[2009] IEEE. Reprinted, with permission, from Xu, Wei; Zhu, Jianguo; Guo, Youguang; Wang, Yi; Zhang, Yongchang; Tan, Longcheng. Equivalent Circuits

Transcription

1 [009] IEEE. Rpintd, with pmission, fom Xu, Wi; Zhu, Jianguo; Guo, Youguang; Wang, Yi; Zhang, Yongchang; Tan, Longchng. Equivalnt Cicuits fo Singld-sidd Lina Induction Motos', Pocdings of IEEE Engy Convsion Confnc & Epo, pp This matial is postd h with pmission of th IEEE. Such pmission of th IEEE dos not in any way imply IEEE ndosmnt of any of th Univsity of Tchnology, Sydny's poducts o svics. Intnal o psonal us of this matial is pmittd. Howv, pmission to pint/publish this matial fo advtising o pomotional puposs o fo cating nw collctiv woks fo sal o distibution must b obtaind fom th IEEE by witing to pubs-pmissions@i.og. By choosing to viw this documnt, you ag to all povisions of th copyight laws potcting it.

2 Equivalnt Cicuits fo Singl-sidd Lina Induction Motos Wi Xu, Jianguo Zhu, Youguang Guo, Yi Wang, Yongchang Zhang School of Elctical, Mchanical and Mchatonic Systms Univsity of Tchnology Sydny Sydny, Austalia Abstact -- Singl-sidd lina induction motos (SLIMs) hav latly bn applid in tanspotation systm taction divs, spcially in th intmdiat spd ang. Thy hav mits such as th ability to t thust on th sconday without mchanical contact, high acclation o dclation, lss whl wa, small tuning cicl adius and flibl oad lin. Th thoy of opation fo ths machins can b dictly divd fom otay induction motos, but sval issus involving th tansvsal dg and th longitudinal nd ffcts, and half-filld slots at th pimay nds, nd to b invstigatd. In this pap, a T modl quivalnt cicuit is poposd which is basd on ondimnsional magntic quations of th ai-gap, wh half-filld slots a considd by an quivalnt pol numb. Thn, it dducs two-ais quivalnt cicuits to study th SLIM dynamic pfomanc. Th thotical analyss hav bn validatd by pimntal sults on SLIM pototyps. Ind Tms Singl-sidd lina induction moto (SLIM), tansvsal dg ffct, longitudinal nd ffct, quivalnt cicuit, coct cofficint, mutual inductanc, sconday sistanc, stady pfomanc, dynamic pfomanc, contol schm. I. NOMENCLATURE ε Pimay winding shot pitch ω Angula fquncy of pow supply τ Pol pitch d Sconday sht thicknss f s Pimay fquncy g Equivalnt ai gap lngth g m Mchanical ai gap lngth l δ Pimay lamination width p Equivalnt numb of pol pais C (s) Tansvsal dg ffct cofficint fo th sconday sistanc C (s) Tansvsal dg ffct cofficint fo th mutual inductanc K (s) Longitudinal nd ffct cofficint fo sconday sistanc K (s) Longitudinal nd ffct cofficint fo mutual inductanc L m Mutual inductanc in two-ais cicuit L m Mutual inductanc in p-phas cicuit L ls Pimay lakag inductanc L l Sconday lakag inductanc Sconday sistanc R II. INTRODUCTION Th stuctu diagam of a singl-sidd lina induction moto (SLIM) is shown in Fig.. Th SLIM s pimay sid can b simply gadd as a otay induction moto (RIM) stato which is cut-opn and olld flat. Th sconday sid is simila to th RIM oto which oftn consists of a sht conducto, such as copp o aluminum, with a solid back ion acting as th tun path of th magntic flu. Th thust Longchng Tan Institut of Elctical Engining Chins Acadmy of Scincs Bijing, China lchtan@mail.i.ac.cn cosponding to th RIM toqu can b poducd by th action btwn th ai gap flu dnsity and th ddy cunt in th sconday sht. Fig.. Stuctu of an SLIM. A tain divn by SLIM, also calld as lina mto, has bn paid attntion by acadmia and industy fo mo than two dcads. By now, th a mo than twnty commcial lina mto lins of about 400 kilomts in th wold, such as th Knndy ai lin in Amica, th lina mto in Japan, th Vancouv light tain in Canada, th Guangzhou subway lin 4 and th Bijing aipot apid tanspot lin in China. Th typical SLIM div stuctu and systm diagams a illustatd in Fig.. It can b sn that th SLIM pimay is hangd blow th dicto, which is supplid by th invt on th vhicl. Th sconday is flattd on th ailway tack, which consists of a 5 mm thick copp/aluminum conductanc flt and a 0 mm thick back ion. Whn th pimay th-phas windings a input with AC cunt, thy can build up ai gap flu linkag and induc ddy cunt in th sconday sht. This ddy cunt will act with afomntiond ai gap flu linkag so as to poduc hoizontal lctomagntic thust that can div th vhicl fowad dictly without fiction btwn whl and tack. Th SLIM s spcial stuctu mans that its pfomanc is a littl diffnt fom that of an RIM. As w know, in th RIM, an accuat quivalnt cicuit modl can b divd asily basd on th symmtical gomty p pol. Unfotunatly, it is not as staightfowad to gain th quivalnt cicuit fo an SLIM mainly fo th following th

3 causs. () As th SLIM pimay movs, a nw flu is continuously dvlopd at th pimay ntanc sid, whil th ai gap flu disappas quickly at th it sid. By th influnc of th suddn gnation and disappaanc of th ai gap pntating flu dnsity, an amount of ddy cunt in anti-diction to th pimay cunt will occu in th sconday sht, which cospondingly affcts th ai gap flu pofil along th longitudinal diction (-ais) as illustatd in Fig. 3. This phnomnon is calld longitudinal nd ffct, which could incas th copp loss, and dcas th mutual inductanc as th vlocity gos up. In th nd, th ffctiv lctomagntic thust will b ducd bcaus of its attnuatd ai gap avag flu linkag [5]. () Th diffnt width btwn pimay lamination and sconday sht can sult in non-unifom flu dnsity, which may incas th sconday quivalnt sistanc. Th phnomnon is namd as tansvsal dg ffct. (3) Fo th cut-opn pimay magntic cicuit, th ist half-filld slots in th pimay nds. Hnc, th-phas magntic cicuits a not symmtical with ach oth. Thphas cunts a not symmtical vn whn citd by th-phas balancd voltags. Th half-filld slots will affct th ai gap flu dnsity distibution so as to sult in som altation in mutual inductanc, lakag inductanc, and sconday quivalnt sistanc. Fig.. Simpl vhicl systm diagams poplld by th SLIM: photo of an SLIM divn vhicl, diagam of th div systm. Fig. 3. Pofils of sconday ddy cunt and ai gap flu linkag. Duing th past sval dcads, plnty of paps hav bn availabl on th SLIM pfomanc analysis, involving stady and dynamic stats. Rf. [] basd on th RIM T- modl quivalnt cicuit povidd on simpl and usful foc function pssion accoding to th sconday ddy cunt avag valu by ngy convsion balanc thom. This function is affctd by th SLIM spd, sconday sistanc, sconday inductanc, and som stuctual paamts, such as th pimay lngth. Th p-phas simplifid modl can b usd to pdict th SLIM output thust and vtical foc convnintly. Rfs. [] and [3] accoding to th sults in [] dducd th two-ais modls (dq o αβ), which can b applid in vcto contol o dict toqu contol to pdict th dynamic pfomanc. Howv, th divation pocss