Flux-linkage equations for 7-winding representation (similar to eq in text)

Transcription

1 lux-lkge eutos fo 7-wg epesetto (sl to e.. text The oe tes e efe s follows: Stto-stto tes: Stto-oto tes: oto-stto tes: =s+os =os =os =-[s+os(+] =os =os(- =-[s+os(+5] =s =os(- =s =-[s+os(+] =os =s+os(- =os(- =os(- =-[s+os(-9] =os(- =os(- =s(- =-[s+os(+5] =s(- =s =-[s+os(-9] =s(- =s+os(- =os(- =s(- =os(- oto-oto tes: =s(- =s = =s(- =s(- = =s(- = = = == == = = =Y == =Y =

2 So the opt fo of the flux lkge eutos e (e. whh, whe expe wth the expessos fo self utul utes, eoe: S os [ S os ( ] [ S os ( 5 os os s s [ S [ S S os ( ] os ( os ( 9] os( os( s( s( [ [ S S S os ( 5] os ( 9] os ( os( os( s( s( os( os( os os( os( os s( s( s Y s s( s( Y (e. -ex

3 Voltge eutos The oltge eutos eelope hee wll hteze the eletoget ys of the syhoous he. Cose the stto ut s t ppes s g. : g. The uet eto the phses, whh s out of the tels fo geeto opeto, poues flux tht s the egte eto of the espete phse xs (uet eto to the tels woul poue flux tht s the poste eto of the espete phse xs.

4 We ssue tht the eutl outo s ot getlly ouple wth y othe ut. We wte oltge euto fo eh of the phse wgs s follows: We y lso wte oltge euto fo the eutl ut s follows: ( ( Now let s look t the oto uts. Thee e fou of the. g. : -Axs el g. : -Axs pe

5 5 g. : -Axs pe g. 5: -Axs el Puttg ll of these eutos togethe tx fo, we he tht: (e. We wte ths oe optly, sl to e..6 text: (e..6 The pe otto o euto ues tes the ge euto oespos to the euto etfe you text y tht ue, wth soe ofto. eelly, the ofto s the to of the lst oltge euto fo the -ut.

6 otto fo Pk s Tsfoto We ese to get the oe euto to stte-spe fo ( x Ax so tht we oe t wth ou etl eutos the pply uel tegto sole the togethe. o ot lose sght tht ths s ou ojete. We ote, howee, tht we he two types of elte stte les the oe eutos: flux lkges ( uets (. We elte oe of the, ths s ot h se flux lkges e expesse s futos of the uets tht poue the. o exple, fo sgle outo, we wte tht = (see lso e. ( t the egg of ths ouet. But e..6 hs etes o. Ag, o pole, se /t=(/t. It s hee tht we u to toule, se the utes tht we e elg wth e, geel, futos of, whh s tself futo of te. Theefoe the utes e futos of te, ffeetto of flux lkges esults expessos lke: t t The ffeetto wth espet to, /t, wll esult te-yg oeffet o the stte le. Whe we eple, e..6, the etes o wth the etes t 6

7 o, the sole fo the etes o ( oe to ot x Ax, we wll ot uet les o the ght-hse tht he te yg oeffets,.e., the oeffet tx A wll ot e ostt. Ths es tht we wll he to el wth ffeetl eutos wth te yg oeffets, whh e geelly oe ffult to sole th ffeetl eutos wth ostt oeffets. Ths pesets soe sgft ffultes, tes of soluto, tht we woul lke to o. We look fo ffeet ppoh. We wll f the ffeet ppoh ot oly soles ths pole ut offes us wth sple ew uestg of syhoous he eletoget ys. The ffeet ppoh s se o the oseto tht ou toule oes fo the utes elte to the stto (phse wgs: Stto self utes Stto-stto utul utes Stto-oto utul utes.e., ll of these he te-yg utes. I oe to llete the toule, we wll pojet the -- uets oto p of xes whh we wll ll the xes o the - xes whh e ottg oote fe of efeee. Although we y spefy the spee of these xes 7

8 to e y spee tht s oeet fo us, we wll geelly spefy t to e syhoous spee. I kg these pojetos, we wt to ot expessos fo the opoets of the stto uets tht e phse wth the xes. Oe sulze the pojeto y thkg of the -- uets s hg susol to IN TIE log the espete xes. The ptue elow llusttes fo the -phse. -xs θ -xs ' g. 6 We osee fo g. 6 tht wll he opoet the -xs eto of osθ opoet the -xs eto of sθ. eoposg the -phse uet the -phse uet the se wy, the g the up, poes us wth: 8

9 k k os s os( os( s( s( Hee, the ostts k k e hose so s to splfy the uel oeffets the geelze KV eutos tht we wll get. We he tsfoe les,, to two les. Ths yels ue-etee syste, eg We uuely tsfo,, to We ot uuely tsfo to,, (uless thee s othe ostt suh s ++= So we ee th uet. We tke ths uet popotol to the zeo-seuee uet: k (-zeo We ote tht, ue le otos, s zeo, theefoe poues o flux. I ft, t s possle to show tht poues o flux whh lks the oto wgs t ll (see Coo s ook, pg. lso Kk Vol III, pg. 6. The plto s tht ue ll otos, se e eulet to, se flux fo oes ot lk wth othe uts, the, poue the ext se flux lkge s,,. We wte ou tsfoto oe optly s: 9

10 o k k k k k k os os( os( k s k s( k s( P P (e.. We y lso opete o oltges fluxes the se wy: P P, (e..7 Ths tsfoto esulte fo the wok oe y Bloel (9, ohety Nkle (96, Pk (99, 9, s esult, s usully lle Pk s tsfoto, the tsfoto tx P s usully lle Pk s tsfoto tx o just Pk s tx. I, Pk s 99 ppe ws ote the ost pott ppe of the lst yes (eh otesue s ppe o syetl opoets., Pk, Two eto theoy of syhoous hes, Tstos of the AIEE,. 8, p. 76-7, 99.. Heyt, S. Vekt, N. Bljepll, Hgh pt ppes powe egeeg, 9-999, NAPS,. See fo teestg ogphy o Pk, wtte y Chles Coo (hself oe of the ost fous powe syste egees ee!, eplte elow, togethe wth stteet tht ws poste to the Poweloe few yes go.

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12 o eet Poweloe susso: The el fouto of ost of the syhoous he theoy tlke toy ws l ppe y eh Egee, Bloel, who ws the fst to popose "two eto theoy" 895. The ohety Nkle pulshe extese lyss of syhoous hes usg two eto theoy ue of ppes etwee At the ehest of Chle Coo (s tol y Chle hself, Pk pulshe thee ppes 98 to 9 ogze the wok of ohety Nkle tx fo tht s wht s est kow toy tes of Pk's Tsfoto. Coo Pk wee ollegues E t tht te. - O lk

13 I Pk s ogl ppe, he use k=/, k=/, k=-/ (he ssue the -xs s leg the -xs; f he woul he ssue the -xs s lggg the -xs, s we he oe, the he woul he h k=/. Howee, thee e two stges wth ths hoe:. The tsfoto s ot othogol. Ths es tht P - P T. If the tsfoto wee othogol (P - =P T, the the powe lulto, whh s T p p T, s lso ge y ( s theefoe lle powe t y you ook. Ths e poe (see e.. text s follows. o oe es..,.7, P T T p P P P ellg tht ( T = T T, the oe s: p T o T T o, we y wte: T T P P P P o. The tsfoe utul utes, whe pe-utze, o ot poe tht jk=kj, plyg tht the pe-ut ute tx s ot syet. Ths peets us fo fg el physl ut to use oelg the tsfoe syste. See text, pg. 88 fo oe o ths. I oe to oeoe these poles, we (Aeso ou ke ffeet hoe of ostts, og to:

14 k, k k The hoe of k, whe pple to e. (-zeo oe, esults : So we see tht the fto s the ultple o ll thee eutos, esultg Pk s tsfoto ( the oe tht we wll use s: os s os( s( os( s( P (.5 Aothe hoe of oeffets s to hoose the s /, /, /, espetely, whh uses the gtue of the - uttes to e eul to tht of the thee-phse uttes. Ths hoe, use y Kk Vol III, es. (6 (wth egto fo oeffet ue to use of leg -xs, s efee to s gtue e, whh we poe elow fo the euto oly (ut ths uses / ultple fot of the powe expesso so s ot powe-t. k os os( os(

15 POO: et =Aos(ωt; =Aos(ωt-; =Aos(ωt- susttute to euto: k Aost os Aos( t os( Aos( t os( k Aos t os os( t os( os( t os( Now use tg etty: os(uos(=(/[ os(u-+os(u+ ] k A os( t os( t os( t os( t os( t os( t k A os( t os( t os( t os( t os( t os( t Now ollet tes ωt-θ ple kets ou wht s left: k A os( t os( t os( t os( t Osee tht wht s the kets s zeo! Theefoe: k A k A os( t os( t Now ote tht fo ka/=a (thus heg gtue e fo the uet, we ust he k=/. E. We ke two oe oets out Pk s tsfoto. st, euse t s othogol, the ese s esy to ot t s just P T, ge expltly s follows: osθ sθ P = [ os (θ π s (θ π os (θ + π s (θ + π ] (.9 5

16 Seo, the gle θ e geelze y hoosg y tl gle y spee, esultg ( ( t whee ɣ s uy le of tegto. Although Pk hose the spee to e the syhoous spee ( so wll we, t e y ostt o yg gul eloty o t y e sttoy. You wll ofte he of the ty efeee fe. The phse ty stes fo the ft tht the gul eloty of the geelze tsfoto s uspefe e selete tly to expete the soluto of the eutos o to stsfy the syste ostts [see Kuse s ook fo oe o geelze efeee fe theoy]. Pk s Tsfoto Apple tovoltge eutos fo 7-wg epesetto Now pefo the Pk s tsfoto o oth ses of the oltge euto (e.. o.6. Note tht we pply P to oly the -- uttes,.e., we lee the --- uttes loe se these uttes e ley o the oto ( the oto-oto utes e ley ostts. Ths es we ee to ultply e. (. o.6 though y tx P U ell (.6 s: whee U s x etty tx. 6

17 ultplyg though y ou tx, we ot: (e..6 P P P P U U U U t e te t e (e. te We ee to expess e. (te tes of -- uttes. I wht follows, we o ths oe te t te. Ou geel poeue wll e to eple the -- uttes wth -- uttes the splfy. te The esest oe s te, so we wll eg wth t. Te : P P U Te : P U Note tht P U Susttuto yels: P U 7

18 P P U P U P U P P Note tht the uppe left-h eleet hs gol tx the le of two othogol tes. P t: If P s ogthogol, the gol hg eul eleets o the gol. P f s You test ths s follows. et A It s esy to show ths s othogol usg A A T =U. The ty ultplyg A A T whee. It s esy to poe s follows. If s gol tx wth ll of ts gol eleets the se, ll the, the =U. The AA T = AUA T = AUA T =AA T =U=. 8

19 Hee, we wll ssue == whh s ey typl of syhoous hes sply ples tht ll phse wgs e eul legth wth the se type of outo, whh s lwys the se. Theefoe te s just: P U P P epetg ou euto (te hee fo oeee. P P P P U U U U t e te ellg wht we he oe so f: TE : TE : P P U t e te 9

20 P U P P Susttutg, we ot: P P U U te te te te e. (te Now we osee tht tes he les ot tes of -- uttes. We wok o te ext (efoe te euse t s ese. Te : Osee tht =[ ] T. Theefoe, whe we ultply P, we get eleets the seo th ows of P eg sle y the se ostt ( the sue. Cose these eleets the seo th ows of P, elow. P os s os( s( os( s( So the pout of the seo ow wth, o of the th ow, wll lue suto of syetl opoets,

21 whh wll e zeo. So the oly o-zeo eleet P wll e the pout of the fst ow of P,.e., the fst eleet of the te eto, whh s (* But ell fo ou ut the oltge euto tes tht: ( ( (** Also, ell tht fo the Pk s tsfoto o=p tht the uet s (pg. : (*** Susttuto of (*** to (** yels: ( ( eplg (* wth ths, we he: P U P (*# whee s the fst eleets s the lst eleets.

