Lazhar Roubache 1, *, Kamel Boughrara 1,Frédéric Dubas 2,andRachidIbtiouen 1

Transcription

1 Progress In Electromgnetics Reserch B, Vol. 77, 85 11, 17 Semi-Anlyticl Modeling of Spoke-Type Permnent-Mgnet Mchines Considering the Iron Core Reltive Permebility: Subdomin Technique nd Tylor Polynomil Lzhr Roubche 1, *, Kmel Boughrr 1,Frédéric Dubs,ndRchidIbtiouen 1 Abstrct This rticle presents novel contribution to the improvement of the nlytic modeling of electricl mchines using two-dimensionl -D) subdomin technique with Tylor polynomil. To vlidte this novel method, the semi-nlyticl model hs been implemented for spoke-type permnentmgnet PM) mchines STPMM). Mgnetosttic Mxwell s equtions re solved in polr coordintes nd in ll prts of the mchine. The globl solution is obtined by using the trditionl boundry conditions BCs), in ddition to new rdil BCs e.g., between the PMs nd the rotor teeth) which re trduced into system of liner equtions ccording to Tylor series expnsion. The mgnetic field clcultions re performed for two different vlues of iron core reltive permebility viz., 1 nd 1,) nd compred to finite-element method FEM) predictions. The results show tht very good greement is obtined. 1. INTRODUCTION The full clcultion of mgnetic field in electricl mchines is the first step for their design nd optimiztion. The methods of mgnetic field prediction cn be clssified in vrious ctegories [1]: Lehmnn s grphicl, numericl equivlent circuit, Schwrz-Christoffel mpping, Mxwell-Fourier Some comprehensive reviews of mgnetic field prediction in electricl mchine cn be found in [1 8], nd their references. Numericl methods i.e., the finite-element, finite-difference, or boundry-element nlysis) [9 11], which cn be clssified s the very ccurte method compred to the rel results with lrge flexibility to vrious geometries, include nonliner nd nonhomogeneous mterils. The most ccurte models re three-dimensionl 3-D) numericl methods. However, these pproches re very time-consuming nd not very suitble for optimiztion nlysis. Nevertheless, in [1, 13], it is possible to optimize electromgnetic systems from numericl methods. Nowdys, in order to reduce the computtion time, hybrid numericl methods cn be developed [14]. The ctul design works re minly bsed on semi-)nlyticl models i.e., equivlent circuit, SC mpping nd Mxwell-Fourier methods) [1]. These pproches hve been proposed under some geometricl nd physicl ssumptions. Mxwell-Fourier methods re one of the most resent semi-nlytic pproches with good ccurte results in D or 3-D electromgnetic performnce clcultion. These models re bsed on the forml resolution of Mxwell s equtions pplied in ech subdomin of the electromgnetic devices. These subdomins cn be divided into two types of subdomins, viz., periodic e.g., ir-gp) nd non-periodic e.g., slots, teeth, tooth-tips... ). In the second type, the generl solution is derived by pplying homogeneous Neumnn BCs. These conditions re obtined from the pproximtion tht the iron prts re considered to be infinitely permeble. In the literture, we find the pplictions of these methods for the nlysis of Received 1 My 17, Accepted 8 July 17, Scheduled 7 July 17 * Corresponding uthor: Lzhr Roubche roubche.lzhr@gmil.com). 1 Ecole Ntionle Polytechnique LRE-ENP), Algiers, 1, Av. Psteur, El Hrrch, BP 18, 16, Algeri. Déprtement ENERGIE, FEMTO-ST, CNRS, Univ. Bourgogne Frnche-Comté, Belfort, Frnce.

2 86 Roubche et l. severl types of electricl mchines, such s [, 7, 8, 15 19] for PM synchronous mchines, [ ] for solid or cge rotor induction motors, nd [3 5] for reluctnce mchines. It is interesting to note tht n overview on the existing semi-)nlyticl models in Mxwell-Fourier methods with the locl/globl sturtion hs been relized in [1], whose some detils nd the dis)dvntges of these techniques cn be found. In [6,3 5], the uthors pproximte the djcent regions e.g., rotor slots/teeth)in the hrmonic modeling technique s one homogeneous region with reltive permebility developed s Fourier series expnsion. In [1], Dubs nd Boughrr developed the first model introducing the iron prts in the mgnetic field clcultion using subdomin technique, where the uthors solved prtil differentil equtions EDPs) of mgnetic potentil vector in Crtesin coordintes in which the subdomins connection is performed directly in both directions i.e., x-ndy-edges). However, to pply this contribution in the modultion of electricl mchines, it should trnsform it to polr coordintes. Thus, the work in this pper tkes prt in the development nd improvement of the subdomin technique on the scientific topic. The im of this pper is to propose new contribution improving -D subdomin technique with Tylor polynomil by focusing on the considertion of iron. The proposed technique involves solution in polr coordintes of Lplce s nd Poisson s equtions in the sttor yoke, sttor slots nd teeth, ir-gp, rotor teeth, PMs, nd rotor nonmgnetic regions. For the rotor i.e., PMs/teeth) nd sttor i.e., slots/teeth), we tke the generl solution considering the nonhomogeneous Neumnn BCs. In ddition to trditionl interfce conditions between two djcent regions represented with rdius vlue nd intervl of ngles e.g., between the ir-gp nd sttor slots/teeth), we dd the considertion of the interfce conditions between two djcent regions represented with ngle vlue nd intervl of rdius e.g., between the PMs nd the rotor teeth). The first type of interfce conditions permits to get system of liner equtions ccording to Fourier series expnsion, nd the second type permits to obtin system of equtions ccording to Tylor series expnsion. All results from the developed semi-nlyticl model re then compred to those found by FEM [6].. PROBLEM DEFINITIONS..1. Mgnetic Field Solution Figure 1 represents the topology of frctionl-slot STPM mchines with the definition of region where Region I represents the ir-gp, Region II the nonmgnetic mteril under rotor mgnetic i.e., under PMs nd rotor teeth), Region III the sttor yoke, Region IV the buried PMs, Region V the rotor Figure 1. Exmple of frctionl-slot STPM mchines i.e., 4-slots/-poles) with the definition of regions.

3 Progress In Electromgnetics Reserch B, Vol. 77, teeth, Region VI the sttor slots, nd Region VII the sttor teeth. The -D semi-nlyticl model is formulted in mgnetic vector potentil A z nd polr coordintes r, θ) with the following ssumptions: the end-effects re neglected i.e., tht the mgnetic vribles re independent of z); the sttor tooth-tips nd rotor bridge re not considered. However, they cn be introduced esily; the sttor slots/teeth, rotor teeth nd PMs hve rdil sides; the current density hs only one component long the z-xis; the electricl conductivities of mterils re ssumed to be null i.e., the eddy-currents induced in the copper/iron/pms re neglected); the PMs demgnetiztion curve is ssumed to be liner; the direction of PMs mgnetiztion is supposed purely tngentil, i.e., M = {,M θ, } In this rticle, we tke into ccount the iron core reltive permebility in sttor i.e., yoke nd teeth) nd rotor i.e., teeth) subdomin regions. The generl EDP issued from mgnetosttic Mxwell s equtions in continuous nd isotopic region cn be expressed, in term of mgnetic potentil vector A z,by ΔA z r, θ) =, in Region I, II, III, V, nd VII 1) ΔA z r, θ) = μ M, in Region IV ) ΔA z r, θ) = μ J z, in Region VI 3) where M is the PMs mgnetiztion, J z the current density in the sttor slots, nd μ the vcuum permebility. The field vectors B = {B r ; B θ ;} nd H = {H r ; H θ ;} in the different regions re coupled by: B = μ H, in Region I, II, nd VI 4) B = μ μ rm H + μ M, in Region IV 5) B = μ μ rch, in Region III, V, nd VII 6) where μ rm is the reltive recoil permebility of PMs, nd μ rc is the reltive permebility of the iron core. Using B = A,ther- ndθ-components of mgnetic flux density re deduced from A z by B r = 1 r A θ nd B θ = A r.1. Generl Solution of Lplce s Eqution with Nonhomogeneous Neumnn BCs In generl cse of electricl mchine nlysis using subdomin technique, the iron prts re considered to hve infinite reltive permebility which leds to homogeneous Neumnn BCs in slots/pms/teeth subdomins), nd Lplce s, Poisson s or Helmholtz s equtions should be solved only in slots/pms/airgp regions [8, ]. In order to dd rotor/sttor teeth in the mgnetic field prediction in electricl mchines, it is necessry to consider the generl solution of Mxwell s equtions with nonhomogeneous Neumnn BCs. In slots/pms/teeth subdomins, we hve to solve the Lplce s eqution in polr coordintes A z r, θ) r + 1 A z r, θ) + 1 A z r, θ) r r r θ = 8) The electricl mchines hve cylindricl form, nd the following periodicity condition between nd π should be dded to solution A z r, θ) θ= = A z r, θ) θ=π 9) Considering the periodicity condition given in Eq. 9), the generl solution of Eq. 8) cn be written s A z r, θ)=d1 +D lnr)+ D1 k r k +D k r k) sink θ θ 1 ))+ D3 k r k +D4 k r k) cosk θ θ 1 )) 1) 7)

4 88 Roubche et l. A Z D ) r θ = Δ A Z = A Z D ) 1 r θ = θ 1 θ R1 R Figure. Slots/PMs/teeth subdomins with nonhomogeneous Neumnn BCs. Figure 3. Description: Air-gp i.e., Region I), nonmgnetic mteril under rotor mgnetic i.e., Region II) nd sttor yoke Region III). where k is positive integer, nd D1 D4 k re the integrtion constnts. As shown in Figure, the slots/pms/teeth subdomins re delimited by r [R 1 ; R ] θ [θ 1 ; θ ] nd re ssocited with the following BCs: Ar, θ) D 1 r) = θ = k D1 k r k + D k r k) 11) θ=θ1 D r) = Ar, θ) θ = θ=θ k D1 k r k + D k r k) cos k)+ k D3 k r k +D4 k r k) sink) 11b) In the cse where D 1 r) = D r) = i.e., homogeneous Neumnn BCs), the derivtion of Eqs. 11) nd 11b) gives k D1 k r k + D k r k) = D1 k = D k = 1) k D3 k r k + D4 k r k) sin k)= k = mπ 1b) where m is positive integer, nd the solution of Eq. 1) cn be simplified s ) A z r, θ) =D1 + D lnr)+ D3 m r mπ + D4m r mπ mπ ) cos θ θ 1) 13) m=1 As explicted in [1], D1 D4 m cn be determined by Fourier series expnsions ssocited with BCs for the rdii R 1 nd R. In the generl cse where the vrition of A z versus θ for θ = θ 1 nd θ = θ is not necessrily null, it should consider the generl solution of Eq. 1), which cn be written with nother form s ) A z = D1 + D lnr)+ B1 m r mπ + B m r mπ mπ ) cos θ θ 1) + m=1 D1 k r k + D k r k) sin kθ θ 1 )) + k mπ D3 k r k + D4 k r k) cos kθ θ 1 )) 14)

5 Progress In Electromgnetics Reserch B, Vol. 77, where B1 m & B m re new integrtion constnts which respectively represent D3 k & D4 k for k = mπ/. It is interesting to note tht the dded Fourier constnts D1 k D4 k cn be determined by Tylor series expnsions ssocited with BCs for θ 1 nd θ, nd some explntions of the obtined equtions re explined in Section.6. The solution of Poisson s equtions is derived by dding the corresponding prticulr solution... Solution of Lplce s Eqution in Regions I, II nd III 1) Air-gp Region I): In the ir-gp, which is n nnulr domin t r [R ; R 3 ] see Figure 3), the solution of Eq. 1) is defined by A zi r, θ) =A1 + A lnr)+ A1n r n + A n r n) sinnθ)+ A3n r n + A4 n r n) cosnθ) 15) n=1 where n is positive integer, nd A1 A4 n re the integrtion constnts in Region I. ) Nonmgnetic Mteril under Rotor Mgnetic Region II): The generl solution in this subdomin hs the sme form s the solution in Region I. Nevertheless, in dding the condition of the finite vlue of A z for r =, the solution cn be written s A zii r, θ) = A5 n r n sinnθ)+ A6 n r n cosnθ) 16) n=1 where A5 n & A6 n re the integrtion constnts in Region II. 3) Sttor Yoke Region III): In the sttor yoke, which is n nnulr domin t r [R 4 ; R ext ]see Figure 3), the generl solution hs the sme form s the solution in Region I. Nevertheless, in dding the Dirichlet s BC for r = R ext, the solution cn be written s ) n r ) n ) r A ziii r, θ) = A7 n sinnθ)+ R n=1 ext R ext n=1 where A7 A8 n re the integrtion constnts in Region III. n=1 n=1 A8 n r R ext ) n ) ) r n cosnθ) 17) R ext.3. Solution of Poisson s Eqution with Nonhomogeneous Neumnn in Regions IV nd VI 1) Buried PMs Region IV): In ech buried PM subdomin j) of Region IV see Figure 1), we hve to solve Eq. ), i.e., Poisson s eqution. Becuse the direction of PMs Mgnetiztion is purely tngentil, Eq. ) cn be reduced to A z r, θ) r + 1 r A z r, θ) r + 1 r A z r, θ) θ = μ M θ r where M θ = M j = 1) j B rm /μ with j vrying from 1 to p poles, nd B rm is the remnent flux density of PMs. As shown in Figure 4), the jth PM is ssocited with nonhomogeneous Neumnn BCs, nd using the method explicted in Section., the solution of Eq. 18) cn be written s A ziv j r, θ) = B1 j + B j lnr) μ M j r k mπ D1 jk r k + D jk r k) sin m=1 k B1 jm r mπ θ α j + 18) ) + B jm r mπ mπ cos θ α j + )) )) D3 jk r k + D4 jk r k) cos k θ α j + )) 19)

6 9 Roubche et l. ) b) Figure 4. Nonhomogeneous Neumnn BCs for ) jth buried PMs nd b) ith sttor slot. where m nd k re the positive integers; B1 j B jm nd D1 jk D4 jk re the integrtion constnts; α j is the ngulr position of the jth PMs; is the PM-opening. ) Sttor Slots Region VI):In ech slot subdomin i) of Region VI, we hve to solve Eq. ), i.e., Poisson s eqution, A z r, θ) r + 1 A z r, θ) + 1 A z r, θ) r r r θ = μ J i ) where J zi is the current density in the ith sttor slot with i vrying from 1 to Q s in which Q s represents the number of sttor slots. Tking into ccount the BCs shown in Figure 4b), the solution of Eq. ) cn be written s A zv Ii r, θ) =C1 i + C i lnr) 1 4 μ J i r + + E1 ik r k + E ik r k) sin k θ γ i + c m=1 )) + C1 im r mπ c k mπ c ) + C im r mπ mπ c cos θ γ i + c )) c E3 ik r k + E4 ik r k) cos k θ γ i + c )) 1) where m nd k re the positive integers; C1 i C im nd E1 ik E4 ik re the integrtion constnts; γ i is the ngulr position of the ith sttor slot; c is the sttor slot-opening Solution of Lplce s Eqution with Nonhomogeneous Neumnn BCs in Regions V nd VII 1) Rotor Teeth Region V): The jth rotor tooth is subdomin region with nonhomogeneous Neumnn BCs s shown in Figure 5). As explicted in Section., the solution of Eq. 1), i.e., Lplce s eqution, cn be written s ) A zv j r, θ) = B3 j + B4 j lnr)+ B3 jl r lπ b + B4 jl r lπ lπ b cos θ β j + b )) b l=1 + D5 jk r k + D6 jk r k) sin k θ β j + b )) + k lπ b D7 jk r k + D8 jk r k) cos k θ β j + b )) )

