Dynamic Stability Evaluation for an Electromagnetic Contactor

Transcription

1 Dynamc Stablty Evaluaton for an Electromagnetc Contactor Cheh-Tsung Ch *1 Chh-Yu Hung * Department of Electrcal Engneerng Chenkuo Technology Unversty No. 1, Cheh Shou N. Rd., Changhua Cty 5, Tawan, R.O.C *1 jh@cc.ctu.edu.tw, * hcy@cc.ctu.edu.tw Abstract: - Ths paper ams at researchng the relatonshp between the dynamc stablty of an electromagnetc contactor (abbrevated as contactor below) and the magnetc energy stored n the contactor. Explored contactors are served as an equvalent nonlnear and tme-varyng nductor. Some basc and mportant characterstcs of the contactor are theoretcally researched and dscussed. Several crtcal results are found durng closng phase and voltage sag events occurred smply by usng the basc defntons of passvty condtons theorem. By means of magnetc crcut analyzng method, a mathematc model and ts numercal computer program of a contactor are frst establshed n our laboratory. Therefore, some typcal and mportant experments such as the dynamc stablty of contactor durng the closng process and voltage-sag events are carred out by usng ths model. Later, these smulaton results are compared wth the expermental results for valdatng the correctness of ths contactor model too. The fnal comparng results are n agreement wth well. For purpose of further to know the voltage senstvty of the expermental contactor, the CBEMA curves of contactor are used and obtaned as well. Fnally, the dynamc stablty of the contactor durng closng process and voltage sags s examned n terms of the energy stored through the smulaton process of the contactor. Lots of smulaton and expermental results make sure that the contactor model s correct and proposed dynamc stablty evaluaton method s effectve. Key-Words: - CBEMA curve, passvty energy, contactor, voltage sag, dynamc stablty 1 Introducton Contactors are devces that are very senstve to power dsturbances lke voltage sags [1][]. Untl now, these devces have been wdely used n many ndustral applcatons. Contactors often act as swtches n a varety of electrcal systems for both power and control purposes. Durng power dsturbances, for example, voltage-sags events, f the accdental events occurred; the contactors have possbly been dsengaged because the power lne voltage s lower than the normal voltage value. As a result of the accdental events produced by the power dsturbances lke voltage sags s outsde the system s control and are random, therefore, the resultng dsengagements may lead to an uncontrolled and possbly expensve process shutdown. In ths paper, for brevty, the voltage sags wll be characterzed by the sag ampltude, duraton, and ts pont-n-wave where a smulated voltage-sag event occurs, and the effect of the voltage sag on the performance of contactor wll be studed and demonstrated the results by usng the CBEMA curves [3]. In addton, there s another unstable state for a contactor often occurs when t s ntated by an ac voltage source and stuated n the closng process. In the normal closng process of contactor, there are two knds of forces, such as the magnetc force and the sprng force, act on the movable part of contactor at the same tme. So that, the movement and the movng drecton of the movable part of contactor that follows the net force or the dfference of these two forces. Indeed, f the electrcal character of contactors s consdered, contactors are nonlnear and tmevaryng elements. There s a general fact descrbes that any nonlnear passve tme-varyng RLC network made of flux-controlled nductors s a stable network. In other words, the energy stored n nductors wll decrease as long as the resstve network absorbs energy. In many applcatons ths process wll force the stored energy to go to zero as the tme tends to nfnty. Hence, for the purpose of nvestgatng the dynamc stablty property of contactor durng closng process and voltage sag event occurred, the characterstcs of contactor stuated n these states wll respectvely be consdered n terms of the energy stored n the contactor by usng the poston versus tme varyng curves and the flux lnkage versus nstantaneous col ISSN: Issue 11, Volume 9, November 1

2 current varyng curves (here, the col current and armature poston are assumed as two ndependent varables). Mathematcal Models Contactors are typcal electromechancal devce. Fg. 1 ndcates that the basc structure of a popular electromagnetc contactor s parttoned nto three portons, such as the electrcal porton, the magnetc energy-converson porton, and the mechancal porton. For smplfyng the theoretcal analyss, the loss ncluded n the magnetc energy-converson porton s assumed to be consdered n the electrc porton and mechancal porton, respectvely. Therefore, the magnetc energy-converson porton can be therefore vewed drectly as a lossless and conservatve sub-system. It s also referred to as a lossless magnetc couplng system. Based on the law of conservaton [4] n physcs, the energy balance relatonshp nsde an electromagnetc contactor system s gven by W = W + W + W (1) e f m h where W e : The nput electrcal energy s suppled by the external voltage source u ( ; W f : The magnetc couplng energy s stored n the magnetc energy-converson sub-system; W m : The total work done s dsspated by the mechancal sub-system; W h : An amount of heatng loss s dsspated n all the system..1 Electrcal model The external voltage source u ( supples to the electrc energy over a tme nterval. By Krchhoff s voltage law and work-energy prncple, we derve equatons () and (3) as follows, dλ u( = r+ () dw e = u (3) where, the symbol, λ, represents the flux lnkage of contactor col. As we known, the external voltage source u ( equals the number of wndngs of col N tmes the dervatve of flux wth respect to tme, that s N( dφ ). Substtute u ( nto (3), we obtan dw e = N dφ (4) When the electromagnetc force F e produced by the magnetc crcut mposed upon the armature and larger than the sprng ant-force, the armature certanly begns movng forward the fxed ron core and the dsplacement s assumed to be dx. The dfferental work done on the armature by the mechancal system can be gven as follows dwm = Fe dx (5) Durng the closng process, the amount of the flux n the magnetc crcut and the col-current value are almost not affected [5]. Ths phenomenon means that the nstantaneous change of total flux and col current approxmates zero. Equaton (5) yelds the followng form. F dx= (6) e dw f The smplfed expresson shown n (6) ndcates the dfferental work done by the electromagnetc force that s determned by the decrease n the energy stored n magnetc couplng feld. The feld energy stored n the lossless magnetc energy converson system commonly equals the stored energy n the nductor. It can be descrbed by Fg. 1. Sketch the equvalent confguraton of an electromagnetc contactor. 1 W f = L( x) (7) Substtutng (7) nto (6), the electromagnetc force dynamcally mposed upon on the armature can be determned by the col current and the dervatve of ISSN: Issue 11, Volume 9, November 1

3 the self-nductance wth respect to armature dsplacement and shown as follows: F e dw f 1 dl( x) = = (8) dx dx In theory, the basc characterstc of self-nductance L (x) vares nversely proportonal wth the armature dsplacement x, so that the electromagnetc force shown n (8) llustrates that the value of electromagnetc force s a functon of the square of col current and the change of selfnductance wth respect to the armature dsplacement. Obvously, the change of electromagnetc force forced on the armature s always a nonlnear value durng the closng process.. Mechancal model The mechansm appearance of the electromagnetc contactor shown n Fg. conssts of an armature and a fxed ron core. One complete closng course of the electromagnetc contactor can bascally be dvded nto two stages: the frst stage s the state transton from the ntal openng poston to the contacts closng state. The movable ron core s normally lnked wth one or trple sets of contacts moves toward the fxed ron core by smple mechansm. If the electromagnetc force overcomes the sprng ant-force, the armature or the movable contacts moves forward to the drecton of fxed ron core. After the movable contacts have engaged wth the fxed contacts, the second stage begns. The armature contnues movng toward the fxed ron core untl they are actually closed wth the fxed ron core. Durng ths stage, the ant-force produced by the return sprng wll be ncorporated nto the tenson force of contacts sprng. Therefore, the mechancal model should be consdered by two stages, respectvely [5,6]...1 Frst stage: ( d A d C ) x d A The symbol d C s the aperture between the movable contacts and the fxed contacts. In contrast, the symbol d A represents the aperture between the movable ron core of armature and the fxed ron core. By usng the Newton s law of moton, the mathematcal model of mechancal system of contactor can be expressed as follows: F = ma (9) The resultant force F s equvalent to the armature mass multples ts acceleraton and t s the addton of three dfferent forces, such as electromagnetc force, sprng ant-force and vscous frcton force. The movng acceleraton tem n (9) can be descrbed as the second order tme dfferentaton of armature dsplacement. Equaton (9) can be rearranged and shown as follows: d x F = Fe F f Fb = m (1) where F : electromagnetc force; e F : counter force produced by return sprng; f F b : vscous frcton force; a : movng acceleraton of armature; m : armature mass; x : armature dsplacement. Fg.. Show the basc mechansm of an electromagnetc contactor. The representaton of an electromagnetc force shown n (8) s a functon of the col current and the armature dsplacement, that s F e = f (, x). The sprng tenson force vares proportonally wth the change of armature dsplacement. Ths ant force s come from the two return sprngs s assumed to be expresses as F f = K1x+ a1, where a 1 s a constant ntal pressure force of the sprng system mposed upon the armature. Furthermore, the vscous frcton force F b, n prncple, s a functon of the movng velocty of armature durng the consdered tme nterval. The representaton of ths tem s dx represented as F b = b, where the symbol b s constant coeffcent. The armature mass s the contrbuton of the mass of movable ron core, m, A ISSN: Issue 11, Volume 9, November 1

4 and the mass of one or three movable contacts mass, m. C.. Second stage: x ( d A dc ) Durng the second stage of mechancal moton, movable contacts would be engaged wth fxed contacts frst, and then the armature contnues to move toward the fxed ron core. As a result of the dsappearance of the contact aperture, ths movng process s also called as the super path of the armature. As stated n the prevous secton, the total counter force mposed upon the armature s the addton of the former workng stage and the tenson force of contacts sprng. It can be wrtten as below: F f = K + 3K ) x+ a 3d (11) ( 1 1+ where K s the elastc coeffcent of three contacts sprng. d s the ntal pressure of contacts sprng forced on the armature. Durng the super path of armature, the movable ron core unquely contnues to move toward the fxed ron core, but not all the armature mechansm. Therefore, the total armature mass s only equvalent to the mass of movable ron core m. A 3 Dynamc Behavor In fact, the dynamc behavor of contactor s energy transton from the electrcal power to the mechancal movement. Almost the basc closng process of contactor, the energy transton process s a nonlnear and tme-varyng as mentoned earler [7]. In the followng, we wll dscuss the correspondng property varatons n the respectve electro-magnet part and mechancal part durng transent events occurred. 3.1 Electro-magnet part Fg. 3 shows the equvalent electrcal crcut of the contactor from the vew pont of ts col. By employng the Krchhoff s voltage laws (KVL), the appled col voltage of contactor should be dropped on each load n the same electrcal loop. Respectvely, the voltage drop terms occur due to the col resstance, col nductance, dλ (= f L ( x, ) ), and the moton voltage, e (= f v ( x, ) ), and so on. It s noted that the latter two terms are always vared wth the poston or dsplacement of movable-part and the value of col current. u( r f L (x,) f V (x,) Fg. 3. Equvalent electrcal crcut as the contactor s energzed by a voltage source. The equatons of moton are obtaned for the contactor operatng n the closng phase can be derved by a straghtforward applcaton of Kchhoff s voltage law on a generalzed coordnates. The contactor can be theoretcally descrbed by a set of four dfferental equatons and gven n the followng: dλ u( = r+ = U d x F = m dx = v 1 Fe = RMS dl dx sn( wt+ϕ) (1) where the symbols used n (1) are defned as follows, respectvely: U RMS : s the root-mean-square value of the appled col voltage source, r : s the resstance of the col, m : s the mass of the movable part, w : s the angular frequency of ac source, t : s the operatng tme, ϕ : s the ntal phase angle of appled AC voltage, F : s the occurred ant-force due to the return f sprngs, F e : s the electromagnetc force, whch acts on the movable part or armature, λ : s the flux lnkage of col, ts value s equal to the col nductance multpled by the nstantaneous col current or the number of col wndngs multpled by the magnetc flux, one can wrte as follows: λ= L ( x) = N φ (13) ISSN: Issue 11, Volume 9, November 1

5 As we known, the movng velocty of armature or movable part equals the tme rate of change of poston, v= dx. By usng the chan rule, lots of valuable nformaton could be found. From () or (1), dl = ( dl dx)( dx ) = v( dl dx), therefore, the loop voltage equaton shown n (7) can be rewrtten as: d dl u ( = r+ L + v (14) dx where the r term represents the ohmc voltage drops across the col, the L d term represents the nductve voltage drop n a col, and the v dl dx term represents the motonal voltage drop produced by the movng velocty of armature and t can also be rewrtten as follows: dl( x) e= v dx (15) = fv ( x, ) Note that the motonal voltage of the movable part can be expressed n terms of ts velocty, nstantaneous col current, and the poston rate of the change of the col nductance. As seen n (15), the motonal voltage e s a functon of the value of the movng velocty or armature poston and the nstantaneous current value whch flows n the col. In the followng, the transent behavors of the contactor worked n the closng phase and the close phase are gong to be studed, respectvely, but the openng phase wll not be dscussed here because the latter phase does not concern wth our nvestgatng topc Closng phase Dfferentates (13) wth tme and the chan rule s ntroduced n basc dfferental theory, the results can be wrtten by dλ d = d = L + [ L( ( ] dl (16) We may nterpret the rght-hand sde of (16) as follows. The term L ( ( d / ) gves the voltage at tme t across a tme-nvarant nductor whose nductance s equal to the number L (. In the second term of (16) the voltage at tme t s proportonal to the current at tmet ; hence t can be nterpreted as the voltage across a lnear tmevaryng resstor whose resstance s equal to L ɺ (. The seres connecton of these elements descrbes the voltage-current relaton of a lnear tme-varyng nductor. Clearly, f L ɺ ( s negatve, we have an actve resstor; thus, t s necessary that L ɺ ( be nonnegatve for the tme-varyng nductor to be passve (namely t means that never delvers power to the outsde world). Accordng to (14), we realze that the relaton between the appled col voltage and nstantaneous col current s tme-varyng and nonlnear durng the closng process of contactor. Consequently, t s a very dffcult for people to accurately predct the exact dynamc behavours of contactor durng ths phase stage Close phase Once the workng status of contactor has entered nto the close phase, the value of the col nductance becomes a constant as well as the motonal voltage s zero. Because that the behavor of the movable part worked n ths phase s statonary, so that, the equaton gven n (14) and (15) can be rearranged and wrtten usng as follows: dλ d = L, = e l (17) A general expresson for the contactor col, as seen n (1), can be further expressed by the followng smplfed form: d u ( = r+ L (18) Clearly, (18) s a one order dfferental equaton and the soluton of the nstantaneous col current certanly s an exponental form: t u ( = (1 e τ ) (19) r Physcally, (19) means that the value of nstantaneous col current should exponentally ncrease and become a constant value after approxmately fve tmes tme constant, τ (= L r ). 3. Mechancal part In general, the mechancal part of the contactor s conssts of movable part, statonary electromagnet, two return sprngs, and one or three contacts sprngs. In fact, the mechansm of contactor s really equvalent to a mass-sprng-damp system. Therefore the dynamc behavours of the mechancal part can ISSN: Issue 11, Volume 9, November 1

6 be naturally represented by a second order ordnary dfferental equaton and lke as follows. d x dx F e = m + B + kx () Most of the workng tme of the contactor, the vscous coeffcent B s small, so that t s reasonable f t s gnored n our research work. For brevty, the sprng tenson term, whch acts on the movable part, s defned as F here. Fnally, () can be smplfed and gven by: F F d x = m = F e f f (1) In case of the contactor col s energzed by an external voltage source, the general expresson of the electromagnetc force can be wrtten as follows: 1 dl( x) F e = () dx Note that the electromagnetc force s a functon of the square nstantaneous value of the col current and the armature poston rate of change of the col nductance. Strctly speakng, the value of the electromagnetc force s dynamcally vared wth the poston of movable part. If there s a movement s produced by the movable part, the value electromagnetc force should change as well. 4 Passvty and Stablty Consder a one-port whch nclude nonlnear and tme-varyng elements. Let us drve the one-port by a generator, as shown n Fg. 4 [8]. From basc physcs then we know that nstantaneous power enterng the one-port at tme t s p( = e ( ( (3) l and the energy delvered to the one-port from tme t to t s t t W ( t, = p( t ) = el ( t ) ( t ) (4) t Snce the explorng target s gong to be a contactor n ths paper and ts characterstc n nature s nonlnear tme-varyng nductor, and the characterstc s gven for each t by a curve smlar t to that shown n Fg. 5, the curves changes as t vares. For convenence, let us assume that at all tmes t the characterstc curve of one port goes through the orgn; thus, the nductor s n the zero state when the flux s zero. In addton, we also assume that at all tmes t the nductor s fluxcontrolled. Furthermore, we may represent the nonlnear tme-varyng nductor by = ˆ ( λ, (5) Note that s an explct functon of both λ and. The voltage across the nductor s gven by Faraday s law as follows: e l dλ = (6) Substtutes (5) and (6) nto (4), the energy delvered by the generator to the nductor from tme t to t can be wrtten as λ( W ( t, = e ( t ) ( t ) = ˆ( λ, dλ (7) t t l λ( t ) Equaton (7) shows that W ( t, s a functon of the flux at the startng tme t and at the observng tme t. If we assume that the state of flux s zero, that s, λ( t ) =, and f we choose the state of zero flux to correspond to zero stored energy, then, recallng that an nductor stores energy but does not dsspate energy, accordng to the energy conservaton the energy stored ε must be equal to the energy delvered by the generator from t to t, that s, W ( t,, must be equal to the energy stored ( ε[ λ( t ), t] = W ( t = λ, ˆ( λ, dλ (8) Note that φ s the dummy varable of ntegraton and that t s consdered as a fxed parameter durng the ntegraton process. Equaton (16) can also be wrtten n the form t W ( t, =ε[ λ(, t] ε[ λ( t ), t ] ε[ λ( t ), t )] t t (9) The frst two terms gve the dfference between the energy stored at tme t and the energy stored at t. The thrd term s the energy delvered by the crcut ISSN: Issue 11, Volume 9, November 1

7 to the agent. Ths term changes the characterstc of the nductor; thus, t s the work done by the electrodynamc for forces durng the changes of the confguraton of the nductor. If the nput of contactor s assumed as a tme varyng nductor, we use a tme-varyng nductance L ( to descrbe ts tme-varyng characterstc. Thus, λ = (3) L( Accordng to (1), the energy delvered to the tmevaryng nductor by a generator from t to t s t W ( t, = L( ( L( t ) ( t ) + Lɺ t t ( ) ( ) t (31) L ɺ ( (35) Generator ( + v( One-port Fg. 4. A one-port s drven by a generator. φ φ ( = ˆ [ φ(, t] where the last term n the rght-hand sde s the energy delvered by the crcut to the agent. Ths term changes the characterstc of the tme-varyng nductor. It s mportant to note that ths last term depends both on the waveform L ɺ ( ) and the waveform ( ). A one-port s sad to be passve f the energy s always nonnegatve, ths s also called the passvty condton. ˆ [ φ(, t] W t, + ε( t ) (3) ( for all ntal tme t, for all tme t t, and for all possble nput waveforms. From (3) passvty requres that the sum of the stored energy at tme t and the energy delvered to the one-port from t to t be nonnegatve under all crcumstances. So that, the passvty condton for tme-varyng nductors requre that for all tme t, and for all possble nductor current and nductor voltage waveforms. From (31) and (3), the condton for passvty s 1 L( t 1 ( + Lɺ ( t ) ( t ) (33) t for all tme t, for all startng tmes t, and for all possble currents ( ). Therefore we can assert that a nonlnear tme-varyng nductor s passve f and only f and L ( (34) Fg. 5. Characterstcs curve of a typcal nonlnear tme-varyng nductor. (Provdng that s tmenvarant for smplcty) An nductve one-port s sad to be locally passve at an operatng pont f the slope of ts characterstc n the λ plane s nonnegatve at the pont. An nductve one-port s sad to be locally actve at an operatng pont f the slope of ts characterstc n the λ plane s negatve at the pont. If the nductve one-port s passve, the nstantaneous power enterng t s nonnegatve. Thus, the energy stored n the network s a nonncreasng functon of tme. In other words, ε ( ε( t ) for all t t (36) We can say that a network s passve f all the elements of the network are passve. And, however, any nonlnear passve tme-varyng network made of flux-controlled nductors s a stable network. 5 Smulaton and Dscusson ISSN: Issue 11, Volume 9, November 1

8 In ths paper, the expermental contactor s manufactured by a domestc and professonal company, Shln. Ths devce s a trpolar contactor and ts type s S-C1L. The col wll be appled an ac voltage source, ther nomnal power frequency s 6 Hz, and ts rated root-mean-square col voltage s V. Rated contacts capacty s 5.5 KW, 4 A, RMS the number of col wndngs are 375, the col resstance s 85 Ω, and the mass of the movable part s.115 Kg. Col current (A) smulaton experment 5.1 Establshng smulaton model In general, the dynamc behavor of the contactor can be predcted by usng the smulaton technque. From the precedng statement n ths paper, we know that the governng equatons of the contactor are bascally composed by the electrcal crcut equatons and mechancal moton equatons. Therefore, fve ndvdual smulaton sub-modules are frstly establshed based on the respectvely governng equatons. At last, a complete contactor smulaton model could be bult by means of combnng the precedng obtaned fve sub-modules; nto a resultng model. The completed smulaton model of contactor s shown n Fg. 6. In order to verfy the correctness of the smulaton model of the contactor, further to compare of the concernng parameters between the expermental results and the smulaton results are essental and necessary work. Hence, the nstantaneous col current, electromagnetc force, counter force and movable-part poston, are all used as the compared parameter tems. In Fg. 6, t s shown that the smulaton results usng contactor model are bascally n agreement wth well the expermental results. To a word, the valdaton of the correctness and precson of the smulaton model of the contactor are successfully done. Ths smulaton model can then be used to smulate and predct the performance of contactor under dfferent operatng assumed workng status. e fa fa L(x) e Magnetc force (N-m) Counter force (N-m) Tme (sec) (a) Tme (sec) (b) smulaton experment Tme (sec) (c) smulaton experment [t1,e] u x x Input voltage Voltage equ. Flux lnkage equ. dl/dx1 v Emf. equ. dl/dx Fe Fe x v Magnetc force Mechancal equ. Fg. 6. Completed smulaton model of the electromagnetc contactor. ISSN: Issue 11, Volume 9, November 1

9 7 x 1-3 (a) 6 1 Poston (N-m) smulaton experment Ampltude(%) Intal voltage phase Tme (sec) (d) Fg. 7. Smulated parameter waveforms by model compared wth those measured by expermental contactor. 3 Fal area Duraton(cycle) (b) 5. CBEMA curves analyss The dynamc performance of contactor s to be studed and expressed by usng CBEMA curves when t s operated under dfferent voltage-sags condtons lke ampltude, duraton, and ntal voltage sags phase. Ths knd of curve, CBEMA, was orgnally ntroduced to represent the computers voltage-tolerance performance for over-voltage and under-voltage. They show mnmum voltage aganst maxmum duraton, whch ensure the non-stop operaton of contactor. Provded that a specal pontn-wave where sag occurs, the boundary between succeed and fal to close the electrcal contacts are measured and plotted, as shown n Fg. 8. The abscssa of Fg. 8 s voltage-sags duraton n cycle, whle the ordnate s voltage-sags ampltude n percentage. CBEMA curves of contactor n terms of ntal voltage sags phase, 45, 9 and 135 are respectvely obtaned, notce that the border of the 45 ntal voltage phase s smlar to one of the 135, but there are not completely dentcal. Ampltude(%) Intal voltage phase Ampltude(%) Ampltude(%) Intal voltage phase Duraton(cycle) (c) Intal voltage phase 9 Fal area 135 Fal area 4 3 Fal area: Duraton(cy)cle Duraton(cycle) (d) Fg. 8. CBEMA curves of the contactor tolerance under phase angle, 45, 9 and 135 where sag occurs. ISSN: Issue 11, Volume 9, November 1

10 5.3 Flux lnkage versus col current varyng curves From the characterstc of the contactor n terms of nput drvng mpedance, t s ndeed an nductor element. As mentoned above, f the nductve oneport s passve, the nstantaneous power enterng t s nonnegatve. Thus, the energy stored n the network s gong to be a nonncreasng functon of tme, as seen n (1). However, any nonlnear passve tme-varyng network made of fluxcontrolled nductors s a stable network. We can say that a network s passve f all the elements of the network are passve. Therefore, we say that the dynamc behavor of contactors s ether stable or unstable can be completely decded by the net energy stored n the contactor durng the dynamcal perod. If the net energy stored s nonncreasng, the dynamc stablty of contactor wll be stuated at stable state. As a matter of fact, n many practcal applcatons, most the characterstc of the contactor s passve, but locally actve at some operatng ponts occur because the slope of the characterstc n the λ plane s negatve. So that the electrcal contacts s whether reman closed or dropped out, completely reles on the energy delvered by the t 1 crcut to the agent Lɺ ( t ) ( t ) durng the t dynamcal perod. If the stored energy wthn ths perod s negatve, the electrcal contacts are antcpated to be dropped out, as shown n Fg. 9; contrarly, f the stored energy wthn ths perod s nonnegatve, the electrcal contacts wll therefore reman closed or dsengagement and re-engagement, as shown n Fg. 1 and 11. In one word, the electrcal contacts are ether reman closed or dropped out after the consdered dynamcal perod past depends on the value of the net stored energy s negatve or nonnegatve wthn the dynamcal perod Poston (mm.) 7 x (a) Tme (sec.) (b) Fg. 9. Col nductance L ɺ ( and poston varyng curves when the voltage sag whose ampltude s 1%, duraton s one cycles, and pont-n-wave where sag occur. Poston (mm.) Flux lnkage (H-A) Col current (A) 7 x (a) Flux lnkage (H-A) Col current (A) Tme (sec.) (b) Fg. 1. Col nductance L ɺ ( and poston varyng curves when the voltage sag whose ampltude s ISSN: Issue 11, Volume 9, November 1

11 Flux lnkage (H-A) Poston (mm.) %, duraton s three cycles, and pont-n-wave where sag occur Col current (A) 7 x (a) Tme (sec.) (b) Fg. 11. Col nductance L ɺ ( and poston varyng curves when the voltage sag whose ampltude s %, duraton s two cycles, and pont-n-wave 9 where sag occur. As commented earler, the dynamc characterstc of the contactor s lke a nonlnear tme-varyng nductor. Nevertheless, for a nonlnear tme-varyng nductor to be passve, ts characterstc (n the λ plane) must pass through the orgn and le n the frst and thrd quadrants n the neghbourhoods of the orgn. Also, f the characterstc of a nonlnear tme-varyng nductor s monotoncally ncreasng and les n the frst and thrd quadrants, t s passve. Fg. 9 shows that the energy stored n the contactor s zero durng the voltage sag occurs because the poston of the movable part never leaves away the electromagnet. However, the characterstc of a nonlnear tme-varyng nductor s monotoncally ncreasng and les n the frst and thrd quadrants, t s passve. Therefore, the electrcal contacts s reman closed durng the voltage sag occurs. Fg. 1 shows that the total energy stored wthn the poston ncreasng part s Joules, whle the total energy stored wthn the poston decreasng part s.6493 Joules durng the voltage sag occurs. The characterstc of a nonlnear tmevaryng nductor s monotoncally ncreasng and les n the thrd quadrants, t s passve. Note that the there s partal characterstc occurs the tme rate of the change of col nductance L ɺ ( s negatve, as shown n Fg. 1(a), where denoted by a dash crcle; namely, the operatng feature of these ponts s locally actve. Consequently, the energy stored n the contactor wthn ths dynamc perod becomes a negatve value and the magnetc crcut produces an nterestng result that the movable part dsengage and then re-engage wthout any dsengagement of the electrcal contacts. In addton, consderng the net energy stored n the contactor durng voltage sag occurs s stll a nonnegatve value; hence, the contactor s to be passve and stable. Fg. 11 shows that the total energy stored wthn the poston ncreasng part s Joules, whle the total energy stored wthn the poston decreasng part s.5413 Joules durng the voltage sag occurs. Snce the characterstc of a nonlnear tme-varyng nductor s monotoncally ncreasng and les n the frst and the thrd quadrants, t should be passve. Note that the there are some operatng ponts n the characterstc occurs the tme rate of the change of col nductance L ɺ ( s negatve, as shown n Fg. 11(a), where s denoted by a dash crcle; namely, the operatng feature of these ponts s locally actve. The sustanng tme of the negatve col nductance L ɺ ( s longer than precedng case. As seen n (37), because of the net energy stored n the contactor durng voltage sags has become a nonnegatve value, and therefore the unntended dsengagement of the electrcal contacts occurs. For ths stuaton, the characterstc of the contactor can be sad to be an actve and the acton s consdered as an unstable operaton. 6 Concluson Up to now, there s nothng works concernng the dynamc stablty of the contactor by researchng of the characterstc of contactor can be found. In ths paper, accordng to the basc defntons of the stablty and passvty condton, we provde a new approach, whch can be used to predct the dynamc ISSN: Issue 11, Volume 9, November 1

12 stablty of contactor n terms of ts characterstc curve n the λ plane. We are successful to obtan some valuable results by usng the establshed smulaton model of the contactor. The contactor model followed by the electromagnetc crcut analyss approaches s establshed. In the followng the contactor voltage sag senstvty s also smulated on ths model and the smulaton results are shown n CBEMA curves. Addtonally, as seen n the characterstc of contactor n the λ plane, provded that there exsts locally actve pont durng closng process or where voltage sag occurs, the fnal state of contacts may be reman closed, dsengagement or dsengagement and reengagement, whch depends only on the net energystorng wthn the dynamc perod. From those smulaton results, we can ensure the dynamc stablty of contactor such as durng closng process or where voltage sags occur, can be predcted by usng the characterstc of a contactor n the λ plane. References: [1] M. F. McGranaghan, D. R. Mueller and M. J. Samotyj, Voltage Sags n Industral Systems, IEEE Transacton on Industral Applcatons, Vol. 9, No., 1993, pp [] A.E. Turner and E. R. Collns, The Performance of AC Contactors durng Voltage Sags, 7 th Internatonal Conf. on Harmoncs and Qualty of Power, 1996, pp [3] R. C. Dugan, M. F. McGranaghan and H. W. Beauty, Electrcal Power System Qualty, New York, McGraw-Hll, [4] A. E. Ftzgerald, C. K. Jr and S. D. Umans, Electrc Machne, McGraw-Hll Book Company, Tawan, [5] A. G. Espnosa, J.-R. R. Ruz and X. A. Morera, A Sensorless Method for Controllng the Closure of a Contactor, IEEE Transacton on. Magnetcs, Vol. 43, No. 1, 7, pp [6] J. Pedra, F. Corcoles and L. Sanz, Study of AC Contactor durng Voltage Sags, Proc. of the 1th Internatonal Conf. on Harmoncs and Qualty of Power, Vol.,, pp [7] H. Lang, G. C. Ma, L. Ca, X. Yong and G. Q. Xe, Research on the Relatonshp Between Contact Breakaway Intal Velocty and Arc duraton, Proc. 49 th IEEE Holm Conf. on Electrcal Contacts, 3, pp [8] A. D. Charles and S. K. Ernest, Basc Crcut Theory, Tawan: Central Book Company, ISSN: Issue 11, Volume 9, November 1