_03_EE394J_2_Sping2_Tansmissin_Linesdc Tansmissin Lines Inductance and capacitance calculatins f tansmissin lines GMR GM L and C matices effect f gund cnductivity Undegund cables Equivalent Cicuit f Tansmissin Lines (Including Ovehead and Undegund) The pwe system mdel f tansmissin lines is develped fm the cnventinal distibuted paamete mdel shwn in Figue i ---> R/2 L/2 i di ---> G C v v dv R/2 L/2 - - <--- i <--- i di < dz > R L G C pe unit length Figue istibuted Paamete Mdel f Tansmissin Line Once the values f distibuted paametes esistance R inductance L cnductance G and capacitance ae knwn (units given in pe unit length) then eithe "lng line" "sht line" mdels can be used depending n the electical length f the line Assuming f the mment that R L G and C ae knwn the elatinship between vltage and cuent n the line may be detemined by witing Kichhff's vltage law (KVL) aund the ute lp in Figue and by witing Kichhff's cuent law (KCL) at the ight-hand nde KVL yields Rdz Ldz i Rdz Ldz i v i v dv i 0 2 2 t 2 2 t This yields the change in vltage pe unit length v z i Ri L t which in phas fm is V ~ ( R jωl)i ~ z Page f 30
_03_EE394J_2_Sping2_Tansmissin_Linesdc KCL at the ight-hand nde yields i i di Gdz ( v dv) Cdz ( v dv) t 0 If dv is small then the abve fmula can be appximated as v di ( Gdz) v Cdz t I ~ ( G jωc)v ~ z i v Gv C which in phas fm is z t Taking the patial deivative f the vltage phas equatin with espect t z yields 2 ~ V 2 z ~ I ( R jωl) z Cmbining the tw abve equatins yields 2 ~ V ~ 2 ~ ( R jωl)( G jωc) V γ V whee γ ( R jωl)( G jωc) α jβ and 2 z whee γ α and β ae the ppagatin attenuatin and phase cnstants espectively The slutin f V ~ is ~ V ( z) Ae γz γz Be A simila pcedue f slving I ~ yields ~ I ( z) γz γz Ae Be Z whee the chaacteistic "suge" impedance Z is defined as Z ( R jωl) ( G jωc) Cnstants A and B must be fund fm the bunday cnditins f the pblem This is usually accmplished by cnsideing the teminal cnditins f a tansmissin line segment that is d metes lng as shwn in Figue 2 Page 2 f 30
_03_EE394J_2_Sping2_Tansmissin_Linesdc Sending End Receiving End Is ---> I ---> Tansmissin Vs Line Segment - <--- Is <--- I z -d V - z 0 < d > Figue 2 Tansmissin Line Segment In de t slve f cnstants A and B the vltage and cuent n the eceiving end is assumed t be knwn s that a elatin between the vltages and cuents n bth sending and eceiving ends may be develped Substituting z 0 int the equatins f the vltage and cuent (at the eceiving end) yields ~ V R ~ A B I R Slving f A and B yields ( A B) ~ ~ VR ZI R VR ZI R A B 2 2 ~ ~ Substituting int the V ( z ) and I ( z) equatins yields Z ~ VS ~ I S ~ ~ VR csh( γd ) Z0I R sinh( γd ) ~ VR ~ sinh( γd ) I R csh( γd ) Z A pi equivalent mdel f the tansmissin line segment can nw be fund in a simila manne as it was f the ff-nminal tansfme The esults ae given in Figue 3 Page 3 f 30
_03_EE394J_2_Sping2_Tansmissin_Linesdc Sending End Receiving End Is ---> I ---> Ys Vs Ys Y V - <--- Is <--- I - z -d z 0 < d > Y S Y R d tanh γ 2 Z Y SR Z sinh ( γd ) Z ( R jωl) γ ( R jωl)( G jωc) ( G jωc) R L G C pe unit length Figue 3 Pi Equivalent Cicuit Mdel f istibuted Paamete Tansmissin Line Shunt cnductance G is usually neglected in vehead lines but it is nt negligible in undegund cables F electically "sht" vehead tansmissin lines the hypeblic pi equivalent mdel simplifies t a familia fm Electically sht implies that d < 005 λ whee wavelength 8 30 ( ) m / s λ 5000 km @ 60 Hz 6000 km @ 50 Hz Theefe electically sht f ε Hz vehead lines have d < 250 km @ 60 Hz and d < 300 km @ 50 Hz F undegund cables the cespnding distances ae less since cables have smewhat highe elative pemittivities (ie ε 2 5 ) Substituting small values f γd int the hypeblic equatins and assuming that the line lsses ae negligible s that G R 0 yields YS jωcd YR and 2 Y SR jωld Then including the seies esistance yields the cnventinal "sht" line mdel shwn in Figue 4 whee half f the capacitance f the line is lumped n each end Page 4 f 30
_03_EE394J_2_Sping2_Tansmissin_Linesdc Rd Ld Cd 2 Cd 2 < d > R L C pe unit length Figue 4 Standad Sht Line Pi Equivalent Mdel f a Tansmissin Line 2 Capacitance f Ovehead Tansmissin Lines Ovehead tansmissin lines cnsist f wies that ae paallel t the suface f the eath T detemine the capacitance f a tansmissin line fist cnside the capacitance f a single wie ve the eath Wies ve the eath ae typically mdeled as line chages ρ l Culmbs pe mete f length and the elatinship between the applied vltage and the line chage is the capacitance A line chage in space has a adially utwad electic field descibed as E ql aˆ Vlts pe mete ε This electic field causes a vltage dp between tw pints at distances a and b away fm the line chage The vltage is fund by integating electic field Vab b E aˆ a ql ε b a V If the wie is abve the eath it is custmay t teat the eath's suface as a pefect cnducting plane which can be mdeled as an equivalent image line chage ql lying at an equal distance belw the suface as shwn in Figue 5 Page 5 f 30
_03_EE394J_2_Sping2_Tansmissin_Linesdc Cnduct with adius mdeled electically as a line chage ql at the cente b B A a h Suface f Eath bi ai h Image cnduct at an equal distance belw the Eath and with negative line chage -ql Figue 5 Line Chage q l at Cente f Cnduct Lcated h Metes Abve the Eath Fm supepsitin the vltage diffeence between pints A and B is Vab b bi ql b bi ql b ai Eρ aˆ Eρi aˆ ε a ai a bi a ai ε If pint B lies n the eath's suface then fm symmety b bi and the vltage f pint A with espect t gund becmes Vag ql ε ai a The vltage at the suface f the wie detemines the wie's capacitance This vltage is fund by mving pint A t the wie's suface cespnding t setting a s that Vg ql ε 2h f h >> The exact expessin which accunts f the fact that the equivalent line chage dps slightly belw the cente f the wie but still emains within the wie is V g q l ε h 2 h 2 Page 6 f 30
_03_EE394J_2_Sping2_Tansmissin_Linesdc The capacitance f the wie is defined as fmula abve becmes ql C which using the appximate vltage V g ε C Faads pe mete f length 2h When seveal cnducts ae pesent then the capacitance f the cnfiguatin must be given in matix fm Cnside phase a-b-c wies abve the eath as shwn in Figue 6 Thee Cnducts Repesented by Thei Equivalent Line Chages a ab b ac Cnduct adii a b c c aai aci Suface f Eath ai abi bi ci Images Figue 6 Thee Cnducts Abve the Eath Supepsing the cntibutins fm all thee line chages and thei images the vltage at the suface f cnduct a is given by Vag ε aai qa a abi qb ab aci qc ac The vltages f all thee cnducts can be witten in genealized matix fm as V V V ag bg cg ε p p p aa ba ca p p p ab bb cb p p p ac bc cc q q q a b c Vabc PabcQabc ε whee Page 7 f 30
_03_EE394J_2_Sping2_Tansmissin_Linesdc paa aai a pab abi ab etc and a is the adius f cnduct a aai is the distance fm cnduct a t its wn image (ie twice the height f cnduct a abve gund) ab is the distance fm cnduct a t cnduct b abi bai is the distance between cnduct a and the image f cnduct b (which is the same as the distance between cnduct b and the image f cnduct a) etc A Matix Appach f Finding C Fm the definitin f capacitance invesin Q CV then the capacitance matix can be btained via abc 2 Pabc C πε If gund wies ae pesent the dimensin f the pblem inceases pptinally F example in a thee-phase system with tw gund wies the dimensin f the P matix is 5 x 5 Hweve given the fact that the line-t-gund vltage f the gund wies is ze equivalent 3 x 3 P and C matices can be fund by using matix patitining and a pcess knwn as Kn eductin Fist wite the V PQ equatin as fllws: Vag Vbg Vcg V vg 0 Vwg 0 ε P P abc vw abc (3x3) (2x3) P abc vw P vw qa qb (3x2) qc (2x2) q v qw Vabc Vvw ε Pabc Pvw abc Pabc vw Qabc Pvw Qvw whee subscipts v and w efe t gund wies w and v and whee the individual P matices ae fmed as befe Since the gund wies have ze ptential then Page 8 f 30
_03_EE394J_2_Sping2_Tansmissin_Linesdc 0 0 ε [ P Q P Q ] vw abc abc vw vw s that [ P Q ] Q vw Pvw vw abc abc Substituting int the V abc equatin abve and cmbining tems yields V abc [ PabcQabc Pabc vwpvw Pvw abcqabc ] [ P abc P abc vw P vw P vw abc ] Q abc ε ε V Q abc abc ' [ P abc ] Q abc s that ε ' ' C V whee C ε [ P ] ' abc abc abc abc Theefe the effect f the gund wies can be included int a 3 x 3 equivalent capacitance matix An altenative way t find the equivalent 3 x 3 capacitance matix ' C abc is t Gaussian eliminate ws 32 using w 5 and then w 4 Aftewad ws 32 ' will have zes in clumns 4 and 5 P abc is the tp-left 3 x 3 submatix Invet 3 by 3 ' P abc t btain ' C abc Cmputing 02 Capacitances fm Matices ' Once the 3 x 3 C abc matix is fund by eithe f the abve tw methds 02 capacitances can ' be detemined by aveaging the diagnal tems and aveaging the ff-diagnal tems f C abc t pduce CS CM CM avg C abc CM CS CS CM CM CS avg C abc has the special symmetic fm f diagnalizatin int 02 cmpnents which yields Page 9 f 30
_03_EE394J_2_Sping2_Tansmissin_Linesdc CS 2CM 0 0 avg C 02 0 CS CM 0 0 0 CS CM The Appximate Fmulas f 02 Capacitances Asymmeties in tansmissin lines pevent the P and C matices fm having the special fm that allws thei diagnalizatin int decupled psitive negative and ze sequence impedances Tanspsitin f cnducts can be used t nealy achieve the special symmetic fm and hence impve the level f decupling Cnducts ae tanspsed s that each ne ccupies each phase psitin f ne-thid f the lines ttal distance An example is given belw in Figue 7 whee the adii f all thee phases ae assumed t be identical a b c a c then b then b a c then b c a c a b then then c b a whee each cnfiguatin ccupies ne-sixth f the ttal distance Figue 7 Tanspsitin f A-B-C Phase Cnducts F this mde f cnstuctin the aveage P matix ( Kn educed P matix if gund wies ae pesent) has the fllwing fm: avg Pabc 6 paa pab pac pbb pbc pcc 6 paa pac pab pcc pbc pbb 6 pbb pab pbc paa pac pcc pcc pac pbc paa pab pbb 6 pbb pbc pab pcc pac paa 6 pcc pbc pac pbb pab paa whee the individual p tems ae descibed peviusly Nte that these individual P matices ae symmetic since ab ba pab pba etc Since the sum f natual lgaithms is the same as the lgaithm f the pduct P becmes ps pm pm avg P abc pm ps pm pm pm ps Page 0 f 30
_03_EE394J_2_Sping2_Tansmissin_Linesdc whee s 3 bb Pcc aai bbi 3 3 a b c Paa P cci p and M 3 ac Pbc abi aci bci 3 3 abac bc Pab P p avg Since P abc has the special ppety f diagnalizatin in symmetical cmpnents then tansfming it yields whee Inveting C p0 0 0 ps 2 pm 0 0 avg P 02 0 p 0 0 ps pm 0 0 0 p2 0 0 ps pm 3 3 3 3 aai bbi cci abiacibci aai bbi cci abacbc p s pm 3 3 3 3 ab c abacbc a b c abi acibci C 0 0 avg P 02 and multiplying by 0 0 avg 02 C 0 πε 0 2 πε yields the cespnding 02 capacitance matix 0 0 0 0 0 p0 ps 2 pm ε 0 0 2 0 0 p ps pm C2 0 0 0 0 p2 ps pm When the a-b-c cnducts ae clse t each the than they ae t the gund then aai bbi cci abi aci bci yielding the cnventinal appximatin 3 abacbc p2 ps pm 3 a b c GM2 p GMR 2 Page f 30
_03_EE394J_2_Sping2_Tansmissin_Linesdc whee GM 2 and GMR 2 ae the gemetic mean distance (between cnducts) and gemetic mean adius espectively f bth psitive and negative sequences Theefe the psitive and negative sequence capacitances becme πε πε 2 2 C C2 Faads pe mete ps pm GM2 GMR2 F the ze sequence tem 3 3 aai bbi cci abi acibci p 0 ps 2 pm 2 3 3 ab c abacbc 3 Expanding yields p 0 3 9 ( aai bbi cci )( abi acibci ) 2 ( )( ) a b ab ac bc ( aai bbi cci )( abi aci bci ) 2 ( )( ) a b ab ac bc 2 2 3 9 ( aai bbi cci )( abi acibci )( bai cai cbi ) ( )( )( ) a b ab ac bc ba ca cb GM0 p 0 3 GMR0 whee ( )( )( ) GM 9 0 aai bbi cci abi aci bci bai cai cbi ( )( )( ) GMR 9 0 a b c ab ac bc ba ca cb The ze sequence capacitance then becmes ε ε C 0 Faads pe mete ps 2 pm 3 GM0 GMR Page 2 f 30
_03_EE394J_2_Sping2_Tansmissin_Linesdc which is ne-thid that f the entie a-b-c bundle by because it epesents the aveage cntibutin f nly ne phase Bundled Phase Cnducts If each phase cnsists f a symmetic bundle f N identical individual cnducts an equivalent adius can be cmputed by assuming that the ttal line chage n the phase divides equally amng the N individual cnducts The equivalent adius is N eq [ NA ]N whee is the adius f the individual cnducts and A is the bundle adius f the symmetic set f cnducts Thee cmmn examples ae shwn belw in Figue 8 uble Bundle Each Cnduct Has Radius A eq 2A Tiple Bundle Each Cnduct Has Radius A eq 3 2 3A Quaduple Bundle Each Cnduct Has Radius A eq 4 3 4A Figue 8 Equivalent Radius f Thee Cmmn Types f Bundled Phase Cnducts Page 3 f 30
_03_EE394J_2_Sping2_Tansmissin_Linesdc 3 Inductance The magnetic field intensity pduced by a lng staight cuent caying cnduct is given by Ampee's Cicuital Law t be I Hφ Ampees pe mete whee the diectin f H is given by the ight-hand ule Magnetic flux density is elated t magnetic field intensity by pemeability μ as fllws: B μh Webes pe squae mete and the amunt f magnetic flux passing thugh a suface is Φ B ds Webes 7 whee the pemeability f fee space is 4π ( 0 ) μ Tw Paallel Wies in Space Nw cnside a tw-wie cicuit that caies cuent I as shwn in Figue 9 Tw cuent-caying wies with adii I I < > Figue 9 A Cicuit Fmed by Tw Lng Paallel Cnducts The amunt f flux linking the cicuit (ie passes between the tw wies) is fund t be μi μ I I dx μ dx Φ x x π Henys pe mete length Fm the definitin f inductance NΦ L I Page 4 f 30
_03_EE394J_2_Sping2_Tansmissin_Linesdc whee in this case N and whee N >> the inductance f the tw-wie pai becmes μ L Henys pe mete length π A und wie als has an intenal inductance which is sepaate fm the extenal inductance shwn abve The intenal inductance is shwn in electmagnetics texts t be μint L int Henys pe mete length 8π F mst cuent-caying cnducts μ int μ s that L int 005µH/m Theefe the ttal inductance f the tw-wie cicuit is the extenal inductance plus twice the intenal inductance f each wie (ie cuent tavels dwn and back) s that L tt μ π μ 2 8π μ π 4 μ π e 4 μ π e 4 It is custmay t define an effective adius eff e 4 0 7788 and t wite the ttal inductance in tems f it as L tt μ Henys pe mete length π eff Wie Paallel t Eath s Suface F a single wie f adius lcated at height h abve the eath the effect f the eath can be descibed by an image cnduct as it was f capacitance calculatins F a pefectly cnducting eath the image cnduct is lcated h metes belw the suface as shwn in Figue 0 Page 5 f 30
