System Normalization and Iron Saturation Based on Generalized Coupled Circuits Analysis as Fundamentals for Electric Machines Modeling Course

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1 Syse Noralzao ad ro Saurao Based o Geeralzed Coupled Crcus Aalyss as Fudaeals for Elecrc Maches Modelg Course Ree Wakeue, Seor Meer, EEE Nee Nee éadre, Fouad Slaou-Hasaou École de Gée, UQAT Dépar. G. élecrque, ENSET École de Gée, UQAT Rouy-Norada, QC, Caada Douala, Caerou Rouy-Norada, QC, Caada ree.wakeue@uqa.ca leadre@gal.co fouad.slaou-hasaou@uqa.ca Asrac Ths paper descres he use of a suale approach o eachg geeralzed agecally coupled elecrc crcus as a roduco chaper o he elecrc aches odelg ad sulao course for power egeerg sudes. The eachg ehodology focuses o soe coo coceps ad fudaeals of elecrcal ache heory such as ache ducaces (self, leakage ad uual), equvale crcus, agec crcus, ro saurao, recprocal per u syse, sae odelg ad sulao, so ha he odelg approach of each ype of classcal ache ca easly e deduced fro he geeral heory esalshed y he chaper o agecally coupled elecrc crcus. Keywords Magecally Coupled crcus; ro saurao; per u syse oralzao, sae space odel, applcao ad sulaos. NTRODUCTON Magecally coupled elecrc crcus are ceral o he operao of rasforers ad elecrcal aches, as specfed y Prof. Krause e al. []. A good udersadg of he heory uderlyg agecally coupled crcus grealy splfes oh he eachg ad he learg of rasforers ad elecrcal aches []-[4]. Therefore, for power egeerg sudes, specal aeo should e pad o eachg agecally coupled crcus, wh a ephass o her elecrcal ache srucure, as he frs chaper of he elecrcal aches course where rase odelg ad sulaos are cluded. The odelg of he equvale elecrc crcus of elecrcal aches s ased o agecally coupled crcus. The odelg ad sulao of elecrcal aches are grea challeges for power syse egeerg sudes aroud he world. Currely, ay uverses aroud he world, he agecally coupled crcus are oly augh he elecrc crcus, whch s a coo udergraduae course for frs year elecrcal egeerg progras []-[4]. Alhough rase ehavors are cluded elecrc crcus course, he sudy of elecrc aches for udergraduae power egeerg sudes s os of e sll led o he seady sae perforaces [5]. Furherore, here are o suffce pedagogcal lks ewee he courses elecrc crcus ad ay udergraduae exooks reag elecrc aches whch geeral do coe agecally coupled crcus [5]-[6]. Very few auhors have realzed he porace o ser he agecally coupled crcus as he frs chaper of her research exook [], [7]-[8]. However, eve hough he agecally coupled crcus developed [], [7] cluded he saurao pheoeo ad sulao aalyss, her elecrcal odels gve lock-dagra for, are o flexle o adap o he roag aches sudy ad corol purposes. As show he prese work, he course o agecally coupled crcus offers eresg opporues o roduce a fudaeals of he elecrcal aches odelg ad sulao. hs paper, geeralzed coceps ha could e appled o all elecrcal aches aalyss such as he sae odel, elecrcal equvale crcus, per-u syse oralzao, ro saurao, uercal sulaos are roduced a course o agecally coupled crcus o faclae he udersadg ad eres of fuure power egeerg sudes coecg he sudy ad aalyss of elecrcal aches. Repeag hese coceps over e whe aalyzg every ype of elecrcal ache (redudacy prcple) wll crease he sude s udersadg of he pedagogcal ojecves o e acheved.. EQUATONS OF GENERAZED MAGNETCAY COUPED CRCUTS Magecally coupled elecrc crcus wh a adjusale uer of cols are show Fg.. The flux equaos for cols are gve () ad reorgazed () whle he volage equaos are defed (3) []: wh: λ N + N R ( N + N + + N ) f λ N + N R ( N + N + + N ) f λ N + N R ( N + N + + N ) f ( l ) ( ) λ λ + λ + + λ λ λ + λ + + λ + + l + λ λ + λ + λ l d d d e r d d d e r d d d e r () () (3) /3/$ EEE

