Exploiting Nonlinearities in MEMS: Applications to Energy ...

Exploiting Nonlinearities in MEMS:

Applications to Energy Harvesting

and RF Communication

Jeremy Scerri

Department of Microelectronics and Nanoelectronics

Faculty of ICT

University of Malta

September 2019

Submitted in partial fulfillment of the requirements for the degree of

Doctor of Philosophy

iii

ABSTRACT

Traditionally, in engineering, nonlinear behaviour is avoided, however engineering

applications that are intended to create a nonlinear relationship between inputs and

outputs also exist. In this thesis, it is shown that exploiting nonlinear phenomena in

MEMS design is instrumental in providing counter intuitive solutions to an

application involving a vibrational energy harvester and another two designs with

applications to communication signal processing.

Vibrational energy harvesters at MEMS scale are generally a challenge since at these

scales resonant frequencies are in the kHz range and this makes them insensitive to

the lower frequencies that are more abundant in the environment. One solution is

to include a nonlinear spring such that the harvester becomes sensitive to

broadband base excitations. In this work, one such broadband harvester is designed

by making use of a ‘quintic’ stiffness, buckling (bistable) spring.

The novel aspect in this work can be attributed to the topological arrangement of

the two buckling beams and the mass. The arrangement allows only the required

beam modes to dominate and together with the designed beam boundary

conditions, it is possible to replace the non-linear partial differential equation model

(resulting from continuum mechanics) with a simpler nonlinear differential

equation. It is demonstrated that this simpler model can still capture the salient

characteristics of the complex buckling behaviour; replacing complex finite element

analysis simulations with simple numerical solutions of differential equations and

hastening the design process. Although the design was constrained geometrically to

satisfy this simpler mathematical model, it is demonstrated that these constraints

do not impinge negatively on the harvesting capabilities. The harvester has a power

destiny of 0.13 mW cm-3 at 3.5g ms-2 at 560 Hz of vibrational excitation.

iv

The second design involves a torsionally vibrating plate which is capable of binary

phase shift keying demodulation. This plate is driven by electrostatic forces and

electrostatics provide signal mixing. The target application is demodulation of

signals encoded according to the 802.15.4 standard which describes a low data rate

BPSK signalling with a carrier frequency of 868 MHz and a chip rate of 300 kchips/s.

It is shown that the torsional plate has a damped resonant frequency of 1.54 MHz

and this being greater than the 3rd harmonic of the data rate recovers the baseband

signal successfully with 20 V peak of actuation voltages. At normal temperature and

pressure, the resulting Q-factor was found to be 60 which narrows the frequency

response and as a result the baseband signal recovered is slightly oscillatory. This

same torsional plate is investigated under higher actuation voltages and it is shown

that when actuation voltages exceed 75 V, nonlinear spring behaviour dominates

the response and chaotic trajectories in phase-space appear. At these higher

voltages, this device can be used for different purposes, for example, as a hardware

random number generation and a chaotic carrier generator. One drawback of using

electrostatics for mixing purposes is that apart from the required pure mixing

components, spurious products also appear. This is due to the quadratic

relationship in the electrostatic interaction and these would need to be filtered out

mechanically. However, it is shown that with a differential electrostatic drive using

the same torsional plate, these spurious products are attenuated and the resulting

plate displacement becomes practically proportional to pure signal mixing. This

relaxes the bandwidth-selectivity trade-off in the mechanical filtering and

consequently relieves some of the dimensional constraints of the torsional plate.

With this possibility, an in-phase/quadrature mixer is designed that is able to

demodulate different quadrature amplitude modulated signals with drive voltage

levels at 17 Vrms, a footprint of around 40,000 µm2 and giving output voltage levels

of 0.18 Vrms for the in-phase and quadrature signals.

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Two features are considered novel in this design; the width of application and the

ability to approximate pure mixing. These features are a result of the adopted

torsional topology.

In the final design, also related to communication signal processing, a MEMS device

is presented that is able to convert a BPSK signal to a simpler amplitude shift keying

modulation scheme. Although the structure involves also rotational motion, the

topology is very different and much more complex than the designs mentioned

previously. A mathematical model was developed and validated against finite

element analysis simulation results and this was used to obtain optimised

dimensions using a hybrid particle swarm optimisation algorithm. The design, with

a footprint of 2.9 mm2, was fabricated and experimentally validated. It was tested

with carrier frequencies ranging from 174 kHz to 1 MHz at a binary phase shift

keying (BPSK) data rate of 6.6 kbps and with carrier amplitudes of 9.7 V, resulting

in an amplitude shift keying (ASK) modulation index of 0.79 at the output sensors

and a power consumption of 2.9 𝜇W.

The novelty of this device is that it provides a MEMS solution for BPSK to ASK

conversion, a function that has always been realised in CMOS as the first stage to

BPSK demodulation. The device is capable of meeting current specification

requirements (data rates, power consumption and footprint) for implantable

medical devices. It is demonstrated that the power consumption is low enough such

that it provides an attractive alternative to CMOS realisations. Moreover, being a

MEMS, has potential for integration with MEMS sensors and harvesters in wireless

sensor network nodes.

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ACKNOWLEDGEMENTS

Throughout these years I have received a great deal of support from my supervisor,

Prof. Ivan Grech, whose expertise and ideas proved invaluable to keep me on track

and motivated.

vii

Dedicated to my family,

my parents and in-laws,

my two sons and especially, my wife Annalisa.

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CONTENTS

Author Contributions .............................................................................................................. xxiii

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

1.1 Motivation .................................................................................................................................. 2

1.1.1 MEMS Integration and RF Signal Processing ............................................................. 3

1.1.2 MEMS for Energy Harvesting ............................................................................................ 6

1.2 Existing Research Problems ................................................................................................ 8

1.3 Proposed Solutions that address the Research Problems ....................................... 8

1.4 Thesis Outline ........................................................................................................................... 9

2. Literature Review ....................................................................................................... 11

2.1 Background Literature on Nonlinearities .................................................................... 11

2.2 Nonlinearities due to the External Fields ..................................................................... 14

2.3 Electrostatic nonlinearities and Signal Mixing ........................................................... 15

2.3.1 Mixers and Image Rejection ............................................................................................ 19

2.3.2 Zero IF mixers or direct downconverters .................................................................. 20

2.3.3 The IQ mixer or Quadrature Downconverter .......................................................... 21

2.3.4 MEMS mixers ......................................................................................................................... 22

2.3.5 Electro-Mechanical Mixing.............................................................................................. 23

2.3.6 Electro-Thermal Mixing .................................................................................................... 25

2.3.7 BPSK to ASK conversion .................................................................................................... 28

2.3.8 Sensing Strategies ............................................................................................................... 29

2.4 Geometric Nonlinearities and Vibrational Energy Harvesting ............................ 30

2.5 Modelling and Validation Approach in nonlinear MEMS ....................................... 32

3. RF frontend functions in MetalMUMPs ............................................................... 33

3.1 A MEMS BPSK Demodulator .............................................................................................. 34

3.1.1 The Mechanical Structure ................................................................................................ 35

3.1.2 Torsional Oscillations of a Plate.................................................................................... 36

3.1.3 Modelling and Analysis ..................................................................................................... 37

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3.1.4 Frequency Content of the Input Force ........................................................................ 37

3.1.5 Mechanical Filtering .......................................................................................................... 39

3.1.6 Current Sensing at the Polysilicon electrode ........................................................... 41

3.1.7 Simulations and Results .................................................................................................... 43

3.1.8 Investigation of Potential Complex Dynamics ........................................................ 46

3.1.9 Development of the Mathematical Model ................................................................. 46

3.1.10 Behaviour by Region of Operation ............................................................................ 51

3.2 Suppression of Spurious Products in an Electrostatic Downconverter ........... 57

3.2.1 Frequency Perspective ....................................................................................................... 58

3.2.2 Prototype Design Dimensions and Simulation Results ........................................ 64

3.2.3 Low-IF IQ mixing ................................................................................................................. 66

3.2.4 Numerical Simulations ...................................................................................................... 72

3.3 Parasitic Insensitive Sensing ............................................................................................. 76

3.4 Conclusions .............................................................................................................................. 76

4. Bistable Vibrational Energy Harvester in SINTEF moveMEMS .................. 78

4.1 Introduction ............................................................................................................................. 78

4.2 Design within SINTEF process constraints ................................................................. 79

4.3 Mathematical Model ............................................................................................................. 82

4.3.1 Design Approach .................................................................................................................. 87

4.4 PZT Harvester Model validation against FEA ............................................................. 89

4.4.1 Validation of the Static Response ................................................................................. 89

4.4.2 Validation of the Harmonic Response ........................................................................ 92

4.4.3 MATLAB Dynamic Response Simulations ................................................................. 95

4.5 Conclusions ........................................................................................................................... 100

5. BPSK to ASK Converter in SOIMUMPs ............................................................... 102

5.1 Introduction .......................................................................................................................... 102

5.2 Design Requirements ........................................................................................................ 103

5.3 Design Approach - constraints and the resulting topology ................................ 105

5.4 Mathematical model .......................................................................................................... 109

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5.4.1 Actuation ............................................................................................................................... 109

5.4.2 Spring Stiffness ................................................................................................................... 113

5.4.3 Static Equilibria and Pull-In ......................................................................................... 116

5.4.4 Mechanical Dynamics ...................................................................................................... 117

5.4.5 Actuation Capacitance and Instantaneous Power .............................................. 123

5.4.6 Displacement Sensing and Complete System Model ........................................... 124

5.4.7 Output ASK Modulation Index and Fringe Capacitance ................................... 129

5.5 Optimisation Towards the Design Objectives ......................................................... 131

5.5.1 Dimensionality and FEA Validation .......................................................................... 132

5.5.2 Dimensional Optimisation using MATLAB ............................................................. 134

5.5.3 Design Validation using MATLAB............................................................................... 139

5.6 Experimental Validation .................................................................................................. 143

5.6.1 Geometric and Capacitive Measurements............................................................... 143

5.6.2 Transient and Modulation Index Measurements ................................................. 149

5.6.3 Device Power Consumption ........................................................................................... 152

5.7 Conclusions ........................................................................................................................... 157

6. Conclusions and Further Work ............................................................................ 159

6.1 Torsional Plate in MetalMUMPs .................................................................................... 160

6.2 Buckling Spring for Broadband Vibrational Energy Harvester ........................ 163

6.3 BPSK to ASK conversion in MEMS ............................................................................... 164

6.4 Further Work ....................................................................................................................... 165

References ........................................................................................................................... 170

Appendices .......................................................................................................................... 187

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LIST OF TABLES

Table 3.1: Breakdown of force components around 0 Hz as in equation (3.5). ........ 39

Table 3.2: Mode types and frequency, Q factor and damped resonant frequency. .. 43

Table 3.3:Parameter values for the Non-linear model ........................................................ 49

Table 3.4: Equilibrium points for the linear and non-linear models ............................. 50

Table 3.5: Design Steps for IQ mixing......................................................................................... 72

Table 4.1: Variables describing the beam motion................................................................. 83

Table 4.2: Fs – y1 Quintic Polynomial coefficients ................................................................. 91

Table 5.1: Design Objectives....................................................................................................... 104

Table 5.2: Design Process ............................................................................................................ 106

Table 5.3: Coefficients of the resulting degree 7 polynomial ........................................ 117

Table 5.4: Linear vs. Nonlinear Spring Stiffness and Overall linearity ...................... 128

Table 5.5: Dimensions Table ...................................................................................................... 132

Table 5.6: Constrained and Unconstrained design specification targets .................. 134

Table 5.7: Constrained functions as targets for design specifications ....................... 136

Table 5.8: Valid Ranges for design dimensions .................................................................. 136

Table 5.9: The final dimensions (µm) and resulting design specifications .............. 139

Table 5.10: The manufactured dimensions in (µm) – as measured............................ 146

Table 5.11: The actual (as manufactured) device specifications ................................. 151

Table 6.1: Broadening of functionality by harnessing cubic nonlinearity ................ 161

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LIST OF FIGURES

Figure 1.1: Photo of an embedded resonator, [6] ................................................................... 4

Figure 1.2: An all-MEMS receiver front end, [9] ...................................................................... 5

Figure 1.3: Multi-band/Multi-mode SDR architecture, [11] ............................................... 5

Figure 2.1: A perfect multiplier followed by a filter ............................................................. 16

Figure 2.2: Phase detector response of an ideal multiplier [68] ..................................... 17

Figure 2.3: (a) DC offset affects ∆ϕ at which no output is obtained and voltage

magnitude. .................................................................................................................................. 18

Figure 2.4: High-Side injection gives Fi = Fd + 2Fif and mirrors the IF spectrum ..... 19

Figure 2.5: The superheterodyne downconvertor ............................................................... 20

Figure 2.6: Frequency folding when FLO = FRF ......................................................................... 21

Figure 2.7: The basic topology of an IQ mixer ........................................................................ 22

Figure 2.8: The MEMS designed by [83] ................................................................................... 25

Figure 2.9: The dome mixer, [85], a) showing actuation b) showing mode shape .. 26

Figure 2.10: Structure used and electrodes for mixing [87]. ............................................ 27

Figure 3.1: The MetalMUMPs layers, smallest gap between conductors is 1.45 µm

......................................................................................................................................................... 33

Figure 3.2: The complete S1 structure showing metal layers in violet ........................ 36

Figure 3.3: Section through S1; view from bottom showing only one tether. ........... 36

Figure 3.4: Schematic diagram of the torsional BPSK demodulator depicting the bias

and excitation scheme required for mixing, filtering and sensing. ...................... 37

Figure 3.5: The spectrum of the electrostatic force generated and the required

mechanical bandwidth for adequate reconstruction. ................................................ 39

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Figure 3.6: FEA result for damping force coefficient against frequency for the

torsional plate taking into account squeezed film effects. ....................................... 41

Figure 3.7: Differential setup for sensing using a DCA........................................................ 42

Figure 3.8: Currents at the outputs for both positive and negative DC biasing ........ 44

Figure 3.9: Displacement against frequency. .......................................................................... 44

Figure 3.10: The displacement has a strong 3rd and 4th harmonic. ................................ 45

Figure 3.11: The system structure .............................................................................................. 47

Figure 3.12: The force curve for static displacements as large as 0.4 µm ................... 47

Figure 3.13: The EPs as a function of Vdc, red lines for unstable, black for stable. ... 50

Figure 3.14: Phase portrait, Poincaré map and spectrum for 726 kHz and Vdc =100 V

......................................................................................................................................................... 53


......................................................................................................................................................... 54


......................................................................................................................................................... 55

Figure 3.17: Autocorrelation of chaotic time series ............................................................. 56

Figure 3.18: Histogram of displacement samples for chaotic time series ................... 57

Figure 3.19: Actuation with one pair of electrodes .............................................................. 59

Figure 3.20: Torque frequency components with a single pair of actuation electrodes

......................................................................................................................................................... 59

Figure 3.21: Proposed torsional plate having both differential drive and sense ..... 60

Figure 3.22: Torque Frequency Components with two pairs of actuation electrodes

......................................................................................................................................................... 61

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Figure 3.23: Section through the proposed structure showing two pairs of actuation

electrodes .................................................................................................................................... 61

Figure 3.24: The whole structure has drive torque proportional to the product v1v2

......................................................................................................................................................... 63

Figure 3.25: The electrical sensing circuitry. .......................................................................... 63

Figure 3.26: The final device showing details of both polysilicon and nickel

electrodes .................................................................................................................................... 64

Figure 3.27: Differential current in nA vs. frequency in kHz. ........................................... 65

Figure 3.28: The core structure that provides actuation and sensing. ......................... 67

Figure 3.29: The complete structure consists of two mixing structures. .................... 68

Figure 3.30: The design steps and their effect on the frequency content.................... 71

Figure 3.31: vi (green), vq (red) and output (blue-atan2) showing the 4 levels

representing [00,01,10,11]. ................................................................................................. 74

Figure 3.32: vi (green), vq (red) and output (Blue-atan2) shows 2 levels representing

[0,0,0,1,1]..................................................................................................................................... 75

Figure 4.1: SINTFEF piezoVolume Process Overview ......................................................... 79

Figure 4.2: The mechanical schematic showing the proof mass M and the two

compliant springs .................................................................................................................... 80

Figure 4.3: Vibrational mode, FEA mesh and layering detail on spring ....................... 81

Figure 4.4: Buckling modes, even modes suppressed by connecting two beams [143]

......................................................................................................................................................... 82

Figure 4.5: The correct buckling sequence, only mode 1 and mode 3 involved. ...... 82

Figure 4.6: Fs - y1 curve for large Q with a mode 2 constrained beam [143] .............. 84

Figure 4.7: Electrical force component in the vertical direction ..................................... 86

xv

Figure 4.8: Using eq. (4.13) to determine h and t ................................................................. 89

Figure 4.9: The Force-Displacement (y1) asymmetric curve obtained using FEA ... 90

Figure 4.10: Strain energy vs. displacement showing a maximum of 25 nJ at the

unstable point ............................................................................................................................ 91

Figure 4.11: Speed-Displacement phase portrait and y1(t) for base acceleration of

0.001g at fi giving 8.8 µm peaks ......................................................................................... 93

Figure 4.12: Current i(t) µA and Power P(t) µW and their respective RMS values in

title. ................................................................................................................................................ 93

Figure 4.13: Output current (i) and power (P) ratios of FEA-to-model RMS ............. 94

Figure 4.14: A high energy orbit producing 0.2 µW of power with spring force-

displacement and equilibria in superposition. ............................................................. 95

Figure 4.15: Trajectories in state-space with B = 9 mN/(m/s) and no inertial frame

acceleration ................................................................................................................................ 98

Figure 4.16: Driving the harvester away from resonance exposes chaotic

trajectories. ................................................................................................................................. 99

Figure 5.1: Block diagram of the converter showing design properties and objectives

...................................................................................................................................................... 104

Figure 5.2: Extract from SOIMUMPs handbook [24], showing the process layers 105

Figure 5.3: Actuation and sense capacitors, solid lines are fixed plates, while dashed

are moving ............................................................................................................................... 107

Figure 5.4: Two rotor designs - a) Radial vs. b) Orthogonal comb fingers .............. 108

Figure 5.5: The final octagonal layout showing comb finger insets and electrical

schematic.................................................................................................................................. 108

Figure 5.6: Octagon dimensions – one side, showing the i th finger ............................ 110

Figure 5.7: Schematic showing actuation with BPSK input ........................................... 112

xvi

Figure 5.8: Torque levels for ASK and BPSK, dashed lines are in-phase, solid in anti-

phase .......................................................................................................................................... 113

Figure 5.9: Linear (left) vs. Non-linear (right) spring designs ...................................... 114

Figure 5.10: H-Fixture that provides control on axial and transverse stiffness, [153]

...................................................................................................................................................... 115

Figure 5.11: Cantilever spring showing transverse and axial displacements ......... 116

Figure 5.12 Finger section showing electric field .............................................................. 123

Figure 5.13: Parameters affecting modulation index, M ................................................. 130

Figure 5.14: Sample run – PSO convergence, verbose and results .............................. 137

Figure 5.15: Narrow vs. broad optimality property .......................................................... 138

Figure 5.16: Response from DE model ................................................................................... 140

Figure 5.17: Displacement (until pull-in) vs. actuation voltage for increasing na . 141

Figure 5.18: Final layout showing SOI layer and connections ...................................... 142

Figure 5.19: Experimental setup .............................................................................................. 144

Figure 5.20: Device microphotograph and laser profilometry on comb................... 145

Figure 5.21: SEM photograph showing cantilever spring width at 8.5 µm .............. 145

Figure 5.22: SEM photograph showing comb gap of 2.55 µm ....................................... 146

Figure 5.23: Capacitance measurements between each electrode ............................. 147

Figure 5.24: Actual measurements vs. linear and cubic stiffness for CS1 and CS2. . 148

Figure 5.25: Optical microscope images showing comb gap change for increasing

voltage ....................................................................................................................................... 149

Figure 5.26: Experimental measurement of transient and its superposition on

output ASK ............................................................................................................................... 150

xvii

Figure 5.27: a) Solid line is simulation, points are experimental b) ASK output signal

for ∆𝑉𝑅𝑀𝑆 = 8.4 V - experimental ............................................................................... 151

Figure 5.28: Actuation current measurement setup ......................................................... 153

Figure 5.29: Current and power consumption for ∆𝑉𝑟𝑚𝑠 = 7.3 V and 𝑓𝑐 = 174 kHz.

...................................................................................................................................................... 153

Figure 5.30: Current and power consumption for ∆𝑉𝑟𝑚𝑠 = 13 V and 𝑓𝑐 = 174 kHz.

...................................................................................................................................................... 154

Figure 5.31: Velocity Squared Signal for a 0.5 kHz data rate and 13 V RMS actuation

...................................................................................................................................................... 155


...................................................................................................................................................... 156

Figure 5.33: Velocity Squared Signal for a 6 kHz data rate and 13 V RMS actuation

...................................................................................................................................................... 156

Figure 5.34: The actuation current (blue), average current (green) and power

(purple) ..................................................................................................................................... 157

Figure 6.1 Time series and histogram for 468 kHz and Vdc =100 V ............................ 160

Figure 6.2: Torque frequency components arising from the -v1 (dotted) and v1 (solid)

pads. ........................................................................................................................................... 162

Figure 6.3: Rotor central weight design: Wanted mode (left) and Unwanted mode

(right) ........................................................................................................................................ 167

Figure 6.4: The design that failed to release anchor supports (encircled) .............. 168

Figure 0.1: The overall SIMULINK block setup ................................................................... 196

Figure 0.2: The modulator block .............................................................................................. 196

Figure 0.3: The MEMS block ....................................................................................................... 197

Figure 0.4: Electrostatics ‘I’ in MEMS block ......................................................................... 197

xviii

Figure 0.5: Electrostatics ‘Q’ in MEMS block ........................................................................ 197

Figure 0.6: Plate Dynamics Angle in MEMS block .............................................................. 198

Figure 0.7: Plate angle to delta Cn block in MEMS block ................................................. 198

Figure 0.8: Sensing side UP block in MEMS block .............................................................. 198

Figure 0.9: ADC and DSP block .................................................................................................. 199

xix

LIST OF ABBREVIATIONS AND ACRONYMS

ADC Analogue to Digital Conversion

AM Amplitude Modulation

ASK Amplitude Shift Keying

BPSK Binary Phase Shift Keying

CMOS Complementary Metal Oxide Semiconductor

DAC Digital to Analogue Conversion

DAE Differential Algebraic Equations

DE Differential Equations

DoF Degree of Freedom

FEA Finite Element Analysis

FSK Frequency Shift Keying

IC Integrated Circuit

IF Intermediate Frequency

IMD Implantable Medical Device

IoT Internet of Things

IQ In-Phase/Quadrature

LO Local Oscillator

MEMS Micro Electro-Mechanical Systems

NTP Normal Temperature and Pressure

OOK On-Off Keying

PDE Partial Differential Equations

xx

PLL Phase Locked Loop

PM Phase Modulation

PSK Phase Shift Keying

PSO Particle Swarm Optimisation

PZT Lead Zirconate Titanate

QAM Quadrature Amplitude Modulation

QPSK Quadrature Phase Shift Keying

RF Radio Frequency

RFID Radio Frequency Identification

RMS Root Mean Square

SEM Scanning Electron Microscope

SFD Squeeze Film Damping

SOI Silicon on Oxide

VEH Vibrational Energy Harvester

WSN Wireless Sensor Network

xxi

LIST OF APPENDICES

Appendix 3.1 Dynamics Simulations – MATLAB Script ................................................... 188

Appendix 3.2 Equilibrium Points – MATLAB Script .......................................................... 193

Appendix 3.3 Simulink Implementation of IQ mixer ........................................................ 196

Appendix 4.1 Transient Response - MATLAB Scripts ...................................................... 200

Appendix 5.1 Resultant Stiffness .............................................................................................. 205

Appendix 5.2 Equilibria – MATLAB Script ............................................................................ 206

Appendix 5.3 - Total inertia of N/2 fingers ........................................................................... 208

Appendix 5.4 Change in fringe Capacitance - MATLAB Script ...................................... 210

Appendix 5.5 Monotonicity in Sensing ................................................................................... 212

Appendix 5.6 Modulation Index, n and Fringe Capacitance - MATLAB Script ........ 213

Appendix 5.7 PSO - MATLAB Scripts ....................................................................................... 214

Appendix 5.8 Dynamics – inputs to output - MATLAB Scripts...................................... 221

xxiii

LIST OF PAPERS

Parts of this dissertation have been published in peer reviewed conferences and

journals:

1. ‘A MEMS BPSK Demodulator - Micromechanical Mixing and Filtering using MetalMUMPs',

9th PRIME Conference, Villach, Austria, pp. 113-116, 2013.

2. 'Versatility provided by an electrostatic torsional microstructure as a consequence of its

complex dynamics', IET Electronic Letters, vol. 50, no. 4, pp. 303-304, 2014.

3. 'Reduced order model for a MEMS PZT vibrational energy harvester exhibiting buckling

bistability', IET Electronic Letters, vol. 51, no. 5, pp. 409-411, 2015.

4. 'Suppression of spurious products in an electrostatic RF MEMS downconverter having

differential drive and sense', 18th Melecon Conference, Limassol, Cyprus, 2016.

5. 'A MEMS Low-IF IQ-Mixer in MetalMUMPS: Modelling and Simulation', ICECS 2017

Proceedings, Batumi, Georgia, 2017.

6. 'Exploiting nonlinearities to improve the linear region in an electrostatic MEMS

demodulator', 14th Conference on PhD Research in Microelectronics and Electronics (PRIME

2018), Prague, Czech Republic, 2018. – This paper received the Gold Leaf Award.

7. 'A MEMS BPSK to ASK Converter', Microelectronics International Journal, Emerald Insight,

Vol. 36 Issue 1, DOI: 10.1108/MI-06-2018-0039, 2019.

8. ‘Dimensional Optimisation of a MEMS BPSK to ASK Converter in SOIMUMPs’, Integration,

the VLSI Journal, Vol. 67, 2019. DOI: 10.1016/j.vlsi.2019.03.002

Author Contributions

The author took the leading role in the writing of all the papers from inception to

design, analysis, mathematical modelling, simulation and experimental validation

(Papers 6 to 8).

Introduction

September 2019 Jeremy Scerri 1

1. INTRODUCTION

This thesis presents MEMS designs with applications to RF communication and

energy harvesting whose feasibility relies on avoiding nonlinearities when

unwanted and exploiting them efficiently when required. The MEMS for RF

communication focuses on a device capable of converting Binary Phase Shift Keying

(BPSK) to Amplitude Shift Keying (ASK) and involves electrostatic actuation which

is nonlinear with voltage. This nonlinearity gives the required frequency mixing. The

MEMS for Energy Harvesting is intended to capture vibrational energy, makes use

of the piezoelectric effect and has a bistable spring for improved bandwidth. In both

designs, the displacement statics and dynamics are carefully controlled by adding

an adequate nonlinear spring.

The origin of nonlinearities in MEMS is due to many factors and due to their small

size, nonlinearities are generally exacerbated. It is common practice to add to the

linear elastic force a force that is proportional to the cube of the displacement, 𝑥3.

This is added even when the system is well within the intrinsic linear stress-strain

relationship. The reason behind this addition would typically be due to the effect of

external nonlinear potentials (e.g. electrostatic force) and geometric effects

Introduction


(e.g. clamped beams, initial curvature). This additive term is enough to change the

behaviour from that involving simple harmonic motion to a Duffing system [1].

Additionally, nonlinearities may arise in practical experimental realisations due to

the manner with which the device is actuated and sensed and the manner with

which it is clamped/bonded to the surrounding material. Damping mechanisms can

also change from linear, that is, proportional to the velocity , to nonlinear.

Whenever it is reasonable to add cubic spring stiffness terms, it is also reasonable

to add a nonlinear damping term such that damping increases with amplitude.

1.1 Motivation

The primary reason for the successful commercialisation of MEMS devices is their

size. The size has a direct impact on cost and also on device power consumption.

Moreover, when structures are scaled down, not all physical phenomena are scaled

down in proportion and this opens up new possibilities in virtually all domains, be

it electrical, thermal and also mechanical.

One area that is benefitting from MEMS technology is the development of Wireless

Sensor Networks (WSN). The miniaturisation of WSN nodes due to MEMS has made

remarkable progress. A sensor node would generally include four subsystems: the

wireless transceiver, the microcontroller, a sensor and the power management

module. Nodes require an energy source and can be powered from ambient sources

through optical cells, piezoelectric crystals, thermoelectric elements or

electromagnetic waves [2]. Meanwhile, communication standards like the ZigBee®

have been well developed to address the typical low data rates and low energy

consumption for sensor nodes. MEMS can offer solutions for the energy harvesting

module, for the sensing module and for the transceiver modules of a sensor node.

Having more than one module in MEMS will potentially aid in keeping to the small

size constraints of WSN nodes [3]. A related specialist area that could benefit from

such integration is that of implantable medical devices (IMD). Such applications

Introduction


have different design constraints and one special requirement is that of maximum

power transfer. The choice of digital modulation scheme is critical to maximise

power transfer. BPSK is usually chosen since it is of constant amplitude and if ASK

is adopted, it is used with a low modulation depth [4].

In [5], the capability to build multiple sensors using a modified CMOS fabrication

process is demonstrated. Building multiple MEMS structures with different

functions on the same substrate is generally called multi-MEMS. Such a fabrication

approach allows for greater integration than having to use multiple ICs each

dedicated to sense different environmental conditions.

The two avenues of investigation in this dissertation are motivated by the possibility

of integrating not only multiple sensors, but also vibrational energy harvesters and

RF signal processors in MEMS.

1.1.1 MEMS Integration and RF Signal Processing

The miniaturization of mechanical vibrating structures lends itself naturally to high

frequency communication applications. RF transceivers make use of building blocks

such as amplifiers, mixers and oscillators (active circuits) and also matching

networks and filters (passive networks). MEMS technology in RF/microwave

systems is playing a big role in component integration. This can be done on-chip in

contrast to, for example, having externally mounted quartz crystals (Fig. 1.1).

Introduction


Figure 1.1: Photo of an embedded resonator, [6]

Although it is nowadays common practice to integrate active circuits on silicon, full

integration of filters was hindered by the expected performance requirements.

When a radio channel - with bandwidths in the order of kHz - needs to be filtered at

a receiver frontend and this channel is centered at GHz frequencies the Quality

Factor (Q) of this filter becomes prohibitively high. Relaxing this constraint can be

achieved with the Superheterodyne receiver, by filtering at RF and then down

converting to a lower frequency and then filtering again at the IF for channel

selection. This reduces the Q factor required but is still difficult to obtain with

inductors and capacitors integrated on silicon. Hence, this is practically done by

filtering off-chip either with a ceramic or quartz crystal, surface acoustic wave

(SAW), and more recently film bulk acoustic resonator (FBAR) filters. These can

achieve Q’s up to 10,000. However, off-chip filtering hinders miniaturization, low

power operation and also low-cost production. With the advent of MEMS

technology, micromechanical filters that had the potential for high Q were proposed

as early as 1992 [7].

Using MEMS structures to replace parts of the traditional RF frontend architectures

has been the subject of investigation in recent years [8], [9], [10]. Fig. 1.2 shows one

such proposed RF frontend which employs MEMS. This is in essence a low-IF radio

and is called a MEMS channel-selectable architecture. It employs an RF image reject

Introduction


filter, a fixed micromechanical resonator LO, and a switchable array of IF

micromechanical mixer filters.

Figure 1.2: An all-MEMS receiver front end, [9]

More recently, Software Defined Radio (SDR) frontend architectures which pass on

most of the analogue signal processing into the digital domain are being put forward

(Fig. 1.3).

Figure 1.3: Multi-band/Multi-mode SDR architecture, [11]

Introduction


However, as can be seen in Fig. 1.3, IQ mixers are still required in analogue form as

ADC/DAC requirements (bandwidth and resolution) and power consumption

requirements are still prohibitive for an entirely digital SDR.

MEMS in RF frontends provide further new possibilities especially when it comes to

combining transceiver stages within a single structure, stages which are

traditionally implemented with separate transceiver modules [12]. A mixer

designed as a MEMS has this potential of incorporating within it other functions,

[13], [14] and [15] and this would also be an asset to RF frontend component

integration and miniaturisation.

The designs described in Chapter 3 and 5 provide MEMS solutions to RF frontend

functions. The designs provide a range of functionalities, from mixer-filters, IQ

mixers, BPSK demodulators and BPSK to ASK conversion. All designs involve a

mechanical structure that is actuated using electrostatics and whose resulting

displacement is sensed through a capacitive gap change.

1.1.2 MEMS for Energy Harvesting

More recently, WSN technological progress has taken on a new urgency as the

Internet of Things (IoT) is becoming one of the underlying modern ‘smart’

application that makes use of distributed remote sensing abilities. Development of

zero-power or power-autonomous technologies able to scavenge energy from the

environment and turn it into electricity will fill a technological gap that is currently

limiting widespread adoption of IoT applications. With this state of affairs, the

miniaturisation of energy harvesters that make use of MEMS technology has great

potential to satisfy the main requirements for IoT, namely, energy-autonomy,

miniaturization and integration.

Furthermore, MEMS devices usually interact with fields and forces that are not

necessarily electromagnetic, such as mechanical, piezoelectric and thermoelectric

Introduction


forces. This breadth in the application domains has also promoted MEMS technology

as a valid option for miniaturisation of energy harvesters [16].

For vibrational energy harvesters, miniaturisation is an issue because of two

reasons. A smaller mass implies smaller kinetic energy and hence less power output

available. Moreover, with smaller dimensions of spring structures, the resonant

frequency will typically be in the kHz region which is much higher than what is

commonly available – a few hundred Hz – in environmental vibrations. There have

been attempts to tackle the latter problem by making use of frequency

up-conversion [17] and non-linear vibrations [18], and also efforts that make use of

compliant structures [19].

Non-linear vibratory systems make use of bistability and multistability. These are

generally desirable properties in mechanical structures used for energy harvesting.

With a plurality of equilibrium points, a vibrational system can achieve broadband

capabilities. In literature, one can find many ways to achieve bistability, [20]. In [21],

a theoretically extensive treatment of the behaviour of buckling beams and their

combination to obtain compliant multistable systems is presented.

Achieving broadband sensitivity in MEMS vibrational energy harvesters is essential,

as without broadband, such small structures would only be sensitive to frequencies

in the kHz region. Having MEMS scale vibrational energy harvesters is instrumental

in the integration and miniaturisation of WSN nodes, however, at MEMS scales these

would need to harness intrinsic, by-design nonlinearities for broadband

sensitivities.

The design described in Chapter 4 achieves broadband capability with the use of a

buckling spring. With this bistable spring, the device becomes sensitive to excitation

frequencies far lower than the natural frequency of the encastré spring and is able

to harvest energy at relatively low excitation frequencies. Moreover, the complex

static and dynamic behaviour of this encastré buckling spring could be described

with a simplified model which reduced simulation time.

Introduction


1.2 Existing Research Problems

In this work, two challenges in MEMS research are investigated; the first is of a

generic nature and targets the limitations and constraints for design optimisation in

MEMS, the second avenue of investigation looks into the potential of integrating RF

frontend functions by adopting MEMS implementations. These two avenues are

described in more detail hereunder:

1. MEMS are, by their nature, cross-domain devices and obtaining a good

understanding of the behaviour of a MEMS device is traditionally achieved

by modelling using finite element techniques in a multi-physics environment.

This however comes at a high computational cost and in many cases, this

prohibits extensive simulation runs in a cross-domain environment. In

practice, high computational cost would in turn hinder the possibility to use

algorithms for design optimisation.

2. Currently, BPSK to ASK convertors that satisfy Implantable Medical Device

(IMD) specifications are realised in CMOS. IMDs consist of multiple

subsystems and apart from the RF frontend, IMDs would also typically have

sensors and energy harvesters. With miniaturisation, many such sensors and

energy harvesters are being successfully implemented in MEMS. The

prospect of having a BPSK to ASK converter also implemented in MEMS

would offer the IMD system designer better integration prospects for IMD

subsystems.

1.3 Proposed Solutions that address the Research Problems

Solutions to these two research challenges are provided as follows:

1. Numerical solutions to differential algebraic equations (DAEs) modelling a

BPSK demodulator were able to show that for high actuation voltages the

Introduction


device can be used for a variety of other functions. This revealed potential

avenues for integration (Chapter 3).

2. A DAE model (Chapter 4) that accurately predicts the energy harvesting

capabilities of a broadband bistable vibrational energy harvester could be

used to:

a. Cut the design and optimisation time drastically.

b. Control the nonlinear stiffness to obtain broadband capabilities and

harness excitation frequencies below its resonant point.

3. Designed, fabricated and validated the DAE model of a BPSK to ASK

converter. This device is capable of conversion at data rates, power

consumption and modulation index relevant to current IMD requirements.

Being in MEMS, it offers an alternative to CMOS implementations and

provides new integration prospects (Chapter 5).

1.4 Thesis Outline

This dissertation investigates how nonlinear behaviour can be captured effectively

in the mathematical model and how nonlinearities can be exploited for effective

MEMS design. Two areas of application are considered (energy harvesting and RF

communications), however, the main area of investigation, which also includes

experimental validation, is the RF communications one.

The writeup is divided in 6 chapters. In the first chapter, the reader is introduced to

electrostatic actuation, sensing and displacement and the nonlinearities arising

from them. Here, the research challenges and the proposed solutions are outlined.

Chapter 2 is a literature review on nonlinearities in general, with particular focus to

the two application areas. Chapter 3 presents a rotational structure designed using

the MetalMUMPs® [22] manufacturing process which could be used for several

communication signal processing functions. Three functions are described, a BPSK

demodulator, a downconverter and an IQ mixer. Analytical models are proposed and

finite element validation is performed to confirm functionality. This section also

Introduction


includes analysis that shows that this device can be used for other purposes

including energy harvesting. Chapter 4 presents a bistable vibrational energy

harvesting device designed using the SINTEF® [23] manufacturing process and

includes an analytical model and numerical simulations. Chapter 5 presents the

design, optimisation process, fabrication and experimental validation of a BPSK to

ASK converter fabricated using the SOIMUMPs® [24] manufacturing process.

Chapter 6 concludes by looking at the results to provide a critical assessment of the

contributions.

Literature Review


2. LITERATURE REVIEW

In this chapter, a review of recent developments on different facets involving

nonlinear manifestations in MEMS and their use is presented. The first section of the

review gives the broader picture and describes the primary root causes for

nonlinearities and their nature and also mathematical modelling approaches. In the

subsequent sub-sections, exploitation of nonlinear behaviour both for RF and

energy harvesting applications is reviewed, as these are the two main areas of

investigation of this thesis.

2.1 Background Literature on Nonlinearities

All devices (evidently in all physical domains) present nonlinear behaviour at large

drives, and the research community, across the whole scientific spectrum, is dealing

with nonlinear mechanics [25], [26]. MEMS are no exception and nonlinear

manifestations in their behaviour is ubiquitous. Nonlinear spring stiffness and

damping mechanisms are exemplified in [27], [28], nonlinear capacitive, resistive

and inductive circuit elements in [29], [30], and nonlinear forces on surfaces, in

fluids and in the electric and magnetic domains in [31], [32], [33], [34].

