IEEE Transactions on Power Apparatus and Systems, Vol. PAS-99,No. 3 May/June 1980

A GENERAL FORMULATION OF IMPEDANCE AND ADMITTANCE OF CABLESA. AMETANI

Doshisha UniversityKyoto, Japan

ABSTRACT

Interest in the analysis of wave propagation char-acteristics and transients associated with cable sys-tems has rapidly increased. In order to answer theneed of the analyst,impedances and admittances of var-ious cables have to be known. This paper describes ageneral formulation of impedances and admittances ofsingle-core coaxial and pipe-type cables. The formu-lation presented here can handle a coaxial cable con-sisting of a core, sheath and armor, a pipe-type cableof which the pipe thickness is finite and an overheadcable, which has not been discussed in the literatureheretofore.

Using the formulation presented in this paper, itnow becomes possible to analyze wave propagation char-acteristics and transients on any type of cable system.

1. INTRODUCTION

The growing use of cable systems and the increas-ing levels of capacity makes the analysis of wave prop-agation characteristics and transients on cable sys-tems an important task. The cases of underground sin-gle-core coaxial cables (SC cables) consisting of acore and sheath, and pipe-type cables (PT cables) ofwhich the pipe thickness is assumed to be infinite havebeen well studied!-5 However, SC cables consisting ofa core, sheath and armor, PT cables of which the pipethickness is finite, and overhead cables have neverbeen studied.

SC cables with a core, sheath and armor are quiteoften seen in the submarine cable case, and, in fact,this author has been asked about the possibility of cal-culating transients on such cables.6 So far, pipe en-closures were assumed to act as complete shields, thusavoiding consideration of earth return currents. Asfar as wave propagation characteristics and transientson inner conductors of the pipe enclosure are concern-ed, the assumption of the infinite thickness of thepipe is quite acceptable. But once the wave propa-gation and transients on the pipe are to be included,all the previous studies are not applicable. Thus, weneed a way of handling voltage and current on the pipe.

An analysis of overhead cables seemed to be over-due, although there is some need for it. This authorhas been asked to calculate transients in a gas insu-lated substation, where a bus and circuit breaker areenclosed in a pipe, and the pipe is overhead7 This canbe considered to be an overhead cable.

Because of the situation explained above, a for-mulation of impedances and admittances, which is ableto deal with an SC cable consisting of a core, sheathand armor, a PT cable having a finite thickness of the,pipe, and an overhead cable, has been developed in thepresent paper. The formulation is carried out in ageneralized manner so as to be able to handle all theabove cases.

F 79 615-6 A paper recawended and approved by theIEEE Insulated Conductors Cacimittee of the IEEE PowerEngineering Society for presentation at the IEEE PESSummer Meeting, Vancouver, British Columbia, Canada,July 15-20, 1979.Manuscript submitted January 26, 1979;made available for printing May 1, 1979.

2. IMPEDANCE AND ADMITTANCE

The impedance and admittance of a cable systemare defined in the two matrix equations.

d (V)/dx = - [Z] * (I)

d(I)/dx = - [Y] *V)

(1)(2)

where (V) and (I) are vectors of the voltages and cur-rents at a distance x along the cable. [Z] and [Y] aresquare matrices of the impedance and admittance.

In general, the impedance and admittance matricesof a cable can be expressed in the following forms.

IZ] = [Zi] + lZp] + [Zc] + Izo][Y] = s .[p1

[P] = [Pi] + IPp] + [Pc] + [Po] }(3)

(4)

where [P] is a potential coefficient matrix, and s=jw.In the above equations, the matrices with subscri-

pt"i"concern an SC cable and the matrices with subscri-pts"p"and "c" are related to a pipe enclosure. The ma-trices with subscript"o"concern cable outer media,i.e.air space and earth. When a cable has no pipe enclo-sure, there exists no matrix with subscripts"p"and "c".

In the formulation presented here, the followingassumptions are made.(1) The displacement currents and dielectric lossesare negligible.(2) Each conducting medium of a cable has constantpermeability.(3) The pipe thickness is greater than the penetrationdepth 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 ImpedanceWhen an SC cable consists of a core, sheath and

armor as shown in Fig.l (a), the impedance is given inthe following form based on the result of Appendix 1.

