Approved by

"L., Paul J. Wilbur

December 1984

.....:_ Department of Mechanical Engineering

.... Colorado State University

Fort Collins, Colorado

'! Q
k

if

' i

1. Report No. 2. Government Accmmon No. 3, Recipient's Catalog No.

,; NASA CR 174810

4 Title and Subtitle 5. Report Data

'"" ION THRUSTERPERFORMANCEMODEL Dec, 1984

.. 6. PerformingOJ_nization Code

:?'/ 7. Author(s) 8. PerformingOrganizationReport No.

_', • John R. Brophy

i :

, 10. Work Unit No.

9. PerformingOrganizationName and Address

__t_ Departmentof MechanicalEngineering

. Colorado State University ll. Contractor GrantNo.

Fort ColIins, Colorado 80523 NGR-06-O02-112

_r' " 13. Type of Report end Period Covered

_,:, "' 12. SponsoringAgency Name and Address Dec. 1, 1983 - Dec. l, 1984

_ NationalAeronauticsand SpaceAdministration 14.SponsoringAgency Code

• _,_ Washington.D.C. 20546

; ' _/',, 15. Supplementary Notes Grant P_nitor - William Kerslake, NASALewis Research Center, Cleveland, Ohio 4413S.

_ - This reportis a reproductionof the Ph.D. Dissertationof John R. Brophy. It is submittedto the

sponsorand to the distributionlist in this form both as a presentationof the technicalmaterial,

i_ and as an indicationof the academicprogramsupportedby the grant.

_: 16. Abstract

A modelof ion thrusterperformanceis developedfor high flux density_cuspedmagneticfield

thrusterdesigns. This model is formulatedin termsof the averageenergyrequiredto producean

.... ion in the dischargechamberplasmaand the fractionof theseions that are extractedto form the

-----_, beam. The directloss of high energy (primary)electronsfrom the plasmato the anode is shownto

:_ have a major eff'.'cton thrusterperformance.The model providessimplealgebraicequations

° , enablingone to calculatethe beam ion energycost. t__.averagedischargechamberplasmaion

____,i"_ energycost,the primaryelectrondensity,the primary-to-Maxwellianelectrondensityratioand

, the Haxwellianelectrontemperature.Experimentsindicatethat the model correctlypredictsthe

_": variationin plasma ion energycost for changesin propellantgas (Ar, Kr and Xe), grid trans- !

:' :_ parencyto neutralatoms beam extractionarea dischargevoltage,and dischargechamberwall

_:._ temperature.

The modeland experimentsindicatethat thrusterperformancemay be describedin termsof only

_ four thrusterconfigurationdependentparametersand two operatingparameters. The modelalso

• suggeststhat improvedperformanceshouldbe exhibitedby thrusterde._ignswhich extracta large

..... fractionof the ions producedin the dischargechamber,whichhave good primaryelectronand

neutralatom containmentand whichoperateat high propellantflow rates. In addition,it suggests

-:_ " that hollowcathodeefficiencybecomesincreasinglyimportantto the dischargechamberperformance

_-_ as the dischargevoltageis reduced. Finally,the utilityof the model in missionanalysiscalcu-

-_ ,: lationsis demonstrated.The model makes it easy to determinewhichchangesIn thrusterdesignor

operatingparametershave the greatesteffecton the payloadfractionand/ormissionduration.

L.:.i'

LL'r: 17i Key Words (Suggestedby Author(s)) 18. Distribution Statement

" !

• ElectrostaticThruster Unclasslfi_,d- Unllmited

DischargeChamberModel

18. Security Classif.(ofthis report) 20. Security Classif'.(of this page) 21. No. of Pages 22. Price*

Unclassified Unclasstfled 133

' ForsalebytheNationalTechnicaInf0rm l atl0nService.Sprmgfleld.VirgmnJa22]61

" NASA-C-168 (Rev. 10.75)

I
_L;I•

C:

d '

• TABLE OF CONTENTS

_' Chapter

I• INTRODUCTION,Qeeeolooleoloeoeeoa l

:=__'". Background eeDolea_eaeooeeloeo 2

' _ Overview 4

_d:'4*_ L II, ION THRUSTEROPERATION............... 7

-,_,,:_ Ill. THEORETICALDEVELOPMENT ............ 14

...... :: ThrusterPerformance ......... 14

,_:,:i Assumptionsand Limitatios ......... 14

- Beam Ion EnergyCost ............ 15

_"__, P1asma Ion EnergyCosc ........ 18

? ,,ak,> .... Calculationof PlasmaProp r i ........ 27

_. PrimaryElectronDensity .... 27

Primary-to-TotalElectronDensity'Ratio" . 31

_: Primary-to-Maxwell•anElectronDensityRatio 32

_"_ Maxwell•anElectronTemperature ....... 32

_ :! Double Ion Production 34

_._, IV. EXPERIMENTALPROCEDURESAND RESULTS ........ 37

.....

.... __: Apparatus oooojoeeooooe_oo•eeo 37

_ ' Procedure 41

_, Experimental ............

,_ii Plasma Ion EnergyCost ............ 43

,o. ExtractedIon Fraction............ 54

....... P•asma Properties 56

'" V MODEL APPLICATIONS 6g

' ThrusterDesign 69

_-:_/.'-.* ThrusterScal_"'i ng .............. . • . • . ........ •

_ LC_ _ _'" "'i NeutralLoss Ratu .......... 77

.......... ' ThrusterTestingWithout'BeamExtraction .... 78

_±,_-, Space PropulsionMissionAnalysis . . . . . . . . 81

CONCLUSIONS 92

0,

• Suggestions' fo; Futu;e'Work ........... ........... g4

REFERENCES g6

APPENDIXA - Theoretical_!culatlon of the

• BaselinePlasma Ion EnergyCost, ¢_ . . . . . lOl

:,. ii
F-_-_!_ ' A I_,_

L.'_'":_,,_.,L,-:'

L i'"

h.

_.... , : Table of Contents(Continued)

P_ e.

: APPENDIXB - Error Analysis ................ 109

APPENDIXC - Nomenclature ................. 123

• ii:

,,L .

.c, ;,:,, '

...... /
fl

L,.

LIST OF FIGURES

! ' Figure Page

.... 1 Ion ThrusterSchematic .............. 8

.. 2 DischargePIasma Power Balance Schematic ..... Ig

3a Ring Cusp Ion SourceSchem_";c -

: ConfigurationI 38

_"", 3b Ring Cusp Ion Source Schematic-

!'_:i ConfigurationII 38

- _: 4a PlasmaIon EnergyCost Curve for

• ConfigurationI • • e • • • • • • • • • • • • • • 44

, ,.ii, 4b PlasmaIon EnergyCost Curve for

High AcceleratorGrid Transparency

_ to NeutralAtoms - ConfigurationI ..... . . . 44 :

•_;! 5a Plasma Ion EnergyCost Curve for Krypton-

,, ConfigurationI 47

__ .,- 5b Plas;_ Ion EnergyCost Curve for Small

._ DiameterIon Beam - ConfigurationI ........ 47

, 6a Plasma Ion Energy Cost Curve fo_ Argon

.,. ,* Com"igurationII • e • • • • o o • • • • • • • • • 50

.:., 6b Plasma Ion EnergyCost Curve for Xenon -

...." ConfigurationII 50

,, 7a Plasma Ion Energy Cost Variationat Low

.., DischargeVoltage - ConfigurationII ....... 51

7b Anode ElectronTemperatureVariationat

Low DischargeVoltage - ConfigurationII ..... 51

_. 8a Effectof DischargeVoltageon the extracted

_r. Ion Fraction- ConfigurationI .......... 55

"t

' • 8b Effectof Propellanton the ExtractedIon

Fraction- ConfigurationI ............ 55

ga PrimaryElectronDensity Variationfor Argon -

ConfigurationII ................. 58
List of Figures (Continued)

k

Figure Page

9b PrimaryElectronDensity Variationfor

•.... Xenon - ConfigurationII ............. 58

i-r _ ,._ IDa Primary-to-TotalElectronDensity Ra_io

i• _,:. for Argon ConfigurationII 61

='

i- ' "

lOb Primary-to-TotalElectronDensityRatio

, ,, fo.-Xenon - ConfigurationII ........... 61

•; ,.'!.. II MaxwellianElectronTemperatureVariatior,

_'_:- for Xenon - ConfigurationII ........... 62

i

i"i,iii_,_o_ _'i 12 IonizationRate Factorfor Xenon ......... 64

13 Correlationof ElectronTemperatures ....... 66

__:-_,_ ._ 14 Doubly-to-SinglyCharged Ion Beam Current Results . 67

• fB

..... _ 15a Effect of on Performance ............ 71

._._!.: 15b Effectof Co on Performance............ 7l

°_ 16a Effectof PropellantFlow Rate on Performance

_i for Small Co 73

16b Effectof PropellantFlow Rate on Performance

! ',' for Large CO 73

,._ 17 Effectof CathodeOperationon Performance

• for Xenon 76

• 18 Payload Fraction,PropellantMass Fractionand

_ :r GeneratorMass FractionVariationwith

._ PropellantUtilization .............. 85

i- i Iga Effectof Power Plant SpecificMass (_)

on Payload Fraction................ 87

F Igb Effectof Power Plant SpecificMass on

: Optimum PropellantUtilization .......... 87

i

20a Effectof Bo on PayloadFraction ......... 87

20b Effectof B° on the PerformanceCurve and

the OptimumPropellantUtilization ........ 89

2la Effectof fB on Payload Fraction ......... 90

V
®

_,c_ Listof Figures (Continued)

'. 21b Effectof fB on the PerformanceCurve and

.... t,le Optimum Propellant Utilization ........ gO

+_+: 22a Effectof Propellanton Payload Fraction ..... 91

+.,,_ 22b Effectof Propellanton the Performance

Curve end the Optimum Propellant Utilization . . . 91

+: A-la lom ation CollisionCross Sectionfor

• Xenon (Ref 50) 104

A-Ib Total ExcitationCollisionCross Section

for Xenon (Ref. 49) ................ 104

A-2 BaselinePlasma Ion EnergyCost Variation

Calculated from Eq. A-7 for Xenon ......... lOS

A-3 Solutionsfor the BaselinePlasmaIon

...., EnergyCost for Xenon 108

+. ,:; B-I Ion Source InstrumentationSchematic Ill

+ **?!

_ B-2 Uncertaintyin the PlasmaIon Energy

+L l ?l' Cost Measurements................. l16

,, B-3 Uncertaintyin the Doubly-to-Singly

,_ Charged Ion Beam CurrentMeasurements....... 122

+,

+++, ,.

i

i.

L :

i

i .

vi

C

_- .... ,.j........... +_ __+,L_. ........
4,

• LIST OF TABLES

i ,_ Table

.. 1 StandardConfigurationParameters.........

:> . 74

_, :• 2 Effectof Vc on _p ......... .......

r 7,.

"!

' / ..

; _ .

, ,, • °

i =i_"

•%
' _ It •

. ,,, .

i* ̧ _/, •

.......... I. INTRODUCTION

-iTh::i; i.

-u,::., ' Electronbombardmention thrustershave been proposedfor both

:,.. primaryand auxiliaryspace propulsionapplicationsfor nearly two-anda-halfdecades. During this time, thrusterperformancerequirements

have variedaccordingto the missionsof current interest. Also during

this time, the developmentof thrusterdesigns capableof meetingthese

requirementshas been largelyexperimental. That is, thrusterdevelopment has generallybeen accomplishedby a procedurein which the design

, o?-; "_i parametersthat influencethrusterperformanceare physicallyvaried

i until an acceptableconfigurationis obtained.

o.; This procedurehas several inherentlimitations First of all, it

can be time consuming,especiallyif a large numberof parametersis

involved. Secondly,an optimum configurationmay not be found. That is,

dE once a configurationcapable of fulfillingthe mission requirementsis

_.._: identified,the procedureis generallyterminatedeven though this may

_-_i not resultin the best configurationpossible. Finally,changes in the

missionsof current interestto those characterizedby different

thrusterrequirementsnecessitatethat the iterativeexperimentalpro-

,?

cedurebe repeated.

' Consequertly, , there is a need for the developmentof an analytical

model which describesthe effectsof thrusterdesignvariablesand

"° operatingparameterson thrusterperformance. Such a model should,as

a minimum,be capableof providingguidancefor the iterativeprocedu_'e
_ 2

_,_,

__,_, described above, and ideally would be capable of describing exactly how

_¢_ a thruster should be designed to achieve a given set of performance

=_ i requt rements

Many models of discharge chamber operation have been developed

over the last 24 years. A brief discussionof these models is given i,,

the next section. In general, the complexity of the processes taking

place in the dischargechamberof an ion thrustertogetherwith the

!

__ relative ease with which new thruster designs can be tested experimentally has resultedin a situationin which theoreticalunderstanding

has lagged experimental developments.

The objectiveof this researchis to improvethe theoretical

understanding of ion thruster operation by providing a simple physical

i

i

model of the processesaffectingthrusterperformance. Additionalcon-

_,t straintson this model are that it shouldbe easy to use, yet general

)

...... }_ enoughto be applicableto a wide rangeof thrusterconfigurationsand

operatingconditions•

= _;J ""i/

....._, Bac.kground

Analyticalmodelingof electronbombardmention thrustershas been

, i

- on going more or less continuouslysince their introductionby Kaufman

: [1,2] in 1960. Milder [3] providesa surveyof modelingeffortsmade

:: through1969. In this surveyhe concludedthat, ...ourknowledgeand

_u

,_1 understandingof the physicsof these plasma:is far from complete."

...._' In addition,he concludedthat the usefulnessof these effortswas

_ _;' limitedby either the simplyflngassumptionsrequiredto make the

_-_ nroblem tractableor the difficultyof applyingless simplifiedmodels.

.

_._. -- ..... _ ,,,, _ ...... _, "_li-- __ "="_'=='=="
Kaufman [4] presentsa discussionof thrustertechnologyas of

1974 in which the latesttheoriesof thrusteroperationare described.

b:

iHe concluded,as had Milder earlier,that the most successfuleffort

! "-_ to date was the semiempiricalapproachproposedby Masek [5] in 1969.

,_ The theoryof Masek is semiempiricalin that it requiresas input a

i, detailedknowledgeof the plasma propertiesinsidethe thrusterdischarge chamber. These propertiesare generallyobtainedusing a

,_ _ Langmuirprobe togetherwith a data reductionprocedurethat is both

i .... "

_ _' tediousand until recently[6,7] of only limitedaccuracy.

F- ,_.

...._, Since 1974 a numberof analyticalmodels have been developed,

. . =

i_ directedtowarddifferentaspects of ion thrusterdischargechamber

_" operation[7-19]. Of these, the models proposedin references8

_ _ through 12 are extensionsof Masek'smodelingtechniquein that they

° _i" requiredetailedLangmuirprobe data as inputs. References16 through

=i' 18 pre_entmodelswhich do not require probe data, but which instead

_: consistof a complex set of equationswhich must be solvediteratively

,i" by a computer. This complexitymakes these modelsdifficultto apply

_r"{L

_ and limitstheir usefulnessin providingguidelinesto improved

thrusterdesigns.

Dischargechambermodels of ion sources for neutralbeam injectors

, have also been proposedrecently[20-22]. The operationof these

sources isinmany respectsvery similarto that of ion sources for

space propulslonapplications. This similaritywas enhancedby the

recentswitch in space thrusterdesign [23,24]to dischargechambers

characterizedby the same high magneticflux density cuspedfields used

in neutral beam injectors. References20 through22 present relatively

simplemodels that providevaluableinsightto dischargechamber
_i_I'_ _

i'i_.._

LI_,_.,_I _ ,;;./i_

.... _, 4

, _'L i

i_i_ operation,but also requireplasma propertydata as inputs Further,

thesemodels &re not developedto the point where the performanceof a

_. given dischargedesigncan be calculateddirectly,

._r:._;_, _: In summary,the situationat the presenttime is remarkably

_;i;_ similarto the way it was in 1969 as describedby Milder [3]. Many

__' additionalmodels have been proposedsince 1969 providinga significant

_-_'_.... improvementin the overallunderstandingof dischargechamber processes,

, however,most of these models still requiredetailedplasmaproperty

._!,.,:_,.,,_°"_ data as inputs. Those that do not are eitherover simplified,result-

_I!_ _- ing in a loss of generalityand usefulness,or too complex to be

_.._:__,_, appliedeasily. Thus, there is a need for the developmentof a model of

•,_ii _'_',_.:..

_:_:,I/:P!", ion thrusterperformancethat is easy to use, does not require plasma

_i_ I:,C data as inputs,and yet is generalenough and accurateenoughto serve

I_,_'_,_L_ _ii;, as a guidelinefor the designof improvedthrusterconfigurations.

._ OvervJew

,,

'

-" This dissertationis organizedin the followingmanner. First a

brief reviewof ion thrusteroperationis given in Chapter II. In this

chapter,the dominantmechanismaffectingthrusterperformanceare

identifiedand discussedqualitatively. Since this report is primarily

concernedwith ion thrusterdischargechamberoperation,detaileddis-

<o, cussionsof other thrustercomponentssuch as the ion accelerator

Ir systemand cathodes,etc., will not be given. These componentswill be

discussedonly in regardto their effect on the operationof the main

i •

dischargechamber,

In Chapter Ill the analyticalderivationof the equationscom-

,_ prising the thrusterperformancemodel proposedin this investigation
L.__,_,_;:e_"A _i_,

,_h

h'u. 5

11,  ̧

.... _i" is presented. This model consistsof a set of algebraicequations,

., each of which describesthe behaviorof a differentperformanceparam-

.... eter. For example, a single equation for the thruster performance

curve (variationof beam ion energycost with propellantutilization

._'. efficiency)is developed. Other equationsare developedthat allow the

_ averagevaluesof the dischargechamberplasmapropertiesto be calcu-

. lated as fu_lctions of the thruster operating point. These plasma

i:';"!'_ i_i"_: propertiesinclude: the primaryelectrondensity,the primary-to-

-::_',_i:Sl_i_ _.:

_=_ ; : Maxwellianelectrondensity ratio,the Maxwellianelectrontemperature,

_:_'_,i_!_;_ In Chapter IV,the experimentalapparatusand proceduresused to

investigate the validity of the proposed model are described. Experir,, _:{_ mental results comparisonof these resultswith the predictionsof the

_"c_:%' 2i

......... _, model, and a discussionof this comparisonare also given in this

.... ; " _' chapter.

:. __i Three applicationsof the thrusterperformancemodel are discussed

: ",,,-:! in Chapter V. The first of these illustratesthe effect of thruster

i-_:_ designand operatingparameterson the standardperformancecurve. The

i!i. second application describes how the model can be used to extrapolate

=_: data taken without ion beam extractionto obtainthe performancecurve

,=_d_,r: applicableto operationwith beam extraction. Finally,an illustration

_ .- of how the model can be applied to mission analysis calculations is

.... _ _ presented. In this application,the effectsof thrusterdesignparameters and propellant utilization efficiency on the deliverable payload

o fractionare discussedfor an earth orbit raisingmission (low earth

; orb'itto geosynchronousearth orbit).

5"
-i_ .

The major conclusionsof thi-_ investigationare summarizedin

• ChapterVl. Sl units are used throughoutthis reportwith the exception that energy is frequently given in units of electron volts. In

.... :' addition,forthermalelectronstheenergyquantitykT, ,_herke is

iI .

_ _ Boltzmann'sconstantand T is the temperaturewill be given as eTM

_ I_ where e is the electronic charge and TM is the electron temperature in
*(.t)' F

_:_ II. ION THRUSTEROPERATION

_i i'_''°'i!i_IJi;_i: shownAschc3netmatically emporary"ring in Fig.cuspI." eThis lectron thruster bombardment consistsionofthruster a cylindrical is

_ osther teel sentructure d by a setbouofndetwo datcolnosely e endsbypaced a circular grids. The steelvoplate lumeboun anddated the by

___ this structureis referredto as the dischargechamber. The two grids at the end of the dischargechamber have a matched matrixof holes

!. throughwhich the ion beam is extracted. The innergrid is referredto

as the screengrid while the outer one is called the acceleratorgrid.

Duringthrusteroperation,neutral propellantgas is injectedinto

/,

• ._" the dischargechamber, The preferredpropellantgas has, for the most

...... part, been mercuryvapor becauseof its large atomicmass, low ioniza-

_ tion energy, and its liquidphase storability. Cesium vapor has also

_ been used. However,at the present time, the propellantsof most in-

;=_'i _ "

......_ terest are the rare gases argon, kryptonand xpnon. In this investiga-

(_ "

: tion,only these rare gases are considered. The propellantgas is

assumed to fill the dischargechamberuniformlyduring thruster

_ , operation.

o A hollowcathode servesas the sourceof electronsfor the dis-

: charge chamber. Hollow cathodeshave replacedboth refractorymetal

and oxide cathodesin ion thrustersprimarilybecausethey exhibitvery

long lifetimesand can be restarteda numberof times even after

exposureto air (an importantcapabilityif the thrusteris to be
GRID

II

' HOLLOW / \'''' "' SCREEN

__ / GRID _. CATHODE '

' J

,.d_ '_ /

_£,_L_-:L "1t

; ? J

o i_ING

fi j.. !

, \%

._, MAGNETIC I NEUTRAUZER " FIELD LINE _ "

. /HOLLOWCATHODE

_='o _ ANODE

SUPPLY I

/ EUT,

°t'.. + SCREEN + ACCEL ANODE

SUPPLY SUPPLY SUPPLY I

TO SPACECRAFT

GROUND

t!

Figure I. Ion Thruster Schematic

..... IIII I1_'_1 II
_?i:_;'_ 9

_,; preflight tested). A detailed discussion of hollow cathodes is given

_ by Siegfried [25].

_,,_._, The electrons emitted by the cathode in Fig. 1 are accelerated by --_;;,i :TI

an electric field adjacent to the cathode into the discharge chamber.

