Transmission Line Constants THURSDAY,NOVEMB MBER17 ,20 11
T r ans mi mi s si onl i nesa r et hec i r c ui t swi t ht hedi s t r i but edc ons t ant ss uc hasr es i s t anc e,i nduc t anc e,c onduc t anc eand c apac i t anc edi s t r i but edal ongt hel engt hoft r ans mi mi s si onl i ne.Thes ec ons t antc onc ent r at edatan ypoi nt .Thei rv al ues ar enor mal l yex pr es sedasperk i l omet erl engt hofl i ne. Con si dert hed i s t r i but edc i r c ui ts ho wni nt heFi g.1 .Theanal y s i soft hi sc i r c ui tc anbema madebyl u mpi mp n gt h e pa r a me me t e r swh i c hi ss ho wni nt h eFi g .2 .
Fig. 1
Fig. 2
Th ea nal y s i so fc i r c u i tb ec ome sea s i e rb yl u mp mp i n gt h ep ar a me me t e r sa st h es amec ur r en ti sp as s i n gt h r ou ghea c h el e mentofr me es i s t anc ea ndi ndu ct a nc e. Si mi l ar l yt heci r c ui t ss ho wni nt h e Fi g.3( a )and( b)ar es ee nt ob e equi v al e ntast h es amev ol t agei sapp l i ed a c r o s se ac hc o nd uc t a nc ea ndc a pa c i t a nc e .
Fig. 3
Bu ti ft h ee l e me me nt sa r edi s t r i b ut e da ss ho wni nt h eFi g .4 .t h ent h ee l e me me nt sc a nn otb el u mp mp edast h es a me me c ur r e nti sn otfl o wi n gt h r o ug hs er i e se l e me me ntan dt h ev o l t a gea cr o ssdi ff er en te l e me me ntc ha nge sf r o mo nes ec t i o nt o ot her . Cons i de rt he e qu i v al entc i r c ui tofa 2 wi r et r an smi s s i on l i ne ha vi ng r es i s t an ce,i nduc t anc e,c apac i t an ce an d c onduc t anc eas s oc i at edwi t hi tass ho wni nt heFi g.4.
Fig. 4
Thel i nei sdi v i dedi n t ol ar genumbe rofs ma l ls ec t i on sofuni tl engt h.Theupperandl o werpar tofeac hs ec t i on c ar r i e ss amec ur r ent .He nc eweha v et heequi v al en tc i r c ui tass ho wni nt heFi g.5. Her e
R1T = R1+R1' ,R2T =R2+R2'
Si mi l ar l y
L1T =L1+L1' ,L2T =L2+L2' et c
Fig. 5
Th er es i s t anc eperuni tl engt ho ft hel i n ei ss a yRwher eR1T =R2T =R3T =R4T. . .=R,o hm/ me t r e .Weh a v eL1T = L1T =L1T =. . .=L,henr y / met r e G1 =G2 =. . .G,mh o/ me t r e C1 =C2=. . . .C,f ar ad/ met r e I nc as eof2wi r et r a ns mi s s i ons y s t e mt h el i n ep ar a me t e r sa r ee x pr e ss edonl o opba si swh i l ei nc as eof3ph as e t r a ns mi s s i o ns y s t e mt hep ar a me t er sar er e pr e se nt e do np erc on du ct o rba s i s . Theph ys i c all engt hoft het r ans mi s si onl i nedoesno ti ndi c at et hatt hel i nei sshor torl ongbuti ti sbas edon c omp ar i s o nwi t ht h ewa v el en gt h( λ)o ft h es ou r c e.
Factor to be Considered while Deciding Transmission Line THURSDAY,NOVEMBER17 ,20 11
Fol l o wi nga r es omeo ft hei mpor t a ntf ac t or st hatne edt ob ec ons i der edwh i l ed ec i di ngt het r ans mi s s i ons y s t e m. 1.T y peandsi z eofaconduc t or . 2.Vol t agel ev el . 3.Li ner egul at i onandc ont r ol ofv ol t age. 4.Effic i en cyoft r ans mi s s i on. 5 .Co r o nal o ss . 6.Powerfl owc apabi l i t yands t abi l i t y 7 .Re qu i r e me nto fc o mp en s at i o n.
