Voltage distribution of 3-unit string with guard ring

With a guard ring, each link pin has two shunt capacitances: one to the earthed metalwork (tower), which takes current away, and one to the guard ring (at line potential), which supplies current. Kirchhoff's current law at each pin gives the voltage distribution.

Data

• 3 identical units, each of capacitance $C$
• Pin-to-metalwork capacitance $C_1 = 0.15C$
• Pin-to-guard-ring capacitance $C_2 = 0.05C$
• Units numbered 1 (top, at cross-arm) to 3 (bottom, at line), with voltages $V_1, V_2, V_3$; line voltage $V = V_1 + V_2 + V_3$

 cross-arm (earth)
    | unit 1  (V1)
    A ---0.15C--> tower ; <--0.05C-- ring
    | unit 2  (V2)
    B ---0.15C--> tower ; <--0.05C-- ring
    | unit 3  (V3)
 line (V) ======= guard ring (V)

KCL at pin A (potential $V_1$ above earth, $V - V_1$ below ring)

Current in from unit 2 + current from ring = current into unit 1 + current to tower:

$$\begin{aligned} CV_2 + 0.05C(V - V_1) &= CV_1 + 0.15CV_1 \\ V_2 &= 1.2V_1 - 0.05V \qquad (1) \end{aligned}$$

KCL at pin B (potential $V_1 + V_2$)

$$\begin{aligned} CV_3 + 0.05C(V - V_1 - V_2) &= CV_2 + 0.15C(V_1 + V_2) \\ V_3 &= 0.2V_1 + 1.2V_2 - 0.05V \qquad (2) \end{aligned}$$

Solving

Substitute (1) into (2):

$$\begin{aligned} V_3 &= 0.2V_1 + 1.2(1.2V_1 - 0.05V) - 0.05V \\ &= 1.64V_1 - 0.11V \end{aligned}$$

Then $V = V_1 + V_2 + V_3$:

$$\begin{aligned} V &= V_1 + (1.2V_1 - 0.05V) + (1.64V_1 - 0.11V) \\ 1.16V &= 3.84V_1 \\ V_1 &= 0.3021V \end{aligned}$$

Hence

$$\begin{aligned} V_2 &= 1.2(0.3021V) - 0.05V = 0.3125V \\ V_3 &= 1.64(0.3021V) - 0.11V = 0.3854V \end{aligned}$$

Check: $0.3021 + 0.3125 + 0.3854 = 1.000$.

Voltage distribution

String efficiency

$$\begin{aligned} \eta &= \frac{V}{3 \times V_3} = \frac{1}{3 \times 0.3854} \\ &= \frac{96}{111} = 0.8649 = 86.5\% \end{aligned}$$

For comparison, without the guard ring ($C_2 = 0$, $m = 0.15$) the efficiency would be about 82.0%. The guard ring improves the voltage distribution.

Answer: $V_1 = 0.302V$, $V_2 = 0.3125V$, $V_3 = 0.385V$; string efficiency ≈ 86.5%.