Important Formulas in Sag and Tension Calculations
Sag and Tension Formulas for Overhead Power Lines
Sag and tension calculations are fundamental to the design and analysis of overhead transmission and distribution lines. These calculations are essential for ensuring adequate ground clearance, maintaining conductor integrity under various loading conditions, and ensuring the overall reliability and safety of power delivery. This document provides a concise overview of the key formulas used in these calculations, presented for clarity and ease of reference.
Catenary Curve
⇒ Describes the curve formed by a conductor suspended between two supports.
$ y = C \cosh\left(\frac{x}{C}\right) – 1 $
Where:
- $ x $ = horizontal distance from the lowest point in span (m)
- $ y $ = vertical distance from the lowest point of span
- $ C $ = Catenary constant
⇒ Defines the catenary constant.
$ C = \frac{H}{W} $
Where:
- $ H $ = Horizontal component of tension (N)
- $ W $ = Distributed load on conductor (N/m), where $ W = mg $ ($ m $ = Unit mass of conductor (kg/m), $ g $ = Gravitational constant of 9.81 m/s²)
Conductor Length
⇒ Conductor length from the lowest point to any other point.
$ L_c(x) = \frac{H}{W} \sinh\left(\frac{Wx}{H}\right) = x \left(1 + \frac{x^2W^2}{6H^2}\right) $
⇒ Total conductor length for a level span.
$ L_c = \frac{2H}{W} \sinh\left(\frac{WS}{2H}\right) = S \left(1 + \frac{S^2W^2}{24H^2}\right) = S + \frac{S^3W^2}{24H^2} $
⇒ Slack of the line.
$ \text{Slack} = L_c – S = \frac{S^3W^2}{24H^2} $
Changes in Conductor Length
⇒ Change in conductor length due to strain-stress.
$ \frac{L_{C-\text{final}} – L_{C-\text{initial}}}{L_{C-\text{initial}}} = \frac{H}{E_{\text{cond}}A_{\text{cond}}} $
Where:
- $ E_{\text{cond}} $ = Modulus of elasticity of the conductor
- $ A_{\text{cond}} $ = Cross-sectional area of the conductor
⇒ Modulus of elasticity for composite conductors.
$ E_{\text{conductor}} = E_{al} \left(\frac{A_{al}}{A_{\text{total}}}\right) + E_{st} \left(\frac{A_{st}}{A_{\text{total}}}\right) $
Where:
- $ E_{al} $ = Modulus of elasticity of aluminum
- $ E_{st} $ = Modulus of elasticity of steel
- $ A_{al} $ = Cross-sectional area of aluminum
- $ A_{st} $ = Cross-sectional area of steel
- $ A_{\text{total}} $ = Total cross-sectional area
⇒ Effective linear thermal coefficient of expansion for composite conductors.
$ \alpha_{\text{conductor}} = \alpha_{al} \left(\frac{E_{al}}{E_{\text{conductor}}}\right)
\left(\frac{A_{al}}{A_{\text{total}}}\right) +
\alpha_{st} \left(\frac{E_{st}}{E_{\text{conductor}}}\right) \left(\frac{A_{al}}{A_{\text{total}}}\right) $
Where:
- $ \alpha_{al} $ = Coefficient of linear thermal expansion of aluminum
- $ \alpha_{st} $ = Coefficient of linear thermal expansion of steel
⇒ 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) $
Where:
- $ \alpha_{\text{cond}} $ = effective linear thermal coefficient of expansion of the composite conductor
- $ t_i $ = initial temperature
- $ t_f $ = final temperature
Conductor State Change Equation
⇒ Conductor state change equation.
$ H_2^3 + H_2^2 \left(\frac{(W_1S)^2AE}{24H_1^2} – H_1 + (t_2 – t_1) \alpha AE\right) –
\frac{(W_2S)^2AE}{24} = 0 $
Where:
- $ H_1 $ = initial conductor tension at initial temperature (N)
- $ W_1 $ = unit weight of conductor at initial temperature (N/m)
- $ \Delta t = t_2 – t_1 $ = temperature change (°C)
- $ E $ = final modulus of elasticity (Pa)
- $ A $ = cross-sectional area conductor (m²)
- $ \alpha $ = coefficient of linear thermal expansion (/°C)
- $ S $ = length of ruling span (or single span length) (m)
Sag-Tension Relationship
⇒ Basic relationship between sag and span length.
$ D = \frac{WS^2}{8H} \rightarrow H = \frac{WS^2}{8D} $
Where:
- $ D $ = Sag
- $ S $ = Span length
⇒ The sag (D) of a conductor can be calculated using the following (more complex) formula, or
approximated
using a parabolic formula:
$D=\frac{H}{W_{total}}(cosh(\frac{S}{2H/W_{total}})-1)$
$D=\frac{WS^{2}}{8H}$
Effect of Ice and Wind
⇒ Volume of ice.
$ V_{\text{ice}} = \pi t (D + t) $
Where:
- $ 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}} = \text{Volume} \times \text{density} = \rho_{\text{ice}} \pi t (D + t) $
Where:
- $ \rho_{\text{ice}} $ = Ice density (kg/m³)
⇒ Weight due to wind per unit length.
$ W_{\text{wind}} = P_{\text{wind}} (D + 2t) $
Where:
- $ P_{\text{wind}} $ = wind pressure
⇒ Total conductor weight.
$ W_{\text{total}} = \sqrt{(W + W_{\text{ice}})^2 + (W_{\text{wind}})^2} $
⇒ Total sag of the conductor.
$ D = \frac{H}{W_{\text{total}}} \left(\cosh\left(\frac{S}{2H/W_{\text{total}}}\right) – 1\right) $
Where:
- $ D $ = Sag
- $ S $ = Span length
⇒ Vertical sag of the conductor.
$ D_v = D \cos \theta $
Where:
- $ \theta $ = blowout angle
Creep
⇒ Creep is the permanent elongation or inelastic stretch of a material under stress.