Influence of the Injection Bias on the Capacitive Sensing of the Test Mass Motion of Satellite Gravity Gradiometers

The performance of the capacitive gap-sensing system plays a critical role in a satellite-based gravity gradiometer that is developed using an electrostatic accelerometer. The capacitive sensing gain mainly depends on the stabilized injection bias amplitude, the gain of the transformer bridge, and the trans-impedance amplifier. Previous studies have indicated that amplitude noise is the main factor influencing the noise of capacitive displacement detection. Analyzing the capacitive gap-sensing system indicates that the amplitude, frequency, phase, and broadband noises of the stabilized injection bias have varying levels of influence on the performance of the detection system. This paper establishes a model to clarify the mentioned effects. The validation of the sub-tests demonstrates that the analysis and evaluation results of various noise coefficients are highly consistent with the model’s predicted outcomes.


Introduction
With the sequential launches and operations of gravity gradient measurement satellites such as GRACE/COCE, it is vital to develop high precision gravity gradient measurements.The effectiveness of the capacitive gap-sensing system, integral to an onboard gravity gradiometer constructed from an electrostatic accelerometer, plays a pivotal role in achieving the performance of the gradiometer.This system is instrumental in both space gravity experiments and Earth gravity field recovery [1][2][3].
Gravity Field and steady-state Ocean Circulation Explorer (GOCE) was the first satellite mission equipped with a gravitational gradiometer [4][5][6].Initiated in 1998, the GOCE mission aimed to achieve a noise level of about 5 mE/ √ Hz in the Electrostatic Gravity Gradiometer (EGG) across a measurement bandwidth (MBW) ranging from 5 mHz to 0.1 Hz [7].The National Aerospace Research Agency (ONERA) has significantly advanced in developing high-precision detectors capable of measurements with a resolution of up to 10 −7 pF/ √ Hz.Additionally, ONERA has examined the sensor output both with and without carrier signals on the capacitance bridge, assessing the impact of stabilized injection bias amplitude on stability [8,9].Weber et al. developed a capacitive gap-sensing system targeting a resolution near 10 −6 pF/ √ Hz, focusing on the 0.1 mHz to 0.1 Hz bandwidth relevant to the Laser Interferometer Space Antenna (LISA).Their findings emphasize the necessity of the relative stability of the stabilized injection bias being better than 2 × 10 −4 / √ Hz [10,11].Based on Armano et al.'s studies, the capacitive detection noise of the orbiting LISA Pathfinder spaceship (LPF) is lower than 1.8 × 10 −6 pF/ √ Hz at frequencies above 1 mHz, whose equivalent displacement detection noise is 2.4 × 10 −9 m/ √ Hz.They found that the amplitude of the carrier wave directly influences the displacement detection noise [12].Dolesi et al. discussed the current design and noise model for the European gravity sensor (GS) in the LISA Technology Package (LTP), featuring a capacitive gap-sensing system with a resolution of 2.4 × 10 −9 m/ √ Hz, a 10 µm bias, and a 100 ppm/ √ Hz stability of the stabilized injection bias [13].In order to achieve a resolution of ( in the frequency band of 5 mHz to 0.1 Hz, the sinusoidal carrier signal employed in the high-precision displacement detection circuit should produce as little residual noise as possible [9,[14][15][16][17][18].In the scope of this research, the stabilized injection bias utilizes a sinusoidal carrier waveform, which aligns fundamentally with the waveform presented in prior scholarly publications.Previous studies evaluated the stabilized injection bias performance using a lock-in amplifier, focusing solely on amplitude noise, but omitting other significant factors.This paper delves into the principles and noise model of the capacitive gap-sensing system, emphasizing the impact of various factors on the stabilized injection bias.We introduce a model to assess the stabilized injection bias's influence on the measurement system's performance, including amplitude, phase, frequency, and broadband noises.This model provides a comprehensive evaluation methodology for assessing the effect of each factor.Finally, we validate the influence of the stabilized injection bias on the displacement detection system through experimental trials, analyzing the contribution of each factor to the total noise under different scenarios.

Analysis of the Impact of the Carrier Wave on Capacitive Position Sensing
Figure 1 illustrates the fundamental principle of the capacitive gap-sensing system in sensors such as accelerometers or gravity gradiometers.The Test Mass (TM), serving as the reference for geodesic motion, must be shielded from external environmental disturbances.Hence, it is encapsulated within an Electrode Housing (EH) and a spacecraft.Multiple channels can be established between TM and EH.For clarity, one sensing channel for the y-ϕ Degree of Freedom (DoF) is shown in Figure 1, while others are omitted for simplicity.The TM and the plates of its EH form capacitances C 1 and C 2 , respectively.When the TM is centrally positioned, the sensing capacitances C 1 and C 2 are equal; otherwise, a differential capacitance ∆C emerges from the capacitors C 1 and C 2 .The introduction of the stabilized injection bias (V d ) into the sensor, capacitor-transformer bridge, preamplifier, and AC amplifier will modulate the differential capacitance ∆C into an amplitude-modulated signal V BR .
Sensors 2024, 24, x FOR PEER REVIEW 2 featuring a capacitive gap-sensing system with a resolution of 2.4 × 10 m/√Hz, a 10 bias, and a 100 ppm/√Hz stability of the stabilized injection bias [13].In order to ach a resolution of (2~6) × 10 m/√Hz in the frequency band of 5 mHz to 0.1 Hz, th nusoidal carrier signal employed in the high-precision displacement detection ci should produce as little residual noise as possible [9,[14][15][16][17][18].In the scope of this resea the stabilized injection bias utilizes a sinusoidal carrier waveform, which aligns fu mentally with the waveform presented in prior scholarly publications.Previous stu evaluated the stabilized injection bias performance using a lock-in amplifier, focu solely on amplitude noise, but omitting other significant factors.This paper delves into the principles and noise model of the capacitive gap-sen system, emphasizing the impact of various factors on the stabilized injection bias.W troduce a model to assess the stabilized injection bias's influence on the measurement tem's performance, including amplitude, phase, frequency, and broadband noises.model provides a comprehensive evaluation methodology for assessing the effect of factor.Finally, we validate the influence of the stabilized injection bias on the disp ment detection system through experimental trials, analyzing the contribution of each tor to the total noise under different scenarios.After demodulating and filtering the amplitude-modulated signal, the data ar quired and archived by the data acquisition system.Our developed system, known a Stabilized Injection Bias Generation System, adeptly generates Vd.It also concurrently duces a phase-adjustable square wave tailored for demodulation purposes, referred the 'Local Oscillator Signal'.The Stabilized Injection Bias Generation System simult ously creates Vd and a phase-adjustable square wave for demodulation.

