Applied Science, Faculty of
Electrical and Computer Engineering, Department of
DSpace
UBCV
Arshadi, Sayed Abbas
Doctor of Philosophy - PhD
University of British Columbia
Resonant power converters have gained popularity in many application, especially where there are strict requirements on efficiency, power density, and Electromagnetic Compability (EMC). Among different types of resonant converters, LLC and CLLC resonant converters have attracted more attention for uni-directional and bi-directional applications, respectively. This is due to the unique features of these types of converters, such as soft switching. Half-bridge and full-bridge resonant structures have are suitable for low to medium power levels, and three-phase structures are considered for higher power levels. Three-phase resonant converters provide high efficiency, low current ripple, good thermal distribution, and small filter and heat sink size (high power density), that makes them an attractive option for high power battery charger applications. Although there has been substantial research and development on single-phase LLC and CLLC resonant converters for low to medium power levels, the literature investigating three-phase structures has been very limited. Issues to be investigated for three phase resonant converters are: 1) passive components tolerances, which leads to unbalanced behavior of the converter, 2) development of a balancing technique to mitigate the unbalanced behavior of the converter, 3) the natural limited gain range of resonant converter structure, 4) poor light load efficiency of the three-phase resonant converter and possible phase shedding strategies to improve it, and also to increase the reliability of the system. In this work, methods to use three-phase LLC and CLLC resonant converters more efficiently are investigated. This includes: 1) investigation of imbalances in three-phase LLC converters and proposing a novel strategy referred to as Trigonometric Current Balancing (TCB) technique to mitigate the unbalanced behavior of the converter under sever unbalanced situation, 2) investigating phase-shedding techniques and strategies to improve the operation of the converter at light load conditions and extend the gain range of the converter, and 3) investigating the behavior of the three-phase CLLC resonant converter for bi-directional power applications.
Three-Phase DC-DC Resonant Converters for High-Power Battery ChargingApplicationsbySayed Abbas ArshadiM. Eng., Isfahan University of Technology, 2012B. Eng., Malik-Ashtar University of Technology, 2009A THESIS SUBMITTED IN PARTIAL FULFILLMENTOF THE REQUIREMENTS FOR THE DEGREE OFDoctor of PhilosophyinTHE FACULTY OF GRADUATE AND POSTDOCTORAL STUDIES(Electrical & Computer Engineering)The University of British Columbia(Vancouver)November 2022© Sayed Abbas Arshadi, 2022The following individuals certify that they have read, and recommend to the Faculty of Graduate and Post-doctoral Studies for acceptance, the dissertation entitled:Three-Phase DC-DC Resonant Converters for High-Power Battery Charging Applicationssubmitted by Sayed Abbas Arshadi in partial fulfillment of the requirements forthe degree of Doctor of Philosophyin Electrical and Computer EngineeringExamining Committee:Dr. Martin Ordonez, Professor, Electrical and Computer Engineering, UBCSupervisorDr. Wilson Eberle, Associate Professor, Electrical and Computer Engineering, UBC OkanaganSupervisory Committee MemberDr. Deepak Gautam, Sr. Power Electronics Engineer, Delta-Q TechnologiesSupervisory Committee MemberDr. Alireza Nojeh, Professor, Electrical and Computer Engineering, UBCUniversity ExaminerDr. Mauricio Ponga de la Torre, Assistant Professor, Mechanical Engineering, UBCUniversity ExaminerAdditional Supervisory Committee Members:Dr. Juri Jatskevich, Professor, Electrical and Computer Engineering, UBCSupervisory Committee MemberiiAbstractResonant power converters have gained popularity in many application, especially where there are strict re-quirements on efficiency, power density, and Electromagnetic Compability (EMC). Among different typesof resonant converters, LLC and CLLC resonant converters have attracted more attention for uni-directionaland bi-directional applications, respectively. This is due to the unique features of these types of converters,such as soft switching. Half-bridge and full-bridge resonant structures have are suitable for low to mediumpower levels, and three-phase structures are considered for higher power levels. Three-phase resonant con-verters provide high efficiency, low current ripple, good thermal distribution, and small filter and heat sinksize (high power density), that makes them an attractive option for high power battery charger applications.Although there has been substantial research and development on single-phase LLC and CLLC resonantconverters for low to medium power levels, the literature investigating three-phase structures has been verylimited. Issues to be investigated for three phase resonant converters are: 1) passive components tolerances,which leads to unbalanced behavior of the converter, 2) development of a balancing technique to mitigate theunbalanced behavior of the converter, 3) the natural limited gain range of resonant converter structure, 4) poorlight load efficiency of the three-phase resonant converter and possible phase shedding strategies to improveit, and also to increase the reliability of the system. In this work, methods to use three-phase LLC andCLLC resonant converters more efficiently are investigated. This includes: 1) investigation of imbalances inthree-phase LLC converters and proposing a novel strategy referred to as Trigonometric Current Balancing(TCB) technique to mitigate the unbalanced behavior of the converter under sever unbalanced situation, 2)investigating phase-shedding techniques and strategies to improve the operation of the converter at light loadconditions and extend the gain range of the converter, and 3) investigating the behavior of the three-phaseCLLC resonant converter for bi-directional power applications.iiiLay SummaryBattery chargers are the power processing stage between energy sources and batteries. In order to be able tocope with newer technological requirements, higher efficiency, higher power density, and enhanced reliabil-ity are expected from this type of power converter. The purpose of this research is to increase the efficiencyof power converters in high power Electric Vehicle (Electric Vehicle (EV)) battery charging applications.Compact size, high efficiency, and high reliability are specific requirements for high power battery chargers.This has been done by efficiently adopting three-phase DC-DC resonant configurations. An analysis on theconverter is done to better understand its behavior. New techniques and modulations are proposed that sup-ports requirements for battery charging. In another part of this work, three-phase CLLC resonant convertersare also studied for bi-directional battery charging applications. In this work, the behavior of the converteris analyzed and the effectiveness of the proposed current-sharing technique is verified.ivPrefaceThis work is based on the research performed at the Department of Electrical and Computer Engineering ofthe University of British Columbia by Sayed Abbas Arshadi, under the supervision of Prof. Martin Ordonez.Chapter 1 contains modified portions of text from all the below-listed publications. Portions of Chapter 2were published in IEEE Transactions on Power Electronics [1]. It is worth mentioning that this paper was ahighlighted paper in the March 2019 Issue of the journal:• S. A. Arshadi, M. Ordonez, W. Eberle, M. A. Saket, M. Craciun and C. Botting, ”Unbalanced Three-Phase LLC Resonant Converters: Analysis and Trigonometric Current Balancing,” ” IEEE Trans.Power Electron., vol. 34, no. 3, pp. 2025-2038, March 2019.Portions of Chapters 3 were published in IEEE Transactions on Power Electronics, and IEEE EnergyConversion Congress and Exposition (ECCE) [2, 3]:• S. A. Arshadi, M. Ordonez, M. Mohammadi and W. Eberle, ”Efficiency improvement of three-phaseLLC resonant converter using phase shedding,” 2017 IEEE Energy Conversion Congress and Exposi-tion (ECCE), Cincinnati, OH, 2017, pp. 3771-3775• S. A. Arshadi, M. Ordonez, W. Eberle, M. Craciun and C. Botting, ”Three-Phase LLC Battery Charger:Wide Regulation and Improved Light-Load Operation,” in IEEE Transactions on Power Electronics,vol. 36, no. 2, pp. 1519-1531, Feb. 2021Portions of Chapters 4 were published in IEEE Transactions on Power Electronics [4]:• S. A. Arshadi, M. Ordonez and W. Eberle, ”Current-Sharing Worst-Case Analysis of Three-PhaseCLLC Resonant Converters,” in IEEE Transactions on Power Electronics, vol. 37, no. 3, pp. 3099-3110, March 2022vAs the first author of the above-mentioned publications, the author of this thesis developed the theoreticalconcepts and wrote the manuscripts, developed simulation and experimental platforms, receiving advice andtechnical support from Prof. Martin Ordonez and his research team. Also, the author received advice forthe practical side of the design from the engineering team at Delta-Q Technologies, Vancouver, BC, Canada(Mr. Marian Craciun, and Mr. Chris Botting).viTable of ContentsAbstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iiiLay Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ivPreface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . vTable of Contents . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . viiList of Tables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xList of Figures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xiList of Abbreviations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xiiiAcknowledgments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xiv1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.1 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.2 Objectives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31.3 Literature Review . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31.3.1 Current Sharing Analysis of Three-phase LLC Resonant Converters and Current Bal-ancing Techniques . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41.3.2 Converter Gain Range and Light-Load Efficiency Issues . . . . . . . . . . . . . . . 51.3.3 Bi-directional Three-Phase CLLC Resonant Converter . . . . . . . . . . . . . . . . 71.4 Contributions of the Work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71.4.1 Current-sharing analysis and Trigonometric Current Balancing Technique for three-phase LLC resonant converters. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8vii1.4.2 Extended Gain Range and Improved Light-Load Efficiency for Battery Charger Ap-plications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81.4.3 Current-sharing analysis of three-phase CLLC resonant converters . . . . . . . . . . 81.5 Dissertation Outline . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92 Unbalanced three-phase LLC resonant converters . . . . . . . . . . . . . . . . . . . . . . . . 112.1 Current-Sharing Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112.1.1 Balanced Circuit . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 142.1.2 Unbalanced Circuit . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 152.1.3 Worst Case Investigation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 152.2 Trigonometric Current Balancing (TCB) Technique . . . . . . . . . . . . . . . . . . . . . . 182.2.1 Law of sines and law of cosines: . . . . . . . . . . . . . . . . . . . . . . . . . . . . 192.2.2 TCB Technique: . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 202.3 Experimental Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 222.4 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 273 Extended Gain Range and Improved Light-Load Efficiency . . . . . . . . . . . . . . . . . . . 313.1 Proposed Phase-Shedding Strategy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 323.1.1 Operating modes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 333.1.2 Phase-shedding strategy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 363.1.3 Power Loss Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 383.2 Design procedure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 443.3 Experimental Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 473.3.1 Operating modes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 493.3.2 Phase-shedding Transition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 504 Unbalanced bi-directional Three-phase CLLC resonant converters . . . . . . . . . . . . . . . 534.1 Equivalent Circuits . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 534.1.1 Balanced Circuit . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 554.1.2 Unbalanced Circuit . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 564.2 Current-Sharing Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 574.2.1 Law of Sines . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57viii4.2.2 Sensitivity Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 584.2.3 Worst-Case Scenarios . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 644.3 Experimental and Simulation Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 674.4 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 725 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 745.1 Conclusions and Contributions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 745.1.1 Trigonometric current balancing technique . . . . . . . . . . . . . . . . . . . . . . 745.1.2 Extended gain range and light-load efficiency improvement . . . . . . . . . . . . . 755.1.3 Bi-directional three-phase CLLC resonant converters . . . . . . . . . . . . . . . . . 755.1.4 Specified Academic Contributions . . . . . . . . . . . . . . . . . . . . . . . . . . . 765.1.5 Future works . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78ixList of TablesTable 2.1 Possible worst case scenarios . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18Table 2.2 Designed parameters of the experimental setup . . . . . . . . . . . . . . . . . . . . . . . 23Table 2.3 Experimentally verified values . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24Table 2.4 Result values . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26Table 2.5 Light-load results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29Table 3.1 Design worst case conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44Table 3.2 Design parameters of the experimental setup . . . . . . . . . . . . . . . . . . . . . . . . 48Table 3.3 Components summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48Table 4.1 ωn, QC, and ∠Zb at different operating points. . . . . . . . . . . . . . . . . . . . . . . . 60Table 4.2 Sign of sensitivity of γ to different resonant components at different operating points. . . 61Table 4.3 Combinations of T% variations in the resonant components . . . . . . . . . . . . . . . . 62Table 4.4 Designed parameters of the experimental setup. . . . . . . . . . . . . . . . . . . . . . . 67Table 4.5 Experimentally verified values of resonant components . . . . . . . . . . . . . . . . . . 67xList of FiguresFigure 1.1 Block diagram illustration of a typical battery charging system. . . . . . . . . . . . . . . 1Figure 1.2 Block diagram illustration of a typical DC-DC resonant converter. . . . . . . . . . . . . 2Figure 1.3 Three-phase resonant converter structures are suitable for higher power levels. . . . . . . 3Figure 1.4 Ideal three-phase LLC resonant converter structure. . . . . . . . . . . . . . . . . . . . . 4Figure 2.1 Resonant components tolerance issue . . . . . . . . . . . . . . . . . . . . . . . . . . . 12Figure 2.2 Effective reflected load . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13Figure 2.3 Equivalent three-phase circuit of the converter. . . . . . . . . . . . . . . . . . . . . . . 13Figure 2.4 Effect of 10% tolerance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16Figure 2.5 The trend in changes in the current values . . . . . . . . . . . . . . . . . . . . . . . . . 18Figure 2.6 Law of sines . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19Figure 2.7 Law of cosines . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20Figure 2.8 Balancing Technique . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21Figure 2.9 Trigonometric Current Balancing (TCB) flowchart. . . . . . . . . . . . . . . . . . . . . 22Figure 2.10 experimental setup . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23Figure 2.11 Simulation results versus experimental results . . . . . . . . . . . . . . . . . . . . . . . 25Figure 2.12 Current-sharing trend prediction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27Figure 2.13 Experimental waveforms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28Figure 2.14 Experimental result bars . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29Figure 2.15 Light-load results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30Figure 3.1 Proposed modified three-phase LLC resonant converter . . . . . . . . . . . . . . . . . . 32Figure 3.2 Proposed phase-shedding strategy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33Figure 3.3 Three-phase equivalent circuit . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34xiFigure 3.4 Equivalent two-phase circuit . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35Figure 3.5 Equivalent single-phase circuit . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36Figure 3.6 Conceptual flowchart . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37Figure 3.7 Transferred power to load . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38Figure 3.8 Transformers volt-second . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41Figure 3.9 Breakdown of losses . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41Figure 3.10 Normalized gain curves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45Figure 3.11 Design rule checking. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46Figure 3.12 Experimental results of the single-phase operation . . . . . . . . . . . . . . . . . . . . 48Figure 3.13 Experimental results of the three-phase operation . . . . . . . . . . . . . . . . . . . . . 49Figure 3.14 Experimental results of the two-phase operation . . . . . . . . . . . . . . . . . . . . . . 49Figure 3.15 Efficiency curves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51Figure 3.16 Transition response . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51Figure 4.1 Resonant components tolerance issue . . . . . . . . . . . . . . . . . . . . . . . . . . . 54Figure 4.2 Equivalent three-phase circuit . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55Figure 4.3 Single-phase equivalent circuit . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55Figure 4.4 The law of sines . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56Figure 4.5 Sensitivities of the impedance to its resonant components . . . . . . . . . . . . . . . . . 58Figure 4.6 Sensitivity of the phase-angles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58Figure 4.7 Two possible worst-case scenarios . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59Figure 4.8 Steps toward identifying the worst current-sharing cases. . . . . . . . . . . . . . . . . . 59Figure 4.9 Impedance phase-angles at different operating points . . . . . . . . . . . . . . . . . . . 60Figure 4.10 Simulation results of all the possible combinations . . . . . . . . . . . . . . . . . . . . 68Figure 4.11 Experimental waveforms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69Figure 4.12 TCB experimental results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69Figure 4.13 Experimental results bars . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70xiiList of AbbreviationsDC Direct CurrentEV Electric VehicleEMI Electromagnetic InterferenceEMC Electromagnetic CompabilityFHA First Harmonic ApproximationHEV Hybrid Electric VehiclePHEV Plug-in Hybrid Electric VehiclePFC Power Factor CorrectionPWM Pulsed Width ModulationRMS Root Mean SquaredSIC Silicon CarbideTCB Trigonometric Current BalancingZCS Zero Current SwitchingZVS Zero Voltage SwitchingxiiiAcknowledgmentsFirst and foremost, I would love to thank my lovely wife Nazgol for all her love and support in every aspectof my life.My sincere gratitude to Prof. Martin Ordonez for his continuous support of my PhD study and trustingin me to be part of his research team.Special thanks to my committee members, Dr. Wilson Eberle and Dr. Deepak Gautam for their feedbackon my research progress.I am grateful to the University of British Columbia (UBC), the Natural Sciences and Engineering Re-search Council (NSERC), and Delta-Q Technologies for supporting my research project. Special thanks toMr. Marian Craciun and Mr. Chris Botting from Delta-Q Technologies.Many thanks to my colleagues and friends at MartinOrdonez Lab.Lastly, I would love to thank my parents for their unconditional love and all of my brothers and sisters.My special thanks to my elder brother Mahdi for his support and encouragement in doing my PhD.xivChapter 1Introduction11.1 MotivationBattery chargers are the power processing stage between energy sources and batteries. In order to be able tocope with newer technological requirements, higher efficiency, higher power density, and enhanced reliabil-ity are expected from this type of power converter. The safety, durability, and performance of batteries arehighly dependent on how they are charged or discharged. For these reasons, much research work has beendone on improving power converters in the past decade [5, 6]. Fig. 1.1 shows the block diagram of a typicalbattery charging system. Isolated DC-DC converters are an essential part of a battery charging system, whichis the focus of this research.Figure 1.1: Block diagram illustration of a typical battery charging system.Resonant power converters have gained popularity in many applications, especially where there are strictrequirements on efficiency, power density, and Electromagnetic Compability (EMC) [7–10]. A resonantconverter is a DC-DC converter, where the input Direct Current (DC) voltage is turned into a square waveby a switch network to feed the resonant tank that effectively filters out harmonics providing sinusoidal likevoltage and current waveform. This in turn feed a transformer that provides voltage scaling and isolation.1Portions of this chapter have been modified from [1–4]1Then, this voltage is rectified and a DC voltage is provided at the output. The power flow is controlled bymodulating the square wave frequency that changes the resonant tank gain. Fig. 1.2, shows the block diagramof a DC-DC resonant converter. Their popularity is due to the unique features of resonant converters, whichinclude Zero Voltage Switching (ZVS) for the primary switches and Zero Current Switching (ZCS) for thesecondary side rectifier diodes. Soft switching allows higher frequency operation, which reduces the size ofpassive components and increases the power density. [11–14]Figure 1.2: Block diagram illustration of a typical DC-DC resonant converter.Half-bridge and full-bridge resonant converters have been studied thoroughly for low and medium powerlevels in the literature [15, 16]. For medium to high power levels, three-phase structures have been investi-gated for several converter topologies. Three-phase structures have several advantages over half-bridge andfull-bridge structures at high power levels. Three-phase structures can achieve better loss distribution, easierthermal management, much lower input and output current ripple, and smaller size for active and passivecomponents and heat sinks [17–22]. Due to these features, the power level can be increased to several kilo-watts (usually 4.5kW is maximum power where a full-bridge structure can be used efficiently), where thehalf-bridge and full-bridge structures cannot be used effectively. Fig. 1.3, shows the different switchingnetworks in a DC-DC converter.Although there has been substantial research and development on half-bridge and full-bridge resonantconverters, the literature investigating thee-phase structures has been limited and is emerging. The challengewith three-phase structures is that all the key advantages of three-phase resonant converters can be severelycompromised due to passive components tolerances. These tolerances lead to undesired current balanceissues in three-phase resonant converters. Tolerances in resonant tank passive components are inevitableand lead to unequal peak currents between phases, uneven temperature distribution, and large output currentripple. The other unexplored problems with three-phase resonant converters include issues with their limitedgain range and also their poor light-load efficiency compared to half-bridge and full-bridge structures.2+-ovinV VHigher PowerInverter Stage Resonant Tanks Transformers RectifierHalf-BridgeFull-BridgeThree-PhaseLrCrFigure 1.3: Three-phase resonant converter structures are suitable for higher power levels.1.2 ObjectivesThree-phase DC-DC resonant converters offer interesting features at higher power levels, which makes themattractive for level 2 and level 3 EV battery charging systems. The purpose of this research is to analyzethe utilization of three-phase LLC and CLLC resonant converters for level 2 and level 3 battery chargingapplications. This analysis includes identifying the limitations of the converters and finding solutions toovercome these problems. The identified and unexplored issues with three-phase resonant converters are:1) issues with current-sharing due to the inevitable tolerance in the resonant components, 2) poor light-loadefficiency of the converter due to the higher number of active switching components, and 3) the limited gainrange of the converter at light-load conditions.The purpose of this research is: 1) Analyzing the current-sharing behavior of the three-phase LLC res-onant converters and finding the worst-cases, 2) Proposing a current balancing technique to mitigate theunbalanced behavior of the converter, 3) Proposing a phase-shedding strategy to improve the light-load ef-ficiency of the converter at lighter loading conditions, 4) Proposing a modulation technique to extended thegain range of the converter for reviving batteries from the dead zone, 5) investigating the three-phase CLLCresonant converters for bi-directional battery charging applications.1.3 Literature ReviewFor high-power converters, interleaved resonant topologies are the most attractive. Interleaved topologiesprovide higher efficiency and higher power density at increased power levels. Different possible solutionsfor implementing interleaved resonant converters were investigated in [23]-[28]. However, in all of thesesolutions, interleaving is achieved either by means of adding auxiliary circuits ([23]-[25]) or with the cost3of losing soft switching ([26]-[28]). One of the possible ways to implement interleaved resonant converteris to use the three-phase topology. By employing a three-phase resonant converter, interleaving can beimplemented without any auxiliary circuit, and the power level of the system can be easily increased to theorder of several kilo Watts. Three-phase resonant converters can result in higher efficiency, lower volumeof the magnetic components, higher reliability, easier thermal management, and an easier implementation.Moreover, interleaved modulation of the phases can result in much lower current ripple in the input andoutput and, consequently, the filter size can also be reduced.In [29]-[31], the principle of operation and the performance of three-phase series-parallel and three-phase LCC resonant converters were discussed. Three-phase LLC resonant converters were studied in [32],[33], and [34]. In [35], a control strategy for a three-phase LLC resonant converter, fed by a Power FactorCorrection (PFC) stage was proposed for an Electric Vehicle (EV) battery charger application were presented.Three-phase LLC resonant converters with integrated magnetics were presented in [36] and [37]. In all theaforementioned work, balanced current operation was assumed and no analysis was done on the behaviorfor unbalanced operating conditions. Fig. 1.4 shows the structure of an ideal three-phase LLC resonantconverter. The key advantages of a three-phase resonant converter under ideal conditions are depicted in thefigure.Figure 1.4: Ideal three-phase LLC resonant converter structure.1.3.1 Current Sharing Analysis of Three-phase LLC Resonant Converters and CurrentBalancing TechniquesA new work on LLC has been discussed using Silicon Carbide (SIC) and balancing transformer [38], anda star-connected transformers with a floating star point [39]. The balancing issues were identified and acompensation approach was preliminary presented based on a basic PID controller. The development of4an in-depth theoretical analysis of the converter under unbalanced conditions remains as a new researchopportunity. The conventional approach for avoiding unbalanced currents in a multi-phase resonant converter(collection of single-phase converters) is to control the phases separately. Unfortunately, such a methodcannot be applied on a three-phase resonant converter because, with different switching frequencies for thephases, interleaved modulation of the phases cannot be achieved. The idea of using three separate PFCstages to balance the circuit has been presented in [40] and [41]. In that work, PFC rectifiers (3 units) havebeen employed to regulate the DC input voltages of each LLC phase. While balancing can be achieved,using 3 separate PFCs or 3 pre-regulators can be cost-prohibitive for some applications. Usually PFC stagescan handle higher power levels compared to the DC-DC stage, as they deal with higher voltages and lowerconduction losses. Therefore, that approach could be redundant in applications where the DC-DC stage dealswith lower output voltage levels.The root cause of unbalanced operation is the tolerance in Cr, Lr, and Lm values of the LLC converter.Normally, 10% change in the passive component values can be expected, leading to detrimental unbal-anced currents [42]-[44]. Unbalanced currents result in unwanted higher output current ripple, unequallydistributed loss and, consequently, temperature in the circuit, higher voltage and current peak values, andhigher conduction losses. Due to this unbalanced characteristics, effectively using three-phase LLC resonantconverter is challenging. While there are some works, the limited body of literature on three-phase LLCconverters demands more analytical tools to model/explain their behavior and solve fundamental problemssuch as current balancing. In this work, imbalances in three-phase LLC converters are investigated and anovel Trigonometric Current Balancing (TCB) technique using phasor analysis are proposed.1.3.2 Converter Gain Range and Light-Load Efficiency IssuesAlthough there has been substantial research and development on single-phase LLC resonant converters,the literature investigating three-phase LLC structures has been limited to the works mentioned in previoussubsection. Other issues to be investigated for three-phase resonant converters are: 1) the natural limitedgain range of the LLC resonant converter structure, and 2) poor light load efficiency of the three-phase LLCresonant converter and possible phase-shedding strategies to improve it, and also to increase the reliabilityof the system.There are typically three main phases in charging a battery: recovery stage (pre-charge), bulk charging(constant current), and absorption (constant voltage). The recovery stage usually happens when the batteryis over-used or has been stored for a long time. In this situation, the battery is in the dead-zone and needs5to be recovered before the bulk charging starts. This is also called the pre-charge stage, and low voltageand low current is required to recover the battery. One of the problems with LLC resonant converters istheir limited gain range, which prevents them from realizing the wide gain range required for reviving adeeply depleted battery in a battery charger. Due to the natural limited gain range of LLC converters, it is achallenge to be able to provide the low voltage and low current needed when designing the converter. Forthat reason, the pre-charge stage is considered to be the most difficult stage in a charging cycle [45]. Theconventional method to overcome this issue in LLC resonant converters is to employ burst mode operationto provide low output voltages at light-load conditions [46]. However, this approach is not desired dueto complicated implementation, high Electromagnetic Interference (EMI), and the very high low-frequencyoutput current and voltage ripples introduced to the battery terminals, which may affects the battery life.Many different modifications on single-phase LLC resonant converters have been investigated to achieve awide voltage gain range of the converter [47]. These works include modifications on resonant tanks [48]-[52],re-configuring primary-side switch network and modulations [53]-[55], integrated topologies [56], [57], andconfigurations on the secondary-side rectifiers [58]-[62]. Other works such as, adopting a variable DC-linkvoltage [63], reconfigurable dual LLC converter [64], and adopting two-level turns-ratio transformer [65],have also been presented to achieve extended gain range charger with single-phase and dual LLC resonantconverters. Although many different solutions are investigated for extending the gain range of single-phaseand dual LLC resonant converters, no work has been presented on three-phase LLC resonant converters.The other issue with three-phase LLC resonant converters is their poor light-load efficiency. Comparedto single-phase structures, three-phase LLC resonant converters provide higher efficiency at heavy load con-ditions by sharing the load through the phases and reducing the current peak values leading to conductionlosses reduction. In the lighter load conditions, the switching and circulating power losses become dom-inant and that results in a poor light-load efficiency for three-phase structures due to the high number ofsemi-conductor devices. In the absorption stage (constant voltage stage), the battery terminal voltage iskept constant and the charge current starts decreasing to very light-load conditions. As the charging currentdecreases and the converter goes to lighter load conditions, the three-phase structure will not be efficientlycharging anymore. The light-load absorption stage happens during much longer time compare to the othercharging stages. This makes it even more important to improve the light-load efficiency of the converter,and the power saving over time will be significant.Some phase-shedding strategies have been presented formulti-phase converters structures in order to improve light load efficiency. However, the modular structureof three-phase LLC converters makes it more difficult to employ phase-shedding strategies and this has not6been investigated in the three-phase LLC literature discussed above.1.3.3 Bi-directional Three-Phase CLLC Resonant ConverterAmong the different resonant converter topologies, CLLC resonant converters have attracted a lot of attentionwhere isolated bi-directional power flow is required [66–70]. CLLC converters offer soft-switching over awide range of output voltages, as well as a high and flat efficiency curve over the whole range of operatingpoints [71], [72]. Due to its symmetric resonant tank structure, bi-directional power flow is achieved whilemaintaining all the desirable features. With these features, the CLLC converter has been examined as one ofthe most commonly used bidirectional DC-DC converter topologies alongside other well-known converters,such as the Dual Active Bridge (DAB) [73].Half-bridge and full-bridge CLLC converters have been thoroughly investigated in the literature [71]-[80]. These works include investigating synchronous rectification [71], design and modeling of the converter[72]-[77], and studying two-stage structures [78]-[80]. However, there has been very limited work on three-phase CLLC resonant converters [81]. In [82], an optimal design methodology of a three-phase CLLCresonant converter has been studied and a modified structure is presented to achieve a higher power level.In that work, ideal resonant components are assumed and the converter is analyzed under an ideal balancedcondition.Similar to three-phase LLC resonant converters, the three-phase CLLC resonant converter offer some fea-tures for higher power level applications. However, this converter also malfunctions in correspondence to theinevitable tolerance in the resonant components. The high number of resonant components in the three-phaseCLLC converter (fifteen) requires a different mathematical approach to understand and predict the behav-ior of the converter under real conditions while considering the tolerances in the components. Therefore,analyzing the converter under the unbalanced condition remains a research challenge to be addressed.1.4 Contributions of the WorkIn order to take advantage of the unique features that the three-phase LLC and CLLC resonant convertersoffer for high power EV battery charging applications, different analysis have been done and various ap-proaches are mixed together to improve the operation of the converters. This dissertation targets identifyingthe root cause of the limitations and problems of the three-phase resonant converters for battery chargingsystems and proposes novel control and design solutions. The main contributions of this work are the fol-lowing:71.4.1 Current-sharing analysis and Trigonometric Current Balancing Technique forthree-phase LLC resonant converters.This work develops a detailed analysis of the current-sharing characteristics of the three-phase LLC resonantconverter to predict unbalanced operation. The theory shows how the tolerance in components affect thebehavior of the converter leading to current balancing issues. The analysis tool presented in this work canpredict the behavior of the converter, prevent worse case scenarios, and determine the severity of unbalancedbehavior. In addition, TCB is proposed to mitigate the unbalanced behavior of the circuit. In this strategy therequired input voltage phase-angles are calculated to achieve balanced phase currents. The switching-anglesof the inverter legs are updated (shifted) based on the calculated values.1.4.2 Extended Gain Range and Improved Light-Load Efficiency for Battery ChargerApplicationsDue to the multi-phase structure of three-phase LLC resonant converters, higher power conversion efficiencyat higher power levels are achieved compared to single-phase structures. However, in lighter load conditions,where the switching losses are dominant, the efficiency drops significantly. Light-load efficiency improve-ment can be achieved by phase-shedding. Due to the modular structure of three-phase LLC resonant con-verters, employing phase-shedding strategies is a challenge. In this work, light load efficiency improvementis studied and achieved by proposing two-phase operation of the converter. This leads to significant powersaving over long time light-load operations, e.g. absorption charging stage in battery charger applications.A modified three-phase LLC resonant converter with a new modulation technique is proposed to achieveboth extended gain range and light-load efficiency improvement. Compared to the conventional three-phaseLLC resonant converter, the proposed topology has two additional small, low power rated diodes. In thismodified three-phase LLC resonant converter, a phase-shedding strategy is proposed to realize the extendedgain range needed for the recovery zone of battery charging without the need for burst mode operation. Thisis achieved by proposing single-phase operation of the circuit. The proposed technique delivers the extendedgain range needed for covering the dead zone of battery charging cycle.1.4.3 Current-sharing analysis of three-phase CLLC resonant convertersIn this work, the three-phase CLLC resonant converter under a real unbalanced condition is analyzed. A newmethodology based on sensitivity analysis is proposed to identify and measure the worst-case unbalancedbehavior of the converter with a given tolerance in the resonant component values. The proposed worst-caseanalysis helps to find the highest tolerable deviation in the component values to keep the converter operating8within an acceptable unbalanced range without requiring any additional balancing techniques. Moreover,the TCB technique, which was proposed for the three-phase LLC converter, is validated to be effective as ameans to mitigate the unbalanced behavior of the converter in three-phase CLLC converters.1.5 Dissertation OutlineIn Chapter 2, a three-phase equivalent circuit based on First Harmonic Approximation (FHA) is derived. Thederived model is further used for analyzing the current-sharing behavior of the three-phase LLC converterunder unbalanced conditions. The analysis tool presented in this work helps predicting the behavior of theconverter and determining the severity of unbalanced behavior. After that, a balancing technique basedon the law-of-sines is developed for balancing the currents. The technique is called TCB. In this strategythe required input voltage phase-angles are calculated to achieve balanced phase currents. The switching-angles of the inverter legs are updated (shifted) based on the calculated values. TCB successfully balancesthe three-phase currents, resulting in significant reduction in the output capacitor current ripple and betterthermal distribution. Without requiring any additional circuit, this technique provides the opportunity tointegrate the unique features of three-phase structures (e.g., high power and low output current ripple) withthe advantages of LLC resonant converter (e.g., high efficiency). Finally, simulation and experimental resultsof a 3kW three-phase LLC resonant converter are provided to show the validity of the analysis.In Chapter 3, a modified three-phase LLC resonant converter with a new phase-shedding technique isproposed to address the regulation and light-load performance issues of the converter. These are achievedby proposing single-phase and two-phase operations of the converter, respectively. Compared to the con-ventional three-phase LLC resonant converter, the proposed topology has two additional small, low powerrated diodes. The proposed technique delivers the whole package of features needed by a three-phase LLCresonant converter in a high-power battery charger application. In this chapter, the proposed modified three-phase LLC resonant converter and phase-shedding strategy are discussed, and later the design procedure andrequirements are discussed. Finally, simulation and experimental results of a 3kW three-phase LLC resonantconverter are provided to show the validity of the proposed analysis.In Chapter 4, the three-phase CLLC resonant converter under a real unbalanced condition is analyzed.A new methodology based on sensitivity analysis is proposed to identify and measure the worst-case un-balanced behavior of the converter with a given tolerance in the resonant component values. After that, theeffectiveness of the TCB technique for the three-phase CLLC converter is also verified. The proposed worst-case analysis methodology is validated by both simulation and experimental results. The results of the 3kW9experimental prototype shows the effectiveness of the proposed analysis for identifying and predicting theworst cases of current-sharing in the converter with a given tolerance in the resonant components. Also, theresults validate the effectiveness of the TCB technique to mitigate the unbalanced behavior of the convertereven at the worst unbalanced cases with 10% component tolerance.Chapter 5, contains the relevant conclusions, contributions, and planned areas of future work. The workcontributes significantly to the applications of level-2 and level-3 battery charging systems by proposingeffective ways to analyzing and employing three-phase LLC and CLLC resonant converters. The contribu-tions are highlighted in four relevant publications in IEEE Transactions journals and international conferencepapers.10Chapter 2Unbalanced three-phase LLC resonantconverters1High power density is one of the main requirements for a high power battery charger converter. In orderto achieve that, an even power loss distribution in the circuit needs to be guaranteed. Fig. 2.1 conceptuallyshows the current-sharing issue in the three-phase LLC resonant converter and the proposed TCB technique.In this section, the current-sharing analysis tool a long with the balancing technique are discussed in details.Experimental and simulation results are provided to validate the proposed work.2.1 Current-Sharing AnalysisAs mentioned earlier, previous works have analyzed three-phase resonant converters with the assumption ofthe balanced circuit. Assuming the balanced condition, these papers have derived a single-phase equivalentcircuit for the converter that can be used to analyze the characteristics of the converter. However, this type ofanalysis cannot be used to investigate the unbalanced behavior of the LLC resonant converter. In this section,a three-phase equivalent circuit of the LLC resonant converter is developed, which can be used to analyzethe characteristics of the converter and provides detailed insight into the effect of Ls, Cs, and Lm values.Therefore, the resulting analysis can be used to evaluate current-sharing even under unbalanced conditions.The three-phase LLC resonant converter, as shown in Fig. 2.1, is formed by a three-phase DC-AC inverter,three resonant tanks, a high-frequency three-phase transformer or three single-phase transformers connectedin star with floating star point, and a three-phase full-bridge rectifier. Similar to half-bridge and full-bridgeresonant converters, the three-phase converter applies square wave voltages to the input of the resonant1Portions of this chapter have been modified from [1]11+-S S1DD5D3Dov2D D641 3S5S2 S4 S6inVLr CrCrCrLrLrLmLmLm±10% ±10%±10%±10%±10%±10%±10%±10%±10%Tolerance IssueV3V3V1V1V2V2120º120º120ºIoIotI1I1I2I2I3I3αβγUnbalanced Currents High OutputCurrent Ripple120º120º120ºI1I2I3Balanced CurrentsBalancingTechnique:(Shifting Legs forBalancing)BasicModulation:(Legs 120º Phase Shifted)Low OutputCurrent RippleIotයයයයTemperatureV3V1V2φ'13φ'21φ'23Figure 2.1: Resonant components tolerance issue: unbalanced circuit results in unbalanced current,not equally distributed loss and heat in the circuit, and high output current ripple. The proposedbalancing technique achieves fully balanced current in an unbalanced circuit and resolves theaforementioned issues.network. The resonant network filters the harmonics of the input voltages in a way that three sinusoidalcurrents appear. This allows using AC analysis techniques for analyzing the behavior of the converter.Considering the filtering characteristics of the resonant tanks, it is possible to approximate the square inputvoltage waveforms with the fundamental component and compute the resulting sine waves of current andvoltage in the resonant tank with good precision.To do so, it is necessary to model the capacitive output rectifier. As shown in Fig. 2.2, the input volt-age of each of the diode bridges is a five-level square waveform, which is essentially the voltage across thesecondary of the transformers. The fundamental harmonic of this voltage can be calculated using Fourierseries as (2.1), where T is the switching period. Assuming pure sinusoidal waveforms, the Root MeanSquared (RMS) of the current through the transformer winding as a function of output current can be calcu-lated as (2.2).Vac,rms =√2π∫ T0vac(t)sin(2πtT)dt =√2πVO (2.1)IO =62π∫ 2π3π3√2Iac,rms(t)sin(ωt)dt =3√2πIac,rms ⇒ Iac,rms = π3√2IO (2.2)12+vO-RO-IPIO IP ac- v +aci :acv :0(1/3).vO(2/3).vOFigure 2.2: Effective load reflected on each of the secondaries windings of the transformers consideringFHA.Using (2.1) and (2.2), the equivalent load reflected to the primary of each transformer can be calculatedas (2.3), where N is the turns ratio of the secondary to primary of the transformers. Assuming a balancedcircuit, the equivalent three-phase circuit of the converter can be extracted as shown in Fig. 2.3.Rac = N2Vac,rmsIac,rms=6π2N2RO (2.3)Considering the KVLs and KCL in the circuit in Fig. 2.3, the set of equations in (2.4) can be extracted.Lr1Lm1Z2Z3Z1Cr1Lr2Lm2Cr2Lr3Lm3Cr3Rac1Rac2Rac3φ2VVVφ1φ3Figure 2.3: Equivalent three-phase circuit of the converter.V2−V1 +Z1I1−Z2I2 = 0 (I)V3−V1 +Z1I1−Z3I3 = 0 (II)I2 + I3 + I1 = 0 (III)(2.4)Where V1, V2, and V3 are the fundamental component of the square wave voltage at the input of each phase,I1, I2, and I3 are the current through the phases, and Z1, Z2, and Z3 are the impedance of each phase which13can be calculated using (2.5) or (2.6).Zt = Lrt jω+1Crt jω+LmtRact jωRact +Lmt jω, t = 1,2,3 (2.5)Zt = (Lrtω− 1Crtω +LmtR2actωR2act +L2mtω2) j+L2mtRactω2R2act +L2mtω2, t = 1,2,3 (2.6)Using the equations in (2.4), three phase currents as a function of input voltages can be calculated as (2.7).I1I2I3= ( 1Z1Z2 +Z2Z3 +Z1Z3 )×Z2 +Z3 −Z3 −Z2−Z3 Z1 +Z3 −Z1−Z2 −Z1 Z1 +Z2V1V2V3 (2.7)The above-mentioned equations and the three-phase equivalent circuit in Fig. 2.3, can be used to find currentsin different phases under unbalanced conditions. In the following, the circuit under balanced and unbalancedconditions is analyzed.2.1.1 Balanced CircuitIn a balanced circuit, where Lr1 = Lr2 = Lr3 = Lr, Cr1 = Cr2 = Cr3 = Cr, Lm1 = Lm2 = Lm3 = Lm, andconsequently Z1 = Z2 = Z3 = Z, (2.7) can be expressed as (2.8).I1I2I3= ( 13Z )2 −1 −1−1 2 −1−1 −1 2V1V2V3 (2.8)Since the amplitude of the input voltages are equal, and they are 120 degrees phase shifted, the relationbetween input voltages can be expressed as (2.9). Using (2.8) and (2.9), the three phase currents in a balancedcircuit can be calculated as (2.10).V1 +V2 +V3 = 0 (2.9)It =VtZ(2.10)Equation (2.10) shows that the currents through the phases are equal and they can be calculated independentlyfrom the impedance values of the other phases. This also shows the validity of the single-phase equivalentmodel for the balanced three-phase LLC resonant converter.142.1.2 Unbalanced CircuitIn the case of an unbalanced circuit, the single-phase equivalent circuit cannot help with the understandingof unbalanced currents. In this case, the three-phase equivalent circuit shown in Fig. 2.3 and equations(2.6) and (2.7) can be used for analyzing the behavior of unbalanced currents through the phases. Thisanalysis is based on two approximations: 1) First Harmonic Approximation (FHA), and 2) Approximatingunbalanced reflected loads to the primary with equal values of Rac. It will be shown that the impedance valueof each phase is less sensitive to the variations in the values of Lmtn and Rac in comparison with Lrtn and Crtn.Moreover, Lmtn and Rac voltages are clamped to the output voltage during most of the operating modes anddo not participate in resonance in most of the operating modes. Therefore, while these approximations makethe analysis simpler, they achieve results that are accurate enough to show the trend in the changes of threephase currents in correspondence to specified tolerances in resonant tank elements. The ultimate purpose ofthe analysis is to find out about the worst case scenario in current-sharing.2.1.3 Worst Case InvestigationAnalyzing the circuit shown in Fig. 2.3, with nine unknown variables, is difficult to start with. In order toreduce the number of unknown variables, the analysis can be started by investigating the tolerance effectof the resonant tank components of each phase on the impedance value of the corresponding phase. Fig.2.4 shows the effect of variations in the value of each resonant element on the impedance of the resonanttank. This figure is based on the the design parameters and is derived using (2.6). These plots show thenormalized values of the phase-angle and amplitude of the impedance versus normalized values of frequencyand normalized values of the resonant components. The impedance values and resonant component valuesare normalized based on the balanced value, and the frequency is normalized based on the resonant frequency( fr) value as (2.11).|Ztn|= |Zt ||Zto| , ∠Ztn = ∠Zt −∠Zto, Lrtn =LrtLrto, Crtn =CrtCrto, Lmtn =LmtLmto, fn =ffr(2.11)Fig. 2.4 (a) shows that by changing the value of Lrt by 10%, the phase-angle of Ztn will be changed upto 5 degrees in the same direction. Fig. 2.4 (b) also shows that changes in the values of Crt have the sameeffect. Moreover, Fig. 2.4 (d) and (e) show that changes in the values of Lrt or Crt will correspond to thesame trend in changes of the impedance amplitude. However, Fig. 2.4 (c) shows that changes in Lmt has anopposite effect on phase-angle of the impedance compared to changes in Lrt and Crt . Changing the value of15Lrtn(a) (b) (c)(d) (e) (f)1.11.05 1.2110.95 0.90.9 0.8 fn Z˚1.1-10-50510-8-6-4-202460.940.960.9811.021.041.060.940.960.981 │Z │1.021.041.06tnCrtn1.11.05 1.2110.95 0.90.9 0.8 fn Z˚1.1-10-50510tntnLmtn1.11.05 1.2110.95 0.90.9 0.8 fn Z˚1.1-10-50510tnLrtn1.11.11.05 1.2110.95 0.90.9 0.8 fn10.940.960.98 │Z │1.021.041.06tnCrtn1.11.11.05 1.2110.95 0.90.9 0.8 fn10.940.960.98 │Z │1.021.041.06tnLmtn1.11.11.05 1.2110.95 0.90.9 0.8 fnFigure 2.4: Effect of 10% tolerance in the values of each of the resonant components of each phase onthe amplitude and phase-angle of the phase. Changes in the values of Lrtn and Crtn show the sametrend of changes in the impedance values. The impedance values are less sensitive to the changesin the value of Lmtn.Lmt by 10% will result in changing the phase-angle of the impedance by less than 2 degrees. This shows thatthe phase-angle of Ztn is less sensitive to changes of Lmt . Moreover, changes in Lmt have the same effect onthe impedance amplitude as Lrt and Crt . However, the trend is not the same over the changes of frequency(Fig. 2.4 (f)). To summarize this information:1. Changes in Lrt and Crt have the same effect on phase-angle and amplitude of the impedance. Hence,considering the definition of resonant frequency, expressed in (2.12), it can be said that changes inthe resonant frequency will affect the phase-angle and the amplitude of the impedance of each phase.It means that, by changing Lrt and Crt in the same direction, which essentially means changing theresonant frequency, the most amount of deviation in the value of impedance will be caused.fr =12π√LrtCrt(2.12)2. Changes in Lmt have an opposite effect on the impedance phase-angle compared to the effect of Lrtand Crt . It means that, by changing the value of Lmt and simultaneously changing the value of Lrt16in the opposite direction, the most deviation in the value of the impedance phase-angle will result.Considering the inductor ratio (m), which is expressed as (2.13), it can be said that changes in thevalue of m will affect the impedance phase-angle. In other words, changing Lmt and Lrt in oppositedirections, which essentially means changing the inductor ratio (m), will result in the most amountof deviation in the value of impedance phase. The changes in Lmt have an opposite effect on theimpedance amplitude in comparison with its effect of changes on the impedance phase-angle. Also,the changes in Lmt have an opposite effect on the impedance amplitude over the frequency range incomparison with Lrt and Crt . Therefore, it can be said that changes in the value of m can have differentimpacts on the impedance values and current-sharing in the system over the frequency range.mt =LmtLrt(2.13)3. The impedance phase-angle and amplitude are much more sensitive to the changes in the values of Lrtand Crt than to the value of Lmt .The worst case in current-sharing is when two of the phases have lower current, and the other phasetakes the highest current. In this situation, the phase with higher current, experiences the highest peakvalues, highest losses, and highest temperature rise. To find out about the worst case in current-sharing, it isnecessary to investigate different possible worst case unbalanced scenarios. Based on the information above,the worst case deviation in the value of the impedance of each phase happens when the resonant frequency( fr) and the inductor ratio (m) of that phase changes. Therefore, based on the possible combination oftolerances in fr and Lmt , four of the possible scenarios can be as shown in Table 2.1, where:fr(min) = 12π√(Lr+10%)(Cr+10%),Lm(min) = Lm−10%fr(max) = 12π√(Lr−10%)(Cr−10%),Lm(max) = Lm +10%(2.14)In order to investigate the corresponding current-sharing of each scenario, Fig. 2.5 shows three phasecurrents as a function of normalized frequency for different scenarios. In these figures, normalized RMSvalues of currents (normalized based on the value of balanced circuit), versus the normalized frequency aredepicted. As it can be seen, the cases in scenarios II and IV seem to be worse than the other two scenarios. In170.8 0.9 1 1.1 1.20.80.911.11.20.80.911.11.20.80.911.11.20.80.911.11.2fnIn In In InIph1Iph2Iph3Scenario I0.8 0.9 1 1.1 1.2fnScenario IIIph1Iph2Iph30.8 0.9 1 1.1 1.2fnIph1Iph2Iph3Scenario III0.8 0.9 1 1.1 1.2fnIph1Iph2Iph3(a) (b) (c) (d)Scenario IVFigure 2.5: The trend in changes in the current values in correspondence with four possible worst casescenarios. For a) and c) one of the phases carries much less current compared to the other twophases. For b) and d) one of the phases carries much higher current compared to the other twophases.these two scenarios, two of the phases have the maximum resonant frequency ( fr) and the other phase has theminimum fr. On the other hand, the behavior of Lm in affecting the current-sharing is quite unpredictable.Comparing Fig. 2.5 (a) and (c), or (b) and (d), it can be concluded that by changing the value of Lm from itsminimum to its maximum, the trend of current curves does not change significantly. This confirms the smalleffect of the tolerances on the values of Lm of different phases on the current-sharing behavior of the circuit.The analytical results of the three-phase circuit show that in the worst case scenario with 10% tolerancein the components, in which two of the phases have maximum resonant frequency and the other phase hasthe minimum resonant frequency, one of the phases may need to handle currents up to 15% more than itsrated current.2.2 Trigonometric Current Balancing (TCB) TechniqueIn previous subsection, it was shown how the tolerance in the values of resonant components can affectthe impedance value of each phase and how it would generate unbalanced currents through the phases.Table 2.1: Possible worst case scenarios of unbalanced impedancesLevelsPhase 1 Phase 2 Phase 3Scenarios fr, Lm fr, Lm fr, LmCase I min,min min,min max,maxCase II min,min max,max max,maxCase III min,max min,max max,minCase IV min,max max,min max,min18α120˚120˚120˚βγI1I1I2I2I3I3(a) (b)Figure 2.6: Law of sines: a) Balanced currents with ideal components, b) Unbalanced currents withreal components with tolerances.As mentioned in Section 1, unbalanced currents in the converter result in loss of some of the features of thethree-phase structure. Unequally distributed losses in the circuit, developing hot spots, and not having perfectoutput current ripple cancellation, meaning much higher output current ripple, are some of the disadvantagesof having unbalanced currents. In this section, the detailed analytical framework developed is used to developa Trigonometric Current Balancing technique in an unbalanced circuit.2.2.1 Law of sines and law of cosines:The law of sines shows the relation among the angles and the lengths of sides of a triangle, which can alsobe used for the current vectors in a three-phase circuit. Considering FHA and using the law of sines, it canbe shown that if the amplitudes of the currents in a three-phase circuit are equal, then they are 120 degreesphase shifted. Accordingly, if the currents are not balanced, it can be concluded that the angles between thecurrent vectors are no longer equal. Moreover, based on the order of the current amplitudes, the order ofangles between the vectors can be realized. Fig. 2.6, shows current vectors in a (a) balanced circuit, and(b) unbalanced circuit, conceptually. Using the law of sines, the relationships between vectors and angles ispresented in (2.15). i f |I1|= |I2|= |I3| ⇐⇒ β = γ = αi f |I1| ̸= |I2| ̸= |I3| ⇐⇒ β ̸= γ ̸= αi f |I1|> |I2|> |I3| ⇐⇒ α > β > γ(2.15)The exact value of the angles between the current vectors can be calculated using the law of cosines. Thelaw of cosines in conjunction with the law of sines can help in finding all sides and angles of a triangular. Ina basic triangular as shown in Fig. 2.7 (a), where the sides are known, the angle between two of sides can becalculated using the law of cosine as follows:cosθ =a2 +b2− c22ab(2.16)19αβγθ I1I2I3c ccabc(a) (b)Figure 2.7: Law of cosines: a) A triangular, b) The triangular in three-phase current vectors.In three-phase current vectors, a triangular, as shown in Fig. 2.7 (b), can be identified. In this triangular, αc,βc, and γc are the complementary angles of the angles between the current vectors. Therefore, the anglesbetween current vectors can be calculated as:α = π− arccos( I21+I22−I232I1I2 )β = π− arccos( I21+I23−I222I1I3 )γ = π− arccos( I22+I23−I212I2I3 )(2.17)2.2.2 TCB Technique:Based on the law of sines, if the angles between the current vectors are corrected to be equal to 120 degrees,the currents will be balanced. Assuming an unbalanced circuit in Fig. 2.3, there are two variables that canbe manipulated by switching the MOSFETs: the amplitudes and the phase-angles of the input voltages.The amplitude of the input voltages can be controlled by Pulsed Width Modulation (PWM), and the phase-angle can be simply controlled by phase shifting the input voltages. Implementing PWM in LLC resonantconverters has some limitations due to the possibility of losing soft switching for the MOSFETs, and ofchanges in the converter gain. In the following, a Trigonometric Current Balancing technique for three-phaseLLC resonant converter based on shifting the phase of the inverter bridge’s switching points is presented.Fig. 2.8 (a) shows the current vectors in an unbalanced circuit with the basic 120 degrees phase-shiftedinput voltage vectors. The idea to correct the angles between of unbalanced current vectors is to change theangles between the input voltage vectors in a way that it results in shifting the current vector angles towards120 degrees. For example, in Fig. 2.8 (a), α is less than 120◦, and β is larger than 120◦. In this case,reducing φ31 and increasing φ12 can result in increasing α and decreasing β (Fig. 2.8 (b)). Fig. 2.9 presentsthe flowchart of the proposed balancing technique. Different steps of the proposed TCB are as follows:1. Initially, the phase angles between the input voltage vectors are 120◦ by default. If the circuit isunbalanced, this will result in unbalanced currents in the circuit.20αβγV1I1I2I3V1V2V2V3V3=120º=120º=120º යයφ21ºººφ21nφ31φ31nφ23φ23n120º ± ε120º ± ε120º ± ε I1I2I3Input Voltages Primary CurrentsInput Voltages Primary Currents(a)(b)Figure 2.8: Balancing Technique: a) basic 120 degrees phase-shifted input voltages, b) Balancing cur-rents with changing the phase-shift of the input voltages.2. The balancing technique is based on balancing the RMS values of the currents. This is because theRMS value is a better representative of the first harmonic of the currents compared to peak or meanrectified values. As mentioned before, the balancing technique is based on FHA and balancing thefundamental harmonic of the currents. Moreover, RMS value of the current is directly related to theconduction losses. Hence, balancing RMS values will result in equal losses through the phases.3. After reading the current RMS values, they are compared to see if the difference between RMS valuesis less than the specified threshold. If they were close enough, the process of reading the values andcomparing them continuously occurs. If the differences in currents are not within the limit, TCB starts.4. The angles between the current vectors (α , β , and γ) are calculated using (2.17).5. New switching angles based on the current vector angles are defined using (2.18).f (φ) :φ ⋆12 = φ12 +(120◦−α)φ ⋆13 = φ13 +(120◦−β )φ ⋆23 = φ23 +(120◦− γ)(2.18)In this equation, φ ⋆ is the new voltage vector angle, and φ is the angle of the previous stage. At this21INITIALRead RMS Values of I1, I2, and I3Calculate CurrentVector anglesα, β, and γ(using eq. (17))Calculate new Voltage Anglesφ12, φ23, and φ13(using eq. (18))Set φ12 = 120°,φ23 = 120°,φ13 = 120°Updateφ12 = φ12 ,φ23 = φ23 ,φ13 = φ13 **** * *|I1-I2| < ε|I1-I3| < εYESNOFigure 2.9: TCB flowchart.point, the voltage vectors are phase-shifted with the same amount of deviation in the current vectorangles from 120 degrees.6. After that the new switching angles are defined, the inverter legs are shifted based on the new switchingpoints, and then the control circuit goes back to step one.2.3 Experimental ResultsIn order to confirm the validity of the analysis and the proposed TCB technique, extensive experimental testsand simulations have been done. A 3kW three phase LLC resonant converter with the proposed control tech-nique has been manufactured and tested under different conditions. Fig 2.10 shows the prototype, and thespecifications of this platform are presented in Table 2.2. This design can be used for a Plug-in Hybrid Elec-tric Vehicle (PHEV) battery charger application. The maximum power and the output current of this converterare 3kW and 10A, respectively. Table 2.3 shows experimentally verified values of the resonant components.The values are corresponding to the worst case scenarios discussed in section II. The experimental and sim-22Output diodesThree-phase InverterOutput CapacitorResonant TanksControl BoardFigure 2.10: 3kW experimental setup. Reduced output filter size due to the output current ripple reduc-tion.Table 2.2: Designed parameters of the experimental setupParameter ValuesVin 400VVout 300−400VPout(max) 3kWIout(max) 10An1/n2 12/3fr 205kHzm 3Lr 20µH±10%Cr 30nF±10%Lm 60µH±10%ulation results are presented in two parts: simulation results of ideal circuit versus the experimental resultsof the real unbalanced circuit; and results of the TCB applied to the unbalanced circuit versus the resultswithout TCB.To get a better understanding of unbalanced conditions and to better describe it, an unbalanced factoris defined as (2.19). This unbalanced factor (U f ) is defined based on the worst potential difference in theconduction losses between the phases. Conduction losses are proportional to the square values of RMScurrents, and can represent the potential temperature difference caused by unbalanced currents. Moreover,the output current ripple value is another factor for describing unbalanced currents. As, there are moreunbalanced currents, the output current ripple will be higher.U f =Max(I2ph1, I2ph2, I2ph3)−Min(I2ph1, I2ph2, I2ph3)I2ph1 + I2ph2 + I2ph3×100% (2.19)As mentioned earlier, the converter has been tested under the worst case unbalanced scenario. Fig.2.11 shows the simulation results of the ideal balanced circuit versus the experimental results of the real23Table 2.3: Experimentally verified values of resonant components. These values are corresponding tothe discussed worst case scenarios.Ideal componentsPhases Lr [µH] Cr [nF] Lm [µH]Phase I 20 30 60Phase II 20 30 60Phase III 20 30 60Real components (Experimental setup)Phases Lr [µH] Cr [nF] Lm [µH]Phase I 23 33.2 59Phase II 18.7 26.8 58.6Phase III 18 26.9 57.5unbalanced circuit. Figs. 2.11 (a) shows the results at the resonant frequency, Figs. 2.11 (b) shows theresults above the resonant frequency, and Figs. 2.11 (c) shows the results below the resonant frequency. Ascan be seen, the output current ripple of the unbalanced circuit is more than double the value in ideal balancedcircuit. That is the main disadvantage of the unbalanced behavior in a three-phase LLC resonant converter.Uneven distribution of currents among the phases is the other disadvantage of the unbalanced behavior ofthe converter, which results in unequally distributed losses and unequally thermal distribution. Dependingon the deviation in the normal values of the currents, this also results in slightly lower efficiency, which iscaused by higher peak currents and higher conduction losses. The exact values of three phase currents, aswell as the efficiencies of each operation mode, are shown in Table 2.4.As mentioned before, the resonant components’ values are corresponding to the scenarios II and IV dis-cussed in previous Section. In order to validate the current-sharing trend prediction ability by the proposedanalysis, the plots in Fig. 2.12 are presented. In this figure, the analytical results of the normalized cur-rent values over the frequency range, versus the experimental results are shown. As it can be seen, boththe experimental and analytical results show the same trend of changes in the values of the currents withcorrespondence with the given resonant components’ values in Table 2.3. As the results show, phase 3 hasthe highest current, and phase 2 has the lowest current.In order to verify the proposed TCB, this technique has been applied to the unbalanced circuit discussedabove. The technique was applied off-line to calculate the angles and verify the technique. Fig. 2.13 showsthe results of the balancing technique versus the results of the unbalanced circuit with basic 120◦ modulation.This figure clearly shows that TCB results in considerable reduction in the output current ripple and balancesthe RMS values of the currents through the phases. In Table 2.4, the experimental results of the current24Figure 2.11: Simulation results of the ideal circuit (left side) versus the results of the real unbalancedcircuit (right side): a) At resonant frequency; b) above resonant frequency; c) below resonantfrequency. Due to components’ tolerances, unbalanced currents are generated in the experimen-tally implemented circuit. As a result, the output current ripple in the experimental results issignificantly higher than the ripple in the ideal circuit. Moreover, the unbalanced behavior re-sults in unequal conduction power loss in the circuit and unequal thermal distribution.values, unbalanced factor (U f ), efficiency, output current ripple, and the phase angles of the input voltagesin the ideal circuit, the unbalanced circuit, and TCB, are presented. The comparison of primary currentsunder different operating modes, unbalanced factor (U f ), and also current ripples for different conditionsare presented in Fig. 2.14. This figure clearly confirms that the proposed method can successfully balance25Table 2.4: Simulation results of the ideal circuit and experimental results of the unbalanced circuitunder different conditions.Below resonant frequencyParameters Ideal Circuit TCB not Active TCB ActivePrimary currents (rms) [A]Iph1 : 6.3 Iph1 : 6.85 Iph1 : 6.44Iph2 : 6.3 Iph2 : 5.23 Iph2 : 6.25Iph3 : 6.3 Iph3 : 7.65 Iph3 : 6.45Output current ripple [A] 1 7 3Unbalanced factor (U f ) 0% 23.5% 2%Efficiency [%] – 96.65 96.9Input voltage vectors anglesφ12 : 120◦ φ12 : 120◦ φ12 : 141◦φ13 : 120◦ φ13 : 120◦ φ13 : 106◦φ23 : 120◦ φ23 : 120◦ φ23 : 113◦At resonant frequencyParameters Ideal Circuit TCB not Active TCB ActivePrimary currents (rms) [A]Iph1 : 6.3 Iph1 : 6.8 Iph1 : 6.25Iph2 : 6.3 Iph2 : 4.7 Iph2 : 6.36Iph3 : 6.3 Iph3 : 7.3 Iph3 : 6.22Output current ripple [A] 1.8 5.3 2.4Unbalanced factor (U f ) 0% 25.6% 1.5%Efficiency [%] – 96.8 97.2Input voltage vectors anglesφ12 : 120◦ φ12 : 120◦ φ12 : 141◦φ13 : 120◦ φ13 : 120◦ φ13 : 98◦φ23 : 120◦ φ23 : 120◦ φ23 : 121◦Above resonant frequencyParameters Ideal Circuit TCB not Active TCB ActivePrimary currents (rms) [A]Iph1 : 5.9 Iph1 : 6.20 Iph1 : 5.85Iph2 : 5.9 Iph2 : 4.89 Iph2 : 5.95Iph3 : 5.9 Iph3 : 6.96 Iph3 : 6.00Output current ripple [A] 2 6 3.3Unbalanced factor (U f ) 0% 22% 1.7%Efficiency [%] – 96.46 96.72Input voltage vectors anglesφ12 : 120◦ φ12 : 120◦ φ12 : 154◦φ13 : 120◦ φ13 : 120◦ φ13 : 100◦φ23 : 120◦ φ23 : 120◦ φ23 : 106◦the currents in an unbalanced three-phase LLC resonant converter, resulting in significant reduction in theoutput capacitor current ripple and even distribution of losses through the circuit. Table 2.4 presents the finalvalues of the voltage vector angles used by TCB to balance the currents. According to this table, the largestdeviation in the value of voltage vector angles happens above the resonant frequency, where TCB applies 34phase shift on the phase 1 input voltage. Moreover, the efficiency is improved using TCB. For example, at26fnInIph30.70.80.911.11.21.40.10.75 1.25100.20.30.50.40.61.3ExperimentalAnalyticalIph2Iph1Figure 2.12: Current-sharing trend prediction with proposed analysis versus the experimental results.The analysis reasonably predicts the trend of changes in the currents with the given components’values.resonant frequency, the efficiency of the real unbalanced circuit is improved by 0.4 percentage points usingTCB, which means around 12W less losses at full load operation.Moreover, in order to show the performance of the proposed TCB in different load levels, the test resultsof light-load and half-load conditions are provided in Fig. 2.15 and Table 2.5. The results validate theperformance of the proposed TCB for different power levels. It is clear that the worst case of unbalancedcurrents happens at full-load condition and the issue of the unbalanced currents is less critical in light-load conditions. The proposed TCB technique can perfectly balance the currents, reduce the output currentripple, and improve the performance of the converter at any load condition. In fact, the results show that theperformance of the proposed TCB doesn’t depend on the load.2.4 SummaryIn this Chapter, an analysis of the current-sharing behavior of three-phase LLC resonant converter in corre-spondence to the resonant components’ tolerances has been presented and discussed. Moreover, a Trigono-metric Current Balancing (TCB) technique to improve the unbalanced behavior of the converter has beenpresented. The proposed TCB manipulates the phase-shift between the inverter legs to balance the currentsthrough the phases. This will result in significant decrease in the output current ripple, which allows the useof smaller output capacitor filter. Moreover, this technique results in equally distributed losses in the circuit,which makes thermal management of the circuit easier. As a result of even current-sharing among phases,27Figure 2.13: Results of the the unbalanced circuit without TCB (left side) versus the results of TCB(right side): a) At resonant frequency; b) above resonant frequency; c) below resonant frequency.Using the proposed TCB, the primary currents are perfectly balanced, and as a result, the outputcurrent ripple is significantly reduced.no component experiences higher temperature or larger current than the expected current peaks and so it ispossible to select components with lower power rating. In order to show the validity of the presented analysisand also the functionality of TCB, a 3kW experimental prototype for PHEV application has been developedand tested. The experimental results show the validity of the presented current-sharing analysis with a goodapproximation, and also the validity of the proposed TCB.28≈045678025201510555(b)Iph1 Iph2 Iph3 UfWithout TCBWith TCB9%30%34%15%21%21.5(c)(a)(A)(A)(A)(%)≈045678Iph1 Iph2 Iph3≈045678Iph1 Iph2 Iph3025201510Uf24(%)025201510Uf20.3(%)8%20%19%5%02468I (ripple)o4(A)2.902468I (ripple)o(A)(A)02468I (ripple)o2.7Without TCBWith TCBWithout TCBWith TCBFigure 2.14: Experimental results: primary currents RMS values and output current ripple values ofthe real circuit without TCB versus with TCB: a) below resonant frequency; b) at resonantfrequency; and c) above resonant frequency.Table 2.5: Experimental results at resonant frequency before and after TCB. The results verify theeffectiveness of the proposed TCB at different load levels.Before TCBParameters Full-load half-load light-loadPrimary currents (rms) [A]Iph1 : 6.8 Iph1 : 3.8 Iph1 : 2.3Iph2 : 4.7 Iph2 : 3.3 Iph2 : 2.7Iph3 : 7.3 Iph3 : 4.5 Iph3 : 2.1Output current ripple [A] 5.3 2.5 2.2Unbalanced factor (U f ) 25.6% 20.5% 17%After TCBParameters Full-load half-load light-loadPrimary currents (rms) [A]Iph1 : 6.26 Iph1 : 3.8 Iph1 : 2.35Iph2 : 6.3 Iph2 : 3.95 Iph2 : 2.45Iph3 : 6.36 Iph3 : 3.85 Iph3 : 2.35Output current ripple [A] 2.4 1.2 1Unbalanced factor (U f ) 1.5% 2.5% 2.8%29Figure 2.15: Experimental results of the the unbalanced circuit without TCB (left side) versus the re-sults of TCB (right side) at resonant frequency at: a) 50% of full-load, and b) 10% of full-load.30Chapter 3Extended Gain Range and ImprovedLight-Load Efficiency1Wide voltage range and flat efficiency curve are critical requirements for high power battery charger con-verters. As mentioned in Chapter 1, Some important issues with three-phase LLC resonant converters are:1) the natural limited gain range of the LLC resonant converter structure, 2) poor light-load efficiency ofthe converter and possible phase-shedding strategies. Fig. 3.1 shows how these two issues limit the perfor-mance of the converter in a battery charger application. As shown, there are typically three main phases incharging a battery: recovery stage (pre-charge), bulk charging (constant current), and absorption (constantvoltage). The recovery stage usually happens when the battery is over-used or has been stored for a longtime. In this situation, the battery is in the dead-zone and needs to be recovered before the bulk chargingstarts. This is also called the pre-charge stage, and low voltage and low current is required to recover thebattery. As mentioned, the other issue with three-phase LLC resonant converters is their poor light-loadefficiency. In the absorption stage (constant voltage stage), the battery terminal voltage is kept constant andthe charge current starts decreasing to very light-load conditions. As the charging current decreases and theconverter goes to lighter load conditions, the three-phase structure will not be efficiently charging anymore.The light-load absorption stage happens for much longer time than other charging stages. This makes it evenmore important to improve the light-load efficiency of the converter, and the power saving over time will besignificant. Modular and interleaved structure of three-phase LLC converters makes it more challenging toemploy phase-shedding strategies with the aim of light-load efficiency improvement.In this Chapter, a modified three-phase LLC resonant converter with a new phase-shedding technique is1Portions of this chapter have been modified from [2, 3]31Battery Voltage and CurrentThree-Phase LLC Converter Modified Three-Phase LLC ConverterExtended gain forrecovering battery.Low Cost Small SizeMinor Change2 Low power-rateddiodesImproved light loadefficiency.Single-PhaseOperationBulk ChargingPre-ChargeTwo-PhaseOperationS1D 5D3Dov2D6D7D8D4D1S3 S5S2 S4 S6inVLr CrLr CrLr Cr+-S1D 5D3Dov2D6D4D1S3 S5S2 S4 S6inVLr CrLr CrLr Cr++-Low efficiencyNot CoveredVoltageCurrentAbsorption Charging Bulk ChargingPre-ChargeAbsorption ChargingHigh PowerChargingLigh Load Operation Ligh Load OperationThree-PhaseOperationThree-PhaseOperationVoltageCurrentImproved EfficiencyNo SacrificeExtended GainExtended GainLow PowerLow VoltageLow PowerLow VoltageHigh PowerChargingTimeFigure 3.1: Proposed modified three-phase LLC resonant converter. Compared to a conventional three-phase LLC converter, the modified topology has two more small, low power rated diodes. Theproposed modified converter provides two-phase and single-phase operations of the circuit. Theseprovide light-load efficiency improvement, and extended gain range for covering the dead zonecharging of a battery, receptively.proposed to address the regulation and light-load performance issues of the converter. These are achievedby proposing single-phase and two-phase operations of the converter, respectively. Compared to the conven-tional three-phase LLC resonant converter, the proposed topology has two additional small, low power rateddiodes as shown in Fig. 3.1. The proposed technique delivers the whole package of features needed by athree-phase LLC resonant converter in a high-power battery charger application.3.1 Proposed Phase-Shedding StrategyThe proposed modified three-phase LLC resonant converter to achieve extended gain range and improvedlight-load operation is shown in Fig. 3.1. As mentioned in the previous section, in the modified convertertwo small low power rated diodes are added to the secondary side of the conventional three-phase LLCresonant converter. The proposed converter has three operating modes: three-phase mode (Fig. 3.2 (a)),two-phase operating mode (Fig. 3.2 (b)), and single-phase mode (Fig. 3.2 (c)). As explained in Fig. 3.2,with single-phase operation the recovery zone of charging is covered, three-phase mode is used for the bulkcharging period, and two-phase operation is for covering the absorption. With these three operating modes,all the charging stages are efficiently covered as shown 3.2 (d). In this section, the three operating modes areanalyzed and explained.32≈ 4.1 V ≈ 2.5 V Battery Voltage(c)(d)VLPTVPREPre-ChargeConstantCurrentConstantVoltageTimeVTICHGICHG : Charge CurrentVT : Battery Termination VoltageVPRE : Pre-charge finished ≈ 0.6 VTVLPT : Low Protection Threshold ≈ 0.25 VTSingle-Phase OperationThree-Phase OperationIPRE : Pre-Charge current ≈ 0.1 ICHGIPRECharge CurrentBulk ChargingAbsorptionDead-zone≈ 1.2 V Two-Phase OperationVoltageCurrent(a)(b)+-S1D5D3Dov2D6D7D8D4D1S3 S5S2 S4S6inVLr1 Cr1Lr2 Cr2Lr3 Cr3+-S1D5D3Dov2D6D7D8D4D1S3 S5S2 S4S6inVLr1 Cr1Lr2 Cr2Lr3 Cr3+-S1D5D3Dov2D6D7D8D4D1S3 S5S2 S4S6inVLr1 Cr1Lr2 Cr2Lr3 Cr3Figure 3.2: Proposed phase-shedding strategy: (a) Three-phase operation to cover bulk charging andhigh power absorption charging; (b) Two-phase operation to cover light-load absorption charging;(c) Single-phase operation to cover the recovering stage, where the battery is fully depleted andlow voltage and low current are required for safe charging; (d) Conceptual charging profile oftypical Li-ion batteries.3.1.1 Operating modesThe three operating modes of the proposed modified converter are shown in Fig. 3.2 (a), (b), and (c). UsingFHA and standard analysis principles for resonant converters, the equivalent circuits of the three operatingmodes of the converter can be derived as in Figs. 3.3, 3.4, and 3.5.Three-phase operationThe three-phase operation happens when the three switching legs are operating phase shifted by 120◦. Inthis situation, the converter operates as a normal three-phase LLC resonant converter as shown in Fig. 3.2(a). In this mode, the two added diodes (D7 and D8) are off. Three-phase operation of LLC resonantconverter is highly desirable for higher power levels. This is because of the unique features provided bythree-phase operation in comparison with conventional half-bridge and full-bridge structures. These featuresare: interleaved operation, significantly reduced output capacitor current ripple, reduced output filter size,more distributed losses, smaller peak current and temperature values, and lower conduction losses at higherpower levels and higher efficiency. Relying on these features, the proposed phase-shedding technique uses33Lr Cr LmLr Cr LmLr Cr LmRacRacRac(b)(a)Lr CrLm .Ro3PReq= .N26__ 20Vin-Vin+-Figure 3.3: Equivalent resonant tank circuit of three-phase operation: (a) resonant tank circuit in oper-ation, (b) equivalent circuit.the three-phase operation of the circuit to cover the high power charging stages of a charging cycle.The gain of the converter in this mode is the same as the gain of any conventional three-phase LLCresonant converter with full-bridge three-phase rectifier. The equivalent circuit of the converter in this modeis as shown in Fig. 3.3 (b) and the gain of the converter is as:M3P(h,Q, fn) =VOVin=1N√(1+ 1h − 1h f 2n )2 +Q23P( fn− 1fn )2(3.1)where N is the transformer turns ratio, h is the inductor ratio, fn is normalized frequency, and Q3P is thequality factor of the three-phase circuit as follows:N =NPNS, h =LmLr, fn =fsfr, fr =12π√Lr.Cr, Q3P =Z3PN2Req,Z3P = ZO =√LrCr. (3.2)Req is the equivalent load reflected to primary and is as shown in Fig. 3.3 (b) as:Req =6π2.Ro3P.N2 (3.3)where Ro3P is the output load at three-phase operation.Two-phase operationTwo-phase operation happens when one of the phases is shut down and the other two phases are operatingcomplementary at 180◦ phase shift. In this case, the converter behaves as a full-bridge LLC resonant con-34(a)(b)1D3Do+v-2D4DLr CrLr CrLmLm.Ro2P.N2+-0Vin-VinR'eq= 8__ 22Lr C /2r2LmFigure 3.4: Equivalent resonant tank circuit of two-phase operation: (a) resonant tank circuit in opera-tion, (b) equivalent circuit.verter. This mode is shown in Fig. 3.2 (b). This mode results in higher efficiency at lighter load conditionscompared to three-phase operation mode. This is due to the fact that at lighter load conditions, the switchinglosses and circulating power losses are dominant. Hence, shedding one of the phases results in lower lossesand higher efficiency. The equivalent circuit of this mode is shown in Fig. 3.4 and the gain of the converteris as follows:M2P(h,Q, fn) =VOVin=1N√(1+ 1h − 1h f 2n )2 +Q22P( fn− 1fn )2(3.4)where:Q2P =Z2PN2Req, Req =8π2.Ro2P.N2, h =LmLr, fr =12π√Lr.Cr, Z2P = 2ZO = 2√LrCr. (3.5)The two-phase operating mode is expected to cover a major part of the absorption stage of charging. In thatstage, the charger is in constant voltage mode and the charger is supposed to provide the maximum voltage(VT ) at the output. So, in the two-phase operating mode, the voltage gain is supposed to cover an outputvoltage fixed at VT at light-load conditions.Single-phase operationThe single-phase operation happens when one of the switching legs is running and the other two legs are off.In this situation, the current path is closed through the ground connection of the transformer in the primary.At the secondary side, diodes D7 and D8 form a full bridge rectifier along with diodes D1 and D2 as shown35(a)(b)1D7Do+v-2D8DLr CrLm.Ro1P.N2R''eq= 8__ 2Lr CrLm+-+-0(1/2).Vin(-1/2).VinFigure 3.5: Equivalent resonant tank circuit of single-phase operation: (a) resonant tank circuit in op-eration, (b) equivalent circuit.in Fig. 3.2 (c). The equivalent circuit of the converter in this mode is as shown in Fig. 3.5 and the gain ofthe converter is as:M1P(h,Q, fn) =VOVin=12N√(1+ 1h − 1h f 2n )2 +Q21P( fn− 1fn )2(3.6)where:Q1P =Z1PN2Req, Req =8π2.Ro1P.N2, h =LmLr, fr =12π√Lr.Cr, Z1P = ZO =√LrCr. (3.7)As can be seen in Fig. 3.5 (a), in the single-phase operating mode, the converter behaves as a half-bridgeLLC resonant converter and that is why the input voltage of the resonant tank in Fig. 3.5 (b) is half thevoltage value in three-phase and two-phase operating modes. Because of the half-bridge operation, the gainof the converter in the single-phase mode is almost half the value of that in the other two operating modes.This feature provides the low voltage and low current needed for covering the dead zone of battery charging.3.1.2 Phase-shedding strategyThe intended outcomes of using a three-phase circuit are to improve power distribution in the circuit and toachieve higher power while maintaining efficiency. At high power levels, the conduction losses dominatethe total losses in the circuit. Conduction losses are proportional to the square value of the peak current.Efficiency at high power levels is improved by reducing the peak conducting current by sharing the currentbetween the three phases in the circuit. For these reasons, the three-phase operation is desirable for bulkcharging and some high power charging portions of the absorption stage. With the three-phase operation, the36STARTDisconnect BatteryMeasure:VBAT, IBATVBAT> VLPTYESNO NO NO NOSingle-PhasemodeConstant Currentat IPREYES YESVBAT> VPREThree-PhasemodeConstant Currentat ICHGVBAT≥ VTThree-PhasemodeConstant Voltageat VTTwo-PhasemodeConstant Voltageat VTIBAT< INYESFigure 3.6: Conceptual flowchart of phase-shedding strategy in the proposed modified three-phase LLCresonant converter.converter is expected to cover the output voltage range from VPRE to VT at different loading conditions (frommaximum current (ICHG) to IN (where the three-phase mode will be switched to two-phase mode).The problem within three-phase converters is that the efficiency drops at low power levels. That is due tothe fact that, at low power levels, the switching losses, gate driving circuit’s losses, and circulating currentslosses dominate the total losses in the power stage. Efficiency improvement in low power levels can beachieved through phase-shedding. Phase-shedding for the three-phase resonant converter is achieved byusing two-phase operation, as discussed earlier. In this operation mode, one of the phases is shut downand carries no load, no circulating current, and no switching and driving losses. That leads to improvedefficiency at light-load conditions and considerable power saving over time. In the two-phase operatingmode, the converter must provide the maximum output voltage (VT ) from IN to very light-load conditions.When the converter operates in the single-phase mode, the gain of the converter is reduced to almost halfthe value compared to the other two modes. With single-phase operation, the low voltage and low currentneeded for the recovery mode can be easily provided. Due to the low current in the recovery mode, the twoadded diodes (D7 and D8) are small low rated diodes. Moreover, since the recovery mode is supposed tohappen at one tenth of the nominal power, the single-phase operation can easily take that power.Considering the above mentioned features of the three operating modes, the phase-shedding strategy forbattery charging is shown in Fig. 3.6. This flowchart shows how the phase-shedding strategy works as it isdefined in Fig. 3.2 (d).37Ipr(t)ILm(t)tt0T/2TIsec(ave)Figure 3.7: The subtraction of the total primary current and magnetizing current reflects the powertransferred to the load.3.1.3 Power Loss AnalysisThe intended outcome of two-phase operation is achieving higher efficiency at low-power absorption charg-ing. The following analysis is to mathematically prove the higher efficiency of two-phase operation com-pared to three-phase operation at low power levels.Conduction LossesThese part of losses are investigated as primary losses and secondary losses:• Primary Conduction Losses: For the primary conduction losses, the RMS values of the primary currentsis needed. Fig. 3.7, conceptually shows how the primary current and the magnetizing current look like. Thepeak value of the magnetizing current in both three-phase mode and two-phase mode are equal and can bedetermined as: ILm(Peak) =NPNS×VO22.Lm× T2 =NPNS×VO28.Lm. fswIpr(t) =√2IRMS sin(ω)−→ Ipr(t0) =√2IRMS sin(φ) =− NVO8Lm fsw(3.8)As depicted in the figure, the difference between the primary and magnetizing current is equal to the trans-ferred current to the secondary side. Therefore:2T∫ T20(ipr(t)− iLm(t))dt = Isec(ave)N (3.9)where, Isec(ave) is the average current in the secondary side of the transformers and is given as in the follow-38ing equation, in the different operating modes:Two−Phase Operation :Isec(ave) = IOThree−Phase Operation :Isec(ave) = 2IO/3(3.10)Now, by solving equation 3.9, the RMS values of the primary currents in both operating modes can be foundas: Two−Phase Operation :I2pr,3P(RMS) = (π28N2 )× I2O + N264×2×L2m× f 2 ×V2OThree−Phase Operation :I2pr,2P(RMS) =49 × ( π28N2 )× I2O + N264×2×L2m× f 2 ×V2O(3.11)As it can be seen in (3.11), there are two terms that determine the total RMS value of the primary currents.The first term is a function of output current (IO), which determines the transferred power. The other term isa function of output voltage, which determines the circulating power generated by the magnetizing inductor.Using these values, the total primary conduction losses of both operating modes can be estimated as:Two−Phase Operation :Pcond,pr : 2× I2pr,2P(RMS)× (RDS,ON +RL +Rw)Three−Phase Operation :Pcond,pr : 3× I2pr,3P(RMS)× (RDS,ON +RL +Rw)(3.12)where, RDS,ON , RL, and Rw are the parasitic resistances of the MOSFETs, inductors, and transformer’swindings, respectively. The AC resistance of the inductors and transformers can also be integrated in thevalues of RL, and Rw. Fig. 3.9 (a), conceptually shows the trend of the primary conduction losses in bothtwo-phase and three-phase operating modes. As it can be seen, there are lower primary conduction lossesin two-phase operation at light load conditions. However, by moving to heavy load the losses in two-phaseoperation increases exponentially.• Secondary Conduction Losses: There are two different losses associated with the secondary side con-39duction losses: 1) losses associated with the voltage drop of diodes, 2) losses associated with the internalresistance of diodes.1) Losses associated with the voltage drop of diodes: To estimate this loss, the average value of thecurrent through diodes needs to be taken into account. The average current through diodes in both operatingmodes can be found as following:Two−Phase Operation : ID(ave) = IO/2Three−Phase Operation : ID(ave) = IO/3(3.13)The above values are derived based on the fact that in the three-phase operation, the peak value of the diodesis equal to IO×π/3. However, in two-phase operation, the peak value of diodes is equal to IO×π/2. Usingthe values in the equation above, the total loss can be estimated as:Two−Phase Operation :Pcond,sec1 : ID(ave)×VD,ON ×4 = 2× IO×VD,ONThree−Phase Operation :Pcond,sec1 : ID(ave)×VD,ON ×6 = 1.9× IO×VD,ON(3.14)Fig. 3.9 (c), conceptually shows the trend of this loss in both operating modes.2) losses associated with the parasitic resistances: For calculating this part of losses, the RMS values ofthe currents through diodes is required. These values can be calculated as:Two−Phase Operation : ID(RMS) = IO×π/4Three−Phase Operation : ID(RMS) = IO×π/6(3.15)The total losses associated with this part in both operating modes can be estimated as:Two−Phase Operation :Pcond,sec2 : 4×RD× I2D,RMS = 2.47×RD× I2OThree−Phase Operation :Pcond,sec2 : 6×RD× I2D,RMS = 1.65×RD× I2O(3.16)40tt0 T/3T/6Vo/3Vo/22Vo/3(a)(b)TTwo-phase mode:Three-phase mode:λ T/2tt0Tλ T/2Figure 3.8: The volt-second of the secondary side of the transformers in: (a) three-phase mode, (b)two-phase mode.Figure 3.9: The breakdown of the total estimated power loss over the output current range at constantoutput voltage: a) primary conduction losses (Pcond,pr), b) secondary conduction losses (Pcond,sec2),c) diodes foraward voltage drop losses (Pcond,sec1), d) transformers core losses (Pcore,Tr), e) induc-tors core losses (Pcore,L), f) switching losses (Psw), and g) the total power loss.In Fig. 3.9 (b), the trend of these losses are depicted, conceptually. By adding the values of (3.14) and(3.16), the total losses associated with secondary conduction losses can be estimated. As it can be seen, thisloss increases exponentially with increasing the load. Moreover, the losses in the two-phase circuit is alwayshigher than in the three-phase circuit. Although, this difference is not considerable at low output currents, itbecomes significant as the converter moves to higher power levels.Magnetic LossesThese losses are associated with the inductors and transformer:41• Transformers’ core Losses: The core losses of a magnetic component is calculated using the followingequation:Pcore = k f e fsw(∆B)βAClm (3.17)where, k f e is a constant, and the typical values of β for ferrite is around 2.6 or 2.7. ∆B is the maximumflux swing, AC is the cross sectional area of the core, and lm is the length of the magnetic path. Except for∆B, other parameters in the equation above are based on the core geometry and are equal in both operatingmodes. The maximum flux swing in both operating modes can be found using the volt-second relation of thetransformers. Fig. 3.8, shows the secondary voltage of the transformers in both operating modes. Therefore:Two−Phase Operation :Bmax =∫ T20 vtrdt2NsA= VO8 fswANsThree−Phase Operation :Bmax =∫ T20 vtrdt2NsA= VO8 fswANs(3.18)As can be seen in the equation above, transformers in both operating modes experience the same flux vari-ations. This means the same value of loss per transformers in both operating modes. However, the totaltransformer core loss for the three-phase operation would be 32 times higher than two-phase operation, asthere are three transformers under operation in three-phase circuit. This can be stated as:Total Pcore,Tr,3P =32Total Pcore,tr,2P (3.19)This losses is almost independent from loading conditions. In Fig. 3.9 (d), these losses are depicted, con-ceptually.• Inductors core Losses: As mentioned, the core losses of magnetic components are calculated using(3.17). In an inductor, the maximum flux swing is proportional to the peak current of the inductor and iscalculated as:Bmax =LIL,peakNA(3.20)42The peak value of the current in the inductors is equal to the peak current of the primary current and is as:Two−Phase Operation :IL,Peak =√2× Ipr,2P(RMS)Three−Phase Operation :IL,Peak =√2× Ipr,3P(RMS)(3.21)As expected, the primary current peak value is higher for two-phase operation at heavier loads comparedto three-phase operation. Using the values in above equations and (3.17), the relationship between the totalinductor core losses in two-phase operation and three-phase operation can be estimated as:Total Pcore,L,2P =23× ( IL,Peak,2PIL,Peak,3P)β ×Total Pcore,L,3P (3.22)Despite the transformers core losses, the total inductor core losses have a different behavior. In the verylight-load conditions, the primary peak current values are equal for both operating modes and that results inhigher inductor loss for the three-phase operation. However, as the load current increases, the peak currentvalue for the two-phase operation exponentially exceeds the value in three-phase operation. This results in ahigher total inductor loss in two-phase operation at heavier load conditions. Fig. 3.9 (e), show the conceptualtrend of this loss.Switching and gate-driver LossesSince the LLC converter provides ZVS at turn-on, there is negligible turn-on losses. The remained sourcesof switching losses are associated with turn-off and gate-drive losses. Turn-off happens at ILm(peak), whichis equal at both operating modes and is as (3.8). Therefore, the switching loss can be estimated as:Psw =12VinILm f tdis =12VinNVO8Lm× tdis (3.23)where, tdis is the discharge time of the MOSFET parasitic capacitance, and is calculated as:tdis =2CeqVinILm(3.24)43As can be seen from equations above, the turn-off losses are independent from the load current. This is thesame for the gate-drive losses. Hence, the total switching losses are equal to the calculated numbers timesthe number of active switches. The relation between the total switching losses in both operating modes canbe stated as:Total Psw,2P =23×Total Psw,3P (3.25)Fig. 3.9 (f), shows the relation between the switching losses of two-phase and three-phase operating modes.The total estimated power losses is derived by adding the different calculated losses mentioned above.Fig. 3.9 (g), shows the total estimated losses within two-phase and three-phase operating modes. Based onthis, a higher efficiency is expected in two-phase operation at light-load conditions.3.2 Design procedureThe goal of this section is to see how the three operating modes of the proposed modified converter areable to cover the design requirements of a battery charger application and to explain the design procedure.Resonant tank design of resonant converters is the most challenging part of the design. The first step for theresonant tank design is looking into the characteristics of the equivalent resonant tanks of the three operatingmodes. As can be seen from (3.2), (3.5), and (3.7), the resonant frequencies of the resonant circuits do notchange for the three-phase, two-phase, and single-phase operating modes of the converter. Moreover, theratio of the parallel inductor value to the series inductor value (h) is equal for all the operating modes. Thesetwo parameters are critical in designing LLC resonant converters and determining the frequency and gainrange of the converter. The gain of the DC-AC inverter bridge, which basically determines the DC gainof the converter at resonant frequency ( fr), is unity for both the three-phase and the two-phase operatingTable 3.1: Design worst case conditionsOperating mode Case Gain Worst case conditionThree-PhaseI M3P,Max QMax @ RO,Min= VTICGHII M3P,Min QMin @ RO,Max=VPREICGHTwo-Phase III M2P,Max QMax @ RO,Min=VTINSingle-PhaseIV M2P,Max QMax @ RO,Min=VPREIPREV M3P,Min QMin @ RO,Max=VLPTIPRE440.4 0.6 0.8 1 1.2 1.400.511.522.53Q = 0.08Q = 0.050.4 0.6 0.8 1 1.2 1.400.511.522.53Q = 0.25Q = 0.17Normalized Frequency fs/fr(a) (b) (c)Two-phase operation@ light load absorption mode (constant voltage)Single-phase operation@ low voltage, low current recovery charging zoneNormalized Frequency fs/fr Normalized Frequency fs/frMV= N VOUT / VIN0.4 0.6 0.8 1 1.2 1.400.511.522.53Q = 0.62Q = 0.40Q = 0.29Q = 0.23Three-phase operation@ Heavy load bulk charging and absorptionfs,minVTVTI IIIVIVIIVPREVPREVLPTFigure 3.10: Normalized gain curves of the different operating modes, where h = 3: (a) Three-phaseoperating mode, (b) Two-phase operating mode, and (c) Single-phase operating mode. The colorshaded areas show the charging states are covered by different operating modes as explained inFig. 3.2 (d), and I, II, III, IV , and V are the worst case design conditions as explained in Table3.1.modes. However, in the single-phase operating mode, the DC-AC stage delivers half the gain compared tothe three-phase and two-phase operating modes, as mentioned in the previous section. Comparing (3.2),(3.5), and (3.7), there are slight differences in the values of the impedance and the equivalent reflected loadof the equivalent resonant circuits of the three operating modes. These two parameters define the qualityfactors of the equivalent resonant tanks’ circuits and have an impact on the gain curves of the converter.So, the design of the converter needs to be optimized based on the minimum and maximum values ofthe quality factor for different operating modes. In order to do that, the worst case scenarios that need to beconsidered in designing the converter are defined as in Table 3.1. These worst case conditions are definedbased on the maximum and minimum gains required in different operating modes. As shown in Fig. 3.2 (d),the three-phase operating mode needs to cover the output voltage from VPRE to VT at heavy-loads to light-load conditions. Then, in the two-phase operating mode, the output voltage will be fixed at VT for light-loadto no-load conditions. The case for the single-phase operating mode is a bit different. In the single-phaseoperating mode, the output voltage varies from VLPT to VPRE at a fixed low output current (IPRE). Table3.1 summarizes this information. In Fig. 3.10, the normalized gain curves of the three operating modesare depicted, based on the design shown in Table 3.2. The gain curves of the different operating modescorrespond to the maximum and minimum values of quality factors as explained in Table 3.1.The LLC resonant converter’s design has been studied in details in the literature [10]-[16]. The designrule check flowchart of the proposed modified converter is shown in Fig. 3.11. Based on the fact that the two-phase and single-phase operations of the converter happen at fairly light-load conditions with low numbers45STARTENDYESYESYESSelect ZO , NSelect h (Eq. 2)CalculateQmax & Qmin for:3P2P1Poperations{Calculate:M3P, Max & M3P, MinAre M3P, Max& M3P, Min = required?Calculate M2P, Max Decrease hIncrease hIs M2P, Max= required?M2P, Max< requiredM3P, Max< required M3P, Min > requiredNONONOYESYESCalculate:M1P, Max & M1P, MinAre M1P, Max& M1P, Min = required?Decrease hIncrease hM1P, Max< required M1P, Min > requiredNONOYESDecrease hIncrease hNOFigure 3.11: Design rule checking.for the quality factors, the gain requirements of those operating modes would possibly be met if the converterwas properly designed for the three-phase operating mode. This can be observed in the plots in Fig. 3.10.However, this also depends on the design requirements, such as VPRE , VLPT , ICHG, IPRE , etc. For that reason,the design needs to be verified for the different operating modes. Based on that, the first step in the designprocedure of the proposed modified three-phase LLC resonant converter is to design a basic three-phase LLCresonant converter. Once the resonant tank is preliminary designed, the design rule checking, as shown inFig. 3.11, needs to be done. In this flowchart, the worst conditions defined in Table 3.1 are targeted. First,the conditions for the three-phase mode are verified, and then it proceeds to the two-phase and single-phasemodes, respectively. Once the requirements for all the worst conditions are met, the design is passed.Turn-on soft switching is the other important design consideration, which is determined with the switch-46on current. Switch-on current is equal to the magnetizing peak current (ILm(Peak)). Discharging the MOS-FET junction capacitor happens at that current and if enough dead-time is ensured, ZVS at turn-on will beachieved (please refer to section II-C-2). As indicated in (3.8), the magnetizing peak current in all the op-erating modes are equal. Therefore, if ZVS is ensured for the three-phase operation, changing the operationmode does not affect the turn-on ZVS.A significant advantage of the three-phase operation is the output capacitor current ripple reduction,which is due to the interleaved operation of the phases. This feature enables the use of a much smaller outputcapacitor (compared to a single-, or two-phase solution), which is particularly beneficial for high currentoutputs since the required capacitance is proportional to the load current. However, when the converterswitches to two-phase and single-phase operation, the ripple cancellation benefit is lost. Output capacitorcurrent ripple for the three-phase and two-phase operating modes are given by (3.26) and (3.27), respectively:∆ic = (1−√32).π3.IO ≈ 0.14× IO (3.26)∆ic =π2.IO ≈ 1.57× IO (3.27)where IO is the output current mean value. As can be seen, for the same output current mean value, the outputcapacitor current ripple in the two-phase operating mode is more than 10 times greater than the ripple in thethree-phase operating mode. Depending on the point where the switch from three-phase mode to two-phasemode happens, the capacitance needs to be sized according to the maximum two-phase load current in agiven design. Hence, the value of the output capacitor must be determined based on the power value that theconverter operating mode is switched to the two-phase mode.As mentioned earlier, the two added diodes D7 and D8 only operate in the single-phase operating mode,where the output current is equal to IPRE , around one-tenth of ICHG. Therefore, these two diodes are rated atmuch smaller values compared to the other six diodes.3.3 Experimental ResultsIn order to verify the extended gain range and improved light-load operation of the proposed phase-sheddingstrategy, a 3kW prototype was developed and tested under different operating conditions. The specificationsof the developed converter are shown in Table 3.2. The converter is designed for charging a Li-ion batterypack consisting of a series connection of 24 battery cells. The nominal output voltage of the converter is 96Vand VLPT and VPRE are 32V and 60V , respectively. The maximum charging current (ICHG) is 30A, and the47Table 3.2: Design parameters of the experimental setupParameter ValuesVin 400VVout,nom 96VPout(max) 3kWVT 100VVPRE 60VVLPT 32VICHG 30AIPRE 3An1/n2 16/3fr 205kHzh 3Lr 20µHCr 30nFLm 60µHFigure 3.12: Experimental results of the single-phase operating mode: Primary currents waveforms.With single-phase operation, the recovery stage of charging can be covered without the need forburst mode operation.pre-charge current IPRE is 3A. Based on this design, the converter is expected to cover the output voltagesof 32V to 100V . Table 3.3, summarizes the important selected components. As can be seen, diodes D1-D6 package is TO-220, however, D7&D8 package is DO-219AD, which is a much smaller surface-mountpackage.Table 3.3: Components summaryComponent Part numberMOSFETs IPP60R190P6Diodes D1-D6 ST PS20120DDiodes D7&D8 V 3FM12Transformers core E42/21/20Inducotrs core RM1448Figure 3.13: Experimental results of the three-phase operating mode: Primary currents waveforms at60V, 79V, and 100V output voltage. The three-phase operation efficiently covers the bulk andabsorption stages of the charging profile.Figure 3.14: Experimental results of the two-phase operating mode: Primary currents waveforms at100V output voltage, and from light-load condition to half load. In this mode, the two phasesare operating at 180◦ phase shift. The two-phase operation mode covers the lighter load partof the absorption stage of charging cycle with a higher efficiency compared to the three-phaseoperating mode (Fig. 3.15)3.3.1 Operating modesConsidering the charging profile in Fig. 3.2 and the design parameters in Table 3.2, the converter will havethe following operating modes and conditions:• Single-phase operating mode: this mode covers the pre-charge stage of charging. It means that theconverter in this mode must be able to provide output voltages from 32V to 60V at an output current of 3A.The experimental results of the single-phase operation are shown in Fig. 3.12. As the experimental resultsshow, the converter is able to cover the pre-charge stage. In this mode of operation, the current through two49of the phases is zero and only one of the phases is operating. Diodes D1, D2, D7, and D8 form the outputrectifier of the converter in this mode. As explained earlier, diodes D7 and D8 are much smaller rated diodescompared to the other six diodes and they only operate in the single-phase operation.• Three-phase operating mode: This mode of the converter must be able to cover the bulk charging andabsorption stages of the charging profile. Referring to Fig. 3.2 and Table 3.2, the three-phase operationshould provide output voltages from 60V to 100V at 30A output current in the bulk charging stage, and100V output voltage at light-load conditions to full-load in the absorption stage. Fig. 3.13 presents theexperimental results of the converter at the three-phase operation. The experimental results validate that theconverter meets the design requirements of bulk and absorption charging in the three-phase operation. Asexplained earlier, in this mode the converter behaves as a normal three-phase LLC resonant converter, anddiodes D7 and D8 will not be conducting.• Two-phase operating mode: When the charging profile goes to the light-load conditions of the absorp-tion stage, the two-phase operating mode is expected to provide higher efficiency. Improved efficiency at thelight-load condition in the absorption stage is important because this part of charging takes much longer thanthe other stages of charging. Hence, the power savings over time are highlighted. The experimental resultsof the two-phase operation are shown in Fig. 3.14. As the results show, in this mode the converter can coverthe light-load to mid-load conditions of the absorption charging. In order to confirm the improved light-loadefficiency of the two-phase operation, the efficiency curves of both three-phase and two-phase operationsare plotted in Fig. 3.15. The results clearly validate the higher efficiency of the two-phase operation at thelight-load absorption stage. The efficiency improvement is around 5% at 100W and almost 1% at 500W .This also verifies the power loss analysis presented in section II-C and the trend in Fig. 3.9.3.3.2 Phase-shedding TransitionThe transition between the operating modes happens in the following sequence:1) Transition from single-phase operation to three-phase operation:This transition happens when the low-power pre-charge has been done and the bulk charging needs tobe started. A non-interrupted power delivery is not required for this transition. This is why burst-modeoperation is allowed for the pre-charge mode. This means that when the pre-charge is done, the converter canbe restarted to start bulk-charging with the three-phase operating mode. This is done through a handshakingprotocol between the Battery Management System (BMS) and the charger. In the case of transition fromsingle-phase operation (pre-charge) to three-phase operation (bulk charging), restarting the converter takes50Power (kW)Efficiency %750 0.5 1 1.5 2.52 380859095100Two-PhaseThree-PhaseFigure 3.15: Efficiency curves of two-phase operation versus three-phase operation. This test has beendone at the maximum output voltage (Vo=100V). Light-load efficiency improvement by two-phase operation is evident.Figure 3.16: Experimental result of the transition from three-phase operation to two-phase operation atlight-load absorption charging.less than few milliseconds, which is is not considered as an issue compared to the total hours of chargingprocess. The stabilized operation of the converter in this transition is inherent.2) Transition from three-phase operation to two-phase operation:This transition happens when the constant-current charging (bulk charging) and most of the constant-voltage (absorption) has been passed, and the converter is entering the low-efficiency light-load absorptioncharging at maximum output voltage. This is where the two-phase operation offers a higher efficiencyas depicted in Fig. 3.15. Despite the transition from pre-charge to bulk-charging mentioned above, herean uninterrupted power delivery is preferred as it may cause some confusions for the BMS. Based on the51analysis provided in the section II-A, the two-phase and three-phase operating modes of the converter havevery similar behavior characteristics at light load conditions, which allows a very soft transition between themodes. The transition is simply done by shutting down one of the phases and phase shifting the other twoswitching legs from 120◦ to 180◦ at the same time. Fig. 3.16, shows the experimental result of the transitionof the converter from three-phase operation to two-phase operation at light-load absorption charging. Theexperimental and simulation results validate the excellent stability of the converter during the transition fromthree-phase operation mode to the two-phase mode. This transition is done at the output current of 3A, andthe output voltage of 100V .52Chapter 4Unbalanced bi-directional Three-phaseCLLC resonant converters1In some certain high-power battery charging applications, a bidirectional power flow is required. In thisChapter, the three-phase bidirectional CLLC resonant converter under a real unbalanced condition is ana-lyzed. Fig. 4.1, shows how the tolerance in the resonant component values can lead in unbalanced currentsand higher output current ripples. A higher tolerance value leads to a more unbalanced behavior. Moreover,the proposed TCB technique is also validated to be effective as a means to mitigate the unbalanced behaviorof the converter in three-phase CLLC converters.4.1 Equivalent CircuitsIn this section, the often neglected tolerance in the components is considered to derive the equivalent circuitof the converter under real unbalanced conditions. Analyzing the unbalanced behavior of the converter is ofsignificant importance because it addresses two main concerns about the converter operation: 1) identifyingthe worst unbalanced operation case with a given tolerance in the component values, and 2) finding themaximum allowed tolerance to keep the converter working within an acceptable unbalanced range.Previous works develop the balanced operation analysis of the converter based on a derived single-phaseequivalent circuit. However, in order to analyze the unbalanced behavior of the converter, a three-phaseequivalent circuit is needed. This can be done using FHA and modeling the capacitive output rectifier. Athree-phase rectifier in a three-phase resonant converter can be modeled with three reflected AC resistances,1Portions of this chapter have been modified from [4]53High Unbalanced Factor (UF)Fifteen resonant componentsPerfect Balanced CurrentsWorst-Case Analysis:Identifying the worst-case valueswith a given tolerance in components.Predicting the effect oftolerance on the target variables.Real CircuitBeyond AcceptableRange (T>X%):Ideal Circuit(T=0%):UF>YUF=0 ∆ioioLow RippletimeoioHigh RippleWorst-case UFwith X% ToleranceHighest OutputCurrent Ripple with X% Tolerancetimei", i#, i$:i", i#, i$:Acceptable Unbalanced Factor (UF)Real CircuitAcceptable Range(0%<T<X%):0<UF<Y ∆i <Zo∆i >ZioAcceptable Rippletimei", i#, i$:∆i (A)oUF(%)Tolerance (T)Tolerance (T)Target variables:Unbalanced Factor (UF):Output Current Ripple:10%5%0% 15%30%0%8410%5%0% 15%BL11±T L21±TLm1±TLm2±TC11±T C21±TL22±TC22±TL23±TC23±TL12±TC12±TLm3±TL13±TC13±T+-S S1DD5D3DV2D D641 3S5S2S4S6PFCVi"Bii#i$PFCiFigure 4.1: Resonant components tolerance issue: unbalanced circuit results in uneven peak currents,unequally distributed heat and losses, and higher output current ripple. Proposed analysis method-ology facilitates predicting the behavior of the converter with fifteen unknown variables and iden-tifying the worst-case values.as presented in Chapter 2. This AC resistance is indicated as Rac and can be derived as:Rac = N2Vac,rmsIac,rms=6π2N2RO (4.1)where N is the transformers’ turns ratios (NP/NS) and RO is the output resistance. Using FHA and reflectingall the components to the primary side, the three-phase equivalent circuit of the converter is provided in Fig.4.2. In this figure, v1, v2, and v3 are the first harmonics of each of the inverter leg voltages and are equal to:vn =2πVin sin(ωst +(n−1)π3 ), n = 1,2,3 (4.2)Similar to the work in Chapter 2, the relation between the phases voltages and currents can be derived as:i1i2i3= ( 1Z1Z2 +Z2Z3 +Z1Z3 )×Z2 +Z3 −Z3 −Z2−Z3 Z1 +Z3 −Z1−Z2 −Z1 Z1 +Z2v1v2v3 (4.3)54L11L21Lm1Z2Z3Z1C11C21L12Lm2C12L13Lm3Cr13Racφ2V1V2V3φ1φ3N2 /N2L22C22Rac N2/N2L23C23Rac N2/N2v1(a) (b)v2v3i2i1i3Z2Z3Z1Figure 4.2: Equivalent three-phase circuit of the converter. This circuit is derived using FHA andmodeling the capacitive output filter: (a) expanded circuit, (b) simplified circuit.+-0VinRacL1 C1N2/N2L2 C2LmFigure 4.3: Single-phase equivalent circuit of the converter. The converter under the ideal balancedcondition can be analyzed using this equivalent circuit. However, this doesn’t provide informationabout the current-sharing behavior of the converter in a real unbalanced circuit.where i1, i2, and i3 are fundamental harmonics of the currents through the phases, and Z1, Z2, and Z3 are theimpedance of each phase and are given by:Zt = L1t jωs +1C1t jωs+Lmt ||(Rac +N2L2t jωs + 1N2C2t jωs ) t = 1,2,3 (4.4)The equivalent circuit in Fig. 4.2, along with 4.3, gives the starting point for analyzing the converter un-der balanced and unbalanced conditions. In the following sub-sections, the circuit is studied under bothconditions.4.1.1 Balanced CircuitA balanced circuit is an ideal case where all related components are equal, i.e. L11 = L12 = L13 = L1,L21 = L22 = L23 = L2, Lm1 = Lm2 = Lm3 = Lm, C11 =C12 =C13 =C1, and C21 =C22 =C23 =C2. Under this55I3I3I1 I1I2I2ααCγγCββCFigure 4.4: The law of sines relates the vectors’ amplitudes to the sines of the angles between thevectors.condition: Z1 = Z2 = Z3 = Z, and equation (4.3) can be simplified as (4.5). [1]i1i2i3= ( 13Z )2 −1 −1−1 2 −1−1 −1 2v1v2v3 (4.5)Given v1, v2, and v3 are balanced three-phase voltages and their sum is equal to zero, the equation can befurther simplified as:it =vtZ, t = 1,2,3 (4.6)The above equation indicates the balanced behavior of the converter with equal three-phase peak currents.This also validates using single-phase equivalent circuits for analyzing three-phase CLLC converters underideal balanced cases. Under the ideal condition, the equivalent circuit shown in Fig. 4.3 can be used to derivethe circuit characteristics such as gain curves.4.1.2 Unbalanced CircuitAs mentioned, the single-phase equivalent circuit works for an ideal balanced circuit and can only be usedfor deriving the general characteristics of the converter. Moreover, the single-phase equivalent circuit is notable to provide any information on the current-sharing analysis of the converter under a real unbalancedcondition. For this reason, the three-phase equivalent circuit in Fig. 4.2 is derived to analyze the converter ina real case scenario.Equation (4.3) is the base of the proposed current-sharing analysis. Analyzing this equation directly withfifteen unknown variables (L1n,L2n, Lmn, C1n, and C2n) requires heavy calculations. Therefore, the analysisis broken into four simple steps, as follows:1. The effect of the variations of each resonant component on the impedance of each phase is studied.562. The effect of the changes in the impedance of each phase on the phase-angles between the currentvectors is analyzed.3. These effects are combined to find the effect of the tolerances in each of the components on the phase-angles between the current vectors.4. The worst cases of the current sharing are identified using law of sines.In the following section, the current-sharing behavior of the converter is analyzed following the above-mentioned steps. Also, the worst cases of the unbalanced current will be identified.4.2 Current-Sharing AnalysisThe ultimate goal in this section is to identify the worst cases of current-sharing with a given tolerance inthe resonant components. This analysis addresses the two questions raised in the introduction about theunbalanced behavior of the three-phase CLLC resonant converters.4.2.1 Law of SinesThe current-sharing analysis is based on FHA. This means that only fundamental components of the currentsand voltages are considered, which are pure sinusoidal waveforms. Based on this, the relation between theprimary current waveforms (i1, i2, and i3) can be defined using the law of sines. The law of sines relates thelengths of a triangle to the sines of its angles. Based on the law of sines, the current waveforms in Fig. 4.4are related as:I1sin(β )=I2sin(γ)=I3sin(α)(4.7)Based on the above equation, it can also be concluded that:i f |I1|= |I2|= |I3| ⇐⇒ β = γ = αi f |I1| ̸= |I2| ̸= |I3| ⇐⇒ β ̸= γ ̸= αi f |I1|> |I2|> |I3| ⇐⇒ α > β > γ(4.8)In a balanced circuit, where |I1|= |I2|= |I3|, the angles α , β , and γ are equal to 120◦. When unbalanced,however, the angles will not be equal anymore. The worst-case scenarios of an unbalanced circuit can beidentified using (4.8) and studying the angles α , β , and γ . In the following, the effect of the tolerances onthese angles is studied with a sensitivity analysis.57|Zt|QCɷ0.00.4-0.10.3-0.20.21.21.11.00.90.8 0.00.1SL2nm=3m=5m=7ZtQCɷ30.420.310.201.21.11.00.90.8 0.00.1SL1nm=3m=5m=7ZtQCɷ0.430.320.2101.21.11.00.90.8 0.00.1SC2nm=3m=5m=7(k)|Zt|QCɷ0.40.40.30.30.20.20.11.21.11.00.90.8 0.00.1SL1nm=3m=5m=7|Zt|QCɷ1.20.40.80.30.40.20.01.21.11.00.90.8 0.00.1SLmnm=3m=5m=7|Zt|QCɷ0.60.40.50.30.30.20.40.20.11.21.11.00.90.8 0.00.1SC1nm=3m=5m=7(a) (b) (c) (d)(f) (g)|Zt|QCɷ0.00.4-0.10.3-0.20.2-0.3-0.41.21.11.00.90.8 0.00.1SC2nm=3m=5m=7(e)(i)QCɷ00.4-10.3-20.2-3-41.21.11.00.90.8 0.00.1SLmnm=3m=5m=7(h)ZtQCɷ10.450.3430.2201.21.11.00.90.8 0.00.1SC1nm=3m=5m=7ZtZtQCɷ0.420.310.201.21.11.00.90.8 0.00.1SL2nm=3m=5m=7Figure 4.5: Sensitivities of the impedance of each phase to its resonant components: a) S|Zt |L1 , b) S|Zt |L2 ,c) S|Zt |Lm , d) S|Zt |C1 , e) S|Zt |C2 , f) S∠ZtL1 , g) S∠ZtL2 , h) S∠ZtLm , i) S∠ZtC1 , k) S∠ZtC2 . These results are the first pieceof information needed for obtaining the total sensitivity of the phase-angles between the currentvectors to the variations in the resonant component values.(a)Sensitivity of α Sensitivity of βSensitivity of γ To variations of |Zt| To variations of Zt(b) (c) (d) (e) (f)(g) (h) (j) (i) (k) (l)(m) (n) (o) (p) (q) (r)Zb |Zb| (Ω)0.50.0-0.51.00 0 1020 3040 50Л/6Л/3Л/2αS|Z3|Zb |Zb| (Ω)0.50.0-0.51.00 0 1020 3040 50Л/6Л/3Л/2βS|Z1|Zb |Zb| (Ω)0.50.0-0.51.00 0 1020 3040 50Л/6Л/3Л/2γS|Z2|Zb |Zb| (Ω)0.20.10.00.30 0 1020 3040 50Л/6Л/3Л/2αS Z1Zb |Zb| (Ω)0.20.10.00.30 0 1020 3040 50Л/6Л/3Л/2βS Z2Zb |Zb| (Ω)0.20.10.00.30 0 1020 3040 50Л/6Л/3Л/2γS Z3Zb |Zb| (Ω)-0.1-0.2-0.30.00 0 1020 3040 50Л/6Л/3Л/2αZb |Zb| (Ω)-0.1-0.2-0.30.00 0 1020 3040 50Л/6Л/3Л/2βS Z3Zb |Zb| (Ω)-0.1-0.2-0.30.00 0 1020 3040 50Л/6Л/3Л/2γS Z1Zb |Zb| (Ω)0.50.0-0.51.0-1.00 0 1020 3040 50Л/6Л/3Л/2αS Z3Zb |Zb| (Ω)0.50.0-0.51.0-1.00 0 1020 3040 50Л/6Л/3Л/2βS Z1Zb |Zb| (Ω)0.50.0-0.51.0-1.00 0 1020 3040 50Л/6Л/3Л/2γS Z2Zb |Zb| (Ω)0.50.0-0.5-1.00 0 1020 3040 50Л/6Л/3Л/2αS|Z1|Zb |Zb| (Ω)0.50.0-0.5-1.00 0 1020 3040 50Л/6Л/3Л/2αS|Z2|Zb |Zb| (Ω)0.50.0-0.5-1.00 0 1020 3040 50Л/6Л/3Л/2βS|Z2|Zb |Zb| (Ω)0.50.0-0.5-1.00 0 1020 3040 50Л/6Л/3Л/2βS|Z3|Zb |Zb| (Ω)0.50.0-0.5-1.00 0 1020 3040 50Л/6Л/3Л/2γS|Z1|Zb |Zb| (Ω)0.50.0-0.5-1.00 0 1020 3040 50Л/6Л/3Л/2γS|Z3|S Z2Figure 4.6: Sensitivity of the phase-angles between the current vectors to the variation of the amplitudeand phase-angles of different phases’ impedance. These results are the second piece of informa-tion needed for obtaining the total sensitivity of the phase-angles between the current vectors tothe variations in the resonant component values.4.2.2 Sensitivity AnalysisThe simple definition of circuit sensitivity is how much a particular circuit characteristic changes as a par-ticular component value varies. Using the sensitivity analysis, the sensitivity of angles between the currentvectors (α , β , and γ) to the variations of each resonant component (L1t ,L2t , Lmt , C1t , and C2t) can be studied.58α αββγ→max γ→minI1I1I1I2I2→min I2→maxI3I3I3α=120ºγ=120ºβ=120ºBalanced Case(a)Worst-CaseScenario I(b)Worst-CaseScenario II(c)Figure 4.7: Two possible worst-case scenarios versus the balanced circuit based on the law of sines: (a)Balanced circuit, where all the angles are equal to 120◦; (b) Worst-case scenario I, where one ofthe angles is increased to its maximum value; (c) Worst-case scenario II, where one of the anglesis reduced to its minimum value.STARTENDObtaining sensitivities of amplitude and phase-angle of the impedance of each phase to its resonant components.(Section 4.2.2 and Fig. 4.5)Obtaining sensitivities of α, β, and γ to amplitude and phase-angle of the impedance of each phase.(Section 4.2.2 and Fig. 4.6)Identifying ωn, QC, and Zb for each of the selectedoperating points. Obtaining the sensitivity of one of the phase-angles (α, β, γ)to all the resonant components (Eq. 4.12), at differentoperating points.Calculating MS of that angle at different operatingpoints. (Eq. 4.20).Finding the operating point with highest MS.Creating the combination of variations in the resonantcomponents toward each of the worst-case scenarios based on the sensitivity signs in that operating point.Figure 4.8: Steps toward identifying the worst current-sharing cases.The equation below is the mathematical definition of circuit sensitivity:Syx =lim∆x−→ 0{∆y/y∆x/x}=xy∂y∂x(4.9)where Syx is the sensitivity of y to the variations of x. The sensitivity is the percentage that the dependentvariable, in this case y, changes relative to the independent variable x. Also, the sign of the sensitivity59(≈ pi/6)(≈ pi/3)(≈ pi/2)bFrequency [kHz]Z [radian]RO=25 [Ω]RO=30 [Ω]RO=35 [Ω]RO=40 [Ω]RO=50 [Ω]1.53.02.52.01.51.00.501.00.50.70.450.0-0.5-1.0-1.5-2.0Operating PointsVo =250VVo =250VVo =300VVo =300VVo =350VVo =350VVo =400VVo =400VFigure 4.9: Impedance phase-angles (∠Zb) at different operating points in Charge-Mode. These valuesare based on the designed parameters in Table 2.2. The values correspond to the operating pointsfrom 250V to 400V as indicated in the designed parameters table. The impedance values fordifferent operating points in Discharge-Mode can also be obtained in the same way.implies whether y increases or decreases in case of increasing x. Based on this definition, the followingsensitivity equations can also be derived:Sy1y2x = Sy1x +Sy2xSyx1 = Syx2 ×Sx2x1(4.10)According to (4.9) and (4.10), the sensitivity of the phase-angles between the current vectors to theTable 4.1: ωn, QC, and ∠Zb at different operating points.Charge Mode OperationOperating Point ωn QC ∠ZbA VB=250V, IB=10A 1.22 0.64 0.7B VB=300V, IB=10A 1 0.53 0.5C VB=350V, PO=3kW 0.8 0.4 0.5D VB=400V, PO=3kW 0.72 0.3 0.6Discharge Mode OperationOperating Point ωn QD ∠ZbE VB=300V, IPFC=7.5A 1 0.54 0.5F VB=400V, IPFC=7.5A 1.3 0.54 0.8860Table 4.2: Sign of sensitivity of γ to different resonant components at different operating points.Operating PointsCharge-Mode Discharge-ModeA B C D E FSγL11% -0.91 -0.17 -0.12 -0.1 -0.05 -0.19SγL21% -0.13 -0.06 +0.01 +0.02 -0.15 -0.17SγLm1% +0.06 +0.07 +0.04 -0.06 +0.06 +0.05SγC11% -0.15 -0.17 -0.2 -0.22 -0.05 -0.07SγC21% -0.09 -0.06 +0.02 +0.03 -0.15 -0.1SγL12% +0.14 +0.08 +0.04 +0.07 -0.05 +0.16SγL22% ≃ 0 -0.06 -0.07 -0.05 +0.07 +0.14SγLm2% +0.03 +0.07 +0.16 +0.3 +0.07 +0.03SγC12% +0.11 +0.08 +0.09 +0.14 -0.05 ≃ 0SγC22% ≃ 0 -0.06 -0.11 -0.09 +0.07 +0.08SγL13% +0.05 +0.08 +0.05 +0.03 +0.11 +0.03SγL23% +0.13 +0.12 +0.06 +0.02 +0.07 +0.04SγLm3% -0.09 -0.13 -0.24 -0.25 -0.14 -0.08SγC13% +0.03 +0.08 +0.11 +0.06 +0.11 +0.06SγC23% +0.08 +0.12 +0.1 +0.05 +0.07 +0.02MSγ 3.28 1.41 1.42 1.49 1.27 1.23variations of the resonant components can be written as:Sα,β ,γL1t = Sα,β ,γ|Zt | ×S|Zt |L1t +Sα,β ,γ∠Zt ×S∠ZtL1tSα,β ,γL2t = Sα,β ,γ|Zt | ×S|Zt |L2t +Sα,β ,γ∠Zt ×S∠ZtL2tSα,β ,γLmt = Sα,β ,γ|Zt | ×S|Zt |Lmt +Sα,β ,γ∠Zt ×S∠ZtLmtSα,β ,γC1t = Sα,β ,γ|Zt | ×S|Zt |C1t +Sα,β ,γ∠Zt ×S∠ZtC1tSα,β ,γC2t = Sα,β ,γ|Zt | ×S|Zt |C2t +Sα,β ,γ∠Zt ×S∠ZtC2tt = 1,2,3 (4.11)where |Zt | and ∠Zt are the amplitude and phase-angles of impedances Zt , respectively. Equation (4.11)indicates that the sensitivity of each phase-angle (α , β , or γ) to the variations of each resonant componentcan be broken up into two sub-effects: 1) sensitivity of amplitude and phase-angle of Zt to the variationsof the resonant component, and 2) sensitivity of the phase-angles (α , β , or γ) to the variations of |Zt | and∠Zt . Next, these two sub-effects will be derived. At the end, the total effect will be analyzed to identify theworst-case scenarios of current-sharing.61Table 4.3: Combinations of T% variations in the resonant components that result in the worst current-sharing behavior at different operating points in Table 4.1.Scenario I Scenario IICharge-Mode Discharge-Mode Charge-Mode Discharge-Mode@A @B @C @D @E @F @A @B @C @D @E @FL11 -T% -T% -T% -T% -T% -T% +T% +T% +T% +T% +T% +T%L21 -T% -T% – +T% -T% -T% +T% +T% – -T% +T% +T%Lm1 +T% +T% +T% -T% +T% +T% -T% -T% -T% +T% -T% -T%C11 -T% -T% -T% -T% -T% -T% +T% +T% +T% +T% +T% +T%C21 -T% -T% – +T% -T% -T% +T% +T% – -T% +T% +T%L12 +T% +T% +T% +T% -T% +T% -T% -T% -T% -T% +T% -T%L22 – -T% -T% -T% +T% +T% – +T% +T% +T% -T% -T%Lm2 +T% +T% +T% +T% +T% +T% -T% -T% -T% -T% -T% -T%C12 +T% +T% +T% +T% -T% – -T% -T% -T% -T% +T% –C22 – -T% -T% -T% +T% +T% – +T% +T% +T% -T% -T%L13 +T% +T% +T% +T% +T% +T% -T% -T% -T% -T% -T% -T%L23 +T% +T% +T% +T% +T% +T% -T% -T% -T% -T% -T% -T%Lm3 -T% -T% -T% -T% -T% -T% +T% +T% +T% +T% +T% +T%C13 +T% +T% +T% +T% +T% +T% -T% -T% -T% -T% -T% -T%C23 +T% +T% +T% +T% +T% +T% -T% -T% -T% -T% -T% -T%Sensitivity of Zt to resonant componentsAccording to (4.11), the effect of variations of the resonant components on the amplitude and phase-angleof Zt are needed. This can be expressed as: S|Zt |x = x|Zt |∂ |Zt |∂xS∠Ztx =x∠Zt∂∠Zt∂x(4.12)where x is any of the resonant components (L1t ,L2t , Lmt , C1t , and C2t). Impedance Zt in (4.4) can be rewrittenas:Zt = (L1tωs− 1C1tωs ) j+ωs Lmt N2−ωs3C2t L2t Lmt N2 +(ωs2C2t Lmt Rac)jωsC2t Rac +(C2t L2tωs2 N2 +C2t Lmtωs2−N2) j (4.13)The objective is to derive the derivatives in (4.12) based on the basic design parameters of CLLC resonantconverters. These parameters are:m =LmL1, ωr =1√L1C1, QC =√L1/C1Rac, ωn =ωsωr(4.14)62where m is the inductance ratio, ωr is the resonant angular frequency, QC is the quality factor, and ωn is thenormalized frequency. The analysis in this article is based on the conventional symmetric design, where:L1 = N2L2 = LrC1 =C2/N2 =Cr(4.15)Based on (4.14) and (4.15), the resonant components can be described and substituted based on the the designparameters in (4.14), as follows: Lr = QCRac/ωrLm = mRacQC/ωrCr = 1/(ωrQCRac)ωs = ωnωr(4.16)The set of steps and calculations to get the sensitivities in (4.12) based on the design parameters and normal-ized frequency is as follows:• First, by using (4.13) as Zt in (4.12), the equation is solved and the sensitivities to the different resonantcomponents (L1t ,L2t , Lmt , C1t , and C2t) are derived.• After that, the parameters L1t ,L2t , C1t , and C2t are substituted with the symmetric parameters Lr andCr as in (4.15).• Finally, the resonant components parameters Lr, Lm, Cr, and switching frequency ωs, are substitutedwith the design parameters as in (4.16).Following the above steps gives the sensitivities of |Zt | and ∠Zt to variations of each resonant componentas a function of the design parameters in (4.14). The results are very long equations, but they are plottedfor different values of ωn, QC, and m in Fig. 4.5. These results are the first piece of information needed forobtaining the total sensitivity of the phase-angles between the current vectors to the variations in the resonantcomponents.Sensitivity of phase-angles to ZtBased on (4.11), the second piece of information needed for obtaining the total sensitivity is the sensitivityof the phase-angles between the current vectors to the variations in the amplitude and phase-angle of the63impedance Zt . These can be expressed as:Sα|Zt | =|Zt |α∂α∂ |Zt | , Sα∠Zt =∠Ztα∂α∂∠ZtSβ|Zt | =|Zt |β∂β∂ |Zt | , Sβ∠Zt =∠Ztβ∂β∂∠ZtSγ|Zt | =|Zt |γ∂γ∂ |Zt | , Sγ∠Zt =∠Ztγ∂γ∂∠Zt, t = 1,2,3 (4.17)Angles α , β , and γ are as depicted in Fig. 4.4 and can be calculated as:α =| ∠i1−∠i2 |β =| ∠i3−∠i2 |γ =| ∠i1−∠i3 |(4.18)where i1, i2, and i3 are calculated as in (4.3). The sensitivities in (4.17) are solved based on the anglesin (4.18) and currents in (4.3). After obtaining the sensitivity equations for Z1, Z2, and Z3, the sensitivitynumbers around the balanced circuit can be calculated by substituting |Z1| = |Z2| = |Z3| = |Zb| and ∠Z1 =∠Z2 = ∠Z3 = ∠Zb. The results are very long equations, which can be plotted as in Fig. 4.6. These resultsalongside the results in Fig. 4.5 provide the necessary information to obtain the total sensitivities in (4.11).As can be seen in Fig. 4.6, the results are rotating for different angles(α , β , and γ). This is self-evidentbecause of the three-phase angles symmetry. Moreover, as can be seen, the sensitivities to |Zt | are constantover the whole range of impedance values, but sensitivities to∠Zt may depend on the value of the impedancephase-angle.4.2.3 Worst-Case ScenariosInitially, the worst-case scenario is defined as when the maximum possible RMS current is experienced. Inthis case, either one or both of the other two phases contributes a much lower current value to supply theoutput load. To experience such an unbalanced behavior, two possible scenarios are defined based on the lawof sines in Fig. 4.7:• Scenario I: One of the angles reaches the highest possible value: Based on (4.7), in this case, the cur-rent vector opposing this angle (in Fig. 4.7-(b), γ is the angle and i2 is the opposing vector) experiencesthe minimum possible absolute value.• Scenario II: One of the angles reaches the lowest possible value: In this case, based on (4.7), the cur-64rent vector opposing this angle (in Fig. 4.7-(c), γ is the angle and i2 is the opposing vector) experiencesthe maximum possible absolute value.Based on this, the worst current-sharing case happens when all the resonant components are varied toeither their maximum or minimum value toward each of the above cases. The steps to obtain the worst-casecombinations of variations in the resonant components are shown in Fig. 4.8. To better understand thesesteps, the analysis is studied for a conventional design of CLLC resonant converters. This is based on thedesign parameters presented earlier. The steps are as follows:1. The first two steps in the flowchart (Fig. 4.8) are to obtain the information in Figs. 4.5 and 4.6.2. The next step is to identify the values of variables ωn, QC, and ∠Zb. As can be seen in Figs. 4.5 and4.6, the sensitivity values are dependent on these variables. Therefore, the values of these variablesare needed to calculate the sensitivity in (4.12) and (4.17). Fig. 4.9 and Table 4.1 show these valuesat four different operating points. These operating points are selected for studying the current-sharingbehavior of the converter.3. Now, the next step is to find the sensitivity of one of the phase-angles (α , β , γ) to all the resonantcomponents. Using the values in Table 4.1 and equations (4.12) and (4.17), all the information forcalculating the total sensitivities in (4.11) is provided. Table 4.2 shows the total sensitivity values forthe under-study converter at different operating points.4. In order to realize which operating point can potentially result in a more unbalanced behavior, a max-imum total sensitivity is defined as:MSγ =3∑t=12∑y=1|SγLyt |+3∑t=1|SγLmt |+3∑t=12∑y=1|SγCyt | (4.19)Basically, this parameter indicates the maximum possible deviation in the angle at each operatingpoint. In other words, if all the resonant component values are varied by one percent in the directionthat causes the angle to either increase or decrease, the deviation in the angle would be equal to MSγ%.In this step, the operating point with the highest MSγ needs to be identified. According to Table 4.2,the operating point with highest MSγ value is point A at the charge-mode.5. Therefore, as pointed out in the previous step, point A in the charge-mode can potentially result inthe worst unbalanced behavior. The next step is to find the combination of variations in resonant65components that results in the worst unbalanced current. This can be done by simply by looking at thesensitivity signs at that operating point for different resonant components.By looking at the signs of sensitivities in Table 4.2, the possible combinations of variations in the resonantcomponents that result in the worst current-sharing behavior at each operating point can be identified. Table4.3 shows the worst-case combinations based on scenarios I and II in Fig. 4.7 and based on 10% tolerance inthe components. As pointed out in the steps above, the worst current-sharing behavior can happen at operat-ing point A in the charge-mode. Therefore, to find the worst-case scenario, the combination corresponding tooperating point A needs to be tested. Based on Table 4.3 and scenario II, the worst current-sharing behaviorhappens when the resonant component values are varied to the following values:L′11 = L11b +T %,L′12 = L12b−T %,L′13 = L13b−T %L′21 = L21b +T %,L′23 = L23b−T %L′m1 = Lm1b−T %,L′m2 = Lm2b−T %,L′m3 = Lm3b +T %C′11 =C11b +T %,C′12 =C12b−T %,C′13 =C13b−T %C′21 =C21b +T %,C′23 =C23b−T %, (4.20)where L1tb,L2tb, Lmtb, C1tb, and C2tb are the ideal values, and T is the maximum tolerance expected for eachresonant component. Based on scenario II, the above values for the resonant tanks would result in the worstcurrent-sharing behavior at operating point A, which in turn is the worst case over the whole operating range.Here is a summary of the above findings:• Evidently, the same combination of variations in the resonant components is not necessarily the worst-case for all the operating points in both charge-mode and discharge-mode.•By observing the MSγ values at different points, it can be seen that some operating points can potentiallyresult in a worse unbalanced behavior. This is associated with angles α , β , and γ being more sensitiveto variations in the resonant components. This can be further expanded by looking at Figs. 4.5 and 4.6.According to the sensitivity plots of angles to impedance values, with a higher impedance value (∠Zb )angles become more sensitive to the changes in the resonant components.• The same can be studied on the changes in the value of quality factors (QC and QD). Accordingto the sensitivity plots of impedance values to resonant components, depending on where the converter isdesigned, not all the plots show the same behavior with changes in quality factor. However, except for S|Zt |Lm ,all the other plots show higher sensitivities for higher quality factors. This means at higher power levels, the66current-sharing becomes worse. For this reason, the current-sharing behavior of the converter in this paperis studied for maximum currents and maximum power modes. Moreover, lower power level means lowercurrents, which reduces the level of importance for current-sharing.Once the set of variations in the resonant components toward the worst case in a particular operatingpoint is identified, the total variation in the angle (γ in this case) can be calculated as follows:∆γ% =3∑t=12∑y=1SγLyt ×∆Lyt%+3∑t=1SγLmt ×∆Lmt%+3∑t=12∑y=1SγCyt ×∆Cyt% (4.21)Table 4.4: Designed parameters of the experimental setup.Parameter ValuesVin 400VVout 250−400VPout(max) 3kWIout(max) 10ANP/NS 12/9fr 185kHzfs 135−225kHzm 3.33L1 15µH±10%L2 8.5µH±10%Lm 50µH±10%C1 50nF±10%C2 89nF±10%4.3 Experimental and Simulation ResultsIn order to verify the proposed current-sharing study, a 3− kW bidirectional CLLC resonant converter isdesigned and implemented experimentally. The design parameters of the prototype are shown in Table 4.4.The design is for a Hybrid Electric Vehicle (HEV) battery charging application, where charging the batteryTable 4.5: Experimentally verified values of resonant components. The resonant components are ac-curately built and measured to give a close enough combination to the worst-case scenario II atcharge-mode operating points A, B, and C.Phase L1t C1t Lmt L2t C2tI 16.4 µH 55nF 46.5 µH 9.35 µH 92 nFII 13.9 µH 45nF 46.2 µH 9.4 µH 95.3 nFIII 13.6 µH 45nF 56 µH 7.7 µH 80 nF67(b)(c)(d)(a) Number of SimulationNumber of SimulationNumber of SimulationNumber of Simulation252015108101254600 0.5 1.0 1.5 2.0 2.5 3.0 ×104UF (%)UF (%)UF (%)UF (%)I! (A)8101246I! (A)91157I! (A)91157I! (A)2016128400 0.5 1.0 1.5 2.0 2.5 3.0 ×1042016128400 0.5 1.0 1.5 2.0 2.5 3.0 ×1042016128400 0.5 1.0 1.5 2.0 2.5 3.0 ×104UFI!(rms)UFI!(rms)UFI!(rms)Scenario IIUF(max)I!(max)UF(max)Scenario II!(min)Scenario IIUF(max)I!(max)UF(max)Scenario II!(min)Scenario IIUF(max)I!(max)UF(max)Scenario II!(min)Scenario IIUF(max)I!(max)UF(max)Scenario II!(min)UFI!(rms)1214810I! (A)UF (%)1050(f)Number of Simulation0 0.5 1.0 1.5 2.0 2.5 3.0 ×104UFI!(rms) Scenario IIUF(max)I!(max)UF(max)Scenario II!(min)(e)12101468I! (A)UF (%)15129630Number of Simulation0 0.5 1.0 1.5 2.0 2.5 3.0 ×104UFI!(rms) Scenario IIUF(max)I!(max)UF(max)Scenario II!(min)Figure 4.10: Simulation results of all the possible combinations of variations in the fifteen resonantcomponent values at charge-mode operating points: a) A, b) B, c) C, d) D, and discharge-modeoperating points: e) E and f) F. In these simulations, the component values are varied to eithertheir maximum or minimum values, forming over 32000 different combinations. Highest UFvalues are experienced at combinations corresponding to scenarios I and II in Table 4.3. Lowestand highest current values in the second phase (I2) happen at scenarios I and II, respectively, aspredicted by the analysis.in Grid to Vehicle (G2V) operation happens at Constant Current (CC), Constant Power (CP), and ConstantVoltage (CV) modes, and discharging the battery in Vehicle to Grid (V2G) operation happens at CP mode.Simulation and experimental results are provided to validate the identification of the current-sharing worst-cases of the three-phase CLLC resonant converter. An Unbalanced Factor (UF) is also defined to better68Charge Mode at (A):Charge Mode at (D): Discharge Mode at (E): Discharge Mode at (F):Charge Mode at (B): Charge Mode at (C):∆i"i# i$i%∆i"i#i$i%∆i"i#i$ i%∆i"i#i$ i%∆i"i# i$ i%∆i"i#i$ i%Figure 4.11: Experimental waveforms of the converter at different operating points. The resonant com-ponents of the converter are accurately built and measured to realize the worst unbalanced be-havior of the converter with a 10% tolerance in the components. The converter behaves as thepredicted unbalanced behavior identified in the analysis.∆i i! i" i#Figure 4.12: Experimental waveforms of the converter at one of the operating modes (charge mode atA) after applying TCB. The results verify the validity of the TCB as a means to mitigate theunbalanced behavior of CLLC resonant components at all operating points.demonstrate and compare the unbalanced behavior of the converter as follows:UF =Max(I21 , I22 , I23 )−Min(I21 , I22 , I23 )I21 + I22 + I23×100% (4.22)where I1, I2, and I3 are the RMS values of the currents through the phases. UF represents the unbalanceddistribution of conduction losses and heat in the circuit. In an ideal balanced circuit, where the currentsthrough the phases are equal, UF is equal to zero. As the converter becomes more unbalanced, the UFpercentage becomes higher and higher.In order to validate the identified worst-case scenarios, the simulation results of all the possible combina-tions of variations in the fifteen resonant components are provided. The plots in Fig. 4.10 are the simulation6930UF(%)2520151050Experimental (No phase-shifting)Predicted TrendMatches the Predicted TrendAfter phase-shiftingAnalytical (No phase-shifting)I II III IV I IICharge Mode(a)(b)Discharge ModeI II III IV I IICharge Mode Discharge Mode76543210Before phase-shiftingAfter phase-shifting∆i (A)oPredicted TrendFigure 4.13: Experimental results versus the predicted analytical results: a) unbalanced factor, and b)output current ripple. The experimental UF results match with the predicted trend and the outputcurrent ripple behaves corresponding to the trend.results of the three-phase equivalent circuit of the converter with all the possible combinations of extremevariations in all the fifteen resonant component values based on 10% tolerance in the components. In thesesimulations, the component values are varied to either their maximum or minimum values, forming over32000 different combinations. The Unbalanced Factor (UF) and current value through the second phase (I2)for each of the combinations are shown. Based on the worst-case analysis in Section III, the highest UF isexpected to happen at scenarios I and II, where one of the angles between the current vectors (in this case γ)reaches its maximum and minimum values, respectively. In these two scenarios, the current vector opposingthis angle (in this case I2) will experience its lowest and highest values, respectively. Based on these plots,the worst-cases at each operating point happens at the combinations in Table 4.3, which are correspondingto scenarios I and II. For instance, in Fig. 4.10 (a), the combination in simulation number 5092 correspondsto the combination of Scenario I at charge-mode and operating point A, which is expected to result in thehighest UF and lowest I2 current value. The simulation result confirms that at that combination, the highestUF and lowest I2 values result. By looking at Figs. 4.10 (b) and (c), it can be seen that almost the samecombination also results in a worst-case condition for operating points B and C at charge-mode. This is alsoin accordance with the information in Table 4.3. Different combinations are expected to result in a worst-70case for the operating point D in charge-mode and the operating points in discharge-mode, which are alsovalidated by the simulation results in Fig. 4.10 (d), (e), and (f). The same pattern can also be observed forthe combinations based on scenario II. Therefore, the simulation results validate the identified worst-casecombinations according to scenarios I and II in Section III.As mentioned at the beginning of this Section, a 3− kW experimental prototype is also developed toexperimentally verify the results of the proposed current-sharing study. In this prototype, the resonant com-ponents are accurately built and measured to give a combination close enough to the worst-case scenario II atcharge-mode operating points A, B, and C. As indicated in the previous section, the worst-case in operatingpoint A in the charge-mode is expected to be the worst-case over the whole range of operating points, as ithas the highest MSγ value. The experimentally verified values of the resonant components in the experimen-tal prototype are shown in Table 4.5. Compared to the ideal values in Table 4.4, the real values are variedbased on the data in Table 4.3 to experimentally observe the most unbalanced behavior of the converter witha 10% tolerance in the components. The experimental results of the converter waveforms are shown in Figs.4.11 and 4.12. In Fig. 4.11, the results at different operating points at both charge-mode and discharge-modeare shown. The deviation in the current values, and therefore the deviation in the current phase angles, are aspredicted in the analysis. Since the converter is intentionally built to realize the worst-case scenario II, theconverter experiences the highest peak values in the phase two current value (i2). The result values are stud-ied with more details following in this section. In Fig. 4.12, the resulted waveforms at one of the operatingpoints after applying TCB is shown. TCB results validated the effectiveness of this technique for three-phaseCLLC resonant converters. TCB balances the RMS values of the phases currents based on correcting thephase-shift between the switching inverter legs [1].The UF value in (4.22) is calculated for all the operating points and the results alongside the values ofoutput current ripple (∆iO) are shown in Fig. 4.13. In Fig. 4.13 (a), the experimental results of UF versusthe analytical results are shown. The analytical results come from calculating the current values accordingto the three-phase equivalent circuit derived in Section II and equation (4.3). As can be seen, there is anerror in estimating the experimental UF values by the analytical results. This difference in the values simplycomes from the two approximations in deriving the circuit model: 1) FHA, and 2) approximating equalreflected load values to the primary (Rac). For that reason, a lower error is observed for the results at resonantfrequency (charge-mode II), while the error becomes larger as moving away from the resonant frequency.Nonetheless, as indicated in the figure, the trend of changes in UF values perfectly matches the predictedtrend by the analysis and the discussion outlined in Section III-C. The UF values are proportional to the71variations in the angle γ indicated in (4.21). Fig. 4.13 (b) shows the output current ripple values at differentoperating points. The output current ripple is mostly determined by the peak values of the currents, andtherefore is more dependent on the harmonics of the currents and the shape of the waveforms. For that reason,the output current ripple value is harder to be calculated and estimated. On the other hand, the UF values arecalculated based on the RMS values of the currents and better represent the fundamental components of thecurrents. Therefore, the predicted trend depicted in Fig. 4.13 matches better with experimental UF valuesthan with output current ripple values. Even so, the changes in the output current ripple over the wholeoperation behave as predicted by the analysis.The phase-shifting techniques is also verified for the unbalanced CLLC converter. As can be seen in Fig.4.12, phase-shifting results in balancing the current values as well as reducing the output current ripple. TheUF values and output current ripple values are significantly reduced by phase-shifting, as indicated in Fig.4.13. phase-shifting was able to correct the unbalanced currents under the worst unbalanced case with a 10%tolerance in the components. This comes with the cost of higher control complexity and requires three high-frequency sampling current sensors. By the proposed worst-case analysis, the worst unbalanced behavior ofthe converter with a given tolerance in the components can be identified. This helps to fully understand andpredict the unbalanced behavior of the converter and find the highest tolerance of the components to keepthe converter working within a certain unbalanced range without requiring any balancing techniques.4.4 ConclusionIn this Chapter, a worst-case analysis of the unbalanced behavior of the three-phase CLLC resonant converterdue to the resonant component tolerances has been presented. The proposed analysis helps to understand andpredict the unbalanced behavior of the converter with a given tolerance in the component values. Given thecomplexity of the three-phase CLLC converter and the high number of unknown variables, a new methodol-ogy was needed to be able to identify the worst current-sharing cases among the very high number of possiblescenarios. The effectiveness of TCB technique was also validated for the converter. TCB was able to correctthe unbalanced currents under the worst unbalanced case with a 10% tolerance in the components. Balancingtechniques come at the cost of higher control complexity and require three high-frequency sampling currentsensors. The proposed analysis in this paper helps to find the highest tolerance in the components to keepthe converter working within a certain range of unbalanced behavior without requiring any balancing tech-niques. Experimental and simulation results of a 3-kW bidirectional three-phase CLLC resonant convertershave been presented. The results validate the identified worst unbalanced cases as well as the predicted trend72of variations in the Unbalanced Factor (UF) and output current ripple (∆iO) with a given tolerance in thecomponents over the whole range of operating points.73Chapter 5Conclusion5.1 Conclusions and ContributionsOver the past decade, the use of battery-powered equipment such as EVs, laptops, and cellphones increaseddrastically. This situation has led to a significant market increase in the batteries sector leading to moreinterest in improving performance. Battery chargers are the power processing stage between energy sourcesand batteries. In order to be able to cope with newer technological requirements, higher efficiency and powerdensity as well as enhanced reliability, are expected from this type of power converters. DC-DC convertersare an essential part of an EV battery charging system. Due to the desired features of LLC and CLLCresonant converters, they are considered as an attractive option for battery charger applications. Single-phase LLC resonant converters are thoroughly studied and vastly used in battery charger applications forlow and medium power levels. For medium to high power level chargers, three-phase resonant convertersare an attractive alternative. Besides the desired features of single-phase resonant converters, such as highefficiency and reliability, three-phase resonant converters provide higher efficiency at increased power level,lower current ripple, better thermal distribution, and smaller filter and heat sink size (higher power density).However, there are some issues and limitations identified with these types of converters which have beenmain obstacles in widespread use of these topologies. In this thesis, some of the issues were identified anddiscussed. The following subsections summarizes contributions of this work.5.1.1 Trigonometric current balancing techniqueIn this work, an analysis of the current-sharing behavior of three-phase LLC resonant converter in corre-spondence to the resonant components’ tolerances was presented and discussed. Moreover, a TrigonometricCurrent Balancing (TCB) technique to improve the unbalanced behavior of the converter was presented.74The proposed TCB manipulates the phase-shift between the inverter legs to balance the currents throughthe phases. This will result in significant decrease in the output current ripple, which allows the use ofsmaller output capacitor filter. Moreover, this technique results in equally distributed losses in the circuit,which makes thermal management of the circuit easier. As a result of even current-sharing among phases,no component experiences higher temperature or larger current than the expected current peaks and so it ispossible to select components with lower power rating. In order to show the validity of the presented anal-ysis and also the functionality of TCB, a 3kW experimental prototype for PHEV application was developedand tested. The experimental results show the validity of the presented current-sharing analysis with a goodapproximation, and also the validity of the proposed TCB.5.1.2 Extended gain range and light-load efficiency improvementA modified three-phase LLC resonant converter with a new phase-shedding strategy was proposed. With theaid of two additional small and low power rated diodes, single-phase operation of the converter is realized.Single-phase operation provides an extended gain range that facilitates providing the low voltage and lowcurrent needed for recovering a fully depleted Li-ion battery. Moreover, two-phase operation of the converterhas been implemented delivering improved light-load efficiency. This leads to significant power savingover time during light-load absorption charging. The design requirements and procedure of the proposedconverter were discussed. In order to show the validity of the modified three-phase LLC resonant converterand the functionality of the proposed phase-shedding strategy, a 3kW experimental prototype for charging a96V PHEV battery pack has been developed and tested under different conditions. The results validate theextended gain range and improved light-load efficiency. The proposed converter is able to efficiently coverall the charging stages of a Li-ion PHEV battery pack.5.1.3 Bi-directional three-phase CLLC resonant convertersCLLC resonant converters are often used where a bidirectional power flow is needed. In this situation, LLCconverters cannot be used efficiently. In this work, three-phase CLLC resonant converters were studied. Aworst-case analysis of the unbalanced behavior of the three-phase CLLC resonant converter due to the res-onant component tolerances has been presented. The proposed analysis helps to understand and predict theunbalanced behavior of the converter with a given tolerance in the component values. Given the complexityof the three-phase CLLC converter and the high number of unknown variables, a new methodology wasneeded to be able to identify the worst current-sharing cases among the very high number of possible scenar-ios. The effectiveness of the TCB technique was also validated for the converter. TCB was able to correct the75unbalanced currents under the worst unbalanced case with a 10% tolerance in the components. Balancingtechniques come at the cost of higher control complexity and require three high-frequency sampling currentsensors. The proposed analysis in this paper helps to find the highest tolerance in the components to keepthe converter working within a certain range of unbalanced behavior without requiring any balancing tech-niques. Experimental and simulation results of a 3-kW bidirectional three-phase CLLC resonant convertershave been presented. The results validate the identified worst unbalanced cases as well as the predicted trendof variations in the Unbalanced Factor (UF) and output current ripple (∆iO) with a given tolerance in thecomponents over the whole range of operating points.5.1.4 Specified Academic ContributionsThe work presented in this dissertation is reported in three journals and one international conference papers.The following publications addressed the proposed methods in this research:• S. A. Arshadi, M. Ordonez, W. Eberle, M. A. Saket, M. Craciun and C. Botting, ”Unbalanced Three-Phase LLC Resonant Converters: Analysis and Trigonometric Current Balancing,” in IEEE Transac-tions Power Electronics, vol. 34, no. 3, pp. 2025-2038, March 2019.• S. A. Arshadi, M. Ordonez, M. Mohammadi and W. Eberle, ”Efficiency improvement of three-phaseLLC resonant converter using phase shedding,” 2017 IEEE Energy Conversion Congress and Exposi-tion (ECCE), Cincinnati, OH, 2017, pp. 3771-3775• S. A. Arshadi, M. Ordonez, W. Eberle, M. Craciun and C. Botting, ”Three-Phase LLC BatteryCharger: Wide Regulation and Improved Light-Load Operation,” in IEEE Transactions on PowerElectronics, vol. 36, no. 2, pp. 1519-1531, Feb. 2021• S. A. Arshadi, M. Ordonez and W. Eberle, ”Current-Sharing Worst-Case Analysis of Three-PhaseCLLC Resonant Converters,” in IEEE Transactions on Power Electronics, vol. 37, no. 3, pp. 3099-3110, March 20225.1.5 Future worksThree-phase resonant converters are a fairly new topic in power electronics. Research on these types ofconverters are still emerging. The work presented in this dissertation opens up some research opportunitiesas following:76• Small-signal modeling and control: The work presented in this dissertation focused on the main operat-ing modes of the system and validation of the proposed modulation techniques. Small-signal modelingand control of the converter remains as a research opportunity. The works to be done are: 1) controland small-signal modeling of three-phase LLC and CLLC resonant converters under unbalanced con-ditions, 2) the behavior and control of the proposed TCB technique, and 3) Dynamic behavior of theproposed phase-shedding strategies proposed in Chapter 3.• Study of the operation of multiple three-phase resonant converters in parallel: Higher power level ap-plications can be covered by having multiple converters working in parallel. The behavior of multiplethree-phase LLC converters in parallel needs to be studied.• Integrated and Planar Magnetics: Many research opportunities are still remained in the filed of magnet-ics for three-phase resonant converters. Integrating the series resonant converters with the transformersand/or integrating the transformers can result in a more efficient and cost effective designs. Also, thecurrent-sharing behavior of the converter can be improved by the better implementation of magnetics.Moreover, planar magnetics which offer lower profile size for the converters can be investigated forthese types of converters.• Synchronous Rectification (SR): At lower voltages and higher current levels, the essence of Syn-chronous Rectification is realized. Usually SR comes with significant challenges in resonant convert-ers. This is also remained as a research opportunity for the researchers in the field.77Bibliography[1] S. A. Arshadi, M. Ordonez, W. Eberle, M. A. Saket, M. Craciun and C. 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Thesis/Dissertation
eng
Electrical and Computer Engineering
Vancouver : University of British Columbia Library
Attribution-NonCommercial-NoDerivatives 4.0 International
Graduate
Ordonez, Martin
Three-phase DC-DC resonant converters for high-power battery charging applications
Text