Per-unitization and Equivalent Circuits

Transcription

1 Pe-untzton n Eulent Ccuts 1. Nomlzton of oltge eutons We ese to nomlze the oltge eutons, tht s, we ese to expess them n pe-unt. The ntges of ong so e: The pe-unt system offes computtonl smplcty by elmntng unts n expessng system unttes s mensonless tos. t elmntes the nee fo usng bty constnts n smplfes some of the mthemtcl expessons so tht they my be expesse n tems of eulent ccuts. The numecl lues of cuents n oltges e elte to the te lues especte of mchne sze. mpences, when gen on the mchne bse, le on eltely now nge so tht eos cn be esly etecte. Thee e seel ffeent possble nomlzton schemes. Wht Aneson n ou (A&) he one (see Appenx C n text) s to cefully compe the mets of ll of these schemes. n ong so, they eelope some cte, guelnes, the most mpotnt of whch s tht the eutons must be nepenent of whethe they e n pu o KS fo both the oltge eutons s well s the powe expesson (powe nnce). Note tht mchne mnufctues, when expessng the mchne t n pe unt, my use ffeent system tht oes not stsfy the powe nnce popety - they use P s ognl tnsfomton (clle n A&, e. (4.), nste of ou P ). The choce me by A& stsfes the boe cte; n ton, the A& choce ensues tht the numecl lues of the pe-unt mpences e the sme s those poe by mnufctues usng the system of nomlzton. 1

2 n most unegute powe system nlyses couses, we len tht peuntzton eues selecton of two bse unttes out of the followng fou:,, Z, n S, n then the bse unttes fo the othe two e compute. The stuton s the sme hee, except tht we lso must el wth spee (o feuency). Ths necessttes tht we must lso select bse fo ethe feuency ( o f) o tme, t. n ton, we wll lso he nee to compute bse unttes ssocte wth flux lnge () n nuctnce ( o ). Ou ppoch wll be to obtn the bses fo the stto se n then the bses fo the oto se. One my note two excellent efeences on the subject of pe-untzng synchonous mchne moels: 1. A. nn, Pe-unt mpence of synchonous mchnes, AEE Tnsctons, 64, Aug., Hs, P. wenson, n J. Stephenson, Pe-unt systems wth specl efeence to electcl mchnes, Cmbge Unesty pess, Cmbge, Engln, 197. Othe efeences tht ess ths subject, beses A&, nclue those by - Sue n P - Conco - Py - Kunu n lso couse notes fom e ello. 1.1 Stto se pe-untzton: We select ou stto-se bses s: : the stto te lne-neutl oltge, ms. S : the stto te pe-phse powe, olt-mps : the geneto te spee, n electcl /sec (= e =77)

3 Then we my compute bses fo the followng 5 unttes: cuent: mpence: tme: t S X 1 Z S (Note we coul he use t but ths woul smply poe ffeent sclng n s theefoe bty). Note tht ou choce of t s the tme eue fo the oto to moe one electcl n. lux lnge: t (Ths comes fom the fct tht t ) t t = = X ω ω nuctnce: = λ = t X t ueston: How oes ou choce of stto bse unttes ffect the pe-unt lues of the - n -xs unttes? Note: lthough - n -xs unttes ssocte wth fcttous oto wnngs, we ew them to be stto unttes. To nswe ths ueston, let n be the ms mgntues of the -phse lne-neutl oltge n -phse lne cuent, espectely. Then the pe-unt phsos e u u u u Now let s nestgte the unttes.

4 4 To begn, note tht the expessons fo nstntneous oltges n cuents fo ech phse e: ) 1 sn( ) 1 sn( ) sn( c b ) 1 sn( ) 1 sn( ) sn( c b Use the P s tnsfomton on the boe to obtn: cos sn cos sn (Ths confms ou concluson t the en of the lst set of notes, mchets, tht, fo blnce contons, the unttes e constnts,.e., C.) Now, pe-untze by ng by n : cos sn cos sn u u u cos sn cos sn u u u Obsee bout the boe tht 1. The pe-unt n oltges e eul to the pe-unt -phse oltge scle by sn n cos, espectely.. The pe-unt n cuents e eul to the pe-unt lne cuent scle by sn n cos, espectely.

