Capacitance Readout Circuits Based on Weakly-Coupled ...

Capacitance Readout Circuits Based onWeakly-Coupled Resonators

by

Siamak Hafizi-Moori

B.Sc., University of Tehran, 1991M.Sc., Tehran Polytechnic, 1995

A THESIS SUBMITTED IN PARTIAL FULFILLMENT OFTHE REQUIREMENTS FOR THE DEGREE OF

DOCTOR OF PHILOSOPHY

in

The Faculty of Graduate and Postdoctoral Studies

(Electrical and Computer Engineering)

THE UNIVERSITY OF BRITISH COLUMBIA

(Vancouver)

April 2016

c© Siamak Hafizi-Moori 2016

Abstract

Capacitive sensors and their associated readout circuits are well known and have

been used in many measurement applications in different industries. Improving

the sensitivity, resolution and accuracy of measuring small capacitance changes

has always been one of the important research topics, especially in recent years

that sensors are becoming smaller in size with lower associated capacitance val-

ues. This thesis focuses proposes a new method for implementing capacitance

readout circuits with higher sensitivity. This is the first time, to our knowledge,

that this method has ever been applied directly in electrical domain for capacitance

measurement applications.

The proposed method, which is based on weakly-coupled-resonators (WCRs)

concept, can achieve considerably (orders of magnitudes) higher sensitivity while

simplifying the analog front end circuitry and reducing the cost. For compari-

son, capacitance-to-frequency conversion readout circuits were chosen, which are

one of the most reliable and best performing designs and also the closest to our

WCR method since both involve shift in natural modes due to capacitance changes.

Analysis and SPICE simulations followed by experiments proved the concept. The

experimental results have shown almost two orders of magnitude higher relative

sensitivity for our two-degree-of-freedom (2DOF) WCR-based system. In the next

step we proposed a novel (named hybrid) method to reduce the measurement er-

ii

Abstract

ror considerably (4 to 6 times lower). Hybrid method is robust and insensitive

to variations in excitation frequency, which is one of the main sources for errors.

We have also analyzed the use of active inductors in our coupled resonators. The

analyses and simulations proved the concept. This opens an avenue towards im-

plementation of WCR-based readout in integrated circuits; specifically applicable

for micro-electro-mechanical systems (MEMS) devices, and even integrating both

MEMS sensors and the readout circuit in the same integrated circuit (IC) package.

Another route on this research was to exploit the insensitivity and robustness of

three-degree-of freedom (3DOF) weakly-coupled resonators to resonant frequency

deviations. Analyses, followed by simulations, proved that applying 3DOF WCR

in sensing differential capacitance changes does not require frequency tracking, yet

has the same sensitivity achieved in 2DOF-based readout circuits.

iii

Preface

I, Siamak Hafizi-Moori, am the principal contributor of all chapters. Dr. Edmond

Cretu, supervisor of the research, has provided guidelines, technical support and

editing assistance on the manuscript.

In the early stages of the project, as a reference for one of the conventional

readout circuits, a capacitance-to-voltage readout circuit was designed and tested

by Ahmed Sharkia and I, which is being presented in Appendix A and helped in

completing the experimental results of the following paper:

E.H. Sarraf, A. Sharkia, S. Moori, M. Sharma and E. Cretu. “High Sensitivity

Accelerometer Operating on the Border of Stability with Digital Sliding Mode

Control”, IEEE Sensors 2013.

A version of chapter 4 has been published. S. Hafizi-Moori and E. Cretu,

“Weakly-coupled resonators in capacitive readout circuits,” Circuits and Systems

I: Regular Papers, IEEE Transactions on, vol. 62, no. 2, pp. 337–346, 2015.

A version of chapter 5 has been submitted to a journal and is under review.

iv

Table of Contents

Abstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ii

Preface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iv

Table of Contents . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . v

List of Tables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ix

List of Figures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xi

List of Acronyms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xvi

Acknowledgments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xviii

Dedication . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xix

1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

1.1 History of Sensors . . . . . . . . . . . . . . . . . . . . . . . . . 1

1.2 Readout Circuits . . . . . . . . . . . . . . . . . . . . . . . . . . 4

1.3 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

1.4 Thesis Outlines . . . . . . . . . . . . . . . . . . . . . . . . . . . 8

v

Table of Contents

2 Capacitive Sensors and Their Associated Readout Circuits . . . . . 11

2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11

2.2 Capacitive Sensors . . . . . . . . . . . . . . . . . . . . . . . . . 12

2.3 Capacitance Readout Circuits . . . . . . . . . . . . . . . . . . . 17

2.3.1 Capacitance to Voltage Converter . . . . . . . . . . . . . 21

2.3.2 Capacitance to Duty Cycle Converter . . . . . . . . . . . 23

2.3.3 Capacitance to Phase Shift Converter . . . . . . . . . . . 26

2.3.4 Capacitance to Frequency Converter . . . . . . . . . . . 29

2.4 Comparison . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34

2.5 Justification . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37

2.6 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37

3 Weakly-Coupled-Resonators as Capacitance Readout Circuits . . . 39

3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39

3.2 Weakly Coupled Resonators . . . . . . . . . . . . . . . . . . . . 39

3.3 Reasons for Proposing WCRs as an Alternative for Readout Cir-

cuits . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44

3.4 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46

4 WCR-Based Readout Circuit Analysis and Performance Estimation 47

4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47

4.2 Theory of Operation . . . . . . . . . . . . . . . . . . . . . . . . 48

4.2.1 Analytical Solution . . . . . . . . . . . . . . . . . . . . 55

4.3 Simulations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65

4.4 Experimental Results . . . . . . . . . . . . . . . . . . . . . . . . 73

4.5 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77

vi

Table of Contents

5 Error Reduction in WCR-Based Capacitance Readout Circuits . . 79

5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79

5.2 Theory of Operation . . . . . . . . . . . . . . . . . . . . . . . . 83

5.2.1 Measurement Sensitivity . . . . . . . . . . . . . . . . . . 85

5.2.2 Measurement Error . . . . . . . . . . . . . . . . . . . . 95

5.3 Simulations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100

5.4 Experimental Results . . . . . . . . . . . . . . . . . . . . . . . . 102

5.5 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107

6 3DOF WCRs in Capacitance Measurement . . . . . . . . . . . . . . 108

6.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 108

6.2 Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109

6.2.1 Differential Perturbation Detailed Analysis . . . . . . . . 114

6.2.2 System Response to Common Mode Excitation . . . . . . 123

6.2.3 Differential Perturbation Analysis in Common Mode Ex-

citation . . . . . . . . . . . . . . . . . . . . . . . . . . . 125

6.3 Circuit Simulations . . . . . . . . . . . . . . . . . . . . . . . . . 132

6.3.1 Single-Sided Excitation, Differential Perturbation Case . 132

6.3.2 Differential Excitation, Differential Perturbation Case . . 137

6.3.3 Common-Mode Excitation, Differential Perturbation Case 138

6.4 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 142

7 Active Inductors in WCRs . . . . . . . . . . . . . . . . . . . . . . . 144

7.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 144

7.2 Real (Nonideal) Inductors . . . . . . . . . . . . . . . . . . . . . 144

7.3 Active Inductors . . . . . . . . . . . . . . . . . . . . . . . . . . 147

vii

Table of Contents

7.4 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 153

8 Conclusions and Further Discussions . . . . . . . . . . . . . . . . . 155

8.1 Research Contributions . . . . . . . . . . . . . . . . . . . . . . . 155

8.2 Prospects and Open Problems . . . . . . . . . . . . . . . . . . . 158

Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 160

Appendix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 171

Appendix A Circuit Simulations and Justification for Using CFC as the

Benchmark . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 171

A.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 171

A.2 CVC Simulation and Implementation Results . . . . . . . . . . . 171

A.3 CDC Simulation Results . . . . . . . . . . . . . . . . . . . . . . 176

A.4 CPC Simulation Results . . . . . . . . . . . . . . . . . . . . . . 179

A.5 CFC Simulation Results . . . . . . . . . . . . . . . . . . . . . . 185

viii

List of Tables

2.1 Capacitance readout circuit methods, a brief comparison. . . . . . 36

3.1 Analogy between mass-spring-damper and RLC coupled oscillators. 46

4.1 Analytical values for 2DOF WCRs at out-of-phase resonance. . . 64

4.2 Comparison table between ∆

∣∣∣ i2i1

∣∣∣and ∆ ff methods of measurement. 67

4.3 Experimental results for both eigenvalue and eigenvector based

methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75

5.1 Experimental results for all three methods. . . . . . . . . . . . . . 106

6.1 Measured values at fixed excitation frequency (mode 2). . . . . . 136

6.2 Simulation results for differential excitation at first mode. . . . . . 137

6.3 Simulation results for differential excitation at third mode. . . . . 137

6.4 Simulation results; common mode excitation at 2nd mode. . . . . 138

7.1 Gyrator-based WCR simulation results (mode 1 excitation). . . . . 153

A.1 CDC circuit simulation data. . . . . . . . . . . . . . . . . . . . . 178

A.2 CPC simulation results, ratio of voltage magnitudes vs. perturbation.184

A.3 CPC simulation results, ratio of voltage magnitudes vs. perturbation.186

ix

List of Tables

A.4 CFC simulation results, output frequency vs. perturbation. . . . . 191

x

List of Figures

1.1 Typical sensor and associated readout circuit. . . . . . . . . . . . 4

1.2 Loci of the dimensionless eigenvalues of the two-pendulum system. 6

1.3 An RLC series weakly coupled resonator system. . . . . . . . . . 8

2.1 Simple capacitor construction and schematic symbol. . . . . . . . 12

2.2 Different arrangements for capacitive sensors. . . . . . . . . . . . 13

2.3 Applications of capacitive sensors. . . . . . . . . . . . . . . . . . 15

2.4 Image of an accelerometer obtained with PolytecMSA−500r . . 16

2.5 Image of accelerometer designed at Georgia Institute of Technology. 17

2.6 Examples of capacitance-to-voltage (CVC) readout circuits. . . . 19

2.7 Differential CVC based on charge integration. . . . . . . . . . . . 22

2.8 An improved CVC readout, based on low duty cycle periodic reset. 23

2.9 Schematic of a CDC with direct configuration. . . . . . . . . . . . 24

2.10 Schematic representation of a CDC with the relaxation oscillator . 25

2.11 Phase shift generated using capacitance in an RC circuit. . . . . . 27

2.12 Phase shift plot for differential RC circuit. . . . . . . . . . . . . . 27

2.13 CPC using zero-crossing detection. . . . . . . . . . . . . . . . . . 29

2.14 CPC using analog multiplier. . . . . . . . . . . . . . . . . . . . . 29

2.15 CFC based on simple Hartley oscillator. . . . . . . . . . . . . . . 30

xi

List of Figures

2.16 Switched-capacitor harmonic oscillator with AGC . . . . . . . . . 31

2.17 CFC based on CVC cascaded with VFC. . . . . . . . . . . . . . . 32

2.18 CFC based on integration and periodic reset. . . . . . . . . . . . . 34

3.1 Lumped-element model of a coupled 2DOF mechanical system . . 41

3.2 Loci of the dimensionless eigenvalues of the two coupled oscillators. 41

3.3 SEM image of a set of coupled gold-foil cantilevers. . . . . . . . . 42

4.1 Two weekly coupled mechanical resonators. . . . . . . . . . . . . 49

4.2 2DOF weekly-coupled series RLC resonators. . . . . . . . . . . . 50

4.3 Two weekly coupled resonators natural frequencies loci. . . . . . 52

4.4 Mode localization in two weekly-coupled-resonators. . . . . . . . 54

4.5 Effect of loss on sensitivity. Coefficient r in (4.25). . . . . . . . . 63

4.6 Relative shift in resonant frequency vs. eigenmode in 2DOF WCRs. 65

4.7 Circuit schematic of 2DOF WCRs for SPICE simulations. . . . . 66

4.8 AC analysis of 2DOF WCRs based on series RLC resonators. . . . 66

4.9 i1 plots, coupled RLC circuit AC analysis with sweeping C2. . . . 68

4.10 i2 plots, coupled RLC circuit AC analysis with sweeping C2 . . . . 68

4.11 Resonant frequency loci veering in 2DOF WCR. . . . . . . . . . 69

4.12 Sensitivity comparison between three different methods. . . . . . 70

4.13 LabVIEW-Multisim co-simulation for 2DOF WCRs. . . . . . . . 71

4.14 LabVIEW-Multisim co-simulation results. . . . . . . . . . . . . . 72

4.15 High-level-block-diagram of proposed capacitance readout. . . . . 73

4.16 Test setup for experimental measurements. . . . . . . . . . . . . . 74

4.17 Sensitivity comparison between simulations and experiments. . . . 76

4.18 Effect of parasitic parameters on frequency response. . . . . . . . 77

xii

List of Figures

5.1 Series RLC two weakly coupled resonators. . . . . . . . . . . . . 79

5.2 Relative shift in resonant frequency vs. eigenmode . . . . . . . . 80

5.3 Bode Plot for Series RLC Resonator . . . . . . . . . . . . . . . . 82

5.4 System high-level-block-diagram. . . . . . . . . . . . . . . . . . 83

5.5 Examples of conventional capacitance measurement methods. . . 84

5.6 Eigenvalue loci veering. . . . . . . . . . . . . . . . . . . . . . . . 85

5.7 Frequency response of the system for three values of perturbation

δ =−0.1%, 0%and 0.1%. . . . . . . . . . . . . . . . . . . . . . 93

5.8 Error comparison and improvement by hybrid method. . . . . . . 96

5.9 Error comparison and improvement by hybrid method. . . . . . . 98

5.10 Amplitudes of I1 and I2 at out-of-phase resonance. . . . . . . . . . 100

5.11 Analytical: linear approximation vs. exact for |I1|/|I2|. . . . . . . 101

5.12 Analytical vs. simulation for |I1|/|I2|. . . . . . . . . . . . . . . . 102

5.13 |I1|/|I2| plot around out-of-phase resonant frequencies. . . . . . . 103

5.14 High-level-block-diagram of proposed capacitance readout. . . . . 104

5.15 Magnitude of v1/v2 around out-of-phase resonance. . . . . . . . . 105

5.16 Measurement error comparison. . . . . . . . . . . . . . . . . . . 106

6.1 3DOF coupled spring-mass system with stiffness perturbation. . . 109

6.2 3DOF weekly coupled series RLC resonators. . . . . . . . . . . . 111

6.3 Frequency shift of all three modes in one-sided perturbation. . . . 112

6.4 Frequency shift of all three modes in differential perturbation. . . 113

6.5 3DOF WCR schematic with differential excitation. . . . . . . . . 114

6.6 3DOF WCR schematic with common mode excitation. . . . . . . 115

6.7 3DOF WCR schematic with differential excitation. . . . . . . . . 115

xiii

List of Figures

6.8 Frequency response; unperturbed 3DOF WCRs under differential

excitation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 119

6.9 3DOF WCR schematic with common mode excitation. . . . . . . 120

6.10 Frequency response; unperturbed 3DOF WCRs under common mode

excitation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122

6.11 3DOF WCR, impact of loss on resonant frequencies. . . . . . . . 123

6.12 Phase plot of ε1 and ε2 at ω2 vs. Q factor. . . . . . . . . . . . . . 130

6.13 Magnitude plot of ε1 and ε2 at ω2 vs. Q factor. . . . . . . . . . . . 132

6.14 Three WCR veering from SPICE simulation. . . . . . . . . . . . 133

6.15 Three WCR, relative sensitivities, mode 1 excitation. . . . . . . . 134



6.18 3WCR, normalized current I2at 2nd mode. . . . . . . . . . . . . . 136

6.19 The effect of quality factor on f2- δ dependence. . . . . . . . . . 139

6.20 I2 magnitude for mode 2, common mode excitation, differential

perturbation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 140

6.21 Effect of Q factor on the sensitivity. . . . . . . . . . . . . . . . . 141

6.22 Effect of Q factor on the magnitude of ε2 (simulation). . . . . . . 142

7.1 Equivalent circuits for a real inductor (from CoilCraft Inc.). . . . . 145

7.2 An example of an active inductor. . . . . . . . . . . . . . . . . . 147

7.3 Realization of a floating inductor using gyrators. . . . . . . . . . . 149

7.4 Realization of a floating inductor using gyrators ,rearranged. . . . 150

7.5 Circuit used in simulation of 2DOF WCR based on active inductors. 151

7.6 Gyrator-based 2DOF WCR simulation at balance. . . . . . . . . . 152

xiv

List of Figures

7.7 Gyrator-based 2DOF WCR simulation for different perturbations. 152

7.8 Relative sensitivity of gyrator-based 2DOF WCR. . . . . . . . . . 153

A.1 CVC readout using charge integration, capacitor driving circuit. . 172

A.2 CVC readout using charge integration, input stage differential am-

plifier, filtration and demodulation. . . . . . . . . . . . . . . . . . 173

A.3 CVC readout using charge integration, output buffer and LP filter. 173

A.4 CVC based on differential charge amplifier, capacitance changes

and output voltage plots. . . . . . . . . . . . . . . . . . . . . . . 174

A.5 CVC based on differential charge amplifier, intermediate nodes

simulation waveforms . . . . . . . . . . . . . . . . . . . . . . . . 175

A.6 Varicap SPICE model. . . . . . . . . . . . . . . . . . . . . . . . 176

A.7 Schematic representation of a CDC with the relaxation oscillator. . 177

A.8 Simulation graph for the CDC readout circuit. . . . . . . . . . . . 178

A.9 Simulation results for the CDC readout circuit. . . . . . . . . . . 179

A.10 CPC readout circuit using charge amplifier. . . . . . . . . . . . . 181

A.11 Improved CPC readout circuit. . . . . . . . . . . . . . . . . . . . 182

A.12 Simulation results for the readout circuit. . . . . . . . . . . . . . . 183

A.13 CPC parametric sweep simulation results. . . . . . . . . . . . . . 184

A.14 CFC readout circuit based on Hartley oscillator. . . . . . . . . . . 185

A.15 Simulation results for the readout circuit. . . . . . . . . . . . . . . 186

A.16 CFC readout circuit based on switched-capacitors. . . . . . . . . . 189

A.17 Simulation results for the readout circuit based on SC. . . . . . . . 190

A.18 Simulation results for the readout circuit based on SC . . . . . . . 191

xv

List of Acronyms

ADC Analog-to-digital converter

ASIC Application specific integrated circuits

CDC Capacitance-to-duty cycle converter

CFC Capacitance-to-frequency converter

CPC Capacitance-to-phase shift converter

CVC Capacitance-to-voltage converter

DOF Degree-of-freedom

ESR Equivalent series resistance

IC Integrated circuit

MEMS Micro-electro-mechanical systems

OPA Operational amplifier

PCB Printed circuit board

PLL Phase-locked loop

VARACTOR Variable-reactance diode

xvi

List of Acronyms

VARICAP Variable capacitor, variable-capacitance diode

VFC Voltage-to-frequency converter

WCR Weakly-coupled-resonator

xvii

Acknowledgments

I would like to express my gratitude to my research supervisor Dr. Edmond Cretu

for his supervision and his professional guidance. He is indeed much more than an

academic supervisor and I would always remember his support and advice. I am

also very thankful of Dr. Shahriar Mirabbasi and Dr. Robert Rohling who helped

me a lot, especially at the beginning and initiation of my research at UBC.

I would like to thank all my Ph.D. examination committee members for their

time and valuable comments and feedback.

xviii

Dedication

To my wife,

my mother and

memory of my father

xix

Chapter 1

Introduction

1.1 History of Sensors

Human life is becoming more and more dependent on the measurement of physical

phenomena. The advancement in science owes a great deal to our ability to measure

the environment around us; as Lord Kelvin aptly puts it: “To measure is to know”.

In the course of history, the methods for measurement have advanced alongside

the advancements in science and engineering, resulted in the use of “sensors”. A

sensor, in its crudest form, is a tool that yields a certain electrical output when

exposed to the a physical phenomenon. This broad definition includes everything

from the first electric thermostat patented by Warren S. Johnson in 1885, to the

most advanced pressure sensors used in high performance cars today.

Sensors have become the default tool for us to measure the properties of our

physical surroundings, and with the growth in the number of their applications,

they have reached a market share of $79.5 billion in 2013, and are expected to

reach nearly $154.4 billion by 2020 [1]. Of course this explosive growth is helped

by the advancements in electronics and IC manufacturing capabilities, which began

from the invention of the transistor in 1947 at Bell Laboratories, and the first im-

plementation of the monolithic IC at Fairchild Semiconductor in 1959. This trend

1

1.1. History of Sensors

has continued to the present day by the introduction of increasingly sophisticated

and specialized sensors.

The reasons for this fast growth in reliance on sensors could be summarized as

follows [2].

• Sensors have an electrical output, which is the most versatile form of signal

carrier that can be used for processing and storing sensor related information.

• Many different back-end circuitry options are available for use with the sen-

sors, resulting in the ability to manufacture the sensor and the signal condi-

tioning subsystem in the same package.

• Given that the output of the sensors can be amplified, there is the possibility

of using active sensors, which do not absorb energy from the process being

measured.

• Sensors can be designed to measure nonelectric quantities through the usage

of appropriate material and techniques (changes in the properties of nonelec-

tric material can be translated to electrical changes, which can be detected in

the electrical domain.

• The sensors output can be displayed, recorded and further processed, to pro-

vide more insight into the nature of the variations of the process being mea-

sured.

The need for measuring different types of physical quantities has led to the devel-

opment of many different sensor types, each of which has its own unique character-

istics. Sensors can be categorized based on their need for power (active or passive),

their output signals (analog or digital), or their mode of operation, e.g. deflection

2

1.1. History of Sensors

type or null type. However, in electronic engineering, it is preferred to categorize

sensors based on the measured electrical quantity (e.g. resistance, capacitance, and

inductance) [2].

Resistive sensors are widely used in measurement applications since one of the

simplest way of mapping the measurand onto electrical variations is through the

equivalent electrical resistance modulation. The outputs of the resistive sensors

are readily available for processing, hence these sensors have simple measurement

circuitry. Also, resistive sensors offer many options with regards to their size,

resistance value, back-end circuitry and AC/DC operation [3]. Resistive sensors

have a high sensitivity in general; however, their resolution is affected by thermal

noise, which means that various environment related factors will influence their

output [3].

Inductive sensors rely on the change of self or mutual inductance of a coil or

set of coils for measurement. These sensors can be used in applications where the

thickness of objects needs to be measured. The detection of change in inductive

sensors can be done only by using AC readout circuits. Because of the effect

of inductance on the neighboring circuitry, proper shielding is desirable for air

cored inductors [3]. This aspect, coupled with the direct relationship between the

physical dimensions of the coil and the quality factor (i.e. higher quality factor

coil needs a lower equivalent series resistance, and consequently require the larger

cross section of winding wire for the same number of turns) , means that these

sensors are usually bulkier than other passive sensor types.

Capacitive sensors are widely used for displacement measurement [3]. Because

of their precise performance, low cost construction, simple structure and versatility,

they are common solutions for measuring variables such as acceleration, humidity,

3

1.2. Readout Circuits

liquid levels etc. Capacitive sensors work on the principle of measuring the capac-

itance between two or more conductors in a dielectric environment [3, 4]. Because

of their desirable characteristics, more emphasis has been placed on the capacitive

sensors in this thesis.

1.2 Readout Circuits

Generally speaking, nearly all sensors are coupled to with an electrical subsystem

in order to measure their respective electrical output. This electronic interface is

known as the readout circuit. The readout circuits are as diverse as the sensors

themselves, but the primary task of all of them is signal conditioning, which in the

IEEE standard 1451.4-2004 is defined as processing of a sensor output signal with

operations such as amplification, compensation, filtering and normalization [5].

Figure 1.1: Block diagram of a typical sensor and the associated readout circuit.

As can be seen in Figure 1.1, the readout circuit is the interface between the

sensor and the rest of the system. It performs signal conditioning tasks with the

power received from the power supply. This thesis specifically considers readout

circuits used for capacitive sensors. The existing capacitance readout circuits use

4

1.3. Motivation

various methods, such as capacitance-to-voltage (CVC), capacitance-to-frequency

(CFC), capacitance-to-phase shift (CPC), capacitance-to-duty cycle (CDC) conver-

sions etc. All of the aforementioned methods utilize analog circuitry for filtering,

amplification and even switching.

There are many challenges for the existing capacitance readout circuit tech-

niques. Typically, factors such as intrinsic noise, switching noise (in circuits based

on switching elements), offset problems and temperature dependency of the cir-

cuit components result in a relatively high number of passive and active circuit

components. In addition, measuring small variations in capacitance (the measured

parameter), is often disturbed by the inherent presence of parasitic capacitances,

which could be even larger than the sensing capacitance. Capacitive MEMS sen-

sors have a typical capacitance in the range of 0.2pF to 1pF, parasitic capacitance

of about 2pF and typical resolution range of 1aF to 10aF. Achieving a high sensitiv-

ity/gain in measurement with a low signal to noise ratio has always been a serious

challenge.

In some applications the gain-bandwidth trade-off becomes another challenge;

the higher the rate of the sensor capacitance changes, the lower the overall gain of

the readout circuit. This tradeoff is not of a huge concern in this project since the

capacitance variation is considered to be quasi-static (that is, very slow relative to

the time constants of the readout circuits).

1.3 Motivation

In order to address some of the aforementioned challenges, especially the high sen-

sitivity and robustness, we searched for an alternate and innovative method with

5

1.3. Motivation

higher sensitivity and inherent simplicity. There has been a very elegant method

for measuring perturbations that has been used in mechanical domain for decades.

This method is based on weakly-coupled resonators (WCRs), and has a long his-

tory in mechanical and acoustic domain. WCRs exhibit an interesting feature re-

lated to the mode localization (energy localization), which is the energy repartition

between the two resonators due to perturbation. Mode localization was examined

in solid state physics applications by P. W. Anderson for the first time [6, 7] which

eventually led him win the Noble prize in physics in 1977.

The behavior of the resonant frequencies as functions of perturbation and the

coupling strengths between the resonators, when plotted, gives two sharply veer-

ing traces with high local curvatures. This behavior, shown in Fig. 1.2, was first

investigated by Pierre [8] who named it loci veering.

Figure 1.2: Loci of the dimensionless eigenvalues of the two-pendulum system interms of the disorder, ∆l; representative mode shapes are shown. (a) The stronginterpendulum coupling case, R = 0.5; neither eigenvalue loci veering nor modelocalization occur. (b) The weak interpendulum coupling case, R =0.025; bothcurve veering and strong localization occur.

An alternative, more intuitive, representation of veering phenomenon is pre-

6

1.3. Motivation

sented in Fig 4.4. It is generally accepted that the eigenvalues of the WCRs system

represent the resonant frequencies of the system. As a result the term eigenvalue

loci veering phenomena has been used to describe a range of similar behaviors in

disordered structures in the mechanical and MEMS field [9, 10, 11, 12, 13, 14, 15].

This thesis offers an innovative method for capacitance measurement based

on weakly-coupled resonators (WCRs), which is proven to have more sensitivity

and circuit simplicity. It will be shown that WCR-based readout circuit can reach

several orders of magnitude higher sensitivity than other state-of-the art methods

(e.g. capacitance-to-frequency method). On the other hand, there is a challenge in

matching component values for both resonators. The higher the sensitivity of WCR

method, the more the negative impact of mismatch on the correct reading. Another

challenge with WCRs is the bandwidth of the perturbation since the theory assumes

implicitly a quasi-static perturbation. In this thesis we also study the effect of the

losses on the sensitivity of the system, which has not been offered in the previous

mechanical/MEMS researches.

The simplest WCR in the electrical domain consists of two series/parallel RLC

circuits coupled through a capacitor /inductor. This thesis proposes to use the

WCRs as an alternative for a capacitance readout circuit. Figure 2 shows the con-

figuration of the WCR fundamental circuit examined in this thesis. Using such an

arrangement for capacitance readout circuits results in lower number of compo-

nents required, contributing to the low cost, low power and high reliability of these

circuits. Moreover, as will be shown in the following chapters, the relative sensi-

tivity of a WCR arrangement is much higher than the existing comparable readout

circuit methodologies.

7

1.4. Thesis Outlines

RLC

CC

Vs

R L C

Figure 1.3: An RLC series weakly coupled resonator system.

1.4 Thesis Outlines

With the above mentioned information in mind, the main contributions of this the-

sis include: the use of WCRs for capacitive measurement (the first application

of WCRs principles in the electrical domain in this direction); achieving a much

higher measurement sensitivity compared to the existing capacitance readout cir-

cuit methodologies; proposing a method for minimizing the susceptibility of the

readout circuit to the excitation frequency errors; utilizing three WCRs for differ-

ential capacitive measurements, yielding thus a lower dependance on the excitation

frequency and consequently a more robust readout circuit; and finally examining

the possibility of using active inductors in a WCR arrangement for a capacitance

readout circuit (with the potential of future single-die integration of the capacitance

readout technique).

The next chapters are structured as follows. Chapter 2 presents a representa-

tive, but by no means exhaustive, literature review of the state-of-the-art in sensor

readout circuit technology. This will include overviews of various types of sensors

and different methodologies used for readout circuits, narrowing down to the read-

out circuits used for capacitance measurement. A justification of choosing CFC

8


method as a reference for comparison with our proposed WCR-based method is

presented at the end of this chapter. The details of various state-of-the-art readout

circuits mentioned in this chapter, along with simulation results, are presented in

Appendix A.

Chapter 3 begins by giving a more detailed and historical introduction to WCRs,

and enumerating their various conventional uses. It then continues by formally

proposing the use of WCRs as the alternative method for capacitance readout cir-

cuits. The justification for such a proposal is given and finally the reader is pre-

sented with the research question.

Chapter 4 presents the theoretical analysis and simulation results examining

the use of the WCR methodology for the readout circuit. This is then followed by

the sensitivity analysis as well as the simulation and practical circuit implementa-

tion results. These results are then compared with the conventional CFC method,

showing the full extent of the sensitivity improvement.

Chapter 5 examines the capacitance measurement error problem for the WCR

methodology. It then proposes a method to minimize the measurement error, by

using a combination of the CFC and WCR methods, resulting in a more robust

readout circuit.

Chapter 6 proposes the use of three-degree-of-freedom WCRs in the readout

circuit to perform differential capacitive measurement in a robust manner. This

chapter also studies the effect of losses (quality factor) on the sensitivity. It shows

the trade-offs between quality factor (Q), dynamic range of measurable perturba-

tion and sensitivity. The analytical and simulation results are provided and com-

pared for such an arrangement.

Chapter 7 explores the use of an active inductor (in the form of an op-amp-

9


based circuit) as alternatives for bulky passive inductors in implementing of WCR

methodology by theoretical analysis and simulation. This is helpful toward inte-

grating a complete WCR-based readout circuit on a chip. This chapter is followed

by final discussions, and outlining further avenues of research for the future in

chapter 8.

10

Chapter 2

Capacitive Sensors and Their

Associated Readout Circuits

2.1 Introduction

This chapter introduces capacitive sensors and their associated readout circuits.

The fundamentals of capacitive sensing are presented in §2.2, where various ca-

pacitive sensor configurations are depicted together with different ways of cate-

gorizing such. In addition , the benefits and limitations of capacitive sensors are

examined in detail.

Section 2.3 begins by defining what readout circuits are and different categories

they fall into. Subsequent subsections are then devoted to examining each of the

methods with more in-depth explanations for various configuration where neces-

sary. The related simulations are presented in Appendix A. This chapter continues

with a justification for choosing one of the capacitance readout circuit methodolo-

gies as a benchmark for comparison with our proposed WCR method. A summary

of this chapter is presented in the last section.

11

2.2. Capacitive Sensors

2.2 Capacitive Sensors

The past decades have seen a burgeoning attention to the use of capacitive sensors

for sensing and detecting physical quantities such as pressure, rotational angles,

linear displacement and acceleration [2]. As their name suggests, capacitive sen-

sors rely on a capacitance change in order to measure the desired quantity.

A capacitor in its simplest form consists of two conductive plates, separated by

a dielectric, as shown in Fig 2.1.

d

d

y

x

(a) (b)Figure 2.1: Simple capacitor, (a) construction, (b) schematic symbol and electricfield.

The distance between the plates, the plate overlapping area and the dielectric

substance are the critical parameters in any capacitor, determining the capacitance

value.

If we consider the parallel-plate capacitor model and neglect the fringe field

effects, the capacitance can be calculated using

C = ε0εrAd, (2.1)

12


where C is the capacitance, ε0 = 8.85 pF/m is the dielectric constant for vacuum,

εr is the relative dielectric constant, A = xy is the overlapping plate area, and d is

the distance between the plates.

Various arrangements for capacitors used in capacitive sensors are shown in

Fig 2.2.

(a) (b) (c)

(d)

z

d

d

C 1

C 2

(e)

C 1 C 2

d

z

z0

(f)

Figure 2.2: Different arrangements for capacitive sensors based on: (a,b) variationof area, (c) variation of gap between plates, (d) dielectric change, (e) differentialvariation in the gap, and ( f ) a differential variation in the area.

Using variable capacitors as sensors poses some difficulties. One of the first

problems with regards to such usage is the fringe effect present in parallel plate

capacitors. Although fringe effects are considered negligible in many instances,

this is only acceptable when the distance between the plates is far smaller than the

13


size of the plates.

Additionally there needs to be appropriate shielding for the capacitive sensor

plates and the wires connected to them, to reduce capacitive interference. However,

shielding wires to prevent capacitive interference introduces a new capacitance in

parallel with the sensor (parasitic capacitance). This in turn results in a loss of

sensitivity, as the change in the sensor capacitance only changes a part of the overall

capacitance. Also, relative movement between the wires and the dielectric could

introduce errors, caused by changes in the capacitor geometry.

Another important matter is the quality of the dielectric used in the capacitor.

There should be a constant and high electrical insulation between the plates. If

the insulation is poor, then there will be a leakage resistance in parallel with the

capacitor that affects the overall capacitance. This results in the impedance being

affected by a factor other than the capacitance, which renders the measurement

methods ineffective and prone to errors. Dielectrics with high conductivity (such

as water) could be affected by thermal interference generated because of the power

passing through their effective resistance and generating heat.

In general, the capacitive sensors are categorized into variable capacitors and

differential capacitors. In variable capacitive sensors one or more of the above

mentioned parameters change based on the measured phenomenon, where as in

differential capacitive sensors, the values of two capacitors simultaneously change

in opposite directions by the physical variable to be measured.

Despite the limitations mentioned above, capacitive sensors enjoy several ad-

vantages including: low power consumption, wide operating temperature range,

value dependency mainly on the geometry and less on the material properties, high

resolution and easy for fabrication.

14


As a result, capacitive sensors have a wide variety of applications, including

but not limited to the measurement of displacement, force, pressure, acceleration,

angular velocity. Moreover, the recent rapid growth in human/machine interface

has given rise to the application in touch screens in many personal communication

devices, such as mobile phones and tablets. Another area of interest is in medical

instrumentation, where accurate measurement of signals from the patients body is

of great importance. Fig. 2.3 shows some of these applications, e.g. measuring

inertia in aviation, tilt and inclination in dams, trains and off-shore platforms, and

seismic and vibration in highrises and race cars [16].

Figure 2.3: Applications of capacitive sensors (e.g. accelerometers and gyrosin navigation, aviation, race car data acquisition, oil and gas, seismic) and someMEMS based capacitive sensors fabrication [16].

15


This figure also shows some samples of capacitive sensors designed by Coli-

brys.

One area of recent advancement in capacitive sensing is designing sensors

based on micro-electro-mechanical systems (MEMS). Although MEMS are not

investigated in detail in this thesis, a capacitive interface is the common configura-

tion for them, due to better power efficiency and increased sensitivity [17]. Fig. 2.4

shows an image of a MEMS capacitive accelerometer designed by Dr. Elie Sarraf

at University of British Columbia (UBC) [18].

Figure 2.4: Image of IMOMBCEHS0903 accelerometer obtained withPolytecMSA−500r.

This capacitive sensor has two sets of differential capacitors, one gap varying

and one area varying. The typical value for the gap-varying capacitance in this

design is about 2 pF with a dynamic range of -10g to 10g, a noise floor of approx-

imately 4 µg/√

Hz, and a gain of 23.4 mV/g for the whole system including the

related readout circuit.

16

2.3. Capacitance Readout Circuits

Another example of such sensors is shown in Fig. 2.5. The plates that form the

differential capacitors are also shown on the image [19].

Figure 2.5: Image of accelerometer designed at Georgia Institute of Technology[19].

No matter where the capacitive sensors are used, or the amount of capacitance

change they create, there is a need for a mechanism to measure this change and

translate it into useable output. In literature this mechanism is known as a readout

circuit. The next section introduces the readout circuits and their various types in

more detail.

2.3 Capacitance Readout Circuits

As mentioned in the previous section, all sensors, including capacitive ones, rely

on a mechanism to measure a physical variable (measurand) and translate it to a

suitable signal to be used for further processing through filtering and amplification.

This mechanism, known as the readout circuit, is a broad topic by itself; as there are

as many readout circuits as there are sensors themselves. This section of the thesis

examines capacitance readout circuits in more detail categorizing them based on

their circuit configuration, time sampling and feedback.

17


For either variable capacitance or differential capacitance types of sensors,

there are many types of readout circuits available. Some of the most common

are presented in Fig. 2.6[2]. Generally, a readout circuit designed for a differen-

tial capacitive sensor can also be used for a single variable capacitive sensor. The

most common way is to replace of the sensor capacitors with a fixed capacitor

(usually called reference capacitor). Differential readout circuits typically utilize

the difference between the two amplifier outputs in their circuit configuration. Fig.

2.6 presents examples of simple single-ended and differential readout circuits. Fig.

2.6(a) is a simple charge amplifier. Cx is the sensor capacitor and C f is the feedback

capacitor. R f provides the DC bias current for the operational amplifier (op-amp)

input stage. Assuming Cx is an area-varying capacitive sensor with parameter x as

the ratio of the change in the area:

Cx = ε0εrA0(1+ x)

d

and

Vo =−Cx

C fVs

which shows the output voltage is proportional to the capacitance changes. It

was assumed that R f is large enough to have a negligible effect in the output voltage

for the bandwidth of interest. In case a gap-varying capacitive sensor was used, it

would be helpful to swap Cx and C f to get a linear relationship between Vo and x,

the gap-variation measurand.

18


Vs

Cx

Cf

Rf

Vo

Vs

Cx1

Cf

Rf

Vo

¡Vs

Cx2

Vs

Cx

Cf

Rf

Vo

R1

R2

Cx2

C4

R

Vo

Cx1

C3

R

R

R

Vs

R

Vo

R R

R

Vs

Z1

Z3

Z2

Z4

(a) (b) (c)

(d) (e)

Figure 2.6: Examples of capacitance-to-voltage converter (CVC) readout circuits.(a) single-ended charge amplifier for single variable capacitor (b) single-endedcharge amplifier for differential capacitive sensor. (c) bridge amplifier for sin-gle variable capacitive sensor. (d) differential amplifier for differential capacitivesensor. (e) instrumentation amplifier for differential capacitive sensor.

The readout circuit shown in Fig. 2.6(b) is very similar to the circuit of Fig.

2.6(a). It is rearranged to accommodate for a differential capacitive sensor. Figure

2.6(c) is a pseudo-bridge version of Fig. 2.6(a). The output of the circuit can be

written as [2]:

Vo =VsR1/R2−Z3/Z1

1+R1/R2,

where Z1and Z3 are the total impedances of Cx and C f || R f , respectively.

19


The configurations of Fig. 2.6(d) and (e) are based on pseudo-bridges and

differential amplifiers. Figure 2.6(e) has an additional stage to convert the output

to single-ended, a stage known as instrumentation amplifier. The output voltage

for these differential configurations is:

Vo =Vs

(Z3

Z1− Z4

Z2

).

Differential readout circuits are less susceptible to common mode and power

supply noises; they typically have larger input signal range, and generally a bet-

ter resolution. Moreover, differential readout circuits have a larger common mode

rejection ratio (CMRR), which makes them more desirable. On the other hand,

compared to single-ended readout circuits, differential readout circuits consume

more current. This is because single-ended readout circuits normally use mini-

mum size and number of transistors. Also, due to the need for more components,

differential readout circuits require a larger silicon footprint in application specific

integrated circuits (ASIC) or on the printed circuit boards (in case of discrete circuit

implementation).

In terms of timing, there are two types of designs available for readout circuits,

continuous and discrete time. Intrinsically, the continuous time design has a higher

resolution since it does not suffer from sampling noise[20]. Discrete time readout

circuits are a better option when dealing with larger resistances, for instance in the

feedback loops. Switched-Capacitor circuits are one way of implementation based

on discrete time operation.

Examining the feedback structure of the sensor and corresponding readout cir-

cuits, two types of structures are present, open-loop and closed-loop. In the open-

20


loop configuration the capacitance change from the sensor is amplified and turned

into a usable signal by the readout circuit and the resulting output is then trans-

ferred to subsequent data displays. The closed-loop configuration exploits a sec-

ondary input to the sensor which mimics the magnitude of the primary capacitance

changed caused by the measured phenomenon. This secondary input is essentially

a negative feedback set by the readout circuit designed to keep the actual value

of capacitance in equilibrium. The true capacitance change can be measured by

monitoring the negative feedback line.

As mentioned earlier, many different methods have been proposed and devel-

oped for capacitance readout circuits. Each of these methods rely on measuring

a parameter change with respect to the capacitance, e.g. voltage, phase shift, fre-

quency etc.

2.3.1 Capacitance to Voltage Converter

The first method examined in this thesis is the capacitance to voltage converter

(CVC), where the capacitance changes are translated into a change in voltage for

further processing [20, 21, 22, 23, 24, 25]. The CVC method on its own has many

different configuration, which are briefly mentioned below.

• CVC method using charge integration, which can be used for both single

and differential capacitance. In this configuration, the capacitance change is

converted to a corresponding electrical charge variation, that is afterwards

converted into a change in voltage using an op-amp. A schematic diagram

of this circuit can be seen in Fig. 2.7 below [21].

21


C 0

f

R0

f Vo

Cx0 +¢Cx

Cf

Rf

Vc

C 0

x0¡¢C 0

x

VB

VC

VD

HINA

D

D0

CD

C 0

D

RD

R0

D

Carrier Sensor CurrentDetector

AMDemodulator

Inst:Amp:

AM ¡Modulator

VA

Figure 2.7: Differential CVC based on charge integration [21].

This method is less susceptible to parasitic capacitance, however it needs a

large value feedback resistor which could be difficult to implement on an IC [20].

The bandwidth of the capacitor in this configuration is from DC to 10 KHz and a

reported resolution of 24 aF is measured if a 12 pF capacitor is used [21].

• CVC method using low duty cycle periodic reset configuration is another

sub category of the CVC method which enjoys low noise, linear capacitance

to voltage transfer function, and low susceptibility to system offset. This

configuration has reached a 0.06% resolution for a 0.8 pF capacitor [20]. A

schematic diagram of this configuration is presented in Fig. 2.8.

22


Cx1

Cf

S1

VoCx2

VDD

S2

S1

S 0

2S 0

1

Cp

Reset Sensing

S1, S0

1

S2, S0

2

Figure 2.8: An improved CVC readout, based on low duty cycle periodic reset[20].

• Other configurations designed to reduce offset and increase resolution, two

of which are chopper stabilized configuration, which is shown to reduce the

input offset effects [22] and a ratio-arm bridge which is a symmetrical and

sensitive circuit, but requires transformer coils [23].

2.3.2 Capacitance to Duty Cycle Converter

In the capacitance to duty cycle converter (CDC) method, the changes in capaci-

tance are translated into changes in the duty cycle of a pulse train. This method has

two main configurations that are listed below.

• CDC method using direct configuration is named as such because there is a

23


direct relationship between the capacitance and the duty cycle. To explain

this further, a CDC readout circuit using direct configuration is presented in

Fig. 2.9.

Vref Vref

VEE

VCC

Cref

CxQ1 Q2

R2

R1

R3

R4 R5

R6

R7

R8

Vthreshold

Vcomp:

V

Figure 2.9: Schematic of a CDC with direct configuration.

The timing of the the duty cycle can be expressed by

T = R(Cx− (a−1)Cr) ln(

Vre f

Vre f −Vth

), (2.2)

where T is the on time of the duty cycle, R is the load resistance, Cx is the

measured capacitance, a= 1+R5/R4 , Cr is the reference capacitance, Vre f is

the reference voltage, and Vth (Vthreshold) is the threshold voltage. It is evident

from (2.2) that there is a direct relationship between T and Cx. The direct

configuration enjoys simplicity, lower power consumption (because of the

24


smaller number of components), and easy linearization on the digital side.

If a modern low voltage/power CMOS implementation is used, then this

configuration achieved a bandwidth of 1 KHz with a 13 bit resolution. The

resolution and bandwidth were limited by the speed of the op-amp, which

has a 3 MHz maximum gain bandwidth and a 13 V/µs slew rate [26].

• CDC method using relaxation-oscillator configuration uses two capacitors

for sensing. These capacitors are multiplexed by diode switches to form an

op-amp based integrator. A schematic of this configuration is presented in

Fig. 2.10.

V0

B

A

C

C1

C2

V5

V3

Rt

R1

R2

D1D2

V2

R3

A1 A2

A3

A4

V4

Figure 2.10: Schematic representation of a CDC with the relaxation oscillator .

The interface presented in Fig. 2.10 detects the ratio of capacitances in the

form of the duty ratio.

D =TH

TH +TL=

C2

C1 +C2,

25


where D is the duty cycle of the signal at V5 port.

This configuration allows for high speed measurements as reported in the lit-

erature. In one test case, a resolution of 60 aF was achieved using a 30 MHz

oscillation frequency with a reference capacitance of 3 pF [27].

2.3.3 Capacitance to Phase Shift Converter

Capacitance as a reactive component creates a phase shift between voltage and

current in a circuit. Assuming the rest of the circuit parameters and values are

constant, the phase shift, in reference to the input voltage, is a function of the

capacitance. As an example in a simple RC circuit shown in Fig. 2.11(a), the

phase is:

∠vo = φ =−arctan(1

ωRCx).

Typically in capacitance measurement, there is a reference capacitor, Cr, which is

the reference for the changes of the sensor capacitance Cx. One of the common

choices for Cr is the value of Cx at rest. An example of this configuration is shown

in Fig. 2.11(b). The phase of the differential voltage at the output, vo, is represented

by:

∠vo = φ = arctan(1

ωRCr)− arctan(

1ωRCx

)≈− ∆C/Cr

ωRCr +1

ωRCr

,

where ∆C =Cx−Cr. This approximation is true when Cx and Cr are close enough

i.e. φ < 6.

26


Cx

R

Cr

Cx

R

R

A sin(!t) A sin(!t)

A sin(!t)

vo

vo = B sin(!t + Á)

Á = ¡ arctan( 1

!RCx

)

(a) (b)

\vo = Á ¼ ¡¢C

!RC2

r

+1

!R

Figure 2.11: Phase shift generated using capacitance in an RC circuit. (a) single-ended. (b) differential.

This phase difference for the circuit in 2.11(b) is plotted in Fig. 2.12 for the

values of Cr = 100nF, R = 10kΩ and five different values of Cx ∈ 80nF, 90nF,

100nF, 110nF, 120nF.

1 10 100 1k 10k 100k-6

-4

-2

0

2

4

6

Pha

se d

iffer

ence

(o )

Frequency (Hz)

Cx

80nF 90nF 100nF 110nF 120nF

Figure 2.12: Phase shift plot for differential RC circuit.

27


The phase difference is almost linear at around -5 dB to -15 dB frequencies (10

Hz to 80 Hz); e.g. the phase shift is approximately 1 per 10 nF of capacitance

change (10% change in capacitance) at around 30 Hz. The common challenge in

capacitance measurement based on the phase shift is the nonlinearity introduced

by arctan function.

There are many different methods for measuring the phase of a signal, or the

phase difference between two signals [28], e.g. direct oscilloscope method, zero-

crossing, three-voltmeter, phase-locked-loops (PLLs), Fourier transform etc. An

example of high level schematic diagram for a capacitance-to-phase-shift converter

(CPC) based on zero-crossing detection is shown in Fig. 2.13. In this circuit, zero

crossings of the signals passing through the sensor and reference capacitors, which

represent the phase of the signal, are detected using comparators. The square wave

at the output of the comparators are used to set or reset the output of a R-S flip-flop.

so the duty cycle of the flip-flop output is proportional to the phase shift between

two signals. Another circuit based on modulation and demodulation is shown in

Fig. 2.14. In this method, the capacitance changes are modulated by the carrier

signal. A multiplier is made of two logarithmic amplifier and an analog summa-

tion, followed by an anti-log amplifier. The output of the anti-log amplifier has

two frequency components. The high frequency components is eliminated by the

low-pass filter (integrator) block. The low frequency component, which contains

the information regarding the phase difference, passes through the integrator. This

phase difference has a one-to-one relationship with the difference in the capaci-

tances (Cx−Cr).

28


Non¡ invertingZero¡ crossingdetector

R

S

Integrator

BufferQVs

Vo

InvertingZero¡ crossingdetector

Cr

Cx

R

R

R1

R1

C

C

D1

D1

A

B

C

D

Q

Figure 2.13: CPC using zero-crossing detection.

LogAmplifier

Integrator

BufferVo

LogAmplifier

Cr

Cx

R

R

V0

sin(!t)

V0

cos(!t)

¡

+

Anti¡ LogAmplifierR

1

R1

R2

Figure 2.14: CPC using analog multiplier.

2.3.4 Capacitance to Frequency Converter

The last readout circuit design method introduced in this section converts the change

in capacitance to frequency, and it is known as capacitance to frequency converter

(CFC). The main distinguishing factor with regards to the CFC method is that it

generally does not need an analog-to-digital converter (ADC), since a simple zero

crossing counter can be used. Fig. 2.15 shows a schematic of a simple CFC read-

out circuit based on a Hartley oscillator. The oscillation frequency is a function of

the capacitance CL with the equation fosc = 1/(2π√

C2LT), where LT = L1 +L2 is

29


the equivalent tank inductance [29].

VCC

R1 R2

R3 R4R5

R6L1 L2

C2

C1

C3

Q1

Figure 2.15: CFC based on simple Hartley oscillator.

Another example based on switched-capacitor oscillator is, illustrated in Fig.

2.16. This CFC design, presented in 1985 [30] , enjoys low complexity, as it does

not need an ADC. The design was based on implementation of a quadrature os-

cillator (two integrated loop circuit) using switched capacitors. The relationship

between the frequency of the oscillation and the capacitor to be measured is:

f0 =Cm

Cfc

2π,

where C is the integrating capacitor shown in Fig. 2.16, Cm , αC is the capacitor

to be measured, and fc is the clock frequency of the SC circuit.

30


¡

+

¡

+

fc fc fc fc

Control

Pulse

Generator

Zero

Crossing

Detector

S/H§

Vref

k²C

®C ®C

C CV1

V2

1=f0

V+

¡

Figure 2.16: Switched-capacitor harmonic oscillator with AGC .

CFC has been applied in many readout circuits for different application re-

quirements and variety of implementation technologies. We are going to briefly

point to some of these applications, without going to the details, to show the broad

usage of CFC method in the literature.

A design based on the relaxation oscillator was presented for a capacitive dig-

ital hygrometer in 1995 [31] but no comparison with other contribution was pre-

sented. The design presented in [32] has the advantage of making the frequency

independent of the power supply in a wide dynamic range. The next contribution,

presented in 1991, is also based on switched capacitors. It has two main stages of

CVC (based on switched-capacitor) followed by a voltage-to-frequency converter

(VFC). It has low power, low cost, and linear capacitance to frequency transform

characteristics [33]. A simplified schematic of the circuit is shown in Fig. 2.17.

The CVC circuit based on SC is shown on the left. VR1 and VR2 are constant refer-

ence voltages. CR and CX are reference and measurand capacitors, respectively. VC

is the output of the CVC stage. The VFC circuit schematic is shown on the right

31


side. The input is VC and the output is VO.

Figure 2.17: CFC based on CVC cascaded with VFC.

The relationship between VC and CX , for the CVC section, is given by:

VC =(CX −CR)

CF(VR2−VR1)+VR1 (2.3)

The relationship between the output frequency fo and VC is:

fo =(C1/C2)(VC−VR1)

VR2−VR1

fC2

(2.4)

where fC is the clock, Φ1 or Φ2 frequency.

The contribution presented in [34] improves the solutions presented in [31, 33,

30] by introducing a digital compensation system. This also uses a CVC followed

by a VFC . This solution boasts low complexity, eliminates the need for an ADC,

and increases linearity. The same authors use the same solutions with some minor

changes in [35] and [36].

A combination of CVC and CFC is used in a humidity and accelerometer sen-

sor presented in the literature [29]. The CFC part uses a Hartley oscillator with

a feedback loop. A comparative study presented in [37] improves the previous

32


works [38, 33, 34] to propose a solution, which not only offers better performance

on frequency to code conversion, but has better electrical characteristics, wide in-

put spectrum range, and a wide high frequency dynamic range. The proposed

design is based on the relaxation oscillator. In another contribution presented in

[39], repetitive charge integration and charge conservation is used to combine both

the CVC and VFC into a single CFC that requires only one op-amp. This design

converts the difference between the capacitance values to an output frequency by

the repeated charge integration method.

The study presented in [40] improves on the methods presented in [36, 39, 41]

to get a more accurate and wider frequency range by saving and accumulating

the residual charges. A more recent study based on relaxation oscillator presents

an active bridge where the frequency is linearly related to capacitive imbalance

[42]. A recent paper on CFCs only presents simulation results, which indicate high

temperature (up to 175C), excellent stability over a wide temperature range and

good accuracy and resolution while not using a complex ADC [43]. The simple

principle of the circuit is based on integration, comparison and periodic reset. A

simplified schematic of this circuit is shown in Fig. 2.18. TG is a transmission gate

which discharges the sensing capacitor CS. The negative and positive inputs of

the operational amplifier (OPA) are biased through constant current I and constant

voltage VWE , respectively.

33

2.4. Comparison

Figure 2.18: CFC based on integration and periodic reset [43].

When Vint , which is the integral of current I offsetted by VWE , becomes greater

than threshold voltage Vth, a one-shot pulse gets generated which in turn discharges

the capacitor CS and resets the output at the same time. The frequency of the one-

shot output pulses are related to the capacitor value by:

f ≈ ICS(Vth−VWE)

(2.5)

The common point about the studies presented above are that they are mainly

geared towards IC design; however, this thesis is focused on more fundamental

circuit theory matters. As a result, the review of CFC designs presented above is

performed more for the purpose of completeness, not for a side by side comparison.

2.4 Comparison

Capacitive sensors, and specifically capacitive-based micro-electro-mechanical sys-

tems (MEMS), have more widespread use in comparison to their piezoelectric

and piezoresistive counterparts, due to larger temperature operating ranges, lower

34

2.4. Comparison

power consumptions and good resolution [17]. Both single ended and differential

capacitive sensing configurations are being commonly used. Nevertheless, design-

ing a reliable and accurate capacitance readout circuit is challenging, especially

for capacitive sensing of the displacement in MEMS structures that require small

structural size and hence very small capacitor values and their relative changes.

For instance, present inertial MEMS sensors require small bandwidths (50 - 100

Hz) with resolutions often reaching aF levels for nominal capacitance in the order

of 0.1 - 2 pF [44]. These small sensing capacitors in the presence of parasitic ca-

pacitance, which is in pF range, along with the interconnect resistance will limit

the measurement resolution and bandwidth of the readout circuit.

The more complex the readout circuit, the larger the risk of introducing para-

sitic elements, leading to a deterioration of the overall sensing performance. This is

valid for custom system-in-a-package capacitive sensing solutions, but even more

for discrete readout circuit alternatives. The need for complex solutions appears in

the context of required added features, e.g. self-calibration, temperature compen-

sation, self-testing and analog-to-digital conversion. Many different approaches

and methods have been introduced for high sensitivity capacitance readout circuits:

capacitance-to-voltage converters (CVC) [45, 41, 20, 21, 24, 25], capacitance-to-

frequency converters (CFC) [39], capacitance-to-duty-cycle converters [26, 46]

and capacitance-to-phase-shift converters (CPC) [47]. Each of these principles

can be implemented through multiple circuit techniques. For example, a CVC

can be implemented using charge integration, chopper stabilized, ratio-arm-bridge,

low duty cycle periodic reset, AM based relaxation oscillator, etc., which are pre-

sented in the above mentioned references. Comparisons between various capac-

itance readout methods are detailed in [48, 49]. Table 2.1 shows a comparison

35

2.4. Comparison

summary between the common methods named above.

Author / Manufacturer Method Performance Parameter(s)Ashrafi et al. [47] CPC Resolution: 0.7fF (32ppm)

Zubair and Tang [38] CPC Resolution: 4.7fF (50ppm), 1.5˚/fFWolffenbuttel [50] CPC Resolution: 0.4fF, 1.5˚/fFHaider et al. [51] CVC Resolution: 1fF, Sensitivity: 1mV/fF

Irvine Sensors [52] CVC Resolution: 4aF/√

HzLotters et al. [21] CVC Resolution: 24aF

Solidus [53] CFC Resolution: 20aF

Table 2.1: Capacitance readout circuit methods, a brief comparison.

The most commonly used capacitance readout circuits are CVCs based on

switched-capacitor charge amplifier and CFCs. The former is insensitive to par-

asitic capacitance at the input of integrating amplifier [54]. The main concern re-

lated to the SC method is the noise associated to the charge injection and clock feed

through that occurs in MOS switches. CFCs are among the highest performance

readout circuits, due to their higher sensitivity and circuit simplicity. Although

they are susceptible to parasitic capacitances and resistances, temperature drifts,

and other sources of variation in the nominal oscillation frequency, but then can be

made more robust by using a differential approach that compares the measured ca-

pacitance to a reference capacitance. The only major drawback of this differential

approach is the slower reaction since the circuit would have to switch between the

sensor and the reference, taking twice as long [54].

Since the focus of our project is not that much related to the most of the com-

monly used capacitance readout circuits methodologies, their simulation details

and analyses are left for Appendix A.

36

2.5. Justification

2.5 Justification

Now that all these readout circuit methods have been introduced, it is evident that

many methods for designing readout circuits exist, each of which being suitable

for specific applications and measurement ranges. It is however important to find

the method most suitable to compare against our proposed WCR method. Histor-

ically the measurement systems based on time or frequency are among the most

reliable methods of measuring systems. The output of these methods can be easily

connected to a digital processing systems. They inherently are closer to digital im-

plementations since they do not require analog-to-digital converters at their output.

This thesis applies the WCR-based principles to the capacitance measurement

problem. While CFC methods exploit a shift in the resonant frequency with the

capacitance change, WCR-based circuits are related to the resonant modes (the

resonant frequencies give the eigenvalues of the linear circuit), but focus rather on

the energy repartition between the existing eigenmodes, and the way this repar-

tition is influenced by a change in capacitance that induces a symmetry-breaking

phenomenon. Nevertheless, the nearest method to the proposed WCR is the CFC,

since both methods rely on exploiting resonance related characteristics. Based on

this knowledge, the CFC method is chosen as the benchmark to which WCR-based

readout circuits will be compared.

2.6 Summary

This chapter has presented an overview of capacitive sensors, and their various

types. Then readout circuits were introduced and various subcategories related to

readout circuits, namely CVC, CDC, CPC and CFC, were examined in some detail

37

2.6. Summary

using relevant literature. A summary and a brief comparison between these meth-

ods was presented. Appendix A goes through more detailed simulation of these

readout circuits. After examining all these methods in detail, the CFC was chosen

as the benchmark for comparison with the WCR, both exploiting circuit resonance

characteristics. The next chapter examines WCRs in general and considers their

usage as an alternative for conventional readout circuits.

38

Chapter 3

Weakly-Coupled-Resonators as

Capacitance Readout Circuits

3.1 Introduction

As mentioned in chapter 1, the physical principles which give WCRs their inter-

esting characteristics, namely mode localization and eigenvalue local veering, have

been analyzed and used in solid state physics, mechanics, acoustics, and for MEMS

devices. This chapter starts by introducing WCRs in more detail and reviewing the

existing literature concerning WCRs, which is mainly in the mechanical field. Then

the chapter studies suitability of WCRs as capacitance readout circuits. To deter-

mine its suitability, the WCR method is judged based on criteria such as sensitivity,

robustness and simplicity.

3.2 Weakly Coupled Resonators

Many different fields exhibit the interesting interplay between resonant frequen-

cies, coupling strength between coupled resonant systems, and perturbation. There

is a phenomenon called mode/energy localization which happens in nearly iden-

tical weakly-coupled resonators. Mode localization in its simplest form happens

39

3.2. Weakly Coupled Resonators

between two identical resonators that are weakly coupled and at least one of the

elements of the resonators gets perturbed. For simplicity, we consider two loss-

less spring-mass resonators in Fig. 3.1. At first assume there is no coupling be-

tween the resonators i.e. kc = 0. We also assume that the resonators are identical

i.e. m1 = m2 = m and ∆k = 0. In this case, both resonators have identical res-

onant/natural frequencies (eigenvalues or normal modes) of ω0 =√

k/m. These

resonators under the same initial and excitation conditions, have the same dis-

placements (eigenvectors or mode shapes). Now we assume that they are coupled

through a weak coupling of spring kc. Once they are coupled, then the system

becomes a second order system and the identical natural frequencies split in two

frequencies (eigenvalues). The gap between these two modes is identified by the

strength of the coupling. The stronger the coupling, the farther apart the normal

modes. If this coupled resonators system is excited, e.g. by an initial condition,

it starts oscillating and the energy will be exchanged between the two halves al-

ternatively and evenly. In other words, the energy gets delocalized in the system.

Now, if there is a perturbation introduced in the system, e.g. by changing the sec-

ond spring constant from k to k+∆k, then the localization phenomenon happens,

and one side will get more energy (magnitude of displacement) than the other side.

This is also called mode localization. The relative change of this mode localization

depends on the relative perturbation δ , ∆k/k. If the coupling is weak enough,

the two eigenvalues (natural modes) are close to each other. In this case, another

phenomenon, called normal mode veering, happens besides the mode localization.

Normal mode veering, which is also called eigenvalue loci veering, is shown

in Fig. 3.2 [12]. The vertical axis is the normalized eigenvalues and the horizon-

tal axis is the relative perturbation δ . The higher the perturbation, the more gap

40


k2 = k +¢k

m1 = m m2 = m

k1 = k kCx1 x2

Figure 3.1: Lumped-element model of a coupled two-degree-of-freedom (2DOF)spring–mass system.

between the eigenvalues. The eigenvalues of the system are closest at δ = 0. The

loci is showing an abrupt change around δ = 0, which is called veering zone. If

the coupling is very week, these curve look like intercept lines, which is deceptive.

This is known as eigenvalue veering or normal mode veering.

Figure 3.2: Loci of the dimensionless eigenvalues of the two coupled oscillators interms of δ [12].

These phenomena of mode localization and veering are well known in the field

of mechanics, acoustics and MEMS. Mode localization have been used for detect-

ing and measuring very small perturbations that are nearly impossible, or much

more difficult, using other methods. As an example, ultrasensitive mass sensing

41


was implemented by this method able to measure a 154pg mass, shown in Fig. 3.3

with almost two orders of magnitude better sensitivity than conventional relative

frequency shift methods [9].

Figure 3.3: Scanning electromicroscopy (SEM) image of the first set of coupledgold-foil cantilevers and SEM image of an attached microsphere (inset, circled)[9].

These aspects were firstly analyzed in the solid state field by Anderson in 1978

[6, 7], which was the fundamental for the energy localization in periodic disordered

structures, which also is called mode localization. The same phenomenon creates

another effect which was named curve veering by Pierre in 1988 [8]. More work

with plates was performed in [55], proving that the natural frequencies of plates

that belong to the same symmetry family exhibit the veering phenomenon when

the plates are subjected to geometrical changes. This led the authors to propose the

42


conditions under which the eigenvalue loci veer do not cross. Another work pre-

sented in [56], improved on the conditions proposed in [57] and expanded them to

include general real-valued eigenvalues utilizing a perturbation method. The rela-

tionship between the eigenvalue loci veering and mode localization was described

in [58]. This work was then continued in [59], and the criteria governing the oc-

currence of veering as well as the relationship between the veering of eigenvalue

loci and eigenvector sensitivity was detailed. The dependency of the loci veering

on one or two parameters in a system has been investigated by [60].

Although the phenomenon of mode localization has been very well known and

used in disciplines such as acoustic and structural dynamics [61, 8, 58, 62] for a

long time, few publications have investigated the application of this phenomenon in

sensing and measurements of perturbations. There are several recent reports show-

ing the application of mode localization in detection and measurement of small

changes that can be considered as perturbation in the system. Chen and Kareem

[63] studied the curve veering of cable-stayed and suspension bridge frequency

loci. Spletzer et al. applied this concept as a method for ultrasensitive mass sens-

ing in coupled microcantilevers [9, 15]. Thiruvenkatanathan et al. reported the

use of mode localization concept in designing higher sensitivity MEMS sensors

[11, 12, 13]. The relative sensitivity of the mode localization method, which is

based on relative shift in eigenvectors of the system, is orders of magnitude higher

that the relative shift in the system resonant frequencies or eigenvalues [11]. This is

shown in Fig. 3.1 and equations (3.1) and (3.2). Equation (3.1) shows the influence

of the stiffness perturbation on relative change in eigenvectors.

43

3.3. Reasons for Proposing WCRs as an Alternative for Readout Circuits

|un−u0n||u0n|

≈∣∣∣∣ ∆k4kc

∣∣∣∣ , n = 1,2, (3.1)

where un and u0n are magnitudes of normalized eigenvectors with and without

perturbation respectively i.e. un = [a1 a2] where a1 and a2 are normalized mag-

nitudes of x1 and x2. ∆k is the change in the stiffness and kcis the coupling stiffness.

Equation (3.2) shows the influence of the same stiffness perturbation on the

relative resonant frequency shift.

| f − f0|| f0|

≈∣∣∣∣∆k2k

∣∣∣∣ , (3.2)

where f and f0 are resonant frequencies with and without perturbation respectively

and k is the nonperturbed stiffness of the springs.

Equations (3.1) and (3.2) show that the relative sensitivity based on the relative

shift in eigenvectors, is approximately k2kc

times greater than of the relative shift in

the resonant frequency. For a week coupling, kc k, k2kc

ratio could become orders

of magnitude larger.

To the best of our knowledge, there has been no literature indicating the use of

mode localization to measure the perturbation, e.g. minute capacitance changes, in

the electrical domain, prior to this research.

3.3 Reasons for Proposing WCRs as an Alternative for

Readout Circuits

The readout circuits and methods explained above, as well as many other state-of-

the-art circuits for capacitance readouts, are valuable and indicate present research

44

3.3. Reasons for Proposing WCRs as an Alternative for Readout Circuits

in terms of technology and implementation. Looking at this problem from a dif-

ferent perspective opens a promising way for designing highly sensitive and low

cost capacitance readout circuits. As it has been proven in the mechanical/MEMS

domain, the mode localization is very sensitive to perturbation, and as capacitance

changes in the circuit can be considered a perturbation, this thesis aims to investi-

gate whether WCRs are a suitable alternative to existing capacitance readout cir-

cuits. By looking at the nature of this method and what has been achieved in the

mechanical field, it is very likely that using this method in electrical domain gives

us a very good sensitivity with considerably less sophisticated analog circuitry. In

terms of the physical implementation, as a digital signal processor is one of the

essential units in almost all readout systems, it is much simpler, though not neces-

sarily easier, to choose a method which is inherently closer or matches better with a

digital domain implementation. The proposed method reduces the amount of ana-

log circuitry and takes advantage of digital processors to measure the capacitance.

Sensitivity and the resolution of this method in electrical domain should be studied

and verified. To apply the mode localization technique to the electrical systems, we

can use the analogy between the coupled resonators in Fig. 3.1 and two coupled

LC circuits. The LC resonators could have either series or parallel configuration.

These parallel and series LC circuits are dual of each other. In more general form

in Fig. 3.1 there are dampers in parallel with each spring which have the same

roles as resistors in the electrical domain. This analogy between mechanical and

electrical domain is shown in Table 3.1. Note that to subscript index s , e.g. in Ls,

stands for “series” and index p stands for “parallel” dual circuit.

45

3.4. Summary

Mechanical Electrical (Series RLC) Electrical (Parallel RLC)Mass (M) Inductance (Ls) Inductance (Cp = Ls)

Damping (C) Resistance (Rs) Resistance (Rp = 1/Rs)Spring constant (1/K) Capacitance (Cs) Capacitance (Lp =Cs)

Displacement (x) Electric Charge (qs) Electric Charge (qp)Force (Fs) Voltage source (Vs) Current source (Is)

Table 3.1: Analogy between mass-spring-damper and RLC coupled oscillators.

3.4 Summary

This chapter first introduced the history and background of weakly coupled res-

onators, veering and mode localization in mechanical and acoustic field. The result

of an application of WCRs applied to MEMS inertial sensors was also presented.

The chapter was then concluded by briefly visiting the possibility of adopting WCR

method in electrical domain to measure small capacitance changes. The following

chapter examines the details of two-degree-of-freedom WCR circuit in capacitance

measurement.

46

Chapter 4

WCR-Based Readout Circuit

Analysis and Performance

Estimation

4.1 Introduction

This chapter presents an innovative and simple way of measuring small capaci-

tance changes. The proposed method can be a breakthrough in capacitance readout

circuits, commonly used for microsensors associated with physical quantities such

as pressure, rotational angles, linear displacement and acceleration [2].

In this chapter we propose a new method, based on weakly-coupled resonators

(WCR), which will be compared with the CFC method. It will be proven ana-

lytically, and shown by both simulations and experimental tests, that this method

has at least one order of magnitude, and ideally three orders of magnitude, better

relative sensitivity in comparison to the resonant frequency shift method.

The WCR-based readout circuit is based on the energy (or mode) localiza-

tion phenomenon. If two identical resonators are weakly coupled and excited by

a harmonic source, the oscillation energy is equally shared between them; mode

47

4.2. Theory of Operation

localization appears when one of them is perturbed relative to the other, leading to

an unbalance of the energy repartition among the two individual resonators. This

symmetry-breaking phenomenon is well known in acoustic resonators and other

mechanical systems [64, 8], as described in chapter 3. Recently, more attention

has been given to the use of WCRs in the mechanical aspect of resonating MEMS

mass sensors. It has been proven that weakly-coupled resonators can achieve al-

most three orders of magnitude higher sensitivity than conventional frequency shift

techniques for perturbation sensing [12, 11, 10, 13, 9]. We are hence transferring

this high sensitivity symmetry-breaking technique to the electrical domain, for the

first time (to our knowledge). We will use it for a simple capacitance readout solu-

tion. The measurement technique requires simple analog circuitry and is suitable

for integration with digital signal processing. The outcome of this chapter was pub-

lished as a journal paper in IEEE transactions on circuits and systems I (TCAS-I)

[65].

4.2 Theory of Operation

In principle, mode-localized or WCRs consist of two, or more, nearly identical cou-

pled resonators. As stated previously, the concept of WCRs has been used for the

detection of very fine changes (perturbations) in mechanical resonators. These type

of vibrating systems are analogous to quantum mechanical systems, e.g. the hy-

drogen molecule [9]. The eigenstates of two identical weakly-coupled resonators,

symmetric and antisymmetric, are very similar to "bonding orbitals" and "anti-

bonding orbitals", respectively. The frequency (eigenvalue) of the antisymmetric

mode is either higher or lower than that of the symmetric eigenmode, depending on

48


whether the coupling factor is positive or negative, respectively. Starting from an

identical set of two resonators, a symmetry-breaking perturbation can be defined as

a slight change in one of resonator’s parameters, e.g. the mass or spring constant in

mechanical systems, or capacitance or inductance value in electrical systems. The

perturbation will lead to both a further separation between the resonant frequencies

associated with the symmetric and antisymmetric modes, and to a redistribution of

the energies (amplitudes) associated with these modes, for a given external exci-

tation. The measurement of the ratio between their oscillation amplitudes (e.g.

displacement or charges amplitudes) is therefore directly correlated with the mag-

nitude of the perturbation. The energy is no longer divided evenly between two

resonators in such a case, and tends to localize more in one of them. An example

of 2DOF WCRs in the mechanical domain is shown in Fig. 4.1 illustrating two

spring-coupled mass-spring resonators.

k2 = k +¢k

m1 = m m2 = m

C1 = C CC C2 = C

k1 = k kCx1 x2

Figure 4.1: Two weekly coupled mechanical resonators.

The low stiffness spring connecting the two masses acts as the weak coupling

in this system. The coupling coefficient for this system, neglecting the damping

effect, is defined as kc/k. An analogous representation of this system, using series

RLC circuits, is shown in Fig. 4.2, assuming that the excitation is applied through

49


a voltage source, and the damping in the coupling is negligible. The coupling

coefficient becomes C/Cc in this case.

R1

= R L1

= L C1

= C L2

= L R2

= RC2

= C +¢C

CC

VS

i1 i2

+

+

+

+

¡

¡

¡

¡

q1

q2

Figure 4.2: 2DOF weekly-coupled series RLC resonators.

The coupled resonant system can be represented by the set of differential equa-

tions shown in 4.1. The most general case, damped and perturbed, is considered

here as it covers all other cases, e.g. undamped or unperturbed.

L1

d2q1dt2 +R1

dq1dt + 1

CC(q1−q2)+

1C1

q1 = vs

L2d2q2dt2 +R2

dq2dt + 1

CC(q2−q1)+

1C2

q2 = 0(4.1)

It is assumed that the circuit is excited with a harmonic sinusoidal voltage

source. Applying a perturbation to this circuit was defined as inducing very small

changes in the value of C2, from C to C+∆C. It was also assumed that the rate of

the perturbation change is slow enough (quasi-stationary signal), is that the tran-

sient parts of the responses, including the effect of initial conditions, do not play a

role. These assumptions were equivalent with the ones already used in the literature

related to the weakly-coupled mechanical resonators. Therefore the transient part

of the response was eliminated and the steady state part of the response was used

to detect the perturbation magnitude.With these assumptions, the initial conditions

and transient responses of the system are ignored. Hence (4.1) can be re-written in

50


Laplace domain as:

L

s2 + RL s+ 1

LC (1+ k) − kLC

− kLC s2 + R

L s+ 1LC (1+ k−δ )

Q1

Q2

=

Vs

0

(4.2)

δ , ∆C/C and k =C/CC are defined as the perturbation and coupling strength

of the system, respectively. Although in in circuits and systems it is more common

to use either node voltages or loop current as the circuit variables, here we use

charges on the capacitors in each resonators to follow the analogy given in Table

3.1.

The loci of the resonant frequencies of the system versus the relative changes

in capacitance C2 (perturbations) are shown in Fig. 4.3 [12]. The plotted results

are the outcome of SPICE simulations using National Instruments Multisim-12r

with the values of L1 = L2 = 10mH, C1 = 100nF, C2 = 100nF+∆C, with −4nF <

∆C < 4nF as perturbation, and three different coupling capacitance, CC, values of

5, 15 and 25µF. There are three pairs of curves shown, each corresponding to

different coupling strengths (different CC values). It will be shown that, since the

coupling coefficient is positive in this case, for a given perturbation value, the upper

branch of the locus is related to the out-of-phase mode (higher natural frequency),

while the lower branch corresponds to the in-phase resonant mode (lower natural

frequency). The middle zone of the graph (around zero perturbation) is called the

transition zone. The shown trend, where the loci of these eigenvalues in weakly-

coupled systems approach each other in the transition zone but do not intersect, is

called loci veering.

51


Figure 4.3: Two weekly coupled resonators natural frequencies loci.

As can be seen, the weaker the coupling is, the narrower the transition zone

and the higher the slopes in the curves around the transition zone are. This sudden

veering and narrowing of the transition zone results in a higher degree of energy

localization, or larger changes in the relative amplitudes of oscillation in the left

and right resonators.

This mode localization or shift in the eigenvector component values around the

transition zone is shown in Fig. 4.4. To show the eigenvectors abrupt changes in the

transition zone, the ratio between i2 and i1 (loop currents) is plotted in Fig. 4.4(a).

For positive perturbations, amplitude of i2 becomes considerably larger than i1 at

out-of-phase resonant excitation. Inversely, i1 gets quite larger than i2 at out-of-

52


phase mode excitation using the external voltage source. In other words, if the

system is excited at in-phase resonant frequency, the energy will be more localized

in the left loop of the circuit for positive perturbations and in the right loop for

negative perturbations. The opposite is true for in-phase excitation - energy will

be localized in the left loop for negative perturbations, or in the right loop for

positive perturbations. This is also shown using mode shape vector orientations

along with eigenvalue loci veering curve plot in Fig. 4.4(b). In this plot, normalized

eigenvalues are shown to be similar to the resonant frequencies loci in Fig. 4.3, as

expected.

The key aspect in achieving high sensitivity to perturbation is the drastic change

in the angle of the mode shape vector in the veering zone. In comparison, the

eigenvalue changes are not as significant.

In the series RLC-based WCRs shown in Fig. 4.2, we assume R1 = R2 = R,

L1 = L2 = L and C1 = C2 = C, which means both resonators are identical in the

absence of any perturbation. The coupling is done via CC ,with CC C being the

condition for a weak coupling.

53


(a)

(b)

Figure 4.4: Mode localization in two weekly-coupled-resonators.

54


A small change in any of these values can be defined as a perturbation, which

slightly pushes the system away from the energy equidistribution. We try to exploit

the high sensitivity in the orientation of the eigenvector around the symmetry-

breaking region, in order to use it as a measure of the applied perturbation (C2

variations). We will show that this technique provides a higher sensitivity than

existing state-of-the-art methods, e.g. monitoring the relative shift in the resonant

frequency. A detailed analytical solution to this second order differential equation,

using Laplace transform, is presented in the next section.

4.2.1 Analytical Solution

Equation 4.2, re-written below, representing the two weakly coupled RLC res-

onator systems, is solved through Laplace-transform techniques.

L

s2 + RL s+ 1

LC (1+ k) −1LC

−1LC s2 + R

L s+ 1LC (1+ k−δ )

Q1

Q2

=

Vs

0

(4.3)

The solutions Q1 and Q2 are therefore given by:

Q1 =1L

s2 + RL s+ 1

LC (1+ k−δ )(s2 + R

L s+ 1LC

(1+2k− δ

2

))(s2 + R

L s+ 1LC

(1− δ

2

))Vs (4.4)

Q2 =1L

1LCC(

s2 + RL s+ 1

LC

(1+2k− δ

2

))(s2 + R

L s+ 1LC

(1− δ

2

))Vs (4.5)

55


Using Partial Fraction Expansion (PFE), the equations can be simplified to

Q1 =1L

(k1

s2 + RL s+ 1

LC (1+2k− δ

2 )+

k2

s2 + RL s+ 1

LC (1− δ

2 )

)Vs (4.6)

Q2 =1L

(k3

s2 + RL s+ 1

LC (1+2k− δ

2 )+

k4

s2 + RL s+ 1

LC (1− δ

2 )

)Vs (4.7)

where

k1 =12

(1+

δ

2k

); k2 =

12

(1− δ

2k

)

k3 =−2k2

(LC)2 ; k4 =2k2

(LC)2

The response of this WCRs system to a Dirac Delta input is given by the inverse

Laplace transform of the above equations:

q1 =

12L

[1

ωd1

(1− δ

2k

)sin(ωd1t)+ 1

ωd2

(1+ δ

2k

)sin(ωd2t)

]e−R2L tu(t)

q2 =1

2L

[1

ωd1sin(ωd1t)− 1

ωd2sin(ωd2t)

]e−R2L tu(t)

(4.8)

where

ωd1 =1√LC

√1− δ

2,ωd2 =

1√LC

√1+2k− δ

2

56


As expected, the impulse response fades out with a time constant of R/2L.

In order to deduce the response of the system to a harmonic voltage excitation,

as in Fig. 4.2, it is useful to decouple the equations set in (4.3). We introduce the

transformed state variables:

P1

P2

,1√2

1− δ

4k 1+ δ

4k

1+ δ

4k −1+ δ

4k

Q1

Q2

(4.9)

or

Q1

Q2

≈ 1√2

1− δ

4k 1+ δ

4k

1+ δ

4k −1+ δ

4k

P1

P2

(4.10)

The set of equations in the new state variables is given by:

Ls2 +Rs+ 1C

(1− δ

2

)−A − δ 3

32Ck2

− δ 3

32Ck2 Ls2 +Rs+ 1C

(1+2k− δ

2

)−B

P1

P2

=1√2

(

1− δ

4k

)Vs(

1+ δ

4k

)Vs

(4.11)

where

A =1

32k2

(2Lδ

2s2 +2Rδ2s+

1C

(4δ

2k+δ3−2δ

2))and

B =1

32k2

(2Lδ

2s2 +2Rδ2s+

1C

(4δ

2k+δ3 +2δ

2)) .

57


With the assumption of very small relative perturbations, even in comparison

with the coupling coefficient (i.e. δ k), we can eliminate the terms containing

higher order terms of δ/k. The resulting simplified and completely decoupled set

of equations is:

(

Ls2 +Rs+ 1C

(1− δ

2

))P1 =

1√2

(1− δ

4k

)Vs(

Ls2 +Rs+ 1C

(1+2k− δ

2

))P2 =

1√2

(1+ δ

4k

)Vs

(4.12)

Each of these equations can now be solved independently of one other, and

the inverse transformation to the physical state variables Q1, Q2 will give their

expression.

Assume vs(t) = Asin(ωt)u(t) or Vs(s) = ω/(s2 +ω2

)The equations then become:

P1(s) = ω√

2L

(1− δ

4k

)1

s2+ RL s+ 1

LC (1− δ

2 )1

s2+ω2

P2(s) = ω√2L

(1+ δ

4k

)1

s2+ RL s+ 1

LC (1+2k− δ

2 )1

s2+ω2

(4.13)

Using the inverse Laplace transform, the function x(t), in time domain, can be

retrieved:

x(t) = xt(t)+ xs(t) (4.14)

where xt(t) and xs(t) are transient and steady state parts of x(t) respectively.

Here, we focus on quasi-static perturbations, after the transient parts of the

responses have faded out. Therefore the transient part of the response can be elim-

inated and the steady state part is used to detect the perturbation magnitude.

58


The equations for P1 and P2 can be re-written as

P1(s) =ω√2L

(1− δ

4k

) k1s+ k2

s2 + RL s+ 1

LC

(1− δ

2

) +−k1s+ k3

s2 +ω2

(4.15)

where

ω2n1 =

1LC

(1− δ

2

)and 2ζ1ωn1 =

RL

k1 =2ζ1ωn1(

ω2n1−ω2

)2+(2ζ1ωn1ω)2

=RL(

1LC

(1− δ

2

)−ω2

)2+(R

L ω)2

k2 =(2ζ1ωn1)

2−ω2(ω2

n1−ω2)(

ω2n1−ω2

)2+(2ζ1ωn1ω)2

=

(RL

)2−ω2(

1LC

(1− δ

2

)−ω2

)(

1LC

(1− δ

2

)−ω2

)2+(R

L ω)2

k3 =ω2

n1−ω2(ω2

n1−ω2)2

+(2ζ1ωn1ω)2=

1LC

(1− δ

2

)−ω2(

1LC

(1− δ

2

)−ω2

)2+(R

L ω)2

P2(s) =ω√2L

(1+

δ

4k

) k′1s+ k′2s2 + R

L s+ 1LC

(1+2k− δ

2

) +−k′1s+ k′3s2 +ω2

(4.16)

where

59


ω2n2 =

1LC

(1+2k− δ

2

)and 2ζ2ωn2 =

RL

k′1 =2ζ2ωn1(

ω2n2−ω2

)2+(2ζ2ωn2ω)2

=RL(

1LC

(1+2k− δ

2

)−ω2

)2+(R

L ω)2

k′2 =(2ζ2ωn2)

2−ω2(ω2

n2−ω2)(

ω2n2−ω2

)2+(2ζ2ωn2ω)2

=

(RL

)2−ω2(

1LC

(1+2k− δ

2

)−ω2

)(

1LC

(1+2k− δ

2

)−ω2

)2+(R

L ω)2

k′3 =ω2

n2−ω2(ω2

n2−ω2)2

+(2ζ2ωn2ω)2=

1LC

(1+2k− δ

2

)−ω2(

1LC

(1+2k− δ

2

)−ω2

)2+(R

L ω)2

Assuming a quasi-static perturbation, P1 and P2 can be simplified and approxi-

mated by the steady-state, harmonic, part of the response:

P1(s) = ω√

2L

(1− δ

4k

)−k1s+k3s2+ω2

P2(s) = ω√2L

(1+ δ

4k

)−k′1s+k′3s2+ω2

(4.17)

or

p1(t) = ω√

2L

(1− δ

4k

)(k3ω

sin(ωt)− k1 cos(ωt))

p2(t) = ω√2L

(1+ δ

4k

)(k′3ω

sin(ωt)− k′1 cos(ωt)) (4.18)

Knowing p1 and p2, q1 and q2 can be solved.

60


q1(t) = 1√

2

[(1− δ

4k

)p1 +

(1+ δ

4k

)p2

]q2(t) = 1√

2

[(1+ δ

4k

)p1 +

(−1+ δ

4k

)p2

] (4.19)

By substituting p1 and p2 in the above equations

q1(t) =1

2L

[(1− δ

2k

)ω2

n1−ω2(ω2

n1−ω2)2

+(R

L ω)2 +

(1+

δ

2k

)ω2

n2−ω2(ω2

n2−ω2)2

+(R

L ω)2

]sin(ωt)

− ω

2LRL

[(1− δ

2k

)1(

ω2n1−ω2

)2+(R

L ω)2 +

(1+

δ

2k

)1(

ω2n2−ω2

)2+(R

L ω)2

]cos(ωt)

(4.20)

q1(t) =1

2L

[ω2

n1−ω2(ω2

n1−ω2)2

+(R

L ω)2 −

ω2n2−ω2(

ω2n2−ω2

)2+(R

L ω)2

]sin(ωt)

+ω

2LRL

[− 1(

ω2n1−ω2

)2+(R

L ω)2 +

1(ω2

n2−ω2)2

+(R

L ω)2

]cos(ωt).

(4.21)

The amplitudes of q1 and q2 have peaks at ωn1 and ωn2, which are natu-

ral/resonant angular frequencies of the system. The following approximation is

used in estimating q1, q2:

(1+

δ

2k

)/

(1− δ

2k

)≈ 1+

δ

kfor δ k.

The steady state solution for the ratio of the two capacitor charges will then be:

61


|q1||q2|

=

∣∣∣∣(1− δ

2k

)∣∣∣∣√√√√√√√√

4k2ω20 +

ω2n2

Q2

(1+(

1+ δ

k

)(1+ 4k2Q2

(1+2k− δ

2 )

))2

4k2ω20 +

ω2n2

Q2

(1+ 4k2Q2

(1+2k− δ

2 )

)2 (4.22)

with the following notations:

ω0 =1√LC

, Q =ω0L

R, ωn2 =

1√LC

√1+2k− δ

2(4.23)

and defining the square root term as:

r ,

√√√√√√√√4k2

L2C2 +R2

L3C

(1+2k− δ

2

)(1+(

1+ δ

k

)(1+ 4k2L

R2C(1+2k− δ

2 )

))2

4k2

L2C2 +R2

L3C

(1+2k− δ

2

)(4k2L

R2C(1+2k− δ

2 )

)2 . (4.24)

Using a rational function curve fit, with a least mean square (LMS) approxi-

mation, r is almost independent of δ which is graphed in Fig. 4.5. In this approx-

imation, with the values for L, C and k are 10 mH, 100 nF and 1/150 respectively,

r becomes:

r ∼=∣∣∣∣0.34171R+0.98−0.00252R+1

∣∣∣∣ (4.25)

Hence, the ratio between capacitor charges in (4.22) is well estimated, for re-

sistor values 0 < R < 100Ω, by:

62


|q1||q2|

==

∣∣∣∣0.34171R+0.98−0.00252R+1

(1− δ

2k

)∣∣∣∣ (4.26)

The parameter r indicates the effect of loss (resonator resistance) on sensitivity.

Note that if loss is negligible then r gets close to unity. If loss is considerable, then

the magnitude of q2 becomes smaller than the magnitude of q1, regardless of the

amount of perturbation. In such cases, the value of r, which is |q1/q2| at balance

(δ = 0) is greater than 1.

Figure 4.5: Effect of loss on sensitivity. Coefficient r in (4.25).

For the values of L = 10mH, R = 0.1Ω, CC = 15µF and C = 10nF (i.e. k =

1/150), this ratio will be

|q1||q2|

= 1.014427 |(1−75δ )| (4.27)

63


This is completely in-line with the values from the exact equation, as shown in

Table 4.1 and Fig. 4.6. The following notations are used:

ωn1 =1√LC

√1− δ

2, ωn2 =

1√LC

√1+2k− δ

2

2ζiωni =RL, ωdi = ωni

√1−ζi

2 ; i = 1,2

where ωn1 and ωn2 , or their corresponding frequencies, fn1 and fn2, are the

resonant frequencies of the natural modes of the WCR system.

δ ωn2(Rad/Sec)∣∣∣q1

q2

∣∣∣− ∣∣∣q1q2

∣∣∣0

ωn2−ωn0ωn0

-1.33E-4 31623.83 -0.0098896 3.333E-05-1.07E-4 31623.62 -0.0079273 2.667E-05

-8E-5 31623.41 -0.0059573 1.999E-05-5.33E-05 31623.2 -0.0039795 1.333E-05-2.67E-05 31622.99 -0.0019937 6.667E-06

0 31622.78 0 02.667E-05 31622.57 0.0020017 -6.667E-065.333E-05 31622.35 0.0040114 -1.333E-05

8E-5 31622.14 0.0060293 -2.000E-051.07E-4 31621.93 0.0080552 -2.667E-051.33E-4 31621.72 0.0100894 -3.334E-05

Table 4.1: Analytical values for 2DOF WCRs at out-of-phase resonance.

In Table 4.1∣∣∣q1

q2

∣∣∣0is the value of

∣∣∣q1q2

∣∣∣ when there is no perturbation (δ = 0).

In Fig. 4.6, the relative shift in the resonant frequency of the out-of-phase mode

(Table 4.1) is plotted with a magnified scale of 100, in order to make it noticeable

in comparison to the relative shift of the ratio of capacitor charges. The sensitivity

based on relative capacitor charge measurement is approximately 300 times higher

64

4.3. Simulations

than the sensitivity based on the resonant frequency shift measurement. This data

was exported from circuit simulations using Multisim-12r.

-0.015 -0.010 -0.005 0.000 0.005 0.010 0.015

-0.010

-0.005

0.000

0.005

0.010

±(%)

¯

¯

¯

q1

q2

¯

¯

¯¡

¯

¯

¯

q1

q2

¯

¯

¯

0

!n2¡!n0

!n0

100£

Figure 4.6: Relative shift in resonant frequency vs. eigenmode in 2DOF WCRs.

4.3 Simulations

A series of simulations were performed with sinusoidal input as forced excitation.

Circuit simulations were conducted in National Instruments Multisim-12r for the

circuit shown in Fig. 4.7. The main reason for choosing Multisimr is the ca-

pability of co-simulation with LabVIEWr, which integrates very well with the

NI-PXI hardware platform used in our experiments. Perturbations of C2 values

were simulated by using a voltage-dependent capacitor (∆C in parallel with C2).

65

4.3. Simulations

When C+∆C = 100nF , the resonators are balanced.

R1

0:1Ð 100mH 100nH

L1 C1 C2 L2 R2

¢C

CC 15¹F

VS

0:1Vp

100nH 100mH 0:1Ð

i1 i2

Figure 4.7: Circuit schematic of 2DOF WCRs for SPICE simulations.

The results of the AC analysis, shown in Fig. 4.8, illustrate the natural frequen-

cies (modes) of the unperturbed circuit, at 5.033 kHz and 5.066 kHz, respectively.

Figure 4.8: AC analysis of 2DOF WCRs based on series RLC resonators.

From a practical perspective, the loop currents, i1 and i2, rather than charges

q1and q2, are taken as the eigenvectors components. The weak coupling is achieved

through CC, which was chosen to be approximately 15 times larger than C1 and C2.

If CC becomes too small, then the two resonators will become strongly coupled; if

66

4.3. Simulations

it is too large, then the amount of energy transferred to the second resonator is not

sufficient to be measured given the existing parasitics and noise sources.

In the next step of the simulation, the circuit was excited using a sine wave

input at one of the resonant frequencies. Perturbations were introduced to C2 us-

ing a voltage-controlled capacitor U1. The simulation results are aligned with the

theoretical analyses for both in-phase and out-of-phase modes, at 5.033 kHz and

5.066kHz respectively. A parametric sweep analysis on C2, from 95nF to 105nF,

shows the effect of perturbations on the natural frequencies and loop current values

(mode shapes). The results are shown in Table 4.2.

δ (%) fn2(Hz)∣∣∣∆u

u0

∣∣∣ ∣∣∣ i2i1

∣∣∣ ∣∣∣ i2i1

∣∣∣− ∣∣∣ i2i1

∣∣∣0

100∣∣∣ fn2− fn0

fn0

∣∣∣-5 5183.38 0.8470 3.2496 2.2752 2.3116-4 5156.96 0.8088 3.0782 2.1037 1.7901-3 5131.13 0.7481 2.8134 1.8389 1.2802-2 5106.31 0.6407 2.3969 1.4225 0.7903-1 5083.48 0.4224 1.7465 0.7721 0.33970 5066.27 0.0000 0.9744 0.0000 0.00001 5058.06 0.4238 0.5012 0.4732 0.16212 5054.65 0.6362 0.3079 0.6666 0.22943 5053.25 0.7383 0.2181 0.7564 0.25704 5052.45 0.7937 0.1685 0.8059 0.27285 5051.85 0.8285 0.1374 0.8370 0.2846

Table 4.2: Comparison table between ∆

∣∣∣ i2i1

∣∣∣and ∆ ff methods of measurement.

Note that for the theoretical eigenvectors, the charges on the capacitors C1 and

C2 were defined as vector components. Both analytical calculations and simula-

tions show that the value of relative change in loop current ratio is very close to

the relative shift in eigenvectors (∆u/u0) [9]. In these simulations a weak coupling

of C/CC = 1/150 and a maximum perturbation of δ = ∆C/C = 0.005 resulted in

67

4.3. Simulations

a sensitivity of approximately 0.6, which is in agreement with the analytical cal-

culations. Simulations show that the variation in the ratio between eigenvector

components is about 30 times larger than the relative frequency shift. The results

of these simulations are shown in Fig. 4.9 and Fig. 4.10.

Figure 4.9: i1 plots, coupled RLC circuit AC analysis with sweeping C2 from 99%to 101% of nominal value.

Figure 4.10: i2 plots, coupled RLC circuit AC analysis with sweeping C2 from 99%to 101% of nominal value.

68

4.3. Simulations

The veering of the resonant frequencies is plotted in Fig. 4.11, while a com-

parison between three methods of indirectly measuring the perturbations shown in

Fig. 4.12.

Figure 4.11: Resonant frequency loci veering in 2DOF WCR.CC = 15µF , R =0.1Ω, C = 100nF , L = 10mH.

Note that the |∆ f/ f | method is up-scaled 100 times to be comparable in sensi-

tivity magnitude with the other two methods.

69

4.3. Simulations

-5.0 -2.5 0.0 2.5 5.0

-1.0

-0.5

0.0

0.5

1.0

1.5

2.0

2.5in-phase mode

The

res

pons

e of

thre

e m

etho

dsto

qua

si-s

tatic

per

turb

atio

ns

¢u1= ju10j¢ jI2=I1j¡ 100£¢f=f0

±(%)

(a)

-5.0 -2.5 0.0 2.5 5.0

-1.0

-0.5

0.0

0.5

1.0

1.5

2.0

2.5out-of-phase mode

±(%)

¢u2= ju20j¢ jI2=I1j100£¢f=f0

The

res

pons

e of

thre

e m

etho

dsto

qua

si-s

tatic

per

turb

atio

ns

(b)

Figure 4.12: Sensitivity comparison between three different methods; (a) compar-ison when exciting the in-phase (symmetric) mode (b) comparison when excitingthe antisymmetric mode.

70

4.3. Simulations

As it can be seen from the plots, for small perturbations the sensitivity of loop

currents ratio is almost equal to the sensitivity of the relative change in mode

shapes. The computation and hardware implementation of currents ratio estima-

tion is easier and requires less processing and memory resources; it is therefore the

adopted method. To further simplify the implementation, |i2/i1| ratio can be used

instead of ∆ |i2/i1|, eliminating unnecessary previous state storage in the computa-

tional algorithm.

Figure 4.13: LabVIEW-Multisim co-simulation for 2DOF WCRs.

The experiments, which will be described in the next section, were conducted

using National Instruments LabVIEWr software, together with a PXI data acqui-

sition system. The above simulations were repeated using a LabVIEW-Multisimr

71

4.3. Simulations

co-simulation framework (Fig. 4.13), to have a smoother transition from simula-

tion to hardware implementation. The results have confirmed the previous results

obtained using Multisim-only simulations.

-1.0 -0.5 0.0 0.5 1.0-0.6

-0.4

-0.2

0.0

0.2

0.4

0.6

±(%)

Com

parin

g re

lativ

e sh

ifts

in r

eson

ant

freq

uenc

y, e

igen

mod

e an

d cu

rren

t rat

io

100¢f=f0¢U= jU0j¢ jI2=I1j

Figure 4.14: Relative shifts for different methods; LabVIEW-Multisim co-simulation result.

The results are illustrated in Fig. 4.14, in a similar format as Fig. 4.12(b).

There is a small difference in the co-simulation results, specifically around -3%

perturbation. This comes from the fact that, for very high-Q (Q > 1000) circuits,

the accuracy of integration in time domain is strongly dependent on the time-step

settings in both LabVIEW and Multisim. Fine tuning this timing requires a vari-

able time step of in the order of ns, which makes the simulation very slow and

memory-intensive (several hours on a system with 8GB of RAM). The AC analy-

sis in Multisim simulations gives more reliable results without extensive simulation

72

4.4. Experimental Results

time.

4.4 Experimental Results

The components used for the hardware implementation were R1 =R2 = 0.1Ω, C1 =

C2 = 100nF, L1 = L2 = 10mH and CC = 15µF. C1 was regarded as a reference,

while C2 corresponded to the sensing capacitance. The perturbation of C2 was

generated by adding extra capacitors in parallel to the initial 100nF value. The

test-bench environment needed a voltage source with very low output impedance

for excitation, and an interface to measure the inductors or capacitors voltages.

Measurement of the analog values was done using a National Instrument PXIe-

1062Q system with a PXI-7854R analog interface module. A high level block

diagram of the proposed instrumentation system is shown in Fig. 4.15.

Figure 4.15: High-level-block-diagram of proposed capacitance readout.

73


The circuit was implemented on both a breadboard and a custom printed circuit

board (PCB), and connected to the PXI module from National Instruments. The

LabView-FPGA software toolkit was used for the data acquisition and processing.

Images of the setup are shown in Fig. 4.16. Three 16-bit analog inputs of the PXI

analog interface were used for reading excitation input, as a reference, and both

inductor-capacitor junction node voltages.

(a) (b)

(c) (d)

Figure 4.16: Test setup for experimental measurements. (a) Circuit connections(b) FPGA based data acquisition (PXI) (c) LabView GUI screen-shot (d) Completesystem connections.

The inputs were connected to the PXI module via coaxial cables with BNC

connectors. The only additional equipment that is needed to set up the experiment

74


was an external power supply, or alternatively a pair of 9V batteries, to power up

the input op-amp buffer.

A single tone sine wave generator and a simple PLL algorithm were imple-

mented using LabView-FPGA to keep the excitation frequency tracking the out-

of-phase resonance frequency, when different perturbations were applied to C2.

The system was also capable of manually controlling the sine wave generator

by breaking the PLL loop, in order to test the designed PLL. In our experiments,

the capacitors voltage amplitudes were considered as components of the voltage

vector (i.e. v = [v1 v2]T ). v0 corresponds to v at balance (i.e. δ = 0) which

relates to the out-of-phase resonant frequency f0. Introducing a small perturbation

leads to a resonant frequency shift by ∆ f and a change in v by ∆v.

The values for ∆ f/ f0 and ∆v/v0 were taken as measures for the relative shift

in resonant frequency and the relative eigenvector shift methods, respectively. The

results of these methods are shown in Table IV for different perturbations, δ , of

C2; both the simulated and the measured sensitivities are plotted in Fig. 4.17.

δ (%) fn2 v1 v2

∣∣∣∆vv0

∣∣∣ 50∣∣∣∆ f

f0

∣∣∣0 5144 1.54059 1.30774 0 0

0.0295 5143.5 1.54481 1.30477 0.00502 0.004860.0586 5143 1.54792 1.30324 0.00823 0.009720.1119 5142.5 1.55792 1.30387 0.01425 0.014580.1948 5141 1.56675 1.30005 0.02300 0.029160.9787 5132 1.69866 1.29276 0.11538 0.11664

Table 4.3: Experimental results for both eigenvalue and eigenvector based methods

The experimentally obtained sensitivities were lower than in simulation due to

non-idealities of the circuit components, e.g. internal series resistance of the in-

ductors and electronic component tolerances. These led to reduced quality factors

75


of the resonators and higher tolerance in their coupling strengths. Nevertheless, as

seen in Fig. 4.17 the capacitance-to-eigenvector shifting method proves to be at

least 50 times better than the capacitance-to-frequency shifting method.

Figure 4.17: Sensitivity comparison between simulations and experiments.

The experimental results using CC = 15µF did not show satisfactory results in

terms of distinguishing between the two resonant peaks (mainly due to the equiv-

alent series resistance ESRs). There are other parasitic parameters and sources for

nonlinearities, e.g. skin effect, core hysteresis etc. [66], which are not significant

in our case due to the low frequency and narrow band of operation.

In these lossy coupled resonators, there is a trade-off between the sensitivity

and detectability of the two resonant modes. To increase the gap between the reso-

76

4.5. Summary

nant frequencies, coupling should increase (i.e. CC should decrease) which in turn

decreases the sensitivity. In these experiments CC was changed to approximately

4µF . Fig. 4.18 shows the effect of this change on the frequency response.

(a) (b)

Figure 4.18: Effect of parasitic parameters on frequency response. (a) CC = 15µF(b) CC = 4µF.

4.5 Summary

The work presented in this chapter expands upon a principle validated already in

mechanical and acoustical engineering. The concepts of eigenvalues veering and

modes localization are fundamental in many physical problems; for instance the

mode localization aspect is reflected in the work of P. W. Anderson leading to the

Nobel prize in physics in 1977 (Anderson Localization) [6].

We have applied this technique, for the first time (to our knowledge), to a

capacitance-to-eigenvector shifting readout circuit and presented its superiority

over a capacitance-to- frequency shifting approach. The circuit is relatively sim-

ple and easy to couple with digital processing circuitry. The theoretical analysis

was complemented by both numerical simulations (SPICE and LabVIEWr) and

experimental measurements on a custom-made PCB level circuit.

77

4.5. Summary

Despite the presence of significant parasitic elements, e.g. ESR of inductors

and capacitors, the simple circuit was able to detect capacitive changes as low

as 30ppm and proved to be at least 50 times more sensitive than the equivalent

frequency-shifting method (Fig. 4.17). This opens up the new possibilities for

more advanced integrated solutions relying on mode localization for readout cir-

cuits of various electrical parameters. The next chapter focuses on the errors and

uncertainties associated with the excitation frequency, which are important in both

CFC and WCR-based methods.

78

Chapter 5

Error Reduction in WCR-Based

Capacitance Readout Circuits

5.1 Introduction

The superiority of the WCR over CFC method, in terms of the relative sensitivity,

has been proven and shown in the previous chapter. A WCR-based readout circuit

with a coupling factor of as low as k = 0.02 can show a sensitivity of at least two

orders of magnitude higher than the equivalent CFC configuration. An example

of such systems, based on coupled series RLC circuits, is shown in Fig. 5.1. The

capacitor CC provides the weak coupling between the two series resonance circuits,

with the coupling coefficient being defined as δ , ∆C/CC. The circuit will be

analyzed and discussed in more detail in the following sections.

Figure 5.1: Series RLC two weakly coupled resonators.

79

5.1. Introduction

As stated in chapter 4, we have applied the WCR concept to capacitance changes

measurement problem [65]. The sensitivities of CFC and WCR methods were an-

alytically calculated. It also has been shown that the ratio between sensitivities

of these methods (i.e. the slopes of the relative shifts in resonant frequency and

eigenmode for CFC and WCR, respectively) is inversely proportional to the cou-

pling strength between the two sections. For a weak coupling coefficient of 1/150,

the relative sensitivity of the WCR method is 300 times more than of the CFC

method. This is illustrated in Fig. 5.2. In this figure, ωn0 is the unperturbed res-

onant angular frequency (either in-phase of out-of-phase excitation mode), while

ωn2 is the resonant angular frequency shifted by the perturbation δ . q1 and q2 are

charges on the capacitance C1 and C2, respectively, and |q1/q2|0 is their ratio at

unperturbed (δ = 0) condition.

-0.015 -0.010 -0.005 0.000 0.005 0.010 0.015

-0.010

-0.005

0.000

0.005

0.010

±(%)

¯

¯

¯

q1

q2

¯

¯

¯¡

¯

¯

¯

q1

q2

¯

¯

¯

0

!n2¡!n0

!n0

100£

Figure 5.2: Relative shifts in resonant frequency of CFC vs. eigenmodes of WCRmethodes.

80

5.1. Introduction

The main difference between these two methods is that CFC is based on eigen-

values (natural frequencies) shifting of the system, while WCR scheme is based on

system eigenmodes shifting, which is 1/k times, k , CCC 1, more sensitive than

CFC method. The previous chapter mainly focused on relative sensitivity improve-

ment in capacitance change measurement but without analyzing its robustness to

excitation frequency errors. Therefore it is important to formally define expres-

sions for measurement errors for both CFC-based and WCR-based readout circuits,

to have a fair comparison between them. This chapter presents accurate and ana-

lytically deduced error expressions for CFC and WCR methods. Furthermore it

exploits these derivation for obtaining a new method with guaranteed lower mea-

surement errors. Typically, such systems based on resonance monitoring require a

feedback loop to keep the frequency of the excitation signal at resonance, e.g. by

using a PLL or some other locking mechanism [67]. There is a tradeoff between

sensitivity and phase noise. Higher sensitivity requires higher quality factors. On

the other hand, a high quality factor resonator has a sharp slope of the phase (φ )

vs. frequency ( f or ω) dependency around resonant frequency ( dφ/dω = 2Q/ω0,

), which in turn makes the system more sensitive and prone to error at the locking

frequency Fig. 5.3.

There are several possible implementations for CFC readout circuits. In our

case we have chosen the same double-resonator circuits for both methods, fre-

quency shift and eigenmodes shift. A high level block diagram of the system is

shown in Fig. 5.4. The linear combination of the CFC and WCR methods was

coined “Hybrid WCR” method. If the value of one of the circuit components

changes, the resonant frequency of the system changes accordingly. The feed-

back loop, including a phase detector and a loop filter, keeps the VCO tracking

81

5.1. Introduction

5026 5028 5030 5032 5034 5036 5038 5040−20

−10

0

10

20

Amplitude(dB)

Q = 500

Q = 100

Q = 20

5026 5028 5030 5032 5034 5036 5038 5040

−50

0

50

Phase

φ()

f (Hz)

Q = 500

Q = 100

Q = 20

Figure 5.3: Bode Plot for Series RLC Resonator

the resonant frequency. There are different methods for tracking and locking to

the resonant frequency of such RLC tanks, either based on amplitude, phase or

a combination of them. The phase information is used in this research for sim-

plicity and ease of implementation. It is also assumed that all the uncertainty

and errors in the circuitry, e.g. thermal and electronic noises, ADC quantization

noise, phase/frequency noise of the VCO etc, eventually show themselves in the

frequency/phase of the VCO output as a deviation from the exact resonant fre-

quency of the resonator. This erroneous frequency locking has a direct impact on

the perturbation estimation using CFC method. It also indirectly affects the WCR

method since all calculations are based on the assumption of the system being ex-

cited at resonance.

Our analysis shows that the errors in these two methods are of comparable

magnitude but of opposite sign. As a solution, we have linearly combined the

results of CFC and WCR methods to mitigate the estimation error.

The rest of this chapter is structured as follows: theory is presented in sec-

tion 5.2 followed by simulation results in section 5.3. Experimental results are

presented in section 5.4.

82


vs

vc

i1 i2fs = f0(1 + ®vc)V CO

i1 i2vs

AmplitudeDetection

LoopFilter

I1 I2Vs

WCRMethod

CFCMethod

fs

HybridWCR Method(this work)

Analog

Digital

PhaseDetector

Figure 5.4: System high-level-block-diagram.

5.2 Theory of Operation

Before discussing the details of the WCR system, it is useful to briefly discuss

the related frequency dependent methods. A simple RLC resonator vs. Colpitts

oscillator is shown in Fig. 5.5 as examples. Fig. 5.5a illustrates the capacitance

measurement based on the RLC resonator by simply associating the resonant fre-

quency and the capacitance C values, f0 =(2π√

LC)−1

. The formula indicates

a one-to-one correspondence between capacitance and resonant frequencies mea-

sured with the help of an excitation voltage Vs. An alternative method for a CFC

based capacitance measurement is to use an active oscillator, e.g. Colpitts oscilla-

tor shown in Fig. 5.5b. The op-amp gain and its positive feedback fulfills both gain

83


and phase conditions for oscillation. The frequency and capacitance relationship

follows f0 =(2π√

LCT)−1 equation, where CT =C1C2/(C1+C2) is the equivalent

resonator tank capacitance. In capacitance measurements it is common to assume

either C1 or C2 taken as reference capacitor while the other one serves as the sens-

ing capacitor.

Vs

R L C

(a) RLC resonator.

+

-

C1C2

Rf

Ri

L

A

(b) Colpitts oscillator.

Figure 5.5: Examples of conventional capacitance measurement methods. (a) RLCresonator and (b) Colpitts oscillator.

In the simple 2DOF WCR circuit presented in Fig. 5.1, the sensing capacitor is

one of the capacitors of the series resonators, which in our case is C2 ,C1+∆C. ∆C

is the sensor capacitance change to be measured. Its normalized value, normalized

perturbation, is defined as δ , ∆C/CC, where CC is the coupling capacitance. The

relative sensitivity of both CFC and WCR methods in detecting the perturbation

is explained in [65]. The CFC method is based on the relative shift in resonant

frequency, while the WCR method is based on the relative shift in ratio of the loop

currents (|I1/I2|).

Both in-phase and out-of-phase resonant frequencies are directly related to the

system eigenvalues, while loop currents (capacitance charges) are related to the

84


eigenvector components. Symmetry breaking phenomenon occurs around the point

of zero perturbation, where the orientation of the eigenvectors changes at a much

faster rate than the complimentary changes in the eigenvalues magnitudes. This is

shown in Fig. 5.6. The region around the zero perturbation is called the veering

zone.

Figure 5.6: Eigenvalue loci veering.

5.2.1 Measurement Sensitivity

Fig. 5.1 shows a simple weakly coupled resonators circuit based on series RLC

circuits. The first goal is to carry out closed loop equations for magnitudes of loop

currents, I1 and I2, and their ratio. This will be solved in two unperturbed and

perturbed cases.

First assume an unperturbed, ∆C = 0, case. The circuit is excited by a single

harmonic voltage signal. The phasor equation for the circuit can be interpreted by

85


equation (5.1).

z11 z12

z21 z22

I1

I2

=

Vm∠0

0

, (5.1)

where

z11 = z22 = R+ jωL+ 1jω

(1C + 1

CC

)and

z12 = z21 =− 1jωCC

.

This can be simplified further by:

1ωC

M

I1

I2

=

Vm∠0

0

, (5.2)

where

M =

m11 m12

m21 m22

,

m11 = m22 = RCω + jLCω2 + 1j (1+ k),

m12 = m21 = jk and k = CCC

.

Solving (5.2) for I1 and I2:

I1

I2

= M−1

Vm∠0

0

Cω, (5.3)

|I1|=VmCω

|det(M)|

√(RCω)2 +(LCω2−1− k)2, (5.4)

and

86


|I2|=1

|det(M)| (kCω)Vm, (5.5)

where

det(M) =(RCω + j(LCω

2−1− k))2

+ k2. (5.6)

Hence ratio of loop currents will become:

∣∣∣∣ I1

I2

∣∣∣∣= 1k

√(RCω)2 +(LCω2−1− k)2. (5.7)

Both |I1/I2| and |I1| have minimum values at:

ωz =

√1+ k√LC

(5.8)

which is in the middle of the two resonant frequencies of ω01 and ω02 with the

values of:

ω01 =1√LC

(5.9)

and

ω02 =

√1+2k√

LC(5.10)

The assumption is that the parameters δ , k and R are small enough so that all

second and higher order terms of these parameters could be neglected.

Equation (5.7) can be simplified by linearization around either of the resonant

frequencies, e.g. ω0 = 1/√

LC:

∣∣∣∣ I1

I2

∣∣∣∣= 2√

LCk

∣∣∣∣ω− 1√LC

∣∣∣∣+ 1k

√R2C

L+ k2 (5.11)

87


Now we consider more general case by assuming the presence of perturbation

∆C on capacitor C2 i.e. C2 =C+∆C. The equation (5.2) becomes:

1ωC

m11 m12

m21 m22 + j δ

1+δ

I1

I2

=

Vm∠0

0

(5.12)

where δ = ∆C/C. Solving (5.12) for I1 and I2 leads to:

I1

I2

= N−1

Vm∠0

0

Cω, (5.13)

where

N , (M+δM) =

m11 m12

m21 m22 + j δ

1+δ

. (5.14)

If perturbation is small enough, δ 1 i.e. (1+δ )−1 ≈ 1−δ , then

|I1|=

√(RCω)2 +(LCω2−1+δ − k)2

|det(N)| CωVm, (5.15)

and

|I2|=kCω

|det(N)|Vm, (5.16)

hence

∣∣∣∣ I1

I2

∣∣∣∣= 1k

√(RCω)2 +(LCω2−1+δ − k)2. (5.17)

88


Similar to the unperturbed case, both |I1/I2| and |I1| have minimum values at:

ωz =

√1+ k−δ√

LC, (5.18)

which is slightly shifted in comparison with the unperturbed case. Unlike the un-

perturbed case, this zero is not in the middle of the two resonant frequencies of:

ω01 =

√1−δ/2√

LC(5.19)

and

ω02 =

√1+2k−δ/2√

LC. (5.20)

The approximation (1+ k−δ/2)1/2≈ 12

[(1+2k−δ/2)1/2 +(1−δ/2)1/2

]is used,

where k 1 and δ 1.

Assuming circuit is symmetrical and all component values, except for C2, are

constant. The function |I1/I2| in (5.17), z(ω,δ ) , |I1/I2|, could be linearized in

the vicinity of δ = δ1 = 0 and ω = ω1 = ((1+2k)/LC)1/2 using Taylor series:

z(ω,δ ) ' z(ω1,δ1)

+∂

∂ωz(ω1,δ1)(ω−ω1)

+∂

∂δz(ω1,δ1)(δ −δ1) . (5.21)

The first order derivatives in (5.21) could be extracted from (5.17).

∂

∂ωz(ω1,δ1) =

1k

R2C2 +2kLCd

' 2k

√LC, (5.22)

89


∂

∂δz(ω1,δ1) =

1d' 1

k, (5.23)

where d ,(

R2CL (1+2k)+ k2

)1/2.

Approximations in equations (5.22) and (5.23) are done based on the R2C/L

k assumption which is true for this circuit.

Using values of δ1 = 0, ω1 = ((1+2k)/LC)1/2 = 31832.897, k' 1/150, RC'

10−9 and LC ' 10−9, (5.22) and (5.23) can be simplified further to:

z(ω1,δ1) =1k

√(RCω1)

2 +(LCω12−1+δ1− k)2

=1k

√R2C

L(1+2k)+ k2 (5.24)

' 1

∂

∂ωz(ω1,δ1)'

2k

√LC = 0.0095 (5.25)

∂

∂δz(ω1,δ1)'

1k= 150 (5.26)

Using these approximations, (5.21) can be re-written in simplified form of:

∣∣∣∣ I1

I2

∣∣∣∣= 1+2√

LCk

(ω−

√1+2k

LC

)+

1k

δ (5.27)

Using component values for the circuit related to this work:

90


∣∣∣∣ I1

I2

∣∣∣∣= 1+0.0095(ω−31832.897)+150δ (5.28)

or

∣∣∣∣ I1

I2

∣∣∣∣= 1+0.0597( f −5066.363)+0.0015∆C (5.29)

where f and ∆C are in Hz and pF respectively. Although the focus of this work

is around out-of-phase mode but with a similar approach around in-phase mode,

ω1 = (LC)−1/2 = 31622.777, the following linearization would be resulted.

∣∣∣∣ I1

I2

∣∣∣∣= 1− 2√

LCk

(ω− 1√

LC

)− 1

kδ , (5.30)

∣∣∣∣ I1

I2

∣∣∣∣= 1+0.0597(5032.921− f )−0.0015∆C. (5.31)

With approximation√

1+2k ≈ 1+ k and√

1+ k ≈ 1+ k/2, both in-phase and

out-of-phase equations for |I1/I2| can be combined to a single equation:

∣∣∣∣ I1

I2

∣∣∣∣= 2√

LCk

∣∣∣∣ω− 1√LC

(√1+ k− δ

2

)∣∣∣∣+ ε. (5.32)

Where ε is the error of linear approximation. This linearization should be very

precise around resonant frequencies and less precise around the corner where two

lines intersecting (ωz). To examine this lets calculate ε at these two frequencies

assuming there is no perturbation. At ω = ωz = ((1+ k−δ )/LC)1/2 :

91


ε =1k

√(RCωz)

2 +(LCωz2−1− k)2

=RC

k√

LC

√1+ k ' 0.047. (5.33)

This value matches with the minimum of the curve in Fig. 5.11. The next point

of interest is at out-of-phase resonant frequency, ω = ω1 = ((1+2k)/LC)1/2 :

ε =1k

√(RCω1)

2 +(LCω12−1− k)2−1

=RC

k√

LC

√1+2k+L(k)2/R2C−1 (5.34)

' 0.001139. (5.35)

As it can be seen, the error at resonant frequency is negligible. This error is the

equivalent of kε = 7.6×10−4% perturbation.

At this point we are going back to 5.17, which calculates the ratio of the loop

current magnitudes as:

∣∣∣∣ I1

I2

∣∣∣∣= 1k

√(RCω)2 +(LCω2−1+δ − k)2. (5.36)

The frequency responses of (5.15), (5.16) and (5.36) are shown in Fig. 5.7.

These parameters are shown for three different cases−one unperturbed and two

opposite perturbation values. For each perturbation value, the loop currents, as

well as any other physical parameters of the system, have two resonant frequen-

cies. For the unperturbed case, δ = 0, loop currents |I1| and |I2| are overlapping

92


at either in-phase or out-of-phase resonant frequencies. In positive perturbation

cases, e.g. ∆C = 100 pF or δ = 0.1% , the system has lower resonant frequen-

cies, with |I1| higher than |I2| for in-phase excitation and lower for out-of-phase

excitation. For negative perturbations, e.g. ∆C =−100 pF or δ =−0.1%, the sit-

uation is reversed. The frequency band of interest is either around the in-phase or

out-of-phase resonant frequencies.

5020 5030 5040 5050 5060 5070 50800

5

|I 1|a

nd|I 2

|(mA)

5020 5030 5040 5050 5060 5070 50800

1

2

|I 1|/

|I 2|

f (Hz)

5020 5030 5040 5050 5060 5070 50800

1

2

5020 5030 5040 5050 5060 5070 50800

1

2

|I1| / |I2| for +100 pF

|I1| / |I2| for no purturbation

|I1| / |I2| for -100 pF

∆C = -100 pF

No Perturbation (∆C = 0)∆C = +100 pF

Figure 5.7: Frequency response of the system for three values of perturbation δ =−0.1%, 0%and 0.1%.

As explained above and, in (5.32) and Fig. 5.7, |I1/I2| has a linear behavior in

either resonant frequencies neighborhood. This can be expressed by:

|I1/I2|= 2k

√LC∣∣∣ω− 1√

LC

(√1+ k− δ

2

)∣∣∣+1

k

(R2C

L (1+2k)+ k2)0.5

.(5.37)

In the case of exciting the circuit at the neighborhood of the out-of-phase reso-

nant frequencies, the (5.37) becomes:

93


∣∣∣∣ I1

I2

∣∣∣∣≈ 1+2k

√LC

(ω−

(1+2k

LC

)0.5)+

δ

2k. (5.38)

As an example for L = 10 mH, C = 100 nF and k = 1/150:

∣∣∣∣ I1

I2

∣∣∣∣= 1+0.0597( f −5066.363)+0.0015∆C, (5.39)

where f and ∆C are given in Hz and pF, respectively. Equation (5.38) is valid for

a wide range of excitation frequencies. It has the maximum error around the zero

of the system, which is not a useful excitation frequency. It is very precise in the

resonant frequency area, which is the focus of this work. For the high sensitivity

WCR method, which is based on the veering phenomenon, the following equation

can be used to backtrack the perturbation δ from the measurement of the loop

currents [65].

∆

∣∣∣∣ I1

I2

∣∣∣∣= δ

2k(5.40)

or

∣∣∣∣ I1

I2

∣∣∣∣= 1+δ/2k, (5.41)

where k ,C/CC 1 is the coupling factor of the circuit.

Similarly, for the CFC method, the following equation is applicable [11, 65]:

∣∣∣∣ ω

ω0−1∣∣∣∣= δ

4(1+2k)≈

δ

4, (5.42)

where ω0 =1√LC

or ω0 =1√LC

√1+2k for in-phase or out-of-phase cases, respec-

94


tively. For the out-of-phase case, (5.42) can be written as:

ω =

√1+2k−δ/2√

LC≈ ω0

(1− δ

4(1+2k)

). (5.43)

Equations (5.40) and (5.42) show that the WCR method has considerably ( 2k

times) higher sensitivity than CFC method. The lower the coupling factor, the

higher the sensitivity of WCR. For instance, for k = 0.02, the WCR method is

almost two orders of magnitude more sensitive than the CFC method.

While higher sensitivity is a desired performance metric, we are also interested

in improving the detection limit of the system. Two distinct type of noises limit

this factor. The thermal noise determined by the resistance in the circuit and the

mismatch between the excitation frequency and the actual resonant frequency. The

low resistance of the RLC circuit makes the thermal noise to be negligible with

respect to the latter source of uncertainty.

5.2.2 Measurement Error

As it is assumed in all the sensitivity calculations, to have a precise estimation of

the perturbation, the circuit has to be excited at the resonant frequency. All equa-

tions (5.37) to (5.43) are only valid at resonant frequencies, either the in-phase or

out-of-phase. For both WCR and CFC based capacitance measurements, it is very

crucial to keep the excitation precisely at resonant frequency and to dynamically

controlling it according to the changes in perturbation. Any mismatch between

the excitation and actual resonance frequencies leads to an erroneous perturbation

calculation.

The objective of this work is to introduce a method that is less sensitive to

95


excitation frequency errors. Interestingly, this simple circuit shows a very unique

characteristic that helps us achieving our goal. Considering an out-of-phase sce-

nario, if the excitation frequency is not exactly locked at resonant frequency, it will

introduce an error in estimating perturbation, δ , in either of these methods. The

errors associated with CFC and WCR methods have almost the same ampli-

tude, but opposite signs. This feature can be exploited to drastically reduce the

reading error by combining the results of both methods. Analytical proof of this

finding is presented below.

There is a one-to-one relationship between δ and ω or, δ and |I1/I2| in CFC

or WCR methods, respectively. Assume there is an error of ωerr , ω0−ω ( ferr ,

f0− f ) in excitation frequency. This causes both CFC and WCR method to estimate

the perturbation with errors. This is illustrated in 5.8.

jI1=I

2j

jI1=I

2j

±f

f (Hz)

±0 + "(±f)

±0

±0

±0+"(±

i)

f (Hz)

±i

f0+ferr

f0

Figure 5.8: Error comparison and improvement by hybrid method.

96


In WCR method, this error causes our system to compute a different |I1/I2|

according to (5.38). This deviation in |I1/I2| propagates to the final estimation of

the perturbation, given by (5.41). In summary:

δ = 2k

(∣∣∣∣ I1

I2

∣∣∣∣+2

√LCk

ωerr−1

)(5.44)

= 2k(∣∣∣∣ I1

I2

∣∣∣∣−1)+4√

LCωerr (5.45)

= 2k(∣∣∣∣ I1

I2

∣∣∣∣−1)+4√

2k+1ω0

ωerr (5.46)

where 2√

LCk ωerr is the propagated error to |I1/I2|. Looking at (5.46), the ulti-

mate error in estimating perturbation δ using WCR method will be defined as:

ε(δi), 4√

2k+1ω0

ωerr. (5.47)

Similarly,in the case of monitoring the resonant frequency shift, ωerr generates

a perturbation estimation error given by (5.43):

δ = 4(1+2k)(

1− ω +ωerr

ω0

)(5.48)

= 4(1+2k)(

1− ω

ω0

)−4

1+2kω0

ωerr. (5.49)

The final error in estimating perturbation using CFC method can be defined as:

ε(δ f ),−41+2k

ω0ωerr. (5.50)

97


The key point here is that the final errors in perturbation estimation in these

two methods, ε(δi) and ε(δ f ), have opposite signs. By combining equations (5.46)

and (5.49) to eliminate ωerr term, an unbiased actual perturbation estimation (δ )

is achievable.

δ =2√

1+2k1+√

1+2k

(2(

1− ω

ω0

)+ k(∣∣∣∣ I1

I2

∣∣∣∣−1))

. (5.51)

Figure 5.9 shows the effects of this hybrid method in correcting the error. The

horizontal axis is the excitation-resonance frequency mismatch ( ferr , ωerr/2π).

As an example, an arbitrary and fixed perturbation of about 0.025 is considered

as the reference point. The ideal perturbation curve would be a horizontal line at

0.025. The two lines crossing each other with opposite slopes are perturbation val-

ues estimated using CFC and WCR methods. The hybrid method result is plotted

and matches very closely with the ideal (no mismatch error) line.

−0.2 −0.1 0 0.1 0.20.01

0.015

0.02

0.025

0.03

0.035

0.04

δ

ferr (Hz)

δaδfδiδh

Figure 5.9: Error comparison and improvement by hybrid method.

98


To illustrate this with an example, assume ∆Ca is the actual capacitance change,

corresponding to actual perturbationδa. With ∆Ca = 25pF or δa = 2.5×10−4, the

resonant frequency will be:

fres =ω2

2π=

3.18309×104

2π= 5066.05Hz. (5.52)

If the excitation frequency is off by +0.1 Hz ( f = 5066.15 Hz), then the esti-

mated perturbations will be:

δ f = 4√

1+2k(

1− ff 0

)= 1.669×10−4, (5.53)

and

δi = 2k(∣∣∣∣ I1

I2

∣∣∣∣−1)=

2150

(1.025−1) = 3.299×10−4, (5.54)

which correspond to ∆C f = 16.69pF (-33% error) and ∆Ci = 32.99pF (32% error)

respectively.

Using the hybrid method, the estimated ∆C, defined as ∆Ch, will become:

∆Ch = (1.0133/2.0133)1/2 (16.69+32.99) = 25.005pF ,

which has 0.02% error with respect to the actual value of 25pF.

99

5.3. Simulations

5.3 Simulations

Several SPICE simulations were performed to illustrate the circuit behavior and

support the analytical results. Fig. 5.10 shows a more detailed view of the peaks of

frequency responses of |I1| and |I2| in the region of out-of phase resonant frequen-

cies for different values of δ . The middle peak is related to the unperturbed case,

δ = 0, for which both loop currents have identical amplitudes at the resonant fre-

quency. This is marked with the vertical arrow on Fig. 5.10. Increasing δ reduces

the resonant frequency, increases the amplitude of I1 and slightly decreases the am-

plitude of I2 (thus increasing |I1|/ |I2|). Conversely, decreasing δ reduces both |I1|

and |I2|. Note that the changes in |I2| are not significant at resonant frequencies for

different values of δ . The dotted lines in Fig. 5.10 are used as a guide to show

the trend of the loop current amplitude peaks As it can be seen in Fig. 5.10, for

the ±90 pF range for ∆C (0.18% change in δ ) the relative changes of resonant fre-

quency and |I1|/ |I2| are 0.06% and 14%, respectively. The higher |I1|/ |I2| makes

the sensing easier for the analog front end.

5065 5065.5 5066 5066.5 5067 5067.5 50684.5

4.6

4.7

4.8

4.9

5

5.1

5.2

5.3

5.4

5.5

|I 1|a

nd|I 2

|(mA)

f (Hz)

|I1||I2|

-δ ↑δ ↑δ = 0

Figure 5.10: Amplitudes of I1 and I2 around out-of-phase resonant frequencies fordifferent perturbation values δ . From left to right ∆Cs are: -90, -60, -30, -10, 0, 10,30, 60 and 90 pF.

100

5.3. Simulations

Another set of simulations show the precision of the linearization in equation

(5.38). The circuit component values for this simulation were chosen to be R1 =

R2 = 0.1Ω, C1 = 100 nF, C2 = 100 nF + ∆C, L1 = L2 = 10 mH and CC = 15 µF.

The frequency response of |I1|/ |I2| is calculated for both the exact and linearized

solutions.

The simulation was performed using MATLAB and the results are plotted in

Fig. 5.11. As it can be seen, both exact and linear approximation curves match very

well in most of the frequency domain, especially close to the resonant frequencies.

The maximum error is in 5050 Hz neighborhood corresponding to the system zero

between the two resonant frequencies, which is not the region of excitation.

5030 5035 5040 5045 5050 5055 5060 5065 50700

0.2

0.4

0.6

0.8

1

1.2

1.4

|I 1|/

|I 2|

f (Hz)

exactlinear approx.

∆C=-100 pF

∆C=+100 pF

∆C=0

Figure 5.11: Analytical: linear approximation vs. exact for |I1|/|I2|.

To validate the analytical expressions, SPICE simulator was used to provide the

results shown in Fig. 5.12. The values used for this simulation are the same as the

ones employed for the analytical estimation. The exact solution, the linearization

and the SPICE numerical simulations all provide results validating the technique

101


used.

5030 5035 5040 5045 5050 5055 5060 5065 50700

0.2

0.4

0.6

0.8

1

1.2

1.4

|I 1|/

|I 2|

f (Hz)

Analytical with ∆C = +100 pF

Simulation with ∆C = +100 pF

Analytical with ∆C = 0

Simulation with ∆C = 0

Analytical with ∆C = -100 pF

Simulation with ∆C = -100 pF

Figure 5.12: Analytical vs. simulation for |I1|/|I2|.

For different δ values, |I1|/ |I2| will have corresponding shift in value main-

taining their dependence on the excitation resonant frequency mismatch. This is

illustrated in Fig. 5.13.

5.4 Experimental Results

The components used for the hardware implementation were R1 = R2 = 0.1Ω,

C1 = C2 = 100nF, L1 = L2 = 10mH and CC = 4.7µF. C1 was considered as a

reference; C2 corresponded to the sensing capacitance. The perturbation of C2 was

generated by adding extra capacitors in parallel to the initial 100nF value. The test-

bench environment needed a voltage source with very low output impedance for

excitation, and an interface to measure the inductors or capacitors voltages. Mea-

surement of the analog values was done using a National Instrument PXIe-1062Q

102


5065 5065.5 5066 5066.5 5067 5067.5 50680.75

0.8

0.85

0.9

0.95

1

1.05

1.1

1.15

1.2

1.25|I 1

|/|I 2

|

f (Hz)

δ = +0.09%

δ = +0.06%

δ = +0.03%

δ = +0.01%

δ = 0.00%

δ = −0.01%

δ = −0.03%

δ = −0.06%

δ = −0.09%

Figure 5.13: |I1|/|I2| around out-of-phase resonant frequencies for different pertur-bation values.

system with a PXI-7854R analog interface module. A high-level block diagram of

the proposed instrumentation system is shown in Fig. 5.14.

The circuit PCB was designed using Altium Designer 12 and connected to the

PXIe-1062Q module from National Instruments. The LabView-FPGA software

toolkit was used for the data acquisition and processing. Three 16-bit analog in-

puts of PXI analog interface were used for reading excitation input, as a reference,

and both inductor-capacitor junction node voltages. The inputs were connected

to the PXI module via coaxial cables and BNC connectors. The only additional

equipment that is needed to set up the experiment was an external power supply, or

alternatively a pair of 9V batteries, to power up the input op-amp buffer.

A single tone sine wave generator and a simple PLL algorithm were imple-

mented using LabView-FPGA to keep the excitation frequency tracking the out-

of-phase resonance frequency, when different perturbations were applied to C2.

103


Figure 5.14: High-level-block-diagram of proposed capacitance readout.

The system was also capable of manually controlling the sine wave generator

by breaking the PLL loop, in order to test the designed PLL. In our experiments,

the capacitors voltage amplitudes were considered as components of the voltage

vector (i.e. v = [v1 v2]T ). v0 corresponds to v at balance (i.e. δ = 0) which

relates to the out-of-phase resonant frequency f0. The values for relative shift in

|v1/v2|, in reference to the same ratio at balance, is plotted vs. excitation frequency

range wide enough to include the second natural mode. These experimental results

are plotted in Fig. 5.15 which are resembling the simulation and analytical results.

As it is shown in Fig. 5.15a the plots are parallel lines that are vertically shifted

proportional to the perturbation value. To show the distinction between the parallel

lines clearer, a closer view of the results is plotted in Fig. 5.15b.

104


5100 5110 5120 5130 5140 5150

0.8

1.0

1.2

1.4

1.6

|V1

/ V2|

frequency (Hz)

δ = 0 % δ = 0.03 % δ = 0.06 % δ = 0.115 % δ = 2.15 %

δ increase

(a) Wider perturbation range.

5110 5120 5130 51400.87

0.90

0.93

0.96

0.99

1.02

1.05

1.08

1.11

1.14

1.17

|V1

/ V2|

frequency (Hz)

δ = 0 % δ = 0.03 % δ = 0.06 % δ = 0.115 %

δ increase

(b) Zoomed in, narrower perturbation range.

Figure 5.15: Magnitude of v1/v2 for excitation frequencies around out-of-phasemode for different perturbations.

Another experiment that shows the effectiveness of the proposed method in

system robustness due to frequency error is performed. In this experiment, a fixed

0.03% perturbation (∆C = 30pF) was applied on C2. Then the excitation frequency

was intentionally swept ±0.05% of the actual out-of-phase resonant frequency.

The results for perturbation estimation based on all three methods of frequency

shift (δ f ), eigenmode shift (δi), and hybrid method (δh) along with the actual per-

turbation (δa) which is a horizontal line at 0.03% are presented in table 5.1 and

plotted in Fig. 5.16. Note that all the values are expressed as percentages.

105


ωerr (%) δ f (%) δi (%) δh (%) δa (%)-0.049 0.240 -0.093 0.072 0.030-0.039 0.201 -0.068 0.065 0.030-0.029 0.160 -0.043 0.057 0.030-0.02 0.121 -0.019 0.050 0.030-0.01 0.081 0.006 0.043 0.030

0 0.040 0.031 0.035 0.0300.01 0.001 0.055 0.028 0.0300.02 -0.039 0.080 0.021 0.0300.029 -0.079 0.104 0.014 0.0300.039 -0.120 0.129 0.006 0.0300.049 -0.159 0.154 -0.001 0.030

Table 5.1: Experimental results for all three methods.

It is clear that the proposed method is considerably closer to the actual pertur-

bation value.

-0.06 -0.04 -0.02 0.00 0.02 0.04 0.06-0.2

-0.1

0.0

0.1

0.2

0.3 δf

δi

δh

δa

pert

urba

tion

(%)

ωerr (%)

Figure 5.16: Measurement error comparison.

106

5.5. Summary

5.5 Summary

Based on the proposal for using WCRs in measuring minute capacitance changes

in this thesis, two sensitivity analyses have been performed - one using WCRs and

the other using CFC for comparison. The sensitivity analysis has focused on the

error in estimated δ due to the excitation-resonance mismatch. We have performed

both exact and linearized theoretical approach, complemented by SPICE numeri-

cal simulations and experiments. The two estimation methods based on CFC and

WCR have a dependence on the excitation-resonance mismatch of almost equal

magnitude but opposite sign. Combining the two measurement techniques to a hy-

brid method allows us to obtain a capacitive sensing method that is both highly

sensitive (due to the WCR method) and robust to excitation resonant mismatches.

The estimated error with the new combined method is at least one order of mag-

nitude smaller than either of the WCR or CFC schemes. In the next chapter we

try to examine the use of three-degree-of-freedom (3DOF) WCRs in capacitance

measurement aiming more simplification in excitation method.

107

Chapter 6

3DOF WCRs in Capacitance

Measurement

6.1 Introduction

In last two chapters we have seen the features and advantages of using 2DOF

WCRs in capacitance readout design. This chapter explores the possibility and

features of exploiting higher (three)-degree-of-freedom WCR for capacitance read-

outs. We have used mechano-electrical analogy to directly transfer the analysis per-

formed on 3WCRs to the electrical domain, in terms of eigenvalues and eigenvec-

tors sensitivities. We then run SPICE simulations to validate the theoretical insight.

The majority of the capacitive sensors in MEMS, especially for accelerometers and

gyroscopes, are based on sensing differential capacitors. The theoretical analysis

proves that differential perturbations in 3DOF WCRs do not require resonance fre-

quency tracking. We have shown this insensitivity to the excitation frequency in

SPICE simulation.

108

6.2. Analysis

6.2 Analysis

As we have seen, a 2DOF coupled resonator has two normal modes. Similarly, a

3DOF resonator, e.g. spring-mass, system shown in Fig. 6.1 with stiffness per-

turbation, has three normal modes [68, 69]. k is the stiffness of each of the three

main spring, kc is the stiffness of the two coupling springs, m is the mass and ∆ki

(i = 1,2, 3) is the perturbation of the spring of the ith resonator. Perturbation can

act either as small mass asymmetries or be reflected in ∆k differences that break

the symmetry of the system.

m m m

kc kc

k1 + ¢k1 k2 + ¢k2 k3 + ¢k3

Figure 6.1: 3DOF coupled spring-mass system with stiffness perturbation.

Since the system is initially symmetric, the analysis of a single perturbation

modifying the first or third mass will be the same. It has been shown that, if the

perturbation modifies the spring stiffness of the first of the third resonator, the

resonant frequencies ωi and mode shapes of the system φi are [69]:

109

6.2. Analysis

ω1 = ω0

(1+

δ

6

), φ1 =

[1 1− δ

3κ1− δ

κ

]T

,

ω2 = ω0

(√1+κ +

δ

4

), φ2 =

[1 − δ

κ−1− δ

2κ

]T

,

ω3 = ω0

(√1+3κ +

δ

12

), φ3 =

[1 −2− δ

6κ1+

δ

2κ

]T

, (6.1)

where ω0 =√

k/m is the natural frequency of each of the individual spring-

mass resonators, κ , kc/k , and δ = ∆k/k.

Another important case corresponds to a differential perturbation, where the

perturbations applied to the first and third spring have the same magnitude but

opposite signs +δ and −δ , respectively. The second (middle) resonator stays un-

perturbed. In this case we have:

ω1 = ω0, φ1 =

[1 1+

δ

3κ1+

δ

κ

]T

,

ω2 = ω0√

1+κ, φ2 =

[1

δ

2κ−1+

δ

2κ

]T

,

ω3 = ω0√

1+3κ, φ3 =

[1 −2+

δ

6κ1− δ

2κ

]T

. (6.2)

As the above equation shows, the resonant frequencies are independent of the

differential perturbation δ (for δ 1). This is due to the fact that the loci of

resonant frequencies of the system with a varying differential perturbation have

extrema points in the veering zone (in the vicinity of δ = 0) for all three modes.

This is very helpful in simplifying the excitation circuit design since there is no

110

6.2. Analysis

need for a sophisticated frequency tracking feedback loop to continuously adjust

the excitation frequency depending on the perturbation value. It is also shown that

in 3DOF WCR system, the sensitivity of the mode shapes in differential perturba-

tion mode is almost twice as the mode shape sensitivity in the case of one-sided

perturbation [68, 69]. This feature makes the 3DOF WCR system attractive to be

used as differential capacitance measurement system in the electrical domain.

A 3DOF WCR circuit based on series RLC is shown in Fig. 6.2.

R1 L1 C1 R2 L2 C2 R3 L3 C3

CC1 CC1

Vs

Figure 6.2: 3DOF weekly coupled series RLC resonators.

The equivalent of the equation sets (6.1) (one-sided perturbation) for this RLC

circuit is:

ω1 = ω0

(1− δ

6

), φ1 =

[1 1− δ

3k1− δ

k

]T

,

ω2 = ω0

(√1+ k− δ

4

), φ2 =

[1 − δ

k−1− δ

2k

]T

,

ω3 = ω0

(√1+3k− δ

12

), φ3 =

[1 −2− δ

6k1+

δ

2k

]T

, (6.3)

and for (6.2) (differential perturbation) this will be:

111

6.2. Analysis

ω1 = ω0, φ1 =

[1 1+

δ

3k1+

δ

k

]T

,

ω2 = ω0√

1+ k, φ2 =

[1

δ

2k−1+

δ

2k

]T

,

ω3 = ω0√

1+3k, φ3 =

[1 −2+

δ

6k1− δ

2k

]T

, (6.4)

where k ,C/CC, δ ,∆C/C, and ω0 = 1/√

LC. As it can be seen the sensitivity

of the resonant frequencies to perturbation δ , for all modes, is zero.

We are expecting a shift in resonant frequency corresponding to the perturba-

tion for single-ended perturbation and no shift in differential perturbation. SPICE

simulations, parameter sweep analysis, verifies this fact as shown in figures 6.3 and

6.4.

3DOF WCRs one-sided perturbationPrinting Time:Tuesday, December 15, 2015, 11:49:10 PM

C3 sweep

Frequency (Hz)

5.02k 5.09k5.03k 5.08k5.05k 5.06k

I1 M

ag

nitu

de

0

600m

100m

500m

200m

400m

300m

δ = − 0.1%

δ = + 0.1%

δ = + 0.05%

δ = 0

δ = − 0.05%

Figure 6.3: Frequency shift of all three modes in one-sided perturbation of a 3DOFWCRs system.

112

6.2. Analysis

3DOF WCRs differential perturbationPrinting Time:Tuesday, December 15, 2015, 11:50:50 PM

C3 sweep

Frequency (Hz)

5.02k 5.09k5.03k 5.08k5.05k 5.06k

I1 M

ag

nitu

de

0

600m

100m

500m

200m

400m

300m

δ = + 0.1%

δ = + 0.1%

δ = + 0.1%

δ = + 0.1%

δ = + 0.1%

Figure 6.4: Frequency shift of all three modes in differential perturbation of a3DOF WCRs system

For one-sided perturbation in Fig. 6.3, the frequency response of loop currents

shows the change in the resonant frequency, while Fig. 6.4 shows almost a fix res-

onant frequencies. This is one of the major advantages of differential perturbation

over one-sided perturbation in 3DOF WCRs.

The sensitivity of mode shapes, defined as the relative changes in the angle

of the mode shape in reference to δ . To define sensitivity, modal vectors will

be plotted in three-dimensional Cartesian space with standard basis and take their

projection on the plane defined by the first and the third axes (akin to X-Z plane in

XYZ coordinate system) [69]. The angle Θ j made by the projection of jth modal

vector with the first axis can be used to define sensitivity of the mode. This way,

the sensitivities for resonant frequencies and mode shapes are defined as:

113

6.2. Analysis

Nω j ,

∣∣∣∣ω j−ω j0

ω j0

∣∣∣∣ , NΘ j ,

∣∣∣∣Θ j−Θ j0

Θ j0

∣∣∣∣ (6.5)

where tanΘ j = a3 j and a3 j is the third element of φ j. Using the above defini-

tion, the sensitivity for mode shapes can be calculated.

NΘ1 =

∣∣∣∣ 4π δ

k

∣∣∣∣ , NΘ2 = NΘ3 =

∣∣∣∣ 2π δ

k

∣∣∣∣ . (6.6)

As we will see in the next section, we calculate the sensitivity simply by the

ratio of the magnitude of the change in mode shape vector over its magnitude prior

to perturbation.

6.2.1 Differential Perturbation Detailed Analysis

To make the analysis of 3DOF WCRs circuit shown in Fig. 6.2 simpler, we can use

superposition technique and split it to the circuits shown in Fig. 6.5 and Fig. 6.6.

R1 L1 C1 R2 L2 C2 R3 L3 C3

CC1 CC2

Vd

2¡

Vd

2

Figure 6.5: 3DOF WCR schematic with differential excitation.

114

6.2. Analysis

R1 L1 C1 R2 L2 C2 R3 L3 C3

CC1 CC2

Vc Vc

Figure 6.6: 3DOF WCR schematic with common mode excitation.

Assuming Vd =Vs and VC =Vs/2, then the superposition of the sources on the

left hand side of the circuits results in Vs, and they cancel each other out on the

right hand side. The circuit components are R1 = R2 = R3 = R, L1 = L2 = L3 = L,

CC1 = CC2 = CC, C1 = C+∆C, C3 = C−∆C, and C2 = C. The perturbation δ is

∆C/C and coupling value is k =C/CC.

In differential case we can simplify the circuit as shown in Fig. 6.7. We first

consider the unperturbed case i.e. δ = 0. Fig. 6.7(a) is the same as Fig. 6.5.

The resonator elements are identical and henceZ1 = Z2 = Z3 = Z where Z( jω) =

R+ j(Lω−1/(Cω)) or Z(s) = R+(Ls−1/(Cs)) in Laplace domain.

CC1 CC2

Vd

2

Z1 Z2 Z3 ¡

Vd

2

(a) Differential Excitation.

CC

Vd

2

Z Z=2

(b) Differential excitation, half-circuit.

Figure 6.7: 3DOF WCR schematic with differential excitation.

By splitting Z2 in two series Z/2 impedances and using the symmetry of the

115

6.2. Analysis

circuit, the circuit can be simplified to half-circuit shown in Fig. 6.7(b).

Vd

2=

(Z +

1sCC

Z2

1sCC

+ Z2

)I1 =

(Z +

Z2+ZCCs

)I1 (6.7)

= Z3+ZCCs2+ZCCs

I1. (6.8)

If we calculate the input admittance Yin ,I1

Vd/2 :

Yin(s) =I1

Vd/2=

1Z

2+ZCCs3+ZCCs

, (6.9)

or

Yin( jω) =1

Z( jω)

2+ jZ( jω)CCω

3+ jZ( jω)CCω. (6.10)

So, the resonant frequencies of the system are the zeros/minima of |Z( jω)| or

|3+ jZ( jω)CCω|. These frequencies are:

ω1 =1√LC

= ω0 (6.11)

ω3 =

√1+3k√

LC=√

1+3kω0. (6.12)

The magnitude of Yin( jω) is:

116

6.2. Analysis

Yin( jω) =1

|Z( jω)||2+ jZ( jω)CCω||3+ jZ( jω)CCω| (6.13)

=

√√√√√√ ωC(1− ω2

ω20

)2+(ωRC)2

(1+2k− ω2

ω20

)2+(ωRC)2(

1+3k− ω2

ω20

)2+(ωRC)2

(6.14)

=ωC√(

1− ω2

ω20

)2+(ωRC)2

√√√√√√1+k2−2k

(1+3k− ω2

ω20

)(

1+3k− ω2

ω20

)2+(ωRC)2

. (6.15)

From the half-circuit in Fig. 6.7(b), I2 can be expressed versus I1 :

I2

I1=

1jωCC

1jωCC

+Z=

22+ jωCCZ

(6.16)

I2 =2k

1+2k− ω2

ω20+ jωRCC

I1 (6.17)

|I2| =2k√(

1+2k− ω2

ω20

)2+(ωRCC)

2|I1| . (6.18)

We are interested in the magnitudes of loop currents I1, I2 and I3 at resonant fre-

quencies ω1 and ω3. Note that due to the circuit symmetry, I3 = I1.

At ω = ω1 = ω0 = 1/√

LC :

Yin( jω0) =I1

(Vd/2)=

ω0Cω0RC

√1+

−5k2

9k2 +(ω0RC)2 (6.19)

I1 = I3 =2

3R(Vd/2) , (6.20)

117

6.2. Analysis

The approximation of ω0RCC k is used here, which is valid by a good margin

(more than 4 orders of magnitude). I2 will be:

_

|I2| =2k√

(2k)2 +(ω0RCC)2|I1| (6.21)

≈ |I1|=2

3R(Vd/2) . (6.22)

At ω = ω3 =√

1+3k/√

LC we have:

Yin( jω0) =I1

(Vd/2)=

ω0√

1+3kC3k

√1+

k2

(ω0RC)2 (6.23)

I1 = I3 ≈ 13R

(Vd/2) , (6.24)

and I2 will become:

|I2|=2k√

(k)2 +(ω0√

1+3kRCC)2|I1|= 2 |I1|=

23R

(Vd/2) . (6.25)

In summary, the differential excitation mode, stimulates mode 1 and mode 3.

The resonant frequencies and magnitudes of the loop currents are:

ω1 = ω0, φ1 =

(Vd

2

)(2

3R

)[1 1 1]T ,

ω3 = ω0√

1+3k, φ3 =

(Vd

2

)(1

3R

)[1 −2 1]T , (6.26)

118

6.2. Analysis

SPICE simulation shown in Fig. 6.8 verify the above analysis.

Figure 6.8: Frequency response of unperturbed 3DOF WCRs circuit with differen-tial excitation.

For common mode excitation using Vc = Vs/2, the circuit can be simplified

as shown in Fig. 6.9. The only difference with the previous case is applying

a common mode voltage VC to both ends of the circuit. This symmetry of the

circuit results in no current passing through the middle resonator i.e. I2 = 0, which

simplifies the circuit to the half-circuit shown in Fig. 6.9(b).

119

6.2. Analysis

i = 1; 2; 3CC1 CC2

Vc

Z1 Z2 Z3

Zi = Ri + j(!Li ¡ 1=!Ci)

(a) Common mode excitation.

CC

VcZ

(b) common mode excitation,half-circuit.

Figure 6.9: 3DOF WCR schematic with common mode excitation.

Solving the circuit in 6.9(b) is simple.

Yin( jω) =I1

Vc=

1Z + 1

jωCC

=1

R+ j(

Lω− 1ω

(1C + 1

CC

))=

Cω

RCω + j(

ω

ω0− (1+ k)

) . (6.27)

The resonant frequency of the circuit with common mode excitation would be

ω2 =

√1+ k√LC

= ω0√

1+ k. (6.28)

The magnitude of loop current I1 can be calculated from:

120

6.2. Analysis

∣∣∣∣ I1

Vc

∣∣∣∣ =Cω√

(RCω)2 +(

ω2

ω20− (1+ k)

)2. (6.29)

At ω = ω2 the magnitude of I1 becomes:

|I1| =Cω√(RCω)2

|Vc| (6.30)

=1R|Vc| . (6.31)

I1 has the same magnitude as I3 but it has opposite phase. As we stated above,

I2 = 0. In summary:

ω1 = ω0, φ1 =Vc

R[1 0 −1]T . (6.32)

SPICE simulation shown in Fig. 6.10 verify the analysis the circuit in mode 2.

121

6.2. Analysis

Figure 6.10: Frequency response of unperturbed 3DOF WCRs circuit with com-mon mode excitation.

As it can be seen from the solutions to the differential and common mode

excitation cases above, the differential excitation stimulates mode 1 and 3, and

the common mode excitation only stimulates mode 2. This could be very helpful,

especially in cases of lossy circuits, where one of the challenges are detecting and

differentiating between the two modes due to the overlapping of the resonance

peaks. This is shown in Fig. 6.11 for a 3DOF WCR circuit with higher resistance

values, 2Ω (Q≈160), for each resonator. As it can be seen, mode 1 and mode 2

frequencies are merging into each other in a way that are not detectable using a

one-sided excitation. The peak frequency is also misleading, since mode 1 and 2

peaks are merged and created only one peak at a frequency between mode 1 and

mode 2 resonant frequencies.

122

6.2. Analysis

Figure 6.11: 3DOF WCR, impact of loss on resonant frequencies.

The common mode excitation, which only excites mode 2, avoids the above

mentioned difficulty. Hence we choose common mode excitation for the rest of the

analysis.

6.2.2 System Response to Common Mode Excitation

We have chosen excitation in mode 2, which can be achieved by common mode

circuit excitation as explained above.We also chose differential perturbation since

it is a common operating mode for most of the capacitive MEMS sensors. More-

over, the resonant frequency in this configuration is independent of perturbation

magnitude.

We start the analysis by introducing the quality factor of our 3DOF WCRs

circuit in mode 2. The peak of I1 magnitude can be calculated from (6.29), which

is rewritten below.

123

6.2. Analysis

∣∣∣∣ I1

Vc

∣∣∣∣ =Cω√

(RCω)2 +(

ω2

ω20− (1+ k)

)2. (6.33)

The peak happens at ω = ω2 = ω0√

1+ k, which is:

∣∣∣∣ I1

Vc

∣∣∣∣max

=Cω2√(RCω2)

2(6.34)

=1R

(6.35)

The magnitude at low or high 3dB frequency, ω3dB, is 1/(√

2R)

. The lower

ω3dB is named ω3dBL and the high side is named ω3dBH .

∣∣∣∣ I1

Vc

∣∣∣∣3dB

=Cω3dB√

(RCω3dB)2 +(

ω23dB

ω20− (1+ k)

)2=

1√2R

(6.36)

Solving this equation for ω3dB give us:

ω3dBL =12

√(RL

)2

+4ω20 −

R2L

(6.37)

ω3dBH =12

√(RL

)2

+4ω20 +

R2L

. (6.38)

The bandwidth of the frequency response is:

124

6.2. Analysis

B.W.= ω3dBH −ω3dBL =RL, (6.39)

and the quality factor is:

Q =ω2

B.W.=

ω0√

1+ kR/L

=1R

√LC

√1+ k. (6.40)

Rearranging it for R results in:

R =

√1+ kQ

√LC

=ω2LQ

. (6.41)

We will use this equation later in this chapter to interpret the loss effect on

sensitivity versus quality factor Q.

6.2.3 Differential Perturbation Analysis in Common Mode

Excitation

As we shown above, if our unperturbed 3DOF WCRs series RLC circuit is excited

in common mode configuration at mode 2 resonant frequency, it results in a zero

value for I2 loop current and an I1 loop current with value represented in (6.27). I3

will have the same magnitude as I1 does, with an opposite phase i.e. I3 = −I1. In

other word, we have an unperturbed system with the following state:

125

6.2. Analysis

I1 =Cω

RCω + j(

ω2

ω20− (1+ k)

)Vc (6.42)

I2 = 0 (6.43)

I3 = −I1. (6.44)

Applying a differential perturbation on C1 and C3 perturbs the system and conse-

quently the loop currents. The differential perturbations can be expressed as:

C1 = C+∆C =C (1+δ ) (6.45)

C2 = C−∆C =C (1−δ ) , (6.46)

where δ = ∆C/C.

In order to compute the new system response we will use perturbation anal-

ysis technique, assuming δ 1. In this analysis, we interpret all the perturbed

loop currents as proportionally related to the unperturbed current I1 i.e. I1 will be

changed to I′1, I2 changes to I

′2, and I3 becomes I

′3. This could be formulated as

shown below.

I′1 = I1(1+ ε1) (6.47)

I′2 = I1(ε2) (6.48)

I′3 = −I1(1+ ε3), (6.49)

where εi (i = 1,2,3) are small values representing the sensitivity of the changes

126

6.2. Analysis

in loop currents to perturbation. We solve the circuit in Fig. 6.6 for loop currents

under the perturbation condition. the main equations will be:

(Z + 1

jωCC− δ

jωC

)I′1− 1

jωCCI′2 =Vc(

2jωCC

+Z)

I′2− 1

jωCC

(I′1 + I

′3

)= 0(

Z + 1jωCC

+ δ

jωC

)I′3− 1

jωCCI′2 =−Vc,

(6.50)

where the following approximations are used.

1C±∆C

=1C(1∓δ ). (6.51)

At resonant frequency, ω2 = ω0√

1+ k, we will have:

Z +1

jω2CC= R, (6.52)

or

Z = R− 1jω2CC

. (6.53)

We assume δ and εi (i = 1,2,3) are small enough that we can eliminate any of their

second order or higher terms and their products. The loop currents with eliminating

term δε1 will become:

I1(1+ ε1)

(R− δ

jω2C

)− I1ε2 =Vc

I1ε2

(2

jω2CC− 1

jω2CC+R)− 1

jω2CC(I1(1+ ε1)− I1(1+ ε3)) = 0

−I1(1+ ε3)(

R+ δ

jω2C

)− I1ε2

1jω2CC

=−Vc

(6.54)

127

6.2. Analysis

which can be more simplified to:

Rε1 =

1jω2

(δ

C + ε2CC

)ε2R+ 1

jω2CC= 1

jω2CC(ε1− ε3)

Rε3 =− 1jω2

(δ

C + ε2CC

) (6.55)

Solving this equation for ε1, ε2 and ε3 , will result in:

ε1 =−ε3 =ε2

2(1+ jω2RCC) (6.56)

ε2 =1

1+ 12 (Rω2CC)

2 + j 12 Rω2CC

(−δ

k

). (6.57)

Applying (6.41) to the above equations with the approximation below

Rω2CC =

√1+ kQk

(6.58)

=1Q

√1k+

1k2 (6.59)

≈ 1Qk

(6.60)

results in:

ε2 =1

1+ 12Q2k2 + j 1

2Qk

(−δ

k

)(6.61)

ε1 =−ε3 =ε2

2

(1+ j

1Qk

)(6.62)

128

6.2. Analysis

For the ideal case, loss-less resonators, these values are:

ε1 = −ε3 =−δ

2k

ε2 = −δ

k

In this ideal case (Q→ ∞) the loop currents appear as spikes with infinite magni-

tudes (Dirac impulses in the frequency domain). In real cases, with very low losses

(very high Q), with the same assumptions for εi, the loop currents become:

I′1 =

Vc

R

(1− δ

2k

)(6.63)

I′2 =

Vc

R

(−δ

k

)(6.64)

I′3 =

Vc

R

(−1− δ

2k

)(6.65)

In other words the mode shape vector for high Q system is:

φ2 =Vc

R

[1− δ

2k− δ

k−(

1+δ

2k

)]T

. (6.66)

Now we consider more realistic case that Q is not desirably high. In this case we

compute the magnitudes of εi in (6.61) and (6.62) are affected by Q. The magni-

129

6.2. Analysis

tudes are:

|ε2| =δ/k√(

1+ 12Q2k2

)2+ 1

4Q2k2

(6.67)

|ε1|= |ε3| =

√1+ 1

Q2k2

2

√(1+ 1

2Q2k2

)2+ 1

4Q2k2

(6.68)

which can be simplified to:

|ε2| =1√

1+ 54Q2k2 +

14Q4k4

(δ

k

)(6.69)

|ε1|= |ε3| =

√1+ 1

Q2k2√4+ 5

Q2k2 +1

4Q4k4

(δ

k

)(6.70)

In our circuit here, the argument of εi , ∠εi, is very small and hence |1+ εi| =

1+ |εi|. This equation is not valid in general for complex numbers. Figure. 6.12

shows the arguments ∠ε1 and ∠ε2 versus quality factor Q.

0 100 200 300 400 500 600 700 800 900 1000−0.01

−0.005

0

0.005

0.01

0.015

0.02

0.025

0.03

Quality Factor

ε 1andε 2

arguments

()

Sensitivity in lossy resonators

ε1ε2

Figure 6.12: Phase plot of ε1 and ε2 at ω2 vs. Q factor.

130

6.2. Analysis

As it is graphed, the arguments are even less than 1 over the range shown for

Q.

In summary, when the circuit is excited in common mode, in the presence of

differential perturbation on C1 and C3 , the loop currents will change slightly to the

below values:

I1 =Vc

R(1+ |ε1|) (6.71)

I2 = −Vc

R|ε2| (6.72)

I3 = −Vc

R(1−|ε1|) (6.73)

where εis are

|ε2| = A(

δ

k

)(6.74)

|ε1|= |ε3| = B(

δ

k

)(6.75)

and coefficients A and B are functions of quality factor Q.

A =1√

1+ 54Q2k2 +

14Q4k4

(6.76)

B =

√1+ 1

Q2k2√4+ 5

Q2k2 +1

4Q4k4

(6.77)

The plot of A and B versus Q, Fig. 6.13, shows changes in A and B dependency to

Q. the higher the Q, the higher the sensitivity i.e. the effect of δ/k is translated to

131

6.3. Circuit Simulations

the loop current amplitudes with a higher coefficient.

0 100 200 300 400 500 600 700 800 900 10000

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Quality Factor

Coeffi

cients

Aan

dB

Effect of loss on sensitivity

AB

Figure 6.13: Magnitude plot of ε1 and ε2 at ω2 vs. Q factor.

This plot suggest two approaches. If we have high enough quality factors, the

same input perturbation δ will reflect into an output perturbation ε2 magnitude

larger than ε1 and ε3 current perturbations. For lower quality factors, nevertheless,

ε2 has a higher sensitivity to variations in the quality factors. In such cases, moni-

toring I1 and I3 seems to be a better solution. If the operating conditions guarantee

a stable Q factor, ε2 magnitude is larger than ε1 and ε3, and monitoring I2 results

into a higher sensitivity.

6.3 Circuit Simulations

6.3.1 Single-Sided Excitation, Differential Perturbation Case

In the following simulations it is assumed that the resonators are identical; per-

turbation is differential and occurs on C1 and C3, while the circuit is excited from

132


one end by the harmonic voltage source Vs (single-sided excitation with differential

perturbation) i.e.

C1 =C+∆C, C2 =C, C3 =C−∆C, CC1 =CC2 =Cc,

R1 = R2 = R3 = R, ,and L1 = L2 = L3 = L,

where k = CCC 1 and δ = ∆C

C 1 are conditions for weak coupling and small

perturbations respectively.

A SPICEr simulation of the circuit is done using Multisimr12 for Fig. 6.2

and the loci of the normal modes are plotted in Fig. 6.14. The values R = 100mΩ,

L = 10mH, C = 100nF, CC = 15µF are used in the simulation.

-0.050 -0.025 0.000 0.025 0.0504900

4950

5000

5050

5100

5150

5200

Nat

ural

freq

uenc

ies

(Hz)

δ

f1 f2 f3

Figure 6.14: Three WCR veering from SPICE simulation.

133


Based on the data collected from the simulations, the following relative shift

functions are also calculated and then graphed vs. perturbation changes. the slope

of the graph represents the sensitivity.

Sω1 =ω1−ω10

ω10, Sω2 =

ω2−ω20

ω20, Sω3 =

ω3−ω30

ω30, (6.78)

Sφ1 =

∣∣∣∣φ1−φ10

φ10

∣∣∣∣ , Sφ2 =

∣∣∣∣φ2−φ20

φ20

∣∣∣∣ , Sφ3 =

∣∣∣∣φ3−φ30

φ30

∣∣∣∣ , (6.79)

where φ j0 and φi ( j = 1,2,3) are the vectors with normalized loop currents

magnitudes as the vector components for the ith normal mode without perturbation

and with perturbation, respectively, e.g. φ1 = [I1 I2 I3] for mode 1 where I1 and

I2 and I3 are normalized magnitudes of the loop currents at 1st normal mode. Sφ i

is the relative shift in φi due to the perturbation δ .

-0.0050 -0.0025 0.0000 0.0025 0.0050

-0.1

0.0

0.1

0.2

0.3

0.4

Rel

ativ

e se

nsiti

vity

δ

100x|∆f / f0|

|Φ - Φ0| / |Φ0|

Figure 6.15: Three WCR, relative sensitivities, mode 1 excitation.

134


These two relative shift functions, Sωi and Sφ i are plotted in Fig. 6.15 to Fig.

6.16 for all three modes of the circuit with the component values stated above.

-0.0050 -0.0025 0.0000 0.0025 0.0050

0.0

0.1

0.2

0.3

0.4

0.5

Rel

ativ

e se

nsiti

vity

δ

100x|∆f / f0|

|Φ - Φ0| / |Φ0|


-0.0050 -0.0025 0.0000 0.0025 0.0050

0.00

0.05

0.10

0.15

0.20

0.25

Rel

ativ

e se

nsiti

vity

δ

100x|∆f / f0|

|Φ - Φ0| / |Φ0|


In these figures the Sωi curves are magnified by a factor of 100 to be better

comparable with Sφ i curves. This eigenmode sensitivities are similar, in magni-

135


tudes, to 2DOF WCR configuration. The advantage is, nevertheless, the invariance

of the resonant frequencies on the differential perturbation magnitude, simplifying

the readout circuit.

We choose the 2nd normal mode as excitation frequency of our system and

keep it fixed regardless of the perturbation value (δ ). The circuit was simulated

and the values for Sφ2 were calculated and then plotted in Fig. 6.18.

δ (%) i1(mA) i2(mA) i3(mA) I1 I2 I3 |φ2−φ02|/|φ02|-0.05 531.041 42.5 493.95 0.73096 0.05850 0.67991 0.04379-0.025 516.56 29.18 497.83 0.719455 0.0406 0.69336 0.0197

0 498.98 23.8 498.93 0.70674 0.03371 0.70667 00.025 479.22 31.21 497.83 0.69281 0.04512 0.71971 0.022240.05 456.96 45.27 493.95 0.67756 0.06712 0.7324 0.05129

Table 6.1: Measured values at fixed excitation frequency (mode 2).

-0.050 -0.025 0.000 0.025 0.050

0.00

0.02

0.04

0.06 |φ2-φ02| / |φ02|

Rel

ativ

e sh

ift in

mod

e sh

ape

δ (%)

Figure 6.18: Three WCR, relative shift in normalized loop current vector under fixexcitation at 2nd mode.

136


These simulation results show that, keeping the excitation frequency at unper-

turbed resonant frequency gives almost the same results (slope) as if the excitation

frequency tracks and locks to the exact resonant frequency

6.3.2 Differential Excitation, Differential Perturbation Case

A simulation based on a differential excitation and differential perturbations on

capacitors C1 and C3 is done. Two out of phase harmonic inputs excite the circuit

from both ends as shown in Fig. 6.5.

The results show that only mode 1 and mode 3 appear in this system. At mode

2 excitation all the loop currents are almost zero. The results are shown in Table

6.2 and 6.3.


0 666.8 666.48 666.79 0.5774 0.5771 0.5774 00.05 714.72 663.64 615.56 0.6197 0.5754 0.5337 0.060820.1 757.78 655.33 563.02 0.6594 0.5702 0.4899 0.12009

Table 6.2: Simulation results for differential excitation at first mode.


0 333.81 665.91 333.84 0.4089 0.8158 0.4090 00.05 321.6 665.78 346.61 0.3938 0.8153 0.4245 0.02160.1 309.86 665.55 359.94 0.3790 0.8140 0.4402 0.0433

Table 6.3: Simulation results for differential excitation at third mode.

The results show a relative sensitivity of almost 150 ( 1k ) and 50 ( 1

2k ) for mode

1 and mode 3, respectively.

137


6.3.3 Common-Mode Excitation, Differential Perturbation Case

Another simulation based on common mode excitation and differential perturba-

tion on C1 and C3 is done, as shown in Fig. 6.6. In this case, system only responds

to excitation mode 2. Loop currents for mode 1 and mode 3 are almost zero. the

results are shown in Table 6.4.


0 1000.0 0.0 1000.0 0.7071 0 0.7071 00.05 958.8 74.6 1032.9 0.6794 0.0529 0.7319 0.06460.1 911.4 146.6 1056.1 0.6498 0.1045 0.7529 0.1277

Table 6.4: Simulation results; common mode excitation at 2nd mode.

These results show a relative sensitivity of about 127 for mode 2 with common

mode excitation. This complies with the sensitivity of the mode shape calculated

in (6.66). The analytical value for the relative sensitivity would be:

Sφ2 =

∣∣∣∣φ2−φ20

φ20

∣∣∣∣=√

ε21 + ε2

2 + ε23√

1+0+1=

√3

2δ

k≈ 129.9δ (6.80)

Repeating all these simulations for resonator series resistances of 1mΩ (Q =

ω2LR ≈300,000) and 100mΩ (Q≈3,000) shows that the sensitivity is robust to Q

variations as predicted by theoretical analysis. There is a trade off between the

quality factor Q, coupling strength k, and the desired dynamic range of measured

δ . The higher the quality factor and the weaker the coupling strength are, the lower

the measurable δ range will be. Fig. 6.19 illustrates the effect of different quality

factors on the dependence of the resonant frequency f2 (mode 2), on the differential

138


perturbation magnitude.

5049.4 5049.6 5049.8 5050.0

0

20

40

60

i 1 (m

A)

frequency (Hz)

δ=−0.05% δ=−0.025% δ=0 δ=0.025% δ=0.05%

R = 1mΩ

(a) Q≈300,000 (R = 1 mΩ).

5046 5048 5050 5052 5054

0.1

0.2

0.3

0.4

0.5

0.6

i 1 (m

A)

frequency (Hz)

δ = −0.05% δ = −0.025% δ = 0 δ = 0.025% δ = 0.05%

R = 100 mΩ

(b) Q≈3,000 (R = 100 mΩ).

Figure 6.19: The effect of quality factor on f2- δ dependence.

139


In the example above, with 1mΩ series resistance, the frequency response is

so sharp, that keeping the excitation frequency fixed for a perturbation range of

±0.05% does not work, Fig. 6.19a. The series resistances of the RLC resonators

are increased by a factor of 100 to 0.1Ω to get measurable values for currents over

the full range of δ , Fig. 6.19b. On the other hand, if the resistance is high ( Q is

low) the sensitivity will decrease. For high quality factors, as shown in Fig. 6.19a,

f2 will slightly vary with δ , possibly requiring a tracking feedback loop for the

proper common mode excitation if the dynamic range of δ is wide. This behavior

contrasts with the lower quality factor as shown in Fig. 6.19b, where f2 is practi-

cally insensitive to δ magnitude. Compared to the 2DOF WCR readout circuits, the

measurement setup is simplified not only because of a fixed excitation frequency

(no need of resonant frequency tracking), but also because the measurement of the

differential δ perturbation reduces to monitoring a single current (I2) amplitude.

Fig 6.20 shows I2 magnitude vs. δ .

Figure 6.20: I2 magnitude for mode 2, common mode excitation, differential per-turbation.

In this figure the vertical axis is normalized by dividing to Vc/R. The slope

140


(sensitivity) compiles with the theory in (6.66). In this simulation Vc = 100 mV,

R = 100 mΩ, and coupling strength is k = 1/150.

Depending on the application, the amount of damping should be calculated

to fulfill both sensitivity and dynamic range trade-off. Moreover, as we have ob-

served in 2DOF WCRs, lower quality factors makes the detection of the resonant

frequencies difficult. This happens due to the fact that both amplitudes and the

resonant frequencies of the loop currents are affected by quality factors values. In

these cases, the coupling strength modifications might be helpful. The stronger

the coupling is, the farther the normal modes are pulled apart, hence the easier the

detection of their resonant frequencies becomes. This on the other hand, reduces

the relative sensitivity to the perturbation to be measured.

Another set of simulations shows the effect of Q on the sensitivity more clearly.

A parameter sweep on Q factor for differential perturbation with common mode

excitation (focused on mode 2) gave us the sensitivity results plotted in Fig. 6.21.

0.008

0.01

0.012

0.014

0.016

/ (V

s/R

)

Q>1000

Q=300

Q=150

0

0.002

0.004

0.006

0.008

-0.015 -0.01 -0.005 0 0.005 0.01 0.015

I 2/ (

δδδδ (%)

Q=60

Q=30

Figure 6.21: Effect of Q factor on the sensitivity (slope of normalized current I2);common mode excitation.

141

6.4. Summary

In this plot I2 is scaled in reference to the value of Vc/R.

As it can be seen, the lower Q the less sensitive the circuit is. The simulation

results plot of magnitude plot of ε2, shown as A, defined in (6.76), on vertical axis

in Fig. 6.22, is inline with analytical results estimated and plotted in Fig. 6.13.

0.8

1

1.2

0

0.2

0.4

0.6

1 10 100 1000 10000 100000 1000000

A

Quality factor (Q)

Figure 6.22: Effect of Q factor on the magnitude of ε2 (simulation).

6.4 Summary

In this chapter we explored the possibility and features of 3DOF WCRs in read-

out circuits. We showed that differential perturbation makes the overall circuitry

simpler, by eliminating the need for frequency tracking of the excitation voltage,

without sacrificing the relative sensitivity. We analytically showed that common

mode excitation at mode 2 resonant frequency simplifies the readout circuit, espe-

cially in the real scenarios with lossy elements. We have used perturbation theory

to obtain analytical estimates for eigenmode sensitivities to differential perturba-

tions. SPICE circuit simulations have validated the theoretical analysis. Practical

trade-offs between coupling strength, quality factor and desired perturbation range

142

6.4. Summary

to be measured have been discussed. In the following chapter, we discuss one of the

possible ways of improving circuit realization for WCRs based on active inductors.

143

Chapter 7

Active Inductors in WCRs

7.1 Introduction

This chapter examines the usage of active inductors (op-amp-based) for imple-

mentation of WCRs. The advantages and disadvantages of this method are also

examined.

The advantages are smaller size, with the possibility of die level integration of

WCR-based capacitance readout circuits.

The trend for the capacitive sensors is towards smaller physical dimensions,

and consequently lower capacitance values. In order to design coupled resonators

for smaller capacitance measurements and yet keep the resonant frequency low

enough to prevent ADC circuit complications, relatively large inductances are re-

quired. Although the circuit is simple, large value inductors are considerably bulky,

especially if a high quality factor is important (associated with a low DC series re-

sistance).

7.2 Real (Nonideal) Inductors

All real inductors have parasitic values causes their behavior deviates from their

ideal constitutive equations. There are different approaches to model a real induc-

144

7.2. Real (Nonideal) Inductors

tor using ideal components in the literature [70, 71, 72, 73]. One of the common

models, that is also used by CoilCraft Inc, one of the manufacturers of the induc-

tors, is shown in Fig. 7.1[73, 72]. One of the main parasitic components is the

equivalent series resistance (ESR) which is the DC resistance of the inductor (R2).

There is an additional frequency-dependent resistance (RVAR1), due to the skin ef-

fect, in series with the ideal inductor (LVAR). The parallel capacitance C causes a

self-resonance. If the self-resonant frequency is f0 then the value of the capacitor

C will be [71]:

C =1

(2π f0)2L.

(a) Model including the low power core losses(RVAR2).

(b) Simplified model without core losses.

Figure 7.1: Equivalent circuits for a real inductor (from CoilCraft Inc.).

R1 is the equivalent series resistance (ESR) of capacitor C. For a typical induc-

tor from CoilCraft the values of the equivalent circuits are:

RVAR1 = k1√

f

RVAR2 = k2√

f

145

7.2. Real (Nonideal) Inductors

LVAR = k3− k4 log(k5 f )

where k1 to k5 are empirical coefficients measured by the manufacturer. As

an example for CoilCraft LPS4018-323 inductor with nominal value of L = k3 =

3.3uH these coefficients are:

k1 = 1.8x10−4, k2 = 0.792, k3 = 3.3, k4 = 0.083, k5 = 9.8x10−6.

In our WCR circuit implementation, we have tried many different inductors

and capacitors, to choose the ones with the least loss (or highest quality factors)

and closest mach. In almost all types of inductors there is a trade-off between in-

ductor loss, physical dimensions and inductance. In our case we wanted to push

the capacitance value of the resonator as low as possible, to be closer to the real

cases of capacitive sensors. On the other hand we had the bandwidth limitation of

the National Instruments data acquisition system; PXI-7854R analog interface of

PXIe-1062Q system has a sampling rate of 750kHz. For practical implementation

of the reconstruction filters, the sampling frequency should be more than theo-

ritical Nyquist frequency; practically 5 to 10 times the excitation (resonant) fre-

quency.To have enough samples in order to measure the peaks and zero-crossing

precise enough, sampling rate should be considerably lower, e.g. 5 to 10 times,

than the highest frequency of the signal. This limitation leads us to pick a reso-

nant frequencies below 10kHz. Unfortunately lower resonant frequencies require

inductors with larger inductance values, and if a low resistance (high quality fac-

tor) inductor is required the inductor will become bulky. To allow for a die level

integration of the WCR techniques, replacing physical coils with active inductors

is a promising solution. It even helps solving the circuit size issue, if we can use

146

7.3. Active Inductors

them in our WCR system as a replacement for the main inductors.

7.3 Active Inductors

Using active components in conjunction with passive RC circuits is a known so-

lution in electronics. The concept has been used in active filter circuits for a long

time [74, 75, 76, 77] . One of the use of these active components and RC circuits

is to implement so called active inductors [78]. Realization of active inductors can

be gyrator gyrators [79] or non-gyrator based .

In this section we would like to study the feasibility of using active inductors

in WCR-based readout circuit. We also try to keep the circuit as simple as possi-

ble as one of the main advantages of our method. A simple gyrator-based active

inductor implementation is shown in Fig. 7.2. The circuit is shown in Fig. 7.2(a)

its equivalent passive circuit is shown in Fig. Fig. 7.2(b).

¡

+

Rs

Rp

Cp

Vi

Cp

Rp Leq = RpRsCp

Rs

Vi

(a) (b)

Figure 7.2: An example of an active inductor, (a) schematic, (b) equivalent circuit.

The input impedance of the active circuit can be calculated easily assuming the

op-amp is ideal i.e. infinite input impedance and gain, and zero output impedance.

Since both input of the op-amp are at the same potential (and is equal to ICpRp):

147


ICp

1sCp

= IRsRs. (7.1)

With a KVL from the input voltage Vi through Rs and considering the voltage

at the negative input of the op-amp is ICpRp:

Vi = IRsRs + ICpRp. (7.2)

By eliminating ICp from 7.1 and 7.2:

Vi = (1+ sRpCp)RsIRs , (7.3)

or

IRs =1

Rs (1+ sRpCp)Vi (7.4)

and

ICp =sCp

(1+ sRpCp)Vi. (7.5)

Knowing total input current is Ii = IRs + ICp , the input impedance Zincan be

written as:

Zin =Vi

Ii=

1+ sRpCp1Rs+ sCp

, (7.6)

Zin =Vi

Ii=

Rs + sRsRpCp

Rs + sRsCp. (7.7)

148


or with a bit of rearrangement, it can be written as:

Zin =Vi

Ii=

(1

Rs + sRsRpCp+

1Rp +

1sCp

)−1

, (7.8)

which is direct representation of the equivalent circuit shown in Fig. 7.2(b)

with the equivalent inductor value of Leq = RsRpCp. As it can be seen, Rs appears

as a resistance in series with the inductor Leq and Rp and Cp, which are in series,

appear in parallel with Leq and Rs branch. This is very close to the real inductor

model discussed in the previous section.

One fundamental limitation in realization of an inductor using simple gyra-

tors (same as our case), is that the inductor has always a grounded terminal. In

our WCR circuit we need at least one of the inductors to be floating. There are

some solutions presented in the literature to make a floating inductor using two

interconnected gyrators. Usually the interconnection happens at the capacitance

connections. Fig. 7.3 shows such a floating active inductor circuit based on our

single-ended gyrator of Fig. 7.2.

¡

+

Rs

Rp

Cp

Vi1 Vi2

U2

R0

p

C 0

p

R0

s

U1Vo1 Vo2

Figure 7.3: Realization of a floating inductor using gyrators.

To simplify the analysis this circuit is rearranged as shown in Fig. 7.4. The

equations for the circuit are:

149


Vi1−Vi2 = Rs (1+ sCpRp) IRs , ICp = sCpRsIRs

Vi2−Vi1 = R′s(1+ sC′pR′p

)I′Rs

, I′Cp= sC′pR′sI

′Rs

(7.9)

Assume R′s = Rs, C′p =Cp and C′s =Cs, then I′Rs=−IRs and I′Cp

= ICp , then:

IRs =1

Rs (1+ sCpRp)(Vi1−Vi2) , (7.10)

and

ICp =sCpRs

Rs (1+ sCpRp)(Vi1−Vi2) . (7.11)

¡

+

Rp

Cp

Vi1

Vi2

U1

U2

Vo1

Vo2

Rs

R0

p

C0

p

R0

s

Figure 7.4: Realization of a floating inductor using gyrators ,rearranged.

Eventually the differential input impedance of the circuit can be calculated as:

150


Zin =Vi1−Vi2

ICp + IRs

=Rs (1+ sCpRp)

1+ sCpRs, (7.12)

which is the same as 7.7 except that this inductor is floating between Vi1 and

Vi2.

To prove this realization works in a WCR system, the same 2DOF circuit in Fig.

4.7 was simulated with the inductors L1 and L2 are replaced with a floating and a

single-ended active inductors respectively as shown in Fig. 7.5. This simulation is

done using Multisim 12r. The values considered in this design are:

Rs = 8Ω, Rp = 100kΩ, Cp = 10nF

U1

LMV2011MA

3

2

4

7

6

C1 10nF

R1 8Ω

R2

100kΩ

V1

1 Vpk

5kHz

0°

R3

.1Ω

V3

12 V

V4

12 V

Vcc

Vcc

Vcc Vcc

-Vcc

-Vcc

-Vcc -Vcc

U2

LMV2011MA

3

2

4

7

6

C5

10nF

R7 8Ω

R8

100kΩ

C6

100nF

U3

LMV2011MA

3

2

4

7

6

C4

10nF

R9 8Ω

R10

100kΩ

C8

2µF

C10

100nF

R13

.1Ω

Vi

Figure 7.5: Circuit used in simulation of 2DOF WCR based on active inductors.

In this circuit, C6 and C10 are considered as the main capacitors of the two

coupled resonator circuits. The parametric sweep of C10 was done in the range of

99nF to 101nF i.e. a ±10% perturbation on C10. The AC simulation was done for

each value of C10 and the loop currents, IR3 and IR13, are plotted in Fig. 7.6 and

7.7. The sensitivity factor is taken as the relative shift in u , I3/I13 at resonance for

each value of C10. The values of this relative shift ,|ui−u0i|/ |u0i| , are presented in

table 7.1 and plotted in Fig. 7.8. The simulation results show that WCR method is

almost 30 times more sensitive than the relative resonance frequency shift method.

151


5000 5200 5400 5600 5800 6000 62000.00

0.01

0.02

0.03

0.04

0.05

0.06

0.07

0.08

0.09

Loop

cur

rent

s (m

A)

frequency (Hz)

IR3

IR13

Figure 7.6: Gyrator-based 2DOF WCR simulation at balance, loop currents fre-quency response.

5400 5600 58000.00

0.01

0.02

0.03

0.04

0.05

0.06

0.07

0.08

0.09

0.10

IR13

Loop

cur

rent

s (m

A)

frequency (Hz)

δ = -0.5%δ = -1%

δ = 0.5%δ = 1%

δ = 0

IR3

Figure 7.7: Gyrator-based 2DOF WCR simulation for different perturbations, loopcurrents frequency response.

152

7.4. Summary

C13 (nF) δ (%) i2(mA) i1(mA) i2(mA) |IR3/IR13| (u1−u01)/|u01|99 -1 5466.1 92.767 69.86 1.3279 0.07186

99.5 -0.5 5458.9 89.556 69.809 1.2829 0.03551100 0 5451.7 86.341 69.693 1.2389 0

100.5 0.5 5444.4 83.148 69.523 1.1960 -0.03463101 1 5437.2 79.997 69.303 1.1543 -0.06826

Table 7.1: Gyrator-based WCR simulation results (mode 1 excitation).

-1.0 -0.5 0.0 0.5 1.0-0.10

-0.05

0.00

0.05

0.10

(u-u0)/u0

10 x ∆f/f0

Rel

ativ

e se

nsiti

vity

δ (%)

Figure 7.8: Relative sensitivity of gyrator-based 2DOF WCR.

7.4 Summary

In this chapter we have introduced a typical model for real inductors, we have

explained the reason behind the need for large inductors for our WCRs, and we

have suggested one alternative based on active inductors. One of the main chal-

lenges related to the active inductors is the design of floating inductors. Our WCR

153

7.4. Summary

uses at least one floating inductor. This work offers a topology for realization

of floating inductors. A WCR circuit based on active inductors, one floating and

one grounded, is suggested and simulated. Simulation results are validating the

theory. The relative sensitivity comparison between frequency method and mode

shapes was presented and confirmed the higher sensitivity of the WCR method.

The following chapter summarizes the contributions, challenges and prospects of

this research.

154

Chapter 8

Conclusions and Further

Discussions

8.1 Research Contributions

The focus of this thesis was on applying innovative methods for capacitance read-

out circuits based on weakly-coupled-resonators, a technique applied in the elec-

trical domain for the first time. Other than the novelty of this method in electrical

domain, WCR-based capacitance readout circuits provide higher sensitivity (theo-

retically three orders of magnitude) over the method based on resonant frequency

shift method which is one of the state-of-the-art methods.

Capacitive sensors are one of the most popular sensors in various industries and

applications. Higher sensitivity capacitance readout circuits are of high interest in

recent years since MEMS capacitive sensors, with smaller capacitance changes,

have become common in the industry. This demand for high sensitivity and more

reliable readout circuits was the main motivations for the present work. The re-

search started with a literature survey of state-of-the art readout circuits in chapter

(2). Some of these circuits were simulated along the path of this project and pre-

sented in more detail in Appendix (A). One of the readout circuits, which is based

155

8.1. Research Contributions

on capacitance-to-voltage conversion was made with discrete component at PCB

level and used experimentally in another project conducted by Dr. Elie Sarraf [80].

The application of 2DOF WCRs has been proposed in chapter (3), accompanied

by a theoretical analysis and proof for its higher sensitivity, followed by validat-

ing simulations and experimental measurements presented in chapter (4). Tracking

and exciting the the resonance frequency accurately is essential to the success of

the 2DOF-WCR method. It is paramount that the resonant frequency is locked by

the excitation source with a high degree of precision. The accuracy of the reso-

nant frequency manifests itself in the sharpness of the resonant frequency peaks.

If circuit losses are present, the sharpness of the resonant frequency peaks is sig-

nificantly reduced, to the effect that no distinct peaks would be distinguishable if

losses are higher than a certain threshold. This of course adversely affects the reso-

nance detection in experiments. Moreover, the high sensitivity achievable in theory

(orders of magnitude better than the CFC method), will be significantly reduced as

shown in chapter 4.

Along the way, we have found an improved, hybrid method, combining fre-

quency shifting and energy localization measurements, as proposed in chapter 5.

This technique is more robust, higher insensitivity to the excitation frequency devi-

ations. In another step further, analytical and simulations studies of readout circuit

based on 3DOF-WCR have been presented in chapter 6. It has been proven that,

similar to the mechanical and MEMS domains, the 3DOF-WCR method can be

used to measure differential perturbations and has the advantages an invariant res-

onant frequency over a certain range of perturbations. This is helpful in simplifying

the circuit even more by eliminating the frequency tracking (e.g. PLL). This also

makes it inherently insensitive to errors in excitation frequency. Another unique

156

8.1. Research Contributions

behavior of 2DOF WCR circuits, under differential perturbation, is that only nat-

ural mode 2 or modes 1 & 3 gets stimulated if they excited by common mode or

differential source, respectively. This behavior becomes very helpful is real (lossy)

systems, where the low quality factor impacts the frequency response by pushing

the resonant peaks toward each other and overlapping them to a point that the cur-

rent peaks are not happening at any of the exact resonant frequencies. We have

proven that common mode excitation is insensitive to these impact of low quality

factor for differential perturbations.

One of the challenges of these WCRs, for relatively low resonant frequencies,

is the bulkiness of the inductors. We have presented the use of active inductors (e.g.

gyrator-based inductors) as an alternative way to reduce the circuit size and possi-

bly implementing the entire readout circuit in as a single integrated circuit. This

eliminates the need for physical inductors, as they are replaced by circuits based

on resistors, capacitors and op-amps which are much easier to be implemented in

integrated circuits. This was shown analytically in chapter 7 along with the related

simulations.

Another advantage of readout circuits based on WCRs is insensitivity to the

changes in the ambient conditions, e.g. temperature and relative humidity, due to

the circuit symmetry. They also require simpler analog circuit. On the other, fully

tuning and matching all resonators is a challenge which is needed to get a good

result from WCR based circuits. There is also a limitation on the bandwidth of the

perturbation relative to the resonant frequency since the perturbation assumed to

be quasi-static throughout this work. Another limitation of 2DOF WCRs, which is

eliminated in 3DOF WCR with differential perturbation, is the need for frequency

tracking system that adds to the system complexity.

157

8.2. Prospects and Open Problems

8.2 Prospects and Open Problems

Since the WCR-based readout circuits are being introduced in electrical domain

for the first time, there are several directions for expanding this research, some of

them being outlined below.

The circuit excitation can be done in different ways or with different wave-

forms. Several questions remain: are there more efficient ways of excitation?

Is differential excitation a good choice for differential capacitive sensors? Does

square wave excitation, binary level, have any advantage over sine wave excita-

tion? These questions are valid for both 2DOF and 3DOF WCR cases.

Another topic for research is to evaluate the effects of higher rate of capac-

itance changes on the circuit. There was a fundamental assumption throughout

this research, in accordance with the other researches related to mode localiza-

tion phenomena, namely the assumption of perturbations being quasi-static. This

assumption simplifies the analyses and simulations. Now that the basics of the

WCR-based readout circuit have been proven, there will be a good research topic

to study the bandwidth of the perturbation δ , or ∆C. This research and all the other

related literature considered quasi-static perturbation. There is an open question if

we go towards dynamic perturbation since it enters into nonlinear analysis due to

the fact that both capacitance and excitation signal are changing in time.

On the application of active inductors, there is still a lot of potential for new

research. Other types of active inductors could be designed, simulated and experi-

mentally tested.

There is a challenge about the inductor size and bulkiness, since the frequency

has to stay relatively low (limitation of an ordinary data acquisition system) while

158

8.2. Prospects and Open Problems

the sensor capacitance is low too. Finding solutions for this problem is another

topic for future work. One solution that we have thought about is to use an en-

velope detector before the data acquisition system. Knowing that in WCR-based

methods, the voltages and/or currents are more important than the instantaneous

signal, an envelope detection technique could be helpful to obtain the magnitudes.

The frequency of this envelope is equal to the rate of the capacitance changes,

which is much lower rate of change than the excitation (resonant) frequency. This

allows the use of smaller inductors.

Applying active inductors or methods like envelope detection explained above

or even combination of those could be another research topic. Considering new

technologies which allow MEMS device and CMOS circuitry on the same die is

very promising in pushing this idea even further to implement a single-chip WCR

based capacitive MEMS sensor and associated readout circuit.

There is also another potential to study a variation of WCR based capacitance

readout that keeps the system always at the balance (non-localized) by adding cali-

brated capacitors from a capacitor bank and adding them to the reference capacitor.

This should result in much larger dynamic range of perturbation.

The last point that we want to mention here is that all these WCR based meth-

ods and their variations can be used towards measuring small inductance too.

159

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170

Appendix A

Circuit Simulations and

Justification for Using CFC as the

Benchmark

A.1 Introduction

This chapter continues the discussion on the various methods for the readout cir-

cuits introduced in §2.3 with more detail. Detailed simulation (and one implemen-

tation) results for each of the readout circuits are presented.

A.2 CVC Simulation and Implementation Results

This section presents the circuit design and simulation results for the CVC readout

circuit. As is was mentioned before, there are various CVC readout circuits. The

one that we are simulating here is based on charge amplifier. This circuit was

implemented using discrete components and was used experimentally in a research

related to the closed loop readout circuits using sliding-mode control [80]. The

circuit schematic used for the simulation is shown in Fig. A.1 to A.3.

171

A.2. CVC Simulation and Implementation Results

N1 N2V_REF_1

VinCMN

CMN CMN

V_REF_2

GND GNDV_CTRL+1

V_CTRL-2

REF3

VCC1

VARICAP

V_CTRL+1

V_CTRL-2

REF3

VCC2

VARICAP

1E-12

C_REF_1Cap

GND

1E-12

C_REF_2Cap

GND

10K

R02Res2

V_BIASVSRC

GND

1nF

-10

C01

Cap

5NEG

5POS

OUT 1IN-2

IN+3

48 BUFFERA

OPAMP

1MEG

R01Res2

GND

COM1

NC2

GND3

V+4 VL 5IN 6V- 7NO 8S1

DG419L

5POS5NEG

GND 5POS 5POS

5NEGCLK

Figure A.1: CVC readout using charge integration, capacitor driving circuit.

172


N1

N2

10pF

C02

Cap

10pF

C03

Cap

5POS

5POS

5NEG

10meg

R03

Res2

10meg

R04

Res2

V1P

V1N

OUT+4IN-

1

IN+8

63

OUT- 5Vocm2

U4

THS4131

GND

5NEG

V2P

V2N

5POS

GND100pF

C05

100pF

C06120pF

C07

120pF

C08

20pF

C09

20pF

C10

36K

R05

120K

R06

120K

R07

OUT1

IN-2

IN+3

48 U1A

OPAMP

OUT7

IN-6

IN+5

48

U1BOPAMP

500R11

500

R13

Res2

500R12

500

R14

GND

5NEG

5POS

o1OUT+ 4

IN-1

IN+8

63

OUT-5Vocm

2

U5

THS4131

5NEG

V3P

V3N

5POS

GND100pF

C11

120pF

C13

120pF

C14

20pF

C15

20pF

C16

24K

R08

220K

R09

220K

R10

16K

R15

GND

OUT1

IN-2

IN+3

48 U6A

TQWFMRAP 100pF

C17 COM1

NC2

GND3

V+4

VL5IN6V-7

NO 8S2

DG419L

o2

GND

5POS

5NEG

5POS

GND

CLK

100pF

C12

V2P

V2N

Figure A.2: CVC readout using charge integration, input stage differential ampli-fier, filtration and demodulation.

8.2K

R18

1nF

C19

3.9K

R17

1.5nF

C18

GND

1.8K

R20

1nF

C21

1.5K

R19

12nF

C20

GND

OUT

o2

5NEG5NEG

5POS5POS

5POS

5NEG

OUT 7IN-6

IN+5

48

BUFFERB

BQHKHMFW

1MEG

R16

GND

OUT 1IN-2

IN+3

48

U9A

EAVJCLHD

OUT7IN-6

IN+5

48

U9B

ARLUJRXR

1MEG

R21

GND

12

ANALOG

Header 2GND

Figure A.3: CVC readout using charge integration, output buffer and LP filter.

In Fig. A.1 there are two voltage-controlled capacitors used to simulate the

173


behavior of a differential capacitive sensor. Two opposite phase voltage sources

are used to control these variable capacitors. The common node of the differential

capacitors, CMN node, is excited by a 1MHz square pulse train with a DC bias.

This DC bias is needed in case of using a sensor such as a MEMS capacitive ac-

celerometer to bias the circuit connected to proof-mass [80]. The signal injected

to common node passes through the differential capacitors and gets amplified by

an op-amp based differential amplifier followed by two stages of second-order

multiple-feedback (MFB) high-pass filters. The modulation with 1MHz carrier

makes the effect of 1/f noise negligible by pushing the information signal away

from 1/f noise frequency region before amplification. The amplified differential

output translates to single-ended using an op-amp and then is demodulated by the

same 1MHz clock. After demodulation the signal goes through two low-pass fil-

ters. The final output will be an amplified voltage proportional to the differential

capacitance changes. The input and output of the circuit are plotted in Fig. A.4 and

simulation results for the middle stages are shown in Fig A.5.

0.000u 100.0u 200.0u 300.0u 400.0u 500.0uTime (Sec)

Out

put (

V)

-1.000

-0.750

-0.500

-0.250

0.000

0.250

0.500

0.750

1.000

1.250

1.500

Del

ta_C

(pF

)

-200.0m

-150.0m

-100.0m

-50.00m

0.000m

50.00m

100.0m

150.0m

200.0m

250.0m

300.0m OutputDelta_C

Figure A.4: CVC based on differential charge amplifier, capacitance changes andoutput voltage plots.

174


0.000u 50.00u 100.0u 150.0u 200.0u 250.0u 300.0u 350.0u 400.0u 450.0u 500.0uTime (Sec)

Diff

eren

tial C

apac

itanc

e V

olta

ge

-50.00m

-40.00m

-30.00m

-20.00m

-10.00m

0.000m

10.000m

20.00m

30.00m

40.00m

50.00m (n1-n2)

(a) Differential capacitor output voltage.


Diff

. Cha

rge

Am

p. O

utpu

t (V

)

-400.0m

-300.0m

-200.0m

-100.0m

0.000m

100.0m

200.0m

300.0m

400.0m v1p-v1n

(b) Output of differential amplifier.


Firs

t Filt

er D

iff. O

utpu

t

-750.0m

-500.0m

-250.0m

0.000m

250.0m

500.0m

750.0m v2p-v2n

(c) First high-pass filter differential output.


2nd

Filt

er D

iff. O

utpu

t

-4.000

-3.000

-2.000

-1.000

0.000

1.000

2.000

3.000

4.000 v3p-v3n

(d) Second high-pass filter differential output.


Sin

gle-

ende

d M

odul

ated

and

Filt

erre

d O

utpu

tSig

nal (

V)

-4.000

-3.000

-2.000

-1.000

0.000

1.000

2.000

3.000

4.000 o1

(e) Output of differential to single-ended circuit prior to de-modulation.


Dem

odul

ated

Sig

nal (

V)

-3.000

-2.000

-1.000

0.000

1.000

2.000

3.000

4.000

5.000 o2

(f) Demodulated signal prior to low-pass filtration.

Figure A.5: CVC based on differential charge amplifier, intermediate nodes simu-lation waveforms

175

A.3. CDC Simulation Results

The SPICE model for variable capacitor (VARICAP) is given by:

!"#$ %

"&!

'(!"&

)*+,-./0

+.1+#-%#,2-%#/20%3333,

/0)),

,-,4

))313

)

5

6+

,

This model is shown in Fig. A.6.

Figure A.6: Varicap SPICE model.

Since a capacitance of 1pF is connected to the reference pin, the result of this

model is C[pF] = 1[V/pF]Vctrl[V].

A.3 CDC Simulation Results

This section presents the circuit design and simulation results for the CDC readout

circuit. The circuit schematic used for the simulation is shown in Fig. A.7 [27]

176


V0

B

A

C

C1

C2

V5

V3

Rt

R1

R2

D1 D2

V2

R3

A1 A2

A3

A4

V4

R4 R5

Figure A.7: Schematic representation of a CDC with the relaxation oscillator.

.

The differential capacitive sensor consist of is C1 = C0−∆C and C2 = C0 +

∆C. The duty cycle of output (V5) is linearly related to the differential capacitance

changes ∆C:

D =TH

TH +TL=

C2

C1 +C2=

12

(1+

∆CC0

),

where TL and TH are high and low time periods of output V5, respectively.

The SPICE simulation were done using the following parameters for the circuit,

and the results are shown in Fig XX.

R1 = R2 = R3 =1kΩ, Rt =125MΩ, R4 =5kΩ, R5 =20kΩ,

C0 =3pF, ∆C as a parameter varies between -0.45pF and 0.45pF.

177


∆C/C0 D-0.15 0.4258-0.1 0.4515

-0.05 0.47590 0.4993

0.05 0.52410.1 0.54730.15 0.5706

Table A.1: CDC circuit simulation data.

-0.2 -0.1 0.0 0.1 0.20.40

0.45

0.50

0.55

0.60

Dut

y C

ycle

C/C0

Figure A.8: Simulation graph for the CDC readout circuit.

178

A.4. CPC Simulation Results

CDC simulation

Time (s)0.0 3.0m500.0µ 2.5m1.0m 2.0m1.5m

Vol

tage

(V

)

-14

14

-9

9

-5

5

0

Figure A.9: Simulation results for the CDC readout circuit.

A.4 CPC Simulation Results

This section presents the circuit design and simulation results for the CPC readout

circuit. In CPC, the change in capacitance is translated into a change in the phase

shift of a sinusoid. Fig. A.10 presents the schematic of a CPC readout circuit [81].

It has two input voltages, Usin(ωt) as the main input and (U/a)sin(ωt +π − φ)

as compensating input. The main input goes through the sensor capacitance. The

value of the capacitance at rest (unperturbed) is C0 and the capacitance changes

(perturbation) subject to measure is ∆C. The compensating input goes through a

compensation capacitor CC. There are two conditions that are fundamental for this

circuit to appropriately function as shown in Fig. A.10. First, the ratio of the main

179


input voltage amplitude to the compensating input voltage amplitude should be

equal to the ratio of CCto C0, a ,CC/C0. Second, the condition ωRtCt 1 should

be met. The op-amp output voltage, Uo, was calculated to be:

U0 =−1Ct

√(∆C+

12

φ 2C0)2 +(φC0)2.Ucos(ωt−θ)

where

θ = arctan(

0.5φ 2C0 +∆CφC0

).

This method however is prone to non-linearity (in the form of tan−1), stray ca-

pacitance1 sensitivity and additional frequency dependent phase shift. An improve-

ment to this circuit was presented in [47] which mainly focused on linearization of

the phase shift using a buffer amplifier. on nonlinear phase shift using a charge

amplifier and differential capacitance. This improved configuration is presented in

Fig. A.11 and alleviates the issues mentioned above using two buffers creating an

almost frequency independent phase shift.

1Stray capacitance is a form parasitic capacitance which affects the phase shift.

180


Ct

C0 +¢C

Uo

Rt

A

CC

CC = aC0

!RtCt >> 1

U

asin(!t + ¼ ¡ Á)

Usin(!t)

Figure A.10: CPC readout circuit using charge amplifier.

The improved circuit has an achievable theoretical resolution of 2 ppm using

ideal components, however the experimented resolution of 0.7 fF in measuring a

22 pF capacitor was achieved which translates to 32 parts per million (ppm). This

is mainly due to practical limitations [47].

181


C

T

R

T

ADC

ComparatorCos(!t)

ADC

Comparator

Sin(!t)

LPF

LPF

DS

DC

VS

VC

VO1

Cx

C0

From

Sinusoidal

Oscillator

R R

R R

R

1

C

1

V

B

= Bsin(!t + ¼ ¡ Ã)

V

A

= Asin(!t)

V

A

V

B

V

SS

V

CC

Figure A.11: Improved CPC readout circuit.

It is shown in this paper how to calculate the perturbation on Cx from measured

and digitized values related to voltages VS and VC (DS and DC).

Cx1

Cx0=

tan(ψ)

n.DC

DS−1,

where, Cx0 and Cx1 are the Cx values before and after perturbation, respectively.

n is a constant related to the ADC reference voltages, and ψ is one of the elements

for the phase (ωt +π−ψ) determined by values of R1 and C1 [47]. The perturba-

tion can be defined as δ ,Cx1/Cx0.

182


The simulation results are shown in Fig A.12.

At balance, Cx1 = 0, with the condition of Cx0A =C0Bcos(ψ), the phase delay

at VO1 node is about π/2. This is one of the key points of tuning the circuit. Note

that ψ is an arbitrary and small value (3.5in this simulation).

0 200 400

-2

0

2

Vol

tage

(V

)

Time (µSec)

VA

VB

(a) Input Signals.

0 200 400

-0.04

-0.02

0.00

0.02

0.04

Vol

tage

(V

)

Time (µS)

VO1

VCC

VSS

(b) VCC and VSS.

0 50 100 150 200 250 300 350 400-12

-10

-8

-6

-4

-2

0

2

4

6 0.03

0.02

0.01

0.00

0.01

0.02

SV

0.03Volta

ge (

V)

Time ( Sec)

CV

(c) VC and VS.

Figure A.12: Simulation results for the readout circuit.

183


The results of a SPICEr simulation with a sweep on parameter δ are presented

in the Table A.2

δ VC/VS

-0.03 0.508333333-0.02 0.341666667-0.01 0.175

0 0.0083333330.01 -0.1583333330.02 -0.3416666670.03 -0.5

Table A.2: CPC simulation results, ratio of voltage magnitudes vs. perturbation.

The slope of the line (gain of the whole system) is cot(ψ) which is approxi-

mately 16.68 and complies with the result of the simulation.

-0.03 -0.02 -0.01 0.00 0.01 0.02 0.03-0.6

-0.4

-0.2

0.0

0.2

0.4

0.6

C

S

VV

2

16.875 0.004760.999

y xR

Regression Line:

Figure A.13: CPC parametric sweep simulation results.

184

A.5. CFC Simulation Results

A.5 CFC Simulation Results

This section presents the simulation for two CFC based readout circuits. The first

circuit schematic used for the simulation is the one that previously shown in Fig.

2.15 which is repeated here in Fig. A.14 for ease of reading. The component values

used for the simulation are R1 = 56kΩ, R2 = 112.2kΩ, R3 = 2.5kΩ, R4 = 500Ω,

R5 = 500Ω, R6 = 1kΩ, C1 = C3 = 10µF, C2 = 100pF, L1 = 200µH, L2 = 50µH

and the transistor Q1 is a typical NPN signal transistor, e.g. 2N2222 [29].

VCC

R1 R2

R3 R4R5

R6L1 L2

C2

C1

C3

Q1

Figure A.14: CFC readout circuit based on Hartley oscillator.

In this oscillator based readout circuit, C2 is assumed to be the sensor capaci-

tance. The SPICE simulations are done with parameter sweep on C2 for the values

between 100pF to 700pF. The simulation results are shown in table A.3 and are

plotted in Fig A.15.

185


C3(pF) fo(kHz)100 905150 758200 664250 600300 553350 512400 482450 455500 433550 415600 398650 381700 369

Table A.3: CPC simulation results, ratio of voltage magnitudes vs. perturbation.

0 100 200 300 400 500 600 700 800300

400

500

600

700

800

900

1000

f o (k

Hz)

C3 (pF)

Figure A.15: Simulation results for the readout circuit.

The second CFC readout circuit that is chosen for simulation is based on

186


switched-mode capacitors [39]. This circuit schematic and the related waveforms

are shown in Fig. A.16. CSEN and CREF are reference and sensing capacitors

respectively. Both of these capacitors are connected to the negative input of an op-

amp, the charge amplifier. These capacitors are driven by CLK and CLKB, which

pump electric charges to these capacitors differentially. CINT is the charge integra-

tion capacitor. During periods of time where SWΦ1 is closed, the op-amp acts as a

unity gain buffer and forces the voltage at the common node between all capacitors

to be at VCOM. When SWΦ2 is closed the op-amp and CINT function as charge inte-

grator and adds electric charges to the previous charges on CINT . The output of this

circuit is VOUT node. A third switch SWRST does the reset function and discharges

the capacitor to make it ready for the next measuring cycle. The circuit has three

main phases:

1. Reset phase: At the beginning of each new cycle CINT gets discharged when

SWRST and SWΦ1 are switched on. VOUT also goes to VCOM in this phase.

2. Pump-in phase: When SWΦ2 is on the rising edge of CLK ( falling edge of

CLKB ) happens, the charge correspondence to the difference of CREF and CSEN is

integrated by CINT . The additional charge that is added to the COUT is:

∆Q = ∆VOUTCINT = (CSEN−CREF)VDD = ∆CVDD (A.1)

So the step increase in VOUT can be expressed as:

∆VOUT =∆C

CINTVDD. (A.2)

3. Toggle phase: To have a proper integration functionality, the charge on CINT

should be conserved from the end of each pump-in phase to the beginning of the

187


next pump-in phase. This happens by turning on SWΦ1 and falling edge of CLK

(rising edge of CLKB).

Pump-in and toggle phases alternating and depending on the value of CSEN in

reference to CREF the charge on CINT , or the voltage VOUT , increases or decreases.

This continues until VOUT reaches a predefined upper limit VREF = VH or lower

limit VREF =VL. The initial value for VOUT is VCOM after each reset phase. Assume

VOUT reaches VREF after n consecutive steps.

VREF −VCOM = n∆C

CINTVDD. (A.3)

On the other hand by definition of n:

n =fclk

fout. (A.4)

So the final equation that shows the relationship between fout and ∆C can be

written as:

∆C =CINTVREF −VCOM

VDD

fout

fclk(A.5)

188


¡

+

CSEN

CREF

¡

+

¡

+

VoutSW©1

SW©2

SWRST

VH

VL

FF

VDD

VDD

CLK

CLKB

CLKD

CSENCINT

Reset

Reset

CLK

CLKB

©1

©2

VCOM

Figure A.16: CFC readout circuit based on switched-capacitors.

The simulation plot for the values of CSEN = 10.5pF, CREF =10pF, CINT =10pF

, fclk =100kHz, VCOM = 0, VDD =3.3V and VREF =VDD/2 is shown in Fig. A.17.

189


0.0 0.2 0.40

2

4

0.0 0.2 0.40

2

4

0.0 0.2 0.40

1

2

0.0 0.2 0.4

0

2

4

0.0 0.2 0.40

2

4

φ 1φ 2

Vou

tR

eset

Clk

Time (mS)

Figure A.17: Simulation results for the readout circuit based on SC.

The results of the simulation with parameter sweep on CSEN for values between

9.5pF to 10.5pF are captured in table A.4 and plotted in Fig. A.18. Note that there

is an error associated with the ratio of VREF to ∆VOUT not being a whole number,

which is discussed in more details in literature [40].

190


δ = ∆CCREF

fout(kHz)0.05 7.1386677650.04 5.80150.03 4.35100.02 2.77630.01 1.2195-0.01 -1.2192-0.02 -2.8261-0.03 -4.3440-0.04 -5.5556-0.05 -7.1511

Table A.4: CFC simulation results, output frequency vs. perturbation.

-0.06 -0.04 -0.02 0.00 0.02 0.04 0.06-10

-5

0

5

10

freq

uenc

y (k

Hz)

δ (=∆C/CREF)

Figure A.18: Simulation results for the readout circuit based on SC

191

Capacitance Readout Circuits Based on Weakly-Coupled ...

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