of th visd foc function is vy coas on th assumption that ddy cunt in th sconday sht dcass fom maimum to zo by ponntial attnuation only in th pimay lngth ang. It only consids th mutual inductanc influncd by th ddy cunt, disgading its ffct on th sconday sistanc. Th analytic sults, such as th mutual vaianc, a with incasing o with th masumnt as th vlocity gos up. Rf. [4] divd an quivalnt cicuit modl fom th polby-pol mthod basd on winding functions of th pimay winding. On tnth od st of diffntial quations a divd to dscib a basic modl fo th fou pol machin, and high od systm of quations can b povidd as th pol numb gos up. Rf. [5] basd on th winding function mthod dividd th SLIM ai gap flu dnsity into th componnts, i.. pimay fundamntal, sconday fundamntal, and sconday ddy cunt pats. Thn it divd ths th-goup function pssions and futh achivd inductanc, sconday sistanc and oth paamts so as to analyz th SLIM pfomanc. Th winding function mthod can study th stady, tansint, and dynamic pfomanc. Howv, th pssions of sconday winding function distibutions a

4 achivd by som hypothsis. Fild thoy was utilizd in dvloping th lumpd paamt LIM modl [6], in which nd ffct, fild diffusion in th sconday sht, skin ffct and back-ion satuation w considd. Howv, th sultant lumpd-paamt modls look vy complicatd fo pactical us of modling and contol. Gias and Faiz dvlopd an quivalnt cicuit by supposing th synchonous wav and th pulsating wav causd by th nd ffct [7]-[8]. Mo pimnts should b mad to vify th simulation invstigation. Sval diffnt modls w dvlopd fom th lctomagntic lation in th ai gap though a Foui-sis appoach [9]-[]. Howv, it quis mo substantial computing tim to gain som usful data, which accuacy dpnds closly on many initially givn paamts. If som ky valus a not initializd ationally, th final solution cannot b achivd du to non-convgnc. This pap psnts th pfomanc analysis of two SLIMs by using a p-phas T-modl quivalnt cicuit divd fom on dimnsional flu dnsity quations []. Among th fiv majo paamts involving pimay sistanc, pimay lakag inductanc, mutual inductanc, sconday sistanc, and sconday inductanc, th mutual inductanc and sconday sistanc a influncd by th dg and nd ffcts gatly which can b coctd by fou lativ cofficints, i.. K (s), K (s), C (s) and C (s). Moov, two-ais quivalnt cicuits (dq o αβ) accoding to th T modl quivalnt cicuit a obtaind using a flu linkag consvation mthod, which a analogous to thos of th RIM in a two-ais coodination systm. Th LIM dynamic paamts, spcially th mutual inductanc and sconday sistanc, can b analyzd by using th fou cofficints. Epimntal vification indicats that both th T modl and th nw two-ais cicuits a asonabl fo dscibing th stady and dynamic pfomanc of th SLIM. Ths two modls can povid good guidanc fo th lctomagntic dsign and contol schm implmntation fo SLIM. III. T-MODEL EQUIVALENT CIRCUIT Fig. 4 shows th SLIM longitudinal sid viw, tansvsal sid viw, and on dimnsional analytical modl. In tms of on-dimnsional analysis w can calculat th phas cunts and citation voltags. Th ai-gap flu linkag can b obtaind using Mawll fild quations and solvd using th compl pow mthod with a confomal tansfomation which consids th ffcts of half-filld slots, yok magntic satuation and back ion sistanc. Using th qual compl pow lationship btwn magntic fild and lctic cicuit, w can obtain sval cicuit paamts, such as mutual inductanc L m, sconday sistanc R, pimay lakag inductanc L ls, sconday lakag inductanc L l, longitudinal nd ffct cofficints K (s) and K (s), and tansvs dg ffct cofficints C (s) and C (s). Thi comphnsiv divation can b found in [] and [3]. Th T-modl quivalnt cicuit is shown in Fig. 5. pimay v it sid y z j sconday sht back ion j g m nty sid d (c) Fig. 4. Stuctu and on-dimnsional analytical modl of SLIM: Longitudinal sid viw, Tansvsal sid viw, (c) Analysis modl. U s R s I s Fig. 5. L ls E m L l I m I K ( s) C( s) Lm T-modl quivalnt cicuit of SLIM. K ( s) C ( s) R / s Th longitudinal nd-ffct cofficints K (s) and K (s) a dnotd by sg K () s = pτ + ( SG) K () s = pτ + ( SG) C + C C C + C C wh C and C a th functions of slip s and goodnss facto G, τ is th pimay pol pitch, p is th numb of quivalnt pol pais calculatd by (p ) p = 4p 3 + ε ( mq) wh p is th actual numb of pol pais, m is th pimay phas numb, q is th numb of coil sids p phas p pol, and ε is th lngth of shot pitch. Th tansvsal nd-ffct cofficints C (s) and C (s) a givn by v () () (3)

5 sg[r [ T] + I m[ T]] C () s = (4) R[ T ] R[ T] + I[ m T] m C () s = (5) I [ T ] wh T is th function of slip, goodnss facto and moto stuctu paamts, and R and I m a th al and imag pats of th compl T, spctivly. H T is pssd by λ T = j + th l (6) [ γ ( γ ) (0.5 δα)] 0.5lδ α wh l δ is half th pimay lamination width, α is th atio of c to τ, and c is half th width of th sconday sht ovhang indicatd in Fig. 4. γ and λ can b obtaind by λ = + th(0.5 lδα ) th[0.5 K( c lδ)] γ (7) γ = (8) + jsg wh K is th function of slip and moto stuctu paamts, and c is th width of th sconday sht. Th fiv paamts in th T-cicuit can b calculatd as blow. Th pimay sistanc is = ρ l W / S (9) Cu av cu wh ρ Cu is th sistivity of copp, l av is half th avag lngth of th pimay winding loop, W is th numb of tuns of th pimay p phas in sis, and S cu is th ffctiv coss-sctional aa of th pimay winding conducto. Th pimay lakag inductanc L ls is lδ λ λ + λ + λ = 0.05 ( + ) (0) s t d Lls W q p p wh λ s is th pimay slot lakag magntic conductanc, λ t th pimay tooth lakag magntic conductanc, λ th pimay winding nd lakag magntic conductanc, and λ d th pimay hamonic lakag magntic conductanc. Th sconday sistanc is composd of thos of conducting sht and back ion bcaus th flu can pntat though th aluminum o copp sht [3], and thn nt th back ion. Th dpth of flu dnsity into back ion d F is d F ρ F = () sω μ F wh ρ F is back ion sistivity, μ F th pmability of back ion, and ω th pimay synchonous angula fquncy. Th sistanc of sconday conducting sht sht is ( WK W ) lδ = 4mρsht () p dτ sht wh ρ sht is th sistivity of th sconday conduction sht, and K w is pimay winding cofficint. Th sistanc of sconday back ion F is ( WK W ) lδ = 4m ρf (3) p d τ F Thfo, th sconday quivalnt sistanc is shtf = + sht F F (4) Th sconday lakag actanc is L = l Bsh ( Kd) π fs (5) s wh f s is th pimay fquncy, B is th function of slip, pimay fquncy, and machin stuctu paamts. Th citing action is lv δ s Lm = 4 mμ 0( WK W ) (6) 4 π f gp s wh V s is th synchonous vlocity of pimay sid, and g is th quivalnt ai gap lngth. Aft th appoimat gulation by th fou cofficints and th quivalnt numb of pol pais, th SLIM T modl quivalnt cicuit indicatd in Fig. 5 is simila with that of th RIM. Th influnc by longitudinal and tansvsal nd ffcts and half-filld slots can b stimatd by th cosponding cofficints. Th visd SLIM modl basd on th quivalnt pol pais can b gadd as th-phas magntic cicuit symmty. Whn K (s)=k (s) =C (s)=c (s)=, i.., whn th longitudinal and tansvs ffcts a nglctd, th cicuit can b simplifid as th sam as that of RIM. Thfo, it is vy convnint to analyz th pfomanc of SLIM in th simila way as that of RIM basd on th cicuit []-[6]. IV. VERIFICATION OF T-MODEL EQUIVALENT CIRCUIT In od to validat th T-modl quivalnt cicuit in Fig. 5, lots of stady stat pfomanc analysis has bn mad in [3]. H two kinds of SLIM pfomanc valuations a givn out, i.. th 000 Japans SLIM and an ac SLIM. Thi main dimnsions a shown in Tabl I. TABLE I MAIN DIMENSIONS OF TWO SLIMS Itms 000 Ac Pol pitch (m) Pol pais 4 3 Pimay lngth/width(m).476/ /0. Sconday width(m) Sconday sht thicknss (mm) 7 5 Ai gap lngth (mm) 9 0 Slot numb 79 4 Fig. 6 shows th 000 SLIM pfomanc cuvs, including thust, slip fquncy, pow facto, fficincy, and cunt []. Th bas spd is 40 km/h. Th a two woking modls in th opating gions, i.. constant

6 cunt blow th bas spd and constant voltag byond th bas. It can b sn that th simulation sults about th pow facto, fficincy, and phas cunt a appoimatly in hamony with th masumnts in []. Pow facto Efficincy 50 Vaiabls Thust(kN) Slip Fquncy(Hz) Spd(km/h) Calculation Masumnt Spd (km/h) Calculation Masumnt Spd (km/h) (c) Fig. 7 shows th ac SLIM pototyp pimntal bnch. It has a oto which is fomd on th im of th lag-adius flywhl whos pimay is fd by a convt. Th load cll is a DC machin which is connctd to th shaft of th SLIM ig by blts. Th DC machin can opat at any dsid spd and load blow th ating valus to povid diffnt opational points. Th masumnt snsos locatd btwn SLIM and DC load can cod th SLIM vlocity, load pow and thust. Fig. 8 plots th thust cuvs in th constant cunt constant fquncy mod o th constant voltag constant fquncy mod. It can b sn claly that th thust is ducd du to th nd ffct, spcially in th high spd gion. Moov, th thust citd by th constant voltag mod dcass much fast than th constant cunt mod bcaus of its fast ai gap flu attnuation. Th thust simulation ags with th masumnts asonably in vaious vlocitis. Fig. 7. Epimntal stup of th ac SLIM [] [6]. 00 Cunt (A) Calculation Masumnt Spd (km/h) (d) Fig. 6. Stady pfomanc of Japans 000 SLIM []: Thust and slip fquncy; Pow facto; (c) Efficincy; and (d) Phas cunt.

7 indicats th SLIM quivalnt cicuits in th dq-ais fam. Diffnt fom thos of th RIM, fou mo ctification cofficints, i.. K, C, K, C, a usd in th mutual inductanc and sconday sistanc banch cicuits. In Fig. 9, ω and ω a th angula fquncis of th pimay and sconday lativ to dq-ais. Rs ω ψ qs Lls Ll ω ψ q K C R uds ids pψ ds K C L m pψ d i d Fig. 8. Stady thusts of th SLIM pototyp: Constant cunt constant fquncy mod (solid lins indicat th thusts without nd ffct, dashd lins th thusts with nd ffcts, oths th masud thusts); Constant voltag constant fquncy mod (solid lins show th thust with nd ffct, oths th masud thusts). V. TWO-AXIS EQUIVALENT CIRCUITS Accoding to th Pak coodinat tansfomation by th pow convsion ul, w can obtain th two ais (dq o αβ) quivalnt cicuits. Th flu linkag mati is ψ ds ψ qs = ψ d ψ q (7) KCL m + Lls 0 KCL m 0 ids 0 KCL m Lls 0 KCL i + m qs