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Hernandez TL parameters (book chap)

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A chapter (13) from a Taylor & Francis 2006 handbook, written by Manuel Reta-Hernandez of Universidad Autonoma de Zacatecas, kept as a downloaded reference in the Transmission Lines folder. It covers short, medium and long line equivalent circuits, AC resistance with skin, temperature and stranding effects, and ampacity. It then treats inductance and capacitance of single-phase and three-phase lines, bundled conductors and earth effects, and conductor tables.

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13 Transmission Line Parameters Manue lReta-H erna´ndez Universida dAuto´noma deZacatecas13.1 Equivalent Circuit ........................................................... 13-1 13.2 Resistanc e......................................................................... 13-2 Frequency Effect.Temperatur eEffect.Spiraling and Bundle Conductor Effect 13.3 Current-Carr yingCapacity (Ampacit y)........................ 13-5 13.4 Inductance andInductiv eReactanc e............................. 13-6 Inductan ceofaSolid, Round, Inﬁnitely Long Conducto r. Internal Inductanc eDue toInternal Magn eticFlux.External Inductan ce.Inductanc eofaTwo-W ireSingle-Phase Line . Inductan ceofaThre e-Phase Line.Inductanc eofTransposed Three-Phase Transmiss ionLines 13.5 Capacitanc eandCapacitive Reactance ........................ 13-14 Capacita nceofaSingle-Solid Conductor.Capacita nce ofaSingle-Phase Line withTwoWires.Capacitanc eofa Three-Phase Line.Capacita nceofStranded Bundle Conduc tors.Capacita nceDue toEarth’sSurface 13.6 Characteristics ofOverhead Conductors .................... 13-28 The power transmission lineisoneofthemajor component sofanelectric power system. Itsmajor function istotranspor telectric energ y,withminimal losses, from thepowe rsourc estotheload centers, usually separated bylong distanc es.Thedesign ofatransmission linedepends onfour electrical parameters: 1.Series resist ance 2.Series inductanc e 3.Shunt capacitance 4.Shunt conductanc e Theseries resistance reliesbasically onthephysical composition oftheconductor atagiven temperatur e. Theseries inductanc eandshunt capacitanc eareprod uced bythepresence ofmagnetic andelectric ﬁelds aroundtheconductors, anddepend ontheir geometrical arrangement. Theshunt conductance isdueto leakage currents ﬂowingacross insulators andair.Asleakage current isconsiderably small compar edto nominal current, itisusually neglected, andtherefore,shunt condu ctance isnormally notconsidered for thetransmission linemodeling . 13.1 Equivalent Circuit Onceevaluated, theline parameters areused tomodel thetransmission line and toperform design calculations. The arrangement oftheparameters (equivalent circuit model) representing theline depends upon thelength oftheline. /C2232006 byTaylor &Francis Group ,LLC. Atransmission lineisdeﬁned asashort-length lineifitslength islessthan 80km(50miles). Inthis case, theshut capacitanc eeffect isnegligible and only theresist anceand inductiv ereactance are considered. Assuming balanc edcondit ions, thelinecanberepresented bytheequivalent circuitofa singlephase withresist ance R,and inductive reactanceXLinseries (series impedance ),asshown in Fig.13.1. Ifthetransmission linehasalength between 80km(50miles) and240km(150 miles), theline isconsidered amedium-length line and itssingle-phase equivalent circuit canberepresented ina nominal pcircuitconﬁgu ration [1].Theshunt capacitanc eofthelineisdivided into twoequal parts, each placedatthesending andreceiving ends oftheline. Figur e13.2 showstheequivalent circuitfora medium-length line. Both short-and medium-length transmission lines useappro ximated lumped-parameter models. However,ifthelineislarger than 240km, themodel must consider parameters uniformly distributed along theline. The appropri ateseries impedance and shunt capacitanc earefound bysolvingthe corresponding differe ntial equations, wher evoltages andcurre ntsaredescribed asafunction ofdistance andtime. Figure 13.3 shows theequivalent circuit foralong line. Thecalculation ofthethreebasic transmission lineparameters ispresented inthefollowingsections [1–7]. 13.2 Resistance TheACresist anceofacondu ctor inatransmission lineisbased onthecalculation ofitsDCresist ance. IfDCcurrent isﬂowing along around cylindrical condu ctor,thecurre ntisuniformly distributed over itscross-section area anditsDCresistance isevaluated by RDC¼rl AVðÞ (13:1) wher er¼conductor resist ivityatagiventemperature (V-m) l¼conductor length (m) A¼conductor cross-secti onarea(m2)XL Is LoadR VsIL FIGURE 13.1 Equivalentcircuitofashort-length transmissionline.XL Is LoadR VsIL IlineYC 2YC 2 FIGURE 13.2 Equivalent circuit ofamedium- length transmission line. LoadVsIL IlineIssin h g l Z tan h (g l/2) 2Yg l g l/2 FIGURE 13.3 Equivalent circuit ofalong-length transmission line. Z¼zl¼equivalent total series impedance (V), Y¼yl¼equivalent total shunt admittance (S), z¼series impedance perunit length (V=m),y¼shunt admittance perunit length (S=m),g¼ﬃﬃﬃﬃﬃﬃﬃﬃ ZYp ¼propagation constant. /C2232006 byTaylor &Francis Group ,LLC. IfACcurre ntisﬂowing ,rather than DCcurrent, thecondu ctor effective resistance ishigherdueto frequency orskin effect. 13.2.1 Frequency Effect The frequency oftheACvoltage produces asecond effect onthecondu ctor resistance due tothe nonuniform distribution ofthecurrent. This phenomenon isknownasskin effect. Asfrequency increases, thecurrent tends togotoward thesurface oftheconductor andthecurre ntdensity decre ases atthecenter.Skin effect reduces theeffectiv ecross-section areaused bythecurre nt,andthus, theeffective resist anceincreases. Also,althoug hinsmall amount, afurther resistanc eincrease occurswhen other curre nt-carr yingcondu ctors arepresent intheimmediate vicinit y.Askin co rrection factor k,obtained by differe ntial equations andBessel functions, isconsidered toreevaluate theACresistanc e.For60Hz,kis estimated around 1.02 RAC¼RACk (13:2) Other variations inresist ancearecaused by .Temperature .Spiraling ofstranded condu ctors .Bundle conductors arrangement 13.2.2 Temperature Effect Theresistivit yofanycondu ctive material varies linearly overanoperating temperature, andtherefore, theresist anceofanyconductor suffers thesame variations. Astemperature rises, thecondu ctor resist anceincreases linearly ,overnormal operating temperatur es,according tothefollowingequation: R2¼R1Tþt2 Tþt1/C18/C19 (13:3) wher eR2¼resistance atsecond temperature t2 R1¼resistance atinitial temperatur et1 T¼temperature coefﬁcient fortheparticular material (8C) Resistiv ity(r)and temperature coefﬁcient (T)constants depend upon theparticular condu ctor material. Table13.1 listsresistiv ityandtemperature coefﬁcients ofsome typical condu ctormaterials [3]. 13.2.3 Spiraling andBundle Conductor Effect There aretwotypesoftransmission linecondu ctors: overhead andundergr ound. Overhead condu ctors, made ofnakedmetal and suspended oninsulators, arepreferred overundergr ound conductors because ofthelower cost and easy maintenanc e.Also,overhead transmission lines usealuminum condu ctors, because ofthelower cost and lighter weightcompar edtocopperconductors, althoug h morecross-section areaisneeded to conduct thesame amount ofcurre nt.There aredifferent types ofcommercia llyavailable aluminum condu ctors: aluminum-c onductor -steel-reinfor ced(ACSR), aluminum-c onductor -alloy-r einfor ced(ACAR), all-aluminum-conductor (AAC),and all-aluminum- alloy -conductor (AAA C). TABLE 13.1 Resistiv ityandTemperature Coefﬁcient ofSome Conductors Mater ial Resistivit yat208C(V-m) Temperatur eCoefﬁcient (8C) Silver 1.59/C210/C08243.0 Annealed copper 1.72/C210/C08234.5 Hard-dra wncopper 1.77/C210/C08241.5 Aluminum 2.83/C210/C08228.1 /C2232006 byTaylor &Francis Group ,LLC. ACSR isone ofthemost used condu ctors intransmission lines. Itconsists ofalternate layers of stranded condu ctors, spiraled inopposite directions tohold thestrands together ,surrounding acoreof steel strands. Figure 13.4 shows anexample ofaluminum andsteel strands combination. Thepurpose ofintrod ucing asteel coreinside thestranded aluminum condu ctors istoobtain ahigh strength-to-weig htratio .Astranded condu ctor offers more ﬂexibilit yandeasier tomanufactur ethan a solid large conductor .However ,thetotal resistance isincreased because theoutside strands arelarger than theinside strands onaccount ofthespiraling [8].Theresist anceofeach wound condu ctor atany layer,perunit length, isbased onitstotal length asfollows: Rcond¼r Aﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ 1þp1 p/C18/C192s V=mðÞ (13:4) wher eRcond¼resist ance ofwound conductor (V)ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ 1þp1 p/C18/C192s ¼length ofwound condu ctor (m) pcond¼lturn 2rlayer¼relative pitch ofwound condu ctor lturn¼length ofoneturn ofthespiral (m) 2rlayer¼diameter ofthelayer(m) Theparallel combinatio nofnconductors, withsame diameter perlayer,givestheresistanc eperlayer asfollows: Rlayer¼1 Pn i¼11 RiV=mÞ ð (13:5) Similarly ,thetotal resist anceofthestranded condu ctor isevaluated bytheparallel comb ination of resist ancesperlayer. Inhigh-voltage transmission lines, theremaybemorethan oneconductor perphase (bundle conﬁg- uration) toincrease thecurrentcapabilit yandtoreducecoronaeffect discharge. Corona effect occurs when thesurface potential gradient ofacondu ctorexceedsthedielectric strength ofthesurroun ding air (30kV=cmduring fairweather), producing ionization intheareaclose tothecondu ctor,withconsequent coronalosses, audible noise, andradio interferenc e.Ascorona effect isafunction ofconductor diameter , lineconﬁ guration, andcondu ctor surface condition, then meteorol ogical condit ions playakeyrolein itsevaluation. Coronalosses under rain orsnow,forinstance, aremuch higherthan indryweather . Coro na,howe ver,canbereduced byincreasi ngthetotal conductor surface .Althoug hcorona losses relyonmeteoro logical conditions, their evaluation takesinto account thecondu ctance between con- ductors andbetween condu ctors andground. Byincreasing thenumber ofcondu ctors perphase, the total cross-section areaincreases, thecurrentcapacit yincreases, andthetotal ACresistance decre ases prop ortionally tothenumber ofconductors perbundle. Conductor bundles maybeapplied toanyAluminum Strands 2 Layers, 30 Conductors Steel Strands 7 Conductors FIGURE 13.4 Stranded aluminum conductor withstranded steel core(ACSR). /C2232006 byTaylor &Francis Group ,LLC. voltage butarealwaysused at345kVand abov e to limit corona. Tomaintain thedistanc ebetween bundle condu ctors along theline, spacersmade ofsteel oraluminum bars areused. Figur e13.5 shows some typical arrangement ofstranded bundle conﬁgurations. 13.3 Current-Carrying Capacity (Ampacity) Inoverhead transmission lines, thecurrent-carr yingcapacit yisdetermined mostly bythecondu ctor resist anceandtheheat dissipated from itssurface [8].Theheat generated inacondu ctor (Joule’seffect) isdissipated fromitssurface area byconvecti onandradiation given by I2R¼S(wcþwr)WðÞ (13:6) wher eR¼condu ctor resistanc e(V) I¼condu ctor curre nt-carr ying(A) S¼conductor surface area(sq.in.) wc¼convection heat loss(W=sq.in.) wr¼radiation heat loss(W=sq.in.) Heatdissipation byconvection isdeﬁned as wc¼0:0128ﬃﬃﬃﬃﬃpvp T0:123 airﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃdcondp DtWðÞ (13:7) wher ep ¼atmospheric pressur e(atm) v ¼windvelocity(ft=s) dcond¼conductor diameter (in.) Tair¼airtemperature (kelvin) Dt¼Tc/C0Tair¼temperature riseofthecondu ctor (8C) Heatdissipation byradiation isobtained from Stefan–Boltzmann lawandisdeﬁned as wr¼36:8ETc 1000/C18/C194 /C0Tair 1000/C18/C194"# W=sq:in: ðÞ (13:8) wher ewr¼radiation heat loss(W=sq.in.) E¼emissivi tyconstant (1fortheabsolute black body and0.5foroxidized copper) Tc¼conductor temperatur e(8C) Tair¼ambient temperature (8C)d dd dd d (a) (b) (c) FIGURE 13.5 Stranded conductors arranged inbundles perphase of(a)two,(b)three, and(c)four. /C2232006 byTaylor &Francis Group ,LLC. Substituting Eqs. (13.7) and (13.8) inEq.(13.6) wecanobtain theconductor ampacit yatgiven temperature s I¼ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ Sw cþwr ðÞ Rr AðÞ (13:9) I¼ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ ﬃ S RDt0:0128ﬃﬃﬃﬃﬃpvp/C0/C1 T0:123 airﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃdcondp þ36:8ET4 c/C0T4 air 10004/C18/C19 !vuutAðÞ (13:10) Some approximated current-car ryingcapacit yforoverhead ACSR andAACsarepresented inthesection ‘‘Characteristics ofOverhead Conductors’ ’[3,9]. 13.4 Inductance andInductive Reactance Acurre nt-carr yingconductor produces conc entric magnetic ﬂux lines around theconductor .Ifthe current varies withthetime, themagnetic ﬂux changes and avoltage isinduced. Theref ore, an inductance ispresent, deﬁned astheratio ofthemagnetic ﬂux linkage andthecurrent. The magnetic ﬂux produc edbythecurre ntintransmission line conductors produces atotal inductance whose magnitude depends onthelineconﬁ guration. Todetermine theinductanc eoftheline, itisnecessar y tocalculate, asinanymagnetic circuitwithpermeabilit ym,thefollowingfactors: 1.Magnetic ﬁeld intensit yH 2.Magnetic ﬁeld densit yB 3.Flux linkage l 13.4.1 Inductance ofaSolid, Round, Inﬁnitely Long Conductor Consider aninﬁnitely long,solid cylindrical conductor withradius r,carryingcurre ntIasshownin Fig.13.6. Ifthecondu ctor ismade ofanonmagnetic material, andthecurre ntisassumed uniformly distributed (noskin effect), then thegenerated internal andexternal magnetic ﬁeld lines areconcentric circlesaroundtheconductor withdirection deﬁned bytheright-hand rule. 13.4.2 Internal Inductance Due toInternal Magnetic Flux Toobtain theinternal inductanc e,amagnetic ﬁeld withradius xinside thecondu ctor oflength lis chosen, asshown inFig.13.7. Thefraction ofthecurre ntIxenclosed intheareaofthecirclechosen isdetermined by Ix¼Ipx2 pr2AðÞ (13:11) II Internal FieldExternal Field r FIGURE 13.6 External andinternal conc entric magnetic ﬂux lines around theconductor . /C2232006 byTaylor &Francis Group ,LLC. Amper e’slawdetermines themagnetic ﬁeld intensit yHx,constant atanypoint along thecircle contouras Hx¼Ix 2px¼I 2pr2xA=mðÞ (13:12) Themagnetic ﬂuxdensit yBxisobtained by Bx¼mHx¼m0 2pIx r2/C18/C19 TðÞ (13:13) wher em¼m0¼4p/C210/C07H=mforanonmagnetic material. The differenti alﬂux dfenclosed inaring ofthickness dxfora1-m length ofconductor and the differe ntial ﬂux linkage dlintherespectiv eareaare df¼Bxdx¼m0 2pIx r2/C18/C19 dxWb=m ðÞ (13:14) dl¼px2 pr2df¼m0 2pIx3 r4/C18/C19 dxWb=m ðÞ (13:15) Theinternal ﬂux linkage isobtained byintegrating thediffer ential ﬂux linkage from x¼0tox¼r lint¼ðr 0dl¼m0 8pIWb=m ðÞ (13:16) Theref ore,thecondu ctor inductanc eduetointernal ﬂux linkage, perunit length, becomes Lint¼lint I¼m0 8pH=mðÞ (13:17) 13.4.3 External Inductance Theexternal inductance isevaluated assuming that thetotal curre ntIisconc entrated attheconductor surface (maximum skin effect). Atanypoint onanexternal magnetic ﬁeld circleofradius y(Fig.13.8), themagnetic ﬁeld intensit yHyandthemagnetic ﬁeld densit yBy,perunit length, are Hy¼I 2pyA=mðÞ (13:18) By¼mHy¼m0 2pI yTðÞ (13:19)df x IrHxIx dx FIGURE 13.7 Internal magnetic ﬂux. /C2232006 byTaylor &Francis Group ,LLC. Thedifferenti alﬂuxdfenclosed inaringofthickness dy,from point D1topoint D2,fora1-m length of condu ctor is df¼Bydy¼m0 2pI ydyWb=m ðÞ (13:20) Asthetotal current Iﬂows inthesurface condu ctor, then thediffer ential ﬂux linkage dlhasthesame magnitude asthediffer ential ﬂux df. dl¼df¼m0 2pI ydyWb=m ðÞ (13:21) Thetotal external ﬂuxlinkage enclosed bythering is obtained byintegrating fromD1toD2 l1/C02¼ðD2 D1dl¼m0 2pIðD2 D1dy y¼m0 2pIlnD1 D2/C18/C19 Wb=m ðÞ (13:22) Ingeneral, thetotal external ﬂux linkage fromthesurface ofthecondu ctor toanypoint D,perunit length, is lext¼ðD rdl¼m0 2pIlnD r/C18/C19 Wb=m ðÞ (13:23) ThesummationoftheinternalandexternalﬂuxlinkageatanypointDpermitsevaluationofthetotal inductanceofthe conductorLtot,perunitlength,asfollows: lintlþlext¼m0 2pI1 4þlnD r/C18/C19/C20/C21 ¼m0 2pIlnD e/C01=4r/C18/C19 Wb=m ðÞ (13:24) Ltot¼lintþlext I¼m0 2plnD GMR/C18/C19 H=mðÞ (13:25) wher eGMR (geometric mean radius) ¼e/C01=4r¼0.7788 r GMR canbeconsidered astheradius ofaﬁctitious condu ctor assumed tohavenointernal ﬂuxbut withthesame inductance astheactual conductor withradius r. 13.4.4 Inductance ofaTwo-Wire Single-Phase Line Now,consider atwo-wiresingle-phase linewithsolid cylindrical condu ctors Aand Bwiththesame radius r,same length l,andseparated byadistanc eD,wher eD>r,andcondu cting thesame curre ntI,as shown inFig.13.9.The currentﬂowsfrom thesource to theload incondu ctor Aand returnsin condu ctor B(IA¼/C0IB). The magnetic ﬂux generated byone conductor links theother condu ctor.The total ﬂux linking condu ctor A,forinstance, hastwocomponent s:(a)theﬂuxgenerated bycondu ctor Aand(b)theﬂux generated byconductor Bwhich links condu ctor A. Asshown inFig.13.10 ,thetotal ﬂux linkage from condu ctors AandBatpoint Pis lAP¼lAAPþlABP (13:26) lBP¼lBBPþlBAP (13:27)Ir dyy D2D1 x FIGURE 13.8 External magnetic ﬁeld. /C2232006 byTaylor &Francis Group ,LLC. wher elAAP¼ﬂux linkage frommagnetic ﬁeld ofcondu ctor Aonconductor Aatpoint P lABP¼ﬂux linkage frommagnetic ﬁeld ofcondu ctor Bonconductor Aatpoint P lBBP¼ﬂux linkage from magnetic ﬁeld ofcondu ctor Bonconductor Batpoint P lBAP¼ﬂux linkage frommagnetic ﬁeld ofcondu ctor Aonconductor Batpoint P Theexpressions oftheﬂux linkages above,perunit length, are lAAP¼m0 2pIlnDAP GMR A/C18/C19 Wb=m ðÞ (13:28) lABP¼ðDBP DBBPdP¼/C0m0 2pIlnDBP D/C18/C19 Wb=m ðÞ (13:29) lBAP¼ðDAP DBAPdP¼/C0m0 2pIlnDAP D/C18/C19 Wb=m ðÞ (13:30) lBBP¼m0 2pIlnDBP GMR B/C18/C19 Wb=m ðÞ (13:31) Thetotal ﬂux linkage ofthesystem atpoint Pisthealgebraic summation oflAPandlBP lP¼lAPþlBP¼lAAPþlABP ðÞ þlBAPþlBBP ðÞ (13:32) lP¼m0 2pIlnDAP GMR A/C18/C19D DAP/C18/C19DBP GMR B/C18/C19D DBP/C18/C19 /C20/C21 ¼m0 2pIlnD2 GMR AGMR B/C18/C19 Wb=m ðÞ (13:33) Ifthecondu ctors havethesame radius, rA¼rB¼r,and thepoint Pisshifted to inﬁnit y,then thetotal ﬂux linkage ofthe system becomes l¼m0 pIlnD GMR/C18/C19 Wb=m ðÞ (13:34) and thetotal inductance perunit length becomesrAX rBD B AIB IA IIBIA XD FIGURE 13.9 External magnetic ﬂux around conductors inatwo- wiresingle-phase line. B (a) (b) PDAPA PDBPDAB lABP lAAPDAPA B FIGURE 13.10 Flux linkage of(a)conductor Aatpoint Pand (b)conductor Bonconductor Aatpoint P.Single-phase system. /C2232006 byTaylor &Francis Group ,LLC. L1-phase system ¼l I¼m0 plnD GMR/C18/C19 H=mðÞ (13:35) Comparing Eqs. (13.25) and (13.35), itcanbeseen that theinductance ofthesingle-phase system is twicetheinductanc eofasinglecondu ctor. Foralinewithstrandedconductors,theinductanceisdeterminedusinganewGMR value namedGMRstranded,evaluatedaccordingtothenumberofconductors.IfconductorsAandBinthe single-phasesystem,areformedbynandmsolidcylindricalidenticalsubconductorsinparallel,respect- ively,then GMR Astranded ¼ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ Yn i¼1Yn j¼1Dijn2vuut (13:36) GMR Bstranded ¼ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ Ym i¼1Ym j¼1Dijm2vuut (13:37) Generally ,theGMR stranded foraparticular cable can befound inconductor tables givenbythe manufactur er. Iftheline condu ctor iscompose dofbundle condu ctors, theinductanc eisreevaluated taking into account thenumber ofbundle condu ctors and theseparation among them. The GMR bundle is introduc edtodetermine theﬁnal inductance value. Assuming thesame separation among bundle condu ctors, theequation forGMR bundle ,uptothree condu ctors perbundle, isdeﬁned as GMR nbundle conductors ¼ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ ﬃ dn/C01GMR strandednp (13:38) wher en¼number ofconductors perbundle GMR stranded ¼GMR ofthestranded condu ctor d¼distanc ebetween bundle condu ctors Forfour condu ctors perbundle withthesame separation between consecutive conductors, the GMR bundle isevaluated as GMR 4bundle conductors ¼1:09ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ d3GMR stranded4p (13:39) 13.4.5 Inductance ofaThree-Phase Line Thederivations fortheinductanc einasingle-phase system canbeextended toobtain theinductance per phase inathree-ph asesystem. Consider athree-phase, three-cond uctor system withsolid cylindrical condu ctors withidentical radius rA,rB,and rC,placedhorizontally withseparation DAB,DBC,and DCA (where D>r)among them. Corr esponding currents IA,IB,andICﬂowalong each condu ctor asshown inFig.13.11 . Thetotal magnetic ﬂuxenclosing condu ctorAatapoint Pawayfrom thecondu ctors isthesum ofthe ﬂux prod uced bycondu ctors A,B,andCasfollows: fAP¼fAAPþfABPþfACP (13:40) wher efAAP¼ﬂux prod uced bycurrent IAonconductor Aatpoint P fABP¼ﬂuxproduc edbycurrentIBonconductor Aatpoint P fACP¼ﬂux produced bycurre ntIConcondu ctor Aatpoint P Considering 1-m length foreach condu ctor,theexpres sions fortheﬂuxesabov eare /C2232006 byTaylor &Francis Group ,LLC. fAAP¼m0 2pIAlnDAP GMR A/C18/C19 Wb=m ðÞ (13:41) fABP¼m0 2pIBlnDBP DAB/C18/C19 Wb=m ðÞ (13:42) fACP¼m0 2pIClnDCP DAC/C18/C19 Wb=m ðÞ (13:43) Thecorresponding ﬂux linkage ofconductor Aatpoint P(Fig.13.12) isevaluated as lAP¼lAAPþlABPþlACP (13:44) having lAAP¼m0 2pIAlnDAP GMR A/C18/C19 Wb=m ðÞ (13:45)A B CfA f B f C DAB DBC DCAX X X FIGURE 13.11 Magnetic ﬂux produc edbyeach conductor inathree-phase system. C DAPB PlAAPlABP lACP PPDAC DAP DAPDCPDBPA A B BC C (a) (b) (c)ADAB FIGURE 13.12 Flux linkage of(a)conductor Aatpoint P,(b)conductor Bonconductor Aatpoint P,and(c) conductor Conconductor Aatpoint P.Three-phase system. /C2232006 byTaylor &Francis Group ,LLC. lABP¼ðDBP DABBBPdP¼m0 2pIBlnDBP DAB/C18/C19 Wb=m ðÞ (13:46) lACP¼ðDCP DACBCPdP¼m0 2pIClnDCP DAC/C18/C19 Wb=m ðÞ (13:47) wher elAP¼total ﬂux linkage ofconductor Aatpoint P lAAP¼ﬂux linkage from magnetic ﬁeld ofconductor Aoncondu ctor Aatpoint P lABP¼ﬂuxlinkage from magnetic ﬁeld ofcondu ctor Boncondu ctor Aatpoint P lACP¼ﬂux linkage frommagnetic ﬁeld ofcondu ctor Conconductor Aatpoint P Substi tuting Eqs. (13.45) throu gh(13.47) inEq.(13.44) and rearranging ,accordingtonatural logarithms law,wehave lAP¼m0 2pIAlnDAP GMR A/C18/C19 þIBlnDBP DAB/C18/C19 þIClnDCP DAC/C18/C19 /C20/C21 Wb=m ðÞ (13:48) lAP¼m0 2pIAln1 GMR A/C18/C19 þIBln1 DAB/C18/C19 þICln1 DAC/C18/C19 /C20/C21 þm0 2pIAlnDAPðÞ þIBlnDBPðÞ þIClnDCPðÞ ½/C138 Wb=m ðÞ (13:49) Thearrangement ofEq.(13.48) into Eq.(13.49) isalgebraically correctaccording tonatural logarithms law.However,asthecalculation ofanynatural logarithm must bedimensionless, thenumerator inthe expres sions ln(1=GMR A),ln(1=DAB),andln(1=DAC)must havethesame dimension asthedenominator . Thesame applies forthedenominator intheexpres sions ln(DAP),ln(DBP),andln(DCP). Assuming abalanc edthree-phase system, wher eIAþIBþIC¼0,andshifting thepoint Ptoinﬁnit yin such away that DAP¼DBP¼DCP,then thesecond partofEq.(13.49) iszero,andtheﬂux linkage of condu ctor Abecomes lA¼m0 2pIAln1 GMR A/C18/C19 þIBln1 DAB/C18/C19 þICln1 DAC/C18/C19 /C20/C21 Wb=m ðÞ (13:50) Similarly ,theﬂuxlinkage expres sions forcondu ctors BandCare lB¼m0 2pIAln1 DBA/C18/C19 þIBln1 GMR B/C18/C19 þICln1 DBC/C18/C19 /C20/C21 Wb=m ðÞ (13:51) lC¼m0 2pIAln1 DCA/C18/C19 þIBln1 DCB/C18/C19 þICln1 GMR C/C18/C19 /C20/C21 Wb=m ðÞ (13:52) Theﬂux linkage ofeach phase condu ctor depends onthethree currents ,andtherefore, theinductance perphase isnotonly oneasinthesingle-phase system. Instead, three differe ntinductanc es(self and mutual condu ctor inductanc es)exist. Calculating theinductance values fromtheequations abov eand arranging theequations inamatrix form wecanobtain thesetofinductance sinthesystem lA lB lC2 43 5¼LAA LAB LAC LBA LBB LBC LCA LCB LCC2 43 5IA IB IC2 43 5 (13:53) wher elA,lB,lC¼total ﬂux linkages ofconductors A,B,andC LAA,LBB,LCC¼self-inductanc esofconductors A,B,andCﬁeld ofcondu ctor Aatpoint P LAB,LBC,LCA,LBA,LCB,LAC¼mutual inductance samong conductors /C2232006 byTaylor &Francis Group ,LLC. Withnine differ entinductanc esinasimple three-phase system theanalysis could bealittle more complicated. However,asingleinductanc eperphase canbeobtained ifthethreecondu ctors are arranged with the same separation among them (symmetrical arrangement), wher e D¼DAB¼DBC¼DCA.Forabalanc edthree-phase system (IAþIBþIC¼0,orIA¼/C0IB/C0IC),theﬂux linkage ofeach conductor ,perunit length, willbethesame. From Eq.(13.50) wehave lA¼m0 2p/C0IB/C0IC ðÞ ln1 GMR A/C18/C19 þIBln1 D/C18/C19 þICln1 D/C18/C19 /C20/C21 lA¼m0 2p/C0IBlnD GMR A/C18/C19 /C0IClnD GMR A/C18/C19 /C20/C21 lA¼m0 2pIAlnD GMR A/C18/C19/C20/C21 Wb=m ðÞ(13:54) IfGMR value isthesame forallcondu ctors (either singleorbundle GMR), thetotal ﬂux linkage expression isthesame forallphases. Therefor e,theequivalent inductance perphase is Lphase¼m0 2plnD GMR phase/C18/C19 H=mðÞ (13:55) 13.4.6 Inductance ofTransposed Three-Phase Transmission Lines Inactual transmission lines, thephase condu ctors cannot maintain symmetrical arrangement along the whole length because ofconstruction considerations, evenwhen bundle condu ctor spacersareused. Withasymmetrical spacing ,theinductanc ewillbediffer entforeach phase, withacorresp onding unbalance dvoltage drop oneach condu ctor.Therefor e,thesingle-phase equivalent circuittorepresent thepower system cannot beused. However,itispossible toassume symmetrical arrangement inthetransmission linebytransposing the phase conductors. In a transposed system, each phase condu ctor occupies thelocation oftheother two phases forone-third ofthetotal linelength asshown inFig.13.13. Inthiscase, theaverage distanc e geometrical mean distanc e(GMD) substitutes distanc eD,and thecalculation ofphase inductanc e derive dforsymmetrical arrangement isstillvalid. Theinductanc eperphase perunit length inatransmission linebecomes Lphase¼m0 2plnGMD GMR phase/C18/C19 H=mðÞ (13:56) Oncetheinductance perphase isobtained, theinductiv ereactance perunit length is XLphase¼2pfLphase¼m0flnGMD GMR phase/C18/C19 V=mðÞ (13:57) A B CC B AA C B l/ 3 l/ 3 l/ 3 FIGURE 13.13 Arrangement ofconductors inatransposed line. /C2232006 byTaylor &Francis Group ,LLC. Forbundle conductors, theGMR bundle value isdetermined, asinthesingle-phase transmission linecase, bythenumber ofconductors, andbythenumber ofcondu ctors perbundle andtheseparation among them. Theexpression forthetotal inductiv ereactanceperphase yields XLphase¼m0flnGMD GMR bundle/C18/C19 V=mðÞ (13:58) wher eGMR bundle ¼(dn/C01GMR stranded )1=nuptothreeconductors perbundle (m) GMR bundle ¼1.09( d4GMR stranded )1=4forfour conductors perbundle (m) GMR phase¼geometric mean radius ofphase conductor ,either solid orstranded (m) GMD ¼ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃDABDBCDCA3p¼geometrical mean distanc eforathree-ph aseline(m) d¼distanc ebetween bundle condu ctors (m) n¼number ofcondu ctor perbundle f¼frequency (Hz) 13.5 Capacitance andCapacitive Reactance Capacitanc eexists among transmission linecondu ctors duetotheir potential differe nce. Toevaluate thecapacitanc ebetween condu ctors inasurrounding medium withpermittiv ity«,itisnecessaryto determine thevoltage between thecondu ctors, andtheelectric ﬁeld stren gthofthesurrounding . 13.5.1 Capacitance ofaSingle-Solid Conductor Consider asolid, cylindrical, long condu ctor withradius r,inafreespace withpermittiv ity«0,and withacharge ofqþcoulom bspermeter ,uniformly distributed onthesurface .There isaconstant electric ﬁeld strength onthesurface ofcylinder (Fig.13.14). The resist ivityofthecondu ctor is assumed tobezero (perfect condu ctor), which results inzero internal electric ﬁeld duetothecharge onthecondu ctor. Thecharge qþproduc esanelectric ﬁeld radial tothecondu ctorwithequipotential surfaces concentric tothecondu ctor.Accordin gtoGauss’ slaw,thetotal electric ﬂuxleaving aclosed surface isequal tothe total charge inside thevolume enclosed bythesurfac e.Therefor e,atanoutside point Pseparated x meters from thecenter oftheconductor ,theelectric ﬁeld ﬂuxdensity andtheelectric ﬁeld intensit yare DensityP¼q A¼q 2pxCðÞ (13:59) rP1P2 + +dx lrx1 x2Electric Field LinesPath of Integration Conductor with Charge q+q FIGURE 13.14 Electric ﬁeld produc edfromasingleconductor . /C2232006 byTaylor &Francis Group ,LLC. EP¼DensityP «¼q 2p«0xV=mðÞ (13:60) wher eDensit yP¼electric ﬂux densit yatpoint P EP¼electric ﬁeld intensity atpoint P A¼surface ofaconc entric cylinder with1-m length andradius x(m2) «¼«0¼10/C09 36p¼permittivity offreespaceassumed fortheconductor (F=m) Thepotential differe nceorvoltage differe ncebetween twooutside points P1andP2withcorrespond- ingdistances x1and x2from thecondu ctor center isdeﬁned byintegrating theelectric ﬁeld intensit y from x1tox2 V1/C02¼ðx2 x1EPdx x¼ðx2 x1q 2p«0dx x¼q 2p«0lnx2 x1/C20/C21 VðÞ (13:61) Then, thecapacitanc ebetween points P1and P2isevaluated as C1/C02¼q V1/C02¼2p«0 lnx2 x1/C20/C21 F=mðÞ (13:62) Ifpoint P1islocated attheconductor surface (x1¼r),andpoint P2islocated atground surface below theconductor (x2¼h),then thevoltage oftheconductor andthecapacitanc ebetween thecondu ctor andgroun dare Vcond¼q 2p«0lnh r/C20/C21 VðÞ (13:63) Ccond/C0ground ¼q Vcond¼2p«0 lnh r/C20/C21 F=mðÞ (13:64) 13.5.2 Capacitance ofaSingle-Phase Line with Two Wires Consider atwo- wiresingle-phase linewithcondu ctors Aand Bwiththesame radius r,separated by adistanc eD>rAand rB.Theconductors areenergized byavoltage sourcesuch that condu ctor Ahas acharge qþandcondu ctor Bacharge q/C0asshown inFig.13.15. The charge oneach condu ctor generates independent electric ﬁelds. Charge qþonconductor A generates avoltage VAB–A between both condu ctors. Similarly ,charge q/C0onconductor Bgenerates avoltage VAB–B between conductors. lr Ar BD B Aq A + −q B rA rB B A+ -D q+ q−q+q−− FIGURE 13.15 Electric ﬁeld produc edfromatwo-wire single-phase system. /C2232006 byTaylor &Francis Group ,LLC. VAB–A iscalculated byintegrating theelectric ﬁeld intensit y,duetothecharge onconductor A,on condu ctor Bfrom rAtoD VAB/C0A¼ðD rAEAdx¼q 2p«0lnD rA/C20/C21 (13:65) VAB–B iscalculated byintegrating theelectric ﬁeld intensit yduetothecharge onconductor Bfrom DtorB VAB/C0B¼ðrB DEBdx¼/C0q 2p«0lnrB Dhi (13:66) Thetotal voltage isthesum ofthegenerated voltages VAB/C0Aand VAB/C0B VAB¼VAB/C0AþVAB/C0B¼q 2p«0lnD rA/C20/C21 /C0q 2p«0lnrB Dhi ¼q 2p«0lnD2 rArB/C20/C21 (13:67) Ifthecondu ctors havethesame radius, rA¼rB¼r,then thevoltage between condu ctors VAB,andthe capacitanc ebetween condu ctors CAB,fora1-m linelength are VAB¼q p«0lnD r/C20/C21 VðÞ (13:68) CAB¼p«0 lnD r/C20/C21 F=mðÞ (13:69) Thevoltage between each condu ctorandground (G)(Fig.13.16) isone-half ofthevoltagebetween thetwo condu ctors. There fore, thecapacitanc efromeither linetoground istwicethecapacitanc ebetween lines VAG¼VBG¼VAB 2VðÞ (13:70) CAG¼q VAG¼2p«0 lnD r/C20/C21 F=mðÞ (13:71) CAG ACBG VAG VBG VAB+q− q−q+q+ −B VBGVAG VABCAG CBG BA − FIGURE 13.16 Capacitance between linetoground inatwo-wire single-phase line. /C2232006 byTaylor &Francis Group ,LLC. 13.5.3 Capacitance ofaThree-Phase Line Consider athree-phase linewiththesame voltage magnitude between phases, andassuming abalanc ed system withabc(positiv e)sequence such that qAþqBþqC¼0.Theconductors haveradii rA,rB,andrC, andthespacebetween condu ctors areDAB,DBC,and DAC(where DAB,DBC,and DAC>rA,rB,and rC). Also,theeffect ofearthandneutral condu ctors isneglected. The expression forvoltages between twoconductors inasingle-phase system canbeextended to obtain thevoltages between condu ctors inathree-phase system. Theexpres sions forVABand VACare VAB¼1 2p«0qAlnDAB rA/C20/C21 þqBlnrB DAB/C20/C21 þqClnDBC DAC/C20/C21 /C20/C21 VðÞ (13:72) VAC¼1 2p«0qAlnDCA rA/C20/C21 þqBlnDBC DAB/C20/C21 þqClnrC DAC/C20/C21 /C20/C21 VðÞ (13:73) Ifthe three-ph ase system has triangular arrangement with equidistant condu ctors such that DAB¼DBC¼DAC¼D,withthesame radii fortheconductors such that rA¼rB¼rC¼r(where D>r),theexpressions forVABand VACare VAB¼1 2p«0qAln" D r# þqBln" r D# þqCln" D D# "# ¼1 2p«0qAln" D r# þqBln" r D# "# VðÞ (13:74) VAC¼1 2p«0qAln" D r# þqBln" D D# þqCln" r D# "# ¼1 2p«0qAln" D r# þqCln" r D# "# VðÞ(13:75) Balanc edline-to-line voltages withsequence abc, expressed interms oftheline-to-neutral voltage are VAB¼ﬃﬃﬃ 3p VANﬀ30/C14and VAC¼/C0 VCA¼ﬃﬃﬃ 3p VANﬀ/C030/C14; wher eVANistheline-to-neutral voltage. There fore,VANcanbeexpressed interms ofVABand VACas VAN¼VABþVAC 3(13:76) andthus, substituting VABand VACfromEqs. (13.67) and(13.68) wehave VAN¼1 6p«0qAln" D r# þqBln" r D# "# þqAln" D r# þqCln" r D# "# "# ¼1 6p«02qAln" D r# þqBþqC! ln" r D# # VðÞ" (13:77) Under balanc edcondit ions qAþqBþqC¼0,or/C0qA¼(qBþqC)then, theﬁnal expres sion fortheline- to-neutral voltage is VAN¼1 2p«0qAlnD r/C20/C21 VðÞ (13:78) /C2232006 byTaylor &Francis Group ,LLC. The positiv esequence capacitanc eperunit length between phase Aandneutral cannow beobtained. Thesame result isobtained forcapacitance between phases BandCtoneutral CAN¼qA VAN¼2p«0 lnD r/C20/C21 F=mðÞ (13:79) 13.5.4 Capacitance ofStranded Bundle Conductors Thecalculation ofthecapacitanc eintheequation abov eisbased on 1.Solid conductors withzeroresistivit y(zero internal electric ﬁeld) 2.Charge uniformly distributed 3.Equilateral spacing ofphase condu ctors Inactual transmission lines, theresist ivityoftheconductors produc esasmall internal electric ﬁeld and therefor e,theelectric ﬁeld attheconductor surface issmaller than theestimated. Howev er,the differe nceisnegligible forpractical purposes. Because ofthepresence ofother charged condu ctors, thecharge distribution isnonuniform, and therefor etheestimated capacitance isdiffer ent. Howev er,this effect isnegligible formost practical calculations. In alinewithstranded condu ctors, thecapacitanc eisevaluated assuming asolid conductor withthesame radius astheoutside radius ofthestranded conductor .This produces anegligible differe nce. Mosttransmission lines donothaveequilateral spacing ofphase condu ctors. This causes differe nces between theline-to-neutral capacitanc esofthethreephases. However,transposing thephase conductors balanc esthesystem resulti nginequal line-to-neutral capacitance foreach phase andisdevel oped inthe followingmanner . Consider atransposed three-phase linewithconductors having thesame radius r,and withspace between conductors DAB,DBC,and DAC,wher eDAB,DBC,and DAC>r. Assuming abcpositive sequence, theexpressions forVABontheﬁrst, second, andthird section ofthe transposed lineare VABfirst¼1 2p«0qAlnDAB r/C20/C21 þqBlnr DAB/C20/C21 þqClnDAB DAC/C20/C21 /C20/C21 VðÞ (13:80) VABsecond ¼1 2p«0qAlnDBC r/C20/C21 þqBlnr DBC/C20/C21 þqClnDAC DAB/C20/C21 /C20/C21 VðÞ (13:81) VABthird¼1 2p«0qAlnDAC r/C20/C21 þqBlnr DAC/C20/C21 þqClnDAB DBC/C20/C21 /C20/C21 VðÞ (13:82) Similarly ,theexpressions forVAContheﬁrst, second,andthird section ofthetransposed lineare VACfirst¼1 2p«0qAlnDAC r/C20/C21 þqBlnDBC DAB/C20/C21 þqClnr DAC/C20/C21 /C20/C21 (13:83) VACsecond ¼1 2p«0qAlnDAB r/C20/C21 þqBlnDAC DBC/C20/C21 þqClnr DAB/C20/C21 /C20/C21 (13:84) VACthird¼1 2p«0qAlnDBC r/C20/C21 þqBlnDAB DAC/C20/C21 þqClnr DBC/C20/C21 /C20/C21 (13:85) Taking theaverage value ofthethreesections, wehavetheﬁnal expres sions ofVABand VACinthe transposed line /C2232006 byTaylor &Francis Group ,LLC. VABtransp ¼VABfirstþVABsecond þVABthird 3 ¼1 6p«0qAlnDABDACDBC r3/C20/C21 þqBlnr3 DABDACDBC/C20/C21 þqClnDACDACDBC DACDACDBC/C20/C21 /C20/C21 VðÞ (13:86) VACtransp ¼VACfirstþVACsecond þVACthird 3 ¼1 6p«0qAlnDABDACDBC r3/C20/C21 þqBlnDACDACDBC DABDACDBC/C20/C21 þqClnr3 DACDACDBC/C20/C21 /C20/C21 VðÞ (13:87) Forabalanc edsystem wher e/C0qA¼(qBþqC),thephase-to-neutral voltage VAN(phase voltage) is VANtransp ¼VABtransp þVACtransp 3 ¼1 18p«02qAlnDABDACDBC r3/C20/C21 þqBþqC ðÞ lnr3 DABDACDBC/C20/C21 /C20/C21 ¼1 6p«0qAlnDABDACDBC r3/C20/C21 ¼1 2p«0qAlnGMD r/C20/C21 VðÞ (13:88) wher eGMD ¼ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃDABDBCDCA3p¼geometrical mean distanc eforathree-ph aseline. Forbundle condu ctors, anequivalent radius rereplac estheradius rofasinglecondu ctor and is determined bythenumber ofcondu ctors perbundle andthespacing ofcondu ctors. Theexpression ofre issimilar toGMR bundle used inthecalculation oftheinductanc eperphase, exceptthattheactual outside radius ofthecondu ctor isused instead oftheGMR phase.Therefor e,theexpression forVANis VANtransp ¼1 2p«0qAlnGMD re/C20/C21 VðÞ (13:89) wher ere¼(dn/C01r)1=n¼equivalent radius foruptothree conductors perbundle (m) re¼1.09 (d3r)1=4¼equivalent radius forfour condu ctors perbundle (m) d¼distanc ebetween bundle conductors (m) n¼number ofconductors perbundle Finally ,thecapacitanc eand capacitiv ereactanc e,perunit length, from phase toneutral canbe evaluated as CANtransp ¼qA VANtransp¼2p«0 lnGMD re/C20/C21 F=mðÞ (13:90) XANtransp ¼1 2pfCANtransp¼1 4pf«0lnGMD re/C20/C21 V=mðÞ (13:91) 13.5.5 Capacitance Due toEarth’s Surface Considering asingle -overhead condu ctor withareturn path throu ghtheearth,separated adistanc eH from earth’ssurface, thecharge oftheearthwould beequal inmagnitude tothatonthecondu ctorbutof opposite sign. Iftheearthisassumed asaperfectly conductiv ehorizontal plane withinﬁnite length, then theelectric ﬁeld lines willgofrom theconductor totheearth,perpendicular totheearth’ssurface (Fig.13.17 ). /C2232006 byTaylor &Francis Group ,LLC. Tocalculate thecapacitanc e,thenegativ echarge oftheearthcanbereplac edbyanequivalent charge ofanimage condu ctor withthesame radius astheoverhead condu ctor,lyingjustbelowtheoverhead condu ctor (Fig.13.18). The same principle canbeextended tocalculate thecapacitanc eperphase ofathree-phase system. Figur e13.19 shows anequilateral arrangement ofidentical single condu ctors forphases A,B,and C carryingthecharges qA,qB,and qCandtheir respectiv eimage condu ctors A0,B0,andC0. DA,DB,andDCareperpendicular distanc esfrom phases A,B,andCtoearth’ssurface .DAA0,DBB0,and DCC0aretheperpendicular distanc esfromphases A,B,andCtotheimage condu ctors A0,B0,andC0. Voltage VABcanbeobtained as VAB¼1 2p«0qAlnDAB rA/C20/C21 þqBlnrB DAB/C20/C21 þqClnDBC DAC/C20/C21 /C0 /C0qAlnDAB0 DAA0/C20/C21 /C0qBlnDBB0 DAB0/C20/C21 /C0qClnDBC0 DAC0/C20/C212 66643 7775VðÞ (13:92)− − −+ Earth's SurfaceH+ + ++++ + q − −−− − FIGURE 13.17 Distribution ofelectric ﬁeld lines fromanoverhead conductor toearth’ssurface. + q+ - −-−− − −− − −+ +++ + ++ + −qq 2H Earth’s Surface Equivalent Earth Charge FIGURE 13.18 Equivalent image conductor representing thecharge oftheearth. /C2232006 byTaylor &Francis Group ,LLC. Asoverhead condu ctors areidentical, then r¼rA¼rB¼rC.Also,astheconductors haveequilateral arrangement, D¼DAB¼DBC¼DCA VAB¼1 2p«0qAln" D r# /C0ln" DAB0 DAA0# ! þqB ln" r D# /C0ln" DBB0 DAB0#! /C0qCln" DBC0 DAC0# "# VÞ ð(13:93) Similarly ,expressions forVBCand VACare VBC¼1 2p«0" /C0qAln" DCA0 DBA0# þqB lnD r# /C0ln" DCB0 DBB0#! þqC ln" r D# /C0ln" DCC0 DBC0#! "# VÞ ð(13:94) VAC¼1 2p«0" qAln" D r# /C0ln" DCA0 DAA0#! /C0qBln" DCB0 DAB0 # þqC ln" r D# /C0ln" DCC0 DAC0#!# VÞ ð(13:95) Thephase voltage VANbecome s,throughalgebraic reduction, VAN¼VABþVAC 3 ¼1 2p«0qAlnD r/C20/C21 /C0ln"ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃDAB0DBC0DCA03p ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃDAA0DBB0DCC03p# ! VÞ ð (13:96) Theref ore,thephase capacitanc eCAN,perunit length, is CAN¼qA VAN¼2p«0 lnh D ri /C0lnﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃDAB0DBC0DCA03p ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃDAA0DBB0DCC03p/C20/C21 F=mðÞ (13:97) Equations (13.79) and(13.97) havesimilar expressions, exceptfortheterm ln((DAB0DBC0DCA0)1=3=(DAA0 DBB0DCC0)1=3)included inEq. (13.97). That term represents theeffect oftheearthonphase capacitanc e,increasing itstotal value. However,thecapacitanc eincrement isreally small, andisusuallyA/H11032 B/H11032C/H11032B A C DA DCqB qAqC −qA −qC −qBImage ConductorsOverhead Conductors DCC/H11032 = 2DCDB DBB/H11032 = 2DBDAA/H11032 = 2DAEarth’s Surface FIGURE 13.19 Arrangement ofimage conductors inathree- phase transmission line. /C2232006 byTaylor &Francis Group ,LLC. TABLE 13.2a Characteristics ofAluminum Cable Steel Reinfor cedConductors (ACSR) Cross-Section Area Diameter Approx.Current- Carrying Capac ityResistanc e(mV/km)60HzReactanc es (Dm ¼1m) TotalAluminum Stran ding Conductor Core DCAC(60Hz) GMR X1 X0 Code (mm2) (kcmil) (mm2)A l=Steel (mm) (mm) Layers (Amper es) 258C2 58C5 08C7 58C (mm ) ( V/km) (MV/km) – 1521 2 776 1407 84 =19 50.80 13.87 4 21.0 24.5 26.2 28.1 20.33 0.294 0.175 Joree 1344 2 515 1274 76 =19 47.75 10.80 4 22.7 26.0 28.0 30.0 18.93 0.299 0.178 Thras her 1235 2 312 1171 76 =19 45.77 10.34 4 24.7 27.7 30.0 32.2 18.14 0.302 0.180 Kiwi 1146 2 167 1098 72 =7 44.07 8.81 4 26.4 29.4 31.9 34.2 17.37 0.306 0.182 Bluebir d 1181 2 156 1092 84 =19 44.75 12.19 4 26.5 29.0 31.4 33.8 17.92 0.303 0.181 Chukar 976 1781 902 84=19 40.69 11.10 4 32.1 34.1 37.2 40.1 16.28 0.311 0.186 Falcon 908 1590 806 54=19 39.24 13.08 3 1 380 35.9 37.4 40.8 44.3 15.91 0.312 0.187 Lapwing 862 1590 806 45=7 38.20 9.95 3 1 370 36.7 38.7 42.1 45.6 15.15 0.316 0.189 Parrot 862 1510 765 54=19 38.23 12.75 3 1 340 37.8 39.2 42.8 46.5 15.48 0.314 0.189 Nuthatch 818 1510 765 45=7 37.21 9.30 3 1 340 38.7 40.5 44.2 47.9 14.78 0.318 0.190 Plover 817 1431 725 54=19 37.21 12.42 3 1 300 39.9 41.2 45.1 48.9 15.06 0.316 0.190 Bobo link 775 1431 725 45=7 36.25 9.07 3 1 300 35.1 42.6 46.4 50.3 14.39 0.320 0.191 Martin 772 1351 685 54=19 36.17 12.07 3 1 250 42.3 43.5 47.5 51.6 14.63 0.319 0.191 Dippe r 732 1351 685 45=7 35.20 8.81 3 1 250 43.2 44.9 49.0 53.1 13.99 0.322 0.193 Pheasa nt 726 1272 645 54=19 35.10 11.71 3 1 200 44.9 46.1 50.4 54.8 14.20 0.321 0.193 Bittern 689 1272 644 45=7 34.16 8.53 3 1 200 45.9 47.5 51.9 56.3 13.56 0.324 0.194 Grack le 681 1192 604 54=19 34.00 11.33 3 1 160 47.9 49.0 53.6 58.3 13.75 0.323 0.194 Bunt ing 646 1193 604 45=7 33.07 8.28 3 1 160 48.9 50.4 55.1 59.9 13.14 0.327 0.196 Finch 636 1114 564 54=19 32.84 10.95 3 1 110 51.3 52.3 57.3 62.3 13.29 0.326 0.196 Blueja y 603 1113 564 45=7 31.95 8.00 3 1 110 52.4 53.8 58.9 64.0 12.68 0.329 0.197 Curlew 591 1033 523 54=7 31.62 10.54 3 1 060 56.5 57.4 63.0 68.4 12.80 0.329 0.198 /C2232006 byTaylor &Francis Group ,LLC. Ortolan 560 1033 525 45=7 30.78 7.70 3 1 060 56.5 57.8 63.3 68.7 12.22 0.332 0.199 Merganse r 596 954 483 30=7 31.70 13.60 2 1 010 61.3 61.8 67.9 73.9 13.11 0.327 0.198 Cardinal 546 954 483 54=7 30.38 10.13 3 1 010 61.2 62.0 68.0 74.0 12.31 0.332 0.200 Rail 517 954 483 45=7 29.59 7.39 3 1 010 61.2 62.4 68.3 74.3 11.73 0.335 0.201 Baldpate 562 900 456 30=7 30.78 13.21 2 960 65.0 65.5 71.8 78.2 12.71 0.329 0.199 Cana ry 515 900 456 54=7 29.51 9.83 3 970 64.8 65.5 72.0 78.3 11.95 0.334 0.201 Ruddy 478 900 456 45=7 28.73 7.19 3 970 64.8 66.0 72.3 78.6 11.40 0.337 0.202 Crane 501 875 443 54=7 29.11 9.70 3 950 66.7 67.5 74.0 80.5 11.80 0.335 0.202 Willet 474 874 443 45=7 28.32 7.09 3 950 66.7 67.9 74.3 80.9 11.25 0.338 0.203 Skimmer 479 795 403 30=7 29.00 12.40 2 940 73.5 74.0 81.2 88.4 11.95 0.334 0.202 Mallard 495 795 403 30=19 28.96 12.42 2 910 73.5 74.0 81.2 88.4 11.95 0.334 0.202 Drake 469 795 403 26=7 28.14 10.36 2 900 73.3 74.0 81.2 88.4 11.43 0.337 0.203 Cond or 455 795 403 54=7 27.74 9.25 3 900 73.4 74.1 81.4 88.6 11.22 0.339 0.204 Cuckoo 455 795 403 24=7 27.74 9.25 2 900 73.4 74.1 81.4 88.5 11.16 0.339 0.204 Tern 431 795 403 45=7 27.00 6.76 3 900 73.4 74.4 81.6 88.8 10.73 0.342 0.205 Coot 414 795 403 36=1 26.42 3.78 3 910 73.0 74.4 81.5 88.6 10.27 0.345 0.206 Buteo 447 715 362 30=7 27.46 11.76 2 840 81.8 82.2 90.2 98.3 11.34 0.338 0.204 Redwing 445 715 362 30=19 27.46 11.76 2 840 81.8 82.2 90.2 98.3 11.34 0.338 0.204 Starling 422 716 363 26=7 26.7 9.82 2 840 81.5 82.1 90.1 98.1 10.82 0.341 0.206 Crow 409 715 362 54=7 26.31 8.76 3 840 81.5 82.2 90.2 98.2 10.67 0.342 0.206 Current capacityevaluated at758Cconductor temperature,258Cairtemperatur e,wind speedof1.4mi=h,andfrequency of60Hz. Sources:Transmission LineReferenceBook345kVandAbove,2nded.,Electric PowerResear chInstitute,PaloAlto,Califor nia,1987. Withpermissio n. Glover,J.D.andSarma,M.S.,PowerSystemAnalysisandDesign,3rded. ,Bro oks=Cole,2002. Withpermis sion. /C2232006 byTaylor &Francis Group ,LLC. TABLE 13.2b Characteristics ofAluminum Cable Steel Reinfor cedConductors (ACSR) Cross-Section Area Diame ter Approx.Current- CarryingCapac ityResistance(mV/km)60HzReactanc es (Dm ¼1m) TotalAluminum Strandin g Cond uctor Core DCAC(60Hz) GMR X1 X0 Code (mm2) (kcmil) (mm2)A l=Steel (mm ) (mm ) Layers (Amper es) 258C2 58C5 08C7 58C (mm ) ( V/km) (MV/km) Stilt 410 716 363 24=7 26.31 8.76 2 840 81.5 82.2 90.2 98.1 10.58 0.343 0.206 Grebe 388 716 363 45=7 25.63 6.4 3 840 81.5 82.5 90.4 98.4 10.18 0.346 0.208 Gannet 393 666 338 26=7 25.76 9.5 2 800 87.6 88.1 96.6 105.3 10.45 0.344 0.208 Gull 382 667 338 54=7 25.4 8.46 3 800 87.5 88.1 96.8 105.3 10.27 0.345 0.208 Flami ngo 382 667 338 24=7 25.4 8.46 2 800 87.4 88.1 96.7 105.3 10.21 0.346 0.208 Scoter 397 636 322 30=7 25.88 11.1 2 800 91.9 92.3 101.4 110.4 10.70 0.342 0.207 Egret 396 636 322 30=19 25.88 11.1 2 780 91.9 92.3 101.4 110.4 10.70 0.342 0.207 Grosbeak 375 636 322 26=7 25.15 9.27 2 780 91.7 92.2 101.2 110.3 10.21 0.346 0.209 Goose 364 636 322 54=7 24.82 8.28 3 770 91.8 92.4 101.4 110.4 10.06 0.347 0.208 Rook 363 636 322 24=7 24.82 8.28 2 770 91.7 92.3 101.3 110.3 10.06 0.347 0.209 Kingbird 340 636 322 18=1 23.88 4.78 2 780 91.2 92.2 101.1 110.0 9.27 0.353 0.211 Swirl 331 636 322 36=1 23.62 3.38 3 780 91.3 92.4 101.3 110.3 9.20 0.353 0.212 Wood Duck 378 605 307 30=7 25.25 10.82 2 760 96.7 97.0 106.5 116.1 10.42 0.344 0.208 Teal 376 605 307 30=19 25.25 10.82 2 770 96.7 97.0 106.5 116.1 10.42 0.344 0.208 Squab 356 605 356 26=7 25.54 9.04 2 760 96.5 97.0 106.5 116.0 9.97 0.347 0.208 Peacock 346 605 307 24=7 24.21 8.08 2 760 96.4 97.0 106.4 115.9 9.72 0.349 0.210 Duck 347 606 307 54=7 24.21 8.08 3 750 96.3 97.0 106.3 115.8 9.81 0.349 0.210 Eagle 348 557 282 30=7 24.21 10.39 2 730 105 .1 105.4 115.8 126.1 10.00 0.347 0.210 Dove 328 556 282 26=7 23.55 8.66 2 730 104 .9 105.3 115.6 125.9 9.54 0.351 0.212 Parakeet 319 557 282 24=7 23.22 7.75 2 730 104 .8 105.3 115.6 125.9 9.33 0.352 0.212 /C2232006 byTaylor &Francis Group ,LLC. Ospr ey 298 556 282 18=1 22.33 4.47 2 740 104.4 105.2 115.4 125.7 8.66 0.358 0.214 Hen 298 477 242 30=7 22.43 9.6 2 670 122.6 122.9 134.9 147.0 9.27 0.353 0.214 Hawk 281 477 242 26=7 21.79 8.03 2 670 122.4 122.7 134.8 146.9 8.84 0.357 0.215 Flick er 273 477 273 24=7 21.49 7.16 2 670 122.2 122.7 134.7 146.8 8.63 0.358 0.216 Pelican 255 477 242 18=1 20.68 4.14 2 680 121.7 122.4 134.4 146.4 8.02 0.364 0.218 Lark 248 397 201 30=7 20.47 8.76 2 600 147.2 147.4 161.9 176.4 8.44 0.360 0.218 Ibis 234 397 201 26=7 19.89 7.32 2 590 146.9 147.2 161.7 176.1 8.08 0.363 0.220 Brant 228 398 201 24=7 19.61 6.53 2 590 146.7 147.1 161.6 176.1 7.89 0.365 0.221 Chickadee 213 397 201 18=1 18.87 3.78 2 590 146.1 146.7 161.0 175.4 7.32 0.371 0.222 Oriol e 210 336 170 30=7 18.82 8.08 2 530 173.8 174.0 191.2 208.3 7.77 0.366 0.222 Linn et 198 336 170 26=7 18.29 6.73 2 530 173.6 173.8 190.9 208.1 7.41 0.370 0.224 Widgeon 193 336 170 24=7 18.03 6.02 2 530 173.4 173.7 190.8 207.9 7.25 0.371 0.225 Merlin 180 336 170 18=1 16.46 3.48 2 530 173.0 173.1 190.1 207.1 6.74 0.377 0.220 Piper 187 300 152 30=7 17.78 7.62 2 500 195.0 195.1 214.4 233.6 7.35 0.370 0.225 Ostrich 177 300 152 26=7 17.27 6.38 2 490 194.5 194.8 214.0 233.1 7.01 0.374 0.227 Gadwall 172 300 152 24=7 17.04 5.69 2 490 194.5 194.8 213.9 233.1 6.86 0.376 0.227 Phoeb e 160 300 152 18=1 16.41 3.28 2 490 193.5 194.0 213.1 232.1 6.37 0.381 0.229 Junco 167 267 135 30=7 16.76 7.19 2 570 219.2 219.4 241.1 262.6 6.92 0.375 0.228 Partridge 157 267 135 26=7 16.31 5.99 2 460 218.6 218.9 240.5 262.0 6.61 0.378 0.229 Waxwing 143 267 135 18=1 15.47 3.1 2 460 217.8 218.1 239.7 261.1 6.00 0.386 0.232 Current capacity evaluated at758Cconductor temperatur e,258Cairtemperatur e,wind speedof1.4mi=h,andfreque ncyof60Hz. Sources:Transmission LineReferen ceBook345kVandAbove,2nded.,Electr icPowerResearchInstitute ,PaloAlto,Califo rnia, 1987. Withperm ission. Glover,J.D.andSarma,M.S., Power SystemAnalys isand Design,3rded.,Brooks=Cole,2002. Withpermission. /C2232006 byTaylor &Francis Group ,LLC. TABLE 13.3a Characteristics ofAll-Aluminum-Conductors (AAC) Cross-Section Area Diamete r Approx.Current - Carrying Capac ityResistance(mV=km)60HzReactances (Dm ¼1m) DCAC(60Hz) GMR XL XC Code (mm2) kcmil orAWG Stranding (mm) Layers (Amper es) 258C2 58C5 08C7 58C (mm) (V=km) (MV=km) Coreopsi s 806.2 1591 61 36.93 4 1380 36.5 39.5 42.9 46.3 14.26 0.320 0.190 Glaldiolus 765.8 1511 61 35.99 4 1340 38.4 41.3 44.9 48.5 13.90 0.322 0.192 Carnation 725.4 1432 61 35.03 4 1300 40.5 43.3 47.1 50.9 13.53 0.324 0.193 Columbi ne 865.3 1352 61 34.04 4 1250 42.9 45.6 49.6 53.6 13.14 0.327 0.196 Narcissus 644.5 1272 61 33.02 4 1200 45.5 48.1 52.5 56.7 12.74 0.329 0.194 Hawthorn 604.1 1192 61 31.95 4 1160 48.7 51.0 55.6 60.3 12.34 0.331 0.197 Marigold 564.2 1113 61 30.89 4 1110 52.1 54.3 59.3 64.3 11.92 0.334 0.199 Larkspur 524 1034 61 29.77 4 1060 56.1 58.2 63.6 69.0 11.49 0.337 0.201 Bluebell 524.1 1034 37 29.71 3 1060 56.1 58.2 63.5 68.9 11.40 0.337 0.201 Goldenr od 483.7 955 61 28.6 4 1010 60.8 62.7 68.6 74.4 11.03 0.340 0.203 Magnolia 483.6 954 37 28.55 3 1010 60.8 62.7 68.6 74.5 10.97 0.340 0.203 Crocus 443.6 875 61 27.38 4 950 66.3 68.1 74.5 80.9 10.58 0.343 0.205 Anemone 443.5 875 37 27.36 3 950 66.3 68.1 74.5 80.9 10.49 0.344 0.205 Lilac 403.1 796 61 26.11 4 900 73.0 74.6 81.7 88.6 10.09 0.347 0.207 Arbutus 402.9 795 37 26.06 3 900 73.0 74.6 81.7 88.6 10.00 0.347 0.207 Nasturtium 362.5 715 61 24.76 4 840 81.2 82.6 90.5 98.4 9.57 0.351 0.209 Violet 362.8 716 37 24.74 3 840 81.1 82.5 90.4 98.3 9.48 0.351 0.209 Orchid 322.2 636 37 23.32 3 780 91.3 92.6 101.5 110.4 8.96 0.356 0.212 Mistletoe 281.8 556 37 21.79 3 730 104.4 105.5 115.8 126.0 8.38 0.361 0.215 Dahlia 281.8 556 19 21.72 2 730 104.4 105.5 115.8 125.9 8.23 0.362 0.216 Syringa 241.5 477 37 20.19 3 670 121.8 122.7 134.7 146.7 7.74 0.367 0.219 Cosmos 241.9 477 19 20.14 2 670 121.6 122.6 134.5 146.5 7.62 0.368 0.219 Canna 201.6 398 19 18.36 2 600 145.9 146.7 161.1 175.5 6.95 0.375 0.224 Tulip 170.6 337 19 16.92 2 530 172.5 173.2 190.1 207.1 6.40 0.381 0.228 Laur el 135.2 267 19 15.06 2 460 217.6 218.1 239.6 261.0 5.70 0.390 0.233 Daisy 135.3 267 7 14.88 1 460 217.5 218 239.4 260.8 5.39 0.394 0.233 Oxlip 107.3 212or(4=0) 7 13.26 1 340 274.3 274.7 301.7 328.8 4.82 0.402 0.239 Phlox 85 168or(3=0) 7 11.79 1 300 346.4 346.4 380.6 414.7 4.27 0.411 0.245 Aster 67.5 133or(2=0) 7 10.52 1 270 436.1 439.5 479.4 522.5 3.81 0.40 0.25 Poppy 53.5 106or(1=0) 7 9.35 1 230 550 550.2 604.5 658.8 3.38 0.429 0.256 Pansy 42.4 #1AWG 7 8.33 1 200 694.2 694.2 763.2 831.6 3.02 0.438 0.261 Iris 33.6 #2AWG 7 7.42 1 180 874.5 874.5 960.8 1047. 9 2.68 0.446 0.267 Rose 21.1 #3AWG 7 5.89 1 160 1391.5 1391.5 1528.9 1666. 3 2.13 0.464 0.278 Peachb ell 13.3 #4AWG 7 4.67 1 140 2214.4 2214.4 2443.2 2652 1.71 0.481 0.289 Current capacity evalua tedat758Cconductor temperatur e,258Cairtemperatur e,wind speed of1.4mi=h,andfreque ncyof60Hz. Sources:TransmissionLineReferenceBook345kVandAbove,2nded.,Electric PowerResear chInstitute,PaloAlto,California,1987. Withpermis sion. Glover,J.D.andSarma ,M.S., PowerSystemAnalysisandDesign,3rded. ,Bro oks=Cole,2002.Withpermission . /C2232006 byTaylor &Francis Group ,LLC. TABLE 13.3b Characteristics ofAll-Aluminum-Conductors (AAC) Cross-Sect ionArea Diameter Approx.Current- CarryingCapacityResistanc e(mV=km)60HzReactanc es (Dm ¼1m) DCAC(60Hz) GMR XL XC Code (mm2) kcmilorAWG Strandin g (mm) Layers (Amper es) 258C2 58C5 08C7 58C (mm) (V=km) (MV=km) EVENSIZES Bluebonnet 1773.3 3500 7 54.81 6 16.9 22.2 23.6 25.0 21.24 0.290 0.172 Trillium 1520.2 3000 127 50.75 6 19.7 24.6 26.2 27.9 19.69 0.296 0.175 Lupine 1266.0 2499 91 46.30 5 23.5 27.8 29.8 31.9 17.92 0.303 0.180 Cowslip 1012.7 1999 91 41.40 5 29.0 32.7 35.3 38.0 16.03 0.312 0.185 Jessamine 887.0 1750 61 38.74 4 33.2 36.5 39.5 42.5 14.94 0.317 0.188 Hawkweed 506.7 1000 37 29.24 3 1030 58.0 60.0 65.5 71.2 11.22 0.339 0.201 Cam elia 506.4 999 61 29.26 4 1030 58.1 60.1 65.5 71.2 11.31 0.338 0.201 Snapdragon 456.3 900 61 27.79 4 970 64.4 66.3 72.5 78.7 10.73 0.342 0.204 Cockscomb 456.3 900 37 27.74 3 970 64.4 66.3 72.5 78.7 10.64 0.343 0.204 Catta il 380.1 750 61 25.35 4 870 77.4 78.9 86.4 93.9 9.78 0.349 0.208 Petunia 380.2 750 37 23.85 3 870 77.4 78.9 86.4 93.9 9.72 0.349 0.208 Flag 354.5 700 61 24.49 4 810 83.0 84.4 92.5 100.6 9.45 0.352 0.210 Verbena 354.5 700 37 24.43 3 810 83.0 84.4 92.5 100.6 9.39 0.352 0.210 Meado wswee t 303.8 600 37 2.63 3 740 96.8 98.0 107.5 117.0 8.69 0.358 0.214 Hyacinth 253.1 500 37 20.65 3 690 116 .2 117.2 128.5 140.0 7.92 0.365 0.218 Zinnia 253.3 500 19 20.60 2 690 116 .2 117.2 128.5 139.9 7.80 0.366 0.218 Goldentuft 228.0 450 19 19.53 2 640 129 .0 129.9 142.6 155.3 7.41 0.370 0.221 Daffodil 177.3 350 19 17.25 2 580 165 .9 166.6 183.0 199.3 6.52 0.379 0.227 Peony 152.1 300 19 15.98 2 490 193 .4 194.0 213.1 232.1 6.04 0.385 0.230 Valerian 126.7 250 19 14.55 2 420 232 .3 232.8 255.6 278.6 5.52 0.392 0.235 Sneezew ort 126.7 250 7 14.40 1 420 232 .2 232.7 255.6 278.4 5.21 0.396 0.235 Current capacityevaluated at758Cconductor temperature,258Cairtemperatur e,wind speedof1.4mi=h,andfrequency of60Hz. Sources:Transmission Line Reference Book 345kVand Above,2nded.,Electr icPowerResear chInstitute ,PaloAlto,California, 1987.Withperm ission. Glover,J.D.andSarma, M.S., Power System Analysis and Design,3rded.,Brooks=Cole,2002. Withpermission. /C2232006 byTaylor &Francis Group ,LLC. neglected, because distances from overhead condu ctors toground arealwaysgreater than distanc es among condu ctors. 13.6 Characteristics ofOverhead Conductors Tables 13.2a and13.2b present typical values ofresist ance,inductiv ereactanceandcapacitance react- ance,perunit length, ofACSR condu ctors. Thesizeofthecondu ctors (cross-section area) isspeciﬁed in square millimeters and kcmil, where acmil isthecross-se ction areaofacircular condu ctor witha diameter of1=1000 in.The tables include also theapproximate curre nt-carr yingcapacit yofthe condu ctors assuming 60Hz,windspeed of1.4mi=h,and conductor and airtemperature sof758C and258C,respectiv ely.Tables 13.3a and13.3b present thecorresponding characteristics ofAACs. References 1.Yamay ee,Z.A. andBala, J.L.Jr.,Elect romechanical EnergyDevicesandPower Systems,JohnWileyand Sons, Inc., NewYork, 1994. 2.Glover,J.D.andSarma, M.S., Power System Analysis and Design ,3rded.,Brooks=Cole, 2002. 3.Steven son, W.D.Jr.,Elements ofPower System Analysis ,4thed.McGra w-Hill, NewYork, 1982. 4.Saadat, H.,Power System Analysis ,McGra w-Hill,Boston, MA, 1999. 5.Gross, Ch.A., Power System Analysis ,John WileyandSons, NewYork, 1979. 6.Gungor ,B.R., Power Systems ,Harcourt Brace Jovanovich, Orlando ,FL,1988. 7.Zaborszky ,J.andRittenhouse, J.W.,Elect ricPower Transmission. ThePower SystemintheSteady State , TheRonald PressCompan y,NewYork, 1954. 8.Barnes, C.C., Power Cables. Their Design andInstallati on,2nded.,Chapman andHall, London, 1966. 9.Electric Power Research Institute, Transmission Line Reference Book 345kVand Above,2nded.,Palo Alto,CA,1987. /C2232006 byTaylor &Francis Group ,LLC. 14 SagandTension of Conductor D.A. Dougl ass Power Deliv eryConsultants ,Inc. Ridley Thra sh Southw ireCompany14.1 Catenar yCables ............................................................... 14-2 LevelSpans .Conduc torLength .Conduc torSlack . Inclined Spans.IceandWindConductor Loads. Conductor Tension Limits 14.2 Approximate Sag-T ension Calculations ......................... 14-9 SagChange with Thermal Elongation.SagChange Due toCombined Thermal andElastic Effects .Sag Change Due toIceLoading 14.3 Numerical Sag-T ension Calculations ........................... 14-14 Stress-Strain Curves.Sag-T ension Tables 14.4 Ruling Span Conce pt.................................................... 14-22 Tension Differ encesforAdjace ntDead-End Spans . Tension Equalizatio nbySuspe nsion Insulators.Ruling Span Calculation.Stringing SagTables 14.5 Line Design Sag-T ension Parameters ........................... 14-25 Catenar yConstants.WindSpan.WeightSpan. Uplift atSuspension Structur es.Tower Spotting 14.6 Conductor Installation .................................................. 14-28 Conductor Stringing Methods.Tension Stringing Equipment andSetup .Sagging Procedure 14.7 Deﬁning Terms .............................................................. 14-39 Theenergized condu ctors oftransmission anddistribution lines must beplacedtototally eliminate the possibilit yofinjur ytopeople. Over head condu ctors, howev er,elongate withtime, temperatur e,and tension, thereb ychanging their original positions after installation. Despite theeffects ofweather andloading onaline, thecondu ctors must remai nat safe distanc esfrombuildings, objects, andpeople orvehicles passing beneath thelineatalltimes. Toensur ethissafety,theshape oftheterrain along theright-of-wa y,theheightandlateral position ofthecondu ctorsuppor tpoints, andtheposition ofthe condu ctor between suppor tpoints under allwind,ice,andtemperatur econdit ions must beknown. Bareoverhead transmission ordistribution condu ctors aretypically quite ﬂexible anduniform in weightalong their length. Because ofthese characteristics, they take theform ofacatenar y(Ehrenberg , 1935; Winkelm ann, 1959) between suppor tpoints. Theshape ofthecatenar ychanges withcondu ctor temperature ,iceandwindloading ,andtime. Toensur eadequate vertical andhorizontal clearance under allweather andelectrical loadings, andtoensur ethat thebreakin gstrength oftheconductor isnot exceeded, thebehaviorofthecondu ctorcatenar yunder allcondit ions must beknownbefor ethelineis designed. Thefuture behavioroftheconductor isdetermined through calculations comm only referred toassag-tension calculations. Sag-tension calculations predictthebehavior ofconductors based onrecommended tension limits under varyingloading conditions. These tension limits specify certain percentages ofthecondu ctor’s /C2232006 byTaylor &Francis Group ,LLC. rated breakin gstrength thatarenottobeexceeded upon installation orduring thelifeoftheline. These condit ions, along withtheelastic and permanent elongation propertiesofthecondu ctor,provide thebasis fordeterminating theamount ofresulting sagduring installation andlong-term operation oftheline. Accurate lydetermined initial saglimits areessential inthelinedesign process. Final sagsandtensions depend oninitial installed sags and tensions and onproperhandling during installation. The ﬁnal sagshape ofconductors isused toselect suppor tpoint heightsandspan lengths sothattheminimum clearance swillbemaintained overthelifeoftheline. Ifthecondu ctor isdamaged ortheinitial sags areincorrect, thelineclearance smaybeviolated orthecondu ctor maybreak during heavyiceor windloadings. 