APPENDIX B Publication P3

Transcription

1 APPENDIX B Publcaton P3 Rpo A.-K., Nnaa A., Aro A. Etatng crcut odl for a dp-bar nducton otor ung t haronc fnt lnt analy. Procdng Intrnatonal Confrnc n Elctrcal Machn, Crt, Grc, Sptbr 26, No. 64, 6 p. 26 Anna-Kaa Rpo, Ao Nnaa, and Antro Aro 5

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3 64 Etatng crcut odl for a dp-bar nducton otor ung t haronc fnt lnt analy Anna-Kaa Rpo, Ao Nnaa, and Antro Aro Abtract A thod for tatng quvalnt crcut odl for dp-bar nducton otor prntd. Th thod bad on th t-haronc fnt lnt analy (FEA). Th paratr of th tudd otor dpnd trongly on th frquncy and a ngl-cag quvalnt crcut th contant paratr prdct th torqu only at th a opraton pont hr th paratr ar dfnd. In th prntd papr, th rotor odlld th ultpl branch and th a crcut odl capabl of dpctng a dr opraton rang of th otor. Indx Tr fnt lnt analy, paratr taton, quvalnt crcut odl T I. INTRODUCTION HE t-tppng fnt-lnt analy accurat and dly appld thod n th tudy and ulaton of lctrcal achn. Elctrcal achn uually oprat n conncton th control crcut, por lctronc, coponnt of lctrcal grd and chanc caung a coplx ntracton. Th coputaton capacty oftn lt th u of coprhnv odl, and a plr analytcal odl for th achn rqurd n any applcaton. Th bac ngl-cag quvalnt crcut odl th contant paratr abl to odl th bhavour of th achn only at crtan opraton pont. Bcau of th n ffct n th rotor bar, th paratr of th crcut odl dpnd on th frquncy. Th valu for th rotor rtanc and laag nductanc vary th ot. Svral propoal hav bn ad n ordr to nclud th n ffct nto th crcut odl, for xapl [], [2]. Uually on or or addtonal rotor branch ar ncludd nto th bac odl, hl th phycal ntrprtaton for th paratr tll prrvd. In th propod papr, a ytatc thod for xtractng th quvalnt crcut paratr fro th to-dnonal t-haronc FEA prntd. For th analytcal odl, th nubr of addtonal branch can b chon dpndng on th rqurnt of th applcaton. Th author ar th th Laboratory of Elctrochanc, Hln Unvrty of Tchnology, FI-25 TKK, Fnland (al: annaaa.rpo@t.f; ao.nnaa@t.f; antro.aro@t.f) II. METHOD OF ANALYSIS A. To-dnonal t-haronc odl A radal-flux cag nducton otor tudd aung a to-dnonal agntc vctor potntal A and currnt dnty J A A( xyt,, ) z. () J J ( xyt,, ) z Th z-ax n th drcton of th haft of th achn. Aung nuodal t varaton, phaor varabl ar ud for th fld oluton. Th tator frquncy qual to th upply frquncy ω. Th rotor frquncy th lp frquncy ω. Th rotor tatonary,.. th ffct of oton ar odlld by aung th lp frquncy n th rotor, only. Aftr th aupton, th quaton for th agntc vctor potntal ( ν ff A) J (2) hr ν ff an ffctv prablty [3] and A th vctorpotntal phaor. Th to-dnonal t-haronc odl can b ud for th cor rgon of a radal-flux achn but t do not odl proprly th agntc fld or ndng conncton n th nd-ndng pac. Th nd ffct ar tan nto account approxatly by couplng th fld quaton th th crcut quaton of th ndng and ung nd-ndng pdanc. Wndng currnt or potntal or both of th ar ud a addtonal varabl. For ntanc, th crcut quaton for a tator pha ( ) N u R L d A S n n n ω n n ω βn Sn hr R n and L n ar th rtanc and nd-ndng nductanc of th n th pha. Th uaton ovr nclud all th col d of pha n. Varabl β n ± dfn hthr a potv or ngatv col d condrd. N n th nubr of turn and S n th cro-ctonal ar of col d. th cor lngth of th achn. Th dtal of cobnng th fld and crcut quaton hav bn dcud aong othr n rfrnc [4]-[6]. A fnt lnt thod ud to olv th fld and crcut varabl nurcally. Sn (3) 7

