ESSENTIAL DC/DC CONVERTERS
© 2006 by Taylor & Francis Group, LLC
ESSENTIAL DC/DC CONVERTERS
Fang Lin Luo Nanyang Tec...
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ESSENTIAL DC/DC CONVERTERS
© 2006 by Taylor & Francis Group, LLC
ESSENTIAL DC/DC CONVERTERS
Fang Lin Luo Nanyang Technological University Singapore
Hong Ye Nanyang Technological University Singapore
© 2006 by Taylor & Francis Group, LLC
The material was previously published in Advanced DC/DC Converters © CRC Press LLC 2003
Published in 2006 by CRC Press Taylor & Francis Group 6000 Broken Sound Parkway NW, Suite 300 Boca Raton, FL 33487-2742 © 2006 by Taylor & Francis Group, LLC CRC Press is an imprint of Taylor & Francis Group No claim to original U.S. Government works Printed in the United States of America on acid-free paper 10 9 8 7 6 5 4 3 2 1 International Standard Book Number-10: 0-8493-7238-0 (Hardcover) International Standard Book Number-13: 978-0-8493-7238-4 (Hardcover) Library of Congress Card Number 2005050494 This book contains information obtained from authentic and highly regarded sources. Reprinted material is quoted with permission, and sources are indicated. A wide variety of references are listed. Reasonable efforts have been made to publish reliable data and information, but the author and the publisher cannot assume responsibility for the validity of all materials or for the consequences of their use. No part of this book may be reprinted, reproduced, transmitted, or utilized in any form by any electronic, mechanical, or other means, now known or hereafter invented, including photocopying, microfilming, and recording, or in any information storage or retrieval system, without written permission from the publishers. For permission to photocopy or use material electronically from this work, please access www.copyright.com (http://www.copyright.com/) or contact the Copyright Clearance Center, Inc. (CCC) 222 Rosewood Drive, Danvers, MA 01923, 978-750-8400. CCC is a not-for-profit organization that provides licenses and registration for a variety of users. For organizations that have been granted a photocopy license by the CCC, a separate system of payment has been arranged. Trademark Notice: Product or corporate names may be trademarks or registered trademarks, and are used only for identification and explanation without intent to infringe.
Library of Congress Cataloging-in-Publication Data Luo, Fang Lin. Essential DC/DC converters / Fang Lin Luo, Hong Ye. p. cm. Includes bibliographical references and index. ISBN 0-8493-7238-0 (alk. paper) 1. DC-to-DC converters. I. Ye, Hong, 1973- II. Title. TK7872.C8L855 2005 621.31'3--dc22
2005050494
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Preface
DC/DC conversion technology is the main branch of Power Electronics and is progressing rapidly. Recent reports indicate that the production of DC/ DC converters occupies the largest percentage of the total turnover of all conversion equipment productions. Based on incomplete statistics, there are more than 500 topologies of DC/DC converters existing. Many new topologies are still created every year. It is a lofty undertaking to write about the large number of DC/DC converters. The authors have sorted these converters into six generations since 2000. This systematical work is very helpful for understanding DC/DC converter evolution and development. The categories are listed below: 1. 2. 3. 4. 5. 6.
The The The The The The
first generation (classical/traditional) converters second generation (multiple quadrant) converters third generation (switched component) converters fourth generation (soft switching) converters fifth generation (synchronous rectifier) converters sixth generation (multiple element resonant power) converters
More than 300 prototypes of the first generation converters have been developed in the past 80 years. The purpose of this book is to provide information about essential DC/ DC converters that is both concise and useful for engineering students and practicing professionals. It is well organized in 410 pages and 200 diagrams to introduce more than 80 topologies of the essential DC/DC converters originally developed by the authors. All topologies are novel approaches and great contributions to modern power electronics. They are sorted in three groups: • The voltage-lift converters • The super-lift converters • The ultra-lift converter The voltage lift technique is a popular method that is widely applied in electronic circuit design. Applying this technique effectively overcomes the effects of parasitic elements and greatly increases the output voltage. Therefore, these DC/DC converters can convert the source voltage into a higher output voltage with high power efficiency, high power density and a simple
© 2006 by Taylor & Francis Group, LLC
structure. It is applied in the periodical switching circuit. Usually, a capacitor is charged during switching-on by certain voltage. This charged capacitor voltage can be arranged on top-up to output voltage during switching-off. Therefore, the output voltage can be increased. A typical example is the sawtooth-wave generator with voltage lift circuit. Voltage lift technique has its output voltage increasing in arithmetic progression, stage by stage. Super lift technique is more powerful than voltagelift technique. The output voltage transfer gain of super-lift converters can be very high, and increases in geometric progression, stage by stage. It effectively enhances the voltage transfer gain in power series. Four series of super-lift converters created by the authors are introduced in this book. Some industrial applications verified their versatile and powerful characteristics. Super-lift technique is an outstanding achievement in DC/DC conversion technology. Ultra-lift technique is another outstanding achievement in DC/DC conversion technology. It combines the characteristics of the voltage-lift and superlift techniques to create the very high voltage transfer gain converter ultra-lift Luo converter. It effectively enhances the voltage transfer gain. This book is organized in seven chapters. The general knowledge of DC/ DC conversion technology is introduced in Chapter 1; and voltage lift converters in Chapter 2. Chapters 3–6 introduce the four series super-lift converters. Chapter 7 introduces the ultra-lift Luo converter. The authors have worked in this research area for long periods and created a large number of outstanding converters: namely Luo converters, which cover all six generations of converters. Super-lift converters are our favorite achievements in our 20-years’ research. Our acknowledgment goes to the executive editor for this book. Fang Lin Luo and Hong Ye Nanyang Technological University Singapore May 2005
© 2006 by Taylor & Francis Group, LLC
Figure Credits The following figures were reprinted from Power Electronics Handbook (M.H. Rashid, Ed.), Chapter 17, Copyright 2001, with kind permission from Elsevier. Chapter 1: Figure 1.23 Figure 1.24 Figure 1.25 Figure 1.26 Figure 1.29 Figure 1.30 Figure 1.34 Figure 1.35 Figure 1.36 Figure 1.37
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Chapter 2: Figure 2.1 Figure 2.4 Figure 2.5 Figure 2.6 Figure 2.7 Figure 2.8 Figure 2.9 Figure 2.20 Figure 2.25 Figure 2.28 Figure 2.54 Figure 2.55 Figure 2.56 Figure 2.63 Figure 2.68 Figure 2.73 Figure 2.76
Authors
Dr. Fang Lin Luo received his bachelor of science (First Class with Honours) in radio-electronic physics at the Sichuan University, Chengdu, China and his Ph.D. in electrical engineering and computer science from Cambridge University, England, in 1986. Dr. Luo was with the Chinese Automation Research Institute of Metallurgy (CARIM), Beijing, as a senior engineer after his graduation from Sichuan University. He was with the Entreprises Saunier Duval, Paris, France as a project engineer in 1981 and 1982. He was with Hocking NDT Ltd, Allen-Bradley IAP Ltd. and Simplatroll Ltd. in England as a senior engineer after he earned his Ph.D. from Cambridge University. He is with the School of Electrical and Electronic Engineering, Nanyang Technological University in Singapore. He is a senior member of IEEE. He has published seven teaching textbooks and 218 technical papers in IEEETransactions, IEEE-Proceedings and other international journals, and various international conferences. His present research interest is in the digital power electronics and DC and AC motor drives with computerized artificial intelligent control (AIC) and digital signal processing (DSP), and DC/AC inverters, AC/DC rectifiers, AC/AC and DC/DC converters. Dr. Luo was the chief editor of the international Power Supply Technologies and Applications Journal in 1998–2003. He is the international editor of Advanced Technology of Electrical Engineering and Energy. He is currently the associate editor of the IEEE Transactions on both power electronics and industrial electronics.
© 2006 by Taylor & Francis Group, LLC
Dr. Hong Ye earned a bachelor degree (First Class with Honors) in 1995 and the master of engineering degree from Xi’an Jiaotong University, China in 1999. She completed her Ph.D. at Nanyang Technological University, Singapore. Dr. Ye was with the R&D Institute, XIYI Company, Ltd., China as a research engineer from 1995 to 1997. She joined Nanyang Technical University in 2003. Dr. Ye is an IEEE Member and has co-authored seven books. She has published more than 48 technical papers in IEEE-Transactions, IEEE-Proceedings and other international journals and various international conferences. Her research interests are in the areas of DC/DC converters, signal processing, operations research and structural biology.
© 2006 by Taylor & Francis Group, LLC
Contents
1 1.1 1.2
1.3
1.4
Introduction ..................................................................................... 1 Historical Review...........................................................................................1 Multiple-Quadrant Choppers ......................................................................2 1.2.1 Multiple-Quadrant Operation.........................................................2 1.2.2 The First-Quadrant Chopper...........................................................3 1.2.3 The Second-Quadrant Chopper......................................................4 1.2.4 The Third-Quadrant Chopper.........................................................5 1.2.5 The Fourth-Quadrant Chopper ......................................................6 1.2.6 The First and Second Quadrant Chopper.....................................7 1.2.7 The Third and Fourth Quadrant Chopper ...................................8 1.2.8 The Four-Quadrant Chopper ..........................................................9 Pump Circuits.................................................................................................9 1.3.1 Fundamental Pumps.......................................................................10 1.3.1.1 Buck Pump.........................................................................10 1.3.1.2 Boost Pump........................................................................10 1.3.1.3 Buck-Boost Pump .............................................................10 1.3.2 Developed Pumps...........................................................................10 1.3.2.1 Positive Luo-Pump ...........................................................12 1.3.2.2 Negative Luo-Pump .........................................................12 1.3.2.3 Cúk-Pump ..........................................................................12 1.3.3 Transformer-Type Pumps ..............................................................14 1.3.3.1 Forward Pump ..................................................................14 1.3.3.2 Fly-Back Pump ..................................................................14 1.3.3.3 ZETA Pump .......................................................................15 1.3.4 Super-Lift Pumps ............................................................................15 1.3.4.1 Positive Super Luo-Pump ...............................................16 1.3.4.2 Negative Super Luo-Pump .............................................16 1.3.4.3 Positive Push-Pull Pump.................................................16 1.3.4.4 Negative Push-Pull Pump...............................................17 1.3.4.5 Double/Enhanced Circuit (DEC)...................................17 Development of DC/DC Conversion Technique ...................................18 1.4.1 The First Generation Converters ..................................................19 1.4.1.1 Fundamental Converters .................................................19 1.4.1.2 Transformer-Type Converters .........................................23 1.4.1.3 Developed Converters .....................................................28 1.4.1.4 Voltage Lift Converters ....................................................32 1.4.1.5 Super Lift Converters.......................................................32 1.4.2 The Second Generation Converters .............................................32 1.4.3 The Third Generation Converters ................................................33
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1.4.3.1 Switched Capacitor Converters......................................33 1.4.3.2 Multiple-Quadrant Switched Capacitor Luo-Converters..................................................................34 1.4.3.3 Multiple-Lift Push-Pull Switched Capacitor Converters ..........................................................................34 1.4.3.4 Multiple-Quadrant Switched Inductor Converters ..........................................................................34 1.4.4 The Fourth Generation Converters ..............................................35 1.4.4.1 Zero-Current-Switching Quasi-Resonant Converters ..........................................................................35 1.4.4.2 Zero-Voltage-Switching Quasi-Resonant Converters ..........................................................................35 1.4.4.3 Zero-Transition Converters .............................................36 1.4.5 The Fifth Generation Converters..................................................36 1.4.6 The Sixth Generation Converters .................................................36 1.5 Categorize Prototypes and DC/DC Converter Family Tree ................37 Bibliography ..........................................................................................................39
2 2.1 2.2
2.3
Voltage-Lift Converters ................................................................ 41 Introduction ..................................................................................................41 Seven Self-Lift Converters..........................................................................42 2.2.1 Self-Lift Cúk Converter ..................................................................44 2.2.1.1 Continuous Conduction Mode.......................................44 2.2.1.2 Discontinuous Conduction Mode ..................................48 2.2.2 Self-Lift P/O Luo-Converter .........................................................50 2.2.2.1 Continuous Conduction Mode.......................................50 2.2.2.2 Discontinuous Conduction Mode ..................................53 2.2.3 Reverse Self-Lift P/O Luo-Converter..........................................54 2.2.3.1 Continuous Conduction Mode.......................................54 2.2.3.2 Discontinuous Conduction Mode ..................................58 2.2.4 Self-Lift N/O Luo-Converter ........................................................59 2.2.4.1 Continuous Conduction Mode.......................................60 2.2.4.2 Discontinuous Conduction Mode ..................................61 2.2.5 Reverse Self-Lift N/O Luo-Converter .........................................62 2.2.5.1 Continuous Conduction Mode.......................................63 2.2.5.2 Discontinuous Conduction Mode ..................................65 2.2.6 Self-Lift SEPIC .................................................................................66 2.2.6.1 Continuous Conduction Mode.......................................66 2.2.6.2 Discontinuous Conduction Mode ..................................69 2.2.7 Enhanced Self-Lift P/O Luo-Converter ......................................70 Positive Output Luo-Converters...............................................................72 2.3.1 Elementary Circuit ..........................................................................74 2.3.1.1 Circuit Description ...........................................................74 2.3.1.2 Variations of Currents and Voltages..............................77 2.3.1.3 Instantaneous Values of Currents and Voltages ..........80
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2.4
2.3.1.4 Discontinuous Mode ........................................................81 2.3.1.5 Stability Analysis ..............................................................83 2.3.2 Self-Lift Circuit ................................................................................85 2.3.2.1 Circuit Description ...........................................................85 2.3.2.2 Average Current IC1 and Source Current IS ...............88 2.3.2.3 Variations of Currents and Voltages..............................88 2.3.2.4 Instantaneous Value of the Currents and Voltages...............................................................................91 2.3.2.5 Discontinuous Mode ........................................................92 2.3.2.6 Stability Analysis ..............................................................94 2.3.3 Re-lift Circuit....................................................................................96 2.3.3.1 Circuit Description ...........................................................96 2.3.3.2 Other Average Currents...................................................98 2.3.3.3 Variations of Currents and Voltages............................100 2.3.3.4 Instantaneous Value of the Currents and Voltages.............................................................................103 2.3.3.5 Discontinuous Mode ......................................................105 2.3.3.6 Stability Analysis ............................................................107 2.3.4 Multiple-Lift Circuits .................................................................... 110 2.3.4.1 Triple-Lift Circuit ............................................................ 111 2.3.4.2 Quadruple-Lift Circuit ................................................... 114 2.3.5 Summary......................................................................................... 118 2.3.6 Discussion....................................................................................... 119 2.3.6.1 Discontinuous-Conduction Mode ................................ 119 2.3.6.2 Output Voltage VO vs. Conduction Duty k...............121 2.3.6.3 Switch Frequency f .........................................................121 Negative Output Luo-Converters...........................................................121 2.4.1 Elementary Circuit ........................................................................124 2.4.1.1 Circuit Description .........................................................125 2.4.1.2 Average Voltages and Currents....................................125 2.4.1.3 Variations of Currents and Voltages............................126 2.4.1.4 Instantaneous Values of Currents and Voltages ........129 2.4.1.5 Discontinuous Mode ......................................................130 2.4.2 Self-Lift Circuit ..............................................................................132 2.4.2.1 Circuit Description .........................................................132 2.4.2.2 Average Voltages and Currents....................................132 2.4.2.3 Variations of Currents and Voltages............................135 2.4.2.4 Instantaneous Value of the Currents and Voltages...138 2.4.2.5 Discontinuous Mode ......................................................139 2.4.3 Re-lift Circuit..................................................................................140 2.4.3.1 Circuit Description .........................................................141 2.4.3.2 Average Voltages and Currents....................................141 2.4.3.3 Variations of Currents and Voltages............................144 2.4.3.4 Instantaneous Value of the Currents and Voltages...147 2.4.3.5 Discontinuous Mode ......................................................149
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2.4.4
Multiple-Lift Circuits ....................................................................151 2.4.4.1 Triple-Lift Circuit ............................................................151 2.4.4.2 Quadruple-Lift Circuit ...................................................155 2.4.5 Summary.........................................................................................158 2.5 Modified Positive Output Luo-Converters ...........................................162 2.5.1 Elementary Circuit ........................................................................162 2.5.2 Self-Lift Circuit ..............................................................................163 2.5.3 Re-Lift Circuit ................................................................................165 2.5.4 Multi-Lift Circuit ...........................................................................168 2.5.5 Application .....................................................................................170 2.6 Double Output Luo-Converters ..............................................................171 2.6.1 Elementary Circuit ........................................................................173 2.6.1.1 Positive Conversion Path ..............................................174 2.6.1.2 Negative Conversion Path ............................................176 2.6.1.3 Discontinuous Mode ......................................................178 2.6.2 Self-Lift Circuit ..............................................................................181 2.6.2.1 Positive Conversion Path ..............................................181 2.6.2.2 Negative Conversion Path ............................................183 2.6.2.3 Discontinuous Conduction Mode ................................186 2.6.3 Re-Lift Circuit ................................................................................188 2.6.3.1 Positive Conversion Path ..............................................188 2.6.3.2 Negative Conversion Path ............................................191 2.6.3.3 Discontinuous Conduction Mode ................................194 2.6.4 Multiple-Lift Circuit......................................................................196 2.6.4.1 Triple-Lift Circuit ............................................................196 2.6.4.2 Quadruple-Lift Circuit ...................................................202 2.6.5 Summary.........................................................................................208 2.6.5.1 Positive Conversion Path ..............................................208 2.6.5.2 Negative Conversion Path ............................................210 2.6.5.3 Common Parameters......................................................210 Bibliography ........................................................................................................212
3 3.1 3.2
3.3
Positive Output Super-Lift Luo-Converters ............................ 215 Introduction ................................................................................................215 Main Series .................................................................................................216 3.2.1 Elementary Circuit ........................................................................216 3.2.2 Re-Lift Circuit ................................................................................219 3.2.3 Triple-Lift Circuit...........................................................................220 3.2.4 Higher Order Lift Circuit.............................................................222 Additional Series .......................................................................................222 3.3.1 Elementary Additional Circuit....................................................223 3.3.2 Re-Lift Additional Circuit ............................................................227 3.3.3 Triple-Lift Additional Circuit ......................................................228 3.3.4 Higher Order Lift Additional Circuit ........................................230
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3.4
Enhanced Series .........................................................................................231 3.4.1 Elementary Enhanced Circuit .....................................................231 3.4.2 Re-Lift Enhanced Circuit..............................................................233 3.4.3 Triple-Lift Enhanced Circuit........................................................235 3.4.4 Higher Order Lift Enhanced Circuit..........................................237 3.5 Re-Enhanced Series ...................................................................................237 3.5.1 Elementary Re-Enhanced Circuit ...............................................238 3.5.2 Re-Lift Re-Enhanced Circuit........................................................242 3.5.3 Triple-Lift Re-Enhanced Circuit..................................................243 3.5.4 Higher Order Lift Re-Enhanced Circuit....................................245 3.6 Multiple-Enhanced Series.........................................................................246 3.6.1 Elementary Multiple-Enhanced Circuit.....................................249 3.6.2 Re-Lift Multiple-Enhanced Circuit.............................................250 3.6.3 Triple-Lift Multiple-Enhanced Circuit .......................................251 3.6.4 Higher Order Lift Multiple-Enhanced Circuit .........................253 3.7 Summary of Positive Output Super-Lift Luo-Converters ..................254 3.8 Simulation Results .....................................................................................258 3.8.1 Simulation Results of a Triple-Lift Circuit................................258 3.8.2 Simulation Results of a Triple-Lift Additional Circuit ...........258 3.9 Experimental Results ................................................................................259 3.9.1 Experimental Results of a Triple-Lift Circuit ...........................259 3.9.2 Experimental Results of a Triple-Lift Additional Circuit .......259 3.9.3 Efficiency Comparison of Simulation and Experimental Results ....................................................................260 Bibliography ........................................................................................................261
4 4.1 4.2
4.3
4.4
4.5
Negative Output Super-Lift Luo-Converters........................... 263 Introduction ................................................................................................263 Main Series .................................................................................................264 4.2.1 Elementary Circuit ........................................................................264 4.2.2 N/O Re-Lift Circuit ......................................................................268 4.2.3 N/O Triple-Lift Circuit ................................................................270 4.2.4 N/O Higher Order Lift Circuit ..................................................272 Additional Series .......................................................................................273 4.3.1 N/O Elementary Additional Circuit..........................................273 4.3.2 N/O Re-Lift Additional Circuit..................................................277 4.3.3 N/O Triple-Lift Additional Circuit ............................................279 4.3.4 N/O Higher Order Lift Additional Circuit ..............................282 Enhanced Series .........................................................................................283 4.4.1 N/O Elementary Enhanced Circuit ...........................................283 4.4.2 N/O Re-Lift Enhanced Circuit ...................................................285 4.4.3 N/O Triple-Lift Enhanced Circuit..............................................288 4.4.4 N/O Higher Order Lift Enhanced Circuit................................291 Re-Enhanced Series ...................................................................................291 4.5.1 N/O Elementary Re-Enhanced Circuit .....................................291
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4.5.2 N/O Re-Lift Re-Enhanced Circuit .............................................295 4.5.3 N/O Triple-Lift Re-Enhanced Circuit........................................295 4.5.4 N/O Higher Order Lift Re-Enhanced Circuit..........................296 4.6 Multiple-Enhanced Series.........................................................................297 4.6.1 N/O Elementary Multiple-Enhanced Circuit...........................297 4.6.2 N/O Re-Lift Multiple-Enhanced Circuit...................................299 4.6.3 N/O Triple-Lift Multiple-Enhanced Circuit .............................301 4.6.4 N/O Higher Order Lift Multiple-Enhanced Circuit ...............302 4.7 Summary of Negative Output Super-Lift Luo-Converters ................302 4.8 Simulation Results .....................................................................................306 4.8.1 Simulation Results of a N/O Triple-Lift Circuit......................306 4.8.2 Simulation Results of a N/O Triple-Lift Additional Circuit..............................................................................................306 4.9 Experimental Results ................................................................................306 4.9.1 Experimental Results of a N/O Triple-Lift Circuit .................306 4.9.2 Experimental Results of a N/O Triple-Lift Additional Circuit..............................................................................................307 4.9.3 Efficiency Comparison of Simulation and Experimental Results ....................................................................308 4.9.4 Transient Process and Stability Analysis...................................308 Bibliography ........................................................................................................309
5 5.1 5.2
5.3
5.4
5.5
Positive Output Cascade Boost Converters ............................. 311 Introduction ................................................................................................ 311 Main Series .................................................................................................312 5.2.1 Elementary Boost Circuit .............................................................312 5.2.2 Two-Stage Boost Circuit ..............................................................313 5.2.3 Three-Stage Boost Circuit ............................................................315 5.2.4 Higher Stage Boost Circuit ..........................................................317 Additional Series .......................................................................................318 5.3.1 Elementary Boost Additional (Double) Circuit........................318 5.3.2 Two-Stage Boost Additional Circuit...........................................320 5.3.3 Three-Stage Boost Additional Circuit ........................................322 5.3.4 Higher Stage Boost Additional Circuit......................................324 Double Series..............................................................................................325 5.4.1 Elementary Double Boost Circuit...............................................325 5.4.2 Two-Stage Double Boost Circuit.................................................325 5.4.3 Three-Stage Double Boost Circuit ..............................................327 5.4.4 Higher Stage Double Boost Circuit............................................330 Triple Series ................................................................................................331 5.5.1 Elementary Triple Boost Circuit .................................................331 5.5.2 Two-Stage Triple Boost Circuit ...................................................332 5.5.3 Three-Stage Triple Boost Circuit.................................................334 5.5.4 Higher Stage Triple Boost Circuit...............................................337
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5.6
Multiple Series ..........................................................................................337 5.6.1 Elementary Multiple Boost Circuit ............................................338 5.6.2 Two-Stage Multiple Boost Circuit ..............................................339 5.6.3 Three-Stage Multiple Boost Circuit............................................341 5.6.4 Higher Stage Multiple Boost Circuit..........................................343 5.7 Summary of Positive Output Cascade Boost Converters...................343 5.8 Simulation and Experimental Results....................................................345 5.8.1 Simulation Results of a Three-Stage Boost Circuit..................345 5.8.2 Experimental Results of a Three-Stage Boost Circuit ............347 5.8.3 Efficiency Comparison of Simulation and Experimental Results ....................................................................348 5.8.4 Transient Process ...........................................................................348 Bibliography ........................................................................................................349
6 6.1 6.2
6.3
6.4
6.5
6.6
6.7 6.8
Negative Output Cascade Boost Converters............................ 351 Introduction ................................................................................................351 Main Series ................................................................................................351 6.2.1 N/O Elementary Boost Circuit...................................................352 6.2.2 N/O Two-Stage Boost Circuit.....................................................353 6.2.3 N/O Three-Stage Boost Circuit ..................................................355 6.2.4 N/O Higher Stage Boost Circuit ................................................357 Additional Series .......................................................................................358 6.3.1 N/O Elementary Additional Boost Circuit ..............................358 6.3.2 N/O Two-Stage Additional Boost Circuit ................................360 6.3.3 N/O Three-Stage Additional Boost Circuit..............................362 6.3.4 N/O Higher Stage Additional Boost Circuit............................365 Double Series..............................................................................................365 6.4.1 N/O Elementary Double Boost Circuit.....................................365 6.4.2 N/O Two-Stage Double Boost Circuit.......................................366 6.4.3 N/O Three-Stage Double Boost Circuit ....................................368 6.4.4 N/O Higher Stage Double Boost Circuit..................................370 Triple Series ................................................................................................371 6.5.1 N/O Elementary Triple Boost Circuit .......................................371 6.5.2 N/O Two-Stage Triple Boost Circuit .........................................372 6.5.3 N/O Three-Stage Triple Boost Circuit.......................................374 6.5.4 N/O Higher Stage Triple Boost Circuit ....................................377 Multiple Series ...........................................................................................377 6.6.1 N/O Elementary Multiple Boost Circuit ..................................378 6.6.2 N/O Two-Stage Multiple Boost Circuit ....................................378 6.6.3 N/O Three-Stage Multiple Boost Circuit..................................381 6.6.4 N/O Higher Stage Multiple Boost Circuit ...............................381 Summary of Negative Output Cascade Boost Converters.................383 Simulation and Experimental Results....................................................385 6.8.1 Simulation Results of a Three-Stage Boost Circuit..................385
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6.8.2 6.8.3
Experimental Results of a Three-Stage Boost Circuit .............386 Efficiency Comparison of Simulation and Experimental Results ....................................................................388 6.8.4 Transient Process ...........................................................................388 Bibliography ........................................................................................................389
7
Ultra-Lift Luo-Converter ............................................................ 391 Introduction ................................................................................................391 Operation of Ultra-lift Luo-Converter ...................................................392 7.2.1 Continuous Conduction Mode (CCM) ......................................394 7.2.2 Discontinuous Conduction Mode (DCM).................................397 7.3 Instantaneous Values.................................................................................399 7.3.1 Continuous Conduction Mode (CCM) ......................................399 7.3.2 Discontinuous Conduction Mode (DCM).................................401 7.4 Comparison of the Gain to Other Converters’ Gains .........................403 7.5 Simulation Results .....................................................................................403 7.6 Experimental Results ................................................................................403 7.7 Summary .....................................................................................................405 Bibliography ........................................................................................................406 7.1 7.2
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1 Introduction
Conversion technique is a major research area in the field of power electronics. The equipment for conversion techniques have applications in industry, research and development, government organizations, and daily life. The equipment can be divided in four technologies: • • • •
AC/AC AC/DC DC/DC DC/AC
transformers rectifiers converters inverters
According to incomplete statistics, there have been more than 500 prototypes of DC/DC converters developed in the past six decades. All existing DC/DC converters were designed to meet the requirements of certain applications. They are usually called by their function, for example, Buck converter, Boost converter and Buck-Boost converter, and zero current switching (ZCS) and zero voltage switching (ZVS) converters. The large number of DC/DC converters had not been evolutionarily classified until 2001. The authors have systematically classified the types of converters into six generations according to their characteristics and development sequence. This classification grades all DC/DC converters and categorizes new prototypes. Since 2001, the DC/DC converter family tree has been built and this classification has been recognized worldwide. Following this principle, it is now easy to sort and allocate DC/DC converters and assess their technical features.
1.1
Historical Review
DC/DC conversion technology is a major subject area in the field of power engineering and drives, and has been under development for six decades. DC/DC converters are widely used in industrial applications and computer hardware circuits. DC/DC conversion techniques have developed very 1
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2
Essential DC/DC Converters
quickly. Statistics show that the DC/DC converter worldwide market will grow from U.S. $3336 million in 1995 to U.S. $5128 million in the year 2004 with a compound annual growth rate (CAGR) of 9%.* This compares to the AC/DC power supply market, which will have a CAGR of only about 7.5% during the same period. In addition to its higher growth rate, the DC/DC converter market is undergoing dramatic changes as a result of two major trends in the electronics industry: low voltage and high power density. From this investigation it can be seen that the production of DC/DC converters in the world market is much higher than that of AC/DC converters. The DC/DC conversion technique was established in the 1920s. A simple voltage conversion, the simplest DC/DC converter is a voltage divider (such as rheostat, potential–meter, and so on), but it only transfers output voltage lower than input voltage with poor efficiency. The multiple-quadrant chopper is the second step in DC/DC conversion. Much time has been spent trying to find equipment to convert the DC energy source of one voltage to another DC actuator with another voltage, as does a transformer employed in AC/AC conversion. Some preliminary types of DC/DC converters were used in industrial applications before the Second World War. Research was blocked during the war, but applications of DC/DC converters were recognized. After the war, communication technology developed very rapidly and required low voltage DC power supplies. This resulted in the rapid development of DC/DC conversion techniques. Preliminary prototypes can be derived from choppers.
1.2
Multiple-Quadrant Choppers
Choppers are the circuits that convert fixed DC voltage to variable DC voltage or pulse-width–modulated (PWM) AC voltage. In this book, we concentrate on its first function.
1.2.1
Multiple-Quadrant Operation
A DC motor can run in forward running or reverse running. During the forward starting process its armature voltage and armature current are both positive. We usually call this forward motoring operation or quadrant I operation. During the forward braking process its armature voltage is still positive and its armature current is negative. This state is called the forward regenerating operation or quadrant II operation. Analogously, during the reverse starting process the DC motor armature voltage and current are both negative. This reverse motoring operation is called the quadrant III operation. * Figures are taken from the Darnell Group News report on DC-DC Converters: Global Market Forecasts, Demand Characteristics and Competitive Environment published in February 2000.
© 2006 by Taylor & Francis Group, LLC
Introduction
3 V Quadrant II Forward Regenerating
Quadrant I Forward Motoring
I Quadrant III Reverse Motoring
Quadrant IV Reverse Regenerating
FIGURE 1.1 Four-quadrant operation.
During reverse braking process its armature voltage is still negative and its armature current is positive. This state is called the reverse regenerating operation quadrant IV operation. Referring to the DC motor operation states; we can define the multiplequadrant operation as below: Quadrant I operation: forward motoring, voltage is positive, current is positive Quadrant II operation: forward regenerating, voltage is positive, current is negative Quadrant III operation: reverse motoring, voltage is negative, current is negative Quadrant IV operation: reverse regenerating, voltage is negative, current is positive The operation status is shown in the Figure 1.1. Choppers can convert a fixed DC voltage into various other voltages. The corresponding chopper is usually named according to its quadrant operation chopper, e.g., the firstquadrant chopper or “A”-type chopper. In the following description we use the symbols Vin as the fixed voltage, Vp the chopped voltage, and VO the output voltage.
1.2.2
The First-Quadrant Chopper
The first-quadrant chopper is also called “A”-type chopper and its circuit diagram is shown in Figure 1.2a and corresponding waveforms are shown in Figure 1.2b. The switch S can be some semiconductor devices such as BJT, IGBT, and MOSFET. Assuming all parts are ideal components, the output voltage is calculated by the formula, VO =
© 2006 by Taylor & Francis Group, LLC
ton V = kVin T in
(1.1)
4
Essential DC/DC Converters
S VIN
+
iO
-
+ VP
D
L
+ C
Vo
R
-
-
(a) Circuit Diagram
VIN
t VP VO tON
T
t
kT
T
t
VO
(b) Voltage Waveforms FIGURE 1.2 The first-quadrant chopper.
where T is the repeating period T = 1/f, f is the chopping frequency, ton is the switch-on time, k is the conduction duty cycle k = ton/T. 1.2.3
The Second-Quadrant Chopper
The second-quadrant chopper is the called “B”-type chopper and the circuit diagram and corresponding waveforms are shown in Figure 1.3a and b. The The output voltage can be calculated by the formula, VO =
© 2006 by Taylor & Francis Group, LLC
toff T
Vin = (1 − k )Vin
(1.2)
Introduction
5
+
D L
I
VIN S
+ VP
+ C
-
Vo -
(a) Circuit Diagram
VIN
t VP VO tON
T
t
kT
T
t
VO
(b) Voltage Waveforms FIGURE 1.3 The second-quadrant chopper.
where T is the repeating period T = 1/f, f is the chopping frequency, toff is the switch-off time toff = T – ton, and k is the conduction duty cycle k = ton/T. 1.2.4
The Third-Quadrant Chopper
The third-quadrant chopper and corresponding waveforms are shown in Figure 1.4a and b. All voltage polarity is defined in the figure. The output voltage (absolute value) can be calculated by the formula, VO =
© 2006 by Taylor & Francis Group, LLC
ton V = kVin T in
(1.3)
6
Essential DC/DC Converters
–
S
iO
VIN +
+ D
VP
–
L C
Vo
R
–
+
(a) Circuit Diagram
VIN
t VP
tON
T
t
kT
T
t
VO
(b) Voltage Waveforms FIGURE 1.4 The third-quadrant chopper.
where ton is the switch-on time, and k is the conduction duty cycle k = ton/T. 1.2.5
The Fourth-Quadrant Chopper
The fourth-quadrant chopper and corresponding waveforms are shown in Figure 1.5a and b. All voltage polarity is defined in the figure. The output voltage (absolute value) can be calculated by the formula, VO =
toff T
Vin = (1 − k )Vin
(1.4)
where toff is the switch-off time toff = T – ton, time, and k is the conduction duty cycle k = ton/T.
© 2006 by Taylor & Francis Group, LLC
Introduction
7
D L
VIN – + S
+ VP
I – C
–
Vo +
(a) Circuit Diagram
VIN
t VP
VO tON
T
t
kT
T
t
VO
(b) Voltage Waveforms FIGURE 1.5 The fourth-quadrant chopper.
1.2.6
The First and Second Quadrant Chopper
The first and second quadrant chopper is shown in Figure 1.6. Dual quadrant operation is usually requested in the system with two voltage sources V1 and V2. Assume that the condition V1 > V2, and the inductor L is an ideal component. During quadrant I operation, S1 and D2 work, and S2 and D1 are idle. Vice versa, during quadrant II operation, S2 and D1 work, and S1 and D2 are idle. The relation between the two voltage sources can be calculated by the formula, ⎧ kV1 V2 = ⎨ ⎩(1 − k )V1
© 2006 by Taylor & Francis Group, LLC
QI _ operation QII _ operation
(1.5)
8
Essential DC/DC Converters
D2
S1
+ VI
L
–
+
+ D1
S2
VP
–
V2
–
FIGURE 1.6 The first-second quadrant chopper.
D2
S1 L
VI
+
+ D1
S2
VP
+
V2
FIGURE 1.7 The third-fourth quadrant chopper.
where k is the conduction duty cycle k = ton/T. 1.2.7
The Third and Fourth Quadrant Chopper
The third and fourth quadrant chopper is shown in Figure 1.7. Dual quadrant operation is usually requested in the system with two voltage sources V1 and V2. Both voltage polarity is defined in the figure, we just concentrate their absolute values in analysis and calculation. Assume that the condition V1 > V2, the inductor L is ideal component. During quadrant I operation, S1 and D2 work, and S2 and D1 are idle. Vice versa, during quadrant II operation, S2 and D1 work, and S1 and D2 are idle. The relation between the two voltage sources can be calculated by the formula, ⎧ kV1 V2 = ⎨ ⎩(1 − k )V1
QIII _ operation QIV _ operation
where k is the conduction duty cycle k = ton/T.
© 2006 by Taylor & Francis Group, LLC
(1.6)
Introduction
9
D1
S1
+ VI
V2
L
D3
S3
D4
S4
– D2
S2
FIGURE 1.8 The four-quadrant chopper.
TABLE 1.1 The Switches and Diode’s Status for Four-Quadrant Operation Switch or Diode
Quadrant I
Quadrant II
Quadrant III
Quadrant IV
S1 D1 S2 D2 S3 D3 S4 D4 Output
Works Idle Idle Works Idle Idle On Idle V2 +, I2 +
Idle Works Works Idle Idle Idle Idle On V2 +, I2 –
Idle Works Works Idle On Idle Idle Idle V2 –, I2 –
Works Idle Idle Works Idle On Idle Idle V2 –, I2 +
1.2.8
The Four-Quadrant Chopper
The four-quadrant chopper is shown in Figure 1.8. The input voltage is positive, output voltage can be either positive or negative. The switches and diode status for the operation are shown in Table 1.1. The output voltage can be calculated by the formula, ⎧ kV1 ⎪ (1 − k )V ⎪ 1 V2 = ⎨ − kV 1 ⎪ ⎪⎩−(1 − k )V1
1.3
QI _ operation QII _ operation QIII _ operation QIV _ operation
(1.7)
Pump Circuits
The electronic pump is a major component of all DC/DC converters. Historically, they can be sorted into four groups:
© 2006 by Taylor & Francis Group, LLC
10
Essential DC/DC Converters
• • • •
Fundamental pumps Developed pumps Transformer-type pumps Super-lift pumps
1.3.1
Fundamental Pumps
Fundamental pumps are developed from fundamental DC/DC converters just like their name: • Buck pump • Boost pump • Buck-boost pump All fundamental pumps consist of three components: a switch S, a diode D, and an inductor L. 1.3.1.1 Buck Pump The circuit diagram of the buck pump, and some current waveforms are shown in Figure 1.9. Switch S and diode D are alternately on and off. Usually, the buck pump works in continuous operation mode, inductor current is continuous in this case. 1.3.1.2 Boost Pump The circuit diagram of the boost pump, and some current waveforms are shown in Figure 1.10. Switch S and diode D are alternately on and off. The inductor current is usually continuous. 1.3.1.3 Buck-Boost Pump The circuit diagram of the buck-boost pump and some current waveforms are shown in Figure 1.11. Switch S and diode D are alternately on and off. Usually, the buck-boost pump works in continuous operation mode, inductor current is continuous in this case. 1.3.2
Developed Pumps
Developed pumps are created from the developed DC/DC converters just like their name: • Positive Luo-pump • Negative Luo-pump • Cúk-pump
© 2006 by Taylor & Francis Group, LLC
Introduction
11
i1
L
S
i2
+
+
V1
D
R
V2
_
_
i1
0
kT
T
t
kT
T
t
L
D
i2
0
FIGURE 1.9 Buck pump.
i1
i2
+
+
V1
R
V2
S
_
_
i1
0
kT
T
t
T
t
i2
0
FIGURE 1.10 Boost pump.
© 2006 by Taylor & Francis Group, LLC
kT
12
Essential DC/DC Converters i1
i2 S
D
+
–
V1
L
R
V2
–
+
i1
0
kT
T
t
T
t
i2
0
kT
FIGURE 1.11 Buck-boost pump.
All developed pumps consist of four components: a switch S, a diode D, a capacitor C, and an inductor L. 1.3.2.1 Positive Luo-Pump The circuit diagram of the positive Luo-pump and some current and voltage waveforms are shown in Figure 1.12. Switch S and diode D are alternately on and off. Usually, this pump works in continuous operation mode, inductor current is continuous in this case. The output terminal voltage and current is usually positive. 1.3.2.2 Negative Luo-Pump The circuit diagram of the negative Luo-pump and some current and voltage waveforms are shown in Figure 1.13. Switch S and diode D are alternately on and off. Usually, this pump works in continuous operation mode, inductor current is continuous in this case. The output terminal voltage and current is usually negative. 1.3.2.3 Cúk-Pump The circuit diagram of the Cúk pump and some current and voltage waveforms are shown in Figure 1.14. Switch S and diode D are alternately on and off. Usually, the Cúk pump works in continuous operation mode, inductor current is continuous in this case. The output terminal voltage and current is usually negative. © 2006 by Taylor & Francis Group, LLC
Introduction
13 i2 i1
S
C
+
_
L
V1
D
R
V2
_
+
i1
0
kT
T
t
i2
0
kT
t
T
FIGURE 1.12 Positive Luo-pump.
i1
S D
i2
+
L
V1
C
_
R
V2
_
+
i1
0
kT
T
t
i2
0
FIGURE 1.13 Negative Luo-pump.
© 2006 by Taylor & Francis Group, LLC
kT
T
t
14
Essential DC/DC Converters i1
i2
L
C +
+
V1
D
R
V2
S
_
_
i1
0
kT
T
t
i2
0
kT
T
t
FIGURE 1.14 Cúk-pump.
1.3.3
Transformer-Type Pumps
Transformer-type pumps are developed from transformer-type DC/DC converters just like their name: • Forward pump • Fly-Back pump • ZETA pump All transformer-type pumps consist of a switch S, a transformer with the turn ratio N and other components such as diode D (one or more) and capacitor C. 1.3.3.1 Forward Pump The circuit diagram of the forward pump and some current waveforms are shown in Figure 1.15. Switch S and diode D1 are synchronously on and off, and diode D2 is alternately off and on. Usually, the forward pump works in discontinuous operation mode, input current is discontinuous in this case. 1.3.3.2 Fly-Back Pump The circuit diagram of the fly-back pump and some current waveforms are in Figure 1.16. Since the primary and secondary windings of the transformer © 2006 by Taylor & Francis Group, LLC
Introduction
15
+
i1
1:N
i2
D1
+ D2
R
Vo
V in
– Control
S
– i1
0
kT
T
t
i2
0
kT
T
t
FIGURE 1.15 Forward pump.
are purposely arranged in inverse polarities, switch S and diode D are alternately on and off. Usually, the fly-back pump works in discontinuous operation mode, input current is discontinuous in this case. 1.3.3.3 ZETA Pump The circuit diagram of the ZETA pump and some current waveforms are shown in Figure 1.17. Switch S and diode D are alternately on and off. Usually, the ZETA pump works in discontinuous operation mode, input current is discontinuous in this case.
1.3.4
Super-Lift Pumps
Super-lift pumps are developed from super-lift DC/DC converters: • • • • •
Positive super Luo-pump Negative super Luo-pump Positive push-pull pump Negative push-pull pump Double/Enhanced circuit (DEC)
All super-lift pumps consist of switches, diodes, capacitors, and sometimes an inductor. © 2006 by Taylor & Francis Group, LLC
16
Essential DC/DC Converters
i1
i2
1:N
+
+
D N1
N2
C
R
Vo
V in
_ Control
T1
i1
0
kT
T
t
i2
0
kT
T
t
FIGURE 1.16 Fly-back pump.
1.3.4.1 Positive Super Luo-Pump The circuit diagram of the positive super-lift pump and some current waveforms are shown in Figure 1.18. Switch S and diode D1 are synchronously on and off, but diode D2 is alternately off and on. Usually, the positive superlift pump works in continuous conduction mode (CCM), inductor current is continuous in this case. 1.3.4.2 Negative Super Luo-Pump The circuit diagram of the negative super-lift pump and some current waveforms are shown in Figure 1.19. Switch S and diode D1 are synchronously on and off, but diode D2 is alternately off and on. Usually, the negative superlift pump works in CCM, but input current is discontinuous in this case. 1.3.4.3 Positive Push-Pull Pump All push-pull pumps consist of two switches without any inductor. They can be employed in multiple-lift switched capacitor converters. The circuit diagram of positive push-pull pump and some current waveforms are shown in Figure 1.20. Since there is no inductor in the pump, it is applied in
© 2006 by Taylor & Francis Group, LLC
Introduction
17
L i1
. .
i2
1:N
+ Vin
C
R D
S
+ Vo _
_
i1
0
kT
T
t
i2
0
kT
T
t
FIGURE 1.17 ZETA pump.
switched-capacitor converters. The main switch S and diode D1 are synchronously on and off, but the slave switch S1 and diode D2 are alternately off and on. Usually, the positive push-pull pump works in push-pull state continuous operation mode. 1.3.4.4 Negative Push-Pull Pump The circuit diagram of this push-pull pump and some current waveforms are shown in Figure 1.21. Since there is no inductor in the pump, it is often used in switched-capacitor converters. The main switch S and diode D are synchronously on and off, but the slave switch S1 is alternately off and on. Usually, the super-lift pump works in push-pull state continuous operation mode, inductor current is continuous in this case. 1.3.4.5 Double/Enhanced Circuit (DEC) The circuit diagram of the double/enhanced circuit and some current waveforms are in Figure 1.22. The switch is the only other existing circuit part. This circuit is usually applied in lift, super-lift, and push-pull converters. These two diodes are alternately on and off, so that two capacitors are alternately charging and discharging. Usually, this circuit can enhance the voltage doubly or at certain times.
© 2006 by Taylor & Francis Group, LLC
18
Essential DC/DC Converters Iin
D1
D2 IC1
+ L1
C1
+ VC1 –
Vin
+ C2
+
VC2 –
S
R
VO –
–
iin
0
kT
T
t
iC1
0
kT
T
t
FIGURE 1.18 Positive super Luo-pump.
1.4
Development of DC/DC Conversion Technique
According to incomplete statistics, there are more than 500 existing prototypes of DC/DC converters. The main purpose of this book is to catorgorize all existing prototypes of DC/DC converters. This job is of vital importance for future development of DC/DC conversion techniques. The authors have devoted 20 years to this subject area, their work has been recognized and assessed by experts worldwide. The authors classify all existing prototypes of DC/DC converters into six generations. They are • • • • • •
First generation (classical/traditional) converters Second generation (multi-quadrant) converters Third generation (switched-component SI/SC) converters Fourth generation (soft-switching: ZCS/ZVS/ZT) converters Fifth generation (synchronous rectifier SR) converters Sixth generation (multiple energy-storage elements resonant MER) converters
© 2006 by Taylor & Francis Group, LLC
Introduction
19 IC1
Iin S
+
L1
+ VC1 –
C1
–
Vin
– D1
D2
R
VC2
C2
VO +
+
–
iin
0
kT
Tt
t
iC1
0
kT
T
t
FIGURE 1.19 Negative super Luo-pump.
1.4.1
The First Generation Converters
The first-generation converters perform in a single quadrant mode and in low power range (up to around 100 W). Since its development lasts a long time, it has, briefly, five categories: • • • • •
Fundamental converters Transformer-type converters Developed converters Voltage-lift converters Super-lift converters
1.4.1.1 Fundamental Converters Three types of fundamental DC/DC classifications were constructed, these are buck converter, boost converter, and buck-boost converter. They can be derived from single quadrant operation choppers. For example, the buck converter was derived from an A-type chopper. These converters have two main problems: linkage between input and output and very large output voltage ripple.
© 2006 by Taylor & Francis Group, LLC
20
Essential DC/DC Converters I in
D1
D2 I C1
+
+ S1
C1
V C1 –
V in
+ C2
+
V C2 –
S
R
VO –
–
i in
0
kT
T
t
i C1
0
kT
T
t
FIGURE 1.20 Positive push-pull pump.
1.4.1.1.1 Buck Converter The buck converter is a step-down DC/DC converter. It works in firstquadrant operation. It can be derived from a quadrant I chopper. Its circuit diagram, and switch-on and -off equivalent circuit are shown in Figure 1.23. The output voltage is calculated by the formula, VO =
ton V = kVin T in
(1.8)
where T is the repeating period T = 1/f, f is the chopping frequency, ton is the switch-on time, and k is the conduction duty cycle k = ton/T. 1.4.1.1.2 Boost Converter The boost converter is a step-up DC/DC converter. It works in secondquadrant operation. It can be derived from quadrant II chopper. Its circuit diagram, and switch-on and -off equivalent circuit are shown in Figure 1.24. The output voltage is calculated by the formula, VO =
© 2006 by Taylor & Francis Group, LLC
T 1 Vin = V T − ton 1 − k in
(1.9)
Introduction
21 I C1
I in S
+
+ C1
S1
V C1 -
V in
D1
D2
R
V C2
C2
VO +
+
-
i in
0
kT
T
t
i C1
0
kT
t
T
FIGURE 1.21 Negative push-pull pump.
D1
D2 I C1
I C2
+ C1
V C1 –
S
+ C2
V C2 –
FIGURE 1.22 Double/enhanced circuit (DEC).
where T is the repeating period T = 1/f, f is the chopping frequency, ton is the switch-on time, k is the conduction duty cycle k = ton/T.
© 2006 by Taylor & Francis Group, LLC
22
Essential DC/DC Converters i1
S
+
D -
+
iL
-
V1
i2
L
+
VC
VD
-
+
C
R
V2
iC
-
(a) Circuit diagram
i1
L
i2
+
iL
+
+
VC
V1
R
-
-
V2
iC
C
-
(b) Switch-on
L
iL
i2
+
+
VC -
C
R iC
V2 -
(c) Switch-off
FIGURE 1.23 Buck converter.
1.4.1.1.3 Buck-Boost Converter The buck-boost converter is a step down/up DC/DC converter. It works in third-quadrant operation. Its circuit diagram, switch-on and -off equivalent circuit, and waveforms are shown in Figure 1.25. The output voltage is calculated by the formula, VO =
ton k Vin = V T − ton 1 − k in
(1.10)
where T is the repeating period T = 1/f, f is the chopping frequency, ton is the switch-on time, and k is the conduction duty cycle k = ton/T. By using this converter it is easy to obtain the random output voltage, which can be higher or lower than the input voltage. It provides great convenience for industrial applications.
© 2006 by Taylor & Francis Group, LLC
Introduction
23 i1
L
i2
D VD
+
iL
V1
+
+
VC S
R
–
–
iC
C
V2 –
(a) Circuit diagram
i1
i2
L
+
+
iL
V1
+
VC
R
–
–
iC
C
V2 –
(b) Switch on
i1
+
V1 –
L
i2
+
iL
+
VC –
C
R iC
V2 –
(c) Switch off
FIGURE 1.24 Boost converter.
1.4.1.2 Transformer-Type Converters Since all fundamental DC/DC converters keep the linkage from input side to output side and the voltage transfer gain is comparably low, transformertype converters were developed in the 1960s to 1980s. There are a large group of converters such as the forward converter, push-pull converter, fly-back converter, half-bridge converter, bridge converter, and Zeta (or ZETA) converter. Usually, these converters have high transfer voltage gain and high insulation between both sides. Their gain usually depends on the transformer’s turn ratio N, which can be thousands times. 1.4.1.2.1 Forward Converter A forward converter is a transformer-type buck converter with a turn ratio N. It works in first quadrant operation. Its circuit diagram is shown in Figure 1.26. The output voltage is calculated by the formula, VO = kNVin
© 2006 by Taylor & Francis Group, LLC
(1.12)
24
Essential DC/DC Converters
i1
i2
D
S
–
VD
+
–
C
VC
V1
L
–
R
+
V2
iC
iL
+
(a) Circuit diagram
i2 i1 –
–
+
VC
V1
L
–
R
+
iL
iC
C
V2 +
(b) Switch on
i2
–
–
VC L
R
+
V2
iC
C
+
(c) Switch off
FIGURE 1.25 Buck-boost converter.
1:n
+
D1
L +
D2 Vin
C
Vo
R –
Control
T1
FIGURE 1.26 Forward converter.
where N is the transformer turn ratio, and k is the conduction duty cycle k = ton/T. In order to exploit the magnetic ability of the transformer iron core, a tertiary winding can be employed in the transformer. Its corresponding circuit diagram is shown in Figure 1.27.
© 2006 by Taylor & Francis Group, LLC
Introduction
25 1:1:N
+
L
D1
+ C
D2
Vin
Vo
R –
T1
D3
FIGURE 1.27 Forward converter with tertiary winding.
1:N
D1
+
L +
+ T1
V' –
C
Vo
R –
– T2 D2
FIGURE 1.28 Push-pull converter.
1.4.1.2.2 Push-Pull Converter The boost converter works in push-pull state, which effectively avoids the iron core saturation. Its circuit diagram is shown in Figure 1.28. Since there are two switches, which work alternately, the output voltage is doubled. The output voltage is calculated by the formula, VO = 2 kNVin
(1.13)
where N is the transformer turn ratio, and k is the conduction duty cycle k = ton/T. 1.4.1.2.3 Fly-Back Converter The fly-back converter is a transformer type converter using the demagnetizing effect. Its circuit diagram is shown in Figure 1.29. The output voltage is calculated by the formula, VO =
k NVin 1− k
(1.14)
where N is the transformer turn ratio, and k is the conduction duty cycle k = ton/T.
© 2006 by Taylor & Francis Group, LLC
26
Essential DC/DC Converters D1
1:N
+
+ N1
N2
C
R
Vo –
Control
T1
–
FIGURE 1.29 Fly-back converter.
+
1:N C1
D1
L +
T1 C3
R
Vo –
Vin
C2
T2
D2
–
FIGURE 1.30 Half-bridge converter.
1.4.1.2.4 Half-Bridge Converter In order to reduce the primary side in one winding, the half-bridge converter was constructed. Its circuit diagram is shown in Figure 1.30. The output voltage is calculated by the formula, VO = kNVin
(1.15)
where N is the transformer turn ratio, and k is the conduction duty cycle k = ton/T. 1.4.1.2.5 Bridge Converter The bridge converter employs more switches and therefore gains double output voltage. Its circuit diagram is shown in Figure 1.31. The output voltage is calculated by the formula, VO = 2 kNVin
(1.16)
where N is the transformer turn ratio, and k is the conduction duty cycle k = ton/T.
© 2006 by Taylor & Francis Group, LLC
Introduction
27 D1
1:N
+ T1
L +
T2 C1
R
Vo –
C
Vin
T3
D2
T4
–
FIGURE 1.31 Bridge converter.
L1
. .
C1
L2
1:N
+
+ D
Vin
C2
R
Vo –
S –
FIGURE 1.32 Zeta converter.
1.4.1.2.6 ZETA Converter The ZETA converter is a transformer type converter with a low-pass filter. Its output voltage ripple is small. Its circuit diagram is shown in Figure 1.32. The output voltage is calculated by the formula, VO =
k NVin 1− k
(1.17)
where N is the transformer turn ratio, and k is the conduction duty cycle k = ton/T. 1.4.1.2.7 Forward Converter with Tertiary Winding and Multiple Outputs Some industrial applications require multiple outputs. This requirement is easily realized by constructing multiple secondary windings and the corresponding conversion circuit. For example, a forward converter with tertiary winding and three outputs is shown in Figure 1.33. The output voltage is calculated by the formula, VO = kN iVin
© 2006 by Taylor & Francis Group, LLC
(1.18)
28
Essential DC/DC Converters 1:1:n
+
1
O/P 1 n2
Vin
O/P 2 T1
n3 O/P 3
–
FIGURE 1.33 Forward converter with tertiary winding and three outputs.
where Ni is the transformer turn ratio to the secondary winding, i = 1, 2, and 3 respectively, and k is the conduction duty cycle k = ton/T. In principle, this structure is available for all transformer-type DC/DC converters for multiple outputs applications. 1.4.1.3 Developed Converters Developed-type converters overcome the second fault of the fundamental DC/DC converters. They are derived from fundamental converters by the addition of a low-pass filter. The preliminary design was published in a conference in 1977 (Massey and Snyder, 1977). The author designed three types of converters that derived from fundamental DC/DC converters plus a low-pass filter. This conversion technique was very popular between 1970 and 1990. Typical prototype converters are positive output (P/O) Luo-converter, negative output (N/O) Luo-converter, double output (D/O) Luoconverter, Cúk-converter, SEPIC (single-ended primary inductance converter) and Watkins–Johnson converters. The output voltage ripple of all developed-type converters is usually small and can be lower than 2%. In order to obtain the random output voltage, which can be higher or lower input voltage. All developed converters provide ease of application for industry. Therefore, the output voltage gain of all developed converters is VO =
k V 1 − k in
(1.19)
1.4.1.3.1 Positive Output (P/O) Luo-Converter The positive output Luo-converter is the elementary circuit of the series positive output Luo-converters. It can be derived from the buck-boost converter. Its circuit diagram is shown in Figure 1.34. The output voltage is calculated by the Equation (1.19). 1.4.1.3.2 Negative Output (N/O) Luo-Converter The negative output Luo-converter is the elementary circuit of the series negative output Luo-converters. It can also be derived from buck-boost converters. Its
© 2006 by Taylor & Francis Group, LLC
Introduction
29 i Lo
– VC + S
io LO
C
+
+
iL
VI
CO
D
L
R
–
VO –
FIGURE 1.34 Positive output Luo-converter.
iI + VI –
i LO D
S iL
LO
–
–
vC L +
C
iO
CO
R
vO +
FIGURE 1.35 Negative output Luo-converter.
circuit diagram is shown in Figure 1.35. The output voltage is calculated by the Equation (1.19). 1.4.1.3.3 Double Output (D/O) Luo-Converter In order to obtain mirror symmetrical positive plus negative output voltage double output (D/O) Luo-converters were constructed. The double output Luo-converter is the elementary circuit of the series double output Luo-converters. It can also be derived from the buck-boost converter. Its circuit diagram is shown in Figure 1.36. The output voltage is calculated by Equation (1.19). 1.4.1.3.4 Cúk-Converter The Cúk-converter is derived from boost converter. Its circuit diagram is shown in Figure 1.37. The output voltage is calculated by Equation (1.19). 1.4.1.3.5 Single-Ended Primary Inductance Converter The single-ended primary inductance converter (SEPIC) is derived from the boost converter. Its circuit diagram is shown in Figure 1.38. The output voltage is calculated by Equation (1.19).
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30
Essential DC/DC Converters
+
Di
S
LO
C1
+
VI L1
–
V O+
CO
D1
RO
–
– V O– L 11
Di1
C1O
C11
R1O +
L 1O
D11
FIGURE 1.36 Double output Luo-converter.
iL
+
+ L
vC
i LO
–
iO
LO
C
– VO
VI
S
D
CO
R
–
+
FIGURE 1.37 Cúk-converter.
iL +
+ L
VI –
vC C
S
io
–
i L1 L1
+
D
CO
R
vO –
FIGURE 1.38 SEPIC.
1.4.1.3.6 Tapped Inductor Converter These converters are derived from fundamental converters. The circuit diagrams are shown in Table 1.2. The voltage transfer gains are shown in Table 1.3. Here the tapped inductor ratio is n = n1/(n1 + n2).
© 2006 by Taylor & Francis Group, LLC
Introduction
TABLE 1.2 The Circuit Diagrams of the Tapped Inductor Fundamental Converters Switch Tap
Standard Converter
S
S
L
Rail to Tap
Diode to Tap
S
N2
S
N1
N2 N1
Buck
V IN
C
D
L
Boost
V IN
VO
V IN
D
N1
D C
S
VO
V IN
C
N2
VO
V IN
N1 C
VO
C
VO V IN
D
S
D
N1
N2
D
D
N2 C
V IN
VO
VO
C
VO
C
VO
D
N1
V IN
S
C N2
S
S
BuckBoost
D
S
D
V IN
L
C
VO
C V IN
S
N1
S N2
N2 VO
V IN
N1
N1 D C
VO
V IN
N2
D
31
© 2006 by Taylor & Francis Group, LLC
32
Essential DC/DC Converters
TABLE 1.3 The Voltage Transfer Gains of the Tapped Inductor Fundamental Converters Converter
No tap
Switched to tap
Diode to tap
Rail to tap
Buck
k
k n + k(1 − n)
nk 1 + k(n − 1)
k−n k(1 − n)
Boost
1 1− k
n + k(1 − n) n(1 − k )
1 + k(n − 1) 1− k
n−k n(1 − k )
Buck-Boost
k 1− k
k n(1 − k )
nk 1− k
k 1− k
1.4.1.4 Voltage Lift Converters Voltage lift technique is a good method to lift the output voltage, and is widely applied in electronic circuit design. After long-term industrial application and research this method has been successfully used in DC/DC conversion technique. Using this method the output voltage can be easily lifted by tens to hundreds of times. Voltage lift converters can be classed into selflift, re-lift, triple-lift, quadruple-lift, and high-stage lift converters. The main contributors in this area are Dr. Fang Lin Luo and Dr. Hong Ye. These circuits will be introduced in Chapter 2 in detail. 1.4.1.5 Super Lift Converters Voltage lift (VL) technique is a popular method that is widely used in electronic circuit design. It has been successfully employed in DC/DC converter applications in recent years, and has opened a way to design high voltage gain converters. Three series Luo-converters are examples of voltage lift technique implementations. However, the output voltage increases in stage by stage just along the arithmetic progression. A novel approach — super lift (SL) technique — has been developed, which implements the output voltage increasing stage by stage along in geometric progression. It effectively enhances the voltage transfer gain in power-law. The typical circuits are sorted into four series: positive output super-lift Luo-converters, negative output super-lift Luo-converters, positive output cascade boost converters, and negative output cascade boost converters. These circuits will be introduced in Chapters 3 to 6 in detail.
1.4.2
The Second Generation Converters
The second generation converters are called multiple quadrant operation converters. These converters perform in two-quadrant operation and fourquadrant operation with medium output power range (hundreds of Watts
© 2006 by Taylor & Francis Group, LLC
Introduction
33
or higher). The topologies can be sorted into two main categories: first are the converters derived from the multiple-quadrant choppers and/or from the first generation converters. Second are constructed with transformers. Usually, one quadrant operation requires at least one switch. Therefore, a two-quadrant operation converter has at least two switches, and a fourquadrant operation converter has at least four switches. Multiple-quadrant choppers were employed in industrial applications for a long time. They can be used to implement the DC motor multiple-quadrant operation. As the chopper titles indicate, there are class-A converters (one-quadrant operation), class-B converters (two-quadrant operation), class-C converters, classD converters, and class-E (four-quadrant operation) converters. These converters are derived from multi-quadrant choppers, for example, class B converters are derived from B-type choppers and class E converters are derived from E-type choppers. The class-A converter works in quadrant I, which corresponds to the forward-motoring operation of a DC motor drive. The class-B converter works in quadrant I and II operation, which corresponds to the forward-running motoring and regenerative braking operation of a DC motor drive. The class-C converter works in quadrant I and VI operation. The class-D converter works in quadrant III and VI operation, which corresponds to the reverse-running motoring and regenerative braking operation of a DC motor drive. The class-E converter works in four-quadrant operation, which corresponds to the four-quadrant operation of a DC motor drive. In recent years many papers have investigated the class-E converters for industrial applications. Multi-quadrant operation converters can be derived from the first generation converters. For example, multi-quadrant Luo-converters are derived from positive-output Luo-converters and negative-output Luoconverters. The transformer-type multi-quadrant converters easily change the current direction by transformer polarity and diode rectifier. The main types of such converters can be derived from the forward converter, halfbridge converter, and bridge converter.
1.4.3
The Third Generation Converters
The third generation converters are called switched component converters, and are made of either inductors or capacitors, so-called switched-inductor and switched-capacitors. They can perform in two- or four-quadrant operation with high output power range (thousands of Watts). Since they are made of only inductor or capacitors, they are small. Consequently, the power density and efficiency are high. 1.4.3.1 Switched Capacitor Converters Switched-capacitor DC/DC converters consist of only capacitors. Because there is no inductor in the circuit, their size is small. They have outstanding advantages such as low power losses and low electromagnetic interference
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34
Essential DC/DC Converters
(EMI). Since its electromagnetic radiation is low, switched-capacitor DC/DC converters are required in certain equipment. The switched-capacitor can be integrated into an integrated-chip (IC). Hence, its size is largely reduced. Much attention has been drawn to the switched-capacitor converter since its development. Many papers have been published discussing its characteristics and advantages. However, most of the converters in the literature perform a single-quadrant operation. Some of them work in the push-pull status. In addition, their control circuit and topologies are very complex, especially, for the large difference between input and output voltages. 1.4.3.2 Multiple-Quadrant Switched Capacitor Luo-Converters Switched-capacitor DC/DC converters consist of only capacitors. Since its power density is very high it is widely applied in industrial applications. Some industrial applications require multiple quadrant operation, so that, multiple-quadrant switched-capacitor Luo-converters have been developed. There are two-quadrant operation type and four-quadrant operation type, which will be discussed in detail. 1.4.3.3 Multiple-Lift Push-Pull Switched Capacitor Converters Voltage lift (VL) technique is a popular method widely used in electronic circuit design. It has been successfully employed in DC/DC converter applications in recent years, and has opened a way to design high voltage gain converters. Three series Luo-converters are examples of voltage lift technique implementation. However, the output voltage increases stage-by-stage just along the arithmetic progression. A novel approach — multiple-lift push-pull (ml-pp) technique — has been developed that implements the output voltage, which increases stage by stage along the arithmetic progression. It effectively enhances the voltage transfer gain. The typical circuits are sorted into two series: positive output multiple-lift push-pull switched capacitor Luo-converters and negative output multiple-lift push-pull switched capacitor Luo-converters. 1.4.3.4 Multiple-Quadrant Switched Inductor Converters The switched-capacitors have many advantages, but their circuits are not simple. If the difference of input and output voltages is large, many capacitors must be required. The switched-inductor has the outstanding advantage that only one inductor is required for one switched inductor converter no matter how large the difference between input and output voltages is. This characteristic is very important for large power conversion. At the present time, large power conversion equipment is close to using switched-inductor converters. For example, the MIT DC/DC converter designed by Prof. John G. Kassakian for his new system in the 2005 automobiles is a two-quadrant switched-inductor DC/DC converter.
© 2006 by Taylor & Francis Group, LLC
Introduction 1.4.4
35
The Fourth Generation Converters
The fourth generation DC/DC converters are called soft-switching converters. There are four types of soft-switching methods: 1. 2. 3. 4.
Resonant-switch converters Load-resonant converters Resonant-dc-link converters High-frequency-link integral-half-cycle converters
Until now attention has been paid only to the resonant-switch conversion method. This resonance method is available for working independently to load. There are three main categories: zero-current-switching (ZCS), zerovoltage-switching (ZVS), and zero-transition (ZT) converters. Most topologies usually perform in single quadrant operation in the literature. Actually, these converters can perform in two- and four-quadrant operation with high output power range (thousands of Watts). According to the transferred power becomes large, the power losses increase largely. Main power losses are produced during the switch-on and switch-off period. How to reduce the power losses across the switch is the clue to increasing the power transfer efficiency. Soft-switching technique successfully solved this problem. Professor Fred Lee is the pioneer of the soft-switching technique. He established a research center and manufacturing base to realize the zero-current-switching (ZCS) and zero-voltage-switching (ZVS) DC/DC converters. His first paper introduced his research in 1984. ZCS and ZVS converters have three resonant states: over resonance (completed resonance), optimum resonance (critical resonance) and quasi resonance (subresonance). Only the quasiresonance state has two clear cross-zero points in a repeating period. Many papers after 1984 have been published that develop the ZCS quasi-resonantconverters (QRCs) and ZVS-QRCs. 1.4.4.1 Zero-Current-Switching Quasi-Resonant Converters ZCS-QRC equips the resonant circuit on the switch side to keep the switch-on and switch-off at zero-current condition. There are two states: full-wave state and half wave state. Most engineers use the half-wave state. This technique has half-wave current resonance waveform with two zero-cross points. 1.4.4.2 Zero-Voltage-Switching Quasi-Resonant Converters ZVS-QRC equips the resonant circuit on the switch side to keep the switch-on and switch-off at zero-current condition. There are two states: full-wave state and half wave state. Most engineers use the half-wave state. This technique has half-wave current resonance waveform with two zero-cross points.
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36
Essential DC/DC Converters
1.4.4.3 Zero-Transition Converters Using ZCS-QRC and ZVS-QRC largely reduces the power losses across the switches. Consequently, the switch device power rates become lower and converter power efficiency is increased. However, ZCS-QRC and ZVSQRC have large current and voltage stresses. Therefore the device’s current and voltage peak rates usually are 3 to 5 times higher than the working current and voltage. It is not only costly, but also ineffective. Zero-Transition (ZT) technique overcomes this fault. It implements zero-voltage plus zero-current-switching (ZV-ZCS) technique without significant current and voltage stresses.
1.4.5
The Fifth Generation Converters
The fifth generation converters are called synchronous rectifier (SR) DC/DC converters. This type of converter was required by development of computing technology. Corresponding to the development of the micro-power consumption technique and high-density IC manufacture, the power supplies with low output voltage and strong current are widely used in communications, computer equipment, and other industrial applications. Intel , which developed the Zelog-type computers, governed the world market for a long time. Inter-80 computers used the 5 V power supply. In order to increase the memory size and operation speed, large-scale integrated chip (LSIC) technique has been quickly developed. As the amount of IC manufacturing increased, the gaps between the layers became narrower. At the same time, the micro-powerconsumption technique was completed. Therefore new computers such as those using Pentium I, II, III, and IV, use a 3.3 V power supply. Future computers will have larger memory and will require lower power supply voltages, e.g. 2.5, 1.8, 1.5, even if 1.1 V. Such low power supply voltage cannot be obtained by the traditional diode rectifier bridge because the diode voltage drop is too large. Because of this requirement, new types of MOSFET were developed. They have very low conduction resistance (6 to 8 m Ω¸) and forward voltage drop (0.05 to 0.2 V). Many papers have been published since 1990 and many prototypes have been developed. The fundamental topology is derived from the forward converter. Active-clamped circuit, flat-transformers, double current circuit, soft-switching methods, and multiple current methods can be used in SR DC/DC converters.
1.4.6
The Sixth Generation Converters
The sixth generation converters are called multiple energy-storage elements resonant (MER) converters. Current source resonant inverters are the heart of many systems and equipment, e.g., uninterruptible power supply (UPS) and high-frequency annealing (HFA) apparatus. Many topologies shown in the literature are the series resonant converters (SRC) and parallel resonant
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Introduction
37
converters (PRC) that consist of two or three or four energy storage elements. However, they have limitations. These limitations of two-, three-, and/or four-element resonant topologies can be overcome by special design. These converters have been catagorized into three main types: • Two energy-storage elements resonant DC/AC and DC/AC/DC converters • Three energy-storage elements resonant DC/AC and DC/AC/DC converters • Four energy-storage elements (2L-2C) resonant DC/AC and DC/ AC/DC converters By mathematical calculation there are eight prototypes of two-element converters, 38 prototypes of three-element converters, and 98 prototypes of four-element (2L-2C) converters. By careful analysis of these prototypes we find that few circuits can be realized. If we keep the output in low-pass bandwidth, the series components must be inductors and shunt components must be capacitors. Through further analysis, the first component of the resonant-filter network can be an inductor in series, or a capacitor in shunt. In the first case, only alternate (square wave) voltage sources can be applied to the network. In the second case, only alternate (square wave) current sources can be applied to the network.
1.5
Categorize Prototypes and DC/DC Converter Family Tree
There are more than 500 topologies of DC/DC converters. It is urgently necessary to categorize all prototypes. From all accumulated knowledge we can build a DC/DC converter family tree, which is shown in Figure 1.39. In each generation we introduce some circuits to readers to promote understanding of the characteristics.
© 2006 by Taylor & Francis Group, LLC
38
Essential DC/DC Converters Buck Converter Fundamental Circuits
Boost Converter Buck-Boost Converter
1G Classical Converters
Developed Forward Converter Fly-Back Converter Push-Pull Converter Transformer
Half-Bridge Converter Bridge Converter ZETA Converter
Positive Output Luo-Converter Negative Output Luo-Converter Double Output Luo-Converter Cuk-Converter SEPIC Tapped-Inductor Converters
7 Self-Lift Converter Positive Output Luo-Converter Negative Output Luo-Converter
Voltage Lift
Modified P/O Luo-Converter Double Output Luo-Converter Positive Output Cascade Boost Converter
Super-Lift
Negative Output Cascade Boost Converter Positive Output Super-Lift Luo-Converter
DC/DC Converters
Negative Output Super-Lift Luo-Converter 2G Multi-Quadrant Converters
Transformer-type Converters Developed
Multi-Quadrant Luo-Converter Two Quadrants Converter
Switched-Capacitor Converter 3G SwitchedComponent Converters
Four Quadranrts SC Luo-Converter
Multi-Lift
P/O Multi-Lift Push-Pull Luo-Converter
N/O Multi-Lift Push-Pull Luo-Converter Transformer-type Converters Switched-Inductor Converter Four Quadranrts SI Luo-Converter ZCS-QRC ----- Four Quadrants Zero-Current Switching Luo-Converter
4G Soft-Switching Converters 5G Synchoronous Rectifier Converters 6G Multi-Elements Resonant Power Converters
ZVS-QRC ----- Four Quadrants Zero-Voltage Switching Luo-Converter ZTC ----- Four Quadrants Zero-Transition Luo-Converter Flat-Transformer Synchronous Rectifier Converter Synchronous Rectifier Converter with Active Clamp Circuit Double Current Synchronous Rectifier Converter ZCS Synchronous Rectifier Converter ZVS Synchronous Rectifier Converter 2-Elements ∏ − CLL Current Sources on Resonant Inverter 3-Elements Double Gamma-CL Current Source Resonant Inverter 4-Elements Reverse Double Gamma-CL Resonant Power Converter
FIGURE 1.39 DC/DC converter family.
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Introduction
39
Bibliography Cúk, S., Basics of switched-mode power conversion: topologies, magnetics, and control, in Cúk, S., Ed., Advances in Switched-Mode Power Conversion, Irvine, CA, Teslaco, 1995, vol. 2. Erickson, R.W. and Maksimovic, D., Fundamentals of Power Electronics, 2nd ed., Kluwer Academic Publishers, Norwell, MA, 1999. Kassakian, J., Schlecht, M., and Vergese, G., Principles of Power Electronics, AddisonWesley, Reading, MA, 1991. Kazimierczuk, M.K. and Bui, X.T., Class-E DC-DC converters with an inductive impedance inverter, IEEE-Transactions on Power Electronics, 4, 124, 1989. Liu, Y. and Sen, P.C., New class-E DC-DC converter topologies with constant switching frequency, IEEE-Transactions on Industry Applications, 32, 961, 1996. Luo, F.L., Re-lift circuit, a new DC-DC step-up (boost) converter, IEE-Electronics Letters, 33, 5, 1997. Luo, F.L., Positive output Luo-converters, a series of new DC-DC step-up (boost) conversion circuits, in Proceedings of the IEEE International Conference PEDS 1997, Singapore, p. 882. Luo, F.L., Re-lift converter: design, test, simulation and stability analysis, IEE Proceedings on Electric Power Applications, 145, 315, 1998. Luo, F.L., Negative output Luo-converters, implementing the voltage lift technique, in Proceedings of the Second World Energy System International Conference WES 1998, Toronto, Canada, 1998, 253. Luo, F.L. and Ye, H., DC/DC conversion techniques and nine series Luo-converters, in Rashid M.H., Luo F.L., et al., Eds., Power Electronics Handbook, Academic Press, San Diego, 2001, chap. 17. Maksimovic, D. and Cúk, S., Switching converters with wide DC conversion range, IEEE Transactions on Power Electronics, 6, 151, 1991. Massey, R.P. and Snyder, E.C., High voltage single-ended DC/DC converter, in Proceedings of IEEE PESC ’77, Record. 1977. p. 156. Middlebrook, R.D. and Cúk, S., Advances in Switched-Mode Power Conversion, TESLAco, Pasadena, CA, 1981, vols. I and II. Mohan, N., Undeland, T.M., and Robbins, W.P., Power Electronics: Converters, Applications and Design, 3rd ed., John Wiley & Sons, New York, 2003. Oxner, E., Power FETs and Their Applications, Prentice Hall, NJ, 1982. Rashid, M.H., Power Electronics: Circuits, Devices and Applications, 2nd ed., PrenticeHall, New York, 1993. Redl, R., Molnar, B., and Sokal, N.O., Class-E resonant DC-DC power converters: analysis of operations, and experimental results at 1.5 Mhz, IEEE-Transactions on Power Electronics, 1, 111, 1986. Severns, R.P. and Bloom, E., Modern DC-to-DC Switch Mode Power Converter Circuits, Van Nostrand Reinhold Company, New York, 1985. Smedley, K.M. and Cúk, S., One-cycle control of switching converters, IEEE Transactions on Power Electronics, 10, 625, 1995.
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2 Voltage-Lift Converters
The voltage-lift (VL) technique is a popular method that is widely applied in electronic circuit design. Applying this technique effectively overcomes the effects of parasitic elements and greatly increases the output voltage. Therefore, these DC/DC converters can convert the source voltage into a higher output voltage with high power efficiency, high power density, and a simple structure.
2.1
Introduction
VL technique is applied in the periodical switching circuit. Usually, a capacitor is charged during switch-on by certain voltages, e.g., source voltage. This charged capacitor voltage can be arranged on top-up to some parameter, e.g., output voltage during switch-off. Therefore, the output voltage can be lifted higher. Consequently, this circuit is called a self-lift circuit. A typical example is the saw-tooth-wave generator with a self-lift circuit. Repeating this operation, another capacitor can be charged by a certain voltage, which may possibly be the input voltage or other equivalent voltage. The second capacitor charged voltage is also possibly arranged on top-up to some parameter, especially output voltage. Therefore, the output voltage can be higher than that of the self-lift circuit. As usual, this circuit is called re-lift circuit. Analogously, this operation can be repeated many times. Consequently, the series circuits are called triple-lift circuits, quadruple-lift circuits, and so on. Because of the effect of parasitic elements the output voltage and power transfer efficiency of DC-DC converters are limited. Voltage lift technique opens a way to improve circuit characteristics. After long-term research, this technique has been successfully applied to DC-DC converters. Three series Luo-converters are the DC-DC converters, which were developed from prototypes using VL technique. These converters perform DC-DC voltage increasing conversion with high power density, high efficiency, and cheap topology in simple structure. They are different from any other DC-DC stepup converters and possess many advantages including a high output voltage 41
© 2006 by Taylor & Francis Group, LLC
42
Essential DC/DC Converters
with small ripples. Therefore, these converters are widely used in computer peripheral equipment and industrial applications, especially for high output voltage projects. This chapter’s contents are arranged thusly: 1. 2. 3. 4. 5.
2.2
Seven types of self-lift converters Positive output Luo-converters Negative output Luo-converters Modified positive output Luo-converters Double output Luo-converters
Seven Self-Lift Converters
All self-lift converters introduced here are derived from developed converters such as Luo-converters, Cúk-converters, and single-ended primary inductance converters (SEPICs) discussed in Section 1.3. Since all circuits are simple, usually only one more capacitor and diode required that the output voltage be higher by an input voltage. The output voltage is calculated by the formula VO = (
k 1 + 1)Vin = V 1− k 1 − k in
(2.1)
There are seven circuits: • • • • • • •
Self-lift Cúk converter Self-lift P/O Luo-converter Reverse self-lift P/O Luo-converter Self-lift N/O Luo-converter Reverse self-lift Luo-converter Self-lift SEPIC Enhanced self-lift P/O Luo-converter
These converters perform DC-DC voltage increasing conversion in simple structures. In these circuits the switch S is a semiconductor device (MOSFET, BJT, IGBT and so on). It is driven by a pulse-width-modulated (PWM) switching signal with variable frequency f and conduction duty k. For all circuits, the load is usually resistive, i.e., R = VO / I O
© 2006 by Taylor & Francis Group, LLC
Voltage-Lift Converters
43
The normalized load is zN =
R fLeq
(2.2)
where Leq is the equivalent inductance. We concentrate on the absolute values rather than polarity in the following description and calculations. The directions of all voltages and currents are defined and shown in the corresponding figures. We also assume that the semiconductor switch and the passive components are all ideal. All capacitors are assumed to be large enough that the ripple voltage across the capacitors can be negligible in one switching cycle for the average value discussions. For any component X (e.g., C, L and so on): its instantaneous current and voltage are expressed as iX and vX . Its average current and voltage values are expressed as Ix and Vx. The input voltage and current are VO and IO; the output voltage and current are VI and II. T and f are the switching period and frequency. The voltage transfer gain for the continuous conduction mode (CCM) is M=
Vo I I = VI I o
(2.3)
Variation of current iL :
ζ1 =
ΔiL / 2 IL
(2.4)
Variation of current iLO :
ζ2 =
ΔiLo / 2 I Lo
(2.5)
Variation of current iD :
ξ=
ΔiD / 2 ID
(2.6)
Variation of voltage vC :
ρ=
ΔvC / 2 VC
(2.7)
Variation of voltage vC1 :
σ1 =
ΔvC1 / 2 vC 1
(2.8)
Variation of voltage vC2 :
© 2006 by Taylor & Francis Group, LLC
σ2 =
ΔvC 2 / 2 vC 2
(2.9)
44
Essential DC/DC Converters
Variation of output voltage vO :
ε=
ΔVo / 2 Vo
(2.10)
Here ID refers to the average current iD which flows through the diode D during the switch-off period, not its average current over the whole period. Detailed analysis of the seven self-lift DC-DC converters will be given in the following sections. Due to the length limit of this book, only the simulation and experimental results of the self-lift Cúk converter are given. However, the results and conclusions of other self-lift converters should be quite similar to those of the self-lift Cúk-converter.
2.2.1
Self-Lift Cúk Converter
Self-lift Cúk converters and their equivalent circuits during switch-on and switch-off period are shown in Figure 2.1. It is derived from the Cúk converter. During switch-on period, S and D1 are on, D is off. During switchoff period, D is on, S and D1 are off. 2.2.1.1 Continuous Conduction Mode In steady state, the average inductor voltages over a period are zero. Thus VC1 = VCO = VO
(2.11)
During switch-on period, the voltage across capacitors C and C1 are equal. Since we assume that C and C1 are sufficiently large, so VC = VC1 = VO
(2.12)
The inductor current iL increases during switch-on and decreases during switch-off. The corresponding voltages across L are VI and −(VC − VI ) . Therefore, kTVI = (1 − k )T (VC − VI ) Hence, VO = VC = VC1 = VCO = The voltage transfer gain in the CCM is
© 2006 by Taylor & Francis Group, LLC
1 V 1− k
(2.13)
Voltage-Lift Converters
i
45
v
I
+ L
+ V
i C
D
C
v I
LO
i
–
S
D
–
L 1
O
O
–
–
VO
C1
C
+
C
1
R
O
+
(a) Circuit diagram
iI + +
i LO
vC
L
iO
– LO
C
–
– v C1
VI
S
–
+
vO CO
C1
R
+
(b) Switch on iI + +
L
vC
iO
– LO
C
–
– v C1
VI
+
–
vO C1
CO
R
+
(c) Switch off FIGURE 2.1 Self-lift Cúk converter and equivalent circuits. (a) The self-lift Cúk converter. (b) The equivalent circuit during switch-on. (c) The equivalent circuit during switch-off.
M=
VO I 1 = I = VI IO 1 − k
(2.14)
The characteristics of M vs. conduction duty cycle k are shown in Figure 2.2. Since all the components are considered ideal, the power loss associated with all the circuit elements are neglected. Therefore the output power PO is considered to be equal to the input power PIN : VO I O = VI I I
© 2006 by Taylor & Francis Group, LLC
46
Essential DC/DC Converters M 12 10 8 6 4 2
k
0 0
0.2
0.4
0.6
0.8
1
FIGURE 2.2 Voltage transfer gain M vs. k.
Thus, 1 I 1− k O
IL = II = During switch-off, iD = i L
ID =
1 I 1− k O
(2.15)
The capacitor CO acts as a low pass filter so that I LO = I O The current iL increases during switch-on. The voltage across it during switch-on is VI, therefore its peak-to-peak current variation is ΔiL =
kTVI L
The variation ratio of the current iL is ζ1 =
ΔiL / 2 kTVI k(1 − k )2 R kR = = = IL 2IL 2 fL 2 M 2 fL
© 2006 by Taylor & Francis Group, LLC
(2.16)
Voltage-Lift Converters
47
The variation of current iD is ξ = ζ1 =
kR 2 M 2 fL
(2.17)
The peak-to-peak variation of voltage vC is ΔvC =
I L (1 − k )T I O = C fC
(2.18)
The variation ratio of the voltage vC is ρ=
ΔvC / 2 IO 1 = = 2 fCVO 2 fRC VC
(2.19)
The peak-to-peak variation of the voltage vC1 is ΔvC1 =
I LO (1 − k )T I O (1 − k ) = C1 fC1
(2.20)
The variation ratio of the voltage vC1 is σ1 =
ΔvC1 / 2 I O (1 − k ) 1 = = 2 fC1VO 2 MfRC1 VC1
(2.21)
The peak-to-peak variation of the current iLO is approximately:
ΔiLO
1 ΔvC1 T I (1 − k ) = 2 2 2 = O2 LO 8 f LO C1
(2.22)
The variation ratio of the current iLO is approximately: ζ2 =
ΔiLO / 2 I (1 − k ) 1 = O2 = I LO 16 f LO C1 I O 16 Mf 2 LO C1
(2.24)
The peak-to-peak variation of voltage vO and vCO is
ΔvO = ΔvCO
© 2006 by Taylor & Francis Group, LLC
1 ΔiLO T I O (1 − k ) = 2 2 2 = CO 64 f 3 LO C1CO
(2.25)
48
Essential DC/DC Converters
The variation ratio of the output voltage is ε=
ΔvO / 2 I O (1 − k ) 1 = = VO 128 f 3 LO C1COVO 128 Mf 3 LO C1CO R
(2.26)
The voltage transfer gain of the self-lift Cúk converter is the same as the original boost converter. However, the output current of the self-lift Cúk converter is continuous with small ripple. The output voltage of the self-lift Cúk converter is higher than the corresponding Cúk converter by an input voltage. It retains one of the merits of the Cúk converter. They both have continuous input and output current in CCM. As for component stress, it can be seen that the self-lift converter has a smaller voltage and current stresses than the original Cúk converter. 2.2.1.2 Discontinuous Conduction Mode Self-lift Cúk converters operate in the discontinuous conduction mode (DCM) if the current iD reduces to zero during switch-off. As a special case, when iD decreases to zero at t = T, then the circuit operates at the boundary of CCM and DCM. The variation ratio of the current iD is 1 when the circuit works in the boundary state. ξ=
k R =1 2 M 2 fL
(2.27)
Therefore the boundary between CCM and DCM is MB = k
R = 2 fL
kz N 2
(2.28)
where z N is the normalized load R/(fL). The boundary between CCM and DCM is shown in Figure 2.3a. The curve that describes the relationship between MB and z N has the minimum value MB = 1.5 and k = 1/3 when the normalized load z N is 13.5. When M > MB , the circuit operates in the DCM. In this case the diode current iD decreases to zero at t = t1 = [k + (1 – k) m] T where kT < t1< T and 0 < m < 1. Define m as the current filling factor (FF). After mathematical manipulation: m=
© 2006 by Taylor & Francis Group, LLC
M2 1 = ξ k R 2 fL
(2.29)
Voltage-Lift Converters
49
a) Boundary between CCM and DCM
b) The voltage transfer gain M vs. the normalized load at various k FIGURE 2.3 Boundary between CCM and DCM and DC voltage transfer gain M vs. the normalized load at various k. (a) Boundary between CCM and DCM. (b) The voltage transfer gain M vs. the normalized load at various k.
From the above equation we can see that the discontinuous conduction mode is caused by the following factors: • • • •
Switch frequency f is too low Duty cycle k is too small Inductance L is too small Load resistor R is too big
© 2006 by Taylor & Francis Group, LLC
50
Essential DC/DC Converters
In the discontinuous conduction mode, current iL increases during switchon and decreases in the period from kT to (1 – k)mT. The corresponding voltages across L are VI and −(VC − VI ) . Therefore, kTVI = (1 − k )mT (VC − VI ) Hence, VC = [1 +
k ]V (1 − k )m I
(2.30)
Since we assume that C, C1, and CO are large enough, VO = VC = VCO = [1 +
k ]V (1 − k )m I
(2.31)
or VO = [1 + k 2 (1 − k )
R ]V 2 fL I
(2.32)
R 2 fL
(2.33)
The voltage transfer gain in the DCM is MDCM = 1 + k 2 (1 − k )
The relation between DC voltage transfer gain M and the normalized load at various k in the DCM is also shown Figure 2.3b. It can be seen that in DCM, the output voltage increases as the load resistance R is increasing.
2.2.2
Self-Lift P/O Luo-Converter
Self-lift positive output Luo-converters and the equivalent circuits during switch-on and switch-off period are shown in Figure 2.4. It is the self-lift circuit of the positive output Luo-converter. It is derived from the elementary circuit of positive output Luo-converter. During switch-on period, S and D1 are switchon, D is switch-off. During switch-off period, D is on, S and D1 are off. 2.2.2.1 Continuous Conduction Mode In steady state, the average inductor voltages over a period are zero. Thus VC = VCO = VO During switch-on period, the voltage across capacitor C1 is equal to the source voltage. Since we assume that C and C1 are sufficiently large,
© 2006 by Taylor & Francis Group, LLC
Voltage-Lift Converters
51
iI
–
+
S
vC
i LO
+
iO
LO
C D
+ D1
VI
+ vC1 –
iL –
vO CO
C1
R –
L
a) Self–Lift Positive Output Luo–Converter
iI
–
vC
i LO
+ LO
C
+
iO
+ VI iL L
–
+ vC1 –
vO CO
C1
R –
b) The equivalent circuit during switch–on
iI
–
vC
i LO
+ LO
C
+
iO
+ VI iL –
L
+ vC1 –
v C1
CO
R –
c) The equivalent circuit during switch–off FIGURE 2.4 Self-lift positive output Luo-converter and its equivalent circuits. (a) Self-lift positive output Luo-converter. (b) The equivalent circuit during switch-on. (c) The equivalent circuit during switch-off.
VC1 = VI The inductor current iL increases in the switch-on period and decreases in the switch-off period. The corresponding voltages across L are VI and −(VC − VC1 ) .
© 2006 by Taylor & Francis Group, LLC
52
Essential DC/DC Converters
Therefore kTVI = (1 − k )T (VC − VC1 ) Hence, VO =
1 V 1− k I
The voltage transfer gain in the CCM is M=
VO 1 = VI 1 − k
(2.34)
Since all the components are considered ideal, the power loss associated with all the circuit elements are neglected. Therefore the output power PO is considered to be equal to the input power PIN , VO I O = VI I I Thus, II =
1 I 1− k O
The capacitor CO acts as a low pass filter so that I LO = I O The charge of capacitor C increases during switch-on and decreases during switch-off. Q+ = I C −ON kT = I O kT
Q− = I C −OFF (1 − k )T = I L (1 − k )T
In a switch period, Q + = Q−
IL =
k I 1− k O
during switch-off period, iD = iL + iLO Therefore, I D = I L + I LO =
© 2006 by Taylor & Francis Group, LLC
1 I 1− k O
Voltage-Lift Converters
53
For the current and voltage variations and boundary condition, we can get the following equations using a similar method that was used in the analysis of self-lift Cúk converter. Current variations: ζ1 =
1 R 2 M 2 fL
ζ2 =
k R 2 M fLO
ξ=
k R 2 M 2 fLeq
where Leq refers to Leq =
LLO L + LO
Voltage variations: ρ=
k 1 2 fCR
σ1 =
M 1 2 fC1R
ε=
k 1 2 8 M f LO CO
2.2.2.2 Discontinuous Conduction Mode Self-lift positive output Luo-converters operate in the DCM if the current iD reduces to zero during switch-off. As the critical case, when iD decreases to zero at t = T, then the circuit operates at the boundary of CCM and DCM. The variation ratio of the current iD is 1 when the circuit works in the boundary state. ξ=
k R =1 2 M 2 fLeq
Therefore the boundary between CCM and DCM is MB = k
R = 2 fLeq
kz N 2
(2.35)
where z N is the normalized load R / ( fLeq ) and Leq refers to Leq = LLO L + LO . When M > MB , the circuit operates at the DCM. In this case the circuit operates in the diode current iD decreases to zero at t = t1 = [k + (1 – k) m] T, where KT < t1< T and 0 < m < 1, with m as the current filling factor. We define m as: m=
© 2006 by Taylor & Francis Group, LLC
1 M2 = ξ k R 2 fLeq
(2.36)
54
Essential DC/DC Converters
In the discontinuous conduction mode, current iL increases in the switchon period kT and decreases in the period from kT to (1 – k)mT. The corresponding voltages across L are VI and −(VC − VC1 ) . Therefore, kTVI = (1 − k )mT (VC − VC1 ) and VC = VCO = VO
VC1 = VI
Hence, VO = [1 +
k ]V (1 − k )m I
or VO = [1 + k 2 (1 − k )
R ]V 2 fLeq I
(2.37)
So the real DC voltage transfer gain in the DCM is MDCM = 1 + k 2 (1 − k )
R 2 fL eq
(2.38)
In DCM, the output voltage increases as the load resistance R is increasing.
2.2.3
Reverse Self-Lift P/O Luo-Converter
Reverse self-lift positive output Luo-converters and their equivalent circuits during switch-on and switch-off period are shown in Figure 2.5. It is derived from the elementary circuit of positive output Luo-converters. During switch-on period, S and D1 are on, D is off. During switch-off period, D is on, S and D1 are off. 2.2.3.1 Continuous Conduction Mode In steady state, the average inductor voltages over a period are zero. Thus VC1 = VCO = VO
© 2006 by Taylor & Francis Group, LLC
Voltage-Lift Converters
55
iI
+ S
+
vC1
i LO
–
iO
LO
C1
D1
–
D
VI L
–
vO – vC +
iL
R +
CO C
a) Reverse Self–Lift Positive Output Luo–Converter iI
+
vC1
i LO
– LO
C1
+
iO
– VI
vO – vC +
iL L
–
R +
CO C
b) The equivalent circuit during switch–on iI
+
vC1
i LO
– LO
C1
+
iO
– VI
vO iL
–
L
– vC +
C
CO
R +
c) The equivalent circuit during switch–off FIGURE 2.5 Reverse self-lift positive output Luo-converter and its equivalent circuits. (a) Reverse self-lift positive output Luo-converter. (b) The equivalent circuit during switch-on. (c) The equivalent circuit during switch-off.
During switch-on period, the voltage across capacitor C is equal to the source voltage plus the voltage across C1. Since we assume that C and C1 are sufficiently large, VC1 = VI + VC
© 2006 by Taylor & Francis Group, LLC
56
Essential DC/DC Converters
Therefore, VC1 = VI +
1 k VI = V 1− k 1− k I
VO = VCO = VC1 =
1 V 1− k I
(2.39)
The voltage transfer gain in the CCM is M=
VO 1 = VI 1 − k
(2.40)
Since all the components are considered ideal, the power losses on all the circuit elements are neglected. Therefore the output power PO is considered to be equal to the input power PIN , VO I O = VI I I Thus, II =
1 I 1− k O
The capacitor CO acts as a low pass filter so that I LO = I O The charge of capacitor C1 increases during switch-on and decreases during switch-off Q+ = I C1−ON kT Q− = I LO (1 − k )T = I O (1 − k )T In a switch period, Q + = Q−
I C1−ON =
I C −ON = I LO + I C1−ON = I O +
© 2006 by Taylor & Francis Group, LLC
1− k I k O 1− k 1 IO = IO k k
(2.41)
Voltage-Lift Converters
57
The charge of capacitor C increases during switch-off and decreases during switch-on. Q+ = I C −OFF (1 − k )T
Q− = I C −ON kT =
1 I kT k O
In a switch period, Q + = Q−
1− k 1 I = I k C −ON 1 − k O
I C −OFF =
(2.42)
Therefore, I L = I LO + I C −OFF = I O +
1 2−k I = I = IO + I I 1− k O 1− k O
During switch-off, iD = iL − iLO Therefore, I D = I L − I LO = I O The following equations are used for current and voltage variations and boundary conditions. Current variations: ζ1 =
k R , (2 − k ) M 2 fL
ζ2 =
k R 2 M fLO
ξ=
1 R 2 M 2 fLeq
where Leq refers to Leq =
LLO L + LO
Voltage variations: ρ=
1 1 2 k fCR
© 2006 by Taylor & Francis Group, LLC
σ1 =
1 1 2 M fC1R
ε=
1 k 16 M f 2 CO LO
58
Essential DC/DC Converters
2.2.3.2 Discontinuous Conduction Mode Reverse self-lift positive output Luo-converter operates in the DCM if the current iD reduces to zero during switch-off at t = T, then the circuit operates at the boundary of CCM and DCM. The variation ratio of the current iD is 1 when the circuit works in the boundary state. ξ=
k R =1 2 2 M fLeq
Therefore the boundary between CCM and DCM is MB = k
R = 2 fLeq
kz N 2
(2.43)
where z N is the normalized load R / ( fLeq ) and Leq refers to Leq = LLO L + LO . When M > MB , the circuit operates in the DCM. In this case the diode current iD decreases to zero at t = t1 = [k + (1 – k) m] T, where KT < t1 < T and 0 < m < 1. m is the current filling factor. m=
1 M2 = ξ k R 2 fLeq
(2.44)
In the discontinuous conduction mode, current iL increases during switchon and decreases in the period from kT to (1 – k)mT. The corresponding voltages across L are VI and −VC . Therefore, kTVI = (1 − k )mTVC and VC1 = VCO = VO
VC1 = VI + VC
Hence, VO = [1 +
k ]V (1 − k )m I
or R ⎤ ⎡ VO = ⎢1 + k 2 (1 − k ) ⎥VI fL 2 eq ⎦ ⎣
© 2006 by Taylor & Francis Group, LLC
(2.45)
Voltage-Lift Converters
59
iI
i LO
+ vC1 – S
+ VI
C1 iL
–
L
LO
D – vC D1 +
iO – vO CO
C
R
+
a) Self-Lift Negative Output Luo-Converter iI
+ vC1 –
+
C1
VI
iL
–
L
i LO LO
– vC +
iO – vO CO
C
R
+
b) The equivalent circuit during switch-on iI
+ vC1 –
+
C1
VI
iL
–
L
i LO LO
– vC +
iO
C
– CO
R
+
c) The equivalent circuit during switch-off FIGURE 2.6 Self-lift negative output Luo-converter and its equivalent circuits. (a) Self-lift negative output Luo-converter. (b) The equivalent circuit during switch-on. (c) The equivalent circuit during switch-off.
So the real DC voltage transfer gain in the DCM is MDCM = 1 + k 2 (1 − k )
R 2 fL
(2.46)
In DCM the output voltage increases as the load resistance R increases.
2.2.4
Self-Lift N/O Luo-Converter
Self-lift negative output Luo-converters and their equivalent circuits during switch-on and switch-off period are shown in Figure 2.6. It is the self-lift circuit of the negative output Luo-converter. The function of capacitor C1 is to lift the voltage VC by a source voltage VI . S and D1 are on, and D is off during switchon period. D is on, and S and D1 are off during switch-off period.
© 2006 by Taylor & Francis Group, LLC
60
Essential DC/DC Converters
2.2.4.1 Continuous Conduction Mode In the steady state, the average inductor voltages over a period are zero. Thus VC = VCO = VO During switch-on period, the voltage across capacitor C1 is equal to the source voltage. Since we assume that C and C1 are sufficiently large, VC1 = VI . Inductor current iL increases in the switch-on period and decreases in the switch-off period. The corresponding voltages across L are VI and −(VC − VC1 ) . Therefore, kTVI = (1 − k )T (VC − VC1 ) Hence, VO = VC = VCO =
1 V 1− k I
(2.47)
The voltage transfer gain in the CCM is M=
VO 1 = VI 1 − k
(2.48)
Since all the components are considered ideal, the power loss associated with all the circuit elements are neglected. Therefore the output power PO is considered to be equal to the input power PIN , VO I O = VI I I . Thus, II =
1 I 1− k O
The capacitor CO acts as a low pass filter so that I LO = I O . For the current and voltage variations and boundary condition, the following equations can be obtained by using a similar method that was used in the analysis of self-lift Cúk converter. Current variations: ζ1 =
k R , 2 M 2 fL
Voltage variations:
© 2006 by Taylor & Francis Group, LLC
ζ2 =
k 1 2 16 f LO C
ξ=
k R 2 M 2 fL
Voltage-Lift Converters
ρ=
61
k 1 2 fCR
σ1 =
M 1 2 fC1R
ε=
k 1 128 f 3 LO CCO R
2.2.4.2 Discontinuous Conduction Mode Self-lift negative output Luo-converters operate in the DCM if the current iD reduces to zero at t = T, then the circuit operates at the boundary of CCM and DCM. The variation ratio of the current iD is 1 when the circuit works at the boundary state. ξ=
k R =1 2 M 2 fL
Therefore the boundary between CCM and DCM is MB = k
R = 2 fLeq
kz N 2
(2.49)
where Leq refers to Leq = L and z N is the normalized load R / ( fLeq ) . When M > MB , the circuit operates in the DCM. In this case the diode current iD decreases to zero at t = t1 = [k + (1 – k)m]T, where KT < t1 < T and 0 < m < 1. m is the current filling factor and is defined as: m=
M2 1 = ξ k R 2 fL
(2.50)
In the discontinuous conduction mode, current iL increases during switchon and decreases during period from kT to (1 – k)mT. The voltages across L are VI and −(VC − VC1 ) . kTVI = (1 − k )mT (VC − VC1 ) and VC1 = VI
VC = VCO = VO
Hence, VO = [1 +
k ]V (1 − k )m I
© 2006 by Taylor & Francis Group, LLC
or
VO = [1 + k 2 (1 − k )
R ]V 2 fL I
62
Essential DC/DC Converters iI
–
vC
S
+ VI
iL
–
i LO
+ D1
C
L
LO
+
vC1 D –
iO + vO R–
CO
C1
a) Reverse Self-Lift Negative Output Luo-Converter iI
–
v
i LO
+
C
+ VI
iL
–
LO
+ vC1 –
L
iO + vO R–
CO
C1
b) The equivalent circuit during switch-on
iI
– S
+
vC
i LO
+
C1 vC1
VI
iL
–
L
LO
+ –
iO
C1
+ CO
vO R–
c) The equivalent circuit during switch-off FIGURE 2.7 Reverse self-lift negative output Luo-converter and its equivalent circuits. (a) Reverse self-lift negative output Luo-converter. (b) The equivalent circuit during switch-on. (c) The equivalent circuit during switch-off.
So the real DC voltage transfer gain in the DCM is MDCM = 1 + k 2 (1 − k )
R 2 fL
(2.51)
We can see that in DCM, the output voltage increases as the load resistance R is increasing.
2.2.5
Reverse Self-Lift N/O Luo-Converter
Reverse self-lift negative output Luo-converters and their equivalent circuits during switch-on and switch-off period are shown in Figure 2.7. It is derived from the Zeta converter. During switch-on period, S and D1 are on, D is off. During switch-off period, D is on, S and D1 are off.
© 2006 by Taylor & Francis Group, LLC
Voltage-Lift Converters
63
2.2.5.1 Continuous Conduction Mode In steady state, the average inductor voltages over a period are zero. Thus VC1 = VCO = VO The inductor current iL increases in the switch-on period and decreases in the switch-off period. The corresponding voltages across L are VI and −VC . Therefore kTVI = (1 − k )TVC Hence, VC =
k V 1− k I
(2.52)
voltage across C. Since we assume that C and C1 are sufficiently large, VC1 = VI + VC Therefore, VC1 = VI +
k 1 V = V 1− k I 1− k I
VO = VCO = VC1 =
1 V 1− k I
The voltage transfer gain in the CCM is M=
VO 1 = VI 1 − k
(2.53)
Since all the components are considered ideal, the power loss associated with all the circuit elements is neglected. Therefore the output power PO is considered to be equal to the input power PIN , VO I O = VI I I . Thus, II =
1 I 1− k O
The capacitor CO acts as a low pass filter so that I LO = I O . The charge of capacitor C1 increases during switch-on and decreases during switch-off. Q+ = I C1−ON kT
© 2006 by Taylor & Francis Group, LLC
Q− = I C1−OFF (1 − k )T = I O (1 − k )T
64
Essential DC/DC Converters
In a switch period, Q + = Q−
I C1−ON =
1− k 1− k I = I k C −OFF k O
The charge of capacitor C increases during switch-on and decreases during switch-off. Q+ = I C −ON kT
Q− = I C −OFF (1 − k )T
In a switch period, Q + = Q− I C −ON = I C1−ON + I LO =
I C −OFF =
1− k 1 IO + IO = IO k k
k k 1 1 I C −ON = IO = I 1− k 1− k k 1− k O
Therefore, I L = I C −OFF =
1 I 1− k O
During switch-off period, iD = i L
ID = IL =
1 I 1− k O
For the current and voltage variations and the boundary condition, we can get the following equations using a similar method that was used in the analysis of self-lift Cúk converter. Current variations: ζ1 =
k R 2 M 2 fL
ζ2 =
1 R 2 16 M f LO C1
ξ=
k R 2 M 2 fL
Voltage variations: ρ=
1 1 2 k fCR
© 2006 by Taylor & Francis Group, LLC
σ1 =
1 1 2 M fC1R
ε=
1 1 3 128 M f LO C1CO R
Voltage-Lift Converters
65
2.2.5.2 Discontinuous Conduction Mode Reverse self-lift negative output Luo-converters operate in the DCM if the current iD reduces to zero during switch-off. As a special case, when iD decreases to zero at t = T, then the circuit operates at the boundary of CCM and DCM. The variation ratio of the current iD is 1 when the circuit works in the boundary state. ξ=
k R =1 2 2 M fLeq
The boundary between CCM and DCM is MB = k
R = 2 fLeq
kz N 2
where z N is the normalized load R / ( fLeq ) and Leq refers to Leq = L . When M > MB , the circuit operates at the DCM. In this case, diode current iD decreases to zero at t = t1 = [k + (1 – k) m] T where KT < t1 < T and 0 < m < 1 with m as the current filling factor. m=
1 M2 = ξ k R 2 fLeq
(2.54)
In the discontinuous conduction mode, current iL increases in the switchon period kT and decreases in the period from kT to (1 – k)mT. The corresponding voltages across L are VI and −VC . Therefore, kTVI = (1 − k )mTVC and VC1 = VCO = VO
VC1 = VI + VC
Hence, VO = [1 +
© 2006 by Taylor & Francis Group, LLC
k ]V (1 − k )m I
66
Essential DC/DC Converters
or ⎡ R ⎤ VO = ⎢1 + k 2 (1 − k ) ⎥V 2 fL ⎦ I ⎣
(2.55)
The voltage transfer gain in the DCM is MDCM = 1 + k 2 (1 − k )
R 2 fL
(2.56)
It can be seen that in DCM, the output voltage increases as the load resistance R is increasing.
2.2.6
Self-Lift SEPIC
Self-lift SEPIC and the equivalent circuits during switch-on and switch-off period are shown in Figure 2.8. It is derived from SEPIC (with output filter). S and D1 are on, and D is off during switch-on period. D is on, and S and D1 are off during switch-off period. 2.2.6.1 Continuous Conduction Mode In steady state, the average voltage across inductor L over a period is zero. Thus VC = VI . During switch-on period, the voltage across capacitor C1 is equal to the voltage across C. Since we assume that C and C1 are sufficiently large, VC1 = VC = VI In steady state, the average voltage across inductor LO over a period is also zero. VC 2 = VCO = VO
Thus
The inductor current iL increases in the switch-on period and decreases in the switch-off period. The corresponding voltages across L are VI and −(VC − VC1 + VC 2 − VI ) . Therefore kTVI = (1 − k )T (VC − VC1 + VC 2 − VI ) or
© 2006 by Taylor & Francis Group, LLC
Voltage-Lift Converters
67
iI
+
+ L
vC
–
–
vC1
i LO
+
LO
D
C
iO
+
+
VI
vC2
i L1
D1
L1
S
–
–
vO R –
CO
C2
a) Self-Lift Sepic Converter iI
+
+ L
vC
–
–
i LO
vC1 +
iO
LO
C
+
+
VI
vC2
i L1 L1
S
–
–
vO R –
CO
C2
b) The equivalent circuit during switch-on iI
+
+ L
vC
–
–
vC1
i LO
+
iO
LO
C
+
+
VI
vC2
i L1 S
–
L1
–
C2
CO
vO R –
c) The equivalent circuit during switch-off FIGURE 2.8 Self-lift sepic converter and its equivalent circuits. (a) Self-lift sepic converter. (b) The equivalent circuit during switch-on. (c) The equivalent circuit during switch-off.
kTVI = (1 − k )T (VO − VI ) Hence, VO =
1 V = VCO = VC 2 1− k I
(2.57)
The voltage transfer gain in the CCM is M=
VO 1 = VI 1 − k
(2.58)
Since all the components are considered ideal, the power loss associated with all the circuit elements is neglected. Therefore the output power PO is considered to be equal to the input power PIN , VO I O = VI I I . Thus,
© 2006 by Taylor & Francis Group, LLC
68
Essential DC/DC Converters
II =
1 I = IL 1− k O
The capacitor CO acts as a low pass filter so that I LO = I O The charge of capacitor C increases during switch-off and decreases during switch-on. Q− = I C −ON kT
Q+ = I C −OFF (1 − k )T = I I (1 − k )T
In a switch period, Q + = Q−
I C −ON =
1− k 1− k I = I k C −OFF k I
The charge of capacitor C2 increases during switch-off and decreases during switch-on. Q− = I C 2−ON kT = I O kT
Q+ = I C 2−OFF (1 − k )T
In a switch period, Q + = Q−
IC 2−OFF =
k k = I I 1 − k C −ON 1 − k O
The charge of capacitor C1 increases during switch-on and decreases during switch-off. Q+ = I C1−ON kT
Q− = I C1−OFF (1 − k )T
In a switch period, Q + = Q−
I C1−OFF = I C 2−OFF + I LO =
k 1 IO + IO = I 1− k 1− k O
Therefore I C1−ON =
1− k 1 I C1−OFF = I O k k
During switch-off,
© 2006 by Taylor & Francis Group, LLC
I L1 = I C1−ON − I C −ON = 0
Voltage-Lift Converters
69 iD = i L − i L 1
Therefore, ID = I I =
1 I 1− k O
For the current and voltage variations and the boundary condition, we can get the following equations using a similar method that is used in the analysis of self-lift Cúk converter. Current variations: ζ1 =
k R 2 M 2 fL
ζ2 =
k R 16 f 2 LO C2
ξ=
k R 2 M 2 fLeq
where Leq refers to Leq =
LLO L + LO
Voltage variations: ρ=
M 1 2 fCR
σ1 =
M 1 2 fC1R
σ2 =
k 1 2 fC2 R
ε=
k 1 3 128 f LO C2 CO R
2.2.6.2 Discontinuous Conduction Mode Self-lift Sepic converters operate in the DCM if the current iD reduces to zero during switch-off. As a special case, when iD decreases to zero at t = T, then the circuit operates at the boundary of CCM and DCM. The variation ratio of the current iD is 1 when the circuit works in the boundary state. ξ=
k R =1 2 2 M fLeq
Therefore the boundary between CCM and DCM is MB = k
© 2006 by Taylor & Francis Group, LLC
R = 2 fLeq
kz N 2
(2.59)
70
Essential DC/DC Converters
LLO where z N is the normalized load R / ( fLeq ) and Leq refers to Leq = . L +diode LO When M > MB , the circuit operates in the DCM. In this case the current iD decreases to zero at t = t1 = [k + (1 – k) m] T where KT < t1 < T and 0 < m < 1. m is defined as: m=
1 M2 = ξ k R 2 fLeq
(2.60)
In the discontinuous conduction mode, current iL increases during switchon and decreases in the period from kT to (1 – k)mT. The corresponding voltages across L are VI and −(VC − VC1 + VC 2 − VI ) . Thus, kTVI = (1 − k )T (VC − VC1 + VC 2 − VI ) and VC = VI
VC1 = VC = VI
VC 2 = VCO = VO
Hence, VO = [1 +
k ]V (1 − k )m I
or R ⎤ ⎡ VO = ⎢1 + k 2 (1 − k ) V 2 fL eq ⎥ I ⎣ ⎦ So the real DC voltage transfer gain in the DCM is MDCM = 1 + k 2 (1 − k )
R 2 fLeq
(2.61)
In DCM, the output voltage increases as the load resistance R is increasing.
2.2.7
Enhanced Self-Lift P/O Luo-Converter
Enhanced self-lift positive output Luo-converter circuits and the equivalent circuits during switch-on and switch-off periods are shown in Figure 2.9. They are derived from the self-lift positive output Luo-converter in Figure 2.4 with swapping the positions of switch S and inductor L.
© 2006 by Taylor & Francis Group, LLC
Voltage-Lift Converters
71
iI
+
– vC1 +
i LO
C1
LO
L
iO
D1 VI
+
D + vC
S –
–
CO
vO R –
C
FIGURE 2.9 Enhanced self-lift P/O Luo-converter.
During switch-on period, S and D1 are on, and D is off. Obtain: VC = VC1 and ΔiL =
VI kT L
During switch-off period, D is on, and S and D1 are off. ΔiL =
VC − VI (1 − k )T L
So that VC =
1 V 1− k I
The output voltage and current and the voltage transfer gain are VO = VI + VC1 = (1 +
IO =
M = 1+
© 2006 by Taylor & Francis Group, LLC
1 )V 1− k I
1− k I 2−k I
1 2−k = 1− k 1− k
(2.62)
(2.64)
(2.65)
72
Essential DC/DC Converters
Average voltages: VC =
1 V 1− k I
(2.66)
VC1 =
1 V 1− k I
(2.67)
Average currents:
IL =
I LO = I O
(2.68)
2−k I = II 1− k O
(2.69)
Therefore, VO 1 2−k = +1= VI 1 − k 1− k
2.3
(2.70)
Positive Output Luo-Converters
Positive output Luo-converters perform the voltage conversion from positive to positive voltages using VL technique. They work in the first quadrant with large voltage amplification. Five circuits have been introduced in the literature: • • • • •
Elementary circuit Self-lift circuit Re-lift circuit Triple-lift circuit Quadruple-lift circuit
The elementary circuit can perform step-down and step-up DC-DC conversion, which was introduced in previous section. Other positive output Luo-converters are derived from this elementary circuit, they are the self-lift circuit, re-lift circuit, and multiple-lift circuits (e.g., triple-lift and quadruplelift circuits) shown in the corresponding figures. Switch S in these diagrams is a P-channel power MOSFET device (PMOS), and S1 is an N-channel power MOSFET device (NMOS). They are driven by a PWM switch signal with
© 2006 by Taylor & Francis Group, LLC
Voltage-Lift Converters
73
repeating frequency f and conduction duty k. The switch repeating period is T = 1/f, so that the switch-on period is kT and switch-off period is (1 – k)T. For all circuits, the load is usually resistive, R = VO/IO; the combined inductor L = L1L2/(L1 + L2); the normalized load is zN = R/fL. Each converter consists of a positive Luo-pump and a low-pass filter L2-CO, and lift circuit (introduced in the following sections). The pump inductor L1 transfers the energy from source to capacitor C during switch-off, and then the stored energy on capacitor C is delivered to load R during switch-on. Therefore, if the voltage VC is higher the output voltage VO should be higher. When the switch S is turned off, the current iD flows through the freewheeling diode D. This current descends in whole switch-off period (1 – k)T. If current iD does not become zero before switch S turned on again, this working state is defined as the continuous conduction mode (CCM). If current iD becomes zero before switch S turned on again, this working state is defined as the discontinuous conduction mode (DCM). Assuming that the output power is equal to the input power, PO = PIN
or
VO I O = VI I I
The voltage transfer gain in continuous mode is M=
VO I = I VI IO
ξ1 =
ΔiL1 / 2 I L1
ξ2 =
ΔiL 2 / 2 I L2
ζ=
ΔiD / 2 I L1 + I L 2
Variation ratio of current iL1:
Variation ratio of current iL2:
Variation ratio of current iD:
Variation ratio of current iL2+j is χj =
© 2006 by Taylor & Francis Group, LLC
ΔiL 2+ j / 2 I L 2+ j
j = 1, 2, 3, …
74
Essential DC/DC Converters
Variation ratio of voltage vC: ρ=
ΔvC / 2 VC
Variation ratio of voltage vCj: σj =
ΔvCj / 2 VCj
j = 1, 2, 3, 4, …
Variation ratio of output voltage vCO: ε=
2.3.1
ΔvO / 2 VO
Elementary Circuit
Elementary circuit and its switch-on and -off equivalent circuits are shown in Figure 2.10. Capacitor C acts as the primary means of storing and transferring energy from the input source to the output load via the pump inductor L1. Assuming capacitor C to be sufficiently large, the variation of the voltage across capacitor C from its average value VC can be neglected in steady state, i.e., vC(t) ≈ VC, even though it stores and transfers energy from the input to the output. 2.3.1.1 Circuit Description When switch S is on, the source current iI = iL1 + iL2. Inductor L1 absorbs energy from the source. In the mean time inductor L2 absorbs energy from source and capacitor C, both currents iL1 and iL2 increase. When switch S is off, source current iI = 0. Current iL1 flows through the free-wheeling diode D to charge capacitor C. Inductor L1 transfers its stored energy to capacitor C. In the mean time current iL2 flows through the (CO – R) circuit and freewheeling diode D to keep itself continuous. Both currents iL1 and iL2 decrease. In order to analyze the circuit working procession, the equivalent circuits in switch-on and -off states are shown in Figures 2.10b, c, and d. Actually, the variations of currents iL1 and iL2 are small so that iL1 ≈ IL1 and iL2 ≈ IL2. The charge on capacitor C increases during switch off: Q+ = (1 – k)T IL1.
© 2006 by Taylor & Francis Group, LLC
Voltage-Lift Converters
75
iS
– VC +
iC +
VI N
Vs – + VL1
iL1
C
L1
+ VL2 – L2
iD
+
Io
iL2
iCo
D
VD
+
CoR
Vo
–
–
–
a) Circuit Diagram iS
VO
iC
VL2 +
iL1
+
+ VL1
VIN
L2
– iL2 Vo
VD
–
– b) Switch-on VO
iC
VL2 +
L2
iL1 + VL1 –
L1
– iL2 Vo
iD
c) Switch-off VO iL1
+
L1
L2
VD
iL2 Vo
– d) Discontinuos mode
FIGURE 2.10 Elementary circuit of positive output Luo-converter(a) Circuit diagram. (b) Switch-on. (c) Switch-off. (d) Discontinuous mode.
It decreases during switch-on: Q – = kTIL2
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76
Essential DC/DC Converters
In a whole period investigation, Q+ = Q–. Thus, I L2 =
1− k I k L1
(2.71)
Since capacitor CO performs as a low-pass filter, the output current IL2 = IO
(2.72)
These two Equations (2.71) and (2.72) are available for all positive output Luo-converters. The source current is iI = iL1 + iL2 during switch-on period, and iI = 0 during switch-off. Thus, the average source current II is I I = k × iI = k(iL1 + iL 2 ) = k(1 +
1− k ) I L1 = I L1 k
(2.73)
Therefore, the output current is IO =
1− k I k I
(2.74)
VO =
k V 1− k I
(2.75)
Hence, output voltage is
The voltage transfer gain in continuous mode is ME =
VO k = VI 1 − k
(2.76)
The curve of ME vs. k is shown in Figure 2.11. Current iL1 increases and is supplied by VI during switch-on. It decreases and is inversely biased by –VC during switch-off. Therefore, kTVI = (1 − k )TVC The average voltage across capacitor C is VC =
© 2006 by Taylor & Francis Group, LLC
k V = VO 1− k I
(2.77)
Voltage-Lift Converters
77
10
8
ME
6
4
2
0 0
0.2
0.4
k
0..6
0.8
1
FIGURE 2.11 Voltage transfer gain ME vs. k
2.3.1.2 Variations of Currents and Voltages To analyze the variations of currents and voltages, some voltage and current waveforms are shown in Figure 2.12. Current iL1 increases and is supplied by VI during switch-on. It decreases and is inversely biased by –VC during switch-off. Therefore, its peak-to-peak variation is ΔiL1 =
kTVI L1
Considering Equation (2.73), the variation ratio of the current iL1 is ξ1 =
ΔiL1 / 2 kTVI 1− k R = = I L1 2L1 I I 2 ME fL1
(2.78)
Current iL2 increases and is supplied by the voltage (VI + VC – VO) = VI during switch-on. It decreases and is inversely biased by –VO during switchoff. Therefore its peak-to-peak variation is ΔiL 2 =
kTVI L2
(2.79)
Considering Equation (2.72), the variation ratio of current iL2 is ξ2 =
© 2006 by Taylor & Francis Group, LLC
ΔiL 2 / 2 kTVI k R = = I L2 2L2 I O 2 ME fL2
(2.80)
78
Essential DC/DC Converters
S on
VI
O
L1
S off
II DT VI L2
IO
IL1
− VO L1
VI L1
0 IL2
S on
V 1 − L
IL1 kII
S off
T
t
− VO L2
0 IL2
t1
T
t
− VO L2 L2
VI
IO 0 IS
t
VI L
0
t
IS L
VI
II+IO II
II 0
ID
t
− VO L
II+IO
0 ID
t − V L O
IO IO 0 IC
t IL2
0
t
IC1
IO 0
t
-IL1
IL2
0 -IL1
t
-I1 VL1,VL2 VI
VL1,VL2 VI
VI
0
0
-VO
-VO
t
VS
t
-VO VS
VO+VI
VO+VI
VI
VI
0
t
VD
0
t
VDR
VO+VI
VO+VI VO
VO 0
t
0
a
FIGURE 2.12 Some voltage and current waveforms of elementary circuit.
© 2006 by Taylor & Francis Group, LLC
t
b
Voltage-Lift Converters
79
When switch is off, the free-wheeling diode current is iD = iL1 + iL2 and ΔiD = ΔiL1 + ΔiL 2 =
kTVI kTVI kTVI (1 − k )TVO + = = L1 L2 L L
(2.81)
Considering Equation (2.71) and Equation (2.72), the average current in switch-off period is ID = IL1 + IL2 =
IO 1− k
The variation ratio of current iD is ζ=
ΔiD / 2 (1 − k )2 TVO k(1 − k )R k2 R = = = ID 2LI O 2 ME fL ME 2 2 fL
(2.82)
The peak-to-peak variation of vC is ΔvC =
Q + 1− k = TI I C C
Considering Equation (2.77), the variation ratio of vC is ρ=
ΔvC / 2 (1 − k )TI I k 1 = = VC 2CVO 2 fCR
(2.83)
If L1 = L2 = 1 mH, C = CO = 20 μF, R = 10 Ω, f = 50 kHz and k = 0.5, we get ξ1 = ξ2 = 0.05, ζ = 0.025 and ρ = 0.025. Therefore, the variations of iL1, iL2, and vC are small. In order to investigate the variation of output voltage vO, we have to calculate the charge variation on the output capacitor CO, because Q = CO VO and ΔQ = CO ΔvO ΔQ is caused by ΔiL2 and corresponds to the area of the triangle with the height of half of ΔiL2 and the width of half of the repeating period T/2, which is shown in Figure 2.12. Considering Equation (2.79), ΔQ =
© 2006 by Taylor & Francis Group, LLC
1 ΔiL 2 T T kTVI = 2 2 2 8 L2
80
Essential DC/DC Converters
Thus, the half peak-to-peak variation of output voltage vO and vCO is ΔvO ΔQ kT 2VI = = 2 CO 8CO L2 The variation ratio of output voltage vO is ε=
ΔvO / 2 kT 2 VI k 1 = = 2 VO 8CO L2 VO 8 ME f CO L2
(2.84)
If L2 = 1 mH, CO = 20 μF, f = 50 kHz and k = 0.5, we obtain that ε = 0.00125. Therefore, the output voltage VO is almost a real DC voltage with very small ripple. Because of the resistive load, the output current iO(t) is almost a real DC waveform with very small ripple as well, and IO = VO/R. 2.3.1.3 Instantaneous Values of Currents and Voltages Referring to Figure 2.12, the instantaneous values of the currents and voltages are listed below: ⎧ 0 vS = ⎨ ⎩VO + VI
for for
0 < t ≤ kT kT < t ≤ T
⎧V + VO vD = ⎨ I ⎩ 0
for for
0 < t ≤ kT
(2.85)
(2.86)
kT < t ≤ T
⎧ VI v L1 = ⎨ ⎩−VO
for for
0 < t ≤ kT kT < t ≤ T
(2.87)
⎧ VI vL 2 = ⎨ ⎩−VO
for for
0 < t ≤ kT kT < t ≤ T
(2.88)
V ⎧ ⎪i (0) + iL 2 (0) + I t iI = iS = ⎨ L1 L 0 ⎪⎩
iL1
V ⎧ i L 1 ( 0) + I t ⎪⎪ L1 =⎨ VO ⎪iL1 ( kT ) − (t − kT ) L1 ⎪⎩
© 2006 by Taylor & Francis Group, LLC
for for
for for
0 < t ≤ kT kT < t ≤ T
0 < t ≤ kT kT < t ≤ T
(2.89)
(2.90)
Voltage-Lift Converters
iL 2
81
V ⎧ i L 2 ( 0) + I t ⎪⎪ L2 =⎨ VO ⎪iL 2 ( kT ) − (t − kT ) L2 ⎪⎩
for for
0 ⎧ ⎪ VO iD = ⎨ ⎪⎩iL1 ( kT ) + iL 2 ( kT ) − L (t − kT ) V ⎧ i L 2 ( 0) + I t ⎪⎪ L2 iC ≈ ⎨ VO ⎪−iL1 ( kT ) + (t − kT ) L1 ⎪⎩
iCO
0 < t ≤ kT kT < t ≤ T
for for
for for
V ⎧ i L 2 ( 0) + I t − I O ⎪⎪ L2 ≈⎨ VO ⎪−iL1 ( kT ) + (t − kT ) − I O L1 ⎪⎩
0 < t ≤ kT kT < t ≤ T
0 < t ≤ kT kT < t ≤ T
for for
0 < t ≤ kT kT < t ≤ T
(2.91)
(2.92)
(2.93)
(2.94)
where i L 1 ( 0) =
kI O (1 − k )VO − 1− k 2 fL1
iL1 ( kT ) =
kI O (1 − k )VO + 1− k 2 fL1
i L 2 ( 0) = I O −
(1 − k )VO 2 fL2
iL 2 ( kT ) = I O +
(1 − k )VO 2 fL2
2.3.1.4 Discontinuous Mode Referring to Figure 2.10d, we can see that the diode current iD becomes zero during switch off before next period switch on. The condition for discontinuous mode is ζ≥1
© 2006 by Taylor & Francis Group, LLC
82
Essential DC/DC Converters
20 continuous mode 10 k=0.9 5 ME
k=0.7 2 1
k=0.5
0.5 k=0.3 0.2
discontinuous mode k=0.1
0.1 1
2
5
10
20
50
100
200
500 1000
R/fL
FIGURE 2.13 The boundary between continuous and discontinuous modes and the output voltage vs. the normalized load zN = R/fL.
k2 R ≥1 ME 2 2 fL
i.e.,
ME ≤ k
or
z R =k N 2 fL 2
(2.95)
The graph of the boundary curve vs. the normalized load zN = R/fL is shown in Figure 2.13. It can be seen that the boundary curve is a monorising function of the parameter k. In this case the current iD exists in the period between kT and t1 = [k + (1 – k)mE]T, where mE is the filling efficiency and it is defined as: 2
mE =
ME 1 = ζ k2 R 2 fL
(2.96)
Considering Equation (2.95), therefore 0 < mE < 1. Since the diode current iD becomes zero at t = kT + (1 – k)mET, for the current iL, then kTVI = (1 – k)mETVC or VC =
© 2006 by Taylor & Francis Group, LLC
k R VI = k(1 − k ) V (1 − k )mE 2 fL I
Voltage-Lift Converters
83
with R 1 ≥ 2 fL 1 − k and for the current iLO kT(VI + VC – VO) = (1 – k)mETVO Therefore, output voltage in discontinuous mode is VO =
R k VI = k(1 − k ) V (1 − k )mE 2 fL I
with R 1 ≥ 2 fL 1 − k
(2.97)
i.e., the output voltage will linearly increase during load resistance R increasing. The output voltage vs. the normalized load zN = R/fL is shown in Figure 2.13. It can be seen that larger load resistance R may cause higher output voltage in discontinuous conduction mode. 2.3.1.5 Stability Analysis Stability analysis is of vital importance for any converter circuit. Considering the various methods including the Bode plot, the root-locus method in splane is used for this analysis. According to the circuit network and control system theory, the transfer function in s-domain for switch-on and -off are obtained: Gon = {
δVO ( s) sCR } = δVI ( s) on s 3 CCO L2 R + s 2 CL2 + s(C + CO )R + 1
(2.98)
Goff = {
δVO ( s) sCR } = δVI ( s) off s 3 CCO L2 R + s 2 CL2 + s(C + CO )R + 1
(2.99)
where s is the Laplace operator. From Equation (2.98) and Equation (2.99) in Laplace transform it can be seen that the elementary converter is a third order control circuit. The zero is determined by the equations where the numerator is equal to zero, and the poles are determined by the equation where the denominator is equal to zero. There is a zero at original point (0,
© 2006 by Taylor & Francis Group, LLC
84
Essential DC/DC Converters
2.5
x 104
2 ×
1.5
R↑
1
× jωn
0.5 ×
0
×
-0.5 -1
× -jωn
-1.5
×
-2 -2.5 -30
-20
-10
0
10
20
30
10
20
30
(a) Switch-on x 104 2.5 2 1.5
×
R↑
1
× jωn
0.5 ×
×
0 -0.5 -1 -1.5
× -jωn ¥
-2 -2.5 -30
-20
-10
0
(b) Switch-off
FIGURE 2.14 Stability analysis of elementary circuit. (a) Switch-on. (b) Switch-off.
0) and three poles located in the left-hand half plane in Figure 2.14, so that this converter is stable. Since the equations to determine the poles are the equations with all positive real coefficients, according to the Gauss theorem, the three poles are one negative real pole and a pair of conjugate complex poles with negative real part. When the load resistance R increases and tends toward infinity, the three poles move. The real pole goes to the original point and eliminates with the zero. The pair of conjugate complex poles becomes a pair of imaginary poles located on the image axis. Assuming C = CO and L1 = L2 {L = L1 L2/(L1 + L2) or L2 = 2L}, the pair of imaginary poles are
© 2006 by Taylor & Francis Group, LLC
Voltage-Lift Converters
85
s = ±j
C + C0 1 = ±j = ± jω n CC0 L2 CL
for switch on
(2.100)
s = ±j
C + C0 1 = ±j = ± jω n CC0 L2 CL
for switch off
(2.101)
where ω n = 1 CL is the converter normal angular frequency. They are locating on the stability boundary. Therefore, the circuit works in the critical state. This fact is verified by experiment and computer simulation. When R = ∞, the output voltage vO intends to be very high value. The output voltage VO cannot be infinity because of the leakage current penetrating the capacitor CO . 2.3.2
Self-Lift Circuit
Self-lift circuit and its switch-on and -off equivalent circuits are shown in Figure 2.15, which is derived from the elementary circuit. Comparing to Figure 2.10 and Figure 2.15, it can be seen that the pump circuit and filter are retained and there is only one capacitor C1 and one diode D1 more, as a lift circuit is added into the circuit. Capacitor C1 functions to lift the capacitor voltage VC by a source voltage Vin. Current iC1(t) = δ(t) is an exponential function. It has a large value at the power on moment, but it is small in the steady state because VC1 = Vin. 2.3.2.1 Circuit Description When switch S is on, the instantaneous source current is iI = iL1 + iL2 + iC1. Inductor L1 absorbs energy from the source. In the mean time inductor L2 absorbs energy from source and capacitor C. Both currents iL1 and iL2 increase, and C1 is charged to vC1 = VI. When switch S is off, the instantaneous source current is iI = 0. Current iL1 flows through capacitor C1 and diode D to charge capacitor C. Inductor L1 transfers its stored energy to capacitor C. In the mean time, current iL2 flows through the (CO – R) circuit, capacitor C1 and diode D, to keep itself continuous. Both currents iL1 and iL2 decrease. In order to analyze the circuit working procession, the equivalent circuits in switchon and -off states are shown in Figures 2.15b, c and d. Assuming that capacitor C1 is sufficiently large, voltage VC1 is equal to VI in steady state. Current iL1 increases in switch-on period kT, and decreases in switch-off period (1 – k)T. The corresponding voltages applied across L1 are VI and –(VC – VI) respectively. Therefore, kTVI = (1 − k )T (VC − VI )
© 2006 by Taylor & Francis Group, LLC
86
Essential DC/DC Converters iC
+ V L2 –
S + iIN
Vs
–
L2
c iD1
V IN
D
iO iCO
iL2
iD
iL1 L1
+ VO –
R
D1 iC1
C1
CO
(a) circuit diagram iIN
iC
VO
L2
C + V L1 L 1 – V IN C1
V IN
–
+
VO
– VD + iC1
(b) switch on iC
– VO +
+
V L2 –
i L2
L2
C – VD +
VO
L1 V IN
(c) switch off – VC +
+
V L2 –
i L2
L2 i L1
– VD + VO
L1 V IN
(d) discontinuous mode FIGURE 2.15 Self-lift circuit. (a) Circuit diagram. (b) Switch on. (c) Switch off. (d) Discontinuous mode.
© 2006 by Taylor & Francis Group, LLC
Voltage-Lift Converters
87
10
8
MS
6
4
2
0
0.2
0
0.6
0.4
0.8
1
k
FIGURE 2.16 Voltage transfer gain MS vs. k.
VC =
Hence,
1 V 1− k I
(2.102)
Current iL2 increases in switch-on period kT, and decreases in switch-off period (1 – k)T. The corresponding voltages applied across L2 are (VI + VC – VO) and –(VO – VI). Therefore, kT (VC + VI − VO ) = (1 − k )T (VO − VI ) Hence,
1 V 1− k I
(2.103)
I O = (1 − k )I I
(2.104)
VO =
and the output current is
Therefore, the voltage transfer gain in continuous mode is MS =
VO 1 = VI 1 − k
The curve of MS vs. k is shown in Figure 2.16.
© 2006 by Taylor & Francis Group, LLC
(2.105)
88 2.3.2.2
Essential DC/DC Converters Average Current IC1 and Source Current IS
During switch-off period (1 – k)T, current iC1 is equal to (iL1 + iL2), and the charge on capacitor C1 decreases. During switch-on period kT, the charge increases, so its average current in switch-on period is I C1 =
I 1− k 1− k (i L 1 + i L 2 ) = ( I L1 + I L 2 ) = O k k k
(2.106)
During switch-off period (1 – k)T the source current iI is 0, and in the switch-on period kT, iI = iL1 + iL 2 + iC1 Hence,
I I = k ( i L 1 + i L 2 + iC 1 ) = k ( I L 1 + I L 2 + I C 1 ) = k( I L1 + I L 2 )(1 +
I I 1− k 1 ) = k L2 = O k 1− k k 1− k
(2.107)
2.3.2.3 Variations of Currents and Voltages To analyze the variations of currents and voltages, some voltage and current waveforms are shown in Figure 2.17. Current iL1 increases and is supplied by VI during switch-on period kT. It decreases and is reversely biased by –(VC – VI) during switch-off. Therefore, its peak-to-peak variation is ΔiL1 =
kTVI L1
Hence, the variation ratio of current iL1 is ξ1 =
ΔiL1 / 2 kVI T 1− k R = = I L1 2 kL1 I I 2 MS fL1
(2.108)
Current iL2 increases and is supplied by the voltage (VI + VC – VO) = VI in switch-on period kT. It decreases and is inversely biased by –(VC – VI) during switch-off. Therefore its peak-to-peak variation is ΔiL 2 =
kTVI L2
Thus, the variation ratio of current iL2 is ξ2 =
© 2006 by Taylor & Francis Group, LLC
ΔiL 2 / 2 kVI T k R = = I L2 2L2 I O 2 MS fL2
(2.109)
Voltage-Lift Converters
89
S on
VI
L1
O-
L
0 I L2
t
I
1
T VO – V I L2
–
V
kT VI L2
IO
I L1
VO – V I L1
II
0 IL2
–
S off
V
VI L1
S on –
I L1 kI I
S off
t1
T
t
– V O – L2 L 2 VI
VI
DQ
IO 0
0 II =I S
t V
II =I S
t
t) VI + d( L
II
t) d(
+
I
L
II
0 ID
–
t
VO – V I L
0 ID
t – V O L
IO
–V
I
IO 0
t
I D1 IO k
d(t) IO k
0 I C1
0 ID
t
t
d(t)
0
t
d(t)
IO k
I C1 0
0 –(II+I O )
IC1 IO
d(t)
t
V O–V I L
V O –V I L
IL2
I C1
0
t
–I L1
t
I L2
0 –I L1
t
–I1 V L1 ,V L2 VI
V L2 VI
VI
0
0
–(VO –VI)
–VO
t
VS
VS
VO
VO
VI
VI
0
t
–(VO –VI)
0
t
V D1
V D1
VO
VO
t
VI
(V O –VI)
0
t
VD
0
t
VD
VO
VO kV O
kV O 0
t
0
(a)
FIGURE 2.17 Some voltage and current waveforms of self-lift circuit.
© 2006 by Taylor & Francis Group, LLC
t
(b)
90
Essential DC/DC Converters
When switch is off, the free-wheeling diode current is iD = iL1 + iL2 and ΔiD = ΔiL1 + ΔiL 2 =
kTVI k(1 − k )VO = T L L
(2.110)
Considering Equation (2.71) and Equation (2.72), ID = IL1 + IL2 =
IO 1− k
The variation ratio of current iD is ζ=
ΔiD / 2 k(1 − k )2 TVO k R = = ID 2LI O MS 2 2 fL
(2.111)
The peak-to-peak variation of voltage vC is Q + (1 − k )TI L1 1 − k = = kTI I C C C
ΔvC = Hence, its variation ratio is ρ=
ΔvC / 2 (1 − k )2 kI I T k = = VC 2CVI 2 fCR
(2.112)
The charge on capacitor C1 increases during switch-on, and decreases during switch-off period (1 – k)T by the current (IL1 + IL2). Therefore, its peakto-peak variation is ΔvC1 =
(1 − k )T ( I L1 + I L 2 ) I O = C1 fC1
Considering VC1 = VI, the variation ratio of voltage vC1 is σ=
ΔvC1 / 2 IO MS = = VC1 2 fC1VI 2 fC1R
(2.113)
If L1 = L2 = 1mH, C = C1 = CO = 20 μF, R = 40 Ω, f = 50 kHz and k = 0.5, we obtained that ξ1 = 0.1, ξ2 = 0.1, ζ = 0.1, ρ = 0.006 and σ = 0.025. Therefore, the variations of iL1, iL2, vC1 and vC are small. Considering Equation (2.84) and Equation (2.105), the variation ratio of output voltage vO is
© 2006 by Taylor & Francis Group, LLC
Voltage-Lift Converters
ε=
91 ΔvO / 2 kT 2 VI k 1 = = 2 VO 8CO L2 VO 8 MS f CO L2
(2.114)
If L2 = 1 mH, C0 = 20 μF, f = 50 kHz and k = 0.5, ε = 0.0006. Therefore, the output voltage VO is almost a real DC voltage with very small ripple. Because of the resistive load, the output current iO(t) is almost a real DC waveform with very small ripple as well, and IO = VO/R. 2.3.2.4 Instantaneous Value of the Currents and Voltages Referring to Figure 2.17, the instantaneous values of the currents and voltages are listed below: ⎧0 vS = ⎨ ⎩VO
for for
0 < t ≤ kT kT < t ≤ T
⎧V vD = ⎨ O ⎩0
for for
0 < t ≤ kT kT < t ≤ T
⎧0 vD1 = ⎨ ⎩VO
for for
0 < t ≤ kT kT < t ≤ T
VI ⎧ v L1 = v L 2 = ⎨ ⎩−(VO − VI )
for for
(2.115)
(2.116)
(2.117)
0 < t ≤ kT kT < t ≤ T
V ⎧ ⎪i (0) + iL 2 (0) + δ(t) + I t iI = iS = ⎨ L1 L 0 ⎪⎩
for for
(2.118)
0 < t ≤ kT kT < t ≤ T
(2.119)
iL1
V ⎧ i L 1 ( 0) + I t ⎪⎪ L1 =⎨ VO − VI ⎪iL1 ( kT ) − (t − kT ) L1 ⎪⎩
for for
0 < t ≤ kT kT < t ≤ T
(2.120)
iL 2
V ⎧ i L 2 ( 0) + I t ⎪⎪ L2 =⎨ VO − VI ⎪iL 2 ( kT ) − (t − kT ) L2 ⎪⎩
for for
0 < t ≤ kT kT < t ≤ T
(2.121)
© 2006 by Taylor & Francis Group, LLC
92
Essential DC/DC Converters
0 ⎧ ⎪ VO − VI iD = ⎨ (t − kT ) ⎪⎩iL1 ( kT ) + iL 2 ( kT ) − L ⎧δ(t) iD1 = ⎨ ⎩ 0
for for
for for
0 < t ≤ kT kT < t ≤ T
V ⎧ i L 2 ( 0) + I t ⎪⎪ L2 iC = ⎨ VO − VI ⎪iL1 ( kT ) − (t − kT ) L1 ⎪⎩
for for
V ⎧ iL 2 (0) + I t − IO ⎪⎪ L2 =⎨ VO − VI ⎪iL 2 ( kT ) − (t − kT ) − IO L2 ⎪⎩
(2.122)
(2.123)
0 < t ≤ kT kT < t ≤ T
δ(t) ⎧ ⎪ VO − VI iC 1 = ⎨ (t − kT ) ⎪⎩−iL1 ( kT ) − iL 2 ( kT ) + L
iCO
0 < t ≤ kT kT < t ≤ T
for for
for for
(2.124)
0 < t ≤ kT kT < t ≤ T
0 < t ≤ kT kT < t ≤ T
(2.125)
(2.126)
where iL1(0) = k II – k VI/2 f L1 iL1(kT) = k II + k VI/2 f L1 and iL2(0) = IO – k VO/2 f M L2 iL2(kT) = IO + k VO/2 f M L2 2.3.2.5 Discontinuous Mode Referring to Figure 2.15d, we can see that the diode current iD becomes zero during switch off before next period switch on. The condition for discontinuous mode is ζ ≥ 1, i.e.,
© 2006 by Taylor & Francis Group, LLC
k R ≥1 MS 2 2 fL
Voltage-Lift Converters
93
30 continuous mode
20 k=0.9 10
MS
8 k=0.8 5 3 k=0.5
2
k=0.33 1.5 k=0.1 1
discontinuous mode 24.7
13.56
62.5 R/fL
222
842
FIGURE 2.18 The boundary between CCM and DCM and the output voltage vs. the normalized load zN = R/fL.
MS ≤ k
or
z R = k N 2 fL 2
(2.127)
The graph of the boundary curve vs. the normalized load zN = R/fL is shown in Figure 2.18. It can be seen that the boundary curve has a minimum value of 1.5 at k = 1/3 . In this case the current iD exists in the period between kT and t1 = [k + (1 – k)mS]T, where mS is the filling efficiency and it is defined as: mS =
MS 2 1 = ζ k R 2 fL
(2.128)
Considering Equation (2.127), therefore 0 < mS < 1. Since the diode current iD becomes zero at t = kT + (1 – k)mST, for the current iL kTVI = (1 – k)mST(VC – VI) or VC = [1 +
© 2006 by Taylor & Francis Group, LLC
k R ]V = [1 + k 2 (1 − k ) ]V (1 − k )mS I 2 fL I
94
Essential DC/DC Converters
with k
R 1 ≥ 2 fL 1 − k
and for the current iLO kT(VI + VC –VO) = (1 – k)mST(VO – VI) Therefore, output voltage in discontinuous mode is VO = [1 +
k R ]V = [1 + k 2 (1 − k ) ]V (1 − k )mS I 2 fL I
with
k
R 1 ≥ 2 fL 1 − k
(2.129)
i.e., the output voltage will linearly increase while load resistance R increases. The output voltage VO vs. the normalized load zN = R/fL is shown in Figure 2.18. Larger load resistance R causes higher output voltage in discontinuous conduction mode. 2.3.2.6 Stability Analysis Taking the root-locus method in s-plane for stability analysis the transfer functions in s-domain for switch-on and -off are obtained: Gon = {
Goff = {
δVO ( s) sCR } on = 3 2 δVI ( s) s CCO L2 R + s CL2 + s(C + CO )R + 1
(2.130)
δVO ( s) sCR } off = 3 (2.131) 2 δVI ( s) s (C + C1 )CO L2 R + s (C + C1 )L2 + s(C + C1 + CO )R + 1
where s is the Laplace operator. From Equations (2.130) and (2.131) in Laplace transform it can be seen that the self-lift converter is a third order control circuit. The zero is determined by the equation when the numerator is equal to zero, and the poles are determined by the equation when the denominator is equal to zero. There is a zero at origin point (0, 0) and three poles located in the left-hand half plane in Figure 2.19, so that the self-lift converter is stable. Since the equations to determine the poles are the equations with all positive real coefficients, according to the Gauss theorem, the three poles are one negative real pole and a pair of conjugate complex poles with negative real part. When the load resistance R increases and tends towards infinity, the three poles move. The real pole goes to the origin point and eliminates with the zero. The pair of conjugate complex poles becomes a pair of imaginary poles locating on the image axis. Assuming C = C1 = CO and L1 = L2 {L = L1 L2/(L1 + L2) or L2 = 2L}, the pair of imaginary poles are
© 2006 by Taylor & Francis Group, LLC
Voltage-Lift Converters
2.5
95
x 104
2 ×
1.5
R↑
× jω
1 0.5 ×
0
×
-0.5 -1
× -jωn
-1.5
×
-2 -2.5 -30
-20
-10
0
10
20
30
20
30
(a) Switch on x 104 2.5 ×
2 1.5
R↑ →
1
×
j(√3/2)ωn
0.5 ×
0
→
-0.5 -1
→
× -j(√3/2)ω n
-1.5 ×
-2 -2.5 -30
-20
-10
0
10
(b) Switch off FIGURE 2.19 Stability analysis of self-lift circuit. (a) Switch-on. (b) Switch-off.
C + C0 1 = ±j = ± jω n for switch on CC0 L2 CL
(2.132)
C + C1 + C0 3 3 = ±j = ±j ω for switch off (C + C1 )C0 L2 4CL 2 n
(2.133)
s = ±j
s = ±j
© 2006 by Taylor & Francis Group, LLC
96
Essential DC/DC Converters
where ω n = 1 CL is the self-lift converter normal angular frequency. They are locating on the stability boundary. Therefore, the circuit works in the critical state. This fact is verified by experiment and computer simulation. When R = 8, the output voltage vO intends to be a very high value. The output voltage VO cannot be infinity because of the leakage current penetrating the capacitor CO. 2.3.3
Re-Lift Circuit
Re-lift circuit, and its switch-on and -off equivalent circuits are shown in Figure 2.20, which is derived from the self-lift circuit. It consists of two static switches S and S1; three diodes D, D1, and D2; three inductors L1, L2 and L3; four capacitors C, C1, C2, and CO. From Figure 2.10, Figure 2.15, and Figure 2.20, it can be seen that the pump circuit and filter are retained and there are one capacitor C2, one inductor L3 and one diode D2 added into the re-lift circuit. The lift circuit consists of D1-C1-L3D2-S1-C2. Capacitors C1 and C2 perform characteristics to lift the capacitor voltage VC by twice the source voltage VI. L3 performs the function as a ladder joint to link the two capacitors C1 and C2 and lift the capacitor voltage VC up. Current iC1(t) = δ1(t) and iC2(t) = δ2(t) are exponential functions. They have large values at the moment of power on, but they are small because vC1 = vC2 = VI in steady state. 2.3.3.1 Circuit Description When switches S and S1 turn on, the source instantaneous current iI = iL1 + iL2 + iC1 + iL3 + iC2. Inductors L1 and L3 absorb energy from the source. In the mean time inductor L2 absorbs energy from source and capacitor C. Three currents iL1, iL3 and iL2 increase. When switches S and S1 turn off, source current iI = 0. Current iL1 flows through capacitor C1, inductor L3, capacitor C2 and diode D to charge capacitor C. Inductor L1 transfers its stored energy to capacitor C. In the mean time, current iL2 flows through the (CO – R) circuit, capacitor C1, inductor L3, capacitor C2 and diode D to keep itself continuous. Both currents iL1 and iL2 decrease. In order to analyze the circuit working procession, the equivalent circuits in switch-on and -off states are shown in Figure 2.20b, c, and d. Assuming capacitor C1 and C2 are sufficiently large, and the voltages VC1 and VC2 across them are equal to VI in steady state. Voltage vL3 is equal to VI during switch-on. The peak-to-peak variation of current iL3 is ΔiL 3 =
VI kT L3
(2.134)
This variation is equal to the current reduction when it is switch-off. Suppose its voltage is –VL3–off, so
© 2006 by Taylor & Francis Group, LLC
Voltage-Lift Converters –V C +
S i IN
+
97
–
Vs
iC
D1
D2
VI L1
i CO
i L2
iD
i D1
i L1
–
L2
c D
N
V L2
+
i C1
C1
C2
L3
i L2
R
CO
+ VO –
+ V S1 –
S1
(a) circuit diagram – VO + iIN
iL1
V IN
+ V L2 –
C
i L2
L2
VD
VO
+ V L1 –
V L 1 IN C1
V IN i L3
C2
L3
(b) switch on – VO +
+ V L2 –
C
L2
i L1 L1
i L2
VO V IN C1
V L 3 IN
i L3
C2
(c) switch off – VO +
+ V L2 –
C
L2
i L2
VD iL1
VO V L 1 CIN 1
iL3
V IN L3 C 2
(d) discontinuous mode FIGURE 2.20 Re-lift circuit: (a) circuit diagram; (b) switch on; (c) switch off; (d) discontinuous mode.
ΔiL 3 =
VL 3−off (1 − k )T L3
Thus, during switch-off the voltage drop across inductor L3 is
© 2006 by Taylor & Francis Group, LLC
98
Essential DC/DC Converters
VL 3−off =
k V 1− k I
(2.135)
Current iL1 increases in switch-on period kT, and decreases in switch-off period (1 – k)T. The corresponding voltages applied across L1 are VI and –(VC – 2VI – VL3–off). Therefore, kTVI = (1 − k )T (VC − 2VI − VL 3−off ) VC =
Hence,
2 V 1− k I
(2.136)
Current iL2 increases in switch-on period kT, and it decreases in switch-off period (1 – k)T. The corresponding voltages applied across L2 are (VI + VC – VO) and –(VO – 2VI – VL3–off). Therefore, kT (VC + VI − VO ) = (1 − k )T (VO − 2VI − VL 3−off ) Hence,
VO =
2 V 1− k I
(2.137)
IO =
1− k I 2 I
(2.138)
and the output current is
The voltage transfer gain in continuous mode is MR =
VO 2 = VI 1 − k
(2.139)
The curve of MR vs. k is shown in Figure 2.21. 2.3.3.2 Other Average Currents Considering Equation (2.71), I L1 =
and
© 2006 by Taylor & Francis Group, LLC
k k IO = I I 1− k 2
I L 3 = I L1 + I L 2 =
1 I 1− k O
(2.140)
(2.141)
Voltage-Lift Converters
99
20
16
MR
12
8
4
0
0.2
0
0.6
0.4
0.8
1
k
FIGURE 2.21 Voltage transfer gain MR vs. k.
Currents iC1 and iC2 equal to (iL1+ iL2) during switch-off period (1 – k)T, and the charges on capacitors C1 and C2 decrease, i.e., iC1 = iC2 = (iL1+ iL2) =
1 I 1− k O
The charges increase during switch-on period kT, so their average currents are I C1 = I C 2 =
I 1− k 1− k k ( I L1 + I L 2 ) = ( + 1)I O = O k k 1− k k
(2.142)
During switch-off the source current iI is 0, and in the switch-on period kT, it is iI = iL1 + iL 2 + iC1 + iL 3 + iC 2 Hence, I I = kiI = k( I L1 + I L 2 + I C1 + I L 3 + I C 2 ) = k[2( I L1 + I L 2 ) + 2 I C1 ] = 2 k( I L1 + I L 2 )(1 +
© 2006 by Taylor & Francis Group, LLC
I 1 2 1− k ) = 2k L2 = I 1− k k 1− k O k
(2.143)
100
Essential DC/DC Converters S on IL1 K I1 2 0 IL2
S off
(i)
S on IL1
(ii)
kT
T
(iii)
t
(iv)
K I1 2 0 IL2
S off
(i)
(ii)
T1
T
(iv)
IO 0 II=IS
t
(v)
t
0 II=IS
t (v)
II 0 ID
(vi)
(vii)
t
0 ID
t (viii)
IO IO 0 ID1
t
0 ID
d(t)
t
d(t)
0
0
t a
t b
FIGURE 2.22 Some voltage and current waveforms of re-lift circuit.
2.3.3.3 Variations of Currents and Voltages To analyze the variations of currents and voltages, some voltage and current waveforms are shown in Figure 2.22. Current iL1 increases and is supplied by VI during switch-on period kT. It decreases and is reversely biased by –(VC – 2VI – VL3) during switch-off period (1 – k)T. Therefore, its peak-to-peak variation is ΔiL1 =
kTVI L1
Considering Equation (2.140), the variation ratio of current iL1 is ξ1 =
ΔiL1 / 2 kVI T 1 − k R = = I L1 kL1 I I 2 M R fL1
(2.144)
Current iL2 increases and is supplied by the voltage (VI + VC – VO) = VI during switch-on period kT. It decreases and is reversely biased by –(VO –2VI – VL3) during switch-off. Therefore, its peak-to-peak variation is
© 2006 by Taylor & Francis Group, LLC
Voltage-Lift Converters
101
d'(t) IC1
IC1
0
d'(t)
0 t
(i)
t (ii)
(ii) IC IL2
IC1
IO 0 -I1
0
t
-IL2
VL1,VL2 VI
IL2
-IL2
t
VL2 VI
VI
0
0
-(VO-2VI)
-VO
t
t
-(VO-2VI) VS
VS
VO-VI VI
VI
0
0
t
VD1
t
VD1 VO-VI
VO-VI 0 VD
0 VD
t
t
VO
VO
kVO kVO 0
a
0
t
b
t
FIGURE 2.22 (continued)
ΔiL 2 =
kTVI L2
Considering Equation (2.72), the variation ratio of current iL2 is ξ2 =
© 2006 by Taylor & Francis Group, LLC
ΔiL 2 / 2 kTVI k R = = I L2 2L2 I O 2 M R fL2
(2.145)
102
Essential DC/DC Converters
When switch is off, the free-wheeling diode current is iD = iL1 + iL2 and ΔiD = ΔiL1 + ΔiL 2 =
kTVI k(1 − k )VO = T L 2L
(2.146)
Considering Equation (2.71) and Equation (2.72), ID = IL1 + IL2 =
IO 1− k
The variation ratio of current iD is ζ=
ΔiD / 2 k(1 − k )2 TVO k(1 − k )R k R = = = 4LI O 2 M R fL ID M R 2 fL
(2.147)
Considering Equation (2.134) and Equation (2.141), the variation ratio of current iL3 is χ1 =
ΔiL 3 / 2 kVI T k R = = 2 1 I L3 M R fL3 2 L3 IO 1− k
(2.148)
If L1 = L2 = 1 mH, L3 = 0.5 mH, R = 160 Ω, f = 50 kHz and k = 0.5, we obtained that ξ1 = 0.2, ξ2 = 0.2, ζ = 0.1 and χ1 = 0.2. Therefore, the variations of iL1, iL2 and iL3 are small. The peak-to-peak variation of vC is ΔvC =
Q + 1− k k(1 − k ) = TI L1 = TI I C C 2C
Considering Equation (2.136), the variation ratio is ρ=
ΔvC / 2 k(1 − k )TI I k = = VC 4CVO 2 fCR
(2.149)
The charges on capacitors C1 and C2 increase during switch-on period kT, and decrease during switch-off period (1 – k)T by the current (IL1 + IL2). Therefore their peak-to-peak variations are ΔvC1 =
(1 − k )T ( I L1 + I L 2 ) (1 − k )I I = C1 2C1 f
ΔvC 2 =
(1 − k )T ( I L1 + I L 2 ) (1 − k )I I = C2 2C2 f
© 2006 by Taylor & Francis Group, LLC
Voltage-Lift Converters
103
Considering VC1 = VC2 = VI, the variation ratios of voltages vC1 and vC2 are σ1 =
ΔvC1 / 2 (1 − k )I I MR = = VC1 4 fC1VI 2 fC1R
(2.150)
σ2 =
ΔvC 2 / 2 (1 − k )I I MR = = VC 2 4VI C2 f 2 fC2 R
(2.151)
Considering Equation (2.84), the variation ratio of output voltage vO is ε=
ΔvO / 2 kT 2 VI k 1 = = 2 VO 8CO L2 VO 8 M R f CO L2
(2.152)
If C = C1 = C2 = CO = 20 μF, L2 = 1 mH, R = 160 Ω, f = 50 kHz and k = 0.5, we obtained that ρ = 0.0016, σ1 = σ2 = 0.0125, and ε = 0.0003. The ripples of vC, vC1, vC2 and vCO are small. Therefore, the output voltage vO is almost a real DC voltage with very small ripple. Because of the resistive load, the output current iO(t) is almost a real DC waveform with very small ripple as well, and IO = VO/R. 2.3.3.4 Instantaneous Value of the Currents and Voltages Referring to Figure 2.22, the instantaneous current and voltage values are listed below: ⎧0 vS = ⎨ ⎩VO
for for
0 < t ≤ kT kT < t ≤ T
⎧V vD = ⎨ O ⎩0
for for
0 < t ≤ kT
(2.153)
(2.154)
kT < t ≤ T
⎧0 vD1 + vD 2 = ⎨ ⎩VO
for for
0 < t ≤ kT kT < t ≤ T
(2.155)
⎧⎪ VI k vL 3 = ⎨ VI − ⎩⎪ 1 − k
for for
0 < t ≤ kT kT < t ≤ T
(2.156)
VI ⎧⎪ k v L1 = v L 2 = ⎨ −[V − (2 − )V ] ⎪⎩ O 1− k I
© 2006 by Taylor & Francis Group, LLC
for for
0 < t ≤ kT kT < t ≤ T
(2.157)
104
Essential DC/DC Converters
V V ⎧ ⎪i (0) + iL 2 (0) + iL 3 (0) + δ 1 (t) + δ 2 (t) + I t + I t for 0 < t ≤ kT (2.158) iI = iS = ⎨ L1 L L3 ⎪⎩ for kT < t ≤ T 0
iL 3
iL1
V ⎧ i L 1 ( 0) + I t ⎪ L1 ⎪ k =⎨ )V VO − (2 − ⎪ − 1 k I (t − kT ) ( ) − i kT ⎪ L1 L1 ⎩
for for
0 < t ≤ kT kT < t ≤ T
(2.159)
iL 2
V ⎧ i L 2 ( 0) + I t ⎪ L2 ⎪ k =⎨ )V VO − (2 − ⎪ 1 − k I (t − kT ) ( ) − i kT ⎪ L2 L2 ⎩
for for
0 < t ≤ kT kT < t ≤ T
(2.160)
V ⎧ i L 3 ( 0) + I t ⎪ L3 ⎪ k =⎨ V ⎪ 1 − k I (t − kT ) ⎪iL 3 ( kT ) − L 3 ⎩
for for
0 < t ≤ kT kT < t ≤ T
0 ⎧ ⎪ k iD = ⎨ )V VO − (2 − 1 − k I (t − kT ) ⎪iL1( kT ) + iL 2 ( kT ) − L ⎩ ⎧δ (t) + δ 2 (t) iD1 = ⎨ 1 0 ⎩ ⎧δ (t) iD2 = ⎨ 2 ⎩ 0
for for for for
© 2006 by Taylor & Francis Group, LLC
for 0 < t ≤ kT for kT < t ≤ T
0 < t ≤ kT
(2.162)
(2.163)
kT < t ≤ T
0 < t ≤ kT kT < t ≤ T
V ⎧ i L 2 ( 0) + I t ⎪⎪ L2 iC = ⎨ VO − VI ⎪iL1 ( kT ) − (t − kT ) L1 ⎪⎩
(2.161)
(2.164)
for for
0 < t ≤ kT kT < t ≤ T
(2.165)
Voltage-Lift Converters
105
δ 1 (t ) ⎧ ⎪ k iC1 = ⎨ VO − (2 + )V k I (t − kT ) 1 − ⎪−iL1( kT ) − iL 2 ( kT ) + L ⎩
for 0 < t ≤ kT for kT < t ≤ T
iC 2
for 0 < t ≤ kT δ 2 (t ) ⎧ ⎪ k =⎨ VO − (2 + )V 1 − k I (t − kT ) for kT < t ≤ T ⎪−iL1 ( kT ) − iL 2 ( kT ) + L ⎩
iCO
V ⎧ i L 2 ( 0) + I t − I O ⎪ L2 ⎪ k =⎨ )V VO − (2 + ⎪ 1 − k I (t − kT ) − I O ⎪iL 2 ( kT ) − L2 ⎩
for for
(2.166)
(2.167)
0 < t ≤ kT (2.168) kT < t ≤ T
where iL1(0) = k II/2 – k VI/2 f L1, iL1(kT) = k II/2 + k VI/2 f L1, and iL2(0) = IO – k VI/2 f L2, iL2(kT) = IO + k VI/2 f L2, and iL3(0) = IO + k II/2 – k VI/2 f L3, iL3(kT) = IO + k II/2 + k VI/2 f L3. 2.3.3.5 Discontinuous Mode Referring to Figure 2.20d, we can see that the diode current iD becomes zero during switch off before next period switch on. The condition for discontinuous mode is ζ≥1 i.e., k R ≥1 M R 2 fL
or
MR ≤ k
R = k zN fL
(2.169)
The graph of the boundary curve vs. the normalized load zN = R/fL is shown in Figure 2.23. It can be seen that the boundary curve has a minimum value of 3.0 at k = 1/3 . In this case the current iD exists in the period between kT and t1 = [k + (1 – k)mR]T, where mR is the filling efficiency and it is defined as:
© 2006 by Taylor & Francis Group, LLC
106
Essential DC/DC Converters 60 k=0.95
continuous mode
40 k=0.9 20
MR
k=0.8
10 6 4 3
k=0.5 k=0.33 k=0.1
2
27 32
discontinuous mode 50
125 R/fL
444
1684
FIGURE 2.23 The boundary between continuous and discontinuous modes and the output voltage vs. the normalized load zN = R/fL.
mR =
2 1 MR = ζ k R fL
(2.170)
Considering Equation (2.169), therefore 0 < mR < 1. Since the diode current iD becomes zero at t = kT + (1 – k)mRT, for the current iL kTVI = (1 – k)mRT(VC – 2VI – VL3–off) or VC = [2 +
k k k R + ]VI = [2 + + k 2 (1 − k ) ]V 1 − k (1 − k )mR 1− k 4 fL I
with
k
R 2 ≥ fL 1 − k
and for the current iLO kT(VI + VC – VO) = (1 – k)mRT(VO – 2VI – VL3–off) Therefore, output voltage in discontinuous mode is VO = [2 +
k k k R + ]V = [2 + + k 2 (1 − k ) ]V 1 − k (1 − k )mR I 1− k 4 fL I
© 2006 by Taylor & Francis Group, LLC
Voltage-Lift Converters
107
k
with
R 2 ≥ fL 1 − k
(2.171)
i.e., the output voltage will linearly increase during load resistance R increasing. The output voltage vs. the normalized load zN = R/fL is shown in Figure 2.23. Larger load resistance R may cause higher output voltage in discontinuous mode. 2.3.3.6 Stability Analysis Stability analysis is of vital importance for any converter circuit. According to the circuit network and control systems theory, the transfer functions in s-domain for switch-on and -off states are obtained: Gon = {
Goff = {
=
=
δVO ( s) sCR } = δVI ( s) on s 3 L2 CCO R + s 2 L2 C + s(C + CO )R + 1
(2.172)
δVO ( s) } δVI ( s) off s(C1 + C2 ) + s 3 L3C1C2 R sC s 2 C1C2 1 + sCO R
⎛ s(C1 + C2 ) + s 3 L3C1C2 R + sL2 + s 2 L2 CO R ⎞ 2 ⎟ ⎜ sC s C1C2 1 + sCO R ⎟ ⎜ 3 2 ⎜ + s(C1 + C2 ) + s L3C1C2 + R + sL2 + s L2 CO R ⎟ ⎟ ⎜ s 2 C1C2 1 + sCO R ⎝ ⎠ sCR[(C1 + C2 ) + s 2 L3 C1C2 ] ⎛ sC[(C1 + C2 ) + s 2 L3 C1C2 ][R + sL2 + s 2 L2 CO R] + (1 + sCO R)[(C1 + C2 )⎞ ⎜ ⎟ ⎜ + s 2 L C C ] + sC C [R + sL + s 2 L C R] ⎟ ⎝ ⎠ 3 1 2 1 2 2 2 O
(2.173)
where s is the Laplace operator. From Equation (2.172) and Equation (2.173) in Laplace transform we can see that the re-lift converter is a third order control circuit for switch-on state and a fifth order control circuit for switch-off state. For the switch-on state, the zeros are determined by the equation when the numerator of Equation (2.172) is equal to zero, and the poles are determined by the equation when the denominator of Equation (2.172) is equal to zero. There is a zero at the origin point (0, 0). Since the equation to determine the poles is the equation with all positive real coefficients, according to the Gauss theorem, the three poles are: one negative real pole (p3)
© 2006 by Taylor & Francis Group, LLC
108
Essential DC/DC Converters
3
x 104
×
2
→
1 P3
R↑ →
×
0
P1
× jωn
z
P3 -1
→
P2
× -jω n
P2 ×
-2
-3 -30
-20
-10
0
10
20
30
10
20
30
(a) Switch-on 2.5
x 104
2
P1 ×
P3
1.5
×
P1
× jωn
j√(2ω ) °× j0.88ω
1
z1
0.5 P5
R↑
×
0 -0.5
× ° P 4
×
-1.5
P4
×
n
z3
z2
-1
n
P4
× -j0.88ωn
°× -j√(2ω ) n
-jωn
P2
P2
-2 -2.5 -30
-20
-10
0
(b) Switch-off FIGURE 2.24 Stability analysis of re-lift circuit. (a) Switch-on. (b) Switch-off.
and a pair of conjugate complex poles with negative real part (p1,2). The three poles are located in the left half plane in Figure 2.24, so that the re-lift converter is stable. When the load resistance R increases and intends towards infinity, the three poles move. The real pole goes to the origin point and eliminates with the zero. The pair of conjugate complex poles becomes a pair of imaginary poles locating on the imaginary axis. Assuming that all capacitors have same capacitance C, and L1 = L2 {L = L1 L2/(L1 + L2) or L2 = 2L} and L3 = L, Equation (2.172) becomes:
© 2006 by Taylor & Francis Group, LLC
Voltage-Lift Converters
Gon = {
δVO ( s) } = δVI ( s) on
109 1
1 2 s 2 LC + 2
(2.174)
poles for switch on
(2.175)
C + CO s 2 L2 CO + C
=
and the pair of imaginary poles is p1,2 = ± j
C + C0 1 = ±j = ± jω n L2 CC0 LC
where ωn = (LC)–1/2 is the re-lift converter normal angular frequency. For the switch-off state, the zeros are determined by the equation when the numerator of Equation (2.173) is equal to zero, and the poles are determined by the equation when the denominator of Equation (2.173) is equal to zero. There are three zeros: one (z3) at the original point (0, 0) and two zeros (z1,2) on the imaginary axis which are C1 + C2 2 = ±j = ± j 2ω n L3 C1C2 LC
z1,2 = ± j
zeros for switch off (2.176)
Since the equation to determine the poles is the equation with all positive real coefficients, according to the Gauss theorem, the five poles are one negative real pole (p5) and two pairs of conjugate complex poles with negative real parts (p1,2 and p3,4). There are five poles located in the left-hand half plane in Figure 2.24, so that the re-lift converter is stable. When the load resistance R increases and intends towards infinity, the five poles move. The real pole goes to the origin point and eliminates with the zero. The two pairs of conjugate complex poles become two pairs of imaginary poles locating on the imaginary axis. Assuming that all capacitors have same capacitance C, and L1 = L2 {L = L1 L2/(L1 + L2) or L2 = 2L} and L3 = L, Equation (2.173) becomes: Goff = {
=
δVO ( s) C(C1 + C2 ) + s 2 L3 CC1C2 } off = δVI ( s) ⎛ (CC1 + CC2 + C1C2 + s 2 L3 CC1C2 )(1 + s 2 L2 CO )⎞ ⎜ ⎟ ⎜ + (C C + C C + s 2 L C C C ) ⎟ ⎝ ⎠ 3 O 1 2 O 1 O 2
2C 2 + s 2 LC 3 2 + s 2 LC = 4 2 2 2 2 2 3 (3C + s LC )(1 + 2 s LC) + (2C + s LC ) 2 s L C + 8 s 2 LC + 5 2
2
3
and the two pairs of imaginary poles are
© 2006 by Taylor & Francis Group, LLC
(2.177)
110
Essential DC/DC Converters – VC +
S + V – s
iC
+
c
D1
L2
D
D2
VL2 – iL2
D4
VI N
iL1
iL3
L1 iC1
C1
L3 iC2
iL4 C2
R
L4 iC3
C3
+ VO –
CO
D3 + S1 VS1 –
FIGURE 2.25 Triple-lift circuit.
s 2 LC =
−8 ± 64 − 40 6 ⎧−3.225 = −2 ± =⎨ 4 2 ⎩−0.775
poles for switch off, so that p1,2 = ± j 1.8 ωn p3,4 = ± j 0.88 ωn
and
(2.178)
For both states when R tends to infinity all poles are locating on the stability boundary. Therefore, the circuit works in the critical state. From Equation (2.171) the output voltage will be infinity. This fact is verified by the experimental results and computer simulation results. When R = ∞, the output voltage vO tends to be a very high value. In this particular circuit since there is some leakage current across the capacitor CO, the output voltage vO can not be infinity.
2.3.4
Multiple-Lift Circuits
Referring to Figure 2.20a, it is possible to build multiple-lift circuits using the parts (L3-C2-S1-D2) multiple times. For example in Figure 2.25 the parts (L4-C3D3-D4) were added in the triple-lift circuit. Because the voltage at the point of the joint (L4-C3) is positive value and higher than that at the point of the joint (L3-C2), so that we can use a diode D3 to replace the switch (S2). For multiple-lift circuits all further switches can be replaced by diodes. According to this principle, triple-lift circuits and quadruple-lift circuits were built as shown in Figure 2.25 and Figure 2.28. In this book it is not necessary to introduce the particular analysis and calculations one by one to readers. However, their formulas are shown in this section.
© 2006 by Taylor & Francis Group, LLC
Voltage-Lift Converters
111
30
24
MT
18
12
6
0 0
0.2
0.6
0.4
0.8
1
k
FIGURE 2.26 Voltage transfer gain MT vs. k.
2.3.4.1
Triple-Lift Circuit
A triple-lift circuit is shown in Figure 2.25, and it consists of two static switches S and S1; four inductors L1, L2, L3, and L4; and five capacitors C, C1, C2, C3, and CO; and five diodes. Capacitors C1, C2, and C3 perform characteristics to lift the capacitor voltage VC by three times the source voltage VI. L3 and L4 perform the function as ladder joints to link the three capacitors C1, C2, and C3 and lift the capacitor voltage VC up. Current iC1(t), iC2(t), and iC3(t) are exponential functions. They have large values at the moment of power on, but they are small because vC1 = vC2 = vC3 = VI in steady state. The output voltage and current are VO =
3 V 1− k I
(2.179)
IO =
1− k I 3 I
(2.180)
and
The voltage transfer gain in continuous mode is MT =
VO 3 = VI 1 − k
The curve of MT vs. k is shown in Figure 2.26.
© 2006 by Taylor & Francis Group, LLC
(2.181)
112
Essential DC/DC Converters
Other average voltages: VC = VO
VC1 = VC2 = VC3 = VI
Other average currents: IL2 = IO
I L1 =
k I 1− k O
I L 3 = I L 4 = I L1 + I L 2 =
1 I 1− k O
Current variations: ξ1 =
1− k R 2 MT fL1
ξ2 =
χ1 =
k R 2 MT fL2
k R MT 2 fL3
ζ=
χ2 =
k(1 − k )R k 3R = 2 MT fL MT 2 2 fL
k R MT 2 fL4
Voltage variations: ρ=
k 2 fCR
σ2 =
MT 2 fC2 R
σ1 =
MT 2 fC1R
σ3 =
MT 2 fC3 R
The variation ratio of output voltage vC is ε=
k 1 2 8 MT f CO L2
(2.182)
The output voltage ripple is very small. The boundary between continuous and discontinuous conduction modes is MT ≤ k
3R = 2 fL
3kz N 2
(2.183)
This boundary curve is shown in Figure 3.27. Comparing with Equations (2.95), (2.165) (2.169), and (2.183), it can be seen that the boundary curve has a minimum value of MT that is equal to 4.5, corresponding to k = 1/3 .
© 2006 by Taylor & Francis Group, LLC
Voltage-Lift Converters
113
40 continuous mode
MT
30 k=0.9
15
6
k=0.8
k=0.5
4.5 k=0.33 k=0.1 3
discontinuous mode
40 48
75
188
667
R/fL
FIGURE 2.27 The boundary between continuous and discontinuous modes and the output voltage vs. the normalized load zN = R/fL.
In discontinuous mode the current iD exists in the period between kT and [k + (1 – k)mT]T, where mT is the filling efficiency that is mT =
2 1 MT = ζ k 3R 2 fL
(2.184)
Considering Equation (2.183), therefore, 0 < mT < 1. Since the diode current iD becomes zero at t = kT + (1 – k)mTT, for the current iL1 kTVI = (1 – k)mTT(VC – 3VI – VL3–off – VL4–off) or VC = [3 +
with
k
2k k 2k R + ]VI = [3 + + k 2 (1 − k ) ]V 1 − k (1 − k )mT 1− k 6 fL I
3R 3 ≥ 2 fL 1 − k
and for the current iL2 kT(VI + VC – VO) = (1 – k)mTT(VO – 2VI – VL3–off – VL4–off) Therefore, output voltage in discontinuous mode is VO = [3 +
2k k 2k R + ]VI = [3 + + k 2 (1 − k ) ]V 1− k 6 fL I 1 − k (1 − k )mT
© 2006 by Taylor & Francis Group, LLC
114
Essential DC/DC Converters –V
+
Vs
iC
C +
S
+ VL2
–
L2
iL2
– c D iL1
VI N
VL1
D1 L1 iC1
D2 iL3 C1
D4
L3
iL4
iC2
C2
L4 iC3
D6
iL5 C3
+ R
L5
VO –
iC4
C4 CO
D3
D5 S1
+ VS1 –
FIGURE 2.28 Quadruple-lift circuit.
with
k
3R 3 ≥ 2 fL 1 − k
(2.185)
i.e., the output voltage will linearly increase during load resistance R increasing, as shown in Figure 2.27. 2.3.4.2
Quadruple-Lift Circuit
Quadruple-lift circuit shown in Figure 2.28 consists of two static switches S and S1; five inductors L1, L2, L3, L4, and L5; and six capacitors C, C1, C2, C3, C4, and CO; and seven diodes. Capacitors C1, C2, C3, and C4 perform characteristics to lift the capacitor voltage VC by four times the source voltage VI. L3, L4, and L5 perform the function as ladder joints to link the four capacitors C1, C2, C3, and C4 and lift the output capacitor voltage VC up. Current iC1(t), iC2(t), iC3(t), and iC4(t) are exponential functions. They have large values at the moment of power on, but they are small because vC1 = vC2 = vC3 = vC4 = VI in steady state. The output voltage and current are VO =
4 V 1− k I
(2.186)
IO =
1− k I 4 I
(2.187)
and
The voltage transfer gain in continuous mode is
© 2006 by Taylor & Francis Group, LLC
Voltage-Lift Converters
115
40
32
MQ
24
16
8
0
0
0.2
0.6
0.4
0.8
1
k
FIGURE 2.29 Voltage transfer gain MQ vs. k.
MQ =
VO 4 = VI 1 − k
(2.188)
The curve of MQ vs. k is shown in Figure 2.29. Other average voltages: VC = VO;
VC1 = VC2 = VC3 = VC4 = VI
Other average currents: IL2 = IO;
I L1 =
k I 1− k O
I L 3 = I L 4 = LL 5 = I L1 + I L 2 =
1 I 1− k O
Current variations: ξ1 =
1− k R 2 MQ fL1 χ1 =
ξ2 =
k R MQ 2 fL3
© 2006 by Taylor & Francis Group, LLC
k R 2 MQ fL2 χ2 =
ζ=
k R MQ 2 fL4
k(1 − k )R k 2R = 2 MQ fL MQ 2 fL χ3 =
k R MQ 2 fL5
116
Essential DC/DC Converters 60 50 40
continuous mode
k=0.9
30 k=0.8
MQ
20
10 8 6
k=0.5 k=0.33 discontinuous mode
k=0.1 2
54 64
100
250 R/fL
889
FIGURE 2.30 The boundary between continuous and discontinuous modes and the output voltage vs. the normalized load zN = R/fL.
Voltage variations: ρ=
σ2 =
MQ 2 fC2 R
k 2 fCR
σ3 =
σ1 =
MQ 2 fC1R
MQ 2 fC3 R
σ4 =
MQ 2 fC4 R
The variation ratio of output voltage VC is ε=
1 k 2 8 MQ f CO L2
(2.189)
The output voltage ripple is very small. The boundary between continuous and discontinuous modes is MQ ≤ k
2R = 2 kz N fL
(2.190)
This boundary curve is shown in Figure 2.30. Comparing with Equations (2.95), (2.127), (2.169), (2.183), and (2.190), it can be seen that this boundary curve has a minimum value of MQ that is equal to 6.0, corresponding to k = 1/3 .
© 2006 by Taylor & Francis Group, LLC
Voltage-Lift Converters
117
In discontinuous mode the current iD exists in the period between kT and [k + (1 – k)mQ]T, where mQ is the filling efficiency that is
mQ =
2 1 MQ = ζ k 2R fL
(2.191)
Considering Equation (2.190), therefore 0 < mQ < 1. Since the current iD becomes zero at t = kT + (1 – k)mQT, for the current iL1 we have kTVI = (1 – k)mQT(VC – 4VI – VL3–off – VL4–off – VL5–off) or VC = [4 +
3k k 3k R + ]VI = [4 + + k 2 (1 − k ) ]V 1 − k (1 − k )mQ 1− k 8 fL I
with
k
2R 4 ≥ fL 1 − k
and for current iL2 we have kT(VI + VC – VO) = (1 – k)mQT(VO – 2VI – VL3–off – VL4–off – VL5–off) Therefore, output voltage in discontinuous mode is VO = [4 +
3k k 3k R + ]V = [4 + + k 2 (1 − k ) ]V 1 − k (1 − k )mQ I 1− k 8 fL I
with
k
2R 4 ≥ fL 1 − k
(2.192)
i.e., the output voltage will linearly increase during load resistance R increasing, as shown in Figure 2.30.
© 2006 by Taylor & Francis Group, LLC
118 2.3.5
Essential DC/DC Converters Summary
From the analysis and calculation in previous sections, the common formulas for all circuits can be obtained: VO I = I VI IO
M=
L=
L1L2 L1 + L2
zN =
R fL
R=
VO IO
Current variations: ξ1 =
1− k R 2 M fL1
k R 2 M fL2
χj =
k R 2 M fL j + 2
k 1 2 8 M f CO L2
σj =
M 2 fC j R
ξ2 =
(j = 1, 2, 3, …)
Voltage variations: ρ=
k 2 fCR
ε=
(j = 1, 2, 3, 4, …)
In order to write common formulas for the boundaries between continuous and discontinuous modes and output voltage for all circuits, the circuits can be numbered. The definition is that subscript 0 means the elementary circuit, subscript 1 means the self-lift circuit, subscript 2 means the re-lift circuit, subscript 3 means the triple-lift circuit, subscript 4 means the quadruple-lift circuit, and so on. The voltage transfer gain is Mj =
k h( j ) [ j + h( j )] 1− k
j = 0, 1, 2, 3, 4, …
(2.193)
The free-wheeling diode current iD’s variation is ζj =
k [1+ h( j )] j + h( j ) zN 2 Mj2
(2.194)
The boundaries are determined by the condition: ζj ≥ 1 or k [1+ h( j )] j + h( j ) zN ≥ 1 2 Mj2
© 2006 by Taylor & Francis Group, LLC
j = 0, 1, 2, 3, 4, …
(2.195)
Voltage-Lift Converters
119
Therefore, the boundaries between continuous and discontinuous modes for all circuits are Mj = k
1+ h ( j ) 2
j + h( j ) zN 2
j = 0, 1, 2, 3, 4,….
(2.196)
The filling efficiency is 2
Mj 1 2 1 mj = = [1+ h( j )] ζj k j + h( j ) z N
(2.197)
The output voltage in discontinuous mode for all circuits is VO− j = [ j +
j + h( j ) − 1 1− k + k [ 2−u( j )] z ]V 1− k 2[ j + h( j )] N I
j = 0, 1, 2, 3, 4,…. (2.198)
where ⎧0 h( j ) = ⎨ ⎩1
if if
j≥1 j=0
is the Hong Function
(2.199)
Assuming that f = 50 kHz, L1 = L2 = 1 mH, L2 = L3 = L4 = L5 = 0.5 mH, C = C1 = C2 = C3 = C4 = CO = 20 μF and the source voltage VI = 10 V, the value of the output voltage VO with various conduction duty k in continuous mode are shown in Figure 2.31. Typically, some values of the output voltage VO and its ripples in conduction duty k = 0.33, 0.5, 0.75 and 0.9 are listed in Table 2.1. From these data it states the fact that the output voltage of all Luoconverters is almost a real DC voltage with very small ripple. The boundaries between continuous and discontinuous modes of all circuits are shown in Figure 2.32. The curves of all M vs. zN state that the continuous mode area increases from ME via MS, MR, MT to MQ. The boundary of the elementary circuit is a monorising curve, but other curves are not monorising. There are minimum values of the boundaries of other circuits, which of MS, MR, MT and MQ correspond at k = 1/3 . 2.3.6
Discussion
Some important points are vital for particular circuit design. They are discussed in the following sections. 2.3.6.1
Discontinuous-Conduction Mode
Usually, the industrial applications require the DC-DC converters to work in continuous mode. However, it is irresistible that DC-DC converter works
© 2006 by Taylor & Francis Group, LLC
120
Essential DC/DC Converters 120
output voltage, vo,v
100 80
(i)
50 (ii) (iii)
30
(iv)
10
(v)
0.2
0
0.6 0.4 conduction duty k
0.8
1
FIGURE 2.31 Output voltages of all positive output Luo-converters (VI = 10 V).
TABLE 2.1 Comparison among Five Positive Output Luo-Converters Positive Output Luo-Converters
IO
Elementary Circuit
IO =
Self-Lift Circuit
I O = (1 − k )I I
Re-Lift Circuit
IO =
Triple-Lift Circuit Quadruple-Lift Circuit
VO
k = 0.33
VO (VI = 10 V) k = 0.5 k = 0.75
k = 0.9
VO =
k V 1− k I
5V
10 V
30 V
90 V
VO =
1 V 1− k I
15 V
20 V
40 V
100 V
1− k I 2 I
VO =
2 V 1− k I
30 V
40 V
80 V
200 V
IO =
1− k I 3 I
VO =
3 V 1− k I
45 V
60 V
120 V
300 V
IO =
1− k I 4 I
VO =
4 V 1− k I
60 V
80 V
160 V
400 V
1− k I k I
in discontinuous mode sometimes. The analysis in Section 2.3.2 through Section 2.3.5 shows that during switch-off if current iD becomes zero before next period switch-on, the state is called discontinuous mode. The following factors affect the diode current iD to become discontinuous: 1. 2. 3. 4.
Switch frequency f is too low Conduction duty cycle k is too small Combined inductor L is too small Load resistance R is too big
Discontinuous mode means iD is discontinuous during switch-off. The output current iO(t) is still continuous if L2 and CO are large enough.
© 2006 by Taylor & Francis Group, LLC
Voltage-Lift Converters
121
voltage transfer gain M
continuous mode
MQ MT
6
4.5
MR
3 Ms
1.5 ME discontinuous mode
0.5 44.5
27 40.5 54 13.5 normalized load zN = R.fL
FIGURE 2.32 Boundaries between continuous and discontinuous modes of all positive output Luo-converters.
2.3.6.2
Output Voltage VO vs. Conduction Duty k
Output voltage VO is a positive value and is usually greater than the source voltage VI when the conduction duty ratio is k > 0.5 for the elementary circuit, and any value in the range of 0 < k < 1 for self-lift, re-lift, and multiple-lift circuits. Although small k results that the output voltage VO of self-lift and re-lift circuits is greater than VI and 2VI and so on, when k = 0 it results in VO = 0 because switch S is never turned on. If k is close to the value of 1, the ideal output voltage VO should be a very big value. Unfortunately, because of the effect of parasitic elements, output voltage VO falls down very quickly. Finally, k = 1 results in VO = 0, not infinity for all circuits. In this case the accident of iL1 toward infinity will happen. The recommended value range of the conduction duty k is 0 < k < 0.9 2.3.6.3 Switch Frequency f In this paper the repeating frequency f = 50 kHz was selected. Actually, switch frequency f can be selected in the range between 10 kHz and 500 kHz. Usually, the higher the frequency, the lower the ripples.
2.4
Negative Output Luo-Converters
Negative output Luo-converters perform the voltage conversion from positive to negative voltages using VL technique. They work in the third quadrant with large voltage amplification. Five circuits have been introduced. They are
© 2006 by Taylor & Francis Group, LLC
122 • • • • •
Essential DC/DC Converters Elementary circuit Self-lift circuit Re-lift circuit Triple-lift circuit Quadruple-lift circuit
As the positive output Luo-converters, the negative output Luo-converters are another series of DC-DC step-up converters, which were developed from prototypes using voltage lift technique. These converters perform positive to negative DC-DC voltage increasing conversion with high power density, high efficiency, and cheap topology in simple structure. The elementary circuit can perform step-down and step-up DC-DC conversion. The other negative output Luo-converters are derived from this elementary circuit, they are the self-lift circuit, re-lift circuit, and multiplelift circuits (e.g., triple-lift and quadruple-lift circuits) shown in the corresponding figures and introduced in the next sections respectively. Switch S in these diagrams is a P-channel power MOSFET device (PMOS). It is driven by a PWM switch signal with repeating frequency f and conduction duty k. In this book the switch repeating period is T = 1/f, so that the switch-on period is kT and switch-off period is (1 – k)T. For all circuits, the load is usually resistive, i. e., R = VO/IO; the normalized load is zN = R/fL. Each converter consists of a negative Luo-pump and a “Π”-type filter C-LO-CO, and a lift circuit (except elementary circuit). The pump inductor L absorbs the energy from source during switch-on and transfers the stored energy to capacitor C during switch-off. The energy on capacitor C is then delivered to load during switch-on. Therefore, if the voltage VC is high the output voltage VO is correspondingly high. When the switch S is turned off the current iD flows through the freewheeling diode D. This current descends in whole switch-off period (1 – k)T. If current iD does not become zero before switch S is turned on again, we define this working state to be continuous mode. If current iD becomes zero before switch S is turned on again, we define this working state to be discontinuous mode. The directions of all voltages and currents are indicated in the figures. All descriptions and calculations in the text are concentrated to the absolute values. In this paper for any component X, its instantaneous current and voltage values are expressed as iX and vX, or iX(t) and vX(t), and its average current and voltage values are expressed as IX and VX. For general description, the output voltage and current are VO and IO; the input voltage and current are VI and II. Assuming the output power equals the input power, PO = PIN
or
VO I O = VI I I
The following symbols are used in the text of this paper.
© 2006 by Taylor & Francis Group, LLC
Voltage-Lift Converters
123
The voltage transfer gain is in CCM: M=
VO I = I VI IO
ζ=
ΔiL / 2 IL
ξ=
ΔiLO / 2 I LO
Variation ratio of current iL:
Variation ratio of current iLO:
Variation ratio of current iD: ζ=
ΔiD / 2 IL
during switch-off, iD = iL
Variation ratio of current iLj is χj =
ΔiLj / 2 I Lj
j = 1, 2, 3, …
Variation ratio of voltage vC: ρ=
ΔvC / 2 VC
Variation ratio of voltage vCj: σj =
ΔvCj / 2 VCj
j = 1, 2, 3, 4, …
Variation ratio of output voltage vO = vCO: ε=
© 2006 by Taylor & Francis Group, LLC
ΔvO / 2 VO
124
Essential DC/DC Converters iS
VIN
iD
+
VD
–VLO +
–
iL
+ Vs – + VL
D– L
iLO
C
VC +
–
LO
iC
Io iCo Co
– R Vo +
(a) circuit diagram
LO
+ L C
VIN
CO
R
–
(b) switch on D LO C
L
CO
R
(c) switch off
LO L
C
CO
R
(d) discontinuous mode FIGURE 2.33 Elementary circuit: (a) circuit diagram; (b) switch-on; (c) switch-off; (d) discontinuous mode.
2.4.1
Elementary Circuit
The elementary circuit, and its switch-on and -off equivalent circuits are shown in Figure 2.33. This circuit can be considered as a combination of an electronic pump S-L-D-C and a “Π”-type low-pass filter C-LO-CO. The electronic
© 2006 by Taylor & Francis Group, LLC
Voltage-Lift Converters
125
pump injects certain energy to the low-pass filter every cycle. Capacitor C in Figure 2.33 acts as the primary means of storing and transferring energy from the input source to the output load. Assuming capacitor C to be sufficiently large, the variation of the voltage across capacitor C from its average value VC can be neglected in steady state, i.e., vC(t) ≈ VC, even though it stores and transfers energy from the input to the output. 2.4.1.1 Circuit Description When switch S is on, the equivalent circuit is shown in Figure 2.33b. In this case the source current iI = iL. Inductor L absorbs energy from the source, and current iL linearly increases with slope VI/L. In the mean time the diode D is blocked since it is inversely biased. Inductor LO keeps the output current IO continuous and transfers energy from capacitor C to the load R, i.e., iC–on = iLO. When switch S is off, the equivalent circuit is shown in Figure 2.33c. In this case the source current iI = 0. Current iL flows through the freewheeling diode D to charge capacitor C and enhances current iLO. Inductor L transfers its stored energy to capacitor C and load R via inductor LO, i.e., iL = iC–off + iLO. Thus, currents iL decrease. 2.4.1.2 Average Voltages and Currents The output current IO = ILO because the capacitor CO does not consume any energy in the steady state. The average output current is IO = ILO = IC–on
(2.200)
The charge on the capacitor C increases during switch-off: Q+ = (1 – k) T IC–off And it decreases during switch-on: Q– = k T IC–on
(2.201)
In a whole repeating period T,
Q+ = Q–,
I C −off =
k k I C −on = I 1− k 1− k O
Therefore, the inductor current IL is I L = I C −off + I O =
© 2006 by Taylor & Francis Group, LLC
IO 1− k
(2.202)
126
Essential DC/DC Converters
Equation (2.200) and Equation (2.202) are available for all circuit of negative output Luo-converters. The source current is iI = iL during switch-on period. Therefore, its average source current II is I I = k × iI = kiL = kI L =
k I 1− k O
or IO =
1− k I k I
(2.203)
VO =
k V 1− k I
(2.204)
and the output voltage is
The voltage transfer gain in continuous mode is ME =
VO I k = I = VI IO 1 − k
(2.205)
The curve of ME vs. k is shown in Figure 2.34. Current iL increases and is supplied by VI during switch-on. It decreases and is inversely biased by –VC during switch-off, kTVI = (1 − k )TVC
(2.206)
k V 1− k I
(2.207)
Therefore, VC = VO =
2.4.1.3 Variations of Currents and Voltages To analyze the variations of currents and voltages, some voltage and current waveforms are shown in Figure 2.35. Current iL increases and is supplied by VI during switch-on. Thus, its peak-to-peak variation is ΔiL =
© 2006 by Taylor & Francis Group, LLC
kTVI L
Voltage-Lift Converters
127
10
8
ME
6
4
2
0 0
0.2
0.6
0.4
0.8
1
k
FIGURE 2.34 Voltage transfer gain ME vs. k.
Considering Equation (2.202) and Equation (2.205), and R = VO/IO, the variation ratio of the current iL is ζ=
ΔiL / 2 k(1 − k )VI T k(1 − k )R k2 R = = = IL 2LI O 2 ME fL ME 2 2 fL
(2.208)
Considering Equation (2.201), the peak-to-peak variation of voltage vC is ΔvC =
Q− k = TI O C C
(2.209)
The variation ratio of voltage vC is ρ=
ΔvC / 2 kI O T k 1 = = VC 2CVO 2 fCR
(2.210)
Since voltage VO variation is very small, the peak-to-peak variation of current iLO is calculated by the area (B) of the triangle with the width of T/ 2 and height ΔvC/2. ΔiLO =
B 1T k k = TI O = 2 I LO 2 2 2CLO 8 f CLO O
Considering Equation (2.200), the variation ratio of current iLO is
© 2006 by Taylor & Francis Group, LLC
(2.211)
128
Essential DC/DC Converters S on
DT
T
t
− VO LO
VI Lo
IO
A 0 IS
VI
L1
II
0 ILO
IL1
− VO L
VI L
S off
O
kII
S on
V 1 − L
IL
S off
0 IL2
t1
T
t
VI − V O L2 L2
IO t
VI L
0
t
IS L
VI
II+IO II
II 0
ID
t
− VO L
II+IO
0 ID
t − V L O
IO IO 0 ICO
t IL2
0
IC1
IO 0
t
-IL1
t
IL2
0 -IL1
t
-I1 VL1,VL2 VI
VL1,VL2 VI
VI
0
0
-VO
-VO
t
VS
VS
VO+VI
VO+VI
VI
VI
0
t
VD
t
-VO
0
t
VDR
VO+VI
VO+VI VO
VO 0
t
0
a
FIGURE 2.35 Some voltage and current waveforms of elementary circuit.
© 2006 by Taylor & Francis Group, LLC
t b
Voltage-Lift Converters
129
ξ=
ΔiLO / 2 k 1 = I LO 16 f 2 CLO
(2.212)
Since the voltage vC is a triangle waveform, the difference between vC and output voltage VO causes the ripple of current iLO, and the difference between iLO and output current IO causes the ripple of output voltage vO. The ripple waveform of current iLO should be a partial parabola in Figure 2.35 because of the triangle waveform of ΔvC. To simplify the calculation we can treat the ripple waveform of current iLO as a triangle waveform in Figure 2.35 because the ripple of the current iLO is very small. Therefore, the peak-to-peak variation of voltage vCO is calculated by the area (A) of the triangle with the width of T/2 and height ΔiLO/2: ΔvCO =
A 1T k k = I = I CO 2 2 16 f 2 CCO LO O 64 f 3 CCO LO O
(2.213)
The variation ratio of current vCO is ε=
ΔvCO / 2 IO 1 k k = = 3 128 f CCO LO VO 128 f 3 CCO LO R VCO
(2.214)
Assuming that f = 50 kHz, L = LO = 100 μH, C = CO = 5 μF, R = 10 Ω and k = 0.6, we obtain
ΜΕ = 1.5 ζ = 0.16 ζ = 0.03 ρ = 0.12 and ε = 0.0015 The output voltage VO is almost a real DC voltage with very small ripple. Since the load is resistive, the output current iO(t) is almost a real DC waveform with very small ripple as well, and it is equal to IO = VO/R. 2.4.1.4 Instantaneous Values of Currents and Voltages Referring to Figure 2.35, the instantaneous current and voltage values are listed below: ⎧0 vS = ⎨ ⎩VO
for for
⎧V + VO vD = ⎨ I ⎩ 0 ⎧ VI vL = ⎨ ⎩−VO
© 2006 by Taylor & Francis Group, LLC
0 < t ≤ kT kT < t ≤ T for for
for for
0 < t ≤ kT kT < t ≤ T 0 < t ≤ kT kT < t ≤ T
(2.215)
(2.216)
(2.217)
130
Essential DC/DC Converters V ⎧ ⎪i (0) + I t iI = iS = ⎨ L L 0 ⎪⎩
0 < t ≤ kT
for for
kT < t ≤ T
V ⎧ i L ( 0) + I t ⎪ L iL = ⎨ VO ⎪iL ( kT ) − (t − kT ) L ⎩
for for
0 < t ≤ kT kT < t ≤ T
0 ⎧ ⎪ VO iD = ⎨ ⎪⎩iL ( kT ) − L (t − kT )
for for
0 < t ≤ kT
⎧− I C −on iC ≈ ⎨ I ⎩ C −off
for for
0 < t ≤ kT kT < t ≤ T
kT < t ≤ T
(2.218)
(2.219)
(2.220)
(2.221)
where iL(0) = k II – k VI/2 f L iL(kT) = k II + k VI/2 f L Since the instantaneous current iLO and voltage vCO are partial parabolas with very small ripples, they can be treated as a DC current and voltage. 2.4.1.5 Discontinuous Mode Referring to Figure 2.33d, we can see that the diode current iD becomes zero during switch off before next period switch on. The condition for discontinuous mode is ζ≥1 i.e., k2 R ≥1 ME 2 2 fL or ME ≤ k
© 2006 by Taylor & Francis Group, LLC
z R =k N 2 fL 2
(2.222)
Voltage-Lift Converters
131
20 continuous mode 10 k=0.9 5 ME
k=0.7 2 1
k=0.5
0.5 k=0.3 0.2
discontinuous mode k=0.1
0.1 1
2
5
10
20
50
100
200
500 1000
R/fL
FIGURE 2.36 The boundary between continuous and discontinuous modes and output voltage vs. the normalized load zN = √R/fL (elementary circuit).
The graph of the boundary curve vs. the normalized load zN = R/fL is shown in Figure 2.36. It can be seen that the boundary curve is a monorising function of the parameter k. In this case the current iD exists in the period between kT and t1 = [k + (1 – k)mE]T, where mE is the filling efficiency and it is defined as: 2
mE =
ME 1 = ζ k2 R 2 fL
(2.223)
Considering Equation (2.222), therefore 0 < mE < 1. Since the diode current iD becomes zero at t = kT + (1 – k)mET, for the current iL kTVI = (1 – k)mETVC or VC =
k R VI = k(1 − k ) V (1 − k )mE 2 fL I
with
R 1 ≥ 2 fL 1 − k
and for the current iLO kT(VI + VC – VO) = (1 – k)mETVO
© 2006 by Taylor & Francis Group, LLC
132
Essential DC/DC Converters
Therefore, output voltage in discontinuous mode is VO =
k R VI = k(1 − k ) V (1 − k )mE 2 fL I
with
R 1 ≥ 2 fL 1 − k
(2.224)
i.e., the output voltage will linearly increase during load resistance R increasing. The output voltage vs. the normalized load zN = R/fL is shown in Figure 2.36. Larger load resistance R may cause higher output voltage in discontinuous mode.
2.4.2
Self-Lift Circuit
Self-lift circuit, and its switch-on and -off equivalent circuits are shown in Figure 2.37, which is derived from the elementary circuit. It consists of eight passive components. They are one static switch S; two inductors L, LO; three capacitors C, C1, and CO; and two diodes D, D1. Comparing with Figure 2.33 and Figure 2.37, it can be seen that there are only one more capacitor C1 and one more diode D1 added into the self-lift circuit. Circuit C1-D1 is the lift circuit. Capacitor C1 functions to lift the capacitor voltage VC by a source voltage VI. Current iC1(t) is an exponential function δ(t). It has a large value at the moment of power on, but it is small in the steady state because VC1 = VI. 2.4.2.1 Circuit Description When switch S is on, the equivalent circuit is shown in Figure 2.37b. In this case the source current iI = iL + iC1. Inductor L absorbs energy from the source, and current iI linearly increases with slope VI/L. In the mean time the diode D1 is conducted and capacitor C1 is charged by the current iC1. Inductor LO keeps the output current IO continuous and transfers energy from capacitor C to the load R, i.e., iC–on = iLO. When switch S is off, the equivalent circuit is shown in Figure 2.37c. In this case the source current iI = 0. Current iL flows through the free-wheeling diode D to charge capacitor C and enhances current iLO. Inductor L transfers its stored energy via capacitor C1 to capacitor C and load R (via inductor LO), i.e., iL = iC1–off = iC–off + iLO. Thus, current iL decreases. 2.4.2.2 Average Voltages and Currents The output current IO = ILO because the capacitor CO does not consume any energy in the steady state. The average output current: IO = ILO = IC–on The charge of the capacitor C increases during switch-off:
© 2006 by Taylor & Francis Group, LLC
(2.225)
Voltage-Lift Converters
S iIN VI iL
N
133
+ VC1–
+ VD –
iC1
iD D iD1 – VC +
C1
–
L VD1 +
– V + LO iC
LO
iO iLO
C
iCO CO
– R VO +
(a) circuit diagram
LO
+ VIN
L
C1
C
CO
R
–
(b) switch on C1 LO L
C
CO
R
CO
R
(c) switch off C1 LO L
C
(d) discontinuous mode FIGURE 2.37 Self-lift circuit: (a) circuit diagram; (b) switch on; (c) switch off; (d) discontinuous mode.
© 2006 by Taylor & Francis Group, LLC
134
Essential DC/DC Converters Q+ = (1 – k) T IC–off
And it decreases during switch-on: Q– = k T IC–on
(2.226)
In a whole repeating period T, Q+ = Q–. Thus, I C −off =
k k I C −on = I 1− k 1− k O
Therefore, the inductor current IL is IO 1− k
(2.227)
1 I 1− k O
(2.228)
I L = I C −off + I O = From Figure 2.37, I C1−off = I L = and I C1−on =
1− k 1 I C1−off = I O k k
(2.229)
In steady state we can use VC1 = VI Investigate current iL, it increases during switch–on with slope VI/L and decreases during switch-off with slope –(VO – VC1)/L = –(VO – VI)/L. Therefore, kVI = (1 − k )(VO − VI ) or VO =
© 2006 by Taylor & Francis Group, LLC
1 V 1− k I
(2.230)
Voltage-Lift Converters
135
10
8
MS
6
4
2
0
0
0.2
0.6
0.4
0.8
1
k
FIGURE 2.38 Voltage transfer gain MS vs. k.
and I O = (1 − k )I I
(2.231)
The voltage transfer gain in continuous mode is MS =
VO I 1 = I = VI IO 1 − k
(2.232)
The curve of MS vs. k is shown in Figure 2.38. Circuit (C-LO-CO) is a “Π” type low-pass filter. Therefore, VC = VO =
k V 1− k I
(2.233)
2.4.2.3 Variations of Currents and Voltages To analyze the variations of currents and voltages, some voltage and current waveforms are shown in Figure 2.39. Current iL increases and is supplied by VI during switch-on. Thus, its peakto-peak variation is ΔiL =
© 2006 by Taylor & Francis Group, LLC
kTVI L
136
Essential DC/DC Converters S on VI L
IO
VI
L1
II
0 ILO
IL1
− VO – VI L
S off
VI – VO L 1
kII
S on
−
IL
S off
kT VI LO
−
T
t
VO – VI LO
0 IL2
t1
T
t
V VI − O – V L2 L2 I
ΔQ
IO 0 II=IS
t
t) VI + d( L
0
t
II=IS
t)
+
I
V
II
d(
L
II
0 ID
−
t
VO – VI
L
0 ID
t −
VO
L
IO
–
VI
IO 0
t
ID1 IO k
d(t) IO k
0 IC1
0 ID
t
t
d(t)
0
t
d(t)
IO k
IC1 0 -(II+IO)
IC
d(t)
0 t
VO – V-I L
VO – V-I L
IL2
IC1
IO 0
t
-IL1
t
IL2
0 -IL1
t
-I1 VL1,VL2 VI
VL2 VI
VI
0
0 -VO
t
VS
VS
VO
VO
VI
VI
0
t
0
VD1
VD1
VO
VO
(VO-VI)
VI
0
t
VD
t
-(VO-VI)
t
0
t
VD
VO
VO kVO
kVO 0
t
0
a
FIGURE 2.39 Some voltage and current waveforms of self-lift circuit.
© 2006 by Taylor & Francis Group, LLC
t
b
Voltage-Lift Converters
137
Considering Equation (2.227) and R = VO/IO, the variation ratio of the current iL is ζ=
ΔiL / 2 k(1 − k )VI T k(1 − k )R k R = = = IL 2LI O 2 MS fL MS 2 2 fL
(2.234)
Considering Equation (2.226), the peak-to-peak variation of voltage vC is ΔvC =
Q− k = TI O C C
The variation ratio of voltage vC is ρ=
ΔvC / 2 kI O T k 1 = = VC 2CVO 2 fCR
(2.235)
The peak-to-peak variation of voltage vC1 is ΔvC1 =
1 kT = I I C1 C1−on fC O
The variation ratio of voltage vC1 is σ1 =
ΔvC1 / 2 IO M 1 = = S VC1 2 fC1VI 2 fC1R
(2.236)
Considering the Equation (2.211): ΔiLO =
1T k k TI O = 2 I 2 2 2CLO 8 f CLO O
The variation ratio of current iLO is ξ=
ΔiLO / 2 k 1 = 2 I LO 16 f CLO
Considering Equation (2.213): ΔvCO =
B 1T k k = IO = I 2 3 CO 2 2 16 f CCO LO 64 f CCO LO O
© 2006 by Taylor & Francis Group, LLC
(2.237)
138
Essential DC/DC Converters
The variation ratio of current vCO is ε=
ΔvCO / 2 IO 1 k k = = 128 f 3 CCO LO VO 128 f 3 CCO LO R VCO
(2.238)
Assuming that f = 50 kHz, L = LO = 100 μH, C = CO = 5 μF, R = 10 Ω and k = 0.6, we obtain MS = 2.5 ζ = 0.096 ξ = 0.03 ρ = 0.12 and ε = 0.0015 The output voltage VO is almost a real DC voltage with very small ripple. Since the load is resistive, the output current iO(t) is almost a real DC waveform with very small ripple as well, and it is equal to IO = VO/R. 2.4.2.4 Instantaneous Value of the Currents and Voltages Referring to Figure 2.39, the instantaneous values of the currents and voltages are listed below: ⎧ 0 vS = ⎨ ⎩VO − VI
0 < t ≤ kT kT < t ≤ T
for for
0 < t ≤ kT
⎧V vD = ⎨ O ⎩0
for for
kT < t ≤ T
⎧0 vD1 = ⎨ ⎩VO
for for
0 < t ≤ kT kT < t ≤ T
VI ⎧ vL = ⎨ ⎩−(VO − VI )
for for
V ⎧ ⎪i (0) + δ(t) + I t iI = iS = ⎨ L1 L 0 ⎪⎩ V ⎧ i L ( 0) + I t ⎪ L iL = ⎨ V − VI ⎪iL ( kT ) − O (t − kT ) L ⎩
© 2006 by Taylor & Francis Group, LLC
(2.239)
(2.240)
(2.241)
0 < t ≤ kT kT < t ≤ T
for for
for for
(2.242)
0 < t ≤ kT kT < t ≤ T
0 < t ≤ kT kT < t ≤ T
(2.243)
(2.244)
Voltage-Lift Converters
139
0 ⎧ ⎪ VO − VI iD = ⎨ (t − kT ) ⎪⎩iL1 ( kT ) − L ⎧δ(t) iD1 = ⎨ ⎩ 0
for for
0 < t ≤ kT kT < t ≤ T
for for
δ(t) ⎧ ⎪ VO − VI iC 1 = ⎨ (t − kT ) ⎪⎩−iL ( kT ) + L ⎧− I C −on iC ≈ ⎨ I ⎩ C −off
0 < t ≤ kT kT < t ≤ T
for for
0 < t ≤ kT kT < t ≤ T
(2.245)
(2.246)
for for
0 < t ≤ kT kT < t ≤ T
(2.247)
(2.248)
where iL(0) = k II – k VI/2 f L iL(kT) = k II + k VI/2 f L Since the instantaneous current iLO and voltage vCO are partial parabolas with very small ripples, they can be treated as a DC current and voltage. 2.4.2.5 Discontinuous Mode Referring to Figure 2.37d, we can see that the diode current iD becomes zero during switch off before next period switch on. The condition for discontinuous mode is ζ≥1 i.e., k R ≥1 MS 2 2 fL or MS ≤ k
© 2006 by Taylor & Francis Group, LLC
z R = k N 2 fL 2
(2.249)
140
Essential DC/DC Converters
The graph of the boundary curve vs. the normalized load zN = R/fL is shown in Figure 2.40. It can be seen that the boundary curve has a minimum value of 1.5 at k = 1/3 . In this case the current iD exists in the period between kT and t1 = [k + (1 – k)mS]T, where mS is the filling efficiency and it is defined as: mS =
MS 2 1 = ζ k R 2 fL
(2.250)
Considering Equation (2.249), therefore 0 < mS < 1. Since the diode current iD becomes 0 at t = kT + (1 – k)mST, for the current iL kTVI = (1 – k)mST(VC – VI) or VC = [1 +
k R ]V = [1 + k 2 (1 − k ) ]V (1 − k )mS I 2 fL I
k
with
R 1 ≥ 2 fL 1 − k
and for the current iLO kT(VI + VC – VO) = (1 – k)mST(VO – VI) Therefore, output voltage in discontinuous mode is VO = [1 +
k R ]VI = [1 + k 2 (1 − k ) ]V (1 − k )mS 2 fL I
with
k
R 1 ≥ 2 fL 1 − k
(2.251)
i.e., the output voltage will linearly increase during load resistance R increasing. The output voltage VO vs. the normalized load zN = R/fL is shown in Figure 2.40. Larger load resistance R causes higher output voltage in discontinuous mode.
2.4.3
Re-Lift Circuit
Re-lift circuit, and its switch-on and -off equivalent circuits are shown in Figure 2.41, which is derived from the self-lift circuit. It consists of one static switch S; three inductors L, L1, and LO; four capacitors C, C1, C2, and CO; and diodes. From Figure 2.33, Figure 2.37, and Figure 2.41, it can be seen that there are one capacitor C2, one inductor L1 and two diodes D2, D11 added into the re-lift circuit. Circuit C1-D1-D11-L1-C D2 is the lift circuit. Capacitors C1 and C2 perform characteristics to lift the capacitor voltage VC by twice
© 2006 by Taylor & Francis Group, LLC
Voltage-Lift Converters
141
30 continuous mode
20 k=0.9 10
MS
8 k=0.8 5 3 2
k=0.5 k=0.33
1.5 k=0.1 1
13.56
discontinuous mode 24.7
62.5 R/fL
222
842
FIGURE 2.40 The boundary between continuous and discontinuous modes and output voltage vs. the normalized load zN = R/fL (self-lift circuit).
that of source voltage 2VI. Inductor L1 performs the function as a ladder joint to link the two capacitors C1 and C2 and lift the capacitor voltage VC up. Currents iC1(t) and iC2(t) are exponential functions δ1(t) and δ2(t). They have large values at the moment of power on, but they are small because vC1 = vC2 ≅ VI is in steady state. 2.4.3.1 Circuit Description When switch S is on, the equivalent circuit is shown in Figure 2.41b. In this case the source current iI = iL + iC1 + iC2. Inductor L absorbs energy from the source, and current iL linearly increases with slope VI/L. In the mean time the diodes D1, D2 are conducted so that capacitors C1 and C2 are charged by the current iC1 and iC2. Inductor LO keeps the output current IO continuous and transfers energy from capacitors C to the load R, i.e., iC–on = iLO. When switch S is off, the equivalent circuit is shown in Figure 2.41c. In this case the source current iI = 0. Current iL flows through the free-wheeling diode D, capacitors C1 and C2, inductor L1 to charge capacitor C and enhances current iLO. Inductor L transfers its stored energy to capacitor C and load R via inductor LO, i.e., iL = iC1–off = iC2–off = iL1–off = iC–off + iLO. Thus, current iL decreases. 2.4.3.2 Average Voltages and Currents The output current IO = ILO because the capacitor CO does not consume any energy in the steady state. The average output current: IO = ILO = IC–on
© 2006 by Taylor & Francis Group, LLC
(2.252)
142
Essential DC/DC Converters i D1 1
i C1
iD
– V LO
S i IN
–V + S
i L1
D 11
D
C1
LO
iC
iO
+ i LO
i CO
L1
D 10
– C2
VI
C
R VO
CO
N
iL
+
L D2
D1
(a) circuit diagram
LO
+ VIN
C1
L
L1
C2
C
CO
R
–
(b) switch on C2
C1 L1
LO
L
CO
C
R
(c) switch off C1
C2 LO
L1 L
C
CO
R
(d) discontinuous mode
FIGURE 2.41 Re-lift circuit: (a) circuit diagram; (b) switch on; (c) switch off; (d) discontinuous mode.
The charge of the capacitor C increases during switch-off: Q+ = (1 – k) T IC–off And it decreases during switch-on:
© 2006 by Taylor & Francis Group, LLC
Voltage-Lift Converters
143 Q– = k T IC–on
In a whole repeating period T, Q+ = Q–. Thus, I C −off =
k k I = I 1 − k C −on 1 − k O
Therefore, the inductor current IL is I L = I C −off + I O =
IO 1− k
(2.253)
We know from Figure 2.48b that I C1−off = I C 2−off = I L1 = I L =
1 I 1− k O
(2.254)
and I C1−on =
1− k 1 I C1−off = I O k k
(2.255)
I C 2−on =
1− k 1 I C 2−off = I O k k
(2.256)
and
In steady state we can use VC1 = VC 2 = VI and VL1−on = VI
VL1−off =
k V 1− k I
Investigate current iL, it increases during switch-on with slope VI/L and decreases during switch-off with slope –(V O – V C1 – V C2 – V L1–off )/L = – [VO – 2VI – k VI/(1 – k)]/L. Therefore, kTVI = (1 − k )T (VO − 2VI −
© 2006 by Taylor & Francis Group, LLC
k V) 1− k I
144
Essential DC/DC Converters 20
16
MR
12
8
4
0
0
0.2
0.6
0.4
0.8
1
k
FIGURE 2.42 Voltage transfer gain MR vs. k.
or VO =
2 V 1− k I
(2.257)
IO =
1− k I 2 I
(2.258)
and
The voltage transfer gain in continuous mode is MR =
VO I 2 = I = VI IO 1 − k
(2.259)
The curve of MS vs. k is shown in Figure 2.42. Circuit (C-LO-CO) is a “Π” type low-pass filter. Therefore, VC = VO =
2 V 1− k I
(2.260)
2.4.3.3 Variations of Currents and Voltages To analyze the variations of currents and voltages, some voltage and current waveforms are shown in Figure 2.43.
© 2006 by Taylor & Francis Group, LLC
Voltage-Lift Converters
145 S on
S off
(i)
(ii)
IL K I1 2 0 ILO
kT
S on IL1
T
(iii)
K I1 2 0 IL2
t
(iv)
S off
(i)
(ii)
T1
T
(iv)
IO 0 II=I S
0
t
(v)
t
t
II=I S (v) II
0 ID
(vi)
0 ID
t
(vii)
t (viii)
IO IO 0 ID1
t
0 ID
t
0
d(t)
t
d(t)
0
t
a
b
d'(t) IC1
IC1
0
d' (t)
0 t
(i)
t (ii)
(ii) IC
IL2
IC1
IO 0
t
-IL2
-I 1 V L1 ,V L2 VI
IL2
0
-I L2
t
V L2 VI
VI
0
0
-(V O-2V I)
-V O
t
t
-(V O-2V I) VS
VS
V O-V I VI
VI
0
t
V D1
0
t
V D1
VO
V O -V I V O-V I 0 VD
t
0 VD
t
VO
VO
kV O kV O 0
a
t
0
FIGURE 2.43 Some voltage and current waveforms of re-lift circuit.
© 2006 by Taylor & Francis Group, LLC
b
t
146
Essential DC/DC Converters
Current iL increases and is supplied by VI during switch-on. Thus, its peakto-peak variation is ΔiL =
kTVI L
Considering Equation (2.253) and R = VO/IO, the variation ratio of the current iL is ζ=
ΔiL / 2 k(1 − k )VI T k(1 − k )R k R = = = IL 2LI O 2 M R fL M R 2 fL
(2.261)
The peak-to-peak variation of current iL1 is ΔiL1 =
k TVI L1
The variation ratio of current iL1 is χ1 =
ΔiL1 / 2 kTVI k(1 − k ) R = (1 − k ) = I L1 2L1 I O 2 M R fL1
(2.262)
The peak-to-peak variation of voltage vC is ΔvC =
Q− k = TI O C C
The variation ratio of voltage vC is ρ=
ΔvC / 2 kI O T k 1 = = VC 2CVO 2 fCR
(2.263)
The peak-to-peak variation of voltage vC1 is ΔvC1 =
1 kT = I I C1 C1−on fC O
The variation ratio of voltage vC1 is σ1 =
© 2006 by Taylor & Francis Group, LLC
ΔvC1 / 2 IO M 1 = = R VC1 2 fC1VI 2 fC1R
(2.264)
Voltage-Lift Converters
147
Take the same operation, variation ratio of voltage vC2 is σ2 =
ΔvC 2 / 2 IO M 1 = = R 2 fC2VI 2 fC2 R VC 2
(2.265)
Considering the Equation (2.211): ΔiLO =
1T k k TI = I 2 2 2CLO O 8 f 2 CLO O
The variation ratio of current iLO is ξ=
ΔiLO / 2 1 k = 16 f 2 CLO I LO
(2.266)
Considering the Equation (2.213): ΔvCO =
B 1T k k = IO = I 2 3 CO 2 2 16 f CCO LO 64 f CCO LO O
The variation ratio of current vCO is ε=
ΔvCO / 2 IO 1 k k = = 3 3 128 f CCO LO VO 128 f CCO LO R VCO
(2.267)
Assuming that f = 50 kHz, L = LO = 100 μH, C = CO = 5 μF, R = 10 Ω and k = 0.6, we obtain MR = 5
ζ = 0.048 ξ = 0.03 ρ = 0.12
and ε = 0.0015
The output voltage VO is almost a real DC voltage with very small ripple. Since the load is resistive, the output current iO(t) is almost a real DC waveform with very small ripple as well, and it is equal to IO = VO/R. 2.4.3.4 Instantaneous Value of the Currents and Voltages Referring to Figure 2.43, the instantaneous current and voltage values are listed below: 0 ⎧⎪ k vS = ⎨ VO − (2 − )V 1− k I ⎩⎪
© 2006 by Taylor & Francis Group, LLC
for for
0 < t ≤ kT kT < t ≤ T
(2.268)
148
Essential DC/DC Converters
⎧V vD = ⎨ O ⎩0
for for
0 < t ≤ kT kT < t ≤ T
(2.269)
⎧0 vD1 = vD 2 = ⎨ ⎩VO
for for
0 < t ≤ kT kT < t ≤ T
(2.270)
⎧⎪ VI k v L1 = ⎨ − VI ⎩⎪ 1 − k
for for
0 < t ≤ kT kT < t ≤ T
(2.271)
VI ⎧⎪ k vL = ⎨ )V ] −[V − (2 − ⎪⎩ O 1− k I
for for
0 < t ≤ kT
(2.272)
kT < t ≤ T
V V ⎧ ⎪i (0) + I t + δ 1 (t) + δ 2 (t) + iL1 (0) + I t for 0 < t ≤ kT iI = iS = ⎨ L L L1 ⎪⎩ for kT < t ≤ T 0
(2.273)
V ⎧ i L ( 0) + I t ⎪ L ⎪ k iL = ⎨ )V VO − (2 − ⎪ 1 − k I (t − kT ) ⎪⎩iL ( kT ) − L
(2.274)
iL1
V ⎧ i L 1 ( 0) + I t ⎪ L1 ⎪ k =⎨ V ⎪ 1 − k I (t − kT ) − i kT ( ) ⎪ L1 L1 ⎩
0 ⎧ ⎪ k iD = ⎨ )V VO − (2 − 1 − k I (t − kT ) ⎪iL ( kT ) − L ⎩
for for
for for
0 < t ≤ kT kT < t ≤ T
0 < t ≤ kT kT < t ≤ T
for for
0 < t ≤ kT kT < t ≤ T
(2.275)
(2.276)
⎧δ (t) iD1 = ⎨ 1 ⎩ 0
for for
0 < t ≤ kT kT < t ≤ T
(2.277)
⎧δ (t) iD2 = ⎨ 2 ⎩ 0
for for
0 < t ≤ kT kT < t ≤ T
(2.278)
© 2006 by Taylor & Francis Group, LLC
Voltage-Lift Converters
149
iC 1
δ 1 (t ) ⎧ ⎪ k =⎨ )V VO − (2 − 1 − k I (t − kT ) ⎪−iL1 ( kT ) + L ⎩
for for
0 < t ≤ kT kT < t ≤ T
(2.279)
iC 2
δ 2 (t ) ⎧ ⎪ k =⎨ VO − (2 − )V 1 − k I (t − kT ) ⎪−iL1 ( kT ) + L ⎩
for for
0 < t ≤ kT kT < t ≤ T
(2.280)
⎧− I C −on iC = ⎨ I ⎩ C −off
for for
0 < t ≤ kT kT < t ≤ T
(2.281)
where iL(0) = k II – k VI/2 f L iL(kT) = k II + k VI/2 f L and iL1(0) = k II – k VI/2 f L1 iL1(kT) = k II + k VI/2 f L1 Since the instantaneous currents iLO and iCO are partial parabolas with very small ripples. They are very nearly DC current. 2.4.3.5 Discontinuous Mode Referring to Figure 2.41d, we can see that the diode current iD becomes zero during switch off before next period switch on. The condition for discontinuous mode is ζ≥1 i.e., k R ≥1 M R 2 fL or MR ≤ k
© 2006 by Taylor & Francis Group, LLC
R = k zN fL
(2.282)
150
Essential DC/DC Converters 60 k=0.95
continuous mode
40 k=0.9
MR
20 k=0.8 10 6 4 3
k=0.5 k=0.33 k=0.1
2
27 32
discontinuous mode 50
125 R/fL
444
1684
FIGURE 2.44 The boundary between continuous and discontinuous modes and output voltage vs. the normalized load zN = R/fL (re-lift circuit).
The graph of the boundary curve vs. the normalized load zN = R/fL is shown in Figure 2.44. It can be seen that the boundary curve has a minimum value of 3.0 at k = 1/3 . In this case the current iD exists in the period between kT and t1 = [k + (1 – k)mR]T, where mR is the filling efficiency and it is defined as: mR =
2 1 MR = ζ k R fL
(2.283)
Considering Equation (2.282), therefore 0 < mR < 1. Because inductor current iL1 = 0 at t = t1, so that VL1−off =
k V (1 − k )mR I
Since the current iD becomes zero at t = t1 = [k + (1 – k)mR]T, for the current iL kTVI = (1 – k)mRT(VC – 2VI – VL1–off) or VC = [2 +
2k R ]V = [2 + k 2 (1 − k ) ]V (1 − k )mR I 2 fL I
© 2006 by Taylor & Francis Group, LLC
with
k
R 2 ≥ fL 1 − k
Voltage-Lift Converters
151 iC1
D11
iIN
iD
iC2
S –V + S
D12
D
C2 L2 iL2
D10
iO
– VLO +
C1 iC
LO
iLO i CO
L1 iL1
–
C3
VI
C
CO
N
R VO +
L iL
D2
D3 iD3
iD2
D1 iD1
FIGURE 2.45 Triple-lift circuit.
and for the current iLO kT(VI + VC – VO) = (1 – k)mRT(VO – 2VI – VL1–off) Therefore, output voltage in discontinuous mode is VO = [2 +
2k R ]V = [2 + k 2 (1 − k ) ]V (1 − k )mR I 2 fL I
with
k
R 2 ≥ fL 1 − k
(2.284)
i.e., the output voltage will linearly increase during load resistance R increasing. The output voltage vs. the normalized load zN = R/fL is shown in Figure 2.44. Larger load resistance R may cause higher output voltage in discontinuous mode.
2.4.4
Multiple-Lift Circuits
Referring to Figure 2.45, it is possible to build a multiple-lift circuit just only using the parts (L1-C2-D2-D11) multiple times. For example, in Figure 2.16 the parts (L2-C3-D3-D12) were added in the triple-lift circuit. According to this principle, the triple-lift circuit and quadruple-lift circuit were built as shown in Figure 2.45 and Figure 2.48. In this book it is not necessary to introduce the particular analysis and calculations one by one to readers. However, their formulas are shown in this section. 2.4.4.1 Triple-Lift Circuit Triple-lift circuit is shown in Figure 2.45. It consists of one static switch S; four inductors L, L1, L2, and LO; and five capacitors C, C1, C2, C3, and CO; and
© 2006 by Taylor & Francis Group, LLC
152
Essential DC/DC Converters 30
24
MT
18
12
6
0 0
0.2
0.6
0.4
0.8
1
k
FIGURE 2.46 Voltage transfer gain of triple-lift circuit.
diodes. Circuit C1-D1-L1-C2-D2-D11-L2-C3-D3-D12 is the lift circuit. Capacitors C1, C2, and C3 perform characteristics to lift the capacitor voltage VC by three times that of the source voltage VI. L1 and L2 perform the function as ladder joints to link the three capacitors C1, C2, and C3 and lift the capacitor voltage VC up. Current iC1(t), iC2(t), and iC3(t) are exponential functions. They have large values at the moment of power on, but they are small because vC1 = vC2 = vC3 ≅ VI in steady state. The output voltage and current are VO =
3 V 1− k I
(2.285)
IO =
1− k I 3 I
(2.286)
and
The voltage transfer gain in continuous mode is MT = VO / VI =
3 1− k
The curve of MT vs. k is shown in Figure 2.46. Other average voltages: VC = VO
© 2006 by Taylor & Francis Group, LLC
VC1 = VC2 = VC3 = VI
(2.287)
Voltage-Lift Converters
153
Other average currents: I L = I L1 = I L 2 =
ILO = IO
1 I 1− k O
Current variation ratios: k 3R MT 2 2 fL
ζ=
χ1 =
k 1 16 f 2 CLO
ξ=
k(1 − k ) R 2 MT fL1
χ2 =
k(1 − k ) R 2 MT fL2
Voltage variation ratios: ρ=
σ2 =
k 1 2 fCR
σ1 =
MT 1 2 fC2 R
MT 1 2 fC1R
σ3 =
MT 1 2 fC3 R
The variation ratio of output voltage VC is ε=
k 1 3 128 f CCO LO R
(2.288)
The output voltage ripple is very small. The boundary between continuous and discontinuous modes is MT ≤ k
3R = 2 fL
3kz N 2
(2.289)
It can be seen that the boundary curve has a minimum value of MT that is equal to 4.5, corresponding to k = 1/3 . The boundary curve vs. the normalized load zN = R/fL is shown in Figure 2.47. In discontinuous mode the current iD exists in the period between kT and [k + (1 – k)mT]T, where mT is the filling efficiency that is mT =
© 2006 by Taylor & Francis Group, LLC
2 1 MT = ζ k 3R 2 fL
(2.290)
154
Essential DC/DC Converters
40 continuous mode
MT
30 k=0.9
15
6
k=0.8
k=0.5
4.5 k=0.33 k=0.1 3
discontinuous mode
40 48
75
188
667
R/fL
FIGURE 2.47 The boundary between continuous and discontinuous modes and output voltage vs. the normalized load zN = R/fL (triple-lift circuit).
Considering Equation (2.289), therefore 0 < mT < 1. Because inductor current iL1 = iL2 = 0 at t = t1, so that VL1−off = VL 2−off =
k V (1 − k )mT I
Since the current iD becomes zero at t = t1 = [k + (1 – k)mT]T, for the current iL we have kTVI = (1 – k)mTT(VC – 3VI – VL1–off – VL2–off) or VC = [3 +
3k R ]V = [3 + k 2 (1 − k ) ]V (1 − k )mT I 2 fL I
with
k
3R 3 ≥ 2 fL 1 − k
and for the current iLO we have kT(VI + VC – VO) = (1 – k)mTT(VO – 2VI – VL1–off – VL2–off) Therefore, output voltage in discontinuous mode is VO = [3 +
3k R ]V = [3 + k 2 (1 − k ) ]V (1 − k )mT I 2 fL I
© 2006 by Taylor & Francis Group, LLC
with
k
3R 3 (2.291) ≥ 2 fL 1 − k
Voltage-Lift Converters
155 iD11
iD12 iD13
S iIN
–V + S
D10
iC2
D11 iC3
D12
D13
iC1
C1
iD
D
C3
iC4
L3
L2
iL3
iC
LO
iLO
iCO
L1
iL2
iL1
–
C4
VIN
iO
– VLO +
C2
C
CO
R VO +
D4
L iL
iD4
D3 iD3
D2 iD2
D1 iD1
FIGURE 2.48 Quadruple-lift circuit.
i.e., the output voltage will linearly increase during load resistance R increasing. The output voltage vs. the normalized load zN = R/fL is shown in Figure 2.47. We can see that the output voltage will increase when the load resistance R increases. 2.4.4.2 Quadruple-Lift Circuit Quadruple-lift circuit is shown in Figure 2.48. It consists of one static switch S; five inductors L, L1, L2, L3, and LO; and six capacitors C, C1, C2, C3, C4, and CO. Capacitors C1, C2, C3, and C4 perform characteristics to lift the capacitor voltage VC by four times of source voltage VI. L1, L2, and L3 perform the function as ladder joints to link the four capacitors C1, C2, C3, and C4 and lift the output capacitor voltage VC up. Current iC1(t), iC2(t), iC3(t), and iC4(t) are exponential functions. They have large values at the moment of power on, but they are small because vC1 = vC2 = vC3 = vC4 ≅ VI in steady state. The output voltage and current are VO =
4 V 1− k I
(2.292)
IO =
1− k I 4 I
(2.293)
and
The voltage transfer gain in continuous mode is MQ = VO / VI =
© 2006 by Taylor & Francis Group, LLC
4 1− k
(2.294)
156
Essential DC/DC Converters 40
32
MQ
24
16
8
0
0
0.2
0.6
0.4
1
0.8
k
FIGURE 2.49 Voltage transfer gain of quadruple-lift circuit.
The curve of MQ vs. k is shown in Figure 2.49. Other average voltages: VC = VO
VC1 = VC2 = VC3 = VC4 = VI
Other average currents: ILO = IO
I L = I L1 = I L 2 = I L 3 =
1 I 1− k O
Current variation ratios: ζ=
χ1 =
k(1 − k ) R 2 MQ fL1
k 2R MQ 2 fL χ2 =
ξ=
k 1 16 f 2 CLO
k(1 − k ) R 2 MQ fL2
χ3 =
k(1 − k ) R 2 MQ fL3
Voltage variation ratios: ρ=
σ2 =
MQ
1 2 fC2 R
© 2006 by Taylor & Francis Group, LLC
k 1 2 fCR
σ3 =
σ1 = MQ
MQ
1 2 fC1R
1 2 fC3 R
σ4 =
MQ
1 2 fC4 R
Voltage-Lift Converters
157
60 50 40
continuous mode
k=0.9
30 k=0.8
MQ
20
10 8 6
k=0.5 k=0.33 discontinuous mode
k=0.1 2
54 64
100
250 R/fL
889
FIGURE 2.50 The boundary between continuous and discontinuous modes and output voltage vs. the normalized load zN = R/fL (quadruple-lift circuit).
The variation ratio of output voltage VC is ε=
k 1 3 128 f CCO LO R
(2.295)
The output voltage ripple is very small. The boundary between continuous and discontinuous conduction modes is MQ ≤ k
2R = 2 kz N fL
(2.296)
It can be seen that the boundary curve has a minimum value of MQ that is equal to 6.0, corresponding to k = 1/3 . The boundary curve is shown in Figure 2.50. In discontinuous mode the current iD exists in the period between kT and [k + (1 – k)mQ]T, where mQ is the filling efficiency that is mQ =
2 1 MQ = ζ k 2R fL
(2.297)
Considering Equation (2.296), therefore 0 < mQ < 1. Because inductor current iL1 = iL2 = iL3 = 0 at t = t1, so that
© 2006 by Taylor & Francis Group, LLC
158
Essential DC/DC Converters
VL1−off = VL 2−off = VL 3−off =
k V (1 − k )mQ I
Since the current iD becomes zero at t = t1 = kT + (1 – k)mQT, for the current iL we have kTVI = (1 – k)mQT(VC – 4VI – VL1–off – VL2–off – VL3–off) or VC = [4 +
4k R ]VI = [4 + k 2 (1 − k ) ]V (1 − k )mQ 2 fL I
with
k
2R 4 ≥ fL 1 − k
and for current iLO we have kT(VI + VC – VO) = (1 – k)mQT(VO – 2VI – VL1–off – VL2–off – VL3–off) Therefore, output voltage in discontinuous mode is VO = [4 +
4k R ]VI = [4 + k 2 (1 − k ) ]V (1 − k )mQ 2 fL I
k
with
2R 4 ≥ (2.298) fL 1 − k
i.e., the output voltage will linearly increase while load resistance R increases. The output voltage vs. the normalized load zN = R/fL is shown in Figure 2.50. We can see that the output voltage will increase during load resistance while the load R increases.
2.4.5
Summary
From the analysis and calculation in previous sections, the common formulae can be obtained for all circuits: M=
VO I = I VI IO
zN =
R fL
R=
VO IO
Current variation ratios:
© 2006 by Taylor & Francis Group, LLC
ζ=
k(1 − k )R 2 MfL
ξ=
χj =
k(1 − k )R 2 MfL j
(j = 1, 2, 3, …)
k 16 f 2 CLO
Voltage-Lift Converters
159
Voltage variation ratios: ρ=
k 2 fCR
ε=
k 128 f 3 CCO LO R
σj =
M 2 fC j R
(j = 1, 2, 3, 4, …)
In order to write common formulas for the boundaries between continuous and discontinuous modes and output voltage for all circuits, the circuits can be numbered. The definition is that subscript 0 means the elementary circuit, subscript 1 means the self-lift circuit, subscript 2 means the re-lift circuit, subscript 3 means the triple-lift circuit, subscript 4 means the quadruple-lift circuit, and so on. Therefore, the voltage transfer gain in continuous mode for all circuits is Mj =
k h( j ) [ j + h( j )] 1− k
j = 0, 1, 2, 3, 4, …
(2.299)
The variation of the free-wheeling diode current iD is ζj =
k [1+ h( j )] j + h( j ) zN 2 Mj2
(2.300)
The boundaries are determined by the condition: ζj ≥ 1 or k [1+ h( j )] j + h( j ) zN ≥ 1 2 Mj2
j = 0, 1, 2, 3, 4, …
(2.301)
Therefore, the boundaries between continuous and discontinuous modes for all circuits are Mj = k
1+ h ( j ) 2
j + h( j ) zN 2
j = 0, 1, 2, 3, 4, …
(2.302)
The filling efficiency is 2
mj =
© 2006 by Taylor & Francis Group, LLC
Mj 1 2 1 = [1+ h( j )] ζj k j + h( j ) z N
(2.303)
160
Essential DC/DC Converters
120
output voltage, vo, v
100 80
(i)
50 (ii) (iii)
30
(iv)
10 0
(v)
0.2
0.6 0.4 conduction duty k
0.8
1
FIGURE 2.51 Output voltages of all negative output Luo-converters (VI = 10 V).
The voltage across the capacitor C in discontinuous mode for all circuits VC − j = [ j + k [ 2− h( j )]
1− k z ]V 2 N I
j = 0, 1, 2, 3, 4, …
(2.304)
The output voltage in discontinuous mode for all circuits VO− j = [ j + k [ 2− h( j )]
1− k z ]V 2 N I
j = 0, 1, 2, 3, 4, …
(2.305)
where ⎧0 h( j ) = ⎨ ⎩1
if if
j≥1 j=0
is the Hong Function
The voltage transfer gains in continuous mode for all circuits are shown in Figure 2.51. The boundaries between continuous and discontinuous modes of all circuits are shown in Figure 2.52. The curves of all M vs. zN state that the continuous mode area increases from ME via MS, MR, MT to MQ. The boundary of elementary circuit is a monorising curve, but other curves are not monorising. There are minimum values of the boundaries of other curves, which of MS, MR, MT, and MQ correspond at k = 1/3 . Assuming that f = 50 kHz, L = LO = L1 = L2 = L3 = L4 = 100 μH, C = C1 = C2 = C3 = C4 = CO = 5 μF and the source voltage VI = 10 V, the value of the output voltage VO in various conduction duty k are shown in Figure 2.22. Typically, some values of the output voltage VO in conduction duty k = 0.33,
© 2006 by Taylor & Francis Group, LLC
Voltage-Lift Converters
161
voltage transfer gain M
continuous mode
MQ MT
6 4.5
MR
3 Ms 1.5 ME discontinuous mode 0.5 44.5
27 40.5 54 13.5 normalized load zN R/fL
FIGURE 2.52 Boundaries between continuous and discontinuous modes of all negative output Luo-converters.
TABLE 2.2 Comparison among Five Negative Output Luo-Converters Negative Output Luo-Converters
IO
Elementary Circuit
IO =
Self-Lift Circuit
I O = (1 − k )I I
Re-Lift Circuit
IO =
Triple-Lift Circuit Quadruple-Lift Circuit
VO
k = 0.33
VO (VI = 10 V) k = 0.5 k = 0.75
k = 0.9
VO =
k V 1− k I
5V
10 V
30 V
90 V
VO =
1 V 1− k I
15 V
20 V
40 V
100 V
1− k I 2 I
VO =
2 V 1− k I
30 V
40 V
80 V
200 V
IO =
1− k I 3 I
VO =
3 V 1− k I
45 V
60 V
120 V
300 V
IO =
1− k I 4 I
VO =
4 V 1− k I
60 V
80 V
160 V
400 V
1− k I k I
0.5, 0.75, and 0.9 are listed in Table 2.2. The ripple of the output voltage is very small, say smaller than 1%. For example, using the above data and R = 10 Ω, the variation ratio of the output voltage is ε = 0.0025 × k = 0.0008, 0.0012, 0.0019, and 0.0023 respectively. From these data the fact we find is that the output voltage of all negative output Luo-converters is almost a real DC voltage with very small ripple.
© 2006 by Taylor & Francis Group, LLC
162
2.5
Essential DC/DC Converters
Modified Positive Output Luo-Converters
Negative output Luo-converters perform the voltage conversion from positive to negative voltages using VL technique with only one switch S. This section introduces the technique to modify positive output Luo-converters that can employ only one switch for all circuits. Five circuits have been introduced in the literature. They are • • • • •
Elementary circuit Self-lift circuit Re-lift circuit Triple-lift circuit Quadruple-lift circuit
There are five circuits introduced in this section, namely the elementary circuit, self-lift circuit, re-lift circuit, and multiple-lift circuit (triple-lift and quadruple-list circuits). In all circuits the switch S is a PMOS. It is driven by a PWM switch signal with variable frequency f and conduction duty k. For all circuits, the load is usually resistive, R = VO/IO. We concentrate the absolute values rather than polarity in the following descriptions and calculations. The directions of all voltages and currents are defined and shown in the figures. We will assume that all the components are ideal and the capacitors are large enough. We also assume that the circuits operate in continuous conduction mode. The output voltage and current are VO and IO; the input voltage and current are VI and II. 2.5.1
Elementary Circuit
Elementary circuit is shown in Figure 2.10. It is the elementary circuit of positive output Luo-converters. The output voltage and current and the voltage transfer gain are VO =
k V 1− k I
IO =
1− k I k I
ME =
k 1− k
Average voltage: VC = VO
© 2006 by Taylor & Francis Group, LLC
Voltage-Lift Converters
163
iI
i LO
– vC +
+ VI
S
D1 + vC1 D –
C
iL L
–
iO
LO C1
+ R
CO
vO –
(a) Self-lift circuit iI
vC
–
+ VI
i LO +
C
iL L
–
LO
– vC1 +
iO + R
C1
vO –
CO
(b) Switch-on equivalent circuit i LO
i I=0 + VI –
iL
– – vC1 vC C + L +
iO
LO C1
+ CO
R
vO –
(c) Switch-off equivalent circuit FIGURE 2.53 Modified self-lift circuit and its equivalent circuit. (a) Self-lift circuit. (b) Switch-on equivalent circuit. (c) Switch-off equivalent circuit.
Average currents: ILO = IO
2.5.2
IL =
k I 1− k O
Self-Lift Circuit
Self-lift circuit is shown in Figure 2.53. It is derived from the elementary circuit. In steady state, the average inductor voltages over a period are zero. Thus VC1 = VCO = VO
© 2006 by Taylor & Francis Group, LLC
(2.306)
164
Essential DC/DC Converters
The inductor current iL increases in the switch-on period and decreases in the switch-off period. The corresponding voltages across L are VI and −VC . Therefore kTVI = (1 − k )TVC Hence, VC =
k V 1− k I
(2.307)
During switch-on period, the voltage across capacitor C1 are equal to the source voltage plus the voltage across C. Since we assume that C and C1 are sufficiently large, VC1 = VI + VC Therefore, VC1 = VI +
k 1 V = V 1− k I 1− k I
VO = VCO = VC1 =
1 V 1− k I
The voltage transfer gain of continuous conduction mode (CCM) is M=
VO 1 = VI 1 − k
The output voltage and current and the voltage transfer gain are VO =
1 V 1− k I
I O = (1 − k )I I MS =
© 2006 by Taylor & Francis Group, LLC
1 1− k
(2.308)
Voltage-Lift Converters
165
Average voltages: VC = kVO VC1 = VO Average currents: ILO = IO IL =
1 I 1− k O
We also implement the breadboard prototype of the proposed self-lift circuit. NMOS IRFP460 is used as the semiconductor switch. The diode is MR824. The other parameters are VI = 0 ~ 30 V, R = 30 ~ 340 Ω, k = 0.1 ~ 0.9 C = CO = 100 μF and L = 470 μH
2.5.3
Re-Lift Circuit
Re-lift circuit and its equivalent circuits are shown in Figure 2.54. It is derived from the self-lift circuit. The function of capacitors C2 is to lift the voltage vC by source voltage VI, the function of inductor L1 acts like a hinge of the foldable ladder (capacitor C2) to lift the voltage vC during switch off. In steady state, the average inductor voltages over a period are zero. Thus VC1 = VCO = VO Since we assume C2 is large enough and C2 is biased by the source voltage VI during switch-on period, thus VC 2 = VI From the switch-on equivalent circuit, another capacitor voltage equation can also be derived since we assume all the capacitors to be large enough, VO = VC1 = VC + VI The inductor current iL increases in the switch-on period and decreases in the switch-off period. The corresponding voltages across L are VI and −VL−OFF . Therefore kTVI = (1 − k )TVL−OFF
© 2006 by Taylor & Francis Group, LLC
166
Essential DC/DC Converters
iI D 11
S
+ + VI
v C2
i L1 –
L1
+
+ v C1 D
L
iO
LO
D1
C2 iL
i LO
+
C
D 10
–
vC
–
CO
C1
–
vO
R
–
D2
(a) Re-lift circuit iI
vC
–
i LO
+
LO
C +
–
i L1
+ v C2 –
VI iL
iO
L1
+ v C1 –
C2
+ vO
CO
C1
R
–
L
(b) Switch-on equivalent circuit vC
–
iI = 0
i LO
+
LO
C
+
+ VI –
v C2
i L1 –
L1
C2 iL
iO
+ + v C1 –
C1
CO
vO R
–
L
(c) Switch-off equivalent circuit FIGURE 2.54 Modified re-lift circuit. (a) Re-lift circuit. (b) Switch-on equivalent circuit. (c) Switch-off equivalent circuit.
Hence, VL−OFF =
k V 1− k I
The inductor current iL1 increases in the switch-on period and decreases in the switch-off period. The corresponding voltages across L1 are VI and −VL1−OFF . Therefore kTVI = (1 − k )TVL1−OFF
© 2006 by Taylor & Francis Group, LLC
Voltage-Lift Converters
167
Hence, VL1−OFF =
k V 1− k I
From the switch-off period equivalent circuit, VC = VC −OFF = VL−OFF + VL1−OFF + VC 2 Therefore, VC =
k k 1+ k VI + VI + VI = V 1− k 1− k 1− k I VO =
(2.309)
1+ k 2 VI + VI = V 1− k 1− k I
Then we get the voltage transfer ratio in CCM, M = MR =
2 1− k
(2.310)
The following is a brief summary of the main equations for the re-lift circuit. The output voltage and current and gain are VO =
2 V 1− k I
IO =
1− k I 2 I
MR =
2 1− k
Average voltages: VC =
1+ k V 1− k I
VC1 = VCO = VO VC2 = VI
© 2006 by Taylor & Francis Group, LLC
168
Essential DC/DC Converters
iI D 11
S
D 10 D 12 +
v C3
C2
iO
LO
D1
i L1
– +
L1
C3
v C1
L2
+ iL
i LO
+
C
v C2 –
+
VI
vC
–
D
C1
CO
+ R vO
L
–
–
D3
–
D2
FIGURE 2.55 Modified triple-lift circuit.
Average currents: ILO = IO I L = I L1 =
2.5.4
1 I 1− k O
Multi-Lift Circuit
Multiple-lift circuits are derived from re-lift circuits by repeating the section of L1-C1-D1 multiple times. For example, triple-list circuit is shown in Figure 2.55. The function of capacitors C2 and C3 is to lift the voltage vC across capacitor C by twice the source voltage 2VI, the function of inductors L1 and L2 acts like hinges of the foldable ladder (capacitors C2 and C3) to lift the voltage vC during switch off. The output voltage and current and voltage transfer gain are VO =
3 V 1− k I
IO =
1− k I 3 I
and
MT =
© 2006 by Taylor & Francis Group, LLC
3 1− k
(2.311)
Voltage-Lift Converters
169
iI
– D11
S
+ D12 – D13 i L3
D10 –
+
vC4
vC3
C2
+
L1 L2
L3
+
+ vC1
C4 VI –
iO
LO
D1
C
–
C3
+
i LO
+
i L1
vC2
i L2
vC
D
–
vO C1
CO
R –
iL
L
D3
D4
D2
FIGURE 2.56 Modified quadruple-lift circuit.
Other average voltages: 2+k V 1− k I
VC = and VC1 = VO
VC2 = VC3 = VI
Other average currents: ILO = IO I L1 = I L 2 = I L =
1 I 1− k O
The quadruple-lift circuit is shown in Figure 2.56. The function of capacitors C2, C3, and C4 is to lift the voltage vC across capacitor C by three times the source voltage 3VI. The function of inductors L1, L2, and L3 acts like hinges of the foldable ladder (capacitors C2, C3, and C4) to lift the voltage vC during switch off. The output voltage and current and voltage transfer gain are VO = and
© 2006 by Taylor & Francis Group, LLC
4 V 1− k I
170
Essential DC/DC Converters
IO =
1− k I 4 I
MQ =
4 1− k
(2.312)
Average voltages: VC =
3+k V 1− k I
and VC1 = VO VC2 = VC3 = VC4 = VI Average currents: ILO = IO and IL =
k I 1− k O
I L1 = I L 2 = I L 3 + I L + I LO =
2.5.5
1 I 1− k O
Application
A high-efficiency, widely adjustable high voltage regulated power supply (HVRPS) is designed to use these Luo-converters in a high voltage test rig. The proposed HVRPS is shown in Figure 2.57. The HVRPS was constructed by using a PWM IC TL494 to implement closed-loop control together with the modified positive output Luo-converters. Its output voltage is basically a DC value with small ripple and can be widely adjustable. The source voltage is 24 V DC and the output voltage can vary from 36 V to 1000 V DC. The measured experimental results show that the efficiency can be as high as 95% and the source effect ratio is about 0.001 and load effect ratio is about 0.005.
© 2006 by Taylor & Francis Group, LLC
Voltage-Lift Converters
171
iI
vC
–
D11 C2
V I =+24V
S –
vC3
D13
–
+
–
i L1
vC2
L1 + L2
+
vO
vC1 –
C4
D D4
iL
iO
LO
i L2
L3
+
D1
C
+
C3 i L3
D10 vC4
D12
i LO
+
D3
CO
R
–
C1
D2
L R6
R8 V ref
1.5k 6 (Rt) 12 (Vcc )
5 (Ct)
TL494
15
11 16 7
10k 82k
14
8 R4 820
22u
13
10
9
1 (+)
4 2 (–)
10k 0.001u
C5
R2
R1
C6 1k
R3
R7
R5
1.5k
820 R9 510
Rw1 3.9k
FIGURE 2.57 A high voltage testing power supply.
2.6
Double Output Luo-Converters
Mirror-symmetrical double output voltages are specially required in industrial applications and computer periphery circuits. Double output DC-DC Luo-converters can convert the positive input source voltage to positive and negative output voltages. It consists of two conversion paths. Double output Luo-converters perform from positive to positive and negative DC-DC voltage increasing conversion with high power density, high efficiency, and cheap topology in simple structure. Double output DC-DC Luo-converters consist of two conversion paths. Usually, mirror-symmetrical double output voltages are required in industrial applications and computer periphery circuits such as operational amplifiers, computer periphery power supplies, differential servo-motor drives, and some symmetrical voltage medical equipment. In recent years the DCDC conversion technique has been greatly developed. The main objective is
© 2006 by Taylor & Francis Group, LLC
172
Essential DC/DC Converters
to reach a high efficiency, high power density and cheap topology in simple structure. The elementary circuit can perform step-down and step-up DC-DC conversion. The other double output Luo-converters are derived from this elementary circuit, they are the self-lift circuit, re-lift circuit, and multiple-lift circuits (e.g., triple-lift and quadruple-lift circuits). Switch S in these circuits is a PMOS. It is driven by a PWM switch signal with repeating frequency f and conduction duty k. In this paper the switch repeating period is T = 1/ f, so that the switch-on period is kT and switch-off period is (1 – k)T. For all circuits, the loads are usually resistive, i.e., R = VO+/IO+ and R1 = VO–/IO–; the normalized loads are zN+ = R/fL (where L = L1 and L = L1L2 L1 + L2 for elementary circuits) and zN– = R1/fL11. In order to keep the positive and negative output voltages to be symmetrically equal to each other, usually, we purposely select that L = L11 and zN+ = zN–. Each converter has two conversion paths. The positive path consists of a positive pump circuit S-L1-D0-C1 and a “Π”-type filter (C2)-L2-CO, and a lift circuit (except elementary circuits). The pump inductor L1 absorbs energy from source during switch-on and transfers the stored energy to capacitor C1 during switch-off. The energy on capacitor C1 is then delivered to load R during switch-on. Therefore, a high voltage VC1 will correspondingly cause a high output voltage VO+. The negative path consists of a negative pump circuit S-L11-D10-(C11) and a “Π”-type filter C11-L12-C10, and a lift circuit (except elementary circuits). The pump inductor L11 absorbs the energy from source during switch-on and transfers the stored energy to capacitor C11 during switch-off. The energy on capacitor C11 is then delivered to load R1 during switch-on. Hence, a high voltage VC11 will correspondingly cause a high output voltage VO–. When switch S is turned off, the currents flowing though the freewheeling diodes D0 and D10 are existing. If the currents iD0 and iD10 do not fall to zero before switch S is turned on again, we define this working state to be continuous conduction mode. If the currents iD0 and iD10 become zero before switch S is turned on again, we define that working state to be discontinuous conduction mode. The output voltages and currents are VO+, VO– and IO+, IO–; the input voltage and current are VI and II = II+ + II–. Assuming that the power loss can be ignored, PI = PO, or VIII = VO+ IO+ + VO– IO–. For general description, we have the following definitions in continuous mode: The voltage transfer gain in the continuous mode: M+ =
VO+ VI
and
Variation ratio of the diode’s currents:
© 2006 by Taylor & Francis Group, LLC
M− =
VO− VI
Voltage-Lift Converters
173 ΔiD 0 / 2 I D0
ζ+ =
ζ− =
and
ΔiD10 / 2 I L11
Variation ratio of pump inductor’s currents: ξ 1+ =
ΔiL1 / 2 I L1
ζ− =
and
ΔiL11 / 2 I L11
Variation ratio of filter inductor’s currents: ξ2+ =
ΔiL 2 / 2 I L2
ξ− =
and
ΔiL12 / 2 I L12
Variation ratio of lift inductor’s currents: χ j+ =
ΔiL 2+ j / 2
χ j− =
and
I L 2+ j
ΔiL12+ j / 2 I L12+ j
j = 1, 2, 3, …
Variation ratio of pump capacitor’s voltages: ρ+ =
ΔvC1 / 2 VC1
ρ− =
and
ΔvC11 / 2 VC11
Variation ratio of lift capacitor’s voltages: σ j+ =
ΔvC1+ j / 2 VC1+ j
and
σ j− =
ΔvC11+ j / 2 VC11+ j
j = 1, 2, 3, 4, …
Variation ratio of output voltages: ε+ =
2.6.1
ΔvO+ / 2 VO+
and
ε− =
ΔvO− / 2 VO−
Elementary Circuit
The elementary circuit is shown in Figure 2.58. Since the positive Luo-converters and negative Luo-converters have been published, this section can be simplified.
© 2006 by Taylor & Francis Group, LLC
174
Essential DC/DC Converters
S
N
iL IN+
+
C1
L2
– VC1 +
i L2
D20
Vs
iIN VI
VS
i L1
D0
L1
Io +
+ Co
R Vo +
– – iIN – iL 11
+ VC11 –
L11
C11
+ R1 Vo –
i L12
D10
D21
C10
– L12 Io –
FIGURE 2.58 Elementary circuit.
2.6.1.1
Positive Conversion Path
The equivalent circuit during switch-on is shown in Figure 2.59a, and the equivalent circuit during switch-off in Figure 2.59b. The relations of the average currents and voltages are I L2 =
1− k I k L1
and IL2 = IO+
Positive path input current is I I + = k × iI + = k(iL1 + iL 2 ) = k(1 +
1− k ) I L1 = I L1 k
(2.313)
The output current and voltage are I O+ =
1− k I k I+
and
VO+ =
k V 1− k I
The voltage transfer gain in continuous mode is ME + =
VO+ k = VI 1− k
The average voltage across capacitor C1 is VC1 =
© 2006 by Taylor & Francis Group, LLC
k V = VO+ 1− k I
(2.314)
Voltage-Lift Converters
175
S
iIN+ –
iL2
VC1 +
L2
+
+
+ VIN
L1
iL11
CO
VD0
R
–
VO – –
–
(a) switch on i L2
VC1 +
–
L2 L1
iL11
+ CO
i D0
R
VO – –
(b) switch off
–
i L2
VC1 +
L2
+
+ L1
i L11
VD0
CO
–
R
VO – –
(c) discontinuous mode FIGURE 2.59 Equivalent circuits of elementary circuit positive path: (a) switch on; (b) switch off; (c) discontinuous conduction mode.
The variation ratios of the parameters are ξ 1+ =
ΔiL1 / 2 kTVI Δi / 2 kTVI k R 1− k R = = = = and ξ 2+ = L 2 I L1 I L2 2L1 I I + 2 ME fL1 2L2 I O+ 2 ME fL2
The variation ratio of current iD0 is ζ+ =
ΔiD 0 / 2 (1 − k )2 TVO+ k(1 − k )R k2 R = = = 2LI O+ 2 ME fL I D0 ME 2 2 fL
The variation ratio of vC1 is
© 2006 by Taylor & Francis Group, LLC
(2.315)
176
Essential DC/DC Converters
ρ+ =
ΔvC1 / 2 (1 − k )TI I + k 1 = = VC1 2C1VO+ 2 fC1R
The variation ratio of output voltage vO+ is ε+ =
ΔvO+ / 2 1 kT 2 VI k = = 2 8CO L2 VO+ 8 ME f CO L2 VO+
(2.316)
If L1 = L2 = 1 mH, C1 = C0 = 20 μF, R = 10 Ω, f = 50 kHz and k = 0.5, we get ξ1+ = 0.05, ξ2+ = 0.05, ζ+ = 0.05, ρ+ = 0.025, and ε = 0.00125. Therefore, the variations of iL1, iL2 and vC1 are small. The output voltage VO+ is almost a real DC voltage with very small ripple. Because of the resistive load, the output current iO+(t) is almost a real DC waveform with very small ripple as well, and is equal to IO+ = VO+/R. 2.6.1.2 Negative Conversion Path The equivalent circuit during switch-on is shown in Figure 2.60(a) the equivalent circuit during switch-off is shown in Figure 2.60(b). The relations of the average currents and voltages are IO– = IL12
and IO– = IL12 = IC11–on
Since I C11−off =
k k I C11−on = I 1− k 1 − k O−
the inductor current IL11 is I L11 = I C11−off + I O− =
I O− 1− k
So that I I − = k × iI − = kiL11 = kI L11 =
k I 1 − k O−
The output current and voltage are I O− =
© 2006 by Taylor & Francis Group, LLC
1− k I k I−
and
VO− =
k V 1− k I
(2.317)
Voltage-Lift Converters
177 i L12 L12
+
–
– VIN
L11
i L11
iC11
VC11
C10
R1
+
–
VO– +
(a) switch on i D10
i L12 L12
+
–
– VL11
i L11 iC11 L11
C10
VC11
R1
+
–
VO– +
(b) switch off i L12 VD10
L 12
+
–
– VL11
L11
C10
VC11
R1
+
–
VO– +
(c) discontinuous mode FIGURE 2.60 Equivalent circuits of elementary circuit negative path: (a) switch on; (b) switch off; (c) discontinuous conduction mode.
The voltage transfer gain in continuous mode is ME − =
VO− k = VI 1− k
(2.318)
and VC11 = VO− =
k V 1− k I
From Equations (2.314) and (2.318), we can define that ME = ME+ = ME– . The curve of ME vs. k is shown in Figure 2.61. The variation ratios of the parameters are ξ− =
ΔiL12 / 2 k 1 = I L12 16 f 2 C10 L12
© 2006 by Taylor & Francis Group, LLC
and
ρ− =
ΔvC11 / 2 kI O− T k 1 = = VC11 2C11VO− 2 fC11R1
178
Essential DC/DC Converters 10
8
ME
6
4
2
0 0
0.6
0.4
0.2
0.8
1
k
FIGURE 2.61 Voltage transfer gain ME vs. k.
The variation ratio of current iL11 and iD10 is ζ− =
ΔiL11 / 2 k(1 − k )VI T k(1 − k )R1 k 2 R1 = = = I L11 2L11 I O− 2 ME fL11 ME 2 2 fL11
(2.319)
The variation ratio of current vC1O is ε− =
ΔvC10 / 2 I O− k k 1 = = VC10 128 f 3 C11C10 L12 VO− 128 f 3 C11C10 L12 R1
(2.320)
Assuming that f = 50 kHz, L11 = L12 = 0.5 mH, C = CO = 20 μF, R1 = 10 Ω and k = 0.5, obtained ME = 1, ζ = 0.05, ρ = 0.025, ξ = 0.00125 and ε = 0.0000156. The output voltage VO– is almost a real DC voltage with very small ripple. Since the load is resistive, the output current iO–(t) is almost a real DC waveform with very small ripple as well, and it is equal to IO– = VO–/R1. 2.6.1.3 Discontinuous Mode The equivalent circuits of the discontinuous mode’s operation are shown in Figure 2.59c and Figure 2.60c. In order to obtain the mirror-symmetrical double output voltages, select: L= and R = R1. Thus, we define
© 2006 by Taylor & Francis Group, LLC
L1L2 = L11 L1 + L2
Voltage-Lift Converters
179
20 continuous mode 10 k=0.9 5 ME
k=0.7 2 1
k=0.5
0.5 k=0.3 0.2
discontinuous mode k=0.1
0.1 1
2
5
10
20
50
100
200
500 1000
R/fL
FIGURE 2.62 The boundary between continuous and discontinuous modes and the output voltage vs. the normalized load zN = R/fL (elementary circuit).
VO = VO+ = |VO− |
ME = ME + = ME − =
zN = zN+ = zN–
VO k = VI 1 − k
and ζ = ζ+ = ζ–
The free-wheeling diode currents iD0 and iD10 become zero during switch off before next period switch on. The boundary between continuous and discontinuous modes is ζ≥1 i.e., k 2 zN ≥1 ME 2 2 or ME ≤ k
zN 2
(2.321)
The boundary curve is shown in Figure 2.62. In this case the free-wheeling diode’s diode current exists in the period between kT and [k + (1 – k)mE]T, where mE is the filling efficiency and it is defined as:
© 2006 by Taylor & Francis Group, LLC
180
Essential DC/DC Converters
mE =
2 1 2 ME = 2 ζ k zN
(2.322)
Considering the Equation (2.321), therefore, 0 < mE < 1. Since the diode current iD0 becomes zero at t = kT + (1 – k)mET, for the current iL1 kTVI = (1 – k)mETVC or VC1 =
z k VI = k(1 − k ) N VI (1 − k )mE 2
with
zN 1 ≥ 2 1− k
(2.323)
and for the current iL2 kT(VI + VC1 – VO+) = (1 – k)mETVO+ Therefore, the positive output voltage in discontinuous mode is VO+ =
z k VI = k(1 − k ) N VI (1 − k )mE 2
with
zN 1 ≥ 2 1− k
(2.324)
zN 1 ≥ 2 1− k
(2.325)
For the current iL11 we have kTVI = (1 – k)mETVC11 or VC11 =
z k VI = k(1 − k ) N VI (1 − k )mE 2
with
and for the current iL12 we have kT(VI + VC11 – VO–) = (1 – k)mETVO– Therefore, the negative output voltage in discontinuous mode is VO− =
z k V = k(1 − k ) N VI (1 − k )mE I 2
with
We then have VO = VO+ = VO− = k(1 − k )
© 2006 by Taylor & Francis Group, LLC
zN V 2 I
zN 1 ≥ 2 1− k
(2.326)
Voltage-Lift Converters
S
181
VS
D20
C1 –
iIN VI
iIN+
+
iL1
D1
VC1+
+
iL2 D0
L1
Io +
L2
C2
Co
R Vo +
– N
– iIN– iL11
L11
D21
+ VC11 D11 – D10
C11
C10
+ R1 Vo–
iL2 – L12
C12
Io–
FIGURE 2.63 Self-lift circuit.
i.e., the output voltage will linearly increase while load resistance increases. It can be seen that larger load resistance may cause higher output voltage in discontinuous mode as shown in Figure 2.62.
2.6.2
Self-Lift Circuit
Self-lift circuit shown in Figure 2.63 is derived from the elementary circuit. The positive conversion path consists of a pump circuit S-L1-D0-C1, a filter (C2)-L2-CO, and a lift circuit D1-C2. The negative conversion path consists of a pump circuit S-L11-D10-(C11), a “Π”-type filter C11-L12-C10, and a lift circuit D11-C12. 2.6.2.1 Positive Conversion Path The equivalent circuit during switch-on is shown in Figure 2.64a, and the equivalent circuit during switch-off in Figure 2.64b. The voltage across inductor L1 is equal to VI during switch-on, and –VC1 during switch-off. We have the relations: VC1 =
k V 1− k I
Hence, VO = VCO = VC 2 = VI + VC1 = and
© 2006 by Taylor & Francis Group, LLC
1 V 1− k I
182
Essential DC/DC Converters iIN+ –
S
iL2
VC1 +
L2
+ VIN
iL1
C2
L1
+ CO
VC2
R
VO+ –
–
(a) switch on –
iL2
VC1 +
L2 iL1
L1
C2
+
+ VC2 CO –
VO+
R
–
(b) switch off –
iL2
VC1
L2 + L1
VD0 C 2
VC2 C O
VO+
R
–
(c) discontinuous conduction mode
FIGURE 2.64 Equivalent circuits of self-lift circuit positive path: (a) switch on; (b) switch off; (c) discontinuous conduction mode.
VO+ =
1 V 1− k I
The output current is I O+ = (1 − k )I I + Other relations are II+ = k iI+ and
© 2006 by Taylor & Francis Group, LLC
iI+ = IL1 + iC1–on
iC1−off =
k i 1 − k C1−on
Voltage-Lift Converters
183 I L1 = iC1−off = kiI + = I I +
(2.327)
Therefore, the voltage transfer gain in continuous mode is MS + =
VO+ 1 = VI 1− k
(2.328)
The variation ratios of the parameters are ξ 2+ =
ΔiL 2 / 2 k 1 = I L2 16 f 2 C2 L2
ρ+ =
ΔvC1 / 2 (1 − k )I I + 1 = = k VC1 kfC 2 1R V 2 fC1 1− k I
and σ 1+ =
ΔvC 2 / 2 k = VC 2 2 fC2 R
The variation ratio of the currents iD0 and iL1 is ζ + = ξ 1+ =
ΔiL1 / 2 kVI T k R = = I L1 2L1 I I + MS 2 2 fL1
(2.329)
The variation ratio of output voltage vO+ is ε+ =
ΔvO+ / 2 k 1 = VO+ 128 f 3 C2 CO L2 R
(2.330)
If L1 = L2 = 0.5 mH, C1 = C2 = CO = 20 μF, R = 40 Ω, f = 50 kHz, and k = 0.5, obtained that ξ1+ = ζ = 0.1, ρ+ = 0.00625; and σ1+ = 0.00625, ξ2+ = 0.00125 and ε = 0.000004. Therefore, the variations of iL1, vC1, iL2 and vC2 are small. The output voltage VO+ is almost a real DC voltage with very small ripple. Because of the resistive load, the output current iO+(t) is almost a real DC waveform with very small ripple as well, and IO+ = VO+/R. 2.6.2.2 Negative Conversion Path The equivalent circuit during switch-on is shown in Figure 2.65a, and the equivalent circuit during switch-off in Figure 2.65b. The relations of the average currents and voltages are IO− = I L12 = IC11−on
© 2006 by Taylor & Francis Group, LLC
IC11−off =
k 1− k
IC11−on =
k I 1 − k O−
184
Essential DC/DC Converters
S
iL12
iIN– iC12
+ VIN
L11 C12
iL11
–
– VC11 C 10
C11
VO –
R1
+
–
+
(a) switch on iL12
+ VC12 –
L12 iL11
L11
–
– VC11 C 10
iC11
VO –
R1
+
+
(b) switch off i L12
+ VC12 – VD0
i L11
L11
L12
–
– VC11 C 10 +
C11
R1
VO – +
(c) discontinuous conduction mode
FIGURE 2.65 Equivalent circuits of self-lift circuit negative path: (a) switch on; (b) switch off; (c) discontinuous conduction mode.
and I L11 = IC11−off + IO− =
IO − 1− k
We know that I C12−off = I L11 =
1 I 1 − k O−
and
I C12−on =
1− k 1 I C12−off = I O− k k
So that VO− =
© 2006 by Taylor & Francis Group, LLC
1 V 1− k I
and
I O− = (1 − k )I I
(2.331)
Voltage-Lift Converters
185
10
8
MS
6
4
2
0
0.2
0
0.4
0.6 k
0.8
1
FIGURE 2.66 Voltage transfer gain MS vs. k.
The voltage transfer gain in continuous mode: MS − =
VO− 1 = VI 1− k
(2.332)
Circuit (C11-L12-C10) is a “Π” type low-pass filter. Therefore, VC11 = VO− =
k V 1− k I
From Equation (2.328) and Equation (2.332), we define MS = MS+ = MS–. The curve of MS vs. k is shown in Figure 2.66. The variation ratios of the parameters are ξ− =
ρ− =
σ 1− =
ΔiL12 / 2 k 1 = I L12 16 f 2 C10 L12
ΔvC11 / 2 kI O− T k 1 = = VC11 2C11VO− 2 fC11R1 ΔvC12 / 2 I O− M 1 = = S VC12 2 fC12VI 2 fC12 R1
The variation ratio of currents iD10 and iL11 is
© 2006 by Taylor & Francis Group, LLC
186
Essential DC/DC Converters
ζ− =
ΔiL11 / 2 k(1 − k )VI T k(1 − k )R1 R1 k = = = I L11 2L11 I O− 2 MS fL11 MS 2 2 fL11
(2.333)
the variation ratio of current vC10 is ε− =
ΔvC10 / 2 I O− 1 k k = = 128 f 3 C11C10 L12 VO− 128 f 3 C11C10 L12 R1 VC10
(2.334)
Assuming that f = 50 kHz, L11 = L12 = 0.5 μH, C11 = C10 = 20 μF, R1 = 40 Ω and k = 0.5, we obtain MS = 2, ζ– = 0.2, ρ– = 0.006, σ1– = 0.025, ξ– = 0.0006 and ε– = 0.000004. The output voltage VO– is almost a real DC voltage with very small ripple. Since the load is resistive, the output current iO–(t) is almost a real DC waveform with very small ripple as well, and it is equal to IO– = VO–/R1. 2.6.2.3 Discontinuous Conduction Mode The equivalent circuits of the discontinuous conduction mode’s operation are shown in Figure 2.64 and Figure 2.65 Since we select zN = zN+ = zN–, MS = MS+ = MS– and ζ = ζ+ = ζ– The boundary between continuous and discontinuous conduction modes is ζ≥1 or k zN ≥1 MS 2 2 Hence, MS ≤ k
z = 2
kz N 2
(2.335)
This boundary curve is shown in Figure 2.67. Compared with Equation (2.321) and Equation (2.335), this boundary curve has a minimum value of MS that is equal to 1.5 at k = 1/3 . The filling efficiency is defined as: mS = For the current iL1 we have
© 2006 by Taylor & Francis Group, LLC
2 1 2 MS = ζ kz N
(2.336)
Voltage-Lift Converters
187
30 continuous mode
20 k=0.9 10
MS
8 k=0.8 5 3 2
k=0.5 k=0.33
1.5 k=0.1 1
discontinuous mode
13.56
24.7
62.5 R/fL
842
222
FIGURE 2.67 The boundary between continuous and discontinuous conduction modes and the output voltage vs. the normalized load zN = R/fL (self-lift circuit).
kTVI = (1 – k)mS+TVC1 or VC1 =
z k V = k 2 (1 − k ) N VI (1 − k )mS I 2
with
kz N 1 ≥ 2 1− k
(2.337)
Therefore, the positive output voltage in discontinuous mode is VO+ = VC1 + VI = [1 +
z k ]V = [1 + k 2 (1 − k ) N ]VI (1 − k )mS I 2 kz N 1 ≥ 2 1− k
with For the current iL11 we have
kTVI = (1 – k)mST(VC11 – VI) or VC11 = [1 +
© 2006 by Taylor & Francis Group, LLC
z k ]VI = [1 + k 2 (1 − k ) N ]VI (1 − k )mS 2
(2.338)
188
Essential DC/DC Converters
kz N 1 ≥ 2 1− k
with
(2.339)
and for the current iL12 we have kT(VI + VC11 – VO–) = (1 – k)mS–T(VO– – VI) Therefore, the negative output voltage in discontinuous conduction mode is VO− = [1 +
z k ]VI = [1 + k 2 (1 − k ) N ]VI (1 − k )mS 2 kz N 1 ≥ 2 1− k
with
(2.340)
We then have VO = VO+ = VO− = [1 + k 2 (1 − k )
zN ]V 2 I
i.e., the output voltage will linearly increase during load resistance increasing. Larger load resistance causes higher output voltage in discontinuous conduction mode as shown in Figure 2.67.
2.6.3
Re-Lift Circuit
Re-lift circuit shown in Figure 2.68 is derived from self-lift circuit. The positive conversion path consists of a pump circuit S-L1-D0-C1 and a filter (C2)L2-CO, and a lift circuit D1-C2-D3-L3-D2-C3. The negative conversion path consists of a pump circuit S-L11-D10-(C11) and a “Π”-type filter C11-L12-C10, and a lift circuit D11-C12-L13-D22-C13-D12. 2.6.3.1 Positive Conversion Path The equivalent circuit during switch-on is shown in Figure 2.69a, and the equivalent circuit during switch-off in Figure 2.69b. The voltage across inductors L1 and L3 is equal to VI during switch-on, and –(VC1 – VI) during switch-off. We have the relations: VC1 =
1+ k V 1− k I
© 2006 by Taylor & Francis Group, LLC
and
VO = VCO = VC 2 = VI + VC1 =
2 V 1− k I
Voltage-Lift Converters
S
VS
iIN
D20
iIN+ +
– VC1 +
D2
i L1
–
L1
iIN– iL11 D21 D22
Io+
L2
+
iL2
D0
L3
C2
R
Co
Vo +
D3
–
D11
+
L11 C12
D1
C1
C3
VI N
189
D12 + VC11 –
L 13 C13
C11
R1
C10
Vo –
iL12
D10
–
L12 Io–
FIGURE 2.68 Re-lift circuit.
Thus, VO+ =
2 V 1− k I
I O+ =
1− k I 2 I+
and
The other relations are II+ = k iI+
iI+ = IL1 + IL3+ iC3–on + iC1–on
iC1−off =
k i 1 − k C1−on
and I L1 = I L 3 = iC1−off = iC 3−off =
k 1 i = I 2 I+ 2 I+
(2.341)
The voltage transfer gain in continuous mode is MR+ =
© 2006 by Taylor & Francis Group, LLC
VO+ 2 = 1− k VI
(2.342)
190
Essential DC/DC Converters iIN+
S
iL2
– VC1 +
L2
+ VIN
i L1
L1
CO
L3 C2
C3
+ VO+
R
–
–
(a) switch on C3
iL2
– VC1 +
L2 i L1
L1
L3
CO
D0 C2
+ R
VO+ –
(b) switch off
C3
i L2
– VC1
L2 i L1
L1
L3
CO
VD0 C2
+ R
VO+ –
(c) discontinuous mode FIGURE 2.69 Equivalent circuits of re-lift circuit positive path: (a) switch on; (b) switch off; (c) discontinuous mode.
The variation ratios of the parameters are ξ 2+ =
ΔiL 2 / 2 k 1 = I L2 16 f 2 C2 L2
ρ+ =
σ 1+ =
χ 1+ =
ΔiL 3 / 2 kVI T k R = = 2 1 I L3 M R fL3 2 L3 I I + 2
ΔvC1 / 2 (1 − k )TI I 1 = = + k 1 VC1 (1 + k ) fC1R 4C1 V 1− k I
ΔvC 2 / 2 k = 2 fC2 R VC 2
© 2006 by Taylor & Francis Group, LLC
and
σ 2+ =
ΔvC 3 / 2 1 − k I I + MR = = 4 fC3 VI 2 fC3 R VC 3
Voltage-Lift Converters
191
The variation ratio of currents iD0 and iL1 is ζ + = ξ 1+ =
ΔiD 0 / 2 kVI T k R = = I D0 L1 I I + M R 2 fL1
(2.343)
and the variation ratio of output voltage vO+ is ε+ =
ΔvO+ / 2 k 1 = VO+ 128 f 3 C2 CO L2 R
(2.344)
If L1 = L2 = L3 = 0.5 mH, C1 = C2 = C3 = C0 = 20 μF, R = 160 Ω, f = 50 kHz, and k = 0.5, we obtained that ξ1 = ζ+ = 0.2, χ1 = 0.2, σ▫ = 0.0125, ρ = 0.004; and σ1 = 0.00156, ξ2 = 0.0125 and ε = 0.000001. Therefore, the variations of iL1, iL2 and iL3 are small, and the ripples of vC1, vC3 and vC2 are small. The output voltage vO+ (and vCO) is almost a real DC voltage with very small ripple. Because of the resistive load, the output current iO+ is almost a real DC waveform with very small ripple as well, and IO+ = VO+/R. 2.6.3.2 Negative Conversion Path The equivalent circuit during switch-on is shown in Figure 2.70a, and the equivalent circuit during switch-off is shown in Figure 2.70b. The relations of the average currents and voltages are I C11−off =
IO– = IL12 = IC11–on
k k I = I 1 − k C11−on 1 − k O−
and I L11 = I C11−off + I O− =
I C12−off = I C13−off = I L11 =
1 I 1 − k O−
I C13−on =
I O− 1− k
I C12−on =
(2.345) 1− k 1 I C12−off = I O− k k
1− k 1 I C13−off = I O− k k
In steady state we have: VC12 = VC13 = VI
© 2006 by Taylor & Francis Group, LLC
VL13−on = VI
and
VL13−off =
k V 1− k I
192
Essential DC/DC Converters i L12
iIN –
S +
iC12
VIN
iL11
L11
C12
iL13
iC13
L13C13
C11
L iC11 12
–
C10
R1
VO+
–
(a) switch on
C12
L13
L12
iL13 L11
iL12
C13
–
iC11 C10
C11
iL1
R1
VO+
(b) switch off C12
L13
C13
VD10
iL12 L12
–
– VC11 +
L11
C11 C10
R1
VO– +
(c) discontinuous mode FIGURE 2.70 Equivalent circuits of re-lift circuit negative path: (a) switch on; (b) switch off; (c) discontinuous mode.
VO− =
2 V 1− k I
and
I O− =
1− k I 2 I−
The voltage transfer gain in continuous mode is MR− =
VO− I 2 = I− = VI I O− 1 − k
Circuit (C11-L12-C10) is a “Π” type low-pass filter. Therefore, VC11 = VO− =
© 2006 by Taylor & Francis Group, LLC
2 V 1− k I
(2.346)
Voltage-Lift Converters
193
20
16
MR
12
8
4
0 0
0.2
0.6
0.4
0.8
1
k
FIGURE 2.71 Voltage transfer gain MR vs. k.
From Equation (2.342) and Equation (2.346) we define MR = MR+ = MR–. The curve of MR vs. k is shown in Figure 2.71. The variation ratios of the parameters are ξ− =
and
χ 1− =
ΔiL12 / 2 k 1 = 2 I L12 16 f C10 L12
ΔiL13 / 2 kTVI k(1 − k ) R1 = (1 − k ) = 2L13 I O− 2 M R fL13 I L13
ρ− =
ΔvC11 / 2 kI O− T k 1 = = VC11 2C11VO− 2 fC11R1
σ 1− =
ΔvC12 / 2 I O− M 1 = = R VC12 2 fC12VI 2 fC12 R1
σ 2− =
ΔvC13 / 2 I O− M 1 = = R 2 fC13VI 2 fC13 R1 VC13
The variation ratio of the current iD10 and iL11 is ζ− =
ΔiL11 / 2 k(1 − k )VI T k(1 − k )R1 k R1 = = = I L11 2L11 I O− 2 M R fL11 M R 2 fL11
© 2006 by Taylor & Francis Group, LLC
(2.347)
194
Essential DC/DC Converters
The variation ratio of current vC10 is ε− =
ΔvC10 / 2 I O− k k 1 = = VC10 128 f 3 C11C10 L12 VO− 128 f 3 C11C10 L12 R1
(2.348)
Assuming that f = 50 kHz, L11 = L12 = 0.5 mH, C = CO = 20 μF, R1 = 160 Ω and k = 0.5, we obtain MR = 4, ζ– = 0.2, ρ– = 0.0016, σ1– = σ2– = 0.0125, ξ– = 0.00125 and ε– = 10–6. The output voltage VO– is almost a real DC voltage with very small ripple. Since the load is resistive, the output current iO–(t) is almost a real DC waveform with very small ripple as well, and it is equal to IO– = VO–/R1. 2.6.3.3 Discontinuous Conduction Mode The equivalent circuits of the discontinuous conduction mode are shown in Figure 2.69 and Figure 2.70. In order to obtain the mirror-symmetrical double output voltages, we purposely select zN = zN+ = zN– and ζ = ζ+ = ζ–. The freewheeling diode currents iD0 and iD10 become zero during switch off before next period switch on. The boundary between continuous and discontinuous conduction modes is ζ≥1 or k z ≥1 MR 2 N Hence, M R ≤ kz N
(2.349)
This boundary curve is shown in Figure 2.71. Comparing with Equations (2.321), (2.335), and (2.349), it can be seen that the boundary curve has a minimum value of MR that is equal to 3.0, corresponding to k = 1/3 . The filling efficiency mR is mR =
2 1 MR = ζ kz N
So VC1 = [1 +
© 2006 by Taylor & Francis Group, LLC
z 2k ]V = [1 + k 2 (1 − k ) N ]VI (1 − k )mR I 2
(2.350)
Voltage-Lift Converters
195
with kz N ≥
2 1− k
(2.351)
Therefore, the positive output voltage in discontinuous conduction mode is VO+ = VC1 + VI = [2 +
z 2k ]V = [2 + k 2 (1 − k ) N ]VI (1 − k )mR I 2 kz N ≥
with
2 1− k
(2.352)
For the current iL11 because inductor current iL13 = 0 at t = t1, so that VL13−off =
k V (1 − k )mR I
for the current iL11 we have kTVI = (1 – k)mRT(VC11 – 2VI – VL13–off) or VC11 = [2 +
z 2k ]V = [2 + k 2 (1 − k ) N ]VI (1 − k )mR I 2
with
kz N ≥
2 1− k
(2.353)
and for the current iL12 kT(VI + VC11 – VO–) = (1 – k)mRT(VO– – 2VI – VL13–off) Therefore, the negative output voltage in discontinuous conduction mode is VO− = [2 +
z 2k ]VI = [2 + k 2 (1 − k ) N ]VI (1 − k )mR 2
with
So VO = VO+ = VO− = [2 + k 2 (1 − k )
© 2006 by Taylor & Francis Group, LLC
zN ]V 2 I
kz N ≥
2 1− k
(2.354)
196
Essential DC/DC Converters
60 k=0.95
continuous mode
40 k=0.9
MR
20 k=0.8 10 6 4 3
k=0.5 k=0.33 k=0.1
2
27 32
discontinuous mode 50
125 R/fL
444
1684
FIGURE 2.72 The boundary between continuous and discontinuous modes and the output voltage vs. the normalized load zN = R/fL (re-lift circuit).
i.e., the output voltage will linearly increase during load resistance increasing. Larger load resistance may cause higher output voltage in discontinuous mode as shown in Figure 2.72.
2.6.4
Multiple-Lift Circuit
Referring to Figure 2.68, it is possible to build a multiple-lifts circuit only using the parts (L3-D20-C3-D3) multiple times in the positive conversion path, and using the parts (D22-L13-C13-D12) multiple times in the negative conversion path. For example, in Figure 2.73 the parts (L4-D4-C4-D5) and parts (D23L14-C14-D13) were added in the triple-lift circuit. According to this principle, triple-lift circuits and quadruple-lift circuits have been built as shown in Figure 2.73 and Figure 2.76. In this book it is not necessary to introduce the particular analysis and calculations one by one to readers. However, their calculation formulas are shown in this section. 2.6.4.1 Triple-Lift Circuit Triple-lift circuit is shown in Figure 2.73. The positive conversion path consists of a pump circuit S-L1-D0-C1 and a filter (C2)-L2-CO, and a lift circuit D1C2- D2-C3-D3-L3-D4-C4-D5-L4. The negative conversion path consists of a pump circuit S-L11-D10-(C11) and a “Π”-type filter C11-L12-C10, and a lift circuit D11C12-D22-C13-L13-D12-D23-L14-C14-D13.
© 2006 by Taylor & Francis Group, LLC
Voltage-Lift Converters
S
VS
iIN
iIN+ +
D2
D20
C3 i L1
–
– VC1 +
D4
VI N
197
iIN– i L11
L4
L3
D21
D0
C2
+
R
Co
Vo +
D5
–
D11
D12
+
L14
C13
C12
iL2
D3
L13
L11
Io+
L2
C1
C4
L1
D1
D13 + VC11
C11
D23
Vo –
iL12
D10–
D22
R1
C10
–
C14 L12
Io –
FIGURE 2.73 Triple-lift circuit.
2.6.4.1.1 Positive Conversion Path The lift circuit is D1-C2-D2-C3-D3-L3-D4-C4-D5-L4. Capacitors C2, C3, and C4 perform characteristics to lift the capacitor voltage VC1 by three times of source voltage VI. L3 and L4 perform the function as ladder joints to link the three capacitors C3 and C4 and lift the capacitor voltage VC1 up. Current iC2(t), iC3(t), and iC4(t) are exponential functions. They have large values at the moment of power on, but they are small because vC3 = vC4 = VI and vC2 = VO+ in steady state. The output voltage and current are VO+ =
3 V 1− k I
and
I O+ =
1− k I 3 I+
The voltage transfer gain in continuous mode is MT + =
VO+ 3 = 1− k VI
Other average voltages: VC1 =
2+k V 1− k I
© 2006 by Taylor & Francis Group, LLC
VC3 = VC4 = VI
VCO = VC2 = VO+
(2.355)
198
Essential DC/DC Converters
Other average currents: IL2 = IO+
1 1 II+ = I 3 1 − k O+
I L1 = I L 3 = I L 4 =
Current variations: ξ 1+ = ζ + =
k(1 − k )R k 3R = 2 MT fL MT 2 2 fL
χ 1+ =
k 3R MT 2 2 fL3
ξ 2+ =
χ 2+ =
k 1 16 f 2 C2 L2
k 3R MT 2 2 fL4
Voltage variations: ρ+ =
3 2(2 + k ) fC1R
σ 2+ =
MT 2 fC3 R
σ 1+ =
σ 3+ =
k 2 fC2 R
MT 2 fC4 R
The variation ratio of output voltage VC0 is ε+ =
k 1 128 f 3 C2 CO L2 R
(2.356)
2.6.4.1.2 Negative Conversion Path Circuit C12-D11-L13-D22-C13-D12-L14-D23-C14-D13 is the lift circuit. Capacitors C12, C13, and C14 perform characteristics to lift the capacitor voltage VC11 by three times the source voltage VI. L13 and L14 perform the function as ladder joints to link the three capacitors C12, C13, and C14 and lift the capacitor voltage VC11 up. Current iC12(t), iC13(t) and iC14(t) are exponential functions. They have large values at the moment of power on, but they are small because vC12 = vC13 = vC14 ≅ VI in steady state. The output voltage and current are VO− =
3 V 1− k I
and
I O− =
1− k I 3 I−
The voltage transfer gain in continuous mode is MT − = VO− / VI =
© 2006 by Taylor & Francis Group, LLC
3 1− k
(2.357)
Voltage-Lift Converters
199
30
24
MT
18
12
6
0 0
0.2
0.6
0.4
0.8
1
k
FIGURE 2.74 Voltage transfer gain MT vs. k.
From Equation (2.355) and Equation (2.357) we define MT = MT+ = MT–. The curve of MT vs. k is shown in Figure 2.74. Other average voltages: VC11 = VO–
VC12 = VC13 = VC14 = VI
Other average currents: IL12 = IO–
I L11 = I L13 = I L14 =
1 I 1 − k O−
Current variation ratios: k 3R1 MT 2 2 fL11
ζ− =
χ 1− =
k(1 − k ) R1 2 MT fL13
ξ 2− =
k 1 16 f 2 C10 L12
χ 2− =
k(1 − k ) R1 2 MT fL14
Voltage variation ratios: ρ− =
σ 2− =
© 2006 by Taylor & Francis Group, LLC
k 1 2 fC11R1 MT 1 2 fC13 R1
σ 1− =
MT 1 2 fC12 R1
σ 3− =
MT 1 2 fC14 R1
200
Essential DC/DC Converters
The variation ratio of output voltage VC10 is ε− =
k 1 128 f 3 C11C10 L12 R1
(2.358)
2.6.4.1.3 Discontinuous Mode To obtain the mirror-symmetrical double output voltages, we purposely select: L1 = L11 and R = R1. Define: VO = VO+ = VO−
MT = MT + = MT − =
VO 3 = VI 1 − k
zN = zN+ = zN–
ζ = ζ+ = ζ–
and
The free-wheeling diode currents iD0 and iD10 become zero during switch off before next period switch on. The boundary between continuous and discontinuous conduction modes is ζ≥1 Then MT ≤
3kz N 2
(2.359)
This boundary curve is shown in Figure 2.75. Comparing Equation (2.321), Equation (2.335), Equation (2.349), and Equation (2.359), it can be seen that the boundary curve has a minimum value of MT that is equal to 4.5, corresponding to k = 1/3 . In discontinuous mode the currents iD0 and iD10 exist in the period between kT and [k + (1 – k)mT]T, where mT is the filling efficiency that is mT =
2 1 2 MT = ζ 3kz N
(2.360)
Considering Equation (2.359), therefore 0 < mT < 1. Since the current iD0 becomes zero at t = t1 = [k + (1 – k)mT]T, for the current iL1, iL3 and iL4 3kTVI = (1 – k)mTT(VC1 – 2VI) or VC1 = [2 +
© 2006 by Taylor & Francis Group, LLC
z 3k ]VI = [2 + k 2 (1 − k ) N ]VI (1 − k )mT 2
Voltage-Lift Converters
201
40 continuous mode
MT
30 k=0.9
15
6 4.5
k=0.8
k=0.5 discontinuous mode
k=0.33 k=0.1
3
40 48
75
188
667
R/fL
FIGURE 2.75 The boundary between continuous and discontinuous modes and the output voltage vs. the normalized load zN = R/fL (triple-lift circuit).
3kz N 3 ≥ 2 1− k
with
(2.361)
Therefore, the positive output voltage in discontinuous mode is VO+ = VC1 + VI = [3 +
z 3k ]VI = [3 + k 2 (1 − k ) N ]VI (1 − k )mT 2 3kz N 3 ≥ 2 1− k
with
Because inductor current iL11 = 0 at t = t1, so that VL13−off = VL14−off =
k V (1 − k )mT I
Since iD10 becomes 0 at t1 = [k + (1 – k)mT]T, for the current iL11, kTVI = (1 – k)mT–T(VC11 – 3VI – VL13–off – VL14–off) VC11 = [3 +
© 2006 by Taylor & Francis Group, LLC
z 3k ]VI = [3 + k 2 (1 − k ) N ]VI (1 − k )mT 2
(2.362)
202
Essential DC/DC Converters
3kz N 3 ≥ 2 1− k
with
(2.363)
for the current iL12 kT(VI + VC14 – VO–) = (1 – k)mT–T(VO– – 2VI – VL13–off – VL14–off) Therefore, the negative output voltage in discontinuous mode is VO− = [3 +
z 3k ]VI = [3 + k 2 (1 − k ) N ]VI (1 − k )mT 2 3kz N 3 ≥ 2 1− k
with
(2.364)
So VO = VO+ = VO− = [3 + k 2 (1 − k )
zN ]V 2 I
i.e., the output voltage will linearly increase during load resistance increasing, as shown in Figure 2.75. 2.6.4.2
Quadruple-Lift Circuit
Quadruple-lift circuit is shown in Figure 2.76. The positive conversion path consists of a pump circuit S-L1-D0-C1 and a filter (C2)-L2-CO, and a lift circuit D1-C2-L3-D2-C3-D3-L4-D4-C4-D5-L5-D6-C5-S1. The negative conversion path consists of a pump circuit S-L11-D10-(C11) and a “Π”-type filter C11-L12-C10, and a lift circuit D11-C12-D22-L13-C13-D12-D23-L14-C14-D13-D24-L15-C15-D14. 2.6.4.2.1 Positive Conversion Path Capacitors C2, C3, C4, and C5 perform characteristics to lift the capacitor voltage VC1 by four times the source voltage VI. L3, L4, and L5 perform the function as ladder joints to link the four capacitors C2, C3, C4, and C5, and lift the output capacitor voltage VC1 up. Current iC2(t), iC3(t), iC4(t) and iC5(t) are exponential functions. They have large values at the moment of power on, but they are small because vC3 = vC4 = vC5 = VI and vC2 = VO+ in steady state. The output voltage and current are VO+ =
© 2006 by Taylor & Francis Group, LLC
4 V 1− k I
and
I O+ =
1− k I 4 I+
Voltage-Lift Converters
S iIN
VS
D20 D2
iIN+ +
D4
C3
VI N
203
iL1
–
L5
D3
D5
D7
D11
D12
D13
L11
D21
L4
D1
Io+
L2
C1
C5
L3
iIN– iL11
D6
C4
L1
– VC1 +
iL2
D0
C2
+
R
Co
– D14 + VC11 –
L13
C12
C11
R1
C10
C14
D23
+ Vo –
C13
D22
Vo +
– C15
D24
iL12
D10
L12
Io –
FIGURE 2.76 Quadruple-lift circuit.
The voltage transfer gain in continuous mode is MQ+ =
VO+ 4 = 1− k VI
(2.365)
Other average voltages: VC1 =
3+k V 1− k I
VC3 = VC4 = VC5 = VI
VCO = VC2 = VO
Other average currents: IL2 = IO+
I L1 = I L 3 = I L 4 = I L 5 =
1 1 I = I 4 I + 1 − k O+
Current variations: ξ 1+ = ζ + =
χ 1+ =
k(1 − k )R k 2R = 2 MQ fL MQ 2 fL
k 2R MQ 2 fL3
© 2006 by Taylor & Francis Group, LLC
χ 2+ =
k 2R MQ 2 fL4
ξ 2+ =
1 k 16 f 2 C2 L2
χ 3+ =
k 2R MQ 2 fL5
204
Essential DC/DC Converters
Voltage variations: ρ+ =
σ 2+ =
2 (3 + 2 k ) fC1R
MQ 2 fC3 R
σ 3+ =
MQ
σ 1+ = MQ
2 fC4 R
2 fC2 R σ 4+ =
MQ 2 fC5 R
The variation ratio of output voltage VC0 is ε+ =
k 1 3 128 f C2 C0 L2 R
(2.366)
2.6.4.2.2 Negative Conversion Path Capacitors C12, C13, C14, and C15 perform characteristics to lift the capacitor voltage VC11 by four times the source voltage VI. L13, L14, and L15 perform the function as ladder joints to link the four capacitors C12, C13, C14, and C15 and lift the output capacitor voltage VC11 up. Current iC12(t), iC13(t), iC14(t), and iC15(t) are exponential functions. They have large values at the moment of power on, but they are small because vC12 = vC13 = vC14 = vC15 ≅ VI in steady state. The output voltage and current are VO− =
4 V 1− k I
and
I O− =
1− k I 4 I−
The voltage transfer gain in continuous mode is MQ− = VO− / VI =
4 1− k
(2.367)
From Equation (2.365) and Equation (2.367) we define MQ = MQ+ = MQ–. The curve of MQ vs. k is shown in Figure 2.77. Other average voltages: VC10 = VO–
VC12 = VC13 = VC14 = VC15 = VI
Other average currents: IL12 = IO–
© 2006 by Taylor & Francis Group, LLC
I L11 = I L13 = I L14 = I L15 =
1 I 1 − k O−
Voltage-Lift Converters
205
40
32
MQ
24
16
8
0 0
0.2
0.6
0.4
1
0.8
k
FIGURE 2.77 Voltage transfer gain MQ vs. k.
Current variation ratios: ζ− =
χ 1− =
k(1 − k ) R1 2 MQ fL13
k 2 R1 MQ 2 fL11 χ 2− =
k 1 16 f 2 CL12
ξ− =
k(1 − k ) R1 2 MQ fL14
χ 3− =
k(1 − k ) R1 2 MQ fL15
Voltage variation ratios: ρ− =
σ 2− =
MQ
1
2 fC13 R1
k 1 2 fC11R1 σ 3− =
σ 1− = MQ
MQ
1
2 fC12 R1
1
2 fC14 R1
σ 4− =
MQ
1 2 fC15 R1
The variation ratio of output voltage VC10 is ε− =
k 1 3 128 f C11C10 L12 R1
The output voltage ripple is very small.
© 2006 by Taylor & Francis Group, LLC
(2.368)
206
Essential DC/DC Converters 60 50 40
continuous mode
k=0.9
30 k=0.8
MQ
20
10 8 6
k=0.5 k=0.33 discontinuous mode
k=0.1 2
54 64
100
250 R/fL
889
FIGURE 2.78 The boundary between continuous and discontinuous modes and the output voltage vs. the normalized load zN = R/fL (quadruple-lift circuit).
2.6.4.2.3 Discontinuous Conduction Mode In order to obtain the mirror-symmetrical double output voltages, we purposely select: L1 = L11 and R = R1. Therefore, we may define VO = VO+ = VO− and
MQ = MQ+ = MQ− =
VO 4 = VI 1 − k
zN = zN+ = zN–
ζ = ζ+ = ζ–
The free-wheeling diode currents iD0 and iD10 become zero during switch off before next period switch on. The boundary between continuous and discontinuous conduction modes is ζ≥1 or
MQ ≤ 2 kz N
(2.369)
This boundary curve is shown in Figure 2.78. Comparing Equations (2.321), (2.335), (2.349), (2.359), and (2.369), it can be seen that this boundary curve has a minimum value of MQ that is equal to 6.0, corresponding to k = 1/3 .
© 2006 by Taylor & Francis Group, LLC
Voltage-Lift Converters
207
In discontinuous mode the currents iD0 and iD10 exist in the period between kT and [k + (1 – k)mQ]T, where mQ is the filling efficiency that is mQ =
2 1 MQ = ζ 2 kz N
(2.370)
Considering Equation (2.369), therefore 0 < mQ < 1. Since the current iD0 becomes zero at t = t1 = kT + (1 – k)mQT, for the current iL1, iL3, iL4 and iL5 4kTVI = (1 – k)mQT(VC1 – 3VI) VC1 = [3 +
z 4k ]V = [3 + k 2 (1 − k ) N ]VI (1 − k )mQ I 2 2 kz N ≥
with
4 1− k
(2.371)
Therefore, the positive output voltage in discontinuous conduction mode is VO+ = VC1 + VI = [4 +
z 4k ]VI = [4 + k 2 (1 − k ) N ]VI (1 − k )mQ 2 2 kz N ≥
with
4 1− k
(2.372)
Because inductor current iL11 = 0 at t = t1, so that VL13−off = VL14−off = VL15−off =
k V (1 − k )mQ I
Since the current iD10 becomes zero at t = t1 = kT + (1 – k)mQT, for the current iL11 we have kTVI = (1 – k)mQ–T(VC11 – 4VI – VL13–off – VL14–off – VL15–off) So VC11 = [4 +
with
© 2006 by Taylor & Francis Group, LLC
z 4k ]VI = [4 + k 2 (1 − k ) N ]VI (1 − k )mQ 2 2 kz N ≥
4 1− k
(2.373)
208
Essential DC/DC Converters
TABLE 2.3 Comparison among Five Circuits of Double Output Luo-Converters Double Output Luo-Converters
IO
Elementary Circuit
IO =
Self-Lift Circuit
I O = (1 − k )I S
Re-Lift Circuit
IO =
Triple-Lift Circuit Quadruple-Lift Circuit
VO
k = 0.33
k = 0.5
VO(VS = 10 V) k = 0.75 k = 0.9
VO =
k V 1− k S
5V
10 V
30 V
90 V
VO =
1 V 1− k S
15 V
20 V
40 V
100 V
1− k I 2 S
VO =
2 V 1− k S
30 V
40 V
80 V
200 V
IO =
1− k I 3 S
VO =
3 V 1− k S
45 V
60 V
120 V
300 V
IO =
1− k I 4 S
VO =
4 V 1− k S
60 V
80 V
160 V
400 V
1− k I k S
For the current iL12 kT(VI + VC15 – VO–) = (1 – k)mQT(VO– – 2VI – VL13–off – VL14–off – VL15–off) Therefore, the negative output voltage in discontinuous conduction mode is VO− = [4 +
z 4k ]V = [4 + k 2 (1 − k ) N ]VI (1 − k )mQ I 2 2 kz N ≥
with
4 1− k
(2.374)
So VO = VO+ = VO− = [4 + k 2 (1 − k )
zN ]V 2 I
i.e., the output voltage will linearly increase during load resistance increasing, as shown in Figure 2.78.
2.6.5
Summary
2.6.5.1 Positive Conversion Path From the analysis and calculation in previous sections, the common formulae can be obtained for all circuits:
© 2006 by Taylor & Francis Group, LLC
Voltage-Lift Converters
209
M=
VO+ I = I+ VI I O+ L=
VO+ I O+
R fL
R=
ξ 2+ =
k R 2 ME fL2
zN =
L1L2 L1 + L2
for elementary circuits only; L = L1 for other lift circuit’s current variations: ξ 1+ =
1− k R 2 ME fL1
and
for elementary circuit only; ξ 1+ = ζ + =
k(1 − k )R 2 MfL
ξ 2+ =
and
k 1 16 f 2 C2 L2
for other lift circuits ζ+ =
k(1 − k )R 2 MfL
χ j+ =
k R 2 M fL j + 2
(j = 1, 2, 3, …)
Voltage variations are ρ+ =
k 2 fC1R
ε+ =
k 1 8 ME f 2 C0 L2
for elementary circuit only; ρ+ =
M 1 M − 1 2 fC1R
ε+ =
k 1 3 128 f C2 C0 L2 R
for other lift circuits σ 1+ =
k 2 fC2 R
© 2006 by Taylor & Francis Group, LLC
σ j+ =
M 2 fC j +1R
(j = 2, 3, 4, …)
210
Essential DC/DC Converters
2.6.5.2 Negative Conversion Path From the analysis and calculation in previous sections, the common formulae can be obtained for all circuits: M=
VO− I = I− VI I O−
zN − =
R1 fL11
R1 =
VO− I O−
Current variation ratios: ζ− =
k(1 − k )R1 2 MfL11
ξ− =
k 16 f 2 C11L12
χ j− =
k(1 − k )R1 2 MfL j + 2
(j = 1, 2, 3, …)
Voltage variation ratios: ρ− =
k 2 fC11R1
ε− =
k 128 f 3 C11C10 L12 R1
σ j− =
M 2 fC j +11R1
(j = 1, 2, 3, 4, …)
2.6.5.3 Common Parameters Usually, we select the loads R = R1, L = L11, so that we have got zN = zN+ = zN–. In order to write common formulas for the boundaries between continuous and discontinuous modes and output voltage for all circuits, the circuits can be numbered. The definition is that subscript 0 means the elementary circuit, subscript 1 means the self-lift circuit, subscript 2 means the re-lift circuit, subscript 3 means the triple-lift circuit, subscript 4 means the quadruple-lift circuit, and so on. The voltage transfer gain is Mj =
k h( j ) [ j + h( j )] 1− k
j = 0, 1, 2, 3, 4, …
The characteristics of output voltage of all circuits are shown in Figure 2.79. The free-wheeling diode current’s variation is ζj =
k [1+ h( j )] j + h( j ) zN 2 Mj2
The boundaries are determined by the condition: ζj ≥ 1 or
© 2006 by Taylor & Francis Group, LLC
Voltage-Lift Converters
211
120
output voltage, vo, v
100 80
(i)
50 (ii) (iii)
30
(iv)
10
(v)
0.2
0
0.6 0.4 conduction duty k
0.8
1
FIGURE 2.79 Output voltages of all double output Luo-converters (Vi = 10 V).
k [1+ h( j )] j + h( j ) zN ≥ 1 2 Mj2
j = 0, 1, 2, 3, 4, …
Therefore, the boundaries between continuous and discontinuous modes for all circuits are Mj = k
1+ h ( j ) 2
j + h( j ) zN 2
j = 0, 1, 2, 3, 4, …
The filling efficiency is 2
Mj 1 2 1 mj = = [1+ h( j )] j + h( j ) z N ζj k
j = 0, 1, 2, 3, 4, …
The output voltage in discontinuous mode for all circuits VO− j = [ j + k [ 2− h( j )]
1− k z ]V 2 N I
where ⎧0 h( j ) = ⎨ ⎩1
if if
j≥1 j=0
© 2006 by Taylor & Francis Group, LLC
j = 0, 1, 2, 3, 4, …
h(j) is the Hong Function
212
Essential DC/DC Converters
voltage transfer gain M
continuous mode
MQ MT
6 4.5
MR
3 Ms 1.5 ME discontinuous mode 0.5
4.5
27 40.5 54 13.5 normalized load zN R/fL
FIGURE 2.80 Boundaries between continuous and discontinuous modes of all double output Luo-Converters.
The boundaries between continuous and discontinuous modes of all circuits are shown in Figure 2.80. The curves of all M vs. zN state that the continuous mode area increases from ME via MS, MR, MT to MQ. The boundary of elementary circuit is a monorising curve, but other curves are not monorising. There are minimum values of the boundaries of other curves, which of MS, MR, MT, and MQ correspond at k = 1/3 .
Bibliography Chen, X.F., Luo, F.L., and Ye, H., Modified positive output Luo converters, in Proceedings of the IEEE International Conference PEDS’99, Hong Kong, 1999, p. 450. Farkas, T. and Schecht, M.F., Viability of active EMI filters for utility applications, IEEE Transactions on Power Electronics, 9, 328, 1994. Hart, D., Introduction to Power Electronics, Prentice Hall, New York, 1997. Jozwik, J.J. and Kazimerczuk, M.K., Dual sepic PWM switching-mode DC/DC power converter, IEEE Transactions on Industrial Electronics, 36, 64, 1989. Krein, P., Elements of Power Electronics, Oxford University Press, 1998. LaWhite, I.E. and Schlecht, M.F., Active filters for 1 MHz power circuits with strict input/output ripple requirements, IEEE Transactions on Power Electronics, 2, 282, 1987. Luo, F.L., Positive output Luo-converters: voltage lift technique, IEE Proceedings on Electric Power Applications, 146, 415, 1999. Luo, F.L., Negative output Luo-Converters: voltage lift technique, IEE Proceedings on Electric Power Applications, 146, 208, 1999. Luo, F.L., Double output Luo-converters: advanced voltage lift technique, IEE-Proceedings on Electric Power Applications, 147, 469, 2000.
© 2006 by Taylor & Francis Group, LLC
Voltage-Lift Converters
213
Luo, F.L., Six self-lift DC/DC converters: voltage-lift technique, IEEE- Transactions on Industrial Electronics, 48, 1268, 2001. Luo, F.L., Seven self-lift DC/DC converters: voltage-lift technique, IEE Proceedings on Electric Power Applications, 148, 329, 2001. Martins, D.C., Application of the Zeta converter in switch-mode power supplies, in Proceedings of IEEE APEC’93, U.S., 1993, p. 214. Maksimovic, D. and Cúk, S., A general approach to synthesis and analysis of quasiresonant converters, IEEE Transactions on Power Electronics, 6, 127, 1991. Maksimovic, D. and Cúk, S., Constant-frequency control of quasi-resonant converters, IEEE Transactions on Power Electronics, 6, 141, 1991. Massey, R.P. and Snyder, E.C., High voltage single-ended DC-DC converter, in Proceedings of IEEE PESC’77, Record. 1977, p. 156. Newell, W.E., Power electronics-emerging from limbo, in Proceedings of IEEE Power Electronics Specialists Conference, U.S., 1993, p. 6. Walker, J., Design of practical and effective active EMI filters, in Proceedings of Poweron 11, U.S., 1984, p. 1. Zhu, W., Perreault, D.J., Caliskan, V., Neugebauer, T.C., Guttowski, S., and Kassakian, J.G., Design and evaluaion of an active ripple filter with Rogowski-coil current sensing, in Proceedings of the 30th Annual IEEE-PESC’99, U.S., 1999, p. 874.
© 2006 by Taylor & Francis Group, LLC
3 Positive Output Super-Lift Luo-Converters
Voltage lift (VL) technique has been successfully employed in design of DC/ DC converters, e.g., three series Luo-converters. However, the output voltage increases in arithmetic progression. Super-lift (SL) technique is more powerful than VL technique, its voltage transfer gain can be a very large number. SL technique implements the output voltage increasing in geometric progression. It effectively enhances the voltage transfer gain in power series.
3.1
Introduction
This chapter introduces positive output super lift Luo-converters. In order to differentiate these converters from existing VL converters, these converters are called positive output super-lift Luo-converters. There are several subseries: • Main series — Each circuit of the main series has only one switch S, n inductors for nth stage circuit, 2n capacitors, and (3n – 1) diodes. • Additional series — Each circuit of the additional series has one switch S, n inductors for nth stage circuit, 2(n + 1) capacitors, and (3n + 1) diodes. • Enhanced series — Each circuit of the enhanced series has one switch S, n inductors for nth stage circuit, 4n capacitors, and (5n – 1) diodes. • Re-enhanced series — Each circuit of the re-enhanced series has one switch S, n inductors for nth stage circuit, 6n capacitors, and (7n – 1) diodes. • Multiple (j)-enhanced series — Each circuit of the multiple (j times)enhanced series has one switch S, n inductors for nth stage circuit, 2(1 + j)n capacitors and [(3 + 2j)n – 1] diodes. In order to concentrate the voltage enlargement, assume the converters are working in steady state with continuous conduction mode (CCM). The conduction duty ratio is k, switch frequency is f, switch period is T = 1/f, 215
© 2006 by Taylor & Francis Group, LLC
216
Essential DC/DC Converters
the load is resistive load R. The input voltage and current are Vin and Iin, out voltage and current are VO and IO. Assume no power losses during the conversion process, Vin × Iin = VO × IO. The voltage transfer gain is G: G=
3.2
VO Vin
Main Series
The first three stages of positive output super-lift Luo-converters — main series — are shown in Figure 3.1 through Figure 3.3. For convenience, they are called elementary circuits, re-lift circuit, and triple-lift circuit respectively, and are numbered as n = 1, 2, and 3.
3.2.1
Elementary Circuit
The elementary circuit and its equivalent circuits during switch-on and -off are shown in Figure 3.1. The voltage across capacitor C1 is charged to Vin. The current iL1 flowing through inductor L1 increases with voltage Vin during switch-on period kT and decreases with voltage –(VO – 2Vin) during switchoff period (1 – k)T. Therefore, the ripple of the inductor current iL1 is ΔiL1 =
V − 2Vin Vin kT = O (1 − k )T L1 L1
(3.1)
2−k V 1 − k in
(3.2)
VO 2 − k = Vin 1 − k
(3.3)
VO = The voltage transfer gain is G=
The input current iin is equal to (iL1 + iC1) during switch-on, and only equal to iL1 during switch-off. Capacitor current iC1 is equal to iL1 during switchoff. In steady–state, the average charge across capacitor C1 should not change. The following relations are obtained: iin−off = iL1−off = iC1−off
© 2006 by Taylor & Francis Group, LLC
iin−on = iL1−on + iC1−on
kTiC1−on = (1 − k )TiC1−off
Positive Output Super-Lift Luo-Converters Iin
D1
217
D2 IO
+ L1
+ VC1
C1
–
Vin
+ C2
VC2
+ R
–
S
VO –
–
(a) Circuit diagram Iin IO + + Vin
L1
Vin C2
C1
+ VC2
R
–
–
+ VO –
–
(b) Equivalent circuit during switching-on I in
L1
+
VL1
C1 IO – Vin +
Vin
C2
+ VC2
R
–
+ VO –
–
(c) Equivalent circuit during switching-off FIGURE 3.1 Elementary circuit.
If inductance L1 is large enough, iL1 is nearly equal to its average current IL1. Therefore, iin−off = iC1−off = I L1
iin−on = I L1 +
I 1− k I L1 = L1 k k
iC1−on =
1− k I k L1
and average input current I in = kiin−on + (1 − k )iin−off = I L1 + (1 − k )I L1 = (2 − k )I L1 Considering Vin 1 − k 2 VO 1− k 2 =( ) =( ) R I in 2 − k IO 2−k
© 2006 by Taylor & Francis Group, LLC
(3.4)
218
Essential DC/DC Converters Iin
D1
D2
D4
V1
D5 IO
+ VC1 –
+ L1
C1
L2
+ VC3 –
C3
+
Vin
R D3
+ VC2 –
C2
–
S
VO –
+ VC4 –
C4
(a) Circuit diagram Iin
V1
IO
+ Vin L1
C1
+ Vin
+ VC2 L2
C2
–
–
+ VC4 R –
+ C4 V1
C3
–
+ VO –
–
(b) Equivalent circuit during switching-on Iin
L1
C1
V1
L2
C3 IO
+
VL1
Vin
– Vin + C2
VL2 + V1
– V1 + C4
+ VC4
–
R
–
+ VO –
–
(c) Equivalent circuit during switching-off FIGURE 3.2 Re-lift circuit.
the variation ratio of current iL1 through inductor L1 is ξ1 =
ΔiL1 / 2 k(2 − k )TVin k(1 − k )2 R = = I L1 2L1 I in 2(2 − k ) fL1
(3.5)
Usually ξ1 is small (much lower than unity), it means this converter normally works in the continuous mode. The ripple voltage of output voltage vO is ΔvO =
ΔQ IO kT k VO = = C2 C2 fC2 R
Therefore, the variation ratio of output voltage vO is ε=
© 2006 by Taylor & Francis Group, LLC
ΔvO / 2 k = VO 2 RfC2
(3.6)
Positive Output Super-Lift Luo-Converters I in
D1
D2
219
D4
V1
D5
D7
V2
D8
IO +
L1
+
C1
+
VC1
Vin
C3
L2
+
VC3
C5
L3
VC5
_
_
+
_
R
_
D3
D6
+
C2
VC4
_
+
S
+
C4
VC2
VO
C6
_
_
VC6
_
(a) Circuit diagram
I in
V1 +
+
Vin L1
C1
_
+
Vin C2
L2 V1
_
_
IO
V2 +
+
C3
V1
L3 V2
_
_
C4
C6
+
C5
V2
+
+
VC6
VO
R
_
_
_
(b) Equivalent circuit during switch-on
I in +
L1
V L1
C1 _
Vin
Vin
V1
V L2
+
C2
_
L2
+
V1
C3 _
V1
V2
V L3
+
C4
_
L3
C5 _
+
V2
V2
IO +
C6
+
+
VC 6 _
R
VO
_ _
(c) Equivalent circuit during switch-off FIGURE 3.3 Triple-lift circuit.
Usually R is in kΩ, f in 10 kHz, and C2 in μF, this ripple is smaller than 1%. 3.2.2
Re-Lift Circuit
The re-lift circuit is derived from elementary circuit by adding the parts (L2D3-D4-D5-C3-C4). Its circuit diagram and equivalent circuits during switchon and -off are shown in Figure 3.2. The voltage across capacitor C1 is charged to Vin. As described in previous section the voltage V1 across capacitor C2 is V1 =
2−k V 1 − k in
The voltage across capacitor C3 is charged to V1. The current flowing through inductor L2 increases with voltage V1 during switch-on period kT and decreases with voltage –(VO – 2V1) during switch-off period (1 – k)T. Therefore, the ripple of the inductor current iL2 is
© 2006 by Taylor & Francis Group, LLC
220
Essential DC/DC Converters
ΔiL 2 =
V − 2V1 V1 kT = O (1 − k )T L2 L2
(3.7)
2−k 2−k 2 V =( ) Vin 1− k 1 1− k
(3.8)
VO 2−k 2 =( ) Vin 1− k
(3.9)
VO = The voltage transfer gain is
G=
Similarly, the following relations are obtained: ΔiL1 =
Vin kT L1
I L1 =
I in 2−k
ΔiL2 =
V1 kT L2
I L2 = (
I 2−k − 1)I O = O 1− k 1− k
Therefore, the variation ratio of current iL1 through inductor L1 is ξ1 =
ΔiL1 / 2 k(2 − k )TVin k(1 − k ) 4 R = = I L1 2L1 I in 2(2 − k ) 3 fL1
(3.10)
The variation ratio of current iL2 through inductor L2 is ξ2 =
ΔiL 2 / 2 k(1 − k )TV1 k(1 − k )2 TVO k(1 − k )2 R = = = I L2 2L2 I O 2(2 − k )L2 I O 2(2 − k ) fL2
(3.11)
and the variation ratio of output voltage vO is ε=
3.2.3
ΔvO / 2 k = VO 2 RfC4
(3.12)
Triple-Lift Circuit
Triple-lift circuit is derived from re-lift circuit by double adding the parts (L2-D3-D4-D5-C3-C4). Its circuit diagram and equivalent circuits during switchon and -off are shown in Figure 3.3. The voltage across capacitor C1 is charged to Vin . As described before the voltage V 1 across capacitor C 2 is 2 V1 = (2 − k 1 − k )Vin , and voltage V2 across capacitor C4 is V2 = (2 − k 1 − k ) Vin .
© 2006 by Taylor & Francis Group, LLC
Positive Output Super-Lift Luo-Converters
221
The voltage across capacitor C5 is charged to V2. The current flowing through inductor L3 increases with voltage V2 during switch-on period kT and decreases with voltage –(VO – 2V2) during switch-off (1 – k)T. Therefore, the ripple of the inductor current iL2 is V − 2V2 V2 kT = O (1 − k )T L3 L3
(3.13)
2−k 2−k 2 2−k 3 V =( ) V1 = ( ) Vin 1− k 2 1− k 1− k
(3.14)
ΔiL 3 =
VO =
The voltage transfer gain is G=
VO 2−k 3 =( ) Vin 1− k
ΔiL1 =
Vin kT L1
I L1 =
I in 2−k
ΔiL2 =
V1 kT L2
I L2 =
2−k I (1 − k )2 O
ΔiL3 =
V2 kT L3
I L3 =
IO 1− k
(3.15)
Analogously,
Therefore, the variation ratio of current iL1 through inductor L1 is ξ1 =
ΔiL1 / 2 k(2 − k )TVin k(1 − k )6 R = = I L1 2L1 I in 2(2 − k ) 5 fL1
(3.16)
The variation ratio of current iL2 through inductor L2 is ξ2 =
ΔiL 2 / 2 k(1 − k ) 2 TV1 kT (2 − k ) 4 VO k(2 − k ) 4 R = = = 2(2 − k )L2 I O 2(1 − k ) 3 L2 I O 2(1 − k ) 3 fL2 I L2
(3.17)
The variation ratio of current iL3 through inductor L3 is ξ3 =
ΔiL 3 / 2 k(1 − k )TV2 k(1 − k )2 TVO k(1 − k )2 R = = = 2 L3 I O 2(2 − k )L2 I O 2(2 − k ) fL3 I L3
© 2006 by Taylor & Francis Group, LLC
(3.18)
222
Essential DC/DC Converters
and the variation ratio of output voltage vO is ε=
3.2.4
ΔvO / 2 k = VO 2 RfC6
(3.19)
Higher Order Lift Circuit
Higher order lift circuit can be designed by just multiple repeating the parts (L2-D3-D4-D5-C3-C4). For nth order lift circuit, the final output voltage across capacitor C2n is VO = (
2−k n ) Vin 1− k
The voltage transfer gain is G=
VO 2−k n =( ) Vin 1− k
(3.20)
The variation ratio of current iLi through inductor Li (i = 1, 2, 3, …n) is ξi =
ΔiLi / 2 k(1 − k )2( n−i +1) R = I Li 2(2 − k )2( n−i )+1 fLi
(3.21)
and the variation ratio of output voltage vO is ε=
3.3
ΔvO / 2 k = 2 RfC2 n VO
(3.22)
Additional Series
Using two diodes and two capacitors (D11-D12-C11-C12), a circuit called double/ enhance circuit (DEC) can be constructed, which is shown in Figure 3.4, which is same as the Figure 1.22 but with components renumbered. If the input voltage is Vin, the output voltage VO can be 2Vin, or other value that is higher than Vin. The DEC is very versatile to enhance DC/DC converter’s voltage transfer gain. All circuits of positive output super-lift Luo-converters-additional series are derived from the corresponding circuits of the main series by adding a DEC. The first three stages of this series are shown in Figure 3.5 to Figure 3.7.
© 2006 by Taylor & Francis Group, LLC
Positive Output Super-Lift Luo-Converters Iin
D11
223
D12 IO + VC11
+ C11
– Vin
R S
+ VC12 –
C12
–
+ VO –
FIGURE 3.4 Double/enhanced circuit (DEC).
For convenience they are called elementary additional circuit, re-lift additional circuit, and triple-lift additional circuit respectively, and numbered as n = 1, 2, and 3.
3.3.1
Elementary Additional Circuit
This circuit is derived from elementary circuit by adding a DEC. Its circuit and switch-on and -off equivalent circuits are shown in Figure 3.5. The voltage across capacitor C1 is charged to Vin and voltage across capacitor C2 and C11 is charged to V1. The current iL1 flowing through inductor L1 increases with voltage Vin during switch-on period kT and decreases with voltage –(VO – 2Vin) during switch-off (1 – k)T. Therefore, V1 =
2−k V 1 − k in
(3.23)
VL1 =
k V 1 − k in
(3.24)
and
The output voltage is VO = Vin + VL1 + V1 =
3−k V 1 − k in
(3.25)
The voltage transfer gain is G=
© 2006 by Taylor & Francis Group, LLC
VO 3 − k = Vin 1 − k
(3.26)
224
Essential DC/DC Converters Iin
D1
D2
V1 D11
D12 IO
+ VC1
+ L1
C1
+ VC11
C11
–
–
Vin
+ VO
R S
+ VC2 –
C2
–
–
+ VC12 –
C12
(a) Circuit diagram Iin
V1
IO
+ + C11 V1
+ Vin
L1
C1
Vin
C2
+
–
–
+ VC12 R –
C12
V1 –
+ VO –
– (b) Equivalent circuit during switching–on C11 Iin +
IO
L1 – VL1 +
C1
– V1 + V1
– Vin + C2
Vin –
+ V1
C12
+ VC12 –
R
+ VO –
– (c) Equivalent circuit during switching–off
FIGURE 3.5 Elementary additional (enhanced) circuit.
The following relations are derived: iin−off = I L1 = iC11−off + iC1−off =
2IO 1− k
iin−on = iL1−on + iC1−on = I L1 +
IO k
iC1−on =
I 1− k iC1−off = O k k
iC1−off = iC 2−off =
iC 2−off =
I k k i = i = O 1 − k C 2−on 1 − k C11−on 1 − k
iC11−on =
I 1− k i = O k C11−off k
iC12−off =
kI k iC12−on = O 1− k 1− k
iC11−off = I O + iC12−off = I O +
© 2006 by Taylor & Francis Group, LLC
I k iC12−on = O 1− k 1− k
IO 1− k
Positive Output Super-Lift Luo-Converters I In
D1
V1
D2
225
D5 V2 D11
D4
D12
IO +
L1
+
C1
+
L2
VC1
Vin
C3
+
VC3
_
C11
VC11
_
+
_
R
_
D3
+
C2
S
+
C4
V_C2
VO
+
VC4
C12
_
_
VC12
_
(a) Circuit diagram
I in V1 +
+
C1
Vin
L1
_
+
C2 Vin
L2 V1
_
IO
V2 C3
+
+
C4
V1
_
+
C11 V2
+
+
VC12 _
VO
R
_
_
_
C12
V2
_
(b) Equivalent circuit during switching–on C11
I in
L1 _
+
Vin
L2
V1
C1
V1 +
Vin
C2
_
V2
IO
+
V2
_
+
_
C3 +
+
+
V1
V2
_
_
C12 C4
+
VC12 R
VO
_
(c) Equivalent circuit during switching–off FIGURE 3.6 Re-lift additional circuit.
If inductance L1 is large enough, iL1 is nearly equal to its average current IL1. Therefore, iin−off = I L1 =
2IO 1− k
iin−on = I L1 +
IO 2 1 1+ k =( + )I = I k 1 − k k O k(1 − k ) O
Verification: I in = kiin−on + (1 − k )iin−off = (
1+ k 3−k + 2) I O = I 1− k 1− k O
Considering Vin 1 − k 2 VO 1− k 2 =( =( ) ) R I in 2 − k IO 2−k
© 2006 by Taylor & Francis Group, LLC
226
Essential DC/DC Converters
Iin
D1
D2 V1 + VC1 –
+ L1
C1
D4
L2
D5
V2
+ VC3 –
C3
D7
L3
D8 + VC5 –
C5
D12
D11
C11
IO
+ VC11 –
+
Vin
VO
R D3 C2
–
D6
+ VC2 –
+ V C4 –
C4
+ VC6 –
C6
S
–
+ VC12 –
C12
(a) Circuit diagram Iin
V2
V1
V3
IO
+ C1 Vin
L1
+
+
C2
VC2 L2
Vin
C3
–
–
+
+ C4 V1
C5
VC4 L3 –
–
+ C11 VC6
+ C6 V2
+ C12 V3
–
–
–
+ VC12 R –
+ VO –
–
(b) Equivalent circuit during switching-on C11 Iin
L1
C1
V1
L2
C3
V2
L3 C5
+ – Vin + Vin
C2
+ V1
– V1 + C4
–
IO
– V3 + V3 +
+ V2
– V1 +
–
C6
+ C12 V3
VC12 R –
+ VO –
–
–
(c) Equivalent circuit during switching-off
FIGURE 3.7 Triple-lift additional circuit.
The variation of current iL1 is ΔiL1 =
kTVin L1
Therefore, the variation ratio of current iL1 through inductor L1 is ξ1 =
ΔiL1 / 2 k(1 − k )TVin k(1 − k )2 R = = 4L1 I O 4(3 − k ) fL1 I L1
The ripple voltage of output voltage vO is ΔvO =
© 2006 by Taylor & Francis Group, LLC
ΔQ IO kT k VO = = C12 C12 fC12 R
(3.27)
Positive Output Super-Lift Luo-Converters
227
Therefore, the variation ratio of output voltage vO is ε=
3.3.2
ΔvO / 2 k = VO 2 RfC12
(3.28)
Re-Lift Additional Circuit
This circuit is derived from the re-lift circuit by adding a DEC. Its circuit diagram and switch-on and switch-off equivalent circuits are shown in Figure 3.6. The voltage across capacitor C1 is charged to Vin. As described in previous section the voltage across C2 is V1 =
2−k V 1 − k in
The voltage across capacitor C3 is charged to V1 and voltage across capacitor C4 and C11 is charged to V2. The current flowing through inductor L2 increases with voltage V1 during switch-on period kT and decreases with voltage –(VO – 2V1) during switch-off (1 – k)T. Therefore, V2 =
2−k 2−k 2 ) Vin V =( 1− k 1 1− k
(3.29)
k V 1− k 1
(3.30)
and VL2 = The output voltage is VO = V1 + VL 2 + V2 =
2−k 3−k V 1 − k 1 − k in
(3.31)
The voltage transfer gain is G=
VO 2 − k 3 − k = Vin 1 − k 1 − k
ΔiL1 =
Vin kT L1
I L1 =
3−k I (1 − k )2 O
ΔiL2 =
V1 kT L2
I L2 =
2IO 1− k
(3.32)
Analogously,
© 2006 by Taylor & Francis Group, LLC
228
Essential DC/DC Converters
Therefore, the variation ratio of current iL1 through inductor L1 is ξ1 =
ΔiL1 / 2 k(1 − k )2 TVin k(1 − k ) 4 R = = 2 I L1 2( 3 − k )L1 I O 2(2 − k )( 3 − k ) fL1
(3.33)
and the variation ratio of current iL2 through inductor L2 is ξ2 =
ΔiL 2 / 2 k(1 − k )TV1 k(1 − k )2 R = = I L2 4L2 I O 4(3 − k ) fL2
(3.34)
The ripple voltage of output voltage vO is ΔvO =
ΔQ IO kT k VO = = C12 C12 fC12 R
Therefore, the variation ratio of output voltage vO is ε=
3.3.3
ΔvO / 2 k = 2 RfC12 VO
(3.35)
Triple-Lift Additional Circuit
This circuit is derived from the triple-lift circuit by adding a DEC. Its circuit diagram and equivalent circuits during switch-on and -off are shown in Figure 3.7. The voltage across capacitor C1 is charged to Vin. As described in previous section the voltage across C2 is V1 =
2−k V 1 − k in
and voltage across C4 is V2 =
2−k 2−k 2 ) Vin V1 = ( 1− k 1− k
The voltage across capacitor C5 is charged to V2 and voltage across capacitor C6 and C11 is charged to V3. The current flowing through inductor L3 increases with voltage V2 during switch-on period kT and decreases with voltage –(VO – 2V2) during switch-off (1 – k)T. Therefore, V3 =
© 2006 by Taylor & Francis Group, LLC
2−k 2−k 2 2−k 3 ) V1 = ( ) Vin V2 = ( 1− k 1− k 1− k
(3.36)
Positive Output Super-Lift Luo-Converters
229
and VL3 =
k V 1− k 2
(3.37)
The output voltage is VO = V2 + VL 3 + V3 = (
2−k 2 3−k ) V 1 − k 1 − k in
(3.38)
The voltage transfer gain is G=
VO 2−k 2 3−k =( ) 1− k 1− k Vin
(3.39)
Analogously, ΔiL1 =
Vin kT L1
I L1 =
(2 − k )( 3 − k ) IO (1 − k ) 3
ΔiL2 =
V1 kT L2
I L2 =
3−k I (1 − k )2 O
ΔiL3 =
V2 kT L3
I L3 =
2IO 1− k
Considering Vin 1 − k 2 VO 1− k 2 =( =( ) ) R I in 2 − k IO 2−k the variation ratio of current iL1 through inductor L1 is ξ1 =
k(1 − k )3 TVin ΔiL1 / 2 = 2(2 − k )(3 − k )L1IO I L1
(1 − k )3 k(1 − k )3 T k(1 − k )6 R = VO = 2 2(2 − k )(3 − k )L1IO (2 − k ) ( 3 − k ) 2(2 − k )3 ( 3 − k )2 fL1 and the variation ratio of current iL2 through inductor L2 is
© 2006 by Taylor & Francis Group, LLC
(3.40)
230
Essential DC/DC Converters
ξ2 =
ΔiL 2 / 2 k(1 − k )2 TV1 = IL2 2(3 − k )L2 IO
k(1 − k )2 T k(1 − k )4 R (1 − k )2 = VO = 2(3 − k )L2 IO (2 − k )(3 − k ) 2(2 − k )(3 − k )2 fL2
(3.41)
and the variation ratio of current iL3 through inductor L3 is ξ3 =
ΔiL 3 / 2 k(1 − k )TV2 k(1 − k )T 1 − k k(1 − k )2 R = = VO = I L3 4 L3 I O 4 L3 I O 3 − k 4(3 − k ) fL3
(3.42)
The ripple voltage of output voltage vO is ΔvO =
ΔQ IO kT k VO = = C12 C12 fC12 R
Therefore, the variation ratio of output voltage vO is ε=
3.3.4
ΔvO / 2 k = 2 RfC12 VO
(3.43)
Higher Order Lift Additional Circuit
The higher order lift additional circuit is derived from the corresponding circuit of the main series by adding a DEC. For nth order lift additional circuit, the final output voltage is VO = (
2 − k n −1 3 − k ) V 1− k 1 − k in
The voltage transfer gain is G=
VO 2 − k n −1 3 − k =( ) Vin 1− k 1− k
(3.44)
Analogously, the variation ratio of current iLi through inductor Li (i = 1, 2, 3, …n) is ξi =
ΔiLi / 2 k(1 − k )2( n−i +1) R = 2 ( n − i ) +1 2 u( n− i −1) h ( n− i ) I Li fLi 2[2(2 − k )] (2 − k ) (3 − k )
© 2006 by Taylor & Francis Group, LLC
(3.45)
Positive Output Super-Lift Luo-Converters
231
where x>0 is the Hong function x≤0
⎧0 h( x) = ⎨ ⎩1 and ⎧1 u( x) = ⎨ ⎩0
x≥0 is the unit-step function x<0
and the variation ratio of output voltage vO is ε=
3.4
ΔvO / 2 k = 2 RfC12 VO
(3.46)
Enhanced Series
All circuits of positive output super-lift Luo-converters-enhanced series — are derived from the corresponding circuits of the main series by adding a DEC in each stage circuit. The first three stages of this series are shown in Figures 3.5, 3.8, and 3.9. For convenience they are called elementary enhanced circuit, re-lift enhanced circuit, and triple-lift enhanced circuit respectively, and numbered as n = 1, 2 and 3.
3.4.1
Elementary Enhanced Circuit
This circuit is same as the elementary additional circuit shown in Figure 3.5. The output voltage is VO = Vin + VL1 + V1 =
3−k V 1 − k in
(3.25)
The voltage transfer gain is G= The variation of current iL1 is
© 2006 by Taylor & Francis Group, LLC
VO 3 − k = Vin 1 − k
(3.26)
232
Essential DC/DC Converters Iin
D1
D2 V1 D11
D12
D4
D5 V2 D21
D22 IO
+ VC1
+ L1
C1
+ VC11
C11
–
L2
C3
–
+ VC3
+ VC21
C21
–
+ VO
–
Vin
R
–
D3
+ VC2 –
C2
+ VC12 –
C12
S
C4
+ VC4 –
–
+ VC22 –
C22
(a) Circuit diagram Iin
V1
V2
IO
+ C1 Vin
L1
+ C2 Vin
+ C11 V1 –
–
+ C3 VC12 L2
+ C12 V1
–
–
+ C4 VC12
+ C21 V2
+ C22 V2
–
–
+ VC22 R –
–
+ VO –
–
(b) Equivalent circuit during switching-on C11 Iin +
L1 C1
– VL1 +
– V1 + V1
– Vin + C2
Vin –
C12 +
L2 – VL2 + + VC12
V1
–
C21 IO – V2 + C3 V2 – VC12 + C4
–
C22 + V2
+ VC22 R
+ VO
–
–
–
(c) Equivalent circuit during switching-off FIGURE 3.8 Re-lift enhanced circuit.
ΔiL1 =
kTVin L1
Therefore, the variation ratio of current iL1 through inductor L1 is ξ1 =
ΔiL1 / 2 k(1 − k )TVin k(1 − k )2 R = = 4L1 I O 4(3 − k ) fL1 I L1
The ripple voltage of output voltage vO is ΔvO =
© 2006 by Taylor & Francis Group, LLC
ΔQ IO kT k VO = = C12 C12 fC12 R
(3.27)
Positive Output Super-Lift Luo-Converters Iin
D2 V1 D11
D1
D12
233
D5 V2 D21
D4
D22
D8 V3 D31
D7
D32 IO
+ VC1
+ L1
C1
+ VC11
C11
–
L2
+ VC3
C3
–
+ VC21
C21
–
L3
+ VC5
C5
–
+ VC31
C31
–
–
Vin
R D3
+ VC2 –
C2
–
+ VC12 –
C12
D6
+ VC4 –
C4
+ S VC22 –
C22
+ VC6 –
C6
C32
+ VC32 –
+ VO –
(a) Circuit diagram Iin
V1
V2
V3
IO
+ C1 Vin L1
+ C2 Vin –
+ C11 V1 –
+ L2 VC12
+ C12 V1
C3
–
–
+ C21 V2
+ C4 VC12
–
–
+ L3 VC22
+ C22 V2
–
–
+ C31 V3
+ C6 VC22
C5
+ C32 V3
–
–
–
+ VC32R
+ VO
–
–
–
(b) Equivalent circuit during switching–on C11 Iin + Vin –
L1 – VL1 +
C1 – Vin + C2
– V1 +
L2 – VL2 + C3
V1 C12
+ V1
C21
L3
– V2 +
– VL3
+ – VC12 +
VC12
C4
–
–
C31 IO
+ C5
V2 C22
+ V2
– V3 +
+ VC22
– VC22 +
–
C6
–
V3 C32 + V3
+ VC32 R –
+ VO –
–
(c) Equivalent circuit during switching–off FIGURE 3.9 Triple-lift enhanced circuit.
Therefore, the variation ratio of output voltage vO is ε=
3.4.2
ΔvO / 2 k = VO 2 RfC12
(3.28)
Re-Lift Enhanced Circuit
This circuit is derived from the re-lift circuit of the main series by adding the DEC in each stage circuit. Its circuit diagram and switch-on and switchoff equivalent circuits are shown in Figure 3.8. As described in the previous section the voltage across capacitor C12 is charged to VC12 =
3−k V 1 − k in
The voltage across capacitor C3 is charged to VC12 and voltage across capacitor C4 and C21 is charged to VC4,
© 2006 by Taylor & Francis Group, LLC
234
Essential DC/DC Converters
VC 4 =
2−k 2−k 3−k V = V 1 − k C12 1 − k 1 − k in
(3.47)
The current flowing through inductor L2 increases with voltage VC12 during switch-on period kT and decreases with voltage –(VO – VC4 – VC12) during switch-off (1 – k)T. Therefore, ΔiL 2 =
1− k k (VO − VC 4 − VC12 ) V = L2 C12 L2
VO =
(3.48)
3−k 3−k 2 ) Vin VC12 = ( 1− k 1− k
(3.49)
VO 3−k 2 =( ) Vin 1− k
(3.50)
The voltage transfer gain is G= Analogously, ΔiL1 =
Vin kT L1
I L1 =
3−k I (1 − k )2 O
ΔiL2 =
V1 kT L2
I L2 =
2IO 1− k
Therefore, the variation ratio of current iL1 through inductor L1 is ξ1 =
ΔiL1 / 2 k(1 − k )2 TVin k(1 − k ) 4 R = = I L1 2(3 − k )L1 I O 2(2 − k )(3 − k )2 fL1
(3.51)
and the variation ratio of current iL2 through inductor L2 is ξ2 =
ΔiL 2 / 2 k(1 − k )TV1 k(1 − k )2 R = = 4L2 I O 4(3 − k ) fL2 I L2
The ripple voltage of output voltage vO is ΔvO =
© 2006 by Taylor & Francis Group, LLC
ΔQ IO kT k VO = = C22 C22 fC22 R
(3.52)
Positive Output Super-Lift Luo-Converters
235
Therefore, the variation ratio of output voltage vO is ε=
3.4.3
ΔvO / 2 k = VO 2 RfC22
(3.53)
Triple-Lift Enhanced Circuit
This circuit is derived from triple-lift circuit of the main series by adding the DEC in each stage circuit. Its circuit diagram and equivalent circuits during switch-on and -off are shown in Figure 3.9. As described in the previous section the voltage across capacitor C12 is charged to VC12 = (3 − k 1 − k )Vin , 2 and the voltage across capacitor C22 is charged to VC 22 = (3 − k 1 − k ) Vin . The voltage across capacitor C5 is charged to VC22 and voltage across capacitor C6 and C31 is charged to VC6. VC 6 =
2−k 2−k 3−k 2 ( ) Vin V = 1 − k C 22 1 − k 1 − k
(3.54)
The current flowing through inductor L3 increases with voltage VC22 during switch-on period kT and decreases with voltage –(VO – VC6 – VC22) during switch-off (1 – k)T. Therefore,
ΔiL 3 =
1− k k (VO − VC 6 − VC 22 ) VC 22 = L3 L3
(3.55)
3−k 3−k 3 V =( ) Vin 1 − k C 22 1− k
(3.56)
VO 3−k 3 =( ) Vin 1− k
(3.57)
VO = The voltage transfer gain is
G= Analogously, ΔiL1 =
Vin kT L1
I L1 =
(2 − k )( 3 − k ) IO (1 − k ) 3
ΔiL2 =
V1 kT L2
I L2 =
3−k I (1 − k )2 O
© 2006 by Taylor & Francis Group, LLC
236
Essential DC/DC Converters
ΔiL3 =
V2 kT L3
I L3 =
2IO 1− k
Considering Vin 1 − k 2 VO 1− k 2 =( ) =( ) R I in 2 − k IO 2−k the variation ratio of current iL1 through inductor L1 is ξ1 =
k(1 − k )3 TVin ΔiL1 / 2 = I L1 2(2 − k )(3 − k )L1IO
k(1 − k )3 T (1 − k )3 k(1 − k )6 R = VO = 2 2(2 − k )(3 − k )L1IO (2 − k ) ( 3 − k ) 2(2 − k )3 ( 3 − k )2 fL1
(3.58)
The variation ratio of current iL2 through inductor L2 is ξ2 =
ΔiL 2 / 2 k(1 − k )2 TV1 = 2(3 − k )L2 IO IL2
(1 − k )2 k(1 − k )2 T k(1 − k )4 R = VO = 2 2(3 − k )L2 IO (2 − k )(3 − k ) 2(2 − k )(3 − k ) fL2
(3.59)
and the variation ratio of current iL3 through inductor L3 is ξ3 =
ΔiL 3 / 2 k(1 − k )TV2 = IL3 4L3 IO
k(1 − k )T 1 − k k(1 − k )2 R = VO = 4L3 IO 3 − k 4(3 − k ) fL3
(3.60)
The ripple voltage of output voltage vO is ΔvO =
ΔQ IO kT k VO = = C32 C32 fC32 R
Therefore, the variation ratio of output voltage vO is ε=
© 2006 by Taylor & Francis Group, LLC
ΔvO / 2 k = VO 2 RfC32
(3.61)
Positive Output Super-Lift Luo-Converters 3.4.4
237
Higher Order Lift Enhanced Circuit
The higher order lift enhanced circuit is derived from the corresponding circuit of the main series by adding the DEC in each stage circuit. For the n nth order lift enhanced circuit, the final output voltage is VO = (3 − k 1 − k ) Vin . The voltage transfer gain is G=
VO 3−k n =( ) Vin 1− k
(3.62)
Analogously, the variation ratio of current iLi through inductor Li (i = 1, 2, 3, …n) is ξi =
ΔiLi / 2 k(1 − k )2( n−i +1) R = h ( n− i ) I Li 2[2(2 − k )] (2 − k )2( n−i )+1 (3 − k )2 u( n−i −1) fLi
(3.63)
where ⎧0 h( x) = ⎨ ⎩1
x>0 x≤0
is the Hong function
and ⎧1 u( x) = ⎨ ⎩0
x≥0 x<0
is the unit-step function
and the variation ratio of output voltage vO is ε=
3.5
ΔvO / 2 k = VO 2 RfCn2
(3.64)
Re-Enhanced Series
All circuits of positive output super-lift Luo-converters-re-enhanced series — are derived from the corresponding circuits of the main series by adding the DEC twice in each stage circuit. The first three stages of this series are shown in Figure 3.10 to Figure 3.12. For convenience they are named elementary re-enhanced circuits, re-lift reenhanced circuits, and triple-lift re-enhanced circuits respectively, and numbered as n = 1, 2 and 3.
© 2006 by Taylor & Francis Group, LLC
238
Essential DC/DC Converters
Iin
D1
D2 V1 D11
D12 V3 D13
D14
C1
+ VC1
+ VC11
+ VC13
+ L1
C11
–
C13
IO
–
–
Vin
R + VC2 S –
C2
–
+ VC12 –
C12
+ VC14 –
C14
+ VO –
(a) Circuit diagram Iin
V1
V2
IO
+ L1 C1 Vin
+ C11 V1
+C2 Vin
–
–
+C12 V1 –
+ C13 V2
+ VC14 R
+ C14 V2
–
–
–
+ VO –
–
(b) Equivalent circuit during switching-on C13
Iin
C1
L1
+
– Vin +
Vin
C2
–
IO
C11 – V2 + V2
V1 + V1 –
– V1 + C12
C14 + V2 –
+ VC14 R –
+ VO –
(c) Equivalent circuit during switching-off FIGURE 3.10 Elementary re-enhanced circuit.
3.5.1
Elementary Re-Enhanced Circuit
This circuit is derived from the elementary circuit by adding the DEC twice. Its circuit and switch-on and -off equivalent circuits are shown in Figure 3.10. The output voltage is VO = Vin + VL1 + VC12 = The voltage transfer gain is
© 2006 by Taylor & Francis Group, LLC
4−k V 1 − k in
(3.65)
Positive Output Super-Lift Luo-Converters Iin
D1
D2
C1
+ VC1
+ L1
V1 D11
D12
C11
+ VC11
–
D13
D14
C13
+ VC13 –
–
L2
239
D4
D5
C3
+ VC3
D21
D22 + VC21
C21
–
D23
D24
C23
+ VC23 –
–
IO
Vin
R
–
D3
+ VC2 –
C2
C12
+ VC12 –
C14
+ VC14 –
C4
+ VC4 S –
C22
+ VC22 –
+ VC24 –
C24
+ VO –
(a) Circuit diagram Iin
V1
IO
+ +C2 Vin
C1 Vin L1
–
+ C11 V1 –
+ C12 + C13 V1 VC12 –
–
+ C14 VC12
+ +C4 C3 VC14 L2 VC14 –
–
+ C21 VC4
–
+ C22 VC4
–
+ C23 VC22 –
–
+ C24 VC22 –
+ R VC24
+ VO
–
–
–
(b) Equivalent circuit during switching–on Iin +
L1
C1
Vin C2 –
L2
C13 – Vin C11 + V1 + V1 C12 –
– – V1 V2+ + C14 + VC12 –
C3 + VC14 – C4
C23 – C21 VC14 +
– – VC4 +
+
+ VC22
C24 C22 VC4 –
IO
+ + R VC24 –
+ VO –
–
(c) Equivalent circuit during switching–off FIGURE 3.11 Re-lift re-enhanced circuit.
VO 4 − k = Vin 1 − k
(3.66)
VC 2 =
2−k V 1 − k in
(3.67)
VC12 =
3−k V 1 − k in
(3.68)
VL1 =
k V 1 − k in
(3.69)
G= where
and
© 2006 by Taylor & Francis Group, LLC
Iin
D2
V1 D11
D12
D13
D14
C1
+ VC1
C11
+ VC11
C13
+ VC13
D4
D5
C3
+ VC3
D21
D22
D23
D24
+ VC21
C23
+ VC23
D7
D8
C5
+ VC5
D31
D32
D33
D34
+ VC31
C33
+ VC33
240
D1
IO + L1
–
L2
–
–
C21
–
L3
–
–
C31
–
–
–
+ R
Vin
C2
–
+ VC2 –
C12
+ VC12
D3
+ VC14
C14
–
C4
c
+ VC4 –
C22
D6
+ VC22
+ VC24
C24
–
C6
–
+ VC6 S –
C32
+ VC32
+ VC34
C34
–
VO –
–
(a) Circuit diagram
Iin
V1 IO
+ C1 Vin L1
+
+ C2
C11
+
+ C12
Vin
V1
V1
–
–
–
+ C14
C13 VC12
VC12
–
–
+ VC14
L2
+ C21
+
VC14
VC4
VC4
–
–
–
+
C3
–
C4
+ C23
C22
+
+ C24
C5
L3
+
+ C31
C6
+
+ C33
C32
VC22
VC22
VC24
VC24
VC6
VC6
–
–
–
–
–
–
+
+ C34
VC32 VC32
–
R
+
VC34
VO
–
–
–
–
(b) Equivalent circuit during switching-on Iin
L1
C1
Vin
– C11
+ Vin
V1
+
–
C12
VC12 –
VC14
C21
+ C14
+ V1
C3
– VC4 +
+
VC22
+ C4
VC4 C22 –
+ VC22 –
© 2006 by Taylor & Francis Group, LLC
–
– C5
VC24
C31
VC6
–
VC32
+
C34 VC24 –
+
+ C6
VC6 –
IO
+
+
+
(c) Equivalent circuit during switching-off FIGURE 3.12 Triple-lift re-enhanced circuit.
C33
+
C24
VC14 –
L3
–
–
+ VC12
+
V1
C2
–
C23
C32
VC32 –
+
+ R VC34
VO
–
–
Essential DC/DC Converters
–
+
–
L2
C13
Positive Output Super-Lift Luo-Converters
241
The following relations are obtained: iin−off = I L1 = iC11−off + iC1−off =
2IO 1− k
iin−on = iL1−on + iC1−on = I L1 +
iC1−on =
I 1− k iC1−off = O k k
iC1−off = iC 2−off =
iC 2−off =
I k k iC 2−on = iC11−on = O 1− k 1− k 1− k
iC11−on =
I 1− k iC11−off = O k k
iC12−off =
kI k iC12−on = O 1− k 1− k
iC11−off = I O + iC12−off = I O +
I k iC12−on = O 1− k 1− k
IO k
IO 1− k
If inductance L1 is large enough, iL1 is nearly equal to its average current IL1. Therefore, iin−off = I L1 =
2IO 1− k
iin−on = I L1 +
IO 2 1 1+ k =( + )I = I 1 − k k O k(1 − k ) O k
Verification: I in = kiin−on + (1 − k )iin−off = (
1+ k 3−k + 2) I O = I 1− k 1− k O
Considering Vin 1 − k 2 VO 1− k 2 =( ) =( ) R I in 2 − k IO 2−k the variation of current iL1 is ΔiL1 =
kTVin L1
Therefore, the variation ratio of current iL1 through inductor L1 is ξ1 =
© 2006 by Taylor & Francis Group, LLC
ΔiL1 / 2 k(1 − k )TVin k(1 − k )2 R = = 4L1 I O 4(3 − k ) fL1 I L1
(3.70)
242
Essential DC/DC Converters
The ripple voltage of output voltage vO is ΔvO =
ΔQ IO kT k VO = = C14 C14 fC14 R
Therefore, the variation ratio of output voltage vO is ε=
3.5.2
ΔvO / 2 k = VO 2 RfC14
(3.71)
Re-Lift Re-Enhanced Circuit
This circuit is derived from the re-lift circuit of the main series by adding the DEC twice in each stage circuit. Its circuit and switch-on and -off equivalent circuits are shown in Figure 3.11. The voltage across capacitor C14 is 4−k V 1 − k in
(3.72)
4−k 4−k 2 VC14 = ( ) Vin 1− k 1− k
(3.73)
VO 4−k 2 =( ) Vin 1− k
(3.74)
VC14 = By the same analysis VO = The voltage transfer gain is G= Analogously, ΔiL1 =
Vin kT L1
I L1 =
3−k I (1 − k )2 O
ΔiL2 =
V1 kT L2
I L2 =
2IO 1− k
Therefore, the variation ratio of current iL1 through inductor L1 is ξ1 =
ΔiL1 / 2 k(1 − k )2 TVin k(1 − k ) 4 R = = 2 I L1 2( 3 − k )L1 I O 2(2 − k )( 3 − k ) fL1
© 2006 by Taylor & Francis Group, LLC
(3.75)
Positive Output Super-Lift Luo-Converters
243
The variation ratio of current iL2 through inductor L2 is ξ2 =
ΔiL 2 / 2 k(1 − k )TV1 k(1 − k )2 R = = I L2 4L2 I O 4(3 − k ) fL2
(3.76)
The ripple voltage of output voltage vO is ΔvO =
ΔQ IO kT k VO = = C24 C24 fC24 R
Therefore, the variation ratio of output voltage vO is ε=
3.5.3
ΔvO / 2 k = 2 RfC24 VO
(3.77)
Triple-Lift Re-Enhanced Circuit
This circuit is derived from triple-lift circuit of the main series by adding the DEC twice in each stage circuit. Its circuit and switch-on and -off equivalent circuits are shown in Figure 3.12. The voltage across capacitor C14 is VC14 =
4−k V 1 − k in
(3.78)
4−k 2 ) Vin 1− k
(3.79)
The voltage across capacitor C24 is VC 24 = ( By the same analysis VO =
4−k 4−k 3 VC 24 = ( ) Vin 1− k 1− k
(3.80)
VO 4−k 3 =( ) Vin 1− k
(3.81)
The voltage transfer gain is G= Analogously,
© 2006 by Taylor & Francis Group, LLC
244
Essential DC/DC Converters
ΔiL1 =
Vin kT L1
I L1 =
(2 − k )( 3 − k ) IO (1 − k ) 3
ΔiL2 =
V1 kT L2
I L2 =
3−k I (1 − k )2 O
ΔiL3 =
V2 kT L3
I L3 =
2IO 1− k
Considering Vin 1 − k 2 VO 1− k 2 =( ) =( ) R I in 2 − k IO 2−k the variation ratio of current iL1 through inductor L1 is ξ1 =
k(1 − k )3 TVin ΔiL1 / 2 = I L1 2(2 − k )(3 − k )L1IO
k(1 − k )3 T (1 − k )3 k(1 − k )6 R = VO = 2 2(2 − k )(3 − k )L1IO (2 − k ) ( 3 − k ) 2(2 − k )3 ( 3 − k )2 fL1
(3.82)
The variation ratio of current iL2 through inductor L2 is ξ2 =
ΔiL 2 / 2 k(1 − k )2 TV1 = 2(3 − k )L2 IO IL2
(1 − k )2 k(1 − k )2 T k(1 − k )4 R = VO = 2(3 − k )L2 IO (2 − k )(3 − k ) 2(2 − k )(3 − k )2 fL2
(3.83)
The variation ratio of current iL3 through inductor L3 is ξ3 =
ΔiL 3 / 2 k(1 − k )TV2 k(1 − k )T 1 − k k(1 − k )2 R = = VO = 4 L3 I O 4 L3 I O 3 − k 4(3 − k ) fL3 I L3
The ripple voltage of output voltage vO is ΔvO =
© 2006 by Taylor & Francis Group, LLC
ΔQ IO kT k VO = = C34 C34 fC34 R
(3.84)
Positive Output Super-Lift Luo-Converters
245
Therefore, the variation ratio of output voltage vO is ε=
3.5.4
ΔvO / 2 k = VO 2 RfC34
(3.85)
Higher Order Lift Re-Enhanced Circuit
Higher order lift additional circuits are derived from the corresponding circuit of the main series by adding DEC twice in each stage circuit. For the nth order lift additional circuit, the final output voltage is VO = (
4−k n ) Vin 1− k
The voltage transfer gain is G=
VO 4−k n =( ) 1− k Vin
(3.86)
Analogously, the variation ratio of current iLi through inductor Li (i = 1, 2, 3, …n) is ξi =
ΔiLi / 2 k(1 − k )2( n−i +1) R = h ( n− i ) I Li 2[2(2 − k )] (2 − k )2( n−i )+1 (3 − k )2 u( n−i −1) fLi
(3.87)
where ⎧0 h( x) = ⎨ ⎩1
x>0 x≤0
is the Hong function
and ⎧1 u( x) = ⎨ ⎩0
x≥0 x<0
is the unit-step function
and the variation ratio of output voltage vO is ε=
© 2006 by Taylor & Francis Group, LLC
ΔvO / 2 k = VO 2 RfCn 4
(3.88)
246
Essential DC/DC Converters 1 Iin
D1
D2 V1 D11
2..
D12
j D1(2j–1) D1(2j) IO
+ VC1
+ L1
C1
+ VC11
C11
–
+ C1(2j–1)
VC1(2j–1)
–
–
Vin
R S
+ VC2 –
C2
–
+ VC12 –
C12
C12j
+ VC12j –
+ VO –
(a) Circuit diagram Iin
V1
IO
+ C1 Vin
L1
+ C11 V1
+ C2 Vin
+ V1
–
–
C1(2j–1)
+ C12j VC1(2j–1)
–
–
–
+ VC12jR
+ VO –
–
(b) Equivalent circuit during switching-on
Iin + Vin
C11
C1(2j–1)
– V1 + V1
– VC1(2j–1) +
IO
L1 C1
– VL1 +
– Vin + C2
C12 + V1
–
+ VC12 –
C12j
VC12jR
+ VO
–
–
+
–
(c) Equivalent circuit during switching-off FIGURE 3.13 Elementary multiple-enhanced circuit.
3.6
Multiple-Enhanced Series
All circuits of positive output super-lift Luo-converters — multipleenhanced series — are derived from the corresponding circuits of the main series by adding the DEC multiple (j) times in each stage circuit. The first three stages of this series are shown in Figure 3.13 through Figure 3.15. For convenience they are called elementary multiple-enhanced circuits, re-lift multiple-enhanced circuits, and triple-lift multiple-enhanced circuits respectively, and numbered as n = 1, 2, and 3.
© 2006 by Taylor & Francis Group, LLC
D2 V1 D11
D1
D12
D1(2j–1) D1(2j)
D4
D5
D21
D22
D2(2j–1) D2(2j)
C21
+ VC21 C2(2j–1) –
IO + VC1 –
+ L1
C1
+ VC11 C1(2j–1) –
C11
+ VC1(2j–1) –
L2
+ VC3 –
C3
+ VC2(2j–1) –
+
Vin
VO
R
–
D3
+ VC2 –
C2
+ VC12 –
C12
D6 + S VC12j –
C12j
+ VC4 –
C4
C22
+ VC22 –
–
+ VC22j –
C22j
(a) Circuit diagram Iin
V1
IO
+ C1 Vin
L1
+ C2 Vin –
+ C11 V1
+ C1(2j–1) V1
–
+
+ C12j VC1(2j–1)
–
C3
VC12jL2 –
–
+ C4 VC3 –
+ + C2(2j–1) C21 VC4 VC21 –
+ C22j VC2(2j–1)
–
+ VC22jR –
–
+ VO –
–
Positive Output Super-Lift Luo-Converters
Iin
(b) Equivalent circuit during switching-on Iin +
L1
C11
– VL1 + C1
Vin
– Vin + C2
C1(2j–1)
– V1 + V1 +
C12
V1
–
–
L2
VC1(2j–1) + + VC12 –
–
C12j
C21
C2(2j–1)
–VC21+
– VC2(2j–1) +
IO
– VL2 + C3 + VC12j –
– VC3 + C4
+
C22
VC4
+ VC22 –
C22j
VC22jR
+ VO
–
–
+
–
(c) Equivalent circuit during switching-off
© 2006 by Taylor & Francis Group, LLC
247
FIGURE 3.14 Re-lift multiple-enhanced circuit.
D1
D2 V1 D11
D12
D4
D1(2j–1) D1(2j)
D5
D21
D22
D7
D2(2j–1) D2(2j)
D8
D31
D32
D3(2j–1) D3(2j) IO
+ VC1
+ L1
C1
+ VC11 C1(2j–1)
C11
–
+ VC1(2j–1)
–
L2
+ VC3
C3
–
+ VC21 C2(2j–1)
C21
–
+ VC2(2j–1)
–
L3
+ VC5
C5
+ VC31 C3(2j–1)
C31
–
–
+ VC3(2j–1)
–
+
– R
Vin + VC2 –
C2
–
D3
+ VC12 –
C12
C12j
+ VC12j –
C4
+ VC4 –
D6
+ VC22 –
C22
+ S VC22j –
C22j
+ VC6 –
C6
C32
+ VC32 –
248
Iin
C32j
+ VC32j –
VO –
(a) Circuit diagram
Iin V1
IO
+ C1 Vin
L1
+ C11 V1
+ C2 Vin
+ + C12j C1(2j–1) V1 VC1(2j–1)
–
–
–
+
C3
VC12j L2 –
–
+ C4 VC3 –
+ C21 VC4 –
+
+ + C2(2j–1) C22j VC21 VC2(2j–1) –
C5
VC22jL3 –
–
+ C6 VC5
+ C31 VC6
+ + C3(2j–1) C32j VC31 VC3(2j–1)
–
–
–
–
+ VC32j R –
+ VO –
–
(b) Equivalent circuit during switching-on Iin
–
C11
C1(2j–1)
L2
C21
C2(2j–1)
L3
C31
C3(2j–1) IO
VL1
+ C1
V1
Vin
–
– – V1 + Vin + C12
+ C2
V1 –
–
–VC1(2j–1) +
VL2 + C3
+
C12j
+
VC12
VC12j
–
–
– VC21 + – VC3 +
C4
VL3 +
– VC2(2j–1) +
C22
C5 +
C22j
+
+
VC22
VC22j
VC4
–
–
–
(c) Equivalent circuit during switching-off FIGURE 3.15 Triple-lift multiple-enhanced circuit.
© 2006 by Taylor & Francis Group, LLC
– VC31 + – VC5 +
C6
– VC3(2j–1) +
C32
+ VC6 –
+ VC32 –
C32j
+ VC32j R –
+ VO –
Essential DC/DC Converters
+
L1
Positive Output Super-Lift Luo-Converters 3.6.1
249
Elementary Multiple-Enhanced Circuit
This circuit is derived from the elementary circuit of the main series by adding the DEC multiple (j) times. Its circuit and switch-on and -off equivalent circuits are shown in Figure 3.13. The output voltage is j+2−k Vin 1− k
(3.89)
VO j + 2 − k = Vin 1− k
(3.90)
VO = The voltage transfer gain is G= Following relations are obtained: iin−off = I L1 = iC11−off + iC1−off =
2IO 1− k
iin−on = iL1−on + iC1−on = I L1 +
iC1−on =
I 1− k iC1−off = O k k
iC1−off = iC 2−off =
iC 2−off =
I k k iC 2−on = iC11−on = O 1− k 1− k 1− k
iC11−on =
I 1− k iC11−off = O k k
iC12−off =
kI k iC12−on = O 1− k 1− k
iC11−off = I O + iC12−off = I O +
I k iC12−on = O 1− k 1− k
IO k
IO 1− k
If inductance L1 is large enough, iL1 is nearly equal to its average current IL1. Therefore, iin−off = I L1 =
2IO 1− k
iin−on = I L1 +
IO 2 1 1+ k I =( + )I O = k k(1 − k ) O 1− k k
Verification: I in = kiin−on + (1 − k )iin−off = (
1+ k 3−k + 2) I O = I 1− k 1− k O
Considering Vin 1 − k 2 VO 1− k 2 =( ) =( ) R I in 2 − k IO 2−k
© 2006 by Taylor & Francis Group, LLC
250
Essential DC/DC Converters
the variation of current iL1 is ΔiL1 =
kTVin L1
Therefore, the variation ratio of current iL1 through inductor L1 is ξ1 =
ΔiL1 / 2 k(1 − k )TVin k(1 − k )2 R = = I L1 4L1 I O 4( 3 − k ) fL1
(3.91)
The ripple voltage of output voltage vO is ΔvO =
ΔQ IO kT k VO = = C12 j C12 j fC12 j R
Therefore, the variation ratio of output voltage vO is ε=
3.6.2
ΔvO / 2 k = 2 RfC12 j VO
(3.92)
Re-Lift Multiple-Enhanced Circuit
This circuit is derived from the re-lift circuit of the main series by adding the DEC multiple (j) times in each stage circuit. Its circuit diagram and switch-on and switch-off equivalent circuits are shown in Figure 3.14. The voltage across capacitor C12j is VC12 j =
j+2−k Vin 1− k
(3.93)
The output voltage across capacitor C22j is VO = VC 22 j = (
j+2−k 2 ) Vin 1− k
(3.94)
The voltage transfer gain is G= Analogously,
© 2006 by Taylor & Francis Group, LLC
VO j+2−k 2 =( ) Vin 1− k
(3.95)
Positive Output Super-Lift Luo-Converters
251
ΔiL1 =
Vin kT L1
I L1 =
3−k I (1 − k )2 O
ΔiL2 =
V1 kT L2
I L2 =
2IO 1− k
Therefore, the variation ratio of current iL1 through inductor L1 is ξ1 =
ΔiL1 / 2 k(1 − k )2 TVin k(1 − k ) 4 R = = 2 I L1 2( 3 − k )L1 I O 2(2 − k )( 3 − k ) fL1
(3.96)
and the variation ratio of current iL2 through inductor L2 is ξ2 =
ΔiL 2 / 2 k(1 − k )TV1 k(1 − k )2 R = = I L2 4L2 I O 4(3 − k ) fL2
(3.97)
The ripple voltage of output voltage vO is ΔvO =
ΔQ IO kT k VO = = C22 j C22 j fC22 j R
Therefore, the variation ratio of output voltage vO is ε=
3.6.3
ΔvO / 2 k = 2 RfC22 j VO
(3.98)
Triple-Lift Multiple-Enhanced Circuit
This circuit is derived from the triple-lift circuit of the main series by adding the DEC multiple (j) times in each stage circuit. Its circuit and switch-on and -off equivalent circuits are shown in Figure 3.15. The voltage across capacitor C12j is VC12 j =
j+2−k Vin 1− k
(3.99)
j+2−k 2 ) Vin 1− k
(3.100)
The voltage across capacitor C22j is VC 22 j = (
© 2006 by Taylor & Francis Group, LLC
252
Essential DC/DC Converters
Same analysis, VO =
j+2−k j+2−k 3 VC 22 j = ( ) Vin 1− k 1− k
(3.101)
The voltage transfer gain is G=
VO j+2−k 3 =( ) Vin 1− k
(3.102)
Analogously, ΔiL1 =
Vin kT L1
I L1 =
(2 − k )( 3 − k ) IO (1 − k ) 3
ΔiL2 =
V1 kT L2
I L2 =
3−k I (1 − k )2 O
ΔiL3 =
V2 kT L3
I L3 =
2IO 1− k
Considering Vin 1 − k 2 VO 1− k 2 =( ) =( ) R I in 2 − k IO 2−k the variation ratio of current iL1 through inductor L1 is ξ1 =
k(1 − k )3 TVin ΔiL1 / 2 = I L1 2(2 − k )(3 − k )L1IO
k(1 − k )3 T k(1 − k )6 R (1 − k )3 VO = = 2 2(2 − k )(3 − k )L1IO (2 − k ) ( 3 − k ) 2(2 − k )3 ( 3 − k )2 fL1
(3.103)
The variation ratio of current iL2 through inductor L2 is ξ2 =
ΔiL 2 / 2 k(1 − k )2 TV1 = IL2 2(3 − k )L2 IO
k(1 − k )2 T k(1 − k )4 R (1 − k )2 VO = = 2 2(3 − k )L2 IO (2 − k )(3 − k ) 2(2 − k )(3 − k ) fL2
© 2006 by Taylor & Francis Group, LLC
(3.104)
Positive Output Super-Lift Luo-Converters
253
The variation ratio of current iL3 through inductor L3 is ξ3 =
ΔiL 3 / 2 k(1 − k )TV2 k(1 − k )T 1 − k k(1 − k )2 R (3.105) = = VO = I L3 4 L3 I O 4 L3 I O 3 − k 4(3 − k ) fL3
The ripple voltage of output voltage vO is ΔvO =
ΔQ IO kT k VO = = C32 j C32 j fC32 j R
Therefore, the variation ratio of output voltage vO is ε=
3.6.4
ΔvO / 2 k = 2 RfC32 j VO
(3.106)
Higher Order Lift Multiple-Enhanced Circuit
Higher order lift multiple-enhanced circuits can be derived from the corresponding circuit of the main series converters by adding the DEC multiple (j) times in each stage circuit. For the nth order lift additional circuit, the final output voltage is VO = (
j+2−k n ) Vin 1− k
The voltage transfer gain is G=
VO j+2−k n =( ) Vin 1− k
(3.107)
Analogously, the variation ratio of current iLi through inductor Li (i = 1, 2, 3, …n) is ξi =
ΔiLi / 2 k(1 − k )2( n−i +1) R = 2 ( n − i ) +1 2 u( n− i −1) h ( n− i ) I Li 2[2(2 − k )] (2 − k ) (3 − k ) fLi
where ⎧0 h( x) = ⎨ ⎩1
© 2006 by Taylor & Francis Group, LLC
x>0 x≤0
is the Hong function
(3.108)
254
Essential DC/DC Converters
and x≥0 x<0
⎧1 u( x) = ⎨ ⎩0
is the unit-step function
The variation ratio of output voltage vO is ε=
3.7
ΔvO / 2 k = VO 2 RfCn2 j
(3.109)
Summary of Positive Output Super-Lift Luo-Converters
All circuits of positive output super-lift Luo-converters can be shown in Figure 3.16 as the family tree. From the analysis in previous sections, the common formula to calculate the output voltage is presented: 2−k n ⎧ ( ) Vin ⎪ 1− k ⎪ 2 − k n −1 3 − k ) ( )V ⎪( 1 − k in ⎪ 1− k ⎪ 3−k n VO = ⎨ ( ) Vin 1− k ⎪ 4−k n ⎪ ( ) Vin 1− k ⎪ j+2−k n ⎪ ⎪⎩ ( 1 − k ) Vin
main _ series additional _ series enhanced _ series
(3.110)
re - enhanced _ series multiple - enhanced _ series
The voltage transfer gain is 2−k n ⎧ ( ) ⎪ 1− k ⎪ 2 − k n −1 3 − k ) ( ) ⎪( 1− k ⎪ 1− k V ⎪ 3−k n G= O =⎨ ( ) Vin ⎪ 1− k 4−k n ⎪ ( ) 1− k ⎪ j+2−k n ⎪ ⎪⎩ ( 1 − k )
© 2006 by Taylor & Francis Group, LLC
main _ series additional _ series enhanced _ series re - enhanced _ series multiple - enhanced _ series
(3.111)
Positive Output Super-Lift Luo-Converters
Main Series
255
Additional Series
Enhanced Series
Quintuple-Lift Circuit
Quadruple-Lift Additional Circuit
Quintuple-Lift Enhanced Circuit
Quintuple-Lift Reenhanced Circuit
Quintuple-Lift MultipleEnhanced Circuit
Quintuple-Lift Circuit
Quadruple-Lift Additional Circuit
Quintuple-Lift Enhanced Circuit
Quadruple-Lift Reenhanced Circuit
Quadruple-Lift MultipleEnhanced Circuit
Triple-Lift Circuit
Triple-Lift Additional Circuit
Triple-Lift Enhanced Circuit
Triple-Lift Reenhanced Circuit
Triple-Lift MultipleEnhanced Circuit
Relift Circuit
Relift Additional Circuit
Relift Additional Circuit
Relift Reenhanced Circuit
Relift MultipleEnhanced Circuit
Elementary Additional/Enhanced Circuit
Reenhanced Series
Multiple-Enhanced Series
Elementary Reenhanced Circuit
Elementary MultipleEnhanced Circuit
Elementary Positive Output Super-Lift Luo-Converter FIGURE 3.16 The family of positive output super-lift Luo-converters.
In order to show the advantages of super-lift Luo-converters, their voltage transfer gains can be compared to that of a buck converter, G=
VO =k Vin
forward converter, G=
VO = kN Vin
N is the transformer turns ratio
Cúk-converter, G= fly-back converter,
© 2006 by Taylor & Francis Group, LLC
VO k = Vin 1 − k
256
Essential DC/DC Converters
TABLE 3.1 Voltage Transfer Gains of Converters in the Condition k = 0.2 Stage No. (n)
1
Buck converter Forward converter Cúk-converter Fly-back converter Boost converter Positive output Luo-converters Positive output super-lift Luo-converters — main series Positive output super-lift Luo-converters — additional series Positive output super-lift Luo-converters — enhanced series Positive output super-lift Luo-converters — re-enhanced series Positive output super-lift Luo-converters — multiple (j = 4)enhanced series
G=
VO k N = Vin 1 − k
2
3
4
5
n
1.25 2.25
0.2 0.2 N (N is the transformer turns ratio) 0.25 0.25 N (N is the transformer turns ratio) 1.25 2.5 3.75 5 6.25 1.25n 5.06 11.39 25.63 57.67 2.25n
3.5
7.88
17.72
3.5
12.25
42.88
4.75
22.56
107.2
7.25
52.56
381
39.87
89.7
3.5*2.25(n-1)
150
525
3.5n
509
2418
4.75n
2762
20,030
7.25n
N is the transformer turn ratio
boost converter,
G=
VO 1 = Vin 1 − k
and positive output Luo-converters.
G=
VO n = Vin 1 − k
(3.112)
If we assume that the conduction duty k is 0.2, the output voltage transfer gains are listed in Table 3.1. If the conduction duty k is 0.5, the output voltage transfer gains are listed in Table 3.2. If the conduction duty k is 0.8, the output voltage transfer gains are listed in Table 3.3.
© 2006 by Taylor & Francis Group, LLC
Positive Output Super-Lift Luo-Converters
257
TABLE 3.2 Voltage Transfer Gains of Converters in the Condition k = 0.5 Stage No. (n) Buck converter Forward converter Cúk-converter Fly-back converter Boost converter Positive output Luo-converters Positive output super-lift Luo-converters — main series Positive output super-lift Luo-converters — additional series Positive output super-lift Luo-converters — enhanced series Positive output super-lift Luo-converters — re-enhanced series Positive output super-lift Luo-converters — multiple (j = 4)enhanced series
1
2 3
2
3
4
5
n
0.5 0.5 N (N is the transformer turns ratio) 1 N (N is the transformer turns ratio) 2 4 6 8 10 2n 9 27 81 243 3n 5*3(n-1)
5
15
45
135
405
5
25
125
625
3125
5n
7
49
343
2401
16,807
7n
11
121
1331
14,641
16*104
11n
TABLE 3.3 Voltage Transfer Gains of Converters in the Condition k = 0.8 Stage No. (n) Buck converter Forward converter Cúk-converter Fly-back converter Boost converter Positive output Luo-converters Positive output super-lift Luo-converters — main series Positive output super-lift Luo-converters — additional series Positive output super-lift Luo-converters — enhanced series Positive output super-lift Luo-converters — re-enhanced series Positive output super-lift Luo-converters — multiple (j = 4)enhanced series
© 2006 by Taylor & Francis Group, LLC
1
5 6
2
3
4
5
n
0.8 0.8 N (N is the transformer turns ratio) 4 4 N (N is the transformer turns ratio) 5 10 15 20 25 5n 36 216 1296 7776 6n 14,256
11*6(n-1)
11
66
396
2376
11
121
1331
14,641
16*104
11n
16
256
4096
65,536
104*104
16n
26
676
17,576
46*104
12*106
26n
258
Essential DC/DC Converters
1.0A (9.988m, 618m)
0.5A
(9.988m, 218m) (9.988m, 111m) +
+
+
+
0A I(L1)
I(L2)
+ I(L3)
1.0KV (9.988m, 659)
0.5KV
(9.988m, 194) (9.988m, 66)
SEL>> 0V 9.980ms V(R:2)
9.984ms V(D2:2) V(D5:2)
9.988ms
9.992ms
9.996ms
10.000ms
Time
FIGURE 3.17 The simulation results of triple-lift circuit at condition k = 0.5 and f = 100 kHz.
3.8
Simulation Results
To verify the design and calculation results, the PSpice simulation package was applied to these converters. Choosing Vin = 20 V, L1 = L2 = L3 = 10 mH, all C1 to C8 = 2 μF, and R = 30 kΩ, and using k = 0.5 and f = 100 kHz. 3.8.1
Simulation Results of a Triple-Lift Circuit
The voltage values V1, V2 and VO of a triple-lift circuit are 66 V, 194 V, and 659 V respectively and inductor current waveforms are iL1 (its average value IL1 = 618 mA), iL2, and iL3. The simulation results are shown in Figure 3.17. The voltage values are matched to the calculated results.
3.8.2
Simulation Results of a Triple-Lift Additional Circuit
The voltage values V1, V2, V3, and VO of the triple-lift additional circuit are 57 V, 165 V, 538 V, and 910 V respectively and current waveforms are iL1 (its average value IL1 = 1.8 A), iL2, and iL3. The simulation results are shown in Figure 3.18. The voltage values are matched to the calculated results.
© 2006 by Taylor & Francis Group, LLC
Positive Output Super-Lift Luo-Converters
259
2.0A (19.988m, 1.8) (19.988m, 623m) 1.0A
(19.988m, 248m)
0A I(L1)
I(L2)
I(L3)
1.0KV (19.988m, 910) (19.988m, 538) 0.5KV
(19.988m, 165) (19.988m, 57)
SEL>> 0V 19.980ms V(D8:2)
19.984ms V(R:2) V(D2:2)
19.988ms V(D5:2)
19.992ms
19.996ms
20.000ms
Time
FIGURE 3.18 Simulation results of triple-lift additional circuit at condition k = 0.5 and f = 100 kHz.
3.9
Experimental Results
A test rig was constructed to verify the design and calculation results, and compare with PSpice simulation results. The testing conditions were the same: Vin = 20 V, L1 = L2 = L3 = 10 mH, all C1 to C8 = 2 μF and R = 30 kΩ, and using k = 0.5 and f = 100 kHz. The component of the switch is a MOSFET device IRF950 with the rates 950 V/5 A/2 MHz. The values of the output voltage and first inductor current are measured in the following converters.
3.9.1
Experimental Results of a Triple-Lift Circuit
After careful measurement, the current value of IL1 = 0.62 A (shown in channel 1 with 1 A/Div) and voltage value of VO = 660 V (shown in channel 2 with 200 V/Div). The experimental results (current and voltage values) are shown in Figure 3.19, that are identically matched to the calculated and simulation results, which are IL1 = 0.618 A and VO = 659 V shown in Figure 3.17.
3.9.2
Experimental Results of a Triple-Lift Additional Circuit
The experimental results of the current value of IL1 = 1.8 A (shown in channel 1 with 1 A/Div) and voltage value of VO = 910 V (shown in channel 2 with
© 2006 by Taylor & Francis Group, LLC
260
Essential DC/DC Converters
FIGURE 3.19 The experimental results of triple-lift circuit at condition k = 0.5 and f = 100 kHz.
TABLE 3.4 Comparison of Simulation and Experimental Results of a Triple-Lift Circuit Stage No. (n) Simulation results Experimental results
IL1 (A)
Iin (A)
Vin (V)
Pin (W)
VO (V)
PO (W)
η (%)
0.618 0.62
0.927 0.93
20 20
18.54 18.6
659 660
14.47 14.52
78 78
TABLE 3.5 Comparison of Simulation and Experimental Results of a Triple-Lift Additional Circuit Stage No. (n) Simulation results Experimental results
IL1 (A)
Iin (A)
Vin (V)
Pin (W)
VO (V)
PO (W)
η (%)
1.8 1.8
2.7 2.7
20 20
54 54
910 910
27.6 27.6
51 51
200 V/Div) are shown in Figure 3.20 that are identically matched to the calculated and simulation results, which are IL1 = 1.8 A and VO = 910 V shown in Figure 3.18.
3.9.3 Efficiency Comparison of Simulation and Experimental Results These circuits enhanced the voltage transfer gain successfully, but efficiency, particularly, the efficiencies of the tested circuits is 51 to 78%, which is good for high voltage output equipment. Comparison of the simulation and experimental results, which are listed in the Tables 3.4 and 3.5, demonstrates that all results are well identified each other.
© 2006 by Taylor & Francis Group, LLC
Positive Output Super-Lift Luo-Converters
261
FIGURE 3.20 Experimental results of triple-lift additional circuit at condition k = 0.5 and f = 100 kHz.
Usually, there is high inrush current during the initial power-on. Therefore, the voltage across capacitors is quickly changed to certain values. The transient process is very quick in only few milliseconds.
Bibliography Kassakian, J.G., Wolf, H-C., Miller, J.M., and Hurton, C.J., Automotive electrical systems, circa 2005, IEEE Spectrum, 8, 22, 1996. Lander, C.W., Power Electronics, McGraw-Hill, London, 1993. Luo, F.L., Luo-converters, a new series of positive-to-positive DC-DC step-up converters, Power Supply Technologies and Applications, Xi’an, China, 1, 30, 1998. Luo, F.L., Positive output Luo-converters, the advanced voltage lift technique, Electrical Drives, Tianjin, China, 2, 47, 1999. Luo, F.L. and Ye, H., Positive output super-lift Luo-converters, in Proceedings of IEEEPESC’2002, Cairns, Australia, 2002, p. 425. Luo, F.L. and Ye, H., Positive output super-lift converters, IEEE Transactions on Power Electronics, 18, 105, 2003. Pelley, B.R., Tryristor Phase Controlled Converters and Cycloconverters, John Wiley, New York, 1971. Pressman, A.I., Switching Power Supply Design, 2nd ed., McGraw-Hill, New York, 1998. Rashid, M.H., Power Electronics, 2nd ed., Prentice Hall of India Pvt. Ltd., New Delhi, 1995. Trzynadlowski, A.M., Introduction to Modern Power Electronics, Wiley Interscience, New York, 1998. Ye, H., Luo, F.L., and Ye, Z.Z., Widely adjustable high-efficiency high voltage regulated power supply, Power Supply Technologies and Applications, Xi’an, China, 1, 18, 1998. Ye, H., Luo, F.L., and Ye, Z.Z., DC motor Luo-converter-driver, Power Supply World, Guangzhou, China, 2000, p. 15.
© 2006 by Taylor & Francis Group, LLC
4 Negative Output Super-Lift Luo-Converters
Along with the positive output super-lift Luo-converters, negative output (N/O) super-lift Luo-converters have also been developed. They perform super-lift technique as well.
4.1
Introduction
Negative output super-lift Luo-converters are sorted into several sub-series: • Main series — Each circuit of the main series has one switch S, n inductors, 2n capacitors and (3n – 1) diodes. • Additional series — Each circuit of the additional series has one switch S, n inductors, 2(n + 1) capacitors and (3n + 1) diodes. • Enhanced series — Each circuit of the enhanced series has one switch S, n inductors, 4n capacitors and (5n + 1) diodes. • Re-enhanced series — Each circuit of the re-enhanced series has one switch S, n inductors, 6n capacitors and (7n + 1) diodes. • Multiple-enhanced series — Each circuit of the multiple-enhanced series has one switch S, n inductors, 2(n + j + 1) capacitors and (3n + 2j + 1) diodes. All analyses in this section are based on the condition of steady state operation with continuous conduction mode (CCM). The conduction duty ratio is k, switch period T = 1/f (f is the switch frequency), the load is resistive load R. The input voltage and current are Vin and Iin, output voltage and current are VO and IO. Assume no power losses during the conversion process, Vin × Iin = VO × IO. The voltage transfer gain is G: G=
VO Vin 263
© 2006 by Taylor & Francis Group, LLC
264
Essential DC/DC Converters I in IO
S
+
+
C1
L1
V in
V C1
_
_
–
R D1
VO
_
D2
V C2
C2
+
+
(a) Circuit Diagram
I in
IO _
_
+
+
V in
L1
C1
_
V C2
C2
Vin
VO
R
+
_
+
(b) Swich-on C1
+
+
V in _
L1
Vin
V L1
IO _
_
C2
V C2
_
R
VO
+ +
(c) Swich-off FIGURE 4.1 N/O elementary circuit.
4.2
Main Series
The first three stages of negative output super-lift Luo-converters — main series — are shown in Figure 4.1 to Figure 4.3. For convenience they are called elementary circuits, re-lift circuits, and triple-lift circuits respectively, and numbered as n = 1, 2 and 3.
4.2.1 Elementary Circuit N/O elementary circuit and its equivalent circuits during switch-on and switch-off are shown in Figure 4.1. The voltage across capacitor C1 is charged © 2006 by Taylor & Francis Group, LLC
Negative Output Super-Lift Luo-Converters
265
I in S
IO
D3 +
+
C1
L1
+
L2
V C1 _
V in
C3
V C3
V1
_
_
R
_
D1
D2
_
C2
D4
D5
V C2
VO
_ +
C4
V C4 +
+
(a) Circuit Diagram
I in
V in
L2
+
+
L1
C1
Vin
_
C2
V C4
_ _
_
_
V1+Vin
V1
_
IO
+
C3
C4
VO
R
+
+
V1 +
(b) Swich-on
L2
C1
C3
V1 + +
Vin
L1
V
Vin
_
C2 L1
_
IO _ V L2
+ V1+Vin
V1 +
C4
_
_ _ V C4 R +
VO +
(c) Swich-off FIGURE 4.2 N/O re-lift circuit.
to Vin. The current flowing through inductor L1 increases with slop Vin/L1 during switch-on period kT and decreases with slop –(VO – Vin)/L1 during switch-off (1 – k)T. Therefore, the variation of current iL1 is ΔiL1 =
VO =
© 2006 by Taylor & Francis Group, LLC
V − Vin Vin (1 − k )T kT = O L1 L1
(4.1)
1 2−k Vin = ( − 1)Vin 1− k 1− k
(4.2)
266
Essential DC/DC Converters I in D3
S
D6
IO + +
C1
L1
+
V C1 L2
Vin
+
L3
V C3
C3
C5
_
_
V1
V2
R
_
D1
V C5
_
_
D2
D4
_
C2
D5
_
C4
V C2
D7
D8 C6
V C4
+
+
VO
_
+
V C6 +
(a) Circuit Diagram
L2
I in
+
L3
+
C3
C5
V2+Vin
IO
V1+Vin _
L1
_
Vin
C1
_
_
C2
C4
V C6
V2
_
V1
V1
C6
V2
_
_
+
+
V in
_
VO
R
+
+
+
+
(b) Switch-on
C1
L2
V1
C3
V2
L3
C5
V L3
_ + V2+Vin
IO + + V in
_ Vin
L1
C2
_
_
V L2
+
_ V1+Vin
V1 +
C4
_ V2 +
C6
_ _ V C6 +
R
VO +
(c) Switch-off FIGURE 4.3 N/O triple-lift circuit.
The voltage transfer gain is G1 =
VO 2 − k = −1 Vin 1 − k
(4.3)
In steady-state, the average charge across capacitor C1 in a period should be zero. The relations are available: kTiC1−on = (1 − k )TiC1−off
© 2006 by Taylor & Francis Group, LLC
and
iC1−on =
1− k i k C1−off
Negative Output Super-Lift Luo-Converters
267
This relation is available for all capacitor’s current in switch-on and switchoff periods. The input current iin is equal to (iL1 + iC1) during switch-on, and zero during switch-off. Capacitor current iC1 is equal to iL1 during switch-off. iin−on = iL1−on + iC1−on
iL1−off = iC1−off = I L1
If inductance L1 is large enough, iL1 is nearly equal to its average current IL1. Therefore, iin−on = iL1−on + iC1−on = iL1−on +
1− k 1− k 1 )I L1 = I L1 iC1−off = (1 + k k k
and I in = kiin−on = I L1
(4.4)
Further iC 2−on = I O
iC 2−off =
I L1 = iC 2−off + I O =
k I 1− k O
k 1 i I +I = 1 − k C 2−on O 1 − k O
Variation ratio of inductor current iL1 is ξ1 =
ΔiL1 / 2 k(1 − k )TVin k(1 − k ) R = = I L1 2L1 I O G1 2 fL1
(4.5)
Usually ξ1 is small (much lower than unity), it means this converter works in the continuous conduction mode (CCM). The ripple voltage of output voltage vO is ΔvO =
ΔQ IO kT k VO = = C2 C2 fC2 R
Therefore, the variation ratio of output voltage vO is ε=
© 2006 by Taylor & Francis Group, LLC
ΔvO / 2 k = VO 2 RfC2
(4.6)
268
Essential DC/DC Converters
Usually R is in kΩ, f in 10 kHz, and C2 in μF, this ripple is very small. 4.2.2
N/O Re-Lift Circuit
N/O re-lift circuit is derived from N/O elementary circuit by adding the parts (L2-D3-D4-D5-C3-C4). Its circuit diagram and equivalent circuits during switch-on and -off are shown in Figure 4.2. The voltage across capacitor C1 is charged to Vin. As described in previous section the voltage V1 across capacitor C2 is V1 =
1 V 1 − k in
The voltage across capacitor C3 is charged to (V1 + Vin). The current flowing through inductor L2 increases with slop (V1 + Vin)/L2 during switch-on period kT and decreases with slop –(VO – 2V1 – Vin)/L2 during switch-off (1 – k)T. Therefore, the variation of current iL2 is ΔiL 2 =
V − 2V1 − Vin V1 + Vin kT = O (1 − k )T L2 L2
(4.7)
(2 − k )V1 + Vin 2−k 2 = [( ) − 1]Vin 1− k 1− k
(4.8)
VO =
The voltage transfer gain is G2 =
VO 2−k 2 =( ) −1 1− k Vin
(4.9)
The input current iin is equal to (iL1 + iC1 + iL2 + iC3) during switch-on, and zero during switch-off. In steady-state, the following relations are available: iin−on = iL1−on + iC1−on + iL 2−on + iC 3−on iC 4−on = I O
iC 3−off = I L 2 = I O + iC 4−off =
iC 2−on = I L 2 + iC 3−on =
IO 1− k
IO I IO + O = 1− k k k(1 − k )
© 2006 by Taylor & Francis Group, LLC
iC 4−off =
k I 1− k O
iC 3−on =
IO k
iC 2−off =
IO (1 − k )2
Negative Output Super-Lift Luo-Converters
iC1−off = I L1 = I L 2 + iC 2−off =
269
IO IO 2−k + = I 1 − k (1 − k )2 (1 − k )2 O
iC1−on =
2−k I k(1 − k ) O
Thus iin−on = iL1−on + iC1−on + iL 2−on + iC 3−on =
1 3 − 2k ( I L1 + I L 2 ) = I k k(1 − k )2 O
Therefore I in = kiin−on =
3 − 2k I (1 − k )2 O
Since ΔiL1 =
Vin kT L1
I L1 =
2−k I (1 − k )2 O
ΔiL 2 =
V1 + Vin 2 − k kT kT = V 1 − k L2 in L2
I L2 =
1 I 1− k O
Therefore, the variation ratio of current iL1 through inductor L1 is ξ1 =
kTVin ΔiL1 / 2 k(1 − k )2 R = = 2−k I L1 (2 − k )G2 2 fL1 2L I (1 − k )2 1 O
(4.10)
The variation ratio of current iL2 through inductor L2 is ξ2 =
ΔiL 2 / 2 k(2 − k )TVin k(2 − k ) R = = I L2 2L2 I O G2 2 fL2
(4.11)
The ripple voltage of output voltage vO is ΔvO =
ΔQ IO kT k VO = = C4 C4 fC4 R
Therefore, the variation ratio of output voltage vO is ε=
© 2006 by Taylor & Francis Group, LLC
ΔvO / 2 k = VO 2 RfC4
(4.12)
270 4.2.3
Essential DC/DC Converters N/O Triple-Lift Circuit
N/O triple-lift circuit is derived from N/O re-lift circuit by double adding the parts (L2-D3-D4-D5-C3-C4). Its circuit diagram and equivalent circuits during switch-on and -off are shown in Figure 4.3. The voltage across capacitor C1 is charged to Vin. As described in previous section the, voltage V1 across capacitor C2 is V1 = ((2 − k ) (1 − k ) − 1)Vin = (1 1 − k )Vin , and voltage V2 across 2 2 capacitor C4 is V2 = [(2 − k 1 − k ) − 1]Vin = 3 − 2 k (1 − k ) Vin . The voltage across capacitor C5 is charged to (V2 + Vin). The current flowing through inductor L3 increases with slop (V2 + Vin)/L3 during switch-on period kT and decreases with slop –(VO – 2V2 – Vin)/L3 during switch-off (1 – k)T. Therefore, the variation of current iL3 is
(
ΔiL 3 =
)
V − 2V2 − Vin V2 + Vin kT = O (1 − k )T L3 L3
(4.13)
(2 − k )V2 + Vin 2−k 3 = [( ) − 1]Vin 1− k 1− k
(4.14)
VO =
The voltage transfer gain is G3 =
VO 2−k 3 =( ) −1 Vin 1− k
(4.15)
The input current iin is equal to (iL1 + iC1 + iL2 + iC3 + iL3 + iC5) during switchon, and zero during switch-off. In steady state, the following relations are available: iin−on = iL1−on + iC1−on + iL 2−on + iC 3−on + iL 3− on + iC 5− on iC 6−on = I O
iC 5−off = I L 3 = I O + iC 6−off =
iC 4−on = I L 3 + iC 5−on =
IO 1− k
IO I IO + O = k(1 − k ) 1− k k
iC 3−off = I L 2 = I L 3 + iC 4−off =
© 2006 by Taylor & Francis Group, LLC
2−k I (1 − k )2 O
iC 6−off =
k I 1− k O
iC 5−on =
IO k
iC 4−off =
IO (1 − k )2
iC 3−on =
2−k I k(1 − k ) O
Negative Output Super-Lift Luo-Converters
iC 2−on = I L 2 + iC 3−on =
2−k I k(1 − k )2 O
iC1−off = I L1 = I L 2 + iC 2−off =
(2 − k )2 I (1 − k ) 3 O
271
iC 2−off =
2−k I (1 − k ) 3 O
iC1−on =
(2 − k )2 I k(1 − k )2 O
Thus iin−on = iL1−on + iC1−on + iL 2−on + iC 3−on + iL 3−on + iC 5−on =
1 7 − 9k + 3k 2 ( I L1 + I L 2 + I L 3 ) = I k k(1 − k )3 O
Therefore I in = kiin−on =
7 − 9k + 3k 2 2−k 3 I O = [( ) − 1]I O 1− k (1 − k ) 3
Analogously, ΔiL1 =
Vin kT L1
I L1 =
(2 − k )2 I (1 − k ) 3 O
ΔiL 2 =
V1 + Vin 2−k kT = kTVin (1 − k )L2 L2
I L2 =
2−k I (1 − k )2 O
ΔiL 3 =
V2 + Vin 2 − k 2 kT kT = ( ) V 1 − k L3 in L3
I L3 =
IO 1− k
Therefore, the variation ratio of current iL1 through inductor L1 is ξ1 =
ΔiL1 / 2 k(1 − k ) 3 TVin k(1 − k ) 3 R = = 2 I L1 2(2 − k ) L1 I O (2 − k )2 G3 2 fL1
(4.16)
The variation ratio of current iL2 through inductor L2 is ξ2 =
© 2006 by Taylor & Francis Group, LLC
ΔiL 2 / 2 k(1 − k )TVin k(1 − k ) R = = I L2 2L2 I O G3 2 fL2
(4.17)
272
Essential DC/DC Converters
The variation ratio of current iL3 through inductor L3 is ξ3 =
ΔiL 3 / 2 k(2 − k )2 TVin k(2 − k )2 R = = I L3 2(1 − k )L3 I O (1 − k )G3 2 fL3
(4.18)
The ripple voltage of output voltage vO is ΔvO =
ΔQ IO kT k VO = = C6 C6 fC6 R
Therefore, the variation ratio of output voltage vO is ε=
4.2.4
ΔvO / 2 k = 2 RfC6 VO
(4.19)
N/O Higher Order Lift Circuit
N/O higher order lift circuits can be designed by repeating the parts (L2-D3D4-D5-C3-C4) multiple times. For nth order lift circuit, the final output voltage across capacitor C2n is VO = [(
2−k n ) − 1]Vin 1− k
(4.20)
The voltage transfer gain is Gn =
VO 2−k n =( ) −1 Vin 1− k
(4.21)
The variation ratio of current iLi through inductor Li (i = 1, 2, 3, …n) is ξ1 =
ΔiL1 / 2 k(1 − k )n R = I L1 (2 − k )( n−1) Gn 2 fLi
(4.22)
ξ2 =
ΔiL 2 / 2 k ( 2 − k )( 3 − n ) R = I L2 (1 − k )( n− 3 ) Gn 2 fLi
(4.23)
ξ3 =
ΔiL 3 / 2 k ( 2 − k )( n − i + 2 ) R = I L3 (1 − k )( n−i +1) Gn 2 fL3
(4.24)
© 2006 by Taylor & Francis Group, LLC
Negative Output Super-Lift Luo-Converters
273
The variation ratio of output voltage vO is ε=
4.3
ΔvO / 2 k = VO 2 RfC2 n
(4.25)
Additional Series
All circuits of negative output super-lift Luo-converters — additional series — are derived from the corresponding circuits of the main series by adding a double/enhanced circuit (DEC). The first three stages of this series are shown in Figure 4.4 to Figure 4.6. For convenience they are called elementary additional circuits, re-lift additional circuit, and triple-lift additional circuit respectively, and numbered as n = 1, 2 and 3.
4.3.1
N/O Elementary Additional Circuit
This circuit is derived from the N/O elementary circuit by adding a DEC. Its circuit and switch-on and switch-off equivalent circuits are shown in Figure 4.4. The voltage across capacitor C1 is charged to Vin. The voltage across capacitor C2 is charged to V1 and C11 is charged to (V1 + Vin). The current iL1 flowing through inductor L1 increases with slope Vin/L1 during switch-on period kT and decreases with slope –(V1 – Vin)/L1 during switchoff (1 – k)T. Therefore, ΔiL1 =
V1 =
V − Vin Vin kT = 1 (1 − k )T L1 L1
(4.26)
1 2−k − 1)Vin V =( 1 − k in 1− k VL1−off =
k V 1 − k in
The output voltage is VO = Vin + VL1 + V1 =
© 2006 by Taylor & Francis Group, LLC
2 3−k V =[ − 1]Vin 1 − k in 1− k
(4.27)
274
Essential DC/DC Converters I in IO
S
+
C1
L1
+
+
V C1
C 11
V C11
_
Vin
_
_
R
_
D1
D 11
D2
D 12
_
V C12
C 12
V C2
C2
VO
_
+
+
+
(a) Circuit Diagram
I in
C 11
Vin
L1
_
_
+
C1
Vin
_
_
V1+Vin
+
IO
V1 _
+
V C12
C 12
V1
C2
+
_
R
+
VO +
(b) Switch-on
C11 +
C1 +
+
Vin
L1
Vin
V1
_
C2
IO _
_
VL1
_
_
V1+Vin
C12
V1
_
VC12 +
R
+
VO +
(c) Switch-off FIGURE 4.4 N/O elementary additional (enhanced) circuit.
The voltage transfer gain is G1 =
VO 3 − k = −1 Vin 1 − k
(4.28)
Following relations are obtained: iC12−on = I O
© 2006 by Taylor & Francis Group, LLC
iC12−off =
kI O 1− k
Negative Output Super-Lift Luo-Converters
275
I in S
IO
D3 +
L1
+
C1
+
L2
V C1
V in
C3
_
+
C11
V C3
V1
V C11
V2
_
_
_
R
_
D1
D2
D4
_
C2
D5
V C2
_ D11
C4
D12
V C4
+
C 12
V C12
+
+
VO
_
+
(a) Circuit Diagram
C11
I in
_
+
C3
+
V1+Vin
Vin
C1
L1
Vin
_
_
IO _
L2
+
+
V2
V2+Vin
V C12
_
V1
_
C2
_
_
V2 C4
+
C12
VO
R
+
+
V1 +
(b) Switch-on C11 C1 +
+
Vin _
L1
L2
V1 _
Vin C2
V L1
_
V L2
+ V1+Vin _
V1
C4
+
_
+
C3
IO
V2+Vin V2
_ _
_
V2 C12
VC12 R
+
+
VO +
(c) Switch-off FIGURE 4.5 N/O re-lift additional circuit.
iC11−off = I O + iC12−off =
iC 2−off = iC1−off =
IO 1− k
I L1 = iC1−off + iC11−on =
© 2006 by Taylor & Francis Group, LLC
IO 1− k
iC11−on = iC 2−on =
iC1−on = 2IO 1− k
IO k
IO k
276
Essential DC/DC Converters
I in D3
S
D6
IO + +
C1
L1
+
V C1 L2
Vin
V C3
C3
C5
_
V1
V2
D2
_
C2
D4
D5 C4
V C2
V C11
_
_
V3 R
_
D1
C11
V C5
_
_
+
+
L3
+
_
D7
D8
_
V C4
C6
V C6
D11
D12
_
C12
+
+
VO +
V C12 +
+
(a) Circuit Diagram C11 _
+
L2
I in
+
V3
V3+Vin
L3
+
C3 C5
V1+Vin
IO
V2+Vin _
_ +
+
Vin
Vin
C1
L1
_
_
C2
_
V2
_
V1 C4
V1
C6
V2
_
_
V3
V C12
+
C12
_
+
+
(b) Switch-on C11 V1
C1
+
Vin
V L1 L1
_
+
L2
_
Vin C2
V L2
_
V1
C3 +
V2 _
V1+Vin C4
+
L3
C5 _
+
_
V L3 V2+Vin
V2 +
C6
+
_
IO
V3+Vin V3 _
V3 C12 +
_ _
VC12
R
+
(c) Switch-off FIGURE 4.6 N/O triple-lift additional circuit.
iin = I L1 + iC1−on + iC11−on = (
2 1 1 2 + + )I = I 1 − k k k O k(1 − k ) O
Therefore, I in = kiin =
2 3−k IO = [ − 1]I O 1− k 1− k
The variation ratio of current iL1 through inductor L1 is
© 2006 by Taylor & Francis Group, LLC
R
VO +
+
VO +
Negative Output Super-Lift Luo-Converters
ξ1 =
277
ΔiL1 / 2 k(1 − k )TVin k(1 − k ) R = = I L1 4L1 I O 2G1 2 fL1
(4.29)
The ripple voltage of output voltage vO is ΔvO =
ΔQ IO kT k VO = = C12 C12 fC12 R
Therefore, the variation ratio of output voltage vO is ε=
4.3.2
1− k ΔvO / 2 = 2 RfC12 VO
(4.30)
N/O Re-Lift Additional Circuit
The N/O re-lift additional circuit is derived from the N/O re-lift circuit by adding a DEC. Its circuit diagram and switch-on and switch-off equivalent circuits are shown in Figure 4.5. The voltage across capacitor C1 is charged to vin. As described in a previous section the voltage across C2 is V1 =
1 V 1 − k in
The voltage across capacitor C3 is charged to (V1 + Vin), voltage across capacitor C4 is charged to V2 and voltage across capacitor C11 is charged to (V2 + Vin). The current flowing through inductor L2 increases with voltage (V1 + Vin) during switch-on kT and decreases with voltage –(V2 – 2V1 – Vin) during switch-off (1 – k)T. Therefore, ΔiL 2 =
V2 =
V − 2V1 − Vin V1 + Vin kT = 2 (1 − k )T L2 L2
(4.31)
(2 − k )V1 + Vin 3 − 2k 2−k 2 ) − 1]Vin = = [( 2 (1 − k ) 1− k 1− k
and VL 2−off = V2 − 2V1 − Vin = The output voltage is
© 2006 by Taylor & Francis Group, LLC
k( 2 − k ) V (1 − k )2 in
(4.32)
278
Essential DC/DC Converters
VO = V2 + Vin + VL 2 + V1 =
5 − 3k 3−k 2−k V =[ − 1]Vin (1 − k )2 in 1− k 1− k
(4.33)
The voltage transfer gain is G2 =
VO 2 − k 3 − k = −1 Vin 1 − k 1 − k
(4.34)
Following relations are obtained: iC12−on = I O
iC12−off =
iC11−off = I O + iC12−off =
iC 4−off = iC 3−off =
IO 1− k
iC11−on = iC 4−on =
IO 1− k
I L 2 = iC11−off + iC 3−off =
iC 2−on = I L 2 + iC 3−on =
kI O 1− k
iC 3−on =
IO k
iC 2−off =
1+ k I (1 − k )2 O
iC1−on =
3−k I k(1 − k ) O
2IO 1− k
1+ k I k(1 − k ) O
I L1 = iC1−off = I L 2 + iC 2−off =
3−k I (1 − k )2 O
iin = I L1 + iC1−on + iC 2−on + iC 4−on = [
3−k 3−k 1+ k 1 5 − 3k + + + ]I O = I 2 (1 − k ) k(1 − k ) k(1 − k ) k k(1 − k )2 O
Therefore, I in = kiin =
5 − 3k 3−k 2−k I =[ − 1]I O (1 − k )2 O 1− k 1− k
Analogously, ΔiL1 =
IO k
Vin kT L2
© 2006 by Taylor & Francis Group, LLC
I L1 =
3−k I (1 − k )2 O
Negative Output Super-Lift Luo-Converters
ΔiL 2 =
V1 + Vin 2−k kT = kTVin L2 (1 − k )L2
279
I L2 =
2IO 1− k
Therefore, the variation ratio of current iL1 through inductor L1 is ξ1 =
ΔiL1 / 2 k(1 − k )2 TVin k(1 − k ) 2 R = = I L1 2( 3 − k )L1 I O ( 3 − k )G2 2 fL1
(4.35)
The variation ratio of current iL2 through inductor L2 is ξ2 =
ΔiL 2 / 2 k(2 − k )TVin k(2 − k ) R = = I L2 4L2 I O 2G2 2 fL2
(4.36)
The ripple voltage of output voltage vO is ΔvO =
ΔQ IO kT k VO = = C12 C12 fC12 R
Therefore, the variation ratio of output voltage vO is ε=
4.3.3
ΔvO / 2 k = 2 RfC12 VO
(4.37)
N/O Triple-Lift Additional Circuit
This circuit is derived from the N/O triple-lift circuit by adding a DEC. Its circuit diagram and equivalent circuits during switch-on and switch-off are shown in Figure 4.6. The voltage across capacitor C1 is charged to Vin. As described in a previous section the voltage across C2 is V1 =
1 V 1 − k in
and voltage across C4 is V2 =
3 − 2k 3 − 2k V1 = V 1− k (1 − k )2 in
The voltage across capacitor C5 is charged to (V2 + Vin), voltage across capacitor C6 is charged to V3 and voltage across capacitor C11 is charged to (V3 + Vin). The current flowing through inductor L3 increases with voltage
© 2006 by Taylor & Francis Group, LLC
280
Essential DC/DC Converters
(V 2 + Vin ) during switch-on period kT and decreases with voltage –(V3 – 2V2 – Vin) during switch-off (1 – k)T. Therefore, ΔiL 3 =
V3 =
V − 2V2 − Vin V2 + Vin (1 − k )T kT = 3 L3 L3
(4.38)
(2 − k )V2 + Vin 7 − 9k + 3k 2 2−k 3 = ) − 1]Vin Vin = [( 3 1− k (1 − k ) 1− k
and VL 3−off = V3 − 2V2 − Vin =
k( 2 − k ) 2 V (1 − k ) 3 in
(4.39)
The output voltage is VO = V3 + Vin + VL 3 + V2 =
11 − 13k + 4 k 2 3−k 2−k 2 ( ) − 1]Vin Vin = [ (1 − k ) 3 1− k 1− k
(4.40)
The voltage transfer gain is G3 =
VO 2−k 2 3−k =( ) −1 1− k 1− k Vin
(4.41)
Following relations are available: iC12−on = I O
iC12−off =
iC11−off = I O + iC12−off =
iC 6−off = iC 5−off =
IO 1− k
IO 1− k
I L 3 = iC11−off + iC 5−off =
iC 4−on = I L 3 + iC 5−on =
kI O 1− k
iC11−on = iC 6−on =
IO k
iC 5−on =
IO k
iC 4−off =
1+ k I (1 − k )2 O
2IO 1− k
1+ k I k(1 − k ) O
© 2006 by Taylor & Francis Group, LLC
Negative Output Super-Lift Luo-Converters
I L 2 = iC 3−off = I L 3 + iC 4−off =
iC 2−on = I L 2 + iC 3−on =
3−k I (1 − k )2 O
3−k I k(1 − k )2 O
I L1 = iC1−off = I L 2 + iC 2−off =
(3 − k )(2 − k ) IO (1 − k ) 3
281
iC 3−on =
3−k I k(1 − k ) O
iC 2−off =
3−k I (1 − k ) 3 O
iC1−on =
(3 − k )(2 − k ) IO k(1 − k )2
iin = I L1 + iC1−on + iC 2−on + iC 4−on + iC 6−on =[ =
3−k 1+ k 1 (3 − k )(2 − k ) (3 − k )(2 − k ) + + + + ]I O 3 2 2 k(1 − k ) k(1 − k ) k(1 − k ) k (1 − k )
11 − 13k + 4 k 2 IO k(1 − k )3
Therefore, I in = kiin =
11 − 13k + 4 k 2 3−k 2−k 2 IO = [ ( ) − 1]I O 3 1− k 1− k (1 − k )
Analogously, ΔiL1 =
Vin kT L2
I L1 =
(2 − k )( 3 − k ) IO (1 − k ) 3
ΔiL 2 =
V1 + Vin 2−k kT = kTVin (1 − k )L2 L2
I L2 =
3−k I (1 − k )2 O
ΔiL 3 =
V2 + Vin (2 − k )2 kT = kTVin (1 − k )2 L3 L3
I L3 =
2IO 1− k
Therefore, the variation ratio of current iL1 through inductor L1 is ξ1 =
k(1 − k ) 3 TVin ΔiL1 / 2 k(1 − k ) 3 R = = I L1 2(2 − k )(3 − k )L1 I O (2 − k )( 3 − k )G3 2 fL1
(4.42)
and the variation ratio of current iL2 through inductor L2 is ξ2 =
ΔiL 2 / 2 k(1 − k )(2 − k )TV1 k(1 − k )(2 − k ) R = = 2(3 − k )L2 I O (3 − k )G3 2 fL2 I L2
© 2006 by Taylor & Francis Group, LLC
(4.43)
282
Essential DC/DC Converters
and the variation ratio of current iL3 through inductor L3 is ξ3 =
ΔiL 3 / 2 k(2 − k )2 TVin k( 2 − k ) 2 R = = I L3 4(1 − k )L3 I O 2(1 − k )G3 2 fL3
(4.44)
The ripple voltage of output voltage vO is ΔvO =
ΔQ IO kT k VO = = C12 C12 fC12 R
Therefore, the variation ratio of output voltage vO is ε=
4.3.4
ΔvO / 2 k = 2 RfC12 VO
(4.45)
N/O Higher Order Lift Additional Circuit
Higher order N/O lift additional circuits can be derived from the corresponding circuits of the main series by adding a DEC. Each stage voltage Vi (i = 1, 2, … n) is Vi = [(
2−k i ) − 1]Vin 1− k
(4.46)
This means V1 is the voltage across capacitor C2, V2 is the voltage across capacitor C4 and so on. For nth order lift additional circuit, the final output voltage is 3 − k 2 − k n −1 ( ) − 1]Vin 1− k 1− k
(4.47)
VO 3 − k 2 − k n−1 = ( ) −1 Vin 1 − k 1 − k
(4.48)
VO = [ The voltage transfer gain is Gn =
Analogously, the variation ratio of current iLi through inductor Li (i = 1, 2, 3, …n) is ξ1 =
ΔiL1 / 2 k(1 − k )n R = h(1−n) I L1 2 [(2 − k )( n−2 ) (3 − k )]u( n−2 ) Gn fL1
© 2006 by Taylor & Francis Group, LLC
(4.49)
Negative Output Super-Lift Luo-Converters
283
ξ2 =
ΔiL 2 / 2 k(1 − k )( n−2 ) (2 − k ) R = h ( n− 2 ) I L2 2 (3 − k )( n−2 ) Gn 2 fL2
(4.50)
ξ3 =
ΔiL 3 / 2 k(2 − k )( n−1) R = h ( n− 3 ) ( n− 2 ) I L3 Gn 2 fL3 2 (1 − k )
(4.51)
where ⎧0 h( x) = ⎨ ⎩1
x>0 is the Hong function x≤0
and ⎧1 u( x) = ⎨ ⎩0
x≥0 is the unit-step function x<0
and the variation ratio of output voltage vO is ε=
4.4
ΔvO / 2 k = 2 RfC12 VO
(4.52)
Enhanced Series
All circuits of the negative output super-lift Luo-converters — enhanced series — are derived from the corresponding circuits of the main series by adding the DEC into each stage circuit of all series converters. The first three stages of this series are shown in Figures 4.4, 4.7, and 4.8. For convenience they are called elementary enhanced circuits, re-lift enhanced circuits, and triple-lift enhanced circuits respectively, and numbered as n = 1, 2 and 3.
4.4.1
N/O Elementary Enhanced Circuit
This circuit is derived from N/O elementary circuit with adding a DEC. Its circuit and switch-on and switch-off equivalent circuits are shown in Figure 4.4.
© 2006 by Taylor & Francis Group, LLC
284
Essential DC/DC Converters
I in s
D3
+
IO + L1
C1
+ C11
V C1
V in
–
+
V C11
L2
C3
V C3
–
V1
+ C21
V C21
–
–
– R
D1
D11 –
D2 C2
–
D4 –
D5
D21 –
V C12
C4
V C4
D12 C12
V C2 +
+
D22 C22
+
VO +
– V C22 +
(a) Circuit Diagram I in +
+
Vin
L1
C11 + V in –
C1
+
V1+V
in
L2
C3
–
C21
V C3
V2+V
–
–
V1 –
C2
C4
V C12
+
in
V2 C22 –
–
V1 C12
–
IO
+
+
V2
–
–
V C22 R
VO
+
+
+
(b) Switch–on L2
C11
C21 IO
+
C1 Vin
L1
VL1 C2
–
+ Vin + V2+Vin – – V1 C12 – V1 +
VL2 –
C3
VC12 +
C4
+ VC3 + V3+Vin – – V2 C22 – V2 +
– VC22 R +
– VO +
(c) Switch–off FIGURE 4.7 N/O re-lift enhanced circuit.
The output voltage is VO = Vin + VL1 + V1 =
2 3−k V =[ − 1]Vin 1 − k in 1− k
(4.27)
The voltage transfer gain is G1 =
© 2006 by Taylor & Francis Group, LLC
VO 3 − k = −1 Vin 1 − k
(4.28)
Negative Output Super-Lift Luo-Converters
285
Iin s
D3
+
D6 IO
L1
C1
+ VC1
+ L2 VC11 C3
C11
– V1
Vin
–
+ VC3 C21
+ VC21
–
L3
+ VC5
C5
–
C31
– V1
+ VC31 – VO
– R
D1
D11 – VC2
D2 C2
–
D4 D5 – VC12 C4
D12 C12
+
+
D21 D22 – VC4 C22 +
–
D7
D8 C6
VC22 +
D31 – VC6
D32
– + VC32
C32
+
+
(a) Circuit diagram Iin + C1 Vin L1
C11 + Vin – C2
–
+ V1+Vin L2
C3
–
+ C21 VC3
+ V2+Vin L3
–
V1 – V1C12 +
– VC12 +
C5
–
–
V2 – V2 C22 +
C4
+ C31 VC5
– VC22 +
C6
IO
+ V3+Vin – V3 C32 – V3 +
–
–
VC32R
VO
+
+
(b) Switch-on C11
L2
C21
L3
C31 IO
+ + – Vin V2+Vin – C12 L1 VL1 V1 – V1 + C2
VL2 C3
C1
– VC12 + C4
+ + – VC3 V3+Vin – V2 C22 – V2 +
VL3 C5 – VC22 + C6
+ + – VC5 V4+Vin – V3 C32 – V3 +
– VC32 R +
– VO +
(c) Switch-off FIGURE 4.8 N/O triple-lift enhanced circuit.
4.4.2
N/O Re-Lift Enhanced Circuit
The N/O re-lift enhanced circuit is derived from N/O re-lift circuit of the main series by adding the DEC into each stage. Its circuit diagram and switch-on and switch-off equivalent circuits are shown in Figure 4.7. The voltage across capacitor C12 is charged to VC12 =
3 V 1 − k in
(4.53)
The voltage across capacitor C3 is charged to VC12, and the voltage across capacitor C4 and C12 is charged to VC4
© 2006 by Taylor & Francis Group, LLC
286
Essential DC/DC Converters
VC 4 =
2−k 2−k 3−k V = V 1 − k C12 1 − k 1 − k in
(4.54)
The current flowing through inductor L2 increases with voltage VC12 during switch-on kT and decreases with voltage –(VC21 – VC4 – VC12) during switchoff (1 – k)T. Therefore, ΔiL 2 =
V − VC 4 − VC12 kT (V − Vin ) = C 21 (1 − k )T L2 C12 L2 VC 21 = (
(4.55)
3−k 2 ) Vin 1− k
The output voltage is VO = VC 21 − Vin = [(
3−k 2 ) − 1]Vin 1− k
(4.56)
The voltage transfer gain is G2 =
VO 3−k 2 =( ) −1 1− k Vin
(4.57)
Following relations are obtained: iC 22−on = I O
iC 22−off =
iC 21−off = I O + iC 22−off =
iC 4−off = iC 3−off =
IO 1− k
IO 1− k
I L 2 = iC 21−off + iC 3−off =
2IO 1− k
iC12−on = I L 2 + iC 3−on =
1+ k I k(1 − k ) O
© 2006 by Taylor & Francis Group, LLC
kI O 1− k
iC 21−on = iC 4−on =
IO k
iC 3−on =
IO k
iC12−off =
1+ k I (1 − k )2 O
Negative Output Super-Lift Luo-Converters
iC11−off = I L 2 + iC12−off =
iC11−on = iC 2−on =
3−k I (1 − k )2 O
287
iC 2−off =
3−k I (1 − k )2 O
iC1−on =
3−k I k(1 − k ) O
3−k I k(1 − k ) O
I L1 = iC11−off + iC 2−off = 2
3−k I (1 − k )2 O
iin = I L1 + iC1−on + iC11−on + iC12−on + iC 21−on =
4(2 − k ) I k(1 − k )2 O
Therefore, I in = kiin =
(3 − k )2 4(2 − k ) [ I = − 1]I O O (1 − k )2 (1 − k )2
Analogously, ΔiL1 =
Vin kT L2
I L1 = 2
ΔiL 2 =
VC12 − Vin 2+k kT = kTVin (1 − k )L2 L2
I L2 =
3−k I (1 − k )2 O
2IO 1− k
Therefore, the variation ratio of current iL1 through inductor L1 is ξ1 =
ΔiL1 / 2 k(1 − k )2 TVin k(1 − k )2 R = = I L1 4(3 − k )L1 I O 2(3 − k )G2 2 fL1
(4.58)
The variation ratio of current iL2 through inductor L2 is ξ2 =
ΔiL 2 / 2 k(2 + k )TVin k(2 + k ) R = = I L2 4L2 I O 2G2 2 fL2
The ripple voltage of output voltage vO is ΔvO =
© 2006 by Taylor & Francis Group, LLC
ΔQ IO kT k VO = = C22 C22 fC22 R
(4.59)
288
Essential DC/DC Converters
Therefore, the variation ratio of output voltage vO is ε=
4.4.3
ΔvO / 2 k = VO 2 RfC22
(4.60)
N/O Triple-Lift Enhanced Circuit
This circuit is derived from the N/O triple-lift circuit of main series by adding the DEC into each stage. Its circuit diagram and equivalent circuits during switch-on and switch-off are shown in Figure 4.8. The voltage across capacitor C12 is charged to VC12. As described in the previous section the voltage across CC12 is VC12 =
3−k V 1 − k in
and voltage across C4 and CC22 is VC 22 =
3−k 3−k 2 ) Vin V =( 1 − k C12 1− k
The voltage across capacitor C5 is charged to VC22, voltage across capacitor C6 is charged to VC6 VC 6 =
2−k 2−k 3−k 2 ( ) Vin V = 1 − k C 22 1 − k 1 − k
The current flowing through inductor L3 increases with voltage VC22 during switch-on period kT and decreases with voltage –(VC32 – VC6 – VC22) during switch-off (1 – k)T. Therefore, ΔiL 3 =
V − VC 6 − VC 22 kT (V − Vin ) = C 31 (1 − k )T L3 L3 C 22 VC 31 = (
(4.61)
3−k 3 ) Vin 1− k
and VO = VC 31 − Vin = [(
© 2006 by Taylor & Francis Group, LLC
3−k 3 ) − 1]Vin 1− k
(4.62)
Negative Output Super-Lift Luo-Converters
289
The voltage transfer gain is G3 =
VO 3−k 2 =( ) −1 1− k Vin
(4.63)
The following relations are obtained: iC 32−on = I O
iC 32−off = IO 1− k
iC 31−off = I O + iC 32−off =
iC 6−off = iC 5−off =
IO 1− k
I L 3 = iC 31−off + iC 5−off =
2IO 1− k
iC 22−on = I L 3 + iC 5−on =
1+ k I k(1 − k ) O
iC 21−off = iC 4−off = I L 3 + iC 22−off =
3−k I (1 − k )2 O
I L 2 = iC 4−off + iC 21−off = 2
iC12−on = I L 2 + iC 3−on =
3−k I (1 − k )2 O
(3 − k )(2 − k ) IO k(1 − k )2
iC11−off = I L 2 + iC12−off =
kI O 1− k
iC 31−on = iC 6−on =
IO k
iC 6−on = iC 5−on =
IO k
iC 22−off =
1+ k I (1 − k )2 O
iC 4−on =
3−k I k(1 − k ) O
iC 3−on =
3−k I k(1 − k ) O
iC12−off =
(3 − k )(2 − k ) IO (1 − k ) 3
(3 − k )( 4 − 3k ) IO (1 − k ) 3
iC11−on =
(3 − k )( 4 − k ) IO k(1 − k )2
(3 − k )( 4 − k ) IO (1 − k ) 3
iC1−on = iC 2−on =
I L1 = iC11−off + iC1−off = 2
iin = I L1 + iC1−on + iC 2−on + iC12−on + iC 4− on + iC 22− on + iC 6− on =
© 2006 by Taylor & Francis Group, LLC
(3 − k )( 4 − k ) IO k(1 − k )2
2(13 − 12 k + 3k 2 ) IO k(1 − k ) 3
290
Essential DC/DC Converters
Therefore, I in = kiin = 2
13 − 12 k + 3k 2 3−k 3 I O = [( ) − 1]I O 1− k (1 − k ) 3
Analogously: 2( 4 − k )(3 − k ) IO (1 − k ) 3
ΔiL1 =
Vin kT L2
I L1 =
ΔiL 2 =
V1 + Vin 2−k kT = kTVin (1 − k )L2 L2
I L2 = 2
ΔiL 3 =
V2 + Vin (2 − k )2 kT = kTVin (1 − k )2 L3 L3
I L3 =
3−k I (1 − k )2 O
2IO 1− k
Therefore, the variation ratio of current iL1 through inductor L1 is ξ1 =
k(1 − k ) 3 TVin ΔiL1 / 2 k(1 − k ) 3 R = = I L1 4( 4 − k )(3 − k )L1 I O 2( 4 − k )(3 − k )G3 2 fL1
(4.64)
and the variation ratio of current iL2 through inductor L2 is ξ2 =
ΔiL 2 / 2 k(1 − k )(2 − k )TV1 k(1 − k )(2 − k ) R = = I L2 4(3 − k )L2 I O 2(3 − k )G3 2 fL2
(4.65)
and the variation ratio of current iL3 through inductor L3 is ξ3 =
ΔiL 3 / 2 k(2 − k )2 TVin k( 2 − k ) 2 R = = 4(1 − k )L3 I O 2(1 − k )G3 2 fL3 I L3
(4.66)
The ripple voltage of output voltage vO is ΔvO =
ΔQ IO kT k VO = = C32 C32 fC32 R
Therefore, the variation ratio of output voltage vO is ε=
© 2006 by Taylor & Francis Group, LLC
ΔvO / 2 k = VO 2 RfC32
(4.67)
Negative Output Super-Lift Luo-Converters 4.4.4
291
N/O Higher Order Lift Enhanced Circuit
Higher order N/O lift enhanced circuit is derived from the corresponding circuit of the main series by adding the DEC in each stage. Each stage final voltage VCi1 (i = 1, 2, … n) is VCi 1 = (
3−k i )V 1 − k in
(4.68)
For nth order lift enhanced circuit, the final output voltage is VO = [(
3−k n ) − 1]Vin 1− k
(4.69)
The voltage transfer gain is Gn =
VO 3−k n =( ) −1 Vin 1− k
(4.70)
The variation ratio of output voltage vO is ε=
4.5
ΔvO / 2 k = 2 RfCn2 VO
(4.71)
Re-Enhanced Series
All circuits of negative output super-lift Luo-converters — re-enhanced series — are derived from the corresponding circuits of the main series by adding the DEC twice in each stage circuit. The first three stages of this series are shown in Figure 4.9 through Figure 4.11. For convenience they are called elementary re-enhanced circuits, re-lift re-enhanced circuits, and triple-lift re-enhanced circuits respectively, and numbered as n = 1, 2, and 3.
4.5.1
N/O Elementary Re-Enhanced Circuit
This circuit is derived from the N/O elementary circuit by adding the DEC twice. Its circuit and switch-on and switch-off equivalent circuits are shown in Figure 4.9. The voltage across capacitor C1 is charged to Vin. The voltage across capacitor C12 is charged to VC12. The voltage across capacitor C13 is charged to VC13.
© 2006 by Taylor & Francis Group, LLC
292
Essential DC/DC Converters
I in IO
s
+
+ L1
C1
+
V C1
C11
+
V C11
–
C13
V C13
–
–
–
Vin
R D1
–
D2
D11 –
C2
V C2
D13 – V C12
D12 C12
+
D14
+
–
C14
VO
V C14
+
+
(a) Circuit Diagram I in +
V in
+
L1
C1
C11 + V in – C2
–
IO
+
V1+V
in
C13
–
V2+V1 –
–
V1
C14
V2 –
V1
–
V C14 R
VO
– C12
+
+
V2
+
+
(b) Switch-on C13 IO +
C1 V in
L1
VL1 C2
–
+ C11 + Vin VC11 + V2+V1 – – – V1 V2 C14 – – V1 V2 + C12 +
–
–
VC14 R
VO
+
+
(c) Switch-off FIGURE 4.9 N/O elementary re-enhanced circuit.
VC13 =
4−k V 1 − k in
(4.72)
The output voltage is VO = VC13 − Vin = [ The voltage transfer gain is
© 2006 by Taylor & Francis Group, LLC
4−k − 1]Vin 1− k
(4.73)
Negative Output Super-Lift Luo-Converters
293
Iin s + L1 C1
+ VC1
C11
–
IO
+ D3 VC13 L2 C3
+ VC11 C13 –
+ VC3
–
C21
+ VC21 C23
–
–
+ VC23
D1
D2 C2
–
D11 – VC2 +
D13 D14 – VC12 C14 +
D12 C12
–
D3
D4 C4
VC14 +
D21 – VC4 +
D22 C22
R
– VO
–
+
–
Vin D23 D24 – VC22 C24 +
VC24 +
(a) Circuit Diagram Iin +
Vin
L1
C1
C11 + Vin – C2
–
+ V2+V1 L2 C3 – V2 – – VC14 C14 V2 + +
+ V1+Vin C13 – V1 – C12 V1 +
+ C21 VC3 –
C4
+ VC3+V3 C23 – V3 – C22 V3 +
+ V4+V3 – V4 C24 – V4 +
IO –
–
VC24R
VO
+
+
(b) Switch-on
L2
C13
C23 IO
C1 L1
VL1 C2
+ C11 + Vin VC11 + V2+V1 – – – V1 V2 C14 – – V1 V2 + C12 +
–
VL2
C3
VC14 +
C4
+ C21 + VC3 VC21 + V3+V4 – – – V3 V4 C24 – – V3 V4 + C22 +
–
–
VC24 R
VO
+
+
(c) Switch-off FIGURE 4.10 N/O re-lift re-enhanced circuit.
G1 =
VO 4 − k = −1 Vin 1 − k
(4.74)
The ripple voltage of output voltage vO is ΔvO =
ΔQ IO kT k VO = = C14 C14 fC14 R
Therefore, the variation ratio of output voltage vO is ε=
© 2006 by Taylor & Francis Group, LLC
ΔvO / 2 k = VO 2 RfC14
(4.75)
IO + VC1
L1
C1
+ VC11
C11
+ D3 VC13 L2
C13
–
–
C3
–
+ VC3
+ VC21
C21
C23
–
–
+ D6 VC23
L3
+ VC5
C5
+ VC31
C31
–
–
+ VC33
C33
–
–
–
Vin
R D1
D11 – VC2
D2 C2
–
D13 – VC12
D12 C12
+
D14 C14
–
D4
+
D21 – VC4
D5 C4
VC14 +
D23 – VC22
D22 C22
+
D24
–
C24
VC24
D7
D8 C6
D31 – VC6 +
+
+
D33 – VC32
D32 C32
294
s +
D34
VO
–
C34
+
VC34
+
+
(a) Circuit Diagram Iin
+
+ C11
L2
V1+Vin
V2+V1
–
–
V1
V2
+
C1 Vin
C3
+
C13
+ VC3 –
+
C21
C5
+
C23 VC3 +V3
L3 V4+V3
–
–
V3
V4
IO + VC5 –
C31
+
+
C33 VC5 +V5
V5+V6
–
– C2 –
VC14
–
–
V1
V2 C12
+
–
C14
+ C4
+
–
–
V3
V4 C22
+
–
C24
V6 C34
VC34
–
–
+
V5
V6
V5
VC24 + C6
+
+
C32
–
–
–
Vin
L1
R
VO +
+
(b) Switch-on L3
C23
C33 IO
C1
L1
VL1
+ C11 Vin – V1 –
+ VC11
+ C12
V2+V1
V2
C14
– V2 +
VL2
–
C3
–
–
V1 C2
+
+ C21 VC3 – V3 –
VC14 +
+
C24
V4
+ C22
+
(c) Switch-off
FIGURE 4.11 N/O triple-lift re-enhanced circuit. © 2006 by Taylor & Francis Group, LLC
VL3
–
VC21 V3+V4 – V4 –
V3 C4
+
C5
–
+ C31 VC5 – V5 –
VC24
V6
+ C6
+ C32
+
+
–
VC31 V5+V6 – V6 – V6 +
C34
– VC34 R +
– VO +
Essential DC/DC Converters
L2
C13
Negative Output Super-Lift Luo-Converters 4.5.2
295
N/O Re-Lift Re-Enhanced Circuit
The N/O re-lift re-enhanced circuit is derived from the N/O re-lift circuit by adding the DEC twice in each stage. Its circuit diagram and switch-on and switch-off equivalent circuits are shown in Figure 4.10. The voltage across capacitor C13 is charged to VC13. As described in the previous section the voltage across C13 is VC13 =
4−k V 1 − k in
VC 23 = (
4−k 2 ) Vin 1− k
Analogously, (4.76)
The output voltage is VO = VC 23 − Vin = [(
4−k 2 ) − 1]Vin 1− k
(4.77)
The voltage transfer gain is G2 =
VO 4−k 2 =( ) −1 Vin 1− k
(4.78)
The ripple voltage of output voltage vO is ΔvO =
ΔQ IO kT k VO = = C24 C24 fC24 R
Therefore, the variation ratio of output voltage vO is ε=
4.5.3
ΔvO / 2 k = VO 2 RfC24
(4.79)
N/O Triple-Lift Re-Enhanced Circuit
This circuit is derived from N/O triple-lift circuit by adding the DEC twice in each stage circuit. Its circuit diagram and equivalent circuits during switch-on and switch-off are shown in Figure 4.11. The voltage across capacitor C13 is
© 2006 by Taylor & Francis Group, LLC
296
Essential DC/DC Converters
VC13 =
4−k V 1 − k in
The voltage across capacitor C23 is VC 23 = (
4−k 2 ) Vin 1− k
Analogously, the voltage across capacitor C33 is VC 33 = (
4−k 3 ) Vin 1− k
(4.80)
The output voltage is VO = VC 33 − Vin = [(
4−k 3 ) − 1]Vin 1− k
(4.81)
The voltage transfer gain is G3 =
VO 4−k 3 =( ) −1 1− k Vin
(4.82)
The ripple voltage of output voltage vO is ΔvO =
ΔQ IO kT k VO = = C34 C34 fC34 R
Therefore, the variation ratio of output voltage vO is ε=
4.5.4
ΔvO / 2 k = VO 2 RfC34
(4.83)
N/O Higher Order Lift Re-Enhanced Circuit
Higher order N/O lift re-enhanced circuits can be derived from the corresponding circuits of the main series by adding the DEC twice in each stage circuit. Each stage final voltage VCi3 (i = 1, 2, … n) is VCi 3 = (
© 2006 by Taylor & Francis Group, LLC
4−k i )V 1 − k in
(4.84)
Negative Output Super-Lift Luo-Converters
297
For nth order lift additional circuit, the final output voltage is VO = VCn 3 − Vin = [(
4−k n ) − 1]Vin 1− k
(4.85)
The voltage transfer gain is Gn =
VO 4−k n =( ) −1 Vin 1− k
(4.86)
The variation ratio of output voltage vO is ε=
4.6
ΔvO / 2 k = 2 RfCn 4 VO
(4.87)
Multiple-Enhanced Series
All circuits of negative output super-lift Luo-converters — multipleenhanced series are derived from the corresponding circuits of the main series by adding the DEC multiple (j) times in each stage circuit. The first three stages of this series are shown in Figure 4.12 to Figure 4.14. For convenience they are called elementary multiple-enhanced circuits, relift multiple-enhanced circuits, and triple-lift multiple-enhanced circuits respectively, and numbered as n = 1, 2, and 3.
4.6.1
N/O Elementary Multiple-Enhanced Circuit
This circuit is derived from the N/O elementary circuit by adding the DEC multiple (j) times. Its circuit and switch-on and switch-off equivalent circuits are shown in Figure 4.12. The voltage across capacitor C12j-1 is VC12 j −1 =
j+2−k Vin 1− k
(4.88)
The output voltage is VO = VC12 j −1 − Vin = [
© 2006 by Taylor & Francis Group, LLC
j+2−k − 1]Vin 1− k
(4.89)
298
Essential DC/DC Converters 1
2..
j
Iin IO
s
+
L1
+ VC1
C1
C1(2j–1)
+ VC11
C11
–
+ VC1(2j–1) –
–
Vin
D1
D11 – VC2 +
D2 C2
–
D12 C12
D1(2j–1) D12j – C12j VC12 +
R
– VO
–
+
VC12j +
(a) Circuit Diagram I in +
Vin
+ C11 + V in –
C1
L1
V1+V
in
V1
C2
C12j
V C12j
+
V1
–
–
–
V C12
C12 –
–
V C1(2j–1)
–
–
IO
+
C1(2j–1)
R
+
VO +
+
(b) Switch-on C11
IO
+ C1
Vin
C1(2j–1)
L1
VL1 C2
–
+ Vin + V2+Vin – – V1 C12 – V1 +
– VC1(2j–1) – VC12 +
C12j
–
–
VC12j R
VO
+
+
(c) Switch-off FIGURE 4.12 N/O elementary multiple-enhanced circuit.
The voltage transfer gain is G1 =
VO j + 2 − k = −1 Vin 1− k
The ripple voltage of output voltage vO is ΔvO =
© 2006 by Taylor & Francis Group, LLC
ΔQ IO kT k VO = = C12 j C12 j fC12 j R
(4.90)
Negative Output Super-Lift Luo-Converters
299
Iin D3
s
IO + VC1
+ L1
C1
+ C1(2j–1) + VC11 VC1(2j–1) – –
C11
–
L2
+ VC3
C3
+ C2(2j–1) VC21
C21
–
–
+ VC2(2j–1) –
–
Vin D1
D2 C2
–
D11 – VC2
D12 C12
D1(2j–1) D12j – C12j V C12
+
D4
–
D5 C4
VC12j +
+
D21 – VC4
D22 C22
+
D2(2j–1) D22j – C22j VC22 +
R
VO
–
+
VC22j +
(a) Circuit Diagram Iin + C11 + Vin –
C1 Vin
L1
C2 –
+ C1(2j–1) V1+Vin – – VC12 V1 C12 – + V1 +
+ VC1(2j–1) L2 –
+ VC3 C21 –
C3 – VC12j +
C12j
C4
+
+ VC2(2j–1) – C22j
C2(2j–1)
VC21 – V2 – C22 V2 +
– VC22 +
IO
– VC22j
– R VO
+
+
(b) Switch-on C11
C21
L2
C1(2j–1)
C2(2j–1) IO
+ Vin
C1 L1
–
VL1 C2
+ Vin + V2+Vin – – V1 C12 – V1 +
VL2 C3
– VC1(2j–1) – VC12
C12j
– VC12j
+
+
C4
+ VC3 + – V2 – V2 +
VC21
–
C22
– VC2(2j–1) – VC22 +
C22j
–
–
VC22j R
VO
+
+
(c) Switch-off
FIGURE 4.13 N/O re-lift multiple-enhanced circuit.
Therefore, the variation ratio of output voltage vO is ε=
4.6.2
ΔvO / 2 1− k = 2 RfC12 j VO
(4.91)
N/O Re-Lift Multiple-Enhanced Circuit
The N/O re-lift multiple-enhanced circuit is derived from the N/O re-lift circuit by adding the DEC multiple (j) times into each stage. Its circuit diagram and switch-on and switch-off equivalent circuits are shown in Figure 4.13. The voltage across capacitor C22j-1 is VC 22 j −1 = (
© 2006 by Taylor & Francis Group, LLC
j+2−k 2 ) Vin 1− k
(4.92)
300
D3
s
D6 IO
+ L1
C1
+ VC1
+ C1(2j–1) VC11
C11
–
–
+ VC3
+ L2
VC1(2j–1) –
C3
+ C2(2j–1) VC21
C21
L3
VC2(2j–1) –
–
–
+ VC5
+ C5
C31
+ C3(2j–1) VC31 –
–
+ VC3(2j–1) –
–
Vin D1
D2 C2
–
D1–1
D12
VC2
C12
D1(2j–1) D12j – C12j VC12
+
D4
–
+
D2–1
D5
VC12j
C4
VC4
+
D2(2j–1) D22j – C22j VC22
D22 C22
+
–
D7
VC22j
+
D8
D3–1
D32
C6
VC6
C32
D3(2j–1) D32j – C32j VC32
+
+
R
VO
–
+
VC32j
+
+
(a) Circuit Diagram +
L1 Vin
+ C1
C11 + Vin – C2
–
V1+Vin
C21
+
L3
+ VC5 –
+ C31
C22j
+
–
–
–
VC32
– C32 V3 +
C6
VC3(2j–1)
–
V3
– VC22j +
IO
+
C3(2j–1)
VC31 –
–
VC22
– C22 V2 +
C4
C5
+ VC2(2j–1)
–
V2
– VC12j +
C12j
C2(2j–1)
VC21 –
–
VC12
V1C12 – V1 +
+
+ VC3 –
L2
VC1(2j–1)
–
–
C3
+
C1(2j–1)
C32j
VC32j
+
VO
R
+
+
(b) Switch-on C11
C1(2j–1)
C2(2j–1)
C21
L2
L3
C31
C3(2j–1)
L1
VL1
+ Vin – V1 –
+
V2+Vin C12
V1 C2
–
+V – C1(2j–1) – VC12
VL2
C3
C12j +
+
+
VC3 – V2 –
– VC12j
+
+
V2 C4
VC21 C22
–
+
VC2(2j–1) –
VC22 +
+
(c) Switch-off
FIGURE 4.14 N/O triple-lift multiple-enhanced circuit. © 2006 by Taylor & Francis Group, LLC
VL3
–
C5
–
VC5 – V3 –
VC22j C22j +
+
V3 C6
+
+
VC31 C32
–
+
VC3(2j–1) –
–
VC32 +
– VC32j R
– VO
C32j +
+
Essential DC/DC Converters
IO C1
Negative Output Super-Lift Luo-Converters
301
The output voltage is VO = VC 22 j −1 − Vin = [(
j+2−k 2 ) − 1]Vin 1− k
(4.93)
The voltage transfer gain is G2 =
VO j+2−k 2 =( ) −1 Vin 1− k
(4.94)
The ripple voltage of output voltage vO is ΔvO =
ΔQ IO kT k VO = = C22 j C22 j fC22 j R
Therefore, the variation ratio of output voltage vO is ε=
4.6.3
ΔvO / 2 k = 2 RfC22 j VO
(4.95)
N/O Triple-Lift Multiple-Enhanced Circuit
This circuit is derived from N/O triple-lift circuit by adding the DEC multiple (j) times in each stage circuit. Its circuit diagram and equivalent circuits during switch-on and switch-off are shown in Figure 4.14. The voltage across capacitor C32j-1 is VC 32 j −1 = (
j+2−k 3 ) Vin 1− k
(4.96)
The output voltage is VO = VC 32 j −1 − Vin = [(
j+2−k 3 ) − 1]Vin 1− k
(4.97)
The voltage transfer gain is G3 =
© 2006 by Taylor & Francis Group, LLC
VO j+2−k 3 =( ) −1 Vin 1− k
(4.98)
302
Essential DC/DC Converters
The ripple voltage of output voltage vO is ΔvO =
ΔQ IO kT k VO = = C32 j C32 j fC32 j R
Therefore, the variation ratio of output voltage vO is ε=
4.6.4
ΔvO / 2 k = VO 2 RfC32 j
(4.99)
N/O Higher Order Lift Multiple-Enhanced Circuit
The higher order N/O lift multiple-enhanced circuit is derived from the corresponding circuit of the main series by adding the DEC multiple (j) times in each stage circuit. Each stage final voltage VCi2j-1 (i = 1, 2, … n) is VCi 2 j −1 = (
j+2−k i ) Vin 1− k
(4.100)
For nth order lift multiple-enhanced circuit, the final output voltage is VO = [(
j+2−k n ) − 1]Vin 1− k
(4.101)
The voltage transfer gain is Gn =
VO j+2−k n =( ) −1 Vin 1− k
(4.102)
The variation ratio of output voltage vO is ε=
4.7
ΔvO / 2 k = VO 2 RfCn2 j
(4.103)
Summary of Negative Output Super-Lift Luo-Converters
All circuits of the negative output super-lift Luo-converters as a family can be shown in Figure 4.15. From the analysis in previous sections the common formula to calculate the output voltage can be presented: © 2006 by Taylor & Francis Group, LLC
Negative Output Super-Lift Luo-Converters 2−k n ⎧ [( ) − 1]Vin ⎪ 1− k ⎪ 2 − k n −1 3 − k ) ( ) − 1]Vin ⎪[( 1− k ⎪ 1− k ⎪ 3−k n VO = ⎨ [( ) − 1]Vin 1− k ⎪ 4−k n ⎪ [( ) − 1]Vin 1− k ⎪ j+2−k n ⎪ [( ) − 1]Vin 1− k ⎩⎪
303
main _ series additional _ series enhanced _ series
(4.104)
re - enhanced _ series multiple - enhanced _ series
The corresponding voltage transfer gain is 2−k n ⎧ ( ) −1 ⎪ 1− k ⎪ 2 − k n −1 3 − k ) ( )−1 ⎪( 1− k ⎪ 1− k V ⎪ 3−k n G= O =⎨ ( ) −1 Vin ⎪ 1− k 4−k n ⎪ ( ) −1 1− k ⎪ j+2−k n ⎪ ⎪⎩ ( 1 − k ) − 1
main _ series additional _ series enhanced _ series
(4.105)
re - enhanced _ series multiple - enhanced _ series
In order to show the advantages of N/O super-lift converters, their voltage transfer gains can be compared to that of buck converters, G=
VO =k Vin
G=
VO = kN Vin
G=
VO k = Vin 1 − k
G=
VO kN = Vin 1 − k
Forward converters, (N is the transformer turn’s ratio)
Cúk-converters,
fly-back converters,
© 2006 by Taylor & Francis Group, LLC
(N is the transformer turn’s ratio)
304
Essential DC/DC Converters Main Series
Additional Series
Enhanced Series
Quintuple-Lift Circuit
Quintuple-Lift Additional Circuit
Quintuple-Lift Enhanced Circuit
Quintuple-Lift Reenhanced Circuit
Quintuple-Lift MultipleEnhanced Circuit
Quadruple-Lift Circuit
Quadruple-Lift Additional Circuit
Quadruple-Lift Enhanced Circuit
Quadruple-Lift Reenhanced Circuit
Quadruple-Lift MultipleEnhanced Circuit
Triple-Lift Circuit
Triple-Lift Additional Circuit
Triple-Lift Enhanced Circuit
Triple-Lift Reenhanced Circuit
Triple-Lift MultipleEnhanced Circuit
Relift Circuit
Relift Additional Circuit
Relift Enhanced Circuit
Relift Reenhanced Circuit
Relift MultipleEnhanced Circuit
Elementary Reenhanced Circuit
Elementary MultipleEnhanced Circuit
Elementary Additional/Enhanced Circuit
Reenhanced Series
Multiple-Enhanced Series
Negative Output Elementary Super-Lift Luo-Converter FIGURE 4.15 The family of negative output super-lift Luo-converters.
boost converters, G=
VO 1 = Vin 1 − k
and negative output Luo-converters G=
VO n = Vin 1 − k
(4.106)
If we assume the conduction duty k is 0.2, the output voltage transfer gains are listed in Table 4.1, if the conduction duty k is 0.5, the output voltage transfer gains are listed in Table 4.2, and if the conduction duty k is 0.8, the output voltage transfer gains are listed in Table 4.3.
© 2006 by Taylor & Francis Group, LLC
Negative Output Super-Lift Luo-Converters
305
TABLE 4.1 Voltage Transfer Gains of Converters in the Condition k = 0.2 Stage No. (n) Buck converter Forward converter Cúk-converter Fly-back converter Boost converter Negative output Luo-converters Negative output super-lift converters — main series Negative output super-lift converters — additional series
1
2
3
4
5
n
1.25 1.25
0.2 0.2N (N is the transformer turn’s ratio) 0.25 0.25N (N is the transformer turn’s ratio) 1.25 2.5 3.75 5 6.25 1.25n 4.06 10.39 24.63 56.67 2.25 n-1
2.5
6.88
16.72
38.87
3.5∗2.25(n–1)-1
88.7
TABLE 4.2 Voltage Transfer Gains of Converters in the Condition k = 0.5 Stage No. (n)
1
2
3
4
5
n
Buck converter Forward converter Cúk-converter Fly-back converter Boost converter Negative output Luo-converters Negative output super-lift converters — main series Negative output super-lift converters — additional series
0.5 0.5N (N is the transformer turn’s ratio) 1 N (N is the transformer turn’s ratio) 2 2 4 6 8 10 2n 2 8 26 80 242 3n-1 4 14 44 134 404 5∗3(n–1)-1
TABLE 4.3 Voltage Transfer Gains of Converters in the Condition k = 0.8 Stage No. (n)
1
Buck converter Forward converter Cúk-converter Fly-back converter Boost converter Negative output Luo-converters Negative output super-lift converters — main series Negative output super-lift converters — additional series
0.8 0.8N (N is the transformer turn’s ratio) 4 4N (N is the transformer turn’s ratio) 5 5 10 15 20 25 5n 5 35 215 1295 7775 6n-1
© 2006 by Taylor & Francis Group, LLC
10
2
65
3
395
4
2375
5
14,255
n
11∗6(n–1)-1
306
4.8
Essential DC/DC Converters
Simulation Results
To verify the design and calculation results, PSpice simulation package was applied to these converters. Choosing Vin = 20 V, L1 = L2 = L3 = 10 mH, all C1 to C8 = 2 μF, and R = 30 k, and using k = 0.5 and f = 100kHz. 4.8.1
Simulation Results of a N/O Triple-Lift Circuit
The voltage values V1, V2, and VO of a N/O triple-lift circuit are –46 V, –174 V, and –639 V respectively and current waveforms iL1 (its average value IL1 = 603 mA), iL2, and iL3. The simulation results are shown in Figure 4.16. The voltage values are matched to the calculated results.
4.8.2
Simulation Results of a N/O Triple-Lift Additional Circuit
The voltage values V1, V2, V3, and VO of a N/O triple-lift additional circuit are –38 V, –146 V, –517 V, and –889 V respectively and current waveforms iL1 (its average value IL1 = 1.79 A), iL2, and iL3. The simulation results are shown in Figure 4.17. The voltage values are matched to the calculated results.
4.9
Experimental Results
A test rig was constructed to verify the design and calculation results, and compare with PSpice simulation results. The testing conditions are the same: Vin = 20 V, L1 = L2 = L3 = 10 mH, all C1 to C8 = 2 μF and R = 30 k, and using k = 0.5 and f = 100kHz. The component of the switch is a MOSFET device IRF950 with the rates 950 V/5 A/2 MHz. The output voltage and the first diode current values are measured in the following converters.
4.9.1
Experimental Results of a N/O Triple-Lift Circuit
After careful measurement, the current value of IL1 = 0.6 A (shown in channel 1 with 1 A/Div) and voltage value of VO = –640 V (shown in channel 2 with 200 V/Div) are obtained. The experimental results (current and voltage values) in Figure 4.18 are identically matched to the calculated and simulation results, which are IL1 = 0.603 A and VO = –639 V shown in Figure 4.16.
© 2006 by Taylor & Francis Group, LLC
Negative Output Super-Lift Luo-Converters
307
1.0A (9.99m, 603m) 0.5A
(9.99m, 205m) (9.99m, 55m)
0A I(L1)
I(L2)
I(L3)
0V -250V
(9.99m, -46) (9.99m, -174)
-500V SEL>>
(9.99m, -639)
9.980ms V(C4:2)
9.985ms V(C2:2)
9.990ms
V(R:2)
9.995ms
10.000ms
Time
FIGURE 4.16 Simulation results of a N/O triple-lift circuit at condition k = 0.5 and f = 100 kHz.
2.0A (19.99m, 1.79) (19.99m, 610m) 1.0A
(19.99m, 206m)
0A 0V
I(L1)
I(L2)
I(L3) (19.99m, -38) (19.99m, -146)
-0.5KV SEL>> -1.0KV 19.980ms V(D6:1)
(19.99m, -517) (19.99m, -889) 19.985ms V(D2:1)
V(R:2)
19.990ms
19.995ms
20.000ms
V(C6:2)
Time
FIGURE 4.17 Simulation results of a N/O triple-lift additional circuit at condition k = 0.5 and f = 100 kHz.
4.9.2
Experimental Results of a N/O Triple-Lift Additional Circuit
The experimental results (voltage and current values) are identically matched to the calculated and simulation results as shown in Figure 4.19. The current value of IL1 = 1.78 A (shown in channel 1 with 1 A/Div) and voltage value of VO = –890 V (shown in channel 2 with 200 V/Div) are obtained. The experimental results are identically matched to the calculated and simulation results, which are IL1 = 1.79 A and VO = –889 V shown in Figure 4.17.
© 2006 by Taylor & Francis Group, LLC
308
Essential DC/DC Converters
FIGURE 4.18 Experimental results of a N/O triple-lift circuit at condition k = 0.5 and f = 100 kHz.
FIGURE 4.19 Experimental results of a N/O triple-lift additional circuit at k = 0.5 and f = 100 kHz.
4.9.3
Efficiency Comparison of Simulation and Experimental Results
These circuits enhanced the voltage transfer gain successfully, but efficiency, particularly the efficiencies of the tested circuits are 51 to 78%, which is good for high voltage output equipment. Comparison of the simulation and experimental results, which are listed in the Table 4.4 and Table 4.5, demonstrates that all results are well identified with each other.
4.9.4
Transient Process and Stability Analysis
Usually, there is high inrush current during the first power-on. Therefore, the voltage across capacitors is quickly changed to certain values. The transient process is very quick lasting only a few milliseconds.
© 2006 by Taylor & Francis Group, LLC
Negative Output Super-Lift Luo-Converters
309
TABLE 4.4 Comparison of Simulation and Experimental Results of a N/O Triple-Lift Circuit Stage No. (n) Simulation results Experimental results
IL1 (A)
Iin (A)
Vin (V)
Pin (W)
|VO|(V)
PO (W)
η (%)
0.603 0.6
0.871 0.867
20 20
17.42 17.33
639 640
13.61 13.65
78.12 78.75
TABLE 4.5 Comparison of Simulation and Experimental Results of a N/O Triple-Lift Additional Circuit Stage No. (n) Simulation results Experimental results
IL1 (A)
Iin (A)
Vin (V)
Pin (W)
|VO|(V)
PO (W)
η (%)
1.79 1.78
2.585 2.571
20 20
51.7 51.4
889 890
26.34 26.4
51 51
Bibliography Baliga, B.J., Modern Power Devices, New York, John Wiley & Sons, 1987. Luo, F.L., Advanced voltage lift technique — negative output Luo-converters, Power Supply Technologies and Applications, Xi’an, China, 3, 112, 1998. Luo, F.L. and Ye, H., Negative output super-lift converters, IEEE Transactions on Power Electronics, 18, 268, 2003. Luo, F.L. and Ye, H., Negative output super-lift Luo-converters, in Proceedings of IEEEPESC’2003, Acapulco, Mexico, 2003, p. 1361. Mitchell, D.M., DC-to-DC Switching Regulator Analysis, New York: McGraw-Hill, 1988. Ye, H., Luo, F.L., and Ye, Z.Z., Practical circuits of Luo-converters, Power Supply Technologies and Applications, Xi’an, China, 2, 19, 1999.
© 2006 by Taylor & Francis Group, LLC
5 Positive Output Cascade Boost Converters
Super-lift technique increases the voltage transfer gain in geometric progression. However, these circuits are a bit complex. This chapter introduces a novel approach — the positive output cascade boost converter — that implements the output voltage increasing in geometric progression, but with simpler structure. They also effectively enhance the voltage transfer gain in power-law.
5.1
Introduction
In order to sort these converters differently from existing voltage-lift (VL) and super-lift (SL) converters, these converters are entitled positive output cascade boost converters. There are several subseries: • Main series — Each circuit of the main series has one switch S, n inductors, n capacitors, and (2n – 1) diodes. • Additional series — Each circuit of the additional series has one switch S, n inductors, (n + 2) capacitors, and (2n + 1) diodes. • Double series — Each circuit of the double series has one switch S, n inductors, 3n capacitors, and (3n – 1) diodes. • Triple series — Each circuit of the triple series has one switch S, n inductors, 5n capacitors, and (5n – 1) diodes. • Multiple series — Each multiple series circuit has one switch S and a higher number of capacitors and diodes. In order to concentrate the super-lift function, these converters work in the steady state with the condition of continuous conduction mode (CCM). The conduction duty ratio is k, switching frequency is f, switching period is T = 1/f, the load is resistive load R. The input voltage and current are Vin and Iin, output voltage and current are VO and IO. Assume no power losses during the conversion process, Vin × Iin = VO × IO. The voltage transfer gain is G: 311
© 2006 by Taylor & Francis Group, LLC
312
Essential DC/DC Converters
G=
5.2
VO Vin
Main Series
The first three stages of positive output cascade boost converters — main series — are shown in Figure 5.1 to Figure 5.3. For convenience they are called elementary boost converter, two-stage circuit, and three-stage circuit respectively, and numbered as n = 1, 2, and 3.
5.2.1
Elementary Boost Circuit
The elementary boost converter is the fundamental boost converter introduced in Chapter 1 (see Figure 1.24). Its circuit diagram and its equivalent circuits during switch-on and switch-off are shown in Figure 5.1. The voltage across capacitor C1 is charged to VO. The current iL1 flowing through inductor L1 increases with voltage Vin during switch-on period kT and decreases with voltage –(VO – Vin) during switch-off period (1 – k)T. Therefore, the ripple of the inductor current iL1 is ΔiL1 =
V − Vin Vin kT = O (1 − k )T L1 L1
(5.1)
1 V 1 − k in
(5.2)
VO 1 = Vin 1 − k
(5.3)
VO R
(5.4)
VO = The voltage transfer gain is G= The inductor average current is
I L1 = (1 − k )
The variation ratio of current iL1 through inductor L1 is ξ1 =
© 2006 by Taylor & Francis Group, LLC
kTVin ΔiL1 / 2 k R = = I L1 (1 − k )2L1VO / R 2 fL1
(5.5)
Positive Output Cascade Boost Converters iIN
L1
313
D1
+ VIN –
+ VC1 –
S
C1 R
iO + VO –
(a) Circuit diagram iIN
L1
+ C1
VIN –
+ VC1 R –
iO + VO –
(b) Switching-on iIN
L1 iO
VL1
+ VIN –
C1
+ VC1 –
R
+ VO –
(c) Switching-off FIGURE 5.1 Elementary boost converter.
Usually ξ1 is small (much lower than unity), which means this converter works in the continuous mode. The ripple voltage of output voltage vO is ΔvO =
ΔQ IO kT k VO = = C1 C1 fC1 R
Therefore, the variation ratio of output voltage vO is ε=
ΔvO / 2 k = 2 RfC1 VO
(5.6)
Usually R is in kΩ, f in 10 kHz, and C1 in μF, the ripple is smaller than 1%. 5.2.2
Two-Stage Boost Circuit
The two-stage boost circuit is derived from elementary boost converter by adding the parts (L2-D2-D3-C2). Its circuit diagram and equivalent circuits
© 2006 by Taylor & Francis Group, LLC
314
Essential DC/DC Converters iIN
L1
L2
D1 V1
+ VIN –
D2 C1
+ VC1 –
D3
C2
S
+ VC2R –
iO + VO –
(a) Circuit diagram iIN
V1
+ VIN –
+ VC1 L2 C2 –
L1 C1
+ VC2 R –
iO + VO –
(b) Equivalent circuit during switching-on iIN + VIN –
L1
L2
VL1
VL2 + VC1 C2 –
C1
+ VC2 R –
iO + VO –
(c) Equivalent circuit during switching-off FIGURE 5.2 Two-stage boost circuit.
during switch-on and switch-off are shown in Figure 5.2. The voltage across capacitor C1 is charged to V1. As described in previous section the voltage V1 across capacitor C1 is V1 =
1 V 1 − k in
The voltage across capacitor C2 is charged to VO. The current flowing through inductor L2 increases with voltage V1 during switching-on period kT and decreases with voltage –(VO – V1) during switch-off period (1 – k)T. Therefore, the ripple of the inductor current iL2 is ΔiL 2 =
V − V1 V1 kT = O (1 − k )T L2 L2
(5.7)
1 1 2 V1 = ( ) Vin 1− k 1− k
(5.8)
VO = The voltage transfer gain is
© 2006 by Taylor & Francis Group, LLC
Positive Output Cascade Boost Converters
G=
315
VO 1 2 =( ) Vin 1− k
(5.9)
Analogously, ΔiL1 =
Vin kT L1
I L1 =
IO (1 − k )2
ΔiL2 =
V1 kT L2
I L2 =
IO 1− k
Therefore, the variation ratio of current iL1 through inductor L1 is ξ1 =
ΔiL1 / 2 k(1 − k ) 2 TVin k(1 − k ) 4 R = = I L1 2L1 I O 2 fL1
(5.10)
the variation ratio of current iL2 through inductor L2 is ξ2 =
ΔiL 2 / 2 k(1 − k )TV1 k(1 − k )2 R = = I L2 2L2 I O 2 fL2
(5.11)
and the variation ratio of output voltage vO is ε=
5.2.3
ΔvO / 2 k = VO 2 RfC2
(5.12)
Three-Stage Boost Circuit
The three-stage boost circuit is derived from the two-stage boost circuit by double adding the parts (L2-D2-D3-C2). Its circuit diagram and equivalent circuits during switch-on and switch-off are shown in Figure 5.3. The voltage across capacitor C1 is charged to V1. As described previously, the voltage V1 across capacitor C1 is V1 =
1 V 1 − k in
and voltage V2 across capacitor C2 is V2 = (
© 2006 by Taylor & Francis Group, LLC
1 2 ) Vin 1− k
316
Essential DC/DC Converters D3 iIN
L1
L2
D1 V1
+ VIN –
D2 C1
L3
V2
D5
D4
+ VC1 –
iO
+ VC2 –
C2
S
C3
+ R VC3 –
+ VO –
(a) Circuit diagram iIN + VIN –
L1C1
V1
V2
+ VC1 L2C2 –
+ VC2 L3C3 –
+ VC3 R –
iO + VO –
(b) Equivalent circuit during switching-on iIN
L1
V1
+ VIN
V2
L2
VL1
VL2
C1
+ VC1 C2 –
–
L3 VL3 + VC2 C3 –
+ VC3 R –
iO + VO –
(c) Equivalent circuit during switching-off FIGURE 5.3 Three-stage boost circuit.
The voltage across capacitor C3 is charged to VO. The current flowing through inductor L3 increases with voltage V2 during switching-on period kT and decreases with voltage –(VO – V2) during switch-off (1 – k)T. Therefore, the ripple of the inductor current iL3 is V − V2 V2 kT = O (1 − k )T L3 L3
(5.13)
1 1 2 1 3 V2 = ( ) V1 = ( ) Vin 1− k 1− k 1− k
(5.14)
ΔiL 3 =
VO =
The voltage transfer gain is G= Analogously,
© 2006 by Taylor & Francis Group, LLC
VO 1 3 =( ) 1− k Vin
(5.15)
Positive Output Cascade Boost Converters
317
ΔiL1 =
Vin kT L1
I L1 =
IO (1 − k ) 3
ΔiL2 =
V1 kT L2
I L2 =
IO (1 − k )2
ΔiL3 =
V2 kT L3
I L3 =
IO 1− k
Therefore, the variation ratio of current iL1 through inductor L1 is ξ1 =
ΔiL1 / 2 k(1 − k ) 3 TVin k(1 − k )6 R = = I L1 2L1 I O 2 fL1
(5.16)
The variation ratio of current iL2 through inductor L2 is ξ2 =
ΔiL 2 / 2 k(1 − k ) 2 TV1 k(1 − k ) 4 R = = I L2 2L2 I O 2 fL2
(5.17)
The variation ratio of current iL3 through inductor L3 is ξ3 =
ΔiL 3 / 2 k(1 − k )TV2 k(1 − k )2 R = = 2 L3 I O 2 I L3 fL3
(5.18)
and the variation ratio of output voltage vO is ε=
5.2.4
ΔvO / 2 k = 2 RfC3 VO
(5.19)
Higher Stage Boost Circuit
Higher stage boost circuit can be designed by just multiple repeating of the parts (L2-D2-D3-C2). For nth stage boost circuit, the final output voltage across capacitor Cn is VO = ( The voltage transfer gain is
© 2006 by Taylor & Francis Group, LLC
1 n ) Vin 1− k
318
Essential DC/DC Converters
G=
VO 1 n =( ) Vin 1− k
(5.20)
the variation ratio of current iLi through inductor Li (i = 1, 2, 3, …n) is ξi =
ΔiLi / 2 k(1 − k ) 2( n−i +1) R = I Li 2 fLi
(5.21)
and the variation ratio of output voltage vO is ε=
5.3
ΔvO / 2 k = 2 RfCn VO
(5.22)
Additional Series
All circuits of positive output cascade boost converters — additional series — are derived from the corresponding circuits of the main series by adding a DEC. The first three stages of this series are shown in Figure 5.4 to Figure 5.6. For convenience they are called elementary additional circuits, two-stage additional circuits, and three-stage additional circuits respectively, and numbered as n = 1, 2, and 3.
5.3.1
Elementary Boost Additional (Double) Circuit
This elementary boost additional circuit is derived from elementary boost converter by adding a DEC. Its circuit and switch-on and switch-off equivalent circuits are shown in Figure 5.4. The voltage across capacitor C1 and C11 is charged to V1 and voltage across capacitor C12 is charged to VO = 2 V1. The current iL1 flowing through inductor L1 increases with voltage Vin during switching-on period kT and decreases with voltage –(V1 – Vin) during switching-off (1 – k)T. Therefore, ΔiL1 =
V − Vin Vin kT = 1 (1 − k )T L1 L1 V1 =
© 2006 by Taylor & Francis Group, LLC
1 V 1 − k in
(5.23)
Positive Output Cascade Boost Converters iIN
L1
+
319
D1
D11
D12
+ VC1 –
+ VC11 C1 –
C11
iO
VIN
C12
S –
+ R VC12 –
+ VO
+ VC12 R –
+ VO –
–
(a) Circuit diagram iIN
L1
V1 iO
+
C1
VIN –
+ C11 VC1 –
+ C12 VC11 –
(b) Equivalent circuit during switching-on
iIN + VIN
C11
L1 VL1 C1
– + + VC11 VC1 C12
–
–
+ VC12 R –
iO + VO –
(c) Equivalent circuit during switching-off FIGURE 5.4 Elementary boost additional (double) circuit.
The output voltage is VO = 2V1 =
2 V 1 − k in
(5.24)
The voltage transfer gain is G=
VO 2 = Vin 1 − k
(5.25)
and iin = I L1 =
2 I 1− k O
The variation ratio of current iL1 through inductor L1 is
© 2006 by Taylor & Francis Group, LLC
(5.26)
320
Essential DC/DC Converters ΔiL1 / 2 k(1 − k )TVin k(1 − k )2 R = = I L1 4L1 I O 8 fL1
ξ1 =
(5.27)
The ripple voltage of output voltage vO is ΔQ IO kT k VO = = C12 C12 fC12 R
ΔvO =
Therefore, the variation ratio of output voltage vO is ε=
5.3.2
ΔvO / 2 k = 2 RfC12 VO
(5.28)
Two-Stage Boost Additional Circuit
The two-stage additional boost circuit is derived from the two-stage boost circuit by adding a DEC. Its circuit diagram and switch-on and switch-off equivalent circuits are shown in Figure 5.5. The voltage across capacitor C1 is charged to V1. As described in the previous section the voltage V1 across capacitor C1 is V1 =
1 V 1 − k in
The voltage across capacitor C2 and capacitor C11 is charged to V2 and voltage across capacitor C12 is charged to VO. The current flowing through inductor L2 increases with voltage V1 during switch-on period kT and decreases with voltage –(V2 – V1) during switch-off period (1 – k)T. Therefore, the ripple of the inductor current iL2 is ΔiL2 =
V1 V − V1 kT = 2 (1 − k )T L2 L2
(5.29)
1 1 2 ) Vin V1 = ( 1− k 1− k
(5.30)
V2 = The output voltage is
VO = 2V2 =
© 2006 by Taylor & Francis Group, LLC
2 1 2 V1 = 2( ) Vin 1− k 1− k
(5.31)
Positive Output Cascade Boost Converters iIN
L1
L2
D1
321
D3
D11
D12
+ VC11 –
+ D2 VIN
+ VC1 –
C1 –
S
+ VC2 –
C2
iO C11 C12
+
R VC12 –
+ VO –
(a) Circuit diagram iIN
V1
+ VIN
+ L2 VC1 –
C1
L1
V2 C2
+ C11 V2 –
+ C12 VC11 –
+ VC12 R –
iO + VO –
–
(b) Equivalent circuit during switching-on iIN
L1 VL1
+ VIN –
L2
V1
VL2 + VC1 C2 –
C1
V2
C11 –V + C11 + V2 C12 –
+ VC12 R –
iO + VO –
(c) Equivalent circuit during switching-off FIGURE 5.5 Two-stage boost additional circuit.
The voltage transfer gain is G=
VO 1 2 = 2( ) Vin 1− k
ΔiL1 =
Vin kT L1
I L1 =
2 I (1 − k )2 O
ΔiL2 =
V1 kT L2
I L2 =
2IO 1− k
(5.32)
Analogously,
Therefore, the variation ratio of current iL1 through inductor L1 is
© 2006 by Taylor & Francis Group, LLC
322
Essential DC/DC Converters
ξ1 =
ΔiL1 / 2 k(1 − k )2 TVin k(1 − k ) 4 R = = I L1 4L1 I O 8 fL1
(5.33)
and the variation ratio of current iL2 through inductor L2 is ξ2 =
ΔiL 2 / 2 k(1 − k )TV1 k(1 − k )2 R = = I L2 4L2 I O 8 fL2
(5.34)
The ripple voltage of output voltage vO is ΔvO =
ΔQ IO kT k VO = = C12 C12 fC12 R
Therefore, the variation ratio of output voltage vO is ε=
5.3.3
ΔvO / 2 k = 2 RfC12 VO
(5.35)
Three-Stage Boost Additional Circuit
This circuit is derived from the three-stage boost circuit by adding a DEC. Its circuit diagram and equivalent circuits during switch-on and switch-off are shown in Figure 5.6. The voltage across capacitor C1 is charged to V1. As described previously the voltage V1 across capacitor C1 is V1 = (1 1 − k )Vin , 2 and voltage V2 across capacitor C2 is V2 = (1 1 − k ) Vin . The voltage across capacitor C3 and capacitor C11 is charged to V3. The voltage across capacitor C12 is charged to VO. The current flowing through inductor L3 increases with voltage V2 during switch-on period kT and decreases with voltage –(V3 – V2) during switch-off (1 – k)T. Therefore, ΔiL3 =
V − V2 V2 kT = 3 (1 − k )T L3 L3
(5.36)
and V3 =
1 1 2 1 3 ) V1 = ( ) Vin V =( 1− k 2 1− k 1− k
(5.37)
The output voltage is VO = 2V3 = 2(
© 2006 by Taylor & Francis Group, LLC
1 3 ) Vin 1− k
(5.38)
Positive Output Cascade Boost Converters
D3
I in
L1
D1
L2
V1
323 L3
V2
D5 V3 D11
D12 +
D4
C11
IO
V C11 _
D2
+
R
Vin
+
+
_
C1
C2
V C1
S
VC2
C3
+
C12
V C3
_
_
+
VC12
_
+
VO _
_
(a) Circuit diagram
I in
V1 +
+
Vin
L1
C1
V2 +
L2
C2
VC1
V3 L3
C2
+
IO C12
C11
+
V2
_
V3
_
_
+
+
VC12
_
VO
R
_ V3
_
_
(b) Equivalent circuit during switch-on
I in +
L1
L2
V1
V L1
+
V L2
V1
V in
L3
V2 +
C2
_
V2 _
V L3
C3
C11
V3 _ +
V3 _
V3
IO
+
C12
+
V C12 _
_
+
R
VO _
(b) Equivalent circuit during switch-off FIGURE 5.6 Three-stage boost additional circuit.
The voltage transfer gain is G=
VO 1 3 = 2( ) Vin 1− k
ΔiL1 =
Vin kT L1
I L1 =
2 I (1 − k ) 3 O
ΔiL2 =
V1 kT L2
I L2 =
2 I (1 − k )2 O
ΔiL3 =
V2 kT L3
I L3 =
2IO 1− k
(5.39)
Analogously:
© 2006 by Taylor & Francis Group, LLC
324
Essential DC/DC Converters
Therefore, the variation ratio of current iL1 through inductor L1 is ξ1 =
ΔiL1 / 2 k(1 − k ) 3 TVin k(1 − k )6 R = = I L1 4L1 I O 8 fL1
(5.40)
and the variation ratio of current iL2 through inductor L2 is ξ2 =
ΔiL 2 / 2 k(1 − k ) 2 TV1 k(1 − k ) 4 R = = I L2 4L2 I O 8 fL2
(5.41)
and the variation ratio of current iL3 through inductor L3 is ξ3 =
ΔiL 3 / 2 k(1 − k )TV2 k(1 − k )2 R = = 4 L3 I O 8 I L3 fL3
(5.42)
The ripple voltage of output voltage vO is ΔvO =
ΔQ IO kT k VO = = C12 C12 fC12 R
Therefore, the variation ratio of output voltage vO is ε=
5.3.4
ΔvO / 2 k = 2 RfC12 VO
(5.43)
Higher Stage Boost Additional Circuit
Higher stage boost additional circuits can be designed by repeating the parts (L2-D2-D3-C2) multiple times. For nth stage additional circuit, the final output voltage is VO = 2(
1 n ) Vin 1− k
The voltage transfer gain is G=
VO 1 n = 2( ) Vin 1− k
(5.44)
Analogously, the variation ratio of current iLi through inductor Li (i = 1, 2, 3, …n) is
© 2006 by Taylor & Francis Group, LLC
Positive Output Cascade Boost Converters
ξi =
ΔiLi / 2 k(1 − k ) 2( n−i +1) R = I Li 8 fLi
325
(5.45)
and the variation ratio of output voltage vO is ε=
5.4
ΔvO / 2 k = VO 2 RfC12
(5.46)
Double Series
All circuits of the positive output cascade boost converter — double series — are derived from the corresponding circuits of the main series by adding a DEC in each stage circuit. The first three stages of this series are shown in Figures 5.4, 5.7, and 5.8. For convenience to explain, they are called elementary double circuits, two-stage double circuits, and three-stage double circuits respectively, and numbered as n = 1, 2, and 3.
5.4.1
Elementary Double Boost Circuit
From the construction principle, the elementary double boost circuit is derived from the elementary boost converter by adding a DEC. Its circuit and switch-on and switch-off equivalent circuits are shown in Figure 5.4, which is the same as the elementary boost additional circuit.
5.4.2
Two-Stage Double Boost Circuit
The two-stage double boost circuit is derived from the two-stage boost circuit by adding a DEC in each stage circuit. Its circuit diagram and switch-on and switch-off equivalent circuits are shown in Figure 5.7. The voltage across capacitor C1 and capacitor C11 is charged to V1. As described in the previous section, the voltage V1 across capacitor C1 and capacitor C11 is V1 = (1 1 − k )Vin . The voltage across capacitor C12 is charged to 2V1. The current flowing through inductor L2 increases with voltage 2V1 during switch-on period kT and decreases with voltage –(V2 – 2V1) during switchoff period (1 – k)T. Therefore, the ripple of the inductor current iL2 is ΔiL2 =
© 2006 by Taylor & Francis Group, LLC
2V1 V − 2V1 (1 − k )T kT = 2 L2 L2
(5.47)
326 I in
Essential DC/DC Converters
L1
D1
V1
D11
L2
D12
D21
D3
D22
+
D2
+
V C11
C11
V in
C21
_
+
_
C1
IO
+
V C12
C12
_
+
VC21 R
+
S
+
V C1
_
C2
+
V C2
C22
_
_
_VC22
VO _
(a) Circuit diagram I in
V1
+
Vin
C1
L1
L2 2V 1
+
V1
V2
+
C12
+ _
_
2V 1
V1
C2
+
+
V2 _
_
C 11
IO
_
C22
+
VO
_
V2
C21
+
VC22
R
_
_
(b) Equivalent circuit during switch-on
I in +
L1
C11
V1 _
VL1
V1
+
C1
V C1 _
_
L2
2V 1
C12
_
V L2
+
C21
V2
+
+
VC12
V2
_
C2
_
V2
IO +
C22
+
+
R
VC12
VO
_
V in
_
(c) Equivalent circuit during switch-off FIGURE 5.7 Two-stage boost double circuit.
V2 =
2 1 2 V = 2( ) Vin 1− k 1 1− k
(5.48)
The output voltage is VO = 2V2 = (
2 2 ) Vin 1− k
(5.49)
The voltage transfer gain is G=
© 2006 by Taylor & Francis Group, LLC
VO 2 2 =( ) Vin 1− k
(5.50)
Positive Output Cascade Boost Converters
327
Analogously, ΔiL1 =
Vin kT L1
I L1 = (
ΔiL2 =
V1 kT L2
I L2 =
2 2 ) I 1− k O
2IO 1− k
Therefore, the variation ratio of current iL1 through inductor L1 is ξ1 =
ΔiL1 / 2 k(1 − k )2 TVin k(1 − k ) 4 R = = I L1 8L1 I O 16 fL1
(5.51)
and the variation ratio of current iL2 through inductor L2 is ξ2 =
ΔiL 2 / 2 k(1 − k )TV1 k(1 − k )2 R = = I L2 4L2 I O 8 fL2
(5.52)
The ripple voltage of output voltage vO is ΔvO =
ΔQ IO kT k VO = = C22 C22 fC22 R
Therefore, the variation ratio of output voltage vO is ε=
5.4.3
ΔvO / 2 k = VO 2 RfC22
(5.53)
Three-Stage Double Boost Circuit
This circuit is derived from the three-stage boost circuit by adding a DEC in each stage circuit. Its circuit diagram and equivalent circuits during switchon and -off are shown in Figure 5.8. The voltage across capacitor C1 and capacitor C11 is charged to V1. As described earlier the voltage V1 across capacitor C1 and capacitor C11 is V1 = (1 1 − k )Vin , and voltage V2 across capac2 itor C2 and capacitor C12 is V2 = 2(1 1 − k ) Vin . 2 The voltage across capacitor C22 is 2V2 = (2 1 − k ) Vin . The voltage across capacitor C3 and capacitor C31 is charged to V3. The voltage across capacitor C12 is charged to VO. The current flowing through inductor L3 increases with
© 2006 by Taylor & Francis Group, LLC
L1
D1
V1
D11
D12
L2
2V1
D3
V2 D21
Vin
VC11
C11
C1
C21
D4
_
+
_
VC12
C12
_
D22
C31
_
C2
_
R
C22
_
_
S
+
VC2
IO
VC31
VC21
+
+
VC1
D32 +
+
+
D2
+
D5 V3 D31
328
L3
2V2
Iin
C3
_VC22
+
C32
VC3
+
VC32
_
+
VO _
_
(a) Circuit diagram Iin
V1
+
L1
Vin
C1
+
+
V1 _
_
2V1 +
C12
L2 2V1
V1
C11
V2
C2
+
_
+
V2 _
_
2V2 +
C22
L3 2V2
V2
C21
C3
+
+
V3 _
_
_
IO
V3 C32
_
C31
+
+
VC32
V3
VO
R
_
_
(b) Equivalent circuit during switch-on
+
Vin
L1
C11
V1 _
VL1 +
C1
_
V C1 _
V1
2V1 +
C12
L2
_
VL2 + VC12 _
C21
V2
C2
+ V2 _
V2
2V2 +
C22
L3
_
VL3 + VC22 _
C3
IO
C31
V3
+ V3 _
V3
+
C32
+
VC32 _
R
+
VO _
(c) Equivalent circuit during switch-off FIGURE 5.8 Three-stage boost double circuit. © 2006 by Taylor & Francis Group, LLC
Essential DC/DC Converters
Iin
Positive Output Cascade Boost Converters
329
voltage V2 during switch-on period kT and decreases with voltage –(V3 – 2V2) during switch-off (1 – k)T. Therefore, ΔiL3 =
V − 2V2 2V2 (1 − k )T kT = 3 L3 L3
(5.54)
and V3 =
2V2 4 V = (1 − k ) (1 − k ) 3 in
(5.55)
The output voltage is VO = 2V3 = (
2 3 ) Vin 1− k
(5.56)
The voltage transfer gain is G=
VO 2 3 =( ) Vin 1− k
(5.57)
Analogously, ΔiL1 =
Vin kT L1
I L1 =
8 I (1 − k ) 3 O
ΔiL2 =
V1 kT L2
I L2 =
4 I (1 − k )2 O
ΔiL3 =
V2 kT L3
I L3 =
2IO 1− k
Therefore, the variation ratio of current iL1 through inductor L1 is ξ1 =
ΔiL1 / 2 k(1 − k ) 3 TVin k(1 − k )6 R = = I L1 16L1 I O 128 fL1
(5.58)
and the variation ratio of current iL2 through inductor L2 is ξ2 =
ΔiL 2 / 2 k(1 − k ) 2 TV1 k(1 − k ) 4 R = = I L2 8L2 I O 32 fL2
© 2006 by Taylor & Francis Group, LLC
(5.59)
330
Essential DC/DC Converters
and the variation ratio of current iL3 through inductor L3 is ξ3 =
ΔiL 3 / 2 k(1 − k )TV2 k(1 − k )2 R = = 4 L3 I O 8 I L3 fL3
(5.60)
The ripple voltage of output voltage vO is ΔQ I O (1 − k )T 1 − k VO = = C32 C32 fC32 R
ΔvO =
Therefore, the variation ratio of output voltage vO is ε=
5.4.4
ΔvO / 2 k = 2 RfC32 VO
(5.61)
Higher Stage Double Boost Circuit
The higher stage double boost circuits can be derived from the corresponding main series circuits by adding a DEC in each stage circuit. For nth stage additional circuit, the final output voltage is VO = (
2 n ) Vin 1− k
The voltage transfer gain is G=
VO 2 n =( ) Vin 1− k
(5.62)
Analogously, the variation ratio of current iLi through inductor Li (i = 1, 2, 3, …n) is ξi =
ΔiLi / 2 k(1 − k )2( n−i +1) R = I Li 2 ∗ 2 2n fLi
(5.63)
The variation ratio of output voltage vO is ε=
© 2006 by Taylor & Francis Group, LLC
ΔvO / 2 k = VO 2 RfCn2
(5.64)
Positive Output Cascade Boost Converters iIN
D1 V1 D11
L1
C1
331
D12 2V1 D13
C11
D14 iO
C13
C12
VIN
R
C14 S
+ VO –
(a) Circuit Diagram iIN
V1
+ L1
C1
VIN –
2V1
+ C11 + C12 VC1 VC11 – –
+ C13 VC12 –
+ C14 VC13 –
+ VC12 R –
iO + VO –
(b) Equivalent circuit during switching-on
iIN + VIN –
V1
C13 C11
L1 VL1 C1
2V1 –V + + C11 VC1 C12 –
C14 + VC12 –
R
iO + VO –
(c) Equivalent circuit during switching-off FIGURE 5.9 Cascade boost re-double circuit.
5.5
Triple Series
All circuits of P/O cascade boost converters — triple series — are derived from the corresponding circuits of the double series by adding the DEC twice in each stage circuit. The first three stages of this series are shown in Figure 5.9 to Figure 5.11. For convenience they are called elementary triple boost circuit, two-stage triple boost circuit, and three-stage triple boost circuit respectively, and numbered as n = 1, 2 and 3.
5.5.1
Elementary Triple Boost Circuit
From the construction principle, the elementary triple boost circuit is derived from the elementary double boost circuit by adding another DEC. Its circuit
© 2006 by Taylor & Francis Group, LLC
332
Essential DC/DC Converters
and switch-on and -off equivalent circuits are shown in Figure 5.9. The output voltage of first stage boost circuit is V1, V1 = Vin/(1 – k). The voltage across capacitors C1 and C11 is charged to V1 and voltage across capacitors C12 and C13 is charged to VC13 = 2V1. The current iL1 flowing through inductor L1 increases with voltage Vin during switch-on period kT and decreases with voltage –(V1 – Vin) during switch-off (1 – k)T. Therefore, ΔiL1 =
V − Vin Vin kT = 1 (1 − k )T L1 L1 V1 =
(5.65)
1 V 1 − k in
The output voltage is VO = VC1 + VC13 = 3V1 =
3 V 1 − k in
(5.66)
The voltage transfer gain is G=
5.5.2
VO 3 = Vin 1 − k
(5.67)
Two-Stage Triple Boost Circuit
The two-stage triple boost circuit is derived from the two-stage double boost circuit by adding another DEC in each stage circuit. Its circuit diagram and switch-on and -off equivalent circuits are shown in Figure 5.10. As described in the previous section the voltage V1 across capacitors C1 and C11 is V1 = (1 1 − k )Vin . The voltage across capacitor C14 is charged to 3V1. The voltage across capacitors C2 and C21 is charged to V2 and voltage across capacitors C22 and C23 is charged to VC23 = 2V2. The current flowing through inductor L2 increases with voltage 3V1 during switch-on period kT, and decreases with voltage –(V2 – 3V1) during switch-off period (1 – k)T. Therefore, the ripple of the inductor current iL2 is ΔiL2 =
3V1 V − 3V1 (1 − k )T kT = 2 L2 L2
V2 =
© 2006 by Taylor & Francis Group, LLC
3 1 2 ) Vin V1 = 3( 1− k 1− k
(5.68)
(5.69)
Positive Output Cascade Boost Converters iIN
D1 V1 D11
L1
+
D12 2V1 D13
D14 3V1
333
D3 V2 D21 D22 2V2 D23
L2
C13
C11
D24 iO
C23
C21
+ VO
R
VIN
D2 C1
–
C12
–
S
C14
C2 C22
C24
(a) Circuit diagram iIN +
V1
2V1
3V1
V2
2V2 iO
L1 C1
VIN
+ C11 + C12 VC1 VC11 – –
+ C13 VC12 –
+ C14 VC13 –
+ VC14 L2 –
C2
+ C21 + C22 VC2 VC21 – –
+ C23 VC22 –
+ C24 VC23 –
+ R VC24 –
–
+ VO –
(b) Equivalent circuit during switching-on C23 V2 C13
V1 iIN + VIN
C11
L1 VL1 C1
–
3V1
L2
VC21
–
– VC13 +
–V + + C11 VC1 C12 –
2V1C14 + VC12 –
+ C2 VC14 –
+ VC2 –
– VC23 + +
2V2 C 24
C21
+ VC
C22
–
+ VC24 R –
iO + VO –
22
(c) Equivalent circuit during switching-off
FIGURE 5.10 Two-stage boost re-double circuit.
The output voltage is VO = VC 2 + VC 23 = 3V2 = (
3 2 ) Vin 1− k
(5.70)
The voltage transfer gain is G= Analogously,
© 2006 by Taylor & Francis Group, LLC
VO 3 2 =( ) Vin 1− k
(5.71)
334
Essential DC/DC Converters
ΔiL1 =
Vin kT L1
I L1 = (
ΔiL2 =
V1 kT L2
I L2 =
2 2 ) I 1− k O
2IO 1− k
Therefore, the variation ratio of current iL1 through inductor L1 is ξ1 =
ΔiL1 / 2 k(1 − k )2 TVin k(1 − k ) 4 R = = I L1 8L1 I O 16 fL1
(5.72)
and the variation ratio of current iL2 through inductor L2 is ΔiL 2 / 2 k(1 − k )TV1 k(1 − k )2 R = = I L2 4L2 I O 8 fL2
ξ2 =
(5.73)
The ripple voltage of output voltage vO is ΔvO =
ΔQ IO kT k VO = = C24 C24 fC24 R
Therefore, the variation ratio of output voltage vO is ε=
5.5.3
ΔvO / 2 k = 2 RfC24 VO
(5.74)
Three-Stage Triple Boost Circuit
This circuit is derived from the three-stage double boost circuit by adding another DEC in each stage circuit. Its circuit diagram and equivalent circuits during switch-on and -off are shown in Figure 5.11. As described earlier the voltage V2 across capacitors C2 and C11 is V2 = 3V1 = (3 1 − k )Vin , and voltage across capacitor C24 is charged to 3V2. The voltage across capacitors C3 and C31 is charged to V3 and voltage across capacitors C32 and C33 is charged to VC33 = 2V3. The current flowing through inductor L3 increases with voltage 3V2 during switch-on period kT and decreases with voltage –(V3 – 3V2) during switch-off (1 – k)T. Therefore, the ripple of the inductor current iL3 is ΔiL3 =
© 2006 by Taylor & Francis Group, LLC
V − 3V2 3V2 kT = 3 (1 − k )T L3 L3
(5.75)
iIN
L1
D1 V1 D11
D12 2V D 1 13
D3
D5 V3 D31
D32 2V3 D33
C31
C33
C23
C21
D34 iO R
VIN
D2
–
D4
C14
C12
C1
+ VO
C22
C2
– C3
S
C24
C32
C34
(a) Circuit diagram iIN
V1
+ VIN
C1
2V1 C11
C12
3V1
V2
C14
C13
2V2
C2
C21
3V2
C22
C23
C3
C24
L2
L1
V3
2V3 C31
C32
C33
C34
L3
R
–
(b) Equivalent circuit during switching-on C33
iO + VO –
Positive Output Cascade Boost Converters
V2 D21 D22 2V2 D23 D24 3V2 L3
C13
C11
+
L2
D14 3V1
V3 – V + C33 C23
V2
3V2
C21 C13
V1 iIN + VIN –
– VC23 + 3V1
L2
C11
L1 VL1 C1
– VC13 + 2V1C 14 – + + VC11 + VC1 C12 VC12 – –
– VC21 + C22
+ C2 VC14 –
+ VC2 –
2V2 + VC22 –
C24
– VC31 + 2V3
L3 C31 + VC24 –
C3
© 2006 by Taylor & Francis Group, LLC
+ VC34
C32
C34 + VC32 –
–
335
(c) Equivalent circuit during switching-off FIGURE 5.11 Three-stage boost re-double circuit.
R
+ VC3 –
iO + VO –
336
Essential DC/DC Converters
and V3 =
3 1 3 ) Vin V = 9( 1− k 2 1− k
(5.76)
The output voltage is VO = VC 3 + VC 33 = 3V3 = (
3 3 ) Vin 1− k
(5.77)
The voltage transfer gain is G=
VO 3 3 =( ) Vin 1− k
(5.78)
Analogously, ΔiL1 =
Vin kT L1
I L1 =
32 I (1 − k ) 3 O
ΔiL2 =
V1 kT L2
I L2 =
8 I (1 − k )2 O
ΔiL3 =
V2 kT L3
I L3 =
2 I 1− k O
Therefore, the variation ratio of current iL1 through inductor L1 is ξ1 =
ΔiL1 / 2 k(1 − k ) 3 TVin k(1 − k )6 R = = I L1 64L1 I O 12 3 fL1
(5.79)
and the variation ratio of current iL2 through inductor L2 is ξ2 =
ΔiL 2 / 2 k(1 − k ) 2 TV1 k(1 − k ) 4 R = = I L2 16L2 I O 12 2 fL2
(5.80)
and the variation ratio of current iL3 through inductor L3 is ξ3 =
© 2006 by Taylor & Francis Group, LLC
ΔiL 3 / 2 k(1 − k )TV2 k(1 − k )2 R = = 4 L3 I O 12 I L3 fL3
(5.81)
Positive Output Cascade Boost Converters
337
The ripple voltage of output voltage vO is ΔvO =
ΔQ IO kT k VO = = C34 C34 fC34 R
Therefore, the variation ratio of output voltage vO is ε=
5.5.4
ΔvO / 2 k = VO 2 RfC34
(5.82)
Higher Stage Triple Boost Circuit
Higher stage triple boost circuits can be derived from the corresponding circuit of double boost series by adding another DEC in each stage circuit. For nth stage additional circuit, the final output voltage is VO = (
3 n ) Vin 1− k
The voltage transfer gain is G=
VO 3 n =( ) 1− k Vin
(5.83)
Analogously, the variation ratio of current iLi through inductor Li (i = 1, 2, 3, …n) is ξi =
ΔiLi / 2 k(1 − k )2( n−i +1) R = I Li fLi 12 ( n−i +1)
(5.84)
and the variation ratio of output voltage vO is ε=
5.6
ΔvO / 2 k = VO 2 RfCn2
(5.85)
Multiple Series
All circuits of P/O cascade boost converters — multiple series — are derived from the corresponding circuits of the main series by adding DEC multiple
© 2006 by Taylor & Francis Group, LLC
338
Essential DC/DC Converters 1 iIN
D1 V1 D11
L1
+
C1
2...
D12 2V1
C11
j D1(2j–1) D12j (1+j)V1 iO
C1(2j–1)
C12
VIN
R
C12j S
–
+ VO –
(a) Circuit diagram iIN
V1
+ L1 VIN –
C1
2V1
+ C11 + C12 VC1 VC11 – –
+ C13 VC12 –
jV1 C12(j–1)
(j+1)V1
C1(2j–1)
C12j R
iO + VO –
(b) Equivalent circuit during switching-on C1(2j–1)
(1+j)V1
C13 V1 iIN + VIN
C11
L1 VL1 C1
–
– + + VC11 VC1 C12 –
3V1 C12j
2V1 C14 + VC12 –
iO R
+ VO –
(c) Equivalent circuit during switching-off FIGURE 5.12 Cascade boost multiple-double circuit.
(j) times in each stage circuit. The first three stages of this series are shown in Figure 5.12 to Figure 5.14. For convenience they are called elementary multiple boost circuits, two-stage multiple boost circuits, and three-stage multiple boost circuits respectively, and numbered as n = 1, 2, and 3.
5.6.1
Elementary Multiple Boost Circuit
From the construction principle, the elementary multiple boost circuit is derived from the elementary boost converter by adding DEC multiple (j) times in the circuit. Its circuit and switch-on and -off equivalent circuits are shown in Figure 5.12. The voltage across capacitors C1 and C11 is charged to V1 and voltage across capacitors C12 and C13 is charged to VC13 = 2V1. The voltage across capacitors C1(2j-2) and C1(2j-1) is charged to VC1(2j-1) = jV1. The current iL1 flowing through inductor L1 increases with voltage Vin during switch-on period kT and decreases with voltage –(V1 – Vin) during switch-off (1 – k)T. Therefore,
© 2006 by Taylor & Francis Group, LLC
Positive Output Cascade Boost Converters
ΔiL1 =
339
V − Vin Vin kT = 1 (1 − k )T L1 L1 V1 =
1 V 1 − k in
(5.86)
(5.87)
The output voltage is VO = VC1 + VC1( 2 j −1) = (1 + j )V1 =
1+ j V 1 − k in
(5.88)
The voltage transfer gain is G=
5.6.2
VO 1 + j = Vin 1 − k
(5.89)
Two-Stage Multiple Boost Circuit
The two-stage multiple boost circuit is derived from the two-stage boost circuit by adding multiple (j) DECs in each stage circuit. Its circuit diagram and switch-on and -off equivalent circuits are shown in Figure 5.13. The voltage across capacitor C1 and capacitor C11 is charged to V1 = (1 1 − k )Vin . The voltage across capacitor C1(2j) is charged to (1 + j)V1. The current flowing through inductor L2 increases with voltage (1 + j)V1 during switch-on period kT and decreases with voltage –[V2 – (1 + j)V1] during switch-off period (1 – k)T. Therefore, the ripple of the inductor current iL2 is ΔiL2 =
V − (1 + j )V1 1+ j (1 − k )T kTV1 = 2 L2 L2
V2 =
1+ j 1 2 ) Vin V1 = (1 + j )( 1− k 1− k
(5.90)
(5.91)
The output voltage is VO = VC1 + VC1( 2 j −1) = (1 + j )V2 = (
1+ j 2 ) Vin 1− k
(5.92)
The voltage transfer gain is G=
© 2006 by Taylor & Francis Group, LLC
VO 1+ j 2 =( ) Vin 1− k
(5.93)
340
Essential DC/DC Converters
1 iIN
D1 V1 D11
L1
+
2...
1
j
jV1 (1+)jV1 L2 D1(2j–1) D12j
D12
V2
D3
D21
C1(2j–1)
C11
2...
j jV2 (1+j)V2 D2(2j–1) D22j
D22
iO
C2(2j–1)
C21
VIN
R D2 C1
–
C12
–
S
C12j
+ VO
C2 C22
C22j
(a) Circuit diagram iIN + L1
VIN
V1 C1
jV1
+ C11 VC1 –
(1+)jV1
C1(2j–1) C 12j
C12(j–1)
V2 C2
jV2
C21 C22(j–1)
(1+j)V2
C2(2j–1) C 22j
L2
iO + VO –
+ R V –C22j
–
(b) Equivalent circuit during switching-on (1+j)V2 C2(2j–1) C1(2j–1) (1+)jV 1
V1 iIN + VIN
V2 – VC21 +
L2
C11
L1 VL1
C1 + VC12 –
– + + VC11 VC1 C12
C1
–
–
C22j
C21
C2
+ VC2 –
+ VC22 C22
+ VC 22j R –
iO + VO –
–
(c) Equivalent circuit during switching-off FIGURE 5.13 Two-stage boost multiple-double circuit.
The ripple voltage of output voltage vO is ΔvO =
ΔQ IO kT k VO = = C22 j C22 j fC22 j R
Therefore, the variation ratio of output voltage vO is ε=
© 2006 by Taylor & Francis Group, LLC
ΔvO / 2 k = VO 2 RfC22 j
(5.94)
Positive Output Cascade Boost Converters 5.6.3
341
Three-Stage Multiple Boost Circuit
This circuit is derived from the three-stage boost circuit by adding multiple (j) DECs in each stage circuit. Its circuit diagram and equivalent circuits during switch-on and -off are shown in Figure 5.14. The voltage across capacitor C1 and capacitor C11 is charged to V1 = (1 1 − k )Vin . The voltage across capacitor C1(2j) is charged to (1 + j)V1. The voltage V2 across capacitor C2 and capacitor C2(2j) is charged to (1 + j)V2. The current flowing through inductor L3 increases with voltage (1 + j)V2 during switch-on period kT and decreases with voltage –[V3 – (1 + j)V2] during switch-off (1 – k)T. Therefore, ΔiL3 =
V − (1 + j )V2 1+ j kTV2 = 3 (1 − k )T L3 L3
(5.95)
and V3 =
(1 + j )V2 (1 + j ) 2 = V (1 − k ) (1 − k ) 3 in
(5.96)
The output voltage is VO = VC 3 + VC 3( 2 j −1) = (1 + j )V3 = (
1+ j 3 ) Vin 1− k
(5.97)
The voltage transfer gain is G=
VO 1+ j 3 =( ) Vin 1− k
(5.98)
The ripple voltage of output voltage vO is ΔvO =
ΔQ IO kT k VO = = C32 j C32 j fC32 j R
Therefore, the variation ratio of output voltage vO is ε=
© 2006 by Taylor & Francis Group, LLC
ΔvO / 2 k = VO 2 RfC32 j
(5.99)
1 L1
D1
+
V1
D11
2...jV 1 D12
j
(1+j)V1
D1(2j-1)D12j
2... jV2 j
1 L2
D3
C11
V2
D21
D22
C21
D31
j
(1+j)V3 D3(2j-1) D32j
D32
iO
C3(2j-1) R
D4
C12j
C12
D5
C31
D2
C1
2...jV3
1 V3
L3
C2(2j-1)
C1(2j-1) VIN
(1+j)V2
D2(2j-1)D22j
+ VO -
C3
S
C22j
C22
C2
342
iIN
C32
C32j
(a) Circuit Diagram iIN
V1
+ VIN
C1
jV1
C11 C12(j-1) C1(2j-1)
(1+j)V1 C12j
jV2
V2 C2
C21
C22(j-1)
C2(2j-1)
(1+j)V2
L2
L1
V3
C22j
C3
jV3
C31 C32(j-1)
C3(2j-1)
(1+j)V3 C32j
L3
R
-
iO + VO -
(b) Equivalent circuit during switching-on C3(2j-1)
(1+j)V3
V3 C2(2j-1)
C1(2j-1) V1 iIN + VIN -
(1+j)V1
L2
VL1 C1
+ + VC11 VC1 C12 -
C22 C12j + VC12 -
C2
+ VC2 -
© 2006 by Taylor & Francis Group, LLC
+
C31 C22j
C3
+ VC22 -
(c) Equivalent circuit during switching-off FIGURE 5.14 Three-stage boost multiple-double circuit.
VC31
L3
- VC21 +
C11
L1
-
C32j
+ VC3 -
C32
+ VC32 -
R
iO + VO -
Essential DC/DC Converters
V2
(1+j)V2
C21
Positive Output Cascade Boost Converters 5.6.4
343
Higher Stage Multiple Boost Circuit
Higher stage multiple boost circuit is derived from the corresponding circuit of the main series by adding multiple (j) DECs in each stage circuit. For nth stage additional circuit, the final output voltage is VO = (
1+ j n ) Vin 1− k
The voltage transfer gain is G=
VO 1+ j n =( ) Vin 1− k
(5.100)
Analogously, the variation ratio of output voltage vO is ε=
5.7
ΔvO / 2 k = 2 RfCn2 j VO
(5.101)
Summary of Positive Output Cascade Boost Converters
All circuits of the positive output cascade boost converters as a family are shown in Figure 5.15 as the family tree. From the analysis of the previous two sections we have the common formula to calculate the output voltage:
1 n ⎧ ⎪ ( 1 − k ) Vin ⎪ 1 n ) Vin ⎪2 ∗ ( 1 − k ⎪ 2 n ⎪ VO = ⎨ ( ) Vin ⎪ 1− k ⎪ ( 3 )n Vin ⎪ 1− k ⎪ j+1 n ⎪⎩ ( 1 − k ) Vin The voltage transfer gain is
© 2006 by Taylor & Francis Group, LLC
main _ series additional _ series double _ series triple _ series multiple( j ) _ series
(5.102)
344
Essential DC/DC Converters Main Series
Additional Series
Double Series
Triple Series
Multi Series
5 Stage Boost Circuit
5 Stage Additional Boost Circuit
5 Stage Double Boost Circuit
5 Stage Triple Boost Circuit
5 Stage Multiple Boost Circuit
4 Stage Boost Circuit
4 Stage Additional Boost Circuit
4 Stage Double Boost Circuit
4 Stage Triple Boost Circuit
4 Stage Multiple Boost Circuit
3 Stage Boost Circuit
3 Stage Additional Boost Circuit
3 Stage Double Boost Circuit
3 Stage Triple Boost Circuit
3 Stage Multiple Boost Circuit
2 Stage Boost Circuit
2 Stage Additional Boost Circuit
2 Stage Double Boost Circuit
2 Stage Triple Boost Circuit
2 Stage Multiple Boost Circuit
Elementary Additional/Double Boost Circuit
Elementary Triple Boost Circuit
Elementary Multiple Boost Circuit
Positive Output Elementary Boost Converter
FIGURE 5.15 The family of positive output cascade boost converters
1 n ⎧ ⎪ (1 − k ) ⎪ 1 n ) ⎪2 ∗ ( 1− k ⎪ V 2 n ⎪ G= O =⎨ ( ) Vin ⎪ 1 − k ⎪ ( 3 )n ⎪ 1− k ⎪ j+1 n ⎪⎩ ( 1 − k )
main _ series additional _ series double _ series
(5.103)
triple _ series multiple( j ) _ series
In order to show the advantages of the positive output cascade boost converters, we compare their voltage transfer gains to that of the buck converter, G=
© 2006 by Taylor & Francis Group, LLC
VO =k Vin
Positive Output Cascade Boost Converters
345
forward converter, G=
VO = kN Vin
G=
VO k = Vin 1 − k
G=
VO k N = Vin 1 − k
G=
VO 1 = Vin 1 − k
N is the transformer turn ratio
Cúk-converter,
fly-back converter, N is the transformer turn ratio
boost converter,
and positive output Luo-converters G=
VO n = Vin 1 − k
(5.104)
If we assume that the conduction duty k is 0.2, the output voltage transfer gains are listed in Table 5.1; if the conduction duty k is 0.5, the output voltage transfer gains are listed in Table 5.2; if the conduction duty k is 0.8, the output voltage transfer gains are listed in Table 5.3.
5.8 5.8.1
Simulation and Experimental Results Simulation Results of a Three-Stage Boost Circuit
To verify the design and calculation results, the PSpice simulation package was applied to a three-stage boost converter. Choosing Vin = 20 V, L1 = L2 = L3 = 10 mH, all C1 to C8 = 2 μF, and R = 30 kΩ, k = 0.7 and f = 100 kHz. The
© 2006 by Taylor & Francis Group, LLC
346
Essential DC/DC Converters
TABLE 5.1 Voltage Transfer Gains of Converters in the Condition k = 0.2 Stage No. (n) Buck converter Forward converter Cúk-converter Fly-back converter Boost converter Positive output Luo-converters Positive output cascade boost converters — main series Positive output cascade boost converters — additional series Positive output cascade boost converters — double series Positive output cascade boost converters — triple series Positive output cascade boost (j = 3) converters — multiple series
1
2
1.25 1.25
3
4
5
n
0.2 0.2N (N is the transformer turn ratio) 0.25 0.25N (N is the transformer turn ratio) 1.25 2.5 3.75 5 6.25 1.25n 1.563 1.953 2.441 3.052 1.25n
2.5
3.125
3.906
4.882
6.104
2∗1.25n
2.5
6.25
15.625
39.063
97.66
(2∗1.25)n
3.75
14.06
52.73
197.75
741.58
(3∗1.25)n
5
25
125
625
3125
(4∗1.25)n
TABLE 5.2 Voltage Transfer Gains of Converters in the Condition k = 0.5 Stage No. (n) Buck converter Forward converter Cúk-converter Fly-back converter Boost converter Positive output Luo-converters Positive output cascade boost converters main series Positive output cascade boost converters additional series Positive output cascade boost converters double series Positive output cascade boost converters triple series Positive output cascade boost (j = 3) converters — multiple series
1
2
3
4
5
n
—
0.5 0.5N (N is the transformer turn ratio) 1 N (N is the transformer turn ratio) 2 2 4 6 8 10 2n 2 4 8 16 32 2n
—
4
8
16
32
64
—
4
16
64
256
1024
(2∗2)n
—
6
36
216
1296
7776
(3∗2)n
8
64
512
4096
32,768
(4∗2)n
2∗2n
obtained voltage values V1, V2, and VO of a triple-lift circuit are 66 V, 194 V, and 659 V respectively and inductor current waveforms iL1 (its average value IL1 = 618 mA), iL2, and iL3. The simulation results are shown in Figure 5.16. The voltage values match the calculated results.
© 2006 by Taylor & Francis Group, LLC
Positive Output Cascade Boost Converters
347
TABLE 5.3 Voltage Transfer Gains of Converters in the Condition k = 0.8 Stage No. (n)
1
Buck converter Forward converter Cúk-converter Fly-back converter Boost converter Positive output Luo-converters Positive output cascade boost converters — main series Positive output cascade boost converters — additional series Positive output cascade boost converters — double series Positive output cascade boost converters — triple series Positive output cascade boost (j=3) converters — multiple series
5 5
2
3
4
5
n
0.8 0.8N (N is the transformer turn ratio) 4 4N (N is the transformer turn ratio) 5 10 15 20 25 5n 25 125 625 3125 5n 2∗5n
10
50
250
1250
6250
10
100
1000
10,000
100,000
(2∗5)n
15
225
3375
50,625
759,375
(3∗5)n
20
400
8000
160,000
32∗105
(4∗5)n
1.0A
(9.988m, 618m) 0.5A
(9.988m, 218m) (9.988m, 111m)
0A
I(L1)
I(L2)
I(L3)
1.0KV
(9.988m, 659m) (9.988m, 194m) (9.988m, 66m)
0.5KV SEL>> 0V 9.980ms
V(R:2)
9.984ms
V(D2:2)
9.988ms
V(D5:2)
9.992ms
9.996ms
10.000ms
Time
FIGURE 5.16 The simulation results of a three-stage boost circuit at condition k = 0.7 and f = 100 kHz.
5.8.2
Experimental Results of a Three-Stage Boost Circuit
A test rig was constructed to verify the design and calculation results, and compared with PSpice simulation results. The test conditions are still Vin = 20 V, L1 = L2 = L3 = 10 mH, all C1 to C8 = 2 μF and R = 30 kΩ, k = 0.7, and f = 100kHz. The component of the switch is a MOSFET device IRF950 with the rate 950 V/5 A/2 MHz. The measured values of the output voltage and
© 2006 by Taylor & Francis Group, LLC
348
Essential DC/DC Converters
FIGURE 5.17 The experimental results of a three-stage boost circuit at condition k = 0.7 and f = 100 kHz
TABLE 5.4 Comparison of Simulation and Experimental Results of a Triple-Lift Circuit Stage No. (n) Simulation results Experimental results
IL1 (A)
Iin (A)
Vin (V)
Pin (W)
VO (V)
PO (W)
η (%)
0.618 0.62
0.927 0.93
20 20
18.54 18.6
659 660
14.47 14.52
78 78
first inductor current in a three-stage boost converter. After careful measurement, we obtained the current value of IL1 = 0.62 A (shown in channel 1 with 1 A/Div) and voltage value of VO = 660 V (shown in channel 2 with 200 V/ Div). The experimental results (current and voltage values) in Figure 5.17 match the calculated and simulation results, which are IL1 = 0.618 A and VO = 659 V shown in Figure 5.16.
5.8.3
Efficiency Comparison of Simulation and Experimental Results
These circuits enhanced the voltage transfer gain successfully, and efficiently. Particularly, the efficiency of the tested circuits is 78%, which is good for high voltage output equipment. Comparison of the simulation and experimental results is shown in Table 5.4.
5.8.4
Transient Process
Usually, there is high inrush current during the first power-on. Therefore, the voltage across capacitors is quickly changed to certain values. The transient process is very quick taking only a few milliseconds.
© 2006 by Taylor & Francis Group, LLC
Positive Output Cascade Boost Converters
349
Bibliography Chokhawala, R.S., Catt, J., and Pelly, B.R., Gate drive considerations for IGBT modules, IEEE Trans. on Industry Applications, 31, 603, 1995. Czarkowski, D. and Kazimierczuk, M.K., Phase-controlled series-parallel resonant converter, IEEE Trans. on Power Electronics, 8, 309, 1993. Kassakian, J.G., Schlecht, M.F., and Verghese, G.C., Principles of Power Electronics, Addison-Wesley, New York, 1991. Khan, I.A., DC-to-DC converters for electric and hybrid vehicles, in Proceedings of Power Electronics in Transportation, 1994, p. 113. Luo, F.L. and Ye, H., Positive output cascade boost converters, in Proceedings of IEEEIPEMC’2003, Xi’an, China, 866, 2002. Luo, F.L. and Ye, H., Fundamentals of positive output cascade boost converters, submitted to IEEE Transactions on Power Electronics, 18, 1526, 2003. Luo, F. L. and Ye, H., Development of positive output cascade boost converters, submitted to IEEE Transactions on Power Electronics, 18, 1536, 2003. Poon, N.K. and Pong, M.H., Computer aided design of a crossing current resonant converter (XCRC), in Proceedings of IECON’94, Bologna, Italy, 1994, p. 135. Steigerwald, R.L., High-frequency resonant transistor DC-DC converters, IEEE Trans. on Industrial Electronics, 31, 181, 1984.
© 2006 by Taylor & Francis Group, LLC
6 Negative Output Cascade Boost Converters
6.1
Introduction
This chapter introduces negative output cascade boost converters. Just as with positive output cascade boost converters these converters use the superlift technique. There are several sub-series: • Main series — Each circuit of the main series has one switch S, n inductors, n capacitors, and (2n – 1) diodes. • Additional series — Each circuit of the additional series has one switch S, n inductors, (n + 2) capacitors, and (2n + 1) diodes. • Double series — Each circuit of the double series has one switch S, n inductors, 3n capacitors, and (3n – 1) diodes. • Triple series — Each circuit of the triple series has one switch S, n inductors, 5n capacitors, and (5n – 1) diodes. • Multiple series — Multiple series circuits have one switch S and a higher number of capacitors and diodes. The conduction duty ratio is k, switching frequency is f, switching period is T = 1/f, the load is resistive load R. The input voltage and current are Vin and Iin, output voltage and current are VO and IO. Assume no power losses during the conversion process, Vin × Iin = VO × IO. The voltage transfer gain is G: G=
6.2
VO Vin
Main Series
The first three stages of the negative output cascade boost converters — main series — are shown in Figure 6.1 to Figure 6.3. For convenience they are 351
© 2006 by Taylor & Francis Group, LLC
352
Essential DC/DC Converters
C1 iIN
D1 iO –
S
+ VIN
L1
R
–
VO +
(a) Circuit diagram iIN
C1
+ VIN
L1
+ VC1 – R
iO – VO +
–
(b) Equivalent circuit during switching-on C1
iIN +
iO –
+V – C1
VIN
L1
–
R
VO +
(c) Equivalent circuit during switching-off FIGURE 6.1 Elementary boost converter.
called elementary boost converter, two-stage boost circuit and three-stage boost circuit respectively, and numbered as n = 1, 2 and 3.
6.2.1
N/O Elementary Boost Circuit
The N/O elementary boost converter and its equivalent circuits during switch-on and -off are shown in Figure 6.1. The voltage across capacitor C1 is charged to VC1. The current iL1 flowing through inductor L1 increases with voltage Vin during switch-on period kT and decreases with voltage –(VC1 – Vin) during switch-off period (1 – k)T. Therefore, the ripple of the inductor current iL1 is ΔiL1 =
V − Vin Vin (1 − k )T kT = C1 L1 L1 VC1 =
© 2006 by Taylor & Francis Group, LLC
1 V 1 − k in
(6.1)
Negative Output Cascade Boost Converters
VO = VC1 − Vin =
353 k V 1 − k in
(6.2)
The voltage transfer gain is G=
VO 1 = −1 Vin 1 − k
(6.3)
The inductor average current is I L1 = (1 − k )
VO R
(6.4)
The variation ratio of current iL1 through inductor L1 is ξ1 =
kTVin ΔiL1 / 2 k R = = I L1 (1 − k )2L1VO / R 2 fL1
(6.5)
Usually ξ1 is small (much lower than unity), it means this converter works in the continuous mode. The ripple voltage of output voltage vO is ΔvO =
ΔQ IO kT k VO = = C1 C1 fC1 R
Therefore, the variation ratio of output voltage vO is ε=
ΔvO / 2 k = 2 RfC1 VO
(6.6)
Usually R is in kΩ, f in 10 kHz, and C1 in μF, this ripple is smaller than 1%. 6.2.2
N/O Two-Stage Boost Circuit
The N/O two-stage boost circuit is derived from elementary boost converter by adding the parts (L2-D2-D3-C2). Its circuit diagram and equivalent circuits during switch-on and switch -off are shown in Figure 6.2. The voltage across capacitor C1 is charged to V1. As described in the previous section the voltage V1 across capacitor C1 is V1 =
© 2006 by Taylor & Francis Group, LLC
1 V 1 − k in
354
Essential DC/DC Converters
iIN
+ C1 –
D1
+ C2 – L2
S
+ VIN
D3
iO –
R
VO +
D2
L1
–
(a) Circuit diagram iIN
C2 +
C1
VIN
L1
+ VC1 + VC2 – R –
L2
–
iO – VO +
(b) Equivalent circuit during switching-on iIN
C2 C1
+
+ V – C2
+V –
VIN
L2
C1
L1 –
R
iO – VO +
(c) Equivalent circuit during switching-off FIGURE 6.2 Two-stage boost circuit.
The voltage across capacitor C2 is charged to VC2. The current flowing through inductor L2 increases with voltage V1 during switch-on period kT and decreases with voltage –(VC2 – V1) during switch-off period (1 – k)T. Therefore, the ripple of the inductor current iL2 is ΔiL 2 =
V − V1 V1 kT = C 2 (1 − k )T L2 L2
VC 2 =
1 1 2 V =( ) Vin 1− k 1 1− k
VO = VC 2 − Vin = [( The voltage transfer gain is
© 2006 by Taylor & Francis Group, LLC
(6.7)
1 2 ) − 1]Vin 1− k
(6.8)
Negative Output Cascade Boost Converters
G=
355
VO 1 2 =( ) −1 Vin 1− k
(6.9)
Analogously, ΔiL1 =
Vin kT L1
I L1 =
IO (1 − k )2
ΔiL2 =
V1 kT L2
I L2 =
IO 1− k
Therefore, the variation ratio of current iL1 through inductor L1 is ΔiL1 / 2 k(1 − k ) 2 TVin k(1 − k ) 4 R = = I L1 2L1 I O 2 fL1
ξ1 =
(6.10)
the variation ratio of current iL2 through inductor L2 is ξ2 =
ΔiL 2 / 2 k(1 − k )TV1 k(1 − k )2 R = = I L2 2L2 I O 2 fL2
(6.11)
and the variation ratio of output voltage vO is ε=
6.2.3
ΔvO / 2 k = VO 2 RfC2
(6.12)
N/O Three-Stage Boost Circuit
The N/O three-stage boost circuit is derived from the two-stage boost circuit by double adding the parts (L2-D2-D3-C2). Its circuit diagram and equivalent circuits during switch-on and -off are shown in Figure 6.3. The voltage across capacitor C1 is charged to V1. As described previously the voltage VC1 across capacitor C1 is VC1 = (1 1 − k )Vin , and voltage VC2 across capacitor C2 is 2 VC 2 = (1 1 − k ) Vin . The voltage across capacitor C3 is charged to VO. The current flowing through inductor L3 increases with voltage VC2 during switch-on period kT and decreases with voltage –(VC3 – VC2) during switch-off (1 – k)T. Therefore, the ripple of the inductor current iL3 is ΔiL 3 =
© 2006 by Taylor & Francis Group, LLC
VC 2 V − VC 2 kT = C 3 (1 − k )T L3 L3
(6.13)
356
Essential DC/DC Converters
iIN
+ C1 –
D1
+ C2 –
L2
L3 D5
D3
S
+
+ C3 –
D2
VIN
L1
R
D4
–
iO – VO +
(a) Circuit diagram iIN
C3 +
C1
VIN –
L1
C2 + VC1 L 2 –
+ + VC3 – VC2 L 3 R –
iO – VO +
(b) Equivalent circuit during switching-on iIN
C1
+ VC1 C2 –
+ VC2 C3 –
L2
L3
+ VIN –
L1
R
+ VC3 – iO – VO +
(c) Equivalent circuit during switching-off FIGURE 6.3 Three-stage boost circuit.
VC 3 =
1 1 2 1 3 ) VC1 = ( ) Vin V =( 1 − k C2 1− k 1− k
VO = VC 3 − Vin = [(
1 3 ) − 1]Vin 1− k
(6.14)
The voltage transfer gain is G= Analogously,
© 2006 by Taylor & Francis Group, LLC
VO 1 3 =( ) −1 Vin 1− k
(6.15)
Negative Output Cascade Boost Converters
357
ΔiL1 =
Vin kT L1
I L1 =
IO (1 − k ) 3
ΔiL2 =
V1 kT L2
I L2 =
IO (1 − k )2
ΔiL3 =
V2 kT L3
I L3 =
IO 1− k
Therefore, the variation ratio of current iL1 through inductor L1 is ξ1 =
ΔiL1 / 2 k(1 − k ) 3 TVin k(1 − k )6 R = = I L1 2L1 I O 2 fL1
(6.16)
the variation ratio of current iL2 through inductor L2 is ξ2 =
ΔiL 2 / 2 k(1 − k ) 2 TV1 k(1 − k ) 4 R = = I L2 2L2 I O 2 fL2
(6.17)
the variation ratio of current iL3 through inductor L3 is ξ3 =
ΔiL 3 / 2 k(1 − k )TV2 k(1 − k )2 R = = 2 L3 I O 2 I L3 fL3
(6.18)
and the variation ratio of output voltage vO is ε=
6.2.4
ΔvO / 2 k = 2 RfC3 VO
(6.19)
N/O Higher Stage Boost Circuit
N/O higher stage boost circuit can be designed by multiple repetition of the parts (L2-D2-D3-C2). For nth stage boost circuit, the final output voltage across capacitor Cn is VO = [( The voltage transfer gain is
© 2006 by Taylor & Francis Group, LLC
1 n ) − 1]Vin 1− k
358
Essential DC/DC Converters
G=
VO 1 n =( ) −1 Vin 1− k
(6.20)
the variation ratio of current iLi through inductor Li (i = 1, 2, 3, …n) is ξi =
ΔiLi / 2 k(1 − k ) 2( n−i +1) R = I Li 2 fLi
(6.21)
and the variation ratio of output voltage vO is ε=
6.3
ΔvO / 2 k = 2 RfCn VO
(6.22)
Additional Series
All circuits of negative output cascade boost converters — additional series — are derived from the corresponding circuits of the main series by adding a DEC. The first three stages of this series are shown in Figure 6.4 to Figure 6.6. For convenience they are called elementary additional boost circuit, twostage additional boost circuit, and three-stage additional boost circuit respectively, and numbered as n = 1, 2 and 3.
6.3.1
N/O Elementary Additional Boost Circuit
This N/O elementary boost additional circuit is derived from N/O elementary boost converter by adding a DEC. Its circuit and switch-on and switchoff equivalent circuits are shown in Figure 6.4. The voltage across capacitor C1 and C11 is charged to VC1 and voltage across capacitor C12 is charged to VC12 = 2VC1. The current iL1 flowing through inductor L1 increases with voltage Vin during switch-on period kT and decreases with voltage –(VC1 – Vin) during switch-off (1 – k)T. Therefore, ΔiL1 =
V − Vin Vin kT = C1 (1 − k )T L1 L1 VC1 =
© 2006 by Taylor & Francis Group, LLC
1 V 1 − k in
(6.23)
Negative Output Cascade Boost Converters
359
C12
C1 iIN
D11
D1
D12 iO – VO
S
+ VIN
C11
L1
R
+
–
(a) Circuit diagram iIN
C12 + L1
VIN
C1 + C11 VC11 –
+ + VC12 – VC11 R –
iO – VO +
–
(b) Equivalent circuit during switching-on iIN
C12 +V
C1 +
+V – C1
VIN
– C11
C12
+ V – C11
L1
R
iO – VO +
–
(c) Equivalent circuit during switching-off FIGURE 6.4 Elementary boost additional (double) circuit.
The voltage VC12 is VC12 = 2VC1 =
2 V 1 − k in
(6.24)
2 − 1]Vin 1− k
(6.25)
The output voltage is VO = VC12 − Vin = [ The voltage transfer gain is G=
© 2006 by Taylor & Francis Group, LLC
VO 2 = −1 Vin 1 − k
(6.26)
360
Essential DC/DC Converters
The variation ratio of current iL1 through inductor L1 is ΔiL1 / 2 k(1 − k )TVin k(1 − k )2 R = = I L1 4L1 I O 8 fL1
ξ1 =
(6.27)
The ripple voltage of output voltage vO is ΔvO =
ΔQ IO kT k VO = = C12 C12 fC12 R
Therefore, the variation ratio of output voltage vO is ε=
6.3.2
ΔvO / 2 k = 2 RfC12 VO
(6.28)
N/O Two-Stage Additional Boost Circuit
The N/O two-stage additional boost circuit is derived from the N/O twostage boost circuit by adding a DEC. Its circuit diagram and switch-on and switch-off equivalent circuits are shown in Figure 6.5. The voltage across capacitor C1 is charged to VC1. As described in the previous section the voltage VC1 across capacitor C1 is VC1 =
1 V 1 − k in
The voltage across capacitor C2 and capacitor C11 is charged to VC2 and voltage across capacitor C12 is charged to VC12. The current flowing through inductor L2 increases with voltage VC1 during switch-on period kT and decreases with voltage –(VC2 – VC1) during switch-off period (1 – k)T. Therefore, the ripple of the inductor current iL2 is ΔiL 2 =
V − VC1 VC1 kT = C 2 (1 − k )T L2 L2
(6.29)
1 1 2 ) Vin VC1 = ( 1− k 1− k
(6.30)
VC 2 =
VC12 = 2VC 2 =
© 2006 by Taylor & Francis Group, LLC
2 1 2 VC1 = 2( ) Vin 1− k 1− k
Negative Output Cascade Boost Converters
361
C2
C1 iIN
L2
D1
D3
C12
D11
D12 iO –
S
+
D2
VIN
L1
R
C11
–
VO +
(a) Circuit diagram iIN
C12 +
C1
VIN
+ C2 VC1 –
L2
L1
+ C11 VC2 –
+ + VC12 – VC2 R –
–
iO – VO +
(b) Equivalent circuit during switching-on iIN
+
C1
+ VC1 C2 –
+ VC2 C12 – C11
L2
+ VC11 –
VIN L1
–
R
+ VC12 – iO – VO +
(c) Equivalent circuit during switching-off FIGURE 6.5 Two-stage additional boost circuit.
The output voltage is VO = VC12 − Vin = [2(
1 2 ) − 1]Vin 1− k
(6.31)
The voltage transfer gain is G= Analogously,
© 2006 by Taylor & Francis Group, LLC
VO 1 2 = 2( ) −1 Vin 1− k
(6.32)
362
Essential DC/DC Converters
ΔiL1 =
Vin kT L1
I L1 =
2 I (1 − k )2 O
ΔiL2 =
V1 kT L2
I L2 =
2IO 1− k
Therefore, the variation ratio of current iL1 through inductor L1 is ΔiL1 / 2 k(1 − k )2 TVin k(1 − k ) 4 R = = I L1 4L1 I O 8 fL1
ξ1 =
(6.33)
and the variation ratio of current iL2 through inductor L2 is ξ2 =
ΔiL 2 / 2 k(1 − k )TV1 k(1 − k )2 R = = I L2 4L2 I O 8 fL2
(6.34)
The ripple voltage of output voltage vO is ΔvO =
ΔQ IO kT k VO = = C12 C12 fC12 R
Therefore, the variation ratio of output voltage vO is ε=
6.3.3
ΔvO / 2 k = 2 RfC12 VO
(6.35)
N/O Three-Stage Additional Boost Circuit
This N/O circuit is derived from three-stage boost circuit by adding a DEC. Its circuit diagram and equivalent circuits during switch-on and switch-off are shown in Figure 6.6. The voltage across capacitor C1 is charged to VC1. As described previously, the voltage V C 1 across capacitor C 1 is 2 VC1 = (1 1 − k )Vin , and voltage V2 across capacitor C2 is VC 2 = (1 1 − k ) Vin . The voltage across capacitor C3 and capacitor C11 is charged to VC3. The voltage across capacitor C12 is charged to VC12. The current flowing through inductor L3 increases with voltage VC2 during switch-on period kT and decreases with voltage –(VC3 – VC2) during switch-off (1 – k)T. Therefore, ΔiL 3 =
© 2006 by Taylor & Francis Group, LLC
V − VC 2 VC 2 kT = C 3 (1 − k )T L3 L3
(6.36)
Negative Output Cascade Boost Converters
363
C1 iIN
+ VIN
C3
C2
L2
D1
L3
D3
D5
C12
D11
D12
S D2 L1
C11
D4
–
R
iO – VO +
(a) Circuit diagram C12
iIN C1
+ VIN L1
C2 + VC1 L – 2
+ C3 VC2 – L3
+ C11 VC3 –
+ + VC12 – VC11 R –
–
iO – VO +
(b) Equivalent circuit during switching-on iIN
+ VC12 –
+ VC1 –
+
+ C1 VC2 –
+ C2 VC3 –
L2
L3
C12 C3 C11
VIN L1
–
+ VC11 – R
iO – VO +
(c) Equivalent circuit during switching-off FIGURE 6.6 Three-stage additional boost circuit.
and VC 3 =
1 1 2 1 3 V =( ) VC1 = ( ) Vin 1 − k C2 1− k 1− k
(6.37)
The voltage VC12 is VC12 = 2VC 3 = 2(
1 3 ) Vin 1− k
The output voltage is VO = VC12 − Vin = [2(
© 2006 by Taylor & Francis Group, LLC
1 3 ) − 1]Vin 1− k
(6.38)
364
Essential DC/DC Converters
The voltage transfer gain is G=
VO 1 3 = 2( ) −1 1− k Vin
(6.39)
Analogously, ΔiL1 =
Vin kT L1
I L1 =
2 I (1 − k ) 3 O
ΔiL2 =
V1 kT L2
I L2 =
2 I (1 − k )2 O
ΔiL3 =
V2 kT L3
I L3 =
2IO 1− k
Therefore, the variation ratio of current iL1 through inductor L1 is ξ1 =
ΔiL1 / 2 k(1 − k ) 3 TVin k(1 − k )6 R = = I L1 4L1 I O 8 fL1
(6.40)
and the variation ratio of current iL2 through inductor L2 is ξ2 =
ΔiL 2 / 2 k(1 − k ) 2 TV1 k(1 − k ) 4 R = = I L2 4L2 I O 8 fL2
(6.41)
and the variation ratio of current iL3 through inductor L3 is ξ3 =
ΔiL 3 / 2 k(1 − k )TV2 k(1 − k )2 R = = 4 L3 I O 8 I L3 fL3
(6.42)
The ripple voltage of output voltage vO is ΔvO =
ΔQ IO kT k VO = = C12 C12 fC12 R
Therefore, the variation ratio of output voltage vO is ε=
© 2006 by Taylor & Francis Group, LLC
ΔvO / 2 k = VO 2 RfC12
(6.43)
Negative Output Cascade Boost Converters 6.3.4
365
N/O Higher Stage Additional Boost Circuit
The N/O higher stage boost additional circuit is derived from the corresponding circuit of the main series by adding a DEC. For the nth stage additional circuit, the final output voltage is VO = [2(
1 n ) −]Vin 1− k
The voltage transfer gain is G=
VO 1 n = 2( ) −1 Vin 1− k
(6.44)
Analogously, the variation ratio of current iLi through inductor Li (i = 1, 2, 3, …n) is ξi =
ΔiLi / 2 k(1 − k ) 2( n−i +1) R = I Li fLi 8
(6.45)
and the variation ratio of output voltage vO is ε=
6.4
ΔvO / 2 k = 2 RfC12 VO
(6.46)
Double Series
All circuits of N/O cascade boost converters — double series — are derived from the corresponding circuits of the main series by adding a DEC in each stage circuit. The first three stages of this series are shown in Figures 6.4, 6.7, and 6.8. For convenience they are called elementary double boost circuit, two-stage double boost circuit, and three-stage double boost circuit respectively, and numbered as n = 1, 2 and 3.
6.4.1
N/O Elementary Double Boost Circuit
This N/O elementary double boost circuit is derived from the elementary boost converter by adding a DEC. Its circuit and switch-on and switch-off equivalent circuits are shown in Figure 6.4, which is the same as the elementary boost additional circuit.
© 2006 by Taylor & Francis Group, LLC
366
Essential DC/DC Converters
C1 iIN S
+ VIN
C12
D11
D1
D12
C2
L2
D3
C22 D22
C21
C11 L1
D21
R
D2
–
iO – VO +
(a) Circuit diagram C22
iIN C1
+ VIN L1 –
C + 11 VC1 –
C + 12 VC11 –
C2 + VC12 – L2
+ C21 VC2 –
+ + VC22 – VC21 R –
iO – VO +
(b) Equivalent circuit during switching–on iIN
+ VC22 – + VC1 –
+
+ C1 VC12 C11 –
+ C12 VC2 –
C22 C2 C21
VIN + VC11 –
L2
L1
–
+ VC21 – R
iO – VO +
(c) Equivalent circuit during switching–off FIGURE 6.7 Two-stage double boost circuit.
6.4.2
N/O Two-Stage Double Boost Circuit
The N/O two-stage double boost circuit is derived from two-stage boost circuit by adding a DEC in each stage circuit. Its circuit diagram and switchon and switch-off equivalent circuits are shown in Figure 6.7. The voltage across capacitor C1 and capacitor C11 is charged to V1. As described in the previous section the voltage VC1 across capacitor C1 and capacitor C11 is VC1 = (1 1 − k )Vin . The voltage across capacitor C12 is charged to 2VC1. The current flowing through inductor L2 increases with voltage 2VC1 during switch-on period kT and decreases with voltage –(VC2 – 2VC1) during switchoff period (1 – k)T. Therefore, the ripple of the inductor current iL2 is ΔiL 2 =
© 2006 by Taylor & Francis Group, LLC
V − 2VC1 2VC1 kT = C 2 (1 − k )T L2 L2
(6.47)
Negative Output Cascade Boost Converters
VC 2 =
367
2 1 2 ) Vin V = 2( 1 − k C1 1− k
(6.48)
The voltage VC22 is VC 22 = 2VC 2 = (
2 2 ) Vin 1− k
The output voltage is VO = VC 22 − Vin = [(
2 2 ) − 1]Vin 1− k
(6.49)
The voltage transfer gain is G=
VO 2 2 =( ) −1 Vin 1− k
(6.50)
Analogously, ΔiL1 =
Vin kT L1
I L1 = (
ΔiL2 =
V1 kT L2
I L2 =
2 2 ) I 1− k O
2IO 1− k
Therefore, the variation ratio of current iL1 through inductor L1 is ξ1 =
ΔiL1 / 2 k(1 − k )2 TVin k(1 − k ) 4 R = = I L1 8L1 I O 16 fL1
(6.51)
and the variation ratio of current iL2 through inductor L2 is ξ2 =
ΔiL 2 / 2 k(1 − k )TV1 k(1 − k )2 R = = I L2 4L2 I O 8 fL2
The ripple voltage of output voltage vO is ΔvO =
© 2006 by Taylor & Francis Group, LLC
ΔQ IO kT k VO = = C22 C22 fC22 R
(6.52)
368
Essential DC/DC Converters
C1 iIN
D1
C12
D11
D12
C2
L2
D3
D21
C22 D22
L3
C3 D5
D31
C32 D32
S
+
iO – VO +
C11
VIN
L1
C21
D2
–
R
D4
(a) Circuit diagram C32
iIN +
C1
+ C11 VC1 –
VIN L1
+ C12 VC11 –
C2 + VC12 L – 2
+ C21 + C22 VC21 VC2 – –
C3 + VC22 L – 3
+ C31 VC3 –
+ + VC32 – VC31 R –
iO – VO +
–
(b) Equivalent circuit during switching-on iIN
+
+ VC1 –
+ C1 VC12 C11 –
+ C12 VC2 –
+ VC11 –
L2
+ C2 VC22 – C21
+ C22 VC3 –
C32 C3 C31
VIN + VC21–
L3
L1
–
+ VC31 R
iO – VO +
(c) Equivalent circuit during switching-off FIGURE 6.8 Three-stage double boost circuit.
Therefore, the variation ratio of output voltage vO is ε=
6.4.3
ΔvO / 2 k = VO 2 RfC22
(6.53)
N/O Three-Stage Double Boost Circuit
This N/O circuit is derived from the three-stage boost circuit by adding DEC in each stage circuit. Its circuit diagram and equivalent circuits during switchon and switch-off are shown in Figure 6.8. The voltage across capacitor C1 and capacitor C11 is charged to VC1. As described previously, the voltage VC1 across capacitor C1 and capacitor C11 is VC1 = (1 1 − k )Vin , and voltage VC2 across capac2 itor C2 and capacitor C12 is VC 2 = 2(1 1 − k ) Vin . © 2006 by Taylor & Francis Group, LLC
Negative Output Cascade Boost Converters
369
The voltage across capacitor C22 is 2VC 2 = (2 1 − k ) Vin . The voltage across capacitor C3 and capacitor C31 is charged to V3. The voltage across capacitor C12 is charged to VO. The current flowing through inductor L3 increases with voltage V 2 during switch-on period kT and decreases with voltage –(VC3 – 2VC2) during switch-off (1 – k)T. Therefore, 2
ΔiL 3 =
2VC 2 V − 2VC 2 (1 − k )T kT = C 3 L3 L3
(6.54)
2VC 2 4 V = (1 − k ) (1 − k ) 3 in
(6.55)
and VC 3 = The voltage VC32 is VC 32 = 2VC 3 = (
2 3 ) Vin 1− k
The output voltage is VO = VC 32 − Vin = [(
2 3 ) − 1]Vin 1− k
(6.56)
The voltage transfer gain is G=
VO 2 3 =( ) −1 Vin 1− k
Analogously, ΔiL1 =
Vin kT L1
I L1 =
8 I (1 − k ) 3 O
ΔiL2 =
V1 kT L2
I L2 =
4 I (1 − k )2 O
ΔiL3 =
V2 kT L3
I L3 =
2IO 1− k
Therefore, the variation ratio of current iL1 through inductor L1 is
© 2006 by Taylor & Francis Group, LLC
(6.57)
370
Essential DC/DC Converters
ξ1 =
ΔiL1 / 2 k(1 − k ) 3 TVin k(1 − k )6 R = = I L1 16L1 I O 128 fL1
(6.58)
and the variation ratio of current iL2 through inductor L2 is ξ2 =
ΔiL 2 / 2 k(1 − k ) 2 TV1 k(1 − k ) 4 R = = I L2 8L2 I O 32 fL2
(6.59)
and the variation ratio of current iL3 through inductor L3 is ξ3 =
ΔiL 3 / 2 k(1 − k )TV2 k(1 − k )2 R = = 4 L3 I O 8 I L3 fL3
(6.60)
The ripple voltage of output voltage vO is ΔvO =
ΔQ IO kT k VO = = C32 C32 fC32 R
Therefore, the variation ratio of output voltage vO is ε=
6.4.4
ΔvO / 2 k = 2 RfC32 VO
(6.61)
N/O Higher Stage Double Boost Circuit
The N/O higher stage double boost circuit is derived from the corresponding circuit of the main series by adding DEC in each stage circuit. For nth stage additional circuit, the final output voltage is VO = [(
2 n ) − 1]Vin 1− k
The voltage transfer gain is G=
VO 2 n =( ) −1 Vin 1− k
(6.62)
Analogously, the variation ratio of current iLi through inductor Li (i = 1, 2, 3, …n) is ξi =
© 2006 by Taylor & Francis Group, LLC
ΔiLi / 2 k(1 − k )2( n−i +1) R = I Li fLi 2 ∗ 2 2n
(6.63)
Negative Output Cascade Boost Converters
371
and the variation ratio of output voltage vO is ε=
6.5
ΔvO / 2 k = VO 2 RfCn2
(6.64)
Triple Series
All circuits of N/O cascade boost converters — triple series — are derived from the corresponding circuits of the main series by adding DEC twice in each stage circuit. The first three stages of this series are shown in Figure 6.9 to Figure 6.11. For convenience they are called elementary triple boost (or additional) circuit, two-stage triple boost circuit, and three-stage triple boost circuit respectively, and numbered as n = 1, 2 and 3.
6.5.1
N/O Elementary Triple Boost Circuit
This N/O elementary triple boost circuit is derived from the elementary boost converter by adding DEC twice. Its circuit and switch-on and switchoff equivalent circuits are shown in Figure 6.9. The output voltage of the first stage boost circuit is VC1, VC1 = Vin/(1 – k). After the first DEC, the voltage (across capacitor C12) increases to VC12 = 2VC1 =
2 V 1 − k in
(6.65)
After the second DEC, the voltage (across capacitor C14) increases to 3 V 1 − k in
(6.66)
3 − 1]Vin 1− k
(6.67)
VC14 = VC12 + VC1 = The final output voltage VO is equal to VO = VC14 − Vin = [ The voltage transfer gain is G=
© 2006 by Taylor & Francis Group, LLC
VO 3 = −1 Vin 1 − k
(6.68)
372
Essential DC/DC Converters
C12
C1 D11 iIN
D12
C14 D13
D14
D1 +
S
C11
VIN –
R
C13
L1
iO – VO +
(a) Circuit diagram i IN iO + VIN
C 14
–
L1 C1
–
C 11
C 12
C 13
R
VO +
(b) Equivalent circuit during switching-on i IN
+
C 12
C1
C 14
–
VIN –
iO
C 11 C 13
L1
R
VO +
(c) Equivalent circuit during switching-off FIGURE 6.9 Elementary triple boost circuit.
6.5.2
N/O Two-Stage Triple Boost Circuit
The N/O two-stage triple boost circuit is derived from two-stage boost circuit by adding DEC twice in each stage circuit. Its circuit diagram and switch-on and switch-off equivalent circuits are shown in Figure 6.10. As described in the previous section the voltage across capacitor C14 is VC14 = (3 1 − k )Vin . Analogously, the voltage across capacitor C24 is. VC 24 = (
© 2006 by Taylor & Francis Group, LLC
3 2 ) Vin 1− k
(6.69)
Negative Output Cascade Boost Converters
C12
C1 D11 iIN +
C2
C14
D12
D1
373
D13
D14
C11
L2
C24
C22
D3
D21
D23
D22
C13
D24
C23
C21
S
R
VIN D2
–
iO – VO +
L1
(a) Circuit diagram iIN iO
+ C24
VIN –
R
L1
C1
C11
C12
C13
C14
L2
C2
C21
C22
C23
– VO +
(b) Switch on iIN
C1
C12
C14
+ VIN – L1
C11
C13
C2
C22
C24 iO
L2
C21
– VO
C23 R
+
(c) Switch off FIGURE 6.10 Two-stage triple boost circuit.
The final output voltage VO is equal to VO = VC 24 − Vin = [(
3 2 ) − 1]Vin 1− k
(6.70)
The voltage transfer gain is G=
© 2006 by Taylor & Francis Group, LLC
VO 3 2 =( ) −1 Vin 1− k
(6.71)
374
Essential DC/DC Converters
Analogously, ΔiL1 =
Vin kT L1
I L1 = (
ΔiL2 =
V1 kT L2
I L2 =
2 2 ) I 1− k O
2IO 1− k
Therefore, the variation ratio of current iL1 through inductor L1 is ξ1 =
ΔiL1 / 2 k(1 − k )2 TVin k(1 − k ) 4 R = = I L1 8L1 I O 16 fL1
(6.72)
and the variation ratio of current iL2 through inductor L2 is ξ2 =
ΔiL 2 / 2 k(1 − k )TV1 k(1 − k )2 R = = I L2 4L2 I O 8 fL2
(6.73)
The ripple voltage of output voltage vO is ΔvO =
ΔQ IO kT k VO = = C22 C22 fC22 R
Therefore, the variation ratio of output voltage vO is ε=
6.5.3
ΔvO / 2 k = 2 RfC22 VO
(6.74)
N/O Three-Stage Triple Boost Circuit
This N/O circuit is derived from the three-stage boost circuit by adding DEC twice in each stage circuit. Its circuit diagram and equivalent circuits during switch-on and switch-off are shown in Figure 6.11. As described in the previous section the voltage across capacitor C14 is 2 VC14 = (3 1 − k )Vin , and the voltage across capacitor C24 is VC 24 = (3 1 − k ) Vin. Analogously, the voltage across capacitor C34 is VC 34 = (
3 3 ) Vin 1− k
The final output voltage VO is equal to
© 2006 by Taylor & Francis Group, LLC
(6.75)
D11 iIN
D1
C2
C14
D12 D13
D14
C11
L2
D3
D21
D22
C13
C3
C24
C22 D23
D24
L3
C23
C21
C34
C32 D31
D5
C31
D34
D33
D32
R
S
+
iO – VO +
C33
D2
VIN
L1
–
D4
(a) Circuit diagram
iIN
iO
+ VIN –
C34 R
L1
C1
C11
C12
C13
C14
L2
C2
C21
C22
C23
C24
L3
C3
C31
C32
C33
– VO +
Negative Output Cascade Boost Converters
C12
C1
(b) Equivalent circuit during switch-on iIN
C12
C1
C14
C2
C22
C24
C3
C32
C34 iO
+ VIN – L1
C11
C13
L2
C21
C23
L3
C31
– VO
C33 R
+
(c) Equivalent circuit during switch-off
© 2006 by Taylor & Francis Group, LLC
375
FIGURE 6.11 Three-stage triple boost circuit.
376
Essential DC/DC Converters
VO = VC 34 − Vin = [(
3 3 ) − 1]Vin 1− k
(6.76)
The voltage transfer gain is G=
VO 3 3 =( ) −1 1− k Vin
(6.77)
Analogously, ΔiL1 =
Vin kT L1
I L1 =
32 I (1 − k ) 3 O
ΔiL2 =
V1 kT L2
I L2 =
8 I (1 − k )2 O
ΔiL3 =
V2 kT L3
I L3 =
2 I 1− k O
Therefore, the variation ratio of current iL1 through inductor L1 is ξ1 =
ΔiL1 / 2 k(1 − k ) 3 TVin k(1 − k )6 R = = I L1 64L1 I O 12 3 fL1
(6.78)
and the variation ratio of current iL2 through inductor L2 is ξ2 =
ΔiL 2 / 2 k(1 − k ) 2 TV1 k(1 − k ) 4 R = = I L2 16L2 I O 12 2 fL2
(6.79)
and the variation ratio of current iL3 through inductor L3 is ξ3 =
ΔiL 3 / 2 k(1 − k )TV2 k(1 − k )2 R = = 4 L3 I O 12 I L3 fL3
(6.80)
Usually ξ1, ξ2, and ξ3 are small, this means that this converter works in the continuous mode. The ripple voltage of output voltage vO is ΔvO =
© 2006 by Taylor & Francis Group, LLC
ΔQ IO kT k VO = = C32 C32 fC32 R
Negative Output Cascade Boost Converters
377
Therefore, the variation ratio of output voltage vO is ε=
6.5.4
ΔvO / 2 k = VO 2 RfC32
(6.81)
N/O Higher Stage Triple Boost Circuit
The N/O higher stage triple boost circuit is derived from the corresponding circuit of the main series by adding DEC twice in each stage circuit. For nth stage additional circuit, the voltage across capacitor Cn4 is VCn 4 = (
3 n ) Vin 1− k
The output voltage is VO = VCn 4 − Vin = [(
3 n ) − 1]Vin 1− k
(6.82)
The voltage transfer gain is G=
VO 3 n =( ) −1 1− k Vin
(6.83)
Analogously, the variation ratio of current iLi through inductor Li (i = 1, 2, 3, …n) is ξi =
ΔiLi / 2 k(1 − k )2( n−i +1) R = I Li 12 ( n−i +1) fLi
(6.84)
and the variation ratio of output voltage vO is ε=
6.6
ΔvO / 2 k = VO 2 RfCn2
(6.85)
Multiple Series
All circuits of N/O cascade boost converters — multiple series — are derived from the corresponding circuits of the main series by adding DEC multiple
© 2006 by Taylor & Francis Group, LLC
378
Essential DC/DC Converters
(j) times in each stage circuit. The first three stages of this series are shown in Figure 6.12 to Figure 6.14. For convenience they are called elementary multiple boost circuit, two-stage multiple boost circuit, and three-stage multiple boost circuit respectively, and numbered as n = 1, 2 and 3.
6.6.1
N/O Elementary Multiple Boost Circuit
This N/O elementary multiple boost circuit is derived from elementary boost converter by adding a DEC multiple (j) times. Its circuit and switch-on and switch-off equivalent circuits are shown in Figure 6.12. The output voltage of the first DEC (across capacitor C12j) increases to VC12 j =
j+1 V 1 − k in
(6.86)
The final output voltage VO is equal to VO = VC12 j − Vin = [
j+1 − 1]Vin 1− k
(6.87)
The voltage transfer gain is G=
6.6.2
VO j+1 = −1 Vin 1 − k
(6.88)
N/O Two-Stage Multiple Boost Circuit
The N/O two-stage multiple boost circuit is derived from the two-stage boost circuit by adding DEC multiple (j) times in each stage circuit. Its circuit diagram and switch-on and switch-off equivalent circuits are shown in Figure 6.13. As described in the previous section the voltage across capacitor C12j is VC12 j =
j+1 V 1 − k in
Analogously, the voltage across capacitor C22j is. VC 22 j = (
j+1 2 ) Vin 1− k
The final output voltage VO is equal to
© 2006 by Taylor & Francis Group, LLC
(6.89)
Negative Output Cascade Boost Converters
379
1
D 11 i IN
D 12
D 1(2j–1)
D 12j
D1 S
C 11
VIN –
C12j
C 12
C1
+
j
2....
R C 1(2j–1)
L1
iO – VO +
(a) Circuit diagram iIN iO +
C 12j
VIN L1 C1
–
C11
C 1(2j–i) R
C12
– VO +
(b) Equivalent circuit during switching–on iIN
+
C1
C12
C 12j
–
VIN –
iO
C11 C 1(2j–1)
L1
R
VO +
(c) Equivalent circuit during switching–off FIGURE 6.12 Elementary multiple boost circuit.
VO = VC 22 j − Vin = [(
j+1 2 ) − 1]Vin 1− k
(6.90)
The voltage transfer gain is G=
VO j+1 2 =( ) −1 Vin 1− k
The ripple voltage of output voltage vO is
© 2006 by Taylor & Francis Group, LLC
(6.91)
380
Essential DC/DC Converters
1
C1
iIN
+
2...
D1
j
C12
D11
D12
C12j D1(2j–1) D12j
C2 D3
L2
D21
j
2...
1
C22 D22
C22j
D2(2j–1) D22j
S C11
VIN
C1(2j–1)
–
C2(2j–1)
C21
L1
R
iO – VO +
D2
(a) Circuit diagram iIN iO C22j
+ VIN –
R
L1
C1
C11
C12 C1(2j–1)
C12j
L2
C2
C21
C22 C2(2j–1)
– VO +
(b) Switch on iIN
C1
C12
C12j
C2
C22
C22j iO
+ VIN – L1
C1(2j–1)
C11
L2
C21
C2(2j–1) R
– VO +
(c) Switch off FIGURE 6.13 Two-stage multiple boost circuit.
ΔvO =
ΔQ IO kT k VO = = C22 j C22 j fC22 j R
Therefore, the variation ratio of output voltage vO is ε=
© 2006 by Taylor & Francis Group, LLC
ΔvO / 2 k = VO 2 RfC22 j
(6.92)
Negative Output Cascade Boost Converters 6.6.3
381
N/O Three-Stage Multiple Boost Circuit
This N/O circuit is derived from the three-stage boost circuit by adding DEC multiple (j) times in each stage circuit. Its circuit diagram and equivalent circuits during switch-on and switch-off are shown in Figure 6.14. As described in the previous section the voltage across capacitor C12j is VC12 j = ( j + 1 1 − k )Vin , a n d t h e v o l t a g e a c r o s s c a p a c i t o r C 2 2 j i s 2 VC 22 j = ( j + 1 1 − k ) Vin . Analogously, the voltage across capacitor C32j is VC 32 j = (
j+1 3 ) Vin 1− k
(6.93)
The final output voltage VO is equal to
VO = VC 32 j − Vin = [(
j+1 3 ) − 1]Vin 1− k
(6.94)
The voltage transfer gain is
G=
VO j+1 3 =( ) −1 Vin 1− k
(6.95)
The ripple voltage of output voltage vO is
ΔvO =
ΔQ IO kT k VO = = C32 j C32 j fC32 j R
Therefore, the variation ratio of output voltage vO is
ε=
6.6.4
ΔvO / 2 k = VO 2 RfC32 j
(6.96)
N/O Higher Stage Multiple Boost Circuit
The N/O higher stage multiple boost circuit is derived from the corresponding circuit of the main series by adding DEC multiple (j) times in each stage circuit. For nth stage multiple boost circuit, the voltage across capacitor Cn2j is
© 2006 by Taylor & Francis Group, LLC
1
2...
j
2...
1
j
2...
1
j
382
C1 iIN
D1
C 12
D 11
D 12
C 12j D 1(2j–1) D 12j
C2 D3
L2
D 21
C 22 D 22
C 22j
D 2(2j–1) D 22j
C3 D5
L3
C 32
D 31
D 32
C 32j
D 3(2j–1) D 32j
S +
iO
C 11
– C 1(2j–1) C 2(2j–1)
C 21
L1
V IN
C 3(2j–1)
D2
R
VO +
C 31
– D4
(a) Circuit diagram i IN iO
+ C 32j
V IN –
R
L1
C1
C 11
C 12 C 1(2j–1)
C 12j
L2
C2
C 21
C 22 C 2(2j–1)
C 22j
L3
C3
C 31
C 32
C 3(2j–1)
C1
C 12
C 12j
C2
C 22
C 22j
C3
C 32
C 32j iO
+ V IN – L1
C 11
C 1(2j–1)
L2
C 21
C 2(2j–1)
L3
© 2006 by Taylor & Francis Group, LLC
– VO
C 3(2j–1) R
(c) Equivalent circuit during switch-off FIGURE 6.14 Three-stage multiple boost circuit.
C 31
+
+
Essential DC/DC Converters
(b) Equivalent circuit during switch-on
i IN
– VO
Negative Output Cascade Boost Converters
VCn 2 j = (
383
j+1 n ) Vin 1− k
The output voltage is VO = VCn 2 j − Vin = [(
j+1 n ) − 1]Vin 1− k
(6.97)
The voltage transfer gain is
G=
VO j+1 n =( ) −1 Vin 1− k
(6.98)
The variation ratio of output voltage vO is ε=
6.7
ΔvO / 2 k = 2 RfCn2 j VO
(6.99)
Summary of Negative Output Cascade Boost Converters
All the circuits of the N/O cascade boost converters as a family can be shown in Figure 6.15. From the analysis of the previous two sections we have the common formula to calculate the output voltage: 1 n ⎧ ⎪ [( 1 − k ) − 1]Vin ⎪ 1 n ) − 1]Vin ⎪[2 ∗ ( 1− k ⎪ 2 n ⎪ VO = ⎨ [( ) − 1]Vin ⎪ 1− k ⎪ [( 3 )n − 1]Vin ⎪ 1− k j+1 n ⎪ ⎪⎩ [( 1 − k ) − 1]Vin The voltage transfer gain is
© 2006 by Taylor & Francis Group, LLC
main _ series additional _ series double _ series triple _ series multiple( j ) _ series
(6.100)
384
Essential DC/DC Converters Main Series
Additional Series
Double Series
Triple Series
Multiple Series
5 Stage N/O Boost Circuit
5 Stage N/O Additional Circuit
5 Stage N/O Double Circuit
5 Stage N/O Triple Circuit
5 Stage N/O Multiple Circuit
4 Stage N/O Boost Circuit
4 Stage N/O Additional Circuit
4 Stage N/O Double Circuit
4 Stage N/O Triple Circuit
4 Stage N/O Multiple Circuit
3 Stage N/O Boost Circuit
3 Stage N/O Additional Circuit
3 Stage N/O Double Circuit
3 Stage N/O Triple Circuit
3 Stage N/O Multiple Circuit
2 Stage N/O Boost Circuit
2 Stage N/O Additional Circuit
2 Stage N/O Double Circuit
2 Stage N/O Triple Circuit
2 Stage N/O Multiple Circuit
Elementary N/O Triple Circuit
Elementary N/O Multiple Circuit
Elementary N/O Additional/Double Circuit
Elementary Negative Output Boost Converter
FIGURE 6.15 The family of negative output cascade boost converters.
1 n ⎧ ⎪ (1 − k ) − 1 ⎪ 1 n ) −1 ⎪2 ∗ ( − 1 k ⎪ V 2 n ⎪ G= O =⎨ ( ) −1 Vin ⎪ 1 − k ⎪ ( 3 )n − 1 ⎪ 1− k ⎪ j+1 n ⎪⎩ ( 1 − k ) − 1
main _ series additional _ series double _ series
(6.101)
triple _ series multiple( j ) _ series
In order to show the advantages of N/O cascade boost converters, we compare their voltage transfer gains to that of the buck converter, G=
© 2006 by Taylor & Francis Group, LLC
VO =k Vin
Negative Output Cascade Boost Converters
385
forward converter, G=
VO = kN Vin
G=
VO k = Vin 1 − k
G=
VO k N = Vin 1 − k
G=
VO 1 = Vin 1 − k
N is the transformer turn ratio
Cúk-converter,
fly-back converter, N is the transformer turn ratio
boost converter,
and negative output Luo-converters G=
VO n = Vin 1 − k
(6.102)
If we assume that the conduction duty k is 0.2, the output voltage transfer gains are listed in Table 6.1, if the conduction duty k is 0.5, the output voltage transfer gains are listed in Table 6.2, if the conduction duty k is 0.8, the output voltage transfer gains are listed in Table 6.3.
6.8 6.8.1
Simulation and Experimental Results Simulation Results of a Three-Stage Boost Circuit
To verify the design and calculation results, PSpice simulation package was applied to a three-stage boost circuit. Choosing Vin = 20 V, L1 = L2 = L3 = 10 mH, all C1 to C8 = 2 μF and R = 30 kΩ, and using k = 0.7 and f = 100 kHz. The voltage values V1, V2, and VO of a triple-lift circuit are 66 V, 194 V, and 659 V respectively, and inductor current waveforms iL1 (its average value IL1 = 618 mA), iL2, and iL3. The simulation results are shown in Figure 6.16. The voltage values are matched to the calculated results.
© 2006 by Taylor & Francis Group, LLC
386
Essential DC/DC Converters
TABLE 6.1 Voltage Transfer Gains of Converters in the Condition k = 0.2 Stage No. (n) Buck converter Forward converter Cúk-converter Fly-back converter Boost converter Negative output Luo-converters Negative output cascade boost converters — main series Negative output cascade boost converters — additional series Negative output cascade boost converters — double series Negative output cascade boost converters — triple series Negative output cascade boost converters — multiple series (j=3)
1
2
3
4
5
n
1.25 0.25
0.2 0.2N (N is the transformer turn ratio) 0.25 0.25N (N is the transformer turn ratio) 1.25 2.5 3.75 5 6.25 1.25n 0.563 0.953 1.441 2.052 1.25n-1
1.5
2.125
2.906
3.882
5.104
2∗1.25n-1
1.5
5.25
14.625
38.063
96.66
(2∗1.25)n-1
2.75
13.06
51.73
196.75
740.58
(3∗1.25)n-1
4
24
124
624
3124
(4∗1.25)n-1
TABLE 6.2 Voltage Transfer Gains of Converters in the Condition k = 0.5 Stage No. (n)
1
Buck converter Forward converter Cúk-converter Fly-back converter Boost converter Negative output Luo-converters Negative output cascade boost converters — main series Negative output cascade boost converters — additional series Negative output cascade boost converters — double series Negative output cascade boost converters — triple series Negative output cascade boost converters — multiple series (j=3)
0.5 0.5N (N is the transformer turn ratio) 1 N (N is the transformer turn ratio) 2 2 4 6 8 10 2n 1 3 7 15 31 2n-1
6.8.2
2
3
4
5
n
3
7
15
31
63 2∗2n-1
3
15
63
255
1023 (2∗2)n-1
5
35
215
1295
7775 (3∗2)n-1
7
63
511
4095
32767 (4∗2)n-1
Experimental Results of a Three-Stage Boost Circuit
A test rig was constructed to verify the design and calculation results, and compare with PSpice simulation results. The test conditions are still Vin = 20 V, L1 = L2 = L3 = 10 mH, all C1 to C8 = 2 μF and R = 30 kΩ, and using k = 0.7 and f = 100 kHz. The component of the switch is a MOSFET device IRF950
© 2006 by Taylor & Francis Group, LLC
Negative Output Cascade Boost Converters
387
TABLE 6.3 Voltage Transfer Gains of Converters in the Condition k = 0.8 Stage No. (n)
1
Buck converter Forward converter Cúk-converter Fly-back converter Boost converter Negative output Luo-converters Negative output cascade boost converters — main series Negative output cascade boost converters — additional series Negative output cascade boost converters — double series Negative output cascade boost converters — triple series Negative output cascade boost converters — multiple series (j=3)
2
5 4
3
4
5
n
0.8 0.8N (N is the transformer turn ratio) 4 4N (N is the transformer turn ratio) 5 10 15 20 25 5n 24 124 624 3124 5n-1 2∗5n-1
9
49
249
1249
6249
9
99
999
9999
99999
(2∗5)n-1
14
224
3374
50624
759374
(3∗5)n-1
19
399
7999
15999
32 × 105
(4∗5)n-1
1.0A (9.99m, 603m) 0.5A
0A
(9.99m, 205m) (9.99m, 55m) I(L1)
I(L2)
I(L3)
0V -250V
(9.99m, -46) (9.99m, -174)
-500V SEL>>
(9.99m, -639)
9.980ms V(C4:2)
9.985ms V(C2:2)
V(R:2)
9.990ms
9.995ms
10.000ms
Time
FIGURE 6.16 The simulation results of a three-stage boost circuit at condition k = 0.7 and f = 100 kHz.
with the rates 950 V/5 A/2 MHz. We measured the values of the output voltage and first inductor current in the following converters. After careful measurement, the current value of IL1 = 0.62 A (shown in channel 1 with 1 A/Div) and voltage value of VO = 660 V (shown in Channel 2 with 200 V/Div) are obtained. The experimental results (current and voltage values) in Figure 6.17 match the calculated and simulation results, which are IL1 = 0.618 A and VO = 659 V shown in Figure 6.16.
© 2006 by Taylor & Francis Group, LLC
388
Essential DC/DC Converters
FIGURE 6.17 The experimental results of a three-stage boost circuit at condition k = 0.7 and f = 100 kHz.
TABLE 6.4 Comparison of Simulation and Experimental Results of a Triple-Lift Circuit. Stage No. (n) Simulation results Experimental results
6.8.3
IL1 (A)
Iin (A)
0.618 0.62
0.927 0.93
Vin (V)
Pin (W)
VO (V)
PO (W)
η (%)
20 20
18.54 18.6
659 660
14.47 14.52
78 78
Efficiency Comparison of Simulation and Experimental Results
These circuits enhanced the voltage transfer gain successfully, and efficiently. Particularly, the efficiencies of the tested circuits is 78%, which is good for high output voltage equipment. To compare the simulation and experimental results, see Table 6.4. All results are well identified with each other.
6.8.4
Transient Process
Usually, there is high inrush current during the first power-on. Therefore, the voltage across capacitors is quickly changed to certain values. The transient process is very quick taking only a few milliseconds. It is difficult to demonstrate it in this section.
© 2006 by Taylor & Francis Group, LLC
Negative Output Cascade Boost Converters
389
Bibliography Liu, K.H. and Lee, F.C., Resonant switches — a unified approach to improved performances of switching converters, in Proceedings of International Telecommunications Energy Conference, New Orleans, LA, U.S., 1984, p. 344. Luo, F.L. and Ye, H., Fundamentals of negative output cascade boost converters, in Proceedings of IEEE-EPEMC’2003, Xi’an, China, 2003, p. 1896. Luo, F.L. and Ye, H., Development of negative output cascade boost converters, in Proceedings of IEEE-EPEMC’2003, Xi’an, China, 2003, p. 1888. Martinez, Z.R. and Ray, B., Bidirectional DC/DC power conversion using constantfrequency multi-resonant topology, in Proceedings of Applied Power Electronics Conf. APEC’94, Orlando, FL, U.S., 1994, p. 991. Masserant, B.J. and Stuart, T.A., A high frequency DC/DC converter for electric vehicles, in Proceedings of Power Electronics in Transportation, 1994, p. 123. Pong, M.H., Ho, W.C., and Poon, N.K., Soft switching converter with power limiting feature, IEE Proceedings on Electric Power Applications, 146, 95, 1999.
© 2006 by Taylor & Francis Group, LLC
7 Ultra-Lift Luo-Converter
Voltage-lift (VL) technique has been successfully applied to the design of power DC/DC converters. Good examples are the three-series Luo-converters. Using VL technique can produce high-voltage transfer gain. Super-lift (SL) technique has been given much attention since it yields high-voltage transfer gain. This chapter introduces the ultra-lift Luo-converter as a novel approach within the new ultra-lift (UL) technique, which produces even higher voltage transfer gain. Our analysis and calculations illustrate the advanced characteristics of this converter.
7.1
Introduction
Voltage-lift (VL) technique has been widely applied in electronic circuit design. Since the last century it has been successfully applied to the design of power DC/DC converters. Good examples are the three-series Luo-converters. Using VL technique, one can obtain the converter’s voltage transfer gain stage-by-stage in arithmetical series; this gain is higher than that of other classical converters, such as the Buck converter, the Boost converter, and the Buck-Boost converter. Assume the input voltage and current of a DC/DC converter are V1 and I1, the output voltage and current are V2 and I2, and the conduction duty cycle is k. In order to compare these converters’ transfer gains, we use the formulae below: Buck converter
G=
V2 =k V1
(7.1)
Boost converter
G=
V2 1 = V1 1 − k
(7.2)
Buck-Boost converter
G=
V2 k = V1 1 − k
(7.3)
391
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392
Essential DC/DC Converters
G=
Luo-Converters
V2 k h( n ) [ n + h( n)] = V1 1− k
(7.4)
Where n is the stage number and h(n) is the Hong Function. ⎧1 h( n) = ⎨ ⎪⎩0
n=0 n>0
(7.5)
n = 0 for the elementary circuit with the voltage transfer gain. G=
V2 k = V1 1 − k
(7.6)
Super-lift (SL) technique has been paid much more attention because it yields higher voltage transfer gain. Good examples are the super-lift Luoconverters. Using this technique, one can obtain the converter’s voltage transfer gain stage-by-stage in geometrical series. The gain calculation formulae are: V ⎛ j + 2 − k⎞ G= 2 =⎜ V1 ⎝ 1 − k ⎟⎠
n
(7.7)
where n is the stage number and j is the multiple-enhanced number. n = 1 and j = 0 for the elementary circuit, yielding G=
V2 2 − k = V1 1 − k
(7.8)
This chapter introduces the ultra-lift Luo-converter as a novel approach of new technology, ultra-lift (UL) technique, which produces even higher voltage transfer gain. Simulated results verify our analysis and calculations, and illustrate the advanced characteristics of this converter.
7.2
Operation of Ultra-lift Luo-Converter
The circuit diagram is shown in Figure 7.1 (a), which consists of one switch S, two inductors L1 and L2, two capacitors C1 and C2, three diodes, and the
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Ultra-Lift Luo-Converter i1
393
S
D3
+ V1 -
D1
L2
V3
VC1
L1
-
VC2
i L2
+ C i L1 1
i2
D2
R
+ C2
i C1
V2
i C2
+
(a) Circuit diagram i1 VC1
+ V1
L1
i L2
+ C i L1 1
-
i2
L2
V3
-
VC2 + C2
i C1
R i C2
V2 +
(b) Equivalent circuit during switch-on V3
L1
VC1
i L2
+ C i L1 1
i2
L2
+ C2
i C1
-
VC2
R
V2
i C2
+
(c) Equivalent circuit during switch-off (CCM)
V3
L1
VC1 + C1
i L1
i2
L2 i L2 i C1
VC2 + C2
R i C2
V2 +
(d) Equivalent circuit during switch-off (DCM)
FIGURE 7.1 Ultra-Lift Luo-Converter.
load R. Its switch-on equivalent circuit is shown in Figure 7.1 (b). Its switchoff equivalent circuit for the continuous conduction mode (CCM) is shown in Figure 7.1 (c), and its switch-off equivalent circuit for the discontinuous conduction mode (DCM) is shown in Figure 7.1 (d). It is a very simply structured converter in comparison to other converters. As usual, the input voltage and current of the ultra-lift Luo-converter are V1 and I1, the output voltage and current are V2 and I2, the conduction duty cycle is k and the switching frequency is f. Consequently, the repeating period T = 1/f, the switch-on period is kT, and the switch-off period is (1 – k)T. To concentrate the operation process, we assume that all components except load R are ideal ones. Therefore, no power losses are considered during power transformation, i.e., Pin = PO and V1 × I1 = V2 × I2.
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394 7.2.1
Essential DC/DC Converters Continuous Conduction Mode (CCM)
Referring to Figures 7.1 (b) and (c), we see that the current iL1 increases with the slope + V1/L1 during switch-on, and decreases with the slope – V3/L1 during switch-off. In the steady state, the current increment is equal to the decrement in a whole period T. This gives relation below:
kT
V1 V = ( 1 − k )T 3 L1 L1
Thus, VC 1 = V3 =
k V1 1− k
(7.9)
The current iL2 increases with the slope + (V1 – V3)/L2 during switch-on, and decreases with the slope – (V3 – V2)/L2 during switch-off. In the steady state, the current increment is equal to the decrement in a whole period T. We obtain the relationships below:
kT
V2 = VC 2 =
V1 + V3 V − V3 = ( 1 − k )T 2 L2 L2
2−k k 2−k k( 2 − k ) V3 = V1 = V1 1− k 1− k 1− k ( 1 − k )2
(7.10)
The voltage transfer gain is G=
V2 k 2 − k k( 2 − k ) = = V1 1 − k 1 − k ( 1 − k )2
(7.11)
This is much higher than the voltage transfer gains of the VL Luo-converter and SL Luo-converter in equations (7.6) and (7.8). In fact, the gain in (7.11) is the product of those in (7.6) and (7.8). Another advantage is the starting output voltage of zero. The curve of the voltage transfer gain M versus the conduction duty cycle k is shown in Figure 7.2. The relation between input and output average currents is
I2 =
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( 1 − k )2 I1 k( 2 − k )
(7.12)
Ultra-Lift Luo-Converter
395
G
10 8
5 3 1
k 0
0.5
1
FIGURE 7.2 The voltage transfer gain M versus conduction duty cycle k.
The relation between average currents IL2 and IL1 is I L2 = (1 − k )I L1
(7.13)
Other relations: k 1 )I 2 = I2 1− k 1− k
(7.14)
1 1 2 ) I2 I L2 = ( 1− k 1− k
(7.15)
I L2 = ( 1 +
I L1 =
The variation of inductor current iL1 is ΔiL1 = kT
V1 L1
(7.16)
and its variation ratio is
ξ1 =
ΔiL 1 / 2 k( 1 − k )2 TV1 k( 1 − k )2 TR ( 1 − k ) 4 TR = = = I L1 2 L1 I 2 2 L1 M 2( 2 − k ) fL1
(7.17)
The diode current iD1 is the same as the inductor current iL1 during the switching-off period. For the CCM operation, both currents do not descend to zero, i.e.,
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396
Essential DC/DC Converters ξ1 ≤ 1
The variation of inductor current iL2 is
ΔiL2 =
kTV1 ( 1 − k ) L2
(7.18)
and its variation ratio is
ξ2 =
ΔiL 2 / 2 kTV1 kTR ( 1 − k )2 TR = = = I L2 2 L2 I 2 2 L2 M 2( 2 − k ) fL2
(7.19)
The variation of capacitor voltage vC1 is
ΔvC 1 =
ΔQC 1 kTI L 2 kTI 2 = = C1 C1 ( 1 − k )C 1
(7.20)
and its variation ratio is
σ1 =
ΔvC 1 / 2 kTI 2 k( 2 − k ) = = VC 1 2( 1 − k )V3 C 1 2( 1 − k )2 fC 1 R
(7.21)
The variation of capacitor voltage vC2 is
ΔvC 2 =
ΔQC 2 kTI 2 = C2 C2
(7.22)
and its variation ratio is
ε = σ2 =
ΔvC 2 / 2 kTI 2 k = = VC 2 2V2 C 2 2 fC 2 R
(7.23)
From the analysis and calculations, we can see that all variations are very small. For example, if V1 = 10 V, L1 = L2 = 1 mH, C1 = C2 = 1 μF, R = 3000 Ω, f = 50 kHz, and the conduction duty cycle k varies from 0.1 to 0.9, the output voltage variation ratio ε is less than 0.003. The output voltage is very smooth DC voltage with nearly no ripple.
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Ultra-Lift Luo-Converter
397
iL1
(1-k)T t 0
kT
(1-k)mT
T
FIGURE 7.3 Discontinuous inductor current iL1.
7.2.2
Discontinuous Conduction Mode (DCM)
Referring to Figures 1 (b), (c) and (d), we see that the current iL1 increases with the slope + V1/L1 during switch-on, and decreases with the slope −V3/ L1 during switch-off. The inductor current iL1 decreases to zero before t = T, i.e. the current becomes zero before next time the switch is turned on. The current waveform is shown in Figure 7.3. The DCM operation condition is defined as ξ1 ≥ 1 or
ξ1 =
k( 1 − k )2 TR ( 1 − k ) 4 TR = ≥1 2 L1 M 2( 2 − k ) fL1
(7.24)
To obtain the boundary between the CCM and DCM operation conditions, we define the normalized impedance Zn: R fL1
(7.25)
k( 1 − k ) 2 ZN 2
(7.26)
Zn = The boundary equation is G= or
G k( 1 − k ) 2 = ZN 2
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398
Essential DC/DC Converters
G
CCM Region k = 0.9
100
k = 0.8
10
k = 0.67 k = 0.5 k = 0.33
1
k = 0.2 k = 0.1
0.1
DCM Region
1
10
100 1000 Zn
10k
100k
FIGURE 7.4 Boundary between CCM and DCM.
TABLE 7.1 Boundary between CCM and DCM k G G/Zn Zn
0.2 0.5625 0.064 8.8
0.33 1.25 2/27 16.9
0.5 3 1/16 48
0.67 8 1/27 216
0.8 24 0.016 1500
0.9 99 0.0045 22000
The corresponding ZN is ZN =
k( 2 − k ) k( 1 − k )2 2( 2 − k ) / = 2 ( 1 − k )2 (1 − k )4
(7.27)
The curve is shown in Figure 7.4 and Table 7.1. We define the filling factor m to describe the current. For DCM operation, 0<m≤1 In the steady state, the current increment is equal to the decrement in a whole period T. This gives the relation below:
kT
V V1 = ( 1 − k ) mT 3 L1 L1
Thus, VC 1 = V3 =
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k V1 (1 − k )m
(7.28)
Ultra-Lift Luo-Converter
399
Comparing equations (7.9) and (7.28), we find that the voltage V3 is higher during DCM operation since m < 1. Its expression is m=
1 2 L1G 2( 2 − k ) = = 2 ξ 1 k( 1 − k ) TR ( 1 − k ) 4 Z N
(7.29)
The current iL2 increases with the slope + (V1−V3)/L2 during switch-on, and decreases with the slope −(V3 – V2)/L2 during switch-off. In the steady state the current increment is equal to the decrement in a whole period T. We obtain the relationship below:
kT
V1 + V3 V − V3 = ( 1 − k )T 2 L2 L2
V2 = VC 2 =
k( 2 − k ) 2−k V1 V3 = 1− k m( 1 − k ) 2
(7.30)
The voltage transfer gain in DCM is
GDCM =
V2 k( 2 − k ) k( 1 − k ) 2 = = ZN 2 V1 m( 1 − k ) 2
(7.31)
This is higher that the voltage transfer gain during CCM operation since m < 1. We can see that the voltage transfer gain GDCM increases linearly with, and is proportional to, the normalized impedance ZN, and is shown in Figure 7.4.
7.3
Instantaneous Values
Instantaneous values of the voltage and current of each component is very important in describing the converter operation. By referring to Figure 7.1, we can obtain these values in CCM and DCM operations.
7.3.1
Continuous Conduction Mode (CCM)
Referring to Figures 7.1 (b) and (c), we obtain the instantaneous values of the voltage and current of each component in CCM operation below:
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400
Essential DC/DC Converters ⎧ V1 ⎪ I L 1−min + L t ⎪ 1 iL 1 ( t) = ⎨ V ⎪I − 3t ⎪⎩ L 1−max L1
0 ≤ t ≤ kT (7.32) kT ≤ t ≤ T
⎧ V1 − V3 t ⎪ I L 2 −min + L ⎪ 2 iL 2 ( t) = ⎨ V − V1 ⎪I − 2 t ⎪⎩ L 2 −max L2
0 ≤ t ≤ kT (7.33) kT ≤ t ≤ T
⎧ ⎛ V1 V1 − V3 ⎞ + t ⎪ I1− min + ⎜ i1 (t) = is = ⎨ L2 ⎟⎠ ⎝ L1 ⎪ 0 ⎩ ⎧ 0 ⎪ iD 1 ( t) = ⎨ V3 ⎪ I L 1−max − L t 1 ⎩
(7.34)
kT ≤ t ≤ T
0 ≤ t ≤ kT kT ≤ t ≤ T
⎧ ⎛ V1 − V3 ⎪− ⎜ I L 2− min + L2 iC1 (t) = ⎨ ⎝ ⎪ IC1 ⎩
⎞ t⎟ ⎠
0 ≤ t ≤ kT
(7.35)
(7.36)
kT ≤ t ≤ T
⎧− I iC 2 ( t) = ⎨ 2 ⎪⎩ I C 2
0 ≤ t ≤ kT kT ≤ t ≤ T
(7.37)
⎧V v L1 ( t) = ⎨ 1 ⎪⎩V3
0 ≤ t ≤ kT kT ≤ t ≤ T
(7.38)
⎧V − V3 v L2 ( t) = ⎨ 1 ⎩⎪V2 − V3 ⎧ 0 vs = ⎨ ⎩⎪V1 − V3
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0 ≤ t ≤ kT
0 ≤ t ≤ kT kT ≤ t ≤ T 0 ≤ t ≤ kT kT ≤ t ≤ T
(7.39)
(7.40)
Ultra-Lift Luo-Converter
401 ⎧V − V3 vD1 ( t) = ⎨ 1 ⎪⎩ 0
⎧ ⎪V3 − ⎪ vC 1 ( t) = ⎨ ⎪V + ⎪⎩ 3
I L2 t C1
0 ≤ t ≤ kT
IC 1 t C1
kT ≤ t ≤ T
⎧ I2 ⎪ V2 − C t ⎪ 2 vC 2 ( t) = ⎨ I ⎪V + C 2 t ⎪⎩ 2 C 2
7.3.2
0 ≤ t ≤ kT kT ≤ t ≤ T
(7.41)
(7.42)
0 ≤ t ≤ kT (7.43) kT ≤ t ≤ T
Discontinuous Conduction Mode (DCM)
Referring to Figures 7.1 (b), (c) and (d), we obtain the instantaneous values of the voltage and current of each component in DCM operation. Since inductor current iL1 is discontinuous, some parameters have three states with T′ = kT + (1 – k)mT < T. ⎧ V1 t ⎪ L 1 ⎪ ⎪ V iL1 (t) = ⎨ I L1− max − 3 t L1 ⎪ ⎪ 0 ⎪ ⎩ ⎧ V1 − V3 t ⎪ I L 2 −min + L ⎪ 2 iL 2 ( t) = ⎨ V − V1 ⎪I − 2 t ⎪⎩ L 2 −max L2
0 ≤ t ≤ kT kT ≤ t ≤ T ′ T ' ≤ t ≤ kT
0 ≤ t ≤ kT (7.45) kT ≤ t ≤ T
⎧ ⎛ V1 V1 − V3 ⎞ + t ⎪ I1− min + ⎜ i1 (t) = is = ⎨ L2 ⎟⎠ ⎝ L1 ⎪ 0 ⎩
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(7.44)
0 ≤ t ≤ kT kT ≤ t ≤ T
(7.46)
402
Essential DC/DC Converters ⎧ 0 ⎪ V ⎪ iD1 (t) = ⎨ I L1− max − 3 t L1 ⎪ ⎪⎩ 0
0 ≤ t ≤ kT kT ≤ t ≤ T ′ T ' ≤ t ≤ kT
⎧ ⎛ V1 − V3 ⎪− ⎜ I L 2− min + L2 iC1 (t) = ⎨ ⎝ ⎪ IC1 ⎩
⎞ t⎟ ⎠
0 ≤ t ≤ kT
(7.48)
kT ≤ t ≤ T
⎧− I iC 2 ( t) = ⎨ 2 ⎩⎪ I C 2
0 ≤ t ≤ kT kT ≤ t ≤ T
(7.49)
⎧V1 ⎪ vL1 (t) = ⎨V3 ⎪0 ⎩
0 ≤ t ≤ kT kT ≤ t ≤ T ′ T ' ≤ t ≤ kT
(7.50)
⎧V − V3 v L2 ( t) = ⎨ 1 ⎩⎪V2 − V3
0 ≤ t ≤ kT kT ≤ t ≤ T
(7.51)
⎧ 0 ⎪ vs (t) = ⎨V1 − V3 ⎪ V 1 ⎩
0 ≤ t ≤ kT kT ≤ t ≤ T ′ T ' ≤ t ≤ kT
(7.52)
⎧V1 − V3 ⎪ vD1 (t) = ⎨ 0 ⎪ −V 3 ⎩
0 ≤ t ≤ kT kT ≤ t ≤ T ′ T ' ≤ t ≤ kT
(7.53)
⎧ ⎪V3 − ⎪ vC 1 ( t) = ⎨ ⎪V + ⎪⎩ 3
I L2 t C1
0 ≤ t ≤ kT
IC 1 t C1
kT ≤ t ≤ T
⎧ I2 ⎪ V2 − C t ⎪ 2 vC 2 ( t) = ⎨ I ⎪V + C 2 t ⎪⎩ 2 C 2
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(7.47)
(7.54)
0 ≤ t ≤ kT (7.55 kT ≤ t ≤ T
Ultra-Lift Luo-Converter
403
TABLE 7.2 Comparison of Various Converters Gains k Buck Boost Buck-Boost Luo-Converter Super-Lift Luo-Converter Ultra-Lift Luo-Converter
7.4
0.2 0.2 1.25 0.25 0.25 2.25 0.56
0.33 0.33 1.5 0.5 0.5 2.5 1.25
0.5 0.5 2 1 1 3 3
0.67 0.67 3 2 2 4 8
0.8 0.8 5 4 4 6 24
0.9 0.9 10 9 9 11 99
Comparison of the Gain to Other Converters’ Gains
The ultra-lift Luo-converter has been successfully developed using the novel approach of the new technology, ultra-lift (UL). Table 7.2 lists the voltage transfer gains of various converters at k = 0.2, 0.33, 0.5, 0.67, 0.8 and 0.9. The outstanding characteristics of the ultra-lift Luo-converter are very well presented. From the comparison we can obviously see that the ultra-lift Luoconverter has very high voltage transfer gain: G(k)|k = 0.5 = 3, G(k)|k = 0.667 = 8, G(k)|k = 0.8 = 24, G(k)|k = 0.9 = 99.
7.5
Simulation Results
To verify the advantages of the ultra-lift Luo-converter, we apply the PSpice simulation method. We choose the parameter’s values: V1 = 10 V, L1 = L2 = 1 mH, C1 = C2 = 1 μF, R = 3 kΩ, f = 50 kHz and conduction duty cycle k = 0.6 and 0.66. The corresponding output voltage V2 = 52.5 V and 78 V. The first waveform is the inductor’s current iL1, which flows through the inductor L1. The second and third waveforms are the voltage V3 and output voltage V2. These simulation results are identical to the calculation results. These results are shown in Figures 7.5 and 7.6 respectively.
7.6
Experimental Results
To verify the advantages and design of the ultra-lift Luo-converter and compare them with the simulation results, we construct a test rig with these components: V1 = 10 V, L1 = L2 = 1 mH, C1 = C2 = 1 μF, R = 3 kΩ, f = 50
© 2006 by Taylor & Francis Group, LLC
404
Essential DC/DC Converters
0.0A iL1 -0.5A
-1.0A
-I(L1)
0V v3 -50V SEL> -100V
v2 0s 1ms 2ms V(L2:1) V(R:2)
3ms
4ms
5ms
6ms
7ms
8ms
9ms 10ms
Time
FIGURE 7.5 Simulation results for k = 0.6.
0
iL1 v3
-40
-80
0s
1ms
2ms
V( L 2 : 1 ) V ( R: 2 )
3ms -I(L1)
4ms
5ms
6ms
7ms
8ms
9ms
v2
10ms
Time
FIGURE 7.6 Simulation results for k = 0.66.
kHz and conduction duty cycle k = 0.6 and 0.66. The corresponding output voltage V2 = 52 V and 78 V. The first waveform is the inductor’s current iL1, which flows through the inductor L1. The second waveform is the output voltage V2. The experimental results are shown in Figures 7.7 and 7.8 respectively. The test results are identical to those of the simulation results shown in Figures 7.5 and 7.6, and verify both the calculation results and our design.
© 2006 by Taylor & Francis Group, LLC
Ultra-Lift Luo-Converter
405
iL1
v2 FIGURE 7.7 Experimental results for k = 0.6.
iL1
v2 FIGURE 7.8 Experimental results for k = 0.66.
7.7
Summary
The Ultra-lift Luo-converter has been successfully developed using the novel approach of a new technology, the Ultra-lift (UL) technique, which produces especially high voltage transfer gain. It is much higher than that of Voltagelift Luo-converters and Super-lift Luo-converters. This chapter introduces the operation and characteristics of this converter in detail. This converter will be applied in industrial applications entailing high output voltages.
© 2006 by Taylor & Francis Group, LLC
406
Essential DC/DC Converters
Bibliography Jozwik J. J., and Kazimerczuk M. K., Dual Sepic PWM Switching-Mode DC/DC Power Converter, IEEE Transactions on Industrial Electronics, 36, 1, 1989, pp. 6470. Luo F. L., Negative Output Luo-Converters: Voltage Lift Technique, in IEE-EPA Proceedings, 146, 2, 1999, pp. 208-224. Luo F. L., Positive Output Luo-Converters: Voltage Lift Technique, in IEE-EPA Proceedings, 146, 4, 1999, pp. 415-432. Luo F. L., Double Output Luo-Converters: Advanced Voltage Lift Technique, in IEEEPA Proceedings, 147, 6, 2000, pp. 469-485. Luo F. L. and Ye H., Chapter 17, “DC/DC Conversion Techniques and Nine Series Luo-Converters,” in Power Electronics Handbook, Academic Press, San Diego, 2001, pp. 335-406. Luo F. L. and Ye H., Positive Output Super-Lift Luo-Converters, in Proceedings of IEEE International Conference PESC’2002, Cairns, Australia, 2002, pp. 425-430. Luo F. L. and Ye H., Advanced DC/DC Converters, CRC Press, LLC, Boca Raton, 2003. Luo F. L. and Ye H., Negative Output Super-Lift Luo-Converters, in Proceedings of IEEE International Conference PESC’2003, Acapulco, Mexico, 2003, pp. 1361-1366. Luo F. L. and Ye H., Positive Output Super-Lift Converters, in IEEE-Transactions on PEL, 18, 1, 2003, pp. 105-113. Luo F. L. and Ye H., Negative Output Super-Lift Converters, in IEEE-Transactions on PEL, 18, 5, 2003, pp. 1113-1121. Luo F. L. and Ye H., Ultra-Lift Luo-Converter, in IEE-EPA Proceedings, 152, 1, 2005, pp. 27-32. Massey R. P. and Snyder E. C., High Voltage Single-Ended DC-DC Converter, in IEEE PESC, 1977 Record, pp. 156-159. Mohan N., Undeland T. M., and Robbins W. P., Power Electronics: Converters, Applications and Design, 3rd Ed., John Wiley & Sons, New York, 2003.
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