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Hall Thrusters 343 v = Vb . Vd Inserting Eqs. (7.3-9) through (7.3-11) into Eq. (7.3-8) gives T = 2 M Id b2 v c o. e m ̇ a (7.3-11) (7.3-12) Equation (7.3-12) shows that the Hall thruster efficiency is proportional to the ion mass and the discharge current, because these terms dominate the thrust production, and is inversely proportional to the anode mass flow, which dominates the mass utilization efficiency. This equation can be further simplified by realizing that Me I d b = m ̇ i , and that the total mass utilization efficiency can be expressed as m=m ̇i= m ̇i . m ̇ p m ̇ a + m ̇ c The total efficiency then becomes T = 2 b v m o. ( 7 . 3 - 1 3 ) (7.3-14) (7.3-15) This expression contains the usual gamma-squared term associated with beam divergence and multiply charged ion content and also the mass utilization and electrical utilization efficiencies. However, this expression also includes the efficiencies associated with generating beam ions and imparting the discharge voltage to the beam voltage. This shows directly that Hall thruster designs that maximize beam current production and beam energy and that minimize the cathode flow produce the maximum efficiency, provided that the beam divergence and double-ion content are not adversely affected. Expressions like Eq. (7-3-15) appear in the Hall thruster literature [4,7] because they are useful in illustrating how the efficiency depends on the degree to which the thruster converts power supply inputs (such as discharge current and voltage) into the beam current and beam voltage that impart thrust. Understanding each efficiency term is critical to fully optimizing the Hall thruster performance. The efficiency of a Hall thruster is sometimes expressed in terms of the anode efficiency:PDF Image | Fundamentals of Electric Propulsion: Ion and Hall Thrusters
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