ANALYTICAL CALCULATION OF PARALLEL DOU- BLE EXCITATION AND SPOKE-TYPE PERMANENT- MAGNET MOTORS; SIMPLIFIED VERSUS EXACT MODEL

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1 Progress In Electromgnetics Reserch B, Vol. 47, , 13 ANALYTICAL CALCULATION OF PARALLEL DOU- BLE EXCITATION AND SPOKE-TYPE PERMANENT- MAGNET MOTORS; SIMPLIFIED VERSUS EXACT MODEL Kmel Boughrr 1, *, Thierry Lubin, Rchid Ibtiouen 3, nd Mohmed N. Benlll 1 1 Lbortoire de l Energie et des Systèmes Intelligents LESI), Université de Khemis-Milin, Route de Theniet El-hd, Khemis- Milin 445, Algeri Groupe de Recherche en Electrotechnique et Electronique de Nncy, Université de Lorrine, GREEN, EA 4366, Vndoeuvre-lès-Nncy F- 5456, Frnce 3 Ecole Ntionle Polytechnique LRE-ENP), Algiers, 1, Av. Psteur, El Hrrch, BP 18, 16, Algeri Abstrct This pper dels with the prediction of mgnetic field distribution nd electromgnetic performnces of prllel double excittion nd spoke-type permnent mgnet PM) motors using simplified SM) nd exct EM) nlyticl models. The simplified nlyticl model corresponds to simplified geometry of the studied mchines where the rotor nd sttor tooth-tips nd the shpe of polr pieces re not tken into ccount. A D nlyticl solution of mgnetic field distribution is estblished. It involves solution of Lplce s nd Poisson s equtions in sttor nd rotor slots, irgp, buried permnent mgnets into rotor slots nd non mgnetic region under mgnets. A comprison between the results issued from the simplified model with those from exct model EM) which represents more relistic geometry with sttor nd rotor tooth-tips nd the shpe of polr pieces) is done to show the ccurcy of the simplified geometry on mgnetic field distribution nd electromgnetic performnces cogging torque, electromgnetic torque, flux linkge, bck-emf, self nd mutul inductnces). The nlyticl results re verified with those issued from finite element method FEM). Received 13 November 1, Accepted 1 December 1, Scheduled 31 December 1 * Corresponding uthor: Kmel Boughrr boughrrkmel@yhoo.fr).

2 146 Boughrr et l. 1. INTRODUCTION Anlyticl models re useful tools for first evlutions of electricl motors performnces nd for the first step of design optimiztion. The im of this pper is to nlyticlly predict the mgnetic field distribution nd electromgnetic performnces of prllel double excittion nd spoke-type PM motors, such s cogging torque, flux linkge, bck-emf, electromgnetic torque, self nd mutul inductnces, nd DC rotor excittion current cpbility for the control of flux linkge. The proposed nlyticl model is bsed on subdomin method. Mny uthors hve proposed nlyticl simplified nd exct models bsed on subdomin method in order to study the sttor slotting effects with or without tooth-tips) on mgnetic field distribution nd electromgnetic performnces under no-lod nd lod conditions) in rdil inset nd surfce-mounted permnent mgnet motors [1 11]. It ws shown tht the ccurcy of subdomin models is higher thn permence models [1] or conforml trnsformtions models [13 15]. However, there re no uthors who pplied simplified nlyticl model for predicting mgnetic field nd electromgnetic performnces in prllel double excittion nd spoke-type PM motors. There re only Lin et l. in [13] who clculted mgnetic field nd cogging torque by conforml mpping with simplified model of spoketype PM motors. Wu et l. [5] hve shown recently tht subdomin model which tkes into ccount the sttor tooth-tips in surfce-mounted permnent mgnet motors gives pproximtely the sme results in terms of electromgnetic performnces s the one which neglects sttor toothtips. This is due to the fct tht there re only tooth-tips in sttor slots for surfce-mounted permnent mgnet motors. For prllel double excittion nd spoke-type PM mchines which re studied here, toothtips re loclized in three regions: sttor slots, rotor DC excittion slots nd mgnet slots s shown in Fig.. As will be shown in this pper, the mutul influence between ll of these tooth-tips cn modify considerbly the electromgnetic performnces. It depends on the dimension of the tooth-tip openings compred to the slot openings. In this pper, n exct nlyticl prediction bsed on subdomin model for the computtion of mgnetic field distribution nd electromgnetic performnces in prllel double excittion nd spoketype tngentil PM mchines with distributed windings integer slot per pole nd per phse mchine is presented. It involves the solution of Poisson s nd Lplce s equtions in sttor slots, buried permnent mgnets plced in slots, rotor double excittion slots, ir gp nd non mgnetic region under permnent mgnets. The nlyticl model

3 Progress In Electromgnetics Reserch B, Vol. 47, developed in this pper, which does not tke into ccount the sttor nd rotor tooth-tips nd the shpe of polr piece, is simplifiction of the exct model EM) presented recently by the uthors [16]. A comprison between the results issued from the simplified model SM) with those from exct model EM) [16] is done to show the effect of the simplified geometry on mgnetic field distribution nd electromgnetic performnces cogging torque, electromgnetic torque, flux linkge, bck-emf, self nd mutul inductnces). It is importnt to note tht only mgnetic field distribution is clculted in [16]. The results obtined with nlyticl models re then compred to those found by the finite element method FEM).. MAGNETIC FIELD SOLUTION IN PARALLEL DOUBLE EXCITATION PM MOTOR Figures 1 nd show the mchine model where region I represents the ir gp, region II the mgnets, region III the sttor slots, region IV non mgnetic mteril under mgnets nd region V the rotor excittion slots. The model is formulted in two-dimensionl polr coordintes with the following ssumptions. The sttor nd rotor cores re ssumed to be infinitely permeble Eddy current effects re neglected The xil length of the mchine is infinite, i.e., end effects re neglected The current density hs only one component long the z-xis The sttor nd rotor slots hve rdil sides The prtil differentil equtions for mgnetic field in term of vector potentil A which hs only one component in the z direction nd is not dependent on the z coordinte, cn be expressed by A =, in regions I nd IV 1) A = µ M, in region II ) A = µ J, in region III 3) A = µ J r, in region V 4) where M is the mgnetiztion of permnent mgnets, J the sttor slots current density, J r the excittion rotor slots current density nd µ the permebility of vcuum. The field vectors B nd H, in the different regions, re coupled by B = µ H, in regions I, III, IV nd V 5)

4 148 Boughrr et l. Armture winding Permnent mgenet excittion current Figure 1. Studied prllel double excittion PM mchine 1/4 of the mchine). Figure. Studied model 1/4 of the mchine).