Nonlinearities in the mechanical domain can be mainly attributed to two sources:

(a) geometric and (b) material. In general, a geometrical nonlinearity is a result of

having a nonlinear stress-strain relation or a large deformation while material

nonlinearities are attributed to the combined or individual load level and load

history. Material nonlinearities are also generally classified as rate dependent and

rate independent.

For modelling purposes, in most cases, simplifying the problem to a linear

differential equation set would still give acceptable solutions and in turn, tackling

Literature Review


the harder nonlinear differential equations could be avoided. However, there will

always be some cases in which such a simplification, that makes use of a linear

differential equation set, results in incorrect solutions. These cases are difficult to

pinpoint beforehand and the discrepancies would only come out when an

experimental validation exercise is performed.

In 1890, Poincare´ placed differential equations in a new light and provided his

geometrical interpretations. He also applied this new technique on celestial

mechanics [35]. The Poincare´ map, a diagram which is produced by sampling the

state space at discrete time intervals, set Poincare´ in a position to reveal chaotic

behaviour in dynamic systems. In a paper by Lorenz [36], the chaotic response was

investigated. In this work, he proposed the now well known ‘Lorenz equations’ to

model the dynamics of the atmosphere and discovered the Lorenz attractor and also

chaos in fluids. Chaotic behaviour in nonlinear systems is nowadays a major area of

study, is also known as the physics of chaos [25] and has also made it to the popular

science bookshelves [37].

For the engineer, nonlinearities are an important design parameter. When a device

is intended to operate in a linear fashion, these nonlinearities are problematic as

they limit the dynamic range of operation [38]. Conversely, nonlinearities can be

exploited for frequency mixing as described in [39], synchronization [40], using

bifurcation points for amplification [41], parametric amplification [42], [43] and

also drive [44], amplifier noise suppression in oscillators [45], [46] and [47] and

mass detection [48]. Particularly in [49], it was shown that the Van der Pol oscillator

[50] could be realised in the mechanical domain by making use of nonlinear

damping.

From the scientific point of view, a MEMS exhibiting nonlinear behaviour is in effect

a realization of the Duffing oscillator, that is, a mechanical system having a spring

stiffness term ∝ 𝑥3, where 𝑥 is displacement. The Duffing oscillator is of great

interest to the scientific community since many systems can be modelled using the

Literature Review


same dynamic equations [51]. The mathematical model is relatively simple and, in

many cases, an analytical solution can be found. Moreover, the Duffing system is able

to reveal the theoretical properties in dynamics that are commonly observed in

experimental studies like memory effects [1] and dynamical switching [52], [53]

and [54].

Understanding the nature of these nonlinearities is of utmost importance [54].

Electrostatic actuation is in itself a nonlinear drive and has been investigated

thoroughly in literature [55], [56]. In [25], [40] and [47], an 𝑥2 term in the damping

and its effect on the nonlinear behaviour is discussed.

Furthermore, even if the stiffness and damping mechanisms are independent of the

stress/strain (a perfectly elastic material), nonlinear behaviour in mechanical

devices still manifests itself and this is captured effectively with the Duffing model.

The origin of this nonlinearity is due to structural constraints [57], [58] and apart

from the cubic stiffness term, in general, it will require a force in proportion to the

displacement squared [59], [60]. Less intuitively are the addition of nonlinear terms

related to inertia [61], [62], [63]. In such cases, the Duffing equation is able to

capture the characteristics such that theoretical predictions agree with

measurements. However, this does not make the mechanical system under

investigation a Duffing oscillator since it is in essence a model fitting exercise.

Nevertheless, the majority of the experimental work (supported with theory) has

been done around the Duffing model problem ([38], [39], [41], [45], [46], [48], [64]),

with the majority of the experiments performed in the driven regime ([38], [60],

[64], [65]). This can be attributed to the simplicity of the model. Theoretically,

modelling a nonlinear problem and taking into account the full complexity of the

system is generally very challenging and as long as a simpler Duffing model captures

the observed behaviour, these complex analytical models are replaced with this

simpler model. Most of the experiments are in the driven regime to get natural

amplification through the system’s Q factor.

Literature Review


The most sophisticated analytical models are based on continuum mechanics. These

result in a system of coupled nonlinear equations which describe the mechanical

dynamics of the device [58]. These coupled nonlinear equations are not dealt with

directly and techniques like the Galerkin procedure are usually employed such that

the system of equations is reduced to a one-dimensional problem [58], [62].

2.2 Nonlinearities due to the External Fields

Several external fields contribute towards the resulting forces which act between

two electrodes that form a capacitor. Casimir’s and Van der Waals’ interactions are

inversely proportional to the third and fourth power of the gap and are relevant only

for very small gaps [66]. The electrostatic force is effectively a result of an external

potential which acts on any suspended mechanical structure in MEMS. This force is

proportional to 1/(𝑑 + 𝑥)2 with 𝑑 being the nominal gap and 𝑥 the deviation away

from the nominal position. If this force acts on a linear spring 𝑘𝑥 and without

considering any other nonlinear effects, the resulting equation of motion (using

power series expansion), would have terms in 𝑥, 𝑥2 and 𝑥3.

Due to these terms, for mechanical devices that employ capacitively coupled

electrodes and hence electrostatic interactions, the stable range of operation will be

limited [33]. In [33], the authors took a generic system made up of a clamped-

clamped gold beam where this beam is capacitively coupled to a statically fixed

electrode and investigated the range of stability. Discrepancies between theory and

observation were reported. They attributed this mismatch between theory and

observation to the metal electrodes structural instabilities.

In [31], electrostatically driven MEMS devices were analysed and these exhibited

bifurcation phenomena which resulted in snapping instabilities. In this work,

model-based servo feedback was used to study the snapping instability. Using

servoing, the range for stable displacement was increased to 67% of the nominal

gap.

Literature Review


In [67], the authors managed to design an actuator which was able to close the gap

in a stable manner up until 80% of the original gap. This is well beyond the voltage

control and charge control limits. The authors used a negative capacitance solution

to achieve this. This entailed the use of a closed loop system that removed the charge

when the actuator capacitance increased. In a second design, the parallel-plate

actuator was stabilized while tipping. In this secondary design, they report a

maximum deflection of 1.4 𝜇m equivalent to 91% of the nominal gap with a voltage

of 3 V.

In [30], an impact resonator was shown to exhibit both chaotic and periodic

oscillations. In this investigation, the effect of: parasitic capacitances, air damping,

the resistors (used for charging and discharging) and the DC voltages on the

experimental results were reported.

In [32], a bistable MEMS oscillator was studied and the experimental results were

compared to theoretical predictions. In this publication, the authors confirmed the

existence of a strange attractor in the designed MEMS structure by comparing model

response to experimental results.

The electrostatic force is also proportional to the drive voltage squared. This

nonlinear property lends itself to achieve frequency mixing and is reviewed

thoroughly in the following section.

2.3 Electrostatic nonlinearities and Signal Mixing

In signal processing, a signal mixer or multiplier is a device that takes two signals as

input and is intended to multiply these signals to produce an output with a different

frequency content. Figure 2.1 shows a perfect multiplier followed by a filter. The

filter rejects the high frequency component if the mixer is intended for down

conversion (receiving end) and rejects low frequencies if it is intended for up

conversion (transmission side).

Literature Review


Figure 2.1: A perfect multiplier followed by a filter

The output port (Intermediate Frequency) of the perfect multiplier contains the sum

and difference of the input signal frequencies, that is, the Radio Frequency (RF) and

the Local Oscillator (LO) as described by equation (2.1);

𝑣𝐼𝐹 =𝐴(𝑡)𝐴𝐿𝑂

2cos((𝜔𝐿𝑂 + 𝜔𝑅𝐹)𝑡 + ∆𝜙) + cos((𝜔𝐿𝑂 − 𝜔𝑅𝐹)𝑡 + ∆𝜙) (2.1)

where 𝐴(𝑡) is the RF signal amplitude, 𝐴𝐿𝑂 is the local oscillator amplitude and ∆𝜙

is the phase difference between RF and LO signals.

The perfect multiplier can be used to generate a signal component at 𝜔𝐿𝑂 + 𝜔𝑅𝐹 and

𝜔𝐿𝑂 − 𝜔𝑅𝐹 and also at DC in proportion to the phase difference ∆𝜙. If 𝜔𝐿𝑂 = 𝜔𝑅𝐹 and

the remaining high frequency term is filtered out, the output would be dependent

only on the phase difference. In doing so, equation (2.1) reduces further to equation

(2.2). This is called a zero IF downconvertor or a direct conversion receiver.

𝑣𝐼𝐹 =𝐴(𝑡)𝐴𝐿𝑂

2cos(∆𝜙) (2.2)

Equation (2.2) implies that 𝑣𝐼𝐹 will be fixed depending on the cosine of the phase

difference. Zero readings for 𝑣𝐼𝐹 will be obtained for phase differences of 𝑛𝜋 2⁄ with

𝑛 = ±1, ±3,… .

Moreover, peak values are obtained for ∆𝜙 = 𝑛𝜋 where 𝑛 = 0, ±1,±2,… . This

relationship is nonlinear and worse still it is not monotonic over 2𝜋 of phase

difference.

Literature Review


The derivative of equation (2.2) gives 𝐾𝑑 =𝑑𝑣𝐼𝐹

𝑑𝑡 as a function of ∆𝜙 as in equation

(2.3).

𝐾𝑑(∆𝜙) ∝ sin (∆𝜙) (2.3)

This gives a maximum sensitivity when the phase difference is 𝜋/2 and if 𝑑 percent

of deviation from linearity is tolerated, the region of ‘linear’ operation ∆𝜙 ± 𝛿𝜙

about ∆𝜙 = 𝜋/2 would be as in equation (2.4) [68].

𝑙𝑖𝑛𝑒𝑎𝑟 𝑟𝑒𝑔𝑖𝑜𝑛 = 𝜋/2 ± √6𝑑

100 (2.4)

The characteristics of the ideal multiplier can hence be depicted as in Figure 2.2,

which is far from an idealistic linear relationship.

Figure 2.2: Phase detector response of an ideal multiplier [68]

Figure 2.3 shows two manifestations resulting from the non-ideal characteristics of

a multiplier. These are the dc offset, which in practice would be a result of mixer

Literature Review


asymmetry, and mixer-induced phase shift resulting from discrepancies in the

electrical length from the LO-to-IF port and the RF-to-IF port.

(a) DC offset (b) Mixer-induced phase shift

Figure 2.3: (a) DC offset affects ∆ϕ at which no output is obtained and voltage magnitude.

(b) The relative phase being affected by mixer induced phase shift. [68]

If the phase difference, ∆𝜙, between RF and LO is not constant and is a function of

time, the RF signal could carry information in the phase change which technique is

known as phase modulation (PM). This is described by (2.5);

𝑣𝑅𝐹(𝑡) = 𝐴𝑐𝑜𝑠(𝜔𝑅𝐹𝑡 + 𝑘𝑝𝑣𝑚(𝑡)) (2.5)

where 𝑘𝑝is the change in carrier phase per volt or phase sensitivity in rad/volt,

𝑣𝑚(𝑡) the message signal, ∆𝜙(𝑡) = 𝑘𝑝𝑣𝑚(𝑡) and 𝜙𝑑 = 𝑘𝑝|𝑣𝑚(𝑡)|𝑚𝑎𝑥 is defined as the

maximum phase deviation from the unmodulated value. Furthermore, in literature,

one can distinguish between Narrowband and Wideband PM with narrowband

associated with 𝜙𝑑 ≤ 0.25.

It can be shown that narrowband PM has a similar frequency spectrum as that of AM

modulation and hence the message signal can be demodulated using a mixer. For

wideband phase demodulation, the PLL or an IQ mixer can be employed.

Literature Review


2.3.1 Mixers and Image Rejection

Undesired signals due to the mixing process which can get into the radio signal path

are called the image frequencies, 𝐹𝑖 . Figure 2.4 shows how an image frequency

reappears in the band of the IF filter superimposed on the desired frequency 𝐹𝑑 .

Figure 2.4: High-Side injection gives Fi = Fd + 2Fif and mirrors the IF spectrum

One solution to this problem is to have several IF stages (Figure 2.5). Choosing the

first IF frequency to be higher than the highest desired frequency 𝐹𝑑 would place the

image frequency 𝐹𝑖 very high and out of band where it can be filtered off and

rejected.

Literature Review


Figure 2.5: The superheterodyne downconvertor

However, this comes at a cost since the IF filter must be very high in frequency with

a high Q factor which is expensive. The frequency of the first LO must also be very

high which is again expensive and more sections of down conversion must be used

in order to get to baseband.

2.3.2 Zero IF mixers or direct downconverters

Zero IF mixers or direct downconverters (homodyne) have already been mentioned

(𝜔𝐿𝑂 = 𝜔𝑅𝐹) but the discussion was limited to having a single harmonic as the RF

signal i.e. the phase difference ∆𝜙(𝑡) was a constant.

When the phase difference is varying like in PM, as the LO frequency approaches the

RF centre frequency, the IF output signal crosses the 0 Hz boundary and its spectrum

is folded back from DC to half the bandwidth, jeopardizing its content as shown in

Figure 2.6.

Literature Review


Figure 2.6: Frequency folding when FLO = FRF

This problem and the fact that the image frequency is translated to baseband

together with the signal of interest makes this setup problematic. Moreover, since

the IF is at DC, the mixer DC offset needs to be dealt with.

2.3.3 The IQ mixer or Quadrature Downconverter

Quadrature downconversion is a technique that mitigates the image frequency

problem by using phase cancellation techniques to cancel the image frequency/s as

opposed to the Superheterodyne method of rejecting the image with filtering. There

are several benefits with direct-conversion receiver designs as opposed to

Superheterodyne design [69]. Similar to the zero IF mixer, the RF frequency is

converted directly to baseband; there are no IF stages or band pass filters required.

The added phase shifts along with DSP processing ultimately allow the image signals

to be cancelled leaving only the desired signals. For a detailed mathematical

treatment the reader is referred to [70]. Figure 2.7 shows the basic topology of an

IQ mixer.

Literature Review


Figure 2.7: The basic topology of an IQ mixer

This topology requires two LOs with 90 degrees phase difference. The DC offset

problem can be mitigated by using a low IF frequency rather than zero IF as this

does not place the band of interest at DC. In theory, using this setup, image

frequencies will be cancelled but in practice there would always be some mismatch

or imbalance of the gain and/or phase in the I/Q paths. Consequently, the image

suppression would be far from complete.

2.3.4 MEMS mixers

As mentioned in Sections 2.3.1 to 2.3.3, there are definite advantages associated

with using a zero IF downconvertor or homodyne receiver over the heterodyne

alternative. Direct conversion requires less hardware (one less mixer than

heterodyne) and hence cost and size are reduced. Another added benefit is that the

homodyne receiver makes use of a low pass filter instead of a high-Q bandpass filter.

One objective of this work is to investigate the effectiveness of replacing discrete

electrical components making up the mixer and filter in a homodyne receiver with

a MEMS structure that serves the purpose of demodulation.

Literature Review


Many receiver stages have been replaced with MEMS parts: antennas [71], switches

[72], [73], filters [74], mixers [75] and LO oscillators [76]. Still, integrating each

component in one chip is challenging [77], [78].

It has been demonstrated [15], [79] that micromechanical resonators can also be

utilised as mixer-filters thus eliminating the need for channel filtering at GHz while

retaining the benefits of high mechanical Q. In RF MEMS literature, the word ‘mixer’

is loosely interchangeable with ‘mixer-filter’, sometimes also referred to as ‘mixlers’.

If the structure intrinsically selects a specific range of frequencies apart from mixing,

it is a mixer-filter. MEMS mixer-filters exploit the nonlinearity of the electrostatic

force with the drive voltage in the electromechanical resonators, down converting

GHz RF input signals to excite MHz mechanical resonance for IF filtering. Mechanical

displacement is then capacitively sensed into an IF output. Mixing and filtering are

achieved simultaneously.

Although software reconfiguration is the ultimate goal in multi-band radios, power

and dynamic range limitations require some of the reconfiguration to take place in

the RF front end [12], [80]. As already shown in Figure 1.2, MEMS mixer filters have

the potential to be key components in future reconfigurable multi-band single-chip

radio.

The following sections will distinguish between electrostatically actuated

electromechanical mixers and thermally actuated electro thermal mixers.

2.3.5 Electro-Mechanical Mixing

The mixer-filter designed in [15], is made up of two identical resonators connected

by a highly resistive coupling beam. The resonators are 18.8 × 8 𝜇𝑚. They vibrate

in the vertical direction when the carrier is on the RF electrode and the local

oscillator applied at the anchors at both ends of the input resonator. It is reported

that resonance is at 𝜔𝑜 = 37 MHz which is the same as the intermediate frequency.

This mixer-filter requires a 200 MHz local oscillator. The reported capacitive gap

Literature Review


between the resonator and the input/output electrodes is 32.5 𝜂𝑚 which ensures a

high electromechanical coupling. Mixing is achieved electrostatically and the air gap

was reduced to one third of what was reported in a previous work on resonators

[81].

The RF signal frequency ranges from 233 to 242 MHz and it is mixed with the LO

signal giving output frequency in the range of 33 MHz to 42 MHz, centered at

37 MHz, which is the resonance frequency and the required IF frequency.

Frequency tuning of the resonators is also possible by using two separate electrodes.

For the output signal, a DC bias is required on the output resonators which generates

the output IF signal on the output electrode. The reported conversion and insertion

loss are 13 dB with through measurement at -72 dBm and mixer output at 85 dBm,

with a noise floor of -100 dBm. The process of fabrication is quite intricate and is

custom designed with specific surfactants to remove residues from the small

capacitive gap.

Downconversion performance has been improved by Koskenvuori and Tittonen

[82], [83] and [84]. Their structure takes advantage of parametric resonance of two

double-ended tuning forks (DETF). The structure has a mechanical coupling beam

similar to that used in the previous work [15]. This isolates the IF output from the

RF input. The LO signal is fed directly to the input DETF structure via a 170 𝜂𝑚

capacitive coupling gap, Figure 2.8.

Literature Review


Figure 2.8: The MEMS designed by [83]

In [82], the same authors present a downconversion method which does away with

the LO input. The side bands are specifically chosen to match two Eigen frequencies

of the same resonator. Such a mixer requires no local oscillator. The latest paper by

the same group [84] shows that by exciting the DETF at twice resonance (instead of

some LO frequency), they would be effectively modulating the spring constant at

twice resonance and the downconversion would happen by ‘self-mixing’.

2.3.6 Electro-Thermal Mixing

Electrothermal mixing is another method of signal mixing. In [85], a dome shaped

resonator is employed to implement a mixer. The advantage of electrothermal

actuation over electrostatic actuation is that it does not require the fabrication of

nanometre capacitive gaps. The diameter of the dome is about 30 µm. When the RF

signal is applied to a gold thin-film resistor with 250 mV DC bias, the Joule heating

causes the out-of-plane deflection. When the AC current through the micro heater

matches the resonator frequency, the heat is modulated and dissipated at a

compatible rate as the mechanical resonance - the resonator has smaller thermal

mass and thus a smaller time constant. Figure 2.9a shows the actuation current path

Literature Review


while Figure 2.9b shows the dome mode shape. Equation (2.6) describes how the

change in temperature Δ𝑇 and hence the expansion is being modulated by the RF

signal.

Δ𝑇 ∝𝑉𝑟𝑓2

𝑅∝(𝑉𝑑𝑐 + 𝑉1 sin(𝜔1𝑡) + 𝑉2 sin(𝜔2𝑡))

2

𝑅 (2.6)

Expanding the squared bracket in equation (2.6) reveals the dependency to sum and

difference frequency components. The resonant frequency of the dome will be

matched to the difference: 𝐹𝑟𝑓 − 𝐹𝐿𝑂 .

The quality factor, Q, varies between 3,000 and 10,000 in vacuum but in air (at NTP)

it is about 100. The resonance frequency, 𝑓𝑟𝑒𝑠 is 12.7 MHz which is purposely

matched to the mechanical resonance.

a) b)

Figure 2.9: The dome mixer, [85], a) showing actuation b) showing mode shape

Literature Review


The local oscillator frequency is set at 60 MHz. The input to the resonator is the sum

of the chosen carrier frequency, 𝑓𝑐 at 72.7 MHz. This configuration takes advantage

of two tones offset from the carrier frequency for the test signals. The key is that the

gold resistor acts as the frequency translator and the coupled resonator works as a

post mixing filter. In this paper, measurement and validation was performed using

optical profilometry.

Subsequently, in [86], a 70 µm × 3 µm gold strip on the resonator was used to serve

as a micro heater and another identical size strip acted as a piezoresistor implanted

in the resonator. This was an improvement since sensing was integrated in the same

device. The membrane deflection causes strain on the doped silicon strip. The

change of the resistance is proportional to the corresponding strain. For the same

-20 dBm drive, the in-band insertion loss is improved from -65 dB to -35 dB when

the resonator is coupled to an operational amplifier.

The most recent work by the same authors on electrothermal mixers is [87].

Figure 2.10 shows the most recent work on MEMS thermal mixer that takes

advantage of the thermal expansion property of a bimorph resonator. Figure 2.10

inset shows a scanning electron microscope (SEM) image of the structure.

Figure 2.10: Structure used and electrodes for mixing [87].

Literature Review


An aluminium strip is deposited on the silicon carbide in a cantilever beam

structure. This structure shows a maximum vibration amplitude of 62 nm at a

resonant frequency of 944.49 kHz. This frequency is achieved when two frequencies

of 𝑓1 at 1200 kHz and 𝑓2 at 255 kHz are applied to the two independent aluminium

electrodes. While 𝑓1 is fixed, 𝑓2 is swept to find the resonance frequency by varying

the difference between the two signals in 10 kHz interval. With a cantilever length

of 200 µm, the resonant frequency lowers to 89.37 kHz.

2.3.7 BPSK to ASK conversion

Due to their simplicity, BPSK and ASK are adopted in many standards that are

implemented for low-cost passive transmitters. These modulation schemes are

popular in standards like the Medical Implant Communication System (MICS) and

Medical Data Service (MEDS) for biomedical applications and also Near Field

Communication (NFC) standards A and B. Such standards are designed for passive

transmitters and constant amplitude modulation schemes (FSK/PSK) are preferred

for energy (and data) transmission. When ASK is adopted, it is common practice to

make use of a modulation index M that is close to unity. This index is defined as the

ratio of smallest to highest modulated carrier amplitude and is purposely kept high

such that the passive listener does not lose power.

A BPSK to ASK converter is generally used as a first step to digital detection. In

particular, for medical implants, ASK is the preferred modulation scheme [4] as its

demodulation requires simple hardware, small size and less power consumption.

However, PSK and FSK offer a constant power RF signal which is preferred for a

passive listener. A BPSK to ASK converter can be used for the frontend of an

implantable device such that BPSK signalling (which has constant power) is used to

carry wireless data (and power) across the skin but this is immediately converted

to ASK for eventual demodulation and processing using simpler hardware.

Literature Review


No literature is found on BPSK to ASK conversion using MEMS. To-date, such a

function can only be realised in manufacturing using CMOS technology. In [88], two

circuits for BPSK to ASK conversion are proposed with the aim of reducing the

components required for such function. The authors trade performance with

simplicity of implementation such that low-cost makes it attractive to RFIDs and

sensor networks. In [89], the same application area is considered and the BPSK to

ASK converter is used as a first stage to the envelope detector. Here again, making

use of a BPSK to ASK converter, as a distinct first stage, is given preference over

alternative options such that complexity and power is kept low. In [90], [91], the

authors present a BPSK demodulator which is based on a BPSK to ASK first stage,

and report that the prototype consumes 228 𝜇W of power with sub-GHz carrier. In

[92], an ultra-low power transceiver with a BPSK downlink for a semi-active RFID

sensor node is presented. Even here, the BPSK signal is first converted to ASK and

then the signal is down converted to baseband with an envelope detector. The

authors break down the power consumption of the downlink, giving 204 𝜇W

dissipation for the BPSK to ASK stage and 69 𝜇W for the envelope detector; this had

sub-GHz carrier and data rates up to 10 Mb/s. To achieve low power consumption

and circuit simplification, all the results reported in [88], [89], [90], [91], [92] are

based on the BPSK to ASK architecture, first introduced in [93], [94].

2.3.8 Sensing Strategies

In order to physically extract the baseband signal from the modulated RF signal, a

way to measure the displacement of a MEMS device is required. This can take many

forms: a cantilever beam, a cantilever beam with an attached plate, a fixed-fixed

microbeam or even a tethered microplate. Currently, there are three popular

displacement sensing methods for MEMS applications: capacitive sensing,

piezoresistive sensing and piezoelectric sensing. Each sensing mechanism can be

built and implemented on a bulk silicon substrate.

Literature Review


The main challenge is in sensing displacement effectively while minimising losses

and footprint. The work in [15] measures the signal by capacitive coupling which

requires voltage driving circuitry. The work in [86] integrates piezoresistive sensing

methods. Piezoresistive sensing measures the change of resistance - which is

proportional to the change of displacement - by measuring current. It drifts with

temperature, requires external driving circuitry and consumes power when there is

no incoming RF signal. In [85] and [87], optical methods are used such as a probe

station and a vibrometer. Optical sensing is in general used only for feasibility

studies. In piezoelectric sensing, the voltage change is measured due to change in

displacement. Strain sensitivity up to 5 𝑉/𝜇𝜖 is achievable. The piezoelectric sensing

method requires no external driving circuitry but at steady-state; the absolute

displacement is not detectable due to the high pass nature of the piezoelectric

material.

2.4 Geometric Nonlinearities and Vibrational Energy Harvesting

Traditionally, the operating principle of a vibrational energy harvester (VEH) relies

on linear resonance. The assumption here is that the input frequency to the device

carries just one harmonic, that is, a fixed frequency. To maximise the energy flow

from the environment to the electric device, the beam parameters can be tuned such

that the modal frequencies, in particular the first, is close, ideally equal, to the base

excitation frequency. Using resonance to amplify the beam oscillations is beneficial,

however, this comes at a cost when it comes to achieving broadband

responsiveness. In practice, in VEH design, damping is kept very low to improve the

gain at resonance but this would generally give a high-quality factor resulting in a

relatively narrow bandwidth.

In contrast, excitations available in the environment are generally not harmonic in

nature; they would have a wide spectral content and can also be time-varying in that

the frequency content changes with time. This makes it very challenging to tune a

Literature Review


linear VEH to match the excitation frequency and results in inefficient transduction

of energy.

As a solution to this problem, initial designs featured tuneable VEHs and made use

of arrays of harvesters such that the full range of input excitations could be made

use of. For tuning, both passive and active designs were used to alter the

fundamental mode to match the dominant frequency of vibration [95], [96], [97],

[98], [99], [100], [101]. It soon became clear that the efficiency for stochastic inputs

or rapidly varying frequency sweeps made energy harvesting unreasonable [95].

This is more so since external power would be required to tune and this has an

impact on the net efficiency. Alternatively, if tuning is not considered, the designs

make use of arrays of harvesters such that at least one device was close to the

dominant vibrational frequency [98], [100], [101]. This however impacted

negatively on the energy density and scalability prospects.

The drive to achieve wider coupling between a harmonic oscillator and the (broad)

excitation frequencies has driven many researchers to exploit nonlinearities to

improve on VEHs efficiency. As mentioned earlier, geometric/structural

nonlinearities arise for large deformations [102]. Furthermore, nonlinearities can

also be the result of nonlinear relationships in the electromechanical coupling

mechanism, for example, in piezoelectricity [103]. However, such nonlinearities are

not easily exploited for energy harvesting purposes as they are not easily modified

and are intrinsic to the process. The intentional introduction of nonlinearities in the

design has been the target of more recent investigations. Nonlinearities by design

means the introduction of a feature such that one could control the nature and

magnitude of the nonlinearity for improved VEH efficiency. Two techniques can be

found in literature; one introduces a controlled magnetic force and the other a

controlled mechanical restoring force [104], [105], [106], [107]. These have been

shown to widen the bandwidth of the VEH and hence improve transduction to tap

realistic, widely available environmental excitations.

Literature Review


Further review of how geometric or structural nonlinearities can be employed to

widen the bandwidth of a VEH can be found in Section 4.1.

2.5 Modelling and Validation Approach in nonlinear MEMS

Predicting static and dynamic response accurately for a system that has several

nonlinear contributions is a rather difficult task. Generally, the approach to arrive at

an adequate mathematical model requires several small steps, each involving a

validation step. A case study that involves RF-MEMS which exemplifies this process

of improvement in small steps is described in [108], [109], [110], [111], [112], [113],

[114] and [115]. In this study, the authors start by investigating pull-in in the

presence of structural (geometric) nonlinearities [108]. In [109] to [111], the

numerical approaches adopted are validated experimentally. In [112] and [113], the

authors shift their focus on the dynamics and in [114] and [115] both the

electrostatic and geometric nonlinearity are considered in the model

simultaneously.

RF frontend functions in MetalMUMPs


3. RF FRONTEND FUNCTIONS

IN METALMUMPS

In this chapter, a mechanical structure that is able to serve several RF frontend

functions is presented. It consists of a plate supported by two tethers and able to

vibrate in torsion. The actuation/drive mechanism is electrostatic in nature and

makes use of the smallest gap possible in the MetalMUMPs fabrication process [22].

The layers and critical dimensions are shown in Figure 3.1.

Figure 3.1: The MetalMUMPs layers, smallest gap between conductors is 1.45 µm

The chapter is divided in two main sections, 3.1 and 3.2; these describe how the

design proposed attains its functionality.

In Section 3.1, the proposed structure is designed to give displacements in response

to BPSK signalling. Displacement sensing is also achieved using the same pivoted

plate. At the end of this section, the magnitude of the actuation signals is investigated

and boundaries for linear and non-linear dynamic characteristics are determined.



In Section 3.2, a similar torsional plate, but having more electrode pairs for actuation

and also for sensing, is designed such that unwanted frequency components

generated by electrostatic mixing are suppressed. This section concludes with an

application to low-IF IQ mixing.

3.1 A MEMS BPSK Demodulator

Electrostatically driven torsional structures have been used extensively since one of

their first successful appearance [116] which involved driving a mirror in the

1980’s. In [116], the plate undergoing torsional vibrations was used both as the

mirror and as the electrostatic actuator.

As shown in [117], the switching speed of such micro-mirrors has come a long way

since then. The first structure considered in this paper has dimensions within the

same order of magnitude of [117]. The major differences are the manufacturing

process, the application, and in [117], the mirror undergoing angular movement has

mechanical stops.

Modelling of electrostatic torsional actuation is described in [118]. One of the results

given in [118] is the maximum angular rotation before pull-in occurs. As described

in Section 3.1.3, the plate dimensions and voltages were selected such that pull-in

does not occur.

A considerable amount of work to achieve signal mixing in the mechanical domain

has also been carried out using CMOS commercial fabrication [119], [120], [121] and

[122]. This process is called CMOS-MEMS. These solutions have the advantage that

they can be embedded within CMOS circuitry. The process involves standard CMOS

fabrication followed by an anisotropic etch. Subsequently, a final release step

involving a combination of DRIE and isotropic silicon etch is used to release the

structure. In [120], it was claimed that mixing for frequencies in the range of 10 MHz

to 3.2 GHz was successfully demonstrated. The electrostatic gap used was 1.3 µm.



In all these CMOS-MEMS structures, the mixing is performed electrostatically and

the output is then mechanically filtered with the cantilever structure.

In [15], mixing is also achieved electrostatically, but here, a clamped-clamped beam

is used. This beam is mechanically coupled with a similar clamped-clamped beam to

achieve filtering. In this case, a polysilicon surface-micromachining process was

used with electrostatic gaps ranging from 325 Å to 1000 Å. Successful mixing at

200 MHz was reported.

The mixing and filtering structure presented here employs a relatively large

capacitive gap at 1.45 µm (Figure 3.1). The target application is BPSK demodulation

encoded as in IEEE 802.15.4 which describes a low data rate wireless personal area

network protocol. This standard has a carrier of 868 MHz with a data signal at

300 kchips/s which constrains the mechanical structure to have a bandwidth of at

least 900 kHz (3rd harmonic of data).

Section 3.1.7 details simulation results that demonstrate the feasibility of such a

structure to demodulate BPSK signals as described in the IEEE 802.15.4 standard.

3.1.1 The Mechanical Structure

The smallest achievable gap between conductors in the MetalMUMPs fabrication

process is 1.45 µm. This is the gap between the nickel layer, hereon referred to as

metal, and the polysilicon layer. The polysilicon is encapsulated within top and

bottom nitride layers.

The smallest gaps achievable in the horizontal directions, between metals and/or

polysilicon, are larger at 5 µm. Hence, a structure that vibrated in the vertical gap

between the metal and the polysilicon was designed. Out of the total electrostatic

gap ‘𝑑’ of 1.45 µm, only 1.1 µm of vertical movement is allowed as there is 0.35 µm

of nitride above the polysilicon.



3.1.2 Torsional Oscillations of a Plate

Figure 3.2 to Figure 3.4 show how actuation and sensing were achieved in the

designed structure (S1). On the actuation side, the biased RF signal at the metal

interacts with the biased LO signal to produce a force on the plate. In turn, this force

drives the plate into torsional oscillations. On the sensing side, a current is generated

at the polysilicon output electrode by interacting with the metal above it which is

DC biased. The output signal is also filtered by the low-pass vibration characteristics

of the plate structure. The overall dimensions of the nitride plate are 60 µm × 35 µm

with the axis of rotation being parallel to the longer side. The area of the polysilicon

electrodes is 250 µm2 and each tether is 15 µm wide.

Figure 3.2: The complete S1 structure showing metal layers in violet

Figure 3.3: Section through S1; view from bottom showing only one tether.



Figure 3.4: Schematic diagram of the torsional BPSK demodulator depicting the bias and excitation

scheme required for mixing, filtering and sensing.

3.1.3 Modelling and Analysis

In the first part of this section, some theoretical background on electrostatic mixing

is presented. This theory is then applied to the mixing of the LO signal and the BPSK

modulated RF signal, and the frequency components of the resulting input force are

discussed. In the second part, the mechanical filtering is used to shape the input

spectrum and in the last section, current sensing at the output electrode is shown to

provide the required data signal.

3.1.4 Frequency Content of the Input Force

The LO frequency is set to be the same as the BPSK carrier frequency so that the

conversion of the modulated signal is performed in one step (direct conversion). In

general, the LO and RF are not necessarily in phase, hence, 𝛼 ≠ 0 in (3.1). The LO

signal is DC biased by 𝑉𝑑𝑐𝐿𝑂 volts and 𝜔 rad/s is the carrier frequency. For BPSK, the

phase in the RF signal has two distinct values and the modulated BPSK signal with

DC bias can be formulated as in (3.2).

𝑣𝐿𝑂(𝑡) = 𝑉𝐿𝑂 cos(𝜔𝑡 + 𝛼) + 𝑉𝑑𝑐𝐿𝑂 (3.1)

and



𝑣𝑅𝐹(𝑡) = 𝑉𝑅𝐹(2𝑑(𝑡) − 1) cos(𝜔𝑡) + 𝑉𝑑𝑐𝑅𝐹 (3.2)

where,

𝑑(𝑡) (∈ [0,1]) represents the binary data signal for integer durations of 𝑛𝑇. For the

lowest data rate specified in the 802.15.4 standard which has a chip rate of

300 kchips/s, the pulse duration would be 𝑇 = 1/300000 s and

𝜔 = 2𝜋(868) 𝑀𝑟𝑎𝑑/𝑠. Denoting the vertical displacement of the electrodes by

𝑥(𝑡) µm, the absolute permittivity in air by 휀 and the effective electrode area in µm,

𝐴𝑒 then the force at the input side would be described by (3.3).

𝐹𝑖𝑛(𝑡) = −휀𝐴𝑒

2(𝑑 + 𝑥(𝑡))2[𝑣𝐿𝑂(𝑡) − 𝑣𝑅𝐹(𝑡)]

2 (3.3)

If we further simplify by assuming that 𝑥(𝑡) ≪ 𝑑, that is, the spring stiffness stays

constant, and letting 𝑉𝐿𝑂 = 𝑉𝑑𝑐𝐿𝑂 = 𝑉𝑅𝐹 = 𝑉𝑑𝑐𝑅𝐹 = 𝑉 and substituting (3.1) and (3.2)

in (3.3), we get (3.4):

𝐹𝑖𝑛(𝑡) = 𝐸 (𝐷2 − 𝐵2

2) 𝑐𝑜𝑠(2𝜔𝑡) + 𝐵𝐷𝑠𝑖𝑛(2𝜔𝑡) + (

𝐷2 + 𝐵2

2) (3.4)

where, 𝐷 = 2𝑑(𝑡) − 1 − 𝑐𝑜𝑠𝛼, 𝐵 = 𝑠𝑖𝑛𝛼 and 𝐸 = −𝜀𝐴𝑒 𝑉

2

2𝑑2.

Equation (3.4) has three terms; the first two have bandwidths centered on 2𝜔 while

the last term is centered on 0 Hz. All these components can be seen in Figure 3.5. We

are interested in the baseband part of the input force as the other components are

very high and will be mechanically filtered out. Table 3.1 further breaks down the

last term as expressed in (3.5):

𝐸[2𝑑2(𝑡) − 2(1 − 𝑐𝑜𝑠𝛼)𝑑(𝑡) + 1 + 𝑐𝑜𝑠𝛼] (3.5)

The bandwidth of the data signal 𝑑(𝑡) is inversely proportional to the pulse duration

𝑇. While theoretically, the bandwidth of the data signal is infinite (pulse train), we



aim to capture information up to its 3rd harmonic which dictates the bandwidth of

the mechanical filtering required (Section 3.1.5).

Table 3.1: Breakdown of force components around 0 Hz as in equation (3.5).

Centre Freq. Term Bandwidth

0 Hz ]cos1)(2[ 2 ++tdE Bandwidth of )(td

0 Hz )(]cos1[2 tdE −− Bandwidth of )(td

Figure 3.5: The spectrum of the electrostatic force generated and the required mechanical

bandwidth for adequate reconstruction.

3.1.5 Mechanical Filtering

For small oscillations, the rectangular tether’s torsional spring constant can be

calculated using (3.6) [123].

𝑘𝑡 = 𝑘1𝐺(2𝑎)3(2𝑒) (3.6)

where, 𝑒 × 𝑎 are the tether section dimensions of 15 × 0.8 µm, 𝐼 the moment of

inertia of the plate about the tether pivot line and 𝐺 the modulus of rigidity of the

plate. The parameter 𝑘1 is a function of the ratio 𝑒/𝑎 and is tabled in [9]. The

dimensions of the plate were selected such that the plate’s natural frequency is in



the region of 𝑓𝑛 =1

2𝜋√𝑘𝑡/𝐼 =2 MHz. As required, for the data rate under

consideration, this is higher than the 3rd harmonic of 900 kHz. A simplified 2nd order

linear model, which includes damping with damping coefficient 𝑏, is described by

(3.7).

𝐼 + 𝑏 + 𝑘𝑡𝜃 = 𝐹𝑖𝑛(𝑡) (3.7)

When damping is taken into consideration, the resonant frequency 𝑓𝑟 would be

lower (see results section). Having selected 𝑓𝑛 at least two times greater than the

3rd harmonic of 𝑑(𝑡) would still provide a substantial safety margin.

In general, damping can be divided in two categories: thermoelastic effects and fluid

damping. The dominant form of damping is determined by the Knudsen number

[124]. In [124], the nonlinear relationship between the viscous damping coefficient,

gap and pressure is clearly seen, especially for gaps less than 0.8 𝜇m. However, in

much of the reviewed literature related to resonators with gaps larger than 0.8 𝜇m,

the effects of damping is assumed to be dominated by a viscous damping coefficient

and thermoelastic effects were ignored.

Due to the squeeze film effect, for low enough frequencies, the air can escape around

the moving plate with little resistance but for high frequencies, the air is held in

position due to its own inertia and it compresses resulting in a spring force.

Air does not move around much and damping gets lower as is shown in Figure 3.6.

All FEA simulations included also squeeze-film damping effects due to the small gaps

involved (1.45 𝜇𝑚). Air at NTP has a mean free path, 𝜆, of 68 nm. This gap and mean

free path give a Knudsen number of 0.047. This means that the continuous

assumption for the fluid medium does not hold as the value of 0.047 is within a

transitional regime [125].

The damping coefficient is a function of frequency and the value adopted for

MATLAB simulations, 𝑏 = 2.83 𝜇N/ms-1, is that occurring at the resonant frequency

of the structure (2 MHz).



Figure 3.6: FEA result for damping force coefficient against frequency for the torsional plate taking

into account squeezed film effects.

From [123], pull-in would occur when 𝜃𝑝 ≈ 0.4404 𝑔/𝐿, with 𝐿 being half the width

of the plate (35/2) µm. Hence, for structure S1, 𝜃𝑝 = 0.027𝑟 , which means that

approximately a maximum vertical displacement of 470 nm is permissible.

3.1.6 Current Sensing at the Polysilicon electrode

At the output side, the polysilicon electrode is moving within the electrostatic field

generated between itself and the metal above which is DC biased by 𝑉𝐷𝐶. This

induces a current at the polysilicon output electrode which can be eventually

converted into voltage via a suitable amplifier.

The distances between the two high frequency inputs (LO and RF) and the output

electrode are small and capacitive parasitic coupling at high frequency is bound to

occur.



Hence, together with the required current generated due to oscillations, a current

proportional to the LO/RF frequency is also generated at the output electrode.

One solution to cancel these parasitic couplings is to use a differential topology as

shown in Figure 3.7, in which case, two output currents, 𝑖𝑎 and 𝑖𝑏, from two

mechanically identical structures are fed into a fully differential charge amplifier.

The only difference in the structures would be that for one structure, 𝑆1𝑎, the Metal-

DC would be biased by +𝑉𝐷𝐶, while for the second, 𝑆1𝑏 , the biasing would be −𝑉𝐷𝐶.

Figure 3.7: Differential setup for sensing using a DCA

The output of both structures is held at virtual ground by the operational amplifier

with resistive feedback (Rf) and output currents 𝑖𝑎 and 𝑖𝑏 go through Rf. The input

virtual ground reduces the effect of parasitics (Cp). Although the analysis presented

here assumes linear dynamic behaviour, it still provides the fundamentals required

such that one can wisely choose the electrical and mechanical parameters. As is

demonstrated in the following section, the setup shown in Figure 3.7 is successful in

demodulating low data rate BPSK signals in simulation.



3.1.7 Simulations and Results

This section presents the results of the FEA simulations using CoventorWare®.

Table 3.2 summarizes the mechanical results.

Table 3.2: Mode types and frequency, Q factor and damped resonant frequency.

Mode Freq. (MHz) Mode Type Q factor / Damped Resonant Freq. (MHz)

1.57 Not Torsional Not required

2.01 Torsional 60 / 1.54




The required mode should be a torsional mode as shown in Figure 3.2. The modes

closest to the one required are not torsional in nature and cannot be excited with

the input actuating signals. A mechanical/electrostatic simulation with driving

voltages of 20 V peak was performed and the generated signals were analysed.

Due to the strong parasitic coupling between the output electrode and the RF/LO

signals, the two current graphs are almost linear in nature. These are the output

currents from S1a and S1b respectively, as shown in Figure 3.8. This linearity breaks

down at resonance due to the larger displacements (Figure 3.9). If these currents

are subtracted, the differential output current required (Figure 3.8a) can be

converted into voltage through the DCA setup shown in Figure 3.7.

For S1, this voltage would contain enough harmonics from 𝑑(𝑡) for successful

reconstruction / demodulation. Even though the DC content is highly attenuated due

to the substantial Q factor, this does not present a problem as the 32-bit chips used

in the 802.15.4 modulation scheme do not have any DC content. The maximum

displacement in this response does not exceed 6 nm which is well within the pull-in

region as determined in [126].



Figure 3.8: Currents at the outputs for both positive and negative DC biasing

Figure 3.9: Displacement against frequency.



The FEA analysis provides sufficient information to characterize the torsional plate

dynamics and also the electrical and electrostatic properties of the setup. With this

information, a linear mathematical model was implemented in MATLAB and the

vertical displacement of the moving electrodes was investigated for the data signal

𝑑(𝑡). Figure 3.10 shows the resulting response. This mathematical model was

validated against FEA data and found to give the same displacements and currents.

Figure 3.10: The displacement has a strong 3rd and 4th harmonic.

Note: Settling barely happens within 1 chip time (1.67 µs)

The maximum displacement in this response does not exceed 6 nm. This is well

within the pull-in region as determined in [126]. However, the sensing current

generated is very small, at around 5 nA. Nevertheless, this design provides a proof-

of-concept and shows that with the MetalMUMPs fabrication process, a torsional

vibratory structure can be built to successfully demodulate a low data rate BPSK

signal with a carrier frequency of 868 MHz and a chip rate of 300 kchips/s.



The drive to include extra functionality in portable devices is always increasing and

one way of providing this is by having structures that can be used for different

applications. This MEMS device is one such structure as is shown in the following

sections.

3.1.8 Investigation of Potential Complex Dynamics

Having a good mathematical model is instrumental in accurate prediction of the

different modes of operation. Modelling the dynamics of MEMS resonators is

extensively covered in literature [127], [128], [129] and [130]. Numerical solutions

that involve distributed models and FEA provide the best description of system

behaviour but are computationally expensive. Assumptions are taken to develop

non-linear systems of equations that are able to give a good enough approximation.

Whenever the modes of vibration are sufficiently different and far apart in their

frequencies, it is common to ignore all modes except the most prominent one.

Further assumptions are taken on modelling the spring behaviour. A linear,

quadratic and cubic spring stiffness term is considered in [127] while in [128] only

the linear and cubic terms are included. Inhomogeneity of the silicon structure and

geometric nonlinearities are often responsible for this behaviour [127]. When it

comes to damping, the behaviour is more complex. As already mentioned in Section

3.1.5, the dominant form of damping is determined by the Knudsen number [124]

and the damping coefficient is a function of frequency which also contributes to

nonlinearities. In this section, a non-linear model is developed and the dynamical

behaviour under different input conditions is investigated.

3.1.9 Development of the Mathematical Model

For the derivation of the mathematical model, the same structure was considered

but actuation was simplified and only one signal was used to drive the mechanism

into oscillation as shown in Figure 3.11.



Figure 3.11: The system structure

Using FEA, the spring nonlinearity for large displacements was investigated and the

resulting force-displacement diagram is shown in Figure 3.12.

Figure 3.12: The force curve for static displacements as large as 0.4 µm

For ‘large’ displacements (order of magnitude of the gap), the spring force could be

described by (3.8) (least squares polynomial fit), while for the BPSK demodulator

region of operation, a linear spring model suffices i.e. (3.9).



𝐹𝑧 = 19000𝑥3 + 10−9𝑥 (3.8)

𝐹𝑧 = 10−9𝑥 (3.9)

Substituting the nonlinear spring model (3.8) and the driving electrostatic force

(3.3) in the dynamic model (3.7) and also converting rotational to linear

displacements, the following model, (3.10), is proposed for the mechanical part:

= −𝑏

𝐼 −

𝑘𝑡𝐼𝑥 −

𝑘3𝐼𝑥3 +

휀𝐴𝑒𝑟2

2𝐼(

𝑣2𝑑 − 𝑥

)2

− (𝑣1

𝑑 + 𝑥)2

(3.10)

where 𝑥 is the vertical displacement at electrode P1 (refer to Figure 3.11), 𝑣1 is the

voltage between electrodes P1 and N1, 𝑣2 is the voltage between electrodes P2 and

N2, 𝑘3 is the cubic spring stiffness, 𝑟 is the distance from the pivot to the centre of

the Nickel electrode and 𝐼 is the moment of inertia of the plate about the pivot. Also,

from Figure 3.11, 𝑣1 = 𝑉𝑑𝑐 + 𝑉𝑎𝑐cos (𝜔𝑡) and 𝑣2 = 𝑉𝑑𝑐.

Model (3.10) is nonlinear due to the cubic stiffness term and the electrostatic forces

and describes only the rotational mode of vibration. The damping coefficient is a

function of frequency as shown in Figure 3.6. For the following analysis, it is

assumed constant as the primary scope for this analysis is to determine different

response modes depending on the location of the static equilibrium points (EPs).

The location of these EPs is independent of the damping coefficient. For easier

identification of the location of these EPs in time domain simulations, a smaller

coefficient of damping than the one determined in Figure 3.6 was used. A separate

damping simulation run was performed at a reduced atmospheric pressure (100 Pa)

and ambient temperature that gave a smaller damping coefficient. The estimated

parameters are listed in Table 3.3.



Table 3.3:Parameter values for the Non-linear model

Parameter Value Units Description

d 1.45 µm Nominal Gap

I 2.71 × 10−21 𝑘𝑔𝑚2 Moment of inertia

Ae 250 𝜇𝑚2 Effective area

b 5 × 10−16 𝑁𝑠𝑚−1 Viscous Damping

k3 19000 𝑁/𝑚3 Cubic Stiffness

kt 10−9 𝑁/𝑚 Linear Stiffness

r 10−5 m Radius of rotation

𝑣1 𝑉𝑑𝑐+ 𝑉𝑎𝑐cos (𝜔𝑡)

volts AC actuation

𝑣2 𝑉𝑑𝑐 volts DC actuation

With the values in Table 3.3, the resulting DE is stiff and confirming that the MATLAB

solution was actually correct was problematic. Using the ‘ode15s’ differential

equation solver in MATLAB the solution time step was determined by trial and error

until repeatability in the solutions was observed. Appendix 3.1 gives the MATLAB

implementation. The static behaviour of the structure was first investigated and this

was done by applying a range of biasing voltages, 𝑉𝑑𝑐 and letting 𝑉𝑎𝑐 = = = 0.

This gives the net force, 𝐹𝑛𝑒𝑡, on the structure as in (3.11).

𝐹𝑛𝑒𝑡 = −𝑘𝑡𝐼𝑥 −

𝑘3𝐼𝑥3 +

휀𝐴𝑒𝑟2𝑉𝑑𝑐

2

2𝐼(

1

𝑑 − 𝑥)2

− (1

𝑑 + 𝑥)2

(3.11)

Solving 𝐹𝑛𝑒𝑡 = 0, the equilibrium points (EPs), x*, were obtained. Furthermore, by

evaluating 𝜕𝐹𝑛𝑒𝑡

𝜕𝑥|𝑥=𝑥∗

the nature of the EP can be classified as stable, unstable or a

saddle point. Comparison of the EPs for both models are listed in Table 3.4.



Table 3.4: Equilibrium points for the linear and non-linear models

The nature of the equilibrium points was analysed further, Appendix 3.2 gives

MATLAB code for this investigation. Figure 3.13b shows how the equilibrium points’

position and nature vary as the DC actuation voltage is increased. The DE’s are

second order and the phase portraits are two dimensional with state vector

𝒑 = [𝑥 ]𝑇 .

Figure 3.13: The EPs as a function of Vdc, red lines for unstable, black for stable.

Note: a) 𝑥 − phase portrait with 𝑉𝑑𝑐 = 20 𝑉 shows 3 EPs, two are saddle and one is stable.

b) Location (x) and type of equilibrium points vs. Vdc.

c) 𝑥 − phase portrait with 𝑉𝑑𝑐 = 110 𝑉 shows 5 EPs, three are saddle and two are stable.

𝑉𝑑𝑐 = 20 𝑉




𝑉𝑑𝑐

= 110 𝑉

𝑉𝑑𝑐

= 110 𝑉

𝑉𝑑𝑐

= 110 𝑉

𝑉𝑑𝑐

= 110 𝑉

a)

a)

a)

a)

b)

b)

b)

b)

c)

b)

b)

b)



When the device was operated in the linear (20 volts) region for BPSK

demodulation, there was only one equilibrium point and two saddle points (Figure

3.13a) with the latter two describing the pull in region. However, when the voltage

exceeds 75 volts, the behaviour changes drastically and the system becomes bistable

(Figure 3.13c). Figure 3.13a and Figure 3.13c show trajectories for different initial

conditions with 𝑉𝑎𝑐 = 0 giving a constant (static) input.

Analysing the dynamic behaviour when the system is harmonically driven, that is,

𝑉𝑎𝑐 ≠ 0, one would find no equilibrium points unless ω = 0. Alternatively, the

non-autonomous system can be reduced to an autonomous one by increasing the

dimensionality by one. Hence, by defining a new state 𝑝3 = 𝜔𝑡, a three state

𝒑 = [𝑥 𝜔𝑡]𝑇 system (3.12) is obtained.

=

(

1

2

3)

=

(

𝑝2

−𝑏

𝐼𝑝2 −

𝑘𝑡𝐼𝑝1 −

𝑘3𝐼𝑝13 +

𝜖𝐴𝑒𝑟2

2𝐼(

𝑉𝑑𝑐𝑑 − 𝑝1

)2

− (𝑉𝑑𝑐 + 𝑉𝑎𝑐𝑐𝑜𝑠(𝑝3)

𝑑 + 𝑝1)2

𝜔 )

(3.12)

3.1.10 Behaviour by Region of Operation

The overall behaviour of the plate is primarily dictated by the DC biasing voltages.

The frequency and amplitude of the driving signal also play a role but as can be seen

in Figure 3.13, the EPs are dependent on Vdc. Solving (3.11) for the EPs can give up

to 7 EPs, two of which are neglected since they lie outside the possible range of

movement. For Vdc < 75 V (region 1), the plate has one stable and two unstable EPs,

the latter two demarcating the pull-in regions. For small displacements, the

dynamics in this region are similar to what one would expect if (3.10) did not have



nonlinear terms. For 75 < Vdc < 188 V (region 2), the plate becomes bistable. The

horizontal position is not stable anymore, and the pull-in phenomenon is more

prominent, with only a maximum displacement of 1 µm at 170 V, before pull-in

occurs. This region can give rise to chaotic dynamics. For Vdc > 188 V (region 3),

there are no stable EPs in the system and the plate snaps to one of the nickel

electrodes.

Chaotic Dynamics and Region 2: Chaotic oscillators and chaotic systems have been

used to generate chaotic carriers to secure communications [131]. Region 2 merits

further investigation. By the Poincaré-Bendixson theorem, a strange attractor can

be ruled out for a 2D system. However, a 2D non-autonomous system can be

regarded as a 3D system (3.12) and this can give rise to chaotic trajectories. In region

2, the system is bistable and with a high enough 𝑉𝑎𝑐, the trajectory will traverse

across the two stable EPs through a saddle at (0,0). This is similar to a Duffing system

and by changing the forcing frequency 𝜔, a route to chaos (Figure 3.14 to Figure

3.16) through period doublings (Figure 3.15) was found. The driving voltage 𝑉𝑑𝑐,

was kept at 100 V which is higher than 75 V. This guaranteed that the system was in

the bistable region and the two stable equilibrium points were at ±0.2 𝜇𝑚 from the

horizontal position.

For the three different driving frequencies investigated, two simulations were

executed. These two simulations only differed slightly in the initial conditions (blue

and red traces in figures). This was done such that when the behaviour is chaotic as

in Figure 3.16, the trajectories deviate, showing the susceptibility to differing initial

conditions. What is interesting in these experiments is the fact that at particular

frequencies, the trajectories took irregular and chaotic paths which had a wide

frequency response.



The simulation time for the results shown in Figure 3.14 to Figure 3.16 was of

0.1 ms (sampled at 300 MHz). The driving frequency was lowered from 726 kHz

until the period doubling phenomenon was encountered at 635 kHz. The frequency

was lowered further and eventually chaotic behaviour was observed at 468 kHz.

Supporting evidence for chaotic behaviour is in the broad spectrum obtained in

Figure 3.16c, in the distribution of points on the Poincaré Map (Figure 3.16b) and

also in the divergence of the blue and red traces (different initial conditions).

Although the response is chaotic, it is not, as defined in [132], extensively chaotic. As

a result, the autocorrelation function for the displacement 𝑥(𝑡) (Figure 3.17) is

significantly different from an impulse function while the distribution (Figure 3.18)

is not quite uniform.

Figure 3.17: Autocorrelation of chaotic time series



Figure 3.18: Histogram of displacement samples for chaotic time series

For true random number generators employed for cryptographic applications, a

uniform distribution is required. This means that unless the sampling frequency is

taken down to below resonance, the time series would still have some periodic

content.

3.2 Suppression of Spurious Products in an Electrostatic

Downconverter

Downconverters are building blocks in Superheterodyne receivers. Down

converting involves mixing and filtering stages and these are conventionally

implemented as two distinct stages. It has already been demonstrated in Section 3.1

with reference to [15], [84], [133], [122] and [134] how mixing and filtering can be

accomplished by a single device in MEMS. Such structures are referred to as mixlers



in [15], [84], [134] and mixer-filters [133], [122], [134]. In [135], a symbol for such

a mixer filter device is also proposed. The main benefit of having two functions

within one structure is that mismatches between the mixing and filtering stages are

eliminated.

One important performance metric for a mixer is the conversion loss (CL). The

reported CL in literature varies wildly with values ranging from 125dB [134] to as

low as 13.8 dB in [135] and even -30 dB (a gain) in [84]. In[135], the steepness of

the pass band was increased by combining several structures which increased the

order of the filter.

Another important distinction found in literature is the process employed. In [122],

[133] and [134], mixers are developed in CMOS-MEMS while [15], [84] use other

MEMS dedicated processes. In [122], it is shown how CMOS-MEMS mixers can be

combined to achieve a Hartley image rejection formation. All these reviews have one

thing in common - mixing is performed electrostatically.

In this section, a novel technique employing a differential electrostatic drive is

presented such that the unwanted frequency components resulting from the

quadratic electrostatic relationship are eliminated without using filtering.

Conventionally, the products are suppressed using filtering [15], [83], [136] but this

requires a high selectivity bandpass filter which imposes a constraint on the

bandwidth. With the elimination of this filter, the bandwidth-selectivity trade-off

can be relaxed.

3.2.1 Frequency Perspective

As described by (3.3) the expression for electrostatic force, 𝐹𝑖𝑛, can be simplified for

small (𝑥(𝑡) ≪ 𝑑) oscillations and letting 𝐾 = −𝜀𝐴𝑒

2𝑑2, (3.3) reduces to (3.13).

𝐹𝑖𝑛 = 𝐾[𝑣𝑟𝑓 − 𝑣𝑙𝑜]2= 𝐾[𝑣𝑟𝑓

2 − 2𝑣𝑟𝑓𝑣𝑙𝑜 + 𝑣𝑙𝑜2 ] (3.13)



This setup (Figure 3.19) was used for actuation of the BPSK demodulator. As

described by (3.13), the resulting torque frequency components on the torsional

plate using a single pair of electrodes would be as in Figure 3.20. For

downconversion to be effective, the frequency of interest would be at 𝑓𝑙𝑜 − 𝑓𝑟𝑓 and

mechanical resonance would need to be designed to coincide with this frequency.

An appropriate mechanical Q factor is also required that needs to satisfy two

conditions: meeting data bandwidth requirements and having sufficient roll-off so

as to supress all the unwanted frequency components that are generated by the

electrostatic interaction.

Figure 3.19: Actuation with one pair of electrodes

Figure 3.20: Torque frequency components with a single pair of actuation electrodes



Alternatively, to achieve perfect mixing, the squared terms in (3.13) would need to

be eliminated in some other way. One method of achieving this (without filtering) is

by subtracting another force 𝐹2, given in (3.14) from (3.13).

𝐹2 = 𝐾[𝑣𝑟𝑓 + 𝑣𝑙𝑜]2= 𝐾[𝑣𝑟𝑓

2 + 2𝑣𝑟𝑓𝑣𝑙𝑜 + 𝑣𝑙𝑜2 ] (3.14)

In this manner, 𝐹𝑛𝑒𝑡 = 𝐹𝑖𝑛 − 𝐹2 = −4𝐾𝑣𝑟𝑓𝑣𝑙𝑜 which results in a perfect product. The

layout, which involves two pairs of actuation electrodes, is shown in Figure 3.21 and

the resulting frequency components are shown in Figure 3.22. The schematic in

Figure 3.23 shows a section through the proposed structure identifying the

respective materials in the MetalMUMPs® process.

Figure 3.21: Proposed torsional plate having both differential drive and sense



Figure 3.22: Torque Frequency Components with two pairs of actuation electrodes

Figure 3.23: Section through the proposed structure showing two pairs of actuation electrodes

This structure produces two counteracting torque components and since the two

sides of the pivot differ only by the sign of 𝑣𝑙𝑜 , the net driving torque, T, would be as

in (3.15).

𝑇(𝑡) = −4𝑟𝐾𝑣𝑙𝑜𝑣𝑟𝑓 (3.15)

Under static conditions, this torque would be acting against the torsional spring

produced at the pivot. Hence, the plate angle 𝜃 would be proportional to the driving

torque. Note that since there are no DC biasing voltages, the plate oscillates about



the horizontal position. Including dynamic terms and neglecting cubic stiffness

terms, a simple differential equation similar to (3.7) can be used as a model.

For sufficient plate displacement and eventual sensing, mechanical resonance of the

plate should still be designed to coincide with the frequency content of the driving

torque such that the mechanical structure acts as a bandpass filter. As described in

Section 3.1.5, the resonant frequency of a plate in torsional oscillations depends on

the modulus of rigidity of the torsional tether, the moment of inertia of the plate and

the pivot section dimensions. The resulting driving torque, eq. (3.15), has frequency

components at 𝑓𝑚 = 𝑓𝑟𝑓 − 𝑓𝑙𝑜 , 𝑓𝑝 = 𝑓𝑟𝑓 + 𝑓𝑙𝑜 as well as a DC component due to phase

differences. Figure 3.22 shows how using a differential drive (subtraction)

eliminates the components at 2𝑓𝑟𝑓 and 2𝑓𝑙𝑜 . Since the unwanted frequency

components are suppressed by the differential drive setup the constraint on the Q

factor of the mechanical filter to filter out these frequencies is relaxed.

Figure 3.24 shows a plan view of the whole structure including the drive and sense

electrodes. The plate undergoing torsional vibrations has six polysilicon pads. The

middle two will be driving the plate as these interact with the nickel beam above as

shown in section in Figure 3.23. The outer polysilicon conductors, H1 and H2, are

the two differential outputs which interact with the DC biasing voltages +V and –V

above them. The horizontal dashed centre line shows the pivot line. Note again that

by placing both a +V and –V above and below the pivot line, it is guaranteed that the

plate oscillates about the horizontal position which is important for pull-in reasons.

In practice, the amplitudes of 𝑣1 and −𝑣1 can be separately adjusted to cater for

mismatch in capacitances on the sensing electrodes. These mismatches can arise

both due to unequal pad areas and also due to any plate curvature induced by

residual stresses.



Figure 3.24: The whole structure has drive torque proportional to the product v1v2

Note: Benefit of symmetries: (a) minimise drive to sense parasitic coupling and (b) ensure

oscillations about horizontal to keep away from pull-in.

Figure 3.25 is the electrical equivalent circuit. The currents 𝑖1 and 𝑖2 are fed into a

differential charge amplifier (DCA) having an output voltage 𝑣𝑜 . Placing the sense

pads on both sides of the drive and using a differential topology provides a

significant advantage in that parasitically coupled high-frequency signals appear as

a common-mode signal at the amplifier’s input and are therefore rejected. This

makes it possible to have only one structure as opposed to [122].

Figure 3.25: The electrical sensing circuitry.

Note: The darker lines on each capacitor show the moving polysilicon pads. They move in anti-phase

on either side of the capacitive bridge. The currents generated are inputs to the fully differential

charge amplifier.



3.2.2 Prototype Design Dimensions and Simulation Results

This mixer can be used as a downconvertor stage within a Superheterodyne

receiver. The design process starts by identifying the required IF, which frequency

should be matched to the resonant frequency of the plate. The bandwidth

requirements need to be met by controlling damping. For the purpose of the

simulation, the chosen IF, dimensions and the resulting Q factor are very close to

what is required for typical AM downconversion to the IF stage. For an IF of 380 kHz

following the same calculations as in Section 3.1, the required plate dimensions are

174 × 110 µm with a tether width of 45 µm for the pivot. Figure 3.26a shows a top

view of the whole structure. The semi-transparent electrodes are the nickel

electrodes. Figure 3.26b shows the plate vibration mode. As can be clearly deduced

from this figure, there are four pads making up the differential voltage pair of 𝑉𝐿𝑂

and −𝑉𝐿𝑂. This was needed such that the plate is allowed to achieve this mode of

oscillation.

a) Top view – 5 nickel electrodes

(transparent)

b) Polysilicon electrodes on vibrating

plate

Figure 3.26: The final device showing details of both polysilicon and nickel electrodes

Using CoventorWare®, the required torsional mode frequency was confirmed at

𝑓𝑟 = 378 kHz. With an RF carrier 𝑣𝑅𝐹 = 25sin (2𝜋𝑓𝑟𝑓𝑡) with 𝑓𝑟𝑓 = 1 MHz, two LO

signals are required: 𝑣𝐿𝑂 = 25 sin (2𝜋(𝑓𝑟𝑓 − 𝑓𝑟)𝑡) and −𝑣𝑙𝑜 . This setup generates a



torque signal at 𝑓𝑟 = 𝑓𝑚 which will drive the system while the other component at

𝑓𝑝 = 1.622 MHz is clearly out-of-band and has no effect on the displacement.

Figure 3.27 shows the differential current (𝑖1 − 𝑖2) peak at resonance with these

drive signals. As frequency increases, it is evident that there is little parasitic

coupling confirming the effectiveness of the differential sense topology.

Figure 3.27: Differential current in nA vs. frequency in kHz.

At resonance, the maximum vertical displacement of the plate is 18.8 nm which is

well away from the pull in region as discussed in Section 3.1.8. Moreover, this

confirms that the behaviour is within the 𝑥 ≪ 𝑑 assumption taken to simplify (3.3).

The FEA simulation was performed under NTP and considered squeeze film effects.

These gave a Q factor of 11 with a 3 dB bandwidth of around 35 kHz. The maximum

differential current at resonance is of 36 nA.

The simulation results show that with an electrostatic drive, the torque content at

2f1 and 2f2 (both resulting from the squared terms in (3.13)) can be eliminated

before filtering and this relaxes the design constraints of the bandpass filter,

specifically the selectivity and bandwidth trade-off. Being less stringent on the Q

factor will potentially cater for larger bandwidth requirements. Moreover, this was



achieved with one single structure as opposed to using two devices for differential

sensing as proposed for the BPSK demodulator.

In the following section, a more complex RF function, an IQ mixer, is implemented.

The design builds upon the findings of this section.

3.2.3 Low-IF IQ mixing

Recently, Software Defined Radio (SDR) architectures which transfer most of the

analogue signal processing into the digital domain are being put forward. However,

IQ mixers are still required in analogue form as ADC/DAC requirements (bandwidth

and resolution) and power consumption requirements are still prohibitive for an

entirely digital SDR [137]. Although the ultimate goal in multi band radios is

software reconfiguration, power and dynamic range limitations require some of the

reconfiguration to take place in the RF frontend [137], [138].

Some undesired signals which can get into the radio signal path due to the mixing

process are those which occur at image frequencies. One solution is to have several

IF stages in a superheterodyne architecture. This comes at a cost as the IF filter must

be high in frequency with a high Q factor. Translating the RF frequency to baseband

directly is also possible which is known as the homodyne or zero-IF architecture.

For phase and frequency modulated signals, this option needs to provide quadrature

outputs since half the bandwidth is folded back about DC which jeopardises its

content. Moreover, a zero-IF architecture suffers from DC offset errors. This

problem is mitigated if rather than translating directly to baseband, a low-IF

architecture is adopted. The signal is then down converted to baseband in the digital

domain. A low-IF setup comprises the advantages of both heterodyne and

homodyne receivers. In literature, one can find two main strategies in MEMS mixer

designs: electrostatically actuated and thermally actuated mixers. The work

presented here describes an electromechanical structure that is actuated and

sensed electrostatically.



To realise an IQ mixer, two structures similar in layout as in Figure 3.26 are

employed, however, some of the electrodes are used for a different purpose. The

layout is shown in Figure 3.28.

a) b) c)

Figure 3.28: The core structure that provides actuation and sensing.

Note: a) Differential sensing of the capacitance gap, 𝑣𝑖 = 𝑣𝑖1 − 𝑣𝑖2 , b) Actuation is through

electrostatic interaction in middle electrodes, which changes the gap by ± xi and ± xq for the in-phase

and quadrature structures respectively, c) The layout for the in-phase signal; a similar structure is

used for the quadrature signal.

The fixed nickel electrode in the middle is fed with the RF signal, 𝑣𝑟𝑓, and interacts

with the two electrodes underneath it (Figure 3.28b) which are fed with the local

oscillator signal, 𝑣𝑙𝑜 , and its inverse −𝑣𝑙𝑜 . The other four fixed nickel electrodes are

connected to a DC supply 𝑉𝑑𝑐 through a high resistance 𝑅 (Figure 3.28a). These

generate the sensing voltages in response to the gap change of the oscillating

grounded polysilicon electrodes on the tethered plate underneath.

Figure 3.28c is only showing one path in a quadrature mixer. For a quadrature

mixer, two such structures are needed as shown in Figure 3.29. Suffixes ‘i’ and ‘q’

denote in-phase and quadrature respectively.



Figure 3.29: The complete structure consists of two mixing structures.

Note: Local oscillator signals provided in quadrature, top path shows the in-phase while the bottom

one shows the quadrature output.

In the analysis performed for the differential drive (eq. (3.13) to (3.15)), it was

assumed that the change in gap, 𝑥, is very small compared to the nominal gap 𝑑. For

the following analysis, this assumption is relaxed such that a more detailed

understanding of the amount of suppression achieved is determined. Considering

solely the in-phase structure, the electrostatic torque, 𝑇𝑖, acting on the plate can be

described with (3.16).



𝑇𝑖 = 𝑟휀𝐴𝑒2((𝑣𝑟𝑓 − 𝑣𝑙𝑜

𝑑 − 𝑥𝑖)2

− (𝑣𝑟𝑓 + 𝑣𝑙𝑜

𝑑 + 𝑥𝑖)2

) (3.16)

The ideal mixer would provide a torque in proportion to the product 𝑣𝑟𝑓𝑣𝑙𝑜.

Substituting 𝑛 = 𝑑/𝑥𝑖 in (3.16), (3.17) is obtained.

𝑇𝑖 = −𝑟𝐾𝑛2

(𝑛−1)2(𝑃(𝑛)𝑣𝑟𝑓

2 + 𝑃(𝑛)𝑣𝑙𝑜2 − 2(2 − 𝑃(𝑛))𝑣𝑙𝑜𝑣𝑟𝑓) (3.17)

where 𝑃(𝑛) = 4𝑛/(𝑛2 + 2𝑛 + 1), and for small displacement 𝑥𝑖 , (𝑛 → ∞), 𝑃(𝑛)

approaches zero giving ideal mixing, (3.18).

lim𝑛→∞

𝑇𝑖 ≈ −4𝑟𝐾(𝑣𝑙𝑜𝑣𝑟𝑓) (3.18)

Sensing the resulting movement of the plate could be achieved in several ways. As

described in the previous section and Figure 3.25, the constant voltage approach

using a differential charge amplifier (DCA) is one option. However, in the

subsequent discussion, an analogue option that attempts to keep a constant charge

on the output capacitances is investigated. This makes use of an appropriately sized

resistor, R, as in Figure 3.28a and Figure 3.29. The design constraints imposed by

this sensing strategy and its drawbacks (compared to a DCA) are subsequently

discussed.

The transfer function relating the quadrature voltage signals to displacement is in

effect a high pass RC filter.

Biasing through a large resistor results in a large RC time constant that

approximates a constant charge condition. Analysis of the RC filter with time varying

C(t) gives (3.19),

𝑖 =1

𝑅𝐶(𝑉𝑑𝑐 − 𝑣𝑖 (1 + 𝑅

𝑑𝐶

𝑑𝑡)) (3.19)



where 𝑣𝑖 represents the voltage at the fixed nickel electrode. In this setup, we can

represent the capacitance between fixed and moving electrodes on both sides (side

1 and side 2) of the differential sensing gaps with 𝐶1,2(𝑥𝑖) =𝜀𝐴

𝑑±𝑥𝑖, with xi(t) obtained

from the solution of the DE representing the dynamics and A the capacitive sensing

area. Substituting 𝐶1,2(𝑥𝑖) in (3.19) gives (3.20) whose solution is for both voltages

𝑣𝑖1,2 across the sensing plates:

𝑣1,2 = −[𝑑 ± 𝑥𝑖𝑅휀𝐴

∓𝑖

𝑑 ± 𝑥𝑖] 𝑣𝑖1,2 + (

𝑑 ± 𝑥𝑖𝑅휀𝐴

)𝑉𝑑𝑐 (3.20)

These relationships model actuation and sensing for the in-phase mixer with 𝑣𝑖 in

Figure 3.29 being 𝑣𝑖1 − 𝑣𝑖2. These would need to be duplicated for the quadrature

mixer to give 𝑇𝑞(∝ 𝑣𝑙𝑜90𝑣𝑟𝑓), 𝑥𝑞 and 𝑣𝑞1,2.

The low-IF downconverting mixer gives the required output content at 𝜔𝑟𝑓 − 𝜔𝑙𝑜.

Figure 3.30 breaks down the design process and constraints from the frequency

domain perspective into four steps. Step 1 shows the electrostatic actuation,

whereby due to the quadratic nature of this phenomenon, four frequency torque

components are generated. In Step 2, by having a large n (x << d) as described in

(3.17) and (3.18), the torque components at 2𝜔𝑙𝑜and 2𝜔𝑟𝑓 are attenuated. Step 3

shows the mechanical bandwidth (torque to displacement) centred around the low

frequency component of the mixing components and Step 4 shows the high pass

filtering effect of the RC sensing stage, the stage where displacement is converted to

voltage.



Figure 3.30: The design steps and their effect on the frequency content

In Step 3, the geometric parameters that regulate mechanical resonance √(k/I) of a

torsional plate can be found in Section 3.1.5. Finite Element Analysis (FEA) in

CoventorWare® confirmed these relationships and an estimate of the viscous

damping coefficient which included squeeze film effects at NTP could be obtained at

the mode of interest. For the sensing step, Step 4, the frequency transfer function

can be obtained from a linearised model of one RC filter, (3.21). From (3.21), the

differential setup output vi approaches ±2𝑥𝑖𝑉𝑑𝑐/𝑑 for large R. A large R is required

such that pole frequency 𝜔𝑅𝐶 = 1/𝑅𝐶 is kept below the frequency of interest thus

maximising the output. Table 3.5 summarises the design procedure.

𝑉𝑖1,2𝑋𝑖

= ±𝑗𝜔𝑉𝑑𝑐

𝑑(𝑗𝜔 + 1/𝑅𝐶) (3.21)



Table 3.5: Design Steps for IQ mixing

Design Step Aim Method

1 Create mixing components Electrostatic drive

2 Suppress even harmonics Keep 𝑛 = 𝑑/𝑥𝑖 large

3 Centre resonance on low IF Dimension plate/tether as in Section

3.1.5

4 Maximise sensing gain Keep 1/𝑅𝐶 below the IF frequency

Employing a large resistance for sensing also generates thermal noise on the four

quadrature output voltages. The resistor can be represented as an ideal resistor in

series with a voltage source, 𝑣𝑛, with noise power spectral density of

𝐺𝑥(𝑓) = 2𝑘𝑇𝑅 V2/Hz where 𝑘 is Boltzman’s constant and 𝑇 is the temperature in

Kelvin. It can be demonstrated that the RMS voltage appearing on the four

quadrature voltages, across the capacitive gap C, is independent of R and equal to

√𝑘𝑇

𝐶 V. This implies that for smaller capacitances the noise voltage increases (RC low

pass filter bandwidth gets wider). This noise level would need to be orders of

magnitude smaller than the RMS levels of the four quadrature output voltage

signals.

3.2.4 Numerical Simulations

The FEA simulations performed in CoventorWare® are the same as those presented

in Section 3.1.3 as the plate dimensions are the same (Figure 3.26). The required

static and dynamic specifications were used to fix the plate dimensions using the

equations in Section 3.1.5. The electrostatic force generated by the input signals,

together with the second order differential equation that modelled the plate

dynamics, were validated against the results obtained using FEA. Once the



mathematical model was validated, the full IQ mixer model was implemented in

SIMULINK (Appendix 3.3) for further analysis.

For simulation, an RF carrier frequency of 𝑓𝑟𝑓 = 1 MHz was selected for both BPSK

and QPSK/4-QAM modulation schemes. The local oscillator frequency was fixed to

𝑓𝑙𝑜 = 622 kHz and the mechanical resonant frequency, 𝑓𝑟 , was designed such that it

coincides with 𝑓𝑟 = 𝑓𝑟𝑓 − 𝑓𝑙𝑜 = 378 kHz = 𝑓𝑖𝑓 , the IF frequency. The bit rate adopted

for both simulations was that of 100 kbps. The FEA viscous damping simulation

included squeeze film effects and, at NTP and at this resonant mode of 378 kHz, gave

a damping factor ζ of 0.045. The RF and LO voltages used were the same at 17 volts

RMS. These actuation voltage levels resulted in a peak vertical displacement of the

plate of 19 nm. This gave n = 1.45 µm / 19 nm = 76 satisfying requirements for Step

2 for the suppression of eve harmonics. Moreover, having such a small peak

displacement (compared to the gap), operation is kept away from the electrostatic

pull in region and also justifies neglecting the spring cubic stiffness behaviour. With

19 nm peak displacement, the plate experiences a peak torque of 1640 pNm. This

amount of torque is in agreement with what (3.18) gives for r = 25 µm, d = 1.45 µm

and Ae = 2500 µm2. These are the dimensions, related to actuation, adopted.

On the sensing side, the sensing area was larger at a A = 5500 µm2. This gives a

sensing capacitance of 𝐶𝑠 = 34 fF. For 𝜔𝑅𝐶 < 𝜔𝑟𝑓 − 𝜔𝑙𝑜 to be satisfied, requires a

biasing resistor of 12.5 MΩ or greater. The ‘constant’ charge sensing setup gives

sensing voltages approaching ±2𝑥𝑖𝑉𝑑𝑐/𝑑 = ±0.26 volts, for 𝑉𝐷𝐶 = 10 volts and a

peak sensing current of 35 nA.

With the dimensions of the sensing area A and gap d, the resulting RMS noise voltage

due to the biasing resistor is several orders of magnitude smaller than the sensing

voltages of 0.26 V. This means that the ‘constant’ charge sensing strategy is a viable

option for this IQ mixer.



Simulations in SIMULINK (Appendix 3.3) with equations (3.18) and (3.20) were

executed. The mechanical static and dynamic behaviour were captured with a

bandpass filter for both the in-phase and the quadrature structures. The voltage

levels of the four quadrature output signals (𝑣𝑖1, 𝑣𝑖2, 𝑣𝑞1and 𝑣𝑞2) agreed with the

±0.26 V limiting levels for large R. The I and Q paths were recombined using the

‘atan2’ function to simulate digital reconstruction of the QAM signal under test.

Figure 3.31 shows simulated outputs for a QPSK signal consisting of a repeating

[00,01,10,11] pattern and Figure 3.32 shows the output for a BPSK signal consisting

of a repeating [0,0,0,1,1] pattern.

Figure 3.31: vi (green), vq (red) and output (blue-atan2) showing the 4 levels representing

[00,01,10,11].



Figure 3.32: vi (green), vq (red) and output (Blue-atan2) shows 2 levels representing [0,0,0,1,1]

An IQ mixer can be considered as a universal demodulator. These simulations show

that this RF frontend function can be obtained by a torsionally oscillating structure

within the fabrication constraints of the MetalMUMPs process. The important design

decisions that lead to the successful generation of the I and Q signals are described

and summarised in Table 3.5.

The use of differential local oscillator signals as well as restraining oscillations to

small amplitudes were instrumental in attenuating quadratic terms resulting from

the electrostatic interaction. This relaxed the design constraints for the mechanical

bandwidth as clarified in Figure 3.22 and Figure 3.30 (Step 2). The trade-offs for

positioning the mechanical resonance on the low-IF frequency while keeping

sufficient capacitance area to satisfy the pole placement requirements for the

sensing RC filter are clearly portrayed in the frequency domain showing the design

steps in Figure 3.30.



3.3 Parasitic Insensitive Sensing

Using a constant charge sensing strategy requires a reasonably high resistance and

has its problems. Realisation of high resistance resistors in a small footprint is

challenging. In [139], a 13 MΩ resistor was developed using a simple differential pair

with diode-connected MOSFETs driven in the subthreshold region of the MOSFETs

in 0.35-µm CMOS. It occupied an area of 105 µm x 110 µm. For the IQ mixer, four are

required with a total area of 46,200 µm2 which is two to three times the size of the

two torsional plates for the IQ mixer. Moreover, using a constant charge sensing

strategy does not cancel out parasitic capacitances.

The sensible way to eliminate parasitic capacitances is by using a DCA for sensing as

in Figure 3.7. Two DCAs are required, one for the in-phase and another one for the

quadrature paths. This will eliminate parasitics and can be implemented in 0.18-µm

CMOS [140] with a similar footprint (42,000 µm2) as the four resistors.

The constant voltage approach, using DCA’s, does not impose the high pass filtering

stage (Step 4 in Figure 3.30) on the required signal and has also the added advantage

of eliminating the parasitic capacitances.

3.4 Conclusions

The use of a torsional vibration structure for BPSK demodulation is innovative. It

was demonstrated that this structure had an undamped resonant frequency of

2 MHz and can successfully demodulate a low data rate BPSK signal with a carrier

frequency of 868 MHz and a chip rate of 300 kchips/s. These specifications satisfied

the ZigBee standard for low data rate, wireless personal area networks.

The same structure was investigated under higher actuation voltages (> 75 V). It

was shown that the device exhibits bistability and it has potential for applications

involving random number generation (RNG) for crypto systems [141] and also as

chaotic carrier generators for secure communications [142].



Furthermore, using a differential drive on the same torsional vibratory plate

configuration, suppression of unwanted frequency components resulting from the

quadratic relationship in the electrostatic force was implemented by controlling the

parameter 𝑛 = 𝑑/𝑥𝑖 . With 𝑛 = 76, suppression was sufficient and this relaxed the Q

factor-bandwidth requirements for the downconverting mixer and also for the IQ

mixer designs. The IQ mixer model was validated with FEA simulations and

implemented in SIMULINK. With these simulations, various QAM signalling schemes

were successfully demodulated.

Bistable Vibrational Energy Harvester in SINTEF moveMEMS


4. BISTABLE VIBRATIONAL

ENERGY HARVESTER IN

SINTEF MOVEMEMS

This work investigates the design of a vibrational energy harvester whose ability to

vibrate for a broad range of input frequencies is attributed to a highly nonlinear

spring. The goal was to design a broadband vibrational energy harvester and to

develop a mathematical model that could replace computationally expensive FEA

simulations. The SINTEF process was used; this process has a PZT layer of 2 𝜇m and

can cater for larger proof masses than is possible with MetalMUMPs. The analysis

builds on modelling techniques developed in Sections 3.1.8 and 3.1.10 which deal

with large displacements and cubic stiffness.

4.1 Introduction

To achieve broadband vibrational sensitivity the design employed a buckling spring.

This buckling spring gives two stable positions or bistability. Bistability and

multistability are desirable properties in mechanical structures used for energy

harvesting. Having a plurality of equilibrium points, a vibrational system can exhibit

small amplitude vibrations about each equilibrium position, however, given enough

energy, it would snap through between the stable equilibria.

The main objective of this broadband vibrational energy harvester (BVEH) design

was that accurate estimates of the mechanical and electrical behaviour could be

obtained from a mathematical model which replaces the complex and time

consuming FEA simulations.



In literature, one can find many ways to achieve bistability [20]. In [21], a

theoretically extensive treatment of the behaviour of buckling beams and their

combination to obtain compliant multistable systems is presented.

In this work, the focus was on systems that are bistable due to buckling of clamped-

clamped beams. Modelling of beam dynamics involves PDEs that give as a solution

the time evolution of the displacement at any point on the beam, 𝑦(𝑥, 𝑡). This is a

continuous system, that is, one having infinite degrees of freedom (DoF). A reduced

order model is one which has a (small) finite number of DoF. The proposed

mechanical design, together with results in [143] and some justified

approximations, make it possible to quickly fix dimensions in the design process and

to accurately predict behaviour with a simpler non-linear 3rd order differential

equation. In [144], the mechanical structure considered is very similar to what is

proposed here and also exhibits asymmetric bistability. However, harvesting in

[144] is achieved electrostatically. Promising experimental results are reported but

no attempt to model with a system of differential equations was performed as a

replacement to FEA.

4.2 Design within SINTEF process constraints

The SINTEF piezoVolume process was adopted. Since it has a relatively thick

piezoelectric layer, this enabled the conversion of mechanical to electrical energy.

The SINTEF process is able to deposit 2 𝜇m of Lead Zirconium Titanate (PZT) which

is a material with one of the highest coupling coefficients. The overall wafer

thickness is of 379 𝜇m as detailed in Figure 4.1.

Figure 4.1: SINTFEF piezoVolume Process Overview



In the structure under study, the PZT layer is operated in the d31 mode. The proposed

structure obtains bistability with a simple compliant mechanism intended to

respond to linear vibrations. Figure 4.2 is the schematic showing the mechanical

structure employed.

Figure 4.2: The mechanical schematic showing the proof mass M and the two compliant springs

In this figure, the two beams holding the proof mass, M, and having an initial stress-

free first buckling mode shape, 𝑦0, are shown. Equation (4.1) describes this initial

shape of the two compliant springs having a mid-span height of ℎ.

𝑦0(𝑥) =ℎ

2(1 − cos (

2𝜋𝑥

𝑙)) (4.1)

where 𝑥 and 𝑙 are as shown in Figure 4.2. The beam displacement is represented by

𝑦(𝑥, 𝑡). The acceleration of the mass, , is made up of two components: 𝑏 the

device’s base acceleration and 1 the acceleration of the beam at mid-span, which



results in = 𝑏 +1. In terms of absolute displacement, this relationship becomes

𝑤 = 𝑤𝑏 + 𝑦1 + 𝑦0.

Figure 4.3 shows the FEA 1st mode shape and frequency, the meshing used for the

FEA analysis and detail of the material layering on the top side of the spring is shown

in the inset. Inset also shows the PZT poling direction, ‘3’ and the main strain

direction, ‘1’.

Figure 4.3: Vibrational mode, FEA mesh and layering detail on spring

Since two compliant beams are used, and they are connected at mid-span, the 2nd

buckling mode (Figure 4.4) is suppressed. Moreover, the ratio 𝑄 =ℎ

𝑡 with 𝑡 being

the beam thickness, is kept greater than 4

√3 ; this guarantees that the beam would

buckle in mode 1 and mode 3 only [143] as shown in Figure 4.5.



Figure 4.4: Buckling modes, even modes suppressed by connecting two beams [143]

Figure 4.5: The correct buckling sequence, only mode 1 and mode 3 involved.

4.3 Mathematical Model

Using Euler-Bernoulli beam theory with flexural rigidity 𝐸𝐼, axial stiffness 𝐸𝐴 and

clamped boundary conditions, the PDE, (4.2) is developed.

𝐸𝐼 (𝜕4𝑦

𝜕𝑥4−𝑑4𝑦0𝑑𝑥4

) + 𝐸𝐴 (𝑠0 − 𝑠

𝑠0)𝜕2𝑦

𝜕𝑥2+ (𝐵1 +𝑀)𝛿 (𝑥 −

𝑙

2) + 𝜌𝐴 = 0 (4.2)

This equation is a 4th order non-linear partial differential equation, from which the

position of every point (infinite DoF) on the buckling beams as time evolves can be

found. Table 4.1 describes the meaning of the major variables.



Table 4.1: Variables describing the beam motion

Variable Description

𝑦(𝑥, 𝑡) Vertical position of any point on the beam

𝑠(𝑡) Length of the beam

𝑦1(𝑡) Mass displacement from 𝑦0(𝑥) at mid-span

𝑤(𝑡) Absolute mass displacement

𝛿(𝑥) Dirac delta function

The δ represents the Dirac delta function, s is the length of arc of the beam and s0 the

as-designed length of arc of the beam. The parameters B, M, A and ρ are the damping

coefficient, mass, beam section area and density respectively.

A simpler model could be found since the proof mass 𝑀 was attached to the

mid-span which eliminated the dependency of the mass’s vertical movement 𝑦1 on

𝑥.

The considerations that led to a simpler model were:

• For small oscillations about the two stable buckled positions, the system

behaves as a linear Spring-Mass-Damper and a second order system was

hence adopted.

• To cater for the nonlinear buckling behaviour linear stiffness of the form,

𝐾𝑦1was replaced with a nonlinear function 𝐹𝑠(𝑦1).

• From FEA static analysis, it was found that at least a 5th order polynomial was

required to model the spring behaviour.

• A force-voltage coupling term was included to model electrical stiffening.

The proposed reduced order model is given in (4.3).



𝑀(1 + 𝑏) + 𝐵1 + 𝐹𝑠(𝑦1) + 𝐹𝑣(𝑣, 𝑦1) = 0 (4.3)

The first term, 𝑀(1 + 𝑏), is the mass inertia due to the base and beam mid-point

accelerations. The inertia of the beams was neglected as it is negligible when

compared to the proof mass, 𝑀. Damping forces are primarily viscous in nature, 𝐵1,

and are assumed to be acting at mid-span and in proportion to the beam mid-span

velocity 1.

The stiffness, 𝐹𝑠(𝑦1), of a buckling spring can be approximated [143] with a three-

segment, piecewise linear function for a high Q as in Figure 4.6.

Figure 4.6: Fs - y1 curve for large Q with a mode 2 constrained beam [143]

However, here, a quintic polynomial as in (4.4) was fitted.

𝐹𝑠(𝑦1) = 𝐾1𝑦1 +𝐾2𝑦12 + 𝐾3𝑦1

3 + 𝐾4𝑦14 + 𝐾5𝑦1

5 (4.4)

Having the spring stiffness expressed as a continuous function is convenient for

dynamic analysis. The coefficients of this polynomial representing the spring

stiffness are estimated from an FEA static analysis simulation run. This polynomial

model has three real roots, two of which represent stable equilibrium positions

while the third is unstable. According to [143], these occur at 𝑦1 = 0 = 𝑦𝑖 and

𝑦1 = 1.99ℎ = 𝑦𝑖𝑖𝑖 for the stable equilibria and 𝑦1 = −4ℎ/3 = 𝑦𝑖𝑖 for the unstable

one. This results in a spring that exhibits asymmetric bistability. In [143], analytical

𝐹𝑠

y1

ybot

fbot

ytop

ftop

yii yiii

yi



approximate expressions for 𝑓𝑡𝑜𝑝, 𝑦𝑡𝑜𝑝, 𝑦𝑏𝑜𝑡 and 𝑓𝑏𝑜𝑡 are provided, from which, the

‘linear’ spring stiffnesses around the two stable equilibria could be estimated. A

better estimate of the stiffness around 𝑦𝑖 and 𝑦𝑖𝑖𝑖 was obtained by taking the

derivative of the Fs - y1 curve at these two positions. With the beam having mode 2

constrained, the approximate analytical expression for 𝐹𝑠(𝑦1) provided in [143] can

be adopted, as reproduced in (4.5).

𝐹𝑠 ≈𝐸𝐼ℎ

𝑙3𝐴 (

𝑦1ℎ)3

+ 𝐴(𝐿 + 𝐶) (𝑦1ℎ)2

+ 𝐴𝐿𝐶 (𝑦1ℎ) + 8𝜋4 − 6𝜋4 (

𝑦1ℎ) (4.5)

where 𝐸 is the modulus of elasticity, 𝐼 = 𝑏𝑡3/12 is the second moment of area of the

beam, with 𝑏 being the SINTEF silicon depth of 377 𝜇m, 𝐴 =3𝜋4𝑄2

2,

𝐿 = −3

2+√

1

4−

4

3𝑄2 and 𝐶 = −

3

2−√

1

4−

4

3𝑄2.

Although (4.5) is an approximation, closed form expressions for the gradient at the

two stable equilibria could be obtained by evaluating 𝐾𝑖 =𝑑𝐹𝑠

𝑑𝑦1|𝑦1=𝑦𝑖

and

𝐾𝑖𝑖𝑖 =𝑑𝐹𝑠

𝑑𝑦1|𝑦1=𝑦𝑖𝑖𝑖

. These give the linearised stiffnesses of the spring at the as-

designed position, 𝐾𝑖 and at the buckled position, 𝐾𝑖𝑖𝑖.

𝐾𝑖 =𝐸𝐼𝜋4

𝑙3(3𝑄2 − 4)

(4.6)

𝐾𝑖𝑖𝑖 = 𝐾𝑖/2 (4.7)

Equations (4.6) & (4.7) are applicable for one beam only and hence for small

oscillations about the stable positions, the respective resonant frequencies would

be 𝑓𝑖 = 1/2𝜋√2𝐾𝑖/𝑀 and 𝑓𝑖𝑖𝑖 = 𝑓𝑖/√2.

The last term in (4.3), 𝐹𝑣(𝑣, 𝑦1), describes the electrical stiffness due to strain of the

PZT layer. Under small field conditions, the behaviour of the PZT material is

described by (4.8).



(𝑇𝐷) = (

𝑐 −𝑒31𝑒31 휀 ) (

𝑆𝐸) (4.8)

The PZT coefficient 𝑒31 is the piezo-stress constant (N/Vm). D is the dielectric

displacement (C/m2), T the developed stress (N/m2) in the mechanical displacement

direction, S the applied strain in the mechanical displacement direction and E the

applied electric field (V/m) across the PZT layer of thickness 𝑡𝑝. The modulus of

elasticity and the absolute permittivity of PZT are represented by c and 휀

respectively.

The term that describes how the electrical field induces stress in the material comes

from the first row of (4.8) and is 𝑇𝑒 = −𝑒31𝐸 with 𝐸 = 𝑣/𝑡𝑝, 𝑣 being the voltage

across the PZT layer of thickness 𝑡𝑝. This stress is in the ‘1’ direction, that is, along

the length of the beam. The vertical component of this stress contributes to a force

on the proof mass as shown in Figure 4.7.

Figure 4.7: Electrical force component in the vertical direction

The angle subtended by the beam with the horizontal, 𝜃, can be considered to be

small since the ratio ℎ/𝑙 is small, hence the vertical force due to electrical field would

be 𝑒31𝐸𝐴𝑠𝜃 where 𝐴𝑠 is the PZT cross sectional area, 𝑡 × 𝑡𝑝 giving the term 𝑒31𝑡𝑣𝜃 =

𝐹𝑣(𝑣, 𝑦1) with 𝑦1 = 𝑙/2 𝑡𝑎𝑛 𝜃 − ℎ.

This term, 𝐹𝑣(𝑣, 𝑦1), describes how a voltage across the PZT layer generates a force

on the proof mass, however the reverse process is also occurring and is the one

required to provide an output current 𝑖 across a load resistor 𝑅𝑙 . From the second

row of (4.8), (4.9) is obtained,

Ɵ

𝑒31𝐸𝐴𝑠

𝑒31𝐸𝐴𝑠𝑠𝑖𝑛𝜃

l

h



𝐷 =𝑞

𝐴𝑝= 𝑒31

𝑠 − 𝑠0𝑠0

+ 휀𝐸 (4.9)

where 𝑞 is the charge, 𝐴𝑝 is the piezo layer capacitive area, 𝑙 × 𝑡, 𝑠0 is the beam arc

length and 𝑠 is the instantaneous beam arc length, both determined using the length

of arc integral to a first order approximation on (4.1). Taking the derivative of (4.9)

to obtain an expression for current and re-arranging gives (4.10).

=1

𝐶𝑝𝐴𝑝𝑒31

𝑠0−𝑣

𝑅𝑙 (4.10)

where 𝐶𝑝 = 휀𝐴𝑝

𝑡𝑝, 𝑣

𝑅𝑙= 𝑖 and

𝑠0≈

𝜋2

2𝑙(𝑦1+𝑦0)1

𝑙+𝜋2

4𝑙𝑦02

.

In summary, the PDE which describes the mechanical behaviour, eq. (4.2), was

replaced with equations (4.3) and (4.4) which is a nonlinear DE describing the

position of the proof mass. Electro-mechanical coupling was included in (4.3) as 𝐹𝑣 .

The DE that describes how the strain on the PZT layer results in an output voltage 𝑣

is given by (4.10).

4.3.1 Design Approach

One of the most important design specifications for a vibrational energy harvester

is the frequency or bandwidth for which the harvester is sensitive to. As shown in

Figure 4.6, the spring stiffness characteristic is asymmetric and this gives different

resonant frequencies at the two stable positions with 𝑓𝑖 > 𝑓𝑖𝑖𝑖. Such an asymmetric

bistable energy harvester would be sensitive to vibrations at 𝑓𝑖 and below.

Fixing the dimensions in the design shown in Figure 4.2 such that it is sensitive to a

particular frequency range can be done by substituting (4.6) in 𝑓𝑖 = 1/2𝜋√2𝐾𝑖/𝑀

to obtain (4.11).



𝑓𝑖 =1

2𝜋√2𝐸𝐼𝜋4(3𝑄2 − 4)

𝑀𝑙3

(4.11)

Substituting 𝐼 =𝑏𝑡3

12 and 𝑀 = 𝜌𝑛𝑎2𝑏 gives (4.12);

𝜌𝑛𝑎2𝑏 =2E𝑏𝜋4(3𝑄2 − 4)

12(2𝜋𝑓𝑖)2(𝑡

𝑙)3

(4.12)

where 𝜌 is the density of silicon and n is the aspect ratio of the proof mass, that is,

length/width. One dimension that can be fixed without affecting the overall

footprint of the harvester is the proof mass’s width ‘𝑎’. This can be fixed to be the

same as the beam length 𝑙 giving (4.13):

𝑓𝑖−2 =

24𝜌

E𝜋2(3𝑄2 − 4)𝑡3. 𝑛𝑙5 (4.13)

Equation (4.13) together with the condition for buckling, that is, 𝑄 = ℎ/𝑡 > 4/√3

can give the dimensions for a set of design specifications. For the SINTEF process,

with E =130.2 GPa and 𝜌 =2311 kg/m3, fixing either 𝑡 or ℎ would in turn fix the

gradient of the straight line 𝑓𝑖−2 vs. 𝑛𝑙5. This procedure could either be used to fix

𝑛𝑙5 if the base excitation frequency is known or vice versa.

Equation (4.13) can be represented graphically as in Figure 4.8. If, for example, the

harvester is required to be used at frequencies up to 1500 Hz (𝑓𝑖), the horizontal line

at 𝑓𝑖−2 can be drawn and an intersection point with any of the straight lines

originating at the origin can be used as a design point. In Figure 4.8, for

𝑓𝑖 = 1500 Hz, if a beam thickness of 5 𝜇𝑚 and initial curvature height ℎ = 28 𝜇𝑚 are

selected, a resulting beam length of 𝑙 = 3 mm and proof mass aspect ratio of 𝑛 =

0.478 guarantee that the required 𝑓𝑖is obtained and that snap through occurs. These

result in a total proof mass of 𝑀 = 3.77 𝜇𝑔.



Figure 4.8: Using eq. (4.13) to determine h and t

4.4 PZT Harvester Model validation against FEA

For model validation purposes, an upper frequency, 𝑓𝑖 , of 1500 Hz was selected

together with a beam thickness 𝑡 of 5 𝜇𝑚 and 𝑄 of 6. Using the design process

described in Section 4.3.1, these guaranteed snap through would occur and required

a beam length 𝑙 = 3.15 mm and an aspect ratio 𝑛 = 0.41. With these parameters, the

proof mass was 3.61 𝜇g, beam moment of inertia 3.792 × 10−21 kgm2. The linear

spring stiffnesses (for 2 beams) around the two stable positions were found to be

𝐾𝑖 = 320 N/m and 𝐾𝑖𝑖𝑖 = 160 N/m from (4.6) and (4.7) respectively. These resulted

in 𝑓𝑖𝑖𝑖 = 1060 Hz.

4.4.1 Validation of the Static Response

Figure 4.9 shows the force-displacement relationship obtained using

CoventorWare® (solid line) for a static analysis simulation. With ℎ = 30 𝜇m, the

total deflection for 𝑦1 is 60 𝜇m. For such a simulation in CoventorWare®, one cannot

solve for an arbitrary initial displacement directly. Instead, one needs to increase



the displacement in small steps from zero and restart the analysis from the resultant

displacement found in the previous simulation run. In this manner, the simulation

will not fail easily because defining the initial displacement resolves the large

non-linearity of buckling.

Figure 4.9: The Force-Displacement (y1) asymmetric curve obtained using FEA

FEA results gave stiffness gradients at 𝑦𝑖 and 𝑦𝑖𝑖𝑖 at 𝐾𝑖 = 329.9 N/m and

𝐾𝑖𝑖𝑖 = 165 N/m. These are sufficiently close to the theoretical estimates.

From FEA modal simulations, the resonant modes at the two stable equilibria were

found to be 𝑓𝑖 = 1585 𝐻𝑧 and 𝑓𝑖𝑖𝑖 = 1100 𝐻𝑧. Both these numbers are within 5% of

the analytical model predictions.

For dynamic predictions on MATLAB, a quintic polynomial was fitted (Figure 4.9 –

dotted line) to the force displacement graph with the coefficients as listed in

Table 4.2.



Table 4.2: Fs – y1 Quintic Polynomial coefficients

Coefficient Value

𝐾1 3.270 × 102

𝐾2 -3.175× 107

𝐾3 1.222 × 1012

𝐾4 -2.137× 1016

𝐾5 1.388 × 1020

The minimum energy, 𝐸𝑚𝑖𝑛, required for snap-through (well-to-well jumping) to

occur is the area under the force-displacement curve between 𝑦𝑖 and 𝑦𝑖𝑖 and can be

calculated using (4.14).

𝐸𝑚𝑖𝑛 = ∫ 𝐹𝑠(𝑦1) 𝑑𝑦1

𝑦𝑖𝑖

0

(4.14)

For this device, as can be seen in Figure 4.10, the amount of energy required for

snap-through is 25 nJ. This means that -neglecting damping - any initial condition

that has a total energy (strain and kinetic) greater than 25 nJ would have enough

potential for well-to-well jumping.

Figure 4.10: Strain energy vs. displacement showing a maximum of 25 nJ at the unstable point



4.4.2 Validation of the Harmonic Response

Validation of the dynamic and electrical characteristics was done only for harmonic

oscillations around 𝑦𝑖 and 𝑦𝑖𝑖𝑖. Confirming the validity when snap-through occurs is

more challenging due to the sensitivity to initial conditions and due to the

computationally intensive FEA simulations required. The following FEA and

MATLAB simulations were all designed to stay away from 𝑦𝑖𝑖, hence, this exercise

will be confirming model validity for harmonic oscillations around 𝑦𝑖, 𝑦𝑖𝑖𝑖 or both

(high energy orbits). Testing of the electrical parameters was done by neglecting

mechanical damping. The simulations involved providing a sinusoidal acceleration

at 𝑓𝑖 = 1585 Hz to the inertial frame of 0.001g (𝑏 in (4.3)) both in MATLAB and in

CoventorWare® and comparing the amplitude of vibrations obtained and also

electrical current and power generated. Figure 4.11 and Figure 4.12

Figure 4.12 show one such harmonic simulation in MATLAB. The phase portrait in this

figure shows the three equilibrium positions marked with an ‘•’. In the steady state,

the oscillations reached a peak displacement of 8.8 𝜇𝑚. Once this peak displacement

was determined, a second simulation in MATLAB was performed with an initial

condition on displacement of 8.8 𝜇𝑚. The initial position of the beam displacement

is marked with a red ‘*’. In this simulation, the oscillations’ peak stayed at 8.8 𝜇m

giving 𝑖𝑅𝑀𝑆 = 0.087 𝜇A and 𝑃𝑅𝑀𝑆 = 0.0023 𝜇𝑊 across a load resistance 𝑅𝑙 of 250 kΩ.

In this simulation, which investigated harmonic behaviour, the value of the load

resistance, 𝑅𝑙 , was chosen arbitrarily as there is no internal resistance (mechanical

damping) to which this needs to be matched for maximum power transfer. For

maximum power transfer the internal resistance (mechanical damping) and the

external load would need to be equalised [145].



Figure 4.11: Speed-Displacement phase portrait and y1(t) for base acceleration of 0.001g at fi giving

8.8 µm peaks

Figure 4.12: Current i(t) µA and Power P(t) µW and their respective RMS values in title.

The same simulation was repeated with FEA in CoventorWare® giving the same

peak oscillation and power output and a slightly higher current of 0.095 𝜇A.

Simulations were executed on an Intel i7-3720QM CPU @ 2.6 GHz with 32 Gb of

RAM. Simulation in MATLAB was performed in seconds while with FEA, results were

obtained after around 2 hours. The degree of match between FEA results and the

MATLAB model results was investigated further.

Figure 4.13 shows how ratios of FEA-to-Model RMS current (if /im) and FEA-to-

Model RMS power (Pf /Pm) varied with resistive load 𝑅𝑙 . These ratios are calculated



for two orbits in the 𝑦1 − 1 state space around 𝑦𝑖 at two different amplitudes

5.66 𝜇𝑚 and 6.17 𝜇𝑚.

Figure 4.13: Output current (i) and power (P) ratios of FEA-to-model RMS

As can be seen in Figure 4.13, the model predictions for power and current are

acceptable for loads below 500 kΩ.

The mathematical model developed was also able to predict behaviour for high

energy orbits which are orbits about the two equilibrium points in the 1 − 𝑦1 space

as shown in Figure 4.14. As can be seen, the high energy orbit was achieved for a low

(compared to 𝑓𝑖 and 𝑓𝑖𝑖𝑖) driving frequency of 560 Hz. Being bistable, the broadband

capabilities were also confirmed. The response was achieved with a base

acceleration of 3.5 g. For this simulation, a load resistance of 1.5 MΩ was used

through which a current of 0.23 𝜇A RMS was driven resulting in a voltage of 0.37 V

RMS and power generation of 0.2 𝜇W.



Figure 4.14: A high energy orbit producing 0.2 µW of power with spring force-displacement and

equilibria in superposition.

When compared to the FEA model, the reduced order model (eq. (4.3), (4.4) and

(4.10)) simulated in MATLAB proved very accurate in predicting both the harmonic

behaviour around 𝑦𝑖 and 𝑦𝑖𝑖𝑖 and also the static behaviour. For the harmonic

behaviour, it gives reasonable estimates for power, current and voltage for different

loads. The estimation error starts exceeding 10% for loads larger than 1 MΩ. The

increase in error for larger resistive load can be attributed to the fact that the

capacitance dominates in the model for larger 𝑅𝑙 (Eq. (4.10)) and the mathematical

model does not include fringe field effects.

As shown in Figure 4.14, for the harmonic orbits around the two stable equilibria,

the harvester generated a power density of 0.13 mW cm-3 at 3.5g ms-2 at 560 Hz and

produced d31 open circuit voltages of 0.6 V.

4.4.3 MATLAB Dynamic Response Simulations

The same MATLAB implementation (Appendix 4.1) was used to investigate

transients and dynamics that included viscous damping. This could be done once the

static and harmonic behaviour of the mathematical model implemented in MATLAB

were investigated and found to be a reasonably accurate alternative to



computationally expensive FEA simulations. The Q-factor of the harvester, which is

dependent on both fluid and structural damping, is a critical parameter that dictates

the transient behaviour. The viscous damping component is commonly controlled

by packaging in a vacuum or controlled atmosphere, however, structural damping

(thermoelastic and anchor losses) is more difficult to control.

Typically, in MEMS, viscous air damping is the dominant damping mechanism [146].

In the energy harvester design, the air gaps involved are very large when compared

to the mean free path of air, giving a high Knudsen number. This implies that the

damping coefficient as a result of air damping can be obtained by making use of a

fluid dynamics simulation in the continuum mechanics regime. This would give the

appropriate Q factor due to gas damping. Further simulations that include anchor

and thermoelastic losses (structural) would give the total Q factor for this design.

Consequently, the dominant damping mechanism would be determined and typical

ranges for damping coefficient could be adopted for transients’ simulations. Actual

determination of the Q factor for this harvester was not performed however the

effect of changes in damping coefficient, B, was investigated. For MEMS devices with

similar dimensions, proof mass and operated at NTP [144], the coefficient of viscous

friction, B, would be in the mN/(m/s) range.

Figure 4.15 shows simulation runs with a damping coefficient, 𝐵, of 9 mN/(m/s) and

no inertial frame acceleration (input) for different initial strain conditions. This

simulation clearly shows the three equilibrium positions, their attractive properties

and the unstable nature of the one at 𝑦𝑖𝑖 = −40 𝜇m.



a)

b)

c)



d)

e)

Figure 4.15: Trajectories in state-space with B = 9 mN/(m/s) and no inertial frame acceleration

The trajectories shown in Figure 4.15c and Figure 4.15d have a large enough initial

strain potential energy such that well-to-well jumping occurs while Figure 4.15e

shows the repulsive (unstable) nature of 𝑦𝑖𝑖.

Figure 4.16 shows phase portrait trajectories when a base excitation frequency of

1547 Hz is introduced on the inertial frame. This frequency was chosen to be slightly

away from resonance, 𝑓𝑖 = 1585 Hz, on purpose such that beats, period doublings

and chaotic behaviour can be demonstrated. The damping coefficient, B, was

lowered to 0.009 mN/(m/s) such that this behaviour manifests itself more easily. In

practice, such a value for B might be achieved by operating under vacuum. The initial

condition for all the three runs in Figure 4.16 was kept the same at [0 µm 0 µm/s].



The reference frame acceleration amplitude is increased in steps of 5g starting at 5g

in Figure 4.16a.

a)

b)

c)

Figure 4.16: Driving the harvester away from resonance exposes chaotic trajectories.

In Figure 4.16a and Figure 4.16b the beat phenomenon is evident. In Figure 4.16c, it

is quite clear that the frequency content of 𝑦1(𝑡) has frequencies at 𝑓𝑖 and 𝑓𝑖𝑖𝑖 and



also frequency content below both resonant points. These low frequencies are due

to the high energy orbits that encompass both stable equilibrium points. This

behaviour is what enables the energy harvester to be responsive to broadband

inertial frame excitations.

4.5 Conclusions

In this work, a buckling nonlinear spring was designed to exhibit asymmetric

bistability. This is a desirable property in energy harvesters as it makes them

sensitive to a broader range of frequencies. This provided dynamics with three

equilibrium points. Three equilibrium points can be modelled with a cubic spring,

however, to improve the fitness with what is observed in FEA (and in theory for a

‘high’ Q value) a quintic spring with three real roots was used.

The model is very accurate in predicting both the harmonic behaviour around 𝑦𝑖 and

𝑦𝑖𝑖𝑖 and also the static behaviour. For the former, it gives reasonable estimates for

power, current and voltage generated for different load values. The estimation error

varies between 5% (𝑅𝑙 < 500 𝑘Ω) and 10% for larger 𝑅𝑙 values. For larger 𝑅𝑙 , the

capacitance dominates in (4.10) and this larger error in the model can be reduced if

fringe field effects are taken into account.

The device was operated in the d31 mode. With electrical damping only, the device,

designed for validation purposes, achieved a power density of 0.13 mW cm-3 @ 3.5g

ms-2 @ 560 Hz and produced d31 open circuit voltages of 0.6 V. Sensitivity to a

frequency this low (560 Hz) without a buckling spring would entail the use of either

longer beams or a larger proof mass which in turn would result in a smaller power

density. The device is responsive to a broad range of frequencies as the two resonant

modes (𝑓𝑖 and 𝑓𝑖𝑖𝑖) are in the 1 kHz to 1.5 kHz range but is also responsive to

frequencies on the 0.5 kHz range.

FEA of buckling spring mechanisms is computationally expensive and is, in practice,

not feasible to perform around the unstable equilibrium point due to solution



convergence issues. The design topology adopted permitted the use of a simple

nonlinear DE model which in turn enabled simulations to be performed in seconds.

This hastens further investigation into design optimisation as several runs with

different geometries could be executed in realistic timeframes. The innovative

aspect in this work is the topology that enabled simpler modelling. This employed

two clamped-clamped buckling springs connected (through the proof mass) from

their mid-span to suppress mode 2 buckling (Figure 4.4). The closest design

topology found in literature is described in [144]. The design in [144], makes use of

four, fixed-pinned buckling beams and these are used in an electrostatic harvester.

The authors limited themselves to FEA, no attempt was performed to amend the

topology such that it is conducive to being analysed with a relatively simple DE as

opposed to FEA.

BPSK to ASK Converter in SOIMUMPs


5. BPSK TO ASK CONVERTER

IN SOIMUMPS

5.1 Introduction

The work presented in this chapter builds upon knowledge gained in the designs,

modelling and simulations carried out in Chapters 3 and 4. The objective here is to

design and build a device that can take a BPSK signal and a carrier as inputs,

demodulate the bit stream and generate an ASK signal that carries the same bit

stream, hence, a BPSK to ASK converter. One of the important metrics that measure

the quality of the ASK output signal is the depth of modulation or alternatively the

modulation index. In the subsequent analysis, the latter metric is used. The

demodulation step still makes use of electrostatic mixing, however, here, a different

MUMPs process was employed: the SOIMUMPs fabrication process.

The SOIMUMPs fabrication process makes use of Silicon-on-Insulator (SOI)

technology. It was selected for designing the BPSK to ASK converter since the main

advantage of having such a converter in MEMS is the potential for integration. As

described in [147], SOI is the best candidate for successful co-integration between

MEMS and CMOS. This book mentions numerous SOI sensors that are termed “CMOS

compatible”. Nevertheless, most of them use layers which are non-inherent in a

standard IC fabrication or use steps which are far too complicated to be

co-integrated with CMOS circuits. However, a few have been actually produced in

standard CMOS processes; these are termed by the author as CMOS-SOI technology.

It has been widely demonstrated that MEMS structures lend themselves easily to

integration of functions [148] and can replace filtering and mixing in transceivers.



Having a BPSK to ASK converter implemented in MEMS can be advantageous for

certain applications like WSN nodes. Miniaturization of WSN nodes is essential for

mass adoption and such nodes are already making use of MEMS energy harvesters

and MEMS sensors. Finding solutions to RF digital detection in MEMS can provide

new avenues to integrate the sensor/s, transceiver and power management

modules in one MEMS fabrication step. For a WSN node, this is beneficial for

lowering costs and miniaturization. In Section 17.4 of [2], a case study, involving a

multi-MEMS structure which combined several MEMS structure functions on one

single substrate, is demonstrated. The result is a MEMS-based multisensory chip

that gives a WSN solution having greater simplicity of operation, lower costs and

simpler utilization for a reliable data collection from a great number of sensors.

The rest of this chapter is organized as follows: Sections 5.2 and 5.3 describe the

design requirements, approach and the topology adopted. Once the general layout

is fixed, the mathematical model governing the static and dynamic behaviour is

derived in Section 5.4. Section 5.5 discusses dimensional optimisation in MATLAB

while Section 5.6 describes experimental validation. Finally, Section 5.7 gives the

concluding remarks and limitations of this work.

5.2 Design Requirements

Implantable medical devices are divided in two categories, those that have only

control functions and those that perform real time monitoring. For control

functions, the required bandwidth is only that of a few kHz. In literature [149], one

can find that carrier frequencies used range from 1 MHz to 10 MHz and the most

popular modulation schemes are ASK (with high modulation index) and On-Off

Keying (OOK). These modulation schemes are employed because of their simplicity,

however, constant amplitude schemes like FSK and PSK are preferred due to the

constant power they provide to the passive receiver. A BPSK to ASK converter can

be used such that wireless transmission is carried out using BPSK signalling but



eventual processing can proceed using ASK modulation. With this application in

mind, the design objectives shown in Figure 5.1 and listed in Table 5.1 are formulated.

Figure 5.1: Block diagram of the converter showing design properties and objectives

Table 5.1: Design Objectives

Design Property Design Objective

Voltage that gives maximum deflection Δ𝑉 ≤ 14 𝑉 RMS

Data rate 𝑓𝑑 > 2 kHz

Carrier frequency 0.1 ≤ 𝑓𝑐 ≤ 1 MHz.

Modulation Index 𝑀 ≤ 0.85 at maximum deflection

ASK Output voltage 𝑣𝑝 > 500 mV RMS.

Low power consumption Requires lowest possible air damping and in

turn lowest capacitive area and 𝐶𝑎.

Linearity between sensing capacitance,

𝐶𝑠 and input voltage, ∆𝑉.

Requires finding the correct balance

between cubic and linear stiffness

Smallest footprint Requires lowest possible values for 𝑟 and 𝑞



5.3 Design Approach - constraints and the resulting topology

The SOIMUMPs fabrication process has its design rules. This process has a silicon

layer with thickness, t, of either 10 µm or 25 µm insulated from the substrate by an

oxide layer, hence named Silicon-on-Oxide (SOI) as in Figure 5.2.

Figure 5.2: Extract from SOIMUMPs handbook [24], showing the process layers

In this work, dimensional optimisation of the geometrical features was performed

after the general topology was arrived at by trial and error while adhering to the

SOIMUMPs design rule constraints. As is discussed in the following paragraphs,

several topology layouts were considered in the initial trial and error process. This

high-level design process was not delegated to machine algorithms. Delegation of

such high-level design functions to machine algorithms is an area which is at its

infancy as reviewed in [150] and experimented with in Autodesk’s Dreamcatcher

Project.

In SOIMUMPs, the dimensions of the 2D SOI mask would define the device’s function

completely if the underlying substrate is etched completely. To make use of a

metaheuristic optimisation algorithm, the intricacies of the physical phenomena

governing the behaviour of this parametric MEMS topology had to be developed and

validated. Validation was performed using FEA with CoventorWare®. Once the

mathematical relationships were validated, a hybrid dimensional optimisation

algorithm was used to satisfy the design objectives while staying within the design

constraints. This overall design approach is summarised in Table 5.2. In this table,

in design step 5, the particle swarm optimisation technique can be tuned such that



an acceptable trade-off can be found between getting results in reasonable time

whilst obtaining a sufficiently good solution[151].

Table 5.2: Design Process

Design Step Details

1 Overall topology was arrived at by intuition while adhering to the SOIMUMPs design rules and best practices as detailed in the SOIMUMPs handbook [24].

2 Mathematical model is developed capturing both statics and dynamics.

3 Validation and refinement of the mathematical model using FEA under different domains.

4 Formalization of the design objectives.

5 Use particle swarm optimisation (PSO) to find a sufficiently good solution in reasonable time.

6

Starting close to good solutions use sequential quadratic programming (SQP) to improve the accuracy of these good solutions through MATLAB’s fmincon function and to confirm the broad optimality property.

7 Validate using MATLAB for dynamics and FEA.

In an electrostatic mixer, mixing is achieved by applying two signals, v1 and v2, to the

plates of a capacitor. One plate is fixed while the other is free to move; this is the

actuation capacitance, CA. The force generated between the plates is proportional to

the voltage squared which force is used to create a displacement. The resulting force

has a component at the sum of frequencies and one at the difference. For the BPSK

to ASK converter, the BPSK signal requires demodulation to baseband first and

hence, the frequencies of the two input signals are kept the same. This results in a

component at double the frequency and one at base band but only the latter is within

the mechanical bandwidth.

On the sensing side, two capacitances are designed in such a way that when one gap

widens, the other gap narrows. The sensing capacitances are CS1 and CS2. The arrows



in Figure 5.3 show the movement directions of the actuation and sensing plates.

With the SOIMUMPs process, it is not possible to create two SOI structures that are

electrically isolated while being mechanically connected. This implies that while v1

and v2 are used to actuate the device, v1 will be detected at S1 and S2.

Unless otherwise stated, the letters ‘a’ and ‘s’ as suffixes to variables denote

actuation side and sensing side respectively.

Figure 5.3: Actuation and sense capacitors, solid lines are fixed plates, while dashed are moving

The structure needs to respond as fast as possible to incoming data. Displacement

has to happen within the time for 1 bit. Hence, settling time will determine the

highest bit rate possible and an underdamped, close-to-critical response is

preferred. Mechanically, this requires compromises on the inertia, spring constant

and damping coefficient. Moreover, capacitive area and displacement for both

actuation and sensing need to be kept in check. For actuation and sensing gaps to

make use of the smallest available gap - 2 µm in SOIMUMPs - a rotational setup was

adopted. If linear motion design was considered, a smaller gap for the stoppers

would have had to be used resulting in design rule violation. With a rotational setup,

the actuation and sensing gaps could be kept at the minimum while the stoppers’

gap are also at the minimum, with the stoppers being positioned at a larger radial

distance. This amplifies the movement such that the stoppers close the gap first.

Moreover, to make use of the smallest gap allowed, the specifications require that

the gaps are in the orthogonal direction. Hence, Figure 5.4(a) would have to make



use of a larger gap, while the layout in Figure 5.4(b) could go down to the smallest

gap.

Figure 5.4: Two rotor designs - a) Radial vs. b) Orthogonal comb fingers

Making use of orthogonal combs implies that under rotation, the capacitive gap does

not remain parallel. For this effect to be negligible (< 0.01% of nominal capacitance),

the radius of the combs has to be greater than 600 µm [152] for a finger length of

100 µm (the maximum length allowed for a 2 µm structure).

Although SOIMUMPs allows for designs of non-Manhattan geometries, the process

also requires that all vertices in the layout are on a 0.25 µm grid. Because of this, the

circular layout was approximated with an octagonal layout (Figure 5.5).

Electrical Schematic: Springs not shown for clarity

Figure 5.5: The final octagonal layout showing comb finger insets and electrical schematic



Figure 5.5 shows the topology for the converter. As discussed, this rotational

octagonal setup was able to respect fabrication constraints while making use of the

smallest possible gap, g, of 2 µm. The rotor is supported with four cantilevers; there

are two stator combs for actuation (electrically connected, 𝑣2) and two stator combs

for sensing (S1 and S2). All combs have finger gaps g and ng as shown in Figure 5.5

insets.

The two actuation combs have a total capacitance of 𝐶𝑎 and electrostatic attraction

from both, drives the rotor anti-clockwise. Both combs close the gap on actuation.

This displacement is detected at the two sensing combs, however, on actuation,

sensing is differential with 𝐶𝑆1 increasing and 𝐶𝑆2 decreasing.

5.4 Mathematical model

In this section, the development of the mathematical model for the whole system,

from inputs to output, is described. A critical review of this model and its use for

optimization is presented in Section 5.5.

For actuation, two combs located on either side of the structure are used. These two

combs are electrically connected to 𝑣2. This signal interacts with 𝑣2 on the rotor

combs. The number of fingers on each stator comb is 𝑁𝑎. Since the two 𝑣2 stator

combs are on opposite sides, the finger gaps are in such a way that the force

generated produces a combined anti-clockwise torque on the rotor to achieve

rotation. As shown in Figure 5.5 inset, there are two gaps that control the generated

force - the force produced by the larger gap should ideally be negligible. The larger

gap ‘𝑛𝑎𝑔𝑎’ is a multiple of the smaller gap ‘𝑔𝑎’ where 𝑛𝑎 – the multiplier – is an

important design parameter.

5.4.1 Actuation

The same finger comb setup as in Figure 5.5 inset diagram was used for actuation.

Hence, the net force per finger 𝛥𝐹𝑓 can be described with (5.1);



∆𝐹𝑓 = 휀𝐴𝑎∆𝑉2

2[

1

(𝑔𝑎 − 𝑥)2−

1

(𝑛𝑎𝑔𝑎 + 𝑥)2] (5.1)

where 𝛥𝑉 = 𝑣1 – 𝑣2, 휀 is the absolute permittivity of air, 𝐴𝑎 is the finger overlap

area and 𝑥 is the linear displacement resulting from rotor rotational displacement

𝜃. The produced torque is due to two combs and is proportional to 2𝑁𝑎𝛥𝐹𝑓 . The

fingers are not at the same distance from the centre of the rotor (Figure 5.6) and

therefore, the force generated by each finger results in a different torque

contribution. An effective distance 𝐷 was found by expressing the total torque of the

rotor, 𝑇𝑎, as a sum as in (5.2) using the dimensions as defined in Figure 5.6.

Figure 5.6: Octagon dimensions – one side, showing the i th finger

𝑇𝑎 = 4 ∑ ∆𝐹𝑓

𝑁𝑎/2

𝑖=1

𝑑𝑖 = 4∆𝐹𝑓 ∑ 𝑎+ 𝑏𝑖 + 𝑐𝑎

𝑁𝑎/2

𝑖=1

= 4∆𝐹𝑓(𝑎 + 𝑐𝑎)𝑁𝑎2+ 4∆𝐹𝑓 ∑ 𝑏𝑖

𝑁𝑎/2

𝑖=1

(5.2)

𝑟 = 𝑙√1 +1

√2

ca bi a

l 2𝑖𝑙

𝑁

α

di

α

β

S2

S1

Sf

ΔFf

Rotor

Centre



A closed form expression for the actuation torque, 𝑇𝑎, can be obtained by evaluating

the sum for 𝑏𝑖 in (5.2). This gives 𝑏𝑖 = 𝑙 𝑐𝑜𝑠𝛼 (𝑁𝑎 − 2𝑖)/𝑁𝑎 and substituting this in

(5.2) gives (5.3),

𝑇𝑎 = 2𝑁𝑎∆𝐹𝑓(𝑎 + 𝑐𝑎 +𝑙

2cos𝛼) (5.3)

with the resulting effective distance of 𝐷 = 𝑎 + 𝑐𝑎 +𝑙

2𝑐𝑜𝑠𝛼, where 𝛼 =

3𝜋

8 for an

octagon and 𝑐𝑎is half the finger length (Figure 5.6).

Two ∆𝑉 signals will be investigated, one for actuation with ASK and the other for

actuation with BPSK (shown in Figure 5.7). Actuation with ASK is achieved by letting

𝑣1 and 𝑣2 as in (5.4) and (5.5),

𝑣1 = 𝑉1 cos(𝜔𝑐𝑡) + 𝑘 (5.4)

𝑣2 = (1 − 𝑑(𝑡))𝑉2cos (𝜔𝑐𝑡 + 𝜙) (5.5)

where 𝜔𝑐 is the carrier angular frequency, 𝑘 is a DC shift on the carrier, 𝜙 is the

phase shift between the ASK signal and the carrier, 𝑉1 and 𝑉2 the respective

amplitudes and 𝑑(𝑡) ∈ [0,1] is the binary data. A more realistic definition for 𝑑(𝑡) is

as in (5.6), where 𝑡𝑑 = 1/𝑓𝑑 with 𝑓𝑑 being the data frequency in Hertz. This definition

removes the sharp transitions.

𝑑(𝑡) =

1 − 𝑒−𝑡5𝑡𝑑 0 ≤ 𝑡 < 𝑡𝑑/2

𝑒−𝑡5𝑡𝑑 𝑡𝑑/2 ≤ 𝑡 < 𝑡𝑑

𝑑(𝑡 − 𝑡𝑑) 𝑡 ≥ 𝑡𝑑

(5.6)

The actuation torque, 𝑇𝑎, can be determined by substituting (5.1) and (5.4) to (5.6)

in (5.3). For small displacements, 𝑥 ≪ 𝑔, 𝑇𝑎 is proportional to ∆𝑉2 and ignoring the

torque components generated at 𝜔𝑐 and above, the ASK torque would be as in (5.7).



𝑇𝐴𝑆𝐾 ∝ 𝑉1𝑉2(1 − 𝑑)𝑐𝑜𝑠𝜙 +𝑉12

2+ 𝑘2 +

(1 − 𝑑)2𝑉22

2 (5.7)

The torque component generated by a BPSK signal can be obtained by switching

(5.5) with (5.8).

𝑣2 = (2𝑑(𝑡) − 1)𝑉2cos (𝜔𝑐𝑡 + 𝜙) (5.8)

This results in (5.9).

𝑇𝐵𝑃𝑆𝐾 ∝ 𝑉1𝑉2(2𝑑 − 1)𝑐𝑜𝑠𝜙 +𝑉12

2+ 𝑘2 +

(2𝑑 − 1)2𝑉22

2 (5.9)

Figure 5.7: Schematic showing actuation with BPSK input

Figure 5.8 shows how 𝑇𝐴𝑆𝐾 and 𝑇𝐵𝑃𝑆𝐾 change with 𝑑 and 𝑘 for 𝜑 = 0 (dashed line)

and 𝜑 = 𝜋 (solid line) if 𝑉1 = 𝑉2. It is evident that the largest change in torque level

– 𝛥𝑇𝑎 – occurs with BPSK input when signal changes phase with respect to the

carrier, in this case, 𝛥𝑇𝐵𝑃𝑆𝐾 = 2𝑉22 (Nm). For ASK, 𝛥𝑇𝐴𝑆𝐾 = 0.5𝑉2

2 (Nm) for an in-

phase carrier and 𝛥𝑇𝐴𝑆𝐾 = 1.5𝑉22 (Nm) for an anti-phase carrier. Moreover, for ASK,



if no carrier is presented at 𝑣1 port giving 𝑣1 = 𝑘 (instead of (5.4)), the rotor is still

actuated but with a 𝛥𝑇𝐴𝑆𝐾 of 0.5𝑉22 (Nm).

Figure 5.8: Torque levels for ASK and BPSK, dashed lines are in-phase, solid in anti-phase

The electrical components 𝑅𝑖 and 𝐶𝑝 (shown dashed in Figure 5.7) are the isolation

resistance and the parasitic capacitance intrinsic in the MEMS device while 𝐿𝑚 and

𝑅𝑚 are external passive components whose relevance to the sensing test set-up is

discussed in Sections 5.4.6 and 5.4.7. Electrically, the same four passive components

(in parallel) are also connected to S2 - this is not shown in Figure 5.7.

5.4.2 Spring Stiffness

Two cantilever spring designs were considered, with the first one having an anchor

in the middle of the rotor as in Figure 5.9 left. This produced a linear relationship

between torque and displacement. The second design – the one adopted – was also

a cantilever spring, however, the anchor was outside of the rotor (Figure 5.9 right).

With this design, the cantilevers not only offered transverse stiffness but also axial



stiffness due to elongation. As a result, the torque-displacement characteristics are

non-linear as this spring has a cubic stiffness term. This offered more design

flexibility as is discussed in Section 5.4.6.

Figure 5.9: Linear (left) vs. Non-linear (right) spring designs

For the linear stiffness case, four octagon vertices were at mid-span of a four

clamped-clamped beam springs of length 2𝑟. In this case, the spring force generated

at mid-span of each beam is Tr/4r with 𝑇𝑟 being the restoring toque. Equation (5.10)

gives the deflection, x, at mid-span:

𝑥 = 𝑇𝑟𝑟2/96𝐸𝐼 (5.10)

where I = tw3/12 is the second moment of area of the spring rectangular section, E

the Young’s modulus and, w the spring width and beam depth, t, is fixed by the SOI

thickness. From (5.10), the rotational deflection θ of the rotor would be:

𝜃 = 𝑇𝑟/(4𝑘𝑟2) (5.11)

where k = 192EI / (2r)3 is the transverse stiffness of each clamped-clamped beam

and 4kr2 the resulting rotational linear stiffness.



Alternatively, a hardening spring, that is, a cubic stiffness term together with the

linear term for the total restoring torque is provided by the four springs as in Figure

5.9 (right) described by (5.12),

𝑇𝑟 = 𝑘𝑙𝜃 + 𝑘𝑐𝜃3 (5.12)

where 𝑘𝑙 and 𝑘𝑐 are the linear and cubic stiffness coefficients respectively. Design

optimization would require finding adequate ratios of 𝑘𝑙/𝑘𝑐 . In [153], an H-shaped

spring fixture as in Figure 5.10, was used to control the axial and transverse

stiffnesses, effectively controlling 𝑘𝑙/𝑘𝑐. This option was studied but was not

adopted as it was deducted that the range of 𝑘𝑙/𝑘𝑐 required could be achieved

without the H fixture.

Figure 5.10: H-Fixture that provides control on axial and transverse stiffness, [153]

The ratios for 𝑘𝑙/𝑘𝑐 have to be physically realizable and the alternative spring layout

that provides cubic stiffness is employed as shown in Figure 5.11. In this new layout,

each cantilever spring of length 𝑞 offers transverse stiffness 𝑘𝑡 = 12𝐸𝐼/𝑞3, however,

this layout offers also axial stiffness as the spring is prone to elongation apart from

bending. Hence, a component of axial stiffness 𝑘𝑎 = 𝐸(𝑡𝑤)/𝑞 contributes towards

rotor rotation and the resulting total torque is (5.13),

𝑇𝑟 = 4(𝑘𝑡𝑥 + 𝑘𝑎∆𝑞𝛾)𝑟 (5.13)

where ∆𝑞 is the spring elongation and 𝛾 is the angle subtended by the spring as

shown in Figure 5.11.



Figure 5.11: Cantilever spring showing transverse and axial displacements

Replacing linear with rotational displacement and ∆𝑞 with an approximation of

𝑟𝜃(𝑟𝜃/𝑞)2/2 (see Appendix 5.1) results in (5.14).

𝑇𝑟 = 4𝑘𝑡𝑟2𝜃 + (

2𝑘𝑎𝑟4

𝑞2)𝜃3 (5.14)

5.4.3 Static Equilibria and Pull-In

At equilibrium, 𝑇𝑎 = 𝑇𝑟, (5.15);

2𝑁𝑎∆𝐹𝑓𝐷 = 4𝑘𝑡(𝑟𝜃)𝑟 + 2𝑘𝑎(𝑟𝜃)3𝑟/𝑞2 (5.15)

Replacing 𝑟𝜃 with x, and ∆𝐹𝑓 with (5.1) results in (5.16);

4𝑘𝑡𝑥 +2𝑘𝑎𝑥

3

𝑞2− 휀𝐴𝑎𝑁𝑎 (

𝐷

𝑟)∆𝑉2 [

1

(𝑔𝑎 − 𝑥)2−

1

(𝑛𝑎𝑔𝑎 + 𝑥)2] = 0 (5.16)

Re-arranging this equation gives a polynomial of degree 7 with the coefficients given

in Table 5.3, where 𝑘𝑙𝑙 = 4𝑘𝑡 and 𝑘𝑐𝑙 = 2𝑘𝑎/𝑞2.

Stopper

Stopper

Stopper

Stopper



Table 5.3: Coefficients of the resulting degree 7 polynomial

Degree Coefficient

𝑥0 −𝑔𝑎2𝑝∆𝑉2(𝑛𝑎

2 − 1)

𝑥1 𝑔𝑎4𝑘𝑙𝑙𝑛𝑎

2 − 2𝑔𝑎휀𝐴𝑎𝑁𝑎 (𝐷

𝑟) (𝑛𝑎 + 1)∆𝑉

2

𝑥2 −2𝑔𝑎𝑘𝑙𝑙𝑛𝑎(𝑛𝑎 − 1)

𝑥3 𝑔𝑎2(𝑘𝑐𝑙𝑔𝑎

2𝑛𝑎2 + 𝑘𝑙𝑙(𝑛𝑎

2 − 4𝑛𝑎 + 1))

𝑥4 2𝑔𝑎(𝑛𝑎 − 1)(𝑘𝑙𝑙 − 𝑛𝑎𝑔𝑎2𝑘𝑐𝑙)

𝑥5 𝑘𝑐𝑙𝑔𝑎2(𝑛𝑎

2 − 4𝑛𝑎 + 1) + 𝑘𝑙𝑙

𝑥6 2𝑘𝑐𝑙𝑔𝑎(𝑛𝑎 − 1)

𝑥7 𝑘𝑐𝑙

The solution to this polynomial gives the stable and unstable equilibria – some being

physically inadmissible – for different actuation voltage ∆𝑉. Appendix 5.2 gives

MATLAB code for the stable solutions for varying 𝑛𝑎 and ∆𝑉.

It is worth noting how the pull-in position, 𝑥𝑝𝑖, changes with 𝑛𝑎. For a two-plate

electrostatic actuator, assuming linear spring behaviour, pull-in occurs when

𝑥𝑝𝑖 = 𝑔𝑎/3 if 𝑘𝑎 ≈ 0. In this design, each rotor finger is acted upon from two

opposite sides meaning there is interaction between two gaps. Intuitively, if 𝑛𝑎 ≫ 1,

𝑥𝑝𝑖 → 𝑔𝑎/3 as this would place one plate far away from the rotor finger. This implies

that for maximum travel 𝑛𝑎 should be large. On the contrary for 𝑛𝑎 → 1+,

𝑥𝑝𝑖 < 𝑔𝑎/3. Figure 5.17 shows how 𝑥𝑝𝑖 changes with gap multiplier 𝑛𝑎.

5.4.4 Mechanical Dynamics

Analysis of dynamics requires determination of rotor inertia and damping forces.

The rotor is octagonal in shape and a closed form expression of the inertia of one



side could be derived. With reference to Figure 5.6, each octagon side contains a

radial support beam of width S1, a side strut of width 𝑆2 and 𝑁/2 fingers of width 𝑆𝑓.

These three components have moment of inertia 𝐽1, 𝐽2, and 𝐽𝑓 , and masses 𝑚1, 𝑚2

and 𝑁𝑚𝑓/2 respectively. They are rotationally symmetric and contribute to the total

rotor inertia 𝐽 (5.17).

𝐽 = 8(𝐽1 + 𝐽2 + 𝐽𝑓) (5.17)

For the radial support beam, only half of the width needs be considered, that is, 𝑆1/2,

which gives 𝐽1 as in (5.18);

𝐽1 =1

12

𝑚1

2((𝑆12)2

+ 𝑟2) +𝑚1

2(𝑟

2)2

(5.18)

where the first term is inertia of a cube about the edge and the second term is due

to the parallel axis shift theorem. Similarly, for the side strut, 𝐽2 is as in (5.19).

𝐽2 =1

12𝑚2(𝑆2

2 + 𝑙2) + 𝑚2 (𝑟2 − (

𝑙

2)2

) (5.19)

The total inertia (5.20) of 𝑁/2 fingers is also found by summing the individual finger

inertias taking into consideration that each had a different amount of ‘shift’ from the

origin (Appendix 5.3),

𝐽𝑓 =(𝑁𝑎𝑚𝑓𝑎

+ 𝑁𝑠𝑚𝑓𝑠)

4𝑓(𝛽)[(𝑆𝑓2 + 4𝑐2

12)𝑓(𝛽) + 4(𝑎2 + 𝑐2 + 2𝑎𝑐)

+ 6(𝑎 + 𝑐)𝑙𝑐𝑜𝑠𝛼 +7𝑙2 cos2 𝛼

3]

(5.20)

where 𝑓(𝛽) =𝛽2

sin2(𝛽/2) and 2𝛼 + 𝛽 = 𝜋.



Having an analytical closed form expression for 𝐽, the resonant frequency 𝜔𝑛 for

small oscillations (such that cubic stiffness is negligible) could be evaluated with

(5.21).

𝜔𝑛 = √4𝑘𝑡𝑟2

𝐽=2𝑟

𝑞√𝐸𝑡𝑤3

𝐽 (5.21)

With the capacitive gaps that could be achieved in SOIMUMPs, mechanical damping

is dominated by squeeze film damping (SFD) [154], [155]. In this case, the gas is

forced to squeeze in and out of the space between fingers. This squeeze film effect is

much more important than intrinsic material damping. When the characteristic flow

lengths become small, such that they become comparable to the molecular mean

free path of the gas, the condition is called rarefied gas flow. As a measure of the

amount of rarefaction, the Knudsen number, which is a ratio of the molecular mean

free path 𝜆 to the characteristic length of the flow h is used, (5.22).

𝐾𝑛 =𝜆

ℎ (5.22)

Using this measure, for 𝐾𝑛 < 0.001 gas flows can be treated in the classical

framework of continuum fluid mechanics [154], that is, Reynolds equation with

no-slip boundary condition. In the range 0.001 < 𝐾𝑛 < 0.1, Reynolds equation is still

valid but slip boundary condition must be applied and for larger 𝐾𝑛, statistical

approaches are required. For SOIMUMPs with the smallest gap possible and at

normal temperature and pressure, 𝐾𝑛 = 0.033 meaning Reynolds equation is still

valid but flows need to be corrected for the slip at the walls. The simplest way of

achieving this is by replacing the viscosity, 𝜇, by an effective viscosity

𝜇𝑒𝑓𝑓 = 𝜇/(1 + 6𝐾𝑛) in the Reynolds equation [156]. This is subject to active

research and one can find several expressions for 𝜇𝑒𝑓𝑓, usually of the form

𝜇/(1 + 𝑓(𝐾𝑛)), [154] and [155]. One of the most recent expressions, which is able to



capture squeeze film damping behaviour even with high Knudsen numbers, is given

in [157]. It is reproduced here for convenience, (5.23) and (5.24),

𝜇𝑒𝑓𝑓 =𝜉

6Π𝜇 (5.23)

where 𝜉 = √𝜋/(2𝐾𝑛), which is sometimes referred to as the inverse Knudsen

number [158] and Π defined as in (5.24),

Π =𝜉

6+2 − 𝛼

√𝜋𝑙𝑛 (

1

𝜉+ 2.18) +

𝛼

0.642+(1 − 𝛼)(𝜉 + 2.395)

2 + 1.12𝛼𝜉−1.26 + 10𝛼𝜉

1 + 10.98𝜉

+𝑒−

𝜉5

8.77

(5.24)

with 𝛼 being the tangential momentum accommodation coefficient. In [158], it is

shown that this equation is not sensitive to the choice of 𝛼 if chosen in the region

0 ≤ 𝛼 ≤ 1.

In SFD, the damping force 𝐹𝑑 consists of two parts [125], [154]: the viscous and

elastic damping forces (5.25),

𝐹𝑑 = 𝑏𝑙 + 𝑏𝑘𝑧 (5.25)

where 𝑏𝑙 and 𝑏𝑘 are the coefficients for viscous and elastic damping forces

respectively and 𝑧 represents linear displacement. A dimensionless number that

compares these two damping mechanisms is called the squeeze number 𝜎 defined

as:

𝜎 =12𝜇𝑒𝑓𝑓𝜔𝐿

2

𝑝ℎ0 (5.26)

where 𝐿 is the characteristic length, ℎ0 is the distance between the parallel plates, 𝑝

and 𝜇𝑒𝑓𝑓 are the air pressure and dynamic viscosity respectively. There exists a

cut-off squeeze number, 𝜎𝑐 , (5.27) where the viscous and elastic damping forces are



equal. For 𝜎 < 𝜎𝑐, the viscous damping is dominant while for 𝜎 > 𝜎𝑐, the elastic

damping is dominant [125],

𝜎𝑐 = 𝜋2 [1 + (

𝑊

𝐿)2

] (5.27)

with 𝑊 being the plate width which for SOIMUMPs can be either 10 µm or 25 µm.

Equating 5.24 and 5.25, one gets the cut-off frequency as in (5.28).

𝜔𝑐𝑜 =𝜋2 [1 + (

𝑊𝐿 )

2

] ℎ02𝑝

12𝜇𝑒𝑓𝑓𝑊2

(5.28)

Considering NTP and the SOIMUMPs’ constraints, the lowest value for this cut-off

frequency would be 5.9 MHz (@ 𝑊 = 25 𝜇𝑚, 𝐿 = 100 𝜇𝑚 and ℎ0 = 2 𝜇𝑚). The data

rate dictates the mechanical bandwidth and the target is in the low kHz range as

discussed in Section 5.2. This means that operation will be well below the cut-off

frequency and hence the dominant SFD force is due to viscous damping only, that is,

𝑏𝑙. Expressions for 𝑏𝑙 for 𝜔 ≪ 𝜔𝑐𝑜 are given in [125] and [154], as in (5.29),

𝑏𝑙 = 𝜇𝑒𝑓𝑓𝐿 (𝑊

ℎ0)3

𝛽 (5.29)

where 𝛽 = 1 − 0.58(𝑊/𝐿) is a dimensionless factor related to the aspect ratio of the

plates. For a strip plate, that is, 𝑊 ≪ 𝐿, 𝛽 → 1. Adapting (5.38) to include the wider

(𝑛𝑔) and narrower (𝑔) gaps in both actuation and sensing combs gives the total

linear viscous damping coefficient in 𝑁𝑠/𝑚,

𝑏𝑇 = 2𝑁𝑎𝑙𝑜𝑎 [𝜇𝑒𝑓𝑓𝑎 (𝑡

𝑔𝑎)3

+𝜇𝑒𝑓𝑓𝑎𝑛 (𝑡

𝑛𝑎𝑔𝑎)3

] 𝛽𝑎

+ 2𝑁𝑠𝑙𝑜𝑠 [𝜇𝑒𝑓𝑓𝑠 (𝑡

𝑔𝑠)3

+𝜇𝑒𝑓𝑓𝑠𝑛 (𝑡

𝑛𝑠𝑔𝑠)3

] 𝛽𝑠

(5.30)



where the first bracketed term is the contribution from the actuation combs and the

second bracketed term from the sensing combs, 𝑙𝑜 is the rotor to stator comb finger

overlap length. The effective viscosity for the smaller actuation, larger actuation,

smaller sensing and larger sensing gaps, are denoted by 𝜇𝑒𝑓𝑓𝑎, 𝜇𝑒𝑓𝑓𝑎𝑛, 𝜇𝑒𝑓𝑓𝑠 and

𝜇𝑒𝑓𝑓𝑠𝑛 respectively. These are different since the Knudsen number depends on the

gaps. 𝛽𝑎 and 𝛽𝑠 are 1 − 0.58(𝑡/𝑙𝑜𝑎) and 1 − 0.58(𝑡/𝑙𝑜𝑠) respectively. If 𝑛 ≫ 1, the

contribution from the wider gaps can be neglected.

Equation (5.25) refers to linear displacement and forces. Transforming linear to

rotational motion and changing forces to torques in (5.25) gives the total rotational

viscous damping coefficient in 𝑁𝑚/𝑟𝑎𝑑 𝑠−1, (5.31).

𝑏 = 𝑟2𝑏𝑇 (5.31)

With equations (5.30) and (5.31), SFD behaviour can be captured and the

mechanical dynamic model can be developed (5.32).

𝐽 + 𝑏 + 4𝑘𝑡𝑟2𝜃 + (

2𝑘𝑎𝑟4

𝑞2)𝜃3 = 𝑇𝑎 (5.32)

For this application, the actuation torque, 𝑇𝑎, could be modelled with a step function.

For the resulting displacement to follow the step input as closely as possible, critical

damping is required, meaning a quality factor, 𝑄, of 0.5. For small oscillations and

neglecting the cubic stiffness, such a response can be achieved if 𝑏 = 𝑏𝑐𝑟 (5.33). The

settling time (5% criterion), 𝑡𝑠, would be as in (5.34) giving a maximum data rate of

𝑓𝑑 = 1/2𝑡𝑠 Hz.

𝑏𝑐𝑟 = 4√𝑘𝑡𝑟2𝐽 =4𝑟

𝑞√𝐸𝑡𝑤3𝐽 (5.33)

𝑡𝑠 ≈ 4(2𝐽

𝑏) (5.34)



A state space model for the mechanical system can be obtained by defining state

vector 𝒑 = [𝑥 ]𝑇 . Substituting 𝑥 for 𝑟𝜃 and 𝒑 in (5.32) results in (5.35).

=

[

𝑝2

−𝑏

𝐽𝑝2 −

4𝑘𝑡𝑟2

𝐽𝑝1 −

2𝑘𝑎𝑟2

𝑞2𝐽𝑝13 +

𝑁𝑎𝐷휀𝐴𝑎𝑟

𝐽∆𝑉2

(𝑛𝑎𝑔𝑎 + 𝑝1)2 − (𝑔𝑎 − 𝑝1)

2

((𝑔𝑎 − 𝑝1)(𝑛𝑎𝑔𝑎 + 𝑝1))2 ]

(5.35)

5.4.5 Actuation Capacitance and Instantaneous Power

The actuation capacitance per rotor finger, 𝐶𝑓𝑎(𝑥), is made up of the four

components shown in Figure 5.12, giving the total actuation capacitance 𝐶𝑎(𝑥) =

2𝑁𝑎𝐶𝑓𝑎(𝑥) = 2𝑁𝑎([𝐶𝑔(𝑥) + 𝐶𝑓𝑔(𝑥)] + [𝐶𝑛𝑔(𝑥) + 𝐶𝑓𝑛𝑔(𝑥)]).

Figure 5.12 Finger section showing electric field

In literature, many different formulae for computing fringing fields can be found. In

[159], the authors make a comparative exercise and propose an improved formula,

replicated here for convenience (5.36),

𝐶(𝑔) = 휀𝑡

𝑔[1 +

𝑔

𝜋𝑡+𝑔

𝜋𝑡𝑙𝑛 (

2𝜋𝑡

𝑔) +

𝑔

𝜋𝑡𝑙𝑛(1 +

2𝑆𝑓

𝑔+ 2√

𝑆𝑓

𝑔+𝑆𝑓2

𝑔2)] (5.36)

where t is the thickness of the finger, 𝑔 is the gap and 𝑆𝑓 is the finger width. This

gives the total capacitance, C, per unit length. Adapting this for both the smaller

and the larger gaps gives (5.37),



𝐶𝑎(𝑥) = 2𝑁𝑎𝐶𝑓𝑎(𝑥) = 2𝑁𝑎𝐶(𝑔𝑎 − 𝑥) + 𝐶(𝑛𝑎𝑔a + 𝑥)𝑙𝑜𝑎 (5.37)

where 𝑙𝑜𝑎is the finger overlap length.

With this capacitance model, the instantaneous actuating current, 𝑖𝑎(𝑡), flowing

from 𝑣2 to 𝑣1 could be obtained. This current is the sum flowing through 𝐶𝑎(𝑡) and

the isolation resistance, 𝑅𝑖, the resistance present between the two electrodes. This

is described with (5.38),

𝑖𝑎(t) =𝑑

𝑑𝑡(𝐶𝑎(t)∆𝑉(𝑡)) + ∆𝑉(𝑡)/𝑅𝑖 = [𝐶𝑎(t)∆(t) + ∆𝑉(𝑡)𝑎(𝑡)] + ∆𝑉(𝑡)/𝑅𝑖 (5.38)

from which, the instantaneous actuation power could be computed with (5.39).

𝑃𝑎(t) = 𝑖𝑎(t)∆𝑉(𝑡) (5.39)

Equation (5.38) can be combined in matrix form with (5.35) by defining another

state, 𝑝3 = 𝑖𝑎(𝑡), giving (5.40).

3 = 𝐶𝑎∆ + 𝑎∆𝑉 + ∆(2𝑎 + 1/𝑅𝑖) (5.40)

From (5.38), it can be seen that the absolute/nominal values for the actuation

capacitance and voltages are contributing to instantaneous power dissipation,

however, their rate of change is also a contributor.

Using the state space model (5.35) and 𝐶𝑎(𝑥) as in (5.37), a numerical solution to 𝑎

can be found.

5.4.6 Displacement Sensing and Complete System Model

The rotor is rotationally symmetric and the capacitance on the sensing combs varies

according to 𝑥(𝑡). The sensing comb finger layout is similar to the actuation comb

layout, however, a different gap 𝑔𝑠, gap multiplier 𝑛𝑠 and number of fingers 𝑁𝑠 are

defined. Hence, the actuation capacitance per rotor finger, 𝐶𝑓𝑎(𝑥, 𝑛𝑎 , 𝑔𝑎) in (5.37)



will be replaced with 𝐶𝑓𝑠(𝑥, 𝑛𝑠, 𝑔𝑠) for the sensing capacitance per rotor finger. The

total capacitance per sensor comb is expressed in (5.41),

𝐶𝑠1(𝑥) = 𝑁𝑠𝐶𝑓𝑠(𝑥) = 𝑁𝑠𝐶(𝑔𝑠 − 𝑥) + 𝐶(n𝑠𝑔𝑠 + 𝑥)𝑙𝑜𝑠 (5.41)

for the closing gap sensor, while the opening gap sensing capacitance would be

𝐶𝑆2(𝑥) = 𝐶𝑆1(−𝑥). From (5.36), an expression for the fringe capacitance only can

be extracted (5.42),

𝐶𝑓𝑟𝑖𝑛𝑔𝑒(𝑔) =휀

𝜋[1 + 𝑙𝑛 (

2𝜋𝑡

𝑔) + 𝑙𝑛(1 +

2𝑆𝑓

𝑔+ 2√

𝑆𝑓

𝑔+𝑆𝑓2

𝑔2)] 𝑙𝑜𝑠

(5.42)

from which the total fringe capacitance per rotor finger would be as in (5.43).

𝐶𝑓𝑟𝑖𝑛𝑔𝑒 = 𝐶𝑓𝑟𝑖𝑛𝑔𝑒(𝑔 − 𝑥) + 𝐶𝑓𝑟𝑖𝑛𝑔𝑒(𝑛𝑔 + 𝑥) (5.43)

Using (5.43), it can be shown (Appendix 5.4) that the fringe capacitances are a

substantial part of the overall capacitance for typical dimensions used in MEMS,

however, the change in fringe capacitances over the full range of motion

(0 < 𝑥 < 𝑔/3) is negligible (<3%). This means that a simpler equation can be used

to give a nominal (fixed) fringe capacitance, 𝐶𝑓𝑜, across the whole range of motion

by putting 𝑥 = 0 (5.44).

𝐶𝑓𝑟𝑖𝑛𝑔𝑒 ≈ C𝑓𝑜 = 𝐶𝑓𝑟𝑖𝑛𝑔𝑒(𝑔) + 𝐶𝑓𝑟𝑖𝑛𝑔𝑒(𝑛𝑔) (5.44)

This simplifies (5.41) to give (5.45). This simplification is of benefit for further

analysis and also for the optimisation process.

𝐶𝑠1(𝑥) = 𝑁𝑠휀𝑡𝑙𝑜𝑠 1

𝑛𝑠𝑔𝑠 + 𝑥+

1

𝑔𝑠 − 𝑥 + 𝐶𝑓𝑜 (5.45)

It is required that both 𝐶𝑠1(𝑥) and 𝐶𝑠2(𝑥) (= 𝐶𝑠1(−𝑥)) are monotonic over the full

range of motion. Hence, their respective minimum must not occur within

0 < 𝑥 < 𝑔𝑠/3. Using (5.45), this requires that 𝑛𝑠 > 5/3 (Appendix 5.5).



The change in sensing capacitive gap modulates the current, 𝑖𝑠(𝑡), passing through

it. Several methods could be employed to measure this current and in Figure 5.7, a

passive solution as a test set-up, is shown. This is the same ‘constant’ charge

approach adopted in Section 3.2 for the IQ mixer design with the difference that the

parasitic capacitance (cables and probes) is eliminated using an external inductor.

From this figure, the sensing capacitive reactance, 𝑋𝐶𝑠 , is in series with the total

impedance, 𝑍𝑇 , made up of four components in parallel. The output voltage 𝑣𝑝 is

obtained by potential division using 𝑋𝐶𝑠 and 𝑍𝑇 .

The impedance 𝑍𝑇 is made up of the isolation resistance, 𝑅𝑖, the probes/cabling

parasitic capacitance, 𝐶𝑝, an external inductor, 𝐿𝑚, intended to resonate (at 𝑓𝑐) with

the parasitic capacitance while 𝑅𝑚 is a resistive load. This load would dictate the

gain (potential division) and also the cut-off frequency of the resulting high-pass

filter.

The sensing current and the potential division setup (Figure 5.7) is described by

(5.46) or (5.47).

𝑖𝑠(t) =𝑑

𝑑𝑡(𝐶𝑠(𝑡)(𝑣1(𝑡) − 𝑣𝑝(𝑡))) (5.46)

𝑖𝑠(t) = 𝑝𝐶𝑝 +𝑅𝑚 + 𝑅𝑖𝑅𝑚𝑅𝑖

𝑣𝑝 +1

𝐿𝑚∫𝑣𝑝𝑑𝑡 (5.47)

Defining another 5 states as 𝑝4 = 𝜔𝑐𝑡, 𝑝5 = 𝜔𝑑𝑡, 𝑝6 = 𝑖𝑠, 𝑝7 = 𝑣𝑝 and 𝑝8 = 𝑝,

substituting these in (5.47) and combining them with (5.35) and (5.40), the

complete state space model for the system can be described by (5.48),



=

[

𝑝2

−𝑏

𝐽𝑝2 −

4𝑘𝑡𝑟2

𝐽𝑝1 −

2𝑘𝑎𝑟2

𝑞2𝐽𝑝13 +

𝑁𝑎𝐷휀𝐴𝑎𝑟

𝐽∆𝑉2

(𝑛𝑎𝑔𝑎 + 𝑝1)2 − (𝑔𝑎 − 𝑝1)

2

((𝑔𝑎 − 𝑝1)(𝑛𝑎𝑔𝑎 + 𝑝1))2

𝐶𝑎∆ + 𝑎∆𝑉 + ∆(2𝑎 + 1/𝑅𝑖)

𝜔𝑐

𝜔𝑑

(𝐶𝑝

𝐶𝑝 + 𝐶𝑠) (

𝐶𝑠𝐶𝑝𝐿𝑚

− 𝑠)𝑝7 + (𝐶𝑠(𝑅𝑚 + 𝑅𝑖)

𝐶𝑝𝑅𝑚𝑅𝑖− 2𝑠)𝑝8 + 2𝐶1 + 𝑠𝑣1 + 𝐶𝑠1

𝑝8

1

𝐶𝑝6 −

1

𝐶𝑝𝐿𝑚𝑝7 −

𝑅𝑚 + 𝑅𝑖𝐶𝑝𝑅𝑚𝑅𝑖

𝑝8]

(5.48)

where 𝒑 = [𝑥 𝑖𝑎 𝜔𝑐𝑡 𝜔𝑑𝑡 𝑖𝑠 𝑣𝑝 𝑝]𝑇

By defining 𝜔𝑐𝑡 and 𝜔𝑑𝑡 as states, the system of equations becomes autonomous and

mathematical tools for autonomous systems can be employed. Moreover, this

enables the MEMS device model to be represented in a more compact form. It is

quite clear that the system is nonlinear and time-varying, with 𝑏, ∆𝑉, 𝑣1, 𝐶𝑎 and 𝐶𝑠

being functions of time and/or other states. This model is used to find numerical

solutions in MATLAB (Appendix 5.8).

The equations relating the input voltage difference, ∆𝑉, to the sensing capacitances

𝐶𝑠1 and 𝐶𝑠2 are listed in Table 5.4. The relationships ∆𝐹𝑓(∆𝑉), 𝑥(∆𝐹𝑓) and 𝐶𝑠1(𝑥) and

𝐶𝑠2(𝑥) result in the composite functions 𝐶𝑠1(𝑥(∆𝐹𝑓(∆𝑉))) and 𝐶𝑠2(𝑥(∆𝐹𝑓(∆𝑉))). The

gradient of these composite functions can be controlled by using a cubic stiffness

spring rather than a linear spring. The graphs in red in Table 5.4, show the cubic



stiffness option. The reason for having a higher nonlinear behaviour for 𝐶𝑠1 than for

𝐶𝑠2 can be understood by observing that ∆𝐹𝑓(∆𝑉) has increasing positive gradient

like 𝐶𝑠1 while 𝐶𝑠2 has decreasing negative gradient for 𝑥 > 0.

Controlling the ratio of linear-to-cubic stiffness to linearise the gradient of the

composite relationship is dealt with in Section 5.5.2 which describes the process of

optimising the design parameters.

Table 5.4: Linear vs. Nonlinear Spring Stiffness and Overall linearity

Equation No. and Description Nature of relationship

Actuation voltage to Electrostatic

force

5.1: ∆𝐹𝑓(∆𝑉)

Electrostatic force to Linear

Tangential Displacement

5.11: 𝑥(∆𝐹𝑓) – linear stiffness

5.14: 𝑥(∆𝐹𝑓) – cubic stiffness

Linear Tangential Displacement

to Sensing Capacitances

5.45: 𝐶𝑠1(𝑥) and 𝐶𝑠2(𝑥) =

𝐶𝑠1(−𝑥)

∆𝐹𝑓

∆𝐹𝑓

∆𝐹𝑓

∆𝐹𝑓

𝑥

𝑥

𝑥

𝑥 𝐶𝑠1

𝐶𝑠1

𝐶𝑠1

𝐶𝑠1

∆𝑉

∆𝑉

∆𝑉

∆𝑉

∆𝐹𝑓

∆𝐹𝑓

∆𝐹𝑓

∆𝐹𝑓

𝐶𝑠2

𝑥



Composite Relationship

𝐶𝑠1(𝑥(∆𝐹𝑓(∆𝑉)))

Composite Relationship

𝐶𝑠2(𝑥(∆𝐹𝑓(∆𝑉)))

5.4.7 Output ASK Modulation Index and Fringe Capacitance

The modulation index 𝑀 is defined as the ratio of smallest to highest modulated

carrier amplitude. In this section the dependency of 𝑀 on sensing parameters is

determined. With reference to the schematic shown in Figure 5.7, when the

inductance, 𝐿𝑚 and parasitic capacitance 𝐶𝑝 resonate, their reactances cancel out;

when this happens, 𝑍𝑇 = 𝑅𝑇 , whose value would be primarily dictated by the load

resistor 𝑅𝑚 as the isolation resistance 𝑅𝑖 is typically very high. In the frequency

domain, the resulting transfer function would be as in(5.49).

𝑣𝑝

𝑣1=

𝑅𝑚𝑅𝑚 + 𝑗𝑋𝐶𝑠

(5.49)

The modulation index would then be;

𝑀 =|𝑣𝑝𝐴𝑣1|

|𝑣𝑝𝐵𝑣1|

⁄ = √𝑅𝑚2 + 𝑋𝐶𝑠𝐵

2

𝑅𝑚2 + 𝑋𝐶𝑠𝐴2 = √

𝑅𝑚2 + 𝑋𝐶𝑠𝐵2

𝑅𝑚2 + 𝑒2𝑋𝐶𝑠𝐵2 (5.50)



where the suffixes ‘A’ and ‘B’ here denote two different sensing gaps with ‘A’ being

the larger gap resulting in 𝑋𝐶𝑠𝐴 = 𝑒𝑋𝐶𝑠𝐵 or equivalently 𝐶𝑠𝐵 = 𝑒𝐶𝑠𝐴 where 𝑒 > 1.

For the case where 𝐶𝑠𝐴 is the nominal gap capacitance (𝑥 = 0) and 𝐶𝑠𝐵 is larger

(0 < 𝑥 <𝑔𝑠

3), using the simpler model for the sensing capacitance (5.45), 𝑒 would be

as in (5.51).

𝑒 =𝑁𝑠휀𝑡𝑙𝑜𝑠

1𝑛𝑠𝑔𝑠 + 𝑥

+1

𝑔𝑠 − 𝑥 + 𝐶𝑓𝑜

𝑁𝑠휀𝑡𝑙𝑜𝑠 1

𝑛𝑠𝑔𝑠+1𝑔𝑠 + 𝐶𝑓𝑜

(5.51)

Defining 𝐶𝑟 = 𝐶𝑓𝑜/ [ 𝑁𝑠휀𝑡𝑙𝑜𝑠 1

𝑛𝑠𝑔𝑠+

1

𝑔𝑠] and 𝑓 = 3𝑥/𝑔𝑠 for 0 < 𝑓 < 1, (5.52) is

obtained.

𝑒 =𝐶𝑟(𝑓

2 + 3𝑓(𝑛𝑠 − 1)) − 9𝑛𝑠(𝐶𝑟 + 1)

(𝑓 − 3)(𝐶𝑟 + 1)(𝑓 + 3𝑛𝑠)

(5.52)

Equation (5.52) gives 𝑒, which is unitless and is a function of 𝐶𝑟 , the ratio of fringe

capacitance to nominal capacitance, 𝑓 which is the displacement 𝑥 in terms of the

maximum, that is, 𝑥 = 𝑔𝑠/3 and also 𝑛𝑠. This equation together with (5.50) describe

how the selected design affects the modulation index - Figure 5.13.

Figure 5.13: Parameters affecting modulation index, M

increasing 𝐶𝑟

increasing 𝐶𝑟

increasing 𝐶𝑟

increasing 𝐶𝑟

𝐶𝑟

𝐶𝑟

𝐶𝑟

𝐶𝑟



The graphs in Figure 5.13 are asymptotic to 𝑀 = 2/3 for large 𝑛 and zero fringe

capacitance, which comes about from the maximum displacement allowable, 𝑥𝑝𝑖 (at

pull-in). Equations (5.52) and (5.50) are implemented in MATLAB script as given in

Appendix 5.6.

In this section, the relationships between the geometric parameters and the

underlying physical phenomena were presented in detail. As can be observed, there

is a large number of design variables whose effect on the required specifications is

very often contradictory and hence, a compromise is required. One parameter that

would need to be selected wisely is the gap multiplier 𝑛.

The following bulleted list gives some observations of the direct effect this

parameter has on some of the design requirements.

• 𝑛𝑎 ≫ 1 guarantees 𝑥𝑚𝑎𝑥 → 𝑔𝑎/3

• 𝑛𝑎 ≫ 1 keeps voltages low if and only if the number of fingers, 𝑁𝑎, is kept the

same. For the latter to be satisfied, a larger footprint is required which in turn

increases the inertia.

• For the same footprint, if 𝑛𝑎 or 𝑛𝑠 are increased, rotor would have less fingers

lowering the inertia 𝐽.

• If the inertia is kept the same and 𝑛𝑎 or 𝑛𝑠 are increased, viscous friction 𝑏

due to SFD decreases giving a longer settling time 𝑡𝑠.

• 𝑛𝑠 must be greater than 5/3 for monotonicity in sensing.

• Increasing 𝑛𝑠 improves the modulation index M.

5.5 Optimisation Towards the Design Objectives

The determination of the geometric dimensions that satisfied the design objectives

for such a complex structure required a computational approach which was carried

out in MATLAB. The first steps involved confirming that MATLAB functions which

modelled static behaviour were in fact accurate – when compared to FEA – over the

whole range of motion. In particular, functions that represented the electrostatic



force-to-torque-to-displacement, the actuation and sensing capacitances (including

fringe capacitances) and also simpler checks like the accuracy of expressions

derived for the moment of inertia, required validation using FEA. For FEA, two

software packages were used: CoventorWare® which is a software package

dedicated to MEMS processes and Autodesk Inventor, the latter being used only for

mechanical simulations.

5.5.1 Dimensionality and FEA Validation

Table 5.5 lists all the 15 variables that control the device’s dimensions. Only the load

resistance is an external passive component. The constraint type column gives the

more severe of two types: ‘SOIMUMPs’ which refers to the SOIMUMPs fabrication

process and ‘Functional’ which refers to a constraint resulting from analysis and/or

design requirements.

Table 5.5: Dimensions Table

Variable Symbol Comments Type

Actuation gap 𝑔𝑎 ≥ 2 𝜇𝑚 SOIMUMPs Actuation gap multiplier 𝑛𝑎 > 2 guarantees pull-in close to 𝑔𝑎/3 Functional Actuation finger length 2𝑐𝑎 < 100 𝜇𝑚 if 𝑆𝑓 < 6 𝜇𝑚 SOIMUMPs Actuation finger overlap 0.9(2𝑐𝑎) Fixed at 90% finger length Functional Sensing gap 𝑔𝑠 ≥ 2 𝜇𝑚 SOIMUMPs Sensing gap multiplier 𝑛𝑠 > 5/3 for monotonicity Functional Sensing finger length 2𝑐𝑠 < 100 𝜇𝑚 if 𝑆𝑓 < 6 𝜇𝑚 SOIMUMPs Sensing finger overlap 0.9(2𝑐𝑠) Fixed at 90% finger length Functional Finger Width 𝑆𝑓 ≥ 2 𝜇𝑚 but fixed at 𝟐 𝝁𝒎 Functional

Octagon radius 𝑟 Around 600 𝜇𝑚 or greater (parallel

plates) Functional

Radial beam width 𝑆1 Fixed at 𝟏𝟒 𝝁𝒎 Functional Side Strut width 𝑆2 Fixed at 𝟐𝟎 𝝁𝒎 Functional Spring length 𝑞 Direct effect on overall size Functional Spring Width 𝑤 > 6 𝜇𝑚 since > 100 𝜇𝑚 length SOIMUMPs Load Resistance 𝑅𝑚 Filtering/potential division Functional

As described next, some of these variables (shown in bold in Table 5.5) were fixed

from the outset. The finger width was fixed at the smallest possible to allow for the



maximum capacitive area to be achieved. This capacitance could still be controlled

by the gap and gap multipliers. The octagon side and radial beam widths were fixed

such that no unwanted modes of vibration could manifest themselves. Fixing of

these variables served also the purpose of dimensionality reduction. In effect, the

optimisation problem was now reduced to a 10-dimensional one. From these ten

design parameters, only 𝑅𝑚, the load resistance, is an electrical parameter; the rest,

𝑔𝑎, 𝑛𝑎 , 𝑐𝑎, 𝑔𝑠, 𝑛𝑠 , 𝑐𝑠, 𝑟, 𝑞 and 𝑤 are all geometric in nature and they define the

converter geometry completely.

A 3D solid model of the rotor was designed using Autodesk Inventor employing

parametric dimensioning. Parametric dimensioning allowed for changes to be

affected quickly while respecting the required geometric constraints. This 3D model

was used to perform a corner simulation and to confirm the accuracy of the total

rotor inertia, J, the resonant frequency, 𝜔𝑛, force-to-rotational displacement, 𝜃(∆𝐹),

and the actuation and sensing capacitances 𝐶𝑎 and 𝐶𝑠 ((5.37) and (5.41))

relationships. The latter two relationships were subsequently evaluated in

CoventorWare® by applying 0 V at the stator combs and increasing rotor voltage

(in steps of 0.1 V) in a ‘MemElectro’ parametric study. The displacement and change

in capacitance were found to be in agreement with the solution to (5.16), (5.37) and

(5.41).

The change in fringe capacitance due to change in gap, up to pull-in, was also

confirmed to be negligible. Equation (5.16), which gives the equilibrium (stable and

unstable) position for different ∆𝑉, was solved numerically in MATLAB. It is a 7th

order polynomial and attention was given to extract only the stable solution up until

pull-in. The numerical solution obtained in MATLAB for pull-in was also validated

against FEA results for corner dimensions.

The damping coefficient equations (5.30) and (5.31) were tested for validity using

CoventorWare®. FEA simulations at NTP were performed which included SFD. The

tangential momentum accommodation coefficient, 𝛼, was taken at 0.6 [157] and the



mean free path, 𝜆 = 68 𝑛𝑚; these gave a matching damping coefficient at the mode

of interest equivalent to a Q-factor of 3.

With a valid mathematical model implemented in MATLAB, the problem of

satisfying the design specifications could be addressed.

5.5.2 Dimensional Optimisation using MATLAB

Two types of optimisation targets were used: constrained and unconstrained. Those

falling under the unconstrained category were 𝑟, 𝑞 and 𝐶𝑎 for a small footprint and

low moment of inertia. However, some targets, like output voltage 𝑣𝑝, were required

to be above (or below) a certain limit – constrained - and it was not important to

keep trying to increase when this limit was exceeded. Table 5.6 lists the design

specification targets selected, their type and the relevant equations.

Table 5.6: Constrained and Unconstrained design specification targets

Design

Specification Target Type Relevant Equation/s

𝑟 Smallest Unconstrained -

𝑞 Smallest Unconstrained -

𝐶𝑎 Smallest Unconstrained (5.37) 𝑓𝑑 > 2 kHz Constrained (5.34), (5.31), (5.17)

𝑀 ≤ 0.8 Constrained (5.50), (5.52)

𝑥𝑝𝑖 ≥ 𝑔𝑎/4 Constrained (5.16)

Δ𝑉𝑝𝑖 ≤ 14 𝑉 Constrained (5.16)

∆𝐶𝑠1 ≥ 50 fF Constrained (5.54)

𝑣𝑝 > 500 mV RMS Constrained (5.49)

𝐶𝑙𝑖𝑛 ≤ 0.01 Constrained (5.53)

For 𝑥𝑝𝑖, the pull-in displacement, to be as close as possible to 𝑔𝑎/3, a lower limit of

𝑔𝑎/4 was imposed. The parameter 𝐶𝑙𝑖𝑛 is calculated as in (5.53).



𝐶𝑙𝑖𝑛 = 𝑠𝑡𝑑𝑒𝑣 (𝑑2𝐶𝑠𝑑∆𝑉2

)|∆𝑉>4

(5.53)

This is a measure of how far 𝐶𝑠(∆𝑉) is from perfect linearity for the range

4 < ∆𝑉 < ∆𝑉𝑝𝑖, where 𝑠𝑡𝑑𝑒𝑣 signifies standard deviation. If 𝐶𝑠(∆𝑉) is perfectly

linear, then 𝑑2𝐶𝑠

𝑑∆𝑉2= 0 giving 𝑠𝑡𝑑𝑒𝑣 (

𝑑2𝐶𝑠

𝑑∆𝑉2) = 0 as well. The inclusion of this design

specification was instrumental in finding the correct transverse to axial spring

stiffness ratios, hence, finding the correct nonlinear spring that improved overall

linearity. The carrier frequency for optimisation simulations was chosen at

𝑓𝑐 = 1 MHz.

Particle Swarm Optimization was adopted since it falls under the evolutionary

techniques which are more capable of solving multi-objective functions as described

in [160] and PSO gave positive results with minimum iterations in [151]. The PSO

routine was implemented in MATLAB (Appendix 5.7).

From Table 5.6, the objective function to be minimised is then the linear

combination:

𝑓(𝑔𝑎, 𝑛𝑎 , 𝑐𝑎, 𝑔𝑠, 𝑛𝑠 , 𝑐𝑠, 𝑟, 𝑞, 𝑤, 𝑅𝑚) = 𝑤1𝑟 + 𝑤2𝑞 + 𝑤3𝐶𝑎 + 𝑤𝑑∑0.5(𝐹𝑖 + 1)

7

𝑖=1

(5.54)

where [𝑤1 𝑤2 𝑤3 𝑤𝑑] are positive weights in proportion to the importance of each

individual variable. Equal importance is achieved with [𝑤1 𝑤2 𝑤3 𝑤𝑑] ≈

[1667 2183 5.5611 1]. These numbers, which give equal importance, scale the

optimisation surface equally and are determined by taking the reciprocal of the

expected value of each variable. The functions 𝐹𝑖 are listed in Table 5.7 where sgn

signifies the sign function and the ranges for design variables are listed in

Table 5.8.



Table 5.7: Constrained functions as targets for design specifications

Function Name Function

𝐹1 𝑠𝑔𝑛(2000 − 𝑓𝑑)

𝐹2 𝑠𝑔𝑛(𝑀 − 0.8)

𝐹3 𝑠𝑔𝑛(𝑔𝑎/4 − 𝑥𝑝𝑖)

𝐹4 𝑠𝑔𝑛(∆𝑉𝑝𝑖 − 14)

𝐹5 𝑠𝑔𝑛(50 × 10−15 − ∆𝐶𝑠1)

𝐹6 𝑠𝑔𝑛(0.5 − 𝑣𝑝)

𝐹7 𝑠𝑔𝑛(𝐶𝑙𝑖𝑛 − 0.01)

Table 5.8: Valid Ranges for design dimensions

Parameter Min Max

𝑔𝑎 2 𝜇𝑚 6 𝜇𝑚

𝑛𝑎 2 6

2𝑐𝑎 2*(30 𝜇𝑚) 2*(50 𝜇𝑚)

𝑔𝑠 2 𝜇𝑚 6 𝜇𝑚

𝑛𝑠 5/3 6

2𝑐𝑠 2*(30 𝜇𝑚) 2*(50 𝜇𝑚)

𝑟 500 𝜇𝑚 700 𝜇𝑚

𝑞 400 𝜇𝑚 500 𝜇𝑚

𝑤 6 𝜇𝑚 15 𝜇𝑚

𝑅𝑚 1 k Ω 1000 k Ω

Figure 5.14 presents a sample run. It shows how the objective function is minimised

in twenty iterations for a PSO run starting with approximately equal linear weights

and also gives the resulting geometric dimensions and resulting design specification

values. PSO has several controlling parameters with the most important being the

swarm size (100 particles were used), maximum number of iterations (set at 100),



inertia (ranged from 0.4 to 0.9) and acceleration factor (set at 2). Number of runs

was set to 5 and the best run would be given as output. The parallel processing

toolbox was not made use of and execution took around 10 minutes on an Intel i7-

3720QM CPU @ 2.6 GHz with 32 Gb of RAM. Appendix 5.7 provides the MATLAB

scripts.

By changing the objective function weights, several realisable solutions could be

found with the PSO technique. These PSO solutions were all valid however given that

the SOIMUMPs tolerances are of 0.25 𝜇𝑚 and the smallest allowable features are of

2 𝜇𝑚, it was imperative to select only PSO solutions that remained ‘optimal’ within

the whole tolerance range to guarantee a broadly optimal solution. Instead of

testing the acceptable solutions obtained from PSO, by performing a Monte Carlo

simulation, an alternative approach was used.

Figure 5.14: Sample run – PSO convergence, verbose and results

PSO solutions were tested for broad optimality by feeding PSO solutions to

MATLAB’s fmincon function with the ‘SQP’ option and using also the MultiStart

function. Essentially, this involved a second optimisation process that starts by

generating a uniformly distributed, random set of start points close



(< 10% change) to the optimal solution given by PSO. Those that always converged

on the same point were discarded indicating a ‘narrowly optimal case’.

Specifically, if the PSO stage gives a set, 𝑆, of 𝑛 satisfactory solutions, that is,

𝑆 = 𝑆1 𝑆2 𝑆3… 𝑆𝑛, uniformly distributed noise (±5%) is added to each of these

solutions to generate 𝑁 different starting points per solution. Hence, the sets of

initial points 𝐼 = 𝑆1𝑁 𝑆2𝑁 𝑆3𝑁 … 𝑆𝑛𝑁 are generated to be used for the

subsequent local search. Minimisation (using SQP) of the objective function is

performed starting from these different sets.

If a high percentage of runs from an initial point set, say 𝑆1𝑁, re-converge on 𝑆1, this

solution is considered ‘narrowly optimal solution’ and 𝑆1 is discarded. Figure 5.15

shows this concept in one-dimension. If the value of the objective function is highly

sensitive to changes in the dimension, a minor change in the dimension can easily

result in not satisfying the objective threshold. In this way, a PSO solution that was

broadly optimal within SOIMUMPs tolerances could be selected. The final

dimensions and resulting design targets selected for manufacturing are listed in

Table 5.9.

Figure 5.15: Narrow vs. broad optimality property



Table 5.9: The final dimensions (µm) and resulting design specifications

Final

Dimensions

𝒈𝒂 𝒏𝒂 𝒄𝒂 𝒈𝒔 𝒏𝒔 𝒄𝒔 𝒓 𝒒 𝒘 𝑹𝒎

2 3 50 2 3 50 598 458 9 700 kΩ

Final Design

Specifications

𝑪𝒂 𝒇𝒅 𝑴 𝒙𝒑𝒊 ∆𝑽𝒑𝒊 ∆𝑪𝒔𝟏 𝒗𝒑 𝑪𝒍𝒊𝒏

2.0 pF 3.3 kHz 0.77 0.32ga 11 V 250 fF 0.5 V 0.01

5.5.3 Design Validation using MATLAB

Further simulations were performed with the final dimensions both with FEA and

also in MATLAB (ode23s and ode15i solvers) to include dynamics (Appendix 5.8).

Figure 5.16 shows the dynamic response for actuation with BPSK. The BSPK signal,

𝑣2, and the carrier signal, 𝑣1 used are described in (5.55) and (5.56).

𝑣2 = 7.6𝑠𝑔𝑛(sin(2𝜋(3300𝑡)) cos (2𝜋(174000𝑡)) (5.55)

and

𝑣1 = 7.6cos (2𝜋(174000𝑡)) (5.56)

These voltage signals give an out of phase ∆𝑉 of 15.2 𝑉𝑝𝑘 (10.75 𝑉𝑟𝑚𝑠). Figure 5.16a

gives the resulting rotor displacement (solid line) for one cycle of 𝑓𝑑 . The dashed line

is ∆𝑉 for one cycle. BPSK and carrier are out of phase for the first 0.15 ms and

in-phase from 0.15 ms to 0.3 ms.



Figure 5.16: Response from DE model

Figure 5.16b is the velocity - displacement phase portrait for 0.3 ms, a one on/off

cycle. It can be seen that when signals are out of phase, equilibrium is shifted to

0.5 𝜇𝑚. This simulation had a carrier frequency, 𝑓𝑐 , of 174 kHz which is not

completely out-of-band with respect to the mechanical bandwidth and this

manifests itself as a visible ripple on the actuation cycle in Figure 5.16b. In Figure

5.16c, the change in sensing capacitances in both sensing combs is shown and Figure

5.16d shows the output voltage, 𝑣𝑝, which carries the ASK signal with a modulation

index 𝑀 = 0.83 from sensor 1 (S1).

Figure 5.17 shows how the pull-in voltage changes for 1.5 ≤ 𝑛𝑎 ≤ 3.5 in steps of 0.5;

the design point selected, 𝑛𝑎 = 3, is the solid line. As can be seen, pull-in occurs at

𝑥 = 0.64 𝜇𝑚 (≈ 𝑔𝑎/3) with 10.95 Vrms. This means that with 10.75 Vrms, the

simulated response shown in Figure 5.17 is very close to pull-in.

Actuation

Release



Figure 5.17: Displacement (until pull-in) vs. actuation voltage for increasing na

Figure 5.18a shows the complete SOI layer for one device with all the polysilicon

tracks for connections. Figure 5.18b shows the complete layout of the IC with each

IC having six devices. The bottom three devices had an alternative anchoring

solution which failed to release completely the rotor and could not be used.

Figure 5.18c is a close up of the rotor showing details of the gold layer which was

used to improve the conductivity of the polysilicon tracks. The trench etching can

also be seen in blue.

Increasing n

Increasing n

Pull-In Markers

Increasing n

Simulation at 10.75 Vrms

Increasing n



Figure 5.18: Final layout showing SOI layer and connections

Note: a) One device having total area of 2.9 mm2, b) The whole IC layout with 6 devices and c)

Close-up of one rotor

a) b) Top

Bottom

c)



5.6 Experimental Validation

The design made use of geometric features that are at the limits of the SOIMUMPs

process capabilities. The process constraints that limited reducing further the

power consumption and increasing further the data rate are related to the maximum

feature length and minimum feature size as given in Table 2.2 in the SOIMUMPs

handbook [24]. This constraint gives unlimited feature length for feature sizes

greater than 6 𝜇m while limiting the feature length to a maximum of 100 𝜇m for the

minimum feature size allowable of 2 𝜇m. Since finger lengths and widths were at

these limits and spring width was slightly greater than 6 𝜇m, it was imperative that

thorough visual inspection and dimensional measurements were performed. Actual

sizing of the device could be obtained from these measurements and simulation

response with these manufactured dimensions would be compared with

experimental response.

5.6.1 Geometric and Capacitive Measurements

The manufactured device was found to have two measurements which were at the

lowest of the process tolerance range. The finger widths were found to be smaller

than designed by 0.5 𝜇𝑚 having an average width of 1.5 𝜇𝑚. This resulted in larger

comb gaps at 2.5 𝜇𝑚 (larger by 25%) which gave 𝑛 = 2.6 (still > 5/3). The springs

cantilever width 𝑤 was also found smaller at 8.5 𝜇𝑚 instead of 9 𝜇𝑚.

The experimental setup is shown in Figure 5.19 and device microphotograph in

Figure 5.20, while Figure 5.21 and Figure 5.22 show scanning electron microscope

photographs with widths of cantilever spring and fingers respectively.



Figure 5.19: Experimental setup

MEMS Electrically Grounded Using

Carbon Tape S1 S2

v1

v2



Figure 5.20: Device microphotograph and laser profilometry on comb

Figure 5.21: SEM photograph showing cantilever spring width at 8.5 µm



Figure 5.22: SEM photograph showing comb gap of 2.55 µm

Table 5.10 shows the manufactured dimensions (shaded found different from

designed). The finger lengths (2𝑐𝑎, 2𝑐𝑠), octagon radius (𝑟) and spring length (𝑞)

were found to be as designed.

Table 5.10: The manufactured dimensions in (µm) – as measured

Manufactured

Dimensions

𝒈𝒂 𝒏𝒂 𝒄𝒂 𝒈𝒔 𝒏𝒔 𝒄𝒔 𝒓 𝒒 𝒘

2.5 2.6 50 2.5 2.6 50 598 458 8.5

Capacitance measurements were performed with an LCR meter (Agilent E4980A) at

2 MHz with 256-point averaging. Measurements were repeatable to within 1 fF.

Figure 5.23 gives the measured values of the three capacitances, one for actuation

𝐶𝑎 (between 𝑣1rotor comb and 𝑣2 stator combs), and the two for sensing,

𝐶𝑠1(between 𝑣1rotor comb and 𝑆1 stator comb) and 𝐶𝑠2(between 𝑣1rotor comb and

𝑆2 stator comb). It also shows the parasitics between the three stators which are

negligible.



Figure 5.23: Capacitance measurements between each electrode

Simulations with these (manufactured) dimensions were run to evaluate the

expected behaviour. In simulation, with the manufactured dimensions, the sensing

capacitances 𝐶𝑠1 and 𝐶𝑠2 at no-displacement gave 945 fF with 𝐶𝑎 at 1890 fF. These

had a 5% error when compared to the measured values (Figure 5.23). This was

investigated further and it was found that the rotor, which is suspended from the

four springs, was sagging such that the rotor fingers were on average 1.4 𝜇𝑚 below

the plane of the stator combs which reduced the effective area. A laser profilometer

(Sensofar S neox 3D optical profiler), with an accuracy set at

0.1 𝜇𝑚 was used to determine this sag. Figure 5.20 shows a false colour image

captured with the profilometer. In this image the blue colour is 2 𝜇𝑚 below the red

colour.

The mathematical model was updated such that the SOI thickness for electrostatic

modelling only reflected this new average overlap thickness of 23.6 𝜇𝑚 instead of

25 𝜇𝑚 and the model gave 900 fF for sensing and 1800 fF for actuation. This

confirmed once again the accuracy of the fringe field model selected (5.42). Once the

mathematical model was in agreement with the no-displacement capacitances, the

device was statically actuated for a range of voltages while at the same time the



sensing capacitances were measured. These measurements are shown in

Figure 5.24.

Figure 5.24: Actual measurements vs. linear and cubic stiffness for CS1 and CS2.

Linearity, 𝐶𝑙𝑖𝑛, was still satisfactory at less than 0.01, however, the gradient of

𝐶𝑠(Δ𝑉) and consequently ∆𝐶𝑠1 was smaller due to the larger gaps.

With the actual dimensions, pull-in occurred at 13.00 V in simulation and at 13.78 V

in the lab. Figure 5.25 shows close-up on the sensing comb with optical microscope

under different actuation voltages with the one in Figure 5.25d being almost at

pull-in.

a) 0 volts: smaller gap = 2.5 𝜇𝑚

b) 4.6 volts: smaller gap = 2.3 𝜇𝑚



c) 9.2 volts: smaller gap = 2.0 𝜇𝑚

d) 13.7 volts: smaller gap = 1.7 𝜇𝑚

Figure 5.25: Optical microscope images showing comb gap change for increasing voltage

5.6.2 Transient and Modulation Index Measurements

As regards transient characteristics and the damping model adopted in simulation

(with the manufactured dimensions), a Q-factor of 3 was obtained. This agreed with

the transient characteristics observed in the lab (Figure 5.26a). The transient has a

settling time (2% criterion) of 150 𝜇𝑠. Figure 5.26b shows inputs 𝑣1 (the carrier

signal - (5.4)) in blue and 𝑣2 (the BPSK signal - (5.8)) in yellow transitioning on the

green digital signal. Their amplitudes are of 6 V, giving an effective peak when in

anti-phase of 12 V which in turn is equivalent to ∆𝑉𝑟𝑚𝑠 = 8.4 V. The optimal value

for load resistance, 𝑅𝑚, with the manufactured dimensions was found to be 1 MΩ in

simulation (larger than 700 kΩ -

Table 5.9) and this resistive load was adopted for experimentation. Figure 5.26c

shows the resulting output ASK signal, 𝑣𝑝, which signal shows the transient

(encircled) and has a modulation index, 𝑀, of 0.96. This transient/settling time

resulted in the possibility to convert at 3.3 kHz (6.6 kbps) and possibly higher.



Figure 5.26: Experimental measurement of transient and its superposition on output ASK

Figure 5.27a gives the simulated vs. lab measurements readings of the modulation

index for increasing actuation voltage ∆𝑉. There is a slight mismatch in modulation

index between simulations and experimental values at the higher voltage levels.

This is related to the mismatch in experimental and simulation pull-in voltage. The

last experimental point at pull-in (represented with a cross) could not be measured.

Figure 5.27b shows the output ASK signal from S1 having a modulation index

M = 0.96, for ∆𝑉𝑅𝑀𝑆 = 8.4 V at a data rate of 3.3 kHz (6.6 kbps) and a carrier

frequency 𝑓𝑐 of 174 kHz. An 𝐿𝑚 of 3.3 mH was used to mitigate the effect of the probe

and coaxial cable parasitic capacitances, 𝐶𝑝, of 260 pF. These resonate at the carrier

frequency of 174 kHz = 1/2𝜋√𝐿𝑚𝐶𝑝.



a) b)

Figure 5.27: a) Solid line is simulation, points are experimental b) ASK output signal for

∆𝑉𝑅𝑀𝑆 = 8.4 V - experimental

Table 5.11 gives the actual (as manufactured) specifications. When these are

compared to the design specifications achievable with the optimal dimensions

(

Table 5.9), it is clear that there is some degradation in performance due to the finger

and spring widths not being exactly as designed. Nonetheless, the manufactured

device’s specifications are still within the original design specifications (Table 5.6).

This is attributed to the optimisation refinement step that chose a solution that was

broadly optimal to minimise the sensitivity to manufacturing tolerances.

Table 5.11: The actual (as manufactured) device specifications

Device

Specifications

𝑪𝒂 𝒇𝒅 𝑴 𝒙𝒑𝒊 ∆𝑽𝒑𝒊 ∆𝑪𝒔𝟏 𝒗𝒑 𝑪𝒍𝒊𝒏

1.8 pF 3.3 kHz 0.79 0.32ga 13.78 V 50 fF 0.5 V 0.0098

A metric that can be used to evaluate the accuracy of the model would need to

capture both the static and dynamic characteristics that have a direct effect on the

design specifications. The cubic spring stiffness, the actuation and sensing

capacitances and their change upon rotor movement together with the fringe



capacitance have a direct effect on the modulation index, 𝑀. In view of this, 𝑀 was

chosen to form part of the fitness metric. The dynamics of the model have a direct

effect on the achievable data rate due to the settling time, 𝑡𝑠. On a physical level, this

depends on the damping and damping model adopted. Taking this into account, a

simple metric that is still capable of capturing the overall complex behaviour can be

defined as in (5.57),

𝐹 = √∑(𝑀𝑒𝑖−𝑀𝑠𝑖

)2 /𝑁 + 𝑌(𝑡𝑠𝑒 − 𝑡𝑠𝑠)

2𝑁

𝑖=1

(5.57)

where the suffixes ‘e’ and ‘s’ denote experimental and simulated respectively. The

first term is the summation of the ‘error squared’ in 𝑀 values shown in Figure 5.27a

(with N being the number of points) and the second term is the ‘error squared’ of

the settling time. For standardisation purposes (weighing), the correct value of 𝑌

would need to be ( 𝑡⁄ )2 ,that is, ratio of averages squared. This is a standardised

Euclidean distance metric, hence, the smaller, the better the accuracy of the model.

This metric was evaluated on the actual dimensions found on the prototype device

(Table 5.11) and gave a value of F = 0.0163 which indicates a very small error

between simulation and experimental results.

5.6.3 Device Power Consumption

The manufactured device’s power consumption was measured and compared with

simulation results. For simulation purposes, the total dissipative power was

obtained by adding the electrical average power calculated from (5.39) and the

mechanical average power by integrating the damping force term in (5.35).

The current going into the actuation combs, 𝑖𝑎, was measured using a

transimpedance amplifier (OP275GP op-amp) with a 100 kΩ feedback resistor, 𝑅𝑓 .

Readings for power consumption for different data rates at a carrier frequency of



174 kHz were taken and compared to results in simulation. The measurement setup

is shown in Figure 5.28.

Figure 5.28: Actuation current measurement setup

Figure 5.29: Current and power consumption for ∆𝑉𝑟𝑚𝑠 = 7.3 V and 𝑓𝑐 = 174 kHz.



Figure 5.30: Current and power consumption for ∆𝑉𝑟𝑚𝑠 = 13 V and 𝑓𝑐 = 174 kHz.

As can be seen in Figure 5.29 and Figure 5.30, for these experiments, data rates up

to 6 kHz (12 kbps) (above the design point of 3.3 kHz (6.6 kbps)) were tested. Going

beyond the 3.3 kHz meant that the mechanical dynamics would not have enough

time to reach the steady state. However, the output ASK signal was not compromised

completely and at high data rates (up to 12 kbps) the two output ASK levels could

still be easily distinguished even though the transient would not have completely

settled.

The mathematical model gives a linear relationship for data rates up to around

1.5 kHz (3 kbps) for both actuation voltages considered. Up to 1.5 kHz, when

∆𝑉𝑟𝑚𝑠= 7.3 V, the dissipative power follows a gradient of 373 nW/kHz and when

∆𝑉𝑟𝑚𝑠= 13 V, the dissipative power follows a gradient of 1200 nW/kHz. For both

levels of actuation voltage, the intercept of this linear section (at the lowest data

rate) is the power dissipated across the isolation resistance 𝑅𝑖. The dotted line in

cyan represents the power dissipated electrically in 𝑅𝑖; this power dissipation is

constant for all data rates. The black dotted line represents the power dissipated



due to mechanical damping. This increases as the data rate increases. if and only if

the displacement settles completely within the time of 1-bit.

In simulation, the total power dissipated was obtained by adding the electrical and

mechanical dissipation. The simulation total however, does not keep increasing at

the same rate of the solid black line which depicts the total power dissipation for the

linear case. This is because when the data rate exceeds 2 kHz, the settling time (2%

criterion) of the mechanical transient is less than the time of 1-bit. At around

𝑓𝑑 = 2 kHz, the data bit changes state before steady state is reached and for data rates

above this point, linearity starts to break down. This explains the deviation from

linearity (black solid line) of both the simulated (blue line) and the measured (red

asterisk) power dissipation.

This can be explored in more detail by looking at the velocity squared signal in

Figure 5.31 to Figure 5.33. In Figure 5.31 and Figure 5.32, mechanical displacement

settles completely in the time for 1-bit and proportionality in the mechanical power

dissipation (area per second) is maintained. However, at 6 kHz (Figure 5.33) the

input data rate is high and mechanical displacement does not settle within the time

for 1-bit, hence, giving a larger power dissipation albeit at a smaller rate of increase.





Figure 5.33: Velocity Squared Signal for a 6 kHz data rate and 13 V RMS actuation

Moreover, for 7.3 V < ∆𝑉𝑟𝑚𝑠 < 13 V, the power was also found to be linear with

∆𝑉𝑟𝑚𝑠 and hence the ratio of 100 nW/kHz/V of ∆𝑉𝑟𝑚𝑠 at 𝑓𝑐 = 174 kHz can describe

power consumption for data rates below 1.5 kHz (3 kbps). At higher data rates, up

to 6 kHz (12 kbps), this ratio is lower at around 64 nW/kHz/V.

Figure 5.34 shows oscilloscope signals for an actuation signal of ∆𝑉𝑟𝑚𝑠 = 7.3 V. The

data signal (data rate of 6 kbps) is shown in red for reference. The blue signal is the

current, 𝑖𝑎, while the green signal is the same 𝑖𝑎 averaged over several cycles. The



power signal is shown in purple. This signal has a non-zero average which gives the

total dissipative power of the converter.

