A Generalized Magnetic Circuit Modeling Approach for Design of Surface Permanent Magnet Machines

Transcription

1 ourc:ieee ranaction on Indutrial Elctronic, Vol. 59, o., pp Yar of Publication: I:78-46 Publihr:Intitut of Elctrical and Elctronic Enginr (IEEE) DI:.9/IE..65 IEEE. Pronal u of thi matrial i prmittd. Prmiion from IEEE mut b obtaind for all othr u, in any currnt or futur mdia, including rprinting/rpublihing thi matrial for advrtiing or promotional purpo, crating nw collctiv work, for ral or rditribution to rvr or lit, or ru of any copyrightd componnt of thi work in othr work. A nralizd Magntic Circuit Modling Approach for Dign of urfac Prmannt Magnt Machin Min-Fu Hih, Mmbr, IEEE, and You-Chiuan Hu Abtract - hi papr propo a gnralizd quivalnt magntic circuit modl for th dign of prmannt-magnt (PM) lctric machin. Convntional approach hav bn applid to PM machin dign but may b inufficintly accurat or gnralizd without taking pol-lot count into conidration. hi would rult in rduction of dimnioning accuracy at th initial dign tag. Alo, magntic aturation i oftn ignord or compnatd by corrction factor in implifid modl inc it i difficult to dtrmin th flux in individual tator tth. In thi papr, th flux producd by tator winding currnt and prmannt magnt can b calculatd accuratly and rapidly uing th dvlopd modl, taking aturation into account. A nw modling tchniqu for PM pol i propod o that th magntic circuit i applicabl to any pol-lot combination. hi aid machin dimnioning without th nd for computationally xpniv finit lmnt analyi (FEA). A 54 kw PM machin i firt dignd uing th propod mthod, and thn vrifid with FEA. Anothr 35 W machin i ubquntly dignd, manufacturd and validatd by both FEA and xprimnt. h comparion dmontrat th ffctivn of th propod modl. Indx rm magntic analyi; magntic circuit; prmannt magnt machin; magntic aturation; finit lmnt LI F YMBL Bm,n Bnw Br Cm(n) Dn- Dn+ F F,n fring flux dnity in n-th magnt gmnt nw flux dnity rmannt flux dnity of magnt ffctiv magnt factor ampling point in B-H curv abolut MMF at nod of magntic circuit MMF in n-th tooth/gmnt fring prmanc Manucript rcivd Dcmbr, ; rvid April 8 ; accptd for publication ovmbr, 9. hi work wa upportd in part by th aiwan ational cinc Council undr rant C97--E-6-68MY, and C98-34-E-6-. Copyright IEEE. Pronal u of thi matrial i prmittd. Howvr, prmiion to u thi matrial for any othr purpo mut b obtaind from th IEEE by nding a rqut to pubprmiion@i.org. M. F. Hih i with th Dpartmnt of ytm and aval Mchatronic Enginring, ational Chng Kung Univrity, ainan 7, aiwan (phon: Ext ; fax: ; -mail: mfhih@mail.ncku.du.tw). Y. C. Hu i with th Dpartmnt of ytm and aval Mchatronic Enginring, ational Chng Kung Univrity, ainan 7, aiwan (mail: f48934@nckualumni.org.tw). Hm,n Hnw Lt m rro l g m r,n,n t,n rl Wopn x µ µiron μiron r,n μiron,n μiron t,n μnw μ*nw µr θq θm θm θ θr (n) θ θl (n) θl'(n) ϕ ϕg,n ϕm,n ϕnw ϕr,n ϕ,n ϕt,n magntic intnity of n-th magnt gmnt nw magntic intnity tack lngth (ffctiv axial lngth) numbr of pol numbr of lot rotor radiu (to magnt ba) lakag rluctanc of gap btwn ho air-gap rluctanc magnt rluctanc in radial dirction rluctanc of n-th rotor yok rluctanc of n-th tator yok rluctanc of n-th tooth radiu to lot outr dg lot opning width rang of fring prmanc vacuum prmability iron rlativ prmability rlativ prmability of n-th rotor yok rlativ prmability of n-th tator yok rlativ prmability of n-th tooth nw rlativ prmability modifid prmability matrial rlativ prmability ffctiv portion of magnt in a lot pitch angular pan of -magnt in a lot pitch angular pan of -magnt in a lot pitch opning gap btwn two magnt rmaindr of rlativ lot angl θl (n) lot pitch rlativ lot angl to a rfrnc θl (n) ranging btwn ±8 E flux producd from abolut MMF flux in n-th air-gap magnt rmannt flux nw flux flux in n-th rotor yok flux in n-th tator yok flux flowing through n-th tooth I. IDUCI ar-arth prmannt-magnt (PM) bruhl machin po th fatur of high fficincy, high torqu/powr

2 dnity and low maintnanc. h advantag mak thm xcllnt candidat for variou application, uch a lctric vhicl [-3], wind turbin [4-7] and marin nrgy convrtr [8]. h quivalnt magntic circuit modl (EMCM) i a common tchniqu for th analyi and dign of lctric machin by linking th matrial charactritic to th machin bhavior [9-4]. h magntic fild charactritic can b tchnically obtaind uing lctric circuit principl,.g., Kirchhoff Voltag Law (KVL) and Kirchhoff Currnt Law (KCL). Magntic aturation hould alo b conidrd a it will incra lo and dgrad machin prformanc. Du to xciv implification, convntional EMCM oftn lack th ability to accuratly prdict th flux aturation and machin prformanc with any pol-lot count. Corrction factor bad on xprinc may b rquird for compnation. hrfor, EMCM i oftn ud for prliminary dign, with confirmation or modification rquiring othr approach (.g., finit lmnt analyi, FEA) [5]. n th othr hand, om approach may b ufficintly accurat but limitd to crtain pol-lot combination. FEA can dirctly calculat th flux pattrn, but th ntir proc i computationally xpniv. Changing th dign paramtr (dimnion, lot numbr, winding arrangmnt) in FEA oftn rquir th modl to b rcontructd. For application whr low cogging torqu i advantagou (.g., mall wind turbin gnrator), incraing th lowt common multipl (LCM) of th PM gnrator i a popular olution. Howvr, thi mak it difficult to rduc th computational tim by xploiting machin ymmtry. akn togthr, th factor account for FEA bing unuitabl for th prliminary dign tag of lctric machin. From th abov dicuion, a univral EMCM i ncary for rapid, accurat machin dign of gnral PM machin. Mor dlicat dvic modl hould b mployd in ordr to improv accuracy. Modling tchniqu alo nd to b amlioratd,.g., a circular ntwork that can phyically rprnt th bhavior of th flux in a machin. In particular, a rliabl modl for prmannt magnt i a ky to gnralizing th modl. racing th hitory, th firt rlatd work found wa offrd by otr [6] in 94, whr th analogy btwn th lctrical and magntic circuit wa initially rcognizd. In [7-8], magntic circuit wr dvlopd for analyi of lfand mutual inductanc in an lctrical circuit. h machin charactritic wr computd via th lctrical circuit but not th magntic circuit. Carpntr [9] linkd th dvic modl of a tranformr, which could b th firt tim th quivalnt magntic circuit wa applid for analyi without an lctrical circuit. Campbll [] applid EMCM to dign an axial fild PM machin by auming an infinit prmability lwhr than th air-gap. ady [] built a modl for PM machin with implification, th modl of which covrd on pol pitch. Vlado [-3] conductd a ri of tudi to dvlop th EMCM for induction machin and PM machin, linking th magntomotiv forc (MMF) of th tator winding to th main ntwork. onlinar componnt (.g., matrial aturation) wr alo conidrd. Howvr, th main circuit wa tratd a a linar machin intad of circular, which could rult in omiion of th coupling btwn th firt tooth and th lat tooth. ordon [4] applid th EMCM to ynchronou machin, whil Law t al [] and Buch t al [5] dvlopd an EMCM in fild rgulatd rluctanc machin, whr ntir machin wr rprntd by on quivalnt pol and an analogou linar machin modl. Hamid t al [6] propod a mthod for th dynamic imulation of induction machin with an analogou circular lctrical ntwork, which could, on th othr hand, b complicatd to calculat. Millr t al [7] and Hanlman [5] alo contributd to EMCM, lading th approach into practical application of PM motor dign. Chng t al [8] propod a varying-ntwork EMCM for 6/4- pol doubly alint PM motor and conidrd th rlativ poition of th rotor with rpct to th tator. Kano t al [9] improvd th EMCM by dvloping a thr dimnional ntwork for a novl 3-D PM machin. Howvr, th ntwork wr complx and limitd to u in PM machin with ymmtric contruction. Chn t al [3] xtndd th modl to a ingl-pha flux-witching PM motor, whr th analyi wa limitd to two-pol or ymmtric tructur. Han t al [3] alo rportd work on magntic circuit for PM motor. o um up, th abov mntiond litratur rarly conidrd th dtaild modl for prmannt magnt pol - contribution of orth and outh pol magnt to th flux in ach tooth (and o th flux linkag) for any rlativ rotor poition to th tator. hi impli that thy may not b ufficintly gnralizd for any pol-lot combination. In thi papr, a gnralizd, accurat and tim-ffctiv EMCM for dign and analyi of urfac PM machin i propod, taking pol-lot count and flux aturation into account. h modl mploy a circular configuration that covr all th magntic loop on all th tator tth to nhanc th accuracy. hi nur that th flux pattrn of machin with any pol-lot count and winding can b calculatd. In thi modl, th MMF producd by th winding ar computd to dtrmin th matrial prmability (oprating point) according to th B-H curv o that th flux can b accuratly obtaind. hi calculation i prformd imultanouly for all th MMF ourc, including th rotor magnt. h MMF ourc of th magnt ar tratd in a picwi fahion, i.. th magnt ar virtually modularizd into vral gmnt to aociat with tator lot for fitting into th magntic ntwork. hi gmntd PM modl (dtaild latr) i a maor contribution of thi papr. hi allow th propod EMCM to b applicabl to any pol-lot combination and to b mor gnralizd than convntional approach. Conquntly, th flux in diffrnt location of th machin can b accuratly calculatd and th location whr magntic aturation occur can alo b prdictd. By intgrating th abov fatur, thi olution can thu provid rapid and accurat machin dign. imulation of a 54 kw machin and xprimnt on a 35W machin ar ud to vrify th dvlopd modl.

