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262 Chapter 6 In low-pressure Type B and C cathodes, the sheath potentials are much greater than the 3.6 V calculated for the NSTAR discharge cathode. For example, in Fig. 6-9, the NSTAR solution for the electron temperature at the far right of the graph is in excess of 7 torr, while NEXIS and other large orifice cathodes are closer to the left side of the graph, between 1 and 2 torr. The sheath potential found by solving Eq. (6.4-16) for the NEXIS electron temperature is over 7 V, and so relatively few plasma electrons return to the emitter and do little heating. Most of the insert heating in lower pressure (on the order of 1 to 2 torr), lower internal plasma density cathodes is from ion bombardment of the insert surface due to the higher sheath voltage. The insert plasma density can now be found from Eq. (6.4-8). The ion current term is given by Ii = nonee ive V , (6.4-17) where no is the neutral density, < ive > is the ionization reaction rate coefficient, V is the volume, and ne is the average plasma density over the insert volume. Remembering that the plasma density in the random electron flux equation is evaluated at the plasma edge, Eq. (6.4-8) can be solved using the above equations to produce an expression for the average plasma density: RI2 5T I e 2eV s e T + , (6.4-18) e A e s e V + n o e v e V ( U + s ) where fn is the edge-to-average plasma density ratio. Since the electrons in the insert plasma are Maxwellian, the value of fn can be estimated from the potential difference between the center and the edge: fn = ne e ( axis s ) TeV , (6.4-19) ne where the potential on axis axis must come from measurements or two- dimensional (2-D) codes. The plasma density calculated from Eq. (6.4-18) for the NSTAR discharge cathode at a constant xenon gas flow of 3.7 sccm, using the electron temperature from the radial diffusion model (Fig. 6-9), the sheath potential from the power balance model (Fig. 6-10), and a measured on-axis plasma potential of about 8.5 V [6], is shown in Fig. 6-11. Good agreement with the plasma density measurements made by a miniature scanning probe in this cathode [28] is obtained, and a nearly linear dependence on discharge ne = kT 1/2 f n T e e 2 mPDF Image | Fundamentals of Electric Propulsion: Ion and Hall Thrusters
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