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 Rs 1 i i B where: Rs – 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 Rd Rs K1 1 dt E C i PRZEGLĄD ELEKTROTECHNICZNY (Electrical Review), ISSN 0033-2097, R. 86 NR 1/2010 where: Rd – dynamic resistance of the arc, K1 – factor of proportionality. The above relationship was shaped to obtain dynamic resistance Rd equal to resistance Rs 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 Rd 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. ui Uz Iz 2 1.5 i 1 u 0.5 t0 0 R 0 S1 Rs x K1 d dt t y x x y Rd 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