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EXPERIMENTAL ANALYSI S OF A FAST ACTI NG CI RCUI T BREAKER MECHAKI SMELECTRI CAL ASPECTSRaymond J. Raj ot t e - M chel G. Dr ouetDi r ecti on Sci ences de baseI nst i t ut de Recherche de 1' Hydr o- QuebecVarennes, QuEbec. Canada.

ABSTRACT

An experi ment al anal ysi s of a f ast el ectr omagne-t i c ci r cui t br eaker dr i vi ng mechani sm has been perf or-med. The devi ce ut i l i zes the r epul si on f orce pr oducedon a metal di sc subj ect ed t o a t r ansi ent magneti c f i el dThe spat i al di str i but i on, i n the di sc, of t he dr i vi ngf i el d was det erm ned usi ng magneti c pr obes l ocat ed tt he sur f ace of t he di sc or i mbedded w t hi n. The accel e-r ati on of t he di sc was al so measured usi ng a pi ezo-el ect r i c aceel er ometer. A pul sed power suppl y ci r cui tw t h or w t hout crow- bar ener gi zes t he magnet coi l .The experi ment al r esul t s ar e f ound to agr ee ver ywel l w t h mor e ref i ned t heor et i cal anal ysi s present ed

i n t hi s paper . Some r ecommendat i ons ar e made w t h r e-gard t o the opt i m zati on of both t he geometr y of t hedr i vi ng mechani sm and the char acter i st i cs of t he powersuppl y used t o energi ze thi s dri vi ng mechani sm

I NTRODUCTI ONThe i nter est f or hi gh speed sw t chi ng mechani smf or DC or synchr onous AC i nt er r upt i on of el ect r i calci r cui t i s evi denced by t he many pr ogr ess repor t s pu-bl i shed i n recent year s (e. g. 1, 2, 3, 4) . I n most cases,f ast openi ng of t he cont act s i s achi eved usi ng a

f ast act i ng el ect r omechani cal devi ce whi ch ut i l i zes t her epul si on f orce produced on a metal di sc subj ected to at ransi ent magnet i c fi el d. he worki ng pr i nci pl e of sucha mechani sm i s i l l ust r at ed on Fi g. 1. The di sc s hapedmagnet coi l i s ener gi zed by di schar gi ng t he capaci t orbank i n t he coi l . A t r ansi ent magnet i c f i el d i s t husproduced whi ch i nduces eddy curr ent s i n t he di sc l oca-t ed cl ose t o the magnet coi l . There occur sn el ectr o-magnet i c i nter acti on, bet ween t he i nduced cur r ent andt he sour ce magneti c f i el d, whi ch r esul t s i n a r epul si onof t he di sc car r yi ng t he i nduced cur r ent . The di sc i saccel erated, i n the axi al di recti on, w t h the movi ngcontact and, i n t hi s way, t he separat i on of t he cont actsi s achi eved.I n t he past , t he desi gn and devel opment of t hi s

t ype of hi gh speed dr i vi ng mechani sm has been basedmai nl y on approxi mate el ect r i cal and mechani cal t heo-r i es and t he data deri ved f r om l aborat ory model s andprot otypes. However, i n a recent paper , Basu and, Sri vas-

Paper T 74 408-1, r ec ommende d and approvedby the I EEE SwtchgearCommttee of the IEEE Power Eng neeri ng societyor presentation at the IEEEPES Summer Meet i ng Energy Resources C o d . , Anahei m Cal., July 14-19,1974. Manuscri pt submt tedFebruary 1 1974; mad e avail abl e for printing April29,1974.

t ava have pr esent ed a theoreti cal anal ysi s of cer t ai nel ect r i cal , t hermal and mechani cal aspect s of a fastacti ng ci r cui t br eaker mechani sm of t he type i l l ustr a-t ed on Fi g.1.

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Fi g. 1 The el ectr omagnet i c dr i vi ng mechani smAn exper i ment al anal ysi s of s uchdevi ce has beenunder t aken her e. Measurement s of t he spat i al di st r i bu-t i ons, i n t he di sc, of t he t ransi ent magnet i c f i el d,i nduced vol t age and curr ent have been perf ormed as af unct i on of ( a) t he ampl i t ude and f r equency of t he cur-r ent i n t he dr i vi ng coi l and ( b) t he di st ance betweent he di sc and the coi l . The measurements were made usi ngw r e l oops l ocated on t he sur f ace of t he di sc or i mbed-ded w t hi n. Especi al l y at ower r equenci es, argedi scr epanci es wer e f ound bet ween our measur ements andt he theoret i cal val ues accor di ng to Basu and Sr i vast a-va5. I n order t o resol ve t hese di screpanci es a t heor e-t i cal anal ysi s of t he pr obl em was undert aken whi ch l edto a more r ef i ned model of t henteraction.Furthermore,very good agr eement was obtai ned bet ween t he theoret i -cal and the measured di st r i but i ons of t he cur r ent andof t he magnet i c and el ectr i c f i el ds i n t he di sc.

