---
title: "Graded string: mutual capacitances for uniform voltage"
subject: "Power System Analysis I"
chapter: 2
key: "semester_4/power_system_analysis_i:graded-string-mutual-capacitance"
source: https://www.bctengineer.com/q/semester_4/power_system_analysis_i/graded-string-mutual-capacitance
site: BCT Engineering
---

# Graded string: mutual capacitances for uniform voltage

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

- Asked in: 2075 Baisakh, 6 marks
- Video: [METHODS OF IMPROVING STRING EFFICIENCY| EQUALIZATION OF POTENTIAL ACROSS EACH UNIT](https://www.youtube.com/watch?v=j3M68eE4dao) — EEE TV
- 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/graded-string-mutual-capacitance
- Whole chapter: https://www.bctengineer.com/md/semester_4/power_system_analysis_i/chapter-2

## Question

Find the mutual capacitances of the remaining discs if the mutual capacitance of the top disc is 7C of a graded string of 4 insulators having a uniform voltage across each disc and the capacitance of each pin to earth is C.

## Short answer

The remaining discs need mutual capacitances of 8C, 10C and 13C (from top to line end) for uniform voltage across each disc. This capacitance grading achieves 100% string efficiency.

## Answer

In capacitance grading, each disc gets a different capacitance. The disc nearer the line, which carries more charging current, gets a larger capacitance, so that every disc has the same voltage $v$.

#### Condition at each junction
Number the discs 1 (top, at the cross-arm) to 4 (line end), with capacitances $C_1 \dots C_4$. Let the pin-to-earth capacitance be $C$.

The junction below disc $k$ is at voltage $kv$ above earth. Current through disc $(k+1)$ = current through disc $k$ + current to earth:

$$
\omega C_{k+1}v = \omega C_k v + \omega C(kv) \;\Rightarrow\; C_{k+1} = C_k + kC
$$

#### Calculation (given $C_1 = 7C$)
$$
\begin{aligned}
C_2 &= C_1 + 1\cdot C = 7C + C = 8C\\
C_3 &= C_2 + 2C = 8C + 2C = 10C\\
C_4 &= C_3 + 3C = 10C + 3C = 13C
\end{aligned}
$$

| Disc (from top) | Mutual capacitance | Voltage |
|---|---|---|
| 1 | 7C (given) | $V/4$ |
| 2 | 8C | $V/4$ |
| 3 | 10C | $V/4$ |
| 4 (line end) | 13C | $V/4$ |

**Answer:** The remaining discs need 8C, 10C and 13C. The string efficiency is then 100 %.

Grading needs discs of different sizes, which makes spares and maintenance difficult. In practice, guard rings are preferred.

## Check yourself

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

1. What is the aim of capacitance grading?
   A. To maximise the voltage of the top disc
   B. To make the voltage across every disc equal
   C. To reduce the number of discs
   D. To remove earth capacitance
   Answer: B. Each disc gets a capacitance chosen so every disc carries V/n. (Key point: Aim of grading)

2. What is the relation between successive disc capacitances?
   A. C(k+1) = C(k) - kC
   B. C(k+1) = C(k) + kC
   C. C(k+1) = k C(k)
   D. C(k+1) = C(k) + C
   Answer: B. The junction below disc k is at kv; the pin to earth draws omega C k v, which disc k+1 must supply additionally. (Key point: Recurrence)

3. With C1 = 7C, what are C2, C3, C4?
   A. 8C, 9C, 10C
   B. 7C, 7C, 7C
   C. 8C, 10C, 13C
   D. 8C, 11C, 15C
   Answer: C. C2 = 7C + C, C3 = 8C + 2C, C4 = 10C + 3C. (Key point: Capacitance values)

4. Which disc needs the largest capacitance and why?
   A. The top disc, because it is at earth
   B. The line-end disc, because it carries more charging current
   C. The middle disc, for balance
   D. All discs equal
   Answer: B. Current accumulates toward the line, so the line-end disc needs larger capacitance. (Key point: Which disc gets largest)

## Related in Power System Analysis I

- [Voltage across string discs is not uniform](https://www.bctengineer.com/question.md?key=semester_4%2Fpower_system_analysis_i%3Astring-voltage-not-uniform) — Power System Analysis I, chapter 2, asked 3×
- [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
- [String of 4 insulators at 66 kV, ratio 10](https://www.bctengineer.com/question.md?key=semester_4%2Fpower_system_analysis_i%3Astring-4-insulators-66kv-ratio-10) — Power System Analysis I, chapter 2
- [Line voltage from disc voltages of 4-disc string](https://www.bctengineer.com/question.md?key=semester_4%2Fpower_system_analysis_i%3Aline-voltage-from-disc-voltages) — Power System Analysis I, chapter 2
- [Voltage distribution on 4-disc string at 33 kV](https://www.bctengineer.com/question.md?key=semester_4%2Fpower_system_analysis_i%3Astring-4-discs-33kv) — 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×

## Related in other subjects

- [Line-to-pin capacitances for uniform string voltage](https://www.bctengineer.com/question.md?key=semester_8%2Fhigh_voltage_engineering%3Agrading-ring-five-insulators) — 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
- [Insulator discs per string, 180 MW over 200 km](https://www.bctengineer.com/question.md?key=semester_8%2Ftransmission_and_distribution_system_design%3Ainsulator-discs-180mw-200km) — Transmission and Distribution System Design, chapter 3
- [Tower top potential and insulator string voltage](https://www.bctengineer.com/question.md?key=semester_8%2Fhigh_voltage_engineering%3Atower-top-potential-insulator-voltage) — High Voltage Engineering, chapter 3, asked 2×