HV cables
Inductance and reactance of single-core cables, trefoil and flat
- where
- inductance per phase, trefoil (mH/km)
- internal inductance constant, 0.05 for a solid conductor (mH/km)
- axial spacing between adjacent cables (mm)
- conductor diameter (mm)
- average inductance per phase, flat formation (mH/km)
- reactance per phase (Ω/km), using L_t or L_f
- frequency (Hz)
Explain this equation
Gives the series inductance and reactance per phase of three single-core cables in trefoil or flat formation, for voltage drop, fault level and load flow models.
K is the internal inductance from flux inside the conductor, 0.05 mH/km for a solid round conductor. The log term is the external flux linking the conductor, set by spacing s and conductor diameter d. For flat formation s is multiplied by ∛2, the geometric mean of the spacings s, s and 2s.
Inductance of a transposed three-phase group is μ₀/2π times the log of the geometric mean distance over conductor radius; μ₀/2π is 0.2 mH/km, and 2s/d is the same as s/r. X = 2πfL, with 10⁻³ converting mH to H.
Touching trefoil cables usually give about 0.10 to 0.13 Ω/km at 50 Hz; flat formation adds about 0.015 Ω/km. Doubling the spacing adds 0.2 ln 2 ≈ 0.14 mH/km, about 0.044 Ω/km.
Use the conductor diameter for d, not the cable diameter. With both-ends bonded sheaths the effective reactance is a little lower than this.
Positive-sequence series inductance and reactance of three single-core cables with sheaths single-point bonded or cross-bonded (no circulating sheath current).
Enter your values to see the result.
Good to know
Basis: inductance of a transposed three-phase group, L = μ0/(2π)·[1/4 + ln(GMD/r)] per metre = 0.05 + 0.2 ln(GMD/r) mH/km, i.e. 0.2 ln(2s/d) external plus K internal. K = 0.05 mH/km (μ0/8π) is exact for a solid round conductor; the Electric Cables Handbook tabulates slightly higher K for stranded conductors (about 0.064 for 7 wires down to about 0.051 for 61 wires and more). For flat formation the geometric mean distance is ∛2·s ≈ 1.26s; the value is the average of the three phases (untransposed flat cables are unbalanced). A three-core cable can be treated as trefoil with s = distance between core centres.
Common mistake
Using the cable overall diameter for d (it is the conductor diameter), or the centre-to-outer spacing in flat formation instead of the adjacent spacing.
Limits
Ignores sheath currents: with both-ends bonded sheaths the effective reactance is lower and resistance higher. Ignores armour and nearby steel. Power frequency only.
Sources
- Moore, G.F. (ed.) Electric Cables Handbook, 3rd edn, Blackwell Science / Wiley
- IEC 60287-1-1:2023 Electric cables – Calculation of the current rating – Part 1-1: Current rating equations (100 % load factor) and calculation of losses
- Grainger, J.J. and Stevenson, W.D. Power System Analysis (inductance of transposed three-phase lines, GMD/GMR)
For competent high-voltage engineers. Results are calculations from the published methods shown and depend entirely on the data you enter. They are not a substitute for a system study, manufacturer data, the applicable standards and Engineering Recommendations, or your organisation’s safety rules. Never use a result to justify working on or near live HV equipment.