IEPC-2009-265

Pulsed Plasmoid Propulsion: The ELF Thruster IEPC-2009-265 31 th International Electric Propulsion Conference, Ann Arbor, Michigan September 20-24, 2009 John Slough and David Kirtley MSNW, Redmond, WA 98052, USA and Thomas Weber University of Washington, Seattle, WA 98195, USA The Electrodeless Lorentz Force (ELF) thruster creates a high-density, magnetized plasmoid known as a Field Reversed Configuration (FRC) employing a Rotating Magnetic Field (RMF). The RMF driven azimuthal currents, coupled with the enhanced axial magnetic field gradient produced by the FRC inside the flux preserving conical thruster, produce a large axial J θ x B r force that accelerates the plasmoid to high velocity. The ELF thruster has the potential to produce highly variable thrust and specific impulse at high efficiency in a single compact, lightweight thruster and PPU package. Presented are an ELF thruster primer, a thorough discussion detailing background physics, operating principles, as well as results from the initial set of experiments. Nomenclature a = Hills Vortex radial scaling parameter A = magnetic vector potential (subscripts r, θ, z denote cylindrical components) B = magnetic field vector (subscripts r, θ, z denote cylindrical components) B b , B bias = magnitude of preexisting (vacuum) axial magnetic field B ω = amplitude of rf rotating magnetic field ΔB z = axial magnetic field change due to RMF driven currents (superscript M denotes maximum) B ext = magnitude of magnetic field external (radially) to the FRC β = plasma pressure normalized to external (vacuum) field C = capacitance value δ = classical skin depth = (2η/μ 0 ω) 1/2 E = electric field vector (subscripts r, θ, z denote cylindrical components) e = unit of electron charge E k = propellant kinetic energy E k_RMF = kinetic energy derived from electromagnetic input E k_th = kinetic energy derived from conversion of plasma thermal energy E ion = ionization energy E Ω = energy input from Ohmic heating ε = Hall scaling parameter f = specific force (subscripts r, θ, z denote cylindrical components) F = force vector (subscripts r, θ, z denote cylindrical components) FRC = field reversed configuration φ = magnetic flux ψ = poloidal magnetic flux 1 The 31st International Electric Propulsion Conference, University of Michigan, USA September 20 – 24, 2009 Ψ = poloidal flux function γ = ratio of electron angular frequency to electron-ion collision frequency I = current (subscripts r, θ, z denote cylindrical components) Isp = propellant specific impulse j = current density (subscripts r, θ, z denote cylindrical components) k = Boltzmann’s constant η = plasma resistivity η e = thruster efficiency λ = ratio of plasma radius to classical skin depth δ l ac = acceleration (thruster) length μ 0 = magnetic permeability in vacuum m = electron mass ν ei = electron-ion collision frequency n = plasma density n e = electron density N = electron line density p = plasma pressure θ = azimuthal cylindrical coordinate Q = circuit quality factor r = radial cylindrical coordinate r p = plasma radius r s = magnetic separatrix radius RMF = rotating magnetic field T = plasma temperature T e = electron temperature τ = RMF pulse length ω = angular frequency of RMF ω ce = electron angular frequency in rotating field ω ci = ion angular frequency in rotating field z = axial cylindrical coordinate z s = separatrix length along axis of symmetry I. Introduction Advancements in lightweight space power systems create a unique need for lightweight, higher power, and high performance electric propulsion systems. The Electrodeless Lorentz Force (ELF) thruster is based on the creation of a high-density, magnetized plasmoid known as a Field Reversed Configuration (FRC) by employing a Rotating Magnetic Field (RMF) for the generation of large azimuthal currents. The RMF driven currents, coupled with the large axial magnetic field gradient produced inside the conically shaped flux-conserving thruster, produce a large axial J θ x B r force that accelerates the plasmoid to high velocity. The axial force is thus overwhelmingly determined by the driven J θ and resultant B r rather than thermal expansion forces, maximizing thrust efficiency. The ELF thruster is electrodeless and the plasmoid propellant is magnetically isolated so that thermal and chemical wall interactions are negligible, maximizing lifetime. Unlike other pulsed inductive or field reversed devices, the large azimuthal current (up to 20 k A) is generated with an RF wave in the form of a steady rotating magnetic field in the r-θ plane. Power requirements can be easily met with high efficiency, light-weight modern solid-state power technology. Unlike other electromagnetic thrusters, the propellant is completely uncoupled from the driving and confining fields so no complex magnetic detachment is required. Higher internal plasma temperatures and densities significantly reduce ionization losses over traditional Electric Propulsion, while plasma expansion minimizes thermal and frozen flow losses maximizing total efficiencies. High plasma density, magnetic isolation, and simple magnetic geometry minimize thruster footprint. Additionally, lightweight materials and novel PPU advancements available to pulsed power systems will enable drastically lighter thruster and PPU systems. 2 The 31st International Electric Propulsion Conference, University of Michigan, USA September 20 – 24, 2009 For these reasons, it is believed that an ELF-based thruster system should have significant specific power and performance benefits over current electric propulsion devices. Based on the current laboratory results, the ELF would enable a range of high-power propulsion missions in the 10-100 k W class. Operation on Xenon will provide maximum efficiency and T/P and compatibility with existing spacecraft technologies. The ability to operate on Air would enable very eccentric orbit propulsion, refuelable orbital transfer vehicles, and even direct drag makeup for extremely low orbits. It is expected that the same operational flexibility would extend to operation on complex, combustible, and liquid fuels. Finally, extending this technology to the higher densities and powers demonstrated in lab experiments, there are mission applications in high-altitude, air-breathing, hypersonic flight and beamed-energy upper stage propulsion that are not feasible with current technologies. rz x BJF θ = RMF generated plasma current Steady magnetic field J θ 1 2 3 rz x BJF θ = RMF generated plasma current Steady magnetic field J θ 1 2 3 Figure 1. ELF Thruster Operation: (1) Rotating Magnetic Fields (RMF) form high-density, FRC plasmoid (2) FRC grows and accelerates driven by RMF generated currents & steady field (3) FRC expands as ejected, converting any remaining thermal energy into directed energy The program goal of the current research is to build and characterize an ELF thruster prototype operating in a regime similar to current electric propulsion systems in terms of power and operational variability. In the initial experiments efficient FRC formation and ejection was demonstrated over a large range of power, thrust, and specific impulse levels. A discussion of these initial results will be presented, as well as a fuller exposition of the physical principles behind plasmoid propulsion with the ELF thruster. .Section II provides a detailed discussion of Rotating Magnetic Field and Field Reversed Configuration physics pertinent to thruster applications. Additionally, a complete description and comparison of Helicon and RMF plasmas is provided. Finally, Section II presents the critical parameters and geometry required to apply these technologies in a realistic thruster system. Section III provides a complete description of the Electrodeless Lorentz Force thruster hardware, electrical configuration, and operation. A summary of typical operating characteristics and major milestones is given. Finally, a detailed exposition shows that a coherent, high-velocity plasmoid is being formed and ejected into a downstream chamber. Section IV discusses thruster scaling, efficiency estimates, and realistic thruster considerations. II. The ELF Thruster – Principles of Operation A. Background The Electrodeless Lorentz Force (ELF) Thruster is based on the large axial body force exerted on a plasmoid arising from the large azimuthal currents driven by a transverse rotating magnetic field in the presence of a gradient magnetic field (see Fig. 1). The current arises from the synchronous motion of the electrons magnetized to the rotating field lines. With all electrons participating in the azimuthal rotation, the resultant current can be large enough to completely cancel and reverse a preexisting steady axial field and thereby create an isolated magnetic structure (plasmoid) commonly referred to as a Field Reversed Configuration 1 . The repetitive generation of this plasmoid its subsequent acceleration out of the thruster that provides the high Isp, thrust and efficiency that characterizes the operation of the ELF thruster. 3 EXTERNAL FIELD FRC CLOSED POLOIDAL FIELD R – null radius r s – separatrix radius r c – coil radius x s –r s /r c EXTERNAL FIELD FRC CLOSED POLOIDAL FIELD R – null radius r s – separatrix radius r c – coil radius x s –r s /r c Figure 2. Schematic of an FRC confined in single turn coil. There are essentially two aspects of the ELF thruster that require some background and discussion in order to appreciate the transformational nature The 31st International Electric Propulsion Conference, University of Michigan, USA September 20 – 24, 2009 of this new method of propulsion when compared to other electric propulsion devices. One concerns the generation mechanism and properties of the FRC plasmoid, and the second is the optimal deployment and wide application of the device to a number of unique space propulsion missions. The intent here is to address the first with a description of the FRC, the methods of formation, and in particular the novel method of generation employed in the ELF thruster. A Field Reversed Configuration (FRC) plasmoid consists of a closed field line, fully ionized plasma confined by a large azimuthal self current (see Fig. 2). This plasma diamagnetic current flows opposite to the coil currents producing the external axial magnetic field. Typically FRC plasmoids are formed in a cylindrical coil with a fast (< 10 μs), and large (100’s k A) pulsed inductive discharge resulting in a stable, well-confined plasmoid that is neutral to translation. A simple conical coil can then be employed to produce the magnetic gradient desired for rapid ejection of the FRC for propulsion. The steeper the coil pitch (field gradient) and the shorter the length of the cone, the faster the more rapid will be the FRC acceleration and ejection. Typically this demands a very rapid and large flux change in order to generate a sufficiently large induced current. This method thus inherently requires a high voltage pulse power system for operation. As an aside it can be noted that the pulsed inductive thruster can be thought of in this way as the limit where the cone angle reaches 90°. Propitiously, there is another method for the generation of the FRC that does not rely on inductive techniques. The same azimuthal currents can be caused to arise without the rapid magnetic flux change of pulsed induction by employing a Rotating Magnetic Field (RMF) 2 where the rotating field lines lie in a plane transverse to the axis in a cylindrical geometry (see Fig. 3). J θ ⋅ ⋅ × × × × Loop pair for Horizontal field Component in blue External loop antenna pair for vertical component of Rotating Magnetic Field J θ ⋅ ⋅ × × × × Loop pair for Horizontal field Component in blue External loop antenna pair for vertical component of Rotating Magnetic Field Figure 3. Schematic of the cross section of the plasma column and RMF lines of force. The coil set of axial conductors employed to generate the RMF are also shown. These two orthogonal sets, carrying sinusoidal currents phased 90 ° apart, produce an m=1 rotating magnetic field of constant amplitude . The technique of generating azimuthal electron currents in a plasma column by means of rotating magnetic fields was first investigated by Blevin & Thonemann in 1962 3 . The principles of their technique can best be understood as follows. Consider that a transverse rotating magnetic field completely penetrates a cylindrical plasma column as in Fig. 3. Provided that the angular frequency ω, of the rotating field lies between the ion and electron cyclotron frequencies (ω ci and ω ce ) calculated with reference to the amplitude of the rotating field, B ω , and provided that the electron collision frequency is much less than the electron cyclotron frequency, the electrons can be considered as 'tied' to the lines of force of the rotating field. With the assumption that the electron collisions are relatively infrequent (ν e << ω ce ), they will circulate synchronously with the angular frequency ω, whereas the ions (at least over the time of interest) have no net azimuthal motion. The electrons thus form a steady azimuthal current (in fact, the Hall current). In most systems, Hall currents are inhibited by electric polarization fields. In the situation under discussion, however, charge separation does not occur because of the azimuthal symmetry. It should be noted that the frequency condition for current drive: ω ci < ω < ω ce (1) as well as the antenna geometry for generating the RMF are not unique to this application. In fact, the same antenna configuration and RF frequency requirements (Eq. 1) are found for the propagation of the helicon wave as well (see Fig. 4) 4,5 . The singular difference between the helicon discharge and the full electron entrainment found with FRC generation is the magnitude of B ω . Both rely on the m=1 transverse mode penetrating the plasma and coupling to the electrons. This is accomplished with the rotating field in the case of a two phase antenna as shown in 4 The 31st International Electric Propulsion Conference, University of Michigan, USA September 20 – 24, 2009 Fig. 4, or the right hand circularly rotating component of the oscillating field produced by a single m=1 “saddle coil” antenna commonly employed for helicon discharges. For the particular choice of field geometry and rotation frequency given in the above description, the appropriate form of Ohm's law is E = ηJ + -en(j x B) (2) Steady-state solutions satisfying this equation can be obtained in two limits. First, the situation that applies to the helicon discharge is examined where the ηj term in Ohm's law dominates. The basic tendency is then for induced axial plasma currents to screen out the applied rotating field in a distance equal to the classical skin depth δ [= (2η/μ 0 ω) 1/2 ]. It is found, however, that the inclusion of only a small contribution from the jx B term leads to the generation of azimuthal currents in the plasma column and that the presence of these currents, in turn, aid the further penetration of the rotating field. The second limit which leads to the formation of the FRC considers the jx B term in Ohm's law to be dominant. The solution in this limit was originally obtained by Blevin & Thonemann 3 , which describes the complete and almost complete penetration of the rotating field into the plasma column. The questions of accessibility to the steady-state solutions and of arbitrary contributions from the ηj and jx B terms require a numerical treatment of the problem which has been performed by Hugrass and Grimm 6 and confirms the appropriateness of the analysis given here for the limiting conditions. Separatrix L ~ 5r s Rotating Magnetic Field B ω r s r w Field Reversed Configuration (FRC) I 0 cos(ωt) I 0 sin(ωt) Separatrix L ~ 5r s Rotating Magnetic Field B ω r s r w Field Reversed Configuration (FRC) Separatrix L ~ 5r s Rotating Magnetic Field B ω r s r w Field Reversed Configuration (FRC) I 0 cos(ωt) I 0 sin(ωt) I 0 cos(ωt) I 0 sin(ωt) Figure 4. RF antenna coils are positioned radially external to the axial coil as indicated by the arrows. Two m=1 “saddle” coils similar to the conventional antenna employed in helicon discharges are phased at 90 ° produce a constant amplitude rotating field B ω in the electron drift direction. In the limit of strong electron magnetization to the rotating field ( ω ce >> ν ei ), synchronous electron motion produces a large j θ (= neωr) and an FRC is formed. For the basic model and equations consider an infinitely long plasma cylinder of radius r p lying in a uniform axial magnetic field, B b . A uniform transverse magnetic field which rotates about the cylindrical axis is applied to this plasma column. The amplitude, B ω and the angular frequency, ω, of this rotating field are chosen so that Eq. (1) is satisfied. An (r, θ, z) system of co-ordinates is chosen and SI units are used throughout. The following assumptions are made: (i) All quantities are assumed independent of axial location, z. (ii) The ions form a uniformly distributed neutralizing background of fixed, massive positive charges. (iii) The plasma resistivity, η, is taken to be a scalar quantity which is constant in time and uniform in space. In particular, η is assumed to be of the form 2 en m ei ν =η (3) 5 The 31st International Electric Propulsion Conference, University of Michigan, USA September 20 – 24, 2009 where m is the electron mass. e is the electronic charge, n is the uniform electron number density and ν ei is the electron-ion momentum transfer collision frequency. (iv) The electron inertia is neglected; that is, it is assumed that ω << ω ce ,ν ei (v) The displacement current is neglected. This implies that only systems for which ωr p /c, where c is the speed of light, are considered. Within the above constraints, the system is completely described by the self consistent solutions to the following set of equations: Maxwell's equations, j B B E 0 t μ=×∇ ∂ ∂ =×∇ (4a,b) the equation of motion for the electrons, [] 0)( m e ei =ν−×+ − u Bu E (5) where u is the velocity of the electron fluid, and the definition of the current density is J= neu . (6) The solutions for the field quantities must satisfy the appropriate boundary conditions. In particular, for r ~ r p they must reduce to the equations describing the externally applied magnetic field: B r = B ω cos (ωt-θ) (7) B θ = B ω sin (ωt-θ) (8) B z = B b (9) Equations (3), (5) and (6) can be combined to form the appropriate Ohm's law for the problem, ( Bjj E×+η= en 1 ) . (10) By introducing the magnetic vector potential, A, where AB×∇= (11) and making use of the assumption that ∂/∂z = 0, it is straightforward to show that all the field quantities can be derived from the z components of B and A. These components, B z and A z themselves satisfy the following pair of coupled partial differential equations, ⎥ ⎦ ⎤ ⎢ ⎣ ⎡ ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ ∂ ∂ ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ θ∂ ∂ − ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ θ∂ ∂ ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ ∂ ∂ μ +∇ μ η = ∂ ∂ r B A B r A rne 1 A t A zz 0 z 2 0 z (12) ⎥ ⎦ ⎤ ⎢ ⎣ ⎡ ∇ θ∂ ∂ ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ ∂ ∂ −∇ ∂ ∂ ⎟ ⎠ ⎞ ⎜ ⎝ ⎛ θ∂ ∂ μ +∇ μ η = ∂ ∂ z 2 z z 2 z 0 z 2 0 z A r A A r A rne 1 B t B (13) where 6 The 31st International Electric Propulsion Conference, University of Michigan, USA September 20 – 24, 2009 2 2 2 2 r 1 r r rr 1 θ∂ ∂ + ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ ∂ ∂ ∂ ∂ =∇ . The nonlinear terms which appear on the right-hand side of (12) and (13) arise from the Hall term in Ohm's law [Eq. (10)]. B. Helicon Regime To derive the effects of the nonlinear (Hall) terms in the system described by the two coupled Eqs. (12) and (13), the solution is initially sought as an expansion in terms of a small parameter ε that assumes the coefficient of j from the Hall term in Ohm’s law (10) (B ω /ne) is small compared to the coefficient of j from the resistive term (η), or, 1 1 m e B en B ei ce ei << ν ω = ν = η =ε ωω (14) Where Eq. (3) was substituted for η. The regime of small ε represents the helicon limit where the magnitude B ω is sufficiently small that the collision frequency of the electrons effectively decouples the electron motion from entrainment in the RF magnetic field. With B z and A z expanded in terms of ε, and after a good deal of algebra, the system of Eqs. (12) and (13) can be solved. The result for the magnitude of j θ component arising from the rotating magnetic field in this limit is: pp 2 p 2 ren)rr( 2 exp r )r(jω ⎟ ⎠ ⎞ ⎜ ⎝ ⎛ − δ − ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ δ ε−= θ (15) It is noteworthy that the nonlinear Hall term only starts to play a role in the second and higher terms in the ε expansion. From an experimental point of view, it is the difference in magnetic field from the vacuum field at the plasma edge B z (r p ) = B b and the field on axis B z (0) that is easily measured. This field difference, ΔB z , also characterizes the plasma pressure or diamagnetism in radial equilibrium, and is given by: ∫ θθθ =μ=Δ p r 0 0 drj Iwhere IB which from Eq. (15) yields: ren r2 B 2 p 2 ei ce0 z ω ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ δ ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ ν ωδμ =Δ . (16) It can be seen that the current driven is quite small in this regime as not only does it scale with ε 2 , it also scales with typically another small quantity, the square of the ratio of the skin depth to plasma radius. C. FRC Regime We now consider the case where the Hall term is the dominant term in Ohm’s law. In the limit where ηj ~ 0, Ohm's law assumes the form ( Bj E×= en 1 ) (17) 7 The 31st International Electric Propulsion Conference, University of Michigan, USA September 20 – 24, 2009 Employing the same analysis as before, but dropping the resistive terms in Eqs. (12 and (13), it is easily verified that the following equations, which were first obtained by Blevin & Thonemann 3 , are exact solutions: ()θ−ω−= ω tsinr BA z (18) () 22 pbz rren 2 BB−ω−= 1 . (19) Calculation of the magnetic field components, ( θ−ω= θ∂ ∂ = ω tcos B A r 1 B z r ) (20) ( θ−ω= ∂ ∂ −= ωθ tsin B r A B z ) (21) confirms that this solution describes a steady-state situation where the applied rotating field has completely penetrated the plasma column. Furthermore, the expression for the azimuthal current density, ren r B1 j z 0 ω−= ∂ ∂ μ −= θ (22) indicates that all the plasma electrons rotate synchronously with the rotating field where the electron drift velocity at any radial position r is given by rω. For the case where the Hall term is completely dominant and the electrons are all in synchronous rotation, the equivalent expression to Eq. (16) for the change in field due to the Hall current is: Ne 2 )0(B)r(BB 0 zpz M z ω π μ =−=Δ (23) where the superscript M indicates that this represents the maximum possible current (all electrons co-rotating) and thus maximum field swing possible. Here N is the number of electrons pre unit length (i.e. electron line density), and is given by ∫ π= p r 0 drr)r(n2N . (24) Unlike the helicon regime, the presence of the large RMF induced j θ currents in the plasma substantially effects the equilibrium distribution of the plasma. High β MHD equilibria (with no plasma flow) are described by solutions of the pressure balance equation: p∇=×Bj . (25) For axisymmetric equilibria, the pressure balance equation (with negligible toroidal field) can be rewritten in the form: ψ∂ ∂ μ−= ∂ ψ∂ + ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ ∂ ψ∂ ∂ ∂p r zrr 1 r r 2 0 2 2 (26) 8 The 31st