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96 Chapter 4 1 8kTe 1/2 I a = 4 m e n e A a e x p –e /kT e , ( 4 . 2 - 1 1 ) where Aa is the electron loss area and m is the electron mass. Since ambipolar ion and electron flow to the wall was assumed, equate Eqs. (4.2-1) and (4.2-11) and use Te M for the ion acoustic velocity to give the plasma potential relative to the wall: = kTe ln Aa 2M . (4.2-12) e A m Equation (4.2-12) is normally called the floating potential and applies in this case because there are no applied potentials in our ideal thruster to draw a net current. The electron temperature can be found by equating the ion production and loss rates, Eqs. (4.2-1) and (4.2-3), which gives kTe M = 2noV . (4.2-13) ive A The reaction rate coefficient in the denominator depends on the electron temperature, and so this equation can be solved for Te if the discharge chamber volume, neutral pressure, and ion loss area are known. The discharge loss is defined as the power into the plasma divided by the beam current out of the thruster, which is a figure of merit for the efficiency of the plasma-generation mechanism. The discharge loss for this idealized thruster, using Eq. (4.2-2) for the beam current, is then given by P 2n vV *vej d=in= o ie U++ Ib va A Tg U* ive +Tge 2.5kTe+2kTeln A m . 1 As evident in Eq. (4.2-14), the grid transparency (Tg) directly affects the discharge loss, and the input power is distributed between the first term related (4.2-14) Aa 2MPDF Image | Fundamentals of Electric Propulsion: Ion and Hall Thrusters
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