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Hollow Cathodes 289 If the ions are generated by the accelerated electrons within about one local radius downstream of the double-layer flow back through the double layer, the ion current is z+r + z Ii Ie o ino(z)dz. (6.7-9) z Substituting Eq. (6.7-8) into Eq. (6.7-9) and integrating, the ion current through the sheath is I =I Qgas i 1 1 . i e vo (ro + z)(1+ ) Defining the “Langmuir ratio” from Eq. (6.7-7) as RL Ji M , 2Je m (6.7-10) (6.7-11) a stable double layer will form in the location where RL equals one. Using Eq. (6.7-10), the Langmuir ratio is RL = 2Qgas i M 1 1 . (6.7-12) vo m (ro + z)(1+ ) In the cathode plume, Eq. (6.7-12) indicates that the Langmuir ratio decreases monotonically with distance from the cathode. Alternatively, the axial location where the Langmuir condition equals 1 can be found to identify the location of the double layer. This is shown in Fig. 6-34 where the flow rate at which RL =1 is plotted versus distance from the cathode orifice. At flow rates of 10sccm, Fig. 6-34 suggests that a double layer will form over 1-cm downstream from the cathode, consistent with the data shown in Fig. 6-33. The general behavior of the double-layer location moving axially downstream with increasing gas flow has been reported [1] by both visual observations and by probe measurements. The cathode plume is much more complicated than this simple 1-D model suggests, and the condition of increasing electron-Mach numbers predicted under certain situations by the 1-D code may generate double layers or instabilities in the plasma not well described by the fluid codes. The assumed neutral gas expansion behavior and the associated electron current density in the plume were fortuitously picked to give potential and density profiles that came close to matching the experimental data. In reality, the neutral gas likely expands more rapidly than the 45-deg cone assumed in the 1-D model. The lowPDF Image | Fundamentals of Electric Propulsion: Ion and Hall Thrusters
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