Time-Harmonic Modeling of Squirrel-Cage Induction Motors: A Circuit-Field Coupled Approach

Transcription

1 Excrpt from th Procdings of th COMSOL Confrnc 28 Boston Tim-Harmonic Modling of Squirrl-Cag Induction Motors: A Circuit-Fild Coupld Approach R. Escarla-Prz,*, E. Mlgoza 2 and E. Campro-Littlwood Univrsidad Autonoma Mtropolitana - Azcapotzalco, Dpartamnto d Enrgia, Av. San Pablo 8, Col. Rynosa, C.P. 22, Mxico D.F., MEXICO, 2 Instituto Tcnologico d Morlia, Av. Tcnologico 5, Morlia, Mich., C.P. 582, MEXICO. *Corrsponding author: Av. Cuauhtmoc 963, Dpto 44A, Col. Narvart Ponint, Mxico, D.F., MEXICO, r.scarla@i.org Abstract: Finit lmnt modling of thrphas induction machins rquirs th solution of coupld circuit and fild quations. This work aims to solv this problm using a strong coupling approach. Th machin stator is fd through a thr-phas non-idal voltag sourc and th nd winding ffcts ar accountd for with inductancs and rsistancs. Th bars of th doubl squirrl cag of th rotor ar modld as solid conductors and intrconnctd through ndring sctions that hav finit impdanc. Sinc stady-stat is considrd, a tim-harmonic approach has bn usd to prform th machin simulations. Kywords: Induction machins, Circuit-fild Coupling, 2D finit-lmnt analysis.. Introduction Th accurat modling of induction machins rquirs prcis rprsntation of gomtry, matrial proprtis and xcitations. Currnt-fd modling can b radily achivd with classical finit lmnt formulations. Howvr, squirrlcag asynchronous machins ar voltag fd at th stator trminals through xtrnal impdancs and a voltag sourc. In addition, th rotor has a rathr complicatd arrangmnt of bars, which ar short-circuitd with nd-rings of finit impdancs, that nds to b considrd. This lads to a coupld problm whr fild and circuit variabls must b solvd togthr. Thr ar basically two approachs to achiv this goal: a) simultanous solution of both systms of quations (strong coupling) and b) fild and circuit systms ar solvd indpndntly, using thir rspctiv solutions to xcit ach sparat systm (wak coupling). Th thory for strong coupling of fild and circuit systms can b found for instanc in rfrncs [-5]. Publishd xampls of wak coupling approachs ar givn in [6-8]. Nglcting th xtrnal circuit connctions can lad to important rrors in th prdiction of th machin torqu and currnts. Othr difficulty whn daling with induction machins is th small air-gap that rquirs spcial attntion whil obtaining a propr mshing of this domain. This work addrsss ths problms, namly circuit fild coupling and propr air-gap mshing, using th AC/DC modul of COMSOL Multiphysics. As a rsult, a quasi-3d modl can b obtaind with accurat rprsntation of ovrhang ffcts and coupling with xtrnal circuits. Solid and filamntary conductors ar proprly accountd for. Th modling of rotor and stator with mshs of diffrnt dnsity is also prsntd along with th incorporation of rotor motion through modification of rotor conductivitis and rsistancs ( slip rfrring to stator frquncy). Antipriodic boundary conditions ar considrd in this work to rduc th siz of th final numrical modl. Comparison of th full modl againts modls nglcting coupling with xtrnal circuits and ovrhang ffcts clarly shows th importanc of considring quasi-3d modls. A brif discussion about th us of tim-harmonic modls in th dtrmination of initial conditions rquird by transint modls is also givn. Finally, th manuscript shows th calculation of th torqu-slip curv implmntd with th aid of COMSOL s Maxwll strss implmntation. 2. Magntic Fild Equations Th Maxwll s quations that fully dscrib th modling of low-frquncy lctromagntic dvics in two dimnsions ar B E = t B = H = J ()

