16.522, Space Propulsion Lecture 21
Prof. Manuel Martinez-Sanchez Page 1 of 21
16.522, Space Propulsion
Prof. Manuel Martinez-Sanchez
Lecture 21: Electrostatic versus Electromagnetic Thrusters
Ion Engine and Colloid Thrusters are Electrostatic devices, because the electrostatic
forces that accelerate the ions (or droplets) are also directly felt by some electrode,
and this is how the structure receives thrust. We could manipulate the expression for
thrust density in an ion engine to
2
A0a
1
F= E
2
ε , where
a
4V
E=
3d
was the field on the
surface of the extractor electrode. This is the electrostatic pressure. Since
-12
0
=8.85×10 F/mε and E
a
is rarely more than
6
2,000 V/mm = 2 ×10 V/m, we are
limited to electrostatic pressure of about 20 N/m
2
(and due to various inefficiencies
more like 1-2 N/m
2
).
Hall thrusters occupy an intermediate position, and point the way to a higher thrust
density. Ions accelerate electrostatically, but electrons, which see the same (and
opposite) electrostatic force, because the plasma is quasineutral (n
e
=n
i
), are
essentially stopped (axially) by an interposed magnetic field. Of course, the force is
mutual, and so the electrons exert this force on the magnetic assembly (by means of
the azimuthal Hall current they carry). In the end, then ,the structure is pushed
magnetically. To be more precise, we should say that most of the force is
magnetically transmitted. There is still an electrostatic field in the plasma, and so
there will be some electrostatic pressure
2
0n
1
E
2
ε acting on various surfaces. But
because we made the plasma quasineutral, these fields are much weaker than they
are between the grids of an ion engine, and it is a good thing we have the magnetic
mechanism available. In fact, the thrust density of Hall thrusters is about 10 times
higher than that of ion engines, despite the weak electrostatic fields.
More generally, we can ask how much stronger can the force per unit area on some
structure be when it is transmitted magnetically as compared to electrostatically. As
we will see in detail, the counterpart to the “electrostatic pressure” is the “magnetic
pressure”,
2
0
B
2μ
, where B is the field strength and
-6
0
= 1.256x10 Hy/mμ is the
permeability of vacuum. Without recourse to superconductive structures, B can
easily be of the order of 0.1 Tesla (either using coils or permanent magnets), so
2
2
0
B
8,000 N/m
2μ
null , or 400 times the maximum practical electrostatic pressure.
Thrusters that exploit these magnetic forces are called “Electromagnetic” (although
they should be called “Magnetic” by rights). The magnetic field can be external, i.e.,
supplied by coils and not greatly modified by plasma currents, or it may be self-
induced, when plasma currents became large enough. They can also be steady (or at
least slowly varying compared to plasma flow time), or they can be varying very
fast, so as to set up strong induced electromotive forces (transformer effect). A few
examples are:
- Magneto Plasma Dynamic (MPD) thrusters
The most powerful type, with self-induced ,magnetic fields, operates in
steady (or quasi-steady) fashion, and can generate multi-Newton thrust
16.522, Space Propulsion Lecture 21
Prof. Manuel Martinez-Sanchez Page 2 of 21
levels with a few cm. diameter (compared to about 0.1 N for a 30 cm ion
engine, or for a 10 cm Hall thruster).
- Applied field MPD thrusters
Here currents are less strong, so the main part of the B field is external.
Still steady or quasi-steady.
- Pulsed Plasma Thrusters (PPT)
Pulsed Plasma Thrusters (PPT) are very similar in principle to self-field
MPD, but they use a solid propellant (Teflon) which is ablated during each
pulse of operation. These pulses last 10-20 sμ~ only, but are just long
enough that induced emf fields (from
B
=×E
t
?
?
?
nullnull
null
) are still weak. Because
of various practical (mostly thermal) issues, PPT thrusters are not very
efficient <10%
??
??
??~
, but they are simple and robust.
- Pulsed Inductive Thrusters (PIT)
Here the emphasis is on very fast magnetic risetime ( 1-10 sμ~ ) and the
induced emf is used to break down the gas, ionize it, and drive a closed
current loop that exerts the desired magnetic force. They can be thought
of as a one-turn transformer in which the secondary is a plasma ring; the
repulsion between primary and secondary accelerates the plasma away
and pushes the primary coil forward. To avoid dissipating most of the
power in Ohmic losses, the device must be fairly large >0.5m
??
??
??~
and
powerful (MW to GW of instantaneous power).
In the following few lectures we will have time only to explore the self-field
MPD type. We begin with some basic Physics.
Electromagnetic Forces on Plasmas - MPD Thrusters
For a charge q, moving at velocity v
nullnull
in an electric field E
null
and magnetic field B,
nullnull
the
so-called Lorentz force is
()
F=q E+v x B
nullnullnullnullnullnull
(1)
Now, F
null
cannot depend on the rectilinear motion of the observer. For non-relativistic
velocities, B
nullnull
is also independent of motion, and so is the scalar q. Therefore, the field
E
null
must be the different as viewed from different frames of reference. Let E
null
be the
field in the laboratory frame, and E'
nullnullnull
that in another frame moving at u
null
relative to the
first.
Then we must have
16.522, Space Propulsion Lecture 21
Prof. Manuel Martinez-Sanchez Page 3 of 21
()
E + v x B = E'+ v - u x B
nullnullnullnullnullnullnullnullnullnullnull nullnull
so that
E' = E+u x B
nullnullnullnullnullnullnull
(2)
(in particular, for u=v
nullnullnull
the Lorentz force is seen to be purely electrostatic; i.e.,
F=qE'
nullnullnullnull
). Most often the frame at u
null
is chosen to be that moving at the mean mass
velocity of the plasma.
Consider a plasma where there is a number density n
j
of the j
th
type of charged
particle, which has a charge q
j
and moves at mean velocity jv
nullnull
.
The net Lorentz force per unit volume is
()
j
jj
j
f= nq E+v x B
∑
nullnullnullnullnullnull
(3)
and since the plasma is neutral
jj
j
nq =0
∑
:
j
jj
j
f= nqv x B
??
??
??
∑
nullnullnullnullnull
(4)
But, by definition,
j
jj
j
nqv =j
∑
nullnullnull
(5)
where j
null
is the current density vector (A/m
2
). So, finally,
f=j x B
nullnullnullnull
(N/m
3
) (6)
Notice that jv
nullnull
in Equation (5) could be in any frame, including the plasma frame.
Ohm’s Law
In most cases, the dominant contribution to j
null
(Equation (5)) is from electrons, given
their high mobility. In the plasma frame,
e
e e
jj=-env
nullnull nullnull
null (7)
Notice that ev
nullnull
is the electron mean velocity vector, not to be confused with the
mean thermal speed ec . The picture of electron motions is that of a very rapid,
chaotic swarming of electrons back and forth (“going nowhere”), except that the
whole swarm “slowly” drifts at ev
nullnull
.
16.522, Space Propulsion Lecture 21
Prof. Manuel Martinez-Sanchez Page 4 of 21
Typically
e
e
v<<c
nullnull
.
Let us make a crude model of the motion of the electron swarm. The net force on it
per unit volume is
()
ee
e
f=-eE'+v x Bn
nullnullnullnullnullnullnullnull
(8)
where E'
nullnullnull
is used , since we are in the plasma frame.
In steady state, this is balanced by the drag force opposing motion of electrons
relative to the rest of the fluid, which we are assuming to be at rest, and whose
particles have, by comparison, only a very slow thermal motion. To evaluate this
drag, let
e
ν be the effective collision frequency per electron for momentum transfer.
This frequency is defined such that in each collision with a particle of “the rest of the
fluid,” the electron is, on average, scattered by 90
null
, so that its forward momentum
is completely lost. Then the mean drag force per unit volume is
ee
ee
ee e
m
f=-nmv = j
e
ν
ν
nullnullnullnull
(9)
Equating the sum of (8) and (9) to zero,
()
2
e
e
ee
en
j= E'+v B
m ν
×
nullnullnullnullnullnullnullnull
or, since
e
e
j
v=-
en
null
nullnull
,
2
e
ee ee
en e
j= E'- j × B
mmνν
nullnullnullnullnullnullnull
Define the scalar conductivity
2
e
ee
en
=
m
σ
ν
(10)
and the Hall parameter
ee
eB B
= ; =
mB
??
βββ
??
ν
??
nullnull
null
(11)
and we can write the generalized Ohm’s law as
16.522, Space Propulsion Lecture 21
Prof. Manuel Martinez-Sanchez Page 5 of 21
E' = j+ j × σβ
nullnullnullnullnullnull
(12)
where, as given in Equation (2), E' = E+u x B
nullnullnullnullnullnullnull
.
