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Stratified high voltage bushing
Stratified high voltage bushing is composed of several insulation layers separated with very thin "floating" conductors. Their constant but unknown potential results from the capacitive distribution of electrostatic field and may be controlled by variation of their lengths, thickness and permittivity of the dielectric layers. From the technological point of view only the lengths adjustment can be considered as reasonable way of getting the most uniform distribution of radial component Er of electric strength. It assures the best utilization of insulating material and moderates the field across the high voltage bushing.
In theory, the infinite number of free potential electrodes leads to the uniform and continuous distribution of electric field across the bushing (E(r)=const) instead of logarithmic one. In reality, only the finite number of layers can be considered, usually 1012.
The main goal of the example is to model very simple (only 2 layers) high voltage bushing, find the E(r) distribution and compare results with some mathematical formula. The condition of E_{rmax} equality in every layer (E1_{rmax} = E2_{rmax}) gives the possibility to find the length of "floating" conductor and their potential.
Problem type:
An axisymmetrical problem of free potential electrode in electrostatic field. Dirichlet boundary conditions with given potential are placed at zero and high voltage electrodes.
Geometry of the high voltage bushing:
______________________________________zero potential tubular electrode
 
air  < barrier (zero pot. electrode)
 
_
/ \ < 2nd ins. layer
/__2__\ < "floating" electrode
/ \ < 1st ins. layer
_______________/____1____\______________ high voltage tubular electrode
_ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ axis of symmetry
All dimensions are in centimeters. The length (L) of "floating" electrode is to be calculated before the modelling of the bushing arrangement.
 the radius of zero pot. electrode r_{3} = 30 cm
 the radius of the hole in the barrier r_{2} = 6 cm
 the radius of "floating" electrode r_{1} = 4 cm
 the radius of high voltage electrode r_{0} = 2 cm
Given:
The relative permittivity of dielectric material (epoxy resin) is assumed as 5.0 and high voltage potential  110 kV.
In order to be closer with mathematical description of the model, the slope of dielectric edge was considered as perpendicular to the axis of symmetry.
Results of field calculations:
 Three capacities: C_{1}, C_{2}  of the layers and C_{10}  the free potential electrodetoground are calculated by means of integrals (stored energy or surface charge). The condition of charge balance Q_{1} = Q_{2} + Q_{10} leads to the equation for determining the length (L) of free potential electrode.
____________________________________zero potential tubular electrode
  
  
_____ 
   === C10
C2> ===  
________
  
 === C1 
_________________________________high voltage tubular electrode
_ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ axis of symmetry
 The max. value of E(r) along the middlecross section of the bushing is found from the "XYplot" option. It is recommended to notice the E1_{rmax} and E2_{rmax} equality.
 The potential U of "floating" electrode is found by means of "Average surface potential" calculation or by "Local Value" tool.

C_{10}[pF] 
C_{1}[pF] 
C_{2}[pF] 
E_{r1}[kV/cm] 
E_{r2}[kV/cm] 
U[kV] 
Theory 
3.84 
84.64 
68.51 
36.57 
36.57 
59.31 
QF_1 
3.97 
81.37 
70.10 
30.97 
31.16 
57.51 
QF_2 
3.52 
79.68 
75.90 
 
 
 
Remarks:
1. Two ways of determining the capacity value in QuickField were applied. QF_1: from the energy of electrical field (2·W / U^{2}), QF_2 from the total charge at the electrode (Q/U).
2. The difference with theoretical results is caused mainly by limited number of nodes of the QuickField Student Edition.
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