ELECTROMAGNETIC SYSTEM DESIGN FOR WIRELESS POWER

By

JOAQUIN JESUS CASANOVA

A DISSERTATION PRESENTED TO THE GRADUATE SCHOOLOF THE UNIVERSITY OF FLORIDA IN PARTIAL FULFILLMENT

OF THE REQUIREMENTS FOR THE DEGREE OFDOCTOR OF PHILOSOPHY

UNIVERSITY OF FLORIDA

2010

c© 2010 Joaquin Jesus Casanova

2

To my family, for their support and encouragement

3

ACKNOWLEDGMENTS

First and foremost, I’d like to thank Dr. Jenshan Lin for being the most helpful,

understanding, and encouraging advisor a student could ask for. He is one of the rare

professors who will give his students to explore their research on their own, and in

doing so, allows them to truly learn. Thanks are also due to my commitee, Dr. Henry

Zmuda, Dr. Robert Moore, and Dr. Subrata Roy, for their encouragement and insightful

questions. They put me at ease without going easy on me. I owe a debt of gratitude

to Zhen Ning Low for taking the first steps on this project, for his help understanding

power amplifiers, and for his friendship and conversation. He kept me sane. I thank

Jason Taylor, Ashley Trowell, and Raul Chinga for their technical support, guidance, and

friendship in working on this project. Thanks also go out to my parents and my brother,

who always supported me, even if it did seem like my life was nothing but my research to

the exclusion of all else. Finally, I’d like to thank Florida High Tech Corridor and Florida

Department of Environmental Protection for funding and support.

4

TABLE OF CONTENTS

page

ACKNOWLEDGMENTS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4

LIST OF TABLES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8

LIST OF FIGURES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

ABSTRACT . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14

CHAPTER

1 INTRODUCTION TO WIRELESS POWER TRANSFER . . . . . . . . . . . . . 16

2 LOOSELY-COUPLED NEAR FIELD WIRELESS POWER . . . . . . . . . . . . 18

2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 182.2 Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18

2.2.1 Design Equation for Crx . . . . . . . . . . . . . . . . . . . . . . . . 202.2.2 Design Equation for Lout . . . . . . . . . . . . . . . . . . . . . . . . 212.2.3 Design Equation for Cout . . . . . . . . . . . . . . . . . . . . . . . . 212.2.4 Design Equation for Ct . . . . . . . . . . . . . . . . . . . . . . . . . 22

2.3 Tests . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 232.4 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28

3 NEAR-FIELD ELECTROMAGNETIC ANALYSIS . . . . . . . . . . . . . . . . . 30

3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 303.2 Coil Fields . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 303.3 Coil Inductance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 313.4 Coil Parasitics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33

3.4.1 Round Conductor . . . . . . . . . . . . . . . . . . . . . . . . . . . . 333.4.1.1 Skin effect . . . . . . . . . . . . . . . . . . . . . . . . . . 333.4.1.2 Proximity effect . . . . . . . . . . . . . . . . . . . . . . . . 35

3.4.2 Rectangular Conductor . . . . . . . . . . . . . . . . . . . . . . . . . 363.4.2.1 Skin effect . . . . . . . . . . . . . . . . . . . . . . . . . . 373.4.2.2 Proximity effect . . . . . . . . . . . . . . . . . . . . . . . . 40

3.5 Litz Wire . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 423.6 Regulations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 423.7 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43

4 OPTIMAL PRIMARY COIL DESIGN . . . . . . . . . . . . . . . . . . . . . . . . 45

4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 454.2 Planar Wireless Power System . . . . . . . . . . . . . . . . . . . . . . . . 454.3 Coil Design . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 464.4 Testing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47

5

4.5 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 484.6 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49

5 M:N ANALYSIS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52

5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 525.2 Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 525.3 Tests Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55

5.3.1 Verification . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 575.3.2 Receiver Decoupling . . . . . . . . . . . . . . . . . . . . . . . . . . 585.3.3 Impact on Efficiency and Total Received Power . . . . . . . . . . . 63

5.4 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65

6 OPTIMAL PRIMARY COIL DESIGN FOR MULTIPLE COILS . . . . . . . . . . 67

6.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 676.2 Coil Design . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 676.3 System . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 686.4 Testing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 716.5 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 716.6 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73

7 INCLUSION OF FERRITES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74

7.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 747.2 Inductance Estimation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 757.3 Loss Estimation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 787.4 Thickness and Width Effects . . . . . . . . . . . . . . . . . . . . . . . . . 787.5 Experimental Evaluation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 807.6 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83

8 BAYESIAN LOAD/FAULT TRACKING . . . . . . . . . . . . . . . . . . . . . . . 84

8.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 848.2 Technology/Data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 868.3 Theory/Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86

8.3.1 State/Measurement Model . . . . . . . . . . . . . . . . . . . . . . . 868.3.2 Particle Filter Algorithm . . . . . . . . . . . . . . . . . . . . . . . . 87

8.3.2.1 Dataset generation . . . . . . . . . . . . . . . . . . . . . 878.3.2.2 Initialization . . . . . . . . . . . . . . . . . . . . . . . . . . 888.3.2.3 State . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 888.3.2.4 Measurement . . . . . . . . . . . . . . . . . . . . . . . . 888.3.2.5 Update . . . . . . . . . . . . . . . . . . . . . . . . . . . . 898.3.2.6 Estimate . . . . . . . . . . . . . . . . . . . . . . . . . . . 89

8.3.3 Tests . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 908.3.4 Implementation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90

8.4 Simulation Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91

6

8.5 Measured Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 958.6 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96

9 MIDRANGE WIRELESS POWER TRANSFER . . . . . . . . . . . . . . . . . . 103

9.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1039.2 Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104

9.2.1 Coil Design . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1059.2.2 Component Selection . . . . . . . . . . . . . . . . . . . . . . . . . 106

9.2.2.1 Series-parallel . . . . . . . . . . . . . . . . . . . . . . . . 1079.2.2.2 Series-series . . . . . . . . . . . . . . . . . . . . . . . . . 1089.2.2.3 T-network . . . . . . . . . . . . . . . . . . . . . . . . . . . 108

9.3 Preliminary Tests . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1099.3.1 Rectifying Diode Effects . . . . . . . . . . . . . . . . . . . . . . . . 1099.3.2 Frequency and Inductance Effects . . . . . . . . . . . . . . . . . . 1119.3.3 Topology Effects . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1139.3.4 Sensitivity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113

9.4 Synthesis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1189.4.1 50 cm Separation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1189.4.2 1 m Separation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 119

9.5 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123

10 FAR-FIELD WIRELESS POWER TRANSFER . . . . . . . . . . . . . . . . . . . 125

10.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12510.2 Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12710.3 Solution Details . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12810.4 Physical Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 129

10.4.1 Soil . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12910.4.2 Atmosphere . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 130

10.4.2.1 Gaseous water vapor . . . . . . . . . . . . . . . . . . . . 13010.4.2.2 Water droplets . . . . . . . . . . . . . . . . . . . . . . . . 13110.4.2.3 Ice crystals . . . . . . . . . . . . . . . . . . . . . . . . . . 133

10.4.3 Vegetation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13410.5 Results and Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . 135

10.5.1 Atmospheric Loss Estimation for Solar Power Satellite . . . . . . . 13610.5.2 Loss Estimation for Radiofrequency-Harvesting Sensor Under

Vegetation Canopy . . . . . . . . . . . . . . . . . . . . . . . . . . . 13810.5.3 Flux . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 139

10.6 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 139

11 CONCLUSIONS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 142

REFERENCES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 143

BIOGRAPHICAL SKETCH . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 150

7

LIST OF TABLES

Table page

2-1 Design parameters. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23

2-2 Component values. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23

4-1 Summary of system performance. . . . . . . . . . . . . . . . . . . . . . . . . . 49

5-1 Component values for 1 and 2 transmitter systems. . . . . . . . . . . . . . . . . 57

5-2 Maximum Prx and maximum ηc for different M:N arrangements. . . . . . . . . . 63

6-1 Design parameters. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68

6-2 Component values. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70

6-3 Summary of system performance. . . . . . . . . . . . . . . . . . . . . . . . . . 71

7-1 Ferrite properties. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81

7-2 Ferrite experimental evaluation with solenoid coil. . . . . . . . . . . . . . . . . . 82

9-1 Component values. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 110

9-2 Component values. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111

9-3 Component values. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112

9-4 Component values. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 118

9-5 Summary of 1 m tests. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 120

10-1 Parameter values for RTE. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 136

8

LIST OF FIGURES

Figure page

2-1 One-to-one wireless power system block diagram. . . . . . . . . . . . . . . . . 18

2-2 Class E driving circuit for a wireless power system. . . . . . . . . . . . . . . . 19

2-3 Test setup. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24

2-4 From top left, clockwise: Rin, ∠Ztx , drain voltage waveform at RL = 104Ω,and Q. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25

2-5 Received power and total efficiency as a function of RL. . . . . . . . . . . . . . 26

2-6 Received power and total efficiency as a function of RL, and their 95% confidenceintervals. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27

2-7 Efficiency as a function of RL. . . . . . . . . . . . . . . . . . . . . . . . . . . . 28

3-1 Current stick for MQS analysis. . . . . . . . . . . . . . . . . . . . . . . . . . . 31

3-2 Magnetic field components using MQS and MoM techniques of a 1m by 1msquare coil. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32

3-3 Magnetic field magnitude using MQS and MoM techniques of a 1m by 1msquare coil. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32

3-4 Conductor cross section showing field and current in round conductor underskin effect. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34

3-5 Conductor cross section showing field and current in round conductor underproximity effect. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35

3-6 Conductor cross section showing field and current in rectangular conductorunder skin effect. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37

3-7 Conductor cross section showing field and current in rectangular conductorunder proximity effect. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40

4-1 Transmitter test setup. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44

4-2 Coil layout. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45

4-3 Calculated z-directed magnetic field, assuming 1 A current (A/m). . . . . . . . 46

4-4 Field probe measurement (mV). . . . . . . . . . . . . . . . . . . . . . . . . . . 47

4-5 Received power (W) as a function of the location of the center of the receivingcoil. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49

4-6 Power (W) and efficiency (%) at loads from 10 Ω to 2 kΩ. . . . . . . . . . . . . 49

9

5-1 M:N block diagram. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52

5-2 Coil arrangements. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55

5-3 Measured vs. predicted Prx . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57

5-4 Power space plots for two-receiver tests with small receivers. . . . . . . . . . 58

5-5 Power space plots for two-receiver tests with large receivers. . . . . . . . . . . 58

5-6 Power space plot for three-receiver test. . . . . . . . . . . . . . . . . . . . . . 60

5-7 Power vs. efficiency plot for two-receiver tests with small receivers. . . . . . . 60

5-8 Power vs. efficiency plot for two-receiver tests with large receivers. . . . . . . 61

5-9 Power vs. efficiency plot for three-receiver tests with small receivers. . . . . . 61

5-10 Total received power as a function of RL, and its 95% confidence intervals . . 63

5-11 Total efficiency as a function of RL, and its 95% confidence intervals. . . . . . 64

6-1 Coil layout. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68

6-2 Calculated z-directed magnetic field, assuming 1 A current (A/m). . . . . . . . 68

6-3 Transmitter test setup. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69

6-4 Overlap of dual transmitter coils. . . . . . . . . . . . . . . . . . . . . . . . . . . 69

6-5 Received power (W) as a function of the location of the center of the receivingcoil. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71

6-6 Power (W) and efficiency (%) at loads from 75 Ω to 4 kΩ. . . . . . . . . . . . . 71

7-1 Diagram of ferrite shielding. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73

7-2 Empirical µeff predictions (red x) and observations (blue circle). . . . . . . . . 76

7-3 Flux-field hyteresis loop . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76

7-4 Effects of thickness and relative width of ferrite on inductance. . . . . . . . . . 78

7-5 Effects of thickness and relative width of ferrite on resistance. . . . . . . . . . 79

8-1 Generated measurements in (Vin,IDC ) space. . . . . . . . . . . . . . . . . . . . 90

8-2 True (blue) and estimated (red) mode for N=10, with resampling. . . . . . . . 91

8-3 True (blue) and estimated (red) mode for N=100, with resampling. . . . . . . . 92

8-4 True (blue) and estimated (red) mode for N=1000, with resampling. . . . . . . 93

10

8-5 True (blue) and estimated (red) mode for N=10, without resampling. . . . . . . 94

8-6 True (blue) and estimated (red) mode for N=100, without resampling. . . . . . 95

8-7 True (blue) and estimated (red) mode for N=1000, without resampling. . . . . 96

8-8 RMSE of mode and states. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97

8-9 Mode and charge estimate variance, with resampling. . . . . . . . . . . . . . . 97

8-10 Mode and charge estimate variance, without resampling. . . . . . . . . . . . . 98

8-11 Test of different modes in (Vin,IDC ) space, in real system. . . . . . . . . . . . . 98

8-12 True (blue) and estimated (red) mode for N=100, without resampling, in realsystem. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99

8-13 Predicted and observed power, resistance, and input voltage, and DC inputcurrent for N=100, without resampling, in real system. . . . . . . . . . . . . . 99

8-14 True (blue) and estimated (red) mode for N=1000, without resampling, in realsystem. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100

8-15 Predicted and observed power, resistance, and input voltage, and DC inputcurrent for N=1000, without resampling, in real system. . . . . . . . . . . . . . 100

8-16 True (blue) and estimated (red) mode for N=10000, without resampling, inreal system. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101

8-17 Predicted and observed power, resistance, and input voltage, and DC inputcurrent for N=1000, without resampling, in real system. . . . . . . . . . . . . . 101

9-1 Midrange class E series-parallel architecture. . . . . . . . . . . . . . . . . . . 106

9-2 Midrange class E series-series architecture. . . . . . . . . . . . . . . . . . . . 107

9-3 Midrange class E T network architecture. . . . . . . . . . . . . . . . . . . . . . 108

9-4 Diode effects on system performance. . . . . . . . . . . . . . . . . . . . . . . 109

9-5 Frequency and inductance effects on system performance. . . . . . . . . . . . 110

9-6 Topology effects on system performance. . . . . . . . . . . . . . . . . . . . . . 111

9-7 Effect of D, d , and f on total efficiency at N=8. . . . . . . . . . . . . . . . . . 113

9-8 Effect of D, d , and f on received power at N=8. . . . . . . . . . . . . . . . . . 113

9-9 Effect of N, f , and D on total efficiency where D = d . . . . . . . . . . . . . . . 114

9-10 Effect of N, f , and D on received power where D = d . . . . . . . . . . . . . . 114

11

9-11 Coil offset effects on system performance. . . . . . . . . . . . . . . . . . . . . 116

9-12 Efficiency at nominal component values (black line) and 95% confidence intervals.117

9-13 Power at nominal component values (black line) and 95% confidence intervals.118

9-14 50 cm system performance. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 120

9-15 1 m system setup. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121

9-16 1 m system performance. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122

10-1 An example of a radiofrequency (RF) harvesting wireless sensor node [3]. . . 124

10-2 An illustration of the Solar Power Satellite (SPS) concept [61]. . . . . . . . . . 124

10-3 Atmosphere model schematic used in this chapter [63]. . . . . . . . . . . . . . 125

10-4 Canopy model schematic used in this chapter [64]. . . . . . . . . . . . . . . . 125

10-5 Cloud droplet distribution and absorption cross section. . . . . . . . . . . . . . 131

10-6 Rain droplet distribution and absorption cross section. . . . . . . . . . . . . . 132

10-7 Cloud droplet distribution and scattering cross section. . . . . . . . . . . . . . 133

10-8 Rain droplet distribution and scattering cross section. . . . . . . . . . . . . . . 134

10-9 Ice sphere distribution and absorption cross section. . . . . . . . . . . . . . . 135

10-10 Ice sphere distribution and scattering cross section. . . . . . . . . . . . . . . . 136

10-11 Log intensity distribution through cloud. . . . . . . . . . . . . . . . . . . . . . . 137

10-12 Log intensity distribution through rain. . . . . . . . . . . . . . . . . . . . . . . . 138

10-13 Log intensity distribution through ice. . . . . . . . . . . . . . . . . . . . . . . . 139

10-14 Log intensity distribution through vegetation with diffuse and specular lowerboundary. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 140

10-15 Flux profile through different media. . . . . . . . . . . . . . . . . . . . . . . . . 140

12

Abstract of Dissertation Presented to the Graduate Schoolof the University of Florida in Partial Fulfillment of theRequirements for the Degree of Doctor of Philosophy

ELECTROMAGNETIC SYSTEM DESIGN FOR WIRELESS POWER

By

Joaquin Jesus Casanova

May 2010

Chair: Jenshan LinMajor: Electrical and Computer Engineering

Wireless communications technology has freed electronics from communication

cables. The natural next step is to cut the last wire of portable wireless devices,

the power cable. Wireless power systems would permit charging many different

devices equipped with receiving coils, in addition to delivering power through rooftops

and through the atmosphere. The approaches to wireless power transfer can be

categorized as near-field, midrange, and far-field. To date, the latter is still impractical

for consumer applications due to the high power and large antenna requirement

necessary to achieve levels of power comparable to a wall supply. On the other hand,

near-field inductive coupling has more promise as a wireless power technology for

charging battery-operated devices. Midrange power transfer has the most potential

for applications such as vehicle charging and power transmission through walls and

rooftops. Far-field applications include radiofrequency (RF) energy harvesting and

transmission of power from space.

This dissertation presents several apsects of the design and testing of wireless

power systems. Circuit topology and electromagnetic design of a near-field system is

considered, as well as the extension of the system to multiple coils. In addition, the

use of ferrite shielding and detection and estimation algorithms are considered for the

near-field system. The near-field architecture is extended and modified for a midrange

13

system. Finally, far-field power transfer through the atmosphere and the environment are

considered through the numerical solution of the radiative transfer equation.

14

CHAPTER 1INTRODUCTION TO WIRELESS POWER TRANSFER

The large number of battery operated consumer electronics and the associated

tangle of wall-wart chargers has generated interest in designing a single, convenient

charging platform [1]. Wireless battery charging systems would permit charging many

different devices equipped with receiving coils and cut the last wire of portable wireless

devices. Several techniques exist for transmitting power by electromagnetic fields.

They differ primarily by the distance between receiver and transmitter (D) and the

characteristic dimension of the transmitter (d), relative to wavelength (λ), and by their

typical power levels and applications [2]. Far-field, or radiative, power transfer occurs

when the distance between the transmitter and the receiver exceeds the Rayleigh

distance, D > 2d2/λ and d > λ. At nearer distances, (D ' d) is considered midrange.

Where D << d electromagnetic coupling is considered near-field.

Far-field power transfer is impractical for consumer applications due to the high

power and large antenna requirement necessary to achieve levels of power comparable

to a wall supply [2]. Two major applications of radiative wireless power transfer (WPT)

are in ambient radiofrequency (RF) harvesting and the Solar Power Satellite (SPS).

The idea behind the first technique is to convert the radio waves from communications

into power [3, 4]. The SPS is an idea which came about in the late 1960s [5], where the

principle is to collect solar energy in space using a satellite and beam it to a receiving

station on earth.

To date, midrange has shown promise in theory but practical tests at high power

levels are lacking in the literature. This is an appropriate range for applications such

as wireless charging of electric vehicles [6]. Evanescent couplng employs resonant

structures to ensure a strong link between transmitter and receiver [7] in this regime.

On the other hand, near-field inductive coupling has more promise as a consumer-level

wireless power technology. This is the familiar prinicple used in transformers and AC

15

and DC machinery. It can also be exploited for WPT to charge battery-operated devices

[8–11].

Because of this, this dissertation will focus on the design of a near-field system:

Chapters 2-7 consider the near-field system; Chapters 9-10 consider the midrange

and far-field systems, respectively. Chapter 2 discusses a near-field wireless power

system circuit architecture and design rules for this circuit. Chapter 3 discusses

the electromagnetic theory behind calculation of coil properties of inductance and

resistance. A technique for the optimal design of the primary coil is discussed in Chapter

4. The architecture is extended to a system with an arbitrary number of transmitters and

receivers in Chapter 5. Coil design for multiple transmitting coils in parallel is discussed

in Chapter 6. Chapter 7 discusses the use and evaluation of ferrite shielding. Chapter

8 describes the development and testing of a Bayesian tracking algorithm for receiver

discrimination and charge status determination in the near-field system. The extension

of the system and coil design to midrange is presented in Chapter 9. Chapter 10

describes the use of radiative transfer modeling for estimating losses of far-field wireless

power transmission. Chapter 11 presents some concluding remarks.

16

CHAPTER 2LOOSELY-COUPLED NEAR FIELD WIRELESS POWER

2.1 Introduction

Fig. 2-1 shows a block diagram for a generalized wireless power system and the

circuit diagram is shown in Fig. 2-2. The inverter is a class E amplifier [12] driven by

a low-power clock at 240 kHz, followed by a series-parallel impedance transformation

network [13]. Selecting the values of Crx , Lout , Cout , and Ct for optimum performance of

the wireless power system presents a challenge. [14] shows the design methodolgy for

a similar architecture with closed loop control. [15] demonstrates how to choose design

values for a class E without relying on Raab’s waveform equations; however, it involves

numerical root finding. [16] presents a selection technique for an open-loop system that

relies on sweeping component values numerically until the impedance and drain voltage

satisfy certain constraints. While this method successfully finds appropriate component

values, it is time consuming. This chapter derives simple formulas for the optimum

component values by applying the same constraints.

2.2 Analysis

The optimum values for Crx , Lout , Cout , and Ct can be derived by applying several

constraints on the systems response to the variable load resistance RL. In this analysis,

it is assumed that the components are lossless. In addition, assumptions are made

about the class E to allow use of Raab’s equations [17], namely that the transistor is

Figure 2-1. One-to-one wireless power system block diagram.

17

Figure 2-2. Class E driving circuit for a wireless power system.

a perfect switch and that the choke inductance is infinite. Before these derivations it is

necessary to have expressions for receiver impendance, Zrx , input impedance looking

into the transmitter coil, Zin, and impedance looking into Lout , Ztx . Zrx can be expressed

as follows:

Zrx = Rrx + jXrx

= RL||Crx (2–1)

=RL − jωCrxR2L1 + ω2R2LC

2rx

(2–2)

Zin can be found by examining the coupling equations:

V1 = jωL1I1 + jωMI2

V2 = jωMI1 + jωL2I2 (2–3)

where L1,L2, and M, are transmitter coil, receiver coil, and mutual inductance,

respectively. Zin is V1/I1

18

Zin = Rin + jXin

=ω2M2Rrx

R2rx + (ωL2 + Xrx)2

+ j

(ωL1 −

ω2M2(ωL2 + Xrx)

R2rx + (ωL2 + Xrx)2

)(2–4)

Ztx is just Zin with additional series reactance from Cout and Lout :

Ztx = Rtx + jXtx

= Rin + j

(ωLout −

1

ωCout+ Xin

)(2–5)

2.2.1 Design Equation for Crx

Selection of Crx is determined on the basis of efficiency and quality factor Q. If the

real part of Zin is too low compared to the coil parasitics, the system will be inefficient. If

it is too large, it is difficult to get Q high enough for class E operation (about 1.78 [18]).

By forcing the peak real part of Zin to be a specified value (R0), a compromise between

efficiency and Q can be reached. To derive which Crx forces the maximum real part of

Zin to be R0, the RL corresponding to the peak value is found by setting ∂Rin∂RL

to zero. This

yields a polynomial of degree six, where four of the roots are comprised by a double

conjugate pair and can thus be ignored

RL = ± j

ωCrx(2–6)

There are two real roots

RL = ± ωL21− ω2L2Crx

(2–7)

which can be substituted back into Eq. (2–4); then the real part is set to R0

19

Rin = ± M2L2

ωM

ω2CrxL2 − 1= R0 (2–8)

Solving for Crx yields two roots

Crx =R0L2 ± 1

2ωM2

R0ω2L22(2–9)

The negative root gives a Crx which ensures that Zin phase will increase with

increasing RL, which has the desirable effect of lowering power delivery at high load

resistance. This is desirable because in the case of a device being charged, high load

resistance (thousands of Ω) corresponds to fully-charged condition and thus low power

requirement.

2.2.2 Design Equation for Lout

The purpose of Lout is to ensure the circuit has a minimum Q high enough for

proper functioning of the class E. Q is smallest when the real part of Zin is highest, at R0.

Since Lout contributes the largest part of the reactance of Ztx ,

Q ∼ ωLoutR0

(2–10)

Lout is found:

Lout = ω−1QR0 (2–11)

2.2.3 Design Equation for Cout

Cout brings the range of the phase of Ztx to a range which allows ZVS operation

of the class E and maximum efficiency. From [19], this phase range is 40o to 70o . By

setting the minimum phase to a specified value, φ, efficient operation can be achieved.

The location of the minimum phase is where

∂∠(Ztx)∂RL

= 0 (2–12)

which yields a quadratic equation in RL with the roots

20

RL = ±2R0(L2M

)2×

√ωCout(ωL1(k2 − 1)− R0Q)− 1

ωCout(ωL1(k2 − 1)− R0(Q + 2))− 1(2–13)

where k =√M2

L1L2. Substituting Eq. (2–13) into Eq. (2–5) and setting tan(φ) = Xtx

Rtx

and squaring to eliminate the radical in the numerator yields a quadratic equation in

Cout , with two real roots.

Cout = ω−1ωL1(1− k2) + R0(Q + 1± sec(φ))

× R20 (Q2 + 2Q − tan(φ)2) + 2QR0ωL1(1− k2)

+ ω2L21(1− k2)2 + 2ωL1R0(1− k2)−1 (2–14)

The greater root, corresponding to positive φ, yields

Cout =ω−1

ωL1(1− k2) + R0(Q + 1− sec(φ))(2–15)

after simplification.

2.2.4 Design Equation for Ct

Finally, Ct is selected to guarantee zero voltage switching (ZVS) operation of the

class E. From [17], this optimum value of Ct , given a load resistance R is

Ct =2ω−1

(1 + π2

4)R

(2–16)

Since the load resistance in the wireless power system is variable, Ct is selected

based on the maximum R, which for the circuit under consideration is the magnitude of

Ztx as RL increases to infinity. Taking this limit,

21

Table 2-1. Design parameters.Parameter ValueL1 34.58 µHL2 4.05 µHM 1.65 µHR0 7.5 ΩQ 2φ 65o

Table 2-2. Component values.Component Calculated MeasuredCrx 100.55 nF 100.00 nFCout 11.90 nF 11.57 nFCt 15.15 nF 14.55 nFLout 9.95 µH 9.52 µH

R = ωLout +1

ωCout− ω2M2

ωCrxω2CrxL2 − 1

(2–17)

and substituting in the derived values for the other components (Eqs. (2–9), (2–11),

(2–15)), the optimum Ct is found:

Ct =2ω−1

(1 + π2

4)(1 + sec(φ))R0

(2–18)

2.3 Tests

Having derived the optimum component values for a given L1, L2, M, R0, Q, and φ,

this section demonstrates the performance of the system.

A test system was built, consisting of a 16 cm by 18 cm, 13 turn, spiral transmitting

coil, designed by the technique described in [20], and a rectangular 4 cm by 5 cm,

6 turn, receiving coil. Both coils were constructed of 100 strand, 40 AWG Litz wire

to minimize coil parasitics. Fig. 2-3 shows a picture of the test system. Table 2-1

gives values of the inductances L1, L2, M, and design parameters R0, Q, and φ. R0

was chosen as 7.5 Ω based on the total transmitting and receiving coil parasitics

which amounted to about 0.5 Ω. In general, selection of R0 is system-dependent but it

22

Figure 2-3. Test setup. Top is a diagram showing the coils, where red is the receiver andblue is the transmitter. Bottom is a photograph of the coils.

23

Figure 2-4. From top left, clockwise: Rin, ∠Ztx , drain voltage waveform at RL = 104Ω, andQ.

should be at least an order of magnitude higher than the parasitics. How large an R0 is

acceptable depends on the availability of inductors of sufficient size to achieve minimum

Q. Minimum Q and φ values were chosen based on allowable values given in [18, 19],

with additional buffer to tolerate some deviations of real components. Based on the

proposed method and equations in Section 2.2, the optimum component values were

calculated. Table 2-2 gives the calculated component values and also lists the measured

values of the actual components being used. Fig. 2-4 shows the calculated Rin, ∠Ztx ,

drain voltage waveform, and Q, demonstrating that the desired constraints are met using

the component selection formulas.

One of the key challenges for a wireless power system is to have desirable

performance responding to variable load. To evaluate the system performance with

regards to power and efficiency, RL was swept from 60 to 4000 Ω by means of an

electronic load. The DC voltage and current were measured at the load and at the

supply. Supply voltage was 12 V.

24

Figure 2-5. Received power and total efficiency as a function of RL. Circles aremeasured values and the solid line is simulated, using the calculatedcomponent values.

Fig. 2-5 shows the received power and total efficiency versus load resistance, and

the simulated values of power and efficiency, using the ideal, calculated component

values. The system has power delivery of over 3.7 W, and peak efficiency over 66%.

The important feature in Fig. 2-5 is the trend of decreasing received power and total

efficiency with increasing RL, which is guaranteed by the component selection. The ideal

performance is close to the real system. Power and efficiency are lower by 5%-10% in

the actual system, primarily due to the deviation of Lout and Cout in the real components.

Deviations from their ideal values cause the φ and Q to shift which has a substantial

effect on the class E efficiency.

To further investigate the sensitivity to component selection, a Monte Carlo

simulation was run, assuming the components are normally distributed, with means

given by the derived component formulas and with standard deviations, σ, such that 3σ

is the component tolerance. These simulations were carried out at tolerance levels of

25

Figure 2-6. Received power and total efficiency as a function of RL, and their 95%confidence intervals with 5% component tolerance (circles), 10% componenttolerances (pluses), and 20% component tolerances (squares).

5%, 10%, and 20%. Fig. 2-6 shows the 95% confidence intervals for received power

and total efficiency at the three tolerance levels. As can be seen, the power is skewed

low, with tolerances of about +35/-25% at 5% tolerance, +115/-45% at 10% tolerance,

and +425/-70% at 20% tolerance. The efficiency is skewed very high, with tolerances

of about +3.4/-18% at 5% tolerance, +3.6/-38% at 10% tolerance, and +4.3/-70% at

20% tolerance. This skew low in the power confidence intervals and skew high in the

efficiency confidence intervals shows that the system is not optimized for maximum

power delivery but rather efficiency. This makes sense, as all of the constraints (R0, φ,

Q, and ZVS) used for component selection are chosen to maximize the efficiency of the

class E. At tolerance of 5% or less, the confidence interval shows that the performance

is still decent. For greater tolerances, the potential variability of performance is probably

unacceptable for most applications.

26

Figure 2-7. Efficiency as a function of RL. Squares are class E efficiency, pluses arecoupling efficiency, and circles are total efficiency.

Fig. 2-7 shows the amplifier efficiency, coupling efficiency, and total efficiency.

Amplifier efficiency is AC transmitting power over DC input power; coupling efficiency

is DC power at the load over AC transmitted power; and total efficiency is the product

of these two efficiencies. From Fig. 2-7, the coupling efficiency comprises most of the

losses. The coupling efficiency decreases with increasing load resistance because as

RL increases, Rin decreases, while the parasitics remain the same, so more power is

dissipated through the parasitics. The amplifier efficiency peaks where ∠Ztx is in the

40o − 70o range, which by design is in the neighborhood of the phase minimum.2.4 Conclusion

This chapter has presented a set of design equations for optimizing the performance

of a class E amplifier used in inductively coupled wireless power system. By applying

constraints on the real part of the input impedance to the primary coil, the phase of the

input impedance, the minimum Q, and the drain voltage waveform, components can

be selected to guarantee desirable operating characteristics of the system, namely, the

27

ZVS operation of the class E and the trend of decreasing power received with increasing

load resistance. The proposed optimization method was tested in a system composed

of a 16 cm by 18 cm primary coil and a 4 cm by 5 cm secondary coil with a variable

load. The system shows power delivery of over 3.7 W, and peak efficiency over 66%,

in addition to the desirable trend of decreasing power and efficiency with respect to

increasing load resistance.

28

CHAPTER 3NEAR-FIELD ELECTROMAGNETIC ANALYSIS

3.1 Introduction

The performance of a near-field wireless power system depends heavily on the

electromagnetic properties of the primary and secondary coils. In particular, the fields

produced by the coil, the coil inductance, and the coil parasitics are important to know.

This chapter derives analytically these quantities for round and rectangular conductors

for use in system design.

3.2 Coil Fields

At frequencies less than 500 kHz, instead of solving Maxwell’s coupled equations,

it is still sufficiently accurate to calculate the magnetic field using the magnetostatic

solution, that is, the Biot-Savart Law. This is known as the magnetoquasistatic (MQS)

solution [21]. The analytical MQS solution for a line of current is presented here (see

Fig. 3-1). The fields produced by a polygonal coil can be constructed by superposition.

H =1

4π

∫V ′

~J(~r ′)× ir ′r|~r −~r ′|2

dV ′ (3–1)

=I

4π

∫ ξc

ξb

~c × ~adξ

|a|(ξ2 + r 20 )3/2(3–2)

=I

4π

~c × ~a

|a|ξ

r 20 (ξ2 + r 20 )

1/2

∣∣∣ξcξb

(3–3)

=I

4π

~c × ~a

|~c × ~a|2(~a · ~c

|c |−

~a · ~b|b|

)(3–4)

To test the MQS analytical solution, it was compared to a method of moments

(MoM) solution [22] calculated using the Numerical Electromagnetics Code (NEC) [23].

In particular, a single turn, 1 m by 1 m square coil was tested using both techniques.

The fields were calculated at a 2m by 2m plane 5 cm above the coil. While frequency is

not used in MQS, it is in MoM, and the frequency used in this case was 240 kHz.

29

Figure 3-1. Current stick for MQS analysis.

Fig. 3-2 shows the magnitude of the the field components (Hx ,Hy ,HZ ) for both MQS

and MoM techniques. From the plots, it is evident that the distributions are similar. The

magnitude of the difference, in terms of root mean square difference of the x-component

is 6.3239×10−5 A/m, of the y-component is 9.4997×10−5 A/m, and of the z-component

is 8.4331×10−5. The peak field magnitude in both cases is 1 A/m.

3.3 Coil Inductance

The inductance matrix Mij relates the flux from a primary coil i through a secondary

coil j to the primary’s current. [24] presents many inductance formulas for different

geometries. The most general for a filamentary coil is the Neumann formula:

Mij =µ04π

∮ci

∮cj

~ds i · ~ds j

|~Rij |(3–5)

The self inductance is similar:

Mii =µ04π

∮ci

∮ci

~ds i · ~ds i

|~Rii |

∣∣∣|R|≥a/2

+ Lp,ii (3–6)

where Lp,ii is the internal inductance of the conductor and a is the radius.

30

Figure 3-2. Magnetic field components using MQS and MoM techniques of a 1m by 1msquare coil.

Figure 3-3. Magnetic field magnitude using MQS and MoM techniques of a 1m by 1msquare coil.

31

3.4 Coil Parasitics

The effect of the distribution of current with in the conductor is magnified at higher

frequency. Internal resistance and inductance depend on the distribution of current.

There are two main mechanisms for high frequency losses and internal inductance:

the skin effect and proximity effect. The skin effect is when current distribution is

mostly towards the surface of the conductor, raising the resistance. Proximity effect is

when magnetic fields from nearby conductors induce eddy currents on the conductor

surface, increasing the power dissipation. These two effects are discussed in the

context of transformer design in the literature; there are analytical [25, 26] and empirical

[27, 28] treatments of these effects. However, the results published in the literature

are incomplete in that the exact current distributions are not given, and neither is the

effect on conductor internal inductance. In addition, no paper derives these effects for

a conductor of arbitrary rectangular cross section, such as a PCB trace. This section

derives the current distribution and resistance and inductance under skin and proximity

effects for both round and rectangular conductors.

3.4.1 Round Conductor

[29] presents a partial description of the skin effect in round conductor. This section

presents a complete description of skin and proximity effects. Both skin and proximity

effect parasitics for a round conductor are derived through application of Maxwell’s

equations.

3.4.1.1 Skin effect

Fig. 3-4 shows the wire cross section and boundary conditions. Using the current

distribution equation, separation of variables, and the surface electric field and symmetry

32

Figure 3-4. Conductor cross section showing field and current in round conductor underskin effect.

as boundary conditions, the total current can be related to the surface electric field E0:

∇2Jz = jωσµJz (3–7)

α2 = −jωσµ (3–8)

∂2Jz∂r 2

+1

r

∂Jz∂r+ α2Jz = 0 (3–9)

Jz = C1J0(αr) (3–10)

Jz = σE0J0(αr)

J0(αr0)(3–11)

Ez = E0J0(αr)

J0(αr0)(3–12)

Hφ =1

jωµ

∂Ez∂r

(3–13)

=σE0α

J ′0(αr)

J0(αr0)(3–14)∮

~H · d~l = I = 2πr0Hφ(r0) (3–15)

I =σE0α

J ′0(αr)

J0(αr0)(3–16)

where Jz is current, Hφ is φ-directed magnetic field, Ez is z-directed electric field, and I is

total current.

33

Figure 3-5. Conductor cross section showing field and current in round conductor underproximity effect.

The next steps relate the total current to the parasitics. With the definition of internal

impedance,

Z =Ez(r0)

I(3–17)

= R + jωL (3–18)

Rs =1

σδ(3–19)

δ =1√

πf σµ(3–20)

γ =√2r0δ

(3–21)

Rp =Rs√2πr0

(Ber(γ)Bei ′(γ)− Bei(γ)Ber ′(γ)Ber ′2(γ) + Bei ′2(γ)

)(3–22)

Lp =Rs√2πr0

(Ber(γ)Ber ′(γ)− Bei(γ)Bei ′(γ)Ber ′2(γ) + Bei ′2(γ)

)(3–23)

where Bei and Ber are the imaginary and real Kelvin functions.

3.4.1.2 Proximity effect

Similarly, proximity effect can be handled by applying the boundary condition of an

imposed tangential surface magnetic field. Fig. 3-5 shows the wire cross section and

boundary conditions. Beginning with the current distribution equation, the fields can be

related to the real and reactive power. The power terms may be related to the parasitics

34

through the total current:

1

r 2∂2Jz∂θ2

+1

r

∂

∂r

(r∂Jz∂r

)+ α2Jz = 0 (3–24)

Jz = C1J1(αr) cos(θ) (3–25)

Jz = H0J1(αr)

J1(αr0)cos(θ) (3–26)

Ez =H0σ

J1(αr)

J1(αr0)cos(θ) (3–27)

jωσµ~H = ∇× Ez (3–28)

~H =H0α2

( rr

J1(αr)

J1(αr0)sin(θ) + θα

J ′1(αr)

J1(αr0)cos(θ)

)(3–29)

Preal =σ

2

∫ 2π

0

∫ r0

0

|E |2rdrdθ (3–30)

Pimag =ωµ

2

∫ 2π

0

∫ r0

0

|H|2rdrdθ (3–31)

Preal =π

2

H20σ

∫ r0

0

∣∣∣ J1(αr)J1(αr0)

∣∣∣2rdr (3–32)

Pimag =ωµπ

2

H20α2

∫ r0

0

(1

(αr)2

∣∣∣ J1(αr)J1(αr0)

∣∣∣2 + ∣∣∣ J ′1(αr)J1(αr0)

∣∣∣2)rdr (3–33)

Rp =2PrealI 2p

(3–34)

Lp =2PimagωI 2p

(3–35)

3.4.2 Rectangular Conductor

A rectangular conductor of width w and thickness t, such as a printed circuit board

(PCB) trace can be handled with the current distribution PDE in Cartesian coordinates.

Similar derivations as in the previous section can be carried out in order to derive the

parasitics for a rectangular cross section conductor. The following sections derive the

parasitics through application of Maxwell’s equations.

35

Figure 3-6. Conductor cross section showing field and current in rectangular conductorunder skin effect.

