Full text of DOI:10.2514/2.5832

JOURNAL OF PROPULSION AND POWER
–
Vol. 17, No. 4, July August 2001
Computation of Hall Thruster Performance
L. Garrigues
´
Universite Paul Sabatier, 31062 Toulouse Cedex 4, France
I. D. Boyd†
University of Michigan, Ann Arbor, Michigan 48109
and
J. P. Boeuf‡
´
Universite Paul Sabatier, 31062 Toulouse Cedex 4, France
A quasi-neutral one-dimensional hybrid model of a stationary plasma thruster has been used to predict the
performance of a laboratory model thruster tested in the Propulsion Ionique pour Vols Orbitaux, Interpre´tation
et Nouvelles Expe´riences, facility. In this model ions are treated as particles and electrons as a  uid and quasi
neutrality is assumed. We describe the calculated efŽ ciencies, thrust, andspeciŽ c impulse using xenon and krypton
as propellant and their variations as a function of applied voltage and mass  ow rate. Typically, for a voltage of
350 V and an anode mass  ow rate of 5 mg/s, the thrust is on the order of 75 mN for xenon and krypton, and
the global efŽ ciency is 40 and 30%, respectively, for xenon and krypton. Although the model is rather simple,
the predicted trends are in reasonable agreement with the experimental measurements. The question of doubly
charged ions is also examined, and results show that they do not strongly contribute to performance in a stationary
plasma thruster-100 Hall thruster.
Nomenclature
ve; vi® ; v0
=
=
=
=
=
=
=
=
=
electron, ion, and neutral atom velocities, ms
axial position, m; axial velocity, ms
ionicity
single- and double-chargedions, 1, 2
ratio of thrust and ion  ux
thrust loss factor
ion transport time step, s
Dirac function
electron mean energy and average beam ion
energy, eV
1
1
x
; vx
A
B
=
=
=
=
=
=
=
=
=
=
area cross section of the thruster, m2
magnetic Ž eld, T
Z®
®
¯T ; ¯’
°
ion production term, m 3s
C®
d
1
dimension of the channel, m
1
E
electric Ž eld, Vm
electric charge constant, 1:602 10 19 C
t
1
±
e
fi®
g
ion distributionfunction
gravitationalconstant, 9.81 ms
speciŽc impulse, s
singly and doubly charged ion, total ion, and
2
"e; "b
Isp
Ji®
J
i
J
d
´; ´u ; ´a ; ´E
=
total, propellant, acceleration,and beam energy
efŽ ciencies
;
;
discharge currents, A
1
axial electron mobility, m2V 1s
ki kg;®
k
¹
=
=
e
;
;
=
=
=
=
=
ionization from single ion and from ground
m
i
ºi ; ºi®
frequenciesof ionization from singly charged
state and momentum exchange rates, m3s
1
1
k" kg
ion and from ground state, s
;
electron energy loss rates from singly charged
"
ion and from ground state, m3s
1
ºm ; ºwall; m
=
frequenciesof momentum exchange collision
1
and collisions with walls, s
me
M
;
electron mass, 9:1 10 31 kg; neutral atom
mass Xe 131.3 UMA and Kr 83.8 UMA
propellant mass  ow rate at the anode and ion
1
º"
’i®; 80
=
=
total electron energy loss frequency, s
m m®
ion  ux and neutral  ux in the anode plane,
;
i
1
2s
1
m
mass  ow rate at the exhaust, kg s
number of macroparticlescreated in a cell ,
1
N®j N® N®
j
!
=
electron angular cyclotron frequency, s
B
;
;
s
m
simulated, and minimal
ion, plasma, and neutral densities, m
ni®
Pd
rl ; vl; wl
n
N
3
;
;
=
=
=
p
a
Introduction
dissipated power, W
position, m; velocity, ms 1; and weight of a test
particle
(
)
N a stationary plasma thruster SPT , the ions are created and
accelerated in a plasma channel concentratedbetween two con-
centric dielectric cylinders. Xenon or krypton is injected through
the anode located at one end of the channel. The cathode is outside
the channel and provides the electron current entering the channel
I
S®j S®
j
(
)
;
=
=
ion production rate in a cell and maximal ion
max
1
production rate, m 3s
Sg;®
S
;
ion production rates from ground state and from
singly charged ion, m 3s
1
(
)
through the other end of the channel exhaust . Because of the low
gas pressure in the channel and the small dimensionsof the device,
a plasma can be createdin the channel only if the electronsare con-
Ž ned by a magnetic Ž eld. A radial magnetic Ž eld is created in the
channel by external coils and a magnetic circuit. The radial mag-
netic Ž eld is large in the exhaust region and decreases toward the
anode. The large value of the radial magneticŽ eld in the exhaustre-
gion induces a drop of the axial electron velocity. To maintain large
enough plasma conductivity, the axial electric Ž eld must increase
in the exhaust region. Electron heating in this Ž eld leads to strong
ionization of the neutral  ow. The ions, which are not sensitive to
the magneticŽ eld in the channel,are acceleratedby the electricŽ eld
in the exhaust region.
T T®
;
=
=
total, singly and doubly charged ion thrust, N
applied voltage, V
i
Vd
Received 22 January 2000;revisionreceived 13 December 2000;accepted
for publication8 Febuary2001.Copyright 2001by theAmerican Institute
c
of Aeronautics and Astronautics, Inc. All rights reserved.
Postdoctoral Research Associate, Centre de Physique des Plasmas et
Applications de Toulouse, 118 route de Narbonne. Member AIAA.
†Associate Professor, Department of Aerospace Engineering. Senior
Member AIAA.
‡Research Scientist at CNRS, Centre de Physique des Plasmas et Appli-
cations de Toulouse, 118 route de Narbonne. Member AIAA.
772

GARRIGUES, BOYD, AND BOEUF
773
The thrust and speciŽ c impulse of Hall thrusters are well adapted
for north south station keeping of geostationary telecommunica-
All of the ions created from the ground state are created with the
C®
–
neutral atom velocity v0.
includes also the effect of stepwise
tions satellites.The French space agency,Centre National d’Etudes
Spatialeswill launch an experimentaltelecom satellite Stentor with
two Hall thrusters:PPS-1350 and SPT-100.1 New programs involv-
ing SPTs are alsounderconsideration.First, Hall thrustersare being
ionization, with a lost term of single ions in the transport equation
for single-charged ions, and a source term of creation of double
ions in the doubly charged ions transport equation. A Monte Carlo
method is used to estimate this effect and is detailed subsequently.
The resolution of Eq. 1 gives the ion densities, velocities, and
 uxes for each species ®.
( )
considered for telecommunicationsconstellations in low Earth or-
2
(
)
bit the Skybridge program . A Hall thruster, the PPS-1350, will
–
be used for the Earth moon transfer of Smart1 ESA horizons 2000
The plasma density is obtained with the assumption of quasi
neutrality:
scientiŽc program.2
For these new applications, both larger and smaller devices are
required.In parallelto experimentalinvestigations,numericalmod-
els can help explain and optimize the physical processes in Hall
ni® x t
. ; /
( )
3
n
x t
p. ; /
®
3
6
thrusters. One-dimensionaltransient-hybridmodels of the SPT-
100 accelerationchannel have been developed recently.5;6 The pri-
mary components of the model we have developed5 include the
continuityequation for neutralswith a constantvelocityand ioniza-
tion loss term, a microscopic description of the ion transport, and
a  uid description of electrons. Charge neutrality is assumed, and
the axial distribution of the radial magnetic Ž eld proŽ le is given.
The model successfullypredicted the plasma oscillationsobserved
in this device.7 The predictions of this simple quasi-neutralhybrid
The equation of continuity for the neutral atoms with constant ve-
locity v0 and a loss term due to direct ionization gives the neutral
density:
N
x t
a . ; /
N x t
@ a . ; /
@
( )
4
n
x t N x t kg;® x t
p . ; / a . ; / . ; /
v0
i
t
x
@
@
®
The neutral atom density in the anode plane is
model have been qualitatively validated with a more accurate one-
8
( )
5
N t
a .0; / 80=v0
(
)
dimensionalparticle-in-cell PIC Monte Carlo collisionmodel. A
number of improvementsto the model have been added includinga
particle model of ion transportand doubly charged ions to examine
their effect on performance of the SPT-100.
where 80 is the neutral  ux at the anode plane deduced from the
gas mass  ow rate.
Because the plasma is supposed to be quasi neutral, the electric
Ž eld is not obtained from Poisson’s equation. The electric Ž eld is
In this paper, we focus on the prediction of the performance of a
SPT-100with the one-dimensionalmodel. We describe,Ž rst, the ap-
proximations and governing equations of the model; the numerical
techniques and data are discussed next. Then we deŽ ne the perfor-
mance of the Hall thruster. The calculations of the performance as
a function of voltage and mass  ow rate of xenon and krypton for
a SPT-100 are presented in detail and compared with experimental
results. Last are some brief conclusions.
(
deduced from the electron momentum equation with only the drift
)
term and from current continuity:
®
x t
’i . ; /
J t
T . /
( )
6
E x t
. ; /
n®i x t
x t
. ; /¹e. ; /
n
x t e x t
p. ; / ¹e. ; /
®
( )
The ion  uxes and densities come from Eq. 1 ; the relation
Approximations and Governing Equations
d
The one-dimensionalhybridmodel is based on the same assump-
tions as the model of Ref. 5. In the present work doubly charged
ions are included in the simulations. This group of ions is treated in
the same manner as the singly direct ionization of atoms. We use a
creationsourceterm correspondingto directelectronimpact double
ionization from the ground state of xenon and krypton atoms. The
( )
7
Vd
E x t x
. ; / d
0
( )
gives, with Eq. 8 , the total current.
