Important Formulas in Sag and Tension Calculations

Byjaredcuchapin@gmail.com
Sag and Tension Formulas for Overhead Power Lines

Sag and Tension Formulas for Overhead Power Lines

Essential formulas for the design and analysis of overhead transmission and distribution lines — covering catenary geometry, conductor loading, thermal effects, ice and wind, and conductor state change.

01

Catenary Curve

Describes the curve formed by a conductor suspended between two supports.
$ y = C \cosh\left(\dfrac{x}{C}\right) – 1 $

Variables

  • $ x $ — horizontal distance from the lowest point in span (m)
  • $ y $ — vertical distance from the lowest point of span (m)
  • $ C $ — catenary constant
Defines the catenary constant.
$ C = \dfrac{H}{W} $

Variables

  • $ H $ — horizontal component of tension (N)
  • $ W $ — distributed load on conductor (N/m), where $ W = mg $;  $ m $ = unit mass (kg/m),  $ g $ = 9.81 m/s²
02

Conductor Length

Conductor length from the lowest point to any other point.
$ L_c(x) = \dfrac{H}{W} \sinh\!\left(\dfrac{Wx}{H}\right) = x \left(1 + \dfrac{x^2 W^2}{6H^2}\right) $
Total conductor length for a level span.
$ L_c = \dfrac{2H}{W} \sinh\!\left(\dfrac{WS}{2H}\right) = S \left(1 + \dfrac{S^2 W^2}{24H^2}\right) = S + \dfrac{S^3 W^2}{24H^2} $
Slack of the line.
$ \text{Slack} = L_c – S = \dfrac{S^3 W^2}{24H^2} $
03

Changes in Conductor Length

Change in conductor length due to strain-stress.
$ \dfrac{L_{C\text{-final}} – L_{C\text{-initial}}}{L_{C\text{-initial}}} = \dfrac{H}{E_{\text{cond}}\, A_{\text{cond}}} $

Variables

  • $ E_{\text{cond}} $ — modulus of elasticity of the conductor (Pa)
  • $ A_{\text{cond}} $ — cross-sectional area of the conductor (m²)
Modulus of elasticity for composite conductors (e.g. ACSR).
$ E_{\text{conductor}} = E_{al} \left(\dfrac{A_{al}}{A_{\text{total}}}\right) + E_{st} \left(\dfrac{A_{st}}{A_{\text{total}}}\right) $

Variables

  • $ E_{al},\, E_{st} $ — modulus of elasticity of aluminum and steel (Pa)
  • $ A_{al},\, A_{st} $ — cross-sectional areas of aluminum and steel (m²)
  • $ A_{\text{total}} $ — total cross-sectional area (m²)
Effective linear thermal coefficient of expansion for composite conductors.
$ \alpha_{\text{conductor}} = \alpha_{al} \left(\dfrac{E_{al}}{E_{\text{conductor}}}\right)\!\left(\dfrac{A_{al}}{A_{\text{total}}}\right) + \alpha_{st} \left(\dfrac{E_{st}}{E_{\text{conductor}}}\right)\!\left(\dfrac{A_{st}}{A_{\text{total}}}\right) $

Variables

  • $ \alpha_{al} $ — coefficient of linear thermal expansion of aluminum (/°C)
  • $ \alpha_{st} $ — coefficient of linear thermal expansion of steel (/°C)
Change in conductor length due to thermal expansion.
$ L_{C\text{-final}} = L_{C\text{-initial}} \left(1 + \alpha_{\text{cond}}\,(t_i – t_f)\right) $

Variables

  • $ \alpha_{\text{cond}} $ — effective linear thermal coefficient of expansion of the composite conductor (/°C)
  • $ t_i,\, t_f $ — initial and final temperature (°C)
04

Conductor State Change Equation

Cubic equation relating initial and final tension states across a temperature or load change.
$ H_2^3 + H_2^2 \left(\dfrac{(W_1 S)^2 AE}{24H_1^2} – H_1 + \Delta t\,\alpha\, AE\right) – \dfrac{(W_2 S)^2 AE}{24} = 0 $

Variables

  • $ H_1 $ — initial conductor tension at initial temperature (N)
  • $ W_1,\, W_2 $ — unit weight of conductor at initial and final conditions (N/m)
  • $ \Delta t = t_2 – t_1 $ — temperature change (°C)
  • $ E $ — final modulus of elasticity (Pa)
  • $ A $ — cross-sectional area of conductor (m²)
  • $ \alpha $ — coefficient of linear thermal expansion (/°C)
  • $ S $ — ruling span (or single span) length (m)
05

Sag–Tension Relationship

Basic parabolic approximation — relates sag to span length and tension.
$ D = \dfrac{WS^2}{8H} \qquad \longrightarrow \qquad H = \dfrac{WS^2}{8D} $

Variables

  • $ D $ — sag (m)
  • $ S $ — span length (m)
Exact catenary formula for sag — use when sag-to-span ratio is large.
$ D = \dfrac{H}{W_{\text{total}}} \left(\cosh\!\left(\dfrac{S}{2H/W_{\text{total}}}\right) – 1\right) $
06

Effect of Ice and Wind

Volume of ice per unit length of conductor.
$ V_{\text{ice}} = \pi\, t\,(D + t) $

Variables

  • $ V_{\text{ice}} $ — volume of ice per unit length (m³/m)
  • $ D $ — outside diameter of the conductor (m)
  • $ t $ — thickness of the ice (m)
Weight of ice per unit length.
$ W_{\text{ice}} = \rho_{\text{ice}}\,\pi\,t\,(D + t) $

Variables

  • $ \rho_{\text{ice}} $ — ice density (kg/m³), typically 900 kg/m³
Wind load per unit length of conductor.
$ W_{\text{wind}} = P_{\text{wind}}\,(D + 2t) $

Variables

  • $ P_{\text{wind}} $ — wind pressure (N/m²)
Total resultant conductor weight under combined ice and wind loading.
$ W_{\text{total}} = \sqrt{\left(W + W_{\text{ice}}\right)^2 + W_{\text{wind}}^2} $
Total sag under combined loading (catenary formula).
$ D = \dfrac{H}{W_{\text{total}}} \left(\cosh\!\left(\dfrac{S}{2H/W_{\text{total}}}\right) – 1\right) $
Vertical sag component — accounts for wind blowout of the conductor.
$ D_v = D\cos\theta $

Variables

  • $ \theta $ — blowout angle from vertical (°)
07

Creep

Creep is the permanent elongation (inelastic stretch) of a conductor material under sustained mechanical stress. Unlike elastic strain, creep does not recover when the load is removed, and accumulates over the service life of the line. It must be accounted for in long-term sag and tension predictions.

Similar Posts