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Impedance
In genera,l thecable impedance can be calculated in accordance with IEC 60909-2 'Short-circuit currents in three-phase a.c. systems - Part 2: data of
electrical equipment for short-circuit current calculations. This standard give gives appropriate formulae for a variety of single and multi-core cables
with or without metallicsheaths or shields. For situations not covered by IEC 60909, we can use the fundamental equations to derive suitable formula.
Fundamental Equations
For a single conductor, the internal (self) inductance due to its' own magnetic eld is given by:
0
L =
8
0
with the reactance given by:X = L =
8
and:
0 =permeability of free space, 4 10-7 N.A-2
L = self-inductance in H.m-1
X = reactance in.m-1
= angular frequency = 2f
For a second external conductor, the inductance due to the eld produced by this other conductor is given by:
0
d
Le = ln ( )
2 r
and:
Le = inductance due to external conductor , H.m-1
d= distance to external conductor, m
r = radius of the conductor, m
For two parallel conductors, the total inductance of one conductor is given by:
0 0 0
d 1 d
Lt = L + Le = + ln ( ) = ( + ln )
8 2 r 2 4 r
whit the reactance given by:
0
1 d
Xt = ( + ln )
2 4 r
and:
Lt = total inductance of one conductor (cable core), H.m-1
Xt = total reactance of one conductor(cable core),
For multicore cables, it is often the case that the distance between cores varies. For example in three cables in at formation, L1 to L2 and L2 to L3, will
be dierence than L1 to L3. To cater for these dierences, we use a concept of geometric mean spacing (also see Geometric Mean Distance
(https://mycableengineering.com/knowledge-base/geometric-mean-distance)):
3
d = dL1L2 dL2L3 dL1L3
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The above can be extended for cables with more cores, or to nd the average phase to neutral
LOGIN spacing for example.
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Zero Sequence Impedance
The above are valid as positive sequence impedance for a cable. Unfortunately, the calculation of zero sequence is more dicult. Several papers have
been published on this and the use of Carson's equations as a means to calculate the zero sequence impedance is an accepted approach. Deriving
equations utilising Carsons' equations is fairly complex and not strictly necessary here. The end results of such derivates are presented in IEC 60909
and we can use these directly.
In some instances, it is also necessary to consider the soil penetration depthbin the calculation of zero sequence impedance (see IEC 60909 part 3):
1.851
=
0
where:
- equivalent soil penetration depth, m
0 - permeability of free space (= 4107), H.m1
- soil resistivity,m
Cables without metallic sheaths or shields
For three single core cables, in either trefoil or at formation, the positive and negative sequence impedance is given by:
IEC 60909 Typical
Impedance Equation Equations Arrangements
Z1 = RL + j
0
(
1
+ ln
d
)
(10), (12), (18), (26) Single Core
2 4 rL
Cables:
* equation can be applied to single core and multicore cables
*equationcan be applied to three three-phase (with and without neutral), and single phase loads
0 0
1
(11)
Z0 = RL + 3 + j ( + 3 ln )
8 2 4 3 2
r Ld
* for current return through the earth
0
d
LN
3
(13), (19)
1
Z0 = 4RL + j4 ( + ln )
2 4 r Ld
Multicore
* current return through fourth conductor
cables:
(14), (20)
2
0 0
( +j ln )
8 2 d
LN
Z0 = Z(0)11 3
0 0 1
RL + +j ( +ln )
8 2 4 r
L
* for current return through fourth conductor and earth
* Z(0)11 is from equation 11
where:
d -geometric mean spacing (line to line), m
dLN - geometric mean spacing (line to neutral), m
RL - conductor resistance (seeConductor Resistance (https://mycableengineering.com/knowledge-
base/conductor-resistance)),
0 - permeability of free space (= 4107), H.m1
rL - radius of the conductor, m
- equivalent soil penetration depth, m
Cables with metallic sheaths or shields
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For three single core cables, with a metallic sheath or shield, in either trefoil or at formation, the positive and negative sequence impedance is given
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by: BASE%2FIMPEDANCE)
IEC 60909 Typical
Impedance Equation Equations Arrangements
(15) Single Core Cables:
2
0 d
( ln )
2 r
Sm
Z1 = Z(1)10 +
0 d
Rs +j ln
2 r
Sm
* applies to single core only (for multicore use equation 10)
*equationcan be applied to three three-phase (with and without neutral), and single phase loads
* Z(1)10is from equation 10
(16)
2
0 0
(3 +j3 ln )
8 2 3 2
r d
Sm
Z0 = Z(0)11
0 0
Rs +3 +j3 ln
8 2 3 2
r d
Sm
Multicore Cables:
* current return through shield and earth (for single core cables only)
* Z(0)11 is from equation 11
0
(31)
1 r Sm
Z(0)S = RL + 3RS + j ( + 3 ln )
2 4 3 2
r Ld
*current return through screen - three core cables
0 0
(32)
1
Z(o)SE =RL + 3 + j ( + 3 ln )
8 2 4 3 2
r Ld
2
0 0
( +j ln )
8 2 r
Sm
3
0 0
RS + +j ln
8 2 r
Sm
* current return through screen and earth - three core cables
0 dLN
(27)
1
Z(o)N S = RL + j ( + 3 ln )
2 4 3 2
r Ld
r
0 Sm
R +j ln
S
0 dLN 2 d
1 LN
+3 (RN + j ( + ln ))
2 4 rN
0 1 r
Sm
RN +RS +j ( +ln )
2 4 r
N
*current return through fourth conductor (N) and screen - four core cables
(28)
2 2
ZN ZLS +ZS ZLN 2ZLN ZLS ZN S
1
Z(o)N SE =Z(0)11 2
3 ZN ZS ZN S
with
0 0
1
ZN = RN + + j ( + ln )
8 2 4 rN
0 0
ZS = RS + + j ln
8 2 r Sm
0 0
ZL123N = ZLN = 3 + j3 ln
8 2 dLN
0 0
ZL123S = ZLS = 3 + j3 ln
8 2 r Sm
0 0
ZN S = w + j ln
8 2 r Sm
* current return through fourth conductor (N), screen and earth - four core cables
* Z(0)11 is from equation 11
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IEC 60909 Typical
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Impedance Equation BASE%2FIMPEDANCE) Equations Arrangements
where:
d -geometric mean spacing (line to line), m
dLN - geometric mean spacing (line to neutral), m
RL - conductor resistance (seeConductor Resistance (https://mycableengineering.com/knowledge-
base/conductor-resistance)),
RN - neutral (fourth) conductor resistance,
Rs - metallic sheath or screen resistances,m
0 - permeability of free space (= 4107), H.m1
rL - radius of the conductor, m
rN - radius neutral (fourth) conductor, m
rSm - mean radius of the sheath or shield [0.5*(rSi+rSa)], m
- equivalent soil penetration depth, m
Note: the calculation of zero sequence impedance is complicated. Sheaths, armour, the soil, pipes, metal structures and other return paths all
aect the impedance. Dependable values of zero-sequence impedance isbest obtained by measurement on cables once installed.
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