and, from (9),increasing the thrust density requires a reduction of the gap distance d. As noted before, this route is limited by eventual arcing or even by mechanica shorting due to grid warping or imperfections. For thruster diameters of, say, 10-50 cm,gap distances have been kept above 0. 5-1 mm The only other control, at this level of analysis, is offered by increasing the ion can be kept small, higher thrust(Equation 9). In addition to increasing thrust ided d molecular mass, m This allows increased voltages Va(Equation 10), and, prov density, higher molecular mass also reduces the importance of a given ion production cost Ao(See lecture 3), and hence increases the thruster efficiency The effect of ion deceleration past the accelerator grid (either through the use of a decel"grid or by relative elevation of the neutral incorporated in this 1-D model. For the usual geometries, the screen-accelerator gap still controls the ion current(Equation 6 with Va replaced by Vi, and d by da). This is because the mean ion velocity is high(and hence the mean ion density is low ) in the second gap, between the accelerator and the real or virtual decelerator so that no electrostatic choking occurs there. This is schematically indicated in Figure 4 by a break in the slope of the potential at the decelerator. More specifically it can be shown that Equation(6) still controls the current provided that (1R)(4+2R1);R= (for equal gaps, this is satisfied for all r between 0 and 0. 75, for instance at higher R, the second gap limits current). Accepting, then, Equation(6), the thrust is again given by f= mc, where m has not changed, but c is proportional to v,12.Hence we obtain instead of (9) F 8 18。(M a 9 Eo (12) The last form shows that for a given specific impulse(hence given VN), reducing R=VN/Vr increases thrust. It does so by extracting a higher ion current through the flux-limiting first gap Returning to Equation(6), if we imagine a beam with diameter D, we would predict a total beam current of 丌4 e 13) d where P is the so-called "perveance"of the extraction system. Equation(13)shows that this perveance should scale as the dimensionless ratio D, so that for example he same current can be extracted through two systems, one of which is twice the grid sp Itio 16.522, Space P rtinez- sanchez Lecture 13-14 Prof. Manuel martinez Page 6 of 25
16.522, Space Propulsion Lecture 13-14 Prof. Manuel Martinez-Sanchez Page 6 of 25 and, from (9), increasing the thrust density requires a reduction of the gap distance d. As noted before, this route is limited by eventual arcing or even by mechanical shorting due to grid warping or imperfections. For thruster diameters of, say, 10-50 cm., gap distances have been kept above 0.5-1 mm. The only other control, at this level of analysis, is offered by increasing the ion molecular mass, mi. This allows increased voltages Va (Equation 10), and, provided d can be kept small, higher thrust (Equation 9). In addition to increasing thrust density, higher molecular mass also reduces the importance of a given ion production cost ∆φ (See lecture 3), and hence increases the thruster efficiency. The effect of ion deceleration past the accelerator grid (either through the use of a “decel” grid, or by relative elevation of the neutralizer potential) can be easily incorporated in this 1-D model. For the usual geometries, the screen-accelerator gap still controls the ion current (Equation 6 with Va replaced by VT, and d by da). This is because the mean ion velocity is high (and hence the mean ion density is low) in the second gap, between the accelerator and the real or virtual decelerator, so that no electrostatic choking occurs there. This is schematically indicated in Figure 4 by a break in the slope of the potential at the decelerator. More specifically, it can be shown that Equation (6) still controls the current provided that ( ) ( ) 1 2 d 12 12 N a T d V > 1 - R 1+ 2R ; R = d V (11) (for equal gaps, this is satisfied for all R between 0 and 0.75, for instance; at higher R, the second gap limits current). Accepting, then, Equation (6), the thrust is again given by F m c A A = i , where m A i has not changed, but c is proportional to 1 2 VN . Hence we obtain instead of (9) 2 2 32 12 TN T N 1 2 -3 2 2 a a a F8 8 8 VV V V = RR A9 9 d 9 d d 00 0 ⎛⎞ ⎛⎞ = = ⎜⎟ ⎜⎟ ⎝⎠ ⎝⎠ εε ε (12) The last form shows that for a given specific impulse (hence given VN), reducing R=VN/VT increases thrust. It does so by extracting a higher ion current through the flux-limiting first gap. Returning to Equation (6), if we imagine a beam with diameter D, we would predict a total beam current of 1 2 2 32 32 0 TT i 42 e D Ι V = PV 49 m d = ⎛ ⎞ ⎛ ⎞ ⎜ ⎟ ⎜ ⎟ ⎝ ⎠ ⎝ ⎠ π ε (13) where P is the so-called “perveance” of the extraction system. Equation (13) shows that this perveance should scale as the dimensionless ratio 2 2 D d , so that, for example the same current can be extracted through two systems, one of which is twice the size of the other, provided diameter and grid spacing are kept in the same ratio
