Wedepohl_1973_Transient analysis of underground power-transmission systems. System-model and wave-propagation characteristics

Prévia do material em texto

Transient analysis of underground power-transmission
systems
System-model and wave-propagation characteristics
Prof. L. M. Wedepohl, B.Sc.(Eng.), Ph.D., C. Eng., F.I.E.E., and D. J. Wilcox, B.Sc. Tech. M.Sc., Ph.D.
Indexing terms: Modelling, Power transmission, Skin effect, Transients, Transmission-line theory, Underground
cables
ABSTRACT
A mathematical model suitable for the analysis of travelling-wave phenomena in underground power-transmission
systems is presented. The model is developed in terms of a generalised angular frequency, and may therefore be
applied to the solution of steady-state problems or, by means of Fourier-transform techniques, to the solution of
transient problems. The model takes into account skin effect in the conductors and in the soil. It is then shown
how the system model may be analysed using multiconductor-transmission-line theory to give the transient res-
ponse of the cable system. The wave-propagation characteristics are given for the natural modes of a certain
cable system. These characteristics are examined with a view to their implications on transient phenomena.
LIST
V
I
Z
Y
J
H
E
r
A
U)
P
a
e
Y
m
s
n
OF PRINCIPAL SYMBOLS
= conductor-voltage vector
= conductor-current vector
= series-impedance matrix
= shunt-admittance matrix
= current density
= magnetic-field intensity
= electric-field intensity
= radius
= radius of central conductor
= radius over main insulation
= radius over conducting sheath
= outer radius of cable
=
 r3~r2
= angular frequency
= resistivity or charge density
= conductivity
= permittivity
= permeability
= Euler's constant
= cable separation
= number of effective metallic conductors in system
1 INTRODUCTION
The spreading of urban centres, and the ever increasing de-
mand for electrical power within them, is leading to the use
of relatively long cable circuits operating at high voltage.
Under these conditions, we can expect to be faced with a
problem of transient overvoltages being induced in the
conductors of the underground system whenever it is sub-
jected to a sudden disturbance—a switching operation or per-
haps an insulation failure.
The transient stress across the main insulation owing to
such a disturbance may be estimated with enough accuracy
for most practical purposes by ignoring the presence of the
soil and treating the problem as one of classical wave prop-
agation along a coaxial transmission line. However, as this
method ignores the presence of the soil and neighbouring
cables, it precludes the calculation of both sheath transient
voltages and transient induction effects into other cables.
The method of transient analysis proposed here is based on
the theory of wave propagation in multiconductor systems,1
and therefore takes into account all the metallic conductors
in the system as well as the ground itself. In the theory, the
transmission system is defined in terms of a series-
impedance matrix Z and a shunt-admittance matrix Y. It is
shown in the paper how these matrices may be derived in
terms of a generalised angular frequency to include an allow-
ance for the frequency-dependent nature of the parameters.
Paper 6872 P,first received 4lh September and in revised
form 20th November 1972
Prof. Wedepohl is with the Department of Electrical Engineer-
ing & Electronics, University of Manchester Institute of
Science & Technology,PO Box 88,Sackville Street, Manchester
M60 1QD, England, and Dr. Wilcox is with Mons Polytechnic,
Mons, Belgium
PROC. IEE, Vol. 120, No. 2, FEBR UAR Y 1973
P6
The method of analysis may be interpreted in terms of
natural modes of propagation. The characteristics of these
modes, as a function of frequency, are given for a represen-
tative cable system. The essential nature of these charac-
teristics is explained, as they have an important bearing on
the understanding of transient phenomena.
Briefly, the paper forms a basis for further work on the
transient performance of underground-cable systems.
MATHEMATICAL REPRESENTATION OF THE
TRANSMISSION SYSTEM
The analysis is based on the two matrix equations
dx
dl.
dx
= - Y . V
(la)
(lb)
where V and I are vectors of dimension n representing, res-
pectively, the voltages and currents at a distance x along the
cable system containing n metallic conductors. Z and Y are
square matrices of dimension n x n.
2.1 Principal assumptions
Eqns.2 constitute a mathematical model of the transmission
system, and may be formulated on the basis of the following
principal assumptions:
(i) The cable system consists of n effective metallic con-
ductors whose axes are mutually parallel and are paral-
lel to the surface of the Earth.
(ii) The system is longitudinally homogeneous.
(iii) The anticipated attenuation of conductor voltages and
currents is negligible along a length of system com-
parable with its lateral dimensions.
