Calculation of potential gradients for a dielectric slab placed between a sphere and a plane

Abstract

SynopsisSuccessive overrelaxation methods have been used to calculate potential gradients around a spherical high-voltage electrode separated from an earthed plate by a plane or recessed dielectric slab. The distance from the sphere to the earthed plate has been made one half of the sphere radius for most of the calculations, and the ratio of the permittivities of the dielectric slab and the surrounding medium has been varied. For a limited range of slab-thickness/sphere-radius ratios, it is shown that the highest potential gradient in a plane slab may be calculated by assuming that a uniform permittivity exists throughout and by using a fictitious sphere-radius/electrode-separation ratio. For a plane slab having a permittivity 4 times that of the surrounding medium, and for slab a thickness equal to the sphere radius, the highest potential gradient in the medium is found to be nearly 14 times the average gradient in the shortest gap. When the slab thickness is one half of the sphere radius, the highest potential gradient in the medium is still about 9 times the gap average, but this is considerably reduced when a recessed slab is used. The potential-gradient transitions at the point where the surface of the recessed dielectric slab meets the sphere surface have been determined and analysed. List of symbolsa = mesh length for relaxation b = radius of cylindrical region relaxation used for C 2 , C3, h 2 , h 3 , CA 1 , >• = factors used in relaxation process K)D = electric-flux density d = dielectric-slab thickness E = potential gradient e -permittivity of dielectric ej, e 2 -relative permittivities of surrounding medium and dielectric slab, respectively e 3 -relative permittivity of vessel enclosing dielectric slab and medium e R = relative permittivity of space surrounding test vessel g -separation of sphere and earthed plate h = axial height of cylindrical region used for relaxation R = sphere radius r = radial distance from axis of symmetry p = electric resistivity S = €,/€ 2 9 h 6 2 = angles of incidence and refraction of line of electrostatic flux meeting interface V = potential to = acceleration factor for relaxation z = axial distance below an origin placed at centre of sphere

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“…To validate the FEA, electric fields on the top surface of a dielectric plate (ε d ) in contact with a spherical electrode on the top surface and immersed in a dielectric liquid (ε l ) were calculated for the case of the top electrode surface at a constant potential, V , whereas the bottom surface of the dielectric plate was at zero potential (ground). This particular problem was selected for validation of the FEA model because Binns and Randall, Takuma and Kawamoto, and Poli have calculated electric fields on the surface of the solid dielectric by other numerical methods, such as the finite difference method and the charge simulation method . Table lists the values of the normalized maximum electric fields, ( E max R 1 / V ), on the surface of the solid dielectric plate, for several values of ε s = (ε d /ε l ) and R 1 = t , where R 1 is the radius of the top electrode and t is the thickness of the dielectric plate.…”

“…For all three values of the relative dielectric constant, ε s , the maximum electric fields calculated by FEA were in good agreement with the corresponding values reported in the literature. It is useful to note that Binns and Randall, Takuma and Kawamoto, and Poli considered a dielectric plate infinite in in‐plane dimensions, whereas the FEA was conducted with a finite disk with R = 30 mm and R 1 = t = 10 mm. Additional FEA calculations for R = 60 mm and R = 120 mm indicated that there was no significant change in the maximum electric field.…”

“…The top electrode surface was maintained at a potential of 75 kV, whereas the bottom electrode was at zero potential (ground). The validity of the FEA was established by calculating the maximum electric field in a dielectric plate placed between a sphere and a plane electrode and comparing the results with values reported in the literature …”