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264 Chapter 6 cathode is, therefore, heated predominately by electron heating of the insert, with a comparable amount coming from orifice heating (shown in the next section). Similar analysis of Type B cathodes with larger orifices or lower flow rates, and also most Type C cathodes, indicates that ion heating will become the dominant heating mechanism due to the higher electron temperature and larger sheath potential drop at the insert. It is important to recognize that, as the pressure in the hollow cathode is increased, much of plasma heating comes from resistive heating of the current flowing through the partially ionized plasma. The higher the neutral gas background pressure, the greater the contribution of resistive heating. In cathodes with larger orifices that produce lower internal pressures, most of the heating of the insert plasma comes from the emitted electrons being accelerated across the cathode sheath potential. In lower pressure cathodes, the sheath potential is higher and the plasma resistivity is lower, resulting in less joule heating of the plasma but more ion bombardment heating of the insert surface. This is illustrated in Fig. 6-12, which shows the sheath potential and the ion and electron currents impacting the cathode as a function of the resistive joule heating of the plasma. The behavior shown in Fig. 6-12 can be understood by rearranging the equations in the power balance model above. Using Eqs. (6.4-13) and (6.4-14) in the power balance equation [Eq. (6.4-8)] and solving for the sheath potential gives RI2+IU++5T I+(2T + )I 2miexp( /T ) ei 2eVe eVsi me seV s= I I 1 2mi exp( /T ) . (6.4-22) e i me s eV The decrease in the sheath potential observed in Fig 6-12 as the joule heating ( RIe2 ) becomes more significant follows directly from Eq. (6.4-22), because the joule heating term enters with a negative sign. Equation (6.4-13) also shows that a decrease in the sheath potential allows for more of the electron flux to return to the emitter. Finally, if the heat loss, H (T ) , is fixed, Eq. (6.4-12) shows that the increased electron return flux (second term on the right-hand side) must be balanced by a reduced ion flux (first term on the right-hand side). This illustrates how the design and operating conditions of the hollow cathode (sizes, flow, and discharge current) determine which terms dominate in the cathode self-heating.PDF Image | Fundamentals of Electric Propulsion: Ion and Hall Thrusters
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