Graded string: mutual capacitances for uniform voltage

In capacitance grading, each disc gets a different capacitance. The disc nearer the line, which carries more charging current, gets a larger capacitance, so that every disc has the same voltage $v$.

Condition at each junction

Number the discs 1 (top, at the cross-arm) to 4 (line end), with capacitances $C_1 \dots C_4$. Let the pin-to-earth capacitance be $C$.

The junction below disc $k$ is at voltage $kv$ above earth. Current through disc $(k+1)$ = current through disc $k$ + current to earth:

$$\omega C_{k+1}v = \omega C_k v + \omega C(kv) \;\Rightarrow\; C_{k+1} = C_k + kC$$

Calculation (given $C_1 = 7C$)

$$\begin{aligned} C_2 &= C_1 + 1\cdot C = 7C + C = 8C\\ C_3 &= C_2 + 2C = 8C + 2C = 10C\\ C_4 &= C_3 + 3C = 10C + 3C = 13C \end{aligned}$$

Answer: The remaining discs need 8C, 10C and 13C. The string efficiency is then 100 %.

Grading needs discs of different sizes, which makes spares and maintenance difficult. In practice, guard rings are preferred.