International Journal of Arts and Sciences
3(9): 36 - 46 (2010)
CD-ROM. ISSN: 1944-6934
© InternationalJournal.org
Effect of the Transient Ground Impedance on the Return Path
of the Zero Sequence Current
L. Boukelkoul, LRES University of 20th Aug. 1955 - Skikda, Algeria
Abstract: A fault on a power system is an abnormal condition that involves an
electric failure of power system equipment. Short-circuit faults are a major problem.
Generally, they can occur from an insulation failure or due to a sudden overvoltage
caused by lightning impulses or switching devices. The return path of the zero
sequence current through the ground is strongly influenced by the nature and the
resistance of the soil. Usually the ground parameters are poorly known as they have a
wide range variation with some random phenomena, such as seasonal changes and the
amount of water retained. Due to the non uniform distribution of the zero sequence
current, the determination of the earth return impedance is difficult. In this paper a
summary of various techniques of calculation of the earth return impedance is given.
The influence of the return path on an unbalanced one phase to earth fault is shown
and explained.
Keywords: Ground impedance; closed form; sequence impedance; short circuit
Introduction
Electromagnetic transient simulations are very important for the design of insulation
levels of power and communication systems; therefore detailed transmission line
modeling is required. In the case of underground cables, the model parameters are
considerably influenced by the ground. The influence of the lossy ground on the
conductor impedance has been investigated since 1920 (E. Petrache, F. Rachidi 2005).
The return path of the current through the ground demands a special explanation. The
difficulty in determination of the earth return impedance is that it is strongly
dependent on the ground resistance and on the frequency of the magnetic field. Some
theories of determining the earth return impedance are existing; such as Carson’s
formulas for the case of overhead lines and Pollaczek’s for overhead lines and
underground cables . The existing methods used for estimation of the earth return path
is to consider the ground as a homogeneous conductor which is located parallel to the
respective current carrying conductor in a depth dependent of frequency and ground
resistance. Short-circuit faults can occur between phases, or between phases and earth
or both. A one phase to earth short-circuit in a high impedance earthed distribution
system may cause a sufficient voltage rise on a healthy phase elsewhere in the system
that a flashover and short-circuit fault occurs. The three currents of the zero sequence
system, equal in magnitude and direction, are opposed by an impedance, in the case of
cables (by the loops constituted by the three cable cores, lead sheath and earth). To
calculate the short-circuit current; the earth impedance must be taken into account. In
this paper, a review of closed form calculation of ground impedance is given.
Example of short-circuit models is shown and the method of calculating the zero
sequence current is presented.
International Journal of Arts and Sciences
3(9): 36 - 46 (2010)
CD-ROM. ISSN: 1944-6934
© InternationalJournal.org
1 – Conductor internal impedance
The telegrapher’s equation governing a transmission line is as follows (F. A. U.
Compos 2002):
∂v(x, jω )
= − Zi (x, jω )
∂x
(1)
∂i(x, jω )
= −Yv(x, jω )
∂x
Where
v is the voltage
i is the current
x is the longitudinal distance.
Z is the longitudinal impedance (x and ω dependent)
Y is the transversal admittance (x and ω dependent).
ω is the frequency
j = −1
For insulated wires, the longitudinal impedance Z can be written in the following
form:
(2)
Z = j ωL + Z i + Z g
Where
L is the per-unit length longitudinal inductance of the cable.
µ r 
L = 0 ln e 
2π  ri 
Z i is the per-unit length internal impedance of the conductor.
Z g is the per-unit length ground impedance.
r e and r i are the outer and the inner radius of the cable.
The transverse admittance is:
(G + jωC )Yg
Y=
(G + jωC ) + Yg
Where:
C is the per-unit length transverse capacitance of the cable.
2πε 0 ε r
C=
ln(re / ri )
G is the per-unit length transverse conductance of the cable.
C
G = σi
ε rε 0
(3)
(4)
(5)
(6)
σ i is the conductivity of the insulating jacket.
ε r is the relative permittivity of the insulating jacket.
The conductance in cables can be neglected, the transverse admittance becomes:
jω C Y g
(7)
Y=
jω C + Y g
International Journal of Arts and Sciences
3(9): 36 - 46 (2010)
CD-ROM. ISSN: 1944-6934
© InternationalJournal.org
At low frequencies, the major contribution to the transverse admittance comes from
the capacitance. On the contrary, at higher frequencies the ground admittance
contribution becomes significant (E. Petrache, F. Rachidi, 2005).
The derivation of the internal impedance can be done through the knowledge of the
distribution of the current in a cylindrical conductor with radius r, conductivity σ and
relative permeability µ c . The current can be calculated by solving the Bessel
differential equation given by the following relation:
∂ 2 I 1 ∂I
(8)
+
− jωµ c µ 0σI = 0
∂r 2 r ∂r
The general solution of (8) is:
(9)
I (rγ ω ) = AJ 0 (rγ ω ) + BN 0 (rγ ω )
Where:
J 0 (rγ ω ) is the Bessel function of 1st kind of zero order.
N 0 (rγ ω ) is the Bessel function of 2nd kind of zero order.
A and B are constants to be determined.
γ ω = jω µ 0 (σ + jω ε 0 ε r )
Since the current in the conductor is finite when the radius is zero; so the constant B is
taken to be zero. The solution then for the internal impedance is given by the
following formula [7,8]:
γ ω J 0 (γ ω ri )
(10)
Zi =
2π ⋅ riσJ 1 (γ ω ri )
In alternating current Z i is the sum of a resistance R and an internal inductance as
follows:
(11)
Z i = R + jL' ω
An approximation of (6) for low frequency will give (N. Theetayi, R. Thottppillil
2007):
R + jL' ω ri ωµσ J 0 ri jωµσ jπ / 4
(12)
≅
e
R0
2
J 1 ri jωµσ
(
(
)
)
This will give a good approximation for numerical solution.
If ri >> ωµσ we can obtain:
1  r ωµσ 
R
≅ 1+  i

12 
2
R0

L' ω 1  ri ωµσ 
≅

2 
2
R0

4
(13)
2
If ri << ωµσ we can obtain:
(14)
ri
R
=
R0 2 ωµσ
2 – Ground impedance formulation
The best known model for the ground impedance self and mutual was proposed by
Pollaczek (N. Theethayi, R. Thotapillil 2006):
International Journal of Arts and Sciences
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CD-ROM. ISSN: 1944-6934
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Z g self =
Z g mut
 2h  
jωµ0   re 
 N 0   − N 0   + J s 
2π   p 
 p 
(15)

 x 2 + 4h 2 
jωµ0   x 
+ J 
 N0   − N 0 
=
m


2π   p 
p




+∞
Where: J s =
e −2h
∫λ+
λ + 1/ p
2
−∞
∞
and J m =
e −(h + y )
∫λ+
−∞
λ2 +1 / p 2
λ2 +1 / p 2
λ + 1/ p
2
2
2
(16)
e jxλ dλ
e jxλ dλ p = 1 /
jωµσ is the complex depth of the skin
effect layer, h is the depth and x is the distance between conductors.
The Pollaczek expression is a low approximation in the sense that it can be used when
frequency of incident pulse satisfies ω << σ g / ε g . Because of the low frequency
approximation, the equations (15) and (16) don’t include the permittivity of the
ground.The integral form of the expressions (15) and (16) results from the application
of the Fourier integral transform on the generalized wave equation in the ground and
in the air. The parameter λ in the relations (15) and (16) is the transformed space
variable and mathematically represents the frequency of the Fourier spectrum. The
Pollaczek integral is highly oscillatory, that it does not possess analytic closed form
solutions. There have been several proposals for the approximation of such integral
based on logarithmic formula. Although the accuracy obtained from these proposals is
acceptable, so far there is no general criterion for the validation of such formulation in
a broad range of frequency, conductivity and separation between conductors (J. A.
Stratton 1961). Alternatively, one can use the asymptotic expressions. However, those
expressions are valid for specific frequency ranges, in addition to the presence of the
well-known discontinuities (T. F. R. Martins 2005).
3 – Review of ground impedance
All complexities associated with the integral of pollaczek have led researchers to
develop closed form approximations. For a wide range of frequencies as long as the
transmission line approximation is valid, Sunde (F. A. U. Compos 2002,
G.K.Papagiannis 2005) proposed an expression for the mutual inductance of two
buried cables located at the same depth h and separated by a distance x, which is
given by:
 N (γ x ) − N γ x 2 + 4h 2 
0 g
 0 g

