A conductor is composed of seven identical copper strands, each having a radius 'R'. Then what is self GMD (Geometric Mean Distance) of the conductor?

Self GMD (Geometric Mean Distance) of a Conductor:

Definition: The self GMD (Geometric Mean Distance) of a conductor is a measure used in the calculation of inductance and capacitance of transmission lines. It is defined as the equivalent distance from the center of one strand of a conductor to the center of all strands, considering the mutual effects of all strands within the conductor. For a multi-strand conductor, this calculation involves determining the geometric mean of all possible distances between the strands.

Given Problem: The conductor is composed of seven identical copper strands, each having a radius R. We are to calculate the self GMD of the conductor.

Step-by-Step Solution:

1. Structure of the Conductor:

• The conductor is made of seven strands of copper. One strand is at the center, and six strands are symmetrically arranged around it in a circular pattern.
• Each strand has a radius of R.
• Let us denote the center of the conductor as the origin (O).

2. Understanding Self GMD:

• Self GMD is the geometric mean of all the distances between the strands of the conductor.
• For a multi-strand conductor, it includes the self-distance of each strand (radius R) and the mutual distances between the strands.
• The formula for self GMD is:

Self GMD = e^{(1/N2)Σ(ln(d))}

• Here, N is the total number of strands, and d is the distance between each pair of strands (including self-distance).

3. Calculation of Self GMD:

• The conductor has seven strands, so N = 7.
• Each strand has a self-distance of R (natural log of radius).
• The mutual distances between the strands depend on their geometric arrangement.

4. Arrangement of Strands:

• One strand is at the center.
• Six strands are arranged in a circular pattern around the center, each at a distance of 2R from the center.
• Mutual distances between the outer strands are calculated based on the geometry of the hexagon formed by the outer strands.
• Distance between two adjacent outer strands = 2R × sin(60°) = √3R.

5. Formula for Self GMD:

• For a seven-strand conductor, the self GMD can be derived using the following formula (after considering all mutual distances):

Self GMD = e^{(1/7²)Σ(ln(d))} = 2.177 R

Final Answer: The self GMD of the conductor is 2.177 R.

Important Information

To further understand the analysis, let’s evaluate the other options:

Option 1: 2.645 R

This option is incorrect because the value of self GMD for a seven-strand conductor is not 2.645 R. This value may represent a different configuration or an erroneous calculation.

Option 3: 2.141 R

This option is close to the correct answer but still incorrect. The exact value of self GMD for the seven-strand configuration is 2.177 R. A slight variation in the arrangement of the strands or an approximation might lead to this value, but it is not the accurate result.

Option 4: 1.21 R

This option is incorrect because the value is significantly lower than the actual self GMD for a seven-strand conductor. This might be a miscalculation or a value corresponding to a different type of conductor arrangement.

Conclusion:

The self GMD of a conductor composed of seven identical copper strands, each having a radius R, is 2.177 R. This value is derived based on the geometric arrangement of the strands and the logarithmic mean of the distances involved.