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ION THRUSTER PERFORMANCE MODEL
PREPARED FOR
LEWIS RESEARCH CENTER
NATIONAL AERONAUTICS AND SPACE ADMINIS'TRATION
GRANT NGR-06-002-112
John R. Brophy
Approved by
Paul 3 . Wi lbur
December 1984
Department o f Mechanical Engi n e e r i ng Col orado S t a t e U n i v e r s i t y
F o r t C o l l i n s , Colorado
'For sale by the National Technical Information Service, Springfield, Virginia 22161
NASAC-168 (Rev. 10-75) i
- 1. Report No.
NASA CR 174810 2. Government Accession No. 3. Recipient's Catalog No.
4. Title and Subtitle
ION THRUSTER PERFORMANCE MODEL
7. Author(s)
John R. Brophy
9. Performing Organization Name and Address
Department of Mechanical Engineering Colorado State Univers i ty For t Col 1 ins, Colorado 80523
. 12. Sponsoring Agency Name and Address
National Aeronautics and Space Administration Washington, D. C. 20546
5. Report Date Dec. 1984
6. Perform~ng Organization Code
8. Performing Organization Report No.
10. Work Unit No.
11. Contract or Grant No.
NGR-06-002-112 13. Type of Report and Period Covered Dec. 1, 1983 - Dec. 1, 1984
14. Sponsoring Agency Code
l5 Supplementary Notes Grant Monitor - W i l l iam Kerslake, NASA Lewis Research Center, Cleveland, Ohio 441 35. This repor t i s a reproduction o f the Ph.D. D isser ta t ion o f John R. Brophy. It i s submitted t o the sponsor and t o the d i s t r i b u t i o n l i s t i n t h i s form both as a presentation o f the technical mater ia l , and as an ind icat ion o f the academic program supported by the grant.
16. Abstract A model of i on thruster performance i s developed f o r high f l u x density,cusped magnetic f i e l d
thruster designs. This model i s formulated i n terms o f the average energy required t o produce an ion i n the discharge chamber plasma and the f r a c t i o n o f these ions tha t are extracted t o form the beam. The d i r e c t loss o f h igh energy (primary) electrons from the plasma t o the anode i s shown t o have a major e f f e c t on thruster performance. The model provides simple algebraic equations enabling one t o ca lcu la te the beam i o n energy cost, t he average discharge chamber plasma i o n energy cost, the primary e lec t ron density, the primary-to-Maxwell i a n electron density r a t i o and the t4axwell i a n electron temperature. Experiments ind icate tha t the model correct1 y predicts the va r ia t i on i n plasma i o n energy cost f o r changes i n propel lant gas (Ar, Kr and Xe), g r i d trans- parency t o neutral atoms, beam ext rac t ion area, discharge voltage, and discharge chamber wal l temperature.
The model and experiments ind icate t h a t th rus te r performance may be described i n terms of on ly four thruster conf igurat ion dependent parameters and two operating parameters. The model also suggests t h a t improved performance should be exh ib i ted by thruster designs which ext rac t a l a rge f r a c t i o n o f the ions produced i n the discharge chamber, which have good primary electron and neutral atom containment and which operate a t h igh propel lant flow rates. I n addit ion, i t suggests t h a t hollow cathode e f f i c iency becomes increasingly important t o the discharge chamber performance as the discharge voltage i s reduced. Final ly, the u t i l i t y o f the model i n mission analysis cal cu- l a t i o n s i s demonstrated. The model makes i t easy t o determine which changes i n thruster design o r operating parameters have the greatest e f f e c t on the payload f r a c t i o n and/or mission duration.
17. Key Words (Suggested by Author(s))
Elec t ros ta t i c Thruster Discharge Chamber Model
18. Distribution Statement
Unclassi f ied - Unlimited
19. Security Classif. (of this report)
Unclassif ied 20. Security Classif. (of this page)
Unclassif ied 21. No. of Pages
133 22 Price*
TABLE OF CONTENTS
Chapter Page
I . INTRODUCTION . . . . . . . . . . . . . . . . . . . . 1 Background . . . . . . . . . . . . . . . . . . . 2 Overview . . . . . . . . . . . . . . . . . . . . 4
I1 . I O N THRUSTER OPERATION . . . . . . . . . . . . . . . 7
I11 . THEORETICAL DEVELOPMENT' . . . . . . . . . . . . . . 14 Thruster Performance . Model . . . . . . . . . . . 14 . . . . . . . . . Assum t i ons and L imi ta t ions 14 e Beam on Energy Cost . . . . . . . . . . . . . 15
Plasma Ion Energy Cost . . . . . . . . . . . . 18 . . . . . . . . Cal cu l a t i o n o f P l asma Propert ies 27 . . . . . . . . . . . Primary E l ect ron Density 27 . . . Primary-to-Total Electron Density Rat io 31 P r i mary-to-Maxwel 1 i an Electron Density Rat io . 32 . . . . . . . Maxwellian Electron Temperature 32 Double I o n Production . . . . . . . . . . . . 34
I V . EXPERIMENTAL PROCEDURES AND RESULTS . . . . . . . . 37 . . . . . . . . . . . . . . . . . . . . Apparatus 37 . . . . . . . . . . . . . . . . . . . . Procedure 41 Experimental Resul t s . . . . . . . . . . . . . . 43
Plasma I o n Energy Cost . . . . . . . . . . . . 43 . . . . . . . . . . . . Extracted I o n F rac t i on 54 . . . . . . . . . . . . . . Plasma Propert ies 56
. . . . . . . . . . . . . . . . . V . MODEL APPLICATIONS 69 . . . . . . . . . . . . . . . . . Thruster Design 69 . . . . . . . . . . . . . . . Thruster Scal i ng 77 . . . . . . . . . . . . . . Neutral Loss Rate 77 Thruster Test ing Without Beam Ext rac t ion . . . . 78 . . . . . . . . Space Propulsion Mission Analysis 81
. . . . . . . . . . . . . . . . . . . . V I . CONCLUSIONS 92 . . . . . . . . . . . Suggestions f o r Future Work 94
. . . . . . . . . . . . . . . . . . . . . . . . . REFERENCES 96
APPENDIX A . Theoret ica l Ca lcu la t ion o f t h e * . . . . . Basel ine Plasma Ion Energy Cost. E 101 P
Tab1 e o f Contents (Continued)
& APPENDIX B . Error Analysis . . . . . . . . . . . . . . . . . . 109
APPENDIX C . Nomenclature . . . . . . . . . . . . . . . . . 123
LIST OF FIGURES
Page
. . . . . . . . . . . . . . Ion Thruster Schematic 8
Discharge Plasma Power Balance Schematic . . . . . 19
Ring Cusp Ion Source Schematic - . . . . . . . . . . . . . . . . . . Configuration I 38
Ring Cusp Ion Source Schematic - . . . . . . . . . . . . . . . . . Configuration 11 38
Plasma Ion Energy Cost Curve for . . . . . . . . . . . . . . . . . . Configuration I 44
Plasma Ion Energy Cost Curve for High Accelerator Grid Transparency t o Neutral Atoms - Configuration I . . . . . . . . 44
Plasma Ion Energy Cost Curve fo r Krypton - . . . . . . . . . . . . . . . . . . Configuration I 47
Plasma Ion Energy Cost Curve for S~iiall . . . . . . . . Diameter Ion Beam - Configuration I 47
Plasma Ion Energy Cost Curve fo r Argon . . . . . . . . . . . . . . . . . Configuration I1 50
Plasma Ion Energy Cost Curve for Xenon - . . . . . . . . . . . . . . . . . Configuration XI 50
Plasma Ion Energy Cost Variation a t Low Discharge Voltage - Configuration I1 . . . . . . . 51
Anode Electron Temperature Variation a t . . . . . Low Discharge Voltage - Configuration I1 51
Effect of Discharge Voltage on the extracted Ion Fraction - Configuration I . . . . . . . . . . 55
Effect of Propellant on the Extracted Ion . . . . . . . . . . . . Fraction - Configuration I 55
Primary Electron Density Variation fo r Argon - . . . . . . . . . . . . . . . . . Configuration I1 58
L i s t o f Figures (Continued)
Page Figure
9b Primary E l ectron Density Var iat ion for Xenon - Configurat ion I 1 . . . . . . . . . . . . . 58
P r i mary-to-Total E l ectron Density Ratio f o r Argon - Configuration I 1 . . . . . . . . . . . 61
Primary-to-Total E l ectron Density Ratio f o r Xenon - Configurat ion I 1 . . . . . . . . . . . 61
Maxwell i a n Electron ~ernperature Var ia t ion f o r Xenon - Configurat ion I1 . . . . . . . . . . . 62
Ion iza t ion Rate Factor f o r Xenon . . . . . . . . . 64
Correlat ion o f Electron Temperatures . . . . . . . 66
Doubly-to-Singly Charged I on Beam Current Results . 67
E f fec t o f fB on Performance . . . . . . . . . . . . 71
E f fec t o f Co on Performance . . . . . . . . . . . . 71
Effect of Propel 1 ant Flow Rate. on Performance f o r Small Co . . . . . . . . . . . . . . . . . . . 73
Effect o f Propel lant Flow Rate on Performance for Large Co . . . . . . . . . . . . . . . . . . . 73
Effect o f Cathode Operation on Perforniance f o r Xenon. . . . . . . . . . . . . . . . . . . . . 76
Pay1 oad Fraction, Propel 1 ant Mass Fract ion and Generator Mass Fract ion Var iat ion w i t h Propel lant U t i l i z a t i o n . . . . . . . . . . . . . . 85
Effect o f Power Plant Spec i f ic Mass (a) on Payload Fract ion . . . . . . . . . . . . . . . . 87
Effect o f Power Plant Speci f ic Mass on Optimum Propel lant U t i l i z a t i o n . . . . . . . . . . 87
Effect o f Bo on Payload Fract ion . . . . . . . . . 87
Effect of Bo on the Performance Curve and
the Optimum Propel l a n t U t i l ' i za t i on . . . . . . . . 89
Effect o f fg on Payload Fract ion . . . . . . . . . 90
L i s t of Figures (Continued)
F i gure
21 b
Page
E f f e c t o f fg on t h e Performance Curve and
the Optimum Propel1 an t U t i 1 i z a t i o n . . . . . . . . 90
Ef fec t o f Prope l lan t on Payload Frac t ion . . . . . 91
Effect o f Prope l lan t on t h e Performance Curve and the Optimum Prope l lan t U t i l i z a t i o n . . . 91
I o n i z a t i o n C o l l i s i o n Cross Sect ion f o r . . . . . . . . . . . . . . . . . . Xenon (Ref. 50) 104
Tota l E x c i t a t i o n Col l i s i o n Cross Section . . . . . . . . . . . . . . . . f o r Xenon (Ref. 49) 104
Base1 i n e Plasma I o n Energy Cost Va r ia t i on Calculated from Eq. A-7 f o r Xenon . . . . . . . . . 105
So lu t ions f o r t he Base1 i n e Plasma Ion Energy Cost f o r Xenon . . . . . . . . . . . . . . . 108
. . . . . . . I o n Source Inst rumentat ion Schemati c 11 1
Uncerta inty i n the Plasma I o n Energy . . . . . . . . . . . . . . . . . Cost Measurements 116
Uncerta inty i n t he Doubly-to-Si ng l y . . . . . . . Charged I o n Beam Current Measurements 122
LIST OF TABLES
Tab1 e Page
1 Standard Configuration Parameters . . . . . . . . . 69 *
2 Ef fec t o f V, on E . . . . . . . . . . . . . . . . 74 P
I . I N'TRODUCTI ON
Electron bombardment i o n th rus te rs have been proposed f o r both
primary and a u x i l i a r y space propulsion app l i ca t i ons fo r near ly two-and-
a-ha1 f decades. During t h i s time, th rus te r performance requirements
have var ied according t o t h e missions of cu r ren t i n t e r e s t . Also dur ing
t h i s time, the development o f t h rus te r designs capable of meeting these
requi rements has been 1 argely experimental . That i s , t h rus te r develop-
ment has general ly been accomplished by a procedure i n which the design
parameters t h a t i n f l uence t h r u s t e r performance a re physical l y var ied
u n t i l an acceptabl e conf igurat ion i s obtained.
This procedure has several inherent 1 im i ta t i ons . F i r s t of a l l , i t
can be time consuming,especially i f a l a r g e number o f parameters i s
involved. Secondly, an optimum conf igura t ion may no t be found. That i s ,
once a conf igurat ion capable o f f u l f i l 1 i n g the mission requirements i s
i den t i f i ed , the procedure i s general ly terminated even though t h i s may
not r e s u l t i n the best conf igura t ion possible. F ina l l y , changes i n the
missions of cu r ren t i n t e r e s t t o those character ized by d i f f e r e n t
t h r u s t e r requi rements necessi t a t e t h a t the i t e r a t i v e experimental pro-
cedure be repeated.
Consequently, there i s a need f o r the development of an ana ly t i ca l
model which describes the e f f e c t s o f t h r u s t e r design var iables and
operat ing parameters on t h r u s t e r performance. Such a model should, as
a minimum, be capable o f p rov id ing guidance fo r t h e i t e r a t i v e procedure
described above, and ideally would be capable of describing exactly how
a thruster should be designed to achieve a given se t of performance
requirements . Many models of discharge chamber operation have been developed
over the l a s t 24 years. A brief discussion of these models is given in
the next section. In general, the complexity of the processes taking
place i n the discharge chamber of an ion thruster together w i t h the
relat ive ease with which new thruster designs can be tested experi-
mentally has resulted in a situation i n which theoretical understanding
has 1 agged experimental developments . The objective of this research i s to improve the theoretical
understanding of ion thruster operation by providing a simple physical
model of the processes affecting thruster performance. Additional con-
s t ra in ts on this model are that i t should be easy to use, yet general
enough to be appl icabl e to a wide range of thruster configurations and
operating conditions.
Background
Analytical model i ng of el ectron bombardment ion thrusters has been
on going more or 1 ess continuously since their introduction by Kaufman
[1 ,2] i n 1960. Milder [3] provides a survey of modeling efforts made
through 1969. In this survey he concluded that, ". . .our knowledge and
understanding of the physics of these plasmas i s f a r from complete."
In addition, he concluded that the usefulness of these efforts was
1 i~iiited by ei ther the simplyfing assumptions required to make the
probleni tractable or the diff icul ty of applying 1 ess simp1 if ied models.
Kaufman [4] presents a d iscussion of t h r u s t e r technology as o f
1974 i n which the l a t e s t theor ies of t h r u s t e r operat ion are described.
He concluded, as had Mi lder e a r l i e r , t h a t t he most successful e f f o r t
t o date was t h e semiempirical approach proposed by Masek [5] i n 1969.
The theory of Masek i s semiempirical i n t h a t i t requires as i n p u t a
d e t a i l e d knowledge of t he plasma proper t ies i n s i d e the t h r u s t e r d i s -
charge chamber. These proper t ies are general l y obtained using a
Langmuir probe together w i th a data reduct ion procedure t h a t i s both
tedious and u n t i l r ecen t l y [6,7] of o n l y l i m i t e d accuracy.
Since 1974 a nurriber of a n a l y t i c a l models have been developed,
d i rec ted toward d i f f e ren t aspects of i o n t h r u s t e r discharge chamber
opera t ion [7-191. Of these, the models proposed i n references 8
through 12 a re extensions of Masek's modeling technique i n t h a t they
requ i re d e t a i l ed Langmui r probe data as inputs. References 16 through
18 present models which do no t r e q u i r e probe data, but which instead
cons is t of a complex s e t o f equations which must be solved i t e r a t i v e l y ,
by a computer. This complexity makes these models d i f f i c u l t t o apply
and 1 i m i t s t h e i r usefu l ness i n p rov id ing guide1 i nes t o improved
t h r u s t e r designs.
Discharge chamber models of i o n sources f o r neut ra l beam i n j e c t o r s
have a1 so been proposed r e c e n t l y [20-221. The opera t ion of these
sources i s inmany respects very s i m i l a r t o t h a t of i o n sources fo r
space propuls ion app l ica t ions . This s i m i l a r i t y was enhanced by the
recent swi tch i n space t h r u s t e r design [23,24] t o discharge chambers
character ized by the same high magnetic f l u x dens i ty cusped f i e l d s used
i n neut ra l beam i n j e c t o r s . References 20 through 22 present re1 a t i ve ly
simple models t h a t provide valuable i n s i g h t t o discharge chamber
operation, b u t a1 so require plasma property data as inputs. Further,
these models are not developed to the point where the performance of a
given discharge design can be cal cul ated direct ly . In summary, the situation a t the present time i s remarkably
similar to the way i t was in 1969 as described by Milder [3]. Many
addi tional models have been proposed since 1969 providing a significant
improvement in the overall understanding of discharge chamber processes,
however, most of these models s t i 11 require detai 1 ed plasma property
data as inputs . Those that do not are e i ther over simp1 i f i ed, resul t-
ing in a loss of generality and usefulness, or too complex to be
appl ied easily. Thus, there i s a need for the development of a model of
ion thruster performance that i s easy to use, does not require plasma
data as inputs, and ye t i s general enough and accurate enough to serve
as a guide1 ine for the design of improved thruster configurations.
Overview
This dissertation i s organized i n the following manner. F i rs t , a
brief review of ion thruster operation i s given in Chapter 11. In t h i s
chapter, the dominant mechanism affecting thruster performance are
identified and discussed qualitatively. Since t h i s report i s primarily
concerned with ion thruster discharge chamber operation, detailed dis-
cussions of other thruster components such as the ion accelerator
system and cathodes, e tc . , will not be given. These components will be
discussed only in regard to their effect on the operation of the main
discharge chamber.
In Chapter I11 the analytical derivation of the equations com-
pri sing the thruster performance model proposed in th i s investigation
i s presented. Th is model cons is ts o f a se t o f a lgebraic equations,
each o f which describes the behavior o f a d i f f e r e n t performance param-
e ter . For example, a s ing le equation f o r the t h r u s t e r performance
curve ( v a r i a t i o n o f beam i o n energy cos t w i t h p rope l l an t u t i l i z a t i o n
e f f i c i e n c y ) i s developed. Other equations are developed t h a t a1 low the
average values o f the discharge chamber plasma proper t ies t o be calcu-
l a t e d as fu t l c t ions o f the t h r u s t e r operat ing po in t . These plasma
p rope r t i es inc lude: the pr imary e lec t ron densi ty , the pr imary-to-
Maxwell i an e lec t ron dens i t y r a t i o , the Maxwell i an e lec t ron temperature,
and the r a t i o o f doubly- to-s ingly charged i o n cur ren ts i n the beam.
I n Chapter I V Y the experimental apparatus and procedures used t o
i nves t i ga te the v a l i d i t y o f t he proposed model are described. Experi-
mental resu l t s , comparison o f these r e s u l t s w i t h the p red i c t i ons o f the
model, and a d iscussion o f t h i s comparison are a l so given i n t h i s
chapter.
Three appl i c a t i o n s of the t h r u s t e r performance model are discussed
i n Chapter V. 'The f i r s t o f these i l l u s t r a t e s the e f f e c t o f t h r u s t e r
design and operat ing parameters on the standard performance curve. The
second a p p l i c a t i o n describes how the model can be used t o ex t rapo la te
data taken w i thou t i o n beam e x t r a c t i o n t o ob ta in the performance curve
appl i cab le t o operat ion w i t h beam ex t rac t i on . F i n a l l y , an i l l u s t r a t i o n
o f how the model can be app l i ed t o mission ana lys is ca l cu la t i ons i s
presented. I n t h i s appl i c a t i o n , the e f f e c t s o f t h r u s t e r design param-
e t e r s and p rope l l an t u t i l i z a t i o n e f f i c i e n c y on the del i ve rab le payload
f r a c t i o n are discussed f o r an e a r t h o r b i t r a i s i n g miss ion ( low ea r th
o r b i t t o geosynchronous e a r t h o r b i t ) .
The major conclusions o f t h i s invest igat ion are sumnarized in-
Chapter V I . S I un i t s are used throughout t h i s repor t w i th the excep-
t i o n t h a t energy i s f requent ly given i n un i t s o f e lec t ron vo l t s . I n
addi t ion, f o r thermal electrons the energy quant i ty kT, where k i s
Boltzmann's constant and T i s the temperature w i l l be given as eTM
where e i s the e lec t ron ic charge and TM i s the e lec t ron temperature i n
eV.
11. ION THRUSTER OPERATION
A contemporary " r i n g cusp" e l e c t r o n bombardment i o n t h r u s t e r i s
shown schemat ica l ly i n Fig. 1. Th is t h r u s t e r cons is ts o f a c y l i n d r i c a l
s tee l s t ruc tu re bounded a t one end by a c i r c u l a r s tee l p l a t e and a t t he
o t h e r end by a se t o f two c l o s e l y spaced g r i ds . The volume bounded by
t h i s s t ruc tu re i s r e f e r r e d t o as t h e discharge chamber. The two g r i d s
a t t h e end o f t he discharge chaniber have a matched ma t r i x o f holes
through which the i o n beam i s ext racted. The inner g r i d i s r e f e r r e d t o
as the screen g r i d w h i l e the ou ter one i s c a l l e d t h e accelerator g r i d .
During t h r u s t e r operation, neut ra l propel 1 ant gas i s i n jec ted i n t o
t h e discharge chamber. The p re fe r red p rope l l an t gas has, f o r t he most
pa r t , been mercury vapor because o f i t s l a r g e atomic mass, low ion iza-
t i o n energy, and i t s 1 i q u i d phase s t o r a b i l i ty . Cesium vapor has a l so
been used. However, a t t he present time, t he prope l lan ts o f most i n -
t e r e s t are t h e r a r e gases argon, krypton and xenon. I n t h i s inves t iga-
t ion,only these r a r e gases are considered. The p rope l l an t gas i s
assumed t o f i l l t h e discharge chamber un i fo rm ly dur ing t h r u s t e r
operat ion.
A hol low cathode serves as the source o f e lec t rons f o r t he d i s -
charge chamber. Hol low cathodes have replaced both r e f r a c t o r y metal
and ox ide cathodes i n i o n th rus te rs p r i m a r i l y because they e x h i b i t very
long 1 i f e t i m e s and can be res ta r ted a number o f t imes even a f t e r
exposure t o a i r (an important c a p a b i l i t y i f the t h r u s t e r i s t o be
ANODE POTENTIAL DISCHARGE CHAMBER SHELL
/
ACCELERATOR
NEUTRALIZER HOLLOW CATHODE
TO SPACECRAFT GROUND
Figure 1 . Ion Thruster Schematic
prefl i g h t tested). A detailed discussion of hollow cathodes is given
by Siegfried 1251.
The electrons emitted by the cathode in Fig. 1 are accelerated by
an electric field adjacent t o the cathode into the discharge chamber.
This electric field i s established by biasing the discharge chamber
walls (with the exception of the screen grid) 30 t o 50 volts positive
of the cathode by means of an external DC power supply (called the
di scharge or anode supply). Electrons which have undergone thi s accel-
eration are called primary electrons. The energy of these primary
electrons i s determined by the voltage difference appl ied between the
anode and cathode. The magnitude of this voltage difference is chosen
so that the collision cross section for ionization by the primary
electrons i s large while a t the same time the collision cross section
for the production of mu1 t iply charged ions by the primary electrons i s
small . A second group of electrons originates from the inelastic inter-
action of the primaries w i t h the neutral propellant atoms. These -- interactions reduce the energy of the primary electrons. In this in-
vestigation, an electron i s no longer considered a primary electron i f
i t has had at least one inelastic collision. Ionization, the inelastic
coll ision process of primary interest here, results in the release of
low energy secondary electrons. Primary electrons, t h a t have had their
energy degraded by inelastic coll i sions, and secondary electrons
released by ionization, thermal ize t o form an electron population with a
nearly Maxwell ian energy distribution (characterized by a temperature on
the order of a few eV). I t i s possible for both electron populations
t o e x i s t s imultaneously i n the discharge chamber due t o the low i n t e r -
a c t i o n r a t e between the pr imary and Maxwellian e lec t rons [26].
At t y p i c a l discharge chamber neut ra l atom d e n s i t i e s (%I O1 * ~ m - ~ ) ,
t h e i o n i z a t i o n mean f r e e pa th f o r pr imary e lec t rons i n neut ra l atoms i s
on t h e order o f meters wh i l e t y p i c a l discharge chamber dimensions are
on t h e order o f tens o f centimeters. For t h i s reason, a magnetic f i e l d
i s employed t o r e s t r i c t t he d i r e c t access o f pr imary e lec t rons t o the
anode. The magnetic f i e l d f o r the t h r u s t e r o f F ig. 1 i s created by
t h e r i n g s o f magnets o f a1 t e r n a t i n g p o l a r i t y loca ted along the back and
s ides o f t h e discharge chamber. The con f i gu ra t i on and s t rength o f t he
magnetic f i e l d has a subs tant ia l e f f e c t on t h r u s t e r operat ion and con-
sequently has been the subject o f numerous s tud ies [6,7,13,24,27-331.
Contemporary t h r u s t e r designs are operated w i t h t h e e n t i r e discharge
chamber housing (except f o r the screen g r i d ) a t anode p o t e n t i a l . Mag-
n e t i c f i e l d l i n e s a t the f i e l d cusps, thus, terminate on anode p o t e n t i a l
surfaces. T h i s a l lows e lec t rons t o be l o s t t o t h e anode by t r a v e l 1 i n g
along magnetic f i e l d l i n e s as we l l as by d i f f u s i n g across them. To
adequately r e s t r i c t t h e f l ow o f pr imary e lec t rons along the f i e l d l i n e s
t o t h e anode, magnetic f l u x dens i t i es on the order o f 0.1 t e s l a a re
requ i red .
he- discharge chamber magnetic f i e l d reduces the probab i l i ty t h a t
a pr imary e l e c t r o n w i l l be co l l ec ted by t h e anode w i thout f i r s t having
had an i n e l a s t i c c o l l i s i o n w i t h a neut ra l p rope l l an t atom. Th i s prob-
a b i l i t y i s a f u n c t i o n o f t he t h r u s t e r size, t he discharge chamber mag-
n e t i c f i e l d conf igura t ion , the cathode l o c a t i o n and the neu t ra l atom
densi ty . As the neut ra l dens i ty decreases, t h e probabi l i t y t h a t a
11
primary e lec t ron w i l l be l o s t t o the anode without having an i n e l a s t i c
c o l l i s i o n increases.
For the hypothet ical case o f a zero neut ra l atom density, t h i s
probabi l i t y i s one. That i s , a l l pr imary e lec t rons emit ted by the
cathode w i l l be co l l ec ted by the anode and none w i l l have i n e l a s t i c
c o l l i s i o n s . I n t h i s case, each primary e lec t ron w i l l , on the average,
t r a v e l a c e r t a i n distance through the discharge chamber before being
co l l ec ted by the anode. This distance i s a charac te r i s t i c o f the
th rus te r geometry, magnetic f i e l d conf igura t ion and cathode locat ion ,
and i s c a l l e d the primary e lec t ron containment length [34]. With t h i s
d e f i n i t i o n , the probabi l i t y o f pr imary e lec t ron loss t o the anode may
be expressed as a func t ion o f the r a t i o o f the primary e lec t ron con-
tainment length t o the mean f r e e path f o r primary electron-neutral atom
i n e l a s t i c c o l l i s i o n s [34]. The loss o f primary e lectrons t o the anode
cons t i t u tes a loss o f discharge energy. Consequently, i o n th rus te r
performance i s s t rong ly dependent on the probabi 1 i ty w i t h which primary
e lectrons are l o s t [34,35].
The plasma produced w i t h i n the discharge chamber w i l l t y p i c a l l y
assume a po ten t ia l a few v o l t s p o s i t i v e o f the anode. Thus, a
po ten t ia l sheath w i l l e x i s t a t a l l plasma boundaries. The magnitude o f
t h i s sheath a t cathode po ten t ia l surfaces depends p r i m a r i l y on the
magnitude o f the discharge voltage app l ied between the cathode and
anode. The sheath p o t e n t i a l a t cathode surfaces i s s u f f i c i e n t l y nega-
t i v e t o r e f l e c t a l l bu t the most energet ic e lectrons i n the t a i l o f the
Maxwell i a n d i s t r i b u t i o n . Consequently, the vast ma jo r i t y o f e lectrons
i n the plasma can leave the discharge chamber on ly a t anode po ten t ia l
surfaces.
O f t he ions produced i n the discharge chamber, some w i l l reach the
discharge chamber wa l l s. 'Those t h a t do, recombine w i t h e lectrons there
and r e t u r n t o t h e plasma as neut ra l atoms. Most o f t h e ions t h a t reach
t h e g r i d system,on the o ther hand, are accelerated by the e l e c t r i c f i e l d
between t h e screen and acce lera tor g r i d t o form the i o n beam. The
f r a c t i o n o f t he t o t a l i o n cur rent produced t h a t i s extracted i n t o the
beam i s c a l l e d the ex t rac ted ion f r a c t i o n . Because i t does l i t t l e good
t o produce ions i n the plasma o n l y t o have them recombine a t the d i s -
charge chamber wa l l s, i t i s c l e a r l y desi rab l e t o have th rus te r designs
i n which the ex t rac ted ion f r a c t i o n i s as l a rge (c lose t o one) as pos-
s i b l e [34]. The e l e c t r i c f i e l d between the g r i d s i s establ ished by
b ias ing t h e t h r u s t e r body p o s i t i v e o f ground p o t e n t i a l (on the order o f
1000 v o l t s ) and b ias ing the accelerator g r i d several hundred v o l t s nega-
t i v e o f ground p o t e n t i a l . The f i n a l v e l o c i t y o f t he ions i n the beam
i s determined by the sum o f the p o s i t i v e p o t e n t i a l appl ied t o the
t h r u s t e r cathode and the d i scharge vo l tage.
Electrons are i n jec ted i n t o the p o s i t i v e i on beam by a cathode
( c a l l e d a n e u t r a l i z e r ) pos i t ioned downstream o f the accelerator g r i d i n
order t o space charge and cur rent neut ra l i z e the beam. The negative
acce lera tor g r i d prevents these e lec t rons from backstreaming t o the
p o s i t i v e t h r u s t e r body.
During t h r u s t e r operat ion, the propel 1 ant gas i n the discharge
chamber i s o n l y p a r t i a l l y ionized, w i t h the i on dens i t y t y p i c a l l y an
order o f magnitude smaller than the. neut ra l atom density. Thus, one
might expect t h e neut ra l f l u x through the g r i d s t o be greater than the
i o n f l u x . This, however, i s no t the case since the neutra l f l u x i s
determined by free-molecular f l ow toward the g r i d s a t a temperature
governed by t h e mean discharge chamber wa l l temperature [36] (% 400K).
The ions, on the o the r hand, tend t o move toward the g r i d system a t the
B o h [37] ve loc i t y , which i s governed by the much higher eTectron
temperature ( t y p i c a l l y 5 eV o r % 58,000K). Thus, the i o n f l u x through
t h e g r i d s i s u s ~ l a l l y g reater than the neu t ra l f l u x even though the i on
dens i t y i s s u b s t a n t i a l l y smaller than the neu t ra l densi ty .
The r a t i o o f t he beam cu r ren t (assuming s i n g l y charged ions on ly )
t o the t o t a l p rope l l an t f l o w r a t e (expressed i n u n i t s o f equivalent
amperes) i s known as t h e p rope l l an t u t i l i z a t i o n ef f ic iency. For a
constant t o t a l propel 1 an t f l o w ra te , increasing the prope l lan t u t i l i za-
t i o n e f f i c i e n c y genera l l y requ i res an increase i n the average energy
requ i red t o produce a beam ion. This v a r i a t i o n o f the beam ion energy
cos t w i t h the p rope l l an t u t i l i z a t i o n i s known as a performance curve.
I t w i l l be shown i n the chapters t h a t f o l l o w t h a t the dominant mechan-
isms a f f e c t i n g t h e performance curve are t h e l o s s o f pr imary e lec t rons
t o t h e anode and t h e ex t rac ted ion f r a c t i o n .
I I I. THEORETICAL DEVELOPMENT
Thruster Performance Model
I n t h i s sect ion, a model i s developed which r e s u l t s d i r e c t l y i n a
s i ng le a lgeb ra i c equat ion f o r t he v a r i a t i o n o f t he beam i o n energy cos t
w i t h t h e p r o p e l l a n t u t i l i z a t i o n e f f i c i e n c y f o r h igh f l u x d e n s i t y cusped
magnetic f i e l d t h rus te rs . Th is i s made poss ib le by fo rmu la t i ng the
model i n terms o f t he average energy requ i red t o produce an i o n i n t he
discharge chamber plasma and t h e f r a c t i o n of ions produced which are
ex t rac ted i n t o the beam. A key fea tu re of the model i s t h a t i t
prov ides a simple technique f o r t he c a l c u l a t i o n of t h e average d i s -
charge plasma i o n energy cos t as a f u n c t i o n of t h e p r o p e l l a n t f l o w r a t e
and p r o p e l l a n t u t i l i z a t i o n e f f i c i e n c y .
Assumptions and L i m i t a t ions
The development o f t he model assumes steady s t a t e discharge
chamber operat ion. The model i s appl i c a b l e t o discharge chambers which
produce low pressure, p a r t i a l l y ion ized, o p t i c a l l y t h i n plasmas i n
which neu t ra l d e n s i t i e s a re i n t he range of 1018 t o 1019 m'3 and plasnia
d e n s i t i e s range from 1016 t o 1017 m'3. I n add i t ion , Maxwel l ian e lec -
t r o n temperatures and pr imary e l e c t r o n energies should range from
1 t o 10 eV and 30 t o 50 eV, respec t ive ly .
E lec t ron energy losses due t o e l a s t i c c o l l i s i o n s w i t h i ons o r
neu t ra l atoms a re neglected. This can be j u s t i f i e d because t h e average
energy l o s s per encounter i s p ropor t iona l t o the r a t i o o f masses, and
t h i s r a t i o i s small (Q Elec t ron energy losses r e s u l t i n g f rom
i n e l a s t i c c o l l i s i o n s w i t h ions are a lso neglected because of t h e low
i o n dens i t y r e l a t i v e t o t h e neut ra l density. This should no t cause
a s i g n i f i c a n t e r ro r , except perhaps a t very h igh p rope l l an t u t i l i z a -
t ions . Only s i n g l y charged ions a re assumed t o be produced, and t h e
ef fect of metastable atomic s ta tes on i o n product ion i s neglected.
Primary e lec t ron thermal i z a t i o n resu l t i ng from co l 1 i sions w i t h t h e
background Maxwel l i a n e l ectrons a1 so i s negl ected. Primary e l ec t ron
behavior i s assumed. t o be l i m i t e d t o e i t h e r i n e l a s t i c c o l l i s i o n s w i t h
neu t ra l atoms o r d i r e c t l o s s t o anode p o t e n t i a l surfaces. A pr imary
e l e c t r o n i s considered t o j o i n the Maxwell i a n e lec t ron populat ion a f t e r
having one i n e l a s t i c co l 1 i s i on .
Elect rons are assumed t o be constrained by the plasma sheaths so
they a r e ab le t o leave t h e plasma on ly a t anode po ten t i a l surfaces.
Ions and photons (emi t ted by the de-exc i ta t ion o f exc i t ed p rope l l an t
atoms) a re assumed t o be l o s t across a1 1 plasma boundaries. The
assumption o f a low pressure discharge imp l ies t h a t ion-e lec t ron r e -
combination should be wa l l cont ro l 1 ed, consequently, vo l ume i o n recombi - na t i on i s negl ected.
The neut ra l atom dens i t y i s assumed t o be uni form throughout t he
discharge chamber, and f r e e ~ i io lecu lar f l ow i s assumed t o apply t o the
neu t ra l atom f l u x through the acce lera tor g r i d system.
Beam I o n Energy Cost
The average energy cos t per beam i o n i s def ined as,
€6 I (JD - JB) VdJB , [eV/beam ion ]
where a l l symbols are defined i n Appendix C. The beam cu r ren t (JB) i n
Eq. 1 i s subtracted from the discharge cu r ren t (JD) SO t h a t t h e energy
t h a t goes i n t o acce lera t ing the beam ions through the discharge voltage
i s n o t charged t o the beam i o n energy cost. The numerator on the
r ight-hand-side of Eq. 1 represents the power used t o operate t h e d i s -
charge chamber.
I n a s i m i l a r manner, t he average energy expended i n c rea t ing ions
i n the discharge chamber plasma may be def ined as,
By analogy t o Eq. 1 , the "JB + JCl1 term i s subtracted from the di,s-
charge cu r ren t so t h a t the energy t h a t goes i n t o acce lera t ing thes,e
ions ou t o f t h e discharge chamber plasma i n t o the chamber wa l l s o r the
beam i s n o t included i n the plasma i o n product ion cost. Rearranging
Eq. 2 y ie lds ,
For steady s t a t e operation, the t o t a l i o n cu r ren t produced (Jp) must be
equal t o the t o t a l i o n cur rent l eav ing the plasma. For i o n t h r u s t e r
discharge chambers, ions can on ly leave the plasma by going t o cathode
po ten t ia l surfaces, anode po ten t ia l surfaces o r by being ex t rac ted i n t o
the beam. Thus, the t o t a l i o n cu r ren t produced i s given by*,
D iv id ing t h i s equation through by Jp y ie lds ,
* Equation 4 i s a statement of c o n t i n u i t y fo r t he ions.
where,
fB s JB/Jp , f = J / J and fA I JA/Jp . c - C P (6)
The f rac t ions fB, fC and fA are, i n order, t h e ex t rac ted i o n f rac t ion ,
t h e f r a c t i o n of i o n cu r ren t produced t h a t goes t o cathode p o t e n t i a l
surfaces, and the f r a c t i o n of i o n cur ren t produced t h a t goes t o anode
po ten t i a l surfaces. Using the d e f i n i t i o n s of fg and fC i n Eq. 3 along
w i t h Eq. 1 y i e l d s ,
E = cBfB - fCVD . P (7
Solv ing t h i s equat ion f o r cB g ives the r e s u l t ,
This equat ion describes the beam i o n energy cos t as a f u n c t i o n o f the
plasma i o n energy cos t ( , t h e extracted i o n f r a c t i o n ( f B ) , t h e
f r a c t i o n of i o n cu r ren t t o cathode po ten t i a l surfaces ( f C ) and t h e d i s -
charge vol tage (VD) . The f i r s t term on the r ight-hand-side o f Eq. 8 represents t h e
energy l oss associated w i t h producing ions i n the discharge chamber and
e x t r a c t i n g on l y a f r a c t i o n o f them i n t o the beam. Ions which a r e no t
ex t rac ted i n t o the beam go t o the wal ls o f the discharge chamber where
they recombine. The r e s u l t i n g atoms must then be re - i on i zed before
they can con t r i bu te t o the beam current . The f a c t o r l / fB may be
i n t e r p r e t e d as the average number o f times t h a t a beam i o n undergoes
i o n i z a t i o n from a neutra l s t a t e before being extracted i n t o the beam.
The second term on the r ight-hand-s ide of Eq. 8 represents t h e
energy wasted i n acce lera t ing plasma ions i n t o i n t e r i o r cathode po-
t e n t i a l surfaces. This process i s undesi rable because i t resu l t s i n
both a discharge energy l oss and i n t h e s p u t t e r eros ion of these
surfaces.
To generate performance curves us ing Eq. 8, one must be abl e t o
specify t h e behavior o f each of t he terms on t h e r ight-hand-s ide o f
t h i s equat ion as a funct ion of t he p r o p e l l a n t u t i l i z a t i o n ef f ic iency.
Plasma I o n Energy Cost
The plasma i o n energy cos t parameter ( E ) appearing i n Eq. 8 and P
defined by Eq. 2 r e f l e c t s a1 1 mechanisms o f energy l oss from the d i s -
charge chamber except f o r the acce lera t ion o f ions o u t o f the plasma
through t h e discharge vol tage. Spec i f i ca l ly , E inc ludes energy losses P
due t o t h e fo l low ing phenomena: d i r e c t pr imary e l e c t r o n l o s s t o the
anode, Maxwell i a n e lec t ron co l 1 e c t i o n by the anode, e x c i t a t i o n o f
neut ra l atoms, e x c i t a t i o n o f i o n i c s ta tes (which w i l l be neglected) and
hol low cathode operat ion. To der ive an expression fo r t he plasma i o n
energy c o s t as a funct ion of t he p rope l l an t u t i l i z a t i o n , a power
balance i s made on the discharge chamber plasma represented i n Fig. 2.
The boundary o f t he con t ro l volume f o r t h i s power balance i s defined by
the plasma sheath edge. The pr imary e lec t rons a re assumed t o be accel-
erated from a cathode reg ion plasma p o t e n t i a l t h a t i s VC v o l t s above
cathode p o t e n t i a l t o t he po ten t i a l o f the bul k plasma which i s assumed
t o be near t h a t o f the anode. I n add i t ion , i f i t i s assumed t h a t on ly
the discharge power supply i s used t o sus ta in the discharge, then the
r a t e a t which energy i s suppl ied t o the d ischarge chamber plasma by the
JE (VD-VC) PRIMARY
JM"M MAXWELLIAN
ELECTRON ELECTRONS LOST
INPUT TO ANODE
JL (VD-Vc) PRIMARY ELECTRONS LOST TO
Figure 2. Disharge Plasma Power Balance Schematic
primary electrons i s given by JE(VD - V C ) The "missing11 power JEVC i s
used t o operate the hollow cathode.
Energy i s l o s t from the plasma p r ima r i l y by the f l u x o f four types
o f energy ca r r i e r s across the plasma boundaries: ions, photons (emitted
by de-excitat ion o f exci ted propel l a n t atoms), Maxwell i an electrons,
and primary electrons. The ions and photons are l o s t t o a l l i n t e r i o r
thruster surfaces whereas the Maxwell ian and primary electrons are
assumed t o be l o s t t o the anode surfaces only. I n steady state, the
ra te of energy supplied t o the plasma must be equal t o the ra te a t
which i t i s l os t , thus,
where the sumnation i s over the set o f exc i ted neutral states.
Div id ing Eq. 9 by the ion production current (Jp) and recognizing t ha t
the emission current (JE) re la ted t o the discharge current by *,
al lows Eq. 9 t o be w r i t t en as,
where Eq. 2 has been used. The r a t e a t which the j t h exci ted state i s
produced (expressed as a current) i s given by,
* Equation 10 i s a statement o f con t i nu i t y f o r the electrons.
length), o: i s the t o t a l i ne l as t i c c o l l i s i o n cross'section for
electrons a t the primary e lec t ron energy and no i s the neutral atom
density. Combining Eqs. 14, 16 and 17 y ie lds ,
The current o f Maxwellian electrons t o the anode may be given as the
sum o f the secondary electrons 1 i berated i n the ion izat ion process and
the thermal ized primary electrons, thus,
Using Eqs. 17 and 19 i n 18 y ie lds ,
(20)
F ina l ly , using Eqs. 2 and 10 and solv ing Eq. 20 f o r E resu l t s in , P
I
The fac to r 1-e -uOnOee may be in terpreted as the probabil i t y t ha t a
primary electron w i l l have an i n e l a s t i c c o l l i s i o n before being co l lec ted
by the anode. This i s analogous t o the " f a s t neutron non-leakage prob-
a b i l i t y " used i n nuclear reactor physics [40].
The neutral densi ty (no) may be expressed i n terms o f the pro-
pel l a n t f l ow ra te and propel lant u t i l i za t i on by equating the r a t e a t
which propel lant enters and leaves the discharge chamber, i .e.,
where and 6 are i n u n i t s o f equivalent amperes". The neu t ra l f low
r a t e from the t h r u s t e r may be expressed using the theory f o r f r e e
molecular f l o w through a sharp-edged o r f i c e [39],
' = ' n e v ~ 4 no o o o g 0
Combining Eqs. 22 and 23 y ie lds ,
where the prope l lan t u t i l i z a t i o n defined by nu t Js/m was used. Thus,
Eq. 21 may be w r i t t e n as,
where,
and,
Equation 25 i s a very simple r e l a t i o n s h i p which can be used t o ca l cu la te
the plasnia i on energy cost as a func t i on o f the p rope l l an t u t i l i z a t i o n .
Experimental resu l t s , which w i l l be presented i n Chapter' I V , i n d i c a t e *
t h a t under many condi t ions the parameters Co and E may be taken t o P be independent o f the propel 1 ant u t il i z a t i o n .
