---
title: "Effect of neglecting earth on capacitance"
subject: "Power System Analysis I"
chapter: 4
key: "semester_4/power_system_analysis_i:earth-effect-horizontal-13m"
source: https://www.bctengineer.com/q/semester_4/power_system_analysis_i/earth-effect-horizontal-13m
site: BCT Engineering
---

# Effect of neglecting earth on capacitance

Power System Analysis I · Chapter 4: Line Parameter Calculations

- Asked in: 2068 Bhadra, 6 marks
- Video: [Effect of Earth on Transmission Line Capacitance | Method of Images Explained Simply | Power System](https://www.youtube.com/watch?v=pTWQwbxxtpA) — Circuit Masterclass
- 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/earth-effect-horizontal-13m
- Whole chapter: https://www.bctengineer.com/md/semester_4/power_system_analysis_i/chapter-4

## Question

For a 3-phase overhead line as shown in the figure below, show the effect of neglecting the presence of earth surface in computing the capacitance of the line. Assume solid, round conductors for each phase. [Figure: conductors a, b, c in a horizontal row 13 m above ground; Dab = Dbc = 6 m, Dca = 12 m; ra = rb = rc = 15 mm.]

## Short answer

Without earth capacitance = 8.940 nF/km, with earth = 9.012 nF/km; neglecting earth gives about 0.8% lower capacitance, a small error when height (13 m) is large relative to spacing.

## Answer

The earth is included by the method of images. For a transposed line,

$$
C_n = \frac{2\pi\varepsilon_0}{\ln\dfrac{D_{eq}}{r} - \ln\dfrac{\sqrt[3]{H_{ab}H_{bc}H_{ca}}}{\sqrt[3]{H_aH_bH_c}}}
$$

Without the earth, the second term is dropped.

```text
   a o-----6 m-----o b-----6 m-----o c
     |13 m         |13 m           |13 m
 =========================================  earth
     |13 m         |13 m           |13 m
   a'o             o b'            o c'
```

**Data:** $r = 0.015$ m, $D_{ab} = D_{bc} = 6$ m, $D_{ca} = 12$ m, $h = 13$ m.

$$
D_{eq} = \sqrt[3]{6 \times 6 \times 12} = 7.5595\ \text{m}, \qquad \ln\frac{7.5595}{0.015} = 6.2225
$$

#### Neglecting earth

$$
C_n = \frac{55.63\times10^{-12}}{6.2225} = 8.940\times10^{-12}\ \text{F/m} = 8.940\ \text{nF/km}
$$

#### Considering earth

| Distance | Value (m) |
|---|---|
| $H_a = H_b = H_c = 2h$ | 26 |
| $H_{ab} = H_{bc} = \sqrt{6^2+26^2}$ | 26.683 |
| $H_{ca} = \sqrt{12^2+26^2}$ | 28.636 |

$$
\begin{aligned}
\sqrt[3]{H_{ab}H_{bc}H_{ca}} &= \sqrt[3]{26.683^2 \times 28.636} = 27.319\ \text{m} \\
\text{Correction} &= \ln\frac{27.319}{26} = 0.04948 \\
C_n &= \frac{55.63\times10^{-12}}{6.2225 - 0.0495} = \frac{55.63\times10^{-12}}{6.1730} \\
&= 9.012\times10^{-12}\ \text{F/m} = 9.012\ \text{nF/km}
\end{aligned}
$$

#### Effect of neglecting earth

$$
\frac{9.012 - 8.940}{8.940}\times100 = 0.80\%
$$

Neglecting the earth gives a capacitance about **0.8% lower** than the true value. The earth slightly increases the capacitance because the opposite-sign image charges lower the potential of each conductor, so more charge is needed for the same voltage. The error is small when the height (13 m) is large compared with the spacing, so the earth effect is usually neglected.

**Answer:** without earth $8.940$ nF/km; with earth $9.012$ nF/km; error from neglecting earth ≈ 0.8%.

## Check yourself

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

1. The effect of earth on capacitance is included by:
   A. The method of images (image charges)
   B. Adding a series resistance
   C. Increasing r
   D. Using skin effect
   Answer: A. Earth is replaced by image conductors with opposite charge. (Key point: Method)

2. Including the earth makes capacitance:
   A. Slightly smaller
   B. Slightly larger
   C. Zero
   D. Unchanged exactly
   Answer: B. Image charges lower the potential for a given charge, so C rises (9.012 vs 8.940 nF/km). (Key point: Direction of change)

3. The distance H_a between a conductor and its own image is:
   A. h = 13 m
   B. h/2
   C. 2h = 26 m
   D. sqrt(13^2+6^2)
   Answer: C. Image is as far below ground as the conductor is above. (Key point: Height)

4. Why is earth effect usually neglected?
   A. It always doubles C
   B. Earth is a perfect insulator
   C. It affects only inductance
   D. Error is small (about 0.8%) when height is large compared to spacing
   Answer: D. The correction term ln(27.3/26) is tiny relative to ln(D_eq/r). (Key point: Why neglect)

## Related in Power System Analysis I

- [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
- [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
- [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×
- [Transposition; single-phase capacitance with earth](https://www.bctengineer.com/question.md?key=semester_4%2Fpower_system_analysis_i%3Atransposition-single-phase-earth-capacitance) — Power System Analysis I, chapter 4
- [Inductance of unsymmetrical transposed line](https://www.bctengineer.com/question.md?key=semester_4%2Fpower_system_analysis_i%3Aunsymmetrical-inductance-10mm) — Power System Analysis I, chapter 4
- [Inductance and capacitance of 4-conductor bundle line](https://www.bctengineer.com/question.md?key=semester_4%2Fpower_system_analysis_i%3Aquad-bundle-220kv-l-c) — Power System Analysis I, chapter 4

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