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366 Chapter 7 7.4.1 Hybrid Hall Thruster Models Hybrid Hall thruster models, such as HPHall [30, 57], utilize a steady-state fluid electron momentum equation and a time-dependent electron energy equation to solve for electron temperature and potentials in the channel and plume. The codes also use time-dependent ion and neutral particle equations to calculate the plasma density and ion velocities on a time scale much larger than the electron time scale. These codes are also used to model Hall thruster transit- time oscillations that are on the order of time scales related to neutral atom and ion motions ( 1 MHz) but cannot capture the effects of electron instabilities that have much higher frequencies. From the steady-state electron momentum equation, an Ohm’s law representation from Eq. (3.6-20) for the electron field is Je =E+ p Je B eiJi, (7.4-1) en where the resistive term has the following form in the magnetic frame of reference: J= J + ||J|| + ^J^, (7.4-2) and the subscripts represent the directions perpendicular, parallel, and transverse (in the E B direction), respectively, to the local magnetic field. Equation 7.4-1 must be separated into the two components of the Je B motion in a manner similar to that in Section 3.6 and solved for the electric field. From current conservation, the electron current is taken to be the difference between the discharge current and the ion current from the particle calculations. Typically, the circuit current is chosen at each time step to satisfy the applied voltage ( = E d ) boundary conditions. 7.4.1.1 Transverse Electron Transport. Writing the perpendicular resistivity in terms of the perpendicular electron mobility, as defined in Eq. (3.6-66), gives = 1 =1+ c2 m2 =1+ c2 m2 , (7.4-3) enμe enμe enμe where the collision time m for momentum transfer is equal to one over the collision frequency (1/ m). The perpendicular electron flux from Ohm’s law, Eq. (7.4-1), can then be written asPDF Image | Fundamentals of Electric Propulsion: Ion and Hall Thrusters
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