Resonant frequency detection and adjustment method for a capacitive transducer with differential transformer bridge

A Ground-Based Electrostatically Suspended Accelerometer

In this study, we have developed an electrostatically suspended accelerometer (ESA) specifically designed for ground use. To ensure sufficient overload capacity and minimize noise resulting from high suspension voltage, we introduced a proof mass design featuring a hollow, thin-walled cylinder with a thin flange fixed at the center, offering the highest surface-area-to-mass ratio compared to various typical proof mass structures. Preload voltage is directly applied to the proof mass via a golden wire, effectively reducing the maximum supply voltage for suspension. The arrangement of suspension electrodes, offering five degrees of freedom and minimizing cross-talk, was designed to prioritize simplicity and maximize the utilization of electrode area for suspension purposes. The displacement detection and electrostatic suspension force were accurately modeled based on the structure. A controller incorporating an inverse winding mechanism was developed and simulated using Simulink. The simulation results unequivocally demonstrate the successful completion of the stable initial levitation process and suspension under ±1g overload.

Research on High-Precision Resonant Capacitance Bridge Based on Multiple Transformers

The Taiji program is dedicated to the detection of middle and low-frequency gravitational waves, targeting the 0.1 mHz to 1 Hz frequency band. The project requires an acceleration residual sensitivity of 3 × 10-15 ms-2/Hz1/2, which necessitates a capacitance sensing resolution of 1 aF/Hz1/2 for the capacitive sensing system within the specified frequency range. The noise level of the resonant bridge significantly influences the resolution. Addressing the challenges in enhancing transformer performance parameters in existing resonant capacitance bridges and the constraints on improving the characteristics of resonant capacitance bridges, this study introduces a novel approach to reduce bridge thermal noise without optimizing existing parameters. The simulation results demonstrate that this scheme can reduce the noise to 0.7 times the original level and further reduce bridge thermal noise when other parameters affecting noise are optimized. This not only mitigates the demands for other performance parameters but also increases the range of maximum acceptable resonant frequency deviations and reduces its sensitivity to such variations. Experimental validation confirms that the proposed scheme effectively reduces noise by 0.7 times and improves the resolution of capacitance sensing to 0.6 aF/Hz1/2, thereby advancing the Taiji program gravitational wave detection capabilities.

A method for high-precision measuring differential transformer asymmetry

The differential transformer is an important component in the front-end electronics of high-precision capacitive position sensing circuits, which are widely employed in space inertial sensors and electrostatic accelerometers. The position sensing offset, one of the space inertial sensors' most critical error sources in the performance range, is dominated by the differential transformer asymmetry and requires a high-precision evaluation. This paper proposes a method to assess differential transformers' asymmetry and realize a prototype circuit to test a transformer sample. The results show that the asymmetry measurement precision can achieve 0.6 ppm for the transformer with an asymmetry level of about -278.2 ppm.

A high precision surface potential imaging torsion pendulum facility to investigate physical mechanism of patch effect

The physical mechanism of the patch effect is still an open question. Thus, a high-precision surface potential mapping facility based on a specially designed electrostatically-controlled torsion pendulum is proposed in this paper. The facility not only features high sensitivity and a two-dimensional mapping function but also adapts to various measurement requirements for centimeter-sized samples. The sensitivity of the torsion pendulum reaches about 2.0 × 10^-14 N m/Hz^1/2 in a frequency range of 1-8 mHz. The temporal variation of the surface potential can be detected at a level of 30 μV/Hz^1/2 with a probe whose surface area is 7 mm². The potential spatial distribution resolution comes to 0.1 mm² at a level of 40 μV with 1 h integration time.

Bias Stability Investigation of a Triaxial Navigation-Compatible Accelerometer with an Electrostatic Spring

The bias stability performance of accelerometers is essential for an inertial navigation system. The traditional pendulous accelerometer usually has a flexible connection structure, which could limit the long-term bias stability. Here, based on the main technologies employed in previous space missions of our group, we developed a terrestrial triaxial navigation-compatible accelerometer. Because there is no mechanical connection between the inertial test mass and the frame, the bias performance relies on the stability of the equivalent electrostatic spring, where further sources are analyzed to get the optimal electrostatic force scheme. To investigate the bias stability under different ranges, the vertical and horizontal measurement ranges are designed at 5 g and ±10 mg, respectively. A low-noise high-voltage levitation scheme is adopted to extend the vertical measurement range from sub-mg to more than earth's 1-g gravity. Finally, the experimental validation results show that the 24-h bias stability of vertical and two horizontal directions come to 13.8 μg, 0.84 μg, and 0.77 μg, respectively.

Noise investigation of an electrostatic accelerometer by a high-voltage levitation method combined with a translation-tilt compensation pendulum bench

A high precision electrostatic accelerometer has widely been employed to measure gravity gradients and detect gravitational waves in space. The high-voltage levitation method is one of the solutions for testing electrostatic accelerometers on the ground, which aims at simultaneously detecting all six-degree-of-freedom movements of the electrostatic accelerometers engineering and flight prototypes. However, the noise performance in the high-voltage levitation test is mainly limited by seismic noise. The combined test of the accelerometer and vibration isolation platform is adopted to improve the detection precision of the high-voltage levitation method. In this paper, a high precision electrostatic accelerometer prototype is developed after designed appropriate mechanical parameters with a test mass weighing 300 g and with an estimated resolution of 2 × 10^-12 m/s²/Hz^1/2 from 0.01 to 0.4 Hz. Such a prototype is tested by the high-voltage levitation method, its measurement noise on the ground is mainly limited by the seismic noise, which is about 5 × 10^-7 m/s²/Hz^1/2 around 0.2 Hz and about 4 × 10^-8 m/s²/Hz^1/2 around 0.1 Hz. A vibration isolation pendulum bench based on the translation-tilt compensation principle is adopted for accelerometer prototype combined tests to suppress the seismic noise, which has a large bench area and the ability to adjust the tilt angle precisely. The measured accelerometer noise of the combined test with the translation-tilt compensation pendulum has reached 3 × 10^-9 m/s²/Hz^1/2 around 0.2 Hz, and it is about two orders of magnitude lower than the measurement noise on the ground. The combined test method provides technical guidance for further improving the noise level of ground test in the future.

Investigation on Stray-Capacitance Influences of Coaxial Cables in Capacitive Transducers for a Space Inertial Sensor

Ultra-sensitive inertial sensors are one of the key components in satellite Earth’s gravity field recovery missions and space gravitational wave detection missions. Low-noise capacitive position transducers are crucial to these missions to achieve the scientific goal. However, in actual engineering applications, the sensor head and electronics unit usually place separately in the satellite platform where a connecting cable is needed. In this paper, we focus on the stray-capacitance influences of coaxial cables which are used to connect the mechanical core and the electronics. Specially, for the capacitive transducer with a differential transformer bridge structure usually used in high-precision space inertial sensors, a connecting method of a coaxial cable between the transformer’s secondary winding and front-end circuit’s preamplifier is proposed to transmit the AC modulated analog voltage signal. The measurement and noise models including the stray-capacitance of the coaxial cable under this configuration is analyzed. A prototype system is set up to investigate the influences of the cables experimentally. Three different types and lengths of coaxial cables are chosen in our experiments to compare their performances. The analysis shows that the stray-capacitance will alter the circuit’s resonant frequency which could be adjusted by additional tuning capacitance, then under the optimal resonant condition, the output voltage noises of the preamplifier are measured and the sensitivity coefficients are also calibrated. Meanwhile, the stray-capacitance of the cables is estimated. Finally, the experimental results show that the noise level of this circuit with the selected cables could all achieve 1–2 × 10⁻⁷ pF/Hz^1/2 at 0.1 Hz.

A torque type full tensor gravity gradiometer based on a flexure-strip suspension

Full tensor gravity gradiometers based on accelerometer pair combination are widely employed in earth resource explorations using gravity gradient measurements. However, the extremely high requirements of accelerometer dynamic range and the scale factor consistency of accelerometer pairs are the two main limitations to further improve their performances. In order to overcome these two extreme challenges, the torque type gravity gradiometer based on the Etövös torsion balance pendulum is re-considered. In this paper, an improved torque type full tensor gravity gradiometer with a flexure-strip suspension is proposed, which balances the mechanical sensitivity and the response time. The proposed gradiometer can be used to measure the full tensor gravity gradient by observing angle variations at three azimuths. The principle and feasibility of the torque type full tensor gravity gradiometer based on a flexure-strip suspension are introduced, and the main noise sources including mechanical thermal noise, position sensing noise, and readout noise are analyzed. A prototype gravity gradiometer with a designed resolution of 2 E/Hz^1/2 at 0.1 Hz is constructed (1 E = 10^-9/s²), and the experimental results indicate that its resolution comes to 3 E/Hz^1/2 at 0.1 Hz, which is mainly limited by the seismic noise. This type of gravity gradiometer can be further improved due to its high potential resolution and independence of matching combination requirement, which allows it to be applied in next generation gravity resource exploration.

