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Transmission_Lines pi model.doc

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_08_ELC4340_Spring13_Transmission_Lines.doc, V130228 Transmission Lines Inductance and capacitance cacuations !or transmission ines. #$, #%, L, and C matrices, e!!ect o! ground conducti&it'. (nderground ca)es. 1. Equivalent Circuit for Transmission Lines (Including Overhead and Underground) T*e po+er s'stem mode !or transmission ines is de&eoped !rom t*e con&entiona distri)uted parameter mode, s*o+n in ,igure 1. + - v R/2 L/2 G C R/2 L/2 i ---
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   _08_ELC4340_Spring13_Transmission_Lines.doc, V130228 Transmission Lines Inductance and capacitance cacuations !or transmission ines. #$, #%, L, and C matrices,e!!ect o! ground conducti&it'. (nderground ca)es. 1.Equivalent Circuit for Transmission Lines (Including Overhead and Underground) T*e po+er s'stem mode !or transmission ines is de&eoped !rom t*e con&entiona distri)uted parameter mode, s*o+n in igure 1. +-vR/2L/2GCR/2L/2i ---><--- i+-v + dvi + di ---><--- i + diR, L, G, C per unit length < > dz igure 1. %istri)uted -arameter #ode !or Transmission Linence t*e &aues !or distri)uted parameters resistance $, inductance L, conductance , andcapacitance are /no+n units gi&en in per unit engt*, t*en eit*er ong ine or s*ort inemodes can )e used, depending on t*e eectrica engt* o! t*e ine.ssuming !or t*e moment t*at $, L, , and C are /no+n, t*e reations*ip )et+een &otage andcurrent on t*e ine ma' )e determined )' +riting irc**o!!5s &otage a+ VL around t*e outer oop in igure 1, and )' +riting irc**o!!5s current a+ CL at t*e rig*t6*and node.VL 'ieds 02222 =++++++− t i Ldz i Rdz dvvt i Ldz i Rdz v ∂ ∂ ∂ ∂  .T*is 'ieds t*e c*ange in &otage per unit engt*, or  t i L Ri z v ∂ ∂ ∂ ∂  −−=  ,+*ic* in p*asor !orm is ( )  I  L  j R  z V  77 ω ∂ ∂  +−=  .CL at t*e rig*t6*and node 'ieds-age 1 o! 30   _08_ELC4340_Spring13_Transmission_Lines.doc, V130228 ( ) ( ) 0 =++++++− t dvvCdz dvvGdz diii ∂ ∂   .I! dv  is sma, t*en t*e a)o&e !ormua can )e approimated as ( ) t vCdz vGdz di ∂ ∂  −−=  , or t vC Gv z i ∂ ∂ ∂ ∂  −−=  , +*ic* in p*asor !orm is ( ) V C   jG  z  I  77 ω ∂ ∂  +−=  .Ta/ing t*e partia deri&ati&e o! t*e &otage p*asor e9uation +it* respect to : 'ieds ( )  z  I  L  j R z V  ∂ ∂ ω ∂ ∂  77 22 +−=  .Com)ining t*e t+o a)o&e e9uations 'ieds ( )( )  V V C   jG L  j R  z V  777 222 γ  ω ω ∂ ∂  =++=  , +*ere ( )( )  β α ω ω γ    jC   jG L  j R  +=++=  , and +*ere γ   , α  , and β   are t*e propagation, attenuation, and p*ase constants, respecti&e'.T*e soution !or V  7  is  z  z   Be Ae z V   γ  γ    − += 7  . simiar procedure !or so&ing  I  7  'ieds o z  z   Z  Be Ae z  I  γ  γ    − +−= 7  ,+*ere t*e c*aracteristic or surge impedance o  Z   is de!ined as  ( )( ) C  jG L j R Z  o ω ω  ++=  .Constants  A  and  B  must )e !ound !rom t*e )oundar' conditions o! t*e pro)em. T*is is usua'accompis*ed )' considering t*e termina conditions o! a transmission ine segment t*at is dmeters ong, as s*o+n in igure 2.-age 2 o! 30   _08_ELC4340_Spring13_Transmission_Lines.doc, V130228 d<>+-+-VsVrIs --->Ir ---><--- Is<--- IrSending EndReceiving EndTransmissionLine Segmentz = 0z = -d igure 2. Transmission Line SegmentIn order to so&e !or constants  A  and  B , t*e &otage and current on t*e recei&ing end is assumedto )e /no+n so t*at a reation )et+een t*e &otages and currents on )ot* sending and recei&ingends ma' )e de&eoped.Su)stituting  z   ; 0 into t*e e9uations !or t*e &otage and current at t*e recei&ing end 'ieds ( ) o R R  Z  B A I  B AV   −−=+= 7 ,7  .So&ing !or  A  and  B  'ieds 27 ,27  Ro R Ro R  I  Z V  B I  Z V  A  +=−= .Su)stituting into t*e 07  z V   and 07  z  I   e9uations 'ieds ( ) ( ) d  I  Z d V V   R RS   γ  γ   sin*7cos*77 0 +=  , ( ) ( ) d  I d  Z V  I   Ro RS   γ  γ   cos*7sin*77 +=  . pi e9ui&aent mode !or t*e transmission ine segment can no+ )e !ound, in a simiar manner asit +as !or t*e o!!6nomina trans!ormer. T*e resuts are gi&en in igure 3.-age 3 o! 30   _08_ELC4340_Spring13_Transmission_Lines.doc, V130228 d<>+-+-VsVrIs --->Ir ---><--- Is<--- IrSending EndReceiving Endz = 0z = -dYsrYsYr o RS   Z d Y Y       == 2tan*  γ    , ( ) d  Z Y  oSR γ   sin*1 =  , ( )( ) C  jG L j R Z  o ω ω  ++=  , ( )( ) C   jG L  j R  ω ω γ    ++=  R ,  L , G , C   per unit engt*igure 3. -i E9ui&aent Circuit #ode !or %istri)uted -arameter Transmission LineS*unt conductance G  is usua' negected in o&er*ead ines, )ut it is not negigi)e in undergroundca)es.or eectrica' s*ort o&er*ead transmission ines, t*e *'per)oic pi e9ui&aent mode simpi!iesto a !amiiar !orm. Eectrica' s*ort impies t*at d < 0.0= λ  , +*ere +a&eengt*  Hz   f   sm r  ε λ  >103 8 = ; =000 /m ? @0 A:, or @000 /m ? =0 A:. T*ere!ore, eectrica' s*ort o&er*ead ines *a&e d <2=0 /m ? @0 A:, and d < 300 /m ? =0 A:. or underground ca)es, t*e correspondingdistances are ess since ca)es *a&e some+*at *ig*er reati&e permitti&ities i.e. =.2 ≈ r  ε   .Su)stituting sma &aues o! γ  d   into t*e *'per)oic e9uations, and assuming t*at t*e ine osses arenegigi)e so t*at G ;  R  ; 0, 'ieds 2 Cd   jY Y   RS  ω  ==  , and  Ld   jY  SR ω  1 =  .T*en, incuding t*e series resistance 'ieds t*e con&entiona s*ort ine mode s*o+n in igure 4,+*ere *a! o! t*e capacitance o! t*e ine is umped on eac* end.-age 4 o! 30
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