Design of a microelectromechanical systems resonant electric field sensor with sensitivity robustness

Abstract

Studies indicate that driving circuit and drift error in resonance frequency influence the sensitivity of micromechanical resonant electric field sensors. This study proposes an electric field sensor with a comb structure for driving and sensing. A dynamic model is built for the microsensor, which analyzes the behavior and sensitivity of the system on different closed-loop self-excited circuits using the averaging method. Theory and simulation results show that with the use of a proportional–integral controller, the sensitivity remains constant regardless of the variations in the resonance frequency of the shielding layer and Q-factor. The sensitivity is higher with a suitable proportional–integral controller than without a proportional–integral controller. If the parameters of the proportional–integral controller fail to satisfy the constraint relationship, output voltage becomes unstable, and the sensitivity is distorted.

Keywords

Electric field microsensor sensitivity robustness averaging method

Introduction

Electric field sensors are used in meteorology, aerospace, and other fields. These sensors have attracted the attention of smart grid industries because of their potential application in measuring power system voltage, detection of ice accretion in high-voltage lines, identification of insulator defects, and lightning warning.

Compared with traditional electric field sensors, microelectromechanical system-based electric field sensors (MEFSs) are small and lightweight, thereby facilitating integration and reducing power consumption. Electrostatic comb-driven MEFSs^1,2 and thermally driven MEFSs^3,4 have been developed using different methods, mostly using strip sensing electrodes and exhibiting similar induction structures. The shutter is placed above the sensing electrodes. The vibration structure on the shutter layer of MEFS must continuously detect resonance. Sensitivity is an important property to consider in sensor design, and previously proposed sensors present disadvantages,^5–7 such as susceptibility of sensitivity to change because of material fatigue and the variations in the working temperature. The changes in sensitivity lead to the changes in output voltage, which causes difficulty in determining whether the change in output voltage is caused by the electric field or by other factors. Sensor performance cannot meet the requirements of high-precision measurement.

This study proposes a two-part design of a resonant electric field sensor: the structural design and the closed-loop control circuit design. First, the sensor includes two critical layers, namely, the resonance shielding layer and the sensing layer. We use the comb capacitance to convert voltage into a static driving force. In addition, comb capacitance and interface circuit can convert the displacement of the resonance beam into output voltage. This type of symmetric capacitance structure reduces the nonlinearity of these conversions. Second, several studies incorporated a self-sustained oscillation loop scheme into microelectromechanical systems resonant sensors. These studies used automatic gain control (AGC) technology to maintain a constant amplitude of vibration in the vibration structure. However, the nonlinear problem of sensitivity was not considered in the design. The proposed MEFS is powered by a self-sustained oscillating loop circuit. We adopted the proportional–integral (PI) controller to maintain the sensitivity of the sensor. We employed the averaging method to address the high degree of nonlinearity of the system. The stability of the system and the nonlinearity of sensitivity with the controller were analyzed and compared. The theoretical and simulation results indicate that the method is feasible and the proposed design exhibits high sensitivity when material fatigue is present and the Q-factor is changed.

Principle of MEFS and Q-factor tuning

Figure 1 shows the operating principle of MEFS. The laterally vibrating grounded shutter periodically shields and exposes the sensing electrodes; thus, an alternating current (AC) proportional to the external electric field strength is generated.⁶

Figure 1.

Operating principle of an electric field microsensor.

Normal electric field $E$ produces charge $Z (t)$ on the sensing electrodes. Gauss’ law states that the amount of the charge is proportional to the exposed area, $S (t)$ , which is expressed as follows

Z (t) = ε \cdot S (t) \cdot | E |

(1)

where $ε$ is the permittivity constant. The periodic movement of the shutter effectively exposes an area to the electric field, which generates current $i_{s} (t)$ that flows into a trans-resistance amplifier. The process is given by

i_{s} (t) = \frac{dZ (t)}{dt} = ε \frac{dS (t)}{dt} | E |

(2)

When the electrostatic force is sinusoidal, equation (2) can be described as follows⁷

i_{s} (t) = ε \frac{dX \sin (ω_{n} t + φ)}{dt} L | E | = ε ω_{n} LX \cos (ω_{n} t + φ) | E |

(3)

where $X$ is the vibration amplitude in the x-axis, $ω_{n}$ is the natural frequency of the vibratory shutter, and $φ$ is the corresponding phase. $L$ is the lumped length of a single sensing electrode in the y-axis. After I–V conversion, amplification, band-pass filtration, re-modulation, and low-pass filtration, current $i_{s}$ is transformed into output voltage $V_{0}$ by

V_{0} = H ω_{n} X | E |

(4)

where $H$ is the gain coefficient related to the circuit parameters. An extremely large $H$ induces the saturation of output voltage $V_{0}$ . Equation (4) indicates that sensitivity is determined by $H$ , $X$ , and $ω_{n}$ . When material fatigue is present and Q-factor is changed, $H$ , $X$ , and $ω_{n}$ product must remain constant. We use the structural design and closed-loop control circuit to achieve this goal.

We can adopt comb capacitance or parallel plate capacitor to achieve the conversion of voltage into static driving force, as well as the conversion of the displacement of the resonance beam into output voltage. The conversion in the parallel plate capacitor is nonlinear;⁸ thus, we choose the type of symmetric capacitance structure to reduce conversions of nonlinearity. The entire resonance shielding layer is shown in Figure 2.

Figure 2.

Shutter layer of the proposed electric field microsensor.

The shutter layer of the proposed new MEFS consists of fixed combs A11, A12, B21, B22, C31, C32, D41, and D42, as well as the transferred frame E5. A11 and A12 are differential drive combs. B21, B22 and C31, C32 are differential combs that detect the electrostatic drive force. When a fabrication error occurs, D41 and D42 are connected to the direct current (DC) voltage; D41 and D42 are combs that adjust the resonant frequency of the vibratory shutter through negative electrostatic stiffness.⁸

When only the electrostatic force of the drive is present in the vibratory shutter, the following dynamic equation is used

\begin{matrix} m \overset{\cdot\cdot}{x} + c \overset{\cdot}{x} + k_{m} x & = \frac{1}{2} \frac{\partial C_{d}}{\partial x} [{(V_{c} + V_{a})}^{2} - {(V_{c} + V_{a})}^{2}] \\ = F_{d} \cos Ω t \end{matrix}

(5)

where $m$ is the total mass of the vibratory shutter, $k_{m}$ is the equivalent stiffness, $c$ is the damping coefficient, and $x$ is the position of the vibratory shutter, which is a function of the independent variable $t$ . $C_{d}$ is the drive capacitor between the fixed combs and the vibratory shutter. $C_{d} = ε N_{d} h \cdot (l + x) / d$ , where $N_{d}$ is the total number of drive combs, $h$ is the thickness of the shutter, $l$ is the overlap length of each pair of combs, and d is the gap between the shutter comb and the fixed drive comb. $V_{c}$ and $V_{a}$ are the DC and AC drive voltages, respectively. $F_{d}$ is the drive electrostatic force, and $Ω$ is the frequency of the electrostatic drive force. The vibration amplitude $X$ of the vibratory shutter can be expressed as

X = \frac{F_{d}}{k_{m}} \frac{1}{\sqrt{{(1 - {(\frac{Ω}{ω_{n}})}^{2})}^{2} +} {(\frac{Ω}{Q ω_{n}})}^{2}}

(6)

When $Ω$ is equal to $ω_{n}$ , the maximum value of $X$ is

X = \frac{F_{d}}{k_{m}} \cdot Q

(7)

Equation (7) indicates that under the resonance states of the vibratory shutter, a larger Q-factor produces high sensitivity. The sensing layer includes two sets of cross-symmetrical electrodes, namely, F61 and F62. This design can eliminate common mode system noise in this sensor. The electrode is shown in Figure 3. The two groups of the same interface circuit are connected to the electrodes F61 and F62.

Figure 3.

Sensing layer of the proposed electric field microsensor.

Self-sustained oscillation loop for MEFS

Self-sustained oscillation loop control for MEFS

The proposed MEFS is vacuum-packaged, and a self-sustained oscillation loop is adopted to power the vibratory shutter. The vibration velocity of the comb is converted into the variation in capacitance, and a transimpedance amplifier is used to convert the variation in capacitance into output voltage. The two similar transimpedance amplifiers are connected to the electrodes F61 and F62. Instrument amplifiers are used to amplify the differential mode signal from the two transimpedance amplifiers. The autogain controller consists of a rectifier, a low-pass filter (LPF), an inverted adder, and a PI controller. Figure 4 shows the equivalent block diagram of the entire system. A transimpedance amplifier replaces the two fully differential amplifiers and instrumentation amplifier circuit. The output signal of the transimpedance amplifier $V_{1}$ is first added to the DC voltage $V_{c}$ and then fed into the driving combs of the MEFS. The same output signal $V_{1}$ also becomes the input signal for the rectifier. After an LPF, the DC voltage $A$ is measured. The difference in the DC voltage between the measured vibration velocity amplitude signal $A$ and the reference voltage $V_{R}$ , which is the external DC voltage, is measured using the inverted adder. The difference in the DC voltage is processed using a PI controller, and the corresponding output DC voltage $V_{c}$ is obtained.

Figure 4.

Block diagram of the self-sustained oscillation loop system.

The proposed MEFS features a second-order spring–mass damper system with a dynamic behavior described by

m \cdot (\overset{\cdot\cdot}{x} + \frac{ω_{n}}{Q} \overset{\cdot}{x} + {ω_{n}}^{2} x) = F_{d} + r (t)

(8)

After power-up, the white noise voltage is generated and becomes larger than the equivalent drive voltage; however, the equivalent drive voltage immediately exceeds the white noise voltage. The influence of the electrostatic force $r (t)$ generated by the white noise voltage can be ignored. Equation (8) simplifies equation (9) into

m \cdot (\overset{\cdot\cdot}{x} + \frac{ω_{n}}{Q} \overset{\cdot}{x} + {ω_{n}}^{2} x) = F_{d}

(9)

To analyze the transient behavior of the system, we define the position of the vibratory shutter as follows^9,10

x (t) = a (t) \cos (θ (t) + ϕ (t))

(10)

where $a (t)$ is the amplitude, $θ (t) + ϕ (t)$ is the instantaneous total phase of the vibratory beam position signal, $θ (t)$ is the corresponding phase generated by the inherent resonance frequency of the vibratory beam $ω_{n}$ with $θ (t) = ω_{n} \cdot t$ , and $ϕ (t)$ is the phase generated by the frequency error between the resonance frequency and the frequency of the energizing electrostatic force. Differentiating equation (10) with respect to time yields the following velocity

\begin{matrix} \overset{\cdot}{x} (t) = & - a (t) \overset{\cdot}{θ} \sin (θ + ϕ) + \overset{\cdot}{a} (t) \cos (θ + ϕ) \\ - a (t) \overset{\cdot}{ϕ} \sin (θ + ϕ) \end{matrix}

(11)

To determine $\overset{\cdot}{a} (t)$ and $\overset{\cdot}{ϕ} (t)$ , the sum of the last two terms is assumed to be 0¹⁰ as shown in the following

\overset{\cdot}{a} (t) \cos (θ + ϕ) - a (t) \overset{\cdot}{ϕ} \sin (θ + ϕ) = 0

(12)

The acceleration is obtained by differentiating equation (11) with respect to time; thus

\overset{\cdot\cdot}{x} (t) = - \overset{\cdot}{a} (t) \overset{\cdot}{θ} \sin (θ + ϕ) - a (t) \overset{\cdot}{θ} (θ + ϕ) \cos (θ + ϕ)

(13)

The driving force $F_{d}$ is simplified into

F_{d} = k_{2} \cdot V_{c} \cdot V_{a} \sin (ω t)

(14)

On the basis of the structure of the rectifier, LPF, and the phase shifter, the entire self-sustained oscillation system without the PI controller (Figure 4) is described as follows

{\begin{matrix} m (\overset{\cdot\cdot}{x} + \frac{ω_{n}}{Q} \overset{\cdot}{x} + {ω_{n}}^{2} x) = k V_{c} \overset{\cdot}{x} \\ \overset{\cdot}{A} = \frac{1}{τ} (| k_{1} \overset{\cdot}{x} | - A) \\ V_{c} = V_{R} - A \end{matrix}

(15)

$K$ is the total coefficient of the voltage converted into force, and $K = k_{1} \cdot k_{2}$ . $τ$ is the corner frequency of the LPF, $A$ denotes the output signal of the LPF, and $V_{R}$ is the direct reference current voltage. Substituting equations (10), (11), and (13) with equation (15) and combining them with equation (12) reorders the terms as follows

{\begin{matrix} \overset{\cdot}{a} (t) \cos (ω_{n} t + ϕ (t)) - a (t) \overset{\cdot}{ϕ} (t) \sin (ω_{n} t + ϕ (t)) = 0 \\ \overset{\cdot}{a} (t) \sin (ω_{n} t + ϕ (t)) + a (t) \overset{\cdot}{ϕ} (t) \cos (ω_{n} t + ϕ (t)) \\ = - a (t) \cdot B \cdot \sin (ω_{n} t + ϕ (t)) \end{matrix}

(16)

By assuming that $B = ((ω_{n} / Q) - ((k (V_{R} - A)) / m))$ , $K = k_{1} \cdot k_{2}$ , the following solution is obtained

\overset{\cdot}{a} (t) = - a (t) \cdot B \cdot \sin^{2} (ω_{n} t + ϕ (t))

(17)

\overset{\cdot}{ϕ} (t) = - B \cdot \sin (ω_{n} t + ϕ (t)) \cos (ω_{n} t + ϕ (t))

(18)

\overset{\cdot}{A} = \frac{1}{τ} (| k_{1} \overset{\cdot}{x} | - A)

(19)

For slowly time-varying systems, some equations are established in accordance with the principle of the average cycle method,¹¹ as follows

\frac{1}{T} \int_{0}^{T} \sin^{2} (ω_{n} t) dt = \frac{1}{T} \int_{0}^{T} \cos^{2} (ω_{n} t) dt = \frac{1}{2}

\frac{1}{T} \int_{0}^{T} \sin (ω_{n} t) \cos (ω_{n} t) dt = 0

\frac{1}{T} \int_{0}^{T} | \sin (ω_{n} t) | dt = \frac{1}{T} \int_{0}^{T} | \cos (ω_{n} t) | dt = \frac{2}{π}

Differential equations (16)–(19) are averaged with respect to $φ$ . These differential equations describe approximately how the displacement amplitude and phase evolve with time. If amplitude $a \geq 0$ , then the equilibria of the averaged systems described by equations (16)–(19) are

{\begin{matrix} {\bar{a}}_{0} (t) = 0 \\ {\bar{A}}_{0} = 0 \end{matrix}

(20)

{\begin{matrix} {\bar{a}}_{1} (t) = (V_{R} - \frac{m ω_{n}}{k_{1} \cdot k_{2} \cdot Q}) \frac{π}{2 k_{1} ω_{n}} \\ {\bar{A}}_{1} = V_{R} - \frac{m ω_{n}}{k_{1} \cdot k_{2} Q} \end{matrix}

(21)

The Jacobian linearization at these equilibria is

D_{x} f ({\bar{a}}_{0}, {\bar{A}}_{0}) = | \begin{matrix} \begin{matrix} - \frac{1}{2} (\frac{ω_{n}}{Q} - \frac{K (V_{R} - A)}{m_{x}}) \\ \frac{k_{1}}{τ} \frac{2}{π} ω_{n} \end{matrix} & \begin{matrix} 0 \\ - \frac{1}{τ} \end{matrix} \end{matrix} |

(22)

D_{x} f ({\bar{a}}_{1}, {\bar{A}}_{1}) = | \begin{matrix} \begin{matrix} - \frac{1}{2} \frac{K}{m} (V_{R} - \frac{m ω_{n}}{k_{1} \cdot k_{2} Q}) \frac{π}{2 k_{1} ω_{n}} \\ \frac{k_{1}}{τ} \frac{2}{π} ω_{n} \end{matrix} & \begin{matrix} 0 \\ - \frac{1}{τ} \end{matrix} \end{matrix} |

(23)

When equation (24) is satisfied, the linearized system near equilibrium $({\bar{a}}_{0}, {\bar{A}}_{0})$ has one unstable eigenvalue, and the AGC loop is excited if $\bar{a} (0) \neq 0$ .¹¹ This condition merely states that the anti-damping added by the AGC loop must overcome the natural damping present in the system

V_{R} > \frac{m ω_{n}}{k_{1} \cdot k_{2} Q}

(24)

The $({\bar{a}}_{1}, {\bar{A}}_{1})$ equilibrium point is stable when equation (24) is satisfied. In this case, the sensitivity $S$ can be written as

S = H \cdot ω_{n} \cdot X = H \cdot (V_{R} - \frac{m ω_{n}}{k_{1} \cdot k_{2} \cdot Q}) \frac{π}{2 k_{1}}

(25)

According to equation (25), the sensitivity $S$ changes with resonant frequency $ω_{n}$ , and the Q-factor $Q$ and the large DC reference voltage $V_{R}$ imply high sensitivity. Without the PI controller, the sensitivity is not robust.

Figure 4 shows a PI controller applied in the amplitude control loop.

{\overset{\cdot}{V}}_{c} = k_{p} ({\overset{\cdot}{V}}_{R} - \overset{\cdot}{A}) + k_{I} (V_{R} - A)

(26)

where $k_{p}$ and $k_{I}$ are the coefficients of the PI controller. The averaged autonomous equations derived from equations (12), (15), and (26) are

\overset{\cdot}{a} (t) = - \frac{1}{2} (\frac{ω_{n}}{Q} a (t) - \frac{k_{2} V_{c}}{m ω_{n}})

(27)

\overset{\cdot}{ϕ} (t) = 0

(28)

\overset{\cdot}{A} = \frac{1}{τ} (\frac{2}{π} k_{1} a (t) - A)

(29)

{\overset{\cdot}{V}}_{c} = - k_{p} \overset{\cdot}{A} + k_{I} (V_{R} - A)

(30)

The new equilibrium of the averaged system described is

\bar{a} (t) = \frac{π V_{R}}{2 k_{1} ω_{n}}

(31)

{\bar{V}}_{c} = \frac{m ω_{n}}{k_{1} \cdot k_{2} \cdot Q}

(32)

\bar{A} = V_{R}

(33)

The characteristic equation for the new self-sustained oscillation system is

\begin{matrix} λ^{3} & + (\frac{ω_{n}}{2 Q} + \frac{1}{τ}) λ^{2} + (\frac{ω_{n}}{2 Q} \frac{1}{τ} + \frac{k}{m ω_{n}} \frac{k_{1} k_{p}}{τ}) λ \\ + \frac{k}{m ω_{n}} \frac{k_{1}}{τ π} (\frac{2 k_{p}}{τ} - k_{I}) = 0 \end{matrix}

