Transients and insulation
Field between coaxial cylinders and utilisation factor
- where
- field at the inner conductor (kV/mm)
- voltage between electrodes (kV)
- inner conductor radius (mm)
- outer electrode inner radius (mm)
- field at the outer electrode (kV/mm)
- Schwaiger utilisation factor (E_mean/E_max)
- lowest possible E_max, at r = R/e (kV/mm)
- base of natural logarithms, 2.718
Explain this equation
Gives the electric stress in a coaxial geometry: single-core cables, GIS busbars and condenser bushings. It also gives Schwaiger’s utilisation factor and the lowest stress possible for the outer size.
V is the voltage between the inner conductor (radius r) and the outer electrode (radius R). The field falls as 1/x from the centre, so E_max sits on the inner conductor and E_min on the outer. η is the mean field V/(R − r) divided by E_max: 1 for a uniform field, lower for a less uniform one.
Gauss’s law gives E(x) = V/(x ln(R/r)). Making r larger reduces the 1/r term but increases 1/ln(R/r); the product is lowest when R/r = e, giving E_opt = eV/R.
Sense check: a 100 kV coaxial gap with r = 10 mm, R = 30 mm gives about 9.1 kV/mm at the conductor and η ≈ 0.55. Doubling V doubles every field value.
Watch out: for a cable use the phase-to-earth voltage and radii over the semiconducting screens.
Electric field stress in a coaxial geometry such as a single-core cable, GIS busbar or bushing, with Schwaiger’s field utilisation factor η = Emean/Emax.
Enter your values to see the result.
Good to know
E(x) = V/(x ln(R/r)). For a fixed outer radius and voltage, Emax is smallest when R/r = e (≈ 2.718), giving Emax = eV/R; this sets the optimum conductor size for GIS and cable design. η = 1 is a uniform field; lower η means a less efficient use of the insulation. For a cable, R is the inner radius of the insulation screen and r the radius over the conductor screen; use the peak or test voltage to neutral as V.
Common mistake
Using line-to-line rms voltage for a cable core-to-screen stress, or the conductor radius without the semiconducting screen.
Limits
Ideal infinite coaxial cylinders, homogeneous single dielectric, electrostatic field. Strand-edge effects, voids, graded (multi-layer) insulation and end effects are not included.
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.