5 1. oto-se pe-untzton: ecll tht n system pe-untzton, we must select sngle powe bse fo the ente system, nepenent of the fct tht some sectons of the system e mgnetclly couple though tnsfomes,.e., we o NOT choose ffeent powe bses fo ffeent ses of tnsfome. The sme estcton pples hee, whee the oto ccut s mgnetclly couple to the stto ccut,.e., the powe bse selecte fo the stto se must lso be the powe bse use on the oto se. Ths s S. n ton, we e eue to select the sme tme (o feuency) bse fo both the stto se n the oto se. Ths s t (o ). On the oto se, we he 1 bse left to choose. o tnsfomes, we typclly choose the 1 emnng bse s the oltge bse (o cuent bse) ccong to the tuns to. Hee, howee, we o not now tuns to, n theefoe we e left wth poblem of wht, n how, to choose. (One text tets the poblem une the ssumpton tht tuns to s nown between stto n oto ccuts - see the text by Py, Powe System ynmcs, pp ) n mng ths choce, poblem esults fom the fct tht stto powe leels e typclly seel tmes the oto powe leels. A& ge n nteestng compson (see pg 95) of typcl stto-se pe-phse powe tng of 1 A n fel wnng tngs of 5, 1A (5w). Wht e ou choces of the one emnng oto-se bse untty n ths cse? Choose oltge bse=te oltge=5, but then the cuent bse s =1E6/5=4 mps, n pe-unt lues of fel cuents wll be ey smll. Choose cuent bse=te cuent=1a, but then oltge bse s =1E6/1=1 olts, n pe-unt lues of fel oltges wll be ey smll. 5

6 Anlogy to tnsfomes: Wth tnsfomes, we choose the bse oltge (o cuent) on se 1, n then we choose the bse oltge (o cuent) on se s tht oltge pouce by the tnsfome on se when the bse oltge on se 1 s pple. We ths becuse we wnte pe-unt ccut of the tnsfome whee the el tnsfome ws elmnte. Anothe wy to thn bout wht we o wth tnsfomes s tht we select cuent bses tht woul pouce the sme mutul flux between the two wnngs,.e., we choose 1 n such tht: 1 pouces 1 (the flux lnge fom cuent 1 n col 1 tht lns col ), pouces 1 (the flux lnge fom cuent n col tht lns col 1) n 1 = 1. Ths s bse on the fct tht 1 = 1 1 n 1 = 1, whee 1 = 1 = s the mutul nuctnce between the two tnsfome cols. f col 1 cetes 1, n of ths, only the mutul flux 1 lns wth col, then the ffeence s the lege flux gen by 1 = 1-1 (e. 1) n llustte n g. 1. ege flux, 1 utul flux, 1 1 g. 1: utul n ege flux Hee, ech of these thee fluxes e gen by 1 =l 1 1, 1 = 1 1, n 1 = 1 (e. ) whee l 1, 1, n, e the lege, self, n mutul nuctnces, espectely. Substtuton of () nto (1) esults n: l 1 1 =

7 n cncelng 1 ges: l 1 = 1-1 =l 1 + whch mples tht the self nuctnce s compse of the lege nuctnce plus the mutul nuctnce. Sml nlyss esults n =l +. c to synchonous mchnes: We cn pply the sme concept to the synchonous mchne s we pple to the tnsfome boe. Tht s, We select the bse cuents fo the fou oto-se wnngs, (, ) to pouce the sme mutul flux n the gp s pouce by the stto-se bse cuent flowng n the coesponng fcttous -xs (-xs) col. We wll begn by pplyng ths e to obtn the bse cuent fo the mn fel wnng. The eson ths s benefcl s tht t wll enble us to eelop eltely smple ccut to epesent ectxs pu unttes n nothe one to epesent utue xs pu unttes. The fom of ths cct wll be tee. (see Pe App C, top of p. 549) n pg. 4 of these notes. se-cuent fo mn fel wnng, ppoch 1: One cn sulze the boe concept fo the cse of the eltonshp between the -wnng n the -wnng, n g.. -wnng -wnng m g. : se cuents n n wnngs We see fom g. tht we select, the fel wnng bse cuent, s 7

8 8 tht cuent when flowng n the -wnng wll pouce mutul flux m eul to the sme mutul flux tht s pouce by cuent flowng n the -wnng. ut how o we compute? om ou peous set of notes (p. 8, mchets ), n lso e. 4. n text, we ee e. (4. ) om ths, we cn see tht (e. ) whee =(/). s ll of the flux pouce by the -wnng, but only pt of ths flux, the mutul flux, lns wth the -wnng. Cll ths flux fom the -wnng tht lns wth the -wnng m, gen by m = m, whee m s the mutul nuctnce ssocte wth ths flux (A& cll t mgnetzng nuctnce).

9 As wth the cse of the tnsfome, the ffeence between the totl flux fom the -wnng n the mutul flux s ttbute to the lege flux, so tht, l = - m (e. 4) Cncelng the cuent, we see tht l = - m =l + m (e. 5) When flows n the -wnng, so tht =, the mutul flux s gen by m = m (e. 6) oong bc t e (), we see tht the flux fom the -wnng tht lns the -xs wnng s just. Ou cte fo selectng sys tht when flows n the -wnng, the mutul flux lnng the -wnng shoul eul the mutul flux fom the - wnng lnng the -wnng when t ces. Thus, we wte tht m = m = (e. 7) An we see tht (e. 8) m n m e genelly poe n (o cn be obtne fom) mnufctue s t fo gen mchne 1. cn be compute s llustte n Exmple 4.1 (whch we eew below), usng the mgnetzton cue, m = -l, whee mnufctue s t sheets contn n l. Theefoe, once s selecte, my be compute. se cuent fo mn fel wnng, ppoch : One my lso eelop elton fo fom the pespecte of the flux lnng the fel wnng,.e., nste of usng e. () fom (4. ), use: (e. 9) Sml to e. (5), the self nuctnce s compse of the lege n the mutul,.e., 1 Note: n KS unts (.e., henes), m s not the sme s,.e., the ecpocl mutuls e not eul. 9

10 =l f + m (e. 1) nspectng e. (9), we see tht the flux fom the -wnng lnng wth the -wnng s, so tht when = n =, we he tht m = (e. 11) n we see tht (e. 1) m whee, s befoe, s obtne pe Exmple 4.1 below, m = -l, n, l e obtne fom mnufctue s t sheet. se cuent fo -wnng: We select the -wnng bse cuent, ccong to the followng cte: We select, the -wnng bse cuent, s tht cuent when flowng n the -wnng wll pouce mutul flux m eul to the sme mutul flux tht s pouce by cuent flowng n the - wnng. Sml nlyss s fo the -wnng esults n (e. 1) m, m We my lso utlze sml poceue between n wnngs to obtn m (e. 14) se cuent fo -wnng: We select -wnng bse cuent, ccong to the followng cte: We select, the -wnng bse cuent, s tht cuent when flowng n the -wnng wll pouce mutul flux m eul to the sme mutul flux tht s pouce by cuent flowng n the - wnng. 1

11 Sml nlyss s fo the -wnng esults n, (e. 15) m m se cuent fo -wnng: We select the -wnng bse cuent, ccong to the followng cte: We select, -wnng bse cuent, s tht cuent when flowng n the -wnng wll pouce mutul flux m eul to the sme mutul flux tht s pouce by cuent flowng n the -wnng. Sml nlyss s fo the -wnng esults n m, (e. 16) m We my lso utlze sml poceue between n wnngs to obtn (e. 17) m Summy: Et. (8) togethe wth ets. (1-17) poe the blty to eelop ny of the eutons gen s (4.54) n the text. These eutons e efee to s the funmentl constnts mong bse cuents n e gen by: m m m m m o exmple, ecllng (8) s m n (1) s, we m cn multply the left-hn-ses togethe n the ght-hn-ses togethe to m obtn: m m. m Now efne the followng fctos: 11

