Modeling of a Field-Modulated Permanent-Magnet Machine

energies Article

Modeling of a Field-Modulated Permanent-Magnet Machine Xianglin Li 1, *, K. T. Chau 2 and Yubin Wang 1 1 2

*

College of Information and Control Engineering, China University of Petroleum, Qingdao 266580, China; [email protected] Department of Electrical and Electronic Engineering, The University of Hong Kong, Hong Kong, China; [email protected] Correspondence: [email protected]; Tel.: +86-15066851211

Academic Editor: Joeri Van Mierlo Received: 4 August 2016; Accepted: 12 December 2016; Published: 19 December 2016

Abstract: In this work, an effective field-modulated permanent-magnet (FMPM) machine was investigated, in which the spoke-magnet outer rotor and open-slot stator were employed. The objective of this paper is to provide the mathematical modeling analysis that was performed for the purpose of control research on this type of FMPM machine. The simulation results by means of finite element analysis (FEA) are given to verify the theoretical analysis and the validity of mathematical model. A prototype machine was also fabricated for experimentation. Both the analytical model and the FEA results are validated by experimental tests on the prototype machine. Keywords: field-modulated permanent magnet (FMPM) machine; d-q frame; modeling; finite element analysis (FEA)

1. Introduction The operation of field-modulated permanent magnet (FMPM) machines relies on the “magnetic gearing effect” resulting from the magnetic field modulation [1], which can be derived from the coaxial magnetic gear by replacing the gear’s high-speed rotor with a stationary armature fed by symmetrical three-phase winding currents [2]. It has been discussed that FMPM machines can develop a high torque density by using the high-speed armature field operation with low armature pole-pairs and slot number while the rotor still rotates at a low speed to transmit a high torque [3–5]. Due to the “magnetic gearing effect” coupled with its compact structure, high torque density, and high efficiency, the FMPM machine is promising for low-speed direct-drive applications such as wind power generation [6], wave energy conversion [7], and electric vehicles [8]. Recently, in order to verify the potential of FMPM machines for industry applications, many improved FMPM topologies have been successively proposed and analyzed [9,10]. Furthermore, in order to improve the power factor, a dual-stator spoke-magnet FMPM machine was developed and its attractive characteristics were demonstrated in [11]. The work of foregoing research on FMPM machines mainly focuses on the analysis of electromagnetic characteristics, such as achieving a high torque or force density, core loss calculation, magnetic circuit optimization, among others [12]. These analysis results have certainly confirmed the advantageous features of FMPM machines on torque capability and efficiency compared to traditional permanent magnet (PM) machines [13]. However, so far, a detailed modeling analysis of these FMPM machines prepared for the purpose of driving control has not been reported. It is known that the direct torque control or field-oriented control based on a synchronous d-q frame are commonly used for PM motor drive [14]. However, due to the “magnetic gearing effect”, the rotational velocities of the PM rotor and armature field in an FMPM machine are different, thus it is necessary to discuss Energies 2016, 9, 1078; doi:10.3390/en9121078

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how to define the synchronous d-q frame. Meanwhile, as a foundation for its driving control system, Energies 2016, 9, 1078 2 of 15 the mathematical modeling of FMPM machines based on the newly defined synchronous d-q frame should be established. different, thus it is necessary to discuss how to define the synchronous d-q frame. Meanwhile, as a For this purpose, based on the previously presented spoke-magnet FMPM machine [15], this foundation for its driving control system, the mathematical modeling of FMPM machines based on paper reports on the modeling analysis performed to build a foundation for its control system. the newly defined synchronous d-q frame should be established. In SectionFor 2, this the purpose, topologybased of the FMPM machine will be introduced, and its operating onproposed the previously presented spoke-magnet FMPM machine [15], this principle is overviewed. In Section 3, the mathematical models developed based on stator frame paper reports on the modeling analysis performed to build a foundation for its control system. In and synchronous frame of will beproposed described, and the finitewill element analysis (FEA) also used for Section 2, thed-q topology the FMPM machine be introduced, and itsis operating steady-state serves as themodels basis developed of mathematical models. To verify principlecharacteristics is overviewed. analysis, In Section which 3, the mathematical based on stator frame and the synchronous d-q frame willabe described,machine and the has finitebeen element analysisand (FEA) is also used for was validity of modeling analysis, prototype fabricated its control system steady-state characteristics analysis, which serves as the basis of mathematical models. To verify the also constructed, and the preliminary experimentation performance is discussed in Section 4. Finally, of modeling analysis, a prototype machine has been fabricated and its control system was somevalidity conclusions are drawn in Section 5. also constructed, and the preliminary experimentation performance is discussed in Section 4. Finally, some conclusionsPrinciple are drawn in Section 5. 2. Topology and Operating 2. Topology and Operating Principle 2.1. Topology

Figure 1 shows the configurations of the proposed FMPM machine, in which the spoke-magnet 2.1. Topology outer rotor and open-slot stator are employed. An aluminum case is set outside the rotor by using Figure 1 shows the configurations of the proposed FMPM machine, in which the spoke-magnet a dovetail groove arrangement to fix laminations for structural reliability. Therotor spoke-magnets outer rotor and open-slot stator arerotor employed. An aluminum case is set outside the by using a on the rotor are magnetized along the tangential direction,for thus the flux-focusing effect can be obtained dovetail groove arrangement to fix rotor laminations structural reliability. The spoke-magnets on to greatly air-gap along flux density. Figure 1c shows the assembly diagram, in which theimprove rotor are the magnetized the tangential direction, thus thepractical flux-focusing effect can be obtained improve the air-gap Figure shows the purposes practical assembly diagram,stability. in there tois greatly a protective cover outsideflux thedensity. rotating rotor1cfor safety and structural which there is a protective cover outside the rotating rotor for safety purposes and structural There are 14 pole-pairs on the rotor and 18 slots on the stator, according to the pole-slot combination are 14[16,17], pole-pairs onthe thestator rotor windings and 18 slots on the stator, according to the pole-slot rules stability. of FMPMThere machines thus need to be wound as 4 pole-pairs, which can combination rules of FMPM machines [16,17], thus the stator windings need to be wound as 4 be termed as an 18-slot/8-pole FMPM machine. That is, the stator pole pitch is 9/4 of the slot pitch. pole-pairs, which can be termed as an 18-slot/8-pole FMPM machine. That is, the stator pole pitch is Hence, the three-phase symmetrical distributed windings consisting of 18 double-layer coils can be 9/4 of the slot pitch. Hence, the three-phase symmetrical distributed windings consisting of 18 adopted, in whichcoils eachcan coilbespan covers slot pitches. In this case, theslot phase-winding is 0.945, double-layer adopted, in two which each coil span covers two pitches. In thisfactor case, the and aphase-winding sinusoidal back-electromotive (EMF)back-electromotive can be developed. factor is 0.945, and force a sinusoidal force (EMF) can be developed. Aluminum case

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(a) Aluminum case Rotor PMs

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Figure 1. Configurations of the proposed field-modulated permanent-magnet (FMPM) machine. (a)

Figure 1. Configurations of the proposed field-modulated permanent-magnet (FMPM) machine. Cross-section; (b) three-dimensional global model; and (c) assembly diagram. (a) Cross-section; (b) three-dimensional global model; and (c) assembly diagram.