_03_EE394J_2_Sping2_Tansmissin_Linesdc Cnduct f adius caying cuent I h Suface f Eath h Nte the image flux exists nly abve the Eath Image cnduct at an equal distance belw the Eath Figue 0 Cuent-Caying Cnduct Abve Eath The ttal flux linking the cicuit is that which passes between the cnduct and the suface f the eath Summing the cntibutin f the cnduct and its image yields F 2h h 2h μ I dx Φ x h dx μ I h x ( 2h ) μ I ( h ) 2 >> a gd appximatin is μi 2h Φ Webes pe mete length h s that the extenal inductance pe mete length f the cicuit becmes Lext μ 2h Henys pe mete length The ttal inductance is then the extenal inductance plus the intenal inductance f ne wie L tt μ 2h μ μ 2h 2h 2 8 2 π π π 4 4 μ e using the effective adius definitin fm befe μ 2h Ltt Henys pe mete length eff Page 6 f 30
_03_EE394J_2_Sping2_Tansmissin_Linesdc Bundled Cnducts The bundled cnduct equivalent adii pesented ealie apply f inductance as well as f capacitance The questin nw is what is the intenal inductance f a bundle? F N bundled cnducts the net intenal inductance f a phase pe mete must decease as because the N intenal inductances ae in paallel Cnsideing a bundle ve the Eath then L tt μ 2h μ μ 2h μ 2h μ e 4 eq 8πN eq 4N eq N eq 2h e 4 N Facting in the expessin f the equivalent bundle adius eq yields N eqe 4 N N N N [ NA ] N e 4N Ne 4 A [ N A ]N eff Thus eff emains e 4 n matte hw many cnducts ae in the bundle The Thee-Phase Case F situatins with multiples wies abve the Eath a matix appach is needed Cnside the capacitance example given in Figue 6 except this time cmpute the extenal inductances athe than capacitances As fa as the vltage (with espect t gund) f ne f the a-b-c phases is cncened the imptant flux is that which passes between the cnduct and the Eath's suface F example the flux "linking" phase a will be pduced by six cuents: phase a cuent and its image phase b cuent and its image and phase c cuent and its image and s n Figue is useful in visualizing the cntibutin f flux linking phase a that is caused by the cuent in phase b (and its image) Page 7 f 30
_03_EE394J_2_Sping2_Tansmissin_Linesdc b a ab bg g ai bg abi bi Figue Flux Linking Phase a ue t Cuent in Phase b and Phase b Image Page 8 f 30
_03_EE394J_2_Sping2_Tansmissin_Linesdc The linkage flux is Φ a (due t I b and I b image) μi b μ I μ I π bg b abi b abi ab bg 2 ab Cnsideing all phases and applying supepsitin yields the ttal flux μ μ Ia aai Ib abi Ic aci Φ a a ab ac μ Nte that aai cespnds t 2h in Figue 0 Pefming the same analysis f all thee phases and ecgnizing that N Φ LI whee N in this pblem then the inductance matix is develped using Φ Φ Φ a b c μ aai a bai ba cai ca abi ab bbi b cbi cb aci ac bci bc cci c I I I a b c Φ abc LabcIabc A cmpaisn t the capacitance matix deivatin shws that the same matix f natual lgaithms is used in bth cases and that Labc μ Pabc μ ε C abc μεc abc This implies that the pduct f the L and C matices is a diagnal matix with μ ε n the diagnal pviding that the eath is assumed t be a pefect cnduct and that the intenal inductances f the wies ae igned If the cicuit has gund wies then the dimensin f L inceases accdingly Recgnizing that the flux linking the gund wies is ze (because thei vltages ae ze) then L can be Kn ' educed t yield an equivalent 3 x 3 matix L abc T include the intenal inductance f the wies eplace actual cnduct adius with eff Cmputing 02 Inductances fm Matices Once the 3 x 3 L ' abc matix is fund 02 inductances can be detemined by aveaging the ' diagnal tems and aveaging the ff-diagnal tems f L abc t pduce Page 9 f 30