5 Progress In Electromgnetics Reserch B, Vol. 47, where B r = µ H r, B θ = µ H θ B = µ µ r H + µ M, in region II 6) where B r = µ µ r H r +µ M r, B θ = µ µ r H θ +µ M θ nd µ r is the reltive recoil permebility of permnent mgnets. Rdil nd circumferentil flux density components re deduced from A by B r = 1 r A θ, B θ = A r.1. Generl Solution of Poisson s Eqution in Sttor Slot Subdomin Region III) In ech slot subdomin i) of region III Fig. 3), we hve to solve Poisson s eqution AIII i r + 1 AIII i + 1 AIII i r r r θ = µ J i 8) where J i is the current density in the slot i. As shown in Fig. 3, the ith sttor slot subdomin where i vries from 1 to Q s Q s is the number of sttor slots) is ssocited with boundry conditions t the bottom nd t ech sides of the slot s AIII i θ θ=α i c = nd AIII i θ θ=αi + c = 9) AIII i r=r4 = 1) r where α i is the ngulr position of the ith slot nd c the slot opening in rdin. 7) AIII i / r= AIII i / θ = α C/ i+ AIIIi = µji C α i III r 4 AIII i / θ= α _ i C/ Rs AII j / θ= AIIj =-µ M θ /r g j R m AII j / θ= R / g j / _ r Figure 3. ith sttor slot subdomin. Figure 4. jth permnent mgnet subdomin.

6 15 Boughrr et l. From bove boundry conditions 9) nd 1), the solution of 8) using the method of seprtion of vribles is AIII i r, θ) = C i, + 1 µ J i r4 ln r) 1 4 µ J i r [ ) ) r c r ] c + C i,m cos r 4 r 4 c m=1 where m is positive integer. θ α i + c )) 11).. Generl Solution of Poisson s Eqution in Permnent Mgnet Subdomin Region II) In ech permnent mgnet subdomin j) of region II Figs. nd 4), we hve to solve Poisson s Eqution ). The mgnetiztion of prllel double excittion motor is considered purely tngentil. Eqution ) is then reduced to AII j r AII j r AII j M θ r r θ = µ 1) r where M θ = M j = 1) j Brem µ. For p poles mchine, j vries from 1 to p nd B rem is the remnence of the mgnets. As shown in Fig. 4, the jth mgnet subdomin region II) is ssocited with the following boundry conditions AII j θ θ=gj = nd AII j θ θ=gj + = 13) where g j is the ngulr position of the jth mgnet nd the mgnet opening in rdin. From bove boundry conditions 13), the generl solution of 1) using the method of seprtion of vribles is given by AII j r, θ) = A5 j, + A6 j, ln r) µ M j r ) + A5 j,m r +A6j,m r cos m=1 θ )) 14).3. Generl Solution of Lplce s Eqution in Airgp Subdomin Region I) The Lplce Eqution 1) in the irgp subdomin region I) which is n nnulr domin delimited by the rdii R m nd R s Fig. ) is given by AI r + 1 r AI r + 1 r AI θ = 15)

7 Progress In Electromgnetics Reserch B, Vol. 47, For the studied mchine with integer slot per pole nd per phse, the periodicity of the problem is π p nd the solution of Eqution 15) is AIr, θ) = + A1n r np + A n r np) sinnp θ) + A3 n r np + A4 n r np) cosnp θ) 16) where n is positive integer..4. Generl Solution of Lplce s Eqution in the Non-mgnetic Subdomin Region IV) The Lplce s Eqution 1) in the non-mgnetic subdomin region IV) is given by AIV r + 1 AIV + 1 AIV r r r θ = 17) The generl solution of 17) is AIV r, θ) = A7n r np + A8 n r np) sin np θ) + A9 n r np + A1 n r np) cos np θ) 18) The mgnetic vector potentil must be finite in region IV when r =. Therefore, the constnts A8 n nd A1 n re equls to zero nd 18) is reduced to AIV r, θ) = r np A7 n sin np θ) + r np A9 n cos np θ) 19).5. Generl Solution of Poisson s Eqution in Rotor Excittion Coil Slot Subdomin Region V) In ech rotor slot subdomin ir) of region V, we hve to solve Poisson s Eqution ) AV ir r + 1 AV ir + 1 AV ir r r r θ = µ J rir ) where Jr ir is the current density in rotor slot ir. As shown in Fig. 5, the irth slot subdomin where ir vries from 1 to N r N r is totl number of rotor excittion slots) is ssocited with the following boundry conditions AV ir θ θ=βir cr AV ir r = nd AV ir θ θ=βir + cr = 1) r=r5 = )

8 15 Boughrr et l. β ir AV / θ = ir ir rir AV = µj R m AV ir/ θ = AV ir/ r= β ir + cr/ Figure 5. irth rotor slot subdomin. r 5 cr β _ ir cr/ where β ir is the ngulr position of the irth slot nd cr the rotor slot opening in rdin. From the bove boundry conditions 1) nd ), the solution of ) using the method of seprtion of vribles is AV ir r, θ) = C1 ir, + 1 µ J fir r5 ln r) 1 4 µ J fir r [ ) ) r cr r ] cr + C1 ir, m cos θ β ir + c ) r 3) r 5 r 5 c r m=1 3. BOUNDARY AND INTERFACE CONDITIONS To determine Fourier series unknown constnts A1 n, A n, A3 n, A4 n, A5 j,, A6 j,, A5 j, m, A6 j, m, A7 n, A9 n, C i,, C i, m, C1 ir,, C1 ir, m, boundry nd interfce conditions should be introduced. The interfce conditions must stisfy the continuity of the rdil component of the flux density nd the continuity of the tngentil component of the mgnetic field. The first condition could be replced by the continuity of A. The interfce conditions between regions IV nd II t R r re where g j θ. AII j R r, θ) = AIV R r, θ) 4) HII θj R r, θ) = HIV θ R r, θ) 5) where g j θ. HIV θr r, θ) = elsewhere. The interfce condition between regions I nd II t R m is AII j R m, θ) = AI R m, θ) 6)

