Statistical & Engineering
Hyperbolic COSH, SINH & TANH via EXP: Overhead Cable Catenary Sag & Length
Evaluate exact hyperbolic cosine COSH(u) = (e^u + e^-u)/2 and hyperbolic sine SINH(u) = (e^u - e^-u)/2 using EXP to compute transmission-line catenary sag and true suspended arc length.
Model true catenary sag & conductor length beyond parabolic shortcuts!
A uniform flexible cable of weight per unit length w (N/m) strung across horizontal span S (m) under horizontal tension H (N) hangs in a true hyperbolic catenary with parameter a = H / w and dimensionless half-span ratio u = S / (2 * a). Using Euler’s exact exponential identities COSH(u) = (EXP(u) + EXP(-u)) / 2 and SINH(u) = (EXP(u) - EXP(-u)) / 2, the exact mid-span sag is d = a * (COSH(u) - 1) and the exact suspended cable length is L = 2 * a * SINH(u)!
On long mountain or river crossings where u > 0.15, the textbook parabolic sag approximation w * S^2 / (8 * H) underpredicts ground-clearance sag—making the exact hyperbolic catenary essential for IEEE 738 clearance safety.
| Span Metric (S = 200 m, H = 15,000 N, w = 15 N/m) | Formula (a = H/w = 1,000 m, u = 0.10) | Exact Value |
|---|---|---|
| True Catenary Mid-Span Sag (d) | a * (COSH(0.10) - 1) | 5.0042 m |
| Parabolic Sag Approximation | w * S^2 / (8 * H) | 5.0000 m |
| Suspended Conductor Arc Length (L) | 2 * a * SINH(0.10) | 200.3335 m |
Exact Catenary Mid-Span Sag (m) via Exponential COSH
Computes a * ((EXP(u) + EXP(-u)) / 2 - 1) where a = B3 / B4 and u = B2 / (2 * a).
=IF(OR(B2 < 0, B3 <= 0, B4 <= 0), "Invalid Span or Tension", ROUND((B3 / B4) * ((EXP(B2 * B4 / (2 * B3)) + EXP(-B2 * B4 / (2 * B3))) / 2 - 1), 4))Exact Suspended Cable Arc Length (m) via Exponential SINH
Computes 2 * a * ((EXP(u) - EXP(-u)) / 2) across horizontal span B2.
=IF(OR(B2 < 0, B3 <= 0, B4 <= 0), "Invalid Span or Tension", ROUND((B3 / B4) * (EXP(B2 * B4 / (2 * B3)) - EXP(-B2 * B4 / (2 * B3))), 4))Try the example
Use the original values or change them to test your understanding.