Effect of the radial magnetic field strength on the argon plasma characteristics of an inductive radiofrequency plasma thruster with a time-varying magnetic field

Abstract

This study investigates an inductively-coupled radiofrequency plasma thruster subjected to a time-varying diverging magnetic field in conjunction with a static radial magnetic field to enhance plasma acceleration. A variable radial magnetic field is generated using an electromagnet to assess its effects on argon plasma characteristics. Static magnetic fields are applied separately and jointly with time-varying magnetic fields to analyze their decoupled and synergistic effects, respectively. Non-acceleration mode results without time-varying magnetic fields show that as the radial magnetic field strength increases such that the electron gyroradius decreases, the electron temperature and plasma potential increases monotonically resulting in steep axial gradients at high magnetic field strengths. As a result, perpendicular electric fields experience monotonic enhancement coincident with a non-monotonic 50% plasma density drop. Additionally, the region where the plasma density is maximal shifts downstream toward the thruster exit as the magnetic field strength increases. At relatively higher magnetic fields, the plasma density drop saturates but the plasma potential and electron temperature continue to rise. Similar trends successfully recur in acceleration mode measurements when both magnetic fields are jointly applied. The key difference between decoupled and synergistic effects seems to be plasma parameter enhancement enabled by the additional electrical power in acceleration mode.

Introduction

Durable high-power electric thrusters are vital to successfully exploring deep space, the final frontier of space exploration. Thus, high-power electric propulsion is featured in the Global Exploration Roadmap [1] as a key technology to enable human spaceflight to Mars. Gridded ion thrusters (GITs) had successfully thrusted the Hayabusa missions [2, 3], Dawn, Dart, and BepiColombo [4,5,6]. Similarly, Hall effect thrusters (HETs) played a central role in the SMART-1 [7] and the Psyche mission [8] is currently being driven by the first-ever interplanetary HETs. Meanwhile, in low earth orbit, HETs are partially performing stationkeeping duties onboard China’s Tiangong Space Station [9] and orbit corrections of V2 mini satellites for SpaceX’s Starlink constellation [10, 11]. Although these two pacesetting propulsion systems have been demonstrated multiple times in space, severe thermal load and plasma-electrode interactions are expected to shorten their lifetimes at higher operating powers beyond 100 kW [11,12,13]. In fact, several solutions have been studied to suppress plasma-induced degradation in both HETs and GITs, such as magnetic shielding technology [14], and multi-cusped magnetic field arrangements employed in the Highly Efficient Multistage Plasma Thruster (HEMP-T) [15] and the Diverging Cusped Field Thruster (DCFT) [16]. However, these technologies are relatively nascent, meaning that their effectiveness is not fully understood beyond 100 kW. These electrostatic thrusters also require ion beam neutralization by oppositely charged electron beams to prevent spacecraft charging. Consequently, the neutralizer can be considered as a single point-of-failure in cathode-based thrusters which could potentially jeopardize costly missions. As a result, electrodeless plasma thrusters (EPT) [17] intended to circumvent direct plasma-metal interactions are under development. EPT concepts use different ionization techniques such as electron cyclotron resonance (ECR) [18], inductively coupled plasma (ICP) [19], and helicon waves [20] for generating high-density plasma in the order of 10¹⁸ m⁻³.

A common plasma acceleration method in EPTs has been to apply diverging magnetic fields expanding axially forming magnetic nozzles [20]. The role of the magnetic nozzle is twofold. First, it confines electrons, reducing their losses to the walls. Second, it serves as a transport mechanism which facilitates energy conversion between charged particles by converting the electron thermal energy into ion kinetic energy via ambipolar electric fields [20]. However, plasma transport in magnetic nozzles tends to be hindered by plasma detachment [21] dictated by the closed loop nature of magnetic field lines and low thruster efficiency [17]. To address low thruster efficiency, alternative methods such as ion cyclotron resonance heating (ICRH)-assisted magnetic nozzles [22] and inductive plasma acceleration [23] have gained traction in electrodeless propulsion. Notable electrodeless inductive plasma accelerators include the pulsed inductive thruster (PIT) [24], the Faraday Accelerator with Radiofrequency Assisted Discharge (FARAD) [25], the Lissajous Helicon Plasma Accelerator (LHPA) [26], and the Radiofrequency (RF) Inductive Plasma Accelerator with a Low aspect ratio (RIPAL) [27]. These accelerators harness a Lorentz force emanating from the interaction between radial magnetic fields and azimuthal plasma currents.

Recently, Sekine et al. [28] found that axially-diverging magnetic nozzles facilitate the formation of upstream crossfield electric fields when subjected to a time-varying magnetic field in the order of tens to hundreds of kilohertz. The in-plane electric field initiates a radial ion drive, causing a plasma density enhancement around the thruster axis. Next, the enhanced density gradient reinforces the Boltzmann electric field leading to field-aligned electrostatic acceleration of unmagnetized ions. However, the radially inward induced electric field does not directly contribute to the axial momentum flux in the thruster, undermining thruster efficiency. To induce electric fields favorable to axial thrust, Sekine et al. [28] proposed replacing the diverging magnetic nozzle with a uniform radial-dominant static magnetic field. The first proof of concept was developed by Yamamura [29] and several experiments have been conducted ever since including thrust measurements and ion flow velocity measurements. One of the unknowns in this thruster is the dependency of plasma parameters on the radial magnetic field strength. It is common knowledge that the radial magnetic field intensity and profile affects axial electron mobility, ionization length, electric field distributions and determines the positions of the ionization and acceleration regions in HETs [30, 31]. Additionally, variable magnetic field strength can induce mode transitions in HETs [32]. A helicon thruster analytical model has predicted significant performance improvement by increasing the magnetic field [33]. In addition, direct thrust measurements of EPTs thrusters have revealed that the thrust scales with increasing magnetic field strength [34, 35]. Nekliudova et al. [36] experimentally and numerically explored how the RF power absorption efficiency by the plasma in an RF inductive plasma source operating at frequencies in the range of 2—13.56 MHz depends on the external magnetic field. They showed that the plasma equivalent resistance non-monotonically depends on the magnetic field strength.

In contrast, this study aims to clarify the effect of a radial-dominant static magnetic field on plasma parameters in an inductive radiofrequency plasma thruster with a time-varying magnetic field. The static magnetic field is applied separately and jointly with the time-varying one to observe both the decoupled and synergistic effects, respectively. The key plasma parameters of interest are the ion density and the plasma potential which are indicators of ion transport and electric field distributions in the thruster, respectively. The paper arrangement is as follows: first, the experimental setup and key plasma diagnostics are presented in Sect. "Experimental Setup". Second, the experimental results in both modes are described in Sect. "Results". Third, a discussion and comparison of results follows in Sect. "Discussion". Finally, conclusions are offered in Sect. "Conclusion".

Experimental setup

Thruster and vacuum facility

Radiofrequency plasmas are generated using a triple-turn coil at 13.56 MHz in an inductively coupled plasma source with a low aspect ratio. Figure 1 shows a schematic diagram of the thruster contiguously attached to a 3 m long and 1.4 m diameter vacuum chamber. The stainless-steel chamber is evacuated by a 1000 l/min rotary pump in conjunction with a 10,000 l/s cryopump giving a base pressure of ~ 1 × 10^–3 Pa as measured using an ionization gauge. The thruster comprises a 30 mm long Pyrex tube with an 80 mm inner diameter. The upstream side of the discharge tube is terminated by a polytetrafluoroethylene (PTFE) backplate encircled by a flat spiral plasma acceleration coil. The argon propellant is fed into the discharge tube via Swagelok tube fittings situated within the backplate at a fixed mass flow rate of 60 sccm, causing the chamber background pressure to be about 3 × 10^–2 Pa. The flow rate is regulated using a mass flow controller. In contrast, the downstream open end of the discharge tube is attached to the vacuum chamber via a PTFE conversion flange. A triple-turn coil (RF coil) is wrapped around the discharge tube and coupled to an RF power supply via an automatic impedance matching box. The output RF power is variable up to 400 W but only 200 W is applied herein. The plasma is accelerated using a flat ten-turn spiral coil (AC coil) connected to a power supply via a T-type impedance matching network (Fig. 2) to transfer maximum power to the plasma. The acceleration power source frequency can vary from 100 to 999 kHz and this study used 156 kHz. The fluctuating current flowing in the acceleration coil is measured using a Pearson current monitor. The manganese-zinc (Mn–Zn) ferrite core channels the magnetic flux generated by an electromagnet into the thruster, allowing it to diverge radially. The water-cooled ferrite core has a 12-mm diameter concentric hole, enabling direct water circulation via a coaxial tube. It is also shielded from plasma heat by a Macor ceramic cover. The variable static magnetic field is generated using an electromagnet (EM) with dimensions of Ø280 mm × 105 mm. The EM contains ~ 180 turns of insulated copper tubes wound on a Ø20 mm bobbin and water cooled to suppress high temperatures induced by applied direct currents (DC). The white lines in Fig. 1 represent the radial magnetic field lines calculated by the finite element method magnetics (FEMM) software using 5A current in the EM. The yellow regions indicate a PTFE backplate (z ~ 0 mm) and conversion flange (z ~ 40 mm). The red lines indicate the vacuum chamber walls.

Fig. 1

A schematic diagram of the inductive RF plasma thruster contiguously attached to a vacuum chamber. Plasma diagnostics are mounted on a bidirectional linear actuator spatially variable in both the axial and radial directions

Fig. 2

Schematic of the acceleration circuit where L_acc, L_p, L_pla, C_S1, C_S2, R_pla, and M are the acceleration coil inductance, parallel inductance, plasma inductance, serial capacitances, plasma resistance and mutual inductance, respectively

Emissive probe

The floating potential, plasma potential, and electron temperature were deduced from emissive probe (EP) measurements [37]. Unlike swept Langmuir probes, EPs simplify time-resolved measurements of the plasma parameters when using the floating-point technique because no voltage sweep is needed [38,39,40]. The EP technique used herein is the floating-point method. Figure 3 shows the EP circuit schematic. During experiments it is mounted on a mobile biaxial motorized stage to collect data in both the axial and radial directions. The EP has a Ø0.15 mm hairpin-shaped thoriated tungsten filament inserted into Ø0.8 mm holes inside a Ø3 mm dual-bore alumina tube. Each bore is filled with seven Ø0.2 mm tungsten wires in electrical contact with the filament and connected in parallel with a resistive divider. To obtain the plasma potential, we measure the floating potential of the cold non-emitting probe and a hot emissive probe, and deduce the electron temperature [37]. The floating potential is measured across one of the 100 Ω resistors in the voltage divider. To obtain the floating potential, the EP is connected to ground through a 1 MΩ resistor inside an isolated oscilloscope. The "hot" floating potential ${V}_{f}^{H}$ indicates the potential with zero net current when a DC heating current ranging from 3 A up to 4 A is flowing through the filament. The DC current (measured as a voltage drop V_heat across a 1 Ω shunt in Fig. 3) is supplied by an electrically floating DC power source. Conversely, the "cold" floating potential ${V}_{f}^{C}$ denotes the floating point without a DC heating current. Under the thin-sheath limit, the cold potential is written as [37]

$${V}_{f}^{C}= {\phi }_{p}-\beta {T}_{e}$$

(1)

where ${\phi }_{p}$ is the plasma potential, ${T}_{e}$ is the electron temperature and $\beta = 5.17$ for Argon propellant gas. In contrast, the hot potential is given as [37]

$${V}_{f}^{H}={\phi }_{p}-\alpha {T}_{e}$$

(2)

where $\alpha$ is a constant factoring in the effect of an emissive sheath [37, 41]. Collectively, Eqns. (1) and (2) form a set of simultaneous equations with two unknowns, namely ${\phi }_{p}$ and ${T}_{e}$. Assuming a planar probe model [37], solving the two equations simultaneously gives the electron temperature as

Fig. 3

Emissive probe schematic whereby the filament is represented by the encircled coil

$${T}_{e}=\frac{{ V}_{f}^{H}-{V}_{f}^{C}}{\beta -\alpha }$$

(3)

Subsequently, ${T}_{e}$ is substituted into Eq. (1) to obtain the plasma potential as

$${\phi }_{p}= {V}_{f}^{C}+\beta \left(\frac{{ V}_{f}^{H}-{V}_{f}^{C}}{\beta -\alpha }\right)$$

(4)

Alternatively, ${\phi }_{p}$ can be estimated using Eq. (2). According to Sheehan et al. [37], errors in measuring ${\phi }_{p}$ using this method stem from the numerical factor $\alpha$ in the relation ${V}_{f}^{H}= {\phi }_{p}-\alpha {T}_{e}$, and the subsequent estimation of ${T}_{e}=\frac{{ V}_{f}^{H}-{V}_{f}^{C}}{5.17-\alpha }$. The highest possible value of ${T}_{e}$ is ${T}_{e}\left(\alpha =0\right)= 0.1934\Delta V$ where $\Delta V={ V}_{f}^{H}-{V}_{f}^{C}$, whereas the lowest possible value is ${T}_{e}\left(\alpha =1.5\right)= 0.2725\Delta V$. Using the deviation method, the uncertainty of the electron temperature is approximately 8% of ΔV. Consequently, the error in ${\phi }_{p}$ is 8% from the error propagation method. We did not detect fluctuations at the RF frequency so their effects on measurements are neglected. The limitation of estimating ${T}_{e}$ from Eq. (3) is that it assumes a Maxwellian electron distribution, which is rarely the case in magnetic nozzles where multiple electron populations may exist. Thus, the extent to which the actual plasma deviates from a Maxwellian distribution is unknown.

Langmuir probe

Plasma potential measurements alone are insufficient to fully understand the static radial magnetic field (SRMF) effect on plasma characteristics in the thruster. Therefore, the ion density is deduced from the ion saturation current (I_is) given by the Bohm current [42] as

$${I}_{is }=0.6{n}_{i}e{u}_{B}{A}_{p}$$

(5)

where ${n}_{i}$ is the ion density, ${u}_{B}=\sqrt{\frac{{T}_{e}}{{M}_{i}}}$ is the Bohm velocity, ${M}_{i}$ is the ion mass, and ${A}_{p}$ is the probe collection area. The Bohm velocity is estimated using ${T}_{e}$ from EP measurements. Measuring ${T}_{e}$ with a Langmuir probe would require a high-frequency voltage sweep. The error in the electron temperature is roughly 8%. The uncertainty in Langmuir probe measurements is ± 20–50% [43]. Assuming a 30% uncertainty in ${I}_{is}$, and using the error propagation method [43], the total uncertainty in ${n}_{i}$ is approximately 30.3%. The ion saturation current is measured using a cylindrical RF compensated Langmuir probe (LP). The RF compensation technique has been extensively explained elsewhere [44] and will only be summarized here since it is unnecessary for measuring I_is. The LP consists of a tungsten electrode (main probe) with Ø0.25 mm × 5 mm dimensions. The main probe is biased at −70 V using batteries to repel plasma electrons while attracting plasma ions. The probe tip is bent at a 90-degree angle so that it is always perpendicular to magnetic field lines, which is a rule of thumb in magnetized plasmas [45]. The LP is mounted on a linear biaxial actuator immersed in vacuum as shown in Fig. 1. The biaxial actuator can sweep both radially and axially, enabling two-dimensional profiling of plasma parameters in the r–z plane of the thruster. The measured voltage is divided by the shunt resistance to give the ion saturation current in accordance with Ohm’s law. Figure 4 shows the equivalent circuit of the RF compensated probe where V_s, V_p, V_b and V_out represent the plasma potential, probe potential, probe bias voltage, and the voltage drop across the shunt resistor R_m, respectively. Z_sh is the equivalent impedance of the sheath resistance (R_sh) in parallel with the sheath capacitance (C_sh) whereas Z_ch and Z_re are the impedances of the series RF chokes and the reference electrode, respectively.

Fig. 4

Equivalent circuit of the RF compensated probe

Static magnetic field distribution

Figure 5 shows a comparison of the measured values of the DC magnetic field (circles) and the simulated magnetic field strength (squares) at r = 30 mm. The estimates were calculated by the FEMM program. The experimental values were measured using a Kanetec Gaussmeter (Model TM-701) with a resolution of 0.01 mT and an indication accuracy of $\pm$ 5%. Clearly, there is a large discrepancy between measured values and estimates at z ≥ 5 mm. The main cause of the discrepancy is presently unclear. Due to the discrepancy, the quantitative analysis of crossfield particle transport cannot be reliably applied to the present results. Therefore, the value of the current results is the qualitative effect of the magnetic field strength on the plasma parameters and their subsequent temporal variations when the time-varying magnetic field is imposed.

Fig. 5

Comparison of the actual magnetic field (circles) measured by a Gaussmeter and the simulated magnetic field strength (squares) calculated by FEMM at r = 30 mm. There is a discrepancy between measured values and estimates at z ≥ 5 mm

Figure 6 shows the simulated two-dimensional profiles of the static magnetic field strength for different values of I_coil. The colormaps were calculated by the FEMM program. The black solid lines indicate the dominantly radial magnetic flux lines.

Fig. 6

Static radial magnetic field strength for (a) I_coil = 5 A, (b) I_coil = 10 A, (c) I_coil = 15 A, (d) I_coil = 20 A, and (e) I_coil = 25 A, calculated by the FEMM program. The black lines indicate the magnetic field lines

Spatiotemporal evolution of the total magnetic field

To actively accelerate plasma, a 156 kHz time-varying magnetic field B(t) was inductively superimposed on the thruster in conjunction with the static magnetic field B₀. During data acquisition, measurements of plasma parameters were conducted over a 50 μs range capturing about 7 periods simultaneously. During data analysis, temporal waveforms were loaded into MATLAB followed by fitting a sine wave to the acceleration coil current I_acc to precisely estimate the period. Subsequently, the signals were split into approximately seven individual cycles with the same duration as the estimated period. The number of individual cycles was determined by the calculated period. Next, the data points for each individual cycle were divided into ten segments. Each segment was ensemble-averaged seven times by combining corresponding segments across all cycles. Figure 7 shows the temporal evolution of I_acc indicated by the solid cyan curve which comprises seven consecutive waveforms extracted from a 50 μs signal. The ten blue circles represent the ensemble-averaged segments of I_acc plotted against the median values within the range of each segment. Figure 8 shows the spatiotemporal evolution of the total magnetic field strength B = B₀ + B(t) at representative times of the applied time-varying magnetic field for I_coil = 0–25 A. The temporal magnetic field was generated by the FEMM program using the values of I_acc = 5 A, 20 A, 8A, −15 A, and −18 A at the respective selected times of t = 0.3 μs, 1.6 μs, 2.9 μs, 4.2 μs, and 5.4 μs, in one period of I_acc. The corresponding values of I_acc were extracted from Fig. 7. The black solid lines indicate the magnetic flux lines.

Fig. 7

Temporal evolution of the spiral acceleration coil current I_acc (solid cyan lines) in one period. The five dashed red lines indicate the specific times of t = 0.3 μs, 1.6 μs, 2.9 μs, 4.2 μs, and 5.4 μs selected to draw Figs. 15, 16 and 17. The blue circles represent the ensemble averaged segments of I_acc plotted against medians of the range of each segment

Fig. 8

Two-dimensional maps of the magnetic field strength for (a) I_coil = 0 A, (b) I_coil = 5A, (c) I_coil = 10 A, (d) I_coil = 15A, (e) I_coil = 20 A, and (e) I_coil = 25 A at representative times of t = 0.3 μs, 1.6 s, 2.9 μs, 4.2 μs, and 5.4 μs in one period of the variation of I_acc. The solid black lines show the calculated magnetic field lines

Experimental conditions

Two types of experiments were conducted separately in this study: the “acceleration (AC) mode” experiments and the “non-acceleration (NA) mode” experiments. In acceleration mode, the static and time-varying magnetic fields are jointly applied (synergised) to have maximum effect on plasma acceleration in the thruster. The time-varying magnetic field B(t) is generated by a 40 A_pp current oscillating at 156 kHz in the spiral coil. On the other hand, only the static magnetic field is present in NA mode experiments to decouple the magnetic fields to distinguish their effects. Spatial data were collected at sixty points (five and twelve points along the radial and axial direction respectively) per magnetic field strength during both experiments. Experimental data were acquired using a Yokogawa DL950 scopecorder equipped with a high-speed four-channel 1 MS/s 16-bit isolation module with high noise immunity. Waveforms were recorded using a sampling rate of 10 MS/s and a time base of 5 μs per division. In all modes, probes began sweeping at z = 10 mm in the discharge chamber, ensuring that there is a 10 mm gap between probes and thruster chamber walls. The 10 mm gap was also maintained in the radial direction with respect to the lateral wall. This gap is especially important for emissive probe measurements in hot mode since the hot emissive filament can essentially damage the Teflon walls. Table 1 summarizes the experimental conditions.

Table 1 Experimental conditions for non-acceleration mode and acceleration mode experiments

Results

Decoupled effects of the static magnetic field

Effect on plasma potential

Figure 9(a) to (f) show the two-dimensional (2D) profiles of plasma potential ϕ_p for I_coil = 0–25 A, respectively. Three basic effects are noticeable from the colormaps. First, ϕ_p monotonically increases with magnetic field strength. Precisely, a maximum potential of about 106 V was obtained for I_coil = 25 A. This is 2.3 times larger than the peak potential with I_coil = 0 A. From left to right, the spatially averaged ϕ_p is approximately 46 V, 52 V, 60 V, 69 V, 76 V, and 85 V. Additionally, ϕ_p is relatively constant radially across Figs. 9(a) – (b), indicative of equipotential magnetic field lines. Interestingly, the radial uniformity appears tilted in Figs. 9(b) – (f), forming diagonal contours with respect to the r–z plane. In other words, as the magnetic field increases, the contour lines follow more closely the magnetic field lines. For better visual clarity, Fig. 10 shows the 1D axial distribution of plasma potential for six different magnetic field strengths taken from the contour plot in Fig. 9 for r = 30 mm.

Fig. 9

Decoupled effects of the static magnetic field strength on plasma potential for (a) I_coil = 0 A, (b) I_coil = 5 A, (c) I_coil = 10 A, (d) I_coil = 15 A, (e) I_coil = 20 A, and (f) I_coil = 25 A. The black lines indicate the magnetic field lines

Fig. 10

Axial distribution of plasma potential for I_coil = 0–25 A at r = 30 mm. The dotted lines are for visual guidance only, whereas error bars represent the uncertainty

Effect on electron temperature

Figure 11(a) to (f) comprise the heatmaps of electron temperature T_e for I_coil = 0–25 A, respectively. Similar to plasma potential, T_e monotonically increases with increasing magnetic field strength. In accordance with the alphabetical order in Fig. 11, the spatially averaged T_e is approximately 7 eV, 11 eV, 16 eV, 19 eV, 21 eV, and 23 eV, respectively. Moreover, the heatmaps show negligible radial variations across Figs. 11(a) – (c), indicative of isothermal magnetic field lines. Like ϕ_p, as the magnetic field increases, the contour lines follow more closely the magnetic field lines in Figs. 11(b) – (f). For simplicity and clarity, Fig. 12 shows a 1D plot of the electron temperature as a function of axial position extracted from the contour plot in Fig. 11 at r = 30 mm.

Fig. 11

Decoupled effects of the static magnetic field on electron temperature for (a) I_coil = 0 A, (b) I_coil = 5 A, (c) I_coil = 10 A, (d) I_coil = 15 A, (e) I_coil = 20 A, and (f) I_coil = 25 A. The black lines represent the magnetic field lines

Fig. 12

Axial distribution of electron temperature for I_coil = 0–25 A at r = 30 mm. The dotted lines are for visual guidance only, whereas error bars represent the uncertainty

Effect on ion density

Figure 13(a) to (f) present the ion density (n_i) colormaps for I_coil = 0–25 A. Clearly, the magnetic field strength influence on ion density is twofold. First, n_i decreases with increasing magnetic field strength: the spatially averaged n_i from left to right is approximately 10.5 × 10¹⁷ m⁻³, 5.3 × 10¹⁷ m⁻³, 2.0 × 10¹⁷ m⁻³, 1.8 × 10¹⁷ m⁻³, 1.9 × 10¹⁷ m⁻³, and 2.0 × 10¹⁷ m⁻³, respectively. Since the averaged n_i is in the order of ~ 2.0 × 10¹⁷ m⁻³ between Fig. 13(c) and (f) and considering the ± 20–50% accuracy [43] of Langmuir probe measurements, it can be assumed that the ion density decay saturates at I_coil = 10 A. Second, the region where n_i is maximal shifts downstream toward the thruster exit as the field strength enhances. To elaborate further, the densest region in Fig. 13(a) is located upstream near the bottom left corner above the ferrite. Interestingly, the application of I_coil = 5 A in the electromagnet coincided with the densest region shrinking radially and being restricted to a small region below r = 27 mm while being spread out between z = 12–32 mm in Fig. 13(b). In Fig. 13(c) and (d) the peak ion density is in the vicinity of the thruster exit at z ~ 30 mm. However, the densest region is located around z = 22 mm in Fig. 13(e) and (f), indicating a slight reversal in the ₋z direction. For simplicity and clarity, Fig. 14 depicts a 1D plot of ion density as a function of axial position taken from the contour plot in Fig. 13 at r = 30 mm.

Fig. 13

Decoupled effects of the static radial magnetic field on ion density for (a) I_coil = 0 A, (b) I_coil = 5 A, (c) I_coil = 10 A, (d) I_coil = 15 A, (e) I_coil = 20 A, and (f) I_coil = 25 A. The black lines denote the contours

Fig. 14

Axial distribution of ion density for I_coil = 0–25 A at r = 30 mm. The dotted lines are for visual guidance only, whereas the error bars represent the total uncertainty

Synergistic effects of static and time-varying magnetic fields

Effect on plasma potential

Figure 15(a) – (f) present the spatiotemporal colormaps of ϕ_p at representative times in one period of I_acc for different field strengths generated using I_coil = 0–25 A while discretely incrementing at 5A intervals. The spatially and temporally averaged ϕ_p level is approximately 45, 50, 59, 80, 94, and 102 V in Fig. 15(a) – (e), respectively. Like NA mode results in Fig. 9, ϕ_p is scaling monotonically with magnetic field strength. However, the spatiotemporal evolution is almost nonexistent across all panels, denoting negligible modulation of ϕ_p by the applied time-varying magnetic field.

Fig. 15

Spatiotemporal evolution of the measured plasma potential for (a) I_coil = 0 A, (b) I_coil = 5 A, (c) I_coil = 10 A, (d) I_coil = 15 A, (e) I_coil = 20 A, and (f) I_coil = 25 A at representative times in one period of the variation of I_acc. The solid white lines indicate the contours

Effect on electron temperature

In Fig. 16(a) – (f), the columns display the spatiotemporal evolution of T_e at representative times in one period of I_acc. Vertically, the rows show T_e for different magnetic field strengths generated using I_coil = 0 – 25 A discretely incremented at 5A steps. Figure 16(a) shows the lowest T_e distribution (5 < T_e ≤ 10 eV). In Fig. 16(b) and (c), T_e is well within 10 < T_e ≤ 15 eV and 15 < T_e ≤ 20 eV, respectively. Finally, T_e falls within 20 < T_e ≤ 25 eV, 25 < T_e ≤ 30 eV, and 30 < T_e ≤ 35 eV in Fig. 16 (d) – (f), respectively. Overall, the synergistic effects on T_e are consistent with the decoupled effects observed earlier in NA mode. As expected, T_e and ϕ_p respond similarly to increasing B₀: both scales monotonically and both show suppressed spatiotemporal evolutions, signifying negligible modulation by the time-varying magnetic field. A possible explanation of the suppressed spatiotemporal variations will be provided later in Sect. "Discussion".

Fig. 16

Suppressed spatiotemporal evolution of the measured electron temperature for (a) I_coil = 0 A, (b) I_coil = 5 A, (c) I_coil = 10 A, (d) I_coil = 15 A, (e) I_coil = 20 A, and (f) I_coil = 25 A at representative times in one period of I_acc. The solid white lines indicate the contours

Effect on ion density

Figure 17(a) to (f) exhibit the spatiotemporal colormaps of n_i at representative times in one cycle of I_acc for varying B_r strengths generated using I_coil = 0 – 25 A while discretely incrementing at 5A intervals. The experimental conditions mirrored those used to measure T_e and ϕ_p. Figure 17(a) without B₀ indicates the maximum level of n_i in the order of ~ 2.5 × 10¹⁸ m⁻³. In the same figure, clear spatiotemporal variations can be seen, indicating considerable modulation by the time-varying magnetic field. Moreover, the densest region is localized upstream near the bottom left corner with a relatively wide distribution across the axial direction. In contrast, n_i sharply declines in Fig. 17(b) upon the application of I_coil = 5 A in the electromagnet. The decline is approximately 50% of the zero-field density. In Fig. 17(c), the density decay appears to be saturated at approximately 2 × 10¹⁷ m⁻³. This saturation at ~ 2 × 10¹⁷ m⁻³ is in good agreement with NA mode measurements presented earlier. Finally, due to high n_i in Fig. 17(a), the spatiotemporal evolution of n_i in Fig. 17(b) – (f) is relatively obscured.

Fig. 17

Spatiotemporal profiles of the ion density at representative times in one period of I_acc for (a) I_coil = 0 A, (b) I_coil = 5 A, (c) I_coil = 10 A, (d) I_coil = 15 A, (e) I_coil = 20 A, and (f) I_coil = 25 A

Discussion

Why electron temperature increases as magnetic field increases

Identifying the factors that drive the increase in electron temperature with an increasing magnetic field is key to understanding the behaviors of other plasma parameters. This is because both the plasma potential and plasma density are dependent on the electron temperature: the analytical expression for the plasma potential is written as ϕ_p = ϕ_f + T_e ln $\left(\sqrt{\frac{{M}_{i}}{2\pi m}}\right)$, whereas the ion density formula is expressed as n_i = $\frac{{I}_{is}}{0.6e{A}_{p}}\sqrt{\frac{{M}_{i}}{{T}_{e}}}$. Therefore, it is imperative that we first clarify the electron temperature dependency on the magnetic field strength. Generally, stronger magnetic fields suppress the cross-field transport of electrons which increase the plasma resistivity [46]. High plasma resistivity reduces the ability of the plasma to conduct cross-field current leading to higher resistive heating. As a result, the energy dissipated through resistive heating is mainly absorbed by the electrons, increasing their thermal energy. Therefore, a large fraction of the RF power is possibly being diverted from ionization to resistive heating in the presence of higher magnetic fields causing the relatively high electron temperatures observed in this study. In other words, electrons gain more thermal energy from the RF field but expend less of it on ionization, resulting in a hotter but less dense plasma. A non-monotonic increase in the plasma equivalent resistance with increasing magnetic field has been experimentally and numerically demonstrated in an ICP source using 13.56 MHz [36].

Why plasma density decreases as magnetic field increases

Elevated electron temperatures

In a high-resistivity ICP plasma, the electron temperature is expected to increase due to ohmic heating [47]. However, the relationship between T_e and n_e​ is generally inverse so the plasma density drop in a hot plasma is unsurprising. The inverse nature can be explained by considering the particle balance and energy balance equations in a global model [48]. The particle balance is mathematically given as.

K_iz(T_e)$\cdot$ n_e $\cdot$ n_g $\cdot$ V = n_e $\cdot$ u_B(T_e)$\cdot$ A_eff (6).

where K_iz(T_e), n_e, n_g, V, u_B, and A_eff are the ionization rate constant, the electron density, neutral density, plasma volume, Bohm velocity and effective plasma area, respectively. On the left-hand side, as T_e increases, K_iz may saturate or even decline if T_e exceeds the optimal ionization range [48]. K_iz typically increases sharply with T_e​ up to around 10 – 15 eV for Argon and then saturates. On the right-hand side, u_B scales with the square root of T_e and high T_e can enhance particle losses in the sheath. Finally, the energy balance couples T_e​ to n_e​ as.

P_rf = e $\cdot$ n_e $\cdot$ u_B(T_e)$\cdot$ A_eff $\cdot$ ε_T(T_e) (7).

where e is the elementary charge and ε_T(T_e) = E_C + E_I + E_E is the total energy lost per electron lost where E_I = 5.2T_e (for Argon), E_E = 2T_e and E_C are the ion kinetic energy loss, the electron kinetic energy loss, and the collisional energy loss, respectively. For a fixed P_rf, ε_T increases with T_e​ requiring n_e​ to decrease to maintain the balance.

Enhanced perpendicular electric fields

Another potential contributor to the plasma density reduction is E × B drifts. Figure 18 shows the representative electric fields (E_⟂) alongside the spatially averaged plasma density plotted as a function of I_coil. The error bars represent the quadratic sum of the standard deviation of spatially averaged data points and uncertainty. The absolute values for E_⟂ were deduced from the slopes of straight lines linearly fitted to the corresponding axial distributions of potential in Fig. 20 (a). It can be seen that electric fields experience monotonic enhancement against magnetic field strength coincident with a plasma density drop. The enhancement of E_⟂ is expected to reinforce E × B drifts in the thruster, exacerbating plasma wall losses and further reducing the plasma density [49]. Plasma density falls sharply between 0 and 5 A before plateauing around 10 A resulting in a nearly 80% reduction of the zero-field value. On the other hand, the relationship between E and B appears reasonably linear. A relatively similar plasma density drop was observed experimentally [50] and in simulations in capacitively coupled RF argon plasma discharges [49, 50]. The decline was attributed to a transition from stochastic heating to ohmic heating in Ref. [50] and enhanced E × B drifts in Ref. [49]. However, the plasma density eventually recovered at higher magnetic fields after initially declining in both studies.

Fig. 18

Enhanced electric fields coincident with an ion density drop, plotted as a function of coil current. Error bars denote the total uncertainty for the ion density and standard deviation for the electric field

The reason why the ion density plateaus for I_coil > 10A (Fig. 18) even though the perpendicular electric field is increasing is unclear. Figure 19 shows the axial profiles of the normalized ion Larmor radii for I_coil = 5 – 25 A estimated using the Bohm velocity as the perpendicular velocity ${v}_{\perp }$. Since electrons are magnetized even at 5A, their profiles are excluded. The dimensionless ion Larmor radius ${r}_{Li}$ is given by:

$${r}_{Li}=\frac{{M}_{i}{v}_{\perp }}{\left|e\right|BL}$$

(8)

where B is the magnetic field strength measured at r = 30 mm, L is the thruster length, e is the elementary charge and ${M}_{i}$ is the ion mass. The dashed black horizontal line in Fig. 19 represents ${r}_{Li}=1$. Below and above this line, ions are magnetized and unmagnetized, respectively. Clearly, the ions are unmagnetized across all coil currents. Therefore, the plateau in ion density drop for I_coil > 10A cannot be attributed to ion magnetization.

Fig. 19

The axial distribution of the normalized ion Larmor radii for I_coil = 5 – 25 A. The black dashed horizontal line represents ${r}_{Li}=1$. The solid lines are for visual guidance only

Comparison of decoupled effects with synergistic effects

Enhancement of plasma potential

This section compares decoupled effects corresponding to NA mode with synergistic effects synonymous with AC mode. For simplicity and clarity, the enhancement effect is best illustrated by using one-dimensional plots. Figure 20(a) and (b) show the radially averaged plasma potential as a function of axial position across the thruster for six different magnetic field strengths in both NA mode and AC mode, respectively. Results show that the trends observed in the NA mode were successfully reproduced in AC mode. Precisely, as the magnetic field intensity increases such that the electron gyroradius decreases, the plasma potential increases in both modes. Further, these large potentials are accompanied by steeper axial gradients at relatively higher magnetic fields. This subtly suggests the intensification of the perpendicular electric field. However, potential values in AC mode are slightly higher especially at stronger magnetic fields. A possible reason why the plume attained a relatively higher potential in AC mode is attributed to the synergistic effects of the magnetic fields. In addition, the synergy between RF plasma generation power (P_rf) and plasma acceleration power (P_acc) is expected to increase electron heating in the plasma, leading to higher electron temperatures and higher plasma potential in AC mode. Although the NA mode plasma potential is slightly lower, the resultant potential gradients in both modes are comparable, hinting at equivalence between the induced electric fields. Overall, the similarity between experimental results with and without a time-varying magnetic field suggests that the effects are in large part due to the static magnetic field.

Fig. 20

Axial profiles of radially-averaged plasma potential in (a) NA mode and (b) AC mode where the dashed black lines represent linear fits. The AC mode profiles of ϕ_p correspond to t = 0.3 μs in Fig. 15

To support the interpretation that the induced electric fields are equal in both modes, Fig. 21 compares the representative E fields plotted at different magnetic field strength for both modes. The absolute values were deduced from the slopes of straight lines linearly fitted to the corresponding radially averaged axial distributions of potential in Fig. 20. Clearly, there are relatively small differences between the two electric fields. The common denominator in both modes is B₀. Hence, the results imply that the static magnetic field can single handedly enhance the electric fields, underscoring its influence on the plasma potential structure in the thruster. Moreover, the relationship between E and B₀ seems reasonably linear in NA mode. In contrast, non-linearity is observed in AC mode. The superimposition of a time-varying magnetic field introduces fluctuations in the plasma, oscillating the induced electric fields and possibly introducing some form of non-linearity. Overall, these findings emphasize that B₀ exercises a huge influence on ϕ_p under current experimental conditions.

Fig. 21

The perpendicular electric field E_⟂ deduced from the measured plasma potential in both NA and AC modes. Error bars denote the standard deviation

Enhancement of ion density

Figure 22(a) to (f) display n_i colormaps in NA mode, whereas Fig. 22(g) to (l) present AC mode results. For demonstration, the AC mode colormap (bottom panel) corresponds to t = 0.3 μs in Fig. 17. Comparison of the colormaps shows that AC mode distributions qualitatively agree with the NC mode results. The key difference seems to be plasma density enhancement in AC mode. On average, n_i in AC mode for I_coil = 0 A in Fig. 22(g) is approximately twice the nominal n_i in NA mode in Fig. 22(a). One reasonable explanation for the discrepancy in ion density magnitude could be the additional ionization enabled by the extra forward power supplied by the acceleration coil in AC mode. When the azimuthal current j_θ is induced upstream, a fraction of the energy transferred to the electrons as j_θE_θ is partially expended on additional ionization [28]. Note that ion density enhancement is generally insignificant in the presence of B₀. In other words, there is a sharp contrast between Fig. 22(a) and (g), but Fig. 22(f) and (l) are relatively indiscernible. Other than that, n_i exhibits a qualitatively similar decline in both modes as the magnetic field increases. Additionally, the trend of peak ion density region shifting downstream toward the thruster exit successfully recurs in both modes. Overall, this result reinforces the notion that static magnetic fields, the common denominator in both measurements, exert a dominant influence on plasma characteristics under the current experimental conditions.

Fig. 22

The top panel shows n_i in NA mode whereby (a) – (f) corresponds to I_coil = 0 – 25 A. Similarly, the bottom panel depicts the distribution of n_i in AC mode whereby (g) – (l) corresponds to I_coil = 0 – 25 A while using 5A steps to increment the current

Field-aligned distributions of plasma potential and electron temperature

Figure 23(a) to (f) illustrate the NA mode colormaps of plasma potential, whereas Fig. 23(g) to (l) depict AC mode distributions. First, the plasma potential enhancement is distinct for higher coil currents (I_coil ≥ 15 A) but it is unclear at lower coil currents. Significant enhancement generally starts at I_coil = 15 A. This enhancement relative to the magnetic field strength is in sharp contrast to the decay in the ion density case in Fig. 22. Moreover, clearcut radial uniformity exists at lower coil currents (Fig. 23(a) – (c)) but there is undeniable radial nonuniformity at higher coil currents (Fig. 23(d) – (f)). This trend is duplicated in AC mode where radial uniformity is crystal clear in Fig. 23(g) – (i) but it is relatively altered in Fig. 23(j) – (l). These radial ϕ_p gradients could be the effect of the radial magnetic field gradient (the static magnetic field is not radially uniform in Figs. 6 and 8).

Fig. 23

The top panel illustrates ϕ_p in NA mode whereby (a) – (f) correspond to I_coil = 0 – 25 A. Similarly, the bottom panel displays ϕ_p distributions in AC mode whereby (g) – (l) correspond to I_coil = 0 – 25 A at 5A increments

Figure 24(a) to (f) and (g) to (l) depict the electron temperature maps for NA mode and AC mode, respectively. Along the magnetic field, the temperature appears to be radially constant across all low current colormaps in NA mode (Fig. 24(a) – (c)), indicating isothermal magnetic field lines. However, some degree of radial non-uniformity emerges at higher currents. Overall, the radial distribution of temperature mirrors that for potential and shows corresponding behaviors in similar modes.

Fig. 24

The top panel includes T_e​ in NA mode whereby (a) – (f) corresponds to I_coil = 0 – 25 A with a 5 A gap. Similarly, the bottom panel exhibits T_e​ distributions in AC mode whereby (g) – (l) corresponds to I_coil = 0 – 25 A while incrementing the current by 5A steps

Remarks on the spatiotemporal evolution of plasma parameters

A point of major interest is the failure to observe significant spatiotemporal variations in both plasma potential and electron temperature in acceleration mode measurements depicted in Figs. 15 and 16. Interestingly, ion density shows considerable temporal variations in Fig. 17, suggesting strong modulation by the time-varying magnetic field. This implies that a large portion of the acceleration forward power is mainly expended on plasma density enhancement instead of inducing strong electric fields for plasma acceleration. As a result, the tiny temporal variations in plasma potential are an indicator of inefficient induction of small oscillating electric fields by the time-varying magnetic field. A possible explanation of this effect is thought to be the disproportionate ratio of static magnetic fields to the time-varying ones. The large disparity between B₀ and B(t) is best illustrated in Fig. 8. Clearly, Fig. 8(a) without B₀ (I_coil = 0 A) has the lowest total magnetic field strength compared with Fig. 8(b) – (f) which comprise B(t) in combination with B₀ for I_coil = 5 – 25 A. Additionally, the flat spiral coil has 10 turns whereas the electromagnet has 180 turns, resulting in another disproportionate 18:1 ratio. Therefore, minimal temporal variations suggest that the time-varying magnetic field exerts little influence under the current experimental conditions. This highlights the necessity of increasing the amplitude of the acceleration coil current in future work to lower the magnetic field ratio in favor of B(t).

Remarks on the impact of decreasing plasma density on thruster performance

Plasma conductivity is directly related to plasma density, and inductive coupling efficiency is proportional to the electron number density at low pressure [51]. Lower plasma density means lower conductivity, which can affect how efficiently the forward power from the spiral coil is coupled into the plasma. With reduced conductivity, the plasma could significantly impede the azimuthal currents induced by the time-varying magnetic field, leading to inefficient coupling. Consequently, the magnitude of the azimuthal current induced by the spiral coil will be reduced. Since the Lorentz force is proportional to the product of the current density and the magnetic field strength, a reduction in the current density essentially diminishes the Lorentz force. This reduces the acceleration of the ions and overall thrust generated by the thruster. Therefore, optimization of the plasma density level is essential to reinforce the Lorentz force in particular and improve the thruster efficiency in general.

Remarks on the role of the parallel electric field

In this thruster the parallel electric fields point in a radial direction (along the magnetic field lines) towards the lateral walls. Therefore, their contribution to the axial momentum flux is negligible. However, they are capable of accelerating charged particles towards the lateral walls, leading to a plasma loss. Furthermore, our experimental data (Fig. 9) shows that the plasma potential is constant along the magnetic field lines indicating negligible parallel electric fields being formed. In contrast, the perpendicular electric field, pointing axially in this thruster, is capable of accelerating ions downstream and contributing to the axial thrust. Therefore, the role of the parallel electric field is insignificant from a plasma acceleration standpoint.

Conclusion

This study investigated the effects of static magnetic fields applied upon an inductively-coupled radiofrequency argon plasma thruster to enhance inductive plasma acceleration by a time-varying magnetic field. Effects on plasma characteristics were examined with and without a time-varying magnetic field to synergise and decouple the effects, respectively. Results without a time-varying magnetic field showed that increasing the magnetic field strength elevates both the plasma potential and electron temperature, creating steep axial gradients at higher magnetic fields. Enhanced perpendicular electric fields were also observed alongside a plasma density drop. At higher magnetic fields, the density decay saturated but other plasma parameters kept rising monotonically. Acceleration mode measurements confirmed similar trends when both magnetic fields were jointly applied. Contrary to expectations, minimal spatiotemporal changes were observed in both the plasma potential and electron temperature, indicating that the strength of the induced electric field does not vary temporally under current experimental conditions. Interestingly, the plasma density experienced considerable modulation by the time-varying magnetic field. This implies that a large portion of the acceleration forward power is mainly expended on plasma density enhancement instead of inducing strong electric fields for plasma acceleration. Therefore, optimization is essential to decrease the expenditure of acceleration power on plasma density enhancement and increase the desirable induction of oscillating electric fields.