Electric power can be transmitted or dis

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1 64 64 Principls of Powr Systm CHAPTER CHAPTER Unrgroun Cabls. Unrgroun Cabls. Construction of Cabls.3 Insulating Matrials for Cabls.4 Classification of Cabls.5 Cabls for 3-Phas Srvic.6 Laying of Unrgroun Cabls.7 Insulation Rsistanc of a Singl-Cor Cabl.8 Capacitanc of a Singl-Cor Cabl.9 ilctric Strss in a Singl-Cor Cabl.0 Most Economical Conuctor Siz in a Cabl. Graing of Cabls. Capacitanc Graing.3 Intrshath Graing.4 Capacitanc of 3-Cor Cabls.5 Masurmnts of C an C c.6 Currnt-Carrying Capacity of Unrgroun Cabls.7 Thrmal Rsistanc.8 Thrmal Rsistanc of ilctric of a Singl-Cor Cabl.9 Prmissibl Currnt Loaing.0 Typs of Cabl Faults. Loop Tsts for Location of Faults in Unrgroun Cabls. Murray Loop Tst.3 Varly Loop Tst Introuction Elctric powr can b transmitt or is tribut ithr by ovrha systm or by unrgroun cabls. Th unrgroun cabls hav srvral avantags such as lss liabl to amag through storms or lightning, low maintnanc cost, lss chancs of faults, smallr voltag rop an bttr gnral apparanc. Howvr, thir major rawback is that thy hav gratr installation cost an introuc insulation problms at high voltags compar with th quivalnt ovrha systm. For this rason, unrgroun cabls ar mploy whr it is impracticabl to us ovrha lins. Such locations may b thickly populat aras whr municipal authoritis prohibit ovrha lins for rasons of safty, or aroun plants an substations or whr maintnanc conitions o not prmit th us of ovrha construction. Th chif us of unrgroun cabls for many yars has bn for istribution of lctric powr in congst urban aras at comparativly low or morat voltags. Howvr, rcnt improvmnts in th sign an manufactur hav l to th vlopmnt of cabls suitabl for us at high voltags. This has ma it possibl to mploy unrgroun cabls for transmission of lctric 64

2 65 Unrgroun Cabls 65 powr for short or morat istancs. In this chaptr, w shall focus our attntion on th various aspcts of unrgroun cabls an thir incrasing us in powr systm.. Unrgr groun Cabls An unrgroun cabl ssntially consists of on or mor conuctors covr with suitabl insulation an surroun by a protcting covr. Although svral typs of cabls ar availabl, th typ of cabl to b us will pn upon th working voltag an srvic rquirmnts. In gnral, a cabl must fulfil th following ncssary rquirmnts : (i) Th conuctor us in cabls shoul b tinn stran coppr or aluminium of high conuctivity. Straning is on so that conuctor may bcom flxibl an carry mor currnt. (ii) Th conuctor siz shoul b such that th cabl carris th sir loa currnt without ovrhating an causs voltag rop within prmissibl limits. (iii) Th cabl must hav propr thicknss of insulation in orr to giv high gr of safty an rliability at th voltag for which it is sign. (iv) Th cabl must b provi with suitabl mchanical protction so that it may withstan th rough us in laying it. (v) Th matrials us in th manufactur of cabls shoul b such that thr is complt chmical an physical stability throughout.. Construction of Cabls Fig.. shows th gnral construction of a 3-conuctor cabl. Th various parts ar : (i) Cors or Conuctors. A cabl may hav on or mor than on cor (conuctor) pning upon th typ of srvic for which it is intn. For instanc, th 3-conuctor cabl shown in Fig.. is us for 3-phas srvic. Th conuctors ar ma of tinn coppr or aluminium an ar usually stran in orr to provi flxibility to th cabl. (ii) Insulatian. Each cor or conuctor is provi with a suitabl thicknss of insulation, th thicknss of layr pning upon th voltag to b withstoo by th cabl. Th commonly us matrials for insulation ar imprgnat papr, varnish cambric or rubbr minral compoun. (iii) Mtallic shath. In orr to protct th cabl from moistur, gass or othr amaging liquis (acis or alkalis) in th soil an atmosphr, a mtallic shath of la or aluminium is provi ovr th insulation as shown in Fig.. (iv) Bing. Ovr th mtallic shath is appli a layr of bing which consists of a fibrous matrial lik jut or hssian tap. Th purpos of bing is to protct th mtallic shath against corrosion an from mchanical injury u to armouring. (v) Armouring. Ovr th bing, armouring is provi which consists of on or two layrs of galvanis stl wir or stl tap. Its purpos is to protct th cabl from mchanical injury whil laying it an uring th cours of hanling. Armouring may not b on in th cas of som cabls. (vi) Srving. In orr to protct armouring from atmosphric conitions, a layr of fibrous

3 66 66 Principls of Powr Systm matrial (lik jut) similar to bing is provi ovr th armouring. This is known as srving. It may not b out of plac to mntion hr that bing, armouring an srving ar only appli to th cabls for th protction of conuctor insulation an to protct th mtallic shath from mchanical injury..3 Insulating Matrials for Cabls Th satisfactory opration of a cabl pns to a grat xtnt upon th charactristics of insulation us. Thrfor, th propr choic of insulating matrial for cabls is of consirabl importanc. In gnral, th insulating matrials us in cabls shoul hav th following proprtis : (i) High insulation rsistanc to avoi lakag currnt. (ii) High ilctric strngth to avoi lctrical brakown of th cabl. (iii) High mchanical strngth to withstan th mchanical hanling of cabls. (iv) Non-hygroscopic i.., it shoul not absorb moistur from air or soil. Th moistur tns to cras th insulation rsistanc an hastns th brakown of th cabl. In cas th insulating matrial is hygroscopic, it must b nclos in a watrproof covring lik la shath. (v) Non-inflammabl. (vi) Low cost so as to mak th unrgroun systm a viabl proposition. (vii) Unaffct by acis an alkalis to avoi any chmical action. No on insulating matrial posssss all th abov mntion proprtis. Thrfor, th typ of insulating matrial to b us pns upon th purpos for which th cabl is rquir an th quality of insulation to b aim at. Th principal insulating matrials us in cabls ar rubbr, vulcanis Inia rubbr, imprgnat papr, varnish cambric an polyvinyl chlori. Unrgroun cabl. Rubbr. Rubbr may b obtain from milky sap of tropical trs or it may b prouc from oil proucts. It has rlativ prmittivity varying btwn an 3, ilctric strngth is about 30 kv/mm an rsistivity of insulation is 0 7 Ω cm. Although pur rubbr has rasonably high insulating proprtis, it suffrs form som major rawbacks viz., raily absorbs moistur, maximum saf tmpratur is low (about 38ºC), soft an liabl to amag u to rough hanling an ags whn xpos to light. Thrfor, pur rubbr cannot b us as an insulating matrial.. Vulcanis Inia Rubbr (V.I.R.). It is prpar by mixing pur rubbr with minral mattr such as zin oxi, r la tc., an 3 to 5% of sulphur. Th compoun so form is roll into thin shts an cut into strips. Th rubbr compoun is thn appli to th conuctor an is hat to a tmpratur of about 50ºC. Th whol procss is call vulcanisation an th prouct obtain is known as vulcanis Inia rubbr. Vulcanis Inia rubbr has gratr mchanical strngth, urability an war rsistant proprty than pur rubbr. Its main rawback is that sulphur racts vry quickly with coppr an for this rason, cabls using VIR insulation hav tinn coppr conuctor. Th VIR insulation is gnrally us for low an morat voltag cabls. 3. Imprgnat papr. It consists of chmically pulp papr ma from woo chippings an imprgnat with som compoun such as paraffinic or napthnic matrial. This typ of insulation has almost suprs th rubbr insulation. It is bcaus it has th avantags of low cost, low capacitanc, high ilctric strngth an high insulation rsistanc. Th only isavantag is that papr is hygroscopic an vn if it is imprgnat with suitabl compoun, it absorbs moistur an thus lowrs th insulation rsistanc of th cabl. For this rason, papr insulat cabls ar always

4 67 Unrgroun Cabls 67 provi with som protctiv covring an ar nvr lft unsal. If it is rquir to b lft unus on th sit uring laying, its ns ar tmporarily covr with wax or tar. Sinc th papr insulat cabls hav th tnncy to absorb moistur, thy ar us whr th cabl rout has a *fw joints. For instanc, thy can b profitably us for istribution at low voltags in congst aras whr th joints ar gnrally provi only at th trminal apparatus. Howvr, for smallr installations, whr th lnghts ar small an joints ar rquir at a numbr of placs, VIR cabls will b chapr an urabl than papr insulat cabls. 4. Varnish cambric. It is a cotton cloth imprgnat an coat with varnish. This typ of insulation is also known as mpir tap. Th cambric is lapp on to th conuctor in th form of a tap an its surfacs ar coat with ptrolum jlly compoun to allow for th sliing of on turn ovr anothr as th cabl is bnt. As th varnish cambric is hygroscopic, thrfor, such cabls ar always provi with mtallic shath. Its ilctric strngth is about 4 kv/mm an prmittivity is.5 to Polyvinyl chlori (PVC). This insulating matrial is a synthtic compoun. It is obtain from th polymrisation of actyln an is in th form of whit powr. For obtaining this matrial as a cabl insulation, it is compoun with crtain matrials known as plasticizrs which ar liquis with high boiling point. Th plasticizr forms a gll an rnrs th matrial plastic ovr th sir rang of tmpratur. Polyvinyl chlori has high insulation rsistanc, goo ilctric strngth an mchanical toughnss ovr a wi rang of tmpraturs. It is inrt to oxygn an almost inrt to many alkalis an acis. Thrfor, this typ of insulation is prfrr ovr VIR in xtrm nviormntal conitions such as in cmnt factory or chmical factory. As th mchanical proprtis (i.., lasticity tc.) of PVC ar not so goo as thos of rubbr, thrfor, PVC insulat cabls ar gnrally us for low an mium omstic lights an powr installations..4 Classification of Cabls Cabls for unrgroun srvic may b classifi in two ways accoring to (i) th typ of insulating matrial us in thir manufactur (ii) th voltag for which thy ar manufactur. Howvr, th lattr mtho of classification is gnrally prfrr, accoring to which cabls can b ivi into th following groups : (i) Low-tnsion (L.T.) cabls upto 000 V (ii) High-tnsion (H.T.) cabls upto,000 V (iii) Supr-tnsion (S.T.) cabls from kv to 33 kv (iv) Extra high-tnsion (E.H.T.) cabls from 33 kv to 66 kv (v) Extra supr voltag cabls byon 3 kv A cabl may hav on or mor than on cor pning upon th typ of srvic for which it is intn. It may b (i) singl-cor (ii) two-cor (iii) thr-cor (iv) four-cor tc. For a 3-phas srvic, ithr 3-singl-cor cabls or thr-cor cabl can b us pning upon th oprating voltag an loa man. Fig.. shows th constructional tails of a singl-cor low tnsion cabl. Th cabl has orinary construction bcaus th strsss vlop in th cabl for low voltags (upto 6600 V) ar gnrally small. It consists of on circular cor of tinn stran coppr (or aluminium) insulat by layrs of * Spcial prcautions hav to b takn to prclu moistur at joints. If th numbr of joints is mor, th installation cost incrass rapily an prohibits th us of papr insulat cabls.

5 68 68 Principls of Powr Systm imprgnat papr. Th insulation is surroun by a la shath which prvnts th ntry of moistur into th innr parts. In orr to protct th la shath from corrosion, an ovrall srving of compoun fibrous matrial (jut tc.) is provi. Singl-cor cabls ar not usually armour in orr to avoi xcssiv shath losss. Th principal avantags of singl-cor cabls ar simpl construction an availability of largr coppr sction..5 Cabls for 3-Phas Srvic In practic, unrgroun cabls ar gnrally rquir to livr 3-phas powr. For th purpos, ithr thr-cor cabl or *thr singl cor cabls may b us. For voltags upto 66 kv, 3-cor cabl (i.., multi-cor construction) is prfrr u to conomic rasons. Howvr, for voltags byon 66 kv, 3-cor-cabls bcom too larg an unwily an, thrfor, singl-cor cabls ar us. Th following typs of cabls ar gnrally us for 3-phas srvic :. Blt cabls upto kv. Scrn cabls from kv to 66 kv 3. Prssur cabls byon 66 kv.. Blt cabls. Ths cabls ar us for voltags upto kv but in xtraorinary cass, thir us may b xtn upto kv. Fig..3 shows th constructional tails of a 3-cor blt cabl. Th cors ar insulat from ach othr by layrs of imprgnat papr. Anothr layr of imprgnat papr tap, call papr blt is woun roun th group insulat cors. Th gap btwn th insulat cors is fill with fibrous insulating matrial (jut tc.) so as to giv circular cross-sction to th cabl. Th cors ar gnrally stran an may b of noncircular shap to mak bttr us of availabl spac. Th blt is covr with la shath to protct th cabl against ingrss of moistur an mchanical injury. Th la shath is covr with on or mor layrs of armouring with an outr srving (not shown in th figur). Th blt typ construction is suitabl only for low an mium voltags as th lctrostatic strsss vlop in th cabls for ths voltags ar mor or lss raial i.., across th insulation. Howvr, for high voltags (byon kv), th tangntial strsss also bcom important. Ths strsss act along th layrs of papr insulation. As th insulation rsistanc of papr is quit small along th layrs, thrfor, tangntial strsss st up **lakag currnt along th layrs of papr insulation. Th lakag currnt causs local hating, rsulting in th risk of brakown of insulation at any momnt. In orr to ovrcom this ifficulty, scrn cabls ar us whr lakag currnts ar conuct to arth through mtallic scrns.. Scrn cabls. Ths cabls ar mant for us upto 33 kv, but in particular cass thir us may b xtn to oprating voltags upto 66 kv. Two principal typs of scrn cabls ar H- typ cabls an S.L. typ cabls. (i) H-typ cabls. This typ of cabl was first sign by H. Hochstatr an hnc th nam. Fig..4 shows th constructional tails of a typical 3-cor, H-typ cabl. Each cor is insulat by layrs of imprgnat papr. Th insulation on ach cor is covr with a mtallic scrn which usually consists of a prforat aluminium foil. Th cors ar lai in such a way that mtallic scrns * Sparat singl-cor cabl for ach phas. ** It is infact a lakag currnt but shoul not b confus with th capacitanc currnt.

6 69 Unrgroun Cabls 69 mak contact with on anothr. An aitional conucting blt (coppr wovn fabric tap) is wrapp roun th thr cors. Th cabl has no insulating blt but la shath, bing, armouring an srving follow as usual. It is asy to s that ach cor scrn is in lctrical contact with th conucting blt an th la shath. As all th four scrns (3 cor scrns an on conucting blt) an th la shath ar at arth potntial, thrfor, th lctrical strsss ar purly raial an consquntly ilctric losss ar ruc. Two principal avantags ar claim for H-typ cabls. Firstly, th prforations in th mtallic scrns assist in th complt imprgnation of th cabl with th compoun an thus th possibility of air pockts or vois (vacuous spacs) in th ilctric is liminat. Th vois if prsnt tn to ruc th brakown strngth of th cabl an may caus consirabl amag to th papr insulation. Sconly, th mtallic scrns incras th hat issipating powr of th cabl. H-Typ Cabls (ii) S.L. typ cabls. Fig..5 shows th constructional tails of a 3-cor *S.L. (sparat la) typ cabl. It is basically H-typ cabl but th scrn roun ach cor insulation is covr by its own la shath. Thr is no ovrall la shath but only armouring an srving ar provi. Th S.L. typ cabls hav two main avantags ovr H-typ cabls. Firstly, th sparat shaths minimis th possibility of cor-to-cor brakown. Sconly, bning of cabls bcoms asy u to th limination of ovrall la shath. Howvr, th isavantag is that th thr la shaths of S.L. cabl ar much thinnr than th singl shath of H-cabl an, thrfor, call for gratr car in manufactur. Limitations of soli typ cabls. All th cabls of abov construction ar rfrr to as soli typ cabls bcaus soli insulation is us an no gas or oil circulats in th cabl shath. Th voltag limit for soli typ cabls is 66 kv u to th following rasons : (a) As a soli cabl carris th loa, its conuctor tmpratur incrass an th cabl com- Th four scrns an la shath ar in lctrical contact an la shath is at arth potntial. * In this arrangmnt, ach cor is sparatly la shath an hnc th nam S.L. cabl.

7 70 70 Principls of Powr Systm poun (i.., insulating compoun ovr papr) xpans. This action strtchs th la shath which may b amag. (b) Whn th loa on th cabl crass, th conuctor cools an a partial vacuum is form within th cabl shath. If th pinhols ar prsnt in th la shath, moist air may b rawn into th cabl. Th moistur rucs th ilctric strngth of insulation an may vntually caus th brakown of th cabl. (c) In practic, vois ar always prsnt in th insulation of a cabl. Morn tchniqus of manufacturing hav rsult in voi fr cabls. Howvr, unr oprating conitions, th vois ar form as a rsult of th iffrntial xpansion an contraction of th shath an imprgnat compoun. Th brakown strngth of vois is consirably lss than that of th insulation. If th voi is small nough, th lctrostatic strss across it may caus its brakown. Th vois narst to th conuctor ar th first to brak own, th chmical an thrmal ffcts of ionisation causing prmannt amag to th papr insulation. 3. Prssur cabls For voltags byon 66 kv, soli typ cabls ar unrliabl bcaus thr is a angr of brakown of insulation u to th prsnc of vois. Whn th oprating voltags ar gratr than 66 kv, prssur cabls ar us. In such cabls, vois ar liminat by incrasing th prssur of compoun an for this rason thy ar call prssur cabls. Two typs of prssur cabls viz oil-fill cabls an gas prssur cabls ar commonly us. (i) Oil-fill cabls. In such typs of cabls, channls or ucts ar provi in th cabl for oil circulation. Th oil unr prssur (it is th sam oil us for imprgnation) is kpt constantly suppli to th channl by mans of xtrnal rsrvoirs plac at suitabl istancs (say 500 m) along th rout of th cabl. Oil unr prssur comprsss th layrs of papr insulation an is forc into any vois that may hav form btwn th layrs. u to th limination of vois, oil-fill cabls can b us for highr voltags, th rang bing from 66 kv upto 30 kv. Oil-fill cabls ar of thr typs viz., singl-cor conuctor channl, singl-cor shath channl an thr-cor fillr-spac channls. Fig..6 shows th constructional tails of a singl-cor conuctor channl, oil fill cabl. Th oil channl is form at th cntr by straning th conuctor wir aroun a hollow cylinrical stl spiral tap. Th oil unr prssur is suppli to th channl by mans of xtrnal rsrvoir. As th channl is ma of spiral stl tap, it allows th oil to prcolat btwn coppr strans to th wrapp insulation. Th oil prssur comprsss th layrs of papr insulation an prvnts th possibility of voi formation. Th systm is so sign that whn th oil gts xpan u to incras in cabl tmpratur, th xtra oil collcts in th rsrvoir. Howvr, whn th cabl tmpratur falls uring light loa conitions, th oil from th rsrvoir flows to th channl. Th isavantag of this typ of cabl is that th channl is at th mil of th cabl an is at full voltag w.r.t. arth, so that a vry complicat systm of joints is ncssary. Fig..7 shows th constructional tails of a singlcor shath channl oil-fill cabl. In this typ of cabl, th conuctor is soli similar to that of soli cabl an is papr insulat. Howvr, oil ucts ar provi in th mtallic shath as shown. In th 3-cor oil-fillr cabl shown in Fig..8, th oil ucts ar locat in th fillr spacs. Ths channls ar compos of prforat mtal-ribbon tubing an ar at arth potntial. Vois ar unintntional spacs in th insulation of cabl fill with air or gas, usually at low prssur.

