Transients and insulation
Field between concentric spheres and utilisation factor
- where
- field at the inner sphere (kV/mm)
- voltage between electrodes (kV)
- outer sphere inner radius (mm)
- inner sphere radius (mm)
- field at the outer sphere (kV/mm)
- Schwaiger utilisation factor
- lowest possible E_max, at r = R/2 (kV/mm)
Explain this equation
Gives the electric stress between concentric spheres, the reference case for spherical HV electrodes, grading balls and terminal shields, with Schwaiger’s utilisation factor.
r is the inner sphere radius and R the inner radius of the outer sphere. The field falls as 1/x², so the stress is concentrated near the inner sphere. η = r/R is the ratio of mean to maximum field.
From Gauss’s law the charge on the inner sphere is set by the capacitance 4πεrR/(R − r), so E(x) = VrR/((R − r)x²). For fixed R and V the maximum field is least when r = R/2, giving E_opt = 4V/R.
Sense check: 100 kV with r = 10 mm and R = 30 mm gives 15 kV/mm at the inner sphere and η = 1/3, a much less uniform field than coaxial cylinders of the same radii.
Watch out: two spheres side by side (a sphere gap) is a different, less uniform geometry and is not covered by this formula.
Electric field stress between concentric spheres, the reference case for spherical HV electrodes and corona shields, with Schwaiger’s utilisation factor.
Enter your values to see the result.
Good to know
E(x) = V r R/((R − r) x²). For fixed outer radius, Emax is smallest when r = R/2, giving 4V/R. Large-radius toroids and spheres on HV terminals are sized with this kind of estimate before a numerical field study.
Common mistake
Using this for an isolated sphere gap (two spheres side by side): that field is much less uniform and needs Schwaiger’s charts or IEC 60052 sphere-gap data.
Limits
Ideal concentric spheres, single homogeneous dielectric. Sphere-to-sphere and sphere-to-plane gaps are not covered here because their utilisation factors come from charts and image-charge series rather than a closed form.
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.