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 2516.522, 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