Effects of the Mutual inductances on the Switched Reluctance Machines

Transcription

1 Europea Assocato for the Developmet of Reewable Eerges, Evromet ad Power Qualty (EA4EPQ) Iteratoal Coferece o Reewable Eerges ad Power Qualty (ICREPQ ) Satago de Compostela (Spa), 8th to 3th March, Effects of the Mutual ductaces o the Swtched Reluctace Maches,3 A. Fleury, R. J. Das, W. R. H. Araúo, A. W. F. V. Slvera, D. A. Adrade, 3 G. C. Rbero Potfíca Uversdade Católca de Goás, Uversdade Federal de Uberlâda, 3 Uversdade Estadual de Goás, Rua São Lus, 634, CEP 74 46, Jardm Petrópols, Goâa, Goás, Brasl Phoe: afleury@terra.com.br Abstract. There are may studes whch dscuss the mutual ductaces the Swtched Reluctace Mache SRM. Most of these studes sad that the fluece of ductace s eglgble. I ths paper the flueces of those ductaces are aalyzed. Ths work presets expermetal measures ad aalyss of these quattes o prototypes of SRM wth varous settgs amg to cotrbute to the kowledge of the subect. The varatos of the ductace profle due to the curret the actve phase, the agular posto of the rotor ad saturato are take to accout. Surfaces are preseted to summarze the behavor of the self ductace ad of the mutual ductaces gotte from dfferet maches. It s preseted the smulato of the SRM geerator mode takg to accout the mutual ductaces. The results are preseted ad dscussed coclusvely, ad the some propostos are made. The results preseted here reveal the relatve dmeso ad the fluece of the mutual ductaces ad show that they have the potetal to dcate the posto of the rotor of the mache, thus avodg the use of electroc sesors allowg sesorless techques. Furthermore, ths artcle hghlghts the mportace of cosderg the mutual ductaces the desg, mathematcal modelg ad smulato of SRM order to mprove de mache performace. Key words Math modelg, Mutual ductaces, Self ductace, Swtched Reluctace Mache.. Itroducto The Swtched Reluctace Mache SRM attracted the scetfc commuty from etes due to the U.S. Ar Force More Electrc Arcraft Program. It meas to replace hydraulc, peumatc ad mechacal devces by electrc power to crease the relablty of the arcraft. Ideed a ege drectly coupled to the turbe shaft adds smplcty ad robustess to electromechacal coverso processes. The SRM has o rotor cols ad behaves well uder varable speed ad extra-hgh speed []. Thus, t s cosdered as starter-geerator coupled to the shaft of a teral combusto ege to provde startg torque ad feed electrcal loads. As a result of pulleys ad belts are elmated, reducg fuel cosumpto. O the other had, due to worldwde demad for alteratve ad reewable eergy, after testg, the Swtched Reluctace Geerator SRG s cosdered promsg for applcatos wd eergy. There are also studes for ts use dverse felds such as electrc vehcles or hybrd, rural electrfcato or as a battery charger self-excted (stadaloe). I other expermets, the SRM was partcularly effectve for fractoal power loads []. Ths paper cotues the research o the effects of mutual ductaces [,3] that s a aspect ot yet cosoldated the scetfc commuty. The fact s that they exst ad affect the mache performace. The questo s whether or ot ts effect must be take to accout. The results preseted here reveal the relatve dmeso of the mutual ductaces ad show that they have the potetal to dcate the mache rotor posto, thus avodg the use of electroc sesors allowg sesorless techques. Furthermore, ths artcle hghlghts the mportace of cosderg the mutual ductaces the desg, mathematcal modelg ad smulato of SRM order to crease the mache performace. Ths matter cocerg the mutual ductaces the swtched relucatace mache s focused some relevat artcles as [5] ad [6].. Math modelg. Nomeclature L Iductace. Phase curret. θ Istataeous agular posto of the rotor. Number of phases of the mache. Order of a phase of the mache. R Resstace of a phase col. v Voltage of a phase col. ω Rotor agular speed. C m Mechacal torque. C emag Electromagetc torque. J Rotatoal erta. D Dyamc frcto coeffcet. W co Co-eergy of a phase. It wll be preset a mathematcal model for the SRM takg to accout the mutual ductaces ad saturato effects. Ths procedure s take because the ductace 3 RE&PQJ, Vol., No., Aprl

2 depeds o the costructve parameters of the mache besdes agular rotor posto ad curret as well. I order to obta a equato for the crcut of the phase, cosder the self ad the mutual ductaces of the phase col, depedg o rotor agular posto (θ) ad o the curret that phase col ( ). Ths col has a teral resstace R. The the voltage at the termals of the col s gve by: v () = R + d F dt L + L k k, k k = wth ragg from to. Here L k =L k, k, s the mutual ductace betwee phases ad k, ad L s the self ductace of the phase. Equato () allows wrtg a set of o lear dfferetal equatos: v v =. v R +ω L where ω = d θ dt Equato () descrbes the electromagetc behavor of the SRG. The mechacal torque depeds o the co-eergy of the phases, o the rotatoal erta ad also depeds o the dyamc frcto coeffcet. Icludg the mechacal relatos equato () becomes: Now let [V], [R], [I], [L] ad order they appear (3), the: ω L. ω L ω L R +ω L. ω L.... ω L ω L L L. L L + L. L..... L L. L v v. v C m =. R +ω L +. () R ω L. ω L ω L R. ω L ω L ω L. +. R W co W co. W co ω D θ. + L L. L L L L. L L L L L. L. J.. ω θ (3) [ I] be the matrces the [ I] = [L] - [V] - [L] - [R][I] (4) Ths equato completely descrbes the state of each phase of the mache at ay tme. It ca be solved by teratve umercal computatoal methods. 3. Measuremets of self ad mutual ductaces Scetfc papers dealg wth the SRM dsregard the mutual couplg betwee phases. Accordg to the authors the mutual ductaces ca be dsregarded the desg of the mache, the smulatos ad evaluatos of the performace of the prototypes, wth o mportat mpact o results because they are small, approxmately percet of the self ductace. But ths paper presets studes where the mutual ductaces are greater tha %, elargg ts relevace. So the relatoshp betwee the self ad mutual ductaces s the pot. From (), uder certa crcumstaces the self ductace ca be wrtte as: V L (5) = R ω e I I ths case all the phase cols are ope, except for phase. The rotor s blocked. Alterate curret of electrcal frequecy exctes the actve phase. V ad I are the rms values for the actve phase voltage ad curret respectvely. For the mutual ductaces, ts equato ca be wrtte as: vk L (6) k = Lk = L v R where ad, respectvely. phase, L k v k ω e s the mutual ductace betwee the phases k L s the self ductace of the actve s the duced voltage the phase k, v s the voltage o the actve phase, R s the resstace of ths phase ad s ts curret. Thus, the relatoshp betwee the mutual ad self ductaces s gve by: L k L = v k ( v R ) Ths rato vares depedg o the curret ad the stataeous agular posto of the rotor. The preseted method to determe the mutual ductaces from the expermetal data s appled whe the rotor s locked ad the phases are ope crcuted, except the oe dested to excte the rotor yoke. Ths method allows studyg the dyamc behavor of the relatos betwee the self ductace ad mutual ductaces prototypes. 3. Expermetal measures of the ductaces The mutual ductaces were measured a SRM wth 8x6 topology ad aother wth 6x4 topology. The data were tabulated ad used a computer model developed wth terpolato programs to determe the values of ductaces ad ther dervatves at ay locked rotor posto for varous curret values. Both maches have optcal sesors to dcate stataeous rotor posto. Fg. shows the test bech. (7) 3 RE&PQJ, Vol., No., Aprl

3 ) ) Fg. 3 shows the mutual ductace L DC obtaed from the procedure descrbed 3.. Mutual ductace L 4 3 x -4 Fg. A test bech bult. As prevously descrbed the procedure used to obta the values of self ad mutual ductaces for each rotor posto cossts o fxg the rotor a desred posto ad the eergzg the respectve phase col proceeds wth the voltage ad curret measuremet of the phase col. Besdes, the duced voltages o the other phases are measured order to apply equatos (5) ad (6) amg calculates the values of self ad mutual ductaces. Wth the obtaed (calculated) values s possble to trace the profles of the self ad mutual ductaces as fucto of rotor agular posto ad curret. For the mache 8x6 a SRM wth 4 phases (A, B, C ad D) the phase D was cosdered as a referece for the measuremets. The curret was appled at the phase D ad chaged from A to 37 A, wth steps of A. The agular posto was chaged from (algmet of poles) up to 3 (msalgmet of the poles), also cremets of two uts. Ths resulted a database that allows costructg surfaces for expermetal ductaces. For a better vsualzato of the behavor of ductaces L (θ,) of the 8x6 mache surfaces were costructed from the data collected. These surfaces ca be see Fg.. As show equato (3), the values of self ad mutual ductaces ad ther dervatves wth respect to the agular posto of the rotor are requred to smulate the behavor of the mache. Fg. 3 presets the behavor of the ductace of phase C whe the measuremet procedure was appled at the phase D (referece phase). Fg. 4 shows the surface of the mutual ductace L DB obtaed through the same method of the prevous fgure. I d u t a c e ( h e r y ) IIductace (hery) Fg. 3 Mutual ductace L DC cosderg the 8x6 SRM x Rotor agular posto ( deg ) - MútuaI Iductace L4 3 4 Rotor agular posto (deg) Fg. 4 Mutual ductace L DB cosderg the 8x6 SRM. The mutual ductace L DA has the same behavor of the ductace L DC (Fg. 3) sce these ductaces are symmetrcal cosderg the 8x6 topology. I order to vestgate the ductaces of the 6x4 topology mache a secod study was coducted wth the same procedures wth a 6x4 prototype. Cosderg a three phase SRM wth phases A, B ad C, Fg. 5 shows the self ductace of phase C (took as referece the measuremet procedure). Self fuctace of the mache 6 x Curret ( A ) 3 3 Curret (A) I d u c t a c e ( h e r y ) x -3 S e l f I d u c t a c e R o t o r a g u l a r p o s t o ( d e g ) C u r r e t ( A ) 35 4 Fg. Self ductace surface for phase D cosderg the 8x6 SRM. IIductace (hery) Rotor agular posto (deg) -5 Fg. 5 Self ductace surface for phase C cosderg the 6x4 SRM The same procedure used for the 8x6 prototype was appled the 6x4 mache. As a three phase mache, the symmetry occurs at the mutual ductaces aroud the referece phase (phase C). 4 6 Curret (A) RE&PQJ, Vol., No., Aprl

4 Fg. 6 shows the mutual ductace L CA obtaed for the 6x4 mache. Mutual Iductace L3 d d d3 v = R + L + L + L3 + dt dt dt L dθ L dθ L3 dθ dt dt dt (7) Fg. 6 Mutual ductace L CA cosderg the 6x4 SRM. Through the aalyss of the expermetal results s possble to verfy that the magtude of the mutual ductace may acheve values that exceed 6% of self ductace. The ext topc presets the smulato results that show the fluece of the mutual ductaces over the electrcal quattes of the SRM. 4. Smulatos The mathematcal model used to smulate the Swtched Reluctace Mache was developed ad preseted secto 3. The correspodg computg model was developed MATLAB/SIMULINK evromet order to compare the results wth ad wthout the mutual ductaces. Ths model s totally descrbed [4]. The 6x4 topology mache was smulated cosderg the mutual ductaces. I a secod tme they were ot. The smulato ra out cosderg the mache at 35 RPM speed ad wth a 48V DC exctato voltage sce ths value s appoted at automotve applcatos (electrc cars ad hybrd vehcles). The smulato tme was 5 secods, tme eough for the mache steady state operato. The voltage profle oe of the three phases s preseted Fg. 7, showg ts behavor wth ad wthout the mutual ductaces. v o l t a g e ( v o l t ) IIductace (hery) x -3 Rotor agular postor (deg) e x c t a t o g e e r a t o P h a s e c o l v o l t a g e s t a d b y Fg. 7 Phase voltage profle behavor for a 6x4 mache wth ad wthout the mutual ductaces. Based o equato (), the phase voltage s: Curret (A) 8 w t h the m u t u a l s w t h o u t t h e m u t u a l s Assumg as v, v, v 3, v 4, v 5, v 6 ad v 7 the terms preset the secod member of equato (7), v, v, ad v 5 are related to the phase col wch are cosdered smulato results show Fg. 7 as a straght le (dsregardg the mutual ductaces). The terms v 3, v 4, v 6 ad v 7 s from the mutual couplg betwee phases. These terms