A General Formulation of Impedance and Admittance of Cables

902IEEE Transactions on Power Apparatus and Systems, Vol. PAS-99,No. 3 May/June 1980 A GENERAL FORMULATION OF IMPEDANCE AND ADMITTANCE OF CABLES A. AMETANI Doshisha University Kyoto, Japan ABSTRACT Interest in the analysis of wave propagation characteristics and transients associated with cable sysIn order to answer the tems has rapidly increased. need of the analyst,impedances and admittances of various cables have to be known. This paper describes a general formulation of impedances and admittances of single-core coaxial and pipe-type cables. The formulation presented here can handle a coaxial cable consisting of a core, sheath and armor, a pipe-type cable of which the pipe thickness is finite and an overhead cable, which has not been discussed in the literature heretofore. Using the formulation presented in this paper, it now becomes possible to analyze wave propagation characteristics and transients on any type of cable system. 2. IMPEDANCE AND ADMITTANCE The impedance and admittance of a cable system are defined in the two matrix equations. d (V)/dx = - [Z] * (I) d(I)/dx = - [Y] *V) where (V) and (I) are vectors of the voltages and cur- (1) (2) rents at a distance x along the cable. [Z] and [Y] are square matrices of the impedance and admittance. In general, the impedance and admittance matrices of a cable can be expressed in the following forms. IZ] [Y] = [Zi] + lZp] + [Zc] + Izo] 1. INTRODUCTION The growing use of cable systems and the increasing levels of capacity makes the analysis of wave propagation characteristics and transients on cable systems an important task. The cases of underground single-core coaxial cables (SC cables) consisting of a core and sheath, and pipe-type cables (PT cables) of which the pipe thickness is assumed to be infinite have been well studied!-5 However, SC cables consisting of a core, sheath and armor, PT cables of which the pipe thickness is finite, and overhead cables have never been studied. SC cables with a core, sheath and armor are quite often seen in the submarine cable case, and, in fact, this author has been asked about the possibility of calculating transients on such cables.6 So far, pipe enclosures were assumed to act as complete shields, thus avoiding consideration of earth return currents. As far as wave propagation characteristics and transients on inner conductors of the pipe enclosure are concerned, the assumption of the infinite thickness of the pipe is quite acceptable. But once the wave propagation and transients on the pipe are to be included, all the previous studies are not applicable. Thus, we need a way of handling voltage and current on the pipe. An analysis of overhead cables seemed to be overdue, although there is some need for it. This author has been asked to calculate transients in a gas insulated substation, where a bus and circuit breaker are enclosed in a pipe, and the pipe is overhead7 This can be considered to be an overhead cable. Because of the situation explained above, a formulation of impedances and admittances, which is able to deal with an SC cable consisting of a core, sheath and armor, a PT cable having a finite thickness of the, pipe, and an overhead cable, has been developed in the present paper. The formulation is carried out in a generalized manner so as to be able to handle all the above cases. = s . [p1 [P] = [Pi] + IPp] + [Pc] + [Po] where [P] is a potential coefficient matrix, and s=jw. } (3) (4) In the above equations, the matrices with subscript"i"concern an SC cable and the matrices with subscripts"p"and "c" are related to a pipe enclosure. The matrices with subscript"o"concern cable outer media,i.e. air space and earth. When a cable has no pipe enclosure, there exists no matrix with subscripts"p"and "c". In the formulation presented here, the following assumptions are made. (1) The displacement currents and dielectric losses are negligible. (2) Each conducting medium of a cable has constant permeability. (3) The pipe thickness is greater than the penetration depth of the pipe wall for the PT cable case. The details will beexplained in the following sections. 