A New Readout Method for a High Sensitivity Capacitance Sensor Based on the Weakly Coupled Resonators

This paper proposes a new readout method for a sensor to detect minute variations in the capacitance. A sensor is based on the weakly coupled electrical resonators that use an amplitude ratio (AR) as an output signal. A new readout scheme with a relatively higher output sensitivity is proposed to measure the relative changes in the input capacitor. A mathematical model is derived to express the readout output as a function of change in the capacitance. To validate the theoretical model, a system is modelled and designed using an industry-standard electronic circuit design environment. SPICE simulation results are presented for a wide range of design parameters, such as varying coupling factors between the two electrical resonators. Sensitivity comparison between the existing and the proposed AR readout is presented to show the effectiveness of the method of detection proposed in this work.


Introduction
In today's world, intelligent electronic circuits [1][2][3] to detect/measure minute capacitance change form an integral part of many micromachined/MEMS sensors [4][5][6]. Often, the key design requirements of such capacitive readouts are high sensitivity, ultra-high resolution (low noise floor), low power consumption, etc. However, designing and implementing a high-performance capacitive readout to precisely measure the minute variations in to the input capacitance pose various challenges. These include intrinsic and extrinsic noise sources, and board-level parasitics, etc. Recently, a capacitance readout method based on the two weakly coupled resonators (WCR) has been used to measure minute variations in a capacitance [2,7]. This research was done in the context of proposing a readout method for capacitance sensors. Our work presents an innovative, and new readout method to measure small capacitance changes in the micromachined/MEMS transducers. A readout method reflects in the higher changes in the output voltage (relatively higher sensitivity) for small incremental changes in the perturbation (capacitance). A readout circuit is called as capacitance-to-amplitude/ amplitude ratio (AR) converter.

Mathematical Model
Consider an RLC circuit, shown in Figure 1. This circuit is used to form two electrically coupled resonators (CR) to measure changes to the input capacitance. In this circuit, the following assumptions are valid: L1 = L2 = L, C1 = C2 = C and R1 = R2 = R. This establishes a symmetry in a circuit. A coupling factor between the two individual RLC circuits is modelled as κ = C/Cc. A change in the input capacitance is ∆C. The normalized value of the capacitance change (in the form of a perturbation) is expressed as δ = ∆C/C. Condition for a coupling is given as Cc >> C, indicating two resonant modes can be reasonably distinguished in the output frequency response. The quantities V1(t) and V2(t) are input voltages, and quantities I1(t) and I1(t) are the output loop currents. Assuming no damping in the system-i.e., R = 0-a set of governing differential equations are represented as [ ] It is assumed that the system is driven with only one voltage source-i.e., V1(s) ≠ 0, and V2(s) = 0. Applying Laplace Transform to Equations (1) and (2) yields the following: Solving further, one can obtain the transfer functions as The denominator in (5) and (6) is the characteristic equation which provide the roots. In other words, the roots or eigen-frequencies, 2 i ω (i = 1, 2) as a function of ∆C are obtained as follows: Using (7) in (5) and (6), and taking the ratio of 1 (j ) H ω and 2 (j ) H ω -i.e., 1 ω ω -provides the charge output amplitude ratio (AR). The amplitude ratio, i AR (i = 1, 2), as a function of ∆C, is then obtained as Figure 2a shows a resulting plot of the output loop currents of a RLC readout circuit model seen in Figure 1. A fixed value of the coupling capacitor Cc is used for the simulation. Note that equal output amplitudes of the loop currents represent the initial symmetry of the system. Figure 2b presents similar plots of the output loop currents for varying values of the coupling capacitor Cc.

SPICE Modeling and Results
Higher effective values of Cc leads to the lower effective value of a coupling between the two resonant circuits. A relatively weaker coupling is reflected by the closed-spacing of the mode-frequencies, as seen from the plots.

A New Readout Method (Measure at ω0 Instead of ωi)
A conventional method of sensing the capacitance perturbation is existent in the literature [2]. Such method relies on the fact that mode-frequencies and amplitude ratio vary as a function of ∆C (Equations (7) and (8)).
AR method of sensing is being pursued owing to the relatively higher sensitivity to the input perturbations in the weakly coupled resonant system. In our work, we propose a new method of sensing in which amplitude/s and/or AR of the jth resonator (j = 1, 2) at the ith mode (i = 1, 2) is recorded at the initial resonant mode-frequency, ω0 (instead of ωi) when perturbation is applied in the system. This can be illustrated from Figure 4 that presents a frequency response of the jth resonator (j = 1, 2) at the first resonant mode. As seen, an arrow indicates an amplitude change in the jth resonator (j = 1, 2) when ∆C is applied in the system and this is the conventional output of a readout. We instead propose to sense the amplitude shifts at the resonance for each applied value of ∆C. This is indicated by dots forming a downward line in Figure 4. A similar readout approach can be utilized with a frequency response of the jth resonator (j = 1, 2) at the second resonant mode. Figure 5a shows output amplitude response of a RLC coupled resonators. It shows how amplitudes of the jth resonator (j = 1, 2) varies as a function of % variation in δ. In the same plot, it also shows how amplitudes of the jth resonator (j = 1, 2) varies as a function of % variation in the δ if the proposed method of readout is used. Clearly, a linear (up to certain range) and higher output is available with the new method of readout. Figure 5b provides a final readout in terms of AR of the system. Two AR readout methods are seen in the plot. A proposed AR readout method shows higher changes in the output for small incremental changes in the capacitance.  Figure 5. Output response of a coupled RLC capacitance readout circuit. (a) Output current amplitude vs. ∆C or δ. Lable a11 is amplitude of resonator 1 at mode 1, a22 is amplitude of resonator 2 at mode 1 (b) Amplitude ratio (AR) readout of RLC coupled resonator circuit. ω0 is the initial readout resonant frequency at mode 1.

Conclusions
A new readout method using an electrically coupled resonator circuits is proposed. Such readout circuit when interfaced with the micromachined transducer can be effective for the enhanced sensitivity output. The same concept of the readout can be extended to the MEMS resonators and other inertial sensors, where overall system can benefit from the high quality factor of the MEMS transducers/sensors. Extended work should involve improving the linearity in the output of the proposed readout method.
Findings: This research received no external funding.

Conflicts of Interest:
The authors declare no conflict of interest.