---
title: "4-unit string with guard ring (10% and 5%)"
subject: "Power System Analysis I"
chapter: 2
key: "semester_4/power_system_analysis_i:guard-ring-4-unit-10-5"
source: https://www.bctengineer.com/q/semester_4/power_system_analysis_i/guard-ring-4-unit-10-5
site: BCT Engineering
---

# 4-unit string with guard ring (10% and 5%)

Power System Analysis I · Chapter 2: Overhead and Underground Transmission

- Asked in: 2075 Bhadra, 8 marks
- Video: [L53:Improving String Efficiency in Hindi | 3 Methods | Power System Series](https://www.youtube.com/watch?v=CT_DKToFwS4) — TechInsight
- Answer checked by Claude Opus 5.5 (high effort) on 2026-10-14
- Page: https://www.bctengineer.com/q/semester_4/power_system_analysis_i/guard-ring-4-unit-10-5
- Whole chapter: https://www.bctengineer.com/md/semester_4/power_system_analysis_i/chapter-2

## Question

A 4 unit insulator string is fitted with a guard ring. The capacitance of the link pins to metal work and guard ring can be assumed to be 10% and 5% of the capacitance of each unit. Determine the voltage distribution and string efficiency.

## Short answer

The voltage distribution from tower to line end is 23.74%, 22.30%, 24.21% and 29.75% of the phase voltage. String efficiency is improved to 84.04%.

## Answer

A guard ring is a metal ring at line potential placed around the bottom of the string. It gives each link pin a capacitance to the line that partly offsets the pin-to-tower (earth) capacitance. Here the pin-to-tower capacitance is $mC = 0.1C$ and the pin-to-guard-ring capacitance is $gC = 0.05C$.

```text
 Tower ─[C]─ A ─[C]─ B ─[C]─ D ─[C]─ Line (V)
             |       |       |
  to tower  0.1C    0.1C    0.1C
  to ring   0.05C   0.05C   0.05C
```

#### Junction equation
Number the discs 1 (tower) to 4 (line end). Let $V$ be the phase voltage and $S_k = V_1+\dots+V_k$. At the junction below disc $k$:

$$
\omega C V_{k+1} + \omega gC(V - S_k) = \omega C V_k + \omega mC S_k
$$

$$
\Rightarrow V_{k+1} = V_k + 0.1S_k - 0.05(V - S_k)
$$

#### Solution
$$
\begin{aligned}
V_2 &= 1.15V_1 - 0.05V\\
V_3 &= V_2 + 0.15(V_1+V_2) - 0.05V = 1.4725V_1 - 0.1075V\\
V_4 &= V_3 + 0.15(V_1+V_2+V_3) - 0.05V = 2.0159V_1 - 0.1811V
\end{aligned}
$$

Adding: $5.6384V_1 - 0.3386V = V$, so $V_1 = 0.2374V$.

| Disc | % of phase voltage $V$ |
|---|---|
| 1 (tower end) | 23.74 % |
| 2 | 22.30 % |
| 3 | 24.21 % |
| 4 (line end) | 29.75 % |

#### String efficiency
$$
\eta = \frac{V}{4V_4} = \frac{1}{4\times 0.29747} = 84.04\ \%
$$

**Answer:** The voltage distribution is 23.74 %, 22.30 %, 24.21 % and 29.75 % of the phase voltage, and the string efficiency is 84.04 %. (Without the guard ring, with $m = 0.1$, the line-end disc would take 32.6 % and the efficiency would be about 76.6 %.)

## Check yourself

The answer's key points as multiple-choice questions, with the correct option and why.

1. At what potential is the guard ring?
   A. Earth potential
   B. Line potential
   C. Half the line voltage
   D. Floating at zero
   Answer: B. The ring is connected to the line conductor, so the pins gain capacitance to the line. (Key point: Guard ring potential)

2. Which equation applies (m = 0.1, g = 0.05)?
   A. V(k+1) = V(k) + 0.1 S_k - 0.05 (V - S_k)
   B. V(k+1) = V(k) + 0.1 S_k + 0.05 (V - S_k)
   C. V(k+1) = V(k) + 0.05 S_k - 0.1 (V - S_k)
   D. V(k+1) = V(k) + 0.15 S_k
   Answer: A. Tower capacitance adds current; guard ring capacitance supplies current and so is subtracted. (Key point: Junction equation)

3. Which disc has the lowest voltage?
   A. Disc 1, at 23.74 %
   B. Disc 2, at 22.30 % of V
   C. Disc 4, at 29.75 %
   D. Disc 3, at 24.21 %
   Answer: B. The distribution is 23.74, 22.30, 24.21, 29.75 %. (Key point: Distribution)

4. What is the string efficiency, and how does it compare with no guard ring (m = 0.1)?
   A. 76.6 %, better than 84.04 %
   B. 84.04 %, same as without ring
   C. 84.04 %, better than about 76.6 %
   D. 92 %, worse than without ring
   Answer: C. The guard ring lowers V4 from 32.6 % to 29.75 %, raising efficiency. (Key point: Efficiency)

## Close variants set in other exams

- [4-unit string with guard ring (10% and 6%)](https://www.bctengineer.com/question.md?key=semester_4%2Fpower_system_analysis_i%3Aguard-ring-4-unit-10-6) — Power System Analysis I, chapter 2, asked 2×

## Related in Power System Analysis I

- [3-unit string with guard ring: redistribution of voltage](https://www.bctengineer.com/question.md?key=semester_4%2Fpower_system_analysis_i%3Aguard-ring-3-units-redistribution) — Power System Analysis I, chapter 2
- [Guard ring sizing for 3-unit string at 20 kV](https://www.bctengineer.com/question.md?key=semester_4%2Fpower_system_analysis_i%3Aguard-ring-3-units-20kv) — Power System Analysis I, chapter 2
- [String efficiency: 66 kV, 4 discs, 10%](https://www.bctengineer.com/question.md?key=semester_4%2Fpower_system_analysis_i%3Astring-efficiency-66kv-4-discs) — Power System Analysis I, chapter 2, asked 2×
- [String efficiency: 33 kV, 3 discs, 10%](https://www.bctengineer.com/question.md?key=semester_4%2Fpower_system_analysis_i%3Astring-efficiency-33kv-3-discs) — Power System Analysis I, chapter 2, asked 2×
- [Maximum voltage of 3-disc string, 17.5 kV per unit](https://www.bctengineer.com/question.md?key=semester_4%2Fpower_system_analysis_i%3Amax-voltage-3-disc-17-5kv) — Power System Analysis I, chapter 2
- [Greatest working voltage of 3-unit string, 20 kV limit](https://www.bctengineer.com/question.md?key=semester_4%2Fpower_system_analysis_i%3Agreatest-working-voltage-3-units) — Power System Analysis I, chapter 2

## Related in other subjects

- [Voltage distribution of 3-unit string with guard ring](https://www.bctengineer.com/question.md?key=semester_8%2Fhigh_voltage_engineering%3Astring-guard-ring-3-unit) — High Voltage Engineering, chapter 5
- [Guard ring capacitances for 5-disc string](https://www.bctengineer.com/question.md?key=semester_8%2Fhigh_voltage_engineering%3Aguard-ring-five-disc-capacitance) — High Voltage Engineering, chapter 5