2.1 Single-Core Coaxial Cable (SC cable) 2.1.1 Impedance When an SC cable consists of a core, sheath and armor as shown in Fig.l (a), the impedance is given in the following form based on the result of Appendix 1. IZ] where = [Zi] = + IZo] (5) [Zi] SC cable internal impedance matrix = [Zil] [IO ]-^-[ O ] [0 [Zi2]@**[ ° ] ° (6) [ °] ° ]...[Zin] li0 ipdne arxoftecbl ue [ZO] impedance matrix of the cable outer medium (earth return impedance) = A paper recawended and approved by the F 79 615-6 IEEE Insulated Conductors Cacimittee of the IEEE Power Engineering Society for presentation at the IEEE PES Summer Meeting, Vancouver, British Columbia, Canada, July 15-20, 1979.Manuscript submitted January 26, 1979; made available for printing May 1, 1979. i11 [lo liZo 12] -[Zo ln] (7) [Zo12] [Z022]. [Zo2n] i [Zo in] [Zo2n].- [Zonn] li All the off-diagonal submatrices of [Zi] are zero. 0018-9510/80/0500-0902$00.75 © 1980 IEEE cl) . [Zij] Fig.Z2m + Z23 (12) ZS3= Z20 If an SC cable consists only of a core.: internal impedance of sheath inner surface Z2i = (sPoVi2/2Jr). sheath and armor. 2) 2 core Zcsl LZcsj Zss5J + (11) sulator 3 (Ej3) armor(p3. 1 An SC cable system A diagonal submatrix expresses the self-impedance matrix of an SC cable. the matrix of eq. C3) (a) SC cable cross-section k-th cable where Zccj Zssj ZCS Zs3 ZS3 - 2Z2m = = Zcsj and ZS3. 1 1.(Io(x4)-Kl(x3) Ko(x4)-II(x3)} =Zsa Za4.2Z2m .1'9'10 (1) z11: internal impedance of core outer surface [Zjj]= Zccj Zcsj Zcaj Zcsj Zssj Zsaj Z I = (sviopj/2T) (8) + (1/x2D1) -{Io(x2) * K1 (x1) Ko(x2)YIl(X1)} insulator impedance where Zcaj Zsaj Zaaj Zccj = core self-impedance = zcs + Zsa + Za4 .903 When the SC cable consists of a core and sheath.2Z3M (2) Z12 : core outer Z12 = (s-popij/2r)+ln(r3/r2) + (3) Z2.(8) is reduced to a 2 x 2 matrix. the submatrix is redued to one element.F -11sheW4P 2.Z3M mutual impedance between the sheath and armor (7) Z3i : inteinal impedance of armor inner surface z3i = + (sloUV3/2ir)* (l/xsD3)*{ Io(xs)*Ki(x6 ) Ko(xs)*Il(x6)} mutual impedance = Zcaj Zii + Z12 + Z2j + (8) Z3m : armor where Zcs = Zsa = Z20 Za4 = Z30 Z23 + z3i + Z34 I Z3m = P3/2irrsr6D3 impedance of armor outer surface (9) Z30 (10) : internal Z30 = + (SPoP3/2T) (l/x6D3) -{Io(xr)Kl(x5) Ko(X6)-I (X5)I impedance (10) Z34 : armor outer insulator . V3. the self-impedance matrix is given by: = Zccj = Zil + Z12 (13) The component impedances per unit length in the above equations are given in the following equaions for an SC cable shown in Fig.l(a). When the SC cable consists of a core. r.(l/x3D2)*{Io(X3)KI(X4) Ko(x3)I1l(X4)} Zssj = sheath self-impedance = Zsa + Za4 - 2Z3m (4) Z2m: sheath mutual impedance Z2m = P2/2irr3r4D2 (5) Z20 : internal impedance of sheath outer surface Z20 = + Zaaj = armor self-impedance = za4 Zcsj = mutual impedance between the core and sheath + (SioI2/27r) (l/x4D2). [Z1j] = Fzccj = (pi.Z2m - 2Z3m (6) Z23 : sheath outer insulator impedance Z23 = Zcaj = mutual impedance between the core and armor = (sipOpii2/2ir)*ln(r5/r4) Zsaj = Za4. If a cable is above a stratified earth.I1(x5)*K1(x6) Xk = aks. [Pojk] = Poik Pojk Pojk Pojk Poik PoJk (25) Poj k Poj k Poj k where Pojk is the space potential coefficient and is given for the case of Fig.(7) is given in the following form.(14) to (16) is the earth return impedance between the j-th and k-th cables.1. When the cable consists of above matrix is reduced to: I a core.2 Pipe-Type Cable (PT Cable) I (26) [Pi] = cable internal potential coefficient matrix = [Pil] [Pi2] [ o 2.2. A diagonal submatrix expresses the potential coefficient matrix of an SC cable. the earth return impedance developed by Nakagawa.904 Z34 = (S)o1Pj3/2Tr)*ln(r7/r6) Il(x2). When a cable system is overhead.1 Impedance The impedance matrix of a PT cable shown in Fig. s = r64iOvi3/P3 A submatrix of the earth return impedance [ZO] in eq. (22) and sheath. sheath and armor as shown in Fig. If the cable consists only of includes only one element. the above matrix is reduced to: = [Zojk] [Zojk Zojk Zojk Zojk (15) (1/27Troc1i)*ln(r3/r2) (1/27wco0j2)*ln(r5/r4) (1/27rToCj 3)*ln(r7/r6) a core = If the SC cable consists only of a core.l (b) by: (1) Overhead cable [P] = [Pi] + [Po] (17) (18) Pojj = (1/2TrrO)*ln(2hj/r7j) (2) Underground cable [P] = [Pi] where Pojk = (l/2Irco)*ln(D2/Dj) 2. the matrix includes only one element. then [Pij] [Pii] = Pcj are (24) given in the follow- The submatrices of [Po] ing form. [Poin] F PI2n] [Ponn] [Pij] [Zoj k] = Zojk Zojk Zojk Pcj+Psj+Paj Psj+Paj Paj Psj+Paj Psj+Paj Paf (21) Zojk Zoik Zojk Zojk Zojk Zojk (14) where Paj Pcj Psj Paj = Paj Paj When the SC cable consists of a core and sheath. [Poin] [Po2n] [PO2] [P022] where Di = Il(x1)-K1(x2) (20) D2 = Il(x4). 4 = r4/II0o2/P2 85 = r54i01i3/P33 . based on the result of Appendix 2. See Appendix 2. 82 = r2/4110"/Pl 8 3 r3_0_12/P2. and when the cable system is ungerground. the [Zojk] = Zojk (16) [Pij] = [Pcj + Psj l Psjl PsiJ (23) Psj Zoik in eqs.KI(x3) Il(x3) Kl(x4) D3 = I1(X6)*K1(xs5) . Thus.K1(xi) - [PO] = potential coefficient matrix of the system in air = --[Po11] [Po12]. [PO] is also zero. the diagonal submatrix is given in the following form.al. 2. [P ] and [Pc] are in eq.2. is given in the same manner as the SC cable case.the impedance given by Pollaczek"2 is used.8'9 (19) (1) Pipe thickness assumed to be infinite [Z] = [Zi] + [Zp] (27) [Pin]i (2) Pipe thickness being finite .(4). where an inner conductor is assumed to be an SC cable.13 can be used. When the SC cabe consists of a core.2 Potential coefficient The admittance matrix of a cable system is evaluated from the potential coefficient matrix as given In the SC cable case. zero. the impedance is given by Carson!' When a cable system is underground.l (a). et. All the off-diagonal submatrices of [Pi] are zer. .* [Zp2n] [Zpin] [Zp2n] 0 - (30) [Zpnn] 0 0 0 .eq.-[Zo] Zo Zo Zo .2djdkcosOjk] -nCn/n (djdk/rpi)n. and when the inner conductor consists only of a core.e.(xj)/Knix1 pipe2P_iCn/{n(l[ippKo(xj)/{xiKj(xi)}+ )1] (34) o/21T)* Q*k + ZC1] ZC2 Qjj = = ln[(rpl/rj). a2 = Il(x1) Kl(x2) x. and is given by3'8: a 0 0 0 [Zp] = pipe internal impedance matrix [Zp11] [Zpl 2]. - (39) x2 Fig.(36). i. these should be omitted when the pipe thickness being assumed infinite.(8).. A diagonal submatrix of [Zi]. ] [ZM2]. is given in eq.' [ 0 (29) [Zpjk] r Zpjk Zpjk Zpjk (33) [0] t[] I [Zin] 0 O 0 0.(33) is reduced to 2x2 matrix.(33) is further reduced to a column matrix in the same manner as explained in the case of [Zi]. ) This is the same for all other impedance and admittance matrices explained in this section.. [Zo] Zo [Zo] [Zo.(29). 2 A PT cable by: = rp2/iaOp/Pp A diagonal submatrix of [ZO] in eq.(8). [Zpln] ] [Zp12] [Zp22]. the last column and row correspond to the pipe conductor. is given in eq.-0 Zpjk = (s LZc] = connection impedance matrix between pipe inner and outer surfaces ] [Zcll] ZC11 '' w=1 where + pp) + xiKn-1.(29) and(30). ( See eqs. eq.inner conductors with respect to the pipe inner surface.(32) is given .(33) is the impedance between the j-th and k. eq. Z k in eq. eq. A submatrix of [Zp].{l-(dj/rpl)2}] [ZCl] [ZC1]*' [IZCI] ZC2 (31) and Qik ln[rpl/!d% + di . Thus.905 [Z] = [Zi] + [Zp] + [Zc] + [Zo] (28) where [ZiI = SC cable internal impedance matrix [Zil] [I [ ° 0 1] [ 0 ] 0 1 0 In eqs.ln(rp3/rp2) Il(x2) Kl(xl) a2A.[Zo] Zo [Zo] [Zo. (11) and (13).0--e Zpjk Zpjk Zpjk Zpjk Zpjk ZpjkJ When an inner conductor consists of a core and sheath.(30). ZC2 Zc3 Cn = [Zo] = earth return impedance matrix xl= rpj/pvO'ip/P~p (36) 1Zo] [Zo]. is given in the fol 1 owi ng form.(31) are given in the following form8'14 (32) [Zci]= Zcl Zc1 Zc Zci Zcl Zcl (37) Zo) Zci Zcl Zcl J ZC1 = ZC3 2zpn ZC2 = Zc3 Zpm ZC3 = Zpo + Zp3 - - where Zpm Zpo = = Pp/(2-nrpirp2Dp) (Svolop/27rx2Dp)j{ Io(x2) K1(xl) + KO(X2)Il1(Xl)} (38) zp3 and D = = (slao/21r). cos(n0jk) I 1 =s [ZC1] [ZC1]'' [ZC1] ZC2 ZC2 ZC2 ' .* Zo A submatrix and the last row and column elements of [Zc] in eq. 2. 1 the frequency of cases. in most on a cable system. The first assumption is constant permeability.906 [Zo) = ZO Zo Zo Zo Zo Zo (40) Zo Zo Zo where Zo in the above matrix is the self earth return impedance of the pipe. Quite offten. One should pay attention to the fact that Carsont and Pollaczek's interest is. displacement currents are negligible as far as low frequencies (less than about lMHz) are concerned. Ppjk = Qjk/27Tcplo row (49) [Pi] + EPp] + [Pc] (42) (b) Overhead cable A submatrix and the last column and of [Pc] in eq.ln(2h/rp3) 3. less than MHz.. these approximations so as to make the limit of applicability clear when the formulation is used.-[Po] PO PO (47) [Po] [Po] [Po]. Thus..Po PO lectric losses are small in comparison with the losses in conducting media of cables and earth. Submatrix [Ppjk] of [Pp].2 Potential coefficient The potential coeffidient matrix of a PT cable shown in Fig..(35). . is given in the following form. the assumption is valid.2. methods proposed in references (3) and (4) can be used.Pc Pc pipe The formulation of impedances and admittances of various cables given in the previous section includes It may be important to discuss some approximations.. 0 (45) [Ppnn] 0 0 0 [Pc] = potential coefficient matrix between pipe inner and outer surfaces [PC] [PC]---[PC] PC [Pc] [PC] [Pc]---[Pc] PC (46) PC. When one needs to take the saturation into account. (21).(44) is given in eq. DISCUSSION [Ppl 2] ** *[Ppln] [PpI2] [Pp22]. A diagonal submatrix of [Pi] in eq. In regard to the second assumption. First of all.(46) are given by: elements [P1I = [Pi] where + [Pp) + [Pc] + [Po] (43) [PC] = PC PC Pc Pc Pc Pc [Pi] = SC cable internal potential coefficient matrix Pc Pc PC = PC I (50) (l/21Tcp2co)*ln(rp3/rp2) [Pill [ 0 I]e [ [ O ] ] 0 0 l (44) 0 1 [Pi?2] ' ' ' [ A submatrix and the last row and column elements of the space potential coefficient matrix[PO] is given in the following form. The third assumption will be discussed later. It. Ppjj = Ppjk Ppjk Ppjk Ppjk in the above equation is the potential coefficient between the j-th and k-th inner conductors Qjj/27rcpiso . In the analysis of transients and wave propagation [PO] = potential coefficient matrix of the in air = [PO] [Po] . and is given in the following equation using Q of eq.(44) and(45). The die- .[Po] PO Po0. the saturation of the pipe or armor can be neglected.. [Ppjk] = Ppjk Ppjk Ppjk Ppjk Ppjk Ppjk (48) (1) Pipe thickness assumed to be infinite [P] = [Pi] + [Pp] (41) (2) Pipe thickness being finite (a) Underground cable [P] = with respect to the pipe inner surface.. a pipe and armor are ferromagnetic..2 is given in the following form..these should be omitted when the pipe thickness being assumed infinite.8'9 In eqs. seems to be rather unusual to have high currents to cause saturation of the pipe or armor. No approximation is made for the impedances and admittances of an SC cable as far as Carson's and Pollaczek's earth impedances and Scheikunoff's cylindrical conductor impedance are concerned.. the major assumptions made for the formulation of impedances and admittances (on page 1 of the paper) should be discussed. however.[Po] Po0 [Po]. [0 0 1 1 0 ] [Pin] 0 0 [PO] = Po Po Po PO PO PO (51) O 0 [Pp] = pipe internal potential coefficient matrix Po Po PO PO = [Ppill oe 11~~~~~~ 0 (l/27rEo). eq.[Pp2n] [Ppmn] [Pp2n] 0 0' . the last column and row corresponding to the pipe conductor.. in most cases. (45). Thus.-. into account the effect of the eccentricity on the inner conductor impedance. and the propagation velocity is lower in the underground case. case.01 0.01 10-. it should be expected that the attenuation of the undeground cable is much higher than that of the overhead cable. 1 0-3 E C: io-2 10-4 E r- .01 1000 0. and of core. KHz Fig. it is clear that the admittance ofan underground cable are much greater than those of an overhead cable. the inner conductor impedance and admittance of a PT cable become the same as those of an SC cable. greater than the penetration depth. /// 10-3 m E l04 *v C~~~~~~~~~~~~~~~~ . 3 Effects of pipe thickness on pipe inner surface impedance 10o 0. 10-4 - a -Fl o-!5 0. The same assumption has been If one needs to take made in references (3) and (4).1 frequency. the formula of the outer surface impedance of the inner conductor given in reference (5) can be used.(34) and (35) and potential coIn that efficient given in eq. 4mm. Thus. Calculated results of admittances of a singlephase SC cable are shown in Fig. the formulation of the impedances of both SC and PT cables is correct only upto about lMHz. One can easily find that the formulation of the impedances and admittances of an SC cable given in this paper is identical to that given in reference (1) for the case of a coaxial cable consisting of a core and sheath. Thus. the formulas of the pipe internal impedance given in eqs.1 10-4 1C5 a. 4 Susceptances (imag. kHz 1 10 S- 0) Fig._ 1o-. sheath and armor.1 1 10 100 1000 frequency.3 shows a comparison of the pipe impedances for the cases of the pipe thickness being finite and infinite. 5 Internal impedances Zcc of SC cables (a) Core and its outer insulator (b) Core. Fig.3 Thus. When the pipe thickness is at the frequency of lkHz.01 0. which is nearly equivalent to the penetration depth at 10 Hz.1 1 --- f0 / /. The second assumption concerns the case of finite pipe thickness. Note that earth return currents are not neglected and that complete shielding is not assumed. If the pipe thickness is smaller than the penetration depth. Two assumptoins are included in the PT cable case. From the results. This assumption introduces negligible error for actual PT cables and for frequencies above 1OHz. Thus.907 formulas of the earth return impedance are not applicable at frequencies higher than about lMHz because the effect of displacement currents is not included in the formulas'.4. Significant differences are observed for the cases of SC cables consisting only of a core.01 10-2 1 0. The first one is that the eccentric cable positions within the pipe do not affect the internal impedances and admittances of the inner conductors (SC cable) and the impedances and admittances between the inner and outer surfaces of the pipe.(49) can not be used.Y22) of SC cables . KHz underground cable 10-5 0. its impedance is almost identical to that for the infinite thickness case in the frequency The pipe thickness is. the assumption of infinite pipe wall thickness may be used. accurate formulas of the impedance and potential coefficient can be derived based on the work done by Tegopoulos and Kriezis.10-4 In l o. of core and sheath.B -A 10 100 frequency. sheath and its outer insulator (c) Core.15 Since these formulas are too complicated for practical usage. sheath. i0. The internal impedances of SC cables are shown in Fig. It is assumed that the pipe thickness will be greater than the penetration depth in the pipe wall. in range shown in the figure. The impedance shows not a significant difference between underground and overhead cables.1 1 10 100 1000 frequency. armor and its outer insulator overhead cable Fig. KHz 1 -2 1-0 0 E 10 6 c 4 j7j'1 /f" 0. Similar results are obtained for the PT cable case. most cases. but only to calculate the impedance and potential coefficients of the pipe. the assumption is valid. It is clear that the impedance for the finite pipe thickness case approaches that for the infinite thickness case.5. an equivalent circuit for impedances is given in Fig. vol. et. Bonneville Power Administration.539-554(1926) 12) F. a pipe-type cable of which the pipe thickness is either infinite or finite. M. Voltages between the core.8) 8) A. x REFERENCES Vc L Vs Is 1) L. armor and outer medium are V12. Contract No. and Prof. sheath and armor. J. Pollaczek: "Uber das Feld einer unendlich langen wechsel stromdurchflossenen Einfachleitung". Part II .' ibid.N.120. G. E. Tegopoulos and E. -(Ic Is) (A-3) Ie= -(Ic + Is+ Ia) For voltage V12 between the core and the sheath.532-579 (1934) 11) J. A.2 (1977. I4 and Is as shown in Fig. R.10) 9) A. and an overhead cable. The formulation presented in this paper can handle a coaxial cable consisting of a core. Wilcox:"Transient analysis of underground power-transmission systems". J.EW-78-C-80-1500.E.1680-1687 (1978) 6) B.5. vol. Bonneville Power Administration.826-833(1977) Fig. Nakagawa.89-95 (1976) V12 V23 Va Ia V34 le. on Power App.877-882 (1973) 3) G. Carson: " Wave propagation in overhead wires with ground return". 12 = -Ic. Also inner and outer surface currents of the sheath and the armor are I2. III . Band 3 (Heft 9). IEEE Trans. A. the following relation for currents are armor and outer medium (earth) by Ic.: " Surge propagation in three phase pipe-type cables.13.. . obtained. R. Report No. sheath and/or armor. V12 = + V12 + AV12 Z11AxIc - z12AxI2 + - z2iAxI2 + - Z2mAXI3 .T. PAS-97. pp. C. M.120. sheath. Report No. & Syst. on Power App. vol. vol. and are V12+AV12. Dixon: Private correspondence (1977.Proc. pp. IEE Japan. of pipe thickness". CONCLUSION A general formulation of the impedances and admittances of single-core coaxial cables and pipe-type cables is given. Appendix 1 Impedance of an SC cable consisting of a core. Bell Syst. et. Schinzinger of University of Californua for his helpful discussion and critical reading of the manuscript.Proceedings of Annual Meeting. Tominaga for his encouragement.1 (1978.: Private correspondence (1977. Ia and le at Define currents flowing into the core.PAS-96. Schelkunoff: " The electromagnetic theory of coaxial transmission line and cylindrical shields". The author also wishes to express his appreciation for financial support by Bonneville Power Administration. Brown and R. ibid.A-l. PAS-95. Tech. pp. Ametani: " Surge propagation characteristics of pipe enclosed underground cable<. & Syst. Numerical results based on this formulation are readily available using BPA's computer program EMTP with subroutine CABLE CONSTANTS. pp.339-360 (1926) 13) M. al.. Ametani and T. A. pp. pp.: " Further studies on wave propagation in overhead lines with ground return ". Ono: " Wave propagation characteristics on a pipe-type cable.Consideration x.908 4. sheath and armor In the case of an SC cable with core. I3.IEE. pp.