---
title: "Cable insulation resistance and capacitance"
subject: "Power System Analysis I"
chapter: 2
key: "semester_4/power_system_analysis_i:cable-insulation-resistance-capacitance"
source: https://www.bctengineer.com/q/semester_4/power_system_analysis_i/cable-insulation-resistance-capacitance
site: BCT Engineering
---

# Cable insulation resistance and capacitance

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

- Asked in: 2074 Bhadra, 3 marks
- Video: [Insulation Resistance and Capacitance - Insulated Cables - GATE Power System](https://www.youtube.com/watch?v=nDiJ_tzaSto) — Ekeeda GATE & ESE
- 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/cable-insulation-resistance-capacitance
- Whole chapter: https://www.bctengineer.com/md/semester_4/power_system_analysis_i/chapter-2

## Question

A single core 2 km long cable has a conductor radius of 13 mm and an insulation thickness of 4.6 mm. If the resistivity of dielectric is 7×10¹² ohm-m, find the insulation resistance and the capacitance per metre length of cable if the dielectric has relative permittivity of 3.2.

## Short answer

Insulation resistance of the 2 km cable is 168.8 MΩ. The capacitance is 0.588 nF per metre.

## Answer

For a single-core cable, the insulation resistance and the capacitance both depend on $\ln(R/r)$, where $R$ is the outer radius of the insulation.

#### Data
- $r = 13$ mm
- $R = 13 + 4.6 = 17.6$ mm
- $l = 2$ km $= 2000$ m
- $\rho = 7\times10^{12}\ \Omega$m
- $\varepsilon_r = 3.2$

$$
\ln\frac{R}{r} = \ln\frac{17.6}{13} = 0.30295
$$

#### Insulation resistance
$$
\begin{aligned}
R_{ins} &= \frac{\rho}{2\pi l}\ln\frac{R}{r}\\
&= \frac{7\times10^{12}}{2\pi\times 2000}\times 0.30295\\
&= 1.688\times10^{8}\ \Omega \approx 168.8\ \text{M}\Omega
\end{aligned}
$$

#### Capacitance per metre
$$
\begin{aligned}
C &= \frac{2\pi\varepsilon_0\varepsilon_r}{\ln(R/r)}
= \frac{2\pi\times 8.854\times10^{-12}\times 3.2}{0.30295}\\
&= 5.876\times10^{-10}\ \text{F/m} = 0.588\ \text{nF/m}
\end{aligned}
$$

**Answer:** The insulation resistance of the 2 km cable is about 168.8 MΩ, and the capacitance is 0.588 nF per metre (about 1.175 µF for the whole 2 km).

## Check yourself

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

1. Which formula gives insulation resistance of a cable of length l?
   A. R = rho l ln(R/r)/(2 pi)
   B. R = rho/(2 pi l) x ln(R/r)
   C. R = 2 pi l / (rho ln(R/r))
   D. R = rho/(2 pi l) x (R - r)
   Answer: B. Leakage flows radially through length l, giving R = rho ln(R/r)/(2 pi l). (Key point: Insulation resistance formula)

2. What is the outer radius of the insulation R here?
   A. 17.6 mm (13 + 4.6)
   B. 4.6 mm
   C. 13 mm
   D. 8.6 mm
   Answer: A. R = conductor radius plus insulation thickness. (Key point: Outer radius)

3. What is the capacitance per metre?
   A. About 1.175 uF/m
   B. About 168.8 nF/m
   C. About 0.588 nF/m
   D. About 5.88 nF/m
   Answer: C. C = 2 pi eps0 eps_r/ln(R/r) = 5.876e-10 F/m. (Key point: Capacitance per metre)

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