---
title: "Transposition; single-phase capacitance with earth"
subject: "Power System Analysis I"
chapter: 4
key: "semester_4/power_system_analysis_i:transposition-single-phase-earth-capacitance"
source: https://www.bctengineer.com/q/semester_4/power_system_analysis_i/transposition-single-phase-earth-capacitance
site: BCT Engineering
---

# Transposition; single-phase capacitance with earth

Power System Analysis I · Chapter 4: Line Parameter Calculations

- Asked in: 2073 Bhadra, 6 marks
- Video: [Effect of Earth on the Capacitance of Overhead Transmission Lines (PGTP-KTU) -Praveen Raj RS.](https://www.youtube.com/watch?v=RB-oKxH5qcc) — praveen raj
- 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/transposition-single-phase-earth-capacitance
- Whole chapter: https://www.bctengineer.com/md/semester_4/power_system_analysis_i/chapter-4

## Question

What is the importance of transposition of high voltage transmission lines? Derive the line capacitance and phase capacitance of a single phase line considering the effect of ground.

## Short answer

Transposition ensures equal average inductance and capacitance per phase, preventing unbalanced voltages and reducing interference. For a single-phase line above ground, by method of images the line-to-line capacitance is C_ab = π ε0 / ln(D/(r√(1+D²/(4h²)))) F/m, and phase capacitance to neutral C_n = 2C_ab.

## Answer

#### Importance of transposition

**Transposition** means exchanging the positions of the phase conductors at regular intervals along the line, so that each phase occupies each position for one-third of the length.

- With unsymmetrical spacing, the flux linkages and charges of the three phases differ, so the phases have **unequal inductance and capacitance**.
- Unequal parameters give unequal voltage drops, so the receiving-end voltages become **unbalanced** even with balanced load.
- Transposition makes the average inductance and capacitance of each phase equal, so the line behaves as a balanced circuit and can be modelled per phase using $D_{eq} = \sqrt[3]{D_{12}D_{23}D_{31}}$.
- It reduces **electromagnetic and electrostatic interference** with nearby telephone and communication lines.

#### Capacitance of a single-phase line considering ground

Let conductors a and b (radius $r$, spacing $D$) be at height $h$ above ground, carrying charges $+q$ and $-q$ per metre. By the **method of images**, the earth is replaced by image conductors a' ($-q$) and b' ($+q$) at depth $h$.

```text
   +q a o------- D -------o b -q
        |                 |
        h                 h
 ===============================  ground
        h                 h
        |                 |
   -q a'o                 o b' +q
   a-a' = 2h,  a-b' = sqrt(4h^2 + D^2)
```

Voltage between a and b due to all four charges, using $V_{ab} = \dfrac{1}{2\pi\varepsilon_0}\sum q_k\ln\dfrac{D_{kb}}{D_{ka}}$:

$$
\begin{aligned}
V_{ab} &= \frac{1}{2\pi\varepsilon_0}\left[q\ln\frac{D}{r} - q\ln\frac{r}{D} - q\ln\frac{\sqrt{4h^2+D^2}}{2h} + q\ln\frac{2h}{\sqrt{4h^2+D^2}}\right] \\
&= \frac{q}{\pi\varepsilon_0}\left[\ln\frac{D}{r} - \ln\frac{\sqrt{4h^2+D^2}}{2h}\right] \\
&= \frac{q}{\pi\varepsilon_0}\ln\frac{D}{r\sqrt{1+D^2/4h^2}}
\end{aligned}
$$

**Line capacitance** (between conductors):

$$
C_{ab} = \frac{q}{V_{ab}} = \frac{\pi\varepsilon_0}{\ln\left(\dfrac{D}{r\sqrt{1+D^2/4h^2}}\right)}\ \text{F/m}
$$

**Phase capacitance** (each conductor to neutral, $V_{an} = V_{ab}/2$):

$$
C_n = 2C_{ab} = \frac{2\pi\varepsilon_0}{\ln\left(\dfrac{D}{r\sqrt{1+D^2/4h^2}}\right)}\ \text{F/m}
$$

Since $\sqrt{1+D^2/4h^2} > 1$, the ground **increases** the capacitance. When $h \gg D$, the term tends to 1 and the result reduces to the usual $C_{ab} = \pi\varepsilon_0/\ln(D/r)$.

## Check yourself

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

1. What does transposition of a line mean?
   A. Interchanging the positions of phase conductors at regular intervals so each occupies each position for one-third of the length
   B. Reversing the direction of current
   C. Adding a ground wire
   D. Bundling conductors
   Answer: A. Each phase takes every position for equal length, equalising average L and C. (Key point: Meaning of transposition)

2. What problem does transposition solve?
   A. Corona loss
   B. Unequal inductance and capacitance of phases, giving unbalanced receiving-end voltages
   C. Skin effect
   D. Insulator failure
   Answer: B. Unsymmetrical spacing gives unequal parameters, causing unbalance even with balanced load; transposition balances the line and reduces interference with telephone lines. (Key point: Why needed)

3. For a single-phase line with charges +q on a and -q on b, what are the image charges?
   A. a' has +q, b' has -q
   B. a' has -q, b' has +q
   C. Both images have +q
   D. Both images have -q
   Answer: B. Each image has the opposite charge to its real conductor. (Key point: Image charges)

4. How is the phase (to neutral) capacitance related to the line capacitance C_ab?
   A. C_n = C_ab/2
   B. C_n = C_ab
   C. C_n = 2 C_ab since V_an = V_ab/2
   D. C_n = 4 C_ab
   Answer: C. Each conductor to neutral sees half the voltage, so the capacitance doubles. (Key point: Phase capacitance)

## Related in Power System Analysis I

- [Single-phase line capacitance, 6 mm, 1.3 m](https://www.bctengineer.com/question.md?key=semester_4%2Fpower_system_analysis_i%3Asingle-phase-capacitance-6mm-1-3m) — Power System Analysis I, chapter 4
- [Capacitance with and without earth effect](https://www.bctengineer.com/question.md?key=semester_4%2Fpower_system_analysis_i%3Acapacitance-with-earth-effect-13m) — Power System Analysis I, chapter 4
- [Effect of earth on line capacitance](https://www.bctengineer.com/question.md?key=semester_4%2Fpower_system_analysis_i%3Aearth-effect-capacitance) — Power System Analysis I, chapter 4, asked 2×
- [Effect of neglecting earth on capacitance](https://www.bctengineer.com/question.md?key=semester_4%2Fpower_system_analysis_i%3Aearth-effect-horizontal-13m) — Power System Analysis I, chapter 4
- [Skin effect in transmission lines](https://www.bctengineer.com/question.md?key=semester_4%2Fpower_system_analysis_i%3Askin-effect-brief) — Power System Analysis I, chapter 4
- [Inductance of a single-phase two-wire line](https://www.bctengineer.com/question.md?key=semester_4%2Fpower_system_analysis_i%3Asingle-phase-inductance-derivation) — Power System Analysis I, chapter 4

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