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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:

$L=\frac{{\mu }_{0}}{8\pi }$

with the reactance given by: $X=\omega L=\frac{\omega {\mu }_{0}}{8\pi }$

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:

${L}_{e}=\frac{{\mu }_{0}}{2\pi }\mathrm{ln}\left(\frac{d}{r}\right)$

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:

${L}_{t}=L+{L}_{e}=\frac{{\mu }_{0}}{8\pi }+\frac{{\mu }_{0}}{2\pi }\mathrm{ln}\left(\frac{d}{r}\right)=\frac{{\mu }_{0}}{2\pi }\left(\frac{1}{4}+\mathrm{ln}\frac{d}{r}\right)$

whit the reactance given by:

${X}_{t}=\omega \frac{{\mu }_{0}}{2\pi }\left(\frac{1}{4}+\mathrm{ln}\frac{d}{r}\right)$

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):

$d=\sqrt[3]{{d}_{L1L2}×{d}_{L2L3}×{d}_{L1L3}}$

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):

$\delta =\frac{1.851}{\sqrt{\omega \frac{{\mu }_{0}}{\rho }}}$

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

${Z}_{1}={R}_{L}+j\omega \frac{{\mu }_{0}}{2\pi }\left(\frac{1}{4}+\mathrm{ln}\frac{d}{{r}_{L}}\right)$

* 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), (26)

Single Core Cables:

Multicore cables:

${Z}_{0}={R}_{L}+3\omega \frac{{\mu }_{0}}{8}+j\omega \frac{{\mu }_{0}}{2\pi }\left(\frac{1}{4}+3\mathrm{ln}\frac{\delta }{\sqrt[3]{{r}_{L}{d}^{2}}}\right)$

* for current return through the earth

(11)

${Z}_{0}=4{R}_{L}+j4\omega \frac{{\mu }_{0}}{2\pi }\left(\frac{1}{4}+\mathrm{ln}\frac{\sqrt{{d}_{LN}^{3}}}{{r}_{L}\sqrt{d}}\right)$

* current return through fourth conductor

(13), (19)

${Z}_{0}={Z}_{\left(0\right)11}-3\frac{{\left(\omega \frac{{\mu }_{0}}{8}+j\omega \frac{{\mu }_{0}}{2\pi }\mathrm{ln}\frac{\delta }{{d}_{LN}}\right)}^{2}}{{R}_{L}+\omega \frac{{\mu }_{0}}{8}+j\omega \frac{{\mu }_{0}}{2\pi }\left(\frac{1}{4}+\mathrm{ln}\frac{\delta }{{r}_{L}}\right)}$

* for current return through fourth conductor and earth
* Z(0)11 is from equation 11

(14), (20)
where:
d   - geometric mean spacing (line to line), m
dLN - geometric mean spacing (line to neutral), m
RL   - conductor resistance (see Conductor Resistance), Ω
μ0  - permeability of free space (= 4π × 10−7), H.m−1
rL    - 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

${Z}_{1}={Z}_{\left(1\right)10}+\frac{{\left(\omega \frac{{\mu }_{0}}{2\pi }\mathrm{ln}\frac{d}{{r}_{Sm}}\right)}^{2}}{{R}_{s}+j\omega \frac{{\mu }_{0}}{2\pi }\mathrm{ln}\frac{d}{{r}_{Sm}}}$

* applies to single core only (for multicore use equation 10)
* equation can be applied to three three-phase (with and without neutral), and single phase loads
* Z(1)10 is from equation 10

(15)

Single Core Cables:

Multicore Cables:

${Z}_{0}={Z}_{\left(0\right)11}-\frac{{\left(3\omega \frac{{\mu }_{0}}{8}+j3\omega \frac{{\mu }_{0}}{2\pi }\mathrm{ln}\frac{\delta }{\sqrt[3]{{r}_{Sm}{d}^{2}}}\right)}^{2}}{{R}_{s}+3\omega \frac{{\mu }_{0}}{8}+j3\omega \frac{{\mu }_{0}}{2\pi }\mathrm{ln}\frac{\delta }{\sqrt[3]{{r}_{Sm}{d}^{2}}}}$

* current return through shield and earth (for single core cables only)
* Z(0)11 is from equation 11

(16)

${Z}_{\left(0\right)S}={R}_{L}+3{R}_{S}+j\omega \frac{{\mu }_{0}}{2\pi }\left(\frac{1}{4}+3\mathrm{ln}\frac{{r}_{Sm}}{\sqrt[3]{{r}_{L}{d}^{2}}}\right)$

*current return through screen - three core cables

(31)

* current return through screen and earth - three core cables

(32)

*current return through fourth conductor (N) and screen - four core cables

(27)

$\begin{array}{rcl}{Z}_{\left(o\right)NSE}& =& {Z}_{\left(0\right)11}-\frac{1}{3}\frac{{Z}_{N}{Z}_{LS}^{2}+{Z}_{S}{Z}_{LN}^{2}-2{Z}_{LN}{Z}_{LS}{Z}_{NS}}{{Z}_{N}{Z}_{S}-{Z}_{NS}^{2}}\end{array}$

with

${Z}_{N}={R}_{N}+\omega \frac{{\mu }_{0}}{8}+j\omega \frac{{\mu }_{0}}{2\pi }\left(\frac{1}{4}+\mathrm{ln}\frac{\delta }{{r}_{N}}\right)$

${Z}_{S}={R}_{S}+\omega \frac{{\mu }_{0}}{8}+j\omega \frac{{\mu }_{0}}{2\pi }\mathrm{ln}\frac{\delta }{{r}_{Sm}}$

${Z}_{L123N}={Z}_{LN}=3\omega \frac{{\mu }_{0}}{8}+j3\omega \frac{{\mu }_{0}}{2\pi }\mathrm{ln}\frac{\delta }{{d}_{LN}}$

${Z}_{L123S}={Z}_{LS}=3\omega \frac{{\mu }_{0}}{8}+j3\omega \frac{{\mu }_{0}}{2\pi }\mathrm{ln}\frac{\delta }{{r}_{Sm}}$

${Z}_{NS}=w\frac{{\mu }_{0}}{8}+j\omega \frac{{\mu }_{0}}{2\pi }\mathrm{ln}\frac{\delta }{{r}_{Sm}}$

* current return through fourth conductor (N), screen and earth - four core cables
* Z(0)11 is from equation 11

(28)

where:
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

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.