MICROMACHINED INDUCTORS AND TRANSFORMERS FOR MINIATURIZEDPOWER CONVERTERS

By

CHRISTOPHER D. MEYER

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

2012

c© 2012 Christopher D. Meyer

2

I dedicate this to my loving family.

3

ACKNOWLEDGMENTS

I would like to thank everyone who has contributed to the success of the work

presented in my dissertation. I thank my adviser, Dr. David Arnold, who provided me

with the opportunity to work on exciting topics in power magnetics and who introduced

me to microfabrication at the University of Florida cleanroom. I thank Dr. Rizwan

Bashirullah who served on my committee and who is developing the very high frequency

power converter circuits that motivated my work. I thank Drs. Yong-Kyu Yoon and Peng

Jiang for their valuable insights while also serving on my committee. I thank Xue Lin

for testing my microinductor within his hybrid boost converter. I thank Christopher

Dougherty for enlightening me on the considerations that affect high frequency converter

designs. I thank Jessica Meloy for her help in wirebonding.

I thank the U.S. Army Research Laboratory (ARL) for funding the project and

my colleagues at ARL for their support. I thank Dr. Brian Morgan not only for leading

the Power for Microsystems project from which my research derived, but also for the

clarity he brought and for his mentoring me. I thank Dr. Sarah Bedair for countless

discussions and for her sage advice contributing to my growth both technically and

professionally. I thank Manrico Mirabelli for his microfabrication assistance and for

sharing his photolithography expertise. I thank James Mulcahy of the cleanroom staff for

maintaining and fixing the tools that were vital to this work. I thank William Benard for

heading the cleanroom and keeping it running smoothly.

I thank my grandfather, whose pride in me inspired me to complete my doctoral

degree. I thank my wife, Jennifer, for her steadfast love. Finally, I would like to thank my

parents for their continuous support and loving devotion.

4

TABLE OF CONTENTS

page

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

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

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

ABSTRACT . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

CHAPTER

1 INTRODUCTION . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

1.1 The Case for Small . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 171.1.1 Distributed On-Chip Power for Microprocessors . . . . . . . . . . . 171.1.2 Mobile Autonomous Microsystems . . . . . . . . . . . . . . . . . . 18

1.2 Switched-Mode Power Converters . . . . . . . . . . . . . . . . . . . . . . 181.3 Impedance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 211.4 High Frequency Benefits and Challenges . . . . . . . . . . . . . . . . . . 221.5 Survey of Existing Microfabricated Inductors and Transformers . . . . . . 231.6 Air-Core Passive Components for Microscale Power Converters . . . . . . 24

2 BACKGROUND . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27

2.1 High Frequency Power Converters . . . . . . . . . . . . . . . . . . . . . . 272.2 Inductors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 282.3 Transformers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30

3 INDUCTOR DESIGN . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35

3.1 Quality Factor Definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . 353.1.1 Quality Factor of Non-Ideal Reactive Components . . . . . . . . . 353.1.2 Quality Factor of Inductor . . . . . . . . . . . . . . . . . . . . . . . 37

3.2 Performance Trilemma . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 383.3 Stacked Planar Spiral Layout . . . . . . . . . . . . . . . . . . . . . . . . . 393.4 Low Frequency Analytical Inductor Model . . . . . . . . . . . . . . . . . . 403.5 Trends and Optimization . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43

3.5.1 Analytical . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 433.5.2 FastHenry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44

3.6 Radio Frequency Effects . . . . . . . . . . . . . . . . . . . . . . . . . . . . 473.6.1 Capacitive Coupling . . . . . . . . . . . . . . . . . . . . . . . . . . 473.6.2 Eddy Currents . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53

3.7 Summary of Inductor Design . . . . . . . . . . . . . . . . . . . . . . . . . 57

5

4 TRANSFORMER DESIGN . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59

4.1 Overview and Goals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 594.2 Maximum Efficiency . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60

4.2.1 From Scattering Parameters . . . . . . . . . . . . . . . . . . . . . . 604.2.2 From Coil Quality Factors and Coupling Coefficient . . . . . . . . . 61

4.3 Layout . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 654.3.1 Turns Ratio . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65

4.4 Performance Under Load . . . . . . . . . . . . . . . . . . . . . . . . . . . 664.4.1 Derivation of Efficiency and Voltage Gain for Arbitrary Load . . . . 664.4.2 Conjugate Impedance Matched Loading . . . . . . . . . . . . . . . 69

4.5 Summary of Transformer Design . . . . . . . . . . . . . . . . . . . . . . . 70

5 FABRICATION . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72

5.1 Process Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 735.1.1 Sequential Layer Removal . . . . . . . . . . . . . . . . . . . . . . . 735.1.2 Ultrasonic Agitation in Solvents . . . . . . . . . . . . . . . . . . . . 74

5.2 Features and Variations on the Process . . . . . . . . . . . . . . . . . . . 765.2.1 Planar Processing . . . . . . . . . . . . . . . . . . . . . . . . . . . 765.2.2 Photoresist as a Structural Element . . . . . . . . . . . . . . . . . . 785.2.3 Substrate Versatility . . . . . . . . . . . . . . . . . . . . . . . . . . 79

5.3 Process Steps . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 795.4 Special Processing Considerations . . . . . . . . . . . . . . . . . . . . . . 82

5.4.1 Sputtering . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 825.4.2 Photolithography . . . . . . . . . . . . . . . . . . . . . . . . . . . . 855.4.3 Electroplating . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 875.4.4 Argon Sputter Etch . . . . . . . . . . . . . . . . . . . . . . . . . . . 895.4.5 Photoresist Skin Removal . . . . . . . . . . . . . . . . . . . . . . . 905.4.6 Copper Seed Etch . . . . . . . . . . . . . . . . . . . . . . . . . . . 91

6 INDUCTOR CHARACTERIZATION . . . . . . . . . . . . . . . . . . . . . . . . . 93

6.1 Equipment and Setup . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 936.2 Inductor Characterization Methods . . . . . . . . . . . . . . . . . . . . . . 94

6.2.1 One-Port Inductor Methods . . . . . . . . . . . . . . . . . . . . . . 946.2.2 Two-Port Inductor Methods . . . . . . . . . . . . . . . . . . . . . . 956.2.3 Inductor Characteristics Obtained from Impedance . . . . . . . . . 97

6.3 One-Port Inductor Characterization . . . . . . . . . . . . . . . . . . . . . . 986.3.1 One-Port Inductors on Pyrex Substrates . . . . . . . . . . . . . . . 98

6.3.1.1 Comparison to model predictions . . . . . . . . . . . . . . 1016.3.1.2 Current rating . . . . . . . . . . . . . . . . . . . . . . . . 1016.3.1.3 Interwinding capacitance . . . . . . . . . . . . . . . . . . 102

6.3.2 One-Port Inductors on Silicon Substrates . . . . . . . . . . . . . . 1056.3.2.1 Copper layer thickness: 10 µm vs. 30 µm . . . . . . . . . 1056.3.2.2 Inductor shape: square vs. circular spirals . . . . . . . . 106

6

6.4 Two-Port Inductor Characterization on Silicon Substrates . . . . . . . . . 1106.4.1 Capacitive Coupling through the Substrate . . . . . . . . . . . . . . 1106.4.2 Winding Losses . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113

6.5 Summary of Inductor Characterization . . . . . . . . . . . . . . . . . . . . 118

7 TRANSFORMER CHARACTERIZATION . . . . . . . . . . . . . . . . . . . . . 119

7.1 Equipment and Setup . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1197.2 Impedance Parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1207.3 Load-Dependent Efficiency and Voltage Gain . . . . . . . . . . . . . . . . 1227.4 Characterization of Transformers with 10 µm Thick Layers . . . . . . . . . 124

7.4.1 Extraction of Nominal Inductances and Resistances . . . . . . . . 1257.4.2 Load-Dependent Performance of 1 : 1 Transformer . . . . . . . . . 1277.4.3 Load-Dependent Performance of 1 : 3.5 Transformer . . . . . . . . 131

7.5 Characterization of Transformer with 30 µm Thick Layers . . . . . . . . . . 1357.6 Summary of Transformer Characterization . . . . . . . . . . . . . . . . . . 140

8 PACKAGING AND TESTING WITH CIRCUITS . . . . . . . . . . . . . . . . . . 142

8.1 Microinductor Wire Bonded to Very High Frequency Boost Converter . . . 1428.1.1 About the Microinductor . . . . . . . . . . . . . . . . . . . . . . . . 1428.1.2 About the Converter and Test Results . . . . . . . . . . . . . . . . 143

8.2 Testing with Commercial Surface-Mount Converter . . . . . . . . . . . . . 1458.2.1 About the Texas Instruments TPS61240 Converter . . . . . . . . . 1468.2.2 Module Design and Processing . . . . . . . . . . . . . . . . . . . . 1468.2.3 Converter Module Testing . . . . . . . . . . . . . . . . . . . . . . . 149

8.3 Summary of Inductor Packaging and Testing within Converter Circuits . . 154

9 CONCLUSION . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 157

9.1 Summary of Work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1579.2 Lessons Learned . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1589.3 Future Work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 160

REFERENCES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 166

BIOGRAPHICAL SKETCH . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 174

7

LIST OF TABLES

Table page

2-1 Literature survey of microinductors . . . . . . . . . . . . . . . . . . . . . . . . . 33

2-2 Literature survey of microtransformers . . . . . . . . . . . . . . . . . . . . . . . 34

3-1 Coefficients for modified Wheeler and current sheet expressions . . . . . . . . 41

5-1 Process parameters for passives fabrication . . . . . . . . . . . . . . . . . . . . 81

5-2 Recipe for acid copper sulfate electroplating bath . . . . . . . . . . . . . . . . . 88

6-1 Comparison of measured inductor performance . . . . . . . . . . . . . . . . . . 99

6-2 Comparison of model-predicted to measured inductor performance . . . . . . . 100

6-3 Performance comparison of inductors with different layer thicknesses . . . . . . 106

6-4 Geometric parameters of square and circular inductors . . . . . . . . . . . . . 107

6-5 Performance comparison of square and circular inductors . . . . . . . . . . . . 107

7-1 Comparison of transformer circuit parameters . . . . . . . . . . . . . . . . . . . 126

8-1 Component sizes in functional converter module . . . . . . . . . . . . . . . . . 149

8

LIST OF FIGURES

Figure page

1-1 Common converter circuits . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20

1-2 Review of microinductors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23

1-3 Review of microtransformers . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25

3-1 Circuit diagram of simple inductor model . . . . . . . . . . . . . . . . . . . . . . 39

3-2 Diagram of spiral dimensions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42

3-3 Trends of inductance to resistance ratio vs. packing density . . . . . . . . . . . 45

3-4 Trends of inductance to resistance ratio vs. outer diameter . . . . . . . . . . . 46

3-5 Trend of inductance vs. vertical gap between stack simulated in FastHenry . . 46

3-6 Diagram of capacitive coupling of traces through substrate . . . . . . . . . . . 49

3-7 Circuit model of inductor with substrate capacitance . . . . . . . . . . . . . . . 49

3-8 Substrate resistance effect on inductor impedance . . . . . . . . . . . . . . . . 50

3-9 Circuit model of inductor with winding and substrate capacitances . . . . . . . 51

3-10 Substrate vs. winding capacitance effect on inductor impedance . . . . . . . . 52

3-11 COMSOL simulations of skin effect in winding cross sections . . . . . . . . . . 55

3-12 Measured effect of eddy currents on inductor impedance . . . . . . . . . . . . 57

4-1 Transformer energy flow diagram . . . . . . . . . . . . . . . . . . . . . . . . . . 61

4-2 Transformer efficiency calculated by quality factors and scattering parameters . 64

4-3 Transformer layout winding diagram . . . . . . . . . . . . . . . . . . . . . . . . 65

4-4 Circuit diagram of two-port network cascaded with shunt load . . . . . . . . . . 67

4-5 Circuit diagram of two-port network cascaded with series load . . . . . . . . . . 68

4-6 Circuit diagram of two-port network with source and load impedances . . . . . 69

5-1 Illustrations of additive process stage . . . . . . . . . . . . . . . . . . . . . . . . 74

5-2 Illustrations of subtractive process stage . . . . . . . . . . . . . . . . . . . . . . 75

5-3 Scanning electron micrograph (SEM) of inductor with 10 µm thick layers . . . . 75

5-4 SEM of inductor with 30 µm thick layers . . . . . . . . . . . . . . . . . . . . . . 76

9

5-5 Cross section diagrams of process additive stage . . . . . . . . . . . . . . . . . 83

5-6 Cross section diagrams of process subtractive stage . . . . . . . . . . . . . . . 84

5-7 Adhesion of copper to photoresist . . . . . . . . . . . . . . . . . . . . . . . . . 85

5-8 Electroplating leakage between features . . . . . . . . . . . . . . . . . . . . . . 86

5-9 Electroplated copper cantilever . . . . . . . . . . . . . . . . . . . . . . . . . . . 89

5-10 Comparison images showing argon sputter etch effect on adhesion . . . . . . . 90

5-11 Photoresist blocking layer formed by argon sputter etch . . . . . . . . . . . . . 91

5-12 Sidewall roughening caused by copper etch . . . . . . . . . . . . . . . . . . . . 92

6-1 SEM images of one-port and two-port inductors . . . . . . . . . . . . . . . . . . 94

6-2 Two-port inductor impedance network . . . . . . . . . . . . . . . . . . . . . . . 95

6-3 Identification of inductor specifications from plots . . . . . . . . . . . . . . . . . 99

6-4 Current rating of inductors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100

6-5 Comparison images of interlayer photoresist . . . . . . . . . . . . . . . . . . . 103

6-6 Comparison of interlayer dielectric effect on impedance of small inductor . . . . 104

6-7 Comparison of interlayer dielectric effect on impedance of large inductor . . . . 104

6-8 Comparison of layer thicknesses for small inductor . . . . . . . . . . . . . . . . 108

6-9 Comparison of layer thicknesses for large inductor . . . . . . . . . . . . . . . . 108

6-10 Comparison of shape of small inductor . . . . . . . . . . . . . . . . . . . . . . . 109

6-11 Comparison of shape of large inductor . . . . . . . . . . . . . . . . . . . . . . . 109

6-12 Pad capacitance diagram . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111

6-13 Shunt capacitance at two ports of inductor . . . . . . . . . . . . . . . . . . . . . 115

6-14 Impedance plots from two-port inductor . . . . . . . . . . . . . . . . . . . . . . 115

6-15 SEM images of inductors with solid vs. filamented traces . . . . . . . . . . . . 116

6-16 SEM images of solid vs. filamented traces . . . . . . . . . . . . . . . . . . . . . 116

6-17 Impedance plots of filamented vs. solid traces . . . . . . . . . . . . . . . . . . 117

6-18 Change in resistance due to filamented vs. solid traces . . . . . . . . . . . . . 117

7-1 Circuit representation of two-port impedance parameters . . . . . . . . . . . . 121

10

7-2 Low frequency transformer model . . . . . . . . . . . . . . . . . . . . . . . . . 122

7-3 SEM images of microfabricated transformers . . . . . . . . . . . . . . . . . . . 125

7-4 Impedance plots of 1 : 1 transformer with 10 µm thick layers . . . . . . . . . . . 126

7-5 Impedance plots of 1 : 3.5 transformer with 10 µm thick layers . . . . . . . . . . 127

7-6 Efficiency of 1 : 1 transformer . . . . . . . . . . . . . . . . . . . . . . . . . . . . 128

7-7 Voltage gain of 1 : 1 transformer . . . . . . . . . . . . . . . . . . . . . . . . . . 128

7-8 Magnitude and phase of matched load impedance for 1 : 1 transformer . . . . . 129

7-9 Efficiency and voltage gain vs. load impedance for 1 : 1 transformer . . . . . . 130

7-10 Efficiency of 1 : 3.5 transformer . . . . . . . . . . . . . . . . . . . . . . . . . . . 132

7-11 Voltage gain of 1 : 3.5 transformer . . . . . . . . . . . . . . . . . . . . . . . . . 132

7-12 Magnitude and phase of matched load impedance for 1 : 3.5 transformer . . . 133

7-13 Efficiency and voltage gain vs. load impedance for 1 : 3.5 transformer . . . . . 134

7-14 SEM image of microtransformer with 30 µm thick layers . . . . . . . . . . . . . 136

7-15 Impedance plots of 1 : 1 transformer with 30 µm thick layers . . . . . . . . . . . 137

7-16 Efficiency of thicker 1 : 1 transformer . . . . . . . . . . . . . . . . . . . . . . . . 138

7-17 Voltage gain of thicker 1 : 1 transformer . . . . . . . . . . . . . . . . . . . . . . 138

7-18 Magnitude and phase of matched load impedance for thicker 1 : 1 transformer 139

7-19 Efficiency and voltage gain vs. load impedance for thicker 1 : 1 transformer . . 141

8-1 Microinductor wire bonded to circuit for testing . . . . . . . . . . . . . . . . . . 143

8-2 Impedance of wire bonded inductor . . . . . . . . . . . . . . . . . . . . . . . . 144

8-3 Measured efficiencies of converter with wire bonded microinductor . . . . . . . 145

8-4 Copper layout of converter module . . . . . . . . . . . . . . . . . . . . . . . . . 147

8-5 Photograph of released copper framework . . . . . . . . . . . . . . . . . . . . . 148

8-6 Photographs of functional converter module . . . . . . . . . . . . . . . . . . . . 149

8-7 Impedance of inductor used in converter module . . . . . . . . . . . . . . . . . 151

8-8 Measured efficiency vs. output current of converter module . . . . . . . . . . . 152

8-9 Boost converter circuit diagram . . . . . . . . . . . . . . . . . . . . . . . . . . . 152

11

8-10 Inductor voltage waveform for several input voltages . . . . . . . . . . . . . . . 153

8-11 Waveforms of inductor voltage for different load currents . . . . . . . . . . . . . 155

8-12 Waveforms of output voltage for different load currents . . . . . . . . . . . . . . 155

9-1 Review of microinductors including new results . . . . . . . . . . . . . . . . . . 159

9-2 Review of microtransformers including new results . . . . . . . . . . . . . . . . 159

9-3 Illustrations of package assembly . . . . . . . . . . . . . . . . . . . . . . . . . . 161

9-4 SEM images of copper sockets . . . . . . . . . . . . . . . . . . . . . . . . . . . 163

9-5 SEM images of teeth contacting to surface-mount component . . . . . . . . . . 163

9-6 Surface-mount resistor and capacitor alongside microinductors . . . . . . . . . 164

9-7 Measured impedances of socketed resistor and capacitor . . . . . . . . . . . . 165

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

MICROMACHINED INDUCTORS AND TRANSFORMERS FOR MINIATURIZEDPOWER CONVERTERS

By

Christopher D. Meyer

May 2012

Chair: David P. ArnoldMajor: Electrical and Computer Engineering

Switched-mode dc-dc power converters are a ubiquitous part of modern, feature-rich

portable electronic devices and are essential for efficiently transferring electrical energy

out of battery sources and into various power-hungry loads, such as microprocessors,

displays, sensors, and communications systems. These converters often comprise

a significant portion of total system size/weight, and the largest offenders are often

the associated power inductors and transformers. A significant reduction in the size

the inductors and transformers would have a transformative effect in enabling new

applications, such as mobile autonomous microsystems.

