ELECTROTEHNICĂ, ELECTRONICĂ, AUTOMATICĂ, 59 (2011), nr. 1
1
Accurate Computation of the Prospective Short Circuit
Currents in Low Voltage Electric Installations
Emil CAZACU, Iosif Vasile NEMOIANU, Maria-Cristina CONSTANTIN∗
Abstract
Short circuit is an abnormal operating regime of electric installations and, by its consequences, represents the
most serious malfunction that can occur in an electric network. Thus, the prospective short circuit current is
taken as reference for a series of electric equipment tests and based on its value the right choice for electric
switching devices are accomplished. This paper accurately predict the maximal prospective short circuit current
in a particular electric system. The analytical and numerical approach relays on the exact solutions of the time
differential equations which model the dynamic process.
Keywords: prospective short-circuit value, baking-capacity, coefficient of impact
1. Introduction
The paper presents the method of
calculating the short circuit current values
and the short circuit regime parameters
using an analytical and numerical approach
and also underlying the importance of
knowing these values. The analytical and
numerical calculus will be accomplished
through a computer program which
computes the result of the differential
equations that model this process (MAPLE®
or MATLAB®). Also, this paper follows the
analysis of short circuit mode parameter
variation, depending on the initial phase ϕ ,
the power generator voltage and also the
line parameters, expressed quantitatively by
the time constant Ta . The final purpose of
this paper is to link certain characteristics of
automatic breakers (meant to protect the
installation from short circuit and overload) to
the electric parameters of the short circuit
regime, thus establishing a better selectivity
in the installation of these parameters.
2. Modeling the short circuit regime
For this paper, we will study the mono
phase short circuit, which takes place due to
an insulation flaw, an accidental maneuver in
the network or when a conductor in a mono
phase network is grounded [1, 2] – Figure 1.
∗
Emil CAZACU, reader; Iosif-Vasile NEMOIANU, lecturer;
Maria-Cristina CONSTANTIN, student; „Politehnica”
University of Bucharest.
Figure 1. Modeling the short circuit phenomenon
Next, in order to be able to present some
physical and technical aspects of short
circuits, we will consider that the current is
supplied by a power source of infinite electric
power [3, 4]. A power source of an infinite
electric power is hypothetical and is
characterized by the fact that its own
impedance is considered approximately null.
Thus, at constant frequency, the voltage at
the terminals has constant amplitude. In
these
conditions,
the
phenomena
determined by the magnetic coupling
between the circuits of the stator and the
rotor of the generating machine, and the
demagnetization effect of the machine,
caused by the constant reaction to the short
circuit, are negligible [1]. Because of this, the
transient process in the damaged circuit will
be characterized by a short circuit current
with two components: a periodic component
ip(t), of constant amplitude, at the circuit’s
constant parameters, which represent a
sinusoidal alternating current with its actual
value depending only on the damaged area’s
impedance, and an aperiodic component
ia (t ) that damps with the time constant
ELECTROTEHNICĂ, ELECTRONICĂ, AUTOMATICĂ, 59 (2011), nr. 1
2
corresponding to the damaged
Thereby, ik (t ) = i p (t ) + ia (t ) .
circuit.
In the case of short circuiting a mono
phase line with sinusoidal voltage:
u = U 2 sin(ωt + ϕ) , according to Figure 1,
by applying Kirchhoff’s second theorem we
get the following expression of the voltage
di
,
equation: u = U 2 sin(ωt + ϕ) = Rl i + Ll
dt
which has the following general solution:
−R
t
U 2
(1)
i( t ) =
sin( ωt + ϕ − ψ k ) + Ke L
Zk
where
2
2
represents the short
circuit impedance of the network, in
absolute value, relative to the short
circuit area,
− ϕ voltage initial phase angle,
ωL
− ψ k = arctan
short circuit impedance
R
argument, in complex form,
− K integration constant which is
calculated using the initial conditions of
the short circuit.
The first term
− Z k = R + ( ωL )
ip( t ) =
U 2
2
R + ( ωL )
2
sin( ωt + ϕ − ψ k )
(2)
corresponds to the periodic component of
the short circuit current and represents the
particular solution for the differential
equation; its amplitude is constant, because
of the fact that the voltage at the terminals of
the power source that provides current for
the short circuit remains at a constant value
even after the effect has occurred.
