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Hollow Cathodes 273 n e = I e2 R 52 I e ke ( T e T e i n ) . ( 6 . 5 - 4 ) noe ve U+ r2 An evaluation of the terms in Eq. (6.5-4) for the orifice region uses the same techniques previously described in Section 6.4 for the insert plasma region. The resistance R is given by Eq. (6.4-9), where the conduction length is now simply the orifice plasma length. The input electron temperature, Tein , is the electron temperature in the insert plasma that comes from the diffusion model used in Section 6.4 or from experimental measurements. The detailed measurements of the plasma density and temperature in the orifice of the NSTAR discharge cathode [31] will be used as a first example to compare with the model predictions. The NSTAR discharge cathode has an orifice diameter of 0.1 cm, and the case of the full-power TH15 operating point with 13 A of discharge current at a xenon gas flow rate of 3.7 sccm will be used. The pressure measured inside the insert region for this case is about 7.8 torr [31]. Assuming simple Poiseuille flow (see Appendix B), the pressure in the orifice is estimated to fall to less than 3 torr by the end of the 0.75-mm- long cylindrical section of the orifice. Assuming a gas temperature of about 2000 K in the orifice, the solution for the electron temperature in the diffusion equation [Eq. (6.5-2)] versus pressure in the orifice is shown in Fig. 6-17. The electron temperature predicted by this model varies by less than 1 eV along the orifice length, and the average in the channel is about 2.3 eV. This value is close to the experimentally measured values of 2.2 to 2.3 eV found in this region [28]. Using this electron temperature, the density in the orifice is calculated from Eq. (6.5-2) and plotted in Fig. 6-18 versus the discharge current for the NSTAR cathode. The agreement with the experimental data [31] taken for two discharge currents at the nominal 3.7-sccm cathode flow rate is also very good. The resistance calculated from Eq.(6.4-9) for the cylindrical orifice length is 0.31 ohms, which, at 13 A, produces a voltage drop in the orifice of about 4 V. This is the same magnitude as the voltage change observed in the experimental data, which illustrates that the potential drop in the hollow cathode orifice is resistive due to the very collisional plasma that exists there in these xenon hollow cathodes. Detailed 2-D calculations, described below, indicate that roughly half of the power deposited in this region (P = 4 V * 13 A) goes to the orifice wall, and the remainder is convected into the discharge chamber by the plasma.PDF Image | Fundamentals of Electric Propulsion: Ion and Hall Thrusters
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