4 64 2 B. Ipdanc atrx Th a to prnt th crcut quaton of th achn n atrx for u (4) Th dnon of th pdanc atrx ar (Q r ) (Q r ), hr th nubr of pha and Q r th nubr of rotor bar. u a voltag vctor and a currnt vctor. Th lctroagntc couplng btn th ndng can b unquly dfnd only for a lnar yt. Th lnarzaton don by olvng th agntc fld n a chon opraton pont and frzng th prablty of th ron cor to th oluton. Any of th fld-crcut forulaton rfrrd to n Scton II.A can b ud to olv th lnarzaton fld. Th couplng pdanc ar obtand by nrtng a currnt on by on n ach pha ndng and rotor bar and coputng th voltag producd by th currnt n all th ndng and bar. Th currnt dnty n (2) ut b xprd a a functon of th total currnt. For ntanc, a currnt n rotor bar gnrat a agntc vctor potntal A accordng to quaton ( ν A)ωσ A ω σ d z z S A S (5) S S hr ν th lnard rluctvty, th lp, ω th upply frquncy and S th cro-ctonal ara of th bar. Th voltag nducd n a tator pha ndng obtand fro (3). Th voltag ovr rotor bar n un n Rω d S A S (6) n Sn hr th Kroncr ybol δ n aur that a rtv tr prnt only f th currnt and voltag ar aocatd th th a conductor. A lnt z n of th pdanc atrx un zn (7) Th rotor voltag quaton (6) not n propr for for th pdanc quaton (4) a t do not nclud th voltag nducd at th nd of th rotor cag. Th ng tr can b tan nto account by applyng Krchhoff la to th clod h of th cag. On pobl thod dcud n [6]. Whn appld, th voltag quaton for th rotor bco r r rr r r u M ( M ). (8) Th lnt of atrc r and rr ar obtand fro (6) and (7). Matrx r aocatd th th nd-ndng pdanc of th rotor cag. Matrx M tranfor th bar voltag to h voltag. C. Equvalnt crcut for a ngl-cag otor Fg. ho th convntonal T quvalnt crcut. It rprnt th couplng btn a tator pha currnt and a ohat fcttou rotor currnt. u R X X Fg.. Sngl-cag quvalnt crcut odl. r X r R r () A thr ar to currnt, th rlatd couplng atrx ha dnon 2 2. Th approach of th tudy to frt coput th larg pdanc atrx ncludng all th couplng a dcrbd n th prvou cton, and thn rduc th larg atrx to a 2 2 atrx aocatd th Fgur. A thod rlatd to th ytrc coponnt ud for th rducton. If th tator ndng ha pha and th rotor bar ar tan to for a poly-pha yt th Q r pha, th potv pha-qunc tator voltag, tator currnt and rotor currnt ar dfnd u r Q r Qr u 2 π(-) 2 π(-) 2 π(-) r Qr. (9) Th ubcrpt rfr to th potv pha qunc. By ubttutng th tator voltag fro (4) n th frt xpron of Eq. (9), th potv pha-qunc voltag obtand a a functon of th pha and bar currnt 2 π(-) r 2 (-) Q π r r u z z () hr z rfr to an lnt of th pdanc atrx that coupl to tator ndng and z r to an lnt that coupl a tator ndng to a rotor bar. Th frt u n () tan to b th voltag nducd by th potv pha-qunc tator currnt and th cond u th voltag nducd by th potv pha-qunc rotor currnt. Bad on th aupton, th quaton rttn n a pl calar for r r u z z () hr th pdanc z and z r ar obtand by dvdng th u contanng th tator currnt by and th u contanng th rotor currnt by r 2(-) z z. (2) Qr 2(-) r r r z z r Th potv pha-qunc rotor voltag calculatd n a lar annr a th tator voltag abov r r rr r u (3) r 8