8 7 Unrgroun Cabls 7 Th oil-fill cabls hav thr principal avantags. Firstly, formation of vois an ionisation ar avoi. Sconly, allowabl tmpratur rang an ilctric strngth ar incras. Thirly, if thr is lakag, th fct in th la shath is at onc inicat an th possibility of arth faults is cras. Howvr, thir major isavantags ar th high initial cost an complicat systm of laying. (ii) Gas prssur cabls. Th voltag rquir to st up ionisation insi a voi incrass as th prssur is incras. Thrfor, if orinary cabl is subjct to a sufficintly high prssur, th ionisation can b altogthr liminat. At th sam tim, th incras prssur proucs raial comprssion which tns to clos any vois. This is th unrlying principl of gas prssur cabls. Fig..9 shows th sction of xtrnal prssur cabl sign by Hochstatr, Vogal an Bown. Th construction of th cabl is similar to that of an orinary soli typ xcpt that it is of triangular shap an thicknss of la shath is 75% that of soli cabl. Th triangular sction rucs th wight an givs low thrmal rsistanc but th main rason for triangular shap is that th la shath acts as a prssur mmbran. Th shath is protct by a thin mtal tap. Th cabl is lai in a gas-tight stl pip. Th pip is fill with ry nitrogn gas at to 5 atmosphrs. Th gas prssur proucs raial comprssion an closs th vois that may hav form btwn th layrs of papr insulation. Such cabls can carry mor loa currnt an oprat at highr voltags than a normal cabl. Morovr, maintnanc cost is small an th nitrogn gas hlps in qunching any flam. Howvr, it has th isavantag that th ovrall cost is vry high..6 Laying of Unrgr groun Cabls Th rliability of unrgroun cabl ntwork pns to a consirabl xtnt upon th propr laying an attachmnt of fittings i.., cabl n boxs, joints, branch connctors tc. Thr ar thr main mthos of laying unrgroun cabls viz., irct laying, raw-in systm an th soli systm.. irct laying. This mtho of laying unrgroun cabls is simpl an chap an is much favour in morn practic. In this mtho, a trnch of about 5 mtrs p an 45 cm wi is ug. Th trnch is covr with a layr of fin san (of about 0 cm thicknss) an th cabl is lai ovr this san b. Th san prvnts th ntry of moistur from th groun an thus protcts th cabl from cay. Aftr th cabl has bn lai in th trnch, it is covr with anothr layr of san of about 0 cm thicknss.

9 7 7 Principls of Powr Systm Th trnch is thn covr with bricks an othr matrials in orr to protct th cabl from mchanical injury. Whn mor than on cabl is to b lai in th sam trnch, a horizontal or vrtical intraxial spacing of atlast 30 cm is provi in orr to ruc th ffct of mutual hating an also to nsur that a fault occurring on on cabl os not amag th ajacnt cabl. Cabls to b lai in this way must hav srving of bituminis papr an hssian tap so as to provi protction against corrosion an lctorlysis. Avantags (i) It is a simpl an lss costly mtho. (ii) It givs th bst conitions for issipating th hat gnrat in th cabls. (iii) It is a clan an saf mtho as th cabl is invisibl an fr from xtrnal isturbancs. isavantags (i) Th xtnsion of loa is possibl only by a compltly nw xcavation which may cost as much as th original work. (ii) Th altrations in th cabl ntwok cannot b ma asily. (iii) Th maintnanc cost is vry high. (iv) Localisation of fault is ifficult. (v) It cannot b us in congst aras whr xcavation is xpnsiv an inconvnint. This mtho of laying cabls is us in opn aras whr xcavation can b on convnintly an at low cost.. raw-in systm. In this mtho, conuit or uct of glaz ston or cast iron or concrt ar lai in th groun with manhols at suitabl positions along th cabl rout. Th cabls ar thn pull into position from manhols. Fig.. shows sction through four-way unrgroun uct lin. Thr of th ucts carry transmission cabls an th fourth uct carris rlay protction connction, pilot wirs. Car must b takn that whr th uct lin changs irction ; pths, ips an offsts b ma with a vry long raius or it will b ifficult to pull a larg cabl btwn th manhols. Th istanc btwn th manhols shoul not b too long so as to simplify th pulling in of th cabls. Th cabls to b lai in this way n not b armour but must b provi with srving of hssian an jut in orr to protct thm whn bing pull into th ucts. Avantags (i) Rpairs, altrations or aitions to th cabl ntwork can b ma without opning th groun. (ii) As th cabls ar not armour, thrfor, joints bcom simplr an maintnanc cost is ruc consirably. (iii) Thr ar vry lss chancs of fault occurrnc u to strong mchanical protction provi by th systm. isavantags (i) Th initial cost is vry high. (ii) Th currnt carrying capacity of th cabls is ruc u to th clos grouping of cabls an unfavourabl conitions for issipation of hat. This mtho of cabl laying is suitabl for congst aras whr xcavation is xpnsiv an inconvnint, for onc th conuits hav bn lai, rpairs or altrations can b ma without opn-

10 73 Unrgroun Cabls 73 ing th groun. This mtho is gnrally us for short lngth cabl routs such as in workshops, roa crossings whr frqunt igging is costlir or impossibl. 3. Soli systm. In this mtho of laying, th cabl is lai in opn pips or troughs ug out in arth along th cabl rout. Th troughing is of cast iron, stonwar, asphalt or trat woo. Aftr th cabl is lai in position, th troughing is fill with a bituminous or asphaltic compoun an covr ovr. Cabls lai in this mannr ar usually plain la covr bcaus troughing affors goo mchanical protction. isavantags (i) It is mor xpnsiv than irct lai systm. (ii) It rquirs skill labour an favourabl wathr conitions. (iii) u to poor hat issipation facilitis, th currnt carrying capacity of th cabl is ruc. In viw of ths isavantags, this mtho of laying unrgroun cabls is rarly us now-aays..7 Insulation Rsistanc of a Singl-Cor Cabl Th cabl conuctor is provi with a suitabl thicknss of insulating matrial in orr to prvnt lakag currnt. Th path for lakag currnt is raial through th insulation. Th opposition offr by insulation to lakag currnt is known as insulation rsistanc of th cabl. For satisfactory opration, th insulation rsistanc of th cabl shoul b vry high. Consir a singl-cor cabl of conuctor raius r an intrnal shath raius r as shown in Fig... Lt l b th lngth of th cabl an ρ b th rsistivity of th insulation. Consir a vry small layr of insulation of thicknss x at a raius x. Th lngth through which lakag currnt tns to flow is x an th ara of X-sction offr to this flow is π x l. Insulation rsistanc of consir layr x ρ π xl Insulation rsistanc of th whol cabl is R R r r x ρ ρ x π xl πl x r r ρ log π l r r This shows that insulation rsistanc of a cabl is invrsly proportional to its lngth. In othr wors, if th cabl lngth incrass, its insulation rsistanc crass an vic-vrsa. Exampl.. A singl-cor cabl has a conuctor iamtr of cm an insulation thicknss of 0 4 cm. If th spcific rsistanc of insulation is Ω-cm, calculat th insulation rsistanc for a km lngth of th cabl. Solution Conuctor raius, r / 0 5 cm Lngth of cabl, l km 000 m Rsistivity of insulation, ρ Ω-cm 5 0 Ω-m Intrnal shath raius, r cm

