240 99 8MB
English Pages 1326 Year 1997
Fundamentals of Power Electronics Instructor's slides
Fundamentals of Power Electronics R. W. Erickson Accompanying material for instructors The materials below are intended to be used by instructors of power electronics classes who have adopted Fundamentals of Power Electronics as a text. These instructors may download and use the files for educational purposes free of charge. Students and others who have purchased the text may also use the slides as an educational supplement to the text. Other uses of these materials is prohibited. All slides copyright R. W. Erickson 1997. The slides for each chapter are contained in a .pdf file. These files can be read using the Adobe Acrobat viewer, available free from the Adobe Acrobat web site. Slides and overhead transpariencies covering the material of the entire book can be produced using these files. Back Introduction ●
Chapter 1. Introduction 98kB
Part 1: Converters in Equilibrium ● ● ● ● ●
Chapter 2. Principles of steadystate converter analysis 126kB Chapter 3. Steadystate equivalent circuit modeling, losses, and efficiency 98kB Chapter 4. Switch realization 201kB Chapter 5. The discontinuous conduction mode 96kB Chapter 6. Converter circuits 283kB
Part 2: Converter Dynamics and Control ● ● ● ● ●
Chapter 7. Ac equivalent circuit modeling 422kB Chapter 8. Converter transfer functions Chapter 9. Controller design 365kB Chapter 10. Ac and dc equivalent circuit modeling of the discontinuous conduction mode 218kB Chapter 11. The current programmed mode 236kB
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Fundamentals of Power Electronics Instructor's slides
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Chapter 12. Basic magnetics theory 196kB Chapter 13. Filter inductor design 67kB Chapter 14. Transformer design 175kB
Part 4: Modern Rectifiers and Power System Harmonics ● ● ● ●
Chapter 15. Power and harmonics in nonsinusoidal systems 91kB Chapter 16. Linecommutated rectifiers 130kB Chapter 17. The ideal rectifier 235kB Chapter 18. Low harmonic rectifier modeling and control
Part 5: Resonant Converters ● ●
Chapter 19. Resonant conversion 325kB Chapter 20. Quasiresonant converters 177kB
Appendices ● ● ●
Appendix 1. RMS values of commonlyobserved converter waveforms 26 kB Appendix 2. Magnetics design tables 26kB Appendix 3. Averaged switch modeling of a CCM SEPIC 41kB
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Revision to Fundamentals of Power Electronics
Fundamentals of Power Electronics First Edition R. W. Erickson Power Electronics Group, University of Colorado at Boulder About the second edition
A new textbook on power electronics converters. This book is intended for use in introductory power electronics courses at the senior and firstyear graduate level. It is also intended as a source for professionals working in power electronics, power conversion, and analog electronics. It emphasizes the fundamental concepts of power electronics, including averaged modeling of PWM converters and fundamentals of converter circuits and electronics, control systems, magnetics, lowharmonic rectifiers, and resonant converters. Publisher and vitals New York: Chapman and Hall, May 1997. Hardback ISBN 0412085410 TK7881.15.E75 1997 7"x10", 791 pages, 929 line illustrations. Note: Chapman and Hall has recently been acquired by Kluwer Academic Publishers Note to instructors: how to obtain a copy
More information regarding contents of book ● ● ●
Complete Table of Contents Abridged Table of Contents: Chapter titles only Preface
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Revision to Fundamentals of Power Electronics ●
Index 101kB searchable with Adobe Acrobat Errata, first printing
Supplementary material for instructors ● ●
Slides Solutions to selected problems
Other supplementary material Proximity effect: computer functions 70kB Ferrite toroid data: Excel 5 spreadsheet Derivation of Gg0, Eqs. (11.84) and (11.85)
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CoPEC
CoPEC Colorado Power Electronics Center University of Colorado, Boulder
About CoPEC Research Publications Students Faculty Courses Textbook: Fundamentals of Power Electronics Power Electronics in the CU Boulder Electrical and Computer Engineering Department Links to Other Power Electronics Sites Updated May 21, 2001.
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Revision to Fundamentals of Power Electronics
Fundamentals of Power Electronics Second Edition Authors: R. W. Erickson and D. Maksimovic University of Colorado, Boulder Publisher: Kluwer Academic Publishers 912 pages ISBN 0792372700 ● ●
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First edition web site To order directly from the publisher Note to instructors: how to obtain desk copies Errata, second edition, first printing Errata, second edition, second printing New Certificate Program in Power Electronics PSPICE circuit files and library Courses at the University of Colorado that use the second edition ECEN 5797 Power ❍
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Electronics 1 ECEN 5807 Power Electronics 2 ECEN 5817 Power Electronics 3
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New material on converter simulation using averaged switch models Major revision of material on current mode control, including tables of transfer functions of basic converters Major revision of material on averaged switch modeling New material covering input filter design and Middlebrook's extra element theorem Improved explanations of the proximity effect and MMF diagrams New section on design of multiplewinding magnetics using the Kg method, including new examples New material on soft switching, including active clamp snubbers, the ZVT full bridge converter, and ARCP Major revision of material on lowharmonic rectifiers, to improve flow and readability. New material on critical conduction mode control Major revision and simplification of the chapter on ac modeling of the discontinuous conduction mode Revised problems, and a solutions manual
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Revision to Fundamentals of Power Electronics
Detailed description of revisions ● ● ●
Contents Preface to the Second Edition Chapter 1 Introduction
Part 1. Converters in Equilibrium There are no substantial changes to the chapters of Part 1. ● ● ● ● ●
Chapter 2 Principles of SteadyState Converter Analysis Chapter 3 SteadyState Equivalent Circuit Modeling, Losses, and Efficiency Chapter 4 Switch Realization Chapter 5 The Discontinuous Conduction Mode Chapter 6 Converter Circuits
Part 2. Converter Dynamics and Control ●
Chapter 7 AC Equivalent Circuit Modeling
Chapter 7 has been revised to improve the logical flow, including incorporation of the First Edition Appendix 3 into the chapter. The treatment of circuit averaging and averaged switch modeling (Section 7.4) has undergone major revision. Other changes include Fig. 7.4 and the related text, and Sections 7.2.2, 7.2.7. ●
Chapter 8 Converter Transfer Functions
Major revisions to Chapter 8 include a new introduction, a new input filter example in Section 8.1.8, and substantial changes to the buckboost converter example of Section 8.2.1 and the material of Sections 8.3 and 8.4. ●
Chapter 9 Controller Design
Only minor changes to Chapter 9 were made. ●
Chapter 10 Input Filter Design
This is an entirely new chapter that treats how input filters modify the transfer functions of a dcdc http://ecewww.colorado.edu/~pwrelect/book/SecEd.html (3 of 6) [25/04/2002 16:42:05]
Revision to Fundamentals of Power Electronics
converter, and how to design an input filter that is adequately damped. The approach is based on Middlebrook's Extra Element Theorem (EET) of Appendix C, although it is possible to teach this chapter without use of the EET. ●
Chapter 11 AC and DC Equivalent Circuit Modeling of the Discontinuous Conduction Mode
This chapter has been entirely revised and simplified. ●
Chapter 12 Current Programmed Control
Treatment of the "more accurate model" in Section 12.3 has undergone a major revision. The explanation is more straightforward, and results are summarized for the basic buck, boost, and buckboost converters. The results of simulation are used to illustrate how current programming changes the converter transfer function. The treatment of discontinuous conduction mode in Section 12.4 has been shortened.
Part 3. Magnetics ●
Chapter 13 Basic Magnetics Theory
The material on the skin and proximity effects has undergone a major revision, to better introduce the concepts of the proximity effect and MMF diagrams. The summary of operation of different magnetic devices has been moved from the filter inductor design chapter into this chapter. ●
Chapter 14 Inductor Design
A new section on design of multiplewinding inductors using the Kg method has been added, including two new examples. The summary of different magnetic devices has been moved to the previous chapter, and the material on winding area optimization (previously in the transformer design chapter) has been moved into this chapter. ●
Chapter 15 Transformer Design
Notation regarding maximum, peak, and saturation flux density has been made more clear. The section on winding area optimization has been moved to the previous chapter.
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Chapter 16 Power and Harmonics in Nonsinusoidal Systems
Information on harmonic standards has been updated. ●
Chapter 17 LineCommutated Rectifiers
There is little change to this chapter. ●
Chapter 18 PulseWidth Modulated Rectifiers
Chapter 18 is a consolidation of Chapters 17 and 18 of the First Edition. The material has been completely reorganized, to improve its flow. A new section 18.2.2 has been added. Section 18.3.3 has been expanded, to better cover critical conduction mode control. The material on threephase rectifier topologies has been streamlined.
Part 5. Resonant Converters ●
Chapter 19 Resonant Conversion
The order of the sections has been changed, to improve readability. Section 19.4 has been modified, to include better explanation of resonant inverter/electronic ballast design, and two examples have been added. The material on the ZVT converter has been moved to Chapter 20. ●
Chapter 20 Soft Switching
A new Section 20.1 compares the turnon and turnoff transitions of diode, MOSFET, and IGBT devices under the conditions of hard switching, zerocurrent switching, and zerovoltage switching. The material on quasiresonant converters is unchanged. Coverage of multiresonant and quasisquarewave switches has been exapanded, and includes plots of switch characteristics. A new Section 20.4 has been added, which covers softswitching techniques. Included in Section 20.4 is an expanded explanation of the ZVT fullbridge converter, new material on activeclamp snubbers, and a short treatment of the auxiliary resonant commutated pole. The material on ac modeling of ZCS quasiresonant converters has been dropped.
Appendices ●
Appendix A RMS Values of Commonly Observed Converter Waveforms
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This appendix is unchanged. ●
Appendix B Simulation of Converters
Appendix B is completely new. It covers SPICE simulation of converters using averaged switch models, including CCM, DCM, and currentprogrammed converters. This material complements the discussions of Chapters 7, 9, 11, 12, and 18. It has been placed in an appendix so that the chapter narratives are not interrupted by the details required to run a simulation program; nonetheless, the examples of this appendix are closely linked to the material covered in the chapters. ❍
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PSPICE circuit files and library used in Appendix B
Appendix C Middlebrook's Extra Element Theorem
This is a completely new appendix that explains the Extra Element Theorem and includes four tutorial examples. This material can be taught in conjunction with Chapter 10 and Section 19.4, if desired. ●
Appendix D Magnetics Design Tables
This appendix is unchanged. Update 12/8/00 rwe
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Fundamentals of Power Electronics
Fundamentals of Power Electronics Second Edition Up To instructors of Power Electronics courses: how to obtain an examination copy Evaluation copies are available on a 60day approval basis. Please submit requests in writing on department letterhead and include the following course information: ● ● ● ● ●
course name and number estimated enrollment semester date the text currently used your decision date
Please direct all requests to the Textbook Marketing Department at Kluwer Academic Publishers at the Norwell (Americas) or Dordrecht (all other countries) offices: ●
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In the Americas: 101 Philip Drive Norwell, MA 02061 USA Telephone: (781) 8716600 Fax: (781) 8716528 Attention: Ulysses Guasch Email: [email protected] All other countries: PO Box 989 3300 AZ Dordrecht The Netherlands Telephone: (0) 31 78 6392 392 Fax: (0) 31 78 6546 474 Email: [email protected]
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Fundamentals of Power Electronics
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Fundamentals of Power Electronics Table of Contents
Fundamentals of Power Electronics R. W. Erickson
Table of Contents Back 1. Introduction 1.1. Introduction to power processing 1.2. Several applications of power electronics 1.3. Elements of power electronics
Part I. Converters in Equilibrium 2. Principles of steady state converter analysis 2.1. Introduction 2.2. Inductor voltsecond balance, capacitor charge balance, and the smallripple approximation 2.3. Boost converter example 2.4. Cuk converter example 2.5. Estimating the output voltage ripple in converters containing twopole lowpass filters 2.6. Summary of key points 3. Steadystate equivalent circuit modeling, losses, and efficiency 3.1. The dc transformer model 3.2. Inclusion of inductor copper loss 3.3. Construction of equivalent circuit model 3.4. How to obtain the input port of the model 3.5. Example: Inclusion of semiconductor conduction losses in the boost converter model 3.6. Summary of key points 4. Switch realization 4.1. Switch applications 4.1.1. Single quadrant switches 4.1.2. Currentbidirectional twoquadrant switches 4.1.3. Voltagebidirectional twoquadrant switch http://ecewww.colorado.edu/~pwrelect/book/contents/TOC.html (1 of 11) [25/04/2002 16:42:25]
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4.1.4. Fourquadrant switches 4.1.5. Synchronous rectifiers
4.2. A brief survey of power semiconductor devices 4.2.1. Power diodes 4.2.2. MetalOxideSemiconductor FieldEffect Transistor (MOSFET) 4.2.3. Bipolar Junction Transistor (BJT) 4.2.4. Insulated Gate Bipolar Transistor (IGBT) 4.2.5. Thyristors (SCR, GTO, MCT)
4.3. Switching loss 4.3.1. Transistor switching with clamped inductive load 4.3.2. Diode recovered charge 4.3.3. Device capacitances, and leakage, package, and stray inductances 4.3.4. Efficiency vs. switching frequency
4.4. Summary of key points 5. The discontinuous conduction mode 5.1. Origin of the discontinuous conduction mode, and mode boundary 5.2. Analysis of the conversion ratio M(D,K) 5.3. Boost converter example 5.4. Summary of results and key points 6. Converter circuits 6.1. Circuit manipulations 6.1.1. Inversion of source and load 6.1.2. Cascade connection of converters 6.1.3. Rotation of threeterminal cell 6.1.4. Differential connection of the load
6.2. A short list of converters 6.3. Transformer isolation 6.3.1. Fullbridge and halfbridge isolated buck converters 6.3.2. Forward converter 6.3.3. Pushpull isolated buck converter 6.3.4. Flyback converter 6.3.5. Boostderived isolated converters 6.3.6. Isolated versions of the SEPIC and the Cuk converter
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6.4.1. Switch stress and utilization 6.4.2. Design using computer spreadsheet
6.5. Summary of key points
Part II. Converter Dynamics and Control 7. AC modeling 7.1. Introduction 7.2. The basic ac modeling approach 7.2.1. Averaging the inductor waveforms 7.2.2. Discussion of the averaging approximation 7.2.3. Averaging the capacitor waveforms 7.2.4. The average input current 7.2.5. Perturbation and linearization 7.2.6. Construction of the smallsignal equivalent circuit model 7.2.7. Results for several basic converters
7.3. Example: A nonideal flyback converter 7.4. Statespace averaging 7.4.1. The state equations of a network 7.4.2. The basic statespace averaged model 7.4.3. Discussion of the statespace averaging result 7.4.4. Example: Statespace averaging of a nonideal buckboost converter
7.5. Circuit averaging and averaged switch modeling 7.5.1. Obtaining a timeinvariant circuit 7.5.2. Circuit averaging 7.5.3. Perturbation and linearization 7.5.4. Averaged switch modeling
7.6. The canonical circuit model 7.6.1. Development of the canonical circuit model 7.6.2. Example: Manipulation of the buckboost converter model into canonical form 7.6.3. Canonical circuit parameter values for some common converters
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7.7. Modeling the pulsewidth modulator 7.8. Summary of key points 8. Converter transfer functions 8.1. Review of Bode plots 8.1.1. Single pole response 8.1.2. Single zero response 8.1.3. Right halfplane zero 8.1.4. Frequency inversion 8.1.5. Combinations 8.1.6. Double pole response: resonance 8.1.7. The lowQ approximation 8.1.8. Approximate roots of an arbitrarydegree polynomial
8.2. Analysis of converter transfer functions 8.2.1. Example: Transfer functions of the boost converter 8.2.2. Transfer functions of some basic dcdc converters 8.2.3. Physical origins of the RHP zero
8.3. Graphical construction of converter transfer functions 8.3.1. Series impedances: addition of asymptotes 8.3.2. Parallel impedances: inverse addition 8.3.3. Another example 8.3.4. Voltage divider transfer functions: division of asymptotes
8.4. Measurement of ac transfer functions and impedances 8.5. Summary of key points 9. Controller design 9.1. Introduction 9.2. Effect of negative feedback on the network transfer functions 9.2.1. Feedback reduces the transfer functions from disturbances to the output 9.2.2. Feedback causes the transfer function from the reference input to the output to be insensitive to variations in the gains in the forward path of the loop
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functions 9.4. Stability 9.4.1. The phase margin test 9.4.2. The relation between phase margin and closedloop damping factor 9.4.3. Transient response vs. damping factor
9.5. Regulator design 9.5.1. Lead (PD) compensator 9.5.2. Lag (PI) compensator 9.5.3. Combined (PID) compensator 9.5.4. Design example
9.6. Measurement of loop gains 9.6.1. Voltage injection 9.6.2. Current injection 9.6.3. Measurement of unstable systems
9.7. Summary of key points 10. Ac and dc equivalent circuit modeling of the discontinuous conduction mode 10.1. DCM averaged switch model 10.2. Smallsignal ac modeling of the DCM switch network 10.3. Generalized switch averaging 10.3.1. DCM buck converter example 10.3.2. Proof of generalized averaged switch modeling
10.4. Summary of key points 11. Current programmed control 11.1. Oscillation for D > 0.5 11.2. A simple firstorder model 11.2.1. Simple model via algebraic approach: buckboost example http://ecewww.colorado.edu/~pwrelect/book/contents/TOC.html (5 of 11) [25/04/2002 16:42:25]
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11.2.2. Averaged switch modeling
11.3. A more accurate model 11.3.1. Current programmed controller model 11.3.2. Example: analysis of CPM buck converter
11.4. Discontinuous conduction mode 11.5. Summary of key points
Part III. Magnetics 12. Basic magnetics theory 12.1. Review of basic magnetics 12.1.1. Basic relations 12.1.2. Magnetic circuits
12.2. Transformer modeling 12.2.1. The ideal transformer 12.2.2. The magnetizing inductance 12.2.3. Leakage inductances
12.3. Loss mechanisms in magnetic devices 12.3.1. Core loss 12.3.2. Lowfrequency copper loss
12.4. Eddy currents in winding conductors 12.4.1. The skin effect 12.4.2. The proximity effect 12.4.3. Magnetic fields in the vicinity of winding conductors: MMF diagrams 12.4.4. Power loss in a layer 12.4.5. Example: power loss in a transformer winding 12.4.6. PWM waveform harmonics
12.5. Summary of key points 13. Filter inductor design
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13.1. Several types of magnetic devices, their BH loops, and core vs. copper loss 13.2. Filter inductor design constraints 13.2.1. Maximum flux density 13.2.2. Inductance 13.2.3. Winding area 13.2.4. Winding resistance
13.3. The core geometrical constant Kg 13.4. A stepbystep procedure 13.5. Summary of key points 14. Transformer design 14.1. Winding area optimization 14.2. Transformer design: basic constraints 14.2.1. Core loss 14.2.2. Flux density 14.2.3. Copper loss 14.2.4. Total power loss vs. Bmax 14.2.5. Optimum flux density
14.3. A stepbystep transformer design procedure 14.4. Examples 14.4.1. Example 1: singleoutput isolated Cuk converter 14.4.2. Example 2: multipleoutput fullbridge buck converter
14.5. Ac inductor design 14.5.1. Outline of derivation 14.5.2. Stepbystep ac inductor design procedure
14.6. Summary
Part IV. Modern Rectifiers, and Power System Harmonics 15. Power and harmonics in nonsinusoidal systems http://ecewww.colorado.edu/~pwrelect/book/contents/TOC.html (7 of 11) [25/04/2002 16:42:25]
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15.1. Average power 15.2. Rootmeansquare (rms) value of a waveform 15.3. Power factor 15.3.1. Linear resistive load, nonsinusoidal voltage 15.3.2. Nonlinear dynamic load, sinusoidal voltage
15.4. Power phasors in sinusoidal systems 15.5. Harmonic currents in threephase systems 15.5.1. Harmonic currents in threephase fourwire networks 15.5.2. Harmonic currents in threephase threewire networks 15.5.3. Harmonic current flow in power factor correction capacitors
15.6. AC line current harmonic standards 15.6.1. US MIL STD 461B 15.6.2. International Electrotechnical Commission standard 555 15.6.3. IEEE/ANSI standard 519
16. Linecommutated rectifiers 16.1. The singlephase full wave rectifier 16.1.1. Continuous conduction mode 16.1.2. Discontinuous conduction mode 16.1.3. Behavior when C is large 16.1.4. Minimizing THD when C is small
16.2. The threephase bridge rectifier 16.2.1. Continuous conduction mode 16.2.2. Discontinuous conduction mode
16.3. Phase control 16.3.1. Inverter mode 16.3.2. Harmonics and power factor 16.3.3. Commutation
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16.4. Harmonic trap filters 16.5. Transformer connections 16.6. Summary 17. The ideal rectifier 17.1. Properties of the ideal rectifier 17.2. Realization of a nearideal rectifier 17.3. Singlephase converter systems incorporating ideal rectifiers 17.4. RMS values of rectifier waveforms 17.4.1. Boost rectifier example 17.4.2. Comparison of singlephase rectifier topologies
17.5. Ideal threephase rectifiers 17.5.1. Threephase rectifiers operating in CCM 17.5.2. Some other approaches to threephase rectification
17.6. Summary of key points 18. Low harmonic rectifier modeling and control 18.1. Modeling losses and efficiency in CCM highquality rectifiers 18.1.1. Expression for controller duty cycle d(t) 18.1.2. Expression for the dc load current 18.1.3. Solution for converter efficiency 18.1.4. Design example
18.2. Controller schemes 18.2.1. Average current control 18.2.2. Feedforward 18.2.3. Current programmed control 18.2.4. Hysteretic control 18.2.5. Nonlinear carrier control
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18.3.1. Modeling the outer lowbandwidth control system 18.3.2. Modeling the inner widebandwidth average current controller
18.4. Summary of key points
Part V. Resonant converters 19. Resonant Conversion 19.1. Sinusoidal analysis of resonant converters 19.1.1. Controlled switch network model 19.1.2. Modeling the rectifier and capacitive filter networks 19.1.3. Resonant tank network 19.1.4. Solution of converter voltage conversion ratio M = V/Vg
19.2. Examples 19.2.1. Series resonant dcdc converter example 19.2.2. Subharmonic modes of the series resonant converter 19.2.3. Parallel resonant dcdc converter example
19.3. Exact characteristics of the series and parallel resonant converters 19.3.1. Series resonant converter 19.3.2. Parallel resonant converter
19.4. Soft switching 19.4.1. Operation of the full bridge below resonance: zerocurrent switching 19.4.2. Operation of the full bridge above resonance: zerovoltage switching 19.4.3. The zero voltage transition converter
19.5. Loaddependent properties of resonant converters 19.5.1. Inverter output characteristics 19.5.2. Dependence of transistor current on load 19.5.3. Dependence of the ZVS/ZCS boundary on load resistance
19.6. Summary of key points 20. Quasiresonant converters
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20.1. The zerocurrentswitching quasiresonant switch cell 20.1.1. Waveforms of the halfwave ZCS quasiresonant switch cell 20.1.2. The average terminal waveforms 20.1.3. The fullwave ZCS quasiresonant switch cell
20.2. Resonant switch topologies 20.2.1. The zerovoltageswitching quasiresonant switch 20.2.2. The zerovoltageswitching multiresonant switch 20.2.3. Quasisquarewave resonant switches
20.3. Ac modeling of quasiresonant converters 20.4. Summary of key points
Appendices Appendix 1. RMS values of commonlyobserved converter waveforms A1.1. Some common waveforms A1.2. General piecewise waveform Appendix 2. Magnetics design tables A2.1. Pot core data A2.2. EE core data A2.3. EC core data A2.4. ETD core data A2.5. PQ core data A2.6. American wire gauge data Appendix 3. Averaged switch modeling of a CCM SEPIC
Index
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Fundamentals of Power Electronics Table of Contents Chapter titles only
Fundamentals of Power Electronics R. W. Erickson
Table of Contents Chapter titles only Back 1. Introduction
Part I. Converters in Equilibrium 2. Principles of steady state converter analysis 3. Steadystate equivalent circuit modeling, losses, and efficiency 4. Switch realization 5. The discontinuous conduction mode 6. Converter circuits
Part II. Converter Dynamics and Control 7. AC modeling 8. Converter transfer functions 9. Controller design 10. Ac and dc equivalent circuit modeling of the discontinuous conduction mode 11. Current programmed control
Part III. Magnetics 12. Basic magnetics theory 13. Filter inductor design 14. Transformer design
Part IV. Modern Rectifiers, and Power System Harmonics 15. Power and harmonics in nonsinusoidal systems 16. Linecommutated rectifiers
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Fundamentals of Power Electronics Table of Contents Chapter titles only
17. The ideal rectifier 18. Low harmonic rectifier modeling and control
Part V. Resonant Converters 19. Resonant conversion 20. Quasiresonant converters
Appendices Appendix 1. RMS values of commonlyobserved converter waveforms Appendix 2. Magnetics design tables Appendix 3. Averaged switch modeling of a CCM SEPIC
Index
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Fundamentals of Power Electronics Preface
Preface Fundamentals of Power Electronics R. W. Erickson Back In many university curricula, the power electronics field has evolved beyond the status of comprising one or two specialtopics courses. Often there are several courses dealing with the power electronics field, covering the topics of converters, motor drives, and power devices, with possibly additional advanced courses in these areas as well. There may also be more traditional powerarea courses in energy conversion, machines, and power systems. In the breadth vs. depth tradeoff, it no longer makes sense for one textbook to attempt to cover all of these courses; indeed, each course should ideally employ a dedicated textbook. This text is intended for use in introductory power electronics courses on converters, taught at the senior or firstyear graduate level. There is sufficient material for a one year course or, at a faster pace with some material omitted, for two quarters or one semester. The first class on converters has been called a way of enticing control and electronics students into the power area via the "back door". The power electronics field is quite broad, and includes fundamentals in the areas of ● ● ● ● ●
Converter circuits and electronics Control systems Magnetics Power applications Designoriented analysis
This wide variety of areas is one of the things which makes the field so interesting and appealing to newcomers. This breadth also makes teaching the field a challenging undertaking, because one cannot assume that all students enrolled in the class have solid prerequisite knowledge in so many areas. Indeed, incoming students may have individual backgrounds in the power, control, or electronics areas, but rarely in all three. Yet it is usually desired to offer the class to upperdivision undergraduate and entering graduate students. Hence, in teaching a class on converters (and in writing a textbook), there are two choices: 1. Avoid the problem of prerequisites, by either (a) assuming that the students have all of the prerequisites and discussing the material at a high level (suitable for an advanced graduate class), or (b) leaving out detailed discussions of the various contributing fields. 2. Attack the problem directly, by teaching or reviewing material from prerequisite areas as it is http://ecewww.colorado.edu/~pwrelect/book/contents/Preface.html (1 of 4) [25/04/2002 16:42:32]
Fundamentals of Power Electronics Preface
needed. This material can then be directly applied to power electronics examples. This approach is suitable for a course in the fourth or fifth year, in which fundamentals are stressed. Approach (2) is employed here. Thus, the book is not intended for survey courses, but rather, it treats fundamental concepts and design problems in sufficient depth that students can actually build converters. An attempt is made to deliver specific results. Completion of core circuits and electronics courses is the only prerequisite assumed; prior knowledge in the areas of magnetics, power, and control systems is helpful but not required. In the power electronics literature, much has been made of the incorporation of other disciplines such as circuits, electronic devices, control systems, magnetics, and power applications, into the power electronics field. Yet the field has evolved, and now is more than a mere collection of circuits and applications linked to the fundamentals of other disciplines. There is a set of fundamentals that are unique to the field of power electronics. It is important to identify these fundamentals, and to explicitly organize our curricula, academic conferences, and other affairs around these fundamentals. This book is organized around the fundamental principles, while the applications and circuits are introduced along the way as examples. A concerted effort is made to teach converter modeling. Fundamental topics covered include: ●
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Fundamentals of PWM converter analysis, including the principles of inductor voltsecond balance and capacitor charge balance, and the smallripple approximation (Chapter 2). Converter modeling, including the use of the dc transformer model, to predict efficiency and losses (Chapter 3). Realization of switching elements using semiconductor devices. One, two, and fourquadrant switches. A brief survey of power semiconductor devices (Chapter 4). An uptodate treatment of switching losses and their origins. Diode stored charge, device capacitances, and ringing waveforms (Chapter 4). Origin and steadystate analysis of the discontinuous conduction mode (Chapter 5). Converter topologies (Chapter 6). The use of averaging to model converter smallsignal ac behavior. Averaged switch modeling (Chapter 7). Converter smallsignal ac transfer functions, including the origins of resonances and right halfplane zeroes. Controltooutput and linetooutput transfer functions, and output impedance (Chapter 8). A basic discussion of converter control systems, including objectives, the system block diagram, and the effect of feedback on converter behavior (Chapter 9). Ac modeling of the discontinuous conduction mode. Quantitative behavior of DCM smallsignal transfer functions (Chapter 10). Currentprogrammed control. Oscillation for D > 0.5. Equivalent circuit modeling (Chapter 11). Basic magnetics, including inductor and transformer modeling, and loss mechanisms in highfrequency power magnetics (Chapter 12). An understanding of what determines the size of power inductors and transformers. Power inductor and transformer design issues (Chapters 13 and 14). Harmonics in power systems (Chapter 15). A modern viewpoint of rectifiers, including harmonics, power factor, and mitigation techniques in conventional rectifiers, and operation of sophisticated lowharmonic rectifiers (Chapters 1618). Analysis and modeling of lowharmonic rectifiers (Chapters 1718). Resonant inverters and dcdc converters: approximate analysis techniques, characteristics of basic converters, and
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Fundamentals of Power Electronics Preface
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loaddependent properties (Chapter 19). Zero voltage switching, zero current switching, and the zerovoltagetransition converter (Chapter 19). Resonant switch converters, including basic operation, efficiency and losses, and ac modeling (Chapter 20).
On teaching averaged converter modeling: I think that this is one of the important fundamentals of the field, and hence we should put serious effort into teaching it. Although we in the academic community may debate how to rigorously justify averaging, nonetheless it is easy to teach the students to average: Just average all of the waveforms over one switching period. In particular, for the continuous conduction mode, average the inductor voltages and capacitor currents over one switching period, ignoring the ripple. That's all that is required, and I have found that students quickly and easily learn to average waveforms. The results are completely general, they aren't limited to SPDT switches, and they can easily be used to refine the model by inclusion of losses, dynamics, and control variations. To model dynamics, it is also necessary to linearize the resulting equations. But derivation of smallsignal models is nothing new to the students they have already seen this in their core electronics classes, as well as in numerous math courses and perhaps also in energy conversion. It isn't necessary to teach fullblown statespace averaging, but I have included an optional (with asterisk) section on this for the graduate students. I personally prefer to initially skip Sections 7.4 and 7.5. After covering Chapters 8 and 9, I return to cover Sections 7.4 and 7.5 before teaching Chapters 10 and 11. Averaging aside, it is also important to teach modeling in a pedagogically sound way. The object is to describe the important properties of the converter, in a simple and clear way. The dc transformer represents the basic function of a dcdc converter, and so the modeling process should begin with a dc transformer having a turns ratio equal to the conversion ratio of the converter. For example, the model of the buckboost converter ought to contain a buck transformer cascaded by a boost transformer, or perhaps the two transformers combined into a single D : D' transformer. This firstorder model can later be refined if desired, by addition of loss elements, dynamic elements, etc. The designoriented analysis methods of R. D. Middlebrook have been well accepted by a significant portion of the power electronics community. While the objective of this text is the introduction of power electronics rather than designoriented analysis, the converter analyses and examples are nonetheless done in a designoriented way. Approximations are often encouraged, and several of the techniques of designoriented analysis are explicitly taught in parts of Chapters 8 and 9. We need to teach our students how to apply our academic theory to realworld, and hence complicated, problems. Designoriented analysis is the missing link. Chapter 8 contains a "review" of Bode diagrams, including resonant responses and right halfplane zeroes. Also included is material on designoriented analysis, in the context of converter transfer functions. The Bode diagram material is covered in more depth than in prerequisite classes. I have found that the material of Chapter 8 is especially popular with continuing education students who are practicing engineers. I recommend at least quickly covering this chapter in lecture. Those instructors who choose to skip some or all of Chapter 8 can assign it as reading, and hold students responsible for the material. In a similar manner, Chapter 9 contains a "review" of classical control systems, in the context of switching regulators. This chapter explicitly makes the connection between the smallsignal converter models http://ecewww.colorado.edu/~pwrelect/book/contents/Preface.html (3 of 4) [25/04/2002 16:42:32]
Fundamentals of Power Electronics Preface
derived in other chapters, and their intended application. Many power area students are unfamiliar with this material, and even controlarea students comment that they learned something from the designoriented approach. Parts III, IV, and V can be covered in any order. Part III includes a review of basic magnetics, a discussion of proximity loss, and an introduction to the issues governing design of magnetic devices. The inclusion of stepbystep design procedures may be somewhat controversial; however, these procedures explicitly illustrate the issues inherent in magnetics design. Student tendencies towards cookbook mentality are mitigated by the inclusion of homework problems that cannot be solved using the given stepbystep procedures. Part IV, entitled "Modern rectifiers," covers the issues of power system harmonics, generation of harmonics by conventional rectifiers, and lowharmonic rectifiers. Chapters 17 and 18 cover lowharmonic rectifiers in depth, including converter analysis and modeling, and rectifier control systems. Resonant converters are treated in Part V. There have been a tremendous number of papers written on resonant converters, most of which are very detailed and complicated. Indeed, the complexity of resonant converter behavior makes it challenging to teach this subject in depth. Two somewhat introductory chapters are included here. Stateplane analysis is omitted, and is left for an advanced graduate class. In Chapter 19, resonant inverters and dcdc converters are introduced and are analyzed via the sinusoidal approximation. Soft switching is described, in the context of both resonant converters and the zerovoltage transition converter. Some resonant network theorems are also presented, which yield insight into the design of resonant inverters with reduced circulating currents, with zerovoltage switching over a wide range of load currents, and with desired output characteristics. Resonant switch converters are introduced and modeled in Chapter 20. Most chapters include both short analysis problems, and longer analysis and/or design problems. References are given at the end of each chapter; these are not intended to be exhaustive bibliographies, but rather are a starting place for additional reading. This text has evolved from course notes developed over thirteen years of teaching power electronics at the University of Colorado, Boulder. These notes, in turn, were heavily influenced by my previous experience as a graduate student at the California Institute of Technology, under the direction of Profs. Slobodan Cuk and R. D. Middlebrook, to whom I am grateful. In addition, I appreciate the numerous helpful technical discussions and suggestions of my colleague at the University of Colorado, Prof. Dragan Maksimovic. I would also like to thank the following individuals for their suggestions: Prof. Arthur Witulski (University of Arizona, Tucson), Prof. Sigmund Singer (TelAviv University, Israel), Dr. Michael Madigan, and Carlos Oliveira. Robert W. Erickson Boulder, Colorado
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Fundamentals of Power Electronics R. W. Erickson
INDEX Air gap in coupled inductor, 502 in flyback transformer, 503 in inductor, 464466, 498, 505, 509 in transformer, 469 AL (mH/1000 turns), 509 American wire gauge (AWG) data, 755756 design examples, 527, 531 Amorphous alloys, 473 AmpereÕs law, 457458 Ampsecond balance (see Capacitor charge balance) Apparent power, 550 Artificial ramp circuit, 415 effect on CPM boost lowharmonic rectifier, 637639 effect on linetooutput transfer function of CCM buck, 437438 effect on smallsignal CCM models, 428438 effect on smallsignal DCM models, 438447 effect on stability of CPM controllers, 414418 Asymptotes (see Bode plots) Audiosusceptibility Gvg(s) (see Linetooutput transfer function) Average current control feedforward, 635636 in lowharmonic rectifier systems, 593598, 634636, 649, 650652 modeling of, 649652 Averaged switch modeling, 239245, 390403 of currentprogrammed CCM converters, 423428 of currentprogrammed DCM converters, 438447 in discontinuous conduction mode, 370390 equivalent circuit modeling of switching loss, 241245 examples nonideal buck converter, 241245 DCM buck converter, 393400 CCM SEPIC, 757762 generalization of, 390403 of ideal CCM switch networks, 242, 377, 757762 of ideal DCM switch networks, 377 of quasiresonant converters, 732737 Average power and Fourier series, 542543 modeled by power source element, 375379, 423428, 438447 in nonsinusoidal systems, 542555 predicted by averaged models, 57 power factor, 546550 sinusoidal phasor diagram, 550551 Averaging approximation, discussion of, 195196, 200202 averaged switch modeling, 239245 basic approach, 198209
1
Fundamentals of Power Electronics R. W. Erickson
capacitor charge balance, 24 circuit, 231245 to find dc component, 6, 16 flyback ac model, 209218 inductor voltsecond balance, 2223 introduction to, 193198 modeling efficiency and loss via, 57 to model rectifier output, 645647 to model 3¿ converters, 611614 of quasiresonant converters ac modeling, 732737 dc analysis, 712728 statespace, 218231 Battery charger, 9, 70 BH loop in an ac inductor, 499500 in a conventional transformer, 153, 500501 in a coupled inductor, 501502 in a filter inductor, 497499 in a flyback transformer, 502503 modeling of, 458460 Bidirectional dcdc converters, 70 Bipolar junction transistor (BJT) breakdown mechanisms in, 8687 construction and operation of, 8287 current crowding, 8586 Darlingtonconnected, 87 idealized switch characteristics, 6566 on resistance, 53, 82 quasisaturation, 8283, 86 storage time, 84 stored minority charge in, 8286 switching waveforms, 8386 Bode plots (see also Harmonic trap filters, sinusoidal approximation) asymptote analytical equations, 275276 CCM buckboost example, 289292 combinations, 272276 complex poles, 276282 frequency inversion, 271272 graphical construction of, 296309 addition, 296301 closedloop transfer functions, 329332 division, 307309 parallel combination, 301307 parallel resonance, 301303 series resonance, 298303 impedance graph paper, 307 nonminimum phase zero, 269271 reactance graph paper, 307 real pole, 263268 real zero, 268269 RHP zero, 269271 transfer functions of buck, boost, buckboost, 292293 Body diode (see MOSFET) 2
Fundamentals of Power Electronics R. W. Erickson
Boost converter (see also Bridge configuration, Pushpull isolated converters) active switch utilization in, 179, 608 averaged switch model, DCM, 380381 circuitaveraged model, 233239 currentprogrammed averaged switch model, CCM, 424425 averaged switch model, DCM, 443444 smallsignal ac model, CCM, 427428, 430431 smallsignal ac model, DCM, 445447 as inverted buck converter, 136137 as lowharmonic rectifier, 594597, 605609, 617, 627634 nonideal analysis of, 4351, 5357 quasiresonant ZCS, 722723 smallsignal ac model CCM, 208210, 251 DCM, 385390 steadystate analysis of, CCM, 2429 DCM, 121125 transfer functions, CCM, 292293 Bridge configuration (dcdc converters) boostderived full bridge, 171172 buckderived full bridge, 154157 buckderived half bridge, 157159 full bridge transformer design example, 528531 minimization of transformer copper loss in, 516517 Bridge configuration (inverters) single phase, 78, 142145, 148150 three phase, 70, 143148 Buckboost converter (see also Flyback converter) 3¿acdc rectifier, 615616, 619 averaged switch model, DCM, 370381 as cascaded buck and boost converters, 138141 currentprogrammed averaged switch model, DCM, 438444 more accurate model, CCM, 430432 simple model, CCM, 419423 smallsignal ac model, DCM, 445447 dc3¿ac inverter, 7172, 615616 DCM characteristics, 115, 127129, 381 as lowharmonic rectifier, 598599 manipulation of ac model into canonical form, 248251 nonideal, statespace averaged model of, 227232 noninverting version, 139, 148149 as rotated threeterminal cell, 141142 smallsignal ac model, CCM, 208210, 251 smallsignal ac model, DCM, 382388 transfer functions, CCM, 289293 transformer isolation in, 166171 Buck converter (see also Bridge configuration, Forward converter, Pushpull isolated converters), 6, 1523, 3435 active switch utilization in, 179 averaged switch model, 239245 currentprogrammed 3
Fundamentals of Power Electronics R. W. Erickson
averaged switch model, CCM, 423427 averaged switch model, DCM, 442447 smallsignal ac model, CCM, 421427, 431438 smallsignal ac model, DCM, 442447 equivalent circuit modeling of, smallsignal ac, CCM, 208210, 251 smallsignal ac, DCM, 385388, 393400 steadystate, CCM, 5153 steadystate, DCM, 380381 as high power factor rectifier single phase, 599 three phase, 614615 multiresonant realization, 729 quasisquarewave resonant realizations, 730731 quasiresonant realizations ac modeling of, 732737 zero current switching, 662663, 712722, 723724 zero voltage switching, 728 smallsignal ac model CCM, 208210, 251 DCM, 385390 steadystate analysis of, CCM, 1722, 23, 3435, 5153 DCM, 111121, 380381 switching loss in, 94101, 241245 employing synchronous rectifier, 7374 transfer functions, CCM, 292293 Buck2 converter, 149, 151 Buck 3¿ inverter (see Voltage source inverter) Canonical circuit model, 245251 via generalized switch averaging, 402403 manipulation into canonical form, 248251 parameters for buck, boost, buckboost, 251 physical development of, 245248 transfer functions predicted by, 247248, 292293 Capacitor ampsecond balance (see Capacitor charge balance) Capacitor charge balance boost converter example, 27 Cuk converter example, 3132 definition, 24 in discontinuous conduction mode, 115 nonideal boost converter examples, 45, 55 Capacitor voltage ripple boost converter example, 2829 buck converter example, 3435 in converters containing twopole filters, 3435 Cuk converter example, 3234 Cascade connection of converters, 138141 Characteristic value a (current programmed mode), 414, 417418, 435436 Charge balance (see Capacitor charge balance) Circuit averaging (see also Averaged switch modeling), 231245 averaging step, 235 boost converter example, 233238
4
Fundamentals of Power Electronics R. W. Erickson
linearization, 235238 obtaining a timeinvariant network, 234235 summary of, 231233 Commutation failure, 574 notching, 575 in 3¿ phase controlled rectifier, 573575 Compensators (see also Control system design) design example, 346354 lag, 343345 lead, 340340, 350351 PD, 340343, 350351 PI, 343345 PID, 345346, 352354 Complex power, 550551 Computer power supply, 89 Computer spreadsheet, design using, 180183 Conduction loss (see Copper loss, Semiconductor conduction loss) Conductivity modulation, 75, 79, 82, 87, 90 Control system design (see also Compensators, Negative feedback), 323368 compensation, 340346 construction of closedloop transfer functions, 326332 design example, 346354 for lowharmonic rectifiers approaches, 634652 modeling, 645652 phase margin test, 333334 vs. closedloop damping factor, 334338 stability, 332339 voltage regulator block diagram, 324325, 328, 347349 design specifications, 339340 Controltooutput transfer function as predicted by canonical model, 248 of CCM buck, boost, and buckboost converters, 292293 of current programmed converters, 422, 427428, 434437, 446 of DCM converters, 387390, 396399 of quasiresonant converters, 733, 736 Conversion ratio M (see also Switch conversion ratio m) of boost, 18, 26, 127, 381 of buck, 18, 120, 381 of buckboost, 18, 128, 381 of Cuk converter, 32, 381 of lossfree resistor networks, 376381 in lowharmonic rectifiers, 593595 modeling of, 4043 of quasiresonant converters, 711, 720723 of parallel resonant converter, 676678, 686689 of SEPIC, 151, 381 of series resonant converter, 671674, 679686 via sinusoidal approximation, 670 Copper loss
5
Fundamentals of Power Electronics R. W. Erickson
allocation of window area to minimize, 513517, 519 high frequency effects skin effect, 475476 proximity effect, 476490 inductor design to meet specified, 503509 low frequency, 474 modeling in converters, 4353 Core loss, 471474, 518 Coupled inductors in Cuk converter, 494495, 501 in multipleoutput buckderived converters, 501502, 511 Crossover frequency, 330334 Cuk converter 3¿acdc converter, 615616 active switch utilization of, 179 as cascaded boost and buck converters, 141 conversion ratio M(D), 32, 381 DCM averaged switch model of, 379381 as lowharmonic rectifier, 597599, 608 as rotated threeterminal cell, 141142 steadystate analysis of, 2934 transformer design example, 524528 with transformer isolation, 176177 Currentfed bridge, 148, 150 Current injection, 359360 Current programmed control, 408451 ac modeling of via averaged switch modeling, CCM, 423428 via averaged switch modeling, DCM, 438447 CCM more accurate model, 428438 CCM simple approximation, 418428 artificial ramp, 414418 controller circuit, 409, 415 controller smallsignal block diagram, 428432 in halfbridge buck converters, 159, 410 in low harmonic rectifiers, 636639 oscillation for D > 0.5, 411418 in pushpull buck converters, 166, 410 Current ripple (see inductor current ripple) Current sense circuit, isolated, 187188 Current source inverter (CSI), 146, 148 Cycloconverter, 1, 72 Damping factor z (see also Qfactor), 277 Dc conversion ratio (see Conversion ratio M) Dc link, 10 Dc transformer model in averaged switch models, 237244, 760762 in canonical model, 245247, 250251 in circuit averaged models, 237238 comparison with DCM model, 377 derivation of, 4043 equivalence with dependent sources, 41 manipulation of circuits containing, 4142, 4849
6
Fundamentals of Power Electronics R. W. Erickson
in a nonideal boost converter, 4849, 56 in a nonideal buck converter, 5253 in smallsignal ac CCM models, 208210 Decibel, 262 Deltawye transformer connection, 582583 Dependent power source (see Power source element) Derating factor, 180 Designoriented analysis, techniques of analytical expressions for asymptotes, 275276 approximate factorization, 285288 doing algebra on the graph, 296309 frequency inversion, 271272 graphical construction of Bode plots, 296309 of closedloop transfer functions, 329332 low Q approximation, 282284 philosophy of, 261, 306307 Differential connection of load polyphase inverter, 143148 singlephase inverter, 142143 Diode antiparallel, 67 characteristics of, 78 fast recovery, 77 forward voltage drop (see also Semiconductor conduction losses), 5357, 77 freewheeling, 67 parallel operation of, 7778 recovered charge Qr, 76, 97100, 692, 729 recovery mechanisms, 7677, 98100 Schottky, 74, 77, 101 soft recovery, 9899 snubbing of, 99 switching loss, 97100, 101103, 692 switching waveforms, 7577, 98100, 101102 zero current switching of, 101103, 690692, 696, 725726 zero voltage switching of, 692696, 725726, 729, 734 Discontinuous conduction mode (DCM) BH loop, effect on, 503504 boost converter example, 121127 buck converter example, 111121 buckboost converter example, 370381 in current programmed converters, 438447 equivalent circuit modeling of, 369381, 438444 in forward converter, 159 in linecommutated rectifiers, 564568, 569570 in lowharmonic rectifiers boost rectifier, single phase, 594597 singleswitch, threephase, 615619 mode boundary in boost rectifier, 594697 vs. K, 111115, 121122, 128 vs. load current and Re, 381 origin of, 111115 in parallel resonant converter, 687689 7
Fundamentals of Power Electronics R. W. Erickson
in PWM converters, 110134, 369407, 438447 in series resonant converter, 681683 smallsignal ac modeling of, 382403 Displacement factor, 548, 550551 Distortion factor (see also Total harmonic distortion), 548550 of singlephase rectifier, 548, 563566 Distributed power system, 9 Doing algebra on the graph (see Graphical construction of Bode plots) Duty ratio complement of, 16 definition of, 1516 EC core data, 754 Eddy currents in magnetic cores, 472 in winding conductors, 474477 EE core data, 753 Effective resistance Re in DCM averaged switch model, 374381 in lossfree resistor model, 374381 in resonant converter models with capacitive filter network, 666668 with inductive filter network, 674676 Emulated resistance Re, 590593 Efficiency, 2 averaged switch modeling, predicted by, 245 of boost converter as lowharmonic rectifier, 632634 nonideal dcdc, 4951, 56 calculation via averaged model, 4951, 56 vs. switching frequency, 103104 Equivalent circuit modeling by canonical circuit model, 245251 of CCM converters operating in steadystate, 4061 of converters having pulsating input currents, 5153 of current programmed switch networks CCM, 423428 DCM, 438447 smallsignal models, 421422, 423428, 445447 of flyback converter, CCM, 168, 216218 of ideal rectifiers, 590593, 608611 of ideal dcdc converters, 4042 of inductor copper loss, 4351 smallsignal models CCM, 207209, 230232 DCM, 382390 current programmed, 421422, 424428, 438447 of switching loss, 241245 of switch networks CCM, 239242 DCM, 370381 of systems containing ideal rectifiers, 602 Equilibrium (see Steady state) Equivalent series resistance (esr) of capacitor, 554555 ETD core data, 754 8
Fundamentals of Power Electronics R. W. Erickson
Evaluation and design of converters, 177183 Experimental techniques measurement of impedances, 312314 measurement of loop gains by current injection, 359360 by voltage injection, 357359 of an unstable system, 360361 measurement of smallsignal transfer functions, 309311 Factorization, approximate approximate roots of arbitrarydegree polynomial, 282288 graphical construction of Bode diagrams, 296309 lowQ approximation, 282284 FaradayÕs law, 456457 Feedback (see Control system design, Negative feedback) Ferrite applications of, 499, 525, 528 core loss, 472, 473474, 518 core tables, 751755 saturation flux density, 459, 473 Fill factor (see Ku) Filter inductor BH loop of, 497, 499 design of derivation of procedure, 503508 stepbystep procedure, 508509 Flux F, 456 Flux density B definition, 456 saturation value Bsat, 458459 Fluxlinkage balance (see Inductor voltsecond balance) Flyback converter (see also Buckboost converter) active switch utilization, 178179 derivation of, 166167 nonideal, ac modeling of, 209218 singleswitch rectifier, 3¿acdc DCM, 623 spreadsheet design example, 180183 steadystate analysis of, 166170 two transistor version, 185186 utilization of flyback transformer, 170171 Flyback transformer, 166167, 170173, 502503, 619 Forced commutation of SCRs, 90 Forward converter (see also Buck converter), 159164 active switch utilization, 179 spreadsheet design example, 180183 steadystate analysis of, 159164 transformer reset mechanisms, 162163 transformer utilization in, 164 two transistor version, 163164 Fourquadrant switches (see Switch) Freewheeling diode, 67 Frequency modulator, 732733 Gate turnoff thyristor (GTO), 92 Generalized switch averaging, 390403
9
Fundamentals of Power Electronics R. W. Erickson
Geometrical constant (see Kg, Kgfe) Graphical construction of Bode plots (see also Bode plots, Designoriented analysis) of converter transfer functions, 307309 division, 307309 of harmonic trap filters, 576582 parallel combinations, 301307 parallel resonance, 301303 of parallel resonant converter, 677 series combinations, 296301 series resonance, 298301 of series resonant converter, 671672 Grounding problems, 312314 Gyrator, 682683 Harmonic correction, 621 Harmonic loss factor FH, 488490 Harmonics in power systems average power vs. Fourier series, 542543 distortion factor, 548 harmonic standards, 555559 neutral currents, 552553 power factor, 546550 rootmeansquare value of waveform, 543546 rectifier harmonics, 548550 in threephase systems, 551555 total harmonic distortion, 548 Harmonic trap filters, 575582 bypass resistor, 580582 parallel resonance in, 577579 reactive power in, 582 Hbridge, 78, 142145, 148150 Holdup time, 601 Hot spot formation, 7778, 8586 Hysteresis loss PH, 471472 Hysteretic control, 639641 Ideal rectifier (see also Low harmonic rectifiers), 590626 in converter systems, 599604 properties of, 590593 realization of single phase, 593599 three phase, 608622 rms values of waveforms in, 604608 IEC555, 556557 IEEE/ANSI standard 519, 557559 Impedance graph paper, 307 Inductor copper loss (see Copper loss) Inductor current ripple in ac inductor, 499500 boost example, 28 buck example, 21 calculation of, 21 in converters containing twopole filters, 3436 Cuk converter example, 3233 in filter inductor, 497499
10
Fundamentals of Power Electronics R. W. Erickson
magnitude vs. DCM, 111113 Inductor design ac inductor design derivation, 531533 stepbystep procedure, 533534 filter inductor design derivation, 503508 stepbystep procedure, 508509 Inductor voltsecond balance boost example, 2526 buck example, 2223 Cuk converter example, 3132 definition, 22 in discontinuous conduction mode, 115 Input port, converter ac modeling of, 203 boost static characteristics, 596597, 639 modeling of, via statespace averaging, 200, 227, 231 steadystate modeling of, 5153 Inrush current, 601602 Insulatedgate bipolar transistor (IGBT) construction and operation of, 8790 current tailing in, 8889, 9697 equivalent circuit, 88 forward voltage drop, modeling of, 89 idealized switch characteristics, 6566 parallel operation of, 89 switching loss in, 9697 Inversion of source and load, 136137 Inverters, 1 high frequency, 659662, 697705 line commutated, 572573 single phase, 78, 6869 sinusoidal analysis of resonant converters, 664670, 697705 three phase, 70, 143148 Iron laminations, 459, 473 K, dimensionless parameter critical value Kcrit(D), 114115, 121122, 128 in current programmed mode analysis, 438 and DCM boundary, 114115, 121122, 128 in linecommutated rectifier analysis, 565566 in steadystate DCM analysis, 120, 127128 Kg, core geometrical constant definition of, 507507, 751 ferrite core tables of, 752755 filter inductor design procedure using, 508509 Kgfe, ac core geometrical constant ac inductor design procedure using, 531534 definition of, 521, 751 ferrite core tables of, 752755 transformer design using, derivation, 517521 examples, 524531 stepbystep procedure, 521524 11
Fundamentals of Power Electronics R. W. Erickson
KL, rectifier dimensionless parameter, 565566 Ku, window utilization factor, 507 LCC resonant converter dependence of transistor current on load, 701702 introduction to, 659661 ZVS/ZCS boundary, 703705 LenzÕs law, 457, 472, 475 Linear ripple approximation (see Small ripple approximation) Linetooutput transfer function Gvg(s) of the buck, boost, and buckboost converters in CCM, 292293 canonical model, as predicted by, 247248 closedloop, 326327, 331332 control system design of, 340, 353354 of currentprogrammed converters, 422, 427428, 437438 of DCM converters, 387388 of quasiresonant converters, 736 Litz wire, 487 Loop gain (see also Control system design, Negative feedback) definition, 327 measurement of, 355361 Lossfree resistor model averaged switch model of discontinuous conduction mode, 374381 ideal rectifier model single phase, 590593 three phase, 608611 Low harmonic rectifiers (see also Ideal rectifiers) controller schemes average current control, 634636 current programmed control, 636640 feedforward, 635636 hysteretic control, 639641 nonlinear carrier control, 641645 modeling of efficiency and losses, 627634 lowbandwidth control loop, 645650 widebandwidth average current control loop, 650652 rms calculations in, 604609 Low Q approximation, 282284 Magnetic circuits, 463466 Magnetic field H, 455456 Magnetic path length lm definition, 461 ferrite core tables, 752755 Magnetics, 453538 ac inductor design, 531534 basic relations, 455462 copper loss, 474 core loss, 471474 ferrite core tables, 752755 filter inductor design, 503509 magnetic circuits, 463466 magnetic devices, types of, 497503 optimizing Bmax to minimize total loss, 520521 optimizing window allocation to minimize copper loss, 513517 12
Fundamentals of Power Electronics R. W. Erickson
proximity effect, 476490 transformer basics, 152154, 466471 transformer design, 517531 Magnetizing current, 152153, 458469 Magnetomotive force (MMF) definition, 455456 MMF diagrams, 479482 Majority carrier devices (see also MOSFET, Schottky diode), 7475 Matrix converter, 7273 Meal length per turn (MLT) definition, 507508 ferrite core tables, 752755 Measurement of transfer functions and loop gains (see Experimental techniques) MILSTD461B, 556 Minority carrier devices (see also Bipolar junction transistor, Diode, Gate turnoff thyristor, Insulatedgate bipolar transistor, MOScontrolled thyristor, Silicon controlled rectifier), 7475 Modulation index, 613614 MOScontrolled thyristor (MCT), 9294 MOSFET body diode, 6768, 7980 conduction loss, modeling of, 5357, 209218 construction and operation of, 7882 on resistance, 5357, 7982 switching loss owing to Cds, 100101 as synchronous rectifier, 7374 terminal capacitances, 81 typical characteristics, 8082 zerovoltage and zerocurrent switching of, 689696 Motor drive system, 910 Multiplying controller (see also Average current control, Current programmed control), 596 Multiresonant switch singleswitch 3¿ buck rectifier, 620621 zerovoltage switching dcdc, 729 Negative feedback (see also Control system design) effects of, on network transfer functions, 326329 objectives of, 193194, 323326 reduction of disturbances by, 327329 reduction of sensitivity to variations in forward gain by, 329 Nonlinear carrier control, 641645 Nonminimumphase zero (see Right halfplane zero) Output characteristics of the parallel resonant converter, 689 of resonant inverters, 699700 of the series resonant converter, 685686 Overshoot, 338340 Parallel resonant converter analysis via sinusoidal approximation, 664670, 674678 dependence of transistor current on load, 702 exact characteristics continuous conduction mode, 686687 control plane, 689 discontinuous conduction mode, 687689 output plane, 689 13
Fundamentals of Power Electronics R. W. Erickson
introduction to, 659660 as a low harmonic rectifier, 597 Permeability m definition, 458460 of free space, m0, 458 relative, mr, 458 Phase asymptotes of complex poles, 279282 inverted forms, 272 of real pole, 266268 of real zero, 269 of RHP zero, 270 Phase control of resonant converters, 659 of threephase rectifiers, 570575 of zerovoltage transition dcdc converter, 696 Phase margin vs. closedloop damping factor, 334338 stability test, 333334 Poles complex, Bode plots of, 276282 the low Q approximation, 282284 real, Bode plots of, 263268 Pot core data, 752 Powdered iron, 459, 473 Power factor (see also Total harmonic distortion, Displacement factor, Distortion factor) definition of, 546550 of bridge rectifier, single phase, 566 of peak detection rectifier, 548550 of phasecontrolled rectifier, three phase, 573 Power sink element (see Power source element) Power source element in averaged switch models current programmed mode, CCM, 423428 current programmed mode, DCM, 438447 discontinuous conduction mode, 370382 definition of, 375377 in ideal rectifier model, 592, 599603, 608610 linearization of, 384385, 425426 in lossfree resistor model, 376379 properties of, 375377 PQ core data, 755 Proximity effect conductor spacing factor h, 478 interleaving, effect on, 485487 layer copper loss, 482483 Litz wire, effect of, 487 MMF diagrams, 479482 PWM waveform harmonics, 487490 simple explanation, 476478 transformer design procedure, accounting for, 519 winding loss, total, 483487
14
Fundamentals of Power Electronics R. W. Erickson
Pulse width modulation (PWM), 68, 17 modulator ac model, 253254 operation of modulator, 252 spectrum of PWM waveform, 194195 Pushpull isolated converters based on boost converter, 173 based on buck converter, 164166, 410 WatkinsJohnson converter, 173 Q factor, 277282 canonical model, predicted by, 309 closedloop, vs. phase margin, 334338 of the CCM buck, boost, and buckboost converters, 293 the low Q approximation, 282284 vs. overshoot, 338339 of parallel resonant circuit, 303 of series resonant circuit, 300301 Quasiresonant converters (see also Multiresonant switch, Quasisquarewave switch) ac modeling of, 732737 singleswitch 3¿ buck rectifiers multiresonant, 620621 zero current switching, 619620 zerocurrent switching dcdc full wave, 723724 half wave, 713723 zerovoltage switching dcdc, 726728 Quasisquarewave converters, 730731 Quasistatic approximation, 661 Quiescent operating point, 196197, 204, 225 Reactance graph paper (see Impedance graph paper) Reactive power definition, 550551 in harmonic trap filters, 582 in phasecontrolled rectifiers, 573 Rectifiers (see also Ideal rectifiers, Low harmonic rectifiers), 1 energy storage in singlephase, 599604 high quality, 541 ideal, 590626 linecommutated phase control of, 570575 singlephase, 548550, 562568 threephase, 568575 threephase transformer connections in, 582584 twelve pulse, 582584 in resonant dcdc converter, 666668, 674676 Regulator system (see also Control system design), 192193, 323326, 634645 Reluctance, 463466 Resonance Bode plots of complex poles, 276282 graphical construction examples, 296309 the lowQ approximation, 282284 parallel resonant network, 301307 series resonant network, 301307 Resonant converters (see also Quasiresonant converters, Multiresonant converters, Quasisquarewave converters, Zero voltage transition converter), 659710 15
Fundamentals of Power Electronics R. W. Erickson
analysis of, via sinusoidal approximation, 664679 LCC, 659661, 701705 parallel, 597, 659660, 674678, 686690, 702 resonant link, 662 series, 659674, 679686, 690695, 702 Resonant link converters, 662 Resonant switches (see Quasiresonant converters, Multiresonant switch, Quasisquarewave converters) Right halfplane zero Bode plot of, 269271 physical origins of, 294295 Ripple, switching, 1719, 111113, 194196 Root mean square value of commonlyobserved converter waveforms, 743750 vs. Fourier series, 543546 of nearideal rectifier currents, table of, 609 of nearideal rectifier waveforms, 604609 Rotation of threeterminal cell, 141142 Saturation of inductors, 462, 465466 of magnetic materials, 458460 of transformers, 153154, 469 Schottky diode, 74, 77, 101 Semiconductor conduction loss boost converter example, 5357 inclusion in ac model, 209218 with synchronous rectifier, 7374 Semiconductor cost, 180 Semiconductor power devices (see also Bipolar junction transistor, Diode, Gate turnoff thyristor, Insulatedgate bipolar transistor, MOScontrolled thyristor, Schottky diode, Silicon controlled rectifier), 62109 charge control of, 75, 8385, 94, 98, 101103 conductivity modulation, 75 majority vs. minority carriers, 7475 realization of switches using, 6274 SEPIC (see Singleended primary inductance converter) Series pass regulator, 45 Series resonant converter analysis via sinusoidal approximation, 664674 dependence of transistor current on load, 702 exact characteristics continuous conduction mode, 679681 control plane, 684 even discontinuous conduction mode, 682 odd discontinuous conduction mode, 681 output plane, 685686 introduction to, 659664 subharmonic modes in, 673674 zerocurrent switching in, 690692 zerovoltage switching in, 692695 Silicon area (see Switch stress) Silicon controlled rectifier (SCR) construction and characteristics of, 8992 equivalent circuit, 90 16
Fundamentals of Power Electronics R. W. Erickson
inverter grade, 91 Silicon steel, 459, 473 Singleended primary inductance converter (SEPIC), 38, 148 averaged switch model of continuous conduction mode, 757762 discontinuous conduction mode, 379381 conversion ratio M(D), 151, 381 inverse of, 151, 176 as lowharmonic rectifier, 597599, 608609 transformer isolation in, 174176 Single quadrant switch definitions, 6364 implementation, 6467 origins of DCM, 110113 Sinusoidal approximation, 663, 664679 Sinusoidal PWM, 612614 Skin effect (see also Proximity effect), 472476 Slope compensation (see Artificial ramp) Small ripple approximation in ac modeling approach, 198, 223224 and average power loss, prediction of, 57 boost example, 25 buck example, 20 Cuk converter example, 3031 definition, 19 in discontinuous conduction mode, 116 failure of, in twopole filters, 3436 Smallsignal ac modeling via averaged switch modeling, 239245 via circuit averaging, 231245 of CCM converters, 193260 of current programmed converters, 418447 of DCM converters, 382403 via generalized switch averaging, 390403 of low harmonic rectifiers, 645652 of quasiresonant converters, 732737 of resonant converters, 678 via statespace averaging, 218231 Snubber networks, 86, 94, 99, 696 Soft switching (see also Zero current switching, Zero voltage switching), 689696 Spacecraft power system, 9 Spreadsheet design of converters, 180183 State equations of a network, 218221 Statespace averaging, 218231 discussion, 223226 example: nonideal buckboost converter, 227231 summary of result, 221222 Steady state inductor current waveform, 22 operating point, 196197, 204, 225 Subharmonic modes of series resonant converter, 673674 number x, 679
17
Fundamentals of Power Electronics R. W. Erickson
Switch averaged modeling of, 239245, 377, 390403 currentbidirectional twoquadrant, 6770 fourquadrant, 7173 ideal SPDT in converters, 46, 1516, 24, 29 ideal SPST, 6263 passive vs. active, 6465, 91 power dissipated by ideal, 6, 17 quasiresonant, 711737 realization of, using semiconductor devices, 6274 singlequadrant, 6467 synchronous rectifier, 7374 voltagebidirectional twoquadrant, 7071 Switch conversion ratio m definition, 392 DCM buck example, 395401 in generalized canonical model, 405 of multiresonant switch, 729 of quasiresonant switches fullwave ZCS, 723724 fullwave ZVS, 727 halfwave ZCS, 713, 720721 halfwave ZVS, 727 smallsignal ac modeling using, 732737 of quasisquarewave switches, 731 Switched mode, 34 Switching frequency converter efficiency vs., 103104 definition of, 16 transformer size vs., 527 Switching harmonics (see also Ripple, switching), 6 removal of via averaging, 194196 Switching loss (see also Soft switching, Zero current switching, Zero voltage switching) averaged switch modeling of, 241245, 259260 with clamped inductive load, 9497 and current tailing, 9697 and device capacitances, 100101 and diode recovered charge, 97100, 101103 effect on converter efficiency, 103104 and ringing waveforms, 101103 and stray inductances, 100101 Switch stress S, 177180 Switch utilization U, 177180 Synchronous rectifier, 7374 Temperature rise in a converter, 2 in magnetics, 752 Thyristor (see Gate turnoff thyristor, MOScontrolled thyristor, Silicon controlled rectifier) Topologies of converters (see also Boost, Bridge configuration, Buck, Buckboost, Cuk converter, Forward converter, Transformerisolated converters, etc.) Cascade connections, 138141 Converter synthesis, 146150
18
Fundamentals of Power Electronics R. W. Erickson
Differential connection of load, 142146 Evaluation and comparison, 177183, 608609 Inversion of source and load, 136137 Lowharmonic singlephase rectifiers, 593599 Lowharmonic threephase rectifiers, 608621 Quasiresonant converters, 724731 Resonant converters, 659664 Rotation of threeterminal cell, 141142 Transformer isolation, 150177 Total harmonic distortion (THD) of currentprogrammed rectifiers, 639 definition, 550 vs. distortion factor, 550551 IEEE519 limits, 559561 of peak detection rectifier, 551552 of singlephase bridge rectifiers, 551552, 566570 of threephase bridge rectifiers, 571572, 575 Transfer functions (see also Bode plots) of the buck, boost, and buckboost converters, 292293 of current programmed converters, 422423, 427, 436438, 446447 of DCM converters, 388390 graphical construction of, 296309 of lowharmonic rectifiers, 649650, 651 measurement of, 309311 predicted by canonical model, 247248, 292293 Transformer connections in threephase rectifiers, 582584 Transformerisolated converters, 150177 boostderived topologies, 171173 Cuk converter, 176177 evaluation and comparison of, 177183 flyback, 166171 forward, 159164 full bridge buckderived, 154157 halfbridge buckderived, 157159 multiple outputs and cross regulation, 151152 pushpull buckderived, 164166 SEPIC, 174176 transformer model, 152154, 466471 use of voltsecond balance in, 153154, 156157 Transformers BH loop in, 153, 500501 design of, derivation of procedure, 517521 examples, 524531 stepbystep procedure, 521524 winding area optimization, 513517 flyback transformer, 166167 leakage inductance, 154, 469471 magnetizing inductance, 152154, 468469 modeling of, 152154, 466471 SEPIC transformer, 174175 voltsecond balance in, 153154, 156157 Triplen harmonics in threephase fourwire networks, 552553 19
Fundamentals of Power Electronics R. W. Erickson
in threephase inverter modulation schemes, 614 in threephase rectifier circuits, 568569 in threephase threewire networks, 553554 Twelvepulse rectifier, 582584 Twoquadrant switches (see Switch) Universalinput rectifiers, 602 Variablespeed ac drive, 78 Voltage conversion ratio (see Conversion ratio M) Voltage injection, 357359 Voltagesource inverter, 70, 146147 Voltsecond balance (see Inductor voltsecond balance) WatkinsJohnson converter, 148, 150, 173 inverse of, 148, 150 isolated pushpull, 173 Window area WA allocation of, to minimize total copper loss, 513517 definition, 506507 ferrite core tables, 752755 Window utilization factor Ku, 507 Wire area AW inductor design, 506507, 509 American wire gauge (AWG) table, 755756 Zerocurrent switching (ZCS), 662 in quasiresonant converters, 712726 in quasisquarewave converters, 730731 in series resonant converter, 690692 in singleswitch threephase lowharmonic rectifiers, 619621 ZCS/ZVS boundary, 702705 Zerovoltage switching (ZVS), 662 in LCC resonant converter, 703705 in multiresonant converters, 729 in quasiresonant converters, 726728 in quasisquarewave converters, 730731 in series resonant converter, 692695 in singleswitch threephase lowharmonic rectifiers, 619621 in zerovoltage transition converter, 695696 ZVS/ZCS boundary, 702705 Zerovoltage transition buckderived converter, 695696
20
http://ecewww.colorado.edu/~pwrelect/book/errata.html
Fundamentals of Power Electronics Errata, first edition Back Errors can be reported for inclusion on this list by sending email to Bob Erickson. Please include page number and a short description. ●
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Page ii, smallprint disclaimer near bottom of page contains two misspelled words Page 10, paragraph beginning "Design of the converter..." "linetotransfer functions" should read "linetooutput transfer functions" Page 49, Eq. (3.19) RHS of equation should be divided by R Page 86, Fig. 4.35(a) Units of horizontal axis should be A Page 172, Fig. 6.36 The bottom of the figure is cut off. Transistor Q4 conducts during the first subinterval of every switching period. Page 180, third line under heading 6.4.2 typo "votlage" should read "voltage" Page 181, Table 6.2 Values of L and C are missing. The values are: L = 25.96 uH, C = 25.0 uF. Page 188, problem 6.8(d) Forward voltage drops of diodes D1 and D2 are equal to 0.7 V. Page 237, second line following Eq. (7.126) "small" should read "smaller" Page 241, third paragraph, first line reads "valiables"  should read "variables" Page 250, Equation (7.144) should read: e(s) = (Vg  V)  sLI/DD' (plus sign should be minus sign) Page 275, Figure 8.15 Phase asymptote for frequencies between 10*f1 and f2/10 should be +90 deg instead of 90 deg. Pages 289293, Equation (8.112) and Table 8.2 Missing factor of D' in Eq. (8.112). Mistake propagates to Eqs. (8.113), (8.121), (8.123), (8.124), (8.127), and Fig. 8.26. Buckboost Gd0 entry in Table 8.2 is also incorrect. Corrected Section 8.2.1 and Table 8.2 are in attached pdf file. Page 300, line following Eq. (8.144) Should read: "Subsitution of Eq. (8.142) into Eq. (8.144) leads to" Page 311, last sentence
http://ecewww.colorado.edu/~pwrelect/book/errata.html (1 of 2) [25/04/2002 16:42:57]
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is missing a period Page 333, fourth line following Eq. (9.20) reads "simple"  should be "simply" Page 347, first line of text Equation reference should read "Eq. (7.149)" Page 366, Problem 9.8, last sentence of paragraph (before (a)) should read "The pulsewidth modulator circuit obeys Eq. (7.149)."  not "(7.132)" Page 385, Table 10.2 g2 for the buckboost converter is equal to 2/MRe Page 419, Figure 11.12(a) ig(t) is the current flowing out of the source vg(t) Page 427, Figure 11.20 Value of ac resistance is inverted. The value should read: R/D^2 Page 488, second paragraph, fourth sentence Change "greater" to "smaller", so that the sentence reads: "The skin depth d is smaller for high frequencies..." Page 516, Equations (14.14) and (14.15) Summation limits should be j = 1 to k Page 519, Equation (14.28) should contain a factor of 4 in denominator Page 534, Equation (14.84) should read: Bpk = Bmax + LIdc/nAc Page 546, Equation (15.14) Power loss should be (rms voltage)^2 / Rshunt Page 548, last sentence change "displacement factor" to "distortion factor" Page 576, end of first sentence after Eq. (16.12) is should read ir Page 637, Equation (18.27) Corrected version is in attached pdf file. Page 654, Figure 18.20, and Problem 18.2 Flyback transformer turns ratio is 1:1 Appendix 3, page 760, Figure A3.3 Polarity of dependent current source should be reversed.
update 10/25/99
http://ecewww.colorado.edu/~pwrelect/book/errata.html (2 of 2) [25/04/2002 16:42:57]
Fundamentals of Power Electronics Solutions to Selected Problems
Fundamentals of Power Electronics R. W. Erickson Solutions to Selected Problems Solutions to selected problems at the end of chapters in the text Fundamentals of Power Electronics are listed below. These are intended as an educational supplement to the text, for the use of students who have purchased the book. All files copyright R. W. Erickson 1998. These files can be read using the Adobe Acrobat viewer, available free from the Adobe Acrobat web site. Up Chapter 2 ● ●
Problem 2.2 144kB. Analysis of a SEPIC in a voltage regulator application. Problem 2.3 168kB. Inductor current and capacitor voltage ripples in the SEPIC.
Chapter 3 ● ●
Problem 3.3 192kB. Modeling efficiency and losses in a buck converter with input filter. Problem 3.6 148kB. Comparison of boost and buckboost converters in a voltage stepup application.
Chapter 4 ●
Problem 4.4 196kB. Switch realization in a converter that is probably unfamiliar.
Chapter 5 ● ●
Problem 5.5 81kB. CCM/DCM boundary for the Cuk converter. Problem 5.6 184kB. Solution of the Cuk converter in DCM.
Chapter 6 ●
Problem 6.1 144kB. Analysis of a tappedinductor boost converter.
http://ecewww.colorado.edu/~pwrelect/book/solutions/solndir.html (1 of 2) [25/04/2002 16:42:59]
Fundamentals of Power Electronics Solutions to Selected Problems
Chapter 8 ●
Sample problem 164kB. A solved problem on transfer function asymptotes, similar to problems 8.1  8.4.
Chapter 12 ●
Problem 12.1 116kB. Magnetic circuit analysis.
Update 11/23/98 rwe
http://ecewww.colorado.edu/~pwrelect/book/solutions/solndir.html (2 of 2) [25/04/2002 16:42:59]
Fundamentals of Power Electronics R. W. Erickson
Proximity Effect Functions The functions below were used to generate Figs. 12.35, 12.36, and 12.38. The functions are written in Microsoft Visual Basic, and were called by an Excel spreadsheet. The functions can be easily modified for other programming languages. Functions Gone and Gtwo implement Eq. (12.73). Function FR(phi, M) is identical to Eq. (12.84). Figure 12.32 is a plot of FR for various values of M and j. Figure 12.33 is a plot of FR(phi, M)/phi. Function Pj(Phi1, M, j, D) is a normalized version of Eq. (12.89), the power loss in harmonic j. Function FH(phi1, M, D) coincides with Eq. (12.90) and (12.91). The series is summed numerically. The copper loss in a winding is evaluated as follows. First, the dc and fundamental components of the current waveform are evaluated using Eq. (12.86). Then the functions FH and FR are evaluated. Finally, the results are plugged into Eq. (12.92). Option Explicit Function cosh(x) cosh = 0.5 * (Exp(x) + Exp(x)) End Function Function sinh(x) sinh = 0.5 * (Exp(x)  Exp(x)) End Function Function Gone(phi) Gone = (sinh(2 * phi) + Sin(2 * phi)) / (cosh(2 * phi) Cos(2 * phi)) End Function Function Gtwo(phi) Gtwo = (sinh(phi) * Cos(phi) + cosh(phi) * Sin(phi)) / (cosh(2 * phi)  Cos(2 * phi)) End Function Function FR(phi, M) FR = phi * (Gone(phi) + 2 / 3 * (M * M  1) * (Gone(phi)  2 * Gtwo(phi))) End Function Function pi() pi = 3.14159265 End Function Function Pj(phi1, M, j, D) Dim sj, Ga sj = Sqr(j) Ga = Gone(phi1 * sj) Pj = (1 / (j * sj)) * (Sin(j * pi() * D)) ^ 2 * (Ga + 2 / 3 * (M * M  1) * (Ga  2 * Gtwo(phi1 * sj))) End Function
1
Fundamentals of Power Electronics R. W. Erickson
Function FH(phi1, M, D) Dim P1, Ptot, Pke, Pko, Peps, j, Pk P1 = Pj(phi1, M, 1, D) Pko = P1 Pke = Pj(phi1, M, 2, D) Pk = Pko + Pke Ptot = P1 Peps = P1 * 0.0002 j = 3 Do Pko = Pj(phi1, M, j, D) j = j + 1 Pke = Pj(phi1, M, j, D) Pk = Pke + Pko Ptot = Ptot + Pk j = j + 1 Loop Until Pk < Peps FH = Ptot / P1 End Function
2
Fundamentals of Power Electronics Robert W. Erickson University of Colorado, Boulder
Fundamentals of Power Electronics
1
Chapter 1: Introduction
Chapter 1: Introduction
1.1.
Introduction to power processing
1.2.
Some applications of power electronics
1.3.
Elements of power electronics Summary of the course
Fundamentals of Power Electronics
2
Chapter 1: Introduction
1.1 Introduction to Power Processing
Power input
Switching converter
Power output
Control input
Dcdc conversion: Acdc rectification: Dcac inversion:
Change and control voltage magnitude Possibly control dc voltage, ac current Produce sinusoid of controllable magnitude and frequency Acac cycloconversion: Change and control voltage magnitude and frequency Fundamentals of Power Electronics
3
Chapter 1: Introduction
Control is invariably required
Power input
Switching converter
Power output
Control input feedforward
feedback Controller reference
Fundamentals of Power Electronics
4
Chapter 1: Introduction
High efficiency is essential
η=
Pout Pin
1
η
1 –1 Ploss = Pin – Pout = Pout η
0.8
0.6
High efficiency leads to low power loss within converter Small size and reliable operation is then feasible Efficiency is a good measure of converter performance
0.4
0.2 0
0.5
1
1.5
Ploss / Pout
Fundamentals of Power Electronics
5
Chapter 1: Introduction
A highefficiency converter
Pin
Converter
Pout
A goal of current converter technology is to construct converters of small size and weight, which process substantial power at high efficiency
Fundamentals of Power Electronics
6
Chapter 1: Introduction
+ –
Devices available to the circuit designer
DT
Resistors
Capacitors
Fundamentals of Power Electronics
Magnetics
7
T
s s linearswitchedmode mode Semiconductor devices
Chapter 1: Introduction
+ –
Devices available to the circuit designer
DT
Resistors
Capacitors
Magnetics
T
s s linearswitchedmode mode Semiconductor devices
Signal processing: avoid magnetics
Fundamentals of Power Electronics
8
Chapter 1: Introduction
+ –
Devices available to the circuit designer
DT
Resistors
Capacitors
Magnetics
T
s s linearswitchedmode mode Semiconductor devices
Power processing: avoid lossy elements
Fundamentals of Power Electronics
9
Chapter 1: Introduction
Power loss in an ideal switch
Switch closed: Switch open:
v(t) = 0
+
i(t)
i(t) = 0
v(t) In either event:
p(t) = v(t) i(t) = 0
– Ideal switch consumes zero power
Fundamentals of Power Electronics
10
Chapter 1: Introduction
A simple dcdc converter example I 10A + Vg 100V
+ –
Dcdc converter
R 5Ω
V 50V –
Input source: 100V Output load: 50V, 10A, 500W How can this converter be realized?
Fundamentals of Power Electronics
11
Chapter 1: Introduction
Dissipative realization
Resistive voltage divider I 10A + + Vg 100V
+ –
50V –
Ploss = 500W
R 5Ω
V 50V –
Pin = 1000W
Fundamentals of Power Electronics
Pout = 500W
12
Chapter 1: Introduction
Dissipative realization Series pass regulator: transistor operates in active region +
I 10A
50V –
+ Vg 100V
+ –
linear amplifier and base driver Ploss ≈ 500W
Vref
R 5Ω
V 50V –
Pin ≈ 1000W
Fundamentals of Power Electronics
–+
Pout = 500W
13
Chapter 1: Introduction
Use of a SPDT switch
I 10A
1
+
+ Vg 100V
2
+ –
vs(t)
R
–
vs(t)
v(t) 50V –
Vg Vs = DVg
switch position:
Fundamentals of Power Electronics
DTs
0 (1–D) Ts
t
1
2
1
14
Chapter 1: Introduction
The switch changes the dc voltage level
vs(t)
Vg Vs = DVg 0
switch position:
Ts = switching period
(1 – D) Ts
DTs 1
D = switch duty cycle 0≤D≤1
t
2
1
fs = switching frequency = 1 / Ts
DC component of vs(t) = average value:
Vs = 1 Ts
Ts
vs(t) dt = DVg 0
Fundamentals of Power Electronics
15
Chapter 1: Introduction
Addition of low pass filter Addition of (ideally lossless) LC lowpass filter, for removal of switching harmonics: i(t)
1
+ Vg 100V
+ –
+
L
2
vs(t)
C
v(t) –
– Pin ≈ 500W
R
Ploss small
Pout = 500W
•
Choose filter cutoff frequency f0 much smaller than switching frequency fs
•
This circuit is known as the “buck converter”
Fundamentals of Power Electronics
16
Chapter 1: Introduction
Addition of control system for regulation of output voltage
Power input
Switching converter
Load +
vg
+ –
i
v H(s)
– transistor gate driver
error signal ve
δ(t)
dTs Ts
Fundamentals of Power Electronics
–+
pulsewidth vc G (s) c modulator compensator
δ
sensor gain
Hv
reference vref input
t
17
Chapter 1: Introduction
The boost converter
2
+
L 1
Vg
+ –
C
R
V –
5Vg 4Vg
V
3Vg 2Vg Vg 0 0
0.2
0.4
0.6
0.8
1
D Fundamentals of Power Electronics
18
Chapter 1: Introduction
A singlephase inverter vs(t) 1
Vg
+ –
+
2
– +
v(t)
–
2
1
load
“Hbridge”
vs(t)
t
Fundamentals of Power Electronics
19
Modulate switch duty cycles to obtain sinusoidal lowfrequency component
Chapter 1: Introduction
1.2 Several applications of power electronics
Power levels encountered in highefficiency converters • less than 1 W in batteryoperated portable equipment • tens, hundreds, or thousands of watts in power supplies for computers or office equipment • kW to MW in variablespeed motor drives • 1000 MW in rectifiers and inverters for utility dc transmission lines
Fundamentals of Power Electronics
20
Chapter 1: Introduction
A computer power supply system
regulated dc outputs +
iac(t)
Dcdc converter
Rectifier vac(t) – ac line input 85265Vrms
Fundamentals of Power Electronics
dc link
21
loads
Chapter 1: Introduction
A spacecraft power system
Dissipative shunt regulator + Solar array
vbus – Battery charge/discharge controllers
Dcdc converter
Dcdc converter
Payload
Payload
Batteries
Fundamentals of Power Electronics
22
Chapter 1: Introduction
A variablespeed ac motor drive system
variablefrequency variablevoltage ac
+ 3øac line 50/60Hz
Rectifier
Inverter vlink – Ac machine Dc link
Fundamentals of Power Electronics
23
Chapter 1: Introduction
1.3 Elements of power electronics
Power electronics incorporates concepts from the fields of analog circuits electronic devices control systems power systems magnetics electric machines numerical simulation
Fundamentals of Power Electronics
24
Chapter 1: Introduction
Part I. Converters in equilibrium Inductor waveforms vL(t)
Averaged equivalent circuit RL
DTs
1
iL(t)
t
2
0
+
Vg – V L
Vg
+ –
R
–
∆iL
Predicted efficiency 100%
–V L DTs
V
I
1
iL(DTs)
I iL(0)
D' : 1
D'Ts –V
switch position:
D' RD
+ –
Vg – V
D' VD
D Ron
0.002
90%
0.01
Ts
80%
t
0.02
70%
0.05
60%
η
50%
RL/R = 0.1
40%
Discontinuous conduction mode
30% 20%
Transformer isolation
10% 0% 0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
D
Fundamentals of Power Electronics
25
Chapter 1: Introduction
Switch realization: semiconductor devices
iA(t)
The IGBT
collector
Switching loss
transistor waveforms
Qr Vg
gate
iL
vA(t) 0
0
emitter
t
Emitter
diode waveforms
iL
iB(t) vB(t)
Gate
0
0 t
n
p
n
n
n
p
area –Qr
n
–Vg
minority carrier injection
tr
p
pA(t)
= vA iA area ~QrVg
Collector
area ~iLVgtr t0
Fundamentals of Power Electronics
26
t1 t2
t
Chapter 1: Introduction
Part I. Converters in equilibrium
2. Principles of steady state converter analysis 3. Steadystate equivalent circuit modeling, losses, and efficiency 4. Switch realization 5. The discontinuous conduction mode 6. Converter circuits
Fundamentals of Power Electronics
27
Chapter 1: Introduction
Part II. Converter dynamics and control Closedloop converter system Power input
Averaging the waveforms
Switching converter
Load
gate drive
+ vg(t) + –
v(t)
R feedback connection
–
δ(t)
compensator pulsewidth vc Gc(s) modulator
δ(t)
v averaged waveform Ts with ripple neglected
voltage reference vref
vc(t)
dTs Ts
actual waveform v(t) including ripple
+ –
transistor gate driver
t
t
t
t
Controller
L
Smallsignal averaged equivalent circuit
+ –
1:D
Vg – V d(t)
D' : 1 +
vg(t)
Fundamentals of Power Electronics
+ –
I d(t)
I d(t)
C
v(t)
R
–
28
Chapter 1: Introduction
Part II. Converter dynamics and control
7.
Ac modeling
8.
Converter transfer functions
9.
Controller design
10.
Ac and dc equivalent circuit modeling of the discontinuous conduction mode
11.
Currentprogrammed control
Fundamentals of Power Electronics
29
Chapter 1: Introduction
Part III. Magnetics n1 : n 2
transformer design
iM(t)
i1(t)
the proximity effect
i2(t)
LM R1
R2
3i
layer 3
–2i 2Φ
layer 2
2i –i
ik(t)
Φ layer 1
Rk
d
current density J
: nk
i
4226
Pot core size
3622
0.1
2616
2616 2213
2213 1811
0.08 0.06
1811
0.04
Bmax (T)
transformer size vs. switching frequency
0.02 0 25kHz
50kHz
100kHz
200kHz
250kHz
400kHz
500kHz
1000kHz
Switching frequency
Fundamentals of Power Electronics
30
Chapter 1: Introduction
Part III. Magnetics
12.
Basic magnetics theory
13.
Filter inductor design
14.
Transformer design
Fundamentals of Power Electronics
31
Chapter 1: Introduction
Part IV. Modern rectifiers, and power system harmonics Pollution of power system by rectifier current harmonics
A lowharmonic rectifier system boost converter i(t)
ig(t) +
iac(t) vac(t)
+
L
vg(t)
D1 Q1
vg(t) multiplier
X
v(t)
R
–
– vcontrol(t)
C
ig(t) Rs
PWM va(t)
v (t) +– err Gc(s) vref(t) = kx vg(t) vcontrol(t) compensator
Harmonic amplitude, percent of fundamental
100%
controller
100% 91%
80%
THD = 136% Distortion factor = 59%
73%
60%
iac(t) +
52%
40%
32% 19% 15% 15% 13% 9%
20% 0% 1
3
5
7
Ideal rectifier (LFR)
9
11
13
15
17
19
Model of the ideal rectifier
vac(t)
2
p(t) = vac / Re Re(vcontrol)
–
+ v(t) –
ac input
Harmonic number
i(t)
dc output vcontrol
Fundamentals of Power Electronics
32
Chapter 1: Introduction
Part IV. Modern rectifiers, and power system harmonics
15.
Power and harmonics in nonsinusoidal systems
16.
Linecommutated rectifiers
17.
The ideal rectifier
18.
Low harmonic rectifier modeling and control
Fundamentals of Power Electronics
33
Chapter 1: Introduction
Part V. Resonant converters The series resonant converter Q1
L
Q3
D1
C
1:n
D3
+ Vg
+ –
R
Q2
–
Q4
D2
V
Zero voltage switching
D4
1
vds1(t)
Q = 0.2
Vg
0.9 Q = 0.2
0.8 0.35
M = V / Vg
0.7
0.75
0.5
0.2 0.1 0
1
0.5
0.4 0.3
Dc characteristics
0.5
0.35
0.6
0.75 1 1.5 2 3.5 5 10 Q = 20
0
1.5
conducting devices:
Q1 Q4 turn off Q1, Q4
X D2 D3
t
commutation interval
2 3.5 5 10 Q = 20
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
1.8
2
F = f s / f0
Fundamentals of Power Electronics
34
Chapter 1: Introduction
Part V. Resonant converters 19. 20.
Resonant conversion Quasiresonant converters
Fundamentals of Power Electronics
35
Chapter 1: Introduction
Chapter 2 Principles of SteadyState Converter Analysis
2.1. Introduction 2.2. Inductor voltsecond balance, capacitor charge balance, and the small ripple approximation 2.3. Boost converter example 2.4. Cuk converter example 2.5. Estimating the ripple in converters containing twopole lowpass filters 2.6. Summary of key points Fundamentals of Power Electronics
1
Chapter 2: Principles of steadystate converter analysis
2.1 Introduction Buck converter 1
SPDT switch changes dc component
+
+
Vg
2
+ –
vs(t)
R
v(t)
–
Switch output voltage waveform
vs(t)
Vg D' Ts
DTs
Duty cycle D: 0≤D≤1
complement D’: D’ = 1  D Fundamentals of Power Electronics
–
0 0 switch position:
2
DTs 1
Ts 2
t 1
Chapter 2: Principles of steadystate converter analysis
Dc component of switch output voltage
vs(t)
Vg = DVg
area = DTsVg 0
0 DTs
Ts
t
Fourier analysis: Dc component = average value vs = 1 Ts
Ts
vs(t) dt 0
vs = 1 (DTsVg) = DVg Ts Fundamentals of Power Electronics
3
Chapter 2: Principles of steadystate converter analysis
Insertion of lowpass filter to remove switching harmonics and pass only dc component L
1
+
Vg
+ –
+
2
vs(t)
C
R
v(t)
–
–
V Vg
v ≈ vs = DVg
0 0
Fundamentals of Power Electronics
4
1
D
Chapter 2: Principles of steadystate converter analysis
Three basic dcdc converters a) +
iL(t) Vg
M(D) = D
0.8 2
+ –
C
R
v
M(D)
Buck
1
L
1
0.6 0.4 0.2
–
0 0
0.2
0.4
0.6
0.8
1
0.6
0.8
1
0.6
0.8
1
D
b)
5
2
M(D) =
iL(t) 1
Vg
4
+
+ –
C
R
v
M(D)
Boost
L
1 1–D
3 2 1
–
0 0
0.2
0.4
D
D
c)
0
+ –
+
2
iL(t)
C
R
v
L –
1
M(D)
Vg
0.4
0 1
Buckboost
0.2
2 3 4
M(D) = 1––DD
5
Fundamentals of Power Electronics
5
Chapter 2: Principles of steadystate converter analysis
Objectives of this chapter
●
●
● ●
●
Develop techniques for easily determining output voltage of an arbitrary converter circuit Derive the principles of inductor voltsecond balance and capacitor charge (ampsecond) balance Introduce the key small ripple approximation Develop simple methods for selecting filter element values Illustrate via examples
Fundamentals of Power Electronics
6
Chapter 2: Principles of steadystate converter analysis
2.2.
Inductor voltsecond balance, capacitor charge balance, and the small ripple approximation Actual output voltage waveform, buck converter iL(t)
1
Buck converter containing practical lowpass filter
L + vL(t) –
2
+ –
Vg
+ iC(t)
C
R
v(t) –
Actual output voltage waveform
actual waveform v(t) = V + vripple(t)
v(t) V
v(t) = V + vripple(t)
Dc component V 0
Fundamentals of Power Electronics
t
7
Chapter 2: Principles of steadystate converter analysis
The small ripple approximation actual waveform v(t) = V + vripple(t)
v(t)
v(t) = V + vripple(t)
V Dc component V 0
t
In a welldesigned converter, the output voltage ripple is small. Hence, the waveforms can be easily determined by ignoring the ripple: vripple > RL + DRon + D'RD
Fundamentals of Power Electronics
31
Chapter 3: Steadystate equivalent circuit modeling, ...
Accuracy of the averaged equivalent circuit in prediction of losses • Model uses average currents and voltages
MOSFET current waveforms, for various ripple magnitudes: i(t)
• To correctly predict power loss in a resistor, use rms values
2I (c) (b) (a)
I
• Result is the same, provided ripple is small
(a) ∆i = 0 (b) ∆i = 0.1 I (c) ∆i = I
Fundamentals of Power Electronics
t
0 DTs
0
Inductor current ripple
1.1 I
MOS FET rms current I
D
32
Average power loss in R on D I2 R on
(1.00167) I (1.155) I
Ts
D
D
(1.0033) D I2 R on (1.3333) D I2 R on
Chapter 3: Steadystate equivalent circuit modeling, ...
Summary of chapter 3 1. The dc transformer model represents the primary functions of any dcdc converter: transformation of dc voltage and current levels, ideally with 100% efficiency, and control of the conversion ratio M via the duty cycle D. This model can be easily manipulated and solved using familiar techniques of conventional circuit analysis. 2. The model can be refined to account for loss elements such as inductor winding resistance and semiconductor onresistances and forward voltage drops. The refined model predicts the voltages, currents, and efficiency of practical nonideal converters. 3. In general, the dc equivalent circuit for a converter can be derived from the inductor voltsecond balance and capacitor charge balance equations. Equivalent circuits are constructed whose loop and node equations coincide with the voltsecond and charge balance equations. In converters having a pulsating input current, an additional equation is needed to model the converter input port; this equation may be obtained by averaging the converter input current. Fundamentals of Power Electronics
33
Chapter 3: Steadystate equivalent circuit modeling, ...
Chapter 4. Switch Realization
4.1. Switch applications Single, two, and fourquadrant switches. Synchronous rectifiers
4.2. A brief survey of power semiconductor devices Power diodes, MOSFETs, BJTs, IGBTs, and thyristors
4.3. Switching loss Transistor switching with clamped inductive load. Diode recovered charge. Stray capacitances and inductances, and ringing. Efficiency vs. switching frequency.
4.4. Summary of key points Fundamentals of Power Electronics
1
Chapter 4: Switch realization
SPST (singlepole singlethrow) switches Buck converter
SPST switch, with voltage and current polarities defined
with SPDT switch: 1
L
iL(t) +
1 +
i
Vg
2
+ –
C
R
V –
v –
with two SPST switches: iA
0
L
A
iL(t) +
+ vA –
All power semiconductor devices function as SPST switches. Fundamentals of Power Electronics
Vg
+ –
– vB
B
R
V
+ iB
2
C
– Chapter 4: Switch realization
Realization of SPDT switch using two SPST switches
●
● ●
●
A nontrivial step: two SPST switches are not exactly equivalent to one SPDT switch It is possible for both SPST switches to be simultaneously ON or OFF Behavior of converter is then significantly modified —discontinuous conduction modes (ch. 5) Conducting state of SPST switch may depend on applied voltage or current —for example: diode
Fundamentals of Power Electronics
3
Chapter 4: Switch realization
Quadrants of SPST switch operation 1 +
i
switch onstate current
A singlequadrant switch example:
v
ONstate: i > 0
–
OFFstate: v > 0 switch offstate voltage
0
Fundamentals of Power Electronics
4
Chapter 4: Switch realization
Some basic switch applications
Singlequadrant switch
switch onstate current
Currentbidirectional twoquadrant switch
switch offstate voltage
switch onstate current
Voltagebidirectional twoquadrant switch
Fundamentals of Power Electronics
switch onstate current
switch offstate voltage
switch onstate current
Fourquadrant switch
switch offstate voltage
5
switch offstate voltage
Chapter 4: Switch realization
4.1.1. Singlequadrant switches
1 +
i
v – 0
Fundamentals of Power Electronics
Active switch: Switch state is controlled exclusively by a third terminal (control terminal). Passive switch: Switch state is controlled by the applied current and/or voltage at terminals 1 and 2. SCR: A special case — turnon transition is active, while turnoff transition is passive. Singlequadrant switch: onstate i(t) and offstate v(t) are unipolar.
6
Chapter 4: Switch realization
The diode • A passive switch i
1 +
i
• Singlequadrant switch: • can conduct positive onstate current
on v
off
v – 0 Symbol
instantaneous iv characteristic
Fundamentals of Power Electronics
7
• can block negative offstate voltage • provided that the intended onstate and offstate operating points lie on the diode iv characteristic, then switch can be realized using a diode
Chapter 4: Switch realization
The Bipolar Junction Transistor (BJT) and the Insulated Gate Bipolar Transistor (IGBT)
BJT C
• An active switch, controlled by terminal C
1 i +
i
• Singlequadrant switch:
v – 0
IGBT
• can conduct positive onstate current
on off
v
1 i +
C
v –
instantaneous iv characteristic
0
Fundamentals of Power Electronics
8
• can block positive offstate voltage • provided that the intended onstate and offstate operating points lie on the transistor iv characteristic, then switch can be realized using a BJT or IGBT
Chapter 4: Switch realization
The MetalOxide Semiconductor Field Effect Transistor (MOSFET) • An active switch, controlled by terminal C i
1 i + C
• Normally operated as singlequadrant switch:
on
v
v
off
– on
0
• can conduct positive onstate current (can also conduct negative current in some circumstances)
(reverse conduction)
• can block positive offstate voltage
Symbol
instantaneous iv characteristic
Fundamentals of Power Electronics
9
• provided that the intended onstate and offstate operating points lie on the MOSFET iv characteristic, then switch can be realized using a MOSFET Chapter 4: Switch realization
Realization of switch using transistors and diodes Buck converter example iA
L
A
iL(t) +
+ vA –
Vg
+ –
– vB
B
C
R
V
+
Switch A: transistor
iB
–
Switch B: diode iA
SPST switch operating points
switch A
on
iB switch B
iL
on switch A
switch B
off Vg
off vA
Switch A
Fundamentals of Power Electronics
iL
–Vg
vB
Switch B 10
Chapter 4: Switch realization
Realization of buck converter using singlequadrant switches iA
+
vA
L
– + – vB +
+ –
Vg
iL(t)
vL(t)
–
iB
iB
iA switch A
on
switch B
iL
on switch A
switch B
off
off
Vg
Fundamentals of Power Electronics
vA
–Vg
11
iL
vB
Chapter 4: Switch realization
4.1.2. Currentbidirectional twoquadrant switches • Usually an active switch, controlled by terminal C
i
1 i
on (transistor conducts)
+
C
v – 0
BJT / antiparallel diode realization
Fundamentals of Power Electronics
v
off
on (diode conducts)
instantaneous iv characteristic
12
• Normally operated as twoquadrant switch: • can conduct positive or negative onstate current • can block positive offstate voltage • provided that the intended onstate and offstate operating points lie on the composite iv characteristic, then switch can be realized as shown
Chapter 4: Switch realization
Two quadrant switches
switch onstate current
i 1 +
on (transistor conducts)
i
v
off
v
switch offstate voltage
– 0
Fundamentals of Power Electronics
on (diode conducts)
13
Chapter 4: Switch realization
MOSFET body diode
i
1 i
on (transistor conducts)
+ off
v
C
v –
on (diode conducts)
0 Power MOSFET characteristics
Fundamentals of Power Electronics
Power MOSFET, and its integral body diode
14
Use of external diodes to prevent conduction of body diode
Chapter 4: Switch realization
A simple inverter iA + Vg
+ –
Q1
v0(t) = (2D – 1) Vg
D1 vA –
L
iL +
+ Vg
+ –
Q2
D2 v B –
C
R
v0 –
iB
Fundamentals of Power Electronics
15
Chapter 4: Switch realization
Inverter: sinusoidal modulation of D
v0(t) = (2D – 1) Vg v0
Sinusoidal modulation to produce ac output:
Vg
D(t) = 0.5 + Dm sin (ωt) D
0 0.5
1
–Vg
The resulting inductor current variation is also sinusoidal:
iL(t) =
Vg v0(t) = (2D – 1) R R
Hence, currentbidirectional twoquadrant switches are required. Fundamentals of Power Electronics
16
Chapter 4: Switch realization
The dc3øac voltage source inverter (VSI)
ia Vg
+ –
ib ic
Switches must block dc input voltage, and conduct ac load current.
Fundamentals of Power Electronics
17
Chapter 4: Switch realization
Bidirectional battery charger/discharger
D1 L +
+ Q1
vbus
vbatt
D2
spacecraft main power bus –
Q2 – vbus > vbatt
A dcdc converter with bidirectional power flow.
Fundamentals of Power Electronics
18
Chapter 4: Switch realization
4.1.3. Voltagebidirectional twoquadrant switches • Usually an active switch, controlled by terminal C i
• Normally operated as twoquadrant switch:
i
1 +
on v
v
C
off
off
(diode blocks voltage)
(transistor blocks voltage)
– 0
BJT / series diode realization
instantaneous iv characteristic
• can conduct positive onstate current • can block positive or negative offstate voltage • provided that the intended onstate and offstate operating points lie on the composite iv characteristic, then switch can be realized as shown • The SCR is such a device, without controlled turnoff
Fundamentals of Power Electronics
19
Chapter 4: Switch realization
Twoquadrant switches
1 i
+
switch onstate current
i
v
on
– 0
v
1 i
+
off
off
(diode blocks voltage)
(transistor blocks voltage)
switch offstate voltage
v
C
– 0
Fundamentals of Power Electronics
20
Chapter 4: Switch realization
A dc3øac buckboost inverter
φa
iL
+ vab(t) –
Vg
– +
φb
+ vbc(t) –
φc
Requires voltagebidirectional twoquadrant switches. Another example: boosttype inverter, or currentsource inverter (CSI). Fundamentals of Power Electronics
21
Chapter 4: Switch realization
4.1.4. Fourquadrant switches
switch onstate current
• Usually an active switch, controlled by terminal C • can conduct positive or negative onstate current switch offstate voltage
Fundamentals of Power Electronics
22
• can block positive or negative offstate voltage
Chapter 4: Switch realization
Three ways to realize a fourquadrant switch
1
1
i
i
1 +
1 i
+
+
v
v
v
–
–
–
+
i
v – 0 0
Fundamentals of Power Electronics
0
23
0
Chapter 4: Switch realization
A 3øac3øac matrix converter 3øac input
3øac output ia
van(t)
+ –
vbn(t)
+ –
ib
+ –
vcn(t)
ic
• All voltages and currents are ac; hence, fourquadrant switches are required. • Requires nine fourquadrant switches
Fundamentals of Power Electronics
24
Chapter 4: Switch realization
4.1.5. Synchronous rectifiers Replacement of diode with a backwardsconnected MOSFET, to obtain reduced conduction loss i
1 +
i
1 i +
1 +
i
C
v
v
v
–
–
–
ideal switch
conventional diode rectifier
Fundamentals of Power Electronics
v
off
on
0
0
0
on (reverse conduction)
MOSFET as synchronous rectifier 25
instantaneous iv characteristic
Chapter 4: Switch realization
Buck converter with synchronous rectifier
iA
+
vA
L
–
iL(t)
Q1
Vg
+ –
– vB +
C C Q2
• MOSFET Q2 is controlled to turn on when diode would normally conduct • Semiconductor conduction loss can be made arbitrarily small, by reduction of MOSFET onresistances
iB
• Useful in lowvoltage highcurrent applications
Fundamentals of Power Electronics
26
Chapter 4: Switch realization
4.2. A brief survey of power semiconductor devices
● ● ● ● ●
● ●
Power diodes Power MOSFETs Bipolar Junction Transistors (BJTs) Insulated Gate Bipolar Transistors (IGBTs) Thyristors (SCR, GTO, MCT) On resistance vs. breakdown voltage vs. switching times Minority carrier and majority carrier devices
Fundamentals of Power Electronics
27
Chapter 4: Switch realization
4.2.1. Power diodes A power diode, under reversebiased conditions: v
+ –
p + + +
{ {
low doping concentration
n
n – –
E
–
v
+
–
depletion region, reversebiased
Fundamentals of Power Electronics
28
Chapter 4: Switch realization
Forwardbiased power diode
v
+ –
i
{
conductivity modulation
n
p +
–
+ + +
n
+
–
+ –
minority carrier injection
Fundamentals of Power Electronics
29
Chapter 4: Switch realization
Typical diode switching waveforms v(t)
t
i(t) tr
0 t
di dt
area –Qr (1)
(2)
Fundamentals of Power Electronics
(3)
(4) 30
(5)
(6)
Chapter 4: Switch realization
Types of power diodes Standard recovery Reverse recovery time not specified, intended for 50/60Hz
Fast recovery and ultrafast recovery Reverse recovery time and recovered charge specified Intended for converter applications
Schottky diode A majority carrier device Essentially no recovered charge Model with equilibrium iv characteristic, in parallel with depletion region capacitance Restricted to low voltage (few devices can block 100V or more) Fundamentals of Power Electronics
31
Chapter 4: Switch realization
Characteristics of several commercial power rectifier diodes Part number
R ated max voltage
Rated avg current
V F (typical)
tr (max)
Fas t recov ery rect i fi ers 1N3913
400V
30A
1.1V
400ns
SD453N25S20PC
2500V
400A
2.2V
2µs
Ul t rafas t recov ery rect i fi ers MUR815
150V
8A
0.975V
35ns
MUR1560
600V
15A
1.2V
60ns
RHRU100120
1200V
100A
2.6V
60ns
MBR6030L
30V
60A
0.48V
444CNQ045
45V
440A
0.69V
30CPQ150
150V
30A
1.19V
S chot t ky rect i fi ers
Fundamentals of Power Electronics
32
Chapter 4: Switch realization
4.2.2. The Power MOSFET Source • Gate lengths approaching one micron
Gate
n
p
n
n
p
nn
n
• Consists of many small enhancementmode parallelconnected MOSFET cells, covering the surface of the silicon wafer • Vertical current flow • nchannel device is shown
Drain Fundamentals of Power Electronics
33
Chapter 4: Switch realization
MOSFET: Off state source
n
p
–
n
• pn junction is reversebiased
n
p
n
• offstate voltage appears across nregion
depletion region nn drain
Fundamentals of Power Electronics
+
34
Chapter 4: Switch realization
MOSFET: on state • pn junction is slightly reversebiased
source
n
p
n
n
p
• drain current flows through n region and conducting channel
channel nn drain
Fundamentals of Power Electronics
n
• positive gate voltage induces conducting channel
drain current
35
• on resistance = total resistances of nregion, conducting channel, source and drain contacts, etc. Chapter 4: Switch realization
MOSFET body diode • pn junction forms an effective diode, in parallel with the channel
source
n
p
n
n
p
Body diode
• negative draintosource voltage can forwardbias the body diode • diode can conduct the full MOSFET rated current
nn
• diode switching speed not optimized —body diode is slow, Qr is large
drain
Fundamentals of Power Electronics
n
36
Chapter 4: Switch realization
Typical MOSFET characteristics
=1 DS
• On state: VGS >> Vth V
V
=2 DS
V
10A
0V
00V
• Off state: VGS < Vth
V DS
ID
=2
• MOSFET can conduct peak currents well in excess of average current rating — characteristics are unchanged
on state 5A
V DS =
off state
1V
V DS = 0.5V
0A 0V
5V
10V
VGS Fundamentals of Power Electronics
37
15V
• onresistance has positive temperature coefficient, hence easy to parallel
Chapter 4: Switch realization
A simple MOSFET equivalent circuit D
• Cgs : large, essentially constant • Cgd : small, highly nonlinear Cgd G
• Cds : intermediate in value, highly nonlinear
Cds
• switching times determined by rate at which gate driver charges/ discharges Cgs and Cgd
Cgs
S
Cds(vds) =
Fundamentals of Power Electronics
C0
Cds(vds) ≈ C0
v 1 + ds V0 38
C '0 V0 vds = vds
Chapter 4: Switch realization
Characteristics of several commercial power MOSFETs
Rated max voltage
Rated avg current
R on
Qg (typical)
IRFZ48
60V
50A
0.018Ω
110nC
IRF510
100V
5.6A
0.54Ω
8.3nC
IRF540
100V
28A
0.077Ω
72nC
APT10M25BNR
100V
75A
0.025Ω
171nC
IRF740
400V
10A
0.55Ω
63nC
MTM15N40E
400V
15A
0.3Ω
110nC
APT5025BN
500V
23A
0.25Ω
83nC
APT1001RBNR
1000V
11A
1.0Ω
150nC
Part number
Fundamentals of Power Electronics
39
Chapter 4: Switch realization
MOSFET: conclusions
● ● ● ● ● ●
●
A majoritycarrier device: fast switching speed Typical switching frequencies: tens and hundreds of kHz Onresistance increases rapidly with rated blocking voltage Easy to drive The device of choice for blocking voltages less than 500V 1000V devices are available, but are useful only at low power levels (100W) Part number is selected on the basis of onresistance rather than current rating
Fundamentals of Power Electronics
40
Chapter 4: Switch realization
4.2.3. Bipolar Junction Transistor (BJT)
Base
• Interdigitated base and emitter contacts
Emitter
• Vertical current flow
n
p
n
n
• minority carrier device • onstate: baseemitter and collectorbase junctions are both forwardbiased
nn
• onstate: substantial minority charge in p and n regions, conductivity modulation
Collector Fundamentals of Power Electronics
• npn device is shown
41
Chapter 4: Switch realization
BJT switching times vs(t)
Vs2
–Vs1
VCC
vBE(t) 0.7V
RL iC(t) iB(t)
vs(t)
+ –
RB + vBE(t) –
–Vs1
+
iB(t) IB1
vCE(t) 0
–
–IB2 vCE(t) VCC IConRon iC(t)
ICon
0 (1) (2) (3)
Fundamentals of Power Electronics
42
(4)
(5)
(6)
(7)
(8)
(9)
t
Chapter 4: Switch realization
Ideal base current waveform
iB(t)
IB1 IBon
0 t –IB2
Fundamentals of Power Electronics
43
Chapter 4: Switch realization
Current crowding due to excessive IB2 Base
Emitter
–IB2
p
– –
n –
+
+
– –
n
–
p
can lead to formation of hot spots and device failure
n
Collector
Fundamentals of Power Electronics
44
Chapter 4: Switch realization
BJT characteristics IC
n
gio e r ve
acti
10A
ation
atur asis
qu
CE
V CE = 5V
saturation region
slope =β
5A
V CE = 200V V = 20V
VCE = 0.5V
• Off state: IB = 0 • On state: IB > IC /β • Current gain β decreases rapidly at high current. Device should not be operated at instantaneous currents exceeding the rated value
VCE = 0.2V
cutoff 0A 0V
5V
10V
15V
IB
Fundamentals of Power Electronics
45
Chapter 4: Switch realization
Breakdown voltages BVCBO: avalanche breakdown voltage of basecollector junction, with the emitter opencircuited
IC increasing IB
BVCEO: collectoremitter breakdown voltage with zero base current
IB = 0 open emitter
BVsus BVCEO
BVCBO
VCE
BVsus: breakdown voltage observed with positive base current In most applications, the offstate transistor voltage must not exceed BVCEO.
Fundamentals of Power Electronics
46
Chapter 4: Switch realization
Darlingtonconnected BJT
• Increased current gain, for highvoltage applications
Q1 Q2
D1
Fundamentals of Power Electronics
• In a monolithic Darlington device, transistors Q1 and Q2 are integrated on the same silicon wafer • Diode D1 speeds up the turnoff process, by allowing the base driver to actively remove the stored charge of both Q1 and Q2 during the turnoff transition
47
Chapter 4: Switch realization
Conclusions: BJT
●
●
●
BJT has been replaced by MOSFET in lowvoltage (> (t1 – t0)
0 t
diode waveforms
iL
iB(t) vB(t)
0
WD ≈
0 t
Vg (iL – iB(t)) dt
area –Qr
switching transition
Abruptrecovery diode: (t2 – t1) 0 • Negative inductor current removes diode stored charge Qr • When diode becomes reversebiased, negative inductor current flows through capacitor C.
0
t
area – Qr
vB(t)
t
0
–V2
• Ringing of LC network is damped by parasitic losses. Ringing energy is lost. Fundamentals of Power Electronics
71
t1
t2
t3
Chapter 4: Switch realization
Energy associated with ringing t3
Recovered charge is
Qr = –
iL(t) dt
vi(t)
V1
t2
t 0
Energy stored in inductor during interval t2 ≤ t ≤ t3 : t3 WL = vL(t) iL(t) dt t2
Applied inductor voltage during interval t2 ≤ t ≤ t3 : di (t) vL(t) = L L = – V2 dt Hence, t3 t3 diL(t) WL = L iL(t) dt = ( – V2) iL(t) dt dt t2 t2
–V2
iL(t)
0
vB(t)
t
0
–V2
W L = 12 L i 2L(t 3) = V2 Qr
t1
Fundamentals of Power Electronics
t
area – Qr
72
t2
t3
Chapter 4: Switch realization
4.3.4. Efficiency vs. switching frequency Add up all of the energies lost during the switching transitions of one switching period: W tot = W on + W off + W D + W C + W L + ...
Average switching power loss is Psw = W tot fsw
Total converter loss can be expressed as Ploss = Pcond + Pfixed + W tot fsw
where
Fundamentals of Power Electronics
Pfixed = fixed losses (independent of load and fsw) Pcond = conduction losses
73
Chapter 4: Switch realization
Efficiency vs. switching frequency
Ploss = Pcond + Pfixed + W tot fsw
Switching losses are equal to the other converter losses at the critical frequency
100% dc asymptote fcrit
90%
Pcond + Pfixed fcrit = W tot
80%
η
This can be taken as a rough upper limit on the switching frequency of a practical converter. For fsw > fcrit, the efficiency decreases rapidly with frequency.
70%
60%
50% 10kHz
100kHz
1MHz
fsw
Fundamentals of Power Electronics
74
Chapter 4: Switch realization
Summary of chapter 4 1. How an SPST ideal switch can be realized using semiconductor devices depends on the polarity of the voltage which the devices must block in the offstate, and on the polarity of the current which the devices must conduct in the onstate. 2. Singlequadrant SPST switches can be realized using a single transistor or a single diode, depending on the relative polarities of the offstate voltage and onstate current. 3. Twoquadrant SPST switches can be realized using a transistor and diode, connected in series (bidirectionalvoltage) or in antiparallel (bidirectionalcurrent). Several fourquadrant schemes are also listed here. 4. A “synchronous rectifier” is a MOSFET connected to conduct reverse current, with gate drive control as necessary. This device can be used where a diode would otherwise be required. If a MOSFET with sufficiently low Ron is used, reduced conduction loss is obtained.
Fundamentals of Power Electronics
75
Chapter 4: Switch realization
Summary of chapter 4 5. Majority carrier devices, including the MOSFET and Schottky diode, exhibit very fast switching times, controlled essentially by the charging of the device capacitances. However, the forward voltage drops of these devices increases quickly with increasing breakdown voltage. 6. Minority carrier devices, including the BJT, IGBT, and thyristor family, can exhibit high breakdown voltages with relatively low forward voltage drop. However, the switching times of these devices are longer, and are controlled by the times needed to insert or remove stored minority charge. 7. Energy is lost during switching transitions, due to a variety of mechanisms. The resulting average power loss, or switching loss, is equal to this energy loss multiplied by the switching frequency. Switching loss imposes an upper limit on the switching frequencies of practical converters.
Fundamentals of Power Electronics
76
Chapter 4: Switch realization
Summary of chapter 4 8. The diode and inductor present a “clamped inductive load” to the transistor. When a transistor drives such a load, it experiences high instantaneous power loss during the switching transitions. An example where this leads to significant switching loss is the IGBT and the “current tail” observed during its turnoff transition. 9. Other significant sources of switching loss include diode stored charge and energy stored in certain parasitic capacitances and inductances. Parasitic ringing also indicates the presence of switching loss.
Fundamentals of Power Electronics
77
Chapter 4: Switch realization
Chapter 5. The Discontinuous Conduction Mode
5.1. Origin of the discontinuous conduction mode, and mode boundary 5.2. Analysis of the conversion ratio M(D,K) 5.3. Boost converter example 5.4. Summary of results and key points
Fundamentals of Power Electronics
1
Chapter 5: Discontinuous conduction mode
Introduction to Discontinuous Conduction Mode (DCM) ●
●
●
●
Occurs because switching ripple in inductor current or capacitor voltage causes polarity of applied switch current or voltage to reverse, such that the current or voltageunidirectional assumptions made in realizing the switch are violated. Commonly occurs in dcdc converters and rectifiers, having singlequadrant switches. May also occur in converters having twoquadrant switches. Typical example: dcdc converter operating at light load (small load current). Sometimes, dcdc converters and rectifiers are purposely designed to operate in DCM at all loads. Properties of converters change radically when DCM is entered: M becomes loaddependent Output impedance is increased Dynamics are altered Control of output voltage may be lost when load is removed
Fundamentals of Power Electronics
2
Chapter 5: Discontinuous conduction mode
5.1. Origin of the discontinuous conduction mode, and mode boundary Buck converter example, with singlequadrant switches L
Q1
continuous conduction mode (CCM)
iL(t) Vg
+ –
C
D1
R
iD(t)
+
iL(t)
V
I
∆iL
–
Minimum diode current is (I – ∆iL) Dc component I = V/R Current ripple is (V – V) Vg DD'Ts ∆iL = g DTs = 2L 2L
0 conducting devices:
DTs
Ts D1
Q1
t Q1
iD(t) I
∆iL
Note that I depends on load, but ∆iL does not. 0
Fundamentals of Power Electronics
3
DTs
Ts
t
Chapter 5: Discontinuous conduction mode
Reduction of load current Increase R, until I = ∆iL
CCMDCM boundary L
Q1
iL(t)
+
iL(t) Vg
+ –
C
D1
R
V
iD(t)
–
I
∆iL 0
Minimum diode current is (I – ∆iL) Dc component I = V/R Current ripple is (V – V) Vg DD'Ts ∆iL = g DTs = 2L 2L
conducting devices:
Ts D1
Q1
t Q1
iD(t)
Note that I depends on load, but ∆iL does not. Fundamentals of Power Electronics
DTs
I
∆iL 0
4
DTs
Ts
t
Chapter 5: Discontinuous conduction mode
Further reduce load current Increase R some more, such that I < ∆iL L
Q1
iL(t)
+
iL(t) Vg
+ –
C
D1
Discontinuous conduction mode
R
V
iD(t)
– I
Minimum diode current is (I – ∆iL) Dc component I = V/R Current ripple is (V – V) Vg DD'Ts ∆iL = g DTs = 2L 2L
0 conducting devices:
t
Ts D2Ts D1
D3Ts
Q1
X
iD(t)
Note that I depends on load, but ∆iL does not. The load current continues to be positive and nonzero. Fundamentals of Power Electronics
DTs D1Ts Q1
0
DTs
Ts
t
D2Ts
5
Chapter 5: Discontinuous conduction mode
Mode boundary I > ∆iL for CCM I < ∆iL for DCM Insert buck converter expressions for I and ∆iL : DVg DD'TsVg < R 2L Simplify: 2L < D' RTs
This expression is of the form
where
Fundamentals of Power Electronics
K < K crit(D) for DCM K = 2L and K crit(D) = D' RTs
6
Chapter 5: Discontinuous conduction mode
K and Kcrit vs. D for K < 1:
2
for K > 1:
K < Kcrit: DCM Kc ( rit D) = 1
1
–D
K > Kcrit: CCM
2
K > Kcrit: CCM
K = 2L/RTs
Kc ( rit D) = 1–D
1
K = 2L/RTs 0
0 0
Fundamentals of Power Electronics
1
D
7
0
1
D
Chapter 5: Discontinuous conduction mode
Critical load resistance Rcrit
Solve Kcrit equation for load resistance R:
where
Fundamentals of Power Electronics
R < Rcrit(D) for CCM for DCM R > Rcrit(D) Rcrit(D) = 2L D'Ts
8
Chapter 5: Discontinuous conduction mode
Summary: mode boundary K > K crit(D) K < K crit(D)
or or
R < Rcrit(D) R > Rcrit(D)
for CCM for DCM
Table 5.1. CCMDCM mode boundaries for the buck, boost, and buckboost converters Converter
K crit(D)
max ( K crit )
0≤D≤1
Buck
(1 – D)
Boost
D (1 – D)2
1 4 27
(1 – D)2
1
Buckboost
Fundamentals of Power Electronics
9
R crit(D) 2L (1 – D)T s 2L D (1 – D) 2 T s
2L (1 – D) 2 T s
min ( Rcrit )
0≤D≤1
2 L Ts 27 L 2 Ts
2 L Ts
Chapter 5: Discontinuous conduction mode
5.2.
Analysis of the conversion ratio M(D,K)
Analysis techniques for the discontinuous conduction mode: Inductor voltsecond balance vL = 1 Ts
Ts
vL(t) dt = 0 0
Capacitor charge balance
iC
= 1 Ts
Ts
iC(t) dt = 0 0
Small ripple approximation sometimes applies: v(t) ≈ V
because ∆v I
Converter steadystate equations obtained via charge balance on each capacitor and voltsecond balance on each inductor. Use care in applying small ripple approximation. Fundamentals of Power Electronics
10
Chapter 5: Discontinuous conduction mode
Example: Analysis of DCM buck converter M(D,K) L
iL(t)
+ vL(t) –
subinterval 1
Vg
+ –
+
iC(t)
C
R
v(t) –
L
Q1
iL(t) +
iL(t) Vg
+ –
D1
C iD(t)
R
subinterval 2
V
L
+ vL(t) –
Vg
+ –
+
iC(t)
C
R
–
v(t) –
iL(t)
L + vL(t) –
subinterval 3
Vg
+ –
C
+
iC(t) R
v(t) –
Fundamentals of Power Electronics
11
Chapter 5: Discontinuous conduction mode
Subinterval 1
iL(t)
vL(t) = Vg – v(t) iC(t) = iL(t) – v(t) / R
L + vL(t) –
Vg
Small ripple approximation for v(t) (but not for i(t)!):
+ –
C
+
iC(t) R
v(t) –
vL(t) ≈ Vg – V iC(t) ≈ iL(t) – V / R
Fundamentals of Power Electronics
12
Chapter 5: Discontinuous conduction mode
Subinterval 2
iL(t)
vL(t) = – v(t) iC(t) = iL(t) – v(t) / R
L
+ vL(t) – Vg
Small ripple approximation for v(t) but not for i(t):
+ –
C
+
iC(t) R
v(t) –
vL(t) ≈ – V iC(t) ≈ iL(t) – V / R
Fundamentals of Power Electronics
13
Chapter 5: Discontinuous conduction mode
Subinterval 3
iL(t)
vL = 0, iL = 0 iC(t) = iL(t) – v(t) / R
L + vL(t) –
Vg
Small ripple approximation:
+ –
C
+
iC(t) R
v(t) –
vL(t) = 0 iC(t) = – V / R
Fundamentals of Power Electronics
14
Chapter 5: Discontinuous conduction mode
Inductor voltsecond balance vL(t) Vg – V D1Ts
D2Ts
D3Ts
0 Ts
t
–V
Voltsecond balance:
vL(t) = D1(Vg – V) + D2( – V) + D3(0) = 0 Solve for V:
V = Vg
Fundamentals of Power Electronics
D1 D1 + D2
note that D2 is unknown
15
Chapter 5: Discontinuous conduction mode
Capacitor charge balance L
node equation:
iL(t)
iL(t) = iC(t) + V / R
v(t)/R +
iC(t)
capacitor charge balance:
C
R
iC = 0
v(t) –
hence
iL = V / R must compute dc component of inductor current and equate to load current (for this buck converter example) Fundamentals of Power Electronics
iL(t) Vg – V L
= I
0
–V L
DTs D1Ts
16
ipk
Ts D2Ts
t
D3Ts
Chapter 5: Discontinuous conduction mode
Inductor current waveform peak current:
iL(t)
Vg – V iL(D1Ts) = i pk = D 1T s L average current:
iL = 1 Ts
ipk
Vg – V L
= I
–V L
Ts
iL(t) dt 0
0
DTs D1Ts
Ts D2Ts
t
D3 Ts
triangle area formula: Ts 0
iL(t) dt = 1 i pk (D1 + D2)Ts 2
iL = (Vg – V)
equate dc component to dc load current:
V = D1Ts (D + D ) (V – V) 1 2 g R 2L
D 1T s (D1 + D2) 2L
Fundamentals of Power Electronics
17
Chapter 5: Discontinuous conduction mode
Solution for V Two equations and two unknowns (V and D2):
V = Vg
D1 D1 + D2
V = D1Ts (D + D ) (V – V) 1 2 g R 2L
(from inductor voltsecond balance)
(from capacitor charge balance)
Eliminate D2 , solve for V :
V = 2 Vg 1 + 1 + 4K / D 21 where K = 2L / RTs valid for K < K crit
Fundamentals of Power Electronics
18
Chapter 5: Discontinuous conduction mode
Buck converter M(D,K) 1.0 K = 0.01
M(D,K) 0.8
K = 0.1 0.6
D
K = 0.5
0.4
M= K≥1
0.2
1+
2 1 + 4K / D 2
for K > K crit for K < K crit
0.0 0.0
0.2
0.4
0.6
0.8
1.0
D
Fundamentals of Power Electronics
19
Chapter 5: Discontinuous conduction mode
5.3. Boost converter example D1 i (t) D
L
i(t)
+ vL(t) – Vg
+
iC(t)
+ –
Q1
C
R
v(t) –
Mode boundary:
Previous CCM soln:
I > ∆iL for CCM I < ∆iL for DCM
Fundamentals of Power Electronics
I=
20
Vg D' 2 R
∆iL =
Vg DTs 2L
Chapter 5: Discontinuous conduction mode
Mode boundary
Vg DTsVg > 2L D' 2R
2L > DD' 2 RTs
where
4 Kcrit ( 13 ) = 27
0.15
for CCM
Kcrit(D)
for CCM
0.1
K > K crit(D) for CCM for DCM K < K crit(D) K = 2L and K crit(D) = DD' 2 RTs
0.05
0 0
0.2
0.4
0.6
0.8
1
D
Fundamentals of Power Electronics
21
Chapter 5: Discontinuous conduction mode
Mode boundary
CCM
0.15
where
K > K crit(D) for CCM for DCM K < K crit(D) K = 2L and K crit(D) = DD' 2 RTs
DCM K < Kcrit
CCM K > Kcrit
0.1
K
(D) K crit
0.05
0 0
0.2
0.4
0.6
0.8
1
D Fundamentals of Power Electronics
22
Chapter 5: Discontinuous conduction mode
Conversion ratio: DCM boost L
i(t)
+ vL(t) –
subinterval 1
Vg
+ –
+
iC(t) C
R
v(t) –
i(t)
D1 i (t) D
L + vL(t) –
Vg
+ –
i(t)
+
iC(t) Q1
C
+ vL(t) –
subinterval 2 R
Vg
v(t)
L
+ –
+
iC(t) C
R
v(t) –
– i(t)
L + vL(t) –
subinterval 3
Vg
+ –
+
iC(t) C
R
v(t) –
Fundamentals of Power Electronics
23
Chapter 5: Discontinuous conduction mode
Subinterval 1
i(t)
vL(t) = Vg iC(t) = – v(t) / R
+ vL(t) – + –
Vg
Small ripple approximation for v(t) (but not for i(t)!):
+
iC(t) C
R
v(t) –
vL(t) ≈ Vg iC(t) ≈ – V / R
Fundamentals of Power Electronics
L
0 < t < D1Ts
24
Chapter 5: Discontinuous conduction mode
Subinterval 2
i(t)
vL(t) = Vg – v(t) iC(t) = i(t) – v(t) / R
L + vL(t) –
Vg
+ –
C
Small ripple approximation for v(t) but not for i(t):
R
v(t) –
vL(t) ≈ Vg – V iC(t) ≈ i(t) – V / R
Fundamentals of Power Electronics
+
iC(t)
D1Ts < t < (D1 +D2)Ts
25
Chapter 5: Discontinuous conduction mode
Subinterval 3
i(t)
vL = 0, i = 0 iC(t) = – v(t) / R
L + vL(t) –
Vg
+ –
C
Small ripple approximation:
R
v(t) –
vL(t) = 0 iC(t) = – V / R
Fundamentals of Power Electronics
+
iC(t)
(D1 +D2)Ts < t < Ts
26
Chapter 5: Discontinuous conduction mode
Inductor voltsecond balance vL(t) Vg D1Ts
D2Ts
D3Ts
0 Ts
t
Vg – V
Voltsecond balance: D1Vg + D2(Vg – V) + D3(0) = 0
Solve for V: D + D2 V= 1 Vg D2
Fundamentals of Power Electronics
note that D2 is unknown
27
Chapter 5: Discontinuous conduction mode
Capacitor charge balance node equation: D1 i (t) D
iD(t) = iC(t) + v(t) / R
capacitor charge balance:
+
iC(t)
iC = 0
C
hence
R
v(t) –
iD = V / R
must compute dc component of diode current and equate to load current (for this boost converter example)
Fundamentals of Power Electronics
28
Chapter 5: Discontinuous conduction mode
Inductor and diode current waveforms i(t)
peak current: Vg i pk = DT L 1 s
L Vg – V
average diode current:
iD
= 1 Ts
0
L
Ts
iD(t) dt
0
0
triangle area formula: Ts
ipk
Vg
DTs D1Ts
iD(t)
Ts D2Ts
t
D3Ts
ipk
iD(t) dt = 1 i pk D2Ts 2
Vg – V L
0
DTs D1Ts
Fundamentals of Power Electronics
29
Ts D2Ts
t
D3Ts
Chapter 5: Discontinuous conduction mode
Equate diode current to load current
average diode current: iD
V g D 1 D 2T s 1 1 = i DT = Ts 2 pk 2 s 2L
equate to dc load current: V g D 1 D 2T s V = R 2L
Fundamentals of Power Electronics
30
Chapter 5: Discontinuous conduction mode
Solution for V Two equations and two unknowns (V and D2): D + D2 (from inductor voltsecond balance) V= 1 Vg D2
V g D 1 D 2T s V = R 2L
(from capacitor charge balance)
Eliminate D2 , solve for V. From voltsec balance eqn: Vg D2 = D1 V – Vg Substitute into charge balance eqn, rearrange terms:
V 2gD 21 V – VVg – =0 K 2
Fundamentals of Power Electronics
31
Chapter 5: Discontinuous conduction mode
Solution for V V 2gD 21 V – VVg – =0 K 2
Use quadratic formula:
V = 1± Vg
1 + 4D 21 / K 2
Note that one root leads to positive V, while other leads to negative V. Select positive root:
V = M(D ,K) = 1 + 1 Vg where valid for
1 + 4D 21 / K 2
K = 2L / RTs K < Kcrit(D)
Transistor duty cycle D = interval 1 duty cycle D1 Fundamentals of Power Electronics
32
Chapter 5: Discontinuous conduction mode
Boost converter characteristics 5
0.0
1
M(D,K)
K=
4
3
K
=
5 0.0
M =
.1
0 K=
2
K≥
1 1–D 1+
for K > K crit
1 + 4D 2 / K 2
for K < K crit
7
4/2
1
Approximate M in DCM: 0 0
0.25
0.5
0.75
1
D
Fundamentals of Power Electronics
33
M≈1+ D 2 K
Chapter 5: Discontinuous conduction mode
Summary of DCM characteristics
Table 5.2. S ummary of CCMDCM characteristics for the buck, boost, and buckboost converters K crit (D)
Converter Buck
(1 – D)
Boost
D (1 – D)2 (1 – D)2
Buckboost with
DCM M(D,K) 2 1 + 1 + 4K / D 2 1 + 1 + 4D 2 / K 2 – D K
K = 2L / RT s.
Fundamentals of Power Electronics
DCM D2(D,K) K M(D,K) D K M(D,K) D K
CCM M(D) D 1 1–D – D 1–D
DCM occurs for K < K crit .
34
Chapter 5: Discontinuous conduction mode
Summary of DCM characteristics
DCM M(D,K)
st
1)
o Bo
st
o bo
– (×
1 K
ck
Bu
• DCM buck and boost characteristics are asymptotic to M = 1 and to the DCM buckboost characteristic • DCM buckboost characteristic is linear
1
• CCM and DCM characteristics intersect at mode boundary. Actual M follows characteristic having larger magnitude
Buck
0 0
0.2
0.4
0.6
0.8
• DCM boost characteristic is nearly linear
D
Fundamentals of Power Electronics
1
35
Chapter 5: Discontinuous conduction mode
Summary of key points 1. The discontinuous conduction mode occurs in converters containing current or voltageunidirectional switches, when the inductor current or capacitor voltage ripple is large enough to cause the switch current or voltage to reverse polarity. 2. Conditions for operation in the discontinuous conduction mode can be found by determining when the inductor current or capacitor voltage ripples and dc components cause the switch onstate current or offstate voltage to reverse polarity. 3. The dc conversion ratio M of converters operating in the discontinuous conduction mode can be found by application of the principles of inductor voltsecond and capacitor charge balance.
Fundamentals of Power Electronics
36
Chapter 5: Discontinuous conduction mode
Summary of key points 4. Extra care is required when applying the smallripple approximation. Some waveforms, such as the output voltage, should have small ripple which can be neglected. Other waveforms, such as one or more inductor currents, may have large ripple that cannot be ignored. 5. The characteristics of a converter changes significantly when the converter enters DCM. The output voltage becomes loaddependent, resulting in an increase in the converter output impedance.
Fundamentals of Power Electronics
37
Chapter 5: Discontinuous conduction mode
Chapter 6. Converter Circuits
• Where do the boost, buckboost, and other converters originate?
6.1. Circuit manipulations 6.2. A short list of converters
• How can we obtain a converter having given desired properties?
6.3. Transformer isolation
• What converters are possible?
6.4. Converter evaluation and design
• How can we obtain transformer isolation in a converter?
6.5. Summary of key points
Fundamentals of Power Electronics
• For a given application, which converter is best? 1
Chapter 6: Converter circuits
6.1. Circuit manipulations L
1
+
Vg
+ –
2
C
R
V –
Begin with buck converter: derived in chapter 1 from first principles • Switch changes dc component, lowpass filter removes switching harmonics • Conversion ratio is M = D
Fundamentals of Power Electronics
2
Chapter 6: Converter circuits
6.1.1. Inversion of source and load Interchange power input and output ports of a converter Buck converter example
V2 = DV1
port 1
L
1
+ + –
port 2
+ 2
V1
V2
–
–
Power flow
Fundamentals of Power Electronics
3
Chapter 6: Converter circuits
Inversion of source and load
Interchange power source and load: port 1
L
1
port 2
+
+ 2
V1
V2
–
–
+ –
Power flow
V1 = 1 V2 D
V2 = DV1
Fundamentals of Power Electronics
4
Chapter 6: Converter circuits
Realization of switches as in chapter 4
• Reversal of power flow requires new realization of switches
port 1
• Transistor conducts when switch is in position 2
L
port 2
+
+
V1
V2
–
–
+ –
Power flow
• Interchange of D and D’ V1 = 1 V2 D'
Inversion of buck converter yields boost converter
Fundamentals of Power Electronics
5
Chapter 6: Converter circuits
6.1.2. Cascade connection of converters
converter 1 Vg
+ –
V1 = M 1(D) Vg
converter 2
+ V1 –
V = M (D) 2 V1
+ V –
D
V1 = M 1(D) Vg V = M 2(D) V1
Fundamentals of Power Electronics
V = M(D) = M (D) M (D) 1 2 Vg
6
Chapter 6: Converter circuits
Example: buck cascaded by boost L1
1
L2
2
+ Vg
+ –
+ 1
2
V1
C1
C2
R
V –
{ {
–
Buck converter
V1 =D Vg V = 1 V1 1 – D Fundamentals of Power Electronics
Boost converter
V = D Vg 1 – D
7
Chapter 6: Converter circuits
Buck cascaded by boost: simplification of internal filter remove capacitor C1 L1
1
L2
2
+ Vg
1
2
+ –
C2
R
V –
combine inductors L1 and L2 L
1
iL
2
+ Vg
+ –
2
1
V
Noninverting buckboost converter
–
Fundamentals of Power Electronics
8
Chapter 6: Converter circuits
Noninverting buckboost converter L
1
iL
2
+ Vg
1
2
+ –
V –
subinterval 1
iL Vg
+ –
subinterval 2 +
+ Vg
V
V iL
–
Fundamentals of Power Electronics
+ –
9
–
Chapter 6: Converter circuits
Reversal of output voltage polarity subinterval 1
subinterval 2
+
iL
noninverting buckboost
Vg
+ –
V
+ Vg
+ –
V iL
–
+
iL
inverting buckboost
Vg
+ –
V –
Fundamentals of Power Electronics
10
iL Vg
+ –
–
+ V –
Chapter 6: Converter circuits
Reduction of number of switches: inverting buckboost subinterval 1 +
iL Vg
subinterval 2
+ –
+
iL
V
Vg
+ –
V
–
–
One side of inductor always connected to ground — hence, only one SPDT switch needed: 1
Vg
+ –
+
2
iL
V
V =– D Vg 1–D
–
Fundamentals of Power Electronics
11
Chapter 6: Converter circuits
Discussion: cascade connections
• Properties of buckboost converter follow from its derivation as buck cascaded by boost Equivalent circuit model: buck 1:D transformer cascaded by boost D’:1 transformer Pulsating input current of buck converter Pulsating output current of boost converter
• Other cascade connections are possible Cuk converter: boost cascaded by buck
Fundamentals of Power Electronics
12
Chapter 6: Converter circuits
6.1.3. Rotation of threeterminal cell
t A
a
eet r h
er mi na l c
el
l
Treat inductor and SPDT switch as threeterminal cell:
b
1
B +
Vg
2
+ –
v c C
–
Threeterminal cell can be connected between source and load in three nontrivial distinct ways: aA bB cC
buck converter
aC bA cB
boost converter
aA bC cB
buckboost converter
Fundamentals of Power Electronics
13
Chapter 6: Converter circuits
Rotation of a dual threeterminal network  t er m i n a l c 1
l
th
re e
el
A capacitor and SPDT switch as a threeterminal cell:
A a
Vg
b B
+
2
+ –
v c C
–
Threeterminal cell can be connected between source and load in three nontrivial distinct ways: aA bB cC
buck converter with LC input filter
aC bA cB
boost converter with LC output filter
aA bC cB
Cuk converter
Fundamentals of Power Electronics
14
Chapter 6: Converter circuits
6.1.4. Differential connection of load to obtain bipolar output voltage dc source
load
converter 1
+ V1
V1 = M(D) Vg
+
–
V Vg
+ –
The outputs V1 and V2 may both be positive, but the differential output voltage V can be positive or negative.
+ V2
V2 = M(D') Vg
V = V1 – V2
–
D
converter 2
Differential load voltage is
–
D'
Fundamentals of Power Electronics
15
Chapter 6: Converter circuits
Differential connection using two buck converters
}
Buck converter 1 1
Converter #1 transistor driven with duty cycle D
+ 2
V1 +
–
V Vg
–
+ –
Converter #2 transistor driven with duty cycle complement D’ Differential load voltage is
2 1
+
V = DVg – D'Vg
V2
Simplify:
–
{
V = (2D – 1) Vg
Buck converter 2
Fundamentals of Power Electronics
16
Chapter 6: Converter circuits
Conversion ratio M(D), differentiallyconnected buck converters V = (2D – 1) Vg M(D) 1
0 0.5
1
D
–1
Fundamentals of Power Electronics
17
Chapter 6: Converter circuits
Simplification of filter circuit, differentiallyconnected buck converters Original circuit
Bypass load directly with capacitor
}
Buck converter 1 1
1
+ 2
2
V1 –
Vg
+
+
V
V
–
+ –
Vg 2
–
+ – 2
1
+
1
V2
{
–
Buck converter 2
Fundamentals of Power Electronics
18
Chapter 6: Converter circuits
Simplification of filter circuit, differentiallyconnected buck converters Combine seriesconnected inductors
Redraw for clarity
1
1
C Vg
2
+ – 2
2
L iL
+
V – 1
R + Vg
V
+ –
– 2 1
Fundamentals of Power Electronics
Hbridge, or bridge inverter Commonly used in singlephase inverter applications and in servo amplifier applications
19
Chapter 6: Converter circuits
Differential connection to obtain 3ø inverter dc source
3øac load
V1 = M(D 1) Vg
+ V1
Vn = 1 (V1 + V2 + V3) 3 Phase voltages are
–
+
converter 1
va
n
D1 + –
–
Vg
converter 2
+ vbn –
v cn
V2
+
–
V3 = M(D 3) Vg
+ V3 –
D3
Fundamentals of Power Electronics
Van = V1 – Vn Vbn = V2 – Vn Vcn = V3 – Vn Control converters such that their output voltages contain the same dc biases. This dc bias will appear at the neutral point Vn. It then cancels out, so phase voltages contain no dc bias.
D2
converter 3
Vn
–
V2 = M(D 2) Vg
+
With balanced 3ø load, neutral voltage is
20
Chapter 6: Converter circuits
3ø differential connection of three buck converters dc source
3øac load + V1
va
n
+
–
+ –
–
Vg
+
v cn +
–
–
V2
Vn
+ vbn –
+ V3 –
Fundamentals of Power Electronics
21
Chapter 6: Converter circuits
3ø differential connection of three buck converters Redraw for clarity: 3øac load
–
va
n
+
dc source
Vg
+ –
Vn
+ vbn –
– v cn +
“Voltagesource inverter” or buckderived threephase inverter
Fundamentals of Power Electronics
22
Chapter 6: Converter circuits
6.2. A short list of converters
An infinite number of converters are possible, which contain switches embedded in a network of inductors and capacitors Two simple classes of converters are listed here: • Singleinput singleoutput converters containing a single inductor. The switching period is divided into two subintervals. This class contains eight converters. • Singleinput singleoutput converters containing two inductors. The switching period is divided into two subintervals. Several of the more interesting members of this class are listed.
Fundamentals of Power Electronics
23
Chapter 6: Converter circuits
Singleinput singleoutput converters containing one inductor • Use switches to connect inductor between source and load, in one manner during first subinterval and in another during second subinterval • There are a limited number of ways to do this, so all possible combinations can be found • After elimination of degenerate and redundant cases, eight converters are found:
dcdc converters buck
boost
buckboost
noninverting buckboost
dcac converters bridge
WatkinsJohnson
acdc converters currentfed bridge Fundamentals of Power Electronics
inverse of WatkinsJohnson 24
Chapter 6: Converter circuits
Converters producing a unipolar output voltage
M(D) = D
1. Buck
M(D) 1
1
+ Vg
+ –
2
M(D) =
2. Boost
V
0.5
–
0
1 1–D
D
0
0.5
1
D
3
1
2
V
1 0
–
Fundamentals of Power Electronics
1
4
+ Vg
0.5
M(D)
2
+ –
0
25
Chapter 6: Converter circuits
Converters producing a unipolar output voltage
3. Buckboost
M(D) = – 1
D 1–D
0.5
1
D
0
0.5
1
D
–1
+
2
0 0
–2
Vg
+ –
V
–3 –4
–
4. Noninverting buckboost
M(D) =
M(D)
D 1–D
M(D)
2
1
4
+ Vg
+ –
2
3
1
2
V 1 0
–
Fundamentals of Power Electronics
26
Chapter 6: Converter circuits
Converters producing a bipolar output voltage suitable as dcac inverters M(D) = 2D – 1
5. Bridge
M(D) 1
1
Vg
2
+ –
+
V –
2
0
1
0.5
1
D
0.5
1
D
–1
M(D) = 2D – 1 D
6. WatkinsJohnson
M(D) 1
2
1
+
1
+
or
0 –1
Vg
+ –
V 1
2
Vg
+ –
2
–
–
Fundamentals of Power Electronics
V
27
–2 –3
Chapter 6: Converter circuits
Converters producing a bipolar output voltage suitable as acdc rectifiers
M(D) =
7. Currentfed bridge
M(D)
1 2D – 1
2 1 0.5
1
D
0.5
1
D
0
1
Vg
2 –1
+ –
+
V –
2
–2
1
8. Inverse of WatkinsJohnson
M(D) =
D 2D – 1
M(D)
1 1
Vg
2
+ –
+ V
2
1
2
or
+ Vg
+ –
0
V
2
–1
–
Fundamentals of Power Electronics
1
–
28
–2
Chapter 6: Converter circuits
Several members of the class of twoinductor converters
M(D) = –
1. Cuk
D 1–D
0
0.5
1
D
0
0.5
1
D
0 –1
+ Vg
1
+ –
2
–2 –3
V
–4
M(D)
–
M(D) =
2. SEPIC
D 1–D
M(D) 4
2
Vg + –
1
+ 3 2
V
1
–
Fundamentals of Power Electronics
0
29
Chapter 6: Converter circuits
Several members of the class of twoinductor converters
M(D) =
3. Inverse of SEPIC 1
D 1–D
M(D) 4
+ Vg
+ –
3 2
2
V
1 0
–
M(D) = D 2
4. Buck 2
0
0.5
1
D
0
0.5
1
D
M(D)
1
Vg
+ –
2
1
V
0.5
2
1
Fundamentals of Power Electronics
+
–
30
0
Chapter 6: Converter circuits
6.3. Transformer isolation Objectives: • Isolation of input and output ground connections, to meet safety requirements • Reduction of transformer size by incorporating high frequency isolation transformer inside converter • Minimization of current and voltage stresses when a large stepup or stepdown conversion ratio is needed — use transformer turns ratio • Obtain multiple output voltages via multiple transformer secondary windings and multiple converter secondary circuits
Fundamentals of Power Electronics
31
Chapter 6: Converter circuits
A simple transformer model Multiple winding transformer i1(t)
n1 : n 2
Equivalent circuit model i1(t)
i2(t)
+
+
+
v1(t)
v2(t)
v1(t)
–
–
–
i1'(t)
n1 : n2
i2(t) +
iM(t) LM
v2(t) – i3(t)
i3(t)
+
+ v3(t) – : n3
v1(t) v2(t) v3(t) n 1 = n 2 = n 3 = ... 0 = n 1i1'(t) + n2i2(t) + n 3i3(t) + ...
v3(t) – : n3 ideal transformer
Fundamentals of Power Electronics
32
Chapter 6: Converter circuits
The magnetizing inductance LM Transformer core BH characteristic
• Models magnetization of transformer core material
B(t) ∝
• Appears effectively in parallel with windings • If all secondary windings are disconnected, then primary winding behaves as an inductor, equal to the magnetizing inductance
v1(t) dt
saturation
slope ∝ LM H(t) ∝ i M (t)
• At dc: magnetizing inductance tends to shortcircuit. Transformers cannot pass dc voltages • Transformer saturates when magnetizing current iM is too large Fundamentals of Power Electronics
33
Chapter 6: Converter circuits
Voltsecond balance in LM The magnetizing inductance is a real inductor, obeying di (t) v1(t) = L M M dt integrate: t 1 iM (t) – iM (0) = v (τ) dτ LM 0 1 Magnetizing current is determined by integral of the applied winding voltage. The magnetizing current and the winding currents are independent quantities. Voltsecond balance applies: in steadystate, iM(Ts) = iM(0), and hence
0= 1 Ts
i1(t) + v1(t)
i1'(t)
n1 : n 2
i2(t) +
iM(t) LM
v2(t)
–
– i3(t) + v3(t) – : n3 ideal transformer
Ts
v1(t) dt 0
Fundamentals of Power Electronics
34
Chapter 6: Converter circuits
Transformer reset • “Transformer reset” is the mechanism by which magnetizing inductance voltsecond balance is obtained • The need to reset the transformer voltseconds to zero by the end of each switching period adds considerable complexity to converters • To understand operation of transformerisolated converters: • replace transformer by equivalent circuit model containing magnetizing inductance • analyze converter as usual, treating magnetizing inductance as any other inductor • apply voltsecond balance to all converter inductors, including magnetizing inductance
Fundamentals of Power Electronics
35
Chapter 6: Converter circuits
6.3.1. Fullbridge and halfbridge isolated buck converters Fullbridge isolated buck converter
Q1
D1
Q3
D3
i1(t)
1 : n
D5
L
iD5(t)
i(t)
+
+
+ Vg
+ –
vT(t)
vs(t)
C
R
v
– – Q2
D2
Fundamentals of Power Electronics
Q4
: n
D4
36
–
D6
Chapter 6: Converter circuits
Fullbridge, with transformer equivalent circuit
Q1
Vg
Q3
D1
+ –
D3
i1(t)
i1'(t)
+
iM(t)
vT(t)
LM
1 : n
D5
L
iD5(t) i(t)
+
+ vs(t)
C
R
v
– – Q2
D2
Q4
: n
D4
ideal
D6
–
iD6(t)
transformer model
Fundamentals of Power Electronics
37
Chapter 6: Converter circuits
Fullbridge: waveforms iM(t)
Vg LM
vT(t)
• During first switching period: transistors Q1 and Q4 conduct for time DTs , applying voltseconds Vg DTs to primary winding
– Vg LM
Vg 0
0 –Vg
• During next switching period: transistors Q2 and Q3 conduct for time DTs , applying voltseconds –Vg DTs to primary winding
i(t) ∆i
I
vs(t)
nVg
nVg 0
iD5(t)
0
i 0.5 i
0.5 i
t
0 0
conducting devices:
DTs Q1 Q4 D5
Ts D5 D6
Fundamentals of Power Electronics
Ts+DTs Q2 Q3 D6
• Transformer voltsecond balance is obtained over two switching periods
2Ts
• Effect of nonidealities?
D5 D6
38
Chapter 6: Converter circuits
Effect of nonidealities on transformer voltsecond balance Voltseconds applied to primary winding during first switching period: (Vg – (Q1 and Q4 forward voltage drops))( Q1 and Q4 conduction time) Voltseconds applied to primary winding during next switching period: – (Vg – (Q2 and Q3 forward voltage drops))( Q2 and Q3 conduction time) These voltseconds never add to exactly zero. Net voltseconds are applied to primary winding Magnetizing current slowly increases in magnitude Saturation can be prevented by placing a capacitor in series with primary, or by use of current programmed mode (chapter 11) Fundamentals of Power Electronics
39
Chapter 6: Converter circuits
Operation of secondaryside diodes D5
L
iD5(t) i(t)
vs(t)
C
R
– D6 vs(t)
• Output filter inductor current divides approximately equally between diodes
iD6(t) nVg 0
0
0.5 i
0.5 i
• Secondary ampturns add to approximately zero
i 0 0
conducting devices:
v –
nVg
iD5(t)
• During second (D’) subinterval, both secondaryside diodes conduct
+
+
DTs Q1 Q4 D5
Ts D5 D6
Fundamentals of Power Electronics
t Ts+DTs
Q2 Q3 D6
2Ts D5 D6
40
• Essentially no net magnetization of transformer core by secondary winding currents Chapter 6: Converter circuits
Voltsecond balance on output filter inductor
i(t)
D5
iD5(t) i(t) + vs(t)
+ C
– D6
∆i
I
L
R
vs(t)
nVg
nVg
v –
iD5(t)
0
0.5 i
0.5 i
i
iD6(t)
0 0
V = vs
0
conducting devices:
DTs Q1 Q4 D5
Ts D5 D6
t Ts+DTs
Q2 Q3 D6
2Ts D5 D6
V = nDVg
M(D) = nD
Fundamentals of Power Electronics
buck converter with turns ratio
41
Chapter 6: Converter circuits
Halfbridge isolated buck converter
Q1
D1 Ca
i1(t)
1 : n
D3
+ –
i(t) +
+
+ Vg
L
iD3(t)
vT(t)
vs(t)
C
R
v
– – Q2
D2
Cb
: n
–
D4
• Replace transistors Q3 and Q4 with large capacitors • Voltage at capacitor centerpoint is 0.5Vg • vs(t) is reduced by a factor of two • M = 0.5 nD
Fundamentals of Power Electronics
42
Chapter 6: Converter circuits
6.3.2. Forward converter n1 : n2 : n3
D2
L + D3
Vg
+ –
C
R
V –
Q1 D1
• Buckderived transformerisolated converter • Singletransistor and twotransistor versions • Maximum duty cycle is limited • Transformer is reset while transistor is off Fundamentals of Power Electronics
43
Chapter 6: Converter circuits
Forward converter with transformer equivalent circuit D2
n1 : n2 : n3
Vg
iM
+
LM + –
i1'
–
+
v1
v2
v3
–
+
–
i1 Q1
+
+
D3
vD3 –
i3
i2
L
C
R
V –
+ vQ1
D1
–
Fundamentals of Power Electronics
44
Chapter 6: Converter circuits
Forward converter: waveforms v1
Vg
• Magnetizing current, in conjunction with diode D1, operates in discontinuous conduction mode
0
n – n 1 Vg 2
iM
n Vg – n1 2 LM
0
0
0
DTs
D 2T s Ts
D 3T s
Q1 D2
D1 D3
D3
Vg LM
vD3
conducting devices:
• Output filter inductor, in conjunction with diode D3, may operate in either CCM or DCM
n3 n 1 Vg
Fundamentals of Power Electronics
t
45
Chapter 6: Converter circuits
Subinterval 1: transistor conducts
n1 : n2 : n3
Vg
+ –
iM
+
LM
i1'
+
v1
v2
v3
–
+
–
Q1 on
Fundamentals of Power Electronics
+
+ –
i1
L
D2 on
i2
vD3 –
i3
C
R
V –
D1 off
46
Chapter 6: Converter circuits
Subinterval 2: transformer reset L
n1 : n2 : n3
Vg
+ –
iM
+
LM
i1'
–
+
v1
v2
v3
–
+
–
i1 Q1 off
+
+
D3 on vD3 –
i3
C
R
V –
i2 = iM n1 /n2 D1 on
Fundamentals of Power Electronics
47
Chapter 6: Converter circuits
Subinterval 3
L
n1 : n2 : n3
iM =0 LM Vg
+ –
i1'
–
+
v1
v2
v3
–
+
–
+
i1
+
+
i2
D3 on vD3 –
i3
C
R
V –
Q1 off D1 off
Fundamentals of Power Electronics
48
Chapter 6: Converter circuits
Magnetizing inductance voltsecond balance v1
Vg 0
n – n 1 Vg 2
iM
Vg LM
conducting devices:
n Vg – n1 2 LM
0
DTs
D2Ts Ts
D3Ts
Q1 D2
D1 D3
D3
t
v1 = D ( Vg ) + D2 ( – Vg n 1 / n 2 ) + D3 ( 0 ) = 0 Fundamentals of Power Electronics
49
Chapter 6: Converter circuits
Transformer reset From magnetizing current voltsecond balance:
v1 = D ( Vg ) + D2 ( – Vg n 1 / n 2 ) + D3 ( 0 ) = 0 Solve for D2: n D2 = n 2 D 1 D3 cannot be negative. But D3 = 1 – D – D2. Hence D3 = 1 – D – D2 ≥ 0 n D3 = 1 – D 1 + 2 ≥ 0 n1 Solve for D D ≤ 1n 1+ 2 n1 Fundamentals of Power Electronics
for n1 = n2:
50
D≤ 1 2
Chapter 6: Converter circuits
What happens when D > 0.5 magnetizing current waveforms, for n1 = n2
iM(t)
D < 0.5
DTs D2Ts D3Ts
iM(t)
D > 0.5
DTs Fundamentals of Power Electronics
t
D2Ts 51
2Ts
t
Chapter 6: Converter circuits
Conversion ratio M(D) L +
+
D2 D3
vD3
C
R
V
– vD3
–
n3 n 1 Vg
n vD3 = V = n 3 D Vg 1
conducting devices:
0
0
DTs
D2Ts Ts
D3 Ts
Q1 D2
D1 D3
D3
Fundamentals of Power Electronics
52
t
Chapter 6: Converter circuits
Maximum duty cycle vs. transistor voltage stress Maximum duty cycle limited to
D≤
1 n 1+ 2 n1
which can be increased by increasing the turns ratio n2 / n1. But this increases the peak transistor voltage:
n max vQ1 = Vg 1 + n 1 2 For n1 = n2
D≤ 1 2
Fundamentals of Power Electronics
and
max vQ1 = 2Vg
53
Chapter 6: Converter circuits
The twotransistor forward converter Q1
D3
D1
L +
1:n
Vg
+ –
D4
C
R
V –
D2
Q2
V = nDVg
Fundamentals of Power Electronics
D≤ 1 2
max vQ1 = max vQ2 = Vg
54
Chapter 6: Converter circuits
6.3.3. Pushpull isolated buck converter Q1 1 : n
Vg
+ –
– vT(t) +
D1
V = nDVg
Fundamentals of Power Electronics
+
+
iD1(t)
vs(t)
– vT(t) + Q2
L
i(t)
–
C
R
V –
D2
0≤D≤1
55
Chapter 6: Converter circuits
Waveforms: pushpull iM(t)
• Used with lowvoltage inputs Vg LM
vT(t)
• Secondaryside circuit identical to full bridge
– Vg LM
Vg
• As in full bridge, transformer voltsecond balance is obtained over two switching periods
0
0 –Vg i(t) ∆i
I
vs(t)
nVg
iD1(t)
• Effect of nonidealities on transformer voltsecond balance?
nVg 0
0
0.5 i
0.5 i
i 0 0
conducting devices:
DTs Q1 D1
Ts D1 D2
Fundamentals of Power Electronics
t Ts+DTs
Q2 D2
2Ts D1 D2
56
• Current programmed control can be used to mitigate transformer saturation problems. Duty cycle control not recommended. Chapter 6: Converter circuits
6.3.4. Flyback converter Q1
buckboost converter:
D1 –
Vg
+ –
L
V +
Q1
construct inductor winding using two parallel wires:
D1 1:1
Vg
+ –
L
– V +
Fundamentals of Power Electronics
57
Chapter 6: Converter circuits
Derivation of flyback converter, cont. Q1
Isolate inductor windings: the flyback converter
D1 –
1:1
Vg
+ –
V
LM
+
Flyback converter having a 1:n turns ratio and positive output:
1:n
Vg
+ –
LM Q1
Fundamentals of Power Electronics
+ D1 C
V –
58
Chapter 6: Converter circuits
The “flyback transformer” transformer model ●
ig
+ –
Vg
i
+
LM
vL
1:n
+
iC
D1
●
C
R
– – Q1
● ● ● ●
v
●
●
A twowinding inductor Symbol is same as transformer, but function differs significantly from ideal transformer Energy is stored in magnetizing inductance Magnetizing inductance is relatively small
Current does not simultaneously flow in primary and secondary windings Instantaneous winding voltages follow turns ratio Instantaneous (and rms) winding currents do not follow turns ratio Model as (small) magnetizing inductance in parallel with ideal transformer
Fundamentals of Power Electronics
59
Chapter 6: Converter circuits
Subinterval 1
transformer model
ig Vg
+ –
i
+
LM
vL
+
iC
1:n
C
R
v
– –
vL = Vg iC = – v R ig = i CCM: small ripple approximation leads to
vL = Vg iC = – V R ig = I
Fundamentals of Power Electronics
60
Chapter 6: Converter circuits
Subinterval 2
transformer model
ig =0 Vg
+ –
+ vL –
i
i/n
– v/n +
+
iC
1:n
C
R
v –
vL = – nv iC = ni – v R ig = 0 CCM: small ripple approximation leads to
vL = – V n iC = nI – V R ig = 0
Fundamentals of Power Electronics
61
Chapter 6: Converter circuits
CCM Flyback waveforms and solution vL
Vg
Voltsecond balance: vL = D (Vg) + D' (– V n) = 0 –V/n
Conversion ratio is M(D) = V = n D Vg D' Charge balance: iC = D (– V ) + D' ( nI – V ) = 0 R R Dc component of magnetizing current is I = nV D'R Dc component of source current is
I/n – V/R
iC
–V/R ig I
0 DTs conducting devices:
D'Ts
t
I g = ig = D (I) + D' (0)
Ts Q1
Fundamentals of Power Electronics
D1
62
Chapter 6: Converter circuits
Equivalent circuit model: CCM Flyback vL = D (Vg) + D' (– V n) = 0 iC = D (– V ) + D' ( nI – V ) = 0 R R I g = ig = D (I) + D' (0)
Vg
+ –
DI
+ DV g –
D'V n
+ –
D'I n
R
D' : n +
I
Ig + –
R
V –
Fundamentals of Power Electronics
V –
1:D
Vg
+
I
Ig
63
Chapter 6: Converter circuits
Discussion: Flyback converter
● ● ● ● ● ●
Widely used in low power and/or high voltage applications Low parts count Multiple outputs are easily obtained, with minimum additional parts Cross regulation is inferior to buckderived isolated converters Often operated in discontinuous conduction mode DCM analysis: DCM buckboost with turns ratio
Fundamentals of Power Electronics
64
Chapter 6: Converter circuits
6.3.5. Boostderived isolated converters • A wide variety of boostderived isolated dcdc converters can be derived, by inversion of source and load of buckderived isolated converters: • fullbridge and halfbridge isolated boost converters • inverse of forward converter: the “reverse” converter • pushpull boostderived converter Of these, the fullbridge and pushpull boostderived isolated converters are the most popular, and are briefly discussed here.
Fundamentals of Power Electronics
65
Chapter 6: Converter circuits
Fullbridge transformerisolated boostderived converter i(t)
L + vT(t) – 1 : n
Q1
D1
io(t)
Q3
+
+ Vg
+ –
vT(t)
C
R
v
– – Q2
: n
Q4
D2
• Circuit topologies are equivalent to those of nonisolated boost converter • With 1:1 turns ratio, inductor current i(t) and output current io(t) waveforms are identical to nonisolated boost converter Fundamentals of Power Electronics
66
Chapter 6: Converter circuits
Transformer reset mechanism • As in fullbridge buck topology, transformer voltsecond balance is obtained over two switching periods.
V/n
vT(t) 0
0 – V/n
• During first switching period: transistors Q1 and Q4 conduct for time DTs , applying voltseconds VDTs to secondary winding.
io(t) I/n
0 DTs conducting devices:
Q1 Q2 Q3 Q4
D'Ts Ts
Q1 Q4 D1
Fundamentals of Power Electronics
I/n
0 DTs Q1 Q2 Q3 Q4
D'Ts Ts
67
Q2 Q3 D2
t
• During next switching period: transistors Q2 and Q3 conduct for time DTs , applying voltseconds –VDTs to secondary winding. Chapter 6: Converter circuits
Conversion ratio M(D)
vL(t)
Vg
Application of voltsecond balance to inductor voltage waveform:
Vg
Vg –V/n
Vg –V/n
vL = D (Vg) + D' (Vg – V / n) = 0 i(t)
Solve for M(D):
I
M(D) = V = n Vg D' DTs conducting devices:
Q1 Q2 Q3 Q4
Q1 Q4 D1
Fundamentals of Power Electronics
Q1 Q2 Q3 Q4
D'Ts Ts
68
t
—boost with turns ratio n
Q2 Q3 D2
Chapter 6: Converter circuits
Pushpull boostderived converter
Q1 1 : n
Vg
i(t)
L
+ – + vL(t) –
D1
– vT(t) +
+
io(t) C
– vT(t) +
R
V –
D2
Q2
M(D) = V = n Vg D'
Fundamentals of Power Electronics
69
Chapter 6: Converter circuits
Pushpull converter based on WatkinsJohnson converter
Q1 1 : n
D1 +
Vg
+ –
C
R
V –
D2
Q2
Fundamentals of Power Electronics
70
Chapter 6: Converter circuits
6.3.6. Isolated versions of the SEPIC and Cuk converter
L1
C1
D1
Basic nonisolated SEPIC
+ Vg
+ –
C2
L2
R
v
Q1 –
L1
Isolated SEPIC
C1
i1 Vg
+ –
D1 +
is
ip
C2
R
v
Q1 – 1:n
Fundamentals of Power Electronics
71
Chapter 6: Converter circuits
Isolated SEPIC ip(t)
L1 i1 Vg
+ –
C1
ip
1:n
i2
Q1
LM = L2
i1
D1 +
is
– i2
C2
R
v
(i1 + i2) / n
is(t)
– 0
ideal transformer model i1(t) I1
M(D) = V = n D D' Vg
i2(t) I2
conducting devices:
Fundamentals of Power Electronics
72
DTs Q1
D'Ts
t
Ts D1
Chapter 6: Converter circuits
Inverse SEPIC C1
Q1
Nonisolated inverse SEPIC
L2 +
Vg
+ –
L1
D1
C2
R
v –
1:n
C1
L2 +
Isolated inverse SEPIC Vg
D1
+ – Q1
Fundamentals of Power Electronics
C2
R
v –
73
Chapter 6: Converter circuits
Obtaining isolation in the Cuk converter L1
L2
Nonisolated Cuk converter
+ C1 Vg
+ –
Q1
D1
C2
R
v –
Split capacitor C1 into series capacitors C1a and C1b
C1a
L1
C1b
L2 +
Vg
+ –
Q1
D1
C2
R
v –
Fundamentals of Power Electronics
74
Chapter 6: Converter circuits
Isolated Cuk converter
Insert transformer between capacitors C1a and C1b
C1a
L1
C1b
L2 +
Vg
+ –
Q1
D1
M(D) = V = n D D' Vg
C2
R
v –
1:n
Discussion • Capacitors C1a and C1b ensure that no dc voltage is applied to transformer primary or secondary windings • Transformer functions in conventional manner, with small magnetizing current and negligible energy storage within the magnetizing inductance Fundamentals of Power Electronics
75
Chapter 6: Converter circuits
6.4. Converter evaluation and design
For a given application, which converter topology is best? There is no ultimate converter, perfectly suited for all possible applications Trade studies • Rough designs of several converter topologies to meet the given specifications • An unbiased quantitative comparison of worstcase transistor currents and voltages, transformer size, etc. Comparison via switch stress, switch utilization, and semiconductor cost Spreadsheet design
Fundamentals of Power Electronics
76
Chapter 6: Converter circuits
6.4.1. Switch stress and switch utilization
• Largest single cost in a converter is usually the cost of the active semiconductor devices • Conduction and switching losses associated with the active semiconductor devices often dominate the other sources of loss
This suggests evaluating candidate converter approaches by comparing the voltage and current stresses imposed on the active semiconductor devices. Minimization of total switch stresses leads to reduced loss, and to minimization of the total silicon area required to realize the power devices of the converter.
Fundamentals of Power Electronics
77
Chapter 6: Converter circuits
Total active switch stress S In a converter having k active semiconductor devices, the total active switch stress S is defined as
Σ Vj I j k
S=
j=1
where Vj is the peak voltage applied to switch j, Ij is the rms current applied to switch j (peak current is also sometimes used). In a good design, the total active switch stress is minimized.
Fundamentals of Power Electronics
78
Chapter 6: Converter circuits
Active switch utilization U
It is desired to minimize the total active switch stress, while maximizing the output power Pload. The active switch utilization U is defined as P U = load S The active switch utilization is the converter output power obtained per unit of active switch stress. It is a converter figureofmerit, which measures how well a converter utilizes its semiconductor devices. Active switch utilization is less than 1 in transformerisolated converters, and is a quantity to be maximized. Converters having low switch utilizations require extra active silicon area, and operate with relatively low efficiency. Active switch utilization is a function of converter operating point. Fundamentals of Power Electronics
79
Chapter 6: Converter circuits
CCM flyback example: Determination of S During subinterval 2, the transistor blocks voltage VQ1,pk equal to Vg plus the reflected load voltage: Vg VQ1,pk = Vg + V / n = D'
1:n
Vg
C
LM
+ –
+
D1
Q1
Transistor current coincides with ig(t). RMS value is P I Q1,rms = I D = load Vg D
V –
ig I
Switch stress S is 0
S = VQ1,pk I Q1,rms = (Vg + V / n) (I D) DTs conducting devices:
Fundamentals of Power Electronics
80
D'Ts
t
Ts Q1
D1
Chapter 6: Converter circuits
CCM flyback example: Determination of U 1:D
Express load power Pload in terms of V and I: Pload = D' V nI
Ig Vg
D' : n +
I
+ –
R
V –
Previouslyderived expression for S:
CCM flyback model
S = VQ1,pk I Q1,rms = (Vg + V / n) (I D)
Hence switch utilization U is U=
Pload = D' D S
Fundamentals of Power Electronics
81
Chapter 6: Converter circuits
Flyback example: switch utilization U(D) 0.4
For given V, Vg, Pload, the designer can arbitrarily choose D. The turns ratio n must then be chosen according to n = V D' Vg D
Single operating point design: choose D = 1/3.
max U = 0.385 at D = 1/3
0.3
U
0.2
0.1
small D leads to large transistor current large D leads to large transistor voltage
0 0
0.2
0.4
0.6
0.8
1
D Fundamentals of Power Electronics
82
Chapter 6: Converter circuits
Comparison of switch utilizations of some common converters Table 6.1. Active switch utilizations of some common dcdc converters, single operating point. Converter
U(D)
max U(D)
max U(D) occurs at D =
Buck
D
1
1
Boost
D' D
∞
0
Buckboost, flyback, nonisolated SEPIC, isolated SEPIC, nonisolated Cuk, isolated Cuk Forward, n1 = n2
D' D
2 = 0.385 3 3
1 3
1 D 2
1 2
Other isolated buckderived converters (fullbridge, halfbridge, pushpull) Isolated boostderived converters (full bridge, pushpull)
D 2 2
1 = 0.353 2 2 1 = 0.353 2 2
D' 2 1+D
1 2
0
Fundamentals of Power Electronics
83
1
Chapter 6: Converter circuits
Switch utilization : Discussion ● ●
●
● ●
Increasing the range of operating points leads to reduced switch utilization Buck converter can operate with high switch utilization (U approaching 1) when D is close to 1 Boost converter can operate with high switch utilization (U approaching ∞) when D is close to 1 Transformer isolation leads to reduced switch utilization Buckderived transformerisolated converters U ≤ 0.353 should be designed to operate with D as large as other considerations allow transformer turns ratio can be chosen to optimize design
Fundamentals of Power Electronics
84
Chapter 6: Converter circuits
Switch utilization: Discussion ●
Nonisolated and isolated versions of buckboost, SEPIC, and Cuk converters U ≤ 0.385 Singleoperatingpoint optimum occurs at D = 1/3 Nonisolated converters have lower switch utilizations than buck or boost Isolation can be obtained without penalizing switch utilization
Fundamentals of Power Electronics
85
Chapter 6: Converter circuits
Active semiconductor cost vs. switch utilization
semiconductor cost = per kW output power
semiconductor device cost per rated kVA voltage derating factor
current derating factor
converter switch utilization
(semiconductor device cost per rated kVA) = cost of device, divided by product of rated blocking voltage and rms current, in $/kVA. Typical values are less than $1/kVA (voltage derating factor) and (current derating factor) are required to obtain reliable operation. Typical derating factors are 0.5  0.75 Typical cost of active semiconductor devices in an isolated dcdc converter: $1  $10 per kW of output power.
Fundamentals of Power Electronics
86
Chapter 6: Converter circuits
6.4.2. Converter design using computer spreadsheet
Given ranges of Vg and Pload , as well as desired value of V and other quantities such as switching frequency, ripple, etc., there are two basic engineering design tasks: • Compare converter topologies and select the best for the given specifications • Optimize the design of a given converter A computer spreadsheet is a very useful tool for this job. The results of the steadystate converter analyses of chapters 16 can be entered, and detailed design investigations can be quickly performed: • Evaluation of worstcase stresses over a range of operating points • Evaluation of design tradeoffs Fundamentals of Power Electronics
87
Chapter 6: Converter circuits
Spreadsheet design example • Input voltage: rectified 230Vrms ±20%
S peci fi cat i o ns maximum input voltage V g
390V
minimum input voltage V g
260V
output voltage V
15V
• Rated load power 200W
maximum load power Pload
200W
• Must operate at 10% load
minimum load power Pload
20W
switching frequency fs
100kHz
• Select switching frequency of 100kHz
maximum output ripple ∆v
0.1V
• Output voltage ripple ≤ 0.1V
• Regulated output of 15V
Compare singletransistor forward and flyback converters in this application Specifications are entered at top of spreadsheet
Fundamentals of Power Electronics
88
Chapter 6: Converter circuits
Forward converter design, CCM n1 : n2 : n3
D2
L + D3
Vg
+ –
C
R
V –
Q1 D1
Design variables reset winding turns ratio n2 / n1
1
turns ratio n3 / n1
0.125
inductor current ripple ∆i
2A ref to sec
Fundamentals of Power Electronics
• Design for CCM at full load; may operate in DCM at light load
89
Chapter 6: Converter circuits
Flyback converter design, CCM
1:n
Vg
+ –
D1 C
LM Q1
V –
Design variables turns ratio n2 / n1
0.125
inductor current ripple ∆i
3A ref to sec
Fundamentals of Power Electronics
+
90
• Design for CCM at full load; may operate in DCM at light load
Chapter 6: Converter circuits
Enter results of converter analysis into spreadsheet (Forward converter example) Maximum duty cycle occurs at minimum Vg and maximum Pload. Converter then operates in CCM, with n D= 1 V n 3 Vg Inductor current ripple is ∆i =
D' V Ts 2L
Solve for L:
L=
D' V Ts 2 ∆i
∆i is a design variable. For a given ∆i, the equation above can be used to determine L. To ensure CCM operation at full load, ∆i should be less than the fullload output current. C can be found in a similar manner. Fundamentals of Power Electronics
91
Chapter 6: Converter circuits
Forward converter example, continued Check for DCM at light load. The solution of the buck converter operating in DCM is n 2 V = 3 Vg n1 1 + 4K / D 2 with K = 2 L / R Ts, and R = V 2 / Pload
These equations apply equally well to the forward converter, provided that all quantities are referred to the transformer secondary side. Solve for D: D=
2 K 2n 3Vg –1 n 1V
in DCM
2
–1
D=
n1 V n 3 Vg
in CCM
at a given operating point, the actual duty cycle is the small of the values calculated by the CCM and DCM equations above. Minimum D occurs at minimum Pload and maximum Vg. Fundamentals of Power Electronics
92
Chapter 6: Converter circuits
More regarding forward converter example
Worstcase component stresses can now be evaluated. Peak transistor voltage is
n max vQ1 = Vg 1 + n 1 2 Rms transistor current is
I Q1, rms =
n3 D n1
I 2 + (∆i) 2 / 3 ≈
n3 DI n1
(this neglects transformer magnetizing current) Other component stresses can be found in a similar manner. Magnetics design is left for a later chapter.
Fundamentals of Power Electronics
93
Chapter 6: Converter circuits
Results: forward and flyback converter spreadsheets Forw ard co nv ert er des i g n, C C M
Fl y back conv ert er desi g n, C C M
Design variables
Design variables
reset winding turns ratio n2 / n1
1
turns ratio n2 / n1
0.125
turns ratio n3 / n1
0.125
inductor current ripple ∆i
3A ref to sec
inductor current ripple ∆i
2A ref to sec
Results
Results
maximum duty cycle D
0.462
maximum duty cycle D
0.316
minimum D, at full load
0.308
minimum D, at full load
0.235
minimum D, at minimum load
0.251
minimum D, at minimum load
0.179
Worstcase stresses
Worstcase stresses
peak transistor voltage v Q1
780V
peak transistor voltage v Q1
510V
rms transistor current iQ1
1.13A
rms transistor current iQ1
1.38A
transistor utilization U
0.226
transistor utilization U
0.284
peak diode voltage v D1
49V
peak diode voltage v D1
64V
rms diode current iD1
9.1A
rms diode current iD1
16.3A
peak diode voltage v D2
49V
peak diode current iD1
22.2A
rms diode current iD2
11.1A
rms output capacitor current iC
1.15A
rms output capacitor current iC
9.1A
Fundamentals of Power Electronics
94
Chapter 6: Converter circuits
Discussion: transistor voltage Flyback converter Ideal peak transistor voltage: 510V Actual peak voltage will be higher, due to ringing causes by transformer leakage inductance An 800V or 1000V MOSFET would have an adequate design margin
Forward converter Ideal peak transistor voltage: 780V, 53% greater than flyback MOSFETs having voltage rating greater than 1000V are not available (in 1995) —when ringing due to transformer leakage inductance is accounted for, this design will have an inadequate design margin Fix: use twotransistor forward converter, or change reset winding turns ratio A conclusion: reset mechanism of flyback is superior to forward Fundamentals of Power Electronics
95
Chapter 6: Converter circuits
Discussion: rms transistor current Forward 1.13A worstcase transistor utilization 0.226
Flyback 1.38A worst case, 22% higher than forward transistor utilization 0.284 CCM flyback exhibits higher peak and rms currents. Currents in DCM flyback are even higher
Fundamentals of Power Electronics
96
Chapter 6: Converter circuits
Discussion: secondaryside diode and capacitor stresses
Forward peak diode voltage 49V rms diode current 9.1A / 11.1A rms capacitor current 1.15A
Flyback peak diode voltage 64V rms diode current 16.3A peak diode current 22.2A rms capacitor current 9.1A Secondaryside currents, especially capacitor currents, limit the practical application of the flyback converter to situations where the load current is not too great. Fundamentals of Power Electronics
97
Chapter 6: Converter circuits
Summary of key points 1. The boost converter can be viewed as an inverse buck converter, while the buckboost and Cuk converters arise from cascade connections of buck and boost converters. The properties of these converters are consistent with their origins. Ac outputs can be obtained by differential connection of the load. An infinite number of converters are possible, and several are listed in this chapter. 2. For understanding the operation of most converters containing transformers, the transformer can be modeled as a magnetizing inductance in parallel with an ideal transformer. The magnetizing inductance must obey all of the usual rules for inductors, including the principle of voltsecond balance.
Fundamentals of Power Electronics
98
Chapter 6: Converter circuits
Summary of key points 3. The steadystate behavior of transformerisolated converters may be understood by first replacing the transformer with the magnetizinginductanceplusidealtransformer equivalent circuit. The techniques developed in the previous chapters can then be applied, including use of inductor voltsecond balance and capacitor charge balance to find dc currents and voltages, use of equivalent circuits to model losses and efficiency, and analysis of the discontinuous conduction mode. 4. In the fullbridge, halfbridge, and pushpull isolated versions of the buck and/or boost converters, the transformer frequency is twice the output ripple frequency. The transformer is reset while it transfers energy: the applied voltage polarity alternates on successive switching periods. Fundamentals of Power Electronics
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Chapter 6: Converter circuits
Summary of key points 5. In the conventional forward converter, the transformer is reset while the transistor is off. The transformer magnetizing inductance operates in the discontinuous conduction mode, and the maximum duty cycle is limited. 6. The flyback converter is based on the buckboost converter. The flyback transformer is actually a twowinding inductor, which stores and transfers energy. 7. The transformer turns ratio is an extra degreeoffreedom which the designer can choose to optimize the converter design. Use of a computer spreadsheet is an effective way to determine how the choice of turns ratio affects the component voltage and current stresses. 8. Total active switch stress, and active switch utilization, are two simplified figuresofmerit which can be used to compare the various converter circuits.
Fundamentals of Power Electronics
100
Chapter 6: Converter circuits
Part II Converter Dynamics and Control
7. 8. 9. 10.
AC equivalent circuit modeling Converter transfer functions Controller design Ac and dc equivalent circuit modeling of the discontinuous conduction mode 11. Current programmed control
Fundamentals of Power Electronics
1
Chapter 7: AC equivalent circuit modeling
Chapter 7. AC Equivalent Circuit Modeling
7.1. Introduction 7.2. The basic ac modeling approach 7.3. Example: A nonideal flyback converter 7.4. Statespace averaging 7.5. Circuit averaging and averaged switch modeling 7.6. The canonical circuit model 7.7. Modeling the pulsewidth modulator 7.8. Summary of key points Fundamentals of Power Electronics
2
Chapter 7: AC equivalent circuit modeling
7.1.
There are disturbances: • in vg(t)
A simple dcdc regulator system, employing a buck converter Power input
Load
Switching converter +
vg(t) + –
v(t)
• in R There are uncertainties: • in element values
R feedback connection
– transistor gate driver δ(t)
compensator pulsewidth vc Gc(s) modulator
δ(t)
+ –
Objective: maintain v(t) equal to an accurate, constant value V.
Introduction
v
voltage reference vref
vc(t)
• in Vg • in R Fundamentals of Power Electronics
dTs Ts
t
t
Controller 3
Chapter 7: AC equivalent circuit modeling
Applications of control in power electronics Dcdc converters Regulate dc output voltage. Control the duty cycle d(t) such that v(t) accurately follows a reference signal vref.
Dcac inverters Regulate an ac output voltage. Control the duty cycle d(t) such that v(t) accurately follows a reference signal vref (t).
Acdc rectifiers Regulate the dc output voltage. Regulate the ac input current waveform. Control the duty cycle d(t) such that ig (t) accurately follows a reference signal iref (t), and v(t) accurately follows a reference signal vref. Fundamentals of Power Electronics
4
Chapter 7: AC equivalent circuit modeling
Objective of Part II Develop tools for modeling, analysis, and design of converter control systems Need dynamic models of converters: How do ac variations in vg(t), R, or d(t) affect the output voltage v(t)? What are the smallsignal transfer functions of the converter? • Extend the steadystate converter models of Chapters 2 and 3, to include CCM converter dynamics (Chapter 7) • Construct converter smallsignal transfer functions (Chapter 8) • Design converter control systems (Chapter 9) • Model converters operating in DCM (Chapter 10) • Currentprogrammed control of converters (Chapter 11) Fundamentals of Power Electronics
5
Chapter 7: AC equivalent circuit modeling
Modeling
• Representation of physical behavior by mathematical means • Model dominant behavior of system, ignore other insignificant phenomena • Simplified model yields physical insight, allowing engineer to design system to operate in specified manner • Approximations neglect small but complicating phenomena • After basic insight has been gained, model can be refined (if it is judged worthwhile to expend the engineering effort to do so), to account for some of the previously neglected phenomena
Fundamentals of Power Electronics
6
Chapter 7: AC equivalent circuit modeling
Neglecting the switching ripple Suppose the duty cycle is modulated sinusoidally: d(t) = D + Dm cos ωmt
The resulting variations in transistor gate drive signal and converter output voltage: gate drive
where D and Dm are constants,  Dm  > y(t)
Substitute into averaged state equations: d X+x(t) K = D+d(t) A 1 + D'–d(t) A 2 X+x(t) dt
+
D+d(t) B 1 + D'–d(t) B 2 U+u(t)
Y+y(t) =
D+d(t) C 1 + D'–d(t) C 2 X+x(t)
+
D+d(t) E 1 + D'–d(t) E 2 U+u(t)
Fundamentals of Power Electronics
83
Chapter 7: AC equivalent circuit modeling
Averaged state equations: perturbation and linearization K
dx(t) dt
first–order ac
= AX + BU + Ax(t) + Bu(t) +
dc terms
A 1 – A 2 X + B 1 – B 2 U d(t)
first–order ac terms +
A 1 – A 2 x(t)d(t) + B 1 – B 2 u(t)d(t) second–order nonlinear terms
Y+y(t)
dc + 1st order ac
= CX + EU + Cx(t) + Eu(t) +
dc terms
C 1 – C 2 X + E 1 – E 2 U d(t)
first–order ac terms +
C 1 – C 2 x(t)d(t) + E 1 – E 2 u(t)d(t) second–order nonlinear terms
Fundamentals of Power Electronics
84
Chapter 7: AC equivalent circuit modeling
Linearized smallsignal state equations
Dc terms drop out of equations. Secondorder (nonlinear) terms are small when the smallsignal assumption is satisfied. We are left with:
K
dx(t) = A x(t) + B u(t) + dt
A 1 – A 2 X + B 1 – B 2 U d(t)
y(t) = C x(t) + E u(t) +
C 1 – C 2 X + E 1 – E 2 U d(t)
This is the desired result.
Fundamentals of Power Electronics
85
Chapter 7: AC equivalent circuit modeling
7.4.4. Example: Statespace averaging of a nonideal buckboost converter ig(t)
D1
Q1
Model nonidealities: +
i(t) vg(t)
+ –
L
state vector
x(t) =
i(t) v(t)
Fundamentals of Power Electronics
C
R
input vector u(t) =
vg(t) VD
86
v(t)
• MOSFET onresistance Ron
–
• Diode forward voltage drop VD
output vector y(t) = ig(t)
Chapter 7: AC equivalent circuit modeling
Subinterval 1 ig(t)
di(t) = vg(t) – i(t) Ron dt dv(t) v(t) C =– R dt ig(t) = i(t)
Ron
L
+
i(t) vg(t)
+ –
L
C
R
v(t) –
L 0 0C
K
– Ron 0 d i(t) = dt v(t) 0 – 1 R dx(t) dt ig(t) y(t)
Fundamentals of Power Electronics
=
i(t) + 10 v(t) 00
vg(t) VD
A1
x(t)
B1
u(t)
10
i(t) + 00 v(t)
vg(t) VD
C1
x(t)
u(t)
87
E1
Chapter 7: AC equivalent circuit modeling
Subinterval 2 di(t) = v(t) – VD dt dv(t) v(t) C =– – i(t) R dt ig(t) = 0
VD
+ –
L
vg(t)
+ –
L
C i(t)
L 0 0C
K
d i(t) = dt v(t) dx(t) dt ig(t) y(t)
Fundamentals of Power Electronics
=
0
1 –1 – 1 R
+
ig(t)
i(t) + v(t)
0 –1 0 0
vg(t) VD
B2
u(t)
A2
x(t)
00
i(t) + v(t)
00
vg(t) VD
C2
x(t)
E2
u(t)
88
R
v(t) –
Chapter 7: AC equivalent circuit modeling
Evaluate averaged matrices
– Ron 0 – DRon D' 0 1 A = DA 1 + D'A 2 = D + D' = 1 1 0 – – D' – 1 – 1 – R R R In a similar manner,
B = DB 1 + D'B 2 =
D – D' 0 0
C = DC 1 + D'C 2 = D 0 E = DE 1 + D'E 2 = 0 0
Fundamentals of Power Electronics
89
Chapter 7: AC equivalent circuit modeling
DC state equations
0=AX+BU Y=CX+EU
0 = 0
or,
– DRon – D'
Ig =
DC solution:
I = V
Ig
Fundamentals of Power Electronics
D' –1 R
D0
1 R 1 + D2 on D' R
1 = R 1 + D2 on D' R 90
I + V
D – D' 0 0
Vg VD
I + V
00
Vg VD
D 1 2 D' R D' R – D 1 D'
Vg VD
D2 D D' 2R D'R
Vg VD
Chapter 7: AC equivalent circuit modeling
Steadystate equivalent circuit DC state equations:
0 = 0
– DRon – D'
Ig =
D' –1 R
D0
I + V
D – D' 0 0
Vg VD
I + V
00
Vg VD
Corresponding equivalent circuit: DRon
D'VD
Ig Vg
+ –
1:D I
+ –
D' : 1 + V
R
–
Fundamentals of Power Electronics
91
Chapter 7: AC equivalent circuit modeling
Smallsignal ac model Evaluate matrices in smallsignal model: V – IRon + VD V – V – IRon + VD A 1 – A 2 X + B1 – B2 U = – V + g = g I I 0 C1 – C2 X + E1 – E2 U = I
Smallsignal ac state equations:
– DRon D' L 0 d i(t) = 0 C dt v(t) – D' – 1 R i g(t) = D 0
i(t) + 00 00 v(t)
Fundamentals of Power Electronics
i(t) + v(t)
D – D' 0 0
vg(t) Vg – V – IRon + VD + d(t) 0 I ➚ vD(t)
vg(t) + 0 d(t) 0 I v➚ D(t)
92
Chapter 7: AC equivalent circuit modeling
Construction of ac equivalent circuit d i(t) = D' v(t) – DRon i(t) + D vg(t) + Vg – V – IRon + VD d(t) dt dv(t) v(t) C = –D' i(t) – + I d(t) R dt
Smallsignal ac equations, in scalar form:
L
i g(t) = D i(t) + I d(t) Corresponding equivalent circuits:
inductor equation L
DRon
+ –
I d(t)
D i(t)
+ –
+ –
vg(t)
d(t) Vg – V + V D – IRon
+ d i(t) – L dt D vg(t)
i g(t)
input eqn
– + i(t)
D' v(t)
capacitor eqn D' i(t)
C I d(t)
dv(t) dt
+
v(t) R
C
v(t)
R
–
Fundamentals of Power Electronics
93
Chapter 7: AC equivalent circuit modeling
Complete smallsignal ac equivalent circuit Combine individual circuits to obtain
i g(t)
+ –
1:D
d(t) Vg – V + V D – IRon
L i(t)
vg(t)
+ –
D' : 1 +
DRon
I d(t)
I d(t) C
v(t)
R
–
Fundamentals of Power Electronics
94
Chapter 7: AC equivalent circuit modeling
7.5. Circuit Averaging and Averaged Switch Modeling ●
● ●
●
Historically, circuit averaging was the first method known for modeling the smallsignal ac behavior of CCM PWM converters It was originally thought to be difficult to apply in some cases There has been renewed interest in circuit averaging and its corrolary, averaged switch modeling, in the last decade Can be applied to a wide variety of converters ● ●
● ●
●
We will use it to model DCM, CPM, and resonant converters Also useful for incorporating switching loss into ac model of CCM converters Applicable to 3ø PWM inverters and rectifiers Can be applied to phasecontrolled rectifiers
Rather than averaging and linearizing the converter state equations, the averaging and linearization operations are performed directly on the converter circuit
Fundamentals of Power Electronics
95
Chapter 7: AC equivalent circuit modeling
Separate switch network from remainder of converter
Power input
Load
Timeinvariant network containing converter reactive elements vg(t)
+ –
C vC(t)
+
L
i1(t)
i2(t)
Switch network
–
+ v2(t) –
Control input
Fundamentals of Power Electronics
v(t) –
port 2
v1(t)
port 1
+
R
iL(t)
–
+
d(t)
96
Chapter 7: AC equivalent circuit modeling
Boost converter example L
Ideal boost converter example
+
i(t) vg(t) + –
C
R
v(t) –
Two ways to define the switch network
(a)
(b)
i1(t) +
i2(t)
i1(t)
+
+
i2(t) +
v1(t)
v2(t)
v1(t)
v2(t)
–
–
–
–
Fundamentals of Power Electronics
97
Chapter 7: AC equivalent circuit modeling
Discussion
●
●
The number of ports in the switch network is less than or equal to the number of SPST switches Simple dcdc case, in which converter contains two SPST switches: switch network contains two ports The switch network terminal waveforms are then the port voltages and currents: v1(t), i1(t), v2(t), and i2(t). Two of these waveforms can be taken as independent inputs to the switch network; the remaining two waveforms are then viewed as dependent outputs of the switch network.
●
Definition of the switch network terminal quantities is not unique. Different definitions lead equivalent results having different forms
Fundamentals of Power Electronics
98
Chapter 7: AC equivalent circuit modeling
Boost converter example Let’s use definition (a): i1(t) +
i2(t) +
v1(t)
v2(t)
–
–
L +
i(t) vg(t) + –
C
R
v(t) –
Since i1(t) and v2(t) coincide with the converter inductor current and output voltage, it is convenient to define these waveforms as the independent inputs to the switch network. The switch network dependent outputs are then v1(t) and i2(t).
Fundamentals of Power Electronics
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Chapter 7: AC equivalent circuit modeling
Obtaining a timeinvariant network: Modeling the terminal behavior of the switch network
Replace the switch network with dependent sources, which correctly represent the dependent output waveforms of the switch network
L i(t) vg(t) + –
+
i1(t) v1(t) + –
i2(t) v2(t) C –
+ R
v(t) –
Switch network
Boost converter example
Fundamentals of Power Electronics
100
Chapter 7: AC equivalent circuit modeling
Definition of dependent generator waveforms L
v1(t)
v2(t)
i(t) vg(t) + –
〈v1(t)〉Ts
+
+
i1(t) v1(t) + –
i2(t) v2(t) C
R
–
0
dTs
i2(t)
Ts
t
The waveforms of the dependent generators are defined to be identical to the actual terminal waveforms of the switch network.
i1(t) 〈i2(t)〉Ts
0
The circuit is therefore electrical identical to the original converter.
0 0
–
Switch network
0 0
v(t)
dTs
Fundamentals of Power Electronics
Ts
t
So far, no approximations have been made. 101
Chapter 7: AC equivalent circuit modeling
The circuit averaging step Now average all waveforms over one switching period: Power input
Load
Averaged timeinvariant network containing converter reactive elements 〈vg(t)〉T
s
+ –
C
+
L
〈vC(t)〉Ts
Averaged switch network
–
+ port 2
port 1
s
s
–
〈v2(t)〉T
s
– Control input
Fundamentals of Power Electronics
〈v(t)〉T
〈i2(t)〉T
s
〈v1(t)〉Ts
R
〈iL(t)〉Ts
–
〈i1(t)〉T +
+
102
d(t)
Chapter 7: AC equivalent circuit modeling
The averaging step The basic assumption is made that the natural time constants of the converter are much longer than the switching period, so that the converter contains lowpass filtering of the switching harmonics. One may average the waveforms over an interval that is short compared to the system natural time constants, without significantly altering the system response. In particular, averaging over the switching period Ts removes the switching harmonics, while preserving the lowfrequency components of the waveforms. In practice, the only work needed for this step is to average the switch dependent waveforms.
Fundamentals of Power Electronics
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Chapter 7: AC equivalent circuit modeling
Averaging step: boost converter example L i(t)
+
i1(t) v1(t) + –
vg(t) + –
+
i2(t) v2(t) C
R
v(t) –
– Switch network
L 〈i(t)〉T 〈vg(t)〉Ts
s
+ –
〈i1(t)〉T
+
+
s
〈v1(t)〉Ts
+ –
〈i2(t)〉Ts
〈v2(t)〉Ts –
C
R
〈v(t)〉Ts –
Averaged switch network Fundamentals of Power Electronics
104
Chapter 7: AC equivalent circuit modeling
Compute average values of dependent sources v1(t) 〈v1(t)〉T 0
Average the waveforms of the dependent sources:
v2(t) s
v1(t)
0 dTs
0
i2(t)
Ts
t
i 2(t)
i1(t)
Ts Ts
= d'(t) v2(t) = d'(t) i 1(t)
Ts Ts
〈i2(t)〉Ts 0
0 0
dTs
Ts
t
L 〈i(t)〉T 〈vg(t)〉Ts
s
+ –
〈i1(t)〉Ts
+
+
d'(t) 〈v2(t)〉Ts
+ –
d'(t) 〈i1(t)〉Ts
〈v2(t)〉T s –
C
R
〈v(t)〉T
s
–
Averaged switch model
Fundamentals of Power Electronics
105
Chapter 7: AC equivalent circuit modeling
Perturb and linearize vg(t)
As usual, let:
Ts
= Vg + vg(t)
d(t) = D + d(t) ⇒ d'(t) = D' – d(t) i(t) T = i 1(t) T = I + i(t) s
v(t) v1(t) i 2(t)
Ts Ts Ts
s
= v2(t)
Ts
= V + v(t)
= V1 + v1(t) = I 2 + i 2(t)
The circuit becomes: L +
I + i(t) Vg + vg(t)
+ –
D' – d(t) V + v(t)
+ –
D' – d(t) I + i(t)
C
R
V + v(t)
– Fundamentals of Power Electronics
106
Chapter 7: AC equivalent circuit modeling
Dependent voltage source D' – d(t) V + v(t) = D' V + v(t) – V d(t) – v(t)d(t) nonlinear, 2nd order V d(t)
+ – + –
Fundamentals of Power Electronics
107
D' V + v(t)
Chapter 7: AC equivalent circuit modeling
Dependent current source D' – d(t) I + i(t) = D' I + i(t) – I d(t) – i(t)d(t) nonlinear, 2nd order
D' I + i(t)
Fundamentals of Power Electronics
I d(t)
108
Chapter 7: AC equivalent circuit modeling
Linearized circuitaveraged model V d(t)
L
+ –
+
I + i(t) Vg + vg(t)
+ –
D' V + v(t)
+ –
D' I + i(t)
I d(t) C
V + v(t)
R
–
L
V d(t)
+ –
D' : 1
+
I + i(t) Vg + vg(t)
+ –
I d(t)
C
R
V + v(t)
–
Fundamentals of Power Electronics
109
Chapter 7: AC equivalent circuit modeling
Summary: Circuit averaging method
Model the switch network with equivalent voltage and current sources, such that an equivalent timeinvariant network is obtained Average converter waveforms over one switching period, to remove the switching harmonics Perturb and linearize the resulting lowfrequency model, to obtain a smallsignal equivalent circuit
Fundamentals of Power Electronics
110
Chapter 7: AC equivalent circuit modeling
Averaged switch modeling: CCM Circuit averaging of the boost converter: in essence, the switch network was replaced with an effective ideal transformer and generators:
2
+ –
+
i(t)
D' : 1
+
I + i(t) V d(t)
1
v(t)
I d(t)
–
V + v(t)
–
Switch network
Fundamentals of Power Electronics
111
Chapter 7: AC equivalent circuit modeling
Basic functions performed by switch network 2
i(t)
+ –
+
D' : 1
+
I + i(t) V d(t)
1
v(t)
I d(t)
V + v(t)
–
– Switch network
For the boost example, we can conclude that the switch network performs two basic functions: • Transformation of dc and smallsignal ac voltage and current levels, according to the D’:1 conversion ratio • Introduction of ac voltage and current variations, drive by the control input duty cycle variations Circuit averaging modifies only the switch network. Hence, to obtain a smallsignal converter model, we need only replace the switch network with its averaged model. Such a procedure is called averaged switch modeling. Fundamentals of Power Electronics
112
Chapter 7: AC equivalent circuit modeling
Averaged switch modeling: Procedure 1. Define a switch network and its terminal waveforms. For a simple transistordiode switch network as in the buck, boost, etc., there are two ports and four terminal quantities: v1, i1, v2, i2.The switch network also contains a control input d. Buck example:
i1(t)
i2(t) +
+
v1(t)
v2(t)
–
–
2. To derive an averaged switch model, express the average values of two of the terminal quantities, for example 〈 v2 〉Ts and 〈 i1 〉Ts, as functions of the other average terminal quantities 〈 v1 〉Ts and 〈 i1 〉Ts . 〈 v2 〉Ts and 〈 i1 〉Ts may also be functions of the control input d, but they should not be expressed in terms of other converter signals.
Fundamentals of Power Electronics
113
Chapter 7: AC equivalent circuit modeling
The basic bucktype CCM switch cell i1(t)
vg(t) + –
+ vCE –
iC
i2(t)
+
+
v1(t)
v2(t)
–
–
L
i(t) + C
R
i1(t)
i2
i2
v(t)
0
v2(t)
Ts
= d(t) i 2(t) = d(t) v1(t)
0
dTs
Ts
t
Ts
t
v1
Ts
v2(t) Ts
0
Fundamentals of Power Electronics
T2
0
v2(t) Ts
i 1(t)
–
Switch network
i 1(t)
T2
114
T2
0
0
dTs
Chapter 7: AC equivalent circuit modeling
Replacement of switch network by dependent sources, CCM buck example Perturbation and linearization of switch network: I 1 + i 1(t) = D I 2 + i 2(t) + I 2 d(t)
Circuitaveraged model L + vg(t) + –
v1(t)
i(t) +
i2(t) i1(t)
+ v (t) – 2
C
R
V2 + v2(t) = D V1 + v1(t) + V1 d(t)
v(t)
I1 + i1
+ –
+
–
–
I2 + i2
1:D
Switch network
V1 + v1
V2 + v2
I2 d
–
–
I2 + i2
1:D
I1 + i1 +
+ –
Resulting averaged switch model: CCM buck converter
+
V1 d
+
I+i
V2 + v2
C
V1 d Vg + vg + –
V1 + v1
I2 d
–
L
–
+ R
V +v
–
Switch network
Fundamentals of Power Electronics
115
Chapter 7: AC equivalent circuit modeling
Three basic switch networks, and their CCM dc and smallsignal ac averaged switch models i2(t) +
+
I1 + i1
I2 + i2
1:D
+ –
i1(t)
+
+
see also
V1 d
v1(t)
v2(t)
V1 + v1
–
–
–
+
i2(t) +
–
I1 + i1
+
Appendix 3 Averaged switch modeling of a CCM SEPIC
I2 + i2
D' : 1
+ –
i1(t)
V2 + v2
I2 d
+
V2 d
v1(t)
v2(t)
V1 + v1
–
–
–
i2(t) +
+
I1 + i1
+
v1(t)
v2(t)
V1 + v1
–
–
–
Fundamentals of Power Electronics
V2 + v2
–
I2 + i2
D' : D
+ –
i1(t)
I1 d
+
V1 d DD'
I2 d DD'
V2 + v2
–
116
Chapter 7: AC equivalent circuit modeling
Example: Averaged switch modeling of CCM buck converter, including switching loss i1(t)
vg(t) + –
iC
+ vCE –
L
i2(t)
+
+
v1(t)
v2(t)
–
–
i(t) + C
R
v(t) –
Switch network
vCE(t)
iC(t)
v1 i2 0
0 tir
tvf
t1
tvr
t2
tif
Ts
Fundamentals of Power Electronics
i 1(t) = i C(t) v2(t) = v1(t) – vCE(t)
117
t
Switch network terminal waveforms: v1, i1, v2, i2. To derive averaged switch model, express 〈 v2 〉Ts and 〈 i1 〉Ts as functions of 〈 v1 〉Ts and 〈 i1 〉Ts . 〈 v2 〉Ts and 〈 i1 〉Ts may also be functions of the control input d, but they should not be expressed in terms of other converter signals. Chapter 7: AC equivalent circuit modeling
Averaging i1(t) vCE(t)
iC(t)
v1 i2 0
0 tir
t1
tvf
tvr
tif
t2
t
Ts
i 1(t) T = 1 Ts s
Ts
i 1(t) dt 0
= i 2(t)
Fundamentals of Power Electronics
Ts
t 1 + t vf + t vr + 1 t ir + 1 t if 2 2 Ts
118
Chapter 7: AC equivalent circuit modeling
Expression for 〈 i1(t) 〉 Given i 1(t) T = 1 s Ts
Ts
i 1(t) dt 0
= i 2(t)
Ts
t 1 + t vf + t vr + 1 t ir + 1 t if 2 2 Ts
Let Then we can write
t 1 + 1 t vf + 1 t vr + 1 t ir + 1 t if 2 2 2 2 d= Ts
i 1(t)
Ts
= i 2(t)
Ts
d + 1 dv 2
t vf + t vr Ts t ir + t if di = Ts dv =
Fundamentals of Power Electronics
119
Chapter 7: AC equivalent circuit modeling
Averaging the switch network output voltage v2(t) vCE(t)
iC(t)
v1 i2 0
0 tir
tvf
t1
tvr
tif
t2
t
Ts
v2(t) T = v1(t) – vCE(t) T = 1 Ts s s
v2(t) v2(t)
Ts
Ts
= v1(t) = v1(t)
Fundamentals of Power Electronics
Ts
– vCE(t) dt + v1(t) 0
Ts
t 1 + 1 t vf + 1 t vr 2 2 Ts
Ts
d – 1 di 2 120
Ts
Chapter 7: AC equivalent circuit modeling
Construction of largesignal averagedswitch model
i 1(t)
Ts
= i 2(t)
Ts
d + 1 dv 2
v2(t)
Ts 1 2
+ 〈v1(t)〉Ts
1 2
di(t) 〈v1(t)〉Ts
+ –
〈i1(t)〉Ts
= v1(t)
dv(t) 〈i2(t)〉Ts
d(t) 〈i2(t)〉Ts
+ d(t) 〈v (t)〉 1 Ts –
–
〈i2(t)〉Ts + 〈v2(t)〉Ts –
〈i1(t)〉Ts
1 2
di(t) 〈v1(t)〉T
s
+ –
1 : d(t)
+ 〈v1(t)〉T s
Ts
d – 1 di 2
1 2
〈v2(t)〉T s
dv(t) 〈i2(t)〉T s
–
Fundamentals of Power Electronics
〈i2(t)〉Ts +
–
121
Chapter 7: AC equivalent circuit modeling
Switching loss predicted by averaged switch model
〈i1(t)〉T
1 2
1 : d(t)
+ –
s
+ 〈v1(t)〉Ts
di(t) 〈v1(t)〉Ts
1 2
dv(t) 〈i2(t)〉T
〈i2(t)〉T
s
+ 〈v2(t)〉Ts
s
–
–
Psw =
Fundamentals of Power Electronics
1 2
dv + di
i 2(t)
122
Ts
v1(t)
Ts
Chapter 7: AC equivalent circuit modeling
Solution of averaged converter model in steady state I1
+ –
+ 1 2
Vg
+ –
I2
1:D
V1
1 2
L
I
+
+
Di V1
D v I2
V2
–
C
R
–
V –
Averaged switch network model
Output voltage:
Efficiency calcuation:
Pin = VgI 1 = V1I 2 D + 12 Dv
D V = D – 12 Di Vg = DVg 1 – i 2D
Pout = VI 2 = V1I 2 D – 12 Di
Di D – Di Pout 2D η= = = Pin D D + 12 Dv 1+ v 2D 1 2
Fundamentals of Power Electronics
123
1–
Chapter 7: AC equivalent circuit modeling
7.6. The canonical circuit model
All PWM CCM dcdc converters perform the same basic functions: • Transformation of voltage and current levels, ideally with 100% efficiency • Lowpass filtering of waveforms • Control of waveforms by variation of duty cycle Hence, we expect their equivalent circuit models to be qualitatively similar. Canonical model: • A standard form of equivalent circuit model, which represents the above physical properties • Plug in parameter values for a given specific converter
Fundamentals of Power Electronics
124
Chapter 7: AC equivalent circuit modeling
7.6.1. Development of the canonical circuit model
1. Transformation of dc voltage and current levels • modeled as in Chapter 3 with ideal dc transformer
Converter model 1 : M(D) + Vg
+ –
V
• effective turns ratio M(D) • can refine dc model by addition of effective loss elements, as in Chapter 3
Fundamentals of Power Electronics
R
–
D Power input
Control input
125
Load
Chapter 7: AC equivalent circuit modeling
Steps in the development of the canonical circuit model
2. Ac variations in vg(t) induce ac variations in v(t)
1 : M(D) +
• these variations are also transformed by the conversion ratio M(D)
Vg + vg(s)
+ –
V + v(s)
R
–
D Power input
Fundamentals of Power Electronics
Control input
126
Load
Chapter 7: AC equivalent circuit modeling
Steps in the development of the canonical circuit model
3. Converter must contain an effective lowpass filter characteristic
He(s) 1 : M(D) + Vg + vg(s)
Zei(s)
+ –
• necessary to filter switching ripple
lowpass
Zeo(s)
V + v(s)
R
filter –
• also filters ac variations
D Power input
Control input
• effective filter elements may not coincide with actual element values, but can also depend on operating point Fundamentals of Power Electronics
Effective
127
Load
Chapter 7: AC equivalent circuit modeling
Steps in the development of the canonical circuit model
e(s) d(s)
He(s) 1 : M(D)
+ –
Vg + vg(s) + –
+ Zei(s)
j(s) d(s)
Effective lowpass
Zeo(s)
V + v(s)
R
filter –
D + d(s)
Power input
Control input
Load
4. Control input variations also induce ac variations in converter waveforms • Independent sources represent effects of variations in duty cycle • Can push all sources to input side as shown. Sources may then become frequencydependent Fundamentals of Power Electronics
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Chapter 7: AC equivalent circuit modeling
Transfer functions predicted by canonical model
e(s) d(s)
He(s) 1 : M(D)
+ –
Vg + vg(s) + –
+ Zei(s)
j(s) d(s)
Effective lowpass
Zeo(s)
V + v(s)
R
filter –
D + d(s)
Power input
Control input
Load
Linetooutput transfer function:
Gvg(s) =
v(s) = M(D) H e(s) vg(s)
Controltooutput transfer function:
Gvd(s) =
v(s) = e(s) M(D) H e(s) d(s)
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Chapter 7: AC equivalent circuit modeling
7.6.2. Example: manipulation of the buckboost converter model into canonical form Smallsignal ac model of the buckboost converter
Vg + vg(s)
+ –
Vg – V d
+ –
1:D
L
D' : 1
Id
+ Id
C
V + v(s)
R
–
• Push independent sources to input side of transformers • Push inductor to output side of transformers • Combine transformers
Fundamentals of Power Electronics
130
Chapter 7: AC equivalent circuit modeling
Step 1
Push voltage source through 1:D transformer Move current source through D’:1 transformer Vg – V d D
+ –
Vg + vg(s)
+ –
L
1:D
D' : 1 + I d D'
Id
C
V + v(s)
R
–
Fundamentals of Power Electronics
131
Chapter 7: AC equivalent circuit modeling
Step 2 How to move the current source past the inductor: Break ground connection of current source, and connect to node A instead. Connect an identical current source from node A to ground, so that the node equations are unchanged. Vg – V d D
+ –
Vg + vg(s)
+ –
Id
node A
1:D
L
D' : 1 +
I d D'
I d D'
C
V + v(s)
R
–
Fundamentals of Power Electronics
132
Chapter 7: AC equivalent circuit modeling
Step 3 The parallelconnected current source and inductor can now be replaced by a Theveninequivalent network:
+ –
Vg + vg(s)
+ –
Id
sLI d D'
1:D
+ –
Vg – V d D
I d D'
L D' : 1 + C
V + v(s)
R
–
Fundamentals of Power Electronics
133
Chapter 7: AC equivalent circuit modeling
Step 4 Now push current source through 1:D transformer. Push current source past voltage source, again by: Breaking ground connection of current source, and connecting to node B instead. Connecting an identical current source from node B to ground, so that the node equations are unchanged. Note that the resulting parallelconnected voltage and current sources are equivalent to a single voltage source. Vg – V d D
+ –
Vg + vg(s)
+ –
Id
DI d D'
sLI d D'
1:D
DI d D'
+ –
node B
L D' : 1 + C
V + v(s)
R
– Fundamentals of Power Electronics
134
Chapter 7: AC equivalent circuit modeling
Step 5: final result Push voltage source through 1:D transformer, and combine with existing inputside transformer. Combine seriesconnected transformers. Vg – V – s LI d(s) D DD'
D' : D
L D' 2
+ –
Vg + vg(s) + –
+
I d(s) D'
C
V + v(s)
R
– Effective lowpass filter
Fundamentals of Power Electronics
135
Chapter 7: AC equivalent circuit modeling
Coefficient of controlinput voltage generator
Voltage source coefficient is:
e(s) =
Vg + V s LI – D D D'
Simplification, using dc relations, leads to
e(s) = – V2 1 – s DL D D' 2 R Pushing the sources past the inductor causes the generator to become frequencydependent.
Fundamentals of Power Electronics
136
Chapter 7: AC equivalent circuit modeling
7.6.3. Canonical circuit parameters for some common converters e(s) d(s)
1 : M(D)
Le
+ –
Vg + vg(s) + –
+
j(s) d(s)
C
V + v(s)
R
–
Table 7.1. Canonical model parameters for the ideal buck, boost, and buckboost converters Converter Buck Boost Buckboost
M(D)
Le
D 1 D'
–D D'
Fundamentals of Power Electronics
L L D' 2
e(s) V D2 V 1 – s 2L D' R
j(s) V R V D' 2 R
L D' 2
– V2 1 – s DL D D' 2 R
– V2 D' R
137
Chapter 7: AC equivalent circuit modeling
7.7. Modeling the pulsewidth modulator
Power input
Switching converter
Load +
vg(t) + –
What is the relation between vc(t) and d(t)?
v(t)
R feedback connection
– transistor gate driver
compensator pulsewidth vc Gc(s) modulator
δ(t) δ(t)
v
voltage reference vref
vc(t)
dTs Ts
–+
Pulsewidth modulator converts voltage signal vc(t) into duty cycle signal d(t).
t
t
Controller
Fundamentals of Power Electronics
138
Chapter 7: AC equivalent circuit modeling
A simple pulsewidth modulator
vsaw(t)
VM
Sawtooth wave
vc(t)
vsaw(t)
generator comparator
analog input
0
–
δ(t)
+
PWM waveform
vc(t)
t
δ(t)
0
Fundamentals of Power Electronics
139
dTs
Ts
2Ts
Chapter 7: AC equivalent circuit modeling
Equation of pulsewidth modulator
For a linear sawtooth waveform:
d(t) =
vc(t) VM
vsaw(t)
VM
for 0 ≤ vc(t) ≤ VM
vc(t)
So d(t) is a linear function of vc(t).
0
t
δ(t)
0
Fundamentals of Power Electronics
140
dTs
Ts
2Ts
Chapter 7: AC equivalent circuit modeling
Perturbed equation of pulsewidth modulator PWM equation:
d(t) =
vc(t) VM
Block diagram:
for 0 ≤ vc(t) ≤ VM Vc + vc(s)
1 VM
Perturb:
vc(t) = Vc + vc(t) d(t) = D + d(t)
pulsewidth modulator
Result:
D + d(t) =
D + d(s)
Dc and ac relations:
Vc + vc(t) VM
Fundamentals of Power Electronics
Vc VM v (t) d(t) = c VM D=
141
Chapter 7: AC equivalent circuit modeling
Sampling in the pulsewidth modulator The input voltage is a continuous function of time, but there can be only one discrete value of the duty cycle for each switching period.
sampler vc
1 VM
d
fs
Therefore, the pulsepulsewidth modulator width modulator samples the control waveform, with sampling rate equal to the switching frequency. In practice, this limits the useful frequencies of ac variations to values much less than the switching frequency. Control system bandwidth must be sufficiently less than the Nyquist rate fs/2. Models that do not account for sampling are accurate only at frequencies much less than fs/2. Fundamentals of Power Electronics
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Chapter 7: AC equivalent circuit modeling
7.8. Summary of key points 1. The CCM converter analytical techniques of Chapters 2 and 3 can be extended to predict converter ac behavior. The key step is to average the converter waveforms over one switching period. This removes the switching harmonics, thereby exposing directly the desired dc and lowfrequency ac components of the waveforms. In particular, expressions for the averaged inductor voltages, capacitor currents, and converter input current are usually found. 2. Since switching converters are nonlinear systems, it is desirable to construct smallsignal linearized models. This is accomplished by perturbing and linearizing the averaged model about a quiescent operating point. 3. Ac equivalent circuits can be constructed, in the same manner used in Chapter 3 to construct dc equivalent circuits. If desired, the ac equivalent circuits may be refined to account for the effects of converter losses and other nonidealities. Fundamentals of Power Electronics
143
Chapter 7: AC equivalent circuit modeling
Summary of key points
4. The statespace averaging method of section 7.4 is essentially the same as the basic approach of section 7.2, except that the formality of the statespace network description is used. The general results are listed in section 7.4.2. 5. The circuit averaging technique also yields equivalent results, but the derivation involves manipulation of circuits rather than equations. Switching elements are replaced by dependent voltage and current sources, whose waveforms are defined to be identical to the switch waveforms of the actual circuit. This leads to a circuit having a timeinvariant topology. The waveforms are then averaged to remove the switching ripple, and perturbed and linearized about a quiescent operating point to obtain a smallsignal model.
Fundamentals of Power Electronics
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Chapter 7: AC equivalent circuit modeling
Summary of key points 6. When the switches are the only timevarying elements in the converter, then circuit averaging affects only the switch network. The converter model can then be derived by simply replacing the switch network with its averaged model. Dc and smallsignal ac models of several common CCM switch networks are listed in section 7.5.4. Switching losses can also be modeled using this approach. 7. The canonical circuit describes the basic properties shared by all dcdc PWM converters operating in the continuous conduction mode. At the heart of the model is the ideal 1:M(D) transformer, introduced in Chapter 3 to represent the basic dcdc conversion function, and generalized here to include ac variations. The converter reactive elements introduce an effective lowpass filter into the network. The model also includes independent sources which represent the effect of duty cycle variations. The parameter values in the canonical models of several basic converters are tabulated for easy reference. Fundamentals of Power Electronics
145
Chapter 7: AC equivalent circuit modeling
Summary of key points
8. The conventional pulsewidth modulator circuit has linear gain, dependent on the slope of the sawtooth waveform, or equivalently on its peaktopeak magnitude.
Fundamentals of Power Electronics
146
Chapter 7: AC equivalent circuit modeling
Chapter 8. Converter Transfer Functions
8.1. Review of Bode plots 8.1.1. 8.1.2. 8.1.3. 8.1.4. 8.1.5. 8.1.6. 8.1.7. 8.1.8.
Single pole response Single zero response Right halfplane zero Frequency inversion Combinations Double pole response: resonance The lowQ approximation Approximate roots of an arbitrarydegree polynomial
8.2. Analysis of converter transfer functions 8.2.1. Example: transfer functions of the buckboost converter 8.2.2. Transfer functions of some basic CCM converters 8.2.3. Physical origins of the right halfplane zero in converters Fundamentals of Power Electronics
1
Chapter 8: Converter Transfer Functions
Converter Transfer Functions 8.3. Graphical construction of converter transfer functions 8.3.1. 8.3.2. 8.3.3. 8.3.4.
Series impedances: addition of asymptotes Parallel impedances: inverse addition of asymptotes Another example Voltage divider transfer functions: division of asymptotes
8.4. Measurement of ac transfer functions and impedances 8.5. Summary of key points
Fundamentals of Power Electronics
2
Chapter 8: Converter Transfer Functions
Designoriented analysis How to approach a real (and hence, complicated) system Problems: Complicated derivations Long equations Algebra mistakes Design objectives: Obtain physical insight which leads engineer to synthesis of a good design Obtain simple equations that can be inverted, so that element values can be chosen to obtain desired behavior. Equations that cannot be inverted are useless for design!
Designoriented analysis is a structured approach to analysis, which attempts to avoid the above problems Fundamentals of Power Electronics
3
Chapter 8: Converter Transfer Functions
Some elements of designoriented analysis, discussed in this chapter • Writing transfer functions in normalized form, to directly expose salient features • Obtaining simple analytical expressions for asymptotes, corner frequencies, and other salient features, allows element values to be selected such that a given desired behavior is obtained • Use of inverted poles and zeroes, to refer transfer function gains to the most important asymptote • Analytical approximation of roots of highorder polynomials • Graphical construction of Bode plots of transfer functions and polynomials, to avoid algebra mistakes approximate transfer functions obtain insight into origins of salient features Fundamentals of Power Electronics
4
Chapter 8: Converter Transfer Functions
8.1. Review of Bode plots Table 8.1. Expressing magnitudes in decibels
Decibels G
dB
= 20 log 10 G
Actual magnitude
Decibels of quantities having units (impedance example): normalize before taking log Z
dB
= 20 log 10
Z Rbase
Magnitude in dB
1/2
– 6dB
1
0 dB
2
6 dB
5 = 10/2
20 dB – 6 dB = 14 dB
10
20dB
1000 = 103
3 ⋅ 20dB = 60 dB
5Ω is equivalent to 14dB with respect to a base impedance of Rbase = 1Ω, also known as 14dBΩ. 60dBµA is a current 60dB greater than a base current of 1µA, or 1mA. Fundamentals of Power Electronics
5
Chapter 8: Converter Transfer Functions
Bode plot of fn Bode plots are effectively loglog plots, which cause functions which vary as fn to become linear plots. Given: f n G = f0 60dB 2 f f0
–40dB/decade
Magnitude in dB is
G
dB
= 20 log 10
f f0
n
= 20n log 10
40dB
f f0
20dB
n
–20dB/decade
=
n=
0dB
• Slope is 20n dB/decade • Magnitude is 1, or 0dB, at frequency f = f0
n=
20 dB/decade
1
–1
–20dB
n –40dB
=
–2
40dB/decade –60dB
0.1f0
f0
f f0
2
10f0
f f0
–1
f f0
–2
f log scale
Fundamentals of Power Electronics
6
Chapter 8: Converter Transfer Functions
8.1.1. Single pole response Simple RC example
Transfer function is 1 v2(s) G(s) = = sC 1 +R v1(s) sC
R + v1(s)
+ –
C
v2(s)
Express as rational fraction:
G(s) =
–
1 1 + sRC
This coincides with the normalized form 1 G(s) = 1 + ωs 0
with Fundamentals of Power Electronics
7
ω0 = 1 RC Chapter 8: Converter Transfer Functions
G(jω) and  G(jω)  Let s = jω:
ω 1– j ω 0 1 G( jω) = = ω 2 ω 1+ ω 1+ j ω 0 0
Im(G(jω))
(jω
)

G(jω)
G( jω) = =
2
Re (G( jω)) + Im (G( jω)) 1 ω 2 1+ ω
2

G
Magnitude is
∠G(jω) Re(G(jω))
0
Magnitude in dB: G( jω)
= – 20 log 10 dB
Fundamentals of Power Electronics
ω 1+ ω 0
2
dB
8
Chapter 8: Converter Transfer Functions
Asymptotic behavior: low frequency For small frequency, ω > f0 :
G( jω) =
ω ω0 >> 1 ω 1+ ω 0
2
2
 G(jω) dB
ω ≈ ω 0
0dB
2
0dB
Then  G(jω)  becomes
G( jω) ≈
1 ω 1+ ω 0
–20dB
–20dB/decade
–1
f f0
–40dB
1 ω ω0
2
=
f f0
–1
–60dB
0.1f0
f0
10f0
f
The highfrequency asymptote of  G(jω)  varies as f1. Hence, n = 1, and a straightline asymptote having a slope of 20dB/decade is obtained. The asymptote has a value of 1 at f = f0 . Fundamentals of Power Electronics
10
Chapter 8: Converter Transfer Functions
Deviation of exact curve near f = f0 Evaluate exact magnitude:
at f = f0: G( jω0) =
G( jω0)
dB
1
ω 1 + ω0 0
= – 20 log 10
2
= 1 2 ω 1 + ω0 0
2
≈ – 3 dB
at f = 0.5f0 and 2f0 : Similar arguments show that the exact curve lies 1dB below the asymptotes.
Fundamentals of Power Electronics
11
Chapter 8: Converter Transfer Functions
Summary: magnitude  G(jω) dB
0dB
3dB
1dB
0.5f0 –10dB
1dB
f0 2f0
–20dB
–20dB/decade
–30dB
f Fundamentals of Power Electronics
12
Chapter 8: Converter Transfer Functions
Phase of G(jω) Im(G(jω))
ω 1– j ω 0 1 G( jω) = = ω 2 ω 1+ ω 1+ j ω 0 0

G
(jω )

G(jω)
∠G(jω) Re(G(jω))
∠G( jω) = tan
–1
∠G( jω) = – tan – 1
ω ω0
Im G( jω) Re G( jω)
Fundamentals of Power Electronics
13
Chapter 8: Converter Transfer Functions
Phase of G(jω)
∠G(jω)
0˚
∠G( jω) = – tan – 1
0˚ asymptote
ω ω0
15˚ 30˚ 45˚
45˚
ω
∠G(jω)
0
0˚
f0
60˚
ω
–45˚
∞
–90˚
0
75˚ –90˚ asymptote 90˚ 0.01f0
0.1f0
f0
10f0
100f0
f
Fundamentals of Power Electronics
14
Chapter 8: Converter Transfer Functions
Phase asymptotes Low frequency: 0˚ High frequency: –90˚ Low and highfrequency asymptotes do not intersect Hence, need a midfrequency asymptote
Try a midfrequency asymptote having slope identical to actual slope at the corner frequency f0. One can show that the asymptotes then intersect at the break frequencies
fa = f0 e – π / 2 ≈ f0 / 4.81 fb = f0 e π / 2 ≈ 4.81 f0
Fundamentals of Power Electronics
15
Chapter 8: Converter Transfer Functions
Phase asymptotes
∠G(jω)
fa = f0 / 4.81
0˚ 15˚
fa = f0 e ≈ f0 / 4.81 fb = f0 e π / 2 ≈ 4.81 f0 –π/2
30˚ 45˚
45˚ f0
60˚ 75˚ 90˚ 0.01f0
0.1f0
f0
fb = 4.81 f0
100f0
f
Fundamentals of Power Electronics
16
Chapter 8: Converter Transfer Functions
Phase asymptotes: a simpler choice fa = f0 / 10
∠G(jω)
0˚ 15˚ 30˚
fa = f0 / 10 fb = 10 f0
45˚
45˚ f0
60˚ 75˚ 90˚ 0.01f0
0.1f0
f0
fb = 10 f0
100f0
f Fundamentals of Power Electronics
17
Chapter 8: Converter Transfer Functions
Summary: Bode plot of real pole 0dB
 G(jω) dB
3dB
1dB 0.5f0
G(s) =
1dB
f0
1 1 + ωs
0
2f0
–20dB/decade
∠G(jω)
0˚
f0 / 10 5.7˚ 45˚/decade 45˚ f0 90˚
5.7˚ 10 f0 Fundamentals of Power Electronics
18
Chapter 8: Converter Transfer Functions
8.1.2. Single zero response Normalized form:
G(s) = 1 + ωs 0 Magnitude:
G( jω) =
ω 1+ ω 0
2
Use arguments similar to those used for the simple pole, to derive asymptotes: 0dB at low frequency, ω > ω0 Phase: ∠G( jω) = tan – 1
ω ω0
—with the exception of a missing minus sign, same as simple pole Fundamentals of Power Electronics
19
Chapter 8: Converter Transfer Functions
Summary: Bode plot, real zero G(s) = 1 + ωs 0
+20dB/decade
2f0 f0
1dB
0.5f0 0dB
 G(jω) dB
1dB
3dB
10 f0
+90˚
5.7˚ f0 45˚ +45˚/decade
∠G(jω)
0˚
5.7˚ f0 / 10
Fundamentals of Power Electronics
20
Chapter 8: Converter Transfer Functions
8.1.3. Right halfplane zero Normalized form:
G(s) = 1 – ωs 0 Magnitude:
G( jω) =
ω 1+ ω 0
2
—same as conventional (left halfplane) zero. Hence, magnitude asymptotes are identical to those of LHP zero. Phase: ∠G( jω) = – tan – 1
ω ω0
—same as real pole. The RHP zero exhibits the magnitude asymptotes of the LHP zero, and the phase asymptotes of the pole Fundamentals of Power Electronics
21
Chapter 8: Converter Transfer Functions
Summary: Bode plot, RHP zero G(s) = 1 – ωs 0
+20dB/decade
2f0 f0
1dB
0.5f0 0dB
 G(jω) dB ∠G(jω)
0˚
1dB
3dB
f0 / 10 5.7˚ 45˚/decade 45˚ f0 90˚
5.7˚ 10 f0 Fundamentals of Power Electronics
22
Chapter 8: Converter Transfer Functions
8.1.4. Frequency inversion Reversal of frequency axis. A useful form when describing mid or highfrequency flat asymptotes. Normalized form, inverted pole: 1 G(s) = ω 1 + s0 An algebraically equivalent form:
G(s) =
s ω0 1 + ωs 0
The invertedpole format emphasizes the highfrequency gain.
Fundamentals of Power Electronics
23
Chapter 8: Converter Transfer Functions
Asymptotes, inverted pole G(s) =
1
0dB
ω 1 + s0
3dB
1dB
1dB
f0
2f0
0.5f0
 G(jω) dB +20dB/decade
∠G(jω)
+90˚
f0 / 10 5.7˚ 45˚/decade +45˚ f0 0˚
5.7˚ 10 f0
Fundamentals of Power Electronics
24
Chapter 8: Converter Transfer Functions
Inverted zero Normalized form, inverted zero: ω G(s) = 1 + s0 An algebraically equivalent form:
1 + ωs 0 G(s) = s ω0 Again, the invertedzero format emphasizes the highfrequency gain.
Fundamentals of Power Electronics
25
Chapter 8: Converter Transfer Functions
Asymptotes, inverted zero ω G(s) = 1 + s0
–20dB/decade
 G(jω) dB
0.5f0 f0 1dB
2f0
3dB
5.7˚
1dB
0dB
10 f0
0˚
f0 –45˚ +45˚/decade
∠G(jω)
–90˚
5.7˚ f0 / 10
Fundamentals of Power Electronics
26
Chapter 8: Converter Transfer Functions
8.1.5. Combinations
Suppose that we have constructed the Bode diagrams of two complexvalues functions of frequency, G1(ω) and G2(ω). It is desired to construct the Bode diagram of the product, G3(ω) = G1(ω) G2(ω). Express the complexvalued functions in polar form:
G1(ω) = R1(ω) e jθ 1(ω) G2(ω) = R2(ω) e jθ 2(ω) G3(ω) = R3(ω) e jθ 3(ω) The product G3(ω) can then be written
G3(ω) = G1(ω) G2(ω) = R1(ω) e jθ 1(ω) R2(ω) e jθ 2(ω) G3(ω) = R1(ω) R2(ω) e j(θ 1(ω) + θ 2(ω))
Fundamentals of Power Electronics
27
Chapter 8: Converter Transfer Functions
Combinations G3(ω) = R1(ω) R2(ω) e j(θ 1(ω) + θ 2(ω))
The composite phase is θ 3(ω) = θ 1(ω) + θ 2(ω)
The composite magnitude is
R3(ω) = R1(ω) R2(ω) R3(ω)
dB
= R1(ω)
dB
+ R2(ω)
dB
Composite phase is sum of individual phases. Composite magnitude, when expressed in dB, is sum of individual magnitudes.
Fundamentals of Power Electronics
28
Chapter 8: Converter Transfer Functions
Example 1:
G0
G(s) =
1 + ωs 1
1 + ωs 2
with G0 = 40 ⇒ 32 dB, f1 = ω1/2π = 100 Hz, f2 = ω2/2π = 2 kHz 40 dB
 G  20 dB
G0 = 40 ⇒ 32 dB  G 
∠G
–20 dB/decade
0 dB
0 dB –20 dB
f1 100 Hz
∠G
–40 dB
f2 2 kHz
0˚ f1/10 10 Hz
–60 dB
f2/10 200 Hz –45˚/decade
–90˚
–90˚/decade
10 Hz
100 Hz
0˚ –45˚
10f1 1 kHz 1 Hz
–40 dB/decade
1 kHz
10f2 20 kHz
–135˚
–45˚/decade 10 kHz
–180˚ 100 kHz
f Fundamentals of Power Electronics
29
Chapter 8: Converter Transfer Functions
Example 2 Determine the transfer function A(s) corresponding to the following asymptotes:
f2
 A  f1
 A0 dB
+20 dB/dec
10f1
∠A
 A∞ dB
+45˚/dec
f2 /10
–90˚
–45˚/dec 0˚
0˚
f1 /10 Fundamentals of Power Electronics
10f2 30
Chapter 8: Converter Transfer Functions
Example 2, continued One solution:
A(s) = A 0
1 + ωs 1 1 + ωs 2
Analytical expressions for asymptotes: For f < f1
A0
s 1+➚ ω
= A0 1 = A0 1
1
s 1+➚ ω 2
s = jω
For f1 < f < f2 A0
1 + ωs ➚ s 1+➚ ω
1
2
Fundamentals of Power Electronics
= A0
s ω1
s = jω
1
ω =A f = A0 ω 0 f1 1
s = jω
31
Chapter 8: Converter Transfer Functions
Example 2, continued For f > f2
A0
1 + ωs ➚ 1 + ωs ➚
1
2
= A0 s = jω
s ω1 s ω2
s = jω
ω f = A 0 ω2 = A 0 2 f1 1
s = jω
So the highfrequency asymptote is
A∞ = A0
f2 f1
Another way to express A(s): use inverted poles and zeroes, and express A(s) directly in terms of A∞
A(s) = A ∞
Fundamentals of Power Electronics
ω 1 + s1 ω 1 + s2 32
Chapter 8: Converter Transfer Functions
8.1.6 Quadratic pole response: resonance Example
G(s) =
L
v2(s) 1 = v1(s) 1 + s L + s 2LC R
Secondorder denominator, of the form
+ v1(s)
+ –
C
R
v2(s) –
1 G(s) = 1 + a 1s + a 2s 2
Twopole lowpass filter example
with a1 = L/R and a2 = LC How should we construct the Bode diagram?
Fundamentals of Power Electronics
33
Chapter 8: Converter Transfer Functions
Approach 1: factor denominator G(s) =
1 1 + a 1s + a 2s 2
We might factor the denominator using the quadratic formula, then construct Bode diagram as the combination of two real poles:
G(s) =
1 1 – ss 1
1 – ss 2
with
s1 = –
a1 1– 2a 2
1–
4a 2 a 21
s2 = –
a1 1+ 2a 2
1–
4a 2 a 21
• If 4a2 ≤ a12, then the roots s1 and s2 are real. We can construct Bode diagram as the combination of two real poles. • If 4 a2 > a12, then the roots are complex. In Section 8.1.1, the assumption was made that ω 0 is real; hence, the results of that section cannot be applied and we need to do some additional work. Fundamentals of Power Electronics
34
Chapter 8: Converter Transfer Functions
Approach 2: Define a standard normalized form for the quadratic case G(s) =
1 1 + 2ζ ωs + ωs 0 0
2
or
G(s) =
1 1 + s + ωs Qω0 0
2
• When the coefficients of s are real and positive, then the parameters ζ, ω0, and Q are also real and positive • The parameters ζ, ω0, and Q are found by equating the coefficients of s • The parameter ω0 is the angular corner frequency, and we can define f0 = ω0/2π • The parameter ζ is called the damping factor. ζ controls the shape of the exact curve in the vicinity of f = f0. The roots are complex when ζ < 1. • In the alternative form, the parameter Q is called the quality factor. Q also controls the shape of the exact curve in the vicinity of f = f0. The roots are complex when Q > 0.5. Fundamentals of Power Electronics
35
Chapter 8: Converter Transfer Functions
The Qfactor In a secondorder system, ζ and Q are related according to
Q= 1 2ζ Q is a measure of the dissipation in the system. A more general definition of Q, for sinusoidal excitation of a passive element or system is
Q = 2π
(peak stored energy) (energy dissipated per cycle)
For a secondorder passive system, the two equations above are equivalent. We will see that Q has a simple interpretation in the Bode diagrams of secondorder transfer functions.
Fundamentals of Power Electronics
36
Chapter 8: Converter Transfer Functions
Analytical expressions for f0 and Q Twopole lowpass filter example: we found that
Equate coefficients of like powers of s with the standard form
Result:
Fundamentals of Power Electronics
G(s) =
G(s) =
v2(s) 1 = v1(s) 1 + s L + s 2LC R
1 1 + s + ωs Qω0 0
2
ω0 1 f0 = = 2π 2π LC Q=R C L 37
Chapter 8: Converter Transfer Functions
Magnitude asymptotes, quadratic form In the form
G(s) =
1 1 + s + ωs Qω0 0
let s = jω and find magnitude:
G → 1 for ω > ω0
0 dB
0 dB
f f0
–20 dB
–2
–40 dB
–40 dB/decade –60 dB
0.1f0
Fundamentals of Power Electronics
38
f0
10f0
f
Chapter 8: Converter Transfer Functions
Deviation of exact curve from magnitude asymptotes G( jω) =
1 ω 1– ω 0
2 2
ω + 12 ω 0 Q
2
At ω = ω0, the exact magnitude is
G( jω0) = Q
G( jω0)
or, in dB:
The exact curve has magnitude Q at f = f0. The deviation of the exact curve from the asymptotes is  Q dB
dB
= Q
dB
 G   Q dB
0 dB f0
–40 dB/decade
Fundamentals of Power Electronics
39
Chapter 8: Converter Transfer Functions
Twopole response: exact curves 0° Q=∞
Q=∞ Q = 10 Q =5 Q=2 Q=1 Q = 0.7 Q = 0.5
Q=5
10dB Q=2 Q=1
45°
Q = 0.7 Q = 0.2
0dB
Q = 0.1
∠G
Q = 0.5
 G dB
10dB
90°
Q = 0.2
135° Q = 0.1
20dB 0.3
0.5
0.7
1
2
3
f / f0
180° 0.1
1
f / f0
Fundamentals of Power Electronics
40
Chapter 8: Converter Transfer Functions
10
8.1.7. The lowQ approximation Given a secondorder denominator polynomial, of the form G(s) =
1 1 + a 1s + a 2s 2
or
G(s) =
1 1 + s + ωs Qω0 0
2
When the roots are real, i.e., when Q < 0.5, then we can factor the denominator, and construct the Bode diagram using the asymptotes for real poles. We would then use the following normalized form: G(s) =
1 1 + ωs 1
1 + ωs 2
This is a particularly desirable approach when Q 0.5
11.2
A simple firstorder model Simple model via algebraic approach Averaged switch modeling
11.3
A more accurate model Current programmed controller model: block diagram CPM buck converter example
11.4
Discontinuous conduction mode
11.5
Summary
Fundamentals of Power Electronics
3
Chapter 11: Current Programmed Control
11.1 Oscillation for D > 0.5
•
The current programmed controller is inherently unstable for D > 0.5, regardless of the converter topology
•
Controller can be stabilized by addition of an artificial ramp
Objectives of this section: •
Stability analysis
•
Describe artificial ramp scheme
Fundamentals of Power Electronics
4
Chapter 11: Current Programmed Control
Inductor current waveform, CCM Inductor current slopes m1 and –m2 iL(t) ic iL(0)
m1
0
– m2
dTs
Fundamentals of Power Electronics
iL(Ts)
Ts
t
5
buck converter vg – v m1 = – m2 = – v L L boost converter vg vg – v m1 = – m2 = L L buck–boost converter vg m1 = – m2 = v L L
Chapter 11: Current Programmed Control
Steadystate inductor current waveform, CPM First interval: i L(dTs) = i c = i L(0) + m 1dTs
Solve for d: i – i (0) d= c L m 1T s Second interval: i L(Ts) = i L(dTs) – m 2d'Ts = i L(0) + m 1dTs – m 2d'Ts
iL(t) ic iL(0)
m1
0
– m2
dTs
iL(Ts)
Ts
t
In steady state:
0 = M 1DTs – M 2D'Ts
M2 D = M 1 D' Fundamentals of Power Electronics
6
Chapter 11: Current Programmed Control
Perturbed inductor current waveform iL(t)
i L(0)
ic
i L(T s)
m1
I L0 + i L(0) IL0
– m2
m1 – m2
Perturbed waveform
dT s
0
D + d T s DTs
Fundamentals of Power Electronics
Ts
7
Steadystate waveform
t
Chapter 11: Current Programmed Control
Change in inductor current perturbation over one switching period magnified view
ic m1
i L(T s)
i L(0)
m1
dT s
– m2
– m2
Steadystate waveform Perturbed waveform
i L(0) = – m 1 dTs i L(Ts) = i L(0) – D D'
i L(Ts) = m 2 dTs m2 i L(Ts) = i L(0) – m1 Fundamentals of Power Electronics
8
Chapter 11: Current Programmed Control
Change in inductor current perturbation over many switching periods i L(Ts) = i L(0) – D D' i L(2Ts) = i L(Ts) – D = i L(0) – D D' D'
2
i L(nTs) = i L((n – 1)Ts) – D = i L(0) – D D' D'
i L(nTs) →
For stability: Fundamentals of Power Electronics
0 ∞
n
when – D < 1 D' when – D > 1 D'
D < 0.5 9
Chapter 11: Current Programmed Control
Example: unstable operation for D = 0.6 α = – D = – 0.6 = – 1.5 D' 0.4
iL(t) ic
2.25i L(0)
i L(0)
IL0
– 1.5i L(0)
0
Ts
Fundamentals of Power Electronics
– 3.375i L(0)
2Ts
10
3Ts
4Ts
t
Chapter 11: Current Programmed Control
Example: stable operation for D = 1/3 α = – D = – 1/3 = – 0.5 D' 2/3 iL(t) ic
i L(0) IL0
– 1 i L(0) 2
0
Ts
Fundamentals of Power Electronics
1 i (0) 4 L
2Ts
11
– 1 i L(0) 8
3Ts
1 i (0) 16 L
4Ts
t
Chapter 11: Current Programmed Control
Stabilization via addition of an artificial ramp to the measured switch current waveform Buck converter L
is(t)
iL(t)
ia(t)
+ Q1 vg(t) + –
C
D1
v(t) –
Measure is(t) switch Rf current ma
+ +
0
ia(t)Rf Artificial ramp + –
Analog comparator
ic(t)Rf Control input
ma
0
Ts
2Ts
t
Now, transistor switches off when
Clock
is(t)Rf
R
Ts
i a(dTs) + i L(dTs) = i c
S Q R Latch
or,
i L(dTs) = i c – i a(dTs)
Currentprogrammed controller
Fundamentals of Power Electronics
12
Chapter 11: Current Programmed Control
Steady state waveforms with artificial ramp i L(dTs) = i c – i a(dTs) (ic – ia(t)) ic
– ma
iL(t) m1
IL0
0
Fundamentals of Power Electronics
– m2
dTs
Ts
13
t
Chapter 11: Current Programmed Control
Stability analysis: perturbed waveform
(ic – ia(t)) ic m1
i L (T
s)
) i L(0 m 1
I L0 + i L(0) IL0
– ma
– m2
Steadystate waveform
– m2
Perturbed waveform
dT s
0
D + d T s DTs
Fundamentals of Power Electronics
Ts 14
t
Chapter 11: Current Programmed Control
Stability analysis: change in perturbation over complete switching periods First subinterval:
i L(0) = – dTs m 1 + m a Second subinterval:
i L(Ts) = – dTs m a – m 2 Net change over one switching period:
m –m i L(Ts) = i L(0) – m 2 + ma 1 a After n switching periods:
m –m m –m i L(nTs) = i L((n –1)Ts) – m 2 + ma = i L(0) – m 2 + ma 1 a 1 a Characteristic value:
m –m α = – m 2 + ma 1 a
Fundamentals of Power Electronics
i L(nTs) →
15
0
when α < 1
∞
when α > 1
n
= i L(0) α n
Chapter 11: Current Programmed Control
The characteristic value α ma 1– m 2 α=– D' + m a D m2
•
For stability, require  α  < 1
•
Buck and buckboost converters: m2 = – v/L So if v is wellregulated, then m2 is also wellregulated
•
A common choice: ma = 0.5 m2 This leads to α = –1 at D = 1, and  α  < 1 for 0 ≤ D < 1. The minimum α that leads to stability for all D.
•
Another common choice: ma = m2 This leads to α = 0 for 0 ≤ D < 1. Deadbeat control, finite settling time
Fundamentals of Power Electronics
16
Chapter 11: Current Programmed Control
11.2 A Simple FirstOrder Model Switching converter
+
vg(t) + –
R
v(t) –
Converter voltages and currents
Current programmed controller ic(t)
Compensator
v(t) +–
d(t)
vref
Fundamentals of Power Electronics
17
Chapter 11: Current Programmed Control
The firstorder approximation i L(t)
Ts
= i c(t)
• Neglects switching ripple and artificial ramp • Yields physical insight and simple firstorder model • Accurate when converter operates well into CCM (so that switching ripple is small) and when the magnitude of the artificial ramp is not too large • Resulting smallsignal relation:
i L(s) ≈ i c(s)
Fundamentals of Power Electronics
18
Chapter 11: Current Programmed Control
11.2.1 Simple model via algebraic approach: CCM buckboost example Q1
D1 +
iL(t) + –
vg(t)
L
C
R
v(t) –
iL(t) ic
v L
vg L
0
Fundamentals of Power Electronics
dTs
Ts 19
t Chapter 11: Current Programmed Control
Smallsignal equations of CCM buck–boost, duty cycle control
d i L(t) L = Dvg(t) + D'v(t) + Vg – V d(t) dt dv(t) v(t) = – D'i L – + I L d(t) C R dt i g(t) = Di L + I L d(t) Derived in Chapter 7
Fundamentals of Power Electronics
20
Chapter 11: Current Programmed Control
Transformed equations
Take Laplace transform, letting initial conditions be zero:
sLi L(s) = Dvg(s) + D'v(s) + Vg – V d(s) v(s) + I L d(s) sCv(s) = – D'i L(s) – R i g(s) = Di L(s) + I L d(s)
Fundamentals of Power Electronics
21
Chapter 11: Current Programmed Control
The simple approximation Now let
i L(s) ≈ i c(s) Eliminate the duty cycle (now an intermediate variable), to express the equations using the new control input iL. The inductor equation becomes:
sLi c(s) ≈ Dvg(s) + D'v(s) + Vg – V d(s) Solve for the duty cycle variations:
d(s) =
sLi c(s) – Dvg(s) – D'v(s)
Fundamentals of Power Electronics
Vg – V 22
Chapter 11: Current Programmed Control
The simple approximation, continued Substitute this expression to eliminate the duty cycle from the remaining equations:
sLi c(s) – Dvg(s) – D'v(s) v(s) sCv(s) = – D'i c(s) – + IL R V –V g
i g(s) = Di c(s) + I L
sLi c(s) – Dvg(s) – D'v(s) Vg – V
Collect terms, simplify using steadystate relations: 2 D sLD D 1 sCv(s) = – D' i c(s) – + v(s) – v (s) R R D'R g D'R 2 D sLD D i g(s) = + D i c(s) – v(s) – v (s) R D'R D'R g
Fundamentals of Power Electronics
23
Chapter 11: Current Programmed Control
Construct equivalent circuit: input port 2 D sLD D i g(s) = + D i c(s) – v(s) – v (s) R D'R D'R g
ig D2 v D'R g vg
+ –
– D'R D2
Fundamentals of Power Electronics
D 1 + sL i c D'R
24
Dv R
Chapter 11: Current Programmed Control
Construct equivalent circuit: output port
2 D sLD D 1 sCv(s) = – D' i c(s) – + v(s) – v (s) R R D'R D'R g
Node Dv R D2 v D'R g
R D
D' 1 – sLD ic D' 2R
Fundamentals of Power Electronics
25
v R
sCv
C
R
Chapter 11: Current Programmed Control
CPM Canonical Model, Simple Approximation
ig
+ vg
+ –
r1
f1(s) i c
g1 v
g 2 vg
f2(s) i c
r2
C
R
v
–
Fundamentals of Power Electronics
26
Chapter 11: Current Programmed Control
Table of results for basic converters
Tabl e 11 . 1
Current programmed mode smallsignal equivalent circuit parameters, simple model
Converter
g1
f1
r1
g2
f2
r2
Buck
D R
D 1 + sL R
– R2 D
0
1
∞
Boost
0
1
∞
1 D'R
D' 1 – sL2 D' R
R
Buckboost
– D R
– D'R D2
2 – D D'R
– D' 1 – sDL D' 2R
R D
D 1 + sL D'R
Fundamentals of Power Electronics
27
Chapter 11: Current Programmed Control
Transfer functions predicted by simple model ig
+ vg
+ –
r1
f1(s) i c
g1 v
f2(s) i c
g 2 vg
r2
C
R
v
–
Controltooutput transfer function
Result for buckboost example
Fundamentals of Power Electronics
Gvc(s) =
v(s) i c(s)
Gvc(s) = – R
28
vg = 0
D' 1+D
= f2 r 2  R  1 sC
1 – s DL D' 2R 1 + s RC 1+D
Chapter 11: Current Programmed Control
Transfer functions predicted by simple model ig
+ vg
+ –
r1
f1(s) i c
g1 v
f2(s) i c
g 2 vg
r2
C
R
v
–
Linetooutput transfer function
Result for buckboost example
Fundamentals of Power Electronics
Gvg(s) =
v(s) vg(s)
ic
2 D Gvg(s) = – 1 – D2
29
= g 2 r 2  R  1 sC =0 1 1 + s RC 1+D
Chapter 11: Current Programmed Control
Transfer functions predicted by simple model ig
+ vg
+ –
r1
f1(s) i c
g1 v
g 2 vg
f2(s) i c
r2
C
R
v
–
Output impedance
Result for buckboost example
Fundamentals of Power Electronics
Z out(s) = r 2  R  1 sC Z out(s) =
30
R 1+D
1 1 + s RC 1+D
Chapter 11: Current Programmed Control
11.2.2 Averaged switch modeling with the simple approximation
i1(t)
vg(t) + –
i2(t) +
+
v1(t)
v2(t)
–
–
L
iL(t) + C
R
v(t) –
Switch network
Averaged terminal waveforms, CCM:
v2(t) i 1(t)
Ts Ts
Fundamentals of Power Electronics
= d(t) v1(t) = d(t) i 2(t)
The simple approximation: Ts
i 2(t)
Ts
≈ i c(t)
Ts
Ts
31
Chapter 11: Current Programmed Control
CPM averaged switch equations
v2(t)
Ts
i 1(t)
Ts
= d(t) v1(t) = d(t) i 2(t)
i 2(t)
Ts
Ts
≈ i c(t)
Ts
Ts
Eliminate duty cycle:
i 1(t) i 1(t)
Ts
= d(t) i c(t)
Ts
v1(t)
Ts
Ts
= i c(t)
= Ts
v2(t) v1(t) v2(t)
Ts
i c(t)
Ts
Ts Ts
= p(t)
Ts
So: •
Output port is a current source
•
Input port is a dependent current sink
Fundamentals of Power Electronics
32
Chapter 11: Current Programmed Control
CPM averaged switch model
〈i1(t)〉T s + 〈vg(t)〉Ts
+ –
〈v1(t)〉T s
〈i2(t)〉Ts 〈 p(t)〉T
L
〈iL(t)〉Ts
+
+
s
〈ic(t)〉Ts
–
〈v2(t)〉Ts –
C
R
〈v(t)〉Ts –
Averaged switch network
Fundamentals of Power Electronics
33
Chapter 11: Current Programmed Control
Results for other converters 〈iL(t)〉Ts
Boost
L +
〈 p(t)〉Ts 〈vg(t)〉Ts
〈ic(t)〉Ts
+ –
C
〈v(t)〉Ts
R
– Averaged switch network Averaged switch network
Buckboost 〈 p(t)〉Ts 〈vg(t)〉Ts
〈ic(t)〉Ts
+ –
+ C
R
〈v(t)〉T
s
L 〈iL(t)〉Ts
Fundamentals of Power Electronics
34
–
Chapter 11: Current Programmed Control
Perturbation and linearization to construct smallsignal model Let
Resulting input port equation:
v1(t) i 1(t) v2(t) i 2(t) i c(t)
Ts Ts Ts Ts Ts
= V1 + v1(t) = I 1 + i 1(t) = V2 + v2(t) = I 2 + i 2(t) = I c + i c(t)
V1 + v1(t) I 1 + i 1(t) = I c + i c(t) V2 + v2(t) Smallsignal result:
V2 Ic I1 i 1(t) = i c(t) + v2(t) – v1(t) V1 V1 V1 Output port equation: î2 = î c
Fundamentals of Power Electronics
35
Chapter 11: Current Programmed Control
Resulting smallsignal model Buck example
i1
i2 +
+ vg
+ –
v1
ic
V2 V1
V – 1 I1
v2
Ic V1
ic
–
v2
–
L + C
R
v
–
Switch network smallsignal ac model
V2 Ic I1 i 1(t) = i c(t) + v2(t) – v1(t) V1 V1 V1
Fundamentals of Power Electronics
36
Chapter 11: Current Programmed Control
Origin of input port negative incremental resistance 〈i1(t)〉T
s
Power source characteristic 〈v1(t)〉Ts 〈i1(t)〉Ts = 〈 p(t)〉Ts Quiescent operating point 1 = – I1 r1 V1
I1
〈v1(t)〉Ts
V1 Fundamentals of Power Electronics
37
Chapter 11: Current Programmed Control
Expressing the equivalent circuit in terms of the converter input and output voltages L
ig
iL
+ vg
+ –
i c D 1 + sL R
2 –D R
Dv R
ic
C
R
v
–
2 D L D i 1(s) = D 1 + s i (s) + v(s) – vg(s) R c R R
Fundamentals of Power Electronics
38
Chapter 11: Current Programmed Control
Predicted transfer functions of the CPM buck converter L
ig
iL
+ vg
+ –
i c D 1 + sL R
2 D – R
Dv R
ic
C
R
v
–
v(s) Gvc(s) = i c(s)
Gvg(s) =
Fundamentals of Power Electronics
v(s) vg(s)
= R  1 sC vg = 0
=0 ic = 0
39
Chapter 11: Current Programmed Control
11.3 A More Accurate Model
●
●
●
●
The simple models of the previous section yield insight into the lowfrequency behavior of CPM converters Unfortunately, they do not always predict everything that we need to know: Linetooutput transfer function of the buck converter Dynamics at frequencies approaching fs More accurate model accounts for nonideal operation of current mode controller builtin feedback loop Converter dutycyclecontrolled model, plus block diagram that accurately models equations of current mode controller
Fundamentals of Power Electronics
40
Chapter 11: Current Programmed Control
11.3.1Current programmed controller model
m 1dT s 2
m 2d'T s 2
madTs
ic
– ma
(ic – ia(t)) 〈iL(t)〉dT
m1
〈iL(t)〉d'T
s
– m2 s
〈iL(t)〉T = d〈iL(t)〉dT + d'〈iL(t)〉d'T s
0
s
dTs
i L(t)
iL(t)
s
Ts
t
m 1 dTs m 2 d'Ts – d' Ts Ts 2 2 d 2Ts d' 2Ts = i c(t) T – m a dTs – m 1 – m2 s 2 2 = i c(t)
Fundamentals of Power Electronics
– m a dTs – d
41
Chapter 11: Current Programmed Control
Perturb Let
i L(t) i c(t)
Ts Ts
Note that it is necessary to perturb the slopes, since these depend on the applied inductor voltage. For basic converters,
= I L + i L(t) = I c + i c(t)
buck converter vg – v m1 = m2 = v L L boost converter vg v – vg m1 = m2 = L L buckboost converter vg m1 = m2 = – v L L
d(t) = D + d(t) m 1(t) = M 1 + m 1(t) m 2(t) = M 2 + m 2(t) It is assumed that the artificial ramp slope does not vary.
Fundamentals of Power Electronics
42
Chapter 11: Current Programmed Control
Linearize I L + i L(t) = I c + i c(t) – M aTs D + d(t) – M 1 + m 1(t) D + d(t) – M 2 + m 2(t) D' – d(t)
2
2
Ts 2
Ts 2
The firstorder ac terms are
D 2Ts D' 2Ts i L(t) = i c(t) – M aTs + DM 1Ts – D'M 2Ts d(t) – m 1(t) – m 2(t) 2 2 Simplify using dc relationships:
D 2T s D' 2Ts i L(t) = i c(t) – M aTs d(t) – m 1(t) – m 2(t) 2 2 Solve for duty cycle variations: 2 2 D D' T Ts s 1 d(t) = i (t) – i L(t) – m 1(t) – m 2(t) M aT s c 2 2
Fundamentals of Power Electronics
43
Chapter 11: Current Programmed Control
Equation of the current programmed controller The expression for the duty cycle is of the general form
d(t) = Fm i c(t) – i L(t) – Fg vg(t) – Fv v(t) Fm = 1/MaTs Table 11.2. Current programmed controller gains for basic converters Converter
Fg
Fv
Buck
D 2T s 2L
1 – 2D T s 2L
Boost
2D – 1 T s 2L
D' 2T s 2L
Buckboost
D 2T s 2L
D' 2T s – 2L
Fundamentals of Power Electronics
44
Chapter 11: Current Programmed Control
Block diagram of the current programmed controller
d(t) = Fm i c(t) – i L(t) – Fg vg(t) – Fv v(t) d
•
Describes the duty cycle chosen by the CPM controller
•
Append this block diagram to the duty cycle controlled converter model, to obtain a complete CPM system model
Fm
vg
– –
+
Fg
iL
+
–
v
Fv
ic Fundamentals of Power Electronics
45
Chapter 11: Current Programmed Control
CPM buck converter model Vg d(t)
L
+ –
1:D
+
i L(t) vg(t)
+ –
I d(t)
C
v(t) R –
d
Tv
Fm
vg
– –
Fv
+
Fg
v
Ti iL
+
– ic
Fundamentals of Power Electronics
46
Chapter 11: Current Programmed Control
CPM boost converter model V d(t)
L
+ –
D' : 1
+
i L(t) vg(t) + –
C
I d(t)
v(t) R –
d
Tv
Fm
vg
– –
Fv
+
Fg
v
Ti iL
+
– ic
Fundamentals of Power Electronics
47
Chapter 11: Current Programmed Control
CPM buckboost converter model Vg – V d(t)
L
D' : 1
+ –
1:D
+
i L(t) vg(t)
+ –
I d(t)
I d(t)
C v(t) R –
d
Tv
Fm
vg
– –
+
Fg
Fv
v
Ti iL
+
– ic
Fundamentals of Power Electronics
48
Chapter 11: Current Programmed Control
11.3.2 Example: analysis of CPM buck converter V d D
L
+ –
1:D
+
i L(t) vg(t)
+ –
I d(t)
Zi(s)
C
v(t) R
{
–
Zo(s)
d
Z o = R  1 sC vg
Fg
Fv
v
Ti
i L(s) = D vg(s) + V2 d(s) Z i(s) D v(s) = i L(s) Z o(s)
–
iL
+
Fundamentals of Power Electronics
– –
+
Z i = sL + R  1 sC
Tv
Fm
ic 49
Chapter 11: Current Programmed Control
Block diagram of entire CPM buck converter
vg
++
iL
D Z i(s)
V D2
Voltage loop gain:
Tv(s) = Fm V2 D Z o(s) Fv D Z i(s) Zo(s)
Current loop gain:
d
Ti(s) = Fm
Tv
Fm V2 D D Z i(s) Ti(s) = 1 + Fm V2 D Z o(s) Fv D Z i(s)
v
– –
+
Fg
Fv Ti iL
Transfer function from îc to îL:
+
–
i L(s) Ti(s) = i c(s) 1 + Ti(s)
ic
Fundamentals of Power Electronics
Tv(s) 1 Z o(s) Fv 1 + Tv(s)
50
Chapter 11: Current Programmed Control
Discussion: Transfer function from îc to îL •
When the loop gain Ti(s) is large in magnitude, then îL is approximately equal to îc. The results then coincide with the simple approximation.
•
The controltooutput transfer function can be written as
i L(s) Ti(s) = i c(s) 1 + Ti(s)
Ti(s) v(s) Gvc(s) = = Z o(s) 1 + Ti(s) i c(s) which is the simple approximation result, multiplied by the factor Ti/(1 + Ti).
Fundamentals of Power Electronics
51
Chapter 11: Current Programmed Control
Loop gain Ti(s) Result for the buck converter:
Ti(s) = K 1 – α 1+α
1 + sRC 2DM a 1 – α 2DM a 1 – α 1+s L + s 2 LC R M2 1 + α M2 1 + α
Characteristic value
m 1 – ma 2 α=– D' + m a D m2
Controller stable for  α  < 1
K = 2L/RTs CCM for K > D’
Fundamentals of Power Electronics
52
Chapter 11: Current Programmed Control
Loop gain 1 + ωs z Ti(s) = Ti0 1 + s + ωs Qω0 0
Q
Ti
Ti0
f0 fc
fz
0dB
Ti 1 + Ti
f
Fundamentals of Power Electronics
53
2
Ti0 = K 1 – α 1+α ωz = 1 RC 1 ω0 = 2DM a 1 – α LC M2 1 + α M2 1 + α Q=R C L 2DM a 1 – α
Chapter 11: Current Programmed Control
Crossover frequency fc High frequency asymptote of Ti is
Ti(s) ≈ Ti0
s ωz ω 20 = Ti0 s 2 sωz ω0
Equate magnitude to one at crossover frequency:
f 20 Ti( j2π fc) ≈ Ti0 =1 2π fc fz Solve for crossover frequency:
M 2 fs fc = M a 2πD
Fundamentals of Power Electronics
54
Chapter 11: Current Programmed Control
Construction of transfer functions i L(s) Ti(s) 1 = ≈ i c(s) 1 + Ti(s) 1 + ωs
c
Ti(s) 1 Gvc(s) = Z o(s) ≈R 1 + Ti(s) 1 + sRC 1 + ωs c —the result of the simple firstorder model, with an added pole at the loop crossover frequency
Fundamentals of Power Electronics
55
Chapter 11: Current Programmed Control
Exact analysis
Gvc(s) = R
Ti0 1 + Ti0
1 Ti0 1 s 1 1 1+s + + 1 + Ti0 ωz 1 + Ti0 ω0 1 + Ti0 Qω0
2
for  Ti0  >> 1,
Gvc(s) ≈ R
1 2 s s 1+ ω + z Ti0 ω 20
Low Q approximation:
Gvc(s) ≈ R
which agrees with the previous slide
1 1 + ωs z
Fundamentals of Power Electronics
1+
sωz Ti0ω 20 56
Chapter 11: Current Programmed Control
Linetooutput transfer function Solution of complete block diagram leads to
Gg0(s) Ti(s) v(s) = i c(s) Z o(s) + vg(s) 1 + Ti(s) 1 + Ti(s) with
Gg0(s) = Z o(s) D Z i(s)
Fundamentals of Power Electronics
1 – V2 FmFg D 1 + V2 FmFvZ o(s) D Z i(s) D
57
Chapter 11: Current Programmed Control
Linetooutput transfer function v(s) Gvg(s) = vg(s)
Gvg(s) =
i c(s) = 0
Gg0(s) = 1 + Ti(s)
Ma 1 (1 – α) 1 2D 2 – M2 2 1 + Ti0 (1 + α) Ti0 1 s 1 1 1+s + + 1 + Ti0 ω0 1 + Ti0 ωz 1 + Ti0 Qω0
2
Poles are identical to controltooutput transfer function, and can be factored the same way:
Gvg(0) Gvg(s) ≈ 1 + ωs 1 + ωs z c Fundamentals of Power Electronics
58
Chapter 11: Current Programmed Control
Linetooutput transfer function: dc gain 2 M 2D a Gvg(0) ≈ –1 K M2 2
for large Ti0
For any Ti0 , the dc gain goes to zero when Ma/M2 = 0.5 Effective feedforward of input voltage variations in CPM controller then effectively cancels out the vg variations in the direct forward path of the converter.
Fundamentals of Power Electronics
59
Chapter 11: Current Programmed Control
11.4 Discontinuous conduction mode •
Again, use averaged switch modeling approach
•
Result: simply replace Transistor by power sink Diode by power source
•
Inductor dynamics appear at high frequency, near to or greater than the switching frequency
•
Smallsignal transfer functions contain a single low frequency pole
•
DCM CPM boost and buckboost are stable without artificial ramp
•
DCM CPM buck without artificial ramp is stable for D < 2/3. A small artificial ramp ma ≥ 0.086m2 leads to stability for all D.
Fundamentals of Power Electronics
60
Chapter 11: Current Programmed Control
DCM CPM buckboost example iL(t) i1(t) +
Switch network
vg(t)
ic
–
–
+
– ma
ipk
+
v2(t)
v1(t) + –
i2(t)
C R
m1
v(t) =
L
v1
–
iL(t)
vL(t)
m2 =
v2
Ts
L
Ts
0
L
t v1(t)
Ts
0
v2(t)
Fundamentals of Power Electronics
61
Ts
Chapter 11: Current Programmed Control
Analysis iL(t)
i pk = m 1d 1Ts v1(t) m1 = L
ic
Ts
– ma
ipk m1
i c = i pk + m ad 1Ts
=
= m 1 + m a d 1T s
d 1(t) =
vL(t)
v1
m2 =
Ts
L
Ts
0
L
t v1(t)
Ts
i c(t)
0
m 1 + m a Ts
Fundamentals of Power Electronics
v2
v2(t)
62
Ts
Chapter 11: Current Programmed Control
Averaged switch input port equation 1 i 1(t) T = s Ts
i 1(t)
t + Ts
i 1(τ)dτ = t
q1 Ts
i1(t) Area q1
1
Ts
ipk
= 2 i pk(t)d 1(t) i 1(t)
i 1(t) i 1(t)
i 1(t)
Ts
Ts
Ts
1 2
2 1
Ts
= m 1d (t)Ts 1 2
= v1(t)
v1(t)
Ts
Ts
=
i2(t)
Li 2c fs ma 1+ m1 1 2
Area q2
2
i 2(t)
Li 2c fs
ma 1+ m1
Fundamentals of Power Electronics
ipk
2
= p(t)
63
Ts
Ts
d1Ts
d2Ts
d3Ts
t
Ts Chapter 11: Current Programmed Control
Discussion: switch network input port
•
Averaged transistor waveforms obey a power sink characteristic
•
During first subinterval, energy is transferred from input voltage source, through transistor, to inductor, equal to 1
W = 2 Li 2pk This energy transfer process accounts for power flow equal to
p(t)
1
Ts
= W fs = 2 Li 2pk fs
which is equal to the power sink expression of the previous slide.
Fundamentals of Power Electronics
64
Chapter 11: Current Programmed Control
Averaged switch output port equation 1 i 2(t) T = s Ts
q2 =
1 2
t + Ts
i 2(τ)dτ = t
Area q1
v1(t) v2(t) p(t)
i 2(t)
i1(t)
i pkd 2Ts
d 2(t) = d 1(t) i 2(t)
q2 Ts
Ts
Ts
=
v2(t)
Ts
i 1(t)
Ts
i2(t) ipk
Ts
=
Fundamentals of Power Electronics
Ts
Ts
Ts
v2(t)
ipk
i 2(t) 1 2
Area q2
Ts
Li 2c (t) fs
m 1 + ma 1
2
= p(t)
Ts
d1Ts
d2Ts
d3Ts
t
Ts 65
Chapter 11: Current Programmed Control
Discussion: switch network output port •
Averaged diode waveforms obey a power sink characteristic
•
During second subinterval, all stored energy in inductor is transferred, through diode, to load
•
Hence, in averaged model, diode becomes a power source, having value equal to the power consumed by the transistor power sink element
Fundamentals of Power Electronics
66
Chapter 11: Current Programmed Control
Averaged equivalent circuit
i 1(t)
Ts
+ v1(t) vg(t)
Ts
+ –
p(t)
i 2(t) Ts
–
+ L
Fundamentals of Power Electronics
– v2(t)
Ts
67
Ts
+ Ts
C
R
v(t)
Ts
–
Chapter 11: Current Programmed Control
Steady state model: DCM CPM buckboost +
P Vg
+ –
R
V –
Solution
V2 = P R P=
1 2
V= PR = I c
LI 2c (t) fs
Ma 1+ M1
Fundamentals of Power Electronics
RL fs Ma 2 1+ M1
2
for a resistive load
2
68
Chapter 11: Current Programmed Control
Models of buck and boost L
Buck
+ vg(t)
Ts
+ –
p(t)
C
Ts
R
v(t)
Ts
–
Boost
L + vg(t)
Ts
+ –
p(t)
C
R
v(t)
Ts
Ts
–
Fundamentals of Power Electronics
69
Chapter 11: Current Programmed Control
Summary of steadystate DCM CPM characteristics Table 11.3. Steadystate DCM CPM characteristics of basic converters Icrit
Converter
M
Buck
Pload – P Pload
Boost
Pload Pload – P
Buckboost
Depends on load characteristic:
1 2
0≤M I crit
for CCM
I < I crit
for DCM 70
Chapter 11: Current Programmed Control
Buck converter: output characteristic with ma = 0
I=
P +P= Vg – V V
P
• with a resistive load, there can be two operating points
Vg V 1– V
• the operating point having V > 0.67Vg can be shown to CPM buck be unstable characteristic
CCM unstable for M > 12
I Ic
with ma = 0 resistive load line I = V/R CCM
1 I 2 c
DCM
B
4P Vg
A
DCM unstable for M > 23 1 2
Vg
Fundamentals of Power Electronics
2 3
Vg
Vg
V 71
Chapter 11: Current Programmed Control
Linearized smallsignal models: Buck
i1
i2
+ vg
+ –
v1
L
iL
+ r1
f1 i c
g 1v2
g 2v1
–
Fundamentals of Power Electronics
f2 i c
r2
v2
–
72
+ C
R
v
–
Chapter 11: Current Programmed Control
Linearized smallsignal models: Boost
iL
L
i1
i2
+ vg
+ –
v1
+ r1
f1 i c
g 1v2
g 2v1
–
Fundamentals of Power Electronics
f2 i c
r2
v2
–
73
+ C
R
v
–
Chapter 11: Current Programmed Control
Linearized smallsignal models: Buckboost
i1
i2
–
+ v1
vg
+ –
r1
f1 i c
g 1v2
g 2v1
f2 i c
r2
v2
+
–
+ C
R
v
– L iL
Fundamentals of Power Electronics
74
Chapter 11: Current Programmed Control
DCM CPM smallsignal parameters: input port Table 11.4. Current programmed DCM smallsignal equivalent circuit parameters: input port
Buck
Boost
f1
g1
Converter
m 1 – ma 1 m 1 + ma 1
2
1 M R 1–M
2
I1 Ic
r1
–R 1–M M2
2 I Ic
m 1 – ma R
M2
Fundamentals of Power Electronics
1
1
M – 1 R M –1
Buckboost
m 1 + ma
2
0
75
I1 Ic
m 2 ma 2–M + 1 m M –1 1+ a m1 m 1 + ma 1 –R M 2 1 – ma m1
Chapter 11: Current Programmed Control
DCM CPM smallsignal parameters: output port
Table 11.5. Current programmed DCM smallsignal equivalent circuit parameters: output port g2
Converter Buck
1 M R 1–M
Boost
Buckboost
f2
2 I Ic
ma m1 2 – M – M m 1 + ma 1
1 M R M –1 ma m1 2M R m 1 + ma 1
Fundamentals of Power Electronics
76
r2
m 1 + ma 1 m 1 – 2M + m a 1
1–M R
2
I2 Ic
R M –1 M
2
I2 Ic
R
Chapter 11: Current Programmed Control
Simplified DCM CPM model, with L = 0 Buck, boost, buckboost all become
+ vg
+ –
r1
g 2vg
g 1v
f1 i c
f2 i c
r2
C
R
v
–
Gvc(s) =
v ic
= vg = 0
Gc0 1 + ωs
v Gvg(s) = vg
p
Gc0 = f2 R  r 2 1 ωp = R  r 2 C Fundamentals of Power Electronics
ic
Gg0 = s 1 + ω =0
p
Gg0 = g 2 R  r 2
77
Chapter 11: Current Programmed Control
Buck ωp Plug in parameters:
m 2 – 3M 1 – M + m a M 2 – M 2 ωp = 1 RC ma 1–M 1–M +M m 2 •
For ma = 0, numerator is negative when M > 2/3.
•
ωp then constitutes a RHP pole. Converter is unstable.
•
Addition of small artificial ramp stabilizes system.
•
ma > 0.086 leads to stability for all M ≤ 1.
•
Output voltage feedback can also stabilize system, without an artificial ramp
Fundamentals of Power Electronics
78
Chapter 11: Current Programmed Control
11.5 Summary of key points 1.
In currentprogrammed control, the peak switch current is(t) follows the control input ic(t). This widely used control scheme has the advantage of a simpler controltooutput transfer function. The linetooutput transfer functions of currentprogrammed buck converters are also reduced.
2.
The basic currentprogrammed controller is unstable when D > 0.5, regardless of the converter topology. The controller can be stabilized by addition of an artificial ramp having slope ma. When ma ≥ 0.5 m2, then the controller is stable for all duty cycle.
3.
The behavior of currentprogrammed converters can be modeled in a simple and intuitive manner by the firstorder approximation 〈 iL(t) 〉Ts ≈ ic(t). The averaged terminal waveforms of the switch network can then be modeled simply by a current source of value ic , in conjunction with a power sink or power source element. Perturbation and linearization of these elements leads to the smallsignal model. Alternatively, the smallsignal converter equations derived in Chapter 7 can be adapted to cover the current programmed mode, using the simple approximation iL(t) ≈ ic(t).
Fundamentals of Power Electronics
79
Chapter 11: Current Programmed Control
Summary of key points 4.
The simple model predicts that one pole is eliminated from the converter linetooutput and controltooutput transfer functions. Current programming does not alter the transfer function zeroes. The dc gains become loaddependent.
5.
The more accurate model of Section 11.3 correctly accounts for the difference between the average inductor current 〈 iL(t) 〉Ts and the control input ic(t). This model predicts the nonzero linetooutput transfer function Gvg(s) of the buck converter. The currentprogrammed controller behavior is modeled by a block diagram, which is appended to the smallsignal converter models derived in Chapter 7. Analysis of the resulting multiloop feedback system then leads to the relevant transfer functions.
6.
The more accurate model predicts that the inductor pole occurs at the crossover frequency fc of the effective current feedback loop gain Ti(s). The frequency fc typically occurs in the vicinity of the converter switching frequency fs . The more accurate model also predicts that the linetooutput transfer function Gvg(s) of the buck converter is nulled when ma = 0.5 m2.
Fundamentals of Power Electronics
80
Chapter 11: Current Programmed Control
Summary of key points 7.
Current programmed converters operating in the discontinuous conduction mode are modeled in Section 11.4. The averaged transistor waveforms can be modeled by a power sink, while the averaged diode waveforms are modeled by a power source. The power is controlled by ic(t). Perturbation and linearization of these averaged models, as usual, leads to smallsignal equivalent circuits.
Fundamentals of Power Electronics
81
Chapter 11: Current Programmed Control
Part III. Magnetics
12. Basic Magnetics Theory 13. Filter Inductor Design 14. Transformer Design
Fundamentals of Power Electronics
1
Chapter 12: Basic Magnetics Theory
Chapter 12. Basic Magnetics Theory
12.1. Review of basic magnetics 12.1.1. Basic relations
12.1.2. Magnetic circuits
12.2. Transformer modeling 12.2.1. The ideal transformer
12.2.3. Leakage inductances
12.2.2. The magnetizing inductance
12.3. Loss mechanisms in magnetic devices 12.3.1. Core loss
12.3.2. Lowfrequency copper loss
12.4. Eddy currents in winding conductors 12.4.1. The skin effect
12.4.4. Power loss in a layer
12.4.2. The proximity effect
12.4.5. Example: power loss in a transformer winding
12.4.3. MMF diagrams
12.4.6. PWM waveform harmonics
Fundamentals of Power Electronics
2
Chapter 12: Basic Magnetics Theory
12.1. Review of basic magnetics 12.1.1. Basic relations
v(t)
Faraday's law
terminal characteristics
core characteristics
i(t) Ampere's law
Fundamentals of Power Electronics
B(t), Φ(t)
3
H(t), F(t)
Chapter 12: Basic Magnetics Theory
Basic quantities Magnetic quantities
Electrical quantities
length l
length l
magnetic field H
electric field E
x1
x2
+
x1
–
x2
+
MMF F = Hl
voltage V = El
surface S with area Ac total flux Φ flux density B
surface S with area Ac
{
Fundamentals of Power Electronics
–
total current I current density J
4
{
Chapter 12: Basic Magnetics Theory
Magnetic field H and magnetomotive force F Magnetomotive force (MMF) F between points x1 and x2 is related to the magnetic field H according to x2
F=
H ⋅ dl
x1
Example: uniform magnetic field of magnitude H
Analogous to electric field of strength E, which induces voltage (EMF) V:
length l
length l
magnetic field H x1
electric field E
x2
+
x1
–
+
MMF F = Hl
Fundamentals of Power Electronics
x2
– voltage V = El
5
Chapter 12: Basic Magnetics Theory
Flux density B and total flux Φ The total magnetic flux Φ passing through a surface of area Ac is related to the flux density B according to
Φ=
B ⋅ dA surface S
Example: uniform flux density of magnitude B
Analogous to electrical conductor current density of magnitude J, which leads to total conductor current I:
Φ = B Ac
surface S with area Ac
surface S with area Ac total flux Φ flux density B
{
Fundamentals of Power Electronics
total current I current density J
6
{
Chapter 12: Basic Magnetics Theory
Faraday’s law Voltage v(t) is induced in a loop of wire by change in the total flux Φ(t) passing through the interior of the loop, according to
v(t) =
dΦ(t) dt
area Ac
flux Φ(t)
For uniform flux distribution, Φ(t) = B(t)Ac and hence
– +
dB(t) v(t) = A c dt
Fundamentals of Power Electronics
{
7
v(t)
Chapter 12: Basic Magnetics Theory
Lenz’s law The voltage v(t) induced by the changing flux Φ(t) is of the polarity that tends to drive a current through the loop to counteract the flux change.
induced current i(t)
Example: a shorted loop of wire • Changing flux Φ(t) induces a voltage v(t) around the loop • This voltage, divided by the impedance of the loop conductor, leads to current i(t) • This current induces a flux Φ’(t), which tends to oppose changes in Φ(t)
Fundamentals of Power Electronics
flux Φ(t)
shorted loop
induced flux Φ'(t) 8
Chapter 12: Basic Magnetics Theory
Ampere’s law The net MMF around a closed path is equal to the total current passing through the interior of the path:
H ⋅ dl = total current passing through interior of path closed path
Example: magnetic core. Wire carrying current i(t) passes through core window. i(t)
H
• Illustrated path follows magnetic flux lines around interior of core
magnetic path length lm
• For uniform magnetic field strength H(t), the integral (MMF) is H(t)lm. So F (t) = H(t) l m = i(t) Fundamentals of Power Electronics
9
Chapter 12: Basic Magnetics Theory
Ampere’s law: discussion • Relates magnetic field strength H(t) to winding current i(t) • We can view winding currents as sources of MMF • Previous example: total MMF around core, F(t) = H(t)lm, is equal to the winding current MMF i(t) • The total MMF around a closed loop, accounting for winding current MMF’s, is zero
Fundamentals of Power Electronics
10
Chapter 12: Basic Magnetics Theory
Core material characteristics: the relation between B and H Free space B
A magnetic core material B
B = µ0 H
µ µ0
H
H
µ0 = permeability of free space
Highly nonlinear, with hysteresis and saturation
= 4π · 107 Henries per meter Fundamentals of Power Electronics
11
Chapter 12: Basic Magnetics Theory
Piecewiselinear modeling of core material characteristics No hysteresis or saturation B
Saturation, no hysteresis B Bsat
B=µH µ = µr µ0
µ
µ = µr µ0
H
H
– Bsat
Typical Bsat = 0.30.5T, ferrite 0.51T, powdered iron 12T, iron laminations
Typical µr = 103  105
Fundamentals of Power Electronics
12
Chapter 12: Basic Magnetics Theory
Units
Table 12.1. Units for magnetic quantities quantity
MKS
unrationalized cgs
conversions
core material equation
B = µ 0 µr H
B = µr H
B
Tesla
Gauss
1T = 104G
H
Ampere / meter
Oersted
1A/m = 4π⋅10 Oe
Φ
Weber
Maxwell
1Wb = 108 Mx 2 1T = 1Wb / m
Fundamentals of Power Electronics
13
3
Chapter 12: Basic Magnetics Theory
Example: a simple inductor Faraday’s law:
Φ
For each turn of wire, we can write
dΦ(t) vturn(t) = dt
core area Ac
i(t) + v(t) –
n turns
core permeability µ
Total winding voltage is dΦ(t) v(t) = n vturn(t) = n dt
core
Express in terms of the average flux density B(t) = Φ(t)/A c dB(t) v(t) = n A c dt
Fundamentals of Power Electronics
14
Chapter 12: Basic Magnetics Theory
Inductor example: Ampere’s law Choose a closed path which follows the average magnetic field line around the interior of the core. Length of this path is called the mean magnetic path length lm.
H i(t) n turns
magnetic path length lm
For uniform field strength H(t), the core MMF around the path is H lm. Winding contains n turns of wire, each carrying current i(t). The net current passing through the path interior (i.e., through the core window) is ni(t). From Ampere’s law, we have H(t) lm = n i(t) Fundamentals of Power Electronics
15
Chapter 12: Basic Magnetics Theory
Inductor example: core material model B Bsat
B=
Bsat
for H ≥ Bsat / µ
µH
for H < Bsat / µ
µ
– Bsat for H ≤ Bsat / µ
H
Find winding current at onset of saturation: substitute i = Isat and H = Bsat/µ into equation previously derived via Ampere’s law. Result is B l I sat = µsatn m Fundamentals of Power Electronics
16
– Bsat
Chapter 12: Basic Magnetics Theory
Electrical terminal characteristics We have:
v(t) = n A c
dB(t) dt
H(t) lm = n i(t)
B=
Bsat
for H ≥ Bsat / µ
µH
for H < Bsat / µ
– Bsat for H ≤ Bsat / µ
Eliminate B and H, and solve for relation between v and i. For  i  < Isat, d H(t) µ n 2 A c di(t) v(t) = µ n A c v(t) = dt lm dt which is of the form di(t) µ n2 Ac v(t) = L L= with dt lm —an inductor For  i  > Isat the flux density is constant and equal to Bsat. Faraday’s law then predicts dBsat —saturation leads to short circuit v(t) = n A c =0 dt Fundamentals of Power Electronics
17
Chapter 12: Basic Magnetics Theory
12.1.2. Magnetic circuits Uniform flux and magnetic field inside a rectangular element:
length l MMF F
+ flux Φ
MMF between ends of element is F =Hl
–
{ H
Since H = B / µ and Β = Φ / Ac, we can express F as l F = µ lA Φ with = R c µ Ac A corresponding model:
+
F Φ
core permeability µ R = µAl c
–
R
Fundamentals of Power Electronics
area Ac
= reluctance of element
R 18
Chapter 12: Basic Magnetics Theory
Magnetic circuits: magnetic structures composed of multiple windings and heterogeneous elements • Represent each element with reluctance • Windings are sources of MMF • MMF → voltage, flux → current • Solve magnetic circuit using Kirchoff’s laws, etc.
Fundamentals of Power Electronics
19
Chapter 12: Basic Magnetics Theory
Magnetic analog of Kirchoff’s current law node
Physical structure Divergence of B = 0
Φ1
Flux lines are continuous and cannot end
Φ3 Φ2
Total flux entering a node must be zero
Magnetic circuit
node
Φ1 = Φ2 + Φ3 Φ3
Φ1 Φ2 Fundamentals of Power Electronics
20
Chapter 12: Basic Magnetics Theory
Magnetic analog of Kirchoff’s voltage law Follows from Ampere’s law:
H ⋅ dl = total current passing through interior of path closed path
Lefthand side: sum of MMF’s across the reluctances around the closed path Righthand side: currents in windings are sources of MMF’s. An nturn winding carrying current i(t) is modeled as an MMF (voltage) source, of value ni(t). Total MMF’s around the closed path add up to zero.
Fundamentals of Power Electronics
21
Chapter 12: Basic Magnetics Theory
Example: inductor with air gap core permeability µc Φ crosssectional area Ac
i(t) + v(t) –
n turns
air gap lg magnetic path length lm
Ampere’s law:
Fc + Fg = n i
Fundamentals of Power Electronics
22
Chapter 12: Basic Magnetics Theory
Magnetic circuit model
+
core permeability µc Φ
–
Rc crosssectional area Ac
i(t) + v(t) –
Fc
n turns
air gap lg
+ n i(t)
+ –
Φ(t)
R g Fg –
magnetic path length lm
Fc + Fg = n i ni=Φ Rc+Rg
Fundamentals of Power Electronics
23
l
R c = µ Ac
c
lg Rg= µ A 0 c
Chapter 12: Basic Magnetics Theory
Solution of model core permeability µc Φ
n turns
Fc
–
Rc
crosssectional area Ac
i(t) + v(t) –
+
+
air gap lg
n i(t)
+ –
Φ(t)
magnetic path length lm
dΦ(t) dt di(t) n2 v(t) = Substitute for Φ: R c + R g dt
Faraday’s law:
v(t) = n
R g Fg –
ni=Φ Rc+Rg
l
R c = µ Ac
c
l
R g = µ gA 0 c
Hence inductance is 2 n L= Rc+Rg
Fundamentals of Power Electronics
24
Chapter 12: Basic Magnetics Theory
Effect of air gap ni=Φ Rc+Rg
L=
Φ = BAc
2
n Rc+Rg
BsatAc
Rc
Φsat = BsatAc I sat =
1
1 Rc+Rg
Bsat A c Rc+Rg n
Effect of air gap: • decrease inductance • increase saturation current • inductance is less dependent on core permeability Fundamentals of Power Electronics
nIsat1
nIsat2
ni ∝ Hc
– BsatAc 25
Chapter 12: Basic Magnetics Theory
12.2. Transformer modeling Two windings, no air gap:
l R = µ mA c
F c = n 1 i1 + n 2 i2
Φ i1(t) + v1(t) –
i2(t) + n2 turns v2(t) –
n1 turns
core
Φ R = n 1 i1 + n 2 i2
Rc
Φ
Magnetic circuit model:
+ n1 i1
Fundamentals of Power Electronics
Fc
– – +
+ –
26
n2 i2
Chapter 12: Basic Magnetics Theory
12.2.1. The ideal transformer Φ
In the ideal transformer, the core reluctance R approaches zero.
+
MMF Fc = Φ R also approaches zero. We then obtain 0 = n 1 i1 + n 2 i2
Fundamentals of Power Electronics
27
Fc
i1
– – +
+ –
n1 i1
Also, by Faraday’s law, v1 = n 1 dΦ dt v2 = n 2 dΦ dt Eliminate Φ : dΦ = v1 = v2 dt n 1 n 2 Ideal transformer equations: v1 v2 = and n 1 i1 + n 2 i2 = 0 n1 n2
Rc
n1 : n2
n2 i2
i2
+
+
v1
v2
–
– Ideal Chapter 12: Basic Magnetics Theory
12.2.2. The magnetizing inductance
Φ R = n 1 i1 + n 2 i2
n1 i1
This equation is of the form
v1
R
n2 i2 n1
Fundamentals of Power Electronics
L mp =
n 21
Fc
– – +
+ –
n2 n 1 i2
i1 +
d imp v1 = L mp dt 2 n with L mp = 1
imp = i1 +
+
with v1 = n 1 dΦ dt
Eliminate Φ: n 21 d n v1 = i1 + 2 i2 R dt n1
Rc
Φ
For nonzero core reluctance, we obtain
n1 : n2
n2 i2
i2 +
n i1 + n 2 i2 1
v2
R
–
– Ideal 28
Chapter 12: Basic Magnetics Theory
Magnetizing inductance: discussion
• Models magnetization of core material • A real, physical inductor, that exhibits saturation and hysteresis • If the secondary winding is disconnected: we are left with the primary winding on the core primary winding then behaves as an inductor the resulting inductor is the magnetizing inductance, referred to the primary winding • Magnetizing current causes the ratio of winding currents to differ from the turns ratio
Fundamentals of Power Electronics
29
Chapter 12: Basic Magnetics Theory
Transformer saturation • Saturation occurs when core flux density B(t) exceeds saturation flux density Bsat. • When core saturates, the magnetizing current becomes large, the impedance of the magnetizing inductance becomes small, and the windings are effectively shorted out. • Large winding currents i1(t) and i2(t) do not necessarily lead to saturation. If 0 = n 1 i1 + n 2 i2 then the magnetizing current is zero, and there is no net magnetization of the core. • Saturation is caused by excessive applied voltseconds
Fundamentals of Power Electronics
30
Chapter 12: Basic Magnetics Theory
Saturation vs. applied voltseconds Magnetizing current depends on the integral of the applied winding voltage:
imp(t) = 1 L mp
n2 n 1 i2
i1 + v1
v1(t) dt
L mp =
n 21
n1 : n2
i2 +
n i1 + n 2 i2 1
v2
R
–
–
Flux density is proportional:
B(t) =
1 n1 A c
Ideal
v1(t) dt
Flux density befcomes large, and core saturates, when the applied voltseconds λ1 are too large, where λ1 =
t2
v1(t) dt t1
limits of integration chosen to coincide with positive portion of applied voltage waveform Fundamentals of Power Electronics
31
Chapter 12: Basic Magnetics Theory
12.2.3. Leakage inductances ΦM i1(t) + v1(t) –
i2(t)
Φl1
Φl2
Φl1 ΦM
Φl2 i2(t)
i1(t) + v1(t) –
Fundamentals of Power Electronics
+ v2(t) –
+ v2(t) –
32
Chapter 12: Basic Magnetics Theory
Transformer model, including leakage inductance
Ll1
i1
Ll2
n1 : n2
i2 +
+
L L v1(t) i (t) = 11 12 d 1 L 12 L 22 dt i2(t) v2(t)
v1
n L mp = n 1 L 12 2
v2
–
–
mutual inductance L 12 =
n1 n2
R
Ideal
n = n 2 L mp 1
primary and secondary selfinductances n L 11 = L l1 + n 1 L 12 2 n2 L 22 = L l2 + n L 12 1 Fundamentals of Power Electronics
effective turns ratio
ne =
L 22 L 11
coupling coefficient
k=
L 12 L 11 L 22
33
Chapter 12: Basic Magnetics Theory
12.3. Loss mechanisms in magnetic devices
Lowfrequency losses: Dc copper loss Core loss: hysteresis loss Highfrequency losses: the skin effect Core loss: classical eddy current losses Eddy current losses in ferrite cores High frequency copper loss: the proximity effect Proximity effect: high frequency limit MMF diagrams, losses in a layer, and losses in basic multilayer windings Effect of PWM waveform harmonics Fundamentals of Power Electronics
34
Chapter 12: Basic Magnetics Theory
12.3.1. Core loss Φ
Energy per cycle W flowing into nturn winding of an inductor, excited by periodic waveforms of frequency f:
W=
i(t) + v(t) –
n turns
v(t)i(t)dt
core area Ac
core permeability µ core
one cycle
Relate winding voltage and current to core B and H via Faraday’s law and Ampere’s law: dB(t) v(t) = n A c H(t) lm = n i(t) dt Substitute into integral:
W=
nA c
dB(t) dt
H(t)l m dt n
one cycle
= A cl m
H dB one cycle
Fundamentals of Power Electronics
35
Chapter 12: Basic Magnetics Theory
Core loss: Hysteresis loss B
W = A cl m
Area
H dB
H dB
one cycle
one cycle
The term Aclm is the volume of the core, while the integral is the area of the BH loop.
H
(energy lost per cycle) = (core volume) (area of BH loop) PH = f A cl m
H dB one cycle
Fundamentals of Power Electronics
Hysteresis loss is directly proportional to applied frequency 36
Chapter 12: Basic Magnetics Theory
Modeling hysteresis loss • Hysteresis loss varies directly with applied frequency • Dependence on maximum flux density: how does area of BH loop depend on maximum flux density (and on applied waveforms)? Emperical equation (Steinmetz equation): PH = K H f B αmax(core volume)
The parameters KH and α are determined experimentally. Dependence of PH on Bmax is predicted by the theory of magnetic domains.
Fundamentals of Power Electronics
37
Chapter 12: Basic Magnetics Theory
Core loss: eddy current loss Magnetic core materials are reasonably good conductors of electric current. Hence, according to Lenz’s law, magnetic fields within the core induce currents (“eddy currents”) to flow within the core. The eddy currents flow such that they tend to generate a flux which opposes changes in the core flux Φ(t). The eddy currents tend to prevent flux from penetrating the core.
flux Φ(t)
eddy current i(t)
Eddy current loss i2(t)R
core Fundamentals of Power Electronics
38
Chapter 12: Basic Magnetics Theory
Modeling eddy current loss • Ac flux Φ(t) induces voltage v(t) in core, according to Faraday’s law. Induced voltage is proportional to derivative of Φ(t). In consequence, magnitude of induced voltage is directly proportional to excitation frequency f. • If core material impedance Z is purely resistive and independent of frequency, Z = R, then eddy current magnitude is proportional to voltage: i(t) = v(t)/R. Hence magnitude of i(t) is directly proportional to excitation frequency f. • Eddy current power loss i2(t)R then varies with square of excitation frequency f. • Classical Steinmetz equation for eddy current loss: PE = K E f 2B 2max(core volume) • Ferrite core material impedance is capacitive. This causes eddy current power loss to increase as f4. Fundamentals of Power Electronics
39
Chapter 12: Basic Magnetics Theory
Total core loss: manufacturer’s data
200k
Hz
kHz
Empirical equation, at a fixed frequency: Pfe = K fe B βmax A c l m
Hz 100k
Hz
50kH
z
0.1
20k
Power loss density, Watts / cm
3
500
Hz
1 1M
Ferrite core material
0.01 0.01
0.1
0.3
Bmax, Tesla Fundamentals of Power Electronics
40
Chapter 12: Basic Magnetics Theory
Core materials
Core type
B sat
Relative core loss
Applications
Laminations iron, silicon steel
1.5  2.0 T
high
5060 Hz transformers, inductors
Powdered cores powdered iron, molypermalloy
0.6  0.8 T
medium
1 kHz transformers, 100 kHz filter inductors
Ferrite Manganesezinc, Nickelzinc
0.25  0.5 T
low
20 kHz  1 MHz transformers, ac inductors
Fundamentals of Power Electronics
41
Chapter 12: Basic Magnetics Theory
12.3.2. Lowfrequency copper loss DC resistance of wire
R=ρ
lb Aw
where Aw is the wire bare crosssectional area, and lb is the length of the wire. The resistivity ρ is equal to 1.724⋅106 Ω cm for softannealed copper at room temperature. This resistivity increases to 2.3⋅106 Ω cm at 100˚C. The wire resistance leads to a power loss of
i(t) R
Pcu = I 2rms R
Fundamentals of Power Electronics
42
Chapter 12: Basic Magnetics Theory
12.4. Eddy currents in winding conductors 12.4.1. The skin effect wire current density
Φ(t) eddy currents
wire
i(t) eddy currents i(t)
Fundamentals of Power Electronics
43
Chapter 12: Basic Magnetics Theory
Penetration depth δ For sinusoidal currents: current density is an exponentially decaying function of distance into the conductor, with characteristic length δ known as the penetration depth or skin depth.
δ=
ρ πµ f
Wire diameter 0.1
#20AWG
penetration depth δ, cm
#30AWG
100
˚C
25˚C 0.01
#40AWG
For copper at room temperature:
δ = 7.5 cm f
0.001 10kHz
100kHz
1MHz
frequency
Fundamentals of Power Electronics
44
Chapter 12: Basic Magnetics Theory
12.4.2. The proximity effect Ac current in a conductor induces eddy currents in adjacent conductors by a process called the proximity effect. This causes significant power loss in the windings of highfrequency transformers and ac inductors.
3i
layer 3
–2i 2Φ
layer 2
A multilayer foil winding, with d >> δ. Each layer carries net current i(t).
2i –i Φ i d
current density J
layer 1
Fundamentals of Power Electronics
45
Chapter 12: Basic Magnetics Theory
Estimating proximity loss: highfrequency limit Let P1 be power loss in layer 1:
P1 = I 2rms Rdc d δ Power loss P2 in layer 2 is: 2 P2 = I 2rms Rdc d + 2I rms Rdc d δ δ = 5P1 Power loss P3 in layer 3 is: 2 P3 = 2I rms Rdc d + 3I rms δ = 13P1
2
–2i 2Φ
layer 2
2i –i Φ
Rdc d δ
layer 1
i d
current density J
Power loss Pm in layer m is: Pm = ((m – 1)2 + m 2) P1
Fundamentals of Power Electronics
3i
layer 3
46
Chapter 12: Basic Magnetics Theory
Total loss in Mlayer winding: highfrequency limit Add up losses in each layer: Pj = M Σ j=1 3 M
Pw
= d >> δ
2M 2 + 1 P1
Compare with dc copper loss: If foil thickness were d = δ, then at dc each layer would produce copper loss P1. The copper loss of the Mlayer winding would be
Pw,dc
d=δ
= M P1
For foil thicknesses other than d = δ, the dc resistance and power loss are changed by a factor of d/δ. The total winding dc copper loss is Pw,dc = M P1 δ d So the proximity effect increases the copper loss by a factor of
Pw d >> δ 1 d FR d >> δ = = 2M 2 + 1 3 δ Pw,dc Fundamentals of Power Electronics
47
Chapter 12: Basic Magnetics Theory
Approximating a layer of round conductors as an effective foil conductor
(a)
(b)
(c)
Conductor spacing factor: n η= π d l 4 lw
(d)
d
lw
Effective ratio of conductor thickness to skin depth:
ϕ= η d δ
Fundamentals of Power Electronics
48
Chapter 12: Basic Magnetics Theory
12.4.3. Magnetic fields in the vicinity of winding conductors: MMF diagrams Twowinding transformer example
core
primary layers
secondary layers
i
–i 2i
Fundamentals of Power Electronics
{
{ i
–2i 3i
49
2i
–i
–2i
–3i
Chapter 12: Basic Magnetics Theory
Transformer example: magnetic field lines m p – m s i = F (x)
H(x) =
F (x) lw
–
F(x) i
i
i
–i
–i
–i
lw
+ x Fundamentals of Power Electronics
50
Chapter 12: Basic Magnetics Theory
Ampere’s law and MMF diagram
i
i
i
–i
–i
–i
MMF F(x) 3i 2i i 0
x
m p – m s i = F (x)
Fundamentals of Power Electronics
H(x) =
51
F (x) lw
Chapter 12: Basic Magnetics Theory
MMF diagram for d >> δ
MMF F(x)
i
–i 2i
–2i 3i
–3i 2i
–2i i
–i
3i 2i i 0
Fundamentals of Power Electronics
x
52
Chapter 12: Basic Magnetics Theory
Interleaved windings
pri
sec
pri
sec
pri
sec
i
–i
i
–i
i
–i
MMF F(x) 3i 2i i 0
Fundamentals of Power Electronics
x
53
Chapter 12: Basic Magnetics Theory
Partiallyinterleaved windings: fractional layers secondary
MMF F(x)
–3i 4
–3i 4
primary
i
i
secondary
i
–3i 4
–3i 4
1.5 i i 0.5 i 0
x
–0.5 i –i –1.5 i Fundamentals of Power Electronics
54
Chapter 12: Basic Magnetics Theory
12.4.4. Power loss in a layer
Assume uniform magnetic fields at surfaces of layer, of strengths H(0) and H(d). Assume that these fields are parallel to layer surface (i.e., neglect fringing and assume field normal component is zero).
layer
Approximate computation of copper loss in one layer
H(0)
H(d)
The magnetic fields H(0) and H(d) are driven by the MMFs F(0) and F(d). Sinusoidal waveforms are assumed, and rms values are used. It is assumed that H(0) and H(d) are in phase.
F(x)
F(d) F(0) 0
Fundamentals of Power Electronics
55
d
Chapter 12: Basic Magnetics Theory
Solution for layer copper loss P Solve Maxwell’s equations to find current density distribution within layer. Then integrate to find total copper loss P in layer. Result is
P = Rdc
ϕ n 2l
F 2(d) + F 2(0) G1(ϕ) – 4 F (d)F (0) G2(ϕ)
where nl = number of turns in layer, Rdc = dc resistance of layer,
(MLT) n 2l Rdc = ρ lw η d
(MLT) = meanlengthperturn, or circumference, of layer.
G1(ϕ) =
sinh (2ϕ) + sin (2ϕ) cosh (2ϕ) – cos (2ϕ)
G2(ϕ) =
sinh (ϕ) cos (ϕ) + cosh(ϕ) sin (ϕ) cosh (2ϕ) – cos (2ϕ)
ϕ= η d δ Fundamentals of Power Electronics
η=
π d nl 4 lw 56
Chapter 12: Basic Magnetics Theory
Winding carrying current I, with nl turns per layer If winding carries current of rms magnitude I, then
F (d) – F (0) = n l I
F (d) = m n l I
layer
Express F(d) in terms of the winding current I, as H(0)
H(d)
The quantity m is the ratio of the MMF F(d) to the layer ampereturns nlI. Then,
F (0) m – 1 = m F (d) Power dissipated in the layer can now be written P = I 2 Rdc ϕ Q'(ϕ, m)
Q'(ϕ, m) = 2m 2 – 2m + 1 G1(ϕ) – 4m m – 1 G2(ϕ) Fundamentals of Power Electronics
57
F(x)
F(d) F(0) 0
d
Chapter 12: Basic Magnetics Theory
Increased copper loss in layer P = ϕ Q'(ϕ, m) 2 I Rdc
m = 15 12 10 8 6
100
5
4
3
2
P 2 I Rdc
1.5
1
10
m = 0.5
1 0.1
1
10
ϕ Fundamentals of Power Electronics
58
Chapter 12: Basic Magnetics Theory
Layer copper loss vs. layer thickness P Pdc
d=δ
= Q'(ϕ, m)
m = 15 12 10
100
8 6
P Pdc
5
ϕ=1
4 3
10
2
Relative to copper loss when d = δ
1.5 1
1
m = 0.5
0.1 0.1
1
10
ϕ Fundamentals of Power Electronics
59
Chapter 12: Basic Magnetics Theory
12.4.5. Example: Power loss in a transformer winding Two winding transformer
primary layers
secondary layers
{ {
Each winding consists of M layers Proximity effect increases copper m= F n pi loss in layer m by the factor P = ϕ Q'(ϕ, m) 2 2 I Rdc
n pi
npi
npi
npi
npi
npi
M
1
Sum losses over all 0 primary layers: M Ppri FR = = 1 Σ ϕ Q'(ϕ, m) Ppri,dc M m = 1 Fundamentals of Power Electronics
x
60
Chapter 12: Basic Magnetics Theory
Increased total winding loss Express summation in closed form:
FR = ϕ G1(ϕ) + 2 M 2 – 1 G1(ϕ) – 2G2(ϕ) 3 number of layers M = 15 12 10 8 7
100
6
5
4 3
FR =
Ppri Ppri,dc
2 1.5 1
10
0.5
1 0.1
1
10
ϕ Fundamentals of Power Electronics
61
Chapter 12: Basic Magnetics Theory
Total winding loss number of layers M = 15
100
12 10 8 7 6 5 4
Ppri Ppri,dc
ϕ=1
10
3 2 1.5 1
1
0.5
0.1 0.1
1
10
ϕ Fundamentals of Power Electronics
62
Chapter 12: Basic Magnetics Theory
12.4.6. PWM waveform harmonics i(t)
Fourier series:
Ipk
∞
i(t) = I 0 +
Σ
j=1
2 I j cos ( jωt)
with
Ij =
2 I pk sin ( jπD) jπ
I 0 = DI pk
Copper loss: Dc Ac
0
2 0
Pdc = I Rdc
Pj = I 2j Rdc
DTs
Ts
t
j ϕ 1 G1( j ϕ 1) + 2 M 2 – 1 G1( j ϕ 1) – 2G2( j ϕ 1) 3
Total, relative to value predicted by lowfrequency analysis:
2ϕ 1 Pcu = D + Dπ 2 D I 2pk Rdc
∞
Σ
j=1
Fundamentals of Power Electronics
sin 2 ( jπD) G1( j ϕ 1) + 2 M 2 – 1 G1( j ϕ 1) – 2G2( j ϕ 1) 3 j j 63
Chapter 12: Basic Magnetics Theory
Harmonic loss factor FH Effect of harmonics: FH = ratio of total ac copper loss to fundamental copper loss ∞
FH =
Σ Pj
j=1
P1
The total winding copper loss can then be written Pcu = I 20 Rdc + FH FR I 21 Rdc
Fundamentals of Power Electronics
64
Chapter 12: Basic Magnetics Theory
Increased proximity losses induced by PWM waveform harmonics: D = 0.5 10 D = 0.5
M = 10
FH
8 6 5 4 3 2 1.5 1 M = 0.5
1 0.1
1
10
ϕ1 Fundamentals of Power Electronics
65
Chapter 12: Basic Magnetics Theory
Increased proximity losses induced by PWM waveform harmonics: D = 0.3 100 D = 0.3
FH
M = 10 8
10
6 5 4 3 2
1.5 1 M = 0.5
1 0.1
1
10
ϕ1 Fundamentals of Power Electronics
66
Chapter 12: Basic Magnetics Theory
Increased proximity losses induced by PWM waveform harmonics: D = 0.1 100 D = 0.1
M = 10 8 6
FH
5
4 3 2 1.5 1
10
M = 0.5
1 0.1
1
10
ϕ1 Fundamentals of Power Electronics
67
Chapter 12: Basic Magnetics Theory
Summary of Key Points 1.
Magnetic devices can be modeled using lumpedelement magnetic circuits, in a manner similar to that commonly used to model electrical circuits. The magnetic analogs of electrical voltage V, current I, and resistance R, are magnetomotive force (MMF) F, flux Φ, and reluctance R respectively.
2.
Faraday’s law relates the voltage induced in a loop of wire to the derivative of flux passing through the interior of the loop.
3.
Ampere’s law relates the total MMF around a loop to the total current passing through the center of the loop. Ampere’s law implies that winding currents are sources of MMF, and that when these sources are included, then the net MMF around a closed path is equal to zero.
4.
Magnetic core materials exhibit hysteresis and saturation. A core material saturates when the flux density B reaches the saturation flux density Bsat.
Fundamentals of Power Electronics
68
Chapter 12: Basic Magnetics Theory
Summary of key points 5.
Air gaps are employed in inductors to prevent saturation when a given maximum current flows in the winding, and to stabilize the value of inductance. The inductor with air gap can be analyzed using a simple magnetic equivalent circuit, containing core and air gap reluctances and a source representing the winding MMF.
6.
Conventional transformers can be modeled using sources representing the MMFs of each winding, and the core MMF. The core reluctance approaches zero in an ideal transformer. Nonzero core reluctance leads to an electrical transformer model containing a magnetizing inductance, effectively in parallel with the ideal transformer. Flux that does not link both windings, or “leakage flux,” can be modeled using series inductors.
7.
The conventional transformer saturates when the applied winding voltseconds are too large. Addition of an air gap has no effect on saturation. Saturation can be prevented by increasing the core crosssectional area, or by increasing the number of primary turns.
Fundamentals of Power Electronics
69
Chapter 12: Basic Magnetics Theory
Summary of key points 8.
Magnetic materials exhibit core loss, due to hysteresis of the BH loop and to induced eddy currents flowing in the core material. In available core materials, there is a tradeoff between high saturation flux density Bsat and high core loss Pfe. Laminated iron alloy cores exhibit the highest Bsat but also the highest Pfe, while ferrite cores exhibit the lowest Pfe but also the lowest Bsat. Between these two extremes are powdered iron alloy and amorphous alloy materials.
9.
The skin and proximity effects lead to eddy currents in winding conductors, which increase the copper loss Pcu in highcurrent highfrequency magnetic devices. When a conductor has thickness approaching or larger than the penetration depth δ, magnetic fields in the vicinity of the conductor induce eddy currents in the conductor. According to Lenz’s law, these eddy currents flow in paths that tend to oppose the applied magnetic fields.
Fundamentals of Power Electronics
70
Chapter 12: Basic Magnetics Theory
Summary of key points 10. The magnetic field strengths in the vicinity of the winding conductors can be determined by use of MMF diagrams. These diagrams are constructed by application of Ampere’s law, following the closed paths of the magnetic field lines which pass near the winding conductors. Multiplelayer noninterleaved windings can exhibit high maximum MMFs, with resulting high eddy currents and high copper loss. 11. An expression for the copper loss in a layer, as a function of the magnetic field strengths or MMFs surrounding the layer, is given in Section 12.4.4. This expression can be used in conjunction with the MMF diagram, to compute the copper loss in each layer of a winding. The results can then be summed, yielding the total winding copper loss. When the effective layer thickness is near to or greater than one skin depth, the copper losses of multiplelayer noninterleaved windings are greatly increased.
Fundamentals of Power Electronics
71
Chapter 12: Basic Magnetics Theory
Summary of key points
12. Pulsewidthmodulated winding currents of contain significant total harmonic distortion, which can lead to a further increase of copper loss. The increase in proximity loss caused by current harmonics is most pronounced in multiplelayer noninterleaved windings, with an effective layer thickness near one skin depth.
Fundamentals of Power Electronics
72
Chapter 12: Basic Magnetics Theory
Chapter 13. Filter Inductor Design
13.1. Several types of magnetic devices, their BH loops, and core vs. copper loss 13.2. Filter inductor design constraints 13.3. The core geometrical constant Kg 13.4. A stepbystep design procedure 13.5. Summary of key points
Fundamentals of Power Electronics
1
Chapter 13: Filter inductor design
13.1. Several types of magnetic devices, their BH loops, and core vs. copper loss A key design decision: the choice of maximum operating flux density Bmax • Choose Bmax to avoid saturation of core, or • Further reduce Bmax , to reduce core losses Different design procedures are employed in the two cases. Types of magnetic devices: Filter inductor
Ac inductor
Conventional transformer
Coupled inductor
Flyback transformer
SEPIC transformer
Magnetic amplifier
Saturable reactor
Fundamentals of Power Electronics
2
Chapter 13: Filter inductor design
Filter inductor CCM buck example B
L
Bsat
i(t)
minor BH loop, filter inductor
+ –
∆Hc Hc0
i(t)
Hc
∆iL
I
BH loop, large excitation 0
DTs
Fundamentals of Power Electronics
Ts
t
3
Chapter 13: Filter inductor design
Filter inductor, cont. core reluctance Rc Φ
• Negligible core loss, negligible proximity loss • Loss dominated by dc copper loss
i(t) + v(t) –
n turns
air gap reluctance
Rg
• Flux density chosen simply to avoid saturation • Air gap is employed • Could use core materials having high saturation flux density (and relatively high core loss), even though converter switching frequency is high Fundamentals of Power Electronics
Rc
n i(t)
4
+ –
Φ(t)
Rg
Chapter 13: Filter inductor design
Ac inductor L
B Bsat
i(t) BH loop, for operation as ac inductor
i(t)
–∆Hc
∆i
∆Hc
Hc
t
∆i
core BH loop
Fundamentals of Power Electronics
5
Chapter 13: Filter inductor design
Ac inductor, cont. • Core loss, copper loss, proximity loss are all significant • An air gap is employed • Flux density is chosen to reduce core loss • A highfrequency material (ferrite) must be employed
Fundamentals of Power Electronics
6
Chapter 13: Filter inductor design
Conventional transformer i1(t)
n1 : n2
i2(t)
B
+
imp(t)
+
v1(t)
Lmp
v2(t)
– v1(t)
λ1 2n 1A c
–
BH loop, for operation as conventional transformer
area λ1
n 1∆i mp lm
Hc
core BH loop
imp(t) ∆imp
t
H(t) = Fundamentals of Power Electronics
7
n imp(t) lm
Chapter 13: Filter inductor design
Conventional transformer, cont. • Core loss, copper loss, and proximity loss are usually significant • No air gap is employed • Flux density is chosen to reduce core loss • A high frequency material (ferrite) must be employed
Fundamentals of Power Electronics
8
Chapter 13: Filter inductor design
Coupled inductor i1(t)
Twooutput forward converter example n1
+
i1
v1 vg
∆i1
I1
+ –
–
n2 turns
+
i2
i2(t) ∆i2
I2
t
B minor BH loop, coupled inductor ∆Hc
v2 –
Hc0
Hc
BH loop, large excitation
Fundamentals of Power Electronics
9
Chapter 13: Filter inductor design
Coupled inductor, cont.
• A filter inductor having multiple windings • Air gap is employed • Core loss and proximity loss usually not significant • Flux density chosen to avoid saturation • Lowfrequency core material can be employed
Fundamentals of Power Electronics
10
Chapter 13: Filter inductor design
DCM Flyback transformer i1(t)
n1 : n2
i1
i1,pk
+
i2
imp Lmp
vg
v i2(t)
+ –
–
t
imp(t)
B
i1,pk
BH loop, for operation in DCM flyback converter
t
Hc n 1i 1,pk R c l m R c+R g
core BH loop
Fundamentals of Power Electronics
11
Chapter 13: Filter inductor design
DCM flyback transformer, cont. • Core loss, copper loss, proximity loss are significant • Flux density is chosen to reduce core loss • Air gap is employed • A highfrequency core material (ferrite) must be used
Fundamentals of Power Electronics
12
Chapter 13: Filter inductor design
13.2. Filter inductor design constraints
Objective: Design inductor having a given inductance L,
L
which carries worstcase current Imax without saturating, and which has a given winding resistance R, or, equivalently, exhibits a worstcase copper loss of
i(t) R
Pcu = Irms2 R
Fundamentals of Power Electronics
13
Chapter 13: Filter inductor design
Assumed filter inductor geometry Φ crosssectional area Ac
i(t) + v(t) –
n turns
air gap lg
l
R c = µ cA c c l
R g = µ gA 0 c
Rc
Solve magnetic circuit: ni = Φ R c + R g
n i(t)
+ –
Φ(t)
Rg For R
Fundamentals of Power Electronics
14
c
>>
R g:
ni ≈ Φ R
g
Chapter 13: Filter inductor design
13.2.1. Constraint: maximum flux density Given a peak winding current Imax, it is desired to operate the core flux density at a peak value Bmax. The value of Bmax is chosen to be less than the worstcase saturation flux density of the core material. From solution of magnetic circuit:
ni = BA cR
g
Let I = Imax and B = Bmax :
lg nI max = BmaxA cR g = Bmax µ 0 This is constraint #1. The turns ratio n and air gap length lg are unknown.
Fundamentals of Power Electronics
15
Chapter 13: Filter inductor design
13.3.2. Constraint: Inductance Must obtain specified inductance L. We know that the inductance is 2 µ0 Ac n2 n L= = Rg lg
This is constraint #2. The turns ratio n, core area Ac, and air gap length lg are unknown.
Fundamentals of Power Electronics
16
Chapter 13: Filter inductor design
13.3.3. Constraint: Winding area Wire must fit through core window (i.e., hole in center of core)
core Total area of copper in window:
wire bare area AW core window area WA
nA W Area available for winding conductors:
K uWA Third design constraint:
K uWA ≥ nA W
Fundamentals of Power Electronics
17
Chapter 13: Filter inductor design
The window utilization factor Ku also called the “fill factor” Ku is the fraction of the core window area that is filled by copper Mechanisms that cause Ku to be less than 1: • Round wire does not pack perfectly, which reduces Ku by a factor of 0.7 to 0.55 depending on winding technique • Insulation reduces Ku by a factor of 0.95 to 0.65, depending on wire size and type of insulation • Bobbin uses some window area • Additional insulation may be required between windings Typical values of Ku : 0.5 for simple lowvoltage inductor 0.25 to 0.3 for offline transformer 0.05 to 0.2 for highvoltage transformer (multiple kV) 0.65 for lowvoltage foilwinding inductor Fundamentals of Power Electronics
18
Chapter 13: Filter inductor design
13.2.4 Winding resistance The resistance of the winding is
R=ρ
lb AW
where ρ is the resistivity of the conductor material, lb is the length of the wire, and AW is the wire bare area. The resistivity of copper at room temperature is 1.724⋅106 Ωcm. The length of the wire comprising an nturn winding can be expressed as
l b = n (MLT) where (MLT) is the meanlengthperturn of the winding. The meanlengthperturn is a function of the core geometry. The above equations can be combined to obtain the fourth constraint:
n (MLT) R=ρ AW Fundamentals of Power Electronics
19
Chapter 13: Filter inductor design
13.3 The core geometrical constant Kg The four constraints:
nI max = Bmax
µ 0 A cn 2 L= lg
lg µ0
K uWA ≥ nA W
n (MLT) R=ρ AW
These equations involve the quantities Ac, WA, and MLT, which are functions of the core geometry, Imax, Bmax , µ0, L, Ku, R, and ρ, which are given specifications or other known quantities, and n, lg, and AW, which are unknowns. Eliminate the three unknowns, leading to a single equation involving the remaining quantities. Fundamentals of Power Electronics
20
Chapter 13: Filter inductor design
Core geometrical constant Kg Elimination of n, lg, and AW leads to
ρL 2I 2max A 2c WA ≥ 2 (MLT) B max RK u • Righthand side: specifications or other known quantities • Lefthand side: function of only core geometry So we must choose a core whose geometry satisfies the above equation. The core geometrical constant Kg is defined as
A 2c WA Kg = (MLT)
Fundamentals of Power Electronics
21
Chapter 13: Filter inductor design
Discussion ρL 2I 2max A 2c WA Kg = ≥ 2 (MLT) B max RK u Kg is a figureofmerit that describes the effective electrical size of magnetic cores, in applications where the following quantities are specified: • Copper loss • Maximum flux density How specifications affect the core size: A smaller core can be used by increasing Bmax ⇒ use core material having higher Bsat R ⇒ allow more copper loss How the core geometry affects electrical capabilities: A larger Kg can be obtained by increase of Ac ⇒ more iron core material, or WA ⇒ larger window and more copper Fundamentals of Power Electronics
22
Chapter 13: Filter inductor design
13.4 A stepbystep procedure The following quantities are specified, using the units noted: Wire resistivity ρ (Ωcm) Peak winding current Imax (A) Inductance L (H) Winding resistance R (Ω) Winding fill factor Ku Core maximum flux density Bmax (T) The core dimensions are expressed in cm: Core crosssectional area Ac (cm2) (cm2) Core window area WA Mean length per turn MLT (cm) The use of centimeters rather than meters requires that appropriate factors be added to the design equations.
Fundamentals of Power Electronics
23
Chapter 13: Filter inductor design
Determine core size ρL 2I 2max Kg ≥ 2 10 8 B max RK u
(cm 5)
Choose a core which is large enough to satisfy this inequality (see Appendix 2 for magnetics design tables). Note the values of Ac, WA, and MLT for this core.
Fundamentals of Power Electronics
24
Chapter 13: Filter inductor design
Determine air gap length µ 0LI 2max 4 lg = 2 10 B max A c
(m)
with Ac expressed in cm2. µ0 = 4π107 H/m. The air gap length is given in meters. The value expressed above is approximate, and neglects fringing flux and other nonidealities.
Fundamentals of Power Electronics
25
Chapter 13: Filter inductor design
AL Core manufacturers sell gapped cores. Rather than specifying the air gap length, the equivalent quantity AL is used. AL is equal to the inductance, in mH, obtained with a winding of 1000 turns. When AL is specified, it is the core manufacturer’s responsibility to obtain the correct gap length. The required AL is given by:
AL =
2 2 max c 2 max
10B LI
A
L = A L n 2 10 – 9
Fundamentals of Power Electronics
(mH/1000 turns)
Units: Ac cm2, L Henries, Bmax Tesla.
(Henries)
26
Chapter 13: Filter inductor design
Determine number of turns n
LI max n= 10 4 Bmax A c
Fundamentals of Power Electronics
27
Chapter 13: Filter inductor design
Evaluate wire size AW ≤
K uW A n
(cm 2)
Select wire with bare copper area AW less than or equal to this value. An American Wire Gauge table is included in Appendix 2. As a check, the winding resistance can be computed:
ρn (MLT) R= Aw
Fundamentals of Power Electronics
28
(Ω)
Chapter 13: Filter inductor design
13.5 Summary of key points 1.
A variety of magnetic devices are commonly used in switching converters. These devices differ in their core flux density variations, as well as in the magnitudes of the ac winding currents. When the flux density variations are small, core loss can be neglected. Alternatively, a lowfrequency material can be used, having higher saturation flux density.
2.
The core geometrical constant Kg is a measure of the magnetic size of a core, for applications in which copper loss is dominant. In the Kg design method, flux density and total copper loss are specified.
Fundamentals of Power Electronics
29
Chapter 13: Filter inductor design
Chapter 14. Transformer Design
Some more advanced design issues, not considered in previous chapter: • • •
• •
n1 : n2
Inclusion of core loss i1(t)
Selection of operating flux density to optimize total loss Multiple winding design: how to allocate the available window area among several windings
+
+
v1(t)
v2(t)
–
–
R1
R2 +
A transformer design procedure
ik(t)
vk(t) –
How switching frequency affects transformer size
Fundamentals of Power Electronics
i2(t)
: nk
1
Rk
Chapter 14: Transformer design
Chapter 14. Transformer Design
14.1. Winding area optimization 14.2. Transformer design: Basic constraints 14.3. A stepbystep transformer design procedure 14.4. Examples 14.5. Ac inductor design 14.6. Summary
Fundamentals of Power Electronics
2
Chapter 14: Transformer design
14.1. Winding area optimization Given: application with k windings having known rms currents and desired turns ratios
v1(t) v2(t) n1 = n2 =
n1 : n2 rms current
rms current
I1
I2
v (t) = nk k
Core Window area WA
rms current
Ik
Core mean length per turn (MLT) Wire resistivity ρ Fill factor Ku
Fundamentals of Power Electronics
3
: nk
Q: how should the window area WA be allocated among the windings? Chapter 14: Transformer design
Allocation of winding area Winding 1 allocation α1WA Winding 2 allocation α2WA
{ { Total window area WA
etc.
0 < αj < 1 α1 + α2 +
Fundamentals of Power Electronics
4
+ αk = 1
Chapter 14: Transformer design
Copper loss in winding j Copper loss (not accounting for proximity loss) is
Pcu, j = I 2j R j Resistance of winding j is
lj Rj = ρ A W, j with
Hence
l j = n j (MLT)
length of wire, winding j
WAK uα j A W, j = nj
wire area, winding j
n 2j i 2j ρ (MLT) Pcu, j = WAK uα j
Fundamentals of Power Electronics
5
Chapter 14: Transformer design
Total copper loss of transformer Sum previous expression over all windings:
Pcu,tot = Pcu,1 + Pcu,2 +
ρ (MLT) + Pcu,k = WAK u
Σ k
j=1
n 2j I 2j αj
Need to select values for α1, α2, …, αk such that the total copper loss is minimized
Fundamentals of Power Electronics
6
Chapter 14: Transformer design
Variation of copper losses with α1
P
cu,k
Copper loss
For α1 = 0: wire of winding 1 has zero area. Pcu,1 tends to infinity
+.. .+ cu, 3
Pcu,tot +
P
P cu,1
For α1 = 1: wires of remaining windings have zero area. Their copper losses tend to infinity
u, 2
There is a choice of α1 that minimizes the total copper loss
Pc
0
Fundamentals of Power Electronics
1
7
α1
Chapter 14: Transformer design
Method of Lagrange multipliers to minimize total copper loss Minimize the function
Pcu,tot = Pcu,1 + Pcu,2 +
ρ (MLT) + Pcu,k = WAK u
Σ k
j=1
n 2j I 2j αj
subject to the constraint
α1 + α2 +
+ αk = 1
Define the function
f (α 1, α 2,
, α k, ξ) = Pcu,tot(α 1, α 2,
where
g(α 1, α 2,
, α k) = 1 –
Σα
, α k) + ξ g(α 1, α 2,
, α k)
k
j=1
j
is the constraint that must equal zero and ξ is the Lagrange multiplier Fundamentals of Power Electronics
8
Chapter 14: Transformer design
Lagrange multipliers continued Result:
Optimum point is solution of the system of equations
ρ (MLT) ξ= WAK u
∂ f (α 1, α 2, , α k,ξ) =0 ∂α 1 ∂ f (α 1, α 2, , α k,ξ) =0 ∂α 2
αm =
j j
= Pcu,tot
∞
Σ nI
j j
An alternate form:
αm =
V mI m
∞
Σ VI
n=1
Fundamentals of Power Electronics
j=1
2
n mI m
n=1
∂ f (α 1, α 2, , α k,ξ) =0 ∂α k ∂ f (α 1, α 2, , α k,ξ) =0 ∂ξ
ΣnI k
9
j j
Chapter 14: Transformer design
Interpretation of result αm =
V mI m
∞
Σ VI
n=1
j j
Apparent power in winding j is V j Ij where
Vj is the rms or peak applied voltage Ij is the rms current
Window area should be allocated according to the apparent powers of the windings
Fundamentals of Power Electronics
10
Chapter 14: Transformer design
Example PWM fullbridge transformer i1(t)
n1 turns
i2(t)
{
} }
I
n2 turns
n2 turns
n2 I n1
i1(t)
i3(t)
• Note that waveshapes (and hence rms values) of the primary and secondary currents are different
0
– i2(t)
n2 I n1
I 0.5I
0.5I 0
i3(t)
I
• Treat as a threewinding transformer
0.5I
0.5I
0 0
Fundamentals of Power Electronics
0
11
DTs
Ts
Ts+DTs
2Ts
Chapter 14: Transformer design
t
Expressions for RMS winding currents
I1 =
I2 = I3 =
1 2Ts
2T s
i 21(t)dt =
0
1 2Ts
2T s 0
n2 I n1
i1(t)
n2 I D n1
0
0
–
i 22(t)dt = 12 I 1 + D i2(t)
n2 I n1
I 0.5I
0.5I
see Appendix 1
0 i3(t)
I 0.5I
0.5I
0 0
Fundamentals of Power Electronics
12
DTs
Ts
Ts+DTs
2Ts
Chapter 14: Transformer design
t
Allocation of window area:
αm =
V mI m
∞
Σ VI
n=1
j j
Plug in rms current expressions. Result:
α1 = 1+
1 1+D D 1
α 2 = α 3 = 12 1+
Fundamentals of Power Electronics
Fraction of window area allocated to primary winding
D 1+D
13
Fraction of window area allocated to each secondary winding
Chapter 14: Transformer design
Numerical example Suppose that we decide to optimize the transformer design at the worstcase operating point D = 0.75. Then we obtain
α 1 = 0.396 α 2 = 0.302 α 3 = 0.302 The total copper loss is then given by
ρ(MLT) 3 Pcu,tot = n jI j WAK u jΣ =1 ρ(MLT)n 22 I 2 = 1 + 2D + 2 D(1 + D) WAK u 2
Fundamentals of Power Electronics
14
Chapter 14: Transformer design
14.2 Transformer design: Basic constraints Core loss
Pfe = K feB βmax A cl m Typical value of β for ferrite materials: 2.6 or 2.7 Bmax is the peak value of the ac component of B(t) So increasing Bmax causes core loss to increase rapidly
This is the first constraint
Fundamentals of Power Electronics
15
Chapter 14: Transformer design
Flux density Constraint #2 v1(t)
Flux density B(t) is related to the applied winding voltage according to Faraday’s Law. Denote the voltseconds applied to the primary winding during the positive portion of v1(t) as λ1:
λ1 =
area λ1
t1
t2
t2
v1(t)dt t1
This causes the flux to change from its negative peak to its positive peak. From Faraday’s law, the peak value of the ac component of flux density is
λ1 Bmax = 2n 1A c
Fundamentals of Power Electronics
To attain a given flux density, the primary turns should be chosen according to
λ1 n1 = 2Bmax A c
16
Chapter 14: Transformer design
t
Copper loss Constraint #3 • Allocate window area between windings in optimum manner, as described in previous section • Total copper loss is then equal to
Pcu =
with
ρ(MLT)n I WAK u
2 2 1 tot
Σ k
I tot =
j=1
nj n1 I j
Eliminate n1, using result of previous slide:
ρ λ 21 I 2tot Pcu = Ku
(MLT) WAA 2c
1 B 2max
Note that copper loss decreases rapidly as Bmax is increased
Fundamentals of Power Electronics
17
Chapter 14: Transformer design
Total power loss 4. Ptot = Pcu + Pfe Power loss Co
Ptot = Pfe + Pcu
fe
ss P Co
ss P c
r lo
ppe
Ptot
re l o
There is a value of Bmax that minimizes the total power loss
u
Pfe = K feB βmax A cl m ρ λ 21 I 2tot Pcu = Ku
Fundamentals of Power Electronics
Optimum Bmax
(MLT) WAA 2c
Bmax
1 B 2max
18
Chapter 14: Transformer design
5. Find optimum flux density Bmax Given that
Ptot = Pfe + Pcu Then, at the Bmax that minimizes Ptot, we can write
dPfe dPtot dPcu = + =0 d Bmax d Bmax d Bmax Note: optimum does not necessarily occur where Pfe = Pcu. Rather, it occurs where
dPfe dPcu =– dBmax dBmax
Fundamentals of Power Electronics
19
Chapter 14: Transformer design
Take derivatives of core and copper loss Pfe = K feB
β max
ρ λ 21 I 2tot Pcu = Ku
A cl m
dPfe β–1 = βK feB max A cl m dBmax
Now, substitute into
ρλ I Bmax = 2K u
2 2 1 tot
Fundamentals of Power Electronics
(MLT) WAA 2c
ρλ 21 I 2tot dPcu =–2 4K u d Bmax
dPfe dPcu =– dBmax dBmax
(MLT) 1 WAA 3c l m βK fe
20
1 B 2max
(MLT) – 3 B max 2 WAA c
and solve for Bmax: 1 β+2
Optimum Bmax for a given core and application
Chapter 14: Transformer design
Total loss Substitute optimum Bmax into expressions for Pcu and Pfe. The total loss is:
Ptot = A cl mK fe
ρλ I 4K u
2 2 1 tot
2 β+2
β β+2
(MLT) WAA 2c
β 2
–
β β+2
β + 2
2 β+2
Rearrange as follows:
WA A c
2(β – 1)/β
(MLT) l
2/β m
β 2
–
β β+2
–
+
β 2
Left side: terms depend on core geometry
Fundamentals of Power Electronics
2 β+2
β+2 β
ρλ 21I 2totK fe
2/β
=
4K u Ptot
β + 2 /β
Right side: terms depend on specifications of the application
21
Chapter 14: Transformer design
The core geometrical constant Kgfe Define
K gfe =
WA A c
2(β – 1)/β
(MLT) l m2/β
β 2
–
β β+2
–
+
β 2
2 β+2
β+2 β
Design procedure: select a core that satisfies
K gfe ≥
ρλ I K fe 2 2 1 tot
4K u Ptot
2/β
β + 2 /β
Appendix 2 lists the values of Kgfe for common ferrite cores Kgfe is similar to the Kg geometrical constant used in Chapter 13: • Kg is used when Bmax is specified • Kgfe is used when Bmax is to be chosen to minimize total loss
Fundamentals of Power Electronics
22
Chapter 14: Transformer design
14.3 Stepbystep transformer design procedure The following quantities are specified, using the units noted: Wire effective resistivity ρ (Ωcm) Total rms winding current, ref to pri Itot (A) Desired turns ratios n2/n1, n3/n1, etc. Applied pri voltsec λ1 (Vsec) Allowed total power dissipation Ptot (W) Winding fill factor Ku Core loss exponent β Core loss coefficient Kfe (W/cm3Tβ) Other quantities and their dimensions: Core crosssectional area Ac Core window area WA Mean length per turn MLT Magnetic path length le Wire areas Aw1, … Peak ac flux density Bmax Fundamentals of Power Electronics
23
(cm2) (cm2) (cm) (cm) (cm2) (T) Chapter 14: Transformer design
Procedure 1.
Determine core size
ρλ 21I 2totK fe
2/β
K gfe ≥
4K u Ptot
β + 2 /β
10 8
Select a core from Appendix 2 that satisfies this inequality. It may be possible to reduce the core size by choosing a core material that has lower loss, i.e., lower Kfe.
Fundamentals of Power Electronics
24
Chapter 14: Transformer design
2.
Evaluate peak ac flux density 1 β+2
2 2 ρλ 8 1I tot (MLT) 1 Bmax = 10 2K u WAA 3c l m βK fe
At this point, one should check whether the saturation flux densityis exceeded. If the core operates with a flux dc bias Bdc, then Bmax + Bdc should be less than the saturation flux density. If the core will saturate, then there are two choices: • Specify Bmax using the Kg method of Chapter 13, or • Choose a core material having greater core loss, then repeat steps 1 and 2
Fundamentals of Power Electronics
25
Chapter 14: Transformer design
3. and 4. Primary turns:
n1 =
Evaluate turns
λ1 10 4 2Bmax A c
Choose secondary turns according to desired turns ratios:
Fundamentals of Power Electronics
n2 = n1
n2 n1
n3 = n1
n3 n1
26
Chapter 14: Transformer design
5. and 6.
Choose wire sizes
Fraction of window area assigned to each winding:
Choose wire sizes according to:
n 1I 1 α1 = n 1I tot n 2I 2 α2 = n 1I tot
α 1K uWA n1 α 2K uWA A w2 ≤ n2 A w1 ≤
n kI k αk = n 1I tot
Fundamentals of Power Electronics
27
Chapter 14: Transformer design
Check: computed transformer model Predicted magnetizing inductance, referred to primary:
n1 : n2 i1(t)
µn A c LM = lm 2 1
iM(t)
i2(t)
LM
Peak magnetizing current:
R1
λ1 i M, pk = 2L M
R2 ik(t)
Predicted winding resistances:
ρn 1(MLT) A w1 ρn (MLT) R2 = 2 A w2 R1 =
Fundamentals of Power Electronics
: nk
28
Rk
Chapter 14: Transformer design
14.4.1
Example 1: Singleoutput isolated Cuk converter Ig 4A
+ vC1(t) –
– vC2(t) +
25 V
+ –
20 A
+
– Vg
I
+
v1(t)
v2(t)
+
V 5V
– –
i1(t)
n:1
i2(t)
100 W
fs = 200 kHz
D = 0.5
n=5
Ku = 0.5
Allow Ptot = 0.25 W
Use a ferrite pot core, with Magnetics Inc. P material. Loss parameters at 200 kHz are Kfe = 24.7 Fundamentals of Power Electronics
β = 2.6 29
Chapter 14: Transformer design
Waveforms v1(t)
VC1
Area λ1
Applied primary voltseconds: λ 1 = DTsVc1 = (0.5) (5 µsec ) (25 V) = 62.5 V–µsec
D'Ts DTs
i1(t)
– nVC2
Applied primary rms current:
I/n
I1 = – Ig i2(t)
2
+ D' I g
2
=4A
Applied secondary rms current: I 2 = nI 1 = 20 A
I
Total rms winding current: I tot = I 1 + 1n I 2 = 8 A
– nIg
Fundamentals of Power Electronics
D nI
30
Chapter 14: Transformer design
Choose core size (1.724⋅10 – 6)(62.5⋅10 – 6) 2(8) 2(24.7) 2/2.6 8 K gfe ≥ 10 4 (0.5) (0.25) 4.6/2.6 = 0.00295 Pot core data of Appendix 2 lists 2213 pot core with Kgfe = 0.0049 Next smaller pot core is not large enough.
Fundamentals of Power Electronics
31
Chapter 14: Transformer design
Evaluate peak ac flux density 1/4.6
(1.724⋅10 – 6)(62.5⋅10 – 6) 2(8) 2 (4.42) 1 Bmax = 10 3 2 (0.5) (0.297)(0.635) (3.15) (2.6)(24.7) 8
= 0.0858 Tesla
This is much less than the saturation flux density of approximately 0.35 T. Values of Bmax in the vicinity of 0.1 T are typical for ferrite designs that operate at frequencies in the vicinity of 100 kHz.
Fundamentals of Power Electronics
32
Chapter 14: Transformer design
Evaluate turns –6 (62.5⋅10 ) n 1 = 10 4 2(0.0858)(0.635) = 5.74 turns
n1 n 2 = n = 1.15 turns In practice, we might select n1 = 5
and
n2 = 1
This would lead to a slightly higher flux density and slightly higher loss.
Fundamentals of Power Electronics
33
Chapter 14: Transformer design
Determine wire sizes Fraction of window area allocated to each winding:
α1 = α2 =
4A 8A 1 5
(Since, in this example, the ratio of winding rms currents is equal to the turns ratio, equal areas are allocated to each winding)
= 0.5
20 A 8A
= 0.5
From wire table, Appendix 2:
Wire areas:
(0.5)(0.5)(0.297) = 14.8⋅10 – 3 cm 2 (5) (0.5)(0.5)(0.297) A w2 = = 74.2⋅10 – 3 cm 2 (1) A w1 =
Fundamentals of Power Electronics
34
AWG #16 AWG #9
Chapter 14: Transformer design
Wire sizes: discussion Primary 5 turns #16 AWG
Secondary 1 turn #9 AWG
•
Very large conductors!
•
One turn of #9 AWG is not a practical solution
Some alternatives •
Use foil windings
•
Use Litz wire or parallel strands of wire
Fundamentals of Power Electronics
35
Chapter 14: Transformer design
Effect of switching frequency on transformer size for this Pmaterial Cuk converter example 4226 0.1
2616
2616 2213
2213 1811
0.08 0.06
1811
0.04
Bmax , Tesla
Pot core size
3622
0.02 0 25 kHz
50 kHz
100 kHz
200 kHz
250 kHz
400 kHz
500 kHz
1000 kHz
Switching frequency
• As switching frequency is increased from 25 kHz to 250 kHz, core size is dramatically reduced Fundamentals of Power Electronics
• As switching frequency is increased from 400 kHz to 1 MHz, core size increases 36
Chapter 14: Transformer design
14.4.2
Example 2 MultipleOutput FullBridge Buck Converter
Q1
D1
Q3
T1
D3
n1 :
I5V : n2
160 V
+ –
100 A
+
D5
+ Vg
i2a(t)
i1(t) v1(t)
5V
–
Q2
D2
Q4
Switching frequency
D6
i2b(t)
–
: n2
D4
I15V
: n3
i3a(t)
+
D7
150 kHz
15 A
15 V
Transformer frequency
75 kHz D8
Turns ratio
110:5:15
Optimize transformer at
D = 0.75
Fundamentals of Power Electronics
i2b(t)
–
: n3
37
Chapter 14: Transformer design
Other transformer design details Use Magnetics, Inc. ferrite P material. Loss parameters at 75 kHz: Kfe = 7.6 W/Tβcm3 β = 2.6 Use EE core shape Assume fill factor of Ku = 0.25
(reduced fill factor accounts for added insulation required in multipleoutput offline application)
Allow transformer total power loss of Ptot = 4 W
(approximately 0.5% of total output power)
Use copper wire, with ρ = 1.724·106 Ωcm
Fundamentals of Power Electronics
38
Chapter 14: Transformer design
Applied transformer waveforms v1(t)
T1
D3
n1 :
: n2
i2a(t)
0
0
D5
+
– Vg
i1(t) v1(t)
n n2 I 5V + 3 I 15V n1 n1
i1(t)
– D4
Area λ1 = Vg DTs
Vg
D6
i2b(t)
0
: n2 : n3
i3a(t) D7
–
i2a(t)
n2 n I 5V + 3 I 15V n1 n1
I5V 0.5I5V 0
D8
i2b(t)
i3a(t)
I15V 0.5I15V
: n3
0 0
Fundamentals of Power Electronics
DTs
39
Ts
Ts+DTs 2Ts
t
Chapter 14: Transformer design
Applied primary voltseconds v1(t)
Vg
Area λ1 = Vg DTs 0
0 – Vg
λ 1 = DTsVg = (0.75) (6.67 µsec ) (160 V) = 800 V–µsec
Fundamentals of Power Electronics
40
Chapter 14: Transformer design
Applied primary rms current i1(t)
n n2 I 5V + 3 I 15V n1 n1
0 –
n2 n I 5V + 3 I 15V n1 n1
n2 n3 I 1 = n I 5V + n I 15V 1 1
Fundamentals of Power Electronics
41
D = 5.7 A
Chapter 14: Transformer design
Applied rms current, secondary windings i2a(t)
I5V 0.5I5V 0
i3a(t)
I15V 0.5I15V 0 0
DTs
Ts
Ts+DTs 2Ts
t
I 2 = 12 I 5V 1 + D = 66.1 A I 3 = 12 I 15V 1 + D = 9.9 A Fundamentals of Power Electronics
42
Chapter 14: Transformer design
Itot RMS currents, summed over all windings and referred to primary
I tot =
Σ
all 5 windings
nj n2 n3 n1 I j = I 1 + 2 n1 I 2 + 2 n1 I 3
= 5.7 A + 5 66.1 A + 15 9.9 A 110 110 = 14.4 A
Fundamentals of Power Electronics
43
Chapter 14: Transformer design
Select core size (1.724⋅10 – 6)(800⋅10 – 6) 2(14.4) 2(7.6) 2/2.6 8 K gfe ≥ 10 4 (0.25) (4) 4.6/2.6 = 0.00937 A2.2
EE core data
From Appendix 2
A
Core type
Geometrical constant
Geometrical constant
(A) (mm)
Kg cm5
Kgfe cmx
Crosssectional area Ac (cm2)
Bobbin winding area WA (cm2)
Mean length per turn MLT (cm)
Magnetic path length lm (cm)
Core weight
(g)
EE22
8.26·103
1.8·103
0.41
0.196
3.99
3.96
8.81
EE30
3
85.7·10
6.7·10
3
1.09
0.476
6.60
5.77
32.4
EE40
0.209
11.8·103
1.27
1.10
8.50
7.70
50.3
EE50
0.909
28.4·103
2.26
1.78
10.0
9.58
116
Fundamentals of Power Electronics
44
Chapter 14: Transformer design
Evaluate ac flux density Bmax Eq. (14.41):
2 2 ρλ 8 1I tot (MLT) 1 Bmax = 10 2K u WAA 3c l m βK fe
1 β+2
Plug in values: 1/4.6
(1.724⋅10 – 6)(800⋅10 – 6) 2(14.4) 2 (8.5) 1 Bmax = 10 2 (0.25) (1.1)(1.27) 3(7.7) (2.6)(7.6) 8
= 0.23 Tesla This is less than the saturation flux density of approximately 0.35 T
Fundamentals of Power Electronics
45
Chapter 14: Transformer design
Evaluate turns Choose n1 according to Eq. (14.42):
n1 =
λ1 10 4 2Bmax A c
To obtain desired turns ratio of
(800⋅10 – 6) n 1 = 10 2(0.23)(1.27) = 13.7 turns 4
110:5:15 we might round the actual turns to
Choose secondary turns according to desired turns ratios:
22:1:3 Increased n1 would lead to
5 n2 = n = 0.62 turns 110 1
• Less core loss • More copper loss
15 n = 1.87 turns n3 = 110 1 Fundamentals of Power Electronics
Rounding the number of turns
• Increased total loss 46
Chapter 14: Transformer design
Loss calculation with rounded turns With n1 = 22, the flux density will be reduced to
(800⋅10 – 6) Bmax = 10 4 = 0.143 Tesla 2 (22) (1.27) The resulting losses will be
Pfe = (7.6)(0.143) 2.6(1.27)(7.7) = 0.47 W (8.5) (1.724⋅10 – 6)(800⋅10 – 6) 2(14.4) 2 8 1 Pcu = 10 4 (0.25) (1.1)(1.27) 2 (0.143) 2 = 5.4 W Ptot = Pfe + Pcu = 5.9 W Which exceeds design goal of 4 W by 50%. So use next larger core size: EE50. Fundamentals of Power Electronics
47
Chapter 14: Transformer design
Calculations with EE50 Repeat previous calculations for EE50 core size. Results: Bmax = 0.14 T, n1 = 12, Ptot = 2.3 W Again round n1 to 22. Then Bmax = 0.08 T, Pcu = 3.89 W, Pfe = 0.23 W, Ptot = 4.12 W Which is close enough to 4 W.
Fundamentals of Power Electronics
48
Chapter 14: Transformer design
Wire sizes for EE50 design Window allocations
α1 =
I1 = 5.7 = 0.396 I tot 14.4
α2 =
n 2I 2 = 5 66.1 = 0.209 n 1I tot 110 14.4
α3 =
n 3I 3 = 15 9.9 = 0.094 n 1I tot 110 14.4
Wire gauges α 1K uWA (0.396)(0.25)(1.78) = = 8.0⋅10 – 3 cm 2 (22) n1 ⇒ AWG #19 αKW (0.209)(0.25)(1.78) A w2 = 2 u A = = 93.0⋅10 – 3 cm 2 (1) n2
A w1 =
⇒ AWG #8 αKW (0.094)(0.25)(1.78) A w3 = 3 u A = = 13.9⋅10 – 3 cm 2 (3) n3 ⇒ AWG #16
Might actually use foil or Litz wire for secondary windings
Fundamentals of Power Electronics
49
Chapter 14: Transformer design
Part IV
Modern Rectifiers and Power System Harmonics
Chapter 15
Power and Harmonics in Nonsinusoidal Systems
Chapter 16
LineCommutated Rectifiers
Chapter 17
The Ideal Rectifier
Chapter 18
Low Harmonic Rectifier Modeling and Control
Fundamentals of Power Electronics
1
Chapter 15: Power and Harmonics in Nonsinusoidal Systems
Chapter 15
Power And Harmonics in Nonsinusoidal Systems
15.1. 15.2. 15.3.
15.4. 15.5. 15.6.
Average power in terms of Fourier series RMS value of a waveform Power factor THD Distortion and Displacement factors Power phasors in sinusoidal systems Harmonic currents in threephase systems AC line current harmonic standards
Fundamentals of Power Electronics
2
Chapter 15: Power and Harmonics in Nonsinusoidal Systems
15.1. Average power Observe transmission of energy through surface S i(t) + Source
+ –
v(t)
Load
–
Surface S
Express voltage and current as Fourier series:
Fundamentals of Power Electronics
∞
Σ V cos nωt – ϕ + Σ I cos nωt – θ
v(t) = V0 + i(t) = I0
n
n=1 ∞
n=1
3
n
n
n
relate energy transmission to harmonics
Chapter 15: Power and Harmonics in Nonsinusoidal Systems
Energy transmittted to load, per cycle T
Wcycle =
v(t)i(t)dt 0
This is related to average power as follows:
Wcycle 1 Pav = = T T
T
v(t)i(t)dt 0
Investigate influence of harmonics on average power: substitute Fourier series T
Pav = 1 T
∞
V0 +
ΣV
n=1
n
cos nωt – ϕ n
∞
I0 +
ΣI
n=1
n
cos nωt – θ n dt
0
Fundamentals of Power Electronics
4
Chapter 15: Power and Harmonics in Nonsinusoidal Systems
Evaluation of integral Orthogonality of harmonics: Integrals of crossproduct terms are zero T
Vn cos nωt – ϕ n
I m cos mωt – θ m dt =
0
0
if n ≠ m
V nI n cos ϕ n – θ n 2
if n = m
Expression for average power becomes ∞
Pav = V0I 0 +
Σ
n=1
V nI n cos ϕ n – θ n 2
So net energy is transmitted to the load only when the Fourier series of v(t) and i(t) contain terms at the same frequency. For example, if the voltage and current both contain third harmonic, then they lead to the average power VI 3 3
2 Fundamentals of Power Electronics
cos ϕ 3 – θ 3 5
Chapter 15: Power and Harmonics in Nonsinusoidal Systems
Example 1 v(t)
1
Voltage: fundamental only Current: third harmonic only
i(t)
0.5
0
0.5
1
1
p(t) = v(t) i(t)
Power: zero average 0.5
Pav = 0
0 0.5 1
Fundamentals of Power Electronics
6
Chapter 15: Power and Harmonics in Nonsinusoidal Systems
Example 2 1
Voltage: third harmonic only Current: third harmonic only, in phase with voltage
v(t), i(t)
0.5
0
0.5
1
p(t) = v(t) i(t)
Power: nonzero average
1 0.5
Pav = 0.5
0 0.5
Fundamentals of Power Electronics
1
7
Chapter 15: Power and Harmonics in Nonsinusoidal Systems
Example 3 Fourier series:
v(t) = 1.2 cos (ωt) + 0.33 cos (3ωt) + 0.2 cos (5ωt) i(t) = 0.6 cos (ωt + 30°) + 0.1 cos (5ωt + 45°) + 0.1 cos (7ωt + 60°) Average power calculation:
Pav =
(1.2)(0.6) (0.2)(0.1) cos (30°) + cos (45°) = 0.32 2 2
Fundamentals of Power Electronics
8
Chapter 15: Power and Harmonics in Nonsinusoidal Systems
Example 3 v(t) 1.0
Voltage: 1st, 3rd, 5th Current: 1st, 5th, 7th
0.5
i(t) 0.0
0.5
1.0
0.6
Power: net energy is transmitted at fundamental and fifth harmonic frequencies
0.4
p(t) = v(t) i(t)
Pav = 0.32
0.2
0.0
Fundamentals of Power Electronics
9 0.2 Chapter 15: Power and Harmonics in Nonsinusoidal Systems
15.2. Rootmeansquare (RMS) value of a waveform, in terms of Fourier series
(rms value) =
1 T
T
v 2(t)dt 0
Insert Fourier series. Again, crossmultiplication terms have zero average. Result is
V 20 +
(rms value) =
∞
Σ
n=1
V 2n 2
•
Similar expression for current
•
Harmonics always increase rms value
•
Harmonics do not necessarily increase average power
•
Increased rms values mean increased losses
Fundamentals of Power Electronics
10
Chapter 15: Power and Harmonics in Nonsinusoidal Systems
15.3. Power factor
For efficient transmission of energy from a source to a load, it is desired to maximize average power, while minimizing rms current and voltage (and hence minimizing losses). Power factor is a figure of merit that measures how efficiently energy is transmitted. It is defined as
(average power) power factor = (rms voltage) (rms current) Power factor always lies between zero and one.
Fundamentals of Power Electronics
11
Chapter 15: Power and Harmonics in Nonsinusoidal Systems
15.3.1. Linear resistive load, nonsinusoidal voltage
Then current harmonics are in phase with, and proportional to, voltage harmonics. All harmonics result in transmission of energy to load, and unity power factor occurs.
In =
Vn R
θn = ϕn
(rms voltage) =
V 20 +
(rms current) =
2 0
∞
Σ
n=1
∞
I +
Σ
n=1
so cos (θ n – ϕ n) = 1
V 2n 2
I 2n = 2
∞ V 20 V 2n 2 + Σ 2 n = 1 2R R
= 1 (rms voltage) R ∞ VI Pav = V0I 0 + Σ n n cos (ϕ n – θ n) 2 n=1 Fundamentals of Power Electronics
12
Chapter 15: Power and Harmonics in Nonsinusoidal Systems
15.3.2. Nonlinear dynamical load, sinusoidal voltage
With a sinusoidal voltage, current harmonics do not lead to average power. However, current harmonics do increase the rms current, and hence they decrease the power factor.
V 1I 1 Pav = cos (ϕ 1 – θ 1) 2
(rms current) =
(power factor) =
2 0
∞
I +
Σ
n=1
I1 2
∞
I 20 + ∑
n=1
I 2n 2
2 n
I 2
cos (ϕ 1 – θ 1)
= (distortion factor) (displacement factor) Fundamentals of Power Electronics
13
Chapter 15: Power and Harmonics in Nonsinusoidal Systems
Distortion factor
Defined only for sinusoidal voltage.
I1 2
(distortion factor) =
∞
I +∑ 2 0
n=1
I 2n 2
=
(rms fundamental current) (rms current)
Related to Total Harmonic Distortion (THD): ∞
(THD) =
ΣI
n=2
2 n
I1
(distortion factor) = Fundamentals of Power Electronics
14
1 1 + (THD) 2 Chapter 15: Power and Harmonics in Nonsinusoidal Systems
Distortion factor vs. THD
Distortion factor
100%
90%
80%
Fundamentals of Power Electronics
15
100%
THD
80%
60%
40%
20%
0%
70%
Chapter 15: Power and Harmonics in Nonsinusoidal Systems
Peak detection rectifier example
Conventional singlephase peak detection rectifier
Typical ac line current spectrum
Harmonic amplitude, percent of fundamental
100%
100% 91%
80%
THD = 136% Distortion factor = 59%
73%
60%
52%
40%
32% 19% 15% 15% 13%
20%
9%
0% 1
3
5
7
9
11
13
15
17
19
Harmonic number Fundamentals of Power Electronics
16
Chapter 15: Power and Harmonics in Nonsinusoidal Systems
Maximum power obtainable from 120V 15A wall outlet
with peak detection rectifier
(ac voltage) (derated breaker current) (power factor) (rectifier efficiency) = (120 V) (80% of 15 A) (0.55) (0.98) = 776 W at unity power factor
(ac voltage) (derated breaker current) (power factor) (rectifier efficiency) = (120 V) (80% of 15 A) (0.99) (0.93) = 1325 W
Fundamentals of Power Electronics
17
Chapter 15: Power and Harmonics in Nonsinusoidal Systems
15.4. Power phasors in sinusoidal systems Apparent power is the product of the rms voltage and rms current It is easily measured —simply the product of voltmeter and ammeter readings Unit of apparent power is the voltampere, or VA Many elements, such as transformers, are rated according to the VA that they can supply So power factor is the ratio of average power to apparent power With sinusoidal waveforms (no harmonics), we can also define the real power P reactive power Q complex power S If the voltage and current are represented by phasors V and I, then
S = VI * = P + jQ with I* = complex conjugate of I, j = square root of –1. The magnitude of S is the apparent power (VA). The real part of S is the average power P (watts). The imaginary part of S is the reactive power Q (reactive voltamperes, or VARs). Fundamentals of Power Electronics
18
Chapter 15: Power and Harmonics in Nonsinusoidal Systems
Example: power phasor diagram The phase angle between the voltage and current, or (ϕ1 – θ1), coincides with the angle of S. The power factor is
Imaginary axis Q s
= S
power factor = P = cos ϕ 1 – θ 1 S
I rm ms
Vr
ϕ1 – θ1
ϕ1 θ1
P
Real axis
ϕ1 – θ1
In this purely sinusoidal case, the distortion factor is unity, and the power factor coincides with the displacement factor.
Fundamentals of Power Electronics
S = VI*
V I
19
Chapter 15: Power and Harmonics in Nonsinusoidal Systems
Reactive power Q The reactive power Q does not lead to net transmission of energy between the source and load. When Q ≠ 0, the rms current and apparent power are greater than the minimum amount necessary to transmit the average power P.
Inductor: current lags voltage by 90˚, hence displacement factor is zero. The alternate storing and releasing of energy in an inductor leads to current flow and nonzero apparent power, but P = 0. Just as resistors consume real (average) power P, inductors can be viewed as consumers of reactive power Q.
Capacitor: current leads voltage by 90˚, hence displacement factor is zero. Capacitors supply reactive power Q. They are often placed in the utility power distribution system near inductive loads. If Q supplied by capacitor is equal to Q consumed by inductor, then the net current (flowing from the source into the capacitorinductiveload combination) isin phase with the voltage, leading to unity power factor and minimum rms current magnitude. Fundamentals of Power Electronics
20
Chapter 15: Power and Harmonics in Nonsinusoidal Systems
Lagging fundamental current of phasecontrolled rectifiers It will be seen in the next chapter that phasecontrolled rectifiers produce a nonsinusoidal current waveform whose fundamental component lags the voltage. This lagging current does not arise from energy storage, but it does nonetheless lead to a reduced displacement factor, and to rms current and apparent power that are greater than the minimum amount necessary to transmit the average power. At the fundamental frequency, phasecontrolled rectifiers can be viewed as consumers of reactive power Q, similar to inductive loads.
Fundamentals of Power Electronics
21
Chapter 15: Power and Harmonics in Nonsinusoidal Systems
15.5. Harmonic currents in three phase systems
The presence of harmonic currents can also lead to some special problems in threephase systems: • In a fourwire threephase system, harmonic currents can lead to large currents in the neutral conductors, which may easily exceed the conductor rms current rating • Power factor correction capacitors may experience significantly increased rms currents, causing them to fail In this section, these problems are examined, and the properties of harmonic current flow in threephase systems are derived: • Harmonic neutral currents in 3ø fourwire networks • Harmonic neutral voltages in 3ø threewire wyeconnected loads
Fundamentals of Power Electronics
22
Chapter 15: Power and Harmonics in Nonsinusoidal Systems
15.5.1. Harmonic currents in threephase fourwire networks ia(t)
a
+ van(t) – ideal 3ø source
n
vcn(t)
+ –
c ic(t) in(t)
+ –
vbn(t)
nonlinear loads
neutral connection ib(t)
b ∞
I ak cos (kωt – θ ak) Fourier series of i a(t) = I a0 + kΣ =1 ∞ line currents and i b(t) = I b0 + Σ I bk cos (k(ωt – 120˚) – θ bk) voltages: k=1 ∞ i c(t) = I c0 + Σ I ck cos (k(ωt + 120˚) – θ ck)
van(t) = Vm cos (ωt) vbn(t) = Vm cos (ωt – 120˚) vcn(t) = Vm cos (ωt + 120˚)
k=1
Fundamentals of Power Electronics
23
Chapter 15: Power and Harmonics in Nonsinusoidal Systems
Neutral current i n(t) = I a0 + I b0 + I c0 + ∞
Σ
k=1
I ak cos (kωt – θ ak) + I bk cos (k(ωt – 120˚) – θ bk) + I ck cos (k(ωt + 120˚) – θ ck)
If the load is unbalanced, then there is nothing more to say. The neutral connection may contain currents having spectrum similar to the line currents. In the balanced case, Iak = Ibk = Ick = Ik and θak = θbk = θck = θk , for all k; i.e., the harmonics of the three phases all have equal amplitudes and phase shifts. The neutral current is then ∞
i n(t) = 3I 0 +
Fundamentals of Power Electronics
Σ
k = 3,6,9,
3I k cos (kωt – θ k)
24
Chapter 15: Power and Harmonics in Nonsinusoidal Systems
Neutral currents ∞
i n(t) = 3I 0 +
Σ
3I k cos (kωt – θ k)
k = 3,6,9,
•
Fundamental and most harmonics cancel out
•
Triplen (triplen, or 0, 3, 6, 9, ...) harmonics do not cancel out, but add. Dc components also add.
•
Rms neutral current is
i n, rms = 3
Fundamentals of Power Electronics
I 20 +
∞
Σ
k = 3,6,9,
25
I 2k 2
Chapter 15: Power and Harmonics in Nonsinusoidal Systems
Example A balanced nonlinear load produces line currents containing fundamental and 20% third harmonic: ian(t) = I1 cos(ωt – θ 1 ) + 0.2 I1 cos(3ωt – θ 3). Find the rms neutral current, and compare its amplitude to the rms line current amplitude.
Solution
(0.2I 1) 2 0.6 I 1 i n, rms = 3 = 2 2 I 21 + (0.2I 1) 2 I I i 1, rms = = 1 1 + 0.04 ≈ 1 2 2 2
• The neutral current magnitude is 60% of the line current magnitude! • The triplen harmonics in the three phases add, such that 20% third harmonic leads to 60% third harmonic neutral current. • Yet the presence of the third harmonic has very little effect on the rms value of the line current. Significant unexpected neutral current flows. Fundamentals of Power Electronics
26
Chapter 15: Power and Harmonics in Nonsinusoidal Systems
15.5.2. Harmonic currents in threephase threewire networks Wyeconnected nonlinear load, no neutral connection: ia(t)
a
+ van(t) – ideal 3ø source
n
vcn(t)
+ –
c ic(t)
+ –
vbn(t)
n'
nonlinear loads
+ vn'n –
in(t) = 0 ib(t)
b
Fundamentals of Power Electronics
27
Chapter 15: Power and Harmonics in Nonsinusoidal Systems
No neutral connection
If the load is balanced, then it is still true that
∞
i n(t) = 3I 0 +
Σ
k = 3,6,9,
3I k cos (kωt – θ k)
But in(t) = 0, since there is no neutral connection. So the ac line currents cannot contain dc or triplen harmonics. What happens: A voltage is induced at the load neutral point, that causes the line current dc and triplen harmonics to become zero. The load neutral point voltage contains dc and triplen harmonics. With an unbalanced load, the line currents can still contain dc and triplen harmonics. Fundamentals of Power Electronics
28
Chapter 15: Power and Harmonics in Nonsinusoidal Systems
Deltaconnected load ia(t)
a
+ van(t) – ideal 3ø source
n
vcn(t)
+ –
+ –
vbn(t)
c ic(t)
deltaconnected nonlinear loads
in(t) = 0 ib(t)
b
• There is again no neutral connection, so the ac line currents contain no dc or triplen harmonics • The load currents may contain dc and triplen harmonics: with a balanced nonlinear load, these circulate around the delta. Fundamentals of Power Electronics
29
Chapter 15: Power and Harmonics in Nonsinusoidal Systems
Harmonic currents in power factor correction capacitors PFC capacitors are usually not intended to conduct significant harmonic currents. Heating in capacitors is a function of capacitor equivalent series resistance (esr) and rms current. The maximum allowable rms current then leads to the capacitor rating:
rated rms voltage Vrms =
esr
C
I rms 2π fC
I 2rms rated reactive power = 2π fC
Fundamentals of Power Electronics
30
Chapter 15: Power and Harmonics in Nonsinusoidal Systems
15.6. AC line current harmonic standards
15.6.1. US MILSTD461B 15.6.2. International Electrotechnical Commission Standard 555 15.6.3. IEEE/ANSI Standard 519
Fundamentals of Power Electronics
31
Chapter 15: Power and Harmonics in Nonsinusoidal Systems
15.6.1. US MILSTD461B
•
For loads of 1kW or greater, no current harmonic magnitude may be greater than 3% of the fundamental magnitude.
•
For the nth harmonic with n > 33, the harmonic magnitude may not exceed (1/n) times the fundamental magnitude.
•
Harmonic limits are now employed by all of the US armed forces. The specific limits are often tailored to the specific application.
•
The shipboard application is a good example of the problems faced in a relatively small standalone power system having a large fraction of electronic loads.
Fundamentals of Power Electronics
32
Chapter 15: Power and Harmonics in Nonsinusoidal Systems
15.6.2. International Electrotechnical Commission Standard 555
• •
• •
First draft of their IEC555 standard:1982. It has since undergone a number of revisions. Enforcement of IEC555 is the prerogative of each individual country, and hence it has been sometimes difficult to predict whether and where this standard will actually be given the force of law. Nonetheless, IEC555 is now enforced in Europe, making it a de facto standard for commercial equipment intended to be sold worldwide. IEC555 covers a number of different types of low power equipment, with differing harmonic limits. Harmonics for equipment having an input current of up to 16A, connected to 50 or 60 Hz, 220V to 240V single phase circuits (two or three wire), as well as 380V to 415V three phase (three or four wire) circuits, are limited.
Fundamentals of Power Electronics
33
Chapter 15: Power and Harmonics in Nonsinusoidal Systems
Lowpower harmonic limits
• •
•
•
In a city environment such as a large building, a large fraction of the total power system load can be nonlinear Example: a major portion of the electrical load in a building is comprised of fluorescent lights, which present a very nonlinear characteristic to the utility system. A modern office may also contain a large number of personal computers, printers, copiers, etc., each of which may employ peak detection rectifiers. Although each individual load is a negligible fraction of the total local load, these loads can collectively become significant.
Fundamentals of Power Electronics
34
Chapter 15: Power and Harmonics in Nonsinusoidal Systems
IEC555: Class A and B
Class A: Balanced threephase equipment, and any equipment which does not fit into the other categories. This class includes low harmonic rectifiers for computer and other office equipment. These limits are given in Table 15.1, and are absolute ampere limits. Class B: Portable tools, and similar devices. The limits are equal to the Table 15.1 limits, multiplied by 1.5.
Fundamentals of Power Electronics
35
Chapter 15: Power and Harmonics in Nonsinusoidal Systems
Class A limits
Table 15.1. IEC555 Harmonic current limits, Class A and certain Class C Odd harmonics Harmonic number Maximum current 3 5 7 9 11 13 15 ≤ n ≤ 39
Fundamentals of Power Electronics
2.30A 1.14A 0.77A 0.40A 0.33A 0.21A 0.15A · (15/n)
36
Even harmonics Harmonic number Maximum curre 2 4 6 8 ≤ n ≤ 40
1.08A 0.43A 0.30A 0.23A · (8/n)
Chapter 15: Power and Harmonics in Nonsinusoidal Systems
IEC555: Class C
Class C: Lighting equipment, including dimmers and gas discharge lamps. The input current harmonics of ballasted lamps with an input power of more than 25W must meet the limits of Table 15.2, expressed as a percent of the fundamental current. If the input power is less than 25W, then Table 15.3 applies. Incandescent lamp fixtures containing phasecontrol dimmers, rated at greater than 600W, must meet the limits of Table 15.1. When testing for compliance, the dimmer must drive a ratedpower lamp, with the phase control set to a firing angle of 90˚±5˚. Incandescent lamp dimmers rated at less than 600W are not covered by the standard. Discharge lamps containing dimmers must meet the limits of both Tables 15.2 and 15.3 at maximum load. Harmonic currents at any dimming position may not exceed the maximum load harmonic currents.
Fundamentals of Power Electronics
37
Chapter 15: Power and Harmonics in Nonsinusoidal Systems
Class C limits
Table 15.2. IEC555 Harmonic current limits, certain Class C Harmonic number
Maximum current, percent of fundamental
2 3 5 7 9 11 ≤ n ≤ 39
2% (30%) · (power factor) 10% 7% 5% 3%
Fundamentals of Power Electronics
38
Chapter 15: Power and Harmonics in Nonsinusoidal Systems
IEC555: Class D Class D: Equipment not covered by Class B or C, which has a special waveshape as defined below. This class is directed at diodecapacitor peakdetection rectifiers. Equipment is placed in class D if the ac input current waveshape lies within the shaded area for at least 95% of the duration of each halfcycle. The center line of the shaded area is set to coincide with the peak of the current waveform. A Ipk sinusoid, and a typical peak detection rectifier waveform, are shown for reference; the sinusoid is not Class D but the peak detection rectifier 0.35 Ipk waveform is. The limits for Class D equipment are given in Table 15.3. 0˚
Fundamentals of Power Electronics
39
θpk – 30˚
θpk
θpk + 30˚
180˚
ωt
Chapter 15: Power and Harmonics in Nonsinusoidal Systems
Class D limits
Table 15.3. IEC555 Harmonic current limits, Class D and certain Class C Odd harmonics
Even harmonics
Harmonic number
Relative limit (mA/W)
Absolute limit (A)
Harmonic number
Relative limit (mA/W)
Absolute limit (A)
3 5 7 9
3.6 2.0 1.5 1.0
2.30A 1.14A 0.77A 0.40A
2 4
1.0 0.5
0.3A 0.15A
11 ≤ n ≤ 39
0.6 · (11/n)
0.33A
Fundamentals of Power Electronics
40
Chapter 15: Power and Harmonics in Nonsinusoidal Systems
15.6.3. IEEE/ANSI Standard 519
•
•
•
•
In 1993, the IEEE published a revised draft standard limiting the amplitudes of current harmonics, IEEE Guide for Harmonic Control and Reactive Compensation of Static Power Converters. Harmonic limits are based on the ratio of the fundamental component of the load current IL to the short circuit current at the point of common (PCC) coupling at the utility Isc. Stricter limits are imposed on large loads than on small loads. The limits are similar in magnitude to IEC555, and cover high voltage loads (of much higher power) not addressed by IEC555. Enforcement of this standard is presently up to the local utility company. The odd harmonic limits are listed in Tables 15.4 and 15.5. The limits for even harmonics are 25% of the odd harmonic limits. Dc current components and halfwave rectifiers are not allowed.
Fundamentals of Power Electronics
41
Chapter 15: Power and Harmonics in Nonsinusoidal Systems
IEEE519 current limits, low voltage systems
Table 15.4. IEEE519 Maximum odd harmonic current limits for general distribution systems, 120V through 69kV Isc/IL
n < 11
11≤n
jg(t) = i g(t)
i g(t)
0.8
Rbase V
Mode boundary: CCM occurs when
Re = 0.33R
=
base
Ts
base
i g(t)
Static input characteristics of CPM boost, with minimum slope compensation:
Li 2c (t) fs vg(t) in DCM vg(t) m a L V – vg(t) + V V v 2g(t)Ts vg(t) –1 + in CCM i c(t) + m a T s V 2LV
Re =
0 0.0
ma = V 2L 21
0.2
0.4
0.6
0.8
1.0
vg(t) V
Chapter 18: Low harmonic rectifier modeling and control
18.3 Control system modeling of high quality rectifiers
Boost converter i2(t)
ig(t) +
iac(t)
L
vg(t)
vac(t)
+
D1 Q1
vC(t)
– vcontrol(t)
vg(t) Multiplier
DC–DC Converter
C
–
ig(t)
d(t)
PWM
v(t)
va(t)
vref1(t) = kxvg(t)vcontrol(t)
+
Load v(t)
–
Rs
X
i(t)
v (t) +– err Gc(s)
Compensator and modulator
–+ vref3
Compensator
Widebandwidth output voltage controller
Widebandwidth input current controller vC(t) Compensator
Two loops: Outer lowbandwidth controller
vref2 – +
Lowbandwidth energystorage capacitor voltage controller
Inner widebandwidth controller
ECEN5807 Power Electronics 2
22
Chapter 18: Low harmonic rectifier modeling and control
18.3.1 Modeling the outer lowbandwidth control system This loop maintains power balance, stabilizing the rectifier output voltage against variations in load power, ac line voltage, and component values The loop must be slow, to avoid introducing variations in Re at the harmonics of the ac line frequency Objective of our modeling efforts: lowfrequency smallsignal model that predicts transfer functions at frequencies below the ac line frequency
ECEN5807 Power Electronics 2
23
Chapter 18: Low harmonic rectifier modeling and control
Large signal model averaged over switching period Ts
〈 ig(t)〉Ts
Ideal rectifier (LFR)
s
〈 p(t)〉T 〈 vg(t)〉T
s
+ –
〈 i2(t)〉T
+ s
C
Re (vcontrol )
〈 v(t)〉T
s
Load
– ac input
dc output vcontrol
Ideal rectifier model, assuming that inner widebandwidth loop operates ideally Highfrequency switching harmonics are removed via averaging Ac linefrequency harmonics are included in model Nonlinear and timevarying ECEN5807 Power Electronics 2
24
Chapter 18: Low harmonic rectifier modeling and control
Predictions of largesignal model Ideal rectifier (LFR)
〈 ig(t)〉Ts
If the input voltage is vg(t) = 2 vg,rms sin ωt
〈 p(t)〉T 〈 vg(t)〉T
s
+ –
Re (vcontrol )
〈 i2(t)〉T
s
+ s
C
〈 v(t)〉T
s
Load
–
Then the instantaneous power is: vg(t)
ac input
dc output vcontrol
2
2 v g,rms p(t) T = = 1 – cos 2ωt s Re(vcontrol(t)) Re(vcontrol(t)) Ts
which contains a constant term plus a secondharmonic term
ECEN5807 Power Electronics 2
25
Chapter 18: Low harmonic rectifier modeling and control
Separation of power source into its constant and timevarying components 〈 i2(t)〉Ts + V 2g,rms – cos 2 2ωt Re
V 2g,rms Re
C
〈 v(t)〉Ts
Load
– Rectifier output port The secondharmonic variation in power leads to secondharmonic variations in the output voltage and current
ECEN5807 Power Electronics 2
26
Chapter 18: Low harmonic rectifier modeling and control
Removal of even harmonics via averaging
v(t)
〈 v(t)〉T
s
〈 v(t)〉T2L
t
T2L =
ECEN5807 Power Electronics 2
1 2
27
2π = π ω ω
Chapter 18: Low harmonic rectifier modeling and control
Resulting averaged model
〈 i2(t)〉T2L + V 2g,rms Re
〈 v(t)〉T2L
C
Load
– Rectifier output port
Time invariant model Power source is nonlinear
ECEN5807 Power Electronics 2
28
Chapter 18: Low harmonic rectifier modeling and control
Perturbation and linearization The averaged model predicts that the rectifier output current is
p(t) i 2(t)
T 2L
=
v(t)
T 2L T 2L
=
v 2g,rms(t) Re(vcontrol(t)) v(t)
= f vg,rms(t), v(t) Let
T 2L
, vcontrol(t))
T 2L
with
v(t) i 2(t)
T 2L T 2L
= V + v(t)
V >> v(t)
= I 2 + i 2(t)
I 2 >> i 2(t)
vg,rms = Vg,rms + vg,rms(t) vcontrol(t) = Vcontrol + vcontrol(t)
ECEN5807 Power Electronics 2
Vg,rms >> vg,rms(t) Vcontrol >> vcontrol(t) 29
Chapter 18: Low harmonic rectifier modeling and control
Linearized result vcontrol(t) I 2 + i 2(t) = g 2vg,rms(t) + j2v(t) – r2 where df vg,rms, V, Vcontrol) g2 =
=
dvg,rms
– 1 = r2
v g,rms = V g,rms
df Vg,rms, v d v
, Vcontrol)
T 2L
=–
T 2L
dvcontrol
ECEN5807 Power Electronics 2
I2 V
v T =V 2L
df Vg,rms, V, vcontrol) j2 =
Vg,rms 2 Re(Vcontrol) V
v control = V control
30
V 2g,rms dRe(vcontrol) =– VR 2e (Vcontrol) dvcontrol
v control = V control
Chapter 18: Low harmonic rectifier modeling and control
Smallsignal equivalent circuit i2
+ r2
j2 vcontrol
g 2 vg,rms
C
v
R
– Rectifier output port
Predicted transfer functions
Controltooutput
v(s) 1 = j2 Rr 2 vcontrol(s) 1 + sC Rr 2
Linetooutput
v(s) 1 = g 2 Rr 2 vg,rms(s) 1 + sC Rr 2
ECEN5807 Power Electronics 2
31
Chapter 18: Low harmonic rectifier modeling and control
Model parameters Tabl e 18. 1
Smallsignal model parameters for several types of rectifier control schemes
Controller type
g2
j2
r2
Average current control with feedforward, Fig. 18.9
0
Pav VVcontrol
V2 Pav
Currentprogrammed control, Fig. 18.10
2Pav VVg,rms
Pav VVcontrol
V2 Pav
Nonlinearcarrier charge control of boost rectifier, Fig. 18.14
2Pav VVg,rms
Pav VVcontrol
V2 2Pav
Boost with hysteretic control, Fig. 18.13(b)
2Pav VVg,rms
Pav VT on
V2 Pav
DCM buck–boost, flyback, SEPIC, or Cuk converters
2Pav VVg,rms
2Pav VD
V2 Pav
ECEN5807 Power Electronics 2
32
Chapter 18: Low harmonic rectifier modeling and control
Constant power load ig(t) 〈 pac(t)〉Ts
+
iac(t)
vg(t)
vac(t)
pload(t) = VI = Pload
i2(t)
+
+
Re
C
–
vC(t)
Pload V + –
v(t) load –
– Ideal rectifier (LFR)
Energy storage capacitor
i(t)
Dcdc converter
Rectifier and dcdc converter operate with same average power Incremental resistance R of constant power load is negative, and is 2 V R=– Pav
which is equal in magnitude and opposite in polarity to rectifier incremental output resistance r2 for all controllers except NLC
ECEN5807 Power Electronics 2
33
Chapter 18: Low harmonic rectifier modeling and control
Transfer functions with constant power load When r2 = –R, the parallel combination r2  R becomes equal to zero. The smallsignal transfer functions then reduce to
j v(s) = 2 vcontrol(s) sC
g v(s) = 2 vg,rms(s) sC
ECEN5807 Power Electronics 2
34
Chapter 18: Low harmonic rectifier modeling and control
18.3.2 Modeling the inner widebandwidth average current controller Averaged (but not linearized) boost converter model, Fig. 7.42: L 〈i1(t)〉T
〈i(t)〉T 〈vg(t)〉Ts
+
+
s
s
+ –
〈v1(t)〉Ts
+ –
〈i2(t)〉Ts
〈v2(t)〉Ts –
C
R
〈v(t)〉T
s
–
Averaged switch network
In Chapter 7, we perturbed and linearized using the assumptions
vg(t)
= Vg + vg(t)
d(t) = D + d(t) ⇒ d'(t) = D' – d(t) i(t) T = i 1(t) T = I + i(t) s
v(t) v1(t) i 2(t)
ECEN5807 Power Electronics 2
Ts
Ts Ts Ts
s
= v2(t)
Ts
= V + v(t)
= V1 + v1(t) = I 2 + i 2(t) 35
Problem: variations in vg, i1 , and d are not small. So we are faced with the design of a control system that exhibits significant nonlinear timevarying behavior.
Chapter 18: Low harmonic rectifier modeling and control
Linearizing the equations of the boost rectifier When the rectifier operates near steadystate, it is true that
v(t)
Ts
= V + v(t)
with
v(t) R0 • PRC can produce M approaching infinity, provided output current is limited to value less than Vg / R0 Fundamentals of Power Electronics
37
Chapter 19: Resonant Conversion
19.3 Exact characteristics of the series and parallel resonant dcdc converters Define
f0 f0 < fs < k+1 k
1 J> γ γ
Fundamentals of Power Electronics
45
Chapter 19: Resonant Conversion
Steadystate solution: type k DCM, even k J = 2k γ Conditions for operation in type k DCM, even k :
f0 fs < k 1 >M> 1 k–1 k+1 Ig = gV
gyrator model, SRC operating in an even DCM:
+
g
Vg
46
+ V
g = 2k γR0
– Fundamentals of Power Electronics
Ig = gVg
– Chapter 19: Resonant Conversion
Control plane characteristics, SRC 1
Q = 0.2
0.9 Q = 0.2
0.8 0.35
M = V / Vg
0.7 0.5
0.35
0.6
0.75
0.5
0.3 0.2 0.1 0
1
0.5
0.4
0.75 1 1.5 2 3.5 5 10 Q = 20
0
1.5 2 3.5 5 10 Q = 20
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
1.8
2
F = f s / f0 Fundamentals of Power Electronics
47
Chapter 19: Resonant Conversion
Mode boundaries, SRC k = 1 DCM 1 0.9 0.8 0.7
0.4
k = 3 DCM
0.3
k=4 DCM
0.2
etc.
0.1
k = 1 CCM
k = 0 CCM
k = 2 CCM
k = 2 DCM
0.5
k = 3 CCM
M
0.6
0 0
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
1.8
2
F Fundamentals of Power Electronics
48
Chapter 19: Resonant Conversion
Output characteristics, SRC above resonance
6
F = 1.05 F = 1.07
5
F = 1.10
4
J
F = 1.01
3
F = 1.15
2
F = 1.30 1
0 0
0.2
0.4
0.6
0.8
1
M
Fundamentals of Power Electronics
49
Chapter 19: Resonant Conversion
Output characteristics, SRC below resonance F = 1.0
J F = .93
F = .96
3
F = .90 2.5
F = .85
2
1.5
F = .75 4 π
k = 1 CCM
F = .5
2 π
k = 1 DCM
1
k = 2 DCM
F = .25 F = .1 0 0
0.2
0.4
0.6
0.8
1
M
Fundamentals of Power Electronics
50
Chapter 19: Resonant Conversion
19.3.2 Exact characteristics of the parallel resonant converter
Q1
D1
Q3
L
1:n
D3
D7
Vg
+ –
C
R D8
Q2
D2
Q4
+
D5
D6
V –
D4
Normalized load voltage and current:
InR0 J= Vg
M= V nVg Fundamentals of Power Electronics
51
Chapter 19: Resonant Conversion
Parallel resonant converter in CCM vs(t)
CCM closedform solution
Vg γ
sin (ϕ) M = 2γ ϕ – γ cos 2
γ ω0t
– Vg iL(t)
– cos – 1 cos
γ γ + J sin 2 2
for 0 < γ < π (
+ cos – 1 cos
γ γ + J sin 2 2
for π < γ < 2π
ϕ=
vC(t)
vC(t) V=
Fundamentals of Power Electronics
52
vC(t)
Ts
Chapter 19: Resonant Conversion
Parallel resonant converter in DCM vC(t)
Mode boundary J > J crit(γ) J < J crit(γ)
for DCM for CCM
J crit(γ) = – 1 sin (γ) + 2
sin 2
γ + 1 sin 2 γ 2 4
DCM equations
ω0t
D5 D8 D6 D7
D6 D7 α
M C0 = 1 – cos (β) J L0 = J + sin (β) cos (α + β) – 2 cos (α) = –1 iL(t) – sin (α + β) + 2 sin (α) + (δ – α) = 2J β+δ=γ M = 1 + 2γ (J – δ)
δ
D5 D8
D5 D8
D5 D8 D6 D7
D6 D7
β γ
I
ω0t
(require iteration) –I
Fundamentals of Power Electronics
53
Chapter 19: Resonant Conversion
Output characteristics of the PRC 3.0 2.5 F = 0.51 0.6
2.0
0.7
J
1.5
0.8 0.9 1.0
1.0 0.5
1.5
1.3
1.2
1.1
F=2 0.0 0.0
0.5
1.0
1.5
2.0
2.5
M Solid curves: CCM Fundamentals of Power Electronics
Shaded curves: DCM 54
Chapter 19: Resonant Conversion
Control characteristics of the PRC with resistive load 3.0 Q=5 2.5
M = V/Vg
2.0
1.5
Q=2
1.0
Q=1
Q = 0.5
0.5
Q = 0.2 0.0 0.5
1.0
1.5
2.0
2.5
3.0
fs /f0
Fundamentals of Power Electronics
55
Chapter 19: Resonant Conversion
19.4 Soft switching Soft switching can mitigate some of the mechanisms of switching loss and possibly reduce the generation of EMI Semiconductor devices are switched on or off at the zero crossing of their voltage or current waveforms: Zerocurrent switching: transistor turnoff transition occurs at zero current. Zerocurrent switching eliminates the switching loss caused by IGBT current tailing and by stray inductances. It can also be used to commutate SCR’s. Zerovoltage switching: transistor turnon transition occurs at zero voltage. Diodes may also operate with zerovoltage switching. Zerovoltage switching eliminates the switching loss induced by diode stored charge and device output capacitances. Zerovoltage switching is usually preferred in modern converters. Zerovoltage transition converters are modified PWM converters, in which an inductor charges and discharges the device capacitances. Zerovoltage switching is then obtained. Fundamentals of Power Electronics
56
Chapter 19: Resonant Conversion
19.4.1 Operation of the full bridge below resonance: Zerocurrent switching Series resonant converter example + Q1
vds1(t)
D1
Q3
L +
iQ1(t) – Vg
C
D3
+ –
vs(t) Q2
D2
Q4
–
is(t)
D4
Operation below resonance: input tank current leads voltage Zerocurrent switching (ZCS) occurs
Fundamentals of Power Electronics
57
Chapter 19: Resonant Conversion
Tank input impedance Operation below resonance: tank input impedance Zi is dominated by tank capacitor. ∠Zi is positive, and tank input current leads tank input voltage.
 Zi 
1 ωC
ωL
R0
Re
f0 Qe = R0 /Re
Zero crossing of the tank input current waveform is(t) occurs before the zero crossing of the voltage vs(t). Fundamentals of Power Electronics
58
Chapter 19: Resonant Conversion
Switch network waveforms, below resonance Zerocurrent switching vs1(t)
Vg
vs(t)
+
t
Q1
vds1(t)
D1
Q3
L
C
D3
+
iQ1(t) – – Vg
vs(t)
is(t) Q2
Ts + tβ 2
tβ
D2
Q4
–
is(t)
D4
t
Ts 2
Conduction sequence: Q1–D1–Q2–D2 Conducting devices:
Q1 Q4
D1 D4
Q2 Q3
“Soft” “Hard” “Hard” turnon of turnoff of turnon of Q1, Q4 Q2, Q3 Q1, Q4
Fundamentals of Power Electronics
Q1 is turned off during D1 conduction interval, without loss
D2 D3 “Soft” turnoff of Q2, Q3
59
Chapter 19: Resonant Conversion
ZCS turnon transition: hard switching vds1(t)
Vg + Q1
vds1(t)
D1
Q3
L +
iQ1(t) –
t ids(t)
vs(t) Q2
Ts + tβ 2
tβ Conducting devices:
C
D3
Q1 Q4
D1 D4
Ts 2
“Soft” “Hard” turnon of turnoff of Q1, Q4 Q 1, Q 4
Fundamentals of Power Electronics
D2
Q4
–
is(t)
D4
t Q2 Q3
D2 D3
Q1 turns on while D2 is conducting. Stored charge of D2 and of semiconductor output capacitances must be removed. Transistor turnon transition is identical to hardswitched PWM, and switching loss occurs. 60
Chapter 19: Resonant Conversion
19.4.2 Operation of the full bridge below resonance: Zerovoltage switching Series resonant converter example + Q1
vds1(t)
D1
Q3
L +
iQ1(t) – Vg
C
D3
+ –
vs(t) Q2
D2
Q4
–
is(t)
D4
Operation above resonance: input tank current lags voltage Zerovoltage switching (ZVS) occurs
Fundamentals of Power Electronics
61
Chapter 19: Resonant Conversion
Tank input impedance Operation above resonance: tank input impedance Zi is dominated by tank inductor. ∠Zi is negative, and tank input current lags tank input voltage.
 Zi 
1 ωC
ωL
R0
Re
f0 Qe = R0 /Re
Zero crossing of the tank input current waveform is(t) occurs after the zero crossing of the voltage vs(t).
Fundamentals of Power Electronics
62
Chapter 19: Resonant Conversion
Switch network waveforms, above resonance Zerovoltage switching
vs1(t)
Vg
vs(t)
+ Q1
t
vds1(t)
D1
Q3
L
C
D3
+
iQ1(t) –
vs(t)
– Vg is(t) Q2
tα
D2
Q4
–
is(t)
D4
t
Ts 2
Conduction sequence: D1–Q1–D2–Q2 Conducting D1 devices: D 4 “Soft” turnon of Q1, Q4
Q1 Q4
D2 D3
Q1 is turned on during D1 conduction interval, without loss
Q2 Q3
“Hard” “Soft” “Hard” turnoff of turnon of turnoff of Q1, Q4 Q2, Q3 Q2, Q3
Fundamentals of Power Electronics
63
Chapter 19: Resonant Conversion
ZVS turnoff transition: hard switching? vds1(t)
Vg + Q1
vds1(t)
D1
Q3
L +
iQ1(t) –
t
C
D3
vs(t)
ids(t) Q2
tα Conducting D1 devices: D 4 “Soft” turnon of Q1, Q4
Q1 Q4
Ts 2
D2
Q4
–
is(t)
D4
t D2 D3
“Hard” turnoff of Q1, Q4
Fundamentals of Power Electronics
Q2 Q3
When Q1 turns off, D2 must begin conducting. Voltage across Q1 must increase to Vg. Transistor turnoff transition is identical to hardswitched PWM. Switching loss may occur (but see next slide). 64
Chapter 19: Resonant Conversion
Soft switching at the ZVS turnoff transition + Q1 D1 C leg
Vg
Q3
vds1(t)
Cleg
–
D3
is(t) +
+ –
vs(t) Q2
D2
Cleg
Cleg D4
– Q4
vds1(t)
Conducting devices:
Turn off Q1, Q4
X D2 D3
t
• Introduce delay between turnoff of Q1 and turnon of Q2.
So zerovoltage switching exhibits low switching loss: losses due to diode stored charge and device output capacitances are eliminated.
Commutation interval
Fundamentals of Power Electronics
to remainder of converter
• Introduce small capacitors Cleg across each device (or use device output capacitances).
Tank current is(t) charges and discharges Cleg. Turnoff transition becomes lossless. During commutation interval, no devices conduct.
Vg
Q1 Q4
L
65
Chapter 19: Resonant Conversion
19.4.3 The zerovoltage transition converter Basic version based on fullbridge PWM buck converter Q3
Q1 D1 C leg
Vg
+ –
Cleg
ic(t)
Lc
D3
+
Q2
D2
Cleg
v2(t)
Cleg D4 Q4
–
v2(t)
Vg
• Can obtain ZVS of all primaryside MOSFETs and diodes
Can turn on Q1 at zero voltage
• Secondaryside diodes switch at zerocurrent, with loss • Phaseshift control Fundamentals of Power Electronics
Conducting devices:
Q2
Turn off Q2 66
X D1
t
Commutation interval
Chapter 19: Resonant Conversion
19.5 Loaddependent properties of resonant converters Resonant inverter design objectives: 1. Operate with a specified load characteristic and range of operating points • With a nonlinear load, must properly match inverter output characteristic to load characteristic
2. Obtain zerovoltage switching or zerocurrent switching • Preferably, obtain these properties at all loads • Could allow ZVS property to be lost at light load, if necessary
3. Minimize transistor currents and conduction losses • To obtain good efficiency at light load, the transistor current should scale proportionally to load current (in resonant converters, it often doesn’t!) Fundamentals of Power Electronics
67
Chapter 19: Resonant Conversion
Topics of Discussion Section 19.5
Inverter output iv characteristics Two theorems • Dependence of transistor current on load current • Dependence of zerovoltage/zerocurrent switching on load resistance • Simple, intuitive frequencydomain approach to design of resonant converter
Examples and interpretation • Series • Parallel • LCC
Fundamentals of Power Electronics
68
Chapter 19: Resonant Conversion
Inverter output characteristics transfer function H(s)
Let H∞ be the opencircuit (RÕ∞) transfer function: vo( jω) vi( jω)
io(t)
ii(t) sinusoidal source + vi(t) –
= H ∞( jω)
resonant network
+ Zo v (t) o
Zi purely reactive
–
resistive load R
R→∞
and let Zo0 be the output impedance (with vi Õ shortcircuit). Then, R vo( jω) = H ∞( jω) vi( jω) R + Z o0( jω)
This result can be rearranged to obtain vo
2
+ io
2
Z o0
2
= H∞
2
vi
2
The output voltage magnitude is: vo
2
= vov *o =
H∞
2
1 + Z o0
with
vi 2
Hence, at a given frequency, the output characteristic (i.e., the relation between vo and io) of any resonant inverter of this class is elliptical.
2
/ R2
R = vo / io
Fundamentals of Power Electronics
69
Chapter 19: Resonant Conversion
Inverter output characteristics
General resonant inverter output characteristics are elliptical, of the form 2
 io  I sc =
vi
d oa l ed  tch  Z o0 a m = R
Z o0
2
vo io + 2 =1 V 2oc I sc
with
H∞
inverter output characteristic
Voc = H ∞
I sc =
H∞
I sc 2
vi Voc 2
vi
Voc = H ∞
vi
 vo 
Z o0
This result is valid provided that (i) the resonant network is purely reactive, and (ii) the load is purely resistive. Fundamentals of Power Electronics
70
Chapter 19: Resonant Conversion
Matching ellipse to application requirements
Electrosurgical generator
 io 
 io 
50Ω
Electronic ballast
inverter characteristic inverter characteristic
2A
40
0W
d
lamp characteristic
loa
t
ma
2kV
 vo 
Fundamentals of Power Electronics
d che
71
 vo 
Chapter 19: Resonant Conversion
Input impedance of the resonant tank network Transfer function H(s)
Z (s) R 1 + o0 R Z o0(s) Z i(s) = Z i0(s) = Z i∞(s) Z o∞(s) 1+ R 1 + Z o∞(s) R 1+
where v Z i0 = i ii
R→0
Z i∞ =
Fundamentals of Power Electronics
vi ii
Effective sinusoidal source + vs1(t) –
72
Resonant network Zi
+ Zo
Purely reactive
Z o0 =
R→∞
i(t)
is(t)
vo – io
vi → short circuit
Z o∞ =
v(t) –
vo – io
Effective resistive load R
vi → open circuit
Chapter 19: Resonant Conversion
Other relations
If the tank network is purely reactive, then each of its impedances and transfer functions have zero real parts: Z = – Z*
Reciprocity Z i0 Z o0 = Z i∞ Z o∞
i0
Z i∞ = – Z Z o0 = – Z Z o∞ = – Z H∞ = – H
Tank transfer function H(s) =
where
H ∞(s) 1+ R Z o0 H∞ = H∞
2
vo(s) vi(s) = Z o0
Fundamentals of Power Electronics
i0 * i∞ * o0 * o∞ * ∞
Hence, the input impedance magnitude is 2 R 1+ Z o0 2 2 * Z i = Z iZ i = Z i0 2 R 1+ Z o∞
R→∞
1 – 1 Z i0 Z i∞ 73
2
2
Chapter 19: Resonant Conversion
Zi0 and Zi∞ for 3 common inverters Series
L
1 ωC
Cs
Z i0(s) = sL + 1 sC s
 Zi∞ 
s
ωL
Zo
Zi
 Zi0 
Z i∞(s) = ∞ f
Parallel
1 ωC
L
Z i0(s) = sL
p
ωL
Zi
Cp
 Zi∞ 
Zo
Z i∞(s) = sL + 1 sC p
 Zi0  f
LCC L
1 ωC + s
Cs
1 ωC
Zi
Cp
Z i0(s) = sL + 1 sC s
p
1 ωC s
Zo
ωL  Zi∞ 
Z i∞(s) = sL + 1 + 1 sC p sC s
 Zi0  f
Fundamentals of Power Electronics
74
Chapter 19: Resonant Conversion
A Theorem relating transistor current variations to load resistance R
Theorem 1: If the tank network is purely reactive, then its input impedance  Zi  is a monotonic function of the load resistance R. l
l l
l
So as the load resistance R varies from 0 to ∞, the resonant network input impedance  Zi  varies monotonically from the shortcircuit value  Zi0  to the opencircuit value  Zi∞ . The impedances  Zi∞  and  Zi0  are easy to construct. If you want to minimize the circulating tank currents at light load, maximize  Zi∞ . Note: for many inverters,  Zi∞  <  Zi0  ! The noload transistor current is therefore greater than the shortcircuit transistor current.
Fundamentals of Power Electronics
75
Chapter 19: Resonant Conversion
Proof of Theorem 1 Previously shown:
Zi
2
= Z i0
2
1+ R Z o0 1+ R Z o∞
á Differentiate: 2
d Zi dR 2
á Derivative has roots at:
= 2 Z i0
2
2
–
1 Z o∞
2 R 1+ Z o∞
2
R
2 2
So the resonant network input impedance is a monotonic function of R, over the range 0 < R < ∞.
(i) R = 0 (ii) R = ∞ (iii) Z o0 = Z o∞ , or Z i0 = Z i∞
Fundamentals of Power Electronics
2
1 Z o0
In the special case  Zi0  =  Zi∞ ,  Zi  is independent of R.
76
Chapter 19: Resonant Conversion
Example:  Zi  of LCC
 Zi 
1 ωC + s
1 ωC
f0
f∞
p
rea
1 ωC
inc
rea s
ing
R
ωL inc
s
gR
sin
• for f < f m,  Zi  increases with increasing R . • for f > f m,  Zi  decreases with increasing R . • at a given frequency f,  Zi  is a monotonic function of R. • It’s not necessary to draw the entire plot: just construct  Zi0  and  Zi∞ .
fm
f
Fundamentals of Power Electronics
77
Chapter 19: Resonant Conversion
Discussion: LCC
 Zi0  and  Zi∞  both represent series resonant impedances, whose Bode diagrams are easily constructed.  Zi0  and  Zi∞  intersect at frequency fm.
1 ωC + s
 Zi 
LCC example f0
1 ωC
f∞
p
ωL
1 ωC
s
 Zi∞   Zi0 
For f < fm
1 2π LC s 1 f∞ = 2π LC sC p 1 fm = 2π LC s2C p f0 =
fm
then  Zi0  <  Zi∞  ; hence transistor current decreases as load current decreases For f > fm
f
then  Zi0  >  Zi∞  ; hence transistor current increases as load current decreases, and transistor current is greater than or equal to shortcircuit current for all R Fundamentals of Power Electronics
L
Zi∞
78
Cs
Cp
L
Zi0
Cs
Cp
Chapter 19: Resonant Conversion
Discussion series and parallel
Series
L
1 ωC
Cs
• Noload transistor current = 0, both above and below resonance.
 Zi∞ 
s
ωL
Zo
Zi
 Zi0 
f
Parallel
• Above resonance: noload transistor current is greater than shortcircuit transistor current. ZVS.
1 ωC
L
p
ωL
Zi
Cp
 Zi∞ 
Zo  Zi0 
f
LCC L
1 ωC + s
Cs
• ZCS below resonance, ZVS above resonance
1 ωC
• Below resonance: noload transistor current is less than shortcircuit current (for f f∞, ZVS occurs for all R. For f < f0, ZCS occurs for all R. For f0 < f < f∞, ZVS occurs for R< Rcrit, and ZCS occurs for R> Rcrit. Note that R =  Zo0  corresponds to operation at matched load with maximum output power. The boundary is expressed in terms of this matched load impedance, and the ratio Zi∞ / Zi0.
 Zi 
1 ωC + s
1 ωC
ZCS ZCS: R>Rcrit ZVS for all R ZVS: R  Zi0 . 8. The output characteristics of all resonant inverters considered here are elliptical, and are described completely by the opencircuit transfer function magnitude  H∞ , and the output impedance  Zo0 . These quantities can be chosen to match the output characteristics to the application requirements.
Fundamentals of Power Electronics
87
Chapter 19: Resonant Conversion
Chapter 20
QuasiResonant Converters
Introduction 20.1
The zerocurrentswitching quasiresonant switch cell 20.1.1 20.1.2 20.1.3
20.2
Waveforms of the halfwave ZCS quasiresonant switch cell The average terminal waveforms The fullwave ZCS quasiresonant switch cell
Resonant switch topologies 20.2.1 20.2.2 20.2.3
The zerovoltageswitching quasiresonant switch The zerovoltageswitching multiresonant switch Quasisquarewave resonant switches
20.3
Ac modeling of quasiresonant converters
20.4
Summary of key points
Fundamentals of Power Electronics
1
Chapter 20: QuasiResonant Converters
The resonant switch concept A quite general idea: 1. PWM switch network is replaced by a resonant switch network 2. This leads to a quasiresonant version of the original PWM converter
Example: realization of the switch cell in the buck converter i1(t)
i2(t) +
vg(t) + –
v1(t)
L
i(t) +
+ Switch cell
–
Fundamentals of Power Electronics
v2(t)
C
R
v(t) –
–
2
PWM switch cell
i1(t) +
i2(t) +
v1(t)
v2(t)
–
–
Chapter 20: QuasiResonant Converters
Two quasiresonant switch cells
i1(t) +
D1
Lr
Q1
i2r(t)
+
i2(t) +
i1(t) +
D1
Lr
i2(t) +
i2r(t) + Q1
v1(t)
v1r(t)
–
–
D2
Cr
v2(t)
v1(t)
v1r(t)
–
–
–
Insert either of the above switch cells into the buck converter, to obtain a ZCS quasiresonant version of the buck converter. Lr vg(t) and Cr are small in value, and their resonant frequency f0 is greater than the switching frequency fs. ω0 1 Fundamentals of Power Electronics
–
Fullwave ZCS quasiresonant switch cell
Halfwave ZCS quasiresonant switch cell
2π L rC r
v2(t)
Cr
Switch network
Switch network
f0 =
D2
=
i1(t)
i2(t) +
+ –
v1(t) –
L
i(t) +
+ Switch cell
v2(t) –
C
R
v(t) –
2π 3
Chapter 20: QuasiResonant Converters
20.1 The zerocurrentswitching quasiresonant switch cell
Tank inductor Lr in series with transistor: transistor switches at zero crossings of inductor current waveform Tank capacitor Cr in parallel with diode D2 : diode switches at zero crossings of capacitor voltage waveform
i1(t) +
Q1
i2r(t)
+
v1(t)
v1r(t)
–
–
D2
Cr
i2(t) + v2(t) –
Switch network Halfwave ZCS quasiresonant switch cell
Twoquadrant switch is required:
Halfwave: Q1 and D1 in series, transistor turns off at first zero crossing of current waveform Fullwave: Q1 and D1 in parallel, transistor turns off at second zero crossing of current waveform Performances of halfwave and fullwave cells differ significantly. Fundamentals of Power Electronics
D1
Lr
4
i1(t) +
D1
Lr
i2r(t) +
i2(t) +
Q1 v1(t)
v1r(t)
–
–
D2
Cr
v2(t) –
Switch network Fullwave ZCS quasiresonant switch cell
Chapter 20: QuasiResonant Converters
Averaged switch modeling of ZCS cells It is assumed that the converter filter elements are large, such that their switching ripples are small. Hence, we can make the small ripple approximation as usual, for these elements:
i 2(t) ≈ i 2(t) v1(t) ≈ v1(t)
Ts Ts
In steady state, we can further approximate these quantities by their dc values:
i 2(t) ≈ I 2 v1(t) ≈ V1
Modeling objective: find the average values of the terminal waveforms 〈 v2(t) 〉Ts and 〈 i1(t) 〉Ts
Fundamentals of Power Electronics
5
Chapter 20: QuasiResonant Converters
The switch conversion ratio µ Lr
A generalization of the duty cycle d(t) The switch conversion ratio µ is the ratio of the average terminal voltages of the switch network. It can be applied to nonPWM switch networks. For the CCM PWM case, µ = d. If V/Vg = M(d) for a PWM CCM converter, then V/Vg = M(µ) for the same converter with a switch network having conversion ratio µ.
Q1
i2r(t) +
+ 〈 v1(t)〉Ts
+ –
v1r(t)
〈 i2(t)〉Ts
v2(t) Cr
D2
–
– Switch network
Halfwave ZCS quasiresonant switch cell
i 2(t) ≈ i 2(t) v1(t) ≈ v1(t)
Ts
µ=
Ts
v2(t) v1r(t)
Ts Ts
=
i 1(t) i 2r(t)
Ts Ts
In steady state:
Generalized switch averaging, and µ, are defined and discussed in Section 10.3.
Fundamentals of Power Electronics
D1
i1(t)
i 2(t) ≈ I 2 v1(t) ≈ V1 6
µ=
V2 I 1 = V1 I 2
Chapter 20: QuasiResonant Converters
20.1.1 Waveforms of the halfwave ZCS quasiresonant switch cell
The halfwave ZCS quasiresonant switch cell, driven by the terminal quantities 〈 v1(t)〉 Ts and 〈 i2(t)〉 Ts.
Waveforms: i1(t) I2
Lr
〈 v1(t)〉Ts
+ –
i1(t) +
D1
Q1
v1r(t)
V1 Lr
i2r(t) + D2
–
Subinterval:
〈 i2(t)〉Ts
v2(t) Cr
1
2
3
θ = ω0t
4
v2(t) Vc1
–
–
Switch network α
Halfwave ZCS quasiresonant switch cell
β
I2 Cr
δ
ξ
X
D2
ω0Ts
Each switching period contains four subintervals Fundamentals of Power Electronics
7
Conducting devices:
Q1 D1 D2
Q1 D1
Chapter 20: QuasiResonant Converters
Subinterval 1 Diode D2 is initially conducting the filter inductor current I2. Transistor Q1 turns on, and the tank inductor current i1 starts to increase. So all semiconductor devices conduct during this subinterval, and the circuit reduces to: Lr
Circuit equations: di 1(t) V1 with i1(0) = 0 = L dt r Solution: where
i1(t)
i 1(t) =
R0 =
V1 V t = ω 0t 1 Lr R0
Lr Cr
+ V1
+ –
v2(t)
This subinterval ends when diode D2 becomes reversebiased. This occurs at time ω0t = α, when i1(t) = I2.
I2
–
i 1(α) = α Fundamentals of Power Electronics
8
V1 = I2 R0
α=
I 2R0 V1
Chapter 20: QuasiResonant Converters
Subinterval 2 Diode D2 is off. Transistor Q1 conducts, and the tank inductor and tank capacitor ring sinusoidally. The circuit reduces to: Lr
i1(t) +
V1
+ –
v2(t)
Cr
I2
The circuit equations are
di 1(ω0t) = V1 – v2(ω0t) dt dv (ω t) C r 2 0 = i 1(ω0t) – I 2 dt
Fundamentals of Power Electronics
i 1(ω0t) = I 2 +
V1 sin ω0t – α R0
v2(ω0t) = V1 1 – cos ω0t – α
ic(t)
–
Lr
The solution is
v2(α) = 0 i 1(α) = I 2
9
The dc components of these waveforms are the dc solution of the circuit, while the sinusoidal components have magnitudes that depend on the initial conditions and on the characteristic impedance R0.
Chapter 20: QuasiResonant Converters
Subinterval 2 continued
Peak inductor current:
I 1pk = I 2 +
V1 R0
i 1(ω0t) = I 2 +
V1 sin ω0t – α R0
v2(ω0t) = V1 1 – cos ω0t – α
This subinterval ends at the first zero crossing of i1(t). Define β = angular length of subinterval 2. Then
i 1(α + β) = I 2 +
V1 sin β = 0 R0
I2
V1 Lr
Subinterval:
1 α
2 β
3 δ
Must use care to select the correct branch of the arcsine function. Note (from the i1(t) waveform) that β > π.
10
θ = ω0t
4 ξ
ω0Ts
Hence
β = π + sin – 1
I 2 R0 sin β = – V1
Fundamentals of Power Electronics
i1(t)
I 2 R0 V1
– π < sin – 1 x ≤ π 2 2
I2