THEORYI n gener al , boundar y val ue probl ems i n whi ch cur-r ent s ar e pr esent are t r eat ed by means of t he magnet i cvect or pot ent i al . The var i ous el ectr omagneti c quant i t i esof i nterest can be der i ved once we know t he expr essi onsgi vi ng t he magneti c vect or pot ent i al . The sol ut i on t ot he boundary val ue pr obl em of a spi r al coi l par al l el t ot he sur f ace of a di sc can be si mpl i f i ed gr eat l y i f wemake the fol l ow ng assumpt i ons whi ch are wel l j ust i f i edf cT our exper i ment al model .

1) The cur r ent i n the coi l f l ows i n the azi mut hal di -r ect i on onl y and ther efor e the vect or pot ent i al anda9


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th e eddy cur ren ts hav e cyl ind ric al symmetry.2 ) The r a d i a ld i s t r i b u t i o n of t h ec u r r e n t n h e l a ts p i r a l c o i l i s uniform in amplitu de and i n phase.3 ) The diam ete r fhe i sc i s much largerh an h e

externa l i am e te r of t h e o i l ; we canhereforeassume, when we cal cu la te hevalu e of thev ec to rp o t e n t i a l , h a t h ed i s c has a n n f i n i t ed i a m e t e rand th a t d g e f f ec t snh e rea l electromagne-t ic dev ice can be neg lec ted .

4 ) The various media are li ne ar , homogeneous an d so-t r o p i c ,

5) The disc is maae 01 nomnagnetic materlal.With theseassum ptions , we a re now ab le oca l cu -l a t e h e v a l u e o f h e v e c t o r p o t e n t i a l n s i d e a ndout-s ide of the .meta1 isc . As s h od i n h e p p e n d i x , ft h ecu r r en t l o win g n h eco i l i s per iod icwith angu-

l a r f requency w theon ly component of th evector po-t e n t i a l A i n h e y l in d r i ca l o o r d in a t e y s t em i s i nt h e 8 d i r e c t i o n as i s t h ec u r r e n t n h eco i l and itm ust sa t i s f y h e f o l lo win g d i f f e r en t i a l eq u a t io n :

n n

where u i s the ermeabi l i ty nd u i s t h e l e c t r i c a lco nd uc ti vi ty of th e medium. The s o l u t i o no f h ed i f f e -r e n t i a l q u a t i o n 1) i s o u t l in ednh e p p en d ix o rtwb d i f f e r e n ts i t u a t i o n s. We f i r s t lo ok a t h e case o fa f i l a p e n t u r r e n t o o pp a r a l l e l o h e u r f a c eof al a r g em e t a lp la t e of th ickn ess t. Then, we extend hes o l u t i o noh e a s e f a f l a t p i r a l o i l above al a r g em e ta lp l a t e . This case i s ofmore prac t ica l n -t e r e s t n o u r n v e s t i g a t i o n .

Equation (A.25) i n h eappepdix g ives hevalueoft h e v e c t o r p o t en t i a l A 3 ( r , z ) a t an y p o i n t n s i d e a n d a tt h esur face of them e ta ld i sc as a fun cti on of th e va-r iousparamet ers of the e lec t romagnet ic system cur ren t ,f r eq u en cy ,d i s t an ce r o m h ed i sc o h e o i l , h i ck -ness o f the i sc , onduct iv i ty of the i scm a t e r i a l ,dimensions nd number of ur nso f h e o i l ) . W willnow show how th ev a r io u se l ec t r o m ag n e t i cquant i t ies o fin t eLes t anb ed e r iv ed r o m h ev ec to rpo te nt ia l A3( r , z ) .

The abso lu te a lue nd he hase of the ur re n td en s i ty J i n h e i sc an e ead i ly b t a in ed s in gOhm s law:

aA3 ( r, )a tJ = UE = - U - j w u A3(r,z) ( 2 )

where E i s t h e l e c t r i c i e l d . The v o l t ag e V inducedalong a ce r t a in p a th s i s genera l ly g iven by:

( 3 )0

For a c i rcu larpa th of rad ius r which i s co ax ia lw i t h h e p i r a lc o i la n dp a r a l l e l o i t , equation 3)becomes

V = j w 27 r A(r ,z) 4 )The induce d oltage V i s a p h y s ica lquant i ty o fg r ea tp r ac t i ca l n t e r e s tb ecau se it canbeeasi ly mea-

sure d xper ime ntal ly by means f a f in ewi r e o o p a tthe u r face of thed i s cor imbedded i n i t as w i l l bed e s c r i b e d n more d e t a i l n h e e x t e c t i o n of t h et e x t . W will b eable o compare d i re c t ly hee x p e r i -mental valu e of themagnitudeand hephase of the n-ducedo l t ag ewi thheheore t ica l a luea lcu la tedusingequat ions 4) and (A.25) withoutany ur ther ma-n ip u la t io n .

Another physica l quant i ty o f in te re s t i s t h e magne-t i c l u xd e n s i t y B. Using Maxwell's equations we f i n dthat f o r h e c a s e o f a x i a l symmetry themagqet ic luxd en s i ty has only r and z components.These quant i t iesa r e o b ta in ed f r o m h e v ec to r p o ten t i a l by t h e r e l a t io n -s h i p s:

Br = - - Aazand

EXPERIMENT

A. ExperimentalrrangementA schematic iagram fhe xperimental rrange-ment used i n hepresen t work i s shown on Fig. 2 . Thes p i r a l magnet c o i l a nd t h ec y l i n d r i c a l metal d i s c are

mounted co ax ia ll y. The8 t u r n s p i r a l c o i l i s 70 mm and178 m, ins i de and ou ts ide iameter , espec t ive ly . I ti s made ofc op pe r e r e of s qu ar ecrosssection(AWG7);I ti s supported nd bound, usin g epoxy re s i n, o a 50 mmt h i ck p h en o l i c p l a t e as shorn on Fig. 1.

I n most of th e measurements escribed elowhec o i l w s ener giz ed sing n sci llat or nd power am-p l i f i e r n s t e a do f a cap aci tor bank. Only i n h i s waywas i t possib leonain ta in o thhemplitude andth e r eq u en cyo f h ecu r r en t n h eco i lwh i l ev a r y in gei ther hed is tance ,b ,be tween he o i l nd hed isc ,the h ickness o f thed i s c , t or he requency of thecu r r en t t se l f .Fu r th e r m o r e , i t made po ss ib le he mea-surement of ' t he ela tiv ep h aseo f h e n d u cedv o l t ag ewi th r e sp ec t o h e cu r r en t s in e wave in t h e co i l .

Fig. 2 . Experimentalrrangement.90


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The copper or al um num di scs used were80 m i ndi ameter , l arge enough compar ed t o t he out si de di ameterof t he coi l ( 178 nun t o ensur e t hat t he measur ement scoul d be compared w t h the pr edi ct i ons of t he theor eti -cal anal ysi s wher e t he assumpti onf an i nf i ni t e di s ci 8made. I n order t o deter m ne the i nf l uence of t he di sct hi ckness, several di scs of di f f er ent t hi cknesses wereused and i n addi t i on t hey coul d be st acked onen t opof t he other ; t he behavi or of a st ack of di scsas notdi f f erent f rom t hat of a si ngl e di sc of t hi ckness equalt o that of t he st ack. Each measur ement w s perf ormedw t h t he di sc hel d st ati onnar y by brackets f i xed on t hecoi l pl ate support .B Measuri ng probes

A s t he curr ent i n the coi l vari es, a tr ansi entmagneti c f i el d i s pr oduced whi ch i nduces an el ect r i cf i el d i n t he di sc whi ch i n t urn resul t s i n t he pr oduc-t i on of eddy cur r ent s. The probl em of measurement oft he el ect r omagnet i c quant i t i es i n t he di scs made si mpl er by t he cyl i ndr i cal symmetr y of t he syst em Toperf orm t hi s measurement we have used ni ne coaxi all oops, ci r cul ar i n shape (20,60,90,120,150,165~180,205and 230 mm i n di ameter ) , l ocat ed i n gr ooves maden t hesur f ace of t he di sc. The si ngl e turn l oops ar e not i nel ect r i cal cont act w t h the di sc and ar e made of copperw r e 0.25 mm i n di ameter . They ar e connect ed t o bothan A . C . vol t meter and a phase meter whi ch al l ow t hemeasurements of t he ampl i t ude and of t he phase of t hevol t age i nduced i n each l oop. The cur r ent i n the coi l ,measur ed by a curr ent t r ansf ormer as shown on Fi g. 2 ,i s used as a r ef erence f or t he phase measurement .

The vol t age measured i s i nduced i n a l oop by t hevar i ati on of t he magnet i c f l ux t hr ough t he sur f ace l i -m t ed by the l oop. As coaxi al l oopsof i ncreasi ng andknown ar eas ar e used i t i s easy to det er m ne the di st r i -but i on i n ampl i t ude and phase of t he el ectr i c anda-gneti c f i el ds f r om t he measur ement s of t he vol t age i n-duced i n t he l oops. Fur t hermore, as shown ear l i er.2 ) , i t i s al so possi bl e to cal cul ate the di str i buti onof t he cur r ent i n t he di sc know ng t he val ue of t heel ectr i c fi el d i n t he di sc and t he conducti vi t y of t hemat er i al of t he di sc.The i nduced vol t age w s measured ei t her at t hesur f ace or i nsi de t he di sc i n a pl ane paral l el t o i t ssur f ace. Thi s was done t o determ ne t he ampl i t ude andt he phaseof t he el ectr omagnet i c waves pr opagati ng i n-si de and out si de of t he di sc. The char acteri st i cs oft he wave propagat i ng i n t he metal were det erm ned byi nsert i ng di scs of var i ous t hi cknesses bet ween the ma-gnet coi l and the di sc beari ng t he measuri ng l oops.

EXPERIMENTAL RESULTSA. Radi al dependenceof t he i nduced vol t age

The radi al dependence of t he i nduced vol t age wasdet er m ned for di f f erent val ues' of ei t her t he di stancebet ween the magnet coi l and the di sc or t he f r equencyof the osci l l at i ng curr ent i n the coi l ; typi cal resul tsare shown i n Fi g. and 4 . Over t he r ange of f r equencyand di st ance st udi ed, we obser ve a ver y good agreementbet ween t he t heoret i cal curves and t he exper i ment alpoi nt s. The ampl i t ude of t he i nduced vol t age i saximumat a r adi us equal t o a val ue bet ween t hat of t he i nsi deand out si de r adi i of t he coi l .t decr eases rapi dl y fora r adi us l arger t han t he out si de r adi us of t he coi lespeci al l y at hi gh f r equenci es. Fur t hermore, as coul dbe expect ed, t he coupl i ng between the di sc and the coi ldecreases as t he di st ance between t he two, b, i ncreases.

. lX0

/ tsM o o

f = 60HzI = IOOA. rms120Disc.= Cu.t=28mm 121 .

Fi g. 3 . Radi al dependence of t he i nduced vol t age. Pa-r amet er : t he di st ance coi l - di sc.

Disc: AI.t = 5 3 m m

t44

-1 50

12010tN i lm V5

M M - u-1Fi g. 4 . Radi al dependence of t he i nduced vol t age. Pa-r ameter: t he fr equency of t he exci t i ng cur r ent .

10tN i lm V5

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The r el ati ve phase i bet ween t he exci t ati on cur r ent t i on of t he di sc, depends on t he val ue, t , of t he diand t he i nduced vol t age Vi i s, however , i ndependent of t hi ckness . Measur ements wer e made i n order t o demonst hat same di st ance b . i n t he regi on of t he di sc cor r es-e the i nf l uence of t ; . t he resul t s are pr esent edn Fi g.pondi ng to the i nsi de and out si de radi i of t he coi l 6 . One measur i ng l oop i s l ocat ed on each si de of t hewher e i t s val ue i s about 1270 (Fi g.. As predi ct ed by di sc, t he thi ckness of whi ch, t , may be vari ed f r om t oEq. 4, t he i nduced vol t age i s f ound experi ment al l y ( Fi g.0 mm The di ameter of bot h w r e l oops i s t he same,4 t o i ncrease w t h f r equency; a smal l dependence on 120 m he i nduced vol t age Vi i s r el ated t o the varf r equency of t he phase1 i s al so noted. The sharpness t i on i n t he magneti c f l ux ent eri ng t he di sc. The vol -of t he dependence of bot h i and ( Vi 1 on r adi us obser - age Vt corr esponds to t he var i ati on i n the magnet i cved i n Fi g.4, as compared to t he r esul t s of Fi g., i s f l ux l eavi ng t he di sc on t he ot her si de. Once agai ndue to t he f act t hat t hose resul t s corr espond to ai t he agreement bet ween t he theor et i cal curves and t heni mum val ue,1 mm f or b. experi ment al poi nt s i s ver yood. I t i s i nteresti ng t onote t hat Vt . and, cor r espondi ngl y, t he ampl i t ude of t hmagneti c f l ux t r ansm t t ed t hr ough the di sc and ther e-f or e not cont r i but i ngto any accel er at i on of t he di sc,decr eases rapi dl y w t h t when t he ski n dept hi s grea-The exper i ment al conf i gur at i on of t he coi l - di sc t er t han t. Thi s obser vat i on w l l be of i mpor t ance i nB Vol t age i nduced i nsi de the di sc

syst em used for t hi s par t i cul ar i nvest i gat i on i s i l l us- t he desi gn of t he el ectr omagnet i c dr i vi ng mechani smt rat ed on Fi g. 5. The measur i ng l oop, of di ameter 20m y s i n a pl ane paral l el t o t he di sc sur f ace and at adi st ance z f rom i t . The phase, w t h respect t o the cur-r ent wave i n the coi l , and the ampl i t ude of t he i nducedvol t age Vi ar e pl ot t ed on Fi g.as a f unct i on of t hatdi st ance z . Very good agr eement bet ween t he t heoret i calcurves and t he exper i ment al poi nts i s al so obtai ned. Thephase l a g 1 of t he el ectr omagnet i c wave i ncr eases l i -near l y w t h t he di st ance i n t he met al and i t s ampl i -t ude decr eases exponent i al l y w t hi ndi cat i ng that, f orour par t i cul ar coi l - di sc geomet r y, t he el ectr omagnet i cwave i nsi de of t he metal di sc behaves l i ke a pl ane wave 8.C. Wave pr opagat i ng thr ough the di sc

The di mensi ons of t he di sc have an i nf l uence ont he coupl i ng bet ween the coi l and the di sc i t sel f . npart i cul ar, t he amount of energy t r ansm t t ed t hrought he di sc, whi ch does not cont r i but e to t he accel era-

Fi g. 6 Vari ati on of t he vol t age i nduced on bot h si desof t he di sc w t h di sc thi ckness.

b=4mmI = 4 0 0 A rms -- D. Measurement s of t he accel erat i on of t he di scMeasurement s of t he accel erat i on of t he di sc wereal so per f ormed. Such measurement s ar e ms t i mpor t ant- f or t he opt i m zat i on of t he ener gy t ransf er f r om t he- parer suppl y to t he di sc. I n thi s part i cul ar study the- by a shaf t s shown on Fi g. . The accel erati on of t he5 di sc* was f r ee to move al t hough i t s mot i on was gui ded- di sc was measured by means of a pi ezoel ect r i c accel er- meter f i xed on t op of t he di sc. The si gnal f r om t he- - accel er omet er , a, was ampl i f i ed and pr esent ed on t heosci l l oscope. The vol t age,, i nduced i n a l oop (120u-o o l L L L 1 ' l l L 110mm z- * Al umnum out si de di ameter 280 mm t hi ckness 38 ran,Fi g. 5. Vol t age nduced nsi de hedi sc. otal wei ght , w t hshaf t , 6 kg.92


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Probe / 1Tek.556

where dr is anelementalvolume of t n ed is c and Re [ ]i n d i c a t e s that we must ake he real p ar t of th e com-plex ns tantaneous current and magne t ic f lux dens i t ies .The azimuthalc ur re n tde ns i ty Je and he adi al magne-t i c f l u xd e n s i t y Br are givenby Eqs. ( 2 ) and (5) res-pec t iv e ly. The nega t iveacce le ra t ion ecordedonFig.8the re f ore resul ts f rom the phase dif fe rence be tween hec ur re n t de ns i ty a nd the ma gne t ic nduc t ion n he d i s c .

The relat ive behavior of he current I i n t h e c o i land nducedvoltage V i a t t h e s u r f a c e of the d i s c (F ig .8) c a nbe n te rp re te d n terms of hephasedif fe renceof 127O between I and V i measured seeFig. 3 i n h ecase of a cont inuo us sine wave.E. Operationof hesystemwith acrow-baron th e powerFig. 7 . Experimentalrrangementsedorheeasure-upply.mentsof the a c c e le ra t ion o f the d i s c .

indiameter) oca ted a t the surface of the d isc a nd hec u r r e n t , I i n h e c o i l were monitored imultaneously.The osc i l l a t o r a nd power a mpl i f i e r ,use dpre v ious ly oe ne rg iz eh eo i l , were replacedyheapac i tor-i g n i t r o nc i r c u i tp r e s e n t e d on Fig. 7. The c i r c u i t wasse t up so that only a half wave ofcurrent was de l ive-r e d o h e c o i l as shown on Fig . 8. The c u r r e n t n h ec o i l was l imited on purpose n order that a l l the ener-gy be de l ivered before he disc had time t o move appre-c i a b l y ( = 1mm . The frequency of th e c i rc ui t was adjus-te d omin imiz e hee xc i ta t ionof he many na tu ra l me-chanical esonance requencies of the di sc.

W have note d , n hepre v ious e c t ion , ha t heeff ic iencyof he ys tem is hampered by th e ac t h att h ea c c e l e r a t i o no f h ed i s c is nega t ive - towards hec o i l - during he second s tage of he current njec t ionTo improve theystem we have onnected ngnitrona c ros sh eo i le rmi na ls . The ign i tro n i s used tos h o r t - c i r c u i t h eco il when th ec o i lc u r r e n t i s a t i t sp e a k ; h i sc r o r b a rp r e v e n t s h ecu rr en t from wingingback hrough ero e.g.Ref. 6 ) . The use of a crow-bari n h ec i r c u i t o f n electromagnetic device as eensuggested in thepas t7, however, in ha t paper , no mea-surementwith a row-bar was r e por te d .

Fig. .Experimentalmeasurement fhe ccelerationof he disc .The r e s u l t sp r e s e n t e d on theosc i l logram Fig. 8)show that t h e c c e l e r a t i o n of t h e i s c i s p o s i t i v e(awayrom the oi l )dur i ng mos t f the u r re n th a l fwave.However, the ccel erati on oes hrough e ro ndbecomes slightlynega t ive owards he nd of the cur-

r e n thalf wave.Computer calculationsof he ns ta n ta -neousva lue of the forc e ac t i ng on t h e d i s c , i n h e c a s eof a cont inuous ine wave, have shown t ha t her e i s asmall ne ga t ive o rc eac t ing on the di sc toward the endof eachcurrent half wave in he pri mary coi l. The ins -ta n ta ne ousva lueof he o ta l o rc e ,F , c t ing on t h edi sc an e computedy numerica l ntegra t ion f heins tantaneousa luethe whol e volumeof

[ ,d i s c

of the orcepe run it volume overt h e d i s c as follows:

Je eJwt] x R e [Br eJwt] r (7)Fig. 9. Time v a r i a t i o n s o f t h e u r r e n t I i n h e o i linducedvoltage a t t h esur fa c e o f the d i s c V iand cce le ra t ion of thedi sc a wit h crow-bar

(loweroscil1ogram)andwith out crow-bar upperosc i l logram).93


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To fa c i l i t a t e comparison, we haveeproduced inFig . ' the sc i l lograms of two successive xperimentsperformedwithandwit hout he crow-bar. The curves onFig.9a,which are very similar i n sh ap e o he eco r -dings fFig. , were obtainedwit hou t crow-bar. heinf luence of th ec r o wb ar . is well demonstrated on Hg.9b. W obse rve hat he crow-bar preven ts he urrentin he co i l , I , f r om decr easi ng oo fas t and consequentlyfrom inducing a reve rse volt age V and a r ev e r se cu r r en tin he is c. However, the most int ere sti ng spe ct oft h e crow-bar i s that (a) it p r ev en t s h eacce l e r a t io n ,a , from becoming negative as i s thecasewithout crow-bar)and b) i t co n t r ib u te s o h egenera t ion of a po-s i t i v ea c c e l e r a t i o n o r a cer tai n ime . Comparing th eareaunder hecurves a1 anda2 w e n o te that, by usinga crow-bar in t h e c i r c u i t , we haveobtained more than af i v e o l d n c r e a s e in t h een erg y ran sfe re d o he mo-v in g d i sc .

CONCLUSION

An exper imenta l and heore t ica l ana lysis o f a f a s tact ing lec trom agne tic ircu it rea ker mechanism hasbeenp re sen ted. The s p a t i a l d i s t r i b u t io n o f h e d r iv in gf i e l d n h ed i s ch a sbeen measured. The experimentalr e su l t s o b ta in ed are i n good agreementwith he heore-t i c a la n a l y s i s ; h e y are of a grea t mpor tance o heengineer o r heoptimi zatio n of both hegeometryofthe dri vin g mechanismand the cha rac ter is t ics of t he po-wer supply used oenergize hisdriving mechanism.a )Rela t ived iametersof hed iscand heco i l .

The measurementspresented on Fig. 3 i n d i c a t e t h a tth e i e ld n h ed i sc em a in s m p o r t an t o r a r ad iu sl a r g e r h an h a to f h eco i l , h e r e f o r e h e diameter ofth ed i scsh o u ldb e a r g e r h an that o f h ec o i l . Howe-v e r as shown on Fig. 4, th e recommended re la ti ve va lu esof the diamet ers should also depend on the frequency ofth e u r r en tp u l s e . A s the requency is i n c r ea sed , h erecommended d ia me ter fhe ischouldendowardsthat of thec o i l ; h i s i s a directconsequenceof hedecrease of the skin depth with he frequency.b)Thickness of th ed i s c .

I t might appear hat only mechani cal strength con-siderations shouldbe elevanthere.Roweversthe hick-n ess o f h e d i sc i s also of great mport ance n he op-t imiza t ion of theenergy ransfer . A s shown on Fig.6,themagnetic luxpropagating hrough hedi sc may beimportant; hispossible ossofenergy,demonstrated byourmeasurements,can eriously educe heeffenciencyofhe ystem. The va lu eo f h e i e ld decreases wi thd i s t a n c e n h e d i s c as shown on Fig. 5 ; thesk indepthc h a r a c t e r i z e s h i sa t t e n u a t i o n . The value of the k indepth, as a func tion of frequency, i s well known f o rd i f f e r en tm e ta l s ; i t shouldbe aken ntoconsiderationwhen deciding on the h ickness o f he d isc .c )Cha rac ter is t ic s of the power supply.

We haveobserved Fig. 3) t h a t h ecoupling of en-e r g yoh e i sc ec r ea se swi thn c r ea s in g i s t an cefrom th e co il . Energy sh ould h erefor e be delive red ra-p i d l y o h e c o i l , b e f o r e h e d i s c has time t o move ap-p r ec i ab lyf r o m h eco i l . This implies hat he requen-cyof hecapaci to r -co i lc i rcu i tshouldbehigh enoughesp ec ia l ly f h ig h e r m in a l v e lo c i t i e s are d es i r ed .We havealsonot ed he nflu enc e of a row-bar i nthe power supply on theef f ic iencyof heenergy rans-f e r ; in ourcas e a fi ve fo ld nc re as e was measured. Theus e of a row-bar is al s o recommended in h ecase ofh igh erminalve loc i t iesbecause i t prevents he ur-

94

r en t f r o mwinginghroughero,husmaintaining ahigh a lue fn terac t ion e tweenheoi lndhedisc ,whi le hed is tancebetween them i s s t i l l small.

The authorsx p r e ssh e i rh an k so Mr MichelGagn6 f o r h i s e c h n i c a l a s s i s t a n c e .

APPENDIX

From Maxw ell's equat ions we know that:

c u r l E = - Ba t (A. 1 )

where E i s t h e e l e c t r i c f i e l d , J is t h e cu r r en t d en s i ty ,B is the magnet ic f lux densi ty and p i s t h e p e r m eah l i -t y of he medium. By de fi ni ti on :

where a i s t h ee l e c t r i c a lco nd uc ti vi ty . Combining A.2)and (A.3) w e obtain:

I f h ec u r r e n t l o w s nc o a x i a lc i r c l e s , J and Ahaveonlyazfmuthalcomponents. W c n expand th e le f t -hand sideofequati on (A.5) in hecy l in d r i ca lco o r d i -na te system and ob ta in the fo l lowi ng d i f feren t ia l equa-t io n :

From Maxwell's f i el d eq ua ti on s, i t i s easy o show th at :

E = - -Aa tI f h e i e l d s are periodicwith ngular requency wthenequat ion (A.5) reduce s o

Applying t h e methodof separa t ion of he var iab le stoolv e qua tio n (A.8) it cane ~ h o w n ~ , ~hat t h em ag n e t i cv ec to rpo ten t ia l o r a f i lament u r ren t oophas hegeneral form:


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Flr9;ons rf4

n 4

Fig. 10. C i r c u l a r o o ppa ra l le l o he a c e of thed i s c .

where A , B, C and D ar e complex fun cti ons of k, J1 (k r )and N1 ( k r )a r e i r s t - o r d e r Bessel func t ionand Neumannfunc t io n e spe c t ive l y a ndwhere k l i s a complex unc-t ion ofk defined as fol lows:

kl = (k2 + j w p 0 (A. 10)Since N1 ( k r )go es o nfi ni ty when r goes ozero,

D muste se t equa l t o zeron qua t ionA.9) .Thisequa t ion i s a p a r t i c u l a r o l u t i o n of t h ed i f f e r e n t i a lequa t ion that w e want tosol ve ; he most genera lsolu-t ioncan be obta ined by int egra t io n of th ep a r t i c u l a rso lu t ion ve r a l l values ofromer o t o n f i n i t ywhich gives :

m

A(r ,z ) = [A' xp klz + B ' exp(-klz)l l(kr)dk (A.11)

where A and B a ree q u a l o A x C and B x C r e s p e c t i -ve ly.

W w l l now applyequa tio n (A.11) to he case ofaf i lamentcurre nt oop of rad ius a (Fig.10)carrying ac ur re n t I and l oc at ed at a di st anc e b above a lar ge me-t a l p l a t e of t h ic k ne s s t andof e l e c t r i c a l c o n d u c t i v i t yu , After avingonsideredhease of thei lamentcurrent oop,we will look a t t he more prac t ica l case ofa th in sp i ra l c o i l a bove a me ta l p la te o f f in i t e h i c k-ne s s .

Le tusdivide he pace round he i lament ur-rent oop nto our egions , ach ofwhich i s homoge-neous ,inear and isot rop ic . The vec t or oten t ia lneachegionasheeneralormfquation (A . 1 1 )witha ppropr ia tecoeff i c ients . The e lec tr i ca l c o n d u c t i -v i t y i s z e ro n l l e g ions xc e p t e g ion 3which i soccupied by them e t a lp l a t e .T h e r e f o r ek l i s e qua l ok in egion 1 2 and 4 .

Sinc e heve c to rpo te n t ia lA(r,z) musthave a f i -n i t ev a l u e na l l e g i o n s , A ' must vanish n egion 1because z goeso lusnf in i ty nd B ' muste madee qua l oz e ro n e g ion 4 s i n c e z goes t o minus infini-t y n h a t e g i o n . The complex functions A ' and B ' maynow bedetermined by the useof heboundaryconditionstha t hema gne t icve c to rpo te n t ia l Ae mus t s a t i s fya tthe n te r fa c e be twee n hediffe rent egions .

Ae ( i ) = A e ( i + 1) (A.12)z Ae ( i ) - ( i + 1) = . - l ~ I 6(r-a) 6(z-b) (A.13)azThe sur face urr ent of the ourc e i s confined oI 6(r-a) (z-b) inhe lane fhe i lament urrentloop and i t i n t r o d u ce s a d i s c o n t i n u i t y n a A e / a z a t z =

b . Furthermore,he e te rmina t ion of th e oef f ic ien ts

A' and B ' r e qu i re s heuse of theFourie r-Besse l nte -g r a l l o

f ( k ' ) = [ dk [ (k) l (kr) l (k ' r ) r d r (A.14)Using the onti nuit y quat ions (A.12)ndA.13)and theour ie r-Besse lnte gra l (A.14) a th eh r e esur face s, we can val uat e he oe f f ic ie n t s A ' and B '

fo r each egion nd he esul t ing xpress ions or heve c to r po te n t i a l n he four r e g ions a re g ive n by:m

J

(k kl )exp k t - (k kl )exp(- klt)(k+k,)' exp kl t (k-kl) exp( -klt )

2 2 2 21 2 exp(-kb)]dk (A.15)

A2(r,z) = l (ka) l(kr) exp(-kb)expz

(k -kl )expklt - (k -kl )exp(- klt)(k+kl)xp kl t -(k-kl)xp(-klt)

2 2 2 22 2 exp (-kz) ] dk (A. 1 6 )

mI

A3(r,z) = L I a k Jl( ka) l(k r) exp(-kb)J[(k+kl) xp kl( z+t ) - (k-kl) xp - kl(z+t)

(k+k1)2 exp k t - (k-k exp(-klt)1 1 2 ]dk ( A . 1 7 )

A4(r ,z ) = J I a Jl( ka) l(k r) exp(-kb)

[2 kl exp k(z +t)

dk (A . 18)(k+k,)2 xp kl t - (k-k,)2 exp(-klt)These are the qua t ions of the ec tor otent ia lproduced by a fil amen t urr entoop a ra l l e loheface ofa l a rge me ta l p la te o f f in i t e h ic kne s s . We will

now us e hese es ul ts oe v a l u a t e h ev e c t o rp o t e n t i a ldue to a th in s p i r a l c o i l n f ro n t o f me ta lplate off i n i t e h i c k n e s s .

Regions n04 I

Fig. 11. S p i r a lc o i lp a r a l l e l t o the a c e of t h ed i s c .95


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Apply ing the p r inc ip l e o f ine a r supe rpos i t ion , themagnetic ec tor otent ia l of a co i l of f i n i t e dimen-s i o n s is obta ined by integra t ion of he magne t ic vec torpot ent ial f a number of ilament urrentoops verthec ros s e c t ionof hec o i l .For a t h i ns p i r a lc o i le xtending rom a 1 o a 2 n he r a d ia l d i r e c t ion (F ig urel l ) , we have:

A3(r,Z) = uJc E Jl(kr) exp(-kb)[ (k+kl)exp kl(z+t) - (k-k 1 exp - kl(z+t)(k+kl)xp klt - (k-kl) exp(-klt)2 1dk (A. 25)

(A. 1 9 )mr

a lWe will onlyconsider hecasewhere hecurrent

dens i ty and the phase of hecurrent i s uniform hrough-o u t h ec r o s s e c t i o no f h ecoi l . The l ine arc u r r e n tdens i ty Jc i n a n - t u r nsp i ra lc o i l ,e a c h u rnc a r ry inga current I i s givenby:

Jc = Ia2 - al (A. 20)I no r d e r oo b t a i n h ev e c t o rpotential produced

by t h e s p i r a l c o i l n h e d i f f e r e n t r e g i o n s o f n t e r e s t ,we m ust epl ace I by Jc in qu at io ns (A.15) t o (A.18)and in tegr a te heseequa t io ns f rom the nner adius oft h e c o i l a 1 t o i t s o u t e r r a d i u s a2. The magneticvec torp o t e n t i a l n r e g i o n 1 becomes:a?A1(r,z) = - a Jl(ka)Jl(kr)exp(-kz)[exp kb +2

(k -kl )ex pklt - (k -kl )exp (-kit)(k+kl)2explt - (k-k,)2 exp (-k lt) (A. 21)

The i n t e g r a l n h e r a d i a l d i r e c t i o n i s a function

2 22-xp (-kb) ] da .dk

of k only,S(k),which i s def inedas ol lows:

2 i ' a J1(ka) da (A. 22)

al

Combining (A.21) and A.22) we obt ai n:

Jl(kr) xp(-kz) exp kb +2

(k -kl )ex pklt - (k -kl )exp( -klt)(k+k,)2 exp k1 - (k-kl)exp(-klt)

2 222 exp(-kb)]dk (A.23)

Simila r ly w e can write f o r h e o t h e r r e g i o n s :

2 Jl( kr) exp(-kb) expkz

(k2 -k12)exp kl t - (k 2 2kl )exp(-klt)(k+k1)2 exp lt - (k-kl) exp(-klt) exp (-kz) ]dk (A. 24)

2 kl e xpk(z+t)[ 2 ldk (A.26)(k+kl) xp kl t - (k-k1l2 exp(-klt)

REFERENCESJ . Bee hler nd L.D. McConnell, A new synchronousc i r c u i tr e a k e ror machinerotection , IEEETrans. PAS, Vol. 92, p. 668-672, Mar./Apr. 973.W. Knauer, R.C. Knechtiand K.T. Lian, A f a s t a c-t i n g DC c i rc u i tb re a ke r , I EEE ConferencePaper C72442-2 Summer Power Mee tin g, San F ra nc is co ,Ju ly9-14, 972.Y. Nitta and N. Kiyokuni,Synchronous ai rb l a s tc i r c u i t r e a k e ro r 1 c yc len t e r r u p t i o n , u j iE l e c t r i c Co. Ltd . Review (Japan) 11 pp. 95-103,1965.A.T. Freeman, Fault curr ents im ited by ul t ra hi ghspeedd.c.breaker ,ElectricalTimes,157,pp. 57-60,16April, 1970.S Basu and K.D. Srivas tava , qna lys is f a fa s tact ing cir cui t bre ake r mechanism , I EEE Trans. PAS,Vo l . 91, p. 1197-1203, May/June 972.C.B. Wheelerand A.E. Dangor, Multiplycrowbarredsolenoids orplasma esearch , J . .Phys. E , 6,pp.332-338, 1973.P.J.Rogers nd H.R. Whitt le , E lec tromagnetica l lyac tua ted,as t-c los ing witch s ing olythene sthe main die lec tr ic , roceedings IEE , 116,p.173-179, Jan. 969.K.H. Panofsky nd M. P h i l l i p s ,C l a s s i c a lE l e c t r i -c i ty andMagnetism. London: Addison-Wesley, 1962,pp. 53-156.P . Hannnond, The ca lcu la t io nof hemagnetic ie ldof otatingmachines ,Proceedings I E E , 109'2, pp.508-515, Ap ri l 1962.P.M. Morse and H. Feshbach, Methods ofTheoreticalPhysics. New York: McGraw H i l l , 1953, . 766.

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