5 64 3 Th coffcnt ar Qr r r { M } Q r 2(-) Qr Qr rr rr r r r { M } Q r 2(-). (4) Equaton () and (3) dfn th rlaton btn th potv pha-qunc tator voltag and tator and rotor currnt. A th quaton hav bn drvd fro th voltag quaton of to agntcally coupld poly-pha ndng havng dffrnt pha nubr, th couplng coffcnt z r and r ar not qual. Th qualty forcd by rcalng (3) z hr r z z rr z z r r rr r r rr r ( r ) z z r r r (5) rr. (6) Equaton () and (5) dfn a pl quvalnt crcut of th ngl-cag nducton otor. Hovr, f laag ractanc ar prfrrd, th rotor quantt hav to b rfrrd to th tator. A tator-rfrrd rotor currnt flong n th tator ndng hould nduc a fundantal coponnt of th ar-gap flux that qual to th fundantal coponnt of th flux nducd by th orgnal rotor currnt. Ung th crtron, a rfrnc coffcnt κ obtand r Φ κ (7) r Φ hr Φ and Φ r ar th fundantal coponnt of th argap flux nducd paratly by th tator and rotor currnt. Th flux coponnt ar ntgratd fro th ar-gap vctor potntal. Th paratr rfrrd to th tator and ard by an apotroph ar r r ' κ z ' z r r z ' κ z. (8) r r z ' κ z rr 2 rr z ' κ z Th coponnt of th quvalnt crcut n Fgur ar obtand fro th pdanc rfrrd to th tator R R { z '} { } r { z '} { } rr { z '} X I z ' z ' X r σ I X I z ' z ' r rr r σ r R R. (9) Th lnarzaton of th achn a don to a gvn opraton pont. Th crcut paratr obtand ar, of cour, aocatd th th a opraton pont. D. Mult-cag odl Th thod abov can b gnrald for a ult-cag achn or to a odl n hch th dp bar ar dvdd n vral ub-conductor. Such a dvon hon n Fg. 2. Fg. 2. Rotor bar dvdd nto thr part. A potv pha-qunc rotor currnt dfnd paratly for ach of th ub-conductor layr. If thr ubconductor pr bar ar ud, th larg pdanc atrx a (3Q r ) (3Q r ) atrx, and (), a an xapl, nclud on u for th tator currnt and thr u for th potv pha-qunc ub-conductor currnt. Th rducton of th larg pdanc atrx lad to a 4 4 couplng atrx. Each rotor currnt paratly rfrrd to th tator ung (7). Th crcut quaton for th thr-cag nducton achn of th for u r (2) r r3 Agan, th 4 4 couplng atrx rprnt an quvalnt crcut of th achn. Hovr, ult-cag achn ar oftn odlld ung a laddr-typ quvalnt crcut. Such a crcut th thr rotor branch prntd n Fg 3. c c2 u r r2 r3 r r2 r3 r r2 r3 Fg. 3. Thr-cag quvalnt crcut odl. To fulfl Krchhoff cond la n th crcut of Fg. 3, th couplng atrx hould b of th for dfnd by (2) or at lat, t hould b pobl to tranfor th yt of quaton (2) o that a couplng atrx of typ (2) obtand. Obvouly, th not pobl for a gnral ca. A 4 4 atrx ha 2 lnt but thr ar only 7 coffcnt on th rght hand d of (2). For th qualty, th atrx on th lft hand d ut b ytrc. 9

6 r r2 (2) r3 c c2 Furthror, th follong contrant hav to b fulflld c c c Th dagonal lnt ar r c r2 c c2 r3 c c2 (22) (23) If th couplng atrx fulfl th contrant, th tranforaton to th laddr yt a traght forard proc. If not, th laddr tructur too rtrctv to dcrb th achn. Whn th couplng atrx (2) contructd a dcrbd n Scton II-B II-D, th condton (22) ar t vry ll, and th laddr-crcut paratr gv alot xactly th a achn charactrtc,.. tator currnt and torqu, a th orgnal FEM oluton. Th xprnc to apply for any dvon of th rotor bar to ub-conductor. Pobl probl ar dcud n Chaptr IV. Fg. 4. Gotry of th tator and rotor lot. Tabl I prnt th ratd valu of th tt otor. TABLE I RATED VALUES FOR THE TEST MOTOR. Frt, th ffct of aturaton ar nglctd hn a FE odl th a rlatv prablty of n th ron tudd. Th odl rfrrd a th lnar FE odl. Hovr, th ddy currnt n th rotor bar ar ncludd and thrfor th paratr chang along th th frquncy. In all th prford coputaton, th tpratur of th tator ndng 98 C