Self GMD of a conductor with three strands each of radius r and touching each other is

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Examoogle · Accepted Answer

Option A: $r(0.778 	imes 2 	imes 2)^{1/3}$

Explanation & Concepts:
The Self Geometric Mean Distance (GMD), also known as Geometric Mean Radius (GMR) or $D_s$, of a conductor bundle is calculated as the n-th root of the product of all self and mutual distances between the strands. For a three-strand arrangement where strands of radius $r$ are touching, the self-GMD is given by $(D_{11} 	imes D_{12} 	imes D_{13} 	imes D_{21} 	imes D_{22} 	imes D_{23} 	imes D_{31} 	imes D_{32} 	imes D_{33})^{1/9}$.

Working Principle: In a stranded conductor, the effective radius is modified by the geometric arrangement. For a single strand, the GMR is $r' = r e^{-1/4} \approx 0.7788r$. When three strands are touching in a bundle, $D_{ii} = r'$ and $D_{ij} = 2r$ for $i 
eq j$. The product of all terms in the $3 	imes 3$ matrix results in $(r' 	imes 2r 	imes 2r) 	imes (2r 	imes r' 	imes 2r) 	imes (2r 	imes 2r 	imes r') = (r')^3 	imes (2r)^6$. Taking the 9th root gives $[(r')^3 	imes (2r)^6]^{1/9} = (r')^{1/3} 	imes (2r)^{2/3} = (r' 	imes 2^2 	imes r^2)^{1/3} = (0.778r 	imes 4r^2)^{1/3} = r(0.778 	imes 4)^{1/3}$.

Key Formulas: $D_s = (D_{11} D_{12} D_{13} D_{21} D_{22} D_{23} D_{31} D_{32} D_{33})^{1/9}$ — Definition of GMR for 3 conductors; $r' = 0.7788r$ — GMR of a single solid conductor

Key Points: GMR represents the equivalent radius of a conductor that would have the same internal flux linkage as the actual stranded conductor.; The distance between touching strands of radius $r$ is $2r$.; Self-GMD considers both the self-inductance of each individual strand and the mutual inductance between strands in the bundle.

Advantages: Reduces internal inductance of the transmission line.; Increases the effective radius, thereby reducing the corona effect.

Disadvantages: Increases mechanical complexity of the line design.; Requires more physical material compared to a single equivalent conductor.

Applications: High Voltage AC Transmission Lines.; Bundle conductor design to mitigate line reactance.

Additional Info: The term 0.778 represents $e^{-1/4}$, which accounts for the internal magnetic flux linkage of a round solid conductor.. Option B, C, and D are incorrect because they misapply the exponent rules for the geometric mean calculation.

Answer: A is correct — The self-GMD is determined by the cubic root of the product of the effective radius and the squares of the distances between strands.

Exam Tip: Always remember that for GMD calculations, the diagonal elements of the distance matrix are $r'$ (GMR of the single strand) and off-diagonal elements are the physical distances between strand centers.

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Option B: $r(0.778 	imes 2 	imes 2 	imes 2)$

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Option C: $r(0.778 	imes 2 	imes 2 	imes 2)^3$

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Option D: $r(0.778 	imes 2 	imes 2)^3$

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