Doubts regarding Grading problem

1. 1. In a string of three insulator units, the capacitance of each unit is C, from each conductor to
ground is C/3, and from each connector to the line conductor is C/5. Calculate the voltage across
each unit as a % of the total voltage. To what value the capacitance between the connector of
the bottom unit and the line has to be increased by a guard ring to make the voltage across it
equal to that across the next higher unit? [31.5%, 28.3%, 40.196%; 0.6833C]
Solution:
Apply KCL at node A
I1+Ia = i2 + Ix
V1(jwC) + v1(jwC/3) = v2(jwC) + (v2+v3)(jwC/5)
20v1 = 18v2 + 3v3 ………(1)
Apply KCL at node B
i2 + Ib = I3 + Iy
20v2 +5v1 = 18v3 ………(2)
Eliminating v3 from the above 2 equations
V2 = 115/128 v1
And , then v3 = 1.276 v1
Hence, v1 + v2 + v3 = V or v1 = 1/3.174 V = 31.5% of V
Similarly, v2 = 28.3% of V

2. And v3 = 40.196% of V
With Grading Ring
The first equation does not change, i.e. 20v1 = 18v2 + 3v3
But it is given that v2 = v3
Therefore v2 = (20/21)v1
Applying KCL at node B,
I2 + Ib = i3 + Iy
V2(jwC) + (v1+ v2)(jwC/3) = v3(jwC) + v3(jwCy)
But v2 = v3
1/3(v1 + v2)C = (v2)Cy
Cy = 0.6833C