---
title: "Capacitive reactance and charging of 2-conductor bundle line"
subject: "Power System Analysis I"
chapter: 4
key: "semester_4/power_system_analysis_i:bundle-capacitance-9mm-40cm-10m"
source: https://www.bctengineer.com/q/semester_4/power_system_analysis_i/bundle-capacitance-9mm-40cm-10m
site: BCT Engineering
---

# Capacitive reactance and charging of 2-conductor bundle line

Power System Analysis I · Chapter 4: Line Parameter Calculations

- Asked in: 2079 Chaitra, 8 marks
- Video: [Power Systems | Lecture-15 | Capacitance of Bundled Conductors](https://www.youtube.com/watch?v=qdTZd_2LnM8) — Engineering Devotion
- 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/bundle-capacitance-9mm-40cm-10m
- Whole chapter: https://www.bctengineer.com/md/semester_4/power_system_analysis_i/chapter-4

## Question

Calculate the capacitive reactance, charging current and charging VAR per phase per km of a three-phase transmission line with bundle conductor as shown in figure below. Given that radius of each conductor is 9 mm and the line is transposed. [Figure: horizontal single-circuit line, phases A, B, C each a two-conductor bundle (a–a', b–b', c–c') with 40 cm between sub-conductors; 10 m between centres of adjacent phases.]

## Short answer

Bundle self-GMD = sqrt(0.009*0.40)=0.06 m; mutual GMD = (10*10*20)^(1/3)=12.599 m. Capacitance C = 2*pi*epsilon0 / ln(12.599/0.06) = 0.0104 µF/km. At 50 Hz, Xc = 1/(ωC) = 3.06e5 Ω·km. Assuming 400 kV line, charging current = 0.755 A/km, reactive power per phase = 174 kVAR/km (523 kVAR/km three-phase).

## Answer

The voltage and frequency are not printed. They are assumed as **400 kV (line), 50 Hz**, the usual data for this two-conductor-bundle configuration. Reactance per km does not depend on the voltage; only the charging current and VAR scale with it.

#### Data
- Sub-conductor radius $r = 9$ mm $= 0.009$ m. Bundle spacing $d = 0.40$ m.
- Phase spacing: 10 m, 10 m, 20 m (horizontal), transposed.
- $V_{ph} = 400/\sqrt3 = 230.94$ kV; $\omega = 314.16$ rad/s.

#### Self-GMD for capacitance
For capacitance the actual radius r is used, not $r'$:
$$
D_{sc} = \sqrt{r\,d} = \sqrt{0.009\times0.40} = 0.06\ \text{m}
$$

#### Mutual GMD
$$
D_{eq} = \sqrt[3]{10\times10\times20} = 12.599\ \text{m}
$$

#### Capacitance and capacitive reactance per phase per km
$$
\begin{aligned}
C &= \frac{2\pi\varepsilon_0}{\ln(D_{eq}/D_{sc})} = \frac{5.563\times10^{-11}}{\ln(12.599/0.06)} = \frac{5.563\times10^{-11}}{5.3470}\\
&= 1.0404\times10^{-11}\ \text{F/m} = 0.010404\ \mu\text{F/km}\\
X_C &= \frac{1}{\omega C} = \frac{1}{314.16\times1.0404\times10^{-8}} = 3.059\times10^5\ \Omega\cdot\text{km}
\end{aligned}
$$

#### Charging current and charging VAR (per km)
$$
\begin{aligned}
I_c &= \frac{V_{ph}}{X_C} = \frac{230940}{3.059\times10^5} = 0.7548\ \text{A/km}\\
Q_{c,ph} &= V_{ph}I_c = 230940\times0.7548 = 174.32\ \text{kVAR/km per phase}\\
Q_{c,3\phi} &= 3\times174.32 = 522.97\ \text{kVAR/km}
\end{aligned}
$$

| Quantity (per phase per km) | Value |
|---|---|
| $D_{sc}$ | 0.06 m |
| $D_{eq}$ | 12.599 m |
| C | 0.010404 μF |
| $X_C$ | 0.3059 MΩ·km |
| $I_c$ (400 kV assumed) | 0.755 A |
| Charging VAR | 174.3 kVAR (523.0 kVAR for 3 phases) |

**Answer:** $X_C \approx 3.06\times10^5\ \Omega$·km per phase. At 400 kV, $I_c \approx 0.755$ A/km and the charging VAR is about 174.3 kVAR per phase per km. (At another line voltage V, scale $I_c$ by $V/400$ and the VAR by $(V/400)^2$.)

## Check yourself

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

1. What is the bundle self-GMD for capacitance with r = 9 mm and d = 0.40 m?
   A. sqrt(0.009 x 0.40) = 0.06 m
   B. sqrt(0.7788 x 0.009 x 0.40) = 0.053 m
   C. 0.40 m
   D. 0.009 m
   Answer: A. Capacitance uses the actual radius r, so Dsc = sqrt(r d) = 0.06 m. (Key point: Radius for capacitance)

2. Phase spacings are 10 m, 10 m, 20 m. What is Deq?
   A. 12.6 m
   B. 13.33 m
   C. 10 m
   D. 20 m
   Answer: A. Deq = cube root of (10 x 10 x 20) = 12.599 m. (Key point: Mutual GMD)

3. What is X_C per phase per km, approximately?
   A. 3.06e5 ohm-km
   B. 3.06e3 ohm-km
   C. 1.04e-11 ohm-km
   D. 5.35 ohm-km
   Answer: A. C = 1.0404e-11 F/m, so X_C = 1/(wC) = 3.059e5 ohm-km. (Key point: Capacitive reactance)

4. How do charging current and VAR vary with line voltage V?
   A. Ic scales with V and VAR with V^2
   B. Ic is independent of V
   C. Ic scales with V^2 and VAR with V
   D. Both are independent of V
   Answer: A. Ic = V_ph/X_C is proportional to V and Q = V_ph Ic to V^2; X_C itself does not depend on voltage. (Key point: Dependence on voltage)

## Close variants set in other exams

- [400 kV line with 3-conductor bundles: reactances and VAR](https://www.bctengineer.com/question.md?key=semester_4%2Fpower_system_analysis_i%3Atriple-bundle-400kv-reactances) — Power System Analysis I, chapter 4, asked 3×

## Related in Power System Analysis I

- [400 kV twin-bundle line: reactances and charging VAR](https://www.bctengineer.com/question.md?key=semester_4%2Fpower_system_analysis_i%3Atwin-bundle-400kv-reactances) — Power System Analysis I, chapter 4, asked 2×
- [Capacitance of double circuit line, 6 mm radius](https://www.bctengineer.com/question.md?key=semester_4%2Fpower_system_analysis_i%3Adouble-circuit-capacitance-6mm) — Power System Analysis I, chapter 4
- [Inductance of twin-bundle line, 16 mm radius](https://www.bctengineer.com/question.md?key=semester_4%2Fpower_system_analysis_i%3Atwin-bundle-150km-16mm) — Power System Analysis I, chapter 4, asked 3×
- [400 kV quadruple bundle line, 20 m spacing](https://www.bctengineer.com/question.md?key=semester_4%2Fpower_system_analysis_i%3Aquad-bundle-400kv-20m) — 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
- [Inductance of 4-conductor bundle line, 6 m spacing](https://www.bctengineer.com/question.md?key=semester_4%2Fpower_system_analysis_i%3Aquad-bundle-200km-12mm-6m) — Power System Analysis I, chapter 4, asked 2×

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