IZ] = [Zi] + IZo] (5)

where

[Zi] = SC cable internal impedance matrix

= [Zil][IO ]-^-[ O ]

[0° [Zi2]@**[ ° ]

[ °] ° ]...[Zin]li0 ipdne arxoftecbl ue

(6)

[ZO] impedance matrix of the cable outermedium (earth return impedance)

= [loi11liZo 12] -[Zo ln]

[Zo12] [Z022]. i[Zo2n] (7)

[Zo in] [Zo2n].- li[Zonn]All the off-diagonal submatrices of [Zi] are zero.

0018-9510/80/0500-0902$00.75 © 1980 IEEE

902

903

core (pi, 1 1, cl)- F -11sheW4P 2,2 2)

sulator 3 (Ej3) armor(p3, V3, C3)(a) SC cable cross-section

k-th cable

Fig. 1 An SC cable systemA diagonal submatrix expresses the self-impedance ma-trix of an SC cable. When the SC cable consists of acore, sheath and armor, the self-impedance matrix isgiven by:

[Zjj]= Zccj Zcsj ZcajZcsj Zssj Zsaj (8)

Zcaj Zsaj Zaajwhere

Zccj = core self-impedance

= zcs + Zsa + Za4 - 2Z2m - 2Z3M

Zssj = sheath self-impedance

= Zsa + Za4 - 2Z3m

Zaaj = armor self-impedance = za4

Zcsj = mutual impedance between the core andsheath

=Zsa + Za4- Z2m - 2Z3m

Zcaj = mutual impedance between the core andarmor

= Za4- Z3M

Zsaj = mutual impedance between the sheath andarmor

= Zcajwhere

Zcs = Zii + Z12 + Z2j

Zsa = Z20 + Z23 + z3i

Za4 = Z30 + Z34 I (10)

When the SC cable consists of a core and sheath,the matrix of eq.(8) is reduced to a 2 x 2 matrix.

[Z1j] = Fzccj Zcsl (11 )LZcsj Zss5J

where

Zccj = ZCS + ZS3 - 2Z2mZssj = Zs3

Zcsj = ZS3- Z2m (12)and

ZS3= Z20 + Z23

If an SC cable consists only of a core, the sub-matrix is redued to one element.

[Zij] = Zccj = Zil + Z12 (13)The component impedances per unit length in the

above equations are given in the following equaionsfor an SC cable shown in Fig.l(a).1'9'10

(1) z11: internal impedance of core outer surface

Z I = (sviopj/2T) (1/x2D1) -{Io (x2) * K1 (x1)+ Ko(x2)YIl(X1)}

(2) Z12 : core outer insulator impedance

Z12 = (s-popij/2r)+ln(r3/r2)(3) Z2;: internal impedance of sheath inner surface

Z2i = (sPoVi2/2Jr).(l/x3D2)*{Io(X3)KI(X4)+ Ko(x3)I1l(X4)}

(4) Z2m: sheath mutual impedanceZ2m = P2/2irr3r4D2

(5) Z20 : internal impedance of sheath outer surface

Z20 = (SioI2/27r) (l/x4D2).(Io(x4)-Kl(x3)+ Ko(x4)-II(x3)}

(6) Z23 : sheath outer insulator impedance

Z23 = (sipOpii2/2ir)*ln(r5/r4)

(7) Z3i : inteinal impedance of armor inner surface

z3i = (sloUV3/2ir)* (l/xsD3)*{ Io(xs)*Ki(x6 )+ Ko(xs)*Il(x6)}

(8) Z3m : armor mutual impedance

Z3m = P3/2irrsr6D3

(9) Z30 : internal impedance of armor outer surface

Z30 = (SPoP3/2T) (l/x6D3)-{Io(xr)Kl(x5)+ Ko(X6)-I (X5)I

(10) Z34 : armor outer insulator impedance

r,

904

Z34 = (S)o1Pj3/2Tr)*ln(r7/r6)

where

Di =

D2 =

D3 =

Xk =83

85 =

Il(x2).K1(xi) - Il(x1)-K1(x2)

Il(x4).KI(x3) - Il(x3) Kl(x4)

I1(X6)*K1(xs5) - I1(x5)*K1(x6)

aks, 82 = r2/4110"/Pl

r3_0_12/P2, 4 = r4/II0o2/P2r54i01i3/P3 ,3 s = r64iOvi3/P3

A submatrix of the earth return impedance [ZO] ineq.(7) is given in the following form.

[Zoj k] = Zojk Zojk Zojk

Zojk Zoik Zojk (14)

Zojk Zojk ZojkWhen the SC cable consists of a core and sheath,

the above matrix is reduced to:

[Zojk] [Zojk Zojk (15)Zojk Zojk

If the SC cable consists only of a core, thematrix includes only one element.

[Zojk] = Zojk (16)

Zoik in eqs.(14) to (16) is the earth return im-pedance between the j-th and k-th cables. When a cablesystem is overhead, the impedance is given by Carson!'When a cable system is underground,the impedance givenby Pollaczek"2 is used. If a cable is above a strati-fied earth, the earth return impedance developed byNakagawa, et.al.13 can be used.

2.1.2 Potential coefficientThe admittance matrix of a cable system is eval-

uated from the potential coefficient matrix as givenin eq.(4). In the SC cable case, [P ] and [Pc] arezero, and when the cable system is ungerground, [PO]is also zero. Thus, based on the result of Appendix 2,

(1) Overhead cable

[P] = [Pi] + [Po] (17)

(2) Underground cable

[P] = [Pi] (18)

where

[Pi] = cable internal potential coefficientmatrix

= [Pil] o

[Pi2] [ (19)

[Pin]i

[PO] = potential coefficient matrix of thesystem in air

= [Po11] [Po12].--- [Poin]

[PO2] [P022] [Po2n] (20)

[Poin] F PI2n] [Ponn]All the off-diagonal submatrices of [Pi] are zer.

A diagonal submatrix expresses the potential coeffi-cient matrix of an SC cable. When the SC cabe consistsof a core, sheath and armor as shown in Fig.l (a), thediagonal submatrix is given in the following form.

See Appendix 2.

[Pij] Pcj+Psj+Paj Psj+Paj Paj

Psj+Paj Psj+Paj Paf (21)

Paj Paj Pajwhere

Pcj = (1/27Troc1i)*ln(r3/r2)Psj = (1/27wco0j2)*ln(r5/r4)Paj = (1/27rToCj 3)*ln(r7/r6) I (22)

When the cable consists of a core and sheath, theabove matrix is reduced to:

[Pij] = [Pcj + Psj Psjl (23)

l Psj PsiJ

If the cable consists only of a core, then [Pij]includes only one element.

[Pii] = Pcj (24)

The submatrices of [Po] are given in the follow-ing form.

[Pojk] = Poik Pojk PojkPojk Poik PoJk

Poj k Poj k Poj k

where Pojk is the space potential coefficientgiven for the case of Fig.l (b) by:

Pojj = (1/2TrrO)*ln(2hj/r7j)

Pojk = (l/2Irco)*ln(D2/Dj) I

(25)

and is

(26)

2.2 Pipe-Type Cable (PT Cable)

2.2.1 ImpedanceThe impedance matrix of a PT cable shown in Fig.2,

where an inner conductor is assumed to be an SC cable,is given in the same manner as the SC cable case.8'9

(1) Pipe thickness assumed to be infinite

[Z] = [Zi] + [Zp] (27)

(2) Pipe thickness being finite

905

[Z] = [Zi] + [Zp] + [Zc] + [Zo]

where

SC cable internal impedance matrix

[Zil] [I 0 1] [ 0 ] 0

[ ° ] [ZM2].' [ 0 1 0

[0] t[] I [Zin] 0

0O 0.0--e 0pipe internal impedance matrix

[Zp11] [Zpl 2].] [Zpln] 0

[Zp12] [Zp22].* [Zp2n] 0

[Zpin] [Zp2n] - [Zpnn] 0

0 0 .- -0 0connection impedance matrix betweeninner and outer surfaces

[Zcll] ZC11] '' ZC1] ZC2

[ZCl] [ZC1]*' [IZCI] ZC2

[ZC1] [ZC1]'' [ZC1] ZC2

ZC2 ZC2 ' . ZC2 Zc3earth return impedance matrix

1Zo] [Zo].. [Zo] Zo[Zo] [Zo..[Zo] Zo

[Zo] [Zo- -[Zo] Zo

Zo Zo .* Zo Zo)

(28) In eqs.(29) and(30), the last column and row cor-respond to the pipe conductor. Thus, these should beomitted when the pipe thickness being assumed infinite.A diagonal submatrix of [Zi], i.e.eq.(29), is given ineq.(8). A submatrix of [Zp], eq.(30), is given in thefol 1 owi ng form.