7 Progress In Electromgnetics Reserch B, Vol. 77, ) b) Figure 5. Nonhomogeneous Neumnn BCs for ) jth rotor nd b) ith sttor teeth. where l nd k re the positive integers; B3 j B4 jl nd D5 jk D8 jk re the integrtion constnts; β j is the ngulr position of the j th rotor teeth; b is the rotor tooth-opening. ) Sttor Teeth Region VII): The ith sttor tooth is subdomin region with nonhomogeneous Neumnn BCs shown in Figure 5b). As explined in Section., the solution of Eq. 1), i.e., Lplce s eqution, cn be written s A zv IIi r, θ) = C3 i + C4 i lnr)+ + + l=1 C3 il r lπ d + C4 il r lπ d ) lπ cos d E5 ik r k + E6 ik r k) sin k θ δ i + d )) E7 ik r k + E8 ik r k) cos k θ δ i + d )) k lπ d θ δ i + d )) where l nd k re the positive integers; C3 i C4 il nd E5 ik E8 ik re the integrtion constnts; δ i is the ngulr position of the ith sttor teeth; d is the sttor tooth-opening..3.. Interfces Conditions between Regions To determine the integrtion constnts in Eqs. 15) 17), 19), nd 1) 3), the BCs t the interfce between the vrious regions should be introduced. The interfces conditions in this model cn be divided into two types. One is over ngle intervl for given rdius vlue {R 1,R,R 3,R 4 }, nd the other is over rdius intervl for given ngle {α j ± /, β j ± b/, γ i ± c/, δ i ± d/}. As the obtined equtions from ech type of these interfces conditions hve the sme form, it is sufficient to show one exmple for ech type. The interfce conditions re: between Region IV, V nd I t r = R : 3) A zi R,θ)=A ziv j R,θ) for α j θ α j + A zi R,θ)=A zv j R,θ) for β j b θ β j + b H θi R,θ)=H θiv j R,θ) for α j θ α j + 4) 5) 6)

8 9 Roubche et l. H θi R,θ)=H θv j R,θ) for β j b θ β j + b 7) between Region IV nd V t α j + / =β j b/ ndα j+1 / =β j + b/ forr [R 1 ; R ]: A ziv j r, α j + ) = A zv j r, β j b ) 8) H rivj r, α j + ) = H rvj r, β j b ) 9) A ziv j+1) r, α j+1 ) = A zv j r, β j + b ) 3) H rivj+1) r, α j+1 ) = H rvj r, β j + b ) 31) The interfce conditions in Eqs. 4) 7) concern regions with different subdomin frequencies which need Fourier series expnsions to stisfy equlities of mgnetic potentil vector nd tngentil mgnetic field. According to Fourier series expnsion, the interfce condition in Eq. 4) gives = B1 j + B j lnr ) μ M j R k mπ D3 jk R k + D4 jkr k B1 jm R mπ +B jm R mπ + + k mπ α j + α j D3 jk R k + D4 jk R k ) α j+ α j D1 jk R k + D jkr k ) α j+ α j cos k θ α j + )) dθ = 1 D1 jk R k +D jk R k ) α j+ α j ) α j+ α j sin k θ α j + )) dθ α j + α j sin k θ α j + )) cos A I R,θ)dθ 3) mπ cos k θ α j + )) mπ cos θ α j + )) dθ θ α j + )) dθ mπ A I R,θ)cos θ α j + )) dθ 33) The interfce condition in Eq. 5) gives B3 j + B4 j lnr ) b k mπ D7 jk R k + D8 jkr k B3 jm R mπ +B4 jm R mπ + b D5 jk R k + D6 jkr k ) β j+ b β j b ) β j+ b β j b cos k θ β j + b )) dθ = 1 b D5 jk R k +D6 jk R k ) β j+ b β j + b sin k θ β j + b )) dθ β j + b β j b A I R,θ)dθ 34) sin k θ β j + b )) mπ cos θ β j + b )) dθ

