## 2 Torsional Thrust Balance Description
### Electrostatic Comb Design and Calibration  
The electrostatic force devices are preferred for calibration because it can generate a very small force in a repeatable and accurate manner both at atmosphere and in vacuum. The use of an electrostatic force requires no direct contact with the torsional balance, and no cabling is required on the stand itself, which makes electrostatic calibration particularly well suited for \(\mu\)N thrust balance [8]. There are two main types of electrostatic devices, namely, electrostatic plates and electrostatic combs. However, the force produced by the electrostatic plates is inversely proportional to the square of the gap between the plates, and even a small change in the gap can lead to apparent force variation [9]. In the torsional balance application, the gap will change during the calibration as the arm deflects; this will result in a relatively large calibration error. However, this gap dependance can be eliminated by using the electrostatic combs as shown in Figure 2. The force produced by the engaged combs is given by [10]. \[F_{e}=\frac{\varepsilon_{0}N_{c}V^{2}}{\pi}\bigg{\{}\frac{c+g}{x_{0}}-\Big{\{} \ln\Big{\{}\Big{[}(c/g+1)^{2}-1\Big{]}(1+2g/c)^{1+\frac{c}{g}}\Big{\}}+\pi d/g \Big{\}}\bigg{\}}, \tag{8}\]  
where \(N_{c}\) is the number of comb pairs, \(\varepsilon_{0}\) is the permittivity, \(2x_{0}\) is the engagement distance, and other structural parameters are defined and shown in Figure 2. It can be found from Equation (8) that the electrostatic force is proportional to the engagement distance, which is beneficial for torsional balance calibration as the error resulting by the gap change is relatively small. The electrostatic combs were used in this work with \(c=d=1\) mm, \(g=0.5\) mm, \(N_{c}=5\), and \(l=14\) mm. The combs were made of aluminum and encased in Polyetheretherketones base. One comb was attached to the arm and served as the grounded part, while its counterpart was mounted on a 3-axis manual linear stage and supplied with a voltage. The grounded comb was also grounded to the vacuum facility through the torsional balance, which ensured there was no wire attached to any part influencing the balance characteristic. The engagement distance and the alignment of the electrostatic combs were determined by the stage linear micrometer (resolution is 0.02 mm). Figure 2: Drawing of the electrostatic combs defining structural parameters. The electrostatic force produced by the electrostatic comb may deviate from the theoretical predicted by Equation (8). Therefore, experiment calibration must be performed prior to application. The calibration progress is illustrated in Figure 3. The grounded comb was placed vertically on the analytical mass balance (Metttler Toledo XS205) with a grounding wire attached to the pallet. The resolution of the balance is 0.01 mg which is equal to 0.098 \(\mu\)N (Local gravitational acceleration is 9.7915 m/s\({}^{2}\)). The variable potential comb assembly was positioned above it through a 3-axis manual linear stage, which can ensure the engagement distance is adjustable with an accuracy of 0.01 mm. The applied voltage was provided by a high voltage power supply (Matsusada ES-5R1.2) and monitored by a differential voltage probe (Tektronix P5205A). The balance was reset to zero to avoid shift prior to each measurement. The voltage and balance reading were recorded after the balance reading was stable for more than 15 s. Each calibration condition was repeated three times to minimize the random error. To minimize the vibration and air flow influencing the calibration results, the electrostatic combs experiment was performed on a gas floating platform and covered by an acrylic box. The force produced by the electrostatic combs at different applied voltage with engagement distance varying is shown in Figure 4a. It can be found that the force is almost constant in the engagement distance range of 2-11 mm, which is beneficial to reduce the calibration error caused by the electrostatic comb arrangement.