Friday, September 2, 2011

Capacitance Grading

       The grading done by using the layers of dielectrics having different permittivities between the core and the sheath is called capacitance grading.
       In intersheath grading, the permittivity of dielectric is same everywhere and the dielectric is said to be homogeneous. But in case of capacitance grading, a composite dielectric is used.
       Let    d1 = Diameter of the dielectric with permittivity ε1
      and     D = Diameter of the dielectric with permittivity ε2
       This is shown in the Fig. 1.
Fig. 1 Capacitance grading

       The stress at a point which is at a distance x is inversely proportional to the distance x and given by,
                   gx = Q/(2πε x)
Hence the stress at point in the inner dielectric is,
                    g1 = Q/(2πε1 x)
Similarly the dielectric stress in the outer dielectric is,
                     g2 = Q/(2πε2 x)
       Hence the total voltage V can be expressed as,
       The stress is maximum at surface of conductor i.e. x =d/2.
       And the stress is maximum at inner surface of dielectric i.e. at x = d1/2.
       Substituting Q interms of V we get,

Key Point : Thus the electric stress is inversely proportional to the permittivities and the inner radii of the dielectrics.
1.1 Condition for Equal Maximum Stress
       Let us obtain the condition under which the maximum values of the stresses in the two regions are equal.
      The maximum stresses are given by,
                g1max = Q/(πε1d)
       and    g2max = Q/(πε2d1)
       Equating the two stresses,
                Q/(πε1d) = Q/(πε2d1)

       Now d1 is greater than d so to satisfy above equation ε2 must be less than ε1.
       Thus the dielectric nearest to the conductor must have the highest permittivity.
Similar for the grading with three dielectrics with permittivities ε1, ε2 and ε3, for equal maximum stress the condition is,

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