Introduction

In-situ resource utilization (ISRU) is the exploitation of resources found on astronomical bodies to reduce the amount of supplies that need to be brought from Earth [1, 2]. Harnessing the vast resources present throughout our solar system would be revolutionary for space exploration, offering the potential to drastically reduce launch costs and mission complexity. The requirement that a spacecraft must be launched with the full mass of propellant to sustain a long-duration mission imposes significant limitations on the scale and scope of space exploration. For this reason, deriving a reliable source of fuel from space resources is arguably the most vital challenge for ISRU.

One area of research for ISRU fuel production is the synthesis of bipropellants, such as hydrocarbon-based fuels from the Martian atmosphere or LOX/LH2 from electrolysis of lunar water [3, 4]. While the development of bipropellant fuels is critical for launch vehicles, the \(\Delta V\) requirements for deep-space missions demand high-specific impulse (Isp) from the propulsion system. Electric propulsion (EP) systems, such as gridded ion thrusters and Hall-effect thrusters, often fulfill the role of high-Isp propulsion systems for deep space exploration. However, these systems typically operate with xenon, krypton, or argon due to their low ionization energy and favorable material compatibility. A natural, abundant, and accessible source of these gases in deep space is not currently known. Hence, it is imperative that EP systems be developed for compatibility with common bipropellant precursors, such as water or carbon dioxide.

One promising approach entails the use of an electrolytic propellant conditioning system to decompose water into oxygen and hydrogen, and then operate a traditional Hall-effect thruster and thermionic hollow cathode using each respective gas [5]. A plethora of concepts exploit radio-frequency (RF) discharges to produce a plasma while avoiding any contact with electrode surfaces [6,7,8,9]. RF-EP concepts have been demonstrated with many propellants, including water [6], air [10], and iodine [11, 12]. Prior efforts at the U.S. Naval Research Laboratory (NRL) have sought to develop RF cathodes as a drop-in replacement for the thermionic hollow cathode in a Hall-effect thruster [13, 14]. Tokyo Metropolitan University has also invested significant effort in developing RF cathodes for gridded-ion and Hall-effect thrusters [15,16,17,18,19]. The advantage of RF-EP concepts is that the burden of material compatibility issues is substantially alleviated, and so the motivation of our work is to improve our understanding of these devices.

The objective of the present paper is to develop an analytical model that describes inductively-coupled plasma (ICP) discharge efficiency. Our model is based on the perspective of a lumped-element equivalent circuit model that treats an ICP as an air-core transformer. We use the transformer circuit model as an ansatz to derive expressions for ICP discharge efficiency based on device geometry and plasma properties. Using our ICP efficiency model, we examine the scaling and behavior of ICP discharges. Our model is extensible to RF-based EP systems that are cylindrical, unmagnetized, and rely on inductive coupling to only the electrons. To demonstrate the application of our ICP efficiency model, we incorporate it into the RF cathode sheath model and global discharge model developed by Georgin & McDonald [14], and compare the trends predicted by both models. The addition of the ICP efficiency model allows Georgin & McDonald’s RF cathode model to capture certain trends that were observed in experiments but not predicted by the original model. While our model is primarily motivated by ISRU and the need to operate EP systems with alternative propellants, our derivations here assume a monatomic gas, such as argon.

Inductively-coupled plasma (ICP) efficiency model

In this section, we provide a review of the phenomenological behavior of ICP sources, derive the generalized efficiency model for an ICP, and discuss the scaling trends for ICP sources.

The ICP transformer model

For the presented analysis, we adopt a similar transformer model equivalent as described by Piejak et al. [20], which considers the discharge as a single-turn secondary circuit of an air-core transformer. A circuit diagram of the transformer model is shown in Fig. 1. The ICP excitation coil is modeled as an inductor, \(L_{coil}\), with finite series resistance, \(R_{coil}\). The coil’s resistance is given by:

$$\begin{aligned} R_{coil} \approx \frac{2\pi \rho _{cu} r_{coil} N}{A_w} \end{aligned}$$
(1)

where \(\rho _{cu}\) is the resistivity of copper, \(r_{coil}\) is the coil radius, N is the number of turns, and \(A_w\) is the cross-sectional area of the conductor’s skin depth. We calculate \(A_w\) as:

$$\begin{aligned} A_w = \pi r_w^2 - \pi \left( r_w - \delta _{Cu}\right) ^2 = \pi \delta _{Cu} \left( 2r_w - \delta _{Cu}\right) \end{aligned}$$
(2)

with the skin depth \(\delta _{Cu}\) calculated as:

$$\begin{aligned} \delta _{Cu} = \sqrt{\frac{2\rho _{cu}}{\mu _0 \omega }} \end{aligned}$$
(3)

The plasma is modeled with three circuit elements. The single-turn secondary “coil” in the transformer model is described by the electron current’s azimuthal path around the discharge chamber, denoted by \(L_{s,plasma}\). Electron inertia also contributes to the plasma’s inductance, and is denoted by \(L_{s,electron}\). The plasma’s resistance is represented as \(R_{s,plasma}\). There are two methods of evaluating the plasma resistance, which will be discussed in the next subsection.

The transformed plasma resistance across the transformer is:

$$\begin{aligned} R_{p,plasma} = kN^2 R_{s,plasma} \end{aligned}$$
(4)

where k is the coupling coefficient of the transformer. For coaxial inductors of equal length, the coupling coefficient can be approximated as \(k \approx r_{dsch}^2/r_{coil}^2\) [20, 21]. Evaluating the inductive elements is not necessary for the ensuing efficiency analysis, though interested readers may find derivations for these components in [20, 22].

Fig. 1
Fig. 1
Full size image

Lumped circuit element equivalent of an ICP discharge. a Transformer circuit model of an ICP. b A series equivalent circuit of the transformer model with the plasma secondary circuit elements transformed into their series equivalents as seen from the primary side. Figures adapted from [20]

ICP efficiency

A list of assumptions for the model include:

  • Inductive mode is dominant, capacitive mode is negligible.

  • No applied DC magnetic field (i.e., no helicon wave heating).

  • Induced current is dominated by electrons; ions do not participate.

  • Analysis is limited to cylindrical coil geometries that are coaxial to a cylindrical discharge chamber.

  • The plasma’s radius is approximately equal to the discharge chamber radius.

  • The working gas is a monatomic gas (we typically assume argon).

The RF discharge efficiency is defined as the power deposited into the plasma divided by the total RF power losses in the system:

$$\begin{aligned} \eta _{icp} = \frac{P_{p,plasma}}{P_{rf}} \end{aligned}$$
(5)

where \(P_{rf}\) consists of three components: power deposited into the plasma (\(P_{p,plasma}\)), stray power coupled into peripheral conductors external to the discharge chamber (\(P_{stray}\)), and resistive losses in the coil (\(P_{coil}\)):

$$\begin{aligned} P_{rf} = P_{p,plasma} + P_{stray} + P_{coil} \end{aligned}$$
(6)

and therefore:

$$\begin{aligned} \eta _{icp} \equiv \frac{P_{p,plasma}}{P_{p,plasma} + P_{stray} + P_{coil}} \end{aligned}$$
(7)

Stray coupling is proportional to the magnetic field induced by the coil, which falls off rather sharply as the square of the distance from the coil, so we neglect this contribution to give:

$$\begin{aligned} \eta _{icp} = \frac{P_{p,plasma}}{P_{p,plasma} + P_{coil}} \end{aligned}$$
(8)

Since the equivalent circuit for the transformer model is essentially a set of series circuit components, the efficiency may also be represented in terms of the resistance:

$$\begin{aligned} \eta _{icp} = \frac{R_{p,plasma}}{R_{p,plasma} + R_{coil}} \end{aligned}$$
(9)
Fig. 2
Fig. 2
Full size image

ICP density regimes. Left: High-density regime where the skin depth is less than the discharge chamber radius, and the RF fields can only penetrate up to the skin depth to induce current. Right: Low-density regime where the skin depth is greater than the discharge chamber radius, and the RF fields can fully penetrate the plasma

We can rearrange this expression into the form \(\eta _{icp} \propto \frac{1}{1 + \kappa }\), where \(\kappa\) is an efficiency parameter that will aid in our discussion of scaling later in the paper.

$$\begin{aligned} \eta _{icp} = \frac{1}{1 + \frac{R_{coil}}{R_{p,plasma}}} \end{aligned}$$
(10)

The plasma is a conductive medium, so the extent of RF field penetration into the plasma is dependent on the plasma’s conductivity, and therefore the electron density. Hence, there are two regimes of ICP operation that depend on the plasma skin depth, as depicted in Fig. 2. In the low-density regime, the skin depth is greater than the discharge chamber radius, and so the entire bulk plasma contributes to its resistivity. In the high-density regime, only a thin shell of the plasma approximately one skin depth thick contributes to its resistivity. The skin depth can be calculated in two ways, depending on the plasma’s collisionality. In a low collisional frequency regime, where the excitation frequency is much less than the electron-neutral collision frequency (\(\nu _m \ll \omega\)):

$$\begin{aligned} \delta _p = \sqrt{\frac{m_e}{e^2 \mu _0 h_r n_e}} \end{aligned}$$
(11)

where \(m_e\) is the electron mass, \(n_e\) is the electron density, \(\mu _0\) is the permeability of vacuum, e is the elementary charge, and \(h_r\) is the electron density scaling factor for the density at the edge of the sheath. In the opposite scenario, where \(\nu _m \gg \omega\), the collisional skin depth is the relevant figure of merit:

$$\begin{aligned} \delta _c = \sqrt{\frac{2 \nu _m}{\omega }} \sqrt{\frac{m_e}{e^2 \mu _0 h_r n_e}} \end{aligned}$$
(12)

where \(\nu _m\) is the effective collision frequency. As an example, when \(p = 15\) mTorr, \(\nu _m \approx 6.78\) MHz. For the most part, we operate in the collisional regime, so we assume \(\delta = \delta _c\). In the following sections, we will derive the expressions for \(\eta _{icp}\) in the low-density and high-density operating regimes.

ICP efficiency - low density regimes (\(\delta > r_{dsch}\))

When the electron density is low, then \(\delta > r_{dsch}\), where \(r_{dsch}\) is the discharge chamber radius. In the low-density situation, the excitation fields are able to fully penetrate the plasma, and the plasma’s resistance is determined by using Faraday’s law to find the induced electric field [22]:

$$\begin{aligned} E_\theta (r) = \frac{j \omega r \mu _0 N I_{rf}}{2h} \end{aligned}$$
(13)

and current density from Ohm’s law:

$$\begin{aligned} J_\theta (r) = j \omega \epsilon _0 E_\theta \kappa _p = j \omega \epsilon _0 \left( \frac{j \omega r \mu _0 N I_{rf}}{2h}\right) \left( 1 - \frac{\omega _{pe}^2}{\omega ^2}\right) = \frac{\epsilon _0 r \mu _0 N I_{rf}}{2h} \left( \omega _{pe}^2 - \omega ^2\right) \end{aligned}$$
(14)

Defining the conductivity as \(\sigma _p = \frac{e^2 h_r n_e}{m_e \nu _m}\), plasma frequency as \(\omega _{pe} = \sqrt{\frac{n_e e^2}{m_e \epsilon _0}}\), and assuming \(\omega _{pe} \gg \omega\), we integrate the current density over the volume to obtain the absorbed power:

$$\begin{aligned} P_{abs} = \int _0^{r_{dsch}} \frac{J_\theta ^2}{\sigma _p} 2 \pi r h \, dr = \int _0^{r_{dsch}} \frac{\pi r^3 \mu _0^2 N^2 e^2 h_r n_e I_{rf}^2 \nu _m}{2 h m_e} \, dr = I_{rf}^2 \frac{\pi e^2 h_r n_e \nu _m \mu _0^2 r_{dsch}^4}{8 m_e h} \end{aligned}$$
(15)

Assuming power takes the form \(P = I^2 R\), then

$$\begin{aligned} R_{s,plasma} = \frac{\pi e^2 h_r n_e \nu _m \mu _0^2 r_{dsch}^4}{8 m_e h} \end{aligned}$$
(16)

The impedance of the plasma reflected across the transformer is:

$$\begin{aligned} R_{p,plasma} = kN^2 \frac{\pi e^2 h_r n_e \nu _m \mu _0^2 r_{dsch}^4}{8 m_e h} = \frac{\pi e^2 h_r n_e \nu _m \mu _0^2 N^2 r_{dsch}^6}{8 m_e h r_{coil}^2} \end{aligned}$$
(17)

where h is the discharge height, \(h_r \approx 0.8 \left( 4 + \frac{r_{dsch}}{\lambda _i}\right) ^{-1/2}\) is the sheath edge scaling factor, \(\lambda _i\) is the ion-neutral mean free path, and \(\sigma _i\) is the combined ionic momentum transfer cross section (often approximately for thermal ion energies of interest \(\sigma _i \approx 10^{-20}\) m\(^2\) [20, 22]). Substituting \(R_{s,plasma}\) back into Eq. 10

$$\begin{aligned} \eta _{icp,LD} = \frac{1}{1 + \frac{2 \pi \rho _{cu} r_{coil} N}{A_w} \frac{8 m_e h r_{coil}^2}{\pi e^2 h_r n_e \nu _m \mu _0^2 N^2 r_{dsch}^6}} = \frac{1}{1 + \frac{16 \rho _{cu} m_e}{e^2 \mu _0^2} \frac{1}{h_r n_e \nu _m} \frac{r_{coil}^3}{N A_w} \frac{h}{r_{dsch}^6}} \end{aligned}$$
(18)

The collisional frequency can be expressed as:

$$\begin{aligned} \nu _m = n_g \sigma _i v_{th} = n_g \sigma _i \sqrt{\frac{8 e T_e}{\pi m_e}} \end{aligned}$$
(19)

Substituting the expression for \(\nu _m\) back into the equation, we arrive at an expression for the ICP efficiency in the low-density regime:

$$\begin{aligned} \eta _{icp,LD} = \frac{1}{1 + \frac{8 \rho _{cu} m_e^{3/2} \sqrt{\pi }}{e^{5/2} \mu _0^2 \sqrt{2}} \frac{r_{coil}^3}{N A_w} \frac{h}{r_{dsch}^6} \frac{1}{h_r n_e n_g \sigma _i \sqrt{T_e}}} \end{aligned}$$
(20)

Here, we arrive at an expression for \(\eta _{icp}\) in the low-density regime that is a function of geometry and input parameters that can be controlled experimentally (e.g., the neutral gas density). This form of \(\eta _{icp,LD}\) conveniently lets us examine scaling, as our objective is to ensure the term in the denominator is as small as possible. Immediately, we can see that there is a very strong dependence on the geometry of the ICP.

ICP efficiency - high density regimes (\(\delta \ll r_{dsch}\))

At a certain input power, the electron density becomes high enough that the plasma begins to screen out the RF fields. When \(\delta \ll r_{dsch}\), the plasma is considered in the high-density regime. The resistance can be found by treating the plasma as a thin, conductive, cylindrical shell:

$$\begin{aligned} R_{s,plasma} = \frac{\pi r_{dsch}}{\sigma _p \delta _c h} = \frac{\pi r_{dsch} m_e \nu _m}{h e^2 h_r n_e} \sqrt{\frac{\omega e^2 \mu _0 h_r n_e}{2 \nu _m m_e}} = \frac{\pi r_{dsch} \sqrt{\omega m_e \nu _m \mu _0}}{h e \sqrt{2 h_r n_e}} \end{aligned}$$
(21)

and so the resistance seen from the primary side is:

$$\begin{aligned} R_{p,plasma} = kN^2 \frac{\pi r_{dsch} \sqrt{\omega m_e \nu _m \mu _0}}{h e \sqrt{2 h_r n_e}} = \frac{N^2 \pi r_{dsch}^3 \sqrt{\omega m_e \nu _m \mu _0}}{h e r_{coil}^2 \sqrt{2 h_r n_e}} \end{aligned}$$
(22)

Substituting Eq. 22 into Eq. 10 gives:

$$\begin{aligned} \eta _{icp,HD} = \frac{1}{1 + \frac{2 \rho _{cu} h e r_{coil}^3 \sqrt{2 h_r n_e}}{A_w N r_{dsch}^3 \sqrt{\omega m_e \nu _m \mu _0}}} \end{aligned}$$
(23)

and again substituting the expression for \(\nu _m\):

$$\begin{aligned} \eta _{icp,HD} = \frac{1}{1 + \frac{(2e)^{3/4} \pi ^{1/4} \rho _{cu}}{m_e^{1/4} \sqrt{\mu _0}} \frac{r_{coil}^3}{A_w N} \frac{h}{r_{dsch}^3} \frac{\sqrt{h_r n_e}}{\sqrt{\omega n_g \sigma _i} T_e^{1/4}}} \end{aligned}$$
(24)

Now we have arrived at an expression for \(\eta _{icp}\) in the high-density regime. In the next section, we will elaborate on the key differences between these two expressions for \(\eta _{icp}\).

ICP efficiency - discussion

In this section, we will discuss how the \(\eta _{icp}\) expressions suggest ICPs should scale. We’ll start the discussion with the low-density regime. To examine scaling, we neglect the physical constants:

$$\begin{aligned} \eta _{icp,LD} \propto \frac{1}{1 + \frac{r_{coil}^3}{N A_w} \frac{h}{r_{dsch}^6} \frac{1}{h_r n_e n_g \sigma _i \sqrt{T_e}}} \end{aligned}$$
(25)

For the sake of discussion, we define several parameters that let us break the efficiency equation into the different contributing components. We define some parameter \(\kappa\) such that

$$\begin{aligned} \eta _{icp} \propto \frac{1}{1 + \kappa }; \quad \kappa \equiv \kappa _c \kappa _g \kappa _p \end{aligned}$$
(26)
$$\begin{aligned} \kappa _c = \frac{r_{coil}^3}{N A_w} \end{aligned}$$
(27)
$$\begin{aligned} \kappa _g = \frac{h}{r_{dsch}^6} \end{aligned}$$
(28)
$$\begin{aligned} \kappa _p = \frac{1}{h_r n_e n_g \sigma _i \sqrt{T_e}} \end{aligned}$$
(29)

With the separate components defined, we can examine how they contribute to efficiency. Unity \(\eta _{icp}\) implies perfect efficiency, and occurs when \(\kappa = 0\). We have broken the \(\kappa\) term into three components: a “coil” contribution (\(\kappa _c\)), a “chamber geometry” contribution (\(\kappa _g\)), and a “plasma” contribution (\(\kappa _p\)). The \(\kappa _c\) term suggests that minimizing the coil radius while maximizing the number of turns leads to improved efficiency, and that the coil radius has a strong influence on performance. The \(\kappa _g\) term suggests that the discharge becomes more efficient when the discharge chamber’s width-to-height ratio (a.k.a. the “aspect ratio”) is very large. In other words, a short chamber with a large diameter is preferable. Since \(\kappa _g \propto r_{dsch}^{-6}\), the chamber radius has a very significant weight on efficiency. The \(\kappa _p\) term suggests that the discharge becomes more efficient as \(n_g\), \(\sigma _i\), \(T_e\), and \(n_e\) grow. The dependence on \(n_e\) is stronger than the other \(\kappa _p\) terms. Since the coil and discharge chamber geometries are intertwined, their contributions trade against each other, and so there exists some optimum balance between the two.

Now we move on to the high-density regime. Again, dropping the constants to examine scaling of \(\eta _{icp,HD}\):

$$\begin{aligned} \eta _{icp,HD} \propto \frac{1}{1 + \frac{r_{coil}^3}{A_w N} \frac{h}{r_{dsch}^3} \frac{\sqrt{h_r n_e}}{\sqrt{\omega n_g \sigma _i} (T_e)^{1/4}}} \end{aligned}$$
(30)

and again utilizing our previously defined \(\kappa\) terms from Eq. 24:

$$\begin{aligned} \kappa _c = \frac{r_{coil}^3}{N A_w} \end{aligned}$$
(31)
$$\begin{aligned} \kappa _g = \frac{h}{r_{dsch}^3} \end{aligned}$$
(32)
$$\begin{aligned} \kappa _p = \frac{\sqrt{h_r n_e}}{\sqrt{\omega n_g \sigma _i} (T_e)^{1/4}} \end{aligned}$$
(33)

In the high-density regime, we retain the same three contributors to efficiency. The \(\kappa _c\) term remains the same, while the \(\kappa _g\) term is less sensitive to \(r_{dsch}\). The \(\kappa _p\) term is very different in the high-density regime. While the weight of \(n_g\), \(\sigma _i\), and \(T_e\) are slightly different, the general trend remains the same (i.e., higher values give better efficiency). However, now we see that \(\kappa \propto \sqrt{n_e}\) for the high-density regime, as opposed to \(\kappa \propto n_e^{-1}\) for the low-density regime. The reason for the change in \(n_e\) dependence is that for the high-density regime, the only part of the plasma participating in the RF circuit lies within the skin depth. While \(R_{p,plasma} \propto n_e\) for the low-density regime, the dependency changes to \(R_{p,plasma} \propto 1/\sqrt{n_e}\) in the high-density regime.

Above a certain density, RF power can no longer penetrate fully into the plasma, so the \(\eta _{icp,LD}\) model becomes invalid. Likewise, below a certain density, the geometry of a “thin shell conductor” no longer becomes physically realistic, so the \(\eta _{icp,HD}\) model becomes invalid. To examine an appropriate region to transition from the low-density to high-density regimes, the expressions for \(\eta _{icp,LD}\) and \(\eta _{icp,HD}\) are plotted as a function of \(n_e\) in Fig. 3. In this example, we used \(r_{dsch} = 25\) mm, \(h = 100\) mm, \(f = 6.78\) MHz, \(N = 5\) turns, and \(p_{dsch} = 25\) mTorr. The solid red vertical line denotes the point where the skin depth is equal to the discharge chamber’s radius, so this point should mark the transition from the low-density to high-density operating regime. Indeed there is good agreement between the two models at this intersection point, as seen in Fig. 3, so we choose this condition to transition between the two efficiency models.

The scaling of our derived efficiencies is rather straightforward to understand from examining the analytical expressions, but it is still instructive to look at plotted trends. The following parametric study shows how \(\eta _{icp}\) varies as a function of \(n_e\) between \(n_e = 10^{15} \, \text {m}^{-3}\) and \(10^{21} \, \text {m}^{-3}\), i.e., how \(\eta _{icp}\) varies as we change the input power. The parameters we adjusted in our study include the discharge chamber radius (\(r_{dsch}\)), discharge chamber height (h), number of coil turns (N), and the discharge chamber pressure (\(p_{dsch}\)). For the study, the coil radius was assumed to have a fixed spacing away from the discharge chamber (\(r_{coil} = r_{dsch} + 5 \, \text {mm}\)), the working gas is argon, the excitation frequency is \(f = 6.78 \, \text {MHz}\), and electron temperature is fixed at \(T_e = 4 \, \text {eV}\). Table 1 shows the parameters used for each study. The parametric study results shown in Fig. 4 reinforce the above discussion on \(\eta _{icp}\) scaling. In our parametric study, we assume that the coil’s axial length occupies the full length of the discharge chamber, so turn-to-turn spacing is implicitly varied here. The effect that coil spacing has on self and mutual inductance of the system is not inherently captured by our model, and is the subject of future work.

Fig. 3
Fig. 3
Full size image

Efficiency plotted for the two different density regimes. The dashed line represents \(\eta _{icp,LD}\), i.e. efficiency for the low-density regime (\(\delta > r_{dsch}\)), and the solid line represents \(\eta _{icp,HD}\), i.e. efficiency for the high-density regime (\(\delta \ll r_{dsch}\)). The vertical red line denotes the point where the skin depth equals the radius of the discharge chamber. At electron densities below the red line, \(\eta _{icp,HD}\) becomes invalid, and at electron densities above the red line, \(\eta _{icp,LD}\) becomes invalid. There is good agreement between the models’ intersection point and the density where skin depth equals discharge chamber radius, so this is a good natural point to transition between the models

Table 1 A table of values used for the parametric study. The study names correspond to each subplot in Fig. 4
Full size table
Fig. 4
Fig. 4
Full size image

Results of a parametric study of \(\eta _{icp}\). a Discharge chamber radius study. b Coil turns study. c Chamber length study. d Discharge chamber pressure study

RF cathode scaling

Since an ICP is a central component of an RF cathode, this technology provides a good demonstration of how our derived expressions for \(\eta _{icp}\) can be applied to assess the impact on an RF propulsion system’s performance. First, we will review RF cathode theory and the method of operation. Next, we will define some efficiency metrics for RF cathodes and restate the expressions derived by Georgin and McDonald [14]. We will then present geometric scaling arguments for RF cathodes. Finally, we will apply the \(\eta _{icp}\) model to RF cathodes to examine how discharge efficiency affects overall cathode performance.

RF cathode theory

An RF cathode operates via “nonambipolar electron emission”, which was first described by Longmier and Hershkowitz [23,24,25]. RF cathodes are comprised of a multi-turn RF coil wrapped around an insulating discharge vessel and a metallic collector electrode that is inserted inside of the discharge vessel. Gas is fed into the discharge tube, and plasma is generated by inductive coupling from the RF coil. For each electron that is extracted from the cathode’s orifice, a corresponding ion must be neutralized on an internal collector to enforce quasineutrality. Unlike thermionic hollow cathodes, there are no emissive surfaces in an RF cathode. Instead, electrons are generated from gas-phase impact ionization driven by the inductive coil. Figure 5 shows a schematic illustration of a typical RF cathode and a description of the method of operation.

Georgin & McDonald recently developed theory for RF cathodes that arises from analogous treatment to an asymmetric double Langmuir probe, finding excellent agreement with experiments [14]. The assumptions in their model included:

  • No \(\nabla T_e\)

  • Heat conduction by electrons is small

  • Ionization is the dominant inelastic collision

  • Energy lost to collisions with ions and neutral particles is low due to \(m_e/m_i\) scaling

  • The RF coupling efficiency is constant as a function of plasma parameters and geometry.

The motivation of our work here is to revisit Assumption #5, which will be discussed in Section IV.D. Georgin & McDonald derived the following expression for the I-V curve of an RF cathode [14]:

$$\begin{aligned} V_{DC} = \gamma T_e \ln \left( \frac{(A \cdot N)I + 1}{1 - I}\right) \end{aligned}$$
(34)

where A is the ratio of the cathode area to orifice area \((A_c/A_o)\), N is the ratio of the electron density at the cathode and orifice surfaces \((n_c/n_o)\), I is the discharge current normalized by the cathode’s saturation current \((I_{DC}/I_{sat})\), and \(\gamma\) is a scaling factor that corrects for distortions in the sheath. Georgin & McDonald also derive an argon global model that results in expressions for \(T_e\) and \(n_e\) as a function of geometry and system inputs [14]. The result of their energy balance gives:

$$\begin{aligned} P_{rf} = \frac{\left(U_{iz} + \frac{5}{2} T_e\right)}{\eta _{rf}} \frac{S}{A_c} I_{sat} \end{aligned}$$
(35)

where S is the cathode body’s surface area, and \(\eta _{rf}\) is the RF coupling efficiency. We will discuss \(\eta _{rf}\) in more detail in Section IV.D. The result of their mass balance gives:

$$\begin{aligned} r_{dsch} \sigma _{iz0} \exp \left( \frac{-U_{iz}}{T_e}\right) \left( \frac{p_{dsch}}{k_B T_n} - \frac{\eta _{rf} P_{rf}}{e\left(U_{iz} + \frac{5}{2} T_e\right)} \frac{1}{S} \sqrt{\frac{m_i}{e T_e}}\right) = 2\sqrt{\frac{m_i}{m_e}} \end{aligned}$$
(36)

where \(\sigma _{iz0}\) is the ionization rate coefficient for argon, \(U_{iz}\) is the ionization energy of argon, \(T_n\) is the neutral gas temperature, \(p_{dsch}\) is the neutral gas pressure in the discharge chamber, and \(m_i\) is the ion mass. The expression for ion saturation current \((I_{sat})\) is:

$$\begin{aligned} I_{sat} = 0.61 A_c n_i \sqrt{\frac{e T_e}{m_i}} \end{aligned}$$
(37)

With known geometry values and system input parameters (e.g., pressure and input power), Eqs. 32, 33, 34 and 35 can be used to determine plasma properties and produce a predicted I-V curve of an RF cathode.

Fig. 5
Fig. 5
Full size image

A diagram illustrating the method of operation for an RFC. (1) An RF coil generates a plasma via inductive coupling. (2) Electrons are extracted from the bulk plasma discharge through the orifice. (3) For each electron that is extracted, a corresponding ion must be neutralized on the internal ion collector to ensure quasineutrality of the plasma

RF cathode performance metrics

The two key RF cathode performance metrics are the saturation current \((I_{sat})\) and the effective voltage \((V_{eff})\). The saturation current is defined as the maximum ion current that the RF cathode’s collector surface can extract from the plasma to enforce quasineutrality when sourcing electrons, and so \(I_{sat}\) also represents the maximum emitted electron current. The effective voltage is also known as the “discharge cost” and represents the amount of power that must be supplied to the plasma to extract electron current, and is given as:

$$\begin{aligned} V_{eff} = \frac{P_{rf}}{I_{sat}} + V_{cc} \end{aligned}$$
(38)

where \(V_{cc}\) is the cathode coupling voltage. Rewriting Eq. 35 gives:

$$\begin{aligned} I_{sat} \simeq \frac{\eta _{rf} P_{rf}}{\left(U_{iz} + \frac{5}{2} T_e\right)} \frac{A_c}{S} \end{aligned}$$
(39)

Substituting the two above equations, we find:

$$\begin{aligned} V_{eff} = \frac{\left(U_{iz} + \frac{5}{2} T_e\right)}{\eta _{rf}} \frac{S}{A_c} + V_{cc} \end{aligned}$$
(40)

\(V_{cc}\) is strongly dependent on many factors external to the RF cathode, e.g., the distance between anode and cathode, background pressure, magnetic field characteristics, etc. For the subsequent analysis, we focus on the first term:

$$\begin{aligned} V_{eff,rf} \propto \frac{\left(U_{iz} + \frac{5}{2} T_e\right)}{\eta _{rf}} \frac{S}{A_c} \end{aligned}$$
(41)

This expression describes the discharge cost attributed solely to the RF discharge. The three contributors to \(V_{eff,rf}\) are the RF cathode geometry \((S/A_c)\), the RF discharge efficiency \((\eta _{rf})\), and the plasma properties \(\left(U_{iz} + \frac{5}{2} T_e\right)\). For the scope of our discussion, we assume that \(\left(U_{iz} + \frac{5}{2} T_e\right)\) is relatively constant and acts as the minimum discharge cost for an RF cathode. We instead look to examine the influence of \(S/A_c\) and \(\eta _{rf}\) on RF cathode performance.

RF cathode geometry scaling

Here we examine how the RF cathode geometry affects discharge performance. The discharge chamber surface area for a cylindrical cathode body is:

$$\begin{aligned} S = 2\pi rh + 2\pi r^2 \end{aligned}$$
(42)

Assuming the collector is a coaxial cylinder that fits within the cathode body:

$$\begin{aligned} A_c = 2\pi rh \end{aligned}$$
(43)

Increasing both r and h will yield a greater \(A_c\), and therefore higher \(I_{sat}\). The ratio of cathode body area to collector surface area is:

$$\begin{aligned} \frac{S}{A_c} & = \frac{2\pi rh + 2\pi r^2}{2\pi rh} \nonumber \\ & = \left( 1 + \frac{r}{h}\right) \end{aligned}$$
(44)

Since \(V_{eff} \propto \frac{S}{A_c}\), we wish to minimize the equation above. The RF cathode scaling follows a similar pattern for ICP efficiency in general: greater radius and shorter length give a lower predicted cathode \(V_{eff}\). While we want to maximize \(A_c\) to increase \(I_{sat}\), we only want to do so by increasing the radius so that the RF cathode has a large aspect ratio.

Updated RF cathode sheath model

The RF coupling efficiency introduced in Eq. 35 is very similar to \(\eta _{icp}\), but with one additional component. The ion collector is comprised of a metallic material, and so coupling to the collector’s metallic structure must be taken into consideration given its proximity to the RF coil. We modify Eq. 7 to include an additional collector coupling term, \(P_{collector}\):

$$\begin{aligned} \eta _{rf} \equiv \frac{P_{plasma}}{P_{plasma} + P_{stray} + P_{collector} + P_{coil}} \end{aligned}$$
(45)

Evaluating \(P_{collector}\) and \(P_{stray}\) is analogous to predicting the heating performance in inductive furnaces. Induction heating problems are commonly tackled using computational techniques, as they are analytically intractable unless making significant assumptions. For our analysis, we assume that \(P_{collector}\) is a constant percentage of the total deposited RF power, and so Eq. 45 may be recast as:

$$\begin{aligned} \eta _{rf} = \eta _{coll} \eta _{icp} \end{aligned}$$
(46)

where \(\eta _{coll}\) is a fixed coupling efficiency value that is equivalent to the ion collector’s RF transparency. We assume that the RF transparency is approximately the ratio of open area to the total area.

Fig. 6
Fig. 6
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A power sweep comparison between the “old” RF cathode model and the “new” RF cathode model that has been updated to include our derived ICP efficiency model. Power is swept from 50W to 350W in 50W increments. a An I-V curve of the RF cathode using the old cathode model. b The effective voltage versus discharge current using the old model. c An I-V curve of the RF cathode using the new model. d The effective voltage versus discharge current using the new model. The red x’s denote the point of minimum \(V_{eff}\), i.e., the most efficient operating point on the I-V curve

Fig. 7
Fig. 7
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A comparison of the updated model’s predicted \(V_{eff}\) versus \(P_{rf}\). The model results and experimental data are normalized against the minimum \(V_{eff}\) value of each respective dataset

Fig. 8
Fig. 8
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Plasma properties for the power sweep. The black trace represents results from Georgin & McDonald’s model. The red trace represents results from the updated model

Fig. 9
Fig. 9
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A flow rate sweep comparison between the old RF cathode model and the new model. Flow rate is varied from 15 SCCM to 20 SCCM in 1 SCCM increments. a An I-V curve of the RF cathode using the old cathode model. b The effective voltage versus discharge current using the old model. c An I-V curve of the RF cathode using the new model. d The effective voltage versus discharge current using the new model. The red x’s denote the point of minimum \(V_{eff}\), i.e., the most efficient operating point on the I-V curve

Fig. 10
Fig. 10
Full size image

Plasma properties for the mass flow rate sweep. The black trace represents results from Georgin & McDonald’s model. The red trace represents results from the updated model

Fig. 11
Fig. 11
Full size image

Results of a power sweep up to very high power. Power is swept from 50W to 2000W in 50W increments. The black trace represents results from Georgin & McDonald’s model. The red trace represents results from the updated model. Results of the study show that for a fixed RF cathode geometry, there exists an optimum performance, and that arbitrarily increasing power to obtain more current will result in diminishing returns

The analytical models developed by Georgin & McDonald [14] were adapted to include the ICP efficiency model derived above. For brevity, we refer to Georgin & McDonald’s model as the “old” model, and the one updated to accommodate \(\eta _{icp}\) as the “new” model. First, Eq. 36 is solved implicitly to determine \(T_e\), which is then used with Eqs. 35 and 37 to determine \(n_e\). Georgin & McDonald’s work assumed a constant \(\eta _{rf}\) of 20%. In the new model, \(\eta _{icp,LD}\) and \(\eta _{icp,HD}\) are found by first making an initial guess of \(\eta _{icp}\) to solve for \(T_e\) and \(n_e\), and then iteratively solving for \(\eta _{icp}\) until the error converges. Based on the known RF transparency of the collector, we approximate \(\eta _{coll} \approx 50\%\). With the properties of the plasma determined, they are substituted into Eq. 34, along with geometry and system input parameters to provide a predicted I-V curve.

Georgin & McDonald conducted a power and flow rate sweep to examine RF cathode behavior. We repeat the power and flow rate sweeps using the new model and compare them against the old model. Figure 6 shows the I-V curves and \(V_{eff}\) curves for the old model (top row) and new model (bottom row). While the results appear very similar, the new model captures variation in the minimum \(V_{eff}\), whereas the old model predicts a constant value regardless of \(P_{rf}\). The ability to capture this \(V_{eff}\) trend is important, as it has previously been observed in experimental data but not predicted with the old model. For instance, experimental data presented by Georgin demonstrates a falling \(V_{eff}\) with increasing \(P_{rf}\) for both argon and water as propellants [14]. Figure 7 shows results from the new model plotted against experimental data reported by Georgin & McDonald [14], where both sets of data are normalized to compare trends. Variation in \(V_{eff}\) is due to the changing \(\eta _{rf}\) as a function of RF power. Figure 8 shows that the predicted plasma properties are very similar between the models but illustrates variation in \(\eta _{rf}\) as a function of \(P_{rf}\).

Figure 9 shows the results of the flow rate sweep with the I-V and \(V_{eff}\) curves for the old model (top row) and new model (bottom row). For the flow rate sweep, the differences between the models are not significant, but the new model does show slightly increased variation in the I-V and \(V_{eff}\) curves with changing flow rate. This behavior is indeed consistent with experimental data presented by Georgin & McDonald [14]. Again, the new model’s behavior is attributed to the variation of \(\eta _{rf}\), as changing flow rate influences \(p_{dsch}\). Figure 10 shows the plasma properties over the flow rate sweep.

To underscore the importance of capturing the ICP efficiency behavior in the RF cathode model, we extended the power sweep up to 2 kW. The results of the high-power sweep for the old and new models are shown in Fig. 11. With increasing power, \(V_{eff}\) rapidly decreases to a minimum value, and then begins to rise at a substantially reduced rate. The inflection point in \(V_{eff}\) corresponds to the low-to-high density transition in the ICP model. The RF coupling efficiency is plotted to illustrate the consistent trends between \(\eta _{rf}\) and \(V_{eff}\) with varying power. For a fixed geometry, increasing power arbitrarily results in diminishing returns for RF cathode performance.

Conclusion

In the presented paper, we have developed an analytical model that describes inductively-coupled plasma (ICP) discharge efficiency. The derived expressions are valid for the low and high electron density regimes, and provide ICP discharge efficiency as a function of device geometry and plasma properties. Using our ICP efficiency model, we discussed scaling and predicted behavior of ICP discharges. The key conclusions of our \(\eta _{icp}\) derivation and parametric sweep are:

  • The plasma’s RF skin depth is an intrinsic limit to the ICP discharge efficiency. When \(\delta\) is large, \(\eta _{icp}\) increases linearly with \(n_e\). When \(\delta\) is smaller than the discharge chamber radius, \(\eta _{icp}\) drops with \(n_e^{-1/2}\).

  • ICP efficiency optimizes towards a wider discharge diameter and shorter discharge length, i.e., a larger discharge chamber aspect ratio is more efficient.

Our ICP efficiency model may be generally applied to predict the behavior of RF-EP systems that are cylindrical, have no applied DC magnetization, and rely on inductive coupling to only the electrons. To demonstrate the application of our ICP efficiency model, we incorporated it into the RF cathode sheath model and global discharge model developed by Georgin & McDonald [14], and compared the trends predicted by both models. Key conclusions from our updated RF cathode model are:

  • Including \(\eta _{icp}\) allows us to capture variation in \(V_{eff}\), as seen in experimental data.

  • For a fixed geometry, \(V_{eff}\) improves with increasing power up to a point, but the transition from low to high electron density regime results in an inflection in the \(\eta _{icp}\) trend, and \(V_{eff}\) will decay at higher input power.

Future work will entail experimental validation of the predicted ICP efficiency and RF cathode models, as well as improvements to the RF cathode analytical model. One specific area of improvement includes the incorporation of frozen-flow losses, which would extend the model’s applicability to molecular propellants. Including models of the coils self and mutual inductance with the plasma would also improve our coil-to-plasma coupling coefficient’s accuracy. Another area of improvement is the development of a model that appropriately handles coupling to the RF cathode’s collector.