Modelling
ling of Plasma in Electric Propulsion
Seminar Report submitted by
Name:
Reg. Number:
Course code:
Semester/Batch:
Mentor:
Deeksha Rao
17ETAS012007
ASC406A
7/2017
Dr. Mahesh K Varpe
B. Tech in Aerospace Engineering
Department of Aerospace Engineering
Ramaiah University of Applied Sciences
University House, Gnanagangothri Campus, New BEL Road, M S R Nagar, Bangalore,
Karnataka, India - 560 054
Declaration Sheet
Student Name
Deeksha Rao
Reg. No
17ETAS012007
Programme
B. Tech. (Aerospace Engineering)
Course Code
ASC406A
Course Title
Seminar
Course Date
10th September 2020
to
Batch 2017
5st March 2021
Declaration
The seminar report submitted herewith is a result of my own investigations and that I
have conformed to the guidelines against plagiarism as laid out in the Student
Handbook. All sections of the text and results, which have been obtained from other
sources, are fully referenced. I understand that cheating and plagiarism constitute a
breach of University regulations and will be dealt with accordingly.
Signature of the
Date
student
Name
Signature
Date
First Examiner
Second Examiner
Mentor
MEC406A-Seminar
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Contents
Declaration Sheet..................................................................................................................... i
LIST OF TABLES ........................................................................................................................ v
LIST OF FIGURES ...................................................................................................................... v
ABSTRACT ................................................................................................................................ 1
1
Introduction and scope of work ..................................................................................... 2
1.1
Introduction: ......................................................................................................... 2
1.2
Motivation: ............................................................................................................ 2
1.3
Definition of Important terms ............................................................................... 3
2
Literature Review ........................................................................................................... 4
2.1
Working of Hall Thruster ....................................................................................... 4
2.2
Different Models of Plasma .................................................................................. 4
2.2.1
Kinetic Model ................................................................................................. 5
2.2.2
Fluid Model .................................................................................................... 5
2.2.3
Hybrid Model ................................................................................................. 6
2.3
3
Design Implications ............................................................................................... 7
Details of the Topic ........................................................................................................ 8
3.1
History ................................................................................................................... 8
3.2
Implications ........................................................................................................... 8
3.3
Working of the Hall Thruster................................................................................. 8
3.4
Different Models of Plasma .................................................................................. 9
3.4.1
Kinetic Model ............................................................................................... 10
3.4.2
Fluid Model .................................................................................................. 11
3.4.3
Hybrid Model ............................................................................................... 12
3.5
Design of Hall thruster ........................................................................................ 12
3.6
Facts and Figures ................................................................................................. 14
3.7
Data Analysis ....................................................................................................... 14
MEC406A-Seminar
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3.8
4
Results ................................................................................................................. 16
Challenges and Opportunities ...................................................................................... 16
4.1
Challenges ........................................................................................................... 16
4.2
Opportunities ...................................................................................................... 16
5
Conclusion and Suggestions for Future Work .............................................................. 18
5.1
Conclusion ........................................................................................................... 18
5.2
Future Work ........................................................................................................ 18
6
REFERENCES ................................................................................................................. 19
MEC406A-Seminar
iii
LIST OF ABBREVIATIONS AND NOMENCLATURE
1D
1 Dimensional
2D
2 Dimensional
3D
3 Dimensional
PIC
Particle in Cell
MCC
Monte Carlo Collision
SEE
Secondary Electron Emission
MHD
Magneto-hydrodynamic
HET
Hall effect thruster
MEC406A-Seminar
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LIST OF TABLES
Table 2-1 .............................................................................................................................. 4
Table 2-2 .............................................................................................................................. 4
Table 2-3 .............................................................................................................................. 5
Table 2-4 .............................................................................................................................. 5
Table 2-5 .............................................................................................................................. 6
Table 2-6 .............................................................................................................................. 7
Table 3-1 Comparison of data............................................................................................ 14
LIST OF FIGURES
Figure 1-1 Plasma................................................................................................................. 2
Figure 1-2 Hall thruster and Plasma modeling .................................................................... 3
Figure 3-1 Schematic Representation of Hall thruster ........................................................ 9
Figure 3-2 Flow chart representing explicit PIC cycle(3) ................................................... 10
Figure 3-3 Coupling of the Fluid solver and electro-dynamic solver (4) ........................... 11
Figure 3-4 Iterative Design process for Hall Thrusters (6) ................................................. 12
Figure 3-5 SPT 100 ............................................................................................................. 14
Figure 3-6 Comparison of Electric Field predicted by authors(7,8) ................................... 15
Figure 3-7 Comparison of Number Density predicted by authors .................................... 15
Figure 3-8 Comparison of Normalized Magnetic Field ...................................................... 15
MEC406A-Seminar
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ABSTRACT
This paper presents an overview of the recent advances made in the field of plasma
modeling in reference to the electric propulsion. Hall thrusters are an advanced plasmapropelled electric propulsion device. Modeling of the plasma discharges in Hall thrusters are
gaining traction in the Engineering Field, as the design of Hall thrusters require the modeling
of the Plasma. Thus, it has become important for the Engineering Community to understand
plasma dynamics involved in the construction and analysis of the Hall thruster. The main
objective is to provide engineers with a comprehensive guide which can help choose the
best model for a given parameters and conditions.
Numerical models have disclosed several physical mechanisms, present in the functioning of
the Hall Thruster. It acts as a bridge between the analytical and experimental studies. There
are 3 different approaches that exist for the creation of numerical model: a kinetic model
which uses kinetic description of the particles at a microscopic level and are described using
particle velocities; a Fluid model which uses macroscopic quantities such as density, energy
and velocity, and assumes plasma to be quasi-neutral and collision less and is set in a
macroscopic level; a Hybrid model which uses best features of both the models, it also leads
to computational efficiency.
Fluid and Hybrid models are quantified for performance parameters and validated against
the measured values. The fluid model yields least error in the performance parameters. This
model is used in design of Hall thruster.
Keywords: Hall Thruster, Plasma Modeling, Kinetic Model, Fluid, Hybrid approach
ASC406A-Seminar
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1
1.1
Introduction and scope of work
Introduction:
Plasma, the fifth state of matter is one of the wonderful states which is has many uses
ranging from electric propulsion to tokomak reactor to Welding. This versatile state of
matter is also found in stars.
Plasma refers to a soup of positive ions, neutrals and electrons, which is quasi-neutral in
nature.
Figure 1-1 Plasma
Modelling this state of matter is multi-physics affair. Modelling of plasma helps us to
mathematically characterize several applications and helps improve performance of the
application.
A few applications of Plasma are: Fusion reactor, Electric propulsion and plasma welding.
Electric Propulsion is the future of propulsion. Plasma modelling is an integral part of Ion
and Hall thrusters
1.2
Motivation:
Electric propulsion is one of the most sought-after technologies in the Space realm. There
are several types of electric propulsion systems. Ion thrusters, Hall thrusters and MPD
thrusters usually use plasma as a means of producing thrust. Plasma propulsion systems
lead to greater exhaust speeds. Hall thrusters are very efficient and competitive electric
propulsion devices for satellites and are currently in used in a number of
telecommunications and government spacecraft. Thus, there is a demand for plasma
modeling and analysis.
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Figure 1-2 Hall thruster and Plasma modeling
There is a plethora of engineering applications which require plasma modeling. Thus,
looking into the wide array of plasma modeling techniques from an Engineering perspective
is the main motivation for conducting this study.
1.3
Definition of Important terms
Debye length: It is a characteristic length over which ions and electrons can be separated in
a plasma
It represents the physical scale of the transition from plasma continuity to individual particle
behaviour.
Larmour Radius: This is the radius of circular motion and charged particle in presence of a
uniform magnetic Field
Quasi-neutrality: It refers to the state of plasma, that at macroscopic level is neutral(i.e.,
density of ions = density of electrons)
Cold Plasma: It refers to the plasma, where ions and neutrals are in a much lower
temperature, when compared to electron temperature
1.4. Issues
Plasmas in electric propulsion devices, even in individual parts of a thruster, can span orders
of magnitude in plasma density, temperature, and ionization fraction. Therefore, models
used to describe the plasma behavior and characteristics in the thrusters must be formed
with assumptions that are valid in the regime being studied.
ASC406A-Seminar
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2
2.1
Literature Review
Working of Hall Thruster
Table 2-1
Sl
Paper
Model Type
Salient Feature
Reference
no.
1
D.M.
Goebel,
I.
Katz, Single
particle
(2008), Fundamentals of Model,
Fluid
electric propulsion: Hall Model
Basics,

2
J.P.
thrusters
Boeuf,
and
and Hall thrusters

parameters
Cathodes
Plume modelling
 Classic
transport
Hall thrusters, Journal of Various
mechanism
121, mechanisms
011101
and
electron
 Kinetic, Fluid and
present in Hall
Hybrid
Thruster,
Basics
Case
[1]
Basic Performance
Hollow
Tutorial: Physics of Hall
Physics,
of
the Basics of Ion
Physics and modeling of Thruster,
Applied
deep
understanding
and ion thrusters Wiley, Working of Hall
Hoboken
A
[2]
Model
studies
2.2
Different Models of Plasma
Table 2-2
Sl no Paper
1
Model type
Tang, Haibin and York, Concepts
Thomas
M.
Salient Features
in
(2015), Plasma physics,
 Numerical models of
different techniques
Introduction to Plasmas sheaths,
 Applications
and
 Case studies
With
Plasma
Reviews
Applications
Propulsion,
ASC406A-Seminar
Dynamics. collisions
in
Reference
[3]
of
Space
Magnetic
4
Fusion and Space Physics ,
Academic Press
2
Gianpiero
Colonna, Kinetic Models,
Antonio D’Angola, Plasma Fluid
Models
Modeling Methods and and
Hybrid
Applications, IOP Plasma approach
 Numerical
techniques
for
[4]
various models
 Stability analysis
Physics Series
2.2.1 Kinetic Model
Table 2-3
Sl no
Paper
Model type
1
Domínguez-Vázquez, A.,
Taccogna, F. y Ahedo, E.
(2018).
Particle modeling of radial
electron dynamics in a
controlled
discharge of a Hall
thruster. Plasma Sources
Science and
Technology, 27(6)
1D(z) PIC MCC
Salient Features
Reference
 Particle Model used
Model
in radial section of
Hall
thruster
in
acceleration region
 Secondary
[5]
electron
emmission
2.2.2 Fluid Model
Table 2-4
Sl
Paper
Model type
Salient features
Reference
no.
1
Kwon
Kybeom, Walker 1D Fluid, Self-
Mitchell L R, and Mavris consistent
Dimitri N, (2011), Selfconsistent,
one-
dimensional analysis of
the Hall effect thruster,
Plasma
ASC406A-Seminar
Sources
Sci.



1D, Self sufficient,
Macroscopic
A
collisionless
electron
diffusion
region
A
collisional
dominant
[7]
electron
diffusion region.
5
Technol. 20 045021
2
Manzell
David,
(), 1D Fluid
Simplified
Numerical
Description
of
SPT

Simplified Model

Easy
implementation
[11]
Operation, IEPC-95-34
2.2.3 Hybrid Model
Table 2-5
Sl no Paper
1
K.
Model type
Hara,
Kolobov
D.
Boyd,
Vladimir,
and 1D
Salient Features
Hybrid Improved performance
(2011), Vlasov
Hybrid-Vlasov Simulation For
[8]
Hall Thrusters, 2nd Annual
Graduate
Reference
Student
Symposium, (Ann Arbor, MI)
2
Mikellides I G, Katz I, Mandell 1D Hybrid
1-D computer model
M J, and Snyder J S (2001), A
has been developed in
1-D Model of the Hall-effect
the interest of
thruster with an Exhaust
investigating plasma
Region,
behavior in the
Proc.
7th
AIAA/ASME/SAE/ASEE
Joint
acceleration channel
Propulsion Conf. (Salt Lake
and region
City, UT) (Washington, DC:
downstream of the
American
Hall-Effect Thruster
Institute
Aeronautics
of
[9]
and
Astronautics)
AIAA-2001-
3505
3
Boeuf, J. P. and Garrigues, 1D Hybrid-PCC
Quasi-neutral plasma
L.,(1998),
column,
Low
frequency
oscillations in a stationary
ASC406A-Seminar
[10]
Electron energy with
6
plasma thruster, Journal of
Applied
Physics,
vol.
no temporal evolution
84,
3541-3554
2.3
Design Implications
Table 2-6
Sl no Paper
1
Model type
Enrico A. De Marco and 2D
Salient features
Reference
magnetic Optimization of Design
Mariano Andrenucci (2008), Circuit,
1D
Hall Thrusters Design and Plasma Model
Optimization,
44th
AIAA/ASME/SAE/ASEE Joint
Propulsion
Conference
[6]
&
Exhibit (Hartford, CT), AIAA
2008-4805
ASC406A-Seminar
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3
3.1
Details of the Topic
History
The Hall thrusters have been in development fro over last 50 years, making it a recent
advancement. Plasma modeling started with gusto in the late 1990s, starting with onedimensional models along the axial direction. Along with that, some of the one-dimensional
models along the radial direction have been used to simulate the effect of SEE under the
impact of high-energy electrons on ceramic walls of the channel on plasma properties. Twodimensional models that account for axial and radial directions often channel and near-field
regions have been developed since the early 2000s to better capture plasma properties and
expansion. More recently, a two-dimensional model along the axial and azimuthal directions
has been proposed (5). Recent advances in this field have lead to the development of
sophisticated tools which has made plasma modelling more user-friendly. Advent of
commercial codes like ANSYS Maxwell and COMSOL Multiphysics, in the field of plasma
modelling has made it easier and more graphic for the easy understanding of the field.
3.2
Implications
Hall thrusters have good Specific Impulse. This leads to less consumption of propellant and it
have very good life time. There no indigenously built hall thruster in India. Thus, in business
point of view, there is market for Hall thrusters in India without any competitors.
Since, Hall thrusters use electricity and Xenon to produce thrust. There are no harmful
Pollutants during emission. Thus, it is eco-friendly.
3.3
Working of the Hall Thruster
Geometry of the Hall thrusters has been shown in the above figure. It consists of 1
cylindrical channel with n electromagnet in the middle and 4 more electromagnets at the
corner of the thrusters. There is anode plate at one of the thrusters and Hollow cathode
perched on top of the other edge of the Hall thrusters. The insides of the channel are coated
with dielectric materials, usually Boron nitride. Nobel Gases are usually used a propellants.
Xenon is the most preferred propellant for the Hall Thruster (1).
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Figure 3-1 Schematic Representation of Hall thruster
Hall thrusters are electromagnetic thrusters, which propels, using the Lorentz Force and the
Hall Effect.
Magnetic Field is applied in radial direction, whereas Electric field is applied in axial
direction. The electric and Magnetic field is applied perpendicular to each other.
Neutral Xenon gas is sent through injector into the channel. On application of the electric
field, plasma is generated. The electrons emitted from the Cathode ionize the gas. The
continually applied Electric field lowers the electron conductivity (2). The electrons are
confined by the magnetic field, whereas the Ions are accelerated to produce the Thrust.
The Ion beam is then neutralized by the Hollow Cathode present on the Outer edge of Hall
Thruster.
3.4
Different Models of Plasma
There are 3 different types of plasma modelling techniques:
1.
Kinetic Model
ASC406A-Seminar
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2.
Fluid Model
3.
Hybrid Model
3.4.1 Kinetic Model
Kinetic Model has 2 different models:
1.
Particle models
2.
Boltzmann Vlasov equations
Particle Models model the single particle motion of one species and then uses those
equations to model the species as the sum of th
the individual particle motion.
For this to be used in a Hall thruster, an electron motion is modelled in electric and
Magnetic field. And then it is extrapolated to include the whole species. This is considered
as Particle in Cell model. It uses Maxwell’s equations and Newtonian physics in equal
measure. Numerical model used for this method is called a leapfrog algorithm. Values of
Current density and charge density are aassumed
ssumed and then uses Maxwell’s equations to
compute electric and magnetic fields at next half time step. In the other half of time step,
the obtained fields are used to calculate position and velocity vectors. Thus, the cycle
continues, until desired results are achieved.
Figure 3-2 Flow chart representing explicit PIC cycle(3)
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Boltzmann Valsov system has 7 dimensions system to be calculated. They are probabilistic in
nature. The Boltzmann’s equation can be used in conjunction with Maxwell’s equations to
get distributions of particles and distribution of properties along axis.
3.4.2 Fluid Model
Fluid models consider the plasma as a continuum. There is no need to resolve Debye length
in Fluid modelling as Fluid Models are of 2 types:
1. Single Fluid Model
2. Two fluid Model
Single fluid model assumes plasma to be a single fluid and uses Navier-Stokes equations.
Densities are replaced by plasma number density. Mass fluxes are replaced by electric and
Magnetic fluxes. Thus, we need 2 different solvers. This is MHD Formulation.
Figure 3-3 Coupling of the Fluid solver and electro-dynamic solver (4)
Fluxes are assumed and then fed to the Fluid dynamic solver, this solver gives, velocity,
pressure, temperature and number density as output. This ouput serves as input for the
electrodynamic solver. This cycle continues until desired result is achieved.
Plasma consists of electrons, Ions and neutrals. Two fluid model considers electrons as a one
fluid and Ions as another; but this doesn’t change the dimensionality of the model.
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3.4.3 Hybrid Model
The hybrid approach is the used to bridge the gap between Kinetic and Fluid approach.
The Ions are modelled using the Kinetic approach. Electrons are modelled using the fluid
approach
This helps with chemical reaction in the Hall thruster. This is computationally better than
the Kinetic Models and Gives better accuracy than the Fluid approach
3.5
Design of Hall thruster
Figure 3-4 Iterative Design process for Hall Thrusters (6)
Hall thruster has a few conditions which need to be fulfilled for selecting the length of the
channel in a HET.

Load initial profiles for the neutral density, neutral velocity, plasma density and ion
velocity.

Calculate initial estimation of the electron temperature profile.
ASC406A-Seminar
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
Solve the neutral and ion equations from the anode (with fixed electron temperature
profile).

Solve for the discharge current by imposing the potential fall from the anode to the
exit.

Solve the electron temperature equation from the exit to the anode.

Compare the electron temperature distribution since convergence.
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3.6
Facts and Figures
Figure 3-5 SPT 100
Thruster operating Conditions:
Mass flow rate: 5.2 kg/s
Inlet neutral Velocity: 175 m/s
Discharge Voltage: 300 V
Discharge Current: 4.5 A
Maximum Magnetic Field:200 G
3.7
Data Analysis
Table 3-1 Comparison of data
The performance Data of SPT 100 was taken from several different papers to quantify the
Measured
Data(11)
Variable
Discharge
Current
Thrust efficiency
Thrust
Specific Impulse
Breathing mode
frequency
Unit
Id
ηt
T
Isp
A
mN
s
4.50
0.50
83.00
1600.00
w
kHz
17.00
Self Consistent(7)
Error (in
Value
%)
4.76
0.49
82.00
1728.00
5.78
2.00
1.20
8.00
1D
Hybrid-Vlasov(8)
Error (in
Value
%)
Hybrid PIC(9)
Error (in
Value
%)
3.94
0.55
78.90
1547.00
12.44
10.00
4.94
3.31
3.76
0.53
79.00
1548.00
16.44
6.00
4.82
3.25
18.00
5.88
20.00
17.65
Hybrid(10)
Error (in
Value
%)
3.70
0.60
90.20
1500.00
Model
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17.78
20.00
8.67
6.25
Electric field V/m
Electric field
50000
45000
40000
35000
30000
25000
20000
15000
10000
5000
0
-5000 0
Hybrid Model
Fluid Model
0.02
0.04
0.06
Axial Distance m
Figure 3-6 Comparison of Electric Field predicted by authors(7,8)
Plasma Densities
Plasma/Ion Number density
14
12
10
8
6
4
2
0
Hybrid Model
Fluid Model
0
0.02
0.04
0.06
Axial Position
Figure 3-7 Comparison of Number Density predicted by authors
Normalized Magentic Field
Variation of Magnetic Field
1.2
1
0.8
0.6
Hybrid Model
0.4
Fluid
0.2
0
0
0.5
1
1.5
x/l
Figure 3-8 Comparison of Normalized Magnetic Field
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3.8
Results

Both Electric and Magnetic Fields are at a maximum at the acceleration zones.

High Magnetic fields are required for the confinement of the electrons

Fluid model shows high density at the start of acceleration zone when compared to
the hybrid model as Fluid considers whole of the Plasma and hybrid considers
electrons and Ions separately
4
4.1
Challenges and Opportunities
Challenges
- Modelling of Plasma Physics
Boltzmann equation is made with a system of 7 dimensions. This shows the complexity in
the field of Plasma physics. Modelling these equations in conjunction with the Maxwell’s
equations involves a lot of hard work and mathematics
- Requirement of High Computational Resources
In a kinetic model to obtain good results, particle of the order 1011 is required. But most
studies use particles of the order 108. This type of study involves a lot of computational
resources.
- Limited Experimental Data
Experimental data available is very limited as building a Hall thruster very expensive and
requires lot of theoretical work.
4.2
Opportunities
- Enhance predictive capabilities of modelling
There have not been many advances in this field. Thus, there are a lot of new avenues to
explore in this theoretical sense of plasma physics
- Advent of high computational power, may aid in complex Plasma modelling
Now the computational Power is increasing exponentially, thus simulations can run for more
iterations in less time. And complex modelling techniques also can be now used.
- Surge in Plasma-related applications.
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The Plasma related applications are now increasing rapidly. New technologies, like plasma
gun, Plasma stealth sheath for Military aircrafts are gaining momentum. Thus, increasing the
importance of plasma modeling.
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5
5.1
1
Conclusion and Suggestions for Future Work
Conclusion
Research is conducted using 1D Models so that the models can be understood at an
preliminary level
2
The data obtained from several papers show that 1D Models are also capable of
providing accurate results necessary for the Design Process (6)
3
Kinetic are accurate, but computationally expensive. Kinetic Solvers are required if we
need to quantify all the mechanisms in the Plasma thruster. Due to unavailability of
data in the field of Kinetic modelling, the performance parameters couldn’t be
compared to available data from Fluid and Hybrid Solvers. However, 2D Kinetic Data is
available.
4
Fluid/Hybrid Solvers are used by Engineers for getting performance parameters
5
Fluid Models can be used with add-ons to accurately predict performance parameters.
Fluid Models accurately predict performance parameters compared to Hybrid Model.
Fluid Models are the computationally least expensive, but they cannot capture micromechanisms of Plasma
6
While Fluid/Hybrid models have readily available solvers, Kinetic solvers are not that
accessible
5.2
Future Work
In an Ideal situation, all the models mentioned, needs to be evaluated under the similar
computational domain and initial conditions, so that it can be compared fairly. This
overview can be coded using MATLAB Codes to obtain better solution and comparison.
There are comparatively less studies undertaken in the 1D Kinetic methods of the Plasma
modelling of Hall thrusters, pertaining to its performance. A study can be carried out to
obtain performance parameters from 1D Kinetic equation for Plasma modelling in Hall
thrusters and its performance.
There is less research in the area of Hall thruster design and optimization, Research can be
conducted in this field as well.
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6
REFERENCES
[1] Goebel, D.M. and I. Katz, (2008), Fundamentals of electric propulsion: Hall and ion
thrusters Wiley, Hoboken
[2] J.P. Boeuf, Tutorial: Physics and modeling of Hall thrusters, Journal of Applied
Physics, 121, 011101
[3] Tang, Haibin and York, Thomas M. (2015), Introduction to Plasmas and Plasma
Dynamics. With Reviews of Applications in Space Propulsion, Magnetic Fusion and
Space Physics, Academic Press
[4] Gianpiero Colonna, Antonio D’Angola, Plasma Modeling Methods and Applications,
IOP Plasma Physics Series
[5] Taccogna, F., Garrigues, L., (2019), Latest progress in Hall thrusters plasma
modelling. Rev. Mod. Plasma Phys. 3, 12 https://doi.org/10.1007/s41614-019-0033-1
[6] Enrico A. De Marco and Mariano Andrenucci (2008), Hall Thrusters Design and
Optimization, 44th AIAA/ASME/SAE/ASEE Joint Propulsion Conference & Exhibit
(Hartford, CT), AIAA 2008-4805
[7] Kwon Kybeom, Walker Mitchell L R, and Mavris Dimitri N, (2011), Self-consistent,
one-dimensional analysis of the Hall effect thruster, Plasma Sources Sci. Technol. 20
045021
[8] K. Hara, D. Boyd, and Kolobov Vladimir, (2011), Hybrid-Vlasov Simulation For Hall
Thrusters, 2nd Annual Graduate Student Symposium, (Ann Arbor, MI)
[9] Mikellides I G, Katz I, Mandell M J, and Snyder J S (2001), A 1-D Model of the Halleffect thruster with an Exhaust Region, Proc. 7th AIAA/ASME/SAE/ASEE Joint
Propulsion Conf. (Salt Lake City, UT) (Washington, DC: American Institute of
Aeronautics and Astronautics) AIAA-2001-3505\
[10] Boeuf, J. P. and Garrigues, L.,(1998), Low frequency oscillations in a stationary
plasma thruster, Journal of Applied Physics, vol. 84, 3541-3554
[11] Manzell David, , Simplified Numerical Description of SPT Operation, IEPC-95-34
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