9 Progress In Electromgnetics Reserch B, Vol. 77, = b + b k mπ b β j + b β j + b D7 jk R k + D8 jkr k ) β j+ b β j + b cos k θ β j + b )) mπ cos θ β j + b )) dθ mπ A I R,θ)cos θ β j + b )) dθ 35) The interfces conditions in Eqs. 6) nd 7) give A R = 1 π n A1n R n 1 A n R n 1 ) 1 = μ π Q r α j + j=1 α j Q r α j + j=1 α j Q r n A3n R n 1 A4 n R n 1 ) 1 = μ π α j + j=1 α j H θiv j R,θ)dθ + 1 π Q r H θiv j R,θ)sinnθ)dθ+ 1 π β j + b j=1 β j b H θiv j R,θ)cosnθ)dθ+ 1 π Q r j=1 β j b Q r H θv j R,θ)dθ 36) β j + b β j + b j=1 β j b H θv j R,θ)sinnθ)dθ 37) H θv j R,θ)cosnθ)dθ 38) The interfce conditions in Eqs. 8) 31) concern two polynomils equtions with different degree, which need Tylor series expnsions round R t such s R t ]R 1 ; R ] to stisfy equlities of mgnetic potentil vector nd rdil mgnetic field. The Tylor polynomil degree is chosen to stisfy the lck of equtions obtined by Fourier series expnsion of previous interfce conditions compred to the number of unknowns in system, so for k vrying from 1 to K r it is enough to tke Tylor polynomil of degree K r 1. Therefore, the interfce condition in Eq. 8) gives A IV j R t,α j + ) = A Vj R t,β j b k A IV j R r k t,α j + ) = k A Vj r k R t,β j b ) From the interfce condition in Eq. 9), we get Hr IV j R t,α j + ) = Hr Vj R t,β j b ) k Hr IV j R r k t,α j + ) = k Hr Vj r k R t,β j b ) From the interfce condition in Eq. 3), we get A IV j+1 R t,α j+1 ) k A IV j+1 R r k t,α j + ) From the interfce condition in Eq. 31), we get Hr IV j+1 R t,α j+1 ) k Hr IV j+1 R r k t,α j + ) ) = A Vj R t,β j + b ) = k A Vj r k R t,β j b ) = Hr Vj R t,β j + b ) = k Hr Vj r k R t,β j b ) 39) 4) 41) 4) 43) 44) 45) 46)

10 94 Roubche et l. There re no simple expressions to define the system obtined by Tylor series expnsions 39) 46). However, the coefficients of the unknowns in this system cn be defined recursively. For exmple, the development of interfce conditions Eq. 39)) nd Eq. 4)) permits to define new coefficients C B1k for the vector potentil of region IV s k = k =1 k = B1 j B1 jmr mπ t cosmπ)+...= A zv j R t,β j b ) Bj mπ B1 jm R mπ t cosmπ)+...= AzV j R t,β j b ) R t R t r Bj mπ mπ B1 1 Rt jm R mπ t cosmπ)+...= A zv j R t R t r R t,β j b ) C B1 = R mπ t cosmπ) C B11 = mπ C B1 R t C B1 = The coefficients C B1 k cn be defined by the recursive formul s: { C B1 = R mπ t cosmπ) C B1 k+1 = mπ +1 k 48) R t C B1 k The other coefficients re deduced with the sme resoning. It is interesting to note tht the interfce conditions for the rdii {R 1,R 3,R 4 } nd for the ngels {γ i ± c/, δ i ± d/} re developed with the sme previous method 3. RESULTS AND VALIDATION The developed semi-nlyticl method for frctionl-slot STPM mchines tking into ccount the iron core reltive permebility is used to determine electromgnetic performnces viz., the mgnetic flux density, the mgnetic flux linkge, the bck EMF, the electromgnetic/cogging/ripple torques, the nonintrinsic unblnced mgnetic forces,... ) whose vrious formuls hve been clrified in [19]. The min dimensions nd prmeters of the frctionl-slot STPM mchines e.g., 4-slots/-poles) with the buried PMs nd single lyer winding re given in Tble 1. Then, semi-nlytic results re verified by -D FEM [6]. For the finite-elements simultion, we hve used 99,199 nodes nd 194,796 elements. Tble 1. Prmeters of the studied mchine. mπ 1 R t Symbol Prmeters Vlue Units) B rm Remnence flux density of PMs 1. T) μ rm Reltive permebility of PMs 1 N c Number of conductors per sttor slot 18 I m Pek phse current 18 A) Q s Number of sttor slots 4 c Sttor slot-opening 7.5 deg) PM-opening 3.7 deg) p Number of pole pirs 11 R ext Rdius of the externl sttor surfce 11 mm) R 4 Outer rdius of sttor slot 1 mm) R 3 Rdius of the sttor outer surfce 79 mm) R Rdius of the rotor inner surfce t the PM surfce 78 mm) R 1 Rdius of the rotor inner surfce t the PM bottom 57 mm) g Air-gp length 1 mm) L u Axil length of the mchine 63 mm) Ω Mechnicl pulse of synchronism 1,1 rpm) C B11 47)

