enegies Aticle Rapid Optimiatin Duble-Stats Switched Reluctance Mt with Equivalent Magnetic Cicuit Wu-Sung Ya ID Depatment Mechanical Autmatin Engineeing, Natinal Kahung Ft Univet Science Technlg, N., Univet Rd., Yancha Dt., Kahung Cit 8, Taiwan; wsa@nkfust.edu.tw; Tel.: +886-7-6 Academic Edit: Fede Blaabjeg Received: 8 Septem 7; Accepted: Oct 7; Publhed: Oct 7 Abstact: The pima bjective f th pape t ceate a methdlg t apidl ptimie duble-stats switched eluctance mt (). An analtical mdel equivalent magnetic cicuits f ai gap eluctances aligned ptins ppsed ptimal peatin pint magnet-mtive fce (MMF) detemined. Genetic algithm (GA) integated ppsed equivalent magnetic cicuit develped f apid ptimiatin t each maimum ati tque t vlume. Cmpaed t cnventinal switched eluctance mt (SRM), an illustated eample a -KW thee-phase -Slt-8-Ple used t veif efficienc ppsed methd. Simplified -D electmagnetic mdels ae analed mulated. Finall, esults analtical calculatins finite-element anals (FEA) ae validated b ppsed mt t shw accuac degned stateg. Kewds: apidl ptimie; duble-stats switched eluctance mt; equivalent magnetic cicuit; electmagnetics anals; genetic algithm; finite-element anals. Intductin In ecent eas, switched eluctance mts (SRMs), as shwn in Figue a, have en used f vaius applicatins such as aespace indust, maine ppuln sstems, linea dives, mining dives, hheld tls, hme utilities, etc. The main easns f ung SRMs in se applicatins ae lw-cst cnstuctin chaacteied b an absence magnets t winding, fault tleant pwe stage degn, high level pefmance ve wide ange speeds. In geneal, SRMs have dawbacks vibatin acustic nes t limit i applicatins. T incease pwe dent SRM equies smalle ai gap t eal fce mt int a highl satuated mde. The maj nefits duble-stats switched eluctance mt (), as shwn in Figue b, in cmpan with cnventinal SRM ae highe tque/pwe dent; highe fce cnven efficienc; highe mtinal fce with lwe adial fce; lage effective ai gap eluctance at ptin. Due t duble-stat mechanical degn, degn cndeatin me cmplicated than that SRM. Theefe, sme studies n degning anals fund ecentl. In [], based n dsecting int inne ute t SRMs, a degn stateg cncentating n pwe ating was ppsed t calculate dimenns. Bth inne ute t SRMs degned sepaatel. In [], an analtical mdel f inductance SRM was develped. Bth cntibutin flu in stat slt cntibutin flu in t slt t inductance wee cndeed. In [], stuctual havi a a cnventinal SRM ung a multi-phc anals was cmpaed. An electmagnetic finite-element methd used t calculate fce dent at vaius pats stat suface in bth machines. In [], mal mdeling anals wee Enegies 7,, 6; di:.9/en6 www.mdpi.cm/junal/enegies
Enegies 7,, 6 Enegies 7,, 6 pesented. A thee-dimennal finite-element methd (FEM) has en used t numeicall calculate tempeatue dtibutin in diffeent pats machine. In [5], degn methd based n ptimiatin mtinal fces, which leads t a high-gade electmechanical eneg cnven pcess. A lcal eaminatin fce denties within thughut a cnventinal SRM shws that majit fce pduced in in adial diectin des des nt nt cntibute t t mtin. In In [6], [6], a a degn ptimiatin a duble-stat switched eluctance machine was ppsed f impving magnetic tque qualit machine. F th pupse, finite element methd alng with espnse suface methdlg was used t ptimie thee paametes machine t maimie tque qualit fact i.e., i.e., aveage tque tque t tque tque ipple ipple ati ati in in machine. machine. In [7], In [7], an effective an effective t t shaping shaping ptimiatin technique f f tque tque ipple ipple eductin was waspesented. Th methd leads t lwe tque pulsatin withut gnifit eductin in aveage tque. has a nvel, ptimied patten magnetic flu paths within electical machine, which ields supei pefmance. As seen in eting studies, tque dent twice that SRM. Duble stats (ute stat inne stat) used t educe adial fce ne duing peatin. The shell tpe t esults in damaticall smalle inetia. Theefe, maintains fault tleance wide speed ange capabilit. Fm efeences mentined abve, numeical slutin FEM applied t vaius chaactetic analses adpted in mst studies, while e ae fewe details ai gap eluctances aligned ptins. Theefe, th th pape pape pesents pesents a mamatic a mamatic mdeling mdeling with equivalent with equivalent magnetic cicuit magnetic t pecel cicuit t btain pecel btain magnetic flu magnetic linkages flu linkages aligned aligned ptins. ptins. Th mdeling Th mdeling used t pvide used t taget pvide equatin taget equatin GA f apidl GA f ptimiatin apidl ptimiatin.. Rt Stat Inne stat Oute stat Segmented t (a) (b) Figue. Piles : (a) switched eluctance mt; (b) duble-stats switched eluctance mt. Figue. Piles : (a) switched eluctance mt; (b) duble-stats switched eluctance mt. Th pape ganied as fllws. In Sectin, equivalent magnetic cicuit cnstucted Th pape t anale ganied ai as gap fllws. flu In linkages Sectin, aligned equivalent magnetic cicuit ptins, cnstucted ptimal peatin t anale pint ai gap flu MMF linkages detemined. aligned Optimal degn ptins, with ptimal GA, including peatin pint ce mateial, MMF winding detemined. setting, Optimal machine degn aangement, with GA, pefmed including in Sectin ce mateial,. The winding mplified setting, D electmagnetic machine aangement, mdels ae analed pefmed in mulated Sectin. in The Sectin mplified. Finall, D electmagnetic cncluns mdels ae dawn ae in analed Sectin 5. mulated in Sectin. Finall, cncluns ae dawn in Sectin 5.. Equivalent Magnetic Cicuit Anals. Equivalent Magnetic Cicuit Anals a nvel SRM that degned t pefm at high tque levels. The css sectin a nvel SRM that degned t pefm at high tque levels. The css sectin a a thee-phase -Slt-8-Ple illustated in Figue. nefits fm tw stats, thee-phase -Slt-8-Ple illustated in Figue. nefits fm tw stats, which which ae made laminated femagnetic mateial ae equipped with cncentated windings. ae made laminated femagnetic mateial ae equipped with cncentated windings. The The ae lcated n intei etei a clindical t. The t fmed b segments ae lcated n intei etei a clindical t. The t fmed b segments laminated femagnetic mateial, which ae held tge ung a nn-femagnetic cage. laminated femagnetic mateial, which ae held tge ung nn-femagnetic cage. The electmagnetic tque SRM based n inductance vaiatin stat winding, The electmagnetic tque SRM based n inductance vaiatin stat winding, which a nnlinea functin stat cuent t ptin. Pacticall, inductance which nnlinea functin stat cuent t ptin. Pacticall, inductance stat winding calculated b measuing flu linkage. Hweve, magnetic flu linkage
Enegies 7,, 6 Enegies 7,, 6 Enegies 7,, 6 stat winding difficult t measue diectl in case. Theefe, th pape stat eplits winding equivalent magnetic calculated cicuit b measuing ai gap t btain flu linkage. flu Hweve, linkage cuve, magnetic which flu a functin linkage stat winding difficult t measue diectl case. Theefe, th pape t stat ptin. winding In geneal, difficult SRM t measue a dubl diectl salient bushless case. mt, Theefe, with cncentated th pape eplits cppe eplits equivalent magnetic cicuit ai gap t btain flu linkage cuve, which a functin equivalent windings n magnetic stat cicuit ples, ai n gap windings t btain magnets flu linkage n cuve, t ples. which Each a functin phase wund t ptin. In geneal, SRM a dubl salient bushless mt, with cncentated cppe t ptin. n diameticall In geneal, ppte SRM a stat dubl ples. salient Ecitatin bushless mt, a phase, with cncentated that, ecitatin cppe windings a pai n windings n stat ples, n windings magnets n t ples. Each phase wund diameticall stat ples, ppte n windings stat ples, magnets causes neaest t pai ples. Each t phase ples n diameticall ppte stat ples. Ecitatin a phase, that, t ecitatin wund align with n diameticall ecited a pai diameticall ppte stat ples. stat Th ppte ples. pduces Ecitatin tque stat ples, a egadless phase, causes that neaest, ecitatin diectin pai at pai ples cuent diameticall in phase t align with ppte winding. ecited stat stat ples, Theefe, causes nl ples. Th unipla neaest pduces pai cuents tque t ae egadless ples equied t align in stat with diectin phases, ecited stat sequential cuent ples. in Th ecitatin pduces based phase winding. tque n Theefe, egadless t ptin nl causes unipla diectin cuents t t cuent tate ae equied in phase align its in stat winding. ples with phases, Theefe, ecited sequential nl unipla stat ples. ecitatin cuents In based ae n t equied fllwing, ptin in stat magnetic causes phases, flu linkages t sequential t tate ecitatin aligned align based its ples n t ptins with ptin ecited causes ae analed, stat t ples. tin tate i fllwing, equivalent align its magnetic ples magnetic with cicuits flu ecited ae deived. linkages stat ples. aligned In fllwing, ptins magneticae fluanaled, linkages aligned i equivalent magnetic ptins cicuits ae analed, ae deived. i equivalent magnetic cicuits ae deived. Figue. Magnetic flu ngle phase t align with ecited stat ple... Calculated Figue Magnetic. Magnetic Flu flu Linkage ngle Aligned phase Ptin t align with ecited stat ple... Calculated Figue shws Magnetic Flu magnetic Linkage flu Aligned ngle Ptin phase t align with ecited stat ple. The equivalent magnetic cicuit Figue descid in Figue, whee R s ce Figue shws magnetic flu ngle phase t align with ecited stat eluctance The equivalent ute magnetic stat; R cicuit ce eluctance Figue descid inne stat; in Figue R, ce whee eluctance ple. The equivalent magnetic cicuit Figue descid in Figue, whee R R s ce s eluctance ce eluctance t; R ute g eluctance stat; ute R stat; ute ce eluctance R ai ce gap; eluctance R gi eluctance inne stat; inne R stat; inne ce eluctance R ai gap; ce F eluctance MMF t; R g t; eluctance ute stat; R g eluctance F i ute MMF ai gap; ute Rinne gi ai eluctance stat; gap; R gi φ eluctance, inne φ, φ ai, gap; φ F inne ae MMF ai magnetic gap; F ute flues MMF stat; each F i ute MMF lp. stat; inne F i stat; MMF φ inne, φ, φstat;, φ aeφ, φ magnetic, φ, flues φ ae eachmagnetic lp. flues each lp. F R g R s F R g R s R s φ R gi φ R s F i F R g F R g R R R s φ R gi R φ F i R R gi F i R R gi R R F F i i R R gi R R φ F i φ gi R R F i gi φ F i φ R gi R R R s Rg F Rg F R s F R g R s R g F Figue. Equivalent magnetic cicuit Figue. Figue. Equivalent magnetic cicuit Figue. In Figue, MMF equatins each lp φ, φ, φ, φ given as, espectivel, In Figue, MMF equatins each lp φ, φ, φ, φ given as, espectivel,
Enegies 7, 7,,, 6 6 F In Figue, MMF equatins + F i = R each + R lps φ φ, + φ, φ, φ given as, espectivel, () ( Rg + Rgi + R )( φ+ φ) + ( Rg + Rgi + R )( φ+ φ) ( (F + F i ) = (R + R s )φ + Rg + R ( F gi R (φ + φ ) + ( R g + R gi + R ) (φ + φ ) () + Fi ) = ( R + Rs ) φ + () ( Rg + Rgi + R )( φ + φ) + ( Rg + Rgi + R )( φ+ φ) (F + F i ) = (R + R s )φ + ( Rg + R gi R (φ + φ ) + () ( F + Fi) = ( R + Rs) φr+ g + R gi + R (φ + φ ) () ( Rg + Rgi + R )( φ + φ) + ( Rg + Rgi + R )( φ + φ) (F + F i ) = (R + R s )φ + ( Rg + R gi R (φ + φ ) + () ( F R g + R gi + R (φ + φ ) + Fi ) = ( R + Rs ) φ + () ( Rg + R (F gi + R + )( F φ i ) + = φ(r ) + ( + R s )φ + Rg + R gi + R (φ + φ ) + ( Rg + Rgi + R )( φ + φ) ) () Assuming φ R g + R gi + R (φ + φ ) = φ= φ= φ= φ, Equatins () () integated as Assuming φ = φ = φ = φ = φ, Equatins () () integated as F + F = R + R φ + R + R + R φ (5) ( i ) ( s ) ( g gi ) (F + F i ) = (R + R s )φ + R g + R gi + R φ (5) The pemeance dtibutin f t t align its ples with ecited stat ples shwn in Figue, whee W The pemeance dtibutin s salient ple width ute stat; W f t t align its ples with salient ple width inne ecited stat ples shwn instat; FigueW, whee blck W s t salient width; ple width g ute ai gap stat; width. W Cndeing salient plefinging width effect, inne stat; pemeances, W blck i.e., t P a width; P ai, g t ai gap t align width. its Cndeing ples with finging ute effect, inne stat pemeances, ples i.e., given P a as, Pespectivel, ai, t t align its ples with ute inne stat ples given as, espectivel, P a = µ μ As Pa = A s (6) (6) g g P ai = µ μ A A Pai = (7) gg whee µ μ vacuum vacuum pemeabilit cefficient; A s A s A aea css ae sectins css sectins salient salient ples ples ute stat ute stat inne stat, inne espectivel. stat, espectivel. Oute stat W s d P a _ f P a g P a _ f h W Rt P ai _ f P ai P ai _ f W Inne stat Figue.. Pemeance dtibutin f f t t t t align align its its ples ples with with ecited stat stat ples. ples. Due t elative pemeabilit cefficient stat ce ing finite, f magnetic flu pasng Due t elative pemeabilit cefficient stat ce ing finite, f magnetic flu pasng thugh ai gap b ai gap width epann, finging effect shuld cndeed. The ciculaac staight-line pemeance mdel ten used f mdeling flu flw in ai gap [8]. thugh ai gap b ai gap width epann, finging effect shuld cndeed. The cicula-ac staight-line pemeance mdel ten used f mdeling flu flw in ai gap [8]. Theefe, t calculate elative pemeabilit cefficient stat ce unde finging effect, th Theefe, t calculate elative pemeabilit cefficient stat ce unde finging effect, th pape pape uses cicula-ac staight-line mdeling t calculate pemeances ( P a _ f P ai _ f ) inne ute stat ces, espectivel, i.e.,
Enegies 7,, 6 5 ( W W )/ s uses cicula-ac staight-line mdeling tμcalculate Lstk μpemeances L π W(P W a_ f s stk P P ai_ f ) inne ute a _ f = d= ln stat ces, espectivel, i.e., π + g π g (8) + (W W s )/ µ P a_ f = L stk g + π d µ [ stk ln + π(w ] ( W W) / μ W s ) Lstk μl π ( W W) stk P (8) ai _ f = d= ln π + g g π g (9) + (W W )/ µ P ai_ f = L stk g + π d = µ [ L stk ln + π(w ] whee L stk thickness ciled licn steel sheet. W ) (9) Due t pemeance dtibutin inne ute π ai gaps ing g paallel cnnectin tpe, eluctances ( R g R gi ) ute inne ai-gaps btained b, espectivel, whee L stk thickness ciled licn steel sheet. Due t pemeance dtibutin inne ute ai gaps ing paallel cnnectin tpe, Rg = eluctances (R g R gi ) ute inne ai-gaps P + P () btained b, espectivel, a _ f a R R g gi = P P () a_ ai _ f + P () ai a The ce eluctances ( R s, R, R ) ute inne stats t btained as, R gi = () espectivel, P ai_ f + P ai The ce eluctances (R s, R, R ) ute ls inne stats t btained Rs = () as, espectivel, μ A R s = µa l () R s = () μ R = l A () µa l R = R = l () μ A () µa whee μ pemeabilit cefficient ce; l s, l, l ae magnetic flu lengths ute stat, whee inne stat, µ pemeabilit t, espectivel; cefficient ce; A l s, l, l ae magnetic flu lengths ute css-sectinal aea segmented t. Then, stat, Equatins inne () () stat, t, substituted espectivel; int Equatin A (5) t calculate css-sectinal magnetic aea linkage segmented ai gap. t. T Then, calculate Equatins MMF () () at align ptin, substituted a magnetic int cicuit Equatin with (5) ngle t calculate ecited magnetic phase given linkage as shwn ai gap. in T Figue calculate 5. Then, MMF ttal at align MMF ptin, a magnetic cicuit with ngle ecited phase given as shwn F t given as: in Figue 5. Then, ttal MMF F t given as: Ft = φ( Rce + Rg) (5) F t = φ(r ce + R g ) (5) whee R ce ttal eluctance in ce R g ttal eluctance ai gap. Then, magnetic whee ptential R ce dp ttal F eluctance in ce R g ttal eluctance ai gap. Then, magnetic ce given b: ptential dp F ce given b: F Rce F ce = F t (6) R t l s s t F t φ R ce R g Figue 5. Magnetic cicuit with ngle ecited phase. Figue 5. Magnetic cicuit with ngle ecited phase.
Enegies Enegies 7, 7,,, 6 6 6 In additin, magnetic ptential dp btained b magnetic field intent fm In additin, magnetic ptential dp btained b magnetic field intent fm mateial B-H cuve multiplied b length magnetic path in ce, i.e., mateial B-H cuve multiplied b length magnetic path in ce, i.e., Fce = Hcel Enegies 7,, 6 ce 6 (7) F ce = H ce l ce (7) Fm Equatins (6) (7), we have In additin, magnetic ptential dp btained b magnetic field intent fm Fm mateial Equatins B-H cuve (6) multiplied (7), b we have length magnetic H celcer path in ce, i.e., t Ft = (8) F F t = H R ce = H l cel cer t (7) (8) Fm Equatins (6) (7), we have R ce H F t celce aigr grrt F t = t t (8) R (9) Rce g Figue 6 used t veif accuac analtical slutin (9) t cmpae numeical slutin FEM. In Figue 6, lage MMF Hai ange grt F clse t magnetic satuatin aea t = (9) stat unde aligned ptin. Theefe, sme R dcepanc g in lage MMF ange Figue 6 fund. Figue In 6 th case, used t ptimal veif MMF accuac ppsed analtical slutin degned (9) t as cmpae abut 5 AT, which numeical lcated slutin at nn-magnetic FEM. In Figue satuatin 6, lage aea. MMF Theefe, ange clse ppsed t magnetic equivalent satuatin magnetic aea cicuit mdel stat Figue unde aligned used ptin. t accuac Theefe, degn sme. dcepanc in lage MMF ange Figue 6 fund. In th case, ptimal MMF ppsed degned as abut 5 AT, which lcated at nn-magnetic.satuatin aea. Theefe, ppsed equivalent magnetic cicuit - mdel Figue used t accuac degn...8 Flu linkage(wb) Flu linkage(wb).6... -..8.6 FEM EMCM FEM EMCM. 5 5 5 5 5. magnet-mtive fce(at) 5 5 5 5 5 Figue 6. 6. Cmpan flu linkage at at aligned magnet-mtive ptin fce(at) tween analtical slutin numeical slutin Figue finite-element 6. Cmpan methd flu linkage (FEM). at aligned ptin tween analtical slutin numeical slutin finite-element methd (FEM)..... Calculated Calculated Magnetic Magnetic Flu Flu Linkage Linkage Unaligned Unaligned Ptin Ptin.. Figue Calculated shws Magnetic magnetic Flu Linkage flu Unaligned path Ptin Figue 7 shws magnetic flu path ngle ngle phase phase at at ptin. ptin. It It fund Figue fund that 7 that shws ai ai gap gap magnetic width width flu length length path ngle phase ptin ptin ae ae lage at lage than than that ptin. that aligned aligned It ptin ptin fund iegula iegula that ai shape shape gap width ai ai gap gap length als als fund. fund. ptin Theefe, Theefe, ae lage it it difficult difficult than that t t calculate calculate aligned ptin eluctance. eluctance. iegula In In th shape th wk, ai wk, cndeing cndeing gap als supeptin supeptin fund. Theefe, it linea linea difficult sstem, sstem, t calculate magnetic magnetic linkages linkages geneated geneated eluctance. b b ute ute In th wk, inne inne cndeing ecited ecited stat stat supeptin ae ae analed analed sepaatel, sepaatel, linea sstem, n, n, magnetic esults esults linkages geneated b ute inne ecited stat ae analed sepaatel, n, esults meged. meged. meged. Figue 7. Magnetic flu ngle phase t with ecited stat ple. Figue Figue 7. 7. Magnetic Magnetic flu flu ngle ngle phase phase t t with with ecited ecited stat stat ple. ple.
Enegies 7,, 6 7 Enegies 7,, 6 7 Cndeing finging effect, cmpletel ptin tween stats t shwn Cndeing finging effect, cmpletel ptin tween stats t t shwn shwn in in Figue 8, whee W a t slt width; P ni _ f finging pemeance ai gap inne stat; Figue in Figue 8, whee 8, whee W a W t slt width; P ni_ f finging pemeance ai gap inne stat; P n_ f a t slt width; P ni _ f finging pemeance ai gap inne stat; finging P n _ f finging pemeance pemeance ai gap ai ute gap stat; ute P stat; na pemeance P na pemeance ai gap t ai gap slt; P t ni pemeance slt; P ni P n _ f finging pemeance ai gap ute stat; P na pemeance ai gap t slt; P ni pemeance ai gap tween ai gap inne tween stat inne t stat slt; t P n slt; pemeance P n pemeance ai gap tween ai gap ute tween stat pemeance ai gap tween inne stat t slt; P n pemeance ai gap tween ute t stat slt. t slt. ute stat t slt. W s W s Oute stat Oute stat P n_ f P n P n _ f P wn _ f P n wp n _ f w w P na h P na h w i P ni w_ if W a W a P ni w i P ni w_ if Rt Rt P ni _ f P ni P ni _ f W W Inne stat Inne stat Figue 8. Pemeances tween stat t at cmpletel ptin. Figue 8. Pemeances tween stat t at cmpletel ptin. Cmpaed t pemeabilit cefficient ai gap at cmpletel ptin, pemeabilit Cmpaed t cefficient pemeabilit μ cefficient in ce in Figue 8 ai ai gap gap at cmpletel assumed at as infinite, ptin, magnetic ptin, pemeabilit cefficient cefficient μ µ in ince cein infigue 88 assumed as infinite, magnetic ptential dp in ce appach e. Theefe, we have bunda cnditin ptential dp in ce appach e. Theefe, we have bunda cnditin magnetic field intent t slts as shwn in Figue 9. Assume that bunda cnditin magnetic field intent t slts as as shwn in in Figue 9. 9. Assume that that bunda cnditin t slt des nt include eciting cuent. Then, we have Laplacian peat A as: t t slt slt des des nt nt include include eciting eciting cuent. cuent. Then, Then, we we have have Laplacian peat peat A as: A as: A A A A = + + = () A = A A A AA A = + + = = () whee A t slt magnetic ptential diectin. whee A t t slt slt magnetic ptential diectin. h h (, ) = (, ) = H h f H h f H (, ) = H (, ) = H, Wa = H, Wa = H (,) W = g a H W (,) = g( ) a Figue 9. Bunda cnditin magnetic field intent t slts. Figue 9. Bunda cnditin magnetic field intent t slts. Hee, t anale bunda cnditins magnetic field, intent ute inne Hee, t anale bunda cnditins magnetic field, intent ute inne stats Hee, t anale used t bunda calculate cnditins flu linkage magnetic t slt. field, intent ute inne stats stats used t calculate flu linkage t slt. used t calculate flu linkage t slt... Calculated Flu Linkage Oute Stat Ecited.. Calculated Flu Linkage Oute Stat Ecited The bunda cnditin magnetic field intent ute in. The magnetic The bunda ptential cnditin diectin magnetic given field as: intent ute stat shwnin in Figue. The magnetic ptential diectin given as: as: nπ nπ A (, ) = An csh ( nπ) cs( nπ ) () A (, ) = n = An cshw nπ a cswa nπ A () (, ) = A n= n csh cs a W () W a n= a W a whee A n detemined b magnetic field intent dtibutin bunda at = h, whee A n detemined b magnetic field intent dtibutin bunda at = h, given as Equatin () Figue, i.e., given as Equatin () Figue, i.e.,
Enegies 7,, 6 8 Enegies Enegies 7, 7,,, 6 6 8 whee A n detemined b magnetic field intent dtibutin bunda at = h, given as Equatin () Figue,, NI i.e., H (, h ) = < < w w H (, h ) N I H, w < (, h ) = w < < Ws + w () (, h = w < s w (), H (, h ) = N ( ) NI H, I s s h = w W W s + sw + w < < < W s + w + + w w w whee N winding tuns tuns ute ute stat stat I I input input cuent. cuent. ( ) h A,, = (, ) = H h f A,, Wa = ( ) A,, = W a a Figue. Bunda cnditin magnetic field intent ute stat. H NI NI w NI w ( h,, ) w W W + w + w s s + w Ws + w + w Figue. Epesn magnetic field intent H H (, (,, h ).. Due t windings stat ing epeatabilit, Equatin () epessed as a peidic functin, i.e., nπ H (, h ) =, n nπ H n (, h ) = a n cs () a cs W n= n a n= Wa () whee whee a n = W a nπ H (, h ) cs d = N I W a W a n, d a W a nπ S () nπ NI a (, ) cs a a n = H h d= S W a W ( a n ) π () S = n nπw W a + n nπ(wa w ) W a (5) w w a The mamatic elatin magnetic n nπw n nπ(w a w) n flu dent n magnetic ptential vect given as: Wa Wa S (5) = + ( w H = w ) A (6) µ The mamatic elatin magnetic flu dent magnetic ptential vect given as: as: whee A defined as cul peat A. Theefe, fm Equatin (6), magnetic H = field ( A) μ intent btained as: (6) as whee [ A defined as cul peat ] nπ nπ A.. nπ nπ nπ H(, ) = Theefe, fm A n nh cs î+ csh n ĵ Equatin µ (6), magnetic field intent btained as: n= W a W a W a W a as: W a (7)
H (, ) = An nh cs i+ csh n j n= μwa Wa Wa Wa Wa (7) As = h, Equatin () equal t (7), thus we have NIW aμs Enegies 7,, 6 An = 9 nπ h (8) n π nh Wa As = h, Equatin () equal t (7), thus we have Fm Equatins (7) (8), magnetic flu dent t flw thugh ce suface veticall given as: A n = N IW a µ S ) (8) n π nh( nπh NI W μs a nπ nπ B (, ) = csh n n= nπ h Wa W a Fm Equatins (7) (8), magnetic nπ nh flu dent t flw thugh (9) ce suface veticall given as: Wa B integating Equatin N Iµ B (, ) (9) = alng diectin, S nπ nπ ) csh magnetic flu n btained as: (9) n= nπnh( nπh W a W a Lstk Wa W a μniw alstks φ= B integating Equatin (9) B(, h) dd = alng diectin, magnetic flu btained as: n=..5... nπ h () n π tanh L stk W a Wa µ φ = B (, h )dd = N IW a L stk S ) () n=..5... n π tanh( nπh.. Calculated Flu Linkage Inne Ecited Stat W a.. Calculated The bunda Flu Linkage cnditin Inne Ecited magnetic Stat field intent inne stat shwn in Figue. The magnetic ptential diectin given as: The bunda cnditin magnetic field intent inne stat shwn in Figue. n nπ h π nπ The magnetic ptential diectin given nas: π Wa Wa Wa Ai (, ) = Bn cs [ e + e e ] n= ( Wa )[ () ] nπ A i (, ) = B n cs e nπ Wa + e nπh Wa e nπ Wa B () n detemined b magnetic field intent W dtibutin bunda at =, which n= a given as Equatin () shwn in Figue, i.e., B n detemined b magnetic field intent dtibutin bunda at =, which NI i given as Equatin () Hi (, shwn ) = in Figue <, < i.e., w w H (, N i I Hi (,) = w < w < < Ws + < w () i (, ) = NI w < s w () (,) H i (, ) = N i H i = i I Ws + w < < Ws + w+ w w w W s + w < < W s + w + w whee N i i winding winding tuns tuns inne inne stat. stat. h ( h ) A, = A (, ) = A, Wa = H, = g( ) W a Figue. Bunda cnditin magnetic field intent inne stat. Figue. Bunda cnditin magnetic field intent inne stat. Enegies 7,, 6 H i (,) NiI w i w i W + wi W + wi + wi NiI w Figue. Epesn magnetic field intent H i (, (,).). Accding t Figue, Equatin () epessed as a peidic functin, i.e., nπ Hi (,) = bn cs n= Wa () whee i W a
Enegies 7,, 6 Accding t Figue, Equatin () epessed as a peidic functin, i.e., nπ H i (, ) = b n cs W n= a () whee b n = W a H W i (, ) cs a ( nπ W a ) d = N i I nπ S i () S i = n nπwi W a + n nπ(wa w i ) W a (5) w i w i Theefe, fm Equatin (6), magnetic field intent btained as: H(, ) = n= B n nπ µ W a [cs nπ W a e nπ Wa e nπh Wa e nπ Wa î + ) ] (6) Wa + e nπh Wa e nπ Wa ĵ n( nπ W a ) ( e nπ As =, Equatin () equal t (6), thus we have B n = N i IW a µ S ( i ) (7) n π e nπh Wa Fm Equatins (6) (7), magnetic flu dent vetical thugh ce suface given as: ( ) N B i (, ) = i Iµ S ( i nπ ) n e nπ Wa + e nπh Wa e nπ Wa (8) n= nπ e nπh W a Wa B integating Equatin (8) alng diectin, magnetic flu btained as: φ i = L stk W a B i (, )dd = µ N i IW a L stk S i + e nπh Wa (9) n=..5... n π e nπh Wa Based n Equatins (9) (), pemeance P na ai gap given as: P na = φ + φ i I(N i + N ) = n=..5... µ W a L stk n π (N i + N ) N S ) + tanh( nπh W a N i S i + e nπh Wa e nπh Wa () The pemeances ai gap tween stat t slt P ni P n given as: P n = µ A s g P ni = µ A g () ()
P μ A g s n = () μ A Pni = () Enegies 7,, 6 g The finging pemeance ai gap inne stat P ni _ f ute stat n _ f P given as: The finging pemeance ai gap inne stat P ni_ f ute stat P n_ f given as: w μlstk μlstk πw Pn _ f = w d= ln µ P n_ f = L π stk g + π d = µ ( L stk ln + πw ) g π + g () + () π g w i wi μ µ P ni_ f = L stk g + d µ ( L stk ln + πw ) Lstk μlstk πwi Pni _ f = d= ln i π g π + g g () + The equivalent magnetic cicuit cmpletel ptin tween stat t in The equivalent magnetic cicuit cmpletel ptin tween stat t in Figue dawn as Figue, whee ttal eluctance R nt given as: Figue 8 dawn as Figue, whee ttal eluctance R nt given as: R nt = + (5) Rnt = P n_ f + P n + P ni_ f + P ni + P P + P P + P P na (5) n _ f n ni _ f ni na F P n _ f P n P n _ f P na P ni _ f P ni P ni _ f F i Figue. Equivalent magnetic cicuit cmpletel ptin tween stat t in Figue. Equivalent magnetic cicuit cmpletel ptin tween stat t in Figue 8. Figue 8. Ttal magnetic ptential fm inne ute ecited stats btained as: Ttal magnetic ptential fm inne ute ecited stats btained as: nπ nπ h nπ nπ nπ nπ Wa Wa Wa A t = An csh cs + Bn cs ( )( e + e e ) nπ = W nπ a Wa W nπ a (6) A t = A n csh cs +B n cs e nπ Wa + e nπh Wa e nπ Wa (6) n= W a W a Figues 5 6 used t veif accuac analtical slutin Equatin (6) t cmpae Figues 5 numeical 6 slutin used fm t veif FEM. accuac analtical slutin Equatin (6) Enegies t cmpae 7,, 6 numeical slutin fm FEM. W a Analtical slutin Numeical slutin Figue 5. Cmpan D ttal magnetic ptential vect tween analtical slutin numeical slutin fm FEM. -.9.8.7 b)
Enegies 7, Figue, 5. 6Cmpan D ttal magnetic ptential vect tween analtical slutin numeical slutin fm FEM. -.9.8.7 Flu linkage(wb).6.5.. EMCM FEM.. 5 5 5 5 5 magnet-mtive fce(at) Figue Figue 6. 6. Cmpan Cmpan flu flu linkages linkages at at cmpletel cmpletel ptin ptin tween tween analtical analtical slutin slutin numeical slutin fm FEM. numeical slutin fm FEM..5. Optimal Opeatin Pint MMF.5. Optimal Opeatin Pint MMF The electmagnetic tque based n inductance vaiatin stat winding, which The electmagnetic a nnlinea functin tque stat cuent based t n ptin. inductance In th vaiatin pape, ptimal stat peatin winding, which pint a nnlinea MMF defined functin b stat maimum cuent tque, t which ptin. a degn In th ule pape, f ated ptimal cuent peatin pint winding MMF tuns. defined Thugh b pevius maimum magnetic tque, cicuit which anals, a degn flu ule linkages f cespnding cuent t winding MMF tuns. aligned Thugh pevius ptins magnetic ae detemined, cicuit anals, as shwn in flu Figue linkages 7. The cespnding utput pwe t MMF aligned calculated ung ptins c-eneg, aewhich detemined, epesented as shwn inas Figue aea 7. enclsed The utput tween pwe aligned calculated ung cuves. The c-eneg, electic eneg which cnveted epesented t tque as b aea enclsed vaiatin tween ceneg. Thus, tque cuves. The detemined electic as: eneg cnveted t tque b vaiatin aligned c-eneg. Thus, tque detemined as: Wc ( i, θ ) T = (7) θ T = W c(i, θ) (7) whee W c c-eneg; θ t ptin; θ i cuent. The maimum tque will whee ccued W c at c-eneg; ptimal θ MMF t which ptin; clse t imaimum cuent. aea change The maimum ati tque c-eneg will as ccued shwn at in ptimal Figue MMF 8. Thus, which num clse t maimum winding tuns aea change ated ati cuent c-eneg detemined as shwn t fit in Figue slt aea 8. Thus, electical num degn specificatin. winding tuns ated cuent detemined t fit slt aea electical Enegies 7, degn, 6 specificatin.. -. aligned ptin Flu linkages(wb).8.6 ptin.. 5 5 5 5 5 MMF(At) Figue 7. Flu linkages cespnding t MMF aligned ptins. Figue 7. Flu linkages cespnding t MMF aligned ptins. 7-6 6 5 n(j)
. 5 5 5 5 5 MMF(At) Enegies 7,, 6 Figue 7. Flu linkages cespnding t MMF aligned ptins. 7-6 6 5 Ceneg vaiatin(j) Maimum ceneg vaiatin Optimal MMF 5 5 5 5 5 MMF(At) Figue 8. 8. Plt Plt vaiatin c-eneg cespnding t MMF.. Optimal Degn. Optimal Degn In th pape, GA integated int ppsed equivalent magnetic cicuit used t In th pape, GA integated int ppsed equivalent magnetic cicuit used t degn, such that ati tque t vlume maimum. Due t maimum degn, such that ati tque t vlume maimum. Due t maimum tque ing pptinal t maimum vaiatin c-eneg, maimum vaiatin tque ing pptinal t maimum vaiatin c-eneg, maimum vaiatin c-eneg given as bjective functin GA: ma ( Δ Wc) = Δ Wc( gen + ) Δ W ( gen). c-eneg given as bjective functin GA: ma( W c ) = W c (gen + ) > W c (gen)... Gemetic Paametes Optimiatin In th pape, an illustated eample a thee-phase -Slt-8-Ple given (Figue 9). The cespnding ntatins ae defined as fllwing. W s ute stat tth width. W inne stat tth width. W t t tth tth utde utde width. width. W i t tth inne width. i t tth inne width. b s ute stat ke width. s ute stat ke width. b si inne stat ke width. hb s si ute inne stat stat tth ke height. width. h s inne ute stat tth height. h t tth height. D ute stat diamete. D a shaft diamete. g ai gap length. L stk thickness ce. The angles (β s β ) stat t ples detemined t fit feable tiangle technique [9] equied t degn. The cnstaints ae given as: () 5 < β <. () 5 < β s <.5. () β > β s. () β + β s < 5. (5) W i ( W ) > W s. (6) W a > W s. (7) Winding tuns stat slt fit ptimal MMF.
() β > βs. () β + βs < 5. (5) W i ( W ) > W s. (6) W a > W s Enegies 7,, 6. (7) Winding tuns stat slt fit ptimal MMF. W s h s b s W i ( ) h h W b si D β β s D a W a g Figue 9. Dimenn diagam ppsed. F F degn degn (Figue (Figue 9), 9), t t mplif mplif degn degn pcedue, pcedue, sme sme paametes paametes ae pedetemined, which ae pe-detemined, which ae ae including including diamete diamete ute ute stat stat D D,, diamete diamete mt mt a a D thickness ciled licn steel sheet L stk, length ai gap g. Based n dimenn a, thickness ciled licn steel sheet L stk, length ai gap g. Based n dimenn Figue Figue 9, 9, elated elated gemetic gemetic equatins equatins ae ae given given as: as: ( W ) s = D s hsβ s W (8) s = b s h s β s (8) ( D W ) s = D s hsβ s W s = b s h s β s (9) (9) ( D h W = D W b s g β = s h s g h ) β (5) (5) ( D D h W a W= a = bs g θt W h s g h ) θ t W (5) Opeatin pcedues GA t detemine paametes ae cnstucted, which cmpsed initialiatin, fitness value, selectin epductin, cssve, mutatin, as shwn in Figue. In peating pcedues GA, initial values ptimiatin ae cndeed as input paametes. Accding t cnstaints mentined abve, initial ppulatin gemetic paametes detemined. With gemetic paametes B-H cuve, fitness value analtical slutin equivalent magnetic cicuit dne, used as pefmance inde f ptimiatin. Then, ppulatin gemetic paametes calculated via selectin epductin, cssve, mutatin. The pcedues epeated within allwed limits until degn equiements ae met. Based n ppsed degn methdlg, ptimal degn ppsed achieved. The evlutin cuve shwn in Figue calculated gemetic paametes ae shwn in Table.
gemetic paametes detemined. With gemetic paametes B-H cuve, fitness value analtical slutin equivalent magnetic cicuit dne, used as pefmance inde f ptimiatin. Then, ppulatin gemetic paametes calculated via selectin epductin, cssve, mutatin. The pcedues epeated within allwed limits until degn equiements ae met. Based n ppsed degn Enegies methdlg, 7,, 6 ptimal degn ppsed achieved. The evlutin cuve 5 shwn in Figue calculated gemetic paametes ae shwn in Table. Stat Geneatin= Input pe-detemined paamete: D, D, g, L, B-H cuve a stk Geneates initial ppulatin: Ws, W, W, Wi, bs, bsi, hs Evaluate Fitness Value Wc ( i, θ ) f ( H) = At, Rnt, Ft,ma θ Check Stpping Citeia Satfied Nt Satfied Selectin Repductin f ( H ) m( H, t+ ) = m( H, t) f Cssve δ ( H ) Ps Pc ( L ) Output paametes: Ws, W, W, Wi, bs, bsi, hs Stp Mutatin δ c f ( L ) f H H m( H, t+ ) m( H, t) P O( H) Pm Geneatin+ Figue. Opeatin pcedues Genetic algithm f duble-stats switched eluctance mt. Figue. Opeatin pcedues Genetic algithm f duble-stats switched eluctance mt. Table. Optimal gemetic paametes (unit: mm). Table. Optimal gemetic paametes (unit: mm). Smbl Quantit Smbl W s ute stat Quantit tth width. W s ute stat tth width. W inne stat tth width.79 W Rt tth utde width 5.99 W i Rt tth inne width.6 b s Oute stat ke width 7.56 b si Inne stat ke width 9.87 h s Oute stat tth height 9.9 h Inne stat tth height 9. h Rt tth height 7.9 D Oute stat diamete D a Shaft diamete g Ai gap length. L stk Thickness ce The ttal num tuns pe slt
