Abstract
In this paper, the limitations of two-dimensional magnetic field configuration for an annular ion thruster are analyzed theoretically and numerically. The two-dimensional magnetic field, which is commonly used in traditional ion thrusters, has been proved to be a very effective way for the confinement of primary electrons and ions. Hence, the confinement characteristics of primary electrons and ions in an annular ion thruster with the two-dimensional magnetic field configuration are first analyzed theoretically. Then, according to the analysis, a two-dimensional magnetic field for an annular ion thruster is designed. On this basis, the limitations of the two-dimensional magnetic field configuration are analyzed and discussed by a three-dimensional (3D) Particle-in-cell Monte Carlo collision (PIC-MCC) model. The results show that if the collision process is not considered, a uniform distribution of primary electrons can be achieved theoretically. However, when considering the collision process, the atom forms a significant impedance to the circumferential motion of primary electrons, thus greatly increasing the loss rate of primary electrons. Finally, it is found that, since primary electrons are not guided and constrained in the circumferential direction under the two-dimensional magnetic field, the discharge uniformity cannot be improved only by increasing the magnetic field intensity.
1. Introduction
Due to significant advantages, electric propulsion has been widely used in scientific and commercial space missions [1–4]. As one of the most widely used electric propulsions, conventional Kaufman-type ion thruster has successfully carried out many space missions due to its high efficiency and specific impulse [5–7]. To further improve its thrust and power to cope with the next deep space exploration, Patterson and Herman [8] proposed a new kind of Kaufman-type ion thruster, i.e., the annular ion thruster [9–11]. The newly proposed annular ion thruster utilizes the double annular anodes design which makes its ion optics easily manufactured. Meanwhile, since the area of anode increase nearly doubled, its input power can increase greatly [12, 13]. However, the cathode of annular ion thruster biases on the one side of the discharge chamber, making its discharge process uneven. This is very harmful to ion optics and can lead to excessive discharge loss [14]. Therefore, how to keep a uniform discharge is quite critical for the application of annular ion thruster [15].
In Kaufman-type ion thruster, the magnetic field is one of the most important parameters for performance. The magnetic field is produced by the permanent magnets off anodes and used to confine primary electrons and plasma. In a well-designed magnetic field, most electrons (including primary electrons and secondary electrons in plasma) will be constrained and lost only at the magnetic tip, while most ions will be extracted by ion optics [16]. Since electrons have a small loss area, they can have a long average confinement length, which makes them uniformly distributed in the discharge chamber. Also, since most ions can be used to produce the thrust, the discharge loss is small. Hence, an excellent magnetic field structure can greatly improve the uniformity of primary electrons and improve other performance of the Kaufman-type ion thruster.
Due to the success of the two-dimensional magnetic field configuration which is commonly used in conventional Kaufman-type ion thruster, in current designs, the two-dimensional magnetic field configuration is also adopted in annular ion thruster. However, under this design, serious nonuniform discharge was also observed [17]. Due to the nonaxisymmetric, it is difficult for the plasma to diffuse uniformly in the discharge chamber. Most plasma mainly concentrates on the cathode side, leading to quite high discharge loss. This means that it may not be suitable to use the two-dimensional magnetic field configuration directly in the annular ion thruster.
Therefore, to better understand the limitations of two-dimensional magnetic field configuration when it is used for the annular ion thruster, its characteristics are analyzed in this paper. First, the confinement characteristics of primary electrons and ions in an annular ion thruster with the two-dimensional magnetic field configuration are analyzed theoretically. Then according to the analysis, a two-dimensional magnetic field for the annular ion thruster is designed and presented. After that, the limitations of the two-dimensional magnetic field configuration are analyzed and discussed by a three-dimensional (3D) Particle-in-cell Monte Carlo collision (PIC-MCC) model. Section 2 presents the confinement characteristics of primary electrons and ions by theoretical analysis. The designed two-dimensional magnetic field is also presented. In Section 3, the 3D PIC-MCC model is presented. Then, the simulation results and the analysis are provided. Finally, the conclusions will be given in Section 4.
2. Theoretical Analysis of Magnetic Confinement of Charged Particles
In this section, the radial and axial distributions of primary electrons and ions are analyzed theoretically. Then, the basic design criteria of the two-dimensional magnetic field configuration and the designed magnetic field for an annular ion thruster will be given according to the theoretical analysis results.
2.1. Distribution Characteristics of Primary Electrons
Figure 1 shows the schematic diagram of an annular ion thruster. It mainly consists of five parts: inner anode, outer anode, bottom anode, cathode, permanent magnet, and ion optics. In Figure 1, the settings of several permanent magnets are just schematic. Table 1 shows the relevant geometric parameters.
In discharge chamber, the primary electrons will be first injected from the hollow cathode. Meanwhile, for the primary electrons, the discharge chamber can be regarded as a space with a reflection boundary. Primary electrons are mainly lost near the tip of the magnetic line perpendicular to the wall. The primary electrons move back and forth along the discharge chamber until they collide with the anode in the effective contact area near the magnetic tip, either by ionization or excitation collision with a neutral gas, or heated by Coulomb interaction with plasma electrons. According to the above characteristics, we try to figure out the circumferential and axial distribution of primary electrons theoretically.
2.1.1. Circumferential Distribution Characteristics of Primary Electrons
First, ignoring the impact of collisions on the motion of the primary electron, the primary electron only loses at the tip of the magnetic field. The loss area at one magnet tip is defined as where is the loss area at the magnet tip, is the length of the magnet, is the magnetic field intensity at the magnet tip, and is the electron temperature in eV.
Suppose there are permanent magnets, then the primary electron current lost at the tip of the magnetic field is equal to where
The primary electron density distribution satisfies the continuity rule, where is the primary electron density, is time, is the diffusion velocity of primary electrons, is the generation rate of primary electrons, and is the loss rate of primary electrons.
As shown in Figure 2, it is assumed that the entry of primary electrons is regarded as the boundary condition, i.e., , while their loss at the tip of the magnetic field can be regarded as a spatial loss rate, i.e., . Assuming that the primary electrons are uniformly distributed in the radial and axial directions, we can simplify Eq. (4) into a one-dimensional equation with the variable .
Ignoring the radial and axial variations of the primary electron density, in polar coordinates, Eq. (4) can be rewritten as where is the compromise radius, , is the outer radius, and is the inner radius; is the circumferential diffusion velocity of primary electrons.
can be expressed as where is the radius of the magnet, and is the height of the thruster discharge chamber.
In steady-state, this one-dimensional equation can be expressed as
If the collision process is ignored, can be regarded as constant and , then, we can get
Finally, we can obtain the circumferential distribution of the primary electrons as follows
To ensure the normal operation of the ion optics, the difference of plasma density upstream of the ion optics should not exceed one order of magnitude, that is, less than 10%. Therefore, here, we mainly focus on the circumferential decay degree of primary electron density, and is set to be 1. Then, four factors affect the circumferential distribution of primary electrons: the number of magnets, the difference between inner and outer radius, the height, and the magnetic field strength.
Let and . Then, the simulation cases and their parameter settings are presented in Appendix A. Table 2 summarizes the values of input parameters for the analysis of circumferential distribution characteristics of primary electrons. Figure 3 shows the circumferential distribution of primary electrons of each case group.
(a) Under different magnetic induction intensity
(b) Under different numbers of permanent magnets
(c) Under different channel widths
(d) Under different channel heights
According to the criterion (not less than 10%) and the results in Figure 3, it can be found that the circumferential distribution of primary electrons has the following characteristics: (1) it is sensitive to the magnetic field and changes nonlinearly. The magnetic field near the anode should be greater than 0.1 T. The stronger the magnetic field is, the better, but the improvement is not obvious when the magnetic field is greater than 0.2 T. (2) It is not sensitive to the number of magnets and changes almost linearly. The less the number of magnets is, the better. (3) It is also sensitive to the channel spacing and changes nonlinearly as well. The channel spacing should be greater than 0.06445 m, the wider the channel is, the better. But when the channel spacing is greater than 0.07861 m, the improvement is not obvious. (4) It is very insensitive to height.
2.1.2. Axial Distribution Characteristics of Primary Electrons
The axial transport characteristics of primary electrons are shown in Figure 4. Similarly, the axial distribution of primary electrons can be expressed as where is the magnet circumference; is the axial diffusion velocity of primary electrons; is the area of ion optics, and
Assuming that is constant and , then, we can get
Since the primary electrons are mainly lost at the magnetic tip, the loss length of the primary electron is discrete in the axial direction, that is, should be expressed as , and
Then, we can get
In Appendix A, Table 3 summarizes the values of input parameters for the analysis of axial distribution characteristics of primary electrons. Figure 5 shows the axial distribution of primary electrons of each case group. It indicates that the decay of the primary electron density in the axial direction is not obvious and is not sensitive to the number of magnets, the difference between inner and outer radius, and the magnetic field strength. Under all factors, the decay of the primary electron density in the axial direction is less than 10%.
(a) Under different magnetic induction intensity
(b) Under different numbers of permanent magnets
(c) Under different channel widths
2.2. Distribution Characteristics of Ions
In this section, we directly give the one-dimensional distribution equations of ions in the circumferential directions and then analyze their distribution characteristics. Note that because the loss of ions is mainly on the ion optics, and the loss on the anode can be negligible, the distribution of ions in the circumferential direction is only analyzed.
First, the circumferential distribution of ions can be expressed as where is the ion density, is the circumferential diffusion velocity of ions, is the production rate of ions, is the loss rate of ions. Here, the production rate of ions is assumed to be constant. The loss of ions on the anode is ignored, and the ion transmittance of ion optics is assumed to be 100%. Then, the loss rate of ion is expressed as where is the charge quantity of ions, is the ion beam current, and
Suppose is also constant, hence
Finally, we can obtain the circumferential distribution of ions as follows
Since the variables , , , , are in the same order of magnitude, one can calculate that based on the values listed in Tables 2 and 3. Furthermore, since is very small, usually equal to 0, the distribution of ions is mainly determined by , that is, the distribution of ions mainly depends on the distribution of primary electrons but not on the diffusion process. Therefore, once the uniformly distributed primary electrons are obtained, the uniformly distributed ions will also be obtained.
2.3. The Design of Two Dimensional Magnetic Field Configuration Based on Theoretical Analysis
Based on the above results, we propose the design criteria of the two-dimensional magnetic field configuration for annular ion thruster in this section. We also draw on the magnetic field design criteria of traditional Kaufman-type ion thrusters. In Ref. [18], Wirz and Goebel found that the magnetic confinement and overall performance of the plasma generally improve with the increase of the maximum closed profile, . The field line topology needs to be carefully designed to effectively utilize the primary electrons and ensure ideal beam flatness, low double ion content, and good performance. The must be high enough to restrict the primary electrons. The must be low enough to allow stable discharge of all desired discharge plasma densities. Finally, Wirz and Goebel proposed that should be around 0.2 T [18].
To ensure the circumferential uniformity of electrons and ions, the following criteria should be satisfied.
Rule 1: the magnetic field near the anode should be large enough to effectively constrain the primary electrons. However, in order to ensure a large enough ionization region (i.e., the region where the magnetic field is less than [18]), the magnetic field of the permanent magnet cannot be too large. Therefore, according to the results in Section 2.1 and Section 2.2, the value 0.2 T is chosen.
Rule 2: the number of magnets should be few enough. However, if the number is too few, it is difficult to form an ideal closed profile, . Therefore, the MAXWELL 3D software [19] is used to simulate the magnetic field formed by different numbers of magnets. The results show that the minimum number of permanent magnets that can form an ideal closed profile is 9.
Rule 3: the channel spacing should be large enough. The designed maximum value 0.121 m is selected.
Rule 4: similar to rule 2, if the height is too high, it is difficult for the permanent magnets to form an ideal closed profile. Hence, the height should not be too large. Here, the designed minimum height 0.168 m is selected.
Based on the above criteria, a two-dimensional magnetic field configuration for the annular ion thruster is designed. Figure 6 shows the simulation results of the designed magnetic field, which is implemented through the MAXWELL 3D software [19]. In the next section, the limitations of two-dimensional magnetic field will be analyzed numerically based on this magnetic field configuration.
3. Numerical Analysis on the Limitations of the 2D Magnetic Field Configuration
In this section, the characteristics of primary electrons under the designed two-dimensional magnetic field configuration are simulated. The simulation model for the primary electrons is part of a 3D discharge chamber model for conventional Kaufman-type ion thrusters developed by Lu et al. [17, 20–23]. Then, the 3D density distribution, tangential average density distribution, different collision frequencies, and the traces of primary electron are presented. According to simulation results, the limitations of the 2D magnetic field structure are analyzed.
3.1. Computational Model
The model is built on the IFE-PIC-MCC method. In this paper, the PIC-MCC module is only used. In the PIC-MCC module, the plasma is set to be quasineutral, so its self-consistent field is ignored. The model is described below.
3.1.1. The PIC Method
In the PIC method, the detailed information of the flow field is obtained by directly tracking the movement of a large number of simulated particles. First, the initial state of the simulated particle must be determined according to the input conditions. In this model, the initial states of the primary electrons include the injected number of each step, the initial position, and the initial velocity. The number and initial position can be determined by the emission current and the position of the cathode top, while the initial velocity is determined by the initial energy of the primary electron.
refers to the current emitted from the cathode, while the discharge current refers to the current consisting of charged particles received from the anode, and beam current refers to the current consisting of ions drawn from the optics. In an ideal ion thruster discharge chamber, all ions are extracted by the optics, while all electrons (including primary electrons and secondary electrons) are absorbed by the anode. Therefore, the discharge current is mainly composed of and the secondary electron current . The number of secondary electrons is equal to the number of ions, hence, .
Therefore, can be calculated from discharge current and beam current [23]:
The initial position of the primary electrons is located at the cathode emitter, and the primary electrons are uniformly distributed. In addition, the sheath thickness at each boundary of the discharge chamber is neglected, hence, the primary electrons are assumed to emit from the cathode sheath edge. Therefore, the initial velocity of the primary electron is the velocity accelerated by the cathode sheath potential. Since the potential drop of the cathode sheath (about 30 V) is much larger than the thermal kinetic energy of the primary electron (about 2 eV), the initial velocity of the primary electron in the axial direction (-direction) is defined as the velocity accelerated by the cathode sheath potential, while the initial velocity in the other two directions ( and direction) conforms to Maxwell distribution [23]: where is the most probable thermal velocity, is a random number between 0 and 1, and is the potential drop of cathode sheath. Then, the trajectory of primary electrons is determined by Newton’s second law and Lorentz force:
The above equation is solved by Boris’s leaping frog algorithm [24]. If primary electrons impact the cathode or the screen grid, they are reflected into the discharge chamber, and the reflection is taken as being specular [25], while, if they impact the anode, they are considered to be absorbed. In addition, if the energy of a primary electron is less than the threshold of ionization collision, it will be also deleted from the simulation domain [25]. The time step for the primary electrons is one-third of the primary electron cyclotron frequency.
3.1.2. The MCC Method
In the MCC method, only four collision types of primary electrons are considered: elastic collision, excitation collision, ionization collision, and Bohm collision [26] When all of these collisions occur, it is assumed that the neutral atoms remain unchanged. If elastic collisions occur, the energy of primary electrons remains unchanged, but the velocity will be reset; if an excitation collision occurs, the primary electron loses 8.32 eV of energy, and then the velocity direction is reset again; if an ionization collision occurs, the energy of both the primary electron and the newly generated secondary electron needs to be restored, which conforms to the following function [26]: where is the energy of newly generated secondary electrons; is the energy of the primary electrons after ionization; is the ionization parameter with a value of 8.7 eV; is the threshold of ionization energy; is the cross section of ionization collision.
The collision frequencies of elastic collision , excitation collision , ionization collision , and Bohm collision [27] are where are the collision cross sections of elastic collision, excitation collision, and ionization collision, respectively (the values of each collision cross section can be referred to Ref. [28]). and represent the modes of the primary electron velocity and magnetic induction intensity, respectively, that is, and .
Furthermore, in the MCC method, the null collision method [28] is introduced. First, the maximum collision probability is calculated in each step where is the maximum collision frequency, .
Second, is compared with a random number . If , some collision is considered to have happened. Then, another random number is generated to determine what kind of collision occurs. If , where is the total collision frequency and , the first collision will occur; if , the second collision will occur, and so on. If , no collision is considered.
3.2. Simulation Setup
In our model, atoms are set to be a uniformly distributed background, and their number density is set to be . Discharge currents and beam currents are 11 A and 1 A, respectively. Hence, the cathode emission current is set to be 10 A. The plasma is assumed to be quasi with equal potential, so the electric field in the discharge chamber is ignored. Due to ignoring the plasma self-consistent electric field, a large mesh length is chosen, which is set to . The time step is set to be less than one-third of the reciprocal of electron cyclotron frequency, which is about 10-11 s. The reference parameters and the reference values of these parameters are shown in Table 4. In the following content, except for the results marked with units, the rest of the simulation results are all normalized results. Table 5 shows its initial normalized geometric parameters. The whole normalized simulation domain is set to . For each case, the time required for the PIC-MCC loop to reach the steady-state is about 6 hours on a 2.8 GHz HP Z620 workstation.
3.3. Simulation Results and Analysis
Here, we analyze the limitations of the two-dimensional magnetic field configuration by the distribution characteristics of the primary electrons, their trajectories, magnetic field streamlines, and the distribution characteristics of various collision frequencies. The results of numerical examples with and without collisions, as well as with different magnetic field intensities, are mainly compared and studied.
3.3.1. Influence of Collision Process on Magnetic Confinement
First, to show the convergence of our method, Figure 7 presents the curve of particle number with iteration steps. It shows that when the number of iteration steps exceeds 100000, the calculation converges.
Figure 8 presents the number density of primary electrons with the collision process. Obviously, the density at the cathode side is much higher than that of the other side. The density of primary electrons upstream of the ion optics on the cathode axis is about 10 times higher than that of the other side. The density of primary electrons is nonaxisymmetric. Another obvious feature is that, because of the mirror effect, electrons form a sharp cone near the anode.
Figure 9 shows the number density of primary electrons without the collision process. Compared with Figure 8, the distribution of primary electrons in Figure 9 is more concentrated in the discharge channel. The density of primary electrons upstream of the ion optics on the cathode axis is about 3 times higher than that of the other side. This means that, without the collision process, the circumferential distribution of primary electrons is more uniform.
Figure 10 shows the distribution of Bohm, elasticity, excitation, and ionization collision frequencies. It can be found that, in all of these four collisions, Bohm collision and elastic collision are dominant.
(a) Bohm collision
(b) Elasticity collision
(c) Excitation collision
(d) Ionization collision
Figure 11 shows the primary electron trajectories. The trajectory shows that the primary electrons only move locally at the cathode and rarely move to the opposite side of the cathode.
Figures 12 and 13 show the primary electron trajectories in local regions. An obvious magnetic mirror effect was observed near the cathode. In the channel far away from the cathode, the collision process was observed.
Figure 14 shows the primary electron trajectories without collision process. The trajectory shows that the primary electrons have obvious circumferential movement. The above results show that the atom forms significant impedance to the circumferential motion of the primary electron. Without the collision process, the primary electrons can form obvious circumferential motion under the confinement of the magnetic mirror. The primary electron circumferential distribution is close to the theoretical prediction in the previous section. However, when the collision process is considered, the primary electrons only move locally at the cathode. The primary electron circumferential distribution deviates far from the theoretical prediction in the previous section.
3.3.2. Limitations of the Two-Dimensional Magnetic Field Configuration
Figures 15 and 16 show the number density of primary electrons under 6 and 10 times the initial magnetic field strength, respectively. With the increase of magnetic field intensity, the density of primary electrons in the channel increases significantly. The distribution of primary electrons in Figures 14 and 15 are also more concentrated in the discharge channel. This means that the confinement of the magnetic field to the primary electron is strengthened. However, the densities of primary electrons upstream of the ion optics on the cathode axis are about 40 and 60 times higher than that of the other side, respectively.
Figures 17 and 18 show the collision rate distribution under 6 and 10 times the initial magnetic field strength, respectively. Obviously, with the increase of magnetic field strength, the primary electron density increases, and the frequency of all kinds of collisions also increases.
(a) Bohm collision
(b) Elasticity collision
(c) Excitation collision
(d) Ionization collision
(a) Bohm collision
(b) Elasticity collision
(c) Excitation collision
(d) Ionization collision
To show the above difference more clearly, we extract 40 surfaces in the discharge chamber and calculate the average density on each surface. The extracted surfaces are shown in Figure 19. The angle of the surface cutting the cathode around the thruster axis is set to be , where the counterclockwise direction is set to the rotation direction. Figure 20 presents the average ionization rate of each case, and Figure 21 presents the average ionization rate of each case. The results show that, when the collision process is not considered, the circumferential distribution of primary electrons is the most uniform. However, increasing the magnetic field intensity does not improve the uniformity of the primary electron circumferential distribution, but aggravates the nonuniformity.
Figures 22 and 23 present the primary electron trajectories under 6 and 10 times the initial magnetic field strength, respectively. They show that with the increase of magnetic field strength, a large number of primary electrons only move locally near the cathode.
Figure 24 presents the local magnetic field lines near the cathode. Figure 25 presents the 3D views of magnetic field line near cathode at section and section, respectively. It can be seen from the figure that the magnetic field of the two-dimensional configuration cannot guide and constrain the motion of the primary electron in the circumferential direction. Therefore, when the magnetic field intensity increases, more primary electrons will be constrained to move only locally at the cathode.
(a) section
(b) section
4. Conclusions
In this paper, the confinement characteristics of primary electrons and ions in an annular ion thruster with the two-dimensional magnetic field configuration are first analyzed theoretically. Then, according to the analysis, a two-dimensional magnetic field configuration for the annular ion thruster is designed. On this basis, the limitations of the two-dimensional magnetic field structure are analyzed and discussed by a three-dimensional (3D) Particle-in-cell Monte Carlo collision (PIC- MCC) model. The results show that: (1)Both the simulation results and theoretical analysis results show that, only when the collision process is not considered, the primary electrons can be well constrained by the two-dimensional magnetic field. Hence, only in this ideal situation, a uniform distribution of primary electrons can be achieved(2)When considering the collision process, the atom forms a significant impedance to the circumferential motion of the primary electron. Meanwhile, the collision process greatly increases the loss rate of primary electrons and aggravates the nonuniformity, causing that the simulation results are far worse than the theoretical analysis results(3)Because the primary electron lacks guidance and constraint in the circumferential direction under the two-dimensional magnetic field configuration, the uniformity of discharge cannot be improved only by increasing the magnetic field intensity
Finally, to sum up, the circumferential constraint on charged particles must be considered for the magnetic field configuration of annular ion thruster. This will be an important research direction of annular ion thruster in the future.
Appendix
A. Parameter Settings for the Theoretical Analysis
The value of each parameter is referred to Ref. [12]. Table 2 summarizes the values of input parameters for the analysis of circumferential distribution characteristics of primary electrons, and 4 case groups are set, i.e., case groups 1-4. In case group 1, 6 different values of B are set; in case group 2, 6 different values of are set; in case group 3, 6 different values of are set; in case group 4, 6 different values of are set. The calculation results of case groups 1-4 are shown in Figure 3.
Table 3 summarizes the values of input parameters for the analysis of axis distribution characteristics of primary electrons, and 3 case groups are set, i.e., case groups 5-7. In case group 5, 6 different values of are set; in case group 6, 6 different values of are set; in case group 7, 6 different values of are set. The calculation results of case groups 5-7 are shown in Figure 5.
Data Availability
Some or all data, models, or code generated or used during the study are available from the corresponding author by request.
Conflicts of Interest
The authors declare that they have no conflicts of interest.
Acknowledgments
This work was supported by the National Key R & D Program of China (no. 2020YFC2201100), the National Natural Science Foundation of China (nos. 12102082 and 12175032), the National Key R & D Program for Intergovernmental International Scientific and Technological Innovation Cooperation (no. 2021YFE0116000), the Fundamental Research Funds for the Central Universities of China (no. DUT21GJ206), Key Research and Development Project of Liaoning Province (no. 2020JH2/10500003), and the Central Guided Local Science and Technology Development Project (no. 216Z1901G).