Analytical Model of Electron Backstreaming for Ion Thrusters

Analytical equations historically used to predict the onset of electron backstreaming in ion thrusters tend to underestimate significantly the accel grid voltage required to block electron backflow from the beam plasma because they neglect detailed beamlet focusing and space charge effects inside the grid apertures. We present corrected analytical equations that provide a good estimate of the minimum voltage that must be applied to the accel grid to ensure electron backstreaming is negligible. The equations include terms that account for the effects of voltage penetration induced by screen grids, finite thickness accel grids, and positive space charge associated with ions within accel grid apertures. The backstreaming limit is shown to vary significantly with beamlet current because of ion focusing and associated positive space-charge effects in the accel grid aperture. Calculations of the onset of backstreaming for ion optics designs intended for high specific impulse operation are performed using the analytical equations, and comparisons are made to numerical simulations and experimental results. The analytic and numerical methods give reasonably good agreement with the data, suggesting that the appropriate physics has been included. INTRODUCTION To prevent the flow of electrons from the beam plasma through the grids and into the discharge chamber of an ion thruster, it is necessary to apply a sufficiently negative voltage to the accel grid. This eliminates both erroneous ion beam current readings and unwanted beam-supply power losses into the discharge chamber. Enlargement of the accel grid apertures with time due to charge-exchange-induced ion sputtering of the accel grid aperture walls, and the subsequent onset of electron backstreaming at a given accel grid voltage, is one of the primary life limiting mechanism in ion thrusters. In the past, the magnitude of the required anode voltage has been estimated using a simple equation developed by Kaufman. However, Kaufman’s equation has been found to underestimate significantly the required accel grid voltage at higher beam current densities and/or higher specific impulses that are presently of interest for deep space propulsion. An improved model of the accel grid voltage required to prevent backstreaming (called the backstreaming limit) with greater accuracy is desired to enable rapid predictions of this characteristic voltage especially for analysis of grid systems under development where numerical code runs can be too time consuming. If the grid erosion rate is known from short term wear tests or simple sputtering models, an accurate backstreaming model can also facilitate thruster life predictions. It is also noted that accurate predictions of the backstreaming limit will enable operation at lower accel voltages, where charge exchange ion kinetic energies at accel grid impact, and grid wear rates will be lower. Hence grid lifetimes will be greater. While recently developed numerical models can also be used to predict required voltages with good accuracy, these models are sometimes unavailable or inappropriate for a quick evaluation. As an alternative, we have developed an analytical model for performing rapid predictions of electron backstreaming limits. Our analytical model contains both the effects of the applied potentials and the space charge of the ions as they pass through the accel grid aperture in a simple expression. APPROACH Electrons will migrate upstream into the discharge chamber along a path of lowest potential difference from the beam plasma when their upstream kinetic energy exceeds the minimum adverse potential energy decrement they must pass through. For the typical potential profile shown in Fig. 1, the electrons that can backstream would be those with an energy greater than e(Vbp Vsp). The backstreaming current of Maxwellian electrons at temperature Te (in eV) that would pass from the beam plasma through the saddle point is given by