Jacek HORISZNY, Mirosław WOŁOSZYN
Gdansk University of Technology
A Circuital Model of Electric Arc in a Circuit Breaker
Abstract. The paper presents a circuital model of arc in a circuit breaker. The model is intended to be used in simulations of the electric circuits
transients. It was implemented in the ATP simulation program. The article presents also a few simulation examples of simple DC and AC circuit
containing described arc model. The results show good compatibility with curves available in the literature.
Streszczenie. W artykule zaprezentowano obwodowy model łuku elektrycznego w wyłączniku. Jest on przeznaczony do wykorzystania w
symulacjach stanów przejściowych w obwodach elektrycznych. Przedstawiono także wyniki przykładowych symulacji w obwodzie prądu stałego i
przemiennego z wykorzystaniem opisanego modelu. Uzyskano pełną zgodność otrzymanych wyników z przebiegami podawanymi w literaturze
(Obwodowy model łuku elektrycznego w wyłączniku)
Keywords: circuit breaker, electric arc, ATP/EMTP programs
Słowa kluczowe: wyłącznik, łuk elektryczny, programy ATP/EMTP
Introduction
In general the simulation programs for transient
phenomena analysis in electric circuits contain the most
simplified models of switch in the form of ideal element. The
electric arc arising between contacts at the moment of their
separation has a significant influence on current interruption
in the circuit. It can both limit the current and determine the
moment of the current interruption. These phenomena are
ignored in the ideal switch which can cause a significant
difference between numerical simulation and real values.
The article presents a circuital model of the electric arc for
creation of switch models which allow consideration of the
arc’s influence on commutation process.
where: i - arc current, C, - constants, while =0,25÷0,5.
Considering above relations, it was assumed, that the
static characteristics of arc runs according to the following
relation:
u 
(3)
A
i
 1
B
i
where: A, B – constant coefficients. Fig. 5 provides an
example of this function.
For comparison Fig. 5 also shows relation (2) for C=A.
R
Q
Fig. 1. Basic structure of circuit-breaker model
Voltage-current characteristics of arc
It can be assumed that field intensity in arc column of
the long arc, for stabilized conditions of arc burning is
practically constant [2,3]. Hence arc voltage, disregarding
anode and cathode drop, can be expressed by the formula
[4]:
(1)
u  E l
where: E - electric field intensity in arc column, l - arc
column length.
Field intensity can by described as follows [3]:
(2)
74
arc voltage u
b
Circuit-breaker model
Basic structure of a circuit-breaker model is shown in
Fig. 4. It consists of two elements: an ideal switch Q and an
element R modeling the nonlinear resistance of electric arc.
The circuit-breaker model was built with the application
of components available in ATP (Alternative Transients
program) [1] with the following assumptions:
 arc is modelled basing on its static voltage-current
characteristics,
 arc model includes its simplified dynamic characteristics,
 arc resistance depends not only on the arc current but
also on the moment of contacts opening,
 arc suppression takes place when current becomes lower
than the determined threshold,
 re-ignition occurs when the recovery voltage exceeds the
determined threshold.
0
0
a
arc current i
Fig. 2. Comparison of arc functions: a – resulting from formula (3),
b – resulting from formula (2)
Both curves overlap - the more accurately the higher the
value of current is. For small value of currents the curve (2)
ascends unlimitedly, while characteristics (3) reaches its
maximum and follows towards zero. The differences
observed between characteristics (2) i (3) are not
substantial, taking into account that coefficient B in the
formula (3) was intended to reach maximum below the
current of arc suppression. In addition characteristics (3) is
continuous and bounded for every current value what helps
to eliminate problems with numerical instability.
Nonlinear resistance was used to model the arc in
circuit-breaker, and its value was based on formula (3) in
the following way:
(4)
u
A
Rs   
 1
i
i
B
where: Rs – static resistance of the arc.
The arc’s dynamic characteristics depends on rate of
current changes. Therefore the arc model contains an arc
resistance scaling factor, proportional to arc current
derivative:
(5)


d i
Rd  Rs   K1 
 1
dt


E  C  i
PRZEGLĄD ELEKTROTECHNICZNY (Electrical Review), ISSN 0033-2097, R. 86 NR 1/2010
where: Rd – dynamic resistance of the arc, K1 – factor of
proportionality.
The above relationship was shaped to obtain dynamic
resistance Rd equal to resistance Rs given by formula (4)
for constant value of current (derivative equal to 0).
Dependence between arc resistance and duration of
contacts separation was reached by introduction of
resistance scaling factor, related to time:
R  Rd  K 2 t 
(6)
It was assumed that function K2(t) can be described by:
0
t  t0

K 2 t   
 t  t 0   1 t  t 0
(7)
2.5
where:  - coefficient describing rate of resistance increase
resulting from arc elongation, t0 – the moment when
contacts start to separate.
Structure of arc model
a)
i
shows the diagram of the test circuit. It was assumed that
constant supplying voltage (uz=Uz=const.) was switched on
at the time t=0, and opening of the circuit breaker Q took
place at the time t=t0 when the current reached the steady
state. Simulation results are presented in Fig. 5 and 6. Fig.
5 shows arc voltage and arc current for a case when the
current is cut off. Cutting off of the current is possible if arc
characteristics is located above the characteristic of the
circuit. Both cases possible as per the above were
considered: the first one – when the arc characteristic
intersects the characteristics of the circuit and the other
one, when both characteristics have no intersections. Fig. 6
shows switch current in three cases: (a) the switch fails, (b)
the switch operates and the current is not broken, (c) the
switch operates and the current is cut off.
ui
Uz Iz
2
1.5
i
1
u
0.5
t0
0
R
0
S1
Rs
x
K1
d
dt
t
y
x
x y
Rd
0.03
x
y
1 yx y
K2
x y
R
i
Iz
1
0.04
0.05
b
0.5
t0
0
Fig. 3. Arc model: a – circuital part, b – analog part
The arc model shown in Fig. 6 consists of two parts:
circuital and analog. Circuital part (Fig. 6a) includes a nonlinear resistance R and an ideal switch S1. Resistance R
models arc and its present value is computed in the analog
part. Switch S1 breaks the circuit, simulating arc
suppression when arc current drops below certain
threshold. Closing operation takes place at the time of arc
ignition t0. Opening of the switch follows the simultaneous
fulfillment of the relations below:
i  I 0 , t  t0 ,
di
dt
0
where: t0 – the instant of circuit breaker opening, I0 – the
current of arc suppression. The analog part of the model,
shown in Fig. 6b is used to compute the arc resistance
according to the relations presented above.
R1
Q
t[s]
0.02
C1
C2
R2
Fig. 4. The circuit used to test the switch model with arc
The model presented in the previous paragraph was
tested in test cases described below. Results were
compared with the curves quoted in the literature. Fig. 4
0.04
0.05
u
i
Umax I max
uz
i
0.5
uz
0.03
II. AC circuit
Calculations were done for the circuit shown in Fig. 4. It was
assumed that voltage uz is sinusoidal function of time and
was switched on at time t=0. The switch Q opened at time
t0. Resistance and inductance values of the circuit were
accepted to reach the power factor close to unity.
Simulation was carried out for the case when the current in
the circuit was not cut off – switch S1 in Fig. 3a remains
closed. This means that the arc periodically gets
extinguished when current goes through zero and ignites
when voltage between switch contacts reaches proper
value. Results are shown in Fig. 7.
1
L2
0.01
Fig. 6. Switch current in the circuit from Fig. 4 computed for
constant supply for three cases: a - the switch fails, b - the switch
operates and the current is not broken, c - the switch operates and
the current is cut off.
1.5
Test cases
I. DC circuit
L1
a
c
0
(8)
t[s]
0.02
Fig 5. Computed curves in the circuit from Fig. 4 for constant
supply, after opening the switch: i – switch current, u - arc voltage
b)
i
0.01
0
0.02 t0
-0.5
u
0.04
t[s]
0.06
-1
Fig.7. Curves computed in the circuit in Fig. 4 for sinusoidal voltage
supply and power factor close to unity: uz – supplying voltage, i –
switch current, u - arc voltage
PRZEGLĄD ELEKTROTECHNICZNY (Electrical Review), ISSN 0033-2097, R. 86 NR 1/2010
75
The current curve contains sections when value of
current is practically equal to zero. The current falls down to
zero earlier then sinusoidal function and subsequently starts
growing only when the voltage reaches the value of striking
voltage. High spikes of striking voltage and lower spikes of
arc suppression voltage can be visibly noticed in the voltage
curve. Using computed curves of arc voltage and switch
current, which is also arc current, dynamic characteristic of
AC arc was plotted – Fig. 8. The characteristics contains
distinct loops named arc hysteresis loops.
u
0.4 U
max
0.2
-1
-0.5
0
0
0.5
-0.2
i 1
I max
Fig. 8. Dynamic characteristic of arc in the switch model in the
circuit in Fig. 4 computed for alternating current
Calculations in the simulated circuit were repeated for
the case when the inductive reactance dominates the
resistance. Results are shown in Fig. 9. If the arc resistance
is high, the arc voltage is also high in the whole half-period
and becomes a significant component of voltage balance in
the circuit. It changes the amplitude and the shape of the
current curve.
1
u
i
Umax I max
0
i
t0
0.03
0.05
t[s]
0.07
Another simulation was carried out for the case when,
after switch contacts separation, conditions preventing reignition of arc occurred and the current was cut off. It was
achieved by inserting additional condition into the model,
which enabled the switch S1 to open when the inequalities
(8) were fulfilled. It was assumed that opening of the switch
in the circuit in Fig. 4 took place in the steady state of short
circuit, for the maximal value of current. Results are
presented in Fig. 10.
-2
u
i
Umax I max
Fig. 11. Modification of the circuital part of the arc model to
simulate re-strike
The modified model of arc was tested in the circuit shown
in Fig. 4. The results are presented in Fig. 12.
u
i
Umax I max
0
0.02
-1
u
if
i
t[s]
uz
0.04
0.06
-3
Fig 12. Curves computed in the circuit from Fig. 4 for sinusoidal
supply, for inductive circuit and for a case when re-ignition
occurred: uz – supplying voltage, i – switch current, u - voltage
between contacts, if – current expected if switch failed
Summary
Simulation results are in good compatibility with curves
quoted in the literature [2,3]. Implementation of the switch
model as shown here, satisfactorily simulates electric arc
between switch contacts for cases considered. Increase of
simulations’ accuracy can be reached if the model is
extended by the function of break-down strength of the
post-arc column.
REFERENCES
i
t[s]
0.04
uz
0.06
if
u
-3
Fig 10. Curves computed in the circuit from Fig. 4 for sinusoidal
supply, for inductive circuit and for a case when the current is cut
off: uz – supplying voltage, i – switch current, u - voltage between
contacts, if – current expected if switch failed
76
S2
-2
Fig 9. Curves computed in the circuit from Fig. 4 for sinusoidal
supply and for inductive circuit: uz – supplying voltage, i – switch
current, u - arc voltage
0
0.02
-1
R
S1
1
u
-1
1
i
2
-0.5
2
where: t0 – time below which the switch stays opened, U0 –
re-striking voltage. Switch opening is conditioned by
relations (8). The cycle: closing/opening of the switch may
proceed repeatedly.
3
uz
0.5
u  U 0 , t  t0
(9)
-0.4
1.5
From the moment of switch contacts separation the
impact of arc voltage on the current becomes more and
more stronger. The current increasingly differs from a curve
expected for a case with closed switch. At the moment of
current cutting off, strongly damped oscillatory voltage
occurs between switch contacts, overlapping sinusoidal
supplying voltage. This voltage is known as recovery
voltage.
The switch S1 used in the arc model (Fig. 3a) has no
ability to close again after the opening conditioned by
relations (8). To simulate the re-strike, additional switch S2,
controlled by the voltage between its contacts was inserted
– Fig. 11.
Closing of the switch S2 takes place when the following
relations are fulfilled:
[1] ATP Roole Book, Lueven EMTP Center 1987
[2] C i o k . Z .: Procesy łączeniowe w układach
elektroenergetycznych. WNT Warszawa 1983
[3] D z i e r z b i c k i S .: Wyłączniki wysokonapięciowe prądu
przemiennego. WNT Warszawa 1966
[4] K r ó l i k o w s k i C .: Inżynieria łączenia obwodów elektrycznych
wielkich mocy. Wydawnictwo Politechniki Poznańskiej, Poznań
1998
Authors: dr inż. Jacek Horiszny, Politechnika Gdańska, Wydział
Elektrotechniki i Automatyki, ul. Narutowicza 11/12, 80-233 Gdańsk
e-mail: j.horiszny@ely.pg.gda.pl; dr inż. Mirosław Wołoszyn,
Politechnika Gdańska, Wydział Elektrotechniki i Automatyki, ul.
Narutowicza
11/12,
80-233
Gdańsk,
e-mail:
m.woloszyn@ely.pg.gda.pl
PRZEGLĄD ELEKTROTECHNICZNY (Electrical Review), ISSN 0033-2097, R. 86 NR 1/2010