[American Institute of Aeronautics and Astronautics 44th AIAA/ASME/SAE/ASEE Joint Propulsion...

Effect of Cathode Position on Hall-Effect Thruster Performanceand Near-Field Plume Properties

Jason D. Sommerville∗ and Lyon B. King∗

Michigan Technological University, Houghton, Michigan 49931, USA

Thruster efficiency and plume properties, including ion energy distributions and beam current densityprofiles, of a Hall-effect thruster are compared when the cathode is placed in nine radial positions relative tothe thruster and at three cathode mass flow rates. The cathode was mounted such that its orifice normal was ata right angle to the thruster axis. Plume properties were measured at a radius of 250 mm from the intersectionof the thruster axis and the exit plane. Cathode coupling voltage, beam divergence, and voltage utilization arestrongly affected by cathode position.

Nomenclature

Dprobe Diameter of the Faraday probe electrodef The ion energy distribution function (IEDF)Ib Total ion beam currentId Anode supply currentIprobe Current measured by Faraday probej Ion beam current densitym Xenon atomic mass: 131.29 amum Anode mass flowmc Cathode mass flow rateming Ingested propellent mass flow rateR Probe sweep radius: 250 mmr Radial cathode positionT ThrustV Voltagev Ion velocity

Vcg Cathode coupling voltage: the voltage at whichthe cathode floats below ground

Vd Anode supply voltage〈x〉 The expectation (average) value of any quantity x

〈θ〉 Beam divergenceη Thruster efficiencyηadj Measured efficiency corrected for estimate of in-

gested propellentηθ Beam divergence efficiencyηI Current utilization efficiencyηV Voltage utilization efficiencyηVcg Cathode coupling efficiencyηvdf Velocity distribution efficiencyθ Off-thrust-axis angle

I. Introduction

Hall-effect thrusters (HETs) are a class of electric propulsion devices that use electric and magnetic fields to createa plasma and expel the ions at high velocity in order to generate thrust.1 A critical component of the HET is thecathode. The cathode is a plasma source which provides free electrons which serve two purposes. The first purpose isbeam neutralization—sufficient electrons are expelled via the cathode to balance the charge emitted by the ion beam.The second purpose is to provide the “seed” electrons which initialize and sustain the plasma discharge near the exitplane of the HET. The flow of the seed electrons is known as the recycle current.

The process by which the free electrons in the plume of the cathode are coupled to the anode of a Hall thruster andhow this process affects cathode coupling voltage and thruster performance is not well understood. A few researchershave studied the effects of cathode type2 and mass flow rate2, 3 on HET performance. Albaréde, et al. compared threetypes of thermionic orificed cathodes and found only small differences in behavior.2 Both Albaréde, et al. and Tilley,∗ME-EM Dept., 815 R.L. Smith Bldg., 1400 Townsend Dr.

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American Institute of Aeronautics and Astronautics

44th AIAA/ASME/SAE/ASEE Joint Propulsion Conference & Exhibit21 - 23 July 2008, Hartford, CT

AIAA 2008-4996

Copyright © 2008 by Jason D. Sommerville. Published by the American Institute of Aeronautics and Astronautics, Inc., with permission.

et al. observed increases in cathode coupling voltage with increasing flow rate, and Albaréde, et al. also noticed non-monotonic changes in anode current fluctuation frequency with changes in cathode mass flow.2 It has been repeatedlynoted that cathode placement has an effect on thruster performance.3–8 Hofer, et al.4 and Jameson, et al.,8 noticedsignificant performance improvements by placing the cathode in the center of the HET rather than outside, and Beal,Gallimore and Hargus have inferred from probe measurements that cathode placement and number in a cluster ofHETs will affect performance.6 To the best of our knowledge there are two studies available in the literature thatspecifically address the effect of cathode position on thruster performance across a range of cathode locations. In thefirst, by Tilley, et al. the authors note an increase in cathode coupling voltage as the cathode is moved forward of theexit plane.3 In the second, by Walker and Gallimore, the authors note decreasing performance as the cathode is movedaway from the thruster.7 This is in contradiction to our own work in which we see generally improved performancewith increasing cathode separation for cathodes mounted outside of the thruster body.9, 10 The discrepancy may be dueto the difference in thruster size, magnetic field configurations, and the difference in overall scale over which the twoexperiments were performed.

In our continuing effort to understand cathode to HET coupling, we have undertaken a series of experimentsstudying the performance and plume properties of a Hall thruster as a function of radial cathode position and massflow rate. In prior work, we showed a definite change in thrust, cathode coupling voltage and efficiency as a functionof cathode position.9 We showed further that the change in efficiency resulting from the change in coupling voltagewas insufficient to explain the total change in efficiency seen. This experiment is designed to further illuminate howa change in cathode position affects thruster performance. By measuring the thruster performance along with theion current densities and ion energy distribution functions (IEDFs), we can break the efficiency into constituent lossmechanisms and uncover which mechanisms are changing as cathode position changes.

