Computation and Analysis of Propellant and Levitation Forces of a Maglev System Using FEM Coupled to External Circuit Model

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1 J. Elctromagntic Analysis & Applications, 00, : doi:0.436/jmaa Publishd Onlin April 00 ( Computation and Analysis of Propllant and Lvitation Forcs of a Maglv Systm Using FEM Coupld to Extrnal Circuit Modl Shiyou Yang, Lin Yang, Qiang Zhou Collg of Elctrical Enginring, Zhjiang Univrsity, Hangzhou, China. shiyouyang@yahoo.com Rcivd Dcmbr 3 st, 009; rvisd Fbruary th, 00; accptd Fbruary 5 th, 00. ABSTRACT This papr studis th propllant and lvitation forcs of a prototyp maglv systm whr th propllant forcs ar providd by a linar motor systm. For this purpos, th mathmatical modl and mthod using finit lmnt mthod coupld to xtrnal circuit modl is dvlopd. Th dtails of th propllant and lvitation forcs for a prototyp maglv systm undr diffrnt oprating conditions ar invstigatd, and som dirctions ar givn for practical nginring applications. Kywords: Maglv Systm, Propllant Forc, Lvitation Forc, Numrical Computation, Finit Elmnt Mthod. Introduction Du to th xclusiv salint advantag of non-contact surfacs, an vr-incrasing ffort has bn ddicatd to th application of th magntic lvitation tchnology in nginring disciplins such as maglv trains [], maglv carrirs [], maglv barings [3], maglv vibration isolators [4], and so on. On of th ky problms in th us of maglv tchnologis is th accurat and fficint dtrmination of th propllant and lvitation forcs. Strictly spaking, th lctromagntic fild of a maglv systm is a complx thr dimnsional on involving movmnt componnts. To prdict prcisly th transint lctromagntic phnomna of this kind of problms, th complications such as th saturation of iron matrials and th rlativ movmnt of th moving componnts must b takn into account proprly. Morovr, in a maglv systm, th lctromagntic phnomna ar jointly gnratd by th propllant, th lvitation and th guiding systms. Thus, on nds to considr all of ths issus in th numrical simulations. In addition, th us of powr lctronic dvics in th front nd of th linar synchronous motor maks th situation vn mor srious sinc a voltag sourc must also b modld. Thrfor, on nds to rsort to th numrical tchniqus of lctromagntic fild computations. Gnrally spaking, to addrss th aformntiond issus of th lctromagntic problm of a maglv systm, it is ssntial to us thr dimnsional finit lmnt mthods [5]. Howvr, th havy computational burdn for th implmntation of 3D finit lmnt mthod is numrically unaffordabl for most application cass. In this point of viw, th D tim stpping finit lmnt mthod [6] is proposd to coupl to xtrnal circuit modl to study th transint prformancs of a maglv systm with th goal of dvloping an fficint and accurat numrical tool for th computations of th propllant and lvitation forcs of a maglv systm.. Mathmatical Modls and Mthods. Finit Elmnt Modl Th transint lctromagntic filds in th propllant and lvitation systms ar dtrmind using D tim-stpping finit lmnt mthod. To dtrmin th fild distribution at ach tim stp in a Maglv systm, th D transvrs sction spanning on pol pitch of th linar synchronous motor is studid. As shown in Figur, th solution domain is comprisd of th stator winding, th stator cor, of th linar motor; th rotor cor, and th winding of th lvitation magnt/th rotor. With th displacmnt currnt bing nglctd, th fild is govrnd by A(, xyt,) Axyt (,,) daxyt (,,) ( v ) ( v ) J x x y y dt () A A A 0 ab gh 0 whr, is th rluctivity, is th conductivity, J is th Copyright 00 SciRs.

2 Computation and Analysis of Propllant and Lvitation Forcs of a Maglv 7 Systm Using FEM Coupld to Extrnal Circuit Modl Figur. Th Schmatic diagram of th studid modl sourc currnt dnsity, A(x,y,t) is th magntic vctor potntial. For a moving conductor with a vlocity v, th inducd lctric dnsity is da A v B dt t () Using th Galrkin approach to discrtiz (), on obtains whr, [ S]{ A} [ C]{ I} [ T] d { A} (3) dt 0 0 T T D [ S] [ S] ([ N] [ N] [ N] [ N ]) dxdy, x x y y 0 0 T [ ] [ ] [ ] [ ] D T T N N dxdy, N C C N dxdy. 0 0 [ ] [ ] T [ ] Sa b D whr, a is th numbr of th paralll branchs of th winding, N is th numbr of total srial turns in on coil, S b is th slot ara, [N] is th shap function of th finit lmnt mthod.. Extrnal Circuit Modls In gnral, th sourc currnt dnsity in () is unknown. Thus, th xtrnal lctric circuit modl is coupld to th finit lmnt formulation in this papr to considr th voltag sourc. Morovr, th nd ffct of th propllant and lvitation systms is also modld in this circuit formulation. Figur dpicts schmatically th quivalnt circuit modl of th winding of on phas. Mathmatically, th xtrnal circuit modl, including th lctromotiv forc, th lakag inductanc L, and rsistanc R of th windings, is d {} u {} [ R]{} I [ L] {} I (4) dt whr, [u] = [-u f u a u b u c u d ] T, [I] = [i f i a i b i c i d ] T, [R] = diag [r f r a r b r c r d ], [L ] = diag [L f L a L b L c L d ] is th lakag inductanc considring th nd ffct of th machin and lvitation magnt, subscript f dnots th winding of th lvitation systm/th rotor of th linar motor. To xprss in trms of A(x,y,t), on has pl [ C] d f [ A] (5) dt whr L f is th ffctiv lngth of th cor of th propllant and lvitation systms..3 Coupling of Finit Elmnt and Extrnal Circuit Modls Intgrating (3) and (5) as a whol, on rads S 0 CA R I T PL.4 Tmporal Discrtization f da 0 dt 0 L di u dt Applying Crank-Nicolson algorithm to (6) with rspct to th tim variabl, on obtains th following quation st of th tim-stpping finit lmnt coupld to xtrnal circuit modl n n T S C t A T 4 pl I f C L R t t n n T S C t T 4 plf C L R t t A I n n 0 u Rarranging (7) into a symmtric form, on has whr, (6) (7) n n T S C n A h T hr L I C 4plf plf (8) n n T S C n 0 h A hu T hr L I C 4 pl f 4plf pl f U u u n n, h is th siz of th tim stp..5 Tratmnt of th Rlativ Movmnt of Diffrnt Componnts in a Maglv Systm To considr th rlativ movmnt of a maglv systm, th moving boundary is usd [7] in this papr. Th moving Copyright 00 SciRs.

3 7 Computation and Analysis of Propllant and Lvitation Forcs of a Maglv Systm Using FEM Coupld to Extrnal Circuit Modl Figur. Th quivalnt circuit of on phas winding boundaris ar th boundaris conncting th stator and rotor mshs of th propllant systm, and will vary with th movmnt. Th nods in th stator sid and thir countrparts in th rotor sid satisfy ithr priodic or smi-priodic boundary conditions. For xampl, for a spcific rlativ position of th stator and rotor as dpictd in Figur, th moving boundary conditions ar A A cd f A A (9) d.6 Numrical Computation of th Propllant and Lvitation Forcs d Onc th lctromagntic fild is dtrmind in ach tim stp, th corrsponding propllant and lvitation forcs of th maglv systm in that tim instant can b dtrmind from f x x x y x z nx S y x y y y z nyds n z z x z y z z {[ 0( Bx B ) nx0bxbyny 0BxBn z z] S x (0) [ 0( By B ) ny 0BxBynx0ByBn z z] y [ 0( Bz B ) nz 0BxBn z x0bybn z y] z} ds In th cas study, f x is th propllant forc, and f z th lvitation forc. 3. Numrical Rsults To prdict th transint prformancs of a maglv systm, th computr cods using th proposd modls and mthods ar dvlopd by th authors. Th cods ar programmd in Fortran languag. 3. Validation To validat th proposd mathmatical modl and mthod as wll as th computr cods, thy ar firstly usd to comput th no-load propllant and lvitation forcs of a prototyp Maglv train in Shanghai commrcial Maglv Lin. Th msh of total 4784 nods and 997 lmnts as shown in Figur 3 is usd in this cas study. Th diffrncs btwn th computd lvitation and propllant forcs and th xact ons for both forcs ar lss than 5%. Obviously, this cas study has positivly confirmd th robustnss and fasibility of th proposd modl and mthod for solving th lctromagntic filds of a Maglv systm problm. 3. Th Study of Propllant and Lvitation Forcs of a Prototyp Maglv Systm Aftr th accuracy and fasibility of th proposd modl and mthod ar confirmd, th transint prformancs of th propllant and lvitation forcs of a prototyp Maglv systm undr diffrnt oprating conditions ar studid. Firstly, th ffct of th torqu angl of th linar motor on th propllant and lvitation forcs is invstigatd. As shown in Figur 4, th torqu angl is dfind as th angl Figur 3. Th mshs of on pol pitch rgion usd for validations Figur 4. Th schmatic diagram of th torqu angl Copyright 00 SciRs.

