Guard ring capacitances for 5-disc string

A guard (grading) ring is a metal ring connected to the line conductor and surrounding the lower units of the string. It adds capacitance $C_x$ between each link pin and the line. This capacitance supplies, at every pin, the charging current that the pin-to-earth capacitance $C_1$ takes away. Then the same current flows through every disc, and the voltage is equal across all units.

Assumptions

• 5 identical discs, each of self capacitance $C$.
• Pin-to-earth (tower) capacitance at each link pin = $C_1$.
• Pin-to-guard-ring capacitance at the $n$-th pin = $C_n$ (unknown).
• Pins are numbered $n = 1$ to 4 from the cross-arm; the line is at voltage $5V$, so each disc has voltage $V$.

  cross-arm (earth)
     |  disc 1
     +- pin 1 at V    --C1--> earth   <--Cx1-- ring
     |  disc 2
     +- pin 2 at 2V   --C1--> earth   <--Cx2-- ring
     |  disc 3
     +- pin 3 at 3V   --C1--> earth   <--Cx3-- ring
     |  disc 4
     +- pin 4 at 4V   --C1--> earth   <--Cx4-- ring
     |  disc 5
  line (5V) ===== guard ring (at 5V)

Condition for equal voltage

Since all discs have the same voltage $V$ and capacitance $C$, the current through every disc is the same ($\omega C V$). So at each pin, the current leaving to earth must be exactly supplied by the current coming from the guard ring.

At pin $n$ (voltage above earth $= nV$, voltage below line $= (5 - n)V$):

$$\omega C_n (5 - n)V = \omega C_1 (nV)$$ $$\boxed{C_n = \frac{n}{5 - n}\,C_1}$$

In general, for a string of $N$ units: $C_n = \dfrac{n}{N - n}C_1$.

Values for 5 discs

$$\begin{aligned} C_{x1} &= \frac{1}{4}C_1 = 0.25\,C_1 \\ C_{x2} &= \frac{2}{3}C_1 = 0.667\,C_1 \\ C_{x3} &= \frac{3}{2}C_1 = 1.5\,C_1 \\ C_{x4} &= \frac{4}{1}C_1 = 4\,C_1 \end{aligned}$$

Remarks

• The required capacitance increases sharply towards the line end. This is why the guard ring is placed around the lower discs, close to the line.
• With exact values the string efficiency becomes 100%. In practice exact values cannot be obtained, so the voltage is only approximately equalised.