# SITAEI HC1 Low-Current Hollow Cathode
## HC1 Description
### Design Criteria  
The main cathode dimensions derive from a reduced-order numerical model developed at SITAEL and validated throughout the years against experimental data [10]. According to the theoretical model, the plasma formed inside the hollow cathode is ideally divided into three coupled regions: the emitter, the orifice, and the cathode-to-keeper gap. Equations expressing particle, momentum, and energy balances are numerically solved to compute the plasma properties in each region. Current density equations are introduced to estimate the voltage drop at the sheaths formed at the emitter and keeper surfaces for a given discharge current. The model assumes the formation of double sheaths at the boundary between the different regions of the cathode, being characterized by different plasma properties. A dedicated thermal model is coupled with the plasma models to estimate the power exchanges and the temperature profile along the cathode by means of power balance equations [11].  
The model was used to select the cathode dimensions (e.g., emitter internal diameter and length, orifice shape) after analyzing the results for each fixed geometry. A tradeoff was carried out between multiple factors, which include the overall geometrical envelope, the discharge power, the ratio between the emitter heating with respect to orifice heating, the emitter operating temperature, and the operating lifetime. The main criteria followed in the cathode design are listed hereinafter:  
1. To ensure spot-mode operation, characterized by a well-focused and stable cathode plasma plume, in the predicted operating envelope;
2. To ensure a predominant contribution for the self-heating mode coming from the plasma as compared with conduction from the orifice, to limit its temperature, and in turn to limit its erosion;
3. To operate at an emitter temperature guaranteeing the lifetime requirements;
4. To select a reliable heater design in terms of thermal cycling, heating performance, materials compatibility, and properties;  
\begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Model** & **Program** & **Characteristics** & **Type of test** & **Period** \\ \hline DM & Internal & LaB\({}_{6}\) emitter & Cathode characterization & January\(-\)May 2016 \\ \hline DM & Internal & BaO-W emitter & Cathode characterization & May\(-\)June 2016 \\ \hline DM & Internal & BaO-W emitter & Cathode characterization & June 2016\(-\)September 2017 \\ \hline DM & uHETSat & BaO-W emitter & \begin{tabular}{c} Cathode characterization \\ Coupling test with HT100 \\ Cathode automatic cycling (1500 cycles) \\ \end{tabular} & September\(-\)December 2017 \\ \hline EM & uHETSat & \begin{tabular}{c} BaO-W emitter \\ New heater design \\ \end{tabular} & \begin{tabular}{c} Thermal test \\ Coupling test with HT100 \\ Short endurance (350 h) \\ \end{tabular} & January 2018\(-\)July 2018 \\ \hline EQM & uHETSat & \begin{tabular}{c} BaO-W emitter \\ New heater design \\ Improved mechanical structure \\ \end{tabular} & \
\begin{tabular}{c} Coupling test with HT100 \\ Shock and vibration tests \\ Thermal-vacuum test \\ \end{tabular} & March\(-\)October 2019 \\ \hline PFM & uHETSat & Same as EQM &
\begin{tabular}{c} Coupling test with HT100 \\ Shock and vibration tests \\ Thermal-vacuum test \\ \end{tabular} & December 2019\(-\)March 2020 \\ \hline \hline \end{tabular}
\end{table}
Table 2: History of HC1 cathode development.  
To limit the cathode power consumption.  
An example of the model results showing the plasma parameters in the different cathode regions and surface temperatures in reference points of the assembly is reported in Figure 2. The results refer to operation at 0.5 A discharge current, 0.2 mg/s xenon mass flow rate, and with a floating keeper, corresponding to the nominal operating point in coupled configuration with HT100 for the \(\mu\)HETSat mission.