Parametric optimization of a Hall Effect Thruster...

Joint Conference of 30th ISTS, 34th IEPC and 6th NSAT, Kobe-Hyogo, Japan July 4 – 10, 2015

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Parametric optimization of a Hall Effect Thruster magnetic circuit

IEPC-2015-40

Presented at Joint Conference of 30th International Symposium on Space Technology and Science 34th International Electric Propulsion Conference and 6th Nano-satellite Symposium,

Hyogo-Kobe, Japan July 4 – 10, 2015

Alberto Rossi1, Frédéric Messine2, Carole Henaux3 and Satafa Sanogo4

Laplace laboratory, Toulouse, 31071, France

Abstract: Hall Effect thrusters incorporate a magnetic circuit that must generate a specific electromagnetic flux distribution inside and near the outlet of the plasma channel. The first objective of the design process of this type of structure is to obtain a specific magnetic topography in the thruster channel with given magnetic field radial component values and a certain inclination of this field lines. The aim of this work is to develop tools for solving this inverse magnetostatic problem, which are applied to the SNECMA PPS1350® magnetic circuit to obtain a new low-erosion magnetic configuration.

Nomenclature A = magnetic potential vector B = magnetic induction

= target magnetic induction Mat = material Matrix properties

= optimization objective function = target region

x = variable dimensions r,z = coordinates of control points I = variable currents m = slope of the straight q = y-intercept

= inclination angle

I. Introduction

owadays, two types of space propulsion engines exist: the most common ones are the chemical propulsion engines which provide high thrust impulses allowing fast orbit transfers. But this technology requires a

large amount of propellant and is not suitable for interplanetary displacements, whose propellant mass requirements are too high. The second type of propulsion engine is based on electrical propulsion that provide very low but continuous thrust, resulting in huge propellant savings at the cost of longer spacecraft transfer time. The main

1 PhD student, CNES/SNECMA, LAPLACE laboratory, GREM3, [email protected]. 2 Associate Professor, LAPLACE laboratory, GREM3, [email protected]. 3 Associate Professor, LAPLACE laboratory, GREM3, [email protected]. 4 PhD student, UPS, LAPLACE laboratory, GREM3, [email protected].

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advantage of electric thrusters lies in their highly efficient utilization of propellant mass2 which allows deep space mission (e.g. NASA Deep Space 115 and European Smart One mission). The corresponding reduction in necessary propellant supply makes it possible to board a greater portion of useful payload possible. Hall thruster belongs to the electric spacecraft engines typology and it is constituted of a cylindrical plasma channel, an interior anode, an external cathode and a magnetic circuit that generates a primarily radial magnetic field across the plasma channel. An axial electric field is established between the anode—located on the base of the plasma channel—and the Hollow-cathode plasma—fixed outside of the thruster channel. A transverse (radial) magnetic field prevents electrons from this cathode plasma from streaming directly to the anode. Indeed, the electrons curl along the magnetic field and in the ExB azimuthal direction around the channel. The plasma discharge generated by the electrons efficiently ionizes the propellant injected into the channel from the anode zone. The reduced axial electron mobility produced by the transverse magnetic field makes possible the creation of quasi-neutral plasma in the thruster channel. Thus, the discharge voltage (distributed along the channel axis resulting in an axial electric field in the channel) accelerates the ions outside the channel. The external hollow cathode does not only generate the electrons for the discharge, but also provides the electrons to neutralize the ion beam. The details of the channel structure and specially the magnetic field shape determine the thruster’s performance, efficiency, and lifetime3. The Hall Effect thruster’s structure is shown in Fig.1. The impact of the magnetic field is very important to the operation of the engine so the magnetic circuit plays a fundamental role in the thruster’s operations. Our study about the optimization of the thruster magnetic circuit is thus justified. Methods based on topological optimization have already been developed for these structures. The algorithm ATOPTO (Algorithm To Optimize Propulsion with Topology Optimization) has already demonstrated its efficiency4. In this work we try to extend the scope of the algorithm ATOP by adding a new parametric optimization section called ATOPPO. The ATOP algorithm becomes a very versatile optimization tool for Hall Effect thruster magnetic circuits. The rest of the paper is organized as follow: in section II, we present classical Hall Effect thruster magnetic circuits. This structure is the result of the scientists’ experiences. However, the structure had never made use of mathematical optimization methods for its design. Thus, the magnetic circuit of Hall Effect thrusters has remained virtually unchanged since its conceptualization up to the present. New magnetic configurations, along with their characteristics, are also presented here. In Section III, we discuss our optimal design method (ATOPPO optimization algorithm) for this type of magnetostatic structure. In Section IV, we present some numerical application to validate this optimization method: we have applied this algorithm to the magnetic circuit of the commercial SNECMA PPS1350® Hall thruster to obtain a hybrid wall-less/magnetic-shielding (magnetization region shifted outside the thruster ceramic channel) magnetic configuration to reduce the plasma-surface interaction inside the discharge chamber.

II. Hall thruster magnetic circuit design In a Hall Effect thruster the magnetic circuit constitutes more than half of the whole thruster. Consequently the design of this magnetic circuit must be optimized in order to minimize the embedded mass. The main components of this circuit are the coils which produce the magnetic flux and ferromagnetic parts which guide the flux and to shape the flux density. Usually the magnetic circuit includes four (or more) external coils located around the exterior radius of the plasma channel and one internal coil around the interior radius of the plasma channel. All the coils are supplied by the same DC. Two coils located at the rear of the plasma channel can be also used to perform the magnetic topography. This structure is shown in Fig. 1 and 2. In standard configuration the flux density has its maximum at the channel exit plane with a positive (respectively negative) gradient inside (respectively outside) the channel when following the ion flow direction. An example of a flux density distribution along the channel axis is shown in Fig.3. The magnetic field lines have the shape of a lens that is symmetrical with respect to the channel axis. During long time the magnetic topology showed the same characteristics5,7,8:

- The magnetic field gradients inside and outside the channel,

Figure 1: Conventional structure of a Hall thruster


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- The strength of the magnetic field at the center of the channel, - The axial position of the magnetic field amplitude maximum, - The profile width, - The magnetic field lines curvature, - The magnetic lens inclination, - The magnetic field amplitude adjacent to the anode.

Figure 2: Transversal section of a Hall thruster magnetic circuit

Figure 3: Characteristic elements of a magnetic topology of a Hall Effect thruster

By considering now the requirements in terms of power and lifetime new specifications concerning in particular the “erosion of ceramic wall” have to be taken into account. This weakness has its origins in plasma-surface interaction inside the discharge chamber. Thus, to solve this problem it has been proposed to move the ionization zone outside the thruster channel in order to avoid contact between the ions and ceramic material. Thanks to new studies carried on the impact of magnetic topology20,21, new magnetic configurations have been highlighted to improve the efficiency and reduce the erosion of the ceramic walls such as the magnetic shielding configuration17,19. Let’s underline that the magnetic shielding consists in generating a magnetic field perfectly parallel to the ceramic walls of the channel i.e. a magnetic field with no radial component along the walls of the discharge chamber as shown in fig.4.

Ferromagnetic material

Coils

Ceramic channel

Anode and gas injector

Internal pole

Magnetic screens Trim coils

External pole

Ferromagnetic body

Main coils

The axial position of the magnetic field amplitude

maximum

Magnetic field lines

Visualization of the shape of the radial component of the field in the segment [AB]

Zero magnetic field

Axis of symmetry

Zero magnetic field

The magnetic field amplitude adjacent to the anode

Anode

Magnetic lens

A

B

Magnetic field gradient outside the channel

Magnetic field gradient inside the channel

Magnetic field amplitude maximum

Exit plane of the channel


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In the future one can expect even to propose magnetic circuits such as the wall-less circuit (as showed in fig.5) in order to avoid contact between plasma and ceramic walls moving the ionization region outside the thruster channel and reduce also the cost of the engine18.

Figure 4: magnetic shielding configuration

Even if the magnetic topology is invariant by rotation (axisymmetric circuit), the design of the magnetic circuit is very difficult because of the following characteristics:

- In one hand the plasma channel is very large compared to the allowed space for the magnetic circuit. This channel constitutes a significant air gap and lead to a significant flux leakage (flux leakage of the main magnitude of principal flux).

- In another hand the required magnetic topology is not uniform and is characterized by many various zones between the back and the top of the plasma channel.

Consequently, like in many magnetic circuit design methods, the development of an analytical modelling including the main sizes of the circuit is not available. The design of the thruster magnetic circuit has been resulted, for many years, from an iterative “hand approach”: the designer thinks of a first magnetic circuit and performs it by using finite elements software. Many numerical simulations are in this case necessary to converge on the required magnetic topology. It is the reason why a new design methodology based on the resolution of a magnetostatic inverse problem is proposed in this study. This new approach may allow shaping specific parts of a magnetic circuit from several data of magnetic field topology required in the plasma channel.

Figure 5: wall-less magnetic configuration of Hall thruster18.


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A. Optimal design method

Optimal design of electromagnetic systems can be understood and formulated as an inverse problem. Many electromagnetic problems require determining the spatial distribution of an unknown quantity (material distribution in space or the source; e.g., the current density) which produces a specified quantity (the effect, e.g. the magnetic field, the induced current density or the electrodynamic force) in a specified region of the space. These problems belong to the class of inverse problems. Electromagnetic inverse problems can be divided into two classes: Identification problems and synthesis problems. In identification problems the goal is to find a source that produces a really existing effect. In synthesis problems the goal is to find a material distribution that produces a specific effect: these problems are often formulated as an Optimization problem (see Fig. 6).

Two possible methods for resolving the inverse design problem exist:

- In the first one the resolution of the direct problem begins with initial dimensioning—which is based on a combination of the designer past experience and simplified models. Thus the obtained result is compared with existing fixed objectives. Design parameters are subsequently modified to correspond with the designer experience. The process is repeated until the desired result is achieved.

- In the second one the problem can also be approached from the other hand (as an optimization problem, for example) by formulating a design most adapted to generate the desired characteristics. The problem resolution method can call on successive direct problem solutions to guide convergence. This is the method notably used in the case of metaheuristic algorithms and of descent algorithms (used in this work) which are based on gradient evaluation (gradient algorithm, Newton-Raphson algorithm1).

The second approach is better suited to rationale the decisional process which would enable the conception of an electromagnetic device. The design of this type of magnetic structure via optimization based on an analytical model of “reluctance circuit” type is not appropriate. Thus, it was chosen to use an optimization method using a numerical based model of the structure to assess the relevance of a set of parameters13. The developed design method uses a parametric optimization algorithm with a resolution using a finite elements method in order to achieve an optimized geometry. The finite elements model takes as input the geometric parameters of the structure and it will output the values of the field for an optimal current configuration. An evaluation criterion for the optimization function is then calculated to assess the correlation between the field values obtained and the initially target set values of the magnetic field. Assumptions conventionally used of azimuthal homogeneity of the field in the thruster channel (invariance of the Flux density along the circumference of the channel) allow reducing this model to a 2D model, which represents a considerable saving of time because many functions are called for the evaluation criterion and derivatives during the optimization procedure. The optimal solution can be significantly influenced by the starting point that is chosen, so the optimization procedure is started several times with different points. There is also a great

Figure 6: The electromagnetic inverse problem principle


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chance to fall into local minimum points. The objective function to be minimized is represented by the square difference between the magnetic field calculated at each iteration in the channel and the required magnetic field.

B. Variable parameters In a previous work4, topology optimization has already demonstrated his efficiency to design Hall thruster magnetic structures. This type of optimization does not take into account the entire structure but only isolated zones, like for example internal and external poles (see Fig.2) which represent the most critical areas in the magnetic mapping generation. However, it seems interesting to optimize the whole entire structure or maybe more extensive zones of this magnetic circuit. Thus, in this work, a larger zone considering some sizes of the magnetic circuit as variables is optimized. Contrary to topological optimization, parametric optimization needs to start with a given shape of the structure (which plays a fundamental role). For this study, the starting structure is representing by the PPS1350® magnetic circuit (see Fig. 7) because the objective is to reproduce the same magnetic topology of this magnetic circuit but with additional constraints. This work is focused on the optimization of magnetic poles shape, magnetic screens dimensions and also current values in the coils. The variables of the problem are represented in Fig.7 and they are: the control points that define magnetic poles, heights of magnetics screens, horizontal position of external screen and coils currents.

Figure 7: Design domain with variables of the problem

Thus, we have three types of variables: control points (P), dimensions (x) and currents in the coils (I). There are ten control points, three variable dimensions and three variable currents (internal coil, external coil and trim coil current). The goal is to reproduce the same magnetic mapping of the PPS1350® but outside of the channel to obtain a wall-less magnetic configuration and to limit the ceramic erosion. Thus, we impose as constraint that the magnetic


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circuit’s height must be smaller than a certain value so that the ceramic channel can be moved down. As shown in Fig. 8, in our optimization problem constrains are represented by two lines (red lines in Fig.8) and all ferromagnetic materials must remain under these two lines. The lines inclination is gradually manually decreased to obtain better solutions.

Figure 8: Design domain: .

C. Problem formulation

In order to optimize the magnetic field generated by the structure, it is necessary to define a target region in the channel (see Fig. 7 and 8) where the designer forces the magnetic field to have a specific value. It is an inverse problem: from the sizes of the structure find the magnetic field imposed in the target region . This inverse problem is formulated as a minimization of the square of the difference between the magnetic field produced by the structure (where the sizes vary) and the required magnetic field in . The magnetic field produced by the structure is calculated in the target region using a function (named ) that has as input the variables of the structure (P,x,I) and as output the magnetic field values.

, (1)

, (2)

, (3)

where matrix represents the source term6. From equation (3), it can be noted that the magnetic induction is computed from the variables. B is an implicit function of the design parameters.


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(L) : (4)

Where:

is the set of control points

represents the variable dimensions

represents the variable currents

To solve (L) a called ATOPPO optimization code was built. It is based on well-known computational softwares namely MatLab® and FEMM and Fig.9 shows its main architecture.

Figure 9: ATOPPO architecture

Loop


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III. Numerical results

All calculations were performed on a 32 GB of RAM and a 16 cores microprocessor computational server, to solve the (L) optimization problems. The algorithm of Fig. 9 was coded on MatLab® and it used FEMM finite element software to solve Maxwell equation and find magnetic field values. The used version of MatLab® is the 2013b and the FEMM version is the 4.2. To solve (L), it is necessary to define a number of precise points (the name of these points is “measuring points”) in the region, where the designer forces the magnetic field to have a specific value. Thus, area is discretized and function becomes:

(7)

where is the number of points in zone. We take into account 20 measuring points here. The first solution (Fig.10) was obtained for a value of the angle 1 of 25° and a value of the angle 2 of 20°.

Figure 10: Solution of (L) while 1=25°, 2=20°

In Fig.11 we show the magnetic induction values for this first solution of optimization problem. There is a maximal difference of 5% between the target magnetic field and the final obtained magnetic field. This difference is due to the saturation in the external column of the circuit. The algorithm has also optimized currents in the coils and this new current values are stronger than the initial ones. Thus, the external column is magnetically saturated. The solution was to extend the width of the column. The result is reported in Fig.11b. The difference of the maximal value of the radial component of the magnetic field is less than 1% after the correction of the saturation. Other solutions were obtained with smaller angles 1 and 2. In Fig.12, we provide a solution for 1=22° and 2=17°. As starting point for smaller angles resolutions, we choose the shape of optimal solution obtained with the greater angles. Thus we built a step-by-step procedure where the angles values are gradually decreased by choosing as starting point for the next resolution the optimal solution of the last one.

z

r

Starting structure

z

r

After optimization

Initial shape

Optimal shape

r

r

z

z


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Figure 11: Radial magnetic field component in the target region. i) After optimization process with external column saturated. ii) After the saturation correction: the external column has been enlarged.

Figure 12: Solution of (L) while 1=22°, 2 = 17°

IV. Conclusions In this paper, a new optimization based approach to design the magnetic circuits of Hall thrusters has been proposed. This yields a code named ATOPPO. The optimization problem is hard to solve because this is not possible to have some analytical expressions of the magnetic field inside the structure and hence, this involves some numerical computations by using a finite elements method. Thus, this design optimization problem is of black-box type. However, ATOPPO which is based on local search algorithms makes it possible to compute derivatives to provide solutions on this design problem: the minimization in a zone of the square errors between the magnetic field z-

0.55 0.6 0.65 0.7 0.75 0.8 0.85 0.9 0.950.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

b normalized

Br n

orm

aliz

ed

B after optimizationB target

0.4 0.5 0.6 0.7 0.8 0.9 1 1.10

0.2

0.4

0.6

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b normalized

Br n

orm

aliz

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B targetB after limitation of magnetic saturation

a

a

Forme piece polaires aprés optimisation

i ii

Initial shape

Optimal shape

r

r

z

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component and an imposed field. Obviously, the optimal design solutions are only local ones and strongly depend on the starting structure provided to ATOPPO. However, as ATOPPO has some efficiency in solving those problems, it is possible to randomly generate numerous starting structures in order to provide better solutions or to rationally choose the starting structures according to the designer experience. In the future, 3D simulation of some of these designs will be made and from these studies a prototype of a new type of magnetic circuit will be realized.

V. Acknowledgments

This material is based upon work supported by SNECMA (Safran group) and CNES.

References

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19 Richard Hofer, Dan Goebel, Ioannis Mikellides, and Ira Katz, Design of a Laboratory Hall Thruster with Magnetically Shielded Channel Walls, Phase II: Experiments 48th AIAA/ASME/SAE/ASEE Joint Propulsion Conference & Exhibit. July 2012 20 G. Bourgeois, Influence de la topologie magnétique, de la cathode et de la section du canal sur l'accélération des ions dans un propulseur à effet Hall, PhD thesis, Institut de Combustion, Aèrothermique, Réactivité et Environnement, Orléans (2012) 21 Influence of magnetic field and discharge voltage on the acceleration layer features in a Hall effect thruster D. Gawron, S. Mazouffre, N. Sadeghi, A. Héron, Plasma Sources Sci. Technol. 17, 025001 (2008).

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