Electromagnetic Forces in a Permanent Magnet Synchronous Machine with an Eccentric Rotor

Transcription

1 568 Electomagnetc Foces n a Pemanent Magnet Synchonous Machne wth an Eccentc Roto Mcea Dan IUDEN *),**) *) Techncal Unvesty of Cluj-Napoca St. C. Dacovcu n.5, 4 Cluj-Napoca - Romana Tel: , Fax: Emal: Mcea.Iudean@mas.utcluj.o **) Laboatoy of Electoncs, Electotechncs and Systems UTBM LES (Bat. F); Rue They MIEG, 9 Belfot Fance Tel: , Fax: Emal: dan.udean@utbm.f bstact - Electc machney comes n many dffeent types and a stkngly boad ange of szes, fom those lttle machnes that cause cell telephones and pages to vbate to tubne geneatos wth atngs upwads of a Ggawatt. In ths pape we ae wokng wth a pemanents magnets synchonous moto that s used n the tacton of an electcal bcycle. Ou pupose s to analyze the level and the shape of electomagnetc foces that appeas nsde the machne when t s n a good functonng estate and when the oto s faulted. I. Intoducton The electomagnetc foces actng between the oto and stato when the oto s eccentc compang wth the stato have been the pont of nteest and mathematcal models have been developed almost though the whole hstoy of the electcal machnes. The mathematcal models can be appled n analytcal calculatons o numecal analyss. The electomagnetc foces wtch appea n a pemanent magnet synchonous moto ae: - The magnetostctve foce: - the mateals utlsed n the otatng machnes ae magnetostctve only a lttle so we can neglect these foces. - Maxwell s foce: B d df () - Laplace s foce: df dl B () - The foce nduced by the vaaton of the pemeance: dλ F U m () dx II. nalyss The eccentc motons of the oto, ncludng statonay oto dsplacement, can be descbed by whlng motons of the oto. The cylndcal ccula whlng moton means that the centelne of the oto tavels aound the geometcal centelne of the stato n a ccula obt, wth a cetan fequency, known as the whlng fequency, and wth a cetan adus, known as the whlng adus [8]. The often mentoned basc modes, statc

2 Volume 45, Numbe 5, and dynamc eccentcty, ae two specal cases of whlng moton. The statc eccentcty means that the oto dsplacement s statonay compang wth the stato, and the whlng fequency s equal wth zeo. In dynamc eccentcty, the poston of the mnmum a gap otates wth the oto, and the whlng fequency s the same as the mechancal angula speed [7] [8]. Knowledge of the electomagnetc foces at dffeent whlng fequences povdes a good bass fo desgnes and eseaches wth whch to deal wth some of the demands that exst fo the electcal machnes of the next geneaton. The calculaton of the electomagnetc foces has been a vey popula eseach topc dung the last decades [5]. The laws of electomagnetsm ae based on the Maxwell s equatons wtch makes a lnk between the magnetc feld H and the souce of the cuent wth the cuent densty and also between the electomagnetc nducton B and magnetc vecto potental : B ot J ot H dv B B H J (4) Thee ae two basc methods used to calculate the foces actng between the oto and stato, namely methods based on the pncple of the vtual wok and methods based on the Maxwell s stess []. Fo the fst one we pesume that the stuctue has an nfnte length so fo the nducton vecto we have only two components B x and B y smla wth the feld H ; pesumng ths we educed the J vectos and at the component wtch follows z decton [4] [5]. Combnng the equatons we have []: Bx By H ot H z y H x (5) wtch conduce us at the Posson s dffeental equaton: J (6) Eule s theoem poves that the esoluton of ths equaton conssts of mnmzng the enegy functon W, wtch s gven by []: W Ω dxdy (7) J The ntegal s efeed to the ente doman (all the sufac Ω. We dvde ths suface on lttle elements usng the fnte element method (tangulaton) [] [6]. The potental n evey pont of one such element ( s defned such as the potental of thee sums, dependng of the coodnates of the tangle and the coespondng potentals, such as: ( e ) N N N N (8) Mnmzng enegy functon s possble when the patal devatves concenng the potentals of evey node ae equal wth zeo: W (9) Equaton (8) poves that we have a double ntegal on the doman wtch conssts of n elements (e ) n : W ( ( e ( ) n n x x () ( ( ( J dxdy y