(10) Using the above equation.. Dugan. sheath.: -AV12/Ax = (Zll Z12 + Z2i)Ic Z2mI4 Define Zcs by: Zcs = Z11 + Z12 + Z2j (A-4) = eq..Duplication of field tests including the effects of neutral wires and pipe saturation".. Is. Ametani: "Extension of generalized program for line and cable constants in EMTP".840 (1978) 15) J. Ametani: "Generalized program for line and cable constants". V23 and V34 at x.Shells of finite thickness'IEEE Trans. PAS-90. Kriezis: "Eddy current distribution in cylindrical shells of infinite length due to axial currents.al.. Wilcox: " Estimations of transient sheath overvoltages in power cable transmission systems. Part I-Unsaturated pipe". A-1 An equivalent circuit for impedances of an SC cable 5) R. K.A1. - 4) R. W. Wedepohl and D. Purchase Order No. Part II-.1287-1294 (1971) APPENDICES ACKNOWLEDGEMENTS The author would like to thank Prof.7) 10) S. pp. P. J. pp. Then. and V34+AV34 at x = x + Ax.253-260 (1973) 2) L. Schinzinger and A. Paper No.9) 7) B. Wedepohl and D. ibid.70249. Proc..1521-1528 (1973) 14) A. I3 + = -I4 .' ibid.. + Is = -Ie (A-l) (A-2) Is = I2 Ia= I4 I4 = I3 = -(Ic I4) + Is + = I4 - Ie I From the above equations.pp.IEE vol. V23+AV23.120. Rocamora: "Surge propagation in three-phase pipe-type cables. (A-11).(10) (A-7) (A-8) Then.+-zzzzzzzz-IzI - s Ysa sheath a+A I a armor eqs.Ycs VS VC -(V23 + V34) V12 = = Va - V23 (A-1 0) = + VS (za4 + dxi Is LIal where Ycs = = dxI r -Ycs (Ycs+Ysa) -Ysa 0 IIVs -Ysa (Ysa+Yila4)JIVaJ Substituting eqs.(8) Z3m)- ZO Zo zoJ anrd Va. = .(5) - eq.(A(-8).(21) Pa Pc = = Pa Pa d(V)/dx = -[Z]*(I) (A-1 5) . In the same manner. Pa = s/ya4 -eq.(9) earth Zcs = Zsa + Za4 - Fig. (A-12) and (A-13) writh - (A-22) eq.(22) When a cable is overhead. Va = = 34 = e!q. -AV34/AX where (za4 + Zo)Ie .2z2m . / . From the figure. = Rewriting the above equations.ZSmI4 -AIC/AX -AIs/AX = ycsVc -YcsVs = -YcsVc + (A-9) Za4 = Z30 + Z34 Take the earth voltage of zero potential reference. -AVs/Ax = + (A-ll) into + Ysa (A-12) s2lTsoca/1/n(r3/r2) s2isOq22/1n(r5s/r4 ) (A-21) Za4 (zsa - + 2Z3m za4 + Z2m - Zo)Is + 2Z3m (za4 Zo)IC + (zsa Z3m +Zo)Ia - Ya4 = s21rSo0s3/ln(r7/r6) Potential coefficients being inversly related to admittances.Define zsa by: zsa = Z20 + Z23 + Z3i -AV23/AX Z3i)T4 + Z2mIC Z3mIe - Zsa = za4.. Is(14) . sheath and armor is shown in Fig. -AV23/Ax = zsaI4 + IC= YcsAX(Vc s= ycsAx(Vs - Vs) Vc) + + Ic + AIC YsaAX(Vs - Va) + Is + AIs Z2mIc Ia = YsaAx(Va - Vs) + Ya4AxVa + Ia + Ala (A-18 For voltage V314. (A-7). (5) x+AX ~~~X Ax -+a and [Zi] = Zcc Zcs Zca [Zo] = Zo Zo Z1 Vc.4 2Z3m = (A-17) eq.z2m . (A-14) = [Z] is given by: [Z] = [Zi] + [Zo] (A-23) S/ycs.wx oowoxo} ""4 -1 Ycs core Ic+aIC Zcs Zss Zsa Zca Zsa Zaa where Zcc = = Zo Zo Zo (A-16) 2(Z2m = Vs. the potential coefficient matrix is derived in the same manner as the underground cable case. eq.2z3M +* Zo)Ic + (zsa + Za4 .Z3m Appendix 2 Potential coefficient Aneq`uival ent circuit for the admittance of under ground SC cable with a core. where -AVc/Ax = (Zcs + Zsa + Za4 . considering a space admittance being connected in series to Ya4 in Fig.(A-3).A-2. (A-6) z3mIe = eq.. Ia~ Zcs + Zsa + Za4 - - + = TF Va1 Zss Zsa + Za4 2z3m Z2m - Z Zaa. [Pi]'= Pc+Ps+Pa Ps+Pa Pa Ps+Pa Ps+Pa Pa x40.2z3m + Zo )Is + (Za4 (A-13) Z3m + ZO)Ia Finally frpm eqs. A-2 An equivalent circuit for admittances of an SC cable . Ps s/Ysa.A-2.