KCL m 0 KCL m + Ll 0 id 0 KCL m 0 KCL m + Ll iq wh L m is th quivalnt mutual inductanc in th two-ais fam, which is qual to.5l m. In od to simplify th flu mati pssion, th symbol s in th fou fficincis is omittd. W dfin that L mc = KCL m, L s = KCL m + L ls, L = KCL m + L,thn th fou flu linkag pssions a l futh pssd by ψ ds = Lsids + Lmcid ψ qs = Lsiqs + Lmciq (8) ψ d = Lmcids + Li d ψ q = Lmciqs + Li q Fom (8), it can b sn that th fou SLIM flu linkag quations a totally simila to thos of th RIM. Th spcial pfomanc taits sultd fom th longitudinal nd ffct, tansvsal dg ffct, and half-filld slots a asily dscibd by th fou fficint and quivalnt pol pais. Th SLIM analysis pocdu and algoithm in th two-ais coodination a also simila with thos of th RIM. Fig. 9 uqs Rs iqs ψ ds ω pψ qs Lls K C L Ll m ω ψ d pψ q K C R Fig. 9. Equivalnt cicuits of SLIM in dq-ais coodination: d-ais cicuit; q-ais cicuit. VI. ANALYSIS ON FOUR COEFFICIENTS T-modl cicuit paamts fo th 3-phas ac SLIM pimntal pototyp a: R s = 0.45 Ω, L ls =.45 mh, L m = mh, R = 0. Ω, and L l = mh. Dspit th natu of th topology of th cut-opn SLIM, it is found that th quivalnt cicuits acoss all th th phass a appoimatly qual. In od to invstigat th nd ffcts on th mutual inductanc and sconday sistanc convnintly, w can analyz diffnt vaiabl cuvs in th ac SLIM stat-up pocdu. Th pimay input fquncy vais fom Hz to 30 Hz. Fo ach input fquncy th phas cunt changs fom 0 to 30 A. Th fou-cofficint cuvs in diffnt fquncis a shown in Figs. 0 to 3. Figs. 4 and 5 show th cuvs fo th mutual inductanc and sconday sistanc as th citd fquncy gos up. Th thust along with phas cunt and citd fquncy is givn in Fig. 6. Bcaus of nd ffcts, th avag ai-gap flu dnsity dcass with th gowing vlocity, which ducs th mutual inductanc and incass th sconday sistanc [6]. C f s/hz Fig. 0. Cuv of cofficint C vsus slip fquncy i q

8 C..8.4 Rc/Ω f s /Hz Fig.. Cuv of cofficint C vsus slip fquncy 0.95 Fig f s /Hz Cuv of modifid sconday sistanc R c vsus slip fquncy K f s/hz Fig.. Cuv of cofficint K vsus slip fquncy K f s /Hz Fig. 3. Cuv of cofficint K vsus slip fquncy Lmc/mH 0 9 F/N f s /Hz I s /A Fig. 6. Cuv of thust F vsus phas cunt and slip fquncy VII. EXPERIMENT IN INDIRECT ORIENTATION CONTROL SCHEME Fig. 7 shows th indict oto fild contol schm which is simila to an RIM contol. Th a th closd loops, including th spd, d-ais cunt and q-ais cunt loops, which a modifid by th PI gulatos [6]. Fig. 8 shows th simulation and pimntal cuvs of thust and vlocity, which ag wll with ach oth. ω * * I ds + + PI + ω I q * I ds PI I q ω s / INV U qs U βs PI + ω + ω U ds θ U αs I ds I qs /3 U a U b U c I a I b Ic SLIM ω f s/hz Fig. 4. Cuv of modifid mutual inductanc L mc vsus slip fquncy Fig. 7. Indict ointation contol analysis diagam

9 F/N, v/(m/s) F/N, v/(m/s) F F v t/s t/s Fig. 8. Thust and vlocity cuvs (th atio of thust o vlocity is :5 o 5:): Simulation; Epimnt. VIII. v CONCLUSIONS A T-modl quivalnt cicuit fo th SLIM is dvlopd in this pap which taks th longitudinal nd ffct, tansvsal dg ffct, and half-filld slots into considation. Th fou cofficints K (s), K (s), C (s), and C (s) and quivalnt pol pais p can dscib th SLIM pfomanc asonably fo th calculation on thust, pow facto, fficincy, and phas cunt in diffnt woking conditions. Moov, th two-ais quivalnt cicuits (dq o αβ) hav