Electric Power Generation, Transmission, and Distribution

14.1.3 Conductor Slack

The difference between the conductor length,L, and the span length,S, is called slack. The parabolic
equations for slack may be found by combining the preceding parabolic equations for conductor length,
L, and sag,D:

LS¼S^3

w^2

24 H^2



¼D^2

8

3 S



(14:7)

While slack has units of length, it is often expressed as the percentage of slack relative to the span
length. Note that slack is related to the cube of span length for a givenH=wratio and to the square of sag
for a given span. For a series of spans having the sameH=wratio, the total slack is largely determined by
the longest spans. It is for this reason that the ruling span is nearly equal to the longest span rather than
the average span in a series of suspension spans.
Equation (14.7) can be inverted to obtain a more interesting relationship showing the dependence of
sag,D, upon slack,L-S:

D¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

3 S(LS)

8

r

(14:8)

As can be seen from the preceding equation, small changes in slack typically yield large changes in
conductor sag.

14.1.4 Inclined Spans

Inclined spans may be analyzed using essentially the same equations that were used for level spans. The
catenary equation for the conductor height above the low point in the span is the same. However, the
span is considered to consist of two separate sections, one to the right of the low point and the other to
the left as shown in Fig. 14.2 (Winkelmann, 1959). The shape of the catenary relative to the low point is
unaffected by the difference in suspension point elevation (span inclination).
In each direction from the low point, the conductor elevation, y(x), relative to the low point is given by:

y(x)¼

H

w

cosh

w

H

x



 1



¼

wxðÞ^2

2 H

(14:9)

S

S 1

TR

D

DR

XL XR

DL

TL

h

FIGURE 14.2 Inclined catenary span.