# AC/DC Fields Demodulation Methods of Resonant Electric Field Microsensor

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## Abstract

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## 1. Introduction

## 2. Resonant EFM with Coplanar Electrodes

#### 2.1. Operating Principle

_{n}. As illustrated in Figure 2, when the shutter is in its leftmost position, a larger fraction of the electric field lines terminate on the negative sensing electrode than on the positive sensing electrode, and thus the electric field induces more charge on the negative sensing electrode than on the positive one. When the shutter moves to its rightmost position, the situation is reversed. Consequently, as the grounded shutter swings back and forth, it covers the sidewalls of either the positive or negative sensing electrode; then, a differential AC current is generated on the sensing electrodes. The output AC current i

_{s}is calculated according to:

_{0}is the permittivity of free space, E

_{n}is the component of the electric field normal to the sensing electrodes, and A is the effective area of the sensing electrodes. The current i

_{s}is converted to a voltage V

_{o}by an instrumentation amplifier (INA), and the V

_{o}is then amplified and detected.

#### 2.2. Structure and Fabrication

#### 2.3. Differential Induced Charge and Output Signal

_{d}and antisymmetric sinusoidal voltages V

_{a}sin(ω

_{s}t) are applied on both sides of the electrostatic actuators to drive the shutter. The microsensor was designed to work at the resonant frequency ω

_{s}. When the driving voltages V

_{d}± V

_{a}sin(ω

_{s}t) are applied on the microsensor, the differential induced charge on the sensing electrodes can be written as [32]:

_{A}denotes the magnitude of the differential induced charge, θ is the phase difference between the driving signal and the vibration displacement of the shutter, Q

_{0}is the residual charge caused by the incomplete symmetry of the sensing structure due to fabrication error, D

_{0}is the residual charge per kVm

^{−1}, k

_{q}represents the conversion coefficient of the vibration-amplitude-to-charge-variation per kVm

^{−1}, and X

_{r}is the resonant vibration amplitude of the shutter.

_{f}is expressed as:

## 3. Demodulation Methods

#### 3.1. DC Electric Field Demodulation

_{0}(t) by the AC drive signal r(t) = V

_{a}sin(ω

_{s}t) results in:

_{cl}of the low-pass filter shown in Figure 4 must be much lower than the frequency ω

_{s}of the drive signal. Therefore, after the low-pass filter, the result of the synchronous demodulation is given by:

_{q}, X

_{r}, ω

_{s}, R

_{f}, V

_{a}, and θ are definite values. According to the second-order vibration characteristics of the resonant EFM, ideally, the value of θ is π/2 [35]. Equation (6) illustrates that the result of the synchronous demodulation v(t) is linear to the measured DC field E

_{n}. Therefore, the magnitude and the polarity of the measured DC electric field can be obtained by measuring v(t).

#### 3.2. Power Frequency Electric Field Demodulation

_{0}and ω

_{e}are the amplitude and frequency of the power frequency field, respectively. φ is the initial phase. From Equations (2), (3), and (7), the output voltage of the microsensor under the power frequency field is obtained as:

_{e}to ω

_{s}± ω

_{e}. However, the frequencies of the sensor output signal include not only ω

_{s}± ω

_{e}, but also the frequency ω

_{e}of the measured field, which is directly fed through due to the structural asymmetry. Although directly extracting the signal with the frequency ω

_{e}can realize the detection of the power frequency field, this method is easily affected by co-channel interference noise and its harmonic components, and also does not reflect the advantages of the resonant EFM modulation. Hence, in order to improve the signal detection accuracy, we focus on demodulating the signal with the frequencies ω

_{s}± ω

_{e}in the subsequent signal processing. The spectrum of Equation (8) is represented in Figure 5.

_{s}± ω

_{e}, the sensor output first passes a high-pass filter with the cut-off frequency ω

_{h}to eliminate the feedthrough signal at the frequency ω

_{e}, ω

_{h}∈ (ω

_{e}, ω

_{s}−ω

_{e}). Then, the orthogonal correlation is performed.

_{Y}(t) and w

_{X}(t) can be derived as:

_{Y}(t) and w

_{X}(t) contain three frequency components: ω

_{e}, 2ω

_{s}− ω

_{e}, and 2ω

_{s}+ ω

_{e}, respectively. Because we only pay attention to the power frequency field signal with the frequency ω

_{e}, band-pass filters with the center frequency ω

_{e}are selected after the orthogonal correlation, which can also prevent the interference caused by the AC drive-signal feedthrough of the microsensor. The outputs after the band-pass filter can be expressed as:

_{0}. Figure 7 shows the normalized simulation results of X(t) under the different initial phase φ, where the frequency of the power frequency field is 50 Hz, and the resonant frequency of the EFM is 3050 Hz. The simulation results show that X(t) is a sine wave with a frequency of 50 Hz.

_{0}. For the power frequency field detection, since ω

_{e}is much smaller than ω

_{s}, the maximum value of R(t) can be expressed as:

- Because the microsensor output first passes a high-pass filter, the influence of power frequency interference and its harmonic components on the electric field measurements is suppressed.
- Unlike the synchronous demodulation of the DC electric field using a low-pass filter, band-pass filters are used to obtain the power frequency electric field signal, which inhibits the impact of the AC drive-signal feedthrough of the EFM.
- Unlike the synchronous demodulation of the DC electric field, using the HOCBM to finally extract the peak value of R (t) can avoid the change of the microsensor sensitivity caused by the θ variation.

#### 3.3. AC/DC Hybrid Electric Fields Demodulation

_{d}is a DC field, E

_{mi}is the amplitude of the i-th AC field with the frequency ω

_{i}, and φ

_{i}is the initial phase of the i-th AC field. From Equations (2), (3), and (14), the output voltages of the microsensor under the AC/DC hybrid fields are obtained as:

_{L}is added between the orthogonal correlation and the band-pass filter, ω

_{L}∈ (max{ω

_{i}}, ω

_{s}); (b) When demodulating the DC field, the band-pass filters are replaced by low-pass filters with a very low cut-off frequency. Due to the cut-off frequency of the high-pass filter, ω

_{h}∈ (ω

_{i}, ω

_{s}− ω

_{i}), which illustrates ω

_{i}∈ (0, ω

_{s}/2)(i = 1, 2,…, k). Therefore, in theory, the frequency range of the AC/DC hybrid fields that can be demodulated by the HOCLBM is [0, ω

_{s}/2). The output voltage V

_{0}(t) of the microsensor under the AC/DC hybrid fields is demodulated when adopting the HOCLBM, and the two outputs Y

_{L}(t) and X

_{L}(t) of the added low-pass filters are expressed as:

_{L}(t) and X

_{L}(t) contain all frequency components of the measured hybrid fields. After band-pass filters with the center frequency ω

_{i}or low-pass filters with a very low cut-off frequency, the R

_{i}(t) can be derived as:

_{i}∈ (0, ω

_{s}/2), we can obtain:

_{q}X

_{r}R

_{f}ω

_{s}|. This conclusion indicates that the usage of the resonant EFMs and the HOCLBM to detect low-frequency electric fields has great advantages and is very convenient in power systems. Unfortunately, the HOCLBM cannot identify the polarity of the DC field. The normalized simulation results of R

_{i}(t) at different measured field frequencies are shown in Figure 9, illustrating that the frequencies of the R

_{i}(t) outputs are twice the frequencies of the measured AC fields, and that the maximum values of R

_{i}(t) for the measured AC fields at different frequencies are the same, which is consistent with the result obtained by Equation (18).

## 4. Experimental Results and Discussion

#### 4.1. AC/DC Electric Fields Demodulation Verification Test System

_{c}as:

#### 4.2. Testing and Analysis of Signal Characteristics

#### 4.3. Demodulation Results

_{max}slightly decreases. The possible reasons for this include: (a) Since the distance between the cover plate of the package and the EFM is only 1.5 mm, the very small voltage change of the high-voltage meters will give rise to a relatively large electric field variation inside the package; (b) Using different types of high-voltage meters to generate DC fields and AC fields, respectively; and (c) Being affected by the filter order and the bandwidth of the INA.

## 5. Conclusions

_{s}/2). In addition, the sensitivity of the EFM for any frequency component of the AC/DC hybrid fields in the measurable frequency range is a fixed value |k

_{q}X

_{r}R

_{f}ω

_{s}|. The experimental results demonstrated that within electric field ranges of 0–667 kV/m, the uncertainties were 2.4% and 1.5% for the most common DC and 50 Hz power frequency fields, respectively. The frequency characteristic test results of the microsensor were in agreement with the theoretical analysis in the range of 0–1 kHz under different electric field amplitudes. The proposed demodulation methods are mainly used to extract the measured DC field, power frequency field, or AC/DC hybrid fields from the output voltages of the microsensor, which is very helpful in promoting the application of the resonant EFMs in power systems.

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## Appendix A

_{Ui}of the i-th calibration point subsamples of the direct journey is calculated by:

_{Uij}is the j-th measurement value of the i-th calibration point of the direct journey, and ${\overline{Y}}_{Ui}$ is the average value of the direct journey measurement at the i-th calibration point.

_{Di}of the i-th calibration point subsamples of the reverse journey is calculated by:

_{Dij}is the j-th measurement value of the i-th calibration point of the reverse journey, and ${\overline{Y}}_{Di}$ is the average value of the reverse journey measurement at the i-th calibration point.

_{LH,Ui}of the direct journey and the system error ∆Y

_{LH,Di}of the reverse journey are, relative to the reference straight line, calculated by:

_{i}is the value of the reference straight line of the microsensor at the i-th calibration point.

_{LHR,Ui}and the uncertainty of the reverse journey ξ

_{LHR,Di}are given by:

_{FS}is the full-scale output of the microsensor.

_{LHR}can be expressed as:

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**Figure 1.**Operating principle of the resonant electric field microsensor (EFM) with coplanar electrodes.

**Figure 2.**Electric field line distribution of the shutter (s) at different positions: (

**a**) Proximity to sensing electrode (+); (

**b**) Proximity to sensing electrode (−).

**Figure 7.**Normalized simulation results of X(t) under different initial phase φ values, ω

_{e}= 2π·50 rad/s, ω

_{s}= 2π·3050 rad/s.

**Figure 8.**Normalized simulation results of R(t) under different initial phase φ values, ω

_{e}= 2π·50 rad/s, ω

_{s}= 2π·3050 rad/s.

**Figure 9.**Normalized simulation results of R

_{i}(t) at different measured field frequencies, ω

_{s}= 2π·3050 rad/s, φ

_{i}= 0.

**Figure 10.**Verification test system for the AC/DC electric fields demodulation methods: (

**a**) The schematic diagram; (

**b**) The test setup; (

**c**) A cross-section of the packaged EFM.

**Figure 12.**Spectrum of the microsensor output voltage: (

**a**) under a DC electric field; (

**b**) under a 50 Hz power frequency field; and (

**c**) under a 500 Hz AC field.

**Figure 13.**Demodulation results of a 50 Hz power frequency field using the HOCBM: (

**a**) The output of X(t); (

**b**) The output of R(t).

**Figure 14.**The response characteristics of the EFM: (

**a**) The uncertainty of 2.4% for DC fields; (

**b**) The uncertainty of 1.5% for 50 Hz power frequency fields.

Roundtrip | 0 kV/m | 133 kV/m | 267 kV/m | 400 kV/m | 533 kV/m | 667 kV/m |
---|---|---|---|---|---|---|

1st direct journey | −0.034 mV | 1.785 mV | 4.011 mV | 6.344 mV | 8.808 mV | 11.432 mV |