bn divd basd on th T-modl cicuit, which a simila to thos of th RIM cpt fou cofficints usd in th mutual inductanc and sconday quivalnt sistanc. Th cuvs fo th fou cofficints, mutual inductanc, sconday sistanc, and thust in diffnt vlocity diffnt phas cunt a givn out. Moov, th thust and vlocity in indict ointation contol schm basd on two-ais quivalnt cicuits a studid. Th simulations show good colation with th pimnts. Futh pimntal analysis of th fou fficincis, mutual inductanc, and sconday sistanc will b put fowad in th nt stp sach. ACKNOWLEDGMENT Th authos sincly thank Pof. Yaohua Li and D. Jinqi Rn in th Institut of Elctical Engining, Chins Acadmy of Scincs, fo thi gat suggstions on th sach wok of this pap. REFERENCES [] J. Duncan, Lina induction moto quivalnt cicuit modl, IEE Poc., vol. 30, Pat B, no., pp. 5-57, Jan [] Jong-Hyoun Sung, and Kwangh Nam, A nw appoach to vcto contol fo a lina induction moto considing nd ffcts, in Poc. IEEE IAS Annual Mting, 999, pp [3] G. Kang, and K. Nam, Fild-ointd contol schm fo lina induction moto with th nd ffct, IEE Pocding on Elctical Pow Application, vol. 5, no. 6, pp , Nov [4] T. A. Lipo, and T. A. Nondahl, Pol by pol d-q modl of a lina induction machin, IEEE Tans. Pow Appaatus and Systms, vol. PAS-98, no., pp , Ma./Ap [5] Changan Lu, A nw coupld-cicuit modl of a lina induction moto and its application to stady-stat, tansint, dynamic and contol studis, Ph.D. Disstation, Dpatmnt of Elctical Engining, Qun Univsity, Canada, 993. [6] K. Idi, G. E. Dawson, and A. R. Eastham, Modling and pfomanc of lina induction moto with satuabl pimay, IEEE Tans. Ind. Appl., vol. 9, no. 6, pp. 3-8, Nov./Dc [7] J. F. Gias, G. E. Dawson, and A. R. Eastham, A nw longitudinal nd ffct facto fo lina induction motos, IEEE. Tans. Engy Convsion, vol. EC-, no., pp. 5-59, Ma [8] J. Faiz and H. Jafai, Accuat modling of singl-sidd lina induction moto consids nd ffct and quivalnt thicknss, IEEE. Tans. Magn., vol. 36, no. 5, pp , Spt [9] Yoshida, K., Nw tansf-mati thoy of lina induction machins, taking into account longitudinal and tansvs fomagntic nd ffcts, IEE Poc. Elct. Pow Appl., vol. 8, no. 5, pp. 5-36, Spt. 98. [0] Y. Moi, S. Toii, and D. Ebihaa, End ffct analysis of lina induction moto basd on th wavlt tansfom tchniqu, IEEE. Tans. Magn., vol. 35, no. 5, pp , Spt [] Tsuyoshi Higuchi and Sakutao Nonaka, On th dsign of high fficincy lina induction motos fo lina mto, Elctical Engining in Japan, vol. 37, no., pp , Aug. 00. [] Xialing Long, Thoy and Magntic Dsign Mthod of Lina Induction Moto, China: Scinc Publishing Company, pp , 006. [3] Wi Xu, Rsach on th pfomanc of singl-sidd lina induction moto, Ph.D. Disstation, Institut of Elctical Engining, Chins Acadmy of Scincs, China, 008. [4] Wi Xu, Yaohua Li, Guangshng Sun, and Jinqi Rn, Pfomanc study on high pow lina induction moto in tanspotation, in Poc. Int. Conf. Elctical Machins and Systms, Oct. 007, pp [5] Wi Xu, Jianguo Zhu, Longchng Tan, Youguang Guo, Shuhong Wang, and Yi Wang, Optimal dsign of a lina induction moto applid in tanspotation, in Poc. IEEE Int. Conf. Industy Tchnology, Fb. 009, pp [6] Wi Xu, Yaohua Li, Guangshng Sun, and Jinqi Rn, Rsach on nw quivalnt cicuits of singl sid lina induction moto, Pocdings of th China Socity fo Elctical Engining, vol. 9, no. 9, pp , Ma. 009.