Levitation Micro-Systems: Applications to Sensors and Actuators

ISBN-13: 9783030589103 / Angielski / Miękka / 2021 / 186 str.

Kirill Poletkin

Spis treści

1 Introduction to levitation micro-systems 7

1.1 Levitation micro-systems. Classification . . . . . . . . . . . . . . . . . . . . . . . . 8

1.2 Electric levitation micro-systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

1.3 Magnetic levitation micro-systems . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

1.4 Diamagnetic levitation micro-systems . . . . . . . . . . . . . . . . . . . . . . . . . 10

1.5 Superconducting levitation micro-systems . . . . . . . . . . . . . . . . . . . . . . . 12

1.6 Inductive levitation micro-systems . . . . . . . . . . . . . . . . . . . . . . . . . . . 12

1.7 Hybrid levitation micro-systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

1.8 Future Trends . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14

2 Micro-fabrication techniques 17

2.1 Planar coil technology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17

2.2 3D micro-coil technology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19

3 Analytical modelling 21

3.1 Analytical mechanics of micro-electro-mechanical-systems . . . . . . . . . . . . . . 22

3.2 Statement of the problem for modelling . . . . . . . . . . . . . . . . . . . . . . . . 25

3.3 Stability of inductive levitation systems . . . . . . . . . . . . . . . . . . . . . . . . 31

3.4 Modelling of IL-micro-systems based on symmetric designs . . . . . . . . . . . . . 34

4 Quasi-finite element modelling 39

4.1 Statement of problem for modelling . . . . . . . . . . . . . . . . . . . . . . . . . . . 40

4.2 Procedure for the analysis of IL-micro-systems . . . . . . . . . . . . . . . . . . . . 43

4.3 Calculation of the mutual inductance of circular filaments . . . . . . . . . . . . . . 44

4.3.1 The Kalantarov-Zeitlin method . . . . . . . . . . . . . . . . . . . . . . . . . 45

4.3.2 Derivation of Formulas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47

5 Inductive levitation micro-systems 53

5.1 Micro-bearings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53

5.1.1 Design and fabrication . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53

5.1.2 Measurement of stiffness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54

5.1.3 Modelling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57

5.1.4 Coil impedance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61

5.1.5 Levitation height . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63

5.1.6 Lateral Stability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68

5.1.7 Temperature . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70

5.2 Micro-bearings with lowest energy consumption . . . . . . . . . . . . . . . . . . . . 71

5.2.1 Experimental results and further discussion . . . . . . . . . . . . . . . . . . 72

6 Hybrid levitation micro-systems 77

6.1 Micro-actuators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77

6.1.1 Design and micro-machined fabrication . . . . . . . . . . . . . . . . . . . . 77

6.1.2 Experimental Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79

6.1.3 Eddy current simulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82

6.1.4 Analytical model of static linear pull-in actuation . . . . . . . . . . . . . . . 86

6.1.5 Quasi-FEM of static linear pull-in actuation . . . . . . . . . . . . . . . . . . 87

6.1.6 Preliminary analysis of developed models . . . . . . . . . . . . . . . . . . . 89

6.1.7 Comparison with experiment . . . . . . . . . . . . . . . . . . . . . . . . . . 93

6.1.8 A light disc of a 2.4mm diameter . . . . . . . . . . . . . . . . . . . . . . . . 93

6.1.9 Angular pull-in . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97

6.2 Micro-accelerators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103

6.2.1 Operating principle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103

6.2.2 Micro-fabrication . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104

6.2.3 Linear motion due to the gravity . . . . . . . . . . . . . . . . . . . . . . . . 105

6.2.4 Modelling of stable levitation . . . . . . . . . . . . . . . . . . . . . . . . . . 105

6.3 Micro-accelerometers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112

6.3.1 Fabrication . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112

6.3.2 Operating principle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113

6.3.3 Preliminary experimental results . . . . . . . . . . . . . . . . . . . . . . . . 114

6.3.4 Analytical model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 116

6.3.5 The accelerometer equation of motion . . . . . . . . . . . . . . . . . . . . . 117

6.3.6 Static pull-in instability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 119

6.3.7 Dynamic pull-in instability . . . . . . . . . . . . . . . . . . . . . . . . . . . 121

7 Mechanical thermal noise in levitated micro-gyroscopes 123

7.1 Model of an ideal levitated two-axis rate gyroscope . . . . . . . . . . . . . . . . . . 124

7.2 Mechanical thermal noise . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 126

7.3 Johnson noise. Resolution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 127

7.4 Analysis of resolution of reported gyroscopes . . . . . . . . . . . . . . . . . . . . . 129

7.5 Scale factor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 131

7.6 Mechanical thermal noise in vibratory and levitated gyroscopes . . . . . . . . . . . 133

A Mathematical notation 135

B Mutual inductance between two filaments 143

B.1 MATLAB functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 143

B.2 Determination of angular position of the secondary circular filament . . . . . . . . 145

B.3 Presentation of developed formulas via the pair of angles α and β . . . . . . . . . . 146

C Mutual inductance between two filaments 149

C.1 Derivation of a levitated gyroscope model . . . . . . . . . . . . . . . . . . . . . . . 149

C.2 Integral of equation (7.15) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 150

O autorze

Kirill Poletkin obtained his Ph.D. degree at Moscow Aviation Institute (Moscow State Aviation Technological University), Russia in 2007. Since 2016, he is a scientist at Karlsruhe Institute of Technology as a scientist. In 2020, he is also appointed as an assistant professor at Innopolis University. His research interest includes micro- and nano-scales electromechanical devices and processes of the energy transfer within these scales. 

Blurb

This book presents inductive and hybrid levitation micro-systems and their applications in micro-sensors and –actuators. It proposes and discusses analytical and quasi-finite element techniques for modeling levitation micro-systems based on the Lagrangian formalism. In particular, micro-bearings, -actuators, -accelerators and –accelerometers based on inductive levitation are comprehensively described with accompanying experimental measurements.