11 Progress In Electromgnetics Reserch B, Vol. 77, FEM Anlytic μ rc = 1. FEM Anlytic μ rc = 1 B r T) B θ T) Angle mech. degrees) 1 ) Angle mech. degrees)..5.1 B r T) B θ T) FEM Anlytic μrc = 1 -. FEM Anlytic μrc = Angle mech. degrees) b) Angle mech. degrees).4 1 FEM Anlytic μrc = 1. FEM Anlytic μrc = 1 B r T) B θ T) Angle mech. degrees) c) Angle mech. degrees) Figure 6. Rdil nd tngentil components of the mgnetic flux density in the middle of the ir-gp i.e., Region I): ) PMs lone, b) sttor current lone nd c) on lod condition.

12 96 Roubche et l. The wveforms of rdil nd tngentil components of the mgnetic flux density in the vrious regions re computed with finite number of hrmonic terms, viz., N =, M = L =ndk =. The nlytic clcultion of mgnetic flux distribution in ll regions is done with two different vlues of the iron core reltive permebility viz., 1 nd 1,). In ech cse, the reltive permebility is supposed the sme for ll iron prts i.e., sttor yoke, sttor nd rotor teeth). However, it should be noted tht it is possible to tke specil permebility vlue for ech region. In Figure 6, comprison between the numericl results nd semi-nlyticl predictions is shown in term of the mgnetic flux density in the middle of the ir-gp i.e., Region I) for three operting conditions : i) PMs lone, ii) sttor currents lone, nd iii) on lod condition. One cn see tht the diminution of the iron core reltive permebility is ccompnied by decrese in mgnetic flux mplitude. It cn be seen tht very good greement for the rdil nd tngentil components of the mgnetic flux density is obtined. The proposed model is chrcterized by the bility to know the mgnetic flux density in ll mchine regions, such s sttor/rotor yoke, sttor slots, sttor/rotor teeth nd PMs. Figure 7) shows the rdil nd tngentil components of the mgnetic flux density in the middle of the PMs/rotor teeth under pole-pitch for on lod condition. Very good greement is obtined for the tngentil component. For B r T) Mgnet Anlytic Rotor tooth Femm μ rc = Angle mech. degrees).3 B θ T) Mgnet Anlytic Rotor tooth Femm μ rc = Angle mech. degrees) ).3 Anlytic Femm μ rc = 1 Anlytic Femm μ rc = 1. Sttor tooth B r T).1 Sttor Slot B θ T) Sttor Slot Sttor tooth Angle mech. degrees) Angle mech. degrees) b) Figure 7. Rdil nd tngentil components of the mgnetic flux density under tooth-pitch for on lod condition in the middle of ) PMs/rotor tooth nd b) sttor slot/tooth.

13 Progress In Electromgnetics Reserch B, Vol. 77, Shft Mgnet Sttor tooth Yoke Shft Mgnet Sttor tooth Yoke B r T) Anlytic Femm μ rc = Rdius mm) Shft Rotor tooth Sttor slot Yoke B θ T) ) Anlytic Femm μ rc = Rdius mm).6.4. Shft Rotor tooth Sttor slot Yoke B r T). B θ T) Anlytic Femm μ rc = Rdius mm) b) Anlytic Femm μ rc = Rdius mm) Figure 8. Rdil nd tngentil components of the mgnetic flux density for on lod condition t: ) θ = α 1 nd b) θ = γ 3. the rdil component, the results re good with greeble error, the sme order of ccurcy is seen for sttor slot/teeth under tooth-pitch shown in Figure 7b). Another presenttion of obtining results is shown in Figure 8, where the rdil nd tngentil components of the mgnetic flux density for on lod condition re plotted in function of the rdius t given ngle. For the semi-nlytic results, the mgnetic flux density is clculted for ech intervl of rdius by the corresponding solution. The results re obtined for θ = α 1 nd θ = γ 3, nd very good greement is obtined. The sttic torque versus rotor position is presented in Figure 9). The sttor currents re supposed constnt, nd the rotor position is updted to hve nominl rottion speed. As obtined, the mechnicl ngle hs Θ rs =π/p fundmentl period, nd the mximum torque is obtined for Θ rs = π/p. The mximum torque decreses for smll vlue of reltive permebility. It cn be seen tht very good greement is obtined for the two proposed vlues of reltive permebility. Figure 9b) shows the electromgnetic torque wveform versus electricl ngle. At ech instnt, the sttor currents re updted to hve sinusoidl current wveform. The initil rotor position is tken for giving mximum torque vlue, i.e., Θ rs = π/p. The induced bck EMF for no-lod condition is shown in Figure 1. The simultion is done for

