---
title: "GMR of stranded vs solid conductor"
subject: "Power System Analysis I"
chapter: 4
key: "semester_4/power_system_analysis_i:gmr-stranded-lower-than-solid"
source: https://www.bctengineer.com/q/semester_4/power_system_analysis_i/gmr-stranded-lower-than-solid
site: BCT Engineering
---

# GMR of stranded vs solid conductor

Power System Analysis I · Chapter 4: Line Parameter Calculations

- Asked in: 2079 Chaitra, 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/gmr-stranded-lower-than-solid
- Whole chapter: https://www.bctengineer.com/md/semester_4/power_system_analysis_i/chapter-4

## Question

With a suitable example show that the geometric mean radius of a stranded conductor is lower than that of a solid conductor of the same overall diameter.

## Short answer

For a 7-strand conductor with strand radius r, overall radius is 3r. Stranded GMR = 2.1767r, solid GMR of same diameter = 0.7788*3r = 2.3364r. The stranded GMR is about 7% lower due to air gaps reducing effective conducting cross-section, increasing inductance compared to a solid conductor.

## Answer

The **geometric mean radius (GMR)** of a conductor is the radius of a fictitious thin tube that has the same internal plus external flux linkage as the real conductor. For a solid round conductor of radius $R$, $GMR = r' = e^{-1/4}R = 0.7788R$. A stranded conductor of the same overall diameter contains air gaps between strands, so its current is spread over less metal and its effective GMR is smaller.

#### Example: 7-strand conductor

Take a 7-strand conductor, each strand of radius $r$: one central strand and six touching strands around it. Overall radius $R = 3r$.

```text
        (2)   (3)
     (7)   (1)   (4)      1 = centre strand
        (6)   (5)         ring radius = 2r
```

Distances between strand centres:

| Pair type | Distance | Ordered pairs |
|---|---|---|
| Centre to outer | $2r$ | 12 |
| Adjacent outers | $2r$ | 12 |
| Outer to next-but-one | $2\sqrt{3}r$ | 12 |
| Outer to opposite | $4r$ | 6 |
| Self ($r'$) | $0.7788r$ | 7 |

With $n = 7$ strands there are $n^2 = 49$ terms:

$$
\begin{aligned}
D_s &= \left[(0.7788r)^7 (2r)^{24} (2\sqrt{3}r)^{12} (4r)^6\right]^{1/49} \\
&= 2.1767\,r
\end{aligned}
$$

#### Solid conductor of the same overall diameter

$$
GMR_{solid} = 0.7788 \times 3r = 2.3364\,r
$$

#### Comparison

$$
\frac{D_{s,\,stranded}}{GMR_{solid}} = \frac{2.1767r}{2.3364r} = 0.932
$$

So the 7-strand conductor has a GMR about 7% smaller than a solid conductor of the same diameter.

**Why:** the GMR depends on the actual cross-section that carries current. Stranding leaves air gaps, so the current-carrying material sits in a smaller effective area. A smaller GMR means the stranded conductor has a slightly **higher inductance** ($L = 2\times10^{-7}\ln(D/D_s)$) than a solid conductor of the same outer diameter.

**Answer:** $D_s = 2.1767r$ (stranded) < $2.3364r$ (solid).

## Check yourself

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

1. What is the GMR of a solid round conductor of radius R?
   A. 0.7788 R
   B. R
   C. 1.284 R
   D. 0.5 R
   Answer: A. GMR = e^(-1/4) R = 0.7788 R. (Key point: Solid GMR)

2. For a 7-strand conductor (overall radius 3r), how does Ds compare with a solid conductor of the same diameter?
   A. Ds = 2.1767 r, about 7 % less than 2.3364 r
   B. Ds = 2.3364 r, equal to solid
   C. Ds = 3 r, larger than solid
   D. Ds = 0.7788 r, far smaller
   Answer: A. The 49-term geometric mean gives 2.1767 r versus 0.7788 x 3r = 2.3364 r, ratio 0.932. (Key point: 7-strand result)

3. Why is the stranded GMR lower, and what is the effect?
   A. Air gaps leave metal in a smaller effective area, so inductance is slightly higher
   B. Strands carry more current, so inductance is lower
   C. Stranding raises permeability, lowering inductance
   D. Air gaps increase GMR and reduce L
   Answer: A. A smaller GMR increases ln(D/Ds), so L is slightly higher than for the solid conductor. (Key point: Reason and consequence)

## Related in Power System Analysis I

- [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
- [Optimum conductor radius for minimum cable stress](https://www.bctengineer.com/question.md?key=semester_4%2Fpower_system_analysis_i%3Acable-optimum-conductor-radius) — Power System Analysis I, chapter 2
- [Stranded vs bundled conductors](https://www.bctengineer.com/question.md?key=semester_4%2Fpower_system_analysis_i%3Adifferentiate-stranded-bundled-conductors) — Power System Analysis I, chapter 2
- [GMR for inductance vs capacitance](https://www.bctengineer.com/question.md?key=semester_4%2Fpower_system_analysis_i%3Agmr-inductance-vs-capacitance) — Power System Analysis I, chapter 4

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