Numerical Modelling and Optimisation of Radio

Numerical Modelling and Optimisation of Radio-Frequency Ion Thrusters

Martin Felix Closs, Dipl. Masch.-Ing. ETH

In der Fakultät Luft- und Raumfahrttechnik der Universität der Bundeswehr M¨ unchen eingereichte Dissertation zur Erlangung des akademischen Grades eines Doktors der Ingenieurwissenschaften

Vorsitzender: Prof. Dr. Ing. Leonhard Fottner Erster Gutachter: Prof. Dr. rer. nat. Bernd Häusler Zweiter Gutachter: Prof. Dr. Ing. Wolfgang Koschel Dritter Gutachter: Prof. Dr. rer. nat. Rudolf Treumann

M¨ unchen, 15.10.2001

To my parents: Barbara and Felix.

Acklowledgements I would like to thank Astrium GmbH for funding my research and giving an opportunity for fruitful learning. I am also grateful to Professor Bernd Häusler for the support and suggestions he gave me. Furthermore, I would like to thank him and Professor Wolfgang Koschel as well as Professor Rudolf Treumann for taking the time and patience to appraise my thesis. Special thanks goes to Michael Marx and Hans M¨ uller for the numerous fruitful discussions, the help they gave me and the time they spent listening.

Abstract An entirely self-consistent numerical model of radio-frequency ion thrusters was developed. It allows to calculate the behaviour of such devices over wide operating ranges, without prior knowledge of new designs in the form of test data. The intention was to develop a method of calculating the behaviour of an RF ion thruster given its geometry, the type of propellant used and the electric potentials applied to the grid system. The data provided by this calculation should give valuable information for the control of the thruster, electronics design and for system engineering studies. It should also allow the optimisation of a design on a numerical basis in order to reduce the effort and cost required for the manufacturing and test of prototype models. As a difference to current plasma modelling, the development of this model of RF ion thrusters emphasised on trading detailed information with simplicity and computational efficiency in order to allow a fast calculation of entire performance maps–information which is crucial to the engineering process. The model includes a calculation of the gas discharge kinetics inside the thruster for electrons, singly and doubly charged ions and neutral atoms. A Direct Monte Carlo simulation (DSMC) of the neutral gas efflux through the extraction grid system was performed in order to find a method of calculating the neutral gas density in the plasma. Based on these calculations, an estimation of the charge exchange process which causes ion impingement on the accelerator grid was performed in order to account for the loss of thrust caused by charge exchange. A novel approach was taken in calculating the coupling of RF power into the plasma by solving a nonlinear form of Maxwell’s equations. The Finite Element Method was employed to find a solution to these partial differential equations. This approach allows to ascertain the influence of the RF coil geometry and the thruster housing on the overall performance of the ion thruster. Based on information obtained from the discharge calculation and the treatment of the charge exchange process in the grid system, a new more accurate way of calculating the ion engine’s thrust was developed. Due to the computational efficiency of the model, the numerical optimisation of the thruster’s discharge vessel is rendered possible. The contour of the discharge vessel can be optimised on a theoretical basis, reducing the number of prototype tests. Results promise a reduction in RF power of 25 % and a reduction of the mass of the discharge vessel of 40 %, which would lead to a new, more lightweight thruster design.

Zusammenfassung Ein vollständig selbstkonsistentes Modell f¨ ur Radiofrequenz-Ionentriebwerke wurde entwickelt. Es erlaubt die Berechnung des Verhaltens solcher Triebwerke u ¨ber weite Betriebsbereiche, ohne die Notwendigkeit von Messdaten vorhandener Geräte. Das Ziel war es, eine Berechungsmethode zu entwickeln, mit der man das Verhalten der Triebwerke berechnen kann, wenn die Geometrie, die Treibstoffsorte und die Gitterpotentiale gegeben sind. Die berechneten Daten sollten wertvolle Information f¨ ur die Regelung, Elektronikentwicklung und Systembetrachtungen liefern. Das Modell sollte auch zur theoretischen Optimierung des Triebwerks dienen, um den Aufwand und die Kosten f¨ ur den Bau und Test von Prototypen zu reduzieren. Bei der Entwicklung dieses Modells wurde sorgfältig der Nutzen von detaillierten Berechnungen gegen¨ uber der rechnerischen Einfachheit und Effizienz abgewägt um die Berechnung ganzer Kennlinienfelder zu erlauben - Information, die f¨ ur den Ingenieur unerlässlich ist. Das Modell enthält eine Berechnung der Gasentladungskinetik im Triebwerk f¨ ur Elektronen, einfach und doppelt geladene Ionen und Neutralteilchen. Die Neutralgasströmung durch das Extraktionsgittersystem wurde mit der Direct Simulation Monte Carlo (DSMC)Methode simuliert, um ein Berechnungsverfahren f¨ ur den Neutralgasverlust und -druck zu entwickeln. Basierend auf diesen Berechnungen wurde ein Modell des Ladungsaustausches zwischen Neutralteilchen und Ionen im Gittersystem entwickelt, um den dadurch entstehenden Schubverlust zu ber¨ ucksichtigen. Zur Bestimmung der HF-Leistungseinkoppelung in das Plasma wurde ein neuer Ansatz entwickelt, der eine nichtlineare Form der Maxwell’schen Gleichungen verwendet. Zur Lösung dieser Partiellen Differentialgleichungen wurde die Finite-Elemente-Methode herangezogen. Diese Methode erlaubt die Untersuchung beliebiger HF-Spulengeometrieen und Triebwerksgehäuseformen, um deren Einfluss auf die Effizienz des Triebwerks zu studieren. Basierend auf der Berechnung der Gasentladungskinetik und des Ladungsaustausches im Gittersystem wurde eine neue und genauere Methode zur Abschätzung des Triebwerksschubes entwickelt. Da dieses Modell einen verhältnismäßig geringen Rechenaufwand benötigt, wurde die numerische Optimierung des Entladungsgefäßes möglich. Die Form des Gefäßes kann auf einer theoretischen Basis optimiert werden, was die Anzahl nötiger Prototypen stark reduziert. Die Resultate weisen auf eine Mögliche Einsparung der HF-Leistung von 25 % und eine Gewichtsreduktion des Entladungsgefäßes von 40 % hin.

Contents 1 Introduction 1.1 A Brief History of Ion Propulsion . . . . . . . . . . . . . 1.2 Ion Propulsion: an Overview . . . . . . . . . . . . . . . . 1.2.1 Ground Based Applications of Ion Beam Sources 1.3 The Principle of RF Ion Thrusters . . . . . . . . . . . . 1.3.1 Ion Generation . . . . . . . . . . . . . . . . . . . 1.3.2 Ion Extraction and Acceleration . . . . . . . . . . 1.3.3 Beam Neutralisation . . . . . . . . . . . . . . . . 1.3.4 Power Dissipation . . . . . . . . . . . . . . . . . . 1.4 Previous Work . . . . . . . . . . . . . . . . . . . . . . . 1.5 The Scope of this Thesis . . . . . . . . . . . . . . . . . . 1.5.1 Novelties in the RF Ion Thruster Model . . . . . 1.6 Mission Aspects of Electric Propulsion . . . . . . . . . . 1.6.1 Suitable Propellants . . . . . . . . . . . . . . . . 1.6.2 GEO Station Keeping . . . . . . . . . . . . . . . 1.6.3 Orbit Transfer Manoeuvres . . . . . . . . . . . . 1.6.4 Atmospheric Drag Compensation . . . . . . . . .

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2 Basis 2.1 Plasma Physics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1.1 Collisions in the Plasma of an RF Ion Thruster . . . . . . . 2.1.2 The Plasma Sheath . . . . . . . . . . . . . . . . . . . . . . . 2.1.3 Typical Values of Plasma Parameters in an RF Ion Thruster 2.1.4 Plasma Conductivity . . . . . . . . . . . . . . . . . . . . . . 2.2 Rarefied Gas Flow . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3 Electrodynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3.1 Maxwell’s Equations . . . . . . . . . . . . . . . . . . . . . . 2.3.2 Skin Effect . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 RF Ion Thruster Model 3.1 Discharge . . . . . . . . . . . . . . . . . . . . . 3.1.1 General Assumptions and Simplifications 3.1.2 Definition of the System Domain . . . . 3.1.3 Equations of Conservation . . . . . . . . 3.1.4 Fluxes to and from the Systems . . . . . ix

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3.2 3.3

3.4

3.5 3.6

3.1.5 Assembly of the Equations and Numerical Solution . . . . . . . Accelerator Grid Impingement Current . . . . . . . . . . . . . . . . . . 3.2.1 Estimating the Neutralisation Plane . . . . . . . . . . . . . . . . Thrust . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3.1 Charge Exchange Ions and Beam Divergence Neglected . . . . . 3.3.2 Momentum Loss Due to Charge Exchange and Beam Divergence RF Field Coupling and Coil Impedance . . . . . . . . . . . . . . . . . . 3.4.1 Boundary Conditions . . . . . . . . . . . . . . . . . . . . . . . . 3.4.2 Problem Solution . . . . . . . . . . . . . . . . . . . . . . . . . . 3.4.3 Power Dissipated in Coil and Housing . . . . . . . . . . . . . . . 3.4.4 Capacitive Coupling . . . . . . . . . . . . . . . . . . . . . . . . Assembling the Parts of the Model . . . . . . . . . . . . . . . . . . . . Verification of the Model . . . . . . . . . . . . . . . . . . . . . . . . . . 3.6.1 Coil Inductance without Plasma . . . . . . . . . . . . . . . . . . 3.6.2 Absorbed RF Power PRF . . . . . . . . . . . . . . . . . . . . . . 3.6.3 RFG Power Input PRF G . . . . . . . . . . . . . . . . . . . . . . 3.6.4 Accelerator Impingement Current Iacc . . . . . . . . . . . . . . . 3.6.5 RF Generator Current Irf g . . . . . . . . . . . . . . . . . . . . . 3.6.6 Discharge Vessel Temperature Tdv . . . . . . . . . . . . . . . . .

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61 63 69 71 71 72 74 74 76 78 81 84 84 84 84 89 92 98 103

4 Optimising the Discharge Vessel 107 4.1 Operating Conditions Necessary for Optimisation . . . . . . . . . . . . . . 107 4.2 Optimal Shape for the Discharge Vessel . . . . . . . . . . . . . . . . . . . . 108 4.3 Expected Improvements Through Optimisation . . . . . . . . . . . . . . . 112 5 Conclusions and Outlook

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References

130

A: The Finite Element Method

131

B: The Direct Simulation Monte Carlo Method

135

x

Table of Symbols – Aa : anode surface area – Ace : material constant for charge exchange cross section – Adv : surface area of the discharge vessel walls – Ah,scr : cross section area of a single screen hole – Ascr : surface area of the screen grid – Bce : material constant for charge exchange cross section – B: magnetic flux density vector – Bz : z-component of flux density vector – cB,k : Bohm velocity – ce : electron velocity – hce i: mean thermal velocity of electrons – cex : exhaust velocity – ck+ : ion velocity – Ch : coil-to-discharge vessel capacitance – CKn : open orifice area of grid system – Crf g : RFG resonance capacitance – Cs1 : first plasma sheath capacitance – Cs2 : second plasma sheath capacitance – da : accelerator grid hole diameter – dd : decelerator grid hole diameter – dh : grid hole separation distance – Db : back wall diameter of discharge vessel – D: electric flux density vector – Edv : internal thermal energy of discharge vessel – Ee : internal thermal energy of electron gas – E∗ : mean excitation energy of propellant – E0 : electric field at surface of conductor – Ei,k : k th ionisation energy of xenon – E: electric field vector – fM : Maxwellian energy distribution function – hd : grid dish height – hdv : height of discharge vessel – H: magnetic field vector – irf c : RF coil current – =: imaginary part of a complex value – Iacc : accelerator grid impingement current

xi

– Ib : ion beam current – Iex : extracted ion current – Irf g : RFG input current – Isp : propellant specific impulse – Issp : system specific impulse – Itot : total momentum imparted on spacecraft – jc : surface current density on RF coil – Jsc : space charge limited current density – J: current density vector – kB : Boltzmann’s constant – Kn: Knudsen number – Kt : tank performance factor – ld : accelerator to decelerator grid separation – lg : screen to accelerator grid separation – Lrf c : RF coil inductance – m0 : propellant mass – me : electron mass – mel : mass of electric power supply – mpss : propellant storage system mass – mt : propellant tank mass – mtot : total mass of propulsion system at beginning of life – mXe : atomic mass of xenon – M a: Mach number – n∞ : neutral density in vacuum chamber – n0 : neutral number density – nk : number density in region k – nk+ : number density of ions with charge kqe – ne : electron number density – n: normal vector ∂Ωdv – N˙ b,k : number flux of ions with charge kqe in ion beam

– N˙ 0 : neutral number flux – Nc : number of coil turns – N˙ ce : charge exchange rate ∂Ω – N˙ ex,k : number flux of ions with charge kqe extracted through the screen grid

– N˙ k : neutral number flux from region k – Nh : number of grid holes Ω – Nk+ : number of ions with charge kqe in Ω

xii

Ω – N˙ 0→k+ : number flux of ions with charge kqe generated in Ω Ω – N˙ 0→∗ : number flux of excited atoms generated in Ω

– N0Ω : number of neutrals in Ω ∂Ω – N˙ inlet,0 : number flux of neutrals into Ω ∂Ω – N˙ grid,0 : efflux of neutrals through the grid system ∂Ω – N˙ rec,k : number flux of ions with charge kqe recombining on the walls

– pcoll,k : collision probability – p˙k+ : ion momentum flux – p˙l : loss of ion momentum flux – hpi: mean momentum of a beam ion – P0 : electric power density at surface Ω : power loss through excitation – P0→∗

– Ph : power dissipation in the thruster housing Ω – P0→+ : power loss through ionisation ∂Ω – Prec : power loss trough wall recombination

– PRF : absorbed RF power – Prf c : power dissipation in the RF coil – PRF G : RFG input power – qe : elemental charge – rc : coil lead radius – r: configuration space coordinate – rw : discharge vessel wall contour – 30 kNs/kg, which is much larger than ∆v, allowing for the use of equation (1.6). As stated in o lies at the example of the Artemis satellite, the optimal propellant-specific impulse Isp 34 kNs/kg. For this value, we obtain a total propulsion system mass of 70 kg. Of these 70 kg, about 30 kg are propellant. To get an idea of the dependency of mtot on the thrust, we rewrite equation (1.5) for the system-specific impulse at its optimum:

o Issp =

s

γ τη el el . 2 1 + KzRT tM

(1.17)

√ o This shows that Issp is proportional to τ , which in turn decreases with rising thrust, since a given total momentum Itot can be imparted in shorter time with higher thrust. Therefore, it is generally not desired to use an electric propulsion system with high thrust—from the point of view concerning the system mass mtot . On the other hand, one is not entirely free in choosing the burn time τ for a number of reasons: primarily, the burn must take place at a node of the geosynchronous orbit. Thrust applied at a distance from the node is employed less efficiently than thrust close to the node. Therefore, one is 10

“GTO”=Geosynchronous Transfer Orbit, “GEO”=Geosynchronous Equatorial Orbit

1.6. MISSION ASPECTS OF ELECTRIC PROPULSION

27

interested in a short burn arc distributed symmetrically around the node, which in turn calls for a short burn time. East–West or Longitude Station Keeping (EWSK) Due to the asymmetry of Earth’s gravitational potential, satellites in GEO drift in the direction of longitude if they are not positioned at one of the two stable points at 105.3◦ W or 75.1◦ E. The acceleration on the spacecraft in this direction reaches a maximum of about 6.5·10−8 m/s2 around 120◦ E, which amounts to a ∆v of 2 m/s per year, or 30 m/s for a 15year mission. This is more than twenty times less than required for NSSK. An all-electric satellite would use the same tank and electric power system for NSSK and EWSK, so only the masses of the propellant and the thrusters are of significance in this case. With an Isp of 34 kNs/kg, only 1.3 kg of propellant are required to hold the position of Artemis over its 15-year life. This is a drastic example for the importance of the total system mass of electric propulsion systems, since a single 15 mN thruster weighs 1.8 kg. Due to the high Isp , the propellant mass can be of inferior significance for low-∆v manoeuvres—even if the electric power supply and the tank are shared with other thrusters.

1.6.3

Orbit Transfer Manoeuvres

Station keeping tasks require the compensation of an unwanted acceleration of the spacecraft due to orbit perturbations. This is generally accomplished using the same thrusting strategy for ion propulsion as for chemical rockets, with longer and more frequent thrusting because of the lower thrust levels. Orbit transfer manoeuvres however often ask for quite different strategies: while most chemical propulsion systems are fired in an impulsive manner—i.e. the spacecraft’s true anomaly changes only slightly during the short burn— this is not possible when employing ion thrusters, due to their low thrust level. Therefore, low-thrust strategies use continuous burns, which require higher velocity increments ∆v than impulsive orbit transfers (also known as Keplerian transfers). A good example is the transfer between two coplanar circular orbits. The well-known Hohmann transfer, a classical two-burn Keplerian manoeuvre, requires a ∆v of [80]

∆v =

√

r r r ! r 2 2 1 1 1 1 + µ − − − − , r0 atx r0 r1 atx r1

where • atx [m]: semi-major axis of transfer ellipse, • r0 [m]: radius of initial orbit, • r1 [m]: radius of final orbit, • µ = GM [m3 /s2 ]: gravitational constant of orbited celestial body.

(1.18)

28

CHAPTER 1. INTRODUCTION

As an example, we take the transfer from an initial Earth orbit of 300 km altitude to GEO at 35786 km altitude, which requires a ∆v of 3.89 km/s. If we use a constant low thrust burn in flight direction, which results in a spiral transfer, the velocity increment is approximately [80] ∆v = v1 − v0 ,

(1.19)

where v0 and v1 are the respective velocities of the initial and final orbits. In the example above, this results in a velocity requirement of ∆v = 4.65 km/s, about 20 % more than for a Hohmann transfer. GTO to GEO transfer An interesting treatise on low-thrust orbital manoeuvres and their strategies was published by J. E. Pollard [57]. We revert to his example of a GTO to GEO transfer to illustrate the specifications of an ion propulsion system suited for this task. Pollard supposes a spacecraft in a GTO with an apogee altitude of 35786 km, a perigee altitude of 185 km and an inclination of 28.5◦ . The transfer mission is divided into two phases: the first phase is used to increase the semi-major axis of the orbit to the target value of 42164 km in 97 days. During the second phase, the eccentricity and inclination are driven to zero in 23 days. The complete transfer takes 120 days and requires a ∆v of 2.5 km/s, compared to 1.84 km/s required for a Keplerian transfer. If we assume a satellite of 1000 kg bus mass at beginning of life (BOL), a total impulse of 2500 kNs needs to be delivered by the ion propulsion system, with a mean thrust level of 241 mN. The optimisation of Isp is not to be performed on the basis of equations (1.6) and (1.15), which would lead to a system mass of more than 600 kg at a moderate Isp of 13 kNs/kg. Since the satellite’s payload is not operating during the orbit transfer, the power installed for it can be used for thrusting and no separate power supply needs to be installed. The thrusters would be used later for NSSK, so they are already installed as well. The tightest constraints for the propulsion system are set during NSSK, where the spacecraft’s payload is under operation, so the optimisation is performed for this case as illustrated above. Therefore, the transfer from GTO to GEO can be performed at a high propellant-specific impulse and only the additional fuel and tank masses need to be considered. Isp is optimised by minimising mp + mt with help of equations (1.11) and (1.13). For our example, we obtain o Isp = 31 kNs/kg, leading to approximately 178 kg of additional propellant in a tank that is slightly less than 22 kg11 more heavy than if it were only to hold the fuel for NSSK. This stands in opposition to about 670 kg of mass, including the fuel tank, for a chemical propulsion system with Isp = 4 kNs/kg. Interplanetary Transfers Interplanetary orbit transfers are in many ways similar to manoeuvres in orbits around Earth. However, a great difference is made for solar electric propulsion (SEP), where 11 This value was obtained for a titanium alloy (Ti6Al4V) tank; if Kevlar would be employed, the mass increase could be reduced to 13 kg.


29

the power is supplied by the sun. Since the radiated power density decreases with the square of the distance to the sun, thrusting strategies need to take into account the varying availability of electric power. For propulsion systems which are powered by means independent of solar radiation, such as radio-isotope thermal generators (RTG), other constraints need to be considered. The power output of RTG’s is highest at BOL and slowly decreases over time as the radioactive material decays. Since RTG’s or other nuclear sources carry inherent risks and are therefore poorly accepted, SEP seems the most popular option. The article published by Steven Williams and Victoria Coverstone-Carroll gives an illustrative example for strategies concerning electric propulsion missions to Mars [81]. They developed optimal trajectories with different flight durations between 1.5 and 4.5 years, mainly for the case of a sample return mission. The ion propulsion system is not necessarily operated continuously; in fact, most trajectories follow a thrust sequence with consecutive burn and coast arcs, as illustrated in Figure 1.8. Many trajectories are somewhat counter-intuitive; for instance, one might expect the optimal trajectories to arrive at Mars either a node of the planet’s orbit or at perihelion. The former allows for all the thrust being applied in the ecliptic plane, with no requirement for imparting out-of-plane momentum to the spacecraft in order to rendezvous with Mars. The latter takes advantage of the higher solar irradiation at perihelion. Interestingly, Williams and Coverstone-Carroll state that the 1.5-year trajectories result in a compromise, between perihelion and a node. At longer flight times, the results are more intuitive as the optimal trajectory arrives at the ascending node. Generally speaking, finding optimal trajectories for interplanetary SEP missions is a more complicated task as this might be the case for chemical propulsion using impulsive burns.

1.6.4

Atmospheric Drag Compensation

Ion propulsion is very well suited for compensating the drag which originates from the residual atmosphere at low orbit altitudes. The thrust requirements are very low for regular-sized satellites, e.g. approximately 6 · 10−2 mN for a satellite with a cross section of 1 m2 flying at 500 km altitude. Since chemical thrusters have much higher thrust, the drag is compensated with short, quasi-impulsive burns. Due to the comparatively low thrust of ion motors however, corrections can be made more often than with chemical thrusters, which allows for more accurate orbit control. In very low orbits, as the 260 km of altitude planned for the Gravity and Ocean Circulation Explorer (GOCE) mission, the necessary thrust can reach 10 mN [1]. The GOCE mission requires continuous operation of the ion thruster in order to compensate the acceleration due to atmospheric drag to a level acceptable by the gradiometre on board. As the drag needs to be exactly compensated, ion thrusters are virtually the only devices which can perform this task, since they can be throttled down to 1 mN of thrust. Also the average drag of about 5 mN adds up to a loss of momentum of 158 kNs per year, which would require 40 kg of fuel during the same period for an efficient chemical propulsion system with Isp = 4 kNs/kg. With ion propulsion, one can expect at least a five-fold reduction in propellant consumption. Also Earth imaging gains from low altitude orbits. A satellite proposed by Universität der Bundeswehr in

30


Munich and the DLR in Oberpfaffenhofen, Germany [82], would fly a high resolution camera on a circular orbit at an altitude between 300 and 500 km, using ion propulsion for orbit control and atmospheric drag compensation. An autonomous controller which takes advantage of the wide throttling range of ion thrusters operates the engine intermittently, keeping the satellite’s position within a tolerance of 200 m in altitude and within 1 km in flight direction and perpendicularly to the orbit plane. The proposed application would put the thruster on standby between burns to ensure quick response of the device. Since the ion engine requires a minimal amount of propellant during standby operation, a continuous loss of propellant is to be tolerated. New developments in propellant feed systems and the thruster’s gas inlet may however reduce the time required to ignite the thruster to a few seconds. On the other hand, as long as hollow cathodes are utilised as neutralisers, a heating period of several minutes is required to ignite the device. Future developments in RF neutraliser technology might lead to using such electron sources as neutralisers for ion propulsion. Since an RF neutraliser can be ignited within seconds, standby operation would become unnecessary because the entire propulsion system could be ignited virtually at the turn of a switch.


31

Figure 1.8: Example of an optimal Earth-to-Mars trajectory given by Williams and Coverstone-Carroll [81]. Solid lines represent burn arcs and dashed lines are coasting arcs. The two dotted lines denote Earth’s and Mars’s orbits. The time of flight (TOF) is 2.5 years and the consumed propellant mass amounts to 334 kg, which leaves a spacecraft mass of 2148 kg at burn-out (BO).

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Chapter 2 Basis 2.1 2.1.1

Plasma Physics Collisions in the Plasma of an RF Ion Thruster

Since the ions in the discharge vessel are obtained from electron-atom collisions, collision processes are of major interest in deriving a model of the gas discharge. The collisions of interest are summarised as follows: • Elastic electron-atom collisions • Inelastic electron-atom collisions • Volume recombination collisions Collisions between atoms are not of interest, since they are only of significance in transport phenomena of the plasma, which were not modelled during this work. Even when determining neutral gas losses through the grid system, collisions can be neglected as the mean free path is much larger than the geometrical features of the grids. Collisions leading to the recombination of electrons with ions in the volume of the plasma are considered to be rare [32]. Not only that the ion and electron densities are comparably low, making collisions rather unlikely—the fact that three collision partners are necessary for recombination render this process ineffective at the low plasma densities prevailing in the discharge vessel. The three partners are required to satisfy both energy and momentum conservation before and after the collision. At higher densities, it becomes more likely for three particles to meet within a close range, whereas at low densities the role of the third particle is taken over by a photon which is emitted from the atom—so-called radiative recombination. The latter process was however found to have negligible impact on the gas discharge under investigation. Elastic electron-atom collisions are of significance when estimating the electric conductivity of the plasma. Therefore, they will be treated in section 2.1.4. However of major interest are inelastic collisions between electrons and atoms. In a single-atom gas as is Xenon, only ionising and exciting inelastic collisions happen between electrons and neutral atoms. The two processes are sufficiently similar to allow for their 33

34

CHAPTER 2. BASIS

modelling using the same mathematics. The following discourse will treat ionising collisions [78, 30, 42]. Exciting collisions are treated in exactly the same manner using the respective material data. Ionisation through Inelastic Electron-Atom Collisions The probability pcoll,k for an electron passing through a differential volume element dV = dAdx to collide inelastically with a Xe atom and ionise the latter to charge kqe is

pcoll,k =

n0 σk+ (Ee )dV dA

(2.1)

where • n0 [m−3 ]: Neutral gas number density • σk+ (Ee ) [m2 ]: Electron collision cross section for ionisation to charge kqe The ionisation cross section σk+ (Ee ) is a function of the incident electron energy Ee . A thorough review on measuring ionisation cross sections was given by Kieffer and collaborators [35]. Kieffer and Chamberlain collected and compiled cross section data and published them in references [34, 15]. Figure 2.1 shows the total cross section σt from reference [34], which is defined as follows:

σt =

k=Z XXe

kσk+ ,

(2.2)

k=1

where ZXe = 54 denotes the atomic number of Xenon. Figure 2.2 shows the ratio between the cross sections σk+ (Ee ) and σt . The effective ionisation rate in the volume element dV can be calculated by multiplying the number of electrons moving past dA per unit time with the ionisation probability pcoll,k : d2 N˙ k+ = dne (Ee )ce dApcoll,k = dne (Ee )n0 ce σk+ (Ee )dV

(2.3) (2.4)

1 Ee = me c2e 2

(2.5)

with

2.1. PLASMA PHYSICS

35

Figure 2.1: Total ionisation cross section σt of xenon as a function of incident electron energy Ee [34].

36

CHAPTER 2. BASIS

Figure 2.2: The ratio σk+ /σt of xenon as a function of incident electron energy Ee [34].

2.1. PLASMA PHYSICS

37

The cross section functions shown in Figures 2.1 and 2.2 can be approximated by the following expression [78, 42]: Ee − Ei,k Ee − Ei,k σk+ (Ee ) = σmax,k+ exp 1 − ∆Ek ∆Ek

(2.6)

where • ∆Ek = Ee (σmax,k+ ) − Ei,k [J]: Difference between the energy at maximum cross section and ionisation energy With the help of equation (2.6) an analytical expression can be given for (2.4). To obtain an expression for the ionisation rate dN˙ k+ in dV , it is necessary to integrate over all electron energies Ee above Ei,k . Under the assumption of Maxwellian energy distribution at temperature Te for the electrons1 , one obtains dN˙ k+ = ne n0 hσk+ ce i dV

(2.7)

where hσk+ ce i is the ionisation integral for the species k:

hσk+ ce i =

σmax,k+ 3

(kB Te ) 2

r

8 ∆Ek πme

∆Ek kB Te

2∆Ek + Ei,k 1 + 3 ∆Ek 1 + kB Te

Ei,k exp 1 − kB Te

(2.8)

where • kB [J/kgK]: Boltzmann’s constant, • Te [K]: electron temperature. The values for σmax,k+ and Ee (σmax,k+ ) (respectively σ∗ and Ee (σmax,∗ ) for excitation) were determined by finding the best fit between numerically integrated values of hσk+ ce i and the values obtained through equation (2.8) at electron temperatures between 1 and 15 eV, using the cross section from reference [55]. The results are listed in table 2.1.

2.1.2

The Plasma Sheath

The plasma in an RF ion thruster never reaches out to the discharge vessel’s walls, a sheath is always formed between them [10, 78]. The reason for this is the high thermal velocity of the electrons. Due to their small mass and high temperature, electrons move about in the plasma at speeds much higher than the ions, which are generally in thermal equilibrium with the discharge vessel and the neutral gas. This fact causes a high electron 1 Strictly, this assumption does not hold for the plasma in RF ion thrusters. It however yielded satisfactory results. For a more detailed discussion, cf. section 3.1.1

38

CHAPTER 2. BASIS

species σmax,i [10−20 m2 ] Ee (σmax,i ) [eV] singly charged ions 5 53 doubly charged ions 4 100 excited atoms 6.9 21 Table 2.1: Maximum cross sections and energy differences for inelastic electron-atom collisions. current to reach the walls of the discharge vessel during the first instant after ignition of the plasma. As the ions drift towards the walls much slower than the electrons, a net negative current to the walls results, charging the latter negatively. The loss of electrons from the plasma to the walls causes the potential of the plasma to rise, confining a large portion of the electrons inside the plasma because only few of them carry sufficient energy to overcome the potential drop from the plasma to the walls. This potential drop is usually referred to as the plasma floating potential and typically assumes around 30 V in an RF ion thruster. To obtain quantitative knowledge about the plasma sheath, the electric current density balance for the discharge vessel’s walls is expressed for the stationary state:

|Uf |qe −ne exp − kB Te

r

2 1 X kB Te 3/2 qe + k exp − nk+ cB,qe = J 2πme 2 k=1

(2.9)

where • J [A/m2 ]: net current density towards the wall, • cB,k [m/s]: Bohm velocity of ion species k, • mXe [kg]: atomic mass of xenon, • nk+ [1/m3 ]: number density of ions with charge kqe , • qe [C]: elemental charge, • Uf [V]: plasma floating potential. The first term on the left-hand side of equation (2.9) denotes the electron current towards the wall, whereas the second term stands for the ion current. The right-hand side is the net current density which is drawn from the wall. For an electrically isolating wall, J = 0. Since the discharge vessel is coated with graphite from the accelerator grid already after a few hours of operation, it is assumed that the entire inside of the discharge vessel is conductive.

2.1. PLASMA PHYSICS

39

The ions drift towards the wall at a velocity called the Bohm velocity or the ionic velocity of sound cB,k [10]. It is caused by the remainder of the sheath’s electric field, which penetrates into the quasi-neutral region of the plasma. The potential drop UB caused by this penetration equals half the electron temperature measured in eV. Hence, cB,k is

cB,k =

r

kkB Te . mXe

(2.10)

By using the expression for cB,k while solving equation (2.9) for the floating potential Uf , we obtain

kB Te Uf = − ln qe

√

2πme ne

P2 ! 3/2 k n J k+ k=1 − √ . √ emXe qe kB Te

(2.11)

Note that besides the net current to the wall J, the only variable Uf depends on is the electron temperature Te (for a given gas). Te will be determined by the energy balance of the electrons, cf. section 3.1.3. The ion density nk+ se at the sheath edge of the plasma is required to determine the ion flux to the discharge vessel’s walls. It is found in the following discourse: As mentioned, the electric field in the sheath penetrates into the quasi-neutral region of the plasma and causes a potential drop UB of

UB = −

kB Te 2qe

(2.12)

, first from the undisturbed plasma to the plasma sheath. In order to determine n k+ se the electron density ne se at the sheath edge is calculated with help of the Boltzmann distribution function: |U | B ne se = ne b exp − kB Te 1 = ne b exp − 2 where • ne se [1/m3 ]: electron number density at the sheath edge, • ne b [1/m3 ]: electron number density in the bulk of the plasma.

(2.13) (2.14)

40

CHAPTER 2. BASIS

The plasma is assumed to be quasi-neutral up to the sheath edge. Therefore, equation (3.2) can be used in order to determine the ion number densities n1+ and n2+ . If we further assume, that the ratio between n1+ and n2+ is approximately constant in the region where the electric field penetrates into the plasma, the ion number densities become 1 nk+ se = nk+ b exp − 2

(2.15)

where • nk+ b [1/m3 ]: Ion number densities in the bulk of the plasma

2.1.3

Typical Values of Plasma Parameters in an RF Ion Thruster

The following table gives an overview of the typical parameter values one encounters in the plasma of an RF ion thruster. The values are taken from calculations with the model presented in this thesis and correspond to measurements. Neutral gas number density Neutral gas temperature Electron number density Electron temperature Doubly charged ion density Neutral gas mean free path∗ Electron mean free path† Electron mean collision frequency† Debye length Plasma frequency RF field frequency RF field skin depth

0.5 . . . 3.5 · 1019 1/m3 100 . . . 300 ◦ C 1.5 . . . 4 · 1017 1/m3 3 . . . 10 eV 1 . . . 6 · 1015 1/m3 50 . . . 500 mm 0.5 . . . 5 m 2 . . . 8 MHz 20 . . . 60 µm 3 . . . 6 GHz 0.7 . . . 1 MHz 20 . . . 50 mm

Table 2.2: Typical values of plasma parameters in an RF ion thruster. ∗ For momentum transfer self-collisions. † For momentum transfer with neutral particles.

2.1.4

Plasma Conductivity

Calculating the conductivity of the plasma in an RF ion thruster is not a trivial task. The alternating electric field is too strong and the neutral particle density too small to allow for a simple treatment. In the first of two sections, the conductivity for small electric fields is treated, while the conductivity for stronger fields is discussed in the second section. Both treatments are required for the thrusters under investigation, since the electric field varies from exactly zero to as much as 200 V/m—a value which turned out to call for a more thorough treatment. Both treatments neglect the drift of ions, since current flow in the plasma is heavily dominated by the much more mobile electrons. Less straight-forwardly, collisions between

2.1. PLASMA PHYSICS

41

electrons and ions are neglected and only collisions with neutrals are taken to account. It is recommended to include electron-ion collisions into the calculation of the plasma conductivity at ionisation levels above 0.1 percent [67]. In an RF ion thruster, the ionisation level can however reach several percent. On the other hand, the treatment of electron-ion collisions is a formidable task even for weak electric fields. For the case of strong fields, the treatment would go beyond the scope of this work. It is not possible to define a clear collision cross section between ions and electrons, because the far-reaching Coulomb field of the ion and electron cause interaction beyond the actual dimensions of a xenon atom. The electrons follow hyperbolic Keplerian trajectories around the ion rather than colliding with them directly. A treatment including electron-ion collisions would require involving the Fokker-Planck equation, an integro-differential equation which is derived from the Boltzmann equation and includes electron-ion collisions. Nevertheless, the method of calculation for the plasma conductivity developed for the model presented in this thesis yielded satisfactory results, estimating the RF coil current required to sustain a given plasma within 30 percent. It appears that this is the first time such a thorough treatment of the plasma conductivity was conducted for RF ion thrusters. Previous work only included the simple Ohmic treatment presented below [41, 30]. The AC magnetic field present in the thruster was neglected as well. It is rather weak, causing gyration radii of the electrons which are of similar dimensions as the discharge vessel. However, a thorough investigation of the plasma conductivity including electron-ion collisions and the AC magnetic field would be an interesting subject for further research. Marschall [47, pp. 21ff] gave an interesting treatment of the effects when comparing the momentum-transfer collision frequency of the electrons to the RF frequency of the electric field. He discusses the case where the collision frequency is much higher than the RF frequency, which corresponds to a so-called collision dominated plasma, such as the electron gas in a metal. Also the case of rare momentum transfer collisions within the time scale of the field oscillations is treated—this is the case of a nearly collisionless plasma as e.g. the ionosphere. The third case Marschall discusses is the one we find in an RF ion thruster: here the two frequencies are of similar order of magnitude, calling for the treatment which follows below.

The Case of Constant Collision Frequency νm A rather simple treatment of the problem is possible if the collision frequency for momentum transfer between electrons and atoms νm is independent of the electric field strength [67]. This is primarily the case if the thermal velocity of the electrons is much larger than the drift velocity induced by the electric field. This is also the reason why the following treatment will only hold for sufficiently small electric fields which do not impose a drift velocity which can be of significance over the thermal motion. The resulting formula for calculating the plasma conductivity is the one most commonly used. However, it was found that the drift velocity can reach the same order of magnitude as the thermal velocity if this expression for the conductivity is employed. Therefore, a more thorough treatment was conducted during this work.

42

CHAPTER 2. BASIS

For small electric fields, the conservation of momentum for electrons in a single direction can be expressed as hce˙ id =

qe E exp(jωt) − hce id νm , me

(2.16)

where • hce id [m/s]: mean drift velocity of electrons, • E [V/m]: peak value of electric field, • νm [1/s]: collision frequency for momentum transfer, • ω [rad/s]: frequency of electromagnetic fields. The stationary harmonic solution of (2.16) is

hce id =

qe 1 J exp(jωt + ϕ) E exp(jωt) = , me νm + jω n e qe

(2.17)

where J stands for the electron current density and ϕ denotes the phase shift between E and J. Now the conductivity can be written as

σp0 =

J exp(ϕ) ne qe2 1 = . E me νm + jω

(2.18)

The superscript 0 for the conductivity is used to designate that this expression holds for electric fields which approach zero. The collision frequency for momentum transfer νm is obtained in the same manner as the excitation or the ionisation rates in the plasma: νm = n0 hσm ce i,

(2.19)

where σm stands for the effective momentum transfer collision cross section (cf. Figure 2.3). A critical review on measuring σm was conducted by Bederson and Kieffer [7] in 1971. The data used for this work was published in reference [55] and represents a compilation from several independent measurements (cf. ref. [53]). The Case of Constant Mean Free Path Λm If the electric field becomes large enough to induce a drift velocity which cannot be neglected over the thermal velocity of the electrons, one has to abandon the assumption of constant collision frequency and introduce the more rigorous statement of a constant mean free path between momentum transfer collisions [67]. The threshold where this more detailed treatment needs to be performed is marked at the point where the drift velocity hce id exceeds about 10 % of the mean thermal velocity hce ith . Calculating the

2.1. PLASMA PHYSICS

43

180

Momentum transfer cross section [10−16 cm2]

160

140

120

100

80

60

40

20

−3

10

−2

10

−1

0

10

1

10 10 Electron energy [eV]

2

10

3

10

4

10

Figure 2.3: Effective momentum transfer collision cross section σm of xenon as a function of incident electron energy Ee [55]. drift velocity from equation (2.18) showed that hce id ≈ hce ith for many operational states of the ion thruster, clearly invalidating the simple treatment from above. Adopting a constant mean free path leads to a more difficult treatment of the problem which would go beyond the scope of this dissertation [67]. Here, we may only mention the result: √ E

< σp

=

3/2

2πne qe ξ 1/4 33/4 Γ(3/4)

r

Λm , Eme

(2.20)

and

=

σpE

√ 2Γ(5/4) Λm ωqe ne ξ , = −√ E 3Γ(3/4)

(2.21)

44

CHAPTER 2. BASIS

where • ξ=

2mXe mXe +me

[–]: Energy loss factor,

• 10. The gas densities in the region up- and downstream of the orifice are n1 and n2 , respectively, as are the temperatures T1 and T2 . Since collisions are neglected, only the random thermal motion of the particles contributes to the overall flow. The number of particles which migrate from region 1 to region 2 per unit time is

N˙ 1 = n1

r

kB T1 A, 2πmXe

(2.29)

where A stands for the orifice cross sectional area. Accordingly, N˙ 2 particles per second flow from 2 to 1. Therefore, the net flux of particles from 1 to 2 is

N˙ 1−2 = (n1

p

T1 − n2

r

p

T2 )

kB A. 2πmXe

(2.30)

For the derivation of a model for RF ion thrusters, the neutral particle flux through the grid system is of crucial interest. For this case, the calculation can be performed in a very similar manner. We start with free molecular flow of gas through an orifice in a wall of arbitrary thickness [31]:

N˙ 1−2 = (n1

p

T1 − n2

r

p

T2 )

kB CKn , 2πmXe

(2.31)

where

CKn =

5+ 20 +

38t d

4t d

+ 12

πd t 2 d

2

,

(2.32)

with • d [m]: orifice diameter, • t [m]: wall thickness. For t = 0, CKn = π4 d2 , as above. Previous work [21] used a similar, but less accurate expression as equation (2.32) to calculate the neutral gas flow through the grid system. In these calculations, the screen and decelerator grids were neglected entirely and the accelerator grid took over the role of the wall between the region inside the discharge vessel and the vacuum downstream of the grids. DSMC simulations however showed, that taking only the accelerator grid into account can cause significant errors. Therefore,

2.3. ELECTRODYNAMICS

47

a new approach was taken in order to account for all three grids by determining CKn through DSMC simulations. The main thought behind this method is the fact that CKn depends on geometry alone and has the dimension of an area—it can therefore be viewed as the actual open orifice area of the grid system and replaces A in (2.30). This finding was tested with a series of DSMC simulations which showed applicability of this method over wide ranges of gas density and temperature3 . The following table shows values of CKn for various grid systems: CKn [mm2 ] Grid system RIT10 Artemis 1.6 RIT10 EVO 0.71 RIT-XT 0.66 Table 2.3: Values of the open orifice area CKn for various grid systems.

2.3 2.3.1

Electrodynamics Maxwell’s Equations

The basis of electrodynamics is laid by Maxwell’s equations, which describe the behaviour of time-varying electromagnetic fields. In their most general form, they are written as ∂B , ∂t ∂D ∇×H=J+ , ∂t ∇ · D = ρ, ∇ · B = 0, ∇×E=−

(2.33) (2.34) (2.35) (2.36)

where • B [T]: Magnetic induction or flux density • D [A/m2 ]: Electric flux density • E [V/m]: Electric field vector • H [A/m]: Magnetic field vector. 3 It ought to be mentioned that the results may differ if there is a considerable gas density downstream of the grid system. For the calculations of interest, the density outside the thruster was neglected. If the 1 downstream density is to be taken to account, one needs to determine two open orifice areas CKn and 2 CKn , one for each direction of flow. This is caused by the assymetry of the grid system against the flow direction.

48

CHAPTER 2. BASIS

The materials interest in a RF ion thruster can be assumed to exhibit isotropic behaviour—including the plasma, since the influence of magnetic fields is ignored. Therefore, we can write

B = µH and E = D,

(2.37) (2.38)

where µ and respectively are the magnetic permeability and the electric permittivity of the material, which can be assumed to be equal to those of vacuum, µ0 and 0 [67]. Further, the current density J can be expressed by Ohm’s law:

J = σ(E)E,

(2.39)

where σ(E) stands for the conductivity of the material. Note, that the conductivity of the plasma is a function of E, the peak value of the harmonically time-varying E. Strictly speaking, we do not have Ohmic behaviour in this case. If only Ohmic conductors are involved, Maxwell’s equations are usually linear. The E-dependence of the plasma conductivity, however, renders the equations nonlinear. Substituting B for H in equation (2.34) with the help of (2.37) and replacement of J and D as in equations (2.39) and (2.38), leads to

∇ × B = µ0

! ∂E . σ(E)E + 0 ∂t

(2.40)

If we now derive the above equation after time and combine it with (2.33), we obtain ∂E ∂2E + 0 2 ). ∇ × ∇ × E = −µ0 (σ(E) ∂t ∂t

(2.41)

For the case of a harmonically time-varying electric field4 E, this can be written as ! ω2 σ(E) E = 0, ∇×∇×E− 2 1+ c jω0

(2.42)

where • c [m/s]: Vacuum velocity of light. 4 The derivation of σ(E) is not performed in equation (2.41), because E is the peak value of E(t)—i.e. not a function of time.

2.3. ELECTRODYNAMICS

49

The electromagnetic fields in the thruster are assumed to be rotationally symmetric with an azimuthal electric field component Eϕ only5 : E = [0, Eϕ , 0]T . In cylindrical coordinates, ∇ × ∇ × E is ∇×∇×E=

∂ 2 Eϕ 1 ∂Eϕ Eϕ ∂ 2 Eϕ + − 2 + . ∂r2 r ∂r r ∂z 2

(2.43)

By inserting this expression into (2.42) and multiplying the equation with r, we obtain ! ∂ 2 Eϕ ∂Eϕ Eϕ ∂ 2 Eϕ ω 2 σ(E) r + − +r − 2 1+ rEϕ = 0. ∂r2 ∂r r ∂z 2 c jω0

(2.44)

In vector notation, we can write the equation for Eϕ as follows: 2

∇ · (r∇Eϕ ) +

!

ω σ(E) 1+ r − r−1 Eϕ = 0, 2 c jω0

(2.45)

where the nabla operator ∇ is defined as

∂ ∇= ∂r

∂ ∂z

T

.

(2.46)

(2.45) is the equation of interest for modelling the RF power coupling into the plasma of the thruster. It is a partial differential equation of the Helmholtz type describing wave propagation in a medium. Due to the complicated geometry of the ion thruster and because one is interested in modelling arbitrary shapes, equation (2.45) cannot be solved analytically. Thanks to its flexibility, the Finite Element Method offers a suitable way of finding approximate solutions to this equation (cf. Appendix A).

2.3.2

Skin Effect

This short section is mainly thought to correct a common error made in literature [51]. It concerns the calculation of the Ohmic resistance of the RF coil and does not treat the skin effect in the plasma. The latter is entirely accounted for by employing the complete set of Maxwell’s equations, as described above. An electric field E acting tangentially to the surface of a good6 conductor is attenuated exponentially with depth x: x E(x) = E0 exp −(1 + j) , δ

(2.47)

5 So-called capacitive coupling of RF power into the plasma—which would introduce axial and radial electric field components—can be neglected. See section 3.4.4 for details. 6 By “good”, we mean σ ω0 .

50

CHAPTER 2. BASIS

where • E0 [V/m]: Electric field at the surface • δ [m]: Skin depth. For the skin depth δ, we find

δ=

r

2 , ωµ0 σ

(2.48)

with σ for the conductivity of the material. From equation (2.47), we can write

x J = σE0 exp −(1 + j) δ

(2.49)

for the current in the conductor. Therefore, the actual situation can be replaced by a thin sheet of thickness δ of the same material with a constant current density. This is often the basis for calculating the resistance of a wire at high frequency; it is replaced by a tube with the same radius but wall thickness δ. However, such a treatment leads to only half the actual resistance. The reason is, that for calculating the Ohmic resistance one must use the dissipated power density P instead of J. Since P = σJJ, we obtain x P = σ E0 exp −2 . δ 2

(2.50)

Clearly, the power density attenuates twice as quickly as J and one must introduce a sheet thickness of 2δ for calculating the resistance. This will lead to correct results, as was confirmed by measurement7 .

7 To verify this finding, the resistance of the coil of a RIT 10 thruster without housing was calculated and measured: the calculation yielded 181 mΩ, while 185 mΩ were measured.

Chapter 3 RF Ion Thruster Model 3.1 3.1.1

Discharge General Assumptions and Simplifications

Since this is a zero-dimensional model, all plasma parameters are calculated as mean values averaged over the discharge vessel volume. The following general assumptions and simplifications have been made to derive the model of the RF discharge in the thruster: 1. All the RF power absorbed by the plasma is absorbed by the electrons. 2. The electrons either have a Maxwellian or a Druyvesteyn energy distribution. 3. Neutral particles are in thermal equilibrium with the discharge vessel. 4. No energy is transferred to either ions or neutrals by elastic collisions between electrons and ions/neutrals. 5. Radiative heat transfer from the thruster to the space environment can be ignored. 6. The electron velocity is much larger than the ion/neutral velocity due to the high temperature and small mass of the former. 7. Only singly and doubly charged ions need to be taken to account. 8. Excitation of ions is ignored. 9. Volume recombination of electrons with ions is negligible. 10. The ionisation of metastably excited atoms (Penning effect) can be neglected. 11. The entire inner surface of the discharge vessel is electrically conductive. 51

52

CHAPTER 3. RF ION THRUSTER MODEL

Often, the electrons are assumed to be in thermal equilibrium with each other, leading to a Maxwellian energy distribution (cf. assumption N◦ 2). However in the case of the RF plasma, the electrons cannot completely achieve thermal equilibrium, since electrons with an energy higher than the threshold for xenon excitation are lost from the high energy end of the distribution after an inelastic collision. The high energy end of the distribution is not refilled sufficiently to assume thermal equilibrium. This causes the distribution to diminish more quickly at high energies than it is the case with a Maxwellian distribution. This effect was taken into account by introducing two electron temperatures - one for the low energy end of the distribution and a lower temperature for the high energy end. The verification of the model showed, that this deviation from the Maxwellian distribution only had a marginal effect on the overall results. The present model therefore assumes thermal equilibrium and a Maxwellian energy distribution. Another deviation is caused by the strong electric field in the thruster. Two approaches were taken here: in the case where the electron drift velocity is considerably smaller than their thermal velocity, a Maxwellian distribution function was imposed in order to calculate the mean collision frequency. It was kept in mind, that the electric current in the plasma is caused by an anisotropic disturbance of the Maxwellian energy distribution. The energy distribution for strong1 electric fields is of the Druyvesteyn type (cf. section 2.1.4). This distribution is very difficult to introduce into the discharge model, since it requires a priori knowledge of the electric field, which itself is influenced by the discharge. Therefore, this distribution function was only employed in calculating the plasma conductivity for strong electric fields. The satisfying results obtained with this three-fold mix—pure Maxwelllian distribution for calculating the discharge and collision integrals, Maxwellian plus anisotropic disturbance for the conductivity at weak electric fields and Druyvesteyn for strong electric fields—support these simplifying assumptions. Assuming a Maxwellian energy distribution also simplifies the model of the discharge drastically. It implies a solution to the Boltzmann equation, which otherwise would have to be found separately. Solving Boltzmann’s equation is a very difficult and sometimes still impossible task with today’s technology, so a simple model must circumvent this by making approximate assumptions about the velocity distribution of the electrons. The Maxwellian distribution represents the simplest possible case for the plasma under investigation. Assumption N◦ 4 takes into account that the electron mass is more than 2 · 105 times smaller than the atomic mass of xenon. Due to conservation of momentum and energy before and after an elastic collision, only a very small fraction of the electron’s energy can be transferred to the xenon particle. Hence, this energy transfer is neglected in the following discourse. This model for RF ion thrusters is primarily thought for devices in their testing environment, i.e. a vacuum chamber. The reason for this is the easier access to test data of a given thruster for verification of the model. Since the thruster is mounted onto a flange in the vacuum chamber, heat conduction to the massive chamber wall dominates over radiative heat transfer, so the latter can be neglected (cf. assumption N◦ 5). This also simplifies the numerics of the problem significantly, since the discharge vessel temperature only appears in a linear term. 1 “strong” meaning, that the drift velocity is not negligibly small compared to the mean thermal velocity.

3.1. DISCHARGE

53

There are several different processes for volume recombination of electrons with ions. The only process which takes place in a low pressure plasma is radiative recombination, where the excess energy of the collision is radiated away by a photon. The radiative recombination rate is proportional to both the electron and the ion number densities—ne and nk+ , respectively—i.e. approximately proportional to n2e . In contrast, the effective ionisation rate is proportional to the electron density times the neutral density n0 , which is much larger. Under equilibrium operation of the thruster, the ionisation rate is equal to the ion loss rate, which means that volume recombination accounts for only a small part of the ion loss rate. Additionally, radiative volume recombination is not very effective, so its rate is expected to be low (cf. assumption N◦ 9). At discharge pressures above 1 Pa, inelastic collisions between electrons and metastable atoms become abundant and the effective ionisation rate increases significantly. The reason therefore are ions generated from atoms excited to meta-stable states. Since they are already in a higher energy state than the ground state, less energy is required for ionisation, so even low-energy electrons contribute to the ionisation rate, increasing the latter. This is generally referred to as the Penning effect. Since the discharge in the thruster takes place well below 1 Pa, the Penning effect can be neglected (cf. assumption N◦ 10). Already after a few hours of operation, the inner surface of the discharge vessel is coated with a thin film of graphite sputtered from the accelerator grid (cf. assumption N◦ 11). Since the graphite has a low electrical conductivity, it is easily penetrated by the RF field. On the other hand, the film’s conductivity is sufficiently large to carry away the anode current from the plasma without a significant potential drop.

3.1.2

Definition of the System Domain

The system space Ω for which the balance equations are expressed is defined as the internal volume of the discharge vessel. Its boundary ∂Ω is the inner surface of the discharge vessel together with the total area of the screen grid (i.e. surface facing the plasma). For the energy balance of the discharge vessel, the vessel together with the grids are defined as the system Ωdv .

3.1.3

Equations of Conservation

To derive the thruster model, the following equations of conservation must be expressed: • Energy and mass balance of the electrons in the discharge vessel • Mass balance of singly charged ions in the discharge vessel • Mass balance of doubly charged ions in the discharge vessel • Energy and mass balance of the neutral particles in the discharge vessel • Energy balance of the discharge vessel • Charge balance of the plasma (to fulfill quasi-neutrality)

54


Some of these balance equations will turn out to be identical to others, so only a total of six equations need to be solved. Since we are not interested in the temperature of the ions, the energy balance of the latter is ignored. Energy Balance of Electrons The only energy flux to the electron gas is the RF power PRF absorbed by the plasma. There are several major energy fluxes leaving the electron gas: • Loss of electrons through recombination on the discharge vessel walls • Inelastic collisions with neutrals causing excitation • Inelastic collisions with neutrals causing ionisation Hence, the balance equation has the following form: ∂Ω Ω Ω E˙ e = 0 = PRF − Prec − P0→+ − P0→∗ ,

(3.1)

where • Ee [J]: internal energy of the electron gas, ∂Ω • Prec [W]: power loss through electrons recombining on ∂Ω, Ω [W]: power loss through ionising collisions in Ω, • P0→+ Ω • P0→∗ [W]: power loss through collisions with neutrals in Ω causing excitation,

• PRF [W]: RF power absorbed by the plasma. Mass Balance of Electrons - Charge Balance in the Plasma Since every electron carries the elemental charge qe , the mass balance of the electrons can be shown to be identical to the charge balance of the plasma. The latter equation is fairly simple to express:

ne = n1+ + 2n2+ , where • ne [m−3 ]: electron number density, • n1+ [m−3 ]: number density of singly charged ions, • n2+ [m−3 ]: number density of doubly charged ions.

(3.2)

3.1. DISCHARGE

55

Mass Balance of Ions The mass balance of the species k (ions with charge kqe ) can be written as follows: ∂Ω Ω ∂Ω Ω − N˙ rec,k − N˙ ex,k , N˙k+ = 0 = N˙ 0→k+

(3.3)

where Ω • Nk+ [–]: number of ions with charge kqe in Ω, Ω • N˙ 0→k+ [1/s]: flux of ions with charge kqe generated in Ω, ∂Ω • N˙ rec,k [1/s]: flux of ions with charge kqe recombining on the walls, ∂Ω • N˙ ex,k [1/s]: flux of ions with charge kqe extracted through the screen grid.

Energy and Mass Balance of Neutral Particles Since we assume that the neutrals are in thermal equilibrium with the discharge vessel, it can be shown that their energy and mass balances are identical. They can be written as

∂Ω N˙0Ω = 0 = N˙ inlet,0 +

2 X k=1

∂Ω Ω ∂Ω N˙ rec,k − N˙ 0→k+ − N˙ grid,0 ,

(3.4)

where • N0Ω [–]: number of neutrals in Ω, ∂Ω [1/s]: flux of neutrals flowing through the fuel inlet, • N˙ inlet,0 ∂Ω [1/s]: flux on neutrals lost through the grid holes. • N˙ grid,0

Energy Balance of the Discharge Vessel The energy balance of the discharge vessel is identical to an overall balance of the thruster (without its mount), since the thermal capacity of the xenon in the vessel can be neglected. The purpose in calculating this energy balance is to estimate the discharge vessel’s wall temperatures. Since we assume that the neutral atoms are in thermal equilibrium with the discharge vessel, this estimate will also be used to obtain information on the neutral gas temperature T0 , by stating that T0 = Tdv . The equation of interest is:

E˙dv = 0 = PRF −

2 X k=1

∂Ωdv N˙ b,k Ei,k − αdv (Tdv − T∞ ),

(3.5)

56


where • Edv [J]: internal thermal energy of discharge vessel, • Ei,k [J]: k th ionisation energy of xenon, ∂Ωdv • N˙ b,k [1/s]: flux of ions with charge kqe leaving the thruster in the beam,

• Tdv [K]: discharge vessel temperature, • T∞ [K]: mount temperature, • αdv [W/K]: thermal conductivity from the discharge vessel’s walls to the thruster mount.

3.1.4

Fluxes to and from the Systems

Electron Flux to the Discharge Vessel Walls The floating potential Uf of the xenon plasma was derived in section 2.1.2:

kB Te Uf = − ln qe

√

2πme ne

P2 ! 3/2 k n J k+ k=1 − √ . √ emXe qe kB Te

(3.6)

p All electrons with a velocity component c⊥ perpendicular to the wall larger than 2Uf qe /me are lost from the plasma, since these are able to overcome the floating potential. What remains to express is the total flux of these electrons towards the wall. For an electron moving under an angle ϕ to the perpendicular direction, c⊥ is

c⊥ = c cos ϕ,

(3.7)

where c is the electron’s absolute velocity. The flux of electrons under a given direction ϕ can be written as 1 dne (c)c⊥ dAdΩ, d3 N˙ = 4π where • dA [m2 ]: differential area on plasma sheath edge, • dne (c) [1/m3 ]: number density of electrons with velocity c, • dΩ [srad]: differential solid angle of annulus at ϕ.

(3.8)

3.1. DISCHARGE

57

The differential solid angle is dΩ = 2π sin ϕdϕ. Since c =

q

2Ee , me

(3.9)

where Ee denotes the electron’s kinetic energy, we obtain for d3 N˙ : dA d N˙ = 2 3

r

2Ee sin ϕ cos ϕdne (Ee ). me

(3.10)

The number density dne (Ee ) of electrons at energy Ee is expressed by the Maxwellian energy distribution: dne (Ee ) = ne fM (Ee )dEe ,

(3.11)

1 Ee 2 − 32 2 fM (Ee ) = √ (kB Te ) Ee exp − kB Te π

(3.12)

where

is the Maxwellian energy distribution function [31]. With equations (3.10) and (3.11), we can find an expression for the electron flux per unit area by integrating (3.10) in the following manner: dN˙ = ne dA

r

3 2 (kB Te )− 2 × πme Z ∞ Z cos ϕ=√Uf qe /Ee Ee × Ee exp − sin ϕ cos ϕdϕdEe . kB Te Uf qe cos ϕ=1

(3.13)

Solving this integral leads to r dN˙ Uf qe kB Te = ne exp − , dA kB Te 2πme

(3.14)

which corresponds with the electron flux which one would obtain by applying the Boltzmann distribution function to the electrons (cf. equation (2.9)). In order to obtain the power lost to the walls by these electrons, the number flux d3 N˙ is multiplied by the electron’s kinetic energy Ee , which leads to the following expression: dP = ne dA

r

3 2 (kB Te )− 2 × πme Z ∞ Z cos ϕ=√Uf qe /Ee Ee 2 × sin ϕ cos ϕdϕdEe , Ee exp − kB Te Uf qe cos ϕ=1

(3.15)

58


with the solution r dP Uf qe kB Te = ne exp − (Uf qe + 2kB Te ). dA kB Te 2πme

(3.16)

From (3.16) we see that the mean energy carried from the discharge vessel by an electron equals Uf qe + 2kB Te . Only the latter term is brought to the discharge vessel’s walls; the former represents the potential energy which needs to be overcome. However, both term contribute to the heating of the discharge vessel since for each lost electron, one ion reaches the wall with a kinetic energy of Uf qe . Ion Flux to the Discharge Vessel Walls The ions are accelerated towards the discharge vessel walls to their respective Bohm velocity cB,k and reach the latter with the following flux:

where

∂Ω = nk+ se cB,k Adv , N˙ rec,k

(3.17)

• cB,k [m/s]: Bohm velocity for ions with charge kqe , • nk+ se [m−3 ]: ion number densities at the sheath edge. As shown in section 2.1.2, nk+ se = exp − 21 nk+ and we obtain ∂Ω N˙ rec,k

1 = exp − nk+ cB,k Adv . 2

(3.18)

Extraction of Ions Through the Screen Grid Those ions which happen to drift towards a hole of the screen grid are extracted though the grid system. Analogously to above, the flux of ions towards all screen grid holes is 1 ∂Ω ˙ Nex,k = exp − nk+ cB,k Ah,scr Nh , 2

(3.19)

where • Ah,scr [m2 ]: area of a single screen hole, • Nh [–]: number of grid holes. The extracted ion current is then

Iex =

2 X k=1

∂Ω kqe N˙ ex,k .

(3.20)

3.1. DISCHARGE

59

Ionisation by Inelastic Electron-Atom Collisions The xenon ions in the plasma are generated through inelastic collisions of electrons with neutral atoms. The ionisation rate for the species k was derived in section 2.1.1. With equation (2.8) the rate for the entire discharge vessel is found to be

Ω N˙ 0→k+ = ne n0 hσk+ ce iVdv .

(3.21)

The power loss due to ionisation is obtained by multiplying the ionisation rates with their respective threshold energies and summing them up:

Ω P0→+ =

2 X

Ω Ei,k N˙ 0→k+

(3.22)

k=1

Excitation by Inelastic Electron-Atom Collisions The cross sections for excitation of neutrals are similar to the ionisation cross sections, which means that the equations governing collisional excitation are analogous to the ones that describe ionisation. For this reason, the power loss due to excitation is

Ω Ω P0→∗ = E∗ N˙ 0→∗ ,

(3.23)

Ω N˙ 0→∗ = ne n0 hσ∗ ce iVdv .

(3.24)

with

The total excitation cross section σ∗ (as depicted in Figure 3.1) and energy E∗ are summed values over all transitions from one energy level to another [55]. Here, we are only interested in the total power loss over all excitations, so summed values are sufficient. However if one needs to include the ionisation from meta-stable atoms, one is also obliged to express mass conservation for each meta-stable state in order to determine the ionisation rate from these excited atoms. Flux of Neutrals Lost Through the Grid Holes The mean free path Λ0 for neutrals is several hundreds of millimetres, while the characteristic lengths L of the grid system such as hole diameters and separation distances are in the range of millimetres. This means that the Knudsen number Kn is larger than 10, which allows for a free-molecular treatment of the neutral flow through the grid system (cf. section 2.2). The downstream neutral density is very low compared to that inside the discharge vessel and therefore ignored.

60


0.9

0.7

Excitation cross section [10

−16

2

cm ]

0.8

0.6

0.5

0.4

0.3

0.2

0.1

0

1

2

10

10 Electron energy [eV]

3

4

10

10

Figure 3.1: Total excitation cross section σ∗ of xenon as a function of incident electron energy Ee [55]. Hence, the number flux of neutrals through a single set of grid holes is expressed as

N˙ 0 h = n0

r

kB T0 CKn , 2πmXe

(3.25)

where • CKn [m2 ]: open orifice area • T0 [-]: neutral gas temperature With the number of grid holes Nh , the neutral flux through the grids becomes

∂Ω N˙ grid,0 = Nh n 0

r

kB T0 CKn . 2πmXe

(3.26)

3.1. DISCHARGE

61

The geometry factor CKn is found semi-empirically through the Direct Simulation Monte Carlo method (DSMC) [8, 9], see section 2.2 and Appendix B for details. Since we assume thermal equilibrium between the neutrals and the discharge vessel, the neutral gas temperature T0 can be set equal to the discharge vessel temperature Tdv .

3.1.5

Assembly of the Equations and Numerical Solution

This section contains the simplest case for solving the equations which are the base of the discharge model. It is the case where the RF power absorbed by the plasma is given as well as the propellant mass flow. This is also the physically causal solution of the problem, since PRF and m ˙ 0 determine the plasma parameters in the model. Other calculations however are of interest as well: one might want to dictate the thrust provided at given m ˙ 0 or the beam current. Both these calculations represent “backward” solutions in the causal sense and are therefore more complex2 . Nevertheless, the general method is the same. Energy Balance of Electrons Based on equations (2.11), (3.16), (3.22) and (3.23), equation (3.1) can be written as follows:

PRF

r Uf qe kB Te = Aa ne exp − (Uf qe + 2kB Te ) kB Te 2πme ! 2 X − Vdv n0 ne hσk+ ce iEi,k − hσ∗ ce iE∗ .

(3.27)

k=1

Energy and Mass Balance of Ions Equations (3.3), (3.17), (2.10), (3.19) and (3.21) lead to the expression

1 Vdv n0 ne hσk+ ce i = nk+ exp − 2

r

kkB Te Nh Ah,scr + Aa . mXe

(3.28)

Energy and Mass Balance of Neutral Particles With equations (3.4), (3.17), (2.10), (3.21) and (3.26) we obtain 2 X

1 ∂Ω N˙ inlet,0 = Vdv n0 ne hσk+ ce i − Aa nk+ exp − 2 k=1 r kB T0 + CKn Nh n0 , 2πmXe 2

r

kkB Te mXe

! (3.29)

They also include the model for the accelerator grid impingement current derived in section 3.2.

62


with

m ˙0 ∂Ω N˙ inlet,0 = , mXe

(3.30)

where • m ˙ 0 [kg/s]: xenon propellant mass flow into thruster.

Energy Balance of the Discharge Vessel Finally, equations (3.5) and (3.19) yield

PRF

1 = Nh Ah,scr exp − 2

X 2

nk+

r

k=1

kkB Te Ei,k + αdv (Tdv − T∞ ), mXe

(3.31)

∂Ω where we have neglected the difference between the number of extracted ions N˙ ex,k ∂Ωdv and the number of ions N˙ b,k leaving the thruster in the beam. The two values differ only by 1 . . . 2 %, while the actual temperature variation over the discharge vessel introduces much larger differences when being averaged. Secondly, the first term on the right hand side of (3.31) accounts only for about 3 % of the total heat loss of the discharge vessel, ∂Ωdv ∂Ω reducing the error introduced by neglecting the difference between N˙ ex,k and N˙ b,k to less than 0.06 %. Equations (3.27), (3.28), (3.29) and (3.31) are transcendental, which makes a numerical solution necessary.

Numerical Solution In order to obtain a numerical solution for equations (3.31), (3.29), (3.28) and (3.27) they are rewritten, in respective order:

PRF − Nh Ah,scr exp T0 =

− 12

P2

k=1 nk+

q

kkB Te Ei,k mXe

α dv q P2 kkB Te ∂Ω N˙ inlet,0 + Aa exp − 12 n k=1 k+ mXe q n0 = P kB T0 Vdv ne 2k=1 hσk+ ce i + CKn Nh 2πm Xe

+ T∞

(3.32)

(3.33)

3.2. ACCELERATOR GRID IMPINGEMENT CURRENT n1+ exp r1 = 1 −

r2 = 1 −

− 12

q

kB Te mXe

63

Vdv n0 ne hσ1+ ce i q kB Te 1 n2+ exp − 2 mXe

Vdv n0 ne hσ2+ ce i r |U q | kB Te f e −1 r3 = 1 − PRF ne Aa exp − |Uf qe | + 2kB Te kB Te 2πme !! 2 X hσk+ ce iEi,k + hσ∗ ce iE∗ + Vdv n0

(3.34)

(3.35)

(3.36)

k=1

Q(n1+ , n2+ , Te ) = rT r

(3.37)

With r = [r1 , r2 , r3 ]T . The three residuals r1 , r2 and r3 are zero if equations (3.28) and (3.27) are fulfilled correctly. Equation (3.37) defines a cost function Q(n1+ , n2+ , Te ), which is zero if the above equations are satisfied. Starting values for n1+ , n2+ and Te are provided to the numerical solver, which is subsequently required to minimise Q(n1+ , n2+ , Te ) by finding appropriate values for n1+ , n2+ and Te . This is done by a modified version of the MATLAB function fmins. This general optimisation tool uses the Nelder-Mead Simplex method to minimise an arbitrary function of n variables (n = 3 in our case) [17, 50]. The modification lies in the termination requirement of the algorithm. fmins terminates as soon as the iteration steps cause changes in Q or r below a certain threshold. However, in order to solve (3.37), the value of Q must fall below a certain absolute threshold, which is the modified termination requirement implemented.

3.2

Accelerator Grid Impingement Current

The current which is drawn from the accelerator grid is caused by ions impinging onto it, either directly as a result of poor beam focusing or indirectly through a charge-exchange process. Neutrals which flow through the grid can be charged by a passing ion, which in turn is neutralised. Such secondary ions are called charge exchange ions. Since these ions are generated anywhere within the grid system, they rarely follow well-focused trajectories as the primary ions do. Therefore, most of the charge exchange ions impinge on the accelerator grid. The following discourse describes a simple model for estimating the charge exchange process and the resulting accelerator grid current Iacc . It is important to note, that this model is not intended for grid design, since it only calculates the rate at which ions impinge on the accelerator grid. In order to design a grid system, the engineer requires information on the erosion of the grid, which is not calculated. Since the model derived in the context of this thesis is designed for the system level of the thruster, the erosion rate is of secondary interest. Primarily, mass conservation and the currents to and from the grids are of importance here. Figure 3.2 shows the domain (heavy dashed line) in which the charge exchange process is calculated. It is assumed that all secondary ions which originate from this volume hit the accelerator grid (for clarity, the decelerator grid is not shown). The obstruction

64


ts

lg

ln

ta

screen grid accelerator grid dx

plasma A(x)

x

Figure 3.2: Grid geometry and calculation domain through an eventual decelerator grid for charge exchange ions coming from downstream of the grid system is neglected. Only a very small portion of the charge exchange ions originates from this region, at most an order of magnitude smaller than the total flux. The charge exchange rate is calculated in a similar manner as other collision processes, e.g. ionisation or excitation (cf. section 3.1.4). The charge exchange rate dN˙ ce per unit volume dV is dN˙ ce = n0 (x)n1+ (x)σce (c1+ (x))c1+ (x), dV

(3.38)

where • c1+ [m/s]: ion velocity, • σce [m2 ]: charge exchange cross section, • dV = A(x)dx [m3 ]: differential volume element in the grid system (cf. Figure 3.2). For the ion velocity range of interest the charge exchange cross section σce can be expressed by the empirical formula [59] 1/2 σce = Ace − Bce ln c1+ ,

(3.39)

3.2. ACCELERATOR GRID IMPINGEMENT CURRENT

65

where • Ace ≈ 1.49 · 10−9 [m], • Bce ≈ 8.703 · 10−11 [m]: material parameters [72]. Slightly different values for Ace and Bce were given by Bond [11], but delivered less accurate results. The ion velocity c1+ is estimated by introducing a piecewise linear approximation for the electric potential in the grid system: ( ∀x ∈ [0, l1 [ Up + (Uscr − Uacc ) lx1 , U (x) = , l3 x Up + Uscr + ln Uacc − Uacc ln , ∀x ∈ [l1 , l3 ]

(3.40)

where • l1 = ts + lg + ta /2 [m], cf. Figure 3.2 • l3 = ts + lg + ta + ln [m], cf. Figure 3.2 • Up = UB + Uf ts ta td ds da dd lg ld dh 0.5 2.0 0.5 3.2 2.0 3.2 1.1 1.0 3.75 Table 3.1: Grid geometry of the RIT10 Artemis thruster (all values in mm). ln stands for the distance between the downstream side of the accelerator grid and the neutralisation plane. Ions which are generated downstream of this plane—i.e. at x > l3 — are not attracted by the grid system anymore, they follow the beam direction. If they are generated at x < l3 , they impinge on the accelerator grid. For a three grid system, the neutralisation plane is taken to be the downstream side of the decelerator grid. In the case of a two grid system, the value of ln is obtained through a one-dimensional treatment of Poisson’s equation, which generally follows the model developed by Kaufman [30]. See section 3.2.1 for a brief deduction of this model. The linear variation of U with x actually represents a solution to Laplace’s Equation for the potential between infinitely thin, plane electrodes positioned at 0, l1 and l3 , which is certainly a rather rough approximation of the real conditions. However, in this model—which only calculates the charge exchange rate—the potential is solely used to estimate the velocity of the ions in order to determine the charge exchange collision cross section following equation (3.39). The ion velocity depends on the root of U (cf. eq. (3.41)), so errors in estimating the potential are reduced. Furthermore, the charge exchange cross section depends on the logarithm of the ion velocity, which enhances this effect significantly. It was found that the accelerator grid impingement current is estimated within an accuracy that is limited by the precision of the available cross section data. Therefore, little benefit was seen in introducing more accurate assumptions at the cost of drastically increasing the complexity of the model.

66


With U (x), the ion velocity c1+ (x) can be expressed as

c1+ (x) =

s

2U (x)qe . mXe

(3.41)

To estimate the ion density n1+ (x), we assume that n1+ (x) is constant over the entire area A(x). Then, we can write

n1+ (x) =

Ah,scr cB,1 n1+ (0). A(x)c1+ (x)

(3.42)

The neutral density n0 (x) is also estimated by a piecewise linear approximation: ( 4A −N A n0 (0) 1 − scr4Ascrhl2 h,scr x + n∞ , ∀x ∈ [0, l2 [ , n0 (x) = N Ah,scr n0 (0) h4Ascr + n∞ , ∀x ∈ [l2 , l3 ]

(3.43)

where • Ascr [m2 ]: screen grid total area, • l2 = ts + lg + ta [m], cf. Figure 3.2 • n∞ [1/m3 ]: ambient neutral particle density. n0 (0) is obtained from the discharge model. This linear model assumes approximately unidirectional flow of the neutrals after they have passed through the grid system, which causes the factor of 4 found in the denominator of the fractions in the equations above. The cross sectional area for this flow is taken to be the entire screen grid area Ascr , which is the same as the area of the accelerator grid. The loss of neutrals due to charge exchange is neglected, since less than a tenth of the atoms undergo charge exchange collisions. This is also done in order to prevent the charge exchange rate from appearing in the equation. With equations (3.39), (3.40), (3.41), (3.42) and (3.43) and integrating over x from 0 to l3 , we can write equation (3.38) as N˙ ce = Nh Ah,scr cB,1 n1+ (0)n0 (0)(I1 + I2 + I3 ),

(3.44)

where l1

2 n∞ Bce I1 = 1+ − cx Ace − ln d2 (e1 + f1 x) dx n0 (0) 2 0 Z l2 2 n∞ Bce I2 = 1+ − cx Ace − ln d2 (e2 + f2 x) dx n0 (0) 2 l1 Z l3 2 Nh Ah,scr n∞ Bce I3 = + Ace − ln d2 (e2 + f2 x) dx, 4Ascr n0 (0) 2 l2 Z

(3.45) (3.46) (3.47)


67

with 4Ascr − Nh Ah,scr 4Ascr l2 r 2qe d= mXe e1 = Up l1 + ln e2 = Up + Uscr + Uacc ln Uscr − Uacc f1 = l1 Uacc . f2 = − ln c=

(3.48) (3.49) (3.50) (3.51) (3.52) (3.53)

Please note, that the beam cross section A(x) does not appear in these equations. Therefore, the model is mathematically equivalent to the assumption of one-dimensional ion flow, where the grids are replaced through virtual electrodes at positions 0, l1 and l3 . Non withstanding their complexity, all the integrals have analytical solutions. They are given in the following list of equations:

n∞ I1 = 1 + I11 − cI12 n0 (0) n∞ I2 = 1 + I21 − cI22 n0 (0) Nh Ah,scr n∞ I3 = + I31 4Ascr n0 (0) 2 Bce Bce 2 I11 = l1 Ace − ln d z12 + Bce f1 I111 − I112 2 2 Ace − Bce ln d z12 I111 = f1 l1 − e1 ln f2 z11 1 e1 2 2 1 2 I112 = 2 z12 (ln z12 − 1) − z11 (ln z11 − 1) − ln z12 − ln z11 f1 2 2 Bce f1 Bce l12 Bce 2 I12 = Ace − ln d z12 + I121 − I122 2 2 2 2 Ace − Bce ln d (f1 l1 )2 z12 2 I121 = − e1 f1 l1 + e1 ln f13 2 z11 2 1 z 1 I122 = 3 12 ln z12 − − 2e1 z12 (ln z12 − 1) f1 2 2 e21 2 7 2 + ln z12 − ln z11 + 3 ln e1 − 2 2

(3.54) (3.55) (3.56) (3.57) (3.58) (3.59) (3.60) (3.61)

(3.62)

68

CHAPTER 3. RF ION THRUSTER MODEL 2 2 Bce Bce 2 2 I21 = l2 Ace − ln d z22 − l1 Ace − ln d z21 2 2 Bce + Bce f2 I211 − I212 2 Ace − Bce ln d z22 I211 = f2 (l2 − l1 ) − e2 ln f22 z21 1 e2 2 2 I212 = 2 z22 (ln z22 − 1) − z21 (ln z21 − 1) − ln z22 − ln z21 f2 2 2 l12 2 l2 Bce Bce I22 = 2 Ace − ln d2 z22 − Ace − ln d2 z21 2 2 2 2 Bce f2 Bce + I221 − I222 2 2 2 2 Ace − Bce ln d z22 − z21 z22 2 I221 = − 2e2 f2 (l2 − l1 ) + e2 ln f23 2 z21 2 2 1 z 1 1 z − 21 ln z21 − I222 = 3 22 ln z22 − f2 2 2 2 2 e22 2 2 ln z22 − ln z22 − 2e2 z22 (ln z22 − 1) − z21 (ln z21 − 1) + 2 2 2 Bce Bce I31 = l3 Ace − ln d2 z32 − l2 Ace − ln d2 z31 2 2 Bce + Bce f3 I311 − I312 2 Ace − Bce ln d z32 I311 = f3 (l3 − l2 ) − e3 ln f32 z31 1 e3 2 2 I312 = 2 z32 (ln z32 − 1) − z31 (ln z31 − 1) − ln z32 − ln z31 f3 2 z11 = e1 z12 = e1 + f1 l1 z21 = e2 + f2 l1 z22 = e2 + f2 l2 z31 = e2 + f2 l2 z32 = e2 + f2 l3

(3.63) (3.64) (3.65)

(3.66) (3.67)

(3.68)

(3.69) (3.70) (3.71) (3.72) (3.73) (3.74) (3.75) (3.76) (3.77)

The term Nh Ah,scr cB,1 n1+ (0) in equation (3.44) corresponds to the extracted flux of singly charged ions. To simplify (3.44), the extracted ion current Iex —over both singly and doubly charged ions—is used instead of Nh Ah,scr cB,1 n1+ (0):

Iex N˙ ce = n0 (0)(I1 + I2 + I3 ). qe

(3.78)


3.2.1

69

Estimating the Neutralisation Plane

The following discourse is based on a one-dimensional model derived by Kaufman [30]. It uses Poisson’s equation to estimate the position of the neutralisation plane. In the one-dimensional case, Poisson’s equation can be written as ∂ 2 Φ(x) ρ =− , 2 ∂x

(3.79)

where Φ(x) denotes the electric potential at x and ρ stands for the space charge density. The electric permittivity can be assumed equal to that of vacuum, 0 . If one neglects the influence of doubly charged ions, the space charge density becomes ρ = qe (n1+ − ne ).

(3.80)

The electron density is dominated through the Boltzmann distribution function,

ne = nN e exp

! qe (Φ(x) − ΦN ) , kB Te

(3.81)

where the superscript N stands for the neutralisation plane—this is the position xN along x, where ne = n1+ . Since qe (Φ(x) − ΦN ) exceeds kB Te already at short distances from the neutralisation plane, we shall neglect the electron density entirely. The ion density is estimated through conservation of electric charge. In the one-dimensional case treated here, the current density J must be constant over all x, so we can write J = const. = n1+ qe c1+ or J n1+ = . qe c1+

(3.82)

(3.83)

As above, c1+ is calculated using conservation of energy,

c1+ (x) =

s

2 Up + Uscr − Φ(x) qe . mXe

(3.84)

Combining equations (3.80), (3.83) and (3.84), we obtain J ∂ 2 Φ(x) =− 2 ∂x 0 for Poisson’s equation.

r

mXe 2 Up + Uscr − Φ(x) qe

(3.85)

70


The general solution to this is found to be p r q 2 2C2 + C1 J Up + Uscr − Φ(x) − C − C J Up + Uscr − Φ(x) = x + C3 , 2 1 3(C1 J)2

(3.86)

with • C1 = − 10

q

mXe 2qe

• C2 [V2 /m2 ]: first integration constant, • C3 [m]: second integration constant. This is an almost identical deduction as for Child’s Law [16], which treats the spacecharge limited current flow between two infinitely extended parallel plates. To obtain information about ln , we introduce the boundary conditions at the last grid on one side and at the neutralisation plane on the other. The last grid is the accelerator grid in a two-grid system, respectively the decelerator in a three-grid system. We shall call the respective potential encountered at the last grid Ug . The space coordinate is laid such that it is zero at the downstream side of the grid. Therefore, we can write q p p 2 2C2 + C1 J Up + Uscr − Ug − C − C J Up + Uscr − Ug = C3 2 1 3(C1 J)2

(3.87)

for x = 0, i.e. a Dirichlet condition expressing that the potential at the grid is Ug . Downstream of the neutralisation plane, the electric field is assumed to be constantly zero. Due to the continuous nature of the electric field in the ion beam [30, p. 20], this must also be the case just upstream of xN . Therefore, we have a homogeneous Neumann condition at the neutralisation plane: ∂Φ(x) − ∂x

=

r

C2 − C1 J

q

Up + Uscr − Φ(xN ) = 0.

(3.88)

xN

Equations (3.87) and (3.88) are solved for C2 and C3 . With the integration constants introduced into (3.86), we obtain xN as p 2 p xN = 2 Up + Uscr − Ub + Up + Uscr − Ug × 3 sp p Up + Uscr − Ub − Up + Uscr − Ug . C1 J

(3.89)

The beam potential Ub needs to be estimated. Generally, it lies around 20 V. However, it still pays off to estimate Ub instead of xN directly, since Ub only appears under the root

3.3. THRUST

71

and therefore the influence of an error in estimation is much smaller. The distance ln from the accelerator grid to the neutralisation plane is then obtained as follows: ( xN ln = ld + td + xN

for a two-grid system for a three-grid system,

(3.90)

where • ld [m]: distance between accelerator and decelerator grids • td [m]: decelerator grid thickness. It is interesting to investigate the influence of a decelerator grid on ln : the larger ln is, the more charge exchange ions are attracted to the accelerator grid—therefore, a small value of ln is more attractive. A decelerator grid is usually connected to spacecraft ground, i.e. Ug = 0. For the mean current density J of 30 A/m2 which appears in the grid system of the RIT10 Artemis thruster, we obtain a value of 0.68 mm for xN . Since we are calculating ln for a three-grid system, we need to add ld and td to xN to finally obtain ln = 2.18 mm. If we omit the third grid on RIT10 Artemis , we have Ug = −470 V and obtain xN = ln = 3.4 mm, significantly larger than for the three-grid case. This is one of the reasons for using a decelerator grid.

3.3

Thrust

This section deals with the conservation of momentum for the entire thruster. First, the thrust is calculated while neglecting charge exchange ions and the divergence of the ion beam. Second, an estimation of the thrust loss due to charge exchange is given and the beam divergence is included by using an efficiency factor.

3.3.1

Charge Exchange Ions and Beam Divergence Neglected

The thrust F acted upon the ion engine can be written through Newton’s law for the conservation of momentum:

F =

2 X

p˙k+ ,

(3.91)

k=1

where ∂Ω , p˙k+ = ck+ mXe N˙ ex,k

(3.92)

72


with • ck+ [m/s]: velocity of ions of species k in the beam, • p˙k+ [N]: momentum flux caused by the ions with charge kqe . The ion velocity ck+ can be expressed through conservation of energy within the grid system:

ck+ =

s

2(Up + Uscr − Ub )kqe . mXe

(3.93)

With equations (3.19) and (3.93), we can write

p˙k+ = With cB,k =

√

q

1 2mXe (Up + Uscr − Ub )qe kNh Ah,scr exp − cB,k nk+ . 2

(3.94)

kcB,1 , the thrust F becomes

2 q X 1 F = Nh Ah,scr exp − cB,1 2mXe (Up + Uscr − Ub )qe knk+ 2 k=1 q 1 = Nh Ah,scr exp − cB,1 2mXe (Up + Uscr − Ub )qe ne 2

3.3.2

(3.95)

Momentum Loss Due to Charge Exchange and Beam Divergence

The loss in momentum flux p˙l due to the charge exchange process is estimated in the following manner: p˙l ≈ −

Iacc hpi qe

(3.96)

where • Iacc [A]: Accelerator grid current • hpi [N]: Mean momentum of an expelled ion. hpi is defined as P2 p˙k+ hpi = P2k=1 ∂Ω ˙ k=1 Nex,k P2 q knk+ = 2mXe (Up + Uscr − Ub )qe P2 k=1 1/2 n k+ k=1 k

(3.97)

3.3. THRUST

73

If we introduce equation (3.97) into (3.91) as

F =

2 X

p˙k+ − p˙l ,

(3.98)

k=1

we can write q F = ne 2mXe (Up + Uscr − Ub )qe × 1 Iacc Nh Ah,scr exp − cB,1 − P2 . 2 qe k=1 k 1/2 nk+

(3.99)

This expression holds for a perfectly collimated ion beam. The actual thrust will be smaller, due to beam divergence, which needs to be measured. Usually, the ion beam divergence is defined as the opening angle of a cone originating in the thruster which holds 95 % of the ion current. For RF ion thrusters, it is generally found to lie around 12◦ for a flat grid. If the grid is dished3 , the divergence is increased. For a given beam divergence angle αd , we can introduce an efficiency factor ηb < 1 which accounts for the momentum loss due to beam divergence [19]: 1 ηb = (1 + cos αd ). 2

(3.100)

For a dished grid, the divergence angle is replaced as follows:

αd0 = αd + arctan

! Rg2 − h2d , 2hd

(3.101)

where • hd [m]: height of the dish • Rg [m]: grid radius. So finally, we can express the thrust as q F = ηb ne 2mXe (Up + Uscr − Ub )qe × 1 Iacc Nh Ah,scr exp − cB,1 − P2 . 2 qe k=1 k 1/2 nk+

(3.102)

The second factor in this equation is approximately the beam current Ib of the thruster. Therefore, the thrust is proportional to the square root of the voltage between the plasma and the ion beam (first term in (3.102)) and the beam current. 3

The entire grid system is often dished for greater mechanical stiffness.

74


3.4

RF Field Coupling and Coil Impedance

A key design parameter for the RF generator (RFG) is the coil current which is necessary to sustain the gas discharge for given thrust. From the discharge and grid models, we obtain an estimation of the necessary RF power which needs to be fed into the plasma. Since we assume harmonically time-varying fields, the power density dissipated in the plasma can be written as dP 1 =

Numerical Modelling and Optimisation of Radio

Numerical Modelling and Optimisation of Radio

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