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2.2. Operating Principle 2.2. Operating Principle 2.2. Operating Principle Figure 2 shows the open-circuit field distributions of the proposed machine at different rotor Figure 2 shows the open-circuit field distributions of proposed the proposed machine at different rotor Figure 2 shows open-circuit field distributions ofshown the machine atbe different rotor positions obtained bythe FEA. At the initial rotor position, as in Figure 2a, it can seen that the positions obtained by FEA. At the initial rotor position, as shown in Figure 2a, it can be seen that the positions by FEA. At the rotor teeth position, as shown in A1 Figure 2a, it can seen directionsobtained of flux lines through theinitial two stator linked with coil (or coil A2, orbe coil A3)that are the the directions of flux through the stator teeth linked with coil (or coilA2, A2, or coilA3) A3) are the directions of the flux lineslines through theinduced twotwo stator teeth linked coil A1A1 (or same. Thus, total flux linkage in coil A1, coil with A2, and coil A3 coil of phaseorAcoil reaches are the same. Thus, the total flux linkage induced in coil A1, coil A2, and coil A3 of phase A reaches the the same.maximum Thus, thevalue total at flux inducedWhen in coilthe A1, coilrotates A2, and A3 rotor of phase A reaches positive thelinkage initial position. rotor 1/4coil of the pole-pair pitch positive maximum value at the initial position. When the rotor rotates 1/4 of the rotor pole-pair pitch the positive maximum at the the rotor rotates 1/4induced of the rotor pole-pair counterclockwise to thevalue position as initial shownposition. in FigureWhen 2b, the total flux linkage in coil A1, coil counterclockwise to the position as shown in Figure 2b, the total flux linkage induced in coil A1, A1, coil pitch counterclockwise to the position as shown in Figure 2b, the total flux linkage induced in coil A2, and coil A3 will be nearly equal to zero. At rotor position θe = 180°, as shown in Figure 2c, the A2, and coil A3 will be nearly equal to zero. At rotor position θ e = 180°, as shown in Figure 2c, the ◦ coil A2, and willlinked be nearly to coil zero.A2, Atand rotor position asthat shown in Figure 2c, e = 180 , to directions ofcoil fluxA3 lines with equal coil A1, coil A3 areθopposite generated at the directions of of flux flux lines lines linked linked with coil coil and coil A3 are opposite totothat at the the directions with coil A1, A1, coil A2, A2, opposite thatgenerated generated initial position, thus the total flux linkage induced in and coil coil A1, A3 coilare A2, and coil A3 reaches the initial position, thus the total flux linkage induced in coil A1, coil A2, and coil A3 reaches at the initialnegative position,value. thus the total linkage induced in coil coil A2, and coil reaches the the maximum Also, at flux rotor position θe = 270°, asA1, shown in Figure 2d,A3the total flux maximum negative value. Also, at rotor position θ e = 270°, as shown in Figure 2d, the total flux ◦ maximum negative Also, rotor position = 270 , aszero shown in Figure 2d, the flux linkage linkage induced invalue. coil A1, coilatA2, and coil A3θebecomes again. So, when thetotal rotor rotates by linkage induced in coil A1, coil A2, and coil A3 becomes zero again. So, when the rotor rotates by induced in coil A1, coil A2, and coil A3 becomes zero again. So, when the rotor rotates by one pole-pair one pole-pair pitch, a periodically changing flux linkage occurs in stator coils. However, it can be one pole-pair pitch, a periodically changing flux linkage occurs in stator However, it can be pitch, changing fluxthe linkage occurs in stator However, it cancoils. be clearly clearlya periodically seen from Figure 2 that pole-pair number oncoils. the stator is much lower thanseen the from rotor clearly seen from Figure 2 that the pole-pair number on the stator is much lower than the rotor Figure the pole-pair number the stator isthat much thantooth-slot the rotor magnet pole-pairs. magnet2 that pole-pairs. It has beenondiscussed thelower stator alternation makesIt has the magnet pole-pairs. It has been discussed that the the stator tooth-slot alternation makes the been discussed that the stator tooth-slot alternation permeance circumferential air-gap permeance uneven, whichmakes results incircumferential this magneticair-gap field modulation circumferential air-gap permeance uneven, which results in this magnetic field modulation uneven, whichTherefore, results in this magnetic field modulation phenomenon. Therefore, proposed phenomenon. in the proposed FMPM machine, the stator windings can in be the designed and phenomenon. Therefore, in the can proposed FMPM machine, the stator windings can be designed and FMPM machine, the stator windings be designed and wound by 4 pole-pairs to reduce complexity. wound by 4 pole-pairs to reduce complexity. wound by 4 pole-pairs to reduce complexity.

6.43o

A1 6.43o

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q q d (a) d (a) A3

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A3

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Figure 2. Open-circuit field distributions of the proposed 18-slot/8-pole FMPM machine at four Figure 2. Open-circuit field distributions of ofthe at four Figure 2. Open-circuit field distributions theproposed proposed18-slot/8-pole 18-slot/8-pole FMPM FMPM machine at different rotor positions, θe (“ ” represents the direction of flux line). (a) θe = 0°; (b) θ◦ e = 90°; (c) θe◦= four different rotor positions, positions, θθee (“ (“ ”” represents line). (a) (a)θθee==0°; 0 (b) ; (b)θeθ=e 90°; = 90(c) ; θe = represents the the direction of flux line). 180°; and (d) θe = 270°. ◦ ; and (c) θe180°; = 180and = 270◦ . (d)(d) θe =θe270°.

Figure 3a shows the no-load air-gap flux density of the proposed 18-slot/8-pole FMPM machine Figure 3a shows the no-load air-gap flux density of the proposed 18-slot/8-pole FMPM machine Figure 3a shows the1a, no-load air-gap flux density of the 18-slot/8-pole FMPM machine associated with Figure in which the initial condition of proposed mechanical angle is aligned with x-axis. associated with Figure 1a, in which the initial condition of mechanical angle is aligned with x-axis. associated withinFigure in which initial condition effect of mechanical angle aligned x-axis. As depicted [2,13],1a,due to thethefield-modulation produced by isthe statorwith tooth-slot As depicted in [2,13], due to the field-modulation effect produced by the stator tooth-slot As depicted in [2,13], due flux to thedensity field-modulation produced by the stator tooth-slot alternation, alternation, the air-gap consists ofeffect a series of harmonic fields. By using Fourier alternation, the air-gap flux density consists of a series of harmonic fields. By using the Fourier the air-gap flux density consists of a series of harmonic fields. By using Fourier transformation, transformation, the corresponding harmonic spectrum can be analyzed and is shown in Figure 3b. transformation, the corresponding harmonic spectrum can be analyzed and is shown in Figure 3b. corresponding harmonic be PM analyzed and modulated is shown in4Figure 3b. It can be seen It can be seen that the 14spectrum pole-paircan rotor field and pole-pair harmonic fieldthat are It can be seen that the 14 pole-pair rotor PM field and modulated 4 pole-pair harmonic field are the 14 pole-pair rotor PMthese field two and magnetic modulated 4 pole-pair harmonic field are prominent. Meanwhile, prominent. Meanwhile, fields can be further modulated by the 18 stator teeth to prominent. Meanwhile, these two magnetic fields can be further modulated by the 18 stator teeth to these two magnetic fields can be further modulated by the 18 stator teeth to generate distinct generate distinct 24 pole-pair and 22 pole-pair harmonic fields, respectively. In fact, the generate distinct 24 pole-pair and 22 pole-pair harmonic fields, respectively. In fact, 24 pole-pair and 22energy pole-pair harmonic fields, respectively. In fact, FMPM the electromechanical energy the electromechanical conversion in the proposed 18-slot/8-pole machine is undertaken electromechanical energy conversionFMPM in the machine proposedis 18-slot/8-pole is undertaken conversion in the proposed 18-slot/8-pole undertaken byFMPM two setsmachine of harmonics by two sets of harmonics [13]. The first set is the 4 pole-pair stator fundamental and the [13]. PM by two sets of harmonics [13]. The first set is the 4 pole-pair stator fundamental and the PM The first set is the 4 pole-pair stator fundamental and the PM subharmonic resulting from the interaction subharmonic resulting from the interaction of the 14 pole-pair PM rotor and the 18 stator teeth. The resulting fromthe the18interaction of The the 14 pole-pair rotor and the 18 stator teeth. The of thesubharmonic 14 set pole-pair rotor and stator teeth. second set is PM the interaction of the 14 harmonic pole-pair second is thePM interaction of the 14 pole-pair stator magnetomotive force(MMF) second set is theforce interaction of the 14resulting pole-pair stator magnetomotive force(MMF) harmonic stator magnetomotive (MMF) harmonic the modulation 4 pole-pair stator resulting from the modulation of the 4 pole-pair stator from fundamental and theof 18the stator teeth, and the resulting from the modulation of the 4 pole-pair stator fundamental and the 18 stator teeth, and fundamental thePM 18 stator and 14 pole-pair rotor PM field. Due to the co-contribution of the 14 pole-pair and rotor field. teeth, Due to thethe co-contribution of two sets of harmonics, a high torque 14 pole-pair rotor PM field. Due to the co-contribution of two sets of harmonics, a high torque two sets can of harmonics, high can be realized in the FMPM machine. density be realizedain thetorque FMPMdensity machine. density can be realized in the FMPM machine.

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Flux density density (T) (T) Flux

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20 10 15 20 10 15 Number Number of of pole-pairs pole-pairs

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(b) (b) Figure 3. No-load air-gap flux density of the proposed 18-slot/8-pole machine. (a) Waveform; Figure No-loadair-gap air-gapflux fluxdensity densityof ofthe theproposed proposed 18-slot/8-pole 18-slot/8-pole FMPM FMPM (a)(a) Waveform; Figure 3. 3. No-load FMPMmachine. machine. Waveform; and (b) harmonic spectrum. and (b) harmonic spectrum. and (b) harmonic spectrum.

3. Mathematical Modeling 3. Mathematical Modeling In the proposed machine, it can realize magnetic field modulation between rotor 14 pole-pairs In the proposed machine, canmagnetic realize magnetic fielddifferent modulation between rotor 14 pole-pairs and stator 4 pole-pairs, and it both fields have mechanical rotating velocities, and stator 4 pole-pairs, and both magnetic fields have different mechanical rotating velocities, namely, namely, achieving the “magnetic gearing effect”. However, taking the pole ratio Grr into account, achieving the “magnetic gearing effect”. However, taking the pole ratio G into account, both magnetic r both magnetic fields have the same electrical angular velocity ωee. According to the 4 pole-pair field, fields theangle same between electricaladjacent angular slot velocity thethe 4 pole-pair field, the electric e . According the have electric EMFωvectors is 80°;toon other hand, based on the angle 14 ◦ ; on the other hand, based on the 14 pole-pair field, it is 280◦ . between adjacent slot EMF vectors is 80 pole-pair field, it is 280°. That is, whether it is plotted based on a 4 pole-pair or 14 pole-pair field, the That is, whether is plotted based 4 pole-pair 14 pole-pair field, therespectively. same slot EMF star itdiagram same slot EMFitstar diagram canon beaobtained as or shown in Figure 4a,b, Hence, also can be obtained as shown in Figure respectively. Hence, it also indicates that themagnetic “magnetic indicates that the “magnetic gearing4a,b, effect” can make the 4 pole-pair and 14 pole-pair fields effect” together the energy achieving significantly high torque gearing cancontribute make the to 4 pole-pair andconversion, 14 pole-pairthus magnetic fields together contribute to the capability in the proposed FMPM machine. energy conversion, thus achieving significantly high torque capability in the proposed FMPM machine. E Ecc

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Figure Figure 4. 4. Slot Slot electromotive electromotive force force (EMF) (EMF) star star diagram diagram and and d-q d-q axes axes definition definition of of the the proposed proposed Figure 4. Slot electromotive force (EMF) star diagram and d-q axes definition of the proposed 18-slot/8-pole 18-slot/8-pole FMPM FMPM machine. machine. (a) (a) Based Based on on 44 pole-pair pole-pair field; field; and and (b) (b) based based on on 14 14 pole-pair pole-pair field. field. 18-slot/8-pole FMPM machine. (a) Based on 4 pole-pair field; and (b) based on 14 pole-pair field.

