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
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
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 48 848 69 848 98 88 888 7889 9 9497 879 4 4499 746 9746 8674 88466 869 888967 989 69 969 9989 97 848 874 6 7947 8 8 699 4868 69 84 7 76 64 64 84 9944 86 7748 8 797 4 6876 4 46 77 9 499 4967 967 499 686 89 668 4946 7 7 697 9 89789 64446 49744 667 67 8889 94688 8879 68887 464877 79 49 4747 494 47 767 4847 794 494 786 978 4 86864 4 88 879 79 6966 67869 4 9479 4696 8799 799 4687 898 749 98 6 4947 979 679 78 99 896 888 7 6997 9998 6998 74 86497 96 6444 ble : Prmeer vlues @ coefficies 7 τ [s φ Ψ F F [mm i 7664 87664 768498 87664 46 4478 94478 89849 94 9896 98 4 67 447 447 696 7 796 946679 466 9979 9979 449 4769 974 6 47467 4 74 768 94864 847 7 8 4949 949 466 6878 768 767 8 468 847 847 64966 466949 7876 646 9 68 694 94 946 764874 87 449 ble : Prmeer vlues @ coefficies 7 96
ls of he iversi of Criov lecricl gieerig series No 6 τ [s φ Ψ F F [mm i 88 888 78 4 888 79 89 989 49 67 9644 94894 4 7 86 986 4 848 8676 9 74 897 9897 47 76 67499 994 6 89 69 69 449 9 498 878 7 66 468 468 66 448 98 866 8 68467 88 88 84 694 799 796 9 96 6946 6946 846 4 7796 49 899 899 78 4 4866 7746 49 447 47 98 68 4469 8974 467 48767 8767 78 449 86697 8846 69447 8 8 787 66 78749 9668 4 4 477 477 9688 899 9446 46 4 478 478 774 689 9869 4978 6 69 687 687 6786 78 8644 6 7 6988 786 786 48 4887 94 886469 8 49898 678 478 7 6996 7996 79 9 6477 6774 4774 4878 9 4 4977 ble : Prmeer vlues @ 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 chrcerisics @ 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 chrcerisics @ coeffcie soli lie coeffcie sh lie 97
ls of he iversi of Criov lecricl gieerig series No 6 4 6 K 4 m [ms 4 m Figure 4: Dmic sfe cig coefficie vs g m [ms [mm [ms m [ms [ms 948 748 646 6 4768 984 47 87 89 484 648 7874 47 4 6 74 9884 7778 9889 699 ble 4: m f [ms m [ms [ms 948 748 7 478 9884 89 8 8448 466 49 9 m ble : m f 4 6 8 Figure : he cig ime soli lie movig ime m sh lie srig ime sh o lie @ 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 7 7 8 8 9 Figure 6: he cig ime soli lie movig ime m sh lie srig ime sh o lie @ 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 97-796-- Isi [ Hiuc P eoe rboi ele sece rivi fuciore i regim imic elecromgeilor e cure coiuu uleiul Siiific l Isiuului Poliehic Cluj-Noc r 979 9-9 [4 Hiuc he rivig coiio for he irec curre elecromges uleiul Isiuului Poliehic Isi om XXVIIXXXI Fsc - Seci III- 98 6-7 [ Figerl C Kigsle lecric Mchier he Dmics Sics of lecromechicl erg Coversio r iio Mc rw-hill ook Co New York 97 98