MATHEMATICAL SIMULATION OF THE MAGNETIC FIELD OCCURRED BY THE ARMATURE REACTION OF THE SYNCHRONOUS MACHINE

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1 MATHEMATICAL SIMULATION OF THE MAGNETIC FIELD OCCURRED BY THE ARMATURE REACTION OF THE SYNCHRONOUS MACHINE Aleksandrs Mesajevs Andrejs Zvedrs Faculty of Power and Electrcal Engneerng Rga Techncal Unversty 1 Blvd Kronvalda, 1010 Rga, Latva E-al: kbl@nbox.lv, aaazzz@eef.rtu.lv KEYWORDS Matheatcal sulaton, synchronous achne, agnetc feld. ABSTRACT Usng proposed ethodology and atheatcal sulaton paraeters of the arature wndng, agnetc syste geoetry s estated, as well as the saturaton s nfluence on the entoned paraeter s evaluated. Methods are based on agnetc feld atheatcal sulaton, solvng partal dervatves of the dfferental euaton coparatvely to the vector agnetc potental A and usng one of the ost effcent ethods the fnte eleent ethod (FEM) (Voldek 1978). There s theoretcally justfed and descrbed ethod of obtanng the arature wndng phase agnetc-flux lnkage,.e., (t), fro the vector potental whch, as a functon of spatal coordnates A( x, y), s receved by atheatcal sulaton of the statonary agnetc feld. The ethod s based on solvng several statonary agnetc feld euatons. In euatons as a feld source are defned the phase currents nstantaneous values, whch are algned wth the rotor rotaton angle t. Accordng to the classcal synchronous achne tworeacton theory, synchronous reactances X d and X are deterned fro the agnetc feld s fundaental haronc n the ar gap. INTRODUCTION Synchronous achne s nductve reactances and the processes dependng on the deterne the electroagnetc feld,.e., t s spatal dstrbuton and change n te. Ths feld n operatng condtons can be descrbed by Maxwell's euatons. FEM s the ost effcent nuercal ethod, whch easly allows accurate enough to consder those specfc, portant factors such as any geoetrcal shape and sze coplexty of the agnet syste s ndvdual eleents, the ferroagnetc ateral s non-lnear characterstc, as well as the feld sources (any actual dstrbuton of wndng current). As experence shows whle studyng the agnetc feld, t can be assued that the achne s agnetc feld s plane-parallel. It s approprate to substtute the electroagnetc feld euatons syste wth one euaton whch depends on the agnetc vector potental, havng only one (axal) coponent A Az n a plane-parallel feld. In such euaton A s spatal coordnates x, y and te t functon, whch eans that the euaton (1) descrbes space and te alternatng agnetc feld (Voldek 1978) 2 2 A A j a, (1) 2 2 x y where - agnetc pereablty; and j a the external feld source s current densty. Solvng ths type of euaton s rather dffcult, therefore, t s approprate to reduce t to nuber of spler tasks. In ths case, the task s based on the fact that n te varyng process ( t 0 ) can be vewed as a sngle fxed process set of dfferent consecutve te ponts t 1, t2,..., t,..., t n. So, for exaple, f the feld source current densty s n te varyng snusodal functon ja ( t) ja sn t, then euaton (1) ust be solved n tes, each te n the rght sde of that euaton defne oent of te t that corresponds to current denstes oent value ja ( t ) ja snt. Solvng such a task as the results receve vector potentals, whch are essentally a table fored functonal dependence A f (t). A slar approach s used solvng nonstatonary feld euatons, when rotor speed 0. In ths case, t s possble to solve a nuber of agnetostatc feld euatons. Each of the corresponds to dfferent consecutve rotor postons. For the atheatcal sulaton of the agnetc feld and obtanng results the coplex ult-functonal progra QuckFeld (QuckFeld 2009) s used. Software provdes opportuntes for the followng actons: to descrbe the geoetrc odel (or topology) of the object under study; Proceedngs 24th European Conference on Modellng and Sulaton ECMS Andrzej Bargela, Sayed Aza Al Davd Crowley, Eugène J.H. Kerckhoffs (Edtors) ISBN: / ISBN: (CD)

