Three-Layer Control Strategy for LLC Converters: Large-Signal,Small-Signal, and Steady-State OperationbyMehdi MohammadiBSc., Bonyan Institute of Higher Education, Iran, 2010MSc., Isfahan University of Technology, Iran, 2014A THESIS SUBMITTED IN PARTIAL FULFILLMENTOF THE REQUIREMENTS FOR THE DEGREE OFDoctor of PhilosophyinTHE FACULTY OF GRADUATE AND POSTDOCTORAL STUDIES(Electrical and Computer Engineering)THE UNIVERSITY of BRITISH COLUMBIA(Vancouver)July 2019© Mehdi Mohammadi, 2019The following individuals certify that they have read, and recommend to the Faculty of Graduate andPostdoctoral Studies for acceptance, the thesis entitled:Three-Layer Control Strategy for LLC Converters: Large-Signal, Small-Signal, andSteady-State Operationsubmitted by Mehdi Mohammadi in partial fulfillment of the requirements for the degree of Doctor ofPhilosophy in Electrical and Computer Engineering.Examining Committee:Dr. Martin Ordonez, Electrical and Computer Engineering, University of British Columbia (UBC)SupervisorDr. Fariborz Musavi, Engineering and Computer Science, Washington State UniversitySupervisory Committee MemberDr. Sudip Shekhar, Electrical and Computer Engineering, University of British Columbia (UBC)University ExaminerDr. Ryozo Nagamune, Mechanical Engineering, University of British Columbia (UBC)University ExamineriiAbstractResonant converters, particularly LLC converters, feature low switching losses and electromagnetic in-terference (EMI), and high power density and efficiency. As a result, they have been widely used inDC/DC applications. Although LLC converters naturally provide soft switching conditions and there-fore, produce relatively less switching losses, conduction losses in their rectifier have remained a barrierto achieving higher efficiencies. Moreover, the analysis of LLC converters is complicated since theyprocess the electrical energy through a high-frequency resonant tank that causes excessive nonlinear-ity. The issue of this complexity becomes even worse since, in reality, the resonant frequency of suchconverters deviates due to variations in the temperature, operating frequency, load, and manufacturingtolerances. This complexity has caused:a) limited research on large-signal modeling and control of LLC converters to be performed (this leadsto uncertain large-signal transient behavior and sluggish dynamic/recovery response),b) limited insight into small-signal modeling of LLC converters (this often leads to low accuracy),c) unregulated LLC converters not to operate in their optimum operating point (this leads to degradedefficiency and gain),d) conduction losses in the LLC rectifier to remain the main challenge to achieve higher efficiency.To address the above concerns, in this dissertation, a three-layer control strategy is introduced. Basedon the need, all the three layers or just one of them can be used when implementing the LLC converter.The three-layer control strategy produces accurate and fast dynamics during start-up, sudden load orreference changes with near zero voltage overshoot in the start-up, obtains a near zero steady-state errorby employing a second-order average small-signal model valid below, at, and above resonance, improvesefficiency by a new synchronous rectification technique, and also tracks the series resonant frequency inunregulated DC/DC applications.iiiLay SummaryDuring the past years, resonant power conversion has become a trend for processing the electrical en-ergy in numerous applications such as electric vehicles, solar panel structures, server computers, andtelecommunication systems. Although resonant converters feature high efficiency, small size and lowelectromagnetic interference, their use appears to be challenging, mostly because of their complexity ofthe analysis, and cost of implementation. To further increase the impact of resonant power conversionin our daily life, this work introduces the theory of the homopolarity cycle for resonant converters, anddevelops several tools such as the large and small-signal models, and polarity sensing techniques. Theoutcome of developing such theory and tools is a three-layer control strategy that not only addresses thechallenges shortly mentioned above, but also improves the overall performance from the efficiency andcontrol point of views.ivPrefaceThis work is based on research performed at the Department of Electrical and Computer Engineeringof the University of British Columbia by Mehdi Mohammadi, under the supervision of Prof. MartinOrdonez. Chapter 1 contains modified portions of text from all below-listed publications. Portions ofChapter 2 have been published in IEEE Transactions on Power Electronics and IEEE Energy ConversionCongress and Exposition (ECCE) [1–4]:• M. Mohammadi, and M. Ordonez, “Fast Transient Response of Series Resonant Converters UsingAverage Geometric Control,” IEEE Transactions on Power Electronics, vol. 31, no. 9, pp. 6738-6755, Sep. 2016.• M. Mohammadi, and M. Ordonez, “Inrush Current Limit or Extreme Start-Up Response for LLCConverters Using Average Geometric Control,” IEEE Transactions on Power Electronics, vol. 33,no. 1, pp. 777-792, Jan. 2018.• M. Mohammadi, and M. Ordonez, “Fast Transient Response of Series Resonant Converter Usingan Average Large-Signal Model,” IEEE Energy Conversion Congress and Exposition (ECCE),pp. 187-192, Montreal, Canada, Sep. 2015.• M. Mohammadi, and M. Ordonez, “Extreme start-up response of LLC converters using averagegeometric control,” IEEE Energy Conversion Congress and Exposition (ECCE), pp. 1-7, Milwau-kee, WI, USA, 2016.Portions of Chapter 4 have been published in IEEE Transactions on Industrial Electronics, IEEEEnergy Conversion Congress and Exposition (ECCE), and IEEE Applied Power Electronics Conference& Exposition (APEC) [6–8]:• M. Mohammadi, and M. Ordonez, “Synchronous Rectification of LLC Resonant Converters Us-ing Homopolarity Cycle Modulation,” IEEE Transactions on Industrial Electronics, vol. 66, no.3, pp. 1781-1790, Mar. 2019.• M. Mohammadi, N. Shafiei, and M. Ordonez, “LLC Synchronous Rectification Using CoordinateModulation,” IEEE Applied Power Electronics Conference & Exposition (APEC), pp. 848-853,Long Beach, USA, Mar. 2016.v• M. Mohammadi, and M. Ordonez, “LLC synchronous rectification using homopolarity cyclemodulation,” IEEE Energy Conversion Congress and Exposition (ECCE), Cincinnati, OH, USA,pp. 3776-3780, 2017.Portions of Chapter 5 have been published in IEEE Energy Conversion Congress and Exposition(ECCE) [9]:• M. Mohammadi and M. Ordonez, “Resonant LLC bus conversion using homopolarity width con-trol,” IEEE Energy Conversion Congress and Exposition (ECCE), Cincinnati, OH, USA, pp. 225-229, 2017.As the first author of the above-mentioned publications, the author of this thesis developed thetheoretical concepts and wrote the manuscripts, developed simulation and experimental platforms, andreceived advice and technical support from Prof. Martin Ordonez and his research team, in particularfrom Dr. Mohammad Mahdavi, Dr. David Campos Gaona, Dr. Ion Isbasescu, Mr. Ignacio Galiano,and Mr. Franco Degioanni. Also, the author received advice for the practical side of synchronousrectification development from the engineering team of Alpha Technologies (Dr. Rahul Khandekar, Mr.Peter Ksiazek, and Dr. Navid Shafiei).viTable of ContentsAbstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iiiLay Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ivPreface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . vTable of Contents . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . viiList of Tables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xList of Figures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xiAbbreviations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xviiiAcknowledgments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xxDedication . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xxi1 Introduction1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.1 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.2 Literature Review . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31.2.1 Modeling and Control of Resonant Converters . . . . . . . . . . . . . . . . . 31.2.2 Conduction Loss Reduction of Resonant Converters . . . . . . . . . . . . . . 61.2.3 Resonant Frequency Tracking of Resonant Converters . . . . . . . . . . . . . 71.3 Contributions of the Work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91.4 Dissertation Outline . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102 Large-Signal Modeling and Geometric Control of LLC and Series Resonant Converters2 132.1 The Average Large-Signal Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 162.2 The Average Circular Trajectories . . . . . . . . . . . . . . . . . . . . . . . . . . . . 222.3 Transient Analysis and Average Geometric Control Laws . . . . . . . . . . . . . . . . 251Portions of this chapter have been modified from [1–9]2Portions of this chapter have been published in [1–4]vii2.4 Loss Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 342.5 Average Geometric Control Transient Strategy . . . . . . . . . . . . . . . . . . . . . . 382.6 Design Considerations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 402.7 Stability Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 412.8 Experimental and Simulation Results . . . . . . . . . . . . . . . . . . . . . . . . . . . 422.8.1 Average Large-Signal Model Validation . . . . . . . . . . . . . . . . . . . . . 432.8.2 Closed-Loop Experimental Results: AGC type-1 . . . . . . . . . . . . . . . . 442.8.3 Closed-Loop Experimental Results: AGC type-2 . . . . . . . . . . . . . . . . 522.9 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 573 Average Small-Signal Modeling of LLC Converters3 . . . . . . . . . . . . . . . . . . . . 593.1 The Voltage Gain, Homopolarity Cycle and Switching Frequency Relationship . . . . 623.1.1 Voltage Gain Below Resonance . . . . . . . . . . . . . . . . . . . . . . . . . 633.1.2 Voltage Gain At Resonance . . . . . . . . . . . . . . . . . . . . . . . . . . . 653.1.3 Voltage Gain Above Resonance . . . . . . . . . . . . . . . . . . . . . . . . . 663.2 The Behavioral Average Equivalent Circuit: First Stage . . . . . . . . . . . . . . . . . 693.2.1 The Below Resonant Third-Order Behavioral Average Equivalent Circuit . . . 693.2.2 The Above Resonant Third-Order Behavioral Average Equivalent Circuit . . . 733.2.3 The Unified Third-Order Behavioral Average Equivalent Circuit . . . . . . . . 763.3 The Behavioral Average Equivalent Circuit: Second Stage . . . . . . . . . . . . . . . 773.4 The Average Small-Signal Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 793.4.1 The Below Resonant Average Small-Signal Model . . . . . . . . . . . . . . . 803.4.2 The Above Resonant Average Small-Signal Model . . . . . . . . . . . . . . . 823.5 Experimental and Simulation Results . . . . . . . . . . . . . . . . . . . . . . . . . . . 853.5.1 The Steady-State Validating Experimental Results . . . . . . . . . . . . . . . 853.5.2 The Frequency-Domain Experimental and Simulation Results . . . . . . . . . 893.6 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 904 Synchronous Rectification of LLC Converters4 . . . . . . . . . . . . . . . . . . . . . . . 944.1 The Homopolarity Plane . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 974.2 The Proposed HCM Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 994.3 Experimental and Simulation Results . . . . . . . . . . . . . . . . . . . . . . . . . . . 1034.4 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1095 Resonant Frequency Tracking of LLC Converters Using Homopolarity Cycle5 . . . . . 1105.1 Steady-State Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1135.1.1 Resonant Operation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1135.1.2 Below Resonant Operation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1143Portions of this chapter have been modified from [5]4Portions of this chapter have been published in [6–8]5Portions of this chapter have been modified from [9]viii5.1.3 Above Resonant Operation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1155.2 The Proposed Resonant Frequency Tracking Method . . . . . . . . . . . . . . . . . . 1155.3 Experimental and Simulation Results . . . . . . . . . . . . . . . . . . . . . . . . . . . 1185.4 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1226 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1236.1 Conclusions and Contributions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1236.1.1 Large-Signal Modeling and Geometric Control . . . . . . . . . . . . . . . . . 1236.1.2 Average Small-Signal Modeling of LLC Converters . . . . . . . . . . . . . . . 1246.1.3 Synchronous Rectification of LLC Resonant Converters . . . . . . . . . . . . 1246.1.4 Resonant Frequency Tracking of LLC Converters Using Homopolarity Cycle . 1256.1.5 Specific Academic Contributions . . . . . . . . . . . . . . . . . . . . . . . . 1256.2 Future Work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 127Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 129ixList of TablesTable 2.1 The values of the Lyapunov function derivative in different regions of the state-plane 41Table 2.2 The parameters of the LLC converter . . . . . . . . . . . . . . . . . . . . . . . . . 43Table 2.3 The parameters of the series resonant converter (SRC) . . . . . . . . . . . . . . . . 44Table 3.1 The specifications of the LLC converter . . . . . . . . . . . . . . . . . . . . . . . . 89Table 4.1 The LLC converter specifications . . . . . . . . . . . . . . . . . . . . . . . . . . . 106Table 5.1 The parameters of the LLC converter . . . . . . . . . . . . . . . . . . . . . . . . . 120xList of FiguresFigure 1.1 a) The conceptual block diagram of a resonant converter, b) the proposed three-layer control strategy for resonant converters, in particular for LLC converters, c)Regions of operation in a resonant converter. . . . . . . . . . . . . . . . . . . . . 2Figure 2.1 a) The proposed three-layer control strategy where the layer 1 (geometric nonlinearcontrol) is active, b) regions of operations in resonant converters. The active controllayer is highlighted in yellow. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14Figure 2.2 a) The full bridge series resonant converter (SRC), b) the SRC voltage gain versusthe switching frequency under different load conditions, c) the half-bridge LLCconverter, d) the LLC converter voltage-gain versus the switching frequency underdifferent loading conditions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14Figure 2.3 The equivalent circuit of each LLC structure at the resonant frequency: a) Structure1, b) Structure 2. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16Figure 2.4 The LLC converter normalized start-up performance: a) in the three dimensionalstate-plane, b) in time-domain. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20Figure 2.5 The average large-signal model of the LLC converter at the resonant frequency. . . 21Figure 2.6 The ON and OFF state trajectories under two specific initial conditions. . . . . . . 25Figure 2.7 The converter fast start-up response in the state-plane and time-domain. . . . . . . 26Figure 2.8 The start-up inrush current limit performance in the state-plane and time-domain. . 27Figure 2.9 θa1+θa2 versus the normalized reference voltage. . . . . . . . . . . . . . . . . . 28Figure 2.10 The converter transient response to load step-up in the state-plane and time-domain. 29Figure 2.11 The normalized output voltage variation versus the load added to the converter. . . 30Figure 2.12 The converter transient response to load step-down in the state-plane and time-domain. 32Figure 2.13 The normalized output voltage variation versus the removed load from the converter. 33Figure 2.14 The converter transient response to reference voltage decrement (input voltage in-crement) in the state-plane and time-domain. . . . . . . . . . . . . . . . . . . . . 34Figure 2.15 The converter transient response to reference voltage increment (input voltage decre-ment) in the state-plane and time-domain. . . . . . . . . . . . . . . . . . . . . . . 35Figure 2.16 The control algorithms of the proposed AGC controller: a) AGC-type 1, b) AGC-type 2. These algorithms are employed in the control layer 1. . . . . . . . . . . . . 36xiFigure 2.17 The loss effect on the average circular trajectories. . . . . . . . . . . . . . . . . . 37Figure 2.18 Conceptual block diagram and sensing strategy for the control structure. . . . . . . 38Figure 2.19 The different operating regions of the converter in the state-plane. . . . . . . . . . 38Figure 2.20 Experimental results of the average large-signal model for an LLC operating inopen-loop at resonance validating the accuracy of the proposed model: a) the LLCconverter start-up in no-load condition, b) the average large-signal model start-up inno-load condition, c) the LLC converter start-up under a 12Ω resistive load, d) theaverage large-signal model start-up under a 12Ω resistive load, e) the LLC converterstart-up under a 5.5Ω resistive load, f) the average large-signal model start-up undera 5.5Ω resistive load. The average model closely represents the average large-signalbehavior. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45Figure 2.21 Experimental results of the average large-signal model for an LLC operating inopen-loop at resonance validating the accuracy of the proposed model: a) the LLCconverter behavior following a resistive load step-up from 10Ω to 5Ω, b) the averagelarge-signal model behavior following a resistive load step-up from 10Ω to 5Ω, c)the LLC converter behavior following a resistive load step-down from 5Ω to 10Ω, d)the average large-signal model behavior following a resistive load step-down from5Ω to 10Ω. The average model closely represents the average large-signal behavior. 46Figure 2.22 The open-loop simulation results of the SRC at the resonant frequency and its aver-age large-signal model validating the proposed model. a) Start-up in no load con-dition, b) start-up under a 12Ω load, c) load step-up from 24Ω to 12Ω, d) loadstep-down from 12Ω to 24Ω. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47Figure 2.23 Comparative experimental validation of the converter dynamic transient using theproposed AGC versus a linear PI controller: a) in start-up and under 50W resistiveload, AGC is used, b) the PI controller in start-up and under 50W resistive load,c) AGC in start-up under 25W resistive load, d) PI controller in start-up and under25W resistive load, e) AGC in start-up under no load, and f) PI controller in start-upunder no-load condition. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48Figure 2.24 The zoomed in detailed waveforms of the converter with the proposed controllerduring start-up under different loading conditions. a) under a 50W resistive load, b)under a 25W resistive load, c) in no-load condition. . . . . . . . . . . . . . . . . . 49Figure 2.25 Load transient comparative experimental validation using the proposed AGC (usedin the control layer 1) versus a linear PI controller: a) load step-up with the proposedAGC, b) load step-up with the PI controller, c) load step-down with AGC, and d)load step-down with the PI controller. . . . . . . . . . . . . . . . . . . . . . . . . 50xiiFigure 2.26 The zoomed in detailed waveforms of the converter employing the proposed AGCstrategy (used in the control layer 1) and the rotating switching surface (RSS) con-troller: a) after the load step-up with the AGC, b) after the load step-up with theRSS, c) after the load step-down with the AGC, and d) after the load step-downwith the RSS. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50Figure 2.27 Reference change comparative experimental validation with the proposed AGC ver-sus the linear PI controller: a) voltage reference step-up with the proposed AGC, b)voltage reference step-up with the PI, c) voltage reference step-down with the AGC,and d) voltage reference step-down with the PI. . . . . . . . . . . . . . . . . . . . 51Figure 2.28 The zoomed in detailed waveforms of the converter employing the proposed AGCstrategy and the RSS controller: a) after the reference voltage step-up with the AGC,b) after the reference voltage step-up with the RSS, c) after the reference voltagestep-down with the AGC, and d) after the reference voltage step-down with the RSS. 51Figure 2.29 The block diagram of the conventional controller providing the inrush current limitoperation for the LLC converter. . . . . . . . . . . . . . . . . . . . . . . . . . . . 52Figure 2.30 Experimental results of the closed-loop LLC converter, in no-load condition, con-trolled by a) the conventional controller limiting the start-up inrush current, and b)the AGC type-2 limiting the start-up inrush current. . . . . . . . . . . . . . . . . . 52Figure 2.31 The zoomed-in experimental results of the closed-loop LLC converter, in no-loadcondition, controlled by a) the conventional controller limiting the start-up inrushcurrent, and b) the AGC type-2 limiting the start-up inrush current. . . . . . . . . . 53Figure 2.32 The current taken from the input voltage by the LLC converter controlled by theconventional controller in no-load condition and steady-state. . . . . . . . . . . . . 54Figure 2.33 Experimental results of the closed-loop LLC converter, in full load condition, con-trolled by a) the conventional controller limiting the start-up inrush current, and b)the AGC type-2 limiting the start-up inrush current. . . . . . . . . . . . . . . . . . 55Figure 2.34 The zoomed-in experimental results of the closed-loop LLC converter, in full loadcondition, controlled by a) the conventional controller limiting the start-up inrushcurrent, and b) the AGC type-2 limiting the start-up inrush current. . . . . . . . . . 56Figure 2.35 The efficiency curves of the LLC converter controlled by the proposed AGC, andby the conventional controller. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57Figure 3.1 a) The proposed three-layer control strategy where the layer 2 (linear control) isactive, b) regions of operations in resonant converters. The active control layer ishighlighted in yellow. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60Figure 3.2 a) The circuit schematic of the half bridge LLC converter, b) the gain-frequency di-agram, illustrating that the LLC converter can operate in three regions and that mostof the traditional small-signal modeling techniques are performed in the vicinity ofthe resonant frequency. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61xiiiFigure 3.3 Comparison of properties: the proposed, first harmonic approximation (FHA) andempirical based small-signal modeling techniques. . . . . . . . . . . . . . . . . . 61Figure 3.4 The half bridge LLC converter: a) key time-domain waveforms below resonance,b) the corresponding circuit schematics to the first two operating intervals. . . . . . 63Figure 3.5 The half bridge LLC converter: a) key time domain waveforms at resonance, b) thecorresponding circuit schematic of the first operating interval at resonance. . . . . 66Figure 3.6 The half bridge LLC converter: a) key time domain waveforms above resonance, b)the circuit schematic corresponding to the first operating interval above resonance. 67Figure 3.7 The actual and equivalent rectifier current of the LLC converter, operating belowresonance. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71Figure 3.8 The below resonant third-order behavioral average equivalent circuit of the LLCconverter, in half a switching cycle. . . . . . . . . . . . . . . . . . . . . . . . . . 73Figure 3.9 The actual and equivalent rectifier current of the LLC converter, operating aboveresonance. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74Figure 3.10 The circuit seen from the right hand side port of the resonant tank when the LLCconverter operates above resonance. . . . . . . . . . . . . . . . . . . . . . . . . . 76Figure 3.11 The above resonant third-order behavioral average equivalent circuit of the LLCconverter, in half a switching cycle. . . . . . . . . . . . . . . . . . . . . . . . . . 76Figure 3.12 The unified third-order behavioral average equivalent circuit of the LLC converter,in half a switching cycle. This model is valid below, at and above resonance. . . . 77Figure 3.13 The unified second-order behavioral average equivalent circuit of the LLC con-verter. This model is valid below, at and above resonance. . . . . . . . . . . . . . 79Figure 3.14 The LLC converter, operating below resonance: a) the second-order behavioral av-erage equivalent circuit, b) the below resonant average small-signal circuit model,through which the small-signal transfer function is obtained. . . . . . . . . . . . . 80Figure 3.15 The LLC converter, operating above resonance: a) the second-order behavioral av-erage equivalent circuit, b) the above resonant average small-signal circuit model,through which the small-signal transfer function is obtained. . . . . . . . . . . . . 82Figure 3.16 Pole and zero displacement of the below and above resonant small-signal transferfunctions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83Figure 3.17 Theoretical, simulation, and experimental below resonant normalized gain diagrams.The theoretical curves obtained from the analysis of the homopolarity cycle alignswell with the experimental and simulation curves. The accuracy of the FHA curvesdegrade when the switching frequency moves away from the resonant frequency. . 84Figure 3.18 Theoretical, simulation, and experimental above resonant normalized gain diagrams.The theoretical curves obtained from the analysis of the homopolarity cycle alignswell with the experimental and simulation curves. The accuracy of the FHA curvesdegrade when the switching frequency moves away from the resonant frequency. . 85xivFigure 3.19 The below resonant time-domain experimental waveforms of the LLC converter.These waveforms validate the accuracy of the steady-state theoretical analysis. a)full loading condition b) half loading condition. . . . . . . . . . . . . . . . . . . . 86Figure 3.20 The resonant time-domain experimental waveforms of the LLC converter. Thesewaveforms validate the accuracy of the steady-state theoretical analysis. a) fullloading condition b) half loading condition. . . . . . . . . . . . . . . . . . . . . . 87Figure 3.21 The above resonant time-domain experimental waveforms of the LLC converter.These waveforms validate the accuracy of the steady-state theoretical analysis. a)full loading condition b) half loading condition. . . . . . . . . . . . . . . . . . . . 88Figure 3.22 The below resonant experimental and simulation bode diagrams of the proposedsmall-signal model and LLC converter. These bode diagrams validate the accuracyof the theoretical dynamic analysis, with and without the discretization and time-delay effects: a) simulation, under 5.5Ω resistive load, b) simulation, under 10Ωresistive load, c) experimental, under 5.5Ω resistive load, d) experimental, under10Ω resistive load. The discretization and time-delay effects are due to the digitalimplementation of the modulation system. . . . . . . . . . . . . . . . . . . . . . . 91Figure 3.23 The resonant experimental and simulation bode diagrams of the proposed small-signal model and LLC converter. These bode diagrams validate the accuracy ofthe theoretical dynamic analysis, with and without the discretization and time-delayeffects: a) simulation, under 5.5Ω resistive load, b) simulation, under 10Ω resis-tive load, c) experimental, under 5.5Ω resistive load, d) experimental, under 10Ωresistive load. The discretization and time-delay effects are due to the digital imple-mentation of the modulation system. . . . . . . . . . . . . . . . . . . . . . . . . . 92Figure 3.24 The above resonant experimental and simulation bode diagrams of the proposedsmall-signal model and LLC converter. These bode diagrams validate the accuracyof the theoretical dynamic analysis, with and without the discretization and time-delay effects: a) simulation, under 5.5Ω resistive load, b) simulation, under 10Ωresistive load, c) experimental, under 5.5Ω resistive load, d) experimental, under10Ω resistive load. The discretization and time-delay effects are due to the digitalimplementation of the modulation system. . . . . . . . . . . . . . . . . . . . . . . 93Figure 4.1 a) The proposed three-layer control strategy where the layer 3 (efficiency improve-ment) is active, b) regions of operations in resonant converters. The active controllayer is highlighted in yellow. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95Figure 4.2 a) circuit schematic of the half bridge LLC converter b) conceptual comparisonbetween the homopolarity cycle analysis method (low complexity, simple sensors)and traditional FHA (higher complexity). . . . . . . . . . . . . . . . . . . . . . . 95Figure 4.3 Comparison of properties: the proposed HCM, voltage and current sensing syn-chronous rectification techniques. . . . . . . . . . . . . . . . . . . . . . . . . . . 96xvFigure 4.4 Operation of the LLC converter in the homopolarity plane providing informationabout the non-conduction angles of the SRs. . . . . . . . . . . . . . . . . . . . . . 98Figure 4.5 The key time-domain waveforms, synchronous rectification truth table and homopo-larity cycle definition when the LLC converter operates below resonance. . . . . . 99Figure 4.6 The key time-domain waveforms, synchronous rectification truth table and homopo-larity cycle definition when the LLC converter operates at resonance. . . . . . . . 100Figure 4.7 The key time-domain waveforms, synchronous rectification truth table and homopo-larity cycle definition when the LLC converter operates above resonance. . . . . . 100Figure 4.8 The proposed HCM synchronous rectification algorithm. This algorithm is em-ployed in the control layer 3. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101Figure 4.9 Experimental results of the LLC converter operating at resonance under the fullloading condition in time-domain and in the homopolarity plane. The experimentalresults provide validation that the proposed HCM method is successful in detectingand driving the SRs. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103Figure 4.10 Experimental results of the LLC converter operating: a) below resonance and b)above resonance, in time-domain and in the homopolarity plane. The experimen-tal results indicate that the proposed HCM method is successful in detecting anddriving the SRs under full loading condition. . . . . . . . . . . . . . . . . . . . . 104Figure 4.11 Experimental results of the LLC converter operating: a) below resonance and b)above resonance, in time-domain and in the homopolarity plane. The experimen-tal results indicate that the proposed HCM method is successful in detecting anddriving the SRs under light loading condition (10%) . . . . . . . . . . . . . . . . 105Figure 4.12 The simulation results comparing the performance of the proposed HCM methodwith that of the smart SR driver IC under different loading conditions in terms ofsynchronous rectification coverage. . . . . . . . . . . . . . . . . . . . . . . . . . 106Figure 4.13 The key comparative simulation waveforms of the LLC converter under light load-ing condition below, at and above resonance when its SRs are controlled by: a-c)the smart SR driver IC, d-f) the proposed HCM method. Under the light loadingcondition, the synchronous rectification coverage of the proposed HCM method isclose to 100% below, at and above resonance; however, that of the smart SR driverIC is 19% on average. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107Figure 4.14 The efficiency diagram of the LLC converter. . . . . . . . . . . . . . . . . . . . . 108Figure 4.15 The switching action between the layers of the three-layer control strategy. . . . . 108Figure 5.1 a) The proposed three-layer control strategy where the layer 3 (efficiency improve-ment) is active, b) regions of operations in resonant converters. The active controllayer is highlighted in yellow. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111xviFigure 5.2 a) The unregulated half bridge LLC converter with center-tapped synchronous recti-fiers, b) the effect of variations in the resonant capacitance and inductance on the se-ries resonant frequency and voltage gain of the LLC converter (variations of ±10%in the resonant capacitance and inductance result in unwanted resonant frequencyand gain deviations of 20.2% and 11.6%, respectively). . . . . . . . . . . . . . . . 112Figure 5.3 The illustration of the LLC converter operation at resonance: key time-domainwaveforms and homopolarity plane. . . . . . . . . . . . . . . . . . . . . . . . . . 114Figure 5.4 The illustration of the LLC converter operation below resonance: key time-domainwaveforms and homopolarity plane. . . . . . . . . . . . . . . . . . . . . . . . . . 114Figure 5.5 The illustration of the LLC converter operation above resonance: key time-domainwaveforms and homopolarity plane. . . . . . . . . . . . . . . . . . . . . . . . . . 115Figure 5.6 The proposed homopolarity-cycle-based resonant frequency tracking algorithm. . . 116Figure 5.7 The open-loop operation of the LLC converter under the half loading condition intime-domain and homopolarity plane: a) below resonance b) above resonance. . . 117Figure 5.8 The operation of the unregulated LLC converter with the proposed resonant fre-quency tracking method in time-domain and homopolarity plane: a) under 10% offull loading condition, b) under the full loading condition. . . . . . . . . . . . . . 118Figure 5.9 The dynamic response of the unregulated LLC converter with the proposed reso-nant frequency tracking method: a) following a load step-up from 6A to 12A, b)following a load step-down from 12A to 6A . . . . . . . . . . . . . . . . . . . . . 119Figure 5.10 Comparative simulation results of the unregulated LLC converter with the proposedand conventional resonant frequency tracking methods: a) following a load step-upfrom 6A to 12A, b) following a load step-down from 12A to 6A. . . . . . . . . . . 120Figure 5.11 The efficiency diagram of the unregulated LLC converter with the proposed reso-nant frequency tracking methods. The input voltage is 50V . . . . . . . . . . . . . 