HV cables
Conductor a.c. resistance at operating temperature (IEC 60287-1-1)
- where
- d.c. resistance at θ (Ω/km)
- d.c. resistance at 20 °C, IEC 60228 (Ω/km)
- temperature coefficient at 20 °C (1/K)
- conductor temperature (°C)
- skin-effect argument
- frequency (Hz)
- skin-effect factor (IEC 60287-1-1 Table 2)
- skin-effect factor
- proximity-effect factor, yp = F(d_c/s)²[0.312(d_c/s)² + 1.18/(F + 0.27)], F = x_p⁴/(192 + 0.8x_p⁴), x_p² as x_s² with k_p
- a.c. resistance at θ (Ω/km)
Explain this equation
Finds the conductor resistance at its operating temperature and adds the skin and proximity effects to give the a.c. resistance used in ratings, losses and voltage drop.
R₂₀ is the IEC 60228 d.c. resistance at 20 °C and α₂₀ the temperature coefficient (copper 0.00393, aluminium 0.00403 per K). x_s is a measure of how strongly current crowds to the surface at frequency f; k_s and k_p allow for conductor construction. y_s and y_p are the fractional increases from skin and proximity effects.
Metal resistance rises almost linearly with temperature. At a.c., the conductor's own field pushes current outwards (skin effect) and the neighbouring phases' fields distort it further (proximity effect, set by d_c/s). IEC 60287 fits both with x⁴/(192 + 0.8x⁴) expressions; the 10⁻³ turns Ω/km into Ω/m as the formula needs.
Copper at 90 °C is about 27 % above its 20 °C value. For a 630 mm² copper conductor in trefoil, skin and proximity add roughly 6 % and 4 %; for 240 mm² aluminium they add well under 1 %.
The simple IEC expressions are valid for x_s and x_p up to 2.8; very large conductors need segmental construction and the full method.
Conductor d.c. resistance corrected to operating temperature, then the IEC 60287-1-1 skin and proximity effect factors for three single-core cables or a three-core cable, giving the a.c. resistance used in ratings and voltage drop.
Enter your values to see the result.
Good to know
α20 per IEC 60287-1-1 (consistent with IEC 60228 correction factors): copper 3.93 × 10⁻³ /K, aluminium 4.03 × 10⁻³ /K. R20 is the IEC 60228 maximum d.c. resistance for the conductor class and size (e.g. 630 mm² Cu 0.0283 Ω/km, 240 mm² Al 0.125 Ω/km). Proximity: yp = F·(dc/s)²·[0.312(dc/s)² + 1.18/(F + 0.27)] with F = xp⁴/(192 + 0.8xp⁴), for three single-core cables or three-core cables; for flat formation use s = √(s1·s2); for a three-core cable s = dc + insulation thickness ×2 (plus screens). Enter a very large s to ignore proximity. ks and kp are 1 for round solid and most round stranded conductors; segmental (Milliken) conductors have lower values (see IEC 60287-1-1 Table 2).
Common mistake
Using the 20 °C resistance in a voltage-drop or loss calculation at 90 °C (about 27 % low for copper), or ignoring skin effect on large conductors.
Limits
IEC 60287-1-1 simple expressions, valid for xs and xp up to 2.8. Pipe-type and two-core formulas are not included. Sheath and armour eddy currents are not part of the conductor resistance.
Sources
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.