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The Pennsylvania State University The Graduate School

TESTING AND OPTIMIZATION OF A MINIATURE MICROWAVE ION

THRUSTER

A Thesis in Aerospace Engineering by Pierre-Yves Taunay

c 2012 Pierre-Yves Taunay

Submitted in Partial Fulﬁllment of the Requirements for the Degree of

Master of Science

May 2012 The thesis of Pierre-Yves Taunay was reviewed and approved∗ by the following:

Michael M. Micci Professor of Aerospace Engineering and Director of Graduate Studies Thesis Advisor

Sven G. Bil´en Associate Professor of Engineering Design, Electrical Engineering, and Aerospace Engineering Thesis Co-Advisor

George A. Lesieutre Department Head of Aerospace Engineering

∗Signatures are on ﬁle in the Graduate School. Abstract

Ion thrusters are able to provide low thrust and high speciﬁc impulse, making them suitable for station keeping missions and interplanetary travel. One important feature of this type of space propulsion is the emission of charged particles away from the spacecraft, requiring the use of a neutralizer emitting electrons in order to ensure thrust.

This thesis presents the design and testing of a microwave miniature of ion thruster using an electron cyclotron resonance discharge. The propellant (argon or xenon) flows inside a discharge chamber at an operational mass flow rate of 0.15 sccm and is then ionized by the coupling between an oscillating electromagnetic field of frequency 5 GHz fed by a ring-type antenna and a permanent magnetic field created by two concentric magnets. Only 4 W of total absorbed power is required to obtain ionization of the plasma, and 0.4 W to sustain it. The plasma created then is accelerated through a pair of grids that have a given potential across them. Our device can operate either in an ion thruster mode, accelerating the argon or xenon atoms, or in an electron emission mode, accelerating only electrons. In the ion thruster mode the thruster is predicted to produce a thrust of 217 µN with a mass utilization efficiency of 46% and a total efficiency of 75% if the propellant used is argon.

A numerical analysis of both the permanent magnetic fields and electromagnetic radiation inside the discharge chamber was also conducted. It was found that the thickness of the yoke plate to which the magnets are connected does not have any effect on the magnetic field inside the chamber and that the the antenna had to be positioned 2 mm away from the magnets in order to provide the best permanent magnetic field–electromagnetic radiation coupling. The electromagnetic radiation simulation also allowed us to validate the design of the microwave antenna.

iii Table of Contents

List of Figures vii

List of Tables ix

List of Symbols x

Acknowledgments xv

Chapter 1 Introduction 1 1.1 Miniature Microwave Ion Thruster Overview ...... 1 1.2 Previous Research at Penn State ...... 2 1.3 Current Research ...... 4 1.4 Thesis Overview ...... 5

Chapter 2 Theoretical Background 7 2.1 Principles of Rocket Propulsion ...... 7 2.1.1 Vehicle Performance ...... 7 2.1.2 Categorization of Rocket Propulsion ...... 9 2.2 Ion Thruster Physics ...... 13 2.2.1 One-dimensional ﬂow ...... 14 2.2.2 Exhaust neutralization ...... 16 2.3 Ionization of the Gas via ECR Heating ...... 16 2.3.1 Motion of a single charged particle in an electromagnetic ﬁeld . . . . 16 2.3.2 Microwave breakdown of a neutral gas ...... 17 2.3.2.1 General ionization process ...... 17 2.3.2.2 Collisions in a gas ...... 18 2.3.2.3 Losses of electrons in a gas ...... 22 2.3.2.4 Electron cyclotron resonance heating ...... 26 2.4 Transmission Line Theory ...... 26 2.4.1 Propagating wave on a transmission line ...... 26 2.4.2 General load on a transmission line ...... 27

iv Chapter 3 Ion Thruster Design and Theoretical Predictions 29 3.1 Objectives ...... 29 3.2 Overview ...... 30 3.3 Back Plate ...... 30 3.4 Yoke Plate ...... 32 3.4.1 Magnets ...... 32 3.4.2 Inputs ...... 32 3.5 Discharge Chamber ...... 33 3.6 Front Plate and Grids ...... 35 3.6.1 Front plate ...... 35 3.6.2 Discharge characterization grid ...... 35 3.6.3 Extraction grids ...... 35 3.6.3.1 Design methodology ...... 35 3.6.3.2 Ion and electron grid set ...... 36 3.7 Antenna Design Considerations ...... 40 3.8 Theoretical Predictions ...... 40

Chapter 4 Experimental Setup 41 4.1 Facilities and Equipment ...... 41 4.2 Measurement Methods ...... 44 4.3 Microwave Analysis ...... 45

Chapter 5 Results 46 5.1 Experimental Results ...... 46 5.1.1 Original design ...... 46 5.1.2 New design ...... 47 5.1.2.1 Microwave analysis ...... 48 5.1.2.2 Plasma creation ...... 48 5.1.2.3 Ion and electron extraction ...... 50 5.2 Numerical Simulation Results ...... 51 5.2.1 Magnetic ﬁelds ...... 52 5.2.1.1 Equations ...... 52 5.2.1.2 Geometry ...... 52 5.2.1.3 Materials ...... 53 5.2.1.4 Boundary conditions ...... 53 5.2.1.5 Mesh ...... 54 5.2.1.6 Solver ...... 54 5.2.1.7 Results ...... 54 5.2.2 Two dimensional antenna simulation ...... 58 5.2.2.1 Equations ...... 58

v 5.2.2.2 Geometry ...... 59 5.2.2.3 Materials ...... 60 5.2.2.4 Boundary conditions ...... 60 5.2.2.5 Mesh ...... 61 5.2.2.6 Solver ...... 61 5.2.2.7 Results ...... 61

Chapter 6 Conclusion and Recommendations 64

Bibliography 66

vi List of Figures

1.1 MRIT thruster compared to a U.S. quarter ...... 2 1.2 MRIT thruster ﬁring in a vacuum chamber ...... 3 1.3 MMIT thruster compared to a U.S. quarter ...... 4 1.4 MMIT thruster ﬁring in a vacuum chamber ...... 4

2.1 Forces acting on a control volume surrounding a rocket ...... 8 2.2 Schematic of the ideal rocket ...... 10 2.3 Schematic of an ion thruster ...... 14 2.4 Modification of the electric potential due to an accumulation of electric charges between two grids ...... 15 2.5 Types of collisions that can occur in a gas ...... 19 2.6 Collision probability Pc and collision cross-section Q as a function of electron energy (in eV) for various gases ...... 20 2.7 Collision frequency νc as a function of pressure and electron energy (in eV) for various gases ...... 20 2.8 Excitation efficiency hx as a function of electron energy (in eV) for various gases ...... 21 2.9 Ionization efficiency hi as a function of electron energy (in eV) for various gases ...... 21 2.10 A load of impedance ZL connected to a transmission line of impedance Z0 . 27

3.1 CAD representation of the new version of the MMIT ...... 30 3.2 CAD representation of the new version of the MMIT — Section cut . . . . 31 3.3 CAD representation the back plate — Section cut ...... 31 3.4 CAD representation of the front plate ...... 35 3.5 Front view of the accelerator grid ...... 37 3.6 Potential diﬀerence versus the grid distance for an argon atom ...... 38 3.7 Potential diﬀerence versus the grid distance for an electron ...... 38 3.8 Exhaust velocity versus the grid distance for an argon atom ...... 39 3.9 Exhaust velocity versus the grid distance for an electron ...... 39

4.1 Vacuum chamber ...... 42 4.2 Pressure transducer, pressure reader and temperature reader ...... 42

vii 4.3 Schematic of the microwave line ...... 43 4.4 Voltage sources used for ions and electrons extraction ...... 44 4.5 Faraday cup used for beam dispersion characterization and ion density measurement ...... 44

5.1 Disc antenna (left) and ring antenna (right) used in the previous and current versions of the thruster respectively. The graduations are in millimeters . . 47 5.2 Damage due to arcs on the screen grid. The arrows indicate scars due to arcing ...... 47 5.3 Reflected power for the thruster ...... 48 5.4 Argon plasma created by the new design of the MMIT ...... 49 5.5 Minimum power required to create the plasma ...... 49 5.6 Minimum power required to sustain the plasma ...... 50 5.7 Argon plasma created by the new design of the MMIT with the extraction grids on ...... 51 5.8 Damage due to the plasma on the discharge chamber (left) and on the screen grid (right). We can clearly see a metallic coating on the inside of the discharge chamber and the sputtering on the screen grid ...... 51 5.9 Geometry chosen for the magnetic field simulation ...... 53 5.10 Quality of the mesh for the magnetic field simulation ...... 55 5.11 Magnetic flux density (surface plot) and magnetic flux streamlines for all the yoke plate thicknesses. From left to right: 0.5, 0.75, 1 mm on the first line; 1.25, 1.5, 1.75 mm on the second line; and 2.0 2.25 2.5 mm on the third line 56 5.12 Contour plot of the electron cyclotron frequency (in GHz) ...... 57 5.13 Geometry of the electric field simulation ...... 59 5.14 Statistics of the meshed domain for the antenna simulation ...... 61 5.15 Superimposed results from the electromagnetic wave and permanent magnetic field simulation. The black lines correspond to the electric field and the red lines to the magnetic flux density...... 63

viii List of Tables

1.1 MRIT Operational Parameters ...... 3 1.2 MMIT Operational Parameters and Original Predicted Results ...... 5

2.1 Comparison of the thrust and speciﬁc impulse generated by various types of rocket propulsion ...... 13 2.2 Characteristic diﬀusion length for argon atoms and electrons in a cylindrical cavity of height 1.5 cm, radius 1 cm, pressure 5×10−3 Torr, electron energies of 15 eV, and electron resonance frequency of 5 GHz ...... 24

3.1 Dimensions of the magnets for the two diﬀerent versions of the thruster, in inches ...... 32 3.2 Properties of neodymium, samarium–cobalt and alnico magnets ...... 33 3.3 Compressible gas parameters for xenon and argon ...... 34 3.4 Main dimensions of the screen and accelerator grids and of the insulating spacer ...... 37 3.5 Nominal parameters for the new version of the MMIT ...... 40 3.6 Predicted eﬃciencies for the thruster mode for the given nominal parameters 40

5.1 Relative magnetic permeability for the magnetic ﬁeld simulation ...... 54 5.2 Material properties for the two–dimensional antenna simulation ...... 60

ix List of Symbols

A Generic area, m2

2 Ae Exhaust area of a nozzle, m a Radius of a cylindrical tube, m

B~ Magnetic ﬂux density, T

B0 Average ﬂux density of a magnetic bottle, T

Bmax Maximum ﬂux density of a magnetic bottle, T

B~ r Residual magnetic ﬂux density, T

−1 −1 Cp Heat capacity at constant pressure of a propellant, J · kg · K D Diﬀusion coeﬃcient, m2 · s−1