3 3 II. MACHIE MDELI A. Loading on Magntic Circuit In th EMCM, th accuracy of th modl dpnd on th accuracy of individual componnt modl and th coupling btwn th activ componnt (.g., magnt and winding) and paiv componnt (.g., air gap and iron cor). hi rquir a carful tudy on loading of th magntic circuit. In common opration of a PM machin, th magntic loading and lctric loading ar imultanouly applid. h magntic circuit can accommodat flux to a crtain lvl rgardl of th magntic ourc, but thi ability dpnd on th matrial charactritic and gomtry. nrally, for a propr dign, th magntic circuit hould accommodat mor flux than an inappropriat on to achiv th rquird pcification. h oprating point of frromagntic matrial (nonlinar), aociatd with th magntic loading condition, hould b a clo to th pak prmability on th μ-h curv a poibl. hi indicat that th magntic loading hould b dignd around that lvl to maximiz th flux accommodation via th mot conomic magntic intnity without aturating th matrial. Car hould b takn in th analyi to achiv thi conomic dign conidring both th matrial and gomtry. h nitivity of nonlinar matrial i tudid hr to hlp rfin th highly nitiv componnt in th EMCM o that th dign accuracy in th initial tag can b improvd. A impl C-typ magntic circuit, hown in Fig., i ud for thi tudy, th paramtr of which ar dtaild in abl I. W Iron cor W uually ha rlativ pak prmability btwn 4 and 8 [5], [3]. h flux at a rlativ prmability of th man valu (=6) i thrfor ud to normaliz th flux undr diffrnt matrial prmability, a hown in Fig. (a) and (b). In Fig. (b), th prcntag diffrnc for th normalizd flux (bad on μr=6) btwn μr=4 and 8 i.% (.88% for μr =4 and.3% for μr =8). Dpit bing wll known, it i hr concludd that in high prmability opration, th rror producd from incorrct prmability can b ignord, whil a nonlinar matrial can b tratd a linar with it maximum prmability. B. Modling of Prmannt Magnt hi ction dicu th modling of prmannt magnt in a PM machin by conidring tator lot/tth and rotor poition. A prviouly mntiond, thi PM modl nabl th propod EMCM to analyz and dign machin with any pol-lot combination. Flux lakag occur whn th flux producd by a magnt fail to flow through th tator winding and thu contribut no magnitud to th back-emf. h lvl of flux lakag dpnd on th pol-to-lot ratio, lot gomtric fatur, and rotor poition. hr condition ar conidrd hr. Firt, th "flux cancllation" mod account for th mot riou flux lakag, a hown in Fig. 3(a). cond, in th "partial contribution" mod, part of on pol i within a lot pitch. h contribution of flux from th magnt i mitigatd du to th gap btwn th two magnt (magnt opning), a hown in Fig. 3(b). hird, for th "full contribution" mod in Fig. 3(c), th whol lot pitch i covrd by on pol, whr all th flux producd by th magnt i tratd a flowing through th tooth. Who Lquar Air-gap Magnt g lm Lquar Fig. Dfinition of paramtr in a C-typ circuit Matrial with high prmability i xpctd to oprat at high flux dnity. Hr, an itrativ proc i ud to calculat th corrct prmability for th C-typ circuit undr a magntic fild according to th matrial' B-H and μ-h curv for improving th accuracy. ABLE I IMULAI PAAMEE F C-YPE MDEL Paramtr/ymbol Valu Magnt lngth (lm) 5 ooth width (W) Air-gap lngth (g). Ca lngth (Lquar) Magnt mannt flux dnity (Br).3 Corciv forc (Hc) 93 lativ prmability (μr).5 Unit mm mm mm mm ka/m -- FEA imulation (AF Maxwll) i mployd for th flux calculation of th C cor. ypical iron cor matrial (a) (b) Fig. C-typ circuit (with mm air-gap): (a) flux and (b) prcntag diffrnc of normalizd flux bad on μr=6.

4 4 Bad on th prviou analyi, th PM modl i drivd. Conidring a lot pitch θ, it i aumd that th contribution of th magnt to MMF i proportional to th portion of th magnt within θ, a indicatd in Fig. 4. For th lot pitch of th n-th tooth (or gmnt), th opning gap btwn th two magnt i dnotd by θ, th angl for th -pol magnt i rprntd by θ m, whil θ m dnot that of th -pol magnt. h ffctiv portion of th magnt (dnotd a angl θ q ) in a lot pitch can b xprd a C () q m whr C m i dfind a th ffctiv magnt factor for a tooth/gmnt and can b dfind a C m m m, C m () It hould b notd that C m vari with tooth qunc and hnc th rlativ rotor-tator poition. For xampl, C m = (tooth facing orth pol) and C m = - (tooth facing outh pol) rprnt th "full contribution mod", a hown in Fig. 3(c). Likwi, C m = i in th flux "cancllation modl" whil othr valu in th "partial contribution mod." hrfor, it i ncary to calculat C m for vry ingl tooth or gmnt no mattr what pol-lot combination i o that th coil flux linkag can b dtrmind. C m can thrfor b writtn a a function of n-th tooth a follow: C ( ) m Cm n (3) whr n rprnt th tooth or gmnt qunc. By () - (3), thi mthod can covr machin with any pol-lot combination. h valu of C m can b obtaind by th following proc and paramtr. tp calculation of lot pitch h lot pitch in lctrical dgr can b writtn a: 36 m (4) whr m i th numbr of pol and i th numbr of lot. tp calculation of rlativ tooth angl h rlativ lot angl θ l (n) to a rfrnc for th n-th tooth/gmnt in lctrical dgr i dfind a a function of tooth/gmnt qunc n. ( n) ( n), n =,, 3 (5) l For implification, only th rmaindr of θ l (n) dividd by an lctrical priod of 36 i ndd, which can b xprd a ( n) m ( n), 36 o (6) r l tp3 magnt polarity aociatd with tooth/gmnt h ign of θ r (n) in (6) i ud to rcogniz th polarity of th magnt. h -pol magnt i dfind a poitiv; whil th -pol magnt ngativ., ( ) 8 ignr ( n) o r n (7), othrwi tp4 rarrangmnt of rlativ tooth angl Bcau θ l (n) ha a priod of 36 E, it can b tratd within th rang of ±8 E, a givn blow: o o o '( n) m ( n) 8, 36 8 (8) l l In ordr to facilitat th intrprtation, θ l '(n) can b obrvd on th polar-coordinat plan. A hown in Fig. 5, it i found that θ l '(n) i rciprocal in th firt- and third-quadrant (and alo in th cond- and fourth-quadrant). hrfor, an arbitrary additiv of ±8 E do not rult in any chang in th calculation of C m. Flux lakag occur. h flux nly lakag littl portion occurrd, of only magnt fw ratio contribut of th magnt to contribut MMF of th main MMF into main loop. loop. (a) Part of magnt A part of th magnt contribut contribut to th MMF MMF of into main th main loop. (b) Bt condition, Magnt th magnt contribut contribut th th mot maximum to MMF MMF of into main th main loop loop. (c) Fig. 3 Flux ditribution varying with th rotor poition: (a) flux cancllation, (b) partial contribution and (c) full contribution