14 98 Roubche et l. Torque Nm) Anlytic Femm μ rc = Angle mech. degrees) ) Torque Nm) Anlytic Femm μ rc = Electricl ngle degrees) b) Figure 9. ) Sttic torque nd b) electromgnetic torque t Θ rs = π/p for on lod condition with I =18A). 3 Bck EMF V) Anlytic Femm μ rc = Electricl ngle degrees) Figure 1. Induced bck EMF for no-lod condition with I =A). two different vlues of iron core reltive permebility, nd the obtined results confirm the ccurcy of the proposed semi-nlyticl model. Figure 11) shows the evolution of simultion time versus the number of hrmonics. To reduce the computtion time of the semi-nlyticl model, the number of sptil hrmonics N, M, L nd K cn be reduced. However, this leds to reduction in ccurcy. It is seen tht for giving vlue of sptil hrmonics, the RMS error will not be ffected very much by incresing the number of sptil hrmonics. This vlue cn be chosen s n optiml vlue. Figure 11b) shows the evolution of RMS error for the rdil component of the mgnetic flux density in the middle of the ir-gp versus hrmonic order of Regions I, II, nd III i.e., N) for severl vlues of hrmonic order of the other regions. The RMS error is clculted by error = Npc m=1 B Numeric Ir,m / ) B Anlytic Ir,m N pc 49) It cn be seen from Figure 11b) tht the error strts to converge round N = 18, M = L =6nd K =1.

15 Progress In Electromgnetics Reserch B, Vol. 77, Simultion time s) M=6 L=6 K=3 M=6 L=6 K=1 M=4 L=4 K=3 RMS error T) M=6 L=6 K=3 M=6 L=6 K=1 M=4 L=4 K= Number of hrmonics ) Number of hrmonics b) Figure 11. Influence of the number of hrmonics ssocited with Regions I, II nd III i.e., N) onthe rdil component of the mgnetic flux density in the middle of the ir-gp versus numeric simultions for severl vlues of the number of hrmonics ssocited with the other regions: ) the clcultion time nd b) the RMS error. 4. CONCLUSION In this pper, we hve proposed n improved semi-nlyticl model bsed on subdomin technique with Tylor polynomil for prediction open-circuit, rmture rection nd on lod mgnetic field distribution in STPM mchines. The proposed model tkes into ccount the reltive permebility in ll mchine regions including the iron prts. One of the lrgest dvntges of the discussed pproch is its bility to ccount the finite permebility in rotor/sttor teeth nd yoke, with the possibility to get the solution of mgnetic flux in these regions. Currently, only liner PM mteril proprieties re considered. However, nonliner PM mterils could be ccounted by mens of n itertive lgorithm, nd lso the proposed model offers the possibility to tke especil finite permebility for ech PM or rotor/sttor teeth, sme for rotor nd sttor yoke. Semi-nlytic results re in excellent greement with the ones obtined by -DFEM. From these results, we show the ccurcy of the semi-nlyticl model for severl vlues of iron core reltive permebility. APPENDIX A. NOMENCLATURE H rx,h θx Rdil nd tngentil components of the mgnetic flux intensity in x domin where x cn be I, II, III, IV, V, VI, or VII. A zx Mgnetic potentil vector in x domin. α j, Position nd opening width of j th PMs. β j,b Position nd opening width of j th rotor tooth. γ i,c Position nd opening width of i th sttor slot. δ i,d Position nd opening width of i th sttor tooth. n Hrmonic order for ir-gp, nonmgnetic mteril, nd sttor yoke. k, m, l Hrmonic order for slots/pms/teeth. J z Current density. μ Vcuum permebility. μ rc Reltive permebility of iron core. Reltive permebility of PMs. μ rm

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