2 whr th displacmnt currnt and fr chargs 2 hav bn disrgardd. E, B, H ar th lctric fild, th magntic flux dnsity and th magntic fild, rspctivly. Thy ar strictly containd in a plan. J is th currnt dnsity and its dirction is prpndicular to th plan of E, B and H. Th quations () can b combind to giv th following diffusion-typ quation A σ + ( A) t r σδv σ v ( A) = +J d (2) with B= A, whr A is th magntic vctor potntial and is paralll to J, is th magntic prmability of vacuum and r is th rlativ prmability. J is an xtrnal currnt dnsity imposd in conductor rgions and it is also paralll to J. It is uniform at th conductor cross sction. This currnt dnsity is usually unknown sinc its valu dpnds on th magntic vctor potntial and th xtrnal lmnts connctd to th conductor. Conductors hav lngth d with a potntial diffrnc ΔV which is usually unknown as wll. v 2 is a vlocity trm that may account for moving conductors in problms whr th sam gomtry is prsrvd as th conductor changs position. Analytical solution of (2) is difficult or impossibl for rotating lctrical machins du to thir intricat gomtris, propr considration of matrial proprtis and xtrnal lmnts intracting with thm. Numrical approximations ar normally mployd and th Finit Elmnt Mthod [9] is widly usd to obtain high quality solutions. If th static rgions and th moving conductors ar ach modld in thir own rfrnc frams (motion is xplicitly modld), th vlocity trm can b droppd out as follows A σδv σ + ( A) = + J t d 2. Tim-harmonic rprsntation r (3) If all magntic and lctric quantitis hav sinusoidal variation with an lctrical angular spd ω, thy can b convnintly rprsntd in th frquncy domain using th classical phasor concpt of circuit thory. This 2 way, E, B, H and J, ΔV ar now complx quantitis whras / t is substitutd by jω. Equation (2) can thn b rwrittn as jωσ A+ ( A) r σδv σ v ( A) = + J d 3. Induction machin basics (4) Th stator currnts of thr-phas induction machins oprating undr balancd and stady stat conditions produc a magntic fild that rotats at synchronous spd ω s. ω s = ω for two-pol machins. Th rotor of an induction machin movs with a spd ω r that diffrs from th synchronous spd and thrfor voltags and currnts ar inducd in th rotor circuits. Th spd diffrnc can b calculatd as sω s, whr s is known as th rotor slip. Thus, th rotor currnts produc a magntic fild rotating at this slip spd with rspct to th rotor, which in turn is moving at vlocity (-s)ω s. As a rsult, th magntic fild of th rotor currnts rotats at th sam spd of th stator fild. Th intraction of ths two magntic filds producs an lctromagntic torqu. If th machin is xcitd with sinusoidal quantitis, it is possibl to hav a static gomtry in th frquncy domain. This implis that slot ffcts ar disrgardd, but this ffct can b rducd by positioning th rotor in a location whr th lctrical quantitis hav thir avrag valu ovr an lctrical cycl. Th frquncy of th voltags and currnts in th rotor conductors diffrs from th frquncy f in th stator conductors. Th rotor frquncy is proportional to th slip (sf) and this must b rflctd in th machin modling. In standard lctrical machin thory this is achivd by using an quivalnt circuit whr th rotor impdanc is a slip function. Th rprsntation of induction machins with (4) must acknowldg this fact. Thr ar two ways to achiv this goal. On involvs using th classical concpts of lctrical machin thory and th othr on involvs th manipulation of th vlocity trm in (2) (s blow).

3 3. Stator fild quations It is convnint to divid th machin in two sub-domains: stator and rotor. Th stator domain contains th stator cor, th phas conductors and part of th air gap. Th rotor domain contains th rotor cor, th rotor conductors and th rmaining part of th air gap. Th stator winding conductors can usually b considrd with ngligibl ddy-currnt ffcts (filamntary conductors) sinc th dimnsions of thir cross sctions ar comparabl with th skin dpth at powr frquncy. Thus, th stator can b rprsntd with th following st of quations manipulating th σ v ( A) trm as shown in [] with spac and tim phasors. 4. Extrnal circuit conditions Extrnal conditions that cannot b dirctly includd within th fild modl must b incorporatd in ordr to proprly prdict th machin opration. Th stator is connctd to a thr-phas voltag sourc through th ndwinding impdanc. Figur shows th intrconnction of th fild and stator circuit modl. ( A) = J ( A) = ( A) = r (5) Th first quation applis to stator conductor rgions. Air is rprsntd with th scond quation whil iron cor is dscribd by th third quation. A and J vary with ratd frquncy. 3.2 Rotor fild quations Th conductors of squirrl cag typ rotors must considr th ddy currnt ffct. Hnc, th st of quations for th rotor domain ar givn by σ ( s) ΔV jωσ () s A+ ( A) = d ( A) = ( A) = r (6) whr th conductivity is a function of s. Actually σ(s) is proportional to th actual rotor bar conductivity (σ(s)= sσ) and rfrs th rotor quantitis to th stator frquncy. This procdur is fully quivalnt to th drivation of th quivalnt circuit of induction machins in classical lctrical machin thory whr th rotor rsistanc is dividd by s []. An quivalnt way of gtting this rsult is by Figur. Voltag sourc fding th induction machin. r s is th sourc rsistanc whil L oh is th nd-winding inductanc of th stator windings. Th rotor conductors ar also intrconnctd through xtrnal impdancs. For a squirrl cag rotor, th conductors ar solid and short-circuitd through conducting nd rings of finit impdanc. Th quivalnt circuit of th rotor circuit is shown in Figur 2. Th squirrl cag bars ar rprsntd by gray rctangls and thy blong to th fild domain. Th shown rsistancs r r hav all th sam valu and account for th nd-ring sctions that intrconnct th cag bars. Likwis, th inductancs of th cag circuit L r hav all th sam valu. Th nd-ring rsistancs must b dividd by s, just as in th quivalnt circuit of induction machins, for corrct rprsntation of th rotor impdanc at stator frquncy. Notic that th rotor circuit shown in Figur 2 accounts for th anti-priodic boundary conditions of th machin. Only on pol pitch is rquird to rprsnt th whol machin bhavior.

4 frquncy). Equations (7) () coupl th fild modl with th xtrior circuit systm. 6. Numrical modl and rsults Figur 2. Squirrl cag quivalnt circuit with cag bars rprsntd by th fild modl. r r and L r ar th rsistanc and inductanc of th nd-ring sction conncting two cag bars. 5. Currnt and voltag rlationships Th coupling of circuit and fild variabls rquirs xplicit xprssions for th voltags in all machin conductors. Th voltag xprssion for filamntary conductors is asily obtaind from Faraday s law and th voltag drop causd by th filamntary conductor rsistanc: = ir+ Ndjω A (7) whr N is considrd th numbr of srisconnctd conductors in a givn cross sction and jωna th conductor flux linkags pr unit dpth. i is th circulating currnt in ach filamntary conductor. Th currnt dnsity of th N filamntary conductors is givn by Th induction machin considrd in this work is a doubl squirrl cag induction motor. It is a two-pol, 7.5 kw, 38 V, 5 Hz, thrphas star connctd motor. Its gomtry is dpictd in figur 3 and its finit-lmnt msh is shown in figur 4. Lagrang-quadratic lmnts wr us to prform th numrical simulations. Two indpndnt mshs wr first constructd: on for th stator domain and anothr for th rotor rgion (s sub-sctions 3. and 3.2). Thy wr stitchd togthr using th crat pairs capability of COMSOL. Th prpndicular induction currnts, vctor potntial application mod of th AC/DC modul of COMSOL includs th fild quations prsntd in sction 3, allowing th propr spcification of matrial proprtis and boundary conditions. Th machin also offrs anti-priodic boundary conditions which mans that: J Ni = (8) S whr S is th ara occupid by th N srisconnctd filamntary conductors. Th total currnt circulating through a solid conductor can b found by intgrating jωσ () s A σ () s ΔV and from th first quation of (6) ovr L th conductor cross sction: Figur 3. Machin gomtry. i sol sσδv = jsωσ L A dω (9) Solving for ΔV givs ( sol ωσ ) Δ V = rsol i + js AdΩ () whr r sol is th dc rsistanc of th solid conductor (th rsistanc of th conductor at zro Figur 4. Finit-lmnt msh. Ar (, θ) = Ar (, θ) j8 () for th two-machin pol considrd in this work.

