Yuguo CHENG, Gungqing XIA

a PLA 91550 Element 41, Dalian 116023, China

b State Key Laboratory of Structural Analysis for Industrial Equipment, Dalian University of Technology, Dalian 116024, China

c Key Laboratory of Advanced Technology for Aerospace Vehicles of Liaoning Province, Dalian University of Technology, Dalian 116024, China

KEYWORDS Argon and helium propellants;Circuit characteristics;Circuit-fluid model;Flow properties;Pulsed inductive thruster

Abstract The Pulsed Inductive Thruster (PIT) has the advantages of repeatable startup, no corruption and in-situ propellant feed.To study the flow expansion and circuit characteristics of PITs,the circuit-fluid model is developed,and the high temperature thermodynamic and transport models are combined with the circuit-fluid model to predict the critical plasma parameters. The flow fields of initial mass of 2-8 mg and charge voltages of 10-14 kV are simulated. Comparison of the flow fields of argon and helium propellants suggests that,the flow field structures are similar.Slight differences exist on the magnitude of the density and magnetic field,caused by larger velocity in lighter atom case and difference on the ionization gap between adjacent ionization levels. Analysis of the circuit characteristics by the two-dimensional results indicates that the ratio of coil inductance to circuit inductance affects both the rise rate and phase of the plasma current, the larger the ratio,the greater the rise rate and the better the following characteristic. The calculations show that the magnetic energy obtained within the decoupling distance determines the overall performance the thruster can be obtained; self-induced field maintained by the thermal motion after the main pulse leads to the long attenuation process and difference on the total impulse when the angle of conical pylon is varied under constant coil dimension.

1. Introduction

Deep space exploration has been of continuous interests to human kind for decades and in China, the 2020 Mars exploration mission has been proposed,1meanwhile, the Chang’e lunar exploration program is in progressing,2there is a growing demand for advanced propulsion system that can be used in accurate manipulation of the satellite, or as main engine of the probe, etc. After years of research and development, the electric thrusters are gradually taking the role of chemical thrusters in space propulsion.3,4For an ideal electric thruster,at least three qualities are required.The first is repeatable startup and accurately controlled impulse to fulfill specified instructions;the second is little damage to the structures,especially the electromagnetic interference and plume contamination, so that long distance planet travel is possible; the third is the in-situ feed to reduce the propellant and increase the payload on board.Considering all these requirements,a promising choice is the pulsed inductive thruster.5As is reviewed by Polzin,6the kind of propulsion avoids corruption naturally and the in-situ feed is applicable to PITs owing to good propellant compatibility.

In the early studies on the PITs, key parameters affecting internal flow structures were investigated experimentally under relatively low pulsed energy levels. The thruster designed by Dailey and Lovberg7,8possessed an inductive coil of diameter of 0.2 m, the investigation at energy of 285 J on the structure of high density sheet found that ions do not carry current and are accelerated by the electric field. Following the 0.2 m diameter coil, a 0.3 m coil was developed,9into which additional pre-ionization were introduced, and the pulsed energy was elevated to 675 J. Plasma probes were used to measure the current density and magnetic field intensity,and the vectors are multiplied to evaluate the performance.Test results suggested that larger diameter results in better coupling, and the initial uniformity of the gas is crucial to the performance.In the latter development of PITs, the diameters of the most prototypes were extended to 1.0 m,10and the one turn spiral wires11or Marx-generator coil topology12were employed. The pulsed energy was raised to thousands of Joules, and more attentions were paid to the design of long life time pulsed valve and time scale matching between the electric circuit and discharge. The experiment data showed that high initial current rise rate leads to better performance,and incomplete plasma sheet is formed when the stray inductance is increased. The work during this period highlighted the importance of large coil inductance Lcrelative to the circuit inductance Le.

Besides the experimental trials, numerical efforts were performed based on the circuit and fluid models. The onedimensional circuit model assumes the discharge circuit as the primary and plasma as the secondary of the ‘‘transformer”,13and the plasma circuit parameters are calculated by empirical formulas. The model predicted the discharge regime where constant efficiency is held over a wide range of specific impulse,12and the non-dimensional study indicated that under damped circuits are required for greater efficiency.14The fluid description was realized by the MACH code,15and the simulations reproduced two important findings by the experiments on MK I and MK Va prototypes:The first is the critical mass phenomenon16;the second is the higher performance on NH3,which is attributed to the relative low radiation loss of the propellant.17

Despite the efforts mentioned above, several problems remain to be solved to provide more insights into the pulsed inductive acceleration. The first is the spatial characteristics of the transient flow and the differences of varied propellants;the second is the study of circuit characteristics, which concerns on modification of the circuit model by fluid calculations,so that accuracy of the circuit model can be elevated; At last,the effect of varied configurations on the performance should be ascertained when working mechanisms of the thruster are clarified by the previous studies. In this paper, the twodimensional properties of the plasma are studied first, then the plasma circuit characteristics are investigated using the results obtained from the fluid simulations, flow fields under varied angles of conical pylon are analyzed at last.

2. Model description

2.1. Governing equations

The plasma sheet forms during the initial pulse rise,which is a high density region moving forward,the collision frequency of the particles in the sheet are high enough so that the electrons and ions can be assumed to be in local thermodynamic equilibrium state. In this section, the unsteady MagnetoHydroDynamics (MHD) model is employed to depict the flow of transient inductive plasma. The Navier-Stokes equations are revised by the hyperbolic cleaning algebra equation to maintain conservation of magnetic divergence in the calculation,18the continuum, momentum, energy, magnetic inductive and hyperbolic cleaning equations are as follows:

where Vcis the capacitor voltage, Vpis the plasma voltage, Jcis the coil current. The capacitor is of capacitance C=15 μF and discharge voltage V0,the external circuit is with empirical resistance Re=5 mΩ and inductance Le=120 nH if not mentioned. Vpis calculated by RJcand R=90 mΩ is assumed.The employment of constant resistance R is to give high rise rate and microsecond period current pulse,which is not related to actual plasma resistance. The radial magnetic intensity is calculated by Ampere’s law. Eqs. (1)-(6) form the circuitfluid model predicting the performance of PITs.

2.2. Thermodynamic and transport properties

2.2.1. Thermodynamic property

For Argon and Helium plasma,the thermodynamic state deviates from the idea gas when temperature T is high.During the discharge of PIT, number density n, specific internal energy,etc. at temperature greater than 2.0 eV are required, creating a need for self-consistent estimation of high temperature thermodynamic property.The perfect gas law,charge conservation law and Saha equations are iteratively solved to get necessary parameters and numerical procedure is shown schematically in Fig. 1.

The number densities of Ar and He plasmas as temperature is varied are plotted in Fig.2(in Fig.2,1 atm=105Pa).Density of high ionization species increases and low ionization species decreases when the temperature grows, and the electrons maintain at a stable level due to more charged particles generated from high ionization species. Compared with Ref.20of argon propellant at atmospheric pressure, calculations here reproduce the amplitude and variation trend of the curves,and good accuracy is obtained.

Fig. 1 Numerical procedure calculating p, n and T.

2.2.2. Transport property

The transport coefficients are related to the collision and thermodynamic parameters, and affect the energy transportation of PIT especially in the first half period. In order to obtain the electrical conductivities at varied temperature and pressure, the first order approximation solution of the Boltzmann equation is adopted here.21The calculation procedures for collision integrals and transport coefficients are shown schematically in Fig. 3, and numerical schemes are mainly employed that used in Ref.20and each step is evaluated with assigned accuracy to insure convergence of the algorithm.

The electrical conductivities of Ar and He plasmas in local thermodynamic equilibrium state as temperature is varied are shown in Fig.4.The main growth of the value occurs after first order ionization takes place, and is greater at higher pressure when the temperature is high. Compared with Refs.20,22of argon propellant at atmospheric pressure, the results here reproduce the amplitude and oscillation of the value at constant pressure, and correctness of the algorithm is verified.

2.3. Calculation configuration and model validation

2.3.1. Calculation configuration

Calculation domain in rOz coordinate system is shown in Fig.5,enclosed by polygon ABCDEFG,the simulated mechanism is comprised of the coil(AB), conical pylon(FGA) and confining cuff(BC). The following characteristics are specially reproduced: the first is the boundary conditions: boundary AB, where Eq. (6) applies, wall boundaries BC, FGA, outlet boundaries CD, DE, and symmetric axis EF, are adjusted to accommodate the physical properties of actual thrust; the second is the conical pylon, with an angle θ relative to the axis,which simulates the experiment configuration injecting and confining the propellant.

In the following work, the initial mass, charge voltage and conical angle are changed to study the working mechanisms.The initial static assumption is implemented in the calculation,and the gas is uniformly distributed in axial direction within the length of BC, and the temperature is set to be 0.1 eV.Spatial deviation employs the M-AUSMPW+scheme and the third order TVD-RK is adopted in the time advancement.

Fig. 2 Number densities as temperature is varied at different pressures.

Fig. 3 Numerical procedures calculating collision integrals and transport coefficients.

Fig. 4 Electrical conductivities as temperature is varied at different pressures.

Fig. 5 Calculation configuration of the pulsed inductive flow.

2.3.2. Model validation

To enhance credibility of the simulation, the model developed is demonstrated from two aspects:the first is the ability to distinguish strong magnetic discontinuity, meanwhile, to check the computational accuracy, the MHD shock tube problem is employed, the detailed configuration and initial conditions are given in Ref.23Using the numerical scheme employed in present work, the density of the tube ρtubeis shown in Fig. 6.The comparison with CDS and Roe flux splitting methods shows the location and peak value of slow compound wave are captured, and in other contact surfaces, although dissipation exists, the discontinuities can be distinguished. The second validation is prediction of the practical performance of PIT,the comparison of total impulse Itwith Ref.16was performed in Ref.26and good consistency is obtained.

Fig. 6 Comparison of density using scheme in present work,fourth order Compact Difference Scheme (CDS)24 and Roe flux splitting method.25

3. Results and discussion

In this section, the propulsion parameters, transient fields,and circuit characteristics of a virtual thruster are studied.Temporal evolution properties of the flow are described in Sections 3.1 and 3.2. Section 3.3 mainly concerns on the effect of different conical angles on the field. In Sections 3.1 and 3.2,geometric parameters are as follows: LOA=0.1 m (inner radius ri), LOB=0.3 m (outer radius ro), LBC=0.15 m,LBD=0.6 m, LOF=0.2 m, LGF=0.05 m, and consequently θ ≈14°.

3.1. Flow characteristics of Ar and He propellants

The radial magnetic intensity Br0at coil-plasma boundary excited by the pulse generation circuit at V0=14 kV calculated by Eq.(6)is shown in Fig.7.The amplitude after the first half period is adjusted, which attenuates quickly and approaches zero at about 20 μs.The pulsed form insures majority magnetic energy is generated within the first half period.

In the simulation of PITs, the magnetic field is one of the most important considerations, and an accurate algorithm should simulate the intensity properly.Here charateristic magnetic diffusion depth is taken as the reference parameter to determine the cell size, as that done in Ref.16The typical rise time of the excitation current in the work is about 1.55 μs.By using the empirical formular in Ref.,16the depth can is calculated as

where ηdis the electrical diffusivity,and ηd～1000/T3/2in preliminary estimation, then the diffusion depth is about 7.0×10-3-3.9×10-2m for temperature range of 1-10 eV.The results show the depth can be increased as the temperature decreases, and the cell size is set to be within this range. The CFL number is 1×10-5in the calculation.

When analyzing the flow field in the following sections,the two-dimensional distributions at V0=14 kV and moments corresponding to peak intensities at 1.55 μs, second zero at 5.2 μs and zero asymptotic at 20 μs are chosen,if not mentioned.

By employing the algorithm in Ref.27the decoupling distance z0can be evaluated by curve fitting the formula preliminarily:

Fig. 7 Radial magnetic intensity for V0=14 kV at coil-plasma boundary.

3.1.1. Variations of density profiles

The two-dimensional density distributions of Ar and He plasmas,under the conditions of m=4 mg,and t=1.55,5.2 and 20 μs,are given in Fig. 8.The main differences are the density in the core and the velocity during acceleration.Density in the core of Ar plasma is larger than that of He plasma, while the sheet thickness of the latter is greater than the former, as can be seen from Fig. 8. The core density rises to 3.5×10-4kg/m3for Ar and 3.1×10-4kg/m3for He at 5.2 μs and decreases to 1.8×10-4kg/m3for Ar and 1.4×10-4kg/m3for He at 20 μs, showing expansion of particles to low density coil surface region is more prominent at light atom mass case.

The density profile illustrates macroscopic movement of the particles, and the trajectories can be described as follows: in the initial,the gas on the coil surface is ionized by the magnetic energy transmitted via the coil-plasma boundary, generating a high density and temperature front, then the majority charged species are constrained within the plasma sheet developed the pulse rise period, leaving behind a small amount of residual gas.The boundary between the high density sheet and low density surface region is distinct. As the plasma sheet moves forward, the magnetic heating effect weakens, particles on the edge of the sheet moves toward the surface, leading to increment of the sheet thickness and gradual decrease of the density in the core.

The core densities of the sheets as time is varied for m=2,4 and 8 mg of Ar and He plasmas are shown in Fig. 9. The curves give more insights into the difference between the two species. The He plasma moves at a higher velocity than Ar plasma, and the more intense thermal motion of the former leads to lower density during the whole process. Rise rate of the density changes at the moment when reaching the decoupling distance, then the density rises at a lower rate level to maximum value. For Ar plasma, when reaching decoupling distance, the core densities are about 1.5×10-4kg/m3(t=2.4 μs), 3.2×10-4kg/m3(t=3.2 μs), 5.7×10-4kg/m3(t=4.8 μs) for m=2, 4 and 8 mg, and account for 75.0%,89.5%, 86.4% of the maximum values, respectively, showing that the largest density increment is obtained within the decoupling distance.The curves also show that density decreases at a higher rate in larger propellant mass case than in lower mass case.

Fig. 8 Two-dimensional density distributions of Ar (left) and He (right) plasmas at different time.

Fig. 9 Densities in core of plasma sheet for Ar and He plasmas as time is varied at different propellant masses (V0=14 kV).

In these cases,only one high density sheet is generated,and no additional essential sheet emerges during the pulse period,as can be seen from large area of low density region in Fig.8.The results indicate majority gas is expelled and utilized during the first half period and well coupling between the coil and plasma. In the model employed here, detailed current deformation caused by circuit inductance is not considered,the existence of the latter leads to ill coupling between the excitation current and plasma, and is responsible for a second sheet formation, accompanied by lower velocity. By calculating Eq. (6), the effect of varied inductance on the pulsed excitation current can be evaluated preliminarily. Comparison of the coil currents Jcunder different circuit inductances is shown in Fig. 10, which indicates that the circuit inductance affects both the magnitude and rise rate of the current,and the lower the circuit inductance,the higher the rise rate.A useful suggestion generalized from the analysis is that, to improve the performance,inductance of the circuit should be minimized when choosing appropriate circuit materials, so that the pulsed current rise rate is high enough to generate effective plasma sheet and resistive component of the plasma is dominant to insure maximized energy deposition.

Fig. 10 Coil currents as time is varied at different circuit inductances.

The number densities of ion species for Ar and He plasmas at 20 μs are illustrated in Figs.11 and 12.A prominent feature seen from the figures is that number density of high ionization species increases and the low ionization species decreases as axial distance grows, and the trend is clearer when axial variations of each species are compared in Fig. 13. The number density is mainly controlled by the thermodynamic properties,and the distribution indicates the temperature decreases as the axial length increases. The main difference between the two propellants is spatial distribution of Ar plasma is stripe shape,each species occupies a narrow region, while for He plasma,He±dominates space that the mixture occupies, and He2±is a tiny amount compared with He±. The difference can be explained as follows: the ionization energy of He±, which is 54.1 eV, is far greater than the neutral atom, which is 24.6 eV, and the gap between the two energy levels suggests the temperature should be greater than 5.0 eV,so that number density of He2+is comparable to He±.However,the request is beyond the temperature calculated here in the most of region,especially when the plasma moves away from the coil surface,therefore, the flow field of He plasma mainly consists of the electrons and single charged ions. As for the Ar plasma, the ionization energies of the first, second and third levels are 15.8, 27.6 and 40.7 eV, respectively, the gap between adjacent levels is smaller than He,leading to stripe distribution for each species.

3.1.2. Variations of magnetic field

The inductive magnetic field is determined by the velocity and electrical conductivity, as shown in Eq. (4), and the latter is related to the temperature and number density. By analyzing the two-dimensional distribution,the effect of the fluid characteristics on the magnetic field can be analyzed.

Fig. 11 Number densities of ion species for Ar plasma at 20 μs.

Fig. 12 Number densities of ion species for He plasma at 20 μs.

Fig. 13 Number densities of ion species in axial direction at r=0.15 m and t=20 μs.

Fig. 14 Comparison of field intensity under conditions of no convection velocity with convection process included at different time.

In Fig. 14, the magnetic field of neglecting the convection term in Eq.(4)is calculated to compare with the case including the term.Variations in Fig.14 suggests that the existence of the convection makes the magnetic intensity (Br) decrease in a lower rate, which leads to less ohm heating, proportional to（? ×B）2,within the plasma sheet,and is beneficial to improve the propulsion efficiency.

The radial inductive magnetic field intensities at 1.55, 5.2,and 20 μs for Ar and He plasmas are given in Fig. 15. In the initial discharge, the intensity is uniform in radial direction and decreases monotonically in axial direction, and the front boundary of the magnetic field nearly overlaps with the high density front. As time increases, the nonlinear variation is enhanced, caused by reversing of the excitation current and induced field by the plasma, meanwhile, the influence domain of the inductive field is greater than the density, showing efficient ionization of the background gas. The positive high intensity region moves towards the pylon boundary after the initial pulse rise. At 5.2 μs, the aforementioned region coincides with the plasma sheet,and the high value there is mainly caused by large velocity gradient. At 20 μs, the high intensity region is no longer occupied by dense particles, besides the convection of low density residual gas, the magnetic diffusion on the right hand side of Eq.(4)contributes to the high value,for the inverse of electrical conductivity is larger at lower pressure, as shown in Fig. 4.

3.2. Circuit characteristics of pulsed inductive plasma

The propellant is ionized by transient magnetic field excited by pulsed current generation LRC circuit. The plasma is with a load of resistance Rpand inductance Lp. The twodimensional simulation in Section 3.1 provides the necessary information evaluating the plasma circuit characteristics,which has shown that the acceleration processes of the propellants are similar. In this section, the circuit characteristics of Ar propellant are studied. The resistance is calculated by volume integration of the current density as:

where η is the resistivity. Rpas time is varied at V0=14 kV and m=2, 4 and 8 mg is shown in Fig. 16. Three different stages emerge successively can be seen from the curves. The effect of considerable charged particle generation appears first,Rpdecreases from 140 mΩ in the initial quickly to 20 mΩ at about 0.4 μs as the number density and temperature rise. Rpmaintains at a low level on the order of 5 mΩ after 1.5 μs.Combined with the former analysis, this period corresponds to the high ionization degree of the plasma and slow variation of the thermodynamic parameters within the decoupling distance. The third stage symbolizes the gradual divergence of Rp. In Eq. (9), the denominator approaches zero during this period as shown in Fig.7,and value of the numerator is maintained due to the continuous heating and thermal motion.The decrease process of inductive magnetic field lasts a longer period than the excitation source, as can be seen from Fig. 15,where the average value of Brat 5.2 μs is about 1/3 at 1.55 μs, although Br0approaches zero at 5.2 μs. The comparison shows a much slower attenuation rate of the deposited magnetic energy than the input pulsed energy, and the slowly varying resistance beginning at about 1.5 μs can no longer be maintained after about 3 μs.

Analysis on the resistance further confirms the conclusion above that the similarity between the Ar and He propellants,and shows that the flow characteristics is intrinsic under the excitation of this kind of electromagnetic energy, and the similarity is not due to neglecting the time-varying characteristics of the resistance in the pulsed current generation model used in Eq. (6).

Fig. 15 Radial magnetic intensities of Ar (left) and He (right) plasmas at different time.

Fig. 16 Plasma resistances as time is varied at different propellant masses (V0=14 kV).

The plasma resistance, which incorporates the ionization and location information of the sheet, is the measurement of energy absorption capacity, and energy distribution in turn will affect the total plasma current Ip, the mutual inductance M. The correlation of Rp, Ip, and M can be expressed as28:

where Lcis the coil inductance.In the above equations,the circuit characteristics are related to spatial distribution of the flow by the plasma resistance and axial moving distance. M is mainly determined by axial length; Ipis determined by mutual inductance, coil current, and varies negatively with the resistance.In Eq.(10),the distance is calculated by the time integration of axial velocity and in Eq. (11), the resistance is fitted using calculation by Eq. (9). The mutual inductance and plasma current as time is varied at different cases are shown in Fig. 17.

In Fig.17(a), two different descending rates of the curves exist. In the initial, mutual inductance equals to Lc, and is about 200 nH, and maintains at this level before 0.5-0.8 μs,corresponding to initial energy deposition with little macroscopic movement, and relative displacement between the coil and plasma approaches zero. The process is followed by a gradual decrease when axial moving distance of the plasma sheet approaches and exceeds the decoupling distance. Based on the calculation of Eq. (11), the slow decrease of inductive magnetic field can be explained from the circuit point view.The mutual inductance is a continuous variable and decreases exponentially with axial distance,and as the plasma moves far away from the coil,the value will decouple from the excitation current. The principle is applicable to the inductive magnetic field,which makes the interaction between the plasma and coil exists.

Fig. 17 Mutual inductances and currents at different voltages as time is varied (m=4 mg).

Variation of mutual inductance leads to different current evolutions, as shown in Fig.17(b). The plasma current varies synchronously with the circuit current in the initial due to little movement and efficient energy deposition into the plasma. As the mutual inductance decreases, interaction between the coil and plasma weakens,the pulsed energy obtained by the plasma suffers from the loss of amplitude and shift of phase, and separation of the two current curves occurs.The currents at different coil inductances in Fig. 18 shows that following characteristics of the plasma current is enhanced as coil inductance increases,and better coupling is obtained.Then a second approach improves the performance is increasing the coil inductance, besides the method mentioned in Section 3.1.1.

Fig. 18 Plasma currents at different coil inductances(V0=14 kV, m=4 mg).

In Eq. (9), the resistance is integration of the control volume on the numerator, and the integration within the plasma sheet accounts for the majority, as can be from the inductive magnetic intensity distributions in Fig.15 and density distributions in Fig. 8, where the inductive region mainly exists in the sheet during the first half period can be drawn. The same mechanism applies to the evaluation of mutual inductance in Eq. (10) and total plasma current in Eq. (11). Consequently,the circuit characteristics are also related to the shape of the plasma sheet, as is also shown in the one-dimensional model of Ref.28when estimating the resistance.

Since the sheet is with curved boundaries in the front and the back, and is relatively uniformly distributed along the radial profile, the gradient mainly exists in axial direction,where the thermal conduction and particle kinetic movement is prominent. The thickness hcsof the plasma sheet is then an important parameter that can reflect the force exerted on the plasma.In the calculation of thickness,the following rules are adopted:

(1) The thickness is measured at the center of the sheet, so that the comparison with the rectangular configuration in Ref.28can be made.

(2) The back of the sheet is determined at the location where the density is one fifth of the core.

The thicknesses at different initial mass as time is varied are plotted in Fig.19,where the curves with three different slopes,transitioned at ttrans=3.5 μs, 6.5 μs for 4 mg and 5.2 μs,10.1 μs for 8 mg, are shown. By comparing with the density variations in Fig.9,the first transition corresponds to the stage when reaching the decoupling distance, the second transition occurs at the moment reaching maximum density. The accordance suggests that the electromagnetic field not only affects the density, velocity, etc, but also the spatial dimension of the plasma sheet. The main increment of the thickness occurs in the third period, when excitation current weakens. The curve during the first period can be fitted approximately using the following expression:

Fig. 19 Thicknesses of sheet as time is varied at different propellant masses (V0=14 kV).

3.3. Effect of conical pylon angle on flow

In the simulation of PITs,the coil configuration and geometric parameters of the conical pylon are usually assumed to be constant. Once the coil configuration is determined, as a nonenergy output mechanism,the effect of conical pylon is decoupled from the coil. The shape of the pylon influences the performance by the following two ways:

(1) The first is the initial distribution of the propellant,which is a combined effect of feed speed and pylon shape. The initial uniformly distributed gas has been suggested to be desirable to obtain uniform plasma sheet to enhance the performance.6

(2) The second is flow direction around the pylon boundary,the flow will expand in larger or smaller space depending on the angle of pylon.

In this section,the pylon angle is varied to study the disturbance mentioned in the second point, and the coil size and height of the pylon are in accordance with previous analysis.The cases of LGF=0.075 m and LGF=0.025 m,with the corresponding θ ≈7°, 21°, respectively, are studied.

The final impulse are 31.8×10-3, 31.4×10-3,31.2×10-3N˙ss for θ ≈7°, 14°, 21°, respectively. The results show that varied θ within the range have little effect on the main energy deposition period. Thermal expansion is a subordinate process compared with the inductive excitation not only by the time, but also the space.

In Fig. 20, the magnetic field and density, which directly determine the performance,are given the two-dimensional distributions. Structures of the inductive magnetic field in radial and axial direction are similar, indicating similar interaction between the coil and plasma sheet. Slight difference exists in the maximum density, the value is lower at larger space case caused by the lower initial density, and in the inductive magnetic field around the turning point, and is negligible in the thrust integration.

In general, the geometry of the pylon is a minor factor affecting the plasma behavior when the coil configuration is determined in the studied range, and the macroscopic properties are little disturbed by the angle. Besides the propellant mass, the coil configuration and pulsed current are the main factors that determine the performance.Then the most important consideration in designing the conical pylon is the combination with the high pulsed valve, so that the gas can be uniformly distributed on the coil surface and the feed period is compatible with the pulsed current to maximize the electromagnetic energy deposition.

4. Conclusions

The fluid model is introduced to give insights into the flow properties of pulsed inductive thruster,and circuit characteristics of the plasma are studied using the two-dimensional fluid results. In the development of the model, efforts are made to accurately calculate the high temperature thermodynamic and transport properties to obtain self-consistent plasma parameters. The calculations under different input conditions show that distributions of the flow are closely related to the evolution of the pulsed current and coil dimension, and selfinduced field maintained by the thermal motion after the main pulse leads to the long attenuation process. Intense flow parameter variations occur during the initial pulse rise, and the comparison of coil current and plasma current suggests the increase of the ratio of coil inductance to circuit inductance to enhance the initial rise rate of the pulsed current.

The comparison of the Ar and He propellants flows suggests that no prominent difference exists in the macroscopic performance. Slight differences exist on the density and magnetic field distributions caused by larger velocity in lighter atom case and difference on the ionization gap between adjacent ionization levels. The calculations at varied conical pylon angles show negligible effect of thermal expansion in the initial and limited effect on the disturbance of total impulse, and ascertain that the design of coil configuration and circuit parameters is the most important consideration when developing the PITs, once high pulsed valve can work reliably.

Fig. 20 Two-dimensional density and magnetic field distributions at different angles (V0=14 kV, m=4 mg).

Further improvement of the model is the adoption of more reasonable excitation current revised by the time-varying plasma load in the governing equations, which will affect the reversing time and attenuation rate of the current, and consequently, acceleration period of the plasma. Although no significant change in the performance, Differences on the field structures may emerge,like the generation of a second plasma sheet that has not been reproduced in numerical work yet.

Acknowledgments

This work is supported by the National Natural Science Foundation of China (Nos. 11675040 and 11702319).