Design and Experimental Validation of a Micro-Newton Torsional Thrust Balance for Ionic Liquid Electrospray Thruster

Abstract

This work describes the advances on a micro-newton torsional thrust balance for ionic liquid electrospray thruster (ILET) being developed in the National University of Defense Technology. The torsional thrust balance adopts an asymmetric pendulum arm about a flexural pivot, and an electrostatic comb device is used for calibration, which makes the balance compact and allows the measurement of the micro-newton level thrust with high accuracy. To minimize the influence of gravity on the measurement results, a two-dimensional adjustable counterweight mechanism is adapted to balance the entire arm and make its mass center close to the pivot axis. Mechanical oscillations are passively damped with an eddy current damper. A series of experimental studies are carried out in a vacuum chamber; the results provide validation that the balance has good linearity in the range of 2–30 μN, with the resolution better than 0.21 μN and the settling time to a step force is less than 7 s, which can meet the thrust measurement requirements of the ILET operating in the mode of alternating work voltage polarity. In addition, we find that the electrostatic force generated by the power supply wire has a non-negligible influence on thrust measurement results, which needs to be considered in the actual thrust measurement.

1. Introduction

The rapid evolution of miniaturized, automatic, robotized, function-centered devices have redefined space technology, enabling CubeSats/nanosatellites to realize most near-earth space exploration and interplanetary missions. Due to the ability for CubeSats/nanosatellites to be developed at a large-scale for a low cost and in a short period of time, several hundreds of them are launched and applied in the fields of space communication, remote sensing, and space science experiments every year. Recently, a wide variety of applications for advanced missions by a satellite constellation and deep-space exploration have been proposed. All these missions require a robust propulsion system for active orbit control to perform formation flight and orbit transition. Among several options considered for propulsion of CubeSats/nanosatellites, ionic liquid electrospray thruster (ILET) is an attractive alternative. The relevant research on ILET have shown that it not only has the advantage of compact structure, but also has the potential for high specific impulse and efficiency. A novel porous-media borosilicate glass ILET prototype has been developed in the National Defense University of Science and Technology [1]. The specific impulse and thrust are evaluated by measuring the emission current and the time-of-flight data of the beam. However, the emitted ion clusters may fragment into neutrals and small ions in the acceleration and free field region [2,3], and the neutrals cannot be measured by the time-of-flight method. As a result, the indirect thrust measurement does not contribute neutrals to thrust. Therefore, it is urgent to develop a high precision thrust stand to directly measure the thrust to fever the ILET development.

There are many thrust stands that have been proposed and developed for steady state micro thrust measurement. According to their structural topology, they can be divided into pendulum, inverted pendulum type, and torsion pendulum three major types. All these topologies are variations of the spring-mass-damper system, and the primary difference is the influence of the gravity on the dynamic response characteristics of the system [4]. The thrust generated by a single ILET is in the range of several μN to 10 s μN, which is small compared with the mass of the thruster and pendulum arm. Gravity acts as a restoring force in the hanging pendulum and applied force in the inverted pendulum, respectively, which will significantly influence the thrust measurement results. However, in the ideal torsional pendulum, gravity has no effect on thrust measurement due to the motion plane of the arm being always perpendicular to the gravity vector. Therefore, a torsional thrust balance with a thrust measurement range of 2–30 μN is developed for directly measuring ILET thrust. The detailed design and performance of the thrust balance are presented in this paper.

2. Torsional Thrust Balance Description

Figure 1 shows a schematic of the torsional thrust balance. An aluminum frame was fixed to a platform. The pendulum arm was attached to the frame with a flexural pivot, forming a torsional pendulum configuration with rotational axis defined by the pivot. The electrostatic combs were used to generate calibration force. The deflection displacement of the arm was measured by fiber optic displacement sensor at a certain distance from the pivot axis. To damp the oscillation of the arm to quickly stabilize to steady state deflection, an eddy current damper was used. The stopper was used to protect the displacement sensor probe from impact caused by accidental deflection of the arm.

