Optimization of Radio-Frequency Ion Thruster Discharge Chamber Using an Analytical Model

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1 Optimization of Radio-Frquncy on Thrutr Dicharg Chambr Uing an Analytical Modl Emr Turkoz, Murat Clik* Dpartmnt of Mchanical Enginring Bogazici Univrity tanbul, Turky Abtract Th radio frquncy ion thrutr i an impul gnrator to b ud in pac miion. t i a plama bad gnrator which utiliz lctrotatic fild btwn grid to acclrat th ionizd ga out of th thrutr to gnrat thrut. An analytical modl i ud to optimiz th gomtry of th dicharg chambr which contain th inductivly coupld plama. Th modl i implmntd and vrifid with xiting thrutr and th rult of th optimization proc ar prntd. Th rult of our optimization routin validat th rcnt trnd in radio frquncy thrutr dign, which promot hmiphrical dicharg chambr. Kyword lctric propulion, radio-frquncy ion thrutr, inductivly coupld plama. NTRODUCTON Th radio frquncy (RF) ion ngin, which i alo known a RF ion thrutr, i an impul gnrator for mall thrut valu. t i among many plama thrutr dvlopd ovr th lat fw dcad. A an lctric propulion ytm, th RF ion thrutr rli on th acclration of ionizd ga to gnrat thrut. Th ionization i providd by th nrgy carrid by th RF wav in to th dicharg chambr. Th ga fd into th dicharg chambr i Xnon. Th plama thrutr ar dignd to provid a crtain amount of thrut, which dpnd on th typ of miion. Th goal in thrutr dign i to gnrat utainabl plama with th rquird dnity for a pcifid thrut valu, and to rduc th man nrgy pnt on th ionization of nutral ga. Rducing th man ionization nrgy i rlatd to th confinmnt of th particl, which would allow thm to collid with ach othr in a mor dn ara and thu gnrat ion. To undrtand th phyic of th dicharg chambr of an RF ion thrutr, a numrical modl i ndd. Gobl [] dvlopd an analytical 0D modl to calculat RF ion thrutr prformanc and fficincy. Evn though th modl lack patial rolution, it manag to prdict th prformanc of an RF thrutr with good agrmnt. n thi work, th analytical modl dvlopd in [] i coupld with optimization routin to find th optimum chambr dign and thrutr paramtr, which would rult in minimum nrgy lo for maximum *Corrponding author: Murat Clik murat.clik@boun.du.tr amount of ionization. Th dign of th dicharg chambr of RF ion thrutr i till volving. Firt RF ion thrutr ar dvlopd by Univrity of Gin and Atrium GmbH undr th nam of th RT []. Thi family of ion thrutr conit of dign namd according to thir cylindrical chambr diamtr. RT- 0 and RT-5 [3] ar mmbr of thi family, and thy hav 0 and 5 cm dicharg diamtr, rpctivly. Th hap of th dicharg chambr ha volvd from a cylindrical hap to a conical on with RT-XT [4]. Th chmatic of an RF ion thrutr with a conical dicharg chambr i givn in Fig.. n on of th mot rcnt dign, RT-45 [5], th dicharg chambr i hmiphrical. Our optimization chm utiliz th modl prntd in [] and valuat th dign qualitativly in th cour of finding th optimum dign. Thr ar many important paramtr whil valuating th prformanc of an RF ion thrutr. Th currnt upplid to th RF coil gnrat a magntic fild which i inductivly coupld with th plama and confin th ion to prvnt xciv lo to th wall. Th lctron numbr dnity, which i alo calld a th plama dnity, i a ky factor in dtrmining th ionization colliion frqunci. Th crn ara and th potntial diffrnc btwn th acclrator and th crn grid dtrmin th amount of thrut to b gnratd. Th intrtd radr i rfrrd to [] for furthr rading on th ubjct. Fig. : Th chmatic of an RF ion thrutr with a conical dicharg chambr