While the one-dimensional model is important in identifying many of the governing effects and parameters, its quantitative predictive value is limited Three-dimensional effects, such as those of the ratio of extractor to accelerator diameter the finite grid thicknesses, the potential variation across the beam etc(see Fig. 2)are all left out of account. So are also the effects of varying the properties of the upstream plasma such as its sheath thickness, which will vary depending on the intensity of the ionization discharge, for example. Also, for small values of R=VN/Vt, the beam potential(averaged in its cross-section) cannot be expected to approach the deep negative value of the accelerator, particularly for the very flattened hole geometry prevalent when d/D is also small. Thus, the perveance per hole can be expected to be of the functional form dD。ta (14) where the subscripts(s)and (a) identify the screen and accelerator respectively, t is a grid thickness, and vo is the discharge voltage, which in a bombardment ionizer controls the state of the plasma. These dependencies were examined for a 2-grid extractor in an Argon-fueled bombardment thruster in Ref. 7. Some of the salient conclusions of that study will be summarized here (1)Varying the screen hole diameter Ds while keeping constant all the ratios d/Ds, Da/Ds, etc. )has only a minor effect, down to D 0.5mm if the alignment can be maintained. This confirms the dependence upon the ratio (2) The screen thicknesses are also relatively unimportant in the range studied (t/D≈0.2-0.4) (3)Reducing R=VN/V always reduces the perveance, although the effect tends to disappear at large ratios of spacing to diameter(d/Ds), where the effect of the negative accelerator grid has a better chance to be felt by the ions. The value of d/Ds at which r becomes insensitive is greater for the smaller R values (4) For design purposes, when VN and not Vr is prescribed, a modified perveance va (called the"current parameter"in Ref. 7)is more useful. As Equation (13)shows, one would expect this parameter to scale as r-3/2, favoring low values of R(strong accel-decel design). This trend is observed at low R, but due to the other effects mentioned, it reverses for R near unity as shown in Fig. 5. This is especially noticeable at small gap/ diameter ratios when a point of maximum extraction develops at R07-0.8, which can give currents as high as those with R.0. 2. However, as Fig. 5 also shows the low -R portion of the operating curves will give currents which are independent of the gap/diameter ratio(this is in clear opposition to the 1-D prediction of Equation 13). Thus, the current, in this region, is independent of both d and Ds. This opens up a convenient design avenue using low R values: Fix the smallest distance d compatible with good dimensional control, then reduce the diameter ds to the smallest practicable size(perhaps 0.5 mm). This will 16.522, Space Propulsion Lecture 13-14 Prof. Manuel martinez-Sanchez Page 7 of 25
16.522, Space Propulsion Lecture 13-14 Prof. Manuel Martinez-Sanchez Page 7 of 25 While the one-dimensional model is important in identifying many of the governing effects and parameters, its quantitative predictive value is limited. Three-dimensional effects, such as those of the ratio of extractor to accelerator diameter, the finite grid thicknesses, the potential variation across the beam etc. (see Fig. 2) are all left out of account. So are also the effects of varying the properties of the upstream plasma, such as its sheath thickness, which will vary depending on the intensity of the ionization discharge, for example. Also, for small values of R=VN/VT, the beam potential (averaged in its cross-section) cannot be expected to approach the deep negative value of the accelerator, particularly for the very flattened hole geometry prevalent when d/D is also small. Thus, the perveance per hole can be expected to be of the functional form aa s D ssss T d Dt t V P = p , , , ,R, DDDD V ⎛ ⎞ ⎜ ⎟ ⎝ ⎠ (14) where the subscripts (s) and (a) identify the screen and accelerator respectively, t is a grid thickness, and VD is the discharge voltage, which in a bombardment ionizer controls the state of the plasma. These dependencies were examined for a 2-grid extractor in