1.Introduction

Hall thrusters are widely employed to keep the station of satellites,correct the orbit and control the attitude of satellites,and carry out advanced deep space missions because of their simple con figuration of power and structure[1].Generally speaking,a compact structure,light weight,and low power consumption are required for Hall thrusters to complete such tasks[2].Nevertheless,the miniaturization of thrusters will cause their drawbacks to be more pronounced.For example,miniaturization results in less space for the layout of inner magnetic poles,and the surface-area-to-volume ratio becomes larger,leading to severe ceramic erosion.Furthermore,the problem of magnetic saturation in the discharge channel and the overheating of thrusters resulting from the narrow space cannot be ignored.These issues will noticeably affect the performance and lifetime of Hall thrusters[3–5].

Cylindrical Hall thrusters(CHTs),which were proposed by Raitses[6],efficiently decrease the surface-area-tovolume ratio,and settle the issue of magnetic saturation by means of diminishing the size of the central magnetic pole.Both theoretical and experimental studies have been conducted on CHTs[7–10].Currently,a variety of CHTs have been investigated in the United States of America[11,12],Japan,China,and Korea[13,14].The findings indicate that CHTs can operate more quietly than traditional Hall thrusters,with higher utilization of propellant.Although the design of the cylindrical channel reduces the surface-area-to-volume ratio of low-power Hall thrusters,and reduces the interaction of the plasma in the channel,it is still possible to effectively improve the performance of the CHT by reducing the wall ion energy[15,16].

Figure 1.Anode structure pro files.

Recently,we proposed a new type of magnetically insulated anode(MIA)structure for a 50–100W CHT to obtain further improvement in performance and lifetime.As is shown in figure 1(b),the anode and gas distributor are integrated in our CHT.It mainly comprises an anode base and vent hole panel.Compared with traditional CHTs,our proposed MIA structure has a vent hole panel made of magnetic conductive material(pure iron DT4C with the permeability of 14872)instead of stainless steel.It is capable of changing the point of intersection between the magnetic field line in the channel and wall by changing the size of the vent hole panel and the angle between the vent hole panel and the base.By controlling the distribution of magnetic field lines in the channel,the structure can improve thruster performance and reduce wall erosion[17].In this paper,using the MIA structure,we present a detailed study on the different intersection points of magnetic field lines with the channel wall,and their influence on thruster performance and ion energy and current density on the wall.The organization of this paper is as follows:section 2 introduces the characteristics of the magnetic field,section 3 presents the results and discusses the numerical simulations,and the conclusions are provided in section 4.

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2.Magnetic field characteristics

Figure 2.Magnetic circuit con figuration[17].

The magnetic circuit con figuration of our CHT is illustrated in figure 2,including the outer magnetic pole,permanent magnet,and magnetic conductive back panel.The magnetic field maps in figure 3 are simulated with the software Finite Element Method Magnetics(FEMM).A characteristic magnetic field line(CMFL)is a magnetic field line in the‘near-zero’magnetic field zone,and leads to the outermost periphery of the wall surface.Magnetic field lines within the area of this line,the magnetic conductive anode,and the discharge channel wall,all intersect the anode.This part of the magnetic field makes little contribution to ionization.By adopting the MIA structure,the distribution of the magnetic field in the discharge channel is changed,and the CMflcrossing the anode boundary intersects with the discharge channel[18].Through particle-in-cell and Monte Carlo collision(PIC-MCC)simulations,we studied different intersection points of the magnetic field lines with the channel wall,and their influence on thruster performance and ion energy loss on the walls.

Figure 3(Case 1)shows that the magnetic field lines and anode slope are approximately parallel to each other in the magnetic field con figuration with the non-MIA structure.By changing the dimensions of the MIA,we designed three types of magnetic field con figurations,as shown in figure 3(Cases 2,3,and 4),where L is the channel length.The points of intersection of the CMFL(pink curve in figure 3)with the channel wall in Cases 2,3,and 4 are L/3,2L/3,and L,respectively.The red straight line in figure 3 connects the near-anode zone in the discharge channel to the center of the channel exit.The magnetic field intensity at the location of the straight line is used to describe the magnetic field intensity at the center of the channel,and the distribution of the magnetic field within the channel is described accordingly.Figure 4 presents the magnetic field intensity distribution curves from a position adjacent to the near-anode region in the discharge channel to the central cross section of the channel outlet(along the red straight line in figure 3).The z-axis represents the length of the red line,and the vertical coordinate is normalized to the maximum value of Case 1.

Figure 3.Magnetic field con figuration calculated with FEMM software.

3.Results and discussion

3.1.PIC model description

Figure 4.Normalized magnetic field intensity distribution along the red straight line in figure 3.

With the increase in D,the ion energy and ion current density on the ceramic walls decrease rapidly to their minima when the distance from the intersection point to the bottom of the discharge channel is L(i.e.the intersection point is at the outlet of the BN channel).In this case,only low-energy ions bombard the wall;therefore,the life expectancy of such CHTs may increase.

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Here,I = 12.1　is　the　ionization　energy　for　Xe1+,andEj = 8.7.

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The secondary electron emission model is also taken into consideration at the channel walls.The material of the channel wall is BN,and the secondary electron emission model taken in the model is optimized as shown in the following formula[23].

The computational domain applied in our study of a lowpower CHT is illustrated in figure 5.The discharge channel width is 10 mm,and the plume region is 38.5mm in length and 30 mm in width.The numbers 1–4 represent the boundaries of the discharge channel;6 and 9 are the free space boundaries;5 is the magnetic pole;7 and 8 are the cathode boundaries;and 10 is the symmetry boundary.The oblique boundary in the discharge channel at the bottom represents the anode.Due to restricted division of PIC computational meshes,we substitute the zigzag-shaped anode for the oblique anode to meet the simulation requirements and ensure that the magnetic field within the whole channel computational area does not include a high-intensity magnetic field at the MIA.The MIA structure is adjusted to shift the crossover point of the CMflwith the channel wall toward the channel outlet.Thus,the anode boundary shape of the simulation model is also adjusted from the red boundary(Case 1)to the purple boundary(Case 4)during the simulation.During the calculation process,there are about 106particles simulated in this model;the time step was set as 10−11s;the dimensions of the mesh we adopted here was 0.25 × 0.25 mm2in the discharge channel and 0.25 × 1mm2in the plume region;the mass flow rate of propellant(xenon)was ma = 3 sccm;the voltage applied to the anode was Ud = 200V;and the cathode voltage was set to 0V.

Here,εerefers to the energy of the incident electron that hits against the wall surface;W0,Wr,W2,andW1refer to the probability that the incident electron is absorbed,is subject to elastic reflection,hits two real secondary electrons,or hits one real secondary electron,respectively.P0　=　0.5,　α0=　43.5eV,Pr　=　0.5,　αr =　30eV,and　α2=　127.9eV.Moreover,CEX mainly takes place in the plume region,where the high-energy ions collide with neutral gas.We are chie fly concerned with the energy losses caused by the ionization process in the channel and the recombination of ions in the walls.Therefore,CEX is not taken into account.The Coulomb collision between electrons and ions is also not considered in the model due to its small contribution.Furthermore,in order to accelerate the computation process,the number of grids and iterations is reduced by increasing the permittivity of the vacuum by a coefficient of γ2in the model.Considering the charge unbalance,the coefficient γ2is set as 144.Moreover,another coefficient 1/f is added to the mass of heavy particles;then,the time scale will decrease by because the velocity of ions increases.Correspondingly,the cross section of the collision should also be increased by times to maintain the plasma density.In accordance with the discharge parameters of Hall thrusters,we set as 0.04[24].In this study,using the PICMCC simulation model,we analyzed the potential and ionization rate distribution in the discharge channel of our CHT,along with the ion energy and current density on the wall.

Figure 5.Simulation domain of a CHT.

图10为中子探测效率及中子能量分辨率随质子准直器高度的变化关系。由图10可见，随着质子准直器高度的增加，中子探测效率上升较快，中子能量分辨能力略有降低。当质子准直器的高度大于2 cm时，高度方向的反冲角展宽由二极偏转磁铁的磁隙高度决定，质子准直器高度增加时，中子探测效率和中子能量分辨率均变化很小。图11为中子探测效率及中子能量分辨率随质子准直器长度的变化关系，计算中设定质子准直器后端位置，即图1中L2不变，改变前端的位置，即图1中L1。由图11可见，随着质子准直器长度的增加，中子探测效率逐渐提高，中子能量分辨能力逐渐降低。

3.2.Influence of different MIA structures on discharge characteristics

The ionization and acceleration characteristics in the four types of magnetic field con figurations in figure 3 are simulated using the above PIC simulation model.The distributions of ionization rate in the discharge channel are displayed in figure 6(Cases 1 to 4).It is observed that with the continuous increase in distance from the crossover point of the CMfl crossing the anode boundary with the discharge channel to the bottom of the discharge channel(D),the ionization rate in the discharge channel increases.As shown in Cases 1 and 2 in figure 3,upon application of the MIA,a ‘near-zero’magnetic field zone is generated in the near-anode region,and the ionization rate increases.As shown in Cases 2,3,and 4,with the increase in D,the ‘near-zero’magnetic field zone shifts to the channel center.Thus,the ionization zone will gradually shift to the channel center.As we have mentioned above,in order to guarantee that the magnetic field within the channel’s computational area does not include a high-intensity magnetic field at the MIA,the zigzag-shaped anode adopted for computation is actually in front of the actual anode.Therefore,the ionization zone is actually in front of the ‘near-zero’magnetic field as well.

Figure 6.Ionization rate distributions.

Figures 9 and 10 present the normalized results of the ion energy and ion current density on the outer ceramic walls,respectively,based on Case 1.The x-axis indicates the channel length.

3.3.Influence of different MIA structures on the ion energy and ion current density

Figure 7 displays the potential distributions in the computational region in Cases 1 to 4.With the increase in D,the potential in the discharge channel shifts downstream along the channel.Based on thermalized potential theory[25],the potential on a magnetic field line is uniform at low electron temperatures,and thus,the potential at the CMfl intersection is roughly equal to that of the anode.specifically,the region from the crossover point of the magnetic field line to the bottom of the channel is in a high potential state as it rejects ions,and it is favorable for reducing the ion energy and ion current density on the dielectric walls,thus improving the thruster performance and lifetime.Figure 8 presents the distribution curves of the drop in potential along the channel median line based on the numerical results.It can be observed that with the increase in D,the potential on the centerline of the channel increases,and the potential drop moves downward along the channel.

Figures 9 and 10 indicate that when D equals L/3,2L/3,and L,the maximum ion energies are 53%,24%,and 19%of those in the case of no MIA structure(Case 1),respectively;meanwhile,the maximum ion currents are 42%,26%,and 18%of those in Case 1,respectively.Figures 6 and 7 can explain the decrease in ion energy and flux.With the increase in D,the ionization rate in the discharge channel clearly shifts downward to the channel center and the potential drop also tends to shift downward along the channel.Moreover,according to thermalized potential theory,the potential at the inner wall is in a high state,and thus rejects the positively charged ions,reducing the ion energy and ion current density on the ceramic walls.

PIC-MCC simulation combines the particle-in-cell method and Monte-Carlo method,and is often used to analyze the plasma parameters of Hall thrusters.This method can simulate the velocity,energy,and position of electrons,ions,and neutral gas.The model can also be used to study CHTs[17],low-power Hall thrusters[18],two-stage Hall thrusters[19],magnetic mirror effects[20],and power deposition on walls[21].The plasma in Hall thrusters is quasi-neutral,i.e.the electron number density neis locally equal to the ion number density ni.The mean value of the plasma density in the discharge channel is in the order of 1017–1018/m3.The atoms are calculated according to the free molecular flow model,where both atom–wall collisions and electron–atom collisions are taken into consideration while atom–atom collisions and atom–ion collisions are ignored.The differential cross section for ionization by an electron of primary energy Epwith emission of a secondary electron of energyEsis taken to be of the following form[22]:

whereσ(Ep)is the total ionization cross section for allEs,and

3.4.Influence of different MIA structures on performance of low-power CHTs

In PIC calculations,thrust is the sum of the mass of all ions that leave the open boundary within a unit time,and the product of their respective axial velocities.specific impulse is calculated according to thrust and anode mass flow rate.The computational formula is stated as follows:

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Figure 7.Potential distributions.

Figure 8.Normalized potential distributions on the centerline of the channel.

Here,Ispis the specific impulse,mais the anode mass flow rate,T is the thrust,and g is the acceleration of gravity.Table 1 displays the performance parameters,including thrust,anode Efficiency,and specific impulse,of our CHT at the operating parameters of 200 V and 3 sccm after calculation by the PIC-MCC simulation model.

Figure 9.Normalized ion energy distributions on the wall.

From table 1,it can be observed that the thrust,specific impulse,and Efficiency increase upon application of the MIA.When D equals L/3,2L/3,and L,the thrust increases by 12%,34.7%,and 52.0%,respectively;the specific impulse increases by 12.7%,28.8%,and 47.8%,respectively;and the anode Efficiency increases by 3.0%,8.3%,and 9.0%,respectively.In addition,the thrust and Efficiency reach their maxima when the intersection point is at L.In this case,the thrust is 4.1 mN,specific impulse is 1404s,and Efficiency is 24.5%.From Case 1 to Case 2,it can be seen that in addition to the change of the position of the CMflin the channel,the magnetic field intensity near the anode area is greatly reduced indeed.The change of the distribution of magnetic field within the discharge channel may exert some impact on the performance improvement of the thruster.However,such a near-zero magnetic field also exists in the near-anode region in Case 2,Case 3,and Case 4,and thus the performance gain caused by the MIA structure is remarkable.

Figure 10.Normalized ion current density distributions on the wall.

Table 1.Calculation of performance from simulation.

Performance　Case 1　Case 2　Case 3　Case 4 Thrust(mN)　2.7　3.0　3.6　4.1 Anode Efficiency(%)　15.5　18.5　23.8　24.5 specific impulse(s)　950　1071　1224　1404

4.Conclusions

Simulation results demonstrate that with the increase in distance from the intersection of the CMfl crossing the anode boundary with the wall to the bottom of the thruster channel until the channel outlet,the rate of ionization in the discharge channel increases continuously,the main ionization region tends to shift downward to the channel center,the potential drop tends to shift downward along the channel,the ion energy and ion current density on the discharge channel wall gradually decrease,and the thruster performance(thrust,specific impulse,and Efficiency)significantly improves.When the intersection point is at the channel outlet,the ion energy and ion density on the thruster walls are at a minimum.In this case,the thruster performance is optimal.

The study was financially supported by National Natural Science Foundation of China(Grant Nos.51777045 and 51477035)and Shenzhen Technology Project(Project Nos.JCYJ20160226201347750 and JCYJ20150529115038093).

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