16.522, Space Propulsion Prof. Manuel martinez-Sanchez Lecture 21: Electrostatic versus Electromagnetic Thrusters Ion Engine and Colloid Thrusters are Electrostatic devices, because the electrostatic orces that accelerate the ions (or droplets) are also directly felt by some electrode, and this is how the structure receives thrust. We could manipulate the expression for thrust density in an ion engine to Fa2E0Ea, where E,=3 d was the field on the surface of the extractor electrode. This is the electrostatic pressure Since E0=8.85×1012F/ m and ea is rarely more than2,000mm=2×10°Vm, we are limited to electrostatic pressure of about 20 N/m2(and due to various inefficiencies more like 1-2 N/m) Hall thrusters occupy an intermediate position, and point the way to a higher thrust density Ions accelerate electrostatically but electrons, which see the same(and opposite) electrostatic force, because the plasma is quasineutral (ne=ni),are essentially stopped(axially) by an interposed magnetic field. Of course, the force is he azimuthal Hall current they carry). In the end, then, the structure is pushed s of mutual, and so the electrons exert this force on the magnetic assembly(by means of magnetically. To be more precise, we should say that most of the force is magnetically transmitted. There is still an electrostatic field in the plasma, and so there will be some electrostatic pressure =EoE acting on various surfaces. But because we made the plasma quasineutral these fields are much weaker than they are between the grids of an ion engine and it is a good thing we have the magnetic mechanism available. In fact, the thrust density of Hall thrusters is about 10 times higher than that of ion engines despite the weak electrostatic fields More generally, we can ask how much stronger can the force per unit area on some structure be when it is transmitted magnetically as compared to electrostatically. As we will see in detail, the counterpart to the "electrostatic pressure"is the"magnetic H,Where b is the field strength and Ho=1. 256X10-Hy/m is the permeability of vacuum. Without recourse to superconductive structures, B can easily be of the order of 0.1 Tesla (either using coils or permanent magnets),so 2u. =8,000N/m, or 400 times the maximum practical electrostatic pressure Thrusters that exploit these magnetic forces are called Electromagnetic"(although they should be called" Magnetic"by rights). The magnetic field can be external, i.e supplied by coils and not greatly modified by plasma currents or it may be self induced, when plasma currents became large enough. They can also be steady(or at least slowly varying compared to plasma flow time), or they can be varying very ast, so as to set up strong induced electromotive forces(transformer effect). A few examples are: Magneto Plasma Dynamic(MPD thrusters The most powerful type, with self-induced, magnetic fields, operates in steady(or quasi-steady fashion, and can generate multi-Newton thrust 16.522, Space P pessan Lecture 21 Prof. Manuel martinez Page 1 of 21
16.522, Space Propulsion Lecture 21 Prof. Manuel Martinez-Sanchez Page 1 of 21 16.522, Space Propulsion Prof. Manuel Martinez-Sanchez Lecture 21: Electrostatic versus Electromagnetic Thrusters Ion Engine and Colloid Thrusters are Electrostatic devices, because the electrostatic forces that accelerate the ions (or droplets) are also directly felt by some electrode, and this is how the structure receives thrust. We could manipulate the expression for thrust density in an ion engine to 2 A 0a 1 F= E 2 ε , where a 4 V E = 3 d was the field on the surface of the extractor electrode. This is the electrostatic pressure. Since -12 ε0 = 8.85 × 10 F/m and Ea is rarely more than 6 2,000 V/mm = 2 ×10 V/m, we are limited to electrostatic pressure of about 20 N/m2 (and due to various inefficiencies more like 1-2 N/m2 ). Hall thrusters occupy an intermediate position, and point the way to a higher thrust density. Ions accelerate electrostatically, but electrons, which see the same (and opposite) electrostatic force, because the plasma is quasineutral (ne=ni), are essentially stopped (axially) by an interposed magnetic field. Of course, the force is mutual, and so the electrons exert this force on the magnetic assembly (by means of the azimuthal Hall current they carry). In the end, then ,the structure is pushed magnetically. To be more precise, we should say that most