121xviiAbbreviations2D two dimensional3D three dimensionalACT average circular trajectoryAGC average geometric controlCT current transformerEMI electromagnetic interferenceESR equivalent series resistanceFHA first harmonic approximationHCM homopolarity cycle modulationHWM homopolarity width modulationIC integrated circuitKCL Kirchhoff current lawMOSFET metal-oxide-semiconductor field-effect transistorMOT minimum on timePI proportional integralPWM pulse width modulationrms root mean squareRSS rotating switching surfaceRZCT rectifier zero current timeSFC switching frequency calculatorxviiiSR synchronous rectifierSRC series resonant converterZCS zero current switchingZVS zero voltage switchingxixAcknowledgmentsI want to express my deepest gratitude to my research advisor, Prof. Martin Ordonez for his guidance,encouragement, and support throughout my Ph.D. program. I have learned from him great technicalknowledge, invaluable soft skills, and a positive attitude in hard times.Special thanks go to my committee members, Prof. William Dunford, and Dr. Fariborz Musavi for theirfeedback on my research progress.I am grateful to the University of British Columbia (UBC), Natural Sciences and Engineering ResearchCouncil (NSERC), and Alpha Technologies for supporting my research project. The development of arobust synchronous rectification method for LLC converters would not be possible without the Alphateam’s constructive assistance.Many thanks to my colleagues, especially Dr. Mohammad Mahdavi, Dr. David Campos Gaona, Dr.Ion Isbasescu, Mr. Ignacio Galiano, and Mr. Franco Degioanni, for all the informative discussions andtechnical support they have shared with me.Last but not least, I thank my beautiful wife, mother, father, and sister for all their unconditional love,inspiration, and support.xxDedicationTo my familyxxiChapter 1Introduction11.1 MotivationDevelopments in the power conversion technology have tremendously extended the applications ofpower converters to industries and our daily life. Energy harvesting from renewable/alternative re-sources, electric vehicles, aerospace, commercial, industrial, residential, telecommunication, transporta-tion, and utility systems are only part of a long list of power conversion applications. In order for thepower conversion technology to further impact the industry and society, power electronics convertersmust feature high efficiency and power density [10, 11]. In recent years, resonant converters have gainedpopularity in many applications such as power factor corrector systems [12], bi-directional converters asthe interface between the battery packs and renewable sources in hybrid automobiles and photovoltaicsystems [13–15], micro-robot drivers [16], LED drivers [17–20] and battery chargers [21–23]. In Fig.1.1 (a), the block diagram of a typical resonant converter is shown. Resonant power converters, espe-cially series resonant converters (SRCs) and LLC converters, have gained popularity in the past decadeand feature reduced switching losses [8, 24], improved electromagnetic interference (EMI) [25] andefficiency [26], and high conversion density [27]. Although LLC converters demonstrate high degreesof competence in switching loss and EMI reduction, further improvements in their performance arerequired. This calls for solutions concerning the items listed below:• Complexity of the analysis• Large-signal modeling and dynamic response• Small-signal modeling and quiescent region operation• Efficiency (synchronous rectification and resonant frequency tracking)In order to address the above concerns in LLC converters, a three-layer control strategy, as shownin Fig. 1.1 (b), is proposed. The proposed three-layer control strategy is developed based on a new1Portions of this chapter have been modified from [1–9]1 Efficiency improvement: synchronous rectification and resonant frequency tracking using homopolarity cycle Linear Control: based on small-signal modeling using homopolarity cycle{ Nonlinear geometric control:based on large-signal modeling Vin123Resonant tank RL Vo{Resonant converter(a) (b)COvOiCovrefNon-operating regionTransient region (Control: Layer 1)iCoInverter Rectifier(c)Quiescent region (Control: Layer 2) or(Control: Layer 3)Proposed lthree layer control strategyFigure 1.1: a) The conceptual block diagram of a resonant converter, b) the proposed three-layercontrol strategy for resonant converters, in particular for LLC converters, c) Regions of op-eration in a resonant converter.time-domain analysis technique. As shown in the following chapters, through analyzing the converterin time-domain, the complexity of the analysis is significantly reduced. The regions within which eachcontrol layer is used, are shown in a two-dimensional state plane in Fig. 1.1 (c). The state plane has twoaxes: output voltage (horizontal) and output capacitor current (vertical). The green region representsthe large-signal operation and is called transient. In order to enhance the dynamic response of theconverter, particularly under the large-signal variations and the start-up conditions (in the green region),a novel average geometric control (AGC) method based on large-signal modeling is developed. TheAGC improves the overall dynamic performance of the converter, and mainly provides fast large-signaldynamic and inrush current limiting performances. In pulse power applications, it is necessary to limitthe start-up inrush current. In Fig. 1.1 (c), the circular orange-purple region represents the quiescentarea within which the second and third control layers are used.The second control layer is indeed a linear controller that maintains the steady-state error near zero.Designing a linear controller in the quiescent region requires the development of a new small-signalmodeling technique that unlike conventional modeling techniques, predicts the small-signal dynamicbehavior of the converter not only at resonance but also below and above resonance.Although LLC converters can provide soft switching conditions, the complexity of their analysisand conductive losses in the rectifier have remained a barrier to gaining high efficiencies, especiallyin low-voltage-high-current applications. This work also develops a simple synchronous rectificationstrategy (called homopolarity cycle modulation) that eliminates costly current sensors and synchronous2rectifier (SR) low voltage sensing, which is subject to noise. Unlike most synchronous rectification tech-niques that do not perform well under medium and light loading conditions, an almost flat synchronousrectification coverage curve from light to full loading conditions is achieved when using the proposedhomopolarity cycle modulation (HCM) method. The proposed synchronous rectification method is en-abled in the third control layer.In order to reduce the cost, in some DC/DC applications, unregulated LLC converters are used.Unregulated LLC converters, often used as DC transformers, can obtain higher efficiency if the switch-ing and series resonant frequencies are the same. In practice, the series resonant frequency of LLCconverters deviates due to variations in the temperature, operating frequency, load, and manufacturingtolerances. As a result of this deviation, an unregulated LLC converter with a fixed switching frequencycannot operate in its optimum operating condition. This work also proposes a new resonant frequencytracking method, which is based on the analysis of the homopolarity cycle in time-domain. The pro-posed resonant frequency tracking method can be enabled in the third control layer. It is discoveredthat the resonant frequency tracking can be performed by a large-signal voltage-polarity-based sensingtechnique, which has a larger immunity to noise and has a lower cost. Moreover, the proposed reso-nant frequency tracking method features fast convergence time and is capable of driving synchronousrectifiers to even further enhance the efficiency.1.2 Literature ReviewThere has been a vast amount of research into modeling, control, modulation and resonant frequencytracking techniques for resonant converters, and it is an extremely active topic in power electronicsresearch today. Common research goals include accurate converter modeling, control, synchronous rec-tification, and resonant frequency tracking for resonant converters, in particular for SRC and LLC con-verters. The following cited works have been performed for analysis, modeling, control, synchronousrectification, and resonant frequency tracking of resonant converters.1.2.1 Modeling and Control of Resonant ConvertersSmall-signal Modeling of Resonant ConvertersResonant converters, including SRCs and LLC converters, transfer energy through a high-frequency res-onant tank, behaving as a bandpass filter. The components of the resonant tank have zero mean voltageand current values, in a switching cycle, and therefore, unlike pulse width modulation (PWM) con-verters, traditional averaging techniques cannot be directly used for small-signal modeling of resonantconverters [28]. Therefore, more complicated techniques are required for the analysis and modeling ofresonant converters [4].Generally, the first step in obtaining the small-signal model of resonant converters, such as LLCconverters, is the analysis of their large-signal dynamic behavior. The large-signal dynamic behavior ofLLC converters can be expressed through a set of nonlinear discrete state-space equations [29]. Discretestate-space equations are nonlinear, and they must be linearized first before they are used for small-signal3modeling. Linearization of the discrete state-space equations can be conventionally performed either byTaylor or Fourier expansion around the equilibrium point and most of the time in the vicinity of theresonant frequency [30–42]. Usually, linearization through the Taylor expansion produces high-orderequations, and this makes the derivation of the analytical small-signal transfer function complicated[30–35]. Therefore, empirical/numerical methods are commonly employed to obtain the small-signaltransfer function [35]. Extended describing function (EDF) small-signal modeling techniques, on theother side, linearize the nonlinear state-space equations using the Fourier expansion [36–42]. In EDFmethods, the Fourier expansion can be performed by considering only the first harmonic; however, ifhigher accuracy is needed, higher order harmonics can be taken into consideration. Traditional EDFtechniques usually result in high-order transfer functions, which are represented by a set of matricesrather than an equivalent circuit. In more advanced EDF techniques, the small-signal transfer functionis simplified and represented by an equivalent circuit [43, 44]. This is enabled by the analysis of signalenvelopes.Small-signal modeling of LLC converters can also be performed using the communication theory[45]. In this method, the effects of the first, third and seventh harmonics on the dynamic behavior of theLLC converter are separately analyzed, and then their effects are superimposed. In some applications,the switching frequency of LLC converters are kept fixed, and a switched capacitor obtains the voltageregulation. In this type of LLC converters, the amplitude modulation technique, which is based on an-alyzing the dynamic behavior of the switched capacitor, has to be used for extracting the small-signalmodel [46]. In the frequency domain, the load can be modeled as a time-varying resistor. The magnitudeand phase of the time-varying resistor are obtained through an iterative procedure. Therefore, empiricalimplementation is required. Based on the time-varying resistor model, the small-signal transfer functionusing conversion matrix techniques can be obtained [47]. Another small-signal modeling technique, ad-dressed in the literature, is performed through the obtainment of the lumped parameter equivalent circuit,describing the linearized behavior of switched tank elements [48]. Recently, time-domain modeling ofresonant converters, including LLC and series resonant converters, operating at resonance, has been re-ported in literature [1, 3]. Although simplification is achieved via the time-domain analysis, the modelsobtained are only valid when the LLC converter operates at resonance. Therefore, the accuracy of themodels significantly degrades below and above resonance.LLC converters are required to operate below, at and above resonance in order to provide voltageregulation. As discussed, most of the current small-signal modeling techniques for LLC converters aremainly based on empirical approaches or performed in the vicinity of the resonant frequency. Therefore,the accuracy of those models degrades when the switching frequency as the control signal is not equalto the resonant frequency. Therefore, the opportunity to introduce a new analytical time-domain small-signal modeling technique, which is valid below, at and above resonance, and reduces the complexity ofthe analysis is still open.4Large-Signal Modeling and Control of Resonant ConvertersLinear controllers are unable to anticipate a converter large-signal behavior [2]. This results in a poorlarge-signal dynamic performance, mostly affecting the performance during start-up and large-transients[52]. Small-signal modeling techniques cause critical information loss and misrepresent the actual large-signal dynamic behavior of the converter [53]. This happens because small-signal modeling techniquesare usually performed around the target operating point and are dependent on the operating conditions.Although linear controllers based on small-signal modeling techniques feature zero steady-state error,nonlinear controllers have better performance during dynamic transients such as start-up.In addition to linear controllers, resonant converters, including SRCs and LLC converters, can becontrolled using nonlinear control strategies [54–58]. The energy level of the resonant tank can becontrolled by a nonlinear controller tracking the converter trajectory in state-plane [54–56]. A controllaw is obtained through instantaneously monitoring the energy in the resonant tank and calculating theradius of the converter state trajectories. In order to enable a nonlinear controller with two feedbackloops to control the SRC, the piecewise affine concept can be used to investigate the dynamic of theresonant tank [57]. In this method, the outer voltage loop has a relatively large constant time. This maycause the dynamic of the system to be slower and may also affect the start-up response of the converter.A piecewise affine technique was employed with a hybrid SRC controller (dual loop structure) to makethe dynamic response of the converter faster [58].The start-up response is also a key indicator of a controller’s ability to handle large transients,in addition to steady-state performance and the large-signal behavior of the LLC converter followingdisturbances. The output capacitor of a resonant converter behaves similarly to a hefty load, duringstart-up, causing an inrush current. In pulse-power applications, where the start-up occurs frequently,limiting the start-up inrush current is important. Traditional start-up techniques reduce the inrush currentby initiating the converter at maximum frequency and progressively reduce it until the target operatingpoint is achieved. The reason for initiating the converter at the maximum frequency is that the convertervoltage gain is minimum with the highest switching frequency. This process is involved in calculatingthe time constant for the switching frequency reduction. This is complicated and requires optimizationtechniques/algorithms, such as the iteration algorithm [59]. Aside from the fact that the frequency sweeptechnique for inrush current limiting manages the start-up inrush current, it has some disadvantageslisted below:• Operation at a very high switching frequency• Overdesign in magnetic elements and gate driver circuit• Sluggish start-up response• No-load and light load conduction losses due to the circulating current in the primary sideAlternatively, the start-up inrush current of resonant converters can be reduced by nonlinear con-trollers. The nonlinear bang-bang charge control method was presented to improve the large-signal5dynamic response of the LLC converter [60]. This technique involves in setting thresholds on the reso-nant capacitor voltage that results in a model with the possibility of reducing the start-up inrush current.The cost of implementation can be reduced by calculating some state variables instead of sensing themthrough primary side control methods and observer-based controllers [1, 61, 62]. The trajectory controlmethod can also be employed to limit the inrush current by controlling the switching frequency throughanalyzing the converter state-plane [63]. The advantage of this method is that there is no need to obtainthe time constant for switching frequency reduction. However, high-frequency operation of the gatedriver and magnetic components must be considered in the design process. The variable duty cyclestrategy was introduced in order to reduce the initial switching frequency and concerns regarding theoverdesign of gate driver circuit and magnetic components [64]. Although mitigation of the LLC con-verter inrush current is achievable by the frequency sweep method, further improvements are possiblethrough conduction loss reduction, simple drivers with limited frequency operation, and more nearlyoptimal magnetic elements.Outstanding dynamic performances can be achieved for PWM converters by some advanced non-linear controllers, presented in the literature [65–71]. However, for the SRC and LLC converters, theyhave not been achieved yet. For instance, the large-signal dynamic response of the Buck converteris enhanced by the second order boundary and raster surface control laws [65–68]; however, they arenot explored for the SRC and LLC converters. Also, the natural switching surface has been employedfor PWM topologies (buck, boost, inverters, etc.) featuring rapid transient recovery [69–71] but neverstudied in the SRC and LLC converters. Therefore, opportunities to employ such advanced geometrictechniques to capture the actual large-signal nature of SRC and LLC converters and obtain extremelarge-signal dynamic response remain open.1.2.2 Conduction Loss Reduction of Resonant ConvertersAs the use of power electronic converters and demands for high-efficiency converters have grown, it isincreasingly important to take into consideration efficiency, power density and EMI when designing aconverter [72–74]. In the past decade, resonant converters received considerable attention due to theirability to provide soft switching conditions, reduce switching losses and EMI, and improve efficiency[3]. All of these inherent abilities have resulted in improved converter efficiency with the caveat ofincreased system complexity. Among different types of resonant converters, LLC converters are fre-quently used in various applications, such as battery chargers, LED drivers, and fuel cell and solar panelenergy systems [75–79].Although LLC converters can provide soft switching conditions for the inverter switches and rec-tifier diodes, including zero voltage switching (ZVS) and zero current switching (ZCS) conditions, thecomplexity of their analysis and conductive losses in the rectifier have remained a barrier to gaining highefficiencies, especially in low-voltage-high-current applications [8]. The use of SRs has been widely ad-dressed in literature as a means to tackle the reduced efficiency and high conductive losses caused bythe forward voltage of the rectifier diodes [80–90].The main challenge in using SRs is to finding their conduction angles, which is often performed6using high current or noisy voltage measurements [82–88]. By using a new theoretical foundation, theperformance of the LLC converter with the SR is improved and the implementation cost reduced.One of the earliest methods introduced to drive SRs was current-driven synchronous rectification[80, 81]. This method is interesting as it is applicable in most of power electronic topologies, includingLLC resonant converters. The method utilizes the advantage of series connected current transformers(CTs) with SRs in order to detect the current polarity. Positive and negative currents force the CT toturn the SRs on and off, respectively. The time delay and the effect of the CT leakage and magnetizinginductors should be considered when using CTs in high-switching-frequency applications. If the currentdriven synchronous rectification method is modified, it can be used to drive the SRs of an LLC converterthat has a voltage-doubler rectifier [82, 83]. In this method, a CT with only a secondary winding is usedto drive two SRs in the voltage doubler rectifier. Using the CTs on the secondary side of the transformercan be very dissipative since the secondary current is relatively large in applications where SRs areused. In order to reduce the conduction losses when the current driven synchronous rectification methodis used, the CT can be installed on the primary side of the transformer. The primary current includes thereflected current from the transformer secondary to primary and magnetizing current. In order to detectthe zero crossing points of the secondary current, a current compensating winding can be used to cancelthe effect of the magnetizing current [84].The magnitude of the voltage drop across the drain-to-source of SRs can be sensed as an alternativemethod of detecting their conduction angles. Smart synchronous rectifier driver integrated circuits (ICs)were introduced almost two decades ago in order to provide proper sensing and timing control circuitsand drive the SRs [85, 86]. Some challenges that must be addressed when using smart SR driver ICsare the predefined threshold voltage levels, minimum on time (MOT), the effect of temperature on theIC parameters, and the effects of the stray inductance and loading conditions. The very small voltagedrop over the drain-to-source of a SR or its body diode when conducting is significantly affected bynoise making it difficult to detect the zero crossing point. Therefore, in many cases, it is necessary touse zero-crossing noise filters when using the drain-to-source voltage to detect the conduction anglesof SRs [87, 88]. If the effect of the SR stray inductances is not compensated, the detection of the SRconduction angles cannot be optimized. A predictive gate drive method can be used to compensate theeffect of the stray inductance and provide a more stable driving signal for SRs [89]. The SRs in anLLC converter can also be driven using an adaptive control method, assuming that the start of the SRconduction angles equals the rising edge of the inverter voltage [90]. In this adaptive control method,the turn-off instants of the SRs are digitally tuned based on the corresponding voltage drop. From thedescription above, two main measurement methods can be identified for SRs, the current based methodswith CTs and the SR voltage drop methods, each of them presenting some technical challenges.1.2.3 Resonant Frequency Tracking of Resonant ConvertersLLC converters have a natural ability to provide soft switching conditions for primary and secondarysemiconductor elements [91, 92]. Providing soft switching conditions are necessary for the obtainmentof a higher power density [93]. LLC converters are conventionally controlled by the switching frequency7in order to either regulate the output voltage/current or enhance the efficiency. Unlike regulated LLCconverters, the switching frequency of unregulated LLC converters, widely used as DC transformers,is tuned at the series resonant frequency in order to achieve the optimum efficiency [9, 94–96]. Theresonant tank of the LLC converter is made of two inductors (resonant and magnetizing) and a resonantcapacitor. The Values of the resonant inductor and capacitor determine the series resonant frequencyof the LLC converter. The resonant inductance and capacitance are theoretically constant; however, inpractice, variations in the temperature, frequency, load, and manufacturing tolerances cause the resonantcapacitance and inductance, and consequently, the series resonant frequency of the LLC converter todeviate. The amplitude of this deviation ranges from 1% to 25%, and depends on the materials usedand manufacturing methods [97–101]. For instance, if the variations in the resonant capacitance andinductance are ±10%, the resonant frequency deviates from the theoretical series resonant frequencyby 20.2%. This consequently deviates the voltage gain of the LLC converter by 11.6%; one that wassupposed to be equal to 1 and fixed. Therefore, an open-loop LLC converter with a fixed switchingfrequency cannot consistently operate at resonance. This highlights the importance of tracking theseries resonant frequency of LLC converters in unregulated DC-DC applications.In order to enable the unregulated LLC converter to perform as a DC transformer, despite the electri-cal and environmental variations, several interesting approaches have been introduced in the literature.When the LLC converter operates at resonance, the resonant current can be approximately consideredsinusoidal. Therefore, the harmonics of the resonant current are zero. Below and above resonance, theresonant current is distorted. This means that the harmonics of the resonant current are not anymorezero. Therefore, the resonant frequency tracking can be performed by sensing the resonant current andreal-time calculating its harmonics and minimizing them [102]. The resonant frequency tracking canalso be performed by a plant modeling approach in which the drift in the series resonant frequency isdetected by observing the phase or gain relationship of two paired electrical variables [103–106]. Thisstrategy is involved in integration, peak detection, and level shifting in order to track the series reso-nant frequency of the unregulated LLC converter. In some applications, unregulated LLC converters areequipped with SRs to reduce conduction losses. In the presence of SRs, the resonant frequency trackingcan be performed by tuning the gate driving signals of SRs using the universal adaptive driving method,and then comparing the pulse width difference between the gate driving signals of the inverter switchesand SRs [107, 108]. This comparison is later used to reduce the pulse width difference and consequentlyto detect the series resonant frequency. It is also proven that the resonant operation can be achieved forthe LLC converter if it operates in the boundary of the rectifier zero current time (RZCT) [109, 110]. Inthe RZCT method, the series resonant frequency is detected by sensing the transformer secondary sidecurrent and changing the switching frequency to minimize the secondary zero current time. Althoughthe harmonic, plant modeling, synchronous rectification, and RZCT based resonant frequency trackingmethods have presented interesting results, the opportunity to introduce a resonant frequency trackingmethod, which has a lower cost and larger immunity against noise, is still open.81.3 Contributions of the WorkThe goals of this work involve discovering new large and small-modeling techniques, particularly forLLC converters, designing nonlinear controllers based on large-signal modeling to provide outstand-ing large-signal dynamic behavior, modulation techniques for driving synchronous rectifiers in orderto reduce conduction losses, and proposing a resonant frequency tracking method to address the res-onant frequency deviation in LLC converters. The objective is to contribute to modeling, control andmodulation methods in resonant converters to improve large and small-signal performance, to reduceconduction losses and improve efficiency. This work improves the overall performance of LLC resonantconverters and contributes to three important areas in the following ways:• A novel average geometric control (AGC) method based on large-signal modeling is proposed toachieve enhanced dynamic response for LLC and series resonant converters (SRCs), particularlyunder the large-signal variations and start-up conditions. The proposed AGC method has a lowcomputational burden and simple current/voltage sensor requirements to achieve extreme transientperformance. The AGC is also responsible for addressing the start-up inrush current concerns, es-pecially in the pulse power applications. In order to reduce analysis complexities, a new modelfor the LLC converter, called the average large-signal model, is proposed that maps the converterbehavior onto a two dimensional (2D) state-plane. On the 2D state-plane, it is shown that the con-verter behavior can be explained by circular trajectories called the average circular trajectories.The new geometric control law is mathematically derived using average circular trajectories pro-ducing accurate and fast dynamics during start-up, sudden load or reference changes. The AGCprovides a very fast dynamic response, while voltage overshoot is virtually eliminated and alsocan limit the start-up inrush current, while the response time remains fast enough. The SRC is thebasic topology for the LLC converter. It can be shown that the average large-signal model is alsovalid for the SRC. Therefore, the AGC can also control the SRC. The analytical AGC frameworkand theory are validated through experimental and simulation results and benchmarked againstconventional techniques. On average, the dynamic response of the SRC/LLC converter using theproposed AGC method is much faster than that of using conventional techniques, while overshootis virtually eliminated.• The theory of the homopolarity cycle modulation (HCM) is developed to achieve a simple rectifi-cation strategy that eliminates costly current sensors and SR low voltage sensing, which is subjectto noise. The proposed HCM only requires a polarity sensor and employs a simple control algo-rithm. In order to find the conduction angles of the synchronous rectifiers (SRs) used in the LLCconverter, an accurate time-domain analysis, based on volt-amp-second balance principle, is de-veloped. A new definition, called homopolarity cycle, is applied to the time-domain differentialequations of the LLC converter. This results in reduced analysis complexity, and discovery ofthe fact that the polarity of the gate driving signal of the inverter switch and the rectifier voltagecontain all the information required to detect the conduction angles of the SRs. Compared tothe first harmonic approximation (FHA) analysis method, which is load dependent and does not9provide any information about the conduction angles of the SRs, the homopolarity cycle enablesan accurate calculation of the SR conduction angles based on the time during which the polaritiesof the inverter and rectifier voltages are the same. The theoretical analysis has resulted in a verysimple synchronous rectification control strategy. The proposed HCM synchronous rectificationmethod has a low implementation cost since it requires only sensing the polarity of the rectifiervoltage (the transformer secondary voltage), but it does not sense voltage or current levels inthe power metal-oxide-semiconductor field-effect transistors (MOSFETs). Due to the relativelylarge amplitude of the rectifier voltage, the proposed HCM algorithm has good immunity againstnoise. The theoretical analysis is validated through experimental and simulation results. Unlikemost synchronous rectification techniques that do not perform well under medium and light load-ing conditions, an almost flat synchronous rectification coverage curve from light to full loadingconditions is achieved by the proposed HCM method.