SOME ASPECTS OF REPRESENTATION OF INDUCTANCE DISTRIBUTIONS IN DQ0-AXES IN A SALIENT POLE SYNCHRONOUS GENERATOR

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1 Zezyty Prolemowe Mzyy Eletryze Nr 4/ Krzyztof uwe Kele Uverty of Tehology SOME ASPECTS OF REPRESENTATION OF INDUCTANCE DISTRIBUTIONS IN DQ-AXES IN A SAIENT POE SYNCHRONOUS GENERATOR Atrt: Th pper oer let pole yhroou geertor wth mpg r preet the truto of elf- mutul ute the ttor turl referee frme well the rotor referee frme veru the eletrl rotor poto gle. It how the fluee of hgher hrmo otet of the elf- mutul ttor ute truto for ler oler rutl moel o the ute truto -xe.. Itrouto e of rotor mpg rut xe Stt ym multo tue of repetvely the eletrl gle of the rotor eletrl mhe ug rutl moel re poto re elf-ute of ttor very populr [ - 4]. I the e of hgh power wg f elf-ute of fel wg yhroou geertor the multo tue D Q re elf-ute of rotor mpg re ofte rre out o mthemtl moel rut xe repetvely the two xe the rotor referee frme re mutul ute of or le freuetly the ttor rotor turl ttor-to-ttor wg f f f f referee frme. The rutl moel of f f re mutul ute of ttor-to-fel yhroou geertor -xe re wg D D D D D D re mutul ote from the moel the ttor turl ute of ttor wg to mpg referee frme ug Pr trformto. rut x Q Q Q Q Q Q re Mot ofte eletromget prmeter re mutul ute of ttor wg to vlle for hgh-power yhroou geertor mpg rut x Df Df Df fd moel -xe oete to the power fd fd re mutul ute of fel wg ytem [5-7]. Cott ute to mpg rut x Qf Qf Qf truto me th l of mthemtl fq fq fq re mutul ute of fel moel very populr []. I rutl moel of wg to mpg rut x [ yhroou geertor the ttor rotor f D Q ] T vetor of ttor fel wg turl referee frme elf- mutul rotor x urret [ ] T ute tg to out the eletrl vetor of ttor urret r [ f D Q ] T gle of the rotor poto e efe vetor of urret of fel wg rotor the mtrx form [8]: x. f D Q f D Q f D Q f f D D Q Q f f D D Q Q f f D D Q Q f f fd D fq Q Df f D D DQ Q Qf f QD D Q Q Where: r r r r r the mtrx of elf- mutul ute truto of ttor wg r the mtrx of elf- mutul ute truto of rotor wg r r the mtrx of mutul ute truto of ttor rotor wg re e of ttor wg f the fel wg ex D Q Coerg the o-uform ttor rotor r gp le to the ute truto whh epe o the gle poto of the rotor [ 9 - ]. The reltohp etwee the eletrl mehl gle of the rotor poto eterme follow: m p Where: m re eletrl mehl gle of the rotor poto repetvely p the umer of pole pr. The elf- mutul ute of let pole yhroou geertor the ttor rotor turl referee frme the - xe e lulte follow: O the of freuey hrtert pproxmto eterme expermetl