8.Le vel soff aul t satv ar i ousbusbar sandr equi r ementofnewc i r c ui tbr eak er s . 9 .Gr o un di n gn ee d s . 10 .Pr o t ec t i onsc hemesf orne wl i n es . 11.Coor di nat i onofi ns ul at i on. 12 .Mec hani c al des i gna spec t swhi c hi nc l u des t r es sands agc al c ul at i on s,c ompos i t i o nofc ondu ct or ,s pac i ngof c onduc t or ,andconfi gur at i onsf ori ns ul at or s . 13 .De si g no fpo wers y s t e ms t r u c t u r e . 1 4.Ec o no mi c a la s pe c t s .
Types of Conductors THURSDAY,NOVEMBER17 ,20 11
Whent hec ond uc t or sa r eus edi nt r an smi s s i ons y st em f o rbul kp owert r ans f er ,t hent he ys houl df ul fi lf ol l o wi n g r eq ui r e me nt s . 1 .Th eys ho ul dh a v el o wwe i g ht . 2.The ys houl dha vehi ght ens i l eandf at i gues t r engt h. 3.The ymus tha v ehi ghc onduc t i v i t y . 4 .Th eys ho ul dh av el o wc oe ffic i en to fe x pa ns i on ,l o wc or o nal o ss . 5 .Th eys ho ul dh av el e ssr e si s t a nc ea ndl o wc os t . Thusbas eonconduc t i v i t y ,t ens i l es t r engt h,f at i guest r engt h,c or onal os s,l oc alc ondi t i onsandc os t ,c onduc t or s ar es e l e ct e df o rap ar t i c ul a rl i n e.Th ec on du ct or su s edi np r a ct i c ea r ema deu pf r om t h emat e r i a l ss u cha sc o pp er , al umi ni um andt hei ral l o ys . Th ea dv a nt a ge so fu si n ga l u mi n i u mc o nd uc t o r so v erc op pe rc o nd uc t o r sa r eg i v e nb el o w. 1 .Th eyha v el o wc o s t . 2.Les sr es i s t anc eandco r o nal os s . 3 .L es swe i gh t . Butal umi ni um hasl es st ens i l es t r engt h,hi ghc oeffic i entofe xpans i onandl ar gear eawhi c hr es t r i c t si t su s eal one a sac on du ct o r . I nor dert oi nc r eas et het ens i l es t r engt hofaconduc t or ,oneormor ecent r alc onduc t or sofdi ff er entmat er i al sar e us ed.Thes emat er i a l sgi v eh i ght e ns i l es t r en gt h.Thedi ff er en tt y pe sofa l umi n i um c onduc t or sus edi npo wers y s t ems wi t hf ul l f or msoft hei rabbr ev i at i onsar easgi v enbel ow. AAC:- Al l a l umi ni um c onduc t or . AAAC:-Al l al umi ni um al l o yc ond uc t or . ACSR:- Al u mi ni u mc o nd uc t o rwi t hs t e el r ei n f o r c me nt . ACAR:- Al umi ni um c ond uc t o rwi t ha l l o yr ei n f or c ement . No r ma l l yt h ec o nd uc t o r sar est r an de da si tp os s esgr ea t e rfl e x i b i l i t ya ndme c han i c als t r e ng t hasc omp ar edt o s i ngl ewi r esofs amec r os ssec t i onal ar ea.I nst r andedc onduc t or s ,ac ent r al wi r ei ss ur r oundedbys uc ces s i v el a yer sof
wi r esc ont ai ni ng6,12,18,24. . .wi r es .Thec ons ec ut i v el a y er sar es pi r al l edi noppos i t edi r ec t i onss oast oa voi d unwi ndi ng.Thi sal s omak esout err adi usofonel a y erc oi nc i dewi t hi nnerr adi usoft hene xt . Thest r andedconduc t or sar eel ec t r i c al l yi npar al l elandspi r al l edt oget her .Duet ous eofs t r andedconduc t or st he s k i neff ec ti sr educ ed. Theconduc t ors i z ei sdec i dedbas edoni t sc ur r entc ar r y i ngabi l i t ya ndv ol t agel e velonwhi c hi ti swor k i ng.The t ot al numberofc onduc t or si nas t r andofnl a yer sar egi v enb y T o t a ln umb ero fc o nd uc t o r s=1+3 n( 1+n )=3 n2 +3n+1 Ov er al l di ame t e rofs t r andedc o nduc t o rwi t hnl a y erD=( 1+2 n)d He r edi sd i ame t erofe ac hs t r and.7s t r an dc onduc t orwi l lh av eonec ent r als t r an dwi t h6ou t ers t r and seac h.The s i z eo fc o nd uc t o ri ss pe ci fi e db yi t se qu i v a l e ntc op pe rc r o s ss e ct i o na lar ea a nd t h en umb ero fs t r a nd s wi t ht h e di a me t erofeac hs t r and. No w wewi l l d i s c us si nb r i e ft h ec o mmo nl yu sedc o nd uc t o r s . 1. 1Har dDr awnCopperConduct or Th eh ar dd r a wnc op pe rc o nd uc t o r sa r eu s edf o ro v e r h ea dl i n eswh i c hp r o v i d esh i g ht e ns i l es t r e ng t h .Th es e c onduc t o r sha v er el a t i v el yh i ghe rc ondu ct i v i t y ,l ongl i f eandh i ghs c r apv a l ue .Thec oppe rc on duc t or sar eu sedf or di s t r i but i onne t wor kwher el engt hofl i nei sshor tandt her ear emor et appi ngs . 1 . 2St e el Co r e dCo pp erCo nd uc t o r( SCC) Th es t e elc o r e dc o p pe rc o n du c t o r sa r ema debys u r r o u nd i n g as t e e lc o r ewi t ho no rmo r el a y e r so fc o p pe r s t r ands .Duet oaddi t i onofs t eel c or et ens i l es t r engt hofc onduc t ori si nc r eas ed. 1. 3Cadmi um CopperConduct or Wi t hadd i t i onofc ad mi um t her ei si nc r ea sei nt h et ens i l est r engt hofc oppe ra tt heco stofd ec r eas ei nt he c onduc t i v i t y .Thust hes ec onduc t or sc anbeus edf orl ongers pans .Ast ens i l es t r engt hi si nc r eas ed,l ongers panswi t h s ames agi spos s i bl e.Theo t he ra dv ant ag esi nc l udeeas i nes si nj oi nt i ng,mor er e si s t a nc et oat mo spher i cco r r os i on, be t t err es i s t anc et owea ra ndea symac hi nabi l i t y . Thes ec onduc t or sar ec ar r i edb ys mal l ers uppor t sandar es ubj e ct ed t ol owwi ndandi c el oadi ngsduet ot hei rs mal l erdi ame t er . 1. 4CopperWel dCondcut or I nt hi st y peofc onduc t or ,c opperi swel dedont oas t eel wi r ebyhotr ol l i ngandcol ddr awi ngabi l l etofs t eel c oat ed wi t hc opper .Theuni f or mt hi c knes sofc opperi swel ded.Thec onduc t i v i t yoft hi sconduc t orl i esi nt her angeof30t o 60 %o ft h ato fs ol i dc op pe rc o nd uc t o rh a v i ngs amed i a me t er .Th es ear eu s edf o rl o ng ers pa nss uc ha sr i v e rc r o s si n gs . 1 . 5 Al l Al u mi n i u m Co nd uc t o r( AAC) Duet oi nc r eas i ngc os tofc opper ,al umi ni um i su sedi nt r ans mi s si ons y st em.El ec t r ol y t i c al l yr efi nedal umi ni um i s r ol l edanddr a wnhar df orus eascondc ut or .Fo ras pe ci fi cr es i s t anc e,c r os ss ec t i onalar eao fal umi ni um c ondc ut ori s gr eat ert h ant hatofc opperwhi l ei t swei ghti sabout5 0% oft ha to fc oppe rc onduc t or .Thi smak e st r ans por t at i ona nd er e ct i o no fs u chc o nd uc t o r sec o no mi c al .Co r on ae ffe c ti sr ed uc edd uet oi n cr ea s edd i a me t e ro fc o nd uc t o r .Th es e c onduc t o r sar emor eus edi ndi s t r i bu t i onwher et r ans mi s s i onl i nesar es ho r tandv ol t a gear el o wer .Ther ear ec han ces ofi nt e rp has ef aul t sdu et os wi ngi ft hes ec onduc t or sar eempl o y edi nt hear easwh er et her ear ehi ghwi nds .Thi si s be c au sea l umi n i um c on du ct or sar el i gh t e r ,wi t hl a r g ec o nd uc t o ra r e aan dmor es a g. 1 . 6Al u mi n i u m Co nd uc t o rwi t hSt e el Re i n f o r c e me nt
Th e me c ha ni c a ls t r e ng t ht h ati so bt a i n edf r o mc o nd uc t o rma deupf r o ma l la l u mi n i u m.Th i sd i ffic u l t yc a nb e o v er c omeb yaddi ngs t eel c or et ot hec ond uc t o r .Thec r os ss ec t i o noft hi sco ndu ct ori sassho wni nt heFi g.1.
Fig. 1 ACSR conductor
Ass ho wni nt h eFi g .1t h er ear e7s t e els t r an dswh i c hf o r msc en t r a lc or e .Th i si ss ur r o und edbyt wol a y er so f ar o un d3 0a l u mi n i u ms t r a nd s.Fo rag i v e nr e si s t a nc ec o nd uc t o r sofd i ff er e nts t r e ng ht sca nbema deb yt a k i n gd i ff er e nt pr oper t i esofs t eelandal umi ni um ar eas .Thes t eelc or edoesno tc ont r i but et oc onduc t i onofc ur r entpr ac t i c al l y .The c ur r entc ar r y i ngc apac i t yandr es i s t anc eoft hi sconduc t ori sdependentonc onduc t i v i t yofal umi ni um. Th eACSRc on du c t o r sa r emo r ec o mmo nl yu s e da st h e yh a v ef o l l o wi n ga dv a n t a ge s . 1.Duet ohi ghme chani c als t r engt ha ndt ens i l es t r e ngt h,t hel i nes panc anbei nc r eas ed.Thes agi ssmal l .Sos hor t er s uppor t sar er eq ui r edf orl i ne .I ti sal s op os s i bl et oha v el ongers p ansf oragi v ens ag.Du et os mal l ers u ppor t s , br eak downpos si bi l i t yi sl ow.I ns ul at or sandot herfi t t i ngsneededar eal s ol es s. 2 .Th eyha v el o wc o r o nal o s s . 3.Sk i neffec ti sl es s. 4 .The s ec on du ct o r sa r ei n ex pe ns i v ea sc omp ar e dt oc op pe rc o nd uc t o r sh av i n ge qu alr e si s t a nc ewi t ho utr e du ct i oni s effic i enc y ,us ef ul l i f es pananddur abi l i t y . Th edi s adv ant agewi t hACSRc on duc t ori sd i ffic ul tt omak es pl i c esa nddea dends .Ther ei sp os s i bi l i t yo fc or r os i on du et oel ec t r omec hani c al a ct i onbe t weenal umi ni um ands t eel c or e.Thes er v i c ec ondi t i onsdec i dec or r os i onr at e.Th i s i shi gheri ni ndus t r i al andc oas t al ar eas . The compact ed ACSR conduct ororsmoot h body ACSR conduct ori smade by pr ess i ng conv ent i onalACSR c onduc t o rt hr oughdi e st ofl at t e nt heal umi ni um s t r and si nt os eg ment a ls hape .Thes pa ce swi t hi nt hes t r and sa r efi l l ed whi l edi ame t ero fc on duc t ori sr educ ed.Thi sdo esno ta ffe cte l ec t r i c alandmec hani c alpr ope r t i esoft hi sc onduc t or . Th uswi t hs amea l umi n i um ar e a,d i a me t e ro fs t e elc or ei si n cr e as edwh i c hi n cr e as e sme ch an i c als t r e ng t h .The s e c onduc t o r scanbeus edf orl a r gers panl e ng t h s.Thi si ss ho wni nt heFi g.2.
