# Thermal Noise Decoupling of Micro-Newton Thrust Measured in a Torsion Balance

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## Abstract

**:**

^{1/2}to 4.9 μN/Hz

^{1/2}at 0.11 mHz. Additionally, 1 μN square wave modulation of electrostatic force is extracted from the ambiguous thermal drift background of positive temperature coefficient (PTC) heater. The PID-FTR validation is performed with experimental data in thermal noise decoupling, which can guide the design of thermal control and be extended to other physical quantities for noise decoupling.

## 1. Introduction

^{1/2}within 0.1 mHz–1 Hz [3,4]. Accordingly, a more accurate thrust measurement system is in need to measure the variance of the micro-Newton thrusters on the ground, which is hard for the case of an ultra-low thrust-to-weight ratio (TWR) less than 10

^{−9}. Torsion balances have been widely used in exploring the behavior of gravity for their rotational symmetry across the x-y plane, which is suited for measuring torques while minimizing the impact of gravitational weight. At the University of Washington, a low-frequency torsion pendulum whose angle is measured using a Michelson interferometer was designed, which can be applied to tests of gravity and gravitational wave observation [5]. At the RMC Advanced Propulsion and Plasma Exploration Laboratory, a torsional thrust balance has been developed to measure thrust of 1–15 mN levels with high accuracy via the calibration of electrostatic force [6]. In the Institute of Mechanics, Chinese Academy of Sciences, a set of sub-micro-scale thrust measurement systems using a torsion pendulum was designed and successfully applied in the radiofrequency ion thruster test for “Taiji-1”, the first experiment satellite for space detection of gravitational waves in China [7]. The Airbus has adopted a double hanging pendulum balance to characterize the LISA dedicated micro-Newton thrusters based on the principle of differential measurement [8].

## 2. Setups and Methods

#### 2.1. Thrust Balance Setup

#### 2.2. Mathematical Model

#### 2.3. Decoupling Method

- Transfer function;
- Z-domain fit;
- Signal subtraction.

#### 2.4. Algorithm Optimization

#### 2.4.1. Extended State

#### 2.4.2. Regression Learning

## 3. Results and Discussion

#### 3.1. Diurnal Temperature Fluctuation

^{1/2}at 1 mHz, but up to 24 μN/Hz at 0.12 mHz

^{1/2}. The temperature change caused by thermal radiation of the sun amplifies the low frequency noise of the torsion balance below 1 mHz.

^{1/2}after temperature drift correction, reduced to 48 μN/Hz

^{1/2}at 0.012 mHz (24-h), only 1/5th of the original. In addition, FTR has not only a larger amplitude decoupled than LSR does in fitting thermal thrust noise, but also a wider decoupled noise band of 1 mHz larger than 0.1 mHz of LSR. However, for the limitation of the temperature sampling frequency and the modulation transfer amplitude of the position PT1000 located, the noise reduction effect in this experiment is not obvious above 1 mHz, so shorter period temperature fluctuation is needed to testify.

#### 3.2. Space-Time Variation of Temperature

^{1/2}, which provides a reference for the PT1000 sampling frequency. In addition, the heating position during the whole experiment is also a variable. The deformation of torsion balance is the direct cause of the fluctuation of thrust position, and the influence of the heat source is the indirect factor of thrust drift.

#### 3.3. Square Wave Modulations of ESF and PTC

#### 3.3.1. Temperature and Displacement

^{4}s: thrust and temperature move in accordance with each other in the modulated frequency, but also in the larger low-frequency trend; however, after 3 × 10

^{4}s, although the temperature thrust is still correlated at the modulated frequency, the trend at lower frequencies is reversed. This is attributed to the fact that the heat balance of the system is determined not only by the modulated frequency of PTC heating, but also by the larger system environment (including the platform) and the external temperature exchange. For example, PTC heating can cause the torsion balance to fluctuate briefly with the modulation period, but a longer-term trend is determined by the heating and deformation of basement platform. The height of the four corners of the base varies relative to each other because of thermal expansion and cold contraction. Usually, the gravitational weight of torsion pendulum will not produce a displacement, but when the platform tilts 1 μrad, even a torsion bar of 0.1 kg will produce 1 μN as thrust increment, which needs to seek a new balance with flexible axis deformation. Additionally, 0.1 μm difference between the ends of length 0.1 m is enough to cause a horizontal change of 1 μrad. The thermal deformation of the material is extremely difficult to control, and the problem is worse when the heat power of the thruster and other loads is added.

#### 3.3.2. Thermal Noise Decoupling

^{1/2}in the full frequency band; even if the square wave is restored, it can be still locally reduced to 1.5 μN/Hz

^{1/2}at 0.5 mHz. What is different from the diurnal temperature variation, the thermal noise removal effect is more obvious near 10 mHz except at the modulation frequency of 0.29 mHz, while higher frequencies at 50 mHz still have no thermal noise removed. It indicates that the temperature acquisition frequency about 0.1 Hz is appropriate, and that the selection of temperature measurement points plays an important role in establishing the correlation between temperature and thrust drift of the torsion balance. Still, the results need to be verified by the restoration of electrostatic force in time domain again.

#### 3.3.3. Restoration of Electrostatic Force

^{4}s, which disappear again as expected after 2 × 10

^{4}s. Therefore, the proposed process of thermal noise decoupling does not shave all the noise, but only removes the interference in the 0.5 mHz–50 mHz related to temperature fluctuation, which is more real and more effective than the direct low-pass or band-pass filtering of displacement data.

## 4. Conclusions

^{1/2}at 0.1 mHz, which may be retrieved via temperature-displacement two-stage control, including both in situ thermal stabilization with millikelvin precision and active control of residual thermal thrust noise decoupled by regression learning with sub-micro-Newton precision.

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

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**Figure 1.**Overall structure design of torsion balance showing the flexural pivots, magnetic damper, capacitive sensor and ESF. Not shown is the windshield and PT1000 sensor.

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**MDPI and ACS Style**

Cong, L.; Mu, J.; Liu, Q.; Wang, H.; Wang, L.; Li, Y.; Qiao, C.
Thermal Noise Decoupling of Micro-Newton Thrust Measured in a Torsion Balance. *Symmetry* **2021**, *13*, 1357.
https://doi.org/10.3390/sym13081357

**AMA Style**

Cong L, Mu J, Liu Q, Wang H, Wang L, Li Y, Qiao C.
Thermal Noise Decoupling of Micro-Newton Thrust Measured in a Torsion Balance. *Symmetry*. 2021; 13(8):1357.
https://doi.org/10.3390/sym13081357

**Chicago/Turabian Style**

Cong, Linxiao, Jianchao Mu, Qian Liu, Hao Wang, Linlin Wang, Yonggui Li, and Congfeng Qiao.
2021. "Thermal Noise Decoupling of Micro-Newton Thrust Measured in a Torsion Balance" *Symmetry* 13, no. 8: 1357.
https://doi.org/10.3390/sym13081357