2D RF Ion Thruster Modeling with Fluid Plasma and

Analytical Sheath Formulation

Emre Turkoz, Murat Celik

Bogazici University, Istanbul, 34342, Turkeyemre.turkoz@boun.edu.tr

SP2014 2980764

Abstract

This paper discusses the modeling of the inductively coupled plasma (ICP) in RF ion thruster dischargechamber. The model handles the three species, which are electrons, ions and neutrals, as ideal gas andformulates fluid equations to solve for the flow field parameters. Continuity and momentum equations aresolved for all species whereas an energy equation is solved only for electrons. Ions and neutrals are assumedas cold, constant temperature gas species. Electric and magnetic fields inside the discharge chamber areevaluated by solving a magnetic vector potential equation. The model is implemented for a 2D axisymmetricgeometry and equations are discretized with finite volume method in cylindrical coordinates. To deal withthe real-world experiment conditions, a matching circuit model is added, where the coupling between theplasma and the external circuits are handled through approximating the plasma as the secondary of an air-core transformer. The model equations extend up to the presheath region where the plasma is quasi-neutral.For the sheath region an analytical formulation is presented to evaluate the potential drop. The validationof the model is performed using the commercial software, COMSOL. It is demonstrated that the model canbe used to evaluate the performance of various thruster designs accurately.

I. Introduction

Radio-frequency (RF) ion thrusters are plasma-basedimpulse generators for in-space applications. RF ionthrusters consist of a discharge chamber, which con-tains the plasma, and the electric circuitry imple-mented around this discharge chamber to handle thepower deposition and RF heating. Example geome-tries of a conical and a cylindrical shaped dischargechamber can be seen in Fig.1. The necessary en-ergy deposited into the partially ionized gas comesfrom the RF coils, which generates the inductivelycoupled plasma. Screen and accelerator grids can beseen on the right end of the discharge chamber, whichaccelerate ions out of the discharge chamber. Man-ufacturing of these grids is also subject to a greateffort, since a great precision is required in the ma-chining process. The grids can be made of severalmaterials, such as molybdenum and carbon, whereasthe discharge chamber is generally made of dielectric

materials, such as quartz. Neutral xenon gas is fedfrom the left end. Neutral xenon atoms get ionizedwith the energy provided by the RF coils. The out-put from the discharge chamber is neutralized with acathode, which can be seen aiming towards the plumeof the thruster.

The RF ion thrusters are first invented in the 1960sin Germany [1]. Giessen University was the host ofthis invention. After that, Astrium GmbH, a pri-vate German company, has adopted this developmentand managed to build thrusters which can be usedin space missions. The most advanced product ofthese early efforts was RIT-10, which has a 10 cm dis-charge chamber diameter. RIT-10 is space tested in1992. This spacetest was performed on the EURECAcarrier. RIT-10 was incorporated into the EuropeanARTEMIS satellite, which was sent to the space forgeo-stationary communication purposes. RIT-10 islifetime tested for 15,000 hours in 2000. The com-

Graduate Student, Department of Mechanical Engineering, Bogazici UniversityAssistant Professor, Department of Mechanical Engineering, Bogazici University.

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mercially available RIT-10 package is also called asRITA [2].

After the development of the RIT-10 ion thruster,German Space Agency (DARA) has started a projectin 1995 for RIT-15, which has a 15 cm chamber diam-eter and is planned to deliver specific impulse morethan 4000 seconds at 50 mN of thrust [1]. Thisamount of specific impulse enables the applicationarea of large geostationary satellites and platformsfor RF ion thrusters.

Miniaturization of RF ion thrusters are performed inthe late 2000s. Astrium GmbH and their partners inthe academy developed RIT-X, which is built formicropropulsion applications in 2007 [3]. In 2011, re-searchers from Giessen University and Moscow Avia-tion Institute designed a very large ion thruster, RIT-45, which has a discharge chamber diameter of 46.5cm [4]. RIT-45 works with 35 kW power and deliversa specific impulse of 7000 s.

Plasma physics simulations are relatively new in themodeling world, since most of the underlying physicsis subject to further investigation. Numerical treat-ment of inductively coupled plasma is performed nu-merous times in the literature. Electromagnetic heat-ing is the core of the process, which is again elab-orated in many examples. Two application areas,plasma processing and plasma torch modeling, comeforward for the fields that have similar physics as RFion thrusters. Examples in the literature about thesefields are very helpful in building a numerical modelfor RF plasma.

There are few numerical models for ion thrusters ingeneral, and there are models for inductively cou-pled plasma, but the number of models for RF ionthruster is very limited. A leading example [5] isbased on evaluating the discharge loss per ion withan analytical model. The 0D model described in thatwork is simple but successful at predicting the per-formance of ion thrusters in real applications. It laysout the effect of the induced magnetic field due to theRF coils on the ion confinement and discusses factorsthat result in a decrease in the discharge loss per ion.Another recent 0D model [6] indicates a trade-off be-tween mass utilization efficiency and power transferefficiency with increasing gas flow rate.

Additional to the analytical models, there are alsoone or multi-dimensional RF ion thruster dischargechamber modeling studies. A simple transformermodel [7] is first laid out for 1D modeling, assumingthat the thruster is large enough to assume 1D ap-proach could be valid. Then this model is extendedto a 2D model [8] which evaluates the plasma pa-rameters of RF ion thrusters with the help of addi-

tional experimental data specific for the thruster tobe modeled. In that study, the plasma is treated as acontinuum as it is treated in the same way as in thiswork.

Figure 1: Representation of the cylindrical and coni-cal discharge chambers

There are also studies with the kinetic approach, us-ing a PIC (Particle-In-Cell) code to solve for the spa-tial distribution of the plasma parameters. An ex-ample model [9] is developed to evaluate the perfor-mance of the micro RF ion thrusters. A 3D fullykinetic model [10], that requires strong computationpower, is laid out recently for RF ion thrusters.

Using a PIC code is always possible but the usage ofthe fluid approach decreases the computational costdrastically. Plasma must obey the continuum ap-proach for the fluid modeling to be possible. Theinvestigation of the question whether the inductivelycoupled plasma inside the RF ion thruster dischargechamber obeys the continuum approach is alreadyperformed [8]. Therefore a fluid model is developedin this work. The model presented in this workconsists of three main components: Electromagneticmodel, fluid model and the transformer model. Theelectromagnetic model handles the solution of the

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Maxwell equations, the fluid model evaluates the flowof the plasma and the transformer model evaluatesthe matching circuit parameters and most impor-tantly the alternating current magnitude to be sup-plied to the RF coils.

In this study a new numerical model that relies on thefluid assumption for plasma is presented and verifiedwith a commercial software. The governing equationsare solved on a 2D cylindrical and axially symmetricdomain. The details of this model is given in SectionII. The numerical method utilized to solve the modelequations is explained in Section III. Section IV isdedicated for the results and concluding remarks aregiven in Section V.

II. Theory Model

A model, which consists of electromagnetic, fluid, andtransmformer submodels, is previously described in[11]. The equations utilized in this model are listedhere again.

Continuity equations for ion and neutral number den-sities (ni and nn):

nit

+ (nivi) = R (1)

nnt

+ (nnvn) = R (2)

Momentum equations for ion and neutral velocities(vi and vn):

mini

(vit

+ vi vi)

+ k(niTi) = eniE (3)

+ enivi Bminiin(vi vn)miniei(vi ve)

mnnn

(vnt

+ vn vn)

+ k(nnTn) =

mnnnin(vn vi)mnnnen(vn ve) (4)

Electron flux (e) using drift-diffusion:

e = k(neTe)meeff

+ene

meeff( ve, B) (5)

The electron temperature (Te) is evaluated by solvingthe power balance for electrons:

3

2

t(neeTe) +Qe = eEae + Pdep Pcoll (6)

The electromagnetic fields are evaluated by solving

the magnetic vector potential (A) equation:

2A = 0At

(7)

where R denotes the ion generation through firstionization collision and is formulated as R =nnne veQion. The term in angled brackets is theionization reaction rate obtained from [12] for Xenon.In momentum equations ei, en and in denote theelectron-ion, electron-neutral and ion-neutral colli-sion frequencies, respectively. The magnetic vectorpotential is defined so that it satisfies A = B andA = 0. The energy balance equation (6) containsthe heat flux due to conduction, which is formulatedas given in [13]:

Qe =5

2eeTe 5

2

nee2Te

meveffTe (8)

Power deposition is formulated as:

Pdep = |E|2 (9)

The power loss term due to elastic and inelastic col-lisions is formulated as:

Pcoll = nenne veQionUion + nenne veQexcUexc

+

heavyh

2memh

3

2k(Te Th)ehne(10)

where, Uion is the first ionization energy, Uexc is theaverage excitation energy, and the terms in angledbrackets represent the reaction rates. The electric po-tential, , is evaluated by inserting the drift-diffusionapproximation that contains this term into the cur-rent continuity equation, which is j = (ei ee) = 0.

Additional to these equations, a transformer modelto represent the matching circuit is added into thesimulation. This transformer model is implementedas described in [8] and [14].

The sheath potential drop incident on the floatingwalls can be calculated using the zero-current condi-tion on the walls. According to this condition, theion and electron fluxes incident on the wall should beequal:

nsuB = nsce4

exp

(e

kTe

)(11)

where ns denotes the plasma density at the sheathedge, is the sheath potential drop between thewall and the sheath edge, and uB =

kTe/mi is the

Bohm velocity for ions at the sheath edge.

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Figure 2: Representation of the domains solved inthis study. a) ICP discharge confined withing dielec-tric walls for verification. b) ICP discharge in an ionthruster, where an electrostatic grid is placed to theright end of the domain

According to this formulation, the sheath potentialdrop is:

= kTee

ln

(mi

2pime

)(12)

The floating sheaths for the configurations investi-gated in this study are collisionless and low-pressuresheaths. Using the Child-Langmuir Law presented in[15], the sheath thickness can be expressed as:

s =2

3()3/4

(Ji0

)1/2( 2emi

)1/4(13)

where Ji = eniuB denotes the ion current density en-tering the sheath. The sheath potential drop and thesheath thickness allow for the calculation of the elec-tric potential distribution inside the sheath by solvingthe following nonlinear differential equation:

d2

dx2= Ji

0

(2e(x)

mi

)1/2(14)

where the boundary condition at the plasma bound-ary (x = s) is taken to be zero, and the boundary con-dition on the wall (x = 0) is taken to be the sheathpotential drop, . The above equation is a formof the Poissons equation, where the right hand sideis replaced with the net space charge expression as-suming that ne = 0 in regards to the Child-Langmuir

law. The ion density can be calculated as:

ni(x) = ns

[1

(2e(x)

miu2B

)]1/2(15)

The Child-Langmuir approach assumes that the elec-tron number density is negligible compared to the ionnumber density. But an approximation for the elec-tron density can be evaluated using the Boltzmannsrelation:

ne(x) = ns exp

(e(x)

kTe

)(16)

where ns is the plasma density at the sheathedge.

In this study two different configurations are handledas depicted in Fig. 2. The first configuration repre-sents the verification case, where the ICP is confinedwithin dielectric walls. This pseudo-chamber with noinlets is considered to have RF power deposited intothe plasma with ions undergo a recombination andbecome neutrals at the discharge boundaries. Thenonequilibrium ICP discharge is solved with COM-SOL Plasma Module and AETHER software that isdeveloped in the scope of this study, and the resultsfrom both platforms are used for verification.

The second configuration is the representation of anRF ion thruster, where the right dielectric wall is re-placed with grids for acceleration. This configurationis investigated to understand the discharge character-istics of an ion thruster.

III. Numerical Method

The equations presented in Section II. are discretizedon a structured rectangular grid in 2D cylindrical co-ordinates for an axially symmetric domain. Conti-nuity and momentum equations are solved using thefinite volume method. The species are assumed tobehave as ideal gas, and the finite volume methodis applied as presented in [16] and adapted for com-pressible gases as presented in [17]. The remainingequations are discretized according to the second or-der finite differencing scheme and the resulting linearsystems are solved self-consistently.

The second-order finite differencing and finite volumediscretizations on structured rectangular grids in 2Dyield coefficient matrices that have a penta-diagonalsparsity pattern. The coefficient matrices are storedusing the compressed diagonal storage (CDS) method[18]. The resulting linear systems are solved usingILU-GMRES method [19].

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Figure 3: Plasma density at the end of 10 milisec-onds. Due to the losses to the walls, plasma is con-fined in the center region of the domain

The nonlinear equation (14) to evaluate the electricpotential inside the sheath is solved with Newtonsmethod. The Jacobian matrix is evaluated explicitlyby first evaluating the derivative of the discretizationof this equation, which is performed using second or-der finite differencing.

IV. Results

The model explained in previous sections of this workis used to solve a benchmark ICP configuration toverify the results with the Plasma Module of the com-mercial software COMSOL. For the verification, thepreferred discharge chamber is a cylinder made of adielectric, which is 7 cm long and has a diameter of 8cm. RF power is deposited into the plasma throughthe 10 coil windings around the chamber, which ex-tends 5 cm in axial direction. Driving frequency is13.56 MHz. Argon is the type of the gas. Initial pres-sure is 20 mTorr, which corresponds to 3.0E+20 m3

neutral density. There is no neutral gas inlet to thesystem. All the ions that reach the wall go througha recombination process and directed back into thesystem as neutrals. The same configuration is solvedalso with COMSOL and the results are compared.For comparison, two different power deposition val-ues, that result in steady-state solutions, are chosen.These values are 3500 W and 6000 W.

The plasma density distribution obtained fromAETHER at 10 miliseconds is shown in Fig. 3. It isseen that the plasma is confined at the center of thedischarge chamber because of the losses to the walls.It is also seen that the plasma density is slightlyhigher in the regions that are located below the coils(from origin to 0.050 m in axial direction) comparedto regions that do not lay under RF coils (from 0.050m to 0.070 m in axial direction).

The verification is performed by taking the data ontwo lines and comparing the electron number densityvalues along these lines for different power depositionvalues.

Figure 4: Comparison of number density results fromAETHER and COMSOL along the line L1 for 3000 Wand 6500 W.

Figure 5: Comparison of number density results fromAETHER and COMSOL along the line L2 for 3000 Wand 6500 W.

The first line, L1 = |P1P2|, is the center line in radialdirection, which starts at P1 = (0.035, 0.000) and ex-tends up to P2 = (0.035, 0.040). The number densityvalues along the line L2 are presented in Fig. 4. Thesecond line, L2 = |P3P4|, is the center line in axialdirection, which starts at P3 = (0.000, 0.020) and ex-tends up to P4 = (0.070, 0.020). The number densityvalues along the line L2 are presented in Fig. 5.

The RF ion thruster simulations are performed for avery similar configuration in terms of geometry. Anexample thruster configuration, which is very similarto RIT-15LP [20], is modeled. This thruster has 15cm diameter and 7 cm axial length. The coil wrappedaround the cylindrical discharge chamber is repre-sented on the axisymmetric domain with 13 equidis-tant coils. The RF frequency is 2 MHz, which meansthat one RF cycle is 5.0e 07 sec. Two differentpressure levels are tested by applying 4 sccm and 13sccm neutral flow rates for inlet gas.

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Figure 6: Plasma density vs. time for 4 sccm neutral flowrate at different power levels

Five different power deposition cases are tested with250-, 300-, 350-, 400- and 450-W power depositionvalues for 4 sccm neutral Xenon gas inflow. Theplasma densities for these power deposition values aredepicted in Fig. 6. The mean electron temperaturefor these power depositions does not vary significantlyand can be taken as 3.08 eV.

When the Xenon gas mass flow rate and the back-ground neutral pressure are increased, a higher powerdeposition is required to sustain the plasma as ex-pected. With the same discharge chamber geometry,the steady-state temperature is this time much higheralong with the power deposition. To observe the ef-fect of high pressure, the neutral flux is increasedfrom 4 sccm to 13 sccm. The tested power deposi-tion values that yield steady-state solutions are 500-,600- and 700-W. As it is observed with the previousconfiguration, the electron temperature yields aver-age values which are very close for each case. Theaverage temperature change for each case versus timeis presented in Fig. 7. For the 600 W case, the meanelectron temperature is 4.31 eV.

The sheath region is not resolved with the 2D ax-isymmetric numerical model as stated in previous sec-tions. The analytical method outlined in Section II.is applied for the 4.31 eV bulk plasma electron tem-perature and 1e+ 17 m3 presheath plasma density.The sheath width is calculated as 8.4538e 05 me-ters, and the total sheath potential drop is 13.07 V.The electron and ion number density distributions inthe sheath region are given in Fig. 8. The electricpotential distribution in the sheath is presented inFig. 9.

V. Conclusions

A fluid model of the plasma inside the dishcargechamber of an RF ion thruster is built that does notrequire any empirical input.

Figure 7: Mean temperature change vs. time for 13 sccmneutral flow rate at different power levels

Figure 8: Number densities in the sheath region

The results lay out that the electron temperature in-side an RF ion thruster discharge chamber is almostconstant for various power levels at the same pres-sure. The beam current increases with the increasingpower, and this increase can be attributed to the in-crease of the plasma density and the ionization frac-tion.

There is a nonlinear relation between electron tem-perature, chamber dimensions, plasma density andthe beam current. Thruster efficiency should also betaken into account and an optimization study shouldbe carried out to evaluate the best configuration forRF ion thrusters.

The analytical sheath model presented in this studyserves currently only for post-processing purposes.It is used to monitor the change in species numberdensities and electric potential inside the sheath re-gion. A possible coupling of this analytical formula-tion with the heat loss boundary condition of the fluidmodel should be investigated in the future.

An extension of the code can be possible if the ca-pability to capture the electrostatic fields in plasma

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is implemented. This can also lead to the simulationof various types of plasma sources where the nonam-bipolar flow is dominant. The simulation of thesetype of flows could be possible if electrons are solvedseparately from the ions and the electric potentialcan be evaluated by solving the Gauss Law. An at-tempt to these simulations is to be found in [21]. Theinterested reader should also consider reading the dif-ferent sheath phenomena observed in these types ofplasma [22].

Figure 9: Electric potential in the sheath region

Acknowledgement

This research is supported by Turkish Scientific andTechnological Reseach Council (TUBITAK) underprojects 112M862 and 113M244 and partially byBogazici University Scientific Projects Office underproject number BAP-6184. The authors would like tothank Prof. Huseyin Kurt of Istanbul Medeniyet Uni-versity for allowing the usage of the computationalfacilities at Istanbul Medeniyet University and theCOMSOL Multiphysics software for this study.

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IntroductionTheory ModelNumerical MethodResultsConclusions