cosder the mutual ductace fluece ad the smulato result s show Fg. 7 as a dashed le. The frst term (v ) of the secod member (equato 7) s related to the voltage drop at col resstace. As the resstace s a costat the v profle s the same of the phase curret profle preseted Fg. 8 (v s tmes scaled order to mprove vsualzato). x V t e r m ( v o l t s ), ( v o l t ) V t e r m e q u a t o ( 7 ) -8 Fg. 8 Graphcal profle of v term of equato (7) tmes scaled. Ths term exsts durg the exctato ad geerato perod of the mache operato but t does t whe the phase col s ot eergzed. Ths occurs because the coverter swtches are off ad the free-wheelg dodes are reverse polarzato (so, the phase curret s ull). The term v depeds o self ductace (L ) ad phase curret dervatve. Fg. 9 (scaled for better vsualzato of self ductace ad phase curret) shows the profle of the v term. V a d ( v o l t ), x d u c t a c e ( H e r y ), 5 x p h a s e c u r r e t a f a s e ( A ) s V t e r m e q u a t o ( 7 ) Fg. 9 Term v, phase curret ad voltage. V t e r m se l f d u c t a c e p h a s e c u r r e t - It may be oted that the v chages at the flecto pot of the curret curve whe ts dervatve becomes egatve. As v ths term s ull whe the phase s tured off (closg the coverter swtches). V 34 RE&PQJ, Vol., No., Aprl

5 The term v 5 (whch also correspods to self ductace as well as v ad v ) s show Fg. V 5 t e r m a d ( v o l t s ) Fg. Term v 5 ad phase voltage. The agular speed ad curret are always postve, so, the sg of ths term s derved from the sg of the selfductace relatve to agular speed. As the lterature records, the SRM geerate power whe the phase s eergzed egatve dervatve of ductace. Because ths, the v 5 profle s egatve. Ths term gets ull value whe the coverter swtches are off ad whe phase curret s also ull. The terms v, v, e v 5 (prevously aalyzed) do t deped drectly o mutual ductaces. Besdes, t may be observed that terms v, v, e v 5 flueces the voltage durg the exctato ad geerato perods, but there s o terferece whe the coverter swtched are off. Fg. shows the term v 3 compared wth phase voltage. V 3 t e r m a d ( v o l t ) V 5 t e r m e q u a t o ( 7 ) -4 V 3 t e r m e q u a t o ( 7 ) Fg. - Term v 3 ad voltage o the phase col. V 5 t e r m V 3 t e r m The term v 3 flueces the phase voltage whe the coverter swtches are off ad durg the exctato perod, but t seems to have lttle or oe terferece o phase voltage durg the exctato perod. Ideed, the terms that depeds o curret of the ext phase to be eergzed (cosderg the mache operato) lke v 3 e v 6 duce voltage o the referece phase durg the exctato perod ad whe the coverter swtches are off. Also, the terms that deped o curret of the prevous phase (v 4 e v 7 ) duce voltage durg the geerato perod ad whe the coverter swtches are off. I the same way, Fg. compares the term v 4 wth phase voltage. x V 4 t e r m a d ( v o l t ) Fg. Term v 4 ad phase voltage. The term v 4 flueces the voltage durg geerato perod ad whe the coverter swtches are off ad has o terferece durg the exctato perod. The terms v 6 ad v 7 are compared to col voltage fgures 3 ad 4, respectvely. V 6 te r m a d ( v o l t ) V 3 t e r m a d ( v o l t ) V 4 t er m e q u a t o ( 7 ) V 6 t e rm e q u a t o ( 7 ) Fg. 3 Term v 6 ad phase voltage. Fg. 4 Term v 7 ad phase voltage. V 4 t e r m V 6 t e r m V 7 t e r m e q u a t o ( 7 ) V 7 t e r m -8 These are the terms that deped o mutual ductace dervatve (L e L 3 ). The term v 5 deped o self ductace dervatve. These three terms (v 5, v 6 ad v 7 ) composes part of equato (7) wch depeds o agular speed. Thus, the phase voltage varato related to the agular speed also depeds o mutual ductaces behavor. Fgures 5 ad 6 sythesze the comparso betwee smulato results whe the mutual ductaces are cosdered ad whe they do t (cosderg geerator operato) RE&PQJ, Vol., No., Aprl