3.4.2.1 Skin effect

Fig. 3-6 shows the rectangular cross section and boundary conditions used in this

section. The boundary condition is one of a constant tangential electric field. Using the

reactive and real powers within the cross section, and the total current, the parasitics

may be derived. Applying the PDE and separation of variables, where the total current

distribution must be handled by superposition of two solutions, one associated with a

homogeneous x-direction (Jz1) and another associated with a homogeneous y-direction

(Jz2):

36

∇2Jz = −α2Jz (3–36)

∂2Jz∂x2

+∂2Jz∂y 2

+ α2Jz = 0 (3–37)

Jz = Jz1 + Jz2 (3–38)

Jz1 = X (x)Y (y) (3–39)

X (x) = A cos(βx) + B sin(βx) (3–40)

X ′(0) = 0 (3–41)

X (w/2) = 0 (3–42)

X = A cos(βnx) (3–43)

βn = (2n + 1)π

w(3–44)

Y (y) = C cosh(γy) +D sinh(γy) (3–45)

γ2n = β2n − α2 (3–46)

Y ′(0) = 0 (3–47)

Y (t/2)X (x) = σE0 (3–48)

Yn = an cosh(γny) (3–49)

an =

∫ w/2

0σE0 cos(βnx)dx

cosh(γnt/2)∫ w/2

0cos2(βnx)

(3–50)

an =4(−1)n

π(2n + 1) cosh(γnt/2)(3–51)

Jz1 = σE0Σ∞0 an cosh(γny) cos(βnx) (3–52)

37

Jz2 = X (x)Y (y) (3–53)

Y (y) = A cos(δy) + B sin(δy) (3–54)

Y ′(0) = 0 (3–55)

Y (t/2) = 0 (3–56)

Y = A cos(δny) (3–57)

δn = (2n + 1)π

t(3–58)

X (x) = C cosh(εx) +D sinh(εx) (3–59)

ε2n = δ2n − α2 (3–60)

X ′(0) = 0 (3–61)

X (w/2)Y (x) = σE0 (3–62)

Xn = bn cosh(εnx) (3–63)

bn =

∫ t/20

σE0 cos(δny)dy

cosh(εnw/2)∫ t/20cos2(δny)

(3–64)

bn =4(−1)n

π(2n + 1) cosh(εnw/2)(3–65)

Jz2 = σE0Σ∞0 bn cosh(εnx) cos(δny) (3–66)

38

Jz = σE0Σ∞0

4(−1)n

π(2n + 1)(3–67)

×(cosh(γny) cos(βnx)

cosh(γnt/2)+cosh(εnx) cos(δny)

cosh(εnw/2)

)(3–68)

−jωµσHx =∂Jz∂y

(3–69)

jωµσHy =∂Jz∂x

(3–70)

Hx =σE0α2Σ∞0

4(−1)n

π(2n + 1)(3–71)

×(γn sinh(γny) cos(βnx)

cosh(γnt/2)− δn cosh(εnx) sin(δny)

cosh(εnw/2)

)(3–72)

Hy =σE0α2Σ∞0

4(−1)n

π(2n + 1)(3–73)

×(βn cosh(γny) sin(βnx)

cosh(γnt/2)− εn sinh(εnx) cos(δny)

cosh(εnw/2)

)(3–74)

I

4=

∫ t/2

0

∫ t/2

0

Jzdxdy (3–75)

Ip = σE0Σ∞0

16(−1)n

π(2n + 1)

(tanh(γnt/2)w + tanh(εnw/2)t

)(3–76)

Preal =σ

2

∫ 2π

0

∫ r0

0

|E |2rdrdθ (3–77)

Pimag =ωµ

2

∫ 2π

0

∫ r0

0

|H|2rdrdθ (3–78)

Rp =2PrealI 2p

(3–79)

Lp =2PimagωI 2p

(3–80)

3.4.2.2 Proximity effect

Fig. 3-7 shows the rectangular cross section and boundary conditions used in this

section. The boundary condition is one of an arbitrary, constant, tangential, magnetic

field. Using the reactive and real powers within the cross section, and the total current,

the parasitics may be derived. Proximity effect can be handled with the imposition of

39

Figure 3-7. Conductor cross section showing field and current in rectangular conductorunder proximity effect.

tangential magnetic field on the vertical sides, and odd symmetry about the y-axis:

∇2Jz = −α2Jz (3–81)

∂2Jz∂x2

+∂2Jz∂y 2

+ α2Jz = 0 (3–82)

Jz = X (x)Y (y) (3–83)

Y (y) = A cos(γy) + B sin(γy) (3–84)

Y ′(0) = 0 (3–85)

Y (t/2) = 0 (3–86)

Y = A cos(γny) (3–87)

γn = (2n + 1)π

t(3–88)

X (x) = C cosh(βx) +D sinh(βx) (3–89)

β2n = γ2n − α2 (3–90)

X (w/2)Y (x) = σE0 (3–91)

Xn = an sinh(βnx) (3–92)

40

α2Hoy = −∂Jz∂x

∣∣∣x=w/2

(3–93)

an =−α2H0y

∫ t/20

σE0 cos(γny)dy

βn cosh(βnw/2)∫ t/20cos2(γny)

(3–94)

an =α2H0y4(−1)n+1

π(2n + 1)βn cosh(βnw/2)(3–95)

Jz = Σ∞0 an sinh(βnx) cos(γny) (3–96)

I

4=

∫ t/2

0

∫ t/2

0

Jzdxdy (3–97)

Ip = σE0Σ∞0

16(−1)n

π(2n + 1)

(tanh(γnt/2)w + tanh(βnw/2)t

)(3–98)

Preal =σ

2

∫ 2π

0

∫ r0

0

|E |2rdrdθ (3–99)

Pimag =jωµ

2

∫ 2π

0

∫ r0

0

|H|2rdrdθ (3–100)

Rp =2PrealI 2p

(3–101)

Lp =2PimagωI 2p

(3–102)

3.5 Litz Wire

To mitigate the high frequency parasitics described in the previous sections, Litz

wire could be used as the coil conductor [30]. Litz wire is stranded wire where the

strands are insulated from one another. Since the size of the individual conductors is

much less than the skin depth, the skin and proximity effects are minimized. Typically,

Litz wire is specified in terms of the number of strands and the gauge of the individual

wires. For instance, “100/40” Litz wire is 100 strands of 40 AWG wire.

3.6 Regulations

As the application of wireless power considered in this dissertation is largely

consumer electronics, some discussion of the health and human safety aspects, as well

as appropriate federal regulations, is in order. This section summarizes some regulatory

constraints.

41

FCC Part 18 [31] concerns unlicensed intential, unintential, or incidental radiators

for industrial, medical, or scitific (ISM) and non-ISM equipment. According to Part

18, operation within specific search and rescue bands (490-510 kHz, 2170-2194

kHz, 8354-8374 kHz, 121.4-121.6 MHz, 156.7-156.9 MHz, and 242.8-243.2 MHz).

Additionally there are field strength limits for different frequency bands and applications

and conducted emission limits. If a wirelss device is designed to work in non ISM bands

the the field strength limit is 15 µV/m at 300 m and the conducted emission limit is 66-56

dBµV (quasi-peak) and 56-46 dBµV (average)

FCC Part 15 [32] is concerned with radio frequency devices. Specifically, subpart B

concerns unintentional radiators. There are radiated and conducted emission limits. For

conducted emissions, the limit is 66-56 dBµV (quasi-peak) and 56-46 dBµV (average).

Radiated emission limits are (in µV/m), for frequency between 9 and 490 kHZ, 2400

divided by the frequency in kHz, measured at 300 m. For frequency between 490 and

1705 kHZ, 24000 divided by the frequency in kHz, measured at 30 m.

IEEE C95.1 describes limits on field strength, current and specific absorption

rate (SAR) for health and human safety [33]. Exposure limits are defined for two

cases, controlled and uncontrolled environments (the general public). For consumer

electronics, and near-field induction, the primary concern would be magnetic field

strength limits in uncontrolled environments. Maximum permissible exposures (MPEs),

for head and torso, in uncontrolled environments, between 3.35-5000 kHz, are an rms

flux of 0.205 mT and field of 163 A/m. In the limbs, MPEs are an rms flux of 1.13 mT

and field of 900 A/m. Additionally, there are specific absorption rate (SAR) limits: for the

whole body, 0.08 W/kg; for any localized 10 g of tissue, 2 W/kg; and for extremities, 4

W/kg.

3.7 Conclusion

This chapter analytically derived useful relationships for the fields produced by the

coil, the coil inductance, and the coil parasitics for round and rectangular conductors.

42

These relationships can be used to help estimate performance and guide the design

of the coils before construction of a system. Combined with the design equations from

Chapter 2, this provides a basis for an electronic design automation (EDA) code used

throughout the rest of this dissertation.

43

CHAPTER 4OPTIMAL PRIMARY COIL DESIGN

4.1 Introduction

If multiple devices are to be charged simultaneously on the same system, the

transmitting coil must be large enough to accomodate them. This poses a challenge, as

to ensure uniform power delivery to devices, regardless of position, the electromagnetic

field distribution must be even. In particular, the distribution of the z-component of

the magnetic field in the plane of the receiving coils must be as uniform as possible.

Transmitting coils may be designed to produce such fields; one approach is the optimal

hybrid coil design [34], which is demonstrated for as large a coil as 15 cm by 15 cm.

This chapter describes a different technique for coil design (the primary difference being

the parameterization of the coil shape), which is demonstrated for a 20 cm by 20 cm coil.

4.2 Planar Wireless Power System

In this chapter, the system was configured with series-parallel compensation as

described in [13]. The transmitting coil follows, which is in turn inductively coupled to

the receiving coil, a rectangular coil of 6 cm by 8 cm and 6 turns. Both transmitter and

receiver coils were constructed by hand using Litz wire to reduce resistive losses from

proximity and skin effects. The receiving coil is connected to the second half of the

Figure 4-1. Transmitter test setup.

44

0 2 4 6 8 10 12 14 16 18 200

2

4

6

8

10

12

14

16

18

20

x position (cm)

y po

sitio

n (c

m)

Figure 4-2. Coil layout.

transformation network, a parallel capacitor, and followed by a rectifier and a receiver

load. A picture of the test setup is shown in Fig. 4-1.

4.3 Coil Design

The transmitting coil is a rectangular spiral with blunted corners, where the ratio of

the width of a turn to the overall width, f , is defined by:

f = 1− (1− (N − n + 1)/N)k (4–1)

where n is the turn number, counting from the outside, and N is the number of turns,

k is a parameter, and ∆ gives the fraction of of each corner to be removed to blunt the

corners. The spiral geometry is entirely described by free parameters N, k , and ∆; and

the length and width, which are fixed at 20 cm by 20 cm for this example. By sweeping

the parameter values, evaluating the fields, and calculating an objective function, the

45

0 2 4 6 8 10 12 14 16 18 200

2

4

6

8

10

12

14

16

18

20

x position (cm)

y po

sitio

n (c

m)

−150

−100

−50

0

50

100

150

200

250

Figure 4-3. Calculated z-directed magnetic field, assuming 1 A current (A/m).

coil design which produced maximally flat fields for a coil of the specified size was

determined. The analytical magnetoquasistatic (MQS) solution for a line of current (see

Chapter 3) was used to build the fields for the entire current in a plane 1 mm above the

coil. The objective function was chosen as the coefficient of variation (COV, the standard

deviation divided by the mean) of the z-component of the magnetic field. Minimizing

the COV minimizes the relative variations in the field, ensuring a smooth distribution.

The final optimal coil layout is shown in Fig. 4-2, and the corresponding MQS fields are

shown in Fig. 4-3.

4.4 Testing

The transmitting coil was tested in three ways. First, the z-directed magnetic field

was measured with a 6 cm diameter field probe. Second, the receiver position was

varied over the entire transmitter coil area to gauge the uniformity of wireless power

transfer. Finally, the receiver was fixed at the center of the transmitter and the load

resistance was swept from 10 to 2000 Ω.

46

0 2 4 6 8 10 12 14 16 18 200

2

4

6

8

10

12

14

16

18

20

x position (cm)

y po

sitio

n (c

m)

100

120

140

160

180

200

Figure 4-4. Field probe measurement (mV).

The chief figures of merit considered in the test were the DC power supplied to the

amplifier, the AC power transmitted, and the DC power delivered to the resistive load,

in addition to the amplifier, the coupling, and the total efficiency. Amplifier efficiency

is defined as the ratio of transmitted power to supplied power; coupling efficiency

is defined as the ratio of power received by the load to transmitted power; and total

efficiency is defined as the product of the previous two.

4.5 Results

Table 4-1 summarizes the perfomance charcteristics of the coil. The field measurement

results are shown in Fig. 4-4, in terms of the voltage measured on the field probe. The

peak in the lower left corner corresponds to the location of the input leads, and the

peaks in other corners are due to the superposition of fields at corners of the spiral.

The apparent drop-off at the edges is due to the spatial averaging effect of the probe.

A small portion of the probe was outside of the coil, where the field reverses and

47

Table 4-1. Summary of system performance.Size 20 cm by 20 cm

Peak delivered power 11.8 WPeak total efficiency 80.9%

Peak coupling efficiency 88.4%COV 2.2%

Self-inductance 45.00 µHResistance 0.37 Ω

becomes negative, pulling the average down. Aside from this artifact of the field probe

measurement, the general trend of the field matches the MQS calculations.

From Fig. 4-5, the spatial uniformity of the received power is shown. The two

notable peaks match the field peaks at the corners and near the leads, which can be

seen in the field plots. These peaks are relatively small, however, as the maximum

variation shown in the plot is 0.8 W, less than 10% of the mean. The COV is likewise

small, at 2.2%.

Fig. 4-6 shows the results from the variable loading test. As can be seen, the

received power is maximum at 25 Ω and a value of 11.8 W. The maximum total

efficiency is 80.9% at 100 Ω and the maximum coupling efficiency is 88.4% at 250

Ω. The efficiencies are high under a wide range of loads (it should be noted that the

efficiency is lessened slightly in the presence of multiple loads). This demonstrates

the system’s robustness, not only with respect to receiver placement, but also loading

conditions.4.6 Conclusion

This chapter has demonstrated the feasibility of large transmitting coils for open air

inductively coupled power transfer. Large transmitting coils such as this may be used

for wireless charging of multiple battery-powered devices equipped with receiving

coils, such as cellphones and PDAs. The primary challenge in designing such a

coil is achieving an even power delivery regardless of receiver position, in order to

accommodate multiple devices. Such a coil design was achieved through optimization,

and the 20 cm by 20 cm coil was built and tested with a switchmode power amplifier

48

0 2 4 6 8 10 12 14 16 18 200

2

4

6

8

10

12

14

16

18

20

x position (cm)

y po

sitio

n (c

m)

8.6

8.7

8.8

8.9

9

9.1

9.2

9.3

Figure 4-5. Received power (W) as a function of the location of the center of thereceiving coil.

101

102

103

104

0

5

10

15

20

Pow

er (

W)

Input powerTransmitter powerRecieved Power

101

102

103

104

20

40

60

80

100

Rl (Ω)

Effi

cien

cy (

%)

Coupling efficiencyAmplifier efficiencyTotal efficiency

Figure 4-6. Power (W) and efficiency (%) at loads from 10 Ω to 2 kΩ.

49

and a 6 cm by 8 cm receiving coil. It was found to have a maximum efficiency of 80.9%

and a maximum power delivery of 11.8 W. At a fixed load, the power delivery has a

coefficient of variation of 2.2% as the receiving coil’s position is varied on the transmitter,

and the peak spatial variation is less than 10% of the mean power delivery. In general,

the system is robust and efficiency is high, irrespective of receiver placement and

loading conditions. This demonstrated the feasibility of eliminating the last wire of

wireless portable devices to achieve a completely wireless solution.

50

CHAPTER 5M:N ANALYSIS

5.1 Introduction

Regardless of technique, larger transmitting coils require more turns to achieve an

even field distribution, raising the inductance. This is a problem because the amplifier

operation is sensitive to component variation in the transformation network following

the driving circuit. As the inductance of the primary coil increases, the series capacitor

in the network needs to be smaller, and the class E becomes increasingly sensitive

to small variations in the component values, sometimes severely hindering system

performance. To circumvent this problem, the inductance could be lowered by using

two or more primary coils in parallel. This reduces the inductance while still allowing

a large charging area. In addition, having multiple tansmitting coils in parallel reduces

the influence of one load’s power consumption on that of any other load. This chapter

derives and verifies the mathematical description of the coupling between M transmitters

and N receivers and demonstrates the advantages of such a system experimentally.

5.2 Analysis

The mathematical analysis of power transfer in the M:N case can be performed by

applying Kirchoff voltage and current laws to the circuit shown in Fig. 5-1. The primary

coils are numbered 1 through M, and the receiving coils are numbered M + 1 through

M + N. The voltage-current matrix equation is:

ZI = V (5–1)

b=M+N∑b=1

ZabIb = Va (5–2)

where Ib is the current on the bth coil and Va is the voltage on the ath coil, and Zab is

the (a, b)th element of the impedance matrix, defined as

51

Figure 5-1. M:N block diagram.

Zab =

jωLa + Ra for a = b

jωMab otherwise(5–3)

where ω is angular frequency, La and Ra are the self-inductance and parasitic

resistance of the ath coil, and Mab is the mutual inductance between the ath and bth

coils. Relating current and voltage in each of the coils, Vb can be found. For the primary

coils (in parallel), the voltage is the same for all, the input voltage (Vb = Vin). For coils

M + 1 through M + N, Vb = IbZLb, where ZLb is the impedance of the load and any

transformation network attached to the bth coil. The final constraint is that the sum of the

currents in the primary coils must be equal to the input current, Iin = ZinVin. Applying this

to Eq. 5–2,

(Z − ZL)I = 0 (5–4)

where Z is defined as before, I is a vector of the currents, and ZL is defined as

52

ZL =

Zin for 1 ≤ a ≤ M and 1 ≤ b ≤ M

−ZLb for a = b and b > M

0 otherwise

(5–5)

Eq. 5–4 can be solved for Zin by splitting it into several submatrices as follows:

Z − ZL =

ZIII (ZII )T

ZII ZI

(5–6)

I =

III

II

(5–7)

where ZIII has dimensions M ×M; ZII has dimensions N ×M; ZI has dimensions

N×N; III has dimensions M×1; and II has dimensions N×1. Defining ZIV = ZIII+Zin1MM

(where 1MM is an M ×M matrix of ones), and with some manipulations,

ZI II = −ZII III (5–8)

(ZIV − Zin1MM)III = −(ZII )T II (5–9)

Input current Iin is the sum of currents in the transmitting coils, stated mathematically

as (where 11M is a 1 by M vector of ones):

Iin = 11MII (5–10)

Using Eq. 5–8 through 5–10,

[ZIV − (ZII )T (ZI )−1ZII ]III = Zin1MMIII (5–11)

Substituting Vin = Zin1MMIII and using Zin = Vin/Iin:

53

Zin = 1/11M [ZIV − (ZII )T (ZI )−1ZII ]−11M1 (5–12)

Having a closed-form expression for the input impedance allows derivation of the

currents in the individual coils. By subtracting Zin1MMIII from both sides of Eq. 5–11:

III = null(X) (5–13)

X = ZIV − (ZII )T (ZI )−1ZII − Zin1MM (5–14)

II = −(ZI )−1(ZII III ) (5–15)

Now knowing the currents in the transmitter and receiver coils, the power received

by load b may be computed simply as (IIb)2Re(ZLb). These equations are extensible to

different receiver topologies, such as parallel or series capacitors, and nonlinearities

(such as rectifiers, or proximity and skin effects on resistance and inductance) may be

considered as well, through the use of fixed-point iteration.

5.3 Tests Results

To verify the correctness of the preceding equations as well as to demonstrate the

benefit of using multiple primary coils in parallel, simulations and tests were carried out

for the 1:1, 1:2, 1:3, 2:2, and 2:3 cases. For all except the three-receiver cases, two

receiver sizes were considered. In addition, the two-transmitter tests were performed

with the transmitting coils adjacent and separated. Fig. 5-2 shows the eleven different

configurations for the test setup.

The primary coil inductance is 34.44 µH, reduced by half when the two-coil case

is considered. Component selection procedure for the class E was described in [16],

and component values are specified in Table 5-1 (for all cases Ldc was 500 µH and

Lout was 9.5 µH). Notably the values for Cout are higher with the 2 transmitter system.