2 to assgn the edu characterstcs, ncludng varous ferroagnetc ateral agnetzaton curves B f (H ) ; to assgn feld sources - a current densty n wndngs as a functon of spatal coordnates; to assgn the Drchle and/or the Neuan boundary condtons; to solve tasks wth hgh precson; to get a vsual pcture of the feld; to calculate varous electroagnetc feld dfferental and ntegral characterstcs. Ths paper addresses the followng an tasks: to apply and to use the avalable odern software for atheatcal sulatng of the agnetc feld usng nuercal ethods; to llustrate wth exaples the practcal use of ethods, whch uantfably estate the agnetc constructve paraeters of the syste and, above all, the saturaton effect on paraeters of the achne s electroagnetc feld. DETERMINATION OF THE SYNCHRONOUS REACTANCE USING THE SYNCHRONOUS MACHINE S MATHEMATICAL SIMULATION RESULTS As known (Voldek 1978), the agnetc asyetry of the salent pole synchronous achne s rotor ensures reluctances, whch are bgger n uadrature axs ( drecton) than the reactance n the drect axs (d drecton). Therefore, for the synchronous achnes wth rotor s agnetc asyetry, t s approprate to use a two-reacton ethod based on the superposton prncple. Accordng to ths prncple, the drect axs ( d flux) and uadrature axs ( flux), where the agnetc flux s operatng, are utually ndependent. It should be noted, that ths assupton s correct only for achnes wth unsaturated agnetc syste. However, akng addtonal adjustents based on the agnetc feld atheatcal sulaton results; two-reacton ethod can also be used for achnes wth a saturated agnetc syste. Accordng to the two-reacton ethod theory, the synchronous nductve reactances X d and X, can be deterned: Ed X d Id (2) E X I (3) In these euatons, EMF Ed are the EMFs nduced by the fundaental haronc arature wndng of the drect feld and the uadrature feld; I d and I - arature current s drect and uadrature coponents. The effectve value of the EMFs Ed can be deterned usng the followng forulas E E d (4) 4,44 fd1k w1 4,44 f1k w1 (5) where d1 and 1 - fundaental haronc ( 1) apltude values of flux lnkages generated by the drect feld and the uadrature feld, k w1 - the fundaental haroncs wndng factor. The necessary seuence of operatons s the followng: frst, deterne d1 and 1 (see (4) and (5)), whch, n turn, are reured for reactance X d and X deternaton fro (2) and (3). The fgure 1 shows the calculaton regon of the splfed agnetc feld odel for the three-phase ( 3 ) two-pole ( 2 p 2 ) achne wth a nuber of arature slots Z 36. Fgures 1. Dstrbuton of currents n the arature wndng phases for deternaton of the drect axs feld s flux lnkage of the arature n te d ( t) oents t 0 ( and t 30 (, when the rotor rotaton angle (electrc degrees) s 0 and 30

3 The nstantaneous value of current n arature wndngs phases ay be deterned by the followng forulas: A B C I I I cos( t ) cos( t 120 cos( t 240 ) ) where I 2 I - the arature current as the peak value of the feld source. To deterne the flux lnkage d ( t ) wth suffcent accuracy, t s preferably that the te step s chosen sall enough. As experence of nuercal experents shows, the rotor rotaton angle step s approprate to be chosen so, that t z 2, where t z - the arature tooth ptch. The agnetc flux of one pole can be obtaned wth dfferent arature current nstantaneous values accordng to (7) wth the rotor poston angle. After a seres of agnetc feld calculatons for te oents t (6) t ) ( ) ( A A l (7) d ( d 1 2) and flux lnkage wth arature wndng phase as a functon of te t ) ( A wl (8) d ( 1 ) where A 1 and A 2 - the vector potental values on the surface of the arature n ponts 1 and 2 (fg. 1.), w - nuber of turns per phase, l - the achne length n axal drecton. Nuercal haronc analyss of the functon d ( t ) allows to obtan the fundaental haronc apltude value d1 of the flux lnkage s, whch s reured accordng to forula (4) for deternaton of EMF. Slarly to the expresson (5), the EMF E d can be obtaned f s replaced wth 90 (see table 1.) whle sulatng uadrature axs agnetc feld n expresson (6). RESULTS OF THE MAGNETIC FIELDMATHEMATICAL SIMULATION, ITS ANALYSIS AND USE The sulaton of the synchronous achne agnetc feld s ade for the experental three-phase achne wth 2 p 2, n whch a sngle layer full step ( y ) wndng s located n arature 36 slots. Other achne paraeters: f 50 Hz, turn nuber per phase w 60, slots per pole and phase 6, wndng factor kw 0.96, arature rated current I N A. The ferroagnetc eleents of the achne agnetc syste E are ade up fro the electrcal-sheet steel. The an geoetrc densons are: outer daeter of the arature D a 0.212, the arature s nsde daeter D 0.136, the ar gap ; the arature s actve length l ; the pole overlappng factor , the pole ptch p Calculatons were ade for fve arature current values I (0.4; 0.6; 0.8; 1.0; 1.2)I N, where the effects of saturaton on the agnetc feld s character, on paraeters X d and X values are evaluated. To obtan the flux lnkage wth arature wndng s phase for a certan current value as a functon of te, the nstantaneous current value s set accordng to the rotor rotaton angle wth step 5 (electrc degree), whch s atched wth the arature current phase s change along the sae angle. Because of the perodcty and syetry of the functon ( t), t s suffcent f the calculatons are ade n one uarter of the perod of current changes, that s t 90 (electrc degree). The estated by the forula (7) arature wndng slot s current nstantaneous value n relatve unts are gven n table 1. Phase s zone wdth euals to 6 slots for chosen achne. Table 1. Phase current s nstantaneous value n relatve unts for dfferent rotor rotaton angles (electrcal degrees) t( ) * A ( ) * ( B) * ( C) In the above entoned tables angle values are shown for sulaton of drect axs feld cases assung that the A-phase axs concdes wth the pole drect-axs (see also fg 1.). The angle 90 should be used for sulaton of the arature uadrature-axs feld.