Increasing the switching frequency of the power converters offers to reduce the

values of the required passives. However, the expected switching frequencies of

next generation power converters fall into a gap between magnetic film inductors

and transformers operable at < 10 MHz and microwave air-core devices with high

performance at > 1 GHz. In answer, a new class of air-core microinductors and

microtransformers is presented in this document that leveraged microfabrication-enabled

advancements to attain high performance in the desirable very high frequency (VHF)

switching range and to enable fully integrated power management systems in the

smallest possible packages.

In order to design these devices, models were analyzed to uncover the ideal

characteristics for operating in the VHF range. Compared to traditional air-core

13

components, these new ones featured thicker windings and had more intricate windings

for lower loss and higher density. A multilevel microfabrication process was developed

for molding three-dimensional (3D) copper parts with the necessary characteristics of

thickness, minimum feature size, and out-of-plane stacking.

The 3D copper process enabled the microfabrication of inductors with measured

inductance densities up to 170 nH/mm2 and quality factors as great as 33. Transformers

were measured with even greater inductance densities: up to 325 nH/mm2 was

obtained in a configuration for voltage gain of 3.5 with up to 78% efficiency. Performance

figures for both inductors and transformers were shown to outstrip a number of other

microfabricated examples found in the literature, particularly in the frequency range of

10 MHz–1 GHz.

Microfabricated inductors were tested within the circuits of both a prototype

100 MHz switched-mode hybrid boost converter and a commercially-available surface-mount

converter with up to 4 MHz switching frequency. With up to 37% efficiency at a

conversion ratio of 6, the performance of the prototype 100 MHz converter when using

a 14 nH microfabricated inductor largely matched that obtained when a larger 43 nH

surface-mount inductor was used in the same converter at up to 1 mA load current.

A packaging solution was devised for testing with the surface-mount converter. An

embedded multilevel copper module consisting of both an inductor and interconnects

was detached from its silicon fabrication substrate and served as a platform to which a

surface-mount converter and capacitors were soldered.

14

CHAPTER 1INTRODUCTION

Switched-mode dc-dc power converters are a ubiquitous part of modern, feature-rich

portable electronic devices. These power converters are essential for efficiently

transferring electrical energy out of battery sources and into various power-hungry

loads, such as microprocessors, displays, sensors, and communications systems.

The need for power converters arises from the fact that electricity is utilized in

many different forms even within a single system. Often each subsystem has a different

expectation for electrical current (the “rate”), voltage (the “force”), and duty cycle (the

on/off times). Loads may operate erratically or not at all if the source is incapable of

delivering enough electrical current, at a given voltage level, and for a certain amount of

time. Batteries are designed to provide current at a fixed voltage, which may not match

the needs of the loads. The battery voltage also often decreases with higher current

draws or with time as its energy storage is depleted. Power converters provide the

handshaking that is necessary for the sources and loads to interoperate with each other.

One basic role of the dc-dc switched-mode power converter is to accept electrical

power that is input to it at one voltage level and output that power at a different voltage

level. Intelligent control mechanisms within the converter can respond to fluctuations in

source and load conditions to help smooth the delivery of power and prevent levels from

falling out of specification.

Although the term “dc” implies that the input and output voltages of the dc-dc

converter are ideally constant to the outside world, inside the converter are switches

that dynamically reconfigure the electrical current paths of the converter circuit many

thousands to millions of times per second. Power conversion utilizes the reactions of

inductors and transformers to the switch-induced changes in electrical current within

the converter to raise or lower the voltage to desired levels. Such inductive components

possess the characteristic of inducing a voltage difference to resist changes in the level

15

of current passing through them. The fundamental equation describing this behavior is

vL(t) = Ldi

dt, (1–1)

where vL(t) is the voltage induced across the inductor, di/dt is the change in current

through it, and the ratio L is defined as the inductance, measured in henries (H). For an

electrical current initially at 0 and rising to a level I , the energy E stored in the inductor is

E =1

2LI 2. (1–2)

Inductance is generally proportional to the area enclosed by a coiled conductor.

Because physical volume and mass are placed at a premium in portable electronics

applications, small inductors are desired but have correspondingly small inductances.

The inductors and transformers must, for the sake of efficiency however, store enough

energy per switching cycle to overshadow the power lost during conversion. As a result,

the inductive components can comprise a major portion of the entire converter system

size and mass, especially when the rest of the converter circuit is integrated onto a

single, tiny semiconductor chip with nm-scale transistors.

The large inductive components are, due to their size, generally added as discrete

components connected outside the converter package. External connections further

add to the bulk of the system as each component requires its own packaging, pads,

and solder joints. A significant size and weight savings would be obtained if the

inductors and transformers could be integrated with the rest of the converter circuit

within the same package. Bulky external connections could be replaced by much leaner

wire bonds, embedded interconnects, or flip-chip bumps via a System-in-Package

(SiP) approach. Further size savings would be obtained through the System-on-Chip

(SoC) approach of monolithically fabricating inductors and transformers directly on the

integrated circuits chip.

16

To realize these SiP or SoC concepts for power converters, inductive components

must first decrease in size to the point where integration is not cost-prohibitive. One

route to decreasing the inductor/transformer size is to increase the switching frequency

of the converter, so that the current variations, di/dt, are greater and occur more often.

The other means of reducing size is to increase the inductance density of the inductors

and transformers.

The inductors and transformers presented in this document leverage microfabrication

technologies for increased density and are designed to operate at increased frequencies

that have been emerged from innovative integrated-circuit converters. The goal of this

work is to enable fully-integrated high-frequency switched-mode dc-dc power converters

with ultra-miniaturized, high-density inductors and transformers.

1.1 The Case for Small

From the advent of the integrated circuit in the 1950s up to the present day,

electronic systems have been continually packed into rapidly shrinking devices with

ever-greater processing power. Contemporary consumer electronics are marked

by examples of portable computers, mobile phones, and media players in svelte

forms with increasingly convergent functionality. The need for fully integrated power

converters is reaching a critical point, however, as the scaling of power components

has struggled to keep pace with that of data processing and storage. But beyond just

the consumer-driven aesthetic of small for the sake of small, a significant reduction in

the size of power subsystems could also have a transformative effect in enabling new

applications, like mobile autonomous microsystems, and in improving the distribution of

power, such as for microprocessors.

1.1.1 Distributed On-Chip Power for Microprocessors

Modern microprocessors are highly parallel in operation. Facing the upper limits

of using higher clock frequencies to process data quicker, designers have leveraged

the benefits of a continually-shrinking transistor size and are integrating multiple

17

microprocessor copies, or cores, on a single chip. The cores are able to process

data in parallel, but have a tendency to be under-utilized in situations where tasks

require serial processing. Although todays microprocessors receive power that is

supplied by a converter located outside the microprocessor package, the ability to

integrate many power converters on the chip itself could enable individual portions of

the microprocessor circuit to be rapidly turned on and off as needed, reducing power

consumption and increasing the thermal budget of the active portions. On-die power

converters would additionally reduce the complexity of utilizing independent voltage

levels for portions of the microprocessor operating at different frequencies for further

reductions in power consumption [1, 2].

1.1.2 Mobile Autonomous Microsystems

An emerging research effort is focused at developing mobile autonomous

microsystems, tiny robotic devices that can navigate through their environment by

flying, crawling, or hopping. The number and complexity of subsystems envisioned for

these mobile microsystems is staggering. In addition to the actuators for locomotion,

these manmade bugs are expected to contain sensors for situational awareness, logic

blocks for data processing, communications systems for relaying information, and

possibly generators for harvesting energy from the environment. Interoperation of these

subsystems is likely to be a challenge as each is likely to require operation at unique

voltages, currents, and duty cycles for best performance, and will require an advanced

power management system that must furthermore be vanishingly small and light so as

to not interfere with the mobility of the bug [3].

1.2 Switched-Mode Power Converters

A multitude of circuit topologies exist to achieve switched-mode power conversion.

Some provide a step-up from lower input voltage to a higher output voltage, and some

provide a step-down. Some are capable of providing either step-up or step-down

on-the-fly, while others have input and output voltages that are equal to one another but

18

provide isolation to protect the output from high voltage spikes that may occur on the

input side of the circuit. Common switched-mode converter topologies include the boost,

buck, buck-boost, and flyback circuits.

The basic boost converter is drawn in Figure 1-1A. Current flows through the

inductor L when switch Q is closed, and energy is stored in the magnetic field of the

inductor. When Q is opened, a voltage is induced across the inductor to oppose any

sudden reduction in current, pushing current through diode D, onto the output capacitor

C , and out to the load R. The voltage induced across the inductor in this last step is

negative with respect to the reference for vL(t) labelled on Figure 1-1A, meaning that the

voltage across the load is greater than the input voltage Vin. The role of the capacitor C

is to store charge between switching cycles and ensure that the output voltage remains

at a relatively steady value. The conversion ratio M for the boost converter is controlled

by the duty cycle of the switch in its closed position, as plotted in Figure 1-1B. When the

switch is closed for a longer portion of the cycle than it is closed, the conversion ratio of

the converter is larger.

The buck (Figure 1-1C) and buck-boost (Figure 1-1E) circuits operate similarly in

that transient current through charged inductors induce voltages across the inductors

that are utilized to create voltage differences with respect to the input. The duty cycle of

the switch-closed time again provides modulation of the output voltage. As per Figure

1-1D the buck circuit is able to provide an output voltage that is less than the input, while

the buck-boost provides an inverted voltage that can range in magnitude from levels that

are both greater and lesser than the input (Figure 1-1F).

The flyback converter of Figure 1-1G derives from the buck-boost converter, except

that the inductor of the buck-boost is replaced by an isolating transformer. The switch

in Figure 1-1G is positioned for non-inverting output, so the conversion ratio plotted in

Figure 1-1H for a 1 : 1 transformer is the same as that of the buck-boost but opposite in

polarity. A step-up transformer with non-unity gain may be utilized in this circuit to realize

19

+−

+ vL(t) -

L

Q

D

C R

+

Vout-

Vin

A Boost converter circuit

0 0.25 0.5 0.75 10

1

2

3

4

5

Con

vers

ion

Rat

io M

(V

/V)

Duty Cycle D

B Boost conversion ratio

+−

LQ

D C R

+

Vout-

Vin

C Buck converter circuit

0 0.25 0.5 0.75 10

0.5

1

Con

vers

ion

Rat

io M

(V

/V)

Duty Cycle D

D Buck conversion ratio

+− L

Q D

C R

+

Vout-

Vin

E Buck-boost converter circuit

0 0.25 0.5 0.75 1−5

−4

−3

−2

−1

0

Con

vers

ion

Rat

io M

(V

/V)

Duty Cycle D

F Buck-boost conversion ratio

+−

Q

D

C R

+

Vout-Vin

1 : n

G Flyback converter circuit

0 0.25 0.5 0.75 10

1

2

3

4

5

Con

vers

ion

Rat

io M

(V

/V)

Duty Cycle D

H Flyback conversion ratio

Figure 1-1. Common converter circuits and their idealized conversion ratios M asfunctions of switching duty cycle D. Figures adapted from Erickson andMaksimovic [4].

20

more drastic conversion ratios than would be had from the buck-boost. The transformer

additionally provides isolation protection between input and output.

1.3 Impedance

While Equation 1–1 describes the characteristic transient behavior of inductive

components in inducing voltages in opposition to changes in the current passing through

it, an impedance analysis is useful for characterizing the behavior of an inductor when

the changes are sinusoidal or periodic. The impedance Z of an inductor relates the

voltage V across the inductor to the current I passing through it

Z =V

I, (1–3)

where both voltage and current are sinusoidally varying. Other non-sinusoidal periodic

excitations can be considered using Fourier analysis to decompose the signal into a

summation of sinusoidal signals.

When determining impedance, the voltage and current waveforms are represented

by phasors, each being a vector with magnitude equal to the amplitude of the waveform

and with angle equal to the phase difference between the waveform and some common

reference. In complex form, the real part of the impedance represents the in-phase

energy-dissipative (resistive) component and the imaginary part represents the

out-of-phase energy-storage component. The impedance of an ideal inductor with

inductance L at an angular frequency of ω is,

ZL = jωL. (1–4)

While the current through an ideal inductor lags the voltage across it by 90◦ (a quarter

wavelength), resistive and capacitive effects cause frequency-dependent magnitude and

phase relationships.

21

1.4 High Frequency Benefits and Challenges

Because the impedance of an ideal inductor scales with frequency, power

converters with higher switching frequencies can utilize lower-valued and physically

smaller inductors. However, increasing to very high frequency (VHF) switching

(> 10 MHz) also introduces several challenges that could severely limit performance of

these converters if not addressed.

Many magnetic core materials, which are used to increase magnetic induction in

inductors/transformers, are unable to physically switch their magnetization fast enough

in response to a VHF applied field. The time lag between changes in the applied field

and the responding change in magnetic induction in the material leads to power losses

in the core. Designers often utilize magnetic material anisotropy (or sometimes a dc bias

magnetic field) perpendicular to the applied magnetic field in order to improve the high

frequency response time of magnetic materials at the expense of lower permeability [5].

Eddy current generation within electrically conductive materials results in the skin

effect, the confinement of electric and magnetic fields to the materials’ surface at high

frequency excitation. The skin effect limits the effective cross sectional area of both

the electric winding and the magnetic core, leading to greater resistance and lesser

inductance.

In the VHF switching range, the converter circuit design demands components with

inductance and capacitance values that are on the order of the unintended parasitic

inductances and capacitances that inherently occur between components and in the

interconnections between them [6]. The design of the inductors and transformers must

consider the large parasitic capacitance experienced as energy is stored in the electric

field between adjacent conductor traces. This parasitic capacitance limits the maximum

operating frequency and efficiency of the inductor/transformer.

22

1.5 Survey of Existing Microfabricated Inductors and Trans formers

Gardner et al. [7] published in late 2009 a comprehensive review of contemporary

on-chip inductors with magnetic films and evaluated their application to integrated

power converters. The review reached some interesting conclusions. A majority of

the works featured inductors with inductance densities of less than 100 nH/mm2,

calling into question whether the magnetic films are providing sufficient inductance

improvement to warrant their inclusion. Additionally, few results were applicable to high

frequency switching (> 100 MHz). The review identified quality factor as performance

parameter of interest for efficient power conversion, with quality factor of 1 as the

minimum below which inductors acted more like resistors than as intended as energy

storing components [7]. Air-core microinductors, on the other hand, have mostly been

designed for GHz radio frequency (RF) applications. Such devices can attain high quality

factors when suspended above conductive substrates but typically have low inductances

on the order of only several nH.

Ahn, NiFe

Yamaguchi, FeAlO

Sato, FeCoBN

Song, FeZrBAg

Fukuda, NiZn

Wang, NiFe Viala, FeHfN

Flynn, NiFe

Orlando, NiFe

Lee, CoTaZr

Park, Air

Young, Air

Choi, Air

Weon, Air Yoon, Air

1

10

100

1 10 100 1000 10000

Pe

ak

Qu

ali

ty F

ac

tor

Frequency for Peak Quality Factor (MHz)

Figure 1-2. Review of both magnetic-film (shaded in blue) and air-core (shaded in green)microinductors with each plotted in terms of peak quality factor and thefrequency at which the peak quality factor was obtained. Bubble size isproportional to inductance density.[8–22]

23

A survey of existing microinductors, including both magnetic film and air cores,

revealed that there was a significant gap where few microfabricated inductors were

designed for frequencies ranging from tens of MHz up to 1 GHz. The gap was evident

in Figure 1-2, where a number of microinductors were plotted against their peak quality

factor and the frequency at which the peak quality factor was attained. A typical

magnetic film microinductor had an inductance density of about 55 nH/mm2, almost

twice that of the typical air core counterpart, which had about 30 nH/mm2. The situation

was reversed for the peak quality factor where the median air core inductor had a peak

quality factor of 50, far greater than the median magnetic film inductor at a quality factor

of 9.

A similar frequency gap was found amongst the results gathered from works

reporting existing microtransformers as can be seen on the plot in Figure 1-3. Both

magnetic film and air core microfabricated inductors were plotted against maximum

efficiency of power transfer through the transformer and the frequency at which

the maximum efficiency occurred. Most of the works were found to focus only on

transformers with 1 : 1 turns ratios with near-unity voltage/current gain.

1.6 Air-Core Passive Components for Microscale Power Conve rters

Between magnetic-film-core devices operable at < 10 MHz and microwave RF

air-core devices with high performance at > 1 GHz lies a large frequency gap amongst

the reported microinductors and microtransformers. Coincidentally, the expected

switching frequencies of next generation power converters fall right into this gap for

which no inductor/transformer technology can yet claim championship. However,

the development is presented here for a new class of air-core microinductors and

microtransformers that leverage microfabrication-enabled advancements to attain high

performance in this desirable switching frequency range and to enable fully integrated

power management systems in the smallest possible packages.