an Argon-fueled bombardment thruster in Ref. 7. Some of the salient conclusions of that study will be summarized here: (1) Varying the screen hole diameter Ds while keeping constant all the ratios (d/Ds, Da/Ds, etc.) has only a minor effect, down to D 0.5 s ≈ mm if the alignment can be maintained. This confirms the dependence upon the ratio d/Ds. (2) The screen thicknesses are also relatively unimportant in the range studied ( s t/D 0.2 - 0.4 ≈ ). (3) Reducing R=VN/VT always reduces the perveance, although the effect tends to disappear at large ratios of spacing to diameter (d/ Ds), where the effect of the negative accelerator grid has a better chance to be felt by the ions. The value of d/ Ds at which R becomes insensitive is greater for the smaller R values. (4) For design purposes, when VN and not VT is prescribed, a modified perveance 3 2 N I V ⎛ ⎞ ⎜ ⎟ ⎝ ⎠ (called the “current parameter” in Ref. 7) is more useful. As Equation (13) shows, one would expect this parameter to scale as R-3/2, favoring low values of R (strong accel-decel design). This trend is observed at low R, but, due to the other effects mentioned, it reverses for R near unity, as shown in Fig. 5. This is especially noticeable at small gap/diameter ratios, when a point of maximum extraction develops at R~0.7-0.8, which can give currents as high as those with R~0.2. However, as Fig. 5 also shows, the low – R portion of the operating curves will give currents which are independent of the gap/diameter ratio (this is in clear opposition to the 1-D prediction of Equation 13). Thus, the current, in this region, is independent of both d and Ds. This opens up a convenient design avenue using low R values: Fix the smallest distance d compatible with good dimensional control, then reduce the diameter Ds to the smallest practicable size (perhaps 0.5 mm). This will
allow more holes per unit area(if the hole spacing varies in proportion to their size), hence more current per unit grid area. Due to this circumstance Ref. 7 recommends low R designs. (5) The perveance generally increases as Da/Ds increases, with the exception of cases with R near unity when an intermediate D,/D=0.8 is optimum (6)Increasing Vo/VT, which increases the plasma density appears to flatten th contour of the hole sheath( 8), which reduces the focusing of the beam. this results in direct impingement on the screen, and, in turn forces a reduction of the beam current Some appreciation for the degree to which Child-Langmuir's law departs from the observed current extraction capacity of real devices can be obtained from the data for the 30 cm. J-series thruster, as reported for example in Ref.( 9). In this case,we have d=0.5 mm ta=ts=0. 38 mm, Ds=1.9 mm, Da=1. 14 mm, and a total of 14860 holes. We will refer to data in Xe, for VNET/V=0.7 and Vo=31.2 Volts. VBeam=1200 V. Table Ill of Ref (9)then gives a beam current ]8=4.06 A. The correlation given in the same reference for various propellants is aM+25%6 (15) here a is a double-ion correction factor, given as 0.934 for this case, and M is the molecular mass in a. m u. The power of 2.2 instead of 1.5 for the effect of extractio oltage is to be noticed. This correlation yields for our case Ib=5.4 A, on the outer boundary of the error band For these data, if we apply the Child-Langmuir law(Equation 13)to each hole (diameter Ds), and use directly the spacing d=0.5 mm, we obtain a hole current of 3.83 mA, or, in total IB=57.1 A, i.e. 14 times too high. an approximate 3-D correction(Ref. s 10a, b)is to replace d by (d+t 2+D2/4 in Child-Langmuir's equation this gives now IB=8.4 A, still twice the experimental value. It is of interest to see how well the data of Rovang and Wilbur(ref. 7) can be extrapolated to the J thruster. We first use the data in Fig. 6a of Ref (7), which are for Ds=2mm ,R=0. 8 Da/Ds=0.66(lowest value measured), and VD/V=0. 1. Corrections for the actual Da/Ds=0.6 and Vo/VT=0.018 can be approximated from Fig. ' s 5 and (6a) of the same reference. The effects of R=0.7 instead of 0.8, as well as of the slightly different Ds should be small, according to Ref. 7. We obtain in this manner IB=5.2 A which is indeed as accurate as the correlation of Equation(15) Additional data on grid perveance are shown and assessed Ref.(10c)in the context of ion engine scaling To complete this discussion two limiting conditions should be mentioned he (a)Direct ion impingement on screen: At low beam current, the screen collects a very small stray current, which is due to charge-exchange ion-neutral collisions in the accelerating gap: after one such collision, the newly formed low speed ion is easily accelerated into the screen. The screen current takes however, a strong upwards swing when the beam current increases beyond some well defined limit. This is due to interception of the beam edges and 16.522, Space Propulsion Lecture 13-14 Prof. Manuel martinez-Sanchez Page 8 of 25
