---
title: "String efficiency: 4 units, 20% and 5%"
subject: "Power System Analysis I"
chapter: 2
key: "semester_4/power_system_analysis_i:string-efficiency-4-unit-20-5"
source: https://www.bctengineer.com/q/semester_4/power_system_analysis_i/string-efficiency-4-unit-20-5
site: BCT Engineering
---

# String efficiency: 4 units, 20% and 5%

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

- Asked in: 2078 Chaitra, 6 marks
- Video: [String Efficiency Numerical 1 - Mechanical Design of Overhead Lines - Power System Engineering 1](https://www.youtube.com/watch?v=AQWAad1sAR0) — Ekeeda
- 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/string-efficiency-4-unit-20-5
- Whole chapter: https://www.bctengineer.com/md/semester_4/power_system_analysis_i/chapter-2

## Question

Calculate the string efficiency for a 4-unit suspension insulator if the capacitance of each unit to earth and line be 20% and 5% of the self-capacitance of the unit respectively.

## Short answer

Solving junction equations with m=0.2 and g=0.05 gives disc voltages of 19.82%, 19.77%, 24.67%, 35.74% of V from tower to line end. String efficiency = V/(4*Vmax) = 69.96%.

## Answer

Each link pin has capacitance to earth $mC = 0.2C$ and to the line $gC = 0.05C$, where $C$ is the self-capacitance of a unit. Number the discs 1 (tower end) to 4 (line end). Let $V$ be the phase voltage and $S_k = V_1+\dots+V_k$.

```text
 Tower─[C]─A─[C]─B─[C]─D─[C]─Line(V)
           |     |     |
     earth 0.2C each; line 0.05C each
```

#### Junction equation
Current from the line side = current to the tower side + current to earth − current received from the line:

$$
V_{k+1} = V_k + 0.2S_k - 0.05(V - S_k)
$$

$$
\begin{aligned}
V_2 &= 1.25V_1 - 0.05V\\
V_3 &= V_2 + 0.25(V_1+V_2) - 0.05V = 1.8125V_1 - 0.1125V\\
V_4 &= V_3 + 0.25(V_1+V_2+V_3) - 0.05V = 2.8281V_1 - 0.2031V
\end{aligned}
$$

Sum $= V$:

$$
6.8906V_1 - 0.3656V = V \;\Rightarrow\; V_1 = 0.1982V
$$

| Disc | % of $V$ |
|---|---|
| 1 | 19.82 % |
| 2 | 19.77 % |
| 3 | 24.67 % |
| 4 (line end) | 35.74 % |

#### String efficiency
$$
\eta = \frac{V}{4V_4} = \frac{1}{4\times 0.35737} = 69.96\ \%
$$

**Answer:** The string efficiency is 69.96 % (about 70 %).

## Check yourself

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

1. Which junction equation is used for earth ratio 0.2 and line ratio 0.05?
   A. V(k+1) = V(k) + 0.2 S_k + 0.05 (V - S_k)
   B. V(k+1) = V(k) + 0.2 S_k - 0.05 (V - S_k)
   C. V(k+1) = V(k) + 0.25 S_k
   D. V(k+1) = V(k) - 0.2 S_k - 0.05 V
   Answer: B. Earth capacitance draws extra current; line capacitance supplies some, so it is subtracted. (Key point: Junction equation)

2. What is V2 in terms of V1 and V?
   A. V2 = 1.2 V1
   B. V2 = 1.25 V1 - 0.05 V
   C. V2 = 1.25 V1 + 0.05 V
   D. V2 = 1.05 V1 - 0.2 V
   Answer: B. V2 = V1 + 0.2V1 - 0.05(V - V1) = 1.25V1 - 0.05V. (Key point: Second disc)

3. What fraction of V does the line-end disc carry?
   A. About 25 %
   B. About 19.8 %
   C. About 35.74 %
   D. About 29.2 %
   Answer: C. V1 = 0.1982V, giving V4 = 0.3574V. (Key point: Line-end disc share)

4. What is the string efficiency?
   A. 80.2 %
   B. 55.3 %
   C. 84.0 %
   D. 69.96 % (about 70 %)
   Answer: D. Efficiency = 1/(4 x 0.35737) = 69.96 %. (Key point: Efficiency)

## Close variants set in other exams

- [String efficiency: 3 units, 25% and 10%, 25 kV](https://www.bctengineer.com/question.md?key=semester_4%2Fpower_system_analysis_i%3Astring-efficiency-3-unit-25kv) — Power System Analysis I, chapter 2, asked 2×

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- [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
- [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
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- [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×
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