- Sag Definition: Sag in a transmission line is defined as the vertical distance between the highest points of support and the lowest point of the conductor.
- Purpose of Sag: Including appropriate sag protects transmission lines from excessive tension and potential damage, especially under adverse conditions.
- Calculation Methodology: Calculating sag involves understanding the geometric and physical properties of the transmission line, such as span length, conductor weight, and tension.
- Environmental Impact: Wind and ice alter the sag by changing the effective weight of the conductor, requiring recalculations to ensure stability.
- Safety Considerations: Proper sag calculation is vital for maintaining the structural integrity and operational reliability of transmission lines.
What Is Sag in a Transmission Line?
Sag in a transmission line is the vertical distance between the straight chord joining two support points, such as points on transmission towers, and the conductor at a specified position. For equal-height supports under uniform loading, maximum sag occurs at the lowest point near midspan.
A span between supports at the same attachment elevation is a level span. A span with different attachment elevations is an unequal level span.
Consider conductor AOB between equal-level supports A and B. Under uniform loading, the exact flexible-cable shape is a catenary and the lowest point O is at midspan. When sag is small compared with the span, a parabola gives a useful approximation.
In the diagram, S is the maximum vertical sag below the chord AB.
Why Do Overhead Conductors Need Sag?
An overhead transmission line conductor is installed with a specified initial sag and tension. The values are selected together rather than treating one fixed sag as perfect for every load case.
More sag reduces tension but also reduces ground and object clearance. Less sag increases clearance but raises mechanical loads on the conductor, fittings and structures. Design tables and calculations must satisfy both clearance and strength requirements.
If a conductor were installed without enough length for temperature change and environmental loading, tension could exceed allowable limits. Deliberate sag provides the required geometric allowance while preserving statutory clearances under the governing load cases.
For the simple equal-level, uniformly loaded span:
- The conductor forms a shallow curve, and the parabolic approximation is accurate only when sag is small compared with span.
- The exact curve is a catenary; the sag-span curve is approximated as a parabola in the derivation below.
- Tension acts tangentially at every point of the conductor. Its horizontal and vertical components balance the distributed load.
- The horizontal component of tension is constant throughout a span under static equilibrium.
- Total tension reaches its minimum at the lowest point, where it is horizontal. It increases toward each support. The difference is small only for a shallow sag angle.
How to Calculate Sag in a Transmission Line
The parabolic calculation is handled separately for two support arrangements:
- Supports at equal levels
- Supports at unequal levels
These equations assume a flexible conductor, uniform load per unit horizontal length, a shallow parabolic profile and a known horizontal tension. Final line design uses the governing code, conductor data and sag-tension calculation for all required load cases.
Sag calculation for supports at equal levels
Let AOB represent the conductor. A and B are the supports, and O is the lowest point at midspan.
L is the horizontal span AB.
w is the vertical conductor weight per unit horizontal length.
T in the displayed equations is the horizontal tension, also equal to total tension at O.
Select any conductor point P.
x is the horizontal distance from O to P.
y is the vertical height from O to P.
Static force and moment equilibrium for the parabolic approximation gives:
Sag calculation for supports at unequal levels
Let AOB represent the conductor and O its calculated lowest point.
L is the horizontal span.
h is the difference between support elevations.
x1 is the horizontal distance from lower support A to O.
x2 is the horizontal distance from O to upper support B.
T is the horizontal tension.
w is the conductor weight per unit horizontal length.
The geometry and parabolic sag equations give:
After calculating x1 and x2, use the corresponding distances to calculate S1 and S2. For a large elevation difference, the mathematical lowest point can fall outside the physical span, so the sign and location must be checked.
The displayed formulas describe a still-air load case based on the conductor‘s own weight and specified horizontal tension. A complete sag-tension study also accounts for conductor temperature, elastic stretch, long-term creep, wind, ice and the load combinations required by the applicable standard.
How Do Ice and Wind Affect Sag?
Ice and wind change the distributed load vector and the tension required to maintain the span geometry:
- Ice adds mass and increases the conductor’s projected diameter, so it affects both vertical weight and wind force.
- Wind adds a distributed force mainly transverse to the span. It does not change the conductor’s gravitational self-weight.
- Conductor weight and ice weight act vertically downward.
- Combining the horizontal wind force with the vertical conductor-plus-ice weight gives a resultant load per unit length.
- The conductor sags in the plane of the resultant load, which is inclined from vertical when wind force is present.
Let w be the conductor weight per unit length.
wi is the ice weight per unit length.
wi = ice density × ice volume per unit length.
ww is the transverse wind force per unit length.
ww = design wind pressure × iced projected area per unit length.
The resultant distributed load is:
Using the parabolic approximation, sag in the resultant-load plane is:
Its vertical sag component is:
