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Impedance

In general, 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 a 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 an 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 different 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:

Single Core Cables


Three or four cables without metallic sheath or shield (equally loaded).

Positive sequence impedance (phase or neutral):

Z1=RL+jωμ02π14+lndrL        - (10)

Zero sequence impedance, current return through the earth (E):

Z0=RL+3ωμ08+jωμ02π14+3lnδrLd23        - (11)

Zero sequence impedance, current return through fourth conductor (N)

Z0=4RL+j4ωμ02π14+lndLN3rLd        - (13)

Zero sequence impedance, current return through fourth conductor (N) and earth (E)

Z0=Z(0)113ωμ08+jωμ02πlnδdLN2RL+ωμ08+jωμ02π14+lnδrL        - (14)


Multicore Cables


Three or four core cable without metallic sheath or shield (equally loaded).

Positive sequence impedance (phase or neutral):

use Z(1)10, (equation 10) 

Zero sequence impedance, current return through fourth conductor (N, full cross section)

Z0=4RL+j4ωμ02π14+lndrL        - (19)

Zero sequence impedance, current return through fourth conductor (N, reduced cross section)

Z0=RL+3RN+jωμ02π1+4lndLN3rLrN34d        - (23)

Zero sequence impedance, current return through fourth conductor (N, full cross section) and earth (E)

Z0=Z(0)113ωμ08+jωμ02πlnδdLN2RL+ωμ08+jωμ02π14+lnδrL        - (20)

Zero sequence impedance, current return through fourth conductor (N, reduced cross section) and earth (E)

Z0=Z(0)113ωμ08+jωμ02πlnδdLN2RN+ωμ08+jωμ02π14+lnδrN        - (24)


 

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:

Single Core Cables


Three cables with metallic sheath or shield (equally loaded, bonded both ends).

Positive sequence impedance:

Z1=Z(1)10+ωμ02πlndrSm2Rs+jωμ02πlndrSm        - (15)

Zero sequence impedance, current return through shield (S)

no equation given by standard

Zero sequence impedance, current return through shield (S) and earth (E)

Z0=Z(0)113ωμ08+j3ωμ02πlnδrSmd232Rs+3ωμ08+j3ωμ02πlnδrSmd23        - (16)


Multicore Cables


Three or four core cable with metallic sheath or shield (equally loaded, bonded both ends).

Positive sequence impedance:

use Z(1)10, (equation 10) 

Zero sequence impedance, current return through screen (S)

Z0=RL+3RS+jωμ02π14+3lnrSmrLd23        - (31)

Zero sequence impedance, current return through screen (S) and earth (E)

Z0= Z(0)11-3ωμ08+jωμ02πlnδrSm2RS+ωμ08+jωμ02πlnδrSm                - (32)

Zero sequence impedance, current return through fourth conductor (N) and screen (S)

Z0=RL+jωμ02π14+3lndLNrL d23+3RN+jωμ02π14+lndLNrNRS+jωμ02πlnrSmdLNRN+RS+jωμ02π14+lnrSmrN        - (27)

Zero sequence impedance, current return through fourth conductor (N), screen (S) and earth (E)

Z0=Z(0)11-13ZNZLS2+ZSZLN2-2ZLNZLSZNSZNZS-ZNS2                - (28)

with

ZN=RN+ωμ08+jωμ02π14+lnδrN

ZS=RS+ωμ08+jωμ02πlnδrSm

ZL123N=ZLN=3ωμ08+j3ωμ02πlnδdLN

ZL123S=ZLS=3ωμ08+j3ωμ02πlnδrSm

ZNS=wμ08+jωμ02πlnδrSm


Note: the calculation of zero sequence impedance is complicated. Sheaths, armour, the soil, pipes, metal structures and other return paths all affect the impedance. Dependable values of zero-sequence impedance is best obtained by measurement on cables once installed.  

Symbols

d   - geometric mean spacing (line to line), m
dLN - geometric mean spacing (line to neutral), m
RL   - conductor resistance (see Conductor Resistance), Ω
RN - neutral (fourth) conductor resistance, Ω
Rs - metallic sheath or screen resistances, Ωm
μ0  - permeability of free space (= 4π × 10−7), H.m−1
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
δ   - equivalent soil penetration depth, m

 

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