Currents Physical Components (CPC) in Three-Phase Systems with Asymmetrical Voltage

Transcription

1 Leszek S CZARNECKI, Prashaa BHAARAI Schl f Elecrical Egieerig ad Cmuer Scieces, Luisiaa Sae iversiy, Ba Ruge, SA di:115199/ Curres Physical Cmes (CPC) i hree-phase Sysems wih Asymmerical Vlage Absrac Eergy flw relaed hemea i hree-hase ubalaced, liear, ime-ivaria (LI) lads, sulied wih asymmerical, bu siusidal vlage, i hree-wire sysems, are ivesigaed i he aer I is demsraed ha he lad curre ca be decmsed i Curres Physical Cmes (CPC), assciaed wih disicive hysical hemea i he lad I is als shw hw he CPC ca be exressed i erms f he suly vlage ad equivale arameers f he lad A equivale circui f LI lads a asymmerical, bu siusidal suly vlage is reseed as well his decmsii rvides slid fudameals fr defiig wers f such lads Sreszczeie Arykuł rzedsawia wyiki badań ad zjawiskami eergeyczymi w liiwych, czasw-iezmieiczych (LI) dbirikach iezrówważych, zasilaych iesymeryczym, lecz siusidalym aięciem w bwdach rójrzewdwych Pkaza, że rąd akich dbirików mże być rzłży a Składwe Fizycze, jedzaczie swarzysze z kreślymi zjawiskami fizyczymi Pkaza akże, że rądy e mgą być kreśle rzez aięcie zasilaia i aramery rówważe dbirika Przedsawi akże bwód rówważy iezrówważych dbirików LI, zasilaych iesymmeryczym, lecz siusidalym aięciem Rzkład e wrzy slide dsawy dla defiicji mcy akich dbirików (Składwe Fizycze Prądów w bwdach rójfazwych z iesymeryczym aięciem zasilaia) Keywrds: Curre decmsii, ubalaced lads, asymmerical sysems, wer defiiis, wer hery Słwa kluczwe: Rzkład rądu, dbiriki iezrówważe, sysemy iesymerycze, defiicje mcy, eria mcy Irduci Resideial disribui sysems, sysems i cmmercial buildigs r elecrical raci grids ca be regarded frm a uiliy ersecive as slwly ime-varyig aggregaes f maily sigle-hase lads, sulied frm a hree-hase, hree-wire disribui sysem, usually hrugh a rasfrmer i /Y cfigurai, as shw i Fig 1 Fig 1 hree-hase lad cmsed f aggregaes f sigle-hase lads Symbls u ad i i his figure dee hree-hase vecrs f lie--arificial zer vlages ad lie curres, amely u ur, us, u, i ir, is, i Eve AC arc furaces ca be regarded as hree searae sigle-hase arcs furaces, i a cmm cage, ie, hree sigle-hase lads sulied frm a hree-wire sysem Due a eial imbalace, such sysems differ as wer reries frm sysem dmiaed by hree-hase lads, usually mrs r recifiers I culd be a surrisig bservai ha i sie f he fac ha csiderable amu f eergy rduced i wer sysems is disribued jus i sysems as shw i Fig 1, he wer hery eables w heir descrii i wer erms ly he cdii ha he suly vlage is symmerical I culd be regarded as a remarkable deficiecy f he wer hery Csequely, such lads ca be described i wer erms ly arximaely, a he assumi ha he suly vlage is symmerical fruaely, wih he lack f wer defiiis valid a asymmerical vlage, eve he errr f such arximai ca be evaluaed Sudies wers i asymmerical hree-hase sysems have a ceury lg hisry, bu hese sudies are ccluded eve w hey were iiiaed by Seimez [1] ad Ly [3], while he mai mahemaical l fr hese sudies, i a frm f he cce f symmerical cmes, was rvided by Fresque [2] Difficulies wih he develme f he wer hery f asymmerical hree-hase sysems have sared wih he quesi hw selec he defiii f he aare wer he America Isiue f Elecrical Egieers (AIEE) aded [4] i 192 w differe defiiis f he aare wer, amely: arihmeic aare wer: (1) S = S A = R I R + S I S + I ad gemeric aare wer: (2) S = SG = P + Q A debae [6-8] which e f hese w defiiis is righ was icclusive Csequely, bh were sured by he IEEE Sadard Diciary f Elecrical ad Elecrics erms [18] A he same ime, a defiii f his wer suggesed i 1922 by Buchhlz i [5], amely (3) S SB = R+ S + IR+ IS + I was sured by he IEEE Sadard Eveually i was rve i [21] ha he arihmeic ad gemeric defiiis f he aare wer i sysems wih ubalaced lads rvide a icrrec value f he wer facr, while he righ value f his facr a siusidal vlages ad curres is baied ly whe he Buchhlz defiii (3) is used here is csiderable amu f lieraure varius araches descrii wer reries f hree-hase sysems, wih sme resuls ublished eve recely [22-26] ad sudies his subjec are sill cmleed Ms f sudies [9-11, 13-17, 19] have fcused he aei wer defiiis a siusidal suly vlage fruaely, a a wrg defiii f he aare wer S, eve a siusidal vlages ad curres, i was ssible devel eiher he righ defiiis f elecric wers f hree-hase lads r he righ wer equai his issue fr symmerical suly vlages was 4 PRZEGLĄD ELEKROECHNICZNY, ISSN , R 91 NR 6/215

2 eveually slved i [17], hwever, fr asymmerical suly vlages bh he wer defiiis ad he wer equai have ye be develed heir develme is jus he subjec f his aer Pwers i sysems wih ubalaced lads have becme he bjec f ieres i [12], bu sill a a symmerical suly vlage he Curres Physical Cmes (CPC) rvides a cceual frame fr sudies i his aer I is based hree basic ideas: (i) he suly curre f he lad is a cre quaiy i he circui fr he wer hery develme his rerequisie is i a cras araches based a wer as such a cre quaiy (ii) he suly curre decmsii i muually rhgal cmes Orhgaliy makes he rms value f he lad curre ideede muual ieracis f he curre cmes (iii) he suly curre cmes shuld be assciaed wih disicive hysical hemea i he circui his las rerequisie gave he ame his hery: Curres Physical Cmes (CPC) wer hery hese basic ideas f he CPC wer hery were rigially alied [13] sigle-hase LI lads wih siusidal suly vlage ad ex lads wih sequeially icreasig cmlexiy wih resec heir srucure as well as vlages ad curres wavefrms his aer ca be regarded as a ex se i his hery s develme, w alied ubalaced LI lads wih asymmerical, bu siusidal suly vlage Aar frm he CPC arach, he cce f a ubalaced curre ad ubalaced wer are esseial fr hese sudies Origially, he cce f he ubalaced wer was irduced i [17] fr a ubalaced lad wih siusidal, bu symmerical suly vlage his cce, cfied LI lads eraed a such cdiis, is ulied i he fllwig Seci Origial cce f ubalaced wer A equivale circui f liear saiary LI lads as see frm he rimary side f a /Y rasfrmer, as shw i Fig 1, ca have he frm shw i Fig 2 As i was rve i [2] here is ifiie umber f such circuis, equivale wih resec he lad curres Fig 2 Equivale circui f hree-hase lads he hree-hase vecrs f he lad vlages ad curres, ca be exressed i he frm (4) (5) u R() R j j u () us() 2Re S e 2Re{ e } u() S i R() IR j j i () is() 2Re I S e 2Re{ I e } i() I I hese frmulas symbls ad I dee hree-hase vecrs f cmlex rms (crms) values R, S, ad f lie vlages, measured wih resec a arificial zer, ad lie curres I R, I S, ad I Fr hree-hase vecrs f siusidal quaiies, deed geerally by x() ad y(), f he same frequecy, a scalar rduc (6) ( x, y ) 1 () () d x y ad hree-hase rms value (7) x ( x, x ) 1 ( ) ( ) d x x ca be defied [17] he scalar rduc, defied by (6) i he ime-dmai, ca be calculaed i he frequecydmai, havig vecrs f crms values f hese quaiies X ad Y, as fllws (8) ( x, y ) 1 () () d Re{ } x y X Y w vecrs x() ad y() are muually rhgal he cdii ha (9) ( x, y ) Re{ X Y } = ad csequely, hree-hase rms values f such quaiies saisfy he relaishi 2 (1) x + y x y he scalar rduc f he suly vlage ad he lad curre vecrs is equal he acive wer P f he lad, (11) ( u, i) 1 () () d Re{ } = P u i I he wer equai develed i [17] fr LI lads f he srucure shw i Fig 1 a siusidal ad symmerical ad suly vlages, bu asymmerical curres has he frm (12) S P Q D u he aare wer i his equai was defied, accrdig he Buchhlz defiii (3), as he rduc f vlages ad curres hree-hase rms values, equal 2 (13) u 1 ( ) ( ) d R S u u 2 (14) i 1 ( ) ( ) d IR IS I i i Symbls P ad Q i he wer equai (12) dee cmm acive ad reacive wers, which ca be direcly measured a he lad ermials Symbl D u dees he ubalaced wer, defied as 2 (15) D u = A u he symbl A dees he magiude f he ubalaced admiace f he lad, secified i erms f equivale lie--lie admiaces, as fllws j j2 /3 (16) A = Ae ( YS YR YRS), = 1e he ubalaced wer was als defied i IEEE Sd 1459 [24] I was defied as (17) S S ( P ) ( Q ) PRZEGLĄD ELEKROECHNICZNY, ISSN , R 91 NR 6/215 41

3 P ad Q dee he acive ad reacive wers, bu ly f he vlage ad curre symmerical cme f he siive sequece he wer defied by frmulae (17) share ly he adjecive ubalaced, wih ha defied by (15) hese are w differe wers Frmula (17) ca be rearraged a wer equai, sice i eglecs eergy delivered he lad by he egaive sequece cme f vlages ad curres CPC decmsii a asymmerical vlage Aare wer S f sigle-hase lads ad i balaced hree-hase lads wih siusidal vlages ad curres is equal he magiude f he cmlex aare wer S which fr sigle-hase sysems is defied as j (18) S= I Se P jq Whe he lad is ubalaced ad/r vlages are asymmerical r siusidal he he aare wer S is lger he magiude f he cmlex aare wer S fruaely, similariy f symbls fr bh wers may cause cfusi ad eve may lead errrs Sice i is a very cmm cusm f deig he aare wer by S, a clearly differe symbl is used i his aer fr he wer defied as j (19) I P jq C =Ce Als he adjecive aare will be used he quaiy defied by (19) will be referred as a cmlex wer Wih resec acive ad reacive wers P ad Q a he suly vlage u, he ubalaced lad shw i Fig 1 is equivale a balaced lad shw i Fig 3, he cdii ha is hase admiaces are equal P jq C (2) Yb Gb jbb u u Fig 3 Balaced lad, which is equivale he rigial lad wih resec acive ad reacive wers P ad Q Ideed, he cmlex wer C b f such lad is 2 (21) Cb= Ib ( Yb ) Yb u P jq C he suly vlage u is siusidal, bu i ca be asymmerical hus i ca be decmsed i a sum f symmerical vlages f he siive u ad egaive u sequece, s ha j (22) u u u 2Re{( )e } A zer sequece symmerical cme u z ca cause ay curre flw i hree-wire sysems hus i ca be egleced r lie vlages shuld be measured wih resec a arificial zer f he sysem, s he vlage vecr u wuld cai ay zer sequece cme Le us defie ui hree-hase vecrs f he siive ad egaive sequece j 2/ 3 j 2/ 3 (23) 1 e, 1 1 1e j 2/ 3 j 2 / 3 1e 1e shw i 4 Fig 4 i hree-hase vecrs 1 ad 1 he asymmerical suly vlage u ca be exressed wih hese vecrs as (24) u u u j 2Re{( 1 1 )e } (25) R 1 1,, 3 S 1,, Sice Y b i (2) is admiace f a balaced lad, which is equivale he rigial lad wih resec he acive ad reacive wers, i will be referred as he equivale balaced admiace Such a equivale balaced lad draws he curre j j (26) ib ia ir 2Re{ Ib e } 2Re{ Yb e } cmsed f he acive curre j ia Gb u 2Re{ Gb ( + ) e } = (27) j = 2 Re{ Gb ( ) e } ad he reacive curre j ir Bb u(+ /4) = 2Re{ jbb ( + ) e } = (28) j 2 Re{ jbb ( ) e } he remaiig curre f he lad, afer he curre f he balaced lad is subraced, is caused by he lad imbalace j j (29) i ib 2Re{( I Ib)e } iu 2Re{ I ue } Csequely, he lad curre is decmsed i he acive, reacive ad ubalaced curre cmes, such ha (3) i = a i + r i + u i Muual rhgaliy f he acive ad reacive curres resuls frm heir muual hase shif by /2 Orhgaliy f he balaced ad ubalaced curre has be rve Ideed ( ib, iu) = Re{ Ib( I Ib) }= Re{ Yb I Yb Yb } = (31) = Re{ Yb( I Yb )} = = Re{ Yb( C Cb)} = hus hese hree curre cmes are muually rhgal ad csequely (32) i ia ir i u Each curre cme i decmsii (3) is disicively assciaed wih a uique hysical heme i he circui, hus hey ca be regarded as Curres Physical Cmes, (CPC) his decmsii ca be erfrmed, ad he hreehase rms values f each aricular curre ca be measured r calculaed by measuremes f acive ad reacive wers P ad Q a he lad ermials as well as crms values I R, I S, ad I f he lad curres 42 PRZEGLĄD ELEKROECHNICZNY, ISSN , R 91 NR 6/215

4 Mulilyig (32) by he square f he hree-hase rms value u f he suly vlage 2 { i ia ir iu } u he wer equai (33) S P Q D u is baied, wih he ubalaced wer defied as (34) Du = u i u his wer equai is ideical wih eq (12), bu develed wihu he assumi ha he suly vlage is symmerical hus, he suly vlage asymmery des affec he geeral frm f he wer equai f saiary LI lads wih siusidal suly vlages Pwer equai (34) ad he values f he acive, reacive ad ubalaced wers rvide disicive ifrmai hw ermae flw f eergy he lad; he hase-shif bewee he suly vlage ad he lad curre, as well as he lad curre asymmery affec he aare wer S he ubalaced curre u i i frmula (29) is exressed i erms f he lad arameers, hwever I ly fills a ga bewee he lad curre ad is acive ad reacive cmes he same alies he ubalaced wer D u Defiii (34) has aalgy defiii (15) I is ssible calculae is value, bu i ca be used i a desig rcess f a reacive cmesar ha wuld cmesae his wer A deedece f he ubalaced wer he circui arameers is eeded fr ha herefre, le us fid hw he ubalaced curre ad wer deed he circui arameers he acive ad reacive curres i circuis wih symmerical suly vlage are symmerical curres, such ha (35) a r e e j G jb Ye j i i 2Re{( + ) e } 2Re{ e }, accrdig [17], (36) Ye = Ge+ jbe YS+ YR YRS is he equivale admiace f he lad I is a hase admiace f a balaced lad, which is equivale he rigial e wih resec he acive ad reacive wers P ad Q Whe he suly vlage is asymmerical he, accrdig frmulas (27) ad (28) he acive ad reacive curres fllw he vlage asymmery he hase admiace Y b f he equivale balaced lad is differe ha he equivale admiace Y e, because (37) 2 P jq YRSRSYS S YR R b jbb Yb G u u A symmerical suly vlage RS = S = R = u, s ha Y b = Y e If he vlage is asymmerical he admiaces Y e ad Y b differ muually by admiace Y d, (38) d jbd e Yd = G + = Y Yb deede he vlage asymmery his asymmery ca be secified quaiaively by a cmlex cefficie f he suly vlage asymmery, defied as ( ) (39) e j e j j = ae a j e Whe he vlage asymmery is secified by his cefficie, he he differece bewee admiaces Y b ad Y e is equal 2a = [ cs cs( ) cs( )] 1 a 2 2 (4) Yd Y 2 S YR YRS 3 3 Admiace Y d deeds ly he lie--lie admiaces f he lad, bu als he suly vlage asymmery Whe he lad is balaced, ie, Y RS = Y R = Y S = Y e /3, he Y d =, ideedely he suly vlage asymmery Whe he suly vlage is symmerical ad csequely, asymmery cefficie a =, he Y d =, ideedely he lad imbalace herefre, admiace Y d is referred as a vlage asymmery deede ubalaced admiace i his aer Admiace Y d ca have a -zer value ly if he lad is ubalaced ad he suly vlage is asymmerical he vecr f crms values f ubalaced curre I u i he lad suly lies ca be decmsed, as shw i Aedix A, as fllws (41) Iu = Yd 1 A 1 A (42) A = ( YS+ YR YRS) (43) A = ( YS+ YR YRS ) are ubalaced admiaces fr he siive ad he egaive sequece vlages he erm (44) 1 A = J i frmula (41) sads fr a vecr f crms values f symmerical curres f egaive sequece rrial he siive sequece vlage, while (45) 1 A = J sads fr a vecr f crms values f symmerical curres f siive sequece rrial he egaive sequece vlage hus, he vecrs f he acive, reacive ad ubalaced curres, a i, r i ad u i ca be secified i erms f fur admiaces, Y b, Y d, A ad A, which ca be exressed i erms f lie--lie admiaces Y RS, Y S, ad Y R, lie--lie suly vlage rms values ad he cefficie f is asymmery a Equivale circui he rigial ubalaced LI lad sulied wih asymmerical vlage ca be regarded as a arallel ceci f w balaced lads wih hase admiace Y b ad Y d, resecively, ad w symmerical curre surces, ceced as shw i Fig 5, which ijec w hree-hase curres j ad j he balaced lads draw asymmerical curres f he crms value rrial admiace Y b ad Y d hree-hase curres j ad j are symmerical curres rrial he siive ad egaive sequece cmes f he suly vlage u ad u, bu f sie sequece hse vlages, accrdig frmulae (44) ad (45) All arameers f such equivale circui are exressed i erms f lie--lie admiaces Y RS, Y S, ad Y R, f he cfigured equivale circui shw i Fig 2 Such a circui ca be regarded as a equivale circui f ubalaced lads sulied wih siusidal, bu asymmerical vlage I visualizes he cmlex aure f he ubalaced curre u i his aure culd be irreleva whe PRZEGLĄD ELEKROECHNICZNY, ISSN , R 91 NR 6/215 43

5 ly is rms value r/ad he ubalaced wer have be kw Kwledge f his aure culd ye be crucial fr a rcess f desig f a reacive cmesar ha wuld be caable cmesae he ubalaced curre A sudy a ssibiliy f reacive cmesai f such lads is beyd f he sce f his aer, hwever while he vecr f he suly vlage wih resec he arificial zer is j e j6 j = e 88 2e V j e he equivale balaced admiace Y b f he lad i he circui shw i Fig 3 is equal P jq Yb Gb jb b 6 j 6S 2 u hece, he acive curre vecr has he wavefrm Fig 5 Equivale circui f ubalaced lad he balaced brach wih admiace Y b has he acive ad reacive wers equal P ad Q, resecively, because his admiace was calculaed, accrdig frmula (2), jus saisfy such a cdii he brach wih he ubalaced curre i u, has have zer acive ad reacive wers, P ad Q, because hese w wers f he rigial lad are equal, accrdig frmula (2), he wers f he brach wih curre i b, while he whle equivale circui has saisfy he balace ricile wih resec he acive ad reacive wers he ly -zer wer f his brach culd be he ubalaced wer D u Illusrai Le us calculae hysical cmes f he lad curre i he circui shw i Fig 6, wih srgly asymmerical suly vlage ad srgly ubalaced lad he acive ad reacive wers f he lad are equal, resecively, P = 1 kw ad Q = 1 kvar Fig 6 Examle f ubalaced lad wih asymmerical suly vlage Cmlex rms values f he siive ad egaive sequece symmerical cmes, ad, f he suly vlage, calculaed accrdig (25), are equal 1 1 V 3 e 1,, j e 1 1,, j he hree-hase rms values f he suly vlage symmerical cmes are u V u V ad csequely, hree-hase rms value f he suly vlage is 2 2 u u u V j j ia I a Gb e j6 j 1 1 2Re{ e } 2Re{ ( ) } = = 2Re{6( e ) e } = j e j j 2 Re{ 52 9 e e } A j12 2 e he vecr f he reacive curre is j j i r 2Re{ I re } 2Re{ jbb ( ) e } j e j 13 9 j 2 Re{ 52 9 e e } A j3 2 e he ubalaced curre vecr ca be reseed i he frm j e j j75 j i u= 2Re{( I IaI r)e } 2Re{ 952 e e }A j e hree-hase rms values f he curre cmes are equal i G u = A a r b i B u = A b i u IuRIuSI u A he suly curre has he hree-hase rms value R S i = I I I A ad ideed ia ir i u A which cfirms umerical crrecess f he curre decmsii i hysical cmes he lad wer facr = P/S = i a / i = 32 he ubalaced wer is equal D u = u i u kva fid arameers f he equivale circui f he lad le us calculae ubalaced admiaces A ad A, amely 44 PRZEGLĄD ELEKROECHNICZNY, ISSN , R 91 NR 6/215

6 = ( 165 S + R RS ) = [1+ ( 1)] = 1932 j e j S A Y Y Y = ( 15 S + R RS ) = [1+ ( 1)] = 518 j e j S A Y Y Y he cmlex cefficie f he suly vlage asymmery is equal j6 j 33 33e j6 a = ae 5e herefre, he asymmery deede ubalaced admiace is equal 2a 2 2 Yd = [ Y 2 Scs YRcs( ) YRScs( )] = 1 a j 45 = [cs(6 ) j cs(6 12 )] 566e S Wih hese arameers f he equivale circui, he vecr f he ubalaced curre crms values is equal R j45 Iu = Yd A A = 566e 1 1 S j e j165 j 15 j75 193e 518e 95 2e A j e I culd be checked ha he cmlex wer f he brach f he equivale circui wih he ubalaced curre i u is u I u u u C = P jq his cfirms umerical crrecess f calculai f he ubalaced curre, hus crrecess f calculai f he arameers f he equivale circui Frmula (41) shws ha he ubalaced curre is a iricae quaiy balaced admiaces A ad A deed ly he lad arameers, while he asymmery deede ubalaced admiace Y d deeds mrever he suly vlage asymmery Admiaces A ad A ca have differe values ad csequely, deedece f he ubalaced curre he suly vlage siive ad egaive sequece cmes ca be differe Sice he equivale balaced admiace Y b = Y e Y d ad csequely, G b = G e G d ad B b = B e B d, he acive ad reacive wers ca be decmsed i cmes ideede f he vlage asymmery ad deede i, amely (46) P =G b u ( G e G d) u P s P d P s dees he lad acive wer a a symmerical suly vlage, bu wih he same rms value as he asymmerical e he wer P d ccurs because f he suly vlage asymmery, bu i disaears, ideedely f his asymmery, whe he lad is balaced Similarly, he reacive wer (47) Q= B b u ( B e B d) u Q s Q d Q s dees he reacive wer a symmerical suly vlage, while he wer Q d ccurs because f he suly vlage asymmery i resece f he lad imbalace Observe ha he ubalaced curre cais bh siive ad egaive sequece cmes, sice he vecr (48) 1 A Y d I u is a vecr f crms values f he suly curres f he egaive sequece, while he vecr (49) 1 A Y d I u is a vecr f crms values f he siive sequece curres hus, he ubalaced curre ca be exressed i he frm j (5) iu = 2Re{( Iu I u) e } = iu iu s ha, he lad curre ca be decmsed i fur cmes (51) a r u u i i i i i hese cmes are muually rhgal, s ha heir hree-hase rms values saisfy he relaishi (52) a r u u i i i i i he acive curre a i is assciaed exclusively wih ermae eergy rasfer frm he suly surce he lad, meaig wih he lad acive wer P he reacive curre r i is assciaed exclusively wih he hase-shif bewee he suly vlage ad he lad curre, meaig wih he lad reacive wer Q hese w curres are asymmerical curres ad heir asymmery rerduces he asymmery f he suly vlage Curres iu ad i u are symmerical curres, which ccur exclusively due he lad imbalace hey d cribue he acive ad reacive wers P ad Q f he lad, bu ly a icrease f is hree-hase rms value herefre, hese fur cmes f he lad curre ca be regarded as he Curres Physical Cmes (CPC) Mulilyig eq (52) by he square f he hree-hase rms value f he suly vlage, he wer equai is baied i he frm (53) wih (54) Cclusis D 2 S P Q Du Du u u iu Du u i u =, = he aer shws ha he basic ideas f he Curres Physical Cmes wer hery ca be alied ubalaced hree-hase LI lads sulied wih siusidal, bu asymmerical vlage I eables decmsii f he lad curre i rhgal cmes assciaed wih disicive hysical hemea i he circui ad describe he lad i wer erms Resuls reseed i his aer eable remve e f deficiecies f he wer hery f elecrical circuis, amely, he lack f a wer equai i he siuai whe a LI ubalaced lad is sulied wih siusidal, bu asymmerical vlage PRZEGLĄD ELEKROECHNICZNY, ISSN , R 91 NR 6/215 45

7 Aedix Equivale admiaces Le a LI ubalaced lad has a equivale circui as shw i Fig 2 he cmlex wer C f a hree-hase lad is defied as C = I RIR SIS I hece C = RIR SIS I ( I I ) ( I I ) ( I I ) R RS R S S RS R S RS RS S S R R I I I RS RS RS S S S R R R RSRS S S R R Y Y Y = Y Y Y ( P jq ) ( P jq ) ( P jq R ) = P jq RS RS S S R I ca be als exressed direcly as he sum f cmlex wers f hree sigle-hase lads cfigured i as shw i Fig 2, amely C = C C C I I I RS S R RS RS S S R R = YRSRS YS S YR R he cmlex wer fr idividual braches ca be exressed as fllws Sice RS RS RS RS RS RS RS R S R S C I Y = Y ( 2Re{ }) 2 RS RS R S RS ( )( ) 2Re{ } hus, (A1) C 2 Y (2 2 ) = Y (2 u 3 ) Similarly (A2) (A3) RS RS R S RS S S S S S u R C Y Y (2 3 ) R R R R R u S C Y Y (2 3 ) he equivale balaced admiace f he lad, defied by eq (2), ca be exressed wih eqs (A1) (A3) i he frm (A4) C CRS CS CR Yb u u Ye ( Y 2 SR YRS YRS) = Ye Yd u Ye = Ge+ jb e YSYR YRS is he equivale admiace f he lad whe i is sulied wih a symmerical vlage, ad (A5) 3 2 Yd = ( Y 2 SR YR S YRS) Ye u Le us exress his admiace i erms f crms values f symmerical cmes f he siive sequece ad he egaive sequece Sice (A6) R, S, he 2 R 2Re{ } 2 S 2Re{ } 2 2Re{ } he crms values ad have he frm j j e, e herefre, if we dee j( ) j e W We admiace Y d, give by (A5), ca be exressed as S R RS (A7) Yd = 2 Y Re{ W} Y Re{ W}+ Y Re{ W} Whe he suly vlage asymmery is secified by cmlex asymmery cefficie a, he Re{ } j( ) a Re{ e } cs 2 1a ad csequely, he asymmery deede ubalaced admiace Y d ca be rearraged he frm 2a 2 2 Yd = [ Y cs Y cs( ) Y cs( )] 1 a 3 3 he crms value i lie R curre is equal IR YRS( RS) YR( R) (A8) 2 S R RS ad ca be rearraged he frm I Y ( Y Y Y ) (A9) R e R S R R RS S If crms values f lie vlages R, S ad are exressed i erms f symmerical cmes, ie, wih frmula (A7), he frmula (A9) ca be rearraged IR Ye R A R A R (A1) A = ( YS+ YR YRS) (A11) A = ( YS+ YR YRS ) Similarly, he crms value f lies S ad curres ca be reseed i he frm IS Y e S A A I Y e AS AS hese hree crms values f suly lie curres ca be exressed i he vecr frm IR (A12) I I S Ye 1 A 1 A I s ha, he vecr f ubalaced curres is equal Iu = I Ib ( Y e Y b) 1 A 1 A r i ca be rearraged as fllws (A13) Iu = Yd J J (A14) J = 1 A, J = 1 A 46 PRZEGLĄD ELEKROECHNICZNY, ISSN , R 91 NR 6/215

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