This electric field is established by biasing the discharge chamber

walls (with the exception of the screen grid) 30 to 50 volts positive

of the cathode by meansof an external DC power supply (called the

__: discharge or anode supply). Electrons which have undergone this accel-

_ erationare called primaryel_ctrons. The energy of these primary

j.

_ __, electronsis determinedby the voltagedifferenceapplied betweenthe

_-_ _: anode and cathode. The magnitudeof this voltagedifferenceis chosen

_, :_ so that the collisioncross sectionfor ionizationby the primary

}.

'_ electronsis largewhile at the same time the collisioncross section

, for the productionof multiplycharged ions by the primaryelectronsis

small.

'.!

; A secondgroup of electronsoriginatesfrom the inelasticinterz;

, actionof the primarieswith the neutral propellantatoms. These

° interactionsreduce the energyof the primaryelectrons. In this in-

, vestigation,an electronis no longerconsidereda primaryelectronif :

it has had at least one inelasticcollision. Ionization.the inelastic

collisionprocess of primary interesthere, results in the releaseof

, low energy secondaryelectrons. Primaryelectrons,that have had their

. energydegradedby inelasticcollisions,and secondaryelectrons

releasedby ionizatio&thermalizeto form an electronpopulationwith a

..... nearlyMaxwellianenergydistribution(characterizedby a temperatureon

_ the order of a few eV). It is possiblefor both electronpopulations

• ©
t+o,,

c, _.

L '!!"

to exist simultaneouslyin the dischargechamberdue Lo the low intero action rate betweenthe primary and Maxwellianelectrons[26].

rl At typicaldischargechamber neutralatom densities(_IOIBcm'3),

the ionizationmean free path for primaryelectronsin neutralatoms is

_i,. on the order of meterswhile typical dischargechamberdimensionsare

..... _LI on the order of tens of centimeters. For this reason,a magneticfield

. _ ,%: is employedto restrictthe directaccessof primaryelectronsto the

= :i_;: anode The magneticfield for the thrusterof Fig. I is created by

C •

°":'L.... IC_ the rings of magnetsof alternatingpolaritylocatedalong the back and

sides of the dischargechamber. The configurationand strengthof the

_ magneticfield has a substantialeffecton thrusteroperationand con-

...._ <: sequentlyhas been the subjectof numerousstudies[6,7,13,24,27-33].

= ._i_= Contemporarythrusterdesignsare operatedwith the entiredischarge 

:: chamberhousing (exceptfor the screengrid) at anode potential. Magnetic field lines at the field cusps, thus, terminateon anode potential ,

surfaces• This allowselectronsto be lost to the anode by travelling

along magneticfield lines as well as by diffusingacrossthem. To

_: adequatelyrestrictthe flow of primaryelectronsalong the field lines

_. to the anode,magneticflux densitieson the order of O.l tesla are

required.

.. The dischargechambermagneticfield reducesthe probabilitythat

_f_mlL

a primaryelectronwill be collectedby the anode without first having

had an inelasticcollisionwith a neutralpropellantatom. This probability is a functionof the thrustersize, the dischargechambermag-

......... netic fleia cunfiguration,the cathode locationand the neutralatom

: density• As the neutraldensitydecreases,the probabilitythat a

;, ..:..._._.,,,.._,,_,,, .... ,,, ,,_i,r,_, 'i_,m,,,_i'r "i-riI'll" II I
[,

il/

. ll

i

primaryelectronwill be lost to the anode without havingan inelastic

. collisionincreases.

For the hypotheticalcase of a zero neutral atom density,this

'& '., probabilityis one. That is, all primaryelectronsemitted by the

' cathodewill be collectedby the anode and none will have inelastic

collisions. In this case, each primaryelectronwill, on the average,

travela certaindistancethrough the dischargechamber before being

io_ collectedby the anode. This distanceis a characteristicof the

_ thrustergeometry,magneticfield configurationand cathode location, i :;_{!

i°, i_ and is calledthe primaryelectroncontainmentlength [34]. With this

_ definition,the probabilityof primaryelectronloss to the anode may

_ be expressedas a functionof the ratio of the primaryelectroncon-

': tainmentlengthto the mean free path for primaryelectron-neutralatom

inelasticcollisions[34]. The loss of primaryelectronsto the anode , !

: ,, , T .

constitutesa loss of dischargeenergy. Consequently,ion thruster

_ _ performanceis stronglydependenton the probabilitywith which primary

_ electronsare lost [34,35].

The plasmaproducedwithin the dischargechamberwill typically

assume a potentiala few volts positiveof the anode. Thus, a

potentialsheathwill exist at all plasmaboundaries. The magnitudeof

this sheathat cathode potentialsurfacesdependsprimarilyon the

.}

'_i magnitudeof the dischargevoltage appliedbetweenthe cathodeand

anode. The sheathpotentialat cathode surfacesis sufficientlynegative to reflectall but the most energeticelectronsin the tail of the

Maxwelliandistribution. Consequently,the vastmajorityof electrons

in the plasmacan leave the dischargechamberonly at anode potential

surfaces.
Of the ions produced in the discharge chamber, somewill reach the

dischargechamberwalls. Those that do, recombinewith electronsthere

and return to the plasma as neutral atoms. Most of the ions that reach

: the grid system,on the other hand, are acceleratedby the electricfield

_;i betweenthe screenand acceleratorgrid to form the ion beam. The

,, fractionof the total ion currentproducedthat is extractedintothe

_ : : beam is called the extractedion fraction. Because it does littlegood

[_ '; to produce ions in the plasma only to have them recombine at the dis-

! "

!; o,_ chargechamberwalls, it is clearlydesirable to have thrusterdesigns

! ?

o_ in which the extractedion fractionis as large (closeto one) as pos-

_ sible [34]. The electricfield between the grids is establishedby

L

: biasingthe thrusterbody positiveof groundpotential(on the order of

:.i.. lO00 volts) and biasing the accelerator grid several hundred volts negaZ _L\

'L

, rive of groundpotential.The final velocity of the ions in the beam

_ is determinedby the sum of the positivepotentialappliedto the

, thruster cathode and the discharge voltage.

i:

. Electrons are injected into the positive ion beam by a cathode

(called a neutralizer) positioned downstream of the accelerator grid in

;_ _ order to space chargeand currentneutralizethe beam. The negative

accelerator grid prevents these electrons from backstreaming to the

positivethrusterbody.

During thruster operation, the propellant gas in the discharge

chamber is only partiallyionized,with the ion density tyrIcallyan

order of magnitude smaller than the neutral atom density. Thus, one

might expect the neutral flux through the grids to be greater than the

ion flux. This, however, is not the case since the neutral flux is

determinedby free-molecularflow towardthe grids at a temperature

F

_ Q
ii.

_ 13

governedby the mean dischargechamberwall temperature[36] (_ 40OK).

The ions,on the other hand, tend to move toward the grid systemat the

Bohm [37] velocity,which is governedby the much higherelectron

,,_ temperature(typically5 eV or _ 58,000K). Thus, the ion flux through

the grids is usually greater than the neutral flux even though the ioh

_ density is substantiallysmallerthan the neutraldensity.

The ratio of the beam current (assuming singly charged ions only)

,! to the total propellantflow rate (expressedin units of equivalent

i o _ amperes) is known as the propellantutilizationefficiency. For a

°i°"i. constant total propellant flow rate, increasing the propellant utiliza-

_ _'\:_ . tion efficiencygenerallyrequiresan increasein the average energy i

_, requiredto producea beam ion. This variationof the beam ion energy

_.. cost with the propellantutilizationis known as a performancecurve.

It will be shown in the chapters that follow that the dominant mechanIsms affectingthe performancecurve are the loss of primaryelectrons

.... ___ to the anode and the extracted ion fraction.

i
III• THEORETICALDEVELOPMENT

h_' "

_, ThrusterPerformanceModel

In this section, a model is developed which results directly in a

singlealgebraicequationfor the variationof the beam ion energycost

_ with the propellantutilizationefficiencyfor high flux densitycusped

o: magneticfield thrusters. This is made possibleby formulatingthe

i,

,;,_ model in terms of the averageenergy requiredto producean ion in the

_ dischargechamber plasmaand the fractionof ions producedwhich are

• extractedinto the beam• A key featureof the model is that it

, providesa simpletechniquefor the calculationof the averagedischarge plasma ion energycost as a functionof the propellantflow rate

F

_ and propellantutilizationefficiency•

i "

o

'_-_ Assumptionsand Limitations

The developmentof the model assumes steadystate discharge

chamber operation The model is applicableto dischargechamberswhich

•: produce low pressure,partiallyionized,opticallythin plasmas in

which neutral densities are in the range of 101B to lO 19 m"3 and plasma

densitiesrange from I0Is to lO17 m"3. In addition,Maxwellianelectron temperatures and primary electron energies should range from

, I to I0 eV and 30 to 50 eV, respectively.

Electron energy losses due to elastic collisions with ions or

neutralatoms are neglected. This can be justifiedbecause the average

C
r-_ _._, .....r ......_

L _ _i_!_1 _:

_ r

I

energy loss per encounter is proportional to the ratio of masses, and

this ratio is small (_ 10"5), Electron energy losses resulting from

_ inelasticcollisionswith ions are also neglectedbecauseof the low

ion density relativeto the neutraldensity. This should not cause

° o.- a significanterror,except perhapsat very high propellantutiliza- /-

i ions. Only singlycharged ions are assumed to be produced,and the

_ c_ effectof metastableatomic stateson ion productionis neglected.

CI:, Primaryelectronthermalizationresultingfrom collisionswith the

_

; backgroundMaxwellianelectronsalso is neglected. Primaryelectron

behavioris assumed to be lin,itedto either inelasticcollisionswith

_ _ neutralato[_iosr direct loss to anode potentialsurfaces. A primary

,_i electronis consideredto join the Maxwellianelectronpopulationafter

_ havingone inelasticcollision.

_ Electronsare assumed to be constrainedby the plasmasheathsso

r I'_

,. they are able to leave the plasmaonly at anode potentialsurfaces.

_ Ions and photons (emittedby the de-excitationof excited propellant

_ atoms)are assumedto be lost acrossall plasmaboundaries. The

: assumptionof a low pressuredischargeimpliesthat ion-electronrecombinationshouldbe wall controlled,consequently,volume ion recombinaLion is neglected.

• The neutralatom density is assumed to be uniform throughoutthe

dischargechamber,and free molecularflow is assumed to apply to the

neutralatom flux throughthe acceleratorgrid system.

Beam Ion EnergyCost

The averageenergycost per beam ion is definedas,

tB _ (JD " JB) VD/JB , [eV/beamion] (1)

_,_'_"'_'_'_='tL_'-_":--"_'_l'l [iiII _l ' II ' II4 ill .... |-I IlI ........ I III I_l
-- ¢,

.i

,, where all symbols_re defined in AppendixC. The beam current (JB) in

j_.i. Eq. 1 is subtracted from the discharge current (JD) so that the energy

that goes into acceleratingthe beam ions throughthe dischargevoltage

=_i:_ is not charged to the beam ion energy cost. The numerator on the

-_i:;. right-hand-sideof Eq. 1 representsthe power used to operate the discharge chamber.

_!?ii/ In a similarmanner, the average energyexpendedin creatingions

_,,_ in the dischargechamber plasmamay be definedas,

=_::!/:' Cp --[JD " (JB + JC)] VDIJp , [eV/plasmaion]. (2)

:_;_,_- By analogyto Eq. I, the "JB + JC" term is subtractedfrom the dis-

/;: charge currentso that the energy that goes into acceleratingthese

, ::- ionsout of the dischargechamber plasma into the chamberwalls or the

!,

beam is not includedin the plasma ion productioncost. Rearranging

t

_' Eq. 2 yields,

( 0- iVo

For steady state operation, the total ion current produced (dp) must be

--_:,o equal to the total ion current leaving the plasma. For ion thruster

_'_i discharge chambers, ions can only leave the plasma by going to cathode

potentialsurfaces,anode potentialsurfacesor by being extractedinto

¢ the beam. Thus, the total ion current producedis given by*,

JP -- JB + JC + JA ' (4)

Dividingthis equationthrough by Jp yields,

* Equation4 is a statementof continuityfor the ions.
_--_'._'r_'_"_'_--_--._W .......

17

;_, 1 = fB + fc + fA ' (5)

__ where,

_: fB _ JB/JP ' fc _ JC/JP and fA _ JA/JP ' (6)

z

_ The fractionsfB' fc and fA are, in order, the extractedion fraction,

_ . .. the fraction of ion current produced that goes to cathode potential

..... surfaces,and the fractionof ion current producedthat goes to anode

.,_,. pQtentialsurfaces. Using the definitionsof fB and fc in Eq. 3 along

_ with Eq, 1 yields,

,:.i Cp = CBfB - fcVD (7)

_ Z

0 22

_: i Solving this equationfor cB gives the result,

i_: _B = Cp/fB + fcVD/fB ' (8)

._

_ This equationdescribesthe beam ion energy cost as a functionof the

o_. plasma ion energycost (¢p),the extractedion fraction(fB),the

/_ fractionof ion current to cathodepotentialsurfaces(fc)and the discharge voltage (VD).

'. i/

: The first term on _he right-hand-sideof Eq. 8 representsthe

energy loss associated with producing ions in the discharge chamber and

extractingonly a fractionof them into the beam. Ionswhich are not

_, :. extractedinto the beam go to the walls of the dischargechamberwhere

they recombine. The resulting atoms must then be re-ionized before

they can contributeto the beam current. The factor I/fB may be

..... interpreted as the average number of times that a beam ion undergoes

ionizationfrom a neutral state beforebeing extractedinto the beam,
_,_Z _u_r-_-_--_--_v ...... -_

) . '_.

_, The second term on the right-hand-sideof Eq. 8 representsthe

,. energywasted in acceleratingplasma ions into interiorcathode potential surfaces. This process is undesirable because it results in

.(_ both a dischargeenergyloss and in the sputtererosionof these

surfaces.

To generateperformancecurves using Eq. 8, one must be able to

.... :. specifythe behaviorof each of the terms on the right-hand-sideof

_,_...._, this equationas a functionof the propellantutilizationefficiency

°':,_.... Plasma Ion Energy Cost

_ C , .._ _..

_J:

,

,:i_-,,

%,, The plasma ion energycost parameter(¢p)appearingin Eq. 8 and

defined by Eq. 2 reflects all mechanisms of energy loss from the discharge chamberexceptfor the accelerationof ionsout of the plasma

_ .!_. through the dischargevoltage. Specifically,ep includesenergy losses

i_ " .,/' due to the followingphenomena: direct primaryelectronloss to the ,

__. anode, Maxwellian electron collection by the anode, excitation of

' neutralatoms, excitationof ionic states (whichwill be neglected)and

[ ...._,

:_T ., '" hollow cathode operation • To derive an expression for the plasma ion

:' energycost as a functionof the propellantutilization,a power

' balance is made on the dischargechamber plasmarepresentedin Fig. 2.

The boundaryof the controlvolume for this power balance is defined by

,_ .

_._._. the plasmasheathedge. The primaryelectronsare assumed to be accel-

" erated from a cathode region plasmapotentialthat is Vc volts above

• cathodepotentialto the potentialof the bulk plasmawhich is assumed

: to he near that of the anode. In addition,if it is assumed that only

T

the dischargepower supply is used to sustainthe discharge,then the

rate at which energy is supplied to the discharge chamber plasma by the
) •

19

: [

,,_ Figure2. DishargePlasmaPowerBalanceSchematic

},.

b"

1

............ 7.................... -_-.'.77S_-'.7_--:_-'_,,........ ',:':_:_-_-i I -II III .... III llllllI IIII"i
_ 20

, primaryelectronsis given by JE(VD - VC). The "missing"power JEVc is

used to operatethe hollowcathode.

_ Energy is lost from the plasma primarilyby the flux of four types

of energy carriers across the plasma boundaries: ions, photons (emitted

._ by de-excitationof excited propellan_:atoms),Maxwellian electrons,

':_,.,. and primaryelectrons• The ions and photonsare lost to all interior

-- f _r(

._:: thrustersurfaceswhereas the Maxwellianand primaryelectronsare

m-\!,i'¢i'_';'_ ' •

v_r"c,_"' •

_: assumed to be lost to the anode surfacesonly• In steady state,the

_?_', rate of energy suppliedto the plasmamust be equal to the rate at

--:,,,._ which it is lost, thus,

/'_.':" JE (VD'Vc)= dpU+ + ,_JjUj + JMCM + JL(VD-Vc) , (9)

__ '.. where the summation is over the set of excited neutral states.

_ _ .:. DividingEq• g by the ion productioncurrent (dp)and recognizingthat

_.:.i°): the emissioncurrent (JE) relatedto the dischargecurrent by *,

' " ._ dE = JD" (Jc + JB) ' (lO)

_: ,_T allows Eq. g to be written as,

:: _ JjUj JMCM JLVD JEVc JLVc

..... _::: ep : U+ + jp + jp + -_p + jp jp , (II)

_ where Eq. 2 has been used• The rate at which the jth excited state is

produced (expressed as a current) ts given by,

J_u dj = enone <diVe>_ , (12)

• * EquationlO is a statementof continuityfor the electrons.
. _- __v-_--.,qlF

b.

i, 21

!'i

I where <ajVe>representsthe product of jth excitationcollisioncross

_ section and the electron velocity averaged over the entire electron

speed distribution- includingthe primaryelectrons. Similarly,the

ion production current is given by,

Jp = enone <O+Ve>_ . (13)

• SubstitutingEqs. 12 and 13 into Eq. II yields,

JMCM + JLVD JEVc JLVc

Cp=co,_ _ , _p -jr-, (14)

"_ where the parameterco is definedby,

_,<OjVe>Uj

co -=U+ + _ • (15)

<(_+Ve>

_. This parameteraccountsfor the energy that is expendedin ionization

and excitationreactionsand may be calculatedin the mannerdescribed

by Dugan and Sovie [38]. The third term on the right-hand-sideof

Eq. 14 may be written as,

" JLVD JL JEVD JL

J-7= J--;=-¥ ,

where the last stepwas made using Eqs. 2 and lO. The ratio JL/J is

simplythe fractionof primaryelectronsemitted by the cathodewhich

are collectedby the anode before havingany inelasticcollisions. This

fractionmay be given by the survivalequation[39] as,

i

: JL " °ono_e

=e (I;)

where _e is the averagedistancea primaryelectronwould travel in

the dischargechamber beforebeing collectedby the anode - assumingit

had no inelasticcollisions(i.e.the primaryelectroncontainment

o
..... _ 22

!

.' length), oO is the total inelasticcollisioncross sectionfor

• electrons at the primary electron energy and no is the neutral atom

density. CombiningEqs. 14, 16 and 17 yields,

, ! ,

!':ir '' = .0.° no.te Vc Vc no_e

' JMCM _p e"°0 (18) i._,. . _p eo + jp + epe + ep VD _DD '

_T,C'.

The current of Maxwellian electrons to the anode may be given as the

',- sum of the secondaryelectronsliberatedin the ionizationprocessand

_._:i, the thermalized primaryelectrons,thus,

' /

=_ ' JM- JP + (JE " JL) - JP + JE(I'JL/JE) " (Ig)

t_ Using Eqs. 17 and Ig in 18 yields,

;i. ep = e0 +[Jp + dE(l-e-oo'no_e)] _c+M _p[e-Oono_e.(l-e-Oono_e)_] •

(20)

_:>,;, Finally,using Eqs. 2 and lO and solvingEq. 20 for ep results in,

-- kl.VC+, Mj . (21)

!

:i" -aono£ e

The factorl-e may be interpretedas the probabilitythat e

o primary electronwill have an inelasticcollisionbefore being collected

by the anode. This is analogous to the "fast neutrnn non-leakage probability" used in nuclear reactorphysics [40].

• The neutral density (no) may be expressed in terms of the propellant flow rate and propellantutilizationby equatingthe rate at

which propellant enters and leaves the discharge chamber, i.e.,

= JB + no (22)
t+

,+ 23

,,,

where m and n are in units of equivalentamperes+, The neutralflow

rate from the thrustermay be expressedusing the theory for free

,, ....• molecularflow througha sharp-edgedorfice [39],

+..... no = _ noevoAg¢o ' (23)

° "i CombiningEqs. 22 and 23 yields,

_' 4 m(l-nu)

_ ' " I (24)

+•.... no = evoAg ¢o '

•,.:: where the propellantutilizationdefined by nu _ jB/& was used. Thus,

Eq. 21 may be written as,

Cp = _,[p l.e.Co_(l.nu)] -I , (25)

,r

: where,

t

_- 4Oo_e

"I C° _ eVoAg ¢o ' (26)

and,

• _o + CM (27)

_i Cp _ •

1 VC+¢M

•./ - -VD

I Equation25 is a very simplerelationshipwhich can be used to calculate

the plasma ion energy cost as a functionof the propellantutilization.

, ° Experimentalresults,which will be presentedin ChapterIV, indicate

° ' that under many conditions the parametersCo and cp may be taken to

be independent of the propellant utilization.

= ?

i Substitutionof Eq. 25 intoEq, 8 yields the followingsingle

equation describing the performance of a given thruster design,

+

Equation22 is a statementof continuityfor the propellant.

C

l
, ep fcVD

_::_ e B - fB [ l-l e.Co___u_ + "_B • (28)

....:_ For design purposes,the parametersfB and fc in additionto CO and

_p ;:ay be taken to be independent of the utilization and flow rate.

i_: These parametersdo, however,depend stronglyon the thrusterdesign.

i:ii_ Indeed,these four parametersdeterminethe performanceof a given

thruster design.

The quantityCo reflectsthe degree to which primaryelectrons

interactwith neutralatoms. Thus, it is referredto as the primary

electronutilizationfactor. This factordependson the qualityof the

_'_ primaryelectroncontainment(through_e), the quality of the containment of neutralatoms (throughAg, 0o and Vo) and the propellantgas

, properties(throughao and Vo). Recallthat the primaryelectroncon-

._ tainmentlength _e may be interpretedas the average distancea primary

electronwould travelin the dischargechamberbefore being lost to the

anode--assumingit had no inelasticcollisions• Magneticfieldsin all

discharge chamber designs serve the function of increasing this length

Althoughan effectivemeans of determining_e remainsto be developed,

it is believed that this parameter is a function primarily of the

thrustergeometry,magneticfield configurationand cathode location.

Equation25 suggeststhat the plasma ion energycost should,

throughthe factorCo , be a functionof:

_' I. The propellant,which determinesthe total inelasticcollision

...... ,, cross section (Oo)and atomicmass (whichaffects the thermal neutral

velocity, Vo).

2. The wall temperature,which affects the neutralvelocity.
", i;_.

L

...." 3. The transparency of the grids to neutrals (¢o).

4. The area throughwhich the beam is extracted(At).

5. The discharge voltage, which determines the primary electron

I

_' energyand, thus, effects the value of the cross section (_o).

2/_," Th_ baselineplasma ion energycost (_), defined by Eq. 27,

_Z_ dependson a numberof energy lossmechanismsincluding: the relative

: amountof energyexpendedin excitationscomparedto ionizationof

_;:_ neutralatoms throughto, the averageenergy of the Maxwellian _!ectrons

=_:_ which leave the plasma (_M) and the efficiencywith which the hollow

:_::_: cathodeoperates

: The cathodeefficiencyis reflectedin the value of VC which

represents the plasma potential from which the electrons are supplied.

'_:, Inefficientcathodeoperationresults in high valuesof VC and correspondinglypoor overall thrusterperformance. For tr,ermioniccathodes

,,- Vc = O, however,additionalheater power must be suppliedto effect its

operation. For thrustersequippedwith a cathode pole piece/baffle

assembly,VC shouldbe taken as the plasmapotentialin the cathode

_;_ _ dischargeregion (i.e.,the regionbetween the cathodeand the baffle).

_ In this case, the power representedby JEVc goes into both the operation

_ of the hollowcathodeand the operationof the cathoderegiondischarge.

_ The resultinghigh valuesof Vc, in this case,would be expectedto

_:_ producepooreroverall thrusterperformance. Eliminationof the

_..:. separatecathodedischargeregion should,therefore,improvethe per-

_':--__ formance.

The parameter e , defined by Eq. 15, in general is a function of

o

the entire electronenergy distributionincludingthe primaryelectrons.

L

The poy'ameter e_ (which contains eo' see Eq. 27) is, thus, also a

L _7
o

if"

', 26

functionof the electronenergydistribution.For plasmascharacterized

by a Haxwellianplus monoenergeticelectronenergy distribution,a

,_ " simple method for the calculation of c; may be derived. Sample calcu-

@.. lations using this method, along with the appropriate collision cross

:_,? sectiondata, are given in AppendixA. The experimentalresultsgiven

'_ in ChapterIV indicatethat, under many conditions,the baselineplasma

: ion energy cost (e) may be taken to be a constant. The calculations

=: c,, _ given in AppendixA demonstratethat this experimentalobservationis

:__ predictedby the model.

°._ Althoughthe model cannotyet be used to predictthe performance

-o:

.. of completelynew thrusterdesigns,it providesa clear physicalpicture of the phenomenaaffectingthe performance. Equation25 describes

the plasr_'i,on energy cost in terms of the loss of primaryelectronsto

the anode. At high valuesof the neutraldensityparameter,the neutral

• density in the dischargechamberis large,and the probabilityis high

_. that all the primaryelectronswill undergoinelasticcollisionswith

- neutralatoms and nonewill be lost directlyto the anode. In this

case, the dischargechamberwill be producingions for the minimumor

baselineplasmaion energycost, c*,

-T;_i As the beam current is increased(at a constantpropellantflow

r-

_;L;".... '_' rate),the propellant utilizationincreasescausing the neutraldensity

parameter(andthus the neutraldensity)to decrease (see Eq. 24). The

,_.... decreasein neutraldensityincreasesthe likelihoodof a primaryelectron reachingthe anode without havingan inelasticcollision. This

directloss of primaryelectronenergy increasesthe overall plasma ion

. energycost accordingto Eq. 25 and consequentlyincreasesthe beam ion

. energycost accordingto Eq. 2B. The shapeof the performancecurve is

©
" 27

, largely determined by this direct loss of primary electrons. Thus, any

........ design change which decreases the likelihood of the direct loss of

° _,, primaryelectrons(withoutdecreasingthe extractedion fraction)

• shouldimprovethe thruster'sperformance. The probabilityof direct

_,_

_/ primaryelectronloss is determinedthrough the parameterCo. This

parametermay be increasedby either increasing_e' which makes it

_i harderfor primaryelectronsto escapethe plasma,or by making it

-_: harderfor neutralatoms to escape the dischargechamber.

_'_ Calculationof PlasmaProperties

o_. The followinganalysisprovidesa set of very simple algebraic

"_r: equationswhich allow one to calculatethe valuesof the followingdis-

_ii: chargechamber plasmaproperties: the average primaryelectrondensity,

:_

'" _L

the average primary-to-totalelectrondensity ratio, the averageMax-

•_ wellian electrontemperatureand the averageratio of the doubly-tosinglycharged ion current in the beam. Each of these quantitiesmay be

calculatedas a functionof the propellantflow rate and propellant

utilizationusing only informationthat would normallybe availablein

_ the thrusterdesign phase.

PrimaryElectronDensity

This analysisis based on the recognitionthat all of the energy

suppliedto the dischargechamber plasmais suppliedby the primary

electrons. Thus, correctlyaccountingfor the behaviorof the primary

electronsis essentialin determiningthe average plasmaproperties.

• It is assumedthat a primaryelectroncan do only one of two things.

It can either have an inelasticcollisionwith a neutralpropellant
t'_ I

_i_ 1

atom or it can be lost dtrectly to the anode. For the case where the

_.... primaryelectronhas an inelasticcollisionwith a neutral atom, it

'?

,. will produceeitheran ion or an excitedneutralstate. After such an

)--i ,

inelastic collision, the energy of the primary electron is degraded and

o .

._i": it is assumedthat it is subsequentlythermallzedinto the Maxwellian

.._ electron population. The rate at which primary electrons have in-

....: elasticcollisionswith neutralatoms is given by the differencebei. tween the rate at which they are suppliedby the cathode (dE) and the

°'.... rate at which they are lost directlyto the anode (JL), i.e.,

I I

'_- JE " JL --JP + Jex , (29)

I !

; where dp is the ion current producedby primaryelectronsand dex is

- the primary electron induced production rate of excited neutral states

't expressedas a current. The ion currentproducedby primaryelectrons

is given by,

!

Jp = nonpeo+ -V-Vp , (30)

I

.. where o+ is the ionizationcollisioncross section at the primary electron energy. Similarly,the rateof productionof excited state atoms

• induced by primaries is given by,

' Jex'= nonpeVp_ _ o_ (31)

Ci, where aj is the collisioncross section for the jth excitedstate at

the primary electron energy. The fraction of the primary electrono neutralatom inelasticcollisionsthat produce ions is given by,

• I I

. _ nonpeVpa+ l = (32)

' '

JP+Jex nonpeVpa+_ + nonpeVp_ oj

Q
•_i 29 I

I

or,

I !

dp o+

, , - , . (33)

j + a++ oj p Jex

.... : Note that the sum o+ + _ o.jIs just the total inelasticcollisioncross

. ,] I

_-... sectionfor primaryelectron-neutralatom collisionso.o Therefore,

--_,o ...... ,,. ,/_q ,_. Eq. 33 may be written as,

._ .... I I

? : dp 0.+

:. o.... (34) .._ I I i *

..... ,'!° 2:''r JP + dex o'o

MultiplyingEq. 34 by the rate at which primaryelectron-neutralatom

° :: collisions occur (Eq. 29) yields the rate at which ions are produced by

_ :_;i primaryelectrons,i.e.,

i 0'+

_:'__:_' dp = (dE - jL)--, . (35)

o"0

_'": The term in parenthesesin Eq. 35 may be written as,

.! JE " JL = JE (1 - JL/JE) • (36)

i The quantityJL/JE is the fractionof the input primaryelectroncurrent

lost directly to the anode and is given by the survival equation (Eq. 17

written in a slightlydifferentform)_

JL .Co_(l-nu) . (37)

• _E - e

' CombiningEqs. 2 and I0 yields the followlngexpressionfor the cathode

emission current.

Jpcp (38)

dE = VD
!I.

I _= 30 '1

Combining Eqs. 25, 35, 36, 37 and 38 yields the following expression

for the ion current produced by primary electrons,

, .;' , I

_ 1 , Jpep o+

>:o. .... ., _p - , • (39)

v. ,

_ii. Dividingboth sidesof this equationby dp yields an expressionfor

_':. the ratioof ion current producedby primariesto the total ion current

,. produced,

: I * I

dp Cp a+

i;,:, , _pp = , • (40)

::. VDao

I

_!' Since the ratio dp/Jp cannotbe greaterthan one, Eq. 40 providesa

,: _: theoreticallimit for the maximum value of Cp , i.e.,

h

- _.,-

.;_ , VD_o

;. (eP)max , . (41)

'5.,

._.o;- The total ion current producedmay be expressedin terms of the beam

_;_7,... current by using the definitionof the extractedion fraction(Eq.6),

Jp = dB/fg . (42)

In addition, using the definition of the propellant utilization the

beam currentmay be written as,

: = • (43) mn u

•- CombiningEqs 39, 42 and 43 yields,

I

o+

.. d_ _ u r, . (44)

fBVD°o
,.L:O "

_:=,._j

,: Ftnally, equating Eq. 30 to Eq. 44 and solving for the primary electron

•

density (np) yields,

• r;A0,,oVo/

Z _'" ' II '

. np [% v-eBVDO ° (4s)

;_:-?_i:_'j .;j where Eq• 24 was used for the neutraldensity. Equation45 providesan

_' expressionfor the average primaryelectrondensityas a functionof

the propellant utilization. Remarkably, this expression does not

dependon either the propellantmass flow rate or the primary electron

utilization factor, both of which cancelled out in the analysis. The

"i- term in the square bracketsshouldbe roughlya constantfor a given

thruster design, propellant gas and discharge voltage. A reasonable

• :2"

° :': estimatefor eachof the terms in the square bracketshouldbe possible

;"C

:, for a thrusterbeing designed(assuminga method is developedfor the

determination of the extracted ion fraction).

i_.' Primar.v-to-TotalElectronDenslt_/Ratio

Assulningquasi-neutrality,the average ion density (ni) is equal

,. to the averagetotal electrondensity (ni = np +nM). In addition,the

averageion density is relatedto the beam currentby [41].

:- JB = O'6nievbAg¢i " (46)

Solving this equation for ni yields,

dB

, ni = (47)

O.6eVbAg¢i

where ¢i is the transparency of the screen grid to ions and vb is the
4.

'4

, 32

Bohmvelocity. Dividing Eq. 45 by 47 and using Eq. 43 ytelds the

o. followingexpressionfor the average ratio of primary-to-totalelectron

.... dens i ty

" r ]

.... _ _ = O'IBeCpVbAg¢o¢i l (48)

i .... , (:;:!_ ni k .... VpVDfB"V',,_o &(1.n)u "

/

)_ ' : This equationindicatesthat the average prlmary-to-totalelectron

Z o- _'

E :.... ,i_,,':. densityratio should be a functionof the neutraldensityparameter,

_ ....e:i, _(l.nu). As was the case for Eq. 45,the combinationof parametersin

i".,°_" the squarebracketsof Eq. 48 should be roughlya constantfor a given

;.,__- thrusterdesign,propellantand dischargevoltage. Equation48 also

a _ _ indicatesthat, all else being equal, increasingthe extractedion

....._; fractionshould decreasethe primary-to-totalelectrondensity ratio.

,_: ' Primar¥-to-MaxwellianElectronDensityRatio

=_ Once the primary-to-totalelectrondensityratio is calculated,

,.,'

• using Eq. 48 the primary-to-Maxwellianelectrondensity ratio (np/nM)

I_)":: may be calculatedfrom the followingequation,

. :7

":. np np/ni I = = . (49)

.F i

_ • - nt.1 1-np/n i ni/n p 1

Maxwe]lian ElectronTemperature

The total ion current produced(Jp) is the sum of the ion current

produced by primary electrons (J_) and that produced by Maxwellian

_ _ electrons(Jp,M) . (50)
• t_•

i_I 33

ij.

,: The ion current producedby primaryelectronsis given by Eq. 30 and

. that producedby Maxwellianelectronsis given by,

_: JP,M = nonMe <°+Ve>MV- (51)

where the quantity <_+Ve>M represents the product of the electron

_ velocityand ionizationcollisioncross sectionaveragedover the

_',,, Maxwellianelectronenergy distribution. This quantity is the Max-

, wellian electronrate factor for the productionof ion::and , is given

_; the symbolQo ' i.e.,

" o. "_ 4"

:i_':' Qo : <°+Ve>M " (52)

'L CombiningEqs. 50 to 52 yields,

L: I

..... ,,,_ Jp = nonpeVp+o ¥ + nonMeQ+V- . (53)

+

_; Using Eqs. 42 and 43 in Eq. 53 and solvingfor Qo gives,

,,,_ IIlnU _p_ '

!i,I , Q+_ ,e_ nnM VpO+ . (54)

°_, i,_ fBnonM

_',.. , Substitutingfor the neutraldensity using Eq. 24 yields,

.

.....' + = v°AB¢° nu -nP VpO+ . (55)

•>:, Qo 4fBnM ¥ nM

u The Maxwellian electron density, appearing in Eq. 55, may be written as,

nM

..... Substitutingthis into Eq. 55 yields,
I'

•_'a, Equation45 providesan expressionfor the primaryelectrondensity.

Using this in Eq. 57 yields,

,)

• "'r,

l/ The primary-to-Maxwelllanelectrondensity ratiomay be found using

: Eqs. 48 and 49. Substitutingthese equationsinto Eq. 58 yields the

=:lL_.C_,_:_ l".TI' followingexpressionfor the averageMaxwellianelectronrate factor

=_;_;_, ' for Ion productIon

_;' _'l_ :" '' V pO + ,(

k, V_E

D

:OO0+

" l )

7:i//i?i!::: Qo = [ VpVDOo "V ] • (59)

tO.15eC_VoVbAgZ¢o¢i m(l-nu)- l

d:_C. Equation59 providesan expressionfor the Maxwellianelectronioniza-

_;_i_._,? tion rate factoras a functionof the neutraldensity parameter,

u

&' _l_l _I T" _ ) . Once this rate factor has been calculated,_ t can be compared

=:. ,_,, to a tabulationof rate factorsversus electrontemperaturefor the

..... /

_:1l,,:;_ given propellantto determinethe appropriateMaxwellianelectron

N_d_)l'.:__ .- temperature. Equation59 i_Idicatetshat the averageMaxwellianelec-

_:,, ..... , .

1% "

tron temperatureshouldincreasewith decreasingvaluesof the neutral

:: density parameter.

_ DoubleIon Production

In the previousanalysis,the productionof doubly charged ionswas

neglected. In this section,a slmpleformulationfor the ratio of

-._ %

_l& ";_ _Lq' doubly-to-singlycharged ion current in the beam is developed. The pro-

_j ductlonrate of singlycharged ionsexpressedas a current is given by,

)

b'.,
0 7

_m

+ n + +

dp = ( onpPo+ nonM o) e V- , (60)

where Qo is the Maxwlllan ratefactorfor the productionof single

r_ . ,mr, ionsfrom groundstateneutralatoms and P, is the primaryelectron

•U

r,; '-, ratefactoralso for the productionof singleions fromgroundstate

_. neutrals• Similarly, the production rate of doubly charged ions

., r/' ,.,;

._:,", expressedas a currentis givenby,

ii,'__, _ : , d; = (nonMo+++ nonpPo+++n+nM +++ + n+npP++) 2e _ (61)

_",:.,i_=,.:: where n+ is the singlychargedion density,Q_+ and P;+ are the Maxwellian and primary electron rate factors for double ion production

--,r "_ from groundstateneutrals,respectively,and Q+++and P:+ are the cor-

: respondingratefactorsfor doubleion productionfrom groundstate ions.

y:,. DividingEq• 61 by 60 yields,

' *+ _z p_ ) _2n+ (Q_+,np p+4-_)

_'_'_ J;+ 2(Q° +nM no nM (62)

" ,-,,_;:: JP Q, + nM P, Q, + nM P,

Assuming that the doubly charged extracted ion fraction (f;+) is equal

_ to the singlychargedextractedion fraction(f;) impliesthat,

. dp++

+ - + + - + , (63)

° "" JP JBIfB JB

_ ++ _+

i whereJB /JB is the averageratioof doubly-to-singlychargedion cur-

_ rent in the beam. CombiningEqs.62 and 63 yields,

.. 2n+ _ ++

N PO ) _O0 "M

- 

+ + + __ + •

' JB Q;+_PM'MPo Q, + , Po

® . . ©
36

• The singlycharged ion density (n+)may be conservativelyapproximated

by assuming the beam current is made up entirely of singly charged ions,

:_. thus,

JB

n = ni - • (65)

;:_. + O.6eVbAgOi

. '., In addition,using Eq. 43 for the beam currentand Eq. 24 for the

' neutraldensityallows Eq. 64 to be written as,

- ()()_[

'°_:. JB Qo + nM Po nM

_ _ The first term on the right-hand-sideof Eq. 66 representsthe produci" !_I' tion of double ions from ground state neutralatoms This term depends

! ._', on the thrustergeometryonly throughthe primary-to-Maxwellianelec-

_//'i', tron density ratio (np/nM)which is given by Eqs. 48 and 49 and the

_ Maxwellianelectrontemperature(Eq.59). The secondterm in Eq. 66

represents double ion production from singly charged ions. This term

' is stronglydependenton the propellantutilization. The rate factors

++ ++ +

Qo and Q+ may be determinedonce the rate factorQo is calculated

_ from Eq. 59.

./

_
ji_sr'_"7-- _----"_V

,,o ;i'

!.

f_

_2

_' IV. EXPERIMENTALPROCEDURESAND RESULTS

- , (,¢

_. Apparatus

_,.o.'•<_-, For this investigation,the ion sourcesshown schematicallyin

_:i_i Figs. 3a and b were designedand built. Each of these sourcesnormally

_'"_i

_L producesa 12 cm dia. ion beam and providesthe capabilityfor measuring

__.;,:%o the distributionof ion currentsto the beam, screengrid and internal

thrustersurfaces(withthe exceptionof the anode).

%2 ; The magneticfield for the experimentalion source in Fig. 3a is

" establishedthroughthe use of an electromagnetlocatedon the upstream

: centerlineof the dischargechamber and a numberof 1.9 cm x 1.3 cn, x

,_ 0.5 cm samariumcobaltpermanentmagnets. These permanentmagnets are

', arrangedend-to-endto form ring magnets of alternatepolarityin the

J.

manner suggestedby Fig. 3a. The flux densityat the surfaceof the

magnets is 0.27T and the magnets are attachedto the steeldischarge

°_ chamber housing by their own magnetic attraction. This arrangement

Z _'."

a11ows the ion sourcemagneticfield configurationto be alteredquickly

_ • and easily by simplyadding, removingor changingthe positionof the

magnets. Althoughmany differentconfigurationswere tested,the results

obtainedwere all similar,thus, only those obtainedusing the configurations shown in Figs. 3a and 3h will be presented. For these configura-

• tions, the upstreammagnet ring is coveredwith a strip of 0.13 mm thick

r -

: steel insulatedfrom the magnets themselvesby a strip of 0.25 n_,_thick

flexiblemica. This is done so that the surfaceof this strip can be
13 ¢m

It

It

"* it

' t|

° II

• it

• PROPELLANT "II

•, FEED II

I|

_ *_ II

:'_ II

• .: II

' _ r il

_'__'_"::" ELECTRO- =, II

I|

II

_,' II

II

' o. II

...... _' II

II

INSULATION _ 6

_o_. Figure 3a. Ring Cusp Ion Source Schematic- ConfigurationI

PERMANENT

MAGNETRINGS

:, Figure 3b. Ring Cusp Ion SourceSchematic- ConfigurationII

i:
,_r__"_-_'--'r _......._...... .

!__i ̧ _

i t)

t!

L

_ maintainedat anode potentialwhile the rest of the thrusterbody is

• biasednegativeof cathode potential. The downstreammagnet ring is

" uncovered. The magneticflux density at the surfaceof the electro-

_ .. magnet for the configurationof Fig. 3a can be adjustedfrom zero to

approximately0.2 T by adjustingthe magnet currentfrom zero to

i_ 124 A.

_ Themaindischargechamber cathodes,forbot_ configurations,con-

...... sistofseven 0.25cmdia, tungstenwiresconnectedin parallelandsupportL

ed bytwosupport poststhatare electricallyisolatedfromthe thruster

_;_ r body. Each cathodewire is approximately2.8 cm long so the total

_C cathode lengthexposedto the plasma is about 19.6 cm. These _even

short wires in parallelare used to minimizethe voltagedrop across

_; the cathode. A voltagedrop less than 3 v at the maximum heatercurrent was achievedwith this system. The small voltagedrop across the

• cathode resultsin a primaryelectronenergydistributionthat more

, closely resembesthe monoenergeticdistributionproducedby a hollow

_ cathode. The cathodewires were heatedusing direct currentsin the

range 6 to 8 A per wire. Tests on the configurationof Fig. 3a, were

conductedusing argon,kryptonand xenon propellants. Dischargevoltages were varied from 30 to 50 v for argon and 20 to 40 v for krypton

r and xenon. The dischargecurrentwas adjustedthroughthe range of

0.5 to 5 A by controllingthe heatingcurrentthrough the refractory

cathodewires.

Tw_ ion acceleratorsystemswere used in this study. The first

acceleratorsystemconsistedof a set of dished small hole accelerator

grids (SHAG)_,,itha cold grid separationof 0.75 mm and screenand

acceleratorgrid physicalopen area fractionsof 0.68 and 0.30,
f'_ I

' _ 40

ii ' i

.,_"_. respectively. The second system consisted of a set of dished large !

hole accelerator grids (LHAG)with a cold grid separation of 0.75 _,

and screenand acceleratorgrid physicalopen area fractionsof 0.68

and 0.57, respectively. Both accelerator systems were normally masked

° ' to producea 12 cm diameterbeam One seriesof test was conducted,

ii-:irl however,with the SHAG set masked to producea 6 cm diameterbeam. For

_/_: the 12 cm dia. beam tests,flow rates for both argon and kryptonwere

!_._: variedfrom 500 to 1500 mA e_ and for xenon from 250 to lO00 mA eq.

_,.o For the 6 cm dia. beam test, the flow rateswere varied from 125 to 500

mA eq.

ir_:._ The configurationof Fig. 3b is similar to that of Fig. 3a except

that the electromagnet assembly has been replaced by a permanent magnet

_ attachedto the steel backplate. In addition,the sourceof Fig. 3b is

i 2 cm shorterthan the one in Fig. 3a and the upstreammagnet ring is

' positionedonly 2.5 cm from the downstreamend rather than 6 cm. ,

This source was equipped with two Langmuir probe assemblies. The

first probe consistedof a 0.76 mm dia. tantalumwire, 4.32 mm long,

•' _ supported from a quartz tube insulator. This probe was positioned

along the thrustercenterlineapproximatelyhalf way between the cathode

_; assemblyand the screen grid as suggestedin Fig. 3b. The secondprobe

was a square piece of steel, 1 cm on a side and 0.127 _ thick that was

positionedon the surfaceof the upstreammagneticring. This probe

was insulatedfrom the magnet ring with a piece of 0.13 mm thick

flexible mica.

Measurements of the doubly and singly charged ion beam components

were made on the thrustercenterllneusing a collimatingE x B momentum

analyzer. Details of the use of this probe are given elsewhere [42].

!

Q
v,.

, 41

,- Tests were conductedon the configurationof Fig. 3b using both

, argon and xenon propellants. Argon propellantflow rateswere varied

• over the range 350 to 1500 mA eq. at dischargevoltagesof 30 to 50 v.

:_,._:,...,. Xenon propellantflow rates were variedover the range 250 to I000 mA

" eq. at dischargevoltagesof20 to 40 v. Dischargecurrentsfor both

:__.,._ propellantswere variedover 0.5 to 4.0 A.

........ _,.jo-,

_:t_:_' All tests were conducted in a 1.2 m dia. x 4.6 m long vacuum test

_i_--.-_-_-' facility. Indicatedtank pressuresrangedfrom _ 2 x lO-6 Torr with

_--_ no flow to _ 3 x lO-s Torr at a flow rate of 1500 mA eq. of krypton.

_.__ Procedure

The following set of eYperiments, conducted using the thruster

!,,

configuration of Fig. 3a, was designed to test the suitability of the

model developed in Chapter Ill to predict the functional dependence of

the plasma ion energy cost, ep, on the neutral density parameter,

" m(l-nu) . The model predicts that the plasma ion energy cost should

behave according to Eq. 25 with the primary electron utilization factor

:. (CO ) and the baseline plasma ion energy cost (¢p) given by Eqs. 26 and

21, respectively. The value of ep may be determined experimentally

.... through the use of Eq. 2, which is repeatedhere for convenience,

' [JD " (Jc + JB)]VD

ep

dp

Note that the power used to operate the thermionic cathodes is not included in Eq. 2. In order to use Eq. 2, one must be able to measure

each of the parameters on the right-hand-side of the equation. Measurement of the discharge current, discharge voltage and beam current is

& ©

I
,, .;L

+ 42

) +- straight forward, and was accomplished using the experimental instru-

[

mentatton described in Appendix B.

i ' '

)) +./ To measure the ion currentsJc and Jp, the thrusterconfiguration

[ *

i_+-,._, of Fig. 3a was operatedwith only the upstreammagnet ring at anode

_? potential. All other interiordischargechamber surfaces(with the

- exceptionof the cathode supportposts)were biasedapproximately30 v

.+ negativeof cathode potentialto repel the dischargechamber electrons.

T.

At this bias, the currentto these surfacesconsistsonly of the incom-

_ ing ion current. If the ion currentto the cathodesupportposts and

_= cathodewires is neglected,then the ion currentmeasuredin the manner

describedabove is equal to JC'

To determineJp, it is noted that the total ion productioncurrent

is the sum of the ion currentsleavingthe plasma as given by Eq. 4.

_ Since the anode is exposedto the plasmaonly at a magneticfield cusp, ,

the effectivearea for ion loss to this surface is expectedto be less

• than the physicalarea [43-46]. Rough calculationsindicatethat the

_o - ion current to the anode with this configurationshould be less than a

) few precentof the total productioncurrent. Thus, Jp may be approxi-

_,, mated as the sum of the ion currentsto the beam (includingthe impinge-

°_ ment current)and to the negativelybiaseddischargechamber surfaces

(includingthe screengrid).

Completesetsof data consistingof the beam current,propellant

• flow rate, propellantutilizationand total ion productioncurrentwere

collectedover the rangeof operatingconditionsdiscussedearlierwith

the electromagneticcurrent held constantat 57 A for the thrusterof

Fig. 3a. At each conditiontested,the thrusterwas operatedat flow

L.

ratesof 500, 750, lO00 and 1500 mA eq. for argon and krypton

, _, '++
ii!!iit: i"iii 43

_!o _.:.,_ pratropee,llant the diss,acharge nd 250v,o500 ltage, 7_was0 anheld d I00constant 0 mA eq.while for xethenon.discharge At eachcur- flow

_III_/_ITII._, rentwas varied. Increasingthe dischargecurrent,increasesthe beam

_;i:_L: currentand propellantutilization,thus causing the neutraldensity

i;i parameterm(l-nu) to decrease. By operatingin this manner, the full

) range of neutraldensity parametersfrom close to zero to nearly1500

...._, mA eq, could be investigated, Thus, the plasma ion energycost was determinedfrom Eq. 2 for a wide range of operatingconditions. The ex-

_ o_ tracted ion fraction,fB' defined by Eq, 6, was also computedfrom

_ measuredion currentsover this same range of cunditions.

_ A second set of experimentswas performedusing the thruster

o

configurationof Fig. 3b. These tests were designedto investigatethe

_ abilityof the model to predictthe behaviorof variousplasma propertieswith variationsin thrusteroperatingconditions. The procedure

for these tests was the same as for the previousset with the addition

.... that Langmuirprobe measurementsof the plasmapropertieson thethruster

centerlineand at the anode surfacewere made at each operatingpoint

tested. The ratio of doubly to singly chargedion beam currentsalong

the thrustercenterlinewere also measuredat each operatingcondition,

_,,, using E x B probe.

ExperimentalResults

Plasma Ion EnergyCost

Measurementof the plasma ion energycost, for operationof the

thrusterconfigurationof Fig. 3a with argon propellantat a 50 v

dischargeover the range of neutral flow rates from 500 to 1500 mA eq.,

yielded the resultsshown in Fig. 4a. Here, the measuredvaluesof the
_ 44 L

i .

2 [60 _ SHAG OPTICS

, - _ _ . I2:m diD. BEAM

,40 _.,....Ep- ,'_.O.V

> _--C o - 2.1(A eq)'l

_" _ ]20 _ FLOW RATE (mA liq)

• ,_ 0 1500 O\p a zooo

, ),.•. O\o 0 _oo

B 8o

'; _ 20

o- II1

.2 0 i i I I I I !

._ 0.0 .2 .4 .6 .e |.0 1.2 1.4

._ _ NEUTRAL DENSITY PARAMETER. _(|-qu). (A eq. )

:_, Figure 4a. Plasma Ion Energy Cost Curve - ConfigurationI

l_ ARGON

_, _ \, VO - 5O. Ov

i II_D __ LHAO OPTICS

, -" I2cm diD. BEAM

140

120 _ FLOW RATE (mA eq)

m

: _ /O\_ o Iooo

CO " 1.81A uq) "!

. 40

_ 20

I I I I I i i

0.0 .2 .4 .6 .8 1.0 1.2 i.4

NEUTRAL DENSITY PARAMETER, _(l-qu). (k eq.)

Figure4b. Plasma Ion Energy Cost Curve for High AcceleratorGrid

Transparency to Neutral Atoms - Configuration I
qP

i

plasmaion energy cost (Ep) are plottedas a functionof the neutral

density parameter m(1-nu), as suggested by Eq. 25. The solid line in

Fig.4a is the curve given by Eq. 25 when the parametersCO and _ have

i , been selectedto give the best fit to the data. The parameterc_ was

_' taken to be the value of Cp measuredat large valuesof the neutral

density parameter. The justification for this selection can be under-

_ stood by consideringEq. 25, which shows that when Co tn(l-nu)is large,

_ ' the exponentialterm is small comparedto unity and one obtains

• Cp Cp . Having established the value of the baseline plasma ion

:,,, energy cost, the value of the primaryelectronutilizationfactor (Co)

_.:-_ is varied until the best fit is obtained. The agreementbetweenthe

-):

=,v /<_ functionalform of Eq. 25 and the experimentaldata is seen to be quite

:IC good. This indicatesthat the parametersCO and ¢; may be taken to be

independent of the neutral density parameter.

A value of Co = 3.1(A eq.)-I which is applicableto the ion source

, of Fig. 3a operatingat the conditionsdefined in the legendfor Fig.4a

has now been established. New values of primary electron utilization

factor,applicableto other operatingconditions,may now be calculated

from this value using Eq. 26. For example, changing grid sets from SHAG

to LHAG shouldchangethe value of CO throughthe parameter¢o which is

_ the effectivetransparencyof the grids to neutralatoms. This effective transparency parameter may be calculated for each grid set according to the equation,

CsCa

¢0 -

¢S + ¢a (66)

where Cs and Ca are the modified transparencies for the screen and

.- acceleratorgrids,respectively. These modlf_edtransparenciesmay be

./
calculatedas the physicalripenarea fractionof a grid times the

appropriateClausingfactor [47]. For the two grid sets used in this

o .

study,

....... _.... . (Co)SHAG= 0.16, (@o)LHAG= 0.27 (67)

*L

...._ Thus, the new value of the primary electronutilizationfactorappli-

...... T cable to the LHAGoptics with all other conditionsheld constantis

_!:'_!:ii: given by,

- (Co)SHAG , (68)

ireful: (Co)new - (Co)Fig. 4a

i:!: (¢o)LHAG

_ which yields (Co)hew = 1.8 (A eq.)'1

The measuredvaluesof Cp , obtainedunder the same set of conditions defined in the legendof Fig. 4a,exceptfor the change in optics

o_i from SHAG to LHAG, yielded the resultsshown by the data points in

Fig. 4b. The solid line is the predictionof the model based on the

value of Co calculatedfrom Eq. 68. The value of the baselineplasma

ion energy cost was held constantat ¢: = 57 eV since changingthe

_,_, optics should not affect this parameter. The degreeof agreementbetween the data points and the curve in Fig. 4b shows clearly tnat the

model correctlypredictsthe variationin the plasma ion energy cost

with the neutraldensity parameterwhen the grid systemtransparencyto

neutralsis changed.

The same procedureof calculatinga now value of the primary

electronutilizationfactorfrom the old value accordingto Eq. 26 was

followedfor the analysisof the data displayedin Figs. 5a and 5b.

For the data in Fig. 5a, the thrusterwas operatedwith krypton
_L

II10 KRYPTQN

_ _ VO " 40. Ov

laO SHAG OPTICS

G. |2cm die. BEAM

• 140

:' '" _ 120 FLOWRATE (mA Iq)

'. ;!,, ' / CO m II ? (A lq) "1 (_ | I_OO

12 10G0

wa"_0o _ A ?so

_ _ '_ <> 500 in

B0

o,.,i 0

'_.... .' 20

--"' " " '-: NEUTRAl.OENS|TY PARAMETER,m(|'_u), (A IIq.)

: ,,._: Figure5a. Plasma lon Energy Cost Curve for Krypton - ConfigurationI

IBO

_' KRYPTON

.... VO " 40. Ov

_: _ " 180 SHAGOPTICS

o. _ 6c:mdia. BEAH

:; L40

,:.. v 120 (; - 5|,OmV FLOW RATE (mA llq)

• e--'_Co - 22. sCAmq)'l O 500

r_ 375

..... _ _ A 2so

: L

BO

_ _0 ,

• 0 I I I I I , I , I

0.0 ,2 .4 ,e .ll |.O |.2 1,4

NEUTRALOENS]TY PARAMETER._(|"_u), (A eq,)

• Figure 5b. Plasma Ion Energy Cost Curve for Small Diameter Ion

Beam- ConfigurationI
i_i .... ili"

L.

propellantand SHAG optics at a dischargevoltageof 40 v. The primary

electronutilizationfactorcorrespondingto this operatingcondition

,i. was calculatedusing the value of CO obtainedfrom Fig. 4a together

_,_ ,_ with Eq, 26 in the form,

h :.

_ , M/_r

,_:_,,, i (ao) Kr (69)

: ,jw_ (Co)new- , _ (Co)Fie,,4a '

_._i;.._ (%)Ar J MAr

b_,;_:" In this case, both the change in propellantpropertiesand the change

k ',,', ' "

-> : in dischargevoltagemust be accountedfor. That is, (ao)Kr.is the

_"_"_rl total inelasticcollisioncross sectionfor 40 eV primaryelectron-

¢ 4 rl*'= _ refers, in this case, to 50 eV

_ .__ kryptonatom collisions,whereas (%)Ar

,- i_! primaryelectron-argonatom collisions. The cross sectiondata needed

_i_" in this equationwere obtainedfrom de Herr, et. al [48]. The new

_"_ii. value of the primary electron utilization factor calculated from

_ ...... :'- Eq. 69 was CO - 5.7 (A eq.)"I and the correspondingpredictionof the

model is comparedto the measuredvalues in Fig. 5a. A new value of

the baseline plasma ion energy cost (¢p) was also required to fit the

. _._•;.... , data of Fig. 5a since this parameteris a functionof both the dis-

• _ ;_ charge voltageand the propellanttype. This new value was selected

- :_ in the manner suggestedpreviouslyas the measuredpla2ma ion energy

t

r :,. cost at high neutraldensity levels. These data indicatethat the

,i: model works as well for operationwith kryptonpropellantas it does

_,;2,, for argon.

_ From Eq. 26, it is seen that the primaryelectronutilization

factordepends inverselyon the area of the grids throughwhich the

.... beam is extracted, Ag . This area may be varied without changing the

.... . dischargechamber diameterby masking down the screengrid to produce
:': 49

d_

_: ion beams of differentcross sectionalareas, In this case, the screen

,_,:: grid for the SHAG opticsset was masked down from a beam diameterof

..... _. 12cmto oneof6cm. This four fold reduction in beam area should pro-

:_: ". duce a correspondingfour fold increaseir nrimaryelectronutilization

_.

•..... ' factor. TakingCO for Fig. 5a and multiplyingby four yields the new

value of CO = 22.8 (A eq.) "1 . The model prediction for the plasma ion

o,

_., energy cost versus neutraldensity parametercurve,obtainedusing this

" value of CO, is compared to the measured values of these quantities in

w ,, Fig. 5b. Remarkably,the agreementbetween the model and the experiment

_._'_ is excellent. Similar agreement was obtained for operation with argon

.:ii__i using the masked down grid set.

J_/L.!I. neutral Measurements densityparameter of the plaweresmaalso ion emade nergyoncost the vthruster ariationconfiguration with the

oi:

i, of Fig. 3b. Some of the results obtained with this configuration are

_:, shown in Figs. 6a, 6b, 7a and 7b. The data in Fig. 6a was obtained

with argon propellant,a dischargevoltageof 50 v and a range of pro-

_,, pellantflow rates from 350 to 1500 mA eq. Excellentagreementbetween

._..--,_ " the model and the experimentaldata was obtainedby taking_p = 59 eV

: ...._ and CO = 4.0 (A eq.)"I . The value of e_ = 59 eV, obtainedwith this

;- _ . configuration,agreeswell with the value of ¢; = 57 eV obtainedwith

the other configuration for operation with argon at a discharge voltage

o

of 50 v. The slightlyhighervalue of e_ for the configurationof

_'_ Fig. 3b is believedto be the resultof the shorterdischargechamber

length causing a slightly higher fraction of ion current to go to the

• anode surface_nd cathodesupport posts.
Figure6b. Plasma Ion EnergyCost Curve for Xenon - ConfiourationII

©
/_. 1110 ARGON

%_ ] V0 - 30. Ov

160 SHAG OPTICS

_ 4

! 12cm dia. ilEAH

140 '_-i #

,h_ IlL

- _ 100

L-; ,,,

___,:L,.-" 60 FLOW RATE (.A aq)

";-::" O 1SO0

_,_ •

._ " _ f r-i tO00

_, line 40 L_ 7SO

_'_,.['_,_ L'_' O'S "2 .4 .B "S |.0 1.2 1.4

i. _-_-,;---:- NEUTRAL DENSITY PARANETER. a_11-'1_1. (A eq.)

-, Figure 7a. Plasma Ion ErergyCost Variationat Low Discharge

':i Voltage- ConfigurationII

i_ 7 ARGON FLOW RATE (mA eq)

VO " 30. Or 0 ISO0

Q tO00 '_ 8 SHAG OPTICS A 750

: O .900

• 12cm dis. BEAM _ 150

0 I I I I I I I

_, O.O .i .4 .6 ,I 1.0 I.! 1.4

_UTRAL. OENSITY PARANETER, mll'_u), (A eq,)

Figure 7b. Anode ElectronTemperatureVariationat Low Discharge

Voltage - Configuration II
6

_ l [ From the value of CO in Fig. 6a, a new value of Co applicable to

this configuration operating on xenon propellant at a discharge voltage

of 40 v may be calculated using Eq. 26. The results of this calculab

• ., tion, togetherwith the experimentalresults,are given in Fig. 6b.

For these data, the neutraldensityparameterhas been correctedfor

presenceof doublycharged ions in the beam. Again the value of ¢_ was

r'C'

_,:. chosenin the mannerdescribedabove. Clearly,the model agreeswell

o with the measuredresults. Equations25 and 26 have now been demonstratedto describecorrectlythe effectsof variationsin the plasma

,_ ion energycost resultingfrom changesin propellantflow rate, pro-

_ pellantutilization,effectivegrid transparencyto neutralatoms beam

extractionarea, dischargevoltage,and propellantgas (Ar, Kr and Xe).

_ In addition,other experimentsindicatethat the model correctlypre-

'_, dicts the change in thrusterperformanceresultingfrom a changein the

_//_ effectivedischargechamberwall temperature[36]. The model has also

: been shown to work well on line cusp thrustergeometries[35].

,_ For operationat low dischargewltages, however,the situationis

o

quite differentas seen in Fig. 7a. The data in this figurewere taken

using argon propellant,a dischargevoltageof 30 v and the thruster

configurationof Fig. 3b. Here, a systematicdifferencein the data

taken at differentpropellantflow rates is observed. Clearly,a single

equationsuch as Eq. 25 is not sufficientto explain this behaviorif

_ Co and ¢; are taken to be independentof the propellantflow rate and

propellantutilization. This systematicdifferenceof the plasmaion

energycost curves,measuredat differentflow rates,is believedto

: : resultfrom changes in the baselineplasmaion energycost (c_)which

occur at low dischargevoltages.
!';. 53

" ,,. These changes, in turn, result from the dependenceof the baseline

• plasmaion energycost on the average energyof a Maxwellian electron

collectedby the anode. _M (see Eq. 27). The value of _M' however,

, dependson the Maxwellianelectrontemperatureat the anode. Measured

r _,._ valuesof the electrontemperature,made using the Langmuirprobe po-

,. sitionedon the surfaceof the anode magnetring, are shown in Fig. 7b.

_ . These data were taken under operatingconditionsidenticalto those

! used for the collectionof the data in Fig. 7a. The data in Fig. 7b

• clearlyindicatea systematicdifferencein electrontemperatureat the

_ anode for differentpropellantflow rates. For a given value of the

.... neutraldensity parameter,these data show that the electrontemperature

at ";heanode increaseswith increasingpropellantflow rate. Higher

,,i electrontemperaturesat the anode correspondto highervaluesof _;

_ becauseof the increasedloss of energy from the plasma in the form of

Maxwellianelectrons. Because the ratio tM/VD appears in the equation

for E; , changes in CM become increasinglyimportantat low discharge

volrages.

The systematicdifferencein the curvesof electrontemperature

versus neutraldensityparameterobservedin Fig. 7b is not predicted

by the model (as given by Eq. 59). Thus, some additionalmechanismfor

the transferof energyfrom the primaryelectronsto the Maxwellian

electrons,other than that consideredin the model, must becomeimportantat low dischargevoltages. This mechanismis believedto be

the direct thermalizationof the primaryelectronsas the resultof

electron_electroncollisions. The collisioncross section for primary-

_laxwellianelectroncollisionsincreasesrapidlywith decreasing

primaryelectronenergy. This is consistentwith the observationthat
: 54

.... the separationbetweenthe electrontemperaturecurves taken at differentpropellantflow rates (suchas those in Fig. 7b) increaseswith

.o,

" decreasingprimaryelectronenergy (i.e.,dischargevoltage).

i ExtractedIon Fraction

": Aside from the plasma ion energy cost, the other parameterwhich

o has a major affecton thrusterperformanceis the extractedion fraction, fB' It is of little use to create ions efficientlyin the dis-

;- '_ chargechamber if the fractionof these ions extractedinto the beam

_. is small. The extractedion fraction,which is defined in Eq. 6 as the

_' ratio of the beam current to the total ion current produced,was mea-

, sured for both thrusterconfigurationsover the range of operating

conditionsdiscussedearlier.

r.

,' The resultingextractedion fractiondata, obtainedon the

thrusterconfigurationof Fig. 3a, are shown in Figs. 8a and b. In

Fig. 8a, the value of fB are given versusthe neutraldensity parameter

: for operationwith argon propellantat dischargevoltagesof 30 and

• 50 v over a rangeof propellantflow rates from 500 to 1500mA eq.

These data indicatethat the extractedion fraction,while being relatively independentof neutraldensity,does tend to be slightlygreater

at lower dischargevoltages. Data taken under the same conditions,but

at a 40 v dischargevoltage,fall between the two curves showninFig.8a.

-: The extractedion fractionsfor argon propellantare comparedto

.. those for kryptonpropellantat a dischargevoltageof 40 v in Fig.8b,

The values of fB for argon are seen to be generallyslightlyhigher

than those for krypton.
_ 55

: 0.8

,..' - 0.?

,.m0.6 _,,_

" _ 0.5

_._ _ o.q OPEN SYMBOLS VomSOV mlmAeq}

CLOSEDSYMSOt.S yO. 30v 0 500

" • I_°'3 ,_ 75o

ARGON _ I000

• x 0.2 SHAGOPTICS 0 1500

i

:j' 0.1

0 0 ' i I i I I I

_" 02 0.4 0.6 0.8 I.O 1.2 1.4

, r r_' NEUTRAL DENSITY PARAMETER.m(I-,_,). [AIQ]

Figure8a. Effectof DischargeVoltaqe on the ExtractedIon Fractinn

- Configuration I

oI.,

0,?

i 0•5

oZ_0.4, OPEN SYMBOLS ARGON m|mAeq)

'_ CLOSEDSYMBOLS KRYPTON 0 _00

0._

r_ 750 •HAG OFI'ICS ,_ I000

Vo'40V _> iSO0

0.2

0.1

0 _ i i i & t i

0.2 0.4 0.6 0.8 I 0 1.2 1.4

., NEUTRAL DENSITY PARAk'ETERom(I-,_/,). [Aeq]

• Figure8b. Effectof Propellanton the ExtractedIon FractionConfigurationI

©

i
The data in Figs.8a and 8b indicatethat the extractedion

=i fractionis relativelyindependentof the neutra_density parameter.

Other data not presented,indicatethat the extractedion fraction

_.. =.

•, sometimesdecreaseswith increasingbeam currentsfor certainthruster

L

_ configurations. This decreasein fB results from a decreasein the

_: effectivetransparencyof the screengrid to ions as the plasmadensity

=':_'--:_''" ..... is increased. However,the fractionof the total ion current produced

_:_

_¢)__',_u° that iEarlier s directed studtoward ies [32,the33]gridins dicate remai that ns constant. the extractedion fraction,

--_... or the fractionof ions directedtowardthe grids, is stronglydependent

: on the magneticfield configurationand thrustergeometry. The data in

this report indicatethat fB is also a weak functionof the discharge

voltageand propellantgas, but not a functionof the neutraldensity

:_ parameter. Consequently,a simpledesign approachwould be to take fB

to be a constantfor a given dischargechamberdesign (magneticfield

_, _ configuration),propellantand dischargevoltage. Unfortunately,a

method for calculatingthe value of the extractedion fractionfrom the

above designdata remains to be developed.

• PlasmaProperties

Equation45 providesa simpleexpressionfor the primaryelectron

densityas a functionof the propellantutilization. As mentioned

earlier,the combinationof parametersinsidethe square bracketsin

this equationmay be taken to be roughlya constantfor a given propellant,discharge,,oltageand thrusterconfiguration. The value of

this constant,ap_;-opriatefor argon propellant,a dischargevoltageof

50 v and the thrusterconfiguratlonof Fig. 3b, was calculatedt} be

©
Li_'' 57

l.O x I0I° cm"3 . Thus, Eq. 45 becomes,in this case,

' np = (l 0 x I0I°) nu (cm"3) . (70) ,: ' l-nu

' , The value of c* , requiredfor this calculation,was taken to be the

,/ measuredvalue given in Fig. 6a. The value of _p could, however,have

_ been calculatedusing the methoddescribedin AppendixA.

• The volumeof the ion productionregion (_), requiredby Eq. 45,

was determinedfrom a computerdrawn magneticfield map of the dis-

.... _i charge chamber. This map was created by measuringthe magneticflux

densityand directionat regularlyspaced pointsin the discharge

o_L. chamber. The ion productionvolume,defined by this map, was taken to

be the volume in which the magneticflux densitywas 0.005 Tesla or

- less.

The extractedion fraction(fB)was taken to be its measuredvalue

_, obtainedin tests with argon propellant,a dischargevoltageof 50 v

and the thrusterconfigurationof Fig. 3b. The transparencyof the

' o .C

• grids to neutralatoms (¢o)was calculatedfrom Eq. 66. Finally,the

• neutralatom velocity(Vo)was calculatedbased on an assumedeffective

wall temperatureof 400 K.

_. Comparisonof the calculatedvalues of the primaryelectrondensity

from Eq. 45 (in the form of Eq. 70) with the measuredvalues is given

in Fig. 9a. The measuredprimaryelectrondensitieswere obtainedusing

the Langmuirprobe positionedalong the centerlineof the discharge

o chamber. Equations45 and 70, however,calculateessentiallyan average

primaryelectrondesnity. Further,since th._ centerlineprimaryelectron density is expectedto be higher than the averagevalue, the

i,/

" 59

excellentagreementbetween the calculatedand measuredvalues in

., Fig. 9a is somewhatmisleading. This figureclearly indicates,however,

.. that Eq. 45 has the correct functionalform and that the primaryelec-

: _. tron density is indeed relativelyindependentof the propellantflow

_-L_!'_ rate as predictedby the model.

_ A similarcomparisonbetween calculatedand measuredprimary

_ _ electrondensitiesis given in Fig. 9b for operationof the same

'i thrusterconfiguration,but with xenon propellantand a dischargevolt-

:'_ age of 40 v Again, the same concludsionsdrawn from Fig 9a are also

_ applicable to thruster operation on xenon as _ndicated in Fig. 9b. The

->_. observationthat the calculatedaverage valuesare somewhatgreater

than the measuredcenterlinevalues probablyresultsfrom the use of

, _: the measuredvalue of _ = 44 eV in Eq. 45 ratherthan the calculated

,_ value of e_ = 36 eV given in AppendixA.

,_.:, _ An expressionfor the ratio of pri_.ary-to-totalelectrondensity

,_? is given by Eq. 48. The combinationof parametersin the square brac-

', kets in this equationis approximatelya constantfor a given propellant,

:, aischargevoltageand thrusterconfiguration. For operationwith argon

at VD = 50 and the configurationof Fig. 3b, Eq. 48 becomes,

• _ 0.027. (71)

nl m(l-nu)

...... __ In making this calculation,the Bohm velocity(Vb)was calculatedbased

on an electron temperature of 4 eV. The value of the screen grid

transparencyto ions (¢i)was determinedexperimentallyby measuring

the ion currentto the screen grid and t_ the beam, The transparency

is then the ratio of the beam current to the sum of the screen grid and

................. _ ...... .. - _ " _- _'-_'"" - _--_ir_-T _........... rl '_" _ _-_ -_-"_ _'-' - -'_T_---- FT i
beam currents. The measuredscreen grid transparencyto ions was

approximately0.8 comparedto the physicalopen area fractionof the

_ screengrid which was approximately0.68.

t

;_<, Comparisonof Eq. 48 (in the form of Eq. 71) with the experimentali_ : ly measuredvaluesis given in Fig. lOa. This figure indicatesthat for

L ,_.. a given dischargevoltage,propellantand thrusterconfigurationthe

P _ primary-to-totalelectrondensity ratio is only a functionof the neui ;_ tral density parameter. In addition,it indicatesthat Eq. 71 has the

)_ ;:

_=- correctfunctionalform for the variationof this ratio with the

neutraldensit.yparameter.

,>-._

_ ? For operationof the same thrusterconfigurationwith xenon at

•-,. VD = 40 v, the resultsshown in Fig• lOb were obtained. The solid line

, ;_ in this figure correspondsto Eq. 48, where the constantwas calculated

! ' using valuesof the parametersin Eq. 48 that are appropriatefor xenon

_....., propellantand a dischargevoltage of 40 v. FigureslOa and b indicate

_-_,_ that Eq. 48 correctlyaccountsfor changes in the propellantgas, the

dischargevoltage,the neutraldensity parameterand the propellant

flow rate•

Comparisonof calculatedand measuredMaxwellianelectrontempera-

. tures is given in Fig. If. The calculatedelectrontemperatureswere

obtainedusing Eq. 59. For operationwith xenon at VD = 40 and the

° thrusterconfigurationof Fig. 3b, Eq. 59 becomes,

.... + 7.64 x lO"14

: Qo = (m3/s) , (72)

i;_ 226 m(l-nu)-I

F

where the appropriatecross sectiondata were obtainedfrom References

• 49 and 50. Equation72 gives the value of the Maxwellianionization

rate factor for the productionof single ions from neutralatoms as a

.

F'

©

, ;................ ;............. 1 FI I I " I II : " -- - 1
• 30 l| ARGON

_' L Vo " 5D, Ov

, _ . 25

,, ,- .,, " _ CALCULATED

1., Sq. 48

i ,20

= _ FLOWRATE (mA eq)

-:o_, zs \ : Is..

_ :, ,,_/- OS

d,_,L, _>_"_

• '_:':4 O.O0 I I I * J ' '

o _ - ._,.0 ,2 ,4 .8 .ll I,D 1.2 1,4

NEUTRALDENSITY PARAMETER.{a(l-'l_). (A eq.)

_, _::--_. ConfigurationII

_' i_' .30

_ XENON

_ CALCULATED

:i" _ Eq.4;

\ _ . 2o

• I,-

ii " _ FLOWRATE (mA .q)

u_ .15 0 lO00

Z

{2 750

_ SOD

.: _ .10 0 2'10

...._, .011

" ": U d _ • r_ • ' L L

// O.O0

/ " O.O .2 .4 .6 .ll t.O 1.2 1.4

,/'

// NEUTRALDENSITY PARAMETER,_(l-'lu). (A eq.)

• FigurelOb.Primary-to-TotalEectronDensityRatiofor XenonConfiguration I1
,: qj+

' i_ _ ,

t,

,, VD - 40. Ov

c, ,

.. _,_.r.

_' _::' _r' ' _ CALCULATED

,i' 9

....i;i- _ Eq. 59 FLOWRATE (mA ecI)

_: _ 7 O 10O0

--_ ; 0.= I_A [] ,_o

_:..: ,,=," I-I _ soo o-- =_

S: , Z 5

(,

i

!; _ 3

i: 3

, 0 i I I I I I I

- _' 0.0 ._ .4 .§ .8 l.O 1.2 1.4

_\ NEUTRALDENSITY PARAMETER._(l-'lu). CAaq.)

" Figure II. MaxwellianElectronTemperatureVariationfor Xenon -

:: ConfigurationII

o , ""
I . 63

,_i':::, ̧

1..... "'i:i functionof the neutraldensity parameter. The rate factorQ+o is a

_F!I":_: functionof the Maxwellianelectrontemperature,and is given by,

i y Ee .E

L:I +; o

'. Qo _ e.EITM . (73) y ,E

Using Eq. 73, the rate factor Qo may be plotted as a function of the

electrontemperatureas shown in Fig. 12.

_ To determinethe electrontemperatureas a functionof the neutral

densityparameter,the followingprocedureis used. First,the value

+

: of the rate factorQo is calculatedfor a given value of m(1"nu) using

+ is then used to enter the curve in Fig. 12 _._,,:_:: Eq. 72. Thi_ valueof Qo

_?>:_,i! : ,_:i from which the correspondingelectrontemperatureis determined. Re-

_:,_,__ peatingthis proceduregeneratesthe curve of Maxwellianelectron

:.,._, ._ temperatureversusneutraldensity parametershown as the solid line

_2:::_i:::, in Fig. II. The agreementbetweenthe calculatedand measuredelectron

_ ,,,,,::_,-_ '_!_,;:_ temperaturesis consideredto be quite good.

In additionto the Langmuirprobe positionedon the thruster

: o_,_j: centerline,a second probewas positionedon the upstreammagnet ring.

This probe was used to measure the temperatureof the Maxwellianelec-

_ _ i trons and the energyof the primaryelectronsreachingthe anode.

,,_-,, Becausethe probewas positionedin a regionof very high magneticflux

density,the electrontemperaturesdeterminedfrom the probe traces

.!: obtainedwith this probe are probablyonly equal to the electrontemperatureresultingfrom motion along the magneticfield lines. The

electrontemperaturecorrespondingto motion normal to the field lines

"r/1l
64

6

......-. 10"13

. % 10 "14

_," ":+ 151) /

: _ :2' /

J

.,_ h_

.... ; '::: w 10"15

:s- r_ /

/

" "" /

"! [

: 1o-16 I

_- o 1 2 a 4 s . _ e _ 1o

_. MAXWELLIAN' ELECTRON TEMPERATURE. TM. leVI

Figure12. IonizatioRnateFactorfor Xenon
. 65

=_' may or may not be the same. For simplicity,the_e temperaturesare

assumed,in this work, to be the same.

o_.) The measuredelectrontemperature_at the anode were found to be

"iL

i less than the measuredcenterlinetemperaturesby approximatelya

factorof 2/3_ The correlationof electrontemperatureat the anode

_',_ with the centerlineelectrontemperatureis given in Fig. 13. The data

,'_ in this figurewere obtainedwith the thrusterconfigurationof Fig. 3b

o,

• using both argon and xenon propellants. The solid line in this figure

has a slopeof 2/3. This observationthat the electrontemperatureat

'.. the anode is approximately2/3 of the centerlinetemperatureis used in

?i.

_'_ _"r_ AppendixA for the calculationof c; .

d

. The probe tracesobtainedwith the probe positionedon the magnet

i .... ring surfacealso indicatedthe presenceof primaryelectrons. For

_ operationat a dischargevoltageof 50 v, these primaryelectronshad

.... an energyof approximately50 eV as the model suggeststhey should for

_ this case where a refractorycathode is used (VC = 0). Similarly,for

operationat VD = 40 v, the measuredprimaryelectronenergyat the

anode surfacewas _ 40 eV. These probe tracesprovidedirect evidence

• of the loss of primaryelectronsthrough the magneticfield cusp to the

_. anode. Further,the loss rate of these primaryelectronswas seen to

' _ ,. increaseas the neutreldensity parameterdecreasesas the model

_ ;_ predicts.

u

• A comparisonbetweenthe calculated(from Eq. 66) and measured

valuesof the doubly-to-singlycharged ion beam current ratio is given

in Fig. 14. The measuredvaluesare given by the open symbols and the

' calculatedvalues by the solid symbols. The propellantutilization

efficienciesin this figure have been correctedfor the presenceof

©
!°.

c,'

; i_

66

.!

r_

^ 0

>m R O ARGON

rt XENON SLOPE - 2/3

L I ' _ 4 0 '

_.: _ 0 0

•

.:' 3

v '. U

:, mJ

w

Z I

<

'" 0

0 1 2 3 4 5 8 7 8 9 10

_°_.... CENTERLIN£ ELECTRON TEMPERATURE, TM, (IV)

Z

_ Figure 13. Correlationof ElectronTemperatures

F_
'

: : i!" ̧

_• 67

>

oo_,,.

• 30 ;_ii

_m XENON vO - 40v

, Lm .25

_ MEASURED

,v; . 20 FLOW RATE (,A eq) / /

o ]ooo / I

_' LIJ /Y Ll 5OO 41 '

_, ._ 0 zso /

I

r _ " _ _ __

- !

• _ .0,- __.°i___.__.,-l,-°*-" \:^,:u,^,_0 Eq. 66

o Q

• O. OO i i i i i i t i i J

': O.O .2 .4 .6 .il 3.0

PROPELLANT UTILIZATION, "1u

rl

Figure 14. Doubly-to-Singl.vCharged lon Beam Current Results
doubly chargedions accordingto Eq. B-5 given in AppendixB. The

ratiosJB / JB in Fig. 14 were taken to be equal to the valuesof

/' doubly-to-singlycharged ion currentdensitymeasuredon the thruster

- _ centerline. The valuescalculatedfrom Eq. 66, however,correspondto

_ valuesof this ratio averagedover the entire grid area. Since the

_ centerlineratio is expectedto be greaterthan the average value,the

-_i_ data pointsin Fig. 14 are believedto be shiftedup and to the left

_ relativeto the locationsof the correspondingaverage values, The

amountof this shift is unknown. This shift, however,is believedto

i)i account for at least part of the differencebetween the measuredand

_._ calculatedvalues.

..... _ In calculatingthe valuesof JB. /JB using Eq. 66, the collision 

_ cross sectiondata requiredfor the calculationof the parametersQo

i• and P+o were taken from Reference50. The valuesof the parameters

Qo ' Q_' Po ' P+ were obtainedfrom ReferenceII, in which the colli-

_. sion cross sectionswere calculatedusing the classicalcross section

model of Gryzinski[51]. The use of Gryzinskicross sectiondata in

:" the calculationsof JB /JB might also account for some of the difference 

between the measuredand calculatedvalues. The valuesof the Max-

• wellian electrontemperatureand primary-to-Maxwellianelectrondensity

ratio, requiredas inputsto Eq. 66, were calculatedusing Eqs. 4B, 49

,_ and 59.

,. Finally,it is noted that, the valuesof JB++ / JB calculatedfrom

i Eq. 66 are very sensitiveto the value of the electrontemperatureused

in the calculations. Equation66 does, however,indicatethe correct

functionaldependenceof the doubly-to-singlycharged beam current

ratio on thrusteroperatingconditions.
V. MODEL APPLICATIONS

= ,_': ThrusterDesign

o

i: Equation28 (repeatedbelow) providesa single equationdescribing

_:i the performancecurve of a given thrusterdesign

_" _B = _P + fcVD

i i.

. This equation,along with the data presentedin ChapterIV, suggestthat

c

the performance of any thruster design depends on the values of four

•_!

dischargechamber configurationcontrolledparameters:e;, CO, fB and

"_ fc; and two operatingparameters,m and VD. The effectof these

:, parameterson performancewas investigatedanalyticallythrough Eq. 28

by choosingthe followingset of valuesas the standardset, then vary=

": ing themone at a time to determinetheir effect on the traditional

: performancecurves.

.Z

' Tabl e 1

' StandardConfigurationParameters

_ m = I000 mA eq.

....... fB = 0.6

' fc = 0.I

,___ Vb = 50 v

'° Co = 3.0 A eq.

: ep = 50 eV

!

" 0
:, t*-'

; 70

g

: FigureISa shows the effectof the extractedion fractionon

i : .,

i

.__" performance. As expected,this parameterhas a strongeffecton the

i

:, performance. Changes in fB shift the performancecurve up or down, but

_' do not substantiallychange its shape. Clearly,it is desirableto

i 'i have fB be as large (near unity) as possible.

=_F_ The effectof the primaryelectronutilizationfactor (Co)on

i_/ performanceis given in Fig. 15b. This parameteralso has a strong

!] _ effecton the performance. Indeed,it is this parameterwhich primarily

j: ii

o... determinethe shape of the performancecurve,with larger valuesof CO

L"

:' _ correspondingto improvedperformanceand curves with more sharply

!I _ defined "knees." The definitionof Co given in Eq. 26 suggestsa numi= d

..... _, ber of ways in which the value of Co may be increased. For example,it

S may be increasedby using a propellantgas characterizedby a larger

_i ' and a largeratomicmass (result- ,:'_ inelasticcollisioncross section_o'

i_. ing in a lower neutralvelocityVo). The parameterCo may also increase

by decreasing the grid transparency to neutrals, ¢o' This must be done,

.. of course,without increasingthe acceleratorimpingementcurrent. For

thruster designs with non-uniform beam profiles, Tailoring the accele-

:7

io_ ' rator grid hole size to match the radialcurrent densityprofilemight

',i_" be a usefulway to minimize¢o" Also, three grid systemsmight be exalt,h; pectedto have smallervalues of 0o than two grid systems.

Most importantly, CO may be increased by increasing the primary

r: L electroncontainmentlength t_e). This lengthcorrespondsto the

averagedistancea primaryelectronwould travel in the discharge

• _ chamberbefore being collectedby an anode surface providedit had no

inelastic collisions. As mentioned earlier., the primary function of the

• magneticfield in all thrusterdesigns is to increasethis length. In

,_ • ,._ .._ -
_

"7"1
Lj

72

I,

' cuspedfield thrusters,primaryelectronsare lost to the anode though

_ ,. the cusps. Thus, _e may be increasedby decreasingthe numberof cusps

i

at anode potentialor increasingthe flux density at the cusp, but only

_ i,__ up to a point. Reductionsin effectiveanode cusp area below a certain

° _ limit will resultin unstableoperationof the discharge[22].

.... Equation26 also suggeststhat the primaryelectronutilization

.... ii,!i factormay be increasedby maskingdown the area of the grids through

_-_o which the beam is extracted,Ag. However,decreasingAg in this manner

_._ will lead to a large reductionin the extractedion fraction,and,

:i_ therefore,an overall reductionin performance. ._:_

_ The effectof propellantflow rate on performanceis shown in

_-<_.' Figs. 16a and b. In general higher flow rates producebetterperform-

': ance. The maximum flow rate at which the thrustercan be operated,

.._._" however,is limitedby the abiiityof the acceleratorsystemto extract

the ion currentdirectedtoward it. li_eeffectef flow rateon perform-

_j ance is less dramaticfor thrusterdesigns characterizedby larger

,: ,_.... valuesof CO as shown in Fig. 16b. High valuesof CO should,there_re,

. be particularlydesirablein thrustersdesignedto be throttled.

The effectof Cp is merely to shift the performancecurvesup or

down. The amountof the shift increasesfor smallervaluesof fB" For

'! thrustersthat use hollowcathodes,the efficiencyof the cathodeopera6

' tion (characterizedby Vc irhEq. 27) has a strongeffecton the value

_ of _ . In addition,highervaluesof VD, in general,producesmaller

* High dischargevoltages,however,are undersirablefrom

o : valuesof Cp .

_ thrusterlifetimeconsiderations,Consequently,a trade off between

\ _ thrusterperformanceand thrusterlifetimeis necessary.

o ,
,>

L.

, liXi

• 180 ,, 250 ]

160 .

%

jI 120

°

' "_: i U

" :,- 80 il - 2000m^ Ill

U

" lil i' B - .8

ill 60 " *

" _ z VIi " 50. I)v " 0 -i

" 40 Co " 3.13(A eq)

- " m 20

0

7; 0.0 .2 .4 .11 .8 I.O

>_;_ PROPELLANT UTILIZATION. _u

_ Figure 16a. Effect of Propellant Flow Rate on Performance for

SmalI Co

20ii

180 _ - 251_A eq ]

" 140

- ,._ liO

• _ 80 Ill " 2000mA IIq

_B " .8

z 80 fc " 1

z VU - 50. Ov

, o" 40 CO - 15.0(A eq) "1

z (; - 50.0iV

m 2O

_) 0 ' ' I II I I I I I !

0.0 .2 .4 .8 .8 I.O

PROPELLANT UTILIZATION, I u

<- Figure 16b. Effectof PropellantFlow Rate on Performancefor

,_ Large Co

t

U -
_i_: 74

'._; The effectof cathodeoperationon thrusterperformancecan be

_,. investigated analytically using the performance model. Hollow cathode

operationis reflectedin the value of the parameterVc. The lower

ttle value of this parameter the more efficiently the cathode is opera

: : ating. At a fixeddischargevoltage,increasingthe value of Vc

degradesthe thrusterperformancein two ways. First, it increases

_-°__:: e_ directlythroughits appearancein Eq. 27, and second, it decreases

the energyof the primaryelectrons,which increasesthe value of the

_ collisionalloss parameter_o and decreasesthe value of CO. All of

_ _---T_-.

...._ these affectsare accountedfor when Cp is calculatedaccordingto the

:: proceduredescribedin AppendixA and the primaryelectronenergy is

"L-

_'_. given by VD-VC.

_ _ _: The effectof Vc on the baselineplasma ion energycost was calcu-

_ .... ,_ lated for xenon propellantusing the method describedin AppendixA and

;_.' assuming a discharge voltage of 30 v. The results of these calculations

_i_ are given in Table 2.

.... Table 2

_;ii o :T Effect of Vc on e;

).. _ .

_ ' i VD Vc VD-VC ep Co

(Volts) (Volts) (Volts) (eV) (A eq.)"I

30 0 30 36 8.0

!

30 2.5 27.5 42 7.9

30 5.0 25.0 48 7.6

i

_, 30 7.5 22.5 55 7.2

!_ 30 lO.0 20.0 66 6.5

30 12.5 17.5 79 5.2

!

i

i
..... The valuesof _p and CO, from Table 2, were then used in the equation

for thrusterperformance (Eq. 28) along with the values: m = 1000 mAeq.,

i fB = 0.5, fc = 0.I, and VD = 30 v. The performancecurves generatedin

,_., thismanner a_-eshown in Fig. 17. This figure indicatesthat, at a discharge voltage of 30 v, relatively small changes in Vc can produce

_ _L_':_:_ substantialchanges in thrusterperformance. For example,increasing

p L ,.,

_L--__.... " Vc from I0 to 12.5 v causesan increaseof approximately30 eV/beam ion

_.-_' _, in the performancecurve.

_._._._v/,,_, Work done by Siegfried[52] on inert gas hollowcathodesindicated

_,_,_-.: that a 0.6 v increasein the surfacework functionof the cathodein-

__)._o_,,_,, :_;*:i_,o,_,_ sert could producean increasein internalcathodeplasma potential

,., ,_ from 8 to 12 v. The value of Vc would be expectedto increaseby this

/: same amount. In addition,Siegfriednotes that there is apparentlya

_. ; greatersensitivityof the insert to depletionor contaminationof the

._....!, low work functionmaterialfor op,,rationwith argon or xenon as compared

=_.._._ to operationwith mercury. Consequently,on the basis of these considi'

;r _: erations,one would expect the performanceof inert gas ion thrusters

operating at low discharge voltages to be quite sensitive to the con-

?.

_;_. ditionand operationof the hollowcathode.

_! Changes in the fractionof ion currentgoing to cathode potential ; _ _',

_,. surfaces(fc) also tend to shift the performancecurves up or down.

i !_i Recenttrends[23,24] in thrusterdesign to operate the discharge

..... __ chamberbody at anode ratherthan cathode potentialserve to reduce the

value of fc' and consequentlyimprovethe performance. In addition,

L eliminationof the separatecathodedischargeregionin these designs

, servesto decreaseboth fc and VC, which again improvesthe performance.

_ " It should be noted, however, that an increase in the extracted ion
Figure17.Effectof CathodeOperationon Performancefor Xenon
_ i ̧

77

fractionfB is also believedto be an importantfactorin the perform-

:: ance improvementobserved in these new thrusterdesigns.

.... ThrusterScaling

;_.. The dischargechambermodel also suggestsperformancechanges that

might be expccted from the scaling of thruster designs to different

[ i"".:.

_ ,,_ dischargechamber diameters. This may be seen by examiningthe product

_ _ of the primaryelectronutilizationfactorand propellantflow rate.

L '/_: This product is a dimensionlessquantityand is given the symbolBo.

_-_ .,' . J •

._,:..... ,,,.C_o_em

!.-;,"::_r- Bo - Corn- , ,, (74)

_r'_:'_: r_ For constant,averagebeam currentdensities thrusterscalingshould

_:;C _. be accomplishedsuch that the ratio of mass f_ow rate to activegrid

:"_L

area, m/Ag, is constant. Thus, for the same propellantand grid transparencyto neutrals,Eq. 74 suggests that the discharge chamber performance dependsonly on the primaryelectroncontainmentlength,ge" It is

c

• _;.,

:_,._- believedthat this lengthshouldincreasein approximatelydirect proportionto thrusterdiameter,and this suggeststhat largerdiameter

• thrustersshould be more efficientthan smallerone,_. This is not the

!

i_ .... whole story,however. The effectof thrusterdiameteron the extracted

L,,_:_, ion fractionmust also be considered. The above conclusionremains

_-;, true for thrusterscalingaccomplishedin such a way that fB remains

.- constant.

i _'

L_,_ o

Neutral Loss Rate

::° " RewritingEq. 22 yields an expressionfor the neutralatom loss

: ° o rate from the dischargechamber t,,roughthe g_ids,
" ,",.:: ' Substituting this Into Eq. 28, and solving for the neutral loss rate

! _":-,.'--_ yt el ds, [ ]

:"' no " _oo fB_-B'fcVD "

._ i'; = l gn 1 - eP (76)

This equation gives the value of the neutral atom loss rate as a func-

'? tton of the beamion energy cost. For a specified thruster geometry

and discharge voltage, the design parameters Co, e;, fB' fc and Vo may

:_ be taken to be approximatelyconstant. Thus, since the propellantmass

_._ flow rate doesn't appearon the right-hand-sideof Eq. 76, this equa-

'.t

_ r tion predictsthat the neutral loss rate _o is independentof the flow

_,:" r',

t

+ :

..

" rate at a constantvalue of the energy cost per beam ion (SB). This

__'_''m_i; _'_.. same conclusionwas reachedoriginallyby Kaufman [53] in his constant

'............ /i). neutralloss rate theorydevelopedfor low magneticfield strengthdis-

. _11_

_

":'":_ chargechamberdesigns.

"_'_:: ,, ThrusterTestin_without Beam Extraction

-". It is often desirableto compare the performanceof differentdischarge chamber designs that have been operated without ion beam extrac-

_-_:':_?:_,';, tion. Operationwithout beam extractionis often more convenientand

._;._ generallyrequiresthe use of smellervacuum test facilitiesthan

_ £_:.... operation with beam extraction. However, data obtained under these

._ >_.:

. ., conditionsmust be interpretedcarefully. It has been observedthat

_,_ the performance(eVlbeamion) extrapolatedfrom thrusteroperation

..i_i' without beam extractionis generallysignificantlybetterthan the

:'_ performancemeasuredwith beam extraction[24,54,55]. This difference K_,.,

v .'
79

i

in performance may be understood in context of the performance model

_,! developed in this report.

Thrusteroperationwithout beam extractionshouldbe characterized

by higherdischargechamber neutraldensitiesthan operationat the

same propellantflow rate with beam extraction. This is because,

without beam extractionmost of the propellantleavesthe discharge

chamber in the form of neutralatoms at the neutralatom thermal

I .

:' velocity. With beam extraction,however,most of the propellantleaves

:_ in the form of ions at the Bohmvelocity. Therefore, since the total

.:_ propellantflow rate leavingthe thrustermust be the same in each case,

_: the loss rate of neutralsis smallerwith beam extractionthan without

!LL_

_i it, implyinglower neutraldensities. This is primarilya consequence

_i! of the changingeffectivetransparencyof the screen grid. Without

i!i_

,j beam extraction,the effectivescreen grid transparencyis very small

i_iI. [56]since ions tend to be focusedonto the grid webbing. With beam ,i

extraction,the ions tend to be focusedaway from the screen grid

webbingand through the grid apertures. In any case, operationat

higherneutraldensities,for the same thrusterconfiguration,translates into lower plasma ion energycosts accordingto Eq. 25.

The performancemodel developedhereinmay be used to make more

meaningfulextrapolationof data taken without beam extractionto operation with beam extraction. To do this, it is necessar>to experimentally determinethe valuesof the parametersCp,* Co, fB and fc" The

parameters_p and CO are roughlyindependentof the neutraldensity,

i thus, shouldn'tchangedependingon whetheror.not a beam is extracted.

For hollowcathodeequippedthrusterdesigns,however,the baseline

plasma ion energycost includesthe parameterVc, which is an indication
';. 80

• I

f:,

.... of cathode performance. The value of Vc may be a functionof the discharge chamber neutral density and consequently could cause the value

!

; of _ to change for operationwith and without beam extraction. The

P

"_: magnitudeof the change in Vc, if any, is unknown.

w

'_: _ To determine the values of ep and CO for operation without beam

• extraction,somemethod _f estimatingthe total ion current producedas

G'_._',,I, ..

a functionof the dischargechamber neutraldensitymust be used. This

_ _' _ is probablymost easilyaccomplishedusing several ion current probes

io,'__,,-,,-,_'_ positionedat the walls of the dischargechamber in the manner suggested

: 17

i,_?o.('ii} by Poeschel[54]. Ideally,these data shouldbe acquiredover a range

),) _ _. of neutral densities at a constant discharge voltage.

Once the total ion current producedis known as a functionof the

,_°, neutral density, a curve similar to Fig. 4a may be generated, where the

i.... :i' valuesof ep are calculatedusing Eq. 2. From these data, the values ,

iJ

:) of ¢; and Co may then be determinedas those which give the best fit

_ _i of Eq. 25 to the data. Note, e_ may also be calculatedaccordingto

: " .. the procedureoutlinedin AppendixA.

.__ Finally,the valuesof fB and fc may be determinedusing the above

! ....... ..

i probe data and guessinga value of the effectivescreengrid transpari

i-'_: ency appropriatefor the beam extractioncondition. Once the valuesof

o_._: Cp, Co, fB and fc have been determined,the performancecurve for the

beam extraction condition may be approximated using Eq. 28, for any

desired total propellantflow rate.

b'
h"

"' 81 o_

, °

h

i"

Space PropulsionMissionAnalys!s

.,_...... The performance model, developed in this report, is particularly

o

, useful for examiningthe impactof thrusterperformanceon space pro-

..,._.

,.":' " pulsionmissions. It shouldalso be usefulfor studiesmaking compario ,,

sionsof the capabilitiesof differentpropulsionsystemssuch as those

_ done in References57 and 58. The followinganalysisis intendedas an

: illustrationof the kinds of thingsthat can be done using this thruster

_-_='_,*_ performancemodel. Consequently,the orbit transfermission considered

_ " ;.: here has been greatly simplified Solving a more accurate and complii

" .... " cared orbit transferproblemwould not provideany additionalinsight

into the applicabilityof the model to missionanalysis,and may even

" tend to concealthe desired illustration. In the analysisthat follows,

i ,, o

: the effectof thrusterperformanceon the maximum payloadfractionob-

_.o ::'_ tainablefor a low earth orbit to geosynchronousearth orbit transfer

i_" * mission, with a characteristic velocity of 6000m/s, is investigated.

'--_=-,., The rocketequation(Eq, 77) gives the ratio of final spacecraft

o ".

• .;_. mass (Mf) to initialmass (Mi) as a functionof the characteristic

.... : velocityfor the mission (AV),the thrusterexhaustvelocity (u) and

_ ' the thrusterpropellantutilizationefficiency(nu),

I_ : Mf _ -AVlnuu

--- e . (77)

,. MI

Assuming,for simplicity,that the final mass consistsonly of the

._o-: payloadmass (M£)and the mass of the power plant (Mg) then Eq. 77 can

,. be written as,

M_ = -aV/nuU _ (78)

14,1 e - Mi •

o
-_"_',_ • II_

It'

_' 82

I The mass of the power plant is proportionalto the power generated,

.,i[.. Mg = _ Pg , (79)

;I

IF where _ is the specificpower plantmass (Kg/W),and Pg is the

_i:_ generatorpower (W). The generatorpower may be given as,

F_,

_F

_)!... Pg nt u2nu , (80)

'=_::, i_-,

_,_.'._;__;iii!_' L where nt is the thrusterelectricalefficiencyand mp is the total

_i!;.i!_i:._ propulsionsystem propellantmass flow rate (kg/s). For a constant

_C'I:_,!,,I_ exhaustvelocityand propellantefficiency,Eq. 80 may be integratedto

_=._.'.. ;__;:..._ obtain, Mp° u2 nu

Pg- ,, o" , (81)

,...... / " 2nt t

'i% where Mpo is the initialpropellantmass and t is the totalmission

, ,-J

. time. CombiningEqs. 78-80 and recognizingthat,

;[ _,_ Mpo -aV/nuU i- ioii, - l - e (82)

!i _:."." Mi '

:I

k_i:'ii,:!" yields

M_--e = "aVlnuu ¢U2nu /,.I - e"AV/nuU) . (83)

Mi 2nt t

Consideringonly dischargechamber losses, the thrusterelectrical

efficiency may be approximated by,

_C..... nt _ l (84)

____ I+_B/VN '

where VN is the net accelerating voltage. The net accelerating voltage

..... is relatedto the exhaust velocityby,

• , _,,. z..p. ©
_ 83

' e VN = 1⁄2miu2 (85)

• Combining Eqs. 83-85 yields,

."°•M_ = -AVlnuu _u2nu

(

1 + - - e

-V/,

. uU(86)

ii_. .... M-i e 2t /

ii L

:!-.. 0_._ Equation86 gives the payloadmass fractionas a functionof: the

__'_r"" ,"14 characteristicvelocity,the exhaust velocity,the mission time, the

!i'_:_!_::.:i power plant specific,mass, the charge-to-massratioof the beam ions,

i °:::

i_. ._ the average beam ion energy cost and the propellantutilizationeffici-

"_"'_ ency. The average beam ion energycost is related to the propellant

_,, utilizationefficiencythroughthe equationdevelopedin the thruster

_ ! performance model,

_ o ', W

,:,: _ Cp fcVD

_""_: CB-- [ e'B-_°l'nu')'] '' + fB ' (87) ,

i-,. ';.,..?. fB I-

:J/ o:=" ?,e'

" ._ V_. _ "_T:I where Bo is a dimensionlessquantitydescribingthe utilizationof

_,.::_:'%; primaryelectronsin the dischargechamberand is given by Eq. 74.

i _(_. i For a given characteristic velocity, mission time, power plant

:!:"_ ,.,;..i_..," _ specificmass, and propellantutilizationefficiency,the optimum

_:; exhaustvelocitycorrespondingto the maximum payloadfraction,for

F ' ;i:

i _ these conditions,can be determinedfrom Eq. 86. This is accomplished

; .... by settingthe derivativeof Eq. B6 (with respect to u) equal to zero

.._ and numericallysolvingthe resultingequationfor u. Substituting

_ this value of the exhaust velocity back into Eq. 86 gives the value of

the optimum payload f_.'actionuder the specifiedconditions. This procedure may be repeated for different values of propellant utilization,
q , _.

i L,': i

! ' _r'

t h "

r) 11 ": 84 q

L where, for each value of nu, the corresponding beamion energy cost is

calculated from Eq. 87. In this manner, a curve of optimized payload

fraction versus propellant utilization efficiency may be generated for .¢, _,'.

_ ,._ specified values of the characteristic velocity, mission time, power

. plant specific mass and thruster performance parameters, ep, fB' fc"

, ,p.

Bo and VD.

An exampleof this is shown as the solid line in Fig. 18 for a

characteristic velocity of 6000 m/s, a mission time of 200 days and a

J-_- specificpower plant mass of 40 kg/kW. In addition,the thrusterper-

; ' formanceparameterswere taken to be Cp = 50 eV, fB = 0.5, fc --0.I

,, ",

!_.d's Bo = 5 and VD - 40 v with argon as the propellant. Figure18 indicates

i

_ that the curve of optimizedpayloadfractionversuspropellantutiliza- = g :.

_: tion efficiencygoes through a maximum. The propellantutilization

_. efficiency corresponding to this doubly maximized payload fraction in-

.... _ dicatesthe locationon the thrusterperformancecurve at which the

• thrustershould be operatedin order to truly maximizethe payload

• o "'i

_,,,- fraction. In summary,the doublymaximizedpayloadfractionwas de-

!

i ,,: terminedby optimizingthe exhaust velocityto producethe optimum pay-

!' L

: : load fractionfor a given propellantutilizationefficiencyand

r""_ subsequentlyselectingthe propellantuti'lizatiotno obtain the maximum

payloadfractionfrom this set of optimizedpayloadfractions.

'_ To illustratemore clearlywhat is going on, the initialpropellant

mass fraction (Mpo/Mi) and power plantmass fraction(Mg/Mi) are plotted

_" along with the optimizedpayloadfractionin Fig. 18. Note that the

.... sum of the three curves in this figure, at any propellant utilization,

is equal to unity. As the propellantutilizationincreasesthe initial

propellant mass fraction decreases as expected, in addition, the power
':' 85

,, ,r, . •

r.' '"

_ r: _I, _ "r "8 _V-§OOOm/s ARGON Bo-5. 0 I

• /

'; • O¢'40kg/kW _c'O. I V_-40v

/

- e, . M9

.... _L _ Hi

__i:_,,o' ,

3,i. ....

L:, 0"0 I I I I I I I I I I

...... 0.0 .1 .2 .3 .4 .5 .6 .7 .8 .9 1.0

PROPELLANTUT]L]ZATION, "Iu

IJ

....• Ftgure 18. Payload Fraction, Propellant Nass Fraction, and GenePatoP

Hass Fractton Variation with PPopellant Utilization
,,_',,_ plant mass decreasesinitially,due to a decreasingoptimum exhaust

: velocity, then increases dramatically at high propellant utilization

' ¢_T efficiencies,due to the rapid increasein the average beam ion energy

' cost• For the set of conditions chosen for the curves in Fig. 18, the

..... :. doublymaximizedpayloadfractionoccurs at a propellantutilization

.......... of 0.73

-;;,!_;:,:_._ The effectof the specificpower plant mass on the optimizedpay-

_"_", ;r_,,_o load fractionvs. propellantutilizationefficiencycurve is shown in

_4_';o, Fig• 19a. It is interestingto note that, at very low valuesof _ the

,...... _,_. ... .,._.._,,,_, optimizedpayloadfractionis relativelyinsensitiveto large changes

_:_,,:,_;,.::: _i, ,"_, _n the propellantutilization In addition,the propellantutilization

_-....-'_,_ G* , " at which the doubly maximize.ipayloadfractionoccurs is a functionof

;........... ._ ; _. That is, the point at which one should operate on the performance

' curve of a given thrusterdepends on the specificpower plant mass for

" ,, the spacecraft. This is illustratedin Fig Igb, where the optimum

..... propellantutilizationefficlencesfor the specifiedmission parameters

" 'r_l'_" " are indicatedon the performancecurve used in generatingthe curves in

Fig. 19_. It is also interestingto note that, the optimum propellant

°'_ utilizationfor the _ = l kg/kW case occurswell past what might ordindrily be consideredthe "knee"of the performancecurve• In a similar

manner, the mission time can be shown to effectthe optimum propellant

"_ utilization,with longer trip times yieldinghigheroptimum propellant

....% utiIIzatlons.

: _=_ So far, only one thrusterperformancecurve has been consideredand

it has been observed that the optimum propellantutilizationefficiency

" correspondingto the doublymaximizedpayloadfractionis a functionof

the specific power plant mass and the mission time. The effect of the

_r !..

i= .'_; _..... <, -._-_-..... -_.... -_.................. -__ . .--,ca_. _ _-__'_-_.:2" :"T-I.... -,_-_.-_-__-.-__t': _..___

,._. thrusterperformanceparameterson the optimum payloadvs. propellant

o utilizationcurveswill now be investigated.

°4 Figure20a indicatesthe effectof the parameterBo on payload

_ fractioncurves. This parameterprimarilydeterminesthe shape of the

_:,,_, thrusterperformancecurve. As expected,largervaluesof Bo yield

higher maximum payload fractions. The optimum propellant utilization

efficienciesare indicated,for each performancecurve in Fig. 20 b, by

- the verticallines that intersectthe curves.

_.:_. The effectof the extractedion fractionon the payloadfraction

_-_.,:., curves is shown in Fig. 21a. Not surprisingly,higherextractedion

-..,_;a_ fractionsproducelargermaximum payloadfractions. Again, the optimum

;Si_e'.-_'_,

@,_ propellantutilizationsare indicatedfor each performancecurve in

--:_':: Fig. 21b.

Finally,the effectof the propellantgas is indicatedin Fig. 22a.

-:_-; To generatethe curve labelled"argon"in this figure,the following ,

" Z, TI.

.,,_._:_ thrusterperformanceparameterswere used: fB = 0.5, fc = O.l, VD =

-_ii:_t_!. 40 v, Bo = 5.0 and _p = 50 eV. For kryptonand xenon propellants,the

._ _'i valuesof fB and fc were held constant,but, the valuesof VD. Bo and

-__'_Iii,'__ _p were changed to 30 v, 8.2 and 49 eV for krypton,and 30 v, 14.6 and

"ii": 42 eV for xenon. The changes in the valuesof Bo were calculatedusing

,_._:.. the equationsdevelopedin the thrusterperformancemodel. The changes

o_ in the valuesof Cp are consistentwith measuredchanges in this param-

-:_ i eter for a given thrusterdesign operatedon the three propellantgases.

•-,:_,"__.

• Clearly, xenon propellant is superior to either argon or krypton pro-

._ pellants. The performancecurves,along with the correspondingoptimum

_ propellantefficiencies,are given in Fig. 22b for the cases where

these three propellants are used.

i i

• L,

_'.:_, :. . II AV,,SOOOm/s

t.,2OOdaXil

,_"*' f_B'O, lIO ,

. _r_ ,, _ .''; Ol,,4Okglk W

=±_ .' _ .S . . •

-- '_: . II .4 "

"_,_. .. .... 0.3

"; .,3,

: 1'

o_'=,;i,_,_i - .s .e ._ .e .I i.o

: @.-..-._ PROPELLANTUTILIZATION, "iu

' _ '':_,:..._:__.. "_" _ r Figure 21a. Effectof fB on Payload Fraction

B

ARGON

o ._: 4'c-O.: I i ,

\:' . ,c<_: ! _::;v,o .,-o,J / /

a'= ../ /#!

L: '_ *''0L--'4"--"r"i"

_- ! .... L- -rC"-,,-o.,

i ........ "_'-I I !

i I

i .... I Iil ,

0,4 .S ,8 I .? .0 .ii I.O

...

PROPELLANTUTILIZATION. "Iu

__T::- Figure21b. _ffect of fR on the PerformanceCurve and the Optimum

• ropel lant _tiliTation

" ................ "" - ...... : ......... ' _ ' ' _ " ...... ..... _ .............. I_-,"!

. i/-_:_., . _ ............. _ .......... _:-.7_::*__:::?-..... /:t C-- _''-_+ - : ;" : ::_._:_::::.:.:wi_,_ .................. --I

,_<, , _/.-.,-,,_.!:---_,............... - -.............. .......... . . ............,.......... : . _ - ,-' . .

1
---+

r

_+,,_,!.._ _ , 91

U'.l_.

i +, q

J; "' ..v-_,ooo..

,,, *. 150dayI

c("40NS/IsW XENON

' _ i_..-- "

._ ./'I _, I

i

.m .' . _ I

• I , :! , :

r' e

! ,.' ....

.... :... 0,0 I I

L -- 0.0 .I .2 .3 .4 .S .ll .7 .ll .ii l.O

i ,-_._

_, o " PROPELLANT UTILIZATION. "1u

i : "K Figure 22a. Effect of Propellant on Payload Fraction +

, o . ,

L

; e

? I

o3 ,I

_: " , Z'50

_. .

0 200 KRYPTON

; .. _ ^.oo.-.J / /

_. . w iO0 ---" L....../I "_ XENON "r /

Q

m< 5o I

. I I

0 I I I J ii I

4 ._i .6 .7 .8 .li I.O

........ PROPELLAOTNTXLXZATX%ON.

:_ Figure 22b. Effect of Propellant on the Performance Curve and the

Optimum Propellant Utilization
'" VI. CONCLUSIONS

° A simple th_'uster performance model, that has led to an improved

understandingof ion thrusterdischargechamberoperation,has been

developed. This model describesion thrusterperformancein terms of

four parameters: the plasma ion energy cost (¢p), the extracted ion

_o'_ fraction(fB), the ion currentfractionto cathode potentialsurfaces

(fc)and the dischargevoltage (VD).

,_.. The equation developed to describe the behavior of the plasma ion

. energycost agreeswith the resultsof a varietyof experiments. This

_ _o_ equationprovidesan expressionfor the functionaldependenceof the

,_' plasma ion energycost on the neutraldensity parameter,m(l-nu) ,

Oo,_ Experimentsindicategood agreementbetween the predictedfunctional

o-,;

i;_:::,_! form of the model and the experimentaldata. These experimentsalso

._-_,_ suggest that the primary electron utilization factor (Co) and the base2,_,,_., ,_L line plasma ion energycost (e_) are independentof the neutraldensity

_:_ parameterunder many conditions. The model correctlypredictsthe o

o

v •• variation in plasma ion energy cost for changes in: propellant gas,

.... ' grid transparencyto neutralatoms,beam extractionarea, discharge

_..... voltage and effective discharge chamber wall temperature. The model is

applicableto both ring and line cusp thrusterdesigns.

Measurementsof the extractedion fractionindicatethat this

parameter is relatively independent of the neutral density parameter.

The extractedion fractiondoes, however,appear to be a functionof
_;' 93

_ the thrustergeometryand magneticfield configuration,and to a lesser

_,_ extentthe propellantgas and the dischargevoltage. No effective

_!, methodfor calculatingthe extractedion fractionbased on these con-

_:i siderationsis availableat the presenttime.

;: The valuesof severaldischargechamber plasmapropertie_can be

°; easilycalculated,as a functionof thrusteroperatingconditions,

using the equationsdevelopedin the mod_i. These propertiesinclude:

L

.... _ the primary electrondensity,the primary-to-totaland primary-to-

= ";i•: Maxwellianelectrondensity ratiosand the Maxwellianelectrontempera-

_ ture. The valuesof these properties,calculatedby the model,agree

_ well with the correspondingexperimentaldata. An equationwas also

_• developedto calculatethe doubly-to-singlychargedion beam current

_: ratio. Experimentsindicateonly fair agreementbetweenthis equation

*_ and the measureddata. This is believedto be partly the resultof the

_,:i, techniqueused to measure the doubly-to-singlycharged ion beam current

ratio,and partlythe fact that the calculatedvaluesof this ratio are

i. very sensitiveto the input electrontemperature.

For design purposes,thrusterperformancemay be taken to depend

on only four physicalparameters(the primaryelectronutilization

factorCo, the baselineplasmaion energy cost ¢;, the extractedion

fractionfB and the cathode potentialsurface ion fractionfc) and two

operatingparameters(the propellantflow rate m and the discharge

voltageVD). Improvedperformanceshouldbe characterizedby large

extractedion fractions,long primaryelectroncontainmentlengths

(_e),small effectivegrid transparenciesto neutralatoms (¢o)and

operationat high propellantflow rates. The loss of primaryelectrons

to the anode determinesthe shape of the performancecurve to first

Q

• r
,_F T.... _.... _._

94

_,_ order. The model suggeststhat thrusterdesigns characterizedby

.....i : large valuesof Co shouldallow efficientthrottling. In addition,it

i, suggeststhat hollowcathode efficiencybecomes increasinglyimportant

........ to the dischargechamber performanceat low dischargevoltages. . • ,,'c.,

_.i. The thrusterperformancemodel can be very useful for mission

analysiscalculations. The model allowsone to easilyidentifythe

optimum propellant utilization at which the thrus_cr should be operated

for a given mission. In addition,the model makes it easy to determine

which changes in thruster design or operating parameters have the

,/.i

greatesteffecton the payloadfractionand/ormission duration.

_-_!_i;!_ Finally,a simpledirectmethod for the calculationof the base-

..... line plasma ion energycost has been developed• Calculationsof ¢p

_=-_o_ agree well with the measuredvalues

f

. Suggestionsfor FutureWork

;_; '_ Two of the parametersidentifiedin this investigationas having

_" strongeffectson thrusterperformancewere the primary electroncontainmentlength and the extractedion fraction• At the presenttime,

..... however, these parameters cannot be calculated directly given only the

_ r' _" dischargechamber/magneticfield design, propellantgas and discharge

voltage• Consequently,futurework should focus on the developmentof

a method by which the primary electron containment length and the exracted ion fraction can be calculatedbased only on this information.

_r

• The experimentsdescribedin this investigationwere all performed

using a thruster equipped with a filament cathode• The model suggests,

, however,that the performanceof a thrusterequippedwith a hollow

cathodecan be stronglyinfluencedby the operationof the hollow

(i'

L
 CC

o ,,

, -, r 95

cathode. Consequently,there is a need to verifythe predictionsof

the model on a hollow cathode equipped ion thruster.

' Finally,it is of interestto test the validityof the model on

! ,,_._, _.. '.' the older style, low magnetic field strength thruster designs, such as

a.

T the SERT II [63]or the J-Series[64] thrusters. These thrusterdesigns

_ o_ differfrom the cuspedmagneticfield designs consideredin this invesi='_" tlgationin that electrons,in the dischargechamber plasma,can only

! .

i)_: reach the anode surfaceby crossingmagneticfield lines. In the

" ' cusped field designs,however,the majorityof electronsare believedto

j_

reach the anode by going along the field lines. The mechanismfor elec-

_-_ tron loss in the low magneticfield thrusterdesigns is sufficiently

_::! differentfrom that in the cusped field designs that experimentalveri-

•' ficationof the model on these low field strengthdesigns is required.

,_

r_ _

i

i r. _ •

i 'J "

.r,
_"_.i". REFERENCES

l, Kaufman, H. R. and Reader, P. D., "Experimental Performance of Ion

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_' 6. Beattie,j. R., "NumericalProcedurefor AnalyzingLangmuirProbe

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_ 32. Brophy, J. R. and Wilbur,P. J., "The FlexibleMagneticField

_i ' _'_:"....... Thruster,"Journalof Spacecraftand Rockets,Vol* 20, No 6,

_...(: pp. 611-618,1983.

' _,_"_,_:: 33. Brophy, J. R. and Wilbur, P. J., "RecentDevelopmentsin Ion Sources

,_ . for Space Propulsion,"Proc. Int'l Ion EngineeringCongress-

.... .,;. ISIAT '83 and IPAT '83,Kyoto,Japan, pp. 411-422,1983.

_._:_.. 34. Brophy, J. R. and Wilbur, P. J., "SimplePerformanceModel for Ring

.___:.i and Line Cusp Ion Thrusters,"Int'l ElectricPropulsionConF __=_i_ ference Paper IEPC84-68, Tokyo,Japan, 1984.

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_- _ CuspedMagneticField DischargeChambers,"Int'l ElectricPro-

!. , _.t pulsionConferencePaper IEPC 84-70,Tokyo,Japan, L984•

........ 36. Wilbur, P. J. and Brophy,J. R., "The Effectof DischargeWall

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_ _F_ _br__ z Tenuous Plasmas: A GeneralAtom Theoryand DetailedResults

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__ oL 40. Knief,R. A., Nuclear EnergyTechnology,McGraw-Hill,Hemisphere

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York, 1980.
<,, ,.

_:_ 42. Vahrenkamp, R• P•, "Meesurement of Double Charged Ions in the Beam

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!_;_i 1973.

_:_ Jl_,_ r . 43' Hershkowitz,N• , et.al., "PlasmaLeakagethrougha Low B Line Cusp;'

_. PhysicalReview Letters,Vol. 35, No. 5, 1975, pp. 277-280.

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!_._..

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_,.

il 53. Kaufman,H. R. and Cohen,A. J., "MaximumPropellantUtilizationin

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i:

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",[

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',','Z"'I ,,' •

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',Ir,.

'I_,. ̧ :

APPENDIXA

'.' THEORETICALCALCULATIONOF THE BASELINE

.... :. PLASMAION ENERGYCOST, _;

....'; The definitionof the baselineplasma ion energy cost (c_) is given

.... .#;! by Eq. 27, and is repeatedhere for convenience,

;,:_u .'

_-_ "_ _0 + EM

._. • I'(Vc+¢M)/VD

...... __,i The parameter¢o in Eq. A-I accountsfor the energy that is expendedin

iI ionizationand excitationreactionsand is defined in Eq. 15 as,

_.-_, eo _ U+ + . (A-2)

_.. <a+Ve>

o::

_. The bracketsin this equationrepresentthe enclosedproductaveraged

,_ over the entire electronspeed distributionfunction,i.e.,

,." _ o+(Ve)VeF(Ve)dVe

<O+Ve> = 0 _ , (_A-3_)

. _ F(ve) dve

0

%

where F(Ve) represents the entire electron distribution function.

i For a plasma with an electron population characterized by a /.

Maxwelliandistributionof temperatureTM, and a monoenergetic

: ..... (primary)group of energyEp, Eq. A-2 becomes,

i _'.
_,_"_t _"_!"i{iiii_ l _ ,4-1=.1•t''. ,

'I; 6e_:;

I_H . 102

;. n

• = U++ n , (A-4)

:".... '-'p- ' vp + <o'+ if. nM a+ Ve>M

i.

bl " •

! " where < >M representsthe enclosedproductaveragedover the

_,.:_i., Maxwellianspeed distributionfunction.

_ '_i??' "

b;!_ .. The term under the summationsign in Eq. A-4 may be approximated

_..,:_,_.. by consideringonly a singleequivalentlumpedexcited state character-

!'"__':!. o_<*_" Ized by a total excitat_::co:)llisioncross sectionrex and a lumped

_-::> excitationenergy Uex. For rare gases,Uex may be approximatedby [38],

!:. Uex (U_ + U+) , (A-5)

"_':-i:, where Ug is the lowestexciLa_.ioenergy n level. Using this lumped

o_ , excitationapproximationEq. A-4 becomes,

n

'_-_"_ '_ : [_4 Gex Vp + <_exVe>M]Uex i

_" ' " ¢o = U+ + (A-6)

, : np

! ......... " I Vp

_'_,._".X rim _+ <%ve>M

_,_,y._)--.:.,_:_,:,., Sub.stitutingEq. A-6 into A-I yields the followingexpressionfor ¢_,

l:i: [_ aex + <aexVe>M]Uex

......:d_ d' U+ + _M +

_:'"_"i- ' .__ a_.Vp + <_Ve:,M

.- ¢ = nM (A-7)

" 1 - (Vc + _M) I VD

_ The value of ¢_ may be easily calculatedusing Eq. A-7 for a given

u

, C, c_,' ,
lo3

I

, MaxwelIian electrontemperature,primary-to-M_xwellan l electrondensity

_' ratio, discharge voltage and value of Vc. In addition, the average

_:_ energyof a Maxwelllanelectronlost to thc anode (cM) must be known.

...... ,_, ReferenceIg gives this energyas,

_%2:_,;:' =

-, _ ; eM = 2TA + VA • (A-8)

°'_::_., 'i where TA is the electrontemperatureat the anode and VA is the

:_ differencebetweenplasmapotentialand anode potential. Experiments

presented earlier indicated the electron temperature at the anode was

L

_1_ :: approximately2/3 of the centerlinetemperature. Therefore eM will

i/i_;Z! be taken as,

eM = 2( TM) + VA . (A-g)

- Langmuirprobe measurementsalso indicatethat for the thrusterconfig-

,,,. urationstested in this inves_igatlonVA was alwaysapproximately2v

. ,_:' i" thus,this value of VA will be used in these calculationstogetherwith

,_,,,_ Eq. A-g.

.,. _.._ For xenon, the values of the total excitation collision cross

_,.. section requiredby Eq. A-7 were taken from Reference49. Ionization

'_ cross sectiondata were obtainedfrom Reference50. These data, along

_: with polynomialcurve fits used to facilitateinterpolationbetween

''rI_ data pointsare given in Figs. A-la and A-lb.

./;i./.L....: With these data,Eq. A-7 was used to calculatethe valuesof _ as

L_I;.._. a functionof electrontemperaturewith the primary-to-Maxwellian

: electrondensity ratio as a parameter. The resultsof these calcula- o.

tions are given in Fig. A-2 where the values VD=4Ov, Vc=OVand VA=2V

_ were used. From this figure,it is seen that the baselineplasma ion

.! energy cost can vary over a wide range of values, and it is not clear

, . _;_,.,;._-_.;. ....._g, _.=.............................. .._... ........ .
. t_ ,

i:,7 I

104

0 x10-20 i

_: XENON

' " 5

_L

a

_ 2

"' 1

• 0 * I I I I I I I I I

G 20 40 60 80 100

ELECTRON ENERGY, E , (iV)

. FiaureA-la.IonizationCollisionCrossSectionforXenon(Ref,50)

L, 5 - x10"20

_ XENON

S

")_"' / 4

=_.. 3

". ! 2

(

0 v I I I I I I I I I

_ 0 20 40 eO 80 1130

'_ ELECTRON ENERGY, E , (aY)

r

FigureA-Ib,TotalExcitationCollisionCrossSectionforXenon

.-i. (Ref,49)

E Q,

L • "_

i_. 106

h

at this point which value or values of c; are appropriate. This difficulty arises becausethe Maxwellianelectrontemperatureand prlmary-toMaxwellian electron density ratio may not be selected independently as

was done for the calculationsin Fig. A-2.

_ , _: To resolvethis difficultya secondequationfor _; is required.

:_,, This secondequation for Cp may be derived in the following manner.

Equation40 providesan expressionfor the ratio of ion currentproduced by primary electrons to the total ion current produced, i.e.,

Jp VD oo

_::;, :_ The total ion currentproduced,however,is given by Eq. 52 as,

|

"_i:, Jp nonpeVp°+_ + nonMe<°+Ve>M¥ (A-ll)

and the ion currentproducedby primaryelectronsis given by Eq. 30 as,

! I

Jp = nonpeVpO+ _ (A-12)

dividing Eq. A-12 by A-11 yields,

__ = 1 (A-13)

dp n_a+Ve> N

l pO+

_:_: EquatingEq. A-13 to A-IO and solvingfor ¢_ yields the desired second

: equation for the baseline plasma ion energy cost.

: i i

c; = Vo_ nM<O+Ve> o/a+ M . (A-14)

1 + n '

pO+Vp
_"_, 107

* determined from the correct corresThe appropriatevaluesof Ep ,

pondingelectrontemperaturesand densityratios,may be found by

solvingEqs. A-7 and A-14 simultaneouslyfor c_ and the electron

:_ temperaturefor specifiedvaluesof the primary-to-Maxwellianelectron

° densityratio This procedureis most easily accomplishedgraphically,

. as shown in Fig. A-3, where the intersectionof the curvescorresponding

_-_ to the same valuesof np/nM gives both _ and the electrontemperature.

_._ The locus of these intersectionpoints indicatesthe variationof the

_: , baselineplasma ion energy cost with the electrontemperatureand

_.i primary-to-Maxwellianelectrondensity ratio• FigureA-3 indicates

.._ * does not -._ii _ that, under the assumpt';onsused for these calculations,Cp

vary substantiallyover wide variationsin.electrontemperatureand

..... _C:_ np/_4 . This agreeswith the experimentalobservationthat ¢_ is a

constantfor operationwith xenon propellantat a dischargevoltageof

_ 40 v.

._ * is shown along with the calculatedvalues The measuredvalue of Cp

_.i in Fig. A-3. The experimentalvalue,at low electrontemperatures,is

seen to be generallyslightlyhigherthan the calculatedvalues• The

,, most likelyexplanationfor this is a systematicmeasurementerror of

_ the total ion current producedas discussedin AppendixB. Also indi-

_ cated in Fig. A-3 is the theoreticalmaximum value of ¢_ for this case

(as calculatedfrom Eq. 41). Finally,it shouldbe noted that the

• above resultsare somewhatsensitiveto the choiceof {M which is

assumed here to be given by Eq. A-9.
i_  ̧

" _ 108

_ ORIGINAL P,_,C,'__'_:,_,

...... OF POOR QUALITY

i

',! ̧'_/i,

FigureA-3. Solutionsfor the BaselinePlasma Ion EnegrvCost for

Xenon

,4

i'

!:

(
I

_ APPENDIXB

!.

_, ERRORANALYSIS

=_!"r: _ _;" The experimentsdescribedin this investigationrequire the

L '

measurementof several quantities. These quantitiesmay be separated

into four groups accordingto the techniquerequiredto make the meac,:

surement. These four groupsare:

,_.: I. Measurementof ep, Cp* and fB"

'_:o • _ _•: 2. Measurementof the neutraldensityparameter,_(1-nu)

: _ _!! 3. Measurementof the plasmaproperties:nM,, np , and TM .

!i ;: 4 Measurementof doubly charged ions.

L

L...._: The errors associatedwith the measurementsof the quantitiesin each

.!

L

of these groupswill be discussedseparately. In each case, both

systematicand randomerrorswill be discussed.

i :!.

,._; ,

__ ....... Measurementof ep, e_ and fB

__ Determinationof the quantitiesin this group requiresthe measurement of the thrusterelectricaloperatingparametersincluding:the

dischargevoltage,dischargecurrentand beam current. In addition,

_J the total ion currentproduced(Jp)must be measured. The total ion

" currentproduced is given as the sum of the ion currentsleavingthe

plasma, i.e.,

• Jp = JB + JC + JA + Jimp '

The acceleratorgrid impingementcurrent (Jimp)is generallyless than
',v

:,i• 110 :

one percentof the beam current,and can probablybe neglected. In this

investigation,however,the impingementcurrentwas always includedin

the total ion productioncurrent for completeness.

_ Measurementsof the thrusterelectricaloperatingparameterswere

i c_L, made using the ion sourceinstrumentationshown schematicallyin

°_ Fig. B-l. The measurementof the ion current to cathode potentialsurfaces (Jc)was accomplishedby biasing these surfaces30v negativeof

F _ cathode potential. With this bias,electroncollectionat these surfaces

! was eliminatedallowingthe incomingion currentto be measured. The

i !,"

i _._._ ion current to anode potentialsurfaces(JA) and to the cathode support )

' _-_ posts could not be measuredwith this technique. The measuredion pro-

_: ductioncurren_consequently,does not includethese currents.

_i_ Three potentiallysignificantsystematicerrorsassociatedwith

;_ _:_ the measurementsof Jp (and thus Cp and fB) have been identified. These

',> .i)

, are:

_;_.. I. Neglectingthe ion currentto anode potentialsurfacesand the

i .. cathodesupportposts.

_ 2. Secondaryelectronsemittedby ions strikingthe negatively

_: biasedsurfaces.

3. The presenceof doubly charged ions.

The most seriousof these is the neglectof the ions reachingthe

anode surfacesand the cathode supportposts. The error in Jp and Cp

resultingfrom this omissionis difficultto assessaccurately, However, the physicalareaof these surfaceswas 5% and 2%, respectively,

of the total interiorsurface area of the dischargechamber. Thus, one

• might take 7% to be the maximum systematicerror in Jp resultingfrom

t,

the neglectedion currents. Becauseof these neglectedcurrents,the
_"i FigureB-I. IonSourceInstrumentatioSchn ematic

%
,, I12 _

" ; I

:. true value of Jp should be largerthan the measuredvalue. This implies

._ that the true values of Ep and c; shouldbe smaller than the measured

: _o-" valuesaccordingto Eq. 2. A 7% error in Jp can result in approximately

,_t,,, , a 13% error in _ . It shouldbe noted, however,that Reference43

through 46 indicate that the effective anode area for ion collection at

a magneticfield cusp should be less than the physicalarea. Conse-

{ quently,it is believedthat the measuredvaluesof _p and _; are no

_.,i, _ more than 10% greaterthan the true values In addition,the true

!_>, value of fB would be expected to be slightly smaller than the measured

: value as a resultof the neglectedion currents.

i

•:_. Secondaryelectrons,emitted as a resultof ions strikingthe

negatively biased surfaces, produce an error in the measurement of the

"i

_-_' ,:;.

,:; ion currentto those surfacesresultingin measuredvaluesof Jc that

:_ are largerthan the actualvalues• The largestsecondaryelectron

..' yields at low energies,for the ions and metal surfacesof concern in

,, this study,appear to be for argon ions incidenton a clean molybdenum

..... ' surface [41]. The secondary electron yield for low energy (< 100 eV)

,. argon ions strikingmolybdenumis approximately12% [41]. If half of

,. the ions producedstrikeclean molybdenumsurfacesthen the measured

..... value of Jp would be approximately6% greater than the true value as a

, resultof secondaryelectronemission.

It has been observed,however,that the secondaryelectronyield

of a clean surface decreases after exposure to air [59,60]. Reference

: 60 concludes that secondary electron yields are less than 1% for low

intensityand low energy ion incidenton a metal electrodewhich has

been exposed to air and theambient gas pressure is in the mtorr range
]._ 113

I

!i duringoperation• In the course of the experimentsin _his investlga-

"r "_'_ ' tion the dischargechambersurfaceswere frequentlyexposed to air

! "

_ betweentests. It is believedthat the error in the measurementof Jp

!i).: resultingfrom secondaryelectronemissionis less tllanI%.

In the calculationsof Cp, c_ and fB it is assumedthat only singly

charged ionsexist. On the basis of this assumption,a doublycharged

/

ion leaving the plasmais interpretedas two singlycharged ions. The

effect,however,of these doubly charged ionson _p, Cp* and fB should

not be substantialfor two reasons. First,the fractionof doubly-tosinglycharged ions is generallysmall. Second, the energy requiredto

_. produceone doubly charged ion shouldnot be vastlydifferentthan the

.;i;: , energy requiredto producetwo singlycharged ions. The presenceof

_. doublycharged ions does have a significanteffecton the value of the

neutraldensity parameter,however,and this problem is addressedlater

_ in this appendix

'_; Of the three systematicerrorsmentionedabove only the neglectof

_.' the ion currentsto anode surfacesand to the cathodesupport posts

; appearsto be significant. The measuredvaluesof Ep and ¢_ are, therefore, believedto be no more than 10% greaterthan the true valuesas a

• resultof systematicmeasurementerrors.

The followinganalysiswas performedto estimatethe uncertainty

associatedwith the experimentallymeasuredvaluesof _p. The uncertaintyin Cp resultsfrom the uncertaintyin the measurementsof the

independentvariableswhich appearon the right-hand-sideof Eq. 2. In

general,the uncertaintyof a quantityy which is a functionof the

measurableindependentvariablesxl, x2, x3. . . xn is given by [61]_
:':": .... i Ay = _ AXi + _Bx2 L AX2 +...+ AXn , (B-l) L . oL

::::;:i.;;=_ where Axi is the uncertaintyof the ith Independentvarlable, For

._::.,<.,:_i._i,,. the plasma ion energycost equationgiven by*,

" (jD_Jp)VD

_p jp (b-2)

i ._..

;. Eq. B-I becomes,

Aep = AJD) +\@jp Adp + AVD) . (B-3)

_,_ Carryingout the partialdifferentiations,Eq. B-3 becomes,

_ _i_i: A_p = _ppAJp + _p2aJp + _pp- l AVD , (B-4)

:..l_:_:,:.; _:_::c_: TO use Eq. B-4 the uncertainties , in the measurementsof the discharge

_ ;;;i::i!i!i): current (AJD),the dischargevoltage (AVD) and the total ion current ' ,;' , c

'''" ,l.If? .:;: produced(AJp)must be determined. These uncertaintiesresultfrom the

-o.-,.:_ i, ::,_:_ uncertaintiesof the digitalmeters used to make the measurementsand

• any variation in the thruster operating set point which may occur while

the data is being recorded.

Simpson digital panel meters were used to measure the discharge

current,dischargevoltage and beam current. These meters have an

uncertainty of O.l_ plus 1 digit, The ion current to the negatively

_, :: biasedsurfaceswas measuredusing a Kiethlydiyitalmultimeter. This

c, :;

meter has an uncertaintyof 0.5% plus l digit. The variationin the

,,x •

thrusteroperatingset point was taken to be 1 digit on each of the

_ * This form of Eq. 2 is appropriatefor the case where JA may be

neglectedand the impingementcurrent is includedin J_.
i •

,L 115 ./

_': digitalmeters• In addition,3v was added to the uncertaintyin the

" dischargevoltage becauseof the voltagedrop acrossthe thermonic

cathodewires. With these uncertainties,the uncertaintyin ep was

. ,, calculatedusing Eq. B-4. The resultsof these calculationsare shown

L '

: in Fig. B-2 where the verticalerror bars representthe measurement

.... :_; .

,.'. uncertainty. The horizontalerror bars representthe uncertaintyin

L,, ,_ the neutral densityparameterand is discussedin the next section.

_._,'__,,,_,_,, The data in Fig. B-2 is the same as that in Fig. 6a. The uncertainty

=-":": '" in the measuredvaluesof ep range from approximately7 to 11%

.....•!_'_"_.',. , •

= __i Measurementof m(l-nu)

_ : Three systematicerrors exist in the measurementof the neutral

_,_._'i:-:_, density parameter, m(1-nu). The first is the effect of doubly charged i

"%::_. ions on the propellantutilizationefficiency. The second is the use ,

, of a gas flow meter with argon, krypton and xenon gases that has only

•! been calibratedfor air. The third is the backflowof neutralatoms

,:._ from the vacuumchamber intothe dischargechamber throughthe grids.

_,:_ The presenceof doublycharged ions in the beam leads to artific-

•"_:" " ally highmeasurementsof the propellantutilizationand,consequently,

-,_:.... low values of the neutraldensity parameter. Measurementsof the

, . .++ ..+

--.--_;,.,,,._ doubly-to-singlycharged ion beam currentdB ldB were made along the

_' _'_r_ thrustercenterline. This informationcan be used to correctthe

: ......' , •

!_-_;o:.':_:, propellant uti11zationaccordingto the equation•

._,>_. . 1 .++..+

/ +

JB IJB

(nu)c°rrected= (nu)measured I+ "++"+ ' (B-5)

-- " _B /dB

__,_i Since the value of JB IJB at the centerlineis generallyhigherthan

LI _ _ ............ ' ........ "" k,'..... "_
, 116

4
117

r)

,', the average value its use in Eq. B-5 should tend to over correct the

i ,r ..

propellantutlllzatlonslightly. The resultingerror in the neutral

E ....

densityparameteris believedto be small,however,

.... A TeledyneHastlngs-Raydstflow meter with a range of 0 to 50

vl

sccmwas used to measure the propellantflow rate into the thruster.

r The flow meter was calibratedfor air,requiringthe output readingsto

• ' "_ be analyticallycorrectedfor operationwith the gases argon, krypton

! ......... and xenon. The use of the analytical corrections rather than recal i-

_' ;"_:.. bratlngthe flow meter for the differentgases is expectedto introduce

,-_,'r_"

Y:,':. only a very slighterror in the flow rate measurement. Thus, the

........ systematicerrors in the measurementof the neutraldensity parameter

o'o are believedto be negligible.

"___ The backflowof neutralatoms from the vacuumfacilityinto the

., , , ,, ,,

'L, ,t' dischargechamber was calculatedbased on measurementsof the facility

'-.,.,:i_ pressuremade _Im downstreamof the thruster. These calculationswere

;_)_"J:_ used to correctthe total propellantflow rate into the discharge

i--._,_:_,'

i""_L,_" . chamber.

L' "

i

i __ The accuracyof the flow meter is I% of full scale. This turns

_", ._', :

L ...... _' out to be the major uncertaintyin the determinationof the neutral

density parameter. The uncertainty in the neutral density parameter

!

i-m_: is indicatedby the horizontalerror bars in Fig. B-2.

, Plasma Propert$ Measurement

..... '_: Plasmapropertymeasurementswere made using one Langmuirprobe

, . : positionedalong the dischargechambercenterllneand a secondprobe

i.... positioned on a magnel:tc field cusp. Details of the Langmutr probe

........... '----.... _%_..\;.-/,.,;.;.___-:-__ -T:_]....,-_c:_'-,. -'_--:T-:_;;-:_. c'. _... '....,_, . L..
!_;: 118

ii circuitryused are given in Reference62. There are two major sources

4,

of error in using Langmuirprobesto obtainplasma propertydata. The

i

_ ' first of these deals with obtainingthe probe trace itself(i e ,

! measurementof collectedcurrentversus probe voltage). The second

i sourceof error arisesfrom the analysisof the probe trace to obtain

the desired plasmapropertyinformation.

I_: Errors associatedwith the trace includesuch

obtaining probe

i thingsas: the variationin work functionover the probe surfacearea,

secondaryelectronemissionfrom the probe,probe insulatorcontamination, noise in the plasma, electricalnoise in the probe circuitry, i

iiii plasma perturbation perturbation by the byprobe theisprobe, minimized and magnetic by usingfield a probe effects. with a Plasma small

_'!i!_ surfacearea. Biasing the probe negativeof cathodepotentialwhen

_iI! data is not being collectedtends to sputterclean the surfaceand

_/ minimizesthe variationin work functionover the probe surfacearea.

!_ Changingthe probe supportinsulatorfrequently(_ every 6 hours)mini-

_ mizes the problemof insulatorcontamination. Noise in the plasmawas

_;- minimizedby using a D.C. current heatedtherminoiccathode. Magnetic

• field effects were minimizedby placingthe probe on the discharge

chambercenterlinewhere the magneticflux densitywas less than 0.001

tesla. For the probe positionedon the magneticfield cusp, where the

; magneticflux density is approximately0.2 tesla, the magneticfield

effects are substantial. In this case, only the electrontemperature

and primary electronenergydata obtainedfrom these tracesare believed

to be meaningful.

1.
_''1_l':";_': ..... The probe traceswere analyzedusing a non-linearnumericalcurve

"_'_'_'_:.,_,,:_,.,_fittingroutinesimilarto the one describedby Beattie [6]. This

_._:_,'. routineassumesan _lectronpopulationthat is characterizedby a Max-

.ii '' wellian plus mono-energetic(primary)electronenergydistribution.

o 'i.. Probe traces were digitized using an HP-7470A graphics plotter together

. _ with an HP-85 mini-computer. The plottermay be used as a digitizer

....:"'i. when equippedwith a specialdigitizingsight. Accuratedigitizingis

,. crucial to obtaininggood primaryelectroninformation.

..... •i. The errors introduced during probe trace digitizing and analysis

i):- may be determinedby generatingidealizedprobe traceswith known

:. plasma propertiesand then analyzingthese traces in the usual manner.

....: Eight such idealized probe traces were generated and analyzed, covering

a wide range of plasmaconditions. The resultsindicatedthat the data

. ._...... analysiscan accuratelydistinguishprimary-to-Maxwellianelectron

.... densityratios as small as 0.2% providedthe Maxwelllan electrontemper-

•_. ture is low. The most difficulttraces to analyzeare those correspond-

.- ing to low primary-to-Maxwellian electron density ratios in a plasma

r . with a high Maxwellianelectrontemperature. This particularplasma

m_

E

_..... _i:_ condition,however,was not observedexperimentallyin the courseof

" this investigation.The results in ChaptersIll, IV and AppendixA of

this report indicate that when the primary-to-Maxwellian electron

_ : density ratio is small the Maxwellianelectrontemperatureis low and

iI

............... :" ' when the electrontemperatureis high the primary-to-Maxwellianelectron density ratio is large. These types of probe traces are accurately

analyzedwith the data reductionsystemused in this investigation.

The idealized probe trace analysis indicated that under most conditions
120

the following errors may be expected from the digitizing and curve

fitting procedures:

PlasmaPotentional ± 0.3 v

, MaxwellianElectronDensity,nM _ 5%

6i ::

MaxwellianElectronTemperature,TM _ 3%

_ _ + 15%

_-_.- PrimaryElectronDensity,np .

_ Ep + 5%

Primary ElectronEnergy, -

,

i_.. np/nM + 3%

i_'_.i The overalluncertaintyassociatedwith the plasmapropertymeasure-

_ ments are believedto be the following:

-- Plasma Potential _ l.O v

_i!_ MaxwellianElectronDensity,nM _ 20T

_ "_ MaxwellianElectronTemperature,TM + 10%

) i_ PrimaryElectronDensity,np _ 30% i i_, ,

J + IO%

PrimaryElectronEnergy,Ep

_ :'_ np/nM + 30%

,;_

_., . Measurementof Doubl7Charged Ions

Measurementsof doubly charged ions in the beam were made using an

E x B probe similarto that describedin Reference42, ErrorsassociF

ated with these measurementsinclude:the accuracyof the picoammeter

used to read the probe current,selectionof the properprobe deflection

voltageto obtain the maximum signalcorrespondingto the particular

charged specieof interest,and variationin the thrusteroperating

conditionsduring data collection.
/

.. 121

Probe current readings were madeusing a Keithly model 410A

picoammeter with an accuracy of ± 4% of full scale. Selection of the

, proper probe deflectionvoltage is a fairlydifficultthing to do

-, . accuratelybecauseof the slow responsetime of the plcoammeter

'i: (between0.4 and 12s). This is especiallytrue for currentsin the

t_

:_: lowestrangesof the ammeter. Errorsresultingfrom the uncertainty

., in probe deflectionvoltage and variationin thrusteroperatingconditions are believedto introducea maximumerror of 10% in the measured

¢ values of the singleand double ion currents. The total uncertanties

iii in the measuredvaluesof the doubly-to-singlycharged ion current

ratio, resultingfrom the sourcesof error mentionedabove, are indi-

. _._ cated in Fig. B-3 by the verticalerror bars. The horizontalerror

:. bars indicatethe uncertaintyin the correctedpropellantutilization

efficiencyas mentionedearlier.

!I •

/.
Lii_'-- •

]22

•

+m

XENON

VD - 40V

It ...... _ • 25

.. ,,,,,, .20 FLOWRATE (mA eq)

0 ,000 _/I.L

!, _ r'l 7SO

_!:_ _ ." 50°

,!,

o I_, .05

"

O. O0

0.0 .2 .4 .6 .8 ).0

G PROPELLANT UTILIZATION, "1u

o.' FigureB-3. Uncertaintyin the Doublv-to-SlnglvCharaed Ion Beam CurrentHeasurements "

_ r"

E, Qi

u_ .....
J :¢

APPENDI'_C

:i

:' NOMENCLATURE

_-i_.. Ag = Area of grids throughwhich the ion beam is extracted(m2)

,, , Bo =°

,i CO = Primaryelectronutilizationfactordefinedby Eq. 26 (A eq.)-l

,._:. Ep = PrimaryElectronEnergy (eV)

_...... ii_ e = ElectronicCharge (1.6 x lO"19 coul.)

" :_ fA = Fractionof ion currentto anode surfaces

,_ '!_ fB = Extractedion fraction

--/ fc = Fractionof i:,,currentto cathodesurfaces

'_i JA = Ion current to anode potentialsurfaces(A)

!

JB = Ion beam current (A)

, _ J_ = Singly chargedion beam current (A)

++

_ JB = Doublycharged ion beam current (A)

JC = Ion current to cathode potentialsurfaces(A)

JD = Dischargecurrent (A)

• JE = Cathode EmissionCurrent (A)

I

Jex = Total productionrate of excited neutralatoms by primary

® electrons- expressedas a current (A)

" Jimp = Acceleratorgrid ion impingementcurrent (A)

Jj = Productionrate of jth excited state expressedas a current (A)

o JL = Primaryelectroncurrent to the anode (A)

• JM = Maxwellianelectroncurrent to the anode (A)

Jp = Totalioncurrent productionrate expressedas a current (A)
*i .

_' 124

_,'

-" = Ion productionrate by primaryelectronsexpressedas a

_L JP current (A)

J+ = Productionrate of singly charged ions expressedas a

_i_ P current (A)

X_,

++

ii,: ",'_ Jp = Productionrate of doubly charged ions expressedas a

current (A)

J = Ion productionrate by Maxwellianelectronsexpressedas a

p,m current (A)

_" _e = Primaryelectroncontainmentlength (m)

Mf = Final spacecraftmass (kg)

Mg = Generatormass (kg)

._ Mi = Initialspacecraftmass (kg)

M_ = Payloadmass (kg)

Mpo = Initialpropellantmass (kg)

me = ElectronMass (kg)

_ mi = Ion mass (kg ¢

m = Thrusterpropellantflow rate (A eq.)

: • mp = Total spacecraftpropellantflow rate (kg/s)

. ne = Total electrondensity (m"3)

ni = Total ion density (m"3)

._i nM = Maxwellianelectrondensity (m"3)

n = Primaryelectrondensity (m-3) " p

no = Neutralatom density (m"3)

no = Neutralatom loss rate (A eq,)

• Pg = GeneratorPower (W)

+ = Primaryelectronrate factor for ionizationof neutral

.... Po atoms (m3/s)

P++ = Primaryelectronrate factor for double ionizationof

o neutralatoms (m3/s)
.... 125

i,

-,::: ....,, P+++ = Primaryelectronrate factor for the productionof double

_:;iii,i _.. ions from single ions (m3/s)

--iii_Y_ Q_ = Maxwellianelectronrate factor for ionizationof neutral

_,:_,., atoms (ma/s)

-":'_"::"-_,,,, Q + = Maxwellianelectronrate factor for double ionizationof

_;i_i_" neutralatoms (m3/s)

o,g_ ..... ++

"_.;i Q+ = Maxwellianelectronrate factorfor the productionof

' double ions from singleions (m3/s)

TA = Maxwellian electron temperature at anode surface (eV)

,, TM = Maxwellianelectrontemperaturein bulk plasma (eV)

,, t = Mission duration (s)

_, ,, u = Ion exhaust velocity(m/s)

_L Uex Lumpedexcitationenergy (eV)

..'_- U+ = Ionizationenergy (eV)

_._ Uj = Excitationenergy of jthexcitedstate (eV)

i,_, U_ = Lowestexcitationenergy (eV)

• VA = Anode sheath voltage (v)

_ VC = Plasmapotentialfrom which electronsemitted by the cathode

_, are acceleratedto becomeprimaryelectrons

" VD = Discharge(Anode)voltage (v)

_ VN = Net acceleratingvoltage (v)

" V+ = Screen grid supplyvoltage (v)

V. = Acceleratorgrid supplyvoltage (v)

' vb = Bohmvelocity(m/s)

,_ _ ve = Electronvelocity (m/s)

Vp = Primary electron velocity (m/s)

vo = Neutral atom velocity (m/s)

. _ = Volume of ion productionregion (m3)

x = Independent variable

y = Dependentvariable

r,
"!_. 126 t

d

J

: _ = Power plant specificmass (kg/w)

o

_,. AJD = Dischargecurrent uncertainty(A)

AJp = Ion productioncurrent uncertainty(A)

_. ' AV = Characteristicvelocity (m/s)

_,,i.::-,, AVD = Dischargevoltageuncertainty(V)

Ax = Uncertaintyof independentvariable

Ay = Uncertaintyof dependentvariable

_ cB = Average beam ion energy cost (eV/beamion)

'PI"

EM = Averageenergy of Maxwell!anelectronsleavingthe plasma

=_i_ at the anode (eV)

• ..... Cp = Average plasma ion energy cost (eV/plasmaion)

ep Baselineplasma ion energy cost (eV/plasmaion)

== _o = Average plasma ion energy cost consideringionizationand

- excitationprocessesonly (eV)

'ell nt = Thrusterelectricalefficiency

nu Propellantutilizationefficiency

-_'<_,<_" ' Oex = Total excitationcollisioncross section(m2)

",_ ._ !

:;- _ex = Total excitationcollisioncross sectionat the primary

_ electronenergy (m2)

aj = Excitationcollisioncross sectionof the jth state (m2)

- o. = Excitationcollisioncross sectionof the jth state at the

.... 3 primaryelectronenergy (m2)

!

oo = Total inelasticcollisioncross sectionat the primary

r'' electronenergy (m2)

_' : _+ = Ionizationcollisioncross section (m2)

I

d+ = Ionizationcollisioncross sectionat the primaryelectron

energy (m2)

o "

@a = Transparencyof the acceleratorgrid to neutralatoms

¢i = Transparencyof the screengrid to ions

, • ,,_". d.",_ " .. _-.-"
t, •

E

'_ r " ¢S = Transparency of the screen grid to neutral atoms

_ 0o = Effectivetransparencyof the grid systemto neutral atoms

•- < > = Averageover the entireelectronenergy distributionfunction

< >M= Average over the Maxwellian energy distribution function

ap- j-..,:

i . ..

i 'i

" i

:}