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realize transformationfrom fromstator stator reference reference frame frame, thethe d-q d-q axesaxes To To realize thethe transformation frametotorotor rotorreference reference frame, of the proposed machinecan canbe be defined defined according rotating 4 pole-pair fieldfield or 14or pole-pair field asfield of the proposed machine accordingtoto rotating 4 pole-pair 14 pole-pair shown in in in which θf-4 θandand θf-14 θrepresent the mechanical angles of a 4 as shown in Figure Figure4a,b, 4a,b,respectively, respectively, which f -4 f -14 represent the mechanical angles of pole-pair field and a 14 pole-pair field relative to the initial position, respectively. As As shown in in a 4 pole-pair field and a 14 pole-pair field relative to the initial position, respectively. shown Figure 2a, stator a-axis is defined along the central axis of phase A at the initial position where the Figure 2a, stator a-axis is defined along the central axis of phase A at the initial position where the PM flux linkage of phase A reaches the positive maximum value. Then, the initial d-axis is chosen to PM flux linkage of phase A reaches the positive maximum value. Then, the initial d-axis is chosen be consistent with a-axis, and the q-axis is 90° electrical degrees ahead of d-axis. The following to be consistent with a-axis, and the q-axis is 90◦ electrical degrees ahead of d-axis. The following relationship is governed in the proposed machine: relationship is governed in the proposed machine: 1     f 14  r  G 1 f  4  θ = θ = r r  f −14 Gr θ f −4 e  pr r  (1) (1) θe = pθ  r pr  p r r  G =  r Gr ps p  

s

where θr and e are themechanical mechanical and and electrical ofof thethe rotor position, respectively; pr andprpsand where θr and θe θare the electricalangles angles rotor position, respectively; are the pole-pair number of rotor and stator, respectively. It can be found that with the rotation of ps are the pole-pair number of rotor and stator, respectively. It can be found that with the rotation the rotor, the mechanical angles θ f−4 and θ f−14 are different due to the “magnetic gearing effect”. of the rotor, the mechanical angles θf −4 and θf −14 are different due to the “magnetic gearing effect”. However, their electrical angles the same θe. That is,d-q theaxes d-q definition axes definition on 4 However, their electrical angles are are the same as θeas . That is, the basedbased on 4 pole-pair pole-pair field or 14field pole-pair is in fact equivalent if it is treated fromviewpoint the viewpoint of electrical field or 14 pole-pair is in field fact equivalent if it is treated from the of electrical angle. angle. Hence, to reduce confusion and simplify analysis, the following mathematical model is Hence, to reduce confusion and simplify analysis, the following mathematical model is established established based on the rotor electrical angle θe. based on the rotor electrical angle θe . 3.1. Mathematical Model in Stator Reference Frame

3.1. Mathematical Model in Stator Reference Frame

As shown in Figure 5a, the three-phase PM flux linkage of the proposed machine can be As shown in Figure 5a, the three-phase PM flux linkage of the proposed machine can be expressed as:

expressed as:

  cos (θe e )   ψ pma ψ=pmaψ mψ mcos  o m ψ mcos cos (θe e− 120120  o) ψ pmb ψ= pmbψ    ψ  ψ= ψ ψ cos o 120120  o) pmc pmc m m cos (θe e+

(2) (2)

1.5

1.3

1

1.2

Flux-linkage (Wb)

Flux-linkage (Wb)

where ψmψism the peak flux linkage. linkage. where is the peakvalue valueofofthe thephase phase PM PM flux

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Figure 5. PM flux linkage of the proposed machine in different reference frames: (a) stator reference

Figure 5. PM flux linkage of the proposed machine in different reference frames: (a) stator reference frame; (b) d-q reference frame. frame; (b) d-q reference frame.

Then, the total phase flux linkage excited by PMs and phase current on load can be expressed as:

Then, the total phase flux linkage excited by PMs and phase current on load can be expressed as:  L M ab M ac  ia   pma      a   aa         M L M  i    (3) b ba ψ pma ψa L aa M ab bbMac bc   b  i a pmb         c   M ca M cb Lcc   ic    pmc   (3)  ψb  =  Mba Lbb Mbc  ib  +  ψ pmb  ψ M M L i ψ ca cc c where ia, ib, and ic are the phase currents, Laa, Lcbbb, and Lcc are the phase self-inductances as shown in c pmc Figure 6a, and Mab, Mac, Mba, Mbc, Mca, and Mcb are the phase mutual inductances as shown in Figure 6b.

where ia , ib , and ic are the phase currents, Laa , Lbb , and Lcc are the phase self-inductances as shown in Figure 6a, and Mab , Mac , Mba , Mbc , Mca , and Mcb are the phase mutual inductances as shown in Figure 6b.

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Lcc

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(b)

Figure 6. Self- and mutual-inductance of the proposed machine. (a) Self-inductance; and (b) mutual Figure 6. Self- and mutual-inductance of the proposed machine. (a) Self-inductance; and (b) mutual inductance. inductance.

Table 1 lists the characteristic information of inductance. It can be seen that the first and second Table 1components lists the characteristic information of inductance. It canforbearound seen that and harmonic of self-inductance are significant, which account 10.1%the andfirst 6.7% second components of self-inductance are significant, whichofaccount around 10.1% and of its harmonic DC component LDC, respectively. Moreover, the DC component mutualfor inductance MDC, as 6.7% of itsinDC component LDC , respectively. Moreover, the DC component of mutual inductance shown Table 1, is numerically about 16.4% of the self-inductance DC component LDC , which should be takenininto account. Although theabout fourth16.4% harmonic component of self-inductance and the MDC , as shown Table 1, is numerically of the self-inductance DC component LDC , second harmonic component of mutual inductance are not as negligible as all the components, which should be taken into account. Although the fourth harmonic component ofother self-inductance and order harmonic to facilitate the FMPMofmachine modeling, all components are neglected. theinsecond component mutual inductance arethese not asharmonic negligible as all the other components, phase inductance can be approximately as: in Thus, order the to facilitate the FMPM machine modeling, expressed all these harmonic components are neglected. Thus, the phase inductance can be approximately expressed as:  Laa  LDC  Lm1 cos  e   Lm 2 cos  2e  

