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A discussion is presented on the results and predictive capabilities of a two-dimensional (2D) hybrid Hall effect thruster (HET) model. It is well known that classical (collision-induced) cross-field electron transport and energy losses are not sufficient to explain the observed HET characteristics. The 2D, quasineutral, hybrid discharge model uses empirical parameters to describe additional, anomalous electron transport and energy loss phenomena. It is shown that, for properly adjusted empirical parameters, the model can qualitatively reproduce the observed thruster behavior over a large range of operating conditions. The ionization and transit-time oscillations predicted by the model are described, and their consequences on the time-averaged thruster properties are discussed. Finally, the influence of the empirical parameters on the model results is shown, especially on quantities that can be measured experimentally. © 2004 American Institute of Physics.

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... Later, Hagelaar et al. [58] and Bareilles et al. [59] have also observed this ITT instabilities with a 2D axial-radial hybrid code. During a transit-time oscillation, they found that the position of the acceleration zone oscillates, as seen in figure 1.10: ...

... For example, the early models of Boeuf and Garrigues [49] (1D axial) or Fife [46] (2D axial-radial) were the first to observe the breathing mode and ion transittime oscillations, and the latter is still even used in the community. Later, the ion-transit time instabilities were analysed in more details with the axial-radial hybrid model of Hagelaar, Bareilles et al. [58,59], as described previously in section 1.2.2. Finally, the more recent axial-azimuthal model of Kawashima et al. [94] allowed them to give a new description of the on-set for rotating spokes. ...

... to qualitatively reproduce the experimental ionization oscillations and their parameter scalings, along with a good estimation of the thrust [59]. In the thruster chamber, they considered the effect of the walls by using a different empirical formula ν wall = αν ref , Other empirical models can be found in the literature, such as those based on machine-learning developed recently by Jorns [134], but we have selected only a single representative model here for clarity. ...

Link of the defense: https://youtu.be/V7R1WaD9Bxw
In the last decade, the number of satellites orbiting around Earth has grown exponentially. Thanks to their low propellant consumption, more and more electric thrusters are now used aboard these satellites, with the Hall thrusters being one of the most efficient. From the diversity of applications stems the need of widening the thruster power capabilities. However, due to a lack of knowledge on Hall thruster physics, this scaling is currently done empirically, which limits the efficiency of the newly developed thrusters and increases the development time and cost. To overcome this issue, numerical models can be used but a deeper understanding on key phenomena is still needed, more specifically on the electron anomalous transport which should be self-consistently accounted for to properly capture the discharge behaviour.As this transport is related to the azimuthal electron drift instability, an existing 2D Particle-In-Cell code was further developed to simulate this azimuthal direction along with the axial direction in which the ions are accelerated, producing the thrust. Prior to analyse the discharge behaviour, this code has been verified on a benchmark case, with 6 other PIC codes developed in different international research groups. This simplified case was later used to stress-test previous analytical developments to approximate the instability-enhanced electron-ion friction force which represents the contribution of the azimuthal instabilities to the anomalous transport. Then, the neutral dynamics has been included to capture the full self-consistent behaviour of the discharge. We used an artificial scaling technique, increasing the vacuum permittivity, to relax the PIC stability constraints and speed-up the simulations. Thanks to an efficient code parallelisation, we managed to reduce this scaling factor to a small value, hence simulating a case close to reality. The electron-ion friction force was found to be the main contributor to the anomalous transport throughout the whole low-frequency breathing mode oscillations. Finally, the complex interaction between the breathing mode, the ion-transit time instabilities and the azimuthal electron drift instabilities has been studied, with the formation of long-wavelength structures associated with an enhanced anomalous transport.

... However, the enormous difference in the particle mass of electrons and ions (typically, constituted of heavy xenon atoms) means that full PIC simulations of the stationary plasma response require very large simulation times, often unsuitable for research and development studies. On the contrary, hybrid simulation codes relying on a PIC-MCC formulation for heavy species and a fluid formulation for electrons present an attractive trade-off between simulation cost and reliability of results [9][10][11][12][13]. ...

... where l is the index of the cell centers involved, Φ l are the values at those points, and g ml are the geometric factors of cell l with respect to face m. Using equations (12) and (13) and GRMs, the parallel electric current density at the center of a σ=const face m satisfies ( ) where current densities related to heavy species and provided by the I-module are included in ¢ j m . Acting on all the mesh cells, equation (18) leads to the non-square matrix relation ...

... The different terms of the electron momentum equations are evaluated in figure 7 for case C1A. Panel 7(a) shows that the contribution of heavy species to j e through j c , equation (12), is significant in certain regions of the plasma domain. On the contrary, the contribution of heavy species to the perpendicular electron current density, equation (13), is totally negligible: However, this small Φ is crucial to determine the parallel electron current density in a weakly-collisional electron fluid, both in the present magnetized case and in the unmagnetized one [54]. ...

Plasma discharges in electromagnetic thrusters often operate with weakly-collisional, magnetized electrons. Macroscopic models of electrons provide affordable simulation times but require to be solved in magnetically aligned meshes so that large numerical diffusion does not ruin the solution. This work discusses suitable numerical schemes to solve the axisymmetric equations for the electric current continuity and the tensorial Ohm’s law in such meshes, when bounded by the thruster cylindrical or annular chamber. A finite volume method is appropriate for the current continuity equation, even when meshes present singular magnetic points. Finite differences and weighted least squares methods are compared for the Ohm‘s law. The last method is more prone to producing numerical diffusion and should be used only in the boundary cells and requires a special formulation in the boundary faces. In addition, the use of the thermalized potential is suggested for an accurate computation of parallel electron current densities for very high conductivity. The numerical algorithms are tested in a hybrid (particle/fluid) simulation code of a helicon plasma thruster, for different magnetic fields, mesh refinement, and plume lengths. The different contributions to the electric current density are assessed and the formation and relevance of longitudinal electric current loops are discussed.

... Upon comparison with [30], one will notice similarities in the ion velocity and intensity modulations between the BHT-600 and the Stanford Z-70 Hall thrusters: the main ion population achieves maximum velocity in the current trough and the fluorescence intensity is in phase with the current. This pattern is in agreement with other probe-based breathing mode studies [23] and numerical models [16,17,[78][79][80]. For example, Bareilles et al [78] used a two-dimensional (radialaxial) particle-in-cell (PIC) code to study strong ionization oscillations in a Hall thruster similar to the SPT-100. ...

... This pattern is in agreement with other probe-based breathing mode studies [23] and numerical models [16,17,[78][79][80]. For example, Bareilles et al [78] used a two-dimensional (radialaxial) particle-in-cell (PIC) code to study strong ionization oscillations in a Hall thruster similar to the SPT-100. Figures 6 and 7 of that work, along with the associated discussion, identify the progression of events that are generally understood to give rise to breathing mode oscillations. ...

... A clear picture of the breathing mode emerges from this viewpoint, in agreement with simulation-based studies from the literature [16,17,[78][79][80]. (A particularly effective comparison is again made against figure 7 of Bareilles et al [78], noting that the discharge current peak at 99 μs in that study corresponds to ∼16 μs here.) ...

Several techniques have been developed recently for performing time-resolved laser-induced fluorescence (LIF) measurements in oscillating plasmas. One of the primary applications is characterizing plasma fluctuations in devices like Hall thrusters used for space propulsion. Optical measurements such as LIF are nonintrusive and can resolve properties like ion velocity distribution functions with high resolution in velocity and physical space. The goals of this paper are twofold. First, the various methods proposed by the community for introducing time resolution into the standard LIF measurement of electric propulsion devices are reviewed and compared in detail. Second, one of the methods, the sample-hold technique, is enhanced by parallelizing the measurement hardware into several signal processing channels that vastly increases the data acquisition rate. The new system is applied to study the dynamics of ionization and ion acceleration in a commercial BHT-600 Hall thruster undergoing unforced breathing mode oscillations in the 44-49 kHz range. A very detailed experimental picture of the common breathing mode ionization instability emerges, in close agreement with established theory and numerical simulations.

... The mechanism of these large amplitude low frequency volumetric oscillations has been investigated with simple analytical models of prey-predator type for the neutral gas ionization (0D model) [11][12][13][14], as well as with more sophisticated theories [15][16][17]. The BM appears in numerical simulations even within 1D description [16,18,19] and are unambiguously reproduced by 2D models, e. g. [20]. Over time, kinetic [21], hydrodynamic [22] and hybrid [20] models have been developed. ...

... The BM appears in numerical simulations even within 1D description [16,18,19] and are unambiguously reproduced by 2D models, e. g. [20]. Over time, kinetic [21], hydrodynamic [22] and hybrid [20] models have been developed. ...

The dynamics of the ion flux expelled by a 0 .5 kW -class Hall thruster supplied with krypton was examined in a wide range of discharge voltages. A homemade Faraday probe installed onto a rotary arm was used for reconstructing angular profiles of the plasma plume 0.5 m downstream of the thruster exit plane. The time dependence of the ion current was measured along the thruster axis. For investigating the signal dynamics, a Fourier approach as well as methods of nonlinear time series analysis like bifurcation diagrams and recurrence plot techniques were applied, which are of interest for chaotic behavior identification. Along with the well-known breathing mode ( 10—30 kHz ), other characteristic groups of oscillations were also detected. The bifurcation diagram revealed a drastic transition between large and small amplitude oscillating regimes while varying the discharge voltage from 550 to 700 V . In parallel to this transition, recurrent plots display a qualitative change from a periodic (or quasi periodic) oscillating regime to much less predictable dynamics.

... A second remarkable type of quasi-axial oscillations in the low-frequency range are the ion transit time (ITT) oscillations, so called because they are characterized by periods that are roughly equal to the ion residence time in the chamber [8,9,1]. The prevalent theory for these oscillations links them to an acoustic instability in the ion acceleration region that, by moving upstream, creates an inhomogeneity in the ion velocity profile along the z-axis. ...

... Here, a value α t = 0.02 [46] is set in the full domain for all simulation cases, so that the obtained plasma solution is representative of typical SPT-100 HET experimental data in terms of thrust and specific impulse [47,48]. It is noted however that previous works [8,49] suggest an influence on the ITT oscillations of the anomalous electron transport, and that a more accurate treatment may require varying α t spatially and with the thruster operating point. ...

A data-driven modal analysis of plasma oscillations in a SPT-100-like Hall thruster in the 1–120 kHz range is presented. Data are generated by a two-dimensional (axial-radial) hybrid particle-in-cell/ﬂuid simulation code. While proper orthogonal decomposition is unable to successfully uncouple the diﬀerent dynamics, higher order dynamic mode decomposition (HODMD) cleanly isolates the breathing and ion transit time modes. Indeed, the computed HODMD components can be clustered into two distinct groups, enabling the separate reconstruction of the dynamics of the two oscillation modes. It is also shown that each plasma variable exhibits a diﬀerent behavior in each cluster, namely standing or progressive-wave oscillations. Furthermore, their amplitudes and frequencies depend on the thruster operating point, which is analyzed by varying the discharge voltage and the mass ﬂow rate. This work illustrates the potential of data-processing techniques like HODMD to dissecting the complex physics of Hall thrusters and other propulsion devices.

... These step-out profiles have provided good fittings in previous studies. 39,58,61,62 Hereafter, a particular step-out profile is referred to as (α t1 , α t2 ). ...

... The results are in Table III and Fig. 3. Turbulent transport is larger in the plume by nearly one order of magnitude, in line with the existing literature. Also aligned with previous studies for MS-HETs 13 and traditional HETs, 62 there is a moderate change of the turbulence parameters with the operation point. The parameter α t2 features the largest variation, increasing as V s decreases and _ m A increases. ...

Numerical simulations of a magnetically shielded Hall effect thruster with a centrally mounted cathode are performed with an axisymmetric hybrid particle-in-cell/fluid code and are partially validated with experimental data. A full description of the plasma discharge inside the thruster chamber and in the near plume is presented and discussed, with the aim of highlighting those features most dependent on the magnetic configuration and the central cathode. Compared to traditional magnetic configurations, the acceleration region is mainly outside the thruster, whereas high plasma densities and low temperatures are found inside the thruster. Thus, magnetic shielding does not decrease plasma currents to the walls, but reduces significantly the energy fluxes, yielding low heat loads and practically no wall erosion. The injection of neutrals at the central cathode generates a secondary plasma plume that merges with the main one and facilitates much the drift of electrons toward the chamber. Once inside, the magnetic topology is efficient in channeling electron current away from lateral walls. Current and power balances are analyzed to assess performances in detail.

... The fraction of the multiply charged ion density on the axis of the cylindrical Hall thruster was measured as 37%, and this amount is large enough to promote IITSI. Third, previous studies reported that the appearance of the fast ions was accompanied by the observation of a strong ion transit time instability (ITTI) [38][39][40][41] in the 100-500 kHz range. About 350 kHz oscillations were consistently observed in the discharge current during the experiment, and asymmetry in IVDF appeared with a hot ion population near the thruster axis. ...

... About 350 kHz oscillations were consistently observed in the discharge current during the experiment, and asymmetry in IVDF appeared with a hot ion population near the thruster axis. In addition, numerical simulations reported that ITTI broadens IVDF by generating not only fast ions but also very slow ions [40,41], which were also observed in Fig. 2(a). Both IITSI and ITTI can appear with sufficiently high ion velocity and can be significant after the ions undergo a sufficiently high potential drop. ...

We investigated the structure of the ion acceleration region and the shape of the ion velocity distribution function in cylindrical Hall thruster plasmas, using laser-induced fluorescence spectroscopy on Xe II metastable ions. On the thruster axis, the acceleration front is located deeper than a half-length of the discharge channel length, and the acceleration region reaches up to 3 times the discharge channel length (several centimeters) away from the channel exit, regardless of the discharge condition. It is noteworthy that ion acceleration mostly (more than 70%) takes place outside the discharge channel. The ion velocity distribution function is close to a single Gaussian inside the discharge channel. It however becomes substantially asymmetric when moving downstream. Double Gaussian distributions including cold and hot ion groups was in good agreement with the measured ion velocity distributions downstream with an R-squared greater than 0.995.

... Nous présentons dans cette partie des résultats préliminaires pour plusieurs régimes de fonctionnement du propulseur issus de simulations basées sur un modèle 2D hybride. Celui-ci est dérivé d'un modèle hybride décrit dans [Garrigues 2008], [Hagelaar 2002], [Bareilles 2004] et qui est adapté à la géométrie du propulseur ID-HALL. Dans un premier temps, nous résumons brièvement les principes inhérents à ce modèle. ...

... On peut ainsi réécrire l'expression du coefficient de mobilité anormale (équation (I.71)) comme : En accord avec la plupart des simulations déjà effectuées ( [Hagelaar 2002] [Bareilles 2004] [Boeuf 1998] ]), les paramètres ajustables • et g sont choisis tels que : • = 1 et g = 0 à l'intérieur du canal et dans la chambre d'ionisation ; • = 0 et g = 1 en dehors du propulseur. En suivant ces hypothèses, la fréquence de collision effective et la mobilité des électrons en dehors du propulseur sont plus grandes qu'à l'intérieur du canal avec une discontinuité à l'entrée du canal. ...

Dans un propulseur à courant de Hall, la création des ions et leur accélération sont régis par le même phénomène physique. L'idée du propulseur de Hall double étage (DSHT) est de découpler l'ionisation du gaz (poussée) et l'accélération des ions (ISP), de sorte à rendre le système davantage versatile. Les travaux menés durant cette thèse visent à démontrer, grâce à des essais expérimentaux, la pertinence et la faisabilité d'un tel concept. Dans un premier temps, un prototype de DSHT, baptisé ID-HALL, a été conçu et assemblé. Il est constitué d'une source inductive magnétisée insérée dans un tube en céramique et d'un étage d'accélération identique à une barrière magnétique de propulseur simple étage. La source inductive a été optimisée de sorte à réduire le couplage capacitif et à maximiser l'efficacité du transfert de puissance par ajout de pièces en ferrite et diminution de la fréquence RF d'excitation. Dans un deuxième temps, la source inductive du propulseur a été caractérisée indépendamment du propulseur en argon et xénon pour différentes pressions. Le dispositif expérimental a permis notamment de tracer une cartographie 2D de la densité et de la température. Enfin, le propulseur a été monté dans son caisson et des mesures préliminaires (caractéristiques courant-tension, mesures par sonde RPA) ont été menées. En parallèle, des simulations utilisant un modèle hybride 2D ont été effectuées en mode simple et double étage. Elles mettent en évidence un fonctionnement versatile du moteur pour des tensions inférieures à 150 V. A terme, on visera à démontrer que la densité de courant et l'énergie des ions peuvent être, dans certaines conditions, significativement découplées.

... The axial momentum conservation equation 38 can then be rewritten as follows: [55,56,57], the anomalous mobility is enhanced with an electron-wall collision term representing the global wall effect on the plasma. This additional mobility is not taken into account in this approach. ...

... Wall effects are then represented through an additional mobility in the electron drift-diffusion equation. A detailed description of the model can be found in [56]. The present simulation predicts a lower plasma density. ...

With the increased interest in electric propulsion for space applications, a wide variety of electric thrusters have emerged. For many years, Hall effect thrusters have been the selected technology to sustain observation and telecommunication satellites thanks to their advantageous service lifetime, their high specific impulse and high power to thrust ratio. Despite several studies on the topic, the Hall thruster electric discharge remains still poorly understood. With the increase of available computing resources, numerical simulation becomes an interesting tool in order to explain some complex plasma phenomena. In this paper, a fluid model for plasma flows is presented for the numerical simulation of space thrusters. Fluid solvers often exhibit strong hypotheses on electron dynamics via the drift-diffusion approximation. Some of them use a quasi-neutral assumption for the electric field which is not adapted near walls due to the presence of sheaths. In the present model, all these simplifications are removed and the full set of plasma equations is considered for the simulation of low-temperature plasma flows inside a Hall thruster chamber. This model is implemented in the unstructured industrial solver AVIP, efficient on large clusters and adapted to complex geometries. Electrical sheaths are taken into account as well as magnetic field and majors collision processes. A particular attention is paid on a precise expression of the different source terms for elastic an inelastic processes. The whole system of equations with adapted boundary conditions is challenged with a simulation of a realistic 2D r – z Hall thruster configuration. The full-fluid simulation exhibits a correct behavior of plasma characteristics inside a Hall effect thruster. Comparisons with results from the literature exhibit a good ability of AVIP to model the plasma inside the ionization chamber. Finally a specific attention was brought to the analysis of the thruster performances.

... Typical electron temperatures of a few tens of eV can result in sheath potential drops of few tens of volts when strong electron emission from the ceramic walls occurs [9]. Since magnetic field lines are not purely equipotential, the electric potential lines form a concave length meaning that ions generated close to the walls are accelerated towards them [10], as we can see in figure 1. This results in a high flux of energetic ions colliding with the walls and causing a lot of sputtering. ...

... 4a. Not surprisingly, the minimum of anomalous frequency coincides with the position of the maximum of the magnetic field, very close to the exit plane, as already previously noticed (e.g.[10],[39]). Following Ref.[39], outside the channel, we have limited the anomalous frequency to the cyclotron frequency. ...

... 2,15 The most important ones are the low frequency ionization oscillation, which is also called the breathing mode, and the axial transit time oscillations in the acceleration region. 16,17 These two typical axial oscillations have the frequency range 10-20 and 100-500 kHz. 2 It has been confirmed that these two typical axial oscillations lead to the evident fluctuation in the axial electric fields. 18,19 The influence of the AC-driven ECDI due to time-varied axial fields on the Hall thruster has been studied experimentally and numerically by DesJardin et al. 20 The experimental results have suggested that the AC-driven ECDI could contribute to the self-organized erosion patterns in Hall thrusters. ...

A 2D-3V finite-element particle-in-cell model, which is applied to simulate the radial-azimuthal plane near the exit of Hall thrusters, has been presented to investigate the influence of axial oscillation on electron cyclotron drift instability (ECDI) and anomalous cross field electron transports. The simplified theoretical analysis about the ECDI and the anomalous electron transport has been introduced. The uniform and harmonic axial electric fields, which are based on the typical axial oscillations in Hall thrusters, have been considered in the simulations. It is concluded that different constant axial electric fields can influence the properties of instability but cannot significantly change the cross field electron mobility. However, the axial oscillation plays a significant role in the instability, and the electron transports provided that appropriate amplitudes and frequencies are achieved. The equilibrium of the instability is destroyed and reformed with the axial oscillation. The cross field electron transports are enhanced in the range of low amplitudes and frequencies and are suppressed when they are in a high value. In addition, it is observed that the variation of the electron mobility and electron–ion friction force show high consistency with the trend of electron temperature. It is further confirmed that the increase in electron temperature takes responsibility for the enhanced cross field electron transport due to the axial oscillation.

... The plasma evolution may be described by kinetic models for both the electrons and the ions for the most refined descriptions [14,9,29,5,7,26]. Hybrid models [3] rely on a coarser representation of the electrons, this species being described by a fluid model. Finally the ions may also be described by a fluid model [23,22,36]. ...

... The model is purely electrostatic, since only Poisson's equation is solved for and the magnetic induction field map (radial and axial components) is fixed during the simulation, and taken from Ref. [21]. Therefore, possible magnetic field fluctuations that have been shown [22,23] to additionally enhance the electron transport across the magnetic field lines cannot be captured. ...

A truly self-consistent model for the anomalous transport in Hall thrusters requires a fully kinetic model in three dimensions. In this work, a three dimensional particle-in-cell code is presented. This uses cylindrical coordinates, and includes the most relevant types of particle collisions from the point of view of electron transport, and also secondary electron emission from the walls. The code is parallelized using a hybrid Open MultiProcessing / Message Passage Interface architecture. The results of a preliminary simulation featuring an SPT-100 thruster geometry, and a reduced extension along the azimuthal direction, are presented. These confirm the expected bulk properties trend, and show the expected azimuthal fluctuations that are the main responsible of the anomalous electron transport.

... Energy thresholds of the most important electron-induced collisional processes for the species used in the present work are reported in Table I. The anomalous electron transport contribution coming from the azimuthal fluctuations is taken into account as an additional scattering with a prescribed frequency ano = 0.2 16 taken from [22]. ...

This work represents a first attempt to include the complex variety of electron-molecule collisional processes in a full kinetic PIC model of low power Hall thrusters, with particular emphasis on air species propellants for the air-breathing Hall thruster concept. Results show that N2 and O2 rotational and vibrational excitations have a minor role in comparison to dissociation as additional electron energy loss mechanisms. The dissociation leads to very fast atoms that either cross the channel axially or hit the walls, with very small chance of ionization. By using the standard SPT20 size (channel length and cross sectional area), molecular oxygen shows performances slightly worse than Xe, while molecular nitrogen is a very inefficient propellant. Further studies necessitate to investigate the role of vibrational kinetics and metastable electronic states for stepwise ionization.

... Step-out profiles have performed well in previous studies [34][35][36][37]. Fig. 2(d) shows the axial step-out profile used in this work, where the transition from t1 to t2 takes place close to the maximum magnetic field point along the thruster channel midline. ...

Numerical simulations of the HT20k, a high power magnetically-shielded Hall effect thruster, developed by SITAEL, are carried out with HYPHEN, a two-dimensional axisymmetric hybrid PIC/fluid code with a magnetized and diffusive electron transport model. The model includes a phenomenological anomalous transport parameter, which is adjusted with experimental data. A 2D spatial characterization of the plasma discharge, including profiles along the thruster walls and global balances of current and power, is presented, showing the effectiveness of the magnetic shielding topology. The simulation results show little sensitivity to the plasma-wall interaction parameters. The removal of neutral injection through the cathode leads to a weaker coupling voltage and reduces slightly the performance. When CEX collisions are included in the simulations the performance is not noticeably affected. The sensitivity of the simulation results to the downstream global boundary condition for electric current and heat fluxes is analyzed.

... Over the last few decades, a number of different models, using fluid, 11,15,16 kinetic, [17][18][19][20] and hybrid approaches, [21][22][23] have been proposed to explain these phenomena. Fully kinetic particle-in-cell (PIC) models have been recently used to investigate the role of plasma turbulence induced by kinetic instabilities on the cross-field transport of electrons. ...

Nonlinear interaction between kinetic instabilities in partially magnetized plasmas in the presence of multiply charged ion streams is investigated using kinetic simulations. It was observed by Hara and Tsikata [Phys. Rev. E 102, 023202 (2020)] that the axial ion–ion two-stream instability due to singly and doubly charged ion streams, coupled with the azimuthal electron cyclotron drift instability (ECDI), enhances cross-field electron transport. In the present study, it is observed that the addition of triply charged ions (as a third ion species) contributes to damping of the excited modes, leading to a reduction in the cross-field electron transport. The net instability-driven electron transport is shown to be a function not only of the azimuthal modes, such as the ECDI, but of the multiple ion species that dictate the development of additional plasma waves. It is found that trapping of the higher ion charge states within the plasma waves results in broadening of the ion velocity distribution functions.

... These oscillations have also been reproduced numerically; for example, in the work of Bareilles et al. a 2D hybrid particle-in-cell (PIC) code exhibited an instability related to the ion beam that appeared at similar frequencies. However, here it was related to reciprocation of the acceleration region such that high-and low-velocity ion populations form that upset the steady-state potential structure [23]. This is described as a "surf-riding" effect: when the acceleration region is moving with ions, the ions "ride" this region and reach higher velocities. ...

Hall thrusters can support a wide range of instabilities, many of which remain poorly understood yet are known to play a critical role in the fundamental operation of these devices. In this work, the dominant low-frequency oscillation known as the "breathing mode" is investigated. The goal of this study is to use experimental data to inform a simple model of the breathing mode that could yield an intuitive physical and analytical description of the criteria for the onset and growth of the instability. These criteria could serve as invaluable tools in improving the reliability of Hall thrusters. Foremost, an intuitive physical description of the breathing mode can provide insight into the ramifications of operating with the breathing mode. Such a model can reveal where this instability derives energy and thus which part of the thruster's efficiency is suffering as a result of these oscillations. Additionally, growth criteria can potentially provide insight into the operating conditions and thruster design choices that can minimize these oscillations. That is, if a model can definitively relate the growth rate of the breathing mode to high-level operating parameters, thruster designs can be targeted toward quieter operating conditions. Collectively, this knowledge can be used to intelligently optimize new Hall thrusters. Existing theories of the breathing mode are compared to the collected time-averaged and time-resolved laser data. In examining the scaling of the predicted breathing frequency, positive correlation between the experimental values and those predicted by theory is found, albeit with poor sensitivity. However, a comparison of the dynamic properties of the discharge to those assumed/predicted by theory reveal numerous discrepancies. Ultimately two leading theories for the breathing mode, the classical predator-prey model and a resistive instability, are determined to be incompatible with the measured oscillatory behavior. On the other hand, the data suggests a third possibility: a plasma-driven neutral gas instability. This is substantiated by the observation of neutral drift waves in the thruster channel. The classical zero-dimensional predator-prey model is expanded by the inclusion of more fluctuation terms to increase its fidelity in an attempt to reconcile discrepancies with experiment. Of the models considered, none predict linear instability at self-consistent operating conditions. Two alternative models are proposed that either assume the existence of fluctuations in the ionization region length out of phase with fluctuations in ion density, or assume modulation of the upstream neutral gas flow. Both models are shown to be unstable -- an improvement over the traditional predator-prey model of the breathing mode. Using this theoretical and experimental data, a modified theory of the breathing mode is derived in which coupled ionization instabilities lead to modulation of the neutral gas flow upstream of the traditional ionization region in the thruster. This physical description agrees qualitatively with experimental data. The model retains much of the same properties as the predator-prey model, which is widely accepted to be qualitatively correct. Numerical studies of this model are performed and the existence of unstable roots with reasonable real frequencies is verified. A simplified version of this model is derived to produce straightforward analytical expressions for the real frequency and growth rate of the breathing mode. The high-level trends implied by this simplified model are examined and found to be consistent with empirical scaling relationships.

... This allows the process to repeat as the ions are stepped forward in time again. More information on the hybrid model can be found in references [34][35][36][37]. ...

A typical Hall thruster is powered from a DC power supply and operates with a constant discharge voltage. In operation, the discharge current is subject to strong low frequency oscillations (so-called breathing oscillations). Recent studies have shown that not only can these breathing oscillations be correlated with improved performance, but these oscillations can be induced and controlled by modulating the anode voltage. In this work, a systematic experimental study of the plasma flow in a modulated cylindrical Hall thruster was performed to characterize the effect of natural and modulated breathing oscillations on thruster performance. Measurements suggest that modulating the anode voltage in resonance with the natural breathing frequency does increase the thrust, but a corresponding phase alignment of discharge current and discharge voltage causes the efficiency gains to be insignificant. In addition, the outward shift of the acceleration region causes the plasma plume divergence to increase at the resonance condition and thereby, limit the thrust increase. Mechanisms underlying the relative phase between discharge current, ion current, and discharge voltage are investigated experimentally and corroborated with one-dimensional hybrid simulations of the thruster discharge.

... Several advantages can be obtained with hybrid models that combine different approaches for different species within the same computational framework. Although complex methodologies are available and currently used to carry out 2D (see, for instance, [9][10][11][12][13][14][15][16][17][18]) or even 3D simulations (see, for instance, [19,20]) of Hall thruster discharges (see also [21,22] for recent reviews of the literature), 1D models are still very appealing for studying the fundamental mechanisms in the dynamics of the plasma discharge. Indeed, despite the unavoidable lower accuracy and prediction capabilities in comparison with more complex models, they are easier to handle and definitely cheaper from a computational viewpoint. ...

One of the main oscillatory modes found ubiquitously in Hall thrusters is the so-called breathing mode. This is recognized as a relatively low-frequency (10–30 kHz), longitudinal oscillation of the discharge current and plasma parameters. In this paper, we present a synergic experimental and numerical investigation of the breathing mode in a 5 kW-class Hall thruster. To this aim, we propose the use of an informed 1D fully-fluid model to provide augmented data with respect to available experimental measurements. The experimental data consists of two datasets, i.e., the discharge current signal and the local near-plume plasma properties measured at high-frequency with a fast-diving triple Langmuir probe. The model is calibrated on the discharge current signal and its accuracy is assessed by comparing predictions against the available measurements of the near-plume plasma properties. It is shown that the model can be calibrated using the discharge current signal, which is easy to measure, and that, once calibrated, it can predict with reasonable accuracy the spatio-temporal distributions of the plasma properties, which would be difficult to measure or estimate otherwise. Finally, we describe how the augmented data obtained through the combination of experiments and calibrated model can provide insight into the breathing mode oscillations and the evolution of plasma properties.

... Along with the low-frequency BM peak, we found another spectral content at around 300-400 kHz, which is more important for longer azimuthal domain length L y . This could be evidence of medium-frequency (100-500 kHz) ion transit-time instabilities (ITTI), observed experimentally [43,44,45], numerically [46,47,48,16] and studied theoretically [49,50,51,52,53]. They are related to an instability associated with the transit of ions through the acceleration region. ...

Recent simulations and experiments have observed a transition from short to long-wavelength azimuthal instabilities that leads to enhanced electron transport in Hall thrusters. Here we make the hypothesis that this phenomenon stems directly from the interaction between the axial Ion Transit-Time Instability (ITTI), and the azimuthal Electron Drift Instability (EDI). This interaction is studied using 2D axial-azimuthal self-consistent Particle-in-Cell simulations which include a 1D neutral dynamics solver. It is found that a short to long-wavelength transition only occurs if the Breathing-Mode (BM) and ITTI are captured in the simulation, and two distinct instability regions can be distinguished depending on the local ion Mach number. Upstream of the ion sonic point the EDI exhibits an ion-acoustic behaviour, and the associated instability-enhanced electron transport is well described by a previously developed model based on kinetic theory. Downstream of the ion sonic point however, the ITTI significantly changes the local plasma parameters, and this modifies the EDI while increasing the electron transport.

... Le transport × à travers la barrière magnétique est quant à lui décrit par des coefficients empiriques. Ces coefficients de mobilité effective et de pertes d'énergie sont définis dans les références [12] [61] [62] [63]. Dans ce modèle, il est supposé un équilibre entre la force électrique et le gradient de pression le long des lignes de champ magnétique. ...

Contrairement aux propulseurs chimiques servant à la mise à poste, les propulseurs électriques à courant de Hall sont des moteurs de petite taille utilisés pour le maintien à poste des satellites, le changement d'orbite et les missions interplanétaires. Souvent caractérisés par de faibles poussées, ils ont l'avantage d'avoir une vitesse d'éjection et une impulsion spécifique très importantes. Le principe de fonctionnement est basé sur l'ionisation d'un gaz rare (Xe, Kr) par une différence de potentiel appliquée au travers d'une barrière magnétique. La conductivité électronique localement plus faible dans la barrière conduit à créer un champ électrique dans cette région. Les ions sont alors soumis à ce champ et sont donc accélérés à des vitesses pouvant dépasser plusieurs dizaines de km/s. Le champ électrique au niveau de cette barrière est alors responsable de l'accélération des ions et donc, simultanément, de la poussée et de l'impulsion spécifique. Afin de pouvoir agir indépendamment sur ces deux derniers paramètres, un propulseur à courant de Hall double étage (ID-Hall, Inductive Double stage HALL thruster)) a été développé. Le premier étage est l'étage d'ionisation, constitué d'une source plasma indépendante à couplage inductif, et le second étage est l'étage d'accélération constitué de la barrière. À partir de différents outils de mesures (sonde de flux ionique, analyseur à champ retardateur, caméra haute vitesse, sondes courant-tension, anode segmentée, ...) et d'un modèle numérique (HALLIS), nous avons pu caractériser le plasma, ses instabilités, et les performances du propulseur. Malgré la cartographie magnétique singulière de ce propulseur, les caractéristiques en fonctionnement simple étage sont comparables à celles des propulseurs à courant de Hall classiques. En fonctionnement double étage, la source RF affecte de manière significative le transport des électrons dans le propulseur. De plus, d'autres résultats en double étage montrent qu'à basses tensions de décharge, le courant de décharge est inférieur à celui en simple étage. L'énergie des ions extraits est plus élevée en double étage et le courant d'ion présente une diminution avec l'augmentation de la puissance RF mais reste proche de celui en simple étage. Cette étude a été réalisée en Xénon et en Argon. Des oscillations basses fréquences de grandes amplitudes (Breathing Mode) ont été observées expérimentalement, analysées par sonde résolue en temps et comparées à des résultats obtenus par le modèle. D'autres instabilités azimutales (Rotating Spokes) ont aussi été mises en évidence, ainsi qu'étudiées électriquement et par imagerie.[...]

... This allows the process to repeat as the ions are stepped forward in time again. More information on the hybrid model can be found in references [34][35][36][37]. ...

Hall thrusters contain many instabilities which are known to affect thruster performance, with instability strength varying with thruster regime. One such instability, the breathing mode, can dominate the discharge with current oscillation amplitudes equal to the mean current value. Previous works have shown these large amplitude oscillations can be induced by relatively small oscillations in the discharge voltage by resonating with the natural instability around 10-30 kHz. Theoretical works have also shown that thrust of an oscillating plasma thruster can be throttled and increased if ion energy oscillates with the current. Thrust increases when ion energy oscillations are large and in phase with ion current, while thrust decreases when the two are out of phase. A method was developed to measure time-dependent energy of the ions using a retarding potential analyzer. Measurements of the thruster plume demonstrate the required ion energy oscillations in a Cylindrical Hall thruster. These measurements taken with separate measurements of ion energy provide the ion phase angle, and it is shown that the phase can be controlled by slightly altering the frequency of modulations in the thruster.

... Non-classical electron transport plays a critical role in determining the time-dependent electric field profile, and previous hybrid simulations that considered a static anomalous mobility profile were unable to reproduce measured acceleration zone movements. 34 Earlier hybrid simulations that predicted acceleration zone motion while considering only classical and electron-wall collisions in the channel 16,35 are not directly relevant to magnetically shielded thrusters; however, it is possible that neutral depletion in HERMeS could play a role in moving the electric field peak, as in those simulations, if the dominant turbulent transport becomes negligible at the oscillation phase when E z is the strongest and peaked furthest upstream. 22 The տ 50 kHz oscillation frequency in HERMeS at 500-600 V is unusually high for a large thruster given the classical breathing mode frequency scaling with neutral transit time across the ionization region, 16,17 suggesting either that the nature of the mode is different or the ionization length scale is very short due to the high electron temperature. ...

We present time-resolved laser-induced fluorescence measurements of ion velocity distributions in a 12.5 kW Hall Effect Rocket with Magnetic Shielding (HERMeS) operating in both quasi-periodic and aperiodic oscillation regimes. Transfer function averaging in Fourier space is used to obtain useable signal-to-noise ratios and synchronize data traces taken at different laser wavelengths, measurement axes, and positions in the plasma, achieving a measurement bandwidth of ∼ 100 kHz. For breathing-mode like global oscillations, the results are shown to be robust to the choice of either discharge current I d ( t ) or cathode-to-ground voltage V c g ( t ) as the reference waveform input to the transfer function. At discharge voltage V d = 600 V, a nearly periodic, impulsive oscillation in the acceleration zone position was accompanied by a ≳ 100 V peak-to-peak oscillation in the near-plume plasma potential. Smaller amplitude, aperiodic oscillations in the mean ion velocities were detected at V d = 300 V.

... with ω ce = qB m the electron cyclotron frequency and K an empirical parameter. Hagelaar et al. 40 have used this Bohm approximation in the plume region of a two-dimensional hybrid model and managed to qualitatively reproduce the experimental ionization oscillations and their parameter scalings, along with a good estimation of the thrust 44 . In the thruster chamber, they considered the effect of the walls by using a different empirical formula: ...

Understanding anomalous electron transport in E×B discharges remains a key challenge in the development of self-consistent models of these systems. It has been shown that short-wavelength, high-frequency instabilities in the azimuthal E×B direction may be responsible for increased electron transport due to an enhanced electron-ion friction force. Although a theoretical model based on quasi-linear kinetic theory has previously been proposed to describe this friction force, it has so far only undergone limited validation testing. Here, we rigorously assess this theoretical model by comparison with the friction force self-consistently obtained from 2D axial-azimuthal particle-in-cell simulations. The simulation geometry is based on a recently established benchmark configuration for E×B
discharges, and a broad parametric study is performed by varying the magnetic field strength, the discharge current density, and the presence of different neutral collisional processes. Overall, the theory is found to be in very good agreement with the simulation results for all cases studied, verifying the underlying physical mechanisms leading to enhanced electron transport. We demonstrate, however, that the friction force depends sensitively on the shape of the electron velocity distribution function, thus posing significant challenges to fully self-consistent, first principles modeling of anomalous transport in fluid simulations.

... The coupling between refined models of electric propulsion systems and larger scale plume-interaction tools, validated with measurements, is crucial for the electric propulsion community. 199,200 IV. FRONTIERS AND OUTLOOK A. What does the future hold for electric and plasma propulsion technology? ...

There are a number of pressing problems mankind is facing today that could, at least in part, be resolved by space systems. These include capabilities for fast and far-reaching telecommunication, surveying of resources and climate, and sustaining global information networks, to name but a few. Not surprisingly, increasing efforts are now devoted to building a strong near-Earth satellite infrastructure, with plans to extend the sphere of active life to orbital space and, later, to the Moon and Mars if not further. The realization of these aspirations demands novel and more efficient means of propulsion. At present, it is not only the heavy launch systems that are fully reliant on thermodynamic principles for propulsion. Satellites and spacecraft still widely use gas-based thrusters or chemical engines as their primary means of propulsion. Nonetheless, similar to other transportation systems where the use of electrical platforms has expanded rapidly, space propulsion technologies are also experiencing a shift toward electric thrusters that do not feature the many limitations intrinsic to the thermodynamic systems. Most importantly, electric and plasma thrusters have a theoretical capacity to deliver virtually any impulse, the latter being ultimately limited by the speed of light. Rapid progress in the field driven by consolidated efforts from industry and academia has brought all-electric space systems closer to reality, yet there are still obstacles that need addressing before we can take full advantage of this promising family of propulsion technologies. In this paper, we briefly outline the most recent successes in the development of plasma-based space propulsion systems and present our view of future trends, opportunities, and challenges in this rapidly growing field.

... These oscillations can be very powerful with almost 100% of the steady-state values of discharge current and are usually non-stationary and semi-coherent in time. Such strong oscillations may result in unstable thruster operation and cause degradation of the thruster performance, and the reduction of the thruster lifetime [7][8][9][10]. The physical mechanism responsible for breathing oscillations is usually attributed to some sort of ionization instability [11][12][13]. ...

... It is well known that the collisional electron mobility perpendicular to the magnetic field is not sufficient to explain the experimental results, and, as in most hybrid models of Hall thrusters, we use effective electron collision frequencies to describe the role of turbulence or electron-wall interaction on electron transport across the magnetic field. These effective collision frequencies are defined by adjustable coefficients as described in previous studies [29][30][31][32] and summarized below. The effective momentum transfer collision frequency used in the electron mobility perpendicular to the magnetic field is taken as ...

In Hall thrusters, ions are extracted from a quasineutral plasma by the electric field induced by the local drop of electron conductivity associated with the presence of a magnetic barrier. Since the electric field is used both to extract and accelerate ions and to generate the plasma, thrust and specific impulse are not independent in a Hall thruster. There is a need for versatile thrusters that can be used for a variety of maneuvers, i.e., that can operate either at high thrust or at high specific impulse for a given power. The double stage Hall thruster (DSHT) design could allow a separate control of ionization and acceleration, and hence separate control of thrust and specific impulse. In the DSHT configuration, a supplementary plasma source (ionization stage), independent of the applied voltage, is added and placed upstream of the magnetic barrier (acceleration stage). The DSHT concept is also well adapted to the use of alternative propellants, lighter and with a less efficient ionization than xenon. Several designs of double stage Hall thrusters have been proposed in the past, but these attempts were not really successful. In this paper, we present a brief review of the main DSHT designs described in the literature, we discuss the relevance of the DSHT concept, and, on the basis of simple physics arguments and simulation results, we propose a new design, called ID-HALL (Inductive Double stage HALL thruster). In this design, the ionization stage is a magnetized inductively coupled RF plasma. The inductive coil is inside the central cylinder of the thruster and located nearby the acceleration stage. Preliminary modeling results of this DSHT are described.

Low-temperature E×B plasmas are used in various applications, such as Hall thrusters for satellite propulsion, ion sources and magnetron discharges for plasma processing, and negative ion sources for neutral beam injection in fusion. The plasmas in these devices are partially magnetized, meaning that the electrons are strongly magnetized while the ions are not. They are subject to various micro- and macro-instabilities that differ significantly from instabilities in fusion plasmas. These instabilities are often triggered by the large difference in electron and ion drift velocities in the E×B direction. The possibility of maintaining a large electric field in the quasineutral plasma of Hall thrusters despite anomalous electron transport, or the presence of strong double layers associated with the azimuthal rotation of plasma structures (“rotating spokes”) in magnetron discharges and Hall thrusters are examples of the very challenging and exciting physics of E×B devices. The turbulence and instabilities present in E×B plasma devices constitute a major obstacle to the quantitative description of these devices and to the development of predictive codes and are the subject of intense research efforts. In this tutorial, we discuss the key aspects of the physics of low-temperature partially magnetized E×B plasmas, as well as recent advances made through simulations, theory, and experiments in our understanding of the various types of instabilities (such as gradient-drift/Simon-Hoh and lower hybrid instabilities, rotating ionization waves, electron cyclotron drift instability, modified two-stream instability, etc.) that occur in these plasmas.

The modeling of neutral atoms is important for the full-particle simulations of Hall thrusters. In previous studies, researchers have developed various algorithms to model the neutral kinetics. The choice of those algorithms can influence significantly the computational speed, simulation convergence, and physical results. In this work, we perform a full-particle simulation of a typical 1kW-class SPT-100 Hall thruster using four neutral algorithms, including the fixed-neutral algorithm (FNA), the algorithm of direct simulation of Monte Carlo (DSMC), the collisionless-neutral algorithm (CLNA), and the fluid algorithm (FA), to analyze the effects of different neutral iteration approaches on the simulation results. We found that FNA is sensitive to the initial number density of neutrals, and is difficult to converge properly, while the other algorithms not neglecting the atomic dynamics can get stable results. We count the parameters of the thruster, that is, thrust, specific impulse, and plasma density using different neutral algorithms. The time-averaged results match well with those of the experiment. However, the results differ in the time scale due to the low-frequency oscillations in Hall thrusters. We verify that the oscillations are due to the periodic change of neutrals and establish a zero-dimensional model to analyze the properties of the oscillations in the time scale. It indicates that the ratio of ion migration to neutral migration is the essential factor that significantly affects the calculation results. The model also figures out that the direct neutral iteration methods, like DSMC and CLNA, can better simulate the characteristics of discharge fluctuations in Hall thrusters than the quasi-steady-state method, like FA. Finally, we proposed practical suggestions for the selection of the neutral algorithms for the SPT-100 thruster, which can also be generalized to other low- and medium-power Hall thrusters.

It is assumed that the shift of a strong magnetic field region with a positive gradient from exit plane to outside, namely the transit from a normal loaded magnetic field to an aft-loaded one, enhances the multiple ionization process in the magnetically shielded Hall thruster. To confirm this conjecture, a comparative study is carried out numerically with a particle-in-cell method. The simulation results prove that compared with the normal loaded magnetic field, the application of aft-loaded magnetic field enhances the multiple ionization process. This study further analyzes the ionization characteristics of the transition from low-charged ions to high-charged ions under two magnetic field conditions and the influence of the magnetic strength of aft-loaded magnetic field on the multiple ionization characteristics. The study described herein is useful for understanding the discharge characteristics of Hall thruster with an aft-loaded magnetic field.

In order to investigate the temporal characteristics of the plasma in a Hall thruster with a magnetically shielded configuration, a 2-D cylindrical PIC-MCC model of the Hall thruster is established. The plasma parameters and the sputtering erosion related parameters are derived and obtained. The results indicate the designed combination of the magnetic field and the channel walls remains relatively excellent propulsion performance while achieves near-zero erosion of the channel walls when the anode voltage is 350 V. The results can be utilized for the design improvement of new Hall thruster with optimized propulsion performance and prolonged lifetime. KeywordsHall thrusterPIC-MCCPlasmaSputtering erosion

The effects of particle collisions in a Hall thruster on plasma distributions, performance parameters, and anomalous electron transport are analyzed using a two-dimensional axisymmetric hybrid PIC-DSMC (particle-in-cell direct simulation Monte Carlo model) in-house code. The numerical model is validated by analyzing three simulation cases: without molecular collisions, with charge exchange collisions, and with both charge exchange and momentum exchange collisions. Charge exchange collision is one of the physical processes producing low-energy ions. The effects of charge exchange collisions on the anomalous electron transport are investigated by evaluating the ratio of the production rate of charge exchange ion to the total ion production rate.

Investigating the ion dynamics in the emerging bipolar pulse high power impulse magnetron sputtering (BP-HiPIMS) discharge is necessary and important for broadening its industrial applications. Recently, an optimized plasma source operating the BP-HiPIMS with an auxiliary anode and a solenoidal coil is proposed to enhance the plasma flux and energy, named as ACBP-HiPIMS (‘A’－anode, ‘C’－coil). In the present work, the temporal evolutions of the ion velocity distribution functions (IVDF) in BP-HiPIMS and ACBP-HiPIMS discharges are measured using a retarding field energy analyser (RFEA). For the BP-HiPIMS discharge, operated at various positive pulse voltages U+, the temporal evolutions of IVDFs illustrate that there are two high-energy peaks, E1 and E2, which are both lower than the applied U+. The ratio of the mean ion energy Ei,mean to the applied U+ is around 0.55－0.6 at various U+. In ACBP-HiPIMS discharge, the IVDF evolution shows three distinguishable stages which has the similar evolution trend with the floating potential Vf on the RFEA frontplate: (i) the stable stage with two high-energy peaks (E2 and E3 with energy respectively lower and higher than the applied U+ amplitude) when the floating potential Vf is close to the applied positive pulse voltage; (ii) the transition stage with low-energy populations when the Vf drops by ~20 V within ~10 μs; and (iii) the oscillation stage with alternating E2 and E3 populations and ever-present E1 population when the Vf slighly descreases unitl to the end of positive pulse. The comparison of IVDFs in BP-HiPIMS and ACBP-HiPIMS suggests that both the mean ion energy and high-energy ion flux have been effectively improved in ACBP-HiPIMS discharge. The formation of floating potential drop is explored using the Langmuir probe which may be attributed to the establishment of anode double layer structure.

The gas inlet configuration has a direct impact on the neutral density pattern within a miniature ion thruster. We aimed to investigate the impact the gas inlet configuration and neutral density pattern will have on the neutral recycle rate within a miniature ion thruster utilizing a disk shaped antenna. Four inlet configurations were considered for this study, and 3D electromagnetic particle-in-cell simulation was utilized to simulate the plasma inside the discharge chamber. The simulation results indicate a clear shift in neutral density toward the inlet, with the single horizontal inlet configuration having a 45% increase in neutral density in the vicinity of the inlet walls. The neutral recycle rate also experienced a clear shift toward the inlet walls, with the single bottom inlet configuration experiencing a 23% increase in the rate of ion loss near the inlet wall and a similar 22% increase for the single horizontal inlet while the four smaller inlets had a similar rate of neutral recycling throughout. These results are a novelty in this field as they clearly indicate the impact gas inlet and neutral density have on the miniature ion thruster’s performance and open a new area of research to further optimize the gas inlet configuration for miniature ion thrusters.

An extended Kalman filter (EKF) is developed to estimate unobserved states and parameters in plasma dynamical systems. Physical constraints are satisfied by adapting the process and measurement noise covariances to account for consistency between the estimates and the physical processes. First, the EKF is tested using the Lorenz system to demonstrate the robustness of the EKF with sparse measurement data. Then, the capabilities of the EKF are applied to investigate discharge current oscillations in a Hall effect thruster. It is demonstrated that the dynamics of the electron temperature can be estimated using the discharge current fluctuation as the measurement data. The propagation of the uncertainties of such estimates is also quantified.

View Video Presentation: https://doi.org/10.2514/6.2021-3398.vid A data-driven modal analysis of low frequency oscillations in a SPT-100-like Hall thruster in the 1--100 kHz range is presented. The used data were generated with a two-dimensional (axial-radial) particle-in-cell (PIC)/fluid simulation code. While proper orthogonal decomposition is unable to uncouple the different dynamics successfully, (higher order) dynamic mode decomposition (HODMD) cleanly isolates the breathing and ion-transit-time modes. Additionally, it facilitates separating transients from the attractors in the data. Thus, the resulting HODMD components can be clustered into two distinct groups with multiple harmonics, enabling the reconstruction of the dynamics of the breathing and ion-transit modes. We show that each plasma variable exhibits a different behavior in each cluster (global oscillations or traveling-wave oscillations), and their amplitudes and frequencies depend on the thruster operating point. This work illustrates the potential of the applied techniques to dissect more complex physics of Hall thrusters and other devices.

In this paper, the nonlinear interaction between kinetic instabilities driven by multiple ion beams and magnetized electrons is investigated. Electron diffusion across magnetic field lines is enhanced by the coupling of plasma instabilities. A two-dimensional collisionless particle-in-cell simulation is performed accounting for singly and doubly charged ions in a cross-field configuration. Consistent with prior linear kinetic theory analysis and observations from coherent Thomson scattering experiments, the present simulations identify an ion-ion two-stream instability due to multiply charged ions (flowing in the direction parallel to the applied electric field) which coexists with the electron cyclotron drift instability (propagating perpendicular to the applied electric field and parallel to the E×B drift). Small-scale fluctuations due to the coupling of these naturally driven kinetic modes are found to be a mechanism that can enhance cross-field electron transport and contribute to the broadening of the ion velocity distribution functions.

We present time-resolved laser-induced fluorescence measurements of ion velocity distributions in a 12.5 kW Hall Effect Rocket with Magnetic Shielding (HERMeS) operating in both quasi-periodic and aperiodic oscillation regimes. Transfer function averaging in Fourier space is used to obtain useable signal-to-noise ratios and synchronize data traces taken at different laser wavelengths, measurement axes, and positions in the plasma, achieving a measurement bandwidth of ∼100 kHz. For breathing-mode like global oscillations, the results are shown to be robust to the choice of either discharge current Id(t) or cathode-to-ground voltage Vcg(t) as the reference waveform input to the transfer function. At discharge voltage Vd=600 V, a nearly periodic, impulsive oscillation in the acceleration zone position was accompanied by a ≳100 V peak-to-peak oscillation in the near-plume plasma potential. Smaller amplitude, aperiodic oscillations in the mean ion velocities were detected at Vd=300 V.

A direct kinetic (DK) simulation is capable of modeling the nonequilibrium state of plasma as it evolves in the discharge region of a Hall thruster without the numerical noise that is inherent to particle-based methods since the velocity distribution functions are obtained in a deterministic manner. In this work, a hybrid-DK simulation utilizes a quasi-one-dimensional fluid electron model in conjunction with a two-dimensional DK method to simulate neutral atoms and ions in a Hall thruster channel and near-field plume. Instantaneous and time-averaged plasma properties calculated using the hybrid-DK simulation are benchmarked against the results obtained from a two-dimensional hybrid-particle-in-cell (PIC) simulation with an identical fluid electron model. For both high and low-frequency oscillations, the two simulations show good agreement for time-averaged and dynamic plasma properties. Numerical noise tends to randomize plasma oscillations in the PIC simulation results, whereas the DK results exhibit coherent oscillatory behavior.

The formation of self-organized standing wave structures is observed due to the electron cyclotron drift instability (ECDI) driven by a time-varying external electric field and a crossed magnetic field. Using a particle-in-cell simulation, two standing wave mechanisms are identified: the linear mode and the beating mode. In the former, a standing wave emerges as a superposition of two counterpropagating ion acoustic waves predicted by the linear theory of ECDI. On the other hand, in the beating mode, the plasma wave is in resonance with the applied frequency. Nonlinear resonance of such standing waves results in a change in the dominant wavenumber and frequency. Such counterpropagating plasma waves are consistent with the experimental observations using coherent Thomson scattering in a crossed-field plasma discharge.

In the last 30 years, numerical models have revealed different physical mechanisms involved in the Hall thruster functioning leading to a bridge between analytical prediction/empirical intuition and experiments. For this reason, the need for a model to study Hall thruster operation continues to increase. Two basic approaches exist: one based on fluid/hybrid simulation where the velocity distribution of electrons is prescribed and the plasma inside the thruster, considered as quasineutral, is described with macroscopic quantities (density, velocity and energy), with unmagnetized ions being considered as collisionless; the second approach is based on a kinetic description for charged particles where no approximations are made regarding their velocity distributions. Fluid or hybrid approaches offer the advantages of computational efficiency with modest hardware requirements. They are very useful to perform parametric studies but actually the anomalous phenomena believed to be responsible for electron transport across the magnetic field barrier have not been self-consistently modeled using a fluid approach. A kinetic approach is able to better capture phenomena originating on the Debye scale length like the lateral sheaths, E × B electron drift instability, and it is important to explain the anomalous electron transport, but kinetic simulations require very long run times. For the latter, the advances in computer hardware over the past years have allowed researchers to perform simulations under conditions closer and closer to the actual thruster operation. In this review, we will present two approaches, with emphasis on numerical schemes used with assumptions and approximations and the main results obtained. Future directions in the Hall thruster modeling will finally be outlined.

The time-resolved cross-field electron anomalous collision frequency in a Hall thruster is inferred from minimally invasive laser-based measurements. This diagnostic is employed to characterize the relationship between the dominant low-frequency "breathing" oscillations and anomalous electron transport mechanisms. The ion Boltzmann equation combined with a generalized Ohm's law is used to infer key quantities including the ionization rate and axial electric field strength which are necessary in computing the total electron cross-field collision frequency. This is accomplished by numerically integrating functions of velocity moments of the ion velocity distribution function measured with laser-induced fluorescence, in conjunction with current density measurements at a spatial boundary. Estimates of neutral density are used to compute the classical collision frequency profile and the difference in the total collision frequency, and this quantity describes the anomalous collision frequency. This technique reveals the anticipated trends in electron transport: few collisions in the acceleration region but a collision frequency approaching the cyclotron frequency farther downstream. The time-resolved transport profiles indicate that the anomalous collision frequency fluctuates by several orders of magnitude during a breathing cycle. At troughs in the discharge current, classical collisions may dominate; at peaks in the discharge current, anomalous collisions dominate. These results show that the breathing mode and electron transport are directly correlated. This finding is discussed with regard to existing numerical models for the breathing mode and interpretations of anomalous electron transport.

The results of 1-D azimuthal, and 2-D z-θ, particle-in-cell simulations of Hall thruster acceleration and near plume regions are reported. The 1-D simulations show that fast moving electron waves drive coherent, large amplitude, ion acoustic waves. The ion waves become azimuthally coherent too quickly for ion sound to be the cause; the coherence comes from early electron waves. 2-D simulations with only fast moving, main beam, ions result in ion acoustic wave amplitudes much lower than those in the 1-D calculations. The addition of a small amount of ionization in the domain, at a rate consistent with Hall thruster parameters, causes the potential profile to divide into two distinct regions. In the upstream portion of the domain, the potential profile is very steep; in the downstream portion, the potential profile is almost flat. Electron transport in the two regions is discussed.

Symbolic regression is applied to find a data-driven model for the anomalous cross-field electron transport in a Hall effect thruster. This model is formulated in terms of an anomalous electron collision frequency that is related to the cross-field electron transport through a generalized Ohm's law. Empirically determined estimates of this anomalous collision frequency as a function of local plasma parameters from three 1-6 kW class Hall effect thrusters form the training dataset for this investigation. A commercially-available, evolutionary genetic algorithm is applied to regress this dataset and identify models for the anomalous collision frequency that are expressed as symbolic functions of the local plasma properties. It is found that these data-driven models not only fit the training dataset but that they can predict anomalous collision frequency values for a test dataset taken from a fourth thruster not used in the initial regression. Five existing models for the anomalous collision frequency derived from first-principles are applied to the same training and test datasets used for the data-driven model. The estimates of the anomalous collision frequency as a function of local plasma parameters from the data-driven models are shown to exhibit improved quantitative agreement with both datasets compared to the analytical models. These findings are discussed in terms of the physical insight they yield for identifying dominant physical processes that govern electron transport as well as the practical application of using this technique for creating predictive Hall thruster models.

The discharge and plasma plume characteristics of the cylindrical Hall thruster were studied in regimes with external modulations of the applied voltage. It is found that the amplitude and the root-mean-square (RMS) value of the discharge and ion currents increase with the amplitude of the external modulation exhibiting two different regimes. For smaller amplitudes of the modulation voltage, the oscillations amplitude and RMS value of the discharge and ion currents follow the amplitude of modulations approximately linearly. For larger voltage modulations, the amplitude and the RSM value of the discharge and ion currents grow faster and nonlinearly. In the non-linear regime, the discharge and the ion currents demonstrate pronounced dependence on the frequency of the external modulations. Moreover, the RMS value of the ion current is amplified stronger than the RMS value of the discharge current resulting in an increase of the current utilization (on about 5%) and the propellant utilization efficiencies (on about 40%). The thruster efficiency, defined as a product of the current and propellant utilization coefficients, shows an increase of the about 20%. We also present the results of theoretical modelling of a plasma response to driven oscillations in a simplified 1D model of resistive-ionization mode in quasineutral plasma. This modelling demonstrates the nonlinear property of the fundamental breathing mode similar to the experimental results.

There are still many missing elements to complete the physical picture at the basis of the Hall thruster functioning. The origin of the anomalous electron cross-field transport often ascribed to azimuthal electron E × B drift instability remains decoupled from self-consistent ion axial acceleration and radial boundary conditions, at the same time. This study represents the first attempt to correlate the different mechanisms contributing to the electron transport by means of a fully kinetic three-dimensional Particle-in-Cell model. A geometrical scaling scheme has been used to make the simulation possible. This scheme irremediably changes what are some salient characteristics of the discharge, such as the wall interaction and the axial component of the electric field. For this reason, a critical assessment of the effects of reducing dimensions has been addressed. The present paper deals with the physics of discharge channel. Results confirm the occurrence of E × B drift instability along the azimuthal direction. The modulation is almost standing wave: it moves back and forth travelling only a short distance before being axially convected away. In addition, the dielectric floating potential nature of the lateral walls gives to the azimuthal modulation an important radial component creating an oblique pattern in the radial-azimuthal plane. As a consequence, the azimuthal electric field presents a double alternating structure: two phase-opposing waves are present in the first and second half of the radial extension between the two lateral walls. Finally, the effect of secondary electron emission from walls is not sufficient to guarantee the right electron current to neutralize the ion beam, but rather it works as an auxiliary mechanism (together with ion heating and azimuthal rotation) to saturate the electron drift instability leading to smaller amplitude oscillations.

The main physical features and processes determining stationary plasma thrusters (SPTs) performance levels are considered in this paper, including ionization processes and ion dynamics in the accelerating channel, as well as the results of SPT design optimization, factors determining SPT lifetime, and the possibilities of simulating the plasma particle dynamics in the accelerating channel and in its plume.

Closed drift thrusters are reviewed. The publications on these thrusters constitute a large body of information. This article can therefore include only the most prominent theoretical and experimental features of closed drift thrusters. In some regards, this article is also an attempted synthesis of the differing views of these thrusters found in literature, as well as in our own work.
In a closed drift thruster, the electric field that accelerates the ions is established by an electron current that passes through and is impeded by a magnetic field. The precessing electrons in this magnetic field follow a closed drift path giving this thruster its name. Closed drift thrusters are divided into magnetic layer and anode layer types, based both on the geometrical and material differences in the discharge channels of the two types, and on the different physical processes that take place within the discharge plasma.
Considered as a whole, the publications on closed drift thrusters constitute an impressive body of information that, for the most part, was generated in Russia independently of US research on electric thrusters.

Simulations and experimental characterizations of a stationary plasma thruster are compared for four different wall materials to investigate near-wall conductivity dielectric materials and in-wall conductivity conducting materials in such a discharge. Using a one-dimensional transient fluid model that takes into account a possible electron temperature anisotropy, it is shown that electron-wall backscattering plays a crucial role by maintaining a relatively high electron temperature along the magnetic field lines which in turn drives large electron currents toward the walls. The large differences in discharge current observed experimentally for the dielectric materials are qualitatively recovered, confirming that near-wall conductivity results from the combined effects of secondary electron emission and electron backscattering. A clear correlation is found between the appearance of space charge saturation at the walls and a jump of the discharge current observed in experiments when varying the discharge voltage or the magnetic field. The anomalously high values of discharge current observed experimentally with graphite are also correctly recovered in simulations, which highlight a plasma short-circuiting effect resulting from in-wall currents. © 2003 American Institute of Physics.

Stationary Plasma Thrusters (SPTs) are ion sources used for
satellite propulsion. They operate at low gas density and the electrons are
confined by a magnetic field perpendicular to the discharge axis.
Experimental observations and recent simulations (hybrid models) based on a
simple description of electron transport have shown that SPTs are subject
to strong current oscillations due to ionization instabilities. In this
paper we present comparisons between the predictions of the simple hybrid
model and those from a more detailed PIC-MCC (particle-in-cell Monte Carlo
collisions) simulations. The good qualitative agreement between hybrid and
PIC models confirms the interpretation of the low-frequency oscillations
deduced from our previous work.

A comprehensive analysis of measurements supporting the presence of anomalous cross-field electron mobility in Hall plasma accelerators is presented. Nonintrusive laser-induced fluorescence measurements of neutral xenon and ionized xenon velocities, and various electrostatic probe diagnostic measurements are used to locally determine the effective electron Hall parameter inside the accelerator channel. These values are then compared to the classical (collision-driven) Hall parameters expected for a quiescent magnetized plasma. The results indicate that in the vicinity of the anode, where there are fewer plasma instabilities, the electron-transport mechanism is likely elastic collisions with the background neutral xenon. However, we find that in the vicinity of the discharge channel exit, where the magnetic field is the strongest and where there are intense fluctuations in the plasma properties, the inferred Hall parameter departs from the classical value, and is close to the Bohm value of (omega(ce)tau)(eff) approximately 16. These results are used to support a simple model for the Hall parameter that is based on the scalar addition of the electron collision frequencies (elastic collision induced plus fluctuation induced), as proposed by Boeuf and Garrigues [J. Appl. Phys. 84, 3541 (1998)]. The results also draw attention to the possible role of fluctuations in enhancing electron transport in regions where the electrons are highly magnetized.

'Two-dimensional numerical model of plasma flow in a Hall thruster has been made to estimate analytically the ion-loss flux to the walls of an acceleration channel, and to obtain information about desirable configurations for good thruster performance. The model presented herein is comprised of an electron diffusion equation and an ion kinetic equation, which enables one to compute electrostatic potential contours and ion-beam trajectories. In the first step ion-production distribution was assumed. From the results it was found that electric-field distortion, which is a main cause of ion-loss to the channel walls, is induced not only due to the curvature of magnetic field lines, but also due to the radial nonuniformity of ion-production distribution. In the second step, the ion-production distribution was self-consistently determined by combining an energy conservation equation with the previous two basic equations. The results indicate that the shape of ion-production distribution largely changes with the magnetic field geometry, and hence, the field geometry significantly influences the ion-loss flux to the channel walls. The computed ion-loss fraction (a fraction of ions produced that are lost to the walls) ranges from 0.30 to 0.55, and shows good agreement with the measured values. Therefore, this model should be an effective tool in both the design and improvement of Hall thrusters.

Stationary plasma thrusters (SPTs) are advanced propulsion devices that use a gas discharge to ionize and accelerate the propellant. We present simulation results obtained with a two-dimensional hybrid model of an SPT discharge. The model characterizes the ill-understood anomalous electron transport in SPTs by empirical parameters, of which we demonstrate the influence on the simulation results. Although no optimal values for these parameters can clearly be identified, the model predicts many features of the SPT behavior and yields interesting insights in the SPT physics. Experimentally observed electric potential distributions can only be reproduced if the anomalous electron transport is assumed to be stronger outside than inside the SPT channel. The simulations reproduce experimentally measured oscillations at 10–20 kHz and predict additional oscillations at 100–200 kHz. We discuss the dynamics of these oscillations and their influence on the energy distribution of the ion beam leaving the thruster. © 2003 American Institute of Physics.

A stationary plasma thruster is experimentally studied using different optical spectroscopies of xenon ions. Doppler shift in laser induced fluorescence is used for velocity determination while the ion density is determined by emission spectroscopy. These experiments show unambiguously that the ionization and the acceleration zones are spatially distinct inside the thruster channel. Moreover, it is shown that these results can be easily taken into account with a very simple quasineutral stationary one-dimensional model. © 2002 American Institute of Physics.

Stationary plasma thrusters are devices that use crossed electric and magnetic fields to accelerate ions to high velocities. Ions are created by collisional ionization of a propellant gas with electrons injected from a hollow cathode external to the thruster. A major issue is the electron transport through the magnetic field. It is known to exceed considerably the values predicted by the classical theory. Various 2D models have shown that wall collisions, which have often been invoked as the origin of this anomalous transport, are in fact insufficient. Anomalous turbulent transport has to be added to the model to recover an adequate conductivity. In the present paper the first 2D kinetic model that shows that, indeed, plasma turbulence can explain the observed conductivity is presented. Without any free parameter the model is able to reproduce numerous experimental features. At the end of the paper a preliminary theoretical analysis of the observed instability is provided. © 2004 American Institute of Physics.

Stationary plasma thrusters (SPTs) are advanced propulsion devices that use a gas discharge to ionize and accelerate the propellant. We present in detail a two-dimensional model of an SPT discharge. The model combines a particle simulation of neutral atoms and ions with a fluid description of electrons, where the electric field is obtained from imposing quasineutrality. The electron mobility and energy loss are treated in an empirical way and characterized by ad hoc parameters. Typical simulation results are shown. © 2002 American Institute of Physics.

Stationary plasma thrusters are ion thrusters whose properties make them especially suitable for satellite station keeping or orbit transfer. In these thrusters, a magnetic field transverse to the electron flow towards the anode increases the electron collision frequency and makes possible the generation of a plasma at relatively low gas flow and gas density. The decrease of the plasma conductivity due to the magnetic field induces a large electric field in the plasma which accelerates the quasicollisionless ions whose trajectories are not significantly affected by the magnetic field. The purpose of this article is to clarify, using results from a simple model, the electrical properties of these thrusters and the low frequency oscillation regime which has been observed experimentally. The model is based on the assumption of quasineutrality of the plasma column and on a 1D transient hybrid treatment of electron and ion transport in the device. Electrons are considered as a fluid and ions are described with a collisionless kinetic equation. This model provides reasonable estimates of the plasma properties and is able to give a clear picture of the low frequency oscillations, qualitatively close to the experimental observations. © 1998 American Institute of Physics.

A joint programme, involving research laboratories from CNRS (Centre National de le Recherche Scientifique) and ONERA (Office National de Recherches Aérospatiales), was developed in France in connection with the French Space Agency (CNES) and industry (SNECMA) for the understanding of Hall-effect plasma thrusters. Different activities are pursued in parallel: an experimental test of different laboratories' thrusters; the development of diagnostic techniques to characterize the plasma inside and outside the thrusters; and the development of simulation and modelling able to describe characteristics and evaluate the thrusters' performances.
This paper will be focused on diagnostics systems implemented in the PIVOINE facility. Time- and space-resolved measurements of the ion beam energy, distribution electron density and concentration in the plume are performed with a retarding potential analyser (RPA) and Langmuir probes mounted on a 2.5 m movable drive. The thruster can be moved axially to allow a 40×90 cm² exploration of the plume. The investigation of the plasma inside the thruster is made by optical diagnostics. A CCD camera used in fast imaging mode is set outside the tank. The 45° sight axis allows an internal view of the thruster's channel. Furthermore, a spectroscopic analysis is made by focusing the channel's light to a set of optical fibres connected to an imaging spectrometer equipped with a CCD camera. A specific laboratory thruster of 100 mm external diameter called SPT100-ML was studied in more detail, this model being designed to allow the implementation of optical fibres and wall probes diagnostics inside the channel's thruster. The stationary plasma thruster discharge is almost always characterized by low-frequency instabilities of the order of 10 kHz where the discharge current can reach a very high instantaneous level. The variation of the discharge and ion beam flux currents has been related to the spatiotemporal dynamic of the plasma inside the thruster's channel. The main features are explained by a one-dimensional (1D) hybrid model and a 1D particle-in-cell-Monte Carlo model. A new thruster, working at a very low fluctuation level with a low angular divergence ion beam, is now under investigation in connection with SNECMA.

Thesis (Ph.D.)--Massachusetts Institute of Technology, Dept. of Aeronautics and Astronautics, February 1999. Includes bibliographical references (p. 263-268). by John Michale Fife. Ph.D.