A five-sided faceted cathode with a vertex angle of 3π/2 was then simulated as a replacement for the cylindrical cathode with all other parameters in the calculation unchanged. It was found that the reference parameters: Vca= -26.0 kV, B= 0.12 T, and a
' e
J = 500 A/m were not the optimum for this configuration. The spokes were unstable and
Time (s) Curr en t De n sit y (A/m )
a clean oscillation was not observed. Using the results from a previous model [12] developed in ICEPIC, these parameters were changed to Vca= -22.2 kV, B= 0.09 T, and a '
e
J = 326 A/m. It was found that the model developed in ICEPIC and the VORPAL models were comparable. With these parameters the five-sided faceted cathode was able to oscillate at a frequency of 957 MHz. The cylindrical cathode model was also simulated using these same parameters for comparison purposes. Figure 65 shows the RF Bz field, indicating the π-mode operation for the five-sided faceted cathode. Figure 66 shows the mode switching during start up. Initially there is a 650 MHz mode before stable
operation. Figure 67 shows the FFT of the cavity voltage, with a clear peak at 957 MHz. The linear power density at the loaded cavity was calculated to be 1.2 MW/m.
Figure 65: Five-sided faceted cathode model from VORPAL simulation showing the RF B-field and π-mode. This result corresponds to Vca = -22.2 kV, B = 0.09 T, and J’e = 326 A/m.
Figure 66: Five-sided faceted cathode cavity voltage frequency versus time with moving window, showing the startup time of the device at 200 ns, and the mode switching from 650 MHz to the operating frequency (π- mode) 957 MHz from VORPAL.
Figure 67: Five-sided faceted cathode Fast Fourier Transform (FFT), over entire simulation time, of the loaded cavity voltage from VORPAL
simulation. This plot indicates that the π-mode is dominant at the frequency of operation of 957 MHz.
The spoke formation results are shown in Figure 68. Figure 68(a) shows the pre- oscillation state (t= 0.012 ns). Figure 68(b) shows the spokes forming after oscillation at t= 317 ns. These VORPAL results, both cylindrical and faceted cathodes, were compared with a 2-D simulation previously performed in ICEPIC [12]. It was found that the
cylindrical cathode at Vca= -26.0 kV, B= 0.12 T, and a ' e
J = 500 A/m modeled in VORPAL was a more stable simulation with very clean spokes compared to the simulation parameters used in ICEPIC (-22.2 kV, 0.09 T, 326 A/m). However, the
Time (s) F re q u en cy (Hz) Startup Frequency (Hz) F F T Am p li tu d e
cylindrical cathode was simulated at the ICEPIC parameters for comparative purposes. On the other hand, the ICEPIC parameters for the faceted cathodes were the most stable choice for both cases (five-sided and ten-sided). In addition, the startup time for the -22.2 kV, 0.09 T, 326 A/mis shorter for the cylindrical cathode. Figure 69 shows the
comparison among the three models. It is observed that the VORPAL 2D model is closer to the 3D ICEPIC model, in terms of spoke formation. The spokes are much clearer in the 2D VORPAL model than in the 2D ICEPIC model. An analysis of the startup current was also performed. The total emitted linear current density was varied from 0.25J’e to 2J’e as
shown in Figure 64. A graph of the startup time of the device versus the total emitter linear current density is presented. From the curve, it is observed that the five-sided faceted cathode magnetron starts up at 200 ns for the reference parameters (-22.2 kV, 0.09 T, 326 A/m), while the cylindrical cathode at the Vca= -22.2 kV, B= 0.09 T, and a
' e
J = 326 A/m parameters shows a startup time of 150 ns (see Figure 64). Even though it starts up faster, it is not the most stable model. The startup time levels off above 400 A/m. The startup time increases dramatically below 400 A/m and will not start below 250 A/m.
(a) (b)
Figure 68: Five-sided faceted cathode VORPAL simulation results for Vc = -
22.2 kV, B = 0.09 T, and J’e= 326 A/m. The red dots indicate electron macroparticles. Figure 68(a) is at 0.08 ns, before oscillation, and Figure 68(b) is at 317 ns, after oscillation starts and the model is stable.
Figure 69: Comparison of ICEPIC model versus VORPAL model. Vca = -22.2 kV,
B = 0.09 T, and J’e= 326 A/m. The top figures show the cylindrical
cathode model, the middle figures show the five-sided faceted cathode, and the bottom figures show the ten-sided faceted cathode.
Figure 70 shows the total emitted linear current density versus time for the five- sided faceted cathode. These results were taken with no applied B field, so all the emitted current will be collected at the anode. This was completed to check that the total current
injected to the device was correct. In the figure, it is observed that the current averages to approximately 326 A/m, which matches the current input in the model.
Figure 71 shows the anode linear current density versus time when the device is in operation. It was found that there is an instability in the five-sided cathode oscillations. As it can be observed in Figure 71, this instability results in a current spike to the anode
and a subsequent collapse of the spokes. Figure 72 shows the transition of the spokes before the current spike occurs, at the current spike, and after the current spike, when the spokes collapse. For this particular example, the time between 119.8 ns and 143.5 ns was selected. It can be observed that the shape of the spokes changes at the current spike with the electrons forming a more concentrated cloud or clump; then, the clump extends to the anode when the spike occurs. The result is a loss of a large percentage of available electrons. Then, the spokes disconnect and collapse. Following this mechanism, the spokes will form again. As can be seen in Figure 71, this current spike has a periodic
behavior. The spike occurs, approximately every 250 ns to 350 ns. This means that every time the current spike occurs the spokes will collapse and reform again. Since the
geometry of the cathode consists of five sides, the time that it takes the spokes to go around the cathode is 5τRF, where 1 τRF = 1.04 ns corresponds to one facet. Therefore, the periodicity of the instability could be related to a multiple of the RF period. The causes of this instability are mainly attributed to the cathode shape, but further studies are needed to completely understand it. The electron trajectories around the five-sided cathode become distorted and appear to create too many synchronous electrons. These electron clumps give up their energy almost simultaneously, resulting in the current spike. By implementing a ten-sided cathode geometry (increasing the number of facets), this
problem was reduced. Therefore, the remainder of the phase control research work used the ten-sided cathode geometry.
Figure 70: Five-sided faceted cathode continuous total emitted linear
current density versus time with no applied magnetic field (B=0).
Figure 71: Five-sided faceted cathode continuous anode linear current density during device operation versus time. The periodic current spikes are followed by spoke collapse.
(a) (b) (c)
Figure 72: Five-sided faceted cathode showing the transition of the spokes during the current instability. Spokes are shown (a) before current spike at 119.8 ns, (b) at current spike at 121.32 ns, and (c) after spokes collapse at 143.5 ns. Time (s) C u rr en t D en si ty ( A /m ) Time (s) C u rr en t D en si ty ( A /m ) -6