  Lbb  LDC  Lm1 cos  e  120o   Lm 2 cos2  e  120o   −L L aa ≈ L DC  m1 cos(θe ) + Lm2 cos(2θe )  (4)    Lm1 cos  θe  120oo  Lm 2 cos 2  θe  120o  o )}  L ≈ L  Lcc−LLDC cos θ − 120 + L cos 2 θ − 120 ( ) ( { e e m2 m1 DC bb (4)  M ab  M bc  M ca  M ba  Mocb  M ac  M DC o  L ≈ L − L cos θ + 120 + L cos 2 θ ( ) ( {  cc e e + 120 )} m2 m1 DC   where Lm1 and L m2 are the peak values of the first and second harmonic components of Mab = Mbc = Mca = Mba = Mcb = Mac ≈ MDC self-inductance, respectively. where Lm1 and Lm2 are the peak values of the first and second harmonic components of Table 1. Inductance characteristics in stator reference frame. self-inductance, respectively. Self-Inductance Mutual Inductance TablePeak 1. Inductance characteristics in stator reference frame. Phase Angle Value (mH) Phase Angle Peak Value (mH) DC component 28.711 −4.7168 Self-Inductance 2.9° Mutual Inductance First harmonic 2.9022 0.8127 −76.5° Itemsharmonic Second 1.9161 −51.1°Angle Peak Value (mH) Phase6° Angle Peak1.353 Value (mH) Phase Third harmonic 0.2063 9.3° 0.03 −72.7° DCFourth component 28.711 − 4.7168 harmonic 1.2522 11.8° 0.289 71.3°- ◦ ◦ First harmonic 2.9022 2.9 0.8127 −76.5 Fifth harmonic 0.0784 13.6° 0.1454 −13.4° ◦ Second harmonic 1.9161 6◦ 1.353 −51.1 Third harmonic 0.2063 9.3◦ 0.03 −72.7◦ Thus, the voltage equations of the proposed machine in the stator reference frame can be◦written as: ◦ Fourth harmonic 1.2522 11.8 0.289 71.3 ◦ Fifth harmonic 0.0784 13.6 −13.4◦  d   0.1454 Items

a

  ua   Ra 0 0  ia   dt  u    0 R  i    d  b  0machine b Thus, the voltage equations of the  b proposed    b   dt in  the stator reference frame  uc   0 0 Rc  ic    written as: d  c   dψa         ua Ra 0 0 i a dt   dt        dψb  0 are  uphase  0 Rb which  iequal where Ra, Rb, and Rc are the windings, b  +due b  =resistances, dt symmetrical  to  dψc uc 0 Rfrom ic power supply termed as R. The power absorbed by the0windings the can be expressed as: c dt

(5) be can

thus (5)

Pi  ua ia  ubib  ucic (6) where Ra , Rb , and Rc are the phase resistances, which are equal due to symmetrical windings, thus termed as R. The power absorbed by the windings from the power supply can be expressed as:

Pi = u a i a + ub ib + uc ic

(6)

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Substituting Equations (3) and (5) into Equation (6), and neglecting mutual inductances to simplify derivation, the electromagnetic torque of the proposed machine in stator reference frame can be derived as: Te = Tpm + Tr (7) in which:

  dψ pma dψ pmb dψ pmc Tpm = pr i a + ib + ic dθe dθe dθe   pr 2 dL aa dL dLcc Tr = ia + i2b bb + i2c 2 dθe dθe dθe

(8) (9)

where Tpm is the PM torque, and Tr is called the reluctance torque, which is caused by the fluctuation of the phase self-inductance with rotor positions. 3.2. Abc-dq Transformation The vector-control strategy is based on the synchronous rotor frame, which rotates at the synchronous velocity. In order to get the two-phase rotary d-q axes electromagnetic parameters, the traditional Park matrix P3s/2r as shown in Equation (10) can be used for abc-dq transformation. 

P3s/2r

cos(θe ) 2 =  − sin(θe ) 3 1/2

cos(θe − 120o ) − sin(θe − 120o ) 1/2

 cos(θe + 120o )  − sin(θe + 120o )  1/2

(10)

Thus, the PM flux linkage in the d-q reference frame can be derived as:      ψ pma ψ pmd ψm        ψ pmq  = P3s/2r  ψ pmb  =  0  0 ψ pmc ψ pm0 

(11)

It can be seen from Equation (11) that the PM flux linkage in d-axis ψpmd is equal to the peak value of phase PM flux linkage in the stator reference frame. The PM flux linkages in q-axis ψpmq and in 0-axis ψpm0 are equal to zero. To verify the aforementioned analysis, Figure 5b gives the PM flux linkage in the d-q reference frame, which is transformed from the three-phase PM flux linkage in the stator reference frame, as shown in Figure 5a, obtained by using FEA. The average value of ψpmd , ψpmq , and ψpm0 are summarized in Table 2. It can be seen that the results calculated from the math model are consistent with the FEA. Table 2. PM flux linkage in d-q reference frame. FEA: finite element analysis.

Items ψpmd ψpmq ψpm0

Flux Linkage (Wb) From FEA

From Math Model

1.2106 −0.000079 −0.000173

1.2031 0 0

Then, the inductances in d-q axes frame can be described as follows: 

Ld   Lqd L0d

Ldq Lq L0q

  Ld0 L aa   = P Lq0  3s/2r  Mba Mca L0

Mab Lbb Mcb

 Mac  −1 Mbc  P3s/2r Lcc

(12)

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where Ld , Lq , L0 , Ldq , Lqd , Ld0 , L0d , Lq0 , and L0q are the synchronous inductance components in d-q axes frame. Thus, by substituting Equations (4) and (10) into Equation (12), the synchronous inductance Energies 2016, 9, 1078 8 of 15 components in the d-q axes frame can be derived as: LLm2 −LmL1 cos 3e (3θe ) m1 cos Ld = L DC MDC + m2 Ld −LDC M DC 2 2

(13)

Lm 2  Lm1 cos  3e  Lq  LDC  M DC Lm2 − Lm1 cos(3θe ) Lq = L DC − MDC − 2 2 L  L  2 M L0 =0 L DCDC+ 2MDC DC

(14) (14) (15)

LLm1 sin3(3θ m1 sin e  e)  L= Ldq =LdqLqd qd  22

(16)

Ld0 = 2L× L20d L=0d − +LmL2 m2  LLm1 cos cos 3e ( 3θe ) d0  m1 

(17)

 LLmm2 3(e3θ  e) Lq0 = L2q0× 2L0qL0= sin q − 2 sin

(18)