Ceneg vaia h Rt tth height 7.9 D Oute stat diamete.8 Da Shaft diamete g Ai gap length..7l stk Thickness ce The ttal num tuns pe slt Enegies 7,, 6 6.6. -. 6 8 Ceneg vaiatin.9 Figue. Evlutin.8 cuve degned..7.6 Geneatin tuctual Decmptin Diagam Pttpe he ppsed a thee-phase -Slt-8-Ple 6 8 its pwe KW with 8 V i Geneatin e. The steel mateial stat 5CS5HF made b China Steel Cpatin its Figue. Evlutin cuve degned. shwn in Figue. F degn cil winding stat,.5 A/mm cu.... Stuctual Stuctual Decmptin Decmptin Diagam Diagam Pttpe Pttpe t. mm wie diamete ae detemined. One tun with five-cd-thead cncent ing given its slt fill fact calculated as 6%. The detail winding stu e seen in Figue. T veif validit ppsed, a pttpe a /8 DSSR factued, as shwn in Figue. Figue a shws whle view ppsed (Figue b) inne (Figue c) stats have windings. The stat shaft fied t i segmental t (Figue d) b nnmagnetic lat, t back cve t. The t shaft munted n t fnt cve. The mt pwe utput unilateal fm shaft, inne stat fied t back shell. The The ppsed ppsed a thee-phase thee-phase -Slt-8-Ple -Slt-8-Ple its its pwe pwe KW KW with with 8 8 V input input vltage. vltage. The The steel steel mateial mateial stat stat 5CS5HF 5CS5HF made made b b China China Steel Steel Cpatin Cpatin its its B-H B-H cuve cuve shwn shwn in in Figue Figue.. F F degn degn cil cil winding winding stat, stat,.5.5 A/mm A/mm cuent cuent dent dent.. mm mm wie wie diamete diamete ae ae detemined. detemined. One One tun tun with with five-cd-thead five-cd-thead cncentated cncentated winding winding given given its its slt slt fill fill fact fact calculated calculated as as 6%. 6%. The The detail detail winding winding stuctue stuctue seen seen in in Figue Figue.. T T veif veif validit validit ppsed ppsed,, a a pttpe pttpe a a /8 /8 manufactued, manufactued, as shwn as shwn in Figue in Figue.. Figue Figue a a shws shws whle whle view view ppsed ppsed.. The ute (Figue b) inne (Figue c) stats have windings. The stat shaft fied t inne The ute (Figue b) inne (Figue c) stats have windings. The stat shaft fied t stat segmental t (Figue d) b nnmagnetic lat, t back cve t fnt inne stat segmental t (Figue d) b nnmagnetic lat, t back cve t fnt cve. The t shaft munted n t fnt cve. The mt pwe utput unilateal fm cve. The t shaft munted n t fnt cve. The mt pwe utput unilateal fm t shaft, inne stat fied t back shell. t shaft, inne stat fied t back shell. Figue. B-H cuve steel mateial. Figue. B-H cuve steel mateial. Figue. B-H steel mateial.
Enegies 7,, 6 7 Enegies 7,, 6 C 6 7 B 6 7 7 B 6 7 9 B A 8 C A 5 8 7 A 5 Enegies 7,, 6 C 5 B 6 8 5 C 9 A C8 9 C 9 B B C C A A A 7 A B B Figue. Dtibutin phases slt sltnums nums /8. Figue. Dtibutin phases /8. Figue. Dtibutin phases slt nums /8. (a) (a) (b) (b) (c) (d) Figue. (d)(d) t. Figue. Pttpe Pttpe Ppsed Ppsed: :(a) (a)whle whleview; view;(b) (b)ute utestat; stat;(c)(c)inne innestat; stat; t.. Simulatin Results. Simulatin Results (c) ppsed methd, a D FEM(d) T veif efficienc mdel empled t anale T veif efficienc ppsed methd, a D FEM mdel empled t anale chaactetic t cmpaed with cnventinal SRM. As shwn in Table, chaactetic t cmpaed with SRM. Asstat; shwn in Table, Figue. Pttpe Ppsed : (a) whle view; (b) cnventinal ute stat; (c) inne (d) t. paametes cnventinal /8 SRM ae detemined b cmmeciall available pduct. T paametes cnventinal /8 SRM ae detemined b cmmeciall available pduct. effectivel cmpae tques pe vlume cnventinal SRM ppsed, sme T effectivel cmpae tques pe vlume cnventinal SRM ppsed,. Simulatin Results dimenns tw mts with KW ae given as same in th pape, such as equivalent ai sme dimenns tw mts with KW ae given as same in th pape, such as equivalent gap length, diametes stat t, length stack, as shwn in Table. Figue 5 shws ai gap length, diametes stat t, length as shwn in Table. Figue 5t shws T veif efficienc ppsed methd, a stack, D FEM mdel empled anale flu linkages cespnding t MMF cnventinal SRM degned. F SRM flu linkages cespnding t MMF cnventinal SRM degned. F SRM chaactetic diffeence cmpaed with SRM. As shwn in, flut linkages tween aligned cnventinal ptins pptinal t Table, MMF. At ptin, /8 minimum ppsed smalle than pduct. that paametes cnventinal SRM inductance ae detemined b cmmeciall available T SRM. At aligned ptin, maimum SRM ppsed lage than that effectivel cmpae tques pe vlume inductance cnventinal ppsed, sme SRM. Figues 6 7 shw tque piles cnventinal SRM ppsed dimenns tw mts with KW ae given as same in th pape, such as equivalent ai, espectivel. Figue 8 shws aveage tques cnventinal SRM ppsed gap length, diametes stat t, length stack, as shwn in Table. Figue 5 shws
Enegies 7,, 6 8, diffeence flu linkages tween aligned ptins pptinal t MMF. At ptin, minimum inductance ppsed smalle than that Enegies SRM. 7, At, aligned 6 ptin, maimum inductance ppsed lage 8 than Enegies 7,, 6 8 that SRM. Figues 6 7 shw tque piles cnventinal SRM ppsed. The aveage tque geneated b ppsed lage than that cnventinal SRM.,. espectivel. The aveage tque Figuegeneated 8 shws b ppsed aveage tques lage cnventinal than that cnventinal SRM ppsed SRM. Theefe, f same tque utput, less cuent ppsed equied. Theefe, f aveage same tque tque geneated utput, bless ppsed cuent ppsed lage than that equied cnventinal SRM. electical lads als educed. It fund that ppsed accmmdate Theefe, electical f lads same als tque educed. utput, It less cuent fund that ppsed equied accmmdate me cnducts educes heat cncentatin stat. electical me cnducts lads alseduces educed. heat It cncentatin fund that stat. ppsed accmmdate me cnducts educes Table heat. cncentatin Paametes cnventinal stat. SRM ppsed. Table. Paametes cnventinal SRM ppsed. Table. Paametes cnventinal SRM Cnventinal ppsed. Ppsed Paametes Cnventinal Ppsed Paametes /8 SRM /8 Num Cnventinal /8 SRM Ppsed /8 Paametes phases Oute adius Num ute phases stat (mm) /8 SRM /8 Oute Num Length adius phases ai ute gap stat (mm) (mm).. Oute adius ute stat Oute Length adius ai gap t (mm) (mm).66...66 Length Oute ai gap (mm).. OuteLength adius adius t stack t (mm) (mm) (mm).66.66.66.66 Length Rt Length inetia stack stack (mm) ( (mm) kg mm ).9 -.97 - Rt inetia (kg mm Rt inetia ( kg ) mm ).9 -.9 -.97.97 - - Num Num tuns/phases tuns/phases (N) (N) Num Wie Wie diamete tuns/phases diamete (mm) (N) (mm). 5. 5.. 5 5 Wie diamete (mm). 5. 5.5 -.5 - Cnventinal SRM Cnventinal Ppsed SRM Ppsed aligned ptin aligned ptin Flu Flu linkages(wb) linkages(wb).5.5.5.5 ptin ptin 5 5 5 5 MMF(A-t) 5 5 MMF(A-t) Figue 5. Flu linkages cnventinal SRM ppsed at aligned ptins. Figue Figue 5. 5. Flu Flu linkages linkages cnventinal SRM ppsed at at aligned aligned ptins. ptins. 8 8 6 6 Tque(N Tque(N m) m) - - - - -6-6 5 5 5 5 5 5 Rt ptin(deg.) Rt ptin(deg.) Figue 6. Tque pile cnventinal SRM. Figue 6. Tque pile cnventinal SRM. Figue 6. Tque pile cnventinal SRM. 5 5 6 6 8 8
Enegies 7,, 6 9 Enegies 7,, 6 Enegies 7,, 6 9 9 Tque(N m) Tque(N m) 8 8 6 6 - - - - -6-6 -8-8 - - 8 6 8 6 5 5 5 5 5 5 5 Rt ptin(deg.) 5 Rt ptin(deg.) Figue 7. Tque Figue 7. Tque pile ppsed. 7 7 6 6 Cnventinal SRM Ppsed Cnventinal SRM Ppsed 5 5 Aveage Aveage tque(n-m) tque(n-m) 5 6 7 8 9 Cuent(A) 5 6 7 8 9 Cuent(A) Figue 8. Aveage tque f cnventinal SRM ppsed. Figue Figue 8. 8. Aveage tque f f cnventinal cnventinal SRM SRM ppsed ppsed.. 5. Cncluns 5. Cncluns 5. Cncluns Th pape pesents mamatic mdeling with equivalent magnetic cicuit t Th pape pesents mamatic mdeling with equivalent magnetic cicuit t pecel Th pedict pape pesents magnetic a mamatic flu linkages mdeling aligned with equivalent ptins. magnetic GA integatin cicuit t pecel pedict magnetic flu linkages aligned ptins. GA integatin pecel ppsed pedict equivalent magnetic magnetic flucicuit linkages develped alignedf apid ptimiatin ptins. GA integatin t each ppsed equivalent magnetic cicuit develped f apid ptimiatin t each maimum ppsed equivalent ati tque magnetic t vlume cicuit develped. Cmpaed f apid ptimiatin t cnventinal SRM, an tillustated each maimum ati tque t vlume. Cmpaed t cnventinal SRM, an illustated eample maimum -KW ati thee-phase tque-slt-8-ple vlume. Cmpaed used t veif t cnventinal efficienc SRM, an illustated ppsed eample -KW thee-phase -Slt-8-Ple used t veif efficienc ppsed methd. eample Simplified a -KW thee-phase -D electmagnetic -Slt-8-Ple mdels ae analed used t veif mulated. efficienc Finall, esults ppsed methd. Simplified -D electmagnetic mdels ae analed mulated. Finall, esults analtical methd. Simplified calculatins -D electmagnetic FEA ae validated mdels b ae analed ppsed mt mulated. t shw Finall, accuac esults analtical calculatins FEA ae validated b ppsed mt t shw accuac degned analticalstateg. calculatins In fact, fm FEA aeesults, validated b tque ppsed utput mt ppsed t shw accuac abut.5 degned stateg. In fact, fm esults, tque utput ppsed abut.5 times degned that stateg. cnventinal In fact, fm/8 esults, SRM unde tque same utput input cuent. ppsed abut.5 times times that cnventinal /8 SRM unde same input cuent. that cnventinal /8 SRM unde same input cuent. Cnflicts Inteest: N Cnflicts Inteest fund in th pape. Cnflicts Inteest: The N Cnflicts auth declaes Inteest n cnflict fund inteest. th pape. Refeences Refeences. Asga, M.; Afjei, E.; Tkaman, H. New Stateg f Degn Anals Duble-Stat Switched. Asga, M.; Afjei, E.; Tkaman, H. A New Stateg f Degn Anals a Duble-Stat Switched Reluctance Mt: Electmagnetics, FEM, Epeiment. IEEE Tans. Magn. 5, 5, 8888. Reluctance Mt: Electmagnetics, FEM, Epeiment. IEEE IEEE Tans. Tans. Magn. Magn. 5, 5, 5, 5, 8888. 8888. [CssRef]. Radun, A. Analtical Calculatin Switched Reluctance Mt s Unaligned Inductance. IEEE Tans.. Radun, A. A. Analtical Calculatin Switched Switched Reluctance Reluctance Mt s Mt s Unaligned Unaligned Inductance. Inductance. IEEE Tans. IEEE Tans. Magn. Magn. 999, 5, 7 8. 999, Magn. 5, 999, 7 8. 5, 7 8. [CssRef]. Isfahani, A.H.; Fahimi, B. Cmpan Mechanical Vibatin tween Duble-Stat Switched. Isfahani, A.H.; Fahimi, B. Cmpan Mechanical Vibatin tween a Duble-Stat Switched Reluctance Machine Cnventinal Switched Reluctance Machine. IEEE Tans. Magn., 5, Reluctance Machine a Cnventinal Switched Reluctance Machine. IEEE Tans. Magn., 5, 77. 77.
Enegies 7,, 6. Isfahani, A.H.; Fahimi, B. Cmpan Mechanical Vibatin tween a Duble-Stat Switched Reluctance Machine a Cnventinal Switched Reluctance Machine. IEEE Tans. Magn., 5, 77. [CssRef]. Abab, N.; Wang, W.; Lin, C.; Hean, J.; Fahimi, B. Themal Mdeling Anals a Duble-Stat Switched Reluctance Mt. IEEE Tans. Eneg Cnves. 5,, 9 7. [CssRef] 5. Abbaan, M.; Mallem, M.; Fahimi, B. Duble-Stat Switched Reluctance Machines (): Fundamentals Magnetic Fce Anals. IEEE Tans. Eneg Cnves., 5, 589 597. [CssRef] 6. Abbaan, M.; Hanaeinejad, V. Tque Optimiatin Duble-Stat Switched Reluctance Machine. Appl. Mech. Mate.,, 5 5. [CssRef] 7. Tavakkli, M.; Mallem, M. Optimum Rt Shaping f Tque Impvement Duble Stat Switched Reluctance Mt. J. Elect. Eng. Technl., 9, 5. [CssRef] 8. Huang, Y.; Zhu, T.; Dng, J.; Lin, H.; Yang, H.; Cheng, M. Magnetic Equivalent Cicuit Mdeling Ykeless Aial Flu Pemanent Magnet Machine with Segmented Amatue. IEEE Tans. Magn., 5, 8. [CssRef] 9. Liptak, M. Pinciple Degn Fu-Phase Lw-Pwe Switched Reluctance Machine Aimed t Maimum Tque Pductin. J. Elect. Eng., 55, 8. 7 b auth. Licensee MDPI, Basel, Switel. Th aticle an pen access aticle dtibuted unde tems cnditins Ceative Cmmns Attibutin (CC BY) license (http://ceativecmmns.g/licenses/b/./).