Remembering that the gyro frequency (the angular frequency of motion of an
electron orbiting about a perpendicular magnetic field B
nullnull
) is
e
eB
=
m
ω ,
e
=
ω
β
ν
(13)
i.e. , it represents the ratio of gyro frequency to collision frequency; it can be
expected to be high at low pressures and densities, where collisions are rare, and
also at high magnetic field, where the gyro frequency is high. In many plasmas of
interest in MHD or MPD, 1β ~ .
Electromagnetic Work
The rate at which the external fields do work on the charged particles can be
calculated (per unit volume) as
()
jj
jj
j
W= qn E+v × B . v
∑
nullnullnullnullnullnullnullnull
j
jj
j
=E . qn v
∑
nullnullnull
or
W=E . j
nullnull
(14)
where we used
()
jjv × B . v 0≡
nullnullnullnullnullnullnull
. We see here that the magnetic field does not
directly contribute to the total work, since the magnetic force is orthogonal to the
particle velocity; it does, however, modify E
null
or j
null
(depending on boundary conditions),
and through them it does affect W.
This total work goes partly into heating the plasma (dissipation) and partly into
bodily pushing it (mechanical work). To see this, notice that
() ( )
W = E . j = E' - u × B . j = E' . j + j × B . u
nullnull nullnullnullnull nullnullnullnullnullnullnull null nullnullnull
(using
() ()
u × B . j = - j × B . u
nullnullnullnull nullnullnullnull
).
Also, using Ohm’s law
()
2
1 j
E' . j = j+ j × . j = β
σσ
nullnullnullnull nullnullnullnull
where we used
()
j × . j = 0β
nullnullnull
.
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Prof. Manuel Martinez-Sanchez Page 6 of 21
Thus
()
2
j
W= + j × B . u
σ
nullnullnullnull
(15)
The second term of this expression is simply the rate at which the Lorentz force
j × B
nullnullnull
does mechanical work on the plasma moving at u
null
. The first term is always
positive and is the familiar Joule heating (also called Ohmic heating) effect. Notice
how the presence of the magnetic field introduces the possibility of accelerating a
plasma, in addition to the unavoidable heating. In an efficient accelerator, we would
try to maximize
()
j × B . u
nullnullnullnull
at the expense of
2
j σ .
Origin of the magnetic field
The magnetic field can either be provided by external coils, or induced directly by the
currents circulating in the plasma. The general relationship between B
nullnull
and j
null
(in
steady state and without magnetic materials) is Ampere’s law
0
B
j =× ?
μ
nullnull
null
(16)
where
-7
0
=4 ×10μπ (in MKS units) is the permeability of vacuum.
In integral form
0
B
j . dA = .dl
μ
∫∫ ∫
nullnull
nullnull null
null
(17)
which states that the circulation of
0
B
μ
nullnull
around a closed line equals the total current
linked by the loop. When the current is constrained to circulate in metallic wires, the
integral form can be used to provide simple formulae for the field due to various
conductor arrangements. For example, inside a long solenoid carrying a current I,
the field B
nullnull
is nearly constant, and we obtain (see sketch)
16.522, Space Propulsion Lecture 21
Prof. Manuel Martinez-Sanchez Page 7 of 21
0
B
I n = l
μ
nullnull
where n is the number of turns.
Thus
0
n
B= I
l
μ .
The magnetic field also has the essential property of being solenoidal, i.e.,
. B = 0?
nullnull
(18)
(notice that, due to Ampère’s law, j
null
also obeys . j = 0?
null
, which can be seen as a
statement of charge conservation). In regions where no current is flowing we have
× B = 0?
nullnull
as well, so that a magnetic potential can be defined by B=-?ψ
nullnull
. Then,
since . B = 0,?
nullnull
this potential obeys Laplace’s equation
2
=0?ψ (19)
but notice that no such potential exists in a current-carrying plasma. The vector B
nullnull
there must be found by simultaneous solution of Ampère’s and Ohm’s laws (with the
additional constraint . B = 0?
nullnull
).
Consider now a conductive plasma inside a solenoid, so that both an external B
nullnull
field
()
extB
nullnull
and an induced B
nullnull
field
()
indB
nullnull
exist. The first is due to the coil currents, the
second to those in the plasma itself. Suppose the plasma currents are due to the
16.522, Space Propulsion Lecture 21
Prof. Manuel Martinez-Sanchez Page 8 of 21
flow at u
null
of the plasma in the total magnetic field B
nullnull
, while any external electrodes
are short-circuited, so that E=0
null
in the laboratory frame. Then E' = E + u × B = u × B
nullnullnullnullnullnullnullnullnullnull
.
In order of magnitude, E' u Bnull . Neglecting the Hall effect, then, ju Bσ ~ .
The induced field obeys separately its own Ampère’s law
ind
0
B
×=j?
μ
nullnull
null
, where j
null
is the
plasma current density; this is because ext× B = 0?
nullnull
in the plasma (outside the coil
wires). Thus, in order of magnitude
ind 0
B j lμ~
where l is the characteristic distance for variation
ind
B ,i.e., the plasma “size.”
Altogether,
ind 0
B l u B μσ~ (20)
0indB=B +B
nullnullnullnullnullnull
,
so
ind
0
ind 0
B
l u
B+B
μ σ~
This indicates that the field created by the plasma currents becomes comparable to
the external field when the so-called “magnetic Reynolds number”
em 0
R= l uμ σ (21)
becomes of order unity.
For a high power Argon MPD accelerator,
1000 mho/m, u 10,000 m/sec, l 0.1 m,σ null~ null
so
-6 -1 3 4
em
R 10 ×10 × 10 ×10 = 1~
and so, operation with self-induced magnetic field becomes possible. This simplifies
considerably the design, since no heavy and power-consuming external coils are
needed. The order of magnitude also indicates, however, that external field
augmentation may be desirable under some conditions; the issue of self-field vs.
applied field devices is not yet fully resolved.
A Simple Plasma Accelerator
Consider a rectangular channel with two conducting and two insulating walls, as
shown:
16.522, Space Propulsion Lecture 21
Prof. Manuel Martinez-Sanchez Page 9 of 21
A plasma is flowing in the channel at velocity u
null
, and an external electric field is
applied. Ignoring for now the Hall effect, if this E
null
field is larger in magnitude than uB
(the induced Faraday field), a current j
null
will flow,
given by
( )
j= E'= E+u × Bσσ
null nullnullnullnullnullnullnull
in the direction of E
null
.
The Lorentz force f=j × B
null nullnullnull
is then in the forward
direction, as indicated, and we have an accelerator.
On the other hand, if E<uB, the current j
null
flows in
the direction opposite to E
null
. Externally, positive
current flows into the (+) pole of our “battery” and
could be used to recharge it; we have now an MHD
generator, and the battery would probably be
replaced by a passive load. The Lorentz force now
points backwards, so that the fluid has to be forced
to flow by an external pressure gradient (like in a turbine).
Effect of the Hall Parameter
In a moderate pressure plasma β can easily exceed unity. If the construction is the
same as before, Ohm’s law can be represented graphically as shown below:
16.522, Space Propulsion Lecture 21
Prof. Manuel Martinez-Sanchez Page 10 of 21
We can see that the effect is to turn the current and the Lorentz force counter
clockwise by arc tanβ . There is still a forward force (called the “blowing” force), but
also now a transverse force, called the “pumping” force, because its main effect is to
pump fluid towards the cathode wall, creating a transverse pressure gradient (low
pressure at the anode). Basically, the axial (or Hall) current does no useful work, but
it still contributes to the Joule dissipation
2
j σ . Thus, we may want to turn the whole
diagram by tan
-1
β clockwise and have j
null
flow transversally only and f=j × B
nullnullnullnull
point
axially.
But notice that this implies a forward component of the external field E
null
:
Hence, we have to build the electrode wall in such a
way that an axial voltage can be sustained. For
example, it can be made of independent metallic
segments, connected transversally in pairs, and with
insulation between each pair, so that a voltage can also
be applied between each segment and its downstream
neighbour. Unless a lot of ingenuity is used, this
segmented construction complicates the design and
connections greatly.
Self-field coaxial construction
Leaving aside for the moment the question of how to provide the magnetic field, the
simple continuous electrode accelerator can be further simplified by “wrapping it
around” into an annulus, thereby eliminating the insulating walls:
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Prof. Manuel Martinez-Sanchez Page 11 of 21
The magnetic field is now azimuthal, with circular lines of force. The lower part of
this diagram is seen to be equivalent to that for the simple linear channel (with the
Hall effect included). The insulating sidewalls have been replaced by the annular
plasma itself; of course, an insulating backwall is still required.
It can be seen that the current will (because of the skewness due to the Hall effect)
tend to concentrate near the anode downstream edge and near the cathode root.
Also, the pumping force will now tend to produce a highly concentrated jet of
plasma; in fact, a highly luminous central core is usually observed in MPD plumes,
extending from immediately downstream of the cathode for sometimes several
cathode lengths.
Turning now to the question of the magnetic field, notice that the azimuthal B
nullnull
can be
provided by the plasma current flowing in the meridional plane. This can be seen
from Ampere’s law, written in cylindrical coordinates:
x
0r
BB
B1
j= - =
rxx
φφ
??
?
μ
?φ ? ?
(for B
x
=0) (22)
() ()
r
0x
rB rB
B111
j= - =
rrrr
φφ
??
?
μ
?φ ?φ ?
(for B
r
=0) (23)
The direction of B
nullnull
is also that required for acceleration. This can be seen either from
the equations, or from the right hand rule applied to a number of representative j
null
vectors, or from the known fact that parallel wires carrying current in the same
direction ATTRACT each other.
16.522, Space Propulsion Lecture 21
Prof. Manuel Martinez-Sanchez Page 12 of 21
The magnitude B of the magnetic field at a point P can be simply related to the
amount of current I' which crosses the surface S; this surface leans on the ring that
contains P and extends around the cathode tip as shown.
We have, from the integral Ampère’s law,
()
0
I'
BP=
2r
μ
π
(24)
where r is the radial coordinate of P. In particular, I' = I , the total current, for any
point on the insulating backplate, and I' is zero for points like Q, outside the cylinder.
Approximate calculation of thrust
There are two major contributions to the thrust of our coaxial accelerator. One of
them is the familiar integral of the gas pressure over the back-facing surfaces. This
is called the “electrothermal” or “aerodynamic” thrust, and would be the only one
acting in a device where
2
j σ dominates over
( )
j × B . u
null nullnullnull
(or, for that matter, in a
chemical rocket). The other component is the reaction to the Lorentz forces exerted
on the plasma, and is physically applied as a magnetic force on some of the metallic
conductors carrying current to the thruster. For example, looking at the figure, and
assuming for simplicity that the conductors in the back are arranged symmetrically,
we see that at points like R the enclosed current for the loop shown is the total
current I, and so
0
I
B=
2r
μ
π
, in the same direction as inside the engine. Across the
radial wires, B goes to zero, but at least a part of each wire is subject to B, and since
its current is radially outwards, the Lorentz force on it is to the left, i.e., a thrust.
At high efficiencies, the electromagnetic thrust dominates over the electrothermal
thrust. We can calculate it relatively easily with a few assumptions. First, we have
exactly (by action and reaction)
()
EM
vol vol
0
1
F = j × B dV = × B × B dV?
μ
∫∫∫ ∫∫∫
nullnullnullnullnullnullnullnull
(25)
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Prof. Manuel Martinez-Sanchez Page 13 of 21
In general
() ()()
2
× B × B = B B - B 2???
nullnullnullnullnullnullnullnull
(26)
But since B
nullnull
does not vary along its own direction in our case,
()
2
r
B
BB=- l
r
?
nullnullnullnullnull
(no axial component, integrates to zero by symmetry)
2
EM
vol
0
1B
F=- dV
2
??
?
??
μ
??
∫∫∫
null
(27)
and by a version of Gauss’ theorem,
2
EM
Area
0
1B
F=- dA
2μ
∫∫
nullnullnull
(28)
where the integral extends to the surface surrounding the plasma and dA
nullnull
points
outwards from that surface. We are interested only on the axial force, so
2
x
EM
0 A
1B
F=- dA
2μ
∫∫
(29)
where now dA
x
is the projection of each area element normal to the axial direction.
In particular, for any cylindrical surface, dA
x
=0. The only surfaces surrounding the
plasma which face backwards (or forward) are the backplate, the cathode tip and the
anode rim. For the backplate, using (24) with I' = I , we calculate the contribution
()
a
Back plate
c
2
R
2
00a
EM
0cR
IIR11
F=+ r d=ln
22r 4 R
μμ??
π
??
μπ π
??
∫
(30)
where R
a
and R
c
are the anode and cathode radii, and the (+) sign is because the
normal dA
nullnull
to the surface points backwards (i.e., out of the plasma).
The calculation of the cathode tip and anode rim contributions is much more
involved, since we would need to know the distribution of current on these surfaces.
However, for conventionally built thrusters, these contributions have been estimated
to amount to at most 10% of the total (most of the current flows from the cylindrical
part of the anode to the back of the cathode). Thus, if we also neglect the
electrothermal thrust, we have to a reasonable approximation
2
0a
c
IR
Fln
4R
μ
π
null (31)
(Maecker’s law)
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Prof. Manuel Martinez-Sanchez Page 14 of 21
Notice:
(1) F is independent of size
(2) F scales as the square of the total current.
These rules are observed to apply quite well in practice. For example, the enclosed
figure, from experiments at Princeton, shows the quadratic dependence very clearly.
Using (31), the expression for the exit velocity (and hence the specific impulse)
follows easily
2
0a
e
c
RFI
u= ln
4R
mm
??μ
??
π
??
ii
null (32)
The speed of sound at representative points in the flow will scale like
1
M
(M=molecular mass of the gas), and from (32) we see that the Mach number (say,
at exit) must vary like the quantity
2
IM
m
i
(33)
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Prof. Manuel Martinez-Sanchez Page 15 of 21
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Prof. Manuel Martinez-Sanchez Page 17 of 21
This parameter has been found indeed to be the most important scaling parameter
for MPD thrusters. In particular, it has been found that for each geometrical
arrangement, a limiting value of
2
IMm
i
exists beyond which the operation becomes
highly unsteady and erosion of the electrodes increases by orders of magnitude. The
data for many values of I, m
i
and even for different propellants show this limit at
about the same critical value
2
*
M
I
m
??
??
??
i
of the parameter. Representative values for
Argon (M=40 g/mole)
*
I 23,000 Anull
16.522, Space Propulsion Lecture 21
Prof. Manuel Martinez-Sanchez Page 18 of 21
*
m6 g/sec
i
null
giving
*
2
M
I 560 (I in KA, M in g/mole,m in g/sec)
m
??
??
??
i
i
null .
However, this limit can be modified by changes in the configuration of the engine,
and much research is being done to push it to as high values as possible. The reason
is that as this parameter increases, so does the ratio of magnetic pressure
22
0
B2 Iμ null to dynamic pressure
m
mu
M
i
i
~~, and hence, the relative importance of
electromagnetic effects. Thus, high values of
2
IM
m
i
lead to high thruster efficiency,
and (as shown by Equation (32)) high specific impulse.
The electrical characteristics of the accelerator can also be estimated from Equation
(31).
If the thrust efficiency is
2
2
e
1
Fm
mu
1
2
==
I V 2 I V
??
??
??
η
i
i
,
then, using (31) we obtain
2
3
0a
c
R1I
V= ln
2n 4 R
m
??μ
??
π
??
i
(34)
Thus, if η itself varies little with current, the voltage is seen to vary as the cube of
the current. This trend is indeed observed at high currents (see graph, from
Princeton tests). At lower currents, η does go down, and in addition the
electrothermal component of thrust predominates, and the near-electrode voltage
drops become comparable to the voltage needed for acceleration. The net result is a
departure from the
3
VI~ line, towards a linear dependence.
Power Requirements
Consider the 2300 Amp, 6 g/sec example mentioned before, and assume R
c
=1 cm,
R
a
=5 cm. The thrust is then
()
2
-7
5
F =10 × 23000 ln = 85.1 Nt
1
and the jet power is
16.522, Space Propulsion Lecture 21
Prof. Manuel Martinez-Sanchez Page 19 of 21
()
2
2
5
jet
85.1
F
P= = =6.04×10 Wat
2×0.006
2m
i
Considering that the best efficiencies obtained so far are of the order of 50%, the
actual power required is seen to be about 1.2 MW. This is a difficult requirement to
meet in space, unless relatively large nuclear reactors are used for the power
generation. However, one escape clause available is to operate in a pulsed mode,
i.e., with short high power pulses supplied by a pulse-forming network which gets
“recharged” between pulses by a moderately small power supply. An example of
such a pulse and the schematic of the device and its gas and power supplies are
shown in the figures. The pulse-forming network is a capacitor-inductor bank as
shown:
C
L
+
-
RECHARGING
SWITCH
RECHARGING
POWER SUPPLY
BLEED
RESISTOR
DISCHARGE
SWITCH
IMPEDANCE
RESISTOR
MATCHING
MPD
THRUSTER
16.522, Space Propulsion Lecture 21
Prof. Manuel Martinez-Sanchez Page 20 of 21
Its discharge time is 2N LC , where N is the number of stations in the ladder. Its
output impedance is
L
C
, and should ideally be equal or near the input
impedance
2
2
0a
c
R1I
VI ln
24 R
m
??μ
??
ηπ
??
i
null of the thruster to prevent oscillations (ringing). The
mismatch is compensated for by inserting a matching resistor. These pulse-forming
networks work well, but tend to be bulky and heavy.
16.522, Space Propulsion Lecture 21
Prof. Manuel Martinez-Sanchez Page 21 of 21