4 Computation and Analysis of Propllant and Lvitation Forcs of a Maglv 73 Systm Using FEM Coupld to Extrnal Circuit Modl in lctrical dgrs btwn th axs of th magntic filds gnratd by th stator winding (windings A, Z, B in Figur ) and by th rotor winding. Th valu of this angl will dtrmin th initial rlativ position of th stator and rotor, i.., th priodic and smi-priodic boundary conditions of Equation (9). Figurs 5-7 show th computd transint propllant and lvitation forcs in cass of diffrnt torqu angls, and Figur 8 dpicts th rlationship of th avragd propllant and lvitation forcs with torqu angls. It is should b pointd out that Propllant forc(pu).0.0 th forcs as givn in ths figurs ar normalizd to thir ratd valus. From ths numrical rsults, it is obviously that ) Th torqu angl of th linar motor has a significant ffct on both th propllant and lvitation forcs; ) Th avragd propllant and lvitation forcs will chang priodically with th torqu angl. Thrfor, th running stat of a Maglv systm can b controlld by changing th torqu angl of th linar motor.. Propllant forc(pu) Lvitation forc(pu) t (lctric dgr) Figur 5. Th transint propllant and lvitation forcs at = Lvitation forc(pu) Propllant forc(pu) t (lctric dgr) Figur 7. Th transint propllant and lvitation forcs at = Lvitation forc(pu) t (lctric dgr) Figur 6 Th transint propllant and lvitation forcs at = Propllant forc(pu) Lvitation forc(pu) (lctric dgr) Figur 8. Th avragd propllant and lvitation forcs undr diffrnt powr angl Copyright 00 SciRs.

5 74 Computation and Analysis of Propllant and Lvitation Forcs of a Maglv Systm Using FEM Coupld to Extrnal Circuit Modl Sinc th front nd of th linar motor is gnrally a stady lctronic dvic, typical a PWM sourc for high prcis motion control, th high frquncy harmonics ar unavoidabl in nginring applications of a Maglv systm. In this rgard, th ffcts of harmonics on th propllant and lvitation forcs ar thn studid using th proposd numrical modl and mthod. In th computr simulation, th harmonic voltags ar formulatd as V 0.V cos[5( f t)] () h f f whr V f and f f ar, rspctivly, th amplitud and frquncy of th fundamntal voltag of th sourc. In th numrical implmntation, th torqu angl of th linar motor is st to diffrnt valus. Figurs 9 and 0 dpict, rspctivly for torqu angl = 0 and = 0, th diffrncs of th transint propllant and lvitation forcs of th prototyp maglv systm btwn th normal oprating condition and th aggrgation of harmonic voltags and currnts. From ths numrical rsults, it is clar that: ) Th avragd valus of th diffrncs of th transint propllant and lvitation forcs for on priod will approach zro, this mans that th harmonics of th sourcs hav almost no ffct on th stady stat prformancs of th Maglv systm; ) In viw of th transint propllant and lvitation forcs, th aggrgation of harmonic voltags and currnts will rsult in that th forcs oscillat around thir ratd valus, rsulting in dgradation in th transint prformancs of th Maglv systm; 3) Rlativly, th ffct of a small harmonics on th lvitation forcs can b nglctd compard to that on th Figur 9. Th diffrncs of th transint propllant and lvitation forcs btwn th normal oprating condition and th aggrgation of harmonic voltags and currnts at = 0 Figur 0. Th diffrncs of th transint propllant and lvitation forcs btwn th normal oprating condition and th aggrgation of harmonic voltags and currnts at =0 propllant forcs. 