24

Yamaguchi, Air

Laney, Air

Zolfaghari, Air

Ng, Air Aly, Air

Mino, CoZrRe

Kurata, CoFeSiB

Yamaguchi, CoNbZr

Mino, CoZrRe

Xu, NiFe

Sullivan, NiFe

Sullivan, NiFe

Brunet, NiFe

Park, NiFe

Rassel, NiFe

Yun, NiFe

Wang, NiFe

0%

20%

40%

60%

80%

100%

1 10 100 1000 10000

Eff

icie

nc

y

Frequency for Maximum Efficiency

Figure 1-3. Review of both magnetic-film (shaded in blue) and air-core (shaded in green)microtransformers with each plotted in terms of maximum efficiency and thefrequency at which the maximum efficiency was obtained. Bubble size isproportional to voltage gain. [23–40]

In order to design these devices, models are analyzed to uncover the ideal

characteristics for operating in the VHF range. Compared to traditional air-core

components, the devices here feature thicker metal traces arranged into intricate

three-dimensional windings for lower loss and higher density. A multilevel microfabrication

process is developed for molding copper parts with the necessary characteristics of

thickness, minimum feature size, and out-of-plane stacking.

This dissertation has been organized as follows. In Chapter 2 information gathered

from a survey of existing microfabricated inductors and transformers is presented.

Chapter 3 highlights the goals and considerations that motivated the design of

the microinductors. Similarly, the design of the microtransformers is discussed in

Chapter 4 alongside an introduction to the math and methods used to characterize the

microtransformer performance. Presented in Chapter 5 is the fabrication process that

was developed to response to the aforementioned design needs and the shortcomings

of existing processes in meeting these needs. Characterization of the microfabricated

25

inductors and transformers at radio frequencies is covered separately in Chapters 6 and

7 for the inductors and transformers, respectively. Chapter 8 presents the packaging

and testing of microfabricated inductors within a prototype VHF hybrid boost converter

circuit and with a commercial converter chip. Chapter 9 concludes the dissertation with

a summary of the advancements led by this work in filling the gap for microscale power

inductors and transformers at VHF and enabling fully integrated power converters.

26

CHAPTER 2BACKGROUND

This chapter provides background information on existing works that have

contributed to the state of the art of microfabricated inductors and transformers.

Highlighted first are several significant demonstrations of very high frequency switched

mode power converters that have created the possibility for full integration of all

converter components in a single package. Power inductors and transformers from prior

works are then surveyed with attention focused on the challenges and accomplishments

met by each. Quantitative results from the surveyed inductors and transformers are

outlined in tabular form at the end of the chapter along with results from selected

GHz RF air core components for comparison. The works have been selected for their

inclusion of detailed performance characteristics relevant toward enabling integrated

switched mode power converters.

2.1 High Frequency Power Converters

Examples of very high frequency switching power converters are summarized

here to demonstrate the viability of this new breed of converters in providing high

performance with nH-level inductive components. The results from these works provided

an idea of what switching frequencies would be used in next-generation converters and

what size inductors would be required.

Hazucha et al. [2] reported results from a four-phase dc-dc buck converter

implemented in 90nm CMOS and designed to operate at switching frequencies ranging

from 100− 600 MHz. The optimal switching frequency was determined by the size of the

inductors. Four 6.8 nH discrete inductors were soldered onto the package for a switching

frequency of 233 MHz. The authors quoted quality factors for the inductors of Q = 20 at

100 MHz and Q = 30 at 300 MHz. The chip area of the converter was 1.26 mm2. The

converter delivered 0.3 A at 0.9 V from a 1.2 V input with 83% efficiency.

27

Li et al. [41] reported two discontinuous conduction mode dc-dc boost converters

fabricated in standard 0.13 µm CMOS, both utilizing off-chip discrete inductors. One

was a 100 MHz 4-phase boost converter that delivered 240 mW from a 1.2 V supply with

output ranging from 3 − 5 V and peak efficiency of 64%. This converter used four 22 nH

inductors, one per phase, and the CMOS area alone comprised 0.55 mm2. The other

reported converter was a 45 MHz hybrid boost converter delivering 20 mW at 6 − 10 V

also from a 1.2 V supply, with peak efficiency of 37%. The hybrid converter used a single

43 nH inductor, while the CMOS area was 0.17 mm2. The authors stated that both high

switching frequency and discontinuous conduction mode were utilized to reduce the size

of the required off-chip components.

2.2 Inductors

Ahn et al. [22] constructed a 4 mm × 1 mm × 130 µm toroidal inductor on a silicon

wafer via a multilevel metallization process. The inductor consisted of 33 turns of

40 µm-thick copper traces wound around a 30 µm-thick electroplated Ni81Fe19 magnetic

core. This composition of NiFe was cited as being chosen for achieving maximum

permeability, minimum coercivity, minimum anisotropy, and maximum mechanical

hardness. Permeability of the magnetic core was determined at approximately µr = 800

both by vibrating sample magnetometry and magnetic circuit evaluation with a core

of known dimensions. The measured inductance was 400 nH at 10 kHz, but this value

decreased with frequency to a value of approximately 50 nH at 1 MHz. Such applications

for the inductor were listed as sensors, actuators, and power converters.

Yamaguchi et al. [42] demonstrated a 7.6 nH thin-film inductor with quality factor

of 7.4 at 1 GHz intended for use in impedance matching at the front-end receiver

of a 1 GHz mobile communication handset. The square-spiral inductor measured

370 µm × 370 µm and was comprised of 4 turns of 2.8 µm-thick sputter deposited

AlSi windings. After encapsulating the windings with a 3.5 µm-thick insulating layer of

polyimide, a 0.1 µm-thick Fe61Al13O26 magnetic film was sputter deposited over the coils

28

and was patterned by ion milling. The authors acknowledged that simply covering the

inductor with the magnetic film could only double the inductance at best but predicted

the improvement would prove sufficient for commercial use. An inverse relationship was

found in the FeAlO films between the magnetic film resistivity and its resonant frequency,

although high values of each were desired to avoid excess losses at the GHz range. Slits

were created in the magnetic film to inhibit eddy current generation, resulting in a 31%

reduction in the ac resistance at 1 GHz compared to the case of the film without slits.

Sato et al. [11] developed a rectangular spiral inductor for 5 MHz switching

dc-dc converters. The inductors measured 6310 µm × 3466 µm in area and featured

50 µm-thick electroplated copper windings capped with a FeCoBN magnetic thin film.

The magnetic film was deposited by dc magnetron sputtering with four alternating layers

of 1.5 µm FeCoBN and 0.4 µm AlNx to suppress eddy currents. Film permeability was

estimated at 900 up to 300 MHz. The multilayer film was etched in a single wet step

with mixture of phosphoric, acetic, and nitric acids used to dissolve both constitutive

materials at once. Inductance was measured at 370 nH with a peak quality factor

of 15 at 7 MHz. The inductor was tested in 5 MHz switched mode power converters

constructed of discrete components in both boost and buck configurations. The buck

converter produced an output of 3 V from a 5 V input with a peak efficiency of about

82% at an output current up to 500 mA. The boost converter operated with the same

conversion ratio in reverse; an output of 5 V was obtained from a 3 V input. A similar

peak efficiency of about 82% was achieved from the boost converter at 150 mA output

current.

Fukuda et al. [9] fabricated a 6 mm × 6 mm planar square spiral inductor that

was fully encapsulated between two NiZn ferrite magnetic layers. Ferrite composition

was NiO/CuO/ZnO/Fe2O3 in ratios of 16/12/23/49. The lower ferrite layer was first

screen-printed over a silicon wafer and sintered at 900 − 1000◦C to a final thickness

of 40 µm. Relative permeability of the lower ferrite layer was measured at 120. Copper

29

windings were then electroplated 50 µm-thick on top of the lower ferrite layer. The

upper ferrite layer was screen printed on top of the windings and was hardened but not

sintered. Due to the lower relative permeability—measured at 25—the final upper ferrite

layer was deposited more than twice as thick at 100 µm. Characterization of the inductor

revealed an inductance of 1.4 µH with a peak quality factor of 40 at 5 MHz. Magnetic

field analysis by the finite element method indicated that the inclusion of magnetic ferrite

in the spaces between adjacent turns of the coil were beneficial in confining magnetic

flux to the core, minimizing eddy current loss in the copper coil.

Viala et al. [14] reported a square spiral inductor with a density around 90 nH/mm2

and peak quality factor of about 10 at 1.5 GHz with sputtered FeHfN films over spiral

inductors, but only noted a modest increase in inductance of 35% over the air-core case.

The magnetic films were laminated and consisted of ten alternations of 0.1 µm-thick

(Fe97.6Hf2.4)90N10 magnetic and 500 Å-thick SiO2 insulating layers. The authors noted

difficulty in using magnetic films with spirals due to their having both in-plane and

out-of-plane magnetic field components.

Characteristics of the above mentioned inductors were summarized in Table 2-1

alongside those from other significant works.

2.3 Transformers

Mino et al. [23] presented a 3 mm × 4 mm transformer fabricated by a completely

dry process on a silicon substrate and consisting of copper coils wrapped around a

magnetic layer of CoZrRe. The magnetic film was deposited by ion beam sputtering and

was quoted as having a relative permeability > 3000. The copper coils were wrapped

around the core in a primary:secondary ratio of 12 : 3 to obtain associated primary and

secondary inductances of 350 nH and 40 nH, respectively. The microtransformer was

mounted in a ceramic package and tested within a forward converter circuit operating at

32 MHz. Output from the converter was 0.6 V to a 10 Ω load with 10 V source input. The

30

efficiency of the converter was not given but was reportedly “low” due to the low primary

inductance of the transformer.

