Cable insulation resistance and capacitance

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).