1st reverse journey | −0.035 mV | 1.783 mV | 4.006 mV | 6.342 mV | 8.810 mV | 11.426 mV |

2nd direct journey | −0.031 mV | 1.785 mV | 4.011 mV | 6.346 mV | 8.809 mV | 11.426 mV |

2nd reverse journey | −0.031 mV | 1.788 mV | 4.008 mV | 6.344 mV | 8.809 mV | 11.422 mV |

3rd direct journey | −0.035 mV | 1.781 mV | 4.010 mV | 6.343 mV | 8.809 mV | 11.427 mV |

3rd reverse journey | −0.036 mV | 1.789 mV | 4.010 mV | 6.343 mV | 8.810 mV | 11.422 mV |

**Table 2.**The detailed test results of the response characteristics for 50 Hz power frequency fields.

Roundtrip | 0 kV/m | 133 kV/m | 267 kV/m | 400 kV/m | 533 kV/m | 667 kV/m |
---|---|---|---|---|---|---|

1st direct journey | 0.003 mV | 2.233 mV | 4.550 mV | 6.952 mV | 9.487 mV | 12.200 mV |

1st reverse journey | 0 mV | 2.236 mV | 4.550 mV | 6.954 mV | 9.494 mV | 12.196 mV |

2nd direct journey | 0.001 mV | 2.236 mV | 4.543 mV | 6.955 mV | 9.495 mV | 12.202 mV |

2nd reverse journey | 0.002 mV | 2.236 mV | 4.553 mV | 6.953 mV | 9.488 mV | 12.198 mV |

3rd direct journey | 0.001 mV | 2.238 mV | 4.550 mV | 6.954 mV | 9.494 mV | 12.204 mV |

3rd reverse journey | 0.001 mV | 2.241 mV | 4.551 mV | 6.952 mV | 9.492 mV | 12.199 mV |

**Table 3.**The detailed test data of the EFM frequency characteristics under different electric fields.

Frequency (Hz) | 0 kV/m | 133 kV/m | 267 kV/m | 400 kV/m | 533 kV/m | 667 kV/m |
---|---|---|---|---|---|---|

0 | −0.03 mV | 1.79 mV | 4.01 mV | 6.34 mV | 8.81 mV | 11.43 mV |

50 | 0 | 2.24 mV | 4.55 mV | 6.95 mV | 9.49 mV | 12.20 mV |

100 | 0 | 2.28 mV | 4.65 mV | 7.10 mV | 9.70 mV | 12.48 mV |

200 | 0 | 2.27 mV | 4.65 mV | 7.10 mV | 9.70 mV | 12.49 mV |

500 | 0 | 2.25 mV | 4.58 mV | 6.99 mV | 9.54 mV | 12.27 mV |

1000 | 0 | 2.05 mV | 4.19 mV | 6.46 mV | 8.89 mV | 11.53 mV |

Absolute error | 0.03 mV | 0.5 mV | 0.63 mV | 0.75 mV | 0.89 mV | 1.06 mV |

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## Share and Cite

**MDPI and ACS Style**

Yang, P.; Wen, X.; Chu, Z.; Ni, X.; Peng, C.
AC/DC Fields Demodulation Methods of Resonant Electric Field Microsensor. *Micromachines* **2020**, *11*, 511.
https://doi.org/10.3390/mi11050511

**AMA Style**

Yang P, Wen X, Chu Z, Ni X, Peng C.
AC/DC Fields Demodulation Methods of Resonant Electric Field Microsensor. *Micromachines*. 2020; 11(5):511.
https://doi.org/10.3390/mi11050511

**Chicago/Turabian Style**

Yang, Pengfei, Xiaolong Wen, Zhaozhi Chu, Xiaoming Ni, and Chunrong Peng.
2020. "AC/DC Fields Demodulation Methods of Resonant Electric Field Microsensor" *Micromachines* 11, no. 5: 511.
https://doi.org/10.3390/mi11050511