# Cable Impedance Calculations

This article provides details on the calculation of cable impedances - dc resistance, ac resistance and inductive reactance.

## Cable Resistance

The dc and ac resistance of cable conductors can be calculated based on IEC 60287-1 Clause 2.1.

### DC Resistance

The dc resistance of cable conductors is calculated as follows:

$R_{dc} = \frac{1.02 \times 10^{6} \times \rho_{20}}{S} \left[ 1 + \alpha_{20} ( \theta - 20 ) \right] \,$

Where$R_{dc} \,$ is the dc resistance at the conductor operating temperature $\theta$ ($\Omega / m$)

$\rho_{20} \,$ is the resistivity of the conductor material at 20oC ($\Omega . m$).
• For copper conductors, $\rho_{20} \,$ = 1.7241 x 10-8
• For aluminium conductors, $\rho_{20} \,$ = 2.8264 x 10-8
$S \,$ is the cross-sectional area of the conductor (mm2)
$\alpha_{20} \,$ is the temperature coefficient of the conductor material per K at 20oC.
• For copper conductors, $\alpha_{20} \,$ = 3.93 x 10-3
• For aluminium conductors, $\alpha_{20} \,$ = 4.03 x 10-3
$\theta \,$ is the conductor operating temperature (oC)

### AC Resistance

The ac resistance of cable conductors is the dc resistance corrected for skin and proximity effects.

$R_{ac} = R_{dc} \left( 1 + y_{s} + y_{p} \right) \,$

Where$R_{ac} \,$ is the ac resistance at the conductor operating temperature $\theta$ ($\Omega / m$)

$R_{dc} \,$ is the dc resistance at the conductor operating temperature $\theta$ ($\Omega / m$)
$y_{s} \,$ is the skin effect factor (see below)
$y_{p} \,$ is the proximity effect factor (see below)

The skin effect factor is calculated as follows:

$y_{s} = \frac{ x_{s}^{4}} {192 + 0.8 x_{s}^{4}}\,$

Where $x_{s}^{4} = \left( \frac{8 \pi f}{R_{dc}} k_{s} \times 10^{-7} \right)^{2} \,$

$R_{dc} \,$ is the dc resistance at the conductor operating temperature $\theta$ ($\Omega / m$)
$f \,$ is the supply frequency (Hz)
$k_{s} \,$ is a constant (see table below)

Note that the formula above is accurate provided that $x_{s} \leq$ 2.8.

The proximity effect factor varies depending on the conductor geometry. For round conductors, the following formulae apply.

For 2C and 2 x 1C cables:

$y_{p} = \frac{x_{p}^{4}} {192+ 0.8 x_{p}^{4}} \left( \frac{d_{c}} {s} \right)^{2} \times 2.9 \,$

For 3C and 3 x 1C cables:

$y_{p} = \frac{x_{p}^{4}} {192+ 0.8 x_{p}^{4}} \left( \frac{d_{c}} {s} \right)^{2} \left[ 0.312 \left( \frac{d_{c}} {s} \right)^{2} + \frac{1.18}{\frac{x_{p}^{4}} {192+ 0.8 x_{p}^{4}} + 0.27} \right] \,$

Where $x_{p}^{4} = \left( \frac{8 \pi f} {R_{dc}} k_p \times 10^{-7} \right)^{2} \,$

$R_{dc} \,$ is the dc resistance at the conductor operating temperature $\theta$ ($\Omega / m$)
$f \,$ is the supply frequency (Hz)
$d_{c} \,$ is the diameter of the conductor (mm)
$s \,$ is the distance between conductor axes (mm)
$k_{p} \,$ is a constant (see table below)

Note that the formula above is accurate provided that $x_{s} \leq$ 2.8.

For shaped conductors, the proximity effect factor is two-thirds the values calculated above, and with:

$d_{c} \,$ equal to the diameter of an equivalent circular conductor of equal cross-sectional area and degree of compaction (mm)
$s = d_{c} + t \,$ where $t \,$ is the thickness of the insulation between conductors (mm)

Type of Conductor Dried and Impregnated? $k_{s} \,$ $k_{p} \,$
Copper
Round, stranded Yes 1 0.8
Round, stranded No 1 1
Round, segmental - 0.435 0.37
Sector-shaped Yes 1 0.8
Sector-shaped No 1 1
Aluminium
Round, stranded Either 1 1
Round, 4 segment Either 0.28 0.37
Round, 5 segment Either 0.19 0.37
Round, 6 segment Either 0.12 0.37

## Cable Reactance

The series inductive reactance of a cable can be approximated by the following equation:

$X_{c} = 2 \pi f \left[ K + 0.2 \ln \left( \frac{2s}{d_{c}} \right) \right] \times 10^{-3} \,$

Where $X_{c} \,$ is the conductor inductive reactance ($\Omega / km$)

$f \,$ is the supply frequency (Hz)
$s \,$ is the axial spacing between conductors (mm)
$d_{c} \,$ is the diameter of the conductor, or for shaped conductors, the diameter of an equivalent circular conductor of equal cross-sectional area and degree of compaction (mm)
$K \,$ is a constant factor pertaining to conductor formation (see below for typical values)

No. of wire strands in conductor K
3 0.0778
7 0.0642
19 0.0554
37 0.0528
>60 0.0514
1 (solid) 0.0500

For 3C and 3 x 1C cables, the axial spacing parameter $s \,$ depends on the geometry of the conductors: