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DYNAMIC OFFSET CANCELLATION FOR MEMS

ACCELEROMETERS

A DISSERTATION

SUBMITTED TO THE DEPARTMENT OF ELECTRICAL

ENGINEERING

AND THE COMMITTEE ON GRADUATE STUDIES

OF STANFORD UNIVERSITY

IN PARTIAL FULFILLMENT OF THE REQUIREMENTS

FOR THE DEGREE OF

DOCTOR OF PHILOSOPHY

Pedram Lajevardi

May 2012

http://creativecommons.org/licenses/by-nc-nd/3.0/us/

This dissertation is online at: http://purl.stanford.edu/dk850bw2227

© 2012 by Pedram Lajevardi. All Rights Reserved.

Re-distributed by Stanford University under license with the author.

This work is licensed under a Creative Commons Attribution-Noncommercial-No Derivative Works 3.0 United States License.

ii



http://purl.stanford.edu/dk850bw2227

I certify that I have read this dissertation and that, in my opinion, it is fully adequatein scope and quality as a dissertation for the degree of Doctor of Philosophy.

Boris Murmann, Primary Adviser


Roger Howe


Bruce Wooley

Approved for the Stanford University Committee on Graduate Studies.

Patricia J. Gumport, Vice Provost Graduate Education

This signature page was generated electronically upon submission of this dissertation in electronic format. An original signed hard copy of the signature page is on file inUniversity Archives.

iii

Abstract

Today’s Micro-Electromechanical Systems (MEMS) accelerometers suffer from input

offsets that drift with temperature and time. These devices are calibrated once after

fabrication in order to cancel their offset. In order to cancel the offset drift over

temperature, the temperature is swept during the post-fabrication calibration and the

offset is recorded as a function of temperature. However, the residual offset drifts over

the life-time of the device, and cannot be cancelled by a one-time factory calibration.

With the continuously increasing precision requirements of accelerometers, this offset

drift is emerging as an issue.

This dissertation presents a new approach that dynamically measures and cancels

a major part of the offset drift that is due to change in parasitic capacitance of the

bondwires in a system-in-package-type MEMS accelerometer. This approach is based

on modulation of the spring constant of the sensor element by applying a modulating

electrostatic force. A prototype interface IC was fabricated in a 0.18-µm 3-V CMOS

technology, and was packaged and tested with a MEMS sensor element. The CMOS

readout dissipates 3.1 mW, and has a noise-floor of 220 µg/√Hz. The bandwidth of

the interface is 1 kHz with a usable bandwidth of 200 Hz. The full-scale range is 9.14

g. The proposed scheme reduces the bondwire offset of a prototype by a factor of 112

(41dB).

v

Acknowledgements

I should start by thanking my advisor, Professor Boris Murmann. I am very grateful

for his guidance and support during my graduate studies. He has brought out the

best in me, and what I have learned from him, both technically and personally, will

be with me for the years to come.

I thank Professor Roger Howe for his advice on my PhD work, and another project

I was involved with at Stanford. Conversations with him are always very inspiring. I

would like to thank Professor Bruce Wooley for being on my thesis committee, reading

my thesis, and providing me with helpful feedback. Professor John Pauly was very

welcoming when I had questions and I thank him for his advice during this work.

This research was supported by Robert Bosch Research and Technology Center,

and I have benefited from the invaluable collaboration with their IC team in Palo Alto.

Vladimir Petkov has been a great mentor. I would like to thank him for always having

time to discuss my work. His feedbacks and suggestions were crucial for the success

of this work. Christoph Lang was very supportive throughout the project. We had

very fruitful discussions, and I am very grateful for his support and positive attitude.

Sam Kavusi has been a great friend, and has given me helpful advice on numerous

occasions. I would also like to extend special thanks to Chinwuba Ezekwe, Ganesh

Balachnadran, and Johan Vanderhaegen for many discussions about the design and

the layout of different blocks. Xinyu Xing helped me tremendously with CAD and

pdk issues. I thank Arlen Olive for his help during the PCB design and test. I thank

Andrew Cheng for his help with measurements on the shaker table. Fun discussions

vi

with Thomas Rocznik at late hours in the lab helped me gather energy, and continue

work.

Center for Integrated Systems (CIS) is run by great administrators, and I thank

them all. In particular, I would like to thank Ann Guerra and Joseph Little for always

going the extra mile in helping me. Ann has been extremely helpful and kind, and

I thank her for making sure that everything is going very smoothly. Without Joe,

people cannot work in CIS for more than a few seconds. Joe has always been available,

through email if not in person. I thank him for his help on countless occasions. I

would like to thank all members of the Murmann group, past and present: Echere

I., Yangjin O., Manu A., Jason H., Parastoo N., Manar E., Clay D., Jim S., Wei

X., Justin K., Drew H., Yoonyoung C., Donghyun K., Ray N., Alireza D., Noam

D., Ross W., Martin K., Alex G., Vaibhav T., Siddharth S., Ryan B., Douglas A.,

Bill C., Jonathon S., Man-Chia C., Alex O., Mahmoud S.; thank you all for helpful

discussions and your feedbacks on my presentations. I hope we can stay in touch.

I also thank many friends who made sure I am enjoying my days at Stanford.

Mohammad H., Pedram S., Rostam D., Ali F., Pedram M., Shahriar A., Haleh T.,

Reza N., Hossein K., Roozbeh P., Leila Z., Sahar N., Narges B., Amirali K., Bita N.,

Raja J., Omid A., Maryam F., Fernando G., Farshid M., Amin N., and Mehdi M.:

thank you all for wonderful times at Stanford.

I also thank many family members for their support. I thank Shirin, Behnam, and

Mehri Zand especially for hosting me on my first days at Stanford. I thank Shahriar,

Sima, Milad, and Marjan for keeping my company. My phone conversations with

Farzad have always given me great joy and strength, and I shall strive to make them

more frequent! I thank Fardaneh, Hamid, and Pedram K. for their love and support

over the past few years. Especial thanks go to my brother, Payam, who has always

been making sure that I am doing fine. He has been a great mentor and role model

throughout my life, and I have missed him very deeply during my graduate studies.

I thank Laila for being so nice and supportive, and for taking care of me during my

many visits to Boston and Chicago, especially during my internship in Boston. I am

vii

forever indebted to my parents, Soraya and Hosein. Their love and support has been,

and will be forever, a source of strength. I thank them for always encouraging me to

pursue my passion, even when it required being away from them. Finally, I would

like to thank my wife, Arezou, for her love, support, and encouragement. She has

been my biggest fan on my best workdays, and my escape from the gloomy ones. I

thank her for believing in me when I doubted myself.

viii

Contents

Abstract v

Acknowledgements vi

1 Introduction 1

1.1 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

1.2 Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2

1.3 Chapter Organization . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

2 Micromachined Capacitive Accelerometers 4

2.1 Principles of Operation . . . . . . . . . . . . . . . . . . . . . . . . . . 7

2.1.1 Spring Force . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

2.1.2 Force-to-Displacement Transfer Function . . . . . . . . . . . . 9

2.1.3 Sensor Capacitance . . . . . . . . . . . . . . . . . . . . . . . . 10

2.1.4 Force-to-Capacitance Transfer Function . . . . . . . . . . . . . 11

2.1.5 Electrostatic Force . . . . . . . . . . . . . . . . . . . . . . . . 11

2.1.6 Electrostatic Spring Constant . . . . . . . . . . . . . . . . . . 14

2.1.7 Snap-In Voltage . . . . . . . . . . . . . . . . . . . . . . . . . . 16

2.2 Offset in Accelerometers . . . . . . . . . . . . . . . . . . . . . . . . . 20

2.2.1 MEMS Fabrication . . . . . . . . . . . . . . . . . . . . . . . . 21

2.2.2 MEMS Charge Storage . . . . . . . . . . . . . . . . . . . . . . 21

2.2.3 Bondwires . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22

ix

2.2.4 Electronics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24

3 Offset Detection and Cancellation 26

3.1 Prior Work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26

3.1.1 Trimming . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26

3.1.2 Auto-zeroing . . . . . . . . . . . . . . . . . . . . . . . . . . . 27

3.1.3 Chopping . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28

3.1.4 Open-Loop Sensor Modulation . . . . . . . . . . . . . . . . . . 29

3.2 Closed-Loop Sensor Modulation . . . . . . . . . . . . . . . . . . . . . 30

3.3 Spring Constant Modulation . . . . . . . . . . . . . . . . . . . . . . . 33

3.4 Offset Cancellation Loop . . . . . . . . . . . . . . . . . . . . . . . . . 36

3.4.1 Cross-Correlation . . . . . . . . . . . . . . . . . . . . . . . . . 36

3.4.2 Modulation Signal . . . . . . . . . . . . . . . . . . . . . . . . 39

3.4.3 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41

4 Interface Architecture and Design 42

4.1 Accelerometer Interface Challenges . . . . . . . . . . . . . . . . . . . 42

4.1.1 Capacitance Measurement . . . . . . . . . . . . . . . . . . . . 42

4.1.2 Linearity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44

4.1.3 MEMS with a Single Port . . . . . . . . . . . . . . . . . . . . 46

4.2 Interface Design . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46

4.2.1 Multiplexer and Timing Diagram . . . . . . . . . . . . . . . . 49

4.2.2 Capacitance-to-Voltage Converter . . . . . . . . . . . . . . . . 50

4.2.3 Integrator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56

4.2.4 Compensator . . . . . . . . . . . . . . . . . . . . . . . . . . . 59

4.2.5 Quantizer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63

4.3 Interface Design Considerations . . . . . . . . . . . . . . . . . . . . . 65

4.3.1 Settling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65

4.3.2 Sampling Rate . . . . . . . . . . . . . . . . . . . . . . . . . . 66

4.3.3 OTA Design . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67

x

4.3.4 Stability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69

4.4 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70

5 Experimental Results 71

5.1 Interface IC . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71

5.2 Test Setup . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72

5.3 Measured Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75

5.3.1 Output Spectrum . . . . . . . . . . . . . . . . . . . . . . . . . 75

5.3.2 Convergence . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78

5.3.3 Sensitivity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80

5.3.4 Bondwire Deformation . . . . . . . . . . . . . . . . . . . . . . 81

5.3.5 Parasitic Accelerations . . . . . . . . . . . . . . . . . . . . . . 84

5.4 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88

6 Conclusion 89

6.1 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89

6.2 Future Work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90

Bibliography 92

xi

List of Tables

5.1 Breakdown of active area for each block. . . . . . . . . . . . . . . . . 73

5.2 Measured beat signal amplitude at the correlator output, caused by a

parasitic acceleration. . . . . . . . . . . . . . . . . . . . . . . . . . . . 88

xii

List of Figures

1.1 Sensor interface. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2

2.1 Differential capacitive sensor. . . . . . . . . . . . . . . . . . . . . . . 7

2.2 Spring force. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8

2.3 Electrostatic force on one capacitor. . . . . . . . . . . . . . . . . . . . 13

2.4 Electrostatic force on two capacitors. . . . . . . . . . . . . . . . . . . 14

2.5 Sum of spring and electrostatic Forces on one capacitor. . . . . . . . . 17

2.6 Sum of spring and electrostatic forces on two capacitors. . . . . . . . 18

2.7 Amplifier with offset. . . . . . . . . . . . . . . . . . . . . . . . . . . . 20

2.8 Bondwires used in a sensor interface. . . . . . . . . . . . . . . . . . . 23

2.9 Capacitance change, and the corresponding acceleration offset, that is

caused by the movement of a bondwire. . . . . . . . . . . . . . . . . . 24

3.1 Auto-zeroed amplifier. . . . . . . . . . . . . . . . . . . . . . . . . . . 28

3.2 Chopped amplifier. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29

3.3 Chopped amplifier by modulating sensor sensitivity. . . . . . . . . . . 29

3.4 Block diagram of the closed-loop sensor. . . . . . . . . . . . . . . . . 31

3.5 Block diagram of the closed-loop sensor with modulation. . . . . . . . 32

3.6 Sensor element with modulated spring constant. . . . . . . . . . . . . 33

3.7 Block diagram of the closed-loop sensor with spring constant modulation. 36

3.8 Block diagram of the closed-loop sensor with offset cancellation loop. 37

4.1 Capacitance measurement. . . . . . . . . . . . . . . . . . . . . . . . . 43

xiii

4.2 Sensor capacitance. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45

4.3 Sensor element with one or two ports. . . . . . . . . . . . . . . . . . . 47

4.4 Block diagram of the closed-loop sensor. . . . . . . . . . . . . . . . . 48

4.5 Block diagram of the closed-loop accelerometer interface. . . . . . . . 48

4.6 Timing diagram for one cycle of the interface operation. . . . . . . . . 49

4.7 Circuit diagram for the MUX. . . . . . . . . . . . . . . . . . . . . . . 51

4.8 Capacitance-to-voltage converter with CDS. . . . . . . . . . . . . . . 52

4.9 Sensor element, MUX, and C-to-V during different phases. . . . . . . 53

4.10 Half-circuit model of the sensor element, MUX, and C-to-V during

different phases. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54

4.11 Integrator with CDS. . . . . . . . . . . . . . . . . . . . . . . . . . . . 57

4.12 Half-circuit model of the integrator during different phases. . . . . . . 58

4.13 Block diagram of the compensator. . . . . . . . . . . . . . . . . . . . 59

4.14 Differentiator with some DC gain. . . . . . . . . . . . . . . . . . . . . 60

4.15 Half-circuit model of the compensator during different phases. . . . . 61

4.16 Differentiator with two inputs. . . . . . . . . . . . . . . . . . . . . . . 62

4.17 Block diagram of the quantizer. . . . . . . . . . . . . . . . . . . . . . 63

4.18 Circuit diagram of the comparator. . . . . . . . . . . . . . . . . . . . 64

4.19 Differential-mode half-circuit model of the sensor element, MUX, C-

to-V, integrator, and compensator during Φ1B . . . . . . . . . . . . . . 65

4.20 Circuit diagram of the amplifier. . . . . . . . . . . . . . . . . . . . . . 68

4.21 Circuit diagram of the common-mode feedback. . . . . . . . . . . . . 69

5.1 Chip micrograph. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72

5.2 Block diagram of the closed-loop accelerometer with offset cancellation

loop. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73

5.3 A picture of the test setup. . . . . . . . . . . . . . . . . . . . . . . . . 74

5.4 Measured output spectrum (250000-point FFT after Hann windowing)

when the spring constant modulation is off. . . . . . . . . . . . . . . . 76

xiv


when the spring constant modulation is on. . . . . . . . . . . . . . . . 76


when the offset cancellation loop is on. . . . . . . . . . . . . . . . . . 77


on a shaker table with spring constant modulation on. . . . . . . . . . 78


on a shaker table with offset cancellation Loop on. . . . . . . . . . . . 79

5.9 Measured convergence of the offset cancellation loop. . . . . . . . . . 79

5.10 Measuring DC sensitivity. . . . . . . . . . . . . . . . . . . . . . . . . 81

5.11 Images of the bondwires before and after deformation. . . . . . . . . 82

5.12 Output spectrum before and after bondwire deformation with offset

cancellation loop off. . . . . . . . . . . . . . . . . . . . . . . . . . . . 83

5.13 Output spectrum before and after bondwire deformation with offset

cancellation loop on. . . . . . . . . . . . . . . . . . . . . . . . . . . . 83

5.14 Zoomed-in output spectrum before and after bondwire deformation. . 84

5.15 Measured convergence of ODAC code with square wave modulation in

presence of parasitic accelerations. . . . . . . . . . . . . . . . . . . . . 85


with spring constant modulation using a pseudo-random sequence. . . 86


using a pseudo-random sequence for modulation and offset cancellation

loop on. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86

5.18 Measured convergence of ODAC code with pseudo-random sequence

modulation in presence of parasitic accelerations. . . . . . . . . . . . 87

xv

Chapter 1

Introduction

1.1 Motivation

Accelerometers are used everywhere today. They are used in medical applications,

navigation systems, building and structural monitoring, vibration monitoring sys-

tems. In the automotive industry, accelerometers are used to activate various safety

systems, such as airbags, roll-over bars, and electronic stability programs (ESP). Ac-

celerometers also have a growing market in consumer electronics. In many of these

applications, a measurement of DC acceleration is required, and one major issue is

that the sensor offset appears as DC acceleration.

Where low offset is required these sensors are factory calibrated to null their in-

put offset. During the factory calibration the offset of the accelerometer is measured

under a “no acceleration” or “0 g” condition. This offset is then subtracted from

the output of the sensor. This process is also known as trimming (see Section 3.1.1).

However, the residual offset drifts over the lifetime of the devices due to thermal and

mechanical stress and humidity. As a consequence, the offset accuracy of high-end

MEMS accelerometers is specified on the order of 100 mg [1]. However, with continu-

ously increasing precision requirements, it is desirable to develop new techniques that

address post-calibration drift. In this work, we describe a technique that continuously

1

2 CHAPTER 1. INTRODUCTION

measures and minimizes any offset introduced at and beyond the interface between

the MEMS sensor element and the electronic readout.

1.2 Overview

The device considered in this work is a system-in-package type MEMS accelerometer,

which is commonly used in the automotive industry [2]. This accelerometer employs

a capacitive MEMS sensor element and a CMOS interface IC, which are connected

through bondwires as shown in Figure 1.1. The MEMS sensor element converts input

acceleration force to a differential capacitance (see Section 2.1). There is also an

electronic interface, which measures this differential capacitance (see Chapter 4).

Interface IC

Electronics

MEMS Sensor Element

mCSP

CSM

CBP

CBM

Figure 1.1: Sensor interface.

The offset in this accelerometer stems from mismatches in the components of the

readout electronics, the MEMS element, and the mismatch between parasitic capaci-

tances of the bondwires, COFF = CBP −CBM . In practice, the offset of the electronic

1.3. CHAPTER ORGANIZATION 3

circuits is mitigated using correlated double sampling (CDS). The MEMS offset is

calibrated during production test, and is known to drift mainly due to mechanical

stress. However, there are various stress-mitigating techniques to reduce this drift

to acceptable levels [3], [4]. On the other hand, the bondwire parasitics can drift

appreciably due to deformations from thermal and mechanical stress and absorption

of moisture by the package, which changes the permittivity of the dielectric between

the bondwires. For instance, a displacement of the wires by only 1 µm causes an

input offset of 14 mg in our accelerometer. Therefore, the main goal of this work was

to cancel the offset drift due to parasitic capacitance of the bondwires.

At first glance it may seem impossible to distinguish between sensor element capac-

itances and bondwire capacitances, as both of them affect capacitance measurements.

However, once we look more closely we observe that by applying an electrostatic force,

sensor capacitances will change but the bondwire capacitances will remain constant.

This is the main idea behind this work, and the main challenge is that we would like

to measure the bondwire capacitance without changing the sensor capacitance. This

is explained in detail in Section 3.2.

1.3 Chapter Organization

Chapter 2 gives an overview of the MEMS sensor element, and different sources of

offset in a system-in-package type MEMS accelerometer. Chapter 3 describes how

offset can be distinguished from the DC acceleration, and how it can subsequently

be cancelled. Chapter 4 goes into the details of the fabricated interface IC, finally

measurement results are presented in Chapter 5.

Chapter 2

Micromachined Capacitive

Accelerometers

Micromachining is a fabrication process used to build micro-electromechanical sys-

tems (MEMS) or micro-machinery. There are two types of micromachining: bulk

and surface. In bulk micromachining, a silicon substrate is etched deeply to produce

micro-structures inside the substrate. In surface micromachining, micro structures

are built on top of the substrate. These structures are built by deposition and re-

moval of a few very thin layers, 2-5 µm thick, which are sometimes referred to as thin

films. There are two types of deposition layers: sacrificial and structural. The sacri-

ficial layers, usually silicon dioxide, must be in place before the micro-structures are

build, and are later removed. The structural layers, generally polysilicon, define the

micro-structures and are deposited on top of the sacrificial layers. After deposition of

structural layers, the sacrificial layers are selectively etched away to release the micro

structures [5].

Accelerometers are often composed of two parts: a sensor element that converts

acceleration into a physical signal, and an interface IC that measures that physical

signal. Accelerometers are often categorized by the type of the signal they generate:

1. Capacitive: A proof mass is displaced in presence of acceleration, changing

4

5

the capacitance between the proof mass and another plate. This capacitance is then

measured by an electronic circuit [6]-[11]. This is the type of accelerometer that has

been used for this work. The details of operation of the MEMS sensor element are

discussed in the following sections, and the electronics needed to measure the sensor

capacitance are discussed in Chapter 4.

2. Piezoresistive: A proof mass is displaced in presence of acceleration, inducing

strain in the flexures holding the proof mass. This results in a change of the resistance

of the flexure which is measured by an electronic circuit [12], [13], [14].

3. Piezoelectric: A proof mass is displaced in presence of acceleration, inducing

strain in the flexures holding the proof mass. This results in accumulation of charge,

and development of an electric potential difference that is measured by an electronic

circuit [15], [16], [17].

4. Tunneling: A proof mass is displaced in presence of acceleration, changing

the gap between the proof mass and another metal electrode (tunneling tip). The

tunneling current through this gap has an exponential relationship with the gap size.

Often the gap-size is in the order of 10A and a feedback loop is used to maintain the

gap-size fixed. These accelerometer can have sub-µg resolution and bandwidths in

the kHz range [18], [19], [20].

5. Resonant: The resonance frequency of a resonator will change in presence

of acceleration through a number of different mechanisms. The most common one is

that in presence of acceleration a proof mass is displaced, inducing an axial force on

resonant beams. A tensile force increases resonance frequency, and a compressive force

reduces resonance frequency [21], [22]. Another mechanism is that the proof mass

changes the rigidity of the resonant beams, and therefore, their resonance frequency

[23].

The main advantages of resonant accelerometers are the high resolution, good

stability, and the quasi-digital output. The main disadvantage is the low bandwidth,

typically a few hertz.

6. Thermal: The distance between a heat source and a heat sink will change in

6 CHAPTER 2. MICROMACHINED CAPACITIVE ACCELEROMETERS

presence of acceleration. Because the temperature of the heat sink is a function of its

distance from the heat source, acceleration will change the temperature of the heat

sink [24], [25].

7. Optical: A proof mass is displaced in presence of acceleration, changing

the amount of light that enters an optic fiber or a wave guide. The position of the

proof mass is determined from the intensity of the output light that is measured by

a photodetector. Because this way of detecting the position of the proof mass is

electrically passive, sensing the position has minimal effect on the position [26], [27].

8. Electromagnetic: A planar coil is displaced in presence of acceleration,

changing its magnetic coupling to a secondary coil. The induced voltage at the

secondary coil is proportional to the current at the primary coil and the coupling

of two coils. Therefore, by exciting the primary coil with some AC current and

monitoring the secondary voltage the coupling between the two coils can be measured.

[28].

There are other types of accelerometers, which do not fall in any of the above

categories. For example, Liao uses two bondwires, and measures their mutual induc-

tance to determine how much they have moved in presence of acceleration [29]. An

overview of different types of accelerometers can be found in [30].

The accelerometer we consider in this work is a capacitive MEMS accelerometer

that is packaged with a CMOS interface IC. This system-in-package type MEMS

accelerometer, which is shown in Figure 1.1, is commonly used in automotive appli-

cations.

Section 2.1 presents an overview of the principles of operation of a capacitive

MEMS accelerometer, and Section 2.2 describes different sources of offset in a system-

in-package type MEMS accelerometer.

2.1. PRINCIPLES OF OPERATION 7

2.1 Principles of Operation

As mentioned earlier, a capacitive accelerometer is used in this work. In capacitive

accelerometers, there is a proof mass that moves in presence of acceleration. Once it

moves, the capacitance between it and some fixed plate will change. Figure 2.1 shows

a simple structure where the middle plate is loose and can move along the x-axis.

This middle plate is the proof mass and is indicated by m.

k/2

CSP CSM

m

k/2

0 d0−d0x

Figure 2.1: Differential capacitive sensor.

In presence of an input acceleration in the direction of x, the proof mass lags

behind and moves in the opposite direction. The goal is to measure a differential

capacitance, CSP − CSM . As indicated, the nominal gap size for the two capacitors,

CSP and CSM , is d0.

2.1.1 Spring Force

For the structure shown in Figure 2.1, the spring force that is applied to the proof

mass is proportional to the displacement of the proof mass:

F = −k · x (2.1)


where the minus sign indicates that the direction of the force is in the opposite

direction of the displacement. This means that the spring force is a restoring force;

i.e, it forces the proof mass to be at some equilibrium point, the center of the gap

in this case, and therefore, restores the system to equilibrium. This spring force is

plotted versus displacement in Figure 2.2.

Spring Force

Direction of the Force

Force

0

0

Displacement

d0−d0

Figure 2.2: Spring force.

Notice that there is an equilibrium point at the origin where force is zero and the

proof mass can rest. In Figure 2.2 the red arrows indicate the direction of the force;

i.e., on the negative x-axis where force is positive the arrow points to the right, and

on the positive x-axis where force is negative the arrow points to the left. This shows

that the equilibrium at the origin is a stable equilibrium point.

In general, a spring can be nonlinear, i.e. k can be a function of x, which makes


F a nonlinear function of x. The spring constant can be found as:

k = −∂F

∂x(2.2)

In case of a linear spring this equation simplifies to

k = −F

x(2.3)

A discussion on the design of the mechanical spring can be found in (See [31] Pages

7-11).

2.1.2 Force-to-Displacement Transfer Function

The equation governing the spring-mass system shown in Figure 2.1 is as follows:

FEXT = m · a+ b · v + k · x = m · ∂2x

∂t2+ b · ∂x

∂t+ k · x (2.4)

where m is the mass, b is the damping coefficient, k is the spring constant, x is the

displacement, v is the velocity, and a is the acceleration. The damping coefficient, b,

includes both structural damping and the damping from the viscous flow of gas in

the sensor structure. More discussions on the damping mechanisms can be found in

[32], [33] (pages 16-20), and [31] (pages 11-15). Taking the Laplace Transform results

in the following transfer function:

TF−x =x

F=

1

m · s2 + b · s + k(2.5)

This transfer function can be re-written as

TF−x =1m

s2 + ω0

Q· s+ ω2

0

(2.6)


where ω0 is the resonance frequency of the spring-mass system, and Q is the quality

factor:

ω0 =

√

k

m(2.7)

Q =ω0 ·m

b=

√k ·mb

(2.8)

2.1.3 Sensor Capacitance

As mentioned earlier, an external force will displace the proof mass, and this changes

the capacitances C1 and C2 (see Figure 2.1). The capacitors are parallel-plate capac-

itors, and the capacitance is given by:

C =ǫ · Ad

(2.9)

where ǫ is the permittivity of the dielectric material, A is the area of the parallel

plates, and d is the gap size or the distance between the plates. In this work, the

dielectric material is air. The capacitances, CSP and CSM , can be calculated as a

function of distance:

CSP =ǫ · Ad0 − x

(2.10)

CSM =ǫ · Ad0 + x

(2.11)

Notice that these are nonlinear functions of x. Often, we have a differential readout

and we are interested to measure the difference, CSP − CSM , which is still nonlinear

but does not have even harmonics:

CSEN = CSP − CSM = ǫ · A ·(

1

d0 − x− 1

d0 + x

)

= ǫ · A · 2 · xd20 − x2

(2.12)


For very small displacements, x2 << d20, CSEN can be approximated as a linear

function of displacement:

CSEN = ǫ · A · 2 · xd20

=∂CSEN

∂x

∣

∣

∣

∣

x=0

· x (2.13)

where ∂CSEN

∂x|x=0 is the derivative of CSEN around x = 0:

∂CSEN

∂x

∣

∣

∣

∣

x=0

=2 · ǫ ·A

d20(2.14)

2.1.4 Force-to-Capacitance Transfer Function

The most common way of measuring acceleration is to measure the sensor capaci-

tances. Therefore, we are often interested in the force-to-capacitance transfer func-

tion

TF−C =x

F· Cx

=α

m · s2 + b · s+ k(2.15)

where α is the displacement-to-capacitance gain. For small displacements (e.g., in a

closed-loop accelerometer), α is the derivative of the capacitance

α =∂CSEN

∂x

∣

∣

∣

∣

x=0

=2 · ǫ ·A

d20(2.16)

2.1.5 Electrostatic Force

As we apply a voltage, V , to a parallel-plate capacitor, there is an attractive electro-

static force between the plates:

FE =1

2· C · V 2

d(2.17)

where C is the capacitance, d is the gap between the plates. In micromachined

accelerometers, the small gap between the parallel-plate capacitors as well as the

small mass of the loose-plate make this force non-negligible. If the proof mass is in


the middle of the gap, a voltage of 1 V produces an electrostatic acceleration of more

than 50G.

The electrostatic force can also be written as a function of charge. Since the

charge, Q, on a capacitance is equal to C · V , we have

FE =1

2· C

2 · V 2

C · d =1

2· Q2

ǫ ·A (2.18)

Since ǫ and A are constant, equation (2.18) shows that the electrostatic force is only

a function of charge. This is an important result, and we will come back to this when

we discuss different interface architectures in Section 4.1.1.

The net electrostatic force applied to the proof mass is the difference between the

electrostatic forces on each parallel-plate capacitor

FE,NET =1

2· ǫ · A · V 2

1

(d0 − x)2− 1

2· ǫ ·A · V 2

2

(d0 + x)2(2.19)

It turns out, as we will see in Chapter 4, that we are primarily operating the sensor

element in two cases: 1. a voltage is applied to one capacitor while both terminals

of the other capacitor are shorted to ground (e.g., during force feedback phase). 2.

a voltage is applied to both capacitors (e.g., during capacitance measurement and

spring constant modulation). Therefore for the remainder of this section we will

consider these two cases.

In case 1, where the VD voltage is applied to CSM , and zero voltage is applied to

CSP , the electrostatic force is:

FE1 =1

2· ǫ · A · V 2

D

(d0 + x)2(2.20)

Notice that the positive sign of the electrostatic force suggests that the electrostatic

force will not oppose displacement of the middle plate. This is indeed the case. If the

middle plate moves to the right, the electrostatic force becomes stronger and pulls

the plate even further. If the middle plate moves to the left, the electrostatic force


becomes weaker, and the middle plate will move further to the left. This force is

plotted in Figure 2.3. Notice that this force is always positive, and therefore there is

Electrostatic Force


0

Force

0Displacement

d0−d0

Figure 2.3: Electrostatic force on one capacitor.

no equilibrium point.