The second term in the equation is
−R
−R
t
L
−
= I a0 e
t
Ta
(3)
where K = I a 0 is the initial value, at the
moment the short circuit occurs (t = 0) for this
L
component, and Ta =
is the time constant
R
of the damping, whose value is determined
(4)
i0 =i k 0 = i p 0 + I a 0
where:
− i0 = Iˆ sin(ϕ − ψ ) , and
− i = Iˆ sin(ϕ − ψ ) .
p0
p
k
Under these conditions, we get:
I a 0 = io − i p 0 = Iˆ sin(ϕ − ψ ) − Iˆ p sin(ϕ − ψ k ) and
i = Iˆ p sin( ω t + ϕ − ψ k ) +
+ [ Iˆ sin( ϕ − ψ ) − Iˆ p sin( ϕ − ψ k )] e
−
t
Ta
(5)
.
The equations indicate that the aperiodic
component as well as the total short circuit
current value depend on the following two
factors: the moment of the short circuit
occurrence, i.e. the instantaneous value of
the current in the permanent regime at the
moment t = 0, and the connection phase ϕ
of the voltage.
Admitting the case, when the moment of
the short circuit occurrence coincides with
the moment that the permanent regime
current value passes through zero, i.e.:
i0 = Iˆ sin(ωt + ϕ − ψ ) = 0
we
have:
t
ia (t ) = Ke L and represents the aperiodic
component of the short circuit current, which
dampens after the following exponential
equation:
ia ( t ) = Ke
by the short circuited circuit’s parameters.
The initial value of the aperiodic
component is calculated by considering the
fact that in an inductance circuit, the current
remains unmodified when a disturbance
occurs
in
the
functioning
regime.
Considering this physical aspect and the fact
that in a permanent regime, the current in
the damaged circuit has the following value:
i = Iˆ sin(ωt + ϕ − ψ ) , results in the following
equality for an initial moment t = 0 of the
occurrance of the short circuit:
−
ia = I a 0 e
t
Ta
(6)
I a 0 = − Iˆ p sin(ϕ − ψ k ) = Iˆp sin α
= Iˆp sin α ⋅ e
−
t
Ta
, and
i = Iˆp [sin(ωt − α)] + sin α ⋅ e
−
t
Ta
(7)
where α = ψ k − ϕ .
In order to highlight the waveform of the
short circuit current and its aperiodic and
periodic component, we consider a circuit in
which the line impedance up to the point of
short circuit is given by the following
parameters: Rl = 3.425 mΩ and Ll = 0.123 mH
(general low voltage cable [2]). The temporal
ELECTROTEHNICĂ, ELECTRONICĂ, AUTOMATICĂ, 59 (2011), nr. 1
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variation of these functions is presented in
Figure 2 for ϕ = 0 .
ϕ=
3π
4
Figure 2. Graphical representation of the short circuit
current (1), the aperiodic component (3)
and the periodic component (2) at ϕ = 0
In Figure 3, these waveforms are
represented for different values of the initial
phase φ for the circuit’s power source.
ϕ=π
Figure 3. Graphical representations of ik(t), ip(t) and
ia(t) for different values of the initial phase
φ for the circuit’s power source
In Figure 4, we can see the 3D variation
of the short circuit current value and its
dependence on the φ parameter and time.
ϕ=
π
4
Figure 4. 3D representation of the short circuit
current
If α = 0 , i.e.
ϕ=
π
2
, we get: i a = 0 , and
i k = i p = Iˆ p sin ω t . This means that the
aperiodic component disappears.
If α = ψ k − ϕ = π , we get the following equaϕ = ψ
k
2
tions: I a0 = Iˆp and
π
i p = Iˆ p sin( ω t − ) = − Iˆ p cos ω t
2
.
ELECTROTEHNICĂ, ELECTRONICĂ, AUTOMATICĂ, 59 (2011), nr. 1
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Thus, the short circuit current has a value
of: i k = Iˆ p (e
t
−
Ta
− cos ω t ) .
This means that the aperiodic component
disappears.
If α = ψk − ϕ = π , we get the following equa2
ˆ
tions: I a 0 = I p and i p = Iˆ p sin(ωt − π ) = − Iˆ p cos ωt .
2
Thus, the short circuit current has a value
of: i k = Iˆ p ( e
t
Ta
− cos ω t ) .
3. Specific properties of the short
circuit regime
We will study the case where α =
imax = (ishock )scc
= Iˆp [1 + e
−
0 , 01
Ta
−
0 , 01
Ta
π
,
2
I p = I k∞ =
U med
(11)
Rk2 + X k2
where
− I k ∞ is the permanent short circuit
current,
− U med median intake voltage of the
transformer related to the damaged
area,
impedance in absolute value.
In Figure 5, we can see the variation of
the coefficient of impact, depending on the
time constant of the damaged electric
current portion.
− cos ω ⋅ 0,01] =
(8)
].
The maximum value of the short circuit
current appears after a semi period
(t = 0,01 s) after the malfunction occurs. In
the technical language, this value is called
short circuit shock current, which is taken
into consideration for testing the electrodynamic stress of the collector rods,
isolators, breakers, separators, etc. The
equation, that can be written:
(ishock )scc
periodic component of the short circuit
current, whose value is calculated with the
following equation:
− Z k = Rk2 + X k2 , the short circuit area
because, in these conditions, the highest
current intensity values appear, that stress
the electric installations mechanically,
electro-dynamically and thermally. It’s
because of this, that we have to determine
the maximum amplitude of the short circuit
current and its actual value:
= Iˆp [e
(10)
where I p represents the actual value of the
If α = 0 , i.e. ϕ = ψ k , we get: ia = 0 and
ik = i p = Iˆ p sin ωt .
−
(ishock ) = 1,6 Iˆ p
= kshock Iˆp and kshock = [1 + e
−
0, 01
Ta
]
(9)
is called the shock factor at short circuit or
coefficient of impact [3,5].
The value of the shock factor at short
circuit depends on the time constant of the
damaged circuit, i.e. Ta .
For a normal low voltage network
structure, the time constant varies around
the 0.05 s value, to which a shock factor of
approximately 1.6 corresponds. This results
in a current of:
Figure 5. Variation of the coefficient of impact,
depending on the time constant of the
damaged electric current portion
The actual value of the short circuit shock
current is used in practical calculations for
testing the thermal stress of the current lines
and electric equipments and corresponds to
the first semi period. Usually, it can be
computed with the following equation:
t+
Ik =
1
T
T
2
∫i
2
k
⋅dt
(12)
T
t−
2
In this case, the short circuit current ik
represents a complicated function. Because
of this, to simplify, we consider that in the
ELECTROTEHNICĂ, ELECTRONICĂ, AUTOMATICĂ, 59 (2011), nr. 1
semi period for which the calculation is
made, both components remain constant.
This is valid only in the case of the periodic
component, whose amplitude remains, in the
given conditions, permanently constant, as a
consequence of the fact that the voltage at
the power source terminals is constant. The
aperiodic component is not constant, but for
the considered time values and a time
constant Ta = 0,05 s, it varies from Iˆ p to
Iˆ p e
−
0 , 01
0 , 05
= 0,82 Iˆ p , i.e. it decreases less than
20 % of its initial value. Taking the arithmetic
average of the two values, during the first
semi period, we can consider the aperiodic
component as I a = 0 , 91 I p .
In these conditions, for every moment
within the first semi period, the actual values
of the short circuit current are: I pt =
Iˆ p
2
= Ip;
5
industrial type CBs according to
related standards, notably IEC 609472 [6].
For the latter circuit-breakers, there exists
a wide variety of tripping devices which allow
a user to adapt the protective performance of
the circuit-breaker to the particular
requirements of a load. The protective
scheme performance curve comparison in
the case of a clasical thermal magnetic
protective breaker and an electronic one are
shown in Figure 6 and Figure 7, respectively,
where the following terms were used [6]:
• Ir - overload (thermal or long-delay)
relay trip-current setting,
• Im - short circuit (magnetic or shortdelay) relay trip-current setting,
• Ii - short circuit instantaneous relay
trip-current setting,
• Icu - breaking capacity.
iat = I at , and the actual value of the current
is I kt = I pt2 + I at2 = I p 1 + 2 ⋅ 0,82 ≈ 1,60 I p ,
where I p is calculated using the following
equation: I p =
U med
R k2 + X k2
.
4. The correlation of equipment with
the parameters of short circuit
current
In the next section of this paper, we will
also describe some of the fundamental
parameters of automatic circuit-breakers.
Figure 6. Performance curve of a circuit-breaker
thermal-magnetic protective scheme [6]
4.1 Rated current ( I n )
This is the maximum value of the current
that a circuit-breaker, fitted with a specified
overcurrent tripping relay, can indefinitely
carry at an ambient temperature stated by
the manufacturer, without exceeding the
specified temperature limits of the current
carrying parts.
Another parameter is the short circuit
relay trip-current setting ( I m ).
The
short
circuit
tripping
relays
(instantaneous or slightly time-delayed) are
intended to trip the circuit-breaker rapidly on
the occurrence of high values of fault
current. Their tripping threshold I m is:
— either fixed by standards for domestic
type CBs, e.g. IEC 60898, or
— indicated by the manufacturer for
Figure 7. Performance curve of a circuit-breaker
electronic protective scheme [6]
4.2 Rated short circuit breaking capacity
( I cu or I cn )
The short circuit current-breaking rating of
a CB is the highest (prospective) value of
current that the CB is capable of breaking
without being damaged. The value of current
ELECTROTEHNICĂ, ELECTRONICĂ, AUTOMATICĂ, 59 (2011), nr. 1
6
quoted in the standards is the rms value of
the AC component of the fault current, i.e.
the DC transient component (which is always
present in the worst possible case of short
circuit) is assumed to be zero for calculating
the standardized value. This rated value
( I cu ) for industrial CBs and ( I cn ) for
domestic-type CBs is normally given in kA
rms.
Currents I cu (rated ultimate s.c. breaking
capacity) and I cs (rated service s.c. breaking
capacity) are defined in IEC 60947-2
together with a table relating I cs with I cu for
different
categories
of
utilization
A
(instantaneous tripping) and B (time-delayed
tripping).
Tests for proving the rated s.c. breaking
capacities of CBs are governed by
standards, and include:
— operating sequences, comprising a
succession of operations, i.e. closing
and opening on short circuit;
— current
and
voltage
phase
displacement. When the current is in
phase with the supply voltage ( cos ϕ
for the circuit = 1), interruption of the
current is easier than that at any other
power factor.
Breaking a current at low lagging values
of cos ϕ is considerably more difficult to
achieve; a zero power-factor circuit being
(theoretically) the most onerous case.
In practice, all power-system short circuit
fault currents are (more or less) at lagging
power factors, and standards are based on
values commonly considered to be
representative of the majority of power
systems. In general, the greater the level of
fault current (at a given voltage), the lower
the power factor of the fault-current loop, for
example, close to generators or large
transformers.
The table below (Table 1) extracted from
IEC 60947-2 relates standardized values of
cos ϕ to industrial circuit-breakers according
Following an open-time delay-close/open
sequence to test the I cu capacity of a CB,
further tests are made to ensure that:
— the dielectric withstand capability,
— the
disconnection
(isolation)
performance and
— the correct operation of the overload
protection have not been impaired by
the test.
4.3 Category (A or B) and rated short time
withstand current ( I cw )
There are two categories of LV industrial
switchgear, A and B, according to IEC
60947-2:
− those of A category, for which there is
no deliberate delay in the operation of
the “instantaneous” short circuit
magnetic
tripping
device
(see
Figure 8), are generally moulded-case
type circuit-breakers, and
Figure 8. The A Category circuit-breaker [6]
− those of B category for which, in order
to discriminate with other circuitbreakers on a time basis, it is possible
to delay the tripping of the CB, where
the fault-current level is lower than that
of the short-time withstand current
rating ( I cw ) of the CB (see Figure 9).
to their rated I cu .
Table 1. Icu related to power factor (cos φ) of faultcurrent circuit (IEC 60947-2) [6]
Icu
6 kA < Icu ≤10 kA
10 kA < Icu ≤20 kA
20 kA < Icu ≤50 kA
50 kA ≤ Icu
cos ϕ
0,5
0,3
0,25
0,2
Figure 9. The B Category circuit-breaker [6]
ELECTROTEHNICĂ, ELECTRONICĂ, AUTOMATICĂ, 59 (2011), nr. 1
This is generally applied to large opentype circuit-breakers and to certain heavyduty moulded-case types. I cw is the
maximum current that the B category CB can
withstand, thermally and electro-dynamically,
without sustaining damage, for a period of
time given by the manufacturer.
4.4 Rated making capacity (Icm)
Icm is the highest instantaneous value of
current that the circuit-breaker can establish
at rated voltage in specified conditions. In
AC systems, this instantaneous peak value
is related to Icu (i.e. to the rated breaking
current) by the factor k, which depends on
the power factor ( cos ϕ ) of the short circuit
current loop (see Table 2).
Table 2. Relation between rated breaking capacity
and the rated making capacity at different
power-factor values of short circuit current, as
standardized in IEC 60947-2 [6]
Icm
cos ϕ
Icm = kIcu
6 kA < Icu ≤10 kA
10 kA < Icu ≤20 kA
0,5
1,7 x Icu
0,3
2
20 kA < Icu ≤50 kA
50 kA ≤ Icu
0,25
2,1 x Icu
0,2
2,2 x Icu
x Icu
4.5 Rated service short circuit breaking
capacity ( I cs )
The rated breaking capacity ( I cu ) or ( I cn )
is the maximum fault-current a circuitbreaker can successfully interrupt without
being damaged. The probability of such a
current occurring is extremely low, and in
normal circumstances the fault-currents are
considerably less than the rated breaking
capacity ( I cu ) of the CB. On the other hand,
it is important that high currents (of low
probability) be interrupted under good
conditions, so that the CB is immediately
available for reclosure, after the faulty circuit
has been repaired. It is for these reasons,
that a new characteristic ( I cs ) has been
created, expressed as a percentage of I cu ,
as follows: 25, 50, 75, and 100 % for
industrial circuit-breakers [7].
In Europe, it is the industrial practice to
use a k factor of 100 %, so that I cs = I cu .
5. Conclusions
The paper has accomplished a qualitative
and quantitative analysis of the main electric
parameters of short circuit phenomena that
7
occur in low voltage electric networks.
Thereby, this document proposed a circuit
model of the network where the analyzed
malfunction appears. This model allowed an
analytic preset, starting with the differential
equation that describes the variation in time
of the electric current intensity, of its
maximum and actual values, and also of the
transient components in case of a
malfunction. These parameters prove to be
extremely useful, both in appreciation of the
network electro-dynamical and thermal
stresses in the point where the short circuit
occurs, as well as in the selection of the
protection devices, that are meant to isolate
the damage related to the upstream circuit.
In this sense, the correlation between data
from various manufacturers of electric
protection equipment and the value of short
circuit current parameters calculated for a
network with known parameters was
analyzed.
To
know
in
detail
the
characteristics of these devices and to
correctly interpret them, compared with the
analytically pre-evaluated short circuit
parameters, allow a better choice of these
devices, resulting in a better protection of the
network. The study has taken into
consideration the current European rules
and standards (CEI) that regulate the
operation of low voltage devices.
References
[1] HORTOPAN, G., Aparate electrice de
comutaţie,
vol.1+2,
Editura
Tehnică,
Bucureşti, 2000.
[2] DINCULESCU,
P.,
Instalaţii
Electrice
industriale de joasă tensiune, Editura
MatrixRom, Bucureşti, 2003.
[3] MIRCEA, I., Instalaţii şi echipamente
electrice. Ghid teoretic şi practic (ediţia a
doua), Editura Didactică şi Pedagogică,
Bucureşti, 2002.
[4] IGNAT, J., C.G. Popovici, Reţele electrice de
joasă
tensiune,
Editura
MatrixRom,
Bucureşti, 2003.
[5] WATKINS, J., C. Kitcher, Electric installation
calculations, vol. 1 + 2, Elsevier and Newnes
Publishing, 6th Edition, 2006.
[6] GROUP SCHNEIDER, Electric Installation
Guide, Schneider Electric, 2007.
[7] ELECTRICA
S.A,
Normativ
privind
metodologia de calcul al curentilor de
scurtcircuit în retelele electrice cu tensiunea
sub 1 kV, NTE 0006/06/00, 2006.