and rotor 35 C. Th paratr of th ngl-cag crcut odl ar tatd at dffrnt lp. Ung th obtand quvalnt crcut, th torqu vru pd curv ar calculatd. Fg. 5 prnt th curv gvn by th quvalnt crcut th th paratr calculatd at lp.2,.5 and.. Thy ar copard th th torqu obtand fro th t-haronc FEA at dffrnt opraton pont. Clarly, th ngl-cag quvalnt crcut odl prdct th torqu adquatly only nar th a opraton pont hr th paratr ar tatd. Torqu (N) III. RESULTS A 37-W cag-nducton otor tudd. Fg. 4 ho th gotry of th rotor and tator lot Spd (rp) Fg. 5. Torqu vru pd curv for th lnarzd otor. Sold ln:.2, dahd:.5, dottd:, crcl: torqu gvn by FEA. Dvdng th bar nto thr part, a crcut odl th to addtonal rotor branch obtand. Fg. 6 prnt th torqu vru pd curv gvn by th thr-cag quvalnt crcut th paratr calculatd at th thr lp. Th

7 64 5 torqu odlld or accuratly n a dr opraton rang. Torqu (N) Spd (rp) Fg. 6. Torqu vru pd curv for th thr-cag odl (lnar atral). Sold ln:.2, dahd:.5, dottd:, crcl: torqu gvn by FEA. Whn th paratr ar calculatd at th ratd lp th torqu odlld ll only at th pd rang btn th ratd-load and no-load. Th paratr calculatd at lp.5 and. provd or gnral rult. Th paratr of th ngl-cag and thr-cag odl at th ratd opraton pont ar prntd n Tabl II. Th valu of th rtanc and nductanc corrpond to th ral and agnary part of th pdanc prntd n Fg. 3. Th valu of th rtanc ar to b dvdd by th lp hn placd to th crcut odl. Th rotor paratr of th nglcag odl (Fg. ) ar rfrrd a R r and X r. TABLE II PARAMETERS OF THE SINGLE-CAGE AND THREE-CAGE CIRCUIT MODEL AT RATED OPERATION POINT. Nxt, th prablty of th ron allod to dpnd on th flux dnty. Th odl rfrrd a th nonlnar FE odl. Fg. 7 prnt th torqu vru pd curv calculatd ung thr-cag crcut odl th paratr obtand fro th nonlnar FE odl at dffrnt lp. Fg. 8 prnt th corrpondng tator currnt Torqu (N) Spd (rp) Fg. 7. Torqu vru pd curv for th thr-cag odl (nonlnar atral). Sold ln:.2, dahd:.5, dottd:, crcl: torqu gvn by FEA. Currnt (A) Spd (rp) Fg. 8. Stator currnt gvn by th thr-cag odl (nonlnar atral). Sold ln:.2, dahd:.5, dottd:, crcl: torqu gvn by FEA. Snc th aturaton odlld, th valu of th paratr ar affctd by th upply voltag. Fg. 9 ho th agntzng ractanc of th thr-cag crcut a a functon of lp at upply voltag 24, 38, and 45 V. Th voltag valu corrpond to th lnar part of th agntzaton curv, th ratd opraton pont and a trongly aturatd rgon. Magntzng Ractanc (Oh) Slp Fg. 9. Magntzng ractanc a a functon of lp at th upply voltag 24 V (crcl), 38 V (atr) and 45 V (daond). Th aturaton affct otly th valu of th agntzng ractanc. Alo, th valu of th tator laag ractanc X and rotor ractanc X c, th largt ractanc on th rotor d,

8 64 6 dcra hn th upply voltag ncrad. Hovr, th valu of th rotor ractanc X c2 ran alot contant. Th rotor ractanc X r3 ncra lghtly along th th upply voltag, but n practc th chang nglgbl. Th valu of th rtanc ar not nfluncd by th upply voltag, a xpctd. Th torqu gvn by th t-haronc FE analy alo a plfcaton of th torqu producd by th ral nducton otor. Th torqu a a functon of pd ha bn aurd at a rducd voltag of 32 V. Th aurnt ar copard th th rult of th FEA n Fg.. Torqu (N) Spd (rp) Fg.. Maurd (crcl) and ulatd (atr) torqu vru pd curv at th upply voltag 32 V. IV. DISCUSSION Th quvalnt crcut odl th thr rotor branch capabl of odllng th torqu n a d pd rang hn th prablty of th ron aud to b contant. Hovr, n th ca of th nonlnar FE odl th crcut paratr tll dpnd qut trongly on th frquncy. Only f th crcut paratr ar calculatd at larg lp.5., th quvalnt crcut can odl th torqu thn a raonabl accuracy n th pd rang fro tand-tll to no-load. Addng th nubr of th rotor branch up to vn do not gnfcantly to prov th prforanc of th crcut odl. Alo, n th ca of th lnar odl, crcut odl th or than thr rotor branch do not provd gnfcantly bttr rult. Whn coparng th aurd torqu th th ulatd on, th an dffrnc found at 5 rp. Th tharonc analy fal n torqu coputaton f thr a ynchronou torqu at zro pd. Othr th rult ar clo to ach othr, pcally at all lp. Th ffct of th rotor angl a tudd by tatng th paratr for th ratd opraton pont at dffrnt rotor angl. Th dffrnc btn th paratr r found to b nglgbly all. Whn th pdanc atrx contructd th obtand valu for th tator rtanc about % hghr than th on gvn to th FE odl. Th tator rtanc corrctd to corrpond to th orgnal valu by ovng th xc rtanc to th rotor d. Th odfcaton do not chang th prforanc of th crcut odl. Th taton rult ar dffcult to valuat through aurnt nc th paratr obtand fro th convntonal no-load and tand-tll tt ar not coparabl th th paratr of th loadd opraton pont. Hovr, th paratr tat ar phycally raonabl n th n ho thy bhav a a functon of upply voltag. Th rotor rtanc ar not affctd, but th ractanc dcra along th th voltag, a xpctd. At th larg valu of lp or at th lo upply voltag o of th agnary part of th dagonal lnt n (2) ar ohat allr than th adacnt lnt. Thy ar rlatd to th couplng btn th rotor currnt and thrfor th tatd rotor ractanc X r2 and X r3 can hav all ngatv valu. In that ca, thy ar condrd to b zro nc th valu of th rtanc fully donat th contrbuton of th branch. Th phnonon a ubct for furthr tud. V. CONCLUSIONS Th paratr of th dp-bar nducton otor ar gratly nfluncd by th n ffct n th rotor bar. Thrfor th tradtonal ngl-cag odl fal n prdctng th torqu vru pd curv. Th crcut odl th addtonal rotor branch gv bttr rult. Hovr, f th aturaton of th ron odlld, th paratr bco or dpndnt on th frquncy. Th paratr tatd at th larg lp odl th torqu on a dr opraton rang. Th paratr gvn by th prntd thod ar rlatd to th ffctv prablty of th ron. Thrfor thy ar utabl for odllng th tady-tat opraton of th lctrcal achn. Such paratr can b ud n dgn and ulaton of lctrc drv. REFERENCES [] A. C. Sth,,R. C. Haly, and S. Wllaon, A Trannt Inducton Motor Modl Includng Saturaton and Dp Bar Effct, IEEE Tran. on Enrgy Convron, vol., pp. 8-5, March 996. [2] S. D. Sudhoff, D. C. Alprant, B. T. Kuhn, and P. L. Chapan An Inducton Machn Modl for Prdctng Invrtr-Machn Intracton, IEEE Tran. on Enrgy Convron, vol. 7, pp. 23-2, Jun 22. [3] J. Luo, A. Nnaa, A. Aro, On th u of ffctv rluctvt n agntc fld analy of nducton otor fd fro a nuodal voltag ourc, n Proc. of th Intrnatonal Confrnc on Elctrcal Machn, Münchn, 986, vol. 2, pp [4] E. G. Stranga, and K. R. Th, Shadd pol otor dgn and valuaton ung coupld fld and crcut quaton, IEEE Tran. on Magntc, vol. 2, pp , Sptbr 985. [5] D. Shn, G. Munr; J. Coulob; and J. Sabonnadr, Soluton of agntc fld and lctrc crcut cobnd probl, IEEE Tran. on Magntc, vol. 2, pp , Novbr 985. [6] A. Aro, Analy of nducton otor bad on th nurcal oluton of th agntc fld and crcut quaton, Doctoral Th, Acta Polytchnca Scandnavca, Elctrcal Engnrng Sr, no. 59, Hln, 987, 97 p. Avalabl: 2