2.1. Methodology for the Calculation of the Thrust

The dynamics of the torsional balance can be described by the harmonic oscillator equation:

$$\ddot{\theta }(t)+2\zeta {\omega }_n\dot{\theta }(t)+{\omega }_n^2\theta (t)=\frac{F_t(t)L_t}{I},$$

(1)

$$\zeta =\frac{c}{2\sqrt{Ik}},$$

(2)

$${\omega }_n=\sqrt{\frac{k}{I}},$$

(3)

where θ is the deflection angle of the arm, I is the moment of inertia about the rotation axis, F_t(t) is the force produced by the thruster, L_t is the distance between thruster mount position and rotational axis, c is the damping constant, k is effective torsional spring rate, ζ and ω_n are the damping ratio and natural frequency, respectively.

Assume that the thrust generated by the thruster is constant, F_t(t) = T, the solution of Equation (1) is:

$$\theta (t)=\frac{T{L}_t}{k}{e}^{-\zeta {\omega }_{n}t}\sin ({\omega }_n\sqrt{1-{\zeta }^2}t+\phi ),$$

(4)

When the arm reaches a steady state in a short period of time, the new position can be found by taking $t\to \infty$ into Equation (4):

$$\theta (t\to \infty )=\frac{T{L}_t}{k}\Rightarrow T=\frac{k\theta }{L_t},$$

(5)

This equation shows that the thrust T can be determined by measuring θ if k and L_t are known. However, it is not convenient to directly measure the arm deflection angle θ. For small deflections, the small angle approximation is valid and the arm deflection θ can be determined by measuring the linear displacement at a distance L_s away from the rotational axis:

$$\theta =x/L_s,$$

(6)

The linear displacement can be measured more accurately than the deflection angle; for the instance of θ < 4°, the arm position can be determined within 0.1% [5]. The effective torsional spring rate k is determined by the flexural pivot and thruster power supply wire. Due to the hysteresis of pivot and the wire routing method, k is not constant and may vary with the arm deflection. Therefore, Equation (5) can be rewritten as:

$$T=\frac{k(\theta )\theta }{L_t}=\frac{k(\frac{x}{L_s})\frac{x}{L_s}}{L_t}=\frac{f(x)}{L_t}=F(x),$$

(7)

We can see that determining the thrust comes from function F(x) and the measured displacement at steady state position x. The function f(x) can be fitted by exerting a sequence of known force F_e at a certain distance away from the rotational axis and monitoring the corresponding displacement of the arm.

2.2. Torsional Balance Design

To reduce the longitudinal dimension of the torsional balance, an asymmetric arm structure about the rotational axis was adopted. The arm was made of hard aluminum alloy to ensure sufficient structural strength and small moment of inertia. The thruster to be measured will be mounted on the longer end of the arm. The center of the thruster mounting table was designed to be 270 mm away from the rotational axis, and the ratio to the distance between the electrostatic comb attached position and the rotational axis was 2.5. To reduce the influence of the deviation of the center of mass of the entire arm from the rotational axis [5], a counterweight structure that can be orthogonally adjusted in the plane perpendicular to the rotational axis was designed on the arm’s short end. By adjusting the counterweight, the arm can be balanced and make the mass center of the arm as close to the rotational axis as possible.

To make the torsional balance more compact and easier to assemble, only one C-Flex model F10 flexural pivot (torsional spring rate is 42.08 N·μm/rad) was employed to provide the torsional restoring force. It was chosen mainly because of its good compatibility with vacuum, its frictionless rotation, and its low hysteresis. The shift of geometric center of the pivot will be minimal on the order of 0.02% of its diameter for small rotation angles.

2.3. Fiber Optic Displacement Sensor

A small displacement on the order of 10 s micrometers will occur when the desired thrust range is exerted on the torsional thrust balance arm. The Fiber Optic Displacement Sensor (FODS), which is compatible with vacuum, was used to measure the arm deflection displacement. This kind of displacement sensor has been applied in previous torsional balance development, and successfully realized the small displacement measurement [6,7]. In this work, the Philtec. Model RC32 was adopted; its nominal resolution is 0.25 μm in the range of DC-100 Hz. The control unit of the sensor provides an analog voltage output of 0–5 V, which is proportional to the displacement between the measured target and the sensor probe. The manufacturer provides a calibration curve from a range 0.149–3.175 mm, and the linear range is 1.05–1.88 mm with a sensitivity of 2.757 mV/μm. In the experiments of this paper, the distance between the arm and the sensor probe was always kept in the linear measurement range of the sensor.

2.4. 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 μN 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].

$$F_e=\frac{{\epsilon }_0{N}_c{V}^2}{\pi }\left\{\frac{c+g}{x_0}-\left\{\ln \left\{\left[{\left(c/g+1\right)}^2-1\right]{\left(1+2g/c\right)}^{1+\frac{c}{g}}\right\}+\pi d/g\right\}\right\},$$

(8)

where N_c is the number of comb pairs, ε₀ is the permittivity, 2x₀ 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).

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 μN (Local gravitational acceleration is 9.7915 m/s²). 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. We set the engagement distance as 6 mm, the electrostatic force varying with the applied voltage is shown in Figure 4b, and the maximum electrostatic force is about 135 μN at 1000 V. The fitting function for the experimental data is given as:

$$F_e=1.38603\times {10}^{-4}\cdot V^2+0.00646\cdot V+0.62245,$$

(9)

2.5. 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 0.4~0.8 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·μm/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·m², ω_n = 1.955 rad/s, and ζ = 5.526 × 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·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]:

$$F_{e,d}=-2\pi \sigma \dot{\theta }{L}_p{\displaystyle {\int }_0^{r_c}{\displaystyle {\int }_0^{t}r{B}_z^2\left(r,d_p+z\right)}}drdz,$$

(10)

where L_p is the distance between the plate center and the rotational axis of the arm, $\dot{\theta }$ is the angular velocity of the arm, d_p is the gap between the magnet and plate, r_c is the radius of the plate inscribed circle, t and σ are, respectively, the thickness and conductivity of the plate, and B_z is the magnetic flux density in z direction. The integral result of the above equation can be simplified as [12]:

$$F_{e,d}=\alpha \pi r_c^2\sigma t\dot{\theta }{L}_p{\overline{B}}_z^2,$$

(11)

where ${\overline{B}}_z$ is the average magnetic flux density within the equivalent inscribed plate and α is the correction coefficient containing the plate geometric factor. The damping constant generated by the eddy current damper is given as:

$$F_{e,d}{L}_p=\underset{\_}{\alpha \pi r_c^2\sigma t{L}_p^2{\overline{B}}_z^2}\dot{\theta }=\underset{\_}{c_e}\dot{\theta },$$

(12)

According to the calculation method proposed by Bae [12], α is ~0.5 for the square plate. However, the c_e calculated by the author of [13] referring to Bae’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 (σ = 5.7 × 10⁷ S/m) used in this design is 45 mm × 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 ${\overline{B}}_z$ 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.

3. Torsional Thrust Balance Assembly and Calibration

To minimize the influence of gravity on calibration, the arm must be balanced after the grounded comb, the damper plate, the flexure pivot, and the thruster mock are assembled to it. As shown in Figure 8, the arm assembly was placed on a pair of knife edges and balanced by screwing the counterweights in the corresponding direction. Then, the counter-balanced arm was assembled to the frame, and the damper magnet, electrostatic comb device, and FODS were adjusted to the wright position. The torsional thrust balance assembly is shown in Figure 9.

In the actual thrust measurement, the ILET power supply wire will increase the torsional spring rate of the system. In addition to that, the ILET working voltage is usually above 1000 V, and the electrostatic force generated between the wire and torsional balance system may be considerable which may affect the thrust measurement. Therefore, the influences of the wire and the working voltage ought to be considered in calibration. A wire which had been adapted in ILET was arranged along the arm side from the thruster mounting table to the rotational axis and led out from the frame side after bending to a V-shape. The experimental calibration of the torsional balance was carried out in the VTM2400 vacuum chamber of the National University of Defense Technology. The diameter of the vacuum chamber is 2.4 m, and it is equipped with six cryopumps and a turbo pump. Only the turbo pump was used to evacuate the chamber during the calibration; the ambient pressure in the chamber was on the order of 10⁻² Pa. The electrostatic comb was powered by the high-voltage amplifier (Matsusada AMT-10B, slew rate: 360 V/μs) outside of the chamber. The control signal of the amplifier was generated by a digital signal generator. The FODS output signal and the amplifier output were recorded by an oscilloscope.

