Philippe Basset, Associate professor at Université Paris-Est, ESYCOM, ESIEE Paris, France Dimitri Galayko, associate professor at Sorbonne Universités, UPMC Univ., Paris, France Elena Blokhina, research manager at School of Electrical, Electronic and Communications Engineering, University College Dublin, Ireland

• Preface ixIntroduction: Background and Area of Application xiChapter 1. Introduction to Electrostatic Kinetic Energy Harvesting 1Chapter 2. Capacitive Transducers 72.1. Presentation of capacitive transducers 72.2. Electrical operation of a variable capacitor 112.3. Energy and force in capacitive transducers 122.3.1. Energy of a capacitor 122.3.2. Force of the capacitor 142.3.3. Capacitive transducers biased by an electret layer 172.4. Energy conversion with a capacitive transducer 202.5. Optimization of the operation of a capacitive transducer 212.6. Electromechanical coupling 232.7. Conclusions 242.8. Appendix: proof of formula [2.32] for the energy converted in a cycle 24Chapter 3. Mechanical Aspects of Kinetic Energy Harvesters: Linear Resonators 273.1. Overview of mechanical forces and the resonator model 273.1.1. Linear resonator as the main model of the mechanical part 273.1.2. The nature and effect of the transducer force 303.1.3. Remarks on mechanical forces 333.2. Interaction of the harvester with the environment 363.2.1. Power balance of KEHs 363.2.2. Efficiency of KEHs 403.3. Natural dynamics of the linear resonator 423.3.1. Behavior of the resonator with no input 423.3.2. Energy relation for the resonator with no input 443.3.3. Forced oscillator and linear resonance 453.3.4. Periodic external vibrations 493.3.5. Energy relation for a forced resonator 503.4. The mechanical impedance 523.5. Concluding remarks 54Chapter 4. Mechanical Aspects of Kinetic Energy Harvesters: Nonlinear Resonators 554.1. Nonlinear resonators with mechanically induced nonlinearities 554.1.1. Equation of the nonlinear resonator 554.1.2. Free oscillations of nonlinear resonator: qualitative description using potential wells 604.1.3. Free oscillations of nonlinear resonator: semi-analytical approach 624.1.4. Forced nonlinear resonator and nonlinear resonance 634.2. Review of other nonlinearities affecting the dynamics of the resonator: impact, velocity and frequency amplification and electrical softening 684.3. Concluding remarks: effectiveness of linear and nonlinear resonators 71Chapter 5. Fundamental Effects of Nonlinearity 755.1. Fundamental nonlinear effects: anisochronous and anharmonic oscillations 755.2. Semi-analytical techniques for nonlinear resonators 795.2.1. Normalized form of nonlinear resonators 795.2.2. Anharmonic oscillations demonstrated by straightforward expansion 815.2.3. Anisochronous oscillations demonstrated by the LPM 845.2.4. Multiple scales method 885.2.5. Nonlinearity of a general form 915.3. Concluding remarks 95Chapter 6. Nonlinear Resonance and its Application to Electrostatic Kinetic Energy Harvesters 976.1. Forced nonlinear resonator and nonlinear resonance 976.1.1. Analysis of forced oscillations using the multiple scales method 976.1.2. Forced oscillations with a general form of nonlinear force 1026.2. Electromechanical analysis of an electrostatic kinetic energy harvester 1056.2.1. Statement of the problem 1056.2.2. Mathematical model of the constant charge circuit 1066.2.3. Steady-state nonlinear oscillations 1096.2.4. Dynamical effects and bifurcation behavior 1136.2.5. Other conditioning circuits 1156.3. Concluding remarks 119Chapter 7. MEMS Device Engineering for e-KEH 1217.1. Silicon-based MEMS fabrication technologies 1217.1.1. Examples of bulk processes 1227.1.2. Thin-film technology with sacrificial layer 1237.2. Typical designs for the electrostatic transducer 1247.2.1. Capacitive transducers with gap-closing electrode variation 1257.2.2. Strategies on the stopper’s location in gap-closing e-KEH 1287.2.3. Capacitive