(29) [Zpjk] r Zpjk Zpjk Zpjk (33)

Zpjk Zpjk Zpjk

Zpjk Zpjk ZpjkJWhen an inner conductor consists of a core and

sheath, eq.(33) is reduced to 2x2 matrix, and when theinner conductor consists only of a core, eq.(33) isfurther reduced to a column matrix in the same manneras explained in the case of [Zi]. ( See eqs.(8), (11)and (13). ) This is the same for all other impedance

(30) and admittance matrices explained in this section.Za k in eq.(33) is the impedance between the j-th

and k- inner conductors with respect to the pipe in-ner surface, and is given by3'8:

Zpjk = (s o/21T)* [ippKo(xj)/{xiKj(xi)}+ Q*k +pipe2P_iCn/{n(l + pp) + xiKn-1.(xj)/Knix1 )1]

pipe w=1 (34)where

(31)

(32)

Qjj = ln[(rpl/rj).{l-(dj/rpl)2}]

Qik = ln[rpl/!d% + di - 2djdkcosOjk] -nCn/nCn = (djdk/rpi )n, cos(n0jk)

and

xl= 1=sI rpj/pvO'ip/P~p (36)

A submatrix and the last row and column elementsof [Zc] in eq.(31) are given in the following form8'14

[Zci]= Zcl Zc1 Zc

Zci Zcl Zcl

Zci Zcl Zcl J (37)ZC1 = ZC3 - 2zpnZC2 = Zc3 - Zpm

ZC3 = Zpo + Zp3

where

Zpm = Pp/(2-nrpirp2Dp)Zpo = (Svolop/27rx2Dp)j{ Io(x2) K1(xl)

+ KO(X2)Il1(Xl)}zp3 = (slao/21r).ln(rp3/rp2)

by:Fig. 2 A PT cable

D = Il(x2) Kl(xl) - Il(x1) Kl(x2)

x, is given in eq.(36).

x2 = a2A, a2 = rp2/iaOp/Pp

A diagonal submatrix of [ZO] in eq.(32)

[ZiI =

[Zp] =

LZc] =

[Zo] =

and

(38)

(39)

is given

[Zo) = ZO Zo Zo

Zo Zo Zo (40)

Zo Zo Zo

where Zo in the above matrix is the self earth returnimpedance of the pipe.2.2.2 Potential coefficient

The potential coeffidient matrix of a PT cableshown in Fig.2 is given in the following form.8'9

(1) Pipe thickness assumed to be infinite

[P] = [Pi] + [Pp] (41)

(2) Pipe thickness being finite(a) Underground cable

(42)[P] = [Pi] + EPp] + [Pc]

(b) Overhead cable

[P1I = [Pi] + [Pp) + [Pc] + [Po]where

(43)

[Pi] = SC cable internal potential coefficientmatrix

[Pill [ 0 I]e [ O ]

[ 0 1 [Pi?2] ' ' ' [ ]

[0 1 1 0 ] [Pin]

0

0

0 O 0 0

(44)

[Pp] = pipe internal potential coefficientmatrix

oe 11~~~~~~

[Ppill [Ppl 2] ** *[Ppln]

[PpI2] [Pp22]. .[Pp2n]

[Ppmn] [Pp2n] ... [Ppnn]

0

0

0

0 0' 0 0

[Pc] = potential coefficient matrix betweenpipe inner and outer surfaces

[Pc]

[PC] [Pc]---[Pc]

PC PC.....Pc Pc

[PO] = potential coefficient matrix of thein air

= [PO] [Po] ..[Po] Po0

[Po]

[Po]

[Po]- -[Po]

[Po]- -.[Po]

PO Po0....Po

PO

PO

PO

In eqs.(44) and(45), the last column and row cor-responding to the pipe conductor,these should be omit-ted when the pipe thickness being assumed infinite. Adiagonal submatrix of [Pi] in eq.(44) is given in eq.(21). Submatrix [Ppjk] of [Pp], eq. (45), is given inthe following form.

[Ppjk] = Ppjk Ppjk Ppjk

Ppjk Ppjk Ppjk

Ppjk Ppjk Ppjk

(48)

Ppjk in the above equation is the potential co-efficient between the j-th and k-th inner conductorswith respect to the pipe inner surface, and is givenin the following equation using Q of eq.(35).

