---
title: "GMR for inductance vs capacitance"
subject: "Power System Analysis I"
chapter: 4
key: "semester_4/power_system_analysis_i:gmr-inductance-vs-capacitance"
source: https://www.bctengineer.com/q/semester_4/power_system_analysis_i/gmr-inductance-vs-capacitance
site: BCT Engineering
---

# GMR for inductance vs capacitance

Power System Analysis I · Chapter 4: Line Parameter Calculations

- Asked in: 2076 Baisakh, 6 marks
- Video: [GMD AND GMR CONCEPT](https://www.youtube.com/watch?v=z8GE1CoLgHY) — SUCCESS TECHNICAL CAMPUS PATNA
- 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/gmr-inductance-vs-capacitance
- Whole chapter: https://www.bctengineer.com/md/semester_4/power_system_analysis_i/chapter-4

## Question

How does the GMR for inductance calculation differ from the GMR for capacitance calculation? Explain.

## Short answer

For inductance, GMR uses r' = 0.7788r to account for internal magnetic flux; for capacitance, GMR uses the actual outer radius r because the electric field inside is zero. Thus inductance GMR is smaller than capacitance GMR.

## Answer

The **GMR (self-GMD)** used in line parameter calculations is different for inductance and capacitance because inductance depends on the magnetic flux, which exists **inside** the conductor too, while capacitance depends on the electric field, which exists only **outside** the conductor.

#### GMR for inductance

- Current flows through the whole cross-section, so there is magnetic flux inside the conductor as well as outside.
- The internal flux linkage ($\mu_0 I/8\pi$ per metre) is included by replacing the actual radius $r$ with a smaller fictitious radius:
$$
r' = r\,e^{-1/4} = 0.7788\,r
$$
- For a stranded or bundled conductor, the self-GMD uses $r'$ for each strand:
$D_{sL} = (r' \cdot D_{12} \cdots D_{1n} \cdots)^{1/n^2}$; for a 2-bundle, $D_{sL} = \sqrt{r'd}$.

#### GMR for capacitance

- Charge resides on the **surface** of a conductor; the electric field inside a conductor is zero.
- So there is no "internal" term and the **actual outer radius** $r$ is used:
$D_{sC} = (r \cdot D_{12} \cdots)^{1/n^2}$; for a 2-bundle, $D_{sC} = \sqrt{rd}$.

#### Comparison

| Point | Inductance GMR | Capacitance GMR |
|---|---|---|
| Field involved | Magnetic | Electric |
| Field inside conductor | Present | Zero |
| Radius used | $r' = 0.7788r$ | $r$ |
| Solid conductor | $0.7788r$ | $r$ |
| 2-bundle | $\sqrt{0.7788r\,d}$ | $\sqrt{r\,d}$ |
| Value | Smaller | Larger |

**Example:** for $r = 1$ cm, GMR for inductance $= 0.7788$ cm, while for capacitance it is 1 cm. The mutual GMD between phases is the same for both calculations.

## Check yourself

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

1. Why does the GMR for inductance differ from that for capacitance?
   A. Magnetic flux exists inside the conductor but the electric field inside is zero
   B. Capacitance depends on current in the whole cross-section
   C. Inductance depends on surface charge
   D. The mutual GMD is different for the two
   Answer: A. Internal flux adds to inductance, while charge sits on the surface and there is no internal electric field. (Key point: Why they differ)

2. For a solid conductor of radius r, what radius is used for inductance and for capacitance?
   A. r for inductance, 0.7788r for capacitance
   B. 0.7788r for both
   C. 0.7788r for inductance, r for capacitance
   D. r for both
   Answer: C. Inductance uses r' = r e^(-1/4) = 0.7788r; capacitance uses the actual radius r. (Key point: Radius used)

3. For a 2-bundle with spacing d, the self-GMDs for inductance and capacitance are:
   A. sqrt(r d) and sqrt(0.7788 r d)
   B. sqrt(0.7788 r d) and sqrt(r d)
   C. 0.7788 sqrt(r d) for both
   D. sqrt(r d) for both
   Answer: B. Inductance uses r' in place of r; capacitance uses r. (Key point: Two-bundle self-GMD)

4. For r = 1 cm, which statement is correct?
   A. GMR for inductance 1 cm, for capacitance 0.7788 cm
   B. Both are 1 cm
   C. GMR for inductance 0.7788 cm, for capacitance 1 cm
   D. Both are 0.7788 cm
   Answer: C. The inductance GMR is the smaller one, 0.7788 cm, while capacitance uses 1 cm. (Key point: Relative value)

## Related in Power System Analysis I

- [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
- [GMR of 6-strand triangular conductor](https://www.bctengineer.com/question.md?key=semester_4%2Fpower_system_analysis_i%3Agmr-six-strand-triangle) — Power System Analysis I, chapter 4, asked 2×
- [GMR of 5-strand conductor, 0.5 cm strands](https://www.bctengineer.com/question.md?key=semester_4%2Fpower_system_analysis_i%3Agmr-five-strands-0-5cm) — Power System Analysis I, chapter 4, asked 2×
- [GMR of 4-strand conductor in terms of r](https://www.bctengineer.com/question.md?key=semester_4%2Fpower_system_analysis_i%3Agmr-four-strands-r) — Power System Analysis I, chapter 4
- [GMR of stranded vs solid conductor](https://www.bctengineer.com/question.md?key=semester_4%2Fpower_system_analysis_i%3Agmr-stranded-lower-than-solid) — Power System Analysis I, chapter 4

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