In case 2, where the VC voltage is applied to both capacitors, the electrostatic

force is:

FE2 =1

2· ǫ · A · V 2

C ·(

1

(d0 − x)2− 1

(d0 + x)2

)

=2 · ǫ ·A · V 2

C · d0 · x(d0 − x)2 · (d0 + x)2

(2.21)

Again, the positive sign of the electrostatic force indicates that the electrostatic force

will not oppose displacement of the middle plate. If the middle plate moves to the


right, the electrostatic force on the right capacitor becomes stronger and the electro-

static force on the left capacitor becomes weaker. As a result, the middle plate moves

even further to the right. This force is plotted in Figure 2.4. Notice that the equilib-

Electrostatic Force


0

Force

0Displacement

d0−d0

Figure 2.4: Electrostatic force on two capacitors.

rium point at the origin is unstable. Therefore, unlike spring force, electrostatic force

is not a restoring force, and makes the system unstable.

2.1.6 Electrostatic Spring Constant

By considering the equations for the electrostatic force, equations (2.20) and (2.21), we

observe that the electrostatic force is a nonlinear function of displacement. Therefore,

the electrostatic force can be thought of as a nonlinear spring. Moreover, since it is

memoryless, i.e. it is not a function of derivatives of displacement (velocity and

acceleration), there is no effective mass or damping coefficient associated with the


electrostatic force. If we apply the equation (2.2) to electrostatic force, we obtain the

equivalent electrostatic spring constant.

In case 1 (the VD voltage is applied to CSM , and zero voltage is applied to CSP ),

the electrostatic spring constant is:

kE1 = −∂FE1

∂x= − ǫ ·A · V 2

D

(d0 − x)3(2.22)

In case 2 (the VC voltage is applied to both capacitors), the electrostatic spring

constant is:

kE2 = −∂FE2

∂x= −ǫ · A · V 2

C ·(

1

(d0 − x)3+

1

(d0 + x)3

)

= −ǫ · A · V 2C · 2 · d0 · (d20 + 3 · x2)

(d0 − x)3 · (d0 + x)3

(2.23)

In both cases the electrostatic spring constant is negative. Therefore, electrostatic

force can be seen as a negative nonlinear spring, and the presence of electrostatic force

in a spring mass system can be seen as the reduction of the equivalent spring constant

of the system. This is known as the spring-softening effect, and is in agreement with

the observation that spring force is restoring and the electrostatic force is not. If

VD = VC , equation (2.23) can also be written in the form

kE2 = −ǫ ·A · V 2D ·

(

1

(d0 − x)3+

1

(d0 + x)3

)

= kE1 −ǫ · A · V 2

D

(d0 + x)3

(2.24)

which shows that kE2 is equal to equation kE1 minus another term, making it more

negative. Therefore, when a voltage is applied to both capacitors, the electrostatic

spring constant at any point is larger in magnitude (i.e., more negative) than when a

voltage is applied to one of the capacitors.

Changing the spring constant by applying an electrostatic force is utilized in this


work and is discussed in detail in Chapter 3.

2.1.7 Snap-In Voltage

As mentioned in Section 2.1.5, the electrostatic force drives the system toward insta-

bility. It is interesting to know what the maximum voltage is that can be applied

to the system without making it unstable. This is called the snap-in voltage. If a

slightly larger voltage is applied, the proof mass will snap into one of the fixed plates.

At voltages higher than the snap-in voltage there is no stable equilibrium point in the

system. This snap-in voltage puts a limit on the voltage one can apply to the sensor.

Subsequently, this limits the amplitude of the signal that can be read out, and the

amplitude of the force feedback that can be applied to the sensor element.

First we consider case 1, where the VD voltage is applied to CSM , and zero voltage

is applied to CSP . Figure 2.5 shows the sum of spring and electrostatic forces. Notice

that there are two equilibrium points, where the net force is zero and the middle plate

can rest. The first equilibrium point, at x1, is stable while the second equilibrium

point, at x2, is unstable. As the voltage is increased, x1 and x2 move toward one

another. At the snap-in voltage, x1 and x2 coincide, and for voltages beyond that,

there is no equilibrium point.

In order to find the snap-in voltage, we first find the voltage required to have

equilibrium at some displacement, x, by equating the sum of the spring force (equation

(2.1)) and electrostatic force (equation (2.20)) to zero and by solving for the voltage.

We call this VEQ1, which determines the voltage that produces zero electrostatic force

as a function of x:

VEQ1 =

√

2 · k · x · (d0 − x)2

ǫ · A (2.25)

By equating the derivative of equation (2.25) to zero, we can find the maximum


Electrostatic + Spring Force


x1 x2

Force

0

0

Displacement

d0−d0

Figure 2.5: Sum of spring and electrostatic Forces on one capacitor.

voltage for which an equilibrium point exists. This is the snap-in voltage:

VS1 =

√

8

27· k · d30ǫ ·A (2.26)

It should be noted that at this voltage the equilibrium point is at d0/3. Also notice

that when the applied voltage is the snap-in voltage, the electrostatic spring constant

can be obtained by substituting equation (2.26) into equation (2.22)

kE1 (x) |VD=VS1= − 8

27· k · d30

(d0 − x)3(2.27)

At the equilibrium point, x = d0/3 and this becomes:

kE1 (x = d0/3) |VD=VS1= −k (2.28)


This means that when the snap-in voltage is applied, the overall equivalent spring

constant at the equilibrium point is zero, and therefore, the resonance frequency also

drops to zero at the equilibrium point.

Now we consider case 2, where the VC voltage is applied to both capacitors. Figure

2.6 shows the sum of spring and electrostatic forces.

Electrostatic + Spring Force


Force

x1−x1 0

0

Displacementd0−d0

Figure 2.6: Sum of spring and electrostatic forces on two capacitors.

We observe that there are three equilibrium points: the first equilibrium point, at

−x1, is unstable; the second equilibrium point, at the origin, is stable; and the third

equilibrium point, at x1, is also unstable. As the voltage is increased, −x1 and x1

move toward the origin. At the snap-in voltage, the three equilibrium points coincide

at the origin, and for voltages beyond that there is one unstable equilibrium point at

the origin.

Again, in order to find the snap-in voltage, we first find the voltage required

to have equilibrium at some displacement, x, by equating the sum of spring force


(equation (2.1)) and electrostatic force (equation (2.21)) to zero and by solving for

the voltage:

VEQ2 =

√

k

2 · ǫ · A · d0· (d0 − x) · (d0 + x) (2.29)

By equating the derivative of equation (2.29) to zero we can find the snap-in voltage:

VS2 =

√

1

2· k · d30ǫ · A (2.30)

At this voltage the equilibrium point is at the origin, x = 0. When the applied

voltage is the snap-in voltage, the electrostatic spring constant can be obtained by

substituting equation (2.30) into equation (2.23)

kE2 (x) |VC=VS2= − k · d40 · (d20 + 3 · x2)

(d0 − x)3 · (d0 + x)3(2.31)

At the equilibrium point, x = 0, this becomes

kE2 (x = 0) |VC=VS2= −k (2.32)

Again, this means that when the snap-in voltage is applied, the overall equivalent

spring constant at the equilibrium point is zero, and therefore, the resonance frequency

also drops to zero at the equilibrium point.

It should be pointed out that VS2 > VS1, i.e. if a voltage is applied to both

capacitors the snap-in voltage is larger than when a voltage is applied to only one of

the capacitors. This might be counter-intuitive as kE2 < kE1 < 0, i.e. the electrostatic

spring constant at any point is larger when we apply a voltage to both capacitors (see

equation (2.24)). The point is that when a voltage is applied to both capacitors

the equilibrium point is at the origin, while when a voltage is applied to one of the

capacitors the equilibrium point is at d0/3. In other words, even though for any fixed


x, kE2 < kE1 < 0, the relevant comparison here is between kE1|x=d0/3 and kE2|x=0.

−27

8· ǫ · A · V 2

D

d30= kE1|x=d0/3 < kE2|x=0 = −2 · ǫ · A · V 2

D

d30< 0 (2.33)

Therefore, because kE1|x=d0/3 < kE2|x=0 < 0, a larger voltage is required when a

voltage is applied to both capacitors before the electrostatic spring constant cancels

the mechanical spring constant of the sensor element.

2.2 Offset in Accelerometers

In general, offset in a system refers to the output when no input is present. In an

ideal case, we expect a zero offset. Figure 2.7 shows an operational amplifier that

suffers from offset. When the inputs of an amplifier are shorted, i.e. zero input, it will

have an output, which is known as offset. One common source of offset in electronics

circuits is the component mismatch. Also shown in Figure 2.7 is that this offset is

usually modeled as some source at the input of an ideal device.

Ideal VOUT

VOFF

Figure 2.7: Amplifier with offset.

In accelerometers, the offset error is the non-zero output that is measured when

there is no input acceleration. In this section we describe different sources of offset

in the system shown in Figure 1.1.

2.2. OFFSET IN ACCELEROMETERS 21

2.2.1 MEMS Fabrication

The MEMS sensor element used in this work, outputs a differential capacitance. Any

asymmetry in the capacitances or routing produces some offset. For example, after

fabrication the proof mass may not rest at the center of the gap, which means that

the differential capacitance will not be zero in the absence of acceleration. Jiangfeng

Wu reports that the gap sizes could have a mismatch on the order of 0.1 µm for a gap

size of around 1.5 µm [34]. This offset can only be measured when the device is put in

a zero-acceleration environment, and is usually measured during the post-fabrication

calibration. This offset can also drift over the temperature and the lifetime of the

device. The drift over temperature is also calibrated during the factory test. The

drift over the lifetime of the device is mostly due to mechanical stress, and there are

some stress-mitigating techniques to reduce this drift [3], [4].

Another source of offset is mismatch between routing paths. The differential

capacitors are electrically connected to bonding pads on the MEMS chip, and the

routing has a parasitic resistance and capacitance. Any mismatch between these

parasitics translates into some offset at the output of the interface. Unlike the offset

due to misalignment of the proof mass, the offset from mismatch between routing

signals can be measured in presence of input acceleration. This offset is measured by

the technique proposed in this work.

2.2.2 MEMS Charge Storage

Electric charge is trapped in SiO2 or a nitride layer that may be used for the fab-

rication of the MEMS sensor element. While the charge traps are well studied for

RF MEMs switches, they will also affect non-contacting devices such as resonators,

accelerometers and gyroscopes [35].

In many devices an oxide or a nitride layer is engineered for a purpose. For exam-

ple, silicon resonators exhibit a negative linear temperature coefficient of frequency

[36], and the native oxide is used for passive compensation of temperature coefficient,


which is much cheaper than active compensation [35]. Renata Melamud studies a

flexural-mode beam made up of several layers and shows that the young’s modulus of

each layer affects the temperature coefficient of the resonance frequency of the beam.

Then, she engineers an SiO2 layer, whose Young’s modulus has positive temperature

dependance, to compensate the negative temperature coefficient of frequency of the

silicon resonators [36]. It turns out that dry-oxidization (oxygen ambient) will result

in less fixed charge in the dielectric than wet-oxidization (steam grown) [35]. Some

wet-oxidized devices also show mobile charge in the dielectric [35].

The effect of the trapped charge is modeled as some built-in voltage [35], [37].

These charges affect the electrostatic force, and the electrostatic spring constant

(equations (13), (14) in [35]). Because the amount of trapped charges can change

over the lifetime of the device, this offset drifts over time. However, in a force-

balanced system, the trapped charges introduce some offset, which is detected and

cancelled by the method proposed in this work.

2.2.3 Bondwires

As discussed in Chapter 1, the sensor element and the interface IC are connected

with bondwires for many automotive applications[2] (see Figure 1.1). As shown in

Figure 1.1, the parasitic capacitance of the bondwires, CBP and CBM , are in parallel

with the sensor capacitances, CSP and CSM , respectively. Therefore, the parasitic

capacitances of the bond-wires will be part of the capacitance that the electronic

interface measures. Any mismatch between parasitic capacitances of the bondwires,

COFF = CBP − CBM , appears as some offset. Moreover, this offset can drift through

different mechanisms:

1. Mechanical Stress: the package that contains the system shown in Figure

1.1 is subject to mechanical stress, and as a result the package is bent slightly over

the lifetime of the device. To see how problematic this is we consider five bondwires

as shown in Figure 2.8. With the assumption that the bondwires are straight and


placed in a gel (with ǫr = 4) to fix their positions, we calculate the capacitances for

the dimensions given on the figure.

2r = 32 µm

l = 1.7 mm

d = 240 µm

x

CSM CSP

Figure 2.8: Bondwires used in a sensor interface.

Next we move the middle bondwire along x and plot the capacitance change, and

also the corresponding acceleration error in Figure 2.9. We observe that if the middle

bondwire moves by only 1µm the induced offset is around 14 mg.

2. Humidity: what is shown in Figure 1.1 is put in a molded package, and the

package absorbs moisture over the lifetime of the device. The moisture changes the

dielectric permittivity of the package, which changes the bondwire capacitances. The

dielectric permittivity of the package is around 4, while the dielectric permittivity of

water is around 80.

3. Temperature: bondwires will expand as temperature increases, which changes

their capacitance.


Distance the middle wire has moved, x [µm]

Accelerationerror[m

g]

Difference

inCap

acitan

ce[fF]

0.1

0.10.1

1

11

1

10

10

10

10 100

100

100

1000

Figure 2.9: Capacitance change, and the corresponding acceleration offset, that iscaused by the movement of a bondwire.

At first glance it may seem impossible to distinguish between sensor-element ca-

pacitances and bond-wire capacitances, as both of them affect capacitance measure-

ments. However, once we look more closely we observe that by applying an electro-

static force, the sensor capacitances will change, but the bondwire capacitances will

remain constant. In Section 3.2, we will explore how we can use this fact to mea-

sure bond-wire capacitance. By using this method, the offset, and its drift, due to

mismatch between parasitic capacitances of the bondwires is detected and cancelled.

2.2.4 Electronics

Electronic circuits have some offset. Even though all circuits are differential (i.e. no

systematic offset), any mismatch between differential devices will result in some offset.

There is also flicker noise at low frequencies. Both offset and the flicker noise of the

electronic circuits are canceled using correlated-double-sampling (CDS) as described

in Chapter 4. The offset from the electronics blocks are detected by the technique


proposed in this work, however, CDS is implemented mainly to reduce the flicker

noise.

Chapter 3

Offset Detection and Cancellation

Offset has been a problem in electronic circuits for a long time. In this chapter,

we will go over various offset cancellation techniques that are used in electronic cir-

cuits. Section 3.1 is focused on existing techniques, while Section 3.2 describes the

new method that is used in this work to detect and cancel the bondwire offset in

accelerometers.

3.1 Prior Work

In this section we briefly review the known techniques for canceling offset in the elec-

tronic circuits and sensor interfaces. These techniques fall into two categories: static

and dynamic offset cancellation. In static offset cancellation, the offset is measured

once, and is subtracted from the output, while in dynamic offset cancellation the

offset is continuously or periodically being measured.

3.1.1 Trimming

Trimming is the simplest form of offset cancellation. In this method, the offset of

a system is measured and stored once (e.g., after fabrication of the device) and is

cancelled after that. This is a static offset cancellation method, and as such it does

26

3.1. PRIOR WORK 27

not address offset drift. That is, because the offset is measured once for trimming,

any change in offset after trimming is not cancelled, and is seen as offset drift. Offset

can drift with temperature or over time, for example due to the aging of the device.

Another problem of trimming is that every device needs to be trimmed, and this can

be a very costly process. To mitigate some of these problems, trimming is sometimes

not as simple as described here. The offset may be measured at a few selected

temperatures to reduce temperature drift. Moreover, if the offset of the chips that

are fabricated on the same wafer are correlated, only a few chips are measured to

reduce the cost of measurements in trimming.

3.1.2 Auto-zeroing

Auto-zeroing is a dynamic offset cancellation technique that is very similar to trim-

ming. In auto-zeroing, the system continuously goes through cycles of “offset mea-

surement” and “signal propagation”. In other words, in each cycle, first the input

is disconnected and the offset is measured, and then the input is connected and a

valid output is generated with the offset removed. Figure 3.1 shows an auto-zeroed

amplifier. In this circuit, the offset and noise of the amplifier are measured during

Φ1 and stored on the CAZ capacitor. During Φ2 the offset is removed and the signal

is amplified. Because these cycles are repeated continuously, the system is not very

sensitive to offset drifts. Moreover, the low frequency noise of the amplifier, such

as flicker noise, is also reduced. However, in this approach measuring offset requires

nulling the input, which is not possible in an accelerometer.

In discrete-time systems, such as switched-capacitor circuits, this technique is

also called Correlated Double Sampling (CDS) [38], [39]. As we will see in Chapter

4, CDS is used in this work to cancel the offset and flicker noise of Operational

Transconductance Amplifiers (OTAs).

In continuous-time circuits, a variant of the system discussed above should be

used, as the output of the amplifier, shown in Figure 3.1, is only valid during φ2.

28 CHAPTER 3. OFFSET DETECTION AND CANCELLATION

+−

Φ1

Φ1

Φ2

VIN

CAZ

VOUT

VOFF

Figure 3.1: Auto-zeroed amplifier.

One example is what is referred to as a Ping-Pong amplifier ([40], [41]), where there

are two replicas of the circuit shown in Figure 3.1 working in parallel but in opposite

phases, such that during each phase there is an amplifier with a valid output.

3.1.3 Chopping

While trimming and auto-zeroing rely on measuring the offset and subtracting it,

chopping modulates the input signal to some frequency before the offset is introduced,

so the DC offset does not mix with the signal. Figure 3.2 shows the principle of

chopping. First, the input signal is upconverted before the offset is introduced. The

DC offset is introduced in the amplifier, and the signal after the amplifier has some

DC offset. After the amplifier, the signal is demodulated, and as a result, the offset

is now upconverted to some out-of-band frequency and can be low-pass-filtered. In

an accelerometer interface, chopping requires inverting the direction of the input

acceleration, which is impractical.

3.1. PRIOR WORK 29

LPFVIN

VCH

VOUTVOFF

Figure 3.2: Chopped amplifier.

3.1.4 Open-Loop Sensor Modulation

Although conventional chopping is not feasible for accelerometers, there exists an

alternative approach that relies on a sensitivity modulation of the sensor element.

This idea is very similar to chopping, and can be applied to sensor applications

where the input signal is not an electrical signal, which can be chopped easily. By

modulating the sensitivity of a sensor element, we can chop the signal coming out of

the sensor element. By doing so the output of the sensor will not be affected by any

offset that is added after the sensor element. Figure 3.3 shows how this idea works.

In [42] this idea was applied to a magnetic sensor.

LPFCSEN,MOD

COFF

Sensor

VCH

VOUT

Figure 3.3: Chopped amplifier by modulating sensor sensitivity.

In order to see how we can implement the system shown in Figure 3.3, we consider


the force-to-capacitance transfer function of a sensor element (equation (2.15)) which

is rewritten here

HSEN =α

m · s2 + b · s+ k(3.1)

At low frequencies, where the signal band of interest is located, this transfer function

is HSEN = αk. While α is fixed by the MEMS design, we can consider modulat-

ing the spring constant to mimic the setup of Figure 3.3. However, since HSEN is

inversely proportional to k, this results in a nonlinear modulation of the input accel-

eration. Fortunately, this problem can be overcome in the closed-loop read-out that

was employed in this work. Section 3.2 explains this idea.

3.2 Closed-Loop Sensor Modulation

To investigate the closed-loop modulation of sensor parameters further, consider the

generic closed-loop interface shown in Figure 3.4. The input acceleration (FIN ) causes

displacement of a proof mass, which is read out by measuring the differential sense

capacitance, CSEN = CSP − CSN . Bondwire capacitance offset, COFF = CBP −CBN , adds to the sensor capacitance, and the electronics measures the sum of them,

CSEN + COFF , and converts it to a voltage. This output voltage is then fed back to

the sensor to generate some electrostatic-force-feedback (for more information about

a closed-loop architecture see Chapter 4). The feedback block models the voltage to

electrostatic-force transduction. This system has two inputs: input acceleration force,

FIN , and bondwire capacitance offset, COFF . Therefore, the output of the system is

a function of both of these inputs. However, we are interested to measure only FIN ,

and COFF is unwanted. At low frequencies, the system behaves like a linear system,

and we can use superposition to find the output of the system.

SOUT = FIN · HSEN ·HELEC

1 + FB ·HSEN ·HELEC+ COFF · HELEC

1 + FB ·HSEN ·HELEC(3.2)

3.2. CLOSED-LOOP SENSOR MODULATION 31

Sensor Element Electronics

Feedback

FINHSEN HELEC

COFF

CSEN SOUT

FB

Figure 3.4: Block diagram of the closed-loop sensor.

Moreover, at low frequencies the loop-gain is very large and the above equation sim-

plifies to:

SOUT = FIN · 1

FB+ COFF · 1

FB ·HSEN(3.3)

As mentioned earlier, the output component due to COFF is unwanted and the goal is

to minimize it. Equation (3.3) shows that the output component due to COFF depends

on sensor parameters, HSEN , but the output component due to input acceleration

FIN does not depend on HSEN . This is what we use to differentiate between FIN

and COFF . At low frequencies of interest the sensor transfer function is inversely

proportional to spring constant, k

HSEN ≃ α

k(3.4)

where α is the displacement-to-capacitance gain (see Section 2.1.4), and by substi-

tuting equation (3.4) into (3.3) we obtain

SOUT = FIN · 1

FB+ COFF · k

FB · α (3.5)


Note that the output component due to bondwire capacitance offset is proportional

to the spring constant, but the output component due to input acceleration is in-

dependent of the spring constant. Next, consider a system, whose spring constant

is somehow modulated. Such a system would have a sensor transfer function in the

form of

HSEN,MOD ≃ α

k + kM (t)(3.6)

where k is constant and kM (t) is time-varying. When this sensor is placed in the

closed-loop system the block diagram shown in Figure 3.5 is obtained.


Feedback

FIN

HSEN,MOD HELEC

COFF

CSEN SOUT,MOD

FB

Figure 3.5: Block diagram of the closed-loop sensor with modulation.

The output in Figure 3.5 is given by

SOUT,MOD = FIN · 1

FB+ COFF · k + kM (t)

FB · α= FIN · 1

FB+ COFF · k

FB · α + COFF · kM (t)

FB · α

(3.7)

where the first two terms are near DC (and also appeared in equation (3.5)), and the

third term arises when the spring constant is modulated, and is at the modulating

frequency. We observe that the modulated term does not contain the input signal,

and can be used to measure and subsequently suppress COFF . Section 3.3 explains

how the spring constant can be modulated, and Section 3.4 explains how COFF can

be measured, and nulled.

3.3. SPRING CONSTANT MODULATION 33

3.3 Spring Constant Modulation

This section describes how we can modulate the spring constant of the sensor. As

discussed in section 2.1.6, electrostatic force changes the effective spring constant of

the system, an effect known as the spring-softening property of electrostatic force.

In this work we use the spring-softening effect to modulate the spring constant of

the sensor. It should be noted that the spring-softening effect has been used for

other purposes prior to this work. Because the resonance frequency of the system

depends on the spring constant (see equation (2.7)), the apparent change in the

spring constant results in a change in the resonance frequency of the sensor. This is

used by Chinwuba Ezekwe for matching the resonance frequencies on the drive and

sense loops of a gyroscope. This increases the readout signal, and results in a higher

signal-to-noise ratio for a given power consumption [43].

Figure 3.6 shows two equivalent systems. Notice that the outputs of these systems

are the displacements, x1 and x2, and in order to get the output capacitance we must

multiply the transfer functions with the displacement-to-capacitance gain, α. Figure

Sensor Element

FIN x1HSEN,MOD =1

k + kM (t)

(a) Modulated Spring Constant

Sensor ElementFIN

FM (t)

x2

HSEN =1

k

(b) Modulating Spring Constant by ApplyingForce

Figure 3.6: Sensor element with modulated spring constant.

3.6(a) shows a sensor element whose spring constant is the sum of two terms: k is

a constant term, and kM (t) is a time varying term. Figure 3.6(a) does not show

how this system can be made. Figure 3.6(b) illustrates how the system in Figure

3.6(a) can be implemented by applying an electrostatic force. Since we would like to

modulate the spring constant, the modulating electrostatic force that is used should


be like a spring force; i.e. proportional to displacement (equation (2.1)). In order to

derive this, we write the equations governing these systems. For the system in Figure

3.6(a) we have

FIN = m · ∂2x1

∂t2+ b · ∂x1

∂t+ (k + kM (t)) · x1 (3.8)

and for the system in Figure 3.6(b) we have

FIN + FM (t) = m · ∂2x2

∂t2+ b · ∂x2

∂t+ k · x2 (3.9)

Assuming that the input (FIN) and system parameters (m, b, and k) are the same

for both systems, in order for x1 = x2 = x we must have

FM (t) = −kM (t) · x (3.10)

Next, we obtain an electrostatic force that is proportional to the displacement of the

proof mass. Considering the differential parallel-plate capacitors of a sensor (Figure

2.1), the net electrostatic force is given by equation (2.19) and is rewritten here:

FE,NET =1

2· ǫ · A · V 2

1

(d0 − x)2− 1

2· ǫ ·A · V 2

2

(d0 + x)2(3.11)

The voltages on the two caps can be written in terms of common-mode and differential-

mode voltages:

V1 = Vcm +Vd

2(3.12)

V2 = Vcm − Vd

2(3.13)

3.3. SPRING CONSTANT MODULATION 35

By substituting equations (3.12) and (3.13) into equation (3.11) we obtain

FE,NET =1

2· ǫ · A

(

V 2cm +

V 2d

4

)

· 4 · d0 · x+ Vcm · Vd · 2 · (d20 − x2)

(d20 − x2)2

(3.14)

=2 · ǫ · A ·

(

V 2cm +

V 2d

4

)

d20·

xd0

(

1− x2

d20

)2+

ǫ · A · Vcm · Vd

d20·

1 + x2

d20

(

1− x2

d20

)2(3.15)

In a force-balanced system where the proof mass is rebalanced to the center of the

gap, we can assume that x << d0. This assumption simplifies equation (3.15) to

FE,NET

∣

∣

∣

∣

x<<d0

=2 · C0 ·

(

V 2cm +

V 2d

4

)

d0· x

d0+

C0 · Vcm · Vd

d0(3.16)

where the first term is proportional to displacement, and the second term is a constant

electrostatic force. Since we only want to modulate the spring constant we would like

the second term to be zero. By choosing Vd = 0, i.e. by applying only a common-mode

voltage, Vcm, the second term will be zero, and equation 3.16 simplifies to

FE,NET

∣

∣

∣

∣

x<<d0,Vd=0

=2 · C0 · V 2

cm

d20· x (3.17)

In equation (3.17), the electrostatic force is proportional to displacement, and there-

fore, the electrostatic force obtained from applying a common-mode voltage to the

sensor can be used as the modulating force shown in Figure 3.6(b). By applying a

time-varying common-mode voltage, Vcm = VM (t), the spring constant is modulated,

and we have

FM (t) =2 · C0 · VM (t)2

d20· x (3.18)

kM (t) = −2 · C0 · VM (t)2

d20(3.19)

With this spring constant modulation, we obtain the system shown in Figure 4.4,


whose output is given by equation (3.7). The sensor element spring constant is

modulated by applying a modulating force, FM (t), to the sensor element.


Feedback

FIN α

k + kM (t)HELEC

COFF

CSEN SOUT,MOD

FB

VM (t)

Figure 3.7: Block diagram of the closed-loop sensor with spring constant modulation.

3.4 Offset Cancellation Loop

The next step is to extract COFF from the output of this system. To do this we

correlate the output of the system, SOUT , with the modulation signal, VM (t). Based

on this correlation, a corrective signal is fed back in the electronics to cancel the

bondwire offset. This idea is shown in Figure 3.8.

In order to see how the offset cancellation loop works, we overview the cross-

correlation in Section 3.4.2, and then in Section 3.4.2 we discuss what kind of modu-

lation signal we need.

3.4.1 Cross-Correlation

The correlation of two signals is a measure of how similar they are. For continuous

signals, f(t) and g(t), cross-correlation is defined as

(f ⋆ g) (t) =

∫ ∞

−∞

f ∗(τ) · g (t + τ) dτ (3.20)

3.4. OFFSET CANCELLATION LOOP 37


Feedback

FIN

HSEN,MODHELEC

COFF

CSEN SOUT

FB

CorrelatorVM (t)

Figure 3.8: Block diagram of the closed-loop sensor with offset cancellation loop.

where f ∗ is the complex conjugate of f . Similarly, for discrete signals, p[n] and q[n],

cross-correlation is defined as

(p ⋆ q) [n] =

∞∑

m=−∞

p∗[m] · q [n+m] (3.21)

Equation (3.21) shows that the cross-correlation can be implemented with a multi-

plication and a summation. To gain some insight about cross-correlation we consider

two special cases:


Two Sinusoidal Signals with the Same Frequency

First we find the correlation of the a1 sin [2πf1∆tm] signal with a2 sin [2πf1∆tm]

(a1 sin [2πf1∆tm] ⋆ a2 sin [2πf1∆tm]) [0]

= a1a2 ·∑

sin [2πf1∆tm] · sin [2πf1∆tm]

=a1a22

∑

(cos [4πf1∆tm] + 1)

≈ a1a22∆t

∫

(cos [4πf1t] + 1) · dt

≈ a1a2fS2

· sin [4πf1t]4πf1

+a1a2fS

2· t

(3.22)

The important points here are

1. The amplitude of the correlation is proportional to the amplitudes of both

signals, a1 and a2.

2. The first term is bounded, while the second term is unbounded.

Two Sinusoidal Signals with Different Frequencies

Next we find the correlation of the a1 sin [2πf1∆tm] signal with a2 sin [2πf2∆tm]

(a1 sin [2πf1∆tm] ⋆ a2 sin [2πf2∆tm]) [0]

= a1a2 ·∑

sin [2πf1∆tm] · sin [2πf1∆tm]

=a1a22

∑

(cos [4π (f1 − f2)∆tm]− cos [4π (f1 + f2)∆tm])

≈ a1a22∆t

∫

(cos [4π (f1 − f2) t]− cos [4π (f1 + f2) t]) · dt

≈ a1a2fS2

· sin [4π (f1 − f2) t]

4π (f1 − f2)+

a1a2fS2

· sin [4π (f1 + f2) t]

4π (f1 + f2)

(3.23)

The important points here are

1. The amplitude of the correlation is proportional to the amplitudes of both

signals, a1 and a2.


2. There is one term at the difference of frequencies (beat frequency), and another

tone at the sum of the frequencies.

3. Both terms are bounded. However, their amplitudes are inversely proportional

to their frequencies. As a result, if f1 is very close to f2, then f1 − f2 is very

small, and the amplitude of the correlation term at the frequency f1 − f2 is

large.

3.4.2 Modulation Signal

The cross-correlation between the output of the system (equation (3.7)) and the

modulation signal (VM (t)) is given as follows

(SOUT,MOD ⋆ VM (t)) [n] =

((

FIN · 1

FB+ COFF · k

FB · α

)

⋆ VM (t)

)

[n]

+

((

COFF · kM (t)

FB · α

)

⋆ VM (t)

)

[n]

(3.24)

The first term carries information about both the input signal, FIN , and offset, COFF ,

while the second term carries information only about the offset. As we want to

measure only COFF , we want the second term to be much larger than the first term.

Therefore, we want the following:

1. VM (t) should have no overlap in the frequency domain with k or FIN , so that

the first term in equation (3.24) stays bounded. This implies that VM (t) should

have a zero average, because both k and FIN are near DC.

2. The frequency contents of VM (t) and kM (t) should be similar, so that the sec-

ond term in equation (3.24) grows indefinitely. Since kM (t) is proportional to

the square of VM (t), and we want them to have the same frequency content,

we choose VM (t) to be a binary sequence. As a result, kM (t) becomes a similar

binary sequence with a different amplitude. Therefore, their frequency con-

tents will be similar. Binary sequences have the added advantage that they


are very easy to generate with a counter or a Linear-Feedback Shift-Register

(LFSR). In addition, SOUT is a bit-stream, i.e. a binary sequence, and finding

the correlation of two binary sequences is also very convenient.

Moreover, we choose VM (t) such that it has no frequency components in the

signal band of interest. Although accelerometers are used to measure acceleration

(FIN) within a certain bandwidth (e.g., 0-60 Hz in the automotive industry), parasitic

accelerations outside the signal band of interest are typically also present (e.g., due to

vibrations in a car). If such parasitic accelerations appear close to the spring constant

modulation tones (VM (t)), the output of the correlator will be in error. Specifically,

it will contain a component at the beat frequency, i.e. the difference between the

frequency of the modulating signal and the parasitic acceleration. The amplitude of

this beat frequency depends on the amplitudes of both the modulating signal and

the parasitic acceleration. While the amplitude of the parasitic acceleration is fixed,

we can choose a modulating signal that is composed of more tones with smaller

amplitudes. As a result, the information about offset does not rely heavily on any

particular frequency, and is spread over more harmonics.

Assume that VM (t) is periodic with the fMOD frequency, then its fourier series

representation is

VM (t) =

∞∑

m=1

cm · sin (2πmfMODt) (3.25)

For example, if VM (t) is a pulse, then

cm =4m

π(3.26)

and if VM (t) is a pseudo-random sequence with length L and fundamental frequency

fMOD, then

cm =

√L+ 1

L· sinc

(m

L

)

(3.27)

Note that for a pseudo-random sequence, L can be made large such that cm is small.

This will make the modulation scheme less sensitive to parasitic accelerations that


are present.

3.4.3 Summary

The mismatch between parasitic capacitances of the bondwires is a source of offset,

and affects DC measurement of the input acceleration. However, in a closed-loop

interface, modulating the spring constant of the sensor element produces a tone whose

amplitude is proportional to the offset. This tone is used to measure offset. Moreover,

an offset cancellationloop is implemented, which extracts the amplitude of the tone

and adds a corrective signal to cancel the offset.

Electrostatic force is used for modulating the spring constant. Detailed analysis

shows that in a closed-loop system, applying a common-mode voltage to the sensor

capacitors produces the modulating force. Because the force is proportional to the

square of voltage, a binary sequence is used for modulation, so that the frequency

content of the modulating force is the same as the frequency content of the applied

voltage. This is crucial because the offset cancellationloop finds the correlation be-

tween the output, which is proportional to the modulating signal, and the modulating

voltage.

Finally, it is shown that in order to reduce the sensitivity of this modulation

scheme to parasitic accelerations, a pseudo-random sequence can be used. This will

spread the information about offset over more frequency bins, which makes the system

less sensitive to any particular frequency.

Chapter 4

Interface Architecture and Design

While Chapter 2 focused on a capacitive MEMS accelerometer, this chapter is focused

on the electronics that reads the output capacitance of the MEMS sensor. Section

4.1 goes over challenges that are common in accelerometer interface circuits. Section

4.2 goes into details of what is implemented, and Section 4.3 is focused on particular

design considerations for this implementation.

4.1 Accelerometer Interface Challenges

In this section we go over a few known challenges that arise when designing an inter-

face for an accelerometer.

4.1.1 Capacitance Measurement

An mentioned in Section 2.1, for this work we are using a capacitive accelerometer,

which means we need to measure the sensor capacitances in order to measure accel-

eration. As shown in Figure 4.1, there are two ways of measuring capacitance: 1.

applying a voltage to the capacitor and measuring the current, 2. injecting a current

and measuring the voltage. In both cases, there will be some voltage across the ca-

pacitance, which generates some attractive electrostatic force between the plates of

42

4.1. ACCELEROMETER INTERFACE CHALLENGES 43

the capacitors. As discussed in Section 2.1.5, this electrostatic force is not negligible,

and will move the capacitor plates and change the capacitance of the sensor.

+−C

i(ω)

v(ω)

v(ω)

i(ω)= XC =

1

ω · C(a) Measuring capacitance byapplying a voltage.

+

_

C i(ω)v(ω)

v(ω)

i(ω)= XC =

1

ω · C(b) Measuring capacitance byinjecting a current.

Figure 4.1: Capacitance measurement.

As shown in section 2.1, when applying voltages to the two differential capacitors

of a sensor, the net electrostatic force is given by equation (2.19), which is re-written

here:

FE,NET =1

2· ǫ · A · V 2

1

(d0 − x)2− 1

2· ǫ · A · V 2

2

(d0 + x)2(4.1)

There are two common approaches to measuring sensor capacitances, which are in-

sensitive to electrostatic forces that are generated during capacitance measurements:

1. Force-Balanced System: In this system, the proof mass is kept at the

center of the gap, i.e. x = 0, and the same voltage is applied to the two differential

capacitors, i.e. V1 = V2. Under these assumptions the net electrostatic force becomes

FE,NET =1

2· ǫ · A · V 2

1

(d0 − 0)2− 1

2· ǫ · A · V 2

1

(d0 + 0)2= 0 (4.2)

In a force-balanced system, the sensor element is in a negative feedback loop. The

feedback signal is an electrostatic force, which is used to make sure that the proof

mass is kept at the center of the gap. As shown in Figure 3.4, this is the type of

system that is implemented in this work.

44 CHAPTER 4. INTERFACE ARCHITECTURE AND DESIGN

2. Charge-Balanced System: In this system the charges on the two differential

capacitors are kept to be equal. Equation (4.1) can be re-written in terms of the charge

on the capacitors:

FE,NET =1

2· Q2

1

ǫ · A − 1

2· Q2

2

ǫ ·A (4.3)

where A is the area of the capacitor plates and is the same for the two capacitors. As

equation (4.3) shows, the electrostatic force on a parallel-plate capacitor is propor-

tional to the square of the charge on the capacitor, and as long as the charges on the

two capacitors are equal, the net electrostatic forces on the proof mass will be zero.

In a charge-balanced system, the sensor element is connected to an interface that

measures the sensor capacitors while balancing the charge on them. This type of

interface is commonly used in open-loop accelerometers [44].

4.1.2 Linearity

There are a number of sources of nonlinearity in accelerometer interfaces, and we

briefly discuss these sources in this section:

1. Sensor Element: In a sensor element, the springs that are connected to the

proof mass are nonlinear. In general, the linearity of the spring degrades as the range

of motion of the proof mass increases. In force-balanced systems, this linearity is

improved because the proof mass is kept at the center of the gap.

2. Sensor Capacitance: Depending on how the sensor element is designed, the

sensor capacitance may or may not be a linear function of the displacement of the

proof mass. Figure 4.2 shows two sensors. In Figure 4.2(a), the input acceleration

moves the proof mass such that the area between the parallel plates changes, which

makes the change in capacitance a linear function of input acceleration. However, in

Figure 4.2(b) the input acceleration moves the proof mass such that the gap between

the plates changes, and because the capacitance is inversely proportional to the gap

between the plates, capacitance is a nonlinear function of input acceleration. The

latter is the type of sensor element that is used in this work.

4.1. ACCELEROMETER INTERFACE CHALLENGES 45

d0

xx0

C1 =ǫ · t · xd0

(a) Movement of proof masschanges the effective area be-tween the plates.

d

C2 =ǫ ·Ad

(b) Movement of proof masschanges the gap between theplates.

Figure 4.2: Sensor capacitance.

3. Electrostatic Force: The electrostatic force between the parallel plates of a

capacitor was given in equation (2.17) and is re-written here:

FE =1

2· C · V 2

d(4.4)

This electrostatic force is a nonlinear function of the voltage applied to the capacitor

and the gap between the plates. For the capacitor shown in Figure 4.2(a), this

electrostatic force can be written as

FE =1

2· ǫ · t · (x0 − x) · V 2

d20(4.5)

For the capacitor shown in Figure 4.2(b), which is the type of sensor used in this

work, the electrostatic force between the parallel-plates can be written as

FE =1

2· ǫ · A · V 2

(d0 − x)2(4.6)

which is a very nonlinear function of displacement, x. In force-balanced systems, as

shown in Figure 3.4, there is a negative feedback around the sensor. The feedback


signal is electrostatic force, and the feedback block is modeling the voltage to elec-

trostatic force transduction. Any nonlinearity in this block is added to the input

acceleration force, and therefore, will directly affect nonlinearity of the system. To

overcome this problem, we limit the feedback signal to two values. By doing this,

the system does not see the nonlinear relationship shown in equation (4.6) but only

a linear fit between the two points that are used for feedback.

4. Electronics: The electronic interface has some nonlinearity as well. However,

with proper design, the nonlinearity of the electronics will be negligible.

4.1.3 MEMS with a Single Port

The early accelerometer MEMS chips that were used in a force feedback loop had

two ports, as shown in Figure 4.3(a) [6]. One port is used to measure sensor capac-

itor (through CSP and CSN), and a second port is used to apply a force-feedback

(through CFP and CFN). However, for cost reasons, the size of the sensor-elements

is minimized, and as a result, we usually have a sensor-element that has one port as

shown in Figure 4.3(b). In this case, the same port (CSP and CSN capacitors) is used

for both measuring capacitance and applying force-feedback. Having one port, also

avoids complications with the dynamics of the system that can be present with two

ports.

Ted Smith and Mark Lemkin implemented such a system by utilizing time-multiplexing

([7], [8]). In other words, the system operates in two phases. During one phase, the

sensor capacitance is measured, and during a second phase, force feedback is applied.

4.2 Interface Design

Figure 4.4 shows the block diagram of a force-balanced system. In this section we go

over the building sub-blocks of the “Electronics” block that was built into a CMOS

integrated chip.

4.2. INTERFACE DESIGN 47

MEMS Sensor Element

CSP

CSN

CFP

CFN

(a) Sensor element with two ports.

MEMS Sensor Element

CSP

CSN

(b) Sensor element with one port.

Figure 4.3: Sensor element with one or two ports.

There have been a number of publications with an accelerometer interface as

shown in Figure 4.4. Widge Henrion published the first closed-loop Σ∆ accelerome-

ter interface, where he used the second-order sensor-element as the loop filter of the

Σ∆ modulator [6]. Ted Smith designed another Σ∆ closed-loop accelerometer, where

he used a MEMS element with one port (see Section 4.1.3), and time-multiplexed

capacitance measurement and force-feedback [7]. Ted Smith implemented an electri-

cal integrator, and compensated the loop by feeding back the bitstream to the input

of the integrator. Mark Lemkin designed a 3-axis accelerometer with three sepa-

rate Σ∆ loops, where he used only the second-order transfer function of the sensor

element as the loop filter and compensated the loop by implementing an electrical

lead compensation [8]. Vladimir Petkov showed that a second-order electromechani-

cal Σ∆ accelerometer, i.e. with no additional electrical filtering, cannot be thermal

noise limited in the signal band even at high oversampling ratios [45]. For this rea-

son, Vladimir Petkov designed a second-order electrical filter to make the system

thermal-noise limited.



Feedback

FINHSEN HELEC

COFF

CSEN SOUT

FB

Figure 4.4: Block diagram of the closed-loop sensor.

Force feedback was used in all of the above publications, meaning that electrostatic

force was used as the feedback signal. Because the electrostatic force is a nonlinear

function of voltage, a one-bit quantizer is often used in the Σ∆ loop so that a one-

bit DAC, which is inherently linear, is used in the feedback. Jiangfeng Wu designed

a closed-loop Σ∆ accelerometer with a multi-bit quantizer, and used pulse-density-

modulation (PDM) for the actuation signal in order to achieve a linear multi-bit force

feedback [9].

Figure 4.5 shows the circuit blocks that are used in this work to implement a

closed-loop interface. These blocks are described in detail in the next sections. The

sensor block is a MEMS accelerometer chip provided by Bosch corporation, and the

rest of the blocks were designed in a 0.18µm CMOS technology.

Sensor MUX C-to-V Integrator Compensator Comparator

Vctov Vinteg Vcompen SO

Figure 4.5: Block diagram of the closed-loop accelerometer interface.


4.2.1 Multiplexer and Timing Diagram

The multiplexer (MUX) is needed because of the time-multiplexing that exists in

the system. As described in section 4.1.3, the MEMS element has only one port,

and therefore, the capacitance measurement and force feedback are time-multiplexed.

Moreover, we have an additional phase to apply a time-varying common-mode voltage

as described in Section 3.3 to modulate the spring constant of the sensor. Figure 4.6

Capacitance Measurement k ModulationForce Feedback

Φ1A

Φ1A

Φ1B

Φ1B

Φ1Q

Φ1Q

Φ2

Φ2

Φ3R

Φ3R

Φ3K

Φ3K

Φ1

Φ1a

Φ2,3

Φ1A,3R

Figure 4.6: Timing diagram for one cycle of the interface operation.

shows the timing diagram for the system, and all the clock signals that are used in

the interface. There are three main phases: 1. capacitance measurement, 2. force


feedback, and 3. spring constant modulation. There are two sub-phases during

capacitance measurement, Φ1A and Φ1B, which are required to implement Correlated

Double Sampling (CDS). There is also a third sub-phase, Φ1Q, which is needed at the

end of the second phase of CDS for the quantizer to make a decision. The output

of the quantizer determines which side the proof mass has moved to. During the

feedback phase, a voltage is applied to one of the sensor capacitors. This voltage

will generate an attractive electrostatic force, which pushes the proof mass back to

the center of the gap. The spring constant modulation phase also consists of two

sub-phases. During the main sub-phase, Φ3K , a common-mode voltage is applied to

modulate the spring constant of the sensor. However, because the feedback voltage

carries information about the input acceleration, we need to reset the voltage on the

sensor capacitors, during Φ3R, before we modulate the spring constant so that the

spring constant modulation tones are only a function of the offset in the system, and

not the input acceleration.

The MUX block consists of a number of switches that control time-multiplexing

that occurs on the sensor-element. Figure 4.7 shows the switches inside the MUX

block. The way the force feedback is implemented is through two signals, Φ2p and

Φ2n, which control which side the feedback voltage should be applied to. The main

design concern for this block is to ensure that the ON-impedances of the switches are

low enough, i.e. much smaller than the parasitics of the sensor, so that all signals can

settle during each phase.

4.2.2 Capacitance-to-Voltage Converter

The Capacitance-to-Voltage (C-to-V) stage samples the differential sensor capacitance

and converts it to a voltage. Figure 4.8 shows the circuit schematic of the C-to-V

block, the sensor element, and also part of the MUX that is relevant to the capacitance

measurement.

Correlated double sampling is utilized to reduce the flicker noise and offset from


SensorElement

C-to-V

MUX

VDD

VFB

VFBVCM

VCM

VMOD

VMOD

GND

Φ1A

Φ1

Φ1

Φ3K

Φ3K Φ2p

Φ2n

Φ1a

Φ1A,3R

Φ1A,3R

Φ2p = Φ2 · SO

Φ2n = Φ2 · SO

Φ1a = Φ1A + Φ2 + Φ3R + Φ3K

Φ1A,3R = Φ1A + Φ3R

Φ1 = Φ1A + Φ1B

Figure 4.7: Circuit diagram for the MUX.

the OTA in this block. Ideally, the offset from the electronic blocks can be canceled by

the offset cancellation loop that suppresses the offset from the bondwires. However,

the main reason we use CDS is to reduce flicker noise. Correlated double sampling

works in two phases [39]. First, during Φ1a, the noise and offset of the OTA is sampled

on CH capacitors and the voltages on the sensor caps are reset. The circuit schematic

of the sensor element, MUX, and C-to-V during Φ1a is shown in Figure 4.9(a). Note

that this first phase is Φ1a, as opposed to Φ1A. The Φ1a phase is the union of Φ1A, Φ2,

and Φ3 phases. This is because during the Φ2 and Φ3 phases, the C-to-V, integrator,

and compensator blocks are disconnected from the sensor element, and we can start

the reset phase early for these blocks. The reason that we still need the Φ1A phase is

that the sensor capacitors also need to be reset, which cannot happen during Φ2 and

Φ3 phases.

During Φ1B , a common-mode step is applied to the sensor element. In presence


−

+

CtoV

Vctov

CP

CP

RP

RP

CH1

CF1

CF1

CSp

CSn

Φ1a

Φ1a

Φ1a

Φ1a

Φ1a

Φ1aΦ1a

Φ1aΦ1a

Φ1B

Φ1B

Φ1B

Φ1

Φ1

MUXSensor Element

VDD

VCM

VCM

VCM

VCM

VCM

VCMGND

GND

GND

Figure 4.8: Capacitance-to-voltage converter with CDS.

of acceleration, sensor capacitors, CSp and CSn, will not be equal and a common

mode step will result in a differential current going to the C-to-V block. The circuit

schematic of the sensor element, MUX, and C-to-V during Φ1B is plotted in Figure

4.9(b). Since a common-mode voltage is applied to the sensor and there is asymmetry

in the differential-mode half-circuits, we derive the transfer function of this stage by

considering the differential model of the interface shown in Figure 4.10. The C-to-

V block is also known as a switched-capacitor charge integrator, and it basically

integrates the differential charge that comes out of the sensor capacitors in response

to the common-mode voltage step. Applying a common-mode-voltage step to the

sensor element during Φ1B will also change the input common-mode of the OTA,

which is discussed in Section 4.3.3.


−

+

GND

GND

VDD

VCM

VCMVCM

VCM

CP

CP RP

RP

VctovCH1

CF1

CF1

CSp

CSn

(a) C-to-V during Φ1a.

−

+

GND

GND

GND

CP

CPRP

RP

VctovCH1

CF1

CF1

CSp

CSn

(b) C-to-V during Φ1B.

Figure 4.9: Sensor element, MUX, and C-to-V during different phases.

Next, we present equations for the transfer function and noise of the C-to-V block.

These equations are required to design the interface. Assuming that the DC gain of

the amplifier is A, the input capacitance of the amplifier is CA1, the differential

sensor capacitance is ∆CS = CSp −CSn, the nominal sensor capacitors when there is

no acceleration are CSp = CSn = CS0, and the circuit settles completely during the

two phases, the gain of the C-to-V block is given by:

Hctov =Vctov

∆CS= ∆V · A · (CF1 + CP + CAH)

(CF1 + CP + CS0 + CAH) · [(A+ 1) · CF1 + CP + CS0 + CAH ](4.7)

where ∆V is the amplitude of the common-mode-voltage step, which is VDD in this

implementation, and CAH = CA1‖CH1. For large amplifier gain, i.e. A → ∞, and

small input capacitance of the amplifier, i.e. CA1 << CH1, equation (4.7) simplifies

to

Hctov =Vctov

∆CS=

∆V

CF1

· CF1 + CP + CA1

CF1 + CP + CS0 + CA1

(4.8)


To calculate the noise contributions in the C-to-V interface, we consider the differential-

mode half-circuit model of Figure 4.8 with the assumption that the sensor capaci-

tances are about the same, i.e. CSp = CSn = CS0. Figure 4.10 shows the differential-

mode half-circuit model of the C-to-V block during Φ1a and Φ1B phases.

−

+CP

RP

Vn

VctovCH1

CF1

CA1

CL1A

CS0

(a) C-to-V during Φ1a.

−

+CP

RP

Vn

VctovCH1

CF1

CA1CS0CL1B

(b) C-to-V during Φ1B.

Figure 4.10: Half-circuit model of the sensor element, MUX, and C-to-V duringdifferent phases.

During Φ1a, the feedback capacitor, CF1, and the sensor capacitors are reset.

Also, the noise and offset of the OTA is sampled on the CH1 capacitors. Because

the amplifier is in a reset feedback configuration during Φ1a, the feedback factor is

β1A = 1. The load capacitance during Φ1a, CL1A, comes from the output capacitance

of the OTA and the common-mode feedback.


During Φ1B , the feedback factor is

β1B =CF1

CA1 +CA1+CH1

CH1· (CS0 + CP + CF1)

(4.9)

The load capacitance during Φ1B is

CL1B = CL1A + CI2 + CF1 ·(

1− β1B ·(

1 +CA1

CH1

))

(4.10)

where CI2 is the input capacitance of the next stage, the integrator. The noise of the

OTA in the C-to-V block is given by:

V 2OTA1,n =

KT · nf

CL1A + CH1 + CA1

· 1

β21B

+KT · nf

CL1B· 1

β1B(4.11)

The main design parameters here are CH1 and CL1B, which can be set to achieve a

certain noise power from the OTA. The noise of the parasitic resistances of the sensor

element are:

V 2Rp,n =

[

KT

CS0 + CP+ 4KT · RP · fBW · π

2

]

·(

CS0 + CP

CF1

)2

(4.12)

where fBW is the closed-loop bandwidth of the amplifier during Φ1B

fBW = β1B · fu = β1B · ω−3dB

2π· A = β1B · gm1

2π · CL1B(4.13)

and fu is the unity-gain bandwidth of the amplifier, A is the DC gain of the ampli-

fier, ω−3dB is the open-loop bandwidth of the amplifier during Φ1B , and gm1 is the

transconductance of the OTA in the C-to-V block. Notice that there are no parame-

ters in equation (4.12) that we can use to reduce this noise. This noise is partly set by

CS0 and CP capacitors, which are fixed as they are part of the MEMS accelerometer.

The noise in equation (4.12) is also a function of the closed-loop bandwidth of the


amplifier, fBW . However, this bandwidth is set to ensure complete settling of the sig-

nals, and cannot be decreased any further to reduce noise. Finally, although CF1 can

change the noise power at the output of the C-to-V stage, the effective noise referred

to input acceleration is independent of CF1. This is because the transfer function of

the C-to-V stage is inversely proportional to CF1, as shown in equation (4.8).

In order to minimize the input-referred noise of the next blocks, we want to get a

large gain from the C-to-V block. As shown in equation (4.8), the gain of the C-to-V

block is inversely proportional to the feedback capacitance, CF1, and therefore, we

would like to reduce the CF1 to increase the gain of this block. However, the feedback

factor of the circuit during Φ1B is proportional to CF1 (as shown in equation (4.9)),

and we would like to have a large feedback factor for two reasons: 1. the noise of the

C-to-V stage is inversely proportional to the feedback factor (as shown in equation

(4.11)); 2. the closed-loop bandwidth of the amplifier is proportional to the feedback

factor (as shown in equation (4.13)). As a result, if CF1 is reduced, the noise of

the C-to-V stage increases, and more power must be spent to keep the closed-loop

bandwidth constant. Therefore, there is a tradeoff between the power consumption

of the C-to-V and integrator blocks, and their noise contributions.

4.2.3 Integrator

The integrator is added to make sure that this interface is thermal noise limited in

the signal band. In an electromechanical Σ∆, the quantization noise can be reduced

by either increasing oversampling ratio, or the order of the filter. However, Vladimir

Petkov shows in [45] that as sampling rate is increased, a second-order electrome-

chanical Σ∆ loop only asymptotically becomes thermal noise limited. Moreover, in

presence of typical parasitics of sensor elements, the sampling rate should be low

enough so that signals settle completely. As a result, we increase the order of the

loop, by adding an integrator, to reduce the quantization noise in the signal band.

Figure 4.11, shows the block diagram of the integrator. Again, in order to reduce


−

+Vctov Vinteg

VCMVCM

VCMVCM

CI2 CH2

CF2

CF2

Φ1aΦ1a

Φ1aΦ1a

Φ1B

Φ1B

Φ1B

Φ1B

Figure 4.11: Integrator with CDS.

the flicker noise from the OTA, correlated double sampling is used. Next we look

at the transfer function and noise of this block. In order to find the signal transfer

function, we need the half-circuit models of the integrator during two phases as shown

in Figure 4.12.

Figure 4.12(a) shows the differential-mode half-circuit model of the integrator

during Φ1a. During this phase, the input capacitor is reset, and the amplifier noise

and offset is sampled on the CH2 capacitors. In this phase, the amplifier is in a

reset feedback configuration, and therefore, the feedback factor is β1A = 1. The load

capacitance during Φ1a, CL2A, comes from the output capacitance of the OTA and

the common-mode feedback. Figure 4.12(b) shows the differential-mode half-circuit

model of the integrator during Φ1B. During this phase, the output voltage of the C-

to-V stage is added to the voltage on the CF2 capacitors. During Φ1B, the feedback

factor is

β2B =CF2

CA2 +CA2+CH2

CH2· (CI2 + CF2)

(4.14)


−

+

Vn

VintegCI2

CH2

CA2

CL2A

CF2

(a) Integrator during Φ1a.

−

+Vctov

Vn

VintegCI2

CH2

CA2

CL2B

CF2

(b) Integrator during Φ1B.

Figure 4.12: Half-circuit model of the integrator during different phases.

and the load capacitance is

CL2B = CL2A + CI3 + CF2 ·(

1− β2B ·(

1 +CA2

CH2

))

(4.15)

where CI3 is the input capacitance of the next stage, the compensator. The transfer

function of the integrator is

Hinteg =

CI2

CF2

1 +1+

CI2CF2

A−

(

1 + 1A

)

· z−1

(4.16)

Assuming that the amplifier gain is infinite, equation (4.16) simplifies to

Hinteg =

CI2

CF2

1− z−1(4.17)

The input-referred noise of the integrator, i.e. referred to the output of the C-to-V

stage, is given by

V 2OTA2,n =

[

KT · nf

CL2A + CH2 + CA2

+KT · nf

CL2B· β2B

]

·(

1

β2B− CH2 + CA2

CH2

)2

·(

CF2

CI2

)2

(4.18)


The main design parameters to reduce the noise of the OTA in the integrator are

CH2, CL2B, and the ratio CF2

CI2.

4.2.4 Compensator

The Σ∆ loop in this work is a third order loop (a second-order sensor element and an

integrator). Therefore, we need a compensator to ensure stability of this loop. The

detailed connection between the integrator and the compensator is shown in Figure

4.13. The compensator is composed of a feed-forward around the integrator, and two

differentiators.

Vctov

HintegHdiff1

Hdiff2

Vinteg VcompenIntegrator

Compensator

Differentiator

Differentiator

Figure 4.13: Block diagram of the compensator.

The transfer function for the system shown in Figure 4.13 is given by:

Hinteg,compen =Vcompen

Vctov= Hinteg ·Hdiff1 +Hdiff2 (4.19)

The operation of a differentiator is similar to that of the integrator. For the integrator,

the feedback capacitor is not reset, and the charge from the input will accumulate on

the feedback capacitor. Therefore, the output will be the integral of the input. For

the differentiator, the input capacitor is not reset, so the charge that is stored on the

feedback capacitor is the charge that results from some change in the input voltage.

Therefore, the output will be the derivative of the input, with zero gain at DC.


If we cascade an integrator with a differentiator, they will cancel the effect of one

another, and the order of the loop does not change. In order to have both integration

and compensation in the loop, we implemented a differentiator as shown in Figure

4.14. This circuit has two input capacitors: CI31 is not reset and produces the

−

+Vinteg Vcompen

VCM

VCM

VCM

VCM

VCM

VCM

CI31

CI32

CI32

CF3

CF3

Φ1a

Φ1a

Φ1a

Φ1a

Φ1B

Φ1B

Figure 4.14: Differentiator with some DC gain.

differentiation, while CI32 is reset and provides a small DC gain. The CI31 and CI32

capacitors in parallel produce a zero, which is at some frequency that is determined

by the ratio of CI31 and CI32. CDS is not implemented in the compensator because

this block is preceded by the integrator, which has a large gain at DC.

Figure 4.15(a) shows the differential-mode half-circuit model of the circuit during

Φ1a. During this phase, the CI32 and CF3 capacitors are reset. Figure 4.15(b) shows

the differential-mode half-circuit model of the circuit during Φ1B. During this phase,

the input signal goes through the differentiator.


−

+

Vn

Vcompen

CI31

CI32

CF3

CA3

CL3

(a) Compensator during Φ1a.

−

+Vinteg

Vn

Vcompen

CI31

CI32

CF3

CA3

CL3

(b) Compensator during Φ1B.

Figure 4.15: Half-circuit model of the compensator during different phases.

The transfer function of the differentiator is

Hdiff =

CI31

CF3·(

1 + CI32

CI31− z−1

)

1 +1+

CI31CF3

+CI32CF3

A− CI31

CF3· z−1

A

(4.20)

Assuming that the amplifier gain is infinite, equation (4.20) simplifies to

Hdiff =CI31

CF3

·(

1 +CI32

CI31− z−1

)

(4.21)

In order to implement what is shown in Figure 4.13, we use the circuit shown in

Figure 4.14 and add a second set of input capacitors to it. By doing so, one amplifier

is used for both differentiators and the summation. This is shown in Figure 4.16.

The transfer function of the integrator is given by equation (4.17), and the transfer

function of the differentiator is given by equation (4.21). By substituting these into

equation (4.19), the overall transfer function from the input of the integrator to the

output of the compensator becomes:

Hinteg,compen =K1

1− z−1·[

K2 +K3 · z−1 + z−2]

(4.22)


−

+

Vctov

Vinteg

Vcompen

VCM

VCM

VCM

VCM

VCM

VCM

VCM

CI31

CI32

CI32

CFF21

CFF22

CFF22

CF3

CF3

Φ1a

Φ1a

Φ1a

Φ1a

Φ1a

Φ1a

Φ1B

Φ1B

Φ1B

Φ1B

Figure 4.16: Differentiator with two inputs.

where

K1 =CFF21

CF3

(4.23)

K2 = −CI2

CF2

· CI31 + CI32

CFF21

+ 1 +CFF22

CFF21

(4.24)

K3 =CI2

CF2

· CI31

CFF21

+ 2 +CFF22

CFF21

(4.25)

These equations can be used to find the capacitor values that result in a desired


transfer function.

4.2.5 Quantizer

The quantizer takes the output of the compensator, and makes a decision as to

which side the proof mass has moved to, or which side of the sensor has a larger

capacitance. This information is then used in the feedback phase, Φ2, when a force

feedback is applied to the sensor to push the proof mass back to the center of the gap.

As mentioned in Section 4.1.2, a one-bit quantizer is used because the electrostatic

force, which is used for the feedback, is a nonlinear function of voltage, and by

limiting the feedback signal to two values we can mask the nonlinear dependency of

the electrostatic force on voltage.

Figure 4.17 shows the block diagram of the quantizer. The output of the com-

pensator is sensed and is latched by a comparator during the Φ1Q phase. Then the

output of the comparator is buffered and fed to an SR latch so it is held during Φ2

phase when the output of the compensator and comparator are not valid anymore.

The output of the latch is buffered again and is fed to the MUX block, as shown in

Figure 4.7, to be used during Φ2 phase.

Vcompen

SO

BufferBuffer LatchComparator

QR

S

Figure 4.17: Block diagram of the quantizer.

Figure 4.18 shows the circuit schematic of the comparator [46]. It is composed

of a pre-amplifier, and a comparator latch, which has a positive feedback to saturate

the output of the comparator. The comparator works in two phases: during Φ1Q,

the reset signal, ΦRST , is low and the comparator makes a decision; for the rest of


the time, the reset signal is high, which turns off the comparator latch and resets

the internal nodes. One important point is that the offset of this comparator is not

VinVip

Von VopIB

ΦRST

ΦRSTΦRST

ΦRST = Φ1Q

VDD

GND

Figure 4.18: Circuit diagram of the comparator.

very important as there is a very large DC gain before the quantizer, and the input

referred offset would be very small. The large DC gain exists because the acceleration

signal we are interested in is at DC. In contrast, in a band-pass system where the

input signal is not at DC, e.g. a gyroscope, the comparator offset could translate into

a large input-referred offset, which reduces the dynamic range significantly.

4.3. INTERFACE DESIGN CONSIDERATIONS 65

4.3 Interface Design Considerations

The general design flow for this interface is that we first determine the size of the

capacitors, and then we design the amplifiers such that we have complete settling for

all the blocks. Capacitor sizes are determined by considering noise requirements and

signal transfer functions. In this section we go over a few challenges that came up

during the design of this interface.

4.3.1 Settling

To design the amplifiers one should consider the settling of all stages in this system.

Figure 4.19 shows most of the front-end during Φ1B . The important point is that the

C-to-V, integrator, and compensator stages are settling at the same time. Moreover,

the parasitics of the sensor-element, RP and CP , are quite large, and are in fact the

dominant factor in determining how fast the signals settle. This increases the required

transconductance of the three OTAs, and results in a large power consumption.

−

+

−

+

−

+VctovVinteg

VcompenCA3

CL3B

CI31

CI32

CFF21

CFF22

CF3

Vn1

Vn2

Vn3

CI2

CH2

CA2

CF2

CP

RP CH1

CF1

CA1CS0

Figure 4.19: Differential-mode half-circuit model of the sensor element, MUX, C-to-V,integrator, and compensator during Φ1B .


4.3.2 Sampling Rate

The sampling rate, fS, used in this work is 250 KHz. There are a few factors that

can influence what sampling rate is used.

1. Noise: Large sampling rate reduces both quantization noise and thermal

noise in the signal-band. Since the signal bandwidth is fixed, the oversampling ratio

(OSR) is proportional to the sampling frequency. As shown in equation (4.26) [47],

quantization noise in the signal band reduces as the oversampling ratio increases:

NQ ∝(

1

OSR

)( 2L+1

2 )(4.26)

where L is the order of the loop. As for the thermal noise, the total sampled thermal

noise is usually some factor of KT/C in a switched capacitor circuit, which is spread

over the frequency range [0, fS/2]. Therefore, as shown in equation (4.27), the thermal

noise in the signal bandwidth is proportional to sampling frequency:

Nth ∝ KT

C· 2

fS(4.27)

2. Power: One drawback of a larger sampling rate for this interface is that it

increases the power consumption of the circuit blocks. In a typical switched capacitor

circuit with a constant signal bandwidth, the power consumption is only a function of

the required noise level, not the sampling rate, in spite of the fact that large sampling

frequency means that circuit blocks have less time to settle. This is because for a

fixed noise level, increasing sampling rate by a factor of α reduces noise power spectral

density by a factor of α, but capacitor sizes can be reduced by a factor of α, which

means that the circuit becomes α times faster.

However, this argument falls apart for a system shown in Figure 4.19. As described

in section 4.3.1, the dominant factor in settling is the sensor-element parasitics, which

do not scale with the sampling rate. This has two consequences: first, even if the

circuit blocks have infinite bandwidth, the sensor parasitics pose a limit on how large


the sampling rate can be; second, if the transconductance, gm, of all the stages are

increased by some factor, β, the settling time of the system does not become β times

smaller. Therefore, if the sampling rate is increased by a factor of α, then capacitor

sizes can reduce by a factor of α, which makes each circuit block α times faster, but

now gm’s need to increase even further so the overall settling time of the system is

reduced by a factor α.

4.3.3 OTA Design

All OTA’s have the same folded cascode architecture as shown in Figure 4.20. There

are two cascode NMOS transistors to increase the DC gain of the amplifier. All

cascode transistors are minimum size, and the non-dominant poles from these cascode

transistors are at high enough frequencies that do not affect the stability of the

closed-loop amplifiers. The PMOS current sources are cascoded to increase output

impedance of the current source. Both PMOS and NMOS current sources have a

large length and a large overdrive voltage for better matching.

There is a common-mode feedback (CMFB) to set the output common-mode. It

senses the voltage at the output nodes, and adjusts the output common-mode by

changing VCMFB. Since these OTAs are used in switched capacitor circuit blocks,

a passive CMFB as shown in Figure 4.21 is used [48]. In this figure, the nodes Vop

and Von are connected to the output nodes of the OTA. The node VCMFB is the

node that drives the the CMFB NMOS current sources shown in Figure 4.20. Notice

that the C1 and C2 capacitors are always connected to CMFB current sources so the

OTA always has a valid output common-mode. The nodes VOCM,des and VCMFB,des

are generated in the bias circuit (not shown here). VOCM,des is the desired output

voltage, and VCMFB,des is the expected bias voltage for the CMFB current sources.

C1 and C2 capacitors are charged with these known voltages during ΦCM1, and their

charge is shared with C1 and C2 capacitors during ΦCM2. The ratio between C1 and

C3 (or C2 and C4) determine how quickly the CMFB starts up.


Von Vop

VinVip

VCMFB

VBP1

VBP2

VBP3

VBN1

VBN2

VBN3

VDD

GND

Figure 4.20: Circuit diagram of the amplifier.

The voltage swing at the output of all stages is small enough that there is no

slewing. It should be noted that the voltage swing at the output of the C-to-V block

is dominated by the quantization noise, but the output of the integrator is mostly

from the input acceleration. The latter is because the high frequency quantization

noise is filtered by the integrator, and the input acceleration is amplified. To estimate

what the output swings are before designing the amplifiers, simulations were run in

matlab to see the voltage swings at the output of different stages.

When a common-mode step is applied to the sensor element at the beginning of

phase Φ1B , the input common-mode of the OTA in the C-to-V stage drops by around


VonVop

VOCM,des

VCMFB,des

VCMFB

C1 C2C3 C4

ΦCM1

ΦCM1ΦCM1

ΦCM2

ΦCM2ΦCM2

Figure 4.21: Circuit diagram of the common-mode feedback.

250 mV. Therefore, the input transistors of the OTA are chosen to be PMOS so

that when the input common-mode is dropped the tail current source is not shut off.

Moreover, simulations were run to ensure that settling and noise requirements are

not affected by this common-mode change.

4.3.4 Stability

A 1-bit quantizer has a signal dependent gain [47]. Because we have a quantizer in

the loop, the feedback loop is nonlinear, and stability in this system is not very well

defined. A Σ∆ modulator is considered to be stable if its internal states are bounded,

and its limit cycles are not very large [49]. While finding conditions that ensure

stability of this nonlinear system are very difficult, we can gain a lot of insight by

using quasi-linear analysis [50]. Because the quantizer gain, and therefore, the loop

gain changes with the input signal, we look at a root-locus plot to make sure that


the system remains stable as the loop gain changes [51]. To obtain a range for the

quantizer gain, we run simulations over the input range and calculate the effective

gain of the quantizer for different inputs. Then, we can use the gain values with a

root-locus plot to make sure that the system remains stable.

4.4 Summary

The design of an interface IC for a closed-loop interface is reviewed here. The closed-

loop system is based on a force feedback Σ∆ modulator. Because the sensor element

has only one port, time-multiplexing is utilized to perform different operations on

the sensor element. An integrator is added to reduce the quantization noise in the

signal band, and a compensator is design to ensure the stability of the loop. This

interface IC together with the MEMS accelerometer form a closed-loop accelerometer

with spring constant modulation, which induces a spectral replica of COFF at the

modulation frequency.

Chapter 5

Experimental Results

As discussed in Section 3.4, bondwire offset can be canceled in a closed-loop ac-

celerometer interface by an offset cancellation loop as shown in Figure 3.8. A proto-

type system was implemented and tested to validate this concept. An interface IC

was fabricated as a proof of concept, and was tested with a MEMS accelerometer

chip. The digital part of the offset cancellation loop was implemented on a Field

Programmable Gate Array (FPGA) board. This chapter is focused on measurement

results of this prototype. Section 5.1 is focused on the fabricated interface IC. Section

5.2 goes over the test setup, and finally, Section 5.3 goes over measurement results.

5.1 Interface IC

The switched capacitor circuit described in Chapter 4 was fabricated in a 0.18 µm

CMOS technology. This circuit, together with the MEMS accelerometer, forms a

closed-loop accelerometer interface, which is also a Σ∆ modulator. The interface

IC micrograph is shown in Figure 5.1. The main blocks are highlighted on these

figures. The details of the MUX, C-to-V, Integrator, Compensator, and Quantizer

were described in Section 4.2. The “Clock Phase Generation” block, generates all the

timing signals needed for the operation of the interface IC (see Figure 4.6). The input

71

72 CHAPTER 5. EXPERIMENTAL RESULTS

Clock Phase Generation

MUX

C / V

Integrator

Compensator

Quantizer

1 1718

343551

68

63

65

Figure 5.1: Chip micrograph.

to the “Clock Phase Generation” block is a master clock at 32× fS, or 8 MHz. The

chip size is 3mm×3mm, with an active area of 1.35 mm2. The active area includes the

blocks that are highlighted in Figure 5.1, and also a scan-chain, some digital buffers,

and also routing for the clock phases that are not highlighted. Table 5.1 shows the

breakdown of the active area for all the blocks. The power consumption of the chip

is 3.1 mW. As discussed in Section 4.3.1 the power consumption is large because of

the parasitics of the sensor element.

5.2 Test Setup

Figure 5.2 shows the system in Figure 3.8 with more details about the correlator

block. The main goal of the correlator is to extract the spring constant-modulation

tones from the output bitstream, SOUT . The multiplication and summation perform

the correlation as described in Section 3.4. The output bitstream, SOUT , is decimated,

to filter out the high frequency quantization noise that is present at the output of

a Σ∆ loop. DMOD goes through an identical decimation filter, so that both DMOD

5.2. TEST SETUP 73

Block Area mm2 % of Active AreaMUX 0.055 4.07C-to-V 0.344 25.50Integrator 0.247 18.28Compensator 0.240 17.78Quantizer 0.045 3.30Clock Phase Generation 0.184 13.61Clock Routing 0.183 13.52Scan Chain 0.044 3.28Other Digital 0.008 0.66

Table 5.1: Breakdown of active area for each block.

Sensor ElementElectronics

Feedback

FIN

HSEN,MODHELEC

COFF

CSEN SOUT

FB

DecimationFilter

DecimationFilter

Filter

ODAC

DMOD

Figure 5.2: Block diagram of the closed-loop accelerometer with offset cancellationloop.

and SOUT see the same delay before they are multiplied together. A large DC com-

ponent in SOUT is upconverted by the multiplication block and generates a tone at

the modulation frequency at the output of the multiplication block. The “Filter”


block after the summation is added mainly to attenuate this tone. The output of

the filter is converted to an analog signal through the ODAC block (Offset Digital to

Analog Converter). This analog signal is then fed to the input of the integrator in

the interface IC, as described in Section 4.2.3.

The “Sensor Element” block in Figure 5.2 is the accelerometer sensor element,

which was provided by Robert Bosch Corporation. The “Electronics” block is the

interface IC that was fabricated. The CMOS interface IC and the accelerometer chip

were put in the same package and were connected together with bond-wires. Figure

5.3 shows pictures of the package and the test setup. A printed circuit board (PCB)

FPGA Board PCB Board

Package

CMOS ICMEMS

Figure 5.3: A picture of the test setup.

was designed to test the interface, and the packaged interface was placed on the PCB.

The package is shown on the PCB in Figure 5.3. The Interface IC, and the MEMS

chip are also shown in the package. An FPGA board, also shown in Figure 5.3, is

connected to the PCB to read the output bitstream. DMOD is a binary sequence,

which is generated on the FPGA board. The voltage that is applied to the sensor

element, VM (t), is obtained fromDMOD. The decimation, multiplication, summation,

5.3. MEASURED RESULTS 75

and filter are also implemented on an FPGA board. The offset DAC (ODAC) is a

discrete component used on the PCB. In order to make sure that the FPGA board

samples the output at the right time, a special phase is generated by the interface IC

to drive the clock of the FPGA board after being buffered on the PCB.

5.3 Measured Results

This section is focused on measurement results. The main measurement is the output

spectrum. On the output spectrum, we can see the spring constant modulation tones,

and we can also see how these tones are affected by the offset cancellation loop. All

of these measurements are shown in Section 5.3.1. Then, we look at the convergence

offset cancellation loop in Section 5.3.2. Next, we ensure that the offset cancellation

loop is not affecting the input acceleration, and finally we show how the interface

responds to deformation of parasitic bondwires in Section 5.3.5.

5.3.1 Output Spectrum

Figure 5.4 shows the output spectrum of the system, when the spring constant mod-

ulation and the offset cancellation loop are turned off.

The full-scale range is around 9.14 g (1 g = 9.8 m/s2), and the noise floor corre-

sponds to 220 µg√Hz. Also notice that there is no visible flicker noise down to 1 Hz.

In another measurement, the output spectrum was plotted on a spectrum analyzer

down to 0.1 Hz, and there was no increase in the noise floor. Figure 5.5 shows the

output spectrum of the system, when the spring constant modulation is turned on,

but the offset cancellation loop is off.

For this measurement, the spring constant is modulated with a square wave at

244 Hz. This pulse is generated by the FPGA board and is obtained by dividing

the sampling clock that drives the FPGA board. The sampling clock frequency is

250 kHz, and the modulation signal frequency is 250kHz210

= 244 kHz. The amplitude of


Frequency [Hz]

OutputSpectrum

[dBFS]

Offset Cancellation Loop Off-20

-40

-60

-80

-100100 101 102 103 104 105

Figure 5.4: Measured output spectrum (250000-point FFT after Hann windowing)when the spring constant modulation is off.

Frequency [Hz]

OutputSpectrum

[dBFS]


-40

-60

-80

-100100 101 102 103 104 105

Figure 5.5: Measured output spectrum (250000-point FFT after Hann windowing)when the spring constant modulation is on.

these tones correspond to an offset of 750 mg. It should be mentioned that this offset

is mostly coming from the mismatch between the parasitic capacitances of the sensor


element. These parasitics are on the order of 5 pF, while the sensor element outputs

around 10 fF of capacitance change for 1 g of acceleration.

Figure 5.6 shows the output spectrum of the system when the spring constant

modulation and the offset cancellation loop are turned on. Again, for this measure-

41dB

Frequency [Hz]

OutputSpectrum

[dBFS]

Offset Cancellation Loop On-20

-40

-60

-80

-100100 101 102 103 104 105

Figure 5.6: Measured output spectrum (250000-point FFT after Hann windowing)when the offset cancellation loop is on.

ment the spring constant is modulated with a pulse signal at 244 Hz. As highlighted

on the figure, the spring constant modulation tones are suppressed by 41 dB, which

reduces the offset to 6.2 mg. The main reason that offset is not reduced below 6.2 mg

is the bandwidth of the interface. As discussed earlier, ideally we want the SOUT and

VMOD signals to see the same delay before they are multiplied. However, because the

bandwidth of the interface is around 1 kHz, the harmonics of the modulating signal

that are close to 1 kHz see some phase change. This phase change causes some error,

and as a result, the tones are not completely suppressed.

Next we put the accelerometer on a shaker table. Figure 5.7 shows the output

spectrum when the device is put on a shaker table, and the spring constant modulation

is turned on.

78 CHAPTER 5. EXPERIMENTAL RESULTSts

Frequency [Hz]

OutputSpectrum

[dBFS]


-40

-60

-80

-100100 101 102 103 104 105

Fundamental

Harmonics

k-modulation

Figure 5.7: Measured output spectrum (250000-point FFT after Hann windowing)on a shaker table with spring constant modulation on.

The shaker table is set to generate an acceleration force with magnitude of 1 g

at 100 Hz. The shaker table is quite non-linear, and generates some harmonics at

200 and 300 Hz. We can also see the k-modulation tones. Again, when we turn on

the offset cancellation loop, the spring constant modulation tones are suppressed, as

shown in Figure 5.8. The small tone at 17 Hz is from the vibrations caused by the

air conditioning in the building and disappears when measurements are taken after 7

pm, when the air conditioning is off.

5.3.2 Convergence

Another measurement that is important, is a measurement that shows how quickly

the offset cancellation loop settles. We take a measurement where we look at the

input code to the offset DAC (ODAC). As shown in Figure 5.9, the offset cancellation

loop settles in less than one minute. Because the offset drifts over a much longer time,

this convergence is fast enough for tracking the offset drift. It should also be noted

that there is a tradeoff between how quickly the offset cancellation loop converges,


Frequency [Hz]

OutputSpectrum

[dBFS]

Offset cancellation Loop On-20

-40

-60

-80

-100100 101 102 103 104 105

Fundamental

Harmonics

Figure 5.8: Measured output spectrum (250000-point FFT after Hann windowing)on a shaker table with offset cancellation Loop on.

and how noisy the input to the ODAC is. For this measurement, the loop gain was

made small enough, to get a clean (but slow) convergence.

Time [s]

ODAC

Code

20 40 6007000

7400

7800

8200

Figure 5.9: Measured convergence of the offset cancellation loop.


5.3.3 Sensitivity

It is important to ensure that the signal that is fed back to the integrator to cancel the

offset is not suppressing the DC input acceleration. This section describes a series of

measurements we have taken to make sure that this is the case. The DC component

at the output of an accelerometer interface is sensitive to input acceleration and offset.

This is shown in equation (5.1):

ODC = α · (aIN + aOFF ) (5.1)

where ODC is the DC output, α is the sensitivity of the interface, aIN is the DC

input acceleration, and aOFF is the offset. It should be noted that the output of the

interface IC is a bit stream, so ODC is measured as a fraction of the Full-Scale range

(FS).

We want to check that turning on the offset cancellation loop does not change

the DC sensitivity of the system. Therefore, we measure the DC sensitivity of the

accelerometer for two cases: when the offset cancellation loop is on and off. The DC

sensitivity should be the same for these cases. In order to measure the DC sensitivity,

we run two measurements as shown in Figure 5.10.

First, we tilt the accelerometer such that it measures 1 g of gravitational force as

shown in Figure 5.10(a). The output from this measurement is

O1DC = α · (1G+ aOFF ) (5.2)

Then we rotate the accelerometer by 180◦ so it measures -1 g of acceleration, as shown

in Figure 5.10(b). The output from this measurement is

O2DC = α · (−1G+ aOFF ) (5.3)


Accelerom

eter

O1DC = α · (1G+ aOFF )

(a) Measurement 1: measuring 1G ofacceleration.

Accelerom

eter

O2DC = α · (−1G + aOFF )

(b) Measurement 2: measuring−1G of acceleration.

Figure 5.10: Measuring DC sensitivity.

From equations (5.2) and (5.3) we can solve for sensitivity and offset

α =O1DC − O2DC

2G(5.4)

aOFF =O1DC +O2DC

O1DC − O2DC· 1G (5.5)

We measured the DC sensitivity to be 0.1094 FS/g both when the offset cancellation

loop is on and off, which indicates that the offset cancellation loop does not affect

the DC sensitivity to input acceleration. Therefore, the DC signal that is fed back

to the integrator only cancels the offset in the system, and does not suppress input

acceleration.

5.3.4 Bondwire Deformation

The main goal of this work was canceling the offset drift due to parasitic capacitances

of the bondwires. In this section, we are interested to see how this prototype performs

when the offset due to parasitic capacitance of the bondwires changes. To this end,

we deform the bondwires to change their parasitic capacitances. This will change the

offset due to parasitic capacitances of the bondwires.


Figure 5.11(a) shows a closeup view of the MEMS chip, interface IC, and the

bondwires connecting them. Figure 5.11(b) shows the bondwires after they are de-

formed to change their parasitic capacitance. Two green lines with the same size are

drawn on both figures to highlight the change in distance between the bondwires, this

change is roughly 100 µm.

Interface IC

MEMS Chip

Bondwires

(a) Initial bondwires.

Interface IC

MEMS Chip

Bondwires

(b) Deformed bondwires.

Figure 5.11: Images of the bondwires before and after deformation.

Figure 5.12 shows the output spectrum of the system when the offset cancellation

loop is off. As we expect, deforming the bondwires changes the DC component at the

output. This change corresponds to 350 mg of offset.

Figure 5.13 shows the output spectrum of the system with the offset cancellation

loop is turned on. We observe that when the offset cancellation loop is on, the DC

component at the output does not change much. The small change corresponds to 0.7

mg of offset. Figure 5.14 shows a closer view of the DC components in Figures 5.12

and 5.13. This result shows that with the offset cancellation loop, which continuously

detects and cancels the offset, the interface becomes much less sensitive to parasitic


Frequency [Hz]

OutputSpectrum

[dBFS]

Offset Cancellation Loop Off

0 5 10

-20

-40

-60

-80

-100

InitialDeformed

Figure 5.12: Output spectrum before and after bondwire deformation with offsetcancellation loop off.

Frequency [Hz]

OutputSpectrum

[dBFS]

Offset Cancellation Loop On

0 5 10

-20

-40

-60

-80

-100

InitialDeformed

Figure 5.13: Output spectrum before and after bondwire deformation with offsetcancellation loop on.

capacitances of the bondwires.


Frequency [Hz]

OutputSpectrum

[dBFS]

0 0.5

-20

-40

(a) DC component when the offset can-cellation loop is off.

Frequency [Hz]

OutputSpectrum

[dBFS]

0 0.5

-40

-60

(b) DC component when the offset can-cellation loop is on.

Figure 5.14: Zoomed-in output spectrum before and after bondwire deformation.

5.3.5 Parasitic Accelerations

As analyzed in Section 3.4.2, a parasitic acceleration close to the spring constant

modulation tones creates a new tone at the output of the correlator at the beat

frequency. In this section, we put the accelerometer on a shaker table and introduce

a parasitic acceleration with an amplitude of 1 g close to the fundamental frequency

of the square ware. Then, we use a pseudo-random sequence to modulate the spring

constant to reduce the sensitivity to parasitic accelerations.

Figure 5.15 shows the convergence of the ODAC code when a square wave is used

for spring constant modulation. Figure 5.15 shows three measurements: 1. with a 1

g parasitic acceleration at 245 Hz (1 Hz away from the fundamental), 2. with a 1 g

parasitic acceleration at 254 Hz (10 Hz away from the fundamental), and 3. with no

parasitic accelerations.

From this measurement we observe the following, which are consistent with the

theory discussed in Section 3.4.2:


Time [s]

ODAC

Code

Square Wave

0 10 20 30 40 50 606200

6600

7000

7400

7800

8200

8600

No Parasitic

Parasitic 1 Hz awayParasitic 10 Hz away

Figure 5.15: Measured convergence of ODAC code with square wave modulation inpresence of parasitic accelerations.

1. A parasitic acceleration close to the modulation frequency produces a tone at

the beat frequency.

2. The amplitude of the beat frequency is larger for a parasitic acceleration that

is closer to the modulation frequency.

Next, we modulate the spring constant with a pseudo-random sequence (with no

parasitic accelerations). The output spectrum is shown in Figure 5.16. The pseudo-

random sequence is generated on the FPGA board from an LFSR of length 5 registers.

This will generate a pseudo-random sequence of length 31 bits. The fundamental

frequency is around 126 Hz. Next,we turn on the offset cancellation loop again,

and we observe that the spring constant modulations are suppressed by the offset

cancellation loop. The output spectrum in this case is shown in Figure 5.17.


Frequency [Hz]

OutputSpectrum

[dBFS]


-40

-60

-80

-100

-120

100 101 102 103 104 105

Figure 5.16: Measured output spectrum (250000-point FFT after Hann windowing)with spring constant modulation using a pseudo-random sequence.

Frequency [Hz]

OutputSpectrum

[dBFS]

Offset Cancellation Loop On-20

-40

-60

-80

-100

-120

100 101 102 103 104 105

Figure 5.17: Measured output spectrum (250000-point FFT after Hann windowing)using a pseudo-random sequence for modulation and offset cancellation loop on.

Finally, the accelerometer (with pseudo-random modulation) is put on the shaker


table again and a 1 g parasitic acceleration is placed close to the fundamental fre-

quency of the pseudo-random sequence. Figure 5.18 shows the convergence of the

ODAC code when a pseudo-random sequence is used for spring constant modulation.

Again, Figure 5.18 shows three measurements: 1. with a 1 g parasitic acceleration at

127 Hz (1 Hz away from the fundamental), 2. with a 1 g parasitic acceleration at 136

Hz (10 Hz away from the fundamental), and 3. with no parasitic accelerations. Note

that the amplitudes of the tones at the beat frequency have reduced compared to a

square wave, which is consistent with what discussed in Section 3.4.2.

Time [s]

ODAC

Code

PR Sequence with L=31

0 10 20 30 40 50 606200

6600

7000

7400

7800

8200

8600

No Parasitic

Parasitic 1 Hz awayParasitic 10 Hz away

Figure 5.18: Measured convergence of ODAC code with pseudo-random sequencemodulation in presence of parasitic accelerations.

It should also be noted that in the absence of parasitic accelerations, modulation

with a square wave and a pseudo-random sequence both estimate the same offset in

the system (i.e. in both cases the ODAC code converges to the same value). Table

5.2 summarizes the results of these measurements.


Type of the modula-tion signal

Beat signal amplitudewith parasitic accelera-tion 1 Hz away from thefundamental

Beat signal amplitudewith parasitic accelera-tion 10 Hz away from thefundamental

Square Wave 210.0 mg 10 mgPseudo Random Se-quence (L=15)

30.7 mg 2.7 mg

Table 5.2: Measured beat signal amplitude at the correlator output, caused by aparasitic acceleration.

5.4 Summary

An interface IC was fabricated in a 0.18-µm 3-V CMOS technology. Spring constant

modulation is shown with a square wave, and also a pseudo-random sequence. The

offset of a prototype is reduced by a factor of 112 down to 6.2 mg. The DC sensitivity

of the interface is measured both when the offset cancellation loop is on and off, and

it is shown that the sensitivity to input acceleration does not change. Therefore,

the offset cancellation loop does not affect input acceleration, and only cancels the

offset. It is also shown that using a pseudo-random sequence makes the interface less

sensitive to parasitic accelerations.

Chapter 6

Conclusion

6.1 Summary

Applications for accelerometers have grown dramatically over the last couple of decades.

Today, accelerometers are used in a number of safety systems in the automotive indus-

try. Consumer electronics is another area where the accelerometer market is growing.

According to iSuppli, it is expected that the market for mobile phones and consumer

electronics to have a compound annual growth rate (CAGR) of around 16.8% from

2008 to 2013, and will account for b$2.5 or 30% of the total MEMS market in 2013

[52].

Many applications require accurate measurement of the DC acceleration, and

therefore, low offset is crucial for such applications. The main challenge in designing

a low offset accelerometer is to reduce the offset drift over temperature and lifetime

of the device. While post-fabrication calibration could reduce offset drift over tem-

perature, it cannot address offset drift over the lifetime of the device. Therefore,

in this work, we proposed a technique that continuously measures and cancels the

main source of offset drift in system-in-package type accelerometers, which is parasitic

capacitance of the bondwires.

This technique relies on the parametric modulation of the sensor element. In

89

90 CHAPTER 6. CONCLUSION

particular, the spring constant of the sensor element is electrostatically modulated in

a closed-loop architecture. This modulation produces an upconverted replica of the

offset, which can be measured and nulled through an offset cancellation loop. In the

presented proof-of-concept prototype implementation the offset of an accelerometer

was reduced by a factor of 112 down to 6 mg. Moreover, it was shown that the

system has become very insensitive to change in bondwire capacitance. Finally, a

pseudo-random sequence was used for the modulation of the spring constant in order

to reduce sensitivity of the system to parasitic accelerations that can be present in

the frequency range of the modulation.

6.2 Future Work

The future research in inertial sensor interface design can take a number of directions.

The mechanical sensor element affects the performance of the system in a number of

ways:

1. The resolution of the interface is mainly set by the brownian motion of the

air molecules between the MEMS capacitor plates, and the parasitic resistance

from the poly-silicon routing on the MEMS chip.

2. The parasitics of the sensor element limit the settling of voltages on the sensor

element. Therefore, the sampling rate is mainly limited by the time constant

of the parasitics of the sensor element.

3. The parasitics of the sensor element also limit the settling of the circuit blocks

during capacitance measurement. Therefore, the interface IC is designed to have

a certain bandwidth such that it does not deteriorate the settling any further.

This implies that the parasitics of the sensor element dictate a minimum power

consumption in the interface IC.

6.2. FUTURE WORK 91

4. The bandwidth of the system is mainly set by the dynamics of the sensor ele-

ment.

For all of these reasons, the design of the MEMS sensor element and the interface

IC should be done together, so that the overall performance of the sensor can be

optimized.

As discussed in Section 5.3.2, because the offset drift is very slow, the offset

cancellation loop is also designed to be slow. While this loop is fast enough to track

offset drift, it may be too slow for startup in some applications. Another improvement

can target the DAC in the offset cancellation loop. This DAC can be replaced with

a Σ∆ type DAC whose output approximates, on average, the offset to be cancelled.

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