Analysis of the Impact of the Carrier Wave on Capacitive Position Sensing
As Vd enters the sensor and displacement detector, the gain yielded by the stabil injection bias includes the gains of the sensor, preamplifier, AC amplifier, demodula and capacitor-transformer bridge-differential transformer bridge.The output of the pacitive gap sensing system can be articulated with Equation (1).After demodulating and filtering the amplitude-modulated signal, the data are acquired and archived by the data acquisition system.Our developed system, known as the Stabilized Injection Bias Generation System, adeptly generates V d .It also concurrently produces a phase-adjustable square wave tailored for demodulation purposes, referred to as the 'Local Oscillator Signal'.The Stabilized Injection Bias Generation System simultaneously creates V d and a phase-adjustable square wave for demodulation.
As V d enters the sensor and displacement detector, the gain yielded by the stabilized injection bias includes the gains of the sensor, preamplifier, AC amplifier, demodulation, and capacitor-transformer bridge-differential transformer bridge.The output of the capacitive gap sensing system can be articulated with Equation (1).
where K is the transformer coupling coefficient, C a and C p are the resonant capacitors in the differential transformer bridge, respectively, C FB is the feedback capacitor in the pre-amplifier, and G AMP is the total gain of the AC amplifier and bandpass filter.
Since the injection bias is only a superposition gain without frequency shifting, we set the total gain of the transformer coupling coefficient, the differential transformer bridge, the preamplifier, the AC amplifier, and the bandpass filter in the displacement detection circuit as G DIS .The V BR in the displacement detection circuit is only related to the injection bias for fixed values of the differential capacitance ∆C and the G DIS gain.The injection bias is a sine wave with amplitude V S , a period of T 1 , and an angular frequency of ω 1 .V BR (t) can be expressed with Equation (2).
where θ is the fixed phase of the sine wave.The local oscillator signal r(t) is a square wave of amplitude ±V r with a period of T 2 and an angular frequency of ω 2 .The local oscillator signal r(t) can be expanded in the following Fourier series.
The demodulation circuit functions can be described in the time domain primarily as a multiplication of V BR (t) and the local oscillator signal r(t).The result can be expressed as Equation ( 4).
The first and second terms are the differential frequency and sum frequency, respectively.The output of the multiplier passes through a low-pass filter, where the sum of all terms with n > 1 and difference frequency terms are filtered out, leaving only n = 1 difference frequency terms.
Ideally, the V BR (t) and oscillator signal have the same frequency, i.e., ω 1 = ω 2 = ω 0 .In order to ensure the maximum amplitude and maximum signal-to-noise ratio of the output signal, the phase difference θ is usually kept to 0. When using a demodulation circuit with analogue switches, the amplitude of the local oscillator signal can reach 1 V. Therefore, Equation ( 6) can be simplified to Equation (7), while the output signal is only related to the amplitude of V S .
According to Equations ( 1) and ( 7), it is found that although the output signal of the capacitive gap-sensing system undergoes low-frequency fluctuations due to variations in fixed elements such as circuit components, it is more critical to be aware of how the inherent characteristics of the injection bias can also lead to significant low-frequency noise.Consequently, comprehensively understanding the impact of the stabilized injection bias on the capacitive gap-sensing system necessitates a detailed analysis of Equations ( 6) and (7).

The Influence of Multiple Factors of the Injection Bias
The Stabilized Injection Bias Generation System, as depicted in Figure 2, is a shared resource across all channels.It employs a low-power digital-to-analog converter (DAC) to produce the DC signal, which was used as the signal amplitude.This DC signal is split into four analog levels through a specific timing sequence to create a stepped wave.A fourth-order low-pass filter then smooths this stepped wave into the desired carrier signal.Concurrently, the system generates a square wave signal to facilitate the demodulation process.

The Influence of Multiple Factors of the Injection Bias
The Stabilized Injection Bias Generation System, as depicted in Figure 2, is a shared resource across all channels.It employs a low-power digital-to-analog converter (DAC) to produce the DC signal, which was used as the signal amplitude.This DC signal is split into four analog levels through a specific timing sequence to create a stepped wave.A fourth-order low-pass filter then smooths this stepped wave into the desired carrier signal.Concurrently, the system generates a square wave signal to facilitate the demodulation process.However, the components used in the Stabilized Injection Bias Generation System, including voltage and frequency references, are not perfect.These imperfections will inevitably lead to voltage and frequency instabilities, leading to noise in the sine carrier amplitude VS, the phase difference θ between the injection bias and the local oscillator signal, and the injection bias's angular frequency.Additionally, the system inherently introduces broadband noise superimposed onto the injection bias.
To understand how these bias factors affect the performance of the displacement detection system, it is necessary to adjust Equation ( 6) to account for these non-ideal conditions.The ideal scenario is outlined in Table 1, where it is assumed that all other factors remain ideal when one factor is varied for simplicity.Noise bandwidth (NBW) Hz 1 Equivalent gain A dB 0

Amplitude Noise
The non-idealities of the low-power DAC, such as quantization error, clock jitter, power supply noise, and electromagnetic interference, are significant sources of amplitude noise.Additionally, the 1/f noise characteristic of the DAC's output voltage becomes more pronounced under certain conditions, and the noise increases as the amplitude VS increases in such cases.
Assuming that an amplitude noise value S V ~ is superimposed on the amplitude VS, the displacement detection output should be consistent with the calculated value of Equation (6) at different values of amplitude VS.Accordingly, the displacement detection output is However, the components used in the Stabilized Injection Bias Generation System, including voltage and frequency references, are not perfect.These imperfections will inevitably lead to voltage and frequency instabilities, leading to noise in the sine carrier amplitude V S , the phase difference θ between the injection bias and the local oscillator signal, and the injection bias's angular frequency.Additionally, the system inherently introduces broadband noise superimposed onto the injection bias.
To understand how these bias factors affect the performance of the displacement detection system, it is necessary to adjust Equation ( 7) to account for these non-ideal conditions.The ideal scenario is outlined in Table 1, where it is assumed that all other factors remain ideal when one factor is varied for simplicity.

Amplitude Noise
The non-idealities of the low-power DAC, such as quantization error, clock jitter, power supply noise, and electromagnetic interference, are significant sources of amplitude noise.Additionally, the 1/f noise characteristic of the DAC's output voltage becomes more pronounced under certain conditions, and the noise increases as the amplitude V S increases in such cases.
Assuming that an amplitude noise value ∼ V S is superimposed on the amplitude V S , the displacement detection output should be consistent with the calculated value of Equation ( 7) at different values of amplitude V S .Accordingly, the displacement detection output is ∼ V S produces a pair of sidebands on either side of the central frequency of the injection bias in the spectrogram, as the amplitude noise comprises a mixture of noise at several frequencies.Figure 3 shows the simulated results of the capacitive gap-sensing system's output noise power spectral density (PSD) as the noise ratio of amplitude V S varies from 0 to 10 ppm/ √ Hz of the amplitude voltage.amplitude changes from 0 V to 4 V, and the noise ratio of amplitude V 5 ppm/√Hz, and 10 ppm/√Hz.As the amplitude noise factor increases, noise power increases as a quadratic term.Therefore, the amplitude n be minimized to reduce the total noise voltage.

Frequency Noise and Phase Noise
The crystal oscillator frequency in the system provides the main fr of the Stabilized Injection Bias Generation System.Figure 4a displays bility curve of a typical 10 MHz crystal oscillator (8607-BE).It is evide that the stability of the crystal frequency is positively correlated with t (division period), meaning that the longer the sampling time, the high the crystal frequency.Notably, the frequency stability improves by an o with each tenfold increase in the crossover period-a hallmark trait of h tal oscillators.Concurrently, the response time of switching devices an fects within the system's components can diminish the frequency stabil all these factors commonly lead to the production of the pronounced fre noises in the interplay between the injection bias and the local oscillato The frequency difference ∆ between the injection bias and demo cies can be regarded as frequency modulation (FM).The injection bias as follows where  is the maximum frequency difference, and  is the modula the frequency difference.When the modulation index is very small (nar β<<π/2.let VS be 1 V, we have  The output voltage noise PSD of the capacitive gap-sensing system reaches 6.48 × 10 −12 V 2 / Hz, 1.62 × 10 −10 V 2 /Hz, and 6.48 × 10 −10 V 2 /Hz, respectively, when the amplitude changes from 0 V to 4 V, and the noise ratio of amplitude V S is 1 ppm/Hz 0.5 , 5 ppm/ √ Hz, and 10 ppm/ √ Hz.As the amplitude noise factor increases, the displacement noise power increases as a quadratic term.Therefore, the amplitude noise factor should be minimized to reduce the total noise voltage.

Frequency Noise and Phase Noise
The crystal oscillator frequency in the system provides the main frequency reference of the Stabilized Injection Bias Generation System.Figure 4a displays the frequency stability curve of a typical 10 MHz crystal oscillator (8607-BE).It is evident from this curve that the stability of the crystal frequency is positively correlated with the sampling time (division period), meaning that the longer the sampling time, the higher the stability of the crystal frequency.Notably, the frequency stability improves by an order of magnitude with each tenfold increase in the crossover period-a hallmark trait of high-stability crystal oscillators.Concurrently, the response time of switching devices and the parasitic effects within the system's components can diminish the frequency stability.Consequently, all these factors commonly lead to the production of the pronounced frequency and phase noises in the interplay between the injection bias and the local oscillator signal.
Sensors 2024, 24, x FOR PEER REVIEW 6 of 11 Equation ( 14) demonstrates that the spectrum of a narrowband FM wave consists of a central frequency and two side frequency components.The results obtained in the lockin amplifier circuit can be easily confused because the narrowband FM and AM spectra are comparable.
The phase noise θ at the injection bias can be considered as phase modulation (PM).For single-tone phase modulation, the injection bias can be expressed as follows where  rad is the maximum phase shift.The PM spectrum with phase disturbance  rad is identical to the FM spectrum with frequency disturbance  rad.Therefore, PM and FM modulations can be transformed into each other according to the following relationship.
where  is the main frequency of the injection bias.After modulating the phase and frequency noises, the displacement output can be expressed with Equation ( 17).
Thus, the angular frequency difference ∆ω and the phase difference ∆ϑ can be combined to form the integrated frequency noise  .Figure 4b shows the simulated results of the output noise PSD of the capacitive gap-sensing system as the integrated frequency noise  varies from 0 to 100 Hz.When the amplitude varies from 0 V to 4 V, the maximum phase (frequency) variations of 10 Hz, 50 Hz, 80 Hz, and 100 Hz will contribute to the noise power of 1.57 × 10 V /Hz, 1.03 × 10 V /Hz, and 2.52 × 10 V /Hz, respectively.According to Equation ( 9), the displacement noise power increases significantly as the phase (frequency) noise increases.Therefore, phase (frequency) noise should be minimized to reduce the total noise voltage.More seriously, the combined frequency noise  in the Stabilized Injection Bias Generation System positively correlates with the amplitude VS.Reducing the main frequency may optimize both the frequency and phase noises, while it can only be adjusted appropriately considering the performance of the capacitive gapsensing system.The frequency difference ∆ω between the injection bias and demodulation frequencies can be regarded as frequency modulation (FM).The injection bias can be expressed as follows e(t) = V S cos(ω 1 t + β sin ω m t)

Broadband Noise
Sensors 2024, 24, 1188 6 of 11 where β is the maximum frequency difference, and ω m is the modulation frequency of the frequency difference.When the modulation index is very small (narrowband FM), i.e., β << π/2.Let V S be 1 V, we have cos(β sin ω m t) ≈ 1 (10) sin(β sin Therefore, Equation ( 13) demonstrates that the spectrum of a narrowband FM wave consists of a central frequency and two side frequency components.The results obtained in the lock-in amplifier circuit can be easily confused because the narrowband FM and AM spectra are comparable.
The phase noise θ at the injection bias can be considered as phase modulation (PM).For single-tone phase modulation, the injection bias can be expressed as follows where ϑ d rad is the maximum phase shift.The PM spectrum with phase disturbance ϑ d rad is identical to the FM spectrum with frequency disturbance ϑ d rad.Therefore, PM and FM modulations can be transformed into each other according to the following relationship.
where f m is the main frequency of the injection bias.After modulating the phase and frequency noises, the displacement output can be expressed with Equation (15).
Thus, the angular frequency difference ∆ω and the phase difference ∆ϑ can be combined to form the integrated frequency noise f int .Figure 4b shows the simulated results of the output noise PSD of the capacitive gap-sensing system as the integrated frequency noise f int varies from 0 to 100 Hz.
When the amplitude varies from 0 V to 4 V, the maximum phase (frequency) variations of 10 Hz, 50 Hz, 80 Hz, and 100 Hz will contribute to the noise power of 1.57 × 10 −10 V 2 /Hz, 1.03 × 10 −9 V 2 /Hz, and 2.52 × 10 −9 V 2 /Hz, respectively.According to Equation ( 6), the displacement noise power increases significantly as the phase (frequency) noise increases.Therefore, phase (frequency) noise should be minimized to reduce the total noise voltage.More seriously, the combined frequency noise f int in the Stabilized Injection Bias Generation System positively correlates with the amplitude V S .Reducing the main frequency may optimize both the frequency and phase noises, while it can only be adjusted appropriately considering the performance of the capacitive gap-sensing system.

Broadband Noise
The capacitive gap-sensing system can process signals in the frequency range of (f 0 − f LPF , f 0 + f LPF ) Hz, where f LPF is the low-pass filter's cutoff frequency in the demodulation circuit.Due to its device noise, temperature noise, and other noises, a broadband noise n(t) is inevitably superimposed on the output of the Stabilized Injection Bias Generation System.Therefore, when the noise n(t) is superimposed on the (f 0 − f LPF , f 0 + f LPF ) Hz band, n(t) will be demodulated to the (0, 0 + f LPF ) Hz band.Based on the above analysis, the displacement detection output signal is represented with Equation ( 16).

Total Noise
In the non-ideal case, the equivalent displacement output introduces noise ∼ V S , which depends on the amplitude V S of the carrier, the phase difference ∆θ, and the angular frequency difference ∆ω between the carrier and the local oscillator signal.Thus, Equation ( 6) can be expanded to Equation (17).

Amplitude Noise
In the previous section, Figure 2 shows the block diagram of the Stabilized Injection Bias Generation System.The amplitude is set by the DAC, which can be measured directly.A high-precision ADC (LTC2508) is employed in the time domain to acquire the voltage's amplitude.The LTC2508-based acquisition circuit's data processing parameters are shown in Table 2.The amplitude noise's PSD is shown in Table 3 when the amplitude varies from 1 V to 4 V.This paper employs a Hanning-window for the one-sided spectrum to perform the PSD processing based on the Fourier transform.As the amplitude increases, the amplitude noise's PSD increases accordingly, consistent with the theoretical analysis.At 100 mHz, the maximum noise ratio of amplitude V S is 1.29 ppm/ √ Hz; at 5 mHz, the maximum noise ratio of amplitude V S is 2.49 ppm/ √ Hz due to 1/f noise.

Frequency Noise and Phase Noise
Here, we evaluate the combined frequency noise f int using an Agilent E4440A spectrum analyzer, where its parameters are shown in Table 4.The combined frequency noise combines phase and frequency noises using the main frequency.In the RBW range, the combined frequency noise f int versus the amplitude voltage value is shown in Figure 5.The combined frequency noise f int is positively related to the amplitude.By linear fitting, for every 1 V increase in amplitude, the combined frequency noise is correspondingly increased by 19.90 ± 1.57 Hz.The Carrier Wave Parameters Unit T

Resolution bandwidth (RBW) Hz
The conversion factor of the resolution bandwidth to noise bandwidth kn -

Broadband Noise and Total Noise
Here, the total noise is evaluated by a lock-in amplifier, which will stabilized injection bias with different amplitudes.The noise's PSD of the by the lock-in amplifier can evaluate the performance of the stabilized shown in Figure 6. Figure 6 includes the background noise power spectru amplifier, which is 3.3 × 10 V/√Hz (1.089 × 10 −11 V 2 /Hz) over the range mHz.When the frequency tends to 0 Hz, the 1/f noise plays a vital role noise significantly.Moreover, 1/f noise, or flicker noise, can also originat

Broadband Noise and Total Noise
Here, the total noise is evaluated by a lock-in amplifier, which will demodulate the stabilized injection bias with different amplitudes.The noise's PSD of the obtained signals by the lock-in amplifier can evaluate the performance of the stabilized injection bias, as shown in Figure 6.

