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 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 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
Thus B).u The second term of this expression is simply the rate at which the Lorentz force jx B does mechanical work on the plasma moving at u. The first term is always positive and is the familiar Joule heating(also called Ohmic heating) effect. Notice how the presence of the magnetic field introduces the possibility of accelerating a plasma, in addition to the unavoidable heating In an efficient accelerator, we would try to maximize(jxB). u at the expense of i/o Origin of the magnetic field steady state and without magnetic materials)is Ampere's law een b and j(inthe The magnetic field can either be provided by external coils, or induced directly by currents circulating in the plasma. The general relationship betwe where Ho =4X10(in MKS units)is the permeability of vacuum In integral form .A=手Bd (17) which states that the circulation of - around a closed line equals the total current linked by the loop. When the current is constrained to circulate in metallic wires, the integral form can be used to provide simple formulae for the field due to various conductor arrangements. For example, inside a long solenoid carrying a current I the field B is nearly constant and we obtain(see sketch 16.522, Space P pessan Lecture 21 Prof. Manuel martinez Page 6 of 21
16.522, Space Propulsion Lecture 21 Prof. Manuel Martinez-Sanchez Page 6 of 21 Thus ( ) 2 j W = + j × B . u σ G JG G (15) The second term of this expression is simply the rate at which the Lorentz force j × B G JG does mechanical work on the plasma moving at u G . The first term is always positive and is the familiar Joule heating (also called Ohmic heating) effect. Notice how the presence of the magnetic field introduces the possibility of accelerating a plasma, in addition to the unavoidable heating. In an efficient accelerator, we would try to maximize ( ) j × B . u G JG G at the expense of 2 j σ . Origin of the magnetic field The magnetic field can either be provided by external coils, or induced directly by the currents circulating in the plasma. The general relationship between B JG and j G (in steady state and without magnetic materials) is Ampere’s law 0 B j = × ∇ µ JG G (16) where -7 µ π 0 = 4 ×10 (in MKS units) is the permeability of vacuum. In integral form 0 B j . dA = .dl µ ∫∫ ∫ JG GG G v (17) which states that the circulation of 0 B µ JG around a closed line equals the total current linked by the loop. When the current is constrained to circulate in metallic wires, the integral form can be used to provide simple formulae for the field due to various conductor arrangements. For example, inside a long solenoid carrying a current I, the field B JG is nearly constant, and we obtain (see sketch)
B B≈0 In=B where n is the number of turns Thus The magnetic field also has the essential property of being solenoidal, i.e V.B=0 (notice that, due to Ampere's law, j also obeys vj=0, which can be seen as a statement of charge conservation). In regions where no current is flowing we have VxB=0 as well, so that a magnetic potential can be defined by B=-Vy. Then, since VB=0, this potential obeys Laplace's equation (19) but notice that Ich potential exists in a current-carrying plasma. The vector B there must be found by simultaneous solution of Ampere s and ohms laws(with the additional constraint V B=0) Consider now a conductive plasma inside a solenoid, so that both an external B field (Bext)and an induced B field(Bindexist. The first is due to the coil currents, the second to those in the plasma itself. Suppose the plasma currents are due to the 16.522, Space P pessan Lecture 21 Prof. Manuel martinez Page 7 of 21
16.522, Space Propulsion Lecture 21 Prof. Manuel Martinez-Sanchez Page 7 of 21 0 B I n = l µ JG where n is the number of turns. Thus 0 n B= I l µ . The magnetic field also has the essential property of being solenoidal, i.e., ∇. B = 0 JG (18) (notice that, due to Ampère’s law, j G also obeys ∇. j = 0 G , which can be seen as a statement of charge conservation). In regions where no current is flowing we have ∇ × B = 0 JG as well, so that a magnetic potential can be defined by B=-∇ψ JG . Then, since ∇. B = 0, JG this potential obeys Laplace’s equation 2 ∇ ψ = 0 (19) but notice that no such potential exists in a current-carrying plasma. The vector B JG there must be found by simultaneous solution of Ampère’s and Ohm’s laws (with the additional constraint ∇. B = 0 JG ). Consider now a conductive plasma inside a solenoid, so that both an external B JG field ( ) Bext JG and an induced B JG field ( ) Bind JG exist. The first is due to the coil currents, the second to those in the plasma itself. Suppose the plasma currents are due to the
flow at uof the plasma in the total magnetic field B, while any external electrodes are short-circuited, so that E=0 in the laboratory frame. Then E=E+uxB=uxB In order of magnitude e=uB. Neglecting the Hall effect, then ,j-ouB The induced field obeys separately its own Amperes law V j, where is the es). Thus, in order of magnitude s wir ma current density this is because V x Bext =0 in the plasma(outside the coil where I is the characteristic distance for variation bd i e. the plasma size. Altogether, B lu B (20) B=Bo +Bind This indicates that the field created by the plasma currents becomes comparable to the external field when the so-called"magnetic Reynolds numbe en=μolou becomes of order unity. For a high power Argon MPD accelerator, a=1000mho/m,u-10,000m/sec,l=0.1m, Rn-106×102×103×104=1 and so, operation with self-induced magnetic field becomes possible. This simplifies considerably the design, since no heavy and power-consuming external coils are needed The order of magnitude also indicates however, that external field augmentation may be desirable under some conditions; the issue of self-field vs. applied field devices is not yet fully resolved A Simple plasma Accelerator Consider a rectangular channel with two conducting and two insulating walls, as shown 16.522, Space P pessan Lecture 21 Prof. Manuel martinez Page 8 of 21