• Unregulated LLC converters, often used as DC transformers, can operate in their optimum oper-ating point if the switching and series resonant frequencies are the same. In practice, the seriesresonant frequency of LLC converters deviates due to variations in the temperature, operatingfrequency, load, and manufacturing tolerances. As a result of this deviation, an unregulated LLCconverter with a fixed switching frequency cannot operate in its optimum operating condition.This work also proposes a new resonant frequency tracking method, which is based on the analy-sis of the homopolarity cycle in time-domain. Through the analysis of the homopolarity cycle, thecomplexity of the analysis is significantly reduced, and it is discovered that resonant frequencytracking can be performed by a large-signal voltage-polarity-based sensing technique, which hasa larger immunity against noise and is low-cost. The proposed method also features fast conver-gence time and is capable of driving synchronous rectifiers to even further enhance the efficiency.The experimental and simulation results have shown that the efficiency improvement, resonantoperation under different loading conditions, and fast convergence time can all be achieved by theproposed resonant frequency tracking method.1.4 Dissertation OutlineThis work is organized in the following manner:In Chapter 2, the AGC is introduced to particularly control the LLC converter under large-signaltransients. The AGC can also be used to control the SRC since it is the basic topology for the LLC con-verter. This large-signal control method is used in the first control layer. The control layers are shownin Fig. 1.1 (b). The AGC is based on large-signal modeling and improves the dynamic response ofthe closed-loop LLC and SRC converters particularly under the large-signal variations and the start-upconditions. The large-signal geometric averaging significantly simplifies the analysis of the converterdynamic behavior and brings the benefits of the geometric control surfaces to the LLC and SRC con-verters. The new geometric control law is mathematically derived using average circular trajectoriesproducing accurate and fast dynamics during start-up, sudden load or reference changes. It can also10limit the start-up inrush current, that is often essential in pulse power applications. The proposed AGCmethod requires simple voltage and current sensing and has a low computational burden.In Chapter 3, a new average small-signal modeling technique, performed in time-domain, for theLLC converter is introduced. The contents of this chapter can be used to design linear compensators inthe second control layer. The control layers are shown in Fig. 1.1 (b). The analysis of LLC converters iscomplicated since they process the electrical energy through a high-frequency resonant tank that causesexcessive nonlinearity. As a result of this complexity, small-signal modeling of the LLC converteris traditionally performed using empirical methods, iterative simulation approaches or theory limitedto the vicinity of the resonant frequency. Often, such approaches may lead to limited insight (justempirical trends) or low accuracy below and above resonance. The proposed small-signal modelingtechnique is based on the analysis of the homopolarity cycle and accurately predicts not only the small-signal dynamic behavior of the LLC converter at resonance but also below and above resonance. Byusing the homopolarity cycle, the theoretical analysis of the LLC converter is significantly simplifiedto a level that two second-order circuit models express the small-signal dynamic behavior of the LLCconverter. Experimental and simulation results have shown that the proposed small-signal circuit modelscan accurately predict the small-signal dynamic behaviors of the LLC converter from below to aboveresonant operations.In Chapter 4, a new time-domain theoretical analysis, called homopolarity cycle, is introduced thatenables synchronous rectification in LLC converters with the use of a low-cost-polarity-based sensingtechnique. In order to further reduce losses in LLC converters, the use of SRs in the output rectifieris desirable. Analysis and detection of the conduction angles of the SRs used in an LLC converterare the main challenges in synchronous rectification. The homopolarity cycle considerably reduces thecomplexity of the LLC converter analysis, relates the conduction angles of the SRs to the gate drivingsignals of the inverter switches and the polarity of the rectifier voltage, and finally introduces a low-cost-polarity-based sensing technique. A synchronous rectification control algorithm, called HCM, isproposed that only needs to sense the polarity of the rectifier voltage, which is simple to measure andimmune to noise. The proposed HCM is used in the third control layer. The control layers are shownin Fig. 1.1 (b). A simple graphical plane, named homopolarity, is introduced to provide informationabout the SR conduction angles. Unlike the conventional synchronous rectification technique, the ex-perimental and simulation results have shown that the proposed HCM synchronous rectification methodhas a flat synchronous rectification coverage from light to full loading conditions while using a simplesensing strategy.In Chapter 5, a new resonant frequency tracking method, based on the analysis of the homopolaritycycle in time-domain, is introduced for LLC converters. If the switching and series resonant frequenciesof unregulated LLC converters, often used as DC transformers, are the same, they can operate in theiroptimum operating point. In practice, the series resonant frequency of LLC converters deviates dueto variations in the temperature, operating frequency, load, and manufacturing tolerances. As a resultof this deviation, an unregulated LLC converter with a fixed switching frequency cannot operate in itsoptimum operating condition. Through the analysis of the homopolarity cycle, the complexity of the11analysis is significantly reduced, and it is discovered that resonant frequency tracking can be performedby a large-signal voltage-polarity-based sensing technique, which has a larger immunity against noiseand is low-cost. The proposed method also features fast convergence time and is capable of drivingsynchronous rectifiers to even further enhance the efficiency. Experimental and simulation results haveshown that the efficiency improvement, resonant operation under different loading conditions, and fastconvergence time can all be achieved by the proposed resonant frequency tracking method. The pro-posed resonant frequency tracking method can be employed in the third control layer if required. Thecontrol layers are shown in Fig. 1.1 (b).Chapter 6 contains the relevant conclusions, contributions, and planned areas of future work. Thework contributed to the resonant power converter control for DC/DC applications, by proposing a three-layer control strategy. The contributions are highlighted in ten relevant patent and publications in IEEETransactions journals and international conference papers.12Chapter 2Large-Signal Modeling and GeometricControl of LLC and Series ResonantConverters1As discussed in Chapter 1, further improvements in the performance of LLC converters require solutionsaddressing the following concerns: the complexity of the analysis, large-signal modeling and dynamicresponse, small-signal modeling and quiescent region operation, and degraded efficiency due to conduc-tion losses in the rectifier and variations in the resonant frequency. In order to address these concerns,a three-layer control strategy, as shown in Fig. 2.1 (a), is proposed. Depending on the operating region,one of the control layers is active. The operating regions of a resonant converter are shown in Fig. 2.1(b). The proposed three-layer control strategy is developed based on a new time-domain analysis. In thischapter, the analysis is focused on the first control layer. Since the series resonant converter (SRC) isthe basic topology for LLC converters, in this chapter, the large-signal modeling and average geometriccontrol are also developed for the SRC as well. The topologies of the LLC and SRC converters areshown in Fig. 2.2 (a) and (c).Unlike pulse width modulation (PWM) converters, resonant converters transfer power through ahigh-frequency resonant tank making the dynamic and steady-state much more intricate than the PWMconverter counterpart. Such increased complexity makes design and control of resonant converters morechallenging [50, 51]. The traditional strategy to control power converters involves small-signal mod-eling by performing perturbation and linearization. Since this technique only considers the converteroperation around the equilibrium operating point, it cannot provide sufficient information related to thelarge-signal behavior of the system, resulting in a poor response outside the designed operating point[52]. Despite such difficulties, small-signal and frequency analysis have been popular tools in mod-eling, design, and implementation due to their simplicity. Following the small-signal analysis trend,simplified models of resonant converters have been presented based on first harmonic approximation,including transfer functions for both frequency and duty cycle. Another technique that has been used is1Portions of this chapter have been published in [1–4]13 E synchronous rectification and resonant frequency tracking using homopolarity cycle{Proposed lthree layer control strategy(a)vOiCovrefNon-operating regionTransient region (Control: Layer 1)Quiescent region (Control: Layer 2) or(Control: Layer 3)(b)3based on small-signal modeling using homopolarity cycle2 Nonlinear geometric control:based on large-signal modeling 1Figure 2.1: a) The proposed three-layer control strategy where the layer 1 (geometric nonlinearcontrol) is active, b) regions of operations in resonant converters. The active control layer ishighlighted in yellow.(d)(c)VoutCOCrLrn:1SR1vsecRLLmisipirS2S1Vin vinviLr(a) (b)VoutCORLirLrCrS4S1VinS2S3vinv iLrAA’fr1Switching frequency2nVoltage gainHeavy loadLight loadfrSwitching frequencyVoltage gain1Heavy loadLight loadSR4SR3SR2SR1SR4SR3SR2Figure 2.2: a) The full bridge series resonant converter (SRC), b) the SRC voltage gain versusthe switching frequency under different load conditions, c) the half-bridge LLC converter,d) the LLC converter voltage-gain versus the switching frequency under different loadingconditions.sampled data models that are employed to reduce the transient and steady-state errors by dynamicallycontrolling the switching frequency. However, while all these techniques can describe the small-signalbehavior of the converter successfully, they are not able to provide sufficient information about the actuallarge-signal dynamic nature of the system.In addition to linear controllers, some interesting nonlinear control strategies for resonant convertershave been proposed in the literature [54–58]. A nonlinear controller tracking the converter trajectory instate-plane is proposed in [54–56], which controls the energy level of the resonant tank. Therefore, by14monitoring the instantaneous energy into the resonant tank, a control law is obtained based on calculat-ing the radius of the converter state trajectories. By using the piecewise affine concept [57], the SRC ismodeled, and a nonlinear controller is proposed where the dynamic of the resonant tank is investigated.Since the outer voltage loop has a relatively large constant time, the dynamic of the system may beslower, and this mostly affects the start-up response of the converter. A piecewise affine technique [58]was employed with a hybrid SRC controller (dual loop structure) to make the dynamic response of theconverter faster.In addition to steady-state performance and the large-signal behavior of the LLC converter follow-ing disturbances, the start-up response is one of the key indicators of a controller ability to handle largetransients. During start-up, the output capacitor of the LLC converter behaves similarly to a hefty loadthat produces an inrush current. In pulse power applications, where the start-up frequently occurs, lim-iting the start-up inrush current is inevitable. As shown in Fig. 2.2 (d), since the gain of the converter isreduced with high switching frequencies, traditional start-up techniques attempt to reduce inrush currentby initiating the converter at maximum frequency and progressively reduce it (with a time constant) untilthe target operating point is reached. However, calculating the time constant is complex and needs to beoptimized by some techniques/algorithms, such as the iteration algorithm [59]. Although the frequencysweep inrush current limiting technique manages the inrush current, it carries some disadvantages thatare listed as follows: operation at a very high switching frequency that causes overdesigns in magneticelements and the gate driver circuit, sluggish start-up response, and no-load and light load conductionlosses due to the circulating current in the primary side.The start-up inrush current of the LLC converter can be alternatively reduced by nonlinear con-trollers; however, they may not completely resolve issues associated with no-load conduction losses,magnetics overdesign, and start-up performance. Although the frequency sweep method and nonlinearcontrollers, discussed in Chapter 1, are successful in mitigating inrush current, further improvements arepossible. In particular, pertaining to reducing conduction losses, simple drivers with limited frequencyoperation, and more nearly optimal magnetic elements while improving the start-up response.This chapter develops a new average geometric control (AGC) method based on large-signal mod-eling to achieve an enhanced dynamic response for the LLC and SRC converters, particularly under thelarge-signal variations and start-up conditions. The AGC method is active in the control layer 1 andhas a low computational burden and simple current/voltage sensor requirements to achieve fast transientor start-up inrush current limiting performances. In start-up, the AGC essentially provides two start-upperformances: fast start-up dynamic performance, or inrush current limiting performance. Limiting thestart-up inrush current in pulse power applications is necessary. In order to reduce the complexity ofthe analysis, a new model for the LLC and SRC converters, called the average large-signal model, isproposed, that maps the LLC/SRC converter behavior onto a two dimensional (2D) state-plane. On the2D state-plane, it is shown that the LLC/SRC converter behavior can be explained by circular trajecto-ries called the average circular trajectories. According to the way the average circular trajectories areused, the AGC is categorized as AGC-type 1 and AGC-type 2. The AGC-type 1 provides a fast dynamicresponse for the LLC/SRC converter, in start-up and also following other large-signal transients while15Structure ILr CrViniLrLmv iLmirnnvoCo2ninvStructure IILr CriLrLmv iLmirnnvoCo2ninv(b)(a)Figure 2.3: The equivalent circuit of each LLC structure at the resonant frequency: a) Structure 1,b) Structure 2.voltage overshoot is virtually eliminated. The AGC-type 2 limits the start-up inrush current, while the re-sponse time remains fast enough. To attain the fast start-up response or limit the start-up inrush current,the LLC converter does not need to work far from the resonant frequency. This capability eliminates theneed to overdesign magnetic components and the gate driver circuit, and also eliminates the sluggishstart-up response. Moreover, since the AGC has the advantages of an averaging technique, high-speedsensing techniques are not required. For the AGC-type 1 and AGC-type 2, two simple control lawsare obtained that can be implemented by low-cost microcontrollers or digital circuits. In addition toits other advantages, the proposed AGC can also govern the LLC converter in the quiescent area underone condition: that the input voltage of the LLC converter is fixed. Under this condition, the use offrequency or phase shift modulation is not required.2.1 The Average Large-Signal ModelIn the control layer 1 (shown in Fig. 2.1 (a)), the AGC provides a fast dynamic response in the transientregion or start-up inrush current limiting performance for the LLC/SRC converter. In this chapter, thefocus of the analysis is on the LLC converter; however, it can also be used for the SRC. In this section,it is proven that the average large-signal model of the LLC converter can be obtained by analyzing theconverter dynamic behavior at the resonant frequency. The resonant frequency is defined as follows:Fr =12pi√LrCr(2.1)When the switching frequency of the inverter equals the resonant frequency, as shown in Fig. 2.2(d), the converter voltage gain becomes 1/2n for all loading conditions except the no-load condition. Inthis case, the LLC converter has two equivalent structures in every switching cycle, as shown in Fig.2.3. At the resonant frequency, it can be mathematically shown that the LLC converter behaves linearlyon average; this is also experimentally proven in Section 2.8. Therefore, like in any linear system, thelarge-signal dynamic behavior of the LLC converter, working at the resonant frequency, can be analyzedafter applying a step input under no-load condition while all initial conditions are zero. Since applying16a step input to the LLC converter means entering start-up, in order to interpret the large-signal dynamicbehavior of the LLC converter, the LLC converter start-up performance must be analyzed. Therefore,tk is introduced to indicate the instants when the converter structures are switched. At the moment theinverter is turned on, when all initial conditions are zero, k equals zero; it is increased by 1 at the end ofeach switching cycle. tk is defined as follows:tk = kpi√LrCr,k ∈ {0,1,2, ..} (2.2)Half cycle kth (tk < t < tk+1):At tk, the switch S1 is turned on, making SR1 and SR2 turn on. Therefore, a voltage of Vin−nvo(t)drops across the series resonant components Lr and Cr, making them resonate. Moreover, nvO(t) dropsacross Lm, increasing iLm, linearly. The differential equation describing the behavior of Structure I isgiven as follows:−Vin+Lr diLr(t)dt +1Cr∫ ttkiLr(t)dt+ vCr(tk)+nvo(t) = 0 (2.3)In order to solve (2.3), it is necessary to employ the following auxiliary equations:nvo(t) = LmdiLm(t)dt(2.4)nvo(t) =n2CO∫ ttkir(t)ndt+nvo(tk) (2.5)iLr(t) = iLm(t)+ir(t)n(2.6)By applying (2.4), (2.5) and (2.6) to (2.3), the fourth order differential equation describing thebehavior of structure I is obtained as follows:d4ir(t)dt4+1Lr[1Cr+n2CO(1+LrLm)]d2irdt2+n2COCrLrLmir = 0 (2.7)(2.7) is a homogeneous linear differential equation with constant coefficients. The characteristicequation of (2.7) is as follows:R4+1Lr[1Cr+n2CO(1+LrLm)]R2+n2COCrLrLm= 0 (2.8)By considering that, in practice, CO >>Cr the roots of the characteristic equation are obtained asfollows:R2 =− 1LrCeq0(2.9)where,171Ceq=1Cr+n2CO(1+LrLm) (2.10)Now, (2.7) can be solved by using the characteristic equation roots. Therefore, ir(t), vo(t), iLm(t),iLr(t) and vCr(t) can be obtained via the following equations:ir(t) = nVin− vCr(tk)−nvo(tk)[1+(Lr/Lm)]Zeqsin(ωeq(t− tk)) (2.11)vo(t) = nCeqCO[Vin− vCr(tk)−nvo(tk)][1− cos(ωeq(t− tk))]+ vo(tk) (2.12)iLm(t) =n2CeqLmCO[Vin− vCr(tk)−nvo(tk)][(t− tk)− 1ωeq sin(ωeq(t− tk))]+nLmvo(tk)(t− tk) (2.13)iLr(t) =Vin− vCr(tk)−nvo(tk)[1+(Lr/Lm)]Zeqsin(ωeq(t− tk))+ n2CeqLmCO[Vin− vCr(tk)−nvo(tk)][(t− tk)− 1ωeqsin(ωeq(t− tk))]+ nLm vo(tk)(t− tk) (2.14)vCr(t) =CeqCr{Vin− vCr(tk)−nvo(tk)[1+(Lr/Lm)]}[1− cos(ωeq(t− tk))]+n2CeqLmCO[Vin− vCr(tk)−nvo(tk)][ (t− tk)22+1ω2eq[cos(ωeq(t− tk))−1]+ n2Lm vo(tk)(t− tk)2 (2.15)where,ωeq =1√LrCeq(2.16)Zeq =√LrCeq(2.17)Half cycle (k+1)th (tk+1 < t < tk+2):At tk+1, the switch S1 is turned off, S2 is turned on, and SR3 and SR4 turn on. Therefore, a voltageof nvo(t) drops across the series resonant tank and causes the current through the series resonant tank toincrease in the opposite direction, as shown in Fig. 2.3 (b). Moreover, −nvo(t) drops across Lm, causingits current to increase inversely. The differential equation describing the behavior of Structure II is givenas follows:18LrdiLr(t)dt+1Cr∫ ttk+1iLr(t)dt+ vCr(tk+1)−nvo(t) = 0 (2.18)The auxiliary equations that are required to solve (2.18) are given as follows:nvo(t) =−Lm diLm(t)dt (2.19)nvo(t) =n2CO∫ ttk+1ir(t)ndt+nvo(tk+1) (2.20)iLr(t) = iLm(t)− ir(t)n (2.21)By applying (2.19), (2.20), and (2.21) to (2.18), (2.7) is obtained. By solving (2.7) for Structure II,ir(t), vo(t), iLm(t), iLr(t) and vCr(t) can be calculated by using the following equations:ir(t) = nvCr(tk+1)−nvo(tk+1)[1+(Lr/Lm)]Zeqsin(ωeq(t− tk+1)) (2.22)vo(t) = nCeqCO{vCr(tk+1)−nvo(tk+1)[1+(Lr/Lm)]}[1− cos(ωeq(t− tk+1))]+ vo(tk+1) (2.23)iLm(t) =−n2CeqLmCO{vCr(tk+1)−nvo(tk+1)[1+(Lr/Lm)]}[(t− tk+1)− 1ωeqsin(ωeq(t− tk+1))]− nLm vo(tk)(t− tk+1) (2.24)iLr(t) =−vCr(tk+1)−nvo(tk+1)[1+(Lr/Lm)]Zeq sin(ωeq(t− tk+1))− n2CeqLmCO{vCr(tk+1)−nvo(tk+1)[1+(Lr/Lm)]}[(t− tk+1)− 1ωeqsin(ωeq(t− tk+1))]− nLm vo(tk)(t− tk+1) (2.25)vCr(t) =CeqCr{vCr(tk+1)−nvo(tk+1)[1+(Lr/Lm)]}[cos(ωeq(t− tk+1))−1]− n2CeqLmCOCr{vCr(tk+1)−nvo(tk+1)[1+(Lr/Lm)]}{(t− tk+1)22+1ω2eq[cos(ωeq(t− tk+1))−1]}− n2LmCr vo(tk+1)(t− tk+1)2 (2.26)1915-0.5-121.510.502.51050-5-10151050-5-10voniLrnvCrn(a)0 2 4 6 8 10 12 14 16 18 20-202468101200.511.522.5irecnvonNormalized time (tn)(b)Figure 2.4: The LLC converter normalized start-up performance: a) in the three dimensional state-plane, b) in time-domain.Now, by using the state variables obtained above, the open-loop converter start-up dynamic behaviorcan be plotted on a three dimensional (3D) state-plane, as shown in Fig. 2.4 (a) under the no-loadcondition. The state variables used in the 3D state-plane are von, iLrn and vCrn. von, iLrn and vCrn arethe normalized versions of their actual values, which will be introduced later. The trajectory shownin Fig. 2.4 (a) conveys crucial information about the large-signal behavior of the LLC converter, andthat finally results in obtaining a start-up extreme or inrush current limit operation. This trajectory startsfrom (0,0,0) and finally converges to (0,0,2/n), while its movement on the state-plane creates a sphere.The proposed AGC (used in the control layer 1) is a secondary side controller and uses the variablesavailable on the secondary side. This characteristic causes the number of dimensions to decrease by one.In this case, irn and von are used. If the start-up response is plotted by using (2.11), (2.12), (2.22) and(2.23), the waveforms shown in Fig. 2.4 (b) are obtained. In this figure, the red waveform represents theaverage current of irn. The properties presented by the average current of irn and von are similar to thoseof a rectified second order LC circuit.According to the above mathematical analysis, the average large-signal model of the LLC convertercan be introduced as the circuit shown in Fig. 2.5. In this model, the switch SAM models the inverter ofthe LLC converter. If the inverter of the LLC converter is in the on-state and working at the resonantfrequency, SAM is in the on-state. If the inverter is in the off-state, SAM is in the off-state where −Vin/2is applied to the model. −Vin/2 is applied because of the behavior of the inverter switches’ body diodes.DAM1 and DAM2 model the full bridge rectifier of the LLC converter.In the model shown in Fig. 2.5, all the components are known except for the modeled inductor LAM.Therefore, to use the proposed average large-signal model, it is necessary to find the value of LAM.Since the proposed model is an average model, the average current through the output rectifiershould be calculated during the interval between [t0− t1]. t0 and t1 can be obtained through (2.2). From(2.11), the equation of the output rectified current ir in the interval of [t0− t1] can be calculated asfollows:20LCoVinDAM1on offSAMDAM2ir iCoAMvon:1IOAMAMAMAB2Figure 2.5: The average large-signal model of the LLC converter at the resonant frequency.ir(t) = nVinZeqsin(ωeq(t− tk)) (2.27)By averaging (2.27) in the interval of [t0− t1], the following equation is obtained:irav =2Teq∫ t1t0nVinZeqsin(ωeq(t− tk))dt (2.28)Solving the above equation results in the following equation, which is very important in obtainingLAM.irav =4nVinCeqTeq(2.29)The second order differential equation that describes the average large-signal model behavior in theinterval of [t0− t1] while SAM is in the on-state is as follows:LAMn2d2irAMdt2+1COirAM = 0 (2.30)By solving (2.30), irAM is obtained as follows:irAM =Vin/nZAMsin(ωAM(t− t0)) (2.31)where,ωAM =n√LAMCO(2.32)ZAM =1n√LAMCO(2.33)By averaging irAM in the interval of [t0− t1],irAMav =2Teq∫ t1t0Vin/nZAMsin(ωAM(t− t0))dt (2.34)the following equation is obtained:21irAMav =2TeqnVinCO[1− cos(ωAM Teq2 )] (2.35)Since the circuit shown in Fig. 2.5 is the average large-signal model of the LLC converter, (2.29)and (2.35) must be equal. Therefore, by combining (2.29) and (2.35), and also applying (2.10), LAM canbe calculated as follows:LAM =CrCO+n2Cr[1+(Lr/Lm)]pi2Lr[cos−1(1−2 CrCO+n2Cr[1+(Lr/Lm)])]2(2.36)Now that LAM can be calculated by (2.36), the average large-signal model of the LLC converter canbe used to enable the AGC to provide the fast dynamic response or inrush current limit for the LLCconverter. Since the average large-signal model works closely to the resonant frequency, the inrushcurrent limit mode provided by the AGC-type 2, prevents overdesign in magnetic components and thegate driver. Moreover, the natural frequency of LAM and CO is remarkably lower than the operatingswitching frequency, which is a great advantage. Therefore, low-band-width-low-cost sensors can beused.All the above discussion also applies to the SRC. The only difference is that the SRC, shown inFig. 2.2 (a), does not have the transformer and magnetizing inductor. This means that Lm = ∞ andn= 1. Considering that CO is much larger than Cr, LAM for the SRC can be calculated by the followingequation:LAM−SRC =CrCOpi2Lr[cos−1(1−2 CrCO )]2(2.37)2.2 The Average Circular TrajectoriesThe dynamic behavior of the LLC converter can be geometrically analyzed by studying the behavior ofthe average large-signal model trajectories in the state-plane. According to the way the average circulartrajectories are used, a very fast dynamic response or a start-up inrush current limit performance (in thetransient region, highlighted in Fig. 2.1 (b)) can be obtained with low implementation cost and com-plexity. In this section, the procedure for obtaining the converter trajectories (called the average circulartrajectories) is discussed. To provide a framework for the analysis, it is assumed that the converter isunder a constant current load, which is the worst case scenario from the control point of view. Thegeneral differential equation that describes the average large-signal model dynamic behavior, is givenas follows:−uVin2n+LAMn2dirAMdt+1CO∫ t0(irAM− IO)dt+ voAM0 = 0 (2.38)where, u is a function the value of which determines the status of the switch SAM. u is given asfollows:22u=1, i f SAM : ON−1, i f SAM : OFF (2.39)In order to simplify and generalize the analysis, time and all of the voltages and currents are nor-malized using the following factors:vxn = 2nvxVin(2.40)ixn =2nixVinZAM (2.41)tn = 2pitTAM(2.42)where,tAM =2piωAM(2.43)By applying the above normalizing factors to (2.38), the following normalized differential equationis obtained:−u+ dirAMndtn+∫ tn0(irAMn− IOn)+ voAMn0 = 0 (2.44)The current through LAM can be obtained by solving (2.44), which is given as follows:irAMn(tn) = (u− voAMn0)sin(tn)+(irAMn0− Ion)cos(tn)+ Ion (2.45)where, voAMn0 and irAMn0 are the initial conditions. Since iCoAMn(tn) = irAMn(tn)−Ion, the normalizedcurrent through the output capacitor iCoAMn can be obtained as follows:iCoAMn(tn) = (u− voAMn0)sin(tn)+ iCoAMn0cos(tn) (2.46)Now that the current through the output capacitor is given by (2.46), the output voltage of the averagelarge-signal model can be obtained through the following equation, which will finally help the AGC toprovide a start-up extreme or inrush current limit performance.voAMn(tn) = (u− voAMn0)[1− cos(tn)]+ iCoAMn0sin(tn)+ voAMn0 (2.47)(2.47), can be rearranged as follows:voAMn(tn)−u=−(u− voAMn0)cos(tn)+ iCoAMn0sin(tn) (2.48)Since x1sin(x)+ x2cos(x) =√x21+ x22sin[x+ tan−1(x2/x1)], (2.48), can be rewritten as follows:23voAMn(tn)−u=√(u− voAMn0)2+ i2CoAMn0sin[tn+ tan−1(u− voAMn0iCoAMn0)] (2.49)From (2.49),tn+ tan−1(u− voAMn0iCoAMn0) = sin−1(voAMn(tn)−u√(u− voAMn0)2+ i2CoAMn0) (2.50)The derivative of (2.49), is given as follows:dvoAMn(tn)dtn=√i2CoAMn0+(voAMn0−u)2cos[tn+ tan−1(u− voAMn0iCoAMn0)] (2.51)By applying (2.50) to (2.51), another equation to calculate the normalized current through the outputcapacitor can be obtained as follows:iCoAMn(tn) =√i2CoAMn0+(voAMn0−u)2cos[sin−1(voAMn(tn)−u√i2CoAMn0+(u− voAMn0)2)] (2.52)Since cos[sin−1(x)] =√1− x2, (2.52) results in the obtainment of the average circular trajectoriesof the proposed average large-signal model; these are given as follows:iCoAMn(tn)2+[voAMn(tn)−u]2 = i2CoAMn0+(voAMn0−u)2 (2.53)(2.53) represents two sets of circular trajectories: ON and OFF state circular trajectories. It willbe shown that the AGC-type 1 and AGC-type 2 use these circular trajectories to provide an extremedynamic response or a start-up inrush current limit performance for the LLC converter while voltageovershoot is virtually eliminated. When SAM is in the ON state (u = 1), the average large-signal modeloperating point follows the ON state circular trajectory, and when SAM is in the OFF state (u = −1),the operating point follows the OFF state trajectory. The origin of the circles is (vOAMn(tn)− u(tn),0).The radius of the ON state trajectory is r1 =√i2CoAMn0+(voAMn0−1)2, and the radius of the OFF statetrajectory is r2 =√i2CoAMn0+(voAMn0+1)2. This is evidence that the radius of the circular trajectoriesdepends on the initial conditions and the position of SAM. Because, (2.53) represents two sets of trajec-tories (ON and OFF state trajectories), the relationship between these two sets of trajectories should beidentified. If the distance between the ON and OFF state circles is named O1O2, according to (2.53),O1O2 = 2. The radius of the ON and OFF state trajectories are r1 and r2. For vOAMn(0) < 1, (2.54)is satisfied. The result of satisfying (2.54) is that it shows the ON and OFF state trajectories are twointersecting circles as shown in Fig. 2.6. Because of the existence of DAM2, the output voltage cannot benegative. Therefore, the left side of the state-plane depicted in Fig. 2.6 is highlighted to show that theconverter cannot work in that region. Also, the direction of the circles’ rotations is determined in Fig.2.6.As discussed earlier in this section, the average large-signal model shown in Fig. 2.5 is also validfor the SRC. Therefore, by considering that in the SRC, shown in Fig. 2.2 (a), Lm =∞ and n= 1, (2.53)24−3 −2 −1 0 1 2 3−1.5−1−0.500.511.5vQuadrant1Quadrant2Quadrant3 Quadrant4OAMnOFF state TrajectoryON state TrajectoryCoAMni(vCoAMnx1,iCoAMnx1)(vCoAMnx2,iCoAMnx2)Figure 2.6: The ON and OFF state trajectories under two specific initial conditions.is also valid for the SRC.|r1− r2|< O1O2 < r1+ r2 (2.54)2.3 Transient Analysis and Average Geometric Control LawsReferring to the obtained trajectories of the converter model, in this section, the large-signal dynamicresponse of the converter model in transient is discussed, and finally, two control laws, called AGC type-1 and AGC type-2 will be introduced. These control laws are used in the control layer 1, shown in Fig.2.1 (a). For this purpose, six scenarios are considered: the performance of the converter model in start-up (fast response and limited inrush current), load step-up and step-down, and also sudden referencevoltage changes. The AGC type-1 and AGC type-2 provide fast dynamic response in the transientregion and start-up inrush current limiting performance, respectively.In start-up, the converter is in zero initial conditions which means that voAMn(0) and iCoAMn(0) arezero. Figs. 2.7 and 2.8 are utilized to explain the behavior of the converter, controlled by the AGCtype-1 and AGC type-2 in the start-up in both the state-plane and time-domain. In Fig. 2.7, the start-upresponse of the converter, controlled by the AGC type-1, is illustrated. By assuming that at turn-oninstant, the converter is in the no-load condition, at the point Pa1, SAM turns on and causes the converterto follow the ON state trajectory. Because, the final destination for the converter is to reach to thedesired operating point in which the output voltage equals the reference voltage and the average of theoutput capacitor current is zero, the OFF state trajectory that intersects the ON state trajectory and leadsthe converter to reach to this goal should be determined. Therefore, by knowing the desired OFF statetrajectory, the intersection of both trajectories can be obtained. At the point Pa2 (the intersection pointbetween the two trajectories), SAM turns off, however, DAM1 remains on. Then, the converter followsthe OFF state trajectory as shown in Fig. 2.7. Finally, at the point Pa3, the output voltage and outputcapacitor current reach to the reference voltage and zero, respectively without producing overshoot. At25i CoAMnvOAMnOOn state trajectoryOff state trajectoryvOAMntntn1The average output rectifiedcurrent in on stateThe average output rectifiedcurrent in off statestart-up steady stateVrniCoAMni (P )a2-1 2SAM: OnSAM: OffDesired operating pointƟrPPPtv (P )oAMn a2Vrni ,Cona3a2a1Pa1Pa2Pa3a1ra2a1Ɵa2Pa2Pa3Pa1nPa1tnPa2tnPa3CoAMn(vOAMn(Pa2),iCoAMn(Pa2))Figure 2.7: The converter fast start-up response in the state-plane and time-domain.the point Pa2, the output voltage can be calculated by use of the equation below:voAMn(Pa2) =14Vrn(Vrn+2) (2.55)Where Vrn is the normalized reference voltage. Because the circles shown in Fig. 2.7 rotates withthe speed of the angular frequency, calculating θa1 and θa2 provides a preliminary to find the durationof start-up. θa1 and θa2 can be calculated by the following equations.θa1 = cos−1[1− voAMn(Pa2)] (2.56)θa2 = cos−1[1+ voAMn(Pa2)1+Vrn] (2.57)Although usually applying a constant current load in start-up does not happen in practice, its effect26vOAMnOOn state trajectoryOff state trajectorytntn1The average output rectifiedcurrent in on stateThe average output rectifiedcurrent in off statestart-up steady stateVrniCoAMn-1 2Vrni ,ConP’a1P’a2P’a3P’a2P’a3P’a1imiCoAMnvOAMnFigure 2.8: The start-up inrush current limit performance in the state-plane and time-domain.on the settling time can be assessed theoretically. Referring to Fig. 2.5, the diode DAM2 does not allowthe output voltage to be negative. Hence, due to the existence of LAM, applying a constant currentload in start-up causes DAM2 to turn-on. As long as the amplitude of the current through LAM is notequal to the reflected constant current load IO from the secondary to primary, the output voltage remainszero. Therefore, in this situation where SAM is in the on state, a positive voltage of Vin drops over LAMcausing its current to increase linearly. The instant in which the current through LAM equals IO/n can becalculated by use of the following equation.tnPa1 =LAMIOnn(2.58)By taking into account the delay time tnPa1, the time the output voltage takes to reach the desiredoperating point can be calculated by (2.59). As (2.56) and (2.57) describe, θa1 and θa2 do not dependon the output current and just depend on the normalized reference voltage. In Fig. 2.9, θa1+θa2 versusthe normalized reference voltage is plotted.270 0.2 0.4 0.6 0.8 100.1pi0.2pi0.3pi0.4pi0.5pi0.6pi0.7piThe Normalized Output voltageƟ +Ɵ (radian)a1a2Figure 2.9: θa1+θa2 versus the normalized reference voltage.tnPa3 =TAM2pi(θa1+θa2+ tnPa1) (2.59)As mentioned, the AGC-type 2 limits the start-up inrush current for the converter, so that startingfrom a very high switching frequency is no longer required. Therefore, overdesign in the magneticcomponents and gate driver is eliminated. In order to analyze the start-up dynamic behavior of theconverter employing the AGC-type 2, Fig. 2.8 is utilized. In order to limit the start-up inrush current,the current injected into the output capacitor should be limited. Therefore, as shown in Fig. 2.8, acurrent limit im is set on the state-plane to confine the current through the output capacitor to a valuewithin the permitted range. Once the converter is turned on, the operating point starts to follow the ONstate trajectory from the point P′a1. Following the ON state trajectory causes the inverter, or equivalentlySAM, to turn off when the operating point meets the current limit line im. Therefore, the operating pointpursues the OFF state trajectory until it hits the low-current limit, which is 0. At this point, again, theinverter, or equivalently SAM, is turned on. This cycle repeats for as long as the operating point is withinthe OFF state trajectory, illustrated in Fig. 2.8 (a). Once the operating point hits the boundary of theOFF state trajectory, at point P′a2, the inverter is turned off, which makes the operating point follow theOFF state trajectory. Finally, at point P′a3, the operating point reaches the desired operating point, whichis (Vrn,0).The criterion for selecting the current limit im is that the inverter must remain in the ON state for atleast half a cycle of the resonant period. Therefore, im can be chosen by the equation given as follows:Im1−√1− I2m ≥ tan( piTrTAM ) (2.60)28OSAM: OnSAM: OffDesired operating pointOn state trajectoryOff state trajectory1The average output rectifiedcurrent in on stateThe average output rectifiedcurrent in off stateVrn-1Vrn-Δv’ontVrnƟrPΔvtnΔIb1Pb2Pb3Pb2Pb1Pb3b1rb1rb2b1Ɵb2Ɵb3LnbonbtntnPb1 tnPb2 tnPb3tnCnLS-UvOAMniCoAMni ,Con iCoAMnvoAMn(pb2),iCoAMn(Pb2))vOAMnFigure 2.10: The converter transient response to load step-up in the state-plane and time-domain.In Fig. 2.8, the start-up performance of the converter employing the AGC-type 2 in the time-domainis also shown.Another important scenario for analyzing the converter transient is sudden load step-up when theconverter is controlled by the AGC type-1 and works in the desired operating point. To discuss onthis term, Fig. 2.10 can be considered. By abruptly changing the load, the current through the outputcapacitor becomes negative and the operating point drops from (Vrn,0) to (Vrn,−∆ILnb). Therefore, SAMturns on at the point Pb1 and causes the converter pursues the ON state trajectory as shown in Fig. 2.10.At the point Pb2 where is the intersection between the ON and OFF state trajectories, SAM turns off,and the OFF state trajectory is followed until the converter reaches the desired operating point whichis the point Pb3. Depending on the load added to the output, the interval the converter needs to reachthe desired operating point again can be calculated. For this reason, the angles θb1, θb2 and θb3 can becomputed by the following equations.θb1 = cos−1(1−Vrn√∆I2Lnb+(1−Vrn)2) (2.61)29ΔIΔvLnbonb10.90.80.70.60.50.40.30.20.1000.10.20.30.40.50.60.70.80.91V =1rnV =0.75rnV =0.5rnV =0.25rnV =0.1rnFigure 2.11: The normalized output voltage variation versus the load added to the converter.θb2 = cos−1(1−Vrn+(1/4)∆I2Lnb√∆I2Lnb+(1−Vrn)2) (2.62)θb3 = cos−1(1+Vrn− (1/4)∆I2Lnb1+Vrn) (2.63)Now that the angles θb1, θb2 and θb3 have been obtained, the required time for the converter to reachto the desired point after load step-up can be calculated using the equation below.tLS−U =TAM2pi(θb1+θb2+θb3) (2.64)Also, the normalized output voltage variation after load step-up is given in the following:∆vonb = (1−Vrn)−√∆I2Lnb+(1−Vrn)2 (2.65)In Fig. 2.11, the variation of the normalized output voltage versus the load added to the output fordifferent reference voltages is plotted.The next scenario is analyzing the converter, controlled by the AGC type-1 in load step-down. Toanalyze the dynamics of the converter, Fig. 2.12 is utilized which illustrates the dynamic response of theconverter in both the state-plane and time-domain after applying load step-down. After load step-down,the operating point from (Vrn,0) jumps to (Vrn,∆ILnc). In this case, SAM turns off, and the converterpursues the OFF state trajectory as shown in Fig. 2.12. At the intersection of the ON and OFF statetrajectories (the point Pc2), SAM turns on and causes the converter to follow the ON state trajectory.Finally, at the point Pc3, the output voltage meets the desired operating point. The angles θc1, θc2 and30θc3 are given by the following equations.θc1 = cos−1(1+Vrn√∆I2Lnc+(1+Vinn)2) (2.66)θc2 = cos−1(1+Vrn+(1/4)∆I2Lnc√∆I2Lnc+(1+Vinn)2) (2.67)θc3 = cos−1(1−Vrn− (1/4)∆I2Lnc1−Vrn ) (2.68)Therefore, the required time for the converter to reach the desired operating point can be calculatedby use of the following equation.tnLS−D =TAM2pi(θc1+θc2+θc3) (2.69)Also, the output voltage variation after applying load step-down is given by the equation below.∆vOnc =−(1+Vrn)+√∆I2Lnc+(1+Vrn)2 (2.70)In Fig. 2.13, the variation of the normalized output voltage versus load variation for different refer-ence voltages is plotted.The next scenario is the sudden reference and input voltage changes. In this scenario, the converter iscontrolled by the AGC type-1. Because all the voltages are normalized by the input voltage, discussingon the converter dynamic response to the sudden reference voltage change is valid for the sudden inputvoltage change. For this reason, Figs. 2.14 and 2.15 are utilized. First of all, the converter dynamicresponse in abrupt reference voltage decrement (abrupt input voltage increment) is analyzed, and afterthat, the dynamic of the converter model to the sudden reference voltage increment (sudden input voltagedecrement) will be discussed. As shown in Fig. 2.14, after suddenly reducing the reference voltage fromVrnd0 to Vrnd1, SAM turns off and causes the converter to follow the OFF state trajectory. At the pointPd2 the two trajectories intersect each other. Therefore, at this point, SAM turns on which provides anopportunity for the converter to follow the ON state trajectory. Eventually, at the point Pd3 the convertermeets the new desired operating point. The angles θd1 and θd2 are given as follows:θd1 = cos−1[1− vOAMn(Pd2)1−Vrnd1 ] (2.71)θd2 = cos−1[1+ vOAMn(Pd2)1+Vrnd1] (2.72)Where,von(Pd2) =14[(1+Vrnd0)2+(1−Vrnd1)2] (2.73)By Use of (2.71) and (2.72) the interval in which the converter reaches the desired operating point31vOAMnOOn state trajectoryOff state trajectoryVrnThe average output rectifiedcurrent in on stateThe average output rectifiedcurrent in off stateVrn1-1SAM: OffSAM: OnDesired operating pointVrn+Δv’’VrnontnƟrtnΔvPc1ΔIPc2Pc3Pc3Pc1Pc2c2rc2rc1 c3Ɵc1Ɵc2LnconctnPc1 tnPc2 tnPc3t’n C(voAMn(PC2),iCoAMn(Pc2))i ,Con iCoAMniCoAMnvOAMnFigure 2.12: The converter transient response to load step-down in the state-plane and time-domain.after applying the sudden reference voltage decrement can be calculated as follows:tRD =TAM2pi(θd1+θd2) (2.74)In Fig. 2.15, the dynamic response of the converter model in the abrupt reference voltage incrementis illustrated. After sudden increasing the reference voltage from Vrne0 to Vrne1, SAM turns on and causesthe converter to follow the ON state trajectory as shown in Fig. 2.15. At the point Pe2, the intersectionbetween the ON and OFF state trajectories, SAM turns off which enables the converter to follow the OFFstate trajectory. At the point Pe3, the converter reaches the desired operating point. The angles θe1 andθe2 can be calculated by use of the equations below.θe1 = cos−1[1− vOAMn(Pe2)1−Vrne0 ] (2.75)θe2 = cos−1[1+ vOAMn(Pe2)1+Vrne1] (2.76)32 V =1rnV =0.75rnV =0.5rnV =0.25rnV =0.1rn10.90.80.70.60.50.40.30.20.10ΔILnc00.050.10.150.20.250.30.350.4Δv oncFigure 2.13: The normalized output voltage variation versus the removed load from the converter.Where,vOn(Pe2) =14[(1+Vrne1)2+(1−Vrne0)2] (2.77)By use of (2.75) and (2.76), the interval in which the converter converges to the new desired operat-ing point can be calculated as follows.tRI =TAM2pi(θe1+θe2) (2.78)All the previous discussions on the converter average large-signal model enable the AGC laws (type-1 and type-2) to be obtained. If the boundaries of the ON and OFF state trajectories of the convertermodel are shown by σon and σo f f , the proposed controller algorithms according to the current directionin CO can be easily obtained as shown in Fig. 2.16. These algorithms are implemented in the controllayer 1, which is depicted in Fig. 2.1 (a). When the converter operates in the transient region, high-lighted in Fig. 2.1 (b), the control layer 1 and therefore the AGC type-1 or AGC type-2 are active.It is interesting to note that the geometric control law is just a simple circle equation that is easy toimplement, as discussed in the following:σon = i2CoAMn(tn)+ [vOAMn(tn)−1]2− [1−Vrn]2 (2.79)σo f f = i2CoAMn(tn)+ [vOAMn(tn)+1]2− [1+Vrn]2 (2.80)33OOn state trajectoryOff state trajectorytn1The average output rectifiedcurrent in on stateThe average output rectifiedcurrent in off statesteady stateƟtn-1 2SAM: OffSAM: OnDesired operating pointvrefnV rnd1rd1Pd1Vrnd0tnPd2Pd3Pd1Pd2Pd3rd2d1Ɵd2V rnd0Pd2Pd1 Pd3tntnVrnd1Vrnd0Vrnd1Pd1tnPd2tnPd3RDvOAMniCoAMn(voAMn(Pd2),iCoAMn(Pd2))iCoAMnvOAMnFigure 2.14: The converter transient response to reference voltage decrement (input voltage incre-ment) in the state-plane and time-domain.2.4 Loss AnalysisIn Section 2.2, the ACTs of the LLC converter were obtained, while losses were neglected. If lossesproduced by the resistive behavior of the wires, components, and equivalent series resistance (ESR)of the output capacitor are taken into consideration, the average trajectories do not remain circularaltogether. However, in the state-plane, the trajectories still can be approximated as circles since thedistance the operating point travels is small. Therefore, to maintain the AGC effective for providingthe fast dynamic response or inrush current limit performance in the control layer 1, the loss effect34OOn state trajectoryOff state trajectorytn1The average output rectifiedcurrent in on stateThe average output rectifiedcurrent in off statesteady stateƟe1tnPe1-1 2Vrne0SAM: OnSAM: OffDesired operating pointVrne1vrefnre1Vrne0Vrne1tnRIPe1Pe2Pe3Pe2Pe3Pe1re2Ɵe2Pe3Pe1Pe2tntnVrne0Vrne1tnPe2 tnPe3(voAMn(Pe2),iCoAMn(Pe2))vOAMniCoAMniCoAMnvOAMnFigure 2.15: The converter transient response to reference voltage increment (input voltage decre-ment) in the state-plane and time-domain.must be considered. Different loading conditions, including a resistive loading condition, have differenteffects on the trajectories, and analyzing all the trajectories obtained under different loading conditionsis almost impossible. However, in order to design the proposed average geometric controller, activated inthe control layer 1, the start-up dynamic behavior should be considered. In start-up, the operating pointmoves a larger distance. In start-up, the worst loading condition occurs when the LLC converter is inno-load condition or under a constant current load. Under these two loading conditions, the slope of theON state trajectory is larger than those obtained when a resistive load is used. The general differential35YesStartRead ADCNormalizationiCoAMn>0?σ <0?off1NoNoYesTurn offinverterσ <0?onTurn offinverterYesNoTurn oninverterTurn oninverter1(a)YesStartRead ADCNormalizationiCoAMn>0?σ <0?off11NoNoYesTurn oninverterLow Thr.Turn offinverterNoTurn offinverterσ <0?onTurn offinverterYesNoTurn oninverter(b)L H High Thr.Figure 2.16: The control algorithms of the proposed AGC controller: a) AGC-type 1, b) AGC-type2. These algorithms are employed in the control layer 1.equation of the average large-signal model, considering the loss effect, is given as follows:−uVin2n+LAMn2dirAMdt+1CO∫ t0(irAM− IO)dt+ voAM0+RirAM = 0 (2.81)where R represents the resistive behavior of the system. In order to generalize the analysis, (2.81) isnormalized by the factors given in Section 2.2 as follows:−u+ dirAMndtn+∫ tn0(irAMn− IOn)dtn+ voAMn0+RnirAMn = 0 (2.82)where,Rn =RZAM(2.83)Solving (2.82) in time-domain and considering that iCoAMn = irAMn− IOn give iCoAMn as follows:iCoAMn = e−Rn2 tn [u− voAMn0− (Rn/2)(irAMn0+ IOn)ωnsin(ωntn)+ iCoAMn0cos(ωntn)] (2.84)where,ωn =12√4−R2n (2.85)In practice, R, which represents the resistive behavior of the whole system, is much smaller than ZAMmaking Rn << 1. Therefore, according to (2.85), ωn ≈ 1. Moreover, the average voltage dropped acrossRn, compared to u− voAMn0 is negligible. Hence, (2.84) can be approximately rewritten as follows:360 0.5 1 1.5 2-1-0.500.51Quadrant2Quadrant3Quadrant4Quadrant 1didealdVoAMni CoAMnIdeal trajectoryReal trajectoryFigure 2.17: The loss effect on the average circular trajectories.iCoAMn = e−Rn2 tn [(u− voAMn0)sin(tn)+ iCoAMn0cos(tn)] (2.86)Compared to (2.46), the above equation has a coefficient of e−(Rn/2)tn approaching zero over time.This coefficient causes the average trajectories of the LLC converter not to be circular, as shown in Fig.2.17. However, in start-up and in the region where the AGC controls the LLC converter, the trajectoriescan acceptably be considered circular, while their radii are smaller than the ideal trajectories, as shownin Fig. 2.17.In order to obtain the actual radius of the average trajectories r, the averaging technique can beapplied to the coefficient of (2.86) in the first quadrant of state-plane, as follows:k =1pi∫ pi0e−Rn2 tndtn =2piRn(1− e− pi2 Rn) (2.87)Following the procedure introduced in Section 2.2 results in the equation below, which gives theaverage circular trajectories, and which includes the loss effect.iCoAMn(tn)2+[voAMn(tn)− ku]2 = k2[i2CoAMn0+(u− voAMn0)2] (2.88)Considering the loss effect, the boundaries of the ON and OFF state trajectories, used in the controlalgorithms shown in Fig. 2.16 can be obtained as follows:σon = [iCoAMn(tn)]2+[voAMn(tn)− k]2− [k−Vrn]2 (2.89)37PIFull Bridge InverterSRioRLCO+vo-vinvovref+-vSFCeLrCrResonant TankMUXvoioAGCNonlinearcontroller fLimitterThe control structureio01 Init.SFCFeedforward PathVCOswLmn:1 +vsec-SR controllervsecResonant Freq. TrackingFigure 2.18: Conceptual block diagram and sensing strategy for the control structure.vOAMnONon-operating regionTransient regioniCoAMn(V ,0)Desired Operating pointNon-operating regionLh1Lh2Hysteresis loopquiescent regionHysteresis loopQuiescent regionrrnh2rh1Section IISection ISection IIISection IVFigure 2.19: The different operating regions of the converter in the state-plane.σo f f = [iCoAMn(tn)]2+[voAMn(tn)+ k]2− [k+Vrn]2 (2.90)2.5 Average Geometric Control Transient StrategyAs discussed earlier, further improvements in the performance of resonant converters, including LLCsand SRCs, require solutions addressing the following concerns: the complexity of the analysis, large-signal modeling and dynamic response, small-signal modeling and quiescent region operation, and de-38graded efficiency due to conduction losses in the rectifier and variations in the resonant frequency. Thischapter addresses the first and second concerns and so far has introduced two control algorithms thatare used in the first control layer. The proposed three-layer control strategy is illustrated in Fig. 2.1 (a).In this section, the implementation of the proposed large-signal average theory and the AGC lawsare presented. Fig. 2.18 depicts a conceptual diagram of the three layer controller, including the mea-surement of the output voltage, output current, and polarity of the secondary voltage required to resolvetransients with high performance, operate in steady-state, drive the synchronous rectifiers (SRs), trackthe resonant frequency and protect the converter (overload). Detailed discussions on synchronous recti-fication and resonant frequency tracking can be found in Sections 4 and 5. The AGC is enabled in thecontrol layer 1, which is illustrated in Fig. 2.1 (a). The control structure in Fig. 2.18 allows the LLCand SRC converters to operate in geometric mode (AGC) and quiescent mode and it is employed fortesting, comparison, and validation. Fig. 2.19 exhibits the different regions on the state-plane showingthe quiescent region (orange region) and the transient region (blue region). When the converter is withinthe quiescent region, a simple controller (e.g., proportional integral (PI)) can appropriately control theconverter in steady-state. However, when the system operates away from the quiescent region, the tran-sient is resolved with high performance by the AGC laws, directing the operating point towards thetarget. Once the transient is resolved, any simple linear controller can be used to manage the steady andquiescent operation using frequency modulation.As discussed in theory, the AGC forces the converter to follow the circular trajectories to arrivein the target quiescent area. According to (2.79) and (2.80), the average current through the outputcapacitor iCo participates in the control decisions of the AGC. Since the average iCo can be derived byonly sensing the output voltage (iCo =COdvO/dt), there is no need to use a current sensor for measuringiCo. However, if extra accurate performance is needed, a current sensor can be used to measure theaverage iCo rather than estimating it. Moreover, in most cases, the input voltage of the converter isalmost constant and usually has a tiny variation. Therefore, by knowing the value of the input voltage,the input voltage is not necessary to be sensed. However, in applications with the large variation in theinput voltage, the input voltage can be sensed and employed as part of the AGC in (2.79) and (2.80).Once the transient large-signal dynamics are completed with the AGC, the converter enters the small-signal quiescent region. If a simple linear PI controller is employed for steady-state operation, thetransition between AGC to linear can be done with a switching frequency calculator (SFC). The SFCcalculates the appropriate switching frequency to initialize the linear controller according to the gain ofthe SRC. In order to calculate the appropriate switching frequency, the loading condition informationmay be required. Therefore, the current through the load should be measured. The advantages of sensingthe load current are the provided soft transition between the AGC and quiescent regions and protectionagainst short circuit. Also, to prevent chattering in the boundary of the transient and quiescent regions,a hysteresis loop should be employed. As shown in Fig. 2.19, the quiescent region is determined by thecircle Lh1 with the radius of rh1. The outer circle Lh2 with the radius of rh2 determines the band of thehysteresis loop which can be calculated by rh2− rh1.392.6 Design ConsiderationsIn Section 2.1, it was theoretically shown that the LLC/SRC can be modeled with the model shown inFig. 2.5. To benefit the introduced model for controlling the converter in the control layer 1 using theproposed AGC method, the output voltage and the load current must be measured. The derivative ofthe output voltage can be used to measure the current through the output capacitor. Also, because, thecurrent through CO has two frequency components (a higher one and a lower one) and based on theneed to extract the frequency component with the lower frequency, a low pass filter should be used. ρ ,representing the ratio between ωeq and ωAM, can be defined as follows:ρ =ωeqωAM(2.91)By applying (2.10), (2.16), (2.32) and (2.36) to the above equation, ρ can be obtained as follows:ρ =pin1cos−1(1−2 CrCo+n2Cr[1+(Lr/Lm)])(2.92)SinceCo is much larger thanCr, according to (2.92), ρ >> 1, meaning that ωeq >>ωAM. Therefore,based on the necessity of measuring the current with the lower frequency, a simple low pass filter canbe used to obtain the average current through the output capacitor iCoAM. That is why a low-cost sensorcan be employed to extract the average current through the output capacitor. In this case, a second orderlow pass filter having a cut off frequency of (ωeq+ωAM)/2 can be used while ωAM << ωcut < ωeq issatisfied.The derivative of the output voltage gives the current through the output capacitor. Therefore, em-ploying a current sensor to sense iCo is unnecessary; however, if a more accurate measurement is re-quired, a current sensor can be employed. Hence, the following transfer function can be used to obtainiCoAM from vo:G(s) =Cos[1+(s/ωcut)]2(2.93)The phase shift that the second order low pass filter imposes on the system is given as follows:Φ=−2tan−1( 2ρ+1) (2.94)Since, in practice, Φ is small enough, the phase delay caused by the second order low pass filter usedis negligible, making the AGC effective to provide the extreme start-up dynamic response or inrushcurrent limit performance. The conceptual control structure of the proposed AGC controller in thecontrol layer 1 is shown in Fig. 2.18.For steady-state operation around the quiescent region, a linear PI controller is implemented. Thislinear controller is used in the control layer 2, which is shown in Fig. 2.1 (a). The next step is determin-ing the radius of the circles rh1 and rh2 to create a smooth transition between the fast dynamics AGC andsteady operation. Because, the PI controller is asymptotically stable and due to the extreme dynamics40Table 2.1: The values of the Lyapunov function derivative in different regions of the state-planeCoe f f icients/Sections SectionI SectionII SectionIII SectionIV2(vOAMn−Vrn)u((vOAMn−Vrn)iCoAMn) < 0 = 0 > 0 = 0(vOAMn−Vrn)2δ ((vOAMn−Vrn)iCoAMn) = 0 = 0 = 0 = 0d(vOAMn−Vrn)/dt > 0 > 0 < 0 < 02iCoAMn[1−u((vOAMn−Vrn)iCoAMn)] = 0 > 0 = 0 < 0(iCoAMn)2δ ((vOAMn−Vrn)iCoAMn) = 0 = 0 = 0 = 0diCoAMn/dt < 0,> 0 < 0 > 0,< 0 > 0f < 0 < 0 < 0 < 0of the nonlinear controller which are close to the resonant frequency, the radius of the inner circle of thecomparator shown in Fig. 2.19 can be obtained by use of the following equation.rh1 > i+Copn− IOn (2.95)Where,i+Cop ≈2VinpinRL(2.96)In (2.96), RL is the minimum resistance of the load.The radius of the outer circle of the comparator can be selected based on the error of the voltagegain equation. In case that the first harmonic approximation (FHA) is used, it is recommended that theradius of the outer circuit is 0.1Vrn.2.7 Stability AnalysisThe large-signal stability of the AGC controller, used in the control layer 1, can be justified usingthe Lyapunov theorem. According to the control laws shown in Fig. 2.16, the following function(Y : ℜ2→ℜ):Y (vOAMn−Vrn, iCoAMn) = [(vOAMn−Vrn)2u((vOAMn−Vrn)iCoAMn)+ i2CoAMn[1−u((vOAMn−Vrn)iCoAMn)](2.97)can be a Lyapunov-candidate function of the AGC controller with the unique equilibrium pointY (0,0), because:Y (0,0) = 0 (2.98)Y (vOAMn−Vrn, iCoAMn)> 0,∀(vOAMn−Vrn, iCoAMn) 6= (0,0) (2.99)41||(vOAMn−Vrn, iCoAMn)|| → ∞=⇒ Y (vOAMn−Vrn, iCoAMn)→ ∞ (2.100)and,dY (vOAMn−Vrn, iCoAMn)dt=[2(vOAMn−Vrn)u((vOAMn−Vrn)iCoAMn)+(vOAMn−Vrn)2δ ((vOAMn−Vrn)iCoAMn)]d(vOAMn−Vrn)dt+[2iCoAMn[1−u((vOAMn−Vrn)iCoAMn)]− (iCoAMn)2δ ((vOAMn−Vrn)iCoAMn)]diCoAMndt < 0 (2.101)where,u((vOAMn−Vrn)iCoAMn =0 if (vOAMn−Vrn)iCoAMn > 0,1 if (vOAMn−Vrn)iCoAMn ≤ 0. (2.102)and,δ ((vOAMn−Vrn)iCoAMn) =0 if (vOAMn−Vrn)iCoAMn 6= 0,∞ if (vOAMn−Vrn)iCoAMn = 0. (2.103)The condition (2.101), referred to as the monotonicity requirement of the Lyapunovs theorem, isalways satisfied by the control algorithms shown in Fig. 2.16 and it can be geometrically explained, bydividing the state-plane shown in Fig. 2.19 into four sections. In Table 2.1, the values of the coeffi-cients of (2.101) for different sections are presented. Therefore, Y (vOAMn−Vrn, iCoAMn) is globally andasymptotically stable.As discussed earlier in this section, the AGCs are responsible for governing the LLC/SRC in thetransient region. Once the operating point goes within the quiescent region, the control layer 2 is en-abled. In the control layer 2, a linear compensator is used to minimize the steady-state error. In orderto design the linear compensator, working in the quiescent region, the small-signal models developed inSection 3, can be used. In the quiescent region, the stability of the LLC/SRC, controlled by a linear con-troller can be discussed using the transfer functions of the small-signal models and linear compensator.The final step to ensure the stability of the system is in the selection of the hysteresis band. Theinstructions on how to appropriately select the boundaries of the hysteresis loop can be found in Section2.6.2.8 Experimental and Simulation ResultsFurther improvements in the performance of LLC/SRC converters require solutions addressing the fol-lowing concerns: complexity of the analysis, large-signal modeling and dynamic response, small-signalmodeling and quiescent region operation, and degraded efficiency due to conduction losses in the rec-42tifier and variations in the resonant frequency. In this chapter, the first two concerns are addressed, andtwo AGC algorithms are introduced. The algorithms are used in the control layer 1, which is shown inFig. 2.1 (a). The focus of the analysis in this dissertation is on LLC converters, however, the analysisperformed in this section can be applied to the SRC as well.Table 2.2: The parameters of the LLC converterParameter Value DescriptionVin 400v Input voltageVo 48v Output voltagePo 500W Nominal output powerfr 100kHz Resonant frequencyLr 127µH Resonant inductorCr 20nF Resonant capacitorLm 400µH Magnetizing inductorCo 20µF Output capacitorLAM 320µH Modeled inductor2.8.1 Average Large-Signal Model ValidationIn Section 2.1, the average large-signal model of the LLC converter was introduced. It was shown thatthe average large-signal model of the LLC converter can also be valid for the SRC if it is consideredthat in the SRC, Lm = ∞ and n = 1. Because the average large-signal model is intended to be engagedas the core of the proposed AGC controllers (used in the control layer 1), the accuracy of the averagelarge-signal model must be justified. Therefore, in order to validate the average large-signal model, theexperimental and simulation results of the open-loop LLC and SRC converters and their average large-signal models are presented in this section. For simulation, the software PSIM was used. The parametersof the LLC and SRC converters and their average large-signal models can be found in Tables 2.2 and2.3.In Fig. 2.20, the open-loop start-up dynamic behavior of the LLC converter and average large-signalmodel is shown for different loading conditions. As shown in this figure, for all scenarios tested, thedynamic behavior of both the LLC converter and the average large-signal model are the same. Moreover,the current through LAM resembles the average dynamic behavior of ir. In Fig. 2.21, the open-loopdynamic responses of the LLC converter and the average large-signal model, following a resistive loadstep-up from 10Ω to 5Ω, and a resistive load step-down from 5Ω to 10Ω, are shown. Based on theexperimental waveforms, the proposed model presents the average large-signal dynamic behavior of theLLC converter.Fig. 2.22 shows the key simulation waveforms of the SRC and its average large-signal model infour cases: start-up in no load and under the load conditions, load step-up and load step-down. Thesimulation results prove that the average large-signal model precisely follows the open-loop dynamicbehavior of the SRC if the switching frequency equals the resonant frequency. In the waveforms shown43Table 2.3: The parameters of the SRCConverter Parameters ValueVin 48VVO 24Vfr 80KHzLr 195µHCr 20nFLAM−SRC 481µHCO 33µFωeq 7.93Krad/secωcut 3.14Krad/secρ 63.35Φ 3.65in Fig. 2.22, the output voltages of the SRC and average large-signal model are always equal over thetime, and the current through the equivalent inductor follows the average dynamic of the output rectifiedcurrent.2.8.2 Closed-Loop Experimental Results: AGC type-1As discussed earlier in this section, the average large-signal models of the LLC and SRC convertersare almost the same. Since the AGC type-1 is developed based on the average large-signal models ofthe LLC and SRC converters, experimental validation of the AGC type-1 can be done either by theSRC or LLC converter. In this section, in order to validate the theoretical analysis for AGC type-1, theexperimental results of a 50W prototype full bridge SRC are presented. The AGC type-1 is employedin the control layer 1. Also, to compare the effectiveness of the AGC type-1 with a conventional linearcontroller, the experimental results of a SRC controlled with a PI controller are included. The parametersof the SRC and its average large-signal model are shown in Tables 2.3. The radius of the circles Lh1and Lh2 (rh1 and rh2) that are introduced in Section 2.5, are selected 0.03 and 0.05 in the normalizedstate-plane, respectively.One of the main advantages of the proposed AGC type-1 (used in the control layer 1) is its abilityto prevent voltage overshoot following disturbances. Therefore, to compare the performance of theproposed controller with that of a linear controller, this criterion (prevention of voltage overshoot) shouldbe taken into account during the design of the linear controller. The transfer function of the PI controlleris chosen as follows:C(s) =124[0.2+400s] (2.104)Moreover, since the proposed AGC type-1 is a nonlinear controller, it is appropriate to compare theproposed controller with an advanced nonlinear controller. Therefore, the rotating switching surface(RSS) control method [57] (an advanced control method based on a piecewise affine model) is selected.4460 µ sec 60 µ sec(a)(c)(e)(b)(d)(f)Vg1VoirVoir VgVoAMirAMAMirAMVoAMddidealddideal76.8V 76.8VVg1VoirVoAMirAMVgAM76.8V60 µ sec190 µ sec43.2V60 µ sec76.8V190 µ sec43.6VVg1VoirVoAMirAMVgAM72V60 µ sec140 µ sec38.4V74V62.5 µ sec40.2V140 µ secFigure 2.20: Experimental results of the average large-signal model for an LLC operating in open-loop at resonance validating the accuracy of the proposed model: a) the LLC converter start-up in no-load condition, b) the average large-signal model start-up in no-load condition, c)the LLC converter start-up under a 12Ω resistive load, d) the average large-signal modelstart-up under a 12Ω resistive load, e) the LLC converter start-up under a 5.5Ω resistiveload, f) the average large-signal model start-up under a 5.5Ω resistive load. The averagemodel closely represents the average large-signal behavior.The RSS controller controls the SRC by monitoring the converter behavior in state-plane, where theslope of a line determines the instant when switching actions occur. The RSS controller has two controlloops: inner and outer. The outer loop, which is a linear controller, is responsible for providing the slopeof the line in state-plane. The inner loop is a nonlinear controller that controls the switching actions byusing the slope obtained from the outer loop. In order to employ the RSS control method to controlthe SRC, the current through the resonant inductor, the voltage of the resonant capacitor and the outputvoltage must be measured.45(a) (b)(c) (d)Vg1VoirVg1VoirVoAMirAMVgAMVoAMirAMVgAM37.5 µ sec43.2V 50.4V100 µ sec 44.4V37.5 µ sec100 µ sec 50.4V52.8V 33.5 µ sec95 µ sec44.4V 52.8V 33.5 µ sec95 µ sec44.4VFigure 2.21: Experimental results of the average large-signal model for an LLC operating in open-loop at resonance validating the accuracy of the proposed model: a) the LLC converterbehavior following a resistive load step-up from 10Ω to 5Ω, b) the average large-signalmodel behavior following a resistive load step-up from 10Ω to 5Ω, c) the LLC converterbehavior following a resistive load step-down from 5Ω to 10Ω, d) the average large-signalmodel behavior following a resistive load step-down from 5Ω to 10Ω. The average modelclosely represents the average large-signal behavior.At first, the start-up response of the converter will be discussed. In Fig. 2.23, the start-up responseof the converter employing the AGC type-1 and PI controller is presented under a 50W resistive load(Fig. 2.23 (a) and (b)), a 25W resistive load (Fig. 2.23 (c) and (d)) and no-load condition (Fig. 2.23(e) and (f)). In the case that the converter is controlled by the AGC type-1 and loaded with a 50Wresistive load, the output voltage rise time is 155µsec and the output voltage settling time is 175µsec.Under the 25W resistive load, the output voltage rise time is 162µsec and the output voltage settlingtime is 180µsec. For the same transient, the key experimental waveforms of the converter employingthe conventional PI controller are shown in Fig. 2.23 (b), (d) and (f), where the converter is loaded witha 50W resistive load (Fig. 2.23 (b)), 25W resistive load (Fig. 2.23 (d)) and no-load (Fig. 2.23 (f)).In the case that the converter is loaded with the 50W and 25W resistive loads, the output voltage risetime is 23msec, however, its settling time under the 50W resistive load is 33msec and under the 25Wresistive load is 37msec. Comparing the time responses of the two converters exhibits that the proposedgeometric controller in the start-up is around 200 times faster than the conventional PI controller. Also,in the no-load condition, the results show that the output voltage of the converter controlled with theAGC type-1 (used in the control layer 1) has increased 15% over the reference voltage with the transient46600irec(A) irec(A)0 100 200 300 400 500Time (µs)0-55101520020406080100Vo(V) Veq(V)020406080Vo(A) Veq(V)0 1 2 3 4 5Time (ms)0-5510152025irec(A) irec(A)(a)(b)(c)(d)40424446485052Vo(V) Veq(V)6 7 8 9 10Time (ms)0-5510152025irec(A) irec(A)Open loop load step-up instant4244464850525456Vo(V) Veq(V)6 8 10 12 14Time (ms)0-22468irec(A) irec(A)Open loop load step-down instantFigure 2.22: The open-loop simulation results of the SRC at the resonant frequency and its averagelarge-signal model validating the proposed model. a) Start-up in no load condition, b) start-up under a 12Ω load, c) load step-up from 24Ω to 12Ω, d) load step-down from 12Ω to24Ω.time of 200µsec. However, the conventional PI controller in start-up and no-load condition does notshow a good performance since the output voltage converges to near the twice of the input voltage. Inorder to show the converter start-up dynamic response with more details, where the proposed controller(used in the control layer 1) is engaged, the converter zoomed in waveforms are presented in Figs. 2.24(a) through (c) where the converter is loaded with a 50W resistive load, 25W resistive load and no-load,respectively.In the next step, the converter dynamic response to load step-up and step-down is inspected. In Figs.2.25 (a) and (c), the dynamic response of the converter employing the AGC type-1 (used in the controllayer 1) is shown to the resistive load step-up from 25W to 50W and resistive load step-down from50W to 25W , respectively. The converter response time to both load step-up and step-down is 370µsec.After the load step-up, the output voltage undershoot is 19.8V , and after the load step-down, the outputvoltage overshoot is 28.8V . For the same transients, the converter response to the disturbances, wherethe conventional PI controller is used, is shown in Fig. 2.25 (b) and (d) for the load step-up and step-down, respectively. In this case, the converter response time to the load step-up is 20msec and to theload step-down is 17.5msec. The output voltage undershoot after the load step-up is 16.8V , and theoutput voltage overshoot after the load step-down is 30V . Therefore, the AGC type-1 is around 50 timesfaster than the conventional PI controller to responding the disturbances, in addition to the improvedoutput voltage undershoot after the load step-up by 12.5% and output voltage overshoot after the loadstep-down by 5%. In order to show the dynamic response of the converter employing the AGC type-1to the load step-up and step-down with more details, in Figs. 2.26 (a) and (c), the converter zoomed in47(c)(a)t =180µsecs 205 times faster VoirecVgs1t =175µsecs 188 times faster VoirecVgs1(d)(b)t =23msect =37msecsrVoirecVgs1t =33msect =23msecsr VoirecVgs1(e) (f)3.6Vt =200µsecs VoirecVgs1VoirecVgs1Figure 2.23: Comparative experimental validation of the converter dynamic transient using theproposed AGC versus a linear PI controller: a) in start-up and under 50W resistive load,AGC is used, b) the PI controller in start-up and under 50W resistive load, c) AGC in start-up under 25W resistive load, d) PI controller in start-up and under 25W resistive load, e)AGC in start-up under no load, and f) PI controller in start-up under no-load condition.waveforms are presented. For the same transients, the dynamic response of the closed-loop convertercontrolled with the RSS controller is shown in Figs. 2.26 (b) and (d). The response time is 1.7msec, andthe undershoot is 20V after the load step-up. The waveforms shown in Figs. 2.26 (a) and (b) revealsthat the response time of the proposed AGC controller is 4.6 times faster than that of the RSS controller.In Fig. 2.26 (d), the response of the converter controlled by the RSS controller after a load step-downfrom 50W to 25W is shown. The response time of the system after the load step-down is 1.4msec, andthe overshoot is 30V . For the same transient, the response time of the proposed AGC type-1 is 370µsec,and the overshoot is 28.8V .In the next experiment, the converter performance to the sudden reference voltage change was in-vestigated. In Fig. 2.27 (a) and (c), the response of the converter employing the proposed controller(used in the control layer 1) to the sudden reference voltage change from 15V to 24V and from 24V to48(a)t =180µsect =162µsecsrVoirecVgs1(b)t =175µsect =155µsecsrVoirecVgs13.6Vt =200µsecsVoirecVgs1(c)Figure 2.24: The zoomed in detailed waveforms of the converter with the proposed controllerduring start-up under different loading conditions. a) under a 50W resistive load, b) under a25W resistive load, c) in no-load condition.15V is shown where the converter is under a 25Ω resistive load, respectively. The response time of thesystem to the reference voltage step-up is 200µsec and to the reference voltage step-down is 400µsec.In Fig. 2.27 (b) and (d), the response of the converter controlled with the PI controller to the samedisturbances is shown. As it is indicated in Fig. 2.27 (b) and (d), the response time of the converter tothe reference voltage step-up is 34msec and to the reference voltage step-down is 30msec. These resultsshow that, compared to the conventional PI controller, the proposed controller is 175 times faster atresponding to the reference voltage step-up, and 75 times faster at responding to the reference voltagestep-down. In order to show the dynamic response of the converter controlled with the AGC type-1 withmore details, in Figs. 2.28 (a) and (c), the converter zoomed in waveforms are shown after applyingthe sudden reference voltage changes. In Figs. 2.28 (b) and (d), the dynamic responses of the convertercontrolled with the RSS controller are presented after a reference voltage step-up from 15V to 24V , and49(a)t =370µsecs19.8V54 times faster VoirecVgs1(c)t =370µsecs28.8V47 times fasterVoirecVgs1(b)16.8Vt =20msecsVoirecVgs1(d)30Vt =17.5msecsVoirecVgs1Figure 2.25: Load transient comparative experimental validation using the proposed AGC (usedin the control layer 1) versus a linear PI controller: a) load step-up with the proposed AGC,b) load step-up with the PI controller, c) load step-down with AGC, and d) load step-downwith the PI controller.