List of Abbreviations xvi

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Abstract iii

Beknopte samenvatting vii

Résumé xi

List of Abbreviations xvi

List of Symbols xix

Contents xxi

List of Figures xxix

List of Tables xxxv

1 Introduction 1

1.1 Context and motivation . . . . 1

1.1.1 Demands for high-performance output filters . . . . 1

1.1.2 Computer-aided filter design . . . . 6

1.2 Objectives and scopes . . . . 7

1.2.1 Research questions and objectives . . . . 7

xxi

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1.2.2 Research scopes . . . . 7

1.3 Summary of original contributions . . . . 8

1.4 Outline . . . . 10

2 Overview of Output Filters 13 2.1 Control system . . . . 13

2.2 Filter design . . . . 15

2.3 Resonance and damping . . . . 20

2.3.1 Passive damping (PD) . . . . 21

2.3.2 Active damping (AD) . . . . 21

2.3.3 Hybrid damping (HD) . . . . 22

2.4 Modeling of magnetic components . . . . 22

2.4.1 Core losses . . . . 23

2.4.2 Winding losses . . . . 24

2.4.3 Thermal modeling . . . . 28

3 Virtual Circuit Control for VSCs with Output Filters 29 3.1 Introduction . . . . 29

3.2 Motivation of VCC method based on continuous-time systems . . . . 33

3.2.1 Continuous-time circuit models . . . . 33

3.2.2 Transformations between physical and virtual circuits . 35 3.2.3 Continuous-time state feedback control law . . . . 35

3.3 Proposed VCC method for sampled-data systems . . . . 37

3.4 VCC implementation and mathematical model . . . . 40

3.4.1 Delay compensation and observer . . . . 40

3.4.2 Combining DM and CM VCC controls . . . . 44

3.5 Virtual reference circuit design examples . . . . 44

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3.5.1 CM and DM virtual circuits with series resistors (circuits

A and B) . . . . 47

3.5.2 DM virtual circuit with parallel resistor (circuit C) . . . 49

3.5.3 DM virtual circuit with two series resistors (circuit D) . 50 3.6 Simulation and experimental results . . . . 52

3.6.1 Grid without additional grid impedance . . . . 53

3.6.2 Grid with additional grid impedance . . . . 56

3.6.3 Auxiliary control . . . . 58

3.7 Conclusions . . . . 62

4 Hardware and Control Co-Design Method for Actively-Damped Output Filters 65 4.1 Introduction . . . . 65

4.2 Proposed co-design of filter component values and control . . . 68

4.2.1 Ladder network synthesis . . . . 69

4.2.2 Design constraints . . . . 71

4.3 Example of determining LCL -filter component values . . . . . 75

4.3.1 Design example . . . . 75

4.3.2 Experimental results . . . . 79

4.4 Design example of high-order filters with multiobjective optimization . . . . 81

4.4.1 Dimensioning of inductors . . . . 85

4.4.2 Dimensioning of capacitors . . . . 90

4.4.3 Design constraints . . . . 91

4.4.4 Filter design and optimization . . . . 96

4.5 Conclusions . . . 103

5 Frequency-Domain Homogenization of Litz-Wire Winding 107

5.1 Introduction . . . 107

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5.2 Eddy-current losses in litz-wire bundles . . . 108

5.3 Literature review . . . 110

5.4 Present work . . . 113

5.4.1 The effect of the number of layers . . . 115

5.4.2 The effect of the packing pattern . . . 115

5.4.3 Approximating function for complex permeability . . . . 117

5.5 Application examples . . . 119

5.5.1 Litz-wire transformer . . . 120

5.5.2 Litz-wire inductor . . . 123

5.6 Conclusions . . . 125

6 Time-Domain Homogenization of Winding 129 6.1 Introduction . . . 129

6.2 Macroscopic model of eddy currents in multi-conductor windings . . . 131

6.3 RL ladder networks for time-domain simulation . . . 133

6.3.1 Synthesis of RL ladder networks . . . 134

6.3.2 Synthesis results . . . 135

6.4 Time-domain homogenization of windings in FE models . . . . 137

6.4.1 Analysis of computational cost . . . 138

6.4.2 Sparsity of system matrices . . . 143

6.4.3 Time-domain FE results . . . 146

6.5 Conclusions . . . 147

7 Experimental Extraction of Winding Resistance 149 7.1 Introduction . . . 149

7.2 Electrical circuit model of magnetically coupled windings with a

magnetic core . . . 151

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7.2.1 A device with n

w

windings . . . 151 7.2.2 A two-winding device ( n

_w

= 2) and its T-equivalent

circuit . . . 153 7.3 Influence of winding mutual resistance on the extraction of

winding resistance based on two-winding method . . . 155 7.3.1 Two-winding method for core-loss measurement . . . 156 7.3.2 Effect of winding mutual resistance R

^w₁₂

on extracted

winding resistance . . . 157 7.4 Experimental methodology with improved core loss compensation 158 7.4.1 Core-loss resistance compensation . . . 158 7.4.2 Steps of the winding resistance extraction . . . 161 7.5 Measurement and numerical results . . . 163

7.5.1 Measurement based on windings with high winding mutual resistance . . . 164 7.5.2 Improved measurement based on a 1:1 auxiliary

transformer with single-turn windings . . . 166 7.5.3 Accuracy of the measurement . . . 166 7.6 Implications on winding loss errors based on low-frequency

conventional methods . . . 170 7.6.1 Unawareness of the existence of winding mutual resistance170 7.6.2 Unawareness of winding mutual resistance between the

windings used to perform the core-loss measurement . . 172 7.7 Conclusions . . . 173

8 Conclusions and Outlook 175

8.1 Conclusions . . . 175 8.1.1 Design of filter component values and resonant-damping

controller (Chapters 3 and 4) . . . 176 8.1.2 How can a designer estimate the winding losses of a

design? (Chapters 5 and 6) . . . 179

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8.1.3 How can a designer determine the winding losses of a

magnetic device? (Chapter 7) . . . 182

8.2 Outlook and future work . . . 183

8.2.1 Extension of virtual circuit control . . . 183

8.2.2 Further study of high-order filter design . . . 184

8.2.3 Winding model refinement . . . 186

A Solution to the VCC Problem 187 B State-Space Modeling of VCC-Based Grid-Tied VSCs with Output LCL Filters 191 B.1 Continuous-time modeling of LCL filters in natural abc frame . 191 B.1.1 LCL filter . . . 191

B.1.2 Three-phase balanced grid voltage . . . 193

B.1.3 State-space model of considered grid-tied VSC with LCL filter . . . 194

B.2 Continuous-time modeling of LCL filters in DM ( αβ ) and CM ( γ ) frames . . . 194

B.2.1 DM ( αβ ) and CM ( γ ) decomposition . . . 194

B.2.2 CM-filter model ( γ component) . . . 195

B.2.3 DM-filter model ( αβ components) . . . 196

B.3 Sampled-data modeling of LCL filters . . . 197

B.3.1 Filter model . . . 197

B.3.2 Grid model . . . 198

B.3.3 Delta-sigma measurement processing . . . 198

B.3.4 Kalman observer . . . 200

B.3.5 Digital delay . . . 201

B.3.6 Complete control system . . . 202

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C Derivation of Electrical Circuit Model of Magnetically Coupled Multi-Winding Systems with a Magnetic Core 205

Bibliography 211

List of Publications 231

Curriculum Vitae 233