Study on driving noise of quartz flexible accelerometer torquer

The distributed capacitance inside the quartz flexible accelerometer (QFA) coupled the high frequency voltage excitation signal in the differential capacitance detection circuit to the torquer coil, and superimposes the torquer driving current to form the driving noise. In this study, the values of the distributed capacitance inside the QFA were simulated. According to the formation mechanism of the QFA driving noise, the equivalent circuit model of the driving noise is established, and the driving noise characteristics of the detection circuit with single excitation and double excitation source are analyzed. The theoretical and experimental results show that the electric field coupled driving noise transmission system is a first-order system with high-pass characteristics. The driving noise of the single excitation detection circuit is larger than that of the dual excitation detection circuit (DEDC), and the DEDC can reduce the driving noise by 39.77% when the QFA shell is grounded. The equivalent acceleration of the electric field coupled driving noise is between tens of μg to hundreds of mg, which is one of the important noise sources that affect the measurement accuracy of the QFA. A measure was proposed to suppress the high-frequency driving noise by adding a low-pass filter after the sampling output of the driving current, which can reduce the driving noise to 1.85 μg and effectively reduce the influence of the driving noise on the measurement accuracy of the QFA.


Introduction
Quartz flexible accelerometer (QFA) is the main type of accelerometer in current inertial navigation system. Its measurement accuracy and stability have become one of the factors * Author to whom any correspondence should be addressed.
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restricting the further improvement of navigation performance of inertial navigation system [1]. In recent years, the research on the influence of temperature [2][3][4][5], magnetic circuit [6][7][8], pendulum reed structure [9][10][11], calibration error [12][13][14][15][16] and other factors on the performance of QFA has been relatively mature, but the research on the internal components of QFA and readout circuit generated noise during operation is relatively few. Zhou et al [17] designed a single carrier modulated differential capacitance detection circuit, which can effectively eliminate the influence of parasitic resistance and capacitance and theoretically ensure the stability of sensitivity. Wu et al [18] proposed a dynamic modelling method for QFA, which adopted sine wave excitation to obtain phase-frequency characteristics and cross-correlation algorithm to reduce the influence of noise. Gang et al [19] established the internal torquer noise circuit model of the QFA, studied the influence of mechanical thermal noise of the torquer and resonant noise of the torquer coil on the QFA, and found that the influence of resonant noise is greater than that of mechanical thermal noise. Huang et al [20] established the noise model of readout circuit, analyzed the noise of charge-discharge detection circuit and its equivalent acceleration formula. It is found that the noise of charge-discharge method is about 1 µg, and its influence on the high-precision accelerometer cannot be ignored. Ran et al [21] proposed a method to estimate the gain change using modulated signal and feedback signal respectively, which significantly improved the long-term stability of digital accelerometer. Wu et al [22] derived the transfer function of detection noise in the closed-loop system of the accelerometer, and found that the transfer function of detection noise has the characteristics of low frequency attenuation and high frequency amplification. Li et al [23] designed a fully differential detection circuit with automatic gain control to solve the problem that large parasitic capacitance affects the output dynamic range of readout circuit, which makes the detection accuracy of differential capacitance less than 1 fF and effectively improves the detection circuit accuracy of accelerometer. In addition, Lin et al [24] designed an accelerometer detection circuit based on ring diode, with detection sensitivity up to 64.4 mV pF −1 .
In the above study, the effects of noise generated by the internal components of the accelerometer or parasitic capacitance noise of the detection circuit on the performance of the accelerometer are considered separately. However, the influence of the coupled between the internal components and the detection circuit on the performance of the accelerometer is not considered. When the QFA is working, electromagnetic coupling occurs between the torquer coil and the differential capacitance sensor, which results in bidirectional interference. Our previous studies have found that when the driving signal in the torque coil changes rapidly, electromagnetic radiation will be generated and coupled to the differential capacitor plate, which is transmitted to the differential capacitor detection circuit and affects the measurement accuracy of the QFA [25]. In fact, the high frequency voltage excitation signal in the differential capacitor detection circuit will also be transmitted to the torquer coil through electromagnetic radiation to form driving noise, but the related characteristics of driving noise are not clear. Therefore, this paper studies the characteristics of the electric field coupled driving noise of the QFA torquer.

Working principle of the closed-loop system of QFA
QFA is mainly composed of quartz pendulum component, torquer, differential capacitance sensor, shell, etc [26]. Its structure is shown in figure 1(a). Where, the torquer coil is wound on the coil frame and bonded on both sides of the quartz pendulum reed, forming a quartz pendulum component. The magnetic steel, the magnetic pole piece and the compensation ring are glued together to form a magnetic steel component, and the torquer coil is in the magnetic gap formed by the upper and lower magnetic steel components. The torquer coil, yoke iron, magnetic steel components and bellyband form the torquer. The top and bottom surfaces of the quartz pendulum are coated with metal film, which forms a differential capacitance sensor with the upper and lower yoke iron. The upper and lower yoke iron and the bellyband are welded by laser to form an accelerometer head and fixed into the shell by a spacer ring made of insulation material. Finally, a QFA is assembled.
The quartz pendulum component, differential capacitance sensor and torquer inside the QFA together with the external differential capacitance detection circuit, torquer control unit, the torquer drive circuit and the driving current sampling circuit constitute the closed-loop system of the QFA shown in figure 1(b). When the acceleration occurs in the direction of the sensitive axis, the pendulum component deviates from the equilibrium position under the action of inertia moment M a . The deflection angle ∆θ of the quartz pendulum component is reflected by the change value ∆C of the differential capacitance sensor. The differential capacitance detection circuit converts ∆C to voltage signal V C . The V C is converted into a torquer driving current I T through the torquer control unit and torquer drive circuit, and feed back to the torquer coil. The torquer coil is in a constant magnetic field, generating a magnetic torque M b that balances with M a , which makes the pendulum component return to the equilibrium position again. The closed-loop operation of the accelerometer system is realized through the above process [2].
The closed-loop system transfer function H VT ai (s) of the QFA can be expressed as: where, mL is the pendulosity of the pendulum component, θ (s), K s , K a , M (s), K I , K T and K U are the transfer functions of the pendulum component, differential capacitance sensor, differential capacitance detection circuit, torquer control unit, torquer drive circuit, torquer and driving current sampling circuit, respectively. The transfer function θ (s) of the pendulum component is: Where, J is the moment of inertia of the pendulum component around the output shaft, C is the damping coefficient, K is the elastic coefficient. According to equation (1), the torquer driving current I T is proportional to the input acceleration a i . By measuring the output voltage V T of the sampling circuit, the driving current I T of the torquer can be calculated, and then the input acceleration a i can be obtained. The direction of a i depends on the direction of the driving current I T .

The formation mechanism of distributed capacitance inside the QFA
The shell, yoke iron, torquer coil, and differential capacitor plate of the QFA are made of metal materials, and these components are not in contact with each other. When the QFA is working, there is an electric potential difference between different components, forming a distributed capacitance. The distributed capacitance among the components inside the QFA is shown in figure 2(a). Where, C d1 , C d2 and C d3 are the distributed capacitance between the three plates of the differential capacitor and the torquer coil, respectively. C bb is the distributed capacitance between the top and bottom plates of the differential capacitor. C e1 , C e2 and C e3 are the distributed capacitance between the three plates of the differential capacitor and the shell, respectively. C e4 is the distributed capacitance between the shell and the torquer coil. The torquer coil serves as the electrode plate for distributed capacitance, coupling the excitation signal V s of the differential capacitance detection circuit to the sampling resistor, and superimposed with the driving current I T to form the current driving noise I n , resulting in acceleration measurement errors. Based on this mechanism, the equivalent circuit model of the internal electric field coupled driving noise of the QFA shown in figure 2(b) is established to describe the conduction process from excitation signal V s to noise I n . In the model, V s is the AC excitation source used in the differential capacitance detection circuit, and R CS is the current sampling resistor.

Simulation of distributed capacitance in QFA
When analyzing the electric field coupled driving noise characteristics inside the QFA, the value and variation law of each distributed capacitance must be determined first. In this paper, the electrostatic field simulation method is used to analyze and solve the value of the distributed capacitance. According to the assembly structure and materials of a typical QFA, the QFA simulation model shown in figure 3 is established.
The key performance characteristics of QFA are shown in table 1.
The main structural geometry and the relative dielectric constant of the material are shown in tables 2 and 3 respectively.
Voltage excitation was applied to each component according to the actual working condition of the QFA. The voltage excitation settings are shown in table 4.  The value of the distributed capacitance inside the QFA depends on the area directly opposite and the distance between the metal components. When the acceleration in the direction of the sensitive axis changes, only the pendulum component undergoes the deflection shown in figure 4, causing changes in the area directly opposite and distance between some components. Therefore, only the relationship between the value of the distributed capacitance and the deflection angle ∆θ of the pendulum component needs to be discussed.
In this paper, the variation of the differential and distributed capacitance is simulated for the deflection angle ∆θ from 0 mrad to the maximum deflection angle of 1.15 mrad. The simulation accuracy is 1%, and the results are shown in figure 5. The differential capacitance C s1 monotonically  decreases and C s2 monotonically increases with increasing ∆θ. When the deflection angle of the pendulum component is very small, the reduction value of C s1 is approximately equal to the increase value of C s2 . The variation rule of C s1 and C s2 values can be expressed as: where, ∆C is the change value of differential capacitance, C 0 is the initial value of differential capacitance when the pendulum component is in the balance position. The simulation result of C 0 is 33.97 pF. The relationship between the values of C s1 and C s2 and the deflection angle of the pendulum component agrees with the characteristics of the differential capacitance sensor of the QFA [27]. The average deviation of the distributed capacitors C e2 , C e3 and C e4 is about 3.5%, which is slightly larger than the set simulation accuracy. The main reason is that the values of these distributed capacitors are very small, and it is difficult to solve them accurately. The average deviation of other distributed capacitors is less than 1%, which agrees with the set simulation accuracy.
The simulation results indicate that the value of the capacitance is independent of the deflection angle of the pendulum component, which can be explained by the structure of the QFA. When the pendulum component deflects, there is no relative motion among the torquer coil, coil frame and quartz pendulum reed, so the distributed capacitors C d1 and C d2 are independent of the pendulum component deflection angle. When the torquer coil is deflected with the pendulum reed, it is always between the upper and lower yoke iron. If the area directly opposite of the torquer coil and the yoke iron on one side decreases, the area directly opposite with the other side increases, and the increases and decreases equally, so C d3 always remains unchanged. The metal coatings on both sides of the pendulum reed are always relatively static, and C bb is fixed. The yoke iron and the shell are fixed, and C e1 is fixed. In addition, the internal components of the QFA head can only form an area directly opposite the shell through the holes in the binding posts. This area is very small, so the values of C e2 , C e3 and C e4 are extremely small. Therefore, when using the electric field coupled driving noise equivalent circuit model to analyze the driving noise characteristics, the distributed capacitance C e2 , C e3 and C e4 can be ignored and the other distributed capacitance values are fixed values as shown in table 5.

Characteristic analysis of electric field coupled driving noise
According to the equivalent circuit model of electric field coupled driving noise inside the QFA, the transmission system structure of electric field coupled driving noise is related to the type of differential capacitance detection circuit. The common types of differential capacitance detection circuits mainly include capacitance divider, switch capacitance integrator, ring diode and carrier modulation type detection circuit. These detection circuits can be divided into two categories based on the number of external excitation sources: single excitation detection circuit (SEDC) and dual excitation detection circuit (DEDC). The excitation input modes of the two categories of detection circuits are shown in figure 6. The SEDC inputs voltage excitation V s to the static plate of the differential capacitor, and outputs the variation value of the differential capacitor from the top and bottom plates to the detection circuit. The DEDC inputs a pair of reverse voltage excitation V s and −V s to the top and bottom plates of the differential capacitor, and outputs the differential capacitance change value from the static plate to the detection circuit. Furthermore, the QFA shell has two connection modes of grounded and float. The shell grounded is a common electromagnetic shielding method. The float shell is to prevent the internal components from being destroyed in use. The combination of two categories of differential capacitance detection circuit and two connection modes of QFA shell constitute a different structure of the driving noise transmission system.

Driving noise equivalent circuit of SEDC
When SEDC is adopted, the driving noise equivalent circuit is shown in figure 7(a). Where, C e1 in parallel with the excitation source does not affect the driving noise, and can be ignored. The ∆-connection constituted by C d1 , C d2 and C bb is equivalently transformed into Y-connection, as shown in figure 7(b). The equivalent capacitance in the Y-connection can be expressed as: The total capacitance C n of the distributed capacitive network can be expressed as: The transfer function from excitation source V s to driving noise I n−s of SEDC can be expressed as: where, ω s is the frequency of excitation source V s .

Driving noise equivalent circuit of DEDC
When DEDC is adopted, the driving noise equivalent circuit is shown in figure 8(a). The Y-connection constituted by C s1 , C s2 and C d3 is equivalently transformed into ∆-connection. According to the superposition theorem, the equivalent circuit of driving noise is established when the excitation source V s and −V s are enabled separately. This process is shown in figure 8(b). When V s is enabled, −V s is short circuited, point b is grounded. Parallel connection of C bb and C 23 with the excitation source V s does not affect the driving noise. The situation is the same when −V s is enabled, so C bb and C 23 can be ignored. When the shell is floating, the equivalent capacitance in the ∆-connection can be expressed as: The driving noise I n1−dF when the excitation source V s is enabled and the driving noise I n2−dF when −V s is enabled can be expressed as: When the shell is floating, the driving noise I n−dF of the DEDC is the sum of the driving noise when V s and −V s are enabled separately. The transfer function of the excitation source to the driving noise I n−dF can be expressed as: When the shell is grounded and the excitation source V s is enabled, point b is grounded, C e1 and C s2 are in parallel, the equivalent capacitance in the ∆-connection can be expressed as: When the shell is grounded and the excitation source −V s is enabled, point a is grounded, C e1 and C s1 are in parallel, the equivalent capacitance in the ∆-connection can be expressed as: Similarly, when the shell is grounded, the driving noise I n−dG of the DEDC is the sum of the driving noise when V s and −V s are enabled respectively. The transfer function of the excitation source to the driving noise I n−dG can be expressed as:

Characteristics of electric field coupled driving noise
According to the driving noise transfer functions of different detection circuits, the driving noise transfer system is a firstorder system with high-pass characteristics. The gain of the driving noise transfer system is related to the differential capacitance C s1 and C s2 , that is, to the change value ∆C of the differential capacitance. When the driving current sampling resistance R CS = 1 Ω and ∆C is 0 pF and 1 pF respectively, the amplitude-frequency characteristic curve of the driving noise transfer function is shown in figure 9(a). When ∆C = 0 pF, the transmission characteristics of the driving noise of the DEDC are the same in the floating and grounded state of the QFA shell; while ∆C = 1 pF, the shell grounded can reduce the driving noise. ∆C = 0 pF and 1 pF, the transmission characteristics of driving noise of the SEDC are unchanged. When the frequency of the excitation source V s is 100 kHz, the relationship between the magnitude response of the driving noise transmission system and the variation value ∆C of differential capacitance is shown in figure 9(b). Where, the magnitude response of the driving noise transmission system of the SEDC remains unchanged; With the increase of ∆C, the magnitude response of the driving noise transmission system of the DEDC decreases first and then increases. When the shell is floating, the driving noise transmission system of the DEDC reaches the minimum value at ∆C = 0.032 pF. When the shell is grounded, the system reaches the minimum value at ∆C = 0.064 pF. When ∆C < 0.043 pF, the magnitude response of the shell floating is slightly less than that of the shell grounded. When ∆C > 0.043 pF, the magnitude response of the shell grounded is obviously smaller than that of the shell floating. According to the analysis, the following conclusions are drawn: 1. The magnitude response of the driving noise transmission system of the SEDC is greater than that of the driving noise transmission system of the DEDC. 2. When ∆C < 10 pF, the shell grounded can reduce the magnitude response of the driving noise transfer system.

Transmission characteristics of driving noise in closed-loop system of QFA
The closed  figure 10(a). The transfer function from driving noise I n to sampling output V n is:

H Vn
In ( jω s ) = The amplitude-frequency characteristic curves of H Vn In ( jω s ) and H VT ai (s) are shown in figure 10(b). The driving noise is a high-frequency signal of more than 10 kHz has a fixed gain in the closed-loop system. The transfer gain depends on K U , that is, H Vn In ( jω s ) = K U . Therefore, the transfer function from the excitation source V s to the equivalent acceleration a n of the driving noise can be expressed as: where, K an Vn is the transfer function between the sampling output V n and the equivalent acceleration a n of the driving noise, which is equal to the reciprocal of the product of the sampling circuit transfer function K U and the scaling factor K 1 , that is,

Suppression measures of electric field coupled driving noise
According to the transfer function from the excitation source V s to the equivalent acceleration a n of the driving noise, reducing V s can reduce the driving noise, but it will reduce the sensitivity of the differential capacitance detection circuit, Figure 11. The optimized closed-loop system of QFA. which will lead to the reduction of the measurement accuracy of the differential capacitance. Therefore, the driving noise can only be suppressed during the transmission from I n to V n , that is, the driving noise can be suppressed by optimizing the sampling chain of the driving current. The input signal of the sampling chain includes two parts: the torquer driving current I T and the driving noise I n . Where, the frequency of I T is the low-frequency signal within the bandwidth range of the system, and I n is the high-frequency signal above 10 kHz. Therefore, a low-pass filter can be added after the sampling output K U to suppress the driving noise. The optimized closed-loop system of the QFA is shown in figure 11. The design of a fourth-order Butterworth low-pass filter is shown in figure 12(a). The cut-off frequency of the lowpass filter is 1 kHz, the gain is 0 dB, the stopband frequency is 10 kHz, and the stopband attenuation is −80 dB. The amplitude-frequency characteristics of the optimized closedloop system are shown in figure 12(b). The low-pass filter can effectively suppress the high-frequency driving noise, and the bandwidth of the closed-loop system is not affected.

Experimental devices and methods
In this paper, an experimental platform for electric field coupled driving noise inside the QFA was built to measure the value of electric field coupled driving noise and verify the effectiveness of driving noise suppression measures. The schematic and physical map of the experimental platform are shown in figures 13(a) and (b). Where, The DC power supply provides the driving current I T for the torquer to control the deflection of the pendulum component. The LCR meter is used to measure the values of the differential capacitors C s1 and C s2 and to analyze the degree of deflection of the pendulum component. The signal generator is used to simulate the external excitation source of the differential capacitor detection circuit, generate a single or dual excitation signal V s , and input it to the plate of the differential capacitor. The field coupled driving noise sampling output V n has the same frequency as the excitation signal V s , so the field coupled driving noise can be measured using correlation detection technology. The excitation signal V s was used as synchronous demodulation signal, and the sampled output V n of the electric field coupled driving noise was measured by a phase-locked amplifier. This measurement method can detect a specific carrier frequency signal with minimal amplitude. The LI5655 digital phaselocked amplifier used on the platform has a frequency range of 0.5 Hz-3 MHz and can detect voltage signals of 10 nV-1 V, meeting the measurement requirements for driving noise. The oscilloscope is used to monitor the phase of the excitation signal to make sure that the phase difference between the excitation signals V s and −V s in the dual excitation mode is 180 degrees. The switch S 1 is used to switch the connection mode of the QFA shell. S 1 is closed, the shell is grounded.
The electric field coupled driving noise measurement method is as follows: firstly, input driving current I T to the torquer coil, and measure the values of the differential capacitance C s1 and C s2 . I T is fine-tuned so that the change value ∆C of the differential capacitance is less than 3 pF. At this time, the pendulum component is near the equilibrium position. Then disconnect the LCR meter from the QFA to prevent the LCR meter from interfering with the driving noise measurement. Second, the peak-to-peak value is set for the excitation signal of the output signal of the signal generator as 5 V and the DC offset as 2.5 V, which is consistent with the actual value of V s . By changing the connection mode between the excitation signal and the differential capacitor plate, the working state of the QFA with SEDC and DEDC is simulated successively. Two simulations are made for the connection modes of shell grounded and floating by changing the open and closed state of switch S 1 . Under different working conditions of the QFA, the frequency of V s was gradually increased from 10 kHz to 3 MHz. The effective value of the driving noise sampling voltage V n is recorded from the phase-locked amplifier, so as to obtain the value of the electric field coupled driving noise at different V s frequencies.
The validity verification method of driving noise suppression measures is as follows: Firstly, the output of the signal generator is connected to the input of the low-pass filter, the output of the low-pass filter to the phase-locked amplifier, to measure the amplitude-frequency characteristics of the lowpass filter in the 1 Hz-1 MHz frequency band. Then the lowpass filter is connected between the current sampling output and the input of the phase-locked amplifier, to measure the value of the driving noise after passing through the low-pass filter, and to calculate the attenuation rate of the driving noise.

Experimental devices and methods
When the differential capacitance C s1 = 29 pF and C s2 = 34 pF, the relationship between the electric field coupled driving noise I n of the QFA and the excitation source V s frequency measured on the experimental platform is shown in figure 14. The driving noise I n increases with the increase of V s frequency, which is agreed with theoretical analysis. Where, the measured driving noise I n−s of the SEDC is slightly smaller than the theoretical value, and the average relative error between the measured value and the theoretical value is 16%. When the shell floating, the average relative error between the measured value and the theoretical value of the driving noise I n−dF of the DEDC is 9.6%. When the shell is grounded, the average relative error between the measured value and the theoretical value of the driving noise I n−dG of the DEDC is 32.5%. The main reason for the large I n−dG error is that the simulated distributed capacitance C e1 is relatively large, and the measured value of C e1 is 55 pF. The simulation value of C e1 is larger than the actual value due to the slightly larger relative dielectric constant set by the spacer ring and filling glue during simulation. When C e1 is 55 pF, the average relative error between the theoretical value of I n−dG and the measured value is 9.8%. In summary, the error between the measured value and the theoretical value of the electric field coupled driving noise is about 10%. These errors are mainly caused by the difference between the actual value and the simulated value of the distributed capacitance and the measurement error of the experimental instrument. The experimental error is within a reasonable range, and the experimental results are agreed with the theoretical analysis.
By substituting the measured value of electric field coupled driving noise I n into equation (14), the relationship between the equivalent acceleration of electric field coupled driving noise a n and the frequency of excitation source V s is shown in figure 15. When the V s frequency of the excitation source is in the range of 10 kHz-3 MHz, the minimum and maximum driving noise equivalent acceleration a n−s of the SEDC are 1.28 mg and 338.28 mg. When the shell floating, the minimum and maximum driving noise equivalent acceleration a n−dF of the DEDC are 0.088 mg and 27.53 mg. When the shell is grounded, the minimum and maximum driving noise equivalent acceleration a n−dG of the DEDC are 0.049 mg and 14.61 mg.
The measured amplitude-frequency characteristics of a low-pass filter at 1 Hz-1 MHz are shown in figure 16(a). When the effective value of the input signal is 1 V and the frequency is less than 25 kHz, the amplitude-frequency characteristics of the low-pass filter are agreed with the theory. When the input frequency is greater than 25 kHz, the magnitude response of the low-pass filter does not decrease continuously and maintains between −80 dB to −120 dB. The main reason is that the operational amplifier has voltage noise, and the filtered output signal amplitude is extremely small, which is submerged by the voltage noise. The measured results of the low-pass filter show that the attenuation of the low-pass filter to signals with frequencies greater than 10 kHz is greater than −80 dB, which can be used to suppress the electric field coupled driving noise.
The driving noise of the SEDC is larger than that of the DEDC. Therefore, the closed-loop system of the QFA with the SEDC is experimented to verify the suppression effect of the low-pass filter on the electric field coupled driving noise. The low-pass filter was connected between the current sampling output and the lock-in amplifier input, and the filtered electric field coupled driving noise was measured. The driving noise value before and after filtered and the gain of the driving noise are shown in figure 16(b). The V s frequency of the excitation source is in the range of 10 kHz-1 MHz, and the filtered driving noise is about 20 nA. At this time, the equivalent acceleration of the driving noise is 1.85 µg, which has reached the voltage noise range of the operational amplifier. The experimental results show that adding a low-pass filter after the driving current sampling output can better suppress the electric field coupled driving noise.
The research and experimental results show that, depending on the category of differential capacitance detection circuit, the equivalent acceleration of electric field coupled driving noise ranges from tens of µg to hundreds of mg, which is one of the important noise sources affecting the measurement accuracy of the accelerometer. The transmission system of driving noise is a first-order system with high-pass characteristics. The lower the frequency of excitation source used in differential capacitor detection circuit, the smaller the electric field coupled driving noise. When designing the differential capacitance detection circuit, the lower the frequency of the excitation signal should be selected as far as possible. The electric field coupled driving noise of DEDC is less than that of SEDC. When the DEDC is used, the equivalent acceleration of driving noise when the shell is grounded is 39.77% less than that when the shell floating, which is more conducive to reducing the electric field coupled driving noise. The electric field coupled driving noise suppression method proposed in this paper can reduce the driving noise to 1.85ug, which has a good suppression effect. However, this suppression measure increases the size of the accelerometer servo circuit, and is affected by the inherent voltage noise of the operational amplifier. Theoretically, it is difficult for this method to attenuate the driving noise below 1 µg. Therefore, starting from the structure of the accelerometer, the internal distributed capacitance of the accelerometer should be reduced as much as possible by optimizing the structure of the accelerometer, so as to further reduce the electric field coupled driving noise of the QFA.

Conclusion
In this paper, the formation mechanism and transmission characteristics of electric field coupled driving noise in QFA are studied. The value of distributed capacitance in the QFA is solved by electrostatic field simulation, and the electric field coupled driving noise equivalent circuit model is established. The experiment proves the existence of the electric field coupled driving noise of the QFA, and measures the value of the electric field coupled driving noise of different categories of differential capacitance detection circuit and its relationship with the frequency of the excitation source. The results show that the equivalent acceleration of the electric field coupled driving noise ranges from tens of µg to hundreds of mg. When the high precision QFA is developed, the influence of the distributed capacitance in the QFA on the acceleration measurement must be considered. By adding a low-pass filter after the driving current sampling output, the driving noise can be reduced to 1.85 µg, which can effectively reduce the influence of driving noise on the measurement accuracy of the QFA. The results can be used to evaluate the driving noise of the accelerometer and provide reference for improving and optimizing the accelerometer.

Data availability statement
The data cannot be made publicly available upon publication because no suitable repository exists for hosting data in this field of study. The data that support the findings of this study are available upon reasonable request from the authors.