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202 Chapter 5 n0 = 0 ni+1 =(1 )ni + n. (5.3-3) The density that the code uses asymptotically approaches the final density: ( )i If is sufficiently small, approximate results for all upstream densities less then n can be obtained in a single run: ni = 1 (1 )i n. (5.3-5) By saving the intermediate results, only a single run is needed to estimate the performance of an optics design over a wide range of discharge chamber densities. However, since the calculation is fully converged only at the final density, separate runs with different final densities may be necessary to obtain accurate results over the full range of discharge chamber ion densities. A typical CEX-2D calculation takes a few minutes on a personal computer. Ion optic assemblies designed using the CEX-2D code have met the predicted performance very closely [4], illustrating that grid design techniques are very mature. The ion density in the beamlet is obtained in the codes by tracking representative ion trajectories and accounting for charge exchange collisions that alter the ion energy. Ions enter the computational region from the upstream boundary at the Bohm velocity, and their charge density is found by following their trajectories in a stationary electric field. This is in contrast to the time- dependent particle in cell (PIC) technique generally used in plasma physics simulations. An example of ion trajectories calculated by CEX-2D is shown in Fig. 5-6, which shows the computational space with the dimensions given in meters used for three values of beam perveance for half a beamlet in a three-grid configuration. In this figure, ions from the discharge chamber enter from the left and are accelerated by the electric field between the screen and accel grids. The horizontal boundaries represent lines of symmetry such that an ion crossing at these boundaries has another ion coming in from outside the domain. Figure 5-6(a) shows an over-perveance condition representing a beamlet current too high for the applied voltage, or too low a voltage for the plasma density and ion current provided. In this case, ions directly impinge on the upstream face of the accel grid. This situation is considered to be the perveance limit, where i n n =n1 . (5.3-4)PDF Image | Fundamentals of Electric Propulsion: Ion and Hall Thrusters
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