torque andpower factor with outer stator opening ratio so1 and innerstator opening ratio so2. It can be seen that the slot openingratio has a significant influence on the torque and power factorof the DSSA VPM machines, and in this case, the optimal


Fig. 14. Power factor versus so1 and so2 (Bg1m = 1.43 T, pole arc =0.586, and A = 204 A/cm).

Fig. 15. Torque versus ratio of outer stator to whole turns in series perphase (Bg1m = 1.43 T, pole arc = 0.586, Ns = 440 turns, Irms = 11.8 A,Aconductor = 0.78 mm2, and five conductors in one turn).

combination of so1 and so2 for maximum torque are 0.65 and0.63. At the range of the inner and outer stator slot opening ratiofrom 0.5 to 0.7, the power factor increases as the so1 and so2 goup, as shown in Fig. 14.

C. Electrical Loading

The higher limit on electrical loading can be selected bythe rules of thumb at the early design stage, and a moreaccurate higher limit of electrical loading should be estimatedby the thermal computation. Adjusting electrical loading in thefurther stage may be an inevitable work to guarantee that thetemperature rise is under the acceptable range. In other words,the whole electrical loading is limited by the thermal conditionand cooling capability. For the DSSA VPM machines, thereis another special design parameter, i.e., the ratio of the outerstator electrical loading Aouter to whole electrical loading A,which is a relatively flexible parameter under the same wholeelectrical loading.k is defined as the ratio of the outer stator electrical loading

Aouter to the whole electrical loading A. The variations of thetorque and power factor with k in the DSSA VPM machineswith different phase currents, viz., different copper loss, areshown in Figs. 15 and 16. The smaller k leads to shorter outerstator slot depth and larger outer airgap diameter. Meanwhile,

Fig. 16. Power factor versus ratio of outer stator to whole turns in series perphase (Bg1m = 1.43 T, pole arc = 0.586, Ns = 440 turns, J = 1.2 A/mm2,based Irms = 11.8 A, Aconductor = 0.78 mm2, and five conductors in oneturn).

the shorter slot depth always means weaker field modulationeffect, especially when the stator teeth are saturated. Therefore,there are optimal values of k for the maximum torque and powerfactor.

V. SIZING EQUATION OF THE DSSA VPM MACHINES

After selecting the configurations and several design param-eter initial values, the detailed design on DSSA VPM machinesizes can be done.

Sizing equation is widely used to obtain the main sizes ofthe electrical machines, and the sizing equations of varioustopologies are different due to their specific structures andoperation principle. There are numerous papers researchingon the specific sizing equations for novel topologies such asstator-mounted permanent-magnet machines [14], transverseflux circumferential current machines [15], etc. However, sincethese novel and DSSA VPM machines have different principlesand structures, the existing sizing equations are not availablefor the DSSA VPM machines. Hence, the sizing equation forthe DSSA VPM machines is required.

Generally, the total electromagnetic power of the surface-mounted PM machine is

Pe =m

T

T∫0

e(t)i(t)dt (4)

where m is the number of phases, T is the period of the backEMF, 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 linkagewaveforms of the DSSA VPM machines are really sinusoidaleven in single coil, and then, (4) can be rewritten as

Pe =1

2mEmIm cos γ (5)

where γ is the angle shift between the phase EMF and current,and Em and Im are the peak values of the fundamental phaseback EMF and current, respectively.

As illustrated in [8], the reluctance torque of the DSSAVPM prototype accounts for about 4% of the electromagnetic


Fig. 17. Coil and phase flux linkage waveforms of the DSSA VPM machine.

Fig. 18. Phase back EMF waveform of the DSSA VPM machine.

torque, and the reluctance torque can be neglected. Then, theelectromagnetic torque expression can be given by

Te =Pe

Ω=

m

2PrψpmIm cos γ (6)

where Ψpm 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 fluxlinkage can be obtained from

ψpm = kwNsφm (7)

where Φm is the flux per stator pole, Ns is the turns in seriesper phase, and kw is the winding factor.

