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Overvoltages in reactive circuits due to current suppression
Electric Power Systems Research, 21 (1991) 225 - 229
225
Overvoltages in Reactive Circuits Due to Current Suppression D. O'KELLY, A. R. M, SALIH and B. BENSON
Department of Electrical Engineering, University of Bradford, Bradford BD7 1DP (U.K.) (Received February 21, 1991)
ABSTRACT
Serious overvoltages may be generated by current suppression (or current chopping) when switching out shunt reactors or unloaded trans. formers. Normalized overvoltage profiles are determined using (a) a single concentrated resistor and (b) a distributed resistance network to simulate eddy current loss in the magnetic core. These characteristics show that large errors may be present with the single resistor model if there is significant damping. Results are then given for a modified computer model including the effect of hysteresis and multiple current chopping and restriking.
relatively low peak values of c u r r e n t which can be chopped combine to produce innocuous overvoltage factors. The conditions necessary for the generation of dangerous overvoltages are invariably found in switching reactive circuits such as shunt reactors or unloaded transformers. The effects of different system parameters and non-linearity of the magnetic circuit have been examined by several authors, usually with the eddy current loss represented by a single-valued resistor across the inductive branch [2]. In this paper, the eddy current action is modelled accurately by using a multisection ladder network and overvoltage profiles are determined for both linear and non-linear core characteristics.
1. INTRODUCTION
The phenomenon of the generation of dangerous overvoltages across highly inductive circuits by the near-instantaneous interruption of supply cur r ent has been recognized for many years. The application of EHV airblast circuit breakers which could suppress or 'chop' currents up to about 20A highlighted the problem [1, 2]. More recently, vacuum switches and circuit breakers using SF~ as the interrupting medium have been investigated on this type of duty [3, 4]. Flashovers in 3.3kV induction motor terminal boxes have been attributed to cur r ent suppression overvoltages on switching over from one operating mode to anot her during the run-up period or when pole changing. Overvoltages produced by the current-chopping action of low voltage contactors have also been examined [5]. Current suppression may be present when interrupting highly reactive fault cur r ent by, for example, a gas-blast circuit breaker. However, the circuit parameters, together with the 0378-7796/91/$3.50
2. CIRCUIT MODELLING
When the current energizing an inductive circuit, such as an unloaded transformer, is suddenly suppressed, the energy trapped in the system oscillates between the magnetic and electric fields, producing an oscillating voltage across the circuit, which is a t t e n u a t e d by losses. Figure 1 shows a section of a typical coil system. The relationship between electric circuit quantities and electromagnetic field quantities is well known [6] and the coil system may be represented by the parallel combination of a magnetizing inductance L and a lumped-parameter resistance for the core loss (Fig. 2). Most of the published work is based on this equivalent circuit, where the following assumptions have been made: (a) winding resistance is neglected; (b) the leakage inductance of a t urn is negligible and the mutual flux is common to all turns and flows in the magnetic core; :c Elsevier Sequoia/Printed in The Netherlands
226 ~in4~'n 3
infer- i-urn
cor~
suJirch
--~.
leaka~
":i
iol-oo ~2
capactr~ nce ro ear kkl
flux
Fig. 1. Section of a coil system.
s~oi rck X
Cc
7 Fig. 2. Simple (lumped-parameter) electrical equivalent circuit.
Fig. 3. Distributed-parameter electrical equivalent circuit. (r =shunt resistance per unit width of lamination; n = number of 7r-sections.)
