Current sheet formation in inductive pulsed plasma thrusters (IPPTs) is investigated theoretically to determine how non-equilibrium ionization processes influence plasma impermeability to inductive electromagnetic fields and downstream propellant mass. Plasma impermeability to both electromagnetic fields and downstream mass is a prerequisite for efficient IPPT operation. A lumped-element circuit model of an IPPT plasma is modified to include propellant ionization and the electron energy balance under non-equilibrium conditions, neutral gas entrainment via charge exchange collisions, and electromagnetic coupling to a finite skin depth plasma. It is found that current sheets impermeable to both the accelerating fields and downstream mass—presumed to exist in all prior circuit modeling efforts—form only under specific conditions. The dynamics of electron heating during the early portion of the inductive current cycle are identified as the dominant contributors to current sheet formation. A new dimensionless scaling parameter is derived to characterize electron heating relative to inelastic ionization losses, from which it is found that impermeable current sheet formation requires Ohmic heating in the early formation phase to offset ionization losses associated with the entire propellant mass bit. This finding provides a physical explanation and generalization of the semi-empirical requirement on coil current rise rate that is commonly used in the early design phase of IPPTs to ensure current sheet formation.

## I. INTRODUCTION

The formation and acceleration of plasma using inductive electromagnetic pulses has been proposed for spacecraft electric propulsion.^{1,2} Technologies that leverage this principle are generally classified as inductive pulsed plasma thrusters (IPPTs).^{3,4} In their most common form, energy stored across a capacitor is rapidly released as current through an inductive coil. If the current rise rate is sufficiently fast, the resulting electric field can ionize propellant injected in front of the coil, thus forming a plasma.^{1} A high plasma conductivity gives rise to a localized region of current (i.e., current sheet) that opposes the current in the inductive coil.^{2} The resulting Lorentz force accelerates the plasma away from the inductive coil to velocities in excess of 10 s of km/s. Interest in IPPTs for in-space propulsion is primarily motivated by their ability to achieve high specific impulses ($>1000$ s) and thrust efficiencies ($>50$%) without relying on electrodes in direct contact with the plasma.^{4} This last point distinguishes IPPTs from other electric thrusters commonly used today (e.g., gridded-ion and Hall effect thrusters) and may ultimately enable the development of long-lifetime electric propulsion systems compatible with a wider variety of propellant types.

A key physical requirement for high-performance IPPT operation is the formation of a current sheet that is impermeable to both the electromagnetic fields of the coil and downstream propellant mass. Impermeability to the fields is required for good electromagnetic coupling, whereas impermeability to downstream gas is required for good propellant mass utilization. Indeed, much of the early experimental research on IPPTs focused on the structure and formation of the current sheet in front of a planar coil.^{2} It was empirically found that the best performing concepts possessed a large coil diameter, low coil inductance ($Lc$), and high initial charge voltage ($V0$).^{5} The latter two requirements were explained in terms of the initial coil current rise rate, $dIc/dt|t=0\u2248V0/Lc$, which governs the strength of the azimuthal electric field that forms adjacent to the coil. It was hypothesized that a minimum value of $dIc/dt|t=0$ is required to support the initial breakdown of propellant gas.^{6} One particularly important consequence of the requirement on $dIc/dt|t=0$ was that thruster designs tended to favor higher $V0$ and, therefore, higher pulse energies. Technological limitations on thruster component specific energy (energy per unit mass), combined with application-driven constraints on the thruster specific power (power per unit mass), yielded thruster designs with average power levels that were (and remain) too high ($P\u223c0.1$–1 MW) to be compatible with existing spacecraft buses. Concepts that employ pre-ionization schemes to facilitate current sheet formation at lower values of $dIc/dt|t=0$ have been proposed as a means of scaling IPPTs to lower powers^{7,8} with laboratory experiments demonstrating the successful formation of current sheets in a portion of these devices.

Despite the experimental demonstration of current sheet formation in a variety of IPPTs—both with and without pre-ionization—the specific criteria under which a current sheet forms are not well understood from a fundamental physics perspective. This is unlike many other key physical processes whose influence on IPPT propulsion performance is well understood, such as electromagnetic decoupling of the plasma as it moves away from the coil,^{5} losses due to the internal resistance and stray inductance of the drive circuit,^{6} the relationship between the pre-pulse propellant distribution and acceleration profile,^{9,10} and the addition of a static magnetic field.^{11} In fact, many of these processes have been successfully reproduced using theoretical models;^{5,11–13} a cumulative effort that has contributed numerous dimensionless parameters presently used to optimize new thruster designs.^{14} However, there remains a need for fundamental scaling laws for inductive current sheet formation. Beyond IPPTs, a better understanding of this problem could provide deeper insight into current sheets observed in space^{15} and astrophysical plasmas.^{16}

In this paper, we develop a model to theoretically examine how non-equilibrium ionization, a finite skin depth, and heating of the plasma electron population all conspire to influence current sheet formation in IPPTs. Results of the model are used to understand the early-stage formation physics and guide the derivation of a dimensionless scaling parameter that provides a new design criterion for the formation of well-formed currents sheets (i.e., impermeable to both the accelerating fields and downstream propellant mass). Section II contains a detailed description of the model. Section III presents results from the model, first for a single case to build intuition into the dominant physics and then for multiple cases to examine how those physics change with different input parameters. Finally, Sec. IV uses mass utilization as a metric to examine the scaling of current sheet formation.

## II. MODEL FORMULATION

The model presented here is formulated around the lumped-element circuit equations commonly used to analyze IPPT physics and performance.^{4} We add equations that describe propellant ionization and the electron energy balance under non-equilibrium conditions, neutral gas entrainment via charge exchange collisions, and transparency of the current sheet to electromagnetic fields. The addition of these equations will ultimately allow us to examine the link between current sheet formation and propellant mass utilization in pulsed inductive thrusters.

### A. Circuit model

We adopt the circuit models of Martin and Eskridge^{13} and Polzin and Choueiri,^{14} shown schematically in Fig. 1. Here, the thruster is modeled as a capacitor (capacitance $C$) connected in series with the acceleration coil (inductance $Lc$) and switch. Additional series elements of the thruster circuit include a stray inductance ($L0$) and circuit resistance ($Rc$) that each depend on the internal properties of the capacitor, switch, and connections among the various components. $Rc$ also includes the resistance of the acceleration coil. The plasma is modeled as a separate circuit that consists of an inductor (inductance $Lp$) in series with a resistor that represents the internal resistance of the plasma ($Rp$). The acceleration coil acts as a transformer (mutual inductance $M$) that couples energy between the two circuits. We make the common assumption that $Lp=Lc$, which implies that both the coil and plasma can be approximated as flat annuli of current.^{5} We will return to this assumption later in our discussion.

Application of Kirchoff’s law to the above circuit model yields the following ordinary differential equations for the time-evolution of the circuit current ($Ic$) and plasma current ($Ip$),^{14}

and

The voltage across the capacitor ($V$) evolves according to

Here, prime notation denotes the derivative with respect to time (e.g., $y\u2032=dy/dt$). Equations (1)–(3) represent a closed set if $M$ and $Rp$ are taken as fixed quantities. However, $M$ depends on the distance between the plasma and acceleration coil, which generally increases as a function of time. Furthermore, $Rp$ depends on the geometry and temperature of the plasma, which both evolve as energy is deposited into the plasma.

### B. Plasma model

Previous IPPT circuit models assume the plasma conductivity is sufficiently high to prevent penetration of the accelerating fields beyond the width of the plasma. This allows $M$ to be modeled as a function of only the separation distance between the coil and current sheet. Regarding propellant mass utilization, past models have assumed that mass is either contained entirely within the initial current sheet (i.e., slug model^{17}) or that downstream mass becomes fully-entrained within the sheet (i.e., snowplow model^{17}). The consequence of these assumptions is that past models are only valid for cases in which an impermeable current sheet pre-exists, making them unsuitable for studying formation physics. With the goal of understanding current sheet formation and mass utilization, our plasma model seeks to build on these prior efforts in the following ways: (1) provide equations for the time-evolution of $M$ and $Rp$ that account for acceleration and heating of the plasma along with the transparency of the plasma to electromagnetic fields and (2) develop equations for propellant mass entrainment by the plasma for non-equilibrium conditions.