International Electric Propulsion Conference, University of Michigan, USA September 20 – 24, 2009 where ψ is a stream function that is related to the poloidal flux function Ψ p = ψ/2π. Equation (26) is known as the Grad-Shafranov (G-S) equation. It is a fortunate circumstance that the G-S equation possesses an analytic solution for the case where p varies linearly with ψ, as it is observed to do in most FRCs. This solution is referred to as the Solov’ev solution or alternately as the Hill’s vortex solution, and has the form: ( 2 s 222 2 s 2 0 rzar r2 r B −+=ψ) (27) Here r s is the radius of the separatrix in the z = 0 plane (midplane). For the FRC, the separatrix radius is essentially the plasma radius (r s = r p ). Also B 0 = -B z (0,0) and a = R/z s where z s is the position of the neutral point [B z (0,z s ) = 0] on the z axis. A plot of the equilibrium FRC plasma from 2D MHD numerical calculations for the FRC, show flux contours very close to those of the analytical expression given in Eq. (27) as can be seen in Fig. 5d. Also shown in this figure are the magnetic and plasma configurations corresponding to the helicon mode (Fig. 4b) and the transition from the helicon limit to the FRC limit (Fig. 5c). The transition is characterized by a significant driven j θ current and the formation of a high β plasma. This transition can only be described by the numerical solution to Eqs. (12) and (13) 6 . Although it will not be derived here, it is also possible to obtain an equilibrium solution to the G-S equation for this regime as well. An example of the full nonlinear steady solutions to Ohm’s law and the electron equation of motion discussed above is found in Fig. 6. As can be seen, the increase in the magnetization of the electrons characterized by the parameter γ ≡ ω ce /ν ei , results in a rapid transition from small to very large azimuthal current generation. This has a profound effect on the plasma equilibrium, changing rapidly from a plasma column embedded in what is essentially the vacuum field, to a state characterized by the formation of a magnetically isolated, high β plasmoid confined by an axial magnetic field that has been substantially compressed and increased in strength. It can be shown that this transition occurs when the value of γ is increased to be comparable to the ratio of the plasma radius to the skin depth denoted by the parameter λ ≡ r p /δ, or Transition to FRC when ⇒ γ = λ. (28) The plot in Fig. 5 correspond to a λ ~ 3.5. For larger plasmas, or RMF at higher frequency will increase λ and hence γ required for transition into the FRC mode. 9 The 31st International Electric Propulsion Conference, University of Michigan, USA September 20 – 24, 2009 The 31st International Electric Propulsion Conference, University of Michigan, USA September 20 – 24, 2009 10 Figure 5. a) STARTUP: Solenoidal windings create an axial bias field B b inside array of isolated conducting rings which preserve magnetic flux but permit transverse fields from RF antennas. Neutral gas fills chamber. (RF Antennas are not shown for clarity.) b) HELICON MODE: RF antenna produces oscillating transverse m=1 mode where electrons couple to the component rotating in the electron drift direction. A high density plasma of moderate pressure peaked on axis is produced. Plasma flows out freely along field. Electrons are strongly magnetized to steady field but only weakly couple to RF field (B ω << B b ). c) TRANSITION: Appreciably increasing B ω magnetizes electrons in rotating RF field producing large synchronous electron motion ( θ current). Ohmic power flow dramatically increases plasma β. The high β plasma (diamagnetic) current excludes the initial axial flux increasing the field external to the plasma due to the flux conserving rings. (Lenz’s law dictates that the plasma current be mirrored in the flux conserver thus enhancing the magnetic field) . d) RMF MODE: Increasing plasma density with fully synchronous electron motion with the rotating B ω field produces very large azimuthal currents (10s of k A) capable of completely reversing the axial magnetic field. The result is a well confined, closed field plasmoid (FRC) in equilibrium with an external field now many times larger than the initial bias field. D. Thruster Application Up to this point the description of the RF generation of plasma as well as current has been discussed in terms of a steady equilibrium solution. For the application as a thruster requires that some modification must be introduced that will produce an axial disequilibrium and thereby generate a plasma flow or movement and result in a net axial force reacted back on to the thruster or thruster fields. The simplest means to accomplish this is to create an unbalanced