4. Conclusions A modl and mthod for computing th transint propllant and lvitation forcs of a Maglv systm, with rlativ rror bing smallr than 5%, is proposd basd on tim stpping finit lmnt mthod coupling to xtrnal circuit modls. In addition to having th advantags to considr complications such as th rlativ movmnt of diffrnt componnts of th linar lctrical machins, th saturation of iron matrials, th proposd modl and mthod can also tak into account th intraction of an xtrnal voltag sourc which is vry common with th incrasing us of powr lctronics dvics in th front nd of gnral lctromagntic dvics. Basd on th computr simulation of this cas study, for a maglv systm, it is concludd that: ) Th torqu angl of th linar motor has a significant ffct on both th propllant and lvitation forcs; ) Th avragd propllant and lvitation forcs will chang priodically with th torqu angl; 3) Th addition of small harmonics in th sourcs has almost no ffct on th stady prformancs; 4) Howvr, th aggrgation of small harmonic voltags and currnts in th sourcs will rsult in dgradation in th transint prformancs. 5. Acknowldgmnts This work is supportd by th Scinc and Tchnology Dpartmnt of Zhjiang Provinc undr Grant No. 008C30. Copyright 00 SciRs.

6 Computation and Analysis of Propllant and Lvitation Forcs of a Maglv Systm Using FEM Coupld to Extrnal Circuit Modl 75 REFERENCES [] Y. G. Guo, W. Xu, J. G. Zhu, H. Y. Lu and Y. Wang, Dsign and Analysis of a Linar Induction Motor for a Prototyp HTS Maglv Transportation Systm, Procdings of 009 IEEE Intrnational Confrnc on Applid Suprconductivity and Elctromagntic Dvics, Chngdu, 5-7 Sptmbr 009, pp [] M. Morishita, T. Azukizawa, S. Kanda, N. Tamura and T. Yokoyama, A Nw MAGLEV Systm for Magntically Lvitatd Carrir Systm, IEEE Transactions on Vhicular Tchnology, Vol. 38, No. 4, Novmbr 989, pp [3] Y.-M. Choi, M. G. L, D.-G. Gwon and J. Jong, A Nw Magntic Baring Using Halbach Magnt Arrays for a Magntic Lvitation Stag, Rviw of Scintific Instrumnts, Vol. 80, No. 4, April 009. [4] K. Watanab, S. Hara, Y. Kanmitu, T. Haga, K. Yano, T. Mizuno and R. Katamura, Combination of H and PI Control for an Elctromagntically Lvitatd Vibration Isolation Systm, Procdings of th 35 th Confrnc on Dcision and Control, Kob, -3 Dcmbr 996, pp [5] M. Rizzo, A. Savini and J. Turowski, 3-D Finit Elmnt Analysis of a Linar Rluctanc Motor by Using Diffrnc Scalar Potntial, IEEE Transactions on Magntics, Vol. 3, No. 3, May 996, pp [6] L. Y. Xu and W. N. Fu, Evaluation of Third Harmonic Componnt Effcts in Fiv-Phas Synchronous Rluctanc Motor Driv Using Tim Stpping Finit Elmnt Mthod, Procdings of Industry Applications Confrnc, Rom, Vol., 8- Octobr 000, pp [7] R. Y. Tang, Y. Hu, Z. H. Lu, S. Y. Yang and L. J. Maio, Computation of Transint Elctromagntic Torqu in a Turbognrator undr th Cass of Diffrnt Suddn Short Circuits, IEEE Transactions on Magnctics, Vol. 6, No., March 990, pp Copyright 00 SciRs.