3. SIMULATION OF EDDY CURRENT LOSS At p o w e r f r e q u e n c y the flux d i s t r i b u t i o n in a l a m i n a t i o n is n e a r l y u n i f o r m a n d the effective s h u n t r e s i s t a n c e r e p r e s e n t i n g eddy curr e n t loss is R =6N2N~bp/al
(c) the d i s t r i b u t e d c a p a c i t a n c e e i t h e r bet w e e n t u r n s or to e a r t h m a y be c o m b i n e d into a single s h u n t c a p a c i t a n c e a c r o s s the inductive e l e m e n t L; (d) eddy c u r r e n t loss is r e p r e s e n t e d by a resistive b r a n c h R a c r o s s the L - C element. F o r a u n i f o r m l y d i s t r i b u t e d single-layer w i n d i n g with one end e a r t h e d , the e q u i v a l e n t s h u n t c a p a c i t a n c e C is equal to one-third of the t o t a l d i s t r i b u t e d c a p a c i t a n c e to e a r t h plus the net i n t e r - t u r n c a p a c i t a n c e . Lee [7] h a s s h o w n t h a t with a s s u m p t i o n (b) a n o n - u n i f o r m d i s t r i b u t e d c a p a c i t a n c e m a y still be represented by a single l u m p e d c a p a c i t a n c e , alt h o u g h its v a l u e is not n e c e s s a r i l y o n e - t h i r d of the t o t a l d i s t r i b u t e d c a p a c i t a n c e . T h e m a g n e t i c core c o m p r i s e s m a n y laminations w h i c h are thin c o m p a r e d w i t h the surface d i m e n s i o n s a n d e l e c t r i c a l l y i n s u l a t e d from e a c h other. End-effects m a y be n e g l e c t e d a n d the v a l u e of R is u s u a l l y d e t e r m i n e d from the eddy c u r r e n t loss at p o w e r f r e q u e n c y w h i c h a s s u m e s u n i f o r m flux d i s t r i b u t i o n in the l a m i n a t i o n s . H o w e v e r , s w i t c h i n g transients are u s u a l l y in the f r e q u e n c y r a n g e 500 Hz a n d a b o v e a n d the r e s u l t a n t eddy curr e n t a c t i o n m a k e s the flux density v a r y o v e r a l a m i n a t i o n . An a c c u r a t e m e t h o d of i n c l u d i n g this effect is included in the following section.
fl
(1)
At h i g h e r frequencies, the flux d i s t r i b u t i o n in a l a m i n a t i o n v a r i e s w i t h depth. An accur a t e r e p r e s e n t a t i o n of the eddy c u r r e n t phen o m e n o n is o b t a i n e d by u s i n g a ladder n e t w o r k (Fig. 3) for a l a m i n a t i o n . In t e r m s of d i s t r i b u t e d p a r a m e t e r s , r = 2N2NLbp/I
~/m
(2)
T h e a c c u r a c y of r e p r e s e n t a t i o n depends u p o n the n u m b e r of n-sections n. W i t h six sections, the loss w i t h u n i f o r m flux distribution is v e r y n e a r l y t h a t f o u n d w i t h a concent r a t e d r e s i s t a n c e R, b u t if n is less t h a n 6, the eddy c u r r e n t loss is o v e r e s t i m a t e d . It is seen from eqns. (1) a n d (2) t h a t w i t h n = 1 the e q u i v a l e n t l u m p e d r e s i s t a n c e across the L C circuit is o n l y 2 R / 3 .
4. LINEAR PARAMETERS A s s u m i n g c o n s t a n t v a l u e s of L a n d C in the e q u i v a l e n t circuit of Fig. 2, a n a l y t i c a l expressions m a y be derived for the s w i t c h i n g overv o l t a g e due to c u r r e n t chopping. W i t h little damping, v~(t) ~-- e x p ( - ~ t ) [ i c ( L / C ) 1/'~ sin(w, t)
+ IZ cos (p cos(~v, t)l
(~)
227
where ~'~1= (e~02 - ~2)1/2 ('~o = 1/(LC) 1;2,
~ = 1/2RC
C r i t i c a l d a m p i n g is p r e s e n t w h e n Rcrit
=
(1~2)(L/C) ~12
(4)
a n d t h e c o r r e s p o n d i n g v a l u e of the critical a t t e n u a t i o n f a c t o r is ~Xcrit :
]/(nc)1/2 :
U)0
(5)
T h e o v e r v o l t a g e e x p r e s s i o n s for c r i t i c a l a n d o v e r d a m p e d circuit c o n d i t i o n s are n o w non-oscillatory. T h e a t t e n u a t i o n factor, at, m a y be redefined in t e r m s of a n a n g l e 0, w h i c h is r e l a t e d to the n a t u r a l a n g u l a r f r e q u e n c y ~o0 by 0 = U~ot. If a d a m p i n g f a c t o r fl is defined s u c h t h a t exp( -- at) = exp( -- flO) then
fi =
d /(o o = ( 1 / 2 R C ) ( L C ) 1/2
t h a t is, fi
=
(L / C)I/2/2R
(6)
H e n c e , fi is a useful coefficient for defining a t t e n u a t i o n w h e n the c h a r a c t e r i s t i c s a r e normalized. It h a s a v a l u e of u n i t y for critical damping. 4.1. Overvoltage profiles Theoretical normalized overvoltage charact e r i s t i c s a r e s h o w n in Fig. 4, a s s u m i n g the eddy c u r r e n t loss to be r e p r e s e n t e d by (a) a single r e s i s t i v e e l e m e n t R, as s h o w n in Fig. 2;
1.0
(b) a d i s t r i b u t e d n e t w o r k (see Fig. 3). F o r t h e s e results, it w a s a s s u m e d t h a t (a) the initial flux d i s t r i b u t i o n in a l a m i n a t i o n was uniform; (b) the c u r r e n t t h r o u g h t h e s w i t c h w a s suppressed instantaneously; (c) the m a g n e t i z i n g i n d u c t a n c e was constant; (d) the s u p p l y v o l t a g e at t h e i n s t a n t of c h o p p i n g was zero. T h e n o r m a l i z i n g coefficients in Fig. 4 are p.u. v o l t a g e = i t ( L / C ) 1/2
(7a)
and p.u. t i m e = 2~/~o 0 = 2~(LC) 1/2
(7b)
M e s h c u r r e n t a n a l y s i s was used a n d for a t r a n s i e n t s o l u t i o n the s i m u l t a n e o u s first-order d i f f e r e n t i a l e q u a t i o n s were solved by a R u n g e - K u t t a f o u r t h - o r d e r m e t h o d [8]. Solutions w i t h different n u m b e r s of sections s h o w e d t h a t w i t h n = 10 or m o r e sections the r e s u l t s were v i r t u a l l y the same. H e n c e , t e n sections w e r e used for b o t h the c o m p u t e d a n d the e x p e r i m e n t a l r e s u l t s [8], w h i c h w e r e in good a g r e e m e n t . W i t h the a c c u r a t e r e p r e s e n t a t i o n , the flux d e n s i t y across a h a l f - l a m i n a t i o n v a r i e s conside r a b l y as the o v e r v o l t a g e builds up, as s h o w n in Fig. 5, w h e r e p.u. flux d e n s i t y c o r r e s p o n d s to the v a l u e at t = 0. 4.2. F i n i t e t r a n s f o r m e r voltage at current suppression T h e n o r m a l i z e d r e s u l t s in Fig. 4 a s s u m e t h a t the c u r r e n t is c h o p p e d a t the p e a k v a l u e of the s u p p l y c u r r e n t , t h a t is, w h e n the s u p p l y v o l t a g e is zero. If c u r r e n t c h o p p i n g o c c u r s a t
/~=0
].0 3
t~O
flux
o_
densil-~
>o
o
0,5
F i g . 4. N o r m a h z e d o v e r v o l t a g e p r o f i l e s for a r a n g e of fl values: , l u m p e d - p a r a m e t e r m o d e l ( e d d y c u r r e n t repr e s e n t e d b y r e s i s t a n c e R); . , distributed-parameter model.
Fig. 5. F l u x d e n s i t y d i s t r i b u t i o n i n a l a m i n a t i o n a t differe n t i n s t a n t s of t i m e for a n a t t e n u a t i o n f a c t o r /] = 0.5.
228
less than the peak supply current, there is a finite circuit voltage at the instant of current suppression. The solution now requires an initial charge on the capacitor and finite currents flowing in the resistors of the distributed network at cu rr e nt chopping (t = 0). The initial voltage distribution along the distributed network is not linear (this is due to the slight non-uniform flux distribution in the lamination due to eddy current action). Accurate initial conditions may be determined using the computational technique described in ref. 9. However, the assumption of a linear voltage distribution at zero time gives insignificant errors.
4.3. Air-gap cores Shunt reactors often have an air gap in the core to increase the (magnetizing) current. The equivalent circuit (Fig. 3) is readily modified using a concentrated linear inductor across the capacitor C to represent the air gap. The effect of eddy c ur r e nt action is now less dominant and reduces the difference between the voltage profiles using the single-valued resistor R and the distributed-network representation.
5. N O N - L I N E A R
do 0.4
~ i m ~ (p. ~ )
o.'~
F i g . 7. O v e r v o l t a g e p r o f i l e f o r a n o n - l i n e a r m o d e l w i t h current chopping; ~ • , computed values; - x , measured results.
TREATMENT
Hysteresis and saturation of the magnetic circuit considerably reduce the electromagnetic stored energy. Figure 6 shows a typical B / H curve for a transformer steel. The shaded area represents the stored energy available to produce an overvoltage for a value of chopped cu r r en t corresponding to He. The distributednetwork representation of the half-lamination
F i g . 6. F i e l d e n e r g y f o r a m a t e r i a l saturation.
now has non-lineal" inductance elements which are flmctions of current. An accurate computation of the overvoltage profile may be obtained by modifying the program used in §4.1 to have non-linear inductance elements using a computer simulation of the hysteresis loop [10]. An alternative approximate method is to assume a linear decrement of B with H after current chopping (shown as a chain curve in Fig. 6) and use the appropriate linear treatment of §4 to obtain the overvoltage profile. The extra computing time required for the accurate prediction of the overvoltage profile compared with the approximate technique is probably not justified since the B / H curve is often only representative.
with hysteresis
and
Figure 7 shows one typical result [8] which indicates reasonable agreement between computed and measured values.
6. M U L T I P L E
RESTRIKES
AND CHOPPING
The prospective overvoltage produced by current suppression may exceed the gap dielectric strength of the circuit breaker (contactor), causing gap breakdown or a restrike. In practice, multiple restriking may occur during the current-chopping phenomenon due to an insufficiently fast build-up of the gap dielectric strength. A representative computed result is shown in Fig. 8, assuming an arbitrary build-up of gap strength and linear circuit parameters. It is assumed that there is negligible time between arc reignition and the next current chop so t hat the electromagnetic energy is unchanged during the restrike- chop period. The technique can be extended to include non-linearity in the core material.
229
R r ,~'~,o Jr~,~ e
0
rim~
Fig. 8. Multiple restriking and chopping.
7. CONCLUSIONS
The paper gives normalized switching overvoltage profiles due to current suppression in circuits with linear parameters. A comparison of the results obtained with a single damping resistor to represent eddy current action and a distributed network shows that the simple representation is adequate with little damping but significant errors are present with higher attenuation. The modelling technique is then developed to include cores with hysteresis and saturation. Computed results are given for a switch with a slow increase of gap dielectric strength producing multiple restriking and chopping.
NOMENCLATURE
a B b C H ic L 1 N N,, n
half-thickness of lamination flux density, T width of lamination, m lumped capacitor value, F magnetizing force, A/m suppressed or chopped current magnetizing inductance (linear value), H length of magnetic circuit, m number of turns number of laminations in core number of sections in a half lamination
f/ vs fi p (p ~ (o0 (~)1
single-resistor representation of eddy current loss, f~ section resistor (representing eddy current loss), f~ crest supply voltage instantaneous switching overvoltage attenuation coefficient damping factor resistivity angle of voltage wave at current chop, rad supply angular frequency, rad/s natural angular frequency, rad/s natural angular frequency with damping, rad/s
REFERENCES 1 P. Baltensperger, Overvoltages due to the interruption of small inductive currents, CIGRE, 1950, Paper No. 116. 2 A. F. B. Young, Some researches on current chopping in high-voltage circuit-breakers, Proc. Inst. Electr. Eng., Part 2, 100 (1953) 337 - 360. 3 M. Murano, T. Fujii, H. Nishikawa, S. Nishiwaki and M. Okawa, Voltage escalation in interrupting inductive current by vacuum switches, I E E E Trans., P A S 93 (]974) 264 - 271. 4 N. Ueno, H. Toya, Y. Murai and M. Okada, Monte Carlo simulation of overvoltage generation in the inductive current interruption by vacuum interrupters, I E E E Trans., PAS-103 (1984) 498 - 505. 5 B. Mellitt, Transient voltages generated by inductive switching in control circuits, Proc. Inst. Electr. Eng., 121 (1974) 668. 6 G, Kron, Equivalent Circuits of Electric Machinery, Wiley, Chichester, U.K., 1951. 7 T. H. Lee, The effect of current chopping in circuit breakers on networks and transformers-- Part 1. Theoretical considerations, I E E E Trans., PAS-79 (1960) 534 - 544. 8 A. R. M. Salih, Current chopping in reactive circuits, M.Sc. Dissertation, Univ. Bradford, U.K., 1975. 9 D. O'Kelly, Flux penetration and losses in steel plate with sinusoidal magnetisation, Proc. Inst. Electr. Eng., Part B, 127 (1980) 287- 292. 10 D. O'Kelly, Simulation of transient and steady-state magnetisation characteristics with hysteresis, Proc. Inst. Electr. Eng., 124 (1977) 578- 582.
225
Overvoltages in Reactive Circuits Due to Current Suppression D. O'KELLY, A. R. M, SALIH and B. BENSON
Department of Electrical Engineering, University of Bradford, Bradford BD7 1DP (U.K.) (Received February 21, 1991)
ABSTRACT
Serious overvoltages may be generated by current suppression (or current chopping) when switching out shunt reactors or unloaded trans. formers. Normalized overvoltage profiles are determined using (a) a single concentrated resistor and (b) a distributed resistance network to simulate eddy current loss in the magnetic core. These characteristics show that large errors may be present with the single resistor model if there is significant damping. Results are then given for a modified computer model including the effect of hysteresis and multiple current chopping and restriking.
relatively low peak values of c u r r e n t which can be chopped combine to produce innocuous overvoltage factors. The conditions necessary for the generation of dangerous overvoltages are invariably found in switching reactive circuits such as shunt reactors or unloaded transformers. The effects of different system parameters and non-linearity of the magnetic circuit have been examined by several authors, usually with the eddy current loss represented by a single-valued resistor across the inductive branch [2]. In this paper, the eddy current action is modelled accurately by using a multisection ladder network and overvoltage profiles are determined for both linear and non-linear core characteristics.
1. INTRODUCTION
The phenomenon of the generation of dangerous overvoltages across highly inductive circuits by the near-instantaneous interruption of supply cur r ent has been recognized for many years. The application of EHV airblast circuit breakers which could suppress or 'chop' currents up to about 20A highlighted the problem [1, 2]. More recently, vacuum switches and circuit breakers using SF~ as the interrupting medium have been investigated on this type of duty [3, 4]. Flashovers in 3.3kV induction motor terminal boxes have been attributed to cur r ent suppression overvoltages on switching over from one operating mode to anot her during the run-up period or when pole changing. Overvoltages produced by the current-chopping action of low voltage contactors have also been examined [5]. Current suppression may be present when interrupting highly reactive fault cur r ent by, for example, a gas-blast circuit breaker. However, the circuit parameters, together with the 0378-7796/91/$3.50
2. CIRCUIT MODELLING
When the current energizing an inductive circuit, such as an unloaded transformer, is suddenly suppressed, the energy trapped in the system oscillates between the magnetic and electric fields, producing an oscillating voltage across the circuit, which is a t t e n u a t e d by losses. Figure 1 shows a section of a typical coil system. The relationship between electric circuit quantities and electromagnetic field quantities is well known [6] and the coil system may be represented by the parallel combination of a magnetizing inductance L and a lumped-parameter resistance for the core loss (Fig. 2). Most of the published work is based on this equivalent circuit, where the following assumptions have been made: (a) winding resistance is neglected; (b) the leakage inductance of a t urn is negligible and the mutual flux is common to all turns and flows in the magnetic core; :c Elsevier Sequoia/Printed in The Netherlands
226 ~in4~'n 3
infer- i-urn
cor~
suJirch
--~.
leaka~
":i
iol-oo ~2
capactr~ nce ro ear kkl
flux
Fig. 1. Section of a coil system.
s~oi rck X
Cc
7 Fig. 2. Simple (lumped-parameter) electrical equivalent circuit.
Fig. 3. Distributed-parameter electrical equivalent circuit. (r =shunt resistance per unit width of lamination; n = number of 7r-sections.)
3. SIMULATION OF EDDY CURRENT LOSS At p o w e r f r e q u e n c y the flux d i s t r i b u t i o n in a l a m i n a t i o n is n e a r l y u n i f o r m a n d the effective s h u n t r e s i s t a n c e r e p r e s e n t i n g eddy curr e n t loss is R =6N2N~bp/al
(c) the d i s t r i b u t e d c a p a c i t a n c e e i t h e r bet w e e n t u r n s or to e a r t h m a y be c o m b i n e d into a single s h u n t c a p a c i t a n c e a c r o s s the inductive e l e m e n t L; (d) eddy c u r r e n t loss is r e p r e s e n t e d by a resistive b r a n c h R a c r o s s the L - C element. F o r a u n i f o r m l y d i s t r i b u t e d single-layer w i n d i n g with one end e a r t h e d , the e q u i v a l e n t s h u n t c a p a c i t a n c e C is equal to one-third of the t o t a l d i s t r i b u t e d c a p a c i t a n c e to e a r t h plus the net i n t e r - t u r n c a p a c i t a n c e . Lee [7] h a s s h o w n t h a t with a s s u m p t i o n (b) a n o n - u n i f o r m d i s t r i b u t e d c a p a c i t a n c e m a y still be represented by a single l u m p e d c a p a c i t a n c e , alt h o u g h its v a l u e is not n e c e s s a r i l y o n e - t h i r d of the t o t a l d i s t r i b u t e d c a p a c i t a n c e . T h e m a g n e t i c core c o m p r i s e s m a n y laminations w h i c h are thin c o m p a r e d w i t h the surface d i m e n s i o n s a n d e l e c t r i c a l l y i n s u l a t e d from e a c h other. End-effects m a y be n e g l e c t e d a n d the v a l u e of R is u s u a l l y d e t e r m i n e d from the eddy c u r r e n t loss at p o w e r f r e q u e n c y w h i c h a s s u m e s u n i f o r m flux d i s t r i b u t i o n in the l a m i n a t i o n s . H o w e v e r , s w i t c h i n g transients are u s u a l l y in the f r e q u e n c y r a n g e 500 Hz a n d a b o v e a n d the r e s u l t a n t eddy curr e n t a c t i o n m a k e s the flux density v a r y o v e r a l a m i n a t i o n . An a c c u r a t e m e t h o d of i n c l u d i n g this effect is included in the following section.
fl
(1)
At h i g h e r frequencies, the flux d i s t r i b u t i o n in a l a m i n a t i o n v a r i e s w i t h depth. An accur a t e r e p r e s e n t a t i o n of the eddy c u r r e n t phen o m e n o n is o b t a i n e d by u s i n g a ladder n e t w o r k (Fig. 3) for a l a m i n a t i o n . In t e r m s of d i s t r i b u t e d p a r a m e t e r s , r = 2N2NLbp/I
~/m
(2)
T h e a c c u r a c y of r e p r e s e n t a t i o n depends u p o n the n u m b e r of n-sections n. W i t h six sections, the loss w i t h u n i f o r m flux distribution is v e r y n e a r l y t h a t f o u n d w i t h a concent r a t e d r e s i s t a n c e R, b u t if n is less t h a n 6, the eddy c u r r e n t loss is o v e r e s t i m a t e d . It is seen from eqns. (1) a n d (2) t h a t w i t h n = 1 the e q u i v a l e n t l u m p e d r e s i s t a n c e across the L C circuit is o n l y 2 R / 3 .
4. LINEAR PARAMETERS A s s u m i n g c o n s t a n t v a l u e s of L a n d C in the e q u i v a l e n t circuit of Fig. 2, a n a l y t i c a l expressions m a y be derived for the s w i t c h i n g overv o l t a g e due to c u r r e n t chopping. W i t h little damping, v~(t) ~-- e x p ( - ~ t ) [ i c ( L / C ) 1/'~ sin(w, t)
+ IZ cos (p cos(~v, t)l
(~)
227
where ~'~1= (e~02 - ~2)1/2 ('~o = 1/(LC) 1;2,
~ = 1/2RC
C r i t i c a l d a m p i n g is p r e s e n t w h e n Rcrit
=
(1~2)(L/C) ~12
(4)
a n d t h e c o r r e s p o n d i n g v a l u e of the critical a t t e n u a t i o n f a c t o r is ~Xcrit :
]/(nc)1/2 :
U)0
(5)
T h e o v e r v o l t a g e e x p r e s s i o n s for c r i t i c a l a n d o v e r d a m p e d circuit c o n d i t i o n s are n o w non-oscillatory. T h e a t t e n u a t i o n factor, at, m a y be redefined in t e r m s of a n a n g l e 0, w h i c h is r e l a t e d to the n a t u r a l a n g u l a r f r e q u e n c y ~o0 by 0 = U~ot. If a d a m p i n g f a c t o r fl is defined s u c h t h a t exp( -- at) = exp( -- flO) then
fi =
d /(o o = ( 1 / 2 R C ) ( L C ) 1/2
t h a t is, fi
=
(L / C)I/2/2R
(6)
H e n c e , fi is a useful coefficient for defining a t t e n u a t i o n w h e n the c h a r a c t e r i s t i c s a r e normalized. It h a s a v a l u e of u n i t y for critical damping. 4.1. Overvoltage profiles Theoretical normalized overvoltage charact e r i s t i c s a r e s h o w n in Fig. 4, a s s u m i n g the eddy c u r r e n t loss to be r e p r e s e n t e d by (a) a single r e s i s t i v e e l e m e n t R, as s h o w n in Fig. 2;
1.0
(b) a d i s t r i b u t e d n e t w o r k (see Fig. 3). F o r t h e s e results, it w a s a s s u m e d t h a t (a) the initial flux d i s t r i b u t i o n in a l a m i n a t i o n was uniform; (b) the c u r r e n t t h r o u g h t h e s w i t c h w a s suppressed instantaneously; (c) the m a g n e t i z i n g i n d u c t a n c e was constant; (d) the s u p p l y v o l t a g e at t h e i n s t a n t of c h o p p i n g was zero. T h e n o r m a l i z i n g coefficients in Fig. 4 are p.u. v o l t a g e = i t ( L / C ) 1/2
(7a)
and p.u. t i m e = 2~/~o 0 = 2~(LC) 1/2
(7b)
M e s h c u r r e n t a n a l y s i s was used a n d for a t r a n s i e n t s o l u t i o n the s i m u l t a n e o u s first-order d i f f e r e n t i a l e q u a t i o n s were solved by a R u n g e - K u t t a f o u r t h - o r d e r m e t h o d [8]. Solutions w i t h different n u m b e r s of sections s h o w e d t h a t w i t h n = 10 or m o r e sections the r e s u l t s were v i r t u a l l y the same. H e n c e , t e n sections w e r e used for b o t h the c o m p u t e d a n d the e x p e r i m e n t a l r e s u l t s [8], w h i c h w e r e in good a g r e e m e n t . W i t h the a c c u r a t e r e p r e s e n t a t i o n , the flux d e n s i t y across a h a l f - l a m i n a t i o n v a r i e s conside r a b l y as the o v e r v o l t a g e builds up, as s h o w n in Fig. 5, w h e r e p.u. flux d e n s i t y c o r r e s p o n d s to the v a l u e at t = 0. 4.2. F i n i t e t r a n s f o r m e r voltage at current suppression T h e n o r m a l i z e d r e s u l t s in Fig. 4 a s s u m e t h a t the c u r r e n t is c h o p p e d a t the p e a k v a l u e of the s u p p l y c u r r e n t , t h a t is, w h e n the s u p p l y v o l t a g e is zero. If c u r r e n t c h o p p i n g o c c u r s a t
/~=0
].0 3
t~O
flux
o_
densil-~
>o
o
0,5
F i g . 4. N o r m a h z e d o v e r v o l t a g e p r o f i l e s for a r a n g e of fl values: , l u m p e d - p a r a m e t e r m o d e l ( e d d y c u r r e n t repr e s e n t e d b y r e s i s t a n c e R); . , distributed-parameter model.
Fig. 5. F l u x d e n s i t y d i s t r i b u t i o n i n a l a m i n a t i o n a t differe n t i n s t a n t s of t i m e for a n a t t e n u a t i o n f a c t o r /] = 0.5.
228
less than the peak supply current, there is a finite circuit voltage at the instant of current suppression. The solution now requires an initial charge on the capacitor and finite currents flowing in the resistors of the distributed network at cu rr e nt chopping (t = 0). The initial voltage distribution along the distributed network is not linear (this is due to the slight non-uniform flux distribution in the lamination due to eddy current action). Accurate initial conditions may be determined using the computational technique described in ref. 9. However, the assumption of a linear voltage distribution at zero time gives insignificant errors.
4.3. Air-gap cores Shunt reactors often have an air gap in the core to increase the (magnetizing) current. The equivalent circuit (Fig. 3) is readily modified using a concentrated linear inductor across the capacitor C to represent the air gap. The effect of eddy c ur r e nt action is now less dominant and reduces the difference between the voltage profiles using the single-valued resistor R and the distributed-network representation.
5. N O N - L I N E A R
do 0.4
~ i m ~ (p. ~ )
o.'~
F i g . 7. O v e r v o l t a g e p r o f i l e f o r a n o n - l i n e a r m o d e l w i t h current chopping; ~ • , computed values; - x , measured results.
TREATMENT
Hysteresis and saturation of the magnetic circuit considerably reduce the electromagnetic stored energy. Figure 6 shows a typical B / H curve for a transformer steel. The shaded area represents the stored energy available to produce an overvoltage for a value of chopped cu r r en t corresponding to He. The distributednetwork representation of the half-lamination
F i g . 6. F i e l d e n e r g y f o r a m a t e r i a l saturation.
now has non-lineal" inductance elements which are flmctions of current. An accurate computation of the overvoltage profile may be obtained by modifying the program used in §4.1 to have non-linear inductance elements using a computer simulation of the hysteresis loop [10]. An alternative approximate method is to assume a linear decrement of B with H after current chopping (shown as a chain curve in Fig. 6) and use the appropriate linear treatment of §4 to obtain the overvoltage profile. The extra computing time required for the accurate prediction of the overvoltage profile compared with the approximate technique is probably not justified since the B / H curve is often only representative.
with hysteresis
and
Figure 7 shows one typical result [8] which indicates reasonable agreement between computed and measured values.
6. M U L T I P L E
RESTRIKES
AND CHOPPING
The prospective overvoltage produced by current suppression may exceed the gap dielectric strength of the circuit breaker (contactor), causing gap breakdown or a restrike. In practice, multiple restriking may occur during the current-chopping phenomenon due to an insufficiently fast build-up of the gap dielectric strength. A representative computed result is shown in Fig. 8, assuming an arbitrary build-up of gap strength and linear circuit parameters. It is assumed that there is negligible time between arc reignition and the next current chop so t hat the electromagnetic energy is unchanged during the restrike- chop period. The technique can be extended to include non-linearity in the core material.
229
R r ,~'~,o Jr~,~ e
0
rim~
Fig. 8. Multiple restriking and chopping.
7. CONCLUSIONS
The paper gives normalized switching overvoltage profiles due to current suppression in circuits with linear parameters. A comparison of the results obtained with a single damping resistor to represent eddy current action and a distributed network shows that the simple representation is adequate with little damping but significant errors are present with higher attenuation. The modelling technique is then developed to include cores with hysteresis and saturation. Computed results are given for a switch with a slow increase of gap dielectric strength producing multiple restriking and chopping.
NOMENCLATURE
a B b C H ic L 1 N N,, n
half-thickness of lamination flux density, T width of lamination, m lumped capacitor value, F magnetizing force, A/m suppressed or chopped current magnetizing inductance (linear value), H length of magnetic circuit, m number of turns number of laminations in core number of sections in a half lamination
f/ vs fi p (p ~ (o0 (~)1
single-resistor representation of eddy current loss, f~ section resistor (representing eddy current loss), f~ crest supply voltage instantaneous switching overvoltage attenuation coefficient damping factor resistivity angle of voltage wave at current chop, rad supply angular frequency, rad/s natural angular frequency, rad/s natural angular frequency with damping, rad/s
REFERENCES 1 P. Baltensperger, Overvoltages due to the interruption of small inductive currents, CIGRE, 1950, Paper No. 116. 2 A. F. B. Young, Some researches on current chopping in high-voltage circuit-breakers, Proc. Inst. Electr. Eng., Part 2, 100 (1953) 337 - 360. 3 M. Murano, T. Fujii, H. Nishikawa, S. Nishiwaki and M. Okawa, Voltage escalation in interrupting inductive current by vacuum switches, I E E E Trans., P A S 93 (]974) 264 - 271. 4 N. Ueno, H. Toya, Y. Murai and M. Okada, Monte Carlo simulation of overvoltage generation in the inductive current interruption by vacuum interrupters, I E E E Trans., PAS-103 (1984) 498 - 505. 5 B. Mellitt, Transient voltages generated by inductive switching in control circuits, Proc. Inst. Electr. Eng., 121 (1974) 668. 6 G, Kron, Equivalent Circuits of Electric Machinery, Wiley, Chichester, U.K., 1951. 7 T. H. Lee, The effect of current chopping in circuit breakers on networks and transformers-- Part 1. Theoretical considerations, I E E E Trans., PAS-79 (1960) 534 - 544. 8 A. R. M. Salih, Current chopping in reactive circuits, M.Sc. Dissertation, Univ. Bradford, U.K., 1975. 9 D. O'Kelly, Flux penetration and losses in steel plate with sinusoidal magnetisation, Proc. Inst. Electr. Eng., Part B, 127 (1980) 287- 292. 10 D. O'Kelly, Simulation of transient and steady-state magnetisation characteristics with hysteresis, Proc. Inst. Electr. Eng., 124 (1977) 578- 582.