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Thruster Principles 23 T = 1.65 Ib Vb [mN], (2.3-9) where Ib is in amperes and Vb is in volts. Equation (2.3-9) is the basic thrust equation that applies for a unidirectional, singly ionized, monoenergetic beam of ions. The equation must be modified to account for the divergence of the ion beam and the presence of multiply charged ions commonly observed in electric thrusters. The assumption of a monoenergetic ion beam in Eq. (2.3-6) is generally valid for ion thrusters, but is only an approximation for the beam characteristics in Hall thrusters, which will be discussed in Chapter 7. The correction to the thrust equation for beam divergence is straightforward for a beam that diverges uniformly upon exiting from the thruster. For a thruster with a constant ion current density profile accelerated by uniform electric fields, the correction to the force due to the effective thrust-vector angle is simply Ft = cos , (2.3-10) where is the average half-angle divergence of the beam. If the thrust half angle is 10 deg, then cos = 0.985, which represents a 1.5% loss in thrust. If the plasma source is not uniform and/or the accelerator system has curvature, then the thrust correction must be integrated over the beam and grid profiles. For cylindrical thrusters, the correction factor is then Ft = 0r 2 r J(r) cos (r) dr , (2.3-11) Ib where J(r) is the ion current density which is a function of the radius. The ion current density is usually determined from direct measurement of the current distribution in the plume by plasma probes. For a constant value of J(r), Eq. (2.3-11) reduces to Eq. (2.3-10). The second correction applied to the thrust equation for electric thrusters accounts for the presence of multiply charged ion species. If the beam contains both singly charged and doubly charged ions such that the total beam current is Ib = I+ + I++, (2.3-12)PDF Image | Fundamentals of Electric Propulsion: Ion and Hall Thrusters
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