14.1 Catenary Cables Abare-stranded overhead condu ctorisnormally held clear ofobjects, people, andother condu ctors by periodic attachment toinsulators. The elevation differ ences between thesuppor ting structure saffect theshape oftheconductor catenar y.Thecatenar y’sshape hasadistinct effect onthesagandtension ofthecondu ctor,andtherefor e,must bedetermined using well-deﬁned mathematical equations. 14.1.1 Level Spans The shape ofacatenar yisafunction ofthecondu ctor weightperunit length, w,thehorizontal component oftension, H,span length, S,andthemaximum sagofthecondu ctor,D.Conductor sag andspan length areillustrated inFig.14.1 foralevel span. Theexact catenar yequation useshyperbolic functions. Relativ e tothelowpoint ofthecatenar ycurve shown inFig.14.1, theheightofthecondu ctor,y(x),abovethislowpoint isgivenbythefollowing equation: y(x)¼H wcoshw Hx/C16/C17 /C01/C16/C17 ¼w(x2) 2H(14:1) S DL 2 x X axisy (x) H a = H/wY axis T FIGURE 14.1 Thecatenar ycurveforlevel spans. /C2232006 byTaylor&Francis Group ,LLC. Notethatxispositiv eineither direct ionfromthelowpoint ofthecatenar y.Theexpression totherightis anappro ximate parabolic equation based upon aMacLaurin expansion ofthehyperbolic cosine. Foralevelspan, thelowpoint isinthecenter ,andthesag,D,isfound bysubstituting x¼S=2inthe precedingequations. Theexact andapproximate parabolic equations forsagbecome thefollowing: D¼H wcoshwS 2H/C18/C19 /C01/C18/C19 ¼w(S2) 8H(14:2) The ratio,H=w,which appears inallofthepreceding equations, iscommonly referred toasthe catenar yconstant. Anincrease inthecatenar yconstant, havingtheunits oflength, causes thecatenar y curvetobecome shallo werandthesagtodecrease. Although itvaries withcondu ctor temperature ,ice andwindloading ,andtime, thecatenar yconstant typically hasavalue intherange ofseveralthousand feetformost transmission-line catenaries. Theappro ximate orparabolic expres sion issufﬁciently accurate aslong asthesagislessthan 5%of thespan length. Asanexample, consider a1000-ft span ofDrak econductor (w¼1.096 lb=ft)installed at atension of4500 lb.Thecatenar yconstant equals 4106 ft.Thecalculated sagis30.48 ftand30.44 ft using thehyperbolic and approximate equations, respectively.Both estimates indicate asag-to-span ratio of3.4% andasagdifferenc eofonly 0.5in. The horizontal comp onent oftension, H,isequal totheconductor tension atthepoint inthe catenar ywhere thecondu ctor slope ishorizontal. Foralevel span, thisisthemidpoint ofthespan length. Attheends ofthelevel span, thecondu ctortension, T,isequal tothehorizontal component plus thecondu ctor weightperunit length, w,multiplied bythesag,D,asshown inthefollowing: T¼HþwD (14:3) Giventheconditions inthepreceding example calculation fora1000-ft level span ofDrake ACSR, the tension attheattachment points exceeds thehorizontal component oftension by33lb.It iscommon to perform sag-tension calculations using thehorizontal tension compone nt,buttheaverage ofthe horizontal andsuppor tpoint tension isusually listed intheoutput. 14.1.2 Conductor Length Application ofcalculus tothecatenar yequation allowsthecalculation ofthecondu ctor length, L(x), measur edalong theconductor from thelowpoint ofthecatenar yineither direction. Theresulting equation becomes: L(x)¼H wSINHwx H/C16/C17 ¼x1þx2w2ðÞ 6H2/C18/C19 (14:4) Foralevelspan, thecondu ctorlength corresponding tox¼S=2ishalfofthetotal condu ctorlength andthetotal length, L,is: L¼2H w/C18/C19 SINHSw 2H/C18/C19 ¼S1þS2w2ðÞ 24H2/C18/C19 (14:5) The parabolic equation forcondu ctor length canalso beexpressed asafunction ofsag,D,by substitution ofthesagparabolic equation, giving: L¼Sþ8D2 3S(14:6) /C2232006 byTaylor &Francis Group ,LLC. 14.1.3 Conductor Slack Thedifferenc ebetween theconductor length, L,andthespan length, S,iscalled slack. Theparabolic equations forslack may befound bycombining thepreceding parabolic equations forconductor length, L,andsag,D: L/C0S¼S3w2 24H2/C18/C19 ¼D28 3S/C18/C19 (14:7) Whileslack hasunits oflength, itisoften expres sedasthepercentage ofslack relative tothespan length. Notethatslack isrelated tothecube ofspan length foragivenH=wratio andtothesquar eofsag foragivenspan. Foraseries ofspans having thesame H=wratio,thetotal slack islargely determined by thelongest spans. Itisforthisreason thattheruling span isnearly equal tothelongest span rather than theaverage span inaseries ofsuspension spans. Equation (14.7) canbeinvertedtoobtain amore interesting relations hipshowing thedependenc eof sag,D,upon slack, L-S: D¼ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ ﬃ 3S(L/C0S) 8r (14:8) Ascanbeseen from theprecedingequation, small changes inslack typically yieldlarge changes in condu ctor sag. 14.1.4 Inclined Spans Inclined spans may beanalyzed using essentially thesame equations thatwere used forlevel spans. The catenar yequation forthecondu ctor heightabovethelowpoint inthespan isthesame. However,the span isconsideredtoconsistoftwoseparate sections, onetotherightofthelowpoint andtheother to theleftasshown inFig.14.2 (Winkelm ann, 1959). Theshape ofthecatenar yrelative tothelowpoint is unaffected bythediffere nceinsuspension point elevation (span inclination). Ineach direction from thelowpoint, thecondu ctorelevation, y(x), relative tothelowpoint isgivenby: y(x)¼H wcoshw Hx/C16/C17 /C01/C16/C17 ¼wx2ðÞ 2H(14:9) S S1 TRD DR XR XLDLTL h FIGURE 14.2 Inclined catenar yspan. /C2232006 byTaylor&Francis Group ,LLC. Notethat xisconsidered positive ineither direction from thelowpoint. Thehorizontal distance, xL,from theleftsuppor tpoint tothelowpoint inthecatenar yis: xL¼S 21þh 4D/C18/C19 (14:10) Thehorizontal distanc e,xR,from therightsuppor tpoint tothelowpoint ofthecatenar yis: xR¼S 21/C0h 4D/C18/C19 (14:11) wher eS¼horizontal distance between suppor tpoints. h¼vertical distanc ebetween suppor tpoints. Sl¼straig ht-line distanc ebetween suppor tpoints. D¼sagmeasur edvertically fromalinethrou ghthepoints ofcondu ctorsuppor ttoalinetangent tothecondu ctor. Themidpoint sag,D,isapproximately equal tothesaginahorizontal span equal inlength tothe inclined span, Sl. Knowingthehorizonal distanc efrom thelowpoint tothesuppor tpoint ineach direction, the precedingequations fory(x),L,D,andTcanbeapplied toeach sideoftheinclined span. Thetotal condu ctorlength, L,intheinclined span isequal tothesum ofthelengths inthexRandxL sub-span sections: L¼Sþx3 Rþx3 L/C0/C1 w2 6H2/C18/C19 (14:12) Ineach sub-span, thesagisrelative tothecorresponding suppor tpoint elevation: DR¼wx2 R 2HDL¼wx2 L 2H(14:13) orinterms ofsag,D,andthevertical distanc ebetween suppor tpoints: DR¼D1/C0h 4D/C18/C192 DL¼D1þh 4D/C18/C192 (14:14) andthemaximum tension is: TR¼HþwD RTL¼HþwD L (14:15) orinterms ofupper andlower suppor tpoints: Tu¼Tlþwh (14:16) wher eDR¼saginrightsub-span section DL¼saginleftsub-span section TR¼tension inrightsub-span section TL¼tension inleftsub-span section Tu¼tension incondu ctor atupper suppor t Tl¼tension inconductor atlower suppor t /C2232006 byTaylor &Francis Group ,LLC. Thehorizontal conductor tension isequal atboth suppor ts.Thevertical comp onent ofconductor tension isgreat erattheupper suppor tandtheresultant tension, Tu,isalsogreat er. 14.1.5 Ice andWind Conductor Loads Whenaconductor iscoveredwithiceand =orisexposed towind,theeffective condu ctorweightperunit length increases. During occasions ofheavyiceand =orwindload, thecondu ctor catenar ytension increases dramatically along withtheloads onangleanddeadend structur es.Both thecondu ctorandits suppor tscanfailunless these high-tension condit ions areconsidered inthelinedesign. TheNational Electric Safet yCode (NESC) suggests certain comb inations oficeandwindcorrespond- ingtoheavy,medium, andlightloading regions oftheUnited States. Figur e14.3 isamap oftheU.S. indicating those areas (NESC, 1993). Thecomb inations oficeandwindcorresponding toloading region arelisted inTable 14.1. The NESC also suggests that increased conductor loads due tohighwindloads without icebe considered. Figure 14.4 shows thesuggested windpressure asafunction ofgeographical area forthe United States (ASCE Std7–88). Certainutilities inveryheavyiceareas useglazeicethicknesses ofasmuch astwoinches tocalculate icedcondu ctorweight. Similarly ,utilities inregions wher ehurricane windsoccur may usewindloads as highas34lb=ft2. AstheNESC indicates, thedegree oficeandwindloads varies withtheregion. Some areas may have heavy icing ,wher eassome areasmayhaveextre mely highwinds. Theloads must beaccounte dforinthe linedesign process sothey donothaveadetrimental effect ontheline. Some oftheeffects ofboth the indiv idual andcomb ined comp onents oficeandwindloads arediscussed inthefollowing. 14.1.5.1 IceLoading Theformation oficeonoverhead conductors maytakeseveral physical forms (glazeice,rime ice,orwet snow) .Theimpact oflowerdensit yiceformation isusually considered inthedesign oflinesections at highaltitudes. Theformation oficeonoverhead condu ctors hasthefollowinginﬂuence onlinedesign: .Iceloads determine themaximum vertical condu ctorloads thatstructures andfoundations must withstand. .Incombinatio nwithsimultaneous windloads, iceloads alsodetermine themaximum transv erse loads onstructures. MEDIUM MEDIUMLIGHT LIGHT LIGHTHEAVY HEAVY FIGURE 14.3 Iceandwindload areasoftheU.S. /C2232006 byTaylor&Francis Group ,LLC. .Inregions ofheavyiceloads, themaximum sags andthepermanent increase insagwithtime (differenc ebetween initial andﬁnal sags) may beduetoiceloadings. Iceloads foruseindesigning lines arenormally derived onthebasis ofpast experience, code requirements, state regulations, andanalysis ofhistorical weather data. Meanrecurr ence interv alsfor heavyiceloadings areafunction oflocal conditions along various routings. The impact ofvarying assumptions concerning iceloading canbeinvestigated withlinedesign software.TABLE 14.1 Deﬁnitions ofIceandWindLoad forNESC Loading Areas Loading Districts Heavy Medium Light Extreme WindLoading Radial thickness ofice (in.) 0.50 0.25 0 0 (mm) 12.5 6.5 0 0 Horizontal wind pressure (lb=ft2) 4 4 9 SeeFig.14.4 (Pa) 190 190 430 Temperatur e (8F) 0 þ15 þ30 þ60 (8C) /C020 /C010 /C01 þ15 Constan ttobeadded tothe resultant forallconduc tors (lb=ft) 0.30 0.20 0.05 0.0 (N=m) 4.40 2.50 0.70 0.0 BASIC WIND SPEED 70 MPH NOTES:GULF OF MEXICO SPECIAL WIND REGION 90908080 7070708080 7070 70 Tacoma Cheyenne LincolnDes MoinesRapid CityBillingsBismarckDuluth Fargo Minneapolis Davenport Chicago Kansas CityColumbusDetroitLansingBuffalo Pittsburgh Richmond Knoxville Birmingham ShreveportLittle RockSt. Louis JacksonJacksonAtlantaRaleighNorfolk Columbia Tampa MiamiNew OrleansPhoenixAmarillo PACIFIC OCEAN ATLANTIC OCEAN80 80 80 80 100110110110 110 11010080 9070100 1100 50 100ALASKA 110110908070 707070 9090 90 100 0 100 200 SCALE 1: 20,000,000300 400 500 MILES 1. VALUES ARE FASTEST-MILE SPEEDS AT 33 FT (10 M) ABOVE GROUND FOR EXPOSURE CATEGORY C AND ARE ASSOCIATED WITH AN ANNUAL PROBABILITY OF 0.02. 2. LINEAR INTERPOLATION BETWEEN WIND SPEED CONTOURS IS ACCEPTABLE. 3. CAUTION IN THE USE OF WIND SPEED CONTOURS IN MOUNTAINOUS REGIONS OF ALASKA IS ADVISED.110Seattle Salt Lake CitySalem Denver Las Vegas San DiegoSan Francisco Fresno Los Angeles 90 80 70Albuquerque Fort WorthOklahoma CityDodge City FIGURE 14.4 Windpressur edesign values intheUnited States. Maximum recordedwindspeed inmiles/hour . (FromOver end, P.R.andSmith, S.,Impulse TimeMethod ofSagMeasurement ,American Societ yofCivilEngineers. Withpermission.) /C2232006 byTaylor &Francis Group ,LLC. The calculation oficeloads oncondu ctors isnormally done withanassumed glazeicedensit yof 57lb=ft3.Theweightoficeperunit length iscalculated withthefollowingequation: wice¼1:244tD cþt ðÞ (14:17) wher et¼thickness ofice,in. Dc¼condu ctoroutside diameter ,in. wice¼resultan tweightofice,lb=ft The ratio oficedweighttobareweightdepends strongly upon condu ctor diameter .Asshown in Table14.2 forthree differ entcondu ctors coveredwith0.5-in radial glazeice,thisratio ranges from4.8 for#1=0AWGto1.6for1590-kcmil condu ctors. Asaresult, small diameter condu ctors may need to haveahigherelastic modulus andhighertensile strength than large condu ctors inheavyiceandwind loading areastolimit sag. 14.1.5.2 Wind Loading Windloadings onoverhead condu ctors inﬂuence linedesign inanumber ofways: .Themaximum span between structures maybedetermined bytheneed forhorizontal clearance toedge ofright-of-wa yduring moderate winds. .The maximum transv erse loads fortangent and small anglesuspension structure sareoften determined byinfrequent highwind-speed loadings. .Permanent increases inconductor sagmaybedetermined bywindloading inareasoflight iceload. Windpressure load oncondu ctors, Pw,iscommonly speciﬁed inlb=ft2.Therelationship between Pw andwindvelocity isgivenbythefollowing equation: Pw¼0:0025( Vw)2(14:18) wher eVw¼thewindspeed inmiles perhour. The windload per unit length ofcondu ctor isequal tothe wind pressure load, Pw, multiplied bytheconductor diameter (including radial iceofthickness t,ifany),isgiven bythe followingequation: Ww¼PwDcþ2t ðÞ 12(14:19) 14.1.5.3 Combined IceandWind Loading Iftheconductor weightistoinclude both iceandwindloading ,theresultant magnitude oftheloads must bedetermined vectorially .Theweightofaconductor under both iceandwindloading isgiven by thefollowingequation: wwþi¼ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ ﬃ wbþwi ðÞ2þWwðÞ2q (14:20)TABLE 14.2 Ratio ofIcedtoBare Condu ctor Weight WbareþWice ACSR Conductor Dc,in. Wbare,lb=ft Wice,lb=ft Wbare #1=0AWG-6=1‘‘Raven’’ 0.398 0.1451 0.559 4.8 477kcmil-26 =7‘‘Hawk’’ 0.858 0.6553 0.845 2.3 1590 kcmil-54 =19‘‘Falcon’’ 1.545 2.042 1.272 1.6 /C2232006 byTaylor&Francis Group ,LLC. wher ewb¼bare conductor weightperunit length, lb=ft wi¼weightoficeperunit length, lb=ft ww¼windload perunit length, lb=ft ww+i¼resultan toficeandwindloads, lb=ft TheNESC prescrib esasafetyfactor ,K,inpounds perfoot, dependent upon loading district, tobe added totheresultant iceandwindloading when performing sagandtension calculations. Theref ore, thetotal resultant condu ctor weight,w,is: w¼wwþiþK (14:21) 14.1.6 Conductor Tension Limits The NESC recommen dslimits onthetension ofbareoverhead condu ctors asapercentage ofthe condu ctor’srated breaking strength. Thetension limits are:60% under maximum iceandwindload, 33.3% initial unloaded (when installed) at608F,and25% ﬁnal unloaded (after maximum loading has occurred)at608F.Itiscommon, howe ver,forlowerunloaded tension limits tobeused. Exceptinareas experiencing sever eiceloading ,itisnotunusual toﬁndtension limits of60% maximum, 25% unloaded initial, and15% unloaded ﬁnal. This setofspeciﬁcations could easily result inanactual maximum tension ontheorder ofonly 35to40%, aninitial tension of20% andaﬁnal unloaded tension level of 15%. Inthiscase, the15% tension limit issaidtogovern. Transmission-line condu ctors arenormally notcoveredwithice,andwinds ontheconductor are usually much lower than those used inmaximum load calculations. Under such everydaycondit ions, tension limits arespeciﬁed tolimit aeolian vibration tosafelevels.Evenwitheverydaylowertension levelsof15to20%, itisassumed thatvibration contr oldeviceswillbeused inthose sections oftheline that aresubject toseverevibration. Aeolianvibration levels,andthus appro priate unloaded tension limits, varywiththetypeofcondu ctor,theterrain, span length, and theuseofdampers. Special condu ctors, such asACSS, SDC, andVR,exhibit highself-damping properties andmay beinstalled tothefullcodelimits, ifdesir ed. 14.2 Approximate Sag-Tension Calculations Sag-tension calculations, using exacting equations, areusually performed withtheaidofacomputer ; howe ver,withcertainsimpliﬁcations, these calculations canbemade withahandheld calculator .The latter approach allows great erinsightinto thecalculation ofsags andtensions than ispossible with compl excompu terprograms. Equations suitable forsuch calculations, aspresented intheprecedin g section, canbeapplied tothefollowing example: Itisdesire dtocalculate thesagandslack fora600-ft level span of795kcmil-26 =7ACSR ‘‘Drake ’’ condu ctor.Thebareconductor weightperunitlength, wb,is1.094 lb=ft.Thecondu ctorisinstalled with ahorizontal tension comp onent, H,of6300 lb,equal to20% ofitsrated breaking strength of31,500 lb. ByuseofEq.(14.2) ,thesagforthislevel span is: D¼1:094(6002) (8)6300¼7:81ft(2:38m) Thelength ofthecondu ctor between thesuppor tpoints isdetermined using Eq.(14.6) : L¼600þ8(7:81)2 3(600)¼600:27ft(182 :96m) /C2232006 byTaylor &Francis Group ,LLC. Notethat theconductor length depends solely onspan and sag.Itisnotdirectlydependent on condu ctortension, weight,ortemperatur e.Thecondu ctorslack isthecondu ctorlength minus thespan length; inthisexample, itis0.27 ft(0.0826 m). 14.2.1 SagChange with Thermal Elongation ACSR andAACconductors elongate withincreasing condu ctortemperatur e.Therateoflinear thermal expansion forthecomposit eACSR conductor islessthan thatoftheAACconductor because thesteel strands intheACSR elongate atappro ximately halftherateofaluminum. Theeffective linear thermal expansion coefﬁcient ofanon-homogenous condu ctor,such asDrak eACSR, may befound fromthe followingequations (Fink andBeatt y): EAS¼EALAAL ATOTAL/C18/C19 þESTAST ATOTAL/C18/C19 (14:22) aAS¼aALEAL EAS/C18/C19AAL ATOTAL/C18/C19 þaSTEST EAS/C18/C19AST ATOTAL/C18/C19 (14:23) wher eEAL ¼Elastic modulus ofaluminum, psi EST ¼Elastic modulus ofsteel, psi EAS ¼Elastic modulus ofaluminum-steel composit e,psi AAL ¼Area ofaluminum strands, squar eunits AST ¼Area ofsteel strands, squar eunits ATOTAL¼Totalcross-se ctional area, square units aAL ¼Aluminum coefﬁcient oflinear thermal expansion, per8F aST ¼Steel coefﬁcient ofthermal elongation, per8F aAS ¼Composite aluminum-steel coefﬁcient ofthermal elongation, per8F The elastic moduli forsolid aluminum wireis10million psiandforsteel wireis30million psi. Theelastic moduli forstranded wireisreduc ed.Themodulus forstranded aluminum isassumed tobe 8.6million psiforallstrandings. Themoduli forthesteel coreofACSR conductors varies withstranding asfollows: .27.5/C2106forsingle-strand core .27.0/C2106for7-strand core .26.5/C2106for19-strand core Using elastic moduli of8.6and 27.0 million psiforaluminum and steel, respectivel y,theelastic modulus forDrake ACSR is: EAS¼(8:6/C2106)0:6247 0:7264/C18/C19 þ(27:0/C2106)0:1017 0:7264/C18/C19 ¼11:2/C2106psi andthecoefﬁcient oflinear thermal expansion is: aAS¼12:8/C210/C068:6/C2106 11:2/C2106/C18/C190:6247 0:7264/C18/C19 þ6:4/C210/C0627:0/C2106 11:2/C2106/C18/C190:1017 0:7264/C18/C19 ¼10:6/C210/C06=/C14F Iftheconductor temperature changes from areferenc etemperature, TREF,toanother temperature ,T, thecondu ctor length, L,changes inprop ortion totheprod uctoftheconductor’ seffective thermal elongation coefﬁcient, aAS,andthechange intemperatur e,T–TREF,asshown below : LT¼LTREF(1þaAS(T/C0TREF)) (14:24) /C2232006 byTaylor&Francis Group ,LLC. Forexample, ifthetemperature oftheDrak e conductor inthepreceding example increases from 608F (158C)to1678F(758C),then thelength at608Fincreases by0.68ft(0.21 m)from600.27 ft(182.96 m)to 600.95 ft(183.17 m): L(167/C14F)¼600:27(1þ(10:6/C210/C06)(167 /C060))¼600:95ft Ignoring forthemoment anychange inlength duetochange intension, thesagat1678F(758C)may becalculated fortheconductor length of600.95 ft(183.17 m)using Eq.(14.8) : D¼ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ ﬃ 3(600)(0 :95) 8r ¼14:62ft Usingarearrangement ofEq.(14.2) ,thisincreased sagisfound to corr espond toadecreased tension of: H¼w(S2) 8D¼1:094(6002) 8(14 :62)¼3367 lb Ifthecondu ctorwereinextensible, thatis,ifithadaninﬁnite modulus ofelasticit y,then these values ofsagand tension foracondu ctor temperature of1678Fwould becorrect. For anyreal con- ductor ,howe ver,theelastic modulus ofthecondu ctor isﬁnite and changes intension dochange thecondu ctor length. Useoftheprecedingcalculation, therefore,willoverstate theincrease insag. 14.2.2 SagChange Due toCombined Thermal andElastic Effects Withmoduli ofelasticit yaround the8.6million psilevel, typical barealuminum andACSR condu ctors elongate about 0.01% forevery1000 psichange intension. Intheprecedin gexample, theincrease in temperature caused anincrease inlength andsagandadecrease intension, buttheeffect oftension change onlength wasignore d. Asdiscussed later,concentric-lay stranded conductors, particularly non-homogenous condu ctors such asACSR, arenotinextensible. Rather ,they exhibit quite complex elastic and plastic behavior. Initial loading ofcondu ctors results inelongation behavior substantially differ entfrom thatcaused by loading manyyearslater.Also,hightension levelscaused byheavy iceandwindloads cause apermanent increase incondu ctor length, affecting subsequent elongation under various condit ions. Accountingforsuchcomplexstress-strainbehaviorusuallyrequiresasophisticated,computer-aided approach.Forillustrationpurposes,however,theeffectofpermanentelongationoftheconductoronsag andtensioncalculationswillbeignoredandasimpliﬁedelasticconductorassumed.Thisidealizedconductor isassumedtoelongatelinearlywithloadandtoundergonopermanentincreaseinlengthregardlessofloading ortemperature.Forsuchaconductor,therelationshipbetweentensionandlengthisasfollows: LH¼LHREF1þH/C0HREF ECA/C18/C19 (14:25) wher eLH¼Length ofcondu ctor under horizontal tension H LHREF¼Length ofconductor under horizontal reference tension HREF EC¼Elastic modulus ofelasticit yofthecondu ctor,psi A¼Cross-sectional area,in.2 Incalculating sagandtension forextensible condu ctors, itisuseful toaddastep totheprecedin g calculation ofsagandtension forelevated temperatur e.This added stepallows aseparation ofthermal elongation andelastic elongation effects, andinvolves thecalculation ofazero tension length, ZTL, at thecondu ctor temperatur eofinterest, Tcdr. /C2232006 byTaylor &Francis Group ,LLC. This ZTL( Tcdr)isthecondu ctorlength attained ifthecondu ctoristaken down from itssuppor tsand laidonthegroun dwithnotension. Byreducing theinitial tension inthecondu ctortozero,theelastic elongation isalsoreduc edtozero,shortening thecondu ctor.It ispossible, then, forthezerotension length tobelessthan thespan length. Consider thepreceding example forDrake ACSR ina600-ft level span. The initial condu ctor temperature is608F,theconductor length is600.27 ft,and EASiscalculated tobe11.2 million psi. Using Eq.(14.25) ,thereduction oftheinitial tension from 6300 lbtozeroyields aZTL (608F)of: ZTL (60/C14F)¼600:27 1 þ0/C06300 (11:2/C2106)0:7264/C18/C19 ¼599:81ft Keeping thetension atzeroand increasing thecondu ctor temperature to1678Fyields apurel y thermal elongation. Thezero tension length at1678Fcanbecalculated using Eq.(14.24) : ZTL (167/C14F)¼599:81/C16 1þ/C16 10:6/C210/C06/C17/C16 167/C060/C17/C17 ¼600:49ft AccordingtoEqs. (14.2) and(14.8) ,thislength corresp onds toasagof10.5 ftandahorizontal tension of4689 lb.Howev er,thislength wascalculated forzerotension andwillelongate elastically under tension. Theactual condu ctorsag-tension determination requir esaprocess ofiteration asfollows: 1.Asdescribed above,thecondu ctor’szerotension length, calculated at1678F(758C),is600.49 ft, sagis10.5 ft,andthehorizontal tension is4689 lb. 2.Because thecondu ctor iselastic, application ofEq.(14.25) shows thetension of4689 lbwill increase thecondu ctorlength from 600.49 ftto: Ll(167/C14F)¼600:49 1 þ4689/C00 0:7264(11 :2/C2106Þ/C18/C19 ¼600:84ft 3.Thesag,D1(167 8F),corresponding tothislength iscalculated using Eq.(14.8): Dl(167/C14F)¼ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ ﬃ 3(600)(0 :84) 8r ¼13:72ft 4.Using Eq.(14.2), thissagyields anew horizontal tension, H1(167 8F),of: H1¼1:094(6002) 8(13 :7)¼3588 lb Anew trial tension istaken astheaverage ofHandH1,andtheprocess isrepeated. Theresults are described inTable 14.3. TABLE 14.3 Interative Solution forIncreased Conductor Temperature Iteration # Length, Ln,ft Sag, Dn,ft Tension, Hn,lb NewTrialTension, lb ZTL 600.550 11.1 4435 — 1 600.836 13.7 35934435þ3593 2¼4014 2 600.809 13.5 36473647þ4014 2¼3831 3 600.797 13.4 36743674þ3831 2¼3753 4 600.792 13.3 37023702þ3753 2¼3727 /C2232006 byTaylor&Francis Group ,LLC. Notethatthebalanc eofthermal andelastic elongation ofthecondu ctoryields anequilibrium tension ofapproximately 3700 lbsandasagof13.3 ft.Thecalculations oftheprevious section, which ignore d elastic effects, results inlower tension, 3440 lb,andagreater sag,14.7 ft. Slack isequal totheexcessofconductor length overspan length. Theprecedin gtable canbereplaced byaplot ofthecatenar yandelastic curvesonagraph ofslack vstension. Thesolution oc curs atthe intersection ofthetwocurves.Figure 14.5 shows thetension versusslack curvesintersecting atatension of3700 lb,which agrees withthepreceding calculations. 14.2.3 SagChange Due toIceLoading Asaﬁnal example ofsag-tension calculation, calculate thesagandtension forthe600-ft Drake span withtheaddition of0.5inches ofradial iceandadrop incondu ctortemperatur e to08F.Employing Eq. (14.17) ,theweightofthecondu ctor increases by: wice¼1:244t(Dþt) wice¼1:244(0 :5)(1 :108þ0:5)¼1:000lb=ft Asintheprevious example, thecalculation uses thecondu ctor’szero tension length at608F,which is thesame asthat found intheprevious section, 599.81 ft.Theiceloading isspeciﬁed foracondu ctor temperature of08F,sotheZTL(0 8F),using Eq.(14.24) ,is: ZTL (0/C14F)¼599:81[1þ(10:6/C210/C06)(0/C060)]¼599:43ft Asinthecaseofsag-tension atelevated temperatur es,theconductor tension isafunction ofslack and elastic elongation. Theconductor tension andtheconductor length arefound atthepoint ofintersec- tion ofthecatenar yandelastic curves(Fig.14.6).Theintersection ofthecurvesoccurs atahorizontal tension component of12,275 lb,notveryfarfrom thecrude initial estimate of12,050 lbthat ignored elastic effects. Thesagcorresponding tothistension andtheiced condu ctor weightperunit length is9.2ft. Inspite ofdoubling thecondu ctor weightperunit length byadding 0.5in.ofice,thesagofthe condu ctor ismuch lessthan thesagat1678F.This condit ionisgenerally true fortransmission condu ctors where minimum ground clearanc eisdetermined bythe hig htemperature rather than the heavyloading condit ion. Small distribution condu ctors, such asthe1=0AWGACSR inTable 14.1, experience amuch larger ice-to-conductor weightratio (4.8), andthecondu ctor sagunder maximum windandiceload may exceedthesagatmoderately highertemperature s.5000 4500 4000 35003700 Ibs ElasticCatenary 3000 2500 2000 0.5 0.75 1.25 1.5 1 Slack / Elongation, ftTension, Ibs FIGURE 14.5 Sag-tension solution for600-ft span ofDrake at1678F. /C2232006 byTaylor &Francis Group ,LLC. Theprecedingappro ximate tension calculations could havebeen more accurate withtheuseofactual stress-strain curvesandgraphic sag-tension solutions, asdescribed indetail inGraphic Method forSag Tension Calculations forACSR and Other Cond uctors (Aluminum Compan yofAmerica, 1961). This method, althoug haccurate, isveryslow andhasbeen replacedcompletely bycompu tational methods. 14.3 Numerical Sag-Tension Calculations Sag-tension calculations arenormally done numerically and allowtheuser toenter manydiffere nt loading and condu ctor temperatur econdit ions. Both initial and ﬁnal conditions arecalculated and multiple tension constraints canbespeciﬁed. The compl exstress-st rain behavior ofACSR-typecon- ductors canbemodeled numerically ,including both temperature, andelastic andplastic effects. 14.3.1 Stress-Strain Curves Stress-strain curvesforbareoverhead conductor include aminimum ofaninitial curveandaﬁnal curve overarange ofelongations from 0to0.45%. Forcondu ctors consisting oftwomaterials, aninitial and ﬁnal curveforeach isincluded. Creep curvesforvarious lengths oftime aretypically included aswell. Overhead conductors arenotpurely elastic. They stretch withtension, butwhen thetension is reduced tozero,they donotreturn totheir initial length. That is,condu ctors areplastic; thechange incondu ctor length cannot beexpressed withasimple linear equation, asforthepreceding hand calculations. Thepermanent length increasethatoccursinoverhead conductors yields thedifferenc ein initial andﬁnal sag-tension data found inmost computer programs. Figur e14.7 shows atypical stress-strain curvefora26=7ACSR conductor (Aluminum Association, 1974); thecurveisvalid forconductor sizes ranging from266.8 to795kcmil. A795kcmil-26 =7ACSR ‘‘Drak e’’condu ctor hasabreakin gstrength of31,500 lb(14,000 kg)andanarea of0.7264 in.2(46.9 mm2)sothatitfails atanaverage stress of43,000 psi(30kg=mm2).Thestress-strain curveillustrates that when thepercentofelongation atastress isequal to50% ofthecondu ctor’sbreaking strength (21,500 psi), theelongation islessthan 0.3% or1.8ft(0.55 m)ina600-ft (180 m)span. Notethat thecompo nent curvesforthesteel coreand thealuminum stranded outer layers are separated. This separation allowsforchanges intherelative curvelocations asthetemperature ofthe condu ctorchanges. Forthepreceding example, withtheDrake condu ctor atatension of6300 lb(2860 kg),thelength oftheconductor inthe600-ft (180 m)span was found tobe0.27 ftlonger than thespan. This tension corresponds toastressof8600 psi(6.05 kg=mm2).Fromthestress-str aincurveinFig.14.7, thiscorresp onds toaninitial elongation of0.105% (0.63 ft).Asintheprecedin ghand calculation, ifthe condu ctor isreduced tozerotension, itsunstressed length would belessthan thespan length.12,275 Ibs ElasticCatenary Slack / Elon gation, ft09000950010000105001100011500120001250013000Tension, Ibs 0.1 0.2 0.3 0.4 0.5 FIGURE 14.6 Sag-tension solution for600-ft span ofDrake at08Fand0.5in.ice. /C2232006 byTaylor&Francis Group ,LLC. Figur e14.8 isastress-st rain curve(Aluminum Association, 1974) foranall-aluminum 37-strand condu ctor ranging insizefrom250kcmil to1033.5 kcmil. Because thecondu ctor ismade entirelyof aluminum, thereisonly oneinitial andﬁnal curve. 14.3.1.1 Permanent Elongation Onceaconductor hasbeen installed ataninitial tension, itcanelongate further.Suchelongation results from twophenomena: permanent elongation duetohightension levelsresulting from iceandwind loads, and creep elongation under everydaytension levels.These typesofcondu ctor elongation are discussed inthefollowingsections. 14.3.1.2 Permanent Elongation Due toHeavy Loading Both Figs. 14.7 and14.8 indicate thatwhen thecondu ctorisinitially installed, itelongates followingthe initial curvethatisnotastraig htline. Ifthecondu ctortension increases toarelatively highlevel under iceandwindloading ,theconductor willelongate. Whenthewindandiceloads abate, thecondu ctor 35,000 30,000 25,000 20,000Stress, psi 15,000 10,000 5,000 0 .1 .2 .3 Unit Strain, %Initial Composite Initial Steel Final SteelFinal AluminumFinal Composite Initial Aluminum 6 Month Creep 1 Year Creep 10 Year Creep Equations for Curves (X = unit strain in %; Y = stress in psi) : Initial composite Initial Steel Initial Aluminum Final Composite Final Steel Final Aluminum 6 Month Creep 1 Year Creep 10 Year Creep.4 .5 : X = 4.07 × 10−3 + (1.28 × 10−5) Y − (1.18 × 10−10) Y2 + (5.64 × 10−15) Y3 Y = −512 + (8.617 × 104) X − (1.18 × 104) X2 − (5.76 × 10−4) X3 : Y = (37.15 × 103) X : Y = −512 = (4.902 × 104) X − (1.18 × 104) X2 − (5.76 × 104) X3 : Y = (107.55 X −17.65) × 103 : Y = (38.60 X −0.65) × 103 : Y = (68.95 X −17.00) × 103 : Y = (68.75 × 103)X : Y = (60.60 × 103)X : Y = (53.45 × 103)X Test Temperature 70 8F to 758F FIGURE 14.7 Stress-strain curvesfor26=7ACSR. /C2232006 byTaylor &Francis Group ,LLC. elongation willreducealong acurveparallel totheﬁnal curve,butthecondu ctorwillneverreturntoits original length. Forexample, refertoFig.14.8 andassume thatanewly strung 795kcmil-37 strand AAC‘‘Arbutus’ ’ condu ctorhasaneverydaytension of2780 lb.Thecondu ctorareais0.6245 in.2,sotheeverydaystressis 4450 psiandtheelongation is0.062%. Following anextreme lyheavyiceandwindload event, assume that thecondu ctor stressreaches18,000 psi.Whenthecondu ctor tension decre asesback toeveryday levels,thecondu ctorelongation willbepermanently increased bymorethan 0.2%. Also thesagunder everydayconditions willbecorresponding lyhigher,andthetension willbeless.Inmost numerical sag- tension methods, ﬁnal sag-tensions arecalculated forsuch permanent elongation duetoheavy loading condit ions. 14.3.1.3 Permanent Elongation atEveryday Tensions (Creep Elongation) Conductors permanently elongate under tension evenifthetension levelneverexceedseverydaylevels. This permanent elongation caused byeverydaytension levels iscalled creep(Aluminum Compan yof America, 1961). Creep canbedetermined bylong-term laborator ycreep tests, theresults ofwhich are used togenerate creep curves.Onstress-str aingraphs, creepcurvesareusually shown for6-mo ,1-yr,and 10-yrperiods. Figure 14.8 shows these typical creep curvesfora37strand 250.0 through1033.5 kcmil AAC.InFig.14.8 assume thattheconductor tension remai nsconstant attheinitial stressof4450 psi.At theintersection ofthisstress level andtheinitial elongation curve,6-month, 1-year,and10-yearcreep 35,000 30,000 25,000 20,000 15,000 10,000 15,000 0 .1 .2 Unit Strain, %Stress, psi 6 Month Creep1 Year Creep10 Year CreepInitial Aluminum Final Aluminum Equations for Curves (X = unit strain in %; Y = stress in psi): Test Temperature 70 8F to 758FX = −5.31 × 10−3 + (1.74 × 10−5) Y−(6.17 × 10−10) Y2 + (5.05 × 10−14) Y3 Y = 136 + (7.46 × 104) X− (8.51 × 104)X2 + (2.33 × 104)X3 Y = (85.20 X −16.14) × 103 Y = (42.30 × 103)X Y = (38.20 × 103)X Y = (30.60 × 103)XInitial Aluminum: Final Aluminum: 6 Month Creep: 1 Year Creep: 10 Year Creep:.3 .4 .5 FIGURE 14.8 Stress-strain curvesfor37-strand AAC. /C2232006 byTaylor&Francis Group ,LLC. curves,thecondu ctorelongation from theinitial elongation of0.062% increases to0.11%, 0.12%, and 0.15%, respectivel y.Because ofcreepelongation, theresulti ngﬁnal sags aregreat erandthecondu ctor tension islessthan theinitial values. Creepelongation inaluminum conductors isquite predi ctable asafunction oftime andobeys a simple exponential relationship .Thus, thepermanent elongation duetocreep ateverydaytension canbe found foranyperiod oftime after initial installation. Creep elongation ofcopperandsteel conductors is much lessandisnormally ignored . Permanentincreaseinconductorlengthduetoheavyloadoccurrencescannotbepredictedatthetime thatalineisbuilt.Thereasonforthisunpredictabilityis thattheoccurrenceofheavyiceandwindisrandom. Aheavyicestormmayoccurthedayafterthelineisbuiltormayneveroccuroverthelifeoftheline. 14.3.2 Sag-Tension Tables Toillustrate theresult oftypical sag-tension calculations, refer toTables 14.4 throu gh14.9 showing initial andﬁnal sag-tension data for795kcmil-26 =7ACSR ‘‘Drak e’’,795kcmil-37 strand AAC‘‘Arbutus’ ’, and795-kcmil Type16‘‘Drak e=SDC’ ’conductors inNESC lightandheavy loading areasforspans of TABLE 14.4 SagandTension Data for795kcmil-26 =7ACSR ‘‘Drak e’’Conductor Span ¼600ft NESC Heavy Loading Distr ict Creepisnotafactor Final Initial Temp,8F Ice,in. Wind,lb=ft2K,lb=ftResultant Weight, lb=ft Sag, ft Tension, lb Sag, ft Tension, lb 0 0.50 4.00 0.30 2.509 11.14 10153 11.14 10153 5415 Al 5415 Al 4738 St 4738 St 32 0.50 0.00 0.00 2.094 44.54 8185 11.09 8512 3819 Al 4343 Al 4366 St 4169 St /C020 0.00 0.00 0.00 1.094 6.68 7372 6.27 7855 3871 Al 4465 Al 3501 St 3390 St 0 0.00 0.00 0.00 1.094 7.56 6517 6.89 7147 3111 Al 3942 Al 3406 St 3205 St 30 0.00 0.00 0.00 1.094 8.98 5490 7.95 6197 2133 Al 3201 Al 3357 St 2996 St 60 0.00 0.00 0.00 1.094 10.44 4725a9.12 5402 1321 Al 2526 Al 3404 St 2875 St 90 0.00 0.00 0.00 1.094 11.87 4157 10.36 4759 634Al 1922 Al 3522 St 2837 St 120 0.00 0.00 0.00 1.094 13.24 3727 11.61 4248 35Al 1379 Al 3692 St 2869 St 167 0.00 0.00 0.00 1.094 14.29 3456 13.53 3649 0Al 626Al 3456 St 3022 St 212 0.00 0.00 0.00 1.094 15.24 3241 15.24 3241 0Al 0Al 3241 St 3239 St aDesign cond ition. /C2232006 byTaylor &Francis Group ,LLC. 1000 and300ft.Typical tension constraints of15% ﬁnal unloaded at608F,25% initial unloaded at608F, and60% initial atmaximum loading areused. Withmost sag-tension calculation methods, ﬁnal sagsarecalculated forboth heavyice=windload and forcreepelongation. Theﬁnal sag-tension values reportedtotheuserarethose withthegreatest increase insag. 14.3.2.1 Initial vs.Final Sags andTensions Rather than calculate thelinesagasafunction oftime, most sag-tension calculations aredetermined based oninitial andﬁnal loading conditions. Initial sagsandtensions aresimply thesagsandtensions at thetime thelineisbuilt. Final sagsandtensions arecalculated if(1)thespeciﬁed iceandwindloading hasoccurred, and (2)thecondu ctor hasexperienc ed10years ofcreepelongation atacondu ctor temperature of608Fattheuser-speciﬁed initial tension.TABLE 14.5 Tension Differenc esinAdjacent Dead-End Spans Conductor: Drak e 795kcmil-26 =7ACSR Span ¼700ft Area¼0.7264 in.2 Creepisafactor NESC HeavyLoading Distr ict Resultant Weight,lb=ftFinal Initial Temp,8F Ice,in.Wind, lb=ft2K,lb=ft Sag,ft Tension, lb Sag, ft Tension, lb 0 0.50 4.00 0.30 2.509 13.61 11318 13.55 11361 32 0.50 0.00 0.00 2.094 13.93 9224 13.33 9643 /C020 0.00 0.00 0.00 1.094 8.22 8161 7.60 8824 0 0.00 0.00 0.00 1.094 9.19 7301 8.26 8115 30 0.00 0.00 0.00 1.094 10.75 6242 9.39 7142 60 0.00 0.00 0.00 1.094 12.36 5429 10.65 6300a 90 0.00 0.00 0.00 1.094 13.96 4809 11.99 5596 120 0.00 0.00 0.00 1.094 15.52 4330 13.37 5020 167 0.00 0.00 0.00 1.094 16.97 3960 15.53 4326 212 0.00 0.00 0.00 1.094 18.04 3728 17.52 3837 aDesign cond ition. Conductor: Drak e 795kcmil-26 =7ACSR Span ¼1000 ft Area¼0.7264 in.2 Creepisnotafactor NESC HeavyLoading Distr ict Resultant Weight,lb=ftFinal Initial Temp,8F Ice,in.Wind, lb=ft2K,lb=ft Sag, ft Tension, lb Sag, ft Tension, lb 0 0.50 4.00 0.30 2.509 25.98 12116 25.98 12116 32 0.50 0.00 0.00 2.094 26.30 9990 25.53 10290 /C020 0.00 0.00 0.00 1.094 18.72 7318 17.25 7940 0 0.00 0.00 0.00 1.094 20.09 6821 18.34 7469 30 0.00 0.00 0.00 1.094 22.13 6197 20.04 6840 60 0.00 0.00 0.00 1.094 24.11 5689 21.76 6300a 90 0.00 0.00 0.00 1.094 26.04 5271 23.49 5839 120 0.00 0.00 0.00 1.094 27.89 4923 25.20 5444 167 0.00 0.00 0.00 1.094 30.14 4559 27.82 4935 212 0.00 0.00 0.00 1.094 31.47 4369 30.24 4544 aDesign cond ition. /C2232006 byTaylor&Francis Group ,LLC. TABLE 14.6 SagandTension Data for795kcmil-26 =7ACSR ‘‘Drak e’’600-ft Ruling Span Conductor: Drak e 795kcmil-26 =7ACSR Span ¼600ft Area¼0.7264 in.2 Creepisnotafactor NESC Heavy Loading District Resultant Weight, lb=ftFinal Initial Temp,8F Ice,in. Wind,lb=ft2K,lb=ft Sag, ft Tension, lb Sag, ft Tension, lb 0 0.50 4.00 0.30 2.509 11.14 10153 11.14 10153 32 0.50 0.00 0.00 2.094 11.54 8185 11.09 8512 /C020 0.00 0.00 0.00 1.094 6.68 7372 6.27 7855 0 0.00 0.00 0.00 1.094 7.56 6517 6.89 7147 30 0.00 0.00 0.00 1.094 8.98 5490 7.95 6197 60 0.00 0.00 0.00 1.094 10.44 4725a9.12 5402 90 0.00 0.00 0.00 1.094 11.87 4157 10.36 4759 120 0.00 0.00 0.00 1.094 13.24 3727 11.61 4248 167 0.00 0.00 0.00 1.094 14.29 3456 13.53 3649 212 0.00 0.00 0.00 1.094 15.24 3241 15.24 3241 aDesign condition . TABLE 14.7 Stringing SagTable for795kcmil-26 =7ACSR ‘‘Drake ’’600-ft Ruling Span 600-ft Ruling Span Contr olling Design Condition: 15% RBS at608F,NoIceorWind,Final NESC HeavyLoad Distr ict Horiz ontal 6493 6193 5910 5645 5397 5166 4952 4753 4569 Tension, lb 20 30 40 50 60 70 80 90 100 Temp,8FSpans Sag, ft-in. Sag, ft-in. Sag, ft-in. Sag, ft-in. Sag, ft-in. Sag, ft-in. Sag,ft-in. Sag, ft-in. Sag, ft-in. 4 0 0 3-4 3-6 3-8 3-11 4 -1 4-3 4-5 4-7 4-9 4 1 0 3-6 3-9 3-1 1 4 -1 4-3 4-5 4-8 4-1 0 5 -0 420 3 -93 -11 4 -14 -34 -64 -84 -10 5 -15 -3 430 3-11 4 -1 4 -3 4 -6 4 -8 4 -11 5 -1 5 -4 5 -6 4 4 0 4-1 4-3 4-6 4-8 4- 1 1 5-2 5-4 5-7 5-10 4 5 0 4-3 4-6 4-8 4-11 5 -2 5-4 5-7 5-1 0 6 -1 4 6 0 4-5 4-8 4-1 1 5 -2 5-4 5-7 5-1 0 6 -1 6-4 4 7 0 4-8 4-11 5 -1 5-4 5-7 5-1 0 6-1 6 -4 6-7 480 4-10 5 -1 5 -4 5 -7 5 -10 6 -1 6 -4 6 -8 6 -11 4 9 0 5-1 5-4 5-7 5-10 6 -1 6-4 6-8 6-1 1 7 -2 5 0 0 5-3 5-6 5-9 6-1 6- 4 6 -7 6- 1 1 7-2 7-6 5 1 0 5-6 5-9 6-0 6-4 6- 7 6 -11 7 -2 7-6 7-9 5 2 0 5-8 6-0 6-3 6-7 6- 1 0 7-2 7-6 7-9 8-1 530 5-11 6 -2 6 -6 6 -10 7 -1 7 -5 7 -9 8 -1 8 -5 5 4 0 6-2 6-5 6-9 7-1 7- 5 7 -9 8- 1 8 -5 8-9 5 5 0 6-4 6-8 7-0 7-4 7- 8 8 -0 8- 4 8 -8 9-1 5 6 0 6-7 6-11 7 -3 7-7 7-1 1 8 -4 8- 8 9 -0 9-5 570 6-10 7 -2 7 -6 7 -10 8 -3 8 -7 9 -0 9 -4 9 -9 5 8 0 7-1 7-5 7-9 8-2 8- 6 8 -11 9 -4 9-8 10-1 590 7-4 7 -8 8 -1 8 -5 8 -10 9 -3 9 -7 10 -0 10 -5 600 7 -77 -11 8 -48 -99 -19 -69 -11 1 0-4 1 0-9 6 1 0 7-1 8-3 8-7 9-0 9- 5 9 -10 1 0-3 1 0-9 11-2 620 8-1 8 -6 8 -11 9 -4 9 -9 10 -2 10 -7 11 -1 11 -6 630 8 - 8-9 9 -2 9 -71 0-11 0-61 1-01 1-51 1-11 640 8-8 9 -1 9 -6 9 -11 10 -5 10 -10 11 -4 11 -9 12 -3 650 8-11 9 -4 9 -9 10 -3 10 -9 11 -2 11 -8 12 -2 12 -8 660 9-2 9 -7 10 -1 10 -7 11 -1 11 -6 12 -0 12 -6 13 -1 670 9-5 9 -11 10 -5 10 -11 11 -5 11 -11 12 -5 12 -11 13 -5 680 9 -91 0-31 0-81 1-21 1-91 2-31 2-91 3-41 3-10 690 10-0 10 -6 11 -0 11 -6 12 -1 12 -7 13 -2 13 -8 14 -3 700 10-4 10 -10 11 -4 11 -11 12 -5 13 -0 13 -6 14 -1 14 -8 /C2232006 byTaylor &Francis Group ,LLC. TABLE 14.8 Time-Sag Table forStopwatch Method Return ofWave Sag, in.3rdTime, sec5thTime, secSag, in.3rdTime, sec5thTime , secSag, in.3rdTime, sec5thTime, secSag, in.3rdTime, sec5thTime, sec 5 1.9 3.2 55 6.4 10.7 105 8.8 14.7 155 10.7 17.9 6 2.1 3.5 56 6.5 10.8 106 8.9 14.8 156 10.8 18.0 7 2.3 3.8 57 6.5 10.9 107 8.9 14.9 157 10.8 18.0 8 2.4 4.1 58 6.6 11.1 109 9.0 15.0 158 10.9 18.1 9 2.6 4.3 59 6.6 11.1 109 9.0 15.0 159 10.9 18.1 10 2.7 4.6 60 6.7 11.1 110 9.1 15.1 160 10.9 18.2 11 2.9 4.8 61 6.7 11.2 111 9.1 15.2 161 11.0 18.2 12 3.0 5.0 62 6.8 11.3 112 9.1 15.2 162 11.0 18.2 13 3.1 5.2 63 6.9 11.4 113 9.2 15.3 163 11.0 18.4 14 3.2 5.4 64 6.9 11.5 114 9.2 15.4 164 11.1 18.4 15 3.3 5.6 65 7.0 11.6 115 9.3 15.4 165 11.1 18.5 16 3.5 5.8 66 7.0 11.7 116 9.3 15.5 166 11.1 18.5 17 3.6 5.9 67 7.1 11.8 117 9.3 15.6 167 11.2 18.6 18 3.7 6.1 68 7.1 11.9 118 9.4 15.6 168 11.2 18.7 19 3.8 6.3 69 7.2 12.0 119 9.4 15.7 169 11.2 18.7 20 3.9 6.4 70 7.2 12.0 120 9.5 15.8 170 11.3 18.8 21 4.0 6.6 71 7.3 12.1 121 9.5 15.8 171 11.3 18.8 22 4.0 6.7 72 7.3 12.2 122 9.5 15.9 172 11.3 18.9 23 4.1 6.9 73 7.4 12.3 123 9.6 16.0 173 11.4 18.9 24 4.2 7.0 74 7.4 12.4 124 9.6 16.0 174 11.4 19.0 25 4.3 7.2 75 7.5 12.5 125 9.7 16.1 175 11.4 19.0 26 4.4 7.3 76 7.5 12.5 126 9.7 16.2 176 11.4 19.1 27 4.5 7.5 77 7.6 12.6 127 9.7 16.2 177 11.5 19.1 28 4.6 7.6 78 7.6 12.7 128 9.8 16.3 178 11.5 19.2 29 4.6 7.7 79 7.7 12.8 129 9.8 16.3 179 11.5 19.3 30 4.7 7.9 80 7.7 12.9 130 9.8 16.4 180 11.6 19.3 31 4.8 8.0 81 7.8 13.0 131 9.9 16.5 181 11.6 19.4 32 4.9 8.1 82 7.8 13.0 132 9.9 16.5 182 11.6 19.4 33 5.0 8.3 83 7.9 13.1 133 10.0 16.6 183 11.7 19.5 34 5.0 8.4 84 7.9 13.2 134 10.0 16.7 184 11.7 19.5 35 5.1 8.5 85 8.0 13.3 135 10.0 16.7 185 11.7 19.6 36 5.2 8.6 86 8.0 13.3 136 10.1 16.8 186 11.8 19.6 37 5.3 8.8 87 8.1 13.4 137 10.1 16.8 187 11.8 19.7 38 5.3 8.9 88 8.1 13.5 138 10.1 16.9 188 11.8 19.7 39 5.4 9.0 89 8.1 13.6 139 10.2 17.0 189 11.9 19.8 40 5.5 9.1 90 8.2 13.7 140 10.2 17.0 190 11.9 19.8 41 5.5 9.2 91 8.2 13.7 141 10.3 17.1 191 11.9 19.9 42 5.6 9.3 92 8.3 13.8 142 10.3 17.1 192 12.0 19.9 43 5.7 9.4 93 8.3 13.9 143 10.3 17.2 193 12.0 20.0 44 5.7 9.5 94 8.4 14.0 144 10.4 17.3 194 12.0 20.0 45 5.8 9.7 95 8.4 14.0 145 10.4 17.3 195 12.1 20.1 46 5.9 9.8 96 8.5 14.1 146 10.4 17.4 196 12.1 20.1 47 5.9 9.9 97 8.5 14.2 147 10.5 17.4 197 12.1 20.2 48 6.0 10.0 98 8.5 14.2 148 10.5 17.5 198 12.1 20.0 49 6.0 10.1 99 8.6 14.3 149 10.5 17.6 199 12.2 20.3 50 6.1 10.2 100 8.6 14.4 150 10.6 17.6 200 12.2 20.3 51 6.2 10.3 101 8.7 14.5 151 10.6 17.7 201 12.2 20.4 52 6.2 10.4 102 8.7 14.5 152 10.6 17.7 202 12.3 20.5 53 6.3 10.5 103 8.8 14.6 153 10.7 17.8 203 12.3 20.5 54 6.3 10.6 104 8.8 14.7 154 10.7 17.9 204 12.3 20.6 Note:Tocalculate thetime ofreturn ofother waves,multiply thetime insecondsforonewavereturn bythenumber ofwave retur nsor,moresimply ,select thecombination ofvalues from thetable thatrepresent sthenumber ofwaveretur nsdesired .For example, thetime ofretur nofthe8thwaveisthesum ofthe3rdand5th,while forthe10th waveitistwice thetime ofthe5th. Theapproximate formula givingtherelationship between sagandtime isgivenas: D¼12:075T N/C18/C192 (inches ) where D¼sag,in. T¼time, sec N¼number ofretur nwavescoun ted /C2232006 byTaylor&Francis Group ,LLC. 14.3.2.2 Special Aspects ofACSR Sag-Tension Calculations Sag-tension calculations withACSR condu ctors aremore compl exthan such calculations withAAC, AAAC,orACAR conductors. Thecompl exityresults fromthediffer entbehaviorofsteel andaluminum strands inresponse totension and temperature. Steel wiresdonot exhibit creep elongation or plastic elongation inrespo nsetohightensions. Aluminum wiresdocreep andrespond plastically to highstress levels.Also,theyelongate twiceasmuch assteel wiresdoinresponse tochanges intemperatur e. Table 14.10 presents various initial andﬁnal sag-tension values fora600-ft span ofaDrake ACSR condu ctor under heavyloading condit ions. Notethatthetension inthealuminum andsteel compon- ents isshownseparately .Inparticular ,some other useful obser vations are: 1.At608F,without iceorwind,thetension levelinthealuminum strands decre aseswithtime asthe strands permanently elongate duetocreep orheavy loading . 2.Both initially andﬁnally ,thetension level inthealuminum strands decre ases withincreasing temperatur ereaching zero tension at2128Fand1678Fforinitial andﬁnal condit ions, respectiv ely. 3.Atthe hig hesttemperatur e(212 8F),wher eallthetension isinthesteel core,theinitial andﬁnal sag-tensions arenearly thesame, illustrating thatthesteel coredoes notpermanently elongate in response totime orhightension.TABLE 14.9 Typical SagandTension Data 795kcmil-26 =7ACSR ‘‘Drak e,’’300- and1000-ft Spans Conductor: Drak e 795kcmil-26 =7ACSR Span ¼300ft Area¼0.7264 in.2 Creepisafactor NESC HeavyLoading Distr ict Weight, lb=ftFinal Initial Temp, 8F Ice,in.Wind, lb=ft2K,lb=ftSag, ftTension, lbSag, ftTension, lb 30 0.00 9.00 0.05 1.424 2.37 6769 2.09 7664 30 0.00 0.00 0.00 1.094 1.93 6364 1.66 7404 60 0.00 0.00 0.00 1.094 2.61 4725a2.04 6033 90 0.00 0.00 0.00 1.094 3.46 3556 2.57 4792 120 0.00 0.00 0.00 1.094 1.00 3077 3.25 3785 167 0.00 0.00 0.00 1.094 4.60 2678 4.49 2746 212 0.00 0.00 0.00 1.094 5.20 2371 5.20 2371 aDesign condition . Conductor: Drak e 795kcmil-26 =7ACSR Span ¼1000 ft Area¼0.7264 in.2 Creepisafactor NESC Heavy Loading District Weight, lb=ftFinal Initial Temp, 8F Ice,in.Wind, lb=ft2K,lb=ftSag, ftTension, lbSag, ftTension, lb 30 0.00 9.00 0.05 1.424 28.42 6290 27.25 6558 30 0.00 0.00 0.00 1.094 27.26 5036 25.70 5339 60 0.00 0.00 0.00 1.094 29.07 4725a27.36 5018 90 0.00 0.00 0.00 1.094 30.82 4460 28.98 4740 120 0.00 0.00 0.00 1.094 32.50 4232 30.56 4498 167 0.00 0.00 0.00 1.094 34.49 3990 32.56 4175 212 0.00 0.00 0.00 1.094 35.75 3851 35.14 3917 aDesign condition . Note:Calculation sbased on:(1)NESC Light Loading District. (2)Tension Limits: a.Initial Loaded –60% RBS @308F; b.Initial Unloaded –25% RBS @608F;c.Final Unloaded –15% RBS @608F. /C2232006 byTaylor &Francis Group ,LLC. 14.4 Ruling Span Concept Transmission lines arenormally designed inlinesections witheach endofthelinesection terminated by astrain structur ethatallows nolongitudinal (along theline) movement ofthecondu ctor(Winkelman, 1959). Structure swithin each linesection aretypically suspension structure sthatsuppor ttheconductor vertically ,butallow freemovement ofthecondu ctor attachment point either longitudinally ortrans- versely. 14.4.1 Tension Differences forAdjacent Dead-End Spans Table14.11 containsinitial andﬁnal sag-tension data fora700-ft anda1000-ft dead-end span when a Drake ACSR conductor isinitially installed tothesame 6300-lb tension limits at608F.Notethat theTABLE 14.10 Typical SagandTension Data 795kcmil-26 =7ACSR ‘‘Drake, ’’300- and1000-ft Spans Conductor: Drak e 795kcmil-26 =7ACSR =SD Span ¼300ft Area¼0.7264 in.2 Creepisafactor NESC HeavyLoading Distr ict Final Initial Temp,8F Ice,in.Wind, lb=ft2K,lb=ftWeight, lb=ft Sag, ftTension, lb Sag,ftTension, lb 0 0.50 4.00 0.30 2.509 2.91 9695 2.88 9802 32 0.50 0.00 0.00 2.094 3.13 7528 2.88 8188 /C020 0.00 0.00 0.00 1.094 1.26 9733 1.26 9756 0 0.00 0.00 0.00 1.094 1.48 8327 1.40 8818 30 0.00 0.00 0.00 1.094 1.93 6364 1.66 7404 60 0.00 0.00 0.00 1.094 2.61 4725a2.04 6033 90 0.00 0.00 0.00 1.094 3.46 3556 2.57 4792 120 0.00 0.00 0.00 1.094 4.00 3077 3.25 3785 167 0.00 0.00 0.00 1.094 4.60 2678 4.49 2746 212 0.00 0.00 0.00 1.094 5.20 2371 5.20 2371 aDesign cond ition. Conductor: Drak e 795kcmil-26 =7ACSR Span ¼1000 ft Area¼0.7264 in.2 Creepisnotafactor NESC HeavyLoading District Final Initial Temp,8F Ice,in.Wind, lb=ft2K,lb=ftWeight, lb=ft Sag, ftTension, lb Sag, ftTension, lb 0 0.50 4.00 0.30 2.509 30.07 10479 30.07 10479 32 0.50 0.00 0.00 2.094 30.56 8607 29.94 8785 /C020 0.00 0.00 0.00 1.094 24.09 5694 22.77 6023 0 0.00 0.00 0.00 1.094 25.38 5406 23.90 5738 30 0.00 0.00 0.00 1.094 27.26 5036 25.59 5362 60 0.00 0.00 0.00 1.094 29.07 4725a27.25 5038 90 0.00 0.00 0.00 1.094 30.82 4460 28.87 4758 120 0.00 0.00 0.00 1.094 32.50 4232 30.45 4513 167 0.00 0.00 0.00 1.094 34.36 4005 32.85 4187 212 0.00 0.00 0.00 1.094 35.62 3865 35.05 3928 aDesign cond ition. Note:Calculation sbased on:(1)NESC HeavyLoading District. (2)Tension Limits: a.Initial Loaded –60% RBS @08F; b.Initial Unloaded –25% RBS @608F;c.Final Unloaded –15% RBS @608F. /C2232006 byTaylor&Francis Group ,LLC. differe ncebetween theinitial andﬁnal limits at608Fisapproximately 460lb.Eventheinitial tension (equal at608F)differs byalmost 900lbat/C0208Fand600lbat1678F. 14.4.2 Tension Equalization bySuspension Insulators Atatypical suspension structure ,thecondu ctor issuppor tedvertically byasuspension insulator assembly ,butallowe dtomovefreelyinthedirect ionofthecondu ctoraxis. This condu ctormovement ispossible duetoinsulator swingalong theconductor axis. Changes inconductor tension between spans, caused bychanges intemperatur e,load, andtime, arenormally equalized byinsulator swing, eliminating horizontal tension differ ences acrosssuspension structure s. 14.4.3 Ruling Span Calculation Sag-tension canbefound foraseries ofsuspension spans inalinesection byuseoftheruling span concept(Ehrenberg ,1935; Winkelm an,1959). Theruling span (RS) forthelinesection isdeﬁned bythe followingequation: RS¼ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ ﬃ S13þS23þ/C1/C1/C1þSn3 S1þS2þ/C1/C1/C1þSns (14:26)TABLE 14.11 Typical SagandTension Data 795kcmil-T ype16ACSR =SD,300- and1000-ft Spans Conductor: Drak e 795kcmil-T ype16ACSR =SD Span ¼300ft Area¼0.7261 in.2 Creepisafactor NESC Heavy Loading District Final Initial Temp,8F Ice,in.Wind, lb=ft2K,lb=ftWeight, lb=ft Sag, ftTension, lb Sag, ftTension, lb 30 0.00 9.00 0.05 1.409 1.59 9980 1.31 12373 30 0.00 0.00 0.00 1.093 1.26 9776 1.03 11976 60 0.00 0.00 0.00 1.093 1.60 7688 1.16 10589a 90 0.00 0.00 0.00 1.093 2.12 5806 1.34 9159 120 0.00 0.00 0.00 1.093 2.69 4572 1.59 7713 167 0.00 0.00 0.00 1.093 3.11 3957 2.22 5545 212 0.00 0.00 0.00 1.093 3.58 3435 3.17 3877 aDesign condition . Conducto r:Drak e 795kcmil-T ype16ACSR =SD Span ¼1000 ft Area¼0.7261 in.2 Creepisafactor NESC HeavyLoading Distr ict Final Initial Temp,8F Ice,in.Wind, lb=ft2K,lb=ftWeight, lb=ft Sag,ftTension, lb Sag, ftTension, lb 30 0.00 9.00 0.05 1.409 17.21 10250 15.10 11676 30 0.00 0.00 0.00 1.093 15.22 8988 12.69 10779 60 0.00 0.00 0.00 1.093 17.21 7950a13.98 9780 90 0.00 0.00 0.00 1.093 19.26 7108 15.44 8861 120 0.00 0.00 0.00 1.093 21.31 6428 17.03 8037 167 0.00 0.00 0.00 1.093 24.27 5647 19.69 6954 212 0.00 0.00 0.00 1.093 25.62 5352 22.32 6136 aDesign condition . Note:Calculation sbased on:(1)NESC Light Loading District. (2)Tension Limits: a.Initial Loaded –60% RBS @308F; b.Initial Unloaded –25% RBS @608F;c.Final Unloaded –15% RBS @608F. /C2232006 byTaylor &Francis Group ,LLC. wher eRS¼Ruling span forthelinesection containing nsuspension spans S1¼Span length ofﬁrstsuspension span S2¼Span length ofsecondsuspension span Sn¼Span length ofnthsuspension span Alternativ ely,agenerally satisfactor ymethod forestimating theruling span istotake thesum oftheaverage suspension span length plus two-thi rdsofthediffer ence between themaximum span and theaverage span. However,some judgment must beexercised inusing thismethod because a large differenc ebetween theaverage andmaximum span maycause asubstantial error intheruling span value. Asdiscussed, suspension spans aresuppor tedbysuspension insulators that arefree tomove inthedirect ionofthecondu ctoraxis. This freedom ofmovement allows thetension ineach suspension span tobeassumed tobethesame andequal tothat calculated fortheruling span. This assumption isvalid forthesuspension spans andruling span under thesame condit ions oftemperatur eandload, forboth initial andﬁnal sags. Forlevelspans, sagineach suspension span isgivenbytheparabolic sagequation: Di¼w(Si2) 8HRS(14:27) wher eDi¼sagintheithspan Si¼span length oftheithspan HRS¼tension from ruling span sag-tension calculations Thesaginlevel suspension spans mayalsobecalculated using theratio: wher eDRS¼saginruling span Suspension spans varyinlength, thoug htypically notoveralarge range. Conductor temperature during sagging varies overarange considerably smaller than thatused forlinedesign purposes. Ifthesaginanysuspension span exceeds appro ximately 5%ofthespan length, acorrection factor should beadded tothesags obtained fromtheaboveequation orthesagshould becalculated using catenar yEq.(14.29). This correction factor may becalculated asfollows: Correction ¼D2w 6H(14:28) wher eD¼sagobtained from parabolic equation w¼weightofcondu ctor,lb=ft H¼horizontal tension, lb Thecatenar yequation forcalculating thesaginasuspension orstringing span is: Sag¼H wcoshSw 2H/C01/C18/C19 (14:29) wher eS¼span length, ft H¼horizontal tension, lb w¼resultant weight,lb=ft 14.4.4 Stringing SagTables Conductors aretypically installed inlinesection lengths consisting ofmultiple spans. Thecondu ctoris pulled fromthecondu ctor reelatapoint near one strain structure progr essing through travelers attached toeach suspension structure toapoint near thenext strain structure .After stringing ,the /C2232006 byTaylor&Francis Group ,LLC. condu ctor tension isincreased until thesaginoneormoresuspension spans reachestheappro priate stringing sagsbased ontheruling span forthelinesection. Thecalculation ofstringing sagsisbased on thepreceding sagequation. Table 14.13 shows atypical stringing sagtable fora600-ft ruling span ofDrak eACSR withsus- pension spans ranging from400to700ftandconductor temperature sof20–100 8F.Allvalues inthis stringing table arecalculated fromruling span initial tensions, shown inTable14.12 using theparabolic sag equation. 14.5 Line Design Sag-Tension Parameters Inlaying outatransmission line, theﬁrststep istosurveytherouteanddrawupaplan-proﬁl eofthe selected right-of-wa y.The plan-pr oﬁle drawings serveanimpor tant function inlinking togetherTABLE 14.12 Typical SagandTension Data 795kcmil-T ype16ACSR =SD,300- and1000-ft Span Conductor: Drak e 795kcmil-T ype16ACSR =SD Span ¼300ft Area¼0.7261 in.2 Creepisafactor NESC Heavy Loading District Final Initial Temp,8F Ice,in.Wind, lb=ft2K,lb=ftWeight, lb=ft Sag, ftTension, lb Sag, ftTension, lb 0 0.50 4.00 0.30 2.486 2.19 12774 2.03 13757 32 0.50 0.00 0.00 2.074 2.25 10377 1.90 12256 /C020 0.00 0.00 0.00 1.093 .91 13477 .87 14156 0 0.00 0.00 0.00 1.093 1.03 11962 .92 13305 30 0.00 0.00 0.00 1.093 1.26 9776 1.03 11976 60 0.00 0.00 0.00 1.093 1.60 7688 1.16 10589a 90 0.00 0.00 0.00 1.093 2.12 5806 1.34 9159 120 0.00 0.00 0.00 1.093 2.69 4572 1.59 7713 167 0.00 0.00 0.00 1.093 3.11 3957 2.22 5545 212 0.00 0.00 0.00 1.093 3.58 3435 3.17 3877 aDesign Condition Conducto r:Drak e 795kcmil-T ype16ACSR =SD Span ¼1000 ft Area¼0.7261 in.2 Creepisafactor NESC HeavyLoading Distr ict Final Initial Temp,8F Ice,in.Wind, lb=ft2K,lb=ftWeight, lb=ft Sag,ftTension, lb Sag, ftTension, lb 0 0.50 4.00 0.30 2.486 20.65 15089 20.36 15299 32 0.50 0.00 0.00 2.074 20.61 12607 19.32 13445 /C020 0.00 0.00 0.00 1.093 12.20 11205 10.89 12552 0 0.00 0.00 0.00 1.093 13.35 10244 11.56 11832 30 0.00 0.00 0.00 1.093 15.22 8988 12.69 10779 60 0.00 0.00 0.00 1.093 17.21 7950a13.98 9780 90 0.00 0.00 0.00 1.093 19.26 7108 15.44 8861 120 0.00 0.00 0.00 1.093 21.31 6428 17.03 8037 167 0.00 0.00 0.00 1.093 24.27 5647 19.69 6954 212 0.00 0.00 0.00 1.093 25.62 5352 22.32 6136 aDesign condition . Note:Calculation sbased on:(1)NESC HeavyLoading District. (2)Tension Limits: a.Initial Loaded –60% RBS @08F; b.Initial Unloaded –25% RBS @608F;Final Unloaded –15% RBS @608F. /C2232006 byTaylor &Francis Group ,LLC. thevarious stages involvedinthedesign andconstr uction oftheline. These drawings,prepar edbased on theroute survey,show thelocation andelevation ofallnatural andman-made obstacles tobetraversed by,oradjac entto,theprop osed line. These plan-pr oﬁles aredrawntoscale andprovidethebasis for towerspotting andlinedesign work. Oncetheplan-pr oﬁle iscompl eted, oneormoreestimated ruling spans forthelinemay beselected. Based onthese estimated ruling spans and themaximum design tensions, sag-tension data may be calculated providing initial andﬁnal sagvalues. From thisdata, sagtemplates maybeconstructed tothe same scale astheplan-pr oﬁle foreach ruling span, andused tographically spot structures. 14.5.1 Catenary Constants Thesaginaruling span isequal totheweightperunit length, w,times thespan length, S,square d, divided by8times thehorizontal compo nent oftheconductor tension, H.The ratio ofconductor horizontal tension, H,toweightperunit length, w,isthecatenar yconstant, H=w.Foraruling span sag- tension calculation using eightloading conditions, atotal of16catenar yconstant values could be deﬁned, oneforinitial andﬁnal tension under each loading condition. Catenar yconstants canbedeﬁned foreach loading condition ofinterestandareused inanyattempt tolocate structure s.Some typical uses ofcatenar yconstants forlocating structure saretoavoidTABLE 14.13 Typical SagandTension Data 795kcmil-37 Strand AAC‘‘Arbutus, ’’300- and1000-ft Spans Conductor: Arbutus 795kcmil-37 Strands AAC Span ¼300ft Area¼0.6245 in.2 Creepisafactor NESC HeavyLoading Distr ict Final Initial Temp,8F Ice,in.Wind, lb=ft2K,lb=ftWeight, lb=ft Sag,ftTension, lb Sag, ftTension, lb 30 0.00 9.00 0.05 1.122 3.56 3546 2.82 4479 30 0.00 0.00 0.00 0.746 2.91 2889 2.06 4075 60 0.00 0.00 0.00 0.746 4.03 2085a2.80 2999 90 0.00 0.00 0.00 0.746 5.13 1638 3.79 2215 120 0.00 0.00 0.00 0.746 6.13 1372 4.86 1732 167 0.00 0.00 0.00 0.746 7.51 1122 6.38 1319 212 0.00 0.00 0.00 0.746 8.65 975 7.65 1101 aDesign cond ition. Conductor: Arbutus 795kcmil-37 Strands AAC Span ¼1000 ft Area¼0.6245 in.2 Creepisafactor NESC HeavyLoading District Final Initial Temp,8F Ice,in.Wind, lb=ft2K,lb=ftWeight, lb=ft Sag, ftTension, lb Sag, ftTension, lb 30 0.00 9.00 0.05 1.122 44.50 3185 42.85 3305 30 0.00 0.00 0.00 0.746 43.66 2158 41.71 2258 60 0.00 0.00 0.00 0.746 45.24 2085a43.32 2175 90 0.00 0.00 0.00 0.746 46.76 2018 44.89 2101 120 0.00 0.00 0.00 0.746 48.24 1958 46.42 2033 167 0.00 0.00 0.00 0.746 50.49 1873 48.72 1939 212 0.00 0.00 0.00 0.746 52.55 1801 50.84 1860 aDesign cond ition. Note:Calculation sbased on:(1)NESC LightLoading District. (2)Tension Limits: a.Initial Loaded –60% RBS @308F; b.Initial Unloaded –25% RBS @608F;c.Final Unloaded –15% RBS @608F. /C2232006 byTaylor&Francis Group ,LLC. overloading ,assure ground clearance issufﬁcient atallpoints along theright-of-way ,andminimize blow outoruplift under coldweather conditions. Todothis,catenar yconstants aretypically found for:(1) themaximum linetemperature ;(2)heavyiceandwindloading; (3)windblow out; and(4)minimum condu ctor temperature .Under anyofthese loading condit ions, thecatenar yconstant allowssag calculation atanypoint withinthespan. 14.5.2 Wind Span Themaximum windspan ofanystructur eisequal tothedistanc emeasur edfromcenter to center ofthe twoadjac entspans suppor tedbyastructure .The windspan isused todetermine themaximum horizontal forceastructure must bedesigned towithstand under highwindcondit ions. Windspan is notdependent oncondu ctor sagortension, only onhorizontal span length. 14.5.3 Weight Span Theweightspan ofastructure isameasur eofthemaximum vertical forceastructure must bedesigned towithstand. Theweight span isequal tothehorizontal distance between thelowpoints andthevertex oftwoadjac entspans. Themaximum weight span forastructur eisdependent ontheloading condit ion being aminimum forheavy iceandwindload. Whentheelevations ofadjac entstructur esarethesame, thewindandweightspans areequal. 14.5.4 Uplift atSuspension Structures Upliftoccurs when theweightspan ofastructure isnegative .Onsteeply inclined spans, thelowpoint ofsag may fallbeyond thelowersuppor t.This indicates that thecondu ctor intheuphill span isexerting a negativ eorupwar dforceonthelower tower.Theamount ofthisupward forceisequal totheweight ofthe condu ctorfrom thelower tower tothelowpoint inthesag.Iftheupward pulloftheuphill span isgreater than thedown ward loadofthenextadjacent span, actual uplift willbecaused andthecondu ctorwillswing freeofthetower.This usually occursunder minimum temperature condit ions andmust bedealt withby adding weightstotheinsulator suspension string orusing astrain structure (Fig.14.9). 14.5.5 Tower Spotting Givensufﬁciently detailed plan-pr oﬁle drawings, structure heights,wind=weightspans, catenar ycon- stants, andminimum groun dclearance s,structure locations canbechosen such thatgroun dclearance isMin. Sag Max. Sag Uplift at Tower Min. Sag Max. Sag FIGURE 14.9 Conductor uplift. /C2232006 byTaylor &Francis Group ,LLC. maintained andstructure loads areacceptable. This processcanbedone byhand using asagtemplate, plan-pr oﬁle drawing,andstructure heights,ornumerically byoneofseveral comm ercial programs. 14.6 Conductor Installation Installation ofabare overhead condu ctor canpresent complex problems. Carefulplanning and a thorou ghunderstanding ofstringing procedures areneeded topreventdamage totheconductor during thestringing operations. The selection ofstringing sheav es,tensioning method, and measuremen t techniques arecritical factors inobtaining thedesir edcondu ctors sagging results .Conductor stringing andsagging equipment andtechniques arediscussed indetail intheIEEE Guide totheInstall ation of Overhead Transmission Line Conducto rs,IEEE Std.524–1992. Some basic factors concerning installation arecoveredinthissection. Because theterminolog yused forequipment andinstallation procedure sfor overhead conductors varies throughout theutilit yindustr y,alimited glossar yofterms andequipment deﬁnitions excerpted fromIEEE Std.524–1992 isprovided inthechapter appendix. Acomplete glossar y ispresented intheIEEE Guide totheInstallation ofOverhead Transmission Line Cond uctors . 14.6.1 Conductor Stringing Methods There aretwobasic methods ofstringing condu ctors, categorized aseither slack ortension stringing . There areasmanyvariations ofthese methods asthereareorganizations installing condu ctors. The selected method, howev er,depends primarily ontheterrain andcondu ctorsurface damage requirements. 14.6.1.1 Slack orLayout Stringing Method Slack stringing ofcondu ctor isnormally limited tolowervoltage lines andsmaller conductors. The condu ctorreel(s) isplaced onreelstands or‘‘jack stands’ ’atthebeginning ofthestringing location. The condu ctor isunreeled from theshipping reelanddragged along theground bymeans ofa vehicle or pulling device.Whenthecondu ctor isdragged past asuppor tingstructure, pulling isstopped andthe condu ctorplacedinstringing sheav esattached tothestructure .Thecondu ctoristhen reattached tothe pulling equipment andthepullcontinued tothenext structure . This stringing method is typically used during construction ofnewlines inareas wher etheright-of-wa y isreadil yaccessible tovehicles used topullthecondu ctor.However,slack stringing maybeused forrepair ormaintenanc eoftransmission lines where rugged terrain limits useofpulling andtensioning equipment. Itisseldom used inurbanareasorwhere thereisanydanger ofcontact withhigh-voltage condu ctors. 14.6.1.2 Tension Stringing Atension stringing method isnormally emplo yedwhen installing transmission conductors. Using this method, theconductor isunreeledunder tension andisnotallowedtocontact thegroun d.In atypical tension stringing operation, travelers areattached toeach structure. Apilot lineispulled throu ghthe travelers and isused, inturn, topull inheavier pulling line. This pulling line isthen used to pullthecondu ctorfrom thereels andthrou ghthetravelers. Tension iscontrolled onthecondu ctor by thetension puller atthepulling endandthebullwheel tension retarderattheconductor payoutendofthe installation. Tension stringing ispreferred foralltransmission installations. This installation method keeps thecondu ctorofftheground, minimizing thepossibilit yofsurface damage andlimiting problems atroadwa ycrossings. Italsolimits damage totheright-of-wa ybyminimizing heavyvehicular trafﬁc. 14.6.2 Tension Stringing Equipment andSetup Stringing equipment typically includes bullwheel ordrum pullers forback-tensioning theconductor during stringing andsagging; travelers(stringing blocks) attached toeveryphase conductor andshield wireattachment point oneverystructure ;abullwheel orcrawler tractor forpulling theconductor throu ghtravelers; and various other special items ofequipment. Figur e14.10 illustrates atypical stringing and sagging setup forastringing section and therange ofstringing equipment requir ed. /C2232006 byTaylor&Francis Group ,LLC. PULL SITE Guy-WHENREQUIRED TRAVELER GROUNDS BOTH SIDES OFENERGIZED CROSSING MID-SPANSPLICE SITE GRID-WHENREQUIRED GUY-WHENREQUIRED GRID-WHEN REQUIRED GUY-WHENREQUIREDPREVIOUSLY SAGGEDCONDUCTORWHENREQUIRED SAG SEC TION TENSION SIT E2 MILESMAXIMUMBETWE ENTRAVELERGROUNDS-142416 1 156 612109819108 9 13 7 16252221128 188 22 1616616 234 177 1617NOTE: CONDUCTORS TO ANCHORS (1) DELETED FOR CLARITY 17 6 201 4 7BC8 AD 322 167SYN. ROPE 1111 FIGURE 14.10 Tension stringing equipment setup . /C2232006 byTaylor &Francis Group ,LLC. Provision forcondu ctorsplicing during stringing must bemade attension siteormidspan sites toavoid pulling splicesthrou ghthetravelers. During thestringing operation, itisnecessarytouseproper tools togripthestrands oftheconductor evenly toavoiddamaging theouter lay erofwires.Twobasic typesorcategories ofgrips arenormally used intransmission construction. Theﬁrstisatypeofgripreferred toasapocketbook ,suitcase, bolted, etc., that hinges to completely surroun dthecondu ctor and incorporates abailforattaching tothe pulling line. Thesecond typeissimilar toaChinese ﬁnger grip andisoften referred toasabasketor ‘‘Kellem ’’grip.Such agrip,shown inFig.14.11, isoften used because ofitsﬂexibilit yandsmall size, making iteasily pulled throu ghsheavesduring thestringing operation. Whatever typeofgripping device isused, aswivelshould beinstalled between thepulling gripandpulling lineorrunning boardtoallow freerotation ofboth thecondu ctorandthepulling line. Atravelerconsistsofasheav eorpulley wheel enclosed inaframe toallowittobesuspended from structure sorinsulator strings. Theframe must havesome typeoflatching mechanism toallowinsert ion andremoval oftheconductor during thestringing operation. Travelers aredesigned foramaximum safe working load. Alwa ysensur ethat thissafe working load willnotbeexceeded during thestringing operation. Sheavesareoften lined withneopr eneorurethane materials topreventscratching ofconduct- orsinhigh-voltage applications; howe ver,unlined sheav esarealsoavailable forspecial applications. FIGURE 14.11 Basket grip pulling device. 100 40 75 30 1.4 3.5 1.2 3 1.0 2.5 0.8 2 0.6 1.5 0.4 150 20 25 1015 1.0 2.51.5 3.75 Conductor Diameter (D c)2.0Sheave Diameter Groove Radius 52.5 (inches) (cm) (inches)(cm)(inches)(cm)Minimum Sheave Diameter (D s) at Base of Groove Minimum Radius (R g) at Base of Groove 6.252535 Rg, 1 or 2 Layer Rg, 3 Layer Min. RgRg, 4 Layer FIGURE 14.12 Recommended minimum sheav edimensions. /C2232006 byTaylor&Francis Group ,LLC. Travelers used intension stringing must befreerolling andcapable ofwithstanding highrunning or static loads without damage. Proper maintenance isessential. Veryhighlongitudinal tension loads can developontransmission structures ifatraveler should ‘‘freeze’’during tension stringing, possibly causing conductor and =orstructur edamage. Signiﬁcant levels ofrotation resist ancewillalso yield tension differ ences between spans, resulting inincorrect sag. Proper selection oftravelers isimport anttoassur ethat travelers operate correctly during tension stringing andsagging .Thesheav ediameter andthegrooveradius must bematched tothecondu ctor. Figur e14.12 illustrates theminimum sheav ediameter fortypical stringing andsagging operations. Larger diameter sheavesmayberequiredwher eparticularly severeinstallation conditions exist. 14.6.3 Sagging Procedure Itisimpor tant that thecondu ctors beprop erlysagged atthecorrectstringing tension forthedesign ruling span. Aseries ofseveralspans, alinesection, isusually sagged inoneoperation. Toobtain the correctsags andtoinsur ethesuspension insulators hang vertically ,thehorizontal tension inallspans must beequal. Figur es14.13 through 14.18 depict typical parabolic methods andcompu tations required Conductor in Travelers Sag Correction (Typ.) See Detail A For Vector Diagram Of Conductor Tension At Traveler See Detail B For Vector Diagram Of Conductor Tension At Suspension ClampConductor in Suspension Clamps “Deadend” Snub Structure (“Zero” Clipping Offset) “Suspension” Snub Structure (“Zero” Clipping Offset)Suspension SuspensionPlump MarkPlump MarkClipping OffsetClipping Offset Suspension Suspension GuysDetail B Detail A H3 H0 H0H5H0H0H4Y2Y1H1 H0 H2 H3H0 (Y2−Y1)H0 V1 V1V1 VT T1V1 VTVVH4 H3 = H4 + W (Y2 − Y1) Stringing Tensions T Are EqualHorizontal Tensions H0 Are Equal Sagging Tensions T & T1 Are UnequalNOTE: W = Conductor Wt Per Unit Len gthVECTOR DIAGRAM FIGURE 14.13 Clipping offset illustration. /C2232006 byTaylor &Francis Group ,LLC. forsagging condu ctors. Factors thatmust beconsider edwhen sagging condu ctors arecreep elongation during stringing andprestr essing ofthecondu ctor. Creepelongation duringstringing :Upon completion ofcondu ctor stringing ,atime ofuptoseveral days mayelapse befor etheconductor istensioned todesign sag.Since thecondu ctortension during the stringing processisnormally wellbelow theinitial sagging tension, andbecause thecondu ctorremains inthestringing sheavesforonly afewdays orless, anyelongation due tocreepisneglected. The400020 25 30 35 40 45 50 60 70 80 902.5 150 200 250 300 350 400 500BC A Formulas for Equivalent Span Length Equiv. Deadend Span = 2C -A Equiv. Suspension Span = A C SAG S3500 3000 2500 2000 1500 10002 3 4 5 10 15 20 5 7.5 10 12.5 15 20 25 37.5 20 62.5 7525 30 40 50 60 80 100 150 200 250 300 For spans between a suspension and deadend tower, use suspension span correction. Example: Assume span with A = 1000 ft, B = 100 ft if deadend span, correction = 10 ft (see above). If suspension span, correction = 2.5 ft (see above). Equivalent span = 1000 ft + correction. Read chart sag for equivalent span length. Sag is based on parabolic functions. If sag exceeds 5% of span, do not use this chart.Sag is based on parabolic functions. If sag exceeds 5% of span, do not use this chart. Deadend Span Suspension SpanHorizontal Spacing of Supports (A)900 800 700 600 500 400 300 200 100∗ ∗Equivalent Span Correction (Add to Horiz. Spacing to Obtain Equivalent Span Length) Vertical Spacing of Supports (B)100 FIGURE 14.14 Nomograph fordetermining levelspan equivalents ofnon-lev elspans. /C2232006 byTaylor&Francis Group ,LLC. condu ctorshould besagged totheinitial stringing sagslisted inthesagtables. However ,ifthecondu ctor tension isexcessiv elyhighduring stringing ,orthecondu ctor isallowe dtoremai nintheblocks foran extended period oftime, then thecreepelongation may become signiﬁcant andthesagging tables should becorrected prior tosagging . Creepisassumed exponential withtime. Thus, condu ctor elongation during theﬁrst dayunder tension isequal toelongation overthenext week. Usingcreep estimation formulas, thecreep strain can beestimated andadjustments made tothestringing sagtables interms ofanequivalent temperatur e. Also,should thisbecome aconcern,South wire’sWireandCable Technolog yGroupwillbehappy to workwithyoutosolve theprobl em. Prest ressing conductor :Prestressing issometimes used tostabilize theelongation ofacondu ctor for some deﬁned period oftime. The prestressi ngtension isnormally much higherthan theunloaded design tension foracondu ctor.Thedegreeofstabilization isdependent upon thetime maintained atthe Procedure Determine from nomograph the control factor of transit "setup" used in sagging the conductor (see examples on the right). For most accurate results in sagging the conductor this value of control factor should not be below the curve shown below. In all cases a control factor of 1.00 is ideal (For T= t).1000Examples B = 60.0 B=60.09 S=49.19T + B B.MTt = 59.12T=40.09 A = 1400.0 (T - t) = 19.12 9 S = 49.1' A = 1400.09 (T − t) = "B" for horizontal line of sightS = 49.1'1 2 3 4 5 6 7 8 9 10 20 30S = 49.1 /H11032Conductor Sag (S)40 50 60 70 80 90 100 200 300 400 500 600 700 800 900 1000Control factor = 0.99 (From nomograph) Control factor = 0.91 (From nomograph) B=60.0 /H11032 T=40.0 /H11032S=49.19T=59.129 A = 1400.0 /H11032 (T − t) = A tan f (+_B) f = Angle of sight. +f = When angle is above horizontal. −f = When angle is below horizontal. B = Vertical distance between points of support +B = When support ahead is higher. −B = When support ahead is lower.φ (Angle of sight) Control factor = 0.99 (From nomograph)Then (T − t) = 1400.0 (+0.02920) − (+60.0) =19.12 Example 1: When sagging by calculated target setting. (See Fig. 2-17) Example 2: When sagging by horizontal line of sight. (See Fig. 2-18) Example 3: When sagging by calculated angle of sight. (See Fig. 2-18) In example, f = +1840' 21" or tan f = +0.02920 A = 1400.0' B = + 60.0' S = 49.1'900 800 700 600 500 400 300 2001.00 .90 .80 .70 020 40 60 80 95Control Factor 99 B T A T = Distance transit is set below conductor support.SS1907030 50100Control Factor 0.1 0.2 0.3 0.4Control Factor Should Not Lie in Shaded Area 0.5 0.6 0.7 B/A100 90 80 70 60 50 40 30± (T − t) 20 10 9 8 7 6 5 4 3 Control Factor = = 1 − = 2 1S1 S∆S1 ∆S(T−t)2 (4S)2= 60.0 /H11032 t = Corresponding distance target is set below opposite support. S = Conductor sag determined from stringing charts. S1 = Corresponding sag of point of tangency of conductor and line of sight. ∆S = Change of sag "s" Sag is based on parabolic functions. If sag exceeds 5% of span, do not use this chart.∆S1 = Change of sag "S1"· FIGURE 14.15 Nomograph fordetermining control factor forconductor sagging . /C2232006 byTaylor &Francis Group ,LLC. prestress tension. After prestr essing ,thetension ontheconductor isreduced tostringing ordesign tension limits. Atthisreduc edtension, thecreeporplastic elongation ofthecondu ctorhasbeen slowe d, reducing thepermanent elongation duetostrain andcreep foradeﬁned period oftime. Bytensioning a condu ctor tolevelsapproaching 50% ofitsbreaking strength fortimes ontheorder ofaday,creep elongation willbetemporarily halted (Cahill, 1973). This simpliﬁes concerns about creep during subsequent installation butpresents both equipment andsafetyproble ms. 14.6.3.1 Sagging byStopwatch Method Amechanical pulse imparte dtoatensioned condu ctormovesataspeed prop ortional tothesquare root oftension divided byweightperunit length. Byinitiating apulse onatensioned condu ctor and measuring thetime requiredforthepulse tomovetotheneare sttermination, thetension, andthusB TS A METHOD 1: Tan f = METHOD 2: Tan f = f = Angle of sight + f When angle is above horizontal − f When angle is below horizontal t = Vertical distance below support to line of sight. ( See Fig . 2-17 ). T = Vertical distance below support for transit. S = Sag A = Horizontal distance between points of support - obtained from structure list or plan & profile B = Vertical distance between points of support - obtained from plan & profile, tower site data sheets or field measurement. + B when support ahead is higher. − B when support ahead is lower. M = Determined from cure on Fig . 2-17.T+_ B − t B + 2T − S(2+M)A Af (Angle of sight)t METHOD 1 METHOD 2 Tan f = Tan f608F = Tan f908F =f608F = f908F = EXAMPLES: Given: A = 1400.0' S = 49.1' @ 60 8F B = +60.0' S = 51.2' @ 90 8F T = 40.0' T = 59.12' @ 60 8F T = 63.76' @ 90 8F Sag is based on parabolic functions. If sa g exceeds 5% of s pan, do not use this chart.T +_ B − t 40.0 − 60.0 − 59.12 1400.0 +18 40'21" f608F =+18 40' 19" f908F =+18 28' 55" +18 28' 59"= 0.02920 = 0.02589ATan f =B + 2T − S (2 + M) A Tan f608F =60.0 + (40.0)(2) − (49.1) (2+0.019) 1400.0= 0.02919 Tan f908F =60.0 + (40.0) (2) − (51.2) (2+0.027) 1400.0= 0.0258740.0 − 60.0 − 63.76 1400.0 Change in angle f for 58F = (18 40' 21" − 18 28' 59") = 0 8 1' 54"5 30( ) Change in angle f for 58 F = (18 40' 19" − 18 28' 55") = 0 8 1' 54"5 30( ) FIGURE 14.16 Conductor sagging bycalculated angleofsight. /C2232006 byTaylor&Francis Group ,LLC. thesagoftheconductor ,canbedetermined. This stopwa tchmethod (Over endandSmith) hascome into wideuse ev enforlong spans andlarge conductors. Thecondu ctorisstruck asharp blow near onesuppor tandthestopwa tchisstartedsimultaneously . Amechanical wavemovesfromthepoint wher ethecondu ctorwasstruck tothenext suppor tpoint at which itwillbepartially reﬂected. Iftheinitiating blow issharp ,thewavewilltravelupanddown the span manytimes before dyingout. Time -sag tables such astheoneshown inTable 14.14 areavailable from manysourc es.Specially designed sagging stopwa tches arealsoavailable. Thereﬂected wavecanbedetected bylightlytouching thecondu ctorbuttheprocedureismore likely tobeaccurate ifthewaveisboth initiated anddetected withalightropeoverthecondu ctor.Normally , thetime forthereturnofthe3rdor5thwaveismonitor ed. Traditionally ,atransit sagging method hasbeen considered tobemoreaccurate forsagging than the stopwa tchmethod. However,manytransmission-line constructors usethestopwa tchmethod exclu- sively,even withlarge conductors.B tS AT METHOD 1: t = (2 S − T)2EXAMPLES METHOD 1 METHOD 2 t = 2S − T + SM T/S608F = 0.815 M608F = 0.019 2S608F = 98.2' t608F = 59.13' T/S908F = 0.781 M908F = 0.027 2S908F = 102.4' t908F = 63.78'Given: A = 1400.0' B = 60.0' T = 40.0' S = 49.1' @ 608F S = 51.2' @ 908F t = (2 S − T)2 T = 6.325 S608F = 7.007 2 S608F = 14.014 2 S908F = 14310 t908F = 63.76'S908F = 7.155t608F = 59.12' 0.14 0.12 0.10 0.08 0.06 0.04 0.02 0.00 0.0 0.6 0.8 1.0 1.2 1.4 1.6 Ratio "R"CURVE FOR DETERMINING VALUE OF "M" For finding value of target setting "t" see Methods 1 & 2, or angle of sight " f" (See Fig. 2-16).Change in "t" for 5 8F = (63.76 − 59.12) = 0.77' For checking value of sag "S" (see Fig. 2-19).Ratio "R" = (T/S). Ratio "R" = (T/t).M = 2 +2(T/S) − 4 T/SFactor "M"METHOD 2: t = 2S − T + SM t = Vertical distance below support for target. T = Vertical distance below support for transit. S = Sag. A = Horizontal distance between structures - obtained from structure list or plan & profile. B = Vertical distance between points of support - obtained from plan & profile, tower site data sheets or field measurement. M = Determined from curve below. M = 2 + 2(T/t) − 4 T/t5 30( ) Change in "t" for 5 8F = (63.76 − 59.13) = 0.78 /H11032 Sag is based on parabolic functions. If sag exceeds 5% of span, do not use this chart.5 30( ) FIGURE 14.17 Conductor sagging bycalculated target method. /C2232006 byTaylor &Francis Group ,LLC. 14.6.3.2 Sagging byTransit Methods IEEE GuideStd.524–1993 liststhreemethods ofsagging condu ctorwithatransit: ‘‘Calculat edAngleof Sight,’’ ‘‘Calculated Target Method, ’’and ‘‘Horizontal Line ofSight.’’The method best suited toa particular linesagging situation mayvarywithterrain andlinedesign.B TS B.M(Level Sight) A T + B T = S (1 − B/4S)2 = SK T = Vertical distance of transit below lower support for taking level sight. A = Horizontal distance between points of support - obtained from structure list of plan & profile. B = Vertical distance between points of support - obtained from plan & profile, tower site data sheets or field measurement. S = Sag. K = (1 −B/4s)2–Determined from curve below. EXAMPLE A = 1400.0' B = 60.0' S = 49.1' @ 60 8F S = 51.2' @ 90 8F B/S = 60.0/49.1 = 1.22 @60 8F B/S = 60.0 / 51.2 = 1.17 @ 90 8F K = 0.482 @ 60 8F K = 0.501 @ 90 8F T = (49.1) (0.482) = 23.66' @ 60 8F T = (51.2) (0.501) = 25.65' @ 90 8F Change in "T" for 5 8F = (25.65 −23.66) = 0.33'5 30 ( )1.0 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0.0 0.0 0.5 1.0 1.5 2.0 Ratio (B/S)"K" Factor 2.5 3.0 3.5 4.0 Sag is based on parabolic functions. If sa g exceeds 5% of s pan, do not use this chart.For most accurate results, use that part of curve drawn in solid line. FIGURE 14.18 Conductor sagging byhorizontal lineofsight. /C2232006 byTaylor&Francis Group ,LLC. 14.6.3.3 Sagging Accuracy Sagging acondu ctorduring construction ofanewlineorintherecondu ctoring ofaoldlineinvolves many variables thatcanleadtoasmall degreeoferror.IEEE Std.524–1993 suggests thatallsagsbewithin 6in.of thestringing sagvalues. However ,aside from measurement error sduring sagging, errorsinterrain measur ement and variations incondu ctor prop erties, loading conditions, and hardw areinstallation haveledsome utilities toallow upto3ftofmargin inaddition totherequiredminimum ground clearanc e. 14.6.3.4 Clipping Offsets Ifthecondu ctor istobesagged inaseries ofsuspension spans wher ethespan lengths arereasona bly close andwher etheterrain isreasonably level, then thecondu ctorissagged using conventional stringing sagtables andthecondu ctor issimply clipped into suspension clamps thatreplac ethetravelers. IftheTABLE 14.14 Typical SagandTension Data 795kcmil-37 Strand AAC‘‘Arbutus, ’’300- and1000-ft Spans Conductor: Arbutus 795kcmil-37 Strands AAC Span ¼300ft Area¼0.6245 in.2 Creepisafactor Final Initial Temp,8F Ice,in.Wind, lb=ft2K,lb=ftWeight, lb=ft Sag, ftTension, lb Sag,ftTension, lb 0 0.50 4.00 0.30 2.125 3.97 6033 3.75 6383 32 0.50 0.00 0.00 1.696 4.35 4386 3.78 5053 /C020 0.00 0.00 0.00 0.746 1.58 5319 1.39 6055 0 0.00 0.00 0.00 0.746 2.00 4208 1.59 5268 30 0.00 0.00 0.00 0.746 2.91 2889 2.06 4075 60 0.00 0.00 0.00 0.746 4.03 2085a2.80 2999 90 0.00 0.00 0.00 0.746 5.13 1638 3.79 2215 120 0.00 0.00 0.00 0.746 6.13 1372 4.86 1732 167 0.00 0.00 0.00 0.746 7.51 1122 6.38 1319 212 0.00 0.00 0.00 0.746 8.65 975 7.65 1101 aDesign condition . Conductor: Arbutus 795kcmil-37 Strands AAC Span ¼1000 ft Area¼0.6245 in.2 Creepisafactor NESC HeavyLoading District Final Initial Temp,8F Ice,in.Wind, lb=ft2K,lb=ftWeight, lb=ft Sag, ftTension, lb Sag, ftTension, lb 0 0.50 4.00 0.30 2.125 45.11 59.53 44.50 6033 32 0.50 0.00 0.00 1.696 45.80 4679 44.68 4794 /C020 0.00 0.00 0.00 0.746 40.93 2300 38.89 2418 0 0.00 0.00 0.00 0.746 42.04 2240 40.03 2350 30 0.00 0.00 0.00 0.746 43.66 2158 41.71 2258 60 0.00 0.00 0.00 0.746 45.24 2085a43.32 2175 90 0.00 0.00 0.00 0.746 46.76 2018 44.89 2101 120 0.00 0.00 0.00 0.746 48.24 1958 46.42 2033 167 0.00 0.00 0.00 0.746 50.49 1873 48.72 1939 212 0.00 0.00 0.00 0.746 52.55 1801 50.84 1860 aDesign condition . Note:Calculation sbased on:(1)NESC Light Loading District. (2)Tension Limits: a.Initial Loaded –60% RBS @08F; b.Initial Unloaded –25% RBS @608F;c.Final Unloaded –15% RBS @608F. /C2232006 byTaylor &Francis Group ,LLC. condu ctor istobesagged inaseries ofsuspension spans wher espan lengths varywidely ormore comm only,where theterrain issteep ,then clipping offsets may need tobeemplo yedinordertoyield vertical suspension strings after installation. Clipping offsets areillustrated inFig.14.19, showing aseries ofsteeply inclined spans terminated ina ‘‘snub’’structure atthebottom anda‘‘deadend’ ’structur eatthetop.Thevector diagram illustrates a balanc eoftotal condu ctor tension inthetravelers butanimbalanc einthehorizontal component of tension.B TS A METHOD 1: S = ( ) METHOD 2: S = METHOD 1 METHOD 2 Note : When using Method 2, value, "T" should lie between 3/4 "S" & 4/3 "S" S = ( ) S = t = 59.12' t/2 = 29.56' T/2 = 20.0 /H11033 M = 0.061 S608F = 20.0 + 29.56 − S608F = 49.1' EXAMPLES Given: A = 1400.0' T = 40.0' B = 60.0' f = +1840'21/H11033 @ 608F (Field Measured)S = Sag t = Vertical distance below support to line of sight. = T +_ B − A tan f when angle f is above horizontal. = T+_ B + A tan f when angle f is below horizontal. T = Vertical distance below support for transit. B = Vertical distance between points of support - obtanied from plan & profile, tower site data sheets or field measurement. + B when support ahead is higher. − B when support ahead is lower. A = Horizontal distance between points of support - obtained from structure list or plan & profile f = Angle of sight M = Determined from cure on Fig. 2.17.ft T + t 2 2 B t tM 2 2 8+ − T + t 2 2B 2t 2tM 8+ − (59.12) (0.061) Sag is based on parabolic functions. if sa g exceeds 5% of span, do not use this chart.t = 40.0 + 60.0 − 1400.0 tan 1 8 40' 21" = 59.12' t = 7.689 T = 6.325 S608F = 49.1'8 FIGURE 14.19 Conductor sagging forchecking sagS. /C2232006 byTaylor&Francis Group ,LLC. 14.7 Deﬁning Terms Block— Adevicedesigned withoneormoresinglesheav es,awoodormetal shell, andanattachment hook orshackle. When rope isreevedthroughtwoofthese devices, theassembly iscommonly referr edtoasablock andtackle .Asetof4sreferstoablock andtackle arrangement utilizing two 4-inch double-sheav eblocks toobtain four load-bearing lines. Similarly ,asetof5sorasetof6srefers tothesame number ofload bearing lines obtained using two5-inch ortwo6-inch double-sheav e blocks, respectivel y. Synonyms: setof4s,setof5s,setof6s. Bullwheel —A wheel incorporated asanintegral partofabullwheel puller ortensioner togenerate pulling orbraking tension onconductors orpulling lines, orboth, throu ghfriction. Apuller or tensioner normally hasoneormorepairs arranged intandem incorporated initsdesign. Thephysical sizeofthewheels willvaryfordifferent designs, but17-in. (43cm)facewidths anddiameters of5ft (150 cm) arecommon. Thewheels arepowerdrivenorretardedandlined withsingle-ormultiple- grooveneopre neorurethane linings. Frictionisaccomplished byreeving thepulling lineorcondu ctor around thegrooveofeach pair. Clipping-in —The transferring ofsagged condu ctors fromthetravelertotheir permanent suspension positions andtheinstalling ofthepermanent suspension clamps. Synonyms :clamping ,clipping. Clipping offset —A calculated distanc e,measur edalong thecondu ctor fromtheplum mark toa point onthecondu ctoratwhich thecenter ofthesuspension clamp istobeplaced.Whenstringing inroughterrain, clipping offset may berequiredtobalanc ethehorizontal forcesoneach suspension structure. Grip,conductor —A devicedesigned topermit thepulling ofcondu ctor without splicing onﬁttings, eyes, etc.Itpermits thepulling ofacontinuous condu ctorwher ethreading isnotpossible. Thedesigns of these grips varyconsiderably .Grips such astheKlein (Chicago) andCrescen tutilize anopen-sided rigid body withopposing jawsandswinglatch. Inaddition topulling condu ctors, thistypeiscommonly used totension guysand, insome cases, pullwirerope.Thedesign ofthecome-along (pock etbook, suitcase, four bolt, etc.) incorporates abailattached tothebody ofaclamp which folds to completely surro und andenvelope theconductor .Bolts arethen used toclose theclamp andobtain agrip. Synonyms :buffalo ,Chicago grip,come-along ,Crescent, four bolt, grip,Klein, pocke tbook, seven bolt, sixbolt, slip-grip ,suitcase. Line, pilot —Alightweig htline, normally synthetic ﬁber rope,used topullheavierpulling lines which in turn areused topull theconductor .Pilot lines maybeinstalled withtheaidofﬁnger lines orby helicopter when theinsulators andtravelers arehung. Synonyms :lead line, leader ,P-line, strawline. Line, pulling —A high-stre ngth line, normally synthetic ﬁber ropeorwirerope,used topull the conductor .Howev er,onreconstruction jobs wher eaconductor isbeing replaced, theoldcondu ctor often servesasthepulling lineforthenewcondu ctor.Insuch cases, theoldconductor must beclosely examined foranydamage prior tothepulling operations. Synonyms :bullline, hard line, lightline, sock line. Puller ,bullwheel —Adevicedesigned topullpulling lines andcondu ctors during stringing operations. Itnormally incorporates oneormorepairs ofurethane- orneopre ne-lined, powe r-driv en,single-or multiple-gr oovebullwheels wher eeach pairisarranged intandem. Pullingisaccomplished byfriction generated against thepulling linewhich isreeved around thegroovesofapairofthebullwheels. The puller isusually equipped withitsown engine which drivesthebullwheels mechanically ,hydraulic ally, orthrou ghacomb ination ofboth. Some ofthese devicesfunction aseither apuller ortensioner . Synonym :puller . Puller ,drum—A devicedesigned topull aconductor during stringing operations. Itisnormally equipped withitsown engine which drivesthedrum mechanically ,hydraulically ,orthrougha combinatio nofboth. Itmaybeequipped withsynthetic ﬁber ropeorwirerope tobeused asthe /C2232006 byTaylor &Francis Group ,LLC. pulling line. Thepulling lineispayedoutfromtheunit, pulled throu ghthetravelersinthesagsection andattached tothecondu ctor.Theconductor isthen pulled inbywinding thepulling lineback onto thedrum. This unit issometimes used withsynthetic ﬁber rope acting asapilot linetopullheavier pulling lines acrosscanyons,rivers,etc. Synonyms :hoist, singledrum hoist, singledrum winch, tugger . Puller ,reel—Adevicedesigned topullacondu ctorduring stringing operations. Itisnormally equipped withitsown engine which drives thesuppor ting shaft forthereelmechanically ,hydraulical ly,or throughacomb ination ofboth. The shaft, inturn, drivesthereel. The application ofthisunit is essentially thesame asthatforthedrum puller previously described. Some ofthese devicesfunction as either apuller ortensioner . Reelstand —A devicedesigned tosuppor toneormorereelsandhaving thepossibilit yofbeing skid, trailer ,ortruck mounted. These devicesmayaccommodate rope orcondu ctor reelsofvaryingsizes andareusually equipped withreelbrake stopreventthereelsfrom turning when pulling isstopped. They areused foreither slack ortension stringing .The designation ofreeltrailer orreeltruck implies that thetrailer ortruck hasbeen equipped withareelstand (jacks) andmayserveasareel transport orpayou tunit, orboth, forstringing operations. Depending upon thesizes ofthereelstobe carried, thetransport ingvehicles mayrange fromsingle-axle trailers tosemi-trucks withtrailers having multiple axles. Synonyms :reeltrailer ,reeltransporte r,reeltruck. Running board —A pulling devicedesigned topermit stringing more than oneconductor simultan- eously withasinglepulling line. Fordistribution stringing, itisusually made oflightweigh ttubing withtheforward endcurvedgently upward toprovidesmooth transition overpole cross-arm rollers. Fortransmission stringing ,thedeviceiseither made ofsections hinged transv ersely tothedirection of pullorofahard-nose rigid design, both having aﬂexible pendulum tailsuspended from therear.This conﬁguration stops thecondu ctors from twisting together andpermits smooth transition overthe sheavesofbundle travelers. Synonyms :alligator ,bird,birdie,monk eytail,sled. Sagsection —The section oflinebetween snub structure s.Morethan onesagsection may berequiredin ordertoproperly sagtheactual length ofcondu ctor which hasbeen strung . Synonyms :pull, setting, stringing section. Site, pull—The location onthelinewher ethepuller ,reelwinder,andanchors (snubs) arelocated. This sitemayalsoserveasthepullortension siteforthenext sagsection. Synonyms :reelsetup ,tugger setup . Site, tension —The location onthelinewhere thetensioner ,reelstands andanchors (snubs) arelocated. This sitemayalsoserveasthepullortension siteforthenext sagsection. Synonyms :condu ctor payoutstation, payoutsite,reelsetup . Snub structure —A structure located atoneendofasagsection andconsidered asazero point for sagging andclipping offset calculations. Thesection oflinebetween twosuch structure sisthesag section, butmorethan onesagsection mayberequir edinordertosagproperly theactual length of conductor which hasbeen strung . Synonyms :0structure, zerostructure . Tensioner ,bullwheel —Adevicedesigned tohold tension against apulling lineorcondu ctorduring the stringing phase. Normally ,itconsistsofoneormorepairs ofurethane- orneopre ne-lined, powe r brake d,single-ormultiple-gr oovebullwheels wher eeach pair isarranged intandem. Tension is accomplished byfriction generated against thecondu ctorwhich isreevedaround thegroovesofapair ofthebullwheels. Some tensioners areequipped withtheir own engines which retard thebullwheels mechanically ,hydraulic ally,orthroughacomb ination ofboth. Some ofthese devicesfunction as either apuller ortensioner .Other tensioners areonly equipped withfriction-t yperetardation. Synonyms :retarder,tensioner . Tensioner ,reel—A devicedesigned togenerate tension against apulling lineorcondu ctor during the stringing phase. Some areequipped withtheir own engines which retardthesuppor ting shaft for /C2232006 byTaylor&Francis Group ,LLC. thereelmechanically ,hydraulical ly,orthroughacombination ofboth. Theshaft, inturn, retardsthe reel. Some ofthese devicesfunction aseither apuller ortensioner .Other tensioners areonly equipped withfriction typeretardatio n. Synonyms :retarder,tensioner . Traveler —Asheav ecompl etewithsuspension armorframe used separately oringroups andsuspended from structure stopermit thestringing ofcondu ctors. These devicesaresometimes bundled witha center drum orsheav e,andanother traveler,andused tostring morethan onecondu ctor simultan- eously .Forprotection ofcondu ctors that should notbenicke dorscratched, thesheav esareoften lined withnonconduc tiveorsemiconduct iveneopre neorwithnonco nductiv eurethane. Anyoneof these materials actsasapadding orcushion fortheconductor asitpasses overthesheave.Traveler grounds must beused withlined travelers inordertoestablish anelectrical ground. Synonyms :block, dolly ,sheav e,stringing block, stringing sheav e,stringing traveler. Winder reel—A devicedesigned toserveasarecoveryunit forapulling line. Itisnormally equipped withitsown engine which drives asuppor tingshaft forareelmechanically ,hydraulical ly,orthrough a combinatio nofboth. Theshaft, inturn, drivesthereel. Itisnormally used torewindapulling lineas itleavesthebullwheel puller during stringing operations. This unitisnotintended toserveasapuller , butsometimes servesthisfunction where only lowtensions areinvolved. Synonyms :take- upreel. References Cahill, T.,Deve lopment ofLow-Cr eepACSR Conductor ,WireJournal,July1973. Ehrenburg ,D.O.,Transmission Line Catenar yCalculations, AIEE Paper,Committee onPower Trans- mission &Distribution, July1935. Fink, D.G.andBeaty ,H.W.,Standard Handbook forElectricalEngineers ,13th ed.,McGraw-Hill. IEEE Guide totheInstallation ofOverhead Transmission Line Cond uctors ,IEEE Standar d524-1993, IEEE, NewYork, 1993. Graphic Method forSagTension Calculations forACSR andOther Conductors ,Aluminum Compan yof America, 1961. Minimum Design Loads forBuilding sandOther Structures, American Societ yofCivilEngineers Stand- ard,ASCE 7–88. National Electrical Safet yCode, 1993 edition. Overend, P.R.andSmith, S.,Impulse TimeMethod ofSagMeasur ement. Stress-St rain-Creep CurvesforAluminum Overhead ElectricalCond uctors, Aluminum Association, 1974. Winkelman, P.F.,Sag-T ension Computations andField Measur ements ofBonnevi llePower Administra- tion, AIEE Paper 59-900, June1959. /C2232006 byTaylor &Francis Group ,LLC.