_03_EE394J_2_Sping2_Tansmissin_Linesdc LS LM LM avg L abc LM LS LS LM LM LS s that LS 2LM 0 0 avg L 02 0 LS LM 0 0 0 LS LM The Appximate Fmulas f 02 Inductancess Because f the similaity t the capacitance pblem the same ules f eliminating gund wies f tanspsitin and f bundling cnducts apply Likewise appximate fmulas f the psitive negative and ze sequence inductances can be develped and these fmulas ae and μ GM2 L L2 GMR2 μ GM0 L0 3 GMR0 It is imptant t nte that the GM and GMR tems f inductance diffe fm thse f capacitance in tw ways: GMR calculatins f inductance calculatins shuld be made with e 4 eff 2 GM distances f inductance calculatins shuld include the equivalent cmplex depth f mdeling finite cnductivity eath (explained in the next sectin) This effect is igned in capacitance calculatins because the suface f the Eath is nminally at ze ptential Mdeling Impefect Eath The effect f the Eath's nn-infinite cnductivity shuld be included when cmputing inductances especially ze sequence inductances (Nte - psitive and negative sequences ae elatively immune t Eath cnductivity) Because the Eath is nt a pefect cnduct the image cuent des nt actually flw n the suface f the Eath but athe thugh a csssectin The highe the cnductivity the nawe the css-sectin Page 20 f 30
_03_EE394J_2_Sping2_Tansmissin_Linesdc It is easnable t assume that the etun cuent is ne skin depth δ belw the suface f the ρ Eath whee δ metes Typically esistivity ρ is assumed t be 00Ω-m F μ f 00Ω-m and 60Hz δ 459m Usually δ is s lage that the actual height f the cnducts makes n diffeence in the calculatins s that the distances fm cnducts t the images is assumed t be δ 4 Electic Field at Suface f Ovehead Cnducts Igning all the chages the electic field at a cnduct s suface can be appximated by E q ε whee is the adius F vehead cnducts this is a easnable appximatin because the neighbing line chages ae elatively fa away It is always imptant t keep the peak electic field at a cnduct s suface belw 30kV/cm t avid excessive cn lsses Ging beynd the abve appximatin the Makt-Mengele methd pvides a detailed pcedue f calculating the maximum peak subcnduct suface electic field intensity f thee-phase lines with identical phase bundles Each bundle has N symmetic subcnducts f adius The bundle adius is A The pcedue is Teat each phase bundle as a single cnduct with equivalent adius N / N [ ] eq NA 2 Find the C(N x N) matix including gund wies using aveage cnduct heights abve gund Kn educe C(N x N) t C(3 x 3) Select the phase bundle that will have the geatest peak line chage value ( q lpeak ) duing a 60Hz cycle by successively placing maximum line-t-gund vltage Vmax n ne phase and Vmax/2 n each f the the tw phases Usually the phase with the lagest diagnal tem in C(3 by 3) will have the geatest q lpeak 3 Assuming equal chage divisin n the phase bundle identified in Step 2 igne equivalent line chage displacement and calculate the aveage peak subcnduct suface electic field intensity using E q lpeak avg peak N ε 4 Take int accunt equivalent line chage displacement and calculate the maximum peak subcnduct suface electic field intensity using Page 2 f 30
_03_EE394J_2_Sping2_Tansmissin_Linesdc E E ( N A max peak avg peak ) 5 Resistance and Cnductance The esistance f cnducts is fequency dependent because f the esistive skin effect Usually hweve this phenmenn is small f 50-60 Hz Cnduct esistances ae eadily btained fm tables in the ppe units f Ohms pe mete length and these values added t the equivalent-eath esistances fm the pevius sectin t yield the R used in the tansmissin line mdel Cnductance G is vey small f vehead tansmissin lines and can be igned 6 Undegund Cables Undegund cables ae tansmissin lines and the mdel peviusly pesented applies Capacitance C tends t be much lage than f vehead lines and cnductance G shuld nt be igned F single-phase and thee-phase cables the capacitances and inductances pe phase pe mete length ae and ε C ε Faads pe mete length b a μ b L Henys pe mete length a whee b and a ae the ute and inne adii f the caxial cylindes In pwe cables a b is typically e (ie 2783) s that the vltage ating is maximized f a given diamete F mst dielectics elative pemittivity ε 20 2 5 F thee-phase situatins it is cmmn t assume that the psitive negative and ze sequence inductances and capacitances equal the abve expessins If the cnductivity f the dielectic is knwn cnductance G can be calculated using σ G C Mhs pe mete length ε Page 22 f 30
_03_EE394J_2_Sping2_Tansmissin_Linesdc Assumptins SUMMARY OF POSITIVE/NEGATIVE SEQUENCE CALCULATIONS Balanced fa fm gund gund wies igned Valid f identical single cnducts pe phase f identical symmetic phase bundles with N cnducts pe phase and bundle adius A Cmputatin f psitive/negative sequence capacitance whee C / ε GM / GMRC / faads pe mete 3 / ab ac bc GM metes whee ae ab ac bc and whee distances between phase cnducts if the line has ne cnduct pe phase distances between phase bundle centes if the line has symmetic phase bundles GMR C / is the actual cnduct adius (in metes) if the line has ne cnduct pe phase GMR N C / N N A if the line has symmetic phase bundles Cmputatin f psitive/negative sequence inductance μ GM L / / GMRL / henys pe mete whee GM / is the same as f capacitance and f the single cnduct case GMR L / is the cnduct gm (in metes) which takes / 4 int accunt bth standing and the e adjustment f intenal inductance If gm is / 4 nt given then assume e and gm Page 23 f 30
_03_EE394J_2_Sping2_Tansmissin_Linesdc f bundled cnducts N N GMRL / N gm A if the line has symmetic phase bundles Cmputatin f psitive/negative sequence esistance R is the 60Hz esistance f ne cnduct if the line has ne cnduct pe phase If the line has symmetic phase bundles then divide the ne-cnduct esistance by N Sme cmmnly-used symmetic phase bundle cnfiguatins A A A N 2 N 3 N 4 Assumptins ZERO SEQUENCE CALCULATIONS Gund wies ae igned The a-b-c phases ae teated as ne bundle If individual phase cnducts ae bundled they ae teated as single cnducts using the bundle adius methd F capacitance the Eath is teated as a pefect cnduct F inductance and esistance the Eath is assumed t have unifm esistivity ρ Cnduct sag is taken int cnsideatin and a gd assumptin f ding this is t use an aveage cnduct height equal t (/3 the cnduct height abve gund at the twe plus 2/3 the cnduct height abve gund at the maximum sag pint) The ze sequence excitatin mde is shwn belw alng with an illustatin f the elatinship between bundle C and L and ze sequence C and L Since the bundle cuent is actually 3I the ze sequence esistance and inductance ae thee times that f the bundle and the ze sequence capacitance is ne-thid that f the bundle Page 24 f 30
_03_EE394J_2_Sping2_Tansmissin_Linesdc I I 3I I 3I I V I C bundle V I L bundle 3I I I 3I I 3I I L V I C C C V I L L 3I Cmputatin f ze sequence capacitance C 0 ε faads pe mete 3 GMC 0 GMR C0 whee GM C0 is the aveage height (with sag facted in) f the a-b-c bundle abve pefect Eath GM C0 is cmputed using GM 2 2 2 C0 9 i i aa bb cc i ab i ac i bc i metes whee is the distance fm a t a-image i is the distance fm a t b-image and s i aa fth The Eath is assumed t be a pefect cnduct s that the images ae the same distance belw the Eath as ae the cnducts abve the Eath Als GMR 3 2 2 2 C0 9 GMRC / ab ac bc ab metes whee GMR C / ab ac and bc wee descibed peviusly Page 25 f 30
