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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
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
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.
Roger Howe
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.
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
iv
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
5.5 Measured output spectrum (250000-point FFT after Hann windowing)
when the spring constant modulation is on. . . . . . . . . . . . . . . . 76
5.6 Measured output spectrum (250000-point FFT after Hann windowing)
when the offset cancellation loop is on. . . . . . . . . . . . . . . . . . 77
5.7 Measured output spectrum (250000-point FFT after Hann windowing)
on a shaker table with spring constant modulation on. . . . . . . . . . 78
5.8 Measured output spectrum (250000-point FFT after Hann windowing)
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
5.16 Measured output spectrum (250000-point FFT after Hann windowing)
with spring constant modulation using a pseudo-random sequence. . . 86
5.17 Measured output spectrum (250000-point FFT after Hann windowing)
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
xvi
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)
8 CHAPTER 2. MICROMACHINED CAPACITIVE ACCELEROMETERS
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
2.1. PRINCIPLES OF OPERATION 9
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)
10 CHAPTER 2. MICROMACHINED CAPACITIVE ACCELEROMETERS
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)
2.1. PRINCIPLES OF OPERATION 11
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
12 CHAPTER 2. MICROMACHINED CAPACITIVE ACCELEROMETERS
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
2.1. PRINCIPLES OF OPERATION 13
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
Direction of the 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
14 CHAPTER 2. MICROMACHINED CAPACITIVE ACCELEROMETERS
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
Direction of the 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
2.1. PRINCIPLES OF OPERATION 15
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
16 CHAPTER 2. MICROMACHINED CAPACITIVE ACCELEROMETERS
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
2.1. PRINCIPLES OF OPERATION 17
Electrostatic + Spring Force
Direction of the 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)
18 CHAPTER 2. MICROMACHINED CAPACITIVE ACCELEROMETERS
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
Direction of the 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
2.1. PRINCIPLES OF OPERATION 19
(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
20 CHAPTER 2. MICROMACHINED CAPACITIVE ACCELEROMETERS
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,
22 CHAPTER 2. MICROMACHINED CAPACITIVE ACCELEROMETERS
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
2.2. OFFSET IN ACCELEROMETERS 23
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.
24 CHAPTER 2. MICROMACHINED CAPACITIVE ACCELEROMETERS
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
2.2. OFFSET IN ACCELEROMETERS 25
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
30 CHAPTER 3. OFFSET DETECTION AND CANCELLATION
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)
32 CHAPTER 3. OFFSET DETECTION AND CANCELLATION
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.
Sensor Element Electronics
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
34 CHAPTER 3. OFFSET DETECTION AND CANCELLATION
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,
36 CHAPTER 3. OFFSET DETECTION AND CANCELLATION
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.
Sensor Element Electronics
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
Sensor Element Electronics
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:
38 CHAPTER 3. OFFSET DETECTION AND CANCELLATION
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.
3.4. OFFSET CANCELLATION LOOP 39
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
40 CHAPTER 3. OFFSET DETECTION AND CANCELLATION
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
3.4. OFFSET CANCELLATION LOOP 41
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
46 CHAPTER 4. INTERFACE ARCHITECTURE AND DESIGN
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.
48 CHAPTER 4. INTERFACE ARCHITECTURE AND DESIGN
Sensor Element Electronics
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. INTERFACE DESIGN 49
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
50 CHAPTER 4. INTERFACE ARCHITECTURE AND DESIGN
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
4.2. INTERFACE DESIGN 51
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
52 CHAPTER 4. INTERFACE ARCHITECTURE AND DESIGN
−
+
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.
4.2. INTERFACE DESIGN 53
−
+
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)
54 CHAPTER 4. INTERFACE ARCHITECTURE AND DESIGN
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.
4.2. INTERFACE DESIGN 55
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
56 CHAPTER 4. INTERFACE ARCHITECTURE AND DESIGN
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
4.2. INTERFACE DESIGN 57
−
+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)
58 CHAPTER 4. INTERFACE ARCHITECTURE AND DESIGN
−
+
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)
4.2. INTERFACE DESIGN 59
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.
60 CHAPTER 4. INTERFACE ARCHITECTURE AND DESIGN
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.
4.2. INTERFACE DESIGN 61
−
+
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)
62 CHAPTER 4. INTERFACE ARCHITECTURE AND DESIGN
−
+
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
4.2. INTERFACE DESIGN 63
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
64 CHAPTER 4. INTERFACE ARCHITECTURE AND DESIGN
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 .
66 CHAPTER 4. INTERFACE ARCHITECTURE AND DESIGN
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
4.3. INTERFACE DESIGN CONSIDERATIONS 67
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.
68 CHAPTER 4. INTERFACE ARCHITECTURE AND DESIGN
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
4.3. INTERFACE DESIGN CONSIDERATIONS 69
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
70 CHAPTER 4. INTERFACE ARCHITECTURE AND DESIGN
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”
74 CHAPTER 5. EXPERIMENTAL RESULTS
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
76 CHAPTER 5. EXPERIMENTAL RESULTS
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]
Offset Cancellation Loop Off-20
-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
5.3. MEASURED RESULTS 77
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]
Offset Cancellation Loop Off-20
-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,
5.3. MEASURED RESULTS 79
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.
80 CHAPTER 5. EXPERIMENTAL RESULTS
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)
5.3. MEASURED RESULTS 81
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.
82 CHAPTER 5. EXPERIMENTAL RESULTS
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
5.3. MEASURED RESULTS 83
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.
84 CHAPTER 5. EXPERIMENTAL RESULTS
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:
5.3. MEASURED RESULTS 85
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.
86 CHAPTER 5. EXPERIMENTAL RESULTS
Frequency [Hz]
OutputSpectrum
[dBFS]
Offset Cancellation Loop Off-20
-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
5.3. MEASURED RESULTS 87
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.
88 CHAPTER 5. EXPERIMENTAL RESULTS
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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