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Ion Thruster Accelerator Grids 211 V*=V +da(Vp Va) 1 2tatan 1 da e t d, (5.4-5) ma 2 d 2t aa e aa where Vm* indicates the minimum potential with the ion space charge neglected, Va is the applied accel grid potential, the grid dimensional terms are defined in Fig. 5-5, and e is given by Eq. (5.2-3). Equation (5.4-5) provides the dependence on the geometry of the grids, but is only useful if the beam space charge is negligible (very low current density beamlets). The reduction in the magnitude of the minimum beam potential due to the presence of the ion space charge in the beamlet can be estimated [26] using the integral form of Gauss’s law: E dA = 1 dV , (5.4-6) SoV where E is the electric field, dA is the differential surface area element, o is the permittivity of free space, and is the ion charge density within the Gaussian surface which has a surface area S and encloses volume V. This equation is solved first in the beamlet and then in the charge-free space between the beamlet and the accel aperture inside diameter. Then, adding the two potentials together gives the total potential between the grid and the beamlet centerline. Assume that the beamlet has a radius db / 2 inside the accel grid aperture with a radius of da / 2 . Integration of the left-hand side of Eq. (5.4-6) over a cylindrical “Gaussian pillbox” aligned with the beamlet axis yields 2 r E dA= aErrd dz=Er 2 rz, (5.4-7) 00 S where it has been assumed that Er is constant in the axial direction over a distance z. If it is also assumed that the ion charge density is uniform in the volume of the pillbox, the right-hand side of Eq. (5.4-6) can also be integrated to obtain 1 dV= 1 rdrd dz= r2z. (5.4-8) oVoVoPDF Image | Fundamentals of Electric Propulsion: Ion and Hall Thrusters
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