Mathematical Model of Acceleration Stage of Magnetic Inductive Pulsed Plasma Thruster

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1 Mahacal Modl of Acclao Sag of Magc ducv Pulsd Plasa Thus EPC--76 Psd a h d aoal Elcc Poulso Cofc, Wsbad Gay Sb 5, Haf Daas Salh ad Sgy Yuyovch so aoal Aosac Uvsy (Khav Avao su), Khav, 67, Ua, Chalova, 7 Absac: Mahacal odl of acclao chas agc ducv ulsd lasa hus ad ossbl ods of oao a sd. Th ahacal odl basd o quas-o dsoal odl. Plasa s cosdd as a lasa sh wh ow bouday codos. O-coo ago hydodyac odl awo-coo lasa dyac odl a usd as basc quaos Mahacal odl. A B E J T T v q q q δ Π oclau = lcoagc fld vco oal = agc duco = lcc flso = cu dsy = c so chag aalogu = ducvy, uual ducvy = lco ass = o ass = lasa coos dss = lasa bul oo = ssac = = lco au = o au = a lcos vlocy odul = lasa ass flow vlocy = lcos ass flow vlocy = os ass flow vlocy = lasa gy flow dsy = lcos gy flow dsy = os gy flow dsy = uay (val) so = lasa c so MSc, Poulsos ad ow us of saccaf, haf.daas@gal.co Assoca ofsso, Poulsos ad ow us of saccaf, chsa@d4.ha.du Th d aoal Elcc Poulso Cofc, Wsbad, Gay Sb 5,

2 . oduco Puls lasa huss s h so of o coo class of agc lasa dyacs acclaos wh aal foc, whch acclas lasa ad gas h hus, aas bcaus of aco of lasa cus ad agc fld wh saa acls ov sla o Hall's df. aal syc goy aal acclao of lasa ca a lac wo ossbl cass: - Azuh agc fld B wh adal lcc fld E ; - Azuh lcc fld E wh adal agc fld B. Th fs cobao s ossbl wh lcods a ad as oao fgus cos, cylds. Azuh agc fld h s gad by aal cu sh cal lcod (usually cahod). Acclaos wh hs goy ca oa boh saoay ad uls ods. Th scod cobao s ossbl wh solods sys us. Magc fld h us cssay b vaabl bcaus of h fac ha azuh (closd) lcc fld ca b gad by oly vaabl agc fld. Ths a s uoso dsco of ocsss sd agc ducv uls lasa hus, whch las o h las acclao sd abov. Puls ducv acclao od s sd o Fg.. Ma souc of uls acclao s h duco agc col fdg by codss bay. Af codsduco ccu swch-o vaabl cu s gad sh duco. Ths cu ducs vaabl aal syc agc fld, whch u ducs vaabl azuh lcc fld. Th las o gas azuh cu lasa. Plasa aas sh acclao chal fo oz wh ual gas s ozd by gas dschag ud aco Fgu. Magc ducv uls lasa hus sucu. f al of hgh fqucy lcoagc oscllaos. Th aal foc, whch acclas lasa oushus chal, aas as a sul of lasa azuh cu aco wh adal coo of duco agc fld. Thus h acclao od agc ducv uls lasa hus (MPTT) s vy sla o co-aal uls lasa hus oao od. Th oly dffc s bw fld ad cus goy ah cl dffc lays h absc of lcod dschag MPPT (as wll as h absc of lcods dd). Ths fac las h cssy of lcos sso acclao chal sufac au MPPT ay b low. O o sucu l of MPPT show o Fg. s agc ozzl h sys of al cols fdg by dc volag. Ma uos of hs sys s o oc acclao chal wall fo hag ad o bobad. Also agc ozzl ca ga addoal acclav foc lasa. MPPT sucu s o ogaz boh h uls ah quas-saoay oao ods. Th las o as h fdg of duco by susodal volag wh lasa s acclad by saa oos ovg sd hus chal closly o af aoh. Ths od s o oa wh cosa ass flow a as wll as wh Th d aoal Elcc Poulso Cofc, Wsbad, Gay Sb 5,