Fig. 2 Compacted ACSR conductor
Th eex p an de dc o nd uc t o r sa r ema deb ya dd i n gapl a s t i co rfi b r o usma t e r i a lb et we ens t e elc o r ea nda l u mi n i u m s t r ands .Thi si nc r eas esdi amet erofc onduc t orwhi c hr educ esc or onal os sandr adi oi nt er f er enc eate xt r ahi ghv ol t ages . Thi st y peofc onduc t ori ss howni nt heFi g.3. Th efi l l erma t er i al s uc hasp ape rs epa r at est hei nne rs t eel s t r an dsf r o m out eral umi ni um s t r ands .
Fig. 3 Expanded conductor
1 . 7Al l Al u mi n i u m Al l o yCo nd uc t o r( AAAC) Thec on duc t ormadef r om al umi ni um al l o y sar es ui t a bl ei nur ba na r easa st he ypr o vi debe t t ert en si l es t r en gt hand c ondc ut i v i t y .The seal l o y sar ek no wnwi t hdi ff er entname si nv ar i ousc ount r i es .Someft h es eal l o y sar ec os t l yast h ey a r eh ea tt r e at e d.On eo ft h ea l l o y so fa l u mi n i u mi sk n owna ss i l ma l e cwh i c hc o nt a i n s0 . 5 % o ds i l i c o n,0 . 5% o f magnes i um andr es tofal u mi ni um.Du et ot hi st her ei si mpr o v eme nti nconduc t i v i t yandme chani c al s t r e ng t h . 1. 8ACARCondcut or I ns uc hc ond uc t or ,t hec en t r alc or ei smadeupf r o m al umi n i um al l o ywhi c hi ss u r r o undedb yl a y er so fal umi ni um c onduc t o r s .Th ec on du ct i v i t yi sbe t t erands t r engt ht owe i gh tr at i oi sequalt oACSRc onduc t orh av i ngs amedi a me t er . Asc o mp ar e dt oACSR c o n du c t o r ,ACSR c o n du c t o ri ss ma l l e ri ns i z ea ndl o we ri nwe i g htf o rt h es a mee l e c t r i c a l c apac i t y . 1. 9PhosphorBr onzeConduct or Thi st y peo fc ond uc t o ri ss t r ongt hanc op perc ondu ct oran dma yb eus edf orl ongerl i nes pan s.Thec onduc t i v i t yo f s u c hc o n du c t o ri sl o w wh i c hc a nb ei mp r o v e db yu s eo fc a d mi u mc o p pe rc o r e .Ph os p h orb r o nz ei sf o un dt ob e s ui t abl ef orat mos pher escon t ai ni nghar mf ul g as es . 1. 10Al umowel dConduct or I nt hi st y peofc onduc t or ,al umi ni um i swel dedonahi ghs t r engt hs t eelwi r e.Thi si sc os t l i erasc ompar edt ost eel c o r e da l u mi n i u m( SCA)o rA CSR c o n du c t o r .Ar o un d7 5 %c o n du c t o ra r e ai sc o v e r e db ya l u mi n i u m.Th i si su s e di n ear t hwi r es . 1 . 1 1Ga l v a ni z edSt e el Co nd uc t o r s Th i st y pe o fc on du ct o ri ss u i t a bl ef o rl a r g el e ng t hl i ne s pa no ri nr u r ala r e as wh er el o ad r e qu i r e me nti s c ompar at i v el ys mal l er .Th i st y peofc ondu ct orha shi ghs t r engt h.Thec ondu ct orh asl ar ger es i s t anc e,i nduc t anc ea nd v ol t agedr op.Thedi s adv ant agewi t ht hi sconduc t ori si thasshor t erl i f e.