22 Now ell et. (te, epete hee fo oeee: P P U U te te susttute et. (*# to ot te P U te te te te te e. (te A so ow the oly -- les eg e te. So let s wok o te. Te : Te s: P U P (. So we ee to o two thgs:. Ot P tes of the -- uttes.. Expess ll of te tes of uets ste of flux lkges. P To eg ths tsk, ell tht, tke etes of oth ses. Note ffeettg the ght-

23 h-se, we ee to out fo the ft tht P s teepeet. Thus: Solg fo P P P, we ot: P (# But the ght-h se stll hs usg P P o. We elte ths Susttuto to e. (# yels: (. P PP Now we he expesse P tes of the -- uttes. Susttuto of e. (. to e. (. oe yels: P P o P P U te te So we he oplshe ou ojete, whh ws to ot P tes of the -- uttes. et s susttute the oe euto to e. (te P U te te te te e. (te

24 to ot PP te te te te te e. (te Now we ee to oplsh ou ojete, whh s to expess ll of te tes of uets ste of flux lkges. To o ths, let s estgte tes oe t te. et s stt wth te. Te : So te s: Ou gol s to see f we expess ths tes of uets, whh es we wll ee to use utes. et s stt y lookg t the se expesso ut wthout the etes, se we kow how to wte ths usg Pk s tsfoto -- flux lkges. Ths s: P U (e. - Now to wte e. (- tes of the / uets (ste of / flux lkges, ell fo e. (, pg., epete hee fo oeee

25 (e. - tht the eto of / flux lkges o the ght of (e. - s elte though the ute tx to the / uets. Now ell tht the / uets y e elte to the / uets usg the ese Pk Tsfoto og to: P U (e. - Susttuto of (- to (- the wht esults to (-, we he P P U U Pefog the oe tx ultplto, we ot. P P P P Now we ee to go though eh of these fou tx ultpltos. I wll hee ot the etls just ge the esults (ote lso wht follows the efto of tol oeltue fo eh of the fou sutes. But efoe og tht, let s e ouseles of wht the oe ute tes look lke. 5

26 6 (e. Y Y S S S S S S S S S s( s( s s( s( s os( os( os os( os( os s( s( os( os( os ( ] 9 os ( [ 5 os ( [ s( s( os( os( ] 9 os ( [ os ( ] os ( [ s s os os ] 5 os ( [ ] os ( [ os (e. -ex

27 7 Sutx (,: P P whee =S-S, =S+S+(/, =S+S-(/. Sutx (,: P Sutx (,: T P

28 8 Sutx (, (ote tht ths sutx s uhge fo the ogl ute tx: Y Y Usg the efe oeltue oe fo the eleets, we flly he: T Expg Y Y (.

29 9 Cope ths to e. (-ex o pge 6 ( pg. to see ey lge poeet splty. Ase: It s oeet hee to ote fo the oe tx elto tht e ge y: We wll use ths eelopg te elow. Oe e supse fo the oe s tht THE ATIX IS CONSTANT!!! As esult of ths e supse, we y ffeette oth ses to get: T ($ o, whe expe, s:

30 Y Y Susttuto of ($ fo te to e. (te, epete hee fo oeee, te te te te te PP e. (te esults te te te te te T PP e. (te5 We e lost oe! The oly eg te whh ots flux lkges s te.

31 Te : ellg te s: P P we see tht we ee to exp the pout. st, ell tht: P os os( os( s s( s( Also, ell tht ( ( t P P θ = ω(t A ote efully tht P s futo of te euse the gle s futo of t. Theefoe we ee to ffeette P. Ths s ot h esults : P P t s os s( os( Now tkg the pout, we ot: P P s( os(

32 P P s os s( os( s( os( os os( os( / / Note the oe tht ow s ll zeos euse ow P s ll zeos. O the othe h, olu s ll zeos euse the ultplto of ows P y olu of yel su of syetl tes. Ths poes tht: PP Soe oets o spee oltges -ωλ ωλ. These spee oltges togethe out fo the oltges ue the (fxe phse wgs s esult of the sptlly-og get fel fo the oto. They epeset the ft tht flux we ottg syhos wth the oto wll ete oltges the sttoy tue ols. s s( s( P