2 88 Zezyty Prolemowe Mzyy Eletryze Nr 4/4 4 wy [6-4] or o the of the fel moel rete wth the ue of progrm e o the fte elemet metho [5 5] or ege elemet metho [6]. Iute lulto e rre out ug oe ow progrm [6] or ommerl oftwre pge uh Flux Mxwell Oper other ege for D or D moel [5] or ug wely vlle Fte Elemet Metho Mget FEMM oftwre tool [ 7 8] 4. vero t pole to eterme the eletromget prmeter oly for D moel. By the omto of eletromget fel euto repreetg fel moel fferetl euto repreetg rutl moel ug the ommerl oftwre metoe erler Flux Mxwell et. [5]. By me of oe ow progrm evelope motly Mtl/Smul Vew et. evromet [ 6]. O the of the tetolly troue Peuo-Rom Bry Seuee PRBS perturto pple to the extto otrol rut the ole etermto of the yhroou geertor moel prmeter the wer to th elerte terferee [4]. I lytl wy [ 9] or lytl wy y me of the wg futo. The m fou of th pper the fluee of the hgher hrmo otet of the elf- mutul ttor ute truto o the ute truto xe for ler oler mget rut. Other ute truto r r r mtre re ot te to out th pper. Therefore the elf- mutul ute for the trute ttor wg preete th pper re eterme ug FEMM progrm the e lege ute of the ttor wg lulte ug the lytl tehue [ - ]. The lulto of elf- mutul ute re rre out for ler oler moel of let pole yhroou geertor rte: S N 5.5 VA U N 4 V Y N rpm I N 7.9 A oϕ N.8 Q 4 umer of ttor lot wth wthout the rotor ew ftory rotor ew α 5 eul to ttor lot pth umer of pole pr p the rotor h - mpg r horte y two rg. The elf- mutul ttor ute re lulte o the of the rel otruto t o-uform r gp t the perphery of the ttor rotor reeve from the mufturer of yhroou geertor tht re ue freuetly geertor et Pol. Whle lultg the ute truto futo of the eletrl rotor poto gle urret the FEMM D oftwre the ftory gle-lyer wg ple the ttor lot te to out. Detle ly of the elf- mutul ute truto for the S N 5.5 VA let pole yhroou geertor ere [ ]. I the pper the ymmetry of the ttor rotor wg the lerty olerty of the mget rut re ume. Moreover FEM D oftwre lo ele the preetto of the fluee of the α rotor ew o the hgher hrmo otet of the elf- mutul ute. I the e of D progrm FEMM Flux Mxwell et. the ew effet e ote y uvg the tve rotor legth to xl le ther ewe rrgemet log the xl legth. A reult ute truto wth rotor ew e lulte []: α m α m α Where: m m re the ute wth wthout rotor ew repetvely.. Sttor wg elf mutul ute Fgure preet mget flux truto le of the exme 5.5 VA oler let pole yhroou geertor wth mpg r 5 r per pole wth tl rotor poto. A metho of etermg the elf- mutul ute truto FEMM progrm etle preete [4]. Fg.. Mget flux truto le the 5.5-VA let pole yhroou geertor

3 Zezyty Prolemowe Mzyy Eletryze Nr 4/ Fgure how the trute ttor wg elf-ute for ler tt oler moel wthout the ew wth the ew Fg. the trute ttor wg mutul-ute for ler oler moel wthout the ew wth the ew Fg. veru the eletrl gulr poto of the rotor the ptl hrmo otet for the oml ttor uret I N 7.9 A..6 hrmo mgtue the gftly greter th the eo hrmo mgtue of the elf-ute of the ttor wg the reult of the prtpto of the lot lege flux whh eve peetrte the rotor pole pee [] [eg] [eg] v - [%] [eg] v v v v hrmo Fg.. Compro of truto of the ttor wg elf-ute veru the gulr poto of the rotor wth wthout the ew: for ler moel for tt oler moel hrmo otet From the preete ute truto of the elf- mutul ttor t e olue tht: The trouto of the rotor ew α 5 oly reue the mpltue of the hgher hrmo ompoet lyg er the fumetl ompoet to mll extet. The reuto of ptl hrmo truto y the ew of the rotor etle preete [ ] Durg oe rotto of the rotor the twofol hge of the mget permelty the - x -x of the rotor troue gft ute vrto eh ttor wg. A lrge prtpto of the eo v - [%] [eg] v v v v hrmo Fg.. Compro of truto of the ttor wg mutul-ute veru the gulr poto of the rotor wth wthout the ew: for ler moel for tt oler moel hrmo otet From the preete ompro of hgher hrmo otet Fg. t e olue tht the ott ompoet eve hrmo omte wth the petrum the mot gft re: the hrmo for mutul ute wth 5 % relto to ott ompoet. O the of the elf mutul ute truto of the ttor wg mpltue petrum Fg. the followg reltohp e formulte [9 - ]: σ σ σ o o 4 o 4

4 Zezyty Prolemowe Mzyy Eletryze Nr 4/ o o o Detle ly of the truto of the ttor wg elf- mutul ute how Fg. exlue the ott ompoet reltohp 4 5 the vrto owg to the rotor ley preete [ ]. 4. A truto of the elf- mutul ttor ute the -xe The ttor ute truto the rotor referee frme -xe e lulte from Pr trformto of the elf- mutul ute ere y euto 4 5. The ute the -xe mtrx form e expree [8]: B B o 6 Where: o o o B -mtrx of trformto B - - vere mtrx of trformto the mtrx of elf- mutul ute truto of ttor wg wth tg to out the eletrl gle of the rotor poto expreo. Geerlly the ute mtrx the - xe e formulte : 7 Where: re the wg e the -xe zero-euee repetvely the mtrx of elf- mutul ute truto the -xe tg to out the vetor of urret [ ] T eletrl gle of the rotor poto re elf-ute the -xe re mutul ute the -xe. The elf- mutul ute of the ttor wg the mtrx fter the Pr trformto 6 the -xe e expree follow: o o o o o o 8 o o o 9 o o o o o o o o o o o o From the preete euto 4 5 t e olue tht reltohp 8 6 the elf- mutul ute of ttor wg e ere ug the ott ompoet eve hrmo of oe futo. From the trgoometr ette t reult tht the trouto of the ott ompoet 4 5 to expreo 8 6 me tht the elf- mutul ute truto the -xe o ot epe o the eletrl rotor poto gle. Ug the trgoometr ette t e prove tht oly ute hve vlue fferet from zero ot. σ. The elf-ute σ the -xe re uully ume ge uto mhe [] yhroou geertor wth permet mget [4 5].

5 Zezyty Prolemowe Mzyy Eletryze Nr 4/4 4 9 From 8-6 reult tht ll mutul ute xe o ot epe o the gulr poto of the rotor re eul to f the elf- mutul ute of the ttor wg 4 5 ot oly the ott ompoet the me mgtue of the hrmo. I th e of hrmo otet the mutul ute -xe - 6 the ute truto of ther two ompoet re outerphe the ute truto re eul to. I the repreetto of the elf- mutul ute truto of the ttor wg t freuetly ume tht - [7 8 ] the me wy for other phe. I the e of yhroou geertor th umpto very populr. A how th pper the e of the 5.5 VA exme let pole yhroou geertor the umpto - true Fg.. ut the umpto fle eue of gft mpt of the lot lege flux Fg.. It me tht the mgtue of hrmo of mutul lege ute four tme greter th the mgtue of the hrmo of the elf-lege ute. Th reult of the ft tht the mgtue of the hrmo the um of the mgtue of the hrmo σ where σ the lot lege ute for the hrmo Fg. the me wy for other phe. For theoretl oerto t e ume tht the hrmo of the elf-ute eglete. So f the elf- mutul ute ot the ott ompoet 4 5 the mutul ute ot the hrmo of oe futo the from trgoometr ette expreo: 8 9 reult tht ot. ot. reult tht σ reult tht 4 reult tht re the futo of o / 5 6 reult tht re the futo of /. I th e the ute mtrx e expree : 7 If ot the ott ompoet the hrmo 4 5 the from trgoometr ette reltohp: 8 9 reult tht ot. ot. reult tht σ reult tht 4 reult tht re the futo of o / 5 6 reult tht re the futo of /. I th e the elemet of mtrx ute e expree how 7. If ot the ott ompoet the hrmo how Fgure the from trgoometr ette reltohp: 8 9 reult tht ot. ot. reult tht σ reult tht 4 reult tht re the futo of o / 5 6 reult tht re the futo of /. I th e the ute mtrx e ere reltohp 7. If multo tue reure oerg the elf- mutul ute hgher hrmo orer greter th the Pr trformto oe ot elmte the fluee of the rotor poto gle o the ute truto -xe. The me rule pple to urret voltge flux [4 ]. Fgure 4 how ompro of the truto of the elf- mutul ute wth wthout the ew -xe veru the eletrl gulr rotor poto lulte o the of reltohp 4 5 of the elf mutul ute Fg. for the ler moel.

6 9 Zezyty Prolemowe Mzyy Eletryze Nr 4/ [eg] [eg] 4 x [eg].8 e f [eg] [eg] [eg] Fg. 4. Compro of the elf- mutul ute truto the -xe wth wthout the ew for ler moel: ; ; ; ; e ; f e f [eg] [eg] 4 x [eg] [eg] [eg] [eg] Fg. 5. Compro of the elf- mutul ute truto the -xe wth wthout the ew for oler moel: ; ; ; ; e ; f

7 Zezyty Prolemowe Mzyy Eletryze Nr 4/4 4 9 I Fgure 4 ll ute truto - xe wth wthout the rotor ew epe o the eletrl rotor poto gle. The elfute wthout the ew re mlr to wth the ew. The me rule pple to mutul ute wth wthout the rotor ew. Th owg to the ft tht wth the ute truto the mot gft oe re the 4th hrmo tht re oly fluee y the rotor ew to mll extet Fg. [ ]. I Fgure 4 the elf ute the mot gft 6th hrmo tht reult from 4th hrmo the elf- mutul ute of the ttor wg Fg.. Fgure 5 how ompro of the truto of the elf- mutul ute wth wthout the ew -xe veru the eletrl gulr rotor poto lulte o the of reltohp 4 5 of the elf mutul ute Fg. for oler moel. 5. Coluo Th rtle preet the ly of the hgher hrmo otet the truto of the elf- mutul ute the ttor turl referee frme the rotor referee frme for the ler oler moel of the let pole yhroou geertor-rte 5.5 VA. The elf- mutul ute truto of the ttor wg ot eve hrmo. From the ompro of preetg hgher hrmo otet reult tht relto to the ott ompoet the hghet peretge of hrmo our the mutul ute of ttor wg where the hrmo wth wthout rotor ew how 5%. Prtpto of the hrmo of the lot lege flux the mutul ute wth wthout rotor ew whh peetrte the rotor pole pee ofte omtte the lulto t ume tht the mgtue of hrmo of elf- mutul ute re eul to. A how the pper the prtpto of the hrmo very mportt the trformto of the ute truto from the turl referee frme to rotor referee frme. The hrmo of the lot lege flux troue lterte ompoet tht preet the mutul ute -xe wth wthout rotor ew. If the elf- mutul ute of the ttor wg prt from ott ompoet ot the me mgtue of hrmo the umpto very populr the the elfute the -xe re ott fferet from eul to σ oly th e ll mutul ute re eul to. If the mgtue of hrmo fferet from the elf- mutul ute the the elf-ute the -xe re ott fferet from eul to σ ut oly the mutul ute re eul to. Other mutul ute truto epe o the eletrl rotor poto gle. If the elf- mutul ute of the ttor wg ot the ott ompoet the the hgher hrmo th ll the elf mutul ute the -xe epe o the rotor poto gle. Moreover omte 6th hrmo mutul ute omte r hrmo. Thee preee of hrmo re reult of Pr' trformto of the le gft 4th hrmo ompro to the hrmo the elf- mutul ute of the ttor wg. For the tete let pole yhroou geertor o the ute truto the -xe the rotor ew hve lttle effet. Th owg to the ft tht wth the elf- mutul ttor ute truto the mot gft re the 4th hrmo whh re oly fluee y the rotor ew to mll extet. For the tete 55 VA let pole yhroou geertor the olerty ree the mgtue of hrmo mly the mutul ute the ttor turl referee frme. For th reo the olerty ree mly the mgtue of the 6th hrmo. For the other mutul ute the -xe from lytl oerto reult tht for the hrmo the ttor turl referee frme the two ompoet the -xe re outerphe. Due to the ft the hgher hrmo mgtue of the re mlr to the ler oe.

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