A Transmission line conductor has been suspended freely from two towers and has taken the form of a catenary that has c = 487.68m. The span between the two towers is 152 m, and the weight of the conductor is 1160 kg/km. Calculate the length of the conductor. 

Explanation:

Length of a Conductor in a Catenary:

Definition: When a transmission line conductor is suspended freely between two towers, it forms a curve known as a catenary. The shape of the catenary is determined by several factors, including the horizontal tension, the weight of the conductor, and the span length. Calculating the length of the conductor requires applying the catenary equation and understanding the parameters involved.

Given Data:

• Constant of the catenary, c = 487.68 m
• Span between the two towers, L = 152 m
• Weight of the conductor per unit length, w = 1160 kg/km = 1.16 kg/m

Formula:

The length of the conductor S in a catenary is calculated using the following equation:

S = 2c sinh(L / 2c)

Where:

• S = Length of the conductor
• c = Constant of the catenary
• L = Span between the two towers
• sinh = Hyperbolic sine function

Solution:

Step 1: Calculate the value of L / 2c

L / 2c = 152 / (2 × 487.68) = 152 / 975.36 = 0.1558

Step 2: Find the hyperbolic sine of L / 2c

sinh(0.1558) can be calculated using the formula for hyperbolic sine:

sinh(x) = (e^x - e^-x) / 2

Substituting x = 0.1558:

sinh(0.1558) = (e^0.1558 - e^-0.1558) / 2

Using exponential values:

• e^0.1558 ≈ 1.1688
• e^-0.1558 ≈ 0.8557

Therefore:

sinh(0.1558) = (1.1688 - 0.8557) / 2 = 0.3131 / 2 = 0.15655

Step 3: Calculate the length of the conductor

Using the formula:

S = 2c sinh(L / 2c)

Substitute the values:

S = 2 × 487.68 × 0.15655

S = 487.68 × 0.3131

S ≈ 487.68 m

Final Answer: The length of the conductor is approximately 487.68 m.

Correct Option: Option 1

Additional Information

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

Option 2: 152.614 m

This option is incorrect because it represents the horizontal span between the two towers, not the actual length of the conductor. The length of the conductor in a catenary is always greater than the horizontal span due to the sag in the cable.

Option 3: 5.934 m

This option is incorrect because it is significantly smaller than the actual length of the conductor. Such a value does not align with the given data and the catenary equation.

Option 4: 11.9 m

This option is incorrect for similar reasons as Option 3. It is far too small to represent the length of the conductor, given the span and other parameters provided.

Conclusion:

The calculation of the length of a conductor in a catenary is essential for designing and installing transmission lines. The correct answer, as derived, is Option 1: 487.68 m. This value aligns with the given data and the catenary equation. Understanding the principles of catenary curves is crucial for ensuring the structural stability and efficiency of overhead power lines.