COUPLED-INDUCTOR MAGNETICS IN POWER ELECTRONICS Thesis by Zhe Zhang In Partial Fulfillment of the Requirements for the Degree of Doctor of Philosophy California Institute of Technology Pasadena, California 1987 (Submitted October 7, 1986) 11 © 1986 Zhe Zhang All Rights Reserved 111 Acknowledgements I wish to express my deepest gratitude to my advisors, Professor S. Cuk and Professor R. D. Middlebrook, for their encouragement and advice during the five years I have studied at Caltech. I appreciate very much the financial support of the California Institute of Technology by the way of Graduate Teaching Assistantships. In addition, Graduate Re­ search Assistantships supported by the International Business Machines, GTE, Emerson Electric, General Dynamics Corporations, and National Aeronautics and Space Admin­ istration Lewis Research Center are gratefully acknowledged. Most importantly, I thank my wife Guo Zhan, son Zhang Pei, and my parents W. Y. Chang and C. S. Wang. Without their support and understanding my study at Caltech would never have been possible. In addition, I wish to thank Mr. Wang Senlin, my first teacher in electronics, and Mr. Xu Qinglu, my supervisor when I started my first job, for their training and support throughout the years. In the years with the Power Electronics Group, I have learned from the Group much more than I have contributed. I thank all my fellow members of the Power Elec­ tronics Group for all their help and support. lV v Abstract Leakages are inseparably associated with magnetic circuits and are always thought of in three different negative ways: either you have them and you don't want them (transformers), or you don't have them but want them (to limit transformer short circuit currents), or you have them and want them, but you don't have them in the right amount (coupled-inductor magnetic structures). The methods of how to introduce the leakages at appropriate places and in just the right amounts in coupled-inductor magnetic structures are presented here, in order to optimize the performance of switching de-to-de converters. Vl Vll Contents ACKNOWLEDGEMENTS iii ABSTRACT v 1 INTRODUCTION 1 2 MAGNETIC CIRCUITS AND MODELS 7 2.1 COUPLED-INDUCTOR EQUATIONS .. 7 2.1.1 Self Inductance and Mutual Inductance for Two Windings . 7 2.1.2 Coupling Coefficients k, k1 and k2 ....... 9 2.1.3 Coupling Inductor Equations for More Windings 12 2.2 RELUCTANCE MODEL .... 13 2.2.l The Reluctance Concept . 13 2.2.2 Inductance of an Inductor . 14 2.2.3 Air Gaps in Magnetic Circuits 14 2.3 CIRCUIT MODEL .......... 16 2.3.l Deriving the Circuit Model From the Coupled-Inductor Equations 16 2.3.2 Deriving the Circuit Model From the Reluctance Model . 18 2.4 DERIVING THE CIRCUIT MODEL FROM THE RELUCTANCE MODEL .... 19 2.4.1 Duality 19 Vlll 2.4.2 Using Duality to Derive the Circuit Model From the Reluctance Model . 19 3 COUPLED-INDUCTORS AND INTEGRATED MAGNETIC CIRCUITS IN SWITCHING CONVERTERS 25 3.1 SIZE AND WEIGHT REDUCTION BY COUPLING TWO INDUCTORS INTO ONE STRUCTURE . 25 3 .1.1 Two Examples . 25 3.1.2 Magnetic Scaling Law 27 3.2 PERFORMANCE IMPROVEMENT. 