2 . THE MODEL n thi ction, th analytical modl prntd in [] i firt dcribd hortly focuing on th input and th output paramtr. Thn th implmntation i laboratd and th coupling with optimization routin i xplaind. Aftrward, th dign paramtr and objctiv ar clarly tatd. A. Th 0D Analytical Modl Th vry dtaild xplanation of th modl i prntd in []. Thi modl i implmntd in thi work. Th mot important output of th modl i th dicharg lo pr ion, which i th maur of th man nrgy pnt to ioniz on nutral particl in th dicharg chambr. n our implmntation, th modl tak th chambr lngth, chambr ba diamtr, chambr top diamtr, crn tranparncy and optical tranparncy valu a input. A th rult, alongid with th dicharg lo pr ion, our implmntation giv confinmnt factor, plama dnity, ion vlocity and th magntic fild magnitud a th output for ach ma utilization valu. Th dicharg lo pr ion i found by dividing th input powr by th ion bam currnt: P ab η d = () b whr η d i th dicharg lo pr ion, P ab i th input powr to th plama and b i th bam currnt. nput powr i calculatd a follow []: P ab = + U U ++ + *U* + ( + w + b ) (0.5T + ) + a (T + ) () whr U + i th firt ionization potntial, U ++ i th cond ionization potntial and U* i th xcitation potntial. Similarly, + i th ingly ionizd particl production rat, ++ th doubl ionizd particl production rat and * th xcitd nutral production rat. Th potntial valu ar pcific for ach typ of ga ud to gnrat th plama. Th production rat hav imilar xprion. Th ingly ionizd particl production rat i []: + = n n σ + 0 i v V (3) Similarly, th doubl ionizd particl production rat i a follow: = n σ v V (4) And, th xcitd nutral production rat i: * i * = n n σ v V (5) 0 n th quation, n 0 dnot th nutral numbr dnity, n dnot th lctron, and du to quai nutrality, plama numbr dnity. i th lmntary charg, σ +, σ ++ and σ* ar th firt ionization, cond ionization and xcitation cro ction of Xnon, rpctivly, v dnot th lctron vlocity and V i th plama volum. Othr trm that tak part in th calculation of aborbd powr ar th ion currnt to th crn grid ( ), to th wall ( w ), and th ion bam currnt that pa through th grid of th crn ( b ) []. Th formulation of th trm i a follow: ( T ) = niva A (6) w = niva Aw f (7) c b = niva AT (8) n th quation v a dnot th ion acoutic vlocity givn by: kt va = (9) M Alo, f c i th confinmnt factor, which indicat th ion currnt raching th wall dividd by th Bohm currnt. A i th grid ara, and A w i th wall ara. n i i th ion dnity which conit of th um of th ingly and doubly ionizd particl. T i th lctron tmpratur and M i th molar ma of th Xnon atom. Th only rmaining trm in th aborbd powr quation i th floating potntial, which i dfind a []: ϕ = kt M ln (0) πm Our implmntation of th modl rquir om approximation. W aum that th crn grid tranparncy i known and it i takn a a contant valu of 0.8. Th optical tranparncy i actually calculatd uing ion optic phyic. An xampl of ion optic numrical modl i th CEXD cod dvlopd in JPL [6]. An xtnion of thi cod in 3D i alo dvlopd again in JPL with th nam CEX3D [7]. n thi work, an ion optic cod i not utilizd. ntad, a firt ordr approximation prntd in [8] i utilizd to calculat th optical tranparncy: 0 = () ( crn ) + ( accl ) whr crn i crn grid tranparncy and accl i th acclrator grid tranparncy. Farnll [9] laid out th convntional rlationhip btwn th crn and th acclrator grid tranparnci in ion thrutr a follow: accl = 0.36 () crn Thi rlation i utilizd in thi papr and th optical tranparncy i calculatd with th formula prntd. Anothr approximation i prformd for th Clauing factor, which dfin th rducd conductanc of th grid with finit thickn. Thi factor i calculatd uing Mont Carlo tchniqu and it i typically on th ordr of 0.5 []. A an approximation thi valu i takn contant for all grid dign

3 a 0.5. n th rfrnc tudy of thi work [], th coil currnt i takn to b contant a th dicharg lo pr ion valu i aumd to b around 50 V/ion. A imilar approach i obtaind in thi work and a formula i ud to calculat th coil currnt and th magntic fild. To dvlop thi approach, th fact i utilizd that th bam currnt i dirctly proportional to th impul dird to b gnratd. Thrfor th bam currnt valu i hld contant and th coil currnt i changd according to th following formula: η Ω = Δ (3) coupling coil coil lo bam whr coil i th coil currnt, bam i th bam currnt, Ωcoil i th coil impdanc, Δ lo i th dicharg lo pr ion and η coupling i th coupling fficincy of th coil and th plama. Th coil impdanc i convntionally takn to b 50 Ω. Th dicharg lo pr ion valu chang with ma utilization fficincy. A th coil currnt chang, th inducd magntic fild alo chang with th following rlation: B = μ0n coil (4) whr B i th inducd magntic fild, μ 0 i th magntic contant, N i takn contant with th valu 00 turn/mtr. Our implmntation rprnt th modl prntd in [] with om minor diffrnc in paramtr dfinition. Th modl u th balanc of ion production and lo trm in th plama to valuat lctron tmpratur. on production i alo calculatd to valuat th dicharg lo. To account for th ion production with highr accuracy, doubl ionization hould alo b brought into th pictur. Th data of raction rat for doubl ionization for Xnon i obtaind from [9]. To u th data, a nonlinar lat quar fit i prformd on th data point uing th following function tmplat:. T ( C BT + AT ) ) 8T + (5) V V πm 5 V V 0 whr TV i lctron tmpratur in lctron volt, m i th lctron ma, i th lmntary charg valu. Thi form i imilar to th form prntd in [0] which i ud to find th ingl ionization raction rat. Th rult i valuatd a follow: A =.63, B = and C = Th ingl and doubl ionization raction rat along with th xcitation raction rat ar plottd in Fig.. B. mplmntation of th Modl Th modl i implmntd in MATLAB. Th optimization routin i xcutd with thr input paramtr. A conical dicharg chambr i aumd. Th paramtr ar th ba and th crn (grid) diamtr of th conical chambr and th axial lngth. Th objctiv i to minimiz th avrag dicharg lo pr ion for th ma utilization valu ranging btwn 0.7 and Th lowr and uppr bound of th thr paramtr ar dtrmind a follow: Th ba diamtr valu ar ubjct to vary btwn 4 cm and cm whra th crn diamtr i ubjct to vary btwn 8 cm and 8 cm. Th chambr lngth can tak valu btwn 0 cm and 5 cm. Th appropriat valu ar pickd by th optimization routin and th thr valu ar givn a input to th main routin of th algorithm. Th main routin mainly conit of thr ntd loop. Th gomtry calculation ar prformd and th chambr volum and th chambr wall urfac ara ar calculatd. Thn th routin ntr th firt loop ovr ma utilization valu ranging from 0.7 to n thi loop thr i a cond loop for th inducd magntic fild valu. Th inducd magntic fild dpnd on th coil currnt which dpnd on th dicharg lo pr ion. Sinc th inducd magntic fild alo affct th dicharg lo pr ion, th loop monitor th convrgnc of th magntic fild for th dicharg lo pr ion valu valuatd by th third loop. Th third loop i th mot computationally xpniv part of th routin. Th third loop monitor th convrgnc of th confinmnt factor. t tart with th calculation of th nutral numbr dnity which i a function of ma utilization fficincy, grid ara and optical tranparncy. Thn lctron tmpratur i calculatd with a trial-and-rror approach. Valu btwn 0.4 and 5 V ar trid and th valu with th minimum rror i pickd a th lctron tmpratur. Th corrct tmpratur valu rult with a harp dclin in th rror function. Th tmpratur valu ar plottd againt th normalizd rror in an xampl trial and rror ca, which i to b n in Fig. 3. Aftr th lctron tmpratur i dtrmind, th plama conductivity, floating potntial and lctron-nutral colliion frqunci ar ubquntly calculatd. Thn th lctron numbr dnity i calculatd with bam currnt, crn ara, tranparncy valu and lctron tmpratur. For th lat tp th Bohm vlocity i calculatd uing th lctron tmpratur and th convrgnc critria, th confinmnt factor i valuatd a th ratio of th ion vlocity dividd by th Bohm vlocity. Th loop itrat until th confinmnt factor convrg to a valu. Th dpndncy of th important paramtr in th third loop i prntd in Fig. 4. Fig. : Raction rat for Xnon.

4 Fig. 4: Dpndncy chm of th third loop Fig. 3: Normalizd rror and lctron tmpratur. RESULTS A validation of th modl wa ncary bfor obtaining th optimization rult. Th validation i prformd according to th data prntd in []. Aftr th implmntation i validatd, th rult of th optimization ar prntd. A. Validation Th validation i prformd uing th xampl ca prntd in []. Thi xampl conical dicharg chambr ha 0 cm chambr lngth, cm crn diamtr and 5 cm ba diamtr. Th cod i prformd for thr inducd magntic fild valu: 0 G, 6 G and.6 G. Th rult ar plottd in Fig. 5. Th rult match with a high accuracy to tho givn in [] and in accordanc with th xprimntal rult prntd in [] for RT- ion ngin in oprating condition (about 0.9 ma utilization fficincy). Fig. 5 how thr diffrnt ca for th diffrnt inducd magntic fild valu. Th olid lin rprnt th ca with 0 G, i.. no ion confinmnt. A xpctd th dicharg lo pr ion in th opration rang i th hight. Th dahd lin rprnt th 6 G ca and th dottd lin with th lowt dicharg lo pr ion valu rprnt th.6 G ca. Thi rult indicat that th mallt dicharg lo pr ion valu can b obtaind with th minimization of th wall ara to rduc th ionization lo. On th othr hand, th crn diamtr i kpt a larg a poibl. Th poibl bound ar dtrmind according to th applicability of our modl. Sinc th modl i 0D, it aum uniform proprti throughout th plama. Th coil i a limitation whil dtrmining th chambr lngth, inc vry diffrnt chambr lngth with th am coil would rult with diffrnt coil impdanc which ar not covrd by th currnt modl. Othr limitation i on th grid diamtr for th approximation mad whil valuating th tranparncy valu and th Clauing factor. Without addition of an ion optic cod and a Mont Carlo olvr, th crn diamtr would bttr rmain in th currnt bound inc th valu ar availabl from th litratur for th valu. Thi tndncy of th optimization routin indicat that th bt configuration ha th mallt urfac ara and th largt plama volum. Th rcnt dign [5] along with our finding lad to think that a hmiphrical dicharg chambr may hav th bt prformanc charactritic. A hmiphrical dign, which ha th am dicharg volum with th optimum conical dign ha 0.3 cm chambr lngth and radiu. Th rult of th optimum conical dign and th optimum hmiphrical dign ar prntd in Fig. 6 how that th dicharg lo pr ion i highr for th optimum conical dign than th hmiphrical dign which ha th am volum and th am grid diamtr. Th curv rprntd by th dahd lin dnot th dicharg lo pr ion valu for th hmiphrical configuration whra th olid lin rprnt th optimum conical configuration for th am volum. B. Rult of th Optimization Th optimization procdur yild th dign paramtr on th bound. A th rult of th optimization proc, th idal conical chambr configuration within th paramtr bound turn out to b 0 cm long, having 4 cm ba diamtr and 8 cm crn diamtr. Th dicharg lo pr ion againt th ma utilization fficincy i plottd for thi configuration and can b n in Fig. 6. Diffrnt bound yild th am tndncy. n othr word, th dicharg lo pr ion i minimum for th conical dicharg chambr configuration if: Th chambr lngth i a mall a poibl Th ba diamtr i a mall a poibl Th crn diamtr i a larg a poibl Fig. 5: Dicharg lo pr ion vru ma utilization fficincy for thr diffrnt magntic fild

5 V. CONCLUSON Th analytical 0D modl prntd in [] i uccfully implmntd and validatd with th rult prntd in th am work. A lf conitnt itrativ olution of th modl i implmntd. An optimization tudy i prformd to minimiz th avrag dicharg lo pr ion ovr raonabl ma utilization fficinci. Th paramtr to b changd by th optimization routin ar dtrmind a th chambr lngth, ba diamtr and crn diamtr. Th rult of th optimization indicat that th dicharg lo pr ion i minimizd whn th wall ara of th dicharg chambr i th mallt. Thi phnomnon and th latt trnd in thrutr dign [5] lad u to conclud that bttr dign ar poibl with a hmiphrical configuration, inc phr i th gomtry which ha th largt urfac ara to volum ratio. An xampl hmiphrical configuration which ha th am chambr volum with th optimum conical configuration i valuatd and thi claim i validatd. Evn though our implmntation ha an accurat tndncy and capability to b ud for a thrutr dign, om improvmnt ar rquird for bttr accuracy. An ion optic cod can b incorporatd into th implmntation, o th grid tranparncy can bttr b valuatd with th changing grid diamtr. A Mont Carlo olvr may alo prov to b uful to calculat th Clauing factor. Alo th raction rat and th colliion frqunci ar valuatd with th cro-ction data for Xnon. Thi data ar valuatd xprimntally and th rult ar mbddd in th cod. f a diffrnt ga i dird to b ud, th ncary xprimntal data again hould b fd into th cod. Th 0D modl aum contant proprti throughout th chambr volum. Th dvlopmnt of a D modl would prov to b vry uful in thrutr dign to obrv th local variation and valuat th dicrt paramtr valu. REFERENCES [] D. M. Gobl, Analytical dicharg modl for RF ion thrutr, EEE tranaction on plama cinc, vol. 36, no. 5, Octobr 008. [] H. Banr, R. Killingr, H.Litr, and J.Müllr, Dvlopmnt tp of th RF-ion thrutr RT, 7 th ntrnational Elctric Propulion Confrnc, 00, EPC [3] K. H. Lob, H. J. Litr, and H. W. Lob, RT 5 A mdium rang radio-frquncy ion thrutr, nd Europan Spac Propulion Confrnc, 997, 997ESASP G. [4] H. J. Litr, R. Killingr, H. Banr, J. Müllr, R. Kuki, and T. Fröhlich, Evaluation of th prformanc of th advancd 00 mn radio frquncy ion thrutr RT-XT, 38 th Joint Propulion Confrnc, ndianapoli, ndiana, July 7-0, 00, AAA [5] H. W. Lob, D. Fili, G. A. Popov, V. A. Obukhov, V. V. Balahov, A.. Mogulkin, V. M. Murahko, A. N. Ntrnko, and S. Khartov, Dign of high-powr high-pcific impul RF-ion thrutr, 3 nd ntrnational Elctric Propulion Confrnc, 0, Wibadn, Grmany, EPC [6] J. R. Brophy,. Katz, J. E. Polk, and J. R. Andron, Numrical imulation of ion thrutr acclrator grid roion, 38 th Joint Propulion Confrnc and Exhibit, 00, ndianapoli, N, USA, AAA [7] J. R. Andron,. Katz, and D. M. Gobl, Numrical imulation of two-grid ion optic uing a 3D cod, 40 th Joint Propulion Confrnc, 004, Fort Laudrdal, Florida. [8] M. Tay, Two-dimnional numrical modling of Radio-Frquncy ion ngin dicharg, Thi (Ph. D.) Maachutt ntitut of Tchnology, Dpt. of Aronautic and Atronautic, 00. [9] E. W. Bll, Elctron-impact ionization of n and X, Phyical Rviw A, vol. 48, no. 6, Dcmbr 993. [0] D. M. Gobl,. Katz, Fundamntal of Elctric Propulion on and Hall Thrutr, JPL Spac Scinc and Tchnology Sri, March 008. [] H. Litr, R. Kuki, R. Killingr, E. Bonlli, S. Scaranzin, and F. Scortcci, RT- ion ngin dvlopmnt Enduranc tt and lif prdiction, prntd at th 4nd Joint Propulion Conf., Sacramnto, CA, Jul. 9, 006, Papr No. AAA Fig. 6: Dicharg lo pr ion vru ma utilization fficincy for th optimum conical and th corrponding hmiphrical dign which hav th am dicharg volum and th grid diamtr