In a transmission line with supports at different heights, the lowest point of the conductor is located:

Explanation:

Determining the Lowest Point of a Conductor in a Transmission Line with Supports at Different Heights

Introduction: In an electrical transmission system, conductors are suspended between two supports to transmit electricity from one point to another. These conductors are subjected to various forces such as their weight, wind load, and ice load (if applicable). When the supports are at different heights, the conductor does not form a symmetric curve. Instead, the lowest point of the conductor is influenced by the relative heights of the supports and the distribution of these forces.

Such a system can be analyzed using the principles of catenary curves, which describe the shape of a flexible cable or conductor suspended by its ends under its own weight and external forces.

Correct Option Analysis:

The correct option is:

Option 2: Closer to the lower support.

This is correct because, in a transmission line with supports at different heights, the lowest point of the conductor is closer to the lower support. This phenomenon can be explained as follows:

1. Forces Acting on the Conductor:

• The primary force acting on the conductor is its weight, which causes it to sag. This weight acts uniformly along the length of the conductor.
• The tension in the conductor varies along its length, being higher at the higher support and lower at the lower support.

2. Tension Distribution:

• At the higher support, the tension is greater to maintain equilibrium against the downward pull caused by the weight of the conductor.
• At the lower support, the tension is less because the vertical component of the tension at the higher support compensates for the weight of the conductor in between.

3. Sag and the Lowest Point:

• The sag of the conductor is directly related to the tension. Higher tension results in less sag, and lower tension results in more sag.
• Since the tension is lower near the lower support, the conductor sags more in that region, causing the lowest point to shift closer to the lower support.

4. Mathematical Representation:

• The shape of the conductor is approximately a catenary curve, described by the equation:

  y = (T₀/w) × cosh(wx/T₀) - (T₀/w)

  where:
  • y is the vertical displacement (sag) of the conductor.
  • T₀ is the horizontal tension in the conductor.
  • w is the weight of the conductor per unit length.
  • x is the horizontal distance from the lowest point.
• When the supports are at different heights, the horizontal tension component remains constant, but the vertical components vary, leading to an asymmetric catenary curve.

5. Practical Observations:

• In real-world transmission lines, the lowest point is often observed to be closer to the lower support in cases where the supports are at different heights. This has been validated through numerous field observations and engineering analyses.
• The exact position of the lowest point depends on the height difference between the supports, the span length, and the conductor’s mechanical properties.

Additional Information

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

Option 1: Exactly at the midpoint of the span.

This is incorrect. The lowest point of a conductor would be at the midpoint only if the supports are at equal heights, resulting in a symmetric catenary curve. When the supports are at different heights, the curve becomes asymmetric, and the lowest point shifts closer to the lower support.

Option 3: Closer to the higher support.

This is incorrect. The lowest point cannot be closer to the higher support because the tension is greater at the higher support, resulting in less sag in that region. The sag is more pronounced near the lower support, causing the lowest point to shift closer to it.

Option 4: At the higher support.

This is incorrect. The conductor cannot have its lowest point at the higher support because the higher support is, by definition, at a greater height than the lower support. The conductor must sag below the level of the higher support, forming a curve with the lowest point closer to the lower support.

Conclusion:

In a transmission line with supports at different heights, the lowest point of the conductor is located closer to the lower support. This is due to the variation in tension along the conductor and the resulting asymmetric sag. Understanding this concept is essential for designing and analyzing transmission lines to ensure their stability, efficiency, and safety.