(a)t =370µsec19.8Vs VoirecVgs1(c)t =370µsec28.8VsVoirecVgs1(b)(d)0510152025303520 20.5 21 21.5 22Time (msec)0-5510Voirec20Vt =1.7msecs0510152025303540 40.35 40.7 41.05 41.4Time (msec)0-5510Voirec30Vt =1.4msecsFigure 2.26: The zoomed in detailed waveforms of the converter employing the proposed AGCstrategy (used in the control layer 1) and the RSS controller: a) after the load step-up withthe AGC, b) after the load step-up with the RSS, c) after the load step-down with the AGC,and d) after the load step-down with the RSS.50(a)t =200µsecs170 times fasterVoirecVgs1(c)t =400µsecs75 times fasterVoirecVgs1(b)t =34msecsVoirecVgs1(d)t =30msecsVoirecVgs1Figure 2.27: Reference change comparative experimental validation with the proposed AGC ver-sus the linear PI controller: a) voltage reference step-up with the proposed AGC, b) voltagereference step-up with the PI, c) voltage reference step-down with the AGC, and d) voltagereference step-down with the PI.(a)t =200µsecsVoirec(c)t =400µsecsVoirec(b)010203040 Vo120 121 122 123 124 125Time (msec)0-551015irect =5msecs126(d)05101520253035 Vo60 62 64 66 68 70Time (msec)0-551015irect =8.1msecsFigure 2.28: The zoomed in detailed waveforms of the converter employing the proposed AGCstrategy and the RSS controller: a) after the reference voltage step-up with the AGC, b)after the reference voltage step-up with the RSS, c) after the reference voltage step-downwith the AGC, and d) after the reference voltage step-down with the RSS.51vo1/48 +-Descrete-time integratorTs=10µsec 010.1Constant0.1Descrete-time integratorTs=10µsec -110.03z-1++Saturation01VCOGate DriverS1 S270kHz -200kHzFigure 2.29: The block diagram of the conventional controller providing the inrush current limitoperation for the LLC converter.no driving lossesno circulation current in no load48 VVg1VoiCoiLr(b)60 Vcirculation current in no loaddriving lossesVg1VoiCoiLr(a)Vref=48 V Overshoot =12VNo overshoot in no load conditionFigure 2.30: Experimental results of the closed-loop LLC converter, in no-load condition, con-trolled by a) the conventional controller limiting the start-up inrush current, and b) the AGCtype-2 limiting the start-up inrush current.after a reference voltage step-down from 24V to 15V . The response time of the closed-loop SRC afterthe reference voltage step-up is 5msec, and after the reference voltage step-down, is 8.1msec.2.8.3 Closed-Loop Experimental Results: AGC type-2It was discussed that the average-large-signal models of the LLC and SRC converters are almost thesame, and therefore, the operation of the AGC algorithms can be justified either in the LLC converteror SRC. In the previous subsection, the AGC type-1 was applied to the SRC and its operation was52100kHz, no overdesign in driver and magnetic componentst = 340 µsecrt = 437 µsecsVg1VoiCoiLr(a)(b)200kHz, overdesign in driver and magnetic componentsVg1VoiCoiLrInrush current limitedInrush current limitedFigure 2.31: The zoomed-in experimental results of the closed-loop LLC converter, in no-loadcondition, controlled by a) the conventional controller limiting the start-up inrush current,and b) the AGC type-2 limiting the start-up inrush current.experimentally validated. The performance of the AGC type-1 was also compared with the linear PIand RSS controllers. In order to show the way the AGC-type 2 governs the LLC converter start-up toprovide the inrush current limit performance, the experimental results of the closed-loop LLC converterare presented here and compared to those of a conventional controller limiting the start-up inrush current.The conventional start-up inrush current controller turns on the inverter at 200kHz switching frequencyand degrades it until the quiescent region is achieved. The block diagram of the conventional linearcontroller is shown in Fig. 2.29. The sampling time is 10µsec.In Fig. 2.30 (a), the closed-loop start-up dynamic response of the LLC converter utilizing the con-ventional controller under the worst case scenario (which is the no-load condition) is shown. As illus-trated in this figure, the maximum current through the resonant tank is limited to 4.6A. However, avoltage overshoot of 25% is generated, since the output voltage converges to 60V in the quiescent area.In this case, the switching frequency of the converter remains at 200kHz in the quiescent area, whichcreates a circulating current in the power stage that degrades the light load efficiency of the converter.Moreover, the gate driver works at 200kHz switching frequency in the quiescent area that imposes thedriving losses to the system. For the same transient and condition, the closed-loop start-up dynamicresponse of the LLC converter employing the AGC-type 2 is shown in Fig. 2.30 (b). As shown in this5303691215VG1 (V) VG2 (V)time0-0.2-0.4-0.6-0.80.20.40.6iin (A)Time division: 2.5µsecFigure 2.32: The current taken from the input voltage by the LLC converter controlled by theconventional controller in no-load condition and steady-state.figure, the start-up inrush current is significantly limited by the AGC-type 2, where the maximum cur-rent of iLr is limited to 5.5A. Moreover, in this case, the settling and rise times are 437µsec and 340µsec,respectively. Compared to the conventional method, the AGC-type 2 can eliminate the circulation andgate driver losses during no-load condition.In order to have a better insight into the LLC converter closed-loop behavior controlled by theconventional controller and AGC-type 2, under the no-load condition, in Fig. 2.31, the zoomed-inexperimental results of the closed-loop LLC converter are shown.In addition to all its other advantages (such as reduced complexity of analysis owing to the use ofthe average large-signal model, operation at a fixed switching frequency during start-up, which elimi-nates the overdesign of magnetic components and gate driver circuit, and improved large-signal start-updynamic performance), the proposed AGC can be used to control the LLC converter in the quiescentarea under one condition: that the input voltage of the LLC converter is fixed. Under this condition,the use of frequency or phase shift modulation is not required, since working at the resonant frequencyresults in a voltage gain of 1/2n. This means that in the control layer 2, the AGC can be employed.In applications where the input voltage is not strictly fixed, the proposed AGC controller takes the op-erating point to the quiescent area. This improves the large-signal performance of the system. Then alinear controller can be used to obtain a zero steady-state error. A more detailed discussion is providedin Section 2.3. In no-load condition, the proposed AGC can naturally turn off the inverter of the LLCconverter, which eliminates no-load switching, the gate driver and circulation losses. In Figs. 2.30 and2.31, the steady-state performance of the LLC converter controlled by the AGC-type 2, and the con-ventional controller is shown in no-load condition. In order to show how much the power losses can bereduced by using the proposed average geometric controller, in Fig. 2.32, the current taken by the LLC54(a)(b)Vg1VoiCoiLrt = 435 µsecrt = 1.825 msecsVg1VoiCoiLrt = 420 µsecrt = 700 µsecsInrush current limitedStart-up time frameStart-up time frameInrush current limitedFigure 2.33: Experimental results of the closed-loop LLC converter, in full load condition, con-trolled by a) the conventional controller limiting the start-up inrush current, and b) the AGCtype-2 limiting the start-up inrush current.converter controlled by the conventional controller, in the no-load condition and steady-state while theswitching frequency is 200kHz, is shown. As shown in this figure, the root mean square (rms) currenttaken from the input voltage is 0.204A, which creates circulation losses. For switches S1 and S2, themetal-oxide-semiconductor field-effect transistor (MOSFET) is used. At 200kHz switching frequency,the gate driver losses are around 2W . The difference between the conventional controller and the pro-posed AGC-type 1 and AGC-type 2 (in terms of reduced power losses) is that the AGC controllers turnoff the switches of the LLC inverter in no-load condition while the converter is in the quiescent area.In this way, the proposed AGC controller (employed in the control layer 1) can reduce the power con-sumption to almost zero in the no-load condition. In no-load condition, defining efficiency is not useful,as the output power is at all times zero. That is why the power consumption is discussed in the no-loadcondition in order to compare the proposed AGC controllers with the conventional linear controller.In Fig. 2.33 (a), the closed-loop start-up dynamic response of the LLC converter using the conven-tional controller is shown, in full load condition, where the settling and rise times are 1.825msec and435µsec, respectively. In this case, the maximum current of the resonant inductor is limited to 5.6A byturning on the inverter at 200kHz switching frequency. In Fig. 2.33 (b), the same transient is shown forthe LLC converter using the AGC-type 2, where the settling and rise times are 700µsec and 420µsec,55(a)(b)Vg1VoiCoiLrt = 1.825 msecst = 435 µsecr200kHz, overdesign in driver and magnetic componentsInrush current limitedVg1VoiCoiLrt = 700 µsecst = 420 µsecrInrush current limitedFigure 2.34: The zoomed-in experimental results of the closed-loop LLC converter, in full loadcondition, controlled by a) the conventional controller limiting the start-up inrush current,and b) the AGC type-2 limiting the start-up inrush current.respectively. As shown in this figure, the maximum current through the resonant tank is limited to 5.5A,while the switching frequency is maintained at the resonant frequency. Therefore, using the AGC-type2 results in a limited amount of start-up inrush current, similar to the amount produced when using theconventional method. However, the settling time of the LLC converter obtained using the AGC-type 2is 2.6 times faster than that obtained using the conventional controller.The AGC-type 2 provides a faster start-up dynamic response while limiting the start-up inrush cur-rent because the proposed controller increases the ratio of the positive current to the negative currentfor the output capacitor CO in the start-up. This means that the DC component of the current throughthe output capacitor is larger than that obtained when the conventional controller controls the LLC con-verter. In the primary side, however, the resonant current DC component is zero, since there is norectification process. To clarify further, the start-up time frames of the LLC converter when controlledby the conventional controller and the AGC-type 2 are shown in Fig. 2.33. As shown in this figure,the start-up time of the LLC converter controlled by the conventional controller is 38% longer than thestart-up time obtained by the AGC-type 2.In order to have a better insight into the LLC converter closed-loop behavior controlled by the con-ventional controller and AGC-type 2, under full load condition, in Fig. 2.34, the zoomed-in experimental5691929394959610% 15% 20%ConventionalAGCEfficiencyPercentage of Output power/Nominal Power95% 100%Burst mode Frequency modulation modeFigure 2.35: The efficiency curves of the LLC converter controlled by the proposed AGC, and bythe conventional controller.results of the closed-loop LLC converter under full load condition are shown.Under the no-load condition, the traditional controller keeps the system running, thus generatinglosses in the primary. Unlike the traditional controller, AGC can go to true standby mode and turn offthe power stage, resulting in near zero power loss. In both cases, and since the output power is zero,the efficiency cannot be calculated under the no-load condition. Under light loading conditions (below20% of full loading condition), the LLC converter controlled by the AGC shows better efficiency thanwhen it is controlled by the traditional controller as shown in Fig. 2.35. In 10% of the nominal outputpower, the efficiency improvement achieved while using the proposed AGC is 1.2%. Under light loadingconditions, the three-layer controller can be programmed in such a way to remain in the control layer1, even when the operating point of the converter arrives in the quiescent region. Therefore, a naturalburst mode control is achieved by the AGC. As shown in Fig. 2.35, below 20% of the nominal power,the control layer 1 still remains in charge of controlling the converter. Above 20% of the nominal powerand in the quiescent region, the control layer 2 is enabled.2.9 SummaryThe traditional technique to control the power electronic converters for many years has been based onthe linear controllers which use small-signal modeling techniques. Since the small-signal models arejust valid near the desired operating point, the converter response to large transients is usually inade-quate, however, it provides a good behavior in steady-state. This work introduces a three-layer controlstrategy (shown in Fig. 2.1 (a)) that addresses not only the above concern but also the efficiency. In thischapter, a nonlinear control method called average geometric control (AGC) was introduced to providefast dynamic response or inrush current limit performance for the LLC and SRC converters with theavailability of low-cost microprocessors. The proposed AGC is used in the control layer 1. In orderto provide an analytical tool for analyzing the converter large-signal behavior, an average large-signal57model was introduced enabling the average circular trajectories of the LLC and SRC converters to obtaintwo control laws called AGC type-1 and AGC type-2. The start-up inrush current can be convention-ally limited by turning on the inverter at a very high switching frequency and reducing the frequencyuntil the target point is achieved. Utilizing this conventional method requires that the magnetic compo-nents and gate driver circuit to be overdesigned. Moreover, it slows down the start-up response of theLLC converters and cannot eliminate the circulating current in the no-load condition. The advantageof the proposed AGC is that the switching frequency does not need to be increased above the resonantfrequency, which prevents the overdesign of the magnetic components and the gate driver circuit. Onaverage, the experimental results of the proposed geometric control method showed that the converterresponse to disturbances, including load and reference voltage changes and also the start-up, is muchfaster than that of the designed linear PI controller. Moreover, the overshoot after disturbances wasvirtually eliminated. The experimental results also showed a significant improvement in the start-updynamic response of the LLC converter when using the AGC.58Chapter 3Average Small-Signal Modeling of LLCConverters1As discussed in Chapter 1, the performance of LLC converters is improved by addressing concerns aboutthe complexity of the analysis, large-signal modeling and dynamic response, small-signal modelingand quiescent region operation, and degraded efficiency due to conduction losses in the rectifier andvariations in the resonant frequency. In this dissertation, a three-layer control strategy is proposed toaddress these concerns. The three-layer control strategy is shown in Fig. 3.1 (a). In chapter 2, thefirst control layer was introduced in order to address the large-signal modeling and control of LLC andseries resonant converters. The operating regions of the LLC converter are shown in Fig. 3.1 (b). Oncethe transient condition is passed (in transient the control layer 1 is active) and the converter enters thequiescent region, the control layers 2 or 3 or both can be activated. In the control layer 2, a linearcontroller is employed that keeps the steady-state error close to zero. In this chapter, a new averagesmall-signal modeling technique, based on the analysis of the homopolarity cycle, is proposed. Small-signal models are essential in designing linear compensators, employed in the control layer 2.As shown in Fig. 3.2 (a), LLC converters transfer energy through a high-frequency resonant tank,behaving as a bandpass filter. The components of the resonant tank have zero mean voltage and currentvalues, in a switching cycle, and therefore, unlike pulse width modulation (PWM) converters, tradi-tional averaging techniques cannot be directly used for small-signal modeling of LLC converters [28].Therefore, more complicated techniques are required for the analysis and modeling of LLC converters[3].Generally, the first step in obtaining the small-signal model of LLC converters is the analysis oftheir large-signal dynamic behavior. The large-signal dynamic behavior of LLC converters can be ex-pressed through a set of nonlinear discrete state-space equations [29]. Discrete state-space equations arenonlinear, and they must be linearized first before they are used for small-signal modeling. Lineariza-tion of the discrete state-space equations can be conventionally performed either by Taylor or Fourierexpansion around the equilibrium point and most of times in the vicinity of the resonant frequency1Portions of this chapter have been modified from [5]59 synchronous rectification and {Proposed lthree layer control strategy(a)vOiCovrefNon-operating regionTransient region (Control: Layer 1)(b)Quiescent region (Control: Layer 2) or(Control: Layer 3) Linear Control: based on small-signal modeling using homopolarity cycle2 Nonlinear geometric control:31Figure 3.1: a) The proposed three-layer control strategy where the layer 2 (linear control) is active,b) regions of operations in resonant converters. The active control layer is highlighted inyellow.[30–42]. Usually, linearization through the Taylor expansion produces high-order equations, and thismakes the derivation of the analytical small-signal transfer function complicated [30–35]. Therefore,empirical/numerical methods are commonly employed to obtain the small-signal transfer function [35].Extended describing function (EDF) small-signal modeling techniques, on the other side, linearize thenonlinear state-space equations using the Fourier expansion [36–42]. In EDF methods, the Fourier ex-pansion can be performed by considering only the first harmonic; however, if higher accuracy is needed,higher order harmonics can be taken into consideration. Traditional EDF techniques usually result inhigh-order transfer functions, which are represented by a set of matrices rather than an equivalent cir-cuit. In more advanced EDF techniques, the small-signal transfer function is simplified and representedby an equivalent circuit [43, 44]. This is enabled by the analysis of signal envelopes.Small-signal modeling of LLC converters can also be performed using the communication theory[45]. In this method, the effects of the first, third and seventh harmonics on the dynamic behavior of theLLC converter are separately analyzed, and then their effects are superposed. In the frequency domain,the load can be modeled as a time-varying resistor. The magnitude and phase of the time-varyingresistor are obtained through an iterative procedure and requires empirical implementation. Based onthe time-varying resistor model, the small-signal transfer function using conversion matrix techniquescan be obtained [47]. Another small-signal modeling technique, addressed in the literature, is performedthrough the obtainment of the lumped parameter equivalent circuit, describing the linearized behaviorof switched tank elements [48]. Recently, time-domain modeling of resonant converters, including LLCand series resonant converters, operating at resonance, has been reported in literature [1, 3]. Althoughsimplification is achieved via the time-domain analysis, the models obtained are only valid when theLLC converter operates at resonance. Therefore, the accuracy of the models significantly degradesbelow and above resonance.As shown in Fig. 3.2 (b), LLC converters are required to operate below, at and above resonance inorder to provide voltage regulation. As discussed, most of the current small-signal modeling techniquesfor LLC converters are mainly based on empirical approaches or performed in the vicinity of the res-60(a)S1Vin VoCOn:1iLrvinvRLiripS2vsecLmLrCrPort 1 Port 2Fr12nFswGainBelow res.small signal modelAbove res.smallsignal modelAt Res.Small-signal modeling using homopolarity cycle Fullloadhalfloadlight load? ?üAt res.smallsignalmodelTraditionalmethods(b)isecSR1SR2SR3SR4Figure 3.2: a) The circuit schematic of the half bridge LLC converter, b) the gain-frequency di-agram, illustrating that the LLC converter can operate in three regions and that most of thetraditional small-signal modeling techniques are performed in the vicinity of the resonantfrequency.CircuitRepresentionHomopolairty cycle FHA based Empirical basedSimpleAccuracyValidityrangeDerivationOrder of themodelAnalysisdomain2ndTime-domainYessimpleBelow to aboveresonanceHigh in the entirerange of FswComplex>=3rdTime+freq.domainSometimescomplicatedAroundresonanceGood around theresonant freq.---NoAt one operatingpointHigh at eachoperating pointFigure 3.3: Comparison of properties: the proposed, first harmonic approximation (FHA) and em-pirical based small-signal modeling techniques.onant frequency (Fr). Therefore, the accuracy of those models degrades when the switching frequency(Fsw) as the control signal is not equal to the resonant frequency. Therefore, the opportunity to intro-duce a new analytical time-domain small-signal modeling technique, which is valid below, at and aboveresonance, and reduces the complexity of the analysis is still open.In this chapter, a new small-signal modeling technique for LLC converters, which is based on theanalysis of the homopolarity cycle in time-domain, is proposed. The proposed small-signal modeling61technique can be used to design linear compensators (employed in the control layer 2) in the quiescentregion and to keep the steady-state error close to zero. As shown in Fig. 3.2 (b), unlike most of thetraditional techniques, which model the dynamic behavior of the LLC converter around the resonance,the proposed technique accurately models the small-signal dynamic behavior of the LLC converter notonly at resonance, but also below and above resonance, while reducing the complexity of the analy-sis. Two second-order circuit models represent the below, at and above resonant dynamic behaviors,and no simulation, numerical computing, or programming software packages are required to analyzethe proposed models and to obtain their transfer functions. The circuit representation is an advantageas it helps designers/engineers to more effectively understand the small-signal nature of the LLC con-verter. To highlight the advantages of the proposed small-signal modeling technique, in Fig. 3.3, it isconceptually compared with the traditional small-signal modeling techniques. The homopolarity cyclemathematically explains the relationship between the polarities of the inverter voltage vinv and secondaryvoltage/current (vsec or isec) and will be discussed in detail in Section 3.1.3.1 The Voltage Gain, Homopolarity Cycle and Switching FrequencyRelationshipIn order to address the complexity and accuracy concerns when deriving the small-signal model ofthe LLC converter below, at and above resonance, the homopolarity cycle can be introduced to time-domain equations of the LLC converter. The homopolarity cycle describes the volt-amp-second balanceprinciple and simplifies the theoretical analysis in such a way that two second-order circuit models areobtained to illustrate the small-signal dynamic behavior of the LLC converter. These circuit models canbe used to control the LLC converter in the quiescent area where the control layer 2 is activated. Thecontrol layers are shown in Fig. 3.1 (a).In this section, the DC voltage gain of the LLC converter as a function of the homopolarity cycleand switching frequency is obtained. The homopolarity cycle is defined as follows: the ratio of a periodduring which the polarities of the inverter voltage and vsec or isec are the same to half a switching period.The resonant tank of the LLC converter is made of two inductors and one capacitor. In the steady state,the volt-second and amp-second balance conditions are satisfied for the resonant inductors and capacitor,respectively. Therefore, if the resonant tank is considered as a component with two terminals, the volt-amp-second balance condition is satisfied for the resonant tank, in steady-state. The homopolarity cyclemathematically explains the volt-amp-second balance principle in LLC converters.The analysis performed in this chapter is mainly focused on the half-bridge LLC converter; however,it can be easily extended to the full-bridge LLC converter. In the steady state, the resonant capacitor ofthe LLC converter has a DC voltage offset of Vin/2. In order to more effectively explain the derivationof DC voltage gain for each operating region, this section is divided into three subsections.62vinvevseciLr ,iLmisecttttTsw/2HbTsw/2tB0tB1tB2tB3tB4VO-VOπIO/2HbLr Cr irb/nvin LmInterval I: tB0<t<tB1iLr iLmR’LC’OnvoLr Crvin LmInterval II: tB1<t<tB2iLr iLmR’LC’Onvo(a) (b)vin-(Vin/2)-vin+(Vin/2)ΔILmbΔILmb2-ΔILmb2R’L=n2RL , C’O=CO/n2Figure 3.4: The half bridge LLC converter: a) key time-domain waveforms below resonance, b)the corresponding circuit schematics to the first two operating intervals.3.1.1 Voltage Gain Below ResonanceThe LLC converter, working below resonance, has 4 operating intervals in a switching cycle. Sincethe operation of the LLC converter is symmetrical in a switching cycle, the analysis of the first twooperating intervals suffices. The key time domain waveforms and circuit schematics corresponding tothe first two operating intervals of the LLC converter, operating below resonance, are shown in Fig.3.4. As shown in this figure, from tB0 to tB1, vinv− (Vin/2) and vsec or isec have the same polarity (bothare positive). When the LLC converter operates below resonance, the homopolarity cycle is defined asfollows:Hb :=tB1− tB0Tsw/2(3.1)where Tsw is the switching period.The circuit schematic corresponding to the below resonant operation of the LLC converter in IntervalI is shown in Fig. 3.4 (b). The differential equation expressing the below resonant operation of the LLCconverter in Interval I, is given as follows:Vin−nVO = Lr diLr(t)dt + vCr(t) (3.2)Integrating (3.2) from tB0 to tB1 results in the following equation:(Vin−nVO)HbTsw2 = Lr[iLr(tB1)− iLr(tB0)]+∫ tB1tB0vCr(t)dt (3.3)The circuit schematic corresponding to the below resonant operation of the LLC converter in IntervalII is shown in Fig. 3.4 (b). The differential equation expressing the below resonant operation of the LLCconverter in Interval II, is given as follows:63Vin = LrdiLr(t)dt+ vCr(t)+LmdiLm(t)dt(3.4)Integrating (3.4) from tB1 to tB2 results in the following equation:Vin(1−Hb)Tsw2 = Lr[iLr(tB2)− iLr(tB1)]+∫ tB2tB1vCr(t)dt+Lm[iLm(tB2)− iLm(tB1)] (3.5)The resonant current is periodic. Therefore, iLr(tB2) = −iLr(tB0) and iLr(tB2)− iLr(tB0) = ∆ILmb.Also, iLr(tB2)≈ iLr(tB1). Therefore, adding (3.3) to (3.5) results in the following equation:(Vin−nVO)HbTsw2 +Vin(1−Hb)Tsw2= Lr∆ILmb+∫ tB2tB0vCr(t)dt (3.6)From tB0 to tB1, nVO drops across Lm and linearly increases iLm. In this time period, iLm can becalculated using the following equation:iLm(t) =nVOLm(t− tB0)− ∆ILmb2 , (tB0 < t < tB1) (3.7)Below resonance, tB1− tB0 = Tr/2. Tr is the series resonant period and can be calculated using2pi√LrCr. Therefore, ∆ILmb can be obtained using the following equation:∆ILmb =nVOTr2Lm(3.8)Therefore, (3.6) can be rewritten as follows:(Vin−nVO)HbTsw2 +Vin(1−Hb)Tsw2=Vin2Tsw2(3.9)Manipulating the above equation results in an equation expressing the DC voltage gain of the LLCconverter operating below resonance as a function of Hb.VOVin=12nHb(3.10)According to Fig. 3.4 (a), the conduction time period of the rectifier in half a switching cycle equalsHbTsw/2. This conduction time when the LLC converter operates below resonance is half a seriesresonant period (Tr/2). Therefore, (HbTsw)/2 = Tr/2. The ratio of the series resonant frequency Fr tothe switching frequency Fsw, is called K f , and defined as follows:K f :=FrFsw=TswTr(3.11)By applying (3.11) to (3.10), the relationship between the below resonant voltage gain, homopolaritycycle and switching frequency is obtained.VOVin=12nHb=K f2n(3.12)64In the following, the conditions under which the below resonant voltage gain equation given by(3.12) remains accurate is discussed. While deriving the below resonant voltage gain equation of theLLC converter, it was assumed that the variation in the magnetizing current is negligible in Interval II,and that the converter has 4 operating intervals in a switching cycle. To guarantee that the variation inthe magnetizing current in Interval II is negligible, the resonant period (Trb) of the resonance occurredin Interval II in the below resonant operation must be much larger than the largest period that the LLCconverter remains in Interval II. Below resonance, from tB1 to tB2, the LLC converter is in Interval II.According to the above discussion, tB2−tB1 =(1−Hb)(Tsw/2). The largest period within which the LLCconverter operates below resonance in Interval II happens when the LLC converter is required to producethe largest voltage gain. According to (3.12), the largest voltage gain is achieved when the switchingfrequency is minimum. If the switching frequency is minimum, the below resonant homopolarity cycleHb is minimum as well. The minimum below resonant homopolarity cycle Hb−min can be calculated byusing the following equation:Hb−min =Fsw−minFr(3.13)The resonant period of the resonance that occurs in Interval II in the below resonant operation canbe calculated through the following equation:Trb = 2pi√(Lr+Lm)Cr (3.14)In order to guarantee the accuracy of the below resonant voltage gain, even at the minimum switch-ing frequency, the following condition must be satisfied:(1−Hb−min) 12Fsw−min << 2pi√(Lr+Lm)Cr (3.15)As illustrated in Fig. 3.2 (b), as long as the above condition is satisfied, and the quality factorQ is smaller than Qmax (full instructions to calculate Qmax is provided in [49]), in the below resonantoperation, from Fsw−min to Fr, the voltage gain curves converge to each other, and this is expressed bythe proposed below resonant voltage gain equation given by (3.12).3.1.2 Voltage Gain At ResonanceThe LLC converter, working at resonance, has 2 operating intervals in a switching cycle. Since theoperation of the LLC converter is symmetrical in a switching cycle, the analysis of the first operatinginterval suffices. The key time domain waveforms and circuit schematic corresponding to the firstoperating interval of the LLC converter, at resonance, are shown in Fig. 3.5. As shown in this figure,from tR0 to tR1, vinv− (Vin/2) and vsec or isec have the same polarity (both are positive). Since tR1− tR0 =Tsw/2, the homopolarity cycle equals 1 when the LLC converter operates at resonance.The circuit schematic corresponding to the LLC converter, operating at resonance and in Interval I isshown in Fig. 3.5 (b). The differential equation expressing the resonant operation of the LLC converter65ttttVO-VOvinvevseciLr ,iLmisectR0tR1tR2piIO/2Lr Cr irr/nvin LmInterval I: tR0<t<tR1iLr iLmR’LC’Onvo(b)(a)Tsw/2Tr/2vin-(Vin/2)-vin+(Vin/2)R’L=n2RL , C’O=CO/n2Figure 3.5: The half bridge LLC converter: a) key time domain waveforms at resonance, b) thecorresponding circuit schematic of the first operating interval at resonance.in Interval I is given as by (3.2). Integrating (3.2) from tR0 to tR1 results in the following equation:(Vin−nVO)Tsw2 = Lr[iLr(tR1)− iLr(tR0)]+∫ tR1tR0vCr(t)dt (3.16)In practice, Lr < Lm and Lr[iLr(tR1)− iLr(tR0)] = nVO(Lr/Lm)(Tsw/2). Therefore, (3.16) can besimplified as follows:(Vin−nVO)Tsw2 =Vin2Tsw2(3.17)Manipulating the above equation results in an equation, expressing the DC voltage gain of the LLCconverter, operating at resonance:VOVin=12n(3.18)3.1.3 Voltage Gain Above ResonanceThe LLC converter, working above resonance, has 4 operating intervals in a switching cycle. Since theoperation of the LLC converter is symmetrical in a switching cycle, the analysis of the first two operatingintervals suffices. The key time domain waveforms and circuit schematics corresponding to the first twooperating intervals of the LLC converter, operating above resonance, are shown in Fig. 3.6. As shownin this figure, from tA0 to tA1, vinv− (Vin/2) and vsec or isec have the same polarity (both are positive).When the LLC converter operates above resonance, the homopolarity cycle is defined as follows:Ha :=tA1− tA0Tsw/2(3.19)66Lr Cr ira/nvin LmInterval I: tA0<t<tA1iLr iLmR’LC’Onvo(b)ttttvinvevseciLr,iLmisectA0tA1tA2tA3tA4VO-VOvin-(Vin/2)πIO/2Tsw/2HaTsw/2(1-Ha)Tsw/2Lr CrLmInterval II: tA1<t<tA2iLr iLmR’LC’Onvo(a)-vin+(Vin/2)ira/nΔILmaR’L=n2RL , C’O=CO/n2Figure 3.6: The half bridge LLC converter: a) key time domain waveforms above resonance, b)the circuit schematic corresponding to the first