transducers with overlapping electrode motion 1307.3. e-KEHs with an electret layer 133Chapter 8. Basic Conditioning Circuits for Capacitive Kinetic Energy Harvesters 1358.1. Introduction 1358.2. Overview of conditioning circuit for capacitive kinetic energy harvesting 1368.3. Continuous conditioning circuit: generalities 1388.3.1. Qualitative discussion on operation of the circuit 1398.3.2. Analytical model in the electrical domain 1408.4. Practical study of continuous conditioning circuits 1418.4.1. Gap-closing transducer 1418.4.2. Area overlap transducer 1458.4.3. Simple conditioning circuit with diode rectifiers 1488.5. Shortcomings of the elementary conditioning circuits: auto-increasing of the biasing 1498.5.1. Appendix: listing of the Eldo netlist used to obtain the presented plots 152Chapter 9. Circuits Implementing Triangular QV Cycles 1559.1. Energy transfer in capacitive circuits 1559.1.1. Energy exchange between two fixed capacitors 1559.1.2. Case of a voltage source charging a capacitor 1569.1.3. Inductive DC-DC converters 1579.1.4. Use of a variable capacitor 1619.2. Conditioning circuits implementing triangular QV cycles 1639.2.1. Constant-voltage conditioning circuit 1639.2.2. Constant-charge conditioning circuits 1659.2.3. Analysis of the circuit implementing a constant-charge QV cycle 1669.2.4. Practical implementation 1699.3. Circuits implementing triangular QV cycles: conclusion 171Chapter 10. Circuits Implementing Rectangular QV Cycles, Part I 17310.1. Study of the rectangular QV cycle 17310.2. Practical implementation of the charge pump 17810.2.1. Evolution of the harvested energy 18010.3. Shortcomings of the single charge pump and required improvements 18210.3.1. Need for a flyback 18210.3.2. Auto-increasing of the internal energy 18310.4. Architectures of the charge pump with flyback 18410.4.1. Resistive flyback 18410.4.2. Inductive flyback 18510.5. Conditioning circuits based on the Bennet’s doubler 18810.5.1. Introduction of the principle . 18810.5.2. Analysis of the Bennet’s doubler conditioning circuit 19110.5.3. Simulation of a Bennet’s doubler 199Chapter 11. Circuits Implementing Rectangular QV Cycles, Part II 20311.1. Analysis of the half-wave rectifier with a transducer biased by an electret 20311.2. Analysis of the full-wave diode rectifier with transducer biased by an electret 20511.3. Dynamic behavior and electromechanical coupling of rectangular QV cycle conditioning circuits 21011.4. Practical use of conditioning circuits with rectangular QV cycle 21511.5. Conclusion on conditioning circuits for e-KEHs 216Bibliography 217Index 225

Philippe Basset, Associate professor at Université Paris-Est, ESYCOM, ESIEE Paris, France Dimitri Galayko, associate professor at Sorbonne Universités, UPMC Univ., Paris, France Elena Blokhina, research manager at School of Electrical, Electronic and Communications Engineering, University College Dublin, Ireland

• Preface ixIntroduction: Background and Area of Application xiChapter 1. Introduction to Electrostatic Kinetic Energy Harvesting 1Chapter 2. Capacitive Transducers 72.1. Presentation of capacitive transducers 72.2. Electrical operation of a variable capacitor 112.3. Energy and force in capacitive transducers 122.3.1. Energy of a capacitor 122.3.2. Force of the capacitor 142.3.3. Capacitive transducers biased by an electret layer 172.4. Energy conversion with a capacitive transducer 202.5. Optimization of the operation of a capacitive transducer 212.6. Electromechanical coupling 232.7. Conclusions 242.8. Appendix: proof of formula [2.32] for the energy converted in a cycle 24Chapter 3. Mechanical Aspects of Kinetic Energy Harvesters: Linear Resonators 273.1. Overview of mechanical forces and the resonator model 273.1.1. Linear resonator as the main model of the mechanical part 273.1.2. The nature and effect of the transducer force 303.1.3. Remarks on mechanical forces 333.2. Interaction of the harvester with the environment 363.2.1. Power balance of KEHs 363.2.2. Efficiency of KEHs 403.3. Natural dynamics of the linear resonator 423.3.1. Behavior of the resonator with no input 423.3.2. Energy relation for the resonator with no input 443.3.3. Forced oscillator and linear resonance 453.3.4. Periodic external vibrations 493.3.5. Energy relation for a forced resonator 503.4. The mechanical impedance 523.5. Concluding remarks 54Chapter 4. Mechanical Aspects of Kinetic Energy Harvesters: Nonlinear Resonators 554.1. Nonlinear resonators with mechanically induced nonlinearities 554.1.1. Equation of the nonlinear resonator 554.1.2. Free oscillations of nonlinear resonator: qualitative description using potential wells 604.1.3. Free oscillations of nonlinear resonator: semi-analytical approach 624.1.4. Forced nonlinear resonator and nonlinear resonance 634.2. Review of other nonlinearities affecting the dynamics of the resonator: impact, velocity and frequency amplification and electrical softening 684.3. Concluding remarks: effectiveness of linear and nonlinear resonators 71Chapter 5. Fundamental Effects of Nonlinearity 755.1. Fundamental nonlinear effects: anisochronous and anharmonic oscillations 755.2. Semi-analytical techniques for nonlinear resonators 795.2.1. Normalized form of nonlinear resonators 795.2.2. Anharmonic oscillations demonstrated by straightforward expansion 815.2.3. Anisochronous oscillations demonstrated by the LPM 845.2.4. Multiple scales method 885.2.5. Nonlinearity of a general form 915.3. Concluding remarks 95Chapter 6. Nonlinear Resonance and its Application to Electrostatic Kinetic Energy Harvesters 976.1. Forced nonlinear resonator and nonlinear resonance 976.1.1. Analysis of forced oscillations using the multiple scales method 976.1.2. Forced oscillations with a general form of nonlinear force 1026.2. Electromechanical analysis of an electrostatic kinetic energy harvester 1056.2.1. Statement of the problem 1056.2.2. Mathematical model of the constant charge circuit 1066.2.3. Steady-state nonlinear oscillations 1096.2.4. Dynamical effects and bifurcation behavior 1136.2.5. Other conditioning circuits 1156.3. Concluding remarks 119Chapter 7. MEMS Device Engineering for e-KEH 1217.1. Silicon-based MEMS fabrication technologies 1217.1.1. Examples of bulk processes 1227.1.2. Thin-film technology with sacrificial layer 1237.2. Typical designs for the electrostatic transducer 1247.2.1. Capacitive transducers with gap-closing electrode variation 1257.2.2. Strategies on the stopper’s location in gap-closing e-KEH 1287.2.3. Capacitive transducers with overlapping electrode motion 1307.3. e-KEHs with an electret layer 133Chapter 8. Basic Conditioning Circuits for Capacitive Kinetic Energy Harvesters 1358.1. Introduction 1358.2. Overview of conditioning circuit for capacitive kinetic energy harvesting 1368.3. Continuous conditioning circuit: generalities 1388.3.1. Qualitative discussion on operation of the circuit 1398.3.2. Analytical model in the electrical domain 1408.4. Practical study of continuous conditioning circuits 1418.4.1. Gap-closing transducer 1418.4.2. Area overlap transducer 1458.4.3. Simple conditioning circuit with diode rectifiers 1488.5. Shortcomings of the elementary conditioning circuits: auto-increasing of the biasing 1498.5.1. Appendix: listing of the Eldo netlist used to obtain the presented plots 152Chapter 9. Circuits Implementing Triangular QV Cycles 1559.1. Energy transfer in capacitive circuits 1559.1.1. Energy exchange between two fixed capacitors 1559.1.2. Case of a voltage source charging a capacitor 1569.1.3. Inductive DC-DC converters 1579.1.4. Use of a variable capacitor 1619.2. Conditioning circuits implementing triangular QV cycles 1639.2.1. Constant-voltage conditioning circuit 1639.2.2. Constant-charge conditioning circuits 1659.2.3. Analysis of the circuit implementing a constant-charge QV cycle 1669.2.4. Practical implementation 1699.3. Circuits implementing triangular QV cycles: conclusion 171Chapter 10. Circuits Implementing Rectangular QV Cycles, Part I 17310.1. Study of the rectangular QV cycle 17310.2. Practical implementation of the charge pump 17810.2.1. Evolution of the harvested energy 18010.3. Shortcomings of the single charge pump and required improvements 18210.3.1. Need for a flyback 18210.3.2. Auto-increasing of the internal energy 18310.4. Architectures of the charge pump with flyback 18410.4.1. Resistive flyback 18410.4.2. Inductive flyback 18510.5. Conditioning circuits based on the Bennet’s doubler 18810.5.1. Introduction of the principle . 18810.5.2. Analysis of the Bennet’s doubler conditioning circuit 19110.5.3. Simulation of a Bennet’s doubler 199Chapter 11. Circuits Implementing Rectangular QV Cycles, Part II 20311.1. Analysis of the half-wave rectifier with a transducer biased by an electret 20311.2. Analysis of the full-wave diode rectifier with transducer biased by an electret 20511.3. Dynamic behavior and electromechanical coupling of rectangular QV cycle conditioning circuits 21011.4. Practical use of conditioning circuits with rectangular QV cycle 21511.5. Conclusion on conditioning circuits for e-KEHs 216Bibliography 217Index 225

Philippe Basset, Associate professor at Université Paris-Est, ESYCOM, ESIEE Paris, France Dimitri Galayko, associate professor at Sorbonne Universités, UPMC Univ., Paris, France Elena Blokhina, research manager at School of Electrical, Electronic and Communications Engineering, University College Dublin, Ireland

• Preface ixIntroduction: Background and Area of Application xiChapter 1. Introduction to Electrostatic Kinetic Energy Harvesting 1Chapter 2. Capacitive Transducers 72.1. Presentation of capacitive transducers 72.2. Electrical operation of a variable capacitor 112.3. Energy and force in capacitive transducers 122.3.1. Energy of a capacitor 122.3.2. Force of the capacitor 142.3.3. Capacitive transducers biased by an electret layer 172.4. Energy conversion with a capacitive transducer 202.5. Optimization of the operation of a capacitive transducer 212.6. Electromechanical coupling 232.7. Conclusions 242.8. Appendix: proof of formula [2.32] for the energy converted in a cycle 24Chapter 3. Mechanical Aspects of Kinetic Energy Harvesters: Linear Resonators 273.1. Overview of mechanical forces and the resonator model 273.1.1. Linear resonator as the main model of the mechanical part 273.1.2. The nature and effect of the transducer force 303.1.3. Remarks on mechanical forces 333.2. Interaction of the harvester with the environment 363.2.1. Power balance of KEHs 363.2.2. Efficiency of KEHs 403.3. Natural dynamics of the linear resonator 423.3.1. Behavior of the resonator with no input 423.3.2. Energy relation for the resonator with no input 443.3.3. Forced oscillator and linear resonance 453.3.4. Periodic external vibrations 493.3.5. Energy relation for a forced resonator 503.4. The mechanical impedance 523.5. Concluding remarks 54Chapter 4. Mechanical Aspects of Kinetic Energy Harvesters: Nonlinear Resonators 554.1. Nonlinear resonators with mechanically induced nonlinearities 554.1.1. Equation of the nonlinear resonator 554.1.2. Free oscillations of nonlinear resonator: qualitative description using potential wells 604.1.3. Free oscillations of nonlinear resonator: semi-analytical approach 624.1.4. Forced nonlinear resonator and nonlinear resonance 634.2. Review of other nonlinearities affecting the dynamics of the resonator: impact, velocity and frequency amplification and electrical softening 684.3. Concluding remarks: effectiveness of linear and nonlinear resonators 71Chapter 5. Fundamental Effects of Nonlinearity 755.1. Fundamental nonlinear effects: anisochronous and anharmonic oscillations 755.2. Semi-analytical techniques for nonlinear resonators 795.2.1. Normalized form of nonlinear resonators 795.2.2. Anharmonic oscillations demonstrated by straightforward expansion 815.2.3. Anisochronous oscillations demonstrated by the LPM 845.2.4. Multiple scales method 885.2.5. Nonlinearity of a general form 915.3. Concluding remarks 95Chapter 6. Nonlinear Resonance and its Application to Electrostatic Kinetic Energy Harvesters 976.1. Forced nonlinear resonator and nonlinear resonance 976.1.1. Analysis of forced oscillations using the multiple scales method 976.1.2. Forced oscillations with a general form of nonlinear force 1026.2. Electromechanical analysis of an electrostatic kinetic energy harvester 1056.2.1. Statement of the problem 1056.2.2. Mathematical model of the constant charge circuit 1066.2.3. Steady-state nonlinear oscillations 1096.2.4. Dynamical effects and bifurcation behavior 1136.2.5. Other conditioning circuits 1156.3. Concluding remarks 119Chapter 7. MEMS Device Engineering for e-KEH 1217.1. Silicon-based MEMS fabrication technologies 1217.1.1. Examples of bulk processes 1227.1.2. Thin-film technology with sacrificial layer 1237.2. Typical designs for the electrostatic transducer 1247.2.1. Capacitive transducers with gap-closing electrode variation 1257.2.2. Strategies on the stopper’s location in gap-closing e-KEH 1287.2.3. Capacitive transducers with overlapping electrode motion 1307.3. e-KEHs with an electret layer 133Chapter 8. Basic Conditioning Circuits for Capacitive Kinetic Energy Harvesters 1358.1. Introduction 1358.2. Overview of conditioning circuit for capacitive kinetic energy harvesting 1368.3. Continuous conditioning circuit: generalities 1388.3.1. Qualitative discussion on operation of the circuit 1398.3.2. Analytical model in the electrical domain 1408.4. Practical study of continuous conditioning circuits 1418.4.1. Gap-closing transducer 1418.4.2. Area overlap transducer 1458.4.3. Simple conditioning circuit with diode rectifiers 1488.5. Shortcomings of the elementary conditioning circuits: auto-increasing of the biasing 1498.5.1. Appendix: listing of the Eldo netlist used to obtain the presented plots 152Chapter 9. Circuits Implementing Triangular QV Cycles 1559.1. Energy transfer in capacitive circuits 1559.1.1. Energy exchange between two fixed capacitors 1559.1.2. Case of a voltage source charging a capacitor 1569.1.3. Inductive DC-DC converters 1579.1.4. Use of a variable capacitor 1619.2. Conditioning circuits implementing triangular QV cycles 1639.2.1. Constant-voltage conditioning circuit 1639.2.2. Constant-charge conditioning circuits 1659.2.3. Analysis of the circuit implementing a constant-charge QV cycle 1669.2.4. Practical implementation 1699.3. Circuits implementing triangular QV cycles: conclusion 171Chapter 10. Circuits Implementing Rectangular QV Cycles, Part I 17310.1. Study of the rectangular QV cycle 17310.2. Practical implementation of the charge pump 17810.2.1. Evolution of the harvested energy 18010.3. Shortcomings of the single charge pump and required improvements 18210.3.1. Need for a flyback 18210.3.2. Auto-increasing of the internal energy 18310.4. Architectures of the charge pump with flyback 18410.4.1. Resistive flyback 18410.4.2. Inductive flyback 18510.5. Conditioning circuits based on the Bennet’s doubler 18810.5.1. Introduction of the principle . 18810.5.2. Analysis of the Bennet’s doubler conditioning circuit 19110.5.3. Simulation of a Bennet’s doubler 199Chapter 11. Circuits Implementing Rectangular QV Cycles, Part II 20311.1. Analysis of the half-wave rectifier with a transducer biased by an electret 20311.2. Analysis of the full-wave diode rectifier with transducer biased by an electret 20511.3. Dynamic behavior and electromechanical coupling of rectangular QV cycle conditioning circuits 21011.4. Practical use of conditioning circuits with rectangular QV cycle 21511.5. Conclusion on conditioning circuits for e-KEHs 216Bibliography 217Index 225