16.522, Space Propulsion Lecture 21 Prof. Manuel Martinez-Sanchez Page 8 of 21 flow at u G of the plasma in the total magnetic field B JG , while any external electrodes are short-circuited, so that E=0 G in the laboratory frame. Then E' = E + u × B = u × B JJG GGJG GJG . In order of magnitude, E' u B . Neglecting the Hall effect, then, j u B ∼ σ . The induced field obeys separately its own Ampère’s law ind 0 B ∇ × =j µ JG G , where j G is the plasma current density; this is because ∇ × B = 0 ext JG in the plasma (outside the coil wires). Thus, in order of magnitude B j l ind 0 ∼ µ where l is the characteristic distance for variation Bind ,i.e., the plasma “size.” Altogether, B l u B ind 0 ∼ µ σ (20) B=B +B 0 ind JG JG JG , so ind 0 ind 0 B l u B +B ∼ µ σ This indicates that the field created by the plasma currents becomes comparable to the external field when the so-called “magnetic Reynolds number” R = l u em 0 µ σ (21) becomes of order unity. For a high power Argon MPD accelerator, σ ∼ 1000 mho/m , u 10, 000 m/sec, l 0.1 m , so -6 -1 3 4 R 10 × 10 × 10 × 10 = 1 em ∼ and so, operation with self-induced magnetic field becomes possible. This simplifies considerably the design, since no heavy and power-consuming external coils are needed. The order of magnitude also indicates, however, that external field augmentation may be desirable under some conditions; the issue of self-field vs. applied field devices is not yet fully resolved. A Simple Plasma Accelerator Consider a rectangular channel with two conducting and two insulating walls, as shown:
xB Cathode d Front View Side View(through insulating wall) a plasma is flowing in the channel at velocity u, and an external electric field is applied Ignoring for now the hall effect if this e field is larger in magnitude than uB (the induced Faraday field), a current j will flow. given by j=oE=o(E+ux B)in the direction of E he Lorentz force f=jx B is then in the forward direction as indicated and we have an accelerator E On the other hand if e<ub the current j flows in the direction opposite to E. Externally, positive X current flows into the(+)pole of our "battery"and could be used to recharge it; we have now an MHD generator, and the battery would probably be replaced by a passive load The Lorentz force now points backwards, so that the fluid has to be forced to flow by an external pressure gradient (like in a turbine ). Effect of the hall parameter In a moderate pressure plasma B can easily exceed unity. If the construction is the same as before, Ohm's law can be represented graphically as shown below: 16.522, Space P pessan Lecture 21 Prof. Manuel martinez Page 9 of 21
16.522, Space Propulsion Lecture 21 Prof. Manuel Martinez-Sanchez Page 9 of 21 A plasma is flowing in the channel at velocity u G , and an external electric field is applied. Ignoring for now the Hall effect, if this E G field is larger in magnitude than uB (the induced Faraday field), a current j G will flow, given by j = E' = E + u × B σ σ ( ) G JJG GGJG in the direction of E G . The Lorentz force f = j × B G G JG is then in the forward direction, as indicated, and we have an accelerator. On the other hand, if E<uB, the current j G flows in the direction opposite to E G . Externally, positive current flows into the (+) pole of our “battery” and could be used to recharge it; we have now an MHD generator, and the battery would probably be replaced by a passive load. The Lorentz force now points backwards, so that the fluid has to be forced to flow by an external pressure gradient (like in a turbine). Effect of the Hall Parameter In a moderate pressure plasma β can easily exceed unity. If the construction is the same as before, Ohm’s law can be represented graphically as shown below:
d xp dxβ 1B ⊙ x甘 We can see that the effect is to turn the current and the lorentz force counter clockwise by arc tan B. There is still a forward force(called the"blowing"force), but also now a transverse force, called the"pumping"force, because its main effect is to pump fluid towards the cathode wall, creating a transverse pressure gradient (low pressure at the anode). Basically the axial (or Hall) current does no useful work but it still contributes to the Joule dissipation j /6. Thus, we may want to turn the whole diagram by tan B clockwise and have j flow transversally only and f=jx B point But notice that this implies a forward component of the external field x阝 Hence we have to build the electrode wall in such a way that an axial voltage can be sustained For example, it can be made of independent metallic segments, connected transversally in pairs, and with nsulation between each pair, so that a voltage can also be applied between each segment and its downstream neighbour. Unless a lot of ingenuity is used, this UxBI segmented construction complicates the design and onnections greatly Self-field coaxial construction Leaving aside for the moment the question of how to provide the magnetic field the imple continuous electrode accelerator can be further simplified by wrapping it around"into an annulus thereby eliminating the insulating walls 16.522, Space P pessan Lecture 21 Prof. Manuel martinez Page 10 of 21
16.522, Space Propulsion Lecture 21 Prof. Manuel Martinez-Sanchez Page 10 of 21 We can see that the effect is to turn the current and the Lorentz force counter clockwise by arc tanβ . There is still a forward force (called the “blowing” force), but also now a transverse force, called the “pumping” force, because its main effect is to pump fluid towards the cathode wall, creating a transverse pressure gradient (low pressure at the anode). Basically, the axial (or Hall) current does no useful work, but it still contributes to the Joule dissipation 2 j σ . Thus, we may want to turn the whole diagram by tan-1 β clockwise and have j G flow transversally only and f = j × B G GJG point axially. But notice that this implies a forward component of the external field E G : Hence, we have to build the electrode wall in such a way that an axial voltage can be sustained. For example, it can be made of independent metallic segments, connected transversally in pairs, and with insulation between each pair, so that a voltage can also be applied between each segment and its downstream neighbour. Unless a lot of ingenuity is used, this segmented construction complicates the design and connections greatly. Self-field coaxial construction Leaving aside for the moment the question of how to provide the magnetic field, the simple continuous electrode accelerator can be further simplified by “wrapping it around” into an annulus, thereby eliminating the insulating walls: