Line-to-pin capacitances for uniform string voltage

A grading (guard) ring is a metal ring connected to the line conductor and placed around the lower units of the string. It adds a capacitance $C_k$ from the line to each cap/pin junction. If these capacitances are chosen correctly, the charging current through the line-to-pin capacitance supplies exactly the current leaking to earth through the pin-to-earth capacitance, so all units carry the same current and share the voltage equally.

Arrangement

   Cross-arm (earth)
        |
     [Unit 1]   self cap  (same for all)
        +---- J1 --- C ---> earth,  C1 <--- line
     [Unit 2]
        +---- J2 --- C ---> earth,  C2 <--- line
     [Unit 3]
        +---- J3 --- C ---> earth,  C3 <--- line
     [Unit 4]
        +---- J4 --- C ---> earth,  C4 <--- line
     [Unit 5]
        |
   Line conductor (V) ---- grading ring

Junctions $J_1 \dots J_4$ are numbered from the cross-arm. Let the total string voltage be $V$.

Condition for uniform distribution

For equal voltage across each of the 5 units, each unit carries $V/5$, so the voltage of junction $J_k$ above earth is

$$V_k = \frac{k\,V}{5}, \qquad k = 1,2,3,4$$

The voltage between the line and $J_k$ is $V - V_k = \frac{(5-k)V}{5}$.

Since all units have the same self-capacitance and the same voltage, the currents through the units are equal. Applying KCL at $J_k$, the current that enters through the line-to-pin capacitance $C_k$ must equal the current leaving through the pin-to-earth capacitance $C$:

$$\begin{aligned} \omega C_k (V - V_k) &= \omega C\, V_k \\ C_k &= C\,\frac{V_k}{V - V_k} = C\,\frac{k}{5-k} \end{aligned}$$

Values

Answer: $C_1 = 0.25C,\ C_2 = 0.667C,\ C_3 = 1.5C,\ C_4 = 4C$ (numbered from the cross-arm end). The value does not depend on the unit's self-capacitance.

The line-to-pin capacitance must rise sharply towards the line end, which is why the ring is placed close to the units nearest the conductor. In practice exact values are hard to obtain, so the grading ring only improves the distribution and string efficiency, while also reducing corona at the line-end fittings.