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

# Self GMD of 4-strand conductor in terms of r

Power System Analysis I · Chapter 4: Line Parameter Calculations

- Asked in: 2071 Magh, 5 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/self-gmd-four-strands-r
- Whole chapter: https://www.bctengineer.com/md/semester_4/power_system_analysis_i/chapter-4

## Question

Compute the self-GMD of the conductor shown in figure in terms of strand radius r. All the strands are similar. [Figure: four identical touching strands of radius r: three in a horizontal row and a fourth resting on top of the right-hand pair.]

## Answer

The self-GMD 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 $r' = 0.7788r$ (for inductance).

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

Strand 4 sits in the groove between 2 and 3, so 2, 3, 4 form an equilateral triangle of side $2r$. Take centres: 1 at $(0,0)$, 2 at $(2r,0)$, 3 at $(4r,0)$, 4 at $(3r, \sqrt3 r)$.

#### Distances

| Pair | Distance |
|---|---|
| 1-2, 2-3, 2-4, 3-4 | $2r$ |
| 1-3 | $4r$ |
| 1-4 | $\sqrt{(3r)^2 + (\sqrt3 r)^2} = 2\sqrt3 r$ |

Product of the six mutual distances: $(2r)^4 (4r)(2\sqrt3 r) = 16 \times 4 \times 2\sqrt3\, r^6 = 128\sqrt3\,r^6 = 221.7\,r^6$. Each pair appears twice in the $n^2 = 16$ terms.

#### Self-GMD

$$
\begin{aligned}
D_s &= \left[(0.7788r)^4 \times \left((2r)^4 (4r)(2\sqrt3 r)\right)^2\right]^{1/16} \\
&= 1.8453\,r
\end{aligned}
$$

For capacitance (self-distance $r$): $D_s = 1.9644\,r$.

**Answer:** self-GMD $= 1.845r$ (for inductance).

## Check yourself

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

1. The self-GMD of an n-strand conductor is:
   A. n^2-th root of the product of all distances including each strand's r'
   B. n-th root of the product of the mutual distances only
   C. n^2-th root excluding the self terms
   D. average of all distances
   Answer: A. With n strands there are n^2 terms (n self terms r' and n(n-1) mutual distances). (Key point: Self-GMD rule)

2. Strand 4 rests in the groove between strands 2 and 3. What shape do 2, 3, 4 form?
   A. A right triangle with sides 2r, 2r, 4r
   B. An equilateral triangle of side 2r
   C. A straight line
   D. An equilateral triangle of side r
   Answer: B. Touching strands have centre spacing 2r, so the three mutually touching strands form an equilateral triangle. (Key point: Geometry)

3. What is the distance between strand 1 and strand 4?
   A. 4r
   B. 2r
   C. 2*sqrt3 r
   D. 3r
   Answer: C. Strand 1 is at (0,0), strand 4 at (3r, sqrt3 r), distance sqrt(9r^2+3r^2) = 2 sqrt3 r. (Key point: Distance 1-4)

4. Why is each pair's distance counted twice in the 16 terms?
   A. Each strand has two neighbours
   B. Because of the square root
   C. Because the conductor is symmetrical
   D. Distance i-j and j-i both appear in the n^2 products
   Answer: D. Of the 16 terms, 4 are self terms and the 12 others are the 6 pairs each counted twice. (Key point: Why twice)

## Related in Power System Analysis I

- [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 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 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 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