Yamaguchi et al. [33] fabricated a 2.4 × 3.1 mm2 microtransformer with stacked

primary and secondary spiral copper coils sandwiched between multilayered CoNbZr/SiO2

magnetic films on a glass substrate. RF sputtering was used to deposit both the copper

windings and the magnetic films, which were each patterned by a photoresist lift-off

method. Annealing of the magnetic core was performed at 250 ◦C for 1 hour under

vacuum with a rotating magnetic field. The copper traces were deposited 7.5 µm thick

and patterned 100 µm wide with 10 µm spacing. The turns ratio of primary:secondary

coils was 8 : 7.3. A 10 Ω load was attached to the secondary winding for measurement

of the transformer efficiency with 1 V sinusoidal input to the primary in the frequency

range of 1 − 20 MHz. A maximum efficiency of 67% was obtained at 10 MHz, beyond

which point efficiency was said to decrease due to core loss.

Sullivan and Sanders [27] measured the performance of microfabricated power

conversion transformers with areas on the order of 10 mm2 and primary:secondary

turns ratios of 8 : 4. Primary and secondary windings were interleaved in an elongated

spiral (racetrack) and consisted of 20 µm-thick electroplated copper. A multilayer

laminated NiFe/SiO2 material acted as the magnetic core with a relative permeability of

2000. Two designs were fabricated: one had a sandwich configuration with the copper

windings embedded between separate layers of magnetic material, while the other

design featured a closed core that fully enclosed the windings. The sandwich design

was said to not only decrease inductance by a factor of 5 compared to the closed core,

but also produced an “unfavorable field distribution” that further increased losses. A

half-bridge forward converter served as a test bed for the sandwich transformer and

measured 43.4% efficiency with 3.74 W input at 30 V and 1.625 W output at 4.21 V. The

same converter circuit was tested again with a litz wire transformer having the same

inductance as the sandwich transformer but assumed to have no loss. By comparing

31

the difference in efficiency with the litz wire versus the sandwich transformer, the

sandwich transformer efficiency was estimated at 61%. The closed core transformer was

tested with a network analyzer and projected to have an efficiency of 70%. Higher than

expected losses were attributed to hysteresis losses and shorting between layers in the

core.

Brunet et al. [28] presented a 30 mm2 microfabricated transformer consisting of

interleaved, racetrack-shaped primary and secondary coils encapsulated in 4 µm-thick

electroplated Ni81Fe19 magnetic core. The copper coils were electroplated 43 µm thick

and were arranged in a turns ratio of 4 : 2. Electrical characteristics were obtained using

an impedance analyzer. A primary inductance of 0.9 µH was measured to be constant

up to 5 MHz. Leakage inductance was determined at 0.4 µH measuring the primary

inductance while shorting the secondary coil. The transformer was tested in a full-bridge

dc-dc converter at 2 MHz. Converter efficiency was measured at 40% for input voltages

> 2 V. The core was found to saturate at an input voltage of 4.5 V for a maximum output

power of 0.4 W.

Characteristics of the above mentioned inductors were summarized in Table 2-2

alongside those from other significant works.

32

Table 2-1. Literature survey of microinductors.Inductance Peak Q DC

Reference Layout Core Area Inductance density Peak frequency resistancematerial (mm2) (nH) (nH/mm2) Q (MHz) (Ω)

Ahn et al. [22] Toroid Ni81Fe19 4 400 100 1.5 1 0.3Yamaguchi et al. [42] Spiral Fe61Al13O26 0.137 7.6 56 7.4 1000 6.5Sato et al. [11] Racetrack FeCoBN 21.9 370 16.9 15 7Song et al. [13] Racetrack FeZrBAg 146 1000 6.84 25 10 3.95Fukuda et al. [9] Spiral NiZn 36 1400 38.9 40 5 0.67Wang et al. [10] Racetrack NiFe 5.69 160 28.1 6 4 0.261Viala et al. [14] Spiral FeHfN 0.09 10 111 10 1500Flynn et al. [20] Toroid NiFe 10 1940 194 2 2Orlando et al. [21] Toroid NiFe 31.4 500 15.9 20 2 0.095Lee et al. [19] Solenoid CoTaZr 0.88 70.2 79.7 6.5 25 0.67Park and Allen [8] Spiral Air 1.69 37.8 22.4 44 1200 2.76Young et al. [15] Solenoid Air 0.25 14 56 18 820Choi et al. [16] Spiral Air 0.144 4.6 31.8 50 3500Weon et al. [17] Solenoid Air 0.05 2.1 42 78 4000 0.342Yoon et al. [18] Solenoid Air 0.06 1.17 19.5 84 2600

33

Table 2-2. Literature survey of microtransformers.Primary Secondary

Reference Area inductance inductance Coupling Voltage Efficiency Frequency Core(mm2) (nH) (nH) coefficient gain (MHz) material

Mino et al. [23] 12.0 350 40 0.5 0.3∗ 3% 32 CoZrReYamaguchi et al. [33] 7.44 500 450 0.7 67% 10 CoNbZrKurata et al. [24] 1.38 50 50 0.92 1∗ 54% 100−250 CoFeSiBMino et al. [25] 25 820 820 0.93 1∗ 58%∗ 25 CoZrReXu et al. [26] 4 800 800 0.9 0.63 77%∗ 10 Ni80Fe20Sullivan and Sanders [27] 8.42 1380∗ 345 0.5∗ 61% 8 NiFeSullivan and Sanders [27] 11.85 3176∗ 794 0.5∗ 70% 10 NiFeBrunet et al. [28] 29.92 900 225∗ 0.58∗ 40% 2 Ni81Fe19Park and Bu [29] 5.7 440 440 0.85 1∗ 32% 25 Ni80Fe20Rassel et al. [30] 4.95 100 80 0.9 0.9∗ 1% 0.5 NiFeYun et al. [31] 78.4 830 830∗ 0.91 0.9 84%∗ 5 Ni81Fe19Wang et al. [32] 23.7 400 400 0.93 0.89 72% 5 NiFeYamaguchi et al. [33] 7.44 70 65 0.4 30% 10 AirCheung et al. [34] 0.16 8∗ 8∗ 0.75∗ 1∗ 1000 AirCheung et al. [34] 0.25 0.5∗ 12∗ 0.75∗ 5∗ 1000 AirLaney et al. [35] 0.16∗ 1.65 1.65 0.55 1∗ 56% 2500 AirLong [36] 0.16 8.5 8.5 0.84 1∗ 2000∗ AirRibas et al. [37] 0.09 8.6 8.6 0.79 1∗ 32%∗ 10000∗ AirZolfaghari et al. [38] 0.06∗ 11∗ 180∗ 3 29%∗ 1500 AirNg et al. [39] 0.09 2 3 0.8 1.2∗ 60% 8000 AirAly and Elsharawy [40] 0.32 4.79 4.79 0.88 1∗ 56%∗ 2000∗ Air∗Asterisked values were estimated based on the other information gathered from the respective references.

34

CHAPTER 3INDUCTOR DESIGN

This chapter outlines a methodology for designing microinductors for use in

high-frequency switched-mode power supplies. The quality factor is investigated as

a metric for the efficiency of the inductor in storing energy. Three inductor attributes

affecting peak quality factor—inductance, resistance, and maximum operating

frequency—are discussed in terms of the trade-offs in attempting to maximize any

one of these quantities. The stacked planar spiral layout is chosen for the microinductors

with the goal of reaching high quality factor by maximizing inductive coupling while

minimizing electrical resistance. A modeling strategy is presented for optimization

of the design of such inductors and prediction of their performance. The models

provide a method for determining the optimal geometric proportions based on certain

combinations of desired criteria: inductance, size, operating frequency, and maximum

quality factor.

3.1 Quality Factor Definition

The quality factor, Q, of a circuit is a dimensionless quantity that generally provides

a metric of how much energy is stored in a circuit versus how much energy is dissipated

by it. However, the generality of this concept has led to confusion of the definition of

Q amongst researchers since there are many application-specific interpretations and

methods of extraction of Q [38, 43–46]. For example, one traditional use of Q is in

quantifying the selectivity of a resonant filter circuit [47]. In such filtering applications, Q

is defined as the ratio of the circuit resonant frequency to its half-power bandwidth.

3.1.1 Quality Factor of Non-Ideal Reactive Components

In contrast to the single-valued quality factor of resonant circuits, the quality factor

Q of an energy storing circuit component (e.g. inductor or capacitor) quantifies how

much energy is stored in the component versus how much is dissipated by it at each

35

frequency. However, even this alternate definition can take on different meanings to

different communities when the device is operated near its resonant frequency.

The discrepancy in definition arises from the fact that although the impedance of a

device under alternating current (ac), sinusoidal excitation at its resonant frequency is

purely resistive, energy is being stored and transferred within the electrical and magnetic

fields within the device. By definition, the reactive part of the impedance falls to zero at

resonance, and none of the energy stored in the device is available to the external circuit

attached to it. Passive components in power conversion applications, however, need to

store energy from the circuit and then provide that energy back to the circuit. For this

reason, a separate definition of Q is most appropriate for power conversion application

with the property that Q falls to zero at resonance. The simplified definition of Q used for

quantification of power-passive performance is the ratio of the imaginary to the real part

of the complex impedance looking into the device,

Q =ℑ{Z}ℜ {Z} . (3–1)

It can be shown that the above definition of Q for any one-port, energy-storing

component correctly provides a measure of the storage efficiency as seen by the

external circuit. The expected measure for the component under sinusoidal excitation is

the rate energy is stored in and retrieved from the device to the rate energy is dissipated

in it,

Q =Reactive power transfer

Real power transfer. (3–2)

Assuming root mean square (RMS) quantities for voltage and current, the complex

power transfer to/from a component is the product of the ac voltage V across the

component and the complex conjugate of the current I through it, so that Equation 3–2

is rewritten in terms of voltages and currents as

Q =ℑ{

V I}

ℜ{

V I} . (3–3)

36

Applying properties of complex conjugates, the real and imaginary part operators can be

expanded into equivalent algebraic expressions as

Q =

(

V I − V I)

/j2(

V I + V I)

/2, (3–4)

which simplifies through manipulation to,

Q =

(

VI− VI

)

/j2(

VI+ VI

)

/2. (3–5)

The voltage and current terms in Equation 3–5 are arranged so that the impedance

equivalent is readily identified as,

Q =

(

Z − Z)

/j2(

Z + Z)

/2. (3–6)

By properties of complex conjugates, Equation 3–6 is identical to the original formulation

of Q in Equation 3–1 as the ratio of the imaginary to the real part of the complex

impedance of component.

3.1.2 Quality Factor of Inductor

As discussed in Section 3.1.1, the quality factor Q provides a metric of the ac

energy storage efficiency of actual, non-ideal reactive circuit components, such as

microinductors. The formulation of Q as the ratio of imaginary to the real part of the

inductor impedance (as in Equation 3–1) provides a figure of merit that quantifies the

degree to which an inductor acts like an inductor to an attached circuit.

For example, when used with direct current (dc) there is no electrical characteristic

that differentiates an inductor from a trace of wire with some resistance. Although

energy is stored in the magnetic field induced by the current flowing through the inductor

even at dc, in the absence of any variation in current over time, that energy is never put

back into the circuit.

When there are ac fluctuations in the current flowing through the inductor, energy

is stored as the current increases in magnitude to its peak level and is then retrieved

37

from the magnetic field back into the circuit as the current decreases in magnitude.

Some energy is dissipated as heat due to the resistance of the electrical path through

the inductor. At low frequencies, the rate of energy storage/retrieval is less than the

power dissipated by the inductor, and the inductor has consequently low quality. As

the frequency of the current oscillation increases, however, so too does rate of energy

storage/retrieval while the power dissipated remains relatively constant, and the inductor

therefore attains a higher Q. If the inductor is modeled as the serial combination of an

ideal resistor R and an ideal inductor L, the expression for quality factor at an angular

frequency of ω as calculated from Equation 3–1 is

QRL =ωL

R. (3–7)

This simplified expression ignores the changes in resistance that occur at very high

frequencies and also ignores capacitive energy storage in the electric field that invariably

exists in the inductor.

Because of parasitic capacitance, Q diminishes near the self-resonant frequency of

the inductor as more energy is stored in the electric field between traces. By definition,

Q = 0 at resonance as equal amounts of energy are traded between electric and

magnetic fields, and the spiral again appears as a resistor to the circuit.

3.2 Performance Trilemma

From Equation 3–7 the quality factor of a quasi-ideal inductor (ignoring effects

of capacitance) is dependent on its inductance, resistance, and operating frequency.

Ideally, the quality factor would be maximized if the inductance and operating frequency

were maximized and the resistance minimized. In practice, however, all of these

quantities are linked, so that improvements to any one of these three attributes is

often done at the detriment of the other two.

Consider the simple model of the inductor shown in Figure 3-1 with inductance

L, series resistance R, and shunt capacitance C . Self-resonance limits the maximum

38

operating frequency of the inductor and for the case of low resistance is approximately

equal to the natural frequency,

ω0 =

√

1

LC. (3–8)

The above equation clearly shows that increasing inductance directly results in

decreasing resonant frequency. However, attaining higher inductance often entails

increasing the trace length of the inductor winding, resulting in higher capacitance,

which in turn further decreases the resonant frequency. The increased trace length

also increases the resistance of the inductor. Designing inductors must balance these

competing goals to deliver a device that is tailored to the application.

C

R

L

Figure 3-1. Circuit diagram of simple inductor model with inductance L, series resistanceR, and shunt capacitance C .

3.3 Stacked Planar Spiral Layout

In response to the previously mentioned concerns for maximizing inductance while

minimizing resistance and capacitance, the stacked planar spiral layout was selected

for the inductors of this work. The planar spiral layout features conductive traces that

are concentrically wound into a flat spiral as depicted in Figure 3-2. This layout is the

most popular amongst all integrated inductors because it offers high density through

tight spiral packing and it is easy to fabricate via conventional planar microfabrication

steps. When all traces are constrained to only a single plane, however, performance is

limited by poor magnetic coupling between outer and inner windings. As the number of

spiral turns is increased, the separation between inner and outer windings can become

so great that the inner turns contribute more towards increasing the resistance of the

inductor than towards its inductance.

39

To overcome the planar limitation, vertical stacking of planar spirals is used to

increase both the inductance density and the quality factors of the inductors. Because

the planar spiral is by nature wider in diameter than it is thick (given typical conductor

thicknesses), stacking spirals provides excellent magnetic field coupling in the vertical

direction. Assuming perfect coupling, the inductance of a two-layer stacked-spiral

inductor can reach up to four times that of a single layer while the resistance is only

doubled. In this simplified example, the inductance to resistance ratio of the two-layer

device is improved to twice that of a single-layer device.

3.4 Low Frequency Analytical Inductor Model

The low-frequency model of the inductor includes only the electrical resistance

along the length of the trace winding and the magnetic field generated when an

electric current passes through the winding. The current is assumed to flow uniformly

through the cross section of each trace, ignoring current crowding due to interactions

between moving charge carriers. Inductance and resistance are first calculated for a

single-layer winding of uniform trace width and thickness. The values are then extended

as appropriate when two layers are vertically stacked.

By the year 1928 Wheeler [48] had derived by empirical data an expression to

predict the inductance of discrete radio coils. More than 70 years later Mohan et al. [49]

modified the existing expression only slightly to be valid also for microinductors,

Lmw =K1µ0n

2davg

1 + K2p. (3–9)

In the above expression, K1 and K2 are empirically derived values that are specific to

the shape of the spiral (i.e. square, hexagonal, octagonal) and are listed in Table 3-1.

An additional expression was presented in Mohan et al. [49] for calculating inductance

based on a current sheet approximation [50],

Lgmd =µn2davgc12

[

ln

(

c2

ρ

)

+ c3ρ+ c4ρ2

]

, (3–10)

40

Table 3-1. Coefficients for modified Wheeler (Equation 3–9) and current sheet (Equation3–10) expressions [49].

Shape K1 K2 c1 c2 c3 c4Square 2.34 2.75 1.27 2.07 0.18 0.13Hexagonal 2.33 3.82 1.09 2.23 0.00 0.17Octagonal 2.25 3.55 1.07 2.29 0.00 0.19Circle - - 1.00 2.46 0.00 0.20

for which expression the shaped-dependent coefficients (c1, c2, c3, and c4) are provided

not only for square, hexagonal, and octagonal shapes but also for circular. These

coefficients are also listed in Table 3-1. The choice between using the two previously

listed expressions depends on the situation. If a circular layout is desired, the current

sheet expression in Equation 3–10 provides the best accuracy. If rearranging the

expression to solve for a different variable, the modified Wheeler expression in Equation

3–9 is simpler to solve.

The authors of these expressions noted that each had been validated only for

inductors < 100 nH with outer diameters ranging from 100 − 480 µm [49]. As part of

this dissertation work, the inductance values calculated from Equation 3–9 were verified

against magnetoquasistatic simulations with less than 5% error for inductors up to

1050 nH and outer diameters up to 2.5 mm (see Section 6.3.1.1).

All of the other variables in both Equation 3–9 and Equation 3–10 — i.e. the number

of turns n, the packing density p, and the average diameter davg — are obtained from the

geometry of the spiral. The geometry of the spiral can be uniquely specified in terms of

the winding trace width w , the spacing between adjacent winding traces s, the number

of winding turns n, and the outer diameter D. These dimensions are marked on the

diagram of an example spiral in Figure 3-2. Inner (d) and outer (D) diameters were

measured from the centerlines of the innermost and outermost traces, respectively.

The inner diameter d represents the space contained within the spiral that is clear of

41

w s

d

D

Figure 3-2. Diagram of planar spiral layout with n = 3 turns and all other dimensionslabelled.

windings and is calculated as a function of the other dimensions,

d = D − 2 [wn + s (n − 1)] . (3–11)

The average diameter is then simply calculated as

davg =D + d

2. (3–12)

The packing density p represents the fraction of the inductor area that is filled with

windings and is defined as

p =D − dD + d

. (3–13)

Extending the aforementioned inductance and resistance calculations for a single

layer spiral, the total inductance for an inductor with two identical spirals stacked

vertically is calculated in proportion to L0, the inductance of a single-layer spiral from

either Equation 3–9 or 3–10,

Ldc = 2 (1 + k)L0, (3–14)

42

where k is the coupling coefficient representing the portion of shared magnetic flux

linking the top and bottom spirals. The value of k can vary between −1 and 1. If the

spirals are positioned so that no magnetic flux is shared between spirals, k = 0 and the

total inductance is twice that of the single-layer coil. When all flux is shared between

coils, k = 1 and the total inductance is four times that of the single-layer coil. If the

coils are stacked so that the magnetic fluxes of each coil are in opposition, k can have

a negative value as the opposing magnetic fields nullify and reduce the total amount of

flux linking the coils.

The dc resistance of the inductor can be calculated by the familiar expression for

resistance,

Rdc =ρl

wt, (3–15)

where ρ is the electrical resistivity of the trace material, l is the total electrical path length

of the inductor, and t is the thickness of the trace. The total trace length of the stacked

spiral windings can be calculated from the geometry design variables. For the two-layer

stacked square spiral, the total electrical trace length is evaluated as

l = 2[

4nD − (2n − 1)2 (w + s)]

. (3–16)

The trace length for the two-layer stacked circular spiral case is calculated as

l = 2π [nD − n (n − 1) (w + s)] . (3–17)

3.5 Trends and Optimization

3.5.1 Analytical

The expressions listed in Section 3.4 for estimating the low-frequency inductance

and resistance of spirals were used to explore performance trends associated with

sweeping certain design variables. A square spiral shape was assumed for ease in

rapidly iterating layout and modeling. Perfect coupling (k = 1) was assumed for all

cases. The target metric for this simplified analysis was the inductance to resistance

43

ratio L/R, which is proportional to the quality factor at low frequencies (see Equation

3–7).

The first case was to determine the optimal packing density. The spacing between

turns was fixed at s = 10 µm, while the width of the traces was varied. The number of

turns was swept from the minimum number of turns (n = 1) turn up to the maximum

number of turns that could physically be packed within the allotted area. Separate

runs were completed for each different outer diameter D. The results were plotted for

D = 500 µm (Figure 3-3A) and D = 1000 µm (Figure 3-3B). For all outer diameters and

widths, the Ldc/Rdc ratios increased drastically as the number of turns was increased

from 1 but then reached their maximal values at a packing density of approximately

p = 0.4, which is equivalent to the points at which inner diameters were barely greater

than 40% of the outer diameters. Also from these plots, the maximum Ldc/Rdc ratio

increased with increasing trace width up to about w = 50 µm, past which no significant

further increases were recorded.

The plots of Figures 3-3A and 3-3B also suggested that the Ldc/Rdc might also

increase with outer diameter. To test this idea, outer diameters from D = 0.5 mm up

to D = 2.5 mm were evaluated using the optimal widths and packing densities already

learned. For each trial with different outer diameter, the trace spacing was fixed at

s = 10 µm, the trace width was fixed at w = 50 µm, and the number of turns n was

calculated such that the packing density would be approximately p = 0.4. In this setup,

the Ldc/Rdc computed for each outer diameter should represent approximately the peak

values of the curves seen in Figure 3-3. The results from sweeping the outer diameter

are plotted in Figure 3-4 and indicate a linear relationship between the Ldc/Rdc ratio and

the outer diameter of the inductor.

3.5.2 FastHenry

FastHenry is a software program that can solve the magnetoquasistatic inductance

and resistance of a three-dimensional structure using integral equation-based mesh

44

0 0.2 0.4 0.6 0.8 16

8

10

12

14

16

18

20

22

24

26

Packing Density p

L dc/

Rdc

(nH

/Ω)

20 µm30 µm40 µm50 µm60 µm70 µm80 µm

Width w

A Outer diameter D = 500 µm

0 0.2 0.4 0.6 0.8 15

10

15

20

25

30

35

40

45

50

55

Packing Density p

L dc/

Rdc

(nH

/Ω)

20 µm30 µm40 µm50 µm60 µm70 µm80 µm

Width w

B Outer diameter D = 1000 µm

Figure 3-3. Trends of inductance to resistance ratio vs. packing density for various tracewidths and outer diameters.

45

0.5 1 1.5 2 2.520

40

60

80

100

120

140

Outer Diameter D (mm)

L dc/

Rdc

(nH

/Ω)

Figure 3-4. Trends of inductance to resistance ratio vs. outer diameter using w = 50 µm,s = 10 µm, and n such that p ≈ 0.4

10−1

100

101

102

103

80

100

120

140

160

Interlayer spacing (µm)

Indu

ctan

ce (

nH)

4L0

2L0

FastHenry simulation results

k=0

k=1

Figure 3-5. Trend of inductance vs. vertical gap between stack in 1 mm× 1 mm assimulated in FastHenry.

46

analysis combined with a multipole-accelerated iterative solution algorithm [51]. A

variety of stacked inductor designs were simulated in FastHenry, first to validate the

analytical model results and then also to determine the effect of vertical separation

between the two stacked layers on the mutual inductance between them. The same

inductor design was simulated several times in FastHenry but with increasing vertical

layer separation in each simulation trial. Plotted in Figure 3-5 is the low frequency

inductance obtained for a 1 mm × 1 mm with interlayer separation varied from 0.1 −

1000 µm. Drawn on the plot are lines indicating 2× and 4× the inductance L0 that would

be obtained for a single winding layer. For the simulation with two winding layers the

inductance asymptotically approaches 4L0 (k = 1) and 2L0 at the extremes of short

and long separations, respectively. FastHenry simulations indicated that there would be

minimal improvement to the inductance with separations less than 1% of the inductor

diameter. A vertical separation of 10 µm was used for the microfabricated devices as

shorter separations would serve only to detrimentally increase the parasitic capacitance

between layers.

3.6 Radio Frequency Effects

Although direct current (dc) assumptions (e.g. uniform current distribution) enabled

a simplified optimization of the inductor layout (such as diameter and number of

turns), inductors designed for microscale power systems need to operate at such

high frequencies (> 10MHz) that complex electromagnetic behavior alters the

apparent inductances and resistances from the expected dc values. The dominant

electromagnetic effects can be classified as due to capacitive coupling or due to eddy

current generation.

3.6.1 Capacitive Coupling

Capacitive coupling results from variations in voltage potential that exist within

different parts of the inductor. It is especially prominent between the terminal ends of the

inductors where the difference in potential is the greatest. Because the terminal ends

47

are where the inductor is connected to the rest of a circuit or where large landing pads

provide electrical connection to probe tips for measurement and characterization, the

effect of capacitive coupling can be highly dependent on factors that are external to the

design of the inductor.

Intrinsic to the design of the inductor, however, is the capacitive coupling that

occurs between adjacent windings of an inductor. Microinductors intended for GHz RF

applications typically consist of a single winding layer and a metal underpass providing

electrical connection to the innermost turn. The interwinding capacitance of these

single-layer inductors has long been known to be dominated by the areas where the

windings and the underpass overlap [52, 53]. The stacking of windings for greater

inductance density, as in the inductors of this work, would result in even greater levels of

interwinding capacitance due to the significantly increased area of overlap. The general

expression for capacitance between parallel plate electrodes,

C =ǫA

g, (3–18)

shows that in addition to the overlap area between plates A, the other aspects affecting

capacitive coupling are the permittivity ǫ of the material between the plates and the

distance g of the gap between them. The multilevel thick-film fabrication technology

presented in Chapter 5 minimizes the capacitive coupling between upper and lower

winding layers by separating the layers by up to 30 µm and removing all dielectric

material from between layers.

If the inductors are fabricated on a conductive substrate such as silicon, the

substrate creates an additional path for capacitive coupling. To electrically isolate

the inductor from the substrate a thin dielectric layer such as silicon dioxide is often

deposited over the substrate. Considering a scenario where two traces of an inductor at

different voltage potentials sit atop the dielectric layer in close proximity, capacitors are

48

formed with the dielectric layer between each trace and the substrate with the substrate

providing an electrical connection between the two traces as illustrated in Figure 3-6.

Copper Trace Copper Trace

Conductive Substrate

Dielectric Layer

Figure 3-6. Diagram illustrating capacitive coupling through substrate between coppertraces of inductor winding.

Like the interwinding capacitance, the shunt capacitance through the substrate

contributes to resonant behavior as energy oscillates between inductive and capacitive

storage elements. However, the finite resistance of the capacitive link through the

substrate can have a profound effect on the perceived inductor behavior near the

resonance. The substrate resistance can be modeled as a resistor Rc in series with the

capacitance to the substrate Cs , as drawn in the circuit diagram in Figure 3-7.

Cs

Rdc

Ldc

Rc

Figure 3-7. Circuit diagram of inductor model with dc inductance Ldc , series resistancethrough the inductor Rdc , shunt capacitance to the substrate Cs , andresistance along the capacitive path through the substrate Rc .

49

107

108

109

10−9

10−8

10−7

10−6

10−5

Frequency (Hz)

Indu

ctan

ce (

n