12 ,,, ecuse we he the sme powe bse on ll stto n oto ccuts, we obtn: S Then,,, Note tht these -fctos my be consee to be effecte tuns tos. We my lso ee expessons fo the esstnce n nuctnce bses. Note ou ese s to be ble to compute oto-se bses s functon of stto-se bses. The -fctos gen boe wll be ey hny hee. oto-se esstnce bses: ewse, oto-se nuctnce bses:,, ewse, t t,, 1

13 oto-stto mutuls: ou text, pg. 95 efes to HW poblem 4.18 whch sttes tht bse mutuls must be the geometc men of the bse self-nuctnces,.e., 1 1 Thus, we he tht the bse fo the fel wnng to stto wnng mutul tems s gen by (see e n text): Note tht t s not the sme s the bse self nuctnce gen boe. ewse, we get (see e n text, except hee we nclue ):, oto-oto mutuls: Thee e just of them (see e n text, except hee we nclue ): ewse,. Exmple 4.1, pg 97 of text Ths s goo exmple tht you shoul eew cefully. Hee s the fst pt of t (p. 98 contnues wth t)., 1

14 Why e some lues estmte fo cemc stuy? Note ths sttement; Wht s gp lne? Note the use of pe phse oltge. The only thng tht s pehps not too cle s the computton of. wll just eew tht pt of t hee. Computton of : A& me the sttement, om the no-lo mgnetzton cue, the lue of fel cuent coesponng to the te oltge on the -gp lne s 65 A. The open-ccut chctestc o mgnetzton cue plots Somethng popotonl to exctng (fel) cuent on hozontl xs Somethng popotonl to the flux on the etcl xs. une open-ccut contons (the -phse wnng s open). gue below llusttes. 14

15 φ λ A-gp lne ue to stuton of the on ( ecese n pemeblty o ncese n eluctnce) fo hgh =N H=/l g. The -gp lne s the s. elton tht esults f the on hs constnt pemeblty. The sol lne tht bens to the ght s the ctul chctestc tht occus, whch shows tht temnl oltge flls wy fom the -gp lne s the fel cuent s se beyon cetn pont. Ths fllng wy s cuse by stuton of the feomgnetc mtel, esultng fom the ecese n pemeblty une hgh flux contons. gue 4 llusttes mgnetzton cue fo el 1.8 synchonous mchne. The etcl xs s lne-to-lne oltge. 15

16 Ths synch gen s septely excte, n so ts fel cuent f (hee esgnte s oto mps ) s supple fom the mtue of septe C gen. The septe C gen hs fel cuent Cgen (on C gen stto) whch cetes fel flux ϕ. The cuent f, whch s the mtue cuent of the C gen n the fel cuent of the synch gen, nceses wth C gen mtue oltge E, n E =ωϕ, whee ϕ nceses wth Cgen. An so f n Cgen e both nctos of synch gen fel stength. Cgen E f el wnng + - The two mgnetzton cues to the left plot lne-to-lne open cct oltge of the synch gen gnst () Cgen n (b) f. g. 4 Wht s one n Ex. 4.1 (n wht s ctully one n nusty to obtn ), s tht the fel cuent s etemne coesponng to stey-stte te open ccut temnl oltge. Ths oltge s = -te /st(). o Ex. 4.1, ths s =15/st()=866 olts. Ths s the ms oltge, but A& ncte tht we nee the coesponng pe oltge: pe = (866)=1,47.1 olts. ut why o we nee the pe oltge? et s conse ths ueston. 16

17 om fst pge of peous notes ttle chne Eutons, o fom e. (4.11 ) n A&, we he b b c c ut = b = c = une open ccut contons. An mpe cuents = = une stey-stte contons. Theefoe ecll tht the g-wnng moels the -xs flux pouce by the eycuent effects n the oto ung the tnsent peo. ut we e now conseng only the stey-stte conton, =. Theefoe (*) Now ecll fom fst pge of peous notes ttle mchets, o fom e. (4.16 ) n A&, tht = cosθ, n substtuton nto (*) yels cos (**) ffeenttng (**) esults n t sn e t Now ecll the oltge euton fo the -phse: Substtutng (***) nto (#), we obtn n sn (***) (#) e sn ut une open ccut contons, =, n = (mplyng n =) n we he om (#*), we see tht pe e sn (#*) So we choose pont off the mgnetzton cue, fo exmple, A& choose =65A, pe =1,47.1olts (65A s the lue of fel cuent e n pe e 17