54

Figure 5-2. Starting top row, left-to-right: coil arrangements (thick red line is receiver,thin blue line is transmitter), for (a) 1:1 small-rx, (b) 1:2 small-rx, (c) 1:3small-rx, (d) 2:2 small-rx, (e) 2:3 small-rx, (f) 1:1 big-rx, (g) 1:2 big-rx, (h) 2:2big-rx, (i) 2:2 split-tx small-rx, (j) 2:3 split-tx small-rx, and (k) 2:2 split-txbig-rx.

55

Table 5-1. Component values for 1 and 2 transmitter systems.M Rx size Crx (nF) Cout (nF) Ct (nF)1 Small 100 11.5 14.71 Large 53.8 15.3 6.82 Small 100 22.3 27.32 Large 53.8 22.3 18.3

Higher capacitance means the impedance will be less sensitive to component variations

because of the inverse relationship between capacitance and reactance. The derivative

of reactance with respect to capacitance goes as the inverse square of capacitance, so

higher capacitance values means a much lower sensitivity. To mitigate proximity and

skin effects, we used Litz wire for coil windings. The small receivers were all 4 cm by

5 cm rectangular coils of 6 turns, the large receivers were 7 cm by 8 cm with 6 turns,

and the transmitters were 16 cm by 18 cm with 13 turns, designed by the technique

described in [20].

For each transmitter/receiver pairing, the resistive load attached to each receiver

was swept from 60 Ω to 4000 Ω by means of programmable electronic loads. The

resistive load is an approximation of the charge status of a battery; a fully charged

device appears as a large resistive load (thousands of Ω) and an uncharged device

appears as a low resistive load (a handful of Ω). DC received power (Prx ) flow was

measured at the electronic loads.5.3.1 Verification

To verify the accuracy of the equations developed in Section 5.2, simulations were

performed using MATLAB code implementing the analytical treatment of the class E

amplifier by Raab [17] for a load with impedance defined as in Eq. 5–12. La and Mab are

caluculated using a numerical integration of the Neumann formula [24]. The measured

and predicted Prx for each of the M:N cases considered in this chapter are shown in

Fig. 5-3. The predicted vs. observed plots show a one-to-one correspondence, aside

from some spread due to uncertainty in secondary and primary coil positions. For 1:3

there is a partiularly large amount of spread. With three receivers in close proximity to

56

Figure 5-3. Measured vs. predicted Prx , for: (a) 1:1 small-rx, (b) 1:2 small-rx, (c) 1:3small-rx, (d) 2:2 small-rx, (e) 2:3 small-rx, (f) 1:1 big-rx, (g) 1:2 big-rx, (h) 2:2big-rx, (i) 2:2 split-tx small-rx, (j) 2:3 split-tx small-rx, and (k) 2:2 split-txbig-rx. Scale is as indicated in (i) for all subplots.

each other, uncertainties in their relative positions have a more pronounced effect on

predicted power.5.3.2 Receiver Decoupling

To show that having multiple primary coils reduces the influence of one receiver on

the others, we map the loading condition (Rl1,Rl2, ...,RlN) to a corresponding received

power delivery condition (Prx1,Prx2, ...,PrxN), using the data from the electronic load

sweeps. Though it is impossible to fully explore the power delivery space due to the

discrete nature of the tests, looking at this discrete set of loading conditions allows us to

outline the physically realizable power values that can be received by multiple loads on

the same primary coil or coils.

Figs. 5-4 and 5-5 shows this for the two receiver condition. In Fig. 5-4, 1:2 and

2:2 show similar power spaces because the receivers are small and further apart so

they are weakly coupled. Fig. 5-5 demonstrates that when the receiver size is large, for

57

Figure 5-4. Power space plots for two-receiver tests with small receivers.

Figure 5-5. Power space plots for two-receiver tests with large receivers.

58

1:2, the power space is squeezed into a much narrower area, while for 2:2 the power

space is close to a square 10 W on each side. The constricted power space for 1:2

occurs because when one load is large (eg, a fully charged device), it “chokes” power

delivery to the other, low load (eg, an uncharged device). This phenomenon can be

seen in the blue dots (1:2) in Fig. 5-5: when receiver 1 has high load resistance and

receives low power (less than 0.2 W), receiver 2 is limited to less than 0.2 W. This

amounts to the pinched shape of the power space. Such power delivery limitations are

unacceptable. The same plot demonstrates that for 2:2, the power delivered to receiver

2 can still reach about 10 W when receiver 1 has low power, high resistance conditions.

Though a simplification, it can be said that with multiple transmitters, the receivers are

essentially in parallel while with one transmitter they are essentially in series. With a

constant voltage source, power delivery to resistive loads in series is governed by the

total resistance, while loads in parallel receive independent power delivery. Multiple

primary coils parallelizes power delivery.

In the same plots, the effect of split transmitter is also demonstrated. The key

difference for the split transmitter is a reduction in received power, seen as a shifting of

the power space towards the origin. This is because the fringing fields of the primary

coils dissipate into the nearby environment instead of into a neighboring coil.

Fig. 5-6 shows the power space with small receivers for 1:3 and for 2:3 (large

receivers could not be considered for 1:3 because of insufficient room on the transmitter).

Though the difference is less pronounced than that of the N = 2 condition, it is apparent

that the 1:3 power space is more curved, with an upward sweep, while the 2:3 power

space is a distinct rectangular prism. When one receiver is in a high resistance, low

power condition, the power received by the other receivers is less in 1:3 than in 2:3. Fig.

5-6 similarly demonstrates the decoupling effect, only with a split transmitter. The effect

is the same as discussed in the preceding paragraph, and for similar reasons.

59

Figure 5-6. Power space plot for three-receiver test.

Figure 5-7. Power vs. efficiency plot for two-receiver tests with small receivers.

60

Figure 5-8. Power vs. efficiency plot for two-receiver tests with large receivers.

Figure 5-9. Power vs. efficiency plot for three-receiver tests with small receivers.

61

Table 5-2. Maximum Prx and maximum ηc for different M:N arrangements.Arrangement Max Prx (W) Max ηc (fraction)1:2, small-rx 3.44 0.752:2, small-rx 3.88 0.75

2:2, small-rx, split-tx 2.60 0.681:2, large-rx 1.82 0.822:2, large-rx 9.45 0.88

2:2, large-rx, split-tx 7.86 0.871:3, small-rx 1.91 0.742:3, small-rx 3.08 0.74

2:3, small-rx, split-tx 2.40 0.67

5.3.3 Impact on Efficiency and Total Received Power

Transmitted power was measured using a current probe (Agilent N2783A), a voltage

probe (Agilent N2863A), and an oscilloscope (Agilent DSO 5034A), with a measurement

accuracy of 1% and 0.5%, respectively. This corresponds to an accuracy of power

measurement of 1.5%. Due to temperature effects and the effect of transmission delay

on the phase of measurement, the actual accuracy is estimated to be around 5%.

Received power was measured using the DC electronic loads (BK 8500), which have a

(worst-case) accuracy of 0.4% for current and 0.38% for voltage, giving a measurement

accuracy for power of about 0.8%.

Fig. 5-7 shows total received power, Prx , and coupling efficiency (ηc , defined as the

total received power over the transmitted power) for the 2 small receiver tests. It’s clear

from the plot that the impact on efficiency is minimal; the maximum ηc for 1:2 and 2:2 is

0.75 and drops to 0.68 with split transmitters. With large receivers (Fig. 5-8), the effect

of changing from 1:2 to 2:2 is seen as an increase in received power, as the maximum

Prx is increased from 1.82 to 9.45. Likewise, with 3 receivers, Fig. 5-9 demonstrates

that there is also an increase in received power, while the maximum efficiency remains

about the same. Using the split transmitter decreases ηc to 0.67. It seems that using

multiple transmitters that are spatially separated from each other reduces efficiency and

received power as the fringing fields are dissipated into the nearby environment instead

62

Figure 5-10. Total received power as a function of RL, and its 95% confidence intervalswith 5% component tolerance (red lines), 10% component tolerances (bluedashes), and 20% component tolerances (black dash-dot), for both 1:2(left) and 2:2 (right) cases.

of coupling into a neighboring coil. Table 5-2 gives the maximum Prx and ηc for each

test.

To investigate the sensitivity to component variation, a Monte Carlo simulation

was run, assuming the components are normally distributed, with means given by

the derived component formulas and with standard deviations, σ, such that 3σ is the

component tolerance. These simulations were carried out at tolerance levels of 5%,

10%, and 20%, for the 1:2 and 2:2 configurations, using the large receivers. One

receiver was fixed at 500 Ω and the other was swept from 60 to 4000 Ω. Figure 5-10

shows the 95% confidence intervals for total received power at the three tolerance

levels. Figure 5-11 shows the 95% confidence intervals for total efficiency at the three

tolerance levels. As can be seen, the power is skewed low, and with tighter tolerances

for 1:2 than for 2:2. Efficiency is skewed high, with tighter tolerances for the 2:2 system

than for the 2:1. This skew low in the power confidence intervals and skew high in the

63

Figure 5-11. Total efficiency as a function of RL, and its 95% confidence intervals with5% component tolerance (red lines), 10% component tolerances (bluedashes), and 20% component tolerances (black dash-dot), for both 1:2(left) and 2:2 (right) cases.

efficiency confidence intervals shows that the system is not optimized for maximum

power delivery but rather efficiency. This makes sense, as all of the component selection

for the system is done on the basis of efficient operation of the class E. The 2:2 system’s

efficiency is less sensitive to component variation primarily because of Cout which

governs the phase range seen by the class E and thus its efficiency. Cout is larger in

the 2:2 system, therefore its reactance is less sensitive to variations. For total received

power, the 1:2 system is less sensitive than the 2:2 system to component variations,

because the two receivers in the 2:2 system can vary more independently due to the

decoupling effect.5.4 Conclusion

Inductive wireless power transfer between M primary coils coupled to N secondary

coils is derived analytically and demonstrated experimentally for M = 1, 2 and N =

64

1, 2, 3. Using multiple primary coils in parallel has advantages over a single primary coil.

First, the reduced inductance of the transmitting coils makes the amplifier less sensitive

to component variations. Second, with multiple receiving coils, the power delivery to

an individual receiver is less sensitive to changes in the loads attached to other coils,

decoupling receivers from each other. In addition, using multiple transmitters is shown

to increase received power with limited impact on coupling efficiency. The multiple

transmitting coil architecture increases the feasibility and effectiveness of simultaneous

multiple device charging as well as making the amplifier more robust to component

variation.

65

CHAPTER 6OPTIMAL PRIMARY COIL DESIGN FOR MULTIPLE COILS

6.1 Introduction

Chapter 4 presented a technique for coil design that ensures an even field.

However, to maintain even fields over greater areas requires a coil of higher inductance.

From Chapters 2 and 5, this leads to increased sensitivity to component variation.

Chapter 5 develops the theory for multiple transmitting coils, which, among other things,

reduces sensitivity to component variations. Naturally, the next step is to combine the

multiple coil idea with the coil design technique. This chapter presents a coil design

technique for multiple transmitting coils. Specifically, a system with two transmitting coils

in parallel is designed.

6.2 Coil Design

The coil design for two transmitters differs from the system in Chapter 5 in that the

dual-coil system there used two identical coils designed to work individually. Here, they

are considered to be working together to establish a larger area of even fields. The

basic principle behind design for multiple coils is the same as for single coils. That is, the

geometry should be parameterized, and then the parameters can be optimized to give

minimum field variations as measured by the coefficient of variation. Following the same

base design from Chapter 4, where successive turns’ widths are related to the overall

width by f :

f = 1− (1− (N − n + 1)/N)k (6–1)

and with the corners blunted by a fraction δ. Consider a two-coil system with overall

y dimensionW and x dimension L. The two coils overlap in the y direction by some

amount b. A list of coordinates (x , y) is generated by the method in Chapter 4 for the coil

associated with parameters (N, k , ∆) with dimensions L and w :

66

Table 6-1. Design parameters.N 15k 2.64∆ 0.10a 0.40b 9.00 cmW 35.00 cm

w = (W − 2b)/2 (6–2)

The y coordinates are then skewed by an exponent a and offset by b:

y1 = w(y/w)a + b (6–3)

y2 = −w(y/w)a − b (6–4)

This stretches out the coils, so that the turns are spaced further apart in the region

where the two coils overlap. So essentially, the multiple coil design is the same as single

coil design with two additional parameters, skew a and overlap b.

For a two coil system withW=35cm and L=25cm, the optimum parameters are

given Table 6-1. These dimensions were chosen such that the coil could accommodate

a laptop equipped with a receiving coil.

Fig. 6-1 shows the corresponding coils and Fig. 6-2 shows the MQS estimate of the

field distribution.6.3 System

The dual coil system was tested, using the parameters in Table 6-1 and a receiver

of 8cm by 10cm with 6 turns. The component values for the class E were selected

according to the general principles described in Chapter 2 by sweeping the component

values and hand tuning. The values used are given in Table 6-2. Figs. 6-3 and 6-4 show

the coils and circuits.

67

Figure 6-1. Coil layout.

Figure 6-2. Calculated z-directed magnetic field, assuming 1 A current (A/m).

68

Figure 6-3. Transmitter test setup.

Figure 6-4. Overlap of dual transmitter coils.

Table 6-2. Component values.Crx 53.80 nFCout 13.50 nFLout 9.12 µHCt 18.00 nF

69

Table 6-3. Summary of system performance.Size 25 cm by 35 cm

Peak delivered power 5.62 WPeak total efficiency 59.3%

Peak coupling efficiency 75.6%COV 17.8%

Coil 1 self-inductance 50.41 µHCoil 1 resistance 0.44 Ω

Coil 2 self-inductance 51.27 µHCoil 2 resistance 0.44 Ω

Coil 1 and 2 mutual inductance 4.87 µHTotal self-inductance 27.86 µH

Total resistance 0.22 Ω

6.4 Testing

To test the system, the received power, transmitted power, and input power, were

measured at a load resistance of 100 Ω at 5cm grid points over the coils to evaluate the

spatial variability. In addition, the received power, transmitted power, and input power,

were measured at loads of 75, 100, 250, 500, 750, 1000, and 4000 Ω and the receiver

centered on the transmitter.6.5 Results

Fig. 6-5 is a plot of the received power over the area of the coils. This shows the

spatia1 uniformity of the power distribution over the area of the coil (−17.5 < y < 17.5

and −12.5 < x < 12.5). There is more variability in the y direction than in the x direction.

The coefficient of variation is 17.79%. The ratio of receiver area to transmitter area is

about 0.09, whereas is Chapter 2 is was 0.12, which explains the greater variability.

Fig. 6-6 shows the plot of the load resistance response. The response trend is

similar to those seen for systems in Chapters 2, 4, and 5. In general, the efficiencies are

lower than those presented in Chapter 4 due to the lower mutual inductance between

the receiver and transmitter.

Table 6-3 gives a summary of the performance characteristics of the system.

70

Figure 6-5. Received power (W) as a function of the location of the center of thereceiving coil.

Figure 6-6. Power (W) and efficiency (%) at loads from 75 Ω to 4 kΩ.

71

6.6 Conclusion

This chapter has demonstrated the application of the design strategy for a single

coil to multiple coils. This confers the advantages of large coil size and reduced primary

inductance. The primary challenge in designing such a coil is achieving an even power

delivery regardless of receiver position, in order to accommodate multiple devices. Such

a coil design was achieved through optimization, and the 25 cm by 35 cm coil was built

and tested with a class E power amplifier and a 8 cm by 10 cm receiving coil. It was

found to have a maximum efficiency of 59.3% and a maximum power delivery of 5.62

W. At a fixed load, the power delivery has a coefficient of variation of 17.78% as the

receiving coil’s position is varied on the transmitter.

72

CHAPTER 7INCLUSION OF FERRITES

7.1 Introduction

In a near-field wireless power system, if the device under charge is electrically

conductive, or if there are electrically conductive objects underneath the transmitting

coil, the fields generated by the primary will be dissipated instead of contributing to

power transfer. It would be desireable to have an EM shield that would allow or even

enhance power transfer without letting fields dissipate. For instance, a cell phone

equipped with a receiving coil would need a shield between the coil and the battery in

order for there to be effective wireless power transfer.

One way to do this is through use of high magnetic permeability materials, such

as ferrite, backed by copper ([35, 36]). The coil is placed on top of ferrite, which is on

top of the copper. The principle of operation is based on the magnetic field boundary

conditions on the normal and tangential components of the field:

Ht1 = Ht2 (7–1)

Bn1 = Bn2 (7–2)

µ0Hn1 = µrµ0Hn2 (7–3)

where subscript t denotes the tangential component, n denotes the normal

component, material 1 is air, and material 2 is ferrite. Since the tangential field is

Figure 7-1. Diagram of ferrite shielding.

73

continuous, and the normal field drops in magnitude due to the large increase in

permeability across the boundary, the field is guided through the ferrite and reduced in

magnitude. The fields reaching the copper are thus reduced so little power is dissipated

in the copper. So instead of uselessly generating currents in conductive objects behind

the coil, the ferrite guides the field through the coil. In terms of the wireless power circuit

parameters, the inductance is increased and the parasitic resistance is decreased.

This chapter presents several aspects of the use of ferrite for shielding. Sections 7.2

and 7.3 discuss theoretical effects of ferrite properties on the coil inductance and losses.

Section 7.4 uses numerical simulation to establish width and thickness effects of a ferrite

shield on the inductance and resistance. Finally, Section 7.5, presents experimental

evaluation of several commercial ferrites.

7.2 Inductance Estimation

[37] presents a derivation of the effect of a semi-infinite, lossless ferrite substrate on

the inductance of a radially symmetric coil using image theory. The end result is

L =2µ

µ+ 1L0 (7–4)

where L0 is the free-space inductance of the coil. So, in the limit, the ferrite

inductance is twice the free-space inductance. Clearly this sort of impact on the

inductance matrix will require careful consideration of circuit tuning. For ferrites of

finite thickness, but infinite planar extent, the effects on the inductance of radial coils

is explored analytically in [38–40]. However, the real case of a finite ferrite shield

over a finite copper shield is more complicated. This could potentially be handled

by Schwarz-Christoffel mapping [41]. By mapping the semi-infinite solution for the

vector potential A to the finite geometry the inductance in the finite geometry could be

estimated. If the vector potential in the semi-infinite geometry is denoted A(z), and

the Schwarz-Christoffel transform between the original coordinates z and the new

coordinates w is w = g(z), the mapped vector potential is A:

74

A(w) =A(g−1(w)

g′(g−1(w))(7–5)

and the new inductance is

L =

∮CAds

I(7–6)

where C is the path of the coil.

A simple way to evaluate the effect of a ferrite shield on inductance would be to

use an empirical function of the coil properties and the ferrite properties to calculate an

effective permeability, similar to the effective dielectric constant used in microstrip design

[42]. Defining this constant as

µeff =L

L0(7–7)

where L is the inductance with ferrite, and L0 is the inductance in free space. A

simple geometrical parameter for describing the coil is the coil’s length l . The ferrite can

be described by its relative permeability µr and by its thickness h. Using a functional

relationship functional relationship similar to that of the effective dielectric,

µeff =2µr

µr + 1C (7–8)

C = A(1 + Bh

l)n (7–9)

where A, B, and n are fitting parameters. To determine the values to use for these

parameters, the coil inductance of circular coils of varying diameter and number of turns

was measured in free space and over three different ferrites, of µr 125, 2000, and 2100.

Using these measurements, the best-fit A is 0.8128, B is 97.7237, and n is -0.09.

75

Figure 7-2. Empirical µeff predictions (red x) and observations (blue circle).

Figure 7-3. Flux-field hyteresis loop

Fig. 7-2 shows the measured data and values estimated by the best fit function. Of

course, this function is a gross simplification, as it ignores many geometrical effects, but

it provides a quick estimate of a particular ferrite’s effect.

76

7.3 Loss Estimation

Besides the effect on the inductance matrix, a ferrite shield introduces losses due to

two effects. The first is hysteresis losses from the alternating fields. Due to saturation,

the peak flux and field is

B = min(Bs ,µ′Hp) (7–10)

H = Hc +B

µ′ (7–11)

Using these, the hysteresis loss can be estimated one way, approximating the B −H

curve as a parallelogram (Fig. 7-3)

Physt = fHcB (7–12)

where Hc is the coercivity.

Or, using the complex permeability µ′ + jµ′′,

Physt = πf µ′′H2 (7–13)

Conductive losses are the second loss mechanism and are usually less than the

hysteresis losses. These can be estimated according to [43] as:

Pcond =(πhfB)2

6ρ(7–14)

where ρ is resistivity.

7.4 Thickness and Width Effects

A practical consideration for ferrite shielding is the necessary dimension and

necessary permeability of the shield material. To determine what size and permeability

shield would be sufficient, finite-element MQS simulations were run in Ansoft Maxwell

of a square coil resting on a ferrite of thickness h, permeability µ, and relative width F .

F is defined as the ratio of the ferrite width to the coil width. These three variables were

swept and at each (h, µ, F ) point the ratio of the inductance to the free-space inductance

77

Figure 7-4. Effects of thickness and relative width of ferrite on inductance.

was calculated and plotted in Fig. 7-4. This ratio shows the expected increasing trend

with increasing h, µ, and F . This alone is insufficient to determine a requirement on

shield size.

The losses must also be considered. Fig. 7-5 shows the ratio of the resistance to

free space resistance. As F increases, this ratio decreases, approaching 1 because

fewer fringing fields are being dissipated in the copper backing. The effect of F

flattens off around 1.2 so the shield width should be about 1.2 times the coil size.

More concretely, the 1/e calculated folding length for the average of the curves in Fig.

7-4 is 1.08 and in Fig. 7-5 is 1.09. Using 1.2 as a guide provides some degree of safety

margin for design purposes.

For µ =600, a practical value, the effect of h variation is small, so it should be

possible to use a shield as thin as 0.2 mm or 0.3 mm.

78

Figure 7-5. Effects of thickness and relative width of ferrite on resistance.

7.5 Experimental Evaluation

To evaluate the losses and shielding effectiveness of potential commercially

available ferrites, current was run through a solenoid test coil placed over the ferrite or

the ferrite/copper combination. The current and voltage waveforms were captured for

one period to calculate the input impedance as follows:

I = Ipcos(ωt) (7–15)

V = Vpcos(ωt + φ) (7–16)

|Z | = VpIp

(7–17)

φ = arccos( ¯IV ) (7–18)

The power dissipated is:

79

Table 7-1. Ferrite properties.Ferrite µ′ µ′′ Bs (T) Hc (A/m) ρ (Ωm)

FairRite 42 2530 118 0.40 5 5×104

3M 600 710 125 0.273 68 108

Ferrox 3C96 2000 10 0.50 13 5

P =1

2I 2p<(Z) (7–19)

For a solenoid coil, the field strength produced is,

Hp =NIphcoil

(7–20)

where N =18 turns and hcoil , the height of the solenoid, is 20 mm with a diameter

of about 20 mm for the particular test coil. The test results for several ferrites and

thicknesses with and without copper shielding are presented in Table 7-2.

The resistance (R), reactance (X ), power dissipation (P), and inductance (L) were

calculated using the voltage and current waveforms. L0 and R0 are the free-space

inductance and resistance of the solenoid test coil. The most desireable material would

have an R/R0 as close to 1.00 and a L/L0 greater than 1.00, for as thin as possible

ferrite, over a copper backing.

Samples of FairRite 42 were only available in one thickness, 1.00 mm. It shows

similar performance with the copper backing and without. This indicates it is an effective

shielding material. However, 1.00 mm may be too thick and heavy to be used as a shield

in a portable device.

3M 600 samples were obtained at three thicknesses, 0.4 mm, 0.3 mm, and 0.2mm.

It has greater hysteresis losses than the FairRite 42, as its µ′′ and Hc are higher. Its

conductive losses are lower, with its greater resistivity. From the test data, the 0.3 mm

thick 3M 600 should provide adequate shielding.

80

Tabl

e7-

2.Fe

rrite

expe

rimen

tale

valu

atio

nw

ithso

leno

idco

il.Fe

rrite

h(m

m)I p

(A)Hp

(A/m

)P

(W)X

(Ω)R

(Ω)L/L0R

/R0

No

ferr

ite-

2.88

2592

0.51

13.1

90.

111.

001.

00F a

irRite

421.

002.

8025

200.

4615

.71

0.12

1.19

1.09

FairR

ite42

w/C

u1.

002.

8425

560.

4915

.77

0.12

1.20

1.09

3M60

00.

402.

6423

760.

5215

.76

0.15

1.19

1.36

3M60

0w

/Cu

0.40

3.16

2844

1.01

14.9

00.

161.

131.

453M

600

0.30

2.72

2448

0.45

15.8

80.

121.

201.

093M

600

w/C

u0.

302.

3220

880.

3915

.00

0.14

1.14

1.27

3M60

00.

202.

7224

840.

4115

.94

0.11

1.21

1.00

3M60

0w

/Cu

0.20

3.24

2916

0.97

15.0

60.

191.

141.

73Fe

rrox

3C96

4.76

2.56

2304

0.48

15.9

40.

151.

211.

36Fe

rrox

3C96

w/C

u4.

762.

4021

600.

4015

.17

0.14

1.15

1.27

Ferr

ox3C

963.

182.

8425

560.

5816

.06

0.14

1.22

1.27

Ferr

ox3C

96w

/Cu

3.18

3.20

2880

1.09

15.2

50.

211.

161.

91Fe

rrox

3C96

1.59

3.04

2736

0.68

15.0

00.

151.

141.

36Fe

rrox

3C96

w/C

u1.

593.

0827

720.

5913

.68

0.25

1.04

2.27

81

Of the three test ferrites, the Ferrox 3C96 has physical properties indicating the

lowest hysteresis and highest conductive losses. In terms of L/L0, and R/R0, it is the

worst performing at the thicknesses available as samples. Of course, thicker ferrites,

all other things being equal, should have greater power dissipation due to the greater

volume of material. In addition, the copper backing has the effect of lowering L/L0 and

raising R/R0 but its effect on power dissipation in tests appears ferrite and thickness

dependent. Overall, the best practical shielding would be the 3M 600 at a thickness of

0.3 mm.

7.6 Conclusion

This chapter presented several aspects of the use of ferrite for shielding: theoretical

effects of ferrite properties on the coil inductance and losses, numerical simulation to

establish width and thickness effects of a ferrite shield on the coil electrical properties,

and empirical formulas to estimate these effects. In addition, the chapter presented

experimental evaluation of several commercial ferrites. 3M ferrite of µ =710 and

thickness 0.3 mm and 20% wider than the coil would be a good shield for wireless power

applications.

82

CHAPTER 8BAYESIAN LOAD/FAULT TRACKING

8.1 Introduction

A wireless power system could charge multiple devices of different types, simultaneously.

Of course, one concern is metal objects near or on the transmitter. These effectively

short out the transmitted fields. Another concern is devices under charge “fighting” each

other for power as described in Chapter 5, which could lead to instabilities or perhaps

excessive charge times. Naturally, then, it would be desirable to detect the presence of

faults, the number of loads, and their battery charge status or some similar measure.

The problem of load or fault detection is essentially a problem of state estimation

[44]. The state estimation problem posed in this chapter is a combination of discrete and

continuous states. The problem of detection and estimation in the wireless system is

compounded by the fact that only two measurements are available on the transmitter:

DC input current and voltage on the transmitter coil. There are multiple possible states

that are a combination of continuous and discrete random variables: different numbers

of loads and their load currents or charges, or possible fault conditions. In addition,

when multiple loads are present, they influence eachothers’ received power. So, a

detection/estimation scheme for this system should be able to handle discrete and

continuous variables. It should also be robust, because a failed detection of a fault

condition could be dangerous.

There are many qualitative or quantititative methods for this kind of estimation in

industrial processes and other complex systems [45–47]. Three families of techniques

are outlined below, all of which are some form of Bayesian tracking.

The Kalman filter is a powerful tool for estimating the state of a process in the

presence of process and measurement noise. The basic idea is to use the known

system and measurement dynamics and a time series of measurements to estimate

the current state of the system [48]. It is used for continuous variables. At each time

83

step there is a prediction and update step. The prediction step uses the known system

dynamics and the previous estimate of the state to estimate the current state. In

the update step, a Kalman gain is applied which essentially weights the estimate

and the measurement to update the current estimate. The classic Kalman Filter has

requirements of linearity and normality but these can be relaxed somewhat in alternate

versions, such as the Extended Kalman Filter and the Unscented Kalman Filter.

Hidden Markov Models are useful in state estimation when states are from a

discrete set. An Hidden Markov Model consists of a set of finite states, transition

probabilities between theses states, and probability distributions of observation symbols

conditioned on these states [49]. The Hidden Markov Model problem is, given a

sequence of observation symbols, what is the sequence of states that caused these

symbols? The famous Viterbi algorithm is one approach. The important restriction is that

the state space is discrete.

Particle filters are a flexible method of state estimation using Markov Chain Monte

Carlo techniques. The essential idea behind a particle filter is to use random samples

to represent the probability distribution of possible states, updating the samples as

the system evolves with time and with new measurements [50]. It combines elements

of the Kalman Filter and Hidden Markov Model. The fact that it uses samples rather

than a distribution lifts any restriction on normality or even linearity. In addition, it can

be extended to include combinations of discrete and continuous states, hierarchies of

states, and risk-weighted states [51, 52]. This means that improbable but dangerous

states will not be missed. Another key feature is its computational simplicity, making it

easier to implement in a less powerful microcontroller as might be used in a process

monitoring situation. For these reasons, this chapter will focus on the particle filter in a

wireless power system.

84

8.2 Technology/Data

In this chapter, the transmitting coil is a 16 by 18 cm coil with 13 turns and the

receivers are all 4 by 5 cm coils with 6 turns. The transmitting coil is designed according

to the technique described in Chapter 4.

Rather than use test data, this chapter uses simulations of the system with a model

already developed in Matlab, which makes use of classic analytical solutions for the

class E amplifier [17, 19] and numerical integration for the calculation of coil inductances

and mutual inductances [24]. In the system, there is measurement noise (thermal)

and process noise (due to load fluctuations). Components are assumed to have zero

tolerance and the receiving coils’ positions are assumed fixed once placed on the

transmitting coil.

8.3 Theory/Methods

8.3.1 State/Measurement Model

Five possible discrete states are considered: zero through three loads, or fault

mode (metal object on the transmitter). For no load or fault mode, there is only the

discrete state to be estimated. For the 1-3 load cases, the discrete state (number of

loads) as well as the continuous state (charge status of the loads’ batteries) must be

estimated. The governing equation is for the charge in the battery and its time rate of

change, where vectors are used to indicate the possibility of a multiplicity of loads. Given

the DC received power (PDC ), and the fixed regulator output voltage (Vreg),

∂ ~Q

∂t= ~I =

~PDCVreg

(8–1)

Relating charge to equivalent resistance, then discretizing:

∂f (~R)

∂t=

~PDCVreg

(8–2)

∂f

∂R

∂~R

∂t=

~PDCVreg

(8–3)

85

~Rk = ~Rk−1 +

(∂f

∂R

)−1 ~PDC(~Rk−1)

Vreg+ nk−1 (8–4)

where k is time index and n is process noise. The measurement equation is

(Vin, IDC) = h(~Rk) + νk (8–5)

where h is the measurement function and ν is measurement noise. The functional

relationships ~PDC(~Rk−1) and h(~Rk) are known from derivations in previous chapters and

already coded in Matlab. It suffices to say they are nonlinear. f relates charge to load

resistance and is defined as follows for the purposes of this chapter:

Q = Q0R

R0(8–6)

for R ≤ R0

Q = (Q1 −Q0)R − R0R1 − R0

+Q0 (8–7)

for R0 < R ≤ R1

Q = Q1 (8–8)

for R > R1. Q, Q1, and Q0 are in Coulombs. R, R0, and R1 are in Ω. For this

instance, Q0 = 36, Q1 = 3600, R0 = 1, and R1 = 100.

8.3.2 Particle Filter Algorithm

The particle filter algorithm is composed of several steps, detailed in the following

sections.

8.3.2.1 Dataset generation

This first step for load tracking is not technically a particle filter step but is necessary

for tests. A ”truth” dataset is generated using the Monte Carlo Markov Chain method

and state/measurement model previously described. This provides the ”measurements”

86

Vin and IDC to be used and mode (mk ) state values (Rk) to be estimated. This chapter

uses a one-hour sequence of measurements with samples every one second.

8.3.2.2 Initialization

The first step is to generate an initial group of N particles. This is done with a

uniform prior distribution for the discrete modes mi0 and Gaussian distribution for each of

the loads R i0 (i is particle index). At the end of this step, each particle has been assigned

a discrete mode, and if the mode has loads asociated with it, the loads have resistance

values.

8.3.2.3 State

The next step is to update the particles according to the state model. The Markov

transition matrix aij is used at this point to determine if any particle will move to a new

mode. A (0,1) uniform random variable u is generated and the CDF for each current

mode is calculated (the current mode is i ):

Fn = Σj=nj=1aij (8–9)

This is used to determine the next mode. For each u, when

Fn−1 < u < Fn (8–10)

the next mode is mode n. If there is a mode change from time k − 1 to time k , the

states of any loads are re-initialized. For example, if the transition is from no-load to

two-load, the two loads are assigned initial state values as described in Section 8.3.2.2.

8.3.2.4 Measurement

Using the measurement equations, estimates of Vin and IDC are obtained. Then,

using the (assumed known) distributions of measurement noise, and measured Vin

and IDC , the conditional probabilities of each particle i at time k (w ik) are obtained. The

collection of N w ik ’s are then scaled so that they sum to one.

87

w ik = w ik−1p(Vin, IDC |Vin, IDC) (8–11)

= w ik−1Pν

((Vin − Vin, IDC − IDC)|~Θν

)(8–12)

w ik =w ikΣiw

ik

(8–13)

where Pν is the noise pdf and ~Θ is its parameter vector.

8.3.2.5 Update

These w are used to update the particles. Two methods are considered in this

chapter. The first is as described above, keeping the same particles but rescaling their

weights. The second incorporates resampling the particles using the CDF C jk from their

weights

C jk = Σi=ji=1w

ik (8–14)

This is used to resample by generating N instances a (0,1) uniform random variable

u. For each u i , when

C j−1k < u i < C jk (8–15)

the new resampled particle i is the old particle j , and all the resampled particles

weights are considered uniform. This is supposed to allieviate problems of the

distribution degenerating to a single particle with weight 1.

These methods will be referred to as particle filter with and without resampling.

8.3.2.6 Estimate

The estimate at each time step is the weighted (by w ) sum of the particles’ values.

88

mk = Σw ikmik (8–16)

Qk = Σw ikQik (8–17)

= Σw ik f (Rik) (8–18)

The modes are given numerical values as follows: m = 1 is one load, 2 is two loads,

3 is three loads, 4 is no load, and 5 is fault mode.

8.3.3 Tests

The state model, measurement model, and particle filters (with and without

resampling) described above were used with a generated data set. N was set to be

10, 100, or 1000. The prior distribution of modes was assumed uniform (p = 1/5 for

each mode), and transition probabilities between modes were defined aij = 0.996 if i = j ,

and 0.001 otherwise. Process noise and measurement noise were assumed Gaussian,

though it should be noted that this is not a requirement of the particle filter technique.

Noise can come from any arbitrary distribution with a particle filter. The standard

deviation and mean of the prior distribution of the load resistances (for initialization) are

0.1 and 1 Ω. The standard deviation and mean of the process noise are 0.1 and 0.0 Ω.

For Vin, the mean of the measurement noise is 0.0 V and the standard deviation is 0.1 V;

for IDC , the mean of the measurement noise is 0.0 A and the standard deviation is 0.01

A.

8.3.4 Implementation

The particle filter load tracking scheme was implemented in a physical system,

using the test setup as in Chapter 5, with the 4 cm by 5 cm receivers. The component

selection is the same. The MATLAB particle filter code was implemented in C++

and combined with the code used for controlling the electronic loads, DC source,

oscilloscope, and function generator. The Vin and IDC measurements were obtained from

the oscilloscope and DC source, and the electronic loads were programmed to behave

89

according to the piecewise-linear model established earlier. Process noise was 0.1

Ω; voltage noise was 0.5 V; and current noise was 0.03 A. The transition probabilities

between modes were defined aij = 0.9 if i = j , and 0.025 otherwise. The system was

tested in different modes individually, and in a sequence of no load, 1 load, 2 loads, 3

loads, 2 loads, 1 load, 0 load, and fault. For the sequence tests, the particle filter used a

hierarchical implementation, in which first the no load and fault mode are ruled out. If Idc

was greater than 1 A then fault mode was determined. If Idc was less than 0.4 A, and Vin

was greater than 70 V, the system was determined to be in 0 load condition. Otherwise,

the particle filter was as in the MATLAB simulations.

8.4 Simulation Results

Figure 8-1. Generated measurements in (Vin,IDC ) space.

Fig. 8-1 shows the generated “truth” dataset in the (Vin, IDC) space. The points

are color coded to indicate the discrete mode. Notice how close the different modes

90

are in the feature space. Actually, in real wireless power systems tested in the lab, this

separation is greater, so discrimination should be easier.

Figure 8-2. True (blue) and estimated (red) mode for N=10, with resampling.

Figs. 8-2, 8-3, and 8-4 show the estimated and true mode time series (mk and mk)

for N as 10, 100, and 1000. What is the physical interpretation of the “truth” time series?

The system starts in fault mode (say a metal sheet is placed on the transmitting coil),

then a load is placed on the transmitting coil. This load charges, then the system goes

into fault mode again. The metal sheet is removed and the system is in no load state.

One load is placed on the transmitting coil; it charges, then another load is placed; it

charges, then it’s removed. The system is somehow placed in fault mode again, followed

by a long stretch of time with three loads. This is followed by a brief unloading, then

one load, then two loads, then a short unloading, then fault mode. Finally, two loads are

placed on the transmitting coil to charge.

91

Figure 8-3. True (blue) and estimated (red) mode for N=100, with resampling.

The estimated mode matches the true mode more closely as N is increased.

Transitions between modes are missed because none of the particles, after resampling,

are in the new mode, so it is impossible to detect that change. Essentially, with

resampling, the particle’s distribution lacks enough variability to detect sudden changes.

Figs. 8-5,8-6, and 8-7 show the estimated and true mode time series for N as 10,

100, and 1000. In general, without resampling, the estimate is more variable because

all of the particles are there, instead of being resampled. On one hand, this is bad,

because the estimate is ”noisy”. On the other hand, this is good because the increased

variablity allows the filter to capture transitions that the filter with resampling misses.

Plots of the charge state time series are not included as they change not just in

value but also in dimension (1, 2, or 3 loads) over time.

Fig. 8-8 shows the mode and state root mean square error (RMSE) as a function

of N, for both with and without resampling. State RMSE is calculated in terms of the

92

Figure 8-4. True (blue) and estimated (red) mode for N=1000, with resampling.

charge status of the loads, not the load resistances. RMSEs decrease with increasing

N as would be expected. The mode RMSEs are lower for the particle filter without

resampling than with resampling; the state RMSEs are lower with resampling than

without. With resampling, the particles have finer granularity (lower variance in mode

and charge; see Fig. 8-9); without resampling, the particles have coarser granularity

(greater variance in mode and charge; see Fig. 8-10). The implication of this is

that the filter without resampling will be more able to catch the sudden transitions

between modes, but not the gradual charging of the device or devices. With resampling,

transitions are more likely to be missed because it could be that after resampling that all

the particles have the same mode.

93

Figure 8-5. True (blue) and estimated (red) mode for N=10, without resampling.

8.5 Measured Results

Fig. 8-11 shows the predicted and observed (Vin,IDC ) space for all 5 modes, tested

individually, with N=10000. From this it is evident that the state/measurement model is

accurate. The primary difficulty is in detecting transitions between modes.

Figs. 8-12 through 8-17 shows the predicted and observed mode time series, and

the predicted and observed state and measurement variables for N=100, N=1000, and

N=10000. Due to the hierarchical technique, no load and fault (modes 0 and 4) are

detected readily for all N values. However, results are generally poor.

For N=100 and N=1000 (Figs. 8-12 and 8-14) the estimated mode is usually within

2. For N=10000 (Fig. 8-16) the estimate is within 1. The poor accuracy could be due to

the numerous uncertainties in the system that are unknown in the tracking model. For

instance, the true transition probability matrix, process and measurement noise are all

unknown. A more refined technique might not rely on such precise knowledge.

94

Figure 8-6. True (blue) and estimated (red) mode for N=100, without resampling.

The load resistance and power delivery estimates for N=100 and 1000 (Figs. 8-13

and 8-15) show poor performance, due in no small part to the incorrectly estimated

mode. This performance improves for N=1000 (Fig. 8-17) as the mode is more closely

estimated. Without a higher N, the continuous and discrete state tracking accuracies are

probably insufficient. This number of particles (more than 10000) is probably impractical

for an on-board microprocessor.

8.6 Conclusion

In conclusion, the particle filter does work for fault detection/load tracking. However,

the number of particles to achieve satisfactory results is impractical for an on-board

microprocessor. Two changes could be implemented. One is to only do selective

resampling: if the probability mass function of the particles has insufficient entropy (or

some other metric) only then is resampling conducted. This would maintain variability

while avoiding degeneracy of the distribution. The state model could be simplified

95

Figure 8-7. True (blue) and estimated (red) mode for N=1000, without resampling.

which would reduce computational complexity for implementation on an ARM or similar

microprocessor.

In addition to changes for performance, model changes could be implemented.

The charge-resistance model used in this chapter was fairly arbitrary; a measured

charge-resistance curve for a particular device should be used. Different receiver types

and relative positions could be included. In general, the particle filter algorithm is an

effective tool for challenging nonlinear discrimination problems such as wireless load

tracking.

96

Figure 8-8. RMSE of mode and states.

Figure 8-9. Mode and charge estimate variance, with resampling.

97

Figure 8-10. Mode and charge estimate variance, without resampling.

Figure 8-11. Test of different modes in (Vin,IDC ) space, in real system.

98

Figure 8-12. True (blue) and estimated (red) mode for N=100, without resampling, in realsystem.

Figure 8-13. Predicted and observed power, resistance, and input voltage, and DC inputcurrent for N=100, without resampling, in real system.

99

Figure 8-14. True (blue) and estimated (red) mode for N=1000, without resampling, inreal system.

Figure 8-15. Predicted and observed power, resistance, and input voltage, and DC inputcurrent for N=1000, without resampling, in real system.

100

Figure 8-16. True (blue) and estimated (red) mode for N=10000, without resampling, inreal system.

Figure 8-17. Predicted and observed power, resistance, and input voltage, and DC inputcurrent for N=1000, without resampling, in real system.

101

CHAPTER 9MIDRANGE WIRELESS POWER TRANSFER

9.1 Introduction

In previous chapters, the receiver-transmitter distance considered was less

than 5 mm. This is appropriate for the primary application considered, charging

battery-operated electronics. However, for some applications it may be desirable to

extend the receiver/transmitter distance to about the size of the coils themselves.

This is considered midrange coupling. The approach taken in [7, 53] is to use high-Q,

electromagnetically resonant structures to form a strong coupling. The frequencies used

are in excess of 10 MHz and coupling efficiencies of 90% are achieved at distances of

75 cm. In addition, rather than the conventional inductive coupling equations considered

earlier in this dissertation, [7] uses coupled mode theory [54–56]. [53] uses a higher

frequency (10 MHz) and coupled mode theory, relying on the self-capacitance of the

coils to achieve resonance, though total efficiency is low (∼40%), due in part to their

selection of a Colpitts oscillator as the driving circuit. [57] uses this resonant technique

but with lumped capacitors to power an LED at a distance of a few centimeters. The

papers using resonance all rely on a total of four coils for every receiver-transmitter pair:

a transmitting coil, two resonantly coupled intermediary coils, and a receiving coil. This

chapter tries to extend the architecture considered in previous chapters to midrange

distances while maintaining high total efficiency. First, the coupled mode theory analysis

is compared to the inductive coupling analysis. Next, coil design is reconsidered for

midrange. Then, design rules are developed for the series-parallel architecture (and

others) to extend the class E’s utility to midrange coupling. Inductance, frequency,

and circuit topology effects are tested on an actual system and evaluated in terms of

power and efficiency. The ideal topology for midrange is found and tested with regards

to its sensitivity to component tolerances and relative receiver-transmitter positioning.

Ultimately, one system is designed with peak total efficiency of 69.2% and peak received

102

power of 0.94 W and at 25 cm, and another is designed with peak total efficiency of

57.9% and peak received power of 3.78 W at 1 m.

9.2 Analysis

Coupled mode theory for two resonant objects can be described [53] by

da1dt

= jω1a1 − Γ1a1 + jκa2 (9–1)

da2dt

= jω2a2 − Γ2a2 + jκa1 (9–2)

where ai is a state variable of object i , such that its square has units of energy; ωi

is the resonant frequency; Γi is the loss, with units of frequency; and κ is the coupling

coefficient, with units of frequency.

To compare this to the inductive system, state variables, losses, and coupling are

defined as follows:

ai =

√Li2Ii (9–3)

Γi =RpiLi

(9–4)

κ =ωM√L1L2

(9–5)

= ωk (9–6)

Qi =ωiLiRpi

(9–7)

ωi =1√LiCi

(9–8)

where Li s the coil inductance, Ii is the coil current, Rpi is the parasitic resistance, M is

the mutual inductance, Qi is the quality factor, and Ci is the resonant capacitance value.

Strong coupling, necessary for midrange transfer, occurs when

κ√Γ1Γ2

À 1 (9–9)

103

which is equivalent toωM√Rp1Rp2

À 1 (9–10)

9.2.1 Coil Design

Coil design for the midrange system has different considerations than for the

near-field system. Whereas before, the primary concern was even field distribution,

now the objective is maximal efficiency at a large separation distance. To do this,

there must be strong coupling as described above, and the Q must be as high as

possible. This means the inductance should be as high as possible with the minimum

parasitic resistance. So, both receiver and transmitter coils should have the maximum

inductance for a given length of coil. Since we are considering a separation distance (d)

approximately equal to the dimension of the coil (D), D is constrained by the desired

distance. Using the inductance formulas from [58], and for simplicity, assuming DC

resistance, the Q of a planar circular coil is:

Q =ωL

R(9–11)

L =(D/2)2N2

8(D/2) + 11N(2a)(9–12)

R =πDNρ

2πa2(9–13)

L =D2N2

16D + 88Na(9–14)

R =DNρ

2a2(9–15)

Q =2ωa2

ρ

DN

16D + 88Na(9–16)

where ρ is resistivity, N is number of turns, and a is wire radius. For a square coil,

104

Q =ωL

R(9–17)

L = N22D

πµ(ln(D/a)− 0.77401) (9–18)

R =4DNρ

πa2(9–19)

Q =2ωNµa2

ρ(ln(D/a)− 0.77401) (9–20)

Since D is fixed, Q can be increased by increasing a or N. For a given wire radius,

for both coils, Q is an increasing function of N. Intuitively, inductance goes as N2 and

parasitic resistance goes as N, there should be no “optimal” N. The constraint then

becomes the impact of the coil inductance on component selection, ie, the capacitor

sensitivity discussed in Chapter 5.

9.2.2 Component Selection

In this chapter, the receiver and transmitter coils are considered indentical in

geometry and thus inductance to simplify component selection. They are made resonant

through proper capacitor selection:

L1 = L2 (9–21)

C1 = C2 (9–22)

C = (ω2L)−1 (9–23)

where C1 is the resonant transmitter capacitor and C2 is the receiver resonant capacitor.

Because the capacitor selection is done for resonance, the design rules are

different from those derived in Chapter 2. In addition, the desireable performance of

the series-parallel topology for near-field induction (decreasing power delivery with

increasing load) is not present when the receiver capacitor is chosen for resonance.

Thus, three circuit topologies will be considered for the midrange system: the series-parallel,

105

Figure 9-1. Midrange class E series-parallel architecture.

the series-series, and the T network. The following sections detail the design rules for

each.

9.2.2.1 Series-parallel

Design rules for the series-parallel topology in resonance, shown in Fig. 9-1, are

based on similar constraints to those developed in Chapter 2 but with the receiver

capacitor chosen for resonance instead of for obtaining a specific R0 at the maximum

real part of Zin.

Zin = ω4M2C 2RL − jω3M2C (9–24)

Ztx =1

jωCout+ jωLout + ω4M2C 2RL − jω3M2C (9–25)

As before, Lout and Cout are chosen to meet Q and phase requirements, and Ct id

chosen to obtain ZVS:

ωLout = Qω4M2C 2RL + ω3M2C (9–26)

ωCout = (ωLout − ω3M2C(1 + tan(φ)ωCRL))−1 (9–27)

ωCt = 2((1 + π2/4)ω4M2C 2RL)−1 (9–28)

where RL is the load resistance where the pahse is φ.

106

Figure 9-2. Midrange class E series-series architecture.

9.2.2.2 Series-series

Similarly, design rules for the series-series architecture, shown in Fig. 9-2, are

developed:

Zin =ω2M2

RL(9–29)

Ztx =1

jωCout+ jωLout +

ω2M2

RL(9–30)

ωLout =Qω2M2

RL(9–31)

ωCout =(ωLout −

ω2M2

RLtan(φ)

)−1(9–32)

ωCt = 2((1 + π2/4)

ω2M2

RL

)−1(9–33)

9.2.2.3 T-network

The idea behind the T network, shown in Fig. 9-3, is to add an additional degree of

freedom to the design of the midrange series-parallel architecture so that a desireable

impedance response may be obtained. X2 and X3 are general reactances.

107

Figure 9-3. Midrange class E T network architecture.

Zin = ω4M2C 2RL − jω3M2C (9–34)

Ztx = jX3 +( 1jX2+1

Zin

)−1(9–35)

Ztx = jX3 +ω3M2CX2 + jω

4M2RLC2X2

ω4M2RLC 2 + j(X2 − ω3M2C)(9–36)

X3 =k2RL tan(φ)X

22 + kωMX2(X2 − kωM)− k4R2LX2(X2 − kωM)2 + k4R2L

(9–37)

9.3 Preliminary Tests

To evaluate how to best maximize efficiency at midrange distances, the effects of

coil inductance, operating frequency, and circuit topology were investigated. In addition,

the impact of the diode parasitic capacitance was investigated.

For tests in Sections 9.3.1, 9.3.2, and 9.3.3, the DC supply voltage was 6 V and the

transistor used was the IRLR3410 NMOS.

9.3.1 Rectifying Diode Effects

The rectifying diode in the half-wave rectifier contributes some capacitance (10-30

pF). As the operating frequency increases, this becomes increasingly important in terms

of achieving resonance. To demonstrate the diode’s effect, the system was configured

with the series-parallel architecture at 761.79 kHz, with receiver and transmitter coils 30

108

Table 9-1. Component values.Component w/ diode compensation w/o diode compensation

C2 0.614 nF 0.808 nFC1 0.808 nF 0.808 nFCout 10.07 nF 10.07 nFCt 6.63 nF 6.63 nFLout 5.7 µH 5.7 µH

Figure 9-4. Diode effects on system performance.

cm square, with 8 turns, constructed of 420/42 Litz wire. The separation distance was

25 cm. The component selections for two systems are shown in Table 9-1; one system

has compensation for the diode capacitance, and the other does not. The values shown

are measured values.

Fig. 9-4 shows the performance curves for the two systems. The non-compensated

system has worse performance. Peak efficiency is about 20% lower, and the peak

received power is about 0.05 W lower, compared to when the diode is taken into

account.

109

Table 9-2. Component values.Component 240 kHz, 5 turns 757 kHz, 5 turns 761.79 kHz, 8 turns

C2 18.45 nF 1.856 nF 0.614 nFC1 18.45 nF 1.858 nF 0.808 nFCout 70.66 nF 10.06 nF 10.07 nFCt 26.3 nF 10.01 nF 6.63 nFLout 9.68 µH 5.7 µH 5.7 µH

Figure 9-5. Frequency and inductance effects on system performance.

9.3.2 Frequency and Inductance Effects

To evaluate the effects of coil inductance and frequency on power delivery, three

systems were tested. A 240 kHz system, with 30 cm square coils of 5 turns; a 757 kHz

system, with 30 cm coils of 5 turns; and a 761.79 kHz system, with 30 cm coils of 8

turns. The separation distance was 25 cm. Table 9-2 shows the relevant component

values.

Fig. 9-5 shows the performance curves of the three systems. In general, as

inductance and frequency go up (an increase in Q of the coils),the peak efficieny goes

110

Table 9-3. Component values.Component Series-parallel Series-series T-network

C2 0.614 nF 0.808 nF 0.614 nFC1 0.808 nF 0.808 nF 0.808 nFCout 10.07 nF 10.81 nF 8.96 nFLp n/a n/a 1.05 µHCt 6.63 nF 5.71 nF 11.1 nFLout 5.7 µH 5.7 µH 5.7 µH

Figure 9-6. Topology effects on system performance.

up and peak power delivery goes down. This is because the real part of Zin increases

with increasing M and ω. The higher real part is increasingly greater than the parasitics,

leading to higher efficiency, and the higher real part results in lower current and thus

lower power delivery.

The strength of coupling as described by coupled mode theory mentioned in

Equation 9–10 is greater than 1 for only the 761.79 kHz system.

111

9.3.3 Topology Effects

Three topologies were evaluated: series-parallel, series-series, and the T-network,

all at 761.79 kHz with the 8 turn coils as before. The component selections are shown in

Table 9-3.

Fig. 9-6 shows the performance curves for the three topologies. The series-series

has the highest power and efficiency, in addition to preserving the desireable trend

of decreasing power with increasing power delivery. This is because for the resonant

series-series, Zin is a purely real, decreasing function of RL, and the phase of Ztx is

monotonically increasing. For series-parallel and T-network, the impedance response is

as in previous chapters, except the minimum phase is at a much larger load resistance.

Additionally, the topologies which have a parallel Crx have a higher voltage on the load,

potentially beyond the voltage ratings of the rectifier and capacitor. ,9.3.4 Sensitivity

Having established the series-series as the best topology for midrange transfer,

this section performs a sensitivity analysis with regards to: frequency, size, separation

distance, and number of turns; and for the 761.79 kHz system described in the previous

section the position and component tolerances.

M =µN2

π

(D log

((√d2 +D2 +Dd

)2√d2 + 2D2 +D√d2 + 2D2 −D

)

+ d +√d2 +D2 − 2

√d2 + 2D2

)(9–38)

(9–39)

[59]

Ptx =2V 2cc1 + π2/4

RL + Rpω2M2 + Rp(RL + Rp)

(9–40)

Prx =2V 2ccRL1 + π2/4

(RL + Rp)2/k2 + ω2M2

(ω2M2 + Rp(RL + Rp))2(9–41)

112

Figure 9-7. Effect of D, d , and f on total efficiency at N=8.

Figure 9-8. Effect of D, d , and f on received power at N=8.

113

Figure 9-9. Effect of N, f , and D on total efficiency where D = d .

Figure 9-10. Effect of N, f , and D on received power where D = d .

114

ηE = 1 (9–42)

ηc =RL

RL + Rp

(RL + Rp)2/k2 + ω2M2

ω2M2 + Rp(RL + Rp)(9–43)

Using the series-series topology, the coil dimension (D), the separation distance

(d), the frequency (f ), and the number of turns (N), were swept simultaneously. For

every D, d , f ,N point, the maximum received power and total efficiency were calculated.

Fig. 9-7 shows the effects of D, d , and f on total efficiency at N=8. Efficiency is

highest when d ≤ D and increases as f increases.

Fig. 9-8 shows the effects of D, d , and f on received power at N=8. Power is

highest where d is slightly higher than D. This is because the weaker coupling leads to

a smaller real part of Zin, leading to lower efficiency but higher power delivery.

Fig. 9-9 shows the effects of N, f , and D on total efficiency where D = d . Efficiency

increases with increasing f , increasing N, and increasing D. This is because the real

part of Zin increases with f and mutual inductance, and mutual inductance increases

with increasing number of turns and coil dimension.

Fig. 9-10 shows the effects of N, f , and D on received power where D = d .

Power delivery shows the trend of decreasing with increasing N while increasing with

increasing f . As D increases, the highest power delivery occurs at lower N.

The systems’sensitivity to receiver placement was tested by measuring the load

response at different vertical (z) and lateral (x , y ) offsets. The performance curves are

shown in Fig. 9-11. The offset vector for each curve is indicated in the legend. For

example, [0, 7.5, 0] indicates the receiver’s center is displaced 7.5 cm in the y direction

from the transmitter’s center. In general, the efficiency and power delivery decrease

with increasing receiver offset. The worst performance is when the receiver is offset by

[15, 15, 0] (the largest absolute offset): the received power is decreased by 0.5 W and

the efficiency is decreased by 15%. However, these decreases are small enough for all

115

Figure 9-11. Coil offset effects on system performance.

other offsets (less than 0.2 W and 9%) to consider the system robust with regards to

receiver/transmitter placement.

An identical Monte Carlo analysis as in Chapter 2 was performed for the series-series

system. The 95% confidence intervals of efficiency are shown in Fig. 9-12 and of power

are shown in Fig. 9-13. The efficiency confidence intervals exhibit a skew similar to that

in Chapter 2, indicating the component selection is done to optimize efficiency. The

power confidence intervals show high upper bounds and a peak around 100-200 Ω. This

is because as the components vary, the system goes off-resonance. The off resonant

system is like the system from previous chapters, with a peak power delivery rather then

a monotonically decreasing power delivery. The power delivery can be higher (though

the efficiency is lower) because at off-resonance the real part of Zin is smaller than at

resonance.

116

Figure 9-12. Efficiency at nominal component values (black line) and 95% confidenceintervals at 5% (red lines), 10% (blue dash), and 20% (black dash-dot)component tolerances for the midrange series-series system.

Table 9-4. Component values.Component Value

C2 0.371 nFC1 0.372 nFCout 19.3 nFCt 3.02 nFLout 5.7 µH

9.4 Synthesis9.4.1 50 cm Separation

The system was configured with the series-series architecture at 758.1 kHz, with

receiver and transmitter coils 50 cm square, with 8 turns, constructed of 100/40 Litz wire.

The separation distance was 50 cm. The component selections for the system is shown

in Table 9-4. The values shown are measured values.

117

Figure 9-13. Power at nominal component values (black line) and 95% confidenceintervals at 5%(red lines), 10% (blue dash), and 20% (black dash-dot)component tolerances for the midrange series-series system.

Fig. 9-14 shows the performance curves. The peak efficiency is about 52.6% and

the peak power delivery is about 0.13 W. Greater efficiency could be achieved with

better Litz wire, as the coil parasitics for the 50 cm system were relatively high (2.6

Ω, compared to the 30 cm system’s 0.6 Ω). It should be possible to increase the total

efficiency at greater distances by using less lossy coils, with Litz of higher gauge and

greater strand number.9.4.2 1 m Separation

A system with 1 m coil separation was constructed using 1725 strand, 48 AWG Litz

wire to build coils of 1 m square and is shown in Fig. 9-15. The coils were attached to

foam posterboard and hung from the ceiling. Number of turns, frequency, supply and

gate voltages, and duty cycle were varied in an attempt to maximize efficiency. The

118

Tabl

e9-

5.S

umm

ary

of1

mte

sts.

Max

imum

effic

ienc

y(%

)44

.647

.948

.851

.255

.156

.356

.857

.954

.3M

axim

umre

ceiv

edpo

wer

(W)

4.91

4.98

0.82

1.65

5.59

4.57

4.37

3.78

3.84

Turn

s4

44

66

66

68

M(µ

H)

1.83

1.83

1.83

4.10

4.10

4.10

4.10

4.10

7.23

L1

(µH

)69

.13

69.1

369

.13

158.

7415

8.74

158.

7415

8.74

158.

7427

4.70

L2

(µH

)68

.51

68.5

168

.51

160.

6716

0.67

160.

6716

0.67

160.

6726

8.79

f(k

Hz)

730.

0070

6.00

701.

0051

0.34

510.

3451

1.88

511.

8851

3.50

438.

55D

uty

cycl

e(%

)50

5050

5050

5045

4050

Vds

(V)

1212

612

2020

2020

20V

gs(V

)5

65

1010

1010

1010

C2

(nF)

0.60

50.

653

0.65

30.

609

0.60

90.

609

0.60

90.

609

0.50

0C1

(nF)

0.59

00.

651

0.60

70.

609

0.60

90.

609

0.60

90.

609

0.49

0Cout

(nF)

1.62

1.65

6.72

7.58

7.58

7.58

7.58

7.58

8.08

Ct

(nF)

0.99

10.

503

3.30

3.90

3.90

3.90

3.90

3.90

2.20

Lout

(µH

)29

.10

29.1

08.

6218

.46

18.4

618

.46

18.4

618

.46

18.4

6

119

Figure 9-14. 50 cm system performance.

transistor used was the IRF640 NMOS, and the receiver rectifier was a full-bridge using

IR10MQ060N diodes.

Table 9-5 summarizes the results of many tests conducted to find the design

maximizing efficiency for the 1 m system. Increasing number of turns increases the

mutual and self inductances, raising the Q and increasing efficiency; however, since the

inductance is so high, the voltage on the transmitting coil’s leads is high enough to lead

to arcing between adjacent turns. This makes tuning and testing the system for more

than 6 turns difficult. Increasing frequency raises the real part of the impedance seen

by the class E inverter, which would increase efficiency if not for the fact that the coil

parasitics increase as well. Increasing supply (Vds) and gate voltages (Vgs) ensures

the MOSFET switches completely into saturation when on. In addition, lower Vds is

associated with stronger nonlinearity in the output capacitance of the MOSFET, making

120

Figure 9-15. 1 m system setup.

121

Figure 9-16. 1 m system performance.

tuning Ct for ZVS more difficult. Finally, changing the duty cycle can ensure ZVS and

zero derivative switching (ZDS) at load resistances where otherwise this would not be

the case.

Fig. 9-16 shows the performance curves of the best performing design. The peak

efficiency is about 57.9% and the peak power delivery is about 3.78 W. The efficiency

was improved by using less lossy Litz (for the 6 turn 1 m system at 513.50 kHz, the

parasitics are about 2.6 Ω, while with 420 strand, 42 AWG Litz the parasitics were about

twice as high) or by increasing the number of turns.9.5 Conclusion

The near-field wireless power system considered in [16] and [60] was extended to

midrange distances, where the coil size is comparable to the separation distance. The

effects of coil inductances, frequency, circuit topology, and coil positions are tested on a

system. It is found that the series-series topology is best for midrange power transfer;

122

that there is a tradeoff between power and efficiency, where efficiency increases and

power decreases with increasing frequency and coil inductance; and that the resonant

tuning makes the system robust with respect to variations in coil positions. Ultimately,

one system is designed with peak total efficiency of 69.2% and peak received power of

0.94 W and at 25 cm, and another is designed with peak total efficiency of 57.9% and

peak received power of 3.78 W at 1 m.

123

CHAPTER 10FAR-FIELD WIRELESS POWER TRANSFER

10.1 Introduction

Figure 10-1. An example of a radiofrequency (RF) harvesting wireless sensor node [3].

Figure 10-2. An illustration of the solar power satellite (SPS) concept [61].

Far-field, or radiative, power transfer occours when the distance between the

transmitter and the receiver exceeds the Rayleigh distance, D > 2d2/λ, where d is

the characteristic dimension of the transmitter, under the condition that d is larger than

λ. Two major applications of radiative wireless power transfer (WPT) are in ambient

radiofrequency (RF) harvesting (Fig: 10-1) and the Solar Power Satellite (SPS) (Fig.

10-2).

The idea behind the first technique is to convert the radio waves from communications

into power. Since the power levels are low, typical applications include wireless

sensors free from batteries and RFID tags equipped with small computational abilities

124

Figure 10-3. Atmosphere model schematic used in this chapter [63].

Figure 10-4. Canopy model schematic used in this chapter [64].

[3, 4].The SPS is an idea which came about in the late 1960s [5] and has seen received

some recent revival [62]. The principle is to collect solar energy in space using a

geosynchronous satellite with large solar panels and then convert the energy in

microwave form and beam it to a receiving station on earth.

Both RF harvesting and SPS involve radiative transfer through participating media.

In RF harvesting, the participating media is absorbing and scattering vegetation and

urban obstacles. In SPS, the media includes the ionosphere and the ice and water

125

particles in the atmosphere. This chapter will discuss the simplifying assumptions made

and the resulting solutions to the radiative transfer equation as well as the practical

implications, for two example frameworks: transmission through the atmosphere from

space by a solar power satellite, and transmission through a vegetation canopy by

an RF transmitting tower of some kind to an RF-harvesting sensor node. Fig. 10-3

illustrates the simplified example of the former considered in this chapter; Fig. 10-4

illustrates the simplified example of the latter. The framework for analyzing the power

transfer is a finite plane-parallel absorbing and/or scattering medium, with an external

beam of incident flux at the upper boundary. The coordinate system starts at 0 and

extends downward. The lower boundary is reflecting.

10.2 Theory

Each of these frameworks are examined with the radiative transfer equation (RTE).

From [65]:

cos(θ)∂I

∂z= κIb − βI (10–1)

+σs4π

∫ 2π

0

∫ π

0

p(θ,φ, θ′,φ′) (10–2)

× I (θ′,φ′, z) sin(θ′)dθ′dφ′ (10–3)

where I is intensity, β is extinction coefficient, κ is absorption coefficient, and σs is

scattering coefficient. p is the scattering phase function. This can be simplified using

axial symmetry. In addition, since at microwave frequencies, thermal emission is very

low in comparison to the incident power, it is neglected in the following analyses.

µ∂I

∂τ= −I + $

2

∫ +1

−1p(µ,µ′)I (µ′, τ)dµ′ (10–4)

µ = cos(θ) (10–5)

τ =

∫ z

0

β(z ′)dz ′ (10–6)

126

The boundary condition at z = 0 is

I (0,µ) = 0 (10–7)

for −1 ≤ µ ≤ 0, with the incident radiation

I (0,µ) = πF δ(µ − µ0) (10–8)

where µ0 is the direction cosine of the incident beam.

There is a lower boundary, with reflectance. In terms of a boundary condition, this

can be stated as:

I (h,µ) =1

2

∫ 1

0

ρ′′(µ,µ′)I (h,µ′)µ′dµ′ (10–9)

for −1 ≤ µ < 0.

10.3 Solution Details

The solution technique used here is finite difference (in τ ) with Gaussian quadrature

(in µ). In Gaussian quadrature, intensity I is discretized at specific µj , the quadrature

points, and∫Idµ is approximated as a weighted sum over these muj , Σjaj Ij . Applying

finite difference approximation,

µiIi ,k+1 − Ii ,k−12∆τ

+ Ii ,k = Σjajp(µi ,µj)Ij ,k (10–10)

where i and j are angle indices and k is the optical depth discretization index.

For a purely absorbing medium, an isotropically scattering medium, and a Rayleigh

scattering medium, respectively, the phase functions are

p(µi ,µj) = 0 (10–11)

p(µi ,µj) = 1 (10–12)

p(µi ,µj) = 1 +P2(µi)P2(µj)

2(10–13)

127

The upper boundary condition becomes

Ii ,0 = 0 (10–14)

for µi < 0

ai Ii ,0 = F (10–15)

for µi = µ0.

The lower boundary condition, for diffuse reflectance, becomes

R

πΣjaj Ij ,K = Ii ,K (10–16)

for µi < 0 and µj > 0. For specular reflectance, this becomes

RIj ,K = Ii ,K (10–17)

for µi = µ0 and µj = −µ0.

10.4 Physical Properties

Since microwave power transfer schemes described in 10.1 are less than 10GHz,

properties will be considered in this range, specifically at 2.45 GHz. The properties are

considered homogeneous, and the atmospheric thickness is taken as 10 km.

10.4.1 Soil

The microwave reflectance of soil is a topic well-covered in the remote sensing

literature. Two famous papers are [66] and [67]. [66] discusses the effects of roughness,

soil moisture, and angle of incidence on reflectance and [67] formulates a widely-used

model for the dielectric of soil. Their semiempirical model gives the soil dielectric as:

εαsoil = 1 +

ρbρs(εαs − 1) +mβ

v εαfw −mv (10–18)

εs = (1.01 + 0.44ρs)2 − 0.062 (10–19)

128

where α and β are fitting parameters, ρs and ρb are solid and bulk densities, mv is

volume water fraction, εfw is free water dielectric. At a frequency of 2.45 GHz, and 0.3

volume moisture fraction, the dielectric constant of sandy loam soil is about 18 + j3.

While there are multiple ways of varying degrees of complexity to characterize the

soil’s reflectance, the two most straightforward are to treat it as specular or as diffuse.

For specular reflectance, using the Fresnel formula, the reflectance is 0.6112.

When considering the SPS, the reflectance will be considered 0 because the

downwelling beam is striking a rectenna array and not soil.

10.4.2 Atmosphere

The SPS involves radiation through the atmosphere, including the ionosphere.

Experiments [68, 69] show that microwave transmission at high frequency nonlinearly

excites various electrostatic plasma waves; subsequent numerical simulation [70]

of using particle-in-cell modeling shows that there is three wave coupling, where

the transmitted wave serves as a pump wave, a backscattered wave occurs with

a slightly lower frequency, and a much lower frequency electrostatic wave causes

electron heating. Where the geomagnetic field is parallel to the transmission beam, the

electrostatic wave leads to heating which prevents formation of the three-wave coupling

and transmission efficiency is maintained at 90% or higher, while where the field is

perpendicular the three-wave coupling continues periodically, causing most (80%) of

the energy to be consumed in the coupling process. [71] shows that the ionosphere

would have minimal effect on transmission through the atmosphere so the effects of

atmospheric plasmas are not considered here.

For the microwave radiative transfer, there are three atmospheric components which

participate noticably: gaseous water vapor, water droplets, and ice crystals.

10.4.2.1 Gaseous water vapor

[72] gives the dielectric of water vapor as a function of frequency and concentration

as well as measurements. From this the absorbtion coefficient can be deduced. [73]

129

shows the impact of atmospheric water vapor (among other things) on microwave

remote sensing of soil moisture at 19 GHz and above. [74] gives an RT model for

microwave transfer in cirrus clouds. At high GHz frequencies, water vapor is strongly

absorbing enough to be a significant loss at atmospheric concentration, but not at low

GHz. From these studies, we can ignore the effects of water vapor for radiative WPT

through the atmosphere.

10.4.2.2 Water droplets

Water droplets can play a significant role in microwave scattering and absorption.

[63] perform a complex microwave radiative simulation of an evolving cloud using a

different, time varying drop size distribution and a Henley-Greenstein scattering phase

function. [75] examines microwave WPT using a two-stream polarimetric model with

vertical and horizontal polarizations. [76] describes a radiative transfer model using

different exponential drop size distributions for clouds or rain, combined with a frequency

dependent dielectric function. They give scattering and absorbing coefficients at a range

of frequencies.

The drop size distribution is a modified gamma (with units of µm−1/cm3):

p(a) = K1aP exp(−K2aQ) (10–20)

where for clouds the parameters are K1 = 1.26 × 10−13, K2 = 0.75, P = 15, and

Q = 1; for rain K1 = 7.41 × 10−28, K2 = 0.025, P = 10, and Q = 1. They give the

dielectric as

εdrop = 5.5 +82.5

1 + j0.0359/λ(10–21)

= 81.47− j22.27 (10–22)

130

At low frequencies 2πa/λ << 1 so Rayleigh scattering dominates. The scattering

and abssorbing cross sections for Rayleigh are

Csca = πa28

3

∣∣∣m2 − 1m2 + 2

∣∣∣x4 (10–23)

Cabs = −4πa2=(m2 − 1m2 + 2

)x (10–24)

κ =

∫ ∞

0

p(a)Cabs(a)da (10–25)

σs =

∫ ∞

0

p(a)Csca(a)da (10–26)

Numerically integrating over the particle distribution gives values of σs = 1.96×10−15

and κ = 1.10 × 10−8 for clouds and 1.39 × 10−11 and 6.88 × 10−9 for rain, with units in

cm−1.

Figure 10-5. Cloud droplet distribution and absorption cross section.

Figs. 10-5-10-8 show the cross sections and distribution functions for clouds and

rain.

131

Figure 10-6. Rain droplet distribution and absorption cross section.

10.4.2.3 Ice crystals

Atmospheric ice crystals are strong scatterers of microwaves. They occur in a

wide range of shapes including spheres, plates, needles, and dendritic shapes, so

their scattering properties vary widely. [77] examines the single-scattering, polarimetric

effects of different ice crystals using polarimetric Rayleigh scattering and a gamma size

distribution. [78] uses a Discrete Dipole Approximation for polarimetric, azimuthally

dependent ice scattering for many shapes. Using the modified gamma with K1 =

1.99× 10−11, K2 = 0.01, P = 2, and Q = 1, with the dielectric evaluated at 2.45 GHz [79]

εice = 3 +97

1 + j2πf × 57× 10−6(10–27)

= 3− j1.12× 10−4 (10–28)

gives σs = 7.45× 10−14 and κ = 2.07× 10−13 at 2.45 GHz.

132

Figure 10-7. Cloud droplet distribution and scattering cross section.

Figs. 10-9-10-10 show the cross sections and distribution functions for ice.

10.4.3 Vegetation

For the RF-harvesting sensor example considered in this chapter, the primary

concerns are absorbtion and scattering by vegetation. This has been heavily investigated

by the microwave remote sensing community.

Vegetation scatters and absorbs microwaves. Since there is no ”typical” vegetation,

often this is calculated with empirical relationships. [80] computes single-scattering

albedo and optical depth for corn and alfalfa at X (7-12.5 GHz) and Ka bands (20-30GHz)

and shows a dependence of optical depth on plant water content. [81] develops

a microwave dielecric model for vegetative tissues which plays an important role

in subsequent studies of microwave-plant interactions. To relate this dielectric to

absorption, sometimes the water cloud model is used [82]. [83] takes the semiempirical

133

Figure 10-8. Rain droplet distribution and scattering cross section.

approach further by incorporating another biophysical parameter, leaf area index. For

this chapter, a 2m corn canopy will be considered, with τ = 1.8 and $ = 0.08.

Other papers take a more electromagnetically rigorous approach. At L-band, [64]

treats the trunks in a forest stand as dihedral reflectors, and the canopy as volume of

water droplets and incorporates multiple reflections. [84] incorporate scattering from

leaves (disks) and stems (cylinders) oriented randomly in a fully polarimetric simulation.

Measurements and modeling of scattering properties of vegetation with different

leaf/stem geometries are given [85].

10.5 Results and Discussion

The RTE is solved for parameter values as determined from the literature review.

All simulations are done with 32 quadrature points and 8 discretization points in the τ

direction.

All intensities are shown in dBW/sr to highlight differences at low intensity levels.

134

Figure 10-9. Ice sphere distribution and absorption cross section.

Table 10-1. Parameter values for RTE.Parameter Cloud Rain Ice Vegetation

µ0 1.00 1.00 1.00 0.71κ 1.10×10−8 6.88×10−9 2.07×10−13 9.00×10−3

σs 1.96×10−15 1.39×10−11 7.45×10−14 7.83×10−4

$ 1.78×10−7 2.01×10−3 2.60×10−1 8.00×10−2

τh 1.10×10−2 6.90×10−3 2.81×10−7 1.80×100

R 0.00 0.00 0.00 0.61

10.5.1 Atmospheric Loss Estimation for Solar Power Satellite

From the literature review, the dominant methods of media particpation through

the atmosphere are Rayleigh scattering and absorbtion by clouds, rain, and ice. For

all three scatterers, where µ < 0 (upwelling radiation) there is a drop in intensity due

to the non-reflecting lower boundary; comparatively little radiation intensity is directed

upwards, and that which is is due entirely to Rayleigh scattering. At the lower boundary,

135

Figure 10-10. Ice sphere distribution and scattering cross section.

the upwelling radiation (τ = τh and µ < 0) is zero, which is −∞ in log scale so it doesn’t

show up on the log intensity plot.

Fig. 10-11 shows the intensity distribution in a cloudy atmosphere. The single-scattering

albedo is very low (see Table 10-1) so the atmosphere is primarily absorbing.

Fig. 10-12 shows the intensity distribution in a rainy atmosphere. The single-scattering

albedo is small but higher than for clouds as the mode particle size is larger so the

atmosphere is more strongly scattering. This manifests itself in the intensity field as

higher intensity levels off of the main beam compared to for clouds.

Fig. 10-13 shows the intensity distribution in an icy atmosphere. The single-scattering

albedo is much higher than for clouds or rain. However, the magnitude of the σs and

κ are quite small due to the fact that the mass density of particles is 1-2 orders of

magnitude lower than for clouds and rain, and the dielectric for ice is much lower and

less lossy than for liquid water. Hence the optical depth is much lower for the same

136

Figure 10-11. Log intensity distribution through cloud.

vertical distance. This manifests itself in the intensity field as higher intensity levels off of

the main beam, due to scattering, but less loss, due to the dielectric of ice compared to

water.

10.5.2 Loss Estimation for Radiofrequency-Harvesting Sensor Under VegetationCanopy

Vegetation is handled as an absorbing and isotropically scattering media, with

empirical parameters from the literature.

Fig. 10-14 shows the intensity field in the vegetative medium for a diffuse and a

specularly reflecting boundary. Since the optical depth is much higher for a vegetation

canopy, the intensity drop is much higher. For the diffusely reflecting boundary, where

µ < 0, the upwelling radiation is spread out. For the specularly reflecting boundary,

where µ < 0, the upwelling radiation is a distinct beam. The single scattering albedo has

the effect of increasing the off-beam intensity.

137

Figure 10-12. Log intensity distribution through rain.

10.5.3 Flux

Integrating over the solid angles at each z coordinate and normalizing to 1 W/m2

incident gives the flux at each height that would be received. The profiles are shown in

Fig. 10-15. Using these the transmission efficiency can be estimated: 98.9% for clouds,

99.3% for rain, 99.999% for ice, 3.49% for vegetation with specular soil, and 8.96% for

vegetation with diffuse soil.

10.6 Conclusions

This chapter examined wireless power transmission in the microwave regime. The

far-field, radiative modes of wireless power transfer (WPT) through clouds, rain, ice,

and vegetation were studied using a numerical solution of the equation of radiative

transfer. The primary conclusion is that the optical depth due to atmospheric particles is

much lower than that for vegetation canopy. Atmospheric ice has less attentuation than

138

Figure 10-13. Log intensity distribution through ice.

atmospheric water droplets due to the different dielectric constants of frozen and liquid

water. Higher single-scattering albedo increases off-beam intensities.

Ultimately, the SPS cannot be considered impractical due to transmission efficiency.

Other technical challenges such as the satellite construction and launch, and the

efficiency of the receiving array, may limit the project. RF harvesting, relying on low

power levels, is much more tolerant of the low transmission efficiencies associated with

transfer through urban and rural environments. Far-field WPT has applicability, just as

near-field and midrange power transfer.

139

Figure 10-14. Log intensity distribution through vegetation with diffuse and specularlower boundary.

Figure 10-15. Flux profile through different media.

140

CHAPTER 11CONCLUSIONS

A proliferation of battery-operated devices, from cellphones to electric cars, has

created a demand for a wireless charging system. The challenge of wireless power

transfer (WPT) can be handled, broadly speakng, in three regimes. In near-field

WPT, the transmission distance is much less than the characteristic dimension of the

transmitter, and magnetic flux from one coil induces current in the receiver. Mid-range

nonradiative WPT occurs when the transmission distance is one to several times

the characteristic dimension, and power is transferred by means of slowly-decaying

evanescent modes between two high-Q resonant structures. Far-field WPT is radiative in

nature and is at transmission distances greater than the Rayleigh distance.

This dissertation has presented several aspects of the design of a WPT system.

The power electronics, electromagnetic, detection and estimation, and radiative transfer

aspects of wireless power system were considered. Chapter 2 described the theory

behind, and derived design equations for, a class E driving circuit and impedance

transformation network. Chapter 3 derived relevant electromagnetic quantities for a

near-field system. In Chapter 4, a coil design providing even fields was developed.

Chapter 5 extended the system to include multiple transmitting coils in parallel. Coil

design for multiple transmitting coils in parallel is discussed in Chapter 6. Chapter

7 discussed the use and evaluation of ferrite shielding and found a suitable material

candidate. Chapter 8 showed the development and testing of a Bayesian tracking

algorithm for receiver discrimination and charge status determination. The extension

of the system and coil design to midrange was presented in Chapter 9. Chapter 10

described the use of radiative transfer modeling for estimating losses of far-field wireless

power transmission.

141

REFERENCES

[1] L. Collins, “Cutting the cord,” Engineering & Technology, vol. 2, no. 6, pp. 30–33,Jun. 2007.

[2] W. C. Brown, “The history of power transmission by radio waves,” IEEE Trans.Microw. Theory Tech., vol. 32, no. 9, pp. 1230–1242, Sep. 1984.

[3] J. R. Smith, A. P. Sample, P. S. Powledge, S. Roy, and A. Mamishev, “Awirelessly-powered platform for sensing and computation,” Lecture Notes inComputer Science, vol. 4206, p. 495, Sep. 2006.

[4] T. Ungan and L. M. Reindl, “Harvesting low ambient RF sources for autonomousmeasurement systems,” in Proc. IEEE Instrumentation and Measurement Technol-ogy Conf., May 2008, pp. 62–65.

[5] P. E. Glaser, “Power from the sun: Its future,” Science, vol. 162, no. 3856, pp.857–861, Nov. 1968.

[6] T. Imura, H. Okabe, and Y. Hori, “Basic experimental study on helical antennas ofwireless power transfer for electric vehicles by using magnetic resonant couplings,”in IEEE Vehicle Power and Propulsion Conf., 2009, pp. 936–940.

[7] A. Karalis, J. D. Joannopoulos, and M. Soljacic, “Efficient wireless non-radiativemid-range energy transfer,” Annals of Phys., vol. 323, no. 1, pp. 34–48, Jan. 2008.

[8] G. B. Joun and B. H. Cho, “An energy transmission system for an artificial heartusing leakage inductance compensation of transcutaneous transformer,” IEEETrans. Power Electron., vol. 13, no. 6, pp. 1013–1022, Nov. 1998.

[9] Y. Jang and M. M. Jovanovic, “A contactless electrical energy transmission systemfor portable-telephone battery chargers,” IEEE Trans. Ind. Electron., vol. 50, no. 3,pp. 520–527, Jun. 2003.

[10] G. Wang, W. Liu, R. Bashirullah, M. Sivaprakasam, G. Kendir, Y. Ji, M. Humayun,and J. Weiland, “A closed loop transcutaneous power transfer system forimplantable devices with enhanced stability,” in Proc. of the 2004 InternationalSymp. on Circuits and Systems, vol. 4, 2004, pp. 17–20.

[11] J. Acero, D. Navarro, L. Barragan, I. Garde, J. Artigas, and J. Burdio, “FPGA-Basedpower measuring for induction heating appliances using sigma–delta A/Dconversion,” IEEE Trans. Ind. Electron., vol. 54, no. 4, pp. 1843–1852, Aug. 2007.

[12] N. O. Sokal and A. D. Sokal, “Class E – A new class of high-efficiency tunedsingle-ended switching power amplifiers,” IEEE J. Solid-State Circuits, vol. 10, no. 3,pp. 168–176, Jun. 1975.

142

[13] C. S. Wang, G. A. Covic, and O. H. Stielau, “Power transfer capability andbifurcation phenomena of loosely coupled inductive power transfer systems,”IEEE Trans. Ind. Electron., vol. 51, no. 1, pp. 530–533, Feb. 2009.

[14] G. Kendir, W. Liu, G. Wang, M. Sivaprakasam, R. Bashirullah, M. Humayun, andJ. Weiland, “An optimal design methodology for inductive power link with class Eamplifier,” IEEE Trans. Circuits Syst. I, vol. 52, no. 5, pp. 857–866, May 2005.

[15] H. Sekiya, I. Sasase, and S. Mori, “Computation of design values for class Eamplifiers without using waveform equations,” IEEE Trans. Circuits Syst. I, vol. 49,no. 7, pp. 966–978, Jul. 2002.

[16] Z. N. Low, R. A. Chinga, R. Tseng, and J. Lin, “Design and test of a high-powerhigh-efficiency loosely coupled planar wireless power transfer system,” IEEE Trans.Ind. Electron., vol. 56, no. 5, pp. 1801–1812, May 2009.

[17] F. H. Raab, “Idealized operation of the class E tuned power amplifier,” IEEE Trans.Circuits Syst., vol. 24, no. 12, pp. 725–735, Dec. 1977.

[18] N. O. Sokal, “Class E RF power amplifiers,” QEX, vol. 204, pp. 9–20, Jan./Feb.2001.

[19] F. H. Raab, “Effects of circuit variations on the class E tuned power amplifier,” IEEEJ. Solid-State Circuits, vol. 13, no. 2, pp. 239–247, Apr. 1978.

[20] J. J. Casanova, Z. N. Low, J. Lin, and R. Tseng, “Transmitting coil achieving uniformmagnetic field distribution for planar wireless power transfer system,” in Proc. Radioand Wireless Symp., 2009, pp. 530–533.

[21] M. Zahn, Electromagnetic field theory: a problem solving approach. New York, NY:Wiley, 1979.

[22] R. F. Harrington, Field computation by moment methods. Melbourne, FL: KriegerPublishing Co., Inc., 1982.

[23] G. J. Burke and A. J. Poggio, “Numerical electromagnetics code (NEC) methodof moments: Part II: Program description-code,” Lawrence Livermore Laboratory,1981.

[24] F. W. Grover, Inductance Calculations: Working Formulas and Tables. New York,NY: Courier Dover Publications, 2004.

[25] J. A. Ferreira, “Appropriate modelling of conductive losses in the design ofmagnetic components,” in IEEE 21th Annual PESC, 1990, pp. 780–785.

[26] ——, “Improved analytical modeling of conductive losses in magnetic components,”IEEE Trans. Power Electron., vol. 9, no. 1, pp. 127–131, Jan. 1994.

143

[27] X. Nan, C. R. Sullivan, D. Coll, and N. H. Hanover, “An improved calculation ofproximity-effect loss in high-frequency windings of round conductors,” in IEEE 34thAnnual PESC, vol. 2, 2003, pp. 853–860.

[28] X. Nan and C. R. Sullivan, “Simplified high-accuracy calculation of eddy-currentloss in round-wire windings,” in IEEE 35th Annual PESC, vol. 2, 2004, pp. 873–879.

[29] S. Ramo, J. R. Whinnery, and T. Van Duzer, Fields and waves in communicationelectronics. New York, NY: Wiley, 1984.

[30] J. Schutz, J. Roudet, and A. Schellmanns, “Modeling Litz wire windings,” in Conf.Record of the 1997 IEEE Thirty-Second IAS Annual Meeting, vol. 2, 1997, pp.1190–1195.

[31] FCC Rules and Regulations, “FCC part 18,” 2007.

[32] ——, “FCC part 15,” 2007.

[33] IEEE Std C95. 1, “IEEE standard for safety levels with respect to human exposureto radio frequency electromagnetic fields, 3 kHz to 300 GHz,” 1999.

[34] X. Liu and S. Y. R. Hui, “Optimal design of a hybrid winding structure for planarcontactless battery charging platform,” IEEE Trans. Power Electron., vol. 23, no. 1,pp. 455–463, Jan. 2008.

[35] S. C. Tang, S. Y. Hui, and H. S. H. Chung, “Evaluation of the shielding effects onprinted-circuit-board transformers using ferrite plates and copper sheets,” IEEETrans. Power Electron., vol. 17, no. 6, pp. 1080–1088, Nov. 2002.

[36] Y. P. Su, X. Liu, and S. Y. R. Hui, “Mutual inductance calculation of movable planarcoils on parallel surfaces,” in IEEE 39th Annual PESC, 2008, pp. 3475–3481.

[37] W. A. Roshen and D. E. Turcotte, “Planar inductors on magnetic substrates,” IEEETrans. Magn., vol. 24, no. 6, pp. 3213–3216, Nov. 1988.

[38] W. G. Hurley and M. C. Duffy, “Calculation of self and mutual impedances in planarmagnetic structures,” IEEE Trans. Magn., vol. 31, no. 4, pp. 2416–2422, Jul. 1995.

[39] ——, “Calculation of self-and mutual impedances in planar sandwich inductors,”IEEE Trans. Magn., vol. 33, no. 3, pp. 2282–2290, May 1997.

[40] W. G. Hurley, M. C. Duffy, S. O’Reilly, and S. C. O’Mathuna, “Impedance formulasfor planar magnetic structures with spiral windings,” IEEE Trans. Ind. Electron.,vol. 46, no. 2, pp. 271–278, Apr. 1999.

[41] M. Markovic, M. Jufer, and Y. Perriard, “A square magnetic circuit analysis usingSchwarz–Christoffel mapping,” Mathematics and Computers in Simulation, vol. 71,no. 4-6, pp. 460–465, Apr. 2006.

144

[42] E. Hammerstad and O. Jensen, “Accurate models for microstrip computer-aideddesign,” in MTT-S International Microwave Symp. Dig., vol. 80, 1980, pp. 407–409.

[43] Y. Sakaki, M. Yoshida, and T. Sato, “Formula for dynamic power loss in ferrite corestaking into account displacement current,” IEEE Trans. Magn., vol. 29, no. 6 Part 2,pp. 3517–3519, Nov. 1993.

[44] R. Isermann, “Supervision, fault detection and fault diagnosis methods: anintroduction,” Control Eng. Practice, vol. 5, no. 5, pp. 639–652, May 1997.

[45] V. Venkatasubramanian, R. Rengaswamy, K. Yin, and S. N. Kavuri, “A review ofprocess fault detection and diagnosis: Part I: Quantitative model-based methods,”Computers and Chem. Eng., vol. 27, no. 3, pp. 293–311, Mar. 2003.

[46] V. Venkatasubramanian, R. Rengaswamy, and S. N. Kavuri, “A review of processfault detection and diagnosis: Part II: Qualitative models and search strategies,”Computers and Chem. Eng., vol. 27, no. 3, pp. 313–326, Mar. 2003.

[47] V. Venkatasubramanian, R. Rengaswamy, S. N. Kavuri, and K. Yin, “A review ofprocess fault detection and diagnosis: Part III: Process history based methods,”Computers and Chem. Eng., vol. 27, no. 3, pp. 327–346, Mar. 2003.

[48] G. Welch and G. Bishop, “An introduction to the Kalman filter,” University of NorthCarolina at Chapel Hill, 1995.

[49] L. Rabiner and B. Juang, “An introduction to Hidden Markov Models,” IEEE ASSPMag., vol. 3, no. 1 Part 1, pp. 4–16, Jan. 1986.

[50] M. S. Arulampalam, S. Maskell, N. Gordon, T. Clapp, D. Sci, T. Organ, and S. A.Adelaide, “A tutorial on particle filters for online nonlinear/non-Gaussian Bayesiantracking,” IEEE Trans. Signal Process., vol. 50, no. 2, pp. 174–188, Feb. 2002.

[51] V. Verma, J. Langford, and R. Simmons, “Non-parametric fault identification forspace rovers,” in International Symp. on Artificial Intelligence and Robotics inSpace, Jun. 2001.

[52] R. Dearden, T. Willeke, R. Simmons, V. Verma, F. Hutter, and S. Thrun, “Real-timefault detection and situational awareness for rovers: report on the Mars technologyprogram task,” in Proc. 2004 IEEE Aerospace Conf., vol. 2, 2004, pp. 826–840.

[53] A. Kurs, A. Karalis, R. Moffatt, J. D. Joannopoulos, P. Fisher, and M. Soljacic,“Wireless power transfer via strongly coupled magnetic resonances,” Science, vol.317, no. 5834, p. 83, Jul. 2007.

[54] T. Aoki, B. Dayan, E. Wilcut, W. P. Bowen, A. S. Parkins, T. J. Kippenberg, K. J.Vahala, and H. J. Kimble, “Observation of strong coupling between one atom and amonolithic microresonator,” Nature, vol. 443, pp. 671–674, Oct. 2006.

145

[55] V. B. Braginsky, M. L. Gorodetsky, and V. S. Ilchenko, “Quality-factor and nonlinearproperties of optical whispering-gallery modes,” Phys. Lett. A, vol. 137, no. 7-8, pp.393–397, May 1989.

[56] H. A. Haus and W. Huang, “Coupled-mode theory,” Proc. IEEE, vol. 79, no. 10, pp.1505–1518, May 1991.

[57] B. L. Cannon, J. F. Hoburg, D. D. Stancil, and S. C. Goldstein, “Magnetic resonantcoupling as a potential means for wireless power transfer to multiple smallreceivers,” IEEE Trans. Power Electron., vol. 24, no. 7, p. 1819, Jul. 2009.

[58] H. A. Wheeler, “Simple inductance formulas for radio coils,” Proc. of the IRE,vol. 16, no. 10, pp. 1398–1400, Oct. 1928.

[59] G. A. Campbell, “Mutual inductances of circuits composed of straight wires,” Phys.Rev., vol. 5, no. 6, pp. 452–458, Jan. 1915.

[60] J. J. Casanova, Z. N. Low, and J. Lin, “A loosely coupled planar wireless powersystem for multiple receivers,” IEEE Trans. Ind. Electron., vol. 56, no. 8, pp.3060–3068, Aug. 2009.

[61] W. C. Brown, “The solar power satellite as a source of base load electrical power,”IEEE Trans. Power App. Syst., vol. 100, no. 6, pp. 2766–2774, Jun. 1981.

[62] M. Mori, H. Nagayama, Y. Saito, and H. Matsumoto, “Summary of studies on spacesolar power systems of the National Space Development Agency of Japan,” ActaAstronautica, vol. 54, no. 5, pp. 337–345, Mar. 2004.

[63] A. Mugnai and E. A. Smith, “Radiative transfer to space through a precipitatingcloud at multiple microwave frequencies: Part I: Model description,” J. Appl.Meteorology, vol. 27, no. 9, pp. 1055–1073, Sep. 1988.

[64] J. A. Richards, G. Q. Sun, and D. S. Simonett, “L-band radar backscatter modelingof forest stands,” IEEE Trans. Geosci. Remote Sens., vol. 24, no. 4, pp. 487–498,Jul. 1987.

[65] S. Chandrasekhar, Radiative transfer. New York, NY: Courier Dover Publications,1960.

[66] F. T. Ulaby, P. P. Batlivala, and M. C. Dobson, “Microwave backscatter dependenceon surface roughness, soil moisture, and soil texture: Part I: Bare soil,” IEEE Trans.Geosci. Remote Sens., vol. 16, no. 4, pp. 286–295, Apr. 1978.

[67] M. C. Dobson, F. T. Ulaby, M. T. Hallikainen, and M. A. El-Rayes, “Microwavedielectric behavior of wet soil: Part II: Dielectric mixing models,” IEEE Trans.Geosci. Remote Sens., vol. 23, no. 1, pp. 35–46, Jan. 1985.

[68] L. M. Duncan and R. A. Behnke, “Observations of self-focusing electromagneticwaves in the ionosphere,” Phys. Rev. Lett., vol. 41, no. 14, pp. 998–1001, Oct. 1978.

146

[69] N. Kaya, H. Matsumoto, S. Miyatake, I. Kimura, M. Nagatomo, and I. Obayashi,“Nonlinear interaction of strong microwave beam with the ionosphere MINIX rocketexperiment,” Space Solar Power Rev., vol. 6, no. 3, pp. 181–186, Aug. 1986.

[70] H. Usui, H. Matsumoto, R. Gendrin, and T. Nishikawa, “Numerical simulations of athree-wave coupling occurring in the ionospheric plasma,” Nonlinear Processes inGeophysics, vol. 9, no. 1, pp. 1–10, Mar. 2002.

[71] H. Usui, H. Matsumoto, and Y. Omura, “Possible influences of SSPS on the spaceplasma environment,” in Proc. of 2nd International Conf. on Recent Advances inSpace Technologies, Jun.9–11 2005, pp. 34–38.

[72] W. Ho, G. M. Hidy, and R. M. Govan, “Microwave measurements of the liquid watercontent of atmospheric aerosols,” J. Appl. Meteorology, vol. 13, no. 8, pp. 871–879,Dec. 1974.

[73] M. Drusch, E. F. Wood, and T. J. Jackson, “Vegetative and atmospheric correctionsfor the soil moisture retrieval from passive microwave remote sensing data: Resultsfrom the Southern Great Plains Hydrology Experiment 1997,” J. Hydrometeorology,vol. 2, no. 2, pp. 181–192, Apr. 2001.

[74] T. R. Sreerekha, S. Buehler, and C. Emde, “A simple new radiative transfer modelfor simulating the effect of cirrus clouds in the microwave spectral region,” J. Quant.Spectroscopy and Radiative Transfer, vol. 75, no. 5, pp. 611–624, Dec. 2002.

[75] Q. Liu and F. Weng, “A microwave polarimetric two-stream radiative transfer model,”J. Atmospheric Sciences, vol. 59, no. 15, pp. 2396–2402, Aug. 2002.

[76] L. Tsang, J. Kong, E. Njoku, D. Staelin, and Waters, “Theory for microwave thermalemission from a layer of cloud or rain,” IEEE Trans. Antennas Propag., vol. 25,no. 5, pp. 650–657, Sep. 1977.

[77] J. Vivekanandan, V. N. Bringi, M. Hagen, and P. Meischner, “Polarimetric radarstudies of atmospheric ice particles,” IEEE Trans. Geosci. Remote Sens., vol. 32,no. 1, pp. 1–10, Jan. 1994.

[78] K. F. Evans and G. L. Stephens, “Microwave radiative transfer through cloudscomposed of realistically shaped ice crystals: Part I: Single scattering properties,” J.Atmospheric Sciences, vol. 52, no. 11, pp. 2041–2057, Jun. 1995.

[79] R. P. Auty and R. H. Cole, “Dielectric properties of ice and solid DO,” J. Chem.Phys., vol. 20, no. 8, pp. 1309–1314, Aug. 1952.

[80] P. Pampaloni and S. Paloscia, “Microwave emission and plant water content: Acomparison between field measurements and theory,” IEEE Trans. Geosci. RemoteSens., vol. 24, no. 6, pp. 900–905, Nov. 1986.

147

[81] F. T. Ulaby and M. A. El-Rayes, “Microwave dielectric spectrum of vegetation: PartII: Dual-dispersion model,” IEEE Trans. Geosci. Remote Sens., vol. 25, no. 5, pp.550–557, Sep. 1987.

[82] A. J. Graham and R. Harris, “Extracting biophysical parameters from remotelysensed radar data: a review of the water cloud model,” Progress in PhysicalGeography, vol. 27, no. 2, p. 217, Jun. 2003.

[83] S. Paloscia and P. Pampaloni, “Microwave polarization index for monitoringvegetation growth,” IEEE Trans. Geosci. Remote Sens., vol. 26, no. 5, pp. 617–621,Sep. 1988.

[84] M. A. Karam, A. K. Fung, R. H. Lang, and N. S. Chauhan, “A microwave scatteringmodel for layered vegetation,” IEEE Trans. Geosci. Remote Sens., vol. 30, no. 4, pp.767–784, Jul. 1992.

[85] G. Macelloni, S. Paloscia, P. Pampaloni, F. Marliani, and M. Gai, “The relationshipbetween the backscattering coefficient and the biomass of narrow and broad leafcrops,” IEEE Trans. Geosci. Remote Sens., vol. 39, no. 4, pp. 873–884, Apr. 2001.

148

BIOGRAPHICAL SKETCH

Joaquin Casanova was born in Gainesville, Florida. Some stuff happened. Then,

in 2006, he got his bachelor’s in agricultural and biological engineering from University

of Florida, with a focus on agrisystems engineering and a senior project designing

a thermally regulated table for seed germination studies. In 2007, he received the

master’s degree in the same subject for his work on microwave remote sensing of

soil and vegetation. After transferring to the UF Electrical and Computer Engineering

Department, he earned another master’s degree in 2008, designing a three-dimensional

fractal heatsink antenna. Shifting research focus to wireless power transfer, with a brief

aside into high-voltage plasma generation electronics, he obtained his doctorate in 2010.

149