(34)

where $λ$ is the variable of the characteristic equation. All eigenvalues of the linearized averaged system are asymptotically stable, if and only if the stability condition for the new equilibrium is

k_{I} < \frac{k_{p}}{τ}

(35)

On the basis of the indirect method by Lyapunov, the equilibrium of the nonlinear averaged system is determined to be asymptotically stable within a neighborhood if and only if equation (35) is satisfied. For the original exact differential equations, this condition means that the solution of oscillation at the natural frequency of the resonator is stable and attractive. Equation (35) provides a criterion for selecting the parameters $k_{p}$ , $k_{I}$ , and $τ$ . In this case, the sensitivity $S$ can be expressed as

S = H \cdot ω_{n} \cdot X = H \cdot ω_{n} \cdot \frac{π V_{R}}{2 k_{1} ω_{n}} = H \cdot \frac{π V_{R}}{2 k_{1}}

(36)

According to equation (36), the sensitivity $S$ is independent of resonant frequency $ω_{n}$ and Q-factor $Q$ . With the use of a PI controller, the same DC reference voltage $V_{R}$ implies higher sensitivity.

Simulation and experiment

The microsensor is fabricated by deep reactive-ion etching bulk micromachining technology and silicon bonding technology. The structural material is single crystalline silicon with dense boron diffusion, and the substrate material is 7740# glass. The sensor is vacuum-packaged in a 14-DIP metal package. The Q-factor is approximately 1500 in this condition. Figure 5 shows that a simulation model without a PI controller is built with Simulink for the proposed MEFS.

Figure 5.

Self-sustained oscillation loop for the electric field microsensor based on SIMULINK.

The applied input disturbance is external stiffness $k_{e}$ . Table 1 lists the numerical values of the parameters used in the model. The output voltage of the electric field is calculated using correlation demodulation, shown as Scope3 in Figure 5. The total simulation time is 5 s, which is used to generate an electric field range of 0–250 V/m to verify the sensor performance.

Table 1.

Parameters of the electric field microsensor.

Parameter	Value	Parameter	Value
k_m	200 N/m	k ₁	100
M	50.7 µg	k ₂	0.021 µN/V²
Q	1500	$ω_{n}$	2π*10⁴ rad
k_p	3	k_I	20

Table 1 also lists the numerical values of the parameters used in the simulation. The critical value of the DC reference voltage $V_{R}$ is equal to 1.0 V in all numerical values of the parameters. Figure 6 shows that output voltage when $V_{R}$ is equal to 1.0 V and the applied input disturbance stiffness $k_{e}$ is changed from 0 to 80 N/m at 2 s. When equation (25) is not satisfied, the entire system is not convergent; thus, equation (25) is the stability condition of the structural vibration. When $V_{R}$ is larger than the critical value shown in Figure 7, the output voltage is related to the electric field. The larger the reference voltage $V_{R}$ , the higher the sensitivity. This conclusion is consistent with equation (25). The reference voltage can be used to adjust the sensitivity.

Figure 6.

Output voltage with V_R = 1 V and Q = 1500.

Figure 7.

Output voltage with V_R = 2 and 3 V.

Figures 8 and 9 show the output voltage with varying stiffness when the Q-factor is 1500. The increases in sensitivity and nonlinearity are limited with varying stiffness. At Q-factor = 600, the nonlinearity markedly increased with varying stiffness, as shown in Figure 9. According to equation (25), without the PI controller, the larger the Q-factor, the smaller are the sensitivity and the nonlinearity. The simulation results are consistent with equation (25). The dynamic characteristics of output voltage are determined by $τ$ , which is the corner frequency of the LPF. Changing $τ$ from 0.001 to 0.1 does not influence other parameters. The smaller the parameter $τ$ , the better it is for the stiffness disturbance at 2 s, as shown in Figure 10. An extremely large $τ$ results in output voltage overshoot.

