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# Impedance

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' table 7.  This table gives appropriate formulae for single and multi-core cables with or without metallic sheaths or shields.

## 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:

where:
R - conductor resistance (see Conductor Resistance), Ω
μ0 - permeability of free space (= 4π × 10−7), H.m−1
rL - radius of the conductor, m

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

where: dL1L2, dL1L3, dL2l3 are the axial distance between conductors, m

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

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

where:
δ - equivalent soil penetration depth, m
ρ - soil resitivity, Ωm

For four single core cables (L1, L2, L3 and N)  the positive sequence impedance is given by:

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

and with the return current through the neutral:

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

and the return current through both the neutral and earth, with Zo(2) being the impedance calculated from equation (2) :

${Z}_{0}={Z}_{0\left(2\right)}-3\frac{{\left(\omega \frac{{\mu }_{0}}{8}+j\omega \frac{{\mu }_{0}}{2\pi }\mathrm{ln}\frac{\delta }{{d}_{LN}}\right)}^{2}}{R+\omega \frac{{\mu }_{0}}{8}+j\omega \frac{{\mu }_{0}}{2\pi }\left(\frac{1}{4}+\mathrm{ln}\frac{\delta }{{r}_{L}}\right)}$                 (5)

with: ${d}_{LN}=\sqrt[3]{{d}_{L1N}{d}_{L2N}{d}_{L3N}}$

where: dL1N, dL2N, dL3N are the axial distances between each conductor and neutral, 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:

${Z}_{1}={Z}_{1\left(1\right)}+\frac{{\left(\omega \frac{{\mu }_{0}}{2\pi }\mathrm{ln}\frac{d}{{r}_{Sm}}\right)}^{2}}{{R}_{s}+j\omega \frac{{\mu }_{0}}{2\pi }\mathrm{ln}\frac{d}{{r}_{Sm}}}$                 (6)

${Z}_{0}={Z}_{0\left(2\right)}-\frac{{\left(3\omega \frac{{\mu }_{0}}{8}+j3\omega \frac{{\mu }_{0}}{2\pi }\mathrm{ln}\frac{\delta }{\sqrt[3]{{r}_{Sm}{d}^{2}}}\right)}^{2}}{{R}_{s}+3\omega \frac{{\mu }_{0}}{8}+j3\omega \frac{{\mu }_{0}}{2\pi }\mathrm{ln}\frac{\delta }{\sqrt[3]{{r}_{Sm}{d}^{2}}}}$                 (7)

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)