Combining (6) and (7), it yields

Te =m

2PrkwNsφmIm. (8)

For the DSSA VPM machines, the flux per stator pole can beexpressed as

φm = kδαp1πDg1

4PsBg1mLstk (9)

where kδ is the leakage factor, αp1 is the rotor pole arc, Ps isthe number of stator pole pairs, Lstk is the stack length, andBg1m is the peak flux density in the outer airgap excited bymagnets. In [9], a leakage factor is defined for the flux reversalPM machines which have similar operation principle with VPM

machines, and its value is from 0.5 to 0.65. kδ = 0.55 ∼ 0.65for the DSSA VPM machines is obtained from several FEAresults used in the primary design.

The DSSA VPM machine has two airgaps, and the electricalloading A of the DSSA VPM machines is defined as

A =6NsIrms

πDg1=

6Irms

πDg1(Nsinner +Nsouter)

=Ainner +Aouter (10)

where Ainner is the inner stator electrical loading, Aouter isthe outer stator electrical loading, and Nsinner and Nsouter arethe number of turns in series per phase of the inner and outerstators, respectively.

The distribution factor of the DSSA VPM machines shouldconsider not only the phase shift among different coils in theouter or inner stator as regular PM machines do but also theinfluence of the angle αmio between the same phase axis ofthe inner and outer stators as shown in Fig. 3(a). Therefore, thedistribution factor of the DSSA VPM machine is quite differentfrom that of regular PM machines and is expressed as

kdv =sin

(vq αs

2

)q sin

(v αs

2

) sin(2v αio

2

)2 sin

(v αio

2

) (11)

and the pitch factor is

kpv = sin(vps

αw

2

)(12)

where v is the order of harmonics and αs and αw are thephase shift angles between the two adjacent EMF vectors inone phase and electrical angle of coil span for the fundamentalspace harmonic, respectively.

The winding factor for the fundamental harmonics is

kw = kd1kp1. (13)

Combining (5), (8), (9), and (10), the sizing equation isobtained

Pe =

√2

8π3kwkδαp1ABg1m

f

PsD2

g1Lstk cos γ. (14)

Meanwhile, the electromagnetic torque can be calculated by

Te =

√2

16π2kwkδαp1ABg1m

Pr

PsD2

g1Lstk cos γ. (15)

The rotor volume Vr is given by

Vr =πD2

g1

4Lstk =

Te√24 π Pr

PskwkδαpABg1m cos γ

(16)

and then the rotor outer diameter and lamination length can beobtained as ⎧⎨

⎩Dg1 =

(4kLVr

π

) 13

Lstk =(

4Vr

πk2L

) 13

(17)

where kL is the ratio of the outer airgap diameter Dg1 to thestack length Lstk.


Fig. 19. Geometry of the outer stator.

VI. GEOMETRICAL DESIGN

A. Outer Stator Design

Based on the sizing equation illustrated in Section IV, theouter rotor diameter Dg1 can be obtained, which may beadjusted in the following process to make sure that the outerdiameter and machine axial length meet the size limitations.

The stator tooth width tw1 shown in Fig. 19 is given by

tw1 =πDg1α1Bg1m

2PrBt1kstk(18)

where α1 is the pole arc of the outer airgap, Bt1 is the outerstator tooth flux density, and kstk is the stack factor.

Given the required outer stator current density J1 and linearloading Aouter of the outer stator, the outer stator slot areaviable for the conductor is expressed as

Aslot =πDg1Aouter

J1kcu=

[π

(Dg1

2+ h1 + h12 + h13

)2

−π

(Dg1

2+ h12 + h13

)2

− ztw1h1

](19)

and then the outer stator tooth depth is obtained as

h1 = 0.5

⎛⎝ztw1

π−Dg1 − 2h12 − 2h13

+

√(Dg1 + 2h12 + 2h13 −

ztw1

π

)2

+4Dg1A1

J1kcu

⎞⎠ . (20)