3.1. Dynamic Response Characteristic

To clarify the dynamic characteristic of the torsional balance, a step force produced by the ESC was exerted on the arm and the resulting motion was monitored. The electrostatic comb voltage pulse was set to 780 V (82.5 μN) with a pulse width of 40 s. Figure 10 shows how the torsional balance responds to the step force. It can be seen from the figure that the arm reaches a steady-state defection position after about one cycle of oscillation, and the settle time is about 7 s. The ω_n and ζ are 2.02 rad/s and 0.55, respectively, which were estimated by fitting a damped sinusoid to the data as mentioned above. These values are slightly different from those of the atmospheric environment experimental results of 1.955 rad/s and 0.56. According to Equations (2) and (3), the difference is mainly due to the wire increasing the total torsional spring rate of the system. As mentioned above, the polarity of the ILET working voltage will alternate with a period of several seconds. It has been confirmed in the long duration firings that the ILET can operate stably under the polarity alternation with a period of 30 s [14]. For the torsional balance designed in this paper, the settle time response to a step excitation is less than the one-half polarity alternation period, namely 15 s. Therefore, the dynamic response characteristic can meet the thrust measurement requirements of ILET.

3.2. Steady-State Characteristic

The intrinsic drift of the torsional balance has an important effect on long duration thrust measurement. To accurately calibrate the steady-state performance, the primary goal is to estimate the intrinsic drift rate of the torsional balance. Figure 11 shows three repeated measurements for arm displacement over a period of 1000 s at a sample rate of 0.02 s. Both the electrostatic comb and the wire were not applied with voltage excitation. From the figure we can find that the torsional balance drifts towards the FODS probe. The recorded data of each measurement was divided into 50 s segments to estimate the intrinsic drift rate. Each data segment was fitted by a linear function, then we averaged all the slopes, resulting in an average drift rate of 0.0023 μm/s.

The torsional thrust balance was calibrated by applying a sequence of known forces producing and the corresponding arm deflection displacement was recorded. The force was generated by electrostatic combs and the value was determined by the electrostatic comb calibration Equation (9), according to the applied voltage. Figure 12 shows the voltage profile applied to the electrostatic comb. Each step lasts 24 s long which can ensure the arm reaches a stable position. The blue values in the figure are the actual electrostatic force calculated in each voltage step. The arm response to the sequence voltage profile applied to electrostatic comb is shown in Figure 13. The displacements shown in the figure are the average displacements during the last 10 s in each step. It can be found in the figure that due to intrinsic drift, the arm displacement is distinct in the loading and unloading condition.

For clarifying and quantifying the distinctions, the calibration procedure was repeated three times to exclude the influence of random error. Figure 14a shows the electrostatic force-displacement relationship in the loading and unloading process; the standard deviation of three displacement measurements under the same excitation voltage is so small that it is not shown in the figure in the form of error bars. It was found that the arm displacement is only in good agreement at high voltage, while it has a large deviation at low voltage. The main reason for this phenomenon is that the high voltage is in the middle of the calibration history, the time interval between loading and unloading at the same voltage is short, and the intrinsic drift of the arm during this period is small. However, the low voltage is at the beginning and the end of the calibration history, and the time interval is longer, resulting in a larger displacement deviation. We compensated the arm displacement according to the obtained intrinsic drift rate. Figure 14b shows the resulting electrostatic force-displacement; it can be seen that the two curves in the loading and unloading process are in good agreement after compensation. We averaged the data in Figure 14b and the resulting steady-state calibration result is shown in Figure 15. The function of arm displacement and thrust can be determined by linear fitting the data as:

$$F_t=0.47968\cdot \Delta x-0.2445,$$

(13)

To obtain the thrust resolution of the torsional thrust balance, two step voltage sequences of 200–240 V (voltage increment of 4 V) and 740–750 V (voltage increment of 2 V) were applied to the electrostatic comb, respectively. The arm displacement response to these two voltage sequences was recorded and substituted into Equation (13) to calculate the corresponding thrust. Figure 16 shows the voltage profiles and the thrust calculation results; the error bars are the standard deviation of the average force in each step loading. It shows that each thrust step can be clearly distinguished; the green values are the difference between the average values of two adjacent thrust steps. To estimate the thrust resolution, we averaged all the calculated differences, resulting in a thrust resolution of 0.09 μN at ~2.5 μN and 0.21 μN at ~27.8 μN.

3.3. Thrust Noise

The thrust noise of the torsional thrust balance was estimated by converting the data in Figure 11 into thrust by the displacement-thrust fitting Equation (13). Then, the power spectrum of the thrust at corresponding frequency was computed after fast Fourier transform. Then, we averaged the square root of the power spectrum to obtain the power spectrum density at each frequency. Figure 17 shows the calculated thrust noise power spectrum density. It can be observed from the figure that the power spectrum density at 0.6 Hz is 0.1 μN/Hz^1/2, and increases with decreasing frequency, reaching the value of 0.6 μN/Hz^1/2 at 10 mHz. We can also find a spike around 0.4 Hz, which is the natural frequency of the torsional balance.

