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,

                   g_x = 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,

                g_1max = Q/(πε₁d)

       and    g_2max = 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,

And

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