16.522, Space Propulsion Lecture 13-14 Prof. Manuel Martinez-Sanchez Page 8 of 25 allow more holes per unit area (if the hole spacing varies in proportion to their size), hence more current per unit grid area. Due to this circumstance, Ref. 7 recommends low R designs. (5) The perveance generally increases as Da/Ds increases, with the exception of cases with R near unity, when an intermediate D /D 0.8 a s ≈ is optimum. (6) Increasing VD/VT, which increases the plasma density, appears to flatten the contour of the hole sheath(8), which reduces the focusing of the beam. This results in direct impingement on the screen, and, in turn, forces a reduction of the beam current. Some appreciation for the degree to which Child-Langmuir’s law departs from the observed current extraction capacity of real devices can be obtained from the data for the 30 cm. J-series thruster, as reported for example in Ref. (9). In this case, we have d=0.5 mm, ta=ts=0.38 mm, Ds=1.9 mm, Da=1.14 mm, and a total of 14860 holes. We will refer to data in Xe, for VNET/VT=0.7 and VD=31.2 Volts. VBeam=1200 v. Table III of Ref. (9) then gives a beam current JB=4.06 A. The correlation given in the same reference for various propellants is ( )2.2 T B 17.5 V 1000 J = + -25% α M (15) where α is a double-ion correction factor, given as 0.934 for this case, and M is the molecular mass in a.m.u.. The power of 2.2 instead of 1.5 for the effect of extraction voltage is to be noticed. This correlation yields for our case IB=5.4 A, on the outer boundary of the error band. For these data, if we apply the Child-Langmuir law (Equation 13) to each hole (diameter Ds), and use directly the spacing d=0.5 mm, we obtain a hole current of 3.83 mA, or, in total IB=57.1 A, i.e., 14 times too high. An approximate 3-D correction (Ref.’s 10a, b) is to replace d2 by 2 2 s s (d + t ) + D /4 in Child-Langmuir’s equation. This gives now IB=8.4 A, still twice the experimental value. It is of interest to see how well the data of Rovang and Wilbur (Ref. 7) can be extrapolated to the Jthruster. We first use the data in Fig. 6a of Ref. (7), which are for Ds=2mm., R=0.8 Da/Ds=0.66 (lowest value measured), and VD/VT=0.1. Corrections for the actual Da/Ds=0.6 and VD/VT=0.018 can be approximated from Fig.’s 5 and (6a) of the same reference. The effects of R=0.7 instead of 0.8, as well as of the slightly different Ds, should be small, according to Ref. 7. We obtain in this manner IB=5.2 A., which is indeed as accurate as the correlation of Equation (15). Additional data on grid perveance are shown and assessed Ref. (10c) in the context of ion engine scaling. To complete this discussion, two limiting conditions should be mentioned here: (a) Direct ion impingement on screen: At low beam current, the screen collects a very small stray current, which is due to charge-exchange ion-neutral collisions in the accelerating gap: after one such collision, the newly formed low speed ion is easily accelerated into the screen. The screen current takes, however, a strong upwards swing when the beam current increases beyond some well defined limit. This is due to interception of the beam edges, and
since the high energy ions are very effective sputtering agents, results in a very destructive mode of operation. All the perveance values reported in Ref. (7), for instance, are impingement-limited, i. e. correspond to the highest current prior to onset of direct impingement (b)Electron back-streaming: For R values near unity the barrier offered by the accelerator negative potential to the neutralizer electrons becomes weak, and beyond some threshold value of R, electrons return up the accelerator potential to the chamber. This results in screen damage, space charg distortion, and shorting of the neutralizer supply Kaufman (oa) gave the theoretical estimate 0.2 (16) D which was confirmed experimentally in Ref. 7, except that it was found to be a somewhat conservative estimate 4 Ion Production 4.1 Physical Process in Electron Bombardment Ionization Chambers In an electron bombardment ionizer, the neutral gas is partially ionized by an auxiliary dC discharge between conveniently located electrodes. Of these, the anode is the same anode which receives the electrons from the ionization process(see Fig 1). The primary electrons responsible for the ionization of the neutral gas are generated at a separate cathode which can be a simple heated tungsten filament, or for longer endurance, a hollow cathode. The cathode-anode potential difference vo is selected in the vicinity of the peak in the ionization cross-section of the propellant gas, which occurs roughly between three and four times the ionization energy i e. around 30-50 Volts for most gases). The structure of the potential distribution in the sheath near the cathode, and the body of the plasma is nearly equipotential, at a the electron saturation level, and so an electron-retarding voltage drop develops Ionization is due both to the nearly mono-energetic primary electrons(with energies of the order of evD)and to the thermalized secondary electrons themselves. These have typically temperatures(Tm)of a few ev, so that only the high energy tail of the Maxwellian energy distribution is above the ionization energy and can contribute to the process but their number density greatly exceeds that of the primaries and both contributions are, in fact, of the same order. It is therefore desirable to maximize the residence time of both types of electrons in the chamber before they are eventually evacuated by the anode this is achieved by means of a suitable distribution of confining magnetic fields. Fig 's 6(a),(b )and (c)show three types of These will be discussed in more detail later, but we note here that magnetic e nce magnetic configurations, of which only the last two are today of practical importa strengths can vary from about 10 to 1000 Guass, depending on type and location 16.522, Space P pessan Lecture 13-14 Prof. Manuel martinez Page 9 of 25
16.522, Space Propulsion Lecture 13-14 Prof. Manuel Martinez-Sanchez Page 9 of 25 since the high energy ions are very effective sputtering agents, results in a very destructive mode of operation. All the perveance values reported in Ref. (7), for instance, are impingement-limited, i.e., correspond to the highest current prior to onset of direct impingement. (b) Electron back-streaming: For R values near unity, the barrier offered by the accelerator negative potential to the neutralizer electrons becomes weak, and beyond some threshold value of R, electrons return up the accelerator potential to the chamber. This results in screen damage, space charge distortion, and shorting of the neutralizer supply. Kaufman(10a) gave the theoretical estimate max a a a 0.2 R =1- le t exp D D ⎛⎞ ⎛⎞ ⎜⎟ ⎜⎟ ⎝⎠ ⎝⎠ (16) which was confirmed experimentally in Ref. 7, except that it was found to be a somewhat conservative estimate. 4 Ion Production 4.1 Physical Process in Electron Bombardment Ionization Chambers In an electron bombardment ionizer, the neutral gas is partially ionized by an auxiliary DC discharge between conveniently located electrodes. Of these, the anode is the same anode which receives the electrons from the ionization process (see Fig. 1). The primary electrons responsible for the ionization of the neutral gas are generated at a separate cathode, which can be a simple heated tungsten filament, or for longer endurance, a hollow cathode. The cathode-anode potential difference VD is selected in the vicinity of the peak in the ionization cross-section of the propellant gas, which occurs roughly between three and four times the ionization energy (i.e., around 30-50 Volts for most gases). The structure of the potential distribution in the discharge is very unsymmetrical: most of the potential difference VD occurs in a thin sheath near the cathode, and the body of the plasma is nearly equipotential, at a level slightly above that of the anode (typically the anode current density is below the electron saturation level, and so an electron-retarding voltage drop develops). Ionization is due both to the nearly mono-energetic primary electrons (with energies of the order of eVD) and to the thermalized secondary electrons themselves. These have typically temperatures (Tm) of a few eV, so that only the high energy tail of the Maxwellian energy distribution is above the ionization energy and can contribute to the process, but their number density greatly exceeds that of the primaries, and both contributions are, in fact, of the same order. It is therefore desirable to maximize the residence time of both types of electrons in the chamber before they are eventually evacuated by the anode. This is achieved by means of a suitable distribution of confining magnetic fields. Fig.’s 6 (a), (b) and (c) show three types of magnetic configurations, of which only the last two are today of practical importance. These will be discussed in more detail later, but we note here that magnetic field strengths can vary from about 10 to 1000 Guass, depending on type and location