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104 Chapter 4 and are collected by the anode at the cusps, where they remove an energy per particle of e = (2kTe / e + ) , which is given in Eq. (4.2-9) and derived in Appendix C. Analytic models of the discharge chamber performance in ion thrusters have been described in the literature for many years [15–17]. The first comprehensive model of the discharge chamber performance using particle and energy balance equations in ring-cusp thrusters was developed by Brophy and Wilbur [18,19] in 1984. In Brophy’s model, volume-averaged particle and energy balance equations including primary electrons were used to derive expressions for the discharge loss as a function of the mass utilization efficiency in the thruster. Brophy’s model was extended by Goebel [20,21] to include electrostatic ion confinement, primary confinement and thermalization, the anode sheath [22], and hollow cathode effects. This model utilizes magnetic field parameters obtained from a magnetic field solver that accurately models the magnetic boundary. Since the model assumes a uniform plasma in the volume inside the magnetic confinement in the discharge chamber, it is sometimes called a 0-dimensional (0-D) model. The 0-D discharge chamber model to be described here [21] self-consistently calculates the neutral gas density, electron temperature, primary electron density, plasma density, plasma potential, discharge current, and ion fluxes to the boundaries of the discharge chamber. While the assumption of uniform plasma is not particularly accurate near the cathode plume, the majority of the plasma in the discharge chamber is relatively uniform, and the model predictions agree well with experimental results. The 0-D model solves for discharge loss as a function of the mass utilization efficiency, which is useful in plotting performance curves that best characterize the discharge chamber performance. The particle flows and potential distribution in the thruster used in the 0-D model are shown schematically in Fig. 4-9. Mono-energetic primary electrons with a current Ie are assumed to be emitted from the hollow cathode orifice into the discharge chamber, where they ionize the background gas to produce a uniform plasma. Electrons produced in the ionization process and primary electrons that have thermalized with the plasma electrons create a Maxwellian plasma electron population that also contributes to the ionization. Due to the relatively high magnetic field produced by the magnets near the wall, the electron Larmor radius is much smaller than the dimensions of the discharge chamber, and both primary and plasma electrons are considered to be reflected from the boundary region between the magnetic cusps. The primary and plasma electrons can be lost at the magnetic cusps because the magnetic field lines arePDF Image | Fundamentals of Electric Propulsion: Ion and Hall Thrusters
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