The electron mobility is describedusing the classical mobility in
a transverse magnetic Ž eld:
(
contributionof stepwise ionization electronimpact of Xe leading
)
toXe andKr leadingtoKr
isalsoincluded.We neglecttheef-
e
x t
ºm . ; /
( )
8
x t
¹e. ; /
fect of triply chargedions becausethe thresholdsof ionizationfrom
2
2
me
x t
x
ºm . ; / !B . /
ground state are 70 and 90 eV for xenon and krypton,9 respectively.
transportof singlyand doublychargedparticlesis described
The ion
When the electronsionize the neutralsin the regionnearthe exhaust
plane of the thruster, the neutral density decreases to zero. In this
region of low neutral density, the typical maximum plasma density
(
)
by following the trajectoriesof a sample of ions PIC model rather
than by solving numerically the Vlasov equation as in Ref. 5.
)
3
and electron temperature are 1012 cm and 20 eV, respectively.
The assumptions of the model are as follows. 1 There is quasi
)
neutrality of the plasma. 2 Ion transport is described with a one
–
If we estimate the electron ion collisions frequency, we obtain an
)
5
1
dimension in space and velocity collisionless PIC method. 3 The
–
–
electron ion collision frequency of 10 s with an electron ion
cross section given by the Coulomb approximation. The electron
energy is large enough that the electron ion collisions are negligi-
ble the Coulomb cross section varies as 1=
10
electric Ž eld is obtained from the current density, using the elec-
)
tron momentum equation. 4 The electron mobility perpendicu-
–
T2
lar to the magnetic Ž eld is classical and includes the effects of
–
(
)
. The electron ion
e
–
–
(
electron neutral collisions as well as electron wall collisions in
collisions are not sufŽ cient to permit the electron transport in the
axial direction in the zone at the exit of the channel. In the exhaust
region, the momentum exchange frequency decreases to zero and
so does the electronconductivity.We do not obtaina stable solution
)
)
a phenomenological way . 5 The electron distribution function
is Maxwellian, the electron mean energy being obtained from a
simpliŽed energy equation.
for Eq. 6 5 if no otherassumptionis made on the electrontransport.
( )
The hybridmodel is, therefore,based on the followingequations.
The effect of the magnetic Ž eld is negligiblefor the ions, and so the
ion transport equation is given by
To make possible the electron transport in the direction perpen-
dicular to the magnetic Ž eld even when the neutral density goes to
zero, we have added to the electron collision frequency a contri-
bution due to collisions with the walls. This near-wall conductivity
was suggestedby Russian research11 to explainthe anomalouselec-
f ®
f ®
e
f ®
@
i
@
@
Z®
i
i
( )
1
E
C®
vx
t
x
M
@
@
@vx
–
tron transport in the SPT. We account for electron wall collisions
( )
the right-handterm in Eq. 1 takes into accountthe creationof ions
by a sourceterm of direct ionizationof the groundstate.This source
in the simulation by introducinga supplementaryconstant collision
5
–
Sg;®
frequency. It is difŽ cult to estimate the electron wall momentum
term
can be written as
–
interaction frequency ºwall; m . It is clear that the electron wall colli-
Sg;® x t
n
x t N x t kg;® x t
p. ; / a . ; / . ; /
( )
2
. ; /
sion frequencydepends of the electron energy, which varies axially
i

774
GARRIGUES, BOYD, AND BOEUF
but also depends of the unknown axial proŽ le of the Debye poten-
V
tial D . In this conditions it is difŽ cult to predict the real variation
–
of the electron wall collision frequency as a function of the axial
proŽ le. We adjusted this frequencyto obtain a calculated current in
reasonableagreementwith the experimentalresults.12 In the results
–
presented later, the electron wall frequency is constant in the en-
tire channel and has a value of 2 106 s 1. This value permits us
to successfullycompare calculationsand experimentalresults con-
(
cerning the in uence of the applied potential on the current shape
13
)
and amplitude of the oscillations for an SPT-100-ML.
The electron mean energy is obtained from the following
simpliŽed energy equation:
x t
5 @"e. ; /
x t
º". ; /
( )
9
eE x t
. ; /
x t
"e. ; /
x
@
x t
ve. ; /
3
Fig. 2 Ionization and energy loss rates calculated with a Maxwellian
electron distribution function in krypton.
with an energy gain term due to the electric Ž eld and a loss term
–
–
due to electron atom and electron single ion collisions and to the
kg
" . The momentum exchangerate is estimated to be 2:5 10
k
m
13
–
electronsecondaryemissionby electronimpact.The electron atom
13
1
and 2:0 10 m3s for xenon and krypton, respectively.
energylosses correspondto inelasticprocessesincludingionization
–
( )
Equation 9 is solved using a classical fourth-order Runge
–
(
)
single and double . The electron ion energy losses come from the
(
)
Kutta method see Ref. 18 . We use an upwind scheme to solve
ionizationof single-chargedions. We take into accountthe effect of
the secondaryelectronemissionby electronimpactin a phenomeno-
( ) 19
( )
Eq. 4 . In the model of Ref. 5, Eq. 1 was solved with an upwind
schemeand a time-splittingmethod.The upwindscheme is a simple
method but rather diffusive.20 In the calculationspresentedhere, we
used a Monte Carlo particle method to describe the collisonlession
transport.
(
)
logical way more details may be found in Ref. 5 .
We discuss next the numerical techniques and the data we used
to solve this set of equations.
21
The channel is divided into cells. As in the PIC technique,
Data and Numerical Techniques
we consider macroparticles, where each macroparticle represents
(
The numerical technique requires the excitation, ionization sin-
7
(
)
a large number of ions more than 10 . The trajectory of a test
)
gle and double , andmomentumexchangecrosssectionsbyelectron
l
particle number is integrated using Newton’s law:
impact on xenon and krypton atoms. Also needed is the ionization
cross section of Xe and Kr leadingto Xe and to Kr , respec-
drl dvl
( )
10
M
Z e
E
®
vl ;
(
tively, by electron impact. The inelastic cross sections excitation
t
d
t
d
)
and ionization are included in the electron energy loss frequency
( )
of Eq. 9 . The ionizationcross sectionsof the ground state are used
The classic leap-frog scheme is used for the resolution of these
equations.22
( )
to estimate the ion production rates of Eq. 2 and the loss term
of neutral atoms in Eq. 4 . The stepwise ionization cross sections
are used to estimate the stepwise ionization rate. The momentum
exchange cross section is used in the electron mobility [Eq. 8 ].
The inelastic cross sections single direct ionizationand excitation
come from Puech and Mizzi14 for xenon and from data of Date et
al.15 for electron impact on krypton. The doubly direct ionization
( )
t
At each time step 1 of the ion motion, some macroparticlesare
S®
created in each grid cell dependingon the ion productionrate
of
j
( )
the species considered.The number of test particles created in each
cell is determined as follows:
(
)
N®j
S®j Sm®ax Ns® Nm®
(
)
11
(
)
cross sections from the ground state for xenon and krypton come
from Wetzel et al.9 The crosssectionsfor ionizationof Xe and Kr
are deduced from Achenbach et al.16 and Defrance et al.17 Ioniza-
tion and energylossrates are obtainedby assumingthatthe electron
distributionfunctionis Maxwellian.A comparisonof the ionization
The Ž rst term is directly proportional to the production rate. The
difŽ culty comes from the difference between the maximal and the
4
(
)
minimal production rates a factor of 10 or more . The second
term permits us to keep a minimum number of particles in each cell
and to avoid a null ion density in a cell. A weight is assigned to
k
kg;®
k
kg
rates and
and energy loss rates and in xenon and kryp-
"
k "
kg;®
ton are given iin Figs. 1 and 2, where
and
are, respectively,
i
l
S®
and
j
the macroparticle proportional to the ion production rate
i
i
inversely proportional to the number of simulated ions created in
the stepwise ionization rate and the direct ionization rates from the
ground state ® 1 for single direct ionizationand ® 2 for double
direct ionization . The energy loss rate for ionizationof a single ion
is , and the total energy loss rate for electron atom collisions is
N®
the cell
:
j
(
)
S®
j 1
t
Nj®
(
)
12
wl
–
k
"
Random numbers are used to distribute the test particles uniformly
in the cell. The velocity of the macroparticlescreated by ionization
is v0 for ionization of atoms from ground state.
To limit the total number of ions in the simulation we use the
N®
followingprocedure.The
ions are created duringa time interval
j
t
1 , which does not necessarily coincide with the integration time
t
step of the ion trajectories but can be larger. The 1 for ion gener-
ation is chosen large enough so that the total number of simulated
ions leaving the device during this time interval is slightly larger
than the total number of simulated ions generated by ionization.
This avoids the random elimination of low-weight macroparticles
(
)
e.g., see Ref. 23 .
To take into account the stepwise ionization by electron impact
on singly charged ions, after the integration of a single particle test
l
t
P
during 1 , a probability of collision is estimated. In our case,
c
Pc
1, so this probability can be simply written as
Fig. 1 Ionization and energy loss rates calculated with a Maxwellian
electron distribution function in xenon.
(
)
13
Pc
t
ºi 1

GARRIGUES, BOYD, AND BOEUF
775
(
np ki
P
c
r
is compared with a pseudorandom number
a
with ºi
.
and the propellant efŽ ciency ´u giving the ionized fraction of the
)
r
P
uniformly weighted between 0 and 1. If
c, then ionization
neutral mass at the exhaust is
a
occurs, and a new double ion test particle of weight wl is created
l
with the velocity and the position of the test particle .
mi®
m
(
)
´
i
19
The ion density is calculatedon the nodes of the grid by a simple
linear-weightingscheme.24 The mean velocities and  uxes of each
species of ion are also estimated on the nodes of the grid. The ion
mean velocities make it possible to deduce the velocity of the ion
®
We can deŽ ne the beam energy efŽ ciency by the relation
(
)
beam [Eq. 14 ], and the ion  uxesare used for the calculationof the
( )
(
)
electricŽ eld [Eq. 6 ] and to obtain the thrust of the SPT [Eq. 14 ].
In the calculations presented in this paper, typically 25,000 test
particles are used. The computational time is 2 h on a 400-MHz
personal computer to simulate 400 ¹s.
(
)
eU
"b=
´
20
E
d
We can also describe the global efŽ ciency as in the case with only
25
singly charged ions with the expression
Comments on the Boundary Conditions
( )
21
´
´u´a ´E ° 2
As mentionedin Ref. 5, the resultsof the model are dependenton
the boundaryconditionsfor the ion densityat the anode.We impose
a nonzero ion  ux and an ion velocity equal to v0 at the anode.
In the results presented in this paper we impose an ion density of
The thrust loss factor ° includes exhaust beam divergence and
proŽ le losses.3
–
(
)
Then, from relations 17 21 , we can deduce the beam energy
3
1011 cm next to the anode. A parametric study has shown that
efŽ ciency ´E with singly and doubly charged ions:
values of this imposed ion densitylarger or equal to 1011 cm 3 does
not stronglyaffect the total currentof the dischargein the exitplane.
This supplementarycurrent is then on the order of few milliampere
(
(
)
)
eU
´
´ ´a ´u° 2 "b=
22
23
E
d
(
)
which is low comparing to the 4 A of the total current . If a much
lower ion density at the anode is used, then a large anode electric
Ž eld develops in the anode region and the model does not give
realistic results. This problem is partly due to the quasi-neutrality
with
2
®
1
.1 ¯T /2
.1 2¯’ /.1 ¯’/
M
vi
"
b
( )
assumptionand to that the calculatedelectricŽ eld [see Eq. 6 ] goes
2
to inŽ nity when the plasma density goes to zero. The simulation of
this region has been simpliŽ ed neglecting the diffusion term in the
electron momentum equation. In this region of low electric Ž eld,
the diffusion term can permit an inversion of the electric Ž eld and
then a sustainmentof the plasma as observedin PIC simulationsof
Hall thrusters.8 We also underestimate the density of the atoms in
the region at the end of the injector typically by a factor of 10 due
to the small area of the injector hole. The plasma density is then
also underestimatedin this region due to the underestimationof the
ionization source term. Work is underway to improve the model
description in the anode region.
2
®
i
1
Ti®
T ®
writing ¯T
=
1 and ¯’
’
2=’i®
.
If the doubly chargied ions are neglected, we obtain the classical
formula
2
®
vi
1
( )
24
M
"
2
b
Results and Discussion
We performed calculations to study the effects of the propellant
composition xenon vs krypton , of the applied voltage and of the
mass  ow rate on the performanceof a Hall thruster. The simulated
device is the SPT-100-ML. This SPT-100 Hall thruster has been
tested in the Propulsion Ionique pour Vols Orbitaux, Interpre´tation
et Nouvelles Expe´riences PIVOINE facility. More details on the
design of this thruster and on the PIVOINE facility are found by
Be´chu et al.26
(
)
Hall Thruster Performance
The thrust,the speciŽ c impulse,and the totalefŽ ciencycharacter-
T
ize the performanceof the SPT. The thrust , in newtons,represents
(
)
the reaction force to the expansion of the propellantmass. The spe-
I
ciŽ c impulse sp, expressedin seconds,is the durationfor which the
thruster can provide a thrust of 10 N with 1 kg of propellant.
We have neglectedthe effects of neutrals in calculatingthe thrust
because they have a negligible velocity compared to that of ions.
Thus the thrust is the sum of the product of the ion mass  ow rate
at the thruster exhaust multiplied by the ion exhaust mean velocity
for each species:
The axial proŽ le of the radial magnetic Ž eld used in all of the
calculationsis shownin Fig. 3. The cross sectionareaof the thruster,
2
A
, is 40 cm .
The neutral atoms are injected with a velocity of 3 104 cm/s
in the case of xenon and 3:76 104 cm/s in the case of krypton, to
keep the same thermal energy of the emitted atoms.
The SPT produces spontaneous oscillations of the current.7 The
performance of the thruster also oscillates in time and all of the
results presented in this paper are averaged in time.
®vi®
mi
14
Ti®
(
)
T
®
®
where the ion mass  ow rate can be expressed as a function of the
ion  ux at the exhaust of the SPT
®
i
mi®
M A
(
)
’
15
The speciŽ c impulse is deduced from the thrust:
(
)
Isp T mg
=
16
P
d
Jd V
d , canbewritten
andthetotalefŽ ciencyofthethruster,with
as
T 2 m P
(
(
)
)
´
2
17
18
d
The accelerationefŽ ciency ´a is given by
Fig. 3 Axial distribution of the radial magnetic Ž eld used for the sim-
ulations; anode is at x = 0 and the exhaust at x = d.
J
i
J
=
d
´
a

776
GARRIGUES, BOYD, AND BOEUF
Table 1 Calculated efŽ ciency of the SPT100-ML for 5 mg/s
of xenon and for different discharge voltages
Vd , V
´u , %
´a , %
´E , %
´, %
150
200
250
300
350
400
44.1
72.5
85.5
89.7
94.4
96.6
95.6
96.5
95.5
96.3
95.6
95.3
50.5
46.0
46.1
47.6
43.0
40.0
21.3
32.2
37.6
41.0
38.8
36.8
Calculations for Xenon
We present in Table 1 the efŽ ciencies for applied voltage in the
range 150 400 V for a xenon mass  ow rate of 5 mg/s at the anode.
As shown in Table 1, the thrust efŽ ciencyincreaseswith the applied
voltage, with a peak for a voltage of 300 V. In the one-dimensional
model, neitherthe ion beam divergencenor the ion lossesto the wall
are taken into account, and the ° factor is, therefore, set to 1 [see
Fig. 4 Total efŽ ciency as a function of discharge voltage; xenon mass
 ow rate is 5 mg/s: comparison between measurements26 and calcula-
tions.
–
(
)
Eq. 21 ]. In the accelerationzone, where the magneticŽ eld is large,
the electric Ž eld must increase to compensate for the low electron
mobility. The ions are accelerated by this electric Ž eld toward the
exhaust.5 It is clearthat, to minimizeion recombinationon the walls,
the accelerationchannellengthshouldbe kept short.Our model can
not, however, properlydescribethis aspectbecauseion losses to the
walls are not considered. The ion loss on the wall chamber in an
SPT is relativelylow comparedwith the ion losses on the grid of an
electrostaticion thruster.25
We note that the acceleration efŽ ciency is relatively insensitive
to the applied voltage, at around 95%. A similar behavior of the
accelerationefŽ ciency as a function of the applied voltage has been
observed experimentally on a 70-mm thruster with a boron nitride
Fig. 5 Thrust as a function of discharge voltage; xenon mass  ow rate
is 5 mg/s: comparison between measurements23 and calculations.
channel.27 The experimental current ratio
=
J
J
varies from 64 to
d
i
(
)
67% for two xenon mass  ow rates 1.04 and 1.53 mg/s and for
voltage between 150 and 500 V. We also note that beam energy ef-
Ž ciency decreases with the applied voltage. Some experimental re-
Figure 4 shows a comparison between calculations and experi-
mental results. Note that no adjustment of the magnetic Ž eld, or
other external parameters, is made to optimize the global efŽ ciency
of the SPT-100-ML in the experiment of Fig. 4. The thrust efŽ -
ciency is maximum for a channel voltage of 300 V with a value
of 50% in the experiments and 40% in the simulation. For larger
voltages, the engine efŽ ciency decreases due to the decrease of ´E
in the calculations.Results obtained by the model underpredictthe
(
)
sults of the beam efŽ ciencyreach a value around 75% Ref. 28 . We
(
)
underestimated this value in our calculations see Table 1 mainly
due to the strong oscillations of the thruster. In this range of volt-
age, and for large enough magnetic Ž eld in the anode region, some
oscillations with a frequency around 100 kHz appear. The high-
frequency oscillations are associated with a maximum of electric
Ž eld moving at a high speed from the anode to the acceleration
zone.8 The distributionof the ionization source term in the channel
is correlated with the maximum of electric Ž eld. Calculations of
time average spatialdistributionof the ionizationsource term show
a repartition spread out in a zone crossing the drop of the electric
potential. A signiŽ cant fraction of the ions are created between the
middle and the end of the thruster channel. The ion beam average
energyis stronglyaffectedby the ions createdon the last centimeter
of the thruster, which are accelerated by only a part of the applied
potential.
–
–
(
)
thrust of the SPT-100-ML by 14 50% for xenon and 8 30% for
krypton, see next paragraph . We have neglected in our model ion-
(
ization mechanisms other than direct ionization from ground or
)
singly ionized states . The effect of ionization of excited states on
the ion current and on the propellant efŽ ciency could be important.
We have also neglected the dissipated power in the magnetic coils
and in the heating of the hollow cathode. This power dissipation
could be nonnegligibleat low voltages.
Figure 5 shows the variation of the thrust for discharge voltages
between150and400V. As expected,thethrustincreaseswiththeap-
plied voltage.The differencebetween experimentsand calculations
is around 15 mN.
Calculations of electron transport have shown that the electron
distributionfunctionisnotMaxwellianduetotheeffectofsecondary
electron emission by electron impact on ceramics in Hall thrusters
SPTs are attractivebecause they are efŽ cient within the optimum
(
)
see Ref. 29 . Work is underway to have a better description of
–
(
range of speciŽ c impulse for station keeping of satellites 1000
the electron distribution function to better estimate the ionization
) (
)
2000 s Ref. 25 . We obtain a speciŽ c impulse of 1500 s for an
( )
rate that in uences the calculation of the source term [Eq. 2 ].
applied voltage of 350 V in the simulation, and 1700 s is obtained
Effectssuchasthe magneticŽ eldin theregionoutsideofthe thruster,
which still have nonnegligiblevaluesgivingan acceleratingelectric
Ž eld out of the channel as observed by laser-induced  uorescence
preliminary measurements on the SPT-100-ML thruster,30 is also a
way to improve the modeling of the ion beam.
26
in the experiments.
Finally,Fig. 6 shows the total thrust efŽ ciencyas a functionof the
thrust. The maximum calculated efŽ ciency is about 10% less than
the measured one and is obtained for a thrust around 75 mN, that is
15 mN lessthan the thrustat maximum efŽ ciencyin the experiment.
For high thrust, the globalefŽ ciency decreases,as observedin other
SPTs.31
The total efŽ ciency dependsstronglyon the propellantefŽ ciency
(
)
see Table 1 . When the voltage is low, the ionizationof the neutral
 ux is incomplete. For large values of the applied voltage, the frac-
tion of the neutral  ux that is ionized by electrons can be as large
The resultsof calculationof the fractionsof the differentcurrents
J®
are summarizedin Table2.
1 representsthe ioncurrentofsingle-
(
is the doubly charged ion current sum of
(
as 97% in the calculations see Table 1, where ´u is 96.6% for a
i
2
J®
)
channel voltage of 400 V . The strong in uence of the propellant
charged ions and
i
efŽ ciency on the total efŽ ciency has been observed experimentally
the ion currents coming from the ionization of ground state and of
on Israeli and Japanese Hall thrusters.27;31
single-chargedions . As can be seen, the total ion current increases
)

GARRIGUES, BOYD, AND BOEUF
777
Table 2 Calculated ion currents for 5 mg/s of xenon
and for different discharge voltages
Table 4 Calculated efŽ ciencies for 5 mg/s of krypton
Vd , V
´u, %
´a, %
´E , %
´, %
Vd , V
Ji® 1, %
Ji® 2, %
Ji , A
200
250
300
350
400
45.4
66.3
75.8
84.3
85.3
96.6
97.2
97.4
97.0
96.8
44.1
40.8
39.3
36.7
33.3
19.3
26.3
29.0
30.0
27.5
150
200
250
300
350
400
95.6
91.8
89.0
88.9
85.7
84.0
4.4
8.2
11.0
11.1
14.3
16.0
1.7
2.8
3.3
3.5
3.8
3.9
Table 3 EfŽ ciencies obtained with the one-dimensional hybrid
model for xenon and a discharge voltage of 300 V
m, mg/s
´u , %
´a , %
´E , %
´, %
3.0
4.0
5.0
6.0
7.0
74.3
81.6
89.7
95.1
95.1
96.4
96.0
96.0
95.6
95.3
44.3
47.2
47.6
49.3
51.1
31.7
37.0
41.0
44.8
46.3
Fig. 7 Discharge power and thrust as a function of mass  ow rate for
300 V.
Fig. 6 Total efŽ ciency as a function of thrust.
as the voltage increases. This behavior is due to the increase of the
(
)
ionization efŽ ciency see Table 1 . Note that the fraction of Xe
decreases from 95.6 to 84.0% when the voltage varies from 150 to
400 V. The fraction of Xe as expected increases with the voltage
Fig. 8 Total efŽ ciency as a function of discharge voltage in the SPT-
100-ML in krypton; krypton mass  ow rate is 5 mg/s: comparison be-
tween measurements26 and calculations.
–
(
)
4.4 16% . Around 50% of the Xe current comes from stepwise
ionization of Xe ; the contributionof doubly charged ions in these
conditions is low, only few millinewtons on the total thrust. It does
not explainthe underestimationof the calculatedthrustcomparedto
measurements.The differencein magnitude between the ionization
thethrustwiththe mass ow rate.Thislinearbehaviorofthe thrustas
a functionofthemass owratehasbeenobservedexperimentallyfor
differentHall thrusters,such as for small SPT33 and for a laboratory
Hall thruster studied in Israel,34 for example.
(
)
rate of the ground state and that of stepwise ionization see Fig. 1
is compensated for by the difference between the atom and ion
single-chargedensities.
We also studiedthe in uence of the mass  ow rate on the thruster
m
For
7 mg/s, the thrust reaches 100 mN, but we also see that
the dischargepower is rather large, on the order of 1.7 kW. The ion
total current is on the order of 1.7 A for 3 mg/s and reaches 5.5 A
for 7 mg/s.
(
)
performance for a constant channel voltage of 300 V see Table 3 .
The engineefŽ ciencyincreaseswith the mass  ow rate. The fraction
(
of the doubly charged ion current is quasi constant around 12% of
Calculations for Krypton
)
the total current in the range of mass  ow rate studied. The beam
energy efŽ ciency is due to the strong oscillationsof the distribution
of the potentialinsidethe channeland to the repartitionof the source
term in the channel.The accelerationefŽ ciencyis not very sensitive
to the increase of the mass  ow rate of xenon. The propellant efŽ -
ciency increaseswith the mass  ow rate. This is because the source
terms of production of ions are improved when the density of the
Table 4 shows the performanceof the SPT-100-MLwith a 5-mg/s
 owrateofkryptonandcanbecomparedwithTable1corresponding
to xenon. The total efŽ ciency is also represented as a function of
voltage in Fig. 8 for krypton.
The behaviors of the acceleration and beam energy efŽ ciencies
compared to the xenon case are the same. As in the xenon case, the
voltagestronglyin uencesthe propellantefŽ ciency´u . Experimen-
tal results of Komurasaki and Arakawa31 show that the propellant
efŽ ciency of xenon is about two times larger than that of argon.
These experimental results were obtained for voltage that ranged
( )
propellantincreases[see Eq. 2 ]. The propellantefŽ ciency reaches
–
m
95% for
6 7 mg/s, and ´ reaches a value around 46%. Exper-
imental results from Raitses et al.32 give both speciŽ c impulse and
thruster efŽ ciency increasing with the mass  ow rate, for a 70-mm
thruster of a channel length of 2 cm, for a voltage of 300 V, and for
(
from 100 to 250 V and for a lower mass  ow rate than our case less
–
)
than 2 mg/s . The higher propellant efŽ ciency of xenon is related
( )
to its larger ionization rate Figs. 1 and 2 . Moreover, the neutral
xenon  ow rate in the range 1 2.2 mg/s.
Figure 7 shows the thrust and the input power for a xenon mass
 owratebetween3and7mg/s. We can seea quasi-linearvariationof
(
)
. We
M
atom density is lower for Xe than for Kr it varies as 1=

778
GARRIGUES, BOYD, AND BOEUF
Table 5 EfŽ ciencies obtained with the one-dimensional
The calculationsshow the strongin uence of the voltage and the
mass  ow rate for xenon and krypton on the propellant utilization.
The model predicts a thrust of around 75 mN in typical conditions
hybrid model for krypton as a function of mass
 ow rate for a discharge voltage of 300 V
–
(
)
350 V, 5 mg/s for an SPT-100 in xenon, a value about 10 15 mN
m, mg/s
´u , %
´a , %
´E , %
´, %
lower than the measured one. The calculatedthrust in krypton is on
the order of 75 mN in the same conditions. The calculated efŽ cien-
cies are on the order of 40 and 30% in xenon and krypton respec-
3.0
4.0
5.0
6.0
7.0
47.5
64.7
77.4
82.8
86.5
97.4
97.6
97.4
97.3
97.1
40.2
36.6
37.0
38.9
41.4
18.6
23.1
27.9
31.4
34.7
(
tively, in reasonable agreement with the measured values which
)
are about 10% higher . The explanationof differences between the
two gases comes from the higherionizationrate for xenon. The dis-
crepanciesbetween experimentsand calculationsare not due to the
effectof doublychargedions,which playa minor roleas we can see.
It seems that the contribution of ionization of excited states could
play a more important role due to the low thresholdof creation and
ionization of these excited states.
Futureworkconcernsa morereŽ nedmodelingoftheanoderegion
with the diffusion term in the electron momentum equation and
an extension of the computational domain taking into account the
region of the nonnegligiblemagneticŽ eld after the exit plane of the
thruster.
Acknowledgments
This work was performed in the framework of the Groupe-
ment de Recherche Centre National de la Recherche ScientiŽque
(
)
(
)
(
CNRS /Centre National d’Etudes Spatiales CNES /Socie´te´ Na-
)
tionaled’Etudeset deConstructionde Moteurd’Avion SNECMA /
Fig. 9 Thrust as a function of discharge voltage; krypton mass  ow
rate is 5 mg/s: comparison between measurements26 and calculations.
(
)
OfŽ ce National d’Etudes et de RecherchesAe´rospatiales ONERA
2232 “Propulsion a` Plasma pour Syste`mes Spatiaux.” L. Garrigues
wants also to acknowledge the Ž nancial support of CNES. The
authors want to thank the Laboratoire d’Ae´othermique, CNRS
–
have kept the same neutral  ow rates at the anode to compare our
calculations to the experimental results. In the case of krypton, the
UPR9020 in Orle´ans for sharing the experimental data.
globalefŽ ciencyincreaseswith thevoltageto a maximum efŽ ciency
25
References
(
)
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(
)
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–
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the order of 1300 s for
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–
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(
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–
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–
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–
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