can be be observed observedfrom fromEquations Equations(13) (13)and and(14) (14) that and q-axis inductances, Ld and Lq , It can that thethe d- dand q-axis inductances, Ld and Lq, are are not constant, both of which contain a small cosine componentwith withthree threetimes timesvariation variation frequency frequency not constant, both of which contain a small cosine component of the PM flux linkage. Generally, the mutual inductance between the d- and q-axis windings should be zero, because the flux induced by a current in one winding will not link with another winding 90◦ electrical electrical degrees. degrees. However, in the PM machine with salient rotor and displaced in space by 90° teeth, aapart partof ofthe thed-axis d-axiswinding windingflux fluxwill will link with q-axis winding to the stator teeth, link with thethe q-axis winding duedue to the factfact thatthat the the uneven reluctance provides a path fluxthrough throughthe theq-axis q-axiswinding winding [18]. [18]. For For the the proposed uneven reluctance provides a path forfor flux cancan produce a certain “salient effect”. Hence, the d- and q-axis machine, the thespoke-magnet spoke-magnetrotor rotor produce a certain “salient effect”. Hence, the dandmutual q-axis inductance Ldq is not rather, is a small waveform and equal toequal Lqd . to Lqd. mutual inductance Ldqzero; is not zero; itrather, it is sinusoidal a small sinusoidal waveform and To validate the aforementioned equations, Figure 7a shows the self-inductance self-inductance components in and L’ L’00,, which which are are transformed transformed directly directly from from the three-phase inductances in d-q axes frame, L’dd, L’qq,, and the stator frame, as shown in Figure 6, obtained by using FEA. On the other hand, by substituting DC components componentsof ofthe theinductances inductances(L(LDCDC ,M ) and peak value of harmonic components of the DC ,M DCDC ) and thethe peak value of harmonic components of the the inductances Lm2 ) listed in Table into Equations (13)–(15), the self-inductance theaxes d-q inductances (Lm1,(LLm1 m2), listed in Table 1 into1 Equations (13)–(15), the self-inductance in theind-q axes frame, , L , and L can also be calculated and comparatively shown in Figure 7a. Meanwhile, frame, Ld, Lq,Land L 0 can also be calculated and comparatively shown in Figure 7a. Meanwhile, Figure 7b q 0 d Figure 7b compares theinductance mutual inductance components in d-qframe, axes frame, L’dq ,Land Ldq obtained by compares the mutual components in d-q axes L’dq, and dq obtained by the the aforementioned methods, respectively. It can seen fromFigure Figure77that thatthere thereare are some some minor aforementioned twotwo methods, respectively. It can bebe seen from and L’dd,, LLqq and and L’ L’qq, , LL00 and and L’ L’0,0 ,but butthe thevariation variationshape shapeisis nearly nearly the the same. same. differences between Ldd and the average averagevalues valuesare areLdL=d 34.33 = 34.33 mH, mH, L’q =mH, 31.58 mH, Moreover, the mH, Lq =L32.47 mH,mH, L’d = L’ 35.35 mH, L’ q = 31.58 and L0 q = 32.47 d = 35.35 = L’0mH = 19.3 mH as can be seen theare results are in good agreement. =and L’0 L=019.3 as listed inlisted Tablein 3. Table It can3.beItseen that the that results in good agreement. 50

2

Inductance (mH)

Inductance (mH)

Ldq Ld

L'd

40 30

Lq

L'q 20 L'0 10

0

60

1 0

L'dq

-1

L0

120 180 240 Rotor position, θe (o)

300

360

-2

(a)

0

60

120 180 240 Rotor position, θe (o)

300

360

(b)

Figure 7. Waveforms of inductance. (a) Self-inductance in d-q axes frame, Ld, Lq, L0, L’d, L’q, and L’0; Figure 7. Waveforms of inductance. (a) Self-inductance in d-q axes frame, Ld , Lq , L0 , L’d , L’q , and L’0 ; and in d-q d-q axes axes frame, frame, L Ldq and and (b) (b) mutual mutual inductance inductance in and L’ L’dq. . dq

dq

Table 3. Average values of inductance in d-q axes frame. Items L’d (Ld) L’q (Lq) L’0 (L0) L’dq (Ldq)

Average Values of Inductance (mH) From FEA From Math Model From [19] 35.35 34.33 33.68 31.58 32.47 30.72 19.3 19.3 −0.094 0 -

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Table 3. Average values of inductance in d-q axes frame.

Items L’d L’q L’0 L’dq

(Ld ) (Lq ) (L0 ) (Ldq )

Average Values of Inductance (mH) From FEA

From Math Model

From [19]

35.35 31.58 19.3 −0.094

34.33 32.47 19.3 0

33.68 30.72 -

In order to verify the inductance expressions in d-q axes frame, the calculation method of d-q frame inductance, as discussed in [19], can also be adopted, namely: Ld =

ψi cos α − ψ pm

Lq =

Id ψi sin α Iq

(19)

(20)

where ψi is the fundamental component of the total flux linkage considering the armature reaction effect, ψpm is the fundamental component of the flux linkage excited by PMs only, and α is the phase difference between ψi and ψpm . Based on Equations (19) and (20), the d-q frame inductances are Ld = 33.68 mH and Lq = 30.72 mH, as listed in Table 3. It can be seen that the calculated inductances in the d-q reference frame are nearly the same as the results obtained from the proposed mathematical method and the FEA. 3.3. Electromagnetic Torque Based on the preceding analysis, and assuming that the zero-sequence current component is equal to zero, the total flux linkage in d-q reference frame can be defined as: ( ψd = ψ pmd + Ld id + Ldq iq (21) ψq = Lq iq + Ldq id Substituting Equations (11), (13), (14), and (16) into Equation (21), it yields:      Lm2 − Lm1 cos(3θe ) Lm1 sin(3θe )   id + iq  ψd = ψm + L DC − MDC + 2 2      L − Lm1 cos(3θe ) Lm1 sin(3θe )   ψq = L DC − MDC − m2 iq + id 2 2

(22)

In order to prove the accuracy of Equation (22), the flux linkage in the d- and q-axis are calculated by two methods and compared in Figure 8, where “ψd _FEA” and “ψq _FEA” denote the total flux linkage in d- and q-axis, obtained by using FEA, when id =0 control strategy is adopted at the rated current, “ψd _Math model” and “ψq _Math model” denote the total flux linkage in d- and q-axis calculated by Equation (22). It can be seen that the shape and value of the d-q axes frame flux linkage based on these two methods are nearly the same.

In order to prove the accuracy of Equation (22), the flux linkage in the d- and q-axis are calculated by two methods and compared in Figure 8, where “ψd_FEA” and “ψq_FEA” denote the total flux linkage in d- and q-axis, obtained by using FEA, when id=0 control strategy is adopted at the rated current, “ψd_Math model” and “ψq_Math model” denote the total flux linkage in d- and q-axis calculated Equation (22). It can be seen that the shape and value of the d-q axes frame flux linkage Energies 2016,by 9, 1078 10 of 15 based on these two methods are nearly the same. 1.4

ψd_Math model

Flux linkage (Wb)

1.2 ψd_FEA

1 0.8 0.6

ψq_Math model

0.4 0.2

ψq_FEA

0

60

120 180 240 Rotor position, θe (o)

300

360

Figure the flux flux linkage linkage in in d-q d-q axes axes frame frame under under iid ==00control. Figure 8. 8. Waveforms Waveforms of of the control. d

The voltage equations in the d- and q-axis can be written as: The voltage equations in the d- and q-axis can be written as: d

 d  e  q  ud  Rid dψ dtd  − ωe ψq  Rid +  ud = q u  Ri  ddt  e  d q q  dψ dt q   u = Riq + + ωe ψd q dt

(23) (23)

Based on Equation (23), the electromagnetic torque of the proposed machine in d-q reference frame can be derived as:   3 ωe ψd i q − ψq i d Te = 2 ωr  i (24)
3pr h = ψm iq + id iq Ld − Lq + Ldq i2q − i2d 2 = Tpm + Tr where ωr is the rotor mechanical angular velocity. Substituting Equations (1), (13), (14), and (16) into Equation (24) yields: 3 (25) Tpm = Gr ps ψm iq 2   3 3 Tr = Gr ps id iq [ Lm2 − Lm1 cos(3θe )] + Gr ps Lm1 i2q − i2d sin(3θe ) (26) 2 4 Obviously, the reluctance torque Tr is the result of Lm1 , Lm2 due to the fluctuation of phase inductance, which can cause torque ripple. When id = 0 control method is adopted, then the three-phase currents are applied in phase with the back-EMF, which can be expressed as:     i a = Im sin(θe ) ib = Im sin(θe − 120o )    i = I sin(θ + 120o ) c m e

(27)

where Im is the peak value of the phase current. Thus, the phase current in d- and q-axis can be transformed as:    id = 0 (28) iq = Im   i =0 0

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By substituting Equation (28) into Equation (24), the electromagnetic torque can be derived as:   2L Im 3 m1 sin(3θe ) Te = Gr ps ψm Im + 2 2

(29)

It can be seen from Equation (29) that the electromagnetic torque consists of two components, namely a DC component and a sinusoidal component with three times variation frequency of the PM flux linkage. Hence, the torque ripple can be defined as: Kripple =

Temax − Temin L Im × 100% = m1 × 100% Teavg ψm

(30)

where Temax , Temin , and Teavg are the maximum value, the minimum value, and the average value of the electromagnetic torque, respectively. Equation (30) also denotes that the torque ripple is proportional to the peak value of the phase current. By substituting Gr = 3.5, ps = 4, ψm = 1.2031 Wb, Im = 8.3 × 1.414 A, Lm1 = 2.9022 mH into Equation (29), the electromagnetic torque of the proposed 18-slot/8-pole FMPM machine can be calculated and noted as “Te _Math model”, as shown in Figure 9. To verify the aforementioned mathematical analysis, the electromagnetic torque of the proposed machine can also be obtained by using FEA and termed as “Te _FEA”, which is simultaneously described in Figure 9. Table 4 lists the key characteristics of “Te _Math model” and “Te _FEA”. It can be seen that the average value of “Te _Math model” and “Te _FEA” are nearly the same, and the torque ripple of the proposed machine is less than 3%. However, it should be noted that the periodicity of the “Te _Math model” is different from that of the “Te _FEA”. This is mainly due to neglecting the higher-order harmonic components in Energies 2016, 9, 1078 11 of 15 the mathematically calculated inductance. 340

Te (Nm)

320

Te_Math model

300 280

Te_FEA

260 240

0

60

120 180 240 Rotor position, θe (o)

300

360

Figure Figure 9. 9. Electromagnetic Electromagnetic torque torque waveforms. waveforms. Table Table 4. 4. Characteristics Characteristics of of electromagnetic electromagnetic torque. torque. Items Items Temax (Nm) Temin (Nm) T emax(Nm) Teavg (Nm) emin(Nm) T T (Nm) Keavg ripple (%) Kripple (%)

Te_Math Model Te _Math Model 301.8 301.8 293.4 293.4 297.6 297.6 2.8 2.8

Te_FEA Te _FEA 296.1 296.1 291.1 291.1 293.4 293.4 1.7 1.7

4. Experimental Results 4. Experimental Results To validate the mathematical model and associated FEA analysis of the proposed machine, a To validate thehas mathematical model associated FEA the proposed prototype machine been fabricated forand experimentation as analysis shown inofFigure 10. The machine, detailed a prototype machine has been fabricated for experimentation as shown in Figure 10.and Thecompares detailed specifications and main design parameters are listed in Table 5. Figure 11 illustrates specifications and main designwith parameters are listed in Table 5. Figure 11 illustrates and measured compares the measured self-inductance the calculated self-inductance by means of FEA. The the measured self-inductance with the calculated self-inductance by means of FEA. The measured inductance waveforms agree well with the simulated results. The measured average phase inductance waveforms agree with the to simulated results. The average phase self-inductance is about 27.1 mH,well which is near the calculated value 28.7measured mH. self-inductance is about 27.1 mH, which is near to the calculated value 28.7 mH.

To validate the mathematical model and associated FEA analysis of the proposed machine, a prototype machine has been fabricated for experimentation as shown in Figure 10. The detailed specifications and main design parameters are listed in Table 5. Figure 11 illustrates and compares the measured self-inductance with the calculated self-inductance by means of FEA. The measured inductance agree well with the simulated results. The measured average 12 phase Energies 2016, 9,waveforms 1078 of 15 self-inductance is about 27.1 mH, which is near to the calculated value 28.7 mH.

(a)

(b)

(c)

Generator Prototype Machine

(d) Figure 10.Prototype of proposed 18-slot/8-pole FMPM machine. (a) Outer rotor; (b) stator; (c) Figure 10. Prototype of proposed 18-slot/8-pole FMPM machine. (a) Outer rotor; (b) stator; prototype assembly; and (d) experimental setup. (c) prototype assembly; and (d) experimental setup.

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Table 5. Design specifications of the prototype machine. Table 5. Design specifications of the prototype machine.

Parameters Parameters Rated Ratedspeed speed(r/min) (r/min) Back-EMF Back-EMF(V) (V) Rated Ratedcurrent current(A) (A) No. No.ofofrotor rotorpole-pairs pole-pairs No.ofofstator statorteeth teeth No. No.ofofstator statorpole-pairs pole-pairs No. 40

Values Parameters Values Values Parameters Values 172 Outer diameter of rotor (mm) 172 Outer diameter of rotor (mm) 220 220 220 Inner diameter of stator (mm) 220 Inner diameter of stator (mm) 60 60 Air-gap length 8.3 8.3 Air-gap length (mm)(mm) 0.5 0.5 length 14 14 StackStack length (mm)(mm) 300 300 18 18 Iron Iron lamination material lamination material DW470 DW470 4 4 PM material N38SH PM material N38SH Lcc_FEA

Laa_FEA

Lbb_FEA

Lcc_measured

Laa_measured

Lbb_measured

Inductance (mH)

30 20 40 30 20

0

60

120 180 240 Rotor position, θe (o)

300

360

Figure Figure 11. 11. Comparison Comparison of of measured measured and and simulated simulated self-inductances. self-inductances.

Also, the simulation and measured open-circuit back-EMF at the rated speed 172 r/min are Also, the simulation and measured open-circuit back-EMF at the rated speed 172 r/min are shown shown in Figure 12. The phase back-EMF root mean square (RMS) values of simulation and in Figure 12. The phase back-EMF root mean square (RMS) values of simulation and experimental experimental results are 224 V and 220 V, respectively, and the corresponding total harmonic results are 224 V and 220 V, respectively, and the corresponding total harmonic distortion (THD) are distortion (THD) are 3.8% and 3%, respectively. It can be found that the experimental results exhibit a good agreement with the simulation results. 400

0

EMF

EMF (V)

RMS

200 RMS

RMS

0

60

120 180 240 Rotor position, θe (o)

300

360

Figure 11. Comparison of measured and simulated self-inductances.

Also, the simulation and measured open-circuit back-EMF at the rated speed 172 r/min are Energies 2016, 9, 1078 13 of 15 shown in Figure 12. The phase back-EMF root mean square (RMS) values of simulation and experimental results are 224 V and 220 V, respectively, and the corresponding total harmonic 3.8% and 3%, respectively. It can3%, be respectively. found that theIt experimental exhibit a good agreement with distortion (THD) are 3.8% and can be foundresults that the experimental results exhibit simulation results. athe good agreement with the simulation results. 400

RMS

Back EMF

Back EMF (V)

RMS

200 0 -200

RMS

frequency

frequency

-400 0

90

270 180 Rotor position, θe (o)

360

(a)

(b)

Figure 12. Back-EMF waveforms at the rated speed. (a) Finite element analysis (FEA) results; and (b) Figure 12. Back-EMF waveforms at the rated speed. (a) Finite element analysis (FEA) results; experimental results (100 V/div). and (b) experimental results (100 V/div).

Based on the aforementioned modeling analysis, the vector-control strategy can be applied for Based on the aforementioned modeling analysis, the vector-control strategy be applied for the the prototype machine. Figure 13a shows the variations of output torque andcan speed with sudden prototype machine. Figure 13a shows the variations of output torque and speed with sudden increased increased load from 115 Nm to 180 Nm at the rated speed 172 r/min, and the corresponding load from of 115the Nm to 180 Nmcurrent at the rated speedin172 r/min, and variations of the variations three-phase are given Figure 13b. It the can corresponding be seen that the output torque three-phase current are given in Figure 13b. It can be seen that the output torque can rapidly track the can rapidly track the load changes, and the torque ripples at steady state are always very low. load changes, and the torque ripples at steady state are always very low. Energies 2016, 9, 1078 13 of 15 300

Torque (Nm)

Speed

Output torque

200

250 200

150

Load torque

150

100

100

50

50

0

Speed (r/min)

250

Phase current (7.5A/div)

300

0

Time (25ms/div)

(a)

Time (25ms/div)

(b)

Figure 13. Performance characteristics of the prototype machine with sudden increased load at the Figure 13. Performance characteristics of the prototype machine with sudden increased load at the rated current variation. variation. rated speed. speed. (a) (a) Output Output torque torque and and speed speed variations; variations; and and (b) (b) three-phase three-phase current

In the experiments, the input power can be calculated by the measured input voltage and In the experiments, input power can be calculated by the measured voltage and current, current, and the output the power can be calculated by the measured outputinput torque and rotor speed. and the output power can be calculated by the measured output torque and rotor speed. Thus, Thus, the efficiency of the prototype machine can be obtained as shown in Figure 14. It can be the efficiency theprototype prototype efficiency machine can be obtained as shown in Figure 14. It can observed that observed thatofthe improves as the load increases. When thebeoutput torque the prototype efficiency the load increases. When the output torque achievesefficiency 255 Nm, achieves 255 Nm, namelyimproves the rated as phase current 8.3A is reached, the measured prototype namely phase 8.3A is reached, the measured prototype is machine about 0.92. is about the 0.92.rated Under this current rated condition, the measured power factor of theefficiency prototype is Under this rated condition, the measured power factor of the prototype machine is about about 0.87, which is consistent with the conclusion drawn in the dual-stator spoke-array vernier0.87, PM which is consistent the indicating conclusion that drawn the dual-statorrotor spoke-array vernier PM machine machine presented with in [20], theinspoke-magnet arrangement can effectively presentedthe inpower [20], indicating thatproposed the spoke-magnet rotor arrangement can effectively improveand the improve factor of the FMPM machine. Without considering the mechanical power factor of the proposed FMPM machine. Without considering the mechanical and stray losses, stray losses, the transmitted torque per total volume of the prototype machine is up to 22.4 kNm/m3 the transmitted torque per condition. total volume of the prototype machine is up to 22.4 kNm/m3 under the under the naturally cooled naturally cooled condition. Furthermore, in order to achieve perfect operation performance, improved control algorithms Furthermore, in order to achieve operation performance,experimental improved control algorithms are are still under testing for the controlperfect system. The corresponding results will be the still under for testing for the paper. control system. The corresponding experimental results will be the substance substance our future for our future paper. 1

fficiency

0.8 0.6 0.4

improve the power factor of the proposed FMPM machine. Without considering the mechanical and stray losses, the transmitted torque per total volume of the prototype machine is up to 22.4 kNm/m3 under the naturally cooled condition. Furthermore, in order to achieve perfect operation performance, improved control algorithms are still under testing for the control system. The corresponding experimental will be the Energies 2016, 9, 1078 14 ofresults 15 substance for our future paper.

Figure 14. Measured efficiency of prototype machine.

5. Conclusions In this work, the structure, operating principle, and the static characteristics of a new field-modulated PM machine have been investigated. Based on the static characteristics, the mathematical model of flux-linkage, inductance, and electromagnetic torque in the stator reference frame has been built. By using Park’s transformation, the d- and q-components of flux-linkage, inductance, voltage, and electromagnetic torque in the d-q reference frame have been derived. The electromagnetic performances based on the mathematical models are also verified by FEA results. To validate the theoretical analysis, a prototype machineFigure has been built andefficiency tested. The experimental results are in 14. Measured of prototype machine. satisfactory agreement with the predicted results from the FEA and mathematical models. Therefore, it can be concluded that the mathematical modeling analysis of the proposed FMPM machine has formed a foundation for future study in vector-control or direct-torque control for this type of FMPM machines. Acknowledgments: This work was supported in part by the National Natural Science Foundation of China (Projects 51507191 and 51277183), by the Fundamental Research Funds for the Central Universities of China (Project 15CX02113A), and by the Scientific Research Foundation of China University of Petroleum. Author Contributions: Xianglin Li mainly conducted the modeling analysis of the proposed field-modulated permanent-magnet machine and performed the experiments. The manuscript was improved and revised by K. T. Chau and Yubin Wang. All of the authors contributed to the paper writing. Conflicts of Interest: The authors declare no conflict of interest.

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