Investigation of Plasma Detachment from a Magnetic Nozzle
Except where reference is made to the work of others, the work described in this thesis is
my own or was done in collaboration with my advisory committee. This thesis does not
include proprietary or classified information.
William Chancery
Certificate of Approval:
David Cicci
Professor
Aerospace Engineering
Roy Hartfield, Chair
Professor
Aerospace Engineering
Edward Thomas, Jr.
Associate Professor
Physics
Joe F. Pittman
Interim Dean
Graduate School
Investigation of Plasma Detachment from a Magnetic Nozzle
William Chancery
A Thesis
Submitted to
the Graduate Faculty of
Auburn University
in Partial Fulfillment of the
Requirements for the
Degree of
Master of Science
Auburn, Alabama
August 4, 2007
Investigation of Plasma Detachment from a Magnetic Nozzle
William Chancery
Permission is granted to Auburn University to make copies of this thesis at its
discretion, upon the request of individuals or institutions and at
their expense. The author reserves all publication rights.
Signature of Author
Date of Graduation
iii
Thesis Abstract
Investigation of Plasma Detachment from a Magnetic Nozzle
William Chancery
Master of Science, August 4, 2007
(B.S., Auburn University, 2002)
52 Typed Pages
Directed by Roy Hartfield
The Variable Specific Impulse Magnetoplasma Rocket (VASIMR) electric space propul-
sion concept currently under development by the Ad Astra Rocket Company utilizes a mag-
netic field to confine a propellant plasma. The energetic ions in the plasma are converted
into a directed exhaust flow by a magnetic nozzle. The interaction between the diverging
magnetic field and the escaping plasma plume is of interest both because of its technolog-
ical importance in this application and its relationship to naturally occuring phenomena
in the field of space plasma physics. The research presented here was carried out at the
Propulsion Research Center of Marshall Space Flight Center. The focus of this effort is the
measurement of the time dependent effect of a plasma plume on the vacuum magnetic field
geometry in a simulated magnetic nozzle. Data was gathered with magnetic field probes
constructed at the Royal Institute of Technology in Stockholm, Sweden. This data was
analyzed and compared to published theoretical descriptions of plasma detachment from a
magnetic nozzle. The results of that analysis are presented here.
iv
Acknowledgments
Thanks go to Franklin Chang-D??az and the Ad Astra Rocket Company for supporting
this research, Greg Chavers and the Propulsion Research Center at Marshall Space Flight
Center for providing the environment and apparatus, and Einar Tennfors and Nils Brenning
at the Alfv?en Institute of the Royal Institute of Technology in Stockholm for constructing
the probes used.
v
Style manual or journal used AIAA
Computer software used The document preparation package TEX (specifically LATEX)
together with the Auburn style-file aums.sty. The mathematical software Matlab R?was
used for data processing and generating plots.
vi
Table of Contents
List of Figures viii
1 Project Description 1
2 Magnetic Nozzle Theory 5
3 Magnetic Probe Calibration 15
4 Plasma Density Measurements 25
5 Magnetic Field Measurements 27
6 Conclusions and Recommendations 39
Bibliography 42
vii
List of Figures
1.1 A schematic drawing of the VASIMR VX-10 prototype.6 . . . . . . . . . . . 2
1.2 Schematic drawing of the DDEX facility. . . . . . . . . . . . . . . . . . . . . 3
2.1 Single particle trajectory calculation for an ion exiting the VASIMR engine.10 9
2.2 A graphical comparison of the evolution of particle and magnetic field energy
density in an expanding magnetic nozzle. . . . . . . . . . . . . . . . . . . . 11
2.3 Exhaust plume of the VASIMR thruster predicted from MHD simulations.10 14
3.1 Diagram of Bdot probe operating principle. . . . . . . . . . . . . . . . . . . 16
3.2 Diagram depicting the mounting of the probe on the translation stage and
the polarity of the loops viewed from above. . . . . . . . . . . . . . . . . . . 18
3.3 Photograph of Bdot probe mounted on translation stage arm. . . . . . . . . 19
3.4 Equipment used for calibrating the Bdot probes. . . . . . . . . . . . . . . . 20
3.5 Magnetic field strength produced by the Helmholtz coil during the calibration
pulse. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20
3.6 Raw voltage data from the Bdot probe recorded by the digitizer. . . . . . . 21
3.7 Results of numerically integrating the probe voltage displayed in figure 3.6 . 21
3.8 Magnetic field and probe response for Br calibration pulse. . . . . . . . . . 23
3.9 Magnetic field and probe response for Bz calibration pulse. . . . . . . . . . 23
4.1 Radial profile of plasma density at z=43cm in the DDEX exhaust plume.18 26
5.1 ?Br data from twenty consecutive shots on machine axis. . . . . . . . . . . 28
5.2 ?Bz data from twenty consecutive shots on machine axis. . . . . . . . . . . 29
5.3 Calculated magnetic field lines in the DDEX machine. The location of the
radial Bdot probe scan is indicated with a black double arrow. . . . . . . . 31
viii
5.4 Change in magnetic field strength in the axial direction for several plasma
pulses constituting a radial scan. . . . . . . . . . . . . . . . . . . . . . . . . 33
5.5 Radial profile of the change in the axial component of magnetic field strength. 33
5.6 Change in magnetic field strength in the radial direction for several plasma
pulses constituting a radial scan. . . . . . . . . . . . . . . . . . . . . . . . . 34
5.7 Radial profile of the change in the radial component of magnetic field strength. 34
5.8 Integrated flux loop data giving change in total magnetic flux. These are the
same pulses that comprise the radial scan in Figures 5.4 and 5.6. . . . . . . 35
5.9 Small time scale behavior of changing magnetic flux for the same pulses
displayed in Figure 5.8. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36
5.10 The radial scan data of Figure 5.4 rescaled to account for shot to shot vari-
ations in plasma source performance. . . . . . . . . . . . . . . . . . . . . . . 38
6.1 Axial magnetic field strength generated by the DDEX magnet coils. . . . . 40
ix
Chapter 1
Project Description
The Variable Specific Impulse Magnetoplasma Rocket (VASIMR) is a high power elec-
tric space propulsion concept under development by the Ad Astra Rocket Company (AARC)
at Johnson Space Center. The concept for the VASIMR engine was developed by Dr.
Franklin Chang D??az as an outgrowth of research in magnetically contained fusion plasmas.1
The development of the propulsion system was initiated at the Massachussetts Institute of
Technology in the early 1980?s and later transferred to Johnson Space Center when Dr.
Chang D??az joined the Astronaut Corps.2,3 The VASIMR engine utilizes a magnetically
contained plasma heated by radio waves to produce thrust. The rocket design can be split
into three separate magnetic sections or cells. In the forward section neutral propellant
gas is injected and ionized by radio waves from a helicon antenna.4 The bulk of the RF
power is directed to the Ion Cyclotron antenna in the middle section of the engine where
the power is absorbed by the ions.5 The last cell is the magnetic nozzle which converts the
thermal energy of the plasma into a directed exhaust flow. It is here that the reaction force
between the energetic ions and the confining magnetic field coils occurs. This interaction
accelerates the ions with respect to the engine structure and is the ultimate source of the
thrust generated by the rocket. Figure 1.1 presents a schematic view of the VX-10 which
is a laboratory experimental prototype of the VASIMR engine with a total power rating of
10kW.6
ThisthesisconcentratesontherelatedDetachmentDemonstrationExperiment(DDEX)
that took place at Marshall Space Flight Center under the direction of Greg Chavers.7 This
1
Figure 1.1: A schematic drawing of the VASIMR VX-10 prototype.6
experimental effort focused on exhaust plume diagnostics for characterizing plasma detach-
ment from a magnetic nozzle. Plasma detachment is the process whereby the exhaust
trajectory of the propellant plasma diverges from the magnetic field lines in a vacuum. In
the classical Magnetohydrodynamic theory of magnetic plasma confinement the constituent
ions and electrons are constrained to move only along the magnetic field lines and not across
them. Plasma detachment is important to the VASIMR technology because all magnetic
field lines are ultimately closed so the propellant must eventually detach from the magnetic
field lines in order for the engine to generate thrust. The effective nozzle efficiency of the
VASIMR engine can be calculated by determining at what point in the expanding exhaust
plume the ideal Magnetohydrodynamic scenario breaks down.
The plasma detachment experimental arrangement consisted of a small pulsed plasma
source in a large vacuum chamber. The plasma source was a washer stack plasma gun
constructed at the University of Washington. Current carrying coils mounted on the outside
of the vacuum chamber provided the guiding magnetic field. The chamber contained a
2
Figure 1.2: Schematic drawing of the DDEX facility.
translation stage with two degrees of freedom. Diagnostics packages including Langmuir
probes, for measuring the plasma density, and Hall effect magnetic sensors were mounted
to the translation stage. The Bdot loops described in this thesis were also mounted to the
translation stage. Figure 1.2 is a three dimensional rendering of the DDEX setup showing
the plasma source, predicted exhaust cone, translation stage and large vacuum chamber.
For a study of magnetic detachment phenomenon it is important to map the nozzle
magnetic field in vacuum. This determines the shape of the nozzle. The magnetic field due
to a set of current carrying coils can, of course, be calculated. But it is important to be able
to compare that calculation to experimental data. For this purpose a three dimensional Hall
probe sensor was developed for the DDEX project. It is equally important to know how
the magnetic field changes when plasma is created. Those changes will be over a very short
3
timescale, too short for the Hall probes to detect. The purpose of this thesis is to explore the
spatial and temporal structure of this disturbance in the magnetic field with Bdot probes
capable of detecting high frequency (MHz) oscillations. Bdot probes measure changes in
magnetic flux rather than steady state magnetic field strength. For this reason they are the
preferred measurement method for observing changes imposed by the propellant plasma on
the vaccuum magnetic field structure.
Because the plasma detachment process has still not been experimentally verified it
is not yet clear how accurately the current analytical and numerical models predict the
actual nozzle efficiency. Additional magnet coils may be necessary to increase the efficiency
of the magnetic nozzle. These nozzle coils could substantially increase the total mass and
cost of the cryogenic superconducting magnet system. But the necessity and geometry
of such coils is intimately connected with the plasma detachment process. The proposed
MHD scenario8 requires an extended nozzle field to ensure efficient plasma detachment.
However, high frequency instabilities in the plasma may violate the frozen-in ideal MHD
model of plasma detachment.9 In reality, plasma detachment will probably be due to a
combination of field dragging and anomalous diffusion. Simulations of plasma detachment10
have indicated that additional nozzle coils are a worthwhile investment, but insufficient
experimental evidence has been gathered to date to show that the long magnetic nozzle field
generated by additional superconducting coils is neccessary for optimal rocket performance.
The research presented here is expected to clarify the dominant detachment mechanism in a
magnetic nozzle configuration by indicating to what extent the propellant plasma influences
the magnetic field geometry in the exhaust plume.
4
Chapter 2
Magnetic Nozzle Theory
The primary purpose of the magnetic nozzle section of the VASIMR engine is to convert
the ion energy into a directed flow.8 Power is deposited in the ions by radio waves whose
frequency matches the natural ion cyclotron frequency of the propellant ions in the confining
magnetic field. The absorption of these waves increases the ions? velocity perpendicular to
the magnetic field. In plasma physics a distinction is often made between temperature
or particle kinetic energy perpendicular and parallel to the magnetic field. Ion Cyclotron
Heating increases particle kinetic energy perpendicular to the magnetic field. In other
words it increases v?. The magnetic nozzle concept makes use of the conservation of the
magnetic moment of a charged particle, often referred to as the first adiabatic invariant.11
The magnetic moment of a gyrating charged particle, which is typically denoted by the
greek letter ? but is unrelated to the magnetic permeability, is a constant of the motion.
The magnetic moment is defined as the ratio of perpendicular kinetic energy to magnetic
field strength. Perpendicular kinetic energy is related to the component of the ion velocity
perpendicular to the magnetic field direction.
? =
1
2mv
2?
B (2.1)
From this we can see that as the magnetic field strength diminishes in the magnetic
nozzle, the perpendicular velocity of the ions must also diminish. Since the total kinetic
energy of the ions is conserved, this leads to an increase in the velocity parallel to the
magnetic field. In this way the ions are accelerated in the axial direction.
5
This particle acceleration process can also be viewed as simply the conservation of an-
gular momentum. As the magnetic field diminishes, the gyroradius of the charged particles
increases. The gyroradius, or Larmor radius, of a charged particle in a magnetic field is
given by equation 2.2.12
r = mv?qB (2.2)
Since the angular momentum L = mrv? must remain constant, the perpendicular
velocity must decrease as the gyroradius increases. A particle?s gyroradius is inversely
proportional to the magnetic field strength. A particle?s angular momentum is proportional
to the square of the perpendicular velocity and inversely proportional to the magnetic field
strength.
L = m
2v2?
qB (2.3)
So as the magnetic field strength decreases the velocity perpendicular to the magnetic
field vector must also decrease. Again conservation of energy decrees that if the perpendicu-
lar velocity decreases then the parallel velocity must increase and so the ions are accelerated
parallel to the magnetic field.
Both of these explanations of particle acceleration in a diverging magnetic nozzle field
are from the viewpoint of a moving particle in a stationary magnetic field. An alternative
description of the acceleration process is to treat the field generating coils and the particles
as two separate magnetic elements. The rotation of the charged particle in the magnetic
field will be opposite to the currents in the coils and there will therefore be a repulsive force
6
between the two. This leads to an acceleration of the particle with respect to the coil and
an equal and opposite force on the magnetic coil.
After the magnetic nozzle has accelerated the ions in the axial direction the ions must
detach from the magnetic field. If they continued to follow the closed magnetic field lines
they would eventually impact the spacecraft. These ions would cause damage to the space
craft and not contribute to engine thrust. It is well known in the field of space physics that
it is possible for energetic ions to escape from confining magnetic fields. Sometimes this
happens with violent results, such as solar flares. But the process responsible for detachment
in the VASIMR technology is still unclear. There are several ways to look at the problem.
Any of the following conditions can be viewed as a defining criteria for exhaust plasma
detachment.
1. Plasma energy density exceeds magnetic field energy density.
2. Exhaust flow velocity exceeds Alfv?en speed.8
3. Ion orbits exceed magnetic field dimensions.10,13
The Alfv?en speed is the speed at which magnetic disturbances travel in a plasma. Its
influence is analogous to that of the sound speed in a chemical rocket. After the flow
velocity of the propellant has exceeded the Alfv?en speed events downstream of that point in
the exhaust cannot affect events upstream. This is equivalent to choked flow in a de Laval
nozzle. The Alfv?en speed is defined as14
vA = B??
0?
(2.4)
Where ?0 is the permeability of free space and ? is the mass density of the plasma.
7
It will be shown here that the first two conditions for detachment are identical. The
ratio of plasma energy density to magnetic field energy density is often used in the study
of magnetically contained plasmas. This ratio is conventionally labeled ?.11
? = 2?0nkTB2 (2.5)
In this case we will use the energy density of the directed flow of the plasma, which
is a function of ion velocity v, rather than the plasma pressure, which is a function of ion
temperature T.
? = ?0?v
2
B2 (2.6)
When this directed flow energy density becomes greater than the magnetic pressure
(? > 1) then the momentum of the exhaust overcomes the guiding force of the magnetic
field. This is equivalent to condition 1 above. It is easy to show that the conditions ? > 1
and v > vA (which is condition 2 above) are the same. The author is not aware of this
development occuring in any previous publications.
? > 1 ? ?0?v
2
B2 > 1 ?v
2 > B2
?0? ?v >
B?
?0? ?v > vA (2.7)
Evaluation of the third possible defining criteria for plasma detachment, where the ion
Larmor radius exceeds the magnetic field curvature, requires a discrete particle analysis in
a specified confining field geometry. This analysis would be outside the scope of the present
research, but is mentioned here for completeness. The interested reader is directed to the
references10,13 for an example of a numerical simulation of this case. The ions in the exhaust
8
Figure 2.1: Single particle trajectory calculation for an ion exiting the VASIMR engine.10
would very quickly deviate from the magnetic field if they were on their own, as is evident
from the ion orbit calculations reproduced in figure 2.1. However it is important to realize
that the exhaust is quasineutral, or composed of equal numbers of ions and electrons. Single
particle calculations for an electron show that the electrons have a much smaller Larmor
radius in the magnetic field and are more tightly bound to it. Since the ions carry the
majority of the momentum in the exhaust and the electron and ion motions are bound by
their mutual attractions through electric field interactions the electrons must eventually
be pulled along with the ions. This can happen through either cross field diffusion of the
electrons or magnetic field dragging by the exhaust plasma. The current driven instability
postulated by researchers at KTH9 indicates an anomalously fast cross field diffusion rate,
whereas the magnetohydrodynamic scenario advanced by researchers at the University of
Texas8 describes a situation in which the magnetic field is frozen into the exhaust flow.
Each of the three conditions outlined above involves a relationship between the exhaust
plasma and the confining magnetic field. Plasma entering the magnetic nozzle region is well
confined by the magnetic field, but this quality changes over the spatial dimensions of the
plume. There is a simple argument which demonstrates that the conditions for plasma
detachment are present in an expanding magnetic nozzle configuration. If the plasma acts
9
as an ideal magnetohydrodynamic fluid then the total magnetic flux in the exhaust plume is
conserved. The magnetic field is ?frozen in? to the plasma. This means that the magnetic
field strength (or flux density) is inversely related to the cross sectional area as the plume
expands in the axial direction. Using rp to represent the plasma radius, this means
B ?r?2p (2.8)
Because the total particle flux is also conserved, the number density will also vary inversely
with the cross sectional area. This is strictly true only if the axial particle velocity is
constant, so the nozzle region under consideration here is the section after the majority of
the axial particle acceleration has already occured.
n?r?2p (2.9)
The magnetic pressure (or magnetic field energy density) is proportional to the square of the
magnetic field strength.11 All these factors combine to imply that the magnetic pressure will
drop faster than the particle energy density as the plume radius expands. In fact because
the ion kinetic energy (W) is conserved in the expansion region the ratio of plasma energy
density to magnetic pressure (?) is proportional to the plume area.
? ? nWB2 ?B?1 ?r2p (2.10)
This is illustrated graphically in Figure 2.2.
There is yet another process which will lead to particle detachment and that is the
recombination of the ions and electrons in the exhaust plasma. Once this recombination
10
Figure 2.2: A graphical comparison of the evolution of particle and magnetic field energy
density in an expanding magnetic nozzle.
takes place the neutral atoms are no longer affected by the guiding magnetic field. Unlike the
typical ion engine which accelerates propellant ions separately from electrons the VASIMR
concept retains a quasineutral plasma throughout the acceleration process, which means
that there are no space charge limitations on the design. Because of this there is no need to
provide an electron emitter after the ion acceleration section as in the standard ion engine
design. No serious attempt has been made to either predict or measure the recombination
rate in the exhaust plasma. The quick rise in background neutral pressure inside the
vacuum chamber during a plasma firing is the chief experimental difficulty to be overcome in
measuring magnetic nozzle detachment phenomena. This effect is particularly prominent in
charge exchange collisions and makes the issue of recombination rate confusing. Limitations
on available chamber volume and pumping capacity mandate very short plasma pulses.
Estimates of the recombination rate could be made based on the plasma density and electron
11
and ion temperatures in the exhaust region, but it is believed that the recombination rates
will be low enough to not play a significant role in the evolution of the exhaust plume.
As with any rocket system, there will be some divergence of the exhaust flow from the
desired thrust direction. This results in an effective nozzle efficiency which is due to the
half-angle of the exhaust cone it forms. In the case of a magnetic nozzle the half angle
is determined by the magnetic field direction at the point in the plume where the plasma
flow begins to deviate from the vacuum magnetic field lines. The outer parts of the plasma
will be diverted from total axial flow before the plasma becomes sufficiently detached from
the magnetic field. If we assume that the density and velocity of the exhaust plasma are
uniform over the cross section of the exhaust plume, then the equation for nozzle efficiency
is the same as that for a chemical rocket nozzle.
? = 1+cos?2 (2.11)
Where ? is the half angle of the nozzle. The dipole magnetic field from a single current
carrying coil diverges rapidly from the axial direction. It has been suggested8 that additional
current carrying coils may be necessary to increase the efficiency of the magnetic nozzle
configuration. Such coils would force the vacuum magnetic field to expand more slowly in
the nozzle region, thus lowering the field strength to below the level necessary for plasma
detachment while maintaining the flow directivity needed for efficient nozzle operation.
However, the extent to which the presence of the plasma itself modifies the vacuum magnetic
field generated by the plasma confining magnetic coils has not yet been experimentally
determined.
12
In order to analyze the magnetic field perturbations observed in DDEX it will be useful
to compare the data with analytical predictions. The model used here will be that of Ilin
et al.10
Bp
B = ?
rp
ap (2.12)
Where Bp is the magnetic field inside the plasma, B is the vacuum field generated by
the magnetic coils alone, rp is the plasma radius and ap is the magnetic field curvature.
Figure 2.3 displays the results of two different numerical simulations of exhaust plasma
detachment from the VASIMR thruster.10 These simulations illustrate the amount of di-
vergence from the axial direction that is expected from current theoretical and numerical
models of the exhaust plume. The correlation between the two different techniques of numer-
ical modeling, both ideal magnetohydrodynamics and discrete particle analysis, reinforce
the conclusion that detachment will occur in the device. The results of these simulations
agreed well with the analytical predictive formula of Equation 2.12.
13
Figure 2.3: Exhaust plume of the VASIMR thruster predicted from MHD simulations.10
14
Chapter 3
Magnetic Probe Calibration
A Bdot probe constructed by collaborators at the Alfv?en Lab in Stockholm was brought
to the Marshall Space Flight Center to search for evidence of magnetic field bending and
plasma detachment in the magnetic nozzle configuration. The Bdot probe is a simple,
passive device for measuring time varying magnetic fields.15 Bdot probes consist of a loop
or coil of wire, the leads of which are connected to a voltage recording instrument or digitizer.
The operating principle of a Bdot probe is Faraday?s Law of Electromagnetic Induction. A
changing magetic flux through the loop will induce an emf across the leads.12
E = ?d?dt (3.1)
Figure 3.1 illustrates the basic design of a Bdot probe. The probe response is dictated
by the number of turns in the coil and the area of the coil. The probe area includes any
area between the lead wires which attach the loop to the voltage recording instrumentation.
The leads of the probe are usually twisted together to offset this effect.
If the probe loop is small enough in diameter then the assumption can be made that
the magnetic field is the same over the area of the loop and the total magnetic flux through
the loop is simply the magnetic field strength perpendicular to the loop multiplied by the
loop area multiplied by the number of turns the loop is comprised of. If the loop diameter
and number of turns are known then the relative variation in magnetic field strength can
be calculated by integrating the recorded voltage over the time period of interest.
15
Figure 3.1: Diagram of Bdot probe operating principle.
The Bdot loops used in this research were originally constructed for measuring high
frequency fields in the related VX-50 experiment at Johnson Space Center. These electro-
magnetic field probes were constructed at the Alfv?en Laboratory of the Royal Institute of
Technology (KTH) in Stockholm, Sweden. They consisted of three mutually orthogonal
single loops of wire. The loops were 5mm in diameter. A mount was fabricated at the
Propulsion Research Center for attaching a probe to the translation stage in the Detach-
ment Demonstration Experiment. A schematic drawing of the probe mounting is presented
in Figure 3.2. This figure also indicates the polarity of the Bdot loops with respect to the
vacuum chamber coordinate system. Increasing magnetic field strength in the direction
indicated in the Figure produced a positive voltage at the digitizer.
The photograph in Figure 3.3 was taken from the perspective of the translation stage
inside the vacuum chamber. The perspective of the picture is looking ?upstream? towards
theplasmasourceandtheplasmagunintheendofthenarrowercylindricalvacuumchamber
16
section beyond the large flux loop can be seen at the top of the image. The Bdot probe is
mounted on the translation stage arm and sticks out horizontally to the right. There is a
triple Langmuir probe sticking vertically up from the end of the translation stage arm. The
coaxial signal cables are also visible coming from the bottom of the probe mount and going
inside the hollow aluminum box beam of the translation stage arm.
The probe also incorporated two sets of electric field probe tips. The data from those
probe components was recorded during the experimental pulses. The data from the B? loop
was not recorded because of lack of feed throughs for the signal cables. There were five
components in the probe, three Bdot loops and two sets of electric field probe tips. Only
four coaxial signal feedthrus were available on the vacuum chamber, so the B? component
was disregarded because the geometry of the experiment is azimuthally symmetrical so no
variation in the ? direction was expected. The original purpose of the Bdot probe was
to detect high frequency excitations. Such excitations are conjectured to be the result
of a plasma instability that could cause unusually fast diffusion of the plasma across the
magnetic field lines.9 This phenomena could potentially result in much more efficient nozzle
detachment than would be expected from the ideal MHD detachment model. Unfortunately
no high frequency components were observed in the data collected and presented here.
Signal lines were constructed of 50? coaxial cable. Inside the vacuum chamber the
coaxial cable was RG-188 fluoropolymer insulated type. The signal lines ran through BNC
vacuum feed throughs into standard RG-58 coaxial cable and from there to the digitizer
equipment. The complete signal path was tested with a network analyzer after installation
before closing the vacuum chamber. The signal path showed a completely linear phase
response and negligible attenuation. This test is an important part of the experimental
17
Figure 3.2: Diagram depicting the mounting of the probe on the translation stage and the
polarity of the loops viewed from above.
setup because it shows that, had any high frequency instabilities been present, they would
have been observable in the recorded data.
The Bdot probe was calibrated in the following way. The entire calibration configu-
ration (pictured in Figure 3.4) was taken inside the chamber. Clockwise from the left in
that picture are the Helmholtz coil for producing the calibration magnetic field (labeled
1.243Gauss/Amp), Pearson coil for measuring the current through the Helmholtz coil, Tek-
tronics digital oscilloscope for recording the current waveform and a current pulser for
energizing the Helmholtz coil. The items resided just inside the vacuum chamber for this
picture. One of the large diffusion pump ports is visible in the background, as well as the
back edge of the translation stage. The Helmholtz coil was held up to the probe and a
pulse of current was sent through the coils with the current pulser. The current through
the Helmholtz coil also ran through the Pearson current transformer (model 301X) which
was labeled ?0.01 Volts per Amp.? The output of the current transformer was recorded by
18
Figure 3.3: Photograph of Bdot probe mounted on translation stage arm.
the Tektronix TDS 3052 oscilloscope. The data from those voltage waveforms was writ-
ten to ExcelTMSpreadsheet files. This data was later read into Matlab R?for calculations.
Figure 3.5 displays the calibration magnetic field generated by the Helmholtz coils. This
is determined by multiplying the voltage data recorded by the oscilloscope by the calibra-
tion markings on the Pearson current meter and the Helmholtz coil. Figure 3.6 displays
the voltage output from a Bdot loop which was recorded by the digitizing module for the
calibration pulse. This data must be integrated to obtain the magnetic field strength as a
function of time.
The voltage output of the Bdot probe (displayed in Figure 3.6) must be numerically
integrated with respect to time over the calibration pulse period in order to retrieve the
change in magnetic field strength. This is because the voltage across the probe leads is
proportional to the time rate of change in magnetic flux. A linear slope is introduced by
integrating the data and is obvious in Figure 3.7. The linear slope is caused by the random
voltage offset of the digitizer. This can be effectively eliminated by subtracting an offset
19
Figure 3.4: Equipment used for calibrating the Bdot probes.
Figure 3.5: Magnetic field strength produced by the Helmholtz coil during the calibration
pulse.
20
Figure 3.6: Raw voltage data from the Bdot probe recorded by the digitizer.
Figure 3.7: Results of numerically integrating the probe voltage displayed in figure 3.6
21
voltage from all points in the data set. A simple way of determining the offset voltage
is to average over some time period when the voltage should be zero. During calibration
and subsequent data analysis the first 0.2ms of points before the pulse were averaged to
determine the offset to subtract. After this is done the voltage should be scaled according
to the area of the Bdot loops (A) and the number of turns (n). This nA calibration term is
the parameter of interest and was determined by analyzing data from the calibration pulses.
The Swedish collaborators who manufactured these probes made the loops to be 5mm in
diameter. The initial assumption was that each loop was a single turn. After analyzing the
data from the calibration pulses it was realized that this was not a good assumption. The
conclusion reached from the calibration data analysis was that the Br loop consists of one
turn and the Bz loop is two turns. So the calibration factors should be nA = 1.96?10?5m2
for the Br loop and nA = 3.92?10?5m2 for the Bz loop. Figures 3.6 and 3.7 are plots
of the calibration magnetic field and the integrated probe data for both the Br calibration
pulse and the Bz calibration pulse. Note that in Figure 3.8 there is some response of the
Bz loop to magnetic field in the r direction and vice-versa in Figure 3.9. This is because
the magnetic axes of the Br and Bz loops are not entirely orthogonal. In particular, the Bz
loop is more sensitive to changes in Br than the Br loop is to changes in Bz. This is to be
expected since the Bz loop is actually composed of two turns.
An interesting behavior can be observed in the calibration data plot Figures 3.8 and 3.9.
The magnetic field produced by the Helmholtz coil obviously drops to zero after the current
pulse is finished. But the integrated voltage data from the Bdot probe never completely
drops back to zero. This can be explained as an artifact of the numerical integration
procedure combined with the probe sensitivity and digitizer response. The number of turns
22
Figure 3.8: Magnetic field and probe response for Br calibration pulse.
Figure 3.9: Magnetic field and probe response for Bz calibration pulse.
23
in the probe coil was intentionally made small to leave the probe receptive to high frequency
variations in magnetic field strength. A large number of turns increases the self inductance
of a coil which effectively damps high frequency oscillations.16 The small number of turns
means that the probe has a small response or low gain. Since the voltage output of a Bdot
probe is proportional to the time rate of change of the magnetic field strength, a slowly
decaying magnetic field produces a smaller output voltage than a quickly changing one. As
the magnetic field produced by the Helmholtz calibration coil asymptotically approaches
zero after the current pulse the voltage data from the Bdot probe becomes too small to see
over the random offset of the digitizer equipment. The offset subtraction and numerical
integration procedure then produces a profile of magnetic field strength that seems to never
return to the pre-pulse value.
In order to increase the probe response but still keep the small spatial resolution the
number of turns in the loop could be increased to increase the effective area. One negative
effect of this technique is that increasing the number of turns increases the skewing of the
magnetic axis of the probe. Increasing the number of turns has the effect of adding some
probe area in a different direction than the direction intended. Notice in the calibration
graphs (Figures 3.8 and 3.9) that while the double loop Bz seems slightly more responsive
to the Bz calibration pulse, it is also more responsive to the Br calibration pulse. Increasing
the number of turns in the loop also has the effect of increasing the self inductance of the
probes. This generally lowers the probe responsiveness to higher frequency magnetic field
oscillations.16
24
Chapter 4
Plasma Density Measurements
One of the key components of the diagnostic efforts on the DDEX project was deter-
mining the plasma number density in the exhaust cone. Examining the spatial evolution
of the propellant plasma as it travels through the expansion region is an important step
in understanding the detachment process. Two separate measurement techniques formed
the basis of this diagnostic system. These were a multiple frequency microwave interfer-
ometer system17 and a pair of triple langmuir probes that were mounted on the internal
translation stage. The operating principles of both the microwave interferometer and the
langmuir probe are covered in detail in the excellent text by Hutchinson.15 The microwave
interferometer system provides a good absolute measurement of electron number density in
the plasma integrated along the line of sight. The system was located at z=0.4m on the
DDEX vacuum chamber and situated to look directly across the plume. The data from the
microwave interferometer system provides an excellent record of the total amount of plasma
created by the washer stack plasma gun, but does not give detailed information on the ra-
dial distribution of that plasma. The triple langmuir probe serves to measure the electron
density with a very good spatial resolution. Because of its mounting point on the internal
translation stage this probe can be moved across the exhaust cone to provide a profile of
plasma density. However the pulsed nature of the experiment means that a scan must be
performed over multiple shots. The variations in propellant plasma created by the washer
stack gun from shot to shot make the triple probe data alone difficult to interpret. It is
necessary to separate shot to shot variations in plasma density from spatial variations and
25
Figure 4.1: Radial profile of plasma density at z=43cm in the DDEX exhaust plume.18
for this purpose it is useful to employ the microwave interferometer system as an indicator
of overall plasma source performance. In this role the density measured by the triple probe
over the course of several pulses which comprise a radial scan is rescaled by multiplying by
a scaling factor obtained from the interferometer data. The scaling factor for a particular
shot is obtained by dividing the line integrated electron density for that shot by the average
value for the set of shots. The results of such a data analysis algorithm are displayed in
Figure 4.1 for a radial scan. The discrete points are langmuir probe data points rescaled by
interferometer results. The line is a Gaussian fit to the density profile. This radial density
profile will be useful for comparison with the magnetic field perturbation measurements
presented in the next chapter.
26
Chapter 5
Magnetic Field Measurements
After the Bdot loops were calibrated the vaccuum chamber was sealed and pumped
down. Bdot probe data was recorded during each pulse of the plasma source. The transla-
tion stage position was changed between shots to allow a radial mapping of the magnetic
field data. Data from the Bdot loops was numerically integrated to determine a ?B during
the plasma shot.
An initial high frequency signal component is obviously present in the raw probe voltage
data. Numerical integration of the data yields unexpected results. As previously noted the
offset of the digitizer results in a linear plot of some random slope when the signal is
integrated. This slope changes from one shot to the next without any apparent pattern.
This effect is accounted for in the analysis of the data by subtracting an offset voltage from
the raw data before integration.
The initial ?B signal upon integration produces a jump or discontinuity in the linear
plot caused by the digitizer offset. This can be seen in Figures 5.1 and 5.2. Several interest-
ing phenomenon can be observed here. If the plasma drags the magnetic field lines with it
as is theoretically predicted then the magnetic field profile should return to its vacuum state
after the plasma disappears. This is not observed in the experiment. What is observed is
a sudden change in the magnetic field strength, but no clear, complete change back to the
original. One possible explanation for this is that the plasma appears quite suddenly when
it is produced by the washer stack plasma gun, but diffuses out into the chamber over a
longer time scale. Therefore the initial change in magnetic field should occur quickly, but
27
Figure 5.1: ?Br data from twenty consecutive shots on machine axis.
the decay back to the original state will take longer. Since the voltage produced by a Bdot
loop is directly proportional to the rate of change of the magnetic field it is possible that the
decay back to the original magnetic configuration does not produce a large enough probe
voltage to be observable.
An attempt was made to record data for a longer period of time after the plasma pulse
by lowering the digitizer sampling rate. Since the voltage recording equipment allowed for a
finite number of samples a trade off had to be made between temporal resolution and total
data recording time. So for example one could record 20ms of data at 500kHz or 1s of data
at 100kHz. However, no evidence was found for a return of the magnetic field strength to
the vacuum conditions, no matter how long data was recorded.
Another interesting effect is that the slope of the straight line caused by integrating
the digitizer offset sometimes changes after the shot. It?s almost as if the small ?B signal
that is present at the beginning of the shot changes the electronics of the digitizer. One
28
Figure 5.2: ?Bz data from twenty consecutive shots on machine axis.
outcome of all of this is that it is obvious now that numerically integrating the digitized
signal from a Bdot probe is not the best technique for finding the net change in magnetic
field strength. An attempt was made at using a passive RC integration circuit between the
probe and the digitizer. This would integrate the Bdot probe signal before it reaches the
digitizer and obviate the need for a numerical integration of the digitized voltage data. But
this architecture was unproductive since the passive integrator circuit also decreases the
signal strength at the digitizer. Future attempts at measuring the changing magnetic field
would do well to incorporate an active op-amp integration circuit between the probe and
the digitizing hardware.
In doing the radial probe scans the implicit assumption is made that the plasma condi-
tions do not vary from shot to shot. This is not a very good assumption because the pulsed
plasma gun used as a source in this experiment produces variable discharges. Figures 5.1
and 5.2 are a good characterization of the variability of the magnetic field perturbations
29
over a range of experiment pulses. Relying on the repeatability of the data from shot to
shot is not a good technique for measuring spatial profiles of plasma properties in the ex-
haust plume. An alternative technique would include constructing a multiple loop probe
that could simultaneously measure disturbances at several locations. However, increasing
the physical dimensions of the probe raises questions about the perturbation of the tenuous
plasma stream by the measuring instrument. An attempt was made to quantify the vari-
ability of the recorded data by firing multiple plasma pulses while leaving the translation
stage in a fixed position. The results of twenty consecutive pulses taken with the probe on
machine axis are presented in Figures 5.1 and 5.2. The variability in ?B observed in these
shots is used to assign error bars to the radial profile plots in Figures 5.5 and 5.7. This
was done by calculating the standard deviation of the change in magnetic field strength
in both the radial and axial direction for the twenty consecutive shots on machine axis.
The errorbars included in Figures 5.5 and 5.7 reflect a 2? uncertainty in the measured ?B
based on the sequence of measurements on machine axis.19 Multiple shots were not taken
at every radial location due to time constraints. The limited machine time devoted to this
particular investigation coupled with the charging time of the firing capacitor did not allow
for multiple plasma pulses at every point in the scan.
Figure 5.3 displays a cross section of the DDEX vacuum chamber along with the calcu-
lated magnetic field lines created by the magnet coils. The downward curve in the field lines
as they enter the main chamber is a result of the ambient magnetic field of the earth. The
strength of the artificially generated field begins to fall below the earth?s naturally occuring
magnetic field strength as the plasma travels farther from the field generating coils. There-
fore the magnetic field line orientation in the main chamber is determined predominantly
30
Figure 5.3: Calculated magnetic field lines in the DDEX machine. The location of the
radial Bdot probe scan is indicated with a black double arrow.
by the direction of the earth?s magnetic field in Huntsville Alabama. These are the field
lines that should be present under vacuum conditions. The presence of a plasma inside the
chamber will alter this magnetic field topology. The goal of this research is to investigate
to what extent the vacuum fields are altered by the plasma.
It was found that the change in magnetic field strength recorded by the Bdot loops
varied as they were moved radially across the plume. The radial scan took place at an axial
location of z=0.633 meters in the nozzle coordinate system. This location is indicated in
Figure 5.3. The data obtained during a sequence of plasma pulses comprising a radial scan
of the plume is presented in Figures 5.4 and 5.6. The polarity of the data was reversed
during the analysis process so that the coordinate system would coincide with the magnetic
nozzle system. This is more sensible than the probe coordinate system indicated in Figure
3.2 which was determined by the probe mounting orientation.
31
The digitized voltage signal from the Bdot probe was numerically integrated after the
digitizer offset was subtracted out. The result was multiplied by the calibration factors
previously obtained. Figure 5.4 shows the axial magnetic field data from a radial scan
comprised of shots 12 thru 31 from February 24th 2006. The translation stage position was
moved 1 inch between each pulse. This plot demonstrates another interesting characteristic
of the discharge. For the first 4 shots (r=0 to 4 inches from machine axis) the plotted data
is rather smooth. Thereafter some additional sinusoidal component appears in the data. It
is unknown why this would be the case. It might be due to some instability in the plasma
at the edges of the discharge. But more likely it is because the plasma discharge from the
washer stack plasma gun is not very reproducible. From this data ?Bz can be plotted as
a function of radial position in the plume. This was done in Figure 5.5. The same radial
profile analysis was done for the radial component of the magnetic field and is presented in
Figures 5.6 and 5.7.
There were two large Bdot loops, referred to as flux loops, which encircled the entire
plasma column. The integrated ?B profile measured by the small Bdot loops can be
compared with the total change in magnetic flux measured by the flux loop. The best
comparison is with Flux Loop 1 which was closest to the plasma source and about 6 inches
downstream of the location of the radial Bdot probe scan. The location of Flux Loop 1 is
indicated in Figure 5.3. Flux Loop 1 consisted of a single loop 0.5m in diameter, or about
9.8 inches radius. The flux loop polarity was tested to ensure that increasing flux in the
positive z direction (downstream) resulted in a postive voltage at the digitizer. The flux
loop data was integrated to give a total change in magnetic flux for the same pulses that
the radial scan was conducted. The results are shown in Figure 5.8. Figure 5.9 is a close up
32
Figure 5.4: Change in magnetic field strength in the axial direction for several plasma pulses
constituting a radial scan.
Figure 5.5: Radial profile of the change in the axial component of magnetic field strength.
33
Figure 5.6: Change in magnetic field strength in the radial direction for several plasma
pulses constituting a radial scan.
Figure 5.7: Radial profile of the change in the radial component of magnetic field strength.
34
Figure 5.8: Integrated flux loop data giving change in total magnetic flux. These are the
same pulses that comprise the radial scan in Figures 5.4 and 5.6.
of the time period near the plasma gun pulse. Because the flux loop encompassed a large
area with respect to the magnetic field geometry, no attempt was made to reduce this to a
magnetic field strength.
An assumption of cylindrical symmetry must be made in order to compare the inte-
grated ?Bz profile measured by the Bdot probe radial scan to the data from Flux Loop
1. This is probably not a very good assumption because the magnetic field perturbation
should be related to the plasma density and Figure 4.1 clearly shows that the plasma density
is somewhat off-center with respect to the vacuum chamber coordinate system. It would
be preferable to perform the ?B radial scan across the entire plasma plume. This would
permit an examination of the cylindrical symmetry assumption. However, the probe con-
struction and mounting technique did not leave sufficient room to scan the probe tip to the
left of the machine?s centerline.
35
Figure 5.9: Small time scale behavior of changing magnetic flux for the same pulses displayed
in Figure 5.8.
By assuming the total change in flux to be the initial peaks observed in Figure 5.9, the
average total change in flux through flux loop one over the shots comprising the radial scan
is ?3.2?10?5 Weber and the standard deviation (used to estimate error) is 4.5?10?6.
An integral of revolution with the Bdot probe data results in a predicted total change in
magnetic flux of ?9.7?10?4 Weber. This agrees with the flux probe data in the direction
of net change in flux, but differs by more than an order of magnitude. One should keep in
mind the degree of repeatibility in the pulse data. This is again illustrated in Figure 5.9.
Carrying the observed variation in magnetic field change through the cylindrical integration
process shows that the error margin in the determination of total change in flux from the
Bdot loop data is of the same order of magnitude as the result (1?10?3).
The variability in plasma parameters over the course of the radial scan can be accounted
for by scaling the Bdot loop data from the individual shots with another experimental
36
measurement that gives some indication of the overall quantity of interest. In the case of
magnetic field measurements it is logical to use the encompassing flux loop as an indicator
of the shot to shot variations in magnetic field effects. Figure 5.10 displays the results of
multiplying the ?B data from each shot of the radial scan with a scaling factor obtained by
dividing the total change in flux observed from the flux loop by the average change in flux
for the series of shots. This process is analagous to the data reduction algorithm used to
compute a radial electron density profile by combining the data from the Langmuir probe
scan and interferometer system. In that calculation the shot to shot variations in electron
density that occur over the course of a multi-shot radial scan are accounted for by scaling the
triple probe data with the variations in the line integrated electron density obtained from
the microwave interferometer data. In both cases an instrument which effectively samples
the entire cross section of the plume was used to interpret data from an instrument with
a smaller spatial resolution. This technique allows the investigator to differentiate between
global changes and local changes in the parameter of interest and makes possible a study
of the detailed geometry of the system.
The density of the plasma created by the washer stack gun should determine the total
magnetic flux that is carried along with the plasma. This is to be expected from the
theoretical description of the detachment process because a more dense plasma will carry
more magnetic flux along with it.
37
Figure 5.10: The radial scan data of Figure 5.4 rescaled to account for shot to shot variations
in plasma source performance.
38
Chapter 6
Conclusions and Recommendations
One of the most important conclusions that can be drawn from this data is a knowledge
of the effect of the plasma plume on the magnetic field. It is important to know whether
the plasma drags the magnetic field along with it or is transported across the magnetic
field lines. If it drags the field lines with it, then we can expect some sort of magnetic
reconnection phenomena to show itself at some point in the plume. If the plasma drags the
magnetic field along with it then it should be classified as a paramagnetic plasma. This
means that it should cause the magnetic field to increase in the downstream direction and
decrease in the radial direction.
What is actually observed is an increase in the magnetic field strength in the radial
direction for all shots of the radial scan. This does not seem to support the hypothesis that
the magnetic field is dragged along with the plasma.
The axial component of the magnetic field is more difficult to analyze. For positions
closer to the axis, the change in B seems to point in the positive z direction (downstream).
For positions farther from the axis, the change appears to point in the negative direction
(upstream). This is evident in Figure 5.5. This observation could support the argument for
a paramagnetic plasma in the exhaust if the assumption is made that the plasma column is
only about 4 inches in radius at the position of the probe scan. This conclusion is supported
by the measurements of plasma density depicted in Figure 4.1 which show an exhaust plume
radius of about 10 cm.
39
Figure 6.1: Axial magnetic field strength generated by the DDEX magnet coils.
The change in magnetic field should be compared with the predicted magnitude of
the magnetic nozzle field. Figure 6.1 is a plot of axial magnetic field strength versus posi-
tion. This magnetic field strength was calculated from the coil geometries and magnet coil
currents used in the radial scan. Comparing this profile with the Bdot probe data shows
that the change in magnetic field strength is almost equal to the maximum magnetic field
generated at the plasma source. This would seem to indicate that the plasma drags almost
all the magnetic field along with it through the nozzle. This observation does not agree
with the prediction formula presented in Chapter 2.
The next step in this research would be searching for magnetic field perturbations
in the exhaust of the VASIMR prototype VX-100. This is a 100kW version scheduled
for operation beginning late 2007. Bdot probe construction should be optimized for the
expected magnetic field transients in the large vacuum test chamber to be delivered summer
2007. Three axis Bdot probes should be attached to the translation stage in the chamber.
40
A thorough investigation of the magnetic field behavior in the detachment region would
search for both singular changes in magnetic field strength when plasma is applied and high
frequency oscillations in the magnetic field. This means that both small, single loop Bdot
probes such as the ones made at KTH and larger, more sensitive probes for detecting ?B
will be necessary. These more sensitive coils should include active analog integration circuits
in the probe design, such as the simple op-amp integrator suggested by Hutchinson.15 The
data from these probes should be combined with theoretical calculations of the vacuum
magnetic field created by the coil set and experimental determination of field strength and
direction using a three axis Hall probe mounted on a translation stage. The dominant
detachment mechanism in the exhaust of the VASIMR engine still needs to be determined
experimentally.
41
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43