# Plume Divergence and Discharge Oscillations of an Accessible Low-Power Hall Effect Thruster
## 3 Results and Discussion
### Discharge Oscillations  
Discharge current measurements were recorded and processed to determine the mean, peak-to-peak, probability distribution, and spectral density. The mean and peak-to-peak discharge current are plotted as a function of the swept parameters in Figure 8. Figure 8a shows that, in general, discharge current amplitude decreased with increasing discharge voltage. At some operating conditions at the 1.27 mg/s flow condition, this trend did not hold. The authors cannot fully explain this inconsistency; however, it is likely the manifestation of a complex relationship between wave dampening and changes in the position and length of the ionization zone within the thruster due to an increase in the number of background neutrals. Peak-to-peak discharge current oscillations increased with increasing magnetic field strength, while the mean discharge current decreased as magnetic field strength increased as shown in Figure 8b. Figure 8c shows the mean and average maximum and minimum of the thruster discharge current as a function of CFF at an anode flow rate of 1.27 mg/s. The amplitude of the discharge current oscillations was lowest at a CFF of 6% and increased sharply at CFFs below 3% and above 26%. This behavior is consistent with results reported in reference [18] wherein large ion instabilities occur at very low cathode flow rates, and turbulent ion acoustic waves drive increased oscillations at higher cathode flow rates. Figure 7: Current utilization, mass utilization, and beam utilization efficiencies at a probe distance 4.5 TDD as a function of (a) discharge voltage, (b) magnet current, (c) CFF at 4 TDD. Utilizing a simple histogram function on the discharge current data, a probability distribution function (PDF) was obtained. The oscillatory behavior of the discharge current at nearly every thruster operating condition exhibited what will herein be referred to as a Gaussian-like probability distribution. An example of this behavior is shown in Figure 9a,b where the thruster is operating at a discharge voltage of 170 V and an anode flow rate of 1.27 mg/s. Two operating conditions were found to be exceptions to the Gaussian PDF where the discharge current oscillations behaved sinusoidally resulting in a twin-peak PDF, as shown in Figure 9c,d. The operating conditions that resulted in sinusoidal behavior occurred at discharge voltages of 170 V and 180 V while operating at an anode flow rate of 1.08 mg/s. Notably, the sinusoidal discharge behavior corresponds to the highest amplitude in the discharge current oscillations. Thus, there exists a rapid mode shift from a sinusoidal to a Gaussian distribution at an anode flow rate of 1.08 mg/s between discharge voltages of 180 V and 190 V.  
The power spectral density for each of the swept voltages and magnet currents are shown in Figure 10. The characteristic oscillation frequency varied between 60 and 90 kHz for all operating conditions, which was higher than expected based on other similarly sized thrusters [19; 20]. As discharge voltage was increased at the high flow condition, Figure 10a, the strongest oscillation frequency remained largely constant at about 60 kHz, and the spectral strength remained broadband throughout the range of discharge voltages swept. As discharge voltage was increased at the low flow condition, Figure 10b, the oscillation frequency, initially a strong sinusoidal 60 kHz behavior, decreased to a Gaussian-like oscillation centered at approximately 40 kHz. While the sinusoidal behavior appears relatively \"clean\", there was a greater than normal tendency for the thruster to extinguish under these operating conditions. As magnetic field strength was increased, Figure 10c, the strongest oscillation frequency increased monotonically from about 60 kHz to 85 kHz  
Figure 8: The mean and peak-to-peak discharge current as a function of (**a**) discharge voltage, (**b**) magnet current, and (**c**) cathode flow fraction. while also increasing in power.