It is convenient to assume that, the system consists of N
cables each with a cross-section of the type shown in Fig. 1.
Each cable then has two metallic conductors, of which one is
the central conductor and the other is a conducting sheath or
armour. This is the basic type of design normally employed
for high-voltage cables. The same basic construction can be
used to represent a screened communication cable where the
conducting sheath
central
Fig.l
Basic cable construction
253
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group of central conductors may be reasonably represented
by a single equivalent conductor for calculating longitudinal-
induction effects.
There are n = 2N metallic conductors, and the soil in which
the cables are buried constitutes an (n + l)th conductor
which is chosen as the voltage reference for the system.
2.2 Formulation of shunt-admittance matrix
Maxwell's field equations show that, in any electromagnetic
field,
3t ~ ep
where p is the volume charge density at any point. This
equation has the solution p = p0 exp (—a/et), which shows
that, if a charge is introduced into a conducting medium (a *
0), it will dilute itself with a time constant of e/o seconds. In
a good conductor such as copper, this time constant will be
of the order of only 10~19 s, and therefore a charge intro-
duced into the body of a good conductor will be very quickly
displaced onto its surface. The process is not so rapid in the
soil, which is a poor conductor, but, even for a relatively high
soil resistivity, say 1000 fim,the time constant is still quite
small, being of the order of 10~8 s. Thus, if the smallest
unit of time of interest is not less than, say, 10~6 s,the
electric charges in the cable system may be assumed to be
surface charges.
In the present model, it is, indeed, assumed that the smallest
time interval of interest in the transient response is large
enough to justify the assumption that all charges are surface
charges. This greatly simplifies the calculations, as the
instantaneous value of the time-varying lateral electric field
has the same form as the electrostatic field. This latter is
readily deducible from elementary theory.
2.2.1 Admittance submatrices
The shunt admittance submatrix of the ith c?ble has the di-
mension 2 x 2 and is given by
v i -Y
Y2
where
Yl = gi + jw2ire1/ln(r2/r1)
Y2 = S2 +jw277€2/ln(r3/r4)
in which gx and g2 represent the leakage conductances across
the inner and outer insulations, and ex and e2 the correspond-
ing permittivities.
2.2.2 Assembly of admittance Matrix
The admittance matrix Y is assembled from the N sub-
matrices Yj by placing them along the leading diagonal. The
soil acts as an electrostatic shield between cables, and
hence offdiagonal submatrices are null. Thus
2.3 Formulation of the series-impedance matrix
The series-impedance matrix Z is built upfrom N2 sub-
matrices. The N submatrices which fall on the leading di-
agonal of Z are obtained as described in the following Section.
2. 3.1 Leading-diagonal impedance submatrices
The impedance submatrix of each individual cable is con-
structed from seven component impedances. These are
(i) z1} the internal impedance of the inner conductor
(ii) z2, the impedance due to the time-varying magnetic
field in the inner insulation
254
(iii) z3, the inner sheath internal impedance. This impe-
dance is calculated from the voltage drop on the inner
surface of the sheath per unit current which returns
via the inner conductor
(iv) z4, the sheath mutual impedance. This impedance is
given from the voltage drop along the outer (inner)
surface of the sheath per unit current returning through
the inner (outer) conductor. In this case, the outer
conductor is the soil
(v) z5,the outer sheath internal impedance. This is given
from the voltage drop along the outer surface of the
sheath when the current returns through the outer
conductor
(vi) z6, the impedance due to the time-varying flux in the
outer insulation
(vii) z7, the self impedance of the earth-return path.
At low frequencies, zx is equal to the d.c. resistance of the
inner conductor. At higher frequencies, the classical formula
pm
27rr1I1(mr1)
ohms per metre (2)
which takes account of skin effect. An approximate formula,
which is generally more suitable for digital computation, is
deduced in Appendix 8.1, and quoted here:
_ pm
1
 27rr1
coth + - ohms per metre (3)
The maximum error in the resistive (real) part of this im-
pedance is 4% and occurs when | rm^ | = 5. The maximum
error in the reactive component is 5%, and occurs when
Irm^ | = 3*5. Away from these maxima, the formula is very
accurate. This formula is justified by the fact that it general-
ly requires rather less digital computation than that of eqn.
2, to which it is an approximation.
The impedances z2 and z6 present no problem, and are
given by
In(r4/r3)
where /ix and /i2 are,respectively,the magnetic perme-
ability of the inner insulation and the permeability of the
outer insulation.
2.3.2 Sheath impedances
Classical formulas for the three sheath impedances are
given in Appendix 8.1. These formulas involve Bessel func-
tions, and are seen to be rather complicated. The following
three approximate formulas give good accuracy for
( r 3 - r 2 ) / ( r 3 + r 2 ) < l / 8 :
Zo = coth (mi) ^ ohms per metre (4)
27rr2(r2 + r,)
z4 = ——° 7 cosech (mA) ohms per metre
7r(r2 + r3)
(5)
ohms per metre (6)
where m = V(jaj/i/p),p is the sheath resistivity and A =
r3 — r2 (the thickness of the sheath). For cable sheaths
normally encountered in practice,the above condition is well
satisfied, and perfectly adequate accuracy may be expected
from the above simple formulas. The derivation of these
formulas is briefly indicated in Appendix 8.1.
2.3.3 Self impedance of the earth-return path
The self impedance of the earth-return path of a buried
cable has been worked out by Pollaczek,5 and, as with Car-
son's formulas for the overhead line, leads to a rather
Pi?OC. IEE, Vol.120, No. 2, FEBRUAR Y 1973
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involved infinite series. Fortunately, up to quite high fre-
quencies, only the following terms need be taken into account:
•' • z7 = ^ - {— ln(ymr4/2) + 1/2 — 4mh/3} ohms per metre
(7)
where y is Euler 's constant and h is the depth at which the
cable is buried. This expression is very accurate for fre-
quencies for which | mr4| < 0*25. In fact, eqn. 7 should give
high accuracy in all practical cases of interest.
2.3.4 Assembly of the submatrix
From the viewpoint of series impedance, the cable may be
represented by the equivalent circuit shown in Fig. 2. This
central
conductor
Z5+Z6*Z7-Z4
conducting
sheath
earth o
Fig. 2
Impedance equivalent circuit
representation may be deduced by considering the effects of
two circulating currents: one flowing down the central con-
ductor and returning through the sheath, and the other flow-
ing down the sheath and returning through the earth. This
equivalent circuit corresponds to the following impedance
submatrix:
z6 +z7-z,
Zr, + Zc + Z , — Z,
2.3. 5 Assembly of the impedance matrix
The submatrices Zj are assembled along the leading diagonal
of the Z matrix in the same order as for the shunt-admit-
tance matrix. The matrix Z then takes the form
Z =
Z21 Z 2
Z31 Z32
Z13
Z23
The offdiagonal submatrices Zjj are not null,but take account
of the mutual inductance between cables.
The net current in the ith cable returning through the soil,
i.e.the algebraic sum of the conductor and sheath currents,
gives rise to a voltage drop in the soil adjacent to the jth
cable. This voltage drop gives the e.m.f.induced into each
conductor of this cable; i.e.the same e.m.f.is induced in both
conductors. It follows from these remarks that all four
elements of the submatrix Z« must be equal;i.e.
Z =
The elemental impedances ZJJ may be calculated from Pol-
laczek's formulas. Formulas equivalent to those of Pol-
laczek have been deduced in Appendix 8.1 by means of inte-
gral-transform techniques. For |ms«| < 0*25,the mutual
impedance zx between the ith and jth cables is given by
plicated, and the impedance may be obtained directly from
the impedance formula given by eqn. 34; i.e.
\a\ + V(a2 +
 m2)
-^jiVCa2 + m2) - e-'jiVfo2 + m2)
2n
where £•» and 6U are,the moduli of the sum and difference,
respectively, of the depths of the ith and jth cables and XJJ
is the horizontal distance between them. The integral con-
verges rapidly, and is therefore suitable for numerical
evaluation. However, in view of the simplicity of eqn. 8, the
•numerical integration is best reserved for the case where
|mjj| > 025. In this case,the integral is particularly well
behaved.
Finally,by virtue of the reciprocality of mutual impedance,
3 ANALYSIS
The voltages and currents represented by V and I and given
in eqns. 1 are implicitly functions of the generalised angular
frequency u>. For harmonic problems,i.e.problems involving
only one frequency, u> is simply the angular frequency in
question, and V and I represent phasors. For transient prob-
lems, V and I represent the Fourier transforms of the volt-
ages and currents at values of w lying along the path of the
Fourier integral necessary to convert the solution back into
the time domain. In this latter case, u> need not be real, and,
indeed, it is advantageous to choose a path of integration dis-
placed below the real axis2'3 in the u> plane which requires
a complex value of w.
The elimination of the current vector I from eqns. 1 gives
the multiconductor Telegrapher's equation.
d2V
dx2 = ZYV
which has the solution
V = exp (-xV(ZY))V+ + exp (+xV(ZY))V" (9)
for the conductor voltages at a distance of x along the sys-
tem. V+ and V~ are vectors containing the 2n constants of
integration which are deduced from a knowledge of the sys-
tem boundary conditions. The matrix functions may be
efficiently evaluated as a consequence of the eigenvalue
definition.6
The solution for the system currents follows from the above
result by using eqn. la to give
I = Z"1 V(ZY) [exp {-xV(ZY)}v+ - exp {+xV(ZY)}V"] (10)
The distribution vectors for the n natural modes of propa-
gation are given by the n columns of the eigenvector matrix
of ZY and the distribution vectors for the n natural modes
of current propagation are given by the n rows of the in-
verse eigenvector matrix. These vectors permit a physical
indentification of the natural modes.
The propagation coefficients of the n natural modes are
given by the eigenvalues of the matrix V(ZY).
For transient problems,the above solutionsare transformed
back into the time domain by means of the Fourier integ-
ral.2;3 Thus,for example,
2TT — ml ohms per metre (8) V(t) = j - /_+°° V(w) exp (jwt)dw
where S-H is the distance between the ith and jth cables,/! is
the magnetic permeability of the soil,y is Euler's constant
and I represents the sum of the depths of the ith and jth
cables. This formula reduces to that given by eqn. 7 by re-
placing Sji by the cable radius r4 . The formula will,in fact,
be valid up to quite high frequencies in practice—of the
order of 100 kHz for cables buried in the same trench.
At frequencies for which | ms-jjl > 0-25, the full form given
by eqn. 40 may be used. However, this form is rather com-
PROC.IEE, Vol.120, No.2, FEBRUARY 1973
4 NATURAL MODES OF PROPAGATION
The nature of the natural modes of wave propagation in cable
systems will be illustrated by considering a cable circuit
consisting of three identical power cables layed in a flat con-
figuration at a depth of 0 76 m and with a cable separation of
015 m,as shown in Fig. 3. The cable-design specification is
given in Appendix 8.4 (Fig. 3).
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4.1 Characteristics
The attenuation characteristics of the six natural modes of
wave propagation of the present system are shown in Fig. 4
and the velocity characteristics are shown in Fig. 5. The
air
soil
20 n m
15cm I 15cm
75cm
Fig. 3
Cable configuration
10-Or—
0 01
0001
O l -
100 1000
frequency, Hz
10000 100000
Fig. 4
Attenuation characteristics of natural modes of propagation
modal-current vectors at 1000 Hz are given in Table 1. The
modes have been designated a to f in the Figures, and may
be identified with the help of Table 1.
Mode a is a zero-sequence sheath mode, and is seen to have
the characteristics of relatively high attenuation and very
low velocity. The velocity is 13 5 km/ms at 1000 Hz and
increases only slowly with frequency. The low velocity is
due to the high inductive impedance of the soil path, which is
of the order of 100 greater than the inductive impedance of
the cable outer insulation.
Modes b and c are seen from Table 1 to be intersheath
modes. Mode b is energised by injecting unit current into
the sheath of the central cable (sheath 2) and extracting half
of it from sheath 1 and the other half from sheath 3,while
mode 3 is energized by injecting unit current into sheath 1
and extracting it from sheath 3. These two modes are seen
from Fig. 4 to have a quasiconstant attenuation over quite a
wide frequency band (of the order of two decades). This band
is very important, as it tends to dominate the frequency
spectra of many of the transient responses of interest.
160i—
120 —
10 100 1000 10000 100000
frequency, Hz
Fig. 5
Velocity characteristics of natural modes of propagation
Fig. 5 shows that,for frequencies in excess of about 200 Hz,
the velocities of these two modes are constant. A surprising
feature is that these velocities are only of the order of one-
quarter of the natural velocity (165 km/ms) associated with
the cable insulation. A further point of interest is that, at
the higher frequencies,these modes are the least lossy.
These modal characteristics are evidently rather different
in nature to those of overhead transmission lines, where,in
the latter case,all modal velocities,even the ground mode,
are a high fraction of the velocity of light and where low
velocity implies high attenuation.
The nature of the sheath characteristics may be explained as
follows. Consider,for example,the case of the intersheath
mode between the two outer conductors (1 and 3). If the
presence of all the system conductors except these two
sheaths is ignored,this 2-conductor path will have the
impedance
zp = 2RS + 2z i + 2(ze - zm) (11)
where Rs is the sheath resistance, z^ is the impedance due
to the inductance associated with the cable outer insulation,
ze is the self impedance of each cable due to the earth path
TABLE 1
MODE-DISTRIBUTION VECTORS FOR CURRENT
Modal vectors for current
Mode a
Velocity, km/ms 13-5
Attenuation, dB/km 0'2637
b
49-0
0-1096
c
3 8 9
0-0866
d
144-8
00991
e
139-6
0-0945
f
141-7
0-0972
Conductor
1
Sheath 1
Conductor
2
Sheath 2
Conductor
3
Sheath 3
0001 94-3
0337 0
0001 94-2
0-350 0
0001 943
0-337 0
0018 279 0-016 94 0-337 0 0343 180 0503 0
0343 180 0503 0 -0-337180-0-34413 -0505187
0-034 99
-0-662 0
0350 0 -0-6620 0
-0-350180 0-664 193 0
0018 279 0-016 274 0337 0 0343 180 -0503 180
0-343 180-0-503180 -0-337180 -0-34413 0505 7
Frequency = 1000 Hz
256 PROC.IEE, Vol.120, No. 2, FEBRUARY 1973
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and z m is the mutual impedance between them. The impe-
dances ze and z m are, effectively, given by eqns.7 and 8;
Zrv, =
— ln
- I n -ms
1 4
~ — , - mh
1 4
2 " - y mh
where s is the cable separation. Now
and hence the series impedance Zp of the intersheath path
becomes
(12)
(13)
(14)
1
(15)
which is exactly the same series impedance as would be
obtained if the soil in which the cables are buried were re-
placed by air. This will remain valid so long as |ms | < 0'25,
i.e.up to a frequency of 1'7 MHz in the present instance,
which should be quite high enough for most practical transient
studies.
Consider now the shunt admittance yp of the same path; the
earth acts as an electrostatic shield oetween the cables, and
hence
(16)_ 1^ jae27Ty P " 2 In(r4/r3)
The propagation coefficient yp of this pure intersheath mode
is then given from eqns. 15 and 16 as
Now, if the inductance L of the cable outer insulation is
given by
(17)
(18)
and the sheath-earth capacitance C is given by
C = 27T€/ln(r4/r3) (19)
the propagation coefficient yp can be expressed by
yp = jwk[LC{l + Rs/(k2jwL)}]i/2 (20)
where
ln(s/r3)
k = In(r4/r3) (21)
At the higher frequencies,the square-root term in eqn.20
may be expanded as a convergent power series. In particular,
consider the case where the frequency is sufficiently high
that only the first two terms of this expansion need be con-
sidered; then
(22)
(23)
The modal velocity is then given by
U P ~ k 7(LC)
and the attenuation factor is given by
In these equations, l/V(LC) is the natural velocity for the
cable insulation and V(L/C) is the sheath-earth surge
impedance.
At frequencies above Rs/7rk2L,both the modal attenuation and
the modal velocity should be constant and are given by eqns.
24 and 23, respectively. The frequency Rs/7rk2L may be
deduced for the case in hand from the data specified in Ap-
pendix 8.4, and is found to be 250 Hz;the factor k is found to
be 421 . Thus, at frequencies in excess of 250 Hz, eqns. 23
and 24 predict the modal velocity to be constant at up =
PROC. IEE, Vol. 120, No. 2, FEBRUAR Y 1973
39*2 km/ms and the modal attenuation to be constant at aD =
9-98 x 10-6 (0-0866 dB/km). These estimated values are
seen to be in good agreement with those obtained from the
computer results given in Table 1. However, according to the
simple theory presented above, the attenuation should remain
constant as the frequency increases from 250 Hz, whereas it
is seen to remain constant only up to about 10 kHz. The ex-
planation is that the simple theory neglects skin effect which
causes the resistance to increase as the square root of fre-
quency; this is confirmed in Fig. 4, as the attenuations of all
but the ground mode increase half a decade for one decade
increase in frequency at the higher frequencies.
The factor k increases with cable separation, and hence both
the velocity and the attenuationof the intersheath modes de-
crease with increasing cable separation.
Modes d,e and f are seen from Table 1 to have essentially
coaxial natures, with velocities which approach the natural
velocity for the dielectric medium,i.e. 165 km/ms. Mode d
is a zero-sequence-type mode which is energised by inject-
ing unit current into each cable conductor and extracting it
from the corresponding sheath. Mode e attempts to be an
interconductor mode energised by injecting two units of
current into the conductor of cable 2 and extracting half of
this from the conductor of cable 1 and the other half from
the conductor of cable 3. However,much of the flux linking
with the interconductor path also links with the correspond-
ing intersheath path, and therefore induces an intersheath
circulating current which opposes this flux linkage, tending
to convert the interconductor mode into a coaxial mode and
an intersheath mode. At rather low frequencies,the circulat-
ing sheath current is restricted by the relatively high sheath
resistance,and mode d will,indeed,behave as a true inter-
conductor mode. As the frequency increases, the sheath
resistance becomes progressively less and less effective in
supressing the sheath current and, in fact,the Figures show
that the characteristics of mode e merge with those of the
pure coaxial mode e at about 100 Hz. A similar transforma-
tion into a coaxial mode occurs for mode f. However, at the
lower frequencies such that modes e and f retain an essential
interconductor-mode nature,the attenuation (and, incidentally,
the velocity) tends to be lower than is perhaps expected,for
much the same reasons as those given for the intersheath
modes;this is just as well,as these are the modes of normal
power transmission.
5 CONCLUSIONS
A model of a buried-cable system has been developed in
terms of a generalised angular frequency (which may be
real or complex) which may be used for the solution of
steady-state problems, e.g. interference into nearby com-
munication cables due to power-circuit harmonics,or,by
applying Fourier-transform techniques, to the solution of a
wide range of transient problems.
The attenuation and velocity characteristics of a represen-
tative cable system have been presented. The remarkably
low velocities and attenuations of the intersheath modes
have been satisfactorily explained, to the extent that they
may be estimated over quite a wide range of frequencies by
simple 'slide-rule' calculations. The nature of these charac-
teristics, notably those of the intersheath modes, have an
important bearing on the transient behaviour of underground-
cable systems. For example,the velocity of the intersheath
and earth modes will determine the time taken for a system
earth to communicate its presence along the system, and
hence the period of time during which sheath overvoltages
may develop before being checked by the system earths.
Also, the intersheath modes are an important vehicle by
which a disturbance in one cable of the system can be com-
municated to other cables. On the other hand, the attenuation
characteristics will determine the decay of natural system
resonances resulting from an abrupt disturbance.
The study of transient sheath overvoltages and other transi-
ent induction effects form the subject of further work to be
carried out by the authors.
6 ACKNOWLEDGMENTS
The authors wish to acknowledge the facilities granted in the
power-systems laboratory of the Department of Electrical
Engineering & Electronics, University of Manchester Insti-
tute of Science & Technology, and the encouragement and
257
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helpful advice of Prof.Ch.Gregoire of the Faculte Poly-
technic de Mons. The authors wish to thank CEGB head-
quarters for financial help and for helpful discussions with
A.Stalewski and A.Rosen of that organisation.
They also wish to thank the University of Manchester com-
puting service for running the programs used to provide the
results.
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PIPES, L. A.: 'Matrix methods for engineering'
(Prentice-Hall, 1963)
MULLINEUX,N.,and REED, J.R.: 'Calculation of elec-
trical parameters for short and long polyphase trans-
mission lines', Proc.IEE, 1965,112, (4), pp. 741-742
IRVING, J.,and MULLINEUX.N.: 'Mathematics in
physics and engineering' (Academic Press, 1959)
8 APPENDIX
Impedance formulas
8.1 Internal impedance of a solid conductor with
circular cross-section
8.1.1 Approximate formula
The approximate formula proposed here is
z, = —— coth (0*777 mr,) + 0'356 - ^ r ohms per metre
27rr1 7rrf
(25)
This formula is deduced by considering first the formula
z
 = 2^7 c o t h ( m r i )
This simple formula is known to exhibit similar properties
to the exact expression given by eqn. 2. For example, at high
frequencies,this impedance tends to pm/27Trl5which is the
well known skin-effect formula, while at low frequencies it
represents a pure resistance, although not,in fact, equal to
the required value of p/irrf.
This formula may be improved by introducing a degree of
freedom so that
z = 2irr
coth (kmr,) + - £ - (1 - l/2k)
Trrf
(26)
where k is an arbitrary constant. The second term on the
right-hand side corrects the impedance at direct current. At
high frequencies,this formula tends towards the required
skin-effect impendance. The constant k is chosen to optimise
the formula at the lower frequencies.
Expanding eqn. 26 in series form,
z = — { 1 +k(mr1)2/6-k3(mr1)4/90 + . . . ]
•nrf
whereas the corresponding expansion of the classical for-
mula is
zx = -£- {1 + ( m r ^ / 8 - (mr,)V192 + . . . }7rrf
At low frequencies,the higher-order terms of the expansion
may be neglected. Now,mr1 is a complex quantity with a
phase angle of 45°, and hence the second term in the expan-
sion gives an inductive component while the third term gives
a resistive component. Terms are compared, and k is chosen
to give the correct resistive component; thus k3 = 90/192 or
k = 0-777.