3 57 If we consde that J s unfom then the foce s equal at []: F JN dxdy ( e ) n () n In the calculaton of foces and toques n the fnte element analyss of electc devces, the methods based on the Maxwell stess tenso ae commonly used. The electomagnetc foce s obtaned as a suface ntegal [8]: F σ ds () F S S ( B n) B B n ds () whee σ s Maxwell stess tenso, n s unt nomal vecto of the ntegaton suface S and B s magnetc flux densty. In a twodmensonal model, the suface ntegal s educed to a lne ntegal along the a gap [7] [8]. If a ccle of adus s taken as the ntegaton path, the foce s obtaned fom the equaton: F π B Bφ eφ ( B B ) φ dφ e (4) whee B, B φ ae the adal and tangental components of the flux densty. If the soluton wee exact, the calculated foce depends geatly on the choce of the ntegaton adus. Howeve, the foce would be ndependent on the ntegaton adus when vaes wthn the a gap. The most appoxmates esults ae obtaned f the lne ntegal n equaton (4) s tansfomed to a suface ntegal ove the coss secton of the a gap: F B B e S s φ φ ag (5) ( B Bφ ) e ds whee s and ae the oute and nne ad of the a gap and S ag s the coss sectonal aea of the a gap. The dawback of the above method s the assumpton of the otatonal symmety [8]. The moto studed s a pemanent magnet synchonous machne wth an exteo oto [] [9]. The stuctue and the chaactestcs of ths moto ae pesented n Table II. and n Fgue II.: Table II. The chaactestcs Rated powe (W) Maxmum speed (ot/mn) Rated load toque (N.m) 9.55 Maxmum toque (N.m) 5 Rated cuent () 8. Effcency (%) 7 Total mass of actve pats (kg) 5. The mass of the magnets (kg).9 Exteo damete (m).5 The length of the moto (m).54 Numbe of poles pas 6 Numbe of the slots 6 The nducton n a gap (T).69 Fgue II.. Fo calculatng and expessng the foces we ae usng FLUXD pogammng envonment, wtch s usng fnte element theoy [9]. The W pemanent magnet synchonous moto that we ae usng s demandng fo a vey complex netwok of tangulaton so, at the end, we have appoxmately 458 nodes and 64 tangles. III. Results To obseve the behavo n good condton estate of the pemanent magnet synchonous machne we analyze the electomagnetc foces that appea nsde the

4 Volume 45, Numbe 5, 4 57 moto. We obtaned a egulate foce; the egulaty can emphasze the good behavo of the machne (Fgue III.): N Foce E 5.E-.E- The values of the foces actng n the magnets ae ncluded n the N nteval. The medum value of these foces s: F67 N. Ou pupose s to develop some paametes and afte that to use these paametes fo calculatng the elablty and avalablty of the moto. To follow ou goals we faulted the machne to obseve the electomagnetc foces actng nsde the moto. We ceated fou postons of the eccentc oto and we decaled the oto close to the stato. t the begnnng (n good functonng estat we had all aound an a-gap of mm. we moved the oto so that we can have a poston whee thee s a smalle a-gap. In that place we deceased the a-gap wth. mm;.5 mm;. mm and.5 mm (Fgue III.): Machne n good functonng condton stato oto.5e- mm.e-.5e-.e-.5e- s Fgue III.. 4.E- Faulted machne.9mm.8mm Fgue III.. 4.5E- 5.E-.85mm.75mm Tme The unbalanced oto ceates a hghe concentaton of foces n the poston mentoned above. Now we can calculate the magnetc pessue n the magnets. We obseve that the magnetc pessue n the magnets aea has nceased (Table III.): Poston of the oto Table III.. Magnetc pessue [N/mm ] Good functonng estate. Decaled wth. mm.7 Decaled wth.5 mm.7 Decaled wth. mm.79 Decaled wth.5 mm.86 The fact that the magnetc pessue has nceased leads us to obseve the level of electomagnetc foces. Evey step of the oto close to the stato nceases these foces (Fgue III.). The nteval that ncludes these foces s also nceasng. Fo the fst movement of the oto (Fgue III..a) the values of the foces ae ncluded n the nteval of N, fo the second movement (Fgue III..b) we have an nteval of N, the thd movement (Fgue III..c) bngs le level of the foces n the nteval of N and the last movement (Fgue III..d) nceases the foces up to N. the foces that appea n the faulted machne ae not so egula and thee s a peak n the poston whee the oto s close to the stato; thee the magnetc pessue nceases and the electomagnetc foces ae also nceased. In the Table III. we pesent the medum values of the foces and the Fmax Fmn dstoton of these foces τ and Fmax also the level of the foces n pecentage. IV. Concluson Fo calculatng the elablty and avalablty of an tem we need cetan paametes. The goal of ths study s to analyze the level of the electomagnetc foces and of the magnetc pessue untl the tme when the machne s wokng n a pope state. We ae able to obseve that the foces

5 foce (N) foce (N) E.E-.E-.E- 4.E- 5.E- 6.E tme (s) a) b) wll ty to fnd a patten fo a multple types of small electcal motos n whch the level of the electomagnetc foces gves us the state of the machne. Poston of the oto Good functonng estate Decaled wth. mm Decaled wth.5 mm Decaled wth. mm Decaled wth.5 mm Table III.. Medum Foces values [N] Level of the foces [%] Dstoton of the foces τ 67 N %.8 68 N % N % N 4 %.99 7 N 5 % foce (N) 65.E.E-.E-.E- 4.E- 5.E- 6.E foce (N) E tme (s).e.e-.e-.e- 4.E- 5.E- 6.E-.E-.E- tme (s).e- tm e (s) 4.E- 5.E- 6.E- Fgue III.: a) oto decaled. mm; b) oto decaled.5 mm; c) oto decaled. mm; d) oto decaled.5 mm. ncease wth evey step of the oto close to the stato. In the futue pat of ths study we c) d) V. Bblogaphy. Tenhunen,. - nalyss of adal foce as a souce of vbaton n an nducton machne wth skewed oto slots, ICEM, Espoo Fnland, p Belahcen,. - Foce calculaton fo vbaton and nose analyss of electcal machnes, ICEM, Espoo Fnland, p Campeanu,., Iancu, V., Radulescu, M.M. - Maşn în acţonă electce, Edtua Scsul Românesc, Caova, Clenet, S. - Influence de l almentaton électonque su le couple et le compotement vbatoe des machnes synchones à amant pemanents, Thèse soutenue le 4 octobe 99, L Insttut Natonal Polytechnque de Toulouse. 7. Faah, P.S. - Etude de la éducton des vbatons des machnes synchones, 6 jullet 995.à L Insttut Natonal Polytechnque de Genoble. 8. Rojas, C.H., Meleo, M.G., Cabanas, M.F., Donson, M.P., Solaes, J.- nalyss of vbaton modes n the oto of hgh voltage nducton machnes by means of the fnte element method, ICEM, Espoo Fnland, p Tenhunen,. Electomagnetc foces actng between the stato and eccentc cage oto dssetaton fo Docto of Scence n Technology, Helsnk. CEDRT, FLUXD. Veson 7.5. Tutoaux de magnétostatque, de themque pemanent et de themque évolutf d électostatque.