Broadband Noise and Total Noise
Here, the total noise is evaluated by a lock-in amplifier, which will demo stabilized injection bias with different amplitudes.The noise's PSD of the obtain by the lock-in amplifier can evaluate the performance of the stabilized injectio shown in Figure 6. Figure 6 includes the background noise power spectrum of t amplifier, which is 3.3 × 10 V/√Hz (1.089 × 10 −11 V 2 /Hz) over the range of 5 m mHz.When the frequency tends to 0 Hz, the 1/f noise plays a vital role in ele noise significantly.Moreover, 1/f noise, or flicker noise, can also originate intern ticularly from components such as operational amplifiers.The equivalent input n age and current of operational amplifiers are significant sources of 1/f noise, esp low-frequency applications.This internal source of noise is often related to d imperfections in the semiconductor material and varies with the operating con the amplifier.To analyze the noise powers in the unit frequency more accurately When the frequency tends to 0 Hz, the 1/f noise plays a vital role in elevating the noise significantly.Moreover, 1/f noise, or flicker noise, can also originate internally, particularly from components such as operational amplifiers.The equivalent input noise voltage and current of operational amplifiers are significant sources of 1/f noise, especially in low-frequency applications.This internal source of noise is often related to defects and imperfections in the semiconductor material and varies with the operating conditions of the amplifier.To analyze the noise powers in the unit frequency more accurately, the PSDs

Figure 1
Figure 1 illustrates the fundamental principle of the capacitive gap-sensing syste sensors such as accelerometers or gravity gradiometers.The Test Mass (TM), servin the reference for geodesic motion, must be shielded from external environmental dist ances.Hence, it is encapsulated within an Electrode Housing (EH) and a spacecraft.M tiple channels can be established between TM and EH.For clarity, one sensing channe the y-ϕ Degree of Freedom (DoF) is shown in Figure 1, while others are omitted for plicity.The TM and the plates of its EH form capacitances C1 and C2, respectively.W the TM is centrally positioned, the sensing capacitances C1 and C2 are equal; otherwi differential capacitance ΔC emerges from the capacitors C1 and C2.The introduction o stabilized injection bias (Vd) into the sensor, capacitor-transformer bridge, preampl and AC amplifier will modulate the differential capacitance ΔC into an amplitude-m lated signal VBR.

Figure 2 .
Figure 2. Block diagram of the Stabilized Injection Bias Generation System.

Figure 2 .
Figure 2. Block diagram of the Stabilized Injection Bias Generation System.

Figure 3 .
Figure 3.The output voltage noise PSD of the capacitive gap-sensing system.

Figure 3 .
Figure 3.The output voltage noise PSD of the capacitive gap-sensing system.

Figure 4 .
Figure 4. (a) The frequency stability curve of a typical 10 MHz crystal oscillator (8607-BE) and (b) the output voltage noise's PSD of the displacement detection circuit.

Figure 4 .
Figure 4. (a) The frequency stability curve of a typical 10 MHz crystal oscillator (8607-BE) and (b) the output voltage noise's PSD of the displacement detection circuit.

Figure 5 .
Figure 5.The combined frequency noise  versus the amplitude voltage value

Figure 6 .
Figure 6.The background noise of the lock-in amplifier and the noise when se injection bias amplitude to 0 to 4 V.

Figure 5 .
Figure 5.The combined frequency noise f int versus the amplitude voltage value.

Figure 5 .
Figure 5.The combined frequency noise  versus the amplitude voltage value.

Figure 6 .
Figure 6.The background noise of the lock-in amplifier and the noise when setting the injection bias amplitude to 0 to 4 V.

Figure 6 .
Figure 6.The background noise of the lock-in amplifier and the noise when setting the stabilized injection bias amplitude to 0 to 4 V.

Table 1 .
Typical parameters in the carrier wave analysis.

Table 1 .
Typical parameters in the carrier wave analysis.

Table 2 .
LTC2508-based acquisition circuit and data processing parameters.

Table 3 .
Noise's PSD for each amplitude.