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## Resonant sensors based on weakly-coupled resonators

We restrict our study to closed-loop Weakly-Coupled-Resonators (WCRs) based on two resonators with nominally identical natural pulsations and quality factors. The resonators are coupled in a way that there is a force acting on one of the resonators, which depends on the state of the other resonator. Depending on the coupling mechanism and on the driving scheme, the resonators may synchronize and oscillate at the same frequency. This synchronized state can be engineered to be highly sensitive to stiffness mismatch between the resonators. In particular, the phase difference between the two synchronized resonators or the ratio of their oscillation amplitudes provides high-sensitivity measurements of , and passive immunity to environmental fluctuations. A simplified schematic of two coupled resonators is illustrated in Fig. 1.2.4.

### Resonant sensors based on mode-localization oscillators

MEMSWCRs using mode localization, originally derived in [4], are based on two electrostaticallycoupled or mechanically-coupled resonators. A functional representation of a closed-loop of MOLO whose architecture is similar to the one in [15] is shown in Fig. 1.2.6. The gain

denotes the relative coupling strength (i.e., the ratio of the coupling stiffness to the nominal stiffness k) through which the resonators are coupled. The limiter and the phase shifter ensure that the Barkhausen criterion is met in both loops, and the two resonators oscillate at 0, with the same amplitude (XST = YST ) provided = 0. Two oscillation states are possible, being sin(ST ) ±1/Q 1 or cos(ST ) ±1. A small variation of results in a large change of the oscillation amplitudes and a little variation in their phase difference [13]. Thus, only the amplitude ratio R is ever considered as an output metric for such sensors.

#### Optimum output metric in the linear oscillation regime

The parametric sensitivities and FOMs obtained close to = 0 are listed in Table 1.1, considering all architectures in the linear and nonlinear oscillation regimes. In the linear regime, regardless of the output metric, all sensitivities to noise decrease as 1/X0, while sensitivities to mismatch are independent of X0. Consequently, the FOM of any output metric increases as X0, with the expected p2 maximum improvement for architectures based on 2 resonators. This is hardly a surprise, but this result corrects a misconception that the ultimate resolution (i.e. the theoretical resolution of a sensor, taking into account thermomechanical noise only)WCRs may be improved by decreasing the coupling stiffness [37] . Arguably, the best trade-off between metric performance and readout complexity can be obtained with in MILO for = /2 ± /4, which has the best FOM and is quasi-digital. Note that sensitivity to is used as a figure-of-merit by many authors [15, 4, 25], which makes sense when measurement noise is dominant over thermomechanical noise. However, one should be careful in that case to only compare output metrics with the same readout requirements, so that measurement noise can be compared from one architecture to the next. For example, it is often read that amplitude ratio in MOLOs is 1/ more sensitive to than conventional closed-loop frequency measurements.However, this comparison makes little sense as the readout associated with frequency measurements and amplitude measurements are very different. Indeed, they are not likely to have the same measurement noise. Furthermore, a trade-off must always be made between sensitivity and dynamic range: for example, MILOs have the best sensitivity to mismatch among amplitude or amplitude ratio-modulated architectures, but they also have the narrowest Locking-Range (LR) and the smallest dynamic range.

**Optimum output metric in the nonlinear oscillation regime**

Above Xc, the sensitivity to mismatch of all architectures decreases as 1/X2 0 , the sensitivity to noise of R decreases as 1/X3 0 ; while that of and decreases as 1/X0, because of the A-f effect. Consequently, the FOM of and start decreasing with X0, whereas that of R keeps increasing linearly with X0 regardless of which architecture is considered. Moreover, above the critical amplitude, the sensitivity to mismatch of R no longer depends on (in MOLOs) or on Q (in MILOs). Thus, sensitivity may be controlled electrically by adjusting the amplitude of the excitation force, rather than being controlled mechanically as in the linear regime. Another positive trade-off that is the decrease in the sensitivity and the increase of the sensor dynamic range, as illustrated in [19]. These interesting properties of the ratio-metric WCRs may also apply to mode-matched MEMS gyroscopes for which the amplitude variation between two resonators coupled through the Coriolis force is the metric of choice.

**Domain: discrete time vs continuous**

M can be implemented using either Discrete-Time (DT) circuits or Continuous-Time (CT). The key concepts of the Lth order CT and DT modulator architectures are shown in Fig. 2.3.2 and Fig. 2.3.3 respectively. The main difference between them is where the sampling is done. In a DT-M, sampling of the input signal occurs prior to the loop filters (Fig. 2.3.2(a)), whereas for CT-M sampling operation takes place just before the quantizer (Fig. 2.3.3(a)). This is a major advantage of the CT structure since all the errors generated by the non-idealities of the sampling process are shaped by the NTF as for the quantization noise. Furthermore, in a DT-M the loop filter is implemented by Switched-Capacitor (SC) circuits (Fig. 2.3.2(b)), whereas in a CT-M by an transconductance amplifier as OTA-RC (Fig. 2.3.3(b)) or by Gm-C (Fig. 2.3.3(c)) topology. The OTA-RC implementation is more linear than the Gm-C implementation, but it is also power consuming [52]. A trade-off to maintain a good linearity while reducing the power consumption is to perform the first stage as an OTA-RC filter and the following ones as Gm-C filters [53]. The advantages and drawbacks of DT-M and CT-M implementations are discussed in detail in the following subsections.