jωµ 
2
2

Sunde
∞ − 2h λ +γ g
(17)
Zg =
(
)
e
x
cos
λ


2π
d
λ
+
2
∫0 λ2 + γ 2


g


One of the difficulties with equation (17) is that, as the frequency is increased, the
integral term converges slowly leading to longer computation time and possible
truncation errors. Further, it was found that the first two Bessel terms in (17) are
oscillatory when frequencies approach 1 MHz. However, one can say Sunde’s
(
)
International Journal of Arts and Sciences
3(9): 36 - 46 (2010)
CD-ROM. ISSN: 1944-6934
© InternationalJournal.org
expression for ground impedance is more valid in the sense that it uses the full
expressions for propagation constants.
4 – Closed form expressions
A. Infinite earth impedance formula:
The infinite earth model consists in considering the earth as a lossy dielectric
cylinder whose external radius tends to infinite and its internal radius is the one for
the buried conductor. In this context Vance developed the following formula for the
self and mutual ground impedance:
γ ω H 0 (γ g ri )
(18a)
Z gs =
2π ⋅ ri γ g H 1 (γ g ri )
Z gm =
γ ω H 0 (γ g x )
2π ⋅ r j ri γ g H 1 (γ g ri ).H (γ g r j )
(18b)
B. Semlyen and Wedepohl formula:
The self component of the cable ground impedance was conjectured by
Wedepohl and reported by Semlyen (F. A. U. Compos 2002 and G.K.Papagiannis
2005).
jωµ
[ln(re + p / re )]
(19a)
Z gs =
2π
The counterpart of the mutual impedance is expressed by Wedepohl as:
− log(γ g x / 2 p )
jωµ 

Z gm =
(19b)
4h

2π + 0.5 −


3p
C. Saad-Gaabba-Giroux formula:
Saad, Gabba and Giroux obtained the following expressions by approximating the
Pollaczek integrals (T. F. R. Martins 2005):

 N 0 (re / p )
jωµ 

Z gs =
2
(20a)
−2 h / p 
e
2π +
2
2

 4 + re / p
Z gm =
jωµ
2π

2
−2h / p 
 N 0 (re / p ) + + x 2 p 2 e

4
/


(20b)
D. Bridges formula:
Starting from the rigorous scattering solution for a buried cable, Bridges mentions
that his expression for ground impedance has two modes, namely, transmission line
modes and radiation mode. He derived a general expression using the transmission
line approximation and neglecting the displacement current Bridges obtained the
following expressions :
jωµ  r 

(21)
Z gs = −
ln Γ
2π  2 p 
Where Γ = 1.7811
International Journal of Arts and Sciences
3(9): 36 - 46 (2010)
CD-ROM. ISSN: 1944-6934
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E. Wait formula:
Developed a more complex expression derived from electromagnetic theory
considering only transmission line theory (F. A. U. Compos 2002, G.K.Papagiannis
2005).


