Introduction

Against the backdrop of continuous advancements in space propulsion systems toward higher specific impulse, higher precision, and longer operational lifetime, the Radio-Frequency Plasma Thruster (RFPT) has emerged as an important research direction for micro-satellites and deep-space exploration missions due to its electrode-free structure, high thrust-to-weight ratio, and strong adaptability1,2,3. This type of thruster generates plasma by exciting propellant gases with radio-frequency electromagnetic fields and accelerates the plasma through electromagnetic mechanisms to produce thrust. As a result, RFPTs have been widely applied in attitude control, orbit transfer, and deep-space navigation4,5,6,7. However, current RFPTs face challenges such as low ionization rates and insufficient electromagnetic coupling efficiency during practical operation, which constrain their overall propulsion performance8,9. To address these issues, recent research efforts have increasingly focused on enhancing plasma generation and transport processes via externally applied static magnetic fields. Magnetic-field-assisted techniques not only improve energy absorption efficiency and electron density but also effectively suppress wall losses, optimize plasma distribution, and enhance discharge stability, thereby demonstrating strong potential for engineering applications10,11. Elucidating the underlying mechanisms between magnetic field parameters and plasma behavior is therefore of great significance for advancing the practical implementation of RFPT technology.

In this context, both experimental and simulation studies have demonstrated that magnetic field configurations exert a significant influence on RF discharge processes12. Considerable attention has been devoted to improving the ionization efficiency of RFPTs by introducing external static magnetic fields. Sho Ito et al.13 systematically investigated the impact of different permanent magnet (PM) configurations on RFPT performance, revealing that when the magnetic flux density reached 290 mT and the magnetic field peak was located at the RF antenna center, both thrust performance and plasma utilization efficiency were markedly improved. Wen Xiaodong et al.14 further explored the application of magnetically enhanced high-power RF plasma sources in electromagnetic thrusters, highlighting the crucial role of magnetic field structures in plasma confinement. Xiang Ma et al.15 studied the characteristics of RF discharges under non-uniform magnetic fields and found that positioning the antenna in a strong magnetic field region significantly enhanced energy absorption efficiency and electron density. Despite these advances in mechanism understanding and parameter optimization, systematic quantitative analyses of multi-factor coupling effects—such as magnetic field gradients, frequency response, and pressure matching—remain limited.

Motivated by these challenges, this work investigates an RF plasma source with a discharge chamber diameter of 32 mm. A two-dimensional axisymmetric multi-physics model is established to compare plasma density and temperature variations under conditions with and without permanent magnet confinement. Furthermore, the effects of RF power and other parameters on plasma density are examined. Particular emphasis is placed on exploring the synergistic optimization mechanisms of magnetic field gradients, frequency tuning, and pressure matching for ionization efficiency enhancement, thereby providing theoretical guidance for the design of high-performance RF plasma thrusters.

Plasma fluid model

Computational model

Figure 1 illustrates a two-dimensional axisymmetric model of the thruster discharge chamber. The discharge is driven at a radio frequency of 13.56 MHz. The chamber consists of a quartz tube with an inner radius of rd=16 mm and an axial length of La =350 mm. The reference background pressure is set to 0.05 Torr, and the initial gas temperature is 300 K. A multi-turn antenna with azimuthal mode number m = 0 is placed at the midsection of the discharge chamber, with an inter-coil spacing of 20 mm. The applied permanent magnetic field is generated by an alternating arrangement of five N54 permanent magnets and four soft iron pieces. The dimensions of the N54 magnets and the soft iron elements are shown in Fig. 2. The residual magnetic flux density of the N54 magnets is 1.47 T. The physical properties of each component are summarized in Table 1.

Fig. 1
figure 1

Two-dimensional axisymmetric model (a) and three-dimensional structure diagram (b).

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Fig. 2
figure 2

Dimensions of N54 permanent magnets and soft iron.

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Table 1 Physical properties of materials for each part.
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This section provides a systematic description of the multi-physics model, including fluid physics modeling, plasma behavior simulation, electromagnetic field coupling mechanisms, and energy deposition calculations. Figure 3 illustrates the overall framework of the physical processes computed in COMSOL18.

Fig. 3
figure 3

Physical process flow diagram computed in COMSOL.

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Fluid flow module

The discharge pressure simulated in this study is 0.05 Torr (approximately 6.67 Pa ), thus it is necessary to verify the applicability of the fluid model under this pressure condition. The Knudsen number (Kn), which reflects the rarefaction level of gases or plasmas, can be used for verification.

$$\:Kn=\frac{\lambda\:}{L}$$
(1)

\(\:\lambda\:\) is the mean free path of particles in the gas or plasma (unit: m ); \(\:\text{L}\) is the characteristic length of the system (unit: m).

The mean free path \(\:\lambda\:\) is the average distance traveled by a particle between two consecutive collisions. For an ideal gas, the mean free path can be expressed as:

$$\:\lambda\:=\frac{{k}_{B}T}{\sqrt{2}{\pi\:d}^{2}P}$$
(2)

where: \(\:{k}_{B}\) is the Boltzmann constant, \(\:\left(1.380649\times\:{10}^{-23}\hspace{0.33em}\text{J}/\text{K}\right);\text{T}\) is the gas temperature (unit: K); d is the diameter of gas molecules (unit: m); P is the gas pressure (unit: Pa). By substituting the working medium argon and discharge chamber data to calculate \(\:0.01<\text{K}\text{n}\approx\:0.033<0.1\), the behavior of the gas or plasma approaches that of a continuous medium, allowing for the use of fluid dynamics equations for description.

In the COMSOL plasma model, the fluid dynamics of neutral gas (argon) are described by the laminar Navier-Stokes equations, which include the following momentum conservation equation and mass conservation Eq20.

$$\:\begin{array}{c}\rho\:\frac{\partial\:u}{\partial\:t}+\rho\:\left(\nabla\:\bullet\:u\right)u=\nabla\:\bullet\:\left[-p\text{I}+\text{K}\right]+F\end{array}$$
(3)
$$\:\begin{array}{c}\frac{\partial\:u}{\partial\:t}+\nabla\:\bullet\:\left(\rho\:u\right)=0\end{array}$$
(4)

Inlet mass flow boundary condition22

$$\:\begin{array}{c}-{\int\:}_{\partial\:{\Omega\:}}\frac{{\uprho\:}}{{{\uprho\:}}_{\text{s}\text{t}}}\left(\text{u}\bullet\:\text{n}\right)dS={\text{Q}}_{\text{s}\text{c}\text{c}\text{m}}\end{array}$$
(5)

Where \(\:{\uprho\:}\) is the gas density, u is the fluid velocity vector field, K is the viscous stress tensor, and I is the identity tensor; \(\:\nabla\:\bullet\:\text{u}\) represents the divergence of the velocity field, which describes the rate of volume change of the fluid. \(\:{{\uprho\:}}_{\text{s}\text{t}}\) is the standard-state density. The wall boundary condition is no-slip, the inlet is set as a mass flow boundary condition, and the outlet is set as a pressure boundary condition. The background gas temperature is set to room temperature, 300 K.

Plasma module

The Plasma Module calculates electron collision reactions and heavy species reactions in the time domain by importing collision cross-section data, ultimately obtaining the distribution of plasma characteristic parameters. The electron energy distribution function is assumed to follow a Maxwellian distribution.The plasma bulk-phase chemical reactions considered in this study are summarized in Table 2, primarily including key processes such as excitation, ionization, and recombination. The surface reaction mechanisms at the wall boundaries are listed in Table 3, which characterize the recombination behavior of reactive species on the chamber walls.

Table 2 Simulated collisions and reaction Table17.
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Table 3 Surface composite reaction Table17.
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In the theoretical study of Inductively Coupled Plasma (ICP), establishing a comprehensive mathematical model requires the integration of the dynamics of charged particles in the plasma, self-consistent coupling of the electromagnetic fields, and energy transport processes. The physical significance of each governing equation and their interactions within the ICP system will be elaborated in the following.

The variation in electron number density in the plasma follows the basic continuity Eq18. :

$$\:\begin{array}{c}\frac{\partial\:{n}_{e}}{\partial\:t}+\nabla\:\bullet\:{\varGamma\:}_{e}={R}_{e}-\left(u\bullet\:\nabla\:\right){n}_{e}\end{array}$$
(6)
$$\:\begin{array}{c}{\varGamma\:}_{e}=-{\text{D}}_{\text{e}}\nabla\:{\text{n}}_{\text{e}}-{{\upmu\:}}_{\text{e}}{\text{n}}_{\text{e}}E\end{array}$$
(7)

Where \(\:{{\Gamma\:}}_{\text{e}}\) represents the electron flux (including diffusion and drift terms), \(\:{R}_{e}\) denotes the ionization or recombination rate caused by electron-neutral collisions, and the convection term \(\:\left(u\bullet\:\nabla\:\right){n}_{e}\) reflects the influence of background fluid motion on electron transport. uuu represents the macroscopic flow velocity of the plasma, and \(\:-{\text{D}}_{\text{e}}\nabla\:{\text{n}}_{\text{e}}\) is the diffusion flux. Electrons move from regions of high density to low density, and \(\:-{{\upmu\:}}_{\text{e}}{\text{n}}_{\text{e}}\text{E}\) represents the migration or drift flux.

Electromagnetic field module

The electromagnetic behavior of Inductively Coupled Plasma (ICP) is fully described by the following set of Eq19. :

Ampère-Maxwell Law:

$$\:\begin{array}{c}\nabla\:\times\:{\text{H}}_{RF}=J\end{array}$$
(8)

The curl of the magnetic vector potential A is equal to the magnetic field strength \(\:{\text{B}}_{RF}\):

$$\:\begin{array}{c}{\text{B}}_{RF}=\nabla\:\times\:A\end{array}$$
(9)

The current response of the plasma is described by the generalized Ohm’s law:

$$\:\begin{array}{c}J=\sigma\:E+j\omega\:D+\sigma\:v\times\:{\text{B}}_{RF}+{\text{J}}_{e}\end{array}$$
(10)

Under the time-harmonic field approximation, the relationship between the electric field and the magnetic vector potential is simplified to:

$$\:\begin{array}{c}E=-j\omega\:A\end{array}$$
(11)

Where \(\:{\text{J}}_{e}\) is the coil driving current (external excitation source); \(\:{\upsigma\:}\text{E}\) is the plasma induced current (Ohmic current); \(\:\text{j}{\upomega\:}\text{D}\) is the displacement current (high-frequency effect); \(\:{\upomega\:}\) is the RF angular frequency (rad/s); \(\:{\upmu\:}\) is the magnetic permeability (H/m); and \(\:{\upsigma\:}\) is the plasma conductivity (S/m).

This equation reveals the physical mechanism of power coupling in ICP, where the alternating magnetic field generated by the RF coil induces an electric field E that heats the electrons, thereby sustaining the discharge.

Model validation

To validate the effectiveness of the model, a case study was conducted and compared with data from the literature. The case study is based on the work by Zhang, G. L. et al.23 (2019), which investigates a columnar inductively coupled plasma source. The RF power sources used were 2 MHz and 13.56 MHz, and the antenna was an 8-turn hollow copper wire spiral tube. The discharge chamber was a cylindrical quartz vessel (diameter 260 mm, height 370 mm), with a gas pressure of 10 Pa. Using a Langmuir probe located 10 cm from the discharge chamber exit, the electron density data were obtained as shown in Fig. 4. The simulation results exhibit the same trend as the experimental measurements, confirming the feasibility of the simulation model.

Fig. 4
figure 4

Comparison between the measured values of the experimental electron density of Zhang, G. L and the simulation calculation values of COMSOL.

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The mesh setup is shown in Fig. 5. A mapped grid distribution is used for the antenna, with an exponential symmetry distribution, ensuring that the boundary mesh is denser than the mesh in the middle of the antenna to account for the skin effect of high-frequency currents. The discharge chamber boundary is set with a boundary layer mesh for fluid properties, consisting of 5 layers with a stretching factor of 1.5 and a thickness adjustment factor of 116. Inside the discharge chamber, triangular meshes are applied for plasma properties. The remaining domain is set with a standard triangular mesh. Three different mesh scales were used in this study: Mesh 1 has 14,025 elements, Mesh 2 has 17,503 elements, and Mesh 3 has 21,481 elements. The electron density variations at the geometric center of the discharge chamber were calculated for RF input powers of 100 W, 500 W, and 1000 W, with the results shown in Fig. 6. The relative errors are all below 5%.

Fig. 5
figure 5

Grid Settings.

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Fig. 6
figure 6

Grid Sensitivity analysis.

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Calculation results and analysis

Comparison of electron density with and without permanent magnet confinement

The simulation conditions are set as follows: RF frequency is 13.56 MHz, RF input power is 100 W, gas pressure is 0.05 Torr, and propellant flow rate is 5 sccm. Figure 7(a) shows the transient magnetic field \(\:{\text{B}}_{RF}\) of the RF coil at 0.01 s, Fig. 7(b) shows the transient magnetic field \(\:{\text{B}}_{RF}\) of the RF coil under static magnetic field \(\:{\text{B}}_{0}\) at 0.01 s, and Fig. 7(c) represents the static magnetic field \(\:{\text{B}}_{0}\) formed by the permanent magnets and soft iron.

Fig. 7
figure 7

Transient flux density modes \(\:{\text{B}}_{\text{R}\text{F}}\) (a) and (b) generated by the RF coil at 0.01s, as well as steady-state flux density modes \(\:{\text{B}}_{0}\) (c) of the permanent magnet.

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Figure 8(a) and Fig. 8(c) show that under the condition of no permanent magnet confinement, the maximum electron density is 6.64 × 1011/cm3, with an electron temperature of 3.48 eV. These results exhibit the same trend as those in reference24, confirming the validity of the results. Figure 8(b) and Fig. 8(d) show that under permanent magnet confinement, the maximum electron density is 1.58 × 1013/cm3, and the electron temperature is 1.95 eV. The electron density in the magnetically confined case is 23.8 times higher, and the electron temperature under permanent magnet confinement is noticeably 1.5 eV lower than that without confinement. This difference primarily results from the strong constraint of the magnetic field on electron motion. Under permanent magnet confinement, electrons are magnetized and move along helical trajectories (Larmor orbits) in the direction of the magnetic field lines, which significantly reduces the gyroradius and lowers the transverse diffusion coefficient by about five orders of magnitude. This dramatically extends the residence time of electrons within the discharge chamber, increasing the number of effective collisions with neutral particles and promoting the ionization process. The specific confinement mechanisms and plasma behavior will be further analyzed in the following sections.

Under permanent magnet confinement, the diffusion coefficient \(\:{\text{D}}_{\perp\:}\) of electrons across the magnetic field:

$$\:\begin{array}{c}{D}_{e}=\frac{{k}_{B}{T}_{e}}{{m}_{e}{v}_{en}}\end{array}$$
(12)
$$\:\begin{array}{c}{D}_{\perp\:}=\frac{{D}_{e}}{1+{\left(\frac{{\omega\:}_{ce}}{{v}_{en}}\right)}^{2}}\end{array}$$
(13)

Where \(\:{\text{D}}_{\text{e}}\) is the electron diffusion coefficient; \(\:{{\upomega\:}}_{\text{c}\text{e}}\) is the electron gyrofrequency; and \(\:{\text{v}}_{\text{e}\text{n}}\) is the electron-neutral particle collision frequency21. Clearly, the stronger the magnetic field, the more difficult it is for electrons to diffuse across the magnetic field lines, and radial diffusion across the magnetic field is suppressed.

Electron loss primarily comes from wall recombination. Under permanent magnet confinement, the radial electron loss rate \(\:{\text{L}}_{\text{e}}\) is approximately:

$$\:\begin{array}{c}{L}_{e}\approx\:\frac{{n}_{e}}{{\tau\:}_{loss}}\end{array}$$
(14)
$$\:\begin{array}{c}{\tau\:}_{loss}\propto\:\frac{{L}^{2}}{{D}_{\perp\:}}\end{array}$$
(15)

After applying permanent magnet confinement, the electron-neutral collision frequency does not change significantly, while the radial electron loss frequency decreases by approximately five orders of magnitude, leading to an extended electron radial loss time. Consequently, the electron loss rate is significantly reduced, by about three orders of magnitude, which effectively increases the electron density in the system.

The introduction of permanent magnet confinement significantly prolongs the electron residence time, allowing electrons to experience more collisions with neutral particles along the axial direction, thereby enhancing the ionization process. Since ionization reactions have high energy requirements, electrons dissipate more energy in exciting and ionizing neutral gas molecules during collision-induced ionization, rather than maintaining their thermal motion. As a result, the electron temperature under permanent magnet confinement is reduced by approximately 1.5 eV compared to the unconstrained case.

Fig. 8
figure 8

The simulation results of electron density and electron temperature, where (a) is the electron density diagram without permanent magnet constraint, (b) is the electron density diagram without permanent magnet constraint, (c) is the electron temperature diagram without permanent magnet constraint, and (d) is the electron temperature diagram under permanent magnet constraint.

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Effect of RF input power on plasma characteristics

The simulation conditions are set as follows: RF frequency is 13.56 MHz, gas pressure is 0.05 Torr, propellant flow rate is 5 sccm, and RF input power ranges from 50 to 1000 W.

Figures 9 and 10 show that as the RF input power increases, both the electron temperature and electron density exhibit an increasing trend. Over the power range from 50 W to 1000 W, the electron density increases by approximately 13.7 times under permanent magnet confinement, while it increases by about 11.51 times under no magnetic confinement. In contrast, the electron temperature increases by approximately 6% in both conditions. This phenomenon indicates that the input energy is primarily used for the ionization process of the propellant, with a small portion converted into electron thermal energy. The underlying physical mechanism is that higher RF input power leads to a simultaneous increase in coil voltage and current, thereby enhancing the high-frequency electromagnetic field strength generated by the antenna, which significantly improves the ionization efficiency of the propellant gas.

Therefore, in the design of RF plasma thrusters, the introduction of magnetic confinement can effectively increase electron density and ionization efficiency without additional power consumption. Meanwhile, RF power itself can also serve as an important parameter for controlling thrust performance, providing a feasible path for optimizing thruster performance.

Theoretical analysis suggests that the RF power is primarily used to sustain the ionization and excitation processes of the plasma. Under steady-state conditions, the input power is balanced with the energy losses20:

$$\:\begin{array}{c}{\text{P}}_{\text{a}\text{b}\text{s}}={{{\upeta\:}}_{\text{c}\text{o}\text{u}\text{p}}\text{P}}_{\text{r}\text{f}}\end{array}$$
(16)
$$\:\begin{array}{c}{\text{P}}_{\text{a}\text{b}\text{s}}={\stackrel{-}{\text{n}}}_{\text{e}}\bullet\:{\stackrel{-}{\text{n}}}_{\text{g}}\bullet\:{\text{k}}_{\text{i}\text{z}}\left({\text{T}}_{\text{e}}\right)\bullet\:{{\upepsilon\:}}_{\text{e}\text{f}\text{f}}\left({\text{T}}_{\text{e}}\right)\bullet\:V\end{array}$$
(17)

Where \(\:{\stackrel{\text{-}}{\text{n}}}_{\text{e}}\), \(\:{\stackrel{\text{-}}{\text{n}}}_{\text{g}}\) is the volume-averaged electron density and neutral gas density (m− 3), \(\:{\text{k}}_{\text{iz}}\left({\text{T}}_{\text{e}}\right)\) is the ionization rate coefficient (m3/s), \(\:{\text{ε}}_{\text{eff}}\left({\text{T}}_{\text{e}}\right)\) represents the average energy required to produce a pair of ion-electron pairs (including ionization energy, excitation losses, wall losses, etc.), and \(\:\text{V}\) is the plasma volume.

From the variation formula, we can obtain:

$$\:\begin{array}{c}{\stackrel{-}{\text{n}}}_{\text{e}}\propto\:\frac{{\text{P}}_{\text{a}\text{b}\text{s}}}{{\stackrel{-}{\text{n}}}_{\text{g}}\bullet\:{\text{k}}_{\text{i}\text{z}}\left({\text{T}}_{\text{e}}\right)\bullet\:{{\upepsilon\:}}_{\text{e}\text{f}\text{f}}\left({\text{T}}_{\text{e}}\right)\bullet\:\text{V}}\end{array}$$
(18)

It can be inferred that when other parameters (such as \(\:{\stackrel{-}{\text{n}}}_{\text{g}}\), \(\:{T}_{e}\)) change relatively little, \(\:{\stackrel{-}{\text{n}}}_{\text{e}}\) increases approximately linearly.

Figures 11 and 12 show the spatial distribution of electron density and electron temperature along the axial centerline of the discharge chamber, in the direction of the symmetry axis. The results indicate that both electron density and electron temperature reach peak values in the central region of the discharge chamber and gradually decrease along the axial direction towards the sides, exhibiting a distribution pattern symmetric about the geometric center of the discharge chamber. This symmetry reflects the strong central concentration of the RF electromagnetic field excitation and energy coupling in space under stable discharge conditions, and further validates the physical plausibility of the model in terms of spatial distribution.

Fig. 9
figure 9

The influence of RF input power and the presence or absence of permanent magnet constraints on the average electron density of the discharge chamber.

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Fig. 10
figure 10

The influence of RF input power and the presence or absence of permanent magnet constraints on the electron temperature at the geometric center of the discharge chamber.

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Fig. 11
figure 11

The influence of RF input power and the presence or absence of permanent magnet constraints on the electron density along the axis of symmetry.

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Fig. 12
figure 12

The influence of RF input power and the presence or absence of permanent magnet constraints on the electron temperature along the symmetry axis.

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Conclusion

This study develops a radio-frequency plasma fluid model considering both permanent magnet confinement and no magnetic field confinement conditions, systematically exploring the regulatory mechanism of static magnetic fields on plasma characteristics in small RF plasma thrusters. The results indicate that magnetic confinement can significantly suppress the transverse diffusion of electrons and extend their residence time, thereby effectively enhancing the collision frequency between electrons and neutral particles and improving ionization efficiency. Under the operating condition of 100 W RF power, the electron density under permanent magnet confinement reaches 1.58 × 1013 /cm− 3, which is approximately 23.8 times higher than that under no confinement. Further analysis reveals that, within the input power range of 50–1000 W, the electron density increases approximately linearly with power, with a more significant growth under permanent magnet confinement. At the same time, the electron temperature changes little across the power levels, indicating that the input energy is primarily used to drive the ionization process rather than enhancing the thermal motion of electrons. This finding suggests that optimizing the magnetic field structure and appropriately configuring the RF power input can significantly enhance electron density, thereby improving the overall performance of the thruster. This study clarifies the key role of permanent magnet confinement in enhancing ionization efficiency and proposes an optimization design strategy based on magnetic field-power synergistic control. The results provide theoretical support for the application of small RF plasma thrusters under extreme conditions of low power and high ionization density, offering significant engineering practical value and promising prospects, especially for deep-space propulsion missions with stringent requirements for high performance and long lifetime.