Resistance
FRI DAY,NOVEMBER18, 201 1
Th er e si s t a nc eo fat r a ns mi s s i o nl i n ei sani mp or t an tp ar a me t e ra si ti st hema i nc a us eo fp owe rl o s si na t r ans mi s s i onl i ne.I ti sdefi nedast heoppos i t i onoffer edb yt het r ans mi s s i onl i nec onduc t or st ot hefl owofc ur r ent .The r es i s t anc e oft he l i ne i s un i f or ml y di s t r i but ed al ong i t s wh ol el engt h.Butt h ep er f o r manc eo ft he l i n ec an be c on v eni ent l yanal a y sedb yc ons i der i ngt hed i s t r i but edr es i s t an ceasl ump edonea ss ho wni nt heFi g .1. Theeffec t i v er es i s t anc eofac onduc t ori sgi v enb y
Wh er et h epo we rl o s si nc o nd uc t o ri si nwa t t sa n dcu r r e ntfl o wi n gt hr o ug ht hec o nd uc t o ri na mp er e s. Whent hedi s t r i but i onofc ur r enti nac onduc t ori su ni f or mt hent heeffec t i v er es i s t anc ei se qual t od. c .r es i s t anc eof t hec onduc t or .Thi sdi r ec tr es i s t anc ei sgi v enby ,
wher eρ=r es i s t i v i t yofc onduc t or( Ωm) l
=l engt hofc ondu ct or( m)
a=c r os ss ec t i onal ar eaofc onduc t or( m2) Thed. c .r e si s t anc ecomput edf r om t heabo v eequat i onoft hes t r an dedc onduc t or si sg r eat erass p i r al i ngoft he s t r a nd s ma k es t h em l o ng ert h an t h ec o nd uc t o ri t s el f .Th i si n cr e as ei nr e si s t a nc ei st a k en a s1 % f o r3 s t r a nd c onduc t o r sand2% f orc on cen t r i c al l ys t r andedconduc t or s . T emper at ur ei sal s ooneoft hef ac t orwhi c hwi l lc hanget hed. c .r es i s t anc eoft hemet al l i cc onduc t or .I t sv ar i at i on wi t hr es i s t anc ei sassho wni nt heFi g.1.I ti sseent obel i nearo vernor mal r angeofoper at i on.
Fig. 1
Fr o mt h ea bo v eFi g .1i tc anb es e ent ha t
wher eR1,R2 ar et her es i s t anc esi ft hec onduc t oratt emper at ur et andt noCandTi st hec ons t antob t a i nedf r om 1 2i t h egr a ph.Thev al ueofTi s234. 5oCf ora nn ea l e dc op pe r1 00 %c on du ct i v i t y .
Inductance FRIDAY, NOVEMBER 18, 2011
Anal t er nat i v efl uxi sp r oduc edb yt heal t er nat i ngc ur r entwhenfl owi ngt hr oughac onduc t or .Thi sfl uxl i nk swi t ht he c o nd uc t o r .Th ec o nd uc t o rp os s e s s esi n du c t a nc ed ue t ofl u xl i n k ag e.Th efl u xl i n k ag esp era mp er ei sc a l l e dt h e i nd uc t anc e. Th usi n du ct an c ei sgi v e nb y L=ψ/ I wh er e
ψ=Fl u xl i n ka gei nwe be r t u r n s I=Cu r r en ti na mp er e s.
