Equivalent Circuit Model of an Optomechanical MEMS Electric Field Strength Sensor ^†

Abstract

We present a simple equivalent circuit model for the transfer function of an optomechanical MEMS transducer capable of distortion-free electric field strength measurements. This model allows not only to qualitatively understand the characteristics of the transducer but also takes into account parasitic effects and material properties. Such parasitic effects have been observed while evaluating the first results of electric field measurements performed with the sensor. The model helped to identify and diminish these parasitic effects.

1. Introduction

Monitoring the strength of static and quasistatic electric fields (E-field) is key for many scientific and industrial purposes, e.g., improving the reliability of weather forecast, researching charge distributions in cloud systems or safety in the vicinity of high-voltage infrastructure. Despite this high demand, as of yet, there exist no state-of-the-art sensors which are sufficiently capable of fulfilling this task. This is either due to the necessity for grounded components which severely distort the E-field (field mills [1,2]), large temperature dependence (electrooptical sensors [3,4]) or limited bandwidth (antennae [5,6]). Recently, a new transduction scheme for the electric field strength was presented which does not suffer from these drawbacks [7].

This transduction exploits the electrostatic induction occurring in conducting bodies inside an electric field $E_0$. The surface charges arising due to the induction are themselves subject to $E_0$ leading to a force $F_{\mathrm{es}}$ proportional to $E_0^2$. The MEMS sensors are designed such that the electric field is concentrated across a narrow gap separating a moveable silicon mass from a fixed Si part (Figure 1a), which increases $F_{\mathrm{es}}$. The force leads to a displacement of the Si mass which is read out optically.

This is achieved by light flux modulation with an optical shutter (Figure 1b). The optical shutter is composed of a Cr grid deposited on a glass chip and an identical grid etched into the moving mass of the Si chip. Thus, light which is introduced onto the top side of the sensor chip is modulated by the mass displacement caused by the electric field. This kind of readout has proven to be highly sensitive [8,9].

Since the E-feld transduction scheme is new, little is known about its properties. Making use of an equivalent circuit model helps understanding a few basic characteristics and provides a qualitative understanding.

2. Model

The equivalent circuit model of the sensor’s frequency-dependent transfer function $H_{\mathrm{s}}\left(\omega \right)$ presented here encompasses not only the sensor chip but also parts of the measurement setup. These parts are the capacitor plates used to provide the electric field to the MEMS. As depicted in Figure 2, the field plates of the experimental setup and the MEMS chip can be understood as three capacitors connected in series.

The first capacitor comprises one of the aluminum plates and the opposing side face of the device layer of the MEMS chip. The second capacitor C corresponds to the sensing gap of the MEMS chip and the third capacitor encompasses the remaining aluminum plate and opposing face of the MEMS chip. Since the chip is placed in the centre of the setup and the dimensions of the plates are the same, it follows with sufficient accuracy that the external capacitances are the same, i.e., $C_{\mathrm{e}}$. The voltage $U_0$ applied to this voltage divider corresponds to the voltage which generates the electric feld $E_0=U_0/d_{\mathrm{p}}$ to be measured, where $d_{\mathrm{p}}$ is the distance between the field plates. The force picked up by the sensor corresponds to the square of the voltage drop U at C, i.e., $F_{\mathrm{es}}\propto U^2=H_0^2{U}_0^2$, where $H_0$ is the characteristic of the capacitive voltage divider which can be calculated as the fraction of the impedance of the sensing gap $Z_0$ over the total impedance $Z_{\mathrm{tot}}$,

$$H_0=\frac{Z_0}{Z_{\mathrm{tot}}}=\frac{1}{1+2\frac{C}{C_{\mathrm{e}}}},\mathrm{with}{Z}_{\mathrm{tot}}=Z_0+2Z_{\mathrm{e}},\mathrm{and}{Z}_{0,\mathrm{e}}=\frac{1}{\mathrm{i}\omega C_{0,\mathrm{e}}}.$$

(1)

This expression is independent from $\omega$ and, thus, constant. Equation (1) corresponds to the ideal case, i.e., perfect conductance of the Si and infinite resistance of the gap between. Furthermore, it reveals that a low ratio $C/C_{\mathrm{e}}$ is in favour of a large voltage drop across C and, thus, a large force $F_{\mathrm{es}}$.

This approach can also be used to explore what happens, if a parasitic resistance $R_{\mathrm{p}}$ in parallel to C occurs. The impedance of a resistor in parallel with a capacitor is given as $Z=R_{\mathrm{p}}/(\mathrm{i}\omega R_{\mathrm{p}}C+1)$. The corresponding voltage drop over the sensing capacitor is, therefore, proportional to the characteristic

$$H\left(\omega \right)=\frac{Z}{Z_{\mathrm{tot}}}=\frac{1}{\left(1+2\frac{C}{C_{\mathrm{e}}}\right)\left(1+\frac{2}{\mathrm{i}\omega R_{\mathrm{p}}(C_{\mathrm{e}}+2C)}\right)}=\frac{H_0}{1+\frac{{\omega }_{\mathrm{c}}}{\omega }}.$$

(2)

In contrast to the ideal case Equation (1), this expression is frequency dependent. It corresponds to a first order high-pass with corner frequency ${\omega }_{\mathrm{c}}=2/R_{\mathrm{p}}(C_{\mathrm{e}}+2C)$.

It can be seen that a finite parasitic resistance severely affects the E-feld transduction for low frequencies and DC. In particular, the bandwidth becomes clipped for frequencies $\omega <{\omega }_{\mathrm{c}}$. Note that $H\left(\omega \right)$ enters the transfer function quadratically, $A\left(\omega \right)\propto H^2\left(\omega \right)\chi \left(2\omega \right)$, where $\chi$ corresponds to the mechanical characteristic of the MEMS [7,10]. Therefore, the linear increase of $\left|H\right(\omega \left)\right|$ reflects in a quadratic increase in $\left|A\right(\omega \left)\right|$ worsening the circumstance (see Figure 3).

3. Results

The above described parasitic effect was encountered during the first measurements with the sensor. The model helped to identify its source as the plastic chip holder which introduced a resistance across the sensing gap (see Figure 3).

Estimating the capacitances C and $C_{\mathrm{e}}$ from the geometry of MEMS and field plates and the corner frequency $f_{\mathrm{c}}\sim 10$ Hz from the measured data, it was possible to roughly quantify the value of $R_{\mathrm{p}}$. The distance between the plate and the chip face is given by $d_{\mathrm{e}}=6.5$ mm. The mean area of the external capacitor can be estimated by $A_{\mathrm{e}}=3.64$ cm². Therefore, the external capacitance roughly has a value of $C_{\mathrm{e}}\approx {\epsilon }_0{A}_{\mathrm{e}}/d_{\mathrm{e}}=0.49$ pF.

The capacitance C of the sensing gap can be estimated in a similar manner and has around $C\sim 0.13$ pF for a 10 µm gap. This yields a parasitic resistance of $R_{\mathrm{p}}=1/\pi f_{\mathrm{c}}(C_{\mathrm{e}}+2C)\approx 45$ G$\Omega$. Even though this value is quite large, the effect was clearly noticeable. Therefore, the choice of mounting the MEMS chip is crucial for proper measurements.

4. Conclusions

A very simple equivalent circuit model of an optomechanical MEMS electric field sensor was presented. The model also encompasses the effects of a parasitic resistor. This way it was possible to explain and identify the source of the reduced sensitivity of the sensor at low frequencies. The model can be expanded easily to account for, e.g., the finite condictance of Si.