## 2 Torsional Thrust Balance Description
### Eddy Current Damper Design  
To suppress electrochemical degradation of the ILET, the emitter voltage polarity will alternate with a period of several seconds. When the voltage polarity alternates from positive to negative, the thrust will decrease from a value to zero and then increase to another certain value, and vice versa. Therefore, the torsional thrust balance dynamic response characteristic is very important for ILET thrust measurement; the time desired for the balance settling to steady state should be less than the one-half voltage alternation period. A damping ratio of 0.4\(\sim\)0.8 generally gives a good response [11]. In order to obtain the torsional balance damping ratio without additional damping, we excited the torsional balance by an impulse. The FODS output voltage as a function of time is shown in Figure 5. Assuming \(k\) is constant (42.08 N\(\cdot\)um/rad), the moment of inertia, damping ratio, and natural frequency were estimated by fitting a damped sinusoid in the form shown as Equation (4) to the data, resulting in: \(I=0.012\) kg\(\cdot\)m\({}^{2}\), \(\omega_{n}=1.955\) rad/s, and \(\zeta=5.526\times 10^{-4}\), which is much smaller than the desired value range. According to Equation (2), the desired damping ratio can be conveniently achieved by increasing the damping constant. We substituted \(I\), \(k\), and the desired damping ratio into Equation (2), resulting in the desired damping constant of 0.018 \(<c<\) 0.036 N\(\cdot\)s/m.  
The damping can be achieved through an eddy current damper, which is contactless and well compatible with vacuum. The configuration of the eddy current damper used in this work is depicted in Figure 6; it consists of a cylindrical permanent magnet and a square copper plate fixed on the arm's short end. When the plate moves in the magnetic field generated by the permanent magnet, the damping force (Lorentz force) will generate in the plate, making the arm rapidly settle to a stable deflection position. The damping force can be written as [11]:  
\[F_{e,d}=-2\pi\sigma\dot{\theta}L_{p}\int_{0}^{r_{e}}\int_{0}^{t}rB_{z}^{2}\big{(} r,d_{p}+z\big{)}drdz, \tag{10}\]  
where \(L_{p}\) is the distance between the plate center and the rotational axis of the arm, \(\dot{\theta}\) is the angular  
According to the calculation method proposed by [PERSON] [12], \(\alpha\) is \(\sim\)0.5 for the square plate. However, the \(c_{e}\) calculated by the author of [13] referring to [PERSON]'s method is 2.6 times bigger than the experimental result. We believe this is mainly due to the fact that the correction coefficient does not take into account the skin effect. Therefore, the correction coefficient was corrected to 0.2 in this work. The copper plate (\(\sigma=5.7\times 10^{7}\) S/m) used in this design is 45 mm \(\times\) 45 mm and the thickness is 2 mm, and the \(L_{p}\) is 245 mm. We substituted all the parameters to Equation (12) to determine \(\overline{B}_{2}\) following that a field of about 0.09-0.13 T to cover the desired damping constant range is needed. The COMSOL Multiphysics software was used to calculate the magnetic field above the permanent magnet; after varying several parameters referring to the commercial off-the-shelf permanent magnet, a 37.6 mm diameter with 7 mm height NdFeB permanent with a remanence of 1.3 T was settled. The magnet was mounted on a manual linear stage; the magnetic field in the plate can be varied by means of adjusting \(d_{p}\). The simulation and experimental results of \(c_{e}\) as a function of \(d_{p}\) are shown in Figure 7, and the blue labels are the damping ratio calculated from the experimental data. The experiment was performed in the atmospheric environment and without the thruster mock. It was found that the simulation and experimental data follow the same trend, and the simulation results are a little higher than the experimental ones when \(d_{p}>6\) mm, and the opposite is true when \(d_{p}<6\) mm. The \(d_{p}\) is set to 6 mm in the torsional balance calibration experiment and the corresponding damping ratio is 0.56.