3.4. Influence of the Wire Voltage

The steady-state calibration was performed under the condition that the thruster power supply wire was off, so the influence of the electrostatic force generated by the wire was not considered. However, the thruster will be applied with a high voltage (positive or negative) through the power supply wire during thrust measurement, and the resulting electrostatic force exerting on the torsional arm may cause the actual measured displacement to be different from that which only the thrust causes. If this uncorrected displacement is applied to calculate the thrust, it must introduce error. Figure 18 shows the torsional thrust balance response to an electrostatic force applied by the electrostatic comb under the conditions that the wire is powered or not. It can be seen from the figure that the arm displacement with the wire powered is offset Δx_o relative to that when the wire is off, resulting in the displacement reading Δx_r relative to the arm initial position being smaller than its actual deflection displacement Δx_r + Δx_o, which will cause the measured thrust to be less than the actual thrust of the thruster. It can be inferred from Equation (13) that the thrust exerting on the arm has a good linear relationship with the response displacement. Therefore, the actual deflection displacement of the arm during thrust measurement can be equivalent to the superposition of displacements caused by the electrostatic force of the wire and the thrust separately. Therefore, the arm displacement during thrust measurement can be compensated if the influence of the wire voltage on arm displacement is clear.

To quantitate the influence of the wire voltage on the arm displacement, we applied a triangular voltage wave with a peek-to-peek of 5 kV to the wire, with or without thrust exerting on the arm. The thrust was simulated by the electrostatic comb powered with 780 V equivalent to 31.96 μN. Figure 19 shows the voltage waveform and the corresponding arm displacement after intrinsic drift was compensated. It can be seen from the figure that in the first three quarter cycles of the voltage waveform, the arm displacement curves under the condition of with and without simulated thrust are in good agreement. This means that the electrostatic force generated by the wire and the thrust of the thruster are decoupled. However, the displacement curves greatly deviate from each other in the last quarter cycle, and the displacement at −2.5 kV is slightly larger than that at 2.5 kV. In fact, electrostatic force is only related to the voltage amplitude and has nothing to do with the polarity. The reason for the above deviations may be that the intrinsic drift rate increases in that time range. We ignored the wire voltage polarity and averaged the displacements at the same voltage amplitude with or without the simulated thrust. The resulting arm displacement as a function of wire voltage is shown in Figure 20; the error bars in the figure are the standard deviation of the mean value.

We applied a 2 kV voltage to the wire and measured the arm displacement when an equivalent thrust of ~3 μN and ~30 μN was applied. Figure 21 shows the arm displacement; the values in black text are displacement deviations with or without wire voltage at the same thrust. We found that by averaging the deviations, compared with the displacement without wire voltage, the actual displacement reading deviation caused by the wire voltage is −1.9 μm ± 0.07 μm (~0.9 μN) and −2.3 μm ± 0.08 μm (~1.1 μN), respectively. The deviation calculated by the fitting equation shown in Figure 20 is −1.6 μm, and the relative error from the experimental measurement is 26.7% and 38.5%, respectively.

4. Conclusions

In this paper, aiming at the thrust performance test requirements during the development of ILET, a micro-newton torsional thrust balance was developed, and detailed experimental calibrations were carried out to examine its characteristics. The results showed that the torsional thrust balance exhibits good linearity in the range of 2 μN~30 μN, the settle time response to a step force is less than 7 s, and the thrust resolution is 0.09 μN at ~2.5 μN and 0.21 μN at ~27.8 μN. This validated performance shows that the balance can fulfill the thrust measurement requirements of the ILET operating in the mode of alternating work voltage polarity. However, the thrust balance has intrinsic drift, and the average deviation rate within 1000 s is about 0.0023 μm/s. For long duration thrust measurement, the drift must be taken into account. In addition, we found that the electrostatic force produced by the thruster power supply wire has a non-negligible impact on the thrust measurement results in this design. The influence of the wire voltage was quantitatively analyzed, and it was found that a wire with 2 kV can cause 1 μN thrust measurement deviation.