Design of a Carrier Wave for Capacitive Transducer with Large Dynamic Range

Capacitive transducers are widely used in fundamental physics experiments, seismology, Earth or planetary observations, and space scientific and technical applications because of their high precision, simple structure, and compatibility with various measurements. However, in real applications, there is a trade-off between their resolution and dynamic range. Therefore, this paper is aimed at enlarging the dynamic range while ensuring high resolution. In this paper, a noise analysis of a capacitive transducer is presented, which shows that the amplitude noise of the carrier wave is the main limiting factor. Hence, a new method of generating a carrier wave with lower-amplitude noise is proposed in the paper. Based on the experimental verification, it is found that the carrier wave produced through the new method performed significantly better than the typical digital carrier wave when they were compared in the same sensing circuit. With the carrier wave produced through the new method, the dynamic range of the capacitive transducer can reach 120.7 dB, which is 18.3 dB greater than for the typical direct digital synthesis (DDS) method. In addition, the resolution of the carrier wave is mainly limited by the voltage reference components.

Modeling and Analysis of the Noise Performance of the Capacitive Sensing Circuit with a Differential Transformer

Capacitive sensing is a key technique to measure the test mass movement with a high resolution for space-borne gravitational wave detectors, such as Laser Interferometer Space Antenna (LISA) and TianQin. The capacitance resolution requirement of TianQin is higher than that of LISA, as the arm length of TianQin is about 15 times shorter. In this paper, the transfer function and capacitance measurement noise of the circuit are modeled and analyzed. Figure-of-merits, including the product of the inductance L and the quality factor Q of the transformer, are proposed to optimize the transformer and the capacitance measurement resolution of the circuit. The LQ product improvement and the resonant frequency augmentation are the key factors to enhance the capacitance measurement resolution. We fabricated a transformer with a high LQ product over a wide frequency band. The evaluation showed that the transformer can generate a capacitance resolution of 0.11 aF/Hz^1/2 at a resonant frequency of 200 kHz, and the amplitude of the injection wave would be 0.6 V. This result supports the potential application of the proposed transformer in space-borne gravitational wave detection and demonstrates that it could relieve the stringent requirements for other parameters in the TianQin mission.

A method for reducing the adverse effects of stray-capacitance on capacitive sensor circuits

We examine the increase in voltage noise in capacitive sensor circuits due to the stray-capacitance introduced by connecting cables. We have measured and modelled the voltage noise of various standard circuits, and we compare their performance against a benchmark without stray-capacitance that is optimised to have a high signal-to-noise ratio for our application. We show that a factor limiting sensitivity is the so-called noise gain, which is not easily avoided. In our application, the capacitive sensor is located in a metallic vessel and is therefore shielded to some extent from ambient noise at radio frequencies. It is therefore possible to compromise the shielding of the coaxial connecting cable by effectively electrically floating it. With a cable stray-capacitance of 1.8 nF and at a modulation frequency of 100 kHz, our circuit has an output voltage noise a factor of 3 larger than the benchmark.

A Subnano-g Electrostatic Force-Rebalanced Flexure Accelerometer for Gravity Gradient Instruments

A subnano-g electrostatic force-rebalanced flexure accelerometer is designed for the rotating accelerometer gravity gradient instrument. This accelerometer has a large proof mass, which is supported inversely by two pairs of parallel leaf springs and is centered between two fixed capacitor plates. This novel design enables the proof mass to move exactly along the sensitive direction and exhibits a high rejection ratio at its cross-axis directions. Benefiting from large proof mass, high vacuum packaging, and air-tight sealing, the thermal Brownian noise of the accelerometer is lowered down to less than 0.2 ng / Hz with a quality factor of 15 and a natural resonant frequency of about 7.4 Hz . The accelerometer's designed measurement range is about ±1 mg. Based on the correlation analysis between a commercial triaxial seismometer and our accelerometer, the demonstrated self-noise of our accelerometers is reduced to lower than 0.3 ng / Hz over the frequency ranging from 0.2 to 2 Hz, which meets the requirement of the rotating accelerometer gravity gradiometer.

Research and Development of Electrostatic Accelerometers for Space Science Missions at HUST

High-precision electrostatic accelerometers have achieved remarkable success in satellite Earth gravity field recovery missions. Ultralow-noise inertial sensors play important roles in space gravitational wave detection missions such as the Laser Interferometer Space Antenna (LISA) mission, and key technologies have been verified in the LISA Pathfinder mission. Meanwhile, at Huazhong University of Science and Technology (HUST, China), a space accelerometer and inertial sensor based on capacitive sensors and the electrostatic control technique have also been studied and developed independently for more than 16 years. In this paper, we review the operational principle, application, and requirements of the electrostatic accelerometer and inertial sensor in different space missions. The development and progress of a space electrostatic accelerometer at HUST, including ground investigation and space verification are presented.

A Novel Controller Design for the Next Generation Space Electrostatic Accelerometer Based on Disturbance Observation and Rejection

The state-of-the-art accelerometer technology has been widely applied in space missions. The performance of the next generation accelerometer in future geodesic satellites is pushed to 8×10−13m/s2/Hz1/2, which is close to the hardware fundamental limit. According to the instrument noise budget, the geodesic test mass must be kept in the center of the accelerometer within the bounds of 56 pm/Hz1/2 by the feedback controller. The unprecedented control requirements and necessity for the integration of calibration functions calls for a new type of control scheme with more flexibility and robustness. A novel digital controller design for the next generation electrostatic accelerometers based on disturbance observation and rejection with the well-studied Embedded Model Control (EMC) methodology is presented. The parameters are optimized automatically using a non-smooth optimization toolbox and setting a weighted H-infinity norm as the target. The precise frequency performance requirement of the accelerometer is well met during the batch auto-tuning, and a series of controllers for multiple working modes is generated. Simulation results show that the novel controller could obtain not only better disturbance rejection performance than the traditional Proportional Integral Derivative (PID) controllers, but also new instrument functions, including: easier tuning procedure, separation of measurement and control bandwidth and smooth control parameter switching.

Novel Capacitive Sensing System Design of a Microelectromechanical Systems Accelerometer for Gravity Measurement Applications

This paper presents an in-plane sandwich nano-g microelectromechanical systems (MEMS) accelerometer. The proof-mass fabrication is based on silicon etching through technology using inductive coupled plasma (ICP) etching. The capacitive detection system, which employs the area-changing sensing method, combines elementary capacitive pickup electrodes with periodic-sensing-array transducers. In order to achieve a large dynamic range with an ultrahigh resolution, the capacitive detection system employs two periodic-sensing-array transducers. Each of them can provide numbers for the signal period in the entire operating range. The suspended proof-mass is encapsulated between two glass caps, which results in a three dimensional structure. The measured resonant frequency and quality factor (Q) are 13.2 Hz and 47, respectively. The calibration response of a ±0.7 g input acceleration is presented, and the accelerometer system presents a sensitivity of 122 V/g and a noise floor of 30 ng/√Hz (at 1 Hz, and 1 atm). The bias stability for a period of 10 h is 30 μg. The device has endured a shock up to ±2.6 g, and the full scale output appears to be approximately ±1.4 g presently. This work presents a new opportunity for highly sensitive MEMS fabrication to enable future high-precision measurement applications, such as for gravity measurements.

Research on supporting mounts of spheres in measurement of gravitational constant G

The ongoing precision measurement of the gravitational constant G at our group is performed by using two different kinds of methods: time-of-swing method (ToS) and angular acceleration feedback method. In the two methods, the stainless steel spheres are employed as source masses, and the position stability of the spheres is an important parameter, which make suitable mounts for supporting the spheres needed extremely. In this paper, an upgraded three-point mount is introduced and tested in detail. Experimental results show that, for the sphere supported by the three-point mount used in the ToS method, the repeatability, the temperature influence, and the vibration influence are all less than 0.1 μm (about 2 ppm for the value of G). For the sphere supported by the three-point mount used in the AAF method, similar results are obtained, the largest change of the sphere's position is about 0.6 μm, introduced by a temperature change of 1 °C, which also results in an uncertainty of 2 ppm for the value of G.

Implementation of the scale factor balance on two pairs of quartz-flexure capacitive accelerometers by trimming bias voltage

Gravity gradient measurement makes use of the difference between the outputs of pairs of linear accelerometers, which results in cancelling out the common mode accelerations caused by mounting platform and external environment. One of the key technologies is to match the acceleration-to-voltage or acceleration-to-current transfer functions of the pairs of the accelerometers to an extremely high degree of accuracy. The differential signals then make the gravity gradients observable. By using two pairs of the quartz-flexure accelerometers with a capacitive sensing and electrostatic closed-loop control, the electrostatic control bias voltages were trimming remotely and automatically in real time. Each pair of accelerometers was matched individually and then all four accelerometers were finally re-balanced. The experimental results show that the consistency of five digits is achieved at a noise level of ~5×10(-8) g/√Hz (1 g ≈ 9.8 m/s(2)) and the scale factors ranging from 0.25 to 0.32 V/mg. Further improvement to the achieved level of matching is limited by the intrinsic noise of the accelerometers used.