9 Progress In Electromgnetics Reserch B, Vol. 47, where g j θ. The interfce condition between regions I nd V t R m is AI R m, θ) = AV ir R m, θ) 7) where β ir cr θ β ir + cr. The interfce conditions between regions I, V nd II t R m re HI θ R m, θ) = HII θj R m, θ) 8) for g j θ nd HI θr m, θ) = HV θi rr m, θ). For β ir cr θ β ir + cr nd HI θr m, θ) = elsewhere. The interfce conditions between regions I nd III t R s re where α i c θ α i + c. AI R s, θ) = AIII i R s, θ) 9) HI θ R s, θ) = HIII θi R s, θ) 3) where α i c θ α i + c. HI θr s, θ) = elsewhere. Interfce conditions 4) to 3) concern regions with different subdomin frequencies which need Fourier series expnsions to stisfy equlities of vector potentil nd mgnetic field t ech interfce rdius. According to Fourier series expnsion, from 4) we obtin two equtions s = 1 = A5 j, + A6 j, ln R r ) M j µ R r g j A5 j, m R g j AIV R r, θ)dθ 31) ) r Interfce condition 5) gives np µ np µ ) A7n Rr np 1 ) 1 = π ) A9n Rr np 1 ) 1 = π + A6 j, m R ) r AIV R r, θ) cos θ )) dθ 3) p j=1 g j p j=1 g j HII θj R r, θ) sin np θ) dθ 33) HIIθ j R r, θ) cos np θ) dθ 34)

10 154 Boughrr et l. Fourier series expnsion of interfce condition 6) between regions II nd I t rdius R m gives = 1 = A5 j, + A6 j, ln R m ) M j µ R m g j A5 j, m R g j AI R m, θ)dθ 35) ) m + A6 j, m R ) m From interfce condition 7), we obtin AIR r, θ) cos θ )) dθ 36) C1 ir, + 1 µ Jr ir r 5 ln R m ) 1 4 µ Jr ir R m = 1 cr β ir + cr AIR m, θ)dθ 37) β ir cr C1 ir, m Rm r 5 ) cr Rm r 5 ) ) cr = cr β ir + cr β ir cr AIR m, θ) cos θ β ir + cr )) dθ 38) cr Fourier series expnsion of interfce condition 8) gives np A1n Rm np 1 +A n R np 1 ) 1 m = µ π + 1 π N r β ir + cr ir=1 β ir cr p j=1 g j HII θj R m, θ)sinnp θ)dθ HV θir R m, θ) sin np θ) dθ 39)

11 Progress In Electromgnetics Reserch B, Vol. 47, np A3n Rm np 1 +A4 n R np 1 ) 1 m = µ π + 1 π N r β ir + cr ir=1 β ir cr p j=1 g j HII θj R m, θ)cosnp θ)dθ HV θir R m, θ) cos np θ) dθ 4) At rdius R s, Fourier series expnsions of interfce condition 9) gives C i, + 1 µ J i r 4 ln R s ) 1 4 µ J i R s = 1 c C i,m Rs r 4 ) c Rs r 4 α i + c α i c ) ) α i + c = c AI R s, θ)cos c α i c AI R s, θ) dθ 41) Fourier series expnsion of interfce condition 3) gives np A1n Rs np 1 + A n R np 1 ) s µ = 1 π = 1 π Q s α i + c i=1 α i c np A3n Rs np 1 µ Q s α i + c i=1 α i c c θ α i + c )) dθ4) HIII θi R s, θ) sin np θ) dθ 43) + A4 n Rs np 1 ) HIII θi R s, θ) cos np θ) dθ 44) Some developments of Equtions 31) to 44) re given in Appendix A. From Equtions 31) 44) we cn clculte the 14 coefficients A1 n, A n, A3 n, A4 n, A5 j,, A6 j,, A5 j,m, A6 j,m, A7 n, A9 n, C i,, C i,m, C1 ir,, C1 ir, m with given number of hrmonics for n nd m. 4. MAGNETIC FIELD SOLUTION IN SPOKE-TYPE PM MOTOR Spoke-type PM motor nlyticl model is specil cse of prllel double excittion PM motor model, where region V is omitted Fig. 6).

12 156 Boughrr et l. Armtur winding Permnent mgnetic Figure 6. Studied spoke-type PM mchine 1/4 of the mchine). Then, Equtions 37) nd 38) dispper nd 39) nd 4) re modified respectively s follow: np A1n Rm np 1 +A n R np 1 ) 1 m = µ π np A3n Rm np 1 +A4 n R np 1 ) 1 m = µ π p j=1 g j p j=1 g j HII θj R m, θ)sinnp θ)dθ 45) HII θj R m, θ)cosnp θ)dθ46) The other equtions re the sme nd the system of equtions to be solved is now constituted from 1 equtions with 1 unknowns A1 n, A n, A3 n, A4 n, A5 j,, A6 j,, A5 j,m, A6 j,m, A7 n, A9 n, C i, nd C i,m. 5. ELECTROMAGNETIC PERFORMANCES CALCULATION Prediction of globl quntities cogging torque, flux linkge, induced bck-emf, self inductnce, mutul inductnce nd electromgnetic torque), llows the evlution of mchine performnces.

13 Progress In Electromgnetics Reserch B, Vol. 47, Cogging Torque Clcultion According to Mxwell stress tensor method, cogging torque T c computed using the nlyticl expression T c = plur g µ π p BIr R g, θ)bi θ R g, θ) dθ 47) where R g is the rdius of circle plced t the middle of the ir-gp nd Lu is the xil length of the motor. Open-circuit rdil nd tngentil components of the flux density in the middle of ir gp BI r R g, θ) nd BI θ R g, θ) re determined from Equtions ) nd 6). 5.. Flux Linkge nd Bck-EMF Clcultion For slotted structures of PM mchines, computtion of flux linkge nd bck-emf with the method of winding function theory is not suitble. The method bsed on Stokes theorem using the vector potentil in sttor slots is used. First, we determine t given rotor position θ r, the flux over ech slot i of cross section S. We hve supposed tht the current is uniformly distributed over the slot re, so the vector potentil cn be verged over the slot re to represent the coil. For the simplified model, we obtin: ϕ i = Lu S α i + c α i c r 4 R s AIII i r, θ)rdrdθ 48) where S = cr 4 R s) is the surfce of the sttor slots inner rdius R s nd outer rdius r 4 ). The vector potentil AIII i r, θ) is given by 4). The development of 48) gives ϕ i = LuC i, 49) µ J i LuRs+ 4 4 lnr s))r4 R s+4 lnr 4 ) 3)r4 4). For the exct model, we 8r4 +8R s obtin: ϕ i = Lu S α i + c α i c r 4 r 3 AIII i r, θ)rdrdθ 5) where S = cr 4 r 3 ) is the surfce of the sttor slots inner rdius r 3 nd outer rdius r 4 ). In this cse, Eqution 49) is modified with replcing is

14 158 Boughrr et l. R s with r 3. Of course, the vlue of the integrtion constnt C i, in 49) is not the sme for the simplified nd exct models. Under no-lod condition nd for both models J i = ), the flux over ech slot becomes ϕ i = LuC i, 51) The phse flux vector is given by ] [ ψ ψ b ψ c = N c C [ϕ 1 ϕ... ϕ Qs 1 ϕ Qs ] 5) where C is the trnspose of connecting mtrix tht represents the distribution of sttor windings in the slots. The mtrix connection between phse current nd sttor slots for one pole pir is given by [ ] C = ) The studied three phses PM motors re fed with 1 rectngulr phse currents. The current density in sttor slots is defined s J i = N c S CT [ I I b I c ] 54) where N c is the number of conductors nd I, I b, I c re the sttor phse currents. The vector of rotor double excittion current density with N r elements N r is the number of rotor slots) for the studied mchine is defined s J ri r = N f I f S f [ ] 55) where N f is the number of conductors in rotor slot, I f the DC excittion current nd S f the surfce of rotor slot. The surfce or rotor slots is given by S f = crr m r5 ) for the simplified model, nd by S f = crr r 1 ) for the exct model. The three phse bck-emf vector is clculted by [ E E b E c ] = Ω d dθ r [ ψ ψ b ψ c ] 56) where Ω is the rotor ngulr speed. Flux linkge nd bck-emf re lso dependent on the vlue of excittion current.

15 Progress In Electromgnetics Reserch B, Vol. 47, Electromgnetic Torque Clcultion Electromgnetic torque cn be computed from the bck-emf by T em = E I + E b I b + E c I c 57) Ω Eqution 47) cn lso be used to predict electromgnetic torque totl torque) if the open circuit flux density is substituted by the on-lod flux density Self nd Mutul Inductnces Clcultion Self nd mutul inductnces cn be clculted from the mgnetic energy: L = W I 58) L c = W c W W c I I c 59) where W, W c nd W c re the mgnetic energies when the mgnets re not mgnetized nd the mchine is fed with I only, I c only, nd both I nd I c, respectively. For the simplified model, mgnetic energy cn be obtined by: W = Lu Q s r 4 α i + c i=1 R s α i c For the exct model, 6) becomes: W = Lu Q s r 4 α i + c i=1 r 3 α i c 6. RESULTS AND VALIDATION AIII i r, θ)j i rdrdθ 6) AIII i r, θ)j i rdrdθ 61) In order to show the ccurcy of the simplified model versus the exct model which tkes into ccount sttor nd rotor tooth-tips [16], we compre the mgnetic field distribution nd electromgnetic performnces obtined with the two models. Double excittion nd spoke-type permnent mgnet mchines re considered. The nlyticl results re lso compred with those obtined by finite element simultions [17]. The min dimensions nd prmeters of the

16 16 Boughrr et l. Tble 1. Prmeters of simplified model for prllel double excittion nd spoke-type permnent-mgnet motors. Prmeter Symbol Vlue nd unit Mgnet remnence Ferrite) B r.4 T Reltive recoil permebility of mgnet µ r 1. Number of conductors per sttor slot N c 1 Pek phse current I m 1.5 A DC excittion current I f 15 A Number of conductors per rotor slot N f 1 Number of sttor slots Q s 36 Sttor slot opening width c 5 Rotor slot opening width c r 5 Number of pole pirs p 3 Number of rotor excittion slots N r 1 Internl rdius of rotor slot r mm Externl rdius of sttor slot r mm Rdius of the externl sttor surfce R o 74.8 mm Rdius of the sttor outer surfce R s 45.3 mm Rdius of the rotor inner surfce t the mgnet surfce R m 44.8 mm Rdius of the rotor inner surfce t the mgnet bottom R r 15 mm Air-gp length g.5 mm Height of mgnet h m 9.8 mm Height of sttor nd rotor slot h s 9 mm Stck length L u 57 mm Mgnet opening mechnicl degrees) 14 Rotor speed Ω 157 rd/s studied mchines for the simplified model re given in Tble 1. The supplementry geometricl prmeters for the exct model re given in Tble Prllel Double Excittion PM Motors The proposed simplified model SM) contins 14 equtions see ppendix) with 14 unknowns. The exct model EM) which ws presented in [16] is more complex nd contins 6 equtions. The solution of the system of equtions gives the potentil vector nd the flux density in ech subdomin.

17 Progress In Electromgnetics Reserch B, Vol. 47, Rdil nd tngentil components of the flux density due to PM, rotor DC excittion current nd rmture rection current cting together on-lod condition) re given in Figs. 7 nd 8. Differences between results obtined with the two nlyticl models re not importnt for the rdil component of the flux density nd re more importnt for the tngentil component s shown in Fig. 8. Differences on the flux density wveforms between the simplified nd exct nlyticl model depends on the tooth-tips opening compre to Tble. Supplementry prmeters of exct model for prllel double excittion nd spoke-type permnent-mgnet motors. Prmeter Symbol Vlue nd unit Externl rdius of rotor slot r 4.8 mm Externl rdius of PM r 4.8 mm Externl rdius of sttor semi-slot r mm Internl rdius of rotor slot r mm Sttor semi-slot Opening d 4 Rotor semi-slot Opening dr 4 PM semi-slot Opening b 13 Externl rdius of sttor slot r mm Rdius of the rotor inner surfce t the mgnet bottom R r 13 mm Br T) FEM SM) Anlyticl SM) ngle Mechnicl degrees) ) Br T) FEM EM) Anlyticl EM) ngle Mechnicl degrees) b) Figure 7. Rdil component of the flux density for lod condition sttor current, rotor excittion current nd PM) in the q-xis rotor position. ) Simplified model, b) exct model.

18 16 Boughrr et l. Bt T) ngle Mechnicl degrees) ) FEM SM) Anlyticl SM) Bt T) ngle Mechnicl degrees) b) FEM EM) Anlyticl EM) Figure 8. Tngentil component of the flux density for lod condition sttor current, rotor excittion current nd PM) in the q-xis rotor position. ) Simplified model, b) exct model Bt T) FEM S M): no-lod, If= A FEM S M): no-lod, If=15 A FEM S M): on-lod, If=15 A Anlyticl SM): on-lod, If=15 A Anlyticl SM): no -lod, If= A Anlyticl SM): no -lod, If=15 A Bt T) FEM EM): no-lod, If= A FEM EM): no-lod, If=15 A FEM EM): on-lod, If=15 A Anlyticl EM): no -lod, If= A Anlyticl EM): no-lod, If=15 A Anlyticl EM): on-lod, If=15 A ngle Mechnicl degrees) ) ngle Mechnicl degrees) b) Figure 9. Tngentil component of the flux density in the middle of the first mgnet j = 1) t no-lod nd on-lod conditions. ) Simplified model, b) exct model. the slots opening. For the studied exmple, we chose ll the toothtips openings closer to slots openings. In the cse of smll tooth-tips openings compred to slot openings, we obtined significnt differences between the two models not presented here). The results presented here re in very good greement with FEM for both simplified nd exct models. With the nlyticl model, we cn predict the mgnetic field distribution in ll subdomins. Fig. 9 shows the tngentil component of the flux density rdil flux density is null) in the middle of the

19 Progress In Electromgnetics Reserch B, Vol. 47, first PM region j = 1) for no-lod nd lod conditions, nd for two vlues of the DC excittion current. With these results, we cn nlyze the rmture rection nd the DC excittion current effects in the demgnetiztion risk of the mgnets. We cn observe tht the PM re not demgnetized, even under lod condition. As known, the demgnetiztion risk occurs when the flux density in the mgnet is pproximtely less thn.1 T in the direction of mgnetiztion. From comprisons with FEM simultions, we cn observe tht x FEM S M) Anlyticl SM).5 3 x 1-3 FEM EM) Anlyticl EM) L H) 1.5 L H) Rotor position mechnicl degrees) ) Rotor position mechnicl degrees) b) Figure 1. Phse A self-inductnce. ) Simplified model, b) exct model. Lc H) -1 x FEM S M) Anlyticl SM) Rotor position mechnicl degrees) ) Lc H) - x FEM EM) Anlyticl EM) Rotor position mechnicl degrees) Figure 11. Mutul inductnce between phses A nd C. ) Simplified model, b) exct model. b)

20 164 Boughrr et l Anlyticl SM) FEM S M) FEM EM) Anlyticl EM).5.5 Tc Nm) -.5 Tc Nm) Rotor position mechnicl degrees) ) Rotor position mechnicl degrees) b) Figure 1. Cogging torque due to PM lone I f = A). ) Simplified model, b) exct model Bck-EMF V) FEM S M) Anlyticl SM) Bck-EMF V) FEM EM) Anlyticl EM) Rotor position mechnicl degrees) ) Rotor position mechnicl degrees) b) Figure 13. Bck-Emf I f model. = 15 A). ) Simplified model, b) exct nlyticl models SM nd EM) results greed very well in the PM subdomin nd re pproximtely the sme for simplified nd exct models. Self nd mutul inductnces re given in Figs. 1 nd 11. We cn observe very good greement between exct nd simplified nlyticl models nd FEM results. From Fig. 1, we cn determine the vlues of q-xis nd d-xis self-inductnce. The mximum vlue of the selfinductnce corresponds to the q-xis rotor position θ r = 1 ). The miniml vlue of the self-inductnce corresponds to the d-xis rotor

21 Progress In Electromgnetics Reserch B, Vol. 47, Tem Nm) FEM S M) Anlyticl S M) Rotor position mechnicl degrees) ) Tem Nm) FEM EM) Anlyticl EM) Rotor position mechnicl degrees) Figure 14. Electromgnetic torque I f = 15 A). ) Simplified model, b) exct model. b) Averge torque Nm) SM EM DC excittion current A) RMS Bck-EMF V) SM EM DC excittion current A) Figure 15. Averge electromgnetic torque for different DC excittion current vlues. Figure 16. RMS bck-emf for different DC excittion current vlues. position θ r = 4 ). To determine the mutul inductnce, the mchine is fed with two phse sttor currents. The q-xis nd d-xis rotor positions in this cse corresponds to θ r = nd θ r = 5 respectively Fig. 11). It cn be seen from the comprison between simplified nd exct models results Fig. 1) tht we hve the sme wveforms for the self-inductnce with difference of pproximtely.5 mh. As expected, the exct model gives higher vlue of the self-inductnce. This is due to the lower equivlent ir-gp dimension cused by the presence of the tooth-tips. For mutul inductnce Fig. 11), this difference in mplitude is pproximtely. mh. There is smll difference in

22 166 Boughrr et l. mplitude between exct nlyticl model nd exct FEM model s shown in Fig. 11b). This difference is due to the number of hrmonics limittion used in the exct nlyticl model. This limittion is discussed in [16] nd [18]. In control process, rotor DC excittion current cn be set to zero, negtive or positive vlues in order to increse or decrese electromgnetic torque, flux linkge nd bck-emf. Cogging torque is lso dependent on the vlue of the excittion current. We cn observe from Figs. 1, 13 nd 14 tht exct model gives pproximtely the sme mplitude compred to simplified model with different wveform for cogging torque, bck-emf nd electromgnetic torque which is due to the presence of sttor nd rotor tooth-tips for the exct model. The results from exct nd simplified nlyticl models re in very good greement with the results obtined with simplified nd exct FEM models. Using the simplified nd exct nlyticl models, the impct of the DC excittion current I f on the electromgnetic performnces of the studied prllel double excittion PM motor is presented here. Averge torque nd bck-emf control cpbility re shown in Figs. 15 nd 16. The study is done for I f rnging from 5 A to 5 A. We cn observe tht bck-emf nd verge electromgnetic torque increse with DC excittion current increse. Simplified nd exct nlyticl models give pproximtely the sme vlues with smll differences for verge torque for lrge vlues of DC excittion current..8.6 FEM SM) Anlyticl SM).6.4 FEM EM) Anlyticl EM).4 Br T ). -. Br T ) ngle Mechnicl degrees) ) ngle Mechnicl degrees) b) Figure 17. Rdil component of the flux density due to PM lone. ) Simplified model, b) exct model.

23 Progress In Electromgnetics Reserch B, Vol. 47, FEM SM) Anlyticl SM).15.1 FEM EM) Anlyticl EM).5.5 Bt T) Bt T) ngle Mechnicl degrees) ) ngle Mechnicl degrees) Figure 18. Tngentil component of the flux density due to PM lone. ) Simplified model, b) exct model. b) Bt T) FEM SM): no-lod FEM SM): no-lod Anlyticl SM): no-lod Anlyticl SM): no-lod Bt T) FEM EM): no-lod FEM EM): no-lod Anlyticl EM): no-lod Anlyticl EM): no-lod ngle Mechnicl degrees) ) ngle Mechnicl degrees) b) Figure 19. Tngentil component of the flux density t no lod nd on-lod conditions in the middle of the first mgnet j = 1). ) Simplified model, b) exct model 6.. Spoke-type PM Motors Anlyticl simplified model presented in this pper for the spoke-type PM motor contins 1 equtions with 1 unknowns. The exct model presented by the uthors in [16] included equtions. The solution of the system of liner equtions leds to the vector potentil nd flux density in ech subdomin. Rdil nd tngentil components of the flux density due to permnent mgnets cting lone re shown in Figs. 17 nd 18, for simplified nd exct nlyticl models nd for FEM simultions. Both nlyticl models give pproximtely the

24 168 Boughrr et l. sme results for the studied mchine where rotor nd sttor tooth-tips openings re closer to rotor nd sttor slots openings. To study the effect of rmture rection on the demgnetiztion risk of ferrite mgnets, we show in Fig. 19 the tngentil component of the flux density rdil flux density is null) in the middle of the first PM subdomin. As shown, the demgnetiztion risk is voided t no-lod nd on-lod conditions. Simplified nd exct nlyticl models give the sme results. Once gin, nlyticl results re in good greement with those obtined by FEM for both simplified nd exct models. Self nd mutul inductnces vritions with rotor position re L H) 3 x FEM S M) Anlyticl SM) L H) 3.5 x 1-3 FEM EM) Anlyticl EM) Rotor position mechnicl degrees) ) Rotor position mechnicl degrees) Figure. Phse A self inductnce. ) Simplified model, b) exct model. Lc H) - x FEM S M) Anlyticl SM) Rotor position mechnicl degrees) ) Lc H) x 1-4 b) FEM EM) Anlyticl EM) Rotor position mechnicl degrees) Figure 1. Mutul inductnce between phse A nd C. ) Simplified model, b) exct model. b)

25 Progress In Electromgnetics Reserch B, Vol. 47, Tc Nm) FEM SM) Anlyticl SM) Rotor position mechnicl degrees) ) Tc Nm) FEM EM) Anlyticl EM) Rotor position mechnicl degrees) Figure. Cogging torque. ) Simplified model, b) exct model. b) Bck-EMF V) FEM S M) Anlyticl SM) Bck-EMF V) FEM EM) Anlyticl EM) Rotor position mechnicl degrees) ) Rotor position mechnicl degrees) Figure 3. Emf. ) Simplified model, b) exct model. shown in Figs. nd 1. Both results obtined from nlyticl models nd FEM re in excellent greement. From Fig., we cn determine the vlues of q-xis nd d-xis self inductnces. Q-xis self inductnce mximl inductnce) corresponds to θ r = 1 rotor position) nd d-xis self inductnce miniml inductnce) corresponds to θ r = 4. When the mchine is fed with two phse sttor current, q-xis nd d-xis rotor positions re locted t θ r = nd θ r = 5 respectively Fig. 1). It cn be seen from Figs. nd 1 difference of pproximtely.5 mh when we compre the mplitudes of self nd mutul inductnces for simplified nd exct model. The mutul inductnce vrition with rotor position Fig. 1) obtined with the nlyticl exct model presents smll difference with FEM EM). This is due to the limiting number of hrmonics used in the clcultion b)

26 17 Boughrr et l. Tem Nm) FEM S M) Anlyticl S M) Rotor position mechnicl degrees) ) Tem Nm) FEM EM) Anlyticl EM) Rotor position mechnicl degrees) Figure 4. Electromgnetic torque. ) Simplified model, b) exct model. b) s discussed in [16] nd [18]. In Fig., we show tht the pek vlue of cogging torque is smller thn in prllel double excittion mchine. This is due to the bsence of rotor slots DC current excittion) for spoke-type mchine. The results obtined with FEM nd with nlyticl models SM nd EM) re in very good greement. We cn observe tht the cogging torque Fig. b)) obtined with the exct model, presents smller pek vlue nd not the sme wveform thn the one obtined with the simplified nlyticl model Fig. )). This result cn be explin by the presence of sttor nd rotor tooth-tips for the exct model. Anlyticl prediction of bck-emf nd electromgnetic torque re shown in Figs. 3 nd 4. The results re in good greement with those issued from FEM. Slight differences in mplitude nd wveform cn be observed between simplified nd exct model. This is due to the rotor nd sttor slots tooth-tips for the exct model. 7. CONCLUSION In this pper, we hve proposed simplified nlyticl model for prllel double excittion nd spoke-type PM mchines. Compred to our previous work [16], the proposed model doesn t tke into ccount the sttor nd rotor tooth-tips nd the exct shpe of polr pieces. The simplified models need fewer equtions for the predictions of mgnetic field. The proposed nlyticl models hve been used to predict mgnetic field distribution nd electromgnetic performnces for double excittion nd spoke-type PM mchines. The Accurcy of nlyticl models hs been verified with finite element simultions

27 Progress In Electromgnetics Reserch B, Vol. 47, for the ir-gp nd PM subdomins. In comprison with rdil surfce-mounted PM motors where the effect of sttor slot tooth-tips doesn t modify highly the wveform nd mplitude of mgnetic field distribution even when the tooth-tips opening re smller thn the slots opening [6], it is not the cse for prllel double excittion nd spoketype PM motors which hve tooth-tips both in the rotor nd sttor sides. For this type of mchines, the effect of sttor, rotor nd PM tooth-tips cn modify highly the mplitude nd wveform of mgnetic field distribution nd electromgnetic performnces when rotor, sttor nd PM tooth-tips opening is smller thn slots opening, due to the mutul influence between sttor nd rotor slots. The demgnetiztion risk of ferrite mgnets hs been nlyzed with the proposed models. We hve shown tht the DC excittion current nd the rmture rection reduce the flux density in the mgnets but without demgnetiztion risk. APPENDIX A. Fourier series coefficients of generl solution in different regions of prllel double excittion permnent mgnet mchines re determined by resolution of system of equtions. Some of those equtions re detiled s follows. From Eqution 31), we get A5 j, +A6 j, ln R r ) M j µ R r = 1 A7 n R np + 1 Development of Eqution 3) gives: = A5 j,m R r A7 n R np r ) + A6 j,m R r g j r ) g j A9 n R np + A9 n Rr np ) cos np θ)cos g j r ) sinnp θ)dθ g j cosnp θ)dθa1) sinnp θ) cos θ )) dθ θ )) dθ A)

28 17 Boughrr et l. From Eqution 33), we hve: ) np A7n Rr np 1 ) = µ 1 πµ µ r g j p j=1 m=1 A5 j,mr r 1 sin np θ) cos θ )) dθ ) p 1 A6 j, πµ µ r R j=1 r Eqution 34) gives ) np A9n Rr np 1 ) = µ 1 πµ µ r g j p j=1 m=1 g j A5 j,mr r sin np θ)dθ 1 cos np θ) cos θ )) dθ A6 j,mr r A6 j,mr r 1 1 ) ) A3) ) p 1 A6 j, πµ µ r R j=1 r Eqution 35) gives g j cos np θ)dθ A5 j, + A6 j, ln R m ) M j µ R m = 1 A1n Rm np + A n R np ) g j+ m sin np θ) dθ g j + 1 A3n Rm np + A4 n R np ) g j+ m cos np θ) dθ g j A4) A5)

29 Progress In Electromgnetics Reserch B, Vol. 47, Eqution 36) gives A5 j,m R = ) m + A6 j,m R ) m A1n Rm np +A n Rm np ) g j + A3n R np m+a4 n Rm np ) g j Eqution 37) development gives sin np θ) cos θ )) dθ cosnp θ)cos θ )) dθa6) C1 ir, + 1 µ Jr ir r 5 ln R m ) 1 4 µ Jr ir R m = 1 cr + 1 cr A1n Rm np + A n Rm np A3n Rm np + A4 n Rm np Eqution 38) development gives Rm ) ) cr ) Rm cr C1 ir, m r 5 r 5 ) β ir+ cr β ir cr ) β ir+ cr β ir cr sin np θ) dθ cos np θ) dθ A7) β ir + cr = A1n Rm np +A n Rm np ) cr + cr β ir cr sin np θ)cos θ β ir + cr )) dθ cr β ir + cr A3n Rm+A4 np n Rm np ) cosnp θ)cos θ β ir + cr cr β ir cr )) dθ A8)

30 174 Boughrr et l. Eqution 39) development gives np A1n Rm np 1 + A n R np 1 ) m µ = 1 p A5 j,m R 1 m A6 j,m R m π µ µ r j=1 m=1 g j 1 πµ µ r 1 πµ β ir + cr β ir cr cos θ )) sin np θ) dθ N r p j=1 ir=1 m=1 A6 j, R m g j C1 ir, m sin np θ) dθ Rm R m cr r 5 ) cr 1 + Rm r 5 cos θ β ir + cr )) sin np θ) dθ cr ir=1 β ir + cr + 1 N r 1 µ Jr ir r5 + 1 ) πµ R m µ Jr ir R m j=1 m=1 β ir cr Eqution 4) development gives np A3n Rm np 1 + A4 n R np 1 ) m µ = 1 p A5 j,m R 1 m A6 j,m R m π µ µ r g j 1 πµ µ r cos θ )) cos np θ) dθ p j=1 A6 j, R m g j cos np θ) dθ ) ) ) cr sin np θ) dθ A9) 1 )

31 Progress In Electromgnetics Reserch B, Vol. 47, πµ β ir + cr β ir cr N r ir=1 m=1 + 1 N r 1 πµ C1 ir, m Rm R m cr r 5 ) cr + Rm r 5 cos θ β ir + cr )) cos np θ) dθ cr ir=1 µ Jr ir r 5 R m Eqution 41) development gives β ir + cr + 1 ) µ Jr ir R m C i, + 1 µ J i r 4 ln R s ) 1 4 µ J i R s β ir cr ) ) cr cos np θ) dθ A1) = 1 c + 1 c A1n R np s A3n R np s + A n R np s Eqution 4) development gives Rs ) ) c ) Rs c C i,m r 4 r 4 + A4 n R np s ) α i+ c α i c ) α i+ c α i c sin np θ) dθ cos np θ) dθ A11) α i + c = A1n Rs np +A n Rs np ) cos c c α i c α i + c α i c θ α i + c )) sinnp θ) dθ + A3n Rs np +A4 n Rs np ) cos θ α i + c )) cosnp θ)dθ A1) c c

32 176 Boughrr et l. Eqution 43) development gives np A1n Rs np 1 + A n R np 1 ) s µ = 1 Q s Rs C i,m πµ cr s r 4 α i + c α i c + 1 πµ i=1 m=1 ) c + Rs r 4 cos θ α i + c )) sin np θ) dθ c Q s i=1 1 µ J i r4 + 1 ) R s µ J i R s Eqution 44) development gives np A3n Rs np 1 + A4 n R np 1 ) s µ = 1 Q s Rs C i,m πµ cr s r 4 α i + c α i c + 1 πµ i=1 m=1 ) c + α i + c α i c Rs r 4 cos θ α i + c )) cos np θ) dθ c Q s i=1 ) ) c sin np θ) dθ A13) ) ) c 1 µ J i r4 + 1 ) α i + c R s µ J i R s. cos np θ) dθ A14) α i c The system of equtions to solve in prllel double excittion PM motors is constituted by the 14 equtions from A1) to A14) with the unknowns A1 n, A n, A3 n, A4 n, A5 j,, A6 j,, A5 j,m, A6 j,m, A7 n, A9 n, C i,, C i,m, C1 ir,, C1 ir, m. For spoke-type PM motors, Equtions A7) nd A8) re omitted nd Equtions A9) nd A1) re modified s explined in Equtions 45) nd 46). REFERENCES 1. Lubin, T., S. Mezni, nd A. Rezzoug, D nlyticl clcultion of mgnetic field nd electromgnetic torque for surfce-inset permnent mgnet motors, IEEE Trns. Mgnetics., Vol. 48, No. 6, 8 91, June 1.

33 Progress In Electromgnetics Reserch B, Vol. 47, Zhu, Z. Q., L. J. Wu, nd Z. P. Xi, An ccurte subdomin model for mgnetic field computtion in slotted surfce-mounted permnent-mgnet mchines, IEEE Trns. Mgnetics., Vol. 46, No. 4, , April Lubin, T., S. Mezni, nd A. Rezzoug, -D exct nlyticl model for surfce-mounted permnent-mgnet motors with semiclosed slots, IEEE Trns. Mgnetics., Vol. 47, No., , Februry Wu, L. J., Z. Q. Zhu, D. Stton, M. Popescu, nd D. Hwkins, An improved subdomin model for predicting mgnetic field of surfce-mounted permnent mgnet mchines ccounting for tooth-tips, IEEE Trns. Mgnetics., Vol. 47, No. 6, , June Wu, L. J., Z. Q. Zhu, D. Stton, M. Popescu, nd D. Hwkins, Subdomin model for predicting rmture rection field of surfce-mounted permnent-mgnet mchines ccounting for tooth-tips, IEEE Trns. Mgnetics., Vol. 47, No. 4, 81 8, April Wu, L. J., Z. Q. Zhu, D. Stton, M. Popescu, nd D. Hwkins, Anlyticl prediction of electromgnetic performnce of surfcemounted permnent mgnet mchines bsed on subdomin model ccounting for tooth-tips, Electric Power Applictions, IET, Vol. 5, No. 7, , Jin, L., K. T. Chu, Y. Gong, C. Yu, nd W. Li, Anlyticl clcultion of mgnetic field in surfce-inset permnent mgnet motors, IEEE Trns. Mgnetics., Vol. 45, No. 1, , October Bli, H., Y. Amr, G. Brkt, R. Ibtiouen, nd M. Gbsi, Anlyticl modeling of open circuit mgnetic field in wound field nd series double excittion synchronous mchines, IEEE Trns. Mgnetics., Vol. 46, No. 1, , October Jin, L., G. Xu, C. C. Mi, K. T. Chu, nd C. C. Chn, Anlyticl method for mgnetic field clcultion in low-speed permnent-mgnet hrmonic mchine, IEEE Trns. Energy Conversion., Vol. 6, No. 3, 86 87, September Jin, L. nd K. T. Chu, Anlyticl clcultion of mgnetic field distribution in coxil mgnetic gers, Progress In Electromgnetics Reserch, Vol. 9, No. 7, 1 16, Lubin, T., S. Mezni, nd A. Rezzoug, Improved nlyticl model for surfce-mounted PM motors considering slotting effects nd rmture rection, Progress In Electromgnetics Reserch B, Vol. 5, , 1.

34 178 Boughrr et l. 1. Hlioui, S., L. Vido, Y. Amr, M. Gbsi, A. Miroui, nd M. Lécrivin, Mgnetic equivlent circuit model of hybrid excittion synchronous mchine, COMPEL: The Interntionl Journl for Computtion nd Mthemtics in Electricl nd Electronic Engineering, Vol. 7, No. 5, 1 115, Lin, D., P. Zhou, C. Lu, nd S. Lin, Anlyticl prediction of cogging torque for spoke type permnent mgnet mchines, IEEE Trns. Mgnetics., Vol. 48, No., , Februry Wu, L. J., Z. Q. Zhu, D. Stton, M. Popescu, nd D. Hwkins, Comprison of nlyticl models of cogging torque in surfcemounted PM mchines, IEEE Trns. Mgnetics., Vol. 59, No. 6, , June Boughrr, K., D. Zrko, R. Ibtiouen, O. Touhmi, nd A. Rezzoug, Mgnetic field nlysis of inset nd surfce mounted permnent mgnet synchronous motors using Schwrz-Christoffel trnsformtion, IEEE Trns. Mgnetics., Vol. 45, No. 8, , August Boughrr, K., R. Ibtiouen, nd T. Lubin, Anlyticl prediction of mgnetic field in prllel double excittion nd spoke-type permnent-mgnet mchines ccounting for tooth-tips nd shpe of polr pieces, IEEE Trns. Mgnetics., Vol. 48, No. 7, , July Meeker, D. C., Finite Element Method Mgnetics, Version 4., April 1, 9 Build, Gysen, B. L. J., E. Ilhn, K. J. Meessen, J. J. H. Pulides, nd E. A. Lomonov, Modeling of flux switching permnent mgnet mchines with fourier nlysis, IEEE Trns. Mgnetics., Vol. 46, No. 6, , June 1.