II. Experiment

A. Setup

The HET was mounted on a thrust stand, and the cathode was mounted on a linear motion stage and could bemoved from 40 mm to 200 mm radially away from the thrust axis, at a fixed axial distance of 30 mm from the thrusterexit plane (see Figures 1 and 2). A Farday probe and a retarding potential analyzer (RPA) were placed on a boommounted on a rotational stage directly above the intersection of the thrust axis and the exit plane. The probes weremounted at a distance of 250 mm.

A possibly significant difference between this work and the prior work is that the cathode was mounted with itsorifice normal at a 90 degree angle to the thrust axis. This change was done to enable the cathode to be placed at radiallocations closer to the thrust axis (r = 0) than was possible with the cathode mounted with its orifice normal vectorparallel to the thrust axis. The implications of this change will be discussed later.

B. Procedure

The experiment proceded as follows: the cathode was placed at a specific radial position and the cathode massflow set. In each experiment, the thruster was set at an operating voltage of 250 V and a mass flow rate of 4 mg/sof xenon. These values were chose to match prior experiments conducted on cathode position with this thruster.9, 10

A magnet current value of 2.5 A was chosen to match prior values as well. In the earlier studies, this value was theoptimal current value, as determined by minimizing the discharge current for the given voltage and mass flow. In theseexperiments, 2.5 A was slightly below the optimal point, but more importance was placed on matching prior magneticfield strengths, and thus the thruster was run with this current. After the thruster stabilized, thrust, discharge current,and efficiency of the thruster were measured over a period of two minutes. After recording the performance, thecathode was moved to a new location, randomly chosen from the nine position in 20 mm intervals between r = 40 mmand 200 mm.

After all of the performance data were taken at a given cathode mass flow, RPA and Faraday probe sweeps wereperformed at five cathode positions: 40 mm, 60 mm, 80 mm, 120 mm, and 200 mm. In the mc=10 SCCM case, the120 mm position was not acquired. Both probes were mounted at a constant radial distance of 250 mm from the centerof the thruster exit plane. The Faraday probe was swept from -90 to 60 degrees in 2.3 degree increments. At eachincrement, 1000 samples were taken at a rate of 10 kHz and averaged. The RPA was swept in 10 degree incrementsover the same range. At each increment, 5 sweeps from 0 to 300 V were taken with 600 points per sweep.

After all performance and probe data were acquired at each of the cathode positions, the cathode mass flow rate wasadjusted and the experiment repeated. Cathode mass flow rates of 10 SCCM, 5 SCCM, and 2 SCCM were chosen.

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Figure 1: Experimental setup

Figure 2: Photograph of experimental setup

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10 SCCM is the nominal cathode ignition flow rate, and 2 SCCM was the lowest cathode mass flow at which thecathode would stably operate.

C. Equipment

1. Thrust Stand

The thrust stand is a NASA-Glenn style, null displacement, inverted pendulum thrust stand, with automatic lev-eling. The displacement nulling is provided by a solenoid driven by a closed-loop digital controller which measuresthe displacement of the thrust stand using an LVDT. The level is sensed by an electrolytic inclinometer, and controlledby a microstepping motor connected to an 80-threads-per-inch screw. The thrust is directly proportional to the currentprovided to the solenoid. Calibration is provided by a linear fit to a set of known weights applied to the thruster acrossa pulley and controlled via a stepper motor.

The data from the thrust stand, as well as the power supplies is continually monitored by a telemetry dataloggerimplemented in software. Communication with the thrust stand and the power supplies is either by analog input to thedatalogging computer, or via GPIB, depending on the device. Telemetry is recorded at a rate of 1 Hz.

From the recorded thrust, anode mass flow rate, discharge current and discharge voltage, the measured efficiencyis calculated according to

η =T 2

2mIdVd. (1)

Note that m does not include cathode mass flow, and likewise, Id and Vd measure only the current through and voltageto the anode.

Given the thermal drifts in the thrust stand compounded with the error in the linear fit, the error in the thrustmeasurement is estimated at 5%. Propagating the errors of the thrust measurement and the mass flow rate through theefficiency equation, we estimate the error on the efficiency measurements at 9% of the stated values.

2. Faraday Probe

The Faraday probe used is a guard-ring-type planar probe. The probe electrode is a tungsten rod 2.4 mm indiameter. The guard is separated from the probe by an alumina tube with an outer diameter of 4.75 mm. The guardring, made of stainless steel, surrounds the alumina. The outer diameter of the guard ring is 10 mm. Except for theface, the guard ring is spray coated with boron-nitride to reduce amount of current collected by the ring.

The current to the probe electrode passes through a shunt resistor, and the voltage across the resistor is amplifiedby an op-amp circuit, and then measured with a computer-based data acquisition system. The voltage to the guard ringis controlled via a FET operating as a voltage-controlled resistor. Both electrodes are biased at -30 V below ground torepel electrons.

The probe was swept from an angle of -90 degrees to +60 degrees. Beyond 60 degrees the probe would collidewith the cathode. At each interval, the voltage from the shunt resistor amplifier was measured by a computerized dataacquisition system, and converted into a current. Current density was then calculated from the probe current accordingto

j =4Iprobe

πD2probe

. (2)

3. Retarding Potential Analyzer

The RPA used in the experiments is a four-grid design. The first grid is a floating grid, designed to isolate theplasma from the various potentials applied to the remaining grids. The second grid is an electron repeller, which isbiased negative relative to plasma potential in order to prevent electrons from entering the remainder of the probe.The third grid is the ion repeller. This grid is swept across the desired voltage range and allows only those ions withenergies above the grid potential to pass through to the collector. The final grid is a secondary electron suppressiongrid. This is designed to force any secondary electrons ejected from the collector by the impact of an ion to return tothe collector.

The grids on the RPA are made from stainless-steel mesh with 0.139-mm spacing and a 30% open-area fraction.Each grid is spaced 2.54 mm apart from the next. The orifice of the probe is a circle 12.7 mm in diameter. Thisdiameter is maintained through the length of the probe, to the equally sized collector, which is simply a grounded,stainless-steel plate.

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Current to the collector is amplified by current amplifier/current-to-voltage converter manufactured by Femto. Theresulting voltage is measured with the data acquisition system. The sweeper grid is swept from 0 to 300 V using aKeithley 2410 source-meter. The electron repeller and secondary suppression grids are biased with a DC power supplyto -15 V. Test traces were also take at -30 V on centerline, where the plasma potential was expected to be highest. Nodifferences were discernable. From this we concluded that -15 V sufficiently repelled electrons.

To reduce the RPA trace, a cubic spline was fit to all of the data acquired. This process was done semi-automatically.A computer program guessed the best spline to fit the data. However, manual interaction was required to choose anappropriate number of spline knots, and to slightly modify the spline in order to achieve the best visual fit of the splineto the raw data. The spline was then scaled so that its value at 0 eV was 1, and at the maximum sweep value of 300 eVthe value is 0. The negative of the derivative of the normalized spline is the IEDF:

f (V) = −dIspline(V)

dV. (3)

4. Vacuum Facility

All experiments were run in the Xenon Vacuum Facility at Michigan Tech’s Ion Space Propulsion Laboratory. Thefacility is a 4-m-long chamber 2 m in diameter. It is evacuated by two 48-inch cryogenic pumps capable of 60,000 L/seach. The base pressure during these experiments was 5 × 10−6 Torr and operating pressures were 2.4 × 10−5 Torr asmeasured by an ion gauge corrected for xenon and located on the top of the downstream third of the chamber.

5. Hall Thruster

The HET used in this experiment is an Aerojet BPT-2000, 2 kW class thruster.11 The thruster has an outer diameterof ∼100 mm and a channel width of ∼10 mm. It operates at a nominal voltage of 300 V and a mass flow of 5 mg/sxenon. At these conditions its specific impulse is ∼1700 s with ∼50% efficiency.

6. Cathode

The cathode used in these experiments is a laboratory cathode similar to the MIREA cathode used by Albadére.2

It is shown schematically in Figure 3. The cathode consists of a 1-inch-diameter titanium cylinder approximately100 mm long. A 2-mm diameter orifice was drilled in one face. Pressed against this hole is a molybdenum pelletholder which holds a lanthanum-hexaboride (LaB6) emitter. Xenon is introduced into the cathode via a feed tubeattached to the side of the cylinder. Filling the length of the cathode from the pellet holder to the base is a tungstenheater coil which heats the emitter to its operating temperature. Radiation insulation loosely wraps the heater andpellet holder to maximize emitter heating. A keeper electrode is positioned approximately 3 mm outside of the orificeand is used to ignite the cathode discharge. After the main HET discharge was ignited, the keeper was unpowered andallowed to float.

In these experiments, the cathode was placed near and within the ion beam. In early trials it was found, notsurprisingly, that this caused the cathode to heat up. While the temperature of the cathode was not directly monitored,the heater voltage was seen to increase while it was held at constant voltage, a sure indication of a temperatureincrease. Any heating of the cathode causes the emitter pellet to more freely give electrons, which affects cathodecoupling voltage. While this is interesting and may bear further research, in these experiments, we wanted to avoidthis affect and focus on the effect of where the cathode gas and electrons are introduced.

To alleviate the thermal loading of the cathode, we constructed an actively cooled shield. The shield, pictured inFigure 3(c), is a copper fin which is slightly wider than the cathode diameter and extends the length of the cathode.1/4-inch copper tubing is soldered to the base of the fin, through which water flows during operation. The exposedportion of the fin is covered in graphite foil to prevent copper from contaminating the thruster or other equipment.Tests with the heat shield showed significant reduction of the cathode heating, though did not completely alleviatethe problem. Without the shield, moving the cathode from outside the beam at 120 mm to inside the beam at 60 mmresulted in a change in of 0.6 V on a nominally 10.5 V heater voltage. With the shield in place, the same change inposition caused a 0.1 V change in heater voltage. Heater currents were 6 A in both cases.

7. Mass Flow Controllers

Flow of xenon to the thruster and the cathode was controlled by MKS Type 1479a mass flow controllers. Theiraccuracy has been tested by monitoring the rise in pressure of a small calibration tank of known volume while flowing

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(a)

(b) (c)

Figure 3: (a) Schematic representation of the laboratory cathode. (b) Photograph. (c) The heat shield.

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gas into the tank. The mass flow controllers are accurate to 2% of the full scale reading: 200 SCCM (19.5 mg/s xenon)for the anode controller, and 20 SCCM (1.95 mg/s xenon) for the cathode controller.

III. Results

A. Performance

The thruster performance data as a function of cathode position is plotted for the three cathode mass flow rates inFigure 4. The data show a consistent trend with thrust, cathode coupling voltage, and efficiency highest near thrust-centerline, and falling off to ∼120 mm. Beyond this point, these parameters rise slightly. The current, meanwhile,decays rapidly from 40–80 mm, afterwhich it is constant. The decay is more rapid as cathode mass flow decreases.

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Figure 4: Thruster performance as a function of cathode position.

B. Faraday Probe Data

The current density as a function of angle is plotted in Figure 5. The data are plotted on a polar graph. However,note there is a change of units at r = 250 mm. Inside this region, a representation of the thruster and cathode areplotted for reference. Outside, the radial dimension corresponds to current density. Each subfigure groups together thedata taken at the same radial position but different cathode mass flow rates.

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One notes an increased broadening and decreased center-spike magnitude as the cathode is moved away from thethrust axis. There is also a decrease in the overall magnitude of the current densities with decreasing cathode massflow, though this is less pronounced when the cathode is further from the thrust axis.

C. RPA Data

Figure 6 shows a sample I-V curve from the RPA data. Additionally, it shows the spline, the knots of the spline,and the resulting IEDF. The IEDF is plotted on the right-hand y-axis and, additionally, on the color bar. These datawere acquired under the following conditions: mc = 5 SCCM, r = 60 mm, θ = −30◦.

Figure 6: Sample RPA data, spline, and resulting IEDF

Figures 7 through 9 show the IEDFs as a function of angle for each cathode position and mass flow operating pointmeasured. Roughly speaking, it shows the probability of an ion at a given angle having an energy that corresponds tothe radial dimension. To calculate the total probability, one would need to multiply the probability on the IEDF by thenormalized current density at that angle.

The data were taken at 10-degree intervals from -90 degrees to 30 degrees. The remaining data have been inter-polated in the figures. The corresponding current density as measured by the Faraday probe is overlaid. Note that thearrangement of the radial axis is identical to that of Figure 5.

Several of the plots show broad energy distributions at zero degrees. It is possible that these are due to pressureeffects inside the RPA and may not, therefore, represent valid data. This will be further addressed in the discussion.

The most striking feature of these graphs is that as the cathode is moved away from the thruster, the distributionsbecome narrower in energy space. Furthermore, the broad energy distribution at 0 degrees seems to go away as thecathode is moved away from the thrust axis and as cathode mass flow is decreased. When present, the narrow spike inprobability density occurs at approximately 200 V, regardless of cathode position, and extends out to about -60 degrees.It appears to vary slightly with cathode mass flow, moving from ∼200 V to ∼210 V as the mass flow is varied from2 SCCM to 10 SCCM.

Finally, one notes a low energy spike near -90 degrees. However, signal levels here are quite small, and we havelow confidence in the data taken between -80 and -90 degrees.

IV. Discussion

A. Performance

Referring back to Figure 4, we see that trends in the performance data are consistent regardless of cathode massflow. More interestingly, they agree relatively well both in magnitude and trend with a similar experiment conductedwith the cathode oriented parallel to the thrust axis and the tip even with the exit plane9 However, in this orientation,that cathode could not be moved closer than about 100 mm away from the thrust centerline.

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Figure 7: Ion energy distributions and ion current density as a function of angle for cathode mass flow rates of 2 SCCM

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Figure 8: Ion energy distributions and ion current density as a function of angle for cathode mass flow rates of 5 SCCM

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Figure 9: Ion energy distributions and ion current density as a function of angle for cathode mass flow rates of 10SCCM

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It is interesting to see that as the cathode moves across the face, all of the performance parameters significantlyincrease. In particular the discharge current is seen to increase. However, away from the thruster face, dischargecurrent is seen to be constant regardless of cathode position, both in this work and in our prior work.9 This suggeststhat some amount of cathode propellent is being ingested by the thruster and accelerated when the cathode is in theseclose locations. It is likely that propellent ingestion is a function of cathode mount angle. In the prior work, in whichthe cathode mounted with its orifice normal vector parallel to the thrust axis, it was not possible to move the thrustercloser than r = 104 mm. In this work we see a small, but noticeable, change in the discharge current, even beyond100 mm, particularly at the 10 SCCM cathode mass flow rate. This suggests that cathode propellent ingestion occursmore readily with the rotated cathode.

To estimate the amount of ingested cathode propellent, we assume that when the cathode is at its furthest locationno propellent is ingested. For closer cathode locations, any discharge current above this amount is assumed to be dueto ingested propellent. These assumptions seem reasonable, given the flat discharge current vs. radial position profilesseen near r = 200 mm in this research, and the completely constant profiles seen in prior work. Note that the onlyother possible method for discharge current to increase is for the recycle current, the fraction of discharge currentabove what is expelled in the ion beam, to increase. We then assume that all “extra” current is due to singly-ionizedxenon. Following these assumptions, the ingested propellent rate is given by:

ming(r) =me

[Id(r) − Id(200 mm)] . (4)

We can then adjust the efficiencies to account for the ingested propellent. Referring to Equation 1 we see the“adjusted” efficiency is given by

ηadj =T 2

2mIdVd

mm + ming

(5)

= ηmeasm

m + ming. (6)

The adjusted efficiencies are plotted in Figure 10 along with the calculated ingested mass flow. Except for at40 mm, the values calculated seem reasonable, and we believe the adjusted efficiencies to be a more accurate measureof efficiency. At 40 mm, the ingestion flow rate is higher than the cathode mass flow rate, which is obviously incorrect.In this position, the cathode partially blocks the thruster exit, and the thruster operates less stably. Severe oscillationswere noted in the cathode coupling voltage and discharge current. It is likely that in this position much of the excesscurrent is due to increased recycle current. This is further support by measurements of the current utilization efficiency,and is further discussed in Section D.

B. Beam Profile

Perhaps the most striking feature of the beam profiles is that the 5 and 10 SCCM traces at 80 mm, and the 2 SCCMtrace at 120 mm [Figures 5(c) and 5(d)] all exhibit anomalous bulges between -60 and -30 degrees. Our first suppositionwas that this was part of the cathode plume superimposed in the beam profile. However, the IEDFs in Figures 7 through9 do not support this theory, as one would expect to see an increased prevalence of low-energy ions. A second theory isthat the bulge is due to cathode neutrals which have been ingested by the thruster and then ionized. Since the cathodeprovides a non-symmetric source of propellent, it would make sense to see a non-symmetric beam profile. However,the fact that it does not show up in all cases, particularly at the 60 mm case suggests another explanation. A thirdexplanation is that the bulge is due to beam ions that have reflected off of the cathode. But, if that were the case, onewould not expect the 120 mm, 2 SCCM case to show a bulge, particularly since the 80 mm, 2 SCCM case does notexhibit one. Clearly, further data are needed to understand the reason for the bulge in ion current density.

Integrating the current density data to find the total beam current is necessary to calculate the current utilizationefficiency discussed in Section D. The bulges cause problems when integrating the beam profiles to obtain total beamcurrent. For the probe sweep geometry used in this study, total beam current is given by integrating the current densityprofile in spherical coordinates assuming azimuthal symmetry:

Ib = 2π

0∫−90◦

j(θ)R2 sin(|θ|)dθ. (7)

When this is done on the data exhibiting bulges, beam currents in excess of the discharge current are obtained. Theuncorrected beam current (Ib) plots in Figure 11 exhibit this. Of course, it is unlikely that the bulge extends a full

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CM

)

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EfficiencyAdjusted EfficiencyIngested Propellent

(a) mc=2 SCCM

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ted P

ropel

lent

(SC

CM

)

r (mm)


(c) mc=10 SCCM

Figure 10: Cathode propellent ingestion and adjusted efficiency as function of cathode position.

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360 degrees azimuthally. In fact, we can see that it does not, from the portion of the current density profile at off-axisangles greater than zero, at least in the mc = 10 SCCM case at 80 mm [Figure 5(c)].

To explore the effect of the bulge, we manually excised the bulges by substituting a spline which approximatedthe general shape of the non-bulged profiles where the bulge was exhibited. This brought the current down to morereasonable, values, as shown in the corrected Ib plots in Figure 11. However, in the 2 SCCM case at 120 mm, and inthe 5 SCCM case at 80 mm, the beam currents still exceed the discharge currents. Most likely, these bulges, whichare shallower than the 10 SCCM, 80 mm bulge, actually extend in θ beyond what was visible as a departure from thenormal profile and skew a much larger portion of the current density profile.

4

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ent

(A)

Div

ergen

ce (

deg

rees

)

Cathode r (mm)

(c) mc=10 SCCM

IdIb

<β>Ib (corrected)

<β> (corrected)

Figure 11: Beam current, discharge current and beam divergence as a function of cathode position. Beam profilesexhibiting “bulges” have been corrected.

In addition to the total beam current, the beam divergence, which is a measure of the beam focus, is anotherimportant quantity which determines thruster efficiency. Here, beam divergence is quantified by the average off-axisangle of the ions, and it is calculated according to

〈θ〉 =

∫ 0−90◦ θ j(θ)R2 sin(|θ|)dθ

Ib. (8)

Beam divergence is also unduly affected by the bulges, as can be seen in Figure 11. Additionally, one notes that thedivergence is consistently lowest at r = 60 mm. This will be further discussed in Section D.

The decrease in peak beam current as a function of increasing cathode position seen in Figure 5 corresponds withthe decrease in total discharge current and beam current, as shown in Figure 11. More interesting is the increase inbeam divergence. This is consistent with the observations of Hofer, et al. that center-mounted cathodes have improved

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beam divergence over externally mounted ones.5 However, even mounting the cathode at 90 degrees and bringing itclose to the discharge channel shows marked improvement in beam divergence.

It is interesting to see that, bulges aside, as cathode mass flow decreases, the discrepancy between discharge currentand beam current also decreases. This will also be further discussed in Section D.

C. Ion Energy Distribution

As we previously noted, the IEDF at angles with magnitudes less than 60 degrees is seen to broaden in energyspace as the cathode is placed closer to the centerline. The most likely explanation is that the broadening is a resultof cathode propellent ingestion. Because the cathode neutrals are not introduced in the back of the anode, but ratherthe front, they are more likely to undergo ionization further down the potential hill created by the thruster. The addedpopulation of slower, cathode-originating ions serves to broaden the IEDF.

A second theory is that the change in location of the cathode causes a significant change in the potential structureinside the HET. Unfortunately, without difficult internal plasma potential and density measurements, neither theorycan be robustly tested.

D. Efficiency Analysis

With ion current densities and energy distributions it is possible to break down the efficiency into its constituentefficiencies with the goal of identifying the significant causes of change in efficiency. Following, Larson12 and Ross,13

we decompose efficiency into the loss mechanisms shown in Table 1. In the following series of equations, we showhow each of these efficiencies are broken out of the total efficiency.

Efficiency Symbol Definition Description How to MeasureVelocity

Distribution ηvdf〈v〉2

〈v2〉Inefficiency due to the spread in1-D velocity space of the ions

Calculate this efficiency at eachangle from the IEDF. Perform aweighted average of all angles,weighting by current density andsolid-angle.

BeamDivergence ηθ 〈cos(θ)〉2 Ion velocity components not par-

allel to the thrust axis tend tocancel and do not generate thrust

Integrate the Faraday-probe-derived current densities to findthe expectation value of cos(θ)

CurrentUtilization ηI Ib/Id The “recycle” current “leaks”

from the cathode to the anodewithout directly creating thrustin the form of beam ions.

Integrate Faraday-probe-derivedcurrent densities to obtain Ib anddivide by discharge current (Id)measured by power supply

CathodeCoupling ηVcg

Vd + Vcg

VdThe voltage at the cathode floatsbelow ground is not available toaccelerate ions. Note that Vcg isalways negative.

Measure the cathode couplingvoltage

VoltageUtilization ηV

12 m〈v2〉

e(Vd + Vcg)Not all ionization takes place atthe top of the potential hill. Be-cause of this, ions do not receivethe full amount of energy avail-able.

Integrate IEDF and current den-sity data, similar to the processfor ηvdf, and divide by the sumof the measured Vd and Vcg

Table 1: Efficiency loss mechanisms

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η =T 2

2mP(9)

=

12 m〈v〉2

IdVd(10)

=〈v〉2

〈v2〉

12 m〈v2〉

IdVd(11)

=〈v〉2

〈v2〉︸︷︷︸ηvdf

〈cos(θ)〉2︸︷︷︸ηθ

12 m〈v2〉

IdVd(12)

= ηvdfηθem/m

Id︸︷︷︸ηI

12 m〈v2〉

eVd(13)

= ηvdfηθηI

12 m〈v2〉

e(Vd + Vcg)︸︷︷︸ηV

Vd + Vcg

Vd︸︷︷︸ηVcg

(14)

= ηvdfηθηIηVηVcg (15)

No attempt is made to correct for ionization fraction in these equations, or the proceding analysis. That is to say, weassume that all ions are singly charged. Because of this, ηvdf is probably slightly overstated, while ηV is understated.

The various expectation values are calculated by a weighted average according to the following equation

〈x〉 =

∫ 0−90◦

∫ 300 V0 x(V, θ) f (V, θ) j(θ)R2 sin(|θ|)dVdθ∫ 0

−90◦

∫ 300 V0 f (V, θ) j(θ)R2 sin(|θ|)dVdθ

. (16)

Since the data integrated is discrete, the inner integral is first performed at each available angle using Simpson’s rule.The results of these integrations are then linearly interpolated, multiplied by the current densities, and integrated acrossθ.

Strictly speaking, the separation of ηvdf and ηθ in Equation 12 is only valid if the IEDF is independent of angle.This is clearly not the case. However, because the IEDF seems to change most at high angles where the current densityis small, we will procede with the assumption anyway.

The bulges discussed in Section B present a particular problem for evaluating the current utilization efficiency.In the proceding analysis, we have removed the bulges from the current profile, as previously discussed. Since twodata points still display calculated beam currents in excess of the discharge current (refer to Figure 11), we limit theresulting calculation of current utilization efficiency to 1.

The results of this efficiency decomposition are shown in Figure 12, along with the measured efficiency, theingestion-adjusted efficiency, and the total efficiency calculated from probe data. The total efficiency is calculatedby multiplying all of efficiency components according to Equation 15 and is denoted by ηmult. Total efficiencies areplotted with solid lines, while efficiency components are plotted with dashes.

The calculated efficiencies show good agreement with the measured efficiencies, except when mc = 10 SCCMand at all cathode mass flow rates when r = 60 mm. The reasons for these discrepancies are unclear. The voltageutilization efficiency is the dominant loss mechanism, with beam divergence being a close second. The remainingefficiency components are all ∼85% or better. All efficiency components show similar trends as cathode position isvaried, regardless of cathode mass flow. The consistency of the trends suggests that there is no fundamental differencein thruster operation as a function of cathode mass flow rate.

The voltage utilization efficiency increases as the cathode is moved away from the thruster. This is likely explainedby the decrease in cathode propellent ingestion as r increases. Ingested cathode propellent is more likely to be ionizedfurther down the potential hill and across a greater range of potentials, as it comes from outside the thruster, ratherthan the back of the anode. An increase in cathode propellent ingestion would therefore be result decrease in voltageutilization, as the velocity distribution was spread to lower values. The fact that the velocity distribution efficiencytrends in the same fashion as the voltage utilization also supports this theory, as it measures the spread in velocityspace of the beam ions.

The current utilization efficiency is quite high, and relatively flat. It does drop significantly at 40 mm, which inpart explains the consistently poor performance at this point. Given the high, periodic oscillations in cathode coupling

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ηmeasηadj

ηmultηβ

ηIηvdfηV

ηVcg

Figure 12: Efficiency decomposition versus cathode radial location

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voltage observed, it is possible that there was increased electron mobility in the HET channel, which would exhibititself as a decrease in current utilization efficiency. A more interesting trend is seen when comparing the overallcurrent utilization from each of the cathode mass flow rates, as was mentioned in Section B. As cathode mass flowrate increases, current utilization decreases. One possible explanation is that the increased cathode flow results inincreased charge-exchange collisions. The fast neutrals, which must be captured for a completely accurate picture ofcurrent utilization, are not detected by the Faraday probe. On the other hand, if this were the only effect, one wouldexpect the current utilization to be much worse with the cathode close to the thruster, than with it at r = 200 mm. Thisis not the case. But, increased ionization of the cathode propellent may offset this, making the two effects impossibleto disentangle with the current data.

The cathode coupling efficiency trends nicely match the trends in efficiency. This can be better seen in Figure 4,where Vcg (∝ ηVcg) is seen to vary nearly in step with performance. However, cathode coupling efficiencies onlyvary by roughly 5% as the cathode position is changed, while total efficiencies vary by up to 20%. We see that thebeam divergence efficiency shows roughly the same trend as the overall efficiency, and also approximately matches thecoupling voltage efficiency trends. The divergence efficiency exhibits a maximum change of roughly 10% as cathodeposition is changes. It is surprising that even at 40 mm, where the cathode partially blocks the thruster channel, beamdivergence remains good. As previously mentioned, the trend seen here is further supported by the conclusions ofHofer, et al.5 Clearly beam divergence is an important factor in explaining the trend in total efficiency.

V. Conclusion and Future Work

The data from this experiment suggest that there is not a simple relationship between cathode position and thrusterperformance. None-the-less, consistent trends are seen in all of the efficiency components. These trends suggest thatchanges in beam divergence and voltage utilization dominate the changes in total efficiency.

However, a more detailed analysis is certainly required before drawing any firm conclusions. The problems of thebulges of unknown azimuthal extent, which violate the assumption of azimuthal symmetry, as well as the violation ofthe assumption that IEDFs are independent of angle present complications in the present analysis. For future work,we plan to alleviate both problems. For the bulge, we plan acquire 2-D current density measurements, and to asses theeffect of the violation of the assumptions in Equation 12.

This work is part of a larger effort to understand the relationship between the magnetic field of the thruster andcathode position. The magnetic field of the thruster has been simulated and measured experimentally. Comparison ofpast data to the field proved interesting, and we expect similarly interesting results when these data are compared tothe magnetic field.

Acknowledgments

The authors would like to thank Marty Toth for providing quick turn-around in fabricating of various napkin-sketched mounts required to set up this experiment. This work has been supported by the Air Force Office of ScientificResearch.

References1R. G. Jahn, Physics of Electric Propulsion. McGraw-Hill Series in Missile and Space Technology, New York: McGraw-Hill Book Company,

1968.2L. Albarède, V. Lago, P. Lasgorceix, M. Dudeck, A. Burgova, and K. Malik, “Interaction of a hollow cathode stream with a Hall thruster,” in

28th International Electric Propulsion Conference, vol. 03-333, Electric Rocket Propulsion Society, March 17–21 2003.3D. L. Tilley, K. H. de Grys, and R. M. Myers, “Hall thruster-cathode coupling,” in 35th AIAA/ASME/SAE/ASEE Joint Propulsion Conference,

vol. AIAA-99-2865, AIAA, June 20–24 1999.4R. R. Hofer and A. D. Gallimore, “Recent results from internal and very-near-field plasma diagnostics of a high specific impulse Hall

thruster,” in 28th International Electric Propulsion Conference, vol. IEPC-2003-037, March 17–21 2003.5R. R. Hofer, L. K. Johnson, D. M. Goebel, and D. J. Fitzgerald, “Effects of an internally-mounted cathode on Hall thruster plume properties,”

in 42nd AIAA/ASME/SAE/ASEE Joint Propulsion Conference, vol. AIAA-2006-4482, American Institute of Aeronautics and Astronautics, July 9–122006.

6B. E. Beal, A. D. Gallimore, and W. A. Hargus, “The effects of cathode configuration on Hall thruster cluster plume properties,” in 41stAIAA/ASME/SAE/ASEE Joint Propulsion Conference, vol. AIAA-2005-3678, July 10–13 2005.

7M. L. R. Walker and A. D. Gallimore, “Hall thruster cluster operation with a shared cathode,” Journal of Propulsion and Power, vol. 23,pp. 528–536, May 2007.

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8K. K. Jameson, D. M. Goebel, R. R. Hofer, and R. M. Watkins, “Cathode coupling in Hall thrusters,” in 30th International Electric PropulsionConference, vol. IEPC-2007-278, (Florence, Italy), Electric Rocket Propulsion Society, September 17–20 2007.

9J. D. Sommerville and L. B. King, “Effect of cathode position on Hall-effect thruster performance and cathode coupling voltage,” in 43rdAIAA/ASME/SAE/ASEE Joint Propulsion Conference, (Cincinnati, OH), American Institute of Aeronautics and Astronautics, July 8–11 2007.

10J. D. Sommerville and L. B. King, “Effect of cathode position on Hall-effect thruster performance and cathode coupling voltage,” in 30thInternational Electric Propulsion Conference, (Florence, Italy), Electric Rocket Propulsion Society, September 17–20 2007.

11D. King, D. L. Tilley, R. Aadland, K. Nottingham, R. Smith, C. Roberts, V. Hruby, B. Pote, and J. Monheiser, “Development of theBPT family of U.S.-designed hall current thrusters for commercial LEO and GEO applications,” in 34th AIAA/ASME/SAE/ASEE Joint PropulsionConference, vol. AIAA-98-3338, Electric Rocket Propulsion Society, July 13–15 1998.

12C. W. Larson, D. L. Brown, and W. A. Hargus, “Thrust efficiency, energy efficiency and the role of VDF in Hall thruster performanceanalysis,” in 43rd AIAA/ASME/SAE/ASEE Joint Propulsion Conference, (Cincinnati, OH), American Institute of Aeronautics and Astronautics, July8–11 2007.

13J. L. Ross and L. B. King, “Energy efficiency in low voltage Hall thrusters,” in 43rd AIAA/ASME/SAE/ASEE Joint Propulsion Conference,vol. AIAA-2007-5179, American Institute of Aeronautics and Astronautics, July 8–11 2007.

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