29 4 ZERO RIPPLE CONDITIONS FOR TWO WINDING COUPLED- INDUCTORS 31 4.1 SEPARATION OF DC AND AC CURRENTS 33 4.2 USING COUPLED-INDUCTOR EQUATIONS TO FIND THE ZERO RIPPLE CONDITION . 34 4.2.1 Writing the Zero Ripple Condition Using the Coupling Coefficient k 36 4.3 USING THE CIRCUIT MODEL TO FIND THE ZERO RIPPLE CON- DITION. 37 4.3.1 Writing the Zero Ripple Condition Using Coupling Coefficients ki, k2 . 39 4.3.2 Comparison of the Matching Conditions . 41 4.4 USING THE RELUCTANCE MODEL TO FIND THE ZERO RIPPLE CONDITION . 42 4.5 AGREEMENT OF THE THREE METHODS 44 4.5.1 Reluctance Model vs. Circuit Model ... 44 4.5.2 Circuit Model vs. Coupled-Inductor Equations 45 ix 4.6 HARDWARE IMPLICATION OF THE LEAKAGE INDUCTANCE . 46 4.6.1 Using External Leakage Inductor . 46 4.6.2 Using a Leakage Magnetic Path in the Magnetic Structure . 47 4.6.3 Using the Leakages of the Windings Only 49 5 SENSITIVITY AND RESIDUAL RIPPLE 53 5.1 MODELS FOR SENSITIVITY AND RESIDUAL RIPPLE CALCULA­ TION AND THE IMPORTANCE OF HIGH LEAKAGE . 55 5.2 FIRST-ORDER ERRORS AND SENSITIVITY . 58 5.2.l Error Voltage Due to Turns Ratio Errors 59 5.2.2 Error Voltage Due to Air-Gap Errors . 62 5.3 SECOND-ORDER EFFECTS ........ 63 5.3.1 Error Voltage Due to Inductor Copper Losses . 64 5.3.2 Error Voltage Due to Energy Transfer Capacitor 67 5.3.3 Error Voltage Due to Inductor Core Loss .... 70 5.4 ESTIMATING THE RESIDUAL RIPPLE AND SENSITIVITY 70 5.5 AN EXAMPLE . 72 6 EFFECT OF THE ISOLATION TRANSFORMER LEAKAGES IN A COUPLED-INDUCTOR CUK CONVERTER 75 6.1 NO LEAKAGES IN EITHER THE TRANSFORMER OR COUPLED- INDUCTORS ....................... 79 6.2 LEAKAGES IN THE COUPLED-INDUCTORS ONLY 79 6.3 LEAKAGES IN THE TRANSFORMER ONLY ..... 82 6.3.1 Leakage in Transformer Only; Output Capacitor ESR is Zero 82 6.3.2 Leakage in Transformer Only; Output Capacitor ESR is Not Zero 85 x 6.4 LEAKAGES IN BOTH TRANSFORMER AND COUPLED- INDUCTORS ... 88 7 MULTIPLE WINDING STRUCTURES 95 7.1 CONVENTIONAL MULTIPLE WINDING COUPLED-INDUCTOR STRUCTURE 95 7.2 IMPROVED MULTIPLE WINDING COUPLED-INDUCTOR STRUCTURE 97 7.3 MULTIPLE AIR-GAP STRUCTURE . 97 7.4 MULTIPLE AIR-GAP STRUCTURE USING STANDARD EE AND EI- CORES ..................................... 102 8 DESIGN-ORIENTED COUPLED-INDUCTOR ANALYSIS 105 8.1 INTRODUCTION OF A NEW LEAKAGE PARAMETER l ....... 106 8.2 THE ANALYSIS OF THE THREE WINDING GAPED EI-CORE COUPLED- INDUCTORS .................................. 107 8.2.1 Determination of Zero-Current Ripple Conditions . 108 8.2.2 De Saturation Conditions . 112 8.2.3 Derivation of Inductance L . 114 8.3 THE DESIGN EQUATIONS FOR THE THREE WINDING COUPLED- INDUCTORS ..................... 114 8.3.1 Derivation of Important Analytical Relations 115 8.3.2 Toward Analytical Solution in a closed Form 115 8.4 CLOSED FORM SOLUTION FOR EQUAL GAPS .. 122 8.5 "BLOW-UP" PROBLEM IN THE COUPLED-INDUCTOR DESIGN 124 8.6 DESIGN EXAMPLE . 132 xi 9 OTHER EI AND EE-CORE STRUCTURES FOR COUPLED- INDUCTORS 135 9.1 THE El-CORE COUPLED-INDUCTOR STRUCTURE USING A COM- MON SPACER ................................. 136 9.1.l The Reluctance Model and the Circuit Model . 