2 N N R l N f ; ; NN j j j ( R f N + N The followg relaoshps are derved ewee varous varales ad paraeers of he cols: ( j ) (4) + l (5) where: dλ dλ e r + ; e r + dλ ; ; e r + a N N ; j; e a e ; a j j λ a λ; r a r; l al; a ; a ; (7) j (8) Fg.3. Elecrcal equvale crcu for a adjusale uer of agecally coupled cols referred (oserved) fro wdg # ( prary) Fg.. Coupled elecrc crcu wh adjusale uer of wdgs. EQUVAENT CRCUT REFERRED FROM CO The a ojecve whe odelg a elecrcal syse such as agecally coupled crcus for he sudy of elecrcal aches s o derve flux ad volage equaos such ha he equvale crcu whou a galvac lk as llusraed Fg. ca e draw. The Krchhoff esh ad odal laws ca herefore e appled o he equvale crcu odel oaed for aalyss ad he soluo of he physcal agecally coupled crcus. order o acheve hs goal, flux ad volage equaos () ad (3) wll e odfed so ha he curre, flux ad volage of a gve col ca e oserved or referred fro col. The cocep of referred crcu oaed y creag fcous varales ad paraeers allows he galvac lk of Fg. o e reoved ewee cols ad, hus creaes her elecrcal coeco. such a coex, he apere-ours ( N j j) a gve secodary col j ca e aheacally replaced y he fcous prary apere-ours ( N ) so ha he curre he col j s gve y j N N referred fro c s equvale e r j j j l M e e Fg. Equvale crcu of wo agecally coupled crcus whou elecrcal coeco (wh a galvac lk) λ l λ + l λ l + 3 l ( col. r ) e() (6) Accordg o he eergy coservao prcple, he power he fcous crcu ad he acual secodary should e he sae ( e j j e j j ). roducg he aove fcous curre ad volage referred fro col () ad (3), he laer equaos ca e orgazed as gve (6) ad (7) wh he prary ad secodary j oh oor ode (for exaple, Fg., cols ad asor power fro exeral sources). The equvale crcu of he geeralzed agecally coupled crcu of Fg. referred fro col ( he oor ode) s gve Fg.3. s pora o oserve ha Fg. s que close o elecrcal equvale crcu of a varale reluca sychroous ache wh adjusale daper wdgs or ul-roor wdgs duco ache. The agezg ducace s also called he uual ducace sce a. j j V. NORMAZATON OF GENERAZED COUPED CRCUTS: PER UNT SYSTEM A. Defo of Base Values Noralzao schees are frequely used power syses odelg ad sulao. The per-u syses appled for he oralzao of elecrc ache equaos are o ore ofe clear o sudes. The grea advaage of he oralzao schee s ha he syse equaos are he sae depedely fro whch col hey are referred. The classcal agecally coupled crcus such as elecrcal aches ad rasforers are defed y he raed values of her dvdual crcus (): appare power S (VA), volage E N N (V), curre N (A), ad he raed pulsao ω π f (rad/s); where f (Hz) s he oal frequecy of he prary wdg volage e () (col ) suppled y he exeral ework. sgle-phase crcu aalyss, as s he case hs sudy, he ase quaes for a gve col () are defed Tale. Fro hs ale s oserved ha for wo arrary dsc cols ad j: E E a a j jj j j hus:

3 E E E E 3 3 S S S S 3 The aove relaoshp (9) s fudaeal ad geeral elecrcal aches aalyss whe syse oralzao s cocered. saes ha: he per-u oralzao, he VA ases of coupled crcus should e equal. TABE : BASE VARABES OF CO # Quay Base us De. Power S SN [VA] [e] Volage E EN [V] [e] Curre pedace S N [A] [] E E [Ω ] Z [z] ω ω π f [rad/s] [/] T ω [s] [] E λ ET ω [W-urs] [e] λ [H] [e/] Pulsao Te Flux ducace B. Noralzao of Volage Equaos For gve ase volages ad curres, E, E,,, E, for crcus of Fg., he volages equaos (3) of geeralzed coupled crcus ca e orgazed as expressed (0) where he dex (u) dcaes a per u quay: d d u u E e r + ω + ω + u u ω ω du du Eeu r u + ω + ω + ω ω d d E e r + ω + ω + wh: r u u u u u ω ω r r ω ; u E Z E λ ω X Z Z u X u () (9) (0) Accordg o relaoshps (), he recprocy of he uual ducaces should e preserved () (sce ). j j Aoher pora resul oaed fro he per-u syse oralzao s ha uual ducaces should e equal as expressed (3) (derved fro ()). ω j j j ω ω a ;( j) j u E E Z j j j ω j ω ω aj ; ju j E E Z j j j u u u u () (3) C. Noralzed Equaos The volage equaos (3) (S syse) ca e orgazed as gve (4) he recprocal per-u syse (pu) whou dex (u) ad uscule syols for leakage ducaces. The volage equaos (4) are foraed he arx for (5), where J (6) s a order arx wh all eres equal. Wh: d d d e r + l ω ω ω d d d e r + + l + + ω ω ω d d d e r l ω ω ω ( λ ) (4) d d e () R () + R () ω + ω (5) [ T ] ( ) ( ) g g g g R dag r r r dag l l l + J V. SATURATON MODE OF GENERAZED COUPED CRCUTS (6) order o roduce he saurao pheoeo prevous coupled crcus, eglecg he volage drop leakage ducace l ad ressace r, he col # volage (4) yelds: ( ) d d e ω 0 ω (7) Assug a col # susodal pu volage ad curre, (7) ca e wre o coplex for (8): E j ( ω ) j j (8) ω 0 0 (8) he agude of he agec curre oly depeds o he agude of eral volage E ad o o he sae of he crcu (loaded or o). Cosequely, f E s aaed

4 cosa, he uloaded ( 3 0 ) agezg 0 curre wll rea he sae eve f oher wdgs are loaded. Thus: (9) E E 0 0 X 0 (0) E f( ) Usg he o-load agezg characersc 0 0 oaed expereally (o load s coeced o secodary cols of he coupled crcus), he agec saurao curve f( ) gve Fg.4 ca e draw po per po fro (0). 0 ( ) (pu) 0 lear zoe (usauraed) (pu)) Fg.4. f( ) 0 characersc of he uloaded coupled crcus ( pu) The curve Fg.4 ca e easly odeled as polyoal fuco gve y () where paraeers α ad α 4 are copued fro curve fg usg polyf fuco of Mala progra ad he characersc f( ) 0. 0 s he usauraed agec ducace of he crcu whch s oaed fro o-load characersc of Fg.4. e( ) E E + + (pu) () 4 ( α α ) r l e () 0 r l e( ) Fg.5. Equvale crcu of wo agecally coupled crcus (rasforer or per phase equvale crcu of woud-roor duco oor) e( ) r l e( ) 0 + r l e( ) s o oe ha he agezg curre 0 depeds o he srucure of he coupled crcus uder sudy as show y exaples of wo agecally coupled cols of Fgs.5 ad 6 where akes he followg expressos respecvely: 0 ; + 0 (3) V. SATURATED STATE SPACE MODENG OF MAGNETCAY COUPED CRCUTS A. Sae Space Model of Magecally Coupled Crcus e us cosder a dyac lear syse defed y he dffereal equao (4); he so-called sadard for of he sae space odel of a lear dyac syse, a well-kow corol heory suggesed for udergraduae egeerg sudes [9], ca e suarzed as gve (5). The elecrcal arx equao (6) derved fro (5) ca e orgazed sadard sae for (7): () x () dx f( x (), u (),) (4) x () Ax () + Bu () (5) y() Cx() + Du() () λ() d( λ) e () R λ() + ω d( λ( )) ( ω R ) λ() + ( ω ) e() () λ() wh: Where () A ω R ; B ω ; C ; DΟ, T ; [ ] u e λ s he sae vecor, () (6) (7) (8) e he pu corol varale ad A, B, C ad D, he sae arx, he sae corol arx, he sae oupu arx ad he sae oupu corol arx respecvely. B. Sauraed Sae Model of Magecally Coupled Crcus eakages ducaces (4) l ; due o agec losses he ar gap are o cocered wh he saurao pheoeo. The agec saurao ca e ake o accou he sae odel (7) a each sa k y adapg he correspodg value ( ) of he agec ducace copued fro () k arces A ad C (8). The pleeed process o accou for saurao s he followg:. Oa he characersc E f( ) 0 0 f( ) of Fg.4. Oa he curve 0 3. Oa he reacace 0 fro f 0 Fg.6. Equvale crcu of wo agecally coupled crcus (AC geeraors)

5 V. 4. Copue paraeers α ad α4 usg he fuco polyf of Mala progra ad f ( ) 5. Defe a e vecor 0 k usg values for F seps of sulao. 6. Se k 0 ; 0 ad copue arces A, B, C usg crcu paraeers ad seady sae coos λ λ( ) Sulae he sae syse (7) usg he fuco ls (.) of Mala corol oolox oe e sep erval[ k k + ]. 8. Se k ad use curres oaed fro sep 7 o copue fro () k gve y (9) ad 9. Copue sae arces A, B, C 0. Repea sep 7 ul k k F EXAMPE OF NUMERCA APPCATON USNG MATAB order o provde sudes wh a saple applcao, uercal sulaos were perfored o a 60-Hz, 0-V/55-V/55-V, ad 330-VA sgle-phase rasforer wh wo secodary wdgs (Fg.7) whose paraeers are lsed Tale. Two dffere ess were perfored. Frs, he wo secodary wdgs were shor-crcued ad a 50-V DC volage was appled o he prary. he secod es, he DC volage was replaced y a 60-Hz 0-V AC volage. Per u ducace ad ressace arces copued for ase values of Tales ad V are respecvely: TABE : PARAMETERS OF THE TRANSFORMER 0.09 H.5 H 3 l 0.0H N 000 urs.5 H 5000 urs l l 3 r 0 Ohs N r N r k Ohs daa Fg.7. Elecrcal equvale crcu of hree wdgs rasforer TABE V: BASE VARABES OF COS #AND #3 Varale Base Value Power S E 330 VA Volage E E 55 V S E E Z ω ω f E λ ω λ Curre pedace Pulsao π 377 rad/s Flux ducace ; R 6 A 9.67 Ω 0.46 W-ur H (3) TABE : BASE VARABES OF CO # Varale Base Value E E Volage Curre S E Power S E 330 VA pedace Z E Pulsao Flux ducace 3 f 377 ω ω π E λ ω λ 0 V 3 A 36.7 Ω 7 rad/s 0.9 W-ur H Fg.8 The o-load agezg characerscc of he rasforer Fg.9 The o-load agezg characerscc 0 of he rasforer oh cases, sulao of he curre of each wdg ad he agezg curre of he rasforer was requesed. The f

6 o-load saurao characersc of he rasforer s provded Fg.8 whle he curve f( ) 0 s ploed Fg.9. Volages of he secodary wdgs are se o zero (shor-crcu) for he frs es. The prary sulaed curre s show Fg.0 whle he agezg curre ad he secodary curres are gve Fgs. ad respecvely. s o e oed ha he wo secodary wdgs are he sae. Afer he peak due o he sudde appled volage, he secodary curre ecoes zero (DC sgal). For he secod es, he susodal volage yelds susodal curres as show Fgs.3-5. The agezg. The pac curre s gve y 0 3 pheoeo s oserved o varous fgures. of he saurao Fg.4 Magezg curre followg a 0-V susodal volage of he prary wdg wh he secodary wdgs shor-crcu Fg.0. Curre of he rasforer prary followg a 50-V sep chage of he prary volage wh he secodary wdgs shor-crcu Fg. Magezg curre followg a 50-V sep chage of he prary volage wh he secodary wdgs shor-crcu Fg. 5 Secodary wdgs curre of he rasforer followg a 0-V susodal volage of he prary wdg wh he secodary wdgs shorcrcu. V. CONCUSON A suale approach o eachg agecally coupled crcus o power egeerg udergraduae sudes s proposed hs paper a ore geeral approach. The work aeps o show ha he udersadg ad eachg of dyac odelg ad sulao of elecrcal aches ca e grealy proved f specal aeo s gve o he eachg ehodology adoped for agecally coupled crcus, where varous coceps ache heory ca e easly roduced. The paper ephaszes o per-u syse oralzao cocep frequely used power syse heory ad o ofe well explaed o sudes. All fudaeals rased are coo o classcal AC aches ad ca e easly well adaped o her odelg ad sulao. Oly sulao resuls are provdedd hs work. s plaed o carry ou expereal seup o ore assess he prese fraework sudy. REFERENCES Fg. Curre of he rasforer secodary followg a 50-V sep chage of he prary volage wh he secodary wdgs shor-crcu [] P. C. Krause, O. Wasyczuz, S.D. Sudhoff, Aalyss of Elecrcal Machery, EEE Press, New York, 995 [] R. C. Dorf, J. A. Svooda, Elecrc Crcus, 7 h Eo, Joh Wley ad Sos, 006 [3] Jaes W. Nlso, Susa A. Redel, Elecrc Crcus, 6 h Eo PrecHall, 00 [4] J. Davd rw, R. Mark Nels, Basc Egeerg Crcus Aalyss, Joh Wley ad Sos, 005 [5] Sephe J. Chapa, Elecrc Machery Fudaeals, Thrd Eo, McGraw-Hll, 998 [6] S.A. Nasar, Hadook of Elecrc Maches, McGRAW-H, New York, USA, 990 [7] Chee-Mu Og,, Dyac Sulaos of Elecrc Machery usg Mala/Sulk, Prece Hall, 997, [8] P. M Aderso, A. A. Fouad, Power Syse Corol ad Saly, secod eo, EEE Power Egeerg Socey, EEE PRESS, 00 [9] T. Marez-Mar, Sae Space Forulao for Crcus Aalyss EEE Trasacos o Educao, vol.53, No.3, pp , Augus, 00 Fg.3 Curre of he rasforer prary followg a 0-V susodal volage of he prary wdg wh he secodary wdgs shor-crcu