Electron Emission for Electric Propulsion: Reducing Power by Mitigating Space Charge Limits

Transcription

1 40th AIAA/ASME/SAE/ASEE Joint Propulsion Confrnc and Exhibit July 2004, Fort Laudrdal, Florida AIAA Elctron Emission for Elctric Propulsion: Rducing Powr by Mitigating Spac Charg Limits David P. Morris * and Brian E. Gilchrist Univrsity of Michigan, Ann Arbor, MI, Elctron mission is a rquirmnt for most forms of spac lctric propulsion, as wll as many othr spac applications. For many such missions minimizing systm powr is critical to th viability of th mission. Th gnration of lctrons rquirs an amount of powr that varis dpnding on th sourc (cold cathod and thrmionic sourcs ar considrd hr, not hollow cathod or othr sourcs rquiring consumabls) but rgardlss of th sourc, th lctrons must lav as a bam that has sufficint powr (vlocity) to scap th spaccraft. Th spac charg limit rfrs to th maximum currnt that can cross a gap (th shath btwn th spaccraft and th surrounding plasma for xampl) at a givn vlocity, and thus is a practical lowr bound to th bam powr rquird for a givn currnt. For many systms, spcially thos using low-powr lctron sourcs, this mans that powr must b addd to th lctron bam, usually with biasd grids. Th minimization of this lctron bam powr rquirmnt is th topic of this papr. Analytic solutions for th spac charg limit in simpl gomtris hav bn dvlopd up to thr dimnsions, and ths dmonstrat som basic prformanc tradoffs. Lss practical for th analytical approach ar mor complicatd gomtris and th addition of spacing and timing factors. Ths complications, howvr, can allow significant improvmnt in th spac charg limit. Thrfor, using particl-in-cll computr simulations (with th XOOPIC cod dvlopd at Brkly), a numbr of tchniqus for mitigating th spac charg limit hav bn invstigatd, ranging from atypical gomtris to spatial and chronological phasing of mission. Som of th advantags and disadvantags of som tchniqus, and th implications for lctron mission systm dsign, will b prsntd hr. spaccraft mittr grid Efild Efild Efild Wh! Plasma darn. * PhD Studnt, Elctrical Enginring and Spac Systms, 2455 Hayward Profssor, Elctrical Enginring and Spac Systms, 2455 Hayward, AIAA Associat Fllow 1 Amrican Institut of Aronautics and Astronautics Copyright 2004 by th Amrican Institut of Aronautics and Astronautics, Inc. All rights rsrvd.

2 Nomnclatur FE = fild mittr EFES = lctron fild mittr systm SCL = spac charg limit PIC = particl-in-cll Vgt = gat tip voltag ε 0 = prmittivity of fr spac = lctron charg [C] m = lctron mass [kg] T o = lctron mission nrgy [V] D = gap spacing [m] V = gap voltag [V] W = mittr width r b = mittr radius A = mittr ara [m 2 ] s = shath siz [m] J CL (N) = N dimnsional Child-Langmuir currnt limit [A/m 2 ] I. Introduction Th topic of this papr is improvmnt of th fficincy of lctron mission, primarily applid to in-spac applications, via analysis of th spac charg limit and th dvlopmnt of tchniqus to mitigat, improv, or gt around this limit. Any systm mitting lctrons at low nrgy (rlativ to th mittd currnt) potntially facs th spac charg limit, a point at which th intractions of th mittd lctrons pushing on ach othr bcoms sufficint to slow down and rflct th bam. This limit is spcially salint to small spaccraft in-spac lctric propulsion and othr spac applications whr powr spnt on mission is at a prmium. Th primary motivation of this work is for th dvlopmnt of cold cathod or fild ffct lctron mittrs (FEs) bcaus thy ar particularly capabl of high currnt low powr mission, but th rsults ar applicabl to thrmionic and othr typs of mittrs as wll. I will bgin with an ovrviw of th spac charg limit and how it ffcts mission. I will brifly covr analytic solutions to th spac charg limit dvlopd from th arly 1900s to today, ranging from 1 dimnsional to 3 dimnsional, and what ths imply. I will thn covr som of th tchniqus to improv upon ths limits, including mittr sgmntation and dfocus rings. I will show th rsults of ths drivd primarily via PIC simulations in XOOPIC, which I will dscrib as wll. I will also prsnt som xprimntal work don in support of this rsarch. II. Th Spac Charg Limit Th first lctrons in a bam moving at acclratd vlocity from an mittr mov frly to th anod, out into th vacuum, or vn mor idally across th positivly biasd shath btwn th spaccraft and surrounding plasma. Subsqunt lctrons, howvr, s th ngativ fild surrounding ach lctron in th initial portion of th bam. Lik chargs rpl and ths lctrons ar dclratd somwhat. Subsqunt lctrons dclrat vn mor. Blow th spac charg limit, in th stady stat, thr will b a bunching of lctrons that hav dclratd thusly somwhr in th gap btwn th mittr and th anod. Elctrons dclrat as thy approach this potntial minimum, and acclrat onc thy ar past it. Th spac charg limit occurs whn thn bunching rachs sufficint dnsity rlativ to th mission vlocity and othr paramtrs that lctrons com to complt stop. Onc this happns, th dnsity at this cntral point, this "virtual cathod", incrass rapidly and nw lctrons ar rflctd back to th sourc. Bcaus th dvlopmnt of th virtual cathod acclrats onc lctrons ar rflctd, raching th spac charg limit is a catastrophic vnt from an mission point of viw. Th virtual cathod grows larg and th majority of th charg is rflctd back to th sourc. Much mor charg is mittd at psilon blow th spac charg limit than at psilon abov. Thus idntifying and avoiding this spac charg limit is critical. Th following figur illustrats this ffct. 2 Amrican Institut of Aronautics and Astronautics

3 Emittr Wh! Smooth sailing, way blow SCL Emittr Dclrating, but still blow SCL Plasma Emittr darn. Rflction, abov SCL Figur 1. Graphical dpiction of th spac charg limit ffct for lctron mission from a spaccraft cathod into a surrounding plasma/anod for various bam dnsitis- blow, nar, and abov th spac charg limit. 1 Dimnsional Spac Charg Limit Th simplst gomtry for th spac charg limit analysis is an infinit plan. This gomtry was first studid in th arly 1900s by Clmnt Dxtr Child [19ll], and by Irving Langmuir [1913]. Thir rsarch was inspird by lctron mission in vacuum tubs, but th applications ar far raching. Using th continuity quation (aka Kirchoff's currnt law) that th currnt must b qual at vry point btwn th plats, th forc quation for chargd particls in an lctric fild, Poisson's quation, F=ma, and calculus, thy drivd th following (convrtd to MKS units): J 1D = 4ε 0 9D 2 2 m 1 2 V 3 2 [A/m^2] Th basic on dimnsional spac charg limit quation has bn studid and xpandd upon a grat dal in th intrvning yars. In on dimnsion with an initial mission vlocity assumd, th spac-charg currnt limitd condition is givn as follows [Luginsland, 1998]: 3 Amrican Institut of Aronautics and Astronautics

4 I CL ( 1)= 4ε o 9 2 m 3 2 T o D V A T o 3 whr D is th gap spacing, V is th gap voltag, T o is th initial nrgy at which th lctrons ar injctd into th gap (~ gat voltag), is th magnitud of th lctron charg, m is th lctron rst mass, and ε o is th fr spac prmittivity. If V = 0 or V << T o ( shortd anod ) This quation indicats that thr is a factor of 8 improvmnt ovr th classical 1-D Child-Langmuir currnt limit (T o = 0). Thus in th most simpl on dimnsional analysis, an EFES mitting 50V lctrons across a 1V 1.5cm gap (a typical shath siz for a nutral body in low arth orbit [Hastings, 1996]), th spac charg currnt limit is 3mA/cm 2. This is an ffctiv lowr limit to th ability of an FEA to mit charg (assuming that crossing th shath is th primary limitation). So to gt 1 A of mission would rquir a minimum of just ovr 300cm 2 of mitting ara, i.. a squar mittr about 18cm on a sid. No accommodation nd b mad in this analysis for spac charg limits incrasing as th mittr ara incrass, bcaus th 1-D analysis alrady assums th worst cas situation- an infinit sht of charg. Two and Thr Dimnsional Spac Charg Limits Mor dtaild analysis indicats that significant improvmnt can b attaind whn you considr sprading of th mittd lctrons in multipl dimnsions. Luginsland [1996] suggsts that th improvmnt by going to a long, thin mittr (modld as an infinit strip, s figur blow) would b a factor of 2 (Using a 0.5 cm wid mittr, 1.5cm shath). This was first dtrmind using numrical analysis with XOOPIC and MAGIC, and latr vrifid analytically [Lau, 2001]. This would rduc th rquird ara to 150 cm 2 for 1 amp at 50 V mission. W =2R lctrons initial nrgy To mittr and ground plan Voltag=0 gap D anod Voltag=V Figur 2. Th scnario bing analyzd for 2 dimnsional spac charg limits J cl (2) J cl (1) = W D ( W D )2 Also at th Univrsity of Michigan, Lau [2001] xtndd his analytical agrmnt with th abov 2D solution to th thr dimnsional cas as givn blow: J CL (2) J CL (1) 1 + D 4R and J CL (3) J CL (1) 1 + D πw D 2π R solvd for situation whr R >= W/2, and R/D >>1. Ths solutions assum a gap siz and/or magntic fild such that bam sprading is not a factor. Thy also assum that th charg dnsity varis only with th mission dirction, not latrally. In th situation whr R/D < 1, as is likly to b th cas for a spac born EFES, anothr thortical calculation by Humphris [1990] which dos includ bam sprading is givn blow (for a flat circular mittr with radius r b ). 4 Amrican Institut of Aronautics and Astronautics

5 [ ( ) 2 ] J CL ( 3) J CL ( 1) = r 2 b + D 2 2 r b = [ 1 + ( D 2r b ) 2 ] This quation suggsts an improvmnt of 2.75 for a 1cm 2 (r b =0.564cm), 1.5 cm shath cas. Comparing back to our initial xampl this would dictat an EFES systm with about 110cm 2 mission ara, which is a markd improvmnt ovr th original 1D Child-Langmuir 300 cm 2 ara. Th cavat is that this ara must b in isolatd 1cm 2 sctions (spacd such that th gaps btwn mittrs ar larg with rspct to D) for th Humphris quation to apply. Th most advancd of th abov quations considr only bam sprading from intrnal ffcts of th bam, as it movs away from a planar sourc to a planar collctor in a DC fashion. Additional sprading occurs whn xtrnal dvics ar in plac, thus dcrasing th dnsity at th cntr whr th virtual cathod first forms, and incrasing th spac charg limit. Furthr improvmnt can b gaind whn mission is sprad ovr gratr ara with gaps btwn mittrs. Ths ar th sorts of modifications to th initial spac charg limit calculations that will b considrd blow. Ths solutions xploit th fact that for this application th charactr of th bam is irrlvant as long as th lctrons do not rturn to th spaccraft. For lctric propulsion, or solar panl arc supprssion, as long as th lctrons mov away into th surrounding plasma it dos not mattr if thy ar in a tight bam, which allows diffrnt solutions than th mor commonly studid travling wav tub amplifir, particl acclrator, plasma tlvision, or othr trrstrial applications. III. Particl-In-Cll Analysis and th XOOPIC Cod Analytical solutions of mor complicatd gomtris rapidly bcom intractabl, whras a wid varity of arbitrary situations can b simulatd quickly and asily with PIC tchniqus. In a particl-in-cll simulation, a computr tracks th location and vlocity of particls at arbitrary positions, and th lctric and magntic filds gnratd by thos particls and th surrounding structurs on a grid stablishd in th simulation spac. A cyclic itration is don to updat th vlocity, position, and fild valus with vry tim stp, as dscribd by th following figurs. Exy, Bxy Vxy (xy) Wighting E nw, B nw -> F nw Intgration of quations of motion, moving particls. a=f/m; v nw = v old + a dt x nw = x old + v nw dt Δt Intgration of Maxwll's quations on grid. Wighting x nw,v nw -> ρ nw,j nw ρ nw,j nw -> E nw,b nw [Birdsall, 1991] Figur 3. Th clls and grids whr paramtrs ar trackd in PIC simulations, and th cycl usd to itrat thos paramtrs With ach cycl th computr sums all th charg within a fw clls of an intrsction to gnrat fild valus for that intrsction. Thn th filds at th four intrsctions ar avragd or intrpolatd to b applid to ach particl in th cll surroundd by th four intrsctions. Th largr th clls, th gratr th rror, as th filds ar calculatd furthr and furthr away from th particls ffctd. Th largr th tim stp th mor clls th particls pass through bfor th fild ffct is rcalculatd, furthr incrasing th rror. Additionally th numbr of particls usd in th simulation- ach simulatd particl typically rprsnting hundrds to hundrds of thousands of ral particls- ffcts th accuracy of th simulation. On 5 Amrican Institut of Aronautics and Astronautics

6 th othr hand, having larg clls, long tim stps, and fwr particls acclrats th simulation allowing you to tst largr paramtr spacs and mor gomtris. Ths paramtrs ar chosn as a balanc btwn accuracy and simulation spd. Rsults usd hr wr found to b slf consistnt with simulation paramtr variations to within about 10%, which should b sufficint to support th conclusions mad. XOOPIC Particl-In-Cll Simulation Cod Th cod usd for rsults in this thsis is XOOPIC, dvlopd out of Brkly by John Vrboncour. [Vrboncour, 1993] XOOPIC is a 2.5D cod (i.. tracking 2 physical dimnsions and 3 vlocity dimnsions). XOOPIC runs on Unix, Linux, and othr Unix basd systms such as Macintosh OS X. Thr is a vrsion calld OOPIC for th PC as wll. Th cod is widly usd and wll supportd. Th following figur shows th stup for most of th XOOPIC simulations. Th simulation uss radial symmtry along th axis of th mittd bam. simulation spac Emittr Spaccraft axis of symmtry r z bam mirrord spac radial boundary Figur 4. Stup of th simulation spac in XOOPIC. Th simulation is 2.5D, 2 dimnsions in spac, 3 dimnsions in vlocity. Simulations in XOOPIC wr compard to rsults obtaind with a thrmionic mittr in a vacuum chambr. In th xprimntal stup th lctrons wr acclratd away from th mittr with a singl grid a fw millimtrs from that mittr. Th grid was groundd, whil th mittr was biasd ngativ to 210 volts. Th lctrons wr collctd at an anod 10 cntimtrs away. Th following plot shows a comparison btwn simulation and xprimnt for a constant mission voltag and a varying anod voltag. Th agrmnt is rasonabl. Thr is a slop in th xprimnt abov th flattning in th simulation data bcaus th simulation uss a constant 29.5mA bam, whil th currnt in th xprimnt is dtrmind not only by th spac charg limit of th fixd bias btwn th acclration grid and th thrmionic mittr, but also by th lctric fild btwn th anod and th grid which laks through th grid to th spac btwn th mittr and th grid. So as th anod bias is changd, th bam currnt also changs, spcially at larg anod biass. Th jump around -105 V is du to a ranging switch of a voltag snsor on th anod which changs th rsistanc of an altrnat currnt path. Additional sourcs of discrpancy ar th varying physics of currnt collction at th acclration grid, variabl mission from th mittr du to chambr contamination, and misalignmnt of th mittr. Anod/Plasma 6 Amrican Institut of Aronautics and Astronautics

7 Thrmionic mittr Grid Elctrons Anod Coppr Plat ground Figur 5. Schmatic diagram of th xprimntal stup Anod Currnt [ma] Anod Currnt vrsus Anod Voltag 30 Simulation 25 Exprimnt Anod Voltag [V] Figur 6. comparison of mission in a vacuum chambr to xprimntal rsults IV. Effcts of Emittr Siz Th following plot shows variation in th spac charg limit not as th vlocity is varid, but as th mittr width is varid. Th spac charg limit in ma is shown for rfrnc but th ky point is th factor of improvmnt ovr th 1D situation (right sid scal). Lau's analysis agrs with th XOOPIC analysis (and Humphris) for widr mittrs, but as th mittr siz drops down towards th vry small, Humphris analysis is a much bttr prdictorwhich is what in his papr Lau says would b th cas, as his solution was only don for th cas whr th mittr was wid with rspct to th gap siz. 7 Amrican Institut of Aronautics and Astronautics

8 10000 Effct of Emittr Siz on SCL SCL [ma] D SCL 3D Sim 3D Sim/1D SCL Lau Humphris Factor Imp. Ovr 1D Emittr Diamtr [cm] Figur 7. Plot of th spac charg limit as a function of mittr siz. Th agrmnt with Humphris analysis is also takn as support that th XOOPIC simulations ar sufficintly accurat to mak dtrminations about th spac charg limit in various situations. In most of th analysis don th rsults will b prsntd as factors of improvmnt ovr th 1D spac charg limit, or th 3D spac charg limit without th improvmnt tchniqu. An absolut shift of th rsults up or down is not takn to b as significant as a chang in th shap of th solution as th paramtr of intrst is varid. Th important conclusion of ths rsults is that bam sprading from small mittrs allows a vast improvmnt in th spac charg limit ovr wid mittrs. For a spaccraft whr mission powr is th most critical fatur, th bst solution may b a numbr of button sizd mittrs sprad around th spaccraft. Th spac btwn ths mittrs must b sufficintly larg that thir rspctiv bams do not ovrlap nough for th combind dnsity to xcd th SCL, as discussd nxt. 0.1 V. Emittr Spacing A critical qustion in th dsign of an EFES is how clos mittrs can b placd to on anothr. It has alrady bn shown that small mittrs oprating in isolation with a larg amount of room for th bam to sprad bnfit from a gratly improvd SCL pr unit mittr ara rlativ to much largr mittrs. By xtnsion, th bst cas scnario would b a singl tip of a Spindt typ micro-fabricatd mittr. In isolation it would not mit much, but th fficincy pr unit surfac ara of that on tip would b immns. In contrast, th worst cas scnario is a larg planar mittr (in this cas "larg" nd only b a fw cntimtrs wid) whr th SCL is almost idntically th planar 1D Child-Langmuir solution. Thus if th only objctiv wr to maximiz mittd currnt pr mittr ara, th idal solution would b many vry small mittrs widly sprad. What is lss obvious is th tradoff from a systms dsignr point of viw of whthr to us on larg mittr or many smallr ons or whr to choos th balanc in btwn. An EFES dsignr must build a systm that mits th rquird amount of currnt within th physical constraints of th systmpowr, mass, volum, and possibly most significantly, spaccraft surfac ara. Clarly smallr mittrs ar mor fficint, but how far apart must thy b to gain this fficincy? 8 Amrican Institut of Aronautics and Astronautics

9 y x Simulation Domain- groundd conducting walls Emittr Width (w) Elctron Flow Anodtypically groundd Emittr Spacing (s) Elctron Flow Anod Gap (d) Figur 8. 2 Dimnsional x-y simulation stup for analysis of spacing ffcts in XOOPIC. Two mittrs ar placd on th lft wall of th simulation domain a distanc s apart. Thy mit lctrons with an initial vlocity in th x dirction to th right wall. For simplicity, all walls, including th anod, ar hld at ground. Th following plots shows th ffct of proximity on two mittrs, as calculatd using XOOPIC in 2D mod. Currnt is mittd in th x dirction. Th mittrs ar simulatd as infinitly tall (for purposs of currnt calculation, thy ar 1m tall) and only sprading in th y dimnsion is considrd. Th widths and spacings btwn mittrs ar shown in cm. It is apparnt that th ffct of th spacing btwn mittrs lvls off at som point, a point dpndant both on th width of th mittr (w) and th mittr-anod gap (d), but not on th mission vlocity. Th dfault anod gap is 2cm. Th width of th simulation domain is th siz of th mittrs plus th gap btwn plus fiv cntimtrs on th outsid of ach mittr. Normalizd SCL Affct of Emittr Width on Effct of Spacing 0.5cm, 100V 1cm, 100V 2cm, 100V 5cm, 100V Emittr Spacing [cm] Figur 9. This figur shows th spac charg limit as dtrmind by XOOPIC. Th data is normalizd sparatly for ach mittr siz (ach lin) such that zro is th limit whn th mittrs ar adjacnt, and 1 is th maximum SCL attaind at wid sparation (s). Emission is at 100V. Gap (d) is 2cm. Obsrv that th widr th mittr, th soonr additional spacing bcoms irrlvant. 9 Amrican Institut of Aronautics and Astronautics

10 Idal (95%) Spacing [cm] Idal Spacing vrsus Emittr Width Emittr Width [cm] Figur 10. Condnsd from th plot abov, this shows th trnd in idal spacing for a pair of mittrs (th point at which th mittr SCL is 95% of what it would b for ach mittr in isolation) vrsus mittr width. VI. Th Effct of Dfocus Rings Adding a biasd mtal con around a planar circular mittr incrass th amount of currnt that can b mittd at a givn mission vlocity bfor th spac charg limit is rachd. Evn with a groundd ring (zro volt bias), th spac charg limit is improvd. Th gnral xplanation for this improvmnt is that a proximat cylindrical conductor incrass th strngth of th radial lctric fild and thus th radial sprading forc is largr and th bam sprads fastr. Bam sprading rducs th dnsity in th cntr of th bam whr th virtual cathod forms. Th lowr th dnsity th mor currnt it taks for th virtual cathod to form, and thus for th spac charg limit to tak ffct. On can think of th dfocus ring as a mor proximat rfrnc point than infinity for th lctric fild lins, or on can think of th positiv charg forming on th intrior of th ring (mirroring th ngativ charg of th mittd bam) thus incrasing th fild strngth as in a capacitor. Rgardlss th bam is sprad and th spac charg limit is improvd with no powr dissipation by th ring (though th ovrall systm cost can b nonzro dpnding on th cost of biasing th spaccraft rlativ to th surrounding plasma). Th following diagram shows th simulation stup with th dfocus ring in plac. Th nxt figurs show scrnshots of th simulation in progrss. width (w) Emittr Spaccraft XOOPIC simulation spac dfocus ring lngth (l) Elctron Flow gap (g) r axis of symmtry Anod Gap (d) z Anod/Plasma Figur 11. Configuration for mittr simulation with cylindrical symmtry in XOOPIC. Th following figurs illustrat quit clarly th bnfit of a dfocus ring. Th simulation is of 30mA of currnt bing mittd from a 1cm 2 mittr across a 2cm gap at 100V. Th dfocus ring is 0.5 cm long, 0.1cm away from th 10 Amrican Institut of Aronautics and Astronautics

11 mittr, and at a 15 angl off of paralll to th mission bam. Th ring is biasd at zro volts as ar all of th surrounding walls. Without th dfocus ring (lft figur) you gt spac charg limitd flow within 15 ns. With th dfocus ring th bam is clarly blow th spac charg limit vn aftr 150 ns. Figur 12. Improvmnt to mission charactristics via th addition of a groundd ring around th mittr. Th voltag on th ring affcts th bnfit, with highr voltags confrring gratr improvmnt to th spac charg limit. Ngativ voltags worsn th spac charg limit- and in fact this configuration is similar to th bam optics usd in lctron guns whr th objctiv is confin and rgulat th lctron bam in a uniform column [Risr, 1994], xactly opposit to th objctiv of avoiding spac charg limits. On spcific xampl of such is a Pirc Cathod- whr a ring surrounding th mittr at an angl of 67 is usd to crat a prfctly collimatd bam. Thr is a strong dpndnc upon th angl of th dfocus ring in dtrmining it's ffctivnss at allviating spac charg limits. Th following plot shows th rsults of simulations with th abov paramtrs (100V mission across a 2cm gap) with dfocus rings biasd from -100 to +100V at varying angls around th mittr. Th rings ar all 0.1cm away from th mittr and 0.5cm long. Th rsults clarly show that th closr to th mittd bam th rings ar (smallr angls), th mor ffctiv. Th limit occurs whn th ring is clos nough, or th bam sprad far nough, that th bam contacts th ring and th ring collcts charg; at this point it is no longr ffctivly assisting in moving charg away from th spaccraft. A bst cas improvmnt of 160% (factor of 2.5) is shown btwn no 11 Amrican Institut of Aronautics and Astronautics

12 ring and a V ring. Evn with no bias on th ring (a simplr chapr systm), an improvmnt of 40% (factor of 1.4) can b obtaind with a groundd ring. SCL [ma] SCL vrsus Dfocus Ring Angl 90 no ring xp cm, cm, cm, cm, cm, Dfocus Voltag [V] Figur 13. a comparison of th ffct of a dfocus ring on th SCL, vrsus th voltag on th ring, for various ring angls. Th dfocus lngth is 0.5cm, and th ring starts 0.1cm away from th mittr. Th following plot shows th ffct of gap siz and dfocus lngth on th ffctivnss of th dfocus ring. SCL vs Dfocus Gap SCL vs Dfocus Lngth SCL [ma] SCL [ma] Emittr Dfocus Gap [cm] Dfocus Lngth [cm] Figur 14. Effct on th spac charg limit of variation of th dfocus gap and dfocus lngth. Simulations ar don with a 1cm^2 circular mittr mitting at 100V across a 2cm 0V gap. Th ffct of th gap siz and th dfocus lngth is similar- th closr th bam gts to th dfocus, th strongr it's ffct. This occurs both in th cas whr th dfocus is positiond closr to th mittr and th cas whr th dfocus continus out to th point whr th bam has sprad almost to within contact of it. In ithr cas, th gap and lngth ar limitd from xponntial incrass at th zro point by contact btwn th bam and th dfocus. All analysis ar xtndd only to this point, as it obfuscats th rsults to intrprt a scnario whr only part of th currnt is scaping. Th dfocus lngth plot abov nds at 0.6 cm bcaus for ths paramtrs byond that lngth th bam is partially collctd by th dfocus. 12 Amrican Institut of Aronautics and Astronautics

13 Th ffctivnss of a dfocus ring varis invrsly with th siz of th mittr- vry larg mittrs rciv littl bnfit, adding anothr tradoff to th systm lvl dcisions about th sizing and spacing of mittrs. VII. Conclusion For mission scnarios whr th bam powr/vlocity is low rlativ to th bam currnt, th siz and shap and surrounding gomtry of th mittr can hav a significant impact on th fficincy of th mittr. Scnarios such as ths occur, for xampl, on small spaccraft with small thrustr systms using cold cathod fild mittrs. For small spaccraft powr is inhrntly limitd so th fficincy of th mittr is all th mor important. In ths and similar situations dsigning th mission systm with th tchniqus prsntd hr to mitigat spac charg limits could provid ssntial powr savings. Acknowldgmnts This work is bing fundd by a NASA-MSFC GSRP studnt grant. Rfrncs Child, C. D., "Discharg from Hot CaO", Phys. Rv., sr. 1, vol. 32, 1911 K. L. Jnsn, Physics of Plasmas 6, 2241, (1999). Gilchrist B. K., Jnsn, A. Gallimor, J. Sornsa, Fild-Emittr Array Cathods (FEACs) for Spac-Basd Applications: An Enabling Tchnology, whit papr, Th Univrsity of Michigan and Th Naval Rsarch Laboratory, 1999 Langmuir, I. Phys. Rv. Sr. II, 2, 450 (1913) Lau, Y. Y., "A simpl Thory on th Two-Dimnsional Child-Langmuir Law", Phys. Rv. Ltt. 87, , Librman, Lichtnbrg, Principls of Plasma Dischargs and Matrials Procssing, John Wily & Sons, Inc., 1994 Luginsland, J., S. McG, and Y. Y. Lau, IEEE Trans. Plasma Sci., 26, p.p , Luginsland, J. W., Y. Y. Lau, R. M. Gilgnbach, Two-dimnsional Child-Langmuir law, Phys. Rv. Ltt., 77, p. 4668, Marrs, C., Compatibility of fild mission cathod and lctric propulsion tchnologis: thortical and xprimntal prformanc valuations and cathod rquirmnts, Ph.D., Univrsity of Michigan, NRL Plasma Formulary publishd by th Naval Rsarch Laboratory, 1998 dition. NRL/PU/ Vrboncour, J. P., M. V. Alzs, V. Vahdi, C. Birdsall, Simultanous Potntial and Circuit Solution for 1-D Boundd Plasma Particl Simulator Cods, J. Comp. Phys., V. 104, p , 1993 Wingl, R.M., P.L. Pritchtt, "Spac Charg Effcts During th Injction of Dns Elctron Bams Into Spac Plasmas", Journal of Gophysical Rsarch, Vol. 92, No. A6, Pags , Amrican Institut of Aronautics and Astronautics