---
title: "Voltage distribution on 4-disc string at 33 kV"
subject: "Power System Analysis I"
chapter: 2
key: "semester_4/power_system_analysis_i:string-4-discs-33kv"
source: https://www.bctengineer.com/q/semester_4/power_system_analysis_i/string-4-discs-33kv
site: BCT Engineering
---

# Voltage distribution on 4-disc string at 33 kV

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

- Asked in: 2069 Bhadra, 6 marks
- Video: [8  Examples on determining the voltage distribution of insulator string](https://www.youtube.com/watch?v=sYwG153QLuY) — HVdesc
- 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-4-discs-33kv
- Whole chapter: https://www.bctengineer.com/md/semester_4/power_system_analysis_i/chapter-2

## Question

Determine the voltage distribution along a string of 4 disc insulators used in a 33 kV overhead line.

## Short answer

Assuming m = C1/C = 0.1, phase voltage 33/√3 = 19.053 kV. Disc voltages from tower end: 3.77 kV, 4.14 kV, 4.93 kV, 6.22 kV (line end). String efficiency = 19.053/(4×6.215) = 76.64%.

## Answer

No capacitance ratio is given. Assume the usual value: the shunt (pin-to-tower) capacitance $C_1$ is 10 % of the self-capacitance $C$ of a disc, so $m = C_1/C = 0.1$.

#### Data
- Phase voltage across the string: $V = 33/\sqrt3 = 19.053$ kV
- Discs numbered 1 (tower end) to 4 (line end)

```text
 Tower─[C]─A─[C]─B─[C]─D─[C]─Line
           |     |     |
          mC    mC    mC  (to tower)
```

#### Junction equations
Current through the lower disc = current through the upper disc + current to earth:

$$
\begin{aligned}
V_2 &= V_1(1+m) = 1.1V_1\\
V_3 &= V_2 + m(V_1+V_2) = 1.31V_1\\
V_4 &= V_3 + m(V_1+V_2+V_3) = 1.651V_1
\end{aligned}
$$

$$
V = (1+1.1+1.31+1.651)V_1 = 5.061V_1 \;\Rightarrow\; V_1 = \frac{19.053}{5.061} = 3.765\ \text{kV}
$$

#### Voltage distribution
| Disc | Voltage (kV) | % of $V$ |
|---|---|---|
| 1 (tower end) | 3.765 | 19.76 % |
| 2 | 4.141 | 21.73 % |
| 3 | 4.932 | 25.88 % |
| 4 (line end) | 6.215 | 32.62 % |

String efficiency:

$$
\eta = \frac{V}{4V_4} = \frac{19.053}{4\times6.215} = 76.64\ \%
$$

**Answer (with $m = 0.1$):** The disc voltages are 3.77, 4.14, 4.93 and 6.22 kV (tower to line end), and the string efficiency is 76.64 %. The disc nearest the line is the most stressed. For any other $m$, use the same relations $V_2 = (1+m)V_1$, $V_3 = (1+3m+m^2)V_1$, $V_4 = (1+6m+5m^2+m^3)V_1$.

## Check yourself

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

1. When no capacitance ratio is given, which value is assumed here?
   A. m = 1
   B. m = 0.1 (shunt capacitance 10 % of disc)
   C. m = 0.5
   D. m = 10
   Answer: B. The usual assumption is C1 is 10 % of C, so m = 0.1. (Key point: Assumed ratio)

2. For m = 0.1, V4 equals
   A. 1.31 V1
   B. 1.1 V1
   C. 1.651 V1
   D. 5.061 V1
   Answer: C. V4 = V3 + 0.1(V1+V2+V3) = 1.651 V1. (Key point: V4 in terms of V1)

3. How is V1 found?
   A. V1 = V / 5.061 with V = 33/sqrt(3) kV
   B. V1 = 33/4 kV
   C. V1 = V x 5.061
   D. V1 = V / 4
   Answer: A. The sum of disc voltages is 5.061 V1 = 19.053 kV, so V1 = 3.765 kV. (Key point: Finding V1)

4. Which disc has the highest voltage, and what is the efficiency?
   A. Disc 1, 76.64 %
   B. Disc 4 (line end), 76.64 %
   C. Disc 4, 86.77 %
   D. Disc 2, 66.92 %
   Answer: B. The disc nearest the line carries 6.215 kV, and eta = 19.053/(4 x 6.215) = 76.64 %. (Key point: Most stressed disc)

## Related in Power System Analysis I

- [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
- [5-disc string at 66 kV: nearest and farthest disc](https://www.bctengineer.com/question.md?key=semester_4%2Fpower_system_analysis_i%3Afive-discs-66kv-distribution) — 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
- [String of 5 insulators on 100 kV, ratio 0.1](https://www.bctengineer.com/question.md?key=semester_4%2Fpower_system_analysis_i%3Astring-5-insulators-100kv) — Power System Analysis I, chapter 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×
- [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

## 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
- [Voltage rise at cable–overhead line junction](https://www.bctengineer.com/question.md?key=semester_8%2Fhigh_voltage_engineering%3Asurge-cable-overhead-junction) — High Voltage Engineering, chapter 3, asked 2×
- [Peterson coil for 33 kV 60 km line](https://www.bctengineer.com/question.md?key=semester_6%2Fswitchgear_and_protection%3Apeterson-coil-33kv-60km) — Switchgear and Protection, chapter 5, asked 6×
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