Simple Performance Modeling of a Radio-Frequency Ion Thruster

Transcription

1 Smpl Prformanc Modlng of a Rado-Frquncy Ion Thrustr IEPC-7-7 Prsntd at th 3 th Intrnatonal Elctrc Propulson Confrnc, Flornc, Italy Sptmbr 17-, 7 Mchal M. Tsay * and Manul Martnz-Sanchz. Massachustts Insttut of Tchnology, Cambrdg, MA, 139, U.S.A. Abstract: A smpl prformanc modl s dvlopd for an nductvly-coupld plasma (ICP) dscharg, rado-frquncy (RF) on thrustr. Mthods of partcl and nrgy balanc ar utlzd for modlng th ICP dscharg. A lumpd-lmnt crcut analyss, a transformr modl, and a transmsson ln modl ar ncorporatd n th quatons of nrgy balanc for stmatng th absorbd powr by th plasma. In th transformr modl, th short solnod s tratd as a two-turn prmary col and th plasma as a on-turn scondary ar col. Th prmary and mutual nductanc ar approxmatd by thors of sngl-layr cols. Th ICP dscharg modl s succssful n prdctng th rsonant frquncy of th systm wth accuracy wthn.1 MHz. Th rflctd powr of th systm s prdctd wth accuracy wthn 3% and ts trnd s also valdatd. Th anod currnt of th thrustr s stmatd by a smpl on-dmnsonal on xtracton modl, whch ylds accuracy wthn 1% of th masurd valus. k R L σ δ ω ρ m n n n T ν ff ν on σ n u B V total φ φ w Nomnclatur = Boltzmann s constant = Chambr radus = Chambr lngth = Plasma conductvty = Skn dpth = Drvng frquncy = Rflcton coffcnt = Ion mass = Elctron dnsty = Nutral dnsty = Elctron tmpratur = Effctv lastc collson frquncy for lctrons = Ionzaton frquncy = Elctron-nutral total scattrng cross scton = Bohm vlocty = Appld ntr-grd potntal = Ion transparncy of th grd = Potntal drop across th shath at th wall * Doctoral studnt, Dpartmnt of Aronautcs and Astronautcs, mmtsay@mt.du. Profssor, Dpartmnt of Aronautcs and Astronautcs, mmart@mt.du. 1 Th 3 th Intrnatonal Elctrc Propulson Confrnc, Flornc, Italy Sptmbr 17-, 7

2 I. Introducton A. Projct background ICP RF on thrustrs wr frst nvntd at th Gssn Unvrsty of Grmany n th 196s. Snc thn, th Europan Aronautc Dfns and Spac Company (EADS) Spac Transportaton GmbH and ts ndustral support, th Grman Astrum GmbH group, hav adaptd ths concpt and succssfully dvlopd a famly of spac-qualfd RF thrustrs known as th Rado-frquncy Ion Thrustr (RIT) srs. 1 Ths typ of on thrustrs hav snc shard a hstory of succssful flght opratons. ICP RF on thrustrs ar vry smlar to th tradtonal drct-currnt (DC) bombardmnt-typ on thrustrs n trms of thr lctrostatc on acclraton mchansm. Th major dffrnc btwn thm s, unlk thr DC countrparts, th ICP RF on thrustrs utlz lctromagntc flds nducd by hgh-frquncy col currnts to achv and sustan onzaton. Durng such procss, th magntc flds nducd by th RF col and th azmuthal plasma currnt ar coupld, thus rcvng ts nam of nductvly-coupld on thrustr. Du to th absnc of th lf-lmtng ntrnal hot cathod, th ICP RF on thrustrs hav th potntal to achv long opratonal lf whl prsrvng othr favorabl charactrstcs of a typcal DC on thrustr such as hgh spcfc mpuls (>3 sconds), hgh propllant utlzaton (>85%), and hgh lctrc ffcncy (>8%). B. Projct motvaton and objctvs Ths projct s motvatd by th smpl prformanc modl dvlopd for DC on thrustrs by Brophy and Wlbur n 198s. Ths modl, wll known as th Brophy s modl, conssts of a partcl and nrgy balanc calculaton that rlats thrustr prformanc to adjustabl opratonal paramtrs such as mass flow rat and dscharg potntal. 3 Insprd by Brophy s approach, w dcdd to us smlar mthod to modl th ICP RF on thrustr. Our objctv s to dvlop a smpl on-dmnsonal zro-th ordr mathmatcal modl that can captur th physcs of an ICP dscharg, analyz and optmz th matchng ntwork, and ultmatly prdct th thrustr prformanc. Th rsults of our modl ar valdatd wth th xprmntal data provdd by Busk Co., Inc. Th xprmntal subjct, shown n Fg. 1, s a proof-of-concpt vrson of th RF on thrustr dvlopd by Busk. Ths prototyp dsgn rsmbls th confguraton of a Grman RIT srs thrustr wth th scrn grd srvng as anod and xnon as propllant. A capactv matchng ntwork was placd n paralll wth th col crcut. In th xprmnts, sourc frquncy was adjustd to sk rsonanc at ach opratng condton. Fgur 1. Concptual arrangmnt of th ICP RF on thrustr. Fgur dos not bar dmnsonal accuracy. II. Thory Th thory scton contans thr major parts: th ICP dscharg modl, a matchng crcut analyss, and a dscusson on th on xtracton modl. In th ICP dscharg modl, bulk plasma proprts ar calculatd by mans of a partcl balanc and an nrgy balanc. Th matchng crcut analyss s usd for calculatng th RF powr absorbd by th plasma, as wll as for prdctng th rsonant frquncy and th rflctd powr. W nclud a smpl on-dmnsonal on xtracton modl for comparson wth th anod currnt data takn from xprmnts. A. Inductvly-coupld plasma dscharg modl Th ICP dscharg modl s dvlopd basd on a smlar modl publshd n 1994 by Lbrman and Gottscho. In our modl, lctron dnsty s consdrd to b volum-avragd and lctron tmpratur s assumd to b Th 3 th Intrnatonal Elctrc Propulson Confrnc, Flornc, Italy Sptmbr 17-, 7

3 unform. In addton, w assum that lctrons ar Maxwllan dstrbutd and havy partcls ar thrmalzd wth th wall. 1. Partcl balanc and soluton for lctron tmpratur Th lctron tmpratur n an ICP RF on thrustr s usually kpt low (lss than 5V). Ths fatur s vry smlar to th on n a DC on thrustr and s vry attractv bcaus th lctrons ar nrgtc nough to ntat onzaton but do not possss much xcss nrgy to caus sgnfcant powr loss to th anod or a larg bam dvrgnc. Th soluton for lctron tmpratur can b found by mans of a partcl balanc whn th chambr gomtry, wall tmpratur, and background nutral dnsty ar spcfd. Bcaus w usually do not hav drct knowldg of th nutral dnsty, w hav to us tratv mthods for carryng out ths ICP dscharg modl. In th modl dscrbd by Lbrman and Gottscho, th govrnng quaton for calculatng th lctron tmpratur s drvd by assumng that ons and lctrons only rcombn at th wall. Partcl balanc s thn achvd by quatng th onzaton rat to th dffuson loss rat to th walls, 4 ν on = u B d ff = kt m d ff (1) whr ν on s th onzaton frquncy and th rght-hand sd of Eq. 1 dscrbs th rat of partcl loss du to Bohm dffuson. It s a lttl countr-ntutv to us a partcl balanc quaton to solv for lctron tmpratur nstad of dnsty. Howvr, t can asly b obsrvd that th lctron dnsty trm dos not vn appar n our partcl balanc quaton. Th ffctv plasma sz for dffuson, shown n Eq. 1, s wrttn as 4 d ff = RL [ ( RhL LhR )] () whr R s th nnr radus, L s th chambr lngth, h L and h R ar th normalzd axal and radal shath dg dnsts prsntd as 4 h n n =.86 3 ( L / λ ) (.86Lu πγd ) h L R 1 [ ] B 1 [ 4 ( R / λ ) (.8Ru /.45J (.45 γd ) ] = (3) shath, L / = n (4) shath, R n =.8 B 1 ) wth J 1 bng th Bssl functon of th frst knd and th paramtrs λ =1/(n n σ n ), γ=t /T, and D =kt /(m ν n ) rprsntng rspctvly th man-fr-path for ons, th rato btwn lctron and on tmpratur, and th on dffuson coffcnt. Th on-nutral scattrng cross scton rqurd for computng th on-nutral scattrng frquncy, ν n, and th man-fr-path of on s calculatd from Banks formula 5 16 σ n = c r, [m ] (5) whr c r s th rlatv vlocty btwn sngly-chargd ons and background nutrals upon lastc collson drvd from th rducd mass. Rturnng to Eq. 1, now w drv th onzaton frquncy, ν on, from th dfnton of onzaton rat, 6 (6) n = ν n on = n n n f cσ on ( c)4πc dc whr f s th Maxwllan dstrbuton functon and σ on s th onzaton cross scton. Insd our ICP dscharg, th bulk lctron tmpratur s lss than th frst onzaton nrgy (1.1V for Xnon) and only a small group of lctrons possss nrgy xcdng th onzaton thrshold. Thrfor, w can consdr only th lnar part of th onzaton cross scton curv nar th thrshold and thn carry out th ntgraton thr algbracally or numrcally. 6. Plasma conductvty and skn dpth In ICP dschargs, th col-nducd magntc fld dos not pntrat fully nto th plasma. Th spatal dcay constant of a normally ncdnt lctromagntc wav wthn th plasma s oftn rfrrd to as th skn dpth and t s nvrsly rlatd to th drvng frquncy. 6 Th trm skn dpth dos not mply th absnc of magntc fld byond t; t mrly charactrzs th waknng of magntc fld. In th xtrm cass of hgh frquncy and hgh prssur, onzaton only occurs wthn th vry thn skn layr of plasma nar th wall and th dscharg s dffcult to b sustand; ths phnomnon s known as th skn ffct. On th othr hand, larg skn dpth s also unfavorabl bcaus an ncras n magntc fld pntraton corrsponds to nsuffcnt onzaton n th chambr. As th rsult, th skn dpth of a sustanabl ICP dscharg s usually n th ordr of 1/4 to 1/3 of th nnr radus. 3 Th 3 th Intrnatonal Elctrc Propulson Confrnc, Flornc, Italy Sptmbr 17-, 7

4 Although th skn ffct s not strong n our ICP RF on thrustr, t s an ssntal lmnt n modlng th plasma rsstanc and ultmatly can affct th nrgy balanc quaton. In our modl, w drv th xprsson for skn dpth by frst modlng th cor plasma as a conductor wth unform conductvty σ = n ( m ν ff ) (7) Th ffctv lastc collson frquncy, ν ff, ncluds both lctron-nutral scattrng and Coulomb collsons ν ff = ν n ν = cnnσ n cnσ (8) whr c s th man thrmal vlocty of lctrons. If assumng sotropc and Maxwllan-dstrbutd lctrons wth unform tmpratur, th lctronnutral scattrng cross scton can b found as a functon of lctron tmpratur by ntgratng th Maxwllan nrgy dstrbuton functon σ n( T ) = f ( E) Qn( E) cde c (9) whr Q n (E) s th scattrng cross scton of atomc Xnon as a functon of lctron nrgy. In our modl, Eq. 9 s numrcally ntgratd wth th data of atomc cross scton obtand from Rf. 7; th rsult s shown n Fg.. Now, assumng constant conductvty and no Hall ffct, th pntraton of magntc fld can b analyzd by solvn g ts radal dffuson quaton Fgur. Elctron-nutral total scattrng cross scton. B Atomc cross sctons ar wth rspct to lctron nrgy = Dm B (1) and avragd cross sctons ar wth rspct to lctron t tmpratur n unts of V. whr D m =1/(σμ ) s oftn rfrrd to as th coffcnt of magntc fld dffuson or magntc dffusvty. Consdr th complx notaton for th axal magntc fld, B(r,t)=R[(B(r)xp(ωt)], Eq. 1 can b rwrttn as d B db η η η B =, whr η r D m ω dη dη = (11) Bcaus w dfn th skn dpth as th charactrstc dffuson dstanc of th magntc fld, t maks sns to xprss th skn dpth as δ = D m ω = 1 ( ωσμ ) (1) and w can s that as thr th plasma dnsty (thus conductvty) or th frquncy ncrass, th skn dpth dcrass. 3. Enrgy balanc and soluton for lctron dnsty In addton to th partcl balanc, an nrgy balanc s ssntal to charactrz th ICP dscharg. Th lctron dnsty, whch dos not appar n th partcl balanc quaton, can b computd from ths nrgy balanc. W balanc th nrgy by assumng that all th absorbd powr by th plasma s lost du to lastc collson, onzaton and xctaton n th dscharg, and wall losss. 8 = V n ( 3m m ) k( T T ) ( n ν ) abs chambr ' [ ff n n on V ] Pwall P ν (13) whr P abs s th absorbd powr, V chambr s th dscharg volum, ν ff s th ffctv lastc collson frquncy, (n ν on ) s th onzaton rat dscussd n Eq. 6, and V contans th nrgy xpndtur du to onzaton and xctaton. Th trm V s approxmatly two to thr tms of th frst onzaton nrgy V dpndng on th lctron tmpratur. For bttr stmaton, w can conduct Maxwllan-avragng on th onzaton and xctaton rat for a gvn lctron tmpratur. By xamnaton, th powr loss du to lastc collsons (frst trm n th brackt) s rlatvly nsgnfcant whn compard to th V trm; on can thrfor choos to nglct t. Th wall loss trm sn n Eq. 13 s th drct lctron nrgy loss to th wall and s dfnd as 4 Th 3 th Intrnatonal Elctrc Propulson Confrnc, Flornc, Italy Sptmbr 17-, 7

5 P [ A ( W ) A ( W φ ( 1φ ))] n u xp( 1 ) φ (14) wall = wall w grd w B whr A wall s th surfac ara of th dlctrc wall, A grd s th ara of th grd, and φ s th on transparncy of th grd. Th avrag kntc nrgy pr lctron lost to th wall s 9 W = (15) kt Bfor th lctrons surrndr thr nrgy of kt to th wall, som of thr nrgy was xchangd wth th ons whn crossng th shath potntal. As th rsult, th lctrons los thr kntc nrgy at th shath whl th ons gan kntc nrgy. Ths nrgy, carrd by th ons, s consdrd lost whn th ons mpact th wall and rcombn to form nutrals. Th shath potntal across whch th ons acclrat s found by quatng th flux of ons to th on of lctrons, [ m m 8 ( 4xp(.5) )] 5. kt φ (for Xnon) (16) w = kt ln π 77 By xamnng Eq. 13 and 14, w found that th wall loss s actually a vry mportant aspct n th dsgn of on thrustrs bcaus t can dcras th plasma dnsty by takng away a sgnfcant porton of th absorbd powr that could othrws contrbut to onzaton. Wth th us of Eq. 13, f th lctron tmpratur and th absorbd powr ar prscrbd, w can calculat th lctron dnsty. Th undrlyng problm s that th absorbd powr dpnds strongly on both lctron tmpratur and dnsty. In addton to an tratv mthod, a crcut analyss s ndd n ordr to stmat th ffct of chambr plasma on th prmary crcut. By computng th plasma rsstanc and transformng such rsstanc nto th prmary crcut, ts rsstv powr dsspaton can b found. W can thn assum ths rsstv powr loss on th prmary crcut to b th sam as th absorbd powr by th plasma. Ths tchnqu s dscussd n th nxt scton. B. Matchng crcut analyss Th matchng crcut analyss s basd on lumpd-lmnt crcut thory and conssts of a transformr modl and a transmsson ln modl. W frst us th transformr modl to rprsnt th plasma loadng on th prmary crcut, and thn us th transmsson ln modl to calculat th powr dsspaton by ths plasma lmnt, thus compltng th nrgy balanc quaton shown n th prvous scton. Th transmsson ln modl s mportant to our IPC dscharg modl bcaus t can also b usd to compar wth th data on rflctd powr and rsonant frquncy obtand n xprmnts. 1. Transformr modl of an ICP dscharg Th da of modlng an ICP dscharg as a transformr wth th plasma bng a on-turn scondary col s not nw. A modfd vrson of th modl prvously dvlopd by Lbrman and Gottscho 4 s prsntd hr. Fgur 3a shows th crcut dagram of our modl whr a paralll LC ntwork s usd n th prmary crcut for mpdanc matchng. Th capactor s also modld wth an quvalnt srs rsstanc (ESR), but t dos not appar n Fg. 3. Th man objctv of ths transformr modl s to calculat th rsstv powr dsspaton du to plasma loadng, whch s shown n Fg. 3b as lmnt R n th quvalnt crcut. Fgur 3. Crcut dagram of th transformr modl. a) Prmary crcut has capactanc C, col rsstanc R c and col nductanc L c. Plasma has nductanc L p and rsstanc R p. b) Equvalnt crcut. Th chang of nductanc n th prmary crcut du to plasma loadng s L and th chang of rsstanc s R. 5 Th 3 th Intrnatonal Elctrc Propulson Confrnc, Flornc, Italy Sptmbr 17-, 7

6 A major part of any transformr calculaton s to fgur out th corrct rprsntaton for gomtrcal nductanc and mutual nductanc bcaus thy hav profound nflunc on th couplng btwn th prmary and th scondary crcuts. W modl th slf-nductanc of a short N-turn col wth th xprsson for sngl-layr cols n cylndrcal wndng form, 1 6 L. ( D 1)( N )[ ln( 4OD l ) 1 ]( ) c = π w c 1, [H] (17) whr D w s th wndng damtr and l c s th lngth of th col, both n unts of m. Th mutual nductanc s thn approxmatd wth th xprsson for a solnod wth a coaxal crcular flamnt, 1 6 ( D 1) ( D 1) ( l 1) ( Lm =.95N 1N w, w,1 c 1 ), [H] (18) whr th subscrpt 1 and rprsnt th prmary and th scondary col. In our transformr modl, th plasma rsstanc s rlatd to th plasma conductvty and skn dpth, 4 R = l σa = πr σlδ (19) ( ) ( ) p path path whr σ s th conductvty, δ s th skn dpth, R and L ar th chambr nnr radus and lngth. Th slf-nductanc of th plasma conssts of gomtrcal nductanc and nductanc arsng from th lctron nrta. Snc w modl th plasma as a on-turn ar col, ts gomtrcal nductanc can b xprssd n a smlar fashon as Eq. 17. Th wndng radus of our plasma col s takn as /3 of th nnr chambr radus snc th skn dpth of th plasma s obsrvd to b wthn 1/3 of th nnr radus. Th nrta nductanc of th plasma s stmatd as 4 Lp, nrta R p ν ff () whr ν ff s th ffctv lastc collson frquncy of th lctrons. Onc all ths quantts ar calculatd, w can rfr to Fg. 3 and procd to valuat th changs of rsstanc and nductanc on th prmary crcut du to plasma loadng, R L [ R ( ωl ) ] ω Lm Rp p p = (1) [ R ( ωl ) ] ω Lm Lp p p = () Wth th us of complx mpdanc rprsntatons for crcut lmnts shown n Fg. 3b and a transmsson ln modl, w can comput th col currnt I c and fgur out th RMS powr dsspaton du to crcut lmnt R. Ths powr dsspaton s thn consdrd to b th absorbd powr of th plasma and can b usd to complt Eq Transmsson ln modl In hgh-frquncy AC crcuts, voltag and currnt oftn travl at dffrnt phas du to wav rflctons. Impdanc matchng among th sourc, th transmsson ln, and th load s thrfor an mportant dsgn aspct for dlvrng maxmum powr to th load and mnmzng th rflctd powr. Fgur 4 llustrats such mpdanc matchng. In practc, RF powr suppls usually com wth an ntrnal mpdanc of 5Ω and coaxal cabls usually hav a matchng mpdanc of 5Ω as wll. Wth th us of a drctonal couplr at th coaxal cabl nar th sourc, w can masur th rflctd powr du to unmatchd load. Notc hr that th load rfrs to all th crcut lmnts shown n Fg. 3b nstad of th plasma load alon. Snc th load mpdanc Z load, shown n Fg. 4, s a functon of drvng frquncy, w can fnd th frquncy at whch th rflctd powr s at ts mnmum. For our RF crcut, ths condton s known as rsonanc and th corrspondng frquncy s rfrrd to as th rsonant frquncy. Th goals of our crcut modl ar to prdct such rsonant frquncy, th rflctd powr at th rsonant condton, and th powr dsspaton by Fgur 4. Crcut dagram of mpdanc matchng. Th load mpdanc Z load rprsnts th mpdanc of th ntr crcut lmnts show n Fg. 3b. 6 Th 3 th Intrnatonal Elctrc Propulson Confrnc, Flornc, Italy Sptmbr 17-, 7

7 th R lmnt shown n Fg. 3b. Rfrrng to Fg. 4, w can wrt th voltag and th currnt on th transmsson ln n trms of thr forward and backward componnts and th ln s charactrstc mpdanc Z, 11 V ( x) = V xp [ ( ωt kx) ] V xp[ ( ωt kx) ] (3) I ( x) = ( V Z ) xp[ ( ωt kx) ] ( V Z ) xp[ ( ωt kx) ] (4) wth k hr bng th wav numbr. Combnng ths two quatons, w can fnd th rlatonshp btwn th forward and backward voltags by applyng Ohm s law at th load, Z = V I = V V I I = Z V V V V load / x= x= V ( ) ( ) ( ) ( ) = V [( Z Z ) ( Z Z )] load Bcaus th rflctd wav s masurabl wth a drctonal couplr, w can dfn a rflcton coffcnt ρ such that ρ V V = P P = Z Z Z (6) = load ( ) ( ) load load Z whr P and P - ar th forward and backward powr. By combnng Eq. 4 and 5, w can fnd th currnt rachng th load as I = I I = ( V V ) Z = V Z [ 1 ( Zload Z ) ( Zload Z )] (7) Rfrrng to Fg. 3b, w can thn us th currnt dvdr law to fnd th col currnt I c and calculat th RMS powr dsspaton by lmnt R, thus compltng th nrgy balanc quaton n our ICP dscharg modl. Th forward voltag sn n Eq. 7 s rlatd to th obsrvabl forward powr and th sourc mpdanc, P = V (8) ( ) 1, pak Z s whr <P > dscrbs th tm-avragd forward powr and s usually obsrvabl n th powr supply s output scrn. Wth Eq. 7 and 8, w can smply us th forward powr as th adjustabl nput paramtr and carry out our ICP dscharg calculatons. C. Ion xtracton Snc an ICP RF on thrustr rls on an lctrostatc fld for acclratng ons and gnratng thrust, t s ssntal to nvstgat th rlatonshp btwn th appld ntr-grd potntal and th xtractabl on currnt. Th on bam currnt also plays a major rol n our ICP dscharg modl for stablshng th stady-stat chambr prssur, snc w choos th oprabl flow rat, rathr than prssur, as on of our nput paramtrs. In classcal thory, th absolut maxmum of xtractabl on currnt dnsty s dscrbd by th Chld-Langmur spac-charg lmtd currnt, 1 j 9[ ε ( m ) ( )] 1 V 3 l bam = 4 total g (9) whr ε s th prmttvty of fr spac, V total s th appld ntr-grd potntal, and l g s th grd 18 spacng. Th Chld-Langmur currnt assums 16 suffcnt plasma producton n th chambr for 14 facltatng on currnt saturaton. Ths condton, 1 howvr, s rarly rachd n an on thrustr 1 opraton. In practc, th maxmum xtractabl on 8 currnt s domnatd by th mpngmnt-lmtd 6 prvanc, whch s a modfd vrson of th 4 Opratonal Rgm Chld-Langmur currnt Ths prvanc trm s not lmtd by th spac-charg ffct, but by th condton whn th acclrator currnt starts to rs abov 5% of th anod currnt du to drct on Intr-grd Voltag, V mpngmnt on th acclrator grd. Although Impngmnt-Lmtd Thortcal Maxmum usful for dtrmnng th boundars of opraton, th mpngmnt-lmtd prvanc s stll an uppr Fgur 5. Opratonal lmts of on xtracton. Bam Currnt, ma 7 Th 3 th Intrnatonal Elctrc Propulson Confrnc, Flornc, Italy Sptmbr 17-, 7 (5)

8 bound of xtractabl currnt and dos not gv prdctons on th bam currnt whn th plasma producton s blow th nomnal pont. Fgur 5 llustrats th lmts of opraton of a typcal on thrustr that ncluds both th thortcal maxmum (Chld-Langmur currnt) and th mpngmnt-lmtd prvanc obtand from xprmnts. Bng unabl to apply thr thory, an altrnatv approach s takn. For our modl, w approxmat th anod currnt as J a = j, nφ Ag = [ nu B xp( 1 ) ] φ Agrd (3) whr j,n s th on currnt dnsty normal to th scrn grd that obys th Bohm condton, u B s th Bohm vlocty, A g s th grd ara, and φ s th on transparncy of th grd. Th on transparncy s naturally a functon of th plasma dnsty, th lctron tmpratur, and th ntr-grd potntal bcaus t s drctly rlatd to th shath n front of th scrn aprturs. Mor spcfcally, by thr ncrasng th appld potntal or dcrasng th plasma dnsty and th lctron tmpratur, w can ncras th sz of th scrn shath and thus achv an ncras n th on transparncy fracton of th grd. For smplcty, w modl th on transparncy as a lnar functon of th appld ntr-grd potntal alon, whch s xpctd to b th domnant factor n th on xtracton mchansm. Th actual on transparncy functon n our modl s thn obtand mprcally by fttng a st of anod currnt data, 4 φ = V total.174 (31) Notc hr th on transparncy functon strctly dpnds on th grd gomtry and th pattrn of th aprturs, whch mans that th Eq. 31 s not unvrsal. From Eq. 31, th on transparncy of th grd s found to b n th sam ordr of th physcal opnng of th scrn grd (~.4), whch s dmd a rasonabl valu for ths partcular grd st. Th anod currnt dscrbd by Eq. 3 s th on currnt. Bcaus thr s no ntrnal nutralzr n our dscharg chambr, th lctron currnt s forcd to flow from th anod to th xtrnal cathod to nutralz th on bam currnt. If takng account for th acclrator-ntrcptd currnt, w can stmat th on bam currnt as 95% of th anod currnt. Th thrust prformanc of an on thrustr can thn b rlatd to th on bam currnt by Thrust: T mγ c mnγnc n = m ( J bam )( Vnt m ) mn [( nncn 4) φn Agrd ] cn = (3) Spcfc Impuls: Isp = T ( mg ) (33) whr V nt s th anod-to-cathod potntal, c n s th man thrmal vlocty for nutrals, φ n s th physcal grd opnng for nutral scap, m s th mass flow rat, and g s th gravtatonal constant. Th abov prformanc calculatons do not nclud th pnalty rsultd from bam dvrgnc. Th xprmntal data prsntd n ths scton ar obtand from an un-optmzd vrson of th ICP RF on thrustr by Busk; thy do not rprsnt th prformanc of th fnal dsgn. Fgur 6 shows th rsonant frquncy vrsus flow rat for th ICP dscharg wthout th appld on xtracton potntal. As can b sn clarly, our ICP dscharg modl and th ovrall crcut modl work vry wll spcally at flow rats hghr than sccm. Th rsults sn n Fg. 6 ar good rprsntaton on th valdty of our ICP dscharg modl bcaus t contans no on xtracton lmnt and thrfor no data-fttng on th anod currnt. Ths rsult, howvr, must b usd wth cauton as th rsonant frquncy s a snstv quantty and a dvaton of.1 MHz off th rsonant valu can rsult n som ncras n rflctd powr. III. Rsonant Frquncy, MHz Rsults X Flow, sccm Exprmnt Thory Fgur 6. Rsonant frquncy of th ICP dscharg wthout on xtracton. 5W RF powr. 8 Th 3 th Intrnatonal Elctrc Propulson Confrnc, Flornc, Italy Sptmbr 17-, 7

9 Fgur 7 shows th rsult of our complt modl that ncluds on xtracton. Aftr comparng wth xprmntal data, w found that rrors ar wthn 1% or bttr. Ths rrors ar most lkly causd by our crud modl of on xtracton. As xpland prvously, th on xtracton mchansm s controlld by th plasma proprts and th xtracton potntal. Ths quantts affct th shap of th shath n front of th scrn aprtur, thus nflunc th amount and th focusng of th on currnt through th shath. By gnorng th plasma proprts n our on transparncy functon, w lmnat th ffct of plasma dnsty and tmpratur on on xtractablty. Nvrthlss, our on xtracton modl works vry wll as sn n Fg. 7 and s dmd adquat ovrall. Fgur 8 s anothr valdaton for our modl n trms of th rflctd powr of th systm whl th thrustr s undr opraton. As can b sn, w ovr-prdct -3% of th rflctd powr, whch mans that t s qut possbl that w undr-prdct th absorbd powr and thrfor undr-prdct th plasma dnsty. Snc our prdctons on th rsonant frquncy ar rlatvly accurat (Fg. 6), ths problm could b rlatd to th plasma rsstanc trm R p sn n Eq. 19. If w corrct R p by multplyng a factor of.8, th absorbd powr ncrass and th rflctd powr would corrspond to th xprmntal data bttr wth rrors wthn 15% or lss, as sn n Fg. 8. Howvr, ths corrcton would also dcras th valus of on transparncy by about 8% compard to th prvous stmats. Snc w bas ths analyss on data-fttng wth th anod currnt, w cannot b crtan what lvls of on transparncy valus ar mor accurat. Ths ambguty cannot b rsolvd unlss w conduct a mor thorough smulaton on th on xtractablty of th grd. Nvrthlss, th ovrall trnd of our prdctd powr rflcton s vry smlar to th actual on n th sns that as mor ons ar xtractd, lss chambr plasma s prsnt and mor powr s rflctd. Our approach for th modl s thrfor dmd corrct, but mprovmnts on th on transparncy prdcton ar ndd. Anod Currnt, ma Powr Rflcton, % Intr-grd Voltag, V Exprmnt 1 Thory 1 Exprmnt Thory Intr-grd Voltag, V Exprmnt Thory Thory (w/ corrcton) Fgur 7. Prformanc of th ICP RF thrustr at rsonant frquncy. St 1: 63W RF and 1.39 sccm X. St : 51.4W RF and.7 sccm X. Fgur 8. Rflctd powr of th RF crcut. Thrustr opratd at rsonant frquncy. 63W RF and 1.39 sccm X. IV. Concluson A smpl on-dmnsonal prformanc modl s dvlopd for an ICP RF on thrustr. Rsults of th modl, ncludng prdctons of rsonant frquncy, rflctd powr, and anod currnt, ar compard to th xprmntal data. Th ICP dscharg modl and th assocatd crcut analyss ar dmd vald. Th prdctd rsonant frquncs ar accurat to wthn.1mhz. Th on xtracton modl also ylds satsfactory rsult wth accuracy wthn 1%, but t may b too crud by not accountng for th ffct of plasma proprts on on xtractablty. Prdctons on th rflctd powr of th systm ar accurat to wthn 3% wthout thory corrcton and 15% wth corrcton. Th authors ar n th procss of acqurng a mor dtald on xtracton cod, wth whch th prsntd ICP dscharg modl can b mprovd furthrmor and can bcom a powr tool n prdctng th prformanc of an ICP RF on thrustr. 9 Th 3 th Intrnatonal Elctrc Propulson Confrnc, Flornc, Italy Sptmbr 17-, 7

10 Acknowldgmnts Th authors would lk to thank Mr. Kurt Hohman and Dr. Lynn Olson of Busk, Co., Inc. for provdng th xprmntal data. Ths projct s fundd by Busk through NASA SBIR award. Rfrncs 1 Ltr, H.J. and Fl, D., RIT 15S A Rado Frquncy Ion Engn for Hgh Spcfc Impuls Opraton, 37 th AIAA/ASME/SAE/ASEE Jont Propulson Confrnc & Exhbt, 1, AIAA Ltr, H., Kllngr, R., Kuks, R., and Frohlch, T., Extndd Prformanc Evaluaton of EADS ST s mn Rado Frquncy Ion Thrustr, 39 th AIAA/ASME/SAE/ASEE Jont Propulson Confrnc & Exhbt,, AIAA Brophy, J.R. and Wlbur, P.J., Smpl Prformanc Modl for Rng and Ln Cusp Ion Thrustrs, AIAA Journal, Vol. 3, No. 11, 1985, pp Gudmundsson, J.T. and Lbrman, M.A., Magntc nducton and plasma mpdanc n a cylndrcal nductv dscharg, Plasma Sourcs Scnc. Tchnology journal, Vol. 6, 1997, pp Szabo, J., Martnz-Sanchz, M. and Batshchv, O., Fully Kntc Hall Thrustr Modlng, 7 th Intrnatonal Elctrc Propulson Confrnc, Pasadna, CA, 1, IEPC Frdman, A. and Knndy, L.A., Plasma Physcs and Engnrng, Taylor & Francs Books, Inc., 4, pp. 17-7, 199-, , Szabo, J., Fully Kntc Numrcal Modlng of a Plasma Thrustr, Ph.D. dssrtaton, Dpartmnt of Aronautcs and Astronautcs, Massachustts Insttut of Tchnology, Cambrdg, MA, January 1, pp Mtchnr, M. and Krugr Jr., C.H., Partally Ionzd Gass, John Wly & Sons, Inc., Nw York, 1973, rprnt n 1993, pp , Lbrman, M.A. and Lchtnbrg, A.J., Prncpls of Plasma Dschargs and Matrals Procssng, John Wly & Sons, Inc., Nw York, 1994, p Grovr, F.W., Inductanc Calculatons, Workng Formulas and Tabls, D. Van Norstrand Company, Inc., NY, 1946, pp , Kong, J.A., Elctromagntc Wav Thory, nd d., John Wly & Sons, Inc., Nw York, 199, pp Sutton, G. and Bblarz, O., Rockt Propulson Elmnts, 7 th d., John Wly & Sons, Inc., Nw York, 1, pp Rovang, D.C. and Wlbur, P.J., Ion Extracton Capablts of Vry Closly Spacd Grds, AIAA/JSASS/DGLR 16 th Intrnatonal Elctrc Propulson Confrnc, Nov , 198, AIAA , 14 Rovang, D.C. and Wlbur, P.J., Ion Extracton Capablts of Two-Grd Acclrator Systms, Journal of Propulson, Vol. 1, No. 3, 1985, pp Brophy, J.R., Effct of Scrn Grd Potntal on Prvanc, appars n Advancd Ion Thrustr Rsarch, NASA CR (N ), Jan Th 3 th Intrnatonal Elctrc Propulson Confrnc, Flornc, Italy Sptmbr 17-, 7