The mechanical effects of short-circuit currents in - Montefiore

asymmetry, this functional relation may very often be
written in the form:
( ϕ ϕ )
2
1+
m⋅cos
−
dyn 2
0
(5.12) Mx= α ⋅I ⋅
1+
m
The coefficient m is denoted as a function of the
max min
maximum dynamic components M xdynand
M xdyn,
found at the maximum and minimum asymmetries, ϕ 0
and π/2+ϕ0 respectively:
M − M
(5.13) m =
min
M
max min
xdyn xdyn
xdyn
This coefficient is independent of current amplitude I.
It varies slightly between one type of fault and
another, as shown in the table below:
Type of fault m
single-phase 0.783
two-phase 0.800
three-phase on busbar 0.944
three-phase on two busbars 0.831
The calculated phase to phase or phase to earth value
provides a good conservative approximation. Formula
(5.12) is not applicable to median phases 2 and 5 of
the "associated phase" arrangements, see in particular
Figure 5.21, Figure 5.24 and Figure 5.26 In such a
case, the stresses average out to zero, although the
zero value is never reached.
7.2.3.2.2.3. Influence of Time Constant τ
Noting that at the minimum asymmetry, the time
constant does not affect the loading, it may be
deduced that α’ is independent of τ, where:
(5.14) α ( t )
ατ
′ k =
1+
( , t k )
m( τ , t )
This leaves a single variable (α or m) for analysis.
Interestingly, there is something of an analogy
between this factor m and that of IEC 60865 [Ref 2]:
tk τ ⎡ −2
⎤
τ
m = ⋅⎢1−e⎥, deriving from integration of
t k ⎣ ⎦
the square of the DC component of the short-circuit
intensity, as well as the differences due to vibration
mechanics. Indeed, in the case of a simple oscillator
ω, the dynamic response to damped loading
k
85
2 2
⎛ ω ⋅τ
⎞
introduces a factor H ⎜ H = ⎟
2 2
⎝ 1+
ω ⋅τ
⎠
(transmittance). In practice, it is best to seek a
pragmatic dependence in accordance with τ or else
opt for a maximized deterministic choice.
7.2.3.2.2.4. Influence of Clearance Time tk
For rigid structures, we may assume that beyond a
certain duration of the short circuit, the maximum
response has been reached. In the simplest cases, like
the one below, a model of the following type may be
used:
M
- if tk ≥ t peak , then α MAX is constant
- if tkc < tcrete
. then
(5.15)
⎛ ⎛
M ⎜ ⎜
π t
α MAX = α MAX ⋅ 1−
cos
⎜ ⎜
⎝ ⎝ 2t
Moment Mx (N.m)
800
700
600
500
400
300
200
100
k
peak
Variation of the maximum value of the dynamic
answer, in function of short-circuit duration
⎞⎞
⋅⎟⎟
⎟⎟
⎠⎠
0
0,00 0,02 0,04 0,06 0,08 0,10 0,12 0,14
short-circuit duration (secondes)
Figure 5.14 Variation of maximum response
In the general case, modal superposition often renders
the analysis of clearance time influence complex. An
example is given below.
Ne
wto
ns.
m
700
600
500
400
300
200
100
0
Dynamic response of a post insulator
0,00 0,02 0,04 0,06 0,08 0,10
time s
Figure 5.15 Variation of maximum response
t= tpeak