---
title: "Self GMD of 5-strand conductor, 6 mm strands"
subject: "Power System Analysis I"
chapter: 4
key: "semester_4/power_system_analysis_i:self-gmd-five-strands-6mm"
source: https://www.bctengineer.com/q/semester_4/power_system_analysis_i/self-gmd-five-strands-6mm
site: BCT Engineering
---

# Self GMD of 5-strand conductor, 6 mm strands

Power System Analysis I · Chapter 4: Line Parameter Calculations

- Asked in: 2072 Magh, 4 marks
- Video: [8.13 GATE 2022: How to Solve Geometric Mean Radius of Conductor Based Problems | GMR | Power Systems](https://www.youtube.com/watch?v=94uw1lX91gU) — Mritunjay Lectures
- 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/self-gmd-five-strands-6mm
- Whole chapter: https://www.bctengineer.com/md/semester_4/power_system_analysis_i/chapter-4

## Question

Determine the self GMD of the following conductor. Radius of each strand is 6 mm. [Figure: five identical touching strands, two in the top row resting on three in the bottom row.]

## Short answer

For the five-strand configuration with strand radius 6 mm, the self-GMD for inductance is 11.47 mm (approximately 1.91r).

## Answer

The self-GMD (GMR) of an $n$-strand conductor is the $n^2$-th root of the product of all inter-strand distances, with the self-distance taken as $r' = 0.7788r$ for inductance (the usual meaning of self-GMD).

```text
        (4)   (5)
     (1)   (2)   (3)
  touching strands, centre spacing 2r
```

**Data:** $r = 6$ mm, so $2r = 12$ mm, $r' = 0.7788 \times 6 = 4.673$ mm.

#### Distances

Strands 4 and 5 sit in the grooves between 1-2 and 2-3, so the centres lie on a triangular grid of side $2r$.

| Pair | Distance |
|---|---|
| 1-2, 2-3, 4-5, 1-4, 2-4, 2-5, 3-5 | $2r = 12$ mm |
| 1-3 | $4r = 24$ mm |
| 1-5, 3-4 | $2\sqrt3 r = 20.78$ mm |

Check: 7 + 1 + 2 = 10 pairs $= \binom{5}{2}$.

#### Self-GMD

$$
\begin{aligned}
D_s &= \left[(r')^5 \times \left((2r)^7 (4r)(2\sqrt3 r)^2\right)^2\right]^{1/25} \\
&= 1.9114\,r = 1.9114 \times 6 = 11.47\ \text{mm}
\end{aligned}
$$

For capacitance (self-distance $r$ instead of $r'$): $D_s = 2.0094r = 12.06$ mm.

**Answer:** self-GMD (inductance) $= 11.47$ mm $\approx 1.147$ cm.

## Check yourself

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

1. For 5 strands, the self-GMD is:
   A. The 25th root of the product of all distances (including 5 self terms)
   B. The 5th root of the product
   C. The 10th root of the product
   D. The 20th root
   Answer: A. n^2 = 25 terms, so the 25th root. (Key point: Self-GMD root)

2. Strands 1, 2, 3 touch in a row; what is D_13 and D_15?
   A. D_13 = 2r, D_15 = 2r
   B. D_13 = 4r, D_15 = 2*sqrt(3)*r
   C. D_13 = 4r, D_15 = 4r
   D. D_13 = 2*sqrt(3)*r, D_15 = 4r
   Answer: B. 1 and 3 are separated by two touching diameters (4r). Strand 5 sits above the 2-3 groove, giving 2*sqrt(3) r to strand 1. (Key point: Strand geometry)

3. How many distinct pairs are there and how many are 2r?
   A. 10 pairs; 7 are 2r
   B. 5 pairs; 5 are 2r
   C. 10 pairs; 10 are 2r
   D. 15 pairs; 7 are 2r
   Answer: A. C(5,2) = 10 pairs: 7 at 2r, 1 at 4r, 2 at 2*sqrt(3) r. (Key point: Counting pairs)

4. With r = 6 mm, the self-GMD for inductance is:
   A. 11.47 mm (1.9114 r)
   B. 12.06 mm
   C. 6 mm
   D. 4.673 mm
   Answer: A. D_s = 1.9114 r = 11.47 mm; the capacitance value (2.0094 r = 12.06 mm) is larger. (Key point: Result)

## Related in Power System Analysis I

- [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
- [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
- [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
- [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 diamond conductor](https://www.bctengineer.com/question.md?key=semester_4%2Fpower_system_analysis_i%3Aself-gmd-four-strands-diamond) — 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×