(A-3) and (A-10) into eq. (A-10) and eq. - -[Yi].909 -AV12/AX For voltage = ZCSIC + Z2mI4 (A-5) V2. Ic 4X.(V) (A-20) -AVa/x Z3m + Zo) (Ic + (za4 + Zo)Ia IS) (A-11) Substitute eqs.3 = (z20 + Z23 + . (10) as -AIa/Ax Put x -+ = -ysaVs + (Ycs (Ysa + Ysa)Vs Ya4 )Va 0 - YsaVa + (A-19) c 0 in the above equations. If the discussor asked formulas of each component impedance and admittance. Semlyen and D. Clarifications concerning both problems discussed above would be welcome. Paper No. Semlyen's comment that the earth return current is actually quite small. A 78 001-0. As far as Carson's or Pollaczek's earth return impedance is adopted. For such a case. If he asked the derivation of the component impedances and admittances. A. Ametani: The author would like to thank the discussor for his interest in this paper. New York City. in other words. Manuscript received July 30. It is probably in conductors and not in dielectrics where this assumption is considered. Ontario. 1979. is well known and can be found in references 11 to 13. the author is not sure what the discussor meant by his question. a systematic matrix formulation is useful. references 1 and 10 could be the answer for a coaxial cable and reference 3 for a pipe-type cable. In reply to his first comment. namely the earth return mode. Kiguel. The contribution consists in assembling the basic data into matrices defined in (1) and (2). I agree with Prof. Could the author indicate the reference which provides details for the calculation of the cable parameters needed for the computation of the impedance and admittance matrices? . we can not deal with the displacement currents between a conductor and earth. the earth return path can be neglected if one concerns only the propagation modes within the pipe. the ground path can be neglected. . therefore.The author's remark that pipe enclosures of finite thickness do not provide a complete shielding is theoretically correct. "Phase Parameters of Pipe Type Cables". Toronto. but. Reference [A] A. 1979. since all capacitive effects are related to displacement currents. Semlyen pointed correctly. if it is the case that the propagation mode between the pipe and the earth. The complexity of cable layouts tends to obscure the analysis of basic phenomena and. the earth return impedance is to be included in the pipe-type cable case. though in most cases it can be neglected. Only the formula of the earth return impedance is not shown in this paper. for instance if one wants to know the surface voltage of a gas insulated transmission line or bus which is overhead. as shown in reference [A]. and therefore. Ametani's paper on cable impedances and admittances is based on the assumption that such parameters are available between components of the cable. these are given in detail in the present paper. the displacement currents mentioned in the paper is related to the conductor as Prof. therefore. Among the basic assumptions listed by the author. This. we need to include the effect of the earth return path. Manuscript received October 22. it concerns with the earth return impedance. Concerning the second comment. The assumption of neglecting the displacement currents is concerned with the displacement currents between the cable and the earth. the ground return current is actually quite small and. But. presented at the 1978 IEEE PES Winter Meeting.910 Discussion Adam Semlyen (University of Toronto. Canada): Dr. becomes significant. we find that displacement currents are negligible. In reply to the third comment. however.