33 Spee oltges e so e to otst the fo wht y e lle tsfoe oltges, whh e ue s esult of te yg get fel. You y he u oss the oept of spee oltges Physs, whee you opute oltge ue ol of we s t oe though stt get fel, whh se, you y he use the euto Bl whee B s flux esty, l s outo legth, s the opoet of the eloty of the og outo (o og fel tht s ol wth espet to the fel flux eto (o outo. The fst spee oltge te, -ωλ, ppes the euto. The seo spee oltge te, ωλ, ppes the euto. Thus, we see tht the -xs flux uses spee oltge the -xs wg, the - xs flux uses spee oltge the -xs wg. tzgel Kgsley the ook Elet hey poe goo susso of spee oltges Chpte. Now we e posto to ot te. Usg the expessos fo ote the Ase of pge oe, we get:

34 P P whee spee Now ellg e. (te5, (& ; spee PP T te te we susttute (& to ot: te te e. (te5 spee T te te Puttg t ll togethe: te te e. (te6 te te

35 5 et s e-wte the oltge euto e. (te6 y susttutg oplete expessos fo ll etos sutes tes,,,,, s ote oe: Te Te Te Y Y Te Te Now, osee tht eh of the o-zeo eleets of te te s ultple y uet o uet ete, tht tes oth get ultple y etos of uets o uet etes, espetely. Theefoe, we

36 y fol- Te Te to the Tes y og pts of the o-zeo te eleets wth the ppopte tx eleet tes. o exple, we y fol the - te ow of te y lug ow (se we e elg wth the seo euto, olu (se we ee the te tht ultples of te. Note tht se te hs us sg out fot, we o ot lue the us sg of - whe we fol t. The le ow oe llustte ths folg- opeto. The oplete esults of ll fol- opetos e poe wht follows: 6

37 7 Y Y It s of teest to ege the oeg of the les so tht the oltge eutos fo ll -xs wgs e togethe the oltge eutos fo ll -xs wgs e togethe euse ths wll ephsze the pesee o see of the ous ouplgs tht we he. The esult of ths e-oeg of the les s s follows:

38 8 Y Y (e..9

39 Soe osetos out the tsfoe oltge eutos:. The fst tx ges. esste oltge ops. Spee oltge ops, s (tes wth. These s s Ou the - - uts, to epeset the ft tht flux we ottg syhos wth the oto wll ete oltges the sttoy tue ols o ot ou uts physlly lote o the oto, se thee s o oto etwee the ottg flux we the oto wgs. Ae use y uets the fel wgs of the othe xs: the -ut s s use y,, the -ut s s use y,,. The tes e lost ostt, exept fo the s tes the fst tx, ut ee these tes e ptlly ostt se we oly see sll hges. The osty of the tes s the otto eh the Pk s tsfoto.. The les he ee eogze so tht ll - xs uts e togethe ll -xs uts e togethe. Ths kes t esy to osee y ouplg/eouplg etwee ffeet sets of uts. 9

40 . The seo tx ges oltge ue y uet (o flux to. Note tht thee s o ouplg etwee the -xs uts (,, the -xs uts (,,. Ths s euse these two sets of uts e othogol. lly, soe oets out the Pk s tsfoto:. e uets fttous p of wgs fxe o the oto.. These uets poue the se flux s o the,, uets.. o le stey-stte opetg otos, we use = P to show tht the uets the wgs e! The plto of ths s tht: The,, uets fxe spe, yg te, poue the se syhoously ottg get fel s The, uets, yg spe, fxe te! o Kk, Vol. III:

41 s oe,, s (o uets pojete oto et xs. s oe,, s (o uets pojete oto utue xs. See g.6, p.8 of these otes fo llustto of pojeto o to et & utue xs. o ss opetg otos, The,, uets, fxe spe, yg te, poue the se syhoously ottg get fel s The, uets, yg spe, fxe te (C. Kk, lke Pk, hose / s hs oeffet fot of the Pk tx, see e. (.5, pg. of these otes. o ll opetg otos, poue the se o the espete xes s,,. o poues o gp flux. We use ths se guet the otes WgsAxes (pp. - to estlsh tht the self utul utes of otooto tes e ostt. Hee, Kk uses ths guet to poe tuto tht self utul utes of (fttous - & - wgs. Kk s hoe of oeffets hs the tge of gtuee ut the stge of ueul utul (see p. of these otes. Aothe teestg pgph fo Kk Vol. III

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