of the force is magnetically transmitted. There is still an electrostatic field in the plasma, and so there will be some electrostatic pressure 2 0 n 1 E 2 ε acting on various surfaces. But because we made the plasma quasineutral, these fields are much weaker than they are between the grids of an ion engine, and it is a good thing we have the magnetic mechanism available. In fact, the thrust density of Hall thrusters is about 10 times higher than that of ion engines, despite the weak electrostatic fields. More generally, we can ask how much stronger can the force per unit area on some structure be when it is transmitted magnetically as compared to electrostatically. As we will see in detail, the counterpart to the “electrostatic pressure” is the “magnetic pressure”, 2 0 B 2µ , where B is the field strength and -6 µ0 = 1.256x10 Hy/m is the permeability of vacuum. Without recourse to superconductive structures, B can easily be of the order of 0.1 Tesla (either using coils or permanent magnets), so 2 2 0 B 8,000 N/m 2µ , or 400 times the maximum practical electrostatic pressure. Thrusters that exploit these magnetic forces are called “Electromagnetic” (although they should be called “Magnetic” by rights). The magnetic field can be external, i.e., supplied by coils and not greatly modified by plasma currents, or it may be selfinduced, when plasma currents became large enough. They can also be steady (or at least slowly varying compared to plasma flow time), or they can be varying very fast, so as to set up strong induced electromotive forces (transformer effect). A few examples are: - Magneto Plasma Dynamic (MPD) thrusters The most powerful type, with self-induced ,magnetic fields, operates in steady (or quasi-steady) fashion, and can generate multi-Newton thrust
levels with a few cm diameter(compared to about 0. 1 N for a 30 cm ion engine, or for a 10 cm Hall thruster). Applied field mPd thrusters Here currents are less strong so the main part of the b field is external Still steady or quasi-steady Pulsed Plasma Thrusters(PPT) Pulsed Plasma Thrusters(PPt)are very similar in principle to self-field MPD, but they use a solid propellant (Teflon)which is ablated during each ise of operation. These pulses last 10-20 us only but are just long aB enough that induced emf fields(from at=VXE)are still weak Because of various practical (mostly thermal)issues, PPt thrusters are not very efficient <10%, but they are simple and robust. Pulsed Inductive Thrusters (PIT) Here the emphasis is on very fast magnetic risetime(1-10 us )and the induced emf is used to break down the gas, ionize it and drive a closed current loop that exerts the desired magnetic force. They can be thought of as a one-turn transformer in which the secondary is a plasma ring; the repulsion between primary and secondary accelerates the plasma away and pushes the primary coil forward. To avoid dissipating most of the power in Ohmic losses, the device must be fairly large>0.5m and powerful (MW to GW of instantaneous power) In the following few lectures we will have time only to explore the self-field MPD type. We begin with some basic Physics Electromagnetic Forces on Plasmas- MPD Thrusters For a charge g, moving at velocity v in an electric field e and magnetic field B, the So-called Lorentz force is F=QE+VX B (1) Now, F cannot depend on the rectilinear motion of the observer For non-relativistic velocities, B is also independent of motion, and so is the scalar q. therefore, the field E must be the different as viewed from different frames of reference let e be the field in the laboratory frame and ethat in another frame moving at u relative to the Then we must have 16.522, Space Propulsion Lecture 21 Prof. Manuel martinez-s Page 2 of 21
16.522, Space Propulsion Lecture 21 Prof. Manuel Martinez-Sanchez Page 2 of 21 levels with a few cm. diameter (compared to about 0.1 N for a 30 cm ion engine, or for a 10 cm Hall thruster). - Applied field MPD thrusters Here currents are less strong, so the main part of the B field is external. Still steady or quasi-steady. - Pulsed Plasma Thrusters (PPT) Pulsed Plasma Thrusters (PPT) are very similar in principle to self-field MPD, but they use a solid propellant (Teflon) which is ablated during each pulse of operation. These pulses last ∼ 10-20 sµ only, but are just long enough that induced emf fields (from B = ×E t ∂ ∇ ∂ JG G ) are still weak. Because of various practical (mostly thermal) issues, PPT thrusters are not very efficient <10% ⎛ ⎞ ⎜ ⎟ ⎝ ⎠ ∼ , but they are simple and robust. - Pulsed Inductive Thrusters (PIT) Here the emphasis is on very fast magnetic risetime ( ∼ 1 - 10 sµ ) and the induced emf is used to break down the gas, ionize it, and drive a closed current loop that exerts the desired magnetic force. They can be thought of as a one-turn transformer in which the secondary is a plasma ring; the repulsion between primary and secondary accelerates the plasma away and pushes the primary coil forward. To avoid dissipating most of the power in Ohmic losses, the device must be fairly large >0.5m ⎛ ⎞ ⎜ ⎟ ⎝ ⎠ ∼ and powerful (MW to GW of instantaneous power). In the following few lectures we will have time only to explore the self-field MPD type. We begin with some basic Physics. Electromagnetic Forces on Plasmas - MPD Thrusters For a charge q, moving at velocity v JG in an electric field E G and magnetic field B, JG the so-called Lorentz force is F = q E + v x B ( ) G GJG JG (1) Now, F G cannot depend on the rectilinear motion of the observer. For non-relativistic velocities, B JG is also independent of motion, and so is the scalar q. Therefore, the field E G must be the different as viewed from different frames of reference. Let E G be the field in the laboratory frame, and E' JJG that in another frame moving at u G relative to the first. Then we must have
so that E=E+uxB (in particular, for u =v the lorentz force is seen to be purely electrostatic; i F=qE). Most often the frame at u is chosen to be that moving at the mean mass velocity of the plasma Consider a plasma where there is a number density n, of the jon type of charged particle, which has a charge qj and moves at mean velocity Vi The net Lorentz force per unit volume is n, q(E+Vi X and since the plasma is neutral nai=0 =∑ n, q v,x B (4) But, by definition ∑nqv=j where j is the current density vector(A/m2). So, finally f=jxB (N/ Notice that vi in Equation(5)could be in any frame including the plasma frame ohm's Law In most cases, the dominant contribution to j(Equation(5))is from electrons, given their high mobility. In the plasma frame 1。=-en Notice that ve is the electron mean velocity vector, not to be confused with the mean thermal speed Ce. The picture of electron motions is that of a very rapid chaotic swarming of electrons back and forth going nowhere), except that the hole swarm "slowly"drifts at ve 16.522, Space P pessan Lecture 21 Prof. Manuel martinez Page 3 of 21
16.522, Space Propulsion Lecture 21 Prof. Manuel Martinez-Sanchez Page 3 of 21 E + v x B = E' + v - u x B ( ) G JG JG JJG JGG JG so that E' = E + u x B JJG GGJG (2) (in particular, for u=v G JG the Lorentz force is seen to be purely electrostatic; i.e., F = qE' G JJG ). Most often the frame at u G is chosen to be that moving at the mean mass velocity of the plasma. Consider a plasma where there is a number density nj of the jth type of charged particle, which has a charge qj and moves at mean velocity vj JG . The net Lorentz force per unit volume is ( ) j j j j f = n q E + v x B ∑ G GJG JG (3) and since the plasma is neutral j j j ∑nq =0 : j j j j f = n q v x B ⎛ ⎞ ⎜ ⎟ ⎝ ⎠ ∑ G JG JG (4) But, by definition, j j j j ∑nqv = j JG G (5) where j G is the current density vector (A/m2 ). So, finally, f = j x B G GJG (N/m3 ) (6) Notice that vj JG in Equation (5) could be in any frame, including the plasma frame. Ohm’s Law In most cases, the dominant contribution to j G (Equation (5)) is from electrons, given their high mobility. In the plasma frame, e e e j j = -en v GG JG (7) Notice that ve JG is the electron mean velocity vector, not to be confused with the mean thermal speed ce . The picture of electron motions is that of a very rapid, chaotic swarming of electrons back and forth (“going nowhere”), except that the whole swarm “slowly” drifts at ve JG
Typically e<<c et us make a crude model of the motion of the electron swarm the net force on it per unit volume is fe=-eE+ ve X b)ne where E is used, since we are in the plasma frame. In steady state, this is balanced by the drag force opposing motion of electrons relative to the rest of the fluid, which we are assuming to be at rest and whose particles have, by comparison only a very slow thermal motion. To evaluate this drag let ve be the effective collision frequency per electron for momentum transfer. This frequency is defined such that in each collision with a particle of the rest of the fluid, the electron is, on average, scattered by 90, so that its forward momentum is completely lost. Then the mean drag force per unit volume is Equating the sum of (8)and(9)to zero, +VeX or, since j=eTe-e'-e-jxB Define the scalar conductivi (10) and the hall parameter B and we can write the generalized ohms law as 16.522, Space P pessan Lecture 21 Prof. Manuel martinez Page 4 of 21
16.522, Space Propulsion Lecture 21 Prof. Manuel Martinez-Sanchez Page 4 of 21 Typically e v << ce JG . Let us make a crude model of the motion of the electron swarm. The net force on it per unit volume is e e ( ) e f = -e E' + v x B n G JJG JG JG (8) where E' JJG is used , since we are in the plasma frame. In steady state, this is balanced by the drag force opposing motion of electrons relative to the rest of the fluid, which we are assuming to be at rest, and whose particles have, by comparison, only a very slow thermal motion. To evaluate this drag, let e ν be the effective collision frequency per electron for momentum transfer. This frequency is defined such that in each collision with a particle of “the rest of the fluid,” the electron is, on average, scattered by 90D , so that its forward momentum is completely lost. Then the mean drag force per unit volume is e e e e ee e m f = -n m v = j e ν ν G JG G (9) Equating the sum of (8) and (9) to zero, ( ) 2 e e e e e n j = E' + v B m ν × G JJG JG JG or, since e e j v =- en G JG , 2 e ee ee e n e j = E' - j × B m m ν ν G JJG GJG Define the scalar conductivity 2 e e e e n = m σ ν (10) and the Hall parameter e e eB B = ; = m B ⎛ ⎞ β ββ ⎜ ⎟ ⎜ ⎟ ν ⎝ ⎠ JG G (11) and we can write the generalized Ohm’s law as
(12) where as given in Equation (2),E=E+ux B Remembering that the gyro frequency(the angular frequency of motion of an electron orbiting about a perpendicular magnetic field B)is o (13) i. e, it represents the ratio of gyro frequency to collision frequency it can be expected to be high at low pressures and densities where collisions are rare, and also at high magnetic field, where the gyro frequency is high. In many plasmas of interest in MHD or MPD, B-1 Electromagnetic Work The rate at which the external fields do work on the charged particles can be calculated (per unit volume)as W=∑q(E+vxB) where we used(Vx B).V,=0. We see here that the magnetic field does not directly contribute to the total work, since the magnetic force is orthogonal to the particle velocity it does, however, modify Eor j(depending on boundary conditions) and through them it does affect W This total work goes partly into heating the plasma(dissipation) and partly into bodily pushing it(mechanical work). To see this, notice that E-uxB)j=Ej+j 问uxB)j=×B) Also, using Ohms law j=(6+j×p where we used (jxB).j=0 16.522, Space P pessan Lecture 21 Prof. Manuel martinez Page 5 of 21
16.522, Space Propulsion Lecture 21 Prof. Manuel Martinez-Sanchez Page 5 of 21 σ β E' = j+ j × JJG GGG (12) where, as given in Equation (2), E' = E + u x B JJG GGJG . Remembering that the gyro frequency (the angular frequency of motion of an electron orbiting about a perpendicular magnetic field B JG ) is e = eB m ω , e = ω β ν (13) i.e. , it represents the ratio of gyro frequency to collision frequency; it can be expected to be high at low pressures and densities, where collisions are rare, and also at high magnetic field, where the gyro frequency is high. In many plasmas of interest in MHD or MPD, β ∼ 1. Electromagnetic Work The rate at which the external fields do work on the charged particles can be calculated (per unit volume) as ( ) j j j j j W = q n E + v × B . v ∑ G JG JJG JG j j j j = E . q n v ∑G JG or W = E . j G G (14) where we used ( ) v × B . v 0 j j ≡ JG JJG JG . We see here that the magnetic field does not directly contribute to the total work, since the magnetic force is orthogonal to the particle velocity; it does, however, modify E G or j G (depending on boundary conditions), and through them it does affect W. This total work goes partly into heating the plasma (dissipation) and partly into bodily pushing it (mechanical work). To see this, notice that W = E . j = E' - u × B . j = E' . j+ j × B . u ( ) ( ) G G JJGG JG G JJGG G JG G (using ( ) () u × B . j = - j × B . u G JGG GJG G ). Also, using Ohm’s law ( ) 2 1 j E' . j = j + j × . β j = σ σ JJGG G G G G where we used ( ) j × . j = 0 β GGG