5 Equations (7) and () can b radily computd using intgration coupling variabls at appropriat sub-domains. Th rsultant valus of th intgration coupling variabls ar inputtd as a potntial ΔV diffrnc inout in th sub-domain sttings of th rotor bars. Likwis, quation (8) is calculatd as a global scalar xprssion that is latr usd as a sub-domain stting of stator conductor sub-domains. Hnc, th COMSOL intrfac with SPICE circuit lists bcoms availabl and th problm is fully st up. Figur 5 shows th flux plot for ratd oprating conditions. It can b sn that th magntic fild can pntrat wll into th rotor cor bcaus th slip is small and th ffctiv rotor conductivity rflcts this fact (σ(s)= sσ). Figur 6 shows th flux plot for th lockd-rotor oprating condition with ratd voltag and frquncy applid at th stator trminals. In this cas, th ffctiv rotor conductivity quals th actual conductivity of th rotor bars. Now, it can b obsrvd that th magntic fild is only raching th outr parts of th rotor body du to th strongr ddy currnt ffcts in th squirrl cag. A paramtric analysis can b asily prformd by allowing th slip to vary from to (motor opration). Th lctromagntic torqu for ach slip is computd using th Maxwll strss tnsor. Ths rsults ar dpictd in figur 7, whr th xpctd bhavior of th lctromagntic torqu vrsus slip is radily sn. Figur 5. Ratd Oprating Condition. Figur 6. Rotor at standstill and ratd voltag and frquncy applid at th stator trminals. Figur 7. Elctromagntic torqu vrsus slip. Transint simulations of induction machins ar important to prdict non stady-stat oprating conditions. Th dtrmination of initial conditions is important for accurat transint computation. This mans that in most cass an additional transint simulation is rquird. Computational savings can b obtaind by using tim-harmonic solutions sinc th actual solution would b nar thm. Th tim-harmonic solution is not th xact initial condition bcaus tooth harmonics and non-linarity hav not bn takn into account. Finally, it is important to mphasiz that nglcting th ovrhang ffcts, such as th ndring impdancs, lads to larg rrors. This is shown with th calculatd currnt at ratd oprating conditions. A valu of 8.74 A is obtaind whn th nd-ring impdancs ar considrd whil 2.67 A is found whn thy ar dirgardd. This implis a 45% rror which cannot b nglctd, and strsss th nd of

6 quasi-3d modls such as th on prsntd in this work. 7. Conclusions Tim harmonic modling of induction machins is important for prdicting thir stadystat opration. Frromagntic matrials can only b considrd linar at th momnt with COMSOL or using a rathr complicatd procdur to find an quivalnt complx prmability. It has bn found that intrconnction of xtrnal systms to th fild modl is important to avoid larg rrors. Corrct idntification of solid and filamntary conductors must b carrid out to avoid larg problms and to proprly modl ddy currnt ffcts. Th capacity of COMSOL for stitching two mshs of diffrnt dnsitis provids a convnint way of forming quality mshs at th air gap rgions. 8. Rfrncs. P. Lombard and G. Munir, A gnral purpos mthod for lctric and magntic combind problms for 2D, axisymmtric and transint systms, IEEE Trans. on Magntics, 29(2), (993) 2. J. R. Braur, B. E. MacNal, L. A. Larkin, and V. D. Ovrby, Nw mthod of modling lctronic circuits coupld with 3D lctromagntic finit lmnt modls, IEEE Trans. on Magntics, 27(5), (99). 3. S. J. Salon, M. J. DBortoli, and R. Palma, Coupling of transint filds, circuits, and motion using finit lmnt analysis, Journal of Elctromagntic Wavs and Applications, 4(), 77 6 (99). 4. H. D Grsm, K. Hamyr, and T. Wiland, Fild-circuit coupld modls in lctromagntic simulation, Journal of Computational and Applid Mathmatics, 68(-2), (24). 5. P. Zhou, W. N. Fu, D. Lin, S. Stanton, and Z. J. Cnds, Numrical modling of magntic dvics, IEEE Trans. on Magntics, 4(4), (24) 6. G. Bdrosian, A nw mthod for coupling finit lmnt fild solutions with xtrnal circuits and kinmatics, IEEE Trans. on Magntics, 29(2), (993) 7. P. Zhou, D. Lin, W. N. Fu, B. Ionscu, and Z. J. Cnds, A gnral cosimulation approach for coupld fild-circuit problms, IEEE Trans. on Magntics, 42(4), 5 54 (26). 8. E. Lang, F. Hnrott, and K. Hamyr, A circuit coupling mthod basd on a tmporary linarization of th nrgy balanc of th finit lmnt modl, IEEE Trans. on Magntics, 44(6), (28). 9. J. P. A. Bastos and N. Sadowski, Elctromagntic Modling by Finit Elmnt Mthods. Marcl Dkkr Inc. (23).. A. E. Fitzgrald, C. K. Kingsly and S. D. Umans, Elctric Machinry. Mc Graw Hill. (23).. N. Bianchi, Elctrical Machin Analysis using Finit Elmnts. CRC Prss, Taylor & Francis Group. (25). 9. Acknowldgmnts This work was supportd in part by th National Council of Scinc and Tchnology (CONACYT-MEXICO) and in part by th National Systm of Rsarchrs (SNI- MEXICO).