Subs t i t u t i on o f Eq. 25 i n t o Eq. 8 y i e l d s the fo l l ow ing s i n g l e
equation descr ib ing the performance o f a g iven t h r u s t e r design,
t Equation 22 i s a statement o f c o n t i n u i t y f o r the prope l lan t .
For design purposes, the parameters fB and fC i n a d d i t i o n t o Co and *
E may be taken t o be independent of the u t i l i z a t i o n and flow ra te . P
These parameters do, however, depend s t rong ly on the t h r u s t e r design.
Indeed, these f o u r parameters determine the performance of a given
t h r u s t e r design.
The quan t i t y Co r e f l e c t s the degree t o which primary e lectrons
i n t e r a c t w i t h neut ra l atoms. Thus, i t i s re fe r red t o as the primary
e lec t ron u t i l i z a t i o n factor. This fac to r depends on the q u a l i t y of the
pr imary e lec t ron containment ( through r e the qua1 i ty of the contain-
ment of neut ra l atoms (through A , po and vo) and the prope l lan t gas 9
proper t ies (through u: and vo). Recall t h a t t h e primary e lec t ron con-
tainment l eng th re may be in te rp re ted as t h e average distance a primary
e l ect ron would t rave l i n the discharge chan~ber before being l o s t t o the
anode--assuming i t had no i n e l a s t i c c o l l i s ions . Magnetic f i e 1 ds i n a1 1
discharge chamber designs serve the func t i on of increasing t h i s 1 ength.
A1 though an e f fec t i ve means o f determining re remains t o be developed,
i t i s be1 ieved t h a t t h i s parameter i s a func t i on p r i m a r i l y of the
th rus te r geometry, magnetic f i e l d con f igu ra t i on and cathode locat ion .
Equation 25 suggests t h a t the plasma i o n energy cos t should,
through the fac tor Co, be a func t i on o f :
1. The propel lant , which determines the t o t a l i n e l a s t i c c o l l i s i o n
cross sec t ion (0:) and atomic mass (which a f fec ts the thermal neutra l
ve loc i t y , vo) . 2. The wa l l temperature, which a f fec ts the neutra l ve loc i t y .
3. The transparency of t he g r i d s t o neu t ra l s (4,).
4. The area through which t h e beam i s ex t rac ted ( A ). 9
5. The discharge voltage, which determines the pr imary e l ec t ron
energy and, thus, e f fec ts the value of t he cross sec t i on (a;).
The base1 i n e plasma i o n energy cos t (E;), defined by Eq. 27,
depends on a number of energy l o s s mechanisms inc lud ing : t he r e l a t i v e
amount of energy expended i n exc i t a t i ons compared t o i o n i z a t i o n o f
neu t ra l atoms through E,, the average energy o f t he Maxwell i a n e lec t rons
which leave the plasma ( E ~ ) and the ef f ic iency w i t h which the hol low
cathode operates.
The cathode ef f ic iency i s re f l ec ted i n t h e value of VC which
represents the plasma p o t e n t i a l from which the e lec t rons a r e suppl ied.
I n e f f i c i e n t cathode opera t ion r e s u l t s i n h igh values of VC and corres-
pondingly poor ove ra l l t h r u s t e r performance. For thermionic cathodes
VC = 0, however, add i t iona l heater power must be suppl ied t o e f fec t i t s
operat ion. For th rus ters equipped w i t h a cathode po le piece/baff l e
assembly, VC should be taken as t h e plasma p o t e n t i a l i n t he cathode
discharge reg ion ( i .e., t he reg ion between the cathode and t h e ba f f le ) .
I n t h i s case, t he power represented by JEVC goes i n t o both the opera t ion
o f t he hol low cathode and the opera t ion o f the cathode reg ion discharge.
The r e s u l t i n g h igh values of VC, i n t h i s case, would be expected t o
produce poorer ove ra l l t h r u s t e r performance. El i m i na t i on of t he
separate cathode discharge reg ion shoul d, therefore, improve the per-
formance.
The parameter E~ , defined by Eq. 15, i n general i s a func t ion of
t he e n t i r e e lec t ron energy d i s t r i b u t i o n i n c l u d i n g t h e pr imary e lect rons.
The parameter E* (which contains E,, see Eq. 27) i s , thus, a l so a P
funct ion of the e lec t ron energy d i s t r i b u t i o n . For p l asmas character ized
by a Maxwell i a n p lus monoenergetic e lec t ron energy d i s t r i b u t i o n , a
simple method fo r t he c a l c u l a t i o n of E* may be derived. Sample cal cu- P
l a t i o n s us ing t h i s method, along w i t h the appropr ia te c o l l i s i o n cross
sec t ion data, a re g iven i n Appendix A. The experimental r e s u l t s g iven
i n Chapter I V i n d i c a t e tha t , under many condi t ions, the basel ine plasma
i o n energy cos t (E*) may be taken t o be a constant. The ca l cu la t i ons P
g iven i n Appendix A demonstrate t h a t t h i s experimental observat ion i s
predic ted by the model . Although the model cannot y e t be used t o p r e d i c t the performance
of corr~pletely new t h r u s t e r designs, i t provides a c l ear physical p i c -
t u r e of the phenomena a f fec t i ng t h e performance. Equation 25 descr i bes
the plasma i o n energy cos t i n terms of the l oss of pr imary e lect rons t o
the anode. A t h igh values of the neut ra l dens i ty parameter, t h e neut ra l
dens i ty i n the discharge chamber i s large, and the p r o b a b i l i t y i s h igh
t h a t a1 1 the primary e lect rons w i l l undergo i n e l a s t i c c o l l i s i o n s w i t h
neut ra l atoms and none w i l l be l o s t d i r e c t l y t o t h e anode. I n t h i s
case, the discharge chamber w i l l be producing ions f o r the minimum o r
base1 i n e plasma i o n energy cost, E* P*
As the beam cu r ren t i s increased ( a t a constant p rope l l an t f low
r a t e ) , t he propel 1 a n t u t i 1 i z a t i o n increases causing t h e neut ra l dens i t y
parameter (and thus the neut ra l dens i ty ) t o decrease (see Eq. 24). The
decrease i n neutra l densi ty increases the 1 i kel i hood of a primary e lec-
t r o n reaching the anode w i thout having an i n e l a s t i c c o l l i s i o n . This
d i r e c t l oss o f primary e lec t ron energy increases the o v e r a l l plasma i o n
energy cos t according t o Eq. 25 and consequently increases the beam i o n
energy cos t accordi ng t o Eq. 28. The shape of t he performance curve i s
l a r g e l y determined by t h i s d i r e c t l o s s of primary e lectrons. Thus, any
design change which decreases the 1 i kel i hood o f the d i r e c t l o s s o f
primary e lec t rons (w i thout decreasing the extracted i o n f r a c t i o n )
should improve the t h r u s t e r ' s performance. The probabi 1 i ty of d i r e c t
primary e lec t ron l o s s i s determined through the parameter Co. This
parameter may be increased by e i t h e r increasing re, which makes i t
harder fo r pr-imary e lectrons t o escape the plasma, o r by making i t
harder fo r neutra l atoms t o escape the discharge chamber.
Cal cul a t i o n o f Pl asma Propert ies
The fo l l o w i rrg analys is provides a s e t o f very simple a1 gebraic
equations which a l l ow one t o ca l cu la te the values of the fo l lowing d i s -
charge chamber p l asma propert ies: the average primary e lec t ron densi ty ,
the average pr imary- to - to ta l e l ec t ron densi ty r a t i o , t he average Max-
we1 1 i a n e lec t ron temperature and the average r a t i o of t he doubly-to-
s ing l y charged i o n cur rent i n the beam. Each of these q u a n t i t i e s may be
ca lcu la ted as a func t i on o f the propel 1 ant f low r a t e and p rope l l an t
u t i l i z a t i o n us ing on ly in format ion t h a t would normal l y be ava i l able i n
the t h r u s t e r design phase.
Primary E lec t ron Density
This ana lys is i s based on the recogn i t ion t h a t a l l of t he energy
suppl ied t o the discharge chamber plasma i s suppl ied by the primary
e lectrons. Thus, c o r r e c t l y accounting f o r the behavior of t he primary
e l ectrons i s essenti a1 i n determi n i ng the average p l asma proper t ies .
It i s assumed t h a t a primary e lec t ron can do only one of two th ings.
It can e i t h e r have an i n e l a s t i c c o l l i s i o n w i t h a neut ra l p rope l l an t
atom or i t can be lost directly to the anode. For the case where the
primary electron has an inelastic collision with a neutral atom, i t
will produce either an ion or an excited neutral state. After such an
inel ast ic coll ision, the energy of the primary electron i s degraded and
i t i s assumed that i t i s subsequently thermalized into the Maxwellian
electron population. The rate a t which primary electrons have in-
elastic coll isions with neutral atoms i s given by the difference be-
tween the rate a t which they are supplied by the cathode ( JE) and the
rate a t which they are lost directly to the anode ( JL) , i . e m ,
where 56 is the ion current produced by primary electrons and JAx i s
the primary electron induced production rate of excited neutral states
expressed as a current. The ion current produced by primary electrons
i s given by, I
~ , , = n n e o ; + v 0 P P , (30)
where o; i s the ionization collision cross section a t the primary elec-
tron energy. Similarly, the rate of production of excited state atoms
induced by primaries i s given by,
n n e v + l o 1 J;X o p p j
where a : is the collision cross section for the j t h excited state at J
the primary electron energy. The fraction of the primary electron-
neutral atom inelastic coll isions that produce ions i s given by,
Note t h a t t he sum 0; + 1 u I s j u s t t he t o t a l i n e l a s t i c c o l l i s i o n cross J I
sec t ion f o r pr imary e l ectron-neutral atom c o l l i s i ons uo . Therefore,
Eq. 33 may be w r i t t e n as,
M u l t i p l y i n g Eq. 34 by the r a t e a t which pr imary e lect ron-neutra l atom
c o l l i s i o n s occur (Eq. 29) y i e l d s the r a t e a t which ions are produced by
pr imary electrons, i .e., I
The term i n parentheses i n Eq. 35 may be w r i t t e n as,
The q u a n t i t y JL/JE i s the f r a c t i o n o f the i n p u t primary e l e c t r o n cu r ren t
l o s t d i r e c t l y t o the anode and i s given by the su rv i va l equat ion (Eq. 17
w r i t t e n i n a s l i g h t l y d i f f e r e n t form),
Combining Eqs. 2 and 10 y i e l d s the f o l l o w i n g expression f o r the cathode
emission current .
Combining Eqs. 25, 35, 36, 37 and 38 y i e l d s the f o l l o w i n g expression
f o r t he i o n cu r ren t produced by pr imary e lectrons,
D iv id ing both sides o f t h i s equation by Jp y i e l d s an expression fo r
the r a t i o o f i o n cu r ren t produced by pr imar ies t o the t o t a l i o n cur rent
produced,
I
Since the r a t i o Jp/Jp cannot be greater than one, Eq. 40 provides a * .
theore t i ca l l i m i t f o r the maximum value of cp , I .e.,
The t o t a l i o n cu r ren t produced may be expressed i n terms o f the beam
cur rent by using the d e f i n i t i o n o f the ex t rac ted i o n f r a c t i o n (Eq. 6),
I n add i t ion , using the d e f i n i t i o n o f the propel 1 an t u t i l i z a t i o n the
beam cur rent may be w r i t t e n as,
Combining Eqs. 39, 42 and 43 y ie lds ,
F i n a l l y , equating Eq. 30 t o Eq. 44 and so l v ing f o r t h e pr imary e l e c t r o n
dens i t y (np) y i e l d s ,
where Eq. 24 was used f o r t he neut ra l densi ty . Equation 45 provides an
expression fo r the average primary e lec t ron dens i t y as a func t ion o f
t h e p rope l l an t u t i l i z a t i o n . Remarkably, t h i s expression does n o t
depend on e i t h e r t he p rope l l an t mass flow r a t e o r t h e pr imary e l e c t r o n
u t i l i z a t i o n fac to r , both o f which cancel led ou t i n t he ana lys is . The
term i n the square brackets should be roughly a constant f o r a g iven
t h r u s t e r design, propel 1 an t gas and discharge vo l tage. A reasonabl e
est imate f o r each o f the terms i n the square bracket should be poss ib le
fo r a t h r u s t e r being designed (assuming a method i s developed fo r t he
determi na t i on o f the extracted i o n f r a c t i o n ) .
Pr imary-to-Total E lect ron Densi ty Rat io
Assuming quasi -neutra l i ty , t h e average i o n dens i ty (ni ) i s equal
t o the average t o t a l e lec t ron dens i t y (ni = n + nM). I n add i t ion , t he P
average i o n dens i t y i s r e l a t e d t o the beam cu r ren t by [41].
So lv ing t h i s equat ion fo r ni y i e l d s ,
where mi i s t he transparency o f t h e screen g r i d t o ions and vb i s the
Bohm v e l o c i t y . D iv id ing Eq. 45 by 47 and using Eq. 43 y i e l d s the
f o l l owing expression for t h e average r a t i o o f pr imary- to- tota l e lec t ron
dens i t y ,
This equation ind ica tes t h a t . the average pr imary- to - to ta l e lec t ron
dens i ty r a t i o should be a funct ion of t he neutra l dens i ty parameter,
1 n u As was the case fo r Eq. 45, the combination of parameters i n
the square brackets o f Eq. 48 should be roughly a constant f o r a given
t h r u s t e r design, propel 1 a n t and discharge voltage. Equation 48 a1 so
i nd i ca tes tha t , a l l e lse being equal, increasing the extracted i o n
f r a c t i o n should decrease the pr imary- to- tota l e lec t ron dens i ty r a t i o .
Primary-to-Maxwell i a n Electron Density Rat io
Once t h e pr imary- to- tota l e l ect ron densi t y r a t i o i s ca l cu l ated ,
using Eq. 48 the primary-to-Maxwel l i a n e lec t ron dens i ty r a t i o (n /n ) P M
may be ca lcu la ted from the fo l lowing equation,
Maxwell i an El ectron Temperature
The t o t a l i o n cur rent produced (Jp) i s t he sum o f the i o n cu r ren t
produced by primary e lectrons (56) and t h a t produced by Maxwellian
e lectrons (Jp M) . 9
(50)
The i o n current produced by primary electrons i s given by Eq. 30 and
tha t produced by Maxwell i an electrons i s given by,
where the quant i ty <U+Ve>M represents the product o f the electron
ve loc i t y and ion iza t ion col 1 i s i o n cross sect ion averaged over the
Maxwell i a n e l ectron energy d is t r ibu t ion . This quant i ty i s the Max-
we l l i an electron- r a t e fac to r f o r the production o f ions and i s given +
the symbol Q0 , i.e.,
Combining Eqs. 50 t o 52 y ie lds ,
+ Using Eqs. 42 and 43 i n Eq. 53 and solv ing f o r Q0 gives,
Subst i tu t ing f o r the neutral density using Eq. 24 y ie lds ,
The Maxwellian electron density, appearing i n Eq. 55, may be wr i t ten as,
Subst i tu t ing t h i s i n t o Eq. 55 y ie lds ,
Equation 45 provides an expression f o r the pr imary e lec t ron density.
Using t h i s i n Eq. 57 y ie lds ,
The primary-to-Maxwell i an e lec t ron dens i ty r a t i o may be found using
Eqs. 48 and 49. Subs t i t u t i ng these equations i n t o Eq. 58 y i e l d s the
f o l l o w i n g expression f o r the average Maxwellian e lec t ron r a t e factor
Equation 59 provides an expression f o r the Maxwell i a n e lec t ron ioniza-
t i o n r a t e f a c t o r as a funct ion o f t he neutra l dens i t y parameter,
1 . Once t h i s r a t e f a c t o r has been calculated, i t can be compared
t o a tabu la t i on o f r a t e fac to rs versus e lec t ron temperature f o r the
g iven propel 1 ant t o determine the appropr iate Maxwell i a n e lec t ron
temperature. Equation 59, ind ica tes t h a t the average Maxwell i an elec-
t r o n temperature should increase w i t h decreasing val ues o f the neutra l
dens i ty parameter.
Double Ion' Production
I n the previous analysis, the product ion o f doubly charged ipns was
neglected. I n t h i s section, a simple formulat ion f o r the r a t i o o f
d o u b l y - t o - s i ~ g l y charged ion cur rent i n the beam i s developed. The pro-
duct ion r a t e o f s ing l y charged ions expressed as a cur rent i s given by,
where Q: i s the Maxwell i a n r a t e fac to r f o r the product ion o f s ing le
ions from ground s ta te neutra l atoms and P: i s the primary e lec t ron
r a t e fac to r a1 so f o r the production o f s ing le ions from ground s ta te
neutra ls . S imi la r ly , the production r a t e o f doubly charged ions
expressed as a cur ren t i s given by,
++ ++ where n+ i s the s ing l y charged i on density, Qo and Po are the Max-
we l l i an and p r imarye lec t ron r a t e fac to rs f o r double i on production ++ ++
from ground s ta te neutrals, respect ively, and Q+ and P+ are the cor-
responding r a t e fac to rs f o r double i on production from ground s ta te ions.
D iv id ing Eq. 61 by 60 yie lds,
++ Assuming t h a t the doubly charged extracted i on f r a c t i o n ( f B ) i s equal
t o the s ing l y charged extracted i on f r a c t i o n ( f i ) imp1 i e s tha t ,
++ + where J B /JB i s the average r a t i o o f doubly-to-singly charged i on cur-
r e n t i n the beam. C?mbinlng Eqs. 62 and 63 y ie ids ,
n 2(QH+ 4 Pi+) ?+ (Q: + 3 p:) JF o . n,,, - - -
no +
'h + + $ 2 + 2 p+
Qo nM o Q o + % o
The s i n g l y charged ion dens i ty (n+) may be conservat ive ly approximated
by assum-ing the beam cur rent i s made up e n t i r e l y o f s i n g l y charged ions,
thus,
n = n = B + 0 . 6 e ~ ~ A ~ ) ~
I n add i t ion , using Eq. 43 f o r the beam cur rent and Eq. 24 f o r the
neut ra l dens i ty a l lows Eq. 64 t o be w r i t t e n as,
The f i r s t term on the r ight-hand-side of Eq. 66 represents the produc-
t i o n o f double ions from ground s t a t e neut ra l atoms. This term depends
on t h e t h r u s t e r geometry on ly through the primary-to-Maxwell i a n e lec-
t r o n dens i t y r a t i o ( n /n ) which i s given by Eqs. 48 and 49 and the P M
Maxwell i a n e lec t ron temperature (Eq. 59). The second term i n Eq. 66
represents double i o n product ion from s i n g l y charged ions. This term
i s s t rong ly dependent on the p rope l l an t u t i l i z a t i o n . The r a t e fac to rs ++ ++ +
Qo and Q+ may be determined once the r a t e fac tor Q0 i s ca lcu la ted /'
from Eq. 59.
I V . EXPERIMENTAL PROCEDURES AND RESULTS
Apparatus
For t h i s inves t iga t ion , the i on sources shown schematical ly i n
Figs. 3a and b were designed and b u i l t . Each of these sources normally
produces a 12 cm dia. i on beam and provides the c a p a b i l i t y f o r measuring
the d i s t r i b u t i o n o f ion cur rents t o the beam, screen g r i d and i n t e r n a l
t h r u s t e r surfaces, ( w i t h the exception o f the anode).
The magnetic f i e l d f o r the :experimental i on source i n Fig. 3a i s
establ ished through the use o f an electromagnet located on the upstream
c e n t e r l i n e o f the discharge chamber and a number o f 1.9 cm x 1.3 cm x
0.5 cm samarium coba l t permanent magnets. These permanent magnets are
arranged end-to-end t o form r i n g magnets of a l t e r n a t e p o l a r i t y i n the
manner suggested by Fig. 3a. The f l u x dens i ty a t t he surface o f the
magnets i s 0.27T and the magnets are attached t o the s tee l discharge
chamber housing by t h e i r own magnetic a t t r a c t i o n . This arrangement
a1 lows the i o n source magnetic f i e l d conf igura t ion t o be a1 te red q u i c k l y
and e a s i l y by simply adding, removing o r changing the p o s i t i o n o f the
magnets. Although many d i f f e r e n t c o n f i g ~ ~ r a t i o n s were tested, the r e s u l t s
obtained were a l l s im i la r , thus, on l y those obtained using the conf igura-
t i o n s shown i n Figs. 3a and 3b w i l l be presented. For these conf igura-
t ions, the upstream magnet r i n g i s covered w i t h a s t r i p o f 0.13 mill t h i c k
s tee l insu la ted from the magnets themselves by a s t r i p o f 0.25 mm t h i c k
f l e x i b l e mica. This i s done so t h a t the surface o f t h i s s t r i p can be
Figure 3a. Ring Cusp I o n Source Sche~iiatic - Conf igurat ion I
Figure 3b. Ring Cusp I o n Source Schematic - Conf igurat ion I 1
maintained a t anode po ten t i a l w h i l e the r e s t of t he t h r u s t e r body i s
biased negat ive of cathode po ten t i a l . The downstream magnet r i n g i s
uncovered. The magnetic f l ux dens i ty a t the sur face of t he e lec t ro -
magnet fo r t he conf igurat ion of Fig. 3a can be adjusted from zero t o
approximately 0.2 T by ad jus t ing the magnet cu r ren t from zero t o
124 A.
The main discharge chamber cathodes, f o r both conf igurat ions, con - s i s t o f seven 0.25 cm dia. tungsten wires connected i n para l l e l and support-
ed by two support posts t h a t are e l e c t r i c a l l y is01 ated from the th rus te r
body. Each cathode w i re i s approximately 2.8 cm long so the t o t a l
cathode l eng th exposed t o the plasma i s about 19.6 cm. These seven
sho r t wi res i n paral l e l a re used t o minimize the vo l tage drop across
t h e cathode. A vo l tage drop l e s s than 3 v a t t he maximum heater cur-
r e n t was achieved w i t h t h i s system. The small vo l tage drop across the
cathoqe r e s u l t s i n a primary e l e c t r o n energy d i s t r i b u t i o n t h a t more
c l o s e l y resembes the monoenergetic d i s t r i b u t i o n produced by a hol low
cathode. The cathode wires were heated us ing d i r e c t cur ren ts i n the
range 6 t o 8 A per wire. Tests on t h e conf igura t ion o f F ig. 3a, were
conducted us ing argon, krypton and xenon propel 1 ants. Discharge v o l t -
ages were var ied from 30 t o 50 v f o r argon and 20 t o 40 v fo r krypton
and xenon. The discharge cu r ren t was adjusted through the range of
0.5 t o 5 A by c o n t r o l l i n g the heat ing cu r ren t through the r e f r a c t o r y
cathode wires.