5 5 (n-) th tooth tator yok m n th tooth m (n+) th tooth ignr (), n l '( n) C () () m n ign r n, l '( n) ignr () n, l '( n) l '( n) b a l '( n) b a () otor yok ( n ) n ( n ) Fig. 4 Fatur of magnt within a lot pitch In Fig. 5, th imaginary axi dmontrat th polarity of th magnt-pic within a lot pitch. h tatu of C m can b obtaind by obrving th proction on th ral axi. Morovr, th θ l '(n) can b furthr tratd within th ±9 E rang for convninc. o o l '( n) 8, l '( n) 8 l '( n) o l '( n) 8, l '( n) 8 A hown in Fig. 5 and 6, th two boundari, a and b, of th flux partial contribution mod ar givn a: a b ( ) d ( ) c -pol quadrant ( ) Imag o -pol quadrant n (9) ( ) ( ) () () al ( ) Flux Flux -pol quadrant -pol quadrant cancllation cancllation ancllation ara Full contribution ara Flux canc ara Partial ara Partial ar contribution ara contribution ara Fig. 5 h modifid lot angl θ l '(n) in polar-coordinat plan b a ( ) tp5 Dtrmination of C m (n) Finally, C m (n) varying with th tooth qunc and rotor poition can b xprd a Flux cancllation d Partial contribution c Full contribution b Partial contribution a Flux cancllation Fig. 6 Boundari of th mod on th ral-axi C. Modling of nral PM Machin h abov rotor PM modl i incorporatd into th dvlopd EMCM for th dign of PM lctric machin. h flux producd by th tator winding currnt and rotor PM ar calculatd for ach tator tooth, taking aturation into account. h prmability i updatd according to th calculation rult of th prviou round in an itrativ proc. A circular ntwork i tablihd for th PM machin, whr th flux loop and nodal quation (i.. KCL and KVL) ar mployd to olv th complx iu. ) Prmanc matrix All of th componnt in th magntic circuit ar modld in accordanc with th machin' gomtric configuration and matrial charactritic. h dfinition of th paramtr, givn in Fig. 7, ar bad on th gomtry of a lot for gnral PM machin. thr rlatd paramtr includ th vacuum prmability µ, tack lngth (ffctiv axial lngth) L t, numbr of lot and iron rlativ prmability µ iron. Aumption ar mad prior to th dvlopmnt of th propod modl: (a) h prmability of th iron cor i variabl, a xprd uing th variabl rluctanc hown in Fig. 8. hy ar all individually conidrd. (b) h flux flowing through on tator tooth flow through th air gap uniformly within a lot pitch. (c) h rotor i lotl. A dmontratd in Fig. 7 and 8, th air-gap rluctanc g i dfind a: g g ( r l.5 g) L ro m t n (3) h rluctanc of th n-th tooth t,n, th n-th rotor yok r,n, and th n-th tator yok,n (tator and rotor yok gmntd into th numbr of lot) ar rpctivly xprd a ( rl grro lm.5 Wyok_ ) tn, ( iron_ tn, ) (4) W L iron _, t n t t

6 6 r,n ( iron _ r,n ) (rro Wyok _ r ) iron _ r,n Wyok _ r Lt,n ( iron _,n ), (5) (rl Wyok _ ) iron _,n Wyok _ Lt F, t, t,3 l g It hould b notd that th lakag flux btwn th two adacnt ho travl through th air (Fig. 9) o that it i not rlatd to th ho matrial prmability. It i known that th ho can b aily aturatd and prnt a nonlinar bhavior, but th ffct on th ovrall machin prformanc i minor and hnc nglctd hr. F, l g t, m, m m, m r, r, r, Fig. 8 Machin modling: dfinition for magntic componnt. A full-iz nod-dfinition-map of th 4-lot PM machin i tablihd but not dmontratd hr for brvity. Lt Fring path H ho x x (8) whr x i th rang to which th fringing prmanc xtnd. h rang x i not fixd by any othr gomtric contraint and th xact valu chon i not critical inc th contribution of diffrntial prmanc dcra a x incra. h gap lngth i Wopn in thi ca. Bad on (8), th complt lakag rluctanc of th ho gap l, taking into account all four id, can b modld a.7 Wopn.4 x Wopn (9) l x H L H L ho t ho t t, l (6) It hould b notd that th magnt rmannt flux ϕm,n and othr undfind ymbol o far will b addrd latr. h flux lakag btwn two ho on two adacnt ho i conidrd. A can b n in Fig. 9, th lot opning prmanc btwn th two ho (lot opning width Wopn) i ubct to th fringing ffct o that modification with a corrction factor i ndd to nhanc th modling accuracy. For th flux pattrn in on id of th ho, a hown in Fig. 9, th fring prmanc modl of a latral flat with th aociatd paramtr can b xprd a [5, 33] x H ho.7 Wopn.4 x, F, iron _ r,n (rro.5wyok _ r ) Lt whr μiron_t,n i th rlativ prmability of th n-th tooth, μiron_r,n i that of th n-th rotor yok, and μiron_,n i that of th n-th tator yok. h paramtr may altr during th itration proc and thi will b dtaild latr. h rluctanc of th magnt in th radial dirction i lm (7) m m (rro.5 lm ) Lt fring, Wyok _ r D Wopn Fig. 9 Configuration of a ho: flux pattrn btwn ho (on id hown) h ampr-turn in on lot i rprntd by an MMF ourc. All th magntic componnt ud in th magntic ntwork ar hown in Fig. 8, whr F,n rprnt th MMF in th n-th tooth/gmnt. ot that th rluctanc in (4)-(6) ar numbrd (.g.,, to,) by th gmnt hown in Fig. 8, whil th dfinition of th nod ar givn in Fig.. Any lot machin can b modld and th corrponding EMCM can b contructd onc i dtrmind. In thi tudy, th EMCM i xprd uing a matrix which can b olvd by KCL/KVL with th aid of computr programming. h dtail of th modl ar givn in th following drivation. Firt, all of th paramtr bad on th machin gomtry, matrial, and winding layout, a dicud abov, ar arrangd in a matrix of iz (6-) (6-), dfind a "Prmanc Matrix A" (th iz of th matrix i a function of ), givn a: B A C D (6 ) (6 ) () whr i a (5-) (5-) matrix and function of coniting of ixtn ub-matric (Matric B, C and D will b xplaind latr). Matrix i arrangd a follow: Fig. 7 Dfinition for th gomtry of typical lctric machin 6 ( ) (5 ) (5 ) ()