4 Fgure 2 shows the drect axs and uadrature axs agnetc feld s pcture, acured for achne, f the arature current coponents Id and I n relatve unts I d I 1,.e. eual to the rated current. Table 2. EMF Ed fundaental haroncs dependence on the arature currents E d (V ) * I (V ) E Fgures 2. Arature drect reacton ( and uadratureaxs reacton ( agnetc feld pctures for angle n oent t, when the phase A current s at axu value Fgures 3 shows dstrbuton curves of the arature drect axs and uadrature axs agnetc flux denstes Bd and B deterned on the teeth and slots ddle ponts on the arature surface. Flux lnkage curves d ( t ) and ( t ) are obtaned wth calculatons carred out for dfferent rotor rotaton angles, observed wth the arature phase currents angle for oents t ( t ).Varous arature currents * * I d and I values for one half of the perod are shown on Fgure 5. The flux lnkage fundaental haroncs d1 and 1 are deterned durng nuercal haronc analyss of d ( t ) and ( t ). The EMFs Ed values for varous arature currents are calculated (see table 2 fro forulae (4) and (5)). Fgures 3. Aratures drect axs ( and uadrature axs ( agnetc flux denstes dstrbuton n the ar gap (on the arature surface); (angle n electrcal degrees) Magnetc syste s saturaton can be assessed by curves E ( * d f I ) and E ( * f I ) shown on fg. 4. Fgures 4. Arature wndng EMF dependence fro arature currents n relatve unts

5 be estated by the saturaton factors k d and k. Saturaton factors values are dependng on agnetc syste geoetrc densons and on level of saturaton ( k d , k ). Estaton results are also confred by the experental results of exaned achne. Fgures 5. By arature drect reacton ( and uadrature reacton ( nduced flux lnkage wth arature wndngs phase as a functon of te Reactances X d and X calculated fro forulae (2) and (3) are gven n table 3. Table 3 also ndcates the saturaton coeffcents and values k d correspondng to the rated current I * 1. k Table 3. Reactances dependence fro saturaton level nduced by arature currents X ( () * I d ) X X d X k d k CONCLUSIONS The fnte eleent ethod offers treendous opportuntes for deternng the synchronous achne s agnetc feld and fro t dependng characterstcs. Synchronous drect axs reactance X d and uadrature axs reactance X can be correctly deterned fro the agnetc feld s atheatcal sulatng results applyng the prncple of superposton used n the classcal synchronous achne theory. Saturaton effect on paraeters X and X can d REFERENCES Banch N Electrcal achnes analyss usng fnte eleents. Boca Raton. CRC Press. QuckFeld Fnte eleent analyss syste. Verson 5.7. User s gude. Svendborg. Tera Analyss Ltd. Voldek A Electrcal achnes. Lenngrad.. Zvedrs Electroagnetc calculatons n electrcal achnes. Rga.. AUTHOR BIOGRAPHIES ALEKSANDRS MESAJEVS was born 1985 n Latva. In 2008 he graduated fro Rga Techncal Unversty, ganng M.Sc.ng. degree. Presently he s a PhD student. In March 2008 receved certfcate Base and advanced sulaton usng QuckFeld software. Fro 2006 he s workng as a laboratory assstant and scentfc assstant n Electrcal Machnes and Apparatus Departent of Rga Techncal Unversty. ANDREJS ZVIEDRIS was born 1938 n Latva. In 1961 he graduated fro the Rga Polytechncal Insttute (RPI) Faculty of Electrcal and Power Engneerng, ganng the ualfcatons of Engneer n the Electrcal Machnes and Apparatus specalty. In 1970 defended a thess and obtaned a Canddate degree of Techncal Scences. Dr.Sc.ng. degree was conferred to A.Zvedrs n After graduatng the RPI A.Zvedrs has worked as an assstant Professor ( ), and Departent of Electrcal Machnes and Apparatus Head ( ). In 2001 he was elected as Assocate Professor n the Departent of Electrcal Machnes and Apparatus, where work to date. Scentfc actvtes of A. Zvedrs are related to Matheatcal Sulaton of agnetc felds n Electrcal Machnes. The results of hs research are aprobated n ore than 50 scentfc publcatons and techncal reports n Internatonal Scentfc conferences. A. Zvedrs s an Expert n Standardzaton Techncal Cottee of Electrotechncal Ternology of Electrcal Engneerng, a eber of Latvan Unon of Scentsts.