We adopt the simplified geometry shown in Fig. 2. Here, the origin ($z=0$) corresponds to the plane at which neutral gas is injected in front of the thruster face. The acceleration coil is located a distance $za$ from the gas injection plane. The current sheet is modeled as a plasma with uniform density $ns$, width $ws$, and location $zs$, all of which evolve with time. The plasma is assumed to possess an inner radius $ri$ and outer radius $ro$. Radial expansion of the plasma is ignored; thus, $ri\u2032=ro\u2032=0$. Although common to all previous circuit models, the assumption of uniform plasma density represents a considerable simplification compared to the Gaussian-like density profiles that have been observed experimentally.^{2} However, mass entrainment is insensitive to the shape of the density profile so long as temperature gradients within the plasma remain small.

Our plasma model is obtained by integrating equations for the conservation of mass, momentum, and energy over the current sheet volume; a procedure that has been detailed in other publications^{9,11,18} and is not reproduced here for the sake of brevity. We opt instead to describe the key assumptions behind each equation and point the reader to a reference that contains the relevant derivation.

#### 1. Mutual inductance

The mutual inductance between a planar coil and disk conductor has been found to decrease exponentially with the distance between the two elements.^{5} Taking the plasma as the conductor, $M$ may then be written as

where $zc$ is the decoupling length of the coil—a property that mainly depends on the coil dimensions and is typically found using experiments^{5,19} or electromagnetic simulations.^{12,13,19} Note that we have introduced an additional factor of $1\u2212\theta e$ to account for the transparency of the plasma to electromagnetic fields, where $\theta e\u2208[0,1]$. We will refer to $\theta e$ as the electromagnetic transparency of the current sheet. The limit $\theta e\u21920$ corresponds to the common assumption of a perfectly conducting current sheet. The opposite limit, $\theta e\u21921$, implies the plasma is transparent to the electromagnetic fields of the coil and that zero energy is transferred between the two.

The transparency of the current sheet to electromagnetic fields depends on the presence and ability of charge carriers in the plasma to respond to the field fluctuations on timescales that are short compared to the period of the acceleration coil current oscillation. This will itself depend on the collisionality and internal structure of the current sheet. Assuming the plasma is a uniform conductor with skin depth $\delta s$, we use the following simplified model for the electromagnetic transparency:

where $c$ is a constant and $\delta s$ depends on microscopic processes within the plasma (described in more detail in Sec. II B6). For $ws\u226b\delta s$, the charge carriers are able to respond to the incident AC fields and establish an induced plasma current density that fully cancels out those fields, thus $\theta e\u21920$. In the opposite limit, $ws\u226a\delta s$, the incident field is unaffected by the particle response within the plasma; thus, $\theta e\u21921$.

In our subsequent analysis, we choose a value of $c=3$ to give $\u223c5%$ transparency when $ws=\delta s$. This value is consistent with experimental measurements of Dailey and Lovberg^{2} that showed that the magnetic field in the current sheet penetrates a distance that is roughly three times the width of the high density region, or mass layer. The use of a semi-empirical constant is sufficient to examine current sheet formation and mass utilization scaling. More precise models of IPPT behavior will require a detailed examination of the internal structure of the current sheet and penetrating fields; a topic beyond the scope of this paper.

#### 2. Conservation of mass

Propellant mass entrainment is modeled using the non-equilibrium continuity equation for neutral particles in the presence of electron impact ionization and charge exchange collisions. Three-body and dielectric recombination are assumed to be negligible. Convective transport of neutral particles generally occurs on timescales much slower than the period of the discharge circuit allowing neutral continuity to be expressed as

Here, $nn$ is the number density of neutral particles, $Kion$ is the ionization reaction rate, $\sigma cx$ is the charge exchange cross section, $vs$ is the current sheet velocity, and $ns$ is the current sheet number density. We note that $nn(z,t)$ is the only parameter within our model that possesses a dependence on both space and time. Here, the spatial dependence is required to understand how the propellant density in front of the thruster evolves as the current sheet grows and accelerates. The piecewise nature of Eq. (6) ensures that neutral particles are only entrained by the current sheet within the current sheet volume, $z\u2208[zs,zs+ws]$.

Mass conservation requires that the neutral gas particles consumed via ionization and charge exchange appear elsewhere within our model of the system. For ionizing collisions, the neutral particle is transformed into an ion, requiring a source term in the ion continuity equation to balance the ionization loss term in Eq. (6). A new ion is not created in charge exchange collisions, which instead act to transfer momentum from a fast-moving ion to a slow-moving neutral particle. The result is a neutral particle moving at a speed much greater than the injected background neutrals. We denote the density of these “fast” neutrals as $nf$ and model them as fluid distinct from the plasma and “slow” background neutral fluids.

Integration of the plasma (ion or electron) continuity equation over the volume of the uniform current sheet yields

where

is the average density of slow neutrals within the current sheet. Two source terms appear on the right-hand side of Eq. (7). The first represents the creation of an ion–electron pair via ionization of slow background neutral particles. The second term accounts for the creation of an ion–electron pair via ionization of fast neutral particles.

Volume integration of the continuity equation for fast neutral particles gives

Here, we define $\Delta vsf=vs\u2212vf$ as the velocity difference between the current sheet and fast neutral population. $H$ is chosen to denote the Heaviside step function. The first term on the right-hand side represents the creation of fast neutrals via charge exchange collisions between plasma ions and background neutrals. The second term describes the loss of fast neutrals by electron impact ionization. The third term accounts for convective loss of fast neutrals from the trailing boundary of the current sheet—a process we will refer to as “neutral slip.” Examination of Eq. (6) reveals a second useful dimensionless quantity that describes the transparency of the current sheet to upstream propellant mass. The frequency at which a neutral particle at a given location in space is consumed is $(Kion+\sigma cxvs)ns$. The residence time of the moving current sheet in the vicinity of that particle is $ws/vs$. The probability of a neutral particle being consumed by the current sheet may then be written as $\theta m=\theta m,ion\theta m,cx$, where

and

Here, $\theta m,ion$ and $\theta m,cx$ represent the probability that a background particle undergoes an ionization or charge exchange collision during the transit of the current sheet, respectively. The snowplow model for mass entrainment is valid in the limit $\theta m\u21920$. Alternatively, the propagating current sheet is transparent to neutral propellant mass in the limit $\theta m\u21921$.

#### 3. Conservation of momentum

Integration of the plasma fluid (ion + electron) momentum conservation equation(s) over the current sheet volume provides the following expression for the acceleration of the current sheet with an evolving mass:

where the first and second terms on the right-hand side represent the Lorentz and collisional forces on the plasma per unit area per unit atomic mass, respectively. Also introduced in Eq. (12) are the atomic mass of the propellant, $mi$, and the thruster cross-sectional area, $As=\pi (ro2\u2212ri2)$. We note that the Lorentz force term in Eq. (12), whose form is described in detail by Martin and Eskridge,^{13} has been modified to include the effects of finite transparency ($\theta e\u22600$). We have also ignored the force due to the background neutral pressure differential across the current sheet^{18} because it was found to be negligible in all cases of interest.

The collisional force accounts for the transfer of momentum from the current sheet due to charge exchange and ionizing collisions and may be written as

This equation accounts for the transfer of momentum *away* from the ions due to charge exchange collisions as well as the transfer of momentum *to* the ions by ionization of fast neutral particles. The form of this equation assumes the following: (1) background neutrals possess a velocity $vn\u226avs$ and, therefore, do not add significant momentum to the ions through ionization; (2) at the expected ion energies ($\u223c1$–$103$ eV), ion–neutral collisions are dominated by resonant charge exchange due to their large cross section compared to other collision types;^{20} and (3) the momentum transfer cross section for electron–neutral collisions is negligible compared to ion–neutral collisions.

The volume-averaged momentum equation for fast neutral particles takes the following form:

Here, we note that the collisional force on the fast neutral population is equal and opposite to the ion population. In other words, charge exchange collisions are a source of momentum for the fast neutral population whereas ionization is as a momentum sink. Momentum may also be lost via slippage of entrained neutrals from the trailing boundary if $\Delta vsf>0$. Equation (14) neglects the force due to pressure gradients in the fast neutral fluid. This assumption is valid for $(\lambda f/ws)(cs,f/vs)2\u226a1$, where $cs,f=kbTf/mi$ is the fast neutral sound speed and $\lambda f=(ns\sigma cx)\u22121$ is the charge exchange mean free path. The ion kinetic energy is the only source of fast neutral thermal energy, which limits $(cs,f/vs)\u22721$. Thus, for cases where fast neutrals are abundant, $(\lambda f/ws)\u226a1$, the pressure force on the fast neutral fluid is expected to be small compared to the collisional force.

Finally, the location of the current sheet is found from the kinematic equation

where $zs$ represents the distance from the propellant injection plane (Fig. 2).

#### 4. Conservation of energy

The equations presented thus far form a closed model for inductive acceleration and mass entrainment of a current sheet with constant $Rp$, $\delta s$, and $Kion$. These quantities, however, depend on the microscopic processes present within the plasma (Sec. II B 6), which themselves are strongly influenced by the temperature of the electrons, $Te$. This dependence was first examined theoretically by Polzin *et al.*^{18} who added an MHD (ion + electron + magnetic) energy equation to a 1D lumped-element IPPT circuit model. The results of the model showed that Ohmic heating of the plasma generally reduced thruster performance because it provided an alternative energy sink for the system, but that this effect was small for cases where the plasma acceleration time and coil rise time were well-matched. The energy equation of Polzin *et al.* assumes an impermeable current sheet in thermodynamic equilibrium; thus, it is unsuitable for studying current sheet formation and mass entrainment.

The energetics of inductive acceleration and propellant mass entrainment are closely linked to heating and cooling of the electron population. For example, currents induced within the plasma increase $Te$ through Ohmic heating, thereby also increasing $Kion$ and reducing $Rp$ and $\delta s$. Increasing $Te$ during formation would then suggest an improvement in mass entrainment and electromagnetic coupling of the current sheet, effects that will compete with the thermal sink found by Polzin *et al.* Furthermore, electron impact excitation and ionization of neutral particles take energy away from the electrons, decreasing $Te$. The timescales associated with these coupled processes can vary widely across parameter spaces relevant to IPPT plasmas, making a non-equilibrium electron energy balance crucial to understanding current sheet formation, acceleration, and mass entrainment.

We derive an equation for the time-evolution of $Te$ by integrating the electron energy equation^{21} over the current sheet volume for a uniform plasma, neglecting electron kinetic energy terms, and assuming zero heat flux out of the volume. This yields the following equation:

where

and

Here, $Pohm$ represents the power gained through Ohmic heating; $Pen$ is the power lost to both inelastic and elastic collisions with neutral particles, parameterized by an effective ionization energy $\epsilon \u2217ion$; and $Pei$ is the power transferred from electrons to ions via Coulomb collisions of frequency $\nu ei$. All temperatures and powers are expressed in units of eV and eV/s, respectively. Also introduced here is the electron mass, $me$.

The rate of energy transfer between electrons and ions due to Coulomb collisions [Eq. (19)] depends on the ion temperature, $Ti$. We, therefore, also require an ion energy equation to model the time-dependence of $Ti$. Integrating the ion energy equation^{21} over the uniform current sheet volume and assuming zero heat flux from the boundaries yields

Here, $Tn$ is the temperature of the background neutrals. Equation (12) has been used to cancel out terms associated with the rate of change of ion kinetic energy, work done on the current sheet via the Lorentz force, and the contribution of charge exchange and momentum transfer collisions. Ohmic heating does not appear in Eq. (20) because $Rp$ is dominated by the electron collision rate. Coulomb collisions (first term on RHS) and thermal energy from ionized neutrals (second term on RHS) are, therefore, the only sources of thermal energy for the ions.

#### 5. Current sheet width

To close our model, we require an equation that describes the time-evolution of the current sheet width. Previous models have either assumed a constant sheet width^{12} or one that evolves in time due to plasma diffusion.^{5,18} Our model aims to understand how ionization processes impact early formation of the current from a pre-ionized state. With that goal in mind, we take the rate of change of the current sheet width as

where the first term represents increasing width resulting from plasma diffusion with diffusivity $Dp$ and the second term accounts for changes in $ws$ that occur during the formation process.

A complete model of $(ws\u2032)f$ requires knowledge of the internal structure of the current sheet and electromagnetic fields, quantities not available to us in the simplified lumped-element circuit description. Instead, we rely on phenomenological arguments to construct an approximate form of $(ws\u2032)f$. First, we assume that for cases in which $ws\u2265\delta s$, newly created plasma particles accumulate within a distance $z\u2208[zs,zs+\delta s]$. This assumption is based on the supposition that, because the induced plasma current density is strongest within the skin depth layer, this region will possess the highest local electron temperature and, therefore, also the highest ionization rate. The entrainment of background gas by the current sheet, therefore, preferentially deposits particles within the skin depth layer, trending $ws$ to $\delta s$ at a rate proportional to the relative rate of particle addition. However, when $ws<\delta s$, newly created particles accumulate within the region $z\u2208[zs,zs+ws]$, suggesting $(ws\u2032)f=0$ for this condition. The following equation can be used to model the above scenario:

where $H$ is the Heaviside step function and $Ns=wsAsns$ is the total number of ions within the current sheet. Using the chain rule to write $Ns\u2032/Ns$ in terms of our state variables, we arrive at the following equation for the rate of change of the current sheet width:

Examination of the above equation shows that $ws\u2192\delta s$ in the limit $Dp\u21920$ and $ws>\delta s$, provided there exists significant mass entrainment ($ns\u2032/ns\u226b1$). Alternatively, $ns\u2032/ns\u226a1$ when mass entrainment is insignificant; thus, $ws\u2032\u21920$.

#### 6. Microscopic processes

We have introduced a number of parameters that rely on a statistical averaging of the microscopic (particle scale) processes within the plasma, namely, $Rp$, $\delta s$, $Kion$, $\sigma cx$, $\epsilon \u2217ion$, $\nu ei$, and $Dp$. We summarize here the equations used within our model to describe these processes.

Treating the plasma as a conducting annular disk with uniform resistivity $\eta p$, the resistance may be written as

Here, the $min$ function in the denominator describes the width inside which current is distributed. The plasma resistivity is

where $e$ is the fundamental electron charge. $\nu ei$ and $\nu en$ are the electron collision rate with ions and neutrals, respectively. The diffusion coefficient also depends on the electron collision rates via

where diffusion has been taken to be ambipolar to maintain quasi-neutrality within the sheet. Finally, we use the following equation for the skin depth:

where $\omega =1/LeffC$ is the angular frequency of the incident EM field, $Leff=L0+(1\u2212k2)Lc$ is the effective inductance of the circuit at a given instant in time, and $k=M/Lc$ is the time-dependent coupling coefficient.

We adopt the effective ionization energy model of Lieberman and Lichtenberg^{22} to account for electron energy lost to ionization, line radiation, and elastic polarization scattering. Within this model, the effective ionization energy is defined as

where $\epsilon ion$ is the first ionization energy of the propellant gas; $Kex,j$ and $\epsilon ex,j$ are the electron impact excitation reaction rate and energy for the $jth$ transition, respectively; and $Kes$ is the rate of elastic polarization scattering of electrons off neutral atoms.

Collision rates involving electrons ($Kion$, $Kex,j$, $Kes$, $\nu ei$, and $\nu en$) are modeled assuming a Maxwellian electron energy distribution. Rates for ion collisions ($\sigma cx$) assume a cold ion population, or $vs\u226beTi/mi$. The exact form of the equations we use for these parameters is summarized in the Appendix for argon propellant.

### C. Initial conditions and model inputs

Equations (1)–(4), (6), (7), (9), (12), (14)–(16), (20), and (23) form a closed model for the time-evolution of the following state variables: $V$, $Ic$, $Ip$, $M$, $nn$, $ns$, $nf$, $Te$, $Ti$, $zs$, $ws$, $vs$, and $vf$. The initial conditions we use for these variables are described below.

For initial conditions related to the plasma state variables, we presuppose the existence of a stationary pre-ionized plasma [$vs(0)=vf(0)=0$] whose initial width [$ws(0)=ws,0$] and location [$zs(0)=zs,0$] are taken as model inputs. We use a pre-ionization function to establish the properties of the pre-ionized plasma, defined as $\chi i(z)\u2261ni,pi(z)/nn,inj(z)$. Here, $nn,inj(z)$ is the injected neutral density profile prior to pre-ionization and $ni,pi(z)$ the ion density profile following pre-ionization. The initial conditions for the neutral density profile and current sheet plasma density are then given by

and

respectively. We also assume the pre-ionized plasma does not possess fast neutrals [$nf(0)=0$]. The initial conditions on density are, therefore, found using two input functions, $\chi i(z)$ and $nn,0(z)$. Finally, we take as inputs the initial electron and ion temperatures, $Te(0)=Te,0$ and $Ti(0)=Ti,0$, respectively. We note that our model is unable to simulate the initial breakdown of propellant gas; thus, some level of pre-ionization is needed, although this may be arbitrarily small.

We assume the capacitor is initially charged to voltage $V0$ and take $t=0$ to correspond to the moment at which the discharge switch closes, thus $V(0)=V0$, $Ic(0)=0$, $Ip(0)=0$. Finally, $M(0)$ may be found from the following transcendental equation for $k0=M(0)/Lc$:

This equation comes from the requirement that the initial plasma skin depth ($\delta s,0$) be consistent with $M(0)$, where $\delta s$ depends on $M$ through $Leff$.

To summarize, the inputs to our model include variables associated with the thruster circuit ($C$, $Rc$, $L0$, $Lc$), thruster geometry ($ri$, $ro$, $za$, $zc$), propellant gas properties ($mi$, $Kion$, $\epsilon ion$, $Kex,j$, $\epsilon ex,j$, $Kes$, $\sigma cx$), propellant gas profile ($nn,0$, $Tn$), and properties of the pre-ionized plasma ($\chi i$, $ws,0$, $zs,0$, $Te,0$, $Ti,0$). In Secs. III and IV, we will examine how a subset of these variables influence the solution of our model.

### D. Propulsion performance metrics

The early dynamics of current sheet formation and acceleration strongly influences the propulsion performance of an IPPT. Our model is unique in its ability to resolve these dynamics and is, therefore, a useful tool for the discovery of new propulsion performance scaling laws that incorporate the complex physics described above. Here, we present equations for the propulsion performance metrics that we will use throughout the remainder of our analysis.

The impulse bit ($Ibit$) refers to the total impulse delivered by a single thruster pulse, where the total impulse is equal to the integral of thrust over time. With our terminology, this may be written as

Here, the first, second, and third terms on the right-hand side represent contributions of to ions, entrained fast neutrals, and slipped fast neutrals, respectively. The function $Ibit$ can be evaluated at a given instant in time to see how the impulse evolves over the duration of a pulse. The asymptotic value of $Ibit$ represents the total impulse delivered by one pulse.

The specific impulse of a pulsed thruster is defined as

where

is the mass bit, defined as the total propellant introduced to the thruster prior to a pulse.

The thrust efficiency of a pulsed thruster is defined as

where $Etot$ is the total energy input into a single pulse. Here, we included the energy contribution of the pre-ionized plasma, $Epi=ns,0ws,0Ase[(5/2)(Te,0+Ti,0)+\epsilon \u2217ion(Te,0)]$, and have assumed residual energy on the capacitor is not lost and is instead available for subsequent pulses.^{13}

Propellant mass utilization efficiency ($\eta m$) is a metric that describes the fraction of total propellant that is accelerated to the ultimate velocity of the current sheet. It may be calculated from our model using the following equation:

Here, $\eta m$ may also be evaluated at an instant in time to examine how mass utilization evolves throughout a pulse.

Finally, we define the electrical efficiency as the ratio of the total kinetic energy in accelerated propellant to the total input energy. Within our formalism, this can be written as

the first, second, and third terms in the numerator represent the kinetic energy of ions, entrained fast neutrals, and slipped fast neutrals, respectively. Note that $\eta T\u2248\eta m\eta e$ in the limit where $ns\u226bnf$.

## III. RESULTS

In this section, we use the results of our analytical model to build physical intuition of the dynamics that occur during the early formation, acceleration, and mass entrainment of an IPPT current sheet. We begin by examining the model outputs for a single case study. From there, we examine how the propulsion performance metrics vary across multiple case studies to obtain an understanding of the effect of varying levels of pre-ionization and coil inductance.

It is first necessary to define the injected propellant neutral density and pre-ionization profiles used in Eqs. (29) and (30). Results presented throughout the remainder of the paper use the following functional forms for these profiles:

and

Here, we have assumed the injected neutral density possesses a maximum value $nn,0$ at $z=0$, and decreases exponentially with the ratio of the distance from the injection plane to a characteristic distance $zn$. We further assume that the pre-ionization scheme creates a uniform ionization fraction $\chi i,0$ that is localized within a width $ws,0$ immediately adjacent to the injection plane (i.e., $zs,0=0$).

To reduce the total number of free parameters to a more manageable set, we keep the following parameters constant throughout our analysis in Secs. III and IV: $Rc=5$ m$\Omega $, $L0=50$ nH, $ri=3$ cm, $ro=10$ cm, $Ti,0=0.1$ eV, $Tn=0.026$ eV. Furthermore, we consider only argon propellant in the present study ($mi\u224840$ AMU; $\epsilon ion\u224815.76$ eV; $\epsilon ex,1\u224812.14$ eV; $Kion$, $Kex,1$, and $Kes$ given in the Appendix.)

### A. Model outputs for a single case study

We begin our analysis by examining results from a single set of input conditions. Here, we consider an initial capacitor energy of 25 J ($C=2$ $\mu $F, $V0=5$ kV). The acceleration coil is taken to have an inductance $Lc=1$ $\mu $H, is located 2 mm behind the gas injection plane ($za=\u22122$ mm), and has a 4 cm coupling distance ($zc=4$ cm). The injected neutral gas is assumed to have a peak value of $nn,0=4\xd71020$ m$3$ and characteristic decay length $zn=1$ cm. The pre-ionized plasma is characterized by $\chi 0=0.01$, $zs,0=0$ m, $Te,0=20$ eV, and $ws,0=2$ mm. Here, the relatively high value of $Te,0$ is representative of temperatures obtained in pulsed pre-ionization sources with large instantaneous powers.^{23–25} Last, we assume that the switch is opened at the first zero crossing of the coil current and that the residual energy on the capacitor is recovered by the circuit.^{13}

The state variables provided by the solution to our model are shown in Fig. 3 for the input conditions described above. The *x* axis on each panel shows time ($t$) normalized by the characteristic $LC$-period of the coil ($TLC=2\pi LcC\u22488.9$ $\mu $s). Closure of the switch at $t=0$ causes a decrease in $V$ and increase in $Ic$; expected behavior from an RLC circuit. $Ic$ and $V$ are 90$\xb0$ out of phase, in accordance to Eq. (3), and the maximum in $Ic$ occurs at the zero crossing of $V$. Although subtle, neither $V$ nor $Ic$ are exactly sinusoidal in time due to the presence of the plasma. The switch is turned off at the first zero crossing of $Ic$ ($t/TLC\u22481/2$), resulting in energy being recovered from the circuit onto the capacitor.