7 7 h dtail of th ub-matric ( 5 ) containd in Matrix ar givn in th appndix (quation A.-A.), whr all th paramtr can b obtaind from (3) to (9). It i notd that th matric hav diffrnt iz, and th combination of thm will form th Matrix. Matrix B in () i in th form of Boolan function who ign ar dtrmind by th dirction of th MMF ourc du to winding currnt. nrally, Matrix B can b xprd by th following quation: B B B B whr 3 (5 ) () B diag (3) B 3 3 ( ) (4) B [] (5) F F F F F F 3 D F F F F 3 F4 F 4 F 4 F 5 F F 3 F Fig. Machin modling: dfinition for th nod Matrix C i a tranpo matrix of B, whil D i a zro matrix. C B (6) D (7) o contruct th modl, th rlationhip btwn MMF ourc and flux producd nd to b dtrmind. Matrix x i hnc introducd, which i givn a: F x (6 ) (8) whr F indicat th abolut MMF at th nod of th ntwork a dfind in Fig., and ϕ rprnt th flux producd from F. F and ϕ ar rpctivly givn by: (9) 5 (5 ) F F F F (3) h lmnt in matrix ϕ rprnt th ambly of th flux aociatd with th n-th tooth/gmnt, uch a th tooth flux, air-gap flux, tator yok flux and rotor yok flux. h will b dtaild latr. h magnitud of th xtrnal MMF producd from th winding currnt and th magnt can b dfind by a matrix z: (3) 3 4 (6 ) z Z Z Z Z whr Z (3 ) (3) 3 m, m, (33) m, Z Z and whr ϕ m, ϕ m, ar th flux ourc producd from th n-th gmnt of th magnt (addrd in th prviou ction). h quivalnt flux ourc producd from th prmannt magnt for th n-th tooth/gmnt can thn b xprd a mn, r mag m rro.5lm Lt B C ( n) (34) whr τ mag, rprnting, in a pol pitch, th ratio of actual magnt to th pol pitch. hi tak out of account th gap btwn th magnt pol containing no magnt inc thy do not produc flux. Whn C m = or - for a crtain tooth/gmnt, th magnt within that gmnt bcom a fully ddicatd ourc. h MMF ourc F,n producd from th winding currnt in th n-th lot can b xprd a Z 4, which i 4,, (35), Z F F F It hould b notd that th F,n hr (th MMF ourc dtrmind from th winding currnt) and th F (abolut MMF at th nod) in (9) ar diffrnt. Finally, matrix x can b obtaind uing th following quation. x A z (36) whr A - i th invr of Matrix A in (). o dtrmin th matrial oprating point, th flux in variou componnt of th machin i calculatd. h flux flowing through th n-th tooth, ϕ t,n, can b xprd a Fn F n tn,, n,,, tn, F t,, n tn, h flux in th n-th air-gap, ϕ g,n, can b givn by F F (37) n n3 g, n, n,,, (38) g Again, th flux in th n-th tator yok, ϕ,n, i