Figure 5.34: The actuation current (blue), average current (green) and power (purple)

5.7 Conclusions

With this prototype, it was shown that a BPSK signal can be converted to an ASK

signal with a mechanical structure. Only two external electrical passive components

were required to obtain measurable signal levels: an inductor and a load resistor.

The mathematical model developed included several nonlinearities. The overall

static and dynamic behaviour of the mathematical model was tested in simulation

and validated with FEA results. The same mathematical model was used to find

dimensions that satisfied design constraints using a hybrid optimisation technique.

Actual measurements on the manufactured device were found to be in very good

agreement with the simulation results. This is attributed to the rigorous analysis

involving the nonlinear spring design and model, the inclusion of fringe capacitances

in the comb fingers and the estimation of the viscous damping coefficient that

included squeeze film effects with the effective viscosity.



The best (smallest) modulation index for the ASK output signal was of 0.79 (with

0.77 as designed). This was achieved with carrier and BPSK signals at 9.7 volts peak

(∆𝑉𝑟𝑚𝑠 = 13.78) and conversion was successful with data rates higher than the

design point of 𝑓𝑑 = 3.3 kHz (6.6 kbps), and carriers ranging from 100 kHz to 1 MHz.

At the design point, the power consumption follows the ratio of 64 nW/kHz/V of

∆𝑉𝑟𝑚𝑠 at 𝑓𝑐 = 174 kHz which gives 2.9 𝜇W for a data rate of 6.6 kbps at the best

modulation index. The device footprint is only 2.9 mm2. This is far smaller than the

device described in [93] which had an area of 49 mm2. However, in [93], both the

data rate and carrier frequencies where higher.

Conclusions and Further Work


6. CONCLUSIONS AND

FURTHER WORK

Traditionally, device designers steer clear of nonlinear behaviour, however, recently

one can find several devices that make use of nonlinear dynamics that are under

development. In this work, the complexity of interactions between static and

dynamic nonlinear behaviour was modelled using DEs or more generally DAEs.

DAEs were shown to be able to capture all the required physical phenomena

together with the geometric relationships and process constraints.

In the presented designs, several nonlinear phenomena and interactions were

modelled. These included nonlinearities in the drive, sense, damping, fringe

capacitance and stiffness, all of which can be broadly categorised as either

external/field or geometric in nature.

Electrostatic differential drive and displacement sensing through differential

capacitance were two nonlinear phenomena that were exploited for the RF designs.

These provided successful demodulation/mixing and also spurious product

suppression. Mixing using electrostatics is well reported in literature. However,

making use of a differential drive on a torsional plate to suppress unwanted

frequency components and giving almost pure mixing without the use of filtering is

novel. This relaxes design constraints on the mechanical filtering aspect.

One nonlinearity that was exploited throughout all of the presented designs

(including the VEH) was the nonlinear spring stiffness.



6.1 Torsional Plate in MetalMUMPs

As investigated in Section 3.1, at low actuation voltages, the torsional vibratory plate

can be used as a filter and/or a mixer or even as a BPSK demodulator. Target

applications for such a device capable of performing BPSK demodulation would be

low-rate, wireless personal area networks (LR-WPANs) such as those described in

the IEEE 802.15.4 standard onto which the ZigBee standard is based. The designed

MEMS is able to mix the Radio Frequency (RF) and Local Oscillator (LO) signals

electrostatically and also filter this mixed signal prior to electrostatic sensing. The

use of a torsional vibration structure for BPSK demodulation is innovative. It was

demonstrated that this structure had an undamped resonant frequency of 2 MHz

and can successfully demodulate a low data rate BPSK signal with a carrier

frequency of 868 MHz and a chip rate of 300 kchips/s.

For higher actuation voltages (> 75 V), the device exhibits bistability and it has

potential for applications involving hardware random number generation (RNG) for

crypto systems [141] and also as chaotic carrier generators for secure

communications [142]. Although the response in this region is chaotic, it is not, as

defined in [132], extensively chaotic. As a result, the distribution in Figure 6.1 is not

uniform.

Figure 6.1 Time series and histogram for 468 kHz and Vdc =100 V



For RNG, a uniform distribution is required. This means that unless the sampling

frequency is taken down to below resonance, the time series would still have some

periodic content. Another potential application for operation in region 2 would be

for energy harvesting. In [161], it is shown that having an energy harvester

operating in chaotic mode is more effective due to the broad bandwidth of the

dynamics. Such a device could be used for a broader range of functions as listed in

Table 6.1.

Table 6.1: Broadening of functionality by harnessing cubic nonlinearity

Function Behaviour

Filter and/or Mixer Monostable

BPSK demodulator Monostable

Hardware Random Number Generator Bistable/Chaotic

Chaotic carrier generator Bistable/Chaotic

Energy Harvesting Bistable/Chaotic

Sensing of wear/parameter changes Bistable/Chaotic

The same plate had its input electrode configuration changed such that a differential

drive could be used. This suppressed the unwanted products to create a torque

which was almost proportional to pure mixing. To clarify further, with input signals

having frequencies 𝑓1 and 𝑓2, it is evident that if only one side of the drive was used,

that is, (3.13), filtering out content at 2𝑓1 (Figure 6.2) by centering the mechanical

filter on 𝑓𝑚 = 𝑓2 − 𝑓1 would have been very difficult unless the Q factor is very high.



Figure 6.2: Torque frequency components arising from the -v1 (dotted) and v1 (solid) pads.

Upon subtraction, the pairs in the dotted boxes cancel out, f2/f1 is at 3/10. Note that

when 3𝑓1 ≈ 𝑓2 the required content at 𝑓𝑚 = 𝑓2 − 𝑓1 ≈ 2𝑓1. In [133] and [135], the

electrostatic drive does not intrinsically cancel the content at 2𝑓1 and 2𝑓2. As a result,

in [133], the device had to be operated in vacuum to increase the Q factor to 786,

impinging negatively on the bandwidth and increasing costs. Another benefit of

having eliminated 2𝑓1 and 2𝑓2 before filtering is that ratios of 𝑓2/𝑓1 close to 1/3 can

be used as well. This means that for downconversion of an incoming RF signal at 𝑓2,

one needs only select 𝑓1 based upon 𝑓𝑟 = 𝑓𝑚 = 𝑓2 − 𝑓1, relaxing the constraint on

𝑓2/𝑓1.

The IQ mixer described in Section 3.2.3 took advantage of the differential drive and

spurious product suppression, as a result, the results add significant improvement

to the BPSK demodulator described in Section 3.1, in several ways:

a) The BPSK demodulator dealt only with coherent demodulation while the IQ

mixer is applicable for general QAM demodulation and does not require a

fixed phase difference.

b) By using a differential drive, the mechanical Q-factor-bandwidth constraint

was relaxed and as shown in Figure 6.2 provides flexibility in design choices.

c) Differential sensing was implemented on the same plate and hence the

possibility of having mismatch between the differential sensing pads was

minimised.



6.2 Buckling Spring for Broadband Vibrational Energy Harvester

The VEH designed in SINTEF made use of a buckling/bistable spring which required

a quintic stiffness model. It was shown that due to this nonlinear stiffness, the device

could be driven in the chaotic regime making it responsive to a broad range of base

excitation frequencies. In general, smaller VEHs result in higher resonant

frequencies, frequencies which are higher than what is naturally available as

excitation frequencies. However, with a bistable VEH, the structure was responsive

to low frequencies (simulated at 560 Hz), frequencies below resonance (1.06 kHz

and 1.5 kHz), making the MEMS VEH a viable option.

Simulations to verify broadband sensitivity were performed for a wide range of

excitation levels, frequencies and loads but this was always done whilst keeping

away from 𝑦𝑖𝑖 (the unstable equilibrium point) such that snap-through is avoided.

Confirming the validity when snap-through occurs is more challenging due to the

sensitivity to initial conditions and to the computationally intensive FEA simulations

required. Even though direct validation near the unstable equilibrium point were

not performed, confidence in the accuracy of the mathematical model was still

obtained since having modelled accurately the direction field in phase-space

(1 − 𝑦1), this model would be able to give reasonable predictions for trajectories

passing close to 𝑦𝑖𝑖.

Using this mathematical model, as implemented in MATLAB, both the 𝐹𝑠 − 𝑦1curve

(static) and the harmonic orbits are generated in seconds. For comparison, FEA

simulation to obtain the 𝐹𝑠 − 𝑦1curve took around 96 hours (on an 8–core, i7,

2.6 GHz machine with 32 Gb of RAM), and generated 60 Gb of data. On the same

machine, FEA simulation for a harmonic orbit close to a stable equilibrium point

takes around 2 hours. If the design requirements are changed, the FEA simulation

would require a full redesign and re-run while with this model, a whole range of

options can be simulated in seconds.



6.3 BPSK to ASK conversion in MEMS

In the final design involving the BPSK to ASK converter, the spring was designed to

have a controlled amount of axial stiffness and this, together with the transverse

component, also resulted in a net cubic stiffness spring. In this design, the cubic

stiffness was not introduced with particular dynamics in mind but it was used to

linearize the overall static response/gain characteristics related to the output ASK

modulation index. Although counter intuitive, linearizing the static behaviour using

a nonlinear spring was shown to be effective.

Although nonlinear design was shown to be of benefit, tackling all the

considerations inherent in device design, such as fabrication tolerances and design

robustness, was a challenge. This was overcome by having a mathematical model

with which the design optimisation process could be hastened. A MATLAB model

was also developed, validated with CoventorWare®’s FEA results and this enabled

the MATLAB model to be used to iterate through design improvements quickly.

In particular for this design, the mathematical model in MATLAB which had around

15 independent parameters and around 6 design constraints was run iteratively

using a hybrid PSO algorithm. The second step in the hybrid PSO technique sieved

through PSO solutions and gave one which was broadly optimal. This was

instrumental in manufacturing a device which was still functional even though it had

up to 25% error in some crucial dimensions.

Both the size and power consumption of the device make it a potential alternative

to BPSK-to-ASK CMOS realisations. All of the BPSK demodulators referenced in [88]

to [92], make use of a BPSK-to-ASK first stage and only one, [92], goes below a total

of 1 mW (631 𝜇W) of power consumption. Moreover, only [92] gives a breakdown

of the power consumption per stage and the authors report that the BPSK-to-ASK

stage consumed 204 𝜇W. This was tested at 10 Mb/s of data rate. The MEMS BPSK

to ASK converter is not able to reach these high data rates primarily due to the ‘large’

rotor inertia 𝐽 (equation (5.34)) and due to this limitation, it was tested at 6.6 kbps.



However, the power consumption is two orders of magnitude lower (2.9 𝜇W) than

what is reported in [92] and for IMD applications, that are only about transmitting

‘command data packages’ as opposed to applications that require high data rates

(visual prosthesis), a data rate of 6.6 kbps is adequate. In [2], (Section 12.5.3) it was

reported that recently typical power consumption per telemetry channel for

implantable systems with a 1 kHz bandwidth is about 1 to 5 𝜇W for advanced

implantable systems. These specifications are within the range achieved with the

MEMS BPSK-to-ASK converter.

As a result, the designed MEMS BPSK-to-ASK converter satisfies data rates required

for low data rate IMD applications with an attractive low power consumption. This

offers a new ‘all-MEMS’ avenue for system level designers.

6.4 Further Work

Although the torsional plate for BPSK demodulation was investigated using FEA

simulations, no attempt was performed to model damping using squeezed film

equations (considered for the BPSK to ASK converter). Having a complete analytical

model allows for easier and faster dimensional optimisation which can potentially

result in meeting better functional specifications.

When the torsional plate was used for IQ mixing with differential sensing, the IQ

mixer still required two devices, one for the in-phase and one for the quadrature

paths. Further investigation should be performed to find ways to minimise potential

mismatch between the two devices possibly by using different biasing resistors. To

gain more confidence in the dynamic operation of the IQ mixer further studies

involving higher actuation voltages should be performed (similar to Section 3.1.8)

such that the limits for linear stiffness for this size of plate is determined.

The IQ mixer took advantage of differential actuation such that the unwanted

products are suppressed before mechanical filtering and relaxing the Q factor-

bandwidth trade-off in the process. The parameter that is critical to the degree of



suppression is 𝑛 = 𝑑/𝑥𝑖 . Having a complete analytical model (including damping) for

the torsional plate will make it possible to optimise to minimise 𝑛.

The vibrational energy harvester made use of a buckling spring. The VEH spring is

intended to buckle at frequencies in the range of 0.5 kHz. An in-depth study into

fatigue failure should be performed such that the lifetime of such a device is

investigated and maximised.

The torsional plate device in MetalMUMPs and the VEH in SINTEF were not

fabricated and the models were only validated against FEA simulations; fabrication

and experimental validation should be performed.

The methodology adopted to design the BPSK-to-ASK converter made use of some

automation in the optimisation process, however, the degree of automation could

be increased. Obtaining a robust design for a relatively complex MEMS structure

with a moderate degree of automation was achieved with a hybrid optimisation

approach which tested for broad optimality. Robustness can be broadly defined as

the insensitivity of the solution to process variability. This was achieved by sieving

through PSO solutions by using MATLAB’s fmincon and Multistart functions (to

generate start points) and looking at whether this optimisation step led to a cluster

of solutions or not. If this second optimisation step gave a sparse solution set, then,

it meant that it is robust to process variation. The effectiveness of this hybrid

optimisation technique was confirmed when the manufactured device was found to

have critical dimensions that were up to 25% away from nominal, but the device’s

specifications were still satisfied. The first optimisation step involved the use of PSO

techniques and this was selected, as it was reported in literature, that it is adequate

for relatively complex MEMS structures. One feature of the PSO implementation that

made it easier to obtain ‘good’ solutions in ‘reasonable’ time was the distinction

between constrained and unconstrained optimisation targets.



Further work should involve the development of an interface between MATLAB and

3D modelling software such that the final solution could be verified with FEA

seamlessly.

As for the mathematical model, rotor sagging should be investigated and included

such that the correct resultant capacitive area is obtained. At the centre of the rotor,

a square shaped structure was designed to reinforce the four radial support beams.

These four beams together with the square shaped central structure had the

purpose of eliminating unwanted planar modes of vibration and also to shift the

vertical mode of vibration (Figure 6.3) away from the required mode’s resonant

point. The dimensions for the central part of the rotor were arrived at by trial and

error. Alternative topologies, for the central rotor design that satisfy the mode

shifting requirements for unwanted modes and to minimise rotor sagging, should

be investigated. The design (shown at bottom of Figure 5.18b) which has an anchor

in the centre of the rotor that failed to release completely between the substrate and

fingers (encircled in Figure 6.4), can be modified to eliminate this problem. This will

in turn eliminate rotor sagging and the unwanted vertical mode shown in Figure 6.3.

Figure 6.3: Rotor central weight design: Wanted mode (left) and Unwanted mode (right)



Figure 6.4: The design that failed to release anchor supports (encircled)

Alternative functionalities for this device should also be tested to include:

a. ASK detection and OOK to ASK conversion which is desirable for NFC-B

applications.

b. Phase Modulation to Amplitude modulation - analogue conversion.

c. Possible operation under vacuum which could also be used for frequency

shifting or mixing.

d. A QPSK to 4-level ASK, however this requires two devices and a quarter

wavelength delay line for phase discrimination.

An investigation into the possibility of improving the data rate and reducing the

power consumption should be performed by relaxing SOIMUMPs minimum feature



size constraints. This of course would require an alternative fabrication process.

However, such an investigation would give an understanding of how this device

would perform if scaled down. Particular attention would need to be given to

understand how damping phenomena change upon scaling such that the noise

boundary is determined.

As for sensing, the test set-up (inductor 𝐿𝑚 and load 𝑅𝑚, Figure 5.7) was only

intended to provide a way to measure the response with ease. For practical use,

a BPSK-to-ASK convertor would precede an ASK demodulator, hence, it would need

to be interfaced with an ASK demodulator circuit and therefore satisfy the ASK

demodulator input specifications.

The BPSK-to-ASK convertor was able to give ASK with modulation index as low as

0.79, output voltage level of 0.5 V RMS, data rates up to 12 kbps with carrier

frequencies up to 1 MHz. In literature, one can find ASK demodulators that accept

input signal levels as low as 0.25 V [162] and modulation index as high as 0.9 [163],

with footprints ranging from 0.16 mm2 [163] to 0.29 mm2 [164]. All literature

reviewed on ASK demodulators [162], [163], [164], [165], [166] and [167] describe

implementations that are able to work with carriers at 1 MHz or lower and could

handle a data rate of 12 kbps. This means that current ASK demodulators

specifications are in line with what the BPSK-to-ASK convertor provides as output

with the adopted test set-up.

In practice, the probes and coaxial cables used for testing, and which are responsible

for the large parasitic capacitance 𝐶𝑝 of 260 pF, will not be there. This would also

change the required inductance for resonance. With a smaller parasitic capacitance

(in the region of 10 fF), the carrier frequency could be increased into the GHz range

while requiring an inductor in the nH range for resonance to cancel the parasitics.

This would decrease the footprint of the inductor required and is favourable for the

output specifications of the BPSK-to-ASK converter.

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NEURAL INTERFACING CHIPS,” IEEE Int. Symp. Circuits Syst., pp. IV-57., 2004.

Appendices


APPENDICES

APPENDIX 3.1 DYNAMICS SIMULATIONS – MATLAB SCRIPT ....................................................... 188

APPENDIX 3.2 EQUILIBRIUM POINTS – MATLAB SCRIPT ............................................................. 193

APPENDIX 3.3 SIMULINK IMPLEMENTATION OF IQ MIXER ............................................................. 196

APPENDIX 4.1 TRANSIENT RESPONSE - MATLAB SCRIPTS .......................................................... 200

APPENDIX 5.1 RESULTANT STIFFNESS .............................................................................................. 205

APPENDIX 5.2 EQUILIBRIA – MATLAB SCRIPT .............................................................................. 206

APPENDIX 5.3 - TOTAL INERTIA OF N/2 FINGERS ........................................................................... 208

APPENDIX 5.4 CHANGE IN FRINGE CAPACITANCE - MATLAB SCRIPT ......................................... 210

APPENDIX 5.5 MONOTONICITY IN SENSING ...................................................................................... 212

APPENDIX 5.6 MODULATION INDEX, N AND FRINGE CAPACITANCE - MATLAB SCRIPT............ 213

APPENDIX 5.7 PSO - MATLAB SCRIPTS ......................................................................................... 214

APPENDIX 5.8 DYNAMICS – INPUTS TO OUTPUT - MATLAB SCRIPTS .......................................... 221

Appendices


APPENDIX 3.1 DYNAMICS SIMULATIONS – MATLAB

SCRIPT

Main Program:

clear all; close all; w0=1.1407e+07; global wratio; wratio=0.258; %0.258%driving freq = wratio*w0 %0.4 same period,0.35

double period, initcond1=[-4.4815e-7 0.1469 1*w0]; %x x_dot startingomega %initcond1=[-0.003e-6 0 1*w0]; %x x_dot startingomega duration=0.0002; timestep=3.3333e-9; %3.3333e-11 [t,y1] = ode15s('tormixercubedrevb',[0:timestep:duration], initcond1);

initset1x=y1(:,1); initset1y=y1(:,2); initset1z=y1(:,3); initcond2=[-4.5e-7 0.1469 1*w0]; %x x_dot startingomega %initcond2=[0.0e-6 0 1*w0]; %x x_dot startingomega [t1,y2] = ode15s('tormixercubedrevb',[0:timestep:duration], in-

itcond2); initset2x=y2(:,1); initset2y=y2(:,2); initset2z=y2(:,3); % if length(initset1x)>length(initset2x) initset1x=initset1x(1:length(initset2x)) initset1y=initset1y(1:length(initset2y)) initset1z=initset1z(1:length(initset2z)) elseif length(initset1x)<length(initset2x) initset2x=initset1x(1:length(initset1x)) initset2y=initset1y(1:length(initset1y)) initset2z=initset1z(1:length(initset1z)) end %

%Lyapunov Exponent diffx=abs(initset1x-initset2x); logdiffx=log(diffx); diffy=abs(initset1y-initset2y); logdiffy=log(diffy); diffz=abs(initset1z-initset2z); logdiffz=log(diffz); subplot(221);plot(logdiffx); xlabel('t'); ylabel('LE1'); title('LE1 - x'); subplot(222);plot(logdiffy); xlabel('t'); ylabel('LE2'); title('LE2 - y'); subplot(223);plot(logdiffz);

Appendices


xlabel('t');

ylabel('LE3');

title('LE3 - z');

figure;

subplot(221)

plot(initcond1(1),initcond1(2),'r*');grid on;

hold on;

plot(initcond2(1),initcond2(2),'b*');grid on;

hold on;

plot(y1(:,1),y1(:,2),'r-');

hold on;

plot(y2(:,1),y2(:,2),'-b');

xlabel('x(t)');

ylabel('y(t)');

title('Phase Plane Portrait for tormixer -- y(t) vs. x(t)');

subplot(222)


hold on;

plot(initcond2(1),initcond2(3),'b*');grid on;

hold on;

plot(y1(:,1),y1(:,3),'r-');

hold on;

plot(y2(:,1),y2(:,3),'-');

xlabel('x(t)');

ylabel('z(t)');

title('Phase Plane Portrait for tormixer -- z(t) vs. x(t)');

subplot(223)

plot(initcond1(2),initcond1(3),'r*');

hold on;

plot(initcond2(2),initcond2(3),'b*');

hold on;

plot(y1(:,2),y1(:,3),'r-');grid on;

hold on;

plot(y2(:,2),y2(:,3),'-');grid on;

subplot(224)

plot(t,initset1x,'r-');grid on;

hold on;

plot(t,initset2x,':');grid on;

xlabel('t');

ylabel('x(t)');

figure;

plot3(y1(:,1)*1e6,y1(:,2),t*1e3,'r-');grid on;

hold on;

plot3(y2(:,1)*1e6,y2(:,2),t*1e3,'b-');grid on;

hold on;

plot3(initcond1(1)*1e6,initcond1(2),0/1e3,'r*');

hold on;

plot3(initcond2(1)*1e6,initcond2(2),0/1e6,'b*');

Appendices


xlabel('x(\mum)');

ylabel('$\dotx$(m/s)', 'Interpreter','latex');

% ylabel('xdot(m/s)');

zlabel('time (ms)');

title('a) 3D phase portrait');

totalsamples=duration/timestep;

%points=600; %should be related to suspected resonance

poincarestep=round(((2*pi)/(wratio*w0))/timestep);%ceil(totalsam-

ples/points);

points=ceil(totalsamples/poincarestep);

for n=1:points

current=n*poincarestep;

if (current<totalsamples)

xy1(n,:)=[initset1x(current),initset1y(current)];

xy2(n,:)=[initset2x(current),initset2y(current)];

end

end

figure;

%remove transient (9 points always?)

xy1=xy1([9:points-1],:);

xy2=xy2([9:points-1],:);

scatter(xy1(:,1),xy1(:,2),'r.');

hold on;

scatter(xy2(:,1),xy2(:,2),'b.');hold on;

axis([-8e-7 8e-7 -0.8 0.8]);

grid on;

xlabel('x(m)');

ylabel('$\dotx$(m/s)', 'Interpreter','latex');

title('b) Poincare` Map')

plot(-2e-7,0,'Marker','o','MarkerFaceColor','black');

plot(2e-7,0,'Marker','o','MarkerFaceColor','black');

maxim=6.5e-7;

minim=-6.5e-7;

discr=ceil(255*(xy2(:,1) - minim*ones(1,length(xy2))')./((maxim-

minim)*ones(1,length(xy2))'))

[a,b]=hist(discr,128);

a=a/256;

divide=log(a)/(log(256));

divide(divide==-Inf)=100; %get rid of -Inf

entropy=sum(-a.*divide)

%

figure;

for all=1:30:length(y1)

plot(y1(all,1),y1(all,2),'r*');grid on;

axis([-1.5e-7 1.5e-7 -15 15]);

hold on;

plot(y2(all,1),y2(all,2),'b*');

Appendices


%axis([-1.5e-7 1.5e-7 -15 15])

hold on;

drawnow;

%pause(0.002);

clf

%plot(y1(all,1),y1(all,2),'w*');

%hold on;

%plot(y2(all,1),y2(all,2),'w*');

%hold on;

end

xlabel('x(t)');

ylabel('y(t)');

zlabel('z(t)');

title('3D phase portrait of tormixer');

%

figure;

plot(xcorr(initset1x,initset1x));

grid on;

xlabel('Samples');

figure;

hist(initset1x,256);xlabel('x(m)');

ylabel('Samples');

figure;

f=[0:(1/timestep)/(duration/timestep+1):(1/(2*timestep))];

freq=fft(initset1x);

plot(f/1000,(abs(freq(1:length(freq)/2+1,:))));grid on;

ylabel('Displacement Magnitude');

xlabel('Frequency (kHz)');

title('c) Frequency Magnitude Spectrum for Displacement')

xaxis([0 14e2]);

Appendices


The system of DEs function: ‘tormixercubedrevb.m’:

function dy = tormixercubedrevb(t,y)

global wratio;

dy = zeros(3,1);

w0=1.1407e+07;

I = 1.716e-21;

r = 10e-6;

b = 7e-16; %8e-16

w=wratio*w0; %0.24 %possible candidates

kt=1.0e-9;

k3=1.9e4;

E=1.26*8.854e-12;

A=250e-12;

V1=100; %2

V2=100; %2

d=1.45e-6;

gamma=(V1^2)/((2*kt)/(E*r));

dy(1) = y(2);

dy(2) = -(b/I)*y(2)-(kt/I)*y(1)-(k3/I)*y(1)^3+(E*A*(r^2)/(2*I))*(-

((V1+100*cos(y(3)))/(d+y(1)))^2+(V2/(d-y(1)))^2);

dy(3) = w;

Appendices


APPENDIX 3.2 EQUILIBRIUM POINTS – MATLAB SCRIPT

clear all;

close all;

I = 2.713e-21;%8.116e-22;%2.713e-21;%

r = 10e-6;

b = 5e-16;

%w=7.6045e6;

kt=1.0e-9;

k3=1.9e4;

E=1.26*8.854e-12;

A=250e-12;

V1=0.4;

V2=5;

d=1.45e-6;

posV2=0;

posV1=0;

allreal=[0,0];

for V2=0:0.01:1

posV2=posV2+1;

posV1=0;

for V1=0:0.01:1

posV1=posV1+1;

V1c=V1;

V2c=V2;

gamma=(2)/((r^2) * E*A);

a7=gamma*k3;

a6=0;

a5=kt*gamma-(2*(d^2)*gamma*k3);

a4=0;

a3=(d^2)*gamma*((d^2)*k3-2*kt);

a2=V1c^2 -V2c^2;

a1=d*(kt*gamma*d^3 - 2*V1c^2-2*V2c^2);

a0=d^2*(V1c^2-V2c^2);

x=roots([a7 a6 a5 a4 a3 a2 a1 a0])

trueall=0;

for allroots=1:length(x)

if (abs(imag(x(allroots)))==0)&&(abs(real(x(allroots)))<1.45e-6)

%real equilibrium point

trueall=trueall+1;

end

end

if trueall==3

allreal(length(allreal)+1,:)=[V1c,V2c];

end

%Check type of equilibrium point

stablerealwithin=NaN;

unstablerealwithin=NaN;

for allroots=1:length(x)

Appendices


if (abs(imag(x(allroots)))==0) %real equilibrium point

dFdx=-kt+(r^2)*E*A*((V2c^2)/(d-x(allroots))^3 +

(V1c^2)/((d+x(allroots))^3));

if (dFdx<0)

sprintf('Eq. point @ %g is stable',x(allroots))

if (abs(x(allroots))<1.45e-6)

stablerealwithin=x(allroots);

end

elseif (dFdx>0)

sprintf('Eq. point @ %g is unstable',x(allroots))

else

sprintf('Eq. point @ %g is a saddle',x(allroots))

end

end

end

V1axis(posV1,posV2)=V1;

V2axis(posV1,posV2)=V2;

Xstar(posV1,posV2)=stablerealwithin;

end

end

%

subplot(1,2,1);mesh(V1axis,V2axis,Xstar);grid on;

subplot(1,2,2);

contour(V1axis,V2axis,Xstar,500);grid on;

set(gca,'DataAspectRatio',[1 1 1]);

figure;

%

xdist=[-5e-6:0.01e-6:5e-6];

step=0;

start=0;

last=230;

res=1;

steps=(last-start)/res;

eqpoint=NaN(steps,7);

eqtype=NaN(steps,7);

for V2=start:res:last

step=step+1;

%posV2=posV2+1;

%posV1=0;

%for V1=5:2:300

%posV1=posV1+1;

%V1c=V1;

V2c=V2;

V1c=V2;

Appendices


gamma=(2)/((r^2) * E*A); a7=gamma*k3; a6=0; a5=kt*gamma-(2*(d^2)*gamma*k3); a4=0; a3=(d^2)*gamma*((d^2)*k3-2*kt); a2=V1c^2 -V2c^2; a1=d*(kt*gamma*d^3 - 2*V1c^2-2*V2c^2); a0=d^2*(V1c^2-V2c^2);

fx=a7*xdist.^7+a6*xdist.^6+a5*xdist.^5+a4*xdist.^4+a3*xdist.^3+a2*xdis

t.^2+a1*xdist+a0; allroots=roots([a7 a6 a5 a4 a3 a2 a1 a0]); % clf; for y=1:length(allroots) if (abs(imag(allroots(y)))==0)&&(abs(allroots(y))<1.45e-6) %%removes

eq points outside possible region dfdx=-7*a7*allroots(y)^6-6*a6*allroots(y)^5-5*a5*allroots(y)^4-

4*a4*allroots(y)^3-3*a3*allroots(y)^2-2*a2*allroots(y)-a1; if dfdx>=0 % plot(allroots(y),0,'ro'); %red=unstable=1 eqpoint(step,y)=allroots(y); eqtype(step,y)=1; else % plot(allroots(y),0,'go'); %green=stable=0 eqpoint(step,y)=allroots(y); eqtype(step,y)=0; end hold on; end end end figure; for h=1:steps for t=1:7 if eqtype(h,t)==1

scatter([h*res+start],eqpoint(h,t),13.9,'k.','MarkerFaceColor','k');ho

ld on;grid on; elseif eqtype(h,t)==0

scatter([h*res+start],eqpoint(h,t),30,'ks','MarkerFaceColor','k');hold

on;grid on; end end end axis([0 200 -1.5e-6 1.5e-6]); xlabel('V_d_c (V)'); ylabel('x (m)'); title('\itEquilibria positions v.s. Biasing Voltages','FontSize',12) %end

Appendices


APPENDIX 3.3 SIMULINK IMPLEMENTATION OF IQ MIXER

Figure 0.1: The overall SIMULINK block setup

Figure 0.2: The modulator block

Appendices


Figure 0.3: The MEMS block

Figure 0.4: Electrostatics ‘I’ in MEMS block

Figure 0.5: Electrostatics ‘Q’ in MEMS block

Appendices


Figure 0.6: Plate Dynamics Angle in MEMS block

Figure 0.7: Plate angle to delta Cn block in MEMS block

Figure 0.8: Sensing side UP block in MEMS block

Appendices


Figure 0.9: ADC and DSP block

Appendices


APPENDIX 4.1 TRANSIENT RESPONSE - MATLAB SCRIPTS

Main Program:

clear all; close all; global Amp; global w; global Rl; global M; PZTpermittivity=1600*8.854e-12; %F/m

M = 3.45e-6;%3.304945E-06; Rl=15.6e6; Amp=5*9.81; w=2*pi*560; %1350 and 10g snaps initcond1=[42.86e-6 0.00 0]; %x x_dot voltage

duration=0.015; timestep=1e-6; %3.3333e-11 options = odeset('RelTol',1e-15,'AbsTol',[1e-15 1e-15 1e-15]); [t,y1] = ode45('fullpztharmonic',[0:timestep:duration],

initcond1,options); initset1x=y1(:,1); %y1 initset1y=y1(:,2); %y1_dot initset1z=y1(:,3); %voltage e31=-4.1;%-4.1;%-4.1; %PZT specs as in FEA Ap=2*0.003*5e-6; As=5e-6*2e-6; tp=2e-6; y0=30e-6; l=0.003; mechelec=rms((abs((y0-initset1x)./(sqrt((l/2)^2+(y0-

initset1x).^2)))).*((As*e31/tp).*initset1z)) mech=rms(initset1x.*(1.38790871920078E+20*initset1x.^4-

2.13681849422879E+16*initset1x.^3+1.22217715950131E+12*initset1x.^2-

3.17475576007519E+7.*initset1x+3.27875225599993E+2))

figure; subplot(231); plot(-initcond1(1)*10^6,initcond1(2),'r*');grid on; hold on; plot(0,0,'r*');grid on; hold on; plot(-59,0,'r*');grid on; hold on; %plot(initcond2(1),initcond2(2),'b*');grid on; %hold on; plot(-y1(:,1)*10^6,y1(:,2),'r-'); %hold on;

Appendices


xlabel('y_1(um)'); ylabel('y_1_d_o_t'); title('Phase Portrait'); axis([-70 10 -0.15 0.15]);

subplot(232) plot(t,10^6*initset1x,'r-');grid on; xlabel('t'); ylabel('y1(um)'); subplot(233) plot(t,10^6*(initset1z/Rl),':');grid on; xlabel('t'); currentrmsuA=10^6*rms(initset1z/Rl) ylabel('i(uA)'); str = sprintf('i= %f uA',currentrmsuA); title(str); subplot(234) plot(t,10^6*(initset1z.^2/Rl),':');grid on; xlabel('t'); powerrmsuW=10^6*rms(initset1z.^2/Rl) ylabel('P(uW)'); str = sprintf('P= %f uW',powerrmsuW); title(str); subplot(235) plot(t,10^6*initset1z,':');grid on; xlabel('t'); rmsvoltuV=10^6*rms(initset1z) ylabel('V(uV)'); str = sprintf('RMSvolt= %f uV',rmsvoltuV); title(str); Energy=trapz(initset1z.^2/Rl) n=100; figure; tita=(y0-initset1x)./(sqrt((l/2)^2+(y0-initset1x).^2)); %-y(1) to get

deacresing angle as y(1) increases (downward) subplot(511); plot(t(1:n),tita(1:n)); title('tita'); subplot(512); plot(t(1:n),initset1x(1:n)); title('y1'); subplot(513); plot(t(1:n),initset1y(1:n));

Appendices


title('y1dot');

subplot(514); plot(t(1:n-1),diff(initset1y(1:n))); title('y1dotdot');

forcemid=initset1x(1:n).*(1.38790871920078E+20*initset1x(1:n).^4-

2.13681849422879E+16*initset1x(1:n).^3+1.22217715950131E+12*initset1x(

1:n).^2-3.17475576007519E+7*initset1x(1:n)+3.27875225599993E+2); subplot(515); plot(t(1:n),forcemid(1:n)); title('midpoint force fd mechanical only');

figure subplot(511); plot(t(1:n),initset1z(1:n)); title('volt');

subplot(512); plot(t(1:n),initset1z(1:n).*tita(1:n).*(Ap*e31/tp)); title('force due volt');

K1=3.27875225599993E+2; K2=-3.17475576007519E+7; K3=1.22217715950131E+12; K4=-2.13681849422879E+16; K5=1.38790871920078E+20; StrainE=(K5/6).*(y1(:,1).^6)+(K4/5).*(y1(:,1).^5)+(K3/4).*(y1(:,1).^4)

+(K2/3).*(y1(:,1).^3)+(K1/2).*(y1(:,1).^2); KE=0.5*M.*y1(:,2).^2; Store=KE+StrainE; Diss=0; figure; subplot(2,2,1); plot(t,StrainE*1e9);grid on; xlabel('t'); ylabel('Strain Energy(nJ)'); subplot(2,2,2); plot(t,KE*1e9);grid on; xlabel('t'); ylabel('KE(nJ)'); subplot(2,2,3); plot(t,Store*1e9);grid on; xlabel('t'); ylabel('Total Stored E(nJ)'); subplot(2,2,4); Einput=0.5*M.*((Amp/w)*sin(w*t)).^2; plot(t,ones(1,length(t))*1e12*rms(Einput));grid on; xlabel('t'); ylabel('Input rms E (pJ)'); tita=abs((y0-(y1(:,1))))./(sqrt((l/2)^2+(y0-(y1(:,1).^2))))

Appendices


s0=(l+((pi*y0/2)^2)/l); speedS=((pi^2)/(2*l)).*(y1(:,1)+y0).*y1(:,2)/s0; Poutrms=2*1e6*rms((0.5*Rl*(e31*As/tp).^2.*speedS.^2)); figure; xlabel('y_1(um)'); ylabel('y_1_d_o_t'); title('Phase Portrait'); %axis([-70 10 -0.15 0.15]); Forced=1000*(y1(:,1).*(1.38790871920078E+20*y1(:,1).^4-

2.13681849422879E+16*y1(:,1).^3+1.22217715950131E+12*y1(:,1).^2-

3.17475576007519E+7*y1(:,1)+3.27875225599993E+2)) [ax,h1,h2]=plotyy(-y1(:,1)*10^6,y1(:,2),-y1(:,1)*10^6,Forced);grid on;

set(h1, 'linestyle', '-','color','blue') set(h2,'linestyle', '.','color','black') set(ax(1),'YLim',[-0.18 0.18]) set(ax(1),'YTick',[-0.18:0.06:0.18]) set(ax(2),'YLim',[-1.5 1.5]) set(ax(2),'YTick',[-1.5:0.5:1.5]) set(ax,'XLim',[-70 10]) axes(ax(1)); ylabel('y_1_d_o_t(m/s)'); axes(ax(2)); ylabel('Force(mN)'); hold on; plot(0,0,'r*');grid on; hold on; plot(-58.5,0,'r*');grid on; hold on; plot(-initcond1(1)*10^6,initcond1(2),'r*');grid on; xlabel('y_1(um)');

Appendices


The system of DEs function: ‘fullpztharmonic.m’:

function dy = fullpztharmonic(t,y)

dy = zeros(3,1);

global Amp;

global w;

global Rl;

global M;

B =0;%9e-19;%9e-19;%9e-03; %8e-16

Cp=1.72e-10;

e31=-4.1;%-4.1;%-4.1; %PZT specs as in FEA

Ap=2*0.003*5e-6; %2 beams times length l=3mm and beam thickness t=5um

As=5e-6*2e-6;

tp=2e-6; % piezo layer thickness

y0=30e-6;

l=0.003;

s0=(l+((pi*y0/2)^2)/l);

dy(1) = y(2); %positive y(1) means moving downwards from y0 -

according to F-d curve on excel

dy(2) = -(B/M)*y(2)-(y(1)*(1.38790871920078E+20*y(1)^4-

2.13681849422879E+16*y(1)^3+1.22217715950131E+12*y(1)^2-

3.17475576007519E+7*y(1)+3.27875225599993E+2))/M-(abs((y0-

y(1))/(sqrt((l/2)^2+(y0-y(1))^2))))*((As*e31/tp)*y(3))/M-Amp*cos(w*t);

dy(3) = (1/Cp)*(Ap*e31*((((pi^2)/(2*l))*(y(1)+y0)*y(2)))/s0-y(3)/Rl);

Appendices


APPENDIX 5.1 RESULTANT STIFFNESS

A similar approach to that found in [8] is adopted for this approximation. The

diagram shows the axial force, 𝐹𝑎 , the force due to beam extension and also the

transverse force, 𝐹𝑡𝑟𝑎𝑛𝑠, the force due to transverse bending. Assuming that 𝛾 is

small, 𝑞 + ∆𝑞 = √𝑞2 + (𝑟𝜃)2 i.e. ∆𝑞 = √𝑞2 + (𝑟𝜃)2 − 𝑞. The total vertical restoring

force, 𝐹𝑇𝑂𝑇 = 𝐹𝑎𝑠𝑖𝑛𝛾 + 𝐹𝑡 i.e. 𝐹𝑇𝑂𝑇 = 𝐹𝑎𝛾 + 𝐹𝑡 for small angles. Now 𝐹𝑡 = 𝑘𝑡(𝑟𝜃) and

𝐹𝑎 = 𝑘𝑎∆𝑞, where 𝑘𝑡 =12𝐸𝐼

𝑞3 and 𝑘𝑎 =

𝐸𝐴

𝑞 are the transverse and axial stiffnesses

respectively. Hence;

𝐹𝑇𝑂𝑇 = 𝐹𝑎𝛾 + 𝐹𝑡

𝐹𝑇𝑇 = 𝑘𝑎∆𝑞𝛾 + 𝑘𝑡(𝑟𝜃)

𝐹𝑇𝑂𝑇 = 𝑘𝑎(√𝑞2 + (𝑟𝜃)2 − 𝑞)𝛾 + 𝑘𝑡(𝑟𝜃)

And the expression (√𝑞2 + (𝑟𝜃)2 − 𝑞)𝛾 = √(𝑞𝛾)2 + (𝑟𝜃𝛾)2 − 𝑞𝛾 =

√(𝑟𝜃)2 + (𝑟𝜃)2𝛾2 − 𝑟𝜃 which simplifies to (𝑟𝜃)√1 + 𝛾2 − 1 and expanding using

Taylor’s series and taking the first term gives:

(𝑟𝜃)(𝑟𝜃)2

2𝑞2

since 𝛾 =𝑟𝜃

𝑞.

𝐹𝑎

𝐹𝑡𝑟𝑎𝑛𝑠

Appendices


APPENDIX 5.2 EQUILIBRIA – MATLAB SCRIPT

clear all; close all; for n=1.5:0.5:3.5 n eps=8.85e-12;A=2.25e-9; %g=2.5e-6; % actual g=2e-6; % as designed %netF=eps*A*(V1-V2).^2*(1/g^2 - 1/(n*g)^2)/2; %force per finger %plot(t,netF);grid on; N=71;r=598e-6;c=50e-6;l=458e-6; alpha=(67.5)*pi/180; a=r-l*cos(alpha); D=(a+c+(l/2)*cos(alpha)); % totalT=2*N*netF*D; % plot(t,totalT);grid on; E=1.69e11;t=25e-6;q=458e-6; %w=8.4e-6; %actual w=9e-6; %design ka=vpa(E*(t*w)/q); I=vpa(t*w^3/12); kt=vpa(12*E*I/q^3); kll=vpa(4*kt); %linear displacement kcl=vpa(2*ka/q^2); %cubic displacement a7=kcl; a6=vpa(2*g*(n-1)*kcl); a5=vpa(kll+kcl*g^2*((n-1)^2-2*n)); a4=vpa(2*g*(n-1)*(kll-n*g^2*kcl)); a3=vpa(g^2*kll*((n-1)^2-2*n)+n^2*g^4*kcl); a2=vpa(-2*n*g^3*kll*(n-1)); V2=0; data=[0 0 0 0 0 0 0 0]; maxV=16; for V1=9:0.003:11.7 p=((D/r)*2*N*eps*A/2); %p=(2*N*eps*A/2); a1=vpa(n^2*g^4*kll-2*p*g*(n+1)*(V1-V2)^2); a0=vpa(-p*(V1-V2)^2*g^2*(n^2-1)); x=roots([a7 a6 a5 a4 a3 a2 a1 a0])*1e6; trueall=0; for allroots=1:length(x) %if (abs(imag(x(allroots)))==0) %if (abs(imag(x(allroots)))==0)&&(abs(real(x(allroots)))<2) %real

equilibrium point realone=real(x(allroots));

Appendices


%end

realroots=x(find(imag(x)==0));

end

data=[data;[V1 realroots' NaN(1,7-length(realroots))]];

end

if n==3

plot(data([3:length(data)],1),abs(data([3:length(data)],[4])),'LineWid

th',3);

else

plot(data([3:length(data)],1),abs(data([3:length(data)],[4])),'--

','LineWidth',1);

end

axis([0 maxV -g*0e6 g*1e6]);

title('Graph of Equilbrium Points vs. Actuation Voltage');

%xlabel(sprintf('0 v < V < %g v',maxV)); % x-axis label

xlabel('RMS \DeltaV (volts)'); % x-axis label

ylabel('Displacement in \mum'); % y-axis label

grid on;

hold on;

end

axis([9 11.7 0.15 1.15]);

pl=[10.89,0.6468;10.95,0.6398;11.04,0.6264;11.21,0.6033;11.6,0.5605]

plot(pl(:,1),pl(:,2),'ro','markers',8)

plot(10.75,0.5055,'r.','markers',30)

Appendices


APPENDIX 5.3 - TOTAL INERTIA OF N/2 FINGERS

Referring to Figure 5.6, the moment of inertia of one finger about its centroid is

𝐽𝑖 = (1

12)𝑚𝑓(𝑆𝑓

2 + (2𝑐)2). Using the parallel-axis shift theorem this can be translated

to the rotor centre, however, each individual finger is at a different distance.

Moreover, it would be enough to sum N/4 fingers, from octagon vertex to middle of

side strut and double due to symmetry.

An expression for the distance from the centroid of the ith finger to the centre of the

rotor is as in 𝑝𝑖;

𝑝𝑖 = (𝑎 + 𝑏𝑖 + 𝑐)𝑐𝑜𝑠 (2𝛽𝑖

𝑁),

where 𝑏𝑖 = 𝑙𝑐𝑜𝑠𝛼 (𝑁−2𝑖

𝑁)

An expression for the average value of 𝑐𝑜𝑠 (2𝛽𝑖

𝑁) can be found by

4

𝑁∫ 𝑐𝑜𝑠 (

2𝛽𝑖

𝑁) 𝑑𝑖

𝑁/4

0=

2

𝛽𝑠𝑖𝑛 (

𝛽

2). This gives an approximate value for 𝑝𝑖 as follows:

𝑝𝑖 ≈ (𝑎 + 𝑏𝑖 + 𝑐)2

𝛽𝑠𝑖𝑛 (

𝛽

2).

Hence using the parallel-axis shift theorem on the ith finger we get;

𝐽𝑖 = (1


2 + (2𝑐)2) + 𝑚𝑓𝑝𝑖2

and for N/2 fingers;

𝐽𝑓 = 2∑ [(1


2 + (2𝑐)2) + 𝑚𝑓𝑝𝑖2]

𝑁/4

𝑖=1

Expanding this expression and substituting 𝑝𝑖 gives;

Appendices


𝐽𝑓 =𝑁

2[𝑚𝑓

12(𝑆𝑓

2 + (2𝑐)2 +4𝑚𝑓 sin (

𝛽2⁄ )

𝛽2(𝑎2 + 𝑐2 + 2𝑎𝑐)]

+8𝑚𝑓 sin

2 (𝛽2⁄ )

𝛽2∑ [𝑏𝑖

2 + 2(𝑎 + 𝑐)𝑏𝑖]𝑁/4

𝑖=1

with

∑ [𝑏𝑖2] =

7𝑁𝑙2 cos2 𝛼

48

𝑁4

𝑖=1

∑ [2(𝑎 + 𝑐)𝑏𝑖]𝑁/4

𝑖=1=3𝑁(𝑎 + 𝑐)𝑙𝑐𝑜𝑠𝛼

8

giving;

𝐽𝑓 =(𝑁𝑎𝑚𝑓𝑎

+ 𝑁𝑠𝑚𝑓𝑠)

4𝑓(𝛽)[(𝑆𝑓2 + 4𝑐2

12)𝑓(𝛽) + 4(𝑎2 + 𝑐2 + 2𝑎𝑐) + 6(𝑎 + 𝑐)𝑙𝑐𝑜𝑠𝛼

+7𝑙2 cos2 𝛼

3]

where 𝑓(𝛽) =𝛽2

sin2(𝛽/2)

Appendices


APPENDIX 5.4 CHANGE IN FRINGE CAPACITANCE -

MATLAB SCRIPT

This script makes use of (5.36) and (5.37) to calculate the change in fringe

capacitance for a typical change in gap. As is shown in the figure provided for full

travel (1/3 gap) the fringe capacitance changes only by less than 5 fF.

Appendices


clear all;

close all;

eps=8.8500e-12;

t=23.6e-6; %misalignment - gravity?

%% fringing field effect in electrostatic actuators vitaly leus and

david elata

%equation 10 on page 11

%actual measured

g=2.548e-6;

n=2.511;

fw=1.568e-6; %finger width

% %designed

% g=2e-6;

% n=3;

% fw=2e-6; %finger width

fl=87e-6; %finger length

Cfr1=70*eps*(t*(fl)/g)*(1+(g/(pi*t))*(1+log(2*pi*t/g))+...

(g/(pi*t)*log(1+(2*fw)/g+...

2*sqrt((fw)/g + ((fw)/g)^2))));

Cfr2=71*eps*(t*(fl)/(n*g))*(1+((n*g)/(pi*t))*(1+log(2*pi*t/(n*g)))+...

((n*g)/(pi*t)*log(1+(2*fw)/(n*g)+...

2*sqrt((fw)/(n*g) + ((fw)/(n*g))^2))));

total=Cfr1+Cfr2

withoutfringe=71*eps*((t*(fl)/g)+(t*(fl)/(n*g)))

display(sprintf('This is static, fringe changes with moevement'));

x=linspace(0,g/3,100); Cfr1=70*eps*(t*(fl)./(g-x)).*(1+((g-x)/(pi*t)).*(1+log(2*pi*t./(g-

x)))+... ((g-x)/(pi*t).*log(1+(2*fw)./(g-x)+... 2*sqrt((fw)./(g-x) + ((fw)./(g-x)).^2)))); Cfr2=71*eps*(t*(fl)./(n*g+x)).*(1+((n*g+x)./(pi*t)).*(1+log(2*pi*t./(n

*g+x)))+... ((n*g+x)./(pi*t).*log(1+(2*fw)./(n*g+x)+... 2*sqrt((fw)./(n*g+x) + ((fw)./(n*g+x)).^2)))); plot(x/g,1e15*Cfr1);grid on; title('Capacitance (including fringe) on the narrow gaps - (g-x)'); figure;plot(x/g,1e15*Cfr2);grid on; title('Capacitance (including fringe) on the wider gaps - (ng+x)'); withoutfringe=71*eps*((t*(fl)./(g-x))+(t*(fl)./(n*g+x))); figure;plot(x/g,1e15*(Cfr1+Cfr2),x/g,1e15*withoutfringe);grid on; title('Total on Cs1 - with and without fringe'); figure;plot(x/g,1e15*(Cfr1+Cfr2-withoutfringe));grid on; changeinfringe=max(Cfr1+Cfr2-withoutfringe)-min(Cfr1+Cfr2-

withoutfringe); averagefringe=mean(Cfr1+Cfr2-withoutfringe); title(sprintf('Cs1 fringe, with overall change of %g fF and nominal of

%g fF',changeinfringe*1e15,averagefringe*1e15)); xlabel('Normalised Displacement - x/g'); ylabel('Fringe Capacitance (fF)');

Appendices


APPENDIX 5.5 MONOTONICITY IN SENSING

As can be seen in the figure provided, for 0 ≤ 𝑥 ≤ 𝑔𝑠/3, 𝐶𝑠1is monotonic however

𝐶𝑠2 has a minimum occurring within this range.

From (5.45), 𝐶𝑠2can be obtained since 𝐶𝑠2(𝑥) = 𝐶𝑠1(−𝑥).

𝐶𝑠2(𝑥) = 𝑁𝑠휀𝑡𝑙𝑜𝑠 1

𝑛𝑠𝑔𝑠 − 𝑥+

1

𝑔𝑠 + 𝑥 + 𝐶𝑓𝑜

Equating the gradient, 𝑑𝐶𝑠2/𝑑𝑥 to zero to determine the minimum location gives;

𝑑𝐶𝑠2𝑑𝑥

= 𝑘 𝑙𝑛 (𝑔𝑠 + 𝑥

𝑛𝑠𝑔𝑠 − 𝑥) = 0

where 𝑘 is a constant.

This requires that 𝑥 = 𝑔𝑠/2(𝑛𝑠 − 1) and to guarantee that this is true for the whole

range of travel up until 𝑥 = 𝑔𝑠/3, 𝑛𝑠 > 5/3.

Appendices


APPENDIX 5.6 MODULATION INDEX, N AND FRINGE

CAPACITANCE - MATLAB SCRIPT

clear all;

close all;

mrange=0:0.1:1;

x_on_g=1/3; %max disp = 1/3

f=1; %f=1 means x=f*g/3 i.e. max disp

Cnom=0.9e-12;

fc=177000;

Xc1=1/(2*pi*fc*Cnom)

R=33e3;

for m=mrange

n=1.0:0.1:7;

r=(m.*(f^2+3*f*(n-1))-9*(m+1).*n)./((f-3)*(m+1).*(f+3*n)); %WRITEUP

M=sqrt((R^2+((1./r)*Xc1).^2)./(R^2+Xc1.^2));

plot(n,M);hold on;grid on;

end

title(sprintf('M vs. n for increasing Fringe Capacitance, f = %g',f));

LEG=legend(num2str(mrange'),'location','northeast');

LEG.FontSize = 8;

ylabel('Modulation Index - M');

xlabel('Comb fingers gap ratio - n');

n=2.511;

m=195/900;

n=2.511;

f=1;

r=(m*(f^2+3*f*(n-1))-9*(m+1)*n)/((f-3)*(m+1)*(f+3*n));

Cnom=0.9e-12;

fc=177000;

Xc0=1/(2*pi*fc*Cnom)

Xcmax=(1/r)*Xc0

R=33e3;

ModIndex=sqrt((R^2+(Xcmax).^2)./(R^2+Xc0.^2))

Appendices


APPENDIX 5.7 PSO - MATLAB SCRIPTS

Main Program:

tic

clc

clear all

close all

rng default

global t E ep rho xpi ga J Na Ns b datarateactual Cs1min delCs1

Mactual gain MaxV alpha;

ep=8.85e-12;

E=1.69e11;

t=25e-6; %

alpha=(67.5)*pi/180;

rho=2.5e-15*(1e6)^3;

%[ r , q , w, ca , cs, ga, gs, Sf,

na, ns, Rm]

LB=[598e-6 ,400e-6 ,6e-6 ,30e-6, 30e-6, 2e-6 , 2e-6 , 2e-6 , 2

, 1.7 , 1000]; %lower bounds of variables

UB=[700e-6 ,500e-6 ,15e-6 ,50e-6, 50e-6, 6e-6 ,6e-6 , 5e-6 , 6

, 6 , 1000000]; %upper bounds of variables

% pso parameters values

m=length(LB); % number of variables

n=30; % population size

wmax=0.9; % inertia weight

wmin=0.4; % inertia weight

c1=2; % acceleration factor

c2=2; % acceleration factor

% pso main program----------------------------------------------------

start

maxite=20; % maximum number of iteration

maxrun=5; % maximum number of runs need to be

for run=1:maxrun

run

% pso initialization----------------------------------------------

start

for i=1:n

for j=1:m

x0(i,j)=(LB(j)+rand()*(UB(j)-LB(j)));

end

end

x=x0 % initial population

v=0.1*x0; % initial velocity

for i=1:n

f0(i,1)=ofun(x0(i,:));

end

[fmin0,index0]=min(f0);

pbest=x0; % initial pbest

gbest=x0(index0,:); % initial gbest

% pso initialization----------------------------------------------

--end

Appendices


% pso algorithm---------------------------------------------------

start

ite=1;

tolerance=1;

while ite<=maxite && tolerance>10^-12

w=wmax-(wmax-wmin)*ite/maxite; % update inertial weight

% pso velocity updates

for i=1:n

for j=1:m

v(i,j)=w*v(i,j)+c1*rand()*(pbest(i,j)-x(i,j))...

+c2*rand()*(gbest(1,j)-x(i,j));

end

end

% pso position update

for i=1:n

for j=1:m

x(i,j)=x(i,j)+v(i,j);

end

end

% handling boundary violations

for i=1:n

for j=1:m

if x(i,j)<LB(j)

x(i,j)=LB(j);

elseif x(i,j)>UB(j)

x(i,j)=UB(j);

end

end

end

% evaluating fitness

for i=1:n

f(i,1)=ofun(x(i,:));

end

% updating pbest and fitness

for i=1:n

if f(i,1)<f0(i,1)

pbest(i,:)=x(i,:);

f0(i,1)=f(i,1);

end

end

[fmin,index]=min(f0); % finding out the best particle

ffmin(ite,run)=fmin; % storing best fitness

ffite(run)=ite; % storing iteration count

% updating gbest and best fitness

Appendices


if fmin<fmin0 gbest=pbest(index,:); fmin0=fmin; end

% calculating tolerance if ite>100; tolerance=abs(ffmin(ite-100,run)-fmin0) end

% displaying iterative results if ite==1 disp(sprintf('Run Iteration Best particle Objective

fun')); end disp(sprintf('%g %8g %8g

%8.4f',run,ite,index,fmin0)); ite=ite+1; end % pso algorithm------------------------------------------------end fvalue=ofun(gbest); fff(run)=fvalue; rgbest(run,:)=gbest; disp(sprintf('--------------------------------------')); end % pso main program-------------------------------------------------end disp(sprintf('\n')); disp(sprintf('********************************************************

*')); disp(sprintf('Final Results-----------------------------')); [bestfun,bestrun]=min(fff) best_variables=rgbest(bestrun,:) disp(sprintf('********************************************************

*')); toc % PSO convergence characteristic plot(ffmin(1:ffite(bestrun),bestrun),'-k'); xlabel('Iteration'); ylabel('Fitness function value'); title('PSO convergence characteristic') %#####################################################################

##### ret=ofun(best_variables) disp(sprintf('%gum %gum %gum %gum %gum %gum %gum %gum %g %g %gOhms'

,best_variables(1)*1e6,best_variables(2)*1e6,best_variables(3)*1e6,bes

t_variables(4)*1e6,best_variables(5)*1e6,best_variables(6)*1e6,best_va

riables(7)*1e6,best_variables(8)*1e6,best_variables(9),best_variables(

10),best_variables(11))); disp(sprintf('%g*gap %gV J=%g %g %g b=%g %gHz %gfF %gfF M=%g Gain=%g

\r\n' ,xpi/ga, MaxV, J, Na, Ns, b, datarateactual, Cs1min*1e15,

delCs1*1e15, Mactual, gain));

Appendices


The Objective function: ‘ofun.m’

function f=ofun(X)

global t E ep rho xpi ga J Na Ns b datarateactual Cs1min delCs1

Mactual gain MaxV alpha;

ep=8.85e-12;

E=1.69e11;

alpha=(67.5)*pi/180;

rho=2.5e-15*(1e6)^3;

r=X(1);

q=X(2);

w=X(3);

ca=X(4);

cs=X(5);

ga=X(6);

gs=X(7);

Sf=X(8);

na=X(9);

ns=X(10);

Rm=X(11);

% constraints (all constraints must be converted into <=0 type)

% if there is no constraints then comments all c0 lines below

l=r/sqrt(1+1/sqrt(2));

Na=2*ceil(l/(2*Sf+ga+na*ga)); %two octagon sides

Ns=2*ceil(l/(2*Sf+gs+ns*gs)); %two octagon sides

loa=0.9*2*ca;

Aa=t*(loa); % 90% of finger length

a=r-l*cos(alpha);

D=(a+ca+(l/2)*cos(alpha));

ka=(E*(t*w)/q);

I=(t*w^3/12); %second moment of area of spring section

kt=(12*E*I/q^3);

[MaxV,xdisp]=equilibriumpoints(na,ga,D,r,Na,Aa,ep,ka,q,kt);

xpi=max(xdisp); %pull in

los=0.9*2*cs;% 90% of finger length

Cfo=Fringe(gs,Sf,los,t)+Fringe(ns*gs,Sf,los,t); %fringe for sensing

Cs1=Ns*ep*t*los*(1./(ns*gs+xdisp)+1./(gs-xdisp))+Cfo; %Cs1(x) - check

linearity

Cs2=Ns*ep*t*los*(1./(ns*gs-xdisp)+1./(gs+xdisp))+Cfo; %Cs2(x) - check

linearity

S1=10e-6;

S2=10e-6;

m1=rho*r*S1*t;

m2=rho*l*S2*t;

mfs=rho*2*cs*t*Sf;

mfa=rho*2*ca*t*Sf;

J1=(1/12)*(m1/2)*((S1/2)^2+r^2)+(m1/2)*(r/2)^2;

J2=(1/12)*(m2)*((S2)^2+l^2)+(m2)*((r)^2-(l/2)^2);

beta=pi-2*alpha;

Appendices


fbeta=(beta^2)/(sin(beta/2))^2; Jf=((Na*mfa+Ns*mfs)/(4*fbeta))*((Sf^2+4*ca^2)*fbeta/12+4*(a^2+ca^2+2*a

*ca)+6*(a+ca)*l*cos(alpha)+(7*(l*cos(alpha))^2)/3); J=8*(J1+J2+Jf);

betaa=1-0.58*(t/loa);betas=1-0.58*(t/los); bT=2*Na*loa*(effvisco(ga)*(t/ga)^3+effvisco(na*ga)*(t/(na*ga))^3)*beta

a+2*Ns*los*(effvisco(gs)*(t/gs)^3+effvisco(ns*gs)*(t/(ns*gs))^3)*betas

; b=r^2*bT; settlingtimeactual=5*(2*J/b); datarateactual=1/(2*settlingtimeactual);

fc=1e6; Cs1max=max(Cs1); Cs1min=min(Cs1); delCs1=Cs1max-Cs1min; XCsb=1/(2*pi*fc*Cs1max); f=1; Cr=Cfo/(Ns*ep*t*los*(1/(ns*gs)+1/gs)); e=(Cr*(f^2+3*f*(ns-1))-9*ns*(Cr+1))/((f-3)*(Cr+1)*(f+3*ns)); Mactual=sqrt((Rm^2+XCsb^2)/(Rm^2+e^2*XCsb^2));

segment=floor((length(xdisp))/3); V1=linspace(0,0.1*length(Cs1),length(Cs1)); diffV11=diff(V1(segment:2*segment))'; diffV12=diff(V1(floor(length(xdisp))-segment:floor(length(xdisp))))'; gain=Rm/sqrt(Rm^2+XCsb^2); c0=[];c=[]; c0(1)=3000-datarateactual;%<0 - 3000<datarateactual c0(2)=Rm^2*(1-(0.85)^2)+XCsb^2*(1-(0.85*e)^2);%<0 - equivalent to

%ceq(2)=Mactual-0.75; c0(3)=(ga/4)-xpi;%<0 - ga/4<xpi c0(4)=MaxV-14;%<0 - MaxV<14 c0(5)=(100e-15)-delCs1;%<0 c0(6)=0.05-Rm/sqrt(Rm^2+XCsb^2);%<0 - 0.5<Rm/sqrt(Rm^2+XCsb^2) % objective function (minimization) Cfoa=Fringe(ga,Sf,loa,t)+Fringe(na*ga,Sf,loa,t); %fringe for actuation Ca=2*Na*ep*t*loa*(1./(na*ga)+1./(ga))+Cfoa; %Cs1(x) - check linearity of=r/(600e-6)+q/(500e-6)+Ca/(1e-12);%+Mactual; % defining penalty for each constraint for i=1:length(c0) if c0(i)>0 c(i)=1; else c(i)=0; end end penalty=1; % penalty on each constraint violation f=of+penalty*sum(c); % fitness function return

Appendices


The Effective viscosity function: ‘effvisco.m’

function [ueff]=effvisco(g)

lambdi=68e-9; %mean free path at NTP

Kn=lambdi/g;

xi=sqrt(pi)/(2*Kn);

uair=1.81e-5; %alpha=0.6 - W Yang

ueff=(xi*uair/6)/(xi/6+((2-0.6)/sqrt(pi))*log(1/xi+2.18)+0.6/0.642+(1-

0.6)*(xi+2.395)/(2+1.12*0.6*xi)-(1.26+10*0.6*xi)/(1+10.98*xi)+exp(-

xi/5)/8.77);

The Equilibrium Point function: ‘equilibriumpoints.m’

function [MaxV,xdisp]=equilibriumpoints(na,ga,D,r,Na,Aa,ep,ka,q,kt)

kll=(4*kt); %linear displacem,ent

kcl=(2*ka/q^2); %linear displacement

a7=kcl;

a6=(2*ga*(na-1)*kcl);

a5=(kll+kcl*ga^2*((na-1)^2-2*na));

a4=(2*ga*(na-1)*(kll-na*ga^2*kcl));

a3=(ga^2*kll*((na-1)^2-2*na)+na^2*ga^4*kcl);

a2=(-2*na*ga^3*kll*(na-1));

V2=0;

V1=0;

curr=1;

loc=1;

data=[0 0 0 0 0 0 0 0];

n=0;

while (data(curr,loc)>=0)

V1=V1+0.1;

p=((D/r)*2*Na*ep*Aa/2);

a1=(na^2*ga^4*kll-2*p*ga*(na+1)*(V1-V2)^2);

a0=(-p*(V1-V2)^2*ga^2*(na^2-1));

xr=roots([a7 a6 a5 a4 a3 a2 a1 a0])*1e6;

trueall=0;

for allroots=1:length(xr)

%realone=real(x(allroots));

trueall=trueall+1;

realroots=xr(find(imag(xr)==0));

end

data=[data;[V1 realroots' NaN(1,7-length(realroots))]];

[curr,k]=size(data);

if V1==0.1

loc=find(data(curr,[2:k-1])<((ga*1e6)/3)&data(curr,[2:k-1])>=0);

end

end

MaxV=max(data(:,1));

xdisp=(1e-6)*data(1:length(data(:,loc))-1,loc+1);

Appendices


The Fringe Capacitance function: ‘Fringe.m’

function [Cfo]=Fringe(g,Sf,los,t)

global ep;

Cfo=(ep/pi)*(1+log(2*pi*t/g)+log(1+(2*Sf/g)+2*sqrt(Sf/g+(Sf/g)^2)))*lo

s;

Appendices


APPENDIX 5.8 DYNAMICS – INPUTS TO OUTPUT - MATLAB

SCRIPTS

Main Program:

clear all;

close all;

global wd k v1 v2 wc cubicspringON modulatedata phi BPSK b J r g kt n

C Cd Cdd Cp L Ri Rm Cm a V2 V2d V2dd timestep;

b=1.3e-10;

J=(1.85e-15);r=598e-6;

g=2.5e-6;n=2.6;

%g=2e-6;n=3; %parameters designed

data=[0 0 0 0 0];

E=1.69e11;t=25e-6;

w=8.4e-6; %actual device

%w=9e-6; %PARAMETERS DESIGNED

q=458e-6;ka=(E*(t*w)/q);I=(t*w^3/12);

kt=(12*E*I/q^3);

eps=8.85e-12;A=2.25e-9;N=71;Cf=119e-15;

% simulation paramters

simpar=[ 1 7.6 1 3.3e3 1 7.6 0 0

0.0009 0 171.82e3 0]; % 3.3kHz data BPSK on 177kHz as v1 and

fc=simpar(11);

fd=simpar(4);

v1=simpar(2); % data amplitude

v2=simpar(6); % carrier amplitude, 0 switches off carrier,

switchsense=simpar(12);

cubicspringON=simpar(1);% 1 = ON, 0 = OFF

modulatedata=simpar(3); % 1 or 0: 0 gives input as baseband, 1

modulates on wc

wd=2*pi*fd;

BPSK=simpar(5); %1 switches on BPSK and off ASK

k=simpar(7);

phi=simpar(8);

%freq response

freq=[0 0];

duration=3*(1/fd); %duration is integer multiple of data freq. 3x

timestep=(1/fc)/500; %/500

spec=zeros(floor(duration/timestep)+1,1);

wc=2*pi*fc;

initcond1=[0 0 wd wc]; %x x_dot startingomega

[t2,y1] = ode23s('actuationDE',[0:timestep:duration], initcond1);

%%

figure;

subplot(221)


hold on;

plot(y1(:,1)*1e6,y1(:,2),'b');

Appendices


minx=-0.4; maxx=2.5; miny=-0.15; maxy=0.15; xrange=[minx:1e-8:maxx]; yrange=[miny:0.001:maxy]; axis([minx maxx miny maxy]); %xlabel('x(t) in \mum'); xlabel('$x(t)\, in\, \mu m$','interpreter','latex'); ylabel('$\dotx(t)$','interpreter','latex');

title('Phase Plane Portrait for tormixer -- x dot(t) vs. x(t)');

subplot(222) yyaxis left plot(0,initcond1(1),'r*');grid on; hold on; [unwanted wanted]=v1minusv2(t2,1,1); yyaxis right plot(t2*1e3,wanted,'r--'); ylabel('\Delta V(t)'); hold on; yyaxis left plot(t2*1e3,y1(:,1)*1e6,'b'); %y1(:,1) is the displacement ylabel('$x(t)\, in\, \mu m$','interpreter','latex'); xlabel('time in ms'); title(sprintf('x(t) vs. t'));

subplot(223) plot(0,initcond1(2),'r*'); hold on; plot(t2,y1(:,2),'b');grid on; xlabel('time'); ylabel('x dot(t)'); title('x dot(t) vs. t');

subplot(224) plot3(y1(:,1),y1(:,2),t2,'b');grid on; hold on; plot3(initcond1(1),initcond1(2),0,'r*'); xlabel('x(t)'); ylabel('y(t)'); zlabel('time'); title('3D phase portrait of tormixer');

%wanted is del V

instaForce=((7.9224e-22)*(r))*((wanted).^2).*(((n*g+y1(:,1)).^2-(g-

y1(:,1)).^2)./((g-y1(:,1)).*(n*g+y1(:,1))).^2);

figure;

Appendices


title(sprintf('v1 and v2, difference and diff^2')); subplot(141) plot(t2,BPSK*v1*cos(modulatedata*wc*t2+(square(wd*t2)+1)/2*pi)+(1-

BPSK)*cos(modulatedata*wc*t2).*(v1*square(wd*t2)+v1)/2);grid on; subplot(142) V2=v2*cos(wc*t2+phi)+k; plot(t2,V2);grid on; subplot(143); plot(t2,wanted);grid on; subplot(144); plot(t2,wanted.^2);grid on;

x=y1(:,1); Ca=eps*A*2*N*(1./(n*g+x)+1./(g-x))+2*Cf; Cad=diff(Ca)./diff(t2); Cad=[Cad;Cad(length(Cad))]; wantfilt=filter(1/600, [1 1/600-1], wanted); wanted_d=diff(wantfilt)./diff(t2); wanted_d=[wanted_d;wanted_d(length(wanted_d))]; Actuationcurrent=Ca.*wanted_d+Cad.*wantfilt+wantfilt/(310e6); figure;plot(Actuationcurrent);grid on; rmsactuation=rms(Actuationcurrent); title(sprintf('Actuation RMS Current is %g nA',rmsactuation*1e9)); xlabel('Time'); ylabel('Current (nA)');

figure; Cs1=eps*A*N*(1./(n*g+x)+1./(g-x))+Cf; %closing - extra 120fF

parasitics Cs2=eps*A*N*(1./(n*g-x)+1./(g+x))+Cf; %opening - extra 120fF

parasitics graph1=plot(t2*1e3,Cs1*10^15,t2*1e3,Cs2*10^15,'--k');grid on; ylabel('Capacitance (fF)'); xlabel('time (ms)'); legend('Cs1','Cs2'); set(graph1,'LineWidth',2);

if switchsense==1 C=Cs2; %opening elseif switchsense==0 C=Cs1; %closing end

Cd=diff(C)./diff(t2); Cd=[Cd;Cd(length(Cd))]; Cdd=diff(Cd)./diff(t2); Cdd=[Cdd;Cdd(length(Cdd))]; V2d=diff(V2)./diff(t2); V2d=[V2d;V2d(length(V2d))];

Appendices


V2dd=diff(V2d)./diff(t2); V2dd=[V2dd;V2dd(length(V2dd))]; %actuation power sig=simpar(2)*[(1-exp(-t2(1:(length(t2)/6))/6e-6));(exp(-

t2((1:length(t2)/6))/6e-6))]; Ca=((100e-15)/0.721)*(-1e13*C+9)+1.8e-12; wsig=[sig;sig;sig;zeros(1,(length(Ca)-3*length(sig)))']; Cad=diff(Ca)./diff(t2); %newwanted=conv(wsig,(1/1000)*ones(1,1000)); wantedd=diff(wsig)./diff(t2); Pa1=wsig.*Ca.*[wantedd;wantedd(length(wantedd))]; Pa2=wsig.*wsig.*[Cad;Cad(length(Cad))]; Pa3=wsig.*wsig/(210e6); Pat=Pa1+Pa2+Pa3; rmsPat=rms(Pat) rmsPat=rms(Pa1)+rms(Pa2)+rms(Pa3); rmsPa1=rms(Pa1)/rmsPat*100 rmsPa2=rms(Pa2)/rmsPat*100 rmsPa3=rms(Pa3)/rmsPat*100 subplot(2,2,1);plot(Pa1);subplot(2,2,2);plot(Pa2);subplot(2,2,3);plot(

Pa3);subplot(2,2,4);plot(Pat); %%%%%%%%%%%%%%%%%%%%%%%%%

Ri=210e2; Reqvp=0.3; Xs1=1/(2*pi*fc*1060e-15);%1060 fF is nominal sense cap Cs1 Rm=(Reqvp*Xs1/v2)/sqrt(1-(Reqvp/v2)^2); Cm=5e-13;%1e-10; a=(Ri+Rm)/(Ri*Rm);Cp=260e-13;%Cp=260e-12; L=1e-4; %remove - small number to be used with capaciive load only Ct=Cm+Cp; XPX=((2*pi*fc*Ct))*(2*pi*fc*L) s=1/(2*pi*fc*1050e-15); u=1/(2*pi*fc*900e-15); delVp=(s/sqrt(s^2+a^2)-u/sqrt(u^2+a^2))*v2; besta=((s*u)^(1/3))*sqrt(u^(2/3)+s^(2/3)); delVpbest=(s/sqrt(s^2+besta^2)-u/sqrt(u^2+besta^2))*v2; bestRm=Ri/(besta*Ri-1); initcond1=[0 0 0]; %i vp vpdot

[t1,S] = ode23s('sensingDE',[0:timestep:duration], initcond1); region1=floor(1.5*(1/fd)/timestep); large=max(S(region1-500:region1,2))-min(S(region1-500:region1,2)); region2=floor(2*(1/fd)/timestep); small=max(S(region2-500:region2,2))-min(S(region2-500:region2,2)); M=small/large; if M>1

Appendices


M=1/M; end figure; plot(t1,S(:,2),'b');grid on;hold on; currentRMS=rms(S(1:floor((1/fd)/timestep),1)) powerRMS=currentRMS^2*(1/a); display(sprintf('RMS Power across sense resistors is %g

uW',powerRMS*1e6)); rmsmechpowerdamp=(b/r)*mean(y1(:,2).^2); display(sprintf('RMS Power in mechanical [damping x vel^2] is %g

uW',rmsmechpowerdamp*1e6)); % figure; plot(((region1-500)*timestep),max(S(region1-500:region1,2)),'r.',

'markers', 30);hold on; plot(((region1)*timestep),max(S(region1-500:region1,2)),'r.',

'markers', 30);hold on; plot(((region1-500)*timestep),min(S(region1-500:region1,2)),'r.',

'markers', 30);hold on; plot(((region1)*timestep),min(S(region1-500:region1,2)),'r.',

'markers', 30);hold on; plot(((region2-500)*timestep),max(S(region2-500:region2,2)),'r.',

'markers', 30);hold on; plot(((region2)*timestep),max(S(region2-500:region2,2)),'r.',

'markers', 30);hold on; plot(((region2-500)*timestep),min(S(region2-500:region2,2)),'r.',

'markers', 30);hold on; plot(((region2)*timestep),min(S(region2-500:region2,2)),'r.',

'markers', 30); title(sprintf('fc = %g kHz M = %g Pa = %.2f pW V1/V2 = %.2f v fd =

%g',fc/1000,M,rmsmechpowerdamp*1e12,vol,fd)); xlabel('Time (s)'); ylabel('v_p (volts)'); figure;plot(S(:,1)); title('current'); display(sprintf('Linear Damping coeff. experimental = %g',(b)));

ueff=0.84*1.81e-5;%effective viscosity lfinger=100e-6; factor = 1-0.58*(t/lfinger); %squeeze film dampingcoeffmodel = (4*N)*factor*ueff*lfinger*(t/g)^3; b_theory=(r^2)*dampingcoeffmodel; display(sprintf('Linear Damping coeff. theoretical = %g',(b_theory))); gzita=(b_theory/r)/(2*sqrt(4*(r)*kt*(J/r))) %b/(2*sqrt(k*m)); Q=1/(2*gzita);

Appendices


Function ‘actuationDE.m’:

function [dy] = phaseportrait3D(t,y)

global wd cubicspringON wc b J r g kt n;

dy=zeros(4,1);

[wanted unwanted]=v1minusv2(1,y(3),y(4));

dy(1)=y(2);

dy(2)=-(b/J)*y(2)-(4*(r^2/J)*kt)*y(1)-

(738821561082)*cubicspringON*(r^2/J)*y(1)^3+...

((7.9224e-22)*(r/J))*((wanted).^2)*(((n*g+y(1))^2-(g-y(1))^2)/((g-

y(1))*(n*g+y(1)))^2);

dy(3)=wd;

dy(4)=wc;

Function ‘sensingDE.m’:

function dy = sensingDE(t1,y)

global C Cd Cdd V2 V2d V2dd L Cp a timestep;

dy=zeros(3,1);

i=(floor(t1/timestep)+1);

dy(1)=Cp/(Cp+C(i))*((2*Cd(i)*V2d(i)+Cdd(i)*V2(i)+C(i)*V2dd(i))+(C(i)/(

Cp*L)-Cdd(i))*y(2)+((C(i)*a)/Cp-2*Cd(i))*y(3));

dy(2)=y(3);

dy(3)=(1/Cp)*dy(1)-(a/Cp)*y(3)-(1/(Cp*L))*y(2);

Function ‘v1minusv2.m’:

function [formodel forplotting]=v1minusv2(t,wdt,wct)

global wd modulatedata k v1 wc v2 phi BPSK

formodel= [BPSK*v1*cos(modulatedata*wct +(square(wdt )+1)/2*pi)+(1-

BPSK)*cos(modulatedata*wct )*(v1*square(wdt )+v1)/2-(v2*cos(wct+phi

)+k)];

forplotting=[BPSK*v1*cos(modulatedata*wc*t+(square(wd*t)+1)/2*pi)+(1-

BPSK)*cos(modulatedata*wc*t).*(v1*square(wd*t)+v1)/2-

(v2*cos(wc*t+phi)+k)];

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