ELECTRIC FIELD NEAR BUNDLE CONDUCTORS

Transcription

1 Jounal of ELECTICAL ENGINEEING, VOL. 54, NO. 5-6, 3, ELECTIC FIELD NEA BUNDLE CONDUCTOS Danel Maye Vít Veselý The pape deals wth computaton of electc feld dstbuton along the sufaces of a system of paallel conductos wth vaous potentals. The method stats fom ntegal equatons and the elaboated algothm s appled to hv and uhv ovehead lnes wth bundle conductos. The esults allow evaluatng the dange of gvng se to coona (even wth espectng the nfluence of a ough suface of the cable) wth all ts ntefeence effects. The algothm was compaed wth othe aleady publshed methods wth emakable ageement. K e y w o d s: hv and uhv ovehead lnes, bundle conductos, coona, method of ntegal equatons. 1 INTODUCTION Hv and uhv ovehead lnes ae mostly ealsed by bundle conductos. Electc feld stength on the sufaces depends on the potental of patcula conductos, the ad, mutual dstances and qualty of the suface. Exceedng the ctcal value E ct 1 kv/cm gves se to a coona. One of the advantages of usng bundle conductos s educton of ths dschage that, as t s well known, nceases losses along the lne, unfavouably affects nea telecommuncaton devces and poduces acoustc nose. A supsng amount of papes wee publshed on the topc, see the bblogaphc lst [6] compled aleady n Vaous methods have been used fo detemnng the electc feld nea the bundle. Olde papes, see, fo nstance, [], [15] stated fom an analytcal soluton of the electc feld poduced by one conducto and the esultant electc feld of a bundle was calculated usng supeposton. The esults obtaned n ths way ae, howeve, only appoxmate. Othe authos [3], [4] and [8] eplace the patcula conductos n the bundle by a system of sutably located cuent flaments and solve the electc feld. Confomal mappng was used n [13]. Smulated chage method was successfully appled n [11] and [1]. Numecal calculaton based on standad fnte element technques (and ealsed by means of pofessonal codes) would be comfotable, nevetheless, poblems wth geometcal ncommensuablty (small coss-sectons of the conductos vesus the lage mutual dstances and doman contanng a) would lead to usng a stongly nonunfom mesh and enomous numbe of equatons. The task epesents, moeove, an open bounday poblem, whch s often a souce of futhe eos. That s why the method of ntegal equatons (see [5]) may appea advantageous fo solvng ths case; ts fundamentals can also be found n [6] o [14]. The unknown quantty s now the chage densty σ along the suface of patcula conductos. We fomulate the coespondng equaton fo σ and solve t numecally. We ae, howeve, manly nteested n the vecto of the electc feld stength on the suface of patcula conductos. It has only the nomal component E n that s gven by a smple (Coulomb) fomula E n = σ ε. The pncpal advantage of ths appoach conssts n the fact that the numecal computaton of the chage densty s much smple than computaton of the dstbuton of the potental nea the conductos (whch would, of couse, also povde the values of E n ). Whle the dstbuton of the potental n the solved aangement epesents a D poblem, soluton of the chage densty s only a 1D poblem. educton of the dmenson by 1 leads to a sgnfcant smplfcaton of the task. Unlke n the open bounday poblem assocated wth computaton of the potental, the chage s dstbuted along sufaces of the conductos, e n a spatally bounded doman. MATHEMATICAL MODEL OF THE POBLEM Consdeed s homogeneous, lnea and sotopc delectc (a) of pemttvty ε wth a set of n dect paallel conductos wth constant potentals ϕ k, k = 1,..., n.. The electc chage on the sufaces s supposed to be dstbuted contnuously, wth chage densty σ k that does not change along the lengths. The electc feld nea the conductos s obvously two-dmensonal. Potental at a geneal pont B of ths feld s gven by expesson [7] ϕ( B ) = 1 σ k ( (k) 1 ln πε (k) B dl(k), (1) l (k) k = 1,..., n Depatment of Theoy of Electcal Engneeng, Faculty of Electcal Engneeng, Unvesty of West Bohema n Plsen, Sady Pětatřcátníků 14, Plsen, Czech epublc, E-mal: maye@kte.zcu.cz, vvesely@kte.zcu.cz ISSN c 3 FEI STU

2 114 D. Maye V. Veselý: ELECTIC FIELD NEA BUNDLE CONDUCTOS dl k (k) (k) l k k = const. A y (k) - B (k) B B x Fg. 1. The k -th conducto n the set of n paallel conductos. A(x (k) (k),y j ) l k k = const. A y - j B(x,y j ) The potental ϕ would be detemned fom (1) whle the electc feld stength fom elaton E( B ) = gad ϕ( B ). (3) Ths s, howeve, beyond ou nteest. Let us etun to ou case, when B l (k) (Fg. ). The electc feld stength at such a pont has only a nomal component and may be detemned fom fomula E (k) n ( (k) ) = σ k( (k) ) ε. (4) In ths way we solved ou poblem because the value of the electc feld stength decdes about gvng se to coona on the suface of any conducto. Now t emans to cay out numecal soluton of (). Fst we dvde the contou lnes l (k) of coss-sectons of the patcula conductos nto N (k) pats of lengths l (k), = 1,..., N k. When ths dvson s suffcently fne, each pat may be supposed to cay a constant value of the chage densty σ (k) = const ( = 1,..., N k, k = 1,..., n). (k) x B The dstance between the mdponts of pats l (k) s gven as l (l) j (k) (l) j = (x (k) and x (l) j ) + (y (k) y (l) j ). (5) Fg.. To the calculaton of the chage densty on the suface of the k -th conducto. whee l (k) s a smply connected cuve n whch the cosssecton S (k) ntesects the plane of the feld pependcula to the conductos, (k) s the adusvecto of the element dl (k), B s the adusvecto of pont B and (k) B s the dstance of a vaable pont A of the plana cuve l (k) fom pont B outsde the conductos (see Fg. 1). We put, moeove, that lm ϕ( B) =. B Let pont B l (k). Now (1) tansfoms nto the fstknd Fedholm ntegal equaton wth an unknown dstbuton of the chage densty σ k. πε ϕ k = σ k ( (k) 1 ) ln (k) B dl(k), l (k) k = 1,..., n. () Soluton of system () povdes σ k at an abtay pont B l (k). Ths knowledge allows consequent computng of the feld quanttes at any pont outsde the conductos. Equaton () fo the dscetsed model now eads (n ode to avod dvdng by zeo we leave the tem fo = j and k = l n the fom of the defnte ntegal) πε ϕ k = l=1 N (l) =1 j k l + σ (k) σ (l) ln l (k) / (k) 1 (l) j l(l) ln 1 dl(k), k = 1,..., n. (6) As fa as l (k) s a lne segment, calculaton of the ntegal n (6) s easy and ts esult s l (k) / ln 1 (k) dl (k) = l(k) ) (1 ln l(k). (7) In ths manne we obtan a system of algebac equatons whose numbe s m = N (k). (8)

3 Jounal of ELECTICAL ENGINEEING VOL. 54, NO. 5-6, E n mn =1.58 kv/cm 15 1 E n max =14.3 kv/cm Fg. 3. A bundle conducto fo n = 4. Fg. 4. Dstbuton of the nomal component of the electc feld stength along the suface of one conducto of the bundle ( n = 4 ). E n mn =1.58 kv/cm 1 15 E n max =14.3 kv/cm Fg. 5. A bundle conducto fo n = 8. Fg. 6. Dstbuton of the nomal component of the electc feld stength along the suface of one conducto of the bundle ( n = 8 ). whee The system can be ewtten nto a matx fom Aσ = πε ϕ (9) A(m, n) = [a pq ], p, q = 1,..., m, l p ln 1 p q, fo p q a pq = ( ) l p 1 ln lp, fo p = q. Hee σ(m, 1) s the column vecto of chage denstes n the patcula segments and ϕ(m, l) the potentals. The pesented elatons espect the nfluence of all conductos n the system (all bundle conductos n all phases) and possbly the nfluence of the eath. 3 ILLUSTATIVE EXAMPLES The next examples solve, howeve, only the case of conductos n a one phase bundle (e we consde nethe the nfluence of othe phases no of the eath). Detemned s the dstbuton of the electc feld stength on the suface of one conducto n the bundle. a. A bundle conducto n = 4 Fo the conductos placed n vetces of a egula squae (see Fg. 3) we calculated the dstbuton of the nomal component of E n along the suface of one conducto. All conductos n the bundle have the same potental ϕ = 1 kv. Pemetes of all conductos wee dscetsed

4 116 D. Maye V. Veselý: ELECTIC FIELD NEA BUNDLE CONDUCTOS E If we eplace the bundle conducto by a sngle conducto of adus = (ts coss-secton beng the same as the total coss-secton of the bundle conducto), the electc feld stength E n would be dstbuted unfomly and ts value would be E n = nq = = 4.73 kv/m. πε πε. Fg. 7. A massve conducto and equvalent tansmsson cable fo n = 8. If we use a bundle conducto whee the dstance between ndvdual conductos s vey lage (the mutual electostatc nteacton s neglgble), the electc feld stength E n wll agan be unfom and ts value would be E n = Q = = 1.36 kv/m. πε πε.1 b. A bundle conducto n = E n max =11.1 kv/cm We calculated E n on the suface of one conducto aanged at vetces of a egula octagon (Fg. 5). Its dstbuton s n Fg. 6. Othe esults analogous to the pevous case ae summased n Tab. 1. c. Influence of ough suface of the tansmsson cable Fg. 8. Dstbuton of the nomal component of the electc feld stength along the suface of one conducto of the tansmsson cable. Table 1. Values of E n fo vaous aangements of conductos. aangement E n (kv/cm) E n /E n1 bundle conducto E n max = n = 8 (Fg. 5) E n mn = one equvalent conducto E n1 = 9.7 bundle conducto, lage dstance among E n = patcula conductos nto N (1) = = N (4) = 5 lne segments. The esults ae depcted n the pola co-odnates n Fg. 4. The maxmum and mnmum values of the electc feld stength ae E n max = 14.3 kv/cm, E n mn = 1.58 kv/cm. The total electc chage pe unt length of one conducto of the bundle s Q = σds N = σ l = C/m. S =1 The tansmsson (fo nstance alumnum-steel) cable may be taken as an exteme case of the bundle conducto, whose patal conductos ae placed vey close one to anothe. Whle on the suface of a massve conducto the nomal component E n s dstbuted unfomly, n the case of the tansmsson cable ths component changes fom zeo on the ntenal suface (nfluence of sheldng) to ts maxmum E n max on the extenal suface. Obvously E n max > E n. Fgue 7 depcts both the cosssecton of the massve conducto and coss-secton of the cable wth 8 conductos. Fgue 8 contans a gaph n pola co-odnates showng the dstbuton of the electc feld stength on the suface of one cable conducto. Calculatons wee pefomed fo potental ϕ = 1 kv, pemete of each conducto beng dvded nto N = 5 lne segments. Analogously to pevous cases we obtan fo a sngle conducto of the cable E n max = kv/cm, E n mn, Q N = σ l = C/m, =1 fo an equvalent bundle conducto nq E n = = 1.77 kv/cm. πε 8 On the suface of the cable (n compason wth the equvalent massve conducto wth smooth suface) the delectc stess s hghe. The measue of ths ncease may be gven by the ato between the maxmal value of ts nomal component to the magnetc feld stength of the massve conducto: E n max /E n = 1.1. Let us emak that mcoscopc oughness of the suface caused, fo example, by cooson o small wate dops gves also se to coona.

5 Jounal of ELECTICAL ENGINEEING VOL. 54, NO. 5-6, CONCLUSION The pesented algothm fo computaton of delectc stess on the suface of a conducto n a bundle was compaed wth esults obtaned by othe methods. Table contans some esults obtaned by dffeent algothms fo a bundle contanng n = 4, 8 and 1 conductos chaactesed by d/ = 6.99 (d beng the dstance of axes of two neghboung conductos n the bundle and the adus). The pemete of each conducto was then dvded nto N = 3 lne segments. The ageement between patcula methods s outstandng. Table. Compason of esults. E mn /E max n = 4 n = 8 n = 1 the descbed method by Cahen [3] by Tmascheff [13] Acknowledgement Ths wok has been fnancally suppoted fom the gant No. 1/1/141 of the Gant Agency of the Czech epublc. efeences [1] ALLAN,. N. SALMAN, S. K. : Electostatc Felds Undeneath Powe Lnes Opeated at Vey Hgh Voltages, Poc. IEE 11 (1974), [] BÜDELINK,. : Induktvtät und Kapaztät de Statkstom-Feletungen, G. Baun, Kalsuhe, [3] CAHEN, F. PELISSEU,. : L emplo de conducteus en fasceaus pou l amement des lnges a tes haute tenson, Bull. Soc. F. Elect., Mach 1948, pp [4] KING, S. Y. : The Electc Feld nea Bundle Conductos, Poc. IEE 16C (1959), 6. [5] FOO, P. Y. KING, S. Y. : Bundle-Conducto Electc Feld by Integal-Equaton Method, Poc. IEE 13 (1976), [6] MAYE, D. ULYCH, B. : Soluton of D and 3D Electostatc and Magnetostatc Felds, Elektotechn. obzo 69 No. 8 (198), (n Czech) [7] MAYE, D. POLÁK, J. : Methods of Soluton of Electc and Magnetc Felds, SNTL/ALFA, Paha, (n Czech) [8] MIOLJUBOV, N. N. KOSTENKO, M. V. LEVINŠTEJN, M. L. TICHODEJEV, N. N. : Methods of Calculaton of Electostatc Felds, Vysshaja shkola, Moskva, (n ussan) [9] SANDELL, D. H. SHEALY, A. N. WHITE, H. B. : Bblogaphy on Bundled Conductos, IEEE Tans. on Powe Appaatus and Systems, Decembe 1963, No. 69, pp [1] SAMA, M. P. JANISCHEWSKYJ, W. : Electostatc Feld of a System of Paallel Cylndcal Conductos, Tans IEEE PAS-88 (1969), [11] SINGE, H. STEINBIGLE, H. WEISS, P. : A Chage Smulaton Method fo the Calculaton of Hgh Voltage Felds, IEEE Tans, pape T , [1] STEINBIGLE, H. SINGE, H. BEGE, S. : Beechnung de andfeldstäken von Bundelleten n Dehstomsystemen. ETZ-A, 9, 1971, pp [13] TIMASCHEFF, A. S. : Fast Calculaton of Gadents of a Thee-Phase Bundle Conducto Lne wth any Numbe of Subconductos, AIEE Tans., Vol. PAS-9, 1971, pp ; PAS-94, 1975, pp [14] TOZONI, O. V. : Method of Seconday Souces n Electcal Engneeng, Enegja, Moskva, (n ussan) [15] VEVEKA, A. : Hgh-Voltage Engneeng, SNTL/ALFA, Paha, (n Czech) eceved 18 Mach 3 Danel Maye (Pof, Ing, DSc) was bon n Plsen, Czech epublc n 193. He eceved the Ing, PhD and DSc degees n electcal engneeng fom Techncal Unvesty of Pague n 195, 1958 and 1979, espectvely. In 1956 he began hs pofessonal caee as a Seno Lectue and late as a Assocate Pofesso at the Unvesty of West Bohema n Plsen. In 1968 he was apponted Full Pofesso of the Theoy of Electcal Engneeng. Many yeas he has been head of the Insttuton of Theoy of Electcal Engneeng. Hs man teachng and eseach nteests nclude ccut theoy, electomagnetc feld theoy, electcal machnes and hstoy of electcal engneeng. He has publshed 6 books and moe than scentfc papes. He s a Fellow of the IEE, membe ICS, ISTET and UICEE, membe of edtoal boads of seveal ntenatonal jounals and leade of many gant pojects. Vít Veselý (Ing) was bon n Domažlce, Czech epublc n He gaduated n electoncs engneeng fom the Faculty of Electcal Engneeng, Unvesty of West Bohema n Plsen n. Hs eseach nteests le manly n the aea of compute scences.