Ppjj = Qjj/27rcpiso , Ppjk = Qjk/27Tcplo (49)

A submatrix and the last column and row elementsof [Pc] in eq.(46) are given by:

[PC] = PC PC Pc

Pc Pc Pc

Pc Pc PC

PC = (l/21Tcp2co)*ln(rp3/rp2) I

A submatrix and the last row and column elementsof the space potential coefficient matrix[PO] is givenin the following form.

[PO] = Po Po Po

PO PO PO

Po Po PO

PO = (l/27rEo).ln(2h/rp3)

3. DISCUSSION

(51)

(45) The formulation of impedances and admittances ofvarious cables given in the previous section includessome approximations. It may be important to discussthese approximations so as to make the limit of appli-cability clear when the formulation is used.

First of all, the major assumptions made for theformulation of impedances and admittances (on page 1 ofthe paper) should be discussed. The first assumptionis constant permeability. Quite offten, a pipe andarmor are ferromagnetic. It, however, seems to berather unusual to have high currents to cause satura-

(46) tion of the pipe or armor. Thus, in most cases, thesaturation of the pipe or armor can be neglected. Whenone needs to take the saturation into account, methodsproposed in references (3) and (4) can be used. In regardto the second assumption, displacement currents arenegligible as far as low frequencies (less than aboutlMHz) are concerned. In the analysis of transients and

pipe wave propagation on a cable system, the frequency ofinterest is, in most cases, less than 1 MHz. The die-lectric losses are small in comparison with the lossesin conducting media of cables and earth. Thus, theassumption is valid. The third assumption will be

(47) discussed later.No approximation is made for the impedances and

admittances of an SC cable as far as Carson's andPollaczek's earth impedances and Scheikunoff's cylin-drical conductor impedance are concerned. One shouldpay attention to the fact that Carsont and Pollaczek's

(50)

906

0 l

[PC] [PC]---[PC] PC

/-

907

formulas of the earth return impedance are not appli-cable at frequencies higher than about lMHz becausethe effect of displacement currents is not included inthe formulas'.3 Thus, the formulation of the impedancesof both SC and PT cables is correct only upto aboutlMHz.

One can easily find that the formulation of theimpedances and admittances of an SC cable given in thispaper is identical to that given in reference (1) forthe case of a coaxial cable consisting of a core andsheath.

Two assumptoins are included in the PT cable case.The first one is that the eccentric cable positionswithin the pipe do not affect the internal impedancesand admittances of the inner conductors (SC cable) andthe impedances and admittances between the inner andouter surfaces of the pipe. Thus, the inner conductorimpedance and admittance of a PT cable become the sameas those of an SC cable. The same assumption has beenmade in references (3) and (4). If one needs to takeinto account the effect of the eccentricity on the in-ner conductor impedance, the formula of the outer sur-face impedance of the inner conductor given in refer-ence (5) can be used.

The second assumption concerns the case of finitepipe thickness. It is assumed that the pipe thicknesswill be greater than the penetration depth in the pipewall. If the pipe thickness is smaller than the pene-tration depth, the formulas of the pipe internal im-pedance given in eqs.(34) and (35) and potential co-

efficient given in eq.(49) can not be used. In that

1 0-3EC:

._

10-4

o-!

case, accurate formulas of the impedance and potentialcoefficient can be derived based on the work done byTegopoulos and Kriezis.15 Since these formulas are toocomplicated for practical usage, the assumption ofinfinite pipe wall thickness may be used, but only tocalculate the impedance and potential coefficients ofthe pipe. Note that earth return currents are not neg-lected and that complete shielding is not assumed.This assumption introduces negligible error for actualPT cables and for frequencies above 1OHz. Fig.3 showsa comparison of the pipe impedances for the cases ofthe pipe thickness being finite and infinite. It isclear that the impedance for the finite pipe thicknesscase approaches that for the infinite thickness case,at the frequency of lkHz. When the pipe thickness is4mm, which is nearly equivalent to the penetrationdepth at 10 Hz, its impedance is almost identical tothat for the infinite thickness case in the frequencyrange shown in the figure. The pipe thickness is, inmost cases, greater than the penetration depth. Thus,the assumption is valid.

Calculated results of admittances of a single-phase SC cable are shown in Fig.4. From the results,it is clear that the admittance ofan underground cableare much greater than those of an overhead cable. Theimpedance shows not a significant difference betweenunderground and overhead cables. Thus, it should beexpected that the attenuation of the undeground cableis much higher than that of the overhead cable, and thepropagation velocity is lower in the underground case.Similar results are obtained for the PT cable case.

The internal impedances of SC cables are shown inFig.5. Significant differences are observed for thecases of SC cables consisting only of a core, of coreand sheath, and of core, sheath and armor.

io-2

E

r-

0)

S-0.01 0.1 1 10

frequency, kHz

1o-,

10-4

10oFig. 3 Effects of pipe thicknesson pipe inner surface impedance

E1-00

10 6

c

4f0

In

l o- B

j7j'1 /;///

f" -A

/

0.01 0.1 1 10 100 1000

frequency, KHz

10-21

-2

10-3m l04E *v

1C5

a,10-410-4

10-5

0.01 0.1 1 10 100 1000frequency, KHz

underground cable--- overhead cable

Fig. 4 Susceptances (imag.Y22) of SC cables

10-;

1

C~~~~~~~~~~~~~~~~

;01 i0.1

0.01 0.1 1 10 100 1000

frequency, KHz

Fig. 5 Internal impedances Zcc of SC cables

(a) Core and its outer insulator

(b) Core, sheath and its outer insulator

(c) Core, sheath, armor and its outer insulator

- a-Fl5

908

4. CONCLUSION

A general formulation of the impedances and ad-mittances of single-core coaxial cables and pipe-typecables is given. The formulation presented in thispaper can handle a coaxial cable consisting of a core,sheath and/or armor, a pipe-type cable of which thepipe thickness is either infinite or finite, and anoverhead cable. Numerical results based on this for-mulation are readily available using BPA's computerprogram EMTP with subroutine CABLE CONSTANTS.

ACKNOWLEDGEMENTS

The author would like to thank Prof. K. Tominagafor his encouragement, and Prof. R. Schinzinger ofUniversity of Californua for his helpful discussionand critical reading of the manuscript. The authoralso wishes to express his appreciation for financialsupport by Bonneville Power Administration.

REFERENCES

1) L. M. Wedepohl and D. J. Wilcox:"Transient analysisof underground power-transmission systems",Proc.IEEvol.120, pp.253-260 (1973)

2) L. M. Wedepohl and D. J. Wilcox: " Estimations oftransient sheath overvoltages in power cable trans-mission systems,' ibid., vol.120, pp.877-882 (1973)

3) G. W. Brown and R. G. Rocamora: "Surge propagationin three-phase pipe-type cables, Part I-Unsaturatedpipe", IEEE Trans. on Power App. & Syst., PAS-95,pp.89-95 (1976)

4) R. C. Dugan, et. al.: " Surge propagation in threephase pipe-type cables, Part II - Duplication offield tests including the effects of neutral wiresand pipe saturation", ibid.,PAS-96,pp.826-833(1977)

5) R. Schinzinger and A. Ametani: " Surge propagationcharacteristics of pipe enclosed underground cable<,'ibid., PAS-97, pp.1680-1687 (1978)

6) B. Dixon: Private correspondence (1977.9)

7) B. P. A.: Private correspondence (1977.8)

8) A. Ametani: "Generalized program for line and cableconstants", Bonneville Power Administration,Purchase Order No.70249, Report No.2 (1977.10)

9) A. Ametani: "Extension of generalized program forline and cable constants in EMTP", Bonneville PowerAdministration, Contract No.EW-78-C-80-1500, ReportNo.1 (1978.7)

10) S. A. Schelkunoff: " The electromagnetic theory ofcoaxial transmission line and cylindrical shields",Bell Syst. Tech. J., vol.13, pp.532-579 (1934)

11) J. R. Carson: " Wave propagation in overhead wireswith ground return", ibid., vol.5, pp.539-554(1926)

12) F. Pollaczek: "Uber das Feld einer unendlich langenwechsel stromdurchflossenen Einfachleitung",E.N.T.,Band 3 (Heft 9), pp.339-360 (1926)

13) M. Nakagawa, et,al.: " Further studies on wavepropagation in overhead lines with ground return ",Proc.IEE, vol.120, pp.1521-1528 (1973)

14) A. Ametani and T. Ono: " Wave propagation charac-teristics on a pipe-type cable, III - Consideration

of pipe thickness", IEE Japan,Proceedings of AnnualMeeting, Paper No.840 (1978)

15) J. A. Tegopoulos and E. E. Kriezis: "Eddy currentdistribution in cylindrical shells of infinitelength due to axial currents, Part II-- Shells offinite thickness'IEEE Trans. on Power App. & Syst.,PAS-90, pp.1287-1294 (1971)

APPENDICES

Appendix 1 Impedance of an SC cable consisting ofa core, sheath and armor

In the case of an SC cable with core, sheath andarmor, an equivalent circuit for impedances is givenin Fig.A-l.

x

Vc LV12

Vs Is

Va

V23Ia

V34le. -

Fig. A-1 An equivalent circuitof an SC cable

for impedances

Define currents flowing into the core, sheath,armor and outer medium (earth) by Ic, Is, Ia and le atx. Also inner and outer surface currents of the sheathand the armor are I2, I3, I4 and Is as shown in Fig.A-1. Voltages between the core, sheath, armor and outermedium are V12, V23 and V34 at x, and are V12+AV12,V23+AV23, and V34+AV34 at x = x + Ax.

Then, the following relation for currents areobtained.

12 = -Ic, I3 = -I4 , Is = -Ie

Is = I2 + I3 = -(Ic + I4)

Ia= I4 + Is = I4 - Ie

From the above equations,

I4 = -(Ic + Is)Ie= -(Ic + Is+ Ia)

(A-l)

(A-2)I

(A-3)

For voltage V12 between the core and the sheath,

V12 = Z11AxIc - z12AxI2 - z2iAxI2 - Z2mAXI3+ V12 + AV12

.: -AV12/Ax = (Zll + Z12 + Z2i)Ic + Z2mI4

Define Zcs by:

Zcs = Z11 + Z12 + Z2j

Using the above equation,

(A-4) = eq.(10)

909

-AV12/AX = ZCSIC + Z2mI4 (A-5)For voltage V2.3

-AV23/AX = (z20 + Z23 + Z3i)T4 + Z2mIC - Z3mIe,Define zsa by:

zsa = Z20 + Z23 + Z3i (A-6) = eq.(10)Then,

-AV23/Ax = zsaI4 + Z2mIc z3mIe

For voltage V314,

-AV34/AX = (za4 + Zo)Ie - ZSmI4where

Za4 = Z30 + Z34 (A-9) = e

Take the earth voltage of zero potential asreference,

Va = 34

VS = -(V23 + V34) = Va - V23VC = V12 + VS

Substituting eqs.(A-3) and (A-10) into eq.(A

-AVa/x = (za4 - Z3m + Zo) (Ic + IS)+ (za4 + Zo)Ia

Substitute eqs.(A-3), (A-10) and (A-ll) intoeq. (A-7).

-AVs/Ax = (zsa + za4 -+ Za4 - 2Z3m +

In the same manner,

-AVc/Ax = (Zcs + Zsa ++ (zsa + Za4 -

+ (Za4 - Z3m +

Finally frpm eqs.(A-11),x40,

Z2m - 2Z3m + Zo)ICZo)Is + (za4 - Z3m

Za4 - 2z2m - 2z3M +z2m - 2z3m + Zo )IsZO)Ia(A-12) and (A-13) w

d(V)/dx = -[Z]*(I)

where [Z] is given by:

[Z] = [Zi] + [Zo]

and

[Zi] = Zcc Zcs

Zcs Zss

Zca Zsa

(A-1 5) =

Zca , [Zo] = Zo Zo Z1

Zsa Zo Zo Zo

Zaa ZO Zo zoJ(A-16) = eqs.(8) anrd (14)

(A-7)

(A-8)

!q. (10)

(A-1 0)

(-8).

Zsa = za4- Z3m

Appendix 2 Potential coefficientAneq`uival ent circuit for the admittance of under

ground SC cable with a core, sheath and armor is shownin Fig.A-2. From the figure,

IC= YcsAX(Vc - Vs) + Ic + AIC

s= ycsAx(Vs - Vc) + YsaAX(Vs - Va) + Is + AIs

Ia = YsaAx(Va - Vs) + Ya4AxVa + Ia + Ala (A-18

Rewriting the above equations,

-AIC/AX = ycsVc -YcsVs

-AIs/AX = -YcsVc + (Ycs + Ysa)Vs-AIa/Ax = -ysaVs + (Ysa + Ya4 )Va

Put x -+ 0 in the above equations,

dxI = - Ycs 0dxi r

Is -Ycs (Ycs+Ysa) -Ysa

LIal 0 -Ysa (Ysa+Yi

- -[Yi].(V)