3 cosa oz oducvy. Bu hs wo s uoso uls oao od dsco wh gas fd sys ad oz also oa uls gs.. M P P T MATHEMATCA MODE Two-coo lasa dyacs quaos s (f. ) O-coo agc hydodyacs (MHD) quaos s ca b obad o h bas of wocoos lasa dyacs, whch dscbs lasa fod by lcos ad sgl-chag osv os. Twocoos lasa dyacs quaos s cluds: - subsac quaos: wh lcos ad os dss chag u of collsos; - oo quaos:,, () ; () E B, () E B ; (4) wh oo dsy chag u of collsos; - gy quaos: wh, T E T q E gy dsy chag u of collsos. Moo ad gy flow dss o-dssav aoao:,,,, T,, (). (4), (7),, 5 q T,,,, (8) Th d aoal Elcc Poulso Cofc, Wsbad, Gay Sb 5,

4 wh uay so:. (9) O-coo agc hydodyacs quaos s Faly llg, h dsco of ul-coo subsac as o-coo o s o duc h ub of aas (ad cosquly h ub of quaos) oly fo qulbu bw h coos boh by aus ad ass flow vlocs. Accodg o lasa h sc of such qulbu would by h way a h absc of cu coos flow dss dffc. Th aoao of o-coo MHD s h coos, whch s o slfy h dsco oly sval acula cass, cludg also h flows sd agc lasa dyacs huss boh saoay ad uls os. dsy : s ossbl o ass fo coos ass flow vlocs o lasa ass flow vlocy ad cu Suay au s:, (). () T T T. () s ossbl o w fo (), () fo ad :, (). (4) Suay ad dffc of subsac quaos (), () gv wh cosd of (), (): - O-coo MHD subsac quao: ; (5) - O-coo MHD chag cosvao law:. (6) Suay ad dffc (wh dvso by ass h las cas) of oo quaos (), (4) gv wh 4 Th d aoal Elcc Poulso Cofc, Wsbad, Gay Sb 5,

5 Th d aoal Elcc Poulso Cofc, Wsbad, Gay Sb 5, 5 cosd of (), (): - O-coo MHD oo quao: B ; (7) - O-coo MHD Oh's law: B B E J. (8) Plasa oo ad cu dss chag collsos a qual:, (9). () Plasa c so a's chag aalogu J a:, () J. () O ca s ha subsac ad chag cosvao quaos (5), (6) a dd o ach oh. Oly ass aso s sd (5) ad oly chag aso (6). Th lao of o-coo MHD hod us s dd by h fac ha ul (5), (6) oo quao (7) ad Oh's law (8) a o dd os. Boh hs quaos clud vy col fo h dscos of boh ass ad chag aso. Fo al fo coos c sos follows fo (7), (), (4): T, () T. (4) So h ssos fo ad J a: T T, (5)

6 T T J. (6) Suffc slfcao of o-coo MHD ssos s ossbl wh cosd of boh h uvsal oy: ah scfcs of flows agc lasa dyacs huss: - ass ad chag aso alos cossd dcos: - ass aso vlocy ough lss h chag aso o:, (7) ; (8) ; (9) - howv, ass aso vlocy ough lag o b ossbl o cosd:. () Cosdg () s ossbl o slfy h ssos (), (4):, (). () Cosdao of (8), () gvs (5), (6): whl h oo quao (7) ad Oh's law (8) oba h fos: T, () T J, (4) B, (5) J E B B. (6) s ossbl o s ha ud show codos so () cluds oly ass aso dscos. 6 Th d aoal Elcc Poulso Cofc, Wsbad, Gay Sb 5,

7 Th d aoal Elcc Poulso Cofc, Wsbad, Gay Sb 5, 7 c vsa, so J (4) cluds h dscos of boh asos ogh wh lco ssu. Th las fac as h cssy o a saaly lcos gy quao fo lcos au T sach ogh wh lasa gy quao (h suay of aal os) fo suay au T. Th, cosdg h o-quvalcs show bfo, s ossbl o sa:, (7), (8), (9). (4) Thus fo (5), (6) s ossbl o w lasa ad lco coo gy quaos: E q T, (4) E q T, (4) wh lasa gy dsy chag collsos s qual o:, (4) lcos gy flow dsy: T T q 5 5, (44) ad lasa gy flow dsy: T T q 5 5. (45) Cosdg (6) ad glcg all aas chag cu dco (cossd wh ass aso dco) s ossbl o w T q 5, (46) T q 5. (47)

8 Sall ass of lcos s o duc col (4), (47). So s ossbl o w: T 5 T E 5 T T E, (48), (49) O-coo MHD quaos s cludg: - subsac quao (5); - chag cosvao law (6); - oo quao (5); - Oh's law (6); - lasa gy quao (48); - lcos gy quao (49) us b cobd wh agc fld oo quao. Elcc fld chag suffcly flcs agc fld bhavo oly ud lavsc codos. So v o-saoay bu o-lavsc cas s ossbl o us saoay fo of oo quao: c B. (5) al quaos s Mahacs odl of lasa acclao MPPT cluds wo quaos ss: - h quaos, whch dscb lcoagc fld bhavo: - lcc flso E ad agc duco B laos wh scala ad vco oals A : - Poso's quao fo vco oal: - o-coo MHD (5), (6), (5), (6), (48), (49): - subsac quao : A E, (5) B A, (5) A c ; (5) - chag cosvao law ( quas-ual volu):, (54), (55) 8 Th d aoal Elcc Poulso Cofc, Wsbad, Gay Sb 5,

9 - lasa oo quao: T B, (56) - o-coo MHD Oh's law: E B v T B, (57) wh v a lcos vlocy odul; a lco-o aso coss-sco. Plasa acclao quaos s (54) (57) s obad fo (5), (6), (5), (6) fo colly ozd lasa wh cosdao of oo chag lco-os collsos. Quas-o-dso odl us s h so aa P a-by-adus valu: P P, d, (58) wh, aal ad adal coodas cyldcal cooda sys; acclao chal al bouday. Ay vco A dvgc ud aal syy s qual o: gao (58) of (59) gh a scod coo gvs: A A A. (59) A, d A,. (6) Tag o cosdao lasa flow o sulao adal bouday wh o-soc vlocy: T s, (6) w ca w fo (54), (56), (57) usg (58) (6): 9 Th d aoal Elcc Poulso Cofc, Wsbad, Gay Sb 5,

10 T T T B, (6), (6) E B. (64) wh h lao of lcos cocao o al bouday o a-by-adus o; azuh cooda cyldcal cooda sys; lasa lcc coducvy:. (65) v Equao (6) ca b w such fo: T T B. (66) Plasa sh odl Plasa sh odl cosds lasa l f volu, whch occus ay sall a of hus chal lgh. Magc fld ad lasa dsy cofguao sd MPPT a show o h fg.. Fgu. Magc fld ad lasa dsy cofguao sd MPPT. us suos ha a h lasa occus h aal cooda ag:. (67) us also cosd h ssu o sh boudas l qual o zo: Th d aoal Elcc Poulso Cofc, Wsbad, Gay Sb 5,

11 P P. (68) Plasa sh ass M ad oo a qual: M d, (69) M d. (7) Ay aa A valu sd lasa volu also ca b foud l gal of hs aa dsy A () : A A d. (7) Paa A full dvav by ca b w as: d A A d d d A A. (7) Cosdg h fac ha lasa sh aal boudas chags uay a qual o lasa ass flow vlocs: d, (7) s ossbl o w fo (7): A A A d. (74) d Cosdg (5), (68), (74) h ssos (6) ad (66) ca b asfoo h os: d d M T M, (75) M M T A d. (76) gh a of (76) ca b w s of lcc gg (s Ad A): Th d aoal Elcc Poulso Cofc, Wsbad, Gay Sb 5, A d d, d (77)

12 wh h cu sh lasa sh; h cu sd -ag (duco wh = o al col wh ); Plasa sh ad -ag uual ducvy; Plasa sh acual cooda (fg..). Th cobao of (75) (77) gvs lasa sh acclao quao: M d d d, (78) whch us b usogh wh Oh's law fo lasa sh: d d (79) Fo ach al col ad fo duco (s Ad A): d d U (8) ad chag cosvao law: Wh du C, (8) U h volag o duco codss (=) aal col volag souc (); C duco codss caacy; Plasa sh ducvy; -ag ad -ag () uual ducvy; Th ducvy of -ag; Plasa sh ssac; -ag lcc ccu ssac. Cosdg h lao d, (8) Plasa sh ad -ag uual ducvy dvav by ca b foud fo h followg sso: d d. (8) d Egy asf dug acclao ocss Th oduc of (79) ad lasa sh cu gvs h sso: Th d aoal Elcc Poulso Cofc, Wsbad, Gay Sb 5,

13 d d d. (84) Also h oduc of (8) by -ag wh followg suay fo all -ag gvs h sso: d d Th followg syy wll b show la Ad A: Cosdg (86) s ossbl o w: U. (85). (86) d d d d d. (87) Th cobao of (78) ad (8) gvs h sso: d M d, (88) Th suay of (84), (85) wh cosd of (75), (8), (87) ad (88) gvs h gy asf quao: d Wc W W (89) Wh, c, lcc ad agc gs ad lasa sh c gy; W, W, W al cols ow cosuo, ssv gy los scod ad lasa sh c gy los scod: CU, (9), (9) M ; (9) Wc U, (9) Th d aoal Elcc Poulso Cofc, Wsbad, Gay Sb 5,

14 W, (94) W T M. (95) Th ssos (9), (9) wll b cosdd Ad A whl acclao ocss lcc gg coffcs dfo. Egy quaos Th quaos s (78) (8) s vy sla o so calld Asovch's quaos s fo uls lasa huss wh aal syc agc fld. Also l Asovch's o h quaos s (78) (8) us b cold by subsac quao (75) ogh wh lasa ad lcos gy quaos. Egy quaos ca b obad fo (48), (49) cosdg gy los ozao ocsss. Quas-o-dso fo of gy quao lcc gg s s as follows: M d T d d, (96) M d T d d. (97) Zo-aoao Th os coo faus of MPPT oao ca b dosad by slfd (78) (8) quaos s whou cosdao of al ags: d M d, (98) d d d (99) d d U () du C, () d, () d d. () d Th aalyss of (98) () quaos s shows ha cods volag, duco ad lasa sh cus 4 Th d aoal Elcc Poulso Cofc, Wsbad, Gay Sb 5,

15 bhavo s of oscllao au wh dcasg aguds (s Fg. ). Ddg o cods caacy, duco ad lasa sh slf ad uual ducvs h acclao ocss dug o uls ca clud sval o lss ha o oscllaos od. Fo al, h suao show o fg. las o h cas of.5 s-ods of oscllaos wh h volag ad boh h cus chag h sg wo s. Th lgh of acclao chal us b chos cosdg h of ffcv acclao wh cosdabl acclao y as lac. Eal cols aco Th aco of al cols ca sul wo dff ffcs: - o oc acclao chal al wall fo dc aco of ho lasa as wll as fo fas os bobad; - o oduc addv lasa sh acclao. Th fs ffc s dd by aal coo of al cols Fgu. MPPT aas chag dug o uls. agc fld. Th scod ffc s dd by adal coo of agc fld. Ddg o cobao of ad B sgs h sg of addoal acclao ca b dff. Fo al, h cobao show o Fg. 4 dosas h ossbly o ovh suao wh aal B chag h sg h sa addoal acclao as o b osv va acclao chal lgh. So h vaao of al cols goy ad fd volag ca ovd h cobao wh h os ffcv addoal acclao sd agc ducv uls lasa hus. Fgu 4. Colao ad B sg. 5 Th d aoal Elcc Poulso Cofc, Wsbad, Gay Sb 5,

16 Cocluso Th oao od dscbd hgh s o h oly ossbl fo show acclao sucu. Also quassaoay oao od s cosdo b alzd wh haoc volag fo duco ow suly. Gas fdg ad ozao sys h ca oa saoay g. Th ahacal odl of quas-saoay oao od s ow ud dvlo wh Fou ss us dscbg all h aas l h ss of duco fd volag fqucy haocs. APPEDX MPPT lcc gg coffcs (f. ) so of (5) o (64) gvs h sso (d fuh sso wll o b w, cosdg oly azuh dco of cu): A A. (A.) Mullyg (A.) by ad usg h dfo of full dvav by s ossbl o w: d A. (A.) gao of (A.) by h volu occud by cu gvs h gal fo of Oh's law: d d A d d, (A.) wh d d d d. (A.4) So fally w hav: d d d d A d d d d d d. (A.5) f s h volu of lcoag col s ossbl o show h volag do va o loo: d U (A.6) as wll as fo col cu A-loos: 6 Th d aoal Elcc Poulso Cofc, Wsbad, Gay Sb 5,

17 d d. (A.7) W : Esso (A.5) shows h dvso of col fd ow U o agc ow M W ad ssv losss wh: U M W W, (A.8) W d d d, (A.9) W M d A d d d. (A.) Elcc gg sso fo W s wll ow: W. (A.) So w ca w fo -ag ad lasa sh ssac (fg: ): d d d d, d d. (A.4) d d Gal soluo of Poso's quao (5) fo vco oal A s: A 4 d c 4 c d. (A.5) Azuh coo of A ay o s: A 4 c 4 c d cos d cos, (A.6) 7 Th d aoal Elcc Poulso Cofc, Wsbad, Gay Sb 5,

18 wh, azuh os os ad ; h agl bw h os,. f cu occus so saa volus s ossbl o w: A A,, (A.7) wh A, 4 c cos d d d. (A.8) cos Thus (A.8) fo saa volu ca b w as follows: M M W W, (A.9) wh W M d A, Th coaso of (A.8) ad (A.) wh (.) s o w: wh uual ducvy of -volu ad -volu s: Paal gao by d d d. (A.) 8 Th d aoal Elcc Poulso Cofc, Wsbad, Gay Sb 5, M d W, (A.) cos d d d d d c d d (A.) gvs h sso: 8 c cos d d 4 d d d d. (A.) d d d d, (A.)

19 wh Th hod of 4 cos d 6 s d. cos (A.4) s calculao s show Ad B. Muual ducvy of -ag ad lasa sh ca b w fo (A.): ha also as wh 8 c 4 d d d d 8 c d d d d, (A.5). (A.6) 4 d d d d d d d. (A.7) Essos (A.4), (A.), (A.5) ad (A.7) us b usd wh so suosos abou cu dsy adal dsbuo sh ags ad lasa sh. Th sls bu adqua o h co of ocsss a h followg suosos: Cos, ~. (A.8) as h followg quvalcs:, 8, (A.9) 8 c d d d d 4, (A.) 9 Th d aoal Elcc Poulso Cofc, Wsbad, Gay Sb 5,

20 4 c 4 d d d d, (A.) 4. 4 c d d d d (A.) Fuco Fuco calculao usd fo ducvs calculao s dfd by sso (A.4): s ossbl o show fo (A.) ha wh al codo: So 6 s d. (A.) s s h soluo f dffal quao: d d 9 4 (A.4) d d 4 ca b sd by sso:. (A.5) 8!!. (A.6) as ha whl couao fuco ca b fd usg h followg algoh: wh K,, (A.7), (A.8) ad K h ub of suay coo, whch alady gvs h cssay accuacy : K. (A.9) Th d aoal Elcc Poulso Cofc, Wsbad, Gay Sb 5,

21 Th ss (A.6) s hocally absoluly cocdg o h ag of agu (A.4) wh logahcal sgulay a =. Bu whl couao h aaaus o lads o o-cocdc of (A.6) wh. Th followg asyoc ss ca b ush: o wh l! 8 K C C, C C, (A.4) l, (A.4),,. (A.4) fcs Daas, H., ad so, S.Yu., Magc duco uls lasa hus aual fo ss ad dloa dsg, Hadboo. aoal Aosac uvsy ( Khav Avao su), Ua (o b ublshd) Dcovc, YU.., ad Kocoaov D.P., Plasa Mahacal Modlg, d d., aya, Moscow, 99, Chas.. Th d aoal Elcc Poulso Cofc, Wsbad, Gay Sb 5,