29 Jun.,2022

Scientific Article | Traceability calibration of mechanical characteristics of thrust stand is an essential prerequisite to ensure traceability measurement...

Following the protocol, the capacitance gradient and the stiffness of the thrust stand are calibrated. The principle of electrostatic force should be introduced. There will be relative motion Dab between two charged plates under the action of external force F. Moreover, the work W by external force will be converted into electric energy E stored in the capacitor. The potential difference U, the charge of both plates Q and capacitance C can be obtained. The stored energy between two plates can be calculated according to Eq.(1). The work done by external force can be calculated according to Eq.(2).

(1)

(2)

Based on the principle of virtual work13,14,15, W is equal to E.

(3)

The external force changes the distance dz between two plates, then

(4)

Therefore, the electrostatic force can be calculated by voltage as long as the capacitance gradient is calibrated based on Eq.(4). Two circular metal plates are used to form a parallel plate capacitor.

**Figure 2** shows the fitting curve of mean values of capacitance and displacement of plate A and plate B. The average of the capacitance gradient dCab/dDab is 6.2 pF/mm, and the goodness of fit can reach 0.9983. Therefore, according to Eq. (4), the parallel plate capacitor can produce traceable force less than 1 µN (as long as the voltage is less than 17.96 V), and the force value range is ~0.279 mN.

**Figure 3A** shows the displacement change of the thrust stand arm with the change of the electrostatic force. As the applied voltage changes equidistantly, each step displacement becomes larger and larger. It can be seen that the displacement changes symmetrically as the voltage increases and the voltage decreases. **Figure 3B** shows the fitting curve of mean values of the displacement of the thrust stand and the electrostatic force. The stiffness k of the thrust stand is 3.176 N/m based on Eq. (5), and the repeatability is 0.0023 N/m.

(5)

The capacitance of the parallel plate capacitor can be described as Eq.(6) under the condition of ignoring the edge effect.

(6)

(7)

S is the overlap area, the permittivity of vacuum ε0 = 8.85 x 10-12 F/m, and the relative dielectric constant εr = 1 in the air. It can be seen that the capacitance gradient changes with the plate spacing. However, the change is very small when the plate spacing changes very little. The capacitance gradient tends to a constant value. According to the request of the mN magnitude standard force for the thrust stand calibration, the plate diameter of the circular parallel plate capacitor should be within the centimeter magnitude and the distance Dab should be within the millimeter magnitude based on the Eq. (1) and (2) when the output voltage of the SMU is a few hundred volts. The output range of the SMU instrument is ±5 µV- ±1100 V. Hence, the diameters of plates A and B are 6 cm and 4 cm, respectively. The distance Dab between the two plates is 1 mm.

In order to obtain the constant capacitance gradient range, the potential distribution, electric field distribution, capacitance, and other parameters of the parallel plate capacitor with high voltage are calculated using finite element simulation. The potential distribution of parallel plate capacitors is shown in **Figure 4**, whose size is consistent with the actual device. In order to meet the demand of micronewton electrostatic force, the plate spacing is set to 1 mm. **Figure 5** shows the electric field distribution of the parallel plate capacitor. The arrow represents the direction of the electric field, starting from the high voltage and pointing to the grounding terminal. The capacitance of the parallel plate capacitor is simulated by changing plate spacing. Step size is 20 µm and ends at 1.2 mm. As shown in **Figure 6**, the capacitance gradient is a constant value of 9.81 pF/mm, when the plate spacing is between 1 mm and 1.2 mm. According to Eq.(4), when the capacitance gradient is a fixed constant, the magnitude of the electrostatic force has a linear relationship with the square of the applied voltage. **Figure 7** shows the simulated electrostatic force at different voltages, which conforms to the derivation of Eq. (4). When the distance between two plates is 1-1.2 mm, to produce the calibration force of 1-1000 µN, according to the simulation calculation, the voltage range is 14.29-451.57 V.

Finally, the uncertainty of calibration results is evaluated below.

1. Type A evaluation of relative uncertainty

The main sources of uncertainty are air disturbance and ground vibration. According to the Bessel method, the relative uncertainty urel(re) is 0.0724% when the mean value of the five measurements is used as the best estimate of the measurement results.

2. Type B evaluation of relative uncertainty

a. Relative uncertainty of displacement measured by the laser interferometer

The laser interferometer is used to measure the displacement of the thrust stand. The relative uncertainty of laser interferometer without compensation is 0.005%, which can be ignored.

b. Relative uncertainty of voltage applied by SMU instrument

The relative uncertainty of the SMU instrument is 0.012%. Therefore, the relative uncertainty introduced by applied voltage urel(U) is 0.012%.

c. Relative uncertainty of capacitance gradient dCab/dDab

The capacitance gradient dCab/dDab is obtained by fitting the capacitance measured by the capacitor bridge with the distance moved by the motorized linear stage. The capacitive bridge has an accuracy of 5 PPM, whose uncertainty can be ignored. The displacement resolution of the motorized linear stage is 5 µm. As a result, the relative uncertainty of dCab/dDab is 0.16%.

3. Expanded uncertainty

The uncertainty components are independent of each other, summarized in **Table 1**. The relative uncertainty is 0.18%, so the standard uncertainty is 0.0057 N/m. The relative uncertainty introduced by capacitance gradient contributes most. The coverage factor is 3, so the extended uncertainty is 0.0171N /m.

(8)

**Figure 1****: Schematic diagram of parallel plate capacitor calibrating thrust stand. **The parallel plate capacitor generated the electrostatic force as a reference for the calibration of the thrust stand. Please click here to view a larger version of this figure.

**Figure 2****: Relationship between capacitance and plate spacing of parallel plate capacitor.** The slope of the fitting line was the capacitance gradient. Please click here to view a larger version of this figure.

**Figure 3****: Electrostatic force calibrating the thrust stand.** (**A**) The displacement of the thrust stand arm. (**B**) The relationship between electrostatic force and displacement of the thrust stand arm. Please click here to view a larger version of this figure.

**Figure 4****: Potential distribution of parallel plate capacitor.** The Potential distribution was uniform. Please click here to view a larger version of this figure.

**Figure 5****: Electric field distribution in parallel plate capacitor. **The arrows represent the direction of the electric field. Please click here to view a larger version of this figure.

**Figure 6****: Simulation of capacitance variation with plate spacing.** The capacitance decreased with the plate spacing. Please click here to view a larger version of this figure.

**Figure 7****: Simulation of electrostatic force with voltage.** The electrostatic force increased with the voltage applied to the plate capacitor. Please click here to view a larger version of this figure.

**Table 1: Uncertainty of thrust stand calibration.**