The mechanical effects of short-circuit currents in - Montefiore

(3.5)
⎧3
r sin δ
ϕ = ⎨
⎩3
r sin δ
( max + cos δmax
−1
)
( + cos δ −1
)
k
k
for
for
δ
δ
k
k
≥ δ
< δ
max
max
In Figure 3.35 b), the dropper is stretched and blocks the
movement of the main conductor, the span is swinging
nearly horizontal and no drop force occurs; the maxima
in the time history of the forces following the first
maxima are less or equal.
Due to the increase in dropper length the actual
maximum swing-out angle also increases and several
fall down follow, Figure 3.35 c); the second one is
decisive, because the speed is nearly zero at the lowest
point and in the main conductor the mechanical energy
is converted to elastical energy. From tests follow that a
fall of span can be expected for δmax ≥ 60º. In an
arrangement without dropper, a fall of span can only
occur if the span has enough energy that the maximum
swing-out angle δm is greater than 70º, as shown in
Figure 4.2 of [Ref 1]. This leads to the assumption that
the drop force Ff shall be calculated if the following
condition is fulfilled :
(3.6) δmax ≥ 60º and δm ≥ 70°
where δk is to be calculated without dropper with
equation [(*29)] and δmax with equation (3.3).
At the end of the short-circuit current flow, the
electromagnetic energy is converted to potential, kinetic
and elastic energy of the swinging system. When the
dropper is stretched, a part of the mechanical energy of
the main conductor is converted to elastic energy of the
dropper and back due to the energy conservation law,
the energy in the system is always constant. During and
at the end of the fall, the dropper is not stretched and the
mechanical energy in the highest point of the main
conductor movement is converted to elastic energy
during the fall down which gives the drop force Ff. The
effect is the same as described in Section 4.2 of [Ref 1]
for the span having no droppers. This shows, that the
drop force Ff of a span with dropper can be calculated in
the same way as for a span without dropper according to
equation [(*35)].
The maximum horizontal displacement bh of the span
depends also on the actual maximum swing-out angle
on δmax. If δmax ≥ δm the dropper length has no influence
and bh has the same value as for a span without dropper,
also in the case δmax < δm when δm ≥ δ1. For δmax < δm,
δ1 the horizontal displacement is limited by the dropper.
Therefore the maximum horizontal displacement bh can
be determined with the two cases:
– δmax ≥ δm:
the horizontal displacement is obtained as for a span
without dropper according to equation [(*41)]
⎧CFC
Dbc
sin δ1
for δm
≥ δ1
(3.7) b h = ⎨
⎩CFC
Dbc
sin δm
for δm
< δ1
– δmax < δm:
50
the horizontal displacement becomes:
⎧CFC
Dbc
sin δ1
(3.8) b h = ⎨
⎩CFC
Dbc
sin δmax
for
for
δ
δ
max ≥
max <
where CF, CD, δ1 and δm are to be calculated for the span
without dropper according to equations [(*21)], [(*31)],
[(*38)] and [(*39)] and δmax with equation (3.3).
If the short-circuit current flows through the complete
length of the main conductor, the electromagnetic force
per unit length is calculated with equation [(*19)] in the
case of a three-phase system:
( I ′′ )
(3.9)
µ
F ′ 0
= 0,
75
2π
k3
a
lc
l
On the other side, if the current flows through half the
span and then through the dropper it is assumed, that the
span without dropper acts the equivalent
electromagnetic force:
( I ′
)
(3.10)
µ
F ′ 0
= 0,
75
2π
k3
a
lc
2 + ld
l
2
where lc is the main conductor length, ld the dropper
length and l the span length. In single-phase systems,
0, 75 I ′ is to be replaced by ( ) 2
I ′′ .
( ) 2
k3
This extension of the simplified method stated in IEC
60865-1 is compared with many tests. The best
agreement could be achieved under the assumption that
the mass of the droppers is disregarded for the shortcircuit
stress. That means the calculation is to be done
without droppers. The results are given in the annex 8.3
and show sufficient accuracy of the method.
For the calculation of the static tensile force Fst and the
static sag, the mass of the dropper cannot be
disregarded. Because the dropper has a bending
stiffness, one part of its gravitational force acts on the
lower point and the other part including the clamp on
the main conductor. Therefore it is recommended to add
an additional mass in the span equal to the clamp mass
plus half the mass of the dropper.
b) Forces apparatus and insulators
Forces on apparatus and insulators are caused by the
electromagnetic forces between the droppers (cases 1, 3,
5 in Figure 3.20) and the pinch effect between the subconductors
in the dropper (cases 2, 4, 6), and in addition
the stretch due to the movement of the busbar (cases 5,
6). First, the droppers with fixed upper ends are
analysed which give the forces in cases 1. The pinch
effect in cases 2, 4, 6 can be estimated according to the
standard.
2
k2
2
δ
δ
1
1