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Impedance

In genera,l the cable 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 metallic sheaths 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 field is given by:

L=μ08π 

with the reactance given by: X=ωL=ωμ08π

and:
μ0       = permeability of free space, 4π 10-7 N.A-2
L        = self-inductance in H.m-1
X        = reactance in Ω.m-1
ω        = angular frequency = 2πf

For a second external conductor, the inductance due to the field produced by this other conductor is given by:

Le=μ02πlndr

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:

Lt=L+Le=μ08π+μ02πlndr=μ02π14+lndr

whit the reactance given by:

Xt=ωμ02π14+lndr

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 flat formation, L1 to L2 and L2 to L3, will be difference than L1 to L3.  To cater for these differences, we use a concept of geometric mean spacing (also see Geometric Mean Distance):

d=dL1L2×dL2L3×dL1L33

The above can be extended for cables with more cores, or to find the average phase to neutral spacing for example. 

Zero Sequence Impedance

The above are valid as positive sequence impedance for a cable.  Unfortunately, the calculation of zero sequence is more difficult.  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 depth ẟb in the calculation of zero sequence impedance (see IEC 60909 part 3):

δ=1.851ωμ0ρ

where:
δ   - equivalent soil penetration depth, m
μ0  - permeability of free space (= 4π × 10−7), H.m−1
ρ  - soil resistivity, Ωm

 

Cables without metallic sheaths or shields

For three single core cables, in either trefoil or flat formation, the positive and negative sequence impedance is given by:

Impedance Equation IEC 60909 Equations Typical Arrangements

Z1=R+jωμ02π14+lndr

* equation can be applied to single core and multicore cables
* equation can be applied to three three-phase (with and without neutral), and single phase loads

(10), (12), (18)

Single Core Cables:

 

Multicore cables:

Z0=R+3ωμ08+jωμ02π14+3lnδr d23

* for current return through the earth

(11)

 

Z0=4R+j4ωμ02π14+lndLN3rd

* current return through fourth conductor
* equation applied to single core and multicore cables

(13), (19)

Z0=R+3ωμ08+jωμ02π14+3lnδr d233ωμ08+jωμ02πlnδdLN2R+ωμ08+jωμ02π14+lnδrL

* for current return through fourth conductor and earth

(14), (20)
where:
d   - geometric mean spacing, m
R   - conductor resistance (see Conductor Resistance), Ω
μ0  - permeability of free space (= 4π × 10−7), H.m−1
r    - radius of the conductor, m
δ   - equivalent soil penetration depth, m
   

 

Cables with metallic sheaths or shields

For three single core cables, with a metallic sheath or shield, in either trefoil or flat formation, the positive and negative sequence impedance is given by:

Impedance Equation IEC 60909 Equations Typical Arrangements

Z1=Z1(10)+ωμ02πlndrSm2Rs+jωμ02πlndrSm

* equation can be applied to three three-phase (with and without neutral), and single phase loads
Z1(10) from equation (10)

(15)

Single Core Cables:


Multicore Cables:

Z0=Z0(11)3ωμ08+j3ωμ02πlnδrSmd232Rs+3ωμ08+j3ωμ02πlnδrSmd23

* current return through shield and earth
* Z0(11) from equation (11)

(16)

where:
Rs - metallic sheath or screen resistances, Ωm
rSm - mean radius of the sheath or shield, m
Z1(1) and Z0(2) are the impedances given by equations (1) and (2)

   

 

 

 

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