Geometric Mean Distance MCQ Quiz - Objective Question with Answer for Geometric Mean Distance - Download Free PDF

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.

Self GMD or GMR:

• Self GMD is also called GMR. GMR stands for Geometrical Mean Radius. 
• GMR is calculated for each phase separately.
• self-GMD of a conductor depends upon the size and shape of the conductor
• GMR is independent of the spacing between the conductors.

GMD:

• GMD stands for Geometrical Mean Distance.
• It is the equivalent distance between conductors.
• GMD depends only upon the spacing 
• GMD comes into the picture when there are two or more conductors per phase.​

Formula:

• The inductance of the single-phase two-wire line is

\(L = \frac{{{\mu _0}}}{\pi } \times \ln \left( {\frac{{GMD}}{{GMR}}} \right) = \frac{{{\mu _0}}}{\pi } \times \ln \frac{D}{{r'}}\)  H/m

GMD = Mutual Geometric Mean Distance = D

GMR = 0.7788r

r= Radius of the conductor​

• The capacitance between two conductors is

\({C_{ab}} = \frac{{\pi \varepsilon }}{{\ln \frac{D}{r}}}\)  F/m

In the calculation of the capacitance, the inner radius of the conductor not considered

Therefore, The self GMD method is used to evaluate Inductance only.

Important points:

• The inductance of the hollow conductor is less when compared to the solid conductor.
• A bundled conductor reduces the reactance of the electric transmission line.
• By making the bundle conductor, the geometric mean radius (GMR) of the conductor increased.

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.

Self GMD or GMR:

• Self GMD is also called GMR. GMR stands for Geometrical Mean Radius. 
• GMR is calculated for each phase separately.
• self-GMD of a conductor depends upon the size and shape of the conductor
• GMR is independent of the spacing between the conductors.

GMD:

• GMD stands for Geometrical Mean Distance.
• It is the equivalent distance between conductors.
• GMD depends only upon the spacing 
• GMD comes into the picture when there are two or more conductors per phase.​

Formula:

• The inductance of the single-phase two-wire line is

\(L = \frac{{{\mu _0}}}{\pi } \times \ln \left( {\frac{{GMD}}{{GMR}}} \right) = \frac{{{\mu _0}}}{\pi } \times \ln \frac{D}{{r'}}\)  H/m

GMD = Mutual Geometric Mean Distance = D

GMR = 0.7788r

r= Radius of the conductor​

• The capacitance between two conductors is

\({C_{ab}} = \frac{{\pi \varepsilon }}{{\ln \frac{D}{r}}}\)  F/m

In the calculation of the capacitance, the inner radius of the conductor not considered

Therefore, The self GMD method is used to evaluate Inductance only.

Important points:

• The inductance of the hollow conductor is less when compared to the solid conductor.
• A bundled conductor reduces the reactance of the electric transmission line.
• By making the bundle conductor, the geometric mean radius (GMR) of the conductor increased.

Self G.M.D of bundle of 4 conductor

= 1.09 (r's³)^1/4

S = Distance b/w 2 conductor

G.M.D = 1.09 (0.7788 × 2 × 10^-2 × 2³)^1/4

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.

The self GMD of the seven strand conductor is the 49^th root of the 49 distances.

Thus, \({D_S} = {\left( {{{\left( {r'} \right)}^7}{{\left( {D_{12}^2D_{26}^2{D_{14}}{D_{17}}} \right)}^6}{{\left( {2r} \right)}^6}} \right)^{\frac{1}{{49}}}}\)

Substituting the values of various distances,

\(\begin{array}{l} {D_S} = {\left( {{{\left( {0.7788r} \right)}^2}{{\left( {{2^x}{r^2} \times 3 \times {2^2}{r^2} \times {2^2}r \times 2r \times 2r} \right)}^6}} \right)^{\frac{1}{{49}}}}\\ {D_S} = \frac{{2r{{\left( {2\left( {0.7788} \right)} \right)}^{\frac{1}{7}}}}}{{{6^{\frac{1}{{49}}}}}} \end{array}\)

= 2.177r