- YsaVa

(A-19)

c

IIVs

la4)JIVaJ

(A-20)

where

(A-11) Ycs = s2lTsoca/1/n(r3/r2)

Ysa = s2isOq22/1n(r5s/r4 ) (A-21)

Ya4 = s21rSo0s3/ln(r7/r6)+ (zsa+Zo)Ia Potential coefficients being inversly related to(A-12) admittances,

[Pi]'= Pc+Ps+Pa Ps+Pa Pa* Zo)Ic Ps+Pa Ps+Pa Pa (A-22) eq.(21)

(A-13) Pa Pa Pa

rith Pc = S/ycs, Ps = s/Ysa, Pa = s/ya4 (A-23)-eq.(22)

When a cable is overhead, considering a space ad-(A-14) mittance being connected in series to Ya4 in Fig.A-2,

the potential coefficient matrix is derived in thesame manner as the underground cable case.

eq.(5) ~~~Xx+AXeq. (5)- Ax -+a

Vc, Ic 4X,, / ,wx oowoxo} ""4 Ic+aIC

Vs, Is-..+-zzzzzzzz-IzI s

Va, Ia~

Zcc = Zcs + Zsa

Zss = Zsa + Za4

Zcs = Zsa + Za4

+ Za4 - 2(Z2m + Z3m)-

- 2z3m Z =Zaa.4 (A-17)- Z2m - 2Z3m = eq.(9)

-1 Ycs core

sheath- Ysa

TF Va1

armor

earth

Fig. A-2 An equivalent circuit for admittancesof an SC cable

wherea+A I a

910

Discussion

Adam Semlyen (University of Toronto, Toronto, Ontario, Canada):Dr. Ametani's paper on cable impedances and admittances is based onthe assumption that such parameters are available between componentsof the cable. The contribution consists in assembling the basic data intomatrices defined in (1) and (2). The complexity of cable layouts tendsto obscure the analysis of basic phenomena and, therefore, a systematicmatrix formulation is useful. Could the author indicate the referencewhich provides details for the calculation of the cable parametersneeded for the computation of the impedance and admittance matrices?

,The author's remark that pipe enclosures of finite thickness do notprovide a complete shielding is theoretically correct, but, as shown inreference [A], the ground return current is actually quite small and,therefore, the ground path can be neglected.

Among the basic assumptions listed by the author, we find thatdisplacement currents are negligible. It is probably in conductors andnot in dielectrics where this assumption is considered, since all capaci-tive effects are related to displacement currents.

Clarifications concerning both problems discussed above would bewelcome.

Reference

[A] A. Semlyen and D. Kiguel, "Phase Parameters of Pipe TypeCables", Paper No. A 78 001-0, presented at the 1978 IEEE PESWinter Meeting, New York City.

Manuscript received July 30, 1979.

A. Ametani: The author would like to thank the discussor for his inter-est in this paper.

In reply to his first comment, the author is not sure what thediscussor meant by his question. If he asked the derivation of the com-ponent impedances and admittances, references 1 and 10 could be theanswer for a coaxial cable and reference 3 for a pipe-type cable. If thediscussor asked formulas of each component impedance and admittance,these are given in detail in the present paper. Only the formula of theearth return impedance is not shown in this paper. This, however, is wellknown and can be found in references 11 to 13.

Concerning the second comment, I agree with Prof. Semlyen'scomment that the earth return current is actually quite small, andtherefore, the earth return path can be neglected if one concerns onlythe propagation modes within the pipe. But, if it is the case that thepropagation mode between the pipe and the earth, namely the earthreturn mode, becomes significant, for instance if one wants to know thesurface voltage of a gas insulated transmission line or bus which is over-head, we need to include the effect of the earth return path. For such acase, the earth return impedance is to be included in the pipe-type cablecase, though in most cases it can be neglected.

In reply to the third comment, the displacement currents mention-ed in the paper is related to the conductor as Prof. Semlyen pointedcorrectly. The assumption of neglecting the displacement currents isconcerned with the displacement currents between the cable and theearth, in other words, it concerns with the earth return impedance. Asfar as Carson's or Pollaczek's earth return impedance is adopted, we cannot deal with the displacement currents between a conductor and earth.

Manuscript received October 22, 1979.