---
title: "Internal inductance independent of geometry"
subject: "Power System Analysis I"
chapter: 4
key: "semester_4/power_system_analysis_i:internal-inductance-independent-geometry"
source: https://www.bctengineer.com/q/semester_4/power_system_analysis_i/internal-inductance-independent-geometry
site: BCT Engineering
---

# Internal inductance independent of geometry

Power System Analysis I · Chapter 4: Line Parameter Calculations

- Asked in: 2078 Chaitra, 6 marks; 2070 Magh, 5 marks (asked 2 times)
- Video: [Internal Flux Linkage with in conductor, full derivation | electrical power system](https://www.youtube.com/watch?v=yGergiJFLFk) — ELECTRICAL HUB
- 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/internal-inductance-independent-geometry
- Whole chapter: https://www.bctengineer.com/md/semester_4/power_system_analysis_i/chapter-4

## Question

Prove that the inductance of a transmission line due to internal flux linkages is independent of the conductor geometry.

## Short answer

The internal inductance L_int = μ0/(8π) = 0.5×10⁻⁷ H/m contains no radius or spacing terms because r⁴ cancels in the integration; it depends only on conductor permeability, not on conductor size, arrangement, or geometry.

## Answer

**Statement:** the internal inductance of a round conductor is $L_{int} = \frac{\mu_0}{8\pi} = 0.5\times10^{-7}$ H/m. It does not contain the conductor radius or the spacing, so it is independent of the conductor geometry (size and arrangement).

#### Derivation
Consider a long solid round conductor of radius r carrying current I with uniform current density ($\mu_r = 1$).

1. **Current enclosed** within radius x (x < r):
$$
I_x = \frac{x^2}{r^2}I
$$

2. **Field intensity** by Ampere's law:
$$
2\pi x H_x = I_x \;\Rightarrow\; H_x = \frac{xI}{2\pi r^2}, \qquad B_x = \frac{\mu_0 xI}{2\pi r^2}
$$

3. **Flux** in a tubular element of thickness dx, per metre length:
$$
d\phi = B_x\,dx = \frac{\mu_0 xI}{2\pi r^2}dx
$$

4. **Partial flux linkage.** This flux links only $x^2/r^2$ of the current:
$$
d\lambda = \frac{x^2}{r^2}\cdot\frac{\mu_0 xI}{2\pi r^2}dx = \frac{\mu_0 I x^3}{2\pi r^4}dx
$$

5. **Integrate** from 0 to r:
$$
\lambda_{int} = \frac{\mu_0 I}{2\pi r^4}\cdot\frac{r^4}{4} = \frac{\mu_0 I}{8\pi}
$$

6. **Inductance:**
$$
L_{int} = \frac{\lambda_{int}}{I} = \frac{\mu_0}{8\pi} = \frac{4\pi\times10^{-7}}{8\pi} = 0.5\times10^{-7}\ \text{H/m}
$$

#### Why it is independent of geometry
- The $r^4$ in the denominator (from the field and partial-linkage terms) cancels exactly with the $r^4/4$ from the integral. So the **radius drops out**: a thin conductor and a thick conductor have the same internal inductance per metre.
- Internal flux exists only inside the conductor, so it does not depend on the **spacing** to other conductors or on the arrangement (horizontal, vertical, triangular).
- Only the permeability of the conductor material matters. For copper and aluminium $\mu = \mu_0$.

The spacing and radius affect only the **external** inductance, $2\times10^{-7}\ln\frac{D}{r}$. Combining both parts gives the familiar form with GMR:
$$
L = 2\times10^{-7}\left(\frac14 + \ln\frac{D}{r}\right) = 2\times10^{-7}\ln\frac{D}{r'}, \quad r' = re^{-1/4} = 0.7788r
$$

## Check yourself

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

1. What is the internal inductance of a solid round conductor (mu_r = 1)?
   A. 2e-7 ln(D/r) H/m
   B. mu0/(8 pi) = 0.5e-7 H/m
   C. mu0/(2 pi) = 2e-7 H/m
   D. mu0 r/(8 pi) H/m
   Answer: B. Integration gives lambda_int = mu0 I/(8 pi), so L_int = mu0/(8 pi). (Key point: Result)

2. Why does the radius not appear in L_int?
   A. The r^4 in the denominator cancels with r^4/4 from the integral
   B. Radius is assumed 1 m
   C. Thin conductors have no internal flux
   D. Because skin effect removes it
   Answer: A. The 1/r^4 factor from field and partial linkage cancels the r^4 from integrating x^3 up to r. (Key point: Radius drops out)

3. Why is internal inductance independent of spacing to other conductors?
   A. Internal flux exists only inside the conductor itself
   B. Spacing is always very large
   C. Other conductors carry no current
   D. Spacing only affects resistance
   Answer: A. Only flux within the conductor counts; spacing affects only the external inductance. (Key point: Spacing independence)

4. How does the internal part combine with external inductance to give the GMR form?
   A. L = 2e-7 (1/4 + ln D/r) = 2e-7 ln(D/r'), r' = 0.7788 r
   B. L = 2e-7 (ln D/r - 1/4), r' = 1.284 r
   C. L = 2e-7 ln(D/r) with no change
   D. L = 2e-7 (1/2 + ln D/r), r' = 0.6065 r
   Answer: A. 0.5e-7 equals 2e-7 x 1/4, and r' = r e^(-1/4) = 0.7788 r. (Key point: Combination with GMR)

## Related in Power System Analysis I

- [Internal inductance independent of geometry and current](https://www.bctengineer.com/question.md?key=semester_4%2Fpower_system_analysis_i%3Ainternal-inductance-geometry-current) — Power System Analysis I, chapter 4
- [Internal inductance independent of current](https://www.bctengineer.com/question.md?key=semester_4%2Fpower_system_analysis_i%3Ainternal-inductance-independent-current) — Power System Analysis I, chapter 4, asked 3×
- [Inductance of composite-conductor single-phase line](https://www.bctengineer.com/question.md?key=semester_4%2Fpower_system_analysis_i%3Asingle-phase-composite-conductors) — 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×
- [Inductance of 3-conductor bundled line, 6.1 m spacing](https://www.bctengineer.com/question.md?key=semester_4%2Fpower_system_analysis_i%3Ainductance-three-subconductor-bundle-6-1m) — 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

## Related in other subjects

- [Inductance from flux linkages](https://www.bctengineer.com/question.md?key=semester_1%2Fbasic_electrical_engineering%3Ainductance-flux-linkages-derivation) — Basic Electrical Engineering, chapter 4