DYNAMIC ELECTROMECHANIC CHARACTERISTIC OF DC ELECTROMAGNETS WITH NON-NULL STARTING FLUX
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1 ls of he iversi of Criov lecricl gieerig series No 6 DYNMIC COMCHNIC CHCISIC OF DC COMNS WIH NON-N SIN FX uge HNIC ri PŞC og HNIC h schi echicl iversi of Isi ehiuc@eeuisiro; lesc@eeuisiro; bhiuc@eeuisiro bsrc he elecromges behviour for iffere vlues of he g usull is esime from he elecromechic chrcerisic io se se coiios lhough urig movble rmure islceme he rcio mic force hs less vlues Cosierig he DC elecromge wih oull srig flux where he iiil osiio of he movble rmure is o gree wih he mechicl equilibrium like elecromechic eerg coversio ssem wih wo freeom egrees i c efie he olier iffereil equios ssem which escribes is behviour herefore i c esblish he ime evoluio of he mic rcio force he ormlie vlues he rcio force io se se coiios lso usig ormlie vlues which llow comriso bewee hese wo es of chrcerisics Kewors: elecromges se-se mic elecromechic chrcerisics INODCION For he elecromges which re oerig wih oull srig flux NNSF he iiil osiio of he movble rmure mximum g is o gree wih he mechicl equilibrium here is iiil resisce force F he cig of hese elecromges is ossible ol if he ol mgeic flux becuse of he ower sul for he coil elecric circui excees he srig mgeic flux Ψ : F where mes cosrucive cos of he elecromge which efies he vriio of he coil iucce gis g vlues: where is equivle g for he mgeic circui of he elecromge he coil iucce vries from he vlue he vlue > see relio he rivig rsie regime for DC elecromge wih NNSF Fig hs hree ses: - firs se srig se from he coil sulie wih DC volge ill he mome whe he ol mgeic flux hs he srig vlue Ψ ; he movble rmure is sill i he osiio of mximum g he coil iucce is ; - seco se whe he movble rmure goes from he iiil osiio of mximum g o fil osiio of he g where he coil iucce icreses from o ; he e of his se he mechic cio of he elecromge is fiishe he cig ime beig he sum bewee srig ime movig ime m : m - hir se whe here is he se-se elecric coiios for he elecromge coil wih rce movble rmure he coil iucce he vlue Figure : he rsie regime for DC elecromge wih NNSF Cosierig he elecric curre io se-se coiios which flows hrough he coil urs of he elecromge wih NNSF wih he resisce sulie from DC volge I 4 he ol mgeic flux Ψ io se-se coiios ees b he g: I I I M I he ormlie vlue of he ol mgeic flux io se-se coiios Ψ is efie b he relio see Fig K F 94
2 6 hece for he ormlie elecromechic chrcerisic io se se coiios here is he relio: k F 7 where mes he ormlie islceme of he movble rmure see Fig is he movble rmure islceme ssumig efie he vlues of he ol mgeic flux io mic coiios urig movble rmure islceme Ψ or Ψ cosierig is ormlie vlues: or 8 he ormlie elecromechic chrcerisic io mic coiios F is esime b he exressio: F 9 llow he comriso wih he ormlie elecromechic chrcerisic io se se coiios F see relio 7 H DC COMN OPIN WIH NON-N SIN FX INO DYNMIC CONDIIONS ccorig o he firs se of he cig rsie regime of DC elecromge wih o-ull srig flux he ime evoluio of he elecric curre which flows hrough coil urs is give b he relio: e I i hece he srig ime c be efie b he equio: e I i ssumig he vrible exchge: cosierig he elecromge like elecromechic eerg coversio ssem wih wo freeom egrees he islceme of he movble rmure he ol mgeic flux Ψ resecivel φ he o-lier iffereil equios ssem which escribes is behviour urig movble rmure islceme is wrie like [: Ω C where Ω C re coss resec o elecromge is coil cosrucio: C M M K Ω 4 he rmeers hve he sigificce meioe before Wih he oios: he o-lier iffereil equios ssem 4 c be wrie like: Ω [ 6 wih he ormlie vlues of mic ol mgeic flux Ψ : 7 where mes he ormlie movble rmure islceme see relio sig he fiie iffereces meho he equios ssem 6 becomes: Ω [ τ τ 8 for imose iegrio se τ - [s he soluio is: 9 ls of he iversi of Criov lecricl gieerig series No 6
3 ls of he iversi of Criov lecricl gieerig series No 6 τ Ω τ [ 9 he movig ime of he elecromge movble rmure m efies kig io cosierio he coiio: m herefore for iffere oerig coiios he cig ime c c be evlue wih he relio Cosierig he ormlie vlues of he mic mgeic flux Ψ see relio 7 i c evlue he ormlie mic rcio force F he relios 8 9 he ormlie mic elecromechic chrcerisic F which comres wih he ormlie elecromechic chrcerisic io se-se coiios F relio 7 lso i c obi he ormlie curre evoluio i which flows hrough he coil urs urig movble rmure islceme: he ime evoluio of he g i NYSIS OF H NSIN IM HVIO OF H DC COMN WIH NNSF I cosiers rel DC elecromge wih NNSF where: Ω [ r / s [/ s 6[/ s [ mm 4[ mm he volge sul : mes he orml volge sul τ [s φ Ψ F F [mm i ble : Prmeer coefficies 7 τ [s φ Ψ F F [mm i ble : Prmeer coefficies 7 96
4 ls of he iversi of Criov lecricl gieerig series No 6 τ [s φ Ψ F F [mm i ble : Prmeer coefficies 8 he comuio resuls for he soluios 9 cosierig iffere srig coiios iffere vlues for lso iffere vlues of he volge sul iffere vlues for re reore i he bles he curves F show i Fig Fig for iffere vlues of he oulies he ifluece of he volge sul uo mic behviour of he DC elecromge wih NNSF he oscillor effec of he movble rmure islceme for is chge i he cse of F K 4 F i rereses he curves K Fig4 i oes h is suer-ui vlues re smller h he cse of he sme elecromge oerig wih ull srig flux If i esimes he srig ime movig ime m cig ime see he relios for iffere mic oerig coiios of he DC elecromges wih NNSF i obis he vlues reore io bles 4 he curves re show i Fig Fig6 [mm F [ms Figure : evoluio vs ime Comriso bewee coeffcie soli lie coeffcie sh lie If i cosiers he mic sfe cig coefficie K efie b he relio: [ms Figure : Normlie mic chrcerisic vs ime Comriso bewee coeffcie soli lie coeffcie sh lie 97
5 ls of he iversi of Criov lecricl gieerig series No K 4 m [ms 4 m Figure 4: Dmic sfe cig coefficie vs g m [ms [mm [ms m [ms [ms ble 4: m f [ms m [ms [ms m ble : m f Figure : he cig ime soli lie movig ime m sh lie srig ime sh o iffere coeffcies Ceril here is ecresig of he srig movig cig imes whe he volge sul is icresig higher vlues lso heir ecresig hev srig coiios higher vlues Figure 6: he cig ime soli lie movig ime m sh lie srig ime sh o iffere coefficies 4 CONCSIONS his er rooses mhemicl moel o su he rsie regime of he DC elecromges wih NNSF where he re cosiere like elecromechic eerg coversio ssem wih wo freeom egrees escribe b o-lier iffereil equios ssem I efies he ime evoluio urig elecromges rivig of some rmeers which chrcerie he ssembl behviour he movble rmure islceme ormlie mic mgeic flux which llow o esime he ormlie mic elecromechic chrcerisic oulies he volge sul ifluece srig coiios I comres he ormlie elecromechic chrcerisic io se se coiios wih ormlie mic elecromechic chrcerisic i efies he mic sfe cig coefficie imor rmeer whe esigig hese kis of ssemblies for ceri imose mic erformces efereces [ Hiuc he su of he vibrio coiios o DC C elecromges lecromges mic coiios PhD hesis Isi 98 [ urlic Hiuc Hiuc re elecrice e comuie cioe cu elecromgei Cs e iur Veus ISN Isi [ Hiuc P eoe rboi ele sece rivi fuciore i regim imic elecromgeilor e cure coiuu uleiul Siiific l Isiuului Poliehic Cluj-Noc r [4 Hiuc he rivig coiio for he irec curre elecromges uleiul Isiuului Poliehic Isi om XXVIIXXXI Fsc - Seci III [ Figerl C Kigsle lecric Mchier he Dmics Sics of lecromechicl erg Coversio r iio Mc rw-hill ook Co New York 97 98
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