Two i o n acce lera tor systems were used i n t h i s study. The f i r s t
accel ekator system consisted o f a s e t o f dished small ho le accelerator
g r i d s (SHAG) w i t h a c o l d g r i d separat ion of 0.75 mni and screen and
acce lera tor g r i d physical open area f rac t i ons of 0.68 and 0.30,
respect ive ly . The second system consis ted of a s e t of dished l a r g e
hole acce lera tor g r i d s (LHAG) w i t h a c o l d g r i d separat ion o f 0.75 mn,
and screen and acce lera tor g r i d physical open area f rac t i ons o f 0.68
and 0.57, respec t i ve l y . Both accel e r a t o r systems were normal l y masked
t o produce a 12 cm diameter beam. One ser ies o f t e s t was conducted,
however, w i t h the SHAG s e t masked t o produce a 6 cm diameter beam. For
t he 12 cm d ia . beam tes ts , f low ra tes f o r both argon and krypton were
var ied from 500 t o 1500 mA eq. and f o r xenon from 250 t o 1000 mA eq.
For t h e 6 cm d ia . beam tes t , t h e f l o w ra tes were var ied from 125 t o 500
mA eq.
'The con f i gu ra t i on o f Fig. 3b i s s i m i l a r t o t h a t o f Fig. 3a except
t h a t t he electromagnet assembly has been replaced by a permanent magnet
attached t o t h e s tee l backplate. I n add i t ion , t h e source of Fig. 3b i s
2 cm sho r te r than the one i n Fig. 3a and the upstream magnet r i n g i s
pos i t ioned o n l y 2.5 cm from the downstream end r a t h e r than 6 cm.
This source was equipped w i t h two Langmuir probe assembl ies . 'The
f i r s t probe consis ted o f a 0.76 mm d ia. tantalum wire, 4.32 mm long,
supported from a quar tz tube i n s u l a t o r . This probe was pos i t ioned
a1 ong the t h r u s t e r cen te r l i ne approximately ha1 f way between the cathode
assen~bly and t h e screen g r i d as suggested i n Fig. 3b. The second probe
was a square piece of s tee l , 1 cm on a s ide and 0.127 mm t h i c k t h a t was
pos i t ioned on the surface o f t h e upstream magnetic r i n g . This probe
was i nsu la ted from the magnet r i n g w i t h a piece o f 0.13 mm t h i c k
f l e x i b l e mica.
Measurements of t he doubly and s i n g l y charged i o n beam components -b -h
were made on the t h r u s t e r cen te r l i ne us ing a c o l l imat ing E x B momentum
analyzer. De ta i l s o f t h e use of t h i s probe are given elsewhere [42].
Tests were conducted on the conf igura t ion o f F ig. 3b us ing both
argon and xenon prope l lan ts . Argon p rope l l an t f low ra tes were var ied
over the range 350 t o 1500 mA eq. a t discharge voltages o f 30 t o 50 v.
Xenon p rope l l an t f low r a t e s were va r ied over t he range 250 t o 1000 mA
eq. a t discharge vol tages o f 20 t o 40 v. Discharge cur ren ts f o r both
propel 1 an ts were var ied over -0.5 t o 4.0 A.
A1 1 tes ts were conducted i n a 1.2 m d ia. x 4.6 m long vaculJm t e s t
f a c i l i t y . I nd i ca ted tank pressures ranged from .J 2 x 1 Om6 Tor r w i t h
no flow t o .J 3 x 1 0-5 Torr a t a f l ow r a t e of 1500 mA eq. of krypton.
Procedure
The f o l l o w i n g s e t of experiments, conducted us ing the t h r u s t e r
c o n f i g ~ ~ r a t i o n o f F ig . 3a, was designed t o t e s t t h e s u i t a b i l i t y of t he
mod.el developed i n Chapter I 1 1 t o p r e d i c t the func t iona l dependence o f
t h e plasma i o n energy cost, E , on t h e neut ra l dens i ty parameter, P
( I - ~ ) . The model p red i c t s t h a t the plasma i o n energy cos t should
behave according t o Eq. 25 w i t h the pr imary e l e c t r o n u t i l i z a t i o n f a c t o r
(Co) and the base1 i n e plasma i o n energy cos t (E* ) given by Eqs. 26 and P
27, respect ive ly . The value of E may be determined experimental ly P
through the use of Eq. 2, which i s repeated here f o r convenience,
Note t h a t the power used t o operate t h e thermionic cathodes i s no t i n -
cluded i n Eq. 2. I n order t o use Eq. 2, one must be able t o measure
each o f t h e parameters on the r ight-hand-s ide of t he equation. Measure-
ment o f t h e discharge current, discharge vol tage and beam cu r ren t i s
s t r a i g h t forward, and was accompl ished using the exper imnta l i n s t r u -
mentation described i n Appendix B.
To measure t h e i o n cur rents JC and Jp, t he t h r u s t e r con f igu ra t i on
o f F ig . 3a was operated w i t h on ly the upstream magnet r i ng a t anode
po ten t ia l . A l l o the r i n t e r i o r discharge chamber surfaces ( w i t h t h e
exception of t he cathode support posts) were biased appmximately 30 v
negative o f cathode po ten t ia l t o repel t he discharge chamber e lectrons.
A t t h i s bias, t h e cur rent t o these surfaces consists only of t he incom-
i n g i o n cur rent . I f t h e i o n cur rent t o the cathode support posts and
cathode wires i s neglected, then the i o n cur rent measured i n the manner
described above i s equal t o JC.
To determine Jp, i t i s noted t h a t the t o t a l i o n product ion cu r ren t
i s t h e sum o f t h e i on . currents leav ing the. plasma as given by Eq. 4.
Since the anode i s exposed t o the plasma on ly a t a magnetic f i e l d cusp,
the e f f e c t i v e area f o r i o n l oss t o t h i s surface i s expected t o be less
than the physical area [43-461. Rough ca lcu la t ions i nd i ca te t h a t t he
i o n cur rent t o the anode w i t h t h i s conf igura t ion should be less than a
few precent o f t he t o t a l product ion current . Thus, Jp may be approxi-
mated as the sum o f the i o n cur rents t o the beam ( inc lud ing the impinge-
ment cur rent ) and t o the negat ive ly biased discharge chamber surfaces
( inc lud ing the screen g r i d ) .
Complete sets o f data cons is t i ng o f the beam current , propel 1 an t
f l ow rate, p rope l lan t u t i l i z a t i o n and t o t a l i o n product ion cu r ren t were
co l 1 ected over t h e range o f operat ing condi t ions discussed ea r l i e r w i t h
the electromagnetic cur rent held constant a t 57 A f o r the th rus te r of
Fig. 3a. A t each cond i t ion tested, the th rus te r was operated a t flow
ra tes o f 500, 750, 1000 and 1500 mA eq. f o r argon and krypton
propel lan ts , and 250, 500, 750 and 1000 mA eq. f o r xenon. A t each f l ow
ra te , t he discharge vo l tage was held constant w h i l e the discharge cur-
r e n t was var ied. Increasing t h e discharge current , increases the beam
cu r ren t and propel 1 a n t u t i 1 i z a t i o n , thus causing the neut ra l dens i t y
parameter ;(l-nu) t o decrease. By opera t i ng i n t h i s manner, the f u l l
range o f neut ra l dens i t y parameters from c lose t o zero t o near ly 1500
niA eq. could be inves t iga ted . Thus, the plasma i o n energy cos t was de-
termined from Eq. 2 f o r a wide range of opera t ing condi t ions. The ex-
t r a c t e d i o n f r a c t i o n , fg, def ined by Eq. 6, was a lso computed from
measured i o n cur ren ts over t h i s same range o f condi t ions,
A second s e t o f experiments was performed us ing the t h r u s t e r
con f i gu ra t i on of Fig. 3b. These t e s t s were designed t o i nves t i ga te the
abi 1 i t y o f t he 'model t o p r e d i c t t h e behavior o f var ious plasma prop-
e r t i e s w i t h va r ia t i ons i n t h r u s t e r ope ra t i ng condi t ions , The procedure
f o r these t e s t s was the same as f o r t he previous s e t w i t h the a d d i t i o n
t h a t Langmui r probe measurements of t he plasma proper t ies on t h e th rus te r
center1 i n e and a t t h e anode sur face were made a t each operat ing p o i n t
tested. The r a t i o of doubly t o s i n g l y charged i o n beam cur ren ts along
t h e t h r u s t e r c e n t e r l i n e were a lso measured a t each operat ing condi t ion, -f -f
us ing E x B probe.
Experimental Results
Plasma I o n Energy Cost
Measurement o f the plasma i o n energy cost , f o r operat ion o f t he
t h r u s t e r con f i gu ra t i on o f F ig . 3a w i t h argon p rope l l an t a t a 50 v
discharge over t h e range o f neut ra l f l ow ra tes from 500 t o 1500 mA eq.,
y i e l d e d the r e s u l t s shown i n Fig. 4a. Here, t h e measured values of t he
ARGON
VD ' 50. Ov
SHAG OPTICS
l2cm dio. BEAM E; ' 57. OoV
C, - 3.lCA oq)-'
FLOW RATE (mA eq) 0 1500 0 1000
0 1 I
0.0 . 2 - 4 - 6 .B 1.0 1.2 1.4
NEUTRAL OENSITY PARAMETER. i (1 -7,). (A eq. )
Figure 4a. Plasma Ion Energy Cost Curve - Configuration I
ARGON
VD - 50. ov
LHAG OPTICS
l2cm dio. BEAM
FLOW RATE (mA eq) 0 1500 0 1000 A 750
n 0
0 1 I
0.0 - 2 .4 . 6 . 0 1.0 1.2 1.4
NEUTRAL DENS I TY PARAMETER. i (1 -7,). (A oq. )
Figure 4b. Plasma Ion Energy Cost Curve f o r High Acce le ra tor Grid Transparency t o Neutral Atoms - Configurat ion I
plasma i o n energy cos t (E ) a re p l o t t e d as a f m c t i o n of the neutra l P
dens i ty parameter m(l -nu), as suggested by Eq. 25. The sol i d 1 i n e i n
Fig. 4a i s t he curve g iven by Eq. 25 when the perameters Co and E* have P
been selected t o g i v e t h e best f it t o the data. The parameter E* was P
taken t o be the value of E measured a t 1 arge values o f the neutra l P
dens i ty parameter. The j u s t i f i c a t i o n f o r t h i s se lec t i on can be under-
stood by consider ing Eq. 25, which shows t h a t h e n Co ;(l-11,) i s large,
t he exponential term i s small compared t o u n i t y and one obta i ns *
E = E . Having establ ished the value of the base1 i n e plasma i o n P P
energy cost, the value of the pr imary e l e c t r o n u t i l i z a t i o n f a c t o r (Co)
i s var ied u n t i l t h e bes t fit i s obtained. The agreement between the
func t i ona l , form o f Eq. 25 and the experi~i iental data i s seen t o be q u i t e
good. This i nd i ca tes t h a t the parameters Co and E* may be taken t o be P
i ndependent o f t he neu t ra l densi ty parameter.
A value o f Co = 3.1 (A eq.)'' which i s appl icable t o the i o n source
of Fig. 3a opera t ing a t t he cond i t ions def ined i n the legend f o r F ig. 4a
has now been establ ished. New values o f primary e lec t ron u t i l i z a t i o n
fac to r , appl i c a b l e t o o the r opera t ing condit ions, may now be ca l cu l ated
from t h i s val ue us ing Eq. 26. For example, changi ng g r i d se ts from SHAG
t o LHAG should change the value of Co through the parameter (o which i s
t he e f fec t ive transparency of - t he g r i d s t o neutral atoms. This e f fec-
t i v e transparency parameter may be ca lcu la ted f o r each g r i d se t accord-
i n g t o the equation,
where $ and $a a re the modified transparencies f o r t he screen and
accel e ra to r grids, respect ive ly . These modi f ied transparencies may be
ca lcu la ted as t h e physical open area f r a c t i o n o f a g r i d t imes t h e
appropriate Clausing f a c t o r [47]. For t h e two g r i d se ts used i n t h i s
study,
Thus, t h e new value o f t h e primary e lec t ron u t i l i z a t i o n f a c t o r appl i-
cable t o t h e LHAG op t i cs w i t h a l l o the r cond i t ions he ld constant i s
given by,
which yields (Co)new = 1.8 (A eq. ) - I
The measured values of s , obtained under the same se t o f condi- P
t i o n s de f ined i n the legend o f F ig. 4a,except f o r the change i n o p t i c s
f rom SHAG t o LHAG, y ie lded the r e s u l t s shown by the data p o i n t s i n
Fig. 4b. The s o l i d l i n e i s t he p red i c t i on of t he model based on t h e
value o f Co ca lcu la ted from Eq. 68. The value o f the base l ine plasma
i o n energy c o s t was held constant a t c* = 57 eV since changing the P
o p t i c s should no t a f f e c t t h i s parameter. The degree o f agreement be-
tween t h e data po in t s and the curve i n Fig. 4b shows c l e a r l y t h a t t he
model c o r r e c t l y p red i c t s the v a r i a t i o n i n the plasma ion energy c o s t
w i t h the neu t ra l dens i ty parameter when the g r i d system transparency t o
neut ra l s i s changed.
The same procedure o f c a l c u l a t i n g a new value o f t h e pr imary
e l e c t r o n u t i l i z a t i o n f a c t o r from the o l d value according t o Eq. 26 was
followed f o r t h e ana lys is o f the data d isp layed i n Figs. 5a and 5b.
For the data i n Fig. 5a, the t h r u s t e r was operated w i t h krypton
- 0.0 . 2 - 4 . 6 . 8 1.0 1.2 1.4
NEUTRAL OENSI TY PARAMETER. i (1-7,). <A oq. )
100 n c 0 " 160 o E a 0 -I 140 a 5 0 " 120
a " 100
c' U)
8 80 r U
8 so Z W
z ,O 40
Figure 5a. Plasma Ion Energy Cost Curve f o r Krypton - Configuration I
- KRYPTON Vo - 40. Ov - SHAG OPTICS 12cm dio. BEAM
-
- FLOW RATE <mA oq) 0 1500
- 1000 A 750 0 so0
-
-
-
KRYPTON Vo - 40. Ov SHAG OPTICS 6cm dlo. BEAM
- 51. OoV FLOW RATE <mA oq)
C, - 22.8 (A oq) -I 0 500 0 375
NEUTRAL OENSI TY PARAMETER. <I -7,). <A oq.
Figure 5b. Plasma Ion Energy Cost Curve f o r Small Diameter Ion Beam - Configuration I
p r o p e l l a n t and SHAG o p t i c s a t a discharge vol tage o f 40 v. The pr imary
e l e c t r o n u t i l i z a t i o n f a c t o r corresponding t o t h i s opera t ing c o n d i t i o n
was ca lcu la ted us ing the value o f Co obtained from Fig. 4a together
w i t h Eq. 26 i n the form,
I n t h i s case, both the change i n p rope l l an t p rope r t i es and the change
i n discharge vol tage must be accounted f o r . That i s , (u:)~, i s the
t o t a l i n e l a s t i c c o l l i s i o n cross sec t ion f o r 40 eV pr imary e lec t ron -
krypton atom c o l l i s ions , whereas refers, i n t h i s case, t o 50 eV
pr imary electron-argon atom c o l l i s i o n s . The cross sec t ion data needed
i n t h i s equat ion were obtained from de Heer, e t . a1 [48]. The new
value o f t he pr imary e lec t ron u t i l i z a t i o n f a c t o r ca l cu la ted from
Eq. 69 was Co = 5.7 (A eq.)- I and the corresponding p r e d i c t i o n o f t he
model i s compared t o the measured values i n Fig. 5a. A new value o f *
the base1 i n e plasma i o n energy cos t ( E ) was a1 so requ i red t o f i t the P
data o f F ig. 5a since t h i s parameter i s a f u n c t i o n o f bo th the d i s -
charge vo l tage and the p rope l l an t type. This new value was selected
i n t h e manner suggested prev ious ly as the measured plasma i o n energy
cos t a t h igh neut ra l dens i ty l eve l s . These data i n d i c a t e t h a t the
model works as we l l f o r operat ion w i t h krypton p r o p e l l a n t as i t does
f o r argon.
From Eq. 26, i t i s seen t h a t the pr imary e l e c t r o n u t i l i z a t i o n
f a c t o r depends i nve rse l y on the area o f the g r i d s through which the
beam i s ext racted, A . This area may be var ied w i thou t changing the 9
discharge chamber diameter by masking down the screen g r i d t o produce
i o n beams of d i f f e r e n t cross sec t iona l areas. I n t h i s case, t he screen
g r i d f o r the SHAG o p t i c s s e t was masked down from a beam diameter o f
12cm t o oneof 6cm. This four fo ld reduc t ion i n beam area should pro-
duce a corresponding four f o l d increase i n pr imary e lec t ron u t i l i z a t i o n
fac to r . Taking Co f o r Fig. 5a and mu1 t i p l y i n g by fou r y i e l d s the new
value o f Co = 22.8 (A eq.)-I . The model p red i c t i on fo r t he plasma i o n
energy cost versus neut ra l dens i ty parameter curve, obtained us ing t h i s
value o f Coy i s compared t o the measured values o f these q u a n t i t i e s i n
Fig. 5b. Remarkably, t he agreement between the model and the experiment
i s excel l en t . S im i l a r agreement was obtained f o r operat ion w i t h argon
us ing the masked down g r i d set.
Measurements o f t he plasma i o n energy cos t v a r i a t i o n w i t h the
neut ra l dens i ty parameter were a1 so made on the t h r u s t e r conf igura t ion
of Fig. 3b. Some of t he r e s u l t s obta ined w i t h t h i s conf igura t ion are
shown i n Figs. 6a, 6b, 7a and 7b. The data i n Fig. 6a was obtained
w i t h argon prope l lan t , a discharge vo l tage of 50 v and a range o f pro-
p e l l a n t f l ow r a t e s from 350 t o 1500 mA eq. Excel1 en t agreement between *
the model and the experimental data was obtained by tak ing e = 59 eV P
and Co = 4.0 (A eq.)-I . The value o f E* = 59 eV, obtained w i t h t h i s P
conf igurat ion, agrees we1 1 w i t h the value of E* = 57 eV obtained w i t h P
the o the r conf igura t ion f o r opera t ion w i t h argon a t a discharge vol tage
of 50 v. The s l i g h t l y h igher va lue of e* f o r the conf igura t ion of P
Fig. 3b i s be1 ieved t o be the r e s u l t o f the shor te r discharge chamber
length causing a s l i g h t l y h igher f r a c t i o n of i o n cu r ren t t o go t o the
anode surface and cathode support posts.
ARGON vg - 50. Ov SHAG OPTICS 12cn dia. BEAM - 50. oov
FLOW RATE (mA eq)
NEUTRAL DENSI TY PARAMETER. i (1-1,). <A eq. )
F igure 6a. Plasma I o n Energy Cost Curve - Conf igura t ion I 1
XENON VD - 40. Ov SHAG OPTICS 12ca dio. BEAM
w a FLOW RATE <nA eq) - 44. OoV
0 1000 0 710
tA4 100 A So0
0 1 I
0.0 . 2 . 4 .6 . a 1.0 1.2 1.4
NEUTRAL DENSI TY PARAMETER. i (1-7,). <A eq. )
F igure 6b. Plasma I o n Energy Cost Curve f o r Xenon - Conf iaura t ion I 1
ARGON Vg - 30. Ov SHAG OPTICS 12cm dia. BEAM
FLOW RATE <mA oq) 0 1500 0 1000 A 750 0 so0 v 350
NEUTRAL DENS1 TY PARAMETER. i (1 -l,). (A eq. )
F igure 7a. Plasma I o n Energy Cost Va r ia t i on a t Low Discharge Voltage - Conf igura t ion I1
ARGON FLOW RATE (mA eq) Vo - 30. Ov 0 1500
0 1000 0 6 .d SHAG OPTICS A 750
12cm dia. BEAM 0 500 V 350
NEUTRAL OENS I TY PARAMETER. 6 Cl-7,). (A eq. )
F igure 7b. Anode E lec t ron Temperature Va r ia t i on a t Low Discharge Voltage - Conf igura t ion I1
From t h e value of Co i n Fig. 6a, a new value of Co app l icab le t o
t h i s conf igura t ion opera t ing on xenon p rope l l an t a t a discharge vol tage
o f 40 v may be ca l cu la ted us ing Eq. 26. The resu l t s o f t h i s c a l c u l a-
t i o n , together w i t h t h e experimental resu l t s , a re given i n Fig. 6b,
For these data, the neut ra l dens i ty parameter has been corrected f o r
presence of doubly charged ions i n the beam. Again the value o f C* was P
chosen i n t h e manner described above. Clear ly , the model agrees we l l
w i t h the measured results, . Equations 25 and 26 have now been demon-
s t r a t e d t o describe c o r r e c t l y the effects of v a r i a t i o n s i n the plasma
i o n energy c o s t r e s u l t i n g from changes i n p rope l l an t f l o w ra te , pro-
pel 1 a n t u t i 1 i z a t i o n , e f fec t ive g r i d transparency t o neutra l atoms, beam
e x t r a c t i o n area, discharge voltage, and p rope l l an t gas (Ar, Kr and Xe).
I n add i t ion , o the r experiments i n d i c a t e t h a t the model c o r r e c t l y pre-
d i c t s t he change i n t h r u s t e r performance r e s u l t i n g from a change i n the
e f f e c t i v e discharge chamber wa l l temperature [36]. The model has a l so
been shown t o work we1 1 on 1 i ne cusp t h r u s t e r geometries [35].
For opera t ion a t low discharge voltages, however, the s i t u a t i o n i s
q u i t e d i f f e r e n t as seen i n Fig. 7a. The data i n t h i s f i gu re were taken
us ing argon prope l lan t , a discharge vol tage of 30 v and the t h r u s t e r
c o n f i g u r a t i o n o f Fig. 3b. Here, a systematic di f ference i n the data
taken a t d i f f e r e n t p rope l l an t f low ra tes i s observed. C lear ly , a s i n g l e
equat ion such as Eq. 25 i s no t su f f i c i en t t o exp la in t h i s behavior if
Co and E* a r e taken t o be independent of the p rope l l an t f low r a t e and P
p rope l l an t u t i l i z a t i o n . This systematic d i f fe rence of the plasma i o n
energy c o s t curves, measured a t d i f f e r e n t f l o w rates, i s be l ieved t o
r e s u l t f r o m changes i n the base1 i n e plasma i o n energy cos t (E*) which P
occur a t low discharge voltages.
These changes, i n turn, r e s u l t from the dependence of the base1 i ne
plasma i o n energy c o s t on the average energy o f a Maxwell i a n e lec t ron
c o l l e c t e d by t h e anode, EM (see Eq. 27). The value of EM, however,
depends on the Maxwell ian e lec t ron temperature a t the anode. Measured
values of the e lec t ron temperature, made using the Langmui r probe po-
s i t i o n e d on the surface of the anode magnet r i ng , a re shown i n Fig. 7b.
These data were taken under operat ing condi t ions i d e n t i c a l t o those
used f o r t h e c o l l e c t i o n of t he data i n Fig. 7a. 'The data i n Fig. 7b
c l e a r l y i n d i c a t e a systematic d i f f e rence i n e lec t ron temperature a t t h e
anode fo r d i f f e ren t p rope l l an t flow rates. For a given value o f the
neut ra l dens i t y parameter, these data show t h a t . the e lec t ron temperature
a t the anode increases w i t h increasing p rope l l an t f low ra te . Higher
e lec t ron temperatures a t the anode correspond t o h igher values o f E* P
because o f the increased loss o f energy from the plasma i n the form o f
Maxwell ian e lectrons. Because the r a t i o €M/VD appears i n the equation
fo r E* , changes i n EM become increas ing ly important a t low discharge P
vol tages . The systematic d i f fe rence i n the curves of e lec t ron temperature
versus neutra l dens i ty parameter observed i n Fig. 7b i s no t predicted
by the model (as given by Eq. 59). Thus, some add i t iona l mechanism f o r
the t r a n s f e r o f energy from the primary e lectrons t o the Maxwell i a n
electrons, o the r than t h a t considered i n the model, must become i m -
por tan t a t low discharge voltages. This mechanism i s be1 ieved t o be
the d i r e c t thermal i z a t i o n o f the pr imary e lectrons as the r e s u l t of
e l ectron-el ectron col 1 i s ions . 'The col 1 i s i o n cross sec t ion f o r primary-
Maxwellian e lec t ron c o l l i s i o n s increases r a p i d l y w i t h decreasing
pr imary e lec t ron energy. This i s cons is ten t w i th the observat ion t h a t
t h e separat ion between the e lec t ron temperature curves taken a t d i f -
f e r e n t p rope l l an t f low ra tes (such as those i n Fig. 7b) increases w i t h
decreasing pr imary e lec t ron energy ( i .e., discharge vol tage) .
Extracted I o n Frac t ion
Aside from t h e plasma i o n energy cost, the o the r parameter which
has a major a f f e c t on t h r u s t e r performance i s t h e extracted i o n f rac-
t i o n , fB. It i s o f l i t t l e use t o c rea te ions e f f i c i e n t l y i n t h e d i s -
charge chamber i f the f rac t i on of these ions ex t rac ted i n t o t h e beam
i s small. The extracted i o n f r a c t i o n , which i s def ined i n Eq. 6 as t h e
r a t i o of the beam cu r ren t t o the t o t a l i o n c u r r e n t produced, was mea-
sured f o r both t h r u s t e r conf igura t ions over t he range o f ope ra t i ng
cond i t ions discussed e a r l i e r .
The r e s u l t i n g extracted i o n f r a c t i o n data, obtained on t h e
t h r u s t e r con f i gu ra t i on o f Fig. 3a, a re shown i n Figs. 8a and b. I n
Fig. 8a, the value of fB are g iven versus the neu t ra l densl t y parameter
f o r opera t ion w i t h argon p rope l l an t a t discharge vol tages o f 30 and
50 v over a range of p rope l lan t f low ra tes from 500 t o 1500 mA eq.
These data i n d i c a t e t h a t t he ex t rac ted i o n f rac t ion , wh i l e being r e l a -
t i v e l y independent o f neut ra l density, does tend t o be s l i g h t l y g rea te r
a t lower discharge voltages. Data taken under t h e same condi t ions, b u t
a t a 40 v discharge voltage, fa1 1 between the two curves shown i n F ig. 8a.
The ex t rac ted i o n f rac t ions f o r argon p rope l l an t a r e compared t o
those f o r krypton prope l lan t a t a discharge vo l tage of 40 v i n Fig. 8b,
The values of fB f o r argon are seen t o be genera l l y s l i g h t l y h igher
than those f o r krypton.
z OPEN SYMBOLS V0=50V h ImAeq) CLOSED SYMBOLS V0=30V 0 500
0.3 750
9 ARGON A 1000 0.2 SHAG OPTICS 0 1500
0 1 I I I , I I
0 0 . 2 0 . 4 0 . 6 0.8 1.0 1.2 1.4 NEUTRAL DENSITY PARAMETER, in(l-l)u). [ ~ e g ]
Figure 8a. E f f e c t o f Discharge Yo1 tage on the Extracted I o n F rac t i on - Conf igurat ion I
OPEN SYMBOLS ARGON rir(mAeq)
CLOSED SYMBOLS KRYPTON 0 5 0 0 0 750
SHAG OPTICS A 1000 0 1500
1 I t I
0 .2 0 .4 0.6 0 .8 1.0 1.2 1.4
NEUTRAL DENSITY PARAMETER, h ( I - q U ) , [ k q ]
Figure 8b. E f f e c t o f Prope l lan t on the Extracted Ion Frac t ion - Configuration I
The data i n Figs. 8a and 8b ind ica te t h a t the extracted i o n
f rac t ion i s re1 a t i vel y i ndependent of the neutral densi t y parameter.
Other data not presented, ind ica te t h a t the extracted i o n f rac t ion
sometimes decreases w i t h increasing beam currents f o r ce r t a i n thruster
configurations. This decrease i n fB r esu l t s from a decrease i n the
ef fect ive transparency of the screen g r i d t o ions as the plasma density
i s increased. However, the f rac t ion of the t o t a l i on cur rent produced
t ha t i s d i rec ted toward the gr ids remains constant.
Ea r l i e r studies [32,33] ind ica te t ha t the extracted i o n f ract ion,
o r the f rac t ion of ions di rected toward the grids, i s s t rong ly dependent
on the magnetic f i e l d configurat ion and thruster geometry. The data i n
t h i s repor t i nd ica te t ha t fB i s also a weak function of the discharge
voltage and propel lant gas, but not a function of the neutral dens1 t y
parameter. Consequently, a simple design approach would be t o take fB
t o be a constant f o r a given discharge chamber design (magnetic f i e l d
conf igurat ion) , propel 1 ant and discharge vol tage. Unfortunately, a
method f o r ca lcu la t ing the value of the extracted i on f rac t ion from the
above design data remains t o be developed.
P l asma Propert i es
Equation 45 provides a simple expression f o r the primary electron
density as a function o f the propel lant u t i l i z a t i o n . As mentioned
ear l i e r , the combination of parameters ins ide the square brackets i n
t h i s equation may be taken t o be roughly a constant f o r a given pro-
pel 1 ant, discharge vol tage and thruster configuration. The value of
t h i s constant, appropriate f o r argon propel 1 ant, a discharge vol tage of
50 v and the thruster conf igurat ion of Fig. 3b, was calculated t o be
1.0 x 1 0 l 0 cm-3 . Thus, Eq. 45 becomes, i n t h i s case,
The value of E* , requ i red fo r t h i s ca l cu la t i on , was taken t o be the P
measured value g iven i n Fig. 6a. The value of s* could, however, have P
been ca l cu la ted us ing the method described i n Appendix A.
The volume of t h e i o n product ion reg ion (W), requ i red by Eq. 45,
was determined from a computer drawn magnetic f i e l d map o f the d i s -
charge chamber. This map was created by measuring the magnetic f l u x
dens i t y and d i r e c t i o n a t r e g u l a r l y spaced po in ts i n the discharge
chamber. The i o n product ion volume, def ined by t h i s map, was taken t o
be the volume i n which the magnetic f l u x dens i ty was 0.005 Tesla o r
1 ess.
The ex t rac ted i o n f r a c t i o n ( fB) was taken t o be i t s measured value
obta ined i n t e s t s w i t h argon propel 1 ant, a discharge vo l tage o f 50 v
and t h e t h r u s t e r c o n f i g u r a t i o n o f F ig. 3b. The transparency o f t h e
g r i d s t o neu t ra l atoms (4,) was ca l cu la ted from Eq. 66. F i n a l l y , t he
neut ra l atom v e l o c i t y (vo) was ca l cu la ted based on an assumed e f f e c t i v e
wa l l temperature o f 400 K.
Comparison o f t h e ca l cu l ated values of the pr imary e lec t ron dens i t y
f rom Eq. 45 ( i n t he form o f Eq. 70) w i t h the measured values i s g iven
i n Fig. 9a. The measured pr imary e l e c t r o n d e n s i t i e s were obta ined us ing
t h e Langmuir probe pos i t ioned along t h e c e n t e r l i n e o f t h e discharge
chamber. Equations 45 and 70, however, ca l cu la te e s s e n t i a l l y an average
pr imary e l e c t r o n desni t y . Further, s ince t h e center1 i ne primary e l ec-
t r o n dens i t y i s expected t o be h igher than the average value, t h e
x10 9
ARGON vo - 5ov
FLOW RATE (mA oq) 0 1500
1000 A 750 0 500 V 350
Eq. 45
0 1 I I I L I 1 I I I I
0.0 .2 .4 - 6 . 0 1.0 . PROPELLANT UTILIZATION, Tu
Figure 9a. or imary E lec t ron Densi ty Y a r i a t i o n f o r Argon - Conf igurat ion I1
45 r xlOB XENON
cP 30 - FLOW RATE CmA oq)
I- 0 1000 25- 750
W 0
A 500 0 250
g 20- E U W
d 15 -
0.0 . 2 .4 .6 . 0 1.0 PROPELLANT UTILIZATION. Tu
Figure 9b. Primary E lec t ron Density V a r i a t i o n f o r Xenon - Conf igurat ion I 1
excel 1 ent agreement between the ca l cu l ated and measured val ues i n
Fig. 9a i s somewhat misleading. 'This f i gu re c l e a r l y ind ica tes , however,
t h a t Eq. 45 has the co r rec t funct ional form and t h a t the pr imary elec-
t r o n dens i ty i s indeed r e l a t i v e l y independent o f the p rope l l an t f low
r a t e as predicted by the model.
A s i ~ n i 1 a r comparison between ca lcu la ted and measured primary
e lec t ron dens i t i es i s given i n Fig. 9b fo r operat ion of t he same
t h r u s t e r conf igura t ion , bu t w i th xenon prope l lan t and a discharge v o l t -
age o f 40 v. Again, t he same concludsions drawn from Fig. 9a a re a lso
appl i c a b l e t o t h r u s t e r opera t ion on xenon as ind ica ted i n F ig. 9b. The
observat ion t h a t the ca l cu l ated average val ues a re somewhat greater
than the measured center1 i ne values probably r e s u l t s from the use of *
the measured value o f E = 44 eV i n Eq. 45 r a t h e r than the calculated P
value o f E* = 36 eV given i n Appendix A. P
An expression f o r t he r a t i o o f pr imary- to- tota l e lec t ron dens i ty
i s given by Eq. 48. The combination o f parameters i n the square brac-
kets i n t h i s equation i s approximately a constant f o r a g iven propel lant,
discharge vol tage and t h r u s t e r conf igura t ion . For operat ion w i t h argon
a t V,, = 50 and the con f igu ra t i on of Fig. 3b, Eq. 48 becomes,
I n making t h i s ca l cu la t i on , the Bohm v e l o c i t y (vb) was ca lcu la ted based
on an e lec t ron temperature o f 4 eV. The value o f t he screen g r i d
transparency t o ions (mi ) was determi ned experimental 1 y by measuri ng
the i o n cu r ren t t o the screen g r i d and t o the beam. The transparency
i s then the r a t i o o f the beam cu r ren t t o the sum of the screen g r i d and
beam currents. The measured screen g r i d transparency t o ions was
approximately 0.8 compared t o the physical open area f r a c t i o n o f the
screen g r i d which was approximately 0.68.
Comparison of Eq. 48 ( i n the form of Eq. 71 ) w i t h the experimental-
l y measured values i s given i n Fig. 10a. This f i g u r e indicates t h a t f o r
a g iven discharge voltage, p rope l lan t and t h r u s t e r conf igura t ion the
pr imary- to - to ta l e lec t ron dens i ty r a t i o i s on l y a func t i on o f t he neu-
t r a l dens i ty parameter. I n addi t ion, i t ind ica tes t h a t Eq. 71 has the
c o r r e c t func t iona l form f o r the v a r i a t i o n o f t h i s r a t i o w i t h the
neut ra l dens i ty parameter.
For operat ion o f t h e same t h r u s t e r con f igu ra t i on w i t h xenon a t
VD = 40 v, the r e s u l t s shown i n Fig. l o b were obtained. The s o l i d 1 i n e
i n t h i s f i g u r e corresponds t o Eq. 48, where the constant was ca lcu la ted
using values o f the parameters i n Eq. 48 t h a t a re appropr iate f o r xenon
propel 1 a n t and a discharge vol tage o f 40 v. Figures 10a and b i n d i c a t e
t h a t Eq. 48 c o r r e c t l y accounts f o r changes i n the propel 1 an t gas, t he
discharge vol tage, t h e neut ra l dens i ty parameter and the propel 1 ant
f l o w ra te .
Comparison o f ca lcu la ted and measured Maxwell i a n e lec t ron tempera-
tu res i s given i n F ig . 1 1 . The ca l cu l ated e l ect ron temperatures were
obtained using Eq. 59. For operat ion w i t h xenon a t VD = 40 and the
t h r u s t e r con f igu ra t i on o f Fig. 3b, Eq. 59 becomes,
where the appropr iate cross sec t ion data were ob ta i ned from References
49 and 50. Equation 72 gives the value o f the Maxwel l i a n i o n i z a t i o n
r a t e f a c t o r f o r t he product ion o f s i n g l e ions fro111 neut ra l atoms as a
FLOW RATE (mA aq>
0 1000 [3 750 A 500 0 250
XENON VD = 40. Ov
0 0.0 . 2 . 4 .6 . 8 1.0 1.2 1.4
NEUTRAL DENS1 TY PARAMETER. i (1-7,). (A eq. >
F igure 11. Maxwel l ian E lec t ron Temperature V a r i a t i o n f o r Xenon - Conf igura t ion I1
f unc t i on o f t h e neut ra l dens i t y parameter. The r a t e f a c t o r Q: i s a
func t i on o f t h e Maxwell i a n e l ect ron temperature, and i s g iven by,
t Using Eq. 73, t he r a t e fac tor Qo may be p l o t t e d as a funct ion o f the
e l e c t r o n temperature as shown i n F ig. 12.
To determine the e l e c t r o n temperature as a funct ion of the neut ra l
dens i t y parameter, the f o l l o w i n g procedure i s used. F i r s t , t he value t
of t he r a t e fac tor Qo i s cal-culated f o r a g iven value o f m(1-su) us ing
Eq. 72. This value of Q: i s then used t o en te r the curve i n Fig. 12
from which the corresponding e lec t ron temperature i s determined. Re-
peat i ng t h i s procedure generates the curve of Maxwell i an e lec t ron
temperature versus neut ra l dens i t y parameter shown as the sol i d l i n e
i n F ig. 11. The agreement between the ca l cu la ted and measured e lec t ron
temperatures i s considered t o be q u i t e good.
I n a d d i t i o n t o the Langmuir probe pos i t ioned on the t h r u s t e r
center1 ine, a second probe was posi t ioned on t h e upstream magnet r i n g .
This probe was used t o measure the temperature of t he Maxwell i a n e lec-
t rons and the energy of the pr imary e lec t rons reaching the anode.
Because t h e probe was pos i t ioned i n a reg ion o f very h igh magnetic f l u x
densi ty , t h e e lec t ron temperatures determined from the probe t races
obtained- w i t h t h i s probe a r e probably o n l y equal t o t he e lec t ron tem-
perature r e s u l t i n g f r o m motion along the magnetic f i e l d 1 ines. The
e lec t ron temperature corresponding t o motion normal t o the f ie1 d 1 ines
MAXWELL I AN' ELECTRON TEMPERATURE. TM. (aV)
Figure 12. I o n i z a t i o n Rate Factor f o r Xenon
may o r may not be the same. For s i m p l i c i t y , these temperatures are
assumed, i n t h i s work, t o be the same.
The measured e lec t ron temperatures a t the anode were found t o be
1 ess than the measured center1 i n e temperatures by approximately a
fac tor of 2/3. The c o r r e l a t i o n o f e lec t ron temperature a t the anode
w i t h the c e n t e r l i n e e lec t ron temperature i s given i n F ig. 13. The data
i n t h i s f i g u r e were obta ined w i t h the th rus te r conf igura t ion o f F ig. 3b
using both argon and xenon propel lants. The s o l i d l i n e i n t h i s f i gu re
has a s lope of 2/3. This observat ion t h a t the e lec t ron temperature a t
the anode i s a p p d i m a t e l y 2/3 o f the c e n t e r l i n e temperature i s used i n
Appendix A f o r the c a l c u l a t i o n o f E* . P
The probe t races obtained w i t h the probe pos i t ioned on the magnet
r i n g surface a lso i nd i ca ted the presence o f pr imary e lectrons. For
operat ion a t a discharge vol tage o f 50 v, these primary e lec t rons had
an energy o f approximately 50 eV as the model suggests they should f o r
t h i s case where a re f rac to ry cathode i s used (VC = 0). Simi la r ly , f o r
operat ion a t V,, = 40 v, t he measured primary e lec t ron energy a t the
anode sur face was % 40 eV. These probe t races provide d i r e c t evidence
o f the l oss o f primary e lec t rons through the magnetic f i e l d cusp t o the
anode. Further, t he l o s s r a t e o f these primary e lectrons was seen t o
increase as the neut ra l dens i t y parameter decreases as the model
p red ic ts .
A comparison between the ca lcu la ted (from Eq. 66) and measured
values o f the doub ly - to -s ing ly charged i o n beam cu r ren t r a t i o i s given
i n Fig. 14. The measured values a re given by t h e open symbols and the
ca l cu la ted values by t h e sol i d symbols. 'The p rope l l an t u t i l i z a t i o n
e f f i c i e n c i e s i n t h i s f i g u r e have been corrected f o r the presence of
0 1 2 3 4 5 6 7 8 9 10 CENT ERLI NE ELECT RON TEMPERA'TURE. Tw <aV)
Figure 13. Correlation of Electron Temperatures
XENON Vg = 40V
MEASURED FLOW RATE <mA oq)
0 1000 0 750 A so0 0 250
0 oA.-.t - -a e~-' - *\ CALCULATED
Eq. 66
00 I I I I I I I I I I I
0.0 . 2 . 4 . 6 . 8 I. 0
PROPELLANT UTILIZATION. '1,
Figure 14. Doubly-ta-Singly Charged Ion Beam Current Results
Figure 15a shows the ef fect of t he extracted i o n f r a c t i o n on
performance. As expected, t h i s parameter has a st rong ef fect on the
performance. Changes i n fg s h i f t t he performance curve up o r down, b u t
do n o t s u b s t a n t i a l l y change i t s shape. Clear ly , i t i s des i rab le t o
have fB be as l a r g e (near u n i t y ) as possible.
The ef fect of the pr imary e lec t ron u t i l i z a t i o n f a c t o r (Co) on
performance i s given i n F ig. 15b. This parameter a1 so has a st rong
effect on the performance. Indeed, i t i s t h i s parameter which p r i m a r i l y
determine t h e shape of t h e performance curve, w i t h l a r g e r values o f Co
corresponding t o improved performance and curves w i t h more sharply
defined "knees." The d e f i n i t i o n of Co given i n Eq. 26 suggests a num-
ber of ways i n which the value of Co may be increased. For example, i t
may be increased by using a p rope l l an t gas character ized by a l a r g e r
i n e l a s t i c c o l l i s i o n cross sec t ion o ' and a l a r g e r atomic mass ( r e s u l t - o '
i n g i n a lower neutra l v e l o c i t y vo). The parameter Co may a lso increase
by decreasing the g r i d transparency t o neutra ls , m0. This must be done,
of course, w i thout i ncreasi ng the accel e r a t o r impingement current . For
t h r u s t e r designs w i t h non-uniform beam pro f i les , T a i l o r i n g t h e accel e-
r a t o r g r i d ho le s i z e t o match the r a d i a l cur rent dens i ty p r o f i l e might
be a usefu l way t o minimize mo. Also, th ree g r i d systems might be ex-
pected t o have smal ler values of $, than two g r i d systems.
Most important ly , Co may be increased by increasing the primary
e l ect ron containment 1 ength (ee) . This 1 ength corresponds t o the
average d is tance a primary e lec t ron would t r a v e l i n the discharge
chamber before being co l l ec ted by an anode sur face provided i t had no
i n e l a s t i c co l 1 i sions. As mentioned ea r l i er, the primary funct ion of t he
magnetic f i e l d i n a1 1 t h r u s t e r designs i s t o increase t h i s 1 ength. I n
PROPELLANT U T I L I Z A T I O N . 7,
Figure 15a. Effect o f fg on Performance
cir I IOOOmA o q
fe - . 6 Co - I S . O < A e q F 1
f, - . I vo - so. Ov
€;: - SO. OeV
PROPELLANT U T I L I Z A T I O N . 7,
Figure 15b. E f fec t o f Co on Performance
cusped f ie1 d th rus ters , pr imary e lec t rons a re l o s t t o t h e anode through
the cusps. Thus, ee may be increased by decreasing the number of cusps
a t anode p o t e n t i a l o r increasing the f l u x dens i ty a t t h e cusp, b u t on l y
up t o a po in t . Reductions i n e f fec t ive anode cusp area below a c e r t a i n
1 i m i t w i l l r e s u l t i n unstable operat ion o f the discharge [22].
Equation 26 a lso suggests t h a t the primary e l e c t r o n u t i l i z a t i o n
fac to r may be increased by masking down the area of t he gr ids through
which t h e beam i s extracted, A However, decreasing A i n t h i s manner g ' 9
w i l l l ead t o a l a r g e reduct ion i n the ex t rac ted i o n f ract ion, and,
therefore, an o v e r a l l reduc t ion i n performance.
The e f f e c t o f p rope l l an t f low r a t e on performance i s shown i n
Figs. 16a and b. I n general, h igher f low ra tes produce b e t t e r perform-
ance. The maximum f l o w r a t e a t which t h e t h r u s t e r can be operated,
however, i s 1 i m i t ed by the a b i l i t y of t he accel e r a t o r system t o e x t r a c t
t he i o n cu r ren t d i rec ted toward it. The e f f e c t of f l o w r a t e on perform-
ance i s l ess dramatic f o r t h r u s t e r designs character ized by l a r g e r
values of Co as shown i n Fig. 16b. High values o f Co should, therefore,
be p a r t i c u l a r l y des i rab le i n t h rus te rs designed t o be t h r o t t l ed.
The effect of E* i s merely t o s h i f t the performance curves up o r P
down. The amount of t he s h i f t increases f o r smal ler values of fB. For
t h rus te rs t h a t use hol low cathodes, the e f f i c i e n c y o f the cathode opera-
t i o n (character ized by VC i n Eq. 27) has a st rong e f f e c t on the value
of E* . I n addi t ion, h i gher values o f VD, i n general , produce small e r P
val ues o f E* . High d ischarge vol tages, however, a re undersi r a b l e from P
t h r u s t e r 1 i fetime considerat ions, Consequently, a t rade off between
t h r u s t e r performance and t h r u s t e r l i f e t i m e i s necessary.
i - 2 0 0 0 n A o q fe - . 6
f e - - 1 vo - SO. Ov
Co - 3.OCA oq)-' c; - SO. o o v
PROPELLANT U T I L I Z A T I O N . 7,
Figure 16a. Effect of Propellant Flow Rate on Performance for Small C,
ZOO ,-
PROPELLANT U T I L I Z A T I O N . 7,
Figure 16b. Effect of propellant Flow Rate on Performance fo r Large Co
The ef fect of cathode operat ion on t h r u s t e r performance can be
inves t iga ted ana ly t i ca l l y using t h e performance model. Hol low cathode
operat ion i s re f lec ted i n the value of the parameter VC. The lower
the value of t h i s parameter the more e f f i c i e n t l y the cathode i s oper-
a t ing . A t a f i xed discharge voltage, increasing t h e value o f VC
degrades t h e t h r u s t e r performance i n two ways. F i r s t , i t increases
* E d i r e c t l y through i t s appearance i n Eq. 27, and second, i t decreases
P the energy of t he pr imary electrons, which increases the value of the
c o l l i s i o n a l l oss parameter E, and decreases t h e value of Co. A l l of
these affects a re accounted fo r when E* i s ca lcu la ted according t o the P
procedure described i n Appendix A and t h e pr imary e lec t ron energy i s
given by VD-VC.
The effect o f VC on t h e basel ine plasma i o n energy cost was calcu-
l a t e d f o r xenon p rope l l an t us ing the method described i n Appendix A and
assuming a discharge vol tage o f 30 v. The r e s u l t s of these ca lcu la t ions
are given i n Table 2.
vc (Vo l t s ) (Vo l ts )
3 0 0
30 2.5
3 0 5.0
30 7.5
30 1-0.0
30 12.5
Table 2
Ef fec t of VC on E* P
* 'D.-'c E P
(Vol t s ) (ev)
30 3 6
27.5 42
25.0 48
22.5 5 5
20.0 66
17.5 79
The values of E* and Co, from Tab1 e 2, were then used i n the equat ion P
fo r t h r u s t e r performance (Eq. 28) along w i t h the values: = 1000 mA eq.,
fB = 0.5, fC = 0.1 , and V,, = 30 v. The performance curves generated i n
t h i s manner a re shown i n Fig. 17. This f i g u r e ind ica tes t h a t , a t a d i s -
charge vol tage o f 30 v, r e l a t i v e l y small changes i n VC can produce
subs tant i a1 changes i n t h r u s t e r performance. For exampl e, i ncreasi ng
VC from 10 t o 12.5 v causes an increase o f approximately 30 eV/beam i o n
i n t h e performance curve.
Work done by S i e g f r i e d [52] on i n e r t gas hol low cathodes i nd i ca ted
t h a t a 0.6 v increase i n t he sur face work func t i on o f t h e cathode i n -
s e r t could produce an increase i n i n t e r n a l cathode plasma p o t e n t i a l
f rom 8 t o 12 v. The value of VC would be expected t o increase by t h i s
same amount. I n add i t i on , S ieg f r ied notes t h a t t he re i s apparent ly a
g rea te r s e n s i t i v i t y of the i n s e r t t o dep le t i on or contaminat ion of t he
low work funct ion ma te r ia l f o r opera t ion w i t h argon o r xenon as compared
t o opera t ion w i t h mercury. Consequently, on t h e basis o f these consid-
, era t ions , one would expect the performance of i n e r t gas i o n t h r u s t e r s
opera t ing a t low discharge voltages t o be q u i t e s e n s i t i v e t o the con-
d i t i o n and opera t ion of the hol 1 ow cathode.
Changes i n the f r a c t i o n of i o n cu r ren t going t o cathode p o t e n t i a l
surfaces ( f C ) a lso tend t o s h i f t t h e performance curves up o r down.
Recent trends [23,24] i n t h r u s t e r design t o operate t h e discharge
chamber body a t anode r a t h e r than cathode po ten t i a l serve t o reduce t h e
va lue of fC, and consequently improve the performance. I n add i t ion ,
e l i m i n a t i o n o f t he separate cathode discharge reg ion i n these designs
serves t o decrease both fC and VC, which again improves the performance.
It should be noted, however, t h a t an increase i n t he ex t rac ted i o n
no = i - J~ = . (75)
Subs t i t u t i ng t h i s i n t o Eq. 28, and so lv ing f o r t h e neut ra l loss r a t e
This equation gives the value of t h e neut ra l atom loss r a t e as a func-
t i o n of t h e beam i o n energy cost . For a s p e c i f i e d t h r u s t e r geometry *
and discharge voltage, the design parameters Co, E ~ , fB, fC and VD may
be taken t o be approximately constant. Thus, s ince t h e propel l a n t mass
flow r a t e doesn' t appear on the r ight-hand-s ide of Eq. 76, t h i s equa-
t i o n p r e d i c t s t h a t t he neut ra l l o s s r a t e t io i s independent o f the f l ow
r a t e a t a constant value o f t h e energy cos t per beam i o n ( E ~ ) . This
same conclus ion was reached o r i g i n a l l y by Kaufman [53] i n h i s constant
neutra l l oss r a t e theory developed fo r low magnetic f i e l d s t rength d i s -
charge chamber designs.
Thruster Test ing w i thout Beam Ex t rac t i on
I t i s o f ten des i rab le t o compare the performance o f d i f f e r e n t d i s -
charge chamber designs t h a t have been operated w i thout i o n beam extrac-
t i o n . Operation w i thout beam e x t r a c t i o n i s o f t e n more convenient and
general l y requ i res the use of small e r vacuum t e s t fac i 1 i ti es than
operat ion w i t h beam ex t rac t i on . However, data ob ta i ned under these
condi t ions must be i n te rp re ted c a r e f u l l y . It has been observed t h a t
the performance (eV/beam ion ) extrapolated from t h r u s t e r operat ion
w i thout beam e x t r a c t i o n i s genera l ly s i g n i f i c a n t l y b e t t e r than the
performance measured w i t h beam e x t r a c t i o n [24,54,55]. This di f ference
i n performance may be understood i n context of the performance model
developed i n t h i s repor t .
Thruster operat ion w i thout beam ex t rac t i on should be character ized
by higher discharge chamber neutra l dens i t ies than operat ion a t t he
same prope l lan t flow r a t e w i t h beam ext rac t ion . 'This i s because,
w.i thout beam ex t rac t i on most of t he prope l lan t 1 eaves t h e discharge
chamber i n the form of neutra l atoms a t the neut ra l atom thermal
ve loc i t y . With beam ext rac t ion , however, most of t he propel 1 an t 1 eaves
i n t h e form o f ions a t t he Bohm v e l o c i t y . Therefore, s ince t h e t o t a l
p rope l l an t f low r a t e 1 eaving the t h r u s t e r must be the same i n each case,
the l oss r a t e of neut ra ls i s smal ler w i t h beam e x t r a c t i o n than wi thout
it, imply ing lower neutra l dens i t ies . This i s p r i m a r i l y a consequence
of t he changing e f fec t ive transparency of the screen g r i d. W i thout
beam ext rac t ion , t he e f f e c t i v e screen g r i d transparency i s very small
[56] s ince ions tend t o be focused onto the g r i d webbing. With beam
ext rac t ion , t he ions tend t o be focused away from the screen g r i d
webbing and through the g r i d apertures. I n any case, opera t ion a t
h igher neutra l dens i t ies , f o r the same th rus te r conf igurat ion, t rans-
l a t e s i n t o lower plasma i o n energy costs according t o Eq. 25.
The performance model developed herein may be used t o make more
meaningful ex t rapo la t i on o f data taken wi thout beam e x t r a c t i o n t o opera-
t i o n w i t h beam ext rac t ion . To do t h i s , i t i s necessary t o experiment-
a l l y determine the values of t he parameters E* Co, fB and fC. The P'
parameters E* and Co are roughly independent of the neut ra l densi ty , P
thus, s houldn' t change depending on whether o r no t a beam i s extracted.
For hol 1 ow cathode equi pped t h r u s t e r designs, however, the base1 i ne
plasma i o n energy cos t includes the parameter VC, which i s an i n d i c a t i o n
o f cathode performance. The value of VC may be a funct ion of t he d i s -
charge chamber neutra l dens i ty and consequently could cause t h e value
o f E * t o change fo r operat ion w i t h and wi thout beam ext rac t ion . The P
magnitude o f the change i n VC, i f any, i s unknown.
To determine the values of E* and Co fo r opera t ion w i thout beam P
ex t rac t ion , some method of est imat ing the t o t a l i o n cur rent produced as
a funct ion of the discharge chamber neut ra l dens i ty must be used. This
i s probably most e a s i l y accomplished using several i o n cur rent probes
posi t ioned a t the wa l ls o f t he discharge chamber i n t h e manner suggested
by Poeschel [54]. Ideal l y , these data should be acquired over a range
o f neut ra l dens i t ies a t a constant discharge vol tage.
Once t h e t o t a l i o n cu r ren t produced i s known as a func t i on of t he
neutra l densi ty , a curve s i m i l a r t o F ig. 4a may be generated, where the
values o f E are ca lcu la ted using Eq. 2. From these data, t he val ues P
of E* and Co may then be determined as those which g i ve the best f i t P
o f Eq. 25 t o the data. Note, E* may a1 so be ca l cu la ted according t o P
t he procedure out1 i ned i n Appendix A.
F ina l l y , the values of fg and f C may be determined using t h e above
probe data and guessing a value o f t he e f f e c t i v e screen g r i d t ranspar-
ency appropr iate f o r the beam e x t r a c t i o n condi t ion. Once the values of *
€P' Co, fB and fC have been determined, the performance curve fo r t he
beam e x t r a c t i o n cond i t ion may be approximated us ing Eq. 28, f o r any
desi red t o t a l propel 1 ant f low r a t e .
Space Propul s i o n Mission Anal y s i s
The perfwmance model, developed i n t h i s repor t , i s p a r t i c u l a r l y
useful f o r examining t h e impact of t h r u s t e r performance on space pro-
pu l s ion missions. It should a lso be useful f o r s tud ies making compari-
s ions of t h e c a p a b i l i t i e s of d i f f e r e n t propuls ion systems such as those
done i n References 57 and 58. The fo l l ow ing analys is i s intended as an
i l l u s t r a t i o n of t h e kinds of th ings t h a t can be done us ing t h i s t h r u s t e r
performance model. Consequently, t he o r b i t t rans fer miss ion considered
here has been g r e a t l y simp1 i f i e d . So lv ing a more accurate and compl i - cated o r b i t t r a n s f e r problem would no t prov ide any add i t i ona l i n s i g h t
i n t o the a p p l i c a b i l i t y o f t he model t o mission analys is , and may even
tend t o conceal t h e desi red i l l u s t r a t i o n . I n t h e ana lys is t h a t fol lows,
t he e f f e c t o f t h r u s t e r performance on the maximum payload f r a c t i o n ob-
t a i n a b l e fo r a low ea r th o r b i t t o geosynchronous ea r th o r b i t t rans fer
mission, w i t h a c h a r a c t e r i s t i c v e l o c i t y of 6000m/s, i s inves t iga ted .
The rocket equat ion (Eq. 77) gives the r a t i o o f f i n a l spacecraf t
mass (Mf) t o i n i t i a l mass (Mi) as a func t i on of t he c h a r a c t e r i s t i c
v e l o c i t y f o r t he miss ion (AV), t he t h r u s t e r exhaust v e l o c i t y (u) and
the t h r u s t e r p rope l l an t u t i l i z a t i o n e f f i c i ency (nu),
Assuming, f o r s i m p l i c i t y , t h a t t h e f i n a l mass cons is ts o n l y o f t h e
payload mass (Me) and the mass of t he power p l a n t (M ) then Eq. 77 can 9
be w r i t t e n as,
The mass of the power p l a n t i s p ropor t iona l t o t he power generated,
where a i s t h e spec i f i c power p l a n t mass (Kg/W), and Pg i s t he
generator power ( W ) . The generator power may be given as,
where qt i s the t h r u s t e r e l e c t r i c a l e f f i c i e n c y and m i s the t o t a l P
propuls ion system p rope l l an t mass f l o w r a t e (kg/s). For. a constant
exhaust v e l o c i t y and p rope l l an t e f f i c i ency , Eq. 80 may be in tegra ted t o
obtain,
where M i s the i n i t i a l p rope l l an t mass and t i s the t o t a l mission Po
time. Combining Eqs. 78-80 and recogniz ing tha t ,
y i e lds ,
Considering o n l y discharge chamber losses, t he t h r u s t e r e l e c t r i c a l
e f f i c i ency may be approximated by,
where VN i s t he ne t accel e r a t i ng vol tage. The n e t accel e ra t i ng vol tage
i s re1 a ted t o the exhaust vel o c i ty by,
Combining Eqs . 83-85 y ie lds ,
Equation 86 gives the payload mass f r a c t i o n as a f u n c t i o n o f : t h e
c h a r a c t e r i s t i c v e l o c i t y , the exhaust ve loc i t y , t h e miss ion time, the
power p l a n t spec i f i c mass, t he charge-to-mass r a t i o o f t h e beam ions,
t he average beam i o n energy cos t and the p rope l l an t u t i l i z a t i o n e f f i c i -
ency. The average beam i o n energy cos t i s r e l a t e d t o the p rope l l an t
u t i l i z a t i o n e f f i c i e n c y through the equation developed i n the t h r u s t e r
performance model,
where Bo i s a dimensionless q u a n t i t y descr ib ing t h e u t i l i z a t i o n o f
pr imary e lec t rons i n t he discharge chamber and i s g iven by Eq. 74.
For a g iven c h a r a c t e r i s t i c ve loc i t y , mission time, power p l a n t
spec i f i c mass, and p rope l l an t u t i l i z a t i o n e f f i c i ency , the optimum
exhaust v e l o c i t y corresponding t o the maximum pay1 oad f r a c t i o n , f o r
these cond i t ions , can be determined from Eq. 86. This i s accomplished
by s e t t i n g the d e r i v a t i v e of Eq. 86 ( w i t h respect t o u ) equal t o zero
and numerical l y so l v ing t h e r e s u l t i n g equat ion f o r u . S u b s t i t u t i n g
t h i s va lue o f t h e exhaust v e l o c i t y back i n t o Eq. 86 gives the value of
t he optimum payload f r a c t i o n under the spec i f i ed condi t ions. This pro-
cedure may be repeated f o r d i f f e r e n t values o f p rope l l an t u t i l i za t i on ,
where, for each val ue of qu, t h e corresponding beam i o n energy c o s t i s
ca lcu la ted from Eq. 87. I n t h i s manner, a curve o f opt imized payload
f r a c t i o n versus prope l lan t u t i l i z a t i o n e f f i c iency liiay be generated f o r
speci f ied values of t h e c h a r a c t e r i s t i c ve loc i ty , miss ion time, power
* p l a n t spec i f i c mass and t h r u s t e r performance parameters, EP, f B ~ f C s
Bo and V,,.
An example o f t h i s i s shown as the s o l i d l i n e i n Fig. 18 f o r a
c h a r a c t e r i s t i c v e l o c i t y of 6000 m/s, a mission t ime o f 200 days and a
s p e c i f i c power p l a n t mass o f 40 kg/kW. I n add i t ion , t he t h r u s t e r per- *
formance parameters were taken t o be E = 50 eV, fB = 0.5, fC = 0.1, P
Bo = 5 and VD = 40 v w i th argon as the prope l lan t . F igure 18 ind ica tes
t h a t t he curve of opt imized payload f r a c t i o n versus p rope l l an t u t i l i z a -
t i o n e f f i c i e n c y goes through a maximum. The propel 1 an t u t i 1 i za t ion
e f f i c i e n c y corresponding t o t h i s doubly maximized payload f r a c t i o n i n -
d icates the l o c a t i o n on the t h r u s t e r performance curve a t which t h e
t h r u s t e r should be operated i n order t o t r u l y maximize the payload
f r a c t i o n . I n summary, t he doubly maximized payload f r a c t i o n was de-
termined by op t im iz ing the exhaust v e l o c i t y t o produce the optimum pay-
load f r a c t i o n f o r a given p rope l l an t u t i l i z a t i o n e f f i c iency and
subsequently se lec t i ng the propel1 a n t u t i l i z a t i o n t o o b t a i n the maximum
payload f r a c t i o n from t h i s s e t o f opt imized payload f rac t i ons .
To i l l u s t r a t e more c l e a r l y what i s going on, the i n i t i a l p rope l lan t
mass f r a c t i o n ( M /Mi) and power p l a n t mass f r a c t i o n (M /Ma) a re p l o t t e d Po 9 1
along w i t h the opt imized payload f r a c t i o n i n Fig. 18. Note t h a t the
sum o f the th ree curves i n t h i s f i gu re , a t any p rope l l an t u t i l i z a t i o n ,
i s equal t o u n i t y . As the p rope l l an t u t i l i z a t i o n increases the i n i t i a l
p rope l l an t mass f r a c t i o n decreases as expected. I n add i t ion , the power
- I AV=6000m/s ARGON 9,-5. 0
I t-200days fB-0. 5 $ -500~ a=40kg/kW fC-0. 1 Vp-40v /
PROPELLANT UTILIZATION. 12,
Figure 18. Payload Fract ion, Propel 1 an t Mass Fract ion, and Generator Mass Frac t ion V a r i a t i o n w i t h Propel 1 an t U t i 1 i za t ion
p l a n t mass decreases i n i t i a l l y , due t o a decreasing optimum exhaust
ve loc i t y , then increases dramat ica l ly a t h igh p rope l l an t u t i l i z a t i o n
e f f i c i e n c i e s , due t o the r a p i d increase i n the average beam i o n energy
cost . For t h e se t o f cond i t ions chosen fo r the curves i n F ig. 18, t he
doubly maximized payload f r a c t i o n occurs a t a p rope l l an t u t i l i z a t i o n
of 0.73.
The e f f e c t o f the s p e c i f i c power p l a n t mass on the opt imized pay-
load f r a c t i o n vs. p rope l lan t u t i l i z a t i o n e f f i c i ency curve i s shown i n
Fig. 19a. It i s i n t e r e s t i n g t o note that , a t very low values of a the
opt imized payload f r a c t i o n i s r e l a t i v e l y i n s e n s i t i v e t o l a r g e changes
i n t h e p rope l l an t u t i l i z a t i o n . I n add i t ion , t he p rope l l an t u t i l i z a t i o n
a t which t h e doubly maximized payload f rac t i on occurs i s a func t i on of
a. That i s , t h e po in t a t which one should operate on the performance
curve o f a g iven th rus te r depends on the spec i f i c power p lan t mass f o r
t he spacecraft. This i s i l l u s t r a t e d i n Fig. 19b, where the optimum
propel 1 ant u t i l i z a t i o n eff ic iences fo r the spec i f i ed mission parameters
are i nd i ca ted on the performance curve used i n generat ing the curves i n
Fig. 19a. It i s a lso i n t e r e s t i n g t o note tha t , t h e optimum prope l lan t
u t i l i z a t i o n fo r the a = 1 kg/kW case occurs we l l past what might o rd in-
a r i l y be considered the "knee" o f t he performance curve. I n a s i m i l a r
manner, t he ~ i i i s s i o n t ime can be shown t o e f f e c t t h e optimum prope l lan t
u t i l i z a t i o n , w i t h longer t r i p times y i e l d i n g h igher optimum prope l lan t
u t i l i z a t i o n s .
So f a r , on l y one t h r u s t e r performance curve has been considered and
i t has been observed t h a t t he optimum prope l lan t u t i l i z a t i o n e f f i c iency
corresponding t o the doubly maximized payload f r a c t i o n i s a func t i on of
t he s p e c i f i c power p lan t mass and the mission t ime. The e f f e c t of the
0.0 . I . 2 .3 . 4 . 5 - 8 .7 .8 .Q 1.0 PROPELLANT UTILIZATION. 7,
Figure 19a. E f f e c t o f Power P lan t Spec i f i c Mass on Payload Frac t ion
ARGON Fg-0.5 I
PROPELLANT UTILIZATION. 1,
Figure 19h. E f f e c t o f Power P lan t S p e c i f i c Mass on Optimum Prope l l an t U t i l i z a t i o n
t h r u s t e r performance parameters on the opt-imu~ii payload vs. propel 1 an t
u t i l i z a t i o n curves w i l l now be invest igated.
F igure 20a i nd i ca tes the ef fect of the parameter Bo on payload
f r a c t i o n curves. This parameter p r i m a r i l y determines t h e shape of t he
t h r u s t e r performance curve. As expected, l a r g e r values o f Bo y i e l d
h igher maximum payload f rac t ions . The optimum propel 1 an t u t i l i z a t i o n
e f f i c i enc ies are indicated, f o r each performance curve i n Fig. 20 b, by
the v e r t i c a l l i n e s t h a t i n t e r s e c t t he curves.
The ef fect of the extracted i o n f r a c t i o n on the payload f r a c t i o n
curves i s shown i n Fig. 21a. Not su rp r i s i ng l y , h igher ext racted i o n
f rac t i ons produce 1 arger maximum payload f rac t i ons . Agai n, t he optimum
propel 1 an t u t i 1 i zat ions a re i ndicated f o r each performance curve i n
Fig. 21b.
F i n a l l y , the e f f e c t o f t h e p rope l l an t gas i s i nd i ca ted i n Fig. 22a.
To generate the curve l a b e l l e d "argon" i n t h i s f i gu re , t he fo l l ow ing
t h r u s t e r performance parameters were used: fB = 0.5, fC = 0.1 , VD =
40 v, Bo = 5.0 and E* = 50 eV. For krypton and xenon propel lants, the P
values o f fB and fC were he ld constant, but, t he values o f VD. Bo and
* E were changed t o 30 v, 8.2 and 49 eV f o r krypton, and 30 v, 14.6 and
P 42 eV f o r xenon. The changes i n t he values o f Bo were ca lcu la ted using
the equations developed i n the t h r u s t e r perfor~iiance model. 'The changes
i n t h e values o f E* a re cons is ten t w i t h measured changes i n t h i s param- P
e t e r f o r a given t h r u s t e r design operated on the th ree p rope l l an t gases.
C lear ly , xenon prope l lan t i s super io r t o e i t h e r argon o r krypton pro-
pel 1 ants. The performance curves, along w i t h the correspondi ng optimum
p rope l l an t e f f i c i e n c i e s , a re given i n Fig. 22b fo r the cases where
these th ree prope l lan ts a re used.
PROPELLANT UTILIZATION. 7,
Figure 20a. Ef fect o f Bo on Payload F rac t i on
Figure 20b. E f f e c t o f Bo on the Performance Curve and the Optimum Propel 1 ant U t i 1 i z a t i on
0.0 . 1 . Z .3 . 4 .S .6 .7 .B . 9 1.0
PROPELLANT UTILIZATION. 7,
Figure 21a. E f f e c t o f fB on Payload Frac t ion
PROPELLANT UTILIZATION. 1,
Figure 21b. E f fec t o f f on the Performance Curve and the Optimum Prope l lan t 1 t i l i z a t i o n
0.0 .I . 2 .3 . 4 . 5 . 8 .7 . 8 . g 1.0
PROPELLANT UTILIZATION. 7,
F i g u r e 22a. E f f e c t o f P r o p e l l a n t on Payload F r a c t i o n
0 . 4 .S .6 - 7 .8 .Q 1.0
PROPELLANT UTILIZATION. 1,
F i g u r e 22b. E f f e c t o f P r o p e l l a n t on t h e Performance Curve and t he Optimum Propel l a n t U t i l i z a t i o n
V I . CONCLUSIONS
A simple t h r u s t e r performance model, t h a t has l e d t o an improved
understanding of i o n t h r u s t e r discharge chamber operat ion, has been
developed. This model describes i o n th rus te r performance i n terms o f
four parameters: the plasma i o n energy cos t ( E ), the extracted i o n P
f r a c t i o n (fB), t he i o n cu r ren t f r a c t i o n t o cathode po ten t ia l surfaces
( fC) and t h e discharge vol tage (VD).
The equation developed t o describe the behavior o f the plasma i o n
energy cos t agrees w i t h the r e s u l t s of a v a r i e t y of experiments. This
equation provides an expression f o r the func t iona l dependence o f t he
plasma i o n energy cost on t h e neut ra l dens i ty parameter, i ( ~ - ~ ~ ) .
Experiments i n d i c a t e good agreement between the predicted funct ional
form o f t he model and the experimental data. These experiments a lso
suggest t h a t t he primary e l e c t r o n u t i l i z a t i o n fac tor (Co) and the base-
l i n e plasma i o n energy cos t (€*) are independent o f t he neutra l dens i ty P
parameter under many condi t ions. The model c o r r e c t l y p red ic ts the
v a r i a t i o n i n plasma i o n energy cos t f o r changes i n : p rope l lan t gas,
g r i d transparency t o neutra l atoms, beam e x t r a c t i o n area, discharge
vol tage and e f f e c t i v e discharge chamber wa l l temperature. The model i s
app l i cab le t o both r i n g and l i n e cusp th rus te r designs.
Measurements o f the extracted i o n f rac t i on i n d i c a t e t h a t t h i s
parameter i s r e l a t i v e l y independent of the neutra l dens i ty parameter.
The extracted i o n f r a c t i o n does, however, appear t o be a funct ion of
cathode. Consequently, there i s a need t o ve r i f y t he p red i c t i ons o f
the model on a hol low cathode equipped i o n th rus te r .
F i n a l l y , i t i s o f i n t e r e s t t o t e s t the v a l i d i t y of the model on
the 01 der s t y le , low magnetic f i e l d s t rength t h r u s t e r designs, such as
the SERT I 1 [63] o r the J-Series [64] th rus ters . These t h r u s t e r designs
d i f f e r from the cusped magnetic f i e l d designs considered i n t h i s inves-
t i g a t i o n i n t h a t e lect rons, i n the discharge chamber plasma, can on l y
reach the anode surface by crossing magnetic f i e l d l i n e s . I n t he
cusped f i e l d designs, however, t he m a j o r i t y of e lec t rons a r e be1 ieved t o
reach the anode by going along the f i e l d l i n e s . The mechanism f o r elec-
t r o n l o s s i n t he low magnetic f i e l d t h r u s t e r designs i s s u f f i c i e n t l y
d i f f e r e n t from t h a t i n the cusped f i e l d designs t h a t experimental v e r i -
f i c a t i o n o f t h e model on these low f i e l d s t rength designs i s requi red.
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29. Bechtel, R. T., "Performance and Control o f a 30-cm. d ia . Low- Impulse, Kaufman Thruster," AIAA Paper 69-238, 1969.
30. Cohn, A. J., "Onset o f Anomal us D i f f u s i o n i n Electron-Bombardment I o n Thrusters," NASA TN D-3731, 1966.
31. Poeschel, R. L. and Vahrankamp, R., "The Radial Magnetic F i e l d Geometry as an Approach t o Total I o n U t i l i z a t i o n i n Kaufman Thrusters, " AIAA Paper No. 72-481 , 1972.
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Brophy, J . R. and Wilbur, P. J., "Simple Performance Model f o r Ring and L ine Cusp Ion Thrusters," I n t ' l E l e c t r i c Propuls ion Con- ference Paper IEPC 84-68, Tokyo, Japan, 1984.
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Dugan, J. V . and Sovie, R. J., "Volume Ion Product ion Costs i n Tenuous Plasmas: A General Atom Theory and Deta i led Results f o r He1 ium, Argon and Cesium," NASA TN D-40150, 1967.
Lee, J. F., Sears, F. W., and Turcotte, D. L., S t a t i s t i c a l Thermo- dynamics, Addison-Wesl ey, 1973.
40. Knief, R. A., Nuclear Energy Technology, McGraw-Hill , Hemisphere Publ ish ing Co., 1981, pp. 111.
41. Chapman, B., Glow Discharge Processes, John W i l ey & Sons, Inc., New Yo'rk, 1980.
42. Vahrenkamp, R. P., "Measurement of Double Charqed Ions i n the Beam o f a' 30-cm ~ e r c u r y Bombardment Thruster, "~AIAA Paper 73-1 057, 1973.
43. Hershkow'l'tz, N., e t .a l ., "Plasma Leakage through a Low 6 L ine Cusp;' Physical Review Let te rs , Vol . 35, No. 5, 1975, pp. 277-280.
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45. Hershkowi tz, N., e t . a1 . , "E lec t ros ta t i c Self-plugging of a Picket Fence Cusped Magnetic Field," The Physics of Fluids, Vol . 21, NO. 1, 1979, pp. 122-125.
46. Kozima, H., e t .a l . , "On the Leak Width o f L ine and Po in t Cusp Mag- n e t i c Fields," Physics Let ters, Vol. 86A, No. 6, 7, 1981. pp. 373-375.
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48. deHeer, F. J., e t .a l . , "Total Cross Sections o f E lec t ron Scat te r ing by Ne, Ar , Kr and Xe," Journal o f Physics B: Atomic 'and Molecular Physics, Vol. 12, No. 6, 1979, pp. 979-1002.
49. Hayashi, M., "Determi na t ion of El ectron-Xenon Total Exc i ta t i on Cross Sections, from Threshold t o 100 eV, from Experimental Values o f own send's a," Journal of physics D: ~ p p l i e d Physics, Vol. 16, pp. 581-589, 1983.
50. Rapp, D. and Englander-Golden, P., "Total Cross Sections f o r Ion iza- t i o n and Attachment i n Gases by Electron Impact. I. Pos i t i ve Ionizat ion," Journal o f chemical Physics, ~ o l . 34, No. 5, pp. 1464-1479, 1965.
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52. S iegf r ied , D., "Xenon and Argon Hollow Cathode Research," i n "Advanced I o n Thruster Research," P. J. W i l bur ed., NASA CR- 168340, Jan. 1984.
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58. Kaufman, H. R. and Robinson, R. S., " E l e c t r i c Thruster Performance f o r O r b i t Raising and Manuevering," Journal of Spacecraft and Rockets, Vol . 21, No. 2. pp. 180-186, 1984.
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APPENDIX A
THEORETICAL CALCULATION OF THE BASELINE PLASMA I O N ENERGY COST, c i
The d e f i n i t i o n of the basel ine plasma ion energy cost (E*) i s given P
by Eq. 27, and i s repeated here f o r convenience,
The parameter E, i n Eq. A-1 accounts f o r the energy t h a t i s expended i n
i o n i z a t i o n and e x c i t a t i o n react ions and i s def ined i n Eq. 15 as,
The brackets i n t h i s equation represent the enclosed product averaged
over the e n t i r e e lec t ron speed d i s t r i b u t i o n funct ion, i .e.,
where F(ve) represents the e n t i r e e lec t ron d i s t r i b u t i o n funct ion.
For a plasma w i t h an e lectron populat ion characterized by a
Maxwell i a n d i s t r i b u t i o n o f temperature T,,,, and a monoenergetic
(pr imary) &oup o f energy E Eq. A-2 becomes, P '
where < >M represents the enclosed product averaged over t h e
Maxwellian speed d i s t r i b u t i o n func t ion .
The term under the summation s ign i n Eq. A-4 may be approximated
by consider ing o n l y a s ing le equivalent lumped exc i ted s ta te character-
ized by a t o t a l e x c i t a t i o n c o l l i s i o n cross sec t ion oeX and a lumped
e x c i t a t i o n energy Uex. For ra re gases, Uex may be approximated by [38],
where UL i s t h e lowest e x c i t a t i o n energy l e v e l . Using t h i s lumped
e x c i t a t i o n approximation Eq. A-4 becomes,
Subs t i t u t i ng Eq. A-6 i n t o A-1 y i e l d s the f o l l o w i n g expression f o r E*, P
n [l 0;x Vp + 'uexve'~] "ex
U+ + +
94 + <(Jv > l e ~ ; ~ , + e , y
€* = "M (A-7) P
9
1 - (VC + / VD
The value of E* may be e a s i l y ca lcu la ted using Eq. A-7 f o r a given P
Maxwell i a n ' e l e c t r o n temperature, primary-to-Maxwell i a n e l e c t r o n d e n s i t y
r a t i o , discharge vo l tage and value o f Vc. I n a d d i t i o n , t h e average
energy o f a Maxwel l ian e l e c t r o n l o s t t o t he anode ( E ~ ) must be known.
Reference 19 g i ves t h i s energy as,
where TA i s t he e l e c t r o n temperature a t t h e anode and vA i s t he
d i f f e r e n c e between plasma p o t e n t i a l and anode p o t e n t i a l . Experiments
presented e a r l i e r i nd i ca ted t h e e l e c t r o n temperature a t t h e anode was
approximately 2/3 o f the center1 i n e temperature. Therefore, EM wi 11
be taken as,
Langmuir probe measurements a l s o i n d i c a t e t h a t f o r t h e t h r u s t e r c o n f i g -
u r a t i o n s t es ted i n t h i s i n v e s t i g a t i o n VA was always approx imate ly 2v,
thus, t h i s value o f VA w i l l be used i n these c a l c u l a t i o n s toge ther w i t h
Eq. A-9.
For xenon, t he values o f t h e t o t a l e x c i t a t i o n c o l l i s i o n c ross
sec t i on requ i red by Eq. A-7 were taken f rom Reference 49. I o n i z a t i o n
c ross sec t ion data were obta ined from Reference 50. These data, a long
w i t h polynomial curve f i t s used t o f a c i l i t a t e i n t e r p o l a t i o n between
data p o i n t s are g iven i n Figs. A-la and A-lb.
Wi th these data,Eq. A-7 was used t o c a l c u l a t e t h e va lues o f E* as P
a f u n c t i o n o f e l e c t r o n temperature w i t h t h e pr imary-to-Maxwel l ian
e l e c t r o n dens i t y r a t i o as a parameter. The r e s u l t s o f these ca l cu la -
t i o n s a re g iven i n F ig . A-2 where the va lues VD=40v, Vc=Ov and VA=2v
were used. From t h i s f i g u r e , i t i s seen t h a t t he base l ine plasma i o n
energy c o s t can vary over a wide range o f values, and i t i s n o t c l e a r
x10-20 XENON
ELECTRON ENERGY. E , <eV)
. I o n i z a t i o n C o l l i s i o n Cross Sect ion f o r Xenon
XENON
0 I I I 1 I I I I I
0 20 40 60 &O 100
ELECTRON ENERGY, E , <eV)
F igu re A-1 b. Tota l E x c i t a t i o n C o l l i s i o n Cross Sect ion f o r Xenon (Ref. 49)
e" r XENON
ELECTRON TEMPERATURE. TM , <eV)
Figure A-2. Basel ine Plasma I o n Energy Cost Var ia t i on Calculated from Eq.A-7 f o r Xenon
a t t h i s p o i n t which value o r values o f E* are appropr iate. Th is d i f f i - P
c u l t y a r i ses because the Maxwell i a n e l e c t r o n temperature and pr imary-to-
Maxwell ian e l e c t r o n dens i t y r a t i o may n o t be selected independently as
was done f o r the c a l c u l a t i o n s i n F ig. A-2.
To reso lve t h i s d i f f i c u l t y a second equat ion f o r E* i s requi red. * P
T h i s second equat ion f o r E may be der ived i n the f o l l o w i n g manner. P
Equation 40 provides an expression f o r the r a t i o o f i o n cu r ren t pro-
duced by pr imary e lec t rons t o the t o t a l i o n cur ren t produced, i .e.,
The t o t a l i o n cu r ren t produced, however, i s g iven by Eq. 52 as,
and the i o n cu r ren t produced by pr imary e lec t rons i s given by Eq. 30 as,
J; = n n ev a ' v O P P +
d i v i d i n g Eq. A-12 by A-11 y ie lds ,
Equating Eq. A-13 t o A-10 and so l v ing f o r E* y i e l d s the des i red second P
equat ion f o r the basel ine plasma i o n energy cost .
The appropr iate values o f E* , determined from t h e c o r r e c t corres- P
ponding e lec t ron temperatures and dens i t y r a t i o s , may be found by
so l v ing Eqs. A-7 and A-14 simultaneously f o r E* and t h e e lec t ron P
temperature f o r spec i f i ed values o f the primary-to-Maxwell i a n e lec t ron
dens i t y r a t i o . Th is procedure i s most e a s i l y accomplished graph ica l l y ,
as shown i n Fig. A-3, where the i n t e r s e c t i o n o f the curves corresponding
t o the same values of n / g ives bo th E* and t h e e l e c t r o n temperature. P l.l P
The locus o f these i n t e r s e c t i o n po in t s i nd i ca tes the v a r i a t i o n o f the
basel ine plasma i o n energy cos t w i t h t h e e l e c t r o n temperature and
primary-to-Maxwell i a n e lec t ron dens i t y r a t i o . Figure A-3 i nd i ca tes
tha t , under the assumptions used f o r these ca lcu la t ions , E* does not P
vary s u b s t a n t i a l l y over wide v a r i a t i o n s i n . e l e c t r o n temperature and
"1% . This agrees w i t h the experimental observat ion t h a t E* i s a P.
constant f o r operat ion .w i th xenon propel 1 an t a t a discharge vol tage o f
40 v.
The measured value o f E* i s shown along w i t h the ca lcu la ted values P
i n Fig. A-3. The experimental value, a t low e lec t ron temperatures, i s
seen t o be genera l l y s l i g h t l y h igher t h a n the ca l cu la ted values. The
most 1 i k e l y explanat ion f o r t h i s i s a systematic measurement e r r o r o f
the t o t a l i o n cu r ren t produced as discussed i n Appendix B. Also i n d i -
cated i n F ig. A-3 i s the theo re t i ca l maximum value o f E* f o r t h i s case P
(as ca lcu la ted from Eq. 41). F i n a l l y , i t should be noted t h a t the
above r e s u l t s are somewhat sens i t i ve t o the choice o f EM which i s
assumed here t o be given by Eq. A-9.
r XENON
Vo- 40. Ov 0 SOLUTIONS
0 1 2 3 4 5 6 7 8 Q 10 ELECTRON TEMPERATURE, TM . (eV)
F igu re A-3. So lu t i ons f o r t h e Base l ine Plasma I o n Enegry Cost f o r Xenon
APPENDIX B
ERROR ANALYSIS
The experiments described i n t h i s i n v e s t i g a t i o n requ i re the
measurement o f several quan t i t i es . These q u a n t i t i e s may be separated
i n t o f o u r groups according t o the technique requ i red t o make the mea-
surement. These f o u r groups are:
1. Measurement o f P , P* and fB. P P
2. Measurement o f the neut ra l dens i t y parameter, &-nu)
3. Measurement o f the plasma proper t ies : nM , n , and TM . P
4. Measurement o f doubly charged ions.
The e r r o r s associated w i t h the measurements o f the q u a n t i t i e s i n each
o f these groups w i l l be discussed separate ly . I n each case, both
systematic and random e r r o r s w i l l be discussed.
Measurement o f oD, E: and fs
Determination o f the q u a n t i t i e s i n t h i s group requ i res the measure-
ment o f the t h r u s t e r e l e c t r i c a l operat ing parameters inc lud ing : the
discharge voltage, discharge cu r ren t and beam cur ren t . I n add i t ion ,
the t o t a l i on cu r ren t produced (J ) must be measured. The t o t a l i o n P
cu r ren t produced i s given as the sum o f t he i o n cu r ren ts leav ing the
plasma. i .e. ,
The acce lera tor g r i d impingement cu r ren t (Jimp) i s genera l l y l e s s than
one percent o f the beam cur ren t , and can probably be neglected. I n t h i s
inves t iga t ion , however, t he impingement cu r ren t was a1 ways i nc l uded i n
the t o t a l i o n product ion cu r ren t f o r compl eteness.
Measurements of the t h r u s t e r e l e c t r i c a l operat ing parameters were
made using the i o n source inst rumentat ion shown schemat ica l ly i n
Fig. B-1. The measurement of the i o n cu r ren t t o cathode p o t e n t i a l sur-
faces (JC) was accompl ished by b ias ing these surfaces 30v negat ive o f
cathode p o t e n t i a l . With t h i s bias, e l e c t r o n c o l l e c t i o n a t these surfaces
was e l im ina ted a l l ow ing the incoming i o n cu r ren t t o be measured. The
i o n cur ren t t o anode p o t e n t i a l surfaces (JA) and t o the cathode support
posts could n o t be measured w i t h t h i s technique. The measured i o n pro-
duct ion current, consequently, does no t i nc l ude these currents.
Three p o t e n t i a l l y s i g n i f i c a n t systematic e r r o r s associated w i t h
the measurements of J (and thus E and fB) have been i d e n t i f i e d . These P P
are :
1. Neglect ing the i o n cu r ren t t o anode po ten t i a l surfaces and the
cathode support posts.
2. Secondary e lec t rons emit ted by ions s t r i k i n g the negat ive ly
biased surfaces.
3. The presence of doubly charged ions.
The most ser ious o f these i s t he neglect of the ions reaching the
anode surfaces and the cathode support posts. The e r r o r i n J and E P P
r e s u l t i n g from t h i s omission i s d i f f i c u l t t o assess accurate ly . How-
ever, t he physical area o f these surfaces was 5% and 2%, respect ive ly ,
o f the t o t a l i n t e r i o r sur face area o f t he discharge chamber. Thus, one
might take 7% t o be the maximum systematic e r r o r i n J r e s u l t i n g from P
the neglected i o n currents. Because o f these neglected currents, the
Figure B-1. Ion Source Instrumentation Schematic
t r u e value of J should be l a r g e r than the measured value. This imp1 i e s P
t h a t the t r u e values of E and E* should be smal ler than the measured P P
val ues according t o Eq. 2. A 7% e r r o r i n J can resul t i n approximately P
a 13% e r r o r i n E* . It should be noted, however, t h a t Reference 43 P
through 46 i n d i c a t e t h a t t he e f f e c t i v e anode area f o r i o n c o l l e c t i o n a t
a magnetic f i e l d cusp should be less than the physical area. Conse-
quent ly, i t i s bel ieved t h a t the measured values of E and E* are no P P
more than 10% greater than the t r u e values. I n add i t ion , the t r u e
value o f fg would be expected t o be s l i g h t l y smal ler than the measured
value as a r e s u l t of the neglected i o n currents.
Secondary electrons, emi t ted as a resul t of ions s t r i k i n g the
negat ive ly biased surfaces, produce an e r r o r i n the measurement of the
i o n cu r ren t t o those surfaces r e s u l t i n g i n measured values of Jc t h a t
are l a r g e r than the actual values. The l a r g e s t secondary e lec t ron
y i e l d s a t low energies, f o r the ions and metal surfaces of concern i n
t h i s study, appear t o be f o r argon ions i nc iden t on a c lean molybdenum
sur face [41]. The secondary e lec t ron y i e l d f o r low energy (< 100 eV)
argon ions s t r i k i n g molybdenum i s approximately 12% [41]. If hal f of
the ions produced s t r i k e c lean molybdenum surfaces then the measured
value of J would be approximately 6% greater than the t r u e value as a P
resu l t o f secondary e lec t ron emission.
I t has been observed, however, t h a t the secondary e lec t ron y i e l d
o f a c lean surface decreases a f t e r exposure t o a i r [59,60]. Reference
60 concludes t h a t secondary e l e c t r o n y i e l d s are l ess than 1 % f o r low
i n t e n s i t y and low energy i o n i nc iden t on a metal e lec t rode which has
been exposed t o a i r and the ambient gas pressure i s i n the mtor r range
dur ing operation. I n . the course of the experiments i n t h i s inves t iga-
t i o n :he discharge chamber surfaces were f requent ly exposed t o a i r
between tes ts . I t i s be1 ieved t h a t the e r r o r i n the measurement of 3 P
resu l t i n g from secondary e lec t ron emission i s l ess than 1%.
I n the ca lcu la t ions of E , E* and fB i t i s assumed t h a t only s i n g l y P P
charged ions ex i s t . On the basis o f t h i s assumption, a doubly charged
i o n 1 eaving the plasma i s i n te rp re ted as two s i n g l y charged ions. The
effect, however, of these doubly charged ions on E , 6; and fB should P
no t be substant ia l f o r two reasons. F i r s t , the f r a c t i o n of doubly-to-
s i n g l y charged ions i s general l y small. Second, the energy required t o
produce one doubly charged i o n should not be v a s t l y d i f f e r e n t than the
energy required t o produce two s i n g l y charged ions. The presence o f
doubly charged ions does have a s i g n i f i c a n t e f f e c t on the value of the
neut ra l densi ty parameter, however, and t h i s problem i s addressed l a t e r
i n t h i s appendix.
Of the th ree systematic e r r o r s mentioned above on ly the neglect of
the i o n currents t o anode surfaces and t o the cathode support posts
appears t o be s ign i f i can t . The measured values of E and E* are, there- P P
fore, bel ieved t o be no more than 10% greater than the t r u e values as a
resul t of systematic measurement er rors .
The f o l l owi ng analys is was performed t o est imate the uncertai n t y
associated w i t h the experimental ly measured values of E The uncer- P '
t a i n t y i n e r e s u l t s from the uncerta inty i n the measurements of the P
independent variables which appear on the r ight-hand-side of Eq. 2. I n
general , the uncer ta in ty o f a quan t i t y y which i s a funct ion of the
measurabl e i ndependent v a r i abl es xl , x2, xj. . . xn i s given by [61],
where nxi i s the uncer ta in ty o f the i th independent var iable. For
t he plasma i o n energy cos t equat ion given by*,
Eq. B-1 becomes,
Car ry ing o u t the p a r t i a l d i f f e ren t i a t i ons , Eq. B-3 becomes,
To use Eq. B-4 the unce r ta in t i es i n the measurements o f the discharge
c u r r e n t (bJD), the discharge vo l tage (AVD) and the t o t a l i o n cu r ren t
produced (AJ ) must be determined. These unce r ta in t i es r e s u l t from the P
unce r ta in t i es of t he d i g i t a l meters used t o make the measurements and
any v a r i a t i o n i n the t h r u s t e r operat ing s e t p o i n t which may occur w h i l e
the data i s being recorded.
Simpson d i g i t a l panel meters were used t o measure the discharge
current , discharge vo l tage and beam cur ren t . These meters have an
unce r ta in t y o f 0.1 % p lus 1 d i g i t . The i o n cur ren t t o the negat ive ly
biased surfaces was measured us ing a K i e t h l y d i g i t a l mult imeter. This
meter has an unce r ta in t y o f 0.5% p lus 1 d i g i t . The v a r i a t i o n i n t he
t h r u s t e r opera t ing s e t p o i n t was taken t o be 1 d i g i t on each o f the
* This form o f Eq. 2 i s appropr iate f o r the case where JA may be neglected and the impingement cu r ren t i s inc luded i n JB.
d i g i t a l meters. I n add i t ion , 3 v was added t o the unce r ta in t y i n the
discharge vol tage because o f the vol tage drop across the thermonic
cathode wires. With these uncer ta in t ies , t h e unce r ta in t y i n e was P
ca l cu la ted usding Eq. B-4. The r e s u l t s o f these ca l cu la t i ons are shown
i n Fig. B-2 where the v e r t i c a l e r r o r bars represent t he measurement
uncer ta in ty . The ho r i zon ta l e r r o r bars represent the unce r ta in t y i n
the neu t ra l dens i t y parameter and i s discussed i n the next sect ion.
The data i n F ig. B-2 i s t he same as t h a t i n Fig. 6a. The unce r ta in t y
i n t he measured values o f e range from approximately 7 t o 11%. P
Measurement o f m(l -nu)
Three systematic e r r o r s e x i s t i n the measurement of t h e neu t ra l
dens i t y parameter, m(1-nu). The f i r s t i s t he e f f e c t o f doubly charged
ions on t h e p rope l l an t u t i l i z a t i o n e f f i c i e n c y . The second i s t he use
o f a gas f l o w meter w i t h argon, krypton and xenon gases t h a t has on l y
been c a l i b r a t e d f o r a i r . The t h i r d i s t he backf low o f neut ra l atoms
froni the vacuum chamber i n t o the discharge chamber through t h e g r i ds .
The presence o f doubly charged ions i n the beam leads t o a r t i f i c -
a1 l y h igh measurements o f t h e p rope l l an t u t i l i z a t i o n and, consequently,
low values o f the neu t ra l dens i t y parameter. Measurements o f t h e ++ + doubly- to-s ingly charged i o n beam cu r ren t JB /JB were made along the
t h r u s t e r cen te r l ine. Th i s in fo rmat ion can be used t o co r rec t the
propel 1 ant u t i 1 i z a t i o n according t o the equat ion.
++ + Since t h e value o f JB /JB a t t h e cen te r l i n e i s genera l l y h igher than
ARGON VD 50. OV
0 1 I I 1 I I I I
0.0 . 2 . 4 . 6 . 8 1.0 1.2 1 .4
NEUTRAL DENS 1 TY PARAMETER. i (1 -1,). (A mq. )
F igure B-2. Unce r ta in t y i n the Plasma I o n Energy Cost Measurements
the average value i t s use i n Eq. B-5 should tend t o over co r rec t the
p rope l l an t u t i l i z a t i o n s l i g h t l y . The r e s u l t i n g e r r o r i n the neu t ra l
dens i t y parameter i s be1 ieved t o be small, however.
A Teledyne Hastings-Raydst f low meter w i t h a range o f 0 t o 50
sccm was used t o measure the p rope l l an t f l o w r a t e i n t o the th rus te r .
The f l o w meter was c a l i b r a t e d f o r air , r e q u i r i n g the output readings t o
be a n a l y t i c a l l y corrected f o r operat ion w i t h the gases argon, krypton
and xenon. The use o f t h e a n a l y t i c a l co r rec t i ons r a t h e r than r e c a l i -
b r a t i n g the f l o w meter f o r the d i f f e r e n t gases i s expected t o in t roduce
on ly a very s l i g h t e r r o r i n the f l o w r a t e measurement. Thus, t h e
systematic e r r o r s i n the measurement o f t he neut ra l dens i t y para~iieter
are be1 ieved t o be negl i g i b l e .
The backflow of neut ra l atoms from t h e vacuum f a c i l i t y i n t o the
discharge chamber was ca lcu la ted based on measurements o f the f a c i l i ty
pressure made ~ l m downstream o f the th rus te r . These ca l cu la t i ons were
used t o co r rec t the t o t a l p rope l l an t f l o w r a t e i n t o the discharge
chamber.
The accuracy o f the f l o w meter i s 1% o f f u l l scale. This t u rns
ou t t o be the major unce r ta in t y i n the determinat ion o f the neut ra l
dens i t y parameter. The unce r ta in t y i n t he neut ra l dens i ty parameter
i s ind ica ted by the hor izonta l e r r o r bars i n Fig. B-2.
P l asma Property Measurement
Plasma proper ty measurements were made using one Langniuir probe
pos i t ioned along the discharge chamber center1 i n e and a second probe
pos i t ioned on a magnetic f i e l d cusp. De ta i l s o f the Langmuir probe
c i r c u i t r y used are given i n Reference 62. There are two major sources
o f e r r o r i n using Langmuir probes t o obta in plasma property data. The
f i r s t o f these deals w i t h obta in ing the probe t race i t s e l f ( i .e.,
measurement o f co l l ec ted cur rent versus probe vol tage) . The second
source o f e r r o r a r i ses from the ana lys is o f the probe t race t o ob ta in
the desi red plasma property information.
Er rors associated w i t h obta in ing the probe t race inc lude such
th ings as: the v a r i a t i o n i n work func t i on over the probe surface area,
secondary e lec t ron emission from the probe, probe i n s u l a t o r contamina-
t i on , noise i n the plasma, e l e c t r i c a l noise i n the probe c i r c u i t r y ,
plasma per turbat ion by the probe, and magnetic f i e l d e f f e c t s . Plasma
per turbat ion by the probe i s minimized by using a probe w i t h a small
surface area. Biasing the probe negative o f cathode p o t e n t i a l when
data i s no t being co l l ec ted tends t o sput te r clean the surface and
minimizes the v a r i a t i o n i n work func t ion over the probe surface area.
Changing the probe s u ~ p o r t i n s u l a t o r f requent ly ( s every 6 hours) m i n i -
mizes the problem o f i n s u l a t o r contamination. Noise i n the plasma was
minimized by using a D.C. cur rent heated therminoic cathode. Magnetic
f i e l d e f f e c t s were minimized by p lac ing the probe on the discharge
chamber cen te r l i ne where the magnetic f l u x dens i ty was l e s s than 0.001
tes la . For the probe posi t ioned on the magnetic f i e l d cusp, where the
magnetic f l u x dens i ty i s approximately 0.2 tes la , the magnetic f i e l d
e f f e c t s are substant ia l . I n t h i s case, on l y the e lec t ron temperature
and primary e lec t ron energy data obtained from these t races are be1 ieved
t o be meaningful.
The probe t races were analyzed us ing a non-l inear numerical curve
f i t t i n g r o u t i n e s i m i l a r t o the one described by Bea t t i e [6]. This
r o u t i n e assumes an e l e c t r o n populat ion t h a t i s charac.terized by a Max-
we1 1 i a n p lus mono-energetic (pr imary) e lec t ron energy d i s t r i b u t i o n .
Probe t races were d i g i t i z e d using an HP-7470A graphics p l o t t e r together
w i t h an HP-85 mini-computer. 'The p l o t t e r may be used as a d i g i t i z e r
when equipped w i t h a specia l d i g i t i z i n g s igh t . Accurate d i g i t i z i n g i s
c r u c i a l t o ob ta in ing good pr imary e lec t ron in format ion.
The e r r o r s in t roduced dur ing probe t r a c e d i g i t i z i n g and ana lys is
may be determined by generat ing i dea l i zed probe t races w i t h known
plasma p rope r t i es and then analyzing these t races i n the usual manner.
E igh t such idea l i zed probe t races were generated and analyzed, covering
a wide range o f plasma condi t ions. The r e s u l t s ind ica ted t h a t t he data
ana lys is can accura te ly d i s t i n g u i s h primary-to-Maxwel 1 i a n e lec t ron
dens i t y r a t i o s as small as 0.2% provided the Maxwell i an e lec t ron temper-
t u r e i s low. The most d i f f i c u l t t r aces t o analyze are those correspond-
i n g t o low primary-to-Maxwell i an e lec t ron dens i ty r a t i o s i n a plasma
w i t h a high Maxwell ian e lec t ron temperature. Th is p a r t i c u l a r plasma
condi t ion, however, was no t observed exper imenta l ly i n t he course o f
t h i s i nves t i ga t i on . The r e s u l t s i n Chapters 111, I V and Appendix A o f
t h i s r e p o r t i n d i c a t e t h a t when the primary-to-Maxwell i a n e lec t ron
dens i t y r a t i o i s small the Maxwell i a n e lec t ron temperature i s low and
when the e lec t ron temperature' i s h igh ' the primary-to-Maxwel 1 i an e l ec-
t r o n d e n s i t y r a t i o i s la rge . 'These types o f probe t races are accura te ly
analyzed w i t h the data reduct ion system used i n t h i s i nves t i ga t i on .
The i dea l i zed pr.obe t race ana lys is ind ica ted t h a t under most cond i t ions
t 'he f o l l o w i n g e r r o r s may be expected from the d i g i t i z i n g and curve
f i t t i n g procedures:
P l asma Potent ional f 0.3 v
Maxwell I a n E lec t ron Density, nM i: 5%
Maxwell i a n E lec t ron Temperature, TM + - 3%
Primary E lec t ron Density, n P
f 15%
Primary E lec t ron Energy, E P
+ 5%
The o v e r a l l unce r ta in t y associated w i t h .the p l asma proper ty measure-
ments are be1 ieved t o be the fo l l ow ing :
Plasma Po ten t i a l f 1 . 0 ~
Maxwell i a n E lec t ron Density, nM ? 20T
Maxwell i a n E lec t ron Temperature, TM + 10%
Primary E lec t ron Density, n ? 30% P
Primary E lec t ron Energy, E P
+ 10%
Measurement o f Doubly Charged Ions
Measurements o f doubly charged ions i n the bean1 were made using an -f + E x B probe s i m i l a r t o t h a t described i n Reference 42. E r ro rs associ-
a ted w i t h these measurements include: the accuracy o f the picoammeter
used t o read the probe cur ren t , se lec t i on o f t he proper probe d e f l e c t i o n
vol tage t o ob ta in the maxiniu~ii s ignal corresponding t o the p a r t i c u l a r
charged specie o f i n t e r e s t , and v a r i a t i o n i n the t h r u s t e r operat ing
cond i t i ons dur ing data c o l l e c t i o n .
Probe current readings were made using a Ke i th l y model 410A
picoamneter w i th an accuracy o f f 4% o f f u l l scale. Select ion o f the
proper probe de f lec t ion voltage i s a f a i r l y d i f f i c u l t t h i ng t o do
accurately because o f the slow response t ime o f the picoamneter
(between 0.4 and 12s). This i s espec ia l ly t rue f o r currents i n the
lowest ranges o f the amneter. Errors resu l t i v g from the uncerta inty
i n probe def 1 ect ion voltage and va r i a t i on i n thruster operating condi - t i ons are be1 ieved t o introduce a maximum e r ro r o f 10% i n thie measured
values o f the s ingle and double i on currents, The t o t a l uncertanties
i n the measured values o f the doubly-to-singly charged ion current
r a t i o , resu l t i ng from the sources o f e r r o r mentioned above, are i n d i -
cated i n Fig. 6-3 by the v e r t i c q l e r r o r bars. The hor izonta l e r ro r
bars indicate the uncer ta in ty i n the corrected propel l a n t u t i l i r a t i o n
e f f i c i ency as mentioned ear l i e r .
0.00 1 I I I I I I I I I I
0.0 . 2 . 4 . 6 . 8 1.0
PROPELLANT UTILIZATION. 1,
Figure 6-3. Uncer ta i nt.y i n the Doubly- to-Si ng l y Charged I o n Beam Current Measurements
APPENDIX C
NOMENCLATURE
Ag = Area of g r i d s through which the i o n beam i s ex t rac ted (m2)
Bo = mCo
Co = Primary e l e c t r o n u t i l i z a t i o n fac tor defined by Eq. 26 ( A eq.)'l
Ep = Primary E lec t ron Energy (eV)
e = E lec t ron i c Charge (1.6 x coul.)
fA = Frac t ion o f i o n cu r ren t t o anode surfaces
fB = Extracted i o n f r a c t i o n
fC = Frac t ion of i o n cu r ren t t o cathode surfaces
JA = Ion cu r ren t t o anode p o t e n t i a l surfaces (A)
JB = Ion beam c u r r e n t (A)
J: = S ing ly charged i o n beam cu r ren t (A)
J: = Doubly charged i o n beam cu r ren t (A)
JC = Ion cu r ren t t o cathode p o t e n t i a l surfaces (A)
JD = Discharge c u r r e n t (A)
JE = Cathode Emission Current (A) I
= Total product ion r a t e of exc i ted neut ra l atoms by pr imary Jex e lect rons - expressed as a cur ren t (A)
Jimp = Accelerator g r i d i o n impingement cu r ren t (A)
Jj = Product ion r a t e of j t h exc i ted s t a t e expressed as a cu r ren t (A)
JL = Primary e lec t ron cu r ren t t o the anode (A)
JM = Maxwell i a n e l e c t r o n cu r ren t t o t he anode (A)
P = Total i o n cu r ren t product ion r a t e expressed as a cu r ren t (A)
I
P = Ion production r a t e by primary electrons expressed as a
current (A)
J; = Production r a t e of singly charged ions expressed as a
current ( A )
J: = Production r a t e of doubly charged ions expressed as a current ( A )
J = Ion production r a t e by Maxwell ian electrons expressed as a p 9 m current ( A )
'e = Primary electron contai r~ment 1 ength (m)
Mf = Final spacecraft mass (kg)
Mg = Generator mass (kg)
Mi = In i t i a l spacecraft mass (kg)
Ma = Payload mass (kg)
Mpo = In i t i a l propellant mass (kg)
m e = Electron Mass (kg)
m i = Ion mass ( k g
m = Thruster propellant flow r a t e ( A eq.)
1;1 P
= Total spacecraft propel 1 ant flow r a t e (kg/s)
ne = Total electron density (m-3)
i = Total ion density ( n ~ ' ~ )
n M
= Maxwell ian electron density ( n ~ ' ~ )
n P
= Primary electron density (nr3)
no = Neutral atom density (w3)
io = Neutral atom loss r a t e - (A eq.)
Pg = Generator Power ( W )
P: = Primary electron r a t e fac tor fo r ionization of neutral atoms (m3/s)
Ptt = Primary electron r a t e factor fo r double ionization of 0 neutral atoms :(m3/s)
PT = Primary e lec t ron r a t e f a c t o r f o r the product ion o f double ions from s ing le ions (m3/s)
9: = Maxwell ian e lec t ron r a t e f a c t o r fo r i o n i z a t i o n o f neut ra l atoms (m3/s)
QF = Maxwell i a n e lec t ron r a t e f a c t o r fo r double i o n i z a t i o n o f neu t ra l atoms (11131s)
Q? = Maxwell i a n e lec t ron r a t e f a c t o r f o r the product ion o f double ions from s ing le ions (m3/s)
TA = Maxwell i a n e lec t ron temperature a t anode surface (eV)
TM = Maxwell i a n e lec t ron temperature i n bu lk plasma (eV)
t = Mission dura t ion ( s )
u = I o n exhaust v e l o c i t y (m/s)
U = Lumped e x c i t a t i o n energy (eV)
U+ = I o n i z a t i o n energy (eV)
= E x c i t a t i o n energy o f j t h exc i ted s ta te (eV)
U, = Lowest e x c i t a t i o n energy (eV)
VA = Anode sheath vol tage ( v )
VC = Plasma p o t e n t i a l from which e lec t rons emit ted by the cathode are accelerated t o become primary e lec t rons
VD = Discharge (Anode) vol tage (v )
VN = Net acce lera t ing voltage ( v )
V+ = Screen g r i d supply vol tage (v)
V - = Accelerator g r i d supply vol tage (v)
vb = Bohm v e l o c i t y (mls)
ve = E lec t ron v e l o c i t y (m/s)
v = Primary e lec t ron v e l o c i t y (m/s) P
vo = Neutral atom v e l o c i t y (mls)
+ = Volume o f i on product ion region (m3)
x = Independent va r iab le
y = Dependent va r iab le
a = Power p l a n t spec i f i c mass (kglw)
AJ D = Discharge cu r ren t unce r ta in t y (A)
A J p = I o n product ion cu r ren t unce r ta in t y (A)
AV = Charac te r i s t i c v e l o c i t y (mls)
A v ~ = Discharge vol tage unce r ta in t y (V)
Ax = Uncer ta in ty o f independent va r i ab le
A y = Uncer ta in ty o f dependent va r i ab le
EB = Average beam i o n energy cos t (eV1beam ion )
EM = Average energy o f Maxwell i a n e lec t rons leav ing the plasma
a t t h e anode (eV)
cP = Average plasma i o n energy cos t (eV/plasma ion)
6; = Basel ine plasma i o n energy cos t (e~ /p lasma ion)
e = Average plasma i o n energy cos t consider ing i o n i z a t i o n and O e x c i t a t i o n processes o n l y (eV)
nt = Thruster e l e c t r i c a l e f f i c i e n c y
llu = Prope l lan t u t - i l i z a t i o n e f f i c i e n c y
a = Tota l e x c i t a t i o n c o l l i s i o n cross sec t ion (m2) ex
a,' = Tota l e x c i t a t i o n c o l l i s i o n cross sec t ion a t the pr imary ex e l e c t r o n e n e r g y ( m 2 )
a = E x c i t a t i o n c o l l i s i o n cross sec t ion o f t h e j t h s t a t e (m2) j
a' = E x c i t a t i o n c o l l i s i o n cross sec t ion o f t he j t h s t a t e a t t he j pr imary e l e c t r o n energy (m2)
a' = To ta l i n e l a s t i c c o l l i s i o n cross sec t ion a t the pr imary O e l e c t r o n energy (m2)
at = I o n i z a t i o n c o l l i s i o n cross sec t ion (m2)
a' = I o n i z a t i o n c o l l i s i o n cross sec t ion a t the pr imary e lec t ron t energy (ni2)
@a = Transparency o f the acce lera tor g r i d to. neut ra l atoms
@ = Transparency o f the screen g r i d t o ions
4s = Transparency o f the screen g r i d t o neutral atoms
4 = E f f e c t i v e transparency o f the g r i d system t o neutral atoms
< > = Average over the e n t i r e e lec t ron energy d i s t r i b u t i o n funct ion
< >M = Average over the Maxwellian energy d i s t r i b u t i o n funct ion