Thei nduc t anc eofat r ans mi s si onl i nei sa l s oadi s t r i but edpar amet ero vert hel engt hofl i ne.Forc onv eni enc ei n an al y s i si ti st ak enasl umpedass ho wni nt heFi g.1 . Th ef u nd ame nt a le qu at i o nu s edt od efi nei n du ct a nc ei sg i v e nb y , e= / dτ dt
. . . . . . . . . . . . . . . . . . . . . .( 1)
whe r eei si nd uc edv o l t a gei nv ol t san di snu mb ero ffl uxl i n ka geo ft heci r c ui ti nwe be r t u r n s .Th enu mb ero f we be rt u r n si st hepr o du cto fea c hweb ero ffl u xa ndt h en umb ero ft u r n si st h ep r od uc to fea c hwe be ro ffl u xa ndt he numbero ft ur nsoft hec i r c ui tl i nk ed.Eac hl i neoffl uxi smul t i pl i edb yt henumberoft ur nsi tl i nk sandt hes epr oduc t s ar eaddedt ode t er mi net o t al fl uxl i nk age s. I fc ons t antper meabi l i t yi sa s s umedf ort hemedi um i nwh i c ht hemagne t i cfi el di ss e tupt henweh av e Nτ .
. .
α
i
Nτ =Li Thec ons t antofpr opor t i onal i t yi sc al l edi nduc t anc e
.
. .
/ N τ =Li Subs t i t ut i ngt hi sv al uei nt hef undament al equat i onweha ve,
Fo rNn umb ero ft u r n s , e=Ldi / d t
. . . . . . . . . . . . . .( 2)
I fp er meabi l i t yi sn otc ons t an tt henabo v eequat i o nma yal s obeus edbu tt hent hei ndu ct anc ei sn otc ons t an t . Sol v i ngeq uat i o ns( 1)a nd( 2 )wege t , L= / dτ dtH Wi t ht hefl uxl i nk agev ar y i ngl i near l ywi t hc ur r entt hent hemagnet i cc i r c ui thasc ons t antper meabi l i t y . L= / H τi .
. .
iWb t u r n s τ =L
I nt hi se quat i on,ii st hei ns t ant aneousv a l ueofc ur r ent .Sor epr es ent si ns t ant aneousfl uxl i nk ages .ForSi nus oi dal al t er nat i ngc ur r ent ,fl uxl i nk agesar eal s os i nus oi dal .Henc eweha ve ψ =LI
Th ev o l t aged r opd uet ofl uxl i nk agei s V=j ωLI v ol t s V=jωψ
v o l t s
Th emut ua li nd uc t an cebe t weent h et woc i r c ui t si sd efi nedast heflu xl i nk ag eso fonec i r c ui tduet oc ur r e nti nt he s ec ondc i r c ui tperamper eofc ur r enti nt hes ec ondc i r c ui tperamper eofc ur r enti nt hes ec ondc i r c ui t . I fc ur r entpr oduc esψ12 fl uxl i nk ageswi t hc i r c ui t1,Themut ual i nduc t anc ei s M12=ψ / 12I 2 Thev ol t agedr opi nc i r c ui t1i sgi v enby V2=jωM12 I ωψ12 v ol t s 2=j Mut u ali ndu ct an cei si mpor t anti nc ons i d er i n gt hei n fluenc eofpo werl i neso nt el ephonel i n esa ndt hec ou pl i n g bet weenpar al l el powerl i nes .
Capacitance FRI DAY,NOVEMBER18, 201 1
Whenan yt woc ond uc t or sa r es ep ar at edbya ni ns ul a t i ngmat e r i ali twi l lf or m ac ap ac i t o r .I nc as eofo v er hea d t r ans mi s s i onl i neai rac t sasani ns ul at i ngmedi um be t weent woc onduc t or s .Sot h er ewi l l bec apa ci t anc ebe t weent wo c onduc t o r swhi c hi sdefi nedasc har g eperuni tpo t ent i a ldi ff er e nc e Ca pa ci t a nc e,
C=q / v f a r a d
Unl i k er es i s t anc e andi nduc t anc e,t hecapac i t anc e oft r ans mi s s i onl i nei sal s ouni f or ml ydi s t r i but edal ongt he wh ol el e ng t ho ft h el i n e.I tma yb et r ea t e da sau ni f o r ms er i e so fc a pa c i t o r sc o nn ec t e db et we ent h ec on du ct o r sa s s ho wni nt heFi g.1andFi g .2 .
Th ec har geo nt hec o nduc t o r sa tan yp oi nti nc r eas eswi t hi nc r eas ean dde cr e as eo ft hei ns t ant ane ou sv a l ueo f t h ev ol t a gebe t weenc on du ct or satt hatpoi nt .Du et ot hi sac ur r entk no wna sc har gi ngc ur r entfl owst hr o ught he c onduc t o r sev ent houg ht hel i nei sope nc i r c ui t ed.Thev ol t agedr op,e ffic i en cyan dpo werf ac t orofl i nedep endsont he v al ueofc apa ci t anc e.
Capacitance FRI DAY,NOVEMBER18, 201 1
Whenan yt woc ond uc t or sa r es ep ar at edbya ni ns ul a t i ngmat e r i ali twi l lf or m ac ap ac i t o r .I nc as eofo v er hea d t r ans mi s s i onl i neai rac t sasani ns ul at i ngmedi um be t weent woc onduc t or s .Sot h er ewi l l bec apa ci t anc ebe t weent wo c onduc t o r swhi c hi sdefi nedasc har g eperuni tpo t ent i a ldi ff er e nc e Ca pa ci t a nc e,
C=q / v f a r a d
Unl i k er es i s t anc e andi nduc t anc e,t hecapac i t anc e oft r ans mi s s i onl i nei sal s ouni f or ml ydi s t r i but edal ongt he wh ol el e ng t ho ft h el i n e.I tma yb et r ea t e da sau ni f o r ms er i e so fc a pa c i t o r sc o nn ec t e db et we ent h ec on du ct o r sa s s ho wni nt heFi g.1andFi g .2 .
Th ec har geo nt hec o nduc t o r sa tan yp oi nti nc r eas eswi t hi nc r eas ean dde cr e as eo ft hei ns t ant ane ou sv a l ueo f t h ev ol t a gebe t weenc on du ct or satt hatpoi nt .Du et ot hi sac ur r entk no wna sc har gi ngc ur r entfl owst hr o ught he c onduc t o r sev ent houg ht hel i nei sope nc i r c ui t ed.Thev ol t agedr op,e ffic i en cyan dpo werf ac t orofl i nedep endsont he v al ueofc apa ci t anc e.
Proximity Effect FRI DAY,NOVEMBER18, 201 1
Th ec ur r entd i s t r i but i o nma ybenonuni f o r m be caus eo fa no t here ffec tk no wna spr o x i mi t ye ffec t . Cons i de rat wowi r el i nea ss h owni nt heF i g .1.
Fig. 1
Le te ac hoft h el i nec ond uc t o ri sa s sumedt obedi v i dedi nt o3se ct i onsh av i ngequalc r os ss ec t i onalar ea.Thr ee par al l ell oopsar ef or med b yt he pai r sx x ' ,y y 'andz z' .The fl uxl i nk i ngl oop x x 'i sl eas ta nd i ti nc r eas esf ort he r emai ni ngl oops .Thust hei nduc t anc eofi nnerl oopi sl es s.Thust hec ur r entdens i t yi shi ghes tati nneredgesoft he c onduc t or .Duet ot hi snonuni f or m di s t r i but i onofc ur r ent ,t heeffec t i v ec onduc t orr es i s t anc ei nc r eas es .Ast hedi s t anc e be t we ent h ec on du ct o r sgo esonr e du ci n g,t h i sdi s t r i b ut i o no fc u r r en tb ec ome smo r ea ndmo r en on u ni f or m. Fornor mals pa ci ngofo v er headl i n est hi spr o xi mi t yeff ec ti snegl i gi bl e.Fo ru nd er gr ou ndc abl est hi seff ec ti s s i gni fi cantast heconduc t or sar el oc at edc l os et oeac hot her . Th epr o x i mi t ye ffe c tal s od ep en dso nt h es amef a ct o r sa st h ato fs k i ne ff ec t