Plasma currents are generated via the mutual inductance between the coil and pre-ionized plasma. Notably, the rise rate of $Ip$ starts small and eventually grows larger around $t/TLC\u22480.05$. This behavior is due to the fact that the pre-ionized plasma is initially partly transparent to the electromagnetic fields generated by the current pulse ($\theta e,0\u22480.84$, $k0\u22480.16$). A rapid increase in $M$ occurs for $t/TLC\u2208[0,0.05]$, suggesting a rapid decrease in $\theta e$ resulting from changes to the internal state of the plasma. Following a maximum around $t/TLC\u22480.05$, $M$ transitions to an exponential decay as the translating plasma spatially decouples from the coil.

The current sheet plasma parameters evolve substantially throughout the formation and acceleration phases. Starting with species densities [Fig. 3(e)], we observe an exponential increase in both $ns$ and $nf$ during early formation, with ionization reactions being favored compared to charge exchange. Charge exchange reactions quickly subside as time increases, which produces a peak value of $nf$ followed by a decay in $nf$ as fast neutrals ionize and slip away from the current sheet. During this time, $ns$ continually increases due to continued ionization of both background and entrained neutrals. The background neutral and plasma density profiles are shown at different instants in time in Fig. 5(a). Here, we see that for $t/TLC\u2208[0,0.05]$, ionization is the most dominant physical process and there is very little axial motion of the plasma. Despite the importance of ionization in the early stages of the pulse, it is apparent from the neutral density profile that a significant percentage of propellant gas is left behind during the early formation of the current sheet for the conditions considered in this example. The Lorentz force becomes dominant for $t/TLC\u22730.05$ leading to substantial acceleration of the current sheet [Fig. 3(j)]. By this point the increased plasma density has reduced the mass transparency of the current sheet to ionization reactions, entraining a larger fraction of downstream neutrals in the process. A maximum in $ns$ occurs near $t/TLC\u22480.2$, after which diffusion drives a gradual increase in the width of the current resulting in a gradual decrease in $ns$.

The electron temperature exhibits rich dynamics due largely to a competition between inelastic losses and Ohmic heating. A rapid drop in $Te$ occurs immediately after the switch closes, which can be attributed to ionization and radiation losses that occur as electrons within the pre-ionized plasma react with the substantial background neutral population near $z=0$. These losses can be seen in the second term on the right-hand side of Eq. (16). Eventually, the trend reverses as the increasing $Ip$ deposits energy into electrons via Ohmic heating. A second reversal occurs near $t/TLC\u22480.05$, which is followed by a more gradual drop in $Te$. The decrease in $Te$ in this timeframe is driven by inelastic losses associated with the rapid increase in $ns$, losses that are partly balanced by increased Ohmic heating near the maximum in $Ip$. For $t/TLC>0.15$, the Lorentz force has pushed the current sheet into in a region of reduced neutral density ($zs>zn$), allowing Ohmic heating to again dominate inelastic losses, yielding a net heating of the electron population. Eventually, $Te$ plateaus for $t/TLC>0.3$ as the current sheet decouple spatially from the coil. In this region of time, there is a slight increase in $Te$ that results from a balance between Ohmic heating by residual plasma currents and cooling due to diffusive expansion of the current sheet.

Unlike electron temperature, the ion temperature is observed to decrease monotonically throughout the pulse with an asymptote $Ti\u2192Tn$. This suggests that Coulomb heating of the ions is negligible and that ion thermal energy is instead dominated by the introduction of cold ions into the current sheet via ionization of background neutrals.

### B. Multiple case studies

The previous example demonstrates how the early dynamics of a forming current sheet play a key role in entraining propellant gas. Understanding these dynamics is, therefore, critical to achieving high thrust efficiencies in IPPTs. We can speculate that these processes will be strongly influenced by both the properties of the pre-ionized plasma and the rate of ionization relative to the rise rate of the current pulse. Here, we use our model to analyze this dependence in more detail by varying the degree of pre-ionization and coil inductance. We will first examine the influence of these parameters on propulsion performance. From there, we will take a deeper look into the resulting trends by considering the time-evolution of relevant state variables and current sheet transparencies.

Figures 4(a)–4(c) show contour plots of the propulsion efficiency metrics (Sec. II D) obtained from our model using thruster inputs from Sec. III A for $\chi 0\u2208[0.01,1]$ and $Lc\u2208[0.1,10]$ $\mu $H. It is clear from Fig. 4(a) that $\eta T$ generally exhibits a stronger dependence on $Lc$ compared to $\chi 0$. $\eta T$ increases as $Lc$ decreases until $Lc\u22480.3$ $\mu $H, a trend that largely results from the scaling of $\eta m$ with $Lc$ [Fig. 4(b)]. For $Lc<0.3$ $\mu $H, $\eta T$ begins to decrease with $Lc$ due to the onset of electrical inefficiencies [Fig. 4(c)]. For $Lc>0.3$ $\mu $H, we observe a gradual increase of $\eta T$ with $\chi 0$ until $\chi 0\u22480.5$. Eventually, diminishing $\eta T$ is observed for $\chi 0>0.5$, which again can be traced back to electrical inefficiencies in the system [Fig. 4(c)]. Finally, we note that a maximum value of $\eta T\u22480.7$ is found at $Lc=0.5$ $\mu $H and $\chi 0=0.3$.

The results shown in Fig. 4 provide the following insights into IPPT performance:

Increased pre-ionization (i.e., larger $\chi 0$) improves propellant mass utilization at the expense of electrical efficiency.

Decreased current rise times (i.e., smaller $Lc$) also improves propellant mass utilization at the expense of electrical efficiency.

Electrical inefficiencies introduced by decreasing the current rise time are less severe than those introduced by increasing pre-ionization.

The trends in propellant mass utilization and electrical efficiency create a local maximum for thrust efficiency (recall $\eta T\u2248\eta m\eta e$).

We now turn to time-dependent results for various cases highlighted in Fig. 4 to provide deeper physical insight into the observed performance trends.

#### 1. Influence of pre-ionization

To better understand the causal relationship between propulsion performance and pre-ionization of the injected plasma, we examine the time-evolution of the current sheet and neutral propellant densities, seen in Fig. 5. Here, we represent the current sheet as a rectangular function of normalized height $ns/nn,0$ and width $ws$. The panel columns in Fig. 5 show snapshots in time normalized by the characteristic $LC$-period of the coil. The entrainment process is described by the current sheet gaining plasma density as it encounters the spatially decreasing background neutral particle distribution $nn(z,t)$, where the gas adds to the density of the sheet as depicted by a darkening color gradient. The shape of the neutral distribution in the wake of the plasma is given by the dashed curve, which deviates from the initial number density according to how effectively the current sheet incorporates neutral particles. By varying $\chi 0$, we observe drastic differences in spatial propagation of the current sheets due to changing entrainment processes. Lower pre-ionization produces a fast-moving current sheet at the expense of a large population of propellant left behind, indicating a lower mass utilization efficiency as we confirmed in Fig. 4. As we step up the fraction of pre-ionized plasma, we observe a slower-propagating current sheet that experiences more charge exchanges and ionizing interactions with the background neutrals, leading to an increase in propellant mass utilization.

Additional insight can be obtained from these cases by examining the time-dependence of notable state variables and transparencies, shown in Fig. 6. Electron temperature [Fig. 6(d)] highlights the critical role of the electron energy balance during current sheet formation for different pre-ionization fractions. Once the switch closes, low ($\chi 0=0.01$) and partial ($\chi 0=0.1$) pre-ionization cases show an immediate drop in $Te$ as ionization and radiation losses occur. The lower density of the $\chi 0=0.01$ case allows $Te$ to recover faster because Ohmic heating deposits more energy per electron. Unlike the low and partial pre-ionization cases, a slight increase in $Te$ immediately occurs for the fully pre-ionized ($\chi 0=1$) current sheet due to Ohmic heating in the absence of ionization losses. However, $Te$ rapidly drops as the sheet begins to move and interact with the downstream neutrals. Although quenching of the electron population occurs in all three cases as the plasma encounters downstream electrons, the effect is most severe for full pre-ionization. As a consequence, the $\chi 0=1$ case possesses the lowest $Te$ during the period of time in which the majority of mass is entrained in the current sheet [Fig. 6(c)], resulting in the highest effective ion energy cost. Recalling that $vs$ also decreases with increasing $\chi 0$ [Fig. 6(a)], it can be concluded that pre-ionization increases the energy lost to ionization and radiation relative to the kinetic energy of the exhaust. It is largely this phenomenon that explains the decrease in $\eta e$ with increasing $\chi 0$ observed in Fig. 4(c).