8 8 For th flux in th n-th rotor yok ϕr,n F4 n Fn 4, n,,, r,n r,n F5 F4, n r,n r,n (4) For th magnt, th flux dnity in th n-th gmnt of magnt Bm,n i givn by Bm,n F3 n Fn 4 rro Lt m, n,,, (4) h magntic intnity in th n-th gmnt of magnt Hm,n i F3 n Fn 4 (4), n,,, lm In th propod EMCM, th ntwork ar tratd a a function of ; thrfor, th approach can b aily xtndd to gnral machin with any numbr of lot and pol. hr i no nd to rbuild th ntwork for ach particular ca. hi approach can alo b xpandd to othr lctric machin with an appropriat modification. H m,n ) Prmability updat for th nonlinar matrial A picwi mthod i ud to dcrib th B-H profil of th nonlinar matrial for th aturation ffct inc th maurd point providd by manufacturr ar limitd. In thi papr, an intrnal-inrtion mthod i mployd to obtain th nw prmability in ach itration proc. Conidr th condition whr Bnw i locatd btwn two concutiv ampling point (Dn and Dn+) of th nonlinar matrial hytri loop, a hown in Fig.. If th ampling rat of th data i ufficint, th curv btwn Dn and Dn+ can b tratd a a traight lin. h Hnw can thrfor b comprhnivly writtn a B Bn (45) H nw H n H n H n nw Bn Bn In ordr to prvnt divrgnc and incra th tability of th itrating proc, th rplacmnt approach (45) i improvd uing a rfrnc to th lat valu of th componnt. For intanc, bfor fding μnw back to rplac th prviou valu, an avrag of μnw and prviou valu μold i takn. Bn Bnw nw A (43) whr ϕnw rprnt th flux and A i a cro-ction whr ϕnw pa through. h Bnw of th componnt acquir a nw magntic intnity Hnw from th oprating point on th B-H profil. h nw rlativ prmability of th componnt bcom B (44) nw nw H nw Finally, th μnw i fd back to th EMCM to rplac th initial valu of th componnt, thu complting on round of th itration. Mor itration i uually ndd to hlp th Dnw Bn Dn Dn Hy t r i D. onlinar Matrial Conidration For nonlinar matrial, th prmability vari with oprating point. hrfor, th prmability of th primary componnt in th EMCM (i.. th tth (4), rotor yok (5), and tator yok (6)) nd to b updatd frquntly by an itrativ proc. ) Itrativ proc of th EMCM An arbitrary but raonabl prmability of 4 i initially givn to th nonlinar matrial in th EMCM to initiat th calculation, a hown in (4)-(6). Aftr th firt round of calculation, a nw condition for th EMCM can b obrvd and th nw flux ϕnw in th primary componnt can b drivd by (38) and (4)-(4). h nw flux dnity Bnw can conquntly b calculatd uing Dn Bnw p (39) proc convrg. h approach to obtain Hnw from th B-H profil and to updat prmability i addrd blow. l oo Fn Fn, n,,,,n,n F, n,n,n Hn H nw H n Magntic intnity (A/m) Magntic intnity (A/m) Fig. B-H curv of nonlinar matrial * h modifid prmability nw can b xprd a * nw nw old (46) o ummariz th itrativ proc, thi approach for a particular rotor poition i dtaild in Fig.. III. FEA AD EXPEIMEAL VEIFICAI wo machin wr ud for dmontration. h firt machin i a 54 kw 4-pol 6-lot machin dignd with th propod EMCM. hi 54 kw PM machin wa dignd for a wind turbin gnrator (non-dirct driv). For uch an application, larg air gap may b rquird. Hnc, th air gap i modifid to two tim th original dign. For th two air gap, th EMCM wa compard with FEA for both th opn circuit and loadd condition. h othr machin i a 35W 6-pol 4-lot machin, which wa vrifid by both FEA and xprimnt. In thi machin, a givn tator cor wa ud for convninc, and othr part (i.., th tator winding and rotor) wr dignd and contructd uing th propod mthod.

9 9 wprotor rotor poition wp poition Initial Initialprmability prmability (4) (4) Dimnional Paramtr paramtr Fig. Dimnional Fig Primary Primary componnt componnt rluctanc rluctanc Effctiv magnt corrction Effctiv magnt factorfactor Cm Cm corrction (3.5.-5) (3)-(7) (3.5.7) (9) Currnt Currnt wavform () (3.4.) wavform (33) - Calculation inofzzand 3 3 ( ) Calculation &ZZ (34) Calculation of in A matrix A () (3.5.8) Calculation Matrix Winding Winding layout layout Calculation Calculation (3.5.45) (35) ofinzz44 (3.5.46) Calculation of in xmatrix matrix x (36) Calculation ( ) Matrialoprating oprating point Matrial point (37)-(4) Matrial Matrial B-H B-Hprofil profil H updating Updat of H (45) (3.5.55) Error Error chck o chck w prmability w prmability (3.5.56) (46) Y o aturation aturation chck chck Y t flux calculation linkag Flux linkag (38) (3.5.47) Accomplihmnt Accomplihmnt Fig. Itrating proc of th EMCM approach ABLE II PECIFICAI F 54 KW MACHIE prating pd (r.p.m.) utput powr (kw) nrator fficincy (%) nrator input powr (kw) quird output torqu (k-m) Currnt dnity (Arm/mm) atd currnt (Arm) lot filld ratio (%) pn-circuit pha voltag (Vrm) atd/loadd pha voltag (Vrm) Lin-to-lin voltag (Vrm) Pha back-emf contant K (V./radM) ABLE III DEI EUL F 54 KW MACHIE Pol numbr 4 Coppr ara (mm) lot numbr 6 Coppr diamtr (mm) umbr of turn pr coil 3 ooth width (mm) Paralll path ooth hight (mm) Air-gap flux dnity ().96 otor yok width (mm) Air-gap (mm).5 tator yok width (mm) Magnt lngth (mm) 6.9 tack lngth (m) Br ().9 otor radiu rro (mm) Magnt Hcb (ka/m) 883 rl (mm) (dfb) 353 tator outr rad. (mm) Hc (ka/m) lot opning (mm) tator innr rad. (mm) 968 ho hight (mm) lot ara (mm) kw Machin with FEA Vrification ) riginal dign h pcification for th propctiv 54 kw gnrator ar dtaild in abl II. h gnrator i xpctd to offr a pha voltag of Vrm at 6 rpm. h dign rult and matrial charactritic ar dtaild in abl III, whil th ampl FEA modl i prntd in Fig. 3. h winding dign in thi tudy follow th approach in [5], which i wll known and not dtaild hr. A % pol pitch i ud to gap th two adacnt magnt, mitigating th flux lakag ffct. h % pol pitch in thi ca i quivalnt to.9 M, which can b ud to calculat Cm in (). h B-H profil of th 5CC3 oft iron [3] i cratd in MALAB uing an itrativ proc. h back-emf contant wavform i obtaind uing th drivation of th flux linkag, which i collctd from th flux within th tth. Equation (38) i ud to calculat th flux linkag with rpct to th winding. h comparion btwn th dvlopd EMCM and FEA i hown in Fig. 4(a) and (b), whr th formr i without magnt kw and th lattr i with kw. h kwd rult ar imply calculatd by uprimpoing thm with th original (a hown in Fig. 4(a)) and hifting (hift.5 M of th wavform in Fig. 4(b)) th wavform. A can b n, th back-emf contant gnratd from th two approach both match th pcification. A. Fig. 3 h FEA modl for th 54 kw machin.