**Table of contents :**

Acknowledgment

Contents

List of Figures

List of Tables

Notations

Résumé long en français

Introduction

Background and motivation

Research Contribution

Organization of the thesis

**1 MEMS resonant sensors and electronic interface **

1.1 MEMS resonator model

1.2 Resonant MEMS architectures and sensing mode

1.2.1 Resonant sensors based on a single resonator sensors

1.2.1.1 Open Loop sensing mode

1.2.1.2 Closed Loop sensing mode

1.2.2 Duffing nonlinearity effect on single resonator sensors

1.2.3 Open loop vs. closed loop

1.2.4 Discussion

1.2.5 Resonant sensors based on weakly-coupled resonators

1.2.5.1 Resonant sensors based on mutually-injection-locked oscillators

1.2.5.2 Resonant sensors based on mode-localization oscillators

1.2.6 Duffing nonlinearity effect on sensors based on weakly coupled resonators

1.3 Global comparison

1.3.1 Optimum output metric in the linear oscillation regime

1.3.2 Optimum output metric in the nonlinear oscillation regime

1.3.3 Conclusion

1.4 Bilinear amplitude-atrio output metric

1.5 Literature on the analog to digital ratiometric interface

1.5.1 Readout interface requirements

1.5.2 One step approach

1.5.3 Two steps approach

1.5.4 ADC as voltage divider

1.5.5 Demodulator

1.6 Conclusion

**2 Ratiometric analog to digital interface based on sigma delta modulator **

2.1 Introductionn

2.2 Operating principle and resolution of a sigma delta converter

2.2.1 Operating principle

2.2.2 Metrics and definitions

2.2.3 Resolution

2.3 Modulator architecture selection

2.3.1 Domain: discrete time vs continuous

2.3.1.1 Coefficient sizing and trimming with frequency

2.3.1.2 Thermal Noise

2.3.1.3 Jitter

2.3.1.4 Switch non-idealities

2.3.1.5 Power consumption

2.3.1.6 Conclusion

2.3.2 Single-bit vs. multi-bit quantizer

2.3.3 Loop filter order

2.3.4 Loop topology

2.3.5 Conclusion

2.4 Modulator coefficients

2.4.1 Feedback-coefficient b1

2.4.2 Integrator-coefficients a1 and a2

2.4.3 Simulation results and conclusion

2.5 System level consideration and electrical modeling of the analog blocks

2.5.1 Noise

2.5.2 Integrator non-idealities and optimum OTA specifications

2.5.2.1 OTA finite DC gain

2.5.2.2 Limited slew-rate and gain-bandwidth of the OTA

2.5.2.3 Saturation

2.5.3 Specifications of the analog blocks

2.5.4 Model implementation

2.5.5 Simulation results

2.6 Conclusion

**3 Circuit level design of the sigma delta modulator **

3.1 Switch

3.2 Operational transconductance amplifier ’OTA’

3.2.1 OTA design comparison

3.2.2 gm/Id design considerations

3.2.3 Folded cascode OTA design

3.3 Bias circuit

3.4 Common mode feedback control circuit

3.5 Integrator

3.5.1 Parasitic insensitive integrator

3.5.2 1-bit DAC input integrator

3.6 Compartor

3.7 Feed forward coefficients

3.8 Non overlapping clock

3.9 Conclusion

**4 Post layout simulation results **

4.1 Introduction

4.2 Switch

4.3 Operational transconductance amplifier ’OTA’

4.3.1 Nominal simulation

4.3.2 PVT simulation

4.4 Capacitor bank

4.5 Comparator and Non overlapping clock

4.6 Modulator

4.6.1 PVT simulation

4.7 Conclusion

Conclusion and perspectives

Appendices

**A Parametric analysis of nonlinear duffing resonator **

**B Noise shaping **

**C Integrator transfer function **

**D VerilogA **

D.1 OTA

D.2 Comparator

D.3 Save data

**E Temperature aware MILO sustaining electronics **

E.1 Introduction

E.2 Current to voltage temperature aware amplifier

E.2.1 Design consideration

E.2.2 Proposed transimpedance amplifier and its operating principle

E.2.3 PVT simulation results

E.2.3.1 ac analysis

E.2.3.2 Input impedance

E.2.3.3 Linearity

E.2.4 Discussion

E.3 Voltage to voltage temperature aware amplifier

E.3.1 Operating principle

E.3.2 Optimization methodology

E.3.2.1 Input differential-pair biasing condition

E.3.2.2 Chanel length effect

E.3.2.3 Comparison of the proposed amplifier with the ordinary active load amplifier

E.3.3 Discussion

E.3.4 Complementary implementation of the proposed temperature aware amplifier

E.4 A 105 dB temperature aware two stages transimpedance amplifier

E.5 Temperature aware two stage miller compensated DOTA

E.6 Sample and hold peak detector

E.7 Conclusion

**Bibliography **