Overhead lines
Conductor sag: parabola and catenary
- where
- conductor weight per unit length (N/m)
- conductor mass per unit length (kg/m)
- sag by parabola (m)
- span length (m)
- horizontal tension (kN)
- sag by catenary (m)
Explain this equation
Gives the mid-span sag of a conductor between level supports for a known horizontal tension, by the parabola and by the exact catenary. Use it to check ground clearance or compare with a sag chart.
w is the weight per metre, from mass m times standard gravity. S is the span and T the horizontal component of conductor tension, the same at every point in the span; ×1000 converts kN to N. T/w is the catenary constant, the length scale of the hanging curve.
A flexible cable with uniform weight along its length hangs as a catenary, cosh. When the sag is small compared with the span, cosh can be expanded and the curve becomes the parabola wS²/8T. Both give sag in metres.
Sense check: sag rises with the square of span and falls in proportion to tension; doubling the span gives four times the sag. For sag below 5 % of span the two answers agree within about 0.5 %.
Watch out: this does not find the tension itself. Tension changes with temperature, wind and ice; a change-of-state calculation is needed for that.
Mid-span sag of a conductor between level supports for a given horizontal tension, by the parabolic approximation and the exact catenary.
Enter your values to see the result.
Good to know
The parabola is within about 0.5 % of the catenary while sag is below about 5–10 % of span, which covers most distribution spans. For wind or ice, use the resultant loading per metre. Tension at the supports exceeds the horizontal tension by w × sag.
Common mistake
Using the tension at the supports or the conductor breaking load as the horizontal tension, or entering weight in kg/m where N/m is expected (here enter mass; g is applied).
Limits
Level supports, uniform load, known horizontal tension at the condition of interest. Does not solve the change-of-state equation (temperature, creep, elastic stretch) or ruling-span effects; for that use a full sag-tension method (CIGRE TB 324, BS EN 50341).
Sources
For competent high-voltage engineers. Results are calculations from the published methods shown and depend entirely on the data you enter. They are not a substitute for a system study, manufacturer data, the applicable standards and Engineering Recommendations, or your organisation’s safety rules. Never use a result to justify working on or near live HV equipment.