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Ion Thruster Accelerator Grids 197 with respect to the perveance that was specified by the Child–Langmuir equation in which the sheath was considered essentially planar. The problem is that the sheath shape in the screen aperture is not planar, as seen in Fig. 5-4, and the exact shape and subsequent ion trajectories have to be solved by 2-D axi-symmetric codes. However, a modified sheath thickness can be used in the Child–Langmuir equation to approximately account for this effect, which is written as Jmax=4 o 2e VT3/2, (5.2-2) 9 M 2e where VT is the total voltage across the sheath between the two grids and the sheath thickness, , e is given by e (g s)2 ds2 = +t +4. (5.2-3) The grid dimensions in Eq. (5.2-3) are defined in Fig. 5-5. As illustrated in the figure, the sheath is allowed to expand essentially spherically through the screen grid aperture. The sheath thickness e accounts for this non-planar condition and has been found to be useful in predicting the space-charge- limited current in ion thruster grid configurations [1,7]. Note that the value of g is the “hot grid gap” that occurs once the grids have expanded into their final shape during operation at a given beam current and voltage. For xenon ions, Discharge Plasma Sheat (Vp ) ds Vm ts le Beam Plasma Potential Surface (Vbp ) z r Accel Grid (Va ) h lg ld da Fig. 5-5. Non-planar sheath model approximation for a two-grid system.PDF Image | Fundamentals of Electric Propulsion: Ion and Hall Thrusters
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