magnetic configuration where the magnetic gradient in z can be employed to produce the necessary B r for a body (Lorentz) force in the z direction. For the helicon mode this force can be vanishingly small so that the method for axial acceleration must rely on the resistive heating of the plasma and the nozzle effect of the diverging magnetic field at the end of the thruster (see Fig 7a). There appears to be a significant enhancement to the ion flow provided by the collisionless sheath that can form downstream of the helicon antenna in the throat of the exit aperture under the right conditions 7 . The accelerating potential in this case is roughly 3 k T e which enhances the thruster efficiency. It is however still, in essence, an electrothermal thruster. Only when one increases the magnetization parameter γ into the transition regime does the rapidly increasing azimuthal current create significant j θ ×B r at the magnetic nozzle, and provide additional thrust. 0 2 4 6 8 10 1.0 0.8 0.6 0.4 0.2 0 ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ Δ Δ θ θ M M z z I I or B B Helicon Regime (electrothermal) FRC Regime (electromagnetic) η = ν ω =γ ω en B ei ce Eq. (16) Eq. (23) Numerical Solution 0 2 4 6 8 10 1.0 0.8 0.6 0.4 0.2 0 ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ Δ Δ θ θ M M z z I I or B B Helicon Regime (electrothermal) FRC Regime (electromagnetic) η = ν ω =γ ω en B ei ce Eq. (16) Eq. (23) Numerical Solution Figure 6. Normalized change in axial field (azimuthal current) as a function of the electron magnetization parameter γ. The picture here is somewhat analogous to the behavior of the arcjet thruster transition to an MPD thruster as the discharge current is substantially increased, and where the thrust power then increases as I 2 from the purely jx B forces. In the static equilibrium configuration described earlier, the transition phase is transient, rapidly leading to the inevitable formation of the FRC. If placed in a conical thruster configuration with a significant radial magnetic field throughout the RF interaction region (see Fig. 7b) it is possible to have a concurrent rapid loss of plasma out the nozzle end preventing the full reversal of the magnetic field to form the FRC. The rapid loss is aided by the increasing Lorentz force on the plasma currents. This higher power mode of operation is substantially better than the standard helicon mode, but it still receives a substantial fraction of the thrust from electrothermal conversion process. The higher plasma temperature improves ionization efficiency as well as produces higher Isp. Operation in this mode as a thruster was achieved with the High Power Helicon Thruster developed at MSNW 8 . While operation in this regime may have certain advantages, and is no doubt superior to the standard helicon, the maximum thrust, Isp and efficiency is obtained in the FRC regime shown in Fig. 7c and 7d. The simplest operation of a thruster operating in the magnetized electron regime (γ > λ) where an FRC is created, is to employ a conically shaped thruster as in Fig. 7c. The steady bias field B b now contains the field gradient required for a large j θ ×B r , and the flux conserving shell allows the amplification of the B r as the FRC expands and compresses the bias field. As the resultant external field can be several times the initial bias field. The RMF field driven currents can thus directly provide the j θ and as a consequence, most of the the B r of the Lorentz force yielding the same I 2 scaling dependence observed in the electromagnetic MPD thruster. In this case however it is accomplished without electrodes or plasma- wall contact. 11 The 31st International Electric Propulsion Conference, University of Michigan, USA September 20 – 24, 2009 z c B r F jx B z c B r F jx B z c B r F jx B z c B r F jx B B r (vac) a) b) c) d) z c B r F jx B z c B r F jx B z c B r F jx B z c B r F jx B B r (vac) z c B r F jx B z c B r F jx B z c B r F jx B z c B r F jx B z c B r F jx B z c B r F jx B z c B r F jx B z c B r F jx B z c B r F jx B z c B r F jx B z c B r F jx B z c B r F jx B B r (vac) a) b) c) d) Figure 7. a) Helicon Thruster: With no significant diamagnetic current and negligible magnetic gradient, this thruster relies on electrothermal heating and nozzle expansion at exit. Two fluid effects (double layer) enhance Isp, and efficiency. Concerns are efficiency, plasma detachment from thruster fields and beam spread . b) High Power Helicon Thruster : Larger RF field amplitude at lower frequency leads to a much larger high density, high β plasma. Plasma is lost from stationary plasma through axial J θ ×B r as well as electro-thermal expansion. Detachment and beam spread problems greatly reduced . c) Electrodeless Lorentz Force (ELF) Thruster: An even larger, lower frequency rotating field imposes synchronous θ motion of all electrons. The resultant field producing a completely isolated, magnetized plasmoid (FRC). The strong axial J θ ×B r force rapidly drives the plasmoid out of the thruster. FRC expansion during ejection converts remnant thermal energy into directed energy. No detachment issues . d) Magnetically Accelerated Plasmoid (MAP) Thruster: With the FRC formed in ELF, further thrust or Isp can be obtained with peristaltic sequencing of axial array of flux coils. Large J θ ×B r force can be maintained throughout FRC passage enabling neutral gas entrainment significantly increasing thruster efficiency at optimal Isp. 12 The 31st International Electric Propulsion Conference, University of Michigan, USA September 20 – 24, 2009 There is an alternate method to the conical coil for generating the axial magnetic gradient field (i.e. B r ) required for strong j θ ×B r thrust generation, and that is to produce it actively by sequencing an array of theta coils along the length of the thruster (see Fig. 7d) rather than maintain a fixed bias field with flux conserving rings. In this thruster, no bias field is required as the FRC is constantly being kept from the thruster wall by the activation of the coils as it passes resulting in the maximum possible magnetic force. Since the magnetic gradient is actively produced by the difference created by the sequential firing, it can be sustained undiminished throughout the length of the thruster. In the conical thruster the radial magnetic field pressure drops rapidly as the plasma expands and exits. Continually higher voltage operation of each coil is required if high Isp is desired. The results from the Magnetically Accelerated Plasmoid (MAP) thruster, which employed this technique, attest to this fact where Isp’s in the range of 18 to 22 ks were obtained 9 . The formation technique used in these experiments for the FRC was the field reversed technique so that high voltage was employed throughout. With the ability form the FRC with the RMF technique, it would be possible to operate with this thruster geometry at much lower voltage. However a much more promising application would be to employ an array of such coils on the exit of the ELF thruster of Fig. 7c. With the Isp already in the optimal range (3-5 ks), increasing the Isp is probably not required. Instead neutral gas can be introduced into the FRC as it passes down the thruster body shown in Fig. 7d. With the charge exchange cross section much larger than the ionization cross section, the neutrals will be entrained with the electrons passing between neutrals and ions “herding” both along with the FRC. The FRC mass is thus increased but without the large ionization penalty. The effect of this additional mass would normally be a reduction in the velocity of the FRC as momentum is conserved. This is where the ability to continually push on the FRC with the plasmoid accelerator provides a way to maintain the FRC velocity while the mass is added. In this manner it is possible to significantly increase the thrust and efficiency of the thruster without resorting to high Isp. An ELF thruster has not yet been configured for neutral entrainment, but plans are currently underway to test this very promising enhancement. E. Thruster Scaling and Energy Deposition The use of the RMF to generate the plasma and magnetic field currents radically changes both the manner in which energy is introduced into the system as well as the scaling when compared to other EM pulse thrusters. This a critically important difference as it provides for a way to add significant propulsive energy and momentum after the process of plasma acceleration and ejection has been initiated. This is key to providing for higher efficiency as well as providing for thrust enhancement through neutral entrainment. The difference can best be understood by examination and comparison to other EM thrusters. For the PTX 10 , which is based on the generation of the FRC with the conventional field reversed theta pinch, there is energy input to the thruster system only during the time that there is axial flux entering the coil through the coil gap. The energy input is maximum at the start, and vanishes as the typically sinusoidal coil voltage (dφ/dt) goes to zero. At this point, only the conversion of the compressional energy to directional energy by expansion out of the conical coil is possible. The coil magnetic field drops as 1/r 2 in the conical coil, and the FRC currents also fall as 1/r 2 as the diamagnetic FRC currents decrease to maintain the FRC in radial equilibrium (see Eq. 25) during the FRC expansion in the cone. The plasma cross