EECR Energy gained by an electron in one cyclotron gyration, J

Ek Kinetic energy of a single particle, J

Ep Potential energy of a single particle, J

E~ Electric ﬁeld, V · m−1

e Elementary charge, 1.6 × 10−19 C

g Earth gravitational acceleration at sea level, 9.81 m · s−2

H~ Magnetic ﬁeld, A · m−1

hi Ionization eﬃciency

hx Excitation eﬃciency

Ib Beam current, A

x Isp Speciﬁc impulse, s

J~ Current density, A · m−2

−2 Jb Beam current density, A · m

Jo Momentum out of the control volume, N j Beam current density, A · m−2

j Imaginary number such that j2 = −1

−1 k0 Wave number, m

−23 −1 kB Boltzmann constant, 1.38 × 10 J · K L Distance between two grids, m

l Length of a tube, m

M Atomic mass of the propellant, kg

Mw Propellant molecular weight, amu mf Final mass of the rocket, kg

mi Initial mass of the rocket, kg m˙ Mass ﬂow rate, kg · s−1

−1 m˙ i Ion mass ﬂow rate, kg · s

−1 m˙ p Non-ionized propellant mass ﬂow rate, kg · s

−1 Nm Flow rate, mol · s n Generic particle number density, m−3

−3 ne Electron number density, m

P1 Pressure at the inlet of a tube, Pa

P2 Pressure at the outlet of a tube, Pa

Pa Ambient pressure surrounding the rocket, Pa

Pav Average power delivered to a load, W

Pb Beam power of an electric thruster, W

Pc Combustion chamber pressure, K

xi Pc Probability of a collision between an electron and a neutral atom

Pe Exhaust pressure, Pa

Pi Probability of ionization of an atom through inelastic collision

Pin Input power for an electric thruster, W

Pjet Kinetic energy of the exhaust, J

Po Other power sources in an electric thruster, W

Px Probability of excitation of an atom through inelastic collision

PT Total input power of an electric thruster, W Q Mass ﬂow rate, sccm

q Charge of a single particle, C

R¯ Speciﬁc gas constant, J · kg−1 · K−1

R Universal gas constant, J · K−1 · mol−1

rL Larmor radius, m SWR Standing wave ratio

T Thrust, N

Tc Combustion chamber temperature, K

Tc Critical temperature of a gas, K

Te Exhaust temperature, K

Te Electron temperature, K

−1 ue Exhaust velocity, m · s

−1 ueq Equivalent exhaust velocity, m · s V Electric potential, V

Va Electric potential of the grid accelerator, V

Vb Beam voltage, V

Vm Magnetic potential, A

−1 ve Electron velocity, m · s

xii −1 vn Neutral atom velocity, m · s v¯ Average of a Maxwellian distribution of velocities over a set of particles, m · s−1

Z0 Impedance, Ω ~Γ Flux of particles, m−2 · s−1

Γ Voltage reﬂection coeﬃcient

γ Ratio of the speciﬁc heats

Permittivity of a generic medium, F · m−1

−12 −1 0 Permittivity of vacuum, 8.85 × 10 F · m

r Relative permittivity ζ Viscosity of a ﬂuid, Poise

ηE Electrical eﬃciency of an electric thruster

ηM Mass utilization eﬃciency of an electric thruster

ηT Total eﬃciency of an electric thruster

ΘI Characteristic temperature of ionization, K Λ Characteristic diﬀusion length, m

λ Mean free path, m

λD Debye length, m µ Mobility, m2 · V−1 · s−1

µ Magnetic moment, m2 · A

µ Magnetic permeability of a generic medium, H · m−1

−7 −1 µ0 Magnetic permeability of vacuum, 4π × 10 H · m

µr Relative permeability

−1 νc Collision frequency, s ρ Charge density, C · m−3

σ Cross section, m2

σ Electrical conductivity, S · m−1

xiii Φ Degree of ionization of a gas

ω Frequency of an electromagnetic wave, rad · s−1

−1 ωc Cyclotron frequency of a charged particle, rad · s

xiv Acknowledgments

I would like to take this opportunity to thank all the people who helped me throughout my graduate studies. First of all, I sincerely thank Dr. Bilénand Dr. Micci for giving me the opportunity to work on this project, and also for their help, guidance and confidence they had in me. Thank you also to Dr. Vladimir Getman for his help on vacuum technology, to Mr. Robert Capuro for his help in fixing equipment failures, and to Ty Druce for his help in the lab and for monitoring the equipment while I was away. I would also like to thank Tom Trudel, the developer of the MRIT for his help and suggestions about testing the MMIT, and also the students from the Penn State Plasma group, especially Erica Capalungan, Jan Herzog, Jeff Hopkins, Russell Moore, Brian Taylor and anyone I may have missed. Finally I would like to thank my family and friends back home for their support and encouragement for continuing my studies abroad.

xv Chapter 1

Introduction

1.1 Miniature Microwave Ion Thruster Overview

Electric propulsion consists in accelerating ions from a plasma by using an electrostatic field. This type of propulsion allows larger exhaust velocities and, therefore, higher specific impulses than other current types of propulsion. This translates into higher propellant efficiency and also longer on time, which makes electric propulsion suitable for interplanetary missions and also for station keeping of satellites in orbit over a long period. Electric propulsion is an active field of research and in recent years multiple space missions succes- fully made use of ion engines: the European radio-frequency ion thruster RIT flew onboard the EURECA platform in 1992, and has been used onboard the European Space Agency’s large geostationary technology satellite ARTEMIS since 2001 [1]; the NASA NSTAR ion engine, which operated for more than 30,000 hours on the probe Deep Space 1 [2]; and the µ10 ion engine, which operated for 31,400 hours over the course of seven years, on the probe Hayabusa from the JAXA [3].

The miniaturization of the ion engines is investigated as a way to decrease the power required to operate the spacecraft. Indeed, the miniaturization oﬀers advantages:

• less mass is used, allowing more mass to be allotted for the payload, and

• less input power is required for the ionization and acceleration of the propellant. Again this translates into less mass for power converters.

The miniature ion thruster is mainly suited for station keeping of small satellites given the range of thrust provided — of the order of 1 µN to 1 mN — though some scientiﬁc missions 2 plan to use miniature ion engines as means to go to the Moon [4].

Numerous studies have shown the feasibility of the miniaturization: Takao et al. succes- fully improved an ion engine to produce an ion beam current of 10 mA, with an input power of 8 W, a mass flow rate of 0.2 sccm, and a propellant utilization efficiency of 72 % [5][6]; and Koizumi and Kuninaka obtained an ion beam current of 3.3 mA for a mass flow rate of 0.15 sccm and an input power of 1.0 W [7][8], but the propellant utilization efficiency was of 37 %. Additionally, Koizumi and Kuninaka also designed a way to neutralize the ion beam from a thruster by emitting electrons from an other thruster [8]. Both the design of Takao et al. and Koizumi and Kuninaka make use of the coupling between a permanent magnetic field created by samarium–cobalt magnets, and microwave radiation in order to ionize the propellant. The ionization occurs through an electron cyclotron resonance heating.

1.2 Previous Research at Penn State

The MMIT follows on the successful design of the Miniature Radio-frequency Ion Thruster (MRIT), designed and tested by Trudel [9] and studied by Mistoco [10]: Figure 1.1 shows the MRIT compared to a U.S. quarter, Figure 1.2 shows the MRIT ﬁring in a vacuum chamber, and the operational values of the MRIT are summed up in Table 1.1.

Figure 1.1. MRIT thruster compared to a U.S. quarter 3

Figure 1.2. MRIT thruster ﬁring in a vacuum chamber

Table 1.1. MRIT Operational Parameters

Parameter Nominal value Input frequency 1.5 MHz Input RF power 16 W Screen grid voltage 1000 V Acceleration grid voltage –200 V Grid separation distance 0.14 in Active grid diameter 1.0 cm Propellant Ar (purity 99.999%) Propellant flow rate 0.038 sccm Specific impulse 2462 s Thrust 64.8 µN Total efficiency 20% Mass utilization efficiency 41.1% Electric efficiency 22.5% 4

The ﬁrst iteration of the MMIT (Figure 1.3) was designed and built by Lubey [11][12]. Its input parameters and predicted results are presented in Table 1.2. It was tested and was able to produce a plasma (Figure 1.4) but not able to accelerate the ions nor the electrons.

Figure 1.3. MMIT thruster compared to a U.S. quarter

Figure 1.4. MMIT thruster ﬁring in a vacuum chamber

1.3 Current Research

Current research at The Pennsylvania State University on the MMIT focuses on decreasing the power used for the ionization of the propellant, and the propellant mass ﬂow rate. Optimization of these parameters is required in order to minimize the power used to ionize and accelerate the propellant. Another goal of the current research is to make the thruster self-neutralizing: as an ion thruster exhausts ions, it becomes more and more negatively charged. At some point, if no neutralization of the exhaust beam is provided, the exhausted ions are attracted back to the thruster. This is called “beam stalling”. Diﬀerent solutions exist to provide neutralization 5

Table 1.2. MMIT Operational Parameters and Original Predicted Results

Parameter Nominal value Input frequency 4.2 GHz Input microwave power 1 W Screen grid voltage 1500 V Acceleration grid voltage –500 V Grid separation distance 0.02 in Active grid diameter 1.0 cm Propellant Xe Propellant flow rate 0.15 sccm Specific impulse 5540 s Thrust 258 µN Total efficiency 32.2% Mass utilization efficiency 32.1% Electric efficiency 50.0%

of the exhaust beam. The MMIT will eventually be designed to be able to accelerate both ions and electrons separately so they can recombine in the exhaust plume, thereby avoiding having to use a neutralizer. The mass saved by a design without a neutralizer can be allotted to a heavier payload or simply to save some propellant over the lifetime of the satellite on which the thruster is installed. We specifically designed a new thruster based on the design of Lubey and Trudel. A series of modifications in terms of the design were implemented in order to improve the previous design. A plasma was successfully created for a range of mass flow rate (from 0.01 sccm to 1 sccm) and the power required to create and sustain the plasma was monitored at low mass flow rate. Two different sets of grids were designed: one was designed for specifically monitoring the power required for a plasma discharge, while the other one was designed to produce thrust or to extract electrons from the plasma. Numerical simulations were also performed to verify the design of the microwave antenna and to monitor the magnetic field created by a pair of concentric magnets.

1.4 Thesis Overview

The design of the thruster, the experimental setup, and the results —both experimental and numerical— are discussed in the remainder of the thesis. Chapter 2 presents the theory behind an electrostatic thruster with an electron cyclotron resonance (ECR) heating type of ionization as well as plasma theory. Chapter 3 discusses the design and choices made in 6 order to build the thruster and gives predicted values for eﬃciencies and thrust. Chapter 4 presents the experimental apparatus and methods used to test the ion thruster in both its ﬁrst and second iteration. The results of the experiment and of the various numerical simulations are discussed in Chapter 5. Finally Chapter 6 concludes this work with a summary and suggestions for future work. Chapter 2

Theoretical Background

This chapter covers the theory of rocket propulsion in general and, more specifically, of ion thrusters. A general presentation of methods to propel a rocket is presented first, then the physics of ion thrusters are discussed. The mechanisms of the electron cyclotron resonance discharge are also presented from an atomic point of view. Finally, the microwave line theory is briefly discussed.

2.1 Principles of Rocket Propulsion

2.1.1 Vehicle Performance

Rocket propulsion relies on Newton’s third law of action and reaction — for every action there is a reaction in the opposite direction and equal in magnitude — and on the conservation of momentum. Since some mass is being ejected from the rocket, Newton’s third law states that there is a force in the other direction, pushing the rocket. The conservation of momentum on a control volume surrounding the rocket states that the net force acting on the volume is equal to the change in momentum. Figure 2.1 shows all the forces acting on the control volume.

To characterize thrust, we can use Newton’s second law

d (mv) = T + (P − P ) · A . (2.1) dt a e e

The total change of momentum is equal to the momentum out of the control volume minus the momentum into the control volume. The latter is equal to zero since there is no inﬂow. 8

Figure 2.1. Forces acting on a control volume surrounding a rocket

The momentum out of the control volume is given by

Jo =mu ˙ e, (2.2) wherem ˙ is the exhaust mass ﬂow rate and ue the exhaust ﬂow velocity. Therefore, we obtain the following equation for the thrust

T =mu ˙ e + (Pe − Pa) · Ae. (2.3)

We deﬁne the equivalent exhaust velocity to be

P − P u = u + e a · A . (2.4) eq e m˙ e

We also define the specific impulse to be the ratio between the thrust and the flow weight, or how efficiently the propellant is being used:

u I = eq . (2.5) sp g

The conservation of momentum also allows us to obtain the so-called “rocket equation” or “Tsiolkovsky equation”, which links the change in the vehicle velocity to its change in mass and the exhaust velocity of the propellant

mi ∆u = ueq ln . (2.6) mf 9

We see that a high exhaust velocity is a great advantage for a given rocket:

• for the same propellant mass it is possible to have a longer lifetime;

• for the same lifetime, less propellant can be used, leaving more room for a bigger payload; and

• for the same lifetime and same payload mass, less propellant can be used, allowing a smaller launch vehicle.

2.1.2 Categorization of Rocket Propulsion

Rocket propulsion can be achieved several ways: chemically, electrically, and thermally, though the ﬁrst two ones are the most commonly used.

Chemical Propulsion Chemical rockets have been extensively used since the Second World War with the German V-2 and rely on the energy stored in the chemical bonds of a certain propellant fuel and oxidizer. This type of propulsion is said to be “energy limited” — there is only a given chemical energy in the atomic bonds available to heat the products of the chemical reaction. The chemical reaction of the fuel and oxidizer allows the propellant ﬂuid to be heated to high temperature and pressure before being expanded through a nozzle. That expansion creates thrust by converting the thermal energy into kinetic energy. Chemical rockets are able to provide a high thrust, which makes them suitable for carrying payloads from the ground to orbit. Under the assumption of an ideal rocket in steady state, pictured in Figure 2.2, which does not have any losses of any kind, we can derive an equation linking the temperature in the combustion chamber of a chemical rocket and the exhaust temperature, i.e.,

u2 C T = C T + e . (2.7) p c p e 2

We also have the following isentropic relations between temperature and pressure and the speciﬁc heat at constant pressure of the gas

γ−1 T P γ = and (2.8) Te Pc

γ R¯ Cp = , (2.9) γ − 1 Mw 10 where γ is the ratio of speciﬁc heats. We therefore obtain after combining equations Eqs. 2.7, 2.8, and 2.9 v u " γ−1 # u γ R¯ Pe γ ue = t2 · · 1 − . (2.10) γ − 1 Mw Pc

As we can see, the exhaust velocity is proportional to the square root of the temperature in the combustion chamber divided by the molecular weight of the gas considered. There- fore, high temperatures and gases with a small molecular weight provide a high exhaust velocity. One of the limits of that type of rocket is due to structural reasons: it is diﬃcult to manufacture combustion chambers and expansion nozzles able to handle really high temperatures and combustion products that are corrosive. In these cases, cooling with some of the fuel or oxidizer can be used to avoid melting the combustion chamber and nozzle. The other limit of that kind of rocket is the atomic mass of the products of the reaction. Usually products of the reaction are heavy components, which lead to a lower exhaust velocity.

Figure 2.2. Schematic of the ideal rocket

Chemical rockets can be split into three groups: solid, liquid, and hybrid. Solid rockets use a propellant made out of a stable solid of fuel and oxidizer. Once the combustion is ignited it is not possible to stop it. Liquid rockets use a liquid form of a propellant. It can be a monopropellant, for example hydrazine, or a bipropellant, for example the fuel and oxidizer mix of oxygen and hydrogen. Contrary to solid rockets, it is possible to stop and reignite an engine using liquid propellant. These rockets provide also some of the highest speciﬁc impulse. However, storage of liquid propellant gives rise to additional issues. Hybrid rockets combine the features of liquid and solid rockets. The solid part of the rocket is usually the fuel while the liquid one is usually the oxidizer. This type of rocket combines the controllability of the liquid rockets with the simplicity of the solid rockets. 11

Electric propulsion Electric propulsion uses electric energy in order to increase the kinetic energy of the propellant. Electric rockets are said to be “power limited” — there is only a certain quantity of power available on the spacecraft, from the Sun with solar panels, or from any other source, for example a nuclear reactor. That kind of propulsion provides higher speciﬁc impulse than chemical rockets — up to 10,000 s. It is very well suited for space maneuvering, orbit control, and interplanetary missions since electric thrusters are able to work for long periods of time while using very little amount of propellant. Electric propulsion can be broken down in three further categories: electrothermal, electrostatic, and electromagnetic. Electrothermal thrusters rely on some form of electromagnetic energy to heat the propellant, which is then expanded through a nozzle. It is possible to achieve propellant heating the following ways:

• with resistances: the propellant goes through a heating resistance;

• with arcing: the propellant passes between a cathode and an anode which strikes an arc, heating the propellant; and

• with microwave or radio-frequency radiation: the propellant goes through a plasma created by the radiation inside a chamber.

Electrothermal thrusters and chemical rockets are similar in the way they use a heated propellant that is expanded through a nozzle. However, for electrothermal thrusters, a lighter propellant is used and it is possible to achieve high chamber temperatures with techniques such as arcing. This leads to a greater exhaust velocity. Just as chemical rockets do, electrothermal thrusters are limited by the temperatures they can achieve, since too high temperatures would damage either the combustion chamber or the nozzle.

Electrostatic thrusters rely on a propellant that is ionized and then accelerated through an electric field created by a change of potential. This kind of thruster also requires a permanent magnetic field to confine the electrons. It is possible to achieve the ionization of the propellant different ways:

• by bombarding the propellant with electrons;

• by using radio-frequency or microwave radiation; and

• by using field emission, where the propellant is fed through small needles that have a high electric potential. At the tip of those needles, a strong electric field is created, able to rip an electron off a propellant atom. 12

Ion thrusters and Hall thrusters are two types of electrostatic thrusters.

Electromagnetic thrusters use both electric and magnetic ﬁelds to accelerate a plasma. They operate in a steady or transient manner. That type of thruster has some disadvantages compared to other thrusters:

• they need a really high ionization energy,

• they need a strong magnetic ﬁeld,

• the exhaust channel tends to erode, and

• eﬃciency for some thrusters is low.

Electromagnetic thrusters, however, can provide high speciﬁc impulse and also high thrust. Some can also work with a large variety of propellants. For example, a pulsed inductive thruster can provide speciﬁc impulse of the order of two thousand seconds while providing a thrust of the order of one newton.

In order to provide useful comparisons, some efficiencies are defined for electric thrusters. We define here the total efficiency, mass utilization efficiency, and electrical efficiency

2 Pjet T ηT = = , (2.11) Pin 2m ˙ pPin

m˙ i IbM ηM = = , (2.12) m˙ p em˙ p

Pb IbVb ηE = = , (2.13) PT IbVb + Po where Pjet and Pin correspond to the total extracted beam power and to the total input electrical power, respectively;m ˙ i andm ˙ p correspond to the ion mass ﬂow rate and propellant mass ﬂow rate, respectively; and Ib and M correspond to the beam current and atomic mass of the propellant. Eq. 2.12 tells us how much propellant is being ionized in the discharge chamber. Pb and PT correspond to the beam power and the total input power respectively. Po corresponds to the other power sources in the thruster.

Thermal propulsion Finally, thermal rockets rely on an external power source to heat the propellant. It is possible to use either the radiation of the Sun or beamed radiation 13 from a laser on the ground. For the beamed energy rocket, the amount of power required is prohibitive and since the radiation source is on the ground, there are some issues with atmospheric events such as clouds or rain.

Table 2.1 gives a comparison of the thrust-to-weight ratio and specific impulse of different types of propulsion from [13] and [14]. It is important to notice that a high specific impulse is not necessarily required. It depends on the type of mission considered, the burn time, the velocity to be achieved, etc.

Table 2.1. Comparison of the thrust and speciﬁc impulse generated by various types of rocket propulsion

Type of propulsion Thrust to weight ratio Speciﬁc impulse (s) Liquid monopropellant 10−1 – 10−2 150 – 225 Liquid bipropellant 10−2 – 100 300 – 450 Solid propellant 10−2 – 100 210 – 320 Electrothermal thruster 10−4 – 10−2 120 – 300 Electrostatic ion thruster 10−6 – 10−4 1200 – 3500 Pulsed inductive thruster 10−4 2200

2.2 Ion Thruster Physics

An ion thruster is comprised of an ion source followed by two or three grids at different potentials. The potential difference between the grids creates an electric field, which ac- celerates the ions using the Lorentz force. The ion exhaust is then neutralized with some device in order to avoid beam stalling. The principle of an ion thruster is sketched in Figure 2.3. 14

Figure 2.3. Schematic of an ion thruster [15]

2.2.1 One-dimensional ﬂow

We have the following potential and kinetic energies for a charged particle starting at rest and going through the two-grid system:

Ep = q∆V, (2.14)

mu 2 E = e . (2.15) k 2 We also have by conservation of energy applied on a single particle, assuming 100% eﬃciency,

mu 2 e = q∆V, (2.16) 2 which leads to the exhaust velocity

r 2q∆V u = . (2.17) e m

We deﬁne successively the thrust, beam current, and beam power created by this charged particle exhaust

T =mu ˙ e, (2.18) m˙ I = q , (2.19) m mu˙ 2 P = IV = e . (2.20) 2 15

The beam power can be interpreted as the kinetic energy of the exhaust per unit time.

The approach taken here does not take into account the “space charge eﬀect”: more than one charged particle is actually present at any time between the grids as shown in Figure 2.4, creating locally an electric charge, which changes the electric potential. Though it is possible to achieve large exhaust velocities with an ion thruster, the space charge eﬀect is a limitation factor.

Figure 2.4. Modiﬁcation of the electric potential due to an accumulation of electric charges between two grids

It is possible to take this effect into account in our equations in 1D using the Maxwell– Gauss equation ρ ∇ · E~ = . (2.21) 0 We also have the relation between the electric field and the electric potential, assuming time-invariant fields E~ = −∇V. (2.22)

Therefore, we obtain

ρ nq j j r m ∇2V = − = − = − = − · , (2.23) 0 0 0u 0 2q∆V 16 which leads to the following potential:

" # 4 3r j m 3 V = Va − x . (2.24) 2 0 2q

The current density for a given grid spacing is given by Child’s law [15],

r 3 4 2q V 2 j = a , (2.25) 9 0 m L2 which tells us that there is a limit for a given grid spacing due to the space charge eﬀect. This aﬀects the thrust per unit area as well since it can be expressed as [15]

T 8 V 2 = a . (2.26) A 9 0 L

2.2.2 Exhaust neutralization

Since the ion thruster emits positively charged ions, the spacecraft itself accumulates a deficit of charge, making it negative: the ionns are then pulled back and all the benefit from exiting ions is cancelled. This is called beam stalling. To avoid this, it is necessary to neutralize the exhaust by emitting electrons as shown in Figure 2.3. It is possible to use either an electron gun, thermionic emitters such as a tungsten wire in which a current is flowing or use field emission as an electron source.

2.3 Ionization of the Gas via ECR Heating

The MMIT makes use of collisions between neutral atoms and mobile electrons in order to create a plasma. The process involved in our case is called electron cyclotron resonance (ECR).

2.3.1 Motion of a single charged particle in an electromagnetic ﬁeld

In the presence of a uniform magnetic field, a charged particle only has a circular motion perpendicular to the direction of the magnetic field, under the condition that the particle has a non-zero initial velocity in that direction. We define the cyclotron frequency and the Larmor radius, which are the frequency of this circular motion and the radius of the circular path of the particle, respectively, as qB ω = , (2.27) c m 17

mu r = . (2.28) L qB If the particle is in a region where the electric or magnetic field is varying with respect to space (i.e., their gradient is non-zero) or where a uniform electric field exists, the center of the circular motion presented before drifts. The motion of the particle is therefore a superposition of two simple motions: a rotational motion due to the magnetic field, with radius increasing or decreasing depending on the strength of the magnetic field, and a linear motion due to the existence of gradients in the electromagnetic field or the existence of an electric field. Using the approximation that the Larmor radius is small compared to the size of the region where we have a magnetic field gradient, it is possible to obtain the expression for the drift velocity

B~ ×∇B • in the case where ∇B perpendicular to B: ~v = ±v⊥rL B2 ; and

• in the case where the magnetic ﬁeld is curved with a constant radius of curvature

m R~c×B~ 2 1 2 ~v = 2 2 · (vk + v⊥ ). q Rc B 2

2.3.2 Microwave breakdown of a neutral gas

The theory of microwave breakdown of neutral gases can be found in References [15], [16], [17], [18], and [19]. We explain here the basic mechanisms that lead to the creation of a plasma from a neutral gas.

2.3.2.1 General ionization process

It is possible to obtain the concentration of electrons — or amount of ionization — at a certain pressure and temperature for a given characteristic temperature of ionization ΘI [20]: 2 5 Φ T 2 Θ = C exp − I , (2.29) 1 − Φ2 P T where C is a constant depending on the gas considered, and Φ is the degree of ionization. Additional assumptions are that the gas is monoatomic and that the electronic excitation of the gas itself is ignored. One consequence of this equation is that there is always a certain amount of ionization in a gas and therefore some free background electrons. When we apply high-frequency radiation such as microwaves to a neutral gas, these background electrons start oscillating at a velocity that is out of phase by π/2, thus gaining no power on average from the electric field. If we add a permanent magnetic field in the gas, the electrons undergo a gyrating 18 motion along the magnetic field lines at a frequency given by Equation 2.27. When we match the microwave frequencies to the cyclotron frequency, the electrons are able to gain energy from the incoming microwaves. The energy transfer between the oscillating field and the gas is then driven by collisions between electrons and atoms. After a certain number of collisions, an electron can have enough energy to collide inelastically with an atom and produce an ion–electron pair. The electron created from this pair then oscillates because of the electric field, and after enough collisions creates an ion–electron pair by an inelastic collision. If the rate of production of electrons is larger than the rate of loss of electrons, then the process continues by itself and a plasma is created.

2.3.2.2 Collisions in a gas

In order for energy to be transferred from an electromagnetic field to a gas, collisions must occur between electrons or any charged particles and the neutral particles in the gas. These collisions can either be elastic or inelastic for our domain of interest. Other types of energy transfer include superelastic collisions, radiative energy exchange and charge-reactive energy exchange. Only elastic and inelastic collisions are relevant to us. Figure 2.5 shows the main different types of collision in a gas. We will consider only electron-atom and electron-ion collisions in the rest of this work for the following reasons. First, we consider high frequency electric fields in which only electrons are mobile. The ions are too heavy to be able to follow the perturbation induced by the oscillating electric field and can be considered as fixed. Second, the energies required to ionize neutral atoms for collisions between neutral atoms or ions is too high to be of our interest.

Collision mean free path We can deﬁne the mean free path λ of a particle in a gas, which is the average distance of travel between two collisions. In this approach, which is based on kinetic theory, all the particles are considered to be rigid spheres, and the mean free path is 1 λ = , (2.30) nσ where n is the number density of atoms —either neutral or ionized— in m−3 and σ is the cross section in m2. The collision frequency is deﬁned as

ν = nσv,¯ (2.31) 19

Figure 2.5. Types of collisions that can occur in a gas [15] wherev ¯ is the average of Maxwellian velocities over all the particles. The previous equation can be slightly modiﬁed if we have a neutral atom moving slowly toward a region of high-density and fast-moving electrons, such as a plasma [13]:

v λ = n , (2.32) nehσvei ne corresponds to the electron number density, ve to the electron velocities, and vn the neutral particle velocity. Equation 2.32 is referred to as the penetration distance since it is in general used to predict that distance a neutral atom travels into a plasma before being ionized.

Elastic collisions Elastic collisions occur when two particles collide and just exchange kinetic energy. The cross-section for that kind of collision is highly dependent on the energy 20

of the colliding electron. Figure 2.6 describes the probability Pc of a collision between a neutral atom and an electron as well as the cross-section of that collision Q for various electron energies and for various gases. Figure 2.7 shows the elastic collisions frequency for various gases and electron energies.

Figure 2.6. Collision probability Pc and collision cross-section Q as a function of electron energy (in eV) for various gases [16]

Figure 2.7. Collision frequency νc as a function of pressure and electron energy (in eV) for various gases [16]

Inelastic collisions Inelastic collisions correspond to a collision that is able to transfer some kinetic energy of one particle to the internal state of one of the two particles. The 21 creation of a diﬀerent particle system where one of the particles is ionized, for example, is also referred as an inelastic collision. The result of such a collision is an atom that is in an excited state. Usually the atom goes back to its ground state, but if the electron has a suﬃcient energy, the atom has a certain probability to be ionized. That type of collision can be described by the following reactions;

A +e ˜ → A∗ + e for the excitation of the atom, A +e ˜ → A+ + 2e for the ionization of the atom.

If we refer to Px and Pi the probability of an atom to be either excited or ionized respectively, one can define the excitation efficiency hx and the ionization efficiency hi as hx = Px/Pc and hi = Pi/Pc. Figure 2.8 shows the excitation efficiency for various gases vs. the electron energy and Figure 2.9 shows the ionization efficiency vs. the electron energy.

Figure 2.8. Excitation eﬃciency hx as a function of electron energy (in eV) for various gases [16]

Figure 2.9. Ionization eﬃciency hi as a function of electron energy (in eV) for various gases [16] 22

2.3.2.3 Losses of electrons in a gas

Collisions in a gas are the atomic process by which ions are produced. It requires electrons in excited states with enough energy to create ions. However multiple mechanisms can also lead to the loss of electrons in a gas and decrease the probability of creating ions. These processes are namely diﬀusion, attachment, and recombination.

Diffusion of electrons Diffusion occurs when a fluid shows a difference of concentration between two regions. Particles move from the high concentration region to the low concentration region. If we consider the general case of particle diffusion before steady-state, the variation of particle density n can be written as

∂n + ∇ · ~Γ = 0, (2.33) ∂t where ~Γ is the flux of some particles. ~Γ is given by the Fick’s law whenever no electric field is present or no particles are charged, which is also called free diffusion:

~Γ = −D∇n, (2.34) where D is the diﬀusion coeﬃcient and can be obtained with

k T D = b . (2.35) mν

Therefore, combining Equations 2.33 and 2.34, we obtain the so-called diﬀusion equation

∂n = D∇2n. (2.36) ∂t

Assuming that the particles diﬀuse exponentially over time, we have

−νt n(~r, t) = n0(~r, t)e , (2.37) with ν the collision frequency. This leads to the following modification to the diffusion equation ν ∇2n = − n. (2.38) D ν A quick dimensional analysis shows that D is the inverse of the square of the characteristic diffusion length Λ, such that 1 ν = . (2.39) Λ2 D 23

In a cylindrical geometry with a height h and radius r, the solution of the diffusion equation leads to [17] 1 ν π 2 2.4052 = = + . (2.40) Λ2 D h r The presence of ions and electrons in the gas contributes to the change in diffusion if their concentration is high enough so electrostatic forces influence the motion of charged particles. This mechanism is called ambipolar diffusion. If particles are charged and are subject to an external electric field, the flux of these particles can be expressed as

~Γj = n~vj = ±µjnE~ − Dj∇n. (2.41)

In a plasma, the flux of ions and electrons has to be the same so as to avoid a charge imbalance. If Γe and Γi are the flux of electrons and ions due to diffusion, respectively, it is possible to show that Fick’s law is still valid if we use a new diffusion coefficient given by

µiDe + µeDi Da = , (2.42) µi + µe where µ is called the mobility and is given by

| q | µ = . (2.43) mν

It is then possible to determine under which conditions the diﬀusion is ambipolar or free in the plasma for the electrons. We deﬁne the Debye length of a plasma as [21]

1 1 k T 2 k T 2 λ = 0 b e or λ = 7430 b e , (2.44) D ne2 D n with kbTe in J. We also define R as the characteristic length over which the gradients of electron concentration are the largest. In our case it is the radius of the discharge chamber, roughly 10 mm. For R λD or R . λD the diffusion is free [18]. In the case of our ion thruster, with electron temperatures ranging from 1 to 20 eV and electron concentration 16 −3 equal to 10 m we find that λD varies between 0.074 mm and 0.332 mm. Therefore,

R λD, and the diffusion is clearly ambipolar. Ambipolar diffusion is, however, less important than free diffusion of electrons when it comes to microwave breakdown in gases [16] if there are no magnetic fields.

One important part of the design of the MMIT is the use of permanent magnets to 24 contain electrons. The presence of the DC magnetic field affects the diffusion. References [17] and [16] give the following result for the characteristic diffusion length in a cylinder of height h and radius r. In the case where the magnetic field is parallel to the axis of the cylinder, we have 1 ν2 2.4052 π 2 = + , (2.45) Λ2 ν2 + ω2 r h where ω is the cyclotron frequency of the considered particle defined in Equation 2.27. In the case where the magnetic field is perpendicular to the axis of the cylinder, we have

! 1 ν2 π 2 1 2.4052 1 2.4052 = + + . (2.46) Λ2 ν2 + ω2 h 2 r 2 r

q ν2+ω2 Adding a magnetic field effectively increases the diffusion length by a factor of ν2 in the directions perpendicular to the magnetic field [16][17]. Also since both Equations 2.45 and 2.46 involve resonance frequency, the ions and electrons diffuse over two different characteristic lengths. As an example, considering a cylindrical cavity of height 1.5 cm, radius 1 cm, with a pressure of 5 × 10−3 Torr, electron energies of 15 eV, and an electron resonance frequency 6 −1 of 5 GHz, we obtain for argon ν = 8.75 × 10 s from Figure 2.7 and, therefore, Λf = 3.14 mm for the free diffusion, Λk = 4.77 mm for the diffusion in a magnetic field parallel to the cylinder axis, and Λ⊥ = 5.88 mm for the diffusion with a magnetic field perpendicular to the axis of the cylinder. For the ions, we obtain Λf = Λk = Λ⊥ = 3.14 mm. We find indeed that ω = 1.2 × 106 rad · s−1 for the ions, so we have ν ω so Equations 2.45 and 2.46 reduce to the free diffusion equation. Table 2.2 summarizes the results.

Table 2.2. Characteristic diﬀusion length for argon atoms and electrons in a cylindrical cavity of height 1.5 cm, radius 1 cm, pressure 5×10−3 Torr, electron energies of 15 eV, and electron resonance frequency of 5 GHz

Species Λf (mm) Λk (mm) Λ⊥ (mm) Argon 3.14 3.14 3.14 Electron 3.14 4.77 5.88

We see that adding a magnetic field effectively increases the characteristic diffusion length of the electrons and, therefore, decreases the losses by diffusion. However, the ions are not affected in our range of frequencies and energies. 25

Attachment Attachment occurs whenever a free electron is captured by a neutral particle. This happens especially for neutral particles, which have an outer electronic shell nearly ﬁlled, for example an oxygen atom.

Recombination The recombination phenomenon is the capture by a positive ion of an electron to form a neutral atom. In order to conserve momentum and energy, this neutral atom is in an excited state and there is emission of a photon in the case of radiative recombination or of an additional particle in the case of three-body recombination [21]. The rate of change of electron concentration due to recombination is simply proportional to the total number of species: ∂n = −αn2, (2.47) ∂t which can be solved as 1 1 = + αt, (2.48) n(~r, t) n0(~r) where n0(~r) is the initial density distribution.

Wall losses An electron can also be lost from the plasma whenever it hits a wall. The magnetic field is designed so that the electrons are trapped inside a magnetic bottle, or magnetic mirror. We define the magnetic moment of a particle gyrating around the magnetic field lines as 1 mv2 µ = ⊥ , (2.49) 2 B where v⊥ is the velocity perpendicular to the magnetic field direction. µ is a constant for a given particle [21]. In an axisymmetric magnetic field designed so that the gradient of the magnetic field is in the same direction as the magnetic field itself, a charged particle gyrating around the magnetic field lines and traveling from a weak field region to a strong field region sees an increasing magnetic field. The velocity perpendicular to the magnetic field therefore increases to keep the magnetic moment constant. In order to conserve the energy of the particle as well, the velocity parallel to the magnetic field has to decrease. If the magnetic field is strong enough at the throat of the magnetic bottle, the velocity parallel to the field eventually reaches 0, and the particle is reflected back into the plasma. The trapping is not perfect however. If a particle has a velocity such that [13]

r Bmax vk ≥ v⊥ − 1, (2.50) B0 26

where B0 is the average magnetic ﬁeld, then it will escape the magnetic bottle and contribute to wall losses.

2.3.2.4 Electron cyclotron resonance heating

It is possible to add a magnetic field to decrease the loss of electrons by diffusion as we saw in the previous section. The presence of a magnetic field forces the electrons to undergo a cyclotron motion. In addition, if this magnetic field is non-uniform it is possible to “trap” the electrons so they can only rotate and travel around magnetic field lines and collide with incoming gas particles.

Electron energy and power gains in a non-uniform magnetic ﬁeld If the microwave frequency is chosen so as to match the cyclotron frequency, the electrons gain the following energy from each gyro cycle in the non-uniform magnetic ﬁeld [7]:

2 E⊥ EECR = πe ∂B , (2.51) vk ∂s

∂B where ∂s corresponds to the gradient of the flux of the magnetic field along a field line.

2.4 Transmission Line Theory

The MMIT is able to create a plasma by using microwaves. It is therefore important to understand the transmission of power from the microwave generator to the antenna, or a load in general.

2.4.1 Propagating wave on a transmission line

A transmission line such as a coaxial cable connected to a microwave generator will see traveling waves with the following voltage and current [22]

+ −γs − γs V (s) = V0 e + V0 e , (2.52)

+ −γs − γs I(s) = I0 e + I0 e , (2.53) where s is the direction of propagation; e−γs corresponds to a wave that travels forward while eγs corresponds to a wave that travels backward. We deﬁne the characteristic impedance 27

+ + Z0 to be the ratio of the two phasors V0 and I0 so that

+ − V0 V0 Z0 = + = − − . (2.54) I0 I0

2.4.2 General load on a transmission line

The previous section dealt with an open line without any load. If a load of impedance ZL diﬀerent from the impedance Z0 of a wave traveling through the line is attached to the line as shown on Figure 2.10, the ratio of voltage to current still has to be equal to ZL at the load. This gives rise to a wave with an amplitude able to satisfy that condition. This wave is reﬂected back to the source.

Figure 2.10. A load of impedance ZL connected to a transmission line of impedance Z0

We define the voltage reflection coefficient Γ as

Z − Z Γ = L 0 . (2.55) ZL + Z0

If ZL = Z0 the load is said to be matched to the transmission line impedance and we see in this case that Γ = 0, i.e., there is no reﬂected wave. It is also possible to obtain the average power delivered to the load [22]

+ 2 1 | V0 | 2 Pav = 1 − | Γ | . (2.56) 2 Z0

If Γ is equal to 1, then no power is transmitted to the load. If the load is matched to the line impedance then we obtain the maximum power transmission since Γ is equal to 0 in this case. We finally define the standing wave ratio (SWR), which is a measure of the mismatch 28 of a line [22] as 1+ | Γ | SWR = . (2.57) 1− | Γ | For a matched load, the SWR is equal to 1. It increases as the load impedance differs from the transmission line impedance. Chapter 3

Ion Thruster Design and Theoretical Predictions

The previous design of the MMIT had a few ﬂaws. It was not possible to place the microwave antenna precisely enough; the magnets were not attached to an iron yoke plate to close the magnetic circuit; and most of the thruster was assembled with epoxy, which is meant to be permanent. Based on these observations of the previous design, a new ion thruster was proposed. The new design is more modular: it is possible, for example, to change the discharge chamber length and diameter; the shape of the antenna; and also the thickness of the yoke plate on which the magnets are attached. The new design also includes a couple of modiﬁcations in order to correct omissions such as an iron yoke plate not being used or not being able to place the antenna precisely.

3.1 Objectives

The MMIT is a proof-of-concept thruster, which requires a lot of optimization. Therefore, the main design objectives are reusability and modularity. It is possible to change different components of the thruster in order to test different configurations, pick the one that works best, and then optimize that design. Permanent sealing and fastening therefore is avoided, leading to additional issues such as ensuring proper sealing. That issue is addressed by using temporary vacuum grease, which is easy to remove and shape and by using removable fasteners such as nuts and screws that can tighten all the parts of the thruster together. 30

3.2 Overview

The new thruster is made out of ﬁve main parts: the back plate on which the microwave input and gas feed are connected; the yoke plate, which creates a closed magnetic circuit; the discharge chamber in which the plasma is created; the front plate, which serves as a fastener for the ion grids; and the magnets and the antenna, which create the plasma. The back plate, front plate, and discharge chamber are all fabricated out of MACOR, which is a carbide-machinable ceramic. This material has been chosen due to its low outgassing at high vacuum, its electrical and thermal insulating properties, and also high temperature capability — MACOR is used for applications up to 1000 ◦C. The yoke plate is made out of 1018 steel, which is a low-carbon alloy. The extraction grids are usually made out of molybdenum but, due to cost restrictions, stainless steel was used. The microwave and gas inputs have been switched compared to the ﬁrst version of the thruster: the microwave line is now on the axis of revolution of the thruster and the gas input is shifted to the side. Figure 3.1 shows a view of the thruster and Figure 3.2 shows a section cut through the axis of revolution of the thruster.

Figure 3.1. CAD representation of the new version of the MMIT

3.3 Back Plate

Figure 3.3 shows a section cut of the back plate. As we can see in Figure 3.2, the back plate has been designed so that it contains both the yoke plate and the discharge chamber. 31

Figure 3.2. CAD representation of the new version of the MMIT — Section cut

Fastening the back plate with the front plate ensures proper sealing of the thruster. The back plate serves as a support for the gas line and the microwave input. The gas line is epoxied to the back plate whereas the microwave candlestick is inserted through a hole in the back plate and fastened to it. The gas is injected in the expansion chamber through a 1/16- in-diameter stainless steel pipe. In order to minimize the power reﬂected in the microwave line, the candlestick antenna has been inserted through a stainless steel tube. This avoids microwave energy being radiated before being carried into the discharge chamber. The gap we can see between the back plate and yoke plate is the expansion chamber and allows the gas to expand before being fed in the discharge chamber.

Figure 3.3. CAD representation the back plate — Section cut 32

3.4 Yoke Plate

3.4.1 Magnets

The yoke plate serves as a support for the two magnets. These two magnets are magnetized along their axis of revolution. In order to create a magnetic bottle between the two magnets, the magnetic circuit has to be closed. Therefore, the two magnets are concentric and their poles are inverted relative to each other. The yoke plate is made out of a material that has a high relative magnetic permeability, typically low-carbon steel or iron. This last feature allows the magnetic circuit to be closed. Table 3.1 shows the dimensions of the magnets for the initial and new version.

Table 3.1. Dimensions of the magnets for the two diﬀerent versions of the thruster, in inches

Magnet Dimension Version 2.0 Version 1.0 Inner magnet Inner diameter 1/8 1/16 Outer diameter 1/4 1/8 Thickness 1/4 Outer magnet Inner diameter 3/8 1/4 Outer diameter 3/4 3/8 Thickness 1/4

The magnets are made out of an alloy of samarium–cobalt, which has the best features in terms of temperature capability. However, due to cost and simplicity, we initially chose neodymium magnets, which are cheap, strong, and easy to find but have a low temperature capability. The Curie temperature of a SmCo magnet is around 800 ◦C, depending on the alloy, while a NdFeB magnet has a Curie temperature of 300 ◦C. Demagnetization effects start to appear at around 80 ◦C for NdFeB magnets. Table 3.2 presents a quick comparison of material properties for a variety of magnetic alloys. The data are from different manufacturer websites (References [23], [24], [25], and [26]). The temperature coefficient Tc states how much the residual flux density Br is affected by the temperature.

3.4.2 Inputs

Eight holes of diameter 1 mm allow the gas to be fed into the discharge chamber, and a central hole allows the microwave candlestick to be inserted inside the chamber. The central hole diameter depends on the dimensions of the inner magnet. 33

Table 3.2. Properties of neodymium, samarium–cobalt and alnico magnets

Magnet type Maximum operating Curie temperature temperature (◦C) (◦C) Neodymium grade N42 80 310 Samarium–cobalt 300 800 (2–17 alloy) AlNiCo 450–500 815–890

Magnet type Residual ﬂux density Br Temperature coeﬃcient (T) Tc (%/K) Neodymium grade N42 1.3 –0.13 Samarium–cobalt 1.2 –0.035 (2–17 alloy) AlNiCo 1.0 –0.2

3.5 Discharge Chamber

The discharge chamber is a simple cylinder in which the plasma is contained. A Faraday cage made out of stainless steel is fixed around the discharge chamber in order to avoid any outside radiation interfering with the microwave radiation inside the chamber. The first material chosen for this discharge chamber is MACOR, but further iterations could make use of glass quartz to be able to see where the plasma is created and sustained in the chamber. For that particular configuration a metallic mesh coating the discharge chamber would be used as a Faraday cage. The length and inner diameter of the chamber are changeable. The pressure inside the discharge chamber is evaluated using Poiseuille’s law. For a cylindrical tube of radius a and length l, Poiseuille’s law gives the rate Nm at which the gas flows (in moles per second) [13]

π a4 P 2 − P 2 N = 1 2 , (3.1) m 16ζ l RT where R is the universal gas constant, P1 is the upstream pressure, P2 the downstream pressure,ζ the gas viscosity in Poise, and T the gas temperature in K. The upstream pressure therefore can be determined using the relation [13]

0.78QζT l P 2 = P 2 + r , (3.2) 1 2 d4 where the pressures are in Torr, Q is the mass ﬂow rate in sccm, l is the length of the 34

chamber in cm, d is the diameter of the oriﬁce in cm, and Tr = T (K)/Tc where Tc is the critical temperature of the gas. If Tr 1.5, the viscosity ζ in Poise is given by [10]

0.34 ζ = T 0.94, (3.3) A r

1 1 2 where A = Tc 6 M 2 Pc 3 , Pc being the critical pressure. For the other cases, ζ (in Poise) is given by [10] 0.1778 5 −7 ζ = (4.58T − 1.67) 8 × 10 . (3.4) A r Table 3.3 sums up the parameters for xenon and argon. The data are taken from Reference [10].

Table 3.3. Compressible gas parameters for xenon and argon

Parameter Xenon Argon A 0.0151 0.0276 Tc (K) 289.8 151.2

As an example, for the single–hole grid of diameter 2.54 mm and thickness 0.635 mm, argon gas at a temperature of 298 K, a mass flow rate of 0.05 sccm, and neglecting P2, −3 which is the vacuum chamber pressure, we obtain P1 = 1.62 × 10 Torr. For a grid with multiple circular apertures it is possible to apply Poiseuille’s law to each of these apertures. If we have N apertures of the same diameter d through a grid of length l, we have for each aperture 0.78Q ζT l P 2 = P 2 + i r , (3.5) 1 2 d4 where Qi is the mass flow rate going through each aperture. Since all these holes have the same diameter and width, Qi is the same for all aperture. We denote it Qa. Since all these holes can be considered as pipes in parallel, we have the following relation for the mass flow rate N X Q = Qi = NQa. (3.6) i=1 Combining Equations 3.5 and 3.6, we obtain that

0.78QζT l P 2 = P 2 + r . (3.7) 1 2 Nd4 35

3.6 Front Plate and Grids

3.6.1 Front plate

The front plate serves both as a support for the ion optics and for the discharge chamber, while ensuring proper fastening to avoid leaks. Figure 3.4 shows a CAD view of the front plate.

Figure 3.4. CAD representation of the front plate

3.6.2 Discharge characterization grid

A single grid has been designed at ﬁrst in order to monitor the power needed for creating and sustaining a plasma with ECR heating in the discharge chamber. That grid does not accelerate ions or electrons, it is simply designed to trap the gas inside the discharge chamber and has a single hole of diameter 2.54 mm.

3.6.3 Extraction grids

The accelerating grids are one of the main parts of an electrostatic ion thruster. These grids are designed to accelerate charged particles to high exhaust velocity by creating a continuous potential change, as seen in Chapter 2.

3.6.3.1 Design methodology

References [13] and [15] include general design methods for accelerator grids. However, in order to design an optimal grid system for a particular thruster, numerical simulation of 36 ion and electron impigement on the grids is required. Lacking the time to perform such a simulation, a general methodology is outlined below:

• the grids should maximize the transparency in order to extract the maximum number of charged particles while keeping the neutral atoms inside the discharge chamber, Transparency is deﬁned as the ratio between the beam current and total ion current from the discharge chamber that approaches the screen grid.

• the particle beam has to be focused, and

• the size of the apertures in the grid should be of the order of the size of a Child– Langmuir sheath in order to achieve a good focusing.

In order to meet the ﬁrst requirement, a hexagonal pattern has been chosen for the holes in both the screen and accelerator grids. The holes in the accelerator grids have to be smaller than the ones in the screen grid in order to meet the second requirement. The third requirement leads to the Child–Langmuir law derived in Chapter 2 (Equation 2.25). For a circular aperture, we have the following result:

I J = b , (3.8) b πD2 4 where D is the diameter of the circular aperture. Therefore, for a given beam current

Ib, distance between two grids L, and voltage diﬀerence Va, it is possible to calculate the aperture diameter using Equations 2.25 and 3.8.

3.6.3.2 Ion and electron grid set

Both set of grids for ion and electron extraction were chosen to be the same in order to spare materials and speed up the testing. The voltage across the grids are adjusted in order to create the same beam current, chosen to be equal to 5 mA. Figure 3.5 shows a 2D CAD view of the accelerator grid. The hexagonal pattern for the holes is clearly visible on the grid. The six outside holes are used to fasten the grids to the front plate, whereas the hole on the tab is used to attach an electrical connection to the grid.

Materials and dimensions Grids are usually made out of molybdenum and are chemically etched or optically etched with a laser, but due to cost reduction and simplicity we chose to use stainless steel in which holes were drilled. Table 3.4 shows the size of the holes chosen for that set of grids. The chosen size is arbitrary and mainly corresponds to 37

Figure 3.5. Front view of the accelerator grid machinability criteria. The accelerator and screen grids are insulated from each other by a thin layer of Teﬂon placed between the two. Given our design, it is also possible to change the thickness of that space.

Table 3.4. Main dimensions of the screen and accelerator grids and of the insulating spacer

Grid part Material Hole diameter (mm) Thickness (mm) Screen grid Stainless steel 1.613 0.635 Accelerator grid Stainless steel 1.321 0.635 Spacer Teﬂon N/A 0.508

Potential difference Since we defined the beam current and hole diameter, it is possible to calculate the required potential difference to obtain a Child–Langmuir sheath for various distances between the grids and, therefore, to obtain the exhaust velocity; using Equations 3.8 and 2.25, we can obtain s 2 3 9 M L 2 Va = 2 Ib. (3.9) π0 2q D The exhaust velocity is then calculated using Equation 2.17. Figures 3.6, 3.7, 3.8, and 3.9 present the required potential difference and exhaust velocity for different grid distances. Given the material sizes available, we chose a distance equal to 0.508 mm. 38

Figure 3.6. Potential diﬀerence versus the grid distance for an argon atom

Figure 3.7. Potential diﬀerence versus the grid distance for an electron 39

Figure 3.8. Exhaust velocity versus the grid distance for an argon atom

Figure 3.9. Exhaust velocity versus the grid distance for an electron 40

3.7 Antenna Design Considerations

The power transmitted to the electrons in the ECR layer is optimized when the electric field is perpendicular to the magnetic flux density, according to Equation 2.51: this equation involves only the component of the electric field perpendicular to the magnetic field lines,

E⊥. The antenna was inspired by the work of Koizumi and Kuninaka [7] in which an analysis of the electric field is conducted and shows that the electric field is indeed perpendicular to the magnetic flux density lines.

3.8 Theoretical Predictions

Based on the equations set up in Chapter 2, and on the design in Chapter 3, it is possible to predict the eﬃciencies and additional useful quantities for the thruster. The thruster can work in two modes: ion emission or electron emission. The propellant used is argon and is assumed to be single ionized — results for xenon are also presented. Table 3.5 sums up the diﬀerent nominal parameters. Using these parameters and Equations 2.5, 2.11, 2.12, and 2.17, we obtain the results summarized in Table 3.6.

Table 3.5. Nominal parameters for the new version of the MMIT

Parameter Value Beam current Ib (mA) 5 Potential difference (ions) Va (V) 2280 Potential difference (electrons) Va (V) 55 Distance between grids L (mm) 0.508 Single aperture diameter D (mm) 1.3208 Ionization power P0 (W) 1 Propellant mass flow ratem ˙ p (sccm) 0.15

Table 3.6. Predicted eﬃciencies for the thruster mode for the given nominal parameters

Parameter Species Argon Xenon Electron Thrust T (µN) 217 392 0.1 Total efficiency ηT (%) 75.4 75.3 0 Mass utilization efficiency ηM (%) 46 46 0 −1 6 Exhaust velocity ue (m · s ) 105, 000 58, 000 4.4 × 10 5 Specific impulse Isp (s) 10, 715 5, 912 4.47 × 10 Chapter 4

Experimental Setup

In order to create a plasma in the discharge chamber of the MMIT and extract ions or electrons from the plasma, additional equipment is required. We used a vacuum chamber with two pumps, a pressure transducer and a pressure reader; a propellant control system; a microwave radiation control system; and a grid potential control system. We also used various methods to measure meaningful quantities such as the ion beam current. An overview of the equipment used to test the two versions of the thruster is presented in this chapter.

4.1 Facilities and Equipment

The two versions of the thruster have been tested in a vacuum chamber at The Pennsylvania State University located in room 306 in the Electrical Engineering East building (Figure 4.1). This chamber has been used in the past to test various iterations of the MRIT thruster.

The chamber has a diameter of approximatively 0.7 m and a depth of 1 m. Multiple ports on the chamber side allow the diﬀerent inputs to be connected to the thrusters and outputs to be read from sensors inside the chamber. Two pumps ensure evacuation to pressures down to 1 × 10−5 Torr. A BOC Edwards IPUP dry pump gets the chamber down to approximatively 5 × 10−3 Torr. A CTI-Cryogenics Cryo-Torr cryo–pump then evacuates the air down to high-vacuum pressures of 1 × 10−5 Torr. A MKS Series 999 Quatro Multi-sensor pressure transducer coupled with a MKS Series 999 Controller pressure reader provide pressure reading with an accuracy of 0.01 × 10n Torr, where n is the order of the pressure measurement. Both are shown in Figure 4.2. The temperature of the head of the cryo–pump is monitored with a Lakeshore Cryotronics, Inc. Cryogenic Thermometer Model DRC 80 temperature reader (Figure 4.2). The head of the 42

Figure 4.1. Vacuum chamber cryo–pump has to reach temperatures as low as 20 K in order to obtain high vacuum.

Figure 4.2. Pressure transducer, pressure reader and temperature reader

The necessary microwave signal is generated by a Hewlett-Packard 8683D Signal Generator, able to provide a signal between 2.3 and 13.0 GHz. This signal is then amplified by a Hughes 8010H Traveling Wave Tube Amplifier that operates in the range of 4.0 to 8.0 GHz. A Narda bi-directional coupler and two Mini-Circuits PWR-SEN-6G+ power sensors that operate in the range of 16000 MHz allow us to read the forward and reflected power. A Harris Type 306A double–stub tuner is finally used to match the microwave line and the antenna to 50 Ω. The matching is important as we saw in Chapter 2 since it allows a large amount of the forwarded power to be transmitted to the antenna and absorbed by the plasma. All these components are in series and connected with N-type cables. The signal is then sent through a SMA-type KF-40 feed-through rated to 8 GHz inside the chamber. Figure 4.3 shows the 43 schematic of the different components of the microwave line.

Figure 4.3. Schematic of the microwave line

A 1/8-in.-diameter stainless steel pipe and a CF gas feed-through ensure the connection of the argon propellant (purity 99.999%) to the thruster. The final version of the thruster will make use of xenon as a propellant, which is easier to ionize and has a higher atomic mass than argon. However, due to the cost of xenon, argon is first used as propellant. Inside the chamber, a Swagelok fitting is used to interface that piping with the thruster’s 1/16- in.-diameter inlet pipe. A Horiba-Stec SEC-7320 Mass Flow Controller (MFC) calibrated for xenon and allowing mass flow rate of up to 1 sccm as well as a MKS Vacuum Gauge Measurement and Control System Type 146 provide control over the mass flow rate. The DC voltage necessary to extract the ions or electrons from the plasma is provided by a Stanford Research Systems, Inc. Model PS310 high voltage power supply that has maximum DC voltage rating of 1250 V and a Stanford Research Systems, Inc. Model PS350 high voltage power supply that has maximum DC voltage rating of 5000 V. These two sources are connected to the chamber with BNC-type cables and provide a potential of +1500 V for the screen grid and –780 V for the accelerator grid for the extraction of ions or –40 V and +15 V for the screen grid and accelerator grid, respectively, for the extraction of electrons. One important detail about these two high voltage sources is that they shut down whenever a power surge is detected in the line in order to protect the equipment and operator. Therefore, whenever the two grids arc together, it is necessary to restart these DC sources. This can be done either manually or automatically. These sources are internally grounded. All the voltage sources are shown in Figure 4.4. 44

Figure 4.4. Voltage sources used for ions and electrons extraction

4.2 Measurement Methods

Measurements are made with a Langmuir probe and a Faraday cup. The Langmuir probe determines the ion density, the ion temperature, and beam current. The Faraday cup provides a measurement of the current density of the beam. The results from both these probes are compared and allow us to obtain the thrust. The Langmuir probe voltage is placed a few centimeters away from the exit grids in order to avoid interference from both the microwaves and the DC voltage applied to the grids, but close enough so as to provide meaningful results. Its voltage is swept from –50 V to 50 V in increments of 2 V. The Faraday cup used is shown in Figure 4.5. It has a length of 3.5 cm and an aperture of diameter 0.635 cm. It is placed at the same distance from the exit grids as the Langmuir probe, and its position also varies radially, in increments of 5◦. This characterizes the beam dispersion.

Figure 4.5. Faraday cup used for beam dispersion characterization and ion density measurement 45

4.3 Microwave Analysis

In order to start with a value for the initial microwave frequency, an Agilent Technologies ENA Series Network Analyzer E5071C has been used. This network analyzer also provides useful information about the complete microwave line, from the ampliﬁer to the thruster. Using the analyzer, it is possible to easily set the tuner to match the impedance of the microwave line of 50 Ω. Chapter 5

Results

This chapter presents both experimental results and numerical results obtained with COM- SOL Multiphysics. The goal of the experimental results was to obtain the minimum power at which a plasma can be created and sustained in the discharge chamber, and to perform the extraction of either the ions or the electrons from the plasma. The numerical simulations were performed to validate the microwave antenna design and to obtain information on the conﬁguration of the magnetic ﬁeld inside the discharge chamber.

5.1 Experimental Results

5.1.1 Original design

For the original design, a disc antenna (shown in Figure 5.1) has been installed in the thruster for testing. That thruster conﬁguration was able to produce a plasma whenever the two grids arced together or whenever the screen grid arced with the microwave antenna, as shown in Figure 1.4. The ﬁrst start-up procedure for that thruster is the following:

• turn on the microwave signal generator and ampliﬁer;

• turn on the extraction grids; and

• open the gas valve.

As soon as the gas valve is opened, a quick transient regime happens inside the discharge chamber, leading to an arc between the two grids or between the screen grid and the microwave antenna. The energy released by the arc was high enough to create a plasma. The magnetic ﬁeld created by the two magnets was then strong enough to contain the 47

Figure 5.1. Disc antenna (left) and ring antenna (right) used in the previous and current versions of the thruster respectively. The graduations are in millimeters plasma, and the microwaves had sufficient power to sustain it. However, as soon as the grids were turned back on to begin the extraction of one of the species, the plasma quickly disappeared. Different start up procedures and numerous frequencies (from 4.0 to 8.0 GHz) were tried but none of them were able to generate a plasma without an arc. The procedure of starting the plasma with an arc was quite destructive for the different elements of the thruster and the sensors on the microwave line. Figure 5.2 shows the screen grid after a few tests involving arcing. The scars due to the high-energy arcs are clearly visible.

Figure 5.2. Damage due to arcs on the screen grid. The arrows indicate scars due to arcing

5.1.2 New design

The ﬁrst new design makes use of a ring antenna (Figure 5.1) which faces the 8 holes in the yoke plate for the gas input. It is introduced close to the magnets in order to provide an ECR heating zone. 48

5.1.2.1 Microwave analysis

The log-mag plot of the reﬂected power of the thruster over a range of frequencies from 4 to 8 GHz is shown in Figure 5.3. We can clearly see a resonance peak of value –20 dB (Marker 2) at a frequency of 4.96 GHz and an other peak of value –15 dB at a frequency of 7 GHz. This guided our initial choice for a microwave frequency of 5 GHz.

Figure 5.3. Reﬂected power for the thruster

5.1.2.2 Plasma creation

Given our original choice of frequency we managed to create and sustain a plasma as shown in Figure 5.4. Figure 5.5 shows the minimum power required to create the plasma over a range of mass ﬂow rates. This has been characterized by increasing the microwave power from 0 W to a value that would initiate the plasma, by increments of 0.2 W. The grid used for that experiment was the single-hole grid, which does not perform any extraction of any species. The following start-up procedure for that thruster was used:

• turn on the microwave signal generator and ampliﬁer; 49

• open the gas valve; and

• increase the microwave power to the maximum power achievable by the ampliﬁer.

The creation of the plasma via ECR heating requires usually a few seconds after having opened the gas valve.

Figure 5.4. Argon plasma created by the new design of the MMIT

Figure 5.5. Minimum power required to create the plasma

Once the plasma is created, it is possible to decrease the power required to sustain it. The minimum power to sustain the plasma is shown in Figure 5.6 over a given range of 50 mass ﬂow rates. This has been characterized by decreasing the microwave power from its current value to a value that would make the plasma extinguish, by increments of 0.1 W. The single-hole grid was again used for that experiment.

Figure 5.6. Minimum power required to sustain the plasma

5.1.2.3 Ion and electron extraction

We showed in the previous section that a plasma could be generated by ECR heating for different propellant mass flow rates. We then installed a different of grids made for the extraction of ions and electrons. The creation of the plasma was more difficult with that set of grids. The neutral propellant does not have a long enough residence time in the discharge chamber since the screen and acceleration grids provide a larger escape area from the discharge chamber. For that reason the pressure inside the discharge chamber is lower than with the single-hole grid. However, a plasma was successfully obtained by ECR heating, as shown in Figure 5.7. After the creation of the plasma the grids were turned on in order to provide the extraction of the ions. It was quickly found that the saturation current for the high voltage sources was reached before the full potential could be attained. The maximum potential obtained 51

Figure 5.7. Argon plasma created by the new design of the MMIT with the extraction grids on on the screen grid was 100 V and the current in the screen grid was equal to 20.5 mA at for that value of the potential. After a few minutes in this regime, the plasma disappeared and we were not able to create the plasma again. It was found the plasma sputtered the antenna, the magnets, and the screen grid, and deposited the sputtered material on the walls of the discharge chamber, as shown in Figure 5.8.

Figure 5.8. Damage due to the plasma on the discharge chamber (left) and on the screen grid (right). We can clearly see a metallic coating on the inside of the discharge chamber and the sputtering on the screen grid

This issue is so far unresolved and further testing is being conducted in order to ﬁx that problem.

5.2 Numerical Simulation Results

In order to predict and to understand further the physics involved in the thruster itself, numerous numerical simulations have been performed using COMSOL Multiphysics. This software package allows the user to perform simulations of one physics type and also of 52 multiple physics, for example heat transfer and electromagnetic radiation at the same time. We used the software to perform simulations of the magnetic ﬁeld and also of the electric ﬁeld created by the antenna.

5.2.1 Magnetic ﬁelds

The magnets are one important element of the thruster. In order to visualize more easily the magnetic ﬁeld created by the two concentric magnets and to decide where to place the microwave antenna in order to obtain the best coupling between the magnets and microwave radiations.

5.2.1.1 Equations

COMSOL uses Maxwell’s equations in steady-state, with no current. We therefore obtain

∇ · B~ = 0, (5.1)

∇ × H~ = ~0. (5.2)

Equation 5.8 allows the deﬁnition of a magnetic potential Vm so that

H~ = ∇Vm. (5.3)

COMSOL solves for the magnetic potential Vm using the constitutive relation between the magnetic field flux density and the magnetic field.

5.2.1.2 Geometry

Since only the yoke plate, which is made out of iron, and the magnets themselves modify the local magnetic field, all the other parts of the thruster were ignored. Only the yoke plate and the magnets were modeled. The holes in the yoke plate were ignored to further simplify the model. The model is also axisymmetric and therefore only a 2D view is necessary. Finally a fillet has been added to the corners of the magnets in order to avoid numerical complications due to the finite element aspect of the simulation: corners are usually hard for the finite element solvers to treat, so using a fillet avoids discontinuities and improves the performance of the solver. The simulation has been run for different thicknesses of the yoke plate in order to see if that parameter affects the magnetic field created by the two magnets. The thickness was 53 varied between 0.5 mm and 2.5 mm in steps of 0.25 mm. Figure 5.9 shows the geometry chosen for an intermediate thickness.

Figure 5.9. Geometry chosen for the magnetic ﬁeld simulation

5.2.1.3 Materials

Only one parameter is relevant in the magnetic ﬁeld simulation in COMSOL, the relative magnetic permeability µr. The materials involved in the simulation are air, iron, and the magnets. Table 5.1 shows the chosen relative magnetic permeability of all these materials.

5.2.1.4 Boundary conditions

COMSOL offers numerous boundary conditions for each physics setup. For the magnetic field solving, we chose the outside air boundaries to be Magnetic Insulation ones. The magnets are modeled with a Flux conservation boundary. That boundary allows the user to choose between different relations to model the domain — for example, the B–H curve 54

Table 5.1. Relative magnetic permeability for the magnetic ﬁeld simulation

Material Relative magnetic permeability Air 1 Iron 4000 Magnets 1.05

or the magnetization of the materials. We chose the constitutive relation

B~ = B~r + µ0µrH~ (5.4) to model the magnets. B~r is the remanent ﬂux density, which has a value of 1.3 to 1.32 T for −7 the type of magnets we chose. µ0 is the permeability of vacuum and is given by 4π × 10 −1 H · m . µr is the magnetic permeability of the magnet and is deﬁned in Section 5.2.1.3.

5.2.1.5 Mesh

The mesh was chosen to be fine close to the magnets to have a nice resolution of the magnetic field, and coarser away from the magnets where results are not of our interest. A large number of nodes were required near the fillets in order to avoid corner problems. Figure 5.10 shows the statistics of the meshed domain: 73995 elements were generated, the minimum quality of the elements is 0.65 out of 1, and the average quality is 0.98 out of 1. The histogram also shows a few elements that have a low quality but the overall quality of the mesh is high.

5.2.1.6 Solver

COMSOL sets a solver by default for each physics problem. In the magnetic field case, the default solver is a direct matrix inversion method using the PARDISO scheme. However, this solver proves inefficient to solve for magnetic fields. The solver therefore changed to a preconditioned iterative method, using a Multigrid as the preconditioner and the FGMRES method, a variant of the GMRES one (Generalized Minimal Residual Method). The result was assumed to be reached when the residuals were less than a value of 10−4.

5.2.1.7 Results

For a fairly high number of degrees of freedom — 148,226 — COMSOL obtained results fairly quickly: it takes usually only a few seconds for the method to converge. Figure 5.11 55

Figure 5.10. Quality of the mesh for the magnetic field simulation shows the results for all the different thicknesses tried. We see that the thickness has almost no effect in the region of the magnetic field we are interested in. The flux density always has a value between 0.3 and 0.55 T for that region between the two magnets and in front of them, depending how close one is from the magnets. The increase in the thickness affects the magnetic field inside the yoke plate.

Since the yoke plate thickness does not aﬀect the magnetic ﬂux density, it is possible to easily perform a simulation of the electron cyclotron frequency. Using Equation 2.27 ω and the fact that f(Hz) = 2π in COMSOL, we obtained the electron cyclotron frequency pictured in Figure 5.12. We chose to limit the range of frequencies to a range we can attain with the current equipment. We can see that the frequencies vary importantly along a small distance in the vertical direction: a range of 4 GHz is covered along only 1.5 mm. Given the frequency chosen previously, we also see that we have to place our antenna 2 mm away from the magnets. 56

Figure 5.11. Magnetic flux density (surface plot) and magnetic flux streamlines for all the yoke plate thicknesses. From left to right: 0.5, 0.75, 1 mm on the first line; 1.25, 1.5, 1.75 mm on the second line; and 2.0 2.25 2.5 mm on the third line 57 Contour plot of the electron cyclotron frequency (in GHz) Figure 5.12. 58

5.2.2 Two dimensional antenna simulation

In order to verify that the electric field generated by the antenna is indeed perpendicular to the magnetic field lines as required by Equation 2.51, we performed a two dimensional simulation of the electric field inside the discharge chamber of the thruster, and superimposed the results from the magnetic field simulation. Since the geometry of the domain changed a bit we had to perform a new permanent magnetic field simulation. The parameters for that simulation are exactly the same as the parameters from the previous simulation shown in Section 5.2.1.

5.2.2.1 Equations

For an electromagnetic radiation simulation, COMSOL uses Maxwell’s equations

ρ ∇ · E~ = , (5.5) 0

∂B~ ∇ × E~ = − , (5.6) ∂t ∇ · B~ = 0, (5.7)

∂E~ ∇ × µ−1B~ = J~ + . (5.8) ∂t Dividing by µ and taking the curl of Equation 5.6, we obtain

∂ ∂E~ ∇ × µ−1∇ × E~ = − ∇ × µ−1B~ = −J~ − . (5.9) ∂t ∂t

Assuming that we have an harmonic electromagnetic wave, we have

jωt E~ = E~0 (~r) e , (5.10) where j2 = −1. Ohm’s law also gives

J~ = σE,~ (5.11) where σ is the conductivity of the considered medium. By combining Equations 5.9, 5.10, and 5.11, we obtain ∇ × µ−1∇ × E~ − ω2 − jσω E~ = ~0. (5.12) 59

2 2 Finally, using the fact that ω = k0/µ00, with k0 the wave number, we obtain that −1 ~ 2 jσ ~ ~ ∇ × µr ∇ × E − k0 r − E = 0, (5.13) ω0 where µr and r are the relative permeability and permittivity of the medium considered. COMSOL solves for the three components of the electric ﬁeld from Equation 5.13.

5.2.2.2 Geometry

The geometry is more complete than in the previous section. The antenna of the thruster is modeled with its Teﬂon coating and shield casing. It is encased in a hollow cylinder of MACOR and surrounded by a Faraday cage. Finally a simple plate is used to model the grids. Figure 5.13 shows the geometry of the simulation.

Figure 5.13. Geometry of the electric ﬁeld simulation 60

5.2.2.3 Materials

The materials used for this simulation are air; steel for the Faraday cage, grid, and antenna shield; copper for the antenna itself; iron for the yoke plate; Teflon for the insulator of the antenna; MACOR; and the magnets material. All materials except for the last three are implemented in COMSOL. The electromagnetic simulation requires more materials data than the magnetic field simulation: the relative permeability, relative permittivity, and conductivity of the materials are required. COMSOL provides these properties for air, steel, copper and iron but not for Teflon, MACOR, and the magnets. The electrical conductivity of both Teflon and MACOR are set to 0 S · m−1 since they are supposed to be insulators. The electrical conductivity of the magnets is supposed to be equal to the conductivity of nickel since they are both coated with it. Those properties were found using References [27], [28], and [29]. Table 5.2 summarizes the material properties used.

Table 5.2. Material properties for the two–dimensional antenna simulation

Material Relative Relative Electrical permeability permittivity conductivity (S · m−1) Air 1 1 0 Copper 1 1 5.998 × 107 Iron 4000 1 1.12 × 107 MACOR 1 5.8 0 Magnets 1.05 1 1.62 × 107 Steel 1 1 4.032 × 106 Teﬂon 1 2.07 0

5.2.2.4 Boundary conditions

Though the antenna, antenna shield, Faraday cage, grids, and magnets are really good electrical conductors for which the skin depth would be of the order of a few micrometers, we chose not to use the Perfect electrical conductor boundary condition: the results with and without that boundary condition are indeed exactly the same and not using it decreases the complexity of the simulation. For the outermost air boundaries, we chose a Far Field Domain boundary condition as well as a Scattering Boundary Condition set for cylindrical waves. Finally a Port is included at the bottom part of the teﬂon insulator, with a coaxial type of port, wave excitation on, and an input power of 10 W. 61

5.2.2.5 Mesh

The mesh again was chosen to be really ﬁne near the important edges of the simulation model, such as the antenna or magnet edges. Figure 5.14 shows the statistics of the mesh for that particular simulation. The mesh is made of 55,449 elements with a minimum quality of 0.76 out of 1, and an average quality of 0.98 out of 1. The histogram again shows that the mesh has a good quality despite a few elements with lower quality.

Figure 5.14. Statistics of the meshed domain for the antenna simulation

5.2.2.6 Solver

For the electromagnetic simulation, we chose the solver to be a preconditioned iterative method with a Multigrid as the preconditioner and the GMRES method. The result was assumed to be reached when the residuals were less than a value of 10−3.

5.2.2.7 Results

Given the mesh chosen, COMSOL had to solve for 388,475 degrees of freedom for electromagnetic simulation, and 111,064 degrees of freedom for the new magnetic field simulation. The results, however, are obtained again fairly quickly and no more than 10 iterations are 62 required in both cases for the different methods to converge. Figure 5.15 shows the superimposed results of both the electromagnetic and magnetic field simulation. A contour plot is provided with the electron cyclotron frequency and two streamline plots are shown. The black streamlines correspond to the electric field lines whereas the red streamlines correspond to the magnetic flux density streamlines. We see that from the center of the antenna (r = 0 mm) to the outermost part of the antenna (r = 3.7 mm) the electric field is locally perpendicular to the magnetic field. This ensures a good electromagnetic wave–permanent magnetic field coupling and validates the design of the antenna. 63 Superimposed results from the electromagnetic wave and permanent magnetic field simulation. The black lines correspond to Figure 5.15. the electric field and the red lines to the magnetic flux density. Chapter 6

Conclusion and Recommendations

The goal of this work was to improve the design of a miniature microwave ion thruster previously built by Lubey [11] and to perform tests on the new version of the thruster. The new design is more versatile since it is possible to change various parameters such as the length of the discharge chamber, the position of the antenna with respect to the magnets, the thickness of the back yoke plate on which the magnets are attached, and the type of grids. Two sets of grids were designed: one was used to ﬁrst characterize the discharge and prove that a plasma could be created at low microwave power and low propellant mass ﬂow rate while the second one allows the thruster to work in an actual ion thruster mode and provide ion or electron extraction.

Numerical simulations were performed with COMSOL Multiphysics in order to find the optimum position of the microwave antenna and to validate the design of the antenna. It was found that the front part of the antenna had to be placed 2 mm away from the magnets and that it is able to provide an efficient coupling between the electromagnetic radiation and the permanent magnetic field created by the two concentric magnets. For that particular configuration, the analysis of the thruster on a microwave analyzer showed that a frequency of 5 GHz should be used in order to provide an efficient transfer of energy to the electrons to obtain a plasma through an electron cyclotron resonance heating. It was also found that the thickness of the back yoke plate does not affect the magnetic field configuration. Therefore, a plate with the smallest thickness should be used in order to provide more space for the plasma in the discharge chamber.

Testing is currently being conducted at The Pennsylvania State University in order to perform the extraction of either the ions or the electrons for the new version of the MMIT. 65

A plasma was created by ECR heating for a wide range of mass ﬂow rates. The extraction of either the ions or electrons so far is unsuccessful due to current saturation of the grids when a high potential is applied to them.

Future work should focus on the additional optimization of diﬀerent critical parts of the thruster:

• The coupling between the microwave radiation and the permanent magnetic ﬁeld should be optimized by improving the shape and position of the microwave antenna. In order to increase the ion beam current of the thruster, the shape of the antenna can be changed to a disc for low mass ﬂow rate [30].

• Numerical simulation should be conducted to optimize the size of the holes of the screen and acceleration grids to maximize the grids transparency and to provide the best focusing of the ion and electron beam.

• The diameter and length of the discharge chamber could also be improved to avoid sputtering of either the walls of the discharge chamber, the magnets, the antenna or the screen grid.

The ﬁnal version of the thruster should also make use of samarium–cobalt magnets, which are more resistant to high temperatures generated by the plasma and the grids should be made out of molybdenum. Additionally the propellant should be xenon.

If these goals are met the next step for this thruster is to provide self neutralization. A special set of grids that allow the extraction of both electrons and ions at the same time will be designed in order to perform the neutralization of the ion beam. Bibliography

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