6 P o w e r ( w a t t ) Fg. 5 Power geerated by the SRG (5 secods smulato tme) wth ad wthout mutual ductaces. Usg the ductaces expermetal results, there s a expectato about a greater power geerato whe the mutual ductaces are cosdered. The dfferece s about 3 watts agast 78 watts the power geerated by the mache, wch correspods a 8,6% devato. The devato magtude represets a value that pots to the mutual ductaces cosderato. Besdes, computer smulato correspods to the tal process of mache maufacturg. Thus, the correct computg model ca acheve mproved values for the mache desg. The 8x6 mache was also smulated wth ad wthout the mutual ductaces cosderg the prevous behavors show fgures, 3 ad 4. The results was verfed ad they show smlar results related to the 6x4 mache. Fg. 6 shows the power geerated by the 8x6 mache whe the mutual ductaces are cosdered ad whe they do t. p ower ( w a t t ) t me ( s e c ) Fg. 6 Power geerated by the 8x6 mache wth ad wthout the mutual ductace. The devato was 46,6 watts agast 49,5 watts wch correspods a varato of 5,% whe the mutual ductaces are cosdered. 4. Cocluso G e e r a t e d P o w e r w t h a d w t h o u t m u t u a l d u c t a c e s Geerated Power wth ad wthout the mutual ductaces w t h m u t u a l s w t h o u t m u t u a l s Wthout mutuals Wth mutuals Ths paper preseted a aalyss about the mutual ductaces of Swtched Reluctace Geerators. Mutual ductaces profles were preseted potg that they magtude s greater tha expected, cotradctg prevous related studes. The power geerated showed a crease of 8,6% the smulato results of the 6x4 topology mache, whle the 8x6 mache smulato resulted a crease of 5,% the power geerated. Thus, cosderg mutual ductaces of Swtched Reluctace Maches (operatg as geerator) results optmzed desgs ad smulatos. Ths s a mportat pot cosderg smulato as a tal step of mache maufacturg. The graphcal results preseted fgures 7 through 4 pots that mutual ductaces cause a couplg betwee mache phases fluecg the phase s voltages waveforms. Thus, ths behavor ca be used order to mplemet sesorless techques despsg optcal sesors, wch s a exemple of applcato. Oce there s ot a drect method to preset expermetal results that cofrm the creasg power geerato whe mutual ductaces are take to accout, ths study s a step toward ew uderstadg of dyamc behavor of the Swtched Reluctace Mache operatg as geerator. Due to the er features of the ductaces, coclusos about them deped o the cofguratos of the mache tested, but ot deped o ts dmesos, ad so, the results preseted here, gotte from two maches of HP class should be vald for larger maches. Ackowledgemet The authors thak Uversdade Estadual de Goás, Uversdade Federal de Uberlâda ad Potfíca Uversdade Católca de Goás. Refereces [] Das, R.J., Coelho, A., Fleury, A.V.S., About phases depedece a swtched reluctace geerator, ICREPQ 8 Iteratoal Coferece o Reewable Eerges ad Power Qualty, electroc md records, March, -4, 8; : Reewable Eergy ad Power Qualty Joural, ISSN 7-38X, March 8 edto. [] Adrade, D. A., Costa, R.S., Texera, R.S., Fleury, A.V., Eergy Effcecy for Fractoal Power Loads, IEEE Idustry Applcato Socety Magaze, vol, um 6, November/December 6, pp. -, ISSN [3] Fleury, A.V.S., Adrade, D.A., Svera, A.W.F.V., Expermetal Measuremet ad Aalyss of the Self ad Mutual Iductaces Two Dfferet Swtched Reluctace Maches, Iteratoal Coferece o Reewable Eerges ad Power Qualty,, Graada - Spa. Record of the ICREPQ, ; : Reewable Eergy ad Power Qualty Joural, Aprl edto. [4] Das, R. J., Adrade, D. A., Cabral, L. G., Slvera, A. W. F. V., Slvera, A. F. V., Gomes, L. C., Bssoch, C. A., Modelg, smulato ad Comparatve Study Betwee a Sgle-Phase Swtched Reluctace Mache (6x6) ad a Three- Phase Swtched Reluctace Mache, : Reewable Eergy ad Power Qualty Joural, No.9, th May. RE&PQJ-8 ISSN 7-38X. [5] de Paula, P.P.; da Slva, W.M.; Cardoso, J.R.; Nabeta, S.I., Assessmet of the Iflueces of the Mutual Iductaces o Swtched Reluctace Maches Performace, : Electrc Maches ad Drves Coferece, 3. IEMDC'3. IEEE Iteratoal vol 3, page(s): [6] Krshamurthy, M.; Fahm, B.; Edrgto, C.S.; O the Measuremet of Mutual Iductace for aswtched Reluctace Mache, : Power Electrocs Specalsts Coferece, 6. PESC'6, 37th IEEE, Page(s): RE&PQJ, Vol., No., Aprl