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210 Chapter 5 This equation describes the required potential difference between the beam potential and the minimum potential in the beamlet to produce a specified amount of electron backstreaming current relative to the beam current. Note that this equation is independent of the grid geometry because it deals solely with the potential difference between a given value of Vm (independent of how it is produced) and the beam-plasma potential. The required potential difference (Vbp Vm ) between the beam plasma and the minimum voltage in the grids to produce a given ratio of backstreaming current to beam current is plotted from Eq. (5.4-4) in Fig. 5-10 for several values of the beam-plasma electron temperature in a thruster plume with a net accelerating voltage of Vp Vbp = 1500 V . For an electron temperature of 2 eV in the beam, which is consistent with values found in NSTAR thrusters plumes [27], a potential difference between the minimum potential in the beamlet and the beam plasma of only 12.5 V is required to reduce the backstreaming current to 1% of the beam current. The actual minimum potential in the beamlet is determined by the grid geometry, the applied grid potentials, and the beam’s space charge. The minimum potential in the two-grid arrangement shown in Fig. 5-5 was first found without considering space charge effects by an analytic solution to LaPlaces’ equation by Spangenberg [28] for thin grids in vacuum tubes. Spangenberg’s expression was simplified by Williams [26] and Kaufman [1] for most ion thruster grid configurations to 0 −5 −10 −15 −20 −25 −30 −35 −40 e 1% Te = 2 T =3 Te = 4 0.000 0.020 Ratio of Backstreaming Electrons to Forward Ion Currents 0.040 0.060 0.080 0.100 Fig. 5-10. Potential difference between the beam plasma and the beamlet potential minimum required to achieve a given electron backstreaming current-to-forward ion current ratio for several beam electron temperatures. Potnetial Difference (V)PDF Image | Fundamentals of Electric Propulsion: Ion and Hall Thrusters
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