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Introduction
I: Background
Most people have a pretty decent understanding of matter. They know three of
its four states (solid, liquid, and gas). However, many are unaware of the fourth state of
matter, plasma. What separates the four states of matter is the amount of energy required
for each state. Plasma requires the most energy of the four states of matter. In the
plasma state, electrons and ions that make-up the atoms in matter are separated due to the
excess energy. This energy allows particles to overcome their attractive forces. These
particles remain in this state until they lose enough energy. After they lose their energy,
electrons and ions combine and form an atom.
Due to its high enrgy, plasma has some useful applications. One plasma
application is an alternate means of propulsion instead of using conventional chemical
thrusters. One of the most efficient types of plasma propulsion is a Stationary Plasma
Thruster (SPT). A SPT uses a closed electron drift, also known as a Hall thrust, and an
extended acceleration zone. The electrical discharge of a SPT has a cross field (ExB)
configuration with a radial magnetic field perpendicular to an axial electric field. The
electric field is used to accelerate the ions with high velocities. The axial Eletric field
and the radial magnetic are used to increase the path of the electrons entering the SPT
(later explained).
II: Thrust and Impulse
Thrusters are used for interplanetary travel due to the vacuum in space. This
vacuum forces satellites and space shuttles to focus on Newton’s Third Law of motion
(Forces exerted between two bodies are equal magnitude and opposite direction) as a
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means to travel through space. This means that when a shuttle 's thrust is in one
direction, the shuttle will be pushed in the other direction. Thrust is defined as the force
on a vessel as a result of the escape velocity (u) of the mass flow rate ( m˙ ) of the fuel
moving in the opposite direction (equation 1). Since the u of the fuel is constant, the
thrust is a result of the m˙ (more gas released, stronger the thrust).
Thrust = -u
dm
= -um˙
dt
(1)
The fuel efficieny of a thruster can be calculated using the specific impulse (Isp).
The Isp of a thruster can be found by divinding the thrust force by the m˙ and the
gravitational acceleration (g) (equation 2). The Isp displays how long a thruster can
maintain a thrust at the rate it releases its fuel. An ideal thruster would have a long Isp
due to a strong thrust and a small m˙ .
Isp =
Fth
m˙ g
(2)
These factors will be used to show a comparison between two types of thruster,
the SPT and the conventional chemical thruster.
III: Plasma Propulsion vs. Conventional Thrusters
The conventional chemical thruster, expels gas from its fuel supply in order to
propel it through space. This thrust is dependent on the rate the fuel (mass) is expelled
from the thruster. A large m˙ produces a strong thrust. There are a few problems with the
conventional thruster. It can only reach terminal velocities ranging from 2 to 5
kilometers per second and requires a large mass of fuel, which is used up very rapidly. In
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fact, a large percentage of the mass of a space shuttle is due to the mass of the propellant
fuel (a gas).
The SPT does not need of a massive fuel source. The SPT is able to propel
satellites with less fuel than the conventional thruster. The SPT can reach a velocity of
10 to 30 kilometers per second. However, the thrust from the SPT is very small. The
thrust ranges from 0.1 to 1 Newton. This is barely enough force to push a piece of paper
let alone a shuttle with a very large mass. However, the minute thrust produced by a
thruster is sufficient for station keeping a satellite, which is orbiting the earth with a
constant orbit.
Despite having a larger thrust, the chemical thruster has a small Isp (approx. 200s)
and is not able to maintain its thrust as long a time as the SPT (Isp approx. 3000s). The
longer Isp of the SPT enables it to reach a higher terminal velocity (Graph 1).
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Geometry of the Thruster
I: Geometry
Unlike the conventional thruster, whose thrust is dependent on the rate the fuel is
used, the operation of the SPT is determined by the geometry of the thruster. To increase
the thrust, the electric potential (f) or the physical parameters of the plasma thruster may
be increased. By increasing the f, a larger electric field is created (later explained how),
which is used to produce a greater ion exit velocity.
The SPT was designed with the shape of an annular chamber (Fig 1). The back of
the chamber is an anode wall with gas inlets. The inlets are where the gas propellant
(Xenon) is released into the annular chamber (Fig 1).
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The cross-sectional area of the exit plane at the end of the SPT is calculated using
the inner and outer radius of the thruster (equation 3). The inner walls of the thruster are
coated with a ceramic shielding to repel the ions and electrons moving within the
chamber. The center rod, inner wall, and the outer walls of the chamber are wrapped in
magnetic coils (Fig 3). These coils are used to create the magnetic field in the chamber.
A = p (r22 - r12 )
(3)
The exit plane of the thruster has a cathode gun (Fig 3). The cathode gun has two
functions for the SPT. One is to release a primary electron into the annular chamber,
which is used to create a hall thrust. The other use is to fire a neutralization beam at the
ions that accelerated through the exit plane of the thruster. The neutralization beam is
composed of electrons. This function is to prevent the released ions from being attracted
to the chamber.
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II: Magnetic Field
To produce a Hall thrust for the SPT, a radial magnetic field is necessary. To gain
a radial magnetic field, the SPT was designed with an annular shape. This design is a
result of a topological transformation of a permanent magnetic rod (having a magnetic
field pointing in one direction) circled by other permanent magnetic rods (each with a
magnetic field pointing in the opposite direction) (Fig 4a).
To produce a magnetic field without using permanent magnets, Biot-Savart's
law is used. Based on the theory of magnetostatics (Steady currents produce time
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independent magnetic fields), Biot-Savart's law (equation 4) states that a current carrying
wire will produce a magnetic field perpendicular to the current. The rods are then
wrapped with magnetic coils. The direction of the magnetic field is decided based on the
right hand rule (using the right hand, fingers curled around rod represent the direction of
the current while the thumb represents the direction of the magnetic field).
B=
mo
dl ¢ ¥ rˆ
IÚ
4p
r2
(4)
The maximum magnetic field is measured at the end of the rods. The maximum
magnetic field at the end of the center rod will move to the ends of the surrounding rods
(Fig 5).
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After the topological transformation of the outer rods, the annular chamber has
magnetic coils wrapped around the center rod, the inner wall, and the outer wall (Fig 4ab). The currents of both the inner wall and the center rod move in the same direction.
Operation of the Thruster
The SPT propels satellites by ionizing xenon gas particles and accelerating them
through the exit plane of the thruster using the crossing electric and magnetic field of the
thruster. Starting at the exit plane of the thruster, the cathode gun releases two electrons
(primary and neutralization electrons). The primary electrons in the chamber are used to
ionize xenon gas particles released into the chamber from the anode inlets.
However, the velocity of the electrons in the chamber is too fast. The probability the
primary electrons will collide with a gas particle is very small. Instead of trying to slow
the primary electrons, the path the electron travels in the chamber is increased. The
radial magnetic field is used to force the electrons to move azimuthally in the thruster
(Fig 6). The primary electrons remain in their rotational positions until they strike
another particle. Upon collision with a particle, the primary electrons will move closer to
the anode wall where they will be absorbed. The Loramir radius of the electrons
becomes larger as the magnetic field decreases from the electron moving to the back of
the annular chamber. This allows the primary electrons to be more susceptible to
collisions and less likely be found down the chamber (equation 5). However, there will
be a large collection of primary electrons around the exit plane of the thruster where the
Loramir radius is small. The primary electrons will ultimately remain in the annular
chamber until they collide with a xenon gas particle. After the collision, the particle will
immediately ionize and release a secondary electron. The primary and secondary
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electrons will strike the anode wall and be absorbed while the ionized xenon particle will
immediately be accelerated through the exit plane due to the electric field.
rL =
u^
u
= -^
wc e B
me
(5)
After the ions are accelerated through the exit plane, they will be attracted back to
the SPT. To prevent the ions from applying an attractive force on the SPT (defeating the
purpose of the thruster), the cathode gun fires a neutralization beam at the ions. The ions
combine with the electrons (from the neutralization beam) and become a neutral particle.
This prevents the ions from returning to the thruster.
I: Magnetic and Electric Fields
Now important questions arise. How are the magnetic and electric fields created.
The magnetic fields are created as a result of the Boit-Savart law with the metallic coils
(see previous section on magnetism). However, it would be difficult to calculate the
magnitude of the magnetic field using the equation from the Biot-Savarts Law (equation
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3). Instead, equation 6 & 7 are used to calculate the magnetic field.
È 1 ( L - 2t )
ÊR ˆ
ÊR ˆ˘
( L - 2t )
1
2
1
o˜
Í
Á
N1 I1 + N2 I2 = F Í2
+
ln Á ˜ +
ln ÁÁ o ˜˜ ˙˙
2
2
2 +
m
m
r
q
m
t
q
r
m
m
q
R
R
m
m
t
q
Ë
¯
(
)
Ë ri ¯ ˚
Î o r
i
o
i
o r
o
o r
(6)
B=
Fmagn
A
(7)
The creation of the electric field is essential to the SPT. After all, it is the electric
field that accelerates the ions from the thruster with a high velocity. One idea to create
the electric field was to use a biased grate located at the exit plane of the thruster. This
was a practical idea, but the gate would be damaged after prolonged use due to escaping
ions colliding with the gate (Fig 7).
Therefore, the gate would cease to function. An alternate means is necessary to create
the electric field.
The solution to this problem was to use the electric potential already existing in
the thruster to create the electric field. The electric field is a product of a changing
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electric potential. The potential is a result of the circuit connecting from the anode to the
cathode (Fig 8). The current of the potential is conserved throughout the entire thruster.
Using equation 8, the electric field is found knowing the potential.
E = -—f
(8)
The electric potential is related to the number of electrons found at the
exit plane. The increased number of electrons at the exit plane decreases the electric
potential of the chamber. This change produces the electric field that propels the ions out
of the exit plane at high velocities.
Thruster Motion
The thrust from the SPT is a result of different processes occurring in the annular
chamber. To understand the motion of the thruster, five equations were developed to
calculate the electron velocity, the continuity of the system, the ion motion, the mass
change of the system, and the discharge current. When solving these equations, one must
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understand that there will be steady constants such as the mass of an electron, while some
unknowns must be solved for in different equations before they can be used. These next
sections will attempt to explain the origin of the equations used in defining the motion of
the plasma thruster.
I: Electron Motion
The motion of electrons is calculated by beginning with the fluid equation of
motion (equation 9). This equation is the Lorentz force equation with the density of the
plasma (n) multiplied through the equation. The velocity crossing magnetic field of the
equation will go to 0 since the equation is steady state and the particle is moving in the zdirection.
r
r r Æ
r
du
nm
= ne( E + u ¥ B) - kTe— n
dt
(9)
The collision frequency of the electrons is added to the equation (equation 10).
This equation will express the probability the electrons will collide with other particles in
the chamber. The electric field moving in the z direction can be changed into the electric
potential, from equation 8. The electron mobility of the plasma (equation 11) is also
used. From these alterations and setting the fluid equation equal to the vez (motion of
electron in the z direction), the motion of the electron can be calculated (equation 12).
However, there are two unknowns found in this final equation. One of these unknowns is
the electron density (ne) which changes along the thruster length L (it increases from the
anode wall to the exit plane). The other unknown is the electric potential which is used to
calculate the electric field, which also changes along the length of the thruster L (it
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increases from the anode wall to the exit plane).
r
r
r
r r
r
du
nm
= ne(-— f + u ¥ B) - kTe— n - mnvu
dt
e
m=
me v
(10)
(11)
The only way to solve these unknowns and find the vez is to use the other derived
equations that proceed in the next sections. It can already be seen that these equations for
the thruster motion will each be related to one another. There will be a constant need to
refer back to each equation as the channel length changes.
uez = m(
-df
k d(Te ne )
+
)
dz
ene dz
(12)
II: Continuity
The purpose of the continuity equation is to determine whether the density of the
system is conserved. This is necessary since it is impossible to put a certain amount of
particles into the thruster and expel a greater number of particles from the exhaust of the
thruster.
The continuity is derived from the multiplication of the initial density of both the
electrons and neutrons with equation 13 which is the Gaussian profile of the chamber (a
is a constant depending on the gas and the mass change is defined in section four). The
product is the change in the density (nI ) and velocity (vI ) of the ions as they move from
the anode wall (0m) to the exit plane(L). This develops equation 13.
(1 - e -am˙ L )
am˙ L
d(niui )
= nenn b
dz
b = bo
(13)
(14)
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III: Cold Ion Fluid Equation of Motion
The equation of ion motion is based on the equation of motion (equation 9).
Unlike the electron, the ion motion does not need to use the collision frequency of the
ions since ion motion begins after an electron collision. The probability of collision for
an accelerating ion is negligible. The motion of the ions depends on the electric potential
and the neutral density multiplied with the Gauss profile (equation 13).
This gives the ion fluid equation (equation 15).
Mui
dui
df
+e
= - Mno ui b
dz
dz
(15)
IV: Mass Change
The mass change of the system is the rate change in mass as ions are created from
the collisions of neutrals and ions. The rate is conserved. Therefore as the neutron
density (nn ) decreases, the ion density (nIi) and velocity (vI ) will increase. The neutron
velocity (vn) and the cross sectional area (equation 3) are constants. The mass of the
neutron and ion particles is approximately the same. This gives us mass change
(equation 16).
dm
= m˙ = (nnu n + niui )Mi A
dt
(16)
V: Discharge Current
The final equation is based on the circuit that is found in the plasma thruster. This
circuit connects the anode to the cathode. The hollow annular chamber of the SPT
completes the circuit as equal discharge currents leave the cathode and enter the anode.
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The actual discharge current is calculated by adding the velocities of the ions and
electrons then multiplying them by their charge, electron density and the area of the cross
sectional area of the SPT's exit plane (equation 17).
Idis = ene (ui + uez )A
(17)
Conclusion
The different values calculated in each equation are used interchange with other
equations to help define a particular situation in the thruster. At any change of position in
the thruster, the equations must be recalculated since some equations rely on such values
like the ion velocity at a certain position in the thruster.
Conclusion
It is clearly seen that the SPT is more useful than a conventional thruster is. The
only problem with using a SPT is its menial thrust. However, future research is being
performed on the SPT in an attempt to reach its full potential. One goal is to create a
chamber that will give the SPT a higher efficiency.
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Reference
1) Analyses of the anode region of a Hall Thruster channel: Michael Keider, Iain D.
Boyd: Dept. of Aerospace Engineering, Univ. of Michigan, Ann Arbor MI 48109
Isak I. Beilis: Electrical Discharge and Plasma Laboratory, Tel Aviv University, Isreal
2) Main Physical Features and Processes Determining the Performance of Stationary
Plasma Thrusters: Vladimir Kim: Moscow Aviation Institute, Moscow 125810, Russia
3) Modelling the behavior of a Hall current plasma accelerator: J. Ashkanazy,
Propulsion Physics Laboratory, Soreq NRC, Yavne 81800, Isreal. A.Fruchtman:
Center for Technological Education Holon, Holon 58102, Isreal. Y. Raites, N.J.
Fisch: Princeton Plasma Laboratory, Princeton University, Princeton, NJ 08543, USA
4) One-dimensional model of the plasma flow in a Hall thruster: E. Ahedo and P.
Martinez-Cerezo: E.T.S.I. Aeronauticos, Univ. Politecnica, 28040 Madrid, Spain. M.
Martinez-Sanchez: Massachusetts Institute of Technology, Cambridge, Massachusetts
02139
5) Physics of Closed drift Thrusters: V.V. Zhurin: Front Range Fakel, Fort Collins, CO
80524, USA. H.R. Kaufman: Front Range Research, Fort Collins, CO 80524, USA.
R.S. Robinson: Colorado State University, Fort Collins, CO 80523, USA
6) Introduction to Electrodynamics 3rd edition: David J. Griffiths: Reed College,
Prentice Hall, Upper Saddle River, New Jersey 07458
7) Introduction to Plasma Physics and Controlled Fusion(Vol 1: Plasma Physics):
Francis F. Chen: Electrical Engineering Dept., School of Engineering and Applied
Science, Univ. of California, Los Angeles; Los Angeles California. Plenum Press New
York and London
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