It may be noted that an error in the inductive component is
more tolerable than an error in the resistive component, as
the internal inductance will,in practice,tend to be swamped
by the external inductance.
8.2 Internal impedance of a solid sheath with annular
cross-section
8.2.1 Classical formulas4
Z4 = ^ 7 T " / D
K0(mr2)I1(mr3)}/D
(27)
where D = I1(mr3)K1(mr2) — I1(mr2)K1(mr3)
8.2.2 Approximate formulas
7 7 c o t h ( m A ) "
z4 = —,—-, r cosech (mA)
"\T9. + Tl>
zs = ^ T coth (mA) + £27rr3(r2 + r3)
8.2.3 Derivation
The magnetic intensity H and the electric current density J
within the solid tube representing the conducting sheath are
related by the equations4
dr r ( 2 8 a )
(28b)
where r is the radial distance from the sheath axis.
Elimination of first H and then J between these two equations
result in two 2nd-order differential equations, one in H and
the other in J. These equations are, in fact,Bessel's equa-
tions and lead tothe classical result given in eqn. 27.
The approximate formulas are derived on the assumption
that the sheath is thin compared with its mean radius. In
this case, eqn. 28a may be written as
dH 2_
dr (r, + r 3 ) H = J
(29)
Elimination of first H and then J between eqns. 29b and 30
results in two 2nd-order differential equations with constant
coefficients. The solution of these equations with suitable
boundary conditions applied leads to the approximate
formulas.
8.3 Earth-return impedances of buried-cable system
258
The ground is assumed to be a homogeneous medium whose
flat surface divides space into two semi-infinite regions:
soil and air. It is further assumed that the field in the soil
will be insignificantly different from the field that would be
present if the cables were replaced by infinitely thin insu-
lated conductors and their volumes were replaced by soil.
The required results may be deduced from the electric-
field intensity E in the soil when current i flows in an
PROC.IEE, Vol.120, No.2, FEBRUARY 1973
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insulated conductor and returns through the soil. Let this
conductor be buried at a depth h below the surface of the
ground, as shown in Fig. 6. This Figure shows the conductor
in a plane perpendicular to its axis and defines the reference
coordinates x and y.
region 1 air
insulated conductor
Fig. 6
Position of insulated filament with respect to the
co-ordinate axes
The general equation for electromagnetic-wave propagation
is
V2E - V(VE) = (30)
where V2 is the Laplacean operator and E is the electric-
field intensity.
For the present problem, it is reasonable to assume that
VE = 0 in both air and soil, and that displacement currents
may be neglected. These are the usual assumptions, and are
justifiable up to quite high frequencies. Eqn. 30 then simpli-
fies to
V2E = (j (31)
On the basis of assumptions (i)-(iii) in Section 2.1,the elec-
tric force may be taken to be locally invariant in the direc-
tion of propagation, and everywhere to be parallel to the
conductor. Eqn. 31 then simplifies to the following equations
in the two media:
whose co-ordinates are (x,y):
E, = - P 2
m 2 i
2 77
"exp{-(h + y)V(a2 +
 m2)}
m 2 )
2
exp{- |y-h |V(a + m2,)} - exp{-|y + h|V(a2 + m2,)}
m 2 )
exp(jax)dx (33)
If the co-ordinates (x,y) coincide with a second cable,the
mutual impedance z m between the two cables will be given
by
exp{-(h + y)V(a2 + m")}
2TT
exp{-|y-h|V(a2 + m2)}-exp{-|y + h|V(a2 +
 m
2)}
2V(a2 + m2)
exp(jax)dxa (34)
where subscripts have been dropped, as all quantities relate
only to the soil medium. It has also been assumed that ju2
= nx. The formula then corresponds to the integral form
given by Pollaczek.
This integral may be evaluated from a convergent series,
starting with the standard result
( 3 5 )
where T = V(g2 + s2)
Eqn. 34 immediately simplifies to
% = ^ ~ {K0(mR) - K0(mR') + i j
where R = V{x2 + (h -y)2},R' = V{x2 + (h + y)2} and
exp{—(h + y)V(a2 + m2)}
—^—^ y— —I, =
1
exp(iax)da
32E,
3x2
y > 0
32E2 32E,
3x2 3y2
h)y < 0 (32)
where subscripts denote the region to which a quantity re-
lates and 6(x) and 6(y + h) are Dirac functions whose values
are zero except when their arguments are zero; in the latter
case they take the value unity.
A solution for E2 is required that satisfies the following
boundary conditions:
(i) continuity of E at the surface: E 1 = E 2 a t y = 0
(ii) continuity of the vertical component of B at the
interface:
3E, ju, 3E?
-Tr-i = ^ -~ at y = 0
3y M 3Y
(iii) continuity of the horizontal component of H at the
surface:
3E, 3E2
— - = — - at y = 0
3x 3x2
Eqn. 22 may be solved without difficulty by integral-trans-
form techniques to give the electric force at a general point
PROC.IEE, Vol.120, No.2, FEBRUARY 1973
= — 1 Sr ^(a2 + m2) exp{-^V(a2 + m2)} exp(jax)da
m2 I °°
— 2 /0°° aexp{— l>l(a2 + m2)} exp(ja |x|)da
m2)} exp(ja |x|)da[ (36)
" 2I3
Now,from eqn. 35,
exp-{—^V(a2 + m2)}
2V(a 2 + m 2 )
and therefore
I2 = ^ (K0(mR')} =
32
dxdl
r • ,
{K0(mR')| = R ' 2
(mR')
(mR')
2 ( * 2 - i (mR' )
(37)
(38)
The integral I3 may be evaluated by transforming the vari-
able according to
mtR' = - j a |x| + ^V(a2 + m2)
259
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which gives where
+ _}W_ exp(-mtR')jdt (39)
V(t2 - 1) )
The path of integration along the real axis from 0 to QO in the
a-plane transforms into the path C in the t-plane,as shown
in Fig.7.
I m
t plane
Re
radius tends
to infinity
Fig. 7
Path of integration in the complex plane
l/R ^ 1 always
Now mtR' is positive real in the segment of 0, and hence
the line integral along the path C must be zero. The func-
tion has no singularities on or within the closed contour C
— C — Coo, and hence, by Cauchy's theorem, j c = Jc/. Thus
after elementary manipulation, eqn. 39 may be written as
h =
R'2 Jc/R' 2V(1 - t
2 ) -
- 1)( exp(-mtR')dt
exp(-mtR')dt
V(l-t2)
The first part of the integral is easily evaluated by normal
methods; the second part is recognised8 as K2(mR'). The
third part is evaluated by expansion of the exponential func-
tion and integration term by term to give S(mR', |x|, C).
Hence I3 is given by
R'4
R ' 2
ml)
S(mR',
R'2
K2(mR')
R' 2 2! R'
{ 4
5
L(e- J i i i V | i (mR1)4 R 3 l x | 3 3. \e\x\s
2 \ R ' 2 / ) 3 4! L R'6 4 I R'?
>) | x | 3( 2 | x | 3
3 R'3 3 1! R'3
+ 2_ (mR')3 / ^ 2 | x | 3 + 2_ J x | 3 \ + 2_ (mR')s ) J4 | X | .
5 3! \ R'3 3 R ' 3 / 7 5! i R'7
5 V R'5 3 R'3
, 1
R ' 2 / 4 4!
\x\(3
+ —R'4 2 \R '
1 (mR') lx| , 1 (mR')3 [ [ x l / 2
 + 2_ \x[
1 1! R' 3 R'3 L R'3 1 R'
5 5! / R'5 3 \ R' 1 R'
1J
and 8 = v/2 — sin"1 (£/R'). Assembling the above results
gives the solution
zm = tet K0(mR, - K0(mR') + M i K0(mR')
m2R'4
At low frequencies,the impedance will be given by the lead-
ing terms of the series expansion corresponding to eqn. 40.
For cables buried at usual depths (~1 m) and for separations
of the order of 1 m and less,for |mR| < 0 25 only, the follow-
ing terms of the expansion need be taken into account:
- ^ i-ln(ymR'/2) + J - - | (41)
8.4 Cable-design specification
Diameter over copper conductor, cm
Diameter over main insulation, cm
Diameter over sheath, cm
Diameter over outer insulation, cm
D.C. resistance of copper conductor, Si/km
D.C. resistance of lead sheath, J2/km
Core-sheath surge impedance, fi
Sheath-earth surge impedance, Q
= 2-54
= 4-56
= 5-08
= 5-59
= 0034
= 0-436
= 19-4
= 4-6
260 PROC.IEE, Vol. 120, No. 2, FEBRUARY 1973
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