18 coesponng to the te oltge on the -gp lne, n 1,47.1/st()=866olts s the te S lne-to-neutl oltge (coesponng to 866st()=15). Then pe e An fom ths we cn compute 1, (65)(77) m l henes whee the enomnto s compse of t poe by the mnufctue. The est of Ex. 4.1 s just n pplcton of ou pe-untzton fomul. Thee s n nteestng pgph n Appenx C, pg. 55 of you text, to whch wnt to w you ttenton. t sys, Note tht ey element n etemnng the fcto, n hence ll the oto bse unttes, s the lue of (n H). Ths s obtne fom the gp lne of the mgnetzton cue poe by the mnufctue. Unfotuntely, no such t s gen fo ny of the motsseu ccuts. Thus, whle the pu lues of the ous motsseu elements cn be etemne, the coesponng KS t e not nown. poe some comments on cetn sentences n ths pgph: Note tht ey element n etemnng the fcto, n hence ll the oto bse unttes, efes to the fct tht we obtn n fom: Ths s obtne fom the gp lne of the mgnetzton cue poe by the mnufctue, s we he seen boe by usng pe e 18

19 We e ble to get n ths wy becuse we cn ectly contol the cuent, wth no othe ccuts enegze (s esult of the open-ccut, stey-stte contons), n ectly mesue the nuce oltge t the -phse temnls. Unfotuntely, no such t s gen fo ny of the motsseu ccuts. t s not possble to ectly contol the cuents,, n, snce the coesponng ccuts o not he souces. The only wy to enegze these ccuts s tnsent conton, but thee s no wy to poe tnsent conton tht wll lso not enegze othe ccuts, whch woul esult n the mesue temnl oltge beng nuce fom the mutul nuctnce between tself n the othe ccuts s well. Thus, whle the pu lues of the ous motsseu elements cn be etemne, the coesponng KS t e not nown. n exmple 4.1, the text puts n stes by some of the pmetes (,,,,, n ), nctng they wee estmte fo cemc stuy ). Ths s becuse mnufctue s tsheets o not lwys nclue the pmetes fo the motsseu (n g-wnng) ccuts, smply becuse they e h to mesue (bse on the comments of the peous bullet). Howee, t s possble to obtn the pe-unt lue (not the KS lue) of some of the motsseu ccut pmetes (specfclly, the mutul nuctnces), becuse, s we shll see n Secton. below, n pe-unt, ll ect-xs mutuls e eul n ll utue-xs mutuls e eul! n othe wos: -xs mutuls: - wnng mutul, - wnng mutul, - wnng mutul, (s clle X n some texts) Tht s, we wll show tht n pe-unt, -xs mutuls: - wnng mutul, - wnng mutul, - wnng mutul, Tht s, we wll show tht n pe-unt, u u u u u u 19

20 .4 Applyng the bses to oltge eutons: ecll ou oltge euton s wtten n KS unts: n c b n et s nomlze them usng ou chosen bses to obtn the eutons n peunt. The pe-unt eutons shoul ppe s boe when one, except tht eeythng must be n pe-unt. Step 1: eplce ll KS oltges on the left wth the pouct of the pe-unt lue n the bse lue (use fo the fst eutons n,,, fo the lst fou eutons), n eplce ll cuents on the ght wth the pouct of the pe-unt lue n the bse lue (use fo the fst eutons n,,, fo the lst fou eutons). Ths esults n elton sml to e. 4.6 n the text, s follows.