Figure 8.

Output voltage with different design stiffness (Q = 1500).

Figure 9.

Output voltage with different design stiffness (Q = 600).

Figure 10.

Output voltage with varying $τ$ .

When the PI controller is added into the self-sustained oscillation loop for the electric field microsensor as shown in Figure 11, the other parameters remain the same, $τ$ is 0.001 and $k_{p}$ is 1, $k_{I}$ is 10, and equation (35) is satisfied. Figures 12 and 13 show the output voltage with varying stiffness and Q-factor. The sensitivity remains the same with varying Q-factor. The sensitivity and the nonlinearity are nearly zero with varying stiffness. The sensitivity is considered robust. However, the transient performance is not the same when the Q-factor is 1500, as shown in Figure 12. The output voltage overshoot exceeds the Q-factor when the Q-factor is set to 600, as shown in Figure 13. Thus, the sensor with a smaller Q-factor is easier to implement.

Figure 11.

Self-sustained oscillation loop for the electric field microsensor with the PI controller.

Figure 12.

Zoom output voltage ( $Q = 1500$ and $V_{R} = 2 V$ ).

Figure 13.

Zoom output voltage ( $Q = 600$ and $V_{R} = 2 V$ ).

Figure 14 shows the sensitivity with and without the PI controller, with the other parameters remaining the same. Given the same DC reference voltage $V_{R}$ and with the use of a PI controller, a higher sensitivity is obtained. Therefore, we can use the reference voltage to adjust the sensitivity.

Figure 14.

Sensitivity with and without the PI controller.

Given the nonlinearity of the entire system, we use the averaging method to approximate and analyze the behavior of the system. We use two stable equilibrium points of the system as equations (21) and (31)–(33). According to equations (31) and (33), the second stable amplitude of vibration at the equilibrium point $\bar{a} (t)$ is 0.5 µm before 2 s, and the stable signal A is 2 V. The simulation results are in accordance with the theoretical analysis results, as shown in Figures 15 and 16.

Figure 15.

Stable equilibrium points $\bar{a} (t)$ of the system.

Figure 16.

Stable signal $A$ of the self-sustained oscillation loop for the electric field microsensor.

The stability of the output voltage of the system depends on the parameters of the controller and whether equation (35) is satisfied. Figure 17 shows the output voltage when equation (35) is not satisfied. The output voltage does not converge, causing difficulty in determining whether the change in output voltage is caused by the electric field or by other factors.

Figure 17.

Output voltage when equation (35) is not satisfied (k_p = 1, k_I = 1500, and $τ = 0.001$ ).

Conclusion

To design a micromechanical resonant electric field sensor with high sensitivity, we propose a two-part design, including structural design and the measurement and control of circuit designs. On the basis of the theoretical analysis, comb capacitance structure can successfully reduce the nonlinearity of the two conversions. A dynamic model is built for the microsensor, which analyzes the behavior and sensitivity of the system on different closed-loop self-excited circuits using the averaging method. The theory and simulation results show that with the use of a PI controller, the sensitivity remains constant regardless of the variations in the resonance frequency of the shielding layer and the Q-factor. The sensitivity is higher with a suitable PI controller than without a PI controller. If the parameters of the PI controller fail to satisfy the constraint relationship, the output voltage is unstable and the sensitivity is distorted.

Footnotes

Academic Editor: Xiaotun Qiu

Declaration of Conflicting Interests

The author(s) declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Funding

The author(s) disclosed receipt of the following financial support for the research, authorship, and/or publication of this article: This work was supported by the Natural Science Key Fund for Colleges and Universities in Jiangsu Province (No. 13KJB510017), the Jiangsu Province Natural Science Fund (No. BK20131001), and the Open Fund of the Key Laboratory of Testing Technology and Manufacturing Process, Ministry of Education (No. 14zxzk02).

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