_03_EE394J_2_Sping2_Tansmissin_Linesdc Cmputatin f ze sequence inductance μ δ L0 3 Henys pe mete GMR L0 ρ whee skin depth δ metes μ f The gemetic mean bundle adius is cmputed using GMR 3 2 2 2 L0 9 GMRL / ab ac bc metes whee GMR L/ ab ac and bc wee shwn peviusly Cmputatin f ze sequence esistance Thee ae tw cmpnents f ze sequence line esistance Fist the equivalent cnduct esistance is the 60Hz esistance f ne cnduct if the line has ne cnduct pe phase If the line has symmetic phase bundles with N cnducts pe bundle then divide the ne-cnduct esistance by N Secnd the effect f esistive Eath is included by adding the fllwing tem t the cnduct esistance: 7 3 9869 0 f hms pe mete (see Begen) whee the multiplie f thee is needed t take int accunt the fact that all thee ze sequence cuents flw thugh the Eath As a geneal ule C / usually wks ut t be abut 2 picf pe mete L wks ut t be abut mich pe mete (including intenal inductance) / 0 C is usually abut 6 picf pe mete L 0 is usually abut 2 mich pe mete if the line has gund wies and typical Eath esistivity abut 3 mich pe mete f lines withut gund wies p Eath esistivity The velcity f ppagatin is appximately the speed f light (3 x 0 8 m/s) f psitive LC and negative sequences and abut 08 times that f ze sequence Page 26 f 30
_03_EE394J_2_Sping2_Tansmissin_Linesdc Electic Field at Suface f Ovehead Cnducts Igning all the chages the electic field at a cnduct s suface can be appximated by E q ε whee is the adius F vehead cnducts this is a easnable appximatin because the neighbing line chages ae elatively fa away It is always imptant t keep the peak electic field at a cnduct s suface belw 30kV/cm t avid excessive cna lsses Ging beynd the abve appximatin the Makt-Mengele methd pvides a detailed pcedue f calculating the maximum peak subcnduct suface electic field intensity f thee-phase lines with identical phase bundles Each bundle has N symmetic subcnducts f adius The bundle adius is A The pcedue is 5 Teat each phase bundle as a single cnduct with equivalent adius N / N [ ] eq NA 6 Find the C(N x N) matix including gund wies using aveage cnduct heights abve gund Kn educe C(N x N) t C(3 x 3) Select the phase bundle that will have the geatest peak line chage value ( q lpeak ) duing a 60Hz cycle by successively placing maximum line-t-gund vltage Vmax n ne phase and Vmax/2 n each f the the tw phases Usually the phase with the lagest diagnal tem in C(3 by 3) will have the geatest q lpeak 7 Assuming equal chage divisin n the phase bundle identified in Step 2 igne equivalent line chage displacement and calculate the aveage peak subcnduct suface electic field intensity using E q lpeak avg peak N ε 8 Take int accunt equivalent line chage displacement and calculate the maximum peak subcnduct suface electic field intensity using Emax peak Eavg peak ( N ) A Page 27 f 30
_03_EE394J_2_Sping2_Tansmissin_Linesdc 345kV uble-cicuit Tansmissin Line Scale: cm 2 m 57 m 78 m 85 m 76 m 76 m 44 m 229 m at twe and sags dwn 0 m at midspan t 29 m Twe Base uble cnduct phase bundles bundle adius 229 cm cnduct adius 4 cm cnduct esistance 00728 Ω/km Single-cnduct gund wies cnduct adius 056 cm cnduct esistance 287 Ω/km Page 28 f 30
_03_EE394J_2_Sping2_Tansmissin_Linesdc 500kV Single-Cicuit Tansmissin Line Scale: cm 2 m 39 m 5 m 5 m 33 m 30 m 0 m 0 m Cnducts sag dwn 0 m at mid-span Eath esistivity ρ 00 Ω-m Twe Base Tiple cnduct phase bundles bundle adius 20 cm cnduct adius 5 cm cnduct esistance 005 Ω/km Single-cnduct gund wies cnduct adius 06 cm cnduct esistance 30 Ω/km Page 29 f 30
_03_EE394J_2_Sping2_Tansmissin_Linesdc ue Wed Feb 22 Use the left-hand cicuit f the 345kV line gemety given n the pevius page etemine the L C R line paametes pe unit length f psitive/negative and ze sequence Then f a 00km lng segment f the cicuit detemine the P s Q s I s VR and δr f switch pen and switch clsed cases The geneat vltage phase angle is ze Q L absbed P jq I R jωl P 2 jq 2 I 2 200kVms jωc/2 Q C pduced jωc/2 Q C2 pduced V R / δ R 400Ω One cicuit f the 345kV line gemety 00km lng Page 30 f 30