operating interval above resonance.The circuit schematic corresponding to the above resonant operation of the LLC converter in IntervalI is shown in Fig. 3.6 (b). The differential equation expressing the operation of the LLC converter inInterval I is given by (3.2). Integrating (3.2) from tA0 to tA1 results in the following equation:(Vin−nVO)HaTsw2 = Lr[iLr(tA1)− iLr(tA0)]+∫ tA1tA0vCr(t)dt (3.20)The corresponding circuit schematic to the above resonant operation of the LLC converter in IntervalII is shown in Fig. 3.6 (b). The differential equation expressing the operation of the LLC converter inInterval II, is given as follows:−nVO = Lr diLr(t)dt + vCr(t) (3.21)Integration the above equation from tA1 to tA2 results in the following equation:−nVO(1−Ha)Tsw2 = Lr[iLr(tA2)− iLr(tA1)]+∫ tA2tA1vCr(t)dt (3.22)Adding (3.20) to (3.22) results in the following equation:(Vin−nVO)HaTsw2 −nVO(1−Ha)Tsw2= Lr∆ILma+∫ tA2tA0vCr(t)dt (3.23)where ∆ILma = (nVOTsw)/2Lm and (2/Tsw)∫ tA2tA0 vCr(t)dt ≈ 0.5Vin+VO[(1/K f )− 1]/[8nFswCrRL]−Lr(nVO)/Lm. Therefore, (3.23) can be rewritten as follows:(Vin−nVO)HaTsw2 −nVO(1−Ha)Tsw2= {Vin2+VO[(1/K f )−1]8nFswCrRL}Tsw2(3.24)Manipulating (3.24) results in the following equation, expressing the DC voltage gain of the LLC67converter operating above resonance:VOVin=Ha−0.5n+ (1/K f )−18nFswCrRL(3.25)The above equation shows that the voltage gain of the LLC converter depends on not only theswitching frequency (the control signal), but also the above resonant homopolarity cycle Ha and load.Since Ha is indeed a consequence and cannot be controlled directly, it has to be calculated. To calculatethe output voltage or voltage gain, the input voltage, load, and switching frequency are considered asknown parameters. The unknowns are Ha and VO. To compute the unknowns, in addition to (3.25),another equation, as a function of the same unknowns is required.Ideally, PO = Pin, where PO and Pin are the output and input powers, respectively. As a result thefollowing equation can be written:VOVin=IinIO(3.26)where, Iin and IO are the average input and output currents, respectively.As illustrated in Fig. 3.6, the input voltage is connected to the circuit in Interval I, and in this interval,the input current equals the current through the resonant inductor iLr. Iin can be calculated through thefollowing equation:Iin =1Tsw∫ tA1tA0iLrdt =1Tsw∫ tA1tA01niradt+1Tsw∫ tA1tA0iLmdt (3.27)From tA0 to tA1, iLm(t) = (nVO/Lm)(t− tA0)− (∆ILma/2), and ∆ILma = nVO/2LmFsw.As discussed later in this chapter, above resonance and in Interval I, ira = Ipasin(ωrt). Ipa is givenby (3.60). By solving (3.27), the following equation is obtained:Iin =Ipa2pinK f[1− cos(piK fHa)]+ nVOHa(Ha−1)4LmFsw (3.28)By applying (3.28) and (3.60) to (3.26), and considering that VO/IO = RL, the following equation isobtained:VOVin=1− cos(piK fHa)2n[1− cos(piK fHa)]+0.5(1−Ha)sin(piK fHa) +nRLHa(Ha−1)4LmFsw(3.29)By using (3.29) and (3.25), the LLC converter voltage gain and Ha can be calculated. Accordingto (3.25), the minimum gain that the LLC converter can theoretically provide is zero. Therefore, theminimum value of Ha is 0.5. As a result, the theoretical range of the above resonant homopolarity cyclecan be given as follows: 0.5 < Ha < 1.The equations (3.12), (3.18) and (3.25) can all be unified in one equation. In general, the DC voltagegain of the LLC converter, homopolarity cycle and switching frequency can all be calculated using thefollowing equation:68VOVin=12nHb=K f2nfsw < f012nfsw = f0Ha−0.5n+ (1/K f )−18nFswCrRLfsw > f0(3.30)The above equation is fundamental since it will be used for deriving the small-signal circuit modelsof the LLC converter in the quiescent region. The small-signal circuit models of the LLC convertercan be used to design linear compensators (employed in the control layer 2) in the quiescent region forkeeping the steady-state error close to zero.3.2 The Behavioral Average Equivalent Circuit: First StageIn order to accurately predict the small-signal dynamic behavior of the LLC converter from below toabove resonant operations, and to reduce the complexity of the analysis, the homopolarity cycle wasintroduced in Section 3.1. As it will be discussed in the following, by using the homopolarity cycle, thetheoretical analysis is simplified to a level that two second-order circuit models can express the small-signal dynamic behavior of the LLC converter. The second-order circuit models can be used to designlinear compensators, employed in the control layer 2. The control layer 2 is shown in Fig. 3.1 (a).Now that the homopolarity cycle is defined, and the DC voltage gain, switching frequency, and ho-mopolarity cycle relationships are formulated, the process of extracting the average small-signal modelof the LLC converter can be started. The process of obtaining the average small-signal model of theLLC converter, using the homopolarity cycle concept, has three stages. The first two stages are involvedin the obtainment of two behavioral average equivalent circuits. In the third stage, small-signal pertur-bations are applied to the signals of the LLC converter in order to theoretically obtain the small-signalmodel of the LLC converter. In this section, the focus of the analysis is on the first stage, in whichthe order of the LLC converter is reduced to three from four using the behavioral averaging technique,enabled by the homopolarity cycle concept.3.2.1 The Below Resonant Third-Order Behavioral Average Equivalent CircuitThe small-signal model of the LLC converter can be obtained by average behavioral modeling of theelements. In this section, the first element which is behaviorally modeled is the magnetizing inductorLm. The behavioral modeling of the magnetizing inductor requires discussion on its below resonantvoltage and current waveforms. The process of modeling Lm is based on obtaining the average valuesof the magnetizing voltage and current in half a switching cycle. As it will be discussed later in thissection, in half a switching cycle, the average current through Lm is a function of nVO, and nVO isindeed the average voltage over Lm in half a switching cycle. Therefore, below resonance and in halfa switching cycle, it can be concluded that Lm behaves similar to a voltage-dependent current source,on average. Since the current of the voltage-dependent current source depends on its average voltage, it69can be simply modeled by a resistor.If it is considered that the LLC converter is operating in steady state and below resonance, as shownin Fig. 3.4 (a), nVO drops across the magnetizing inductor Lm from tB0 to tB1 and causes iLm to linearlyincrease. In interval I (tB0 < t < tB1), iLm can be calculated using (3.7). According to this equation,iLm(tB0) = −∆ILmb/2. In Section 3.1, it was discussed that tB1− tB0 = Hb(Tsw/2) = Tr/2. Therefore,at t = tB1, the current through the magnetizing inductor becomes ∆ILmb/2. This means that iLm(tB1)−iLm(tB0) = ∆ILmb. ∆ILmb is already calculated using (3.8).In Interval II, as shown in Fig. 3.4 (a), the variation in the magnetizing current (diLm/dt) is almostzero. This means that from tB1 to tB2, the amplitude of iLm remains almost constant at ∆ILmb/2. Theaverage current through the magnetizing inductor in half a switching period can be calculated using thefollowing equation:iLm−avg =2Tsw{∫ tB1tB0[nVOLm(t− tB0)− ∆ILmb2 ]dt+∫ tB2tB1∆ILmb2dt} (3.31)In the above equation,∫ tB1tB0 [(nVO/Lm)(t− tB0)− (∆ILmb/2]dt = 0. Since tB2− tB1 = (1−Hb)(Tsw/2),(3.31) results in the following equation:iLm−avg =nVO(1−Hb)4LmFr(3.32)Below resonance and in Interval I, nVO drops across Lm. In Interval II, diLm/dt ≈ 0. Therefore,the average voltage over Lm in half a switching cycle from tB0 to tB2 equals HbnVO. By dividing theaverage voltage of the magnetizing inductor by its average current, Lm can be modeled by a resistorwhose resistance is given by the following equation:RLmb =4HbLmFr1−Hb (3.33)According to (3.12), Hb = 1/K f . Therefore, another way to write (3.33) is given as follows:RLmb =4LmFrK f −1 (3.34)The latter form to mathematically express RLmb is more appropriate because the actual control signalin the LLC converter is the switching frequency, and K f = Fr/Fsw.The differential equation, describing the below resonant behavior of the LLC converter in IntervalI, is given by (3.2). The below resonant rectifier current, irb, in Interval I, can be calculated using thefollowing equation:irb = n(iLr− iLm) (3.35)By applying (3.35) to (3.2), the following differential equation is obtained:70irbpiIO/2HbIOttB0 tB1 tB2 tB3HbTsw/2=Tr/2Tsw/2piIO/2actual irb equivalent irb (irbe)BAFigure 3.7: The actual and equivalent rectifier current of the LLC converter, operating below res-onance.Vin−nVO = Lrndirb(t)dt+LrdiLm(t)dt+1nCr∫ ttB0irb(t)dt+1Cr∫ ttB0iLm(t)dt+ vCr(tB0) (3.36)In Interval I, nVO drops across Lm. Therefore,diLm(t)dt=nVOLm(3.37)By applying (3.37) to (3.36), the following differential equation is obtained:d4ir(t)dt4+1LrCrd2ir(t)dt2= 0 (3.38)The characteristic equation of the above differential equation is S4 + [1/(LrCr)]S2 = 0. The rootsassociated with the characteristic equation are 0 and ± j1/√LrCr. Therefore, irb, below resonance andin Interval I, can be mathematically expressed in time domain as follows:irb(t) =1nVin− vCr(tB0)− [1+(Lr/Lm)]nVOZrsin(ωr(t− tB0)) (3.39)where,ωr =1√LrCr(3.40)Zr =√LrCr(3.41)From tB1 to tB2, irb = 0. The waveform of the below resonant rectifier current is shown in Fig. 3.7in red. In steady state, the load current, IO, equals the mean value of irb. Therefore, the pick current ofirb can be simply calculated as piIO/(2Hb). This simply states that the coefficient of sin(ωr(t− tB0)) in(3.39) equals piIO/(2Hb) or piIOKF/2. Therefore,irb(t) =piKF2IOsin(ωr(t− tB0)) (3.42)71Below resonance, as shown in Fig. 3.7, irb has discontinuity. This creates nonlinearity. In orderto obtain the third-order behavioral average equivalent circuit of the LLC converter, operating belowresonance, the next step is to solve this discontinuity. This can be performed by adjusting the parametersof the LLC converter in such a way that the dashed green waveform represents the rectifier current belowresonance. The dashed green waveform in Fig. 3.7, is called the below resonant equivalent rectifiercurrent (irbe). In order for this to be correct, the area under irb from tB0 to tB2 must be equal the areaunder irbe in the same period of time. This guarantees the transfer of the same energy to the load side inhalf a switching cycle. The area under irb from tB0 to tB2 is called A and calculated as follows:A=∫ tB2tB0irb(t)dt =∫ tB1tB0piKF2IOsin(ωr(t− tB0))dt (3.43)The above equation results in the following:A=piKF2IO1ωr[1− cos(piKF)] (3.44)The area under irb equals the area under irbe, in half a switching cycle, only if ωr and irb =n(iLr − iLm) are divided by K f or alternatively multiplied into Hb. If ωr is divided by K f , a new an-gular frequency, called ωrab, is obtained as follows:ωrab =1K f√LrCr(3.45)The above process does not change Zr in (3.39). Therefore, by using (3.45), it can be concludedthat Lr andCr are multiplied into KF . Therefore, the below resonant equivalent resonant inductance andcapacitance, can be calculated using the following equations:Lrab := K fLr (3.46)Crab := K fCr (3.47)The current through the output capacitor equals irb− IO. Therefore, the pick to pick variation inthe output voltage (∆VO) of the LLC converter, operating below resonance, can be calculated using thefollowing equation:∆VO =1CO∫ tB1tB0(pi2Hb−1)IOsin(ωr(t− tB0))dt− 1CO∫ tB2tB1IOdt (3.48)The result of the above equation is given as follows:∆VO =1CO(pi2Hb−1) IOωr[1− cos(ωrHbTsw2 )]−1COIO(1−Hb)Tsw2 (3.49)As discussed, A equals B only if ωr and irb are divided by K f or equivalently multiplied into Hb.Therefore, by applying (3.45) to the above equation, the following is obtained:72:1RL voviniLrbnKfRLmbirbeLrab=KFLrCrab=KFCrCOb=KFCOVin/2 vCrb1vCrbTResonant cap. DC voltage offsetFigure 3.8: The below resonant third-order behavioral average equivalent circuit of the LLC con-verter, in half a switching cycle.∆VO =HbCO(pi2−1)2IOωrb− HbCOIOTsw2(3.50)From the above equation, it is clear that the equivalent output capacitance when the LLC converteroperates below resonance equalsCO/Hb or equivalently K fCO. Therefore, the below resonant equivalentoutput capacitor can be defined as follows:COb := K fCO =COHb(3.51)All the above discussions result in a third-order behavioral average equivalent circuit, as shown inFig. 3.8. This equivalent circuit represents the linear high and low-frequency steady state and dynamicbehaviors of the LLC converter, operating below resonant, in half a switching cycle.3.2.2 The Above Resonant Third-Order Behavioral Average Equivalent CircuitIf it is considered that the LLC converter is operating in steady state and above resonance, accordingto Fig. 3.6 (a), nVO drops across the magnetizing inductor Lm from tA0 to tA2 and causes iLm to linearlyincrease. Therefore, the current through the magnetizing inductor can be calculated using the followingequation:iLm(t) =nVOLm(t− tA0)− ∆ILma2 , (tA0 < t < tA2) (3.52)Since tA2− tA0 = Tsw/2, ∆ILma can be calculated using the following equation:∆ILma =nVO2LmFsw(3.53)The average current through the magnetizing inductor of the LLC converter, operating above reso-nance, in half a switching cycle can be calculated using the following equation:2Tsw∫ tA2tA0iLm(t)dt =2Tsw∫ tA2tA0[nVOLm(t− tA0)− ∆ILma2 ]dt (3.54)Solving the above equation reveals that the average current through Lm in half a switching cycle,above resonance, is zero.73iratA0piIO/2HaTsw/2Tsw/2Tr/2tA1tA2 tA3 tA4 tA5IOtactual ira equivalent ira (irae)IpaC DFigure 3.9: The actual and equivalent rectifier current of the LLC converter, operating above reso-nance.The differential equation, describing the above resonant behavior of the LLC converter in Interval I,is given by (3.2). As shown in Fig. 3.6 (b), ira = n(iLr− iLm). By applying this current relationship to(3.2), the following equation is obtained:Vin−nVO = Lrndira(t)dt+LrdiLm(t)dt+1nCr∫ ttA0ira(t)dt+1Cr∫ ttA0iLm(t)dt+ vCr(tA0) (3.55)By considering the fact that diLm/dt = nVO/Lm, solving the above equation gives the rectifier currentof the LLC converter, operating above resonance in Interval I:ira(t) =1nVin− vCr(tA0)− [1+(Lr/Lm)]nVOZrsin(ωr(t− tA0)) (3.56)In Interval II, the following differential equation, describes the dynamic behavior of the LLC con-verter, operating above resonance:−nVO = Lrndira(t)dt+LrdiLm(t)dt+1nCr∫ ttA1ira(t)dt+1Cr∫ ttA1iLm(t)dt+ vCr(tA1) (3.57)Solving the above equation results in the rectifier current in Interval II:ira(t) =1n−vCr(tA1)− [1+(Lr/Lm)]nVOZrsin(ωr(t− tA1)) (3.58)By using (3.56) and (3.58), the actual waveform of rectifier current is illustrated by red in Fig. 3.9.As shown in this figure, the load current IO equals the average of the above resonant rectifier current inhalf a switching cycle. Therefore, IO can be calculated using the following equation:IO =2Tsw[∫ tA1tA0Ipasin(ωr(t− tA0))dt+ Ipa(1−Ha)Tsw2 sin(ωrHTsw2)] (3.59)where Ipa is the pick current of the actual above resonant rectifier current. By using the aboveequation, Ipa can be calculated as follows:74Ipa =IO1K f pi [1− cos(K fHapi)]+ 12(1−Ha)sin(K fHapi)(3.60)According to Fig. 3.9, the actual rectifier current shows nonlinearity. The above resonant rectifiercurrent waveform can be modeled by a linear current waveform that guarantees the transfer of the sameamount of energy to the output in half a switching cycle. The waveform of the above resonant equivalentrectifier current (irae) is shown by green in Fig. 3.9. As illustrated in this figure, the pick current of theequivalent rectifier current equals piIO/2. irae is an equivalent to ira if the area under the actual rectifiercurrent (C) equals the area under irae (D) in half a switching cycle. This condition is satisfied only if ωris divided by K f . In this case a new angular frequency, called ωrab, already given by (3.45), is obtained.Since this equalization does not have any effect on Zr in (3.56) and (3.58), it is concluded that Lr andCr of the LLC converter, operating above resonance, are both multiplied into K f . The above resonantequivalent resonant inductance and capacitance, can be calculated using (3.46) and (3.47), respectively.The voltage gain of the LLC converter, operating above resonance, is given by (3.25). This equationcan be rewritten as follows:Vin(Ha−0.5) =VO[n+ (1/K f )−18nFswCrRL ] (3.61)As shown in Fig. 3.2 (a), the resonant tank of the LLC converter has two ports. One port is connectedto the inverter and another to the rectifier. According to (3.61), the average voltage applied to the lefthand side port of the resonant tank is VinHa−0.5Vin. As discussed at the beginning of Section 3.1, theresonant capacitor of the LLC converter has a DC voltage offset of Vin/2. Therefore, VinHa− 0.5Vinmeans that the input voltage source is scaled by Ha and then is subtracted by the DC voltage offset ofthe resonant capacitor. If the average voltage, applied to the right hand side port of the resonant tank,is shown by VY12, according to (3.61), VY12 equals VO[n+(1/K f )−18nFswCrRL]. By factoring out n, VY12 can berewritten as follows:VY12 = nVO[1+(1/K f )−18FswCrn2RL] (3.62)The above equation explains that nVO is a fraction of VY12. Fig. 3.10 illustrates how nVO is relatedto VY12. As shown in this figure, this relationship can be simply explained by using a resistive voltagedivider. In this case, the resistive network is made of the reflected resistor from the secondary to theprimary side of the transformer (Rprim) and a series resistor with the resonant tank, called Ra. Byconsidering (3.62) and the configuration shown in Fig. 3.10, Ra can be calculated using the followingequation:Ra =1K f−18FswCr(3.63)All the above discussion results in a third-order behavioral average equivalent circuit that explainsthe steady-state and high and low-frequency small-signal dynamic behavior of the LLC converter, oper-75n:1RL voiraeCORanvoRprim=n2RL1+(1/Kf)-18FswCrn2RLnvo[ ]ResonantTankY1Y2Figure 3.10: The circuit seen from the right hand side port of the resonant tank when the LLCconverter operates above resonance.n:1RL voviniLraRairaeLrab=KFLr Crab=KFCrCO1:HaVin/2 vCra1vCraTResonant cap. DC voltage offsetFigure 3.11: The above resonant third-order behavioral average equivalent circuit of the LLC con-verter, in half a switching cycle.ating above resonance, in half a switching cycle. This equivalent circuit is shown in Fig. 3.11.3.2.3 The Unified Third-Order Behavioral Average Equivalent CircuitIn this section, the third-order behavioral average equivalent circuits of the LLC converter, operatingbelow, at and above resonance are obtained. Instead of working with two equivalent circuits, shownin Figs. 3.8 and 3.11, it is more convenient to unify them. The unified third-order behavioral averageequivalent circuit of the LLC converter valid below, at and above resonance is shown in Fig. 3.12. Byconsidering the discussions made in this section and Section 3.1, the parameters of the unified third-order behavioral average equivalent circuit are given as follows:Lur = K fLr (3.64)Cur = K fCr (3.65)CuO = [K f u(K f −1)+u(1−K f )]CO (3.66)Rua =(1/K f )−18FswCru(1−K f ) (3.67)RuLmb =4LmFrK f −11u(K f −1) (3.68)76nuo:1RL voviniLurRuairaLurCurCuORuLmb1:nuiVin/2 vCur1vCurTResonant cap. DC voltage offsetiRmbFigure 3.12: The unified third-order behavioral average equivalent circuit of the LLC converter, inhalf a switching cycle. This model is valid below, at and above resonance.nuo = [1K fu(K f −1)+u(1−K f )]n (3.69)nui = [u(K f −1)+Hau(1−K f )]n (3.70)where, u(x) is the step function, defined as follows:u(x) =1 x> 00 x< 0 (3.71)Depending on the operating region of the LLC converter, x can be either K f −1 or 1−K f .3.3 The Behavioral Average Equivalent Circuit: Second StageThe small-signal dynamic behavior of the LLC converter in all operating areas (i.e., below, at andabove resonance) can be accurately expressed by two simple second-order circuit models when thehomopolarity cycle is introduced to time-domain equations of the LLC converter. These circuit modelscan be used to design linear compensators in the control layer 2. The control layers are shown in Fig.3.1 (a). As discussed, the process of obtaining the average small-signal model of the LLC converter hasthree stages. The first two stages were involved in the obtainment of two average equivalent circuits,using the behavioral averaging technique. The first stage was addressed in Section 3.2, where the unifiedthird-order behavioral average equivalent circuit of the LLC converter, below, at and above resonancewas developed. In this section, the analysis is focused on the second stage in which the unified second-order behavioral average equivalent circuit is obtained. Using this second-order equivalent circuit, twocircuit models, representing the small-signal dynamic behavior of the LLC converter below, at and aboveresonance, are later obtained.In Section 3.2, the operation of the LLC converter, below, at and above resonance was linearized inhalf a switching cycle, and a unified third-order behavioral average equivalent circuit was developed toillustrate this linear behavior. This equivalent circuit demonstrates both high and low-frequency averagedynamic behaviors of the LLC converter. From the control point of view, the high-frequency dynamicbehavior is not of interest, and linear compensators are designed based on the low-frequency dynamic77behavior of a power converter. As it will be shown in Section 5.3, the average response of the LLCconverter operating in the quiescent area to small changes is linear and can be modeled with a second-order system, as shown in Fig. 3.13. According to the basic concepts governing the electrical circuittheory, the dynamic response of a linear circuit in steady-state, fed by a DC source, can be obtained byanalyzing its response to a step input while considering that all the initial conditions are zero from thestep input point of view. The output of the LLC converter and consequently of the unified third-orderequivalent circuit can be connected to any type of load, such as a resistor, current source system, etc.If a step input vs with an amplitude of VS is applied to the third-order behavioral average equivalentcircuit at the beginning of a switching cycle, the following differential equation, called ψ1, can bedefined:ψ1 :=−nuiVS+Lur diLurdt +1Cur∫iLurdt+RuLmbiRmb = 0 (3.72)Since RLmbiRmb = (Rua/nuo)iur+nuovO, the above definition can be rewritten as follows:ψ1 :=−nuiVS+Lur diLurdt +1Cur∫iLurdt+Ruaiurnuo+nuovO = 0 (3.73)The limit of ψ1 as K f approaches 1 is given as follows:limK f→1ψ1 :=−VS+Lur diLurdt +(1Cur+n2uoCuO)∫iLurdt = 0 (3.74)Since CuO >>Cur, the coefficient of the integral in the above equation can be simplified as 1/Cur.Therefore, solving the above differential equation results in the following equation:iLur =VSZusin(ωut) (3.75)where,Zu =√LurCur(3.76)ωu =1√LurCur(3.77)The average of (3.75) in its half a switching period is given as follows:i¯Lur =4VSCurTur(3.78)where, Tur = 2pi√LurCur.If the same step input vs with an amplitude of VS is applied to the second-order behavioral averageequivalent circuit, shown in Fig. 3.13, at the beginning of a switching cycle, the following differentialequation, called ψ2, can be defined:78nuo:1RL voviniamRuairamLuamCuORuLmb1:nuiResonant cap. DC voltage offsetVin/2iRmbFigure 3.13: The unified second-order behavioral average equivalent circuit of the LLC converter.This model is valid below, at and above resonance.ψ2 :=−VSnui+Luam diamdt +Rua(iam− iRmb)+nuovO = 0 (3.79)The limit of ψ2 as K f approaches 1 is given as follows:limK f→1ψ2 :=−VS+Luam diamdt +n2uoCuO∫iamdt = 0 (3.80)Solving the above differential equation results in the following equation:iam =VSZamsin(ωamt) (3.81)where,ωam =nuo√LuamCuO(3.82)Zam = nuo√LuamCuO(3.83)By averaging the above equation in half a period of (3.75), the following equation is obtained:i¯am =2VSCuOn2uoTr[1− cos(pinuo√LurCurLuamCuO)] (3.84)If the circuit shown in Fig. 3.13 is a model of the third-order behavioral average equivalent circuit,(3.78) must be equal with (3.84). By using this condition, Luam can be calculated using the followingequation:Luam =n2uoCurCuOpi2Lur[cos−1(1−2n2uoCurCuO )]2(3.85)3.4 The Average Small-Signal ModelThrough the analysis of the homopolarity cycle, two simple second-order circuit models, representingthe small-signal dynamic behavior of the LLC converter, below, at and above resonance, can be obtained.79(a)(b)Luam:1Kf CO RL vovinnKfVin/2iamRuLmbLuam:1Kf CO RL VO+voVin2nKfvin^nVOfswFr^iamRuLmb ^fswFsw^VO +[ ]Kfn( )2 1RuLmb1RLFigure 3.14: The LLC converter, operating below resonance: a) the second-order behavioral av-erage equivalent circuit, b) the below resonant average small-signal circuit model, throughwhich the small-signal transfer function is obtained.These circuit models can be used to design linear compensators in the control layer 2. The control layersare shown in Fig. 3.1 (a). The process of obtaining the two circuit models has three stages. So far, thefirst two stages are covered and a second-order behavioral average equivalent circuit, demonstrating thelow-frequency dynamic behavior of the LLC converter is developed. In this section, the focus of theanalysis is on the third stage, in which perturbations are applied to the signals of the LLC converter inorder to obtain the average small-signal model of the LLC converter below, at and above resonance.3.4.1 The Below Resonant Average Small-Signal ModelWhen the LLC converter operates below resonance, according to the definition of u(x), provided by(3.71), u(1−K f ) = 0 and u(K f −1) = 1. Therefore, according to (3.64), (3.65), (3.66), (3.67), (3.68),(3.69) and (3.70), the second-order behavioral average equivalent circuit is simplified to the circuitshown in Fig. 3.14 (a). In order to obtain the average small-signal model of the LLC converter, thefollowing perturbations are added to the parameters of the system:fsw = Fsw+ fˆsw (3.86)vin =Vin+ vˆin (3.87)vO =VO+ vˆO (3.88)iam = Iam+ iˆam (3.89)80According to Fig. 3.14 (a), the differential equations, expressing the small-signal dynamic behaviorof the LLC converter, operating below resonance, can be written as follows:diamdt=1Luam(vin− Vin2 −nK fvO) (3.90)dvOdt=1K fCO{ nK fiam− [( nK f )2 1RuLmb+1RL]vO} (3.91)By applying the perturbations given by (3.86), (3.87), (3.88) and (3.89) to (3.90) and (3.91), neglect-ing the second order perturbations, and considering that Iam = (K f /n)(VO/RL)+(n/K f )(VO/RuLmb) andK f = Fr/Fsw, the following equations are obtained:diˆamdt=1Luam[Vin2+ vˆin− nK f (VO+ vˆO)−nVOFrfˆsw] (3.92)dvˆOdt=1K fCO{ nK f(Iam+ iˆam)− [( nK f )2 1RuLmb+1RL](VO+ vˆO)+ [(nK f)21RuLmb+1RL]VOfˆswFsw} (3.93)The above equations describe the average small-signal model of the LLC converter, operating belowresonance, and can be translated to the circuit shown in Fig. 3.14 (b). This equation is essential since itcan be used to design linear compensators in the control layer 2.By using the circuit representation of the below resonant average small-signal model, shown in Fig.3.14 (b), the control-to-output and input-to-output transfer functions are given as follows:vˆofˆsw=K2bK fn LuamS−K1bK fn LuamS[SK fCO+1RL+( nK f )2 1RuLmb]+ nK f(3.94)vˆovˆin=1K fn LuamS[SK fCO+1RL+( nK f )2 1RuLmb]+ nK f(3.95)where,K1b =nVOFr(3.96)K2b = [(nK f)21RuLmb+1RL]VOFsw(3.97)The above transfer functions mathematically describe the small-signal dynamic behavior of the LLCconverter below resonance and can be used in the control layer 2. In the control layer 2, a linearcompensator is employed to keep the steady-state error close to zero. The three-layer control strategy isshown in Fig. 3.1 (a)81(a)n:1RL voviniamRuairaLuamCuO1:HaResonant cap. DC voltage offsetVin/2n:1RL VO+vovin iamRuairaLuamCuO1:HaResonant cap. DC voltage offsetVin/2Vin [(Ha-0.5)Vin -nVO]fswFsw^KfRaVOn(1-Kf)RLfswFr^^^(b)Figure 3.15: The LLC converter, operating above resonance: a) the second-order behavioral av-erage equivalent circuit, b) the above resonant average small-signal circuit model, throughwhich the small-signal transfer function is obtained.3.4.2 The Above Resonant Average Small-Signal ModelWhen the LLC converter operates above resonance, according to the definition of u(x), provided by(3.71), u(1−K f ) = 1 and u(K f −1) = 0. Therefore, according to (3.64), (3.65), (3.66), (3.67), (3.68),(3.69) and (3.70), the second-order behavioral average equivalent circuit is simplified to the circuitshown in Fig. 3.15 (a). In order to obtain the average small-signal model of the LLC converter, theperturbations given by (3.86), (3.87), (3.88), and (3.89) are added to the signals of the converter.According to Fig. 3.15 (a), the differential equations, expressing the small-signal dynamic behaviorof the LLC converter, operating above resonance, can be written as follows:diamdt=1Luam(Havin− Vin2 −Ruaiam−nvO) (3.98)dvOdt=1CuO(niam− 1RL vO) (3.99)By applying the perturbations given by (3.86), (3.87), (3.88) and (3.89) to (3.98) and (3.99), ne-glecting the second order perturbations, and considering that Iam = (1/n)(VO/RL) and K f = Fr/Fsw, thefollowing equations are obtained:diˆamdt=1Luam[Ha(Vin+ vˆin)−Vin2 −[(Ha−12)Vin−nVO] fˆswFsw −Rua(iˆam+Iam)−K fRuaVOn(1−K f )RLfˆswFr−n(VO+ vˆo)](3.100)82f s>f rf s<f rf s=f rf sRHPZ ReImDouble PoleFigure 3.16: Pole and zero displacement of the below and above resonant small-signal transferfunctions.dvˆOdt=1CuO[n(Iam+ iˆam)− 1RL (VO+ vˆO)] (3.101)The above equations describe the average small-signal model of the LLC converter, operating aboveresonance, and can be translated to the circuit shown in Fig. 3.15 (b).By using the circuit representation of the above resonant average small-signal model, shown in Fig.3.15 (b), the control-to-output and input-to-output transfer functions are given as follows:vˆofˆsw=− K1a+K2a1n(LuamS+Rua)(1RL+SCuO)+n(3.102)vˆovˆin=Ha1n(LuamS+Rua)(1RL+SCuO)+n(3.103)where,K1a =1Fsw[(Ha−0.5)Vin−nVO] (3.104)K2a =K fRuaVOn(1−K f )RLFr (3.105)The above transfer functions mathematically describe the small-signal dynamic behavior of the LLC830.60.70.80.911.170 75 80 85 90 95 100HbSwitching Freq. (kHz)0.911.11.21.31.470 75 80 85 90 95 100Normalized GainSwitching Freq. (kHz)FHA accuracy degradesaway from FrFr(a)(b)Experimental@ RL=5.5ΩHomopolarity@ RL=5.5ΩExperimental@ RL=10ΩHomopolarity@ RL=10ΩFHA@ RL=10ΩFHA@ RL=5.5ΩSimulation@ RL=10ΩSimulation@ RL=5.5ΩFigure 3.17: Theoretical, simulation, and experimental below resonant normalized gain diagrams.The theoretical curves obtained from the analysis of the homopolarity cycle aligns well withthe experimental and simulation curves. The accuracy of the FHA curves degrade when theswitching frequency moves away from the resonant frequency.converter above resonance and can be used in the control layer 2. In the control layer 2, a linear compen-sator is employed to keep the steady-state error close to zero. The three-layer control strategy is shownin Fig. 3.1 (a). The denominators of (3.94) and (3.95), and those of (3.102) and (3.103) are the same.Moreover, from the control point of view, the control-to-output transfer function is the most importanttransfer function that linear controllers are often designed based on. Therefore, discussing the polesand zeros of the control-to-output transfer functions suffices. Fig. 3.16 illustrates the conceptual poleand zero displacements of equations (3.94) and (3.102). The system displays a double pole that movesdepending on the switching frequency. As it is shown in the figure, at different operating points thesystem shows different damping ratios and natural frequencies. There is a right half plane zero (RHPZ)at the below resonant frequency operation. The position of the zero is far from the imaginary axis, andit increases when the switching frequency is increased. Therefore, the effect of the RHPZ in the rangeof the frequency that a linear controller is designed is not significant and can be neglected.840.80.850.90.9511.051.195 105 115 125 135HaSwitching Freq (kHz)0.60.70.80.911.195 105 115 125 135Normalized GainSwitching Freq (kHz)FHA accuracy degradesaway from Fr(a)(b)Experimental@ RL=5.5ΩHomopolarity@ RL=5.5ΩExperimental@ RL=10ΩHomopolarity@ RL=10ΩFHA@ RL=10ΩFHA@ RL=5.5ΩSimulation@ RL=10ΩSimulation@ RL=5.5ΩFigure 3.18: Theoretical, simulation, and experimental above resonant normalized gain diagrams.The theoretical curves obtained from the analysis of the homopolarity cycle aligns well withthe experimental and simulation curves. The accuracy of the FHA curves degrade when theswitching frequency moves away from the resonant frequency.3.5 Experimental and Simulation ResultsIn order to reduce the complexity of the analysis and to achieve high accuracy in small-signal modelingof the LLC converter, below, at and above resonance, the homopolarity cycle can be introduced tothe time-domain equations of the LLC converter. The analysis of the homopolarity cycle and averagebehavior of the converter results in two simple second-order circuit models, which demonstrate thesmall-signal dynamic behavior of the LLC converter. In the proposed three layer control strategy, thesmall-signal circuit models can be used to design a linear compensator in the quiescent region. Thecontrol layers are shown in Fig. 3.1 (a). In order to validate the theoretical analysis, in this section,simulation and experimental results of a 650W LLC converter are provided. The specifications of theLLC converter are given in Table 3.1.3.5.1 The Steady-State Validating Experimental ResultsIn Section 3.1, the homopolarity cycle concept was defined, and the voltage gain, homopolarity cycle,and switching frequency relationships were obtained. Since the process of obtaining the average small-signal model of the LLC converter is based on the homopolarity cycle, first, the accuracy of the analysis85(a)HbTsw/2=5.2µsTsw/2=6.25µsVO=59VHb=0.832Kf=96kHz/80kHz=1.2vinv(400V/div)vsec(100V/div)isec(20A/div)vO(50V/div)(b)HbTsw/2=5.2µsTsw/2=6.25µsVO=59VHb=0.832Kf=96kHz/80kHz=1.2vinv(400V/div)vsec(100V/div)isec(20A/div)vO(50V/div)Figure 3.19: The below resonant time-domain experimental waveforms of the LLC converter.These waveforms validate the accuracy of the steady-state theoretical analysis. a) full load-ing condition b) half loading condition.performed in Section 3.1 must be validated.In Figs. 3.17 and 3.18, the theoretical, simulation, and experimental homopolarity cycle and nor-malized voltage gain diagrams obtained from the homopolarity cycle and first harmonic approximationanalysis versus the switching frequency are compared against each other. As shown in these figures,(3.30) predicts the voltage gain of the LLC converter below, at and above resonance and under differ-ent loading conditions with high accuracy. As anticipated, when the switching frequency moves awayfrom