jωµ
(22)
(1 + ∆ ) ln 1.12 
Z gs =
γ
.
r
2π
g
e


1 

 N 0 (− 2γ g h ) + 2γ 2 h 2 
1
g
Where: ∆ =


N 0 (− γ g r ) 
− 2 jγ g h


× (1 − 2γ g h )e
F. Petrache formula:
In his expression, he proposed a logarithmic approximation and claims that it is
the simplest expressions for the ground impedance that is available in literature
(N. Theethayi 2005).
jωµ  1 + γ g .r 
ln
(23)
Z gs = −
2π  γ g .r 
As it is known, in buried cables, the wave propagates and returns mostly within
the soil medium. The knowledge of the ground impedance is of interest, several
researchers developed the above expressions for the ground impedance starting
from fundamentals of electromagnetic theory or modification of expressions
developed earlier.
In the closed form approximation (18), one can observe the dependence only on
the wire radius and the burial depth, unlike Sunde’s (17) and Wait’s (22)
expressions. It can be shown that Bridges’s expression (21) is equivalent to
Sunde’s expression (17). Wait shows if h ⋅ εµω 2 << 1 , his expression is valid for
all frequencies below this limit. Quasi-static and hence transmission line
approximations are valid. This limiting condition is about 5MHz for wires at
depths between 0.5m –1m and for different ground conditions (F. A. U. Compos
2002). Neglecting wire depth may not be a good approximation. Theethayi
proposed a modified empirical logarithmic-exponential expression which is
similar to the equation (20):
  1 + γ g .r  
+ 
ln
jωµ   γ g .r  
Z gs = −
(24)

2π  2
−2h γ g


e

 4 + r 2 γ g2
Expression (24) is proposed to take into account the depth term as a correction
term of the expression (23).
International Journal of Arts and Sciences
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CD-ROM. ISSN: 1944-6934
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5 - Short-circuits to ground faults
Due to the continuous extension of networks and the subsequent increase in the
transmission capacities, the determination of the short-circuit currents to be expected
in a. c. systems is of ever increasing importance for the rating of the plant and
equipment. In the interests of economy and reliability, the planning engineer must
devote more attention than ever before to the problem of short-circuits. In order to
provide him with the necessary analytical techniques, we consider here two kinds of
faults which may occur in three-phase systems.
Two and three-phase short-circuits are more common on underground cables than
single line to ground faults. However it is not easy to distinguish if the short circuit
actually has started with a single line to ground fault. When an external unbalanced
condition, such as a single line to ground fault is imposed on a network, sequence
voltages and currents appear on the network at the point of fault.
6 - Two-phase to ground fault
Having a three-phase system, R, S, T with S & T are short-circuited to earth, the
conditions are as follows:
U S = UT = 0 , I R = 0
We write I R in symmetrical components, we obtain:
I R = I 1R + I 2 R + I 0 ⇒ I 1R = − I 2 R − I 0
Where: The indices 0, 1 and 2 refer to zero sequence, positive sequence and negative
sequence respectively.
Since:
(25)
3U 1R = U R + aU S + a 2U T = U R
3U 2 R = U R + a 2U S + aU T = U R
(26)
3U 0 = U R + U S + U T =
We get:
U 1R = U 2 R = U 0
(27)
−j
UR
2π
3
a=e
The current in the faulted lines is obtained from the relations set out above as follows:
U 1R = U 2 R = U 0 and I R = I 1R + I 2 R + I 0
We get:
E − U 1R U 1R U 1R
(28)
IR =
−
−
=0
Z1
Z2
Z0
U 1R =
E.Z 0 .Z 2
Z 1 .Z 2 + Z 2 Z 2 + Z 1 Z 0
(29)
Where Z 0 , Z 1 , Z 2 are the zero sequence impedance, the positive impedance and the
negative sequence impedance respectively.
After insertion of U1R and multiplying we obtain:
International Journal of Arts and Sciences
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CD-ROM. ISSN: 1944-6934
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E (Z 2 + Z 0 )
Z 1 .Z 2 + Z 2 Z 0 + Z 1 Z 0
E Z0
I 2R =
Z 1 .Z 2 + Z 2 Z 0 + Z 1 Z 0
E Z2
I0 = −
Z 1 .Z 2 + Z 2 Z 0 + Z 1 Z 0
I 1R =
(30a)
(30b)
(30c)
From I S = a 2 I 1R + aI 2 R + I 0 ; I T = aI 1R + a 2 I 2 R + I 0 . We get after substitution:
[(
)
IS =
− j 3 E 1+ a2 Z2 + Z0
Z 1 .Z 2 + Z 2 Z 2 + Z 1 Z 0
IT =
j 3 E [(1 + a ) Z 2 + Z 0 ]
Z 1 .Z 2 + Z 2 Z 2 + Z 1 Z 0
]
(31a)
(31b)
7 - Single-phase to earth fault
The relevant conditions are:
U R = 0, I S = 0,
IT = 0
Therefore 3I 1R = I R + aI S + a 2 I T = I R
(32a)
(32b)
3I 2 R = I R + a 2 I S + aI T = I R
(32c)
3I 1R = I R + I S + I T
= IR
And
I 1R = I 2 R = I 0
In the same way, we get the currents in the faulted line.
E
(33)
I 1R = I 2 R = I 0 =
Z1 . + Z 2 + Z 0
To obtain the sequence series impedance and shunt admittance, matrices are
transformed to symmetrical components with the following relations:
1 1 1 
1 1 1 
1




S R01S2T = 1 a 2 a  and S 0R1S2T = 1 a a 2 
3
1 a a 2 
1 a 2 a 




The sequence impedance and admittance matrices will then be:
Z 0 0 0 
(34)
Z 012 = S 0R1S2T Z S R0 S1 T2 = 0 Z 1 0 
0 0 Z 2 
Where Z is the three-phase, R, S, T impedances:
 Z RR Z RS Z RT 
(35)
Z =  Z SR Z SS Z ST 
 Z TR Z TS Z TT 
The diagonal elements represent the self impedance of the cable and the ground
impedance. The non diagonal elements refer to mutual impedances. As was discussed
International Journal of Arts and Sciences
3(9): 36 - 46 (2010)
CD-ROM. ISSN: 1944-6934
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earlier, the different methods of computing the ground impedance lead almost to the
same result. One can use any closed form to compute the self and the mutual ground
impedance.
8 - Surge impedance
The surge impedance Z s is obtained by the following relation:
Z 01 2
Zs =
Y01 2
(36)
Z 012 and Y 012 are related by the propagation constant:
γ 012 = Z 012Y012
(37)
9- Discussion and conclusions
The curves of the self and mutual transient ground impedance are in all cases
monotonically increasing functions of frequency. Fig. 1 shows a comparison of
different closed form approximations, it can be seen that they provide very similar
results up to 10 MHz. Two different values of soil conductivity were considered for a
cable having 2 cm as an outer radius. The relative permeability and permittivity are
taken as unity and 3 respectively. Fig.2 shows the mutual impedances for three
approximations. Vance’s expressions provide a neglected value from the whole range
of low frequency, however the other expressions of Wedpohl and Saad can be
considered and not to be neglected.
The analysis of the proposed approximations provides very similar results for a wide
range of frequencies. From the results obtained, it is obvious that the value of the
ground transient impedance is very important at early time (tends to infinity). This
high value is due certainly, to neglecting the displacement current. This leads to a
singularity in time domain analysis.
(a)
(b)
Fig.1 Comparison between different closed form expressions for self ground impedance, for
a)
σ g = 0.01 [S / m]
b)
d = 0.5 [m ]
σ g = 0.001 [ S / m] and external radius 2 [cm].
International Journal of Arts and Sciences
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CD-ROM. ISSN: 1944-6934
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Fig.2 Comparison between different closed form expressions for mutual ground impedance for
σ g = 0.01 [ S / m]
d = 0.5 [m]
r e = 2 [cm] and distant 0.5 [m].
(a)
Fig. 3. Sequence Impedances (a)
σ g = 0.1 [S], (b) σ g
(b)
= 0.01 [S],
Sequence impedances are calculated Fig.3, in order, to evaluate short-circuit currents.
It can be seen that zero and positive sequences are approximately similar. This is due
to neglecting the lead sheath. The zero sequence currents flow in the ground, opposed
by its impedance. In case of high impedance, this may produce an overvoltage leading
to a fault. This surge voltage may be caused by a lightning strike in the vicinity or a
previous short-circuit.
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