Coupling of the electromagnetic fields to the current sheet is also strongly influenced by the electron energy dynamics, as shown in Figs. 6(e) and 6(f). The high plasma density that comes with increased pre-ionization results in a smaller skin depth and less transparency to the fields of the coil (smaller $\theta e$). Pre-ionization, therefore, increases the initial coupling coefficient, $k0$. However, for $\chi 0=1$, a rapid increase in $\theta e$ occurs when the current sheet encounters downstream neutrals because the conductivity of the plasma drops rapidly along with $Te$. This effect is not as strong for the low and partial pre-ionization cases; thus, the current sheet that forms from a fully pre-ionized plasma ends up being the most transparent to electromagnetic fields. This effect does not strongly influence $\eta e$ because we have taken a relatively small value of $Rc$ and assumed unspent circuit energy is recovered by the thruster; however, it may become a dominant loss process if either of these two assumptions is relaxed.

Finally, we examine the underlying processes of propellant mass entrainment using the mass transparency parameters shown in Figs. 6(g) and 6(h). The probability of a neutral particle being consumed by the current sheet via ionization or charge exchange exhibits clear delineation for different pre-ionization fractions. As $\chi 0$ increases, $\theta m,ion$ and $\theta m,cx$ both decrease, approaching the snowplow model for mass entrainment as $\theta m\u21920$. The time histories of $\theta m,ion$ and $\theta m,cx$ scale inversely with $ns$ for $\chi 0=0.01$ and $\chi 0=0.1$, which suggests the current sheet density is too low to efficiently entrain downstream propellant. Markedly, different behavior is observed for $\chi 0=1$. In the case of full pre-ionization, the current sheet begins to move away from the coil while entraining nearly all neutral gas it encounters via electron impact ionization. The resulting quench of $Te$ reduces $Kion$ to the point where the sheet becomes partly transparent, after which Ohmic heating restores $Te$ to a level where mass transparency is once again reduced. Evidence of this process can be seen in the shape of the residual neutral gas distribution in Fig. 5.

#### 2. Influence of coil inductance

We examine the influence of current rise time by varying coil inductance in a similar line of analysis. Using four cases, we present the time-evolution of current sheet and neutral propellant densities with an entrainment diagram in Fig. 7, followed by a temporal comparison of state variables in Fig. 8. The common data set from our analysis of a single case study remains represented in green for $Lc=1$ $\mu $H. Across the four data sets shown, we maintain low pre-ionization ($\chi 0=0.01$) to isolate the influence of $Lc$ from our previous results in the section above, yielding the same initial plasma density throughout Figs. 7 and 8.

Figure 7 shows significant differences in current sheet propagation based on neutral particle interactions and momentum conservation. Decreasing current rise time (i.e., decreasing $Lc$) allows for much greater neutral entrainment via charge exchange and ionization. This is best portrayed by the steep slope of $nn(z,t)$ inside of the current sheet corresponding to $Lc=0.1$ $\mu $H, which consumes almost all of the downstream neutrals it encounters past $t/TLC\u22480.1$. As $Lc$ increases, it is clear that the current sheet velocity increases at the expense of reduced neutral entrainment. Above a certain threshold inductance ($Lc=5$ $\mu $H in this case), the current sheet accelerates beyond the coupling distance before any entrainment can occur. The region of low mass utilization in Fig. 4(b), therefore, corresponds to cases where the current rise time exceeds the residence time of the plasma.

Figure 8 shows that the coil inductance has a significant impact on current sheet dynamics. The initial rise rate of $Te$ increases as $Lc$ decreases. This trend is due largely to the fact that Ohmic heating outpaces ionization losses in the early formation phase, especially for faster current rise times. A period of electron cooling is also observed that corresponds roughly to the time period in which the rate of neutral entrainment is greatest. Here, ionization and radiation losses overwhelm Ohmic heating to produce a net cooling effect. The rise in $Te$ for $Lc=0.1$ $\mu $H is significant enough to momentarily increase the plasma conductivity such that $\delta s<ws$, during which time the current sheet decreases in width as particles become entrained in the skin layer. The point at which $\delta s<ws$ also corresponds to the time when the current sheet becomes impermeable to the AC fields of the coil ($\theta e\u21920$). Notably, this condition is only met for the lowest inductance case, resulting in the highest peak value of $k$. Finally, for cases where mass entrainment is observed, the final value of $Te$ is observed to increase as $Lc$ decreases. This indicates that a significant portion of input energy is lost as unrecovered thermal energy in the sheet, thus explaining why $\eta e$ decreases with increasing $Lc$ [Fig. 4(c)]. The importance of dynamic plasma heating and unrecovered thermal energy as a potentially dominant loss process was found by Polzin *et al.*^{18} using an equilibrium plasma model. Our results, therefore, suggest that the conclusions of Polzin *et al.* remain valid for non-equilibrium conditions.

Propellant mass entrainment is again influenced strongly by the time-dependent behavior of $ns$ and $Te$. For $Lc=5$ $\mu $H, the plasma remains transparent to the downstream mass because it is accelerated away from the near-field region of the coil before significant electron heating and ionization can occur. Although Ohmic heating allows the electron temperature to reach a relatively high value of $Te\u223c14$ eV, the plasma remains too tenuous to entrain significant mass. Decreasing $Lc$ increases $Te$ during the early phase of formation leading to partial entrainment of downstream mass, mostly via ionizing collisions. This effect is self-reinforcing in time as entrained propellent produces an increase in $ns$, thus further reducing the mass transparency. As $Lc$ decreases further, the rapid drop in $\theta m$ occurs earlier in time relative to the period of the circuit, eventually leading to the formation of a mass impermeable sheet. Finally, in the limit of very low inductance, the rapid rise in $Te$ combined with the slow (relative to $TLC$) initial acceleration of the plasma produces a current sheet that is largely impermeable to mass throughout the duration of the pulse. Again, ionization appears to be the dominant collisional process across all cases considered here. The increase in $\eta m$ with decreasing $Lc$, shown in Fig. 4(b), is, therefore, the result of a complex, non-linear process that depends on the balance between electron heating and cooling. This suggests that the formation of an impermeable current sheet requires sufficient heating to occur before the plasma spatially decouples from the coil.

## IV. MASS UTILIZATION SCALING

A number of dimensionless scaling parameters have previously been derived from IPPT circuit models to describe the influence of various physical processes on propulsion performance.^{14} Absent from this set is a scaling parameter that describes the early time-evolution of the plasma during the current sheet formation process. Polzin^{6} found that there exists a minimum coil current rise rate, $dIc/dt|t=0=V0/Lc$, needed to induce an azimuthal electric field strong enough to break down the neutral propellant gas. Results from earlier experiments lent support to this argument and provided a semi-empirical value for the minimum threshold. A lower empirical threshold on $dIc/dt|t=0$ was proposed for thruster experiments that employed a pre-ionization scheme; however, it is unclear why the azimuthal electric field should play an important role in cases where the plasma is pre-formed because the physics of gas breakdown are distinct from the physics of current sheet formation. Here, we use our model to seek a dimensionless quantity that generalizes Polzin’s empirical requirement and incorporates additional physical processes associated with the formation phase.

The results presented in Sec. III underscore the importance of energy deposition into the electron population during the early stages of current sheet formation. Ionization of neutral propellant in the absence of an energy source leads to quenching of the electrons and a sharp reduction in the rate of electron impact ionization. The resulting current sheet is either partially or completely transparent to downstream propellant gas depending on the severity of the quenching process, which can be a significant detriment to both mass utilization and thrust efficiency. We can, therefore, anticipate that the efficiency of the formation process will depend strongly on the amount of energy deposited into the electrons relative to the energy needed to ionize the entirety of propellant gas.

Based on the observations above, we introduce the following dimensionless parameter:

where

is the amount of energy deposited in the electron population through Ohmic heating and

the energy required to completely ionize the propellant mass bit. Here, $Tdc$ represents the time it takes for the plasma to decouple from the coil and $\epsilon \xaf\u2217ion$ represents the average ionization energy. Note that $\Omega $ can also be obtained via non-dimensionalization of Eq. (16). We will hereafter refer to $\Omega $ as the formation parameter.

The exact evaluation of Eq. (41) requires a solution to the full time-dependent model described above. However, a number of approximations can be made to simplify this expression to a form that is more amenable to a scaling analysis. The plasma remains near the coil during the early formation stage, allowing the plasma current to be approximated as $Ip\u2248kIc$. We assume the coil current evolves with time according to $Ic\u2248Ic,msin(2\pi t/TLC)$, where $Ic,m=V0C/Lc$ is the maximum coil current and $TLC=2\pi LcC$ the LC-period, both in the limit $L0\u226aLc$ and $k0\u226a1$. Although the use of the uncoupled coil current equation appears restrictive, we find that it provides a good approximation for the scaling of $Eh$ provided $k0\u22720.7$.

Following insertion of the approximations for $Ip$ and $Ic$ into Eq. (41), the integration yields

Here, $k\xaf$ represents the average coupling parameter over the integration period, $T\xafLR=Lc/R\xafp$ is the average L/R-time of the plasma, and $R\xafp$ is the average plasma resistance. The above equation shows that the energy deposited as heat into the electron population scales with the total energy, the coupling parameter squared, and the ratio of the decoupling time and L/R-time. Finally, we note that Eq. (43) ensures $Eh<CV02/2$ provided that the circuit is underdamped ($TLC/T\xafLR<1$), in accordance with our assumptions above.

An approximation for the decoupling time can be obtained from the momentum equation (12). Ignoring collisional drag terms, $Tdc$ roughly scales according to $mbitzc/Tdc2\u223ck\xafLcIc,mIp,m/zc$, where $Ip,m\u223ck\xafIc,m\u223ck\xafV0C/Lc$ is the maximum plasma current. We may then use the following equation:

where

is the dynamic impedance parameter. The dimensionless quantity $\alpha $ is known to be an important parameter for IPPT performance because it governs how well the circuit transfers energy to the plasma before it decouples from the coil.^{14} Its appearance in the scaling of $Tdc$ is, therefore, consistent with this physical understanding.

For cases where $Tdc/TLC\u22730.2$, the term is square brackets in Eq. (43) may be approximated as unity. Insertion of Eq. (44) into Eq. (47) then yields

This equation states that, for cases where the plasma remains coupled to the coil for a significant portion of the LC-period, the energy deposited as heat in the electron population can be expressed as the geometric mean of the total energy and an effective kinetic energy with mass equal to the total mass bit and velocity equal to the decoupling length divided by the L/R-time. This entire term is scaled by $k\xaf$.

Using the above approximation for $Eh$, the dimensionless formation parameter may now be cast in the following compact form:

This equation provides a simple scaling law for the ratio of electron energy obtained from Ohmic heating to the energy required to ionize the entire propellant mass bit. Based on the observations in Sec. III B, we would expect thermal quenching of the electron population to occur for cases where $\Omega \u226a1$, leading to incomplete propellant entrainment and poor mass utilization ($\eta m\u226a1$). In the opposite limit, cases for which $\Omega \u226b1$ would suggest electron heating is sufficient to support full ionization of the injected propellant ($\eta m\u21921$). Notably, $\Omega $ depends on the initial coil current rise rate, $dIc/dt|t=0=V0/Lc$, suggesting a possible generalization of Polzin’s semi-empirical formation criteria.

We now return to our model to: (1) assess the validity of $\Omega $ as a scaling parameter for current sheet formation and (2) determine if a critical value of $\Omega $ exists above which the current sheet can be considered to be “fully” formed. To accomplish this, we examine trends of $\eta m$ vs $V0$ for different input parameters that appear in Eq. (47). Here, we use mass utilization efficiency as a quantitative metric for current sheet formation based on observations from Sec. III. The following set of conditions is used as a base: $za=\u22122$ mm, $Lc=0.2$ $\mu $H, $zc=2$ cm, $C=2$ $\mu $F, and $nn,0=4\xd71020$ m$\u22123$. Each of these parameters is then varied to examine their individual influence on the resulting $\eta m$ vs $V0$ trend. Finally, we note that throughout this analysis, we hold the following inputs constant: $zn=1$ cm, $\chi 0=0.01$, $zs,0=0$ m, $Te,0=20$ eV, and $ws,0=2$ mm.

Results from our analysis are presented in Figs. 9(a)–9(e). In general, $\eta m$ exhibits a sharp increase with $V0$ up until a threshold value, $V\u22170$, after which a gradual increase toward an asymptotic value is observed. $V\u22170$ increases as the acceleration coil is moved further behind the gas injection location [Fig. 9(a)]. This trend can be explained by the decrease in $k\xaf$ that occurs with increasing $|za|$, according to Eq. (31). As $k\xaf$ decreases, an increase in $V0$ is required to maintain a constant $\Omega $. Figure 9(b) shows that $V\u22170$ increases linearly with $Lc$, consistent with the requirement of a minimum coil current rise rate. $V\u22170$ also increases with $zc$ [Fig. 9(c)]; however, the trend is not linear as might be implied from Eq. (47). The non-linear behavior here may be explained by the fact that $k\xaf$ also possesses a dependence on $zc$. Figure 9(d) shows an increasing $V\u22170$ with decreasing $C$, consistent with the scaling of $Eh$ with the square root of the total energy in the circuit. Finally, $V\u22170$ increases with $nn,0$ [Fig. 9(e)], the consequence of an increasing $mbit$.

Combining the data from Figs. 9(a)–9(e) and plotting it as a function of $\Omega $ yields Fig. 9(f). Here, we use Eq. (47) along with the following simplifying approximations to calculate $\Omega $: $k\xaf\u2248k0$, $R\xafp\u2248Rp,0$, and $\epsilon \xaf\u2217ion\u2248\epsilon \u2217ion(Te,0)$. Figure 9(f) indicates that the data roughly collapse onto a single curve of $\eta m$ vs $\Omega $. A spread of $\u223c20%$ is observed that is consistent with the approximations made above and in the derivation of Eq. (47). This result highlights the efficacy of $\Omega $ as a scaling parameter for current sheet formation in IPPTs. Furthermore, the location of the “knee” in the curve of $\eta m$ vs $\Omega $ suggests that there exists a critical value $\Omega \u2217\u22481.7$ above which thrusters should operate to ensure a well-formed current sheet.

The critical value $\Omega \u2217$ provides a possible physical interpretation for Polzin’s coil current rise rate requirement for IPPT current sheet formation.^{6} This requirement was based on data obtained from the Faraday Accelerator with Radio-frequency Assisted Discharge (FARAD) proof-of-concept experiment, which combined a planar IPPT with an RF pre-ionization stage. Although a current sheet was observed to form in this experiment, measurements indicated that the current sheet remained partly transparent to both the EM fields of the coil and downstream propellant mass.^{11} Based on these observations, Polzin concluded that the linear current density rise rate should be greater than the largest rise rate of the FARAD experiment. Expressed in terms of the total coil current rise rate, Polzin’s criterion becomes $dIc/dt|t=0>22$ kA/$\mu $s. We can compare this value to the prediction from our model using Eq. (47) and setting $\Omega =\Omega \u2217$. For this calculation, we use the FARAD experiment circuit parameters ($C=39.2$ $\mu $F, $Lc=90$ nH, $Rp=10$ m$\Omega $), geometry ($ro=10$ cm, $ri=3$ cm, $zc=4$ cm), and propellant properties (argon: $mi=40$ AMU, $\epsilon ion=15.6$ eV). We estimate the mass bit as the total mass contained within the coupling volume in front of the current sheet, $mbit=minn\pi (ro2\u2212ri2)zc$, where $nn\u22487.4\xd71020$ m$\u22123$ is the neutral density that corresponds to the 23 mTorr fill pressure of the experiment. Finally, to simplify the calculation, we assume $\epsilon \xaf\u2217ion\u22482\epsilon ion$ and $k\xaf\u22480.5$. Using these numbers, we find that FARAD would require $dIc/dt|t=0\u226524$ kA/$\mu $s in order to operate at $\Omega \u2265\Omega \u2217$. Polzin’s criteria are, therefore, consistent with our generalized requirement for current sheet formation. The FARAD experiment operated at a maximum value of $\Omega \u22481.5$. Examination of Fig. 9(f) suggests that this value would yield a partially transparent current sheet, in agreement with the experimental findings of Polzin.

The formation parameter and criterion presented above were derived under the assumption that a pre-ionized plasma with a low ionization fraction ($\chi 0\u22720.01$) is present before the current pulse. The small $\chi 0$ assumption can be relaxed by replacing $mbit$ in Eq. (42) with the total mass of non-ionized propellant, $mbit\u2192(1\u2212\xi 0)mbit$. Here, we define $\xi 0$ as the percentage of the propellant mass bit that is ionized prior to the current pulse. In general, $\xi 0$ will increase with $\chi 0$ with additional dependences on the neutral gas and pre-ionized plasma distributions. Note that $mbit$ in Eq. (45) remains unchanged. The formation parameter may then be replaced by $\Omega \u2192\Omega /(1\u2212\xi 0)$. Therefore, increased pre-ionization yields an increased value of $\Omega $. In other words, pre-ionization serves to improve the formation of the current sheet, as shown in Fig. 4(b). However, as we showed in Fig. 4, this improvement may be offset by a corresponding reduction in the electrical efficiency of the thruster.

## V. CONCLUSIONS

We have theoretically examined current sheet formation and acceleration in an IPPT using a lumped-element circuit model. The addition of a finite skin depth and non-equilibrium ionization equations allowed us to study how transparency to electromagnetic fields and downstream mass influence thruster electrical and propellant mass utilization efficiencies. We found that current sheets impermeable to both the coil fields and downstream mass—presumed to exist in all prior circuit modeling efforts—formed only under specific conditions. The dynamics of electron heating during the early portion of the coil current cycle were identified as the dominant contributor to current sheet formation. Here, a balance occurs between Ohmic heating of electrons and cooling due to collisional ionization and excitation of neutral particles. The dependences of skin depth and ionization reaction rate on electron temperature complicate this balance through variations in the time-dependent electrical coupling coefficient and ionization rate, making current sheet formation a tightly coupled process. A new dimensionless scaling parameter was derived to characterize this process from which it was found that impermeable current sheet formation requires a threshold amount of Ohmic heating to offset the electron energy lost through inelastic collisions with the propellant gas.

A number of simplifying assumptions were made throughout the derivation of our model that warrant greater scrutiny in future studies. The properties of the plasma are assumed to be uniform within the current sheet; however, experimental measurements^{2,11} and computational models^{26,27} suggest that a more complex internal structure exists. Furthermore, the equation describing the change in the width of the current sheet in the presence of neutral entrainment was found by invoking phenomenological arguments as opposed to first principles. Last, an arbitrary constant is used to relate the current sheet electromagnetic field transparency to the ratio of skin depth to plasma width. A rigorous test of these assumptions would require a non-equilibrium 1D1T numerical plasma model (e.g., see Ref. 28) combined with experimental studies of field penetration within the current sheet^{11} and is beyond the scope of the present study. Radial expansion and cooling of the plasma are also neglected in our model. This assumption appears justified by recent 2D1T numerical simulations that show two-dimensional effects remain small while the current sheet resides within the coil coupling length.^{27} We note that it may be possible to examine the influence of non-uniformities on mass entrainment using our model along with assumed profiles for the plasma density and electron temperature within the current sheet. Such an approach would require modification of the volume-integrated conservation equations [Eqs. (7), (9), (12), (14), (16), and (20)], yielding an additional factor in the source terms that depends on the prescribed spatial profiles.

The assumption that the coil and plasma inductances are equal ($Lp=Lc$) is common in the literature but deserves greater scrutiny. Implicit to this assumption is that the coil can be approximated as a flat annulus with a uniform current distribution across its face in which case the coil and the plasma are effectively mirror images of one another, possessing the same annular geometry. Depending on how the coil is wound, however, it is possible that $Lc>Lp$ for a given geometry. Coil configurations that make use of multiple, single-turn spirals connected in parallel have been experimentally shown to approach the low-inductance limit of an annular disk.^{5} Deviations from this limit will influence the plasma current rise time; thus, the role of dissimilar inductances should be the focus of future research.

Finally, our species kinetics model neglects higher ionization states and recombination reactions. The dominance of higher ionization states has been predicted by models that invoke Saha equilibrium;^{18,27} however, the reaction timescales are generally too slow compared to the decoupling time for Saha equilibrium to be reached. Nevertheless, it remains likely that small fractions of doubly and triply ionized propellant exist within the current sheet. Recombination is not expected to play a dominant role unless the thermal quench of electrons drives $Te\u22721$ eV. While temperatures this low were not observed in the present analysis, these effects may become important when modeling thruster behavior in the presence of elevated background pressures, such as those that might occur in a ground test facility. The addition of higher ionization states and recombination reactions within our species kinetic model will be pursued in future studies.

## ACKNOWLEDGMENTS

The authors would like to thank Dr. Kurt Polzin, Dr. Kamesh Sankaran, and Dr. Ashley Hallock for several insightful conversations regarding modeling non-equilibrium processes in IPPT plasmas. This work was partially supported by an Air Force Office of Scientific Research Young Investigator Program (Award No. FA9550-20-1-0167).

## AUTHOR DECLARATIONS

### Conflict of Interest

The authors have no conflicts to disclose.

### Author Contributions

**Justin M. Little:** Conceptualization (lead); Data curation (lead); Formal analysis (lead); Investigation (lead); Methodology (lead); Software (lead); Writing – original draft (lead); Writing – review & editing (lead). **Gordon McCulloh:** Data curation (supporting); Formal analysis (supporting); Software (supporting); Visualization (supporting); Writing – original draft (supporting). **Cameron Marsh:** Data curation (supporting); Software (supporting); Visualization (supporting); Writing – original draft (supporting).

## DATA AVAILABILITY

The data that support the findings of this study are available from the corresponding author upon reasonable request.

### APPENDIX: COLLISION AND REACTION RATES

Here, we summarize the equations used to model the collision and reaction rates for the argon propellant used in our analysis. The electron–ion collision frequency is modeled as^{22}

where $\nu ei$ has units of s, $ns$ has units of m$\u22123$, and $Te$ has units of eV, and $ln\u2061\Lambda $ is the Coulomb logarithm.

The electron–neutral collision frequency is modeled as^{29}

where $\sigma en=4\xd710\u221220$ m$2$ is the average electron–neutral momentum transfer cross section.

We adopt the following approximation for the charge exchange cross section:^{30}

where $k1$ and $k2$ are empirical constants and $\Delta Es,n=mi(vs2\u2212vn2)/2$ is the difference in kinetic energy between the current sheet ions and neutral particles. For argon, $k1\u22487.49\xd710\u221210$ m and $k2\u22480.73\xd710\u221210$ m.

The following empirical equations are used to model the argon reaction rates:^{22}

Here, the different reaction rates have units of m$3$/s, and $Te$ has units of eV.

## REFERENCES

*47th AIAA/ASME/SAE/ASEE Joint Propulsion Conference and Exhibit*(AIAA, 2011), p. 6068.