Fr, the accuracy of the FHA analysis degrades. In order to plot the FHA voltage gain curves, theequation introduced by [49] is used.In Fig. 3.19, the time-domain experimental waveforms of the LLC converter, operating below res-onance under full and half loading conditions, are shown. In this case, the switching frequency equals80kHz. As shown in Fig. 3.19 (a) and (b), the period during which the polarities of vinv and vsec, in half aswitching period, are both the same equals 5.2µs. According to the definition of the homopolarity cycle,this period is HbTsw/2. The resonant frequency of the LLC converter is 96kHz and therefore half the res-onant period (Tr/2) equals 5.2µs. Therefore, as theoretically predicted in Section 3.1, HbTsw/2 = Tr/2.86(a)Tr/2=5.2µsTr/2=5.2µsVO=50VHb=Ha=1, Kf=96kHz/96kHz=1 vinv(400V/div)vsec(100V/div)isec(20A/div)vO(50V/div)(b)Tr/2=5.2µsTr/2=5.2µsVO=50Vvinv(400V/div)vsec(100V/div)isec(20A/div)vO(50V/div)Hb=Ha=1, Kf=96kHz/96kHz=1Figure 3.20: The resonant time-domain experimental waveforms of the LLC converter. Thesewaveforms validate the accuracy of the steady-state theoretical analysis. a) full loadingcondition b) half loading condition.In Fig. 3.19 (a) and (b), it is also shown that under the full and half loading conditions, the output volt-age remains the same. According to (3.12), in this scenario, the output voltage must be equal to 59.5V .As shown in Fig. 3.19, the output voltage of the LLC converter was measured 59V under full and halfloading conditions. Therefore, the theoretical analysis performed in Section 3.1.1 is validated.In Fig. 3.20, the time-domain experimental waveforms of the LLC converter, operating at resonantunder full and half loading conditions, are shown. In this case, the switching frequency equals theresonant frequency. As shown in Fig. 3.20 (a) and (b), the period of time during which the polaritiesof vinv and vsec, in half a switching cycle, are both the same equals 5.2µs. According to the definitionof the homopolarity cycle, this period is Tsw/2. The resonant frequency of the LLC converter is 96kHzand therefore half the resonant period (Tr/2) equals 5.2µs. Therefore, as theoretically predicted inSection 3.1, Tsw/2 = Tr/2. In Fig. 3.20 (a) and (b), it is also shown that under the full and half loadingconditions, the output voltage remains the same. According to (3.18), in this scenario, the output voltagemust be equals to 50V . As shown in Fig. 3.20, the output voltage of the LLC converter was measured50V under full and half loading conditions. Therefore, the theoretical analysis performed in Section87(a)HaTsw/2=3.85µsTsw/2=4.15µsVO=40VHa=0.927, Kf=96kHz/120kHz=0.8 vinv(400V/div)vsec(100V/div)isec(20A/div)vO(50V/div)(b)VO=42VHaTsw/2=3.95µsTsw/2=4.15µsHa=0.952, Kf=96kHz/120kHz=0.8 vinv(400V/div)vsec(100V/div)isec(20A/div)vO(50V/div)Figure 3.21: The above resonant time-domain experimental waveforms of the LLC converter.These waveforms validate the accuracy of the steady-state theoretical analysis. a) full load-ing condition b) half loading condition.3.1.2 is validated.In Fig. 3.21, the time-domain experimental waveforms of the LLC converter, operating above res-onant under full and half loading conditions, are shown. In this case, the switching frequency equals120kHz. As shown in Fig. 3.21 (a), the period during which the polarities of vinv and vsec, in half aswitching cycle, are both the same equals 3.85µs. According to the definition of the homopolarity cy-cle, this period is HaTsw/2. The voltage gain of the LLC converter can be theoretically calculated using(3.25). According to (3.25), the output voltage of the converter under the full loading condition must bearound 40V . The output voltage of the LLC converter under the full loading condition was measured40V . As shown in Fig. 3.21 (b), the period during which the polarities of vinv and vsec, in half a switchingcycle, are both the same equals 3.95µs. According to (3.25), the output voltage of the converter underthe half loading condition must be around 42V . The output voltage of the LLC converter under the halfloading condition was measured 42V . Therefore, the theoretical analysis performed in Section 3.1.3 isvalidated.88Table 3.1: The specifications of the LLC converterParameter Value DescriptionVin 400V Nominal input voltageVO 48V Nominal output voltagePO 650W Nominal output powerFr 96kHz Resonant frequencyn 4 Turn ratio of the transformerLr 82µH Resonant inductorLm 240µH Magnetizing inductorCr 33nF Resonant capacitorCO 55µF Output capacitor3.5.2 The Frequency-Domain Experimental and Simulation ResultsIn this chapter, it was theoretically discussed that the average small-signal model (ASSM) of the LLCconverter, below, at and above resonance, can be obtained by using the homopolarity cycle concept andthrough a three-stage process. All the theoretical analysis has resulted in two circuit models, shownin Figs. 3.14 and 3.15. These circuit models describe the small-signal dynamic behavior of the LLCconverter below, at and above resonance and can be used to design linear compensators in the controllayer 2, as shown in Fig. 3.1 (a). In order to validate the accuracy of the proposed ASSMs and theirrelative transfer functions, the bode diagram analysis, under different frequency and loading conditions,is required. This is performed by using an experimental setup and simulation software.In the simulation bode diagrams, the green and blue dashed diagrams belong to the control-to-output bode diagrams of the ASSM and LLC converter, respectively, and the red diagram representsthe control-to-output bode diagram of the ASSM when the discretization and time-delay effects aretaken into consideration. The discretization and time-delay effects happen when the modulation sys-tem of a converter is digitally implemented. In the LLC converter, experimentally implemented, theTMS320F2835 DSP is used to generate the control signal, which is the switching frequency. In order toapply the perturbation and perform the bode-diagram analysis, the frequency analyzer Venable Model3120 is used. The perturbing signal, generated by the frequency analyzer, is applied to the analog-to-digital converter (ADC) of the DSP. In order to keep the experimental tests reliable, the sampling timeof the DSP ADC is synchronized with the control signal. Therefore, two phenomena happen: discretiza-tion and time-delay. Discretization occurs because the ADC is indeed a sample-and-hold system andconverts an analog signal to a discrete one. The time-delay effect is seen because the ADC is synchro-nized with the control signal. This means that at the beginning of every switching cycle, the ADC startsthe conversion and also the switching frequency of the control signal is updated. Since at the beginningof a switching cycle, the ADC starts the converting process, the control signal is updated with the lastswitching frequency value remained in the memory of the ADC. Therefore, a time-delay, which equalsthe period of a switching cycle (Tsw), is created.In Figs. 3.22, 3.23 and 3.24, the comparative simulation and experimental control-to-output bode-89diagrams of the LLC converter, operating below, at and above resonance, and ASSMs under 5.5Ω and10Ω resistive loads are shown. In these figures, the green and blue dashed diagrams represent the bodediagram of the proposed ASSM and LLC converter, respectively, where no discretization and time-delayeffects are present. By considering the green and blue dashed diagrams, as shown in these figures, itis evident that the proposed ASSM accurately predicts the small-signal dynamic behavior of the LLCconverter, operating below, at and above resonance and under different loading conditions. As expected,the discretization and time-delay effects can be seen in the experimental bode diagrams. If the effectsof discretization and time-delay are considered in the theoretical control-to-out transfer functions, thered diagrams in Figs. 3.22, 3.23 and 3.24 are obtained. These diagrams are identical to their relativeexperimental bode diagrams.3.6 SummaryIn this chapter, a new average small-signal modeling technique, based on the homopolarity cycle con-cept, for LLC converters was proposed. Due to the complexity, small-signal modeling for LLC con-verters has been traditionally performed using empirical/simulation methods (yielding limited insightinto the dynamic behavior) or in the vicinity of the resonant frequency (suffering from low accuracybelow and above resonance). The analysis of the homopolarity cycle in time-domain showed that highaccuracy, low complexity, and small-signal circuit representation from below to above resonant opera-tions can all be achieved when deriving the small-signal model of LLC converters. It was also shownthat no simulation, numerical computing, or programming software packages are required to obtain theproposed average small-signal circuit models and their small-signal transfer functions. The circuit mod-els and their transfer functions can be used in the control layer 2 of the three-layer control strategy todesign linear compensators and to keep the steady-state error close to zero. As discussed, the three-layercontrol strategy addresses the large and small-signal modeling and control, and efficiency concerns forLLC converters. In order to validate the theoretical analysis, a 650W LLC converter was practicallyimplemented. Both the steady-state and dynamic operations were justified through experimental andsimulation results. The results showed that the proposed circuit models predict the small-signal dy-namic behavior of the LLC converter with high accuracy from below to above resonant operations.90(d)178o@2kHz36dB@2kHz45dB@4.2kHz77o@4.2kHzProposed ASSMLLC (without discretization effect)ASSM (with discretization effect)Frequency (Hz)0-20-40-60204060100 500 1k 5k 10k0-50-100-15050100150phase(VoltagePlant)+180 phase(VoltagePlant)_(LLC_80kHz_No Dis_5.5R)+180 phase(Vo1)+18020kGain (dB)Phase(a)35dB@2kHz177o@2kHz45dB@4.2kHz100o@4.2kHz76o@4.2kHz(c)35.5dB@2kHz178o@2kHz44dB@4.2kHz73o@4.2kHzFrequency (Hz)0-20-40-60204060100 500 1k 5k 10k0-50-100-1505010015020kGain (dB)Phase36dB@2kHz178o@2kHz46dB@4.1kHz74o@4.1kHz100o@4.1kHz(b)Figure 3.22: The below resonant experimental and simulation bode diagrams of the proposedsmall-signal model and LLC converter. These bode diagrams validate the accuracy of thetheoretical dynamic analysis, with and without the discretization and time-delay effects: a)simulation, under 5.5Ω resistive load, b) simulation, under 10Ω resistive load, c) experimen-tal, under 5.5Ω resistive load, d) experimental, under 10Ω resistive load. The discretizationand time-delay effects are due to the digital implementation of the modulation system.91(d)39dB@6.1kHz66o@6.1kHz27dB@2kHz179o@2kHzProposed ASSMLLC (without discretization effect)ASSM (with discretization effect)0-20-40-60204060100 500 1k 5k 10kFrequency (Hz)0-50-100-1505010015020kGain (dB)Phase(a)27dB@2kHz179o@2kHz41dB@6.1kHz100o@6.1kHz70o@6.1kHz(c)27dB@2kHz179o@2kHz39dB@6.2kHz69o@6.2kHz0-20-40-60204060100 500 1k 5k 10kFrequency (Hz)0-50-100-1505010015020kGain (dB)Phase(b)28dB@2kHz180o@2kHz41dB@6.1kHz97o@6.1kHz65o@6.1kHzFigure 3.23: The resonant experimental and simulation bode diagrams of the proposed small-signal model and LLC converter. These bode diagrams validate the accuracy of the the-oretical dynamic analysis, with and without the discretization and time-delay effects: a)simulation, under 5.5Ω resistive load, b) simulation, under 10Ω resistive load, c) experimen-tal, under 5.5Ω resistive load, d) experimental, under 10Ω resistive load. The discretizationand time-delay effects are due to the digital implementation of the modulation system.92Proposed ASSMLLC (without discretization effect)ASSM (with discretization effect)0-20-40-60204060100 5k 10k0-50-100-1505010015020kGain (dB)Phase500 1kFrequency (Hz)(a)17.44dB@13kHz47o @ 13kHz0o @ 13kHz24.6dB@2kHz178.9o@2kHz(c)0o@13kHz17.6dB@13kHz25.5dB@2kHz178o@2kHz0-20-40-60204060100 500 1k 5k 10kFrequency (Hz)0-50-100-15050100150Gain (dB)Phase20k(b)22.5dB@2kHz178.5o@2kHz23.5dB@6kHz120o @ 6kHz94.5o@6kHz(d)25dB@6kHz64o@6kHz23dB@2kHz179o@2kHzFigure 3.24: The above resonant experimental and simulation bode diagrams of the proposedsmall-signal model and LLC converter. These bode diagrams validate the accuracy of thetheoretical dynamic analysis, with and without the discretization and time-delay effects: a)simulation, under 5.5Ω resistive load, b) simulation, under 10Ω resistive load, c) experimen-tal, under 5.5Ω resistive load, d) experimental, under 10Ω resistive load. The discretizationand time-delay effects are due to the digital implementation of the modulation system.93Chapter 4Synchronous Rectification of LLCConverters1In Chapter 1, it was discussed that the complexity of the analysis, large and small-signal modeling anddynamic response, quiescent region operation, and degraded efficiency due to conduction losses in therectifier and variations in the resonant frequency are the challenges that must be addressed to improvethe performance of LLC converters. In this dissertation, a three layer control strategy is proposed toaddress these challenges. The three-layer control strategy is shown in Fig. 4.1 (a). In chapter 2, thefirst control layer was introduced in order to address the large-signal modeling and control of the LLCand series resonant converters. In chapter 3, two small-signal circuit models of the LLC converter wasintroduced. These circuit models can be used in the second control layer to design linear compensatorsand to keep the steady-state error close to zero. The second control layer is active when the converteroperates in the quiescent region. The operating regions of the LLC converter are shown in Fig. 4.1 (b).In this chapter, the focus of the analysis is on the third control layer where the conduction losses andefficiency concerns are addressed. The third control layer is active when the LLC converter operates inthe quiescent region.A typical LLC resonant converter is shown in Fig. 4.2 (a). Although LLC resonant converters canprovide soft switching conditions for the inverter switches and rectifier diodes, including zero voltageand zero current switching (ZVS and ZCS) conditions, the complexity of their analysis and conductivelosses in the rectifier have remained a barrier to gaining high efficiencies, especially in low-voltage-high-current applications [8]. The use of synchronous rectifiers (SRs) has been widely addressed inliterature as a means to tackle the reduced efficiency and high conductive losses caused by the forwardvoltage of the rectifier diodes [80–90].The main challenge in using SRs is to finding their conduction angles, which is often performedusing high current or noisy voltage measurements [82–88]. One of the earliest methods introducedto drive SRs was current-driven synchronous rectification [80, 81]. This method is interesting as it isapplicable in most of power electronic topologies, including LLC resonant converters. The method1Portions of this chapter have been published in [6–8]94 Efficiency improvement: synchronous rectification and resonant frequency tracking using homopolarity cyclebased on small-signal modeling using homopolarity cycle{Proposed lthree layer control strategy3(a)vOiCovrefNon-operating regionTransient region (Control: Layer 1)(b)Quiescent region or(Control: Layer 3) (Control: Layer 2) 2 Nonlinear geomtric control:1Figure 4.1: a) The proposed three-layer control strategy where the layer 3 (efficiency improve-ment) is active, b) regions of operations in resonant converters. The active control layer ishighlighted in yellow.S1Vin VoCOCrLrn:1iLrvinvvrec RLLmiprimS2SR1SR2SR3SR4(a)Complexity12LoadIndependentInfo about SR ConductionIndependent to Converter parametersNOYES(partially)NOYES(1-H)piNOYESFHAHC(b)isecirFigure 4.2: a) circuit schematic of the half bridge LLC converter b) conceptual comparison be-tween the homopolarity cycle analysis method (low complexity, simple sensors) and tradi-tional FHA (higher complexity).utilizes the advantage of series connected current transformers (CTs) with SRs in order to detect thecurrent polarity. Positive and negative currents force the CT to turn the SRs on and off, respectively.The time delay and the effect of the CT leakage and magnetizing inductors should be considered whenusing CTs in high-switching-frequency applications. If the current driven synchronous rectificationmethod is modified, it can be used to drive the SRs of an LLC resonant converter that has a voltage-doubler rectifier [82, 83]. In this method, a CT with only a secondary winding is used to drive two SRsin the voltage doubler rectifier. Using the CTs on the secondary side of the transformer can be verydissipative since the secondary current is relatively large in applications where SRs are used. In order95Synchronous rectification techniquesCurrent sensingVoltage amplitude sensingHCM: voltage polarity sensingSmall time delay Small time delay Large time delayLarge signal sensing Small signal sensing Small signal sensingLarge noise immunity Small noise immunity Small noise immunityNo zero crossing noise filterSwitching freq. limited due to leakage inductanceZero crossing noise filterrequiredSwitching freq. limited due to MOTNo switching freq. limitZero crossing noise filterrequiredResponse to load:Light to FullResponse to load:Medium to FullResponse to load:Medium to FullFigure 4.3: Comparison of properties: the proposed HCM, voltage and current sensing syn-chronous rectification techniques.to reduce the conduction losses when the current driven synchronous rectification method is used, theCT can be installed on the primary side of the transformer. The primary current includes the reflectedcurrent from the transformer secondary to primary and magnetizing current. In order to detect the zerocrossing points of the secondary current, a current compensating winding can be used to cancel theeffect of the magnetizing current [84].The magnitude of the voltage drop across the drain-to-source of SRs can be sensed as an alternativemethod of detecting their conduction angles. Smart synchronous rectifier driver integrated circuits (ICs)were introduced almost two decades ago in order to provide proper sensing and timing control circuitsand drive the SRs [85, 86]. Some challenges that must be addressed when using smart SR driver ICsare the predefined threshold voltage levels, minimum on time (MOT), the effect of temperature on theIC parameters, and the effects of the stray inductance and loading conditions. The very small voltagedrop over the drain-to-source of a SR or its body diode when conducting is significantly affected bynoise making it difficult to detect the zero crossing point. Therefore, in many cases, it is necessary touse zero-crossing noise filters when using the drain-to-source voltage to detect the conduction angles ofSRs [87, 88]. If the effect of the SRs’ stray inductances is not compensated, the detection of the SRs’conduction angles cannot be optimized. A predictive gate drive method can be used to compensate forthe effect of the stray inductance and provide a more stable driving signal for SRs [89]. The SRs in anLLC resonant converter can also be driven using an adaptive control method, assuming that the start ofthe SRs’ conduction angles equals the rising edge of the inverter voltage [90]. In this adaptive controlmethod, the turn-off instants of the SRs are digitally tuned based on the corresponding voltage drop.From the description above, two main measurement methods can be identified for SRs, the current basedmethods with CTs and the SR voltage drop methods, each of them presenting a number of technical96challenges. The above discussion is summarized in the comparative table shown in Fig. 4.3.This chapter develops the theory of homopolarity cycle modulation (HCM), to achieve a simplerectification strategy that eliminates costly current sensors and SR low voltage sensing, which is subjectto noise. The proposed HCM only requires a polarity sensor, employs a simple control algorithm and isused in the control layer 3. In order to find the conduction angles of the SRs used in the LLC converterdepicted in Fig. 4.2 (a), the theory of the homopolarity cycle is used. As a result of employing thehomopolarity cycle concept, the analysis complexity is reduced and it is discovered that the polarity ofthe gate driving signal vg1 and the rectifier voltage vrec contain all the information required to detect theconduction angles of the SRs. Compared to the first harmonic approximation (FHA) analysis method,which is load dependent and does not provide any information about the conduction angles of the SRs,the homopolarity cycle enables an accurate calculation of the SR conduction angles based on the timeduring which the polarities of the inverter and rectifier voltages are the same. A conceptual comparisonbetween the homopolarity cycle and FHA analysis approaches is illustrated in Fig. 4.2 (b). The theo-retical analysis has resulted in a very simple synchronous rectification control strategy. The proposedHCM synchronous rectification method has a low implementation cost since it requires only sensingthe polarity of the rectifier voltage (the transformer secondary voltage), but it does not sense voltageor current levels in the power metal-oxide-semiconductor field-effect transistors (MOSFETs). Due tothe relatively large amplitude of the rectifier voltage, the proposed HCM algorithm has good immunityagainst noise. The theoretical analysis is validated through experimental and simulation results. Unlikemost synchronous rectification techniques that do not perform well under medium and light loadingconditions, an almost flat synchronous rectification coverage curve from light to full loading conditionsis achieved by the proposed HCM method.The homopolarity cycle theory is discussed in detail in Chapter 3, Section 3.1. As a result, the focusin this chapter is on how to use the homopolarity cycle concept to drive the synchronous rectifiers in theLLC converter.4.1 The Homopolarity PlaneUsing the homopolarity cycle, the complexity of the LLC converter analysis is reduced, and a low-cost-polarity-based sensing technique can be utilized to provide synchronous rectification in LLC converters.A very simple equation calculating the LLC converter voltage gain was obtained in Chapter 3, Section3.1. In this section, the conduction angles of the SRs are related to the gate driving signals of theinverter switches and the polarity of the rectifier voltage using the homopolarity cycle. Furthermore,a tool that graphically shows the LLC converter behavior is introduced: the homopolarity plane. Thehomopolarity plane is not actively involved in the control algorithm; however, it is a second layer ofvalidation performed graphically. In this plane, the movement of the operating point is not continuous;rather it jumps from state to state. The homopolarity plane has two axes: the vertical axis is vinv, and thehorizontal axis is vrec. In this plane, the behavior of the LLC converter can be demonstrated through twogeometric shapes: a line and a rectangle. If the LLC converter operates at resonance, a diagonal lineillustrates the converter steady-state behavior. This diagonal line is named the reference line. Below97vrecvinvVonAbove resonance operationBelow resonance operationReference line at resonance-VonVon-VonVinVin2nReference pointVin2nFigure 4.4: Operation of the LLC converter in the homopolarity plane providing information aboutthe non-conduction angles of the SRs.and above resonance, the LLC converter steady-state behavior is described by a rectangle called theoperating rectangle. The height of the operating rectangle equals the input voltage, and its width equalsthe reflected output voltage to the transformer primary side (nVO). Variations in the input and outputvoltages change the height and width of the operating rectangle and the length of the reference line. Thesteady-state behaviors of the LLC converter in the homopolarity plane in different operating regions areshown in Fig. 4.4.The first step in finding the conduction angles of the SRs in the homopolarity plane is to map thetwo end points of the reference line onto the vrec axis. The point mapped on the right side of thehomopolarity plane is called the reference point. The reference point distance from the origin is Vin/2n.If the LLC converter operates below resonance, the dashed dark yellow operating rectangle illustratesthe LLC converter steady-state behavior in the homopolarity plane. If the LLC converter operates aboveresonance, the dashed blue operating rectangle illustrates the LLC converter steady-state behavior in thehomopolarity plane.According to Fig. 4.5, the non-conduction angle of the synchronous rectifiers can be obtained bythe following equation:θNC =(1−Hb)(Tsw/2)Tsw/2pi (4.1)From (3.30), the voltage gain of the LLC converter operating below resonance is (nVO/Vin)= 1/2Hb.By applying nVO/Vin = 1/2Hb to (4.1), the following equation is obtained:θNC =nVO− (Vin/2)nVOpi (4.2)The advantage of (4.2) is that the non-conduction angle and therefore the conduction angle of theLLC converter can be mathematically expressed. Moreover, (4.2) enables the homopolarity plane tographically demonstrate the non-conduction angles of the LLC converter.981 2Polarity1010SR1&2 SR3&4ONOFFOFFOFFHb=tB1-tB0Tsw/2{polarities are the sametttirtHbvrecvinv-(vin/2)12tB0tB1tB2Tsw/2HbTsw/2OFFOFFOFFONHWMHWMFigure 4.5: The key time-domain waveforms, synchronous rectification truth table and homopo-larity cycle definition when the LLC converter operates below resonance.According to Fig. 4.7, vinv leads vrec above resonance. Therefore, there is a phase shift between vinvand vrec that can be calculated as follows:θd =(1−Ha)(Tsw/2)Tsw/2pi (4.3)The voltage gain of the LLC converter operating above resonance is (nVO/Vin)=Ha−(1/2), if n>>(1/K f −1)/(8nFswCrRL). By substituting (nVO/Vin) = Ha− (1/2) into (4.3), the following equation isobtained:θd =(Vin/2)−nVOVinpi (4.4)The equation (4.4) is important because it mathematically expresses the phase shift between vinv andvrec when the LLC converter operates above resonance. Moreover, (4.4) enables the homopolarity planeto graphically demonstrate the phase shift existing between vinv and vrec above resonance.4.2 The Proposed HCM MethodIt has been shown that the complexity of the LLC converter analysis can be significantly reduced byapplying the homopolarity cycle to the LLC converter time-domain equations. Using the homopolaritycycle and homopolarity plane, the conduction angles of the SRs are related to the gate driving signalsof the inverter switches and polarity of the rectifier voltage. In this section, a synchronous rectificationtechnique, called homopolarity cycle modulation (HCM), for LLC resonant converters is introduced.The proposed HCM synchronous rectification technique only needs to sense the polarity of vrec, whichconsiderably reduces the cost and the effect of noise. The proposed synchronous rectification techniqueis employed in the control layer 3 and is activated when the LLC converter is in the quiescent area.According to the theoretical waveforms shown in Figs. 4.5, 4.6, and 4.7 the relevant SRs startconducting at the moment that the polarities of vinv− (Vin/2) and vrec become the same. Because vinvis controlled by the gate driving signals of the inverter switches (vg1 and vg2), there is no need to sense991 2Polarity11SR1&2 SR3&4ONOFFOFFONHr=tR1-tR0Tsw/2{polarities are the same=1tR0tR1tttHrTsw/2vinv-(vin/2)vrecHWMirt12HrHWMFigure 4.6: The key time-domain waveforms, synchronous rectification truth table and homopo-larity cycle definition when the LLC converter operates at resonance.Ha=tA1-tA0Tsw/2{polarities are the sametA0tA1tA2(1-Ha)(Tsw/2)tttTsw/2HaTsw/2tvinv-(vin/2)vrecir121 2Polarity1010SR1&2 SR3&4OFFOFFOFFONOFFOFFOFFONHaHWMHWMFigure 4.7: The key time-domain waveforms, synchronous rectification truth table and homopo-larity cycle definition when the LLC converter operates above resonance.vinv. The conceptual control algorithm of the proposed HCM synchronous rectification technique isshown in Fig. 4.8. Two low-cost voltage comparators can sense the polarity of the rectifier voltage. Thecomparators should have a threshold voltage in order to guarantee that the zero crossing noise does notaffect driving the SRs.When the LLC converter operates below and at resonance, Hb(Tsw/2) = T0/2 despite the loadingcondition. The period, during which the polarity of vrec is the same as vinv, is shorter than half a switch-ing period. Therefore, an auxiliary mechanism is required to change the polarity of vrec at the rightmoment when the LLC converter operates below resonance. Two monostables can be used to resolvethe above case and to cancel the ringing below and at resonance. The oscillations are due to the strayinductances and parasitic capacitors. This phenomenon is shown in Figs. 4.10 (a) and 4.11 (a). Belowand at resonance, when the rising edges of vg1 and vg2 happen, the polarity of vrec becomes positive andnegative, respectively. The rising edge of vg1 and vg2 triggers the monostables. The monostable time isset as Hb(Tsw/2). At the beginning of the first half cycle, below resonance, vg1 becomes 1 and triggersmonostable 1 (vm1 = 1). At the same time, the polarity of vrec becomes positive. Therefore, SR1 and100Startrising edge vg1?trigger monostable 1nonotrigger monostable 2polarity of vrec?polarity of vrec?11negposyesyesposnegvg1&vm1=1?Turn ON SR1 SR2yesTurn OFF SRsnovg2&vm2=1?Turn OFF SRsTurn ON SR3 SR4yesvm1:output signalof monostable 1vm2:output signalof monostable 2rising edge vg2?noFigure 4.8: The proposed HCM synchronous rectification algorithm. This algorithm is employedin the control layer 3.SR2 are turned on. Once the monostable 1 is expired, its output signal vm1 becomes zero, and therefore,SR1 and SR2 are turned off. In the second half cycle, the same procedure occurs for SR3 and SR4, due tothe symmetrical operation of the LLC converter.According to Fig. 4.5, the LLC converter rectifier conducts from tB0 to tB1 and tB2 to tB3. Since theoperation of the LLC converter is symmetrical in a switching cycle, analyzing the first half switchingcycle is enough. The differential equation describing the LLC converter behavior from tB0 to tB1 is givenby (3.2). By using the Kirchhoff current law (KCL), iLr can be expressed in terms of iLm and ir asfollows:iLr(t) = iLm(t)+1nir (4.5)By applying (4.5) to (3.2), the following equation is obtained:Vin−nVO = Lr diLm(t)dt +Lrndir(t)dt+1Cr∫ tB1tB0iLm(t)dt+1nCr∫ tB1tB0ir(t)dt+ vCr(tB0) (4.6)From tB0 to tB1, SR1 and SR2 conduct. Therefore, the voltage across Lm is nVO. This can be mathe-matically expressed as follows:LmdiLm(t)dt= nVO (4.7)By applying (4.7) to (4.6) and solving the resultant equation, the rectified current ir(t) from tB0 totB1 is obtained as follows:ir(t) =nZ0[Vin− vCr(tB0)− (1+ LrLm )nVO]sin(ω0(t− tB0)) (4.8)101where,Z0 =√LrCr(4.9)ω0 =1√LrCr(4.10)From tB0 to tB1, ir has a semi sinusoidal waveform. This means that at tB1, ir becomes zero again.Therefore,tB1− tB0 = piω0 =T02(4.11)According to the definition of the homopolarity cycle, Hb(Tsw/2) = tB1− tB0 = T0/2.Above resonance, at the beginning of the first half cycle, vg1 becomes 1. This triggers the monostable1. Due to the phase shift existing between vg1 and vrec, vrec is still negative. This forces SR1 and SR2 tobe in the off state. Once the polarity of vrec is positive, SR1 and SR2 are turned on. When vg1 becomeszero, SR1 and SR2 are turned off. Due to the symmetrical operation of the LLC converter, the sameprocedure occurs for SR3 and SR4 when the LLC converter operates above resonance.Using the proposed algorithm, the LLC converter robustness must be determined. In order to dis-cuss the robustness of the LLC converter with the proposed HCM synchronous rectification algorithm(employed in the control layer 3), uncertainties should be considered. The proposed homopolarity cyclemodulation synchronous rectification algorithm is shown in Fig. 4.8. In the first row of the control algo-rithm, it is observed that the system is taking advantages of two monostables. It was already discussedthat when the LLC converter operates below and at resonance, Hb(Tsw/2) = T0/2 despite the loadingcondition. Therefore, two monostables can be used to cancel the ringing below and at resonance. Themonostable time is set to Hb(Tsw/2). Uncertainties that can change Hb(Tsw/2) are the variations in theresonant inductance and capacitance. The resonant period of the LLC converter is determined by theresonant inductance and capacitance, and can be calculated using the following equation:T0 = 2pi√LrCr (4.12)By adding uncertainties to the values of the resonant inductor and capacitor, the following equationis obtained:T0+∆T0 = 2pi√(Lr+∆Lr)(Cr+∆Cr) (4.13)In the above equation, ∆Lr and ∆Cr represent the tolerances of the resonant inductor and capacitor,and ∆T0 represents the change in the resonant period due to ∆Lr and ∆Cr. The above equation can berewritten as follows:T0+∆T0 = 2pi√LrCr√(1+∆LrLr)(1+∆CrCr) (4.14)1025.6µs5.6µs≈50Vvrecvinv400VHr=0.98SR1&2:ON SR3&4:ON SR1&2:ONFigure 4.9: Experimental results of the LLC converter operating at resonance under the full load-ing condition in time-domain and in the homopolarity plane. The experimental results pro-vide validation that the proposed HCM method is successful in detecting and driving theSRs.By relocating T0 to the right side of the above equation and considering the fact that T0 = 2pi√LrCr,the following equation can be obtained:∆T0 = T0√(1+∆LrLr)(1+∆CrCr)−T0 (4.15)By dividing the two sides of the above equation by T0, the following equation is obtained:∆T0T0=√(1+∆LrLr)(1+∆CrCr)−1 (4.16)The uncertainties ∆Lr and ∆Cr cause the resonant frequency T0 to change. The proposed HCMsynchronous rectification technique, employed in the control layer 3, remains robust if ∆T0 is positive.This means that ∆T0/T0 must be positive. If ∆T0/T0 is positive, the expression in the right side of (4.16)must be positive. The above discussion results in the following condition:√(1+∆LrLr)(1+∆CrCr)> 1 (4.17)As long as (4.17) is valid, the LLC converter with the proposed HCM synchronous rectificationtechnique remains robust, stable, and not sensitive to variations in T0. In Section ?? and 4.2, it isdiscussed that the variations in the input voltage and load do not affect the robustness of the proposedHCM synchronous rectification algorithm. This is validated through the experimental results, shown inSection 4.3.4.3 Experimental and Simulation ResultsIt has been shown that the complexity of the LLC converter analysis in the quiescent area can be sig-nificantly reduced and a low-cost-polarity-based sensing technique can be used by introducing the ho-mopolarity cycle. The homopolarity cycle relates the conduction angles of the SRs to the voltage gain of103(a)vrecvinv≈50V330V7µs5.8µsSR1&2:ON SR3&4:ONHb=0.82(b)4µs3.5µs0.5µs≈37Vvrecvinv420VHa=0.87SR1&2:ONSR3&4:ONSR1&2:ONSR3&4:ONFigure 4.10: Experimental results of the LLC converter operating: a) below resonance and b)above resonance, in time-domain and in the homopolarity plane. The experimental resultsindicate that the proposed HCM method is successful in detecting and driving the SRs underfull loading condition.the LLC converter, the gate driving signals of the inverter switches and rectifier voltage. Using the the-oretical analysis, a synchronous rectification technique, called homopolarity cycle modulation (HCM),for LLC resonant converters was proposed in section 4.2. The proposed HCM synchronous rectificationtechnique uses the advantage of a polarity-based sensing technique, which considerably reduces the costand noise effect. The proposed HCM is employed in the control layer 3 and is active when the converteroperates in the quiescent area. In this section, the theoretical analysis developed in this chapter is val-idated by experimental and simulation results of a 1.2kW LLC resonant converter with synchronousrectifiers controlled by the proposed HCM technique. Smart SR driver ICs have been used as a regularsolution for driving SRs. In order to compare the operating range of the proposed HCM technique withthat obtained using a conventional technique (smart SR driver IC) simulation results are presented. Thespecifications of the LLC resonant converter are given in Table 4.1.In Fig. 4.9, the experimental waveforms of the LLC converter operating at resonance and underthe full loading condition in the time-domain and the homopolarity plane are shown. As shown in thisfigure, at resonance, the phase shift between vinv and vrec is zero, and the reference line is observed inthe homopolarity plane. As discussed in section 4.2, when vinv and vrec are positive, SR1 and SR2 are104(a)7µs5.8µsvrecvinv≈50V330VHb=0.82SR1&2:ON SR3&4:ON(b)vrecvinv≈40V400V1.6µs1.44µs0.16µs Ha=0.9SR1&2:ON SR3&4:ONFigure 4.11: Experimental results of the LLC converter operating: a) below resonance and b)above resonance, in time-domain and in the homopolarity plane. The experimental resultsindicate that the proposed HCM method is successful in detecting and driving the SRs underlight loading condition (10%)in the on state and when vinv and vrec are negative, SR3 and SR4 are in the on state. The blue signal inFig. 4.9 represents the final product of the homopolarity width modulation (HWM) signal driving thethe synchronous rectifiers. At resonance, it is theoretically expected that the duty cycle of the HWMsignal equals 1. Since vg1 and vg2 are among the inputs of the proposed HCM algorithm, and they havedead-bands in order not to short the DC input bus during the transition, the width of the HWM signal isa bit smaller, and homopolarity cycle is read smaller than its actual value. This small width reductionin the HWM signal, due to the dead-bands of the gate signals of the inverter switches, creates a naturaldead-band between the gate driving signals of the synchronous rectifiers. This dead-band is crucial inpreventing a short on the DC output bus. This feature is a merit of the proposed HCM synchronousrectification algorithm, employed in the control layer 3. The control layers are shown in Fig. 4.1 (a).In Fig. 4.10 (a) and (b), the experimental waveforms of the LLC converter operating below reso-nance and above resonance under the full loading condition in the time-domain and the homopolarityplane are shown, respectively. Below resonance, as discussed before, the rising edges of vinv and vrecoccur at the same time. This means that their polarities simultaneously become positive. The blue sig-10502040608010010 20 30 40 50 60 70 80 90 100Synchronous rectification coverage (%)Output power (%)HCMSmart SR DriverFigure 4.12: The simulation results comparing the performance of the proposed HCM methodwith that of the smart SR driver IC under different loading conditions in terms of syn-chronous rectification coverage.Table 4.1: The LLC converter specificationsParameter Value DescriptionVin 400V Nominal input voltageVO 48V Nominal output voltagePO 1200W Nominal output powerfr 93kHz Resonant frequencyn 4 Transformer turn ratioLr 25µH Resonant inductorLm 240µH Magnetizing inductorCr 100nF Resonant capacitorCO 100µF Output capacitornal, which is the H signal, is on as long as the polarities of vinv and vrec are the same. Above resonance,as shown in Fig. 4.10 (b), the phase shift θd is properly detected by the proposed HCM method. Thisresults in generating the gate driving signals of the SRs.Most of synchronous rectification techniques fail when the LLC converter operates under mediumand light loading conditions. In order to validate that the proposed HCM synchronous rectificationtechnique can drive the SRs even under the light loading conditions, the key experimental waveformsof the LLC converter operating below and above resonance under 10% of the nominal loading condi-tion are shown in Fig. 4.11. As shown in this figure, the non-conduction angle of the SRs when theLLC converter operates below resonance and θd when the LLC converter operates above resonance aresuccessfully detected and the gate driving signals of the SRs are correctly generated.The performance of the proposed HCM method (employed in the control layer 3) is comparedthrough the simulation results with that of the smart synchronous rectifier driver IC. In order to simulatethe proposed HCM method and the smart SR driver, the simulation software package, PSIM, is used.The smart SR driver IC operates based on sensing the voltage across the SRs’ drain-to-source voltage.106(d)0-12120-0.5-1time (s)00.40.8i secv DS_SR1,2v g_SR1-4time div.: 4µs~98% Body diode is OFFi sectime (s)time div.: 2µs(c)20.5% Body diode is ON0-10100-0.5-100.40.8time (s)time div.: 4µsBody diode is ON17% (a)i secv DS_SRsv g_SRs0-12120-0.5-100.40.8(b)time (s)i secv DS_SRsv g_SRstime div.: 2µs19.5% Body diode is ON0-10100-0.5-100.40.80-0.5-100.40.80-1010time (s)time div.: 3µsi secv DS_SR1,2v g_SR1-4(e)~98% Body diode is OFF0-10100-0.5-100.40.8i secv DS_SR1,2v g_SR1-4time (s)time div.: 3µs(f)~98% Body diode is OFFv DS_SRsv g_SRsFigure 4.13: The key comparative simulation waveforms of the LLC converter under light loadingcondition below, at and above resonance when its SRs are controlled by: a-c) the smartSR driver IC, d-f) the proposed HCM method. Under the light loading condition, the syn-chronous rectification coverage of the proposed HCM method is close to 100% below, atand above resonance; however, that of the smart SR driver IC is 19% on average.The comparison is performed based on the percentage of the synchronous rectification coverage. Asshown in Fig. 4.12, the percentage of the synchronous rectification coverage of the two methods isalmost 98% under medium to full loading conditions. Under 30% of the nominal loading condition, thepercentage of the synchronous rectification coverage obtained using the smart SR driver drops to 20%on average. However, under the same loading condition, the percentage of the synchronous rectificationcoverage obtained using the proposed HCM method remains almost 97%. In order to have a betterunderstanding of what happens for the synchronous rectifiers installed in the LLC converter under thelight loading condition, the key simulation waveforms are illustrated in Fig. 4.13.The efficiency diagrams of the LLC converter with the proposed HCM synchronous rectificationtechnique, with the smart SR driver technique, and without synchronous rectification are plotted in10788909294969810010 20 30 40 50 60 70 80 90 100Proposed HWM tachniqueSmart SRDriverDiodeRectifierImproved eff. due toimproved synch. rectification coveragePercentage of the output power (%)LLC converter’s efficiency (%)Figure 4.14: The efficiency diagram of the LLC converter.01020304050VO0-55101520ir00.40.80 0.5 1 1.5 2 2.5 3t (ms)05101520IoLayer1Layer2Layer2&Layer3Layer1Layer2&Layer3Layer2&Layer3Layer1SR gate driving signalsFigure 4.15: The switching action between the layers of the three-layer control strategy.Fig. 4.14. As shown in this figure, the LLC converter efficiency with the proposed HCM synchronousrectification technique is improved by 3% compared to that with the smart SR driver technique, underlight loading conditions. The efficiency of the LLC converter with the proposed HCM synchronousrectification technique is improved by 5% compared to that without synchronous rectifiers, across theentire operating range. These results show that in the third layer of the proposed three-layer controlstrategy, the conduction losses can be reduced and consequently the efficiency can be improved.As discussed earlier in this dissertation, the three-layer control strategy is able to improve the overall108performance of the LLC converter. In Chapters 2 and 3, the first and second control layers were intro-duced, and in this section, it was discussed that how the efficiency can be improved by the third controllayer. Fig. 4.15 illustrates the switching action between the control layers of the proposed three-layercontrol strategy.4.4 SummaryFurther improvement in the LLC converter performance requires the use of synchronous rectifier (SR).The main challenges in synchronous rectification for LLC converters is to analyze and detect the con-duction angles of the SRs. This chapter proposed a synchronous rectification method, called homopo-larity cycle modulation (HCM), for LLC resonant converters. The proposed HCM is employed in thecontrol layer 3. In the previous chapters, the first and second control layers were introduced. In theselayers, the analysis was focused on the large and small-signal modeling. In this chapter, the third con-trol layer was introduced to address the issue of degraded efficiency due to conduction losses. A newdefinition, called homopolarity cycle, was applied to time-domain equations of the LLC converter. Thehomopolarity cycle significantly reduced the complexity of the LLC converter analysis and related theconduction angles of the SRs to the gate driving signals of the inverter switches and polarity of therectifier voltage. The proposed HCM method uses the advantage of a polarity-based sensing techniquethat reduces cost and the effects of noise. The homopolarity plane, which can be used to graphicallyanalyze the LLC converter and conduction angles of the SRs, was introduced. The theoretical analysishas resulted in a simple HCM algorithm that controls the SRs used in the LLC converter, and that hasa low implementation cost. The proposed HCM method is compared with a conventional synchronousrectification method and validated by experimental results. The results have shown that the proposedHCM synchronous rectification method has a near maximum synchronous rectification coverage fromlight to full loading conditions.109Chapter 5Resonant Frequency Tracking of LLCConverters Using Homopolarity Cycle1In Chapter 1, the main concerns in resonant converters, particularly in LLC converters, were intro-duced, and they are listed as follows: the complexity of the analysis, large-signal modeling and dy-namic response, small-signal modeling and quiescent region operation, and degraded efficiency due toconduction losses in the rectifier and variations in the resonant frequency. This dissertation proposes athree-layer control strategy to address the above concerns. The three-layer control strategy is shown inFig. 5.1 (a). In Chapter 2, the first control layer was discussed. In the first control layer, the averagegeometric control (AGC) enhances the large-signal dynamic response of the closed-loop LLC and seriesresonant converters. In chapter 3, two small-signal circuit models of the LLC converter was introduced.These circuit models can be used in the second control layer to design linear compensators and to keepthe steady-state error close to zero. The second control layer is active when the converter operates in thequiescent region. The operating regions of the LLC converter are shown in Fig. 5.1 (b). In chapter 4,the degradation of the efficiency due to conduction losses was discussed and a new synchronous rectifi-cation technique, called HCM, was introduced. The HCM can be used in the control layer 3 when theconverter operates in the quiescent region. In this chapter, the deviation of the series resonant frequencyin LLC converters and its negative effect on the efficiency are discussed. As it will be shown in thischapter, by using a new resonant frequency tracking technique that is employed in the control layer 3,the unregulated LLC converter can operate in its optimum point. In cost sensitive applications wherevoltage regulation is not required, unregulated LLC converters can be used.LLC converters are conventionally controlled by the switching frequency in order to either regulatethe output voltage/current or enhance the efficiency. Unlike regulated LLC converters, the switching fre-quency of unregulated LLC converters, widely used as DC transformers, is tuned at the series resonantfrequency in order to achieve the optimum efficiency [94–96]. As shown in Fig. 5.2 (a), the resonanttank of the LLC converter is made of Lr, Cr, and Lm. The Values of Lr and Cr determine the seriesresonant frequency of the LLC converter (F0 = 1/(2pi√LrCr)). Lr and Cr are theoretically constant;1Portions of this chapter have been modified from [9]110 Efficiency improvement: synchronous rectification and resonant frequency tracking using homopolarity cyclebased on small-signal modeling using homopolarity cycle{Proposed lthree layer control strategy3(a)vOiCovrefNon-operating regionTransient region (Control: Layer 1)(b)Quiescent region or(Control: Layer 3) (Control: Layer 2) 2 Nonlinear geomtric control:1Figure 5.1: a) The proposed three-layer control strategy where the layer 3 (efficiency improve-ment) is active, b) regions of operations in resonant converters. The active control layer ishighlighted in yellow.however, in practice, variations in the temperature, frequency, load, and manufacturing tolerances causeLr and Cr, and consequently, the series resonant frequency of the LLC converter to deviate. The am-plitude of this deviation ranges from 1% to 25%, and depends on the materials used and manufacturingmethods [97–101]. For instance, as shown in Fig. 5.2 (b), if ∆Cr and ∆Lr (variations in the resonant ca-pacitance and inductance, respectively) are ±10%, the resonant frequency deviates from the theoreticalseries resonant frequency by 20.2%. This consequently deviates the voltage gain of the LLC converterby 11.6%; one that was supposed to be equal to 1 and fixed. Therefore, an open-loop LLC converter witha fixed switching frequency cannot consistently operate at resonance. This highlights the importance oftracking the series resonant frequency of LLC converters in unregulated DC-DC applications.In order to enable the unregulated LLC converter to perform as a DC transformer, despite the electri-cal and environmental variations, several interesting approaches have been introduced in the literature.When the LLC converter operates at resonance, the resonant current, iLr, can be approximately consid-ered sinusoidal. Therefore, the harmonics of iLr are zero. Below and above resonance, iLr is distorted.This means that the harmonics of iLr are not zero anymore. Therefore, the resonant frequency trackingcan be performed by sensing iLr and online calculating its harmonics and minimizing them [102]. Theresonant frequency tracking can also be performed by a plant modeling approach in which the driftin the series resonant frequency is detected by observing the phase or gain relationship of two pairedelectrical variables [103–106]. This strategy is involved in integration, peak detection, and level shiftingin order to track the series resonant frequency of the unregulated LLC converter. In some applications,unregulated LLC converters are equipped with SRs to reduce conduction losses. In the presence ofSRs, the resonant frequency tracking can be performed by tuning the gate driving signals of SRs usingthe universal adaptive driving method, and then comparing the pulse width difference between the gatedriving signals of the inverter switches and SRs [107, 108]. This comparison is later used to reduce thepulse width difference and consequently to detect the series resonant frequency. It is also proven that theresonant operation can be achieved for the LLC converter if it operates in the boundary of the rectifierzero current time (RZCT) [109, 110]. In the RZCT method, the series resonant frequency is detected111(a)(b)-10-10-10-5000510151010 ΔCr (%)ΔLr (%)resonant frequencyvariation (%)-9.09%11.11%Δf0 =20.2%ΔGain=+4%ΔGain=-7.6%ΔGain=0%Goal{S1VinVOCOCrLr n:1iLrvinvvrecLmiprimS2SR1SR2vrecvSR2vSR2RLirvg1Figure 5.2: a) The unregulated half bridge LLC converter with center-tapped synchronous rec-tifiers, b) the effect of variations in the resonant capacitance and inductance on the seriesresonant frequency and voltage gain of the LLC converter (variations of ±10% in the reso-nant capacitance and inductance result in unwanted resonant frequency and gain deviationsof 20.2% and 11.6%, respectively).by sensing the transformer secondary side current and changing the switching frequency to minimizethe secondary zero current time. Although the harmonic, plant modeling, synchronous rectification, andRZCT based resonant frequency tracking methods have presented interesting results, the opportunity tointroduce a resonant frequency tracking method, which has a lower cost and larger immunity againstnoise, is still open.In this chapter, a new resonant frequency tracking method for unregulated LLC converters thatcan be used as DC transformers is proposed. In cost sensitive DC/DC applications, unregulated LLCconverters are widely used. The proposed method is based on the analysis of the homopolarity cycle intime-domain and employed in the control layer 3. The homopolarity cycle mathematically describes thevolt-amp-second balance principle in LLC converters and is discussed in detail in Section 5.1. By usingthe homopolarity cycle in the time-domain equations, the complexity of the analysis is significantlyreduced and it is discovered that the gate driving signal of the inverter switch vg1 and the voltage polarityof the rectifier voltage vrec contain all the information required to track the series resonant frequency ofthe LLC converter. vrec and vg1 are highlighted in Fig. 5.2 (a). vg1 is an internally generated signaland therefore, the polarity of vrec is the only converter parameter that must be sensed. As a result ofthis discovery, a simple resonant frequency tracking algorithm is developed that takes advantage of a112low-cost voltage polarity-based sensing technique (more immune against noise), and that eliminatescostly current or sub-voltage sensing techniques. The proposed resonant frequency tracking algorithmhas a fast convergence time and can also drive the SRs for the obtainment of higher efficiency. Atwo dimensional (2D) plane, called homopolarity, is used to demonstrate the operation of the LLCconverter in steady state. This plane is a graphical tool to prove whether the series resonant frequencyof the unregulated LLC converter is tracked. Experimental and simulation results have shown that theefficiency improvement, resonant operation under different loading conditions, and fast convergencetime can all be achieved by the proposed resonant frequency tracking algorithm.5.1 Steady-State AnalysisThe proposed three-layer control strategy, shown in Fig. 5.1 (a), addresses the following concerns:the complexity of the analysis, large and small-signal modeling and control, and degraded efficiencydue to conduction losses and resonant frequency deviations. Through the analysis of the homopolaritycycle in time-domain, the complexity of the analysis is reduced and a new resonant frequency trackingalgorithm, which takes advantage of a low-cost voltage-polarity based sensing technique (presents betternoise immunity compared with current or sub-voltage sensing techniques) and drives the SRs for theobtainment of a higher efficiency, can be developed. The resonant tank of the LLC converter has twoinductors Lr and Lm, and one capacitor Cr. In steady state, the volt-second and amp-second balancedconditions are satisfied for the inductors and capacitor used in the resonant tank. Therefore, for theresonant tank, the volt-amp-second balance condition is satisfied when the LLC converter operates in thesteady state. The homopolarity cycle mathematically describes the volt-amp-second balance principle,and it is discussed in detail in this section. The resonant tank has two ports: one connected to the invertervoltage vinv and another to the rectifier voltage vrec. vinv is clamped to Vin, and vrec is clamped to ±VO.Therefore, by using the volt-amp-second balanced condition, the voltage gain of the LLC converter, asa function of the homopolarity cycle, is formulated. The theory of the homopolarity cycle is discussedin detail in Chapter 3, and in this section, the volt-amp-second balance principle is briefly discussed.5.1.1 Resonant OperationThe unregulated LLC converter operates as a DC transformer if the switching frequency of the inverterequals the series resonant frequency. The key time-domain waveforms of the LLC converter, operatingat resonance, are depicted in Fig. 5.3. In this figure, the operation of the LLC converter at resonanceis also depicted in the homopolarity plane As discussed in Chapter 4, the homopolarity plane has twoaxes: vrec and vinv. At resonance, the phase shift between vinv and vrec is zero. Therefore, the operatingpoint jumps between the two points R1 and R2. Since the operating point jumps very fast, twice ina switching cycle, a diagonal line, called reference line, is observed in the homopolarity plane. Thehomopolarity plane is important because it is a graphical tool, validating the resonant operation of theLLC converter. If the reference line is observed in the homopolarity plane, it is an indication that theswitching frequency of the unregulated LLC converter is tuned at resonance.113tR0tR1tttHrTsw/2vinv-(vin/2)vrecHWMirtHrvrecvinvReference lineR1R2-Vin/2n Vin/2nVinSamePolarityRegiontR0<t<tR1T0/2VOVin2Vin2-VOFigure 5.3: The illustration of the LLC converter operation at resonance: key time-domain wave-forms and homopolarity plane.tttirtHbvrecvinv-(vin/2)tB0tB1tB2Tsw/2HbTsw/2HWMvrecvinvReference line2BR operationVO-VOB1B2-Vin/2n Vin/2nSamePolarityRegiontB0<t<tB1OppositePolarityRegiontB1<t<tB2VOVin2Vin2-VOFigure 5.4: The illustration of the LLC converter operation below resonance: key time-domainwaveforms and homopolarity plane.5.1.2 Below Resonant OperationThe key time-domain waveforms of the LLC converter, operating below resonance, are depicted inFig. 5.4. In Fig. 5.4, the operation of the LLC converter, below resonance, is also depicted in thehomopolarity plane. Below resonance, the rising edges of vinv and vrec are concurrent; however, thefalling edge of vrec happens earlier than that of vinv in every half a switching cycle. Therefore, theoperating point jumps between the two points B1 and B2. In the time period during which the LLCconverter is in Interval 2, vinv remains constant; however, vrec decreases from |VO| due to the resonancebetween Lr, Lm, and Cr. This creates a hysteresis loop in the homopolarity plane, as shown in Fig. 5.4.In this figure, the reference line is also depicted to highlight the difference between the operation of theLLC converter below and at resonance.114vrecvinvReference lineVOAR operation-VOB1B2B3B4 VinVin/2n-Vin/2ntA0tA1tA2tttTsw/2HaTsw/2tvinv-(vin/2)vrecirHaHWMSamePolarityRegiontA0<t<tA1OppositePolarityRegiontA1<t<tA2VOVin2Vin2-VO(1-Ha)(Tsw/2)Figure 5.5: The illustration of the LLC converter operation above resonance: key time-domainwaveforms and homopolarity plane.5.1.3 Above Resonant OperationThe key time-domain waveforms of the LLC converter, operating above resonance, are depicted inFig. 5.5. In this figure, the above resonant operation of the LLC converter is also depicted in thehomopolarity plane. Above resonance, vinv leads vrec, creating a phase shift. Therefore, the operatingpoint jumps between the four points A1 through A4. In Fig. 5.5, the reference line is also depicted tohighlight the difference between the operation of the LLC converter at and above resonance.5.2 The Proposed Resonant Frequency Tracking MethodIn DC/DC applications where the cost is the primary concern, unregulated LLC converters can be used.A fast resonant frequency tracking method, employed in the third control layer, for unregulated LLCconverters can be obtained, through the analysis of the homopolarity cycle in time-domain. By usingthe homopolarity cycle concept, the complexity of the analysis is reduced and a voltage-polarity basedsensing technique, which is low-cost and immune to noise and enables synchronous rectification forthe obtainment of higher efficiency, can be developed. In Chapter 3, the homopolarity cycle conceptwas discussed in detail, and the voltage gain of the LLC converter as a function of the homopolaritycycle was formulated. This voltage gain equation provides a conceptual idea of how the series resonantfrequency should be tracked. In Section 5.1, the homopolarity plane, a graphical validating tool, wasintroduced. In this section, the proposed resonant frequency tracking method is introduced.vinv is controlled by the gate driving signals of the inverter switches vg1 and vg2. Therefore, thereis no need to sense the inverter voltage since it is internally available in the microcontroller. Hereafter,instead of vinv, vg1 is used in the discussion. The only parameter that must be sensed is the voltagepolarity of vrec. The initial switching period of the LLC converter is determined by the theoreticalresonant period (2pi√LrCr) of the LLC converter. Therefore:T0T = 2pi√LrCr (5.1)115StartRisingedge in vg1?Yesvrecpositive?Tsw[n+1]=Tsw[n]+ΔTswNoYesvSR1=vg1vSR2=vg21Non=0Tsw[0]=T0TvSR1 & vSR2=0Tsw[n+1]=Tsw[n]-ΔTswn=n+1|T[n]-T[n-4]|< ε?1 n=4?YesvSR1=0vSR2=0YesNoNo1n=0T[0]=T[4]Figure 5.6: The proposed homopolarity-cycle-based resonant frequency tracking algorithm.Due to the leakage and stray inductance of the transformer, and PCB tracks and wires, and alsothe ambient temperature, frequency and loading condition, the LLC converter is prone to not operateat resonance as designed. First, it is considered that the LLC converter operates at or below resonance.At and below resonance, the rising edges of vg1 and vrec are concurrent; however, the falling edge ofvrec happens earlier than that of vg1. Therefore, the concurrence of the rising edges of vg1 and vrec isan indication that the LLC converter operates at or below resonance. Below resonance, subtracting theswitching period Tsw[n] by ∆Tsw every sampling time TS causes the converter to approach the seriesresonant frequency. Once the operation at resonance is achieved, further switching period reductionresults in an above resonant operation.If it is considered that the LLC converter operates above resonance, a phase shift can be foundbetween vg1 and vrec. When this phase shift is observed, the operation of the LLC converter aboveresonance is detected. The rising edge of vg1 happens earlier than that of vrec when the LLC converteroperates above resonance. Therefore, if the rising edge of vg1 is detected and the polarity of vrec isnot positive yet, the LLC converter operates above resonance. Once the controller discovers this, itadds ∆Tsw to the switching period Tsw[n] every sampling time TS until the series resonant frequency isachieved. Once the rising edges of vg1 and vrec are concurrent, the LLC converter is at resonance.In steady state, the switching period of the LLC converter slightly fluctuates between T0r andT0r − ∆Tsw. T0r is the real resonant period. In order to determine ∆Tsw, an anticipation of the res-116(a)5.26µs5.8µsVO=5.39VvrecvinvNo observation of the reference line Hb=0.906VO<6.89, due to rectifierlosses(b)vrecvinvNo observation of the reference line 4.13µs3.2µsHa=0.775VO=2.17VVO<3.1, due to rectifierlossesFigure 5.7: The open-loop operation of the LLC converter under the half loading condition intime-domain and homopolarity plane: a) below resonance b) above resonance.onant frequency variation due to the tolerance of the resonant inductor and capacitor must be made.According to Fig. 5.2 (b), a tolerance of ±10% in Cr and Lr causes a pick to pick tolerance of 20.2%in the resonant frequency. By assuming that the LLC converter suffers from a tolerance of ±10% in Crand Lr, according to Fig. 5.2 (d), the minimum resonant period is 0.9×2pi√LrCr, where Lr and Cr arethe designed resonant inductance and capacitance, respectively. Therefore, ∆Tsw can be calculated usingthe following equation:∆Tsw = 1.8kpi√LrCr (5.2)where k is the resonant period fluctuation factor. In order to keep the fluctuation small, it is recom-mended that k equals 0.01 or less. The algorithm of the proposed resonant frequency tracking methodis shown in Fig. 5.6. The proposed resonant frequency tracking method can be employed in the thirdcontrol layer. The control layers are shown in Fig. 5.1 (a).In steady state, at resonance, the gate driving signals of the synchronous rectifiers are synchronized117(a)VO=6.11V5.23µs5.23µsHr=1VO≈6.2V The reference line is observed(b)VO=5.96V VO≈6.2V5.24µs5.24µsHr=1The reference line is observedFigure 5.8: The operation of the unregulated LLC converter with the proposed resonant frequencytracking method in time-domain and homopolarity plane: a) under 10% of full loading con-dition, b) under the full loading condition.with the gate driving signals of the inverter switches vg1 and vg2. Moreover, the switching period isT0r−∆Tsw or T0r. In order to take advantage of the SRs for conduction loss mitigation and, consequently,efficiency enhancement, the series resonant frequency must be detected using the above feature. In orderto guarantee the accuracy of the proposed resonant frequency tracking method, employed in the thirdcontrol layer, the first switching period is compared with the last switching period every fourth iterationof the algorithm. If the difference is smaller than ε , the LLC converter is acceptably operating atresonance. Theoretically, ε equals ∆Tsw. However, in order to bring a continuous operation for the SRs,a larger value for ε should be chosen. It is recommended that ε = 2∆Tsw.5.3 Experimental and Simulation ResultsIn some DC/DC applications voltage regulation is not required and therefore the cost can be lowered byusing unregulated LLC converters. In order to achieve higher efficiency, unregulated LLC converters118(a)40µs(b)36µsFigure 5.9: The dynamic response of the unregulated LLC converter with the proposed resonantfrequency tracking method: a) following a load step-up from 6A to 12A, b) following a loadstep-down from 12A to 6Aare required to operate at resonance. Through the analysis of the homopolarity cycle in time-domain,the complexity of the analysis is reduced, and a new resonant frequency tracking method is developed.The proposed resonant frequency tracking method has fast convergence time and low implementationcost, takes advantage of a large-signal voltage-polarity based sensing technique (prone to noise), iscapable of driving synchronous rectifiers to even more enhance the efficiency, and can be employed inthe third control layer. The three-layer control strategy is shown in Fig. 5.1 (a). In order to validatethe theoretical analysis, in this section, the experimental results of an unregulated LLC converter whoseresonant frequency is tracked by the proposed method are provided. The key parameters of the LLCconverter are given in Table 5.1.An open-loop LLC converter with a fixed switching frequency can operate below or above reso-nance, as shown in Fig. 5.7. In Fig. 5.7 (a), the below resonant operation of the LLC converter isillustrated. As shown in this figure, the below resonant homopolarity cycle Hb equals 0.906. Accordingto (3.30), the output voltage of the LLC converter must be 6.89V ; however, due to the open-loop oper-119Time0123456761218243036420Load current (A)Time div.: 1msOutput voltage (proposed method)Output voltage (conventional method)Load current1.89V45µs1.5msOutput voltage (V)(a)33 times faster01234567Load current (A)Output voltage (V)61218243036420TimeOutput voltage (proposed method)Output voltage (conventional method)Load current1.82ms45 times faster(b)40µsFigure 5.10: Comparative simulation results of the unregulated LLC converter with the proposedand conventional resonant frequency tracking methods: a) following a load step-up from 6Ato 12A, b) following a load step-down from 12A to 6A.Table 5.1: The parameters of the LLC converterParameter Value DescriptionLr 12.7µH Resonant inductorLm 40µH Magnetizing inductorCr 200nF Resonant capacitorCO 150µF Output capacitorn 4 Transformer turn ratioVin 50V Input voltagePOn 72W Nominal output poweration of the LLC converter and losses imposed by the forward voltage of the rectifier diodes, the outputvoltage is limited to 5.39V . In addition, the reference line is not observed in the homopolarity plane,which is a graphical indication that the LLC converter is not operating at resonance. In Fig. 5.7 (b), theoperation of the LLC converter above resonance is illustrated. As shown in this figure, the homopolaritycycle above resonance Ha equals 0.775. According to (3.30), the output voltage of the LLC converter12020 30 40 50 60 70 80 90 1002030405060708090100proposed method (Eff1)Average Eff.=96.04%Output current capacity (%)Efficiency (%)Figure 5.11: The efficiency diagram of the unregulated LLC converter with the proposed resonantfrequency tracking methods. The input voltage is 50V .must be 3.1V ; however, due to the open-loop operation of the LLC converter and losses imposed by theforward voltage of the rectifier diodes, the output voltage is limited to 2.17V . In addition, the referenceline is not observed in the homopolarity plane, which is a graphical indication that the LLC converter isnot operating at resonance.The operation of the LLC converter, whose series resonant frequency is tracked by the proposedmethod, under both light and full loading conditions, is illustrated in Fig. 5.8 (a) and (b), respectively.As shown in this figure, the rising and falling edges of vinv and vrec are concurrent under both lightand full loading conditions, meaning that the homopolarity cycle equals 1 and that the LLC converter isoperating at resonance. This is also graphically validated by the homopolarity plane, where the referenceline is observed. According to Table 5.1, Lr and Cr are designed 12.7µH and 200nF , respectively.Therefore, the theoretical resonant frequency of the LLC converter is 100kHz. As shown in Fig. 5.8(a) and (b), the resonