---
title: "GMR of 4-strand conductor in terms of r"
subject: "Power System Analysis I"
chapter: 4
key: "semester_4/power_system_analysis_i:gmr-four-strands-r"
source: https://www.bctengineer.com/q/semester_4/power_system_analysis_i/gmr-four-strands-r
site: BCT Engineering
---

# GMR of 4-strand conductor in terms of r

Power System Analysis I · Chapter 4: Line Parameter Calculations

- Asked in: 2069 Poush, 4 marks
- Video: [GMD and GMR Calculation | Power Systems | GATE (EE) Exam | Ankit Goyal](https://www.youtube.com/watch?v=Qb3qNGnA6O4) — Kreatryx GATE - EE, ECE & IN by Unacademy
- 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-four-strands-r
- Whole chapter: https://www.bctengineer.com/md/semester_4/power_system_analysis_i/chapter-4

## Question

Calculate the GMR to calculate inductance of the stranded conductor shown in figure below in terms of radius 'r' of an individual strand. [Figure: four identical touching strands arranged in a 2 × 2 square.]

## Short answer

GMR is 1.723 times the strand radius r.

## Answer

The GMR of an $n$-strand conductor is the $n^2$-th root of the product of all inter-strand distances, with each strand's self-distance $r' = 0.7788r$.

```text
     (1)   (2)
     (3)   (4)
  side = 2r, diagonal = 2(sqrt2)r
```

#### Distances

| Pair | Distance |
|---|---|
| 1-2, 1-3, 2-4, 3-4 | $2r$ |
| 1-4, 2-3 | $2\sqrt2 r$ |

By symmetry every strand sees the same set: $r'$, $2r$, $2r$, $2\sqrt2 r$.

#### GMR

$$
\begin{aligned}
D_s &= \left[(r' \cdot 2r \cdot 2r \cdot 2\sqrt2 r)^4\right]^{1/16} = \left(r' \times 8\sqrt2\, r^3\right)^{1/4} \\
&= \left(0.7788 \times 11.314\right)^{1/4} r = (8.811)^{1/4} r \\
&= 1.7229\,r
\end{aligned}
$$

**Answer:** GMR $= 1.723r$ (for capacitance, using $r$ instead of $r'$, it is $1.834r$).

## Check yourself

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

1. In the 2 x 2 square, which distances occur?
   A. Sides 2r and diagonals 2*sqrt2 r
   B. All 2r
   C. Sides r and diagonals sqrt2 r
   D. Sides 4r and diagonals 2r
   Answer: A. Strands touch, so centre spacing is 2r, and the diagonal is 2 sqrt2 r. (Key point: Distances)

2. The GMR for 4 strands is:
   A. 4th root of the product of 4 terms
   B. 16th root of the product of 16 terms
   C. 12th root of the mutual distances
   D. 8th root of the product of 8 terms
   Answer: B. n^2 = 16 terms including the four r' self terms. (Key point: Method)

3. For capacitance, the GMR uses:
   A. r' instead of r
   B. 2r instead of r'
   C. r instead of r' (giving 1.834r)
   D. the same value 1.723r
   Answer: C. No internal flux in the electric field case, so the self term is r. (Key point: Capacitance version)

## Related in Power System Analysis I

- [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×
- [Self GMD of 4-strand conductor, 5 mm strands](https://www.bctengineer.com/question.md?key=semester_4%2Fpower_system_analysis_i%3Aself-gmd-four-strands-5mm) — Power System Analysis I, chapter 4
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- [Self GMD of 4-strand conductor in terms of r](https://www.bctengineer.com/question.md?key=semester_4%2Fpower_system_analysis_i%3Aself-gmd-four-strands-r) — Power System Analysis I, chapter 4
- [Self GMD of 5-strand conductor, 6 mm strands](https://www.bctengineer.com/question.md?key=semester_4%2Fpower_system_analysis_i%3Aself-gmd-five-strands-6mm) — Power System Analysis I, chapter 4
- [Self GMD of 7-strand conductor, 4 mm strands](https://www.bctengineer.com/question.md?key=semester_4%2Fpower_system_analysis_i%3Aself-gmd-seven-strands-4mm) — Power System Analysis I, chapter 4