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

# Self GMD of 4-strand diamond conductor

Power System Analysis I · Chapter 4: Line Parameter Calculations

- Asked in: 2073 Magh, 4 marks
- Video: [GMR of the conductor with 4 strands arranged in diamond configuration](https://www.youtube.com/watch?v=MZH4LqtvppA) — Engameer
- 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-diamond
- Whole chapter: https://www.bctengineer.com/md/semester_4/power_system_analysis_i/chapter-4

## Question

Find the self GMD for inductance and capacitance calculations for the following conductor configuration, given that the radius of each sub-conductor is r. [Figure: four identical touching sub-conductors of radius r arranged in a diamond (one top, two side by side in the middle, one bottom).]

## Short answer

For the diamond four-strand arrangement, self-GMD for inductance (using r′ = 0.7788r) is 1.692 r, and for capacitance (using r) is 1.801 r.

## Answer

The **self-GMD** of a stranded conductor of $n$ strands is the $n^2$-th root of the product of all distances between strands, where each strand's distance to itself is $r' = 0.7788r$ (inductance) or $r$ (capacitance).

```text
          (1)
       (2)   (3)
          (4)
  2-3 = 2r, 1-2 = 1-3 = 2-4 = 3-4 = 2r
  1-4 = 2(sqrt3)r
```

#### Distances

Strands 2 and 3 touch: $D_{23} = 2r$. Strands 1 and 4 each touch both 2 and 3, so 1, 2, 3 (and 2, 3, 4) form equilateral triangles of side $2r$. Then $D_{14} = 2 \times \sqrt{3}r = 2\sqrt{3}r$.

| Strand | Distances to others | Product |
|---|---|---|
| 1 | $2r, 2r, 2\sqrt3 r$ | $8\sqrt3\,r^3$ |
| 2 | $2r, 2r, 2r$ | $8r^3$ |
| 3 | $2r, 2r, 2r$ | $8r^3$ |
| 4 | $2r, 2r, 2\sqrt3 r$ | $8\sqrt3\,r^3$ |

Product of mutual distances $= (2r)^{10}(2\sqrt{3}r)^2 = 12288\,r^{12}$.

#### For inductance

$$
\begin{aligned}
D_{sL} &= \left[(0.7788r)^4 \times 12288\,r^{12}\right]^{1/16} \\
&= 1.6922\,r
\end{aligned}
$$

#### For capacitance

$$
D_{sC} = \left[r^4 \times 12288\,r^{12}\right]^{1/16} = 1.8013\,r
$$

**Answer:** self-GMD for inductance $= 1.692r$; for capacitance $= 1.801r$.

## Check yourself

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

1. For n strands, the self-GMD is the root of what order of the product of distances (including self distances)?
   A. n-th
   B. n^2-th
   C. 2n-th
   D. (n-1)-th
   Answer: B. There are n^2 terms (including n self distances), so the n^2-th root is taken; here 16th root. (Key point: Self-GMD definition)

2. In the diamond, strands 1 and 4 each touch both 2 and 3. What is D_14?
   A. 4r
   B. 2*sqrt(3)*r
   C. 2r
   D. sqrt(3)*r
   Answer: B. Triangles 1-2-3 and 2-3-4 are equilateral with side 2r, so D_14 = 2 x sqrt(3) r. (Key point: Geometry)

3. What is each strand's self distance for inductance versus capacitance?
   A. r' = 0.7788r for inductance, r for capacitance
   B. r for inductance, 0.7788r for capacitance
   C. 2r for both
   D. 0 for both
   Answer: A. Inductance includes internal flux through r'; capacitance uses r. (Key point: Self distance)

4. Which pair of results is correct?
   A. 1.801r for L and 1.692r for C
   B. 1.692r for L and 1.801r for C
   C. 1.692r for both
   D. 2r for both
   Answer: B. D_sL = 1.692r, which is smaller than D_sC = 1.801r because r' < r. (Key point: Results)

## 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 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 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 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
- [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×
- [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