HV cables
Optimum conductor radius for minimum stress (Ri/rc = e)
- where
- optimum radius over the conductor screen (mm)
- radius over the insulation (mm)
- base of natural logarithms, 2.71828
- maximum stress at the optimum (kV/mm)
- rated voltage to earth (kV)
- insulation thickness (mm)
Explain this equation
Finds the conductor-screen radius that gives the lowest maximum stress for a fixed insulation outer radius, and that lowest stress. Useful for judging how far a design is from the stress optimum.
Rᵢ is the fixed outer radius of the insulation. r_c is the radius over the conductor screen. E_max is the stress at the conductor screen and t the insulation thickness left.
E_max = U₀/(r_c ln(Rᵢ/r_c)) is large for a thin conductor (small r_c) and for a thin insulation (ln near zero). Setting its derivative to zero gives ln(Rᵢ/r_c) = 1, so Rᵢ/r_c = e and E_max reduces to U₀/r_c.
For a 132 kV cable (U₀ = 76.2 kV) with Rᵢ = 38 mm, the optimum r_c is about 14 mm and E_max about 5.5 kV/mm. The minimum is shallow: a conductor 30 % larger than the optimum raises E_max by only a few per cent.
Conductor size is normally fixed by current rating, so treat the optimum as a reference point rather than a design target.
For a fixed insulation outer radius, the stress at the conductor screen is lowest when Ri/rc = e (about 2.718). Use it to see how far a design is from the stress optimum.
Enter your values to see the result.
Good to know
Setting d(Emax)/d(rc) = 0 for Emax = U0/(rc ln(Ri/rc)) gives ln(Ri/rc) = 1. At the optimum Emax = U0/rc = e·U0/Ri. The minimum is flat, so modest departures from the optimum cost little stress. In practice the conductor size is set by current rating, and HV cables often have conductors larger than the stress optimum.
Common mistake
Treating the optimum as the design target: thermal rating and the conductor size usually govern, and a larger conductor than the optimum barely raises the stress.
Limits
Same coaxial field assumptions as the stress calculator. Ignores the limits on insulation stress at the insulation screen and the practical minimum thicknesses in IEC 60502-2 / IEC 60840.
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.