10 ) Dign with largr air gap h larg air-gap dign ha a magnt lngth of 3.8 mm to maintain th magnt oprating point. All th rt paramtr rmain th am. A can b n in Fig. 5, th back-emf contant ar vry imilar for th ca with g=.5 mm (th ratio of air-gap lngth to magntic pol arc i around /8) and that with g=3 mm (th ratio of air-gap lngth to magntic pol arc i around /4). For g=3 mm, th propod EMCM alo ha a clo wavform to th FEA. hi account for th ffctivn of th propod mthod in daling with larg air gap. 3) imulation undr ratd load h machin (for both g=.5 mm and g=3 mm) i loadd with th ratd currnt. h comparion of flux dnity btwn th FEA and th propod EMCM i hown in abl IV, whr th maximum flux dnity on tth i calculatd for opncircuit and loadd condition. A can b n, for all th ca, th EMCM match FEA vry wll. h oprating point of th tooth magntic matrial ha bn wll abov th kn ara of th B-H curv undr ratd load, and thi indicat om aturation. Fig. 6 how th itrativ calculation for tooth oprating point whn th machin (g=.5 mm) i loadd uing th dvlopd EMCM and computr program. h dot on th B-H curv rprnt th oprating point, and th final point (calculation compltd) ha a flux dnity of.7, a indicatd in abl IV. hrfor, th propod EMCM i capabl of rflcting th aturation ffct whn th machin i loadd. It hould b notd that th FEA imulation for th loadd ca giv a torqu of 9.8 k-m and 9. k-m for g=.5 mm and g=3 mm, rpctivly (dignd ratd torqu=9.39 k-m). hi vrifi that th machin can b loadd to th ratd point. B. 35 W Machin with FEA and Exprimnt A dmontratd in Fig. 7(a) and (b), a kw angl of 3.75 M i mployd in thi tudy to rduc th cogging torqu. wo rotor (Ca and Ca ) wr mad for xprimnt, on with kw (Ca ) and th othr without kw (Ca ). h winding can b n in Fig. 8. h pcification and dign dtail for th prototyp machin ar givn in abl V. h back-emf of Ca (kw) ar compard for th rult of th dvlopd EMCM, FEA and xprimnt, a hown in Fig. 9(a), whr th kwd wavform of EMCM ar calculatd by hifting that of Ca. h comparion for Ca (no kw) i givn in Fig. 9(b). ABLE IV CMPAI F FLUX DEIY pn Circuit Loadd g=.5 mm, EMCM g=.5 mm, FEA g=3 mm, EMCM g=3 mm, FEA g=.5 mm, EMCM g=.5 mm, FEA g=3 mm, EMCM g=3 mm, FEA Maximum flux dnity on tth () orqu (k.m) (a) (b) Fig. 4 h comparion of th 54kW machin back-emf contant: (a) with kw and (b) without kw. Fig. 5 h comparion of th 54kW machin back-emf contant btwn FEA and propod modl for original and larg air gap dign (without kw). Final point B-H curv prating point itration (ooth) aturation limit Magntic intnity (A/m) Fig. 6 Calculation proc for oprating point of tooth matrial whn machin loadd. h figur wa gnratd uing a computr program. h aturation limit i ur-dfind.

11 (a) (b) Fig. 7 Fatur of th rotor (magnt includd): (a) with 3.75 M kw (Ca ); (b) without kw (Ca ). h kw for (a) i a tp kw which u two axial rotor ction with an angular offt. dvlopd to work with th modl. h accuracy of th propod modl ha bn vrifid by FEA on a 54 kw machin with variou cnario (diffrnt air gap, with ratd load or unloadd), and by both FEA and xprimnt on a 35 W 6-pol 4-lot prototyp. h capability of obtaining th ntir machin flux pattrn by imultanouly taking all th winding MMF into account ha bn dmontratd. h dvlopd modl i ufficintly accurat for th dign of PM machin without th aid of tim-conuming finit lmnt analyi..6 Back-EMF contant (V-/radM) E MCM ABLE V DEI EUL F 35 W PYPE Machin Paramtr lot numbr 4 tack lngth (mm) Pol numbr 6 otor radiu (mm) lot-fill factor (%) 39.5 nrator outr radiu (mm) Air-gap lngth (mm).5 Magnt lngth (mm) 5 ap btwn magnt (om) ooth width (mm) Pha back EMF contant ri coil pr pha.4 (V*c/radM) lot ara (mm ) atd currnt (dc) aftr.75 lot dpth (mm) rctifir (A) atd pd (r.p.m.) Bar coppr wir dia. (mm) Br ().7-. umbr of turn pr coil Magnt (35H) Hc (ka/m) lot opning (mm) h harmonic of th back EMF in Ca ar analyzd, a hown in Fig.. h EMCM approach can uccfully calculat th 5th harmonic but th 7th and 3th harmonic ar additionally gnratd du to th fact that th magnt pol ar arrangd to fit th lot in th EMCM. inc th magnitud of th 5th harmonic i largr than that of th 7th and 3th, th additional harmonic do not ignificantly affct th main wavform. o vrify th dign, xprimnt wr conductd to load th machin at 5 rpm, rpm, 5 rpm and rpm, rpctivly. Fig. how th output powr and torqu vru output rctifid dc currnt (gnrator). h machin can achiv ratd 35 W output at rpm, and th torqu vru currnt i linar with imilar lop for all th ca. h FEA giv lightly highr output powr which may b du to th xcluion of mchanical and iron lo. vrthl, thi again confirm th propod EMCM. IV. CCLUI An quivalnt magntic circuit modl ha bn propod for th dign and analyi of PM lctric machin with any pollot combination. A prmannt-magnt modl ha bn contructd a a function of lot numbr. ubquntly, a circular modl wa dvlopd to includ all th pol and lot for improving th accuracy. An itrating tchniqu to obtain accurat prmability affctd by xtrnal ourc wa alo FEA. -. Exprimnt otor poition ( E) (a).6 E MCM Back-EMF contant (V-/radM) Fig. 8 h tator winding.4.4. Exprimnt FEA otor poition ( E) (b) Fig. 9 Comparion of Back-EMF contant wavform: (a) Ca (kw) and (b) Ca (without kw) n FEA MCM EMCM Exprimnt K K Harmonic Indx, n Fig. Harmonic analy of Back-EMF contant wavform whr K rprnt th firt harmonic of th wavform. ACKWLEDEME h author ar gratful for th computr tim and faciliti providd by th ational Cntr for High-Prformanc Computing. h author would alo lik to thank Yu-Han Yh and Wan-Yu Wu for thir kind aitanc during th rarch priod.

12 (a) (b) Fig. FEA and xprimntal rult: (a) powr v currnt; (b) torqu v currnt. h dahd lin in (a) indicat th dignd currnt, at which th xprimntal output i lightly lowr than ratd 35 W. APPEDIX h ub-matric in matrix ar givn in th following (A.- A.). (A.) ( ) (A.) (A.3) 6 (A.4) 3 7 ( ) ( ) ( ) t t t (A.5) 3 8 ( ) t t t (A.6) 9 ( ) (A.7) (A.8) 8 (A.9) 4 g g g (A.) 5 6 (A.) 7 9 (A.) 8 4 (A.3) m m m (A.4) (A.5) ( ) (A.6)

13 3 (A.7) l t g l l l l t g l l l l l t( ) g( ) l l r l t g (A.8) (A.9) 9 m g m g m( ) g( ) m g (A.) 5 m r5 r r r r m r r r r r( ) r( ) r g r( ) r r ( ) m g ( ) g (A.) EFEECE []. P. hah, A. D. Hirzl and B. Cho, "ranmiionl lctivly alignd urfac-prmannt-magnt BLDC motor in hybrid lctric vhicl," IEEE ranaction on Indutrial Elctronic, vol. 57, no., pp , Fb.. [] K.I. Lakari and A.. Klada, "Intrnal prmannt magnt motor dign for lctric vhicl driv," IEEE ranaction on Indutrial Elctronic, vol. 57, no., pp Jan.. [3] K.. Chau, C.C. Chan and C. Liu, "vrviw of prmannt-magnt bruhl driv for lctric and hybrid lctric vhicl," IEEE ranaction on Indutrial Elctronic, vol. 55, no. 6, pp , Jun. 8. [4]. Zhang, K. ng, M. Vilathgamuwa and D. guyn, "Dign of a robut grid intrfac ytm for PM-bad wind turbin gnrator," IEEE ranaction on Indutrial Elctronic, Digital bct Idntifir:.9/IE [5] A. Parkr, C. g, and L. an, "Fault tolrant control for a modular gnrator-convrtr chm for dirct driv wind turbin," IEEE ranaction on Indutrial Elctronic, Digital bct Idntifir:.9/IE [6] A. Di rlando,. Foglia,. Prini and M. Iacchtti, "Axial flux PM machin with concntratd armatur winding: dign analyi and tt validation of wind nrgy gnrator," IEEE ranaction on Indutrial Elctronic, Digital bct Idntifir:.9/IE [7] J. opann, V. uukann, J. rg and J. Pyrhonn, "Dynamic torqu analyi of a wind turbin driv train including a dirct-drivn prmannt magnt gnrator," IEEE ranaction on Indutrial Elctronic, Digital bct Idntifir:.9/IE..873.

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How, "hr-dimnional lumpdparamtr magntic circuit analyi of ingl-pha flux-witching prmannt-magnt motor," IEEE ranaction on Indutry Application, vol. 44, no. 6, pp. 7-7, ov./dc. 8..H. Han,.M. Jahn and W.L. oong,"a magntic circuit modl for an IPM ynchronou machin incorporating moving airgap and crocoupld aturation ffct," IEEE Intrnational Elctric Machin and Driv Confrnc, Antalya, urky, May 3-5, 7. Charactritic Curv of Magntic Proprti of Magntic tl Cor. Kaohiung, aiwan: China tl Corp., 4. F.W. Cartr, "h magntic fild of th dynamic-lctric machin," Journal IEE, vol.64, pp.5, 96. Min-Fu Hih (M') wa born in ainan, aiwan in 968. H rcivd th BEng dgr in mchanical nginring from ational Chng Kung Univrity (CKU), ainan, aiwan in 99. H thn rcivd th Mc and PhD dgr in mchanical nginring from th Univrity of Livrpool, UK in 996 and, rpctivly. From to 3, h rvd a a rarchr in th Elctric Motor chnology arch Cntr at CKU. In 3, h oind th Dpartmnt of ytm and aval Mchatronic Enginring, CKU a an aitant profor. Hi ara of intrt includ rnwabl nrgy gnration (wav, tidal currnt and wind), lctric propulor, rvo control, and lctric machin dign. In 7, h wa promotd to aociat profor. Dr. Hih i a mmbr of th IEEE Magntic, Indutrial Elctronic, canic Enginring, and Indutrial Application ociti. You-Chiuan Hu rcivd hi BEng and MEng dgr from th Dpartmnt of ytm and aval Mchatronic Enginring of ational Chng Kung Univrity, ainan, aiwan in 4 and 6, rpctivly. H i currntly working toward th PhD dgr in th am dpartmnt. In 9, h pnt ight month in th Dpartmnt of Enginring of Univrity of Cambridg, UK, a an official viitor. Hi currnt rarch intrt includ wind gnrator ytm, prmannt magnt gnrator/motor dign, powr lctronic, lctric machin drivr dign and pot-ambly magntization mthod. H ha ovr 5 publication.