Since the stator yoke is required to support half of the fluxper stator pole, the outer stator yoke depth hy1 can be obtainedby the main magnet flux per stator pole Φm and the designedstator yoke flux density By1. hy1 is given by

hy1 =φm

2kstkBy1Lstk. (21)

After obtaining the outer stator yoke depth hy1, stator slotdepth h1, and outer rotor diameter Dg1, the outer stator diame-ter can be obtained as

Do = Dg1 + 2g + 2hy1 + 2h12 + 2h13 + 2h1. (22)

B. Rotor Design

As shown in Fig. 20, the rotor is sandwiched between twostators, and the magnet flux is driven by magnets through thetwo airgaps to combine the two stators together.

Fig. 20. Flux contour line of a DSSA VPM machine.

Fig. 21. Flux contour line of the spoke rotor configuration. (a) Traditionalspoke-type PM machine. (b) DSSA VPM machine.

The magnet flux circuit of the DSSA VPM machines issimilar with that of traditional spoke-array PM machines asshown in Fig. 21; thus, the magnet flux density in the airgapcan be calculated by⎧⎨

⎩2gHg1m +HmLm = 0; Ampere LawBg1m

πDg1m

4Pr= BmWm; Gauss Law

Bm = Br + u0urHm;

(23)

Bg1m =Br

πDg1

4PrWm+ 2μrg

Lm

. (24)

In order to take the iron bridge effect into account, theremanence Br in (21) should be replaced by the new value Brn

[16], i.e.,

Brn = Br −NbBsatbrWm

(25)

where Nb is the number of bridge of one magnet. From the viewpoint of the electromagnetic structure design, the smaller thebridge width br is, the smaller is the magnet leakage. The lowerlimit of the bridge width br is determined by the mechanicalmachining accuracy.

The magnet width Wm shown in Fig. 22 is obtained from(24) and (25)

Wm =

αp1πDg1

4Pr+ NbdBsatbr

Bg1m

Br

Bg1m− 2μrg

Lm

. (26)

Finally, the inner rotor diameter Dg2 can be given as

Dg2 = Dg1 − 2Wm − 4br. (27)


Fig. 22. Geometry of the rotor.

Fig. 23. Geometry of the inner stator.

C. Inner Stator Design

First, the inner diameter of the rotor is obtained from (27),and the inner rotor pole arc α2 is expressed as

α2 =πDg2 − 2PrLm

πDg2. (28)

Then, the maximum flux density excited by the magnets inthe inner airgap is given by

Bg2m =φm

α2Dg2=

Bg1mα1Dg1

α2Dg2. (29)

Meanwhile, the inner stator teeth width is

tw2 =πDg2α2Bg1m

2PrBt2kstk. (30)

The inner stator yoke thickness shown in Fig. 23 is given by

hy2 =φm

2kstkBy2Lstk. (31)

Based on the same method as the outer stator does, the innerstator teeth depth can be obtained

h2 = 0.5

⎛⎝Dg2 − 2h22 − 2h23 −

ztw2

π

−

√(Dg2 − 2h22 − 2h23 −

ztw2

π

)2

− 4Dg2A2

J2kcu2

⎞⎠ . (32)

Then, the inner stator inner diameter is given by

Di = Dg2 − 2hy2 − 2h12 − 2h13 − 2h2 − 2g. (33)

So far, the main sizes and detailed dimensions of each part ofthe DSSA VPM machines can be obtained. In order to validate

TABLE IVCOMPARISON OF ANALYTICAL METHOD AND FEA

the analytical equations derived in the foregoing section, a FEAmodel of a DSSA VPM machine is built, and its main sizes aregiven in Table II. Table IV summarizes the results predicted bythe FEA and analytical method, and it can be seen that the tworesult arrays match well.

After obtaining the sizing equation and geometrical relation-ship, the detailed design procedure can be given as shown inFig. 24 and summarized as follows.

1) The stator/rotor pole number and several design parame-ters of the DSSA VPM machines are selected as referringto the analysis results in Section IV.

2) Then, the rotor outer diameter Dg1 and lamination lengthLstk can be estimated by the sizing equation built inSection V.

3) The outer diameter of the outer stator can be obtainedby the geometrical relationship formulas presented inSection VI. In order to make sure that both the calculatedDo and stack length Lstk satisfy the size limitations,adjusting the electrical and magnetic loading may berequired.

4) In order to further make sure that the accuracy of thedesign results satisfies the specification demands, FEAverification on flux density, back EMF, and torque isemployed.

5) The initial values are selected based on experience or theanalysis results in Section IV and may not be optimal.Hence, the optimization of the design parameters is re-quired in the whole design procedure.

VII. EXPERIMENTS

A. Prototype

In order to verify the aforementioned analysis results, aDSSA VPM prototype has been built, and its major parametersare summarized in Table V. The exploded view of the DSSAVPM prototype is shown in Fig. 2. It can be seen that theprototype structure consists of several parts including the statorand rotor lamination stack, and auxiliary support part includingrotor support, frame, etc. The active part of the prototype hastwo stators and one rotor sandwiched by the stators. Since thestator teeth work as not only a part of the flux path but also ofthe flux modulator, opened slots are used in the DSSA VPMmachines to enhance the slot effect.

Both rotor and stators are stacked by 0.5-mm-thicknesslaminations, and 44 pieces of magnets (14 mm thickness) areinserted in the rotor lamination. The magnets use N35UHmaterial and are tangentially magnetized. In order to simplifythe stack process, all rotor pole shoes are connected togetherwith the 1-mm-thickness bridges as shown in Fig. 25. The


Fig. 24. Design procedure of the DSSA VPM machines.

TABLE VMAJOR PARAMETERS OF THE PROTOTYPE

bridge can also be considered to be removed to reduce magnetleakage, and all rotor pole shoes are connected to the rotorsupport by screws.

B. Test

The no-load braking torque of the prototype is about100 N · m, and it is found that this value slightly varies withspeed. Hence, this high no-load braking torque mainly at-tributes to the immature processing technology. Fig. 26 showsthe test bed of the prototype. The measured line voltage andcurrent waveforms at rated load are presented in Fig. 27.

Figs. 28 and 29 show the comparison of the amplitude of thefundamental phase back EMF and output torque measured bythe prototype experiments and predicted by FEA. The resultsevaluated by the FEA and experiments match well. The com-parison of the simulation and measured performance indexes atrated conditions has been done and summarized in Table VI.

Fig. 25. Photographs of the prototype. (a) Rotor lamination. (b) Rotor assem-bly. (c) Stator lamination.

Fig. 26. Test bed of the prototype.

VIII. CONCLUSION

This paper has presented the design procedure of the DSSAVPM machines, which includes the design parameter initialvalue setting, analytical sizing equation, key geometrical rela-tionship formulas, and design parameter optimization.

In addition, the feasible combinations of stator/rotor polenumbers of the DSSA VPM machines are given in this paper,


Fig. 27. Measured line voltage and current waveform (curve 1—line voltage;curve 2—line current).

Fig. 28. Fundamental phase back EMF versus rotor speed.

Fig. 29. Output torque versus phase current. The rated phase current is23.8 A.

TABLE VICOMPARISON OF SIMULATION AND MEASURED PERFORMANCE INDEXES

and the performance sensitivity of the DSSA VPM machines toseveral 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

pole pairs for maximum torque increase as the outer diameterincreases, and for a given diameter, the larger the pole ratio, thelower the optimal stator pole pairs are. The variation of optimalstator pole pairs to outer diameter is summarized in this paper.In addition, both the optimal inner and outer stator slot openingratios are around 0.65, which is almost independent fromother design parameters. The sizing equation and geometricrelationship formulas of the DSSA VPM machines have beenbuilt based on the magnetic circuit method and validated by FEA.

REFERENCES

[1] K. Atallah, J. Rens, S. Mezani, and D. Howe, “A novel ‘pseudo’ direct-drive brushless permanent magnet machine,” IEEE Trans. Magn., vol. 44,no. 11, pp. 4349–4352, Nov. 2008.

[2] R. Qu and T. Lipo, “Dual-rotor, radial-flux, toroidally wound, permanent-magnet machines,” IEEE Trans. Ind. Appl., vol. 39, no. 6, pp. 1665–1673,Nov./Dec. 2003.

[3] L. Jian, G. Xu, C. Mi, K. Chau, and C. Chan, “Analytical method formagnetic field calculation in a low-speed permanent magnet harmonicmachine,” IEEE Trans. Energy Convers., vol. 26, no. 3, pp. 862–870,Sep. 2011.

[4] A. Toba and T. Lipo, “Generic torque-maximizing design methodologyof surface permanent-magnet vernier machine,” IEEE Trans. Ind. Appl.,vol. 36, no. 6, pp. 1539–1546, Nov./Dec. 2000.

[5] D. Li and R. Qu, “Sinusoidal back-EMF of vernier permanent magnetmachines,” in Proc. ICEMS, Oct. 2012, pp. 1–6.

[6] R. Qu, D. Li, and J. Wang, “Relationship between magnetic gears andvernier PM machines,” in Proc. IEEE Int. Conf. Elect. Mach. Syst.,Aug. 2011, pp. 1–6.

[7] S. Niu, S. Ho, W. Fu, and L. Wang, “Quantitative comparison of novelvernier permanent magnet machines,” IEEE Trans. Magn., vol. 46, no. 6,pp. 2032–2035, Jun. 2005.

[8] B. Kim and T. Lipo, “Operation and design principles of a PMvernier motor,” IEEE Trans. Ind. Appl., vol. 50, no. 6, pp. 3656–3663,Nov./Dec. 2014.

[9] J. Li, K. Chau, J. Jiang, C. Liu, and W. Li, “A new efficient permanent-magnet vernier machine for wind power generation,” IEEE Trans. Magn.,vol. 46, no. 6, pp. 1475–1478, Jun. 2010.

[10] D. Li, R. Qu, and Z. Zhu, “Comparison of Halhach and dual-sidevernier permanent magnet machines,” IEEE Trans. Magn., vol. 50, no. 2,Feb. 2014, Art. ID. 7019804.

[11] Y. Du, K. Chau, M. Cheng et al., “Design and analysis of linear statorpermanent magnet vernier machines,” IEEE Trans. Magn., vol. 47, no. 10,pp. 4219–4222, Oct. 2011.

[12] E. Spooner and L. Hardock, “Vernier hybrid machines,” Proc. Inst. Elect.Eng.—Elect. Power Appl., vol. 150, no. 6, pp. 655–662, Nov. 2003.

[13] D. Li, R. Qu, and T. Lipo, “High power factor vernier permanent mag-net machines,” IEEE Trans. Ind. Appl., vol. 50, no. 6, pp. 3664–3674,Nov./Dec. 2014.

[14] J. Zhang, M. Cheng, Z. Chen, and W. Hua, “Comparison of stator mountedpermanent magnet machines based on a general power equation,” IEEETrans. Energy Convers., vol. 24, no. 4, pp. 826–834, Dec. 2009.

[15] S. Huang, J. Luo, and T. Lipo, “Evaluation of the transverse flux circum-ferential current machine by the use of sizing equations,” in Proc. Int.Elect. Mach. Drives Conf., May 1997, pp. WB2/15.1–WB2/15.3.

[16] N. Bianchi and T. Jahns, “Design, analysis, control of interior PM syn-chronous machines—Tutorial course notes,” presented at the IEEE IASAnnu. Meeting, Seattle, WA, USA, 2004, pp. 3-11–3-12.

Dawei Li (S’12) was born in China. He received theB.Eng. degree in electrical engineering from HarbinInstitute of Technology, Harbin, China, in 2010. Heis currently working toward the Ph.D. degree inthe School of Electrical and Electronic Engineering,Huazhong University of Science and Technology,Wuhan, China.

His research interests include design and analysisof novel permanent-magnet brushless machines.


Ronghai Qu (S’01–M’02–SM’05) was born inChina. He received the B.E.E. and M.S.E.E. de-grees from Tsinghua University, Beijing, China, in1993 and 1996, respectively, and the Ph.D. degreein electrical engineering from the University ofWisconsin–Madison, Madison, WI, USA, in 2002.

In 1998, he joined the Wisconsin ElectricMachines and Power Electronics Consortiums as aResearch Assistant. He became a Senior ElectricalEngineer with Northland, a Scott Fetzer Company,in 2002. In 2003, he joined the General Electric (GE)

Global Research Center, Niskayuna, NY, USA, as a Senior Electrical Engineerwith the Electrical Machines and Drives Laboratory. Since 2010, he has beena Professor with Huazhong University of Science and Technology, Wuhan,China. He is the author of over 50 published technical papers and is the holderof over 40 patents/patent applications.

Prof. Qu is a full member of Sigma Xi. He has been the recipient ofseveral awards from the GE Global Research Center since 2003, including theTechnical Achievement and Management Awards. He was also the recipient ofthe 2003 and 2005 Best Paper Awards, Third Prize, from the Electric MachinesCommittee of the IEEE Industry Applications Society (IAS) at the 2002 and2004 IAS Annual Meeting, respectively.

Wei Xu (M’09–SM’13) received double B.E. andM.E. degrees in electrical engineering from TianjinUniversity, Tianjin, China, in 2002 and 2005, respec-tively, and the Ph.D. degree in electrical engineeringfrom the Institute of Electrical Engineering, ChineseAcademy of Sciences, Beijing, China, in 2008.

From 2008 to 2012, he held several academicpositions with Australian and Japanese universitiesand companies. Since 2013, he has been a Full Pro-fessor with the School of Electrical and ElectronicEngineering, Huazhong University of Science and

Technology, Wuhan, China. His research topics mainly cover electromagneticdesign and control algorithms of linear/rotary machines, including induction,permanent-magnet, switched reluctance, and other emerging novel structuremachines.

Jian Li (M’10) received the B.E.E. degree fromDalian University of Technology, Dalian, China,in 2005 and the M.S.E.E and Ph.D. degrees fromDong-A University, Busan, Korea, in 2007 and 2011,respectively.

He is currently an Associate Research Profes-sor with the School of Electrical and ElectronicEngineering, Huazhong University of Science andTechnology, Wuhan, China. His research interestsinclude the design and analysis of PM and switchedreluctance machines and electric drives.

Thomas A. Lipo (M’64–SM’71–F’87–LF’00) is anative of Milwaukee, WI, USA. He received theB.E.E. and M.S.E.E. degrees from Marquette Uni-versity, Milwaukee, WI, in 1962 and 1964, respec-tively, and the Ph.D. degree in electrical engineeringfrom the University of Wisconsin, Madison, WI,in 1968.

From 1969 to 1979, he was an Electrical Engineerwith the Power Electronics Laboratory, CorporateResearch and Development, General Electric Com-pany, Schenectady, NY, USA. He became a Professor

of electrical engineering with Purdue University, West Lafayette, IN, USA, in1979, and in 1981, he joined the University of Wisconsin–Madison, Madison,WI, USA, where he served for 28 years as the W. W. Grainger Professorfor Power Electronics and Electrical Machines. He is currently an EmeritusProfessor with the University of Wisconsin.

Dr. Lipo received the Outstanding Achievement Award from the IEEEIndustry Applications Society, the William E. Newell Award from the IEEEPower Electronics Society, and the 1995 Nicola Tesla IEEE Field Awardfrom the IEEE Power Engineering Society for his work. He was elected asa member of the Royal Academy of Engineering (U.K.) in 2002, a memberof the National Academy of Engineering (U.S.) in 2008, and a member ofthe National Academy of Inventors (U.S.) in 2013. In 2014, he was selectedto receive the IEEE Medal for Power Engineering. For the past 40 years, hehas served the IEEE in numerous capacities, including President of the IEEEIndustry Applications Society in 1994.

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