# 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:

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

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

- [math]\rho_{20} \,[/math] is the resistivity of the conductor material at 20
^{o}C ([math]\Omega . m [/math]).- For copper conductors, [math]\rho_{20} \,[/math] = 1.7241 x 10
^{-8} - For aluminium conductors, [math]\rho_{20} \,[/math] = 2.8264 x 10
^{-8}

- For copper conductors, [math]\rho_{20} \,[/math] = 1.7241 x 10

- [math]S \,[/math] is the cross-sectional area of the conductor (mm
^{2}) - [math]\alpha_{20} \,[/math] is the temperature coefficient of the conductor material per K at 20
^{o}C.- For copper conductors, [math]\alpha_{20} \,[/math] = 3.93 x 10
^{-3} - For aluminium conductors, [math]\alpha_{20} \,[/math] = 4.03 x 10
^{-3}

- For copper conductors, [math]\alpha_{20} \,[/math] = 3.93 x 10

- [math]\theta \,[/math] is the conductor operating temperature (
^{o}C)

- [math]\rho_{20} \,[/math] is the resistivity of the conductor material at 20

### AC Resistance

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

- [math]R_{ac} = R_{dc} \left( 1 + y_{s} + y_{p} \right) \, [/math]

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

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

The skin effect factor is calculated as follows:

- [math]y_{s} = \frac{ x_{s}^{4}} {192 + 0.8 x_{s}^{4}}\,[/math]

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

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

Note that the formula above is accurate provided that [math]x_{s} \leq[/math] 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:

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

For 3C and 3 x 1C cables:

- [math]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] \,[/math]

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

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

Note that the formula above is accurate provided that [math]x_{s} \leq[/math] 2.8.

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

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

Type of Conductor | Dried and Impregnated? | [math]k_{s} \,[/math] | [math]k_{p} \,[/math] |
---|---|---|---|

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:

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

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

- [math] f \, [/math] is the supply frequency (Hz)
- [math] s \,[/math] is the axial spacing between conductors (mm)
- [math] d_{c} \, [/math] 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)
- [math] K \, [/math] 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 [math] s \,[/math] depends on the geometry of the conductors:

## References

- IEC 60287-1-1, “Electric cables – Calculation of current rating – Part 1: Current rating equations (100% load factor) and calculation of losses – Section 1: General”, 2006
- G.F. Moore, “Electric Cables Handbook”, Third Edition, 1997, an excellent reference book for cables