21 1 u u u u u u u n u u u u u u u c b n u u u u (e. 4.6 ) Step: o ech of the eutons n the boe, we nee to e though by the oltge bse. o those eutons contnng, we eplce t wth = u ( = e ). Then we o some lgeb on ech euton to expess the coeffcents of ech cuent n cuent ete s pe-untze self o mutul nuctnces. As n exmple, the secon euton s one fo you n the text; hee, wll o the lst euton, coesponng to the -wnng. u u u u ) ( Step : e though by to obtn:

22 u u u The fst tem hs enomnto of. The lst thee tems e not so obous. We ese them to he enomntos of,, n, espectely, whee, fom boe, we ecll, u,, whee, n. Step b: et s multply the enomnto of the lst thee tems by /. Ths esults n: u u u Step c: et s multply the enomnto of the lst two tems by /. Ths esults n: u u u u u Step : ecll the -fctos (pg 96 of text): Substtuton yels: u u u, n u. Step e: We e close now, s we nee =, =, n =( ), espectely, on the enomnto of the lst thee tems. ecll

23 tht = /( ), so we nee to e top n bottom on the enomntos of the lst thee tems by. ong so yels: u u u Step f: An substtutng n esults n: u u u u u Step g: ecllng tht =, =, n =( ), we my wte: u whch esults n u u u u u u u u u u u Step h: Howee, we stll he one poblem. ecll tht we wnt the eutons to be entcl n pu to the fom n KS unts. ut n the lst euton, we stll he, whch oes not ppe n ou KS euton. We cn te ce of t, howee, by ecllng tht =1/t, so tht:

24 1 1 1 u u u 1 1 t 1 1 t 1 1 t t u u t t u u t t u t t u t t u ; u ; u whee =t/t s the nomlze tme. Wth ths lst chnge, we cn wte, fnlly, tht u u u whch s the pe-untze fom of the lst euton n e. (4.6 ). Note tht t s exctly the sme fom s the ognl euton n KS unts. Sml wo cn be one fo the othe eutons (n you shoul ty to o one of the othes youself), esultng n eutons sml to e n you text: u u u u 4

25 5 (e ) Note tht n the boe euton, The u subscpt ws oppe; howee, ll pmetes e n pe-unt. We he oppe the zeo-seuence oltge euton snce we wll be nteeste n blnce contons fo stblty stues. (A system hng thee-phse fult, consee to be, usully, the most seee, s stll blnce system. Ths oes not men tht we cnnot nlyze unblnce fults usng stblty pogms. t s possble to nlyze the effects of unblnce fults on the poste seuence netwo epesente n stblty pogms see Kmb ol, pp. -1). Othewse, e. (4.74 ) s pecsely the sme s et. 4.9 n you text (see the sme euton s 4.9, except fo the wnng nclue, n the notes clle mchets we clle t thee, e. 4.9 ).

26 The eutons e enge to bette sply the couplng n ecouplng between the ous ccuts. Ths couplng cn be well llustte by fgue sml to g. 4. n you text, gen below s g. 4.. Notce tht the couplng between the n wnngs s cptue by X. We he clle ths mutul nuctnce n ou wo boe to emn consstent wth A&. These e physclly-elzble ccuts fo whch K n ech of the 6 ccuts esults n the 6 eutons of 4.74 boe. + n + n + - = X ot conenton:. f the efeence cuent ecton entes the otte temnl of col, the efeence polty of the oltge tht t nuces n the othe col s poste t ts otte temnl. b. f the efeence cuent ecton lees the otte temnl of col, the efeence polty of the oltge tht t nuces n the othe col s negte t ts otte temnl. = = g 4. 6

27 7 Now let s me some efntons: ; N ; Wth these efntons, we ewte et. (4.74 ) n compct notton: N ) ( (e. 4.75) We my sole e. (4.75) fo /t so tht t s n stte-spce fom: N 1 1 ) ( (e. 4.76)

28 . Pe-unt mutuls (See Secton 4.11) A useful obseton egng pe-unt lues of,, n : ecll ou efntons of the -xs -fctos: n tht we eelope (see es. 4.54, pg 11 of these notes): m m m (18) om the fst n fouth expesson n e (18), we he: m Thus, (19) ewse, fom the fst n ffth, n fom the fouth n sxth expessons n e (18), we he: m Thus, n m m om the efntons of the -fctos, n es (19) n (), we he: n m () (1) m An fom e n text (lso see p1 of these notes), we fn 8

29 9,, () whch we obtne by usng the fct tht bse mutuls must be the geometc men of the bse self-nuctnces (see pob 4.18). Now, ecll the elements n the pe-untze oltge eutons s gen by e (see pge 4 of these notes) n ptcul, conse the mutul tems n the lst mtx fo the ect xs. Ths s the uppe left-hn x bloc, n the blue box. These tems, n pu, e by efnton the to of the tem n KS to the ppopte bse. Theefoe: Stto-fel mutul: u. Substtutng fo fom e. () n then fom e. (1) esults n:

30 u u Stto--wnng mpe mutul: m m u. mu Substtutng fo fom e. () n then fom e. (1) esults n: u el--wnng mpe mutul: m u m mu Substtutng fo fom e. () n then n fom e. (1) (usng the n expesson fo n e. (4)) esults n: m mpotnt fct: n pe-unt, ll -xs mutuls e numeclly eul! We wll efne new tem fo them, A, s the pe-unt lue of ny -xs mutul nuctnce, so tht: A mu u u u Also note tht, snce the mutul s the ffeence between the self n the lege, ths mples u -l u = u -l u = u -l u = A The boe eltons e gen n es n 4.18 n you text. We cn go though sml pocess fo the -xs mutuls (fom 4.74, we see tht these e the tems n the lowe ght-hn bloc of the mtx,,, n ). wll lee ths fo you to o. The esult s: m mu

31 1 u u u mu A u -l u = u -l u = u -l u = A The boe eltons e gen by e n you text, except fo the ton of the -tem n the -tem. A n A e ey mpotnt fo wng the eulent ccuts. They e lso mpotnt n elng wth stuton becuse they poe fo the efnton of the pe-unt mutul flux (we wll see ths n ou eelopment of the flux-lnge stte-spce moel). 4. Eulent Ccuts (See Secton 4.11) et s etun to the oltge eutons tht we h befoe we fole n the spee oltge tems. They wee: n n c b Assume ll of the boe s n pe-unt (but we he oppe the u- subscpt).

32 Thee s some ntge to e-wtng these eutons n tems of A n A. o exmple, conse the -xs euton. t s: ecll tht = m +l m = -l et s mofy the -xs oltge euton by ng n subtctng l ( /t) : l l, whch cn be wtten s: l l ] ) [( The ntge to ths s tht, n pe-unt, we ecll tht -l = = A. Theefoe, A l ] [ et s epet ths fo the -xs euton, whch s, fom the mtx euton t the begnnng of ths secton: et s mofy the -wnng oltge euton by ng n subtctng l ( /t): l l whch cn be wtten s: l l ) ( The ntge to ths s tht, n pu, we he -l = = A. Theefoe ) ( A l

33 epetng ths poceue fo the,, n eutons, n then summzng, we obtn: -xs eltons: l [ ] l A [ ] [ ] A l A -xs eltons: l ( ) A l A ( ) l ( ) A We ese to w ccuts tht e chcteze by these eutons. Note: The -xs eltons e couple though the A tems. Ths tem, fo ech euton, my be epesente by sngle cente bnch. The othe tems, fo ech euton, my be epesente s sngle bnches whch fee the cente bnch. Ths esults n the ccut of g 4.5 n you text. Sml esonng esults n the ccut of g. 4.6 n you text. We ew these ccuts below.

34 l [ ] A l A[ ] l [ ] A l l l A ect-xs eulent ccut: The boe s the sme s g. 4.5 n you text l ( ) A l A ( ) l ( ) A l l l A utue-xs eulent ccut: The boe s the sme s g. 4.6 n you text, except we he nclue the -ccut 4

35 The blty to w these ccuts s ect esult of the A n A eltons tht occu only n pe-unt. Theefoe, t s mpotnt to be n the pe-unt system when utlzng these ccuts. An the eul mutuls effect cme s esult of the fct tht we chose ou bse cuents ccong to the followng cte (see p. 7 of these notes): We select the bse cuents fo the fou oto-se wnngs, (, ) to pouce the sme mutul flux n the gp s pouce by the stto-se bse cuent flowng n the coesponng fcttous -xs (-xs) col. See Appenx C of you text, t the top of p. 549, fo tculton of ths fct. These eulent ccuts e useful fo: emembeng the oltge eltons. nng physcl unestnng of eltons between unttes. eng the ltetue, whee you wll see them often. 5