11 74 74 Principls of Powr Systm Insulation rsistanc of cabl is ρ r R log log πl r π Ω 34 M Ω Exampl.. Th insulation rsistanc of a singl-cor cabl is 495 MΩ pr km. If th cor iamtr is 5 cm an rsistivity of insulation is Ω-cm, fin th insulation thicknss. Solution. Lngth of cabl, l km 000 m Cabl insulation rsistanc, R 495 MΩ Ω Conuctor raius, r 5/ 5 cm Rsistivity of insulation, ρ Ω-cm Ωm Lt r cm b th intrnal shath raius. Now, R ρ r log π l r or r 6 π lr log π r ρ or 3 log 0 r /r 0 69 or r /r Antilog 0 69/ 3 or r r 5 5 cm Insulation thicknss r r cm Exampl.3. A singl cor cabl 5 km long has an insulation rsistanc of 0 4 MΩ. Th cor iamtr is 0 mm an th iamtr of th cabl ovr th insulation is 50 mm. Calculat th rsistivity of th insulating matrial. Solution. Lngth of cabl, l 5 km 5000 m Cabl insulation rsistanc, R 0 4 MΩ Ω Conuctor raius, r 0/ 0 mm Intrnal shath raius, r 50/ 5 mm Insulation rsistanc of th cabls is ρ r R log π l r or ρ log 5 π ρ Ωm TUTORIAL PROBLEMS. A singl-cor cabl has a conuctor iamtr of.5 cm an insulation thicknss of. cm. If th spcific rsistanc of insulation is Ω cm, calculat th insulation rsistanc pr kilomtr lngth of th cabl. [305 5 MΩ]. A singl cor cabl 3 km long has an insulation rsistanc of 80 MΩ. If th conuctor iamtr is.5 cm an shath iamtr is 5 cm, calculat th rsistivity of th ilctric in th cabl. [ Ωm] 3. trmin th insulation rsistanc of a singl-cor cabl of lngth 3 km an having conuctor raius 5 mm, insulation thicknss 0 mm an spcific rsistanc of insulation of 5 0 Ωm. [56 MΩ]

12 Unrgroun Cabls 75.8 Capacitanc of a Singl-Cor Cabl A singl-cor cabl can b consir to b quivalnt to two long co-axial cylinrs. Th conuctor (or cor) of th cabl is th innr cylinr whil th outr cylinr is rprsnt by la shath which is at arth potntial. Consir a singl cor cabl with conuctor iamtr an innr shath iamtr (Fig..3). Lt th charg pr mtr axial lngth of th cabl b coulombs an ε b th prmittivity of th insulation matrial btwn cor an la shath. Obviously *ε ε 0 ε r whr ε r is th rlativ prmittivity of th insulation. Consir a cylinr of raius x mtrs an axial lngth mtr. Th surfac ara of this cylinr is π x π x m Elctric flux nsity at any point P on th consir cylinr is x C/m π x Elctric intnsity at point P, E x x ε π x ε π x ε 0 ε volts/m r Th work on in moving a unit positiv charg from point P through a istanc x in th irction of lctric fil is E x x. Hnc, th work on in moving a unit positiv charg from conuctor to shath, which is th potntial iffrnc V btwn conuctor an shath, is givn by : Capacitanc of th cabl is V z/ / z/ / Ex x x πx ε ε πε ε C F m V πε0 ε log r πεoεr log ( / F/m ) π ε r F m 303 log 0 ( / ) ε r 0 F m 4 4 log 0( / ) If th cabl has a lngth of l mtrs, thn capacitanc of th cabl is ε C r l 9 0 F 4 4 log 0 Exampl.4. A singl cor cabl has a conuctor iamtr of cm an intrnal shath iamtr of 8 cm. If imprgnat papr of rlativ prmittivity 4 is us as th insulation, calculat th capacitanc for km lngth of th cabl. Solution. ε Capacitanc of cabl, C r l 9 0 F 4 4 log ( / ) 0 * It may b rcall ε r ε/ε 0 whr ε 0 is th prmittivity of fr spac. In th SI units, ε F/m. 0 r 9 0 r log

13 76 Principls of Powr Systm Hr ε r 4 ; l 000 m 8 cm ; cm Substituting ths valus in th abov xprssion, w gt, C 0 F F µf 4 4 log 0( 8 / ) Exampl.5. Calculat th capacitanc an charging currnt of a singl cor cabl us on a 3-phas, 66 kv systm. Th cabl is km long having a cor iamtr of 0 cm an an imprgnat papr insulation of thicknss 7 cm. Th rlativ prmittivity of th insulation may b takn as 4 an th supply at 50 Hz. Solution. ε Capacitanc of cabl, C r l 9 0 F 4 4 log 0( / ) Hr, ε r 4 ; l 000 m 0 cm; cm Substituting ths valus in th abov xprssion, C 0 F F 0 54 µf 4 4 log 0( 4 / 0) Voltag btwn cor an shath is V ph kv V Charging currnt V ph /X C π f C V ph π A 3.04 A Exampl.6. A 33 kv, 50 Hz, 3-phas unrgroun cabl, 4 km long uss thr singl cor cabls. Each of th conuctor has a iamtr of 5 cm an th raial thicknss of insulation is 0 5 cm. trmin (i) capacitanc of th cabl/phas (ii) charging currnt/phas (iii) total charging kvar. Th rlativ prmittivity of insulation is 3. Solution. ε (i) Capacitanc of cabl/phas, C r l 9 0 F 4 4 log 0( / ) Hr ε r 3 ; l 4 km 4000 m 5 cm ; cm Putting ths valus in th abov xprssion, w gt, 9 C log 0( 3 5 / 5) F (ii) Voltag/phas, V ph V 3 Charging currnt/phas, I C V ph π f C V X ph C π A (iii) Total charging kvar 3V ph I C kvar TUTORIAL PROBLEMS. A singl cor cabl has a conuctor iamtr of cm an intrnal shath iamtr of.8 cm. If th imprgnat papr of rlativ prmittivity 3 is us as insulation, calculat th capacitanc for km lngth of th cabl. [0.8 µf]

14 Unrgroun Cabls 77. Calculat th capacitanc an charging currnt of a singl cor cabl us on 3-phas, 66 kv systm. Th cabl is km long having a cor iamtr of 5 cm an imprgnat papr insulation of thicknss.5 cm. Th rlativ prmittivity of th insulation may b takn as 3.5 an supply at 50 Hz. [0.44 µf ;.74 A] 3. An kv, 50 Hz, singl phas cabl.5 km long, has a iamtr of 0 mm an intrnal shath raius of 5 mm. If th ilctric has a rlativ prmittivity of.4, trmin (i) capacitanc (ii) charging currnt (iii) total charging kvar. [(i) 0.8 µf (ii).84 A (iii) 3.4 kvar].9 ilctric ic Strss ss in a Singl-Cor Cabl Unr oprating conitions, th insulation of a cabl is subjct to lctrostatic forcs. This is known as ilctric strss. Th ilctric strss at any point in a cabl is infact th potntial graint (or *lctric intnsity) at that point. Consir a singl cor cabl with cor iamtr an intrnal shath iamtr. As prov in Art.8, th lctric intnsity at a point x mtrs from th cntr of th cabl is E x volts/m πεo εr x By finition, lctric intnsity is qual to potntial graint. Thrfor, potntial graint g at a point x mtrs from th cntr of cabl is g E x or g volts/m...(i) πεo εr x As prov in Art..8, potntial iffrnc V btwn conuctor an shath is V log πεo ε r volts or πε o ε r V...(ii) log Substituting th valu of from xp. (ii) in xp. (i), w gt, g πεoεr V V volts/m...(iii) log / x log πεo εr x It is clar from xp. (iii) that potntial graint varis invrsly as th istanc x. Thrfor, potntial graint will b maximum whn x is minimum i.., whn x / or at th surfac of th conuctor. On th othr han, potntial graint will b minimum at x / or at shath surfac. Maximum potntial graint is V g max volts/m [Putting x / in xp. (iii)] log Minimum potntial graint is V g min volts/m [Putting x / in xp. (iii)] log * It may b rcall that potntial graint at any point is qual to th lctric intnsity at that point.

15 78 Principls of Powr Systm g g max min V log / V log / Th variation of strss in th ilctric is shown in Fig..4. It is clar that ilctric strss is maximum at th conuctor surfac an its valu gos on crasing as w mov away from th conuctor. It may b not that maximum strss is an important consiration in th sign of a cabl. For instanc, if a cabl is to b oprat at such a voltag that *maximum strss is 5 kv/mm, thn th insulation us must hav a ilctric strngth of atlast 5 kv/mm, othrwis brakown of th cabl will bcom invitabl. Exampl.7. A 33 kv singl cor cabl has a conuctor iamtr of cm an a shath of insi iamtr 4 cm. Fin th maximum an minimum strss in th insulation. Solution. Th maximum strss occurs at th conuctor surfac an its valu is givn by; V g max log Hr, V 33 kv (r.m.s) ; cm ; 4 cm Substituting th valus in th abov xprssion, w gt, 33 g max kv /cm 47 6 kv/cm r.ms. log 4 Th minimum strss occurs at th shath an its valu is giv by ; Altrnativly ; g min V log 33 kv/cm 9 kv/cm r.m.s 4 log 4 g min g max 47 6 /4 9 kv/cm r.m.s. Exampl.8. Th maximum an minimum strsss in th ilctric of a singl cor cabl ar 40 kv/cm (r.m.s.) an 0 kv/cm (r.m.s.) rspctivly. If th conuctor iamtr is cm, fin : (i) thicknss of insulation (ii) oprating voltag Solution. Hr, g max 40 kv/cm ; g min 0 kv/cm ; cm ;? (i) As prov in Art..9, gmax gmin or g max 40 8 cm gmin 0 Insulation thicknss 8 3 cm V (ii) g max log * Of cours, it will occur at th conuctor surfac. Not that unit will b kv/cm. It is bcaus V in th numrator is in kv an in th nominator is in cm.

16 Unrgroun Cabls 79 g log max 40 log 4 V kv kv r.m.s. Exampl.9. A singl cor cabl for us on kv, 50 Hz systm has conuctor ara of cm an intrnal iamtr of shath is 8 cm. Th prmittivity of th ilctric us in th cabl is 3 5. Fin (i) th maximum lctrostatic strss in th cabl (ii) minimum lctrostatic strss in th cabl (iii) capacitanc of th cabl pr km lngth (iv) charging currnt. Solution. Ara of cross-sction of conuctor, a cm 4a iamtr of th conuctor, π cm π Intrnal iamtr of shath, 8 cm (i) Maximum lctrostatic strss in th cabl is V g max kv cm 7 65 kv/cm r.m.s. log log (ii) Minimum lctrostatic strss in th cabl is g min V kv cm 5 kv/cm r.m.s. log 8 8 log ε (iii) Capacitanc of cabl, C r l 9 0 F 4 4 log0 Hr ε r 3.5 ; l km 000 m C F 4 4 log (iv) Charging currnt, I C V X π f C V π A C.0 Most Economical Conuctor Siz in a Cabl It has alray bn shown that maximum strss in a cabl occurs at th surfac of th conuctor. For saf working of th cabl, ilctric strngth of th insulation shoul b mor than th maximum strss. Rwriting th xprssion for maximum strss, w gt, g max V volts/m...(i) log Th valus of working voltag V an intrnal shath iamtr hav to b kpt fix at crtain valus u to sign consirations. This lavs conuctor iamtr to b th only variabl in xp. (i). For givn valus of V an, th most conomical conuctor iamtr will b on for which g max has a minimum valu. Th valu of g max will b minimum whn log / is maximum i.. L log O NM P or log or log (/) 0 or log (/) or (/) 78

17 80 Principls of Powr Systm Most conomical conuctor iamtr is 78. an th valu of g max unr this conition is g max V volts/m [Putting log / in xp. (i)] For low an mium voltag cabls, th valu of conuctor iamtr arriv at by this mtho (i.., V/g max ) is oftn too small from th point of viw of currnt nsity. Thrfor, th conuctor iamtr of such cabls is trmin from th consiration of saf currnt nsity. For high voltag cabls, signs bas on this thory giv a vry high valu of, much too larg from th point of viw of currnt carrying capacity an it is, thrfor, avantagous to incras th conuctor iamtr to this valu. Thr ar thr ways of oing this without using xcssiv coppr : (i) Using aluminium insta of coppr bcaus for th sam currnt, iamtr of aluminium will b mor than that of coppr. (ii) Using coppr wirs stran roun a cntral cor of hmp. (iii) Using a cntral la tub insta of hmp. Exampl.0. Fin th most conomical valu of iamtr of a singl-cor cabl to b us on 50 kv, singl-phas systm. Th maximum prmissibl strss in th ilctric is not to xc 40 kv/cm. Solution. Pak valu of cabl voltag, V kv Maximum prmissibl strss, g max 40 kv/cm (assum pak) Most conomical conuctor iamtr is V cm g max 40 Exampl. Fin th most conomical siz of a singl-cor cabl working on a 3 kv, 3- phas systm, if a ilctric strss of 60 kv/cm can b allow. Solution Phas voltag of cabl 3/ 3 76 kv Pak valu of phas voltag, V kv Max. prmissibl strss, g max 60 kv/cm Most conomical conuctor iamtr is V cm g max 60 Intrnal iamtr of shath, cm Thrfor, th cabl shoul hav a conuctor iamtr of 3.6 cm an intrnal shath iamtr of 9 78 cm.. Graing of Cabls Th procss of achiving uniform lctrostatic strss in th ilctric of cabls is known as graing of cabls. It has alray bn shown that lctrostatic strss in a singl cor cabl has a maximum valu (g max ) at th conuctor surfac an gos on crasing as w mov towars th shath. Th maximum voltag that can b safly appli to a cabl pns upon g max i.., lctrostatic strss at th conuctor surfac. For saf working of a cabl having homognous ilctric, th strngth of i-

18 Unrgroun Cabls 8 lctric must b mor than g max. If a ilctric of high strngth is us for a cabl, it is usful only nar th conuctor whr strss is maximum. But as w mov away from th conuctor, th lctrostatic strss crass, so th ilctric will b unncssarily ovrstrong. Th unqual strss istribution in a cabl is unsirabl for two rasons. Firstly, insulation of gratr thicknss is rquir which incrass th cabl siz. Sconly, it may la to th brakown of insulation. In orr to ovrcom abov isavantags, it is ncssary to hav a uniform strss istribution in cabls. This can b achiv by istributing th strss in such a way that its valu is incras in th outr layrs of ilctric. This is known as graing of cabls. Th following ar th two main mthos of graing of cabls : (i) Capacitanc graing (ii) Intrshath graing. Capacitanc Graing Th procss of achiving uniformity in th ilctric strss by using layrs of iffrnt ilctrics is known as capacitanc graing. In capacitanc graing, th homognous ilctric is rplac by a composit ilctric. Th composit ilctric consists of various layrs of iffrnt ilctrics in such a mannr that rlativ prmittivity ε r of any layr is invrsly proportional to its istanc from th cntr. Unr such conitions, th valu of potntial graint at any point in th iltric is *constant an is inpnnt of its istanc from th cntr. In othr wors, th ilctric strss in th cabl is sam vrywhr an th graing is ial on. How vr, ial graing rquirs th us of an infinit numbr of ilctrics which is an impossibl task. In practic, two or thr ilctrics ar us in th crasing orr of prmittivity ; th ilctric of highst prmittivity bing us nar th cor. Th capacitanc graing can b xplain bautifully by rfrring to Fig..5. Thr ar thr ilctrics of outr iamtr, an an of rlativ prmittivity ε, ε an ε 3 rspctivly. If th prmittivitis ar such that ε > ε > ε 3 an th thr ilctrics ar work at th sam maximum strss, thn, ε ε ε 3 or ε ε ε 3 Potntial iffrnc across th innr layr is * As ε r ε x r k/x whr k is a constant. Potntial graint at a istanc x from th cntr Constant πε0 εr x πε0 ( k / x) x πε0 k This shows that if th conition ε r /x is fulfill, potntial graint will b constant throughout th ilctric of th cabl. g max ; g πε0 ε max ; g πε0 ε 3max πε0 ε3 If g max g max g 3max g max (say), thn, ε ε ε3

19 8 Principls of Powr Systm z/ z / / / V gx πε ε 0 x x gmax log πε0 ε * log πε0 ε Similarly, potntial across scon layr (V ) an thir layr (V 3 ) is givn by ; V g max log V 3 g max log Total p.. btwn cor an arth shath is V V + V + V 3 g max L log + log + log NM If th cabl ha homognous ilctric, thn, for th sam valus of, an g max, th prmissibl potntial iffrnc btwn cor an arth shath woul hav bn V g max log Obviously, V > V i.., for givn imnsions of th cabl, a gra cabl can b work at a gratr potntial than non-gra cabl. Altrnativly, for th sam saf potntial, th siz of gra cabl will b lss than that of non-gra cabl. Th following points may b not : (i) As th prmissibl valus of g max ar pak valus, thrfor, all th voltags in abov xprssions shoul b takn as pak valus an not th r.m.s. valus. (ii) If th maximum strss in th thr ilctrics is not th sam, thn, max 3 V g max log g log g max + + log Th principal isavantag of this mtho is that thr ar a fw high gra ilctrics of rasonabl cost whos prmittivitis vary ovr th rquir rang. Exampl.. A singl-cor la shath cabl is gra by using thr ilctrics of rlativ prmittivity 5, 4 an 3 rspctivly. Th conuctor iamtr is cm an ovrall iamtr is 8 cm. If th thr ilctrics ar work at th sam maximum strss of 40 kv/cm, fin th saf working voltag of th cabl. What will b th valu of saf working voltag for an ungra cabl, assuming th sam conuctor an ovrall iamtr an th maximum ilctric strss? Solution. Hr, cm ;? ;? ; 8 cm ε 5 ; ε 4 ; ε 3 3 ; g max 40 kv/cm Gra cabl. As th maximum strss in th thr ilctrics is th sam, ε ε ε 3 or cm an 3 34 cm O P g max * g max πε0 ε g max πε ε 0 or g max πε0 ε

20 Unrgroun Cabls 83 Prmissibl pak voltag for th cabl L NM L NM g max log + log + log 40 log log + log [ ] kv kv Saf working voltag (r.m.s.) for cabl kv Ungra cabl. Prmissibl pak voltag for th cabl g max 40 log log 8 kv kv Saf working voltag (r.m.s.) for th cabl kv This xampl shows th utility of graing th cabl. Thus for th sam conuctor iamtr () an th sam ovrall imnsion (), th gra cabl can b oprat at a voltag ( ) 8 64 kv (r.m.s.) highr than th homognous cabl an incras of about 47%. Exampl.3. A singl cor la shath cabl has a conuctor iamtr of 3 cm; th iamtr of th cabl bing 9 cm. Th cabl is gra by using two ilctrics of rlativ prmittivity 5 an 4 rspctivly with corrsponing saf working strsss of 30 kv/cm an 0 kv/cm. Calculat th raial thicknss of ach insulation an th saf working voltag of th cabl. Solution. Hr, 3 cm ;? ; 9cm ε 5 ; ε 4 g max 30 kv/cm ; g max 0 kv/cm g max ε ; g max ε g max ε g max ε max ε ε or g g max cm 4 Raial thicknss of innr ilctric cm Raial thicknss of outr ilctric Prmissibl pak voltag for th cabl cm g max g max log + log 30 3log log kv O P O P

21 84 Principls of Powr Systm Saf working voltag (r.m.s.) for th cabl 54.7/ kv Exampl.4. A 66-kV singl-cor la shath cabl is gra by using two ilctrics of rlativ prmittivity 5 an 3 rspctivly; thicknss of ach bing cm. Th cor iamtr is cm. trmin th maximum strss in th two ilctrics. Solution. Fig..6 shows th composit ilctric of a capacitanc gra cabl. Th potntial iffrnc V btwn conuctor an arth shath is givn by ; z / / z / / z/ / V g x + g x z/ / x x + πε ε πε ε x x log + log...(i) πε0 ε ε Now, g max...(ii) πε0 ε Putting th valu of g max π ε 0 ε from xp. (ii) in xp. (i), w gt, V g ε max L O log + log NM ε ε P or g max V L ε O log + log ε NM L NM 0 Hr, cm, 4 cm, 6 cm ; V kv, ε 5, ε 3 3 Substituting th valus, w gt, 53 9 g max kv/cm [log 4+ 53log 64] kv/cm [ ] Similarly, it can b *prov that : g max V Lε O NM ε log + log P 539. kv/cm 435 [ log 4+ log 64] kv/cm 4[ ].3 Intrshath Graing P 0 O P...(iii) In this mtho of cabl graing, a homognous ilctric is us, but it is ivi into various layrs by placing mtallic intrshaths btwn th cor an la shath. Th intrshaths ar hl at suitabl potntials which ar inbtwn th cor potntial an arth potntial. This arrangmnt im- * g max. Putting th valu of g πε0 ε max π ε 0 ε in xp. (i), w gt th xp. (iii).

22 Unrgroun Cabls 85 provs voltag istribution in th ilctric of th cabl an consquntly mor uniform potntial graint is obtain. Consir a cabl of cor iamtr an outr la shath of iamtr. Suppos that two intrshaths of iamtrs an ar insrt into th homognous ilctric an maintain at som fix potntials. Lt V, V an V 3 rspctivly b th voltag btwn cor an intrshath, btwn intrshath an an btwn intrshath an outr la shath. As thr is a finit potntial iffrnc btwn th innr an outr layrs of ach intrshath, thrfor, ach shath can b trat lik a homognous singl cor cabl. As prov in Art..9, Maximum strss btwn cor an intrshath is V g max log V Similarly, g max log V g 3max 3 log Sinc th ilctric is homognous, th maximum strss in ach layr is th sam i.., g max g max g 3max g max (say) V V V 3 log log log As th cabl bhavs lik thr capacitors in sris, thrfor, all th potntials ar in phas i.. Voltag btwn conuctor an arth la shath is V V + V + V 3 Intrshath graing has thr principal isavantags. Firstly, thr ar complications in fixing th shath potntials. Sconly, th intrshaths ar likly to b amag uring transportation an installation which might rsult in local concntrations of potntial graint. Thirly, thr ar consirabl losss in th intrshaths u to charging currnts. For ths rasons, intrshath graing is rarly us. Exampl.5. A singl cor cabl of conuctor iamtr cm an la shath of iamtr 5.3 cm is to b us on a 66 kv, 3-phas systm. Two intrshaths of iamtr 3 cm an 4 cm ar introuc btwn th cor an la shath. If th maximum strss in th layrs is th sam, fin th voltags on th intrshaths. Solution. Hr, cm ; 3 cm ; 4 cm 5.3 cm ; V kv 3 V V g max 8 V 3 log log

23 86 Principls of Powr Systm g max V log * This quation is obtain if w put th valus of q. (ii) in q. (i). V 55 log V g 3max V3 V3 5 3 log log 4 04 V 3 As th maximum strss in th layrs is th sam, g max g max g 3max or 8 V V 04 V 3 V ( 8/ ) V 075 V an V 3 ( 8/ 04) V 7 V Now V + V + V 3 V or V V + 7 V 53 9 or V 53 9/ kv an V 075 V kv Voltag on first intrshath (i.., nar to th cor) V V kv Voltag on scon intrshath V V V kv Exampl.6. A singl-cor 66 kv cabl working on 3-phas systm has a conuctor iamtr of cm an a shath of insi iamtr 5.3 cm. If two intrshaths ar introuc in such a way that th strss varis btwn th sam maximum an minimum in th thr layrs, fin : (i) positions of intrshaths (ii) voltag on th intrshaths (iii) maximum an minimum strss Solution. Hr, cm ; 5.3 cm ; V kv (i) Positions of intrshaths. Suppos that iamtrs of intrshaths ar an cm rspctivly. Lt V, V an V 3 rspctivly b th voltag btwn conuctor an intrshath, btwn intrshath an an btwn intrshath an outr la shath. g max V ; g V max log ; g V3 log 3max log As th maximum strss in th thr layrs is th sam, V V V3...(i) log log log In orr that strss may vary btwn th sam maximum an minimum in th thr layrs, w hav, / / /...(ii) V V V3...*(iii)

24 Unrgroun Cabls 87 From xp. (ii), w gt, [ cm] or / an cm or / 0 6 or ( ) /3 76 cm / ( 76) / 3 8 cm Hnc intrshaths of iamtrs.76 cm an 3.8 cm ar rquir to b us. (ii) Voltag on intrshaths V V + V + V 3 or 53 9 V + V + V [From q. (iii)] V K 4 8 V V 53 9/4 8 6 kv F H an V V kv Voltag on first intrshath V V kv max Voltag on scon intrshath V V V kv max (iii) Strsss in ilctrics V 6 Maximum strss log 76 kv/cm 39 kv/cm log V 6 Minimum strss log kv/cm 8 35 kv/cm log TUTORIAL PROBLEMS. A 33 kv, singl-cor cabl has a conuctor iamtr of cm an insulation of.5 cm. Fin th maximum an minimum strss in th insulation. [47 6 kv/cm (r.m.s.), 9 kv/cm (r.m.s.)]. Fin th conomic siz of a singl-cor cabl working on 0 kv, 3-phas systm. Th maximum prmissibl strss in th ilctric is not to xc 50 kv/cm. [ 43 cm, 3 88 cm] 3. Th innr conuctor of a concntric cabl has a iamtr of 3 cm with insulation of iamtr 8 5 cm. Th cabl is insulat with two matrials having rlativ prmittivitis of 5 an 3 with corrsponing saf working strsss of 38 kv/cm an 6 kv/cm. Calculat th raial thicknss of insulating layrs an th saf working voltag of th cabl. [ 5 cm, 0.6 cm, 46 kv (r.m.s.)] 4. A singl-cor la covr cabl is to b sign for 66 kv to arth. Its conuctor iamtr is cm an its thr insulating matrials hav prmittivitis of 5, 4 an 3 rspctivly with th corrsponing maximum saf working strss of 38 kv/cm (r.m.s.), 6 kv/cm. (r.m.s.) an 0 kv/cm (r.m.s.) rspctivly. Fin th minimum iamtr of la shath. [8 3 cm] 5. A singl-cor 66 kv cabl has a conuctor iamtr of cm an a shath of insi iamtr 5 3 cm. Th cabl has an innr layr of cm thick of rubbr of ilctric constant 4 5 an th rst imprgnat papr of ilctric constant 3 6. Fin th maximum strss in th rubbr an in th papr. [63 kv/cm, 39 5 kv/cm] 6. A singl-cor cabl working on 66 kv on 3-phas systm has a conuctor iamtr of cm an a shath of insi iamtr 5 3 cm. If two intrshaths ar us, fin th bst positions, maximum strss an th voltag on th intrshaths. [ 77 cm; 3 84 cm ; 38 7 kv/cm ; V 4 kv, V 3 9 kv] I

25 88 Principls of Powr Systm.4 Capacitanc of 3-Cor Cabls Th capacitanc of a cabl systm is much mor important than that of ovrha lin bcaus in cabls (i) conuctors ar narr to ach othr an to th arth shath (ii) thy ar sparat by a ilctric of prmittivity much gratr than that of air. Fig..8 shows a systm of capacitancs in a 3-cor blt cabl us for 3-phas systm. Sinc potntial iffrnc xists btwn pairs of conuctors an btwn ach conuctor an th shath, lctrostatic fils ar st up in th cabl as shown in Fig..8 (i). Ths lctrostatic fils giv ris to cor-cor capacitancs C c an conuctor-arth capacitancs C as shown in Fig..8 (ii). Th thr C c ar lta connct whras th thr C ar star connct, th shath forming th star point [S Fig..8 (iii)]. Thy lay of a blt cabl maks it rasonabl to assum quality of ach C c an ach C. Th thr lta connct capacitancs C c [S Fig..9 (i)] can b convrt into quivalnt star connct capacitancs as shown in Fig..9 (ii). It can b asily *shown that quivalnt starcapacitanc C q is qual to thr tims th ltacapacitanc C c i.. C q 3C c. Th systm of capacitancs shown in Fig..8 (iii) rucs to th quivalnt circuit shown in Fig..0 (i). Thrfor, th whol cabl is quivalnt to thr star-connct capacitors ach of capacitanc [S Fig..0 (ii)], * Rfr to Fig..9. Th capacitanc btwn any two conuctors of star an lta connct systm must b th sam. C c + C c C q or C q 3C c