136 9.1.2 The Analysis of the Spacer Core Structure .. 140 9.1.3 The Design Equations for the Spacer Core Structure 144 9.2 THE COUPLED-INDUCTOR STRUCTURE USING EE-CORES 145 9.2.1 The Reluctance Model and the Circuit Model . 145 9.2.2 The Use of EE-Core Structures in Coupled-Inductors With More than Three Windings . 145 10 SENSITIVITY AND RESIDUAL RIPPLE OF THE NEW STRUCTURES 149 10.l THE CIRCUIT MODEL FOR THE SENSITIVITY AND RESIDUAL RIPPLE CALCULATION . 152 10.1.1 The Operating Mode . 152 10.1.2 The Test Mode .... 152 10.2 FIRST-ORDER ERRORS .. 154 10.2.1 Ripple Current Due to Drive Voltage Errors . 154 10.2.2 Ripple Current Due to Air-Gap Errors .... 155 10.3 SECOND-ORDER EFFECTS AND RESIDUAL RIPPLE . 157 10.4 EXAMPLE . 158 10.4.1 Sensitivity . 158 10.4.2 Residual Ripple . 159 Xll 11 DESIGN CONSIDERATIONS AND DESIGN EXAI\.1PLE 161 11.1 DESIGN CONSIDERATIONS . 161 11.1.1 Coupled-Inductors for Different Output Voltages 161 11.1.2 The Selection of the Input Inductance . 162 11.1.3 The "Worst Case" Design for Spacer EI and EE-Core Structures 168 11.1.4 Correction for the Fringing Flux . 169 11.1.5 The Window Area for the Input Winding 170 11.2 150W OFF-LINE SWITCHER EXAMPLE . 171 11.2 .1 Design Requirements and Considerations 171 11.2.2 Design of the Coupled-Inductors . 173 11.2.3 Comparison With Other Magnetic Structures 176 12 CROSS-REGULATION PROBLEM IN SWITCHING CON- VERTERS 179 12.l THE DISCONTINUOUS CONDUCTION MODE (DCM) OF SWITCH- ING CONVERTERS. 180 12.2 CROSS-REGULATION IN MULTIPLE OUTPUT CONVERTERS 182 13 CROSS-REGULATION FOR MULTIPLE OUTPUT FLYBACK CON- VERTERS 185 13.l FLYBACK CONVERTER IN CONTINUOUS CONDUCTION MODE 185 13.2 FLYBACK CONVERTER IN DISCONTINUOUS CONDUCTION MODE ..................................... 188 Xlll 14 CROSS-REGULATION FOR MULTIPLE OUTPUT BUCK TYPE CONVERTERS 193 14.1 THE FAMILY OF BUCK TYPE CONVERTERS ............. 193 14.2 MULTIPLE OUTPUT BUCK TYPE CONVERTERS WITHOUT COUPLED- INDUCTORS . 196 14.3 MULTIPLE OUTPUT BUCK TYPE CONVERTERS WITH COUPLED- INDUCTORS ................ 198 14.3.1 Coupled-Inductors with No Leakage 198 14.3.2 Coupled-Inductors With Leakage Inductances . 198 15 CROSS-REGULATION FOR CUK CONVERTERS 207 15.1 CUK CONVERTER WITH SEPARATE INDUCTORS 207 15.1.1 The Continuous Conduction Mode .. 207 15.1.2 The Discontinuous Conduction Mode. 211 15.2 CUK CONVERTER WITH COUPLED-INDUCTORS . 212 16 CONCLUSIONS 213 REFERENCES 215 A LEAKAGE PARAMETER CHARACTERIZATION AND MEASUREMENT 217 A.I LEAKAGE FLUX, LEAKAGE RELUCTANCE AND LEAKAGE INDUC- TANCE .............. 217 A.2 THE LEAKAGE PARAMETERS 218 A.3 THE MEASURING SETUP . 220 A.4 IS THE LEAKAGE PARAMETER A CONSTANT? . 224 A.4.1 Leakage vs. Winding Configuration ...... 224 XlV A.5 THE MEASUREMENT AND CALCULATION FOR DIFFERENT EI AND EE-CORES . 227 A.6 LEAKAGE PARAMETER SPECIFICATION ................ 227 1 Chapter 1 INTRODUCTION The switched mode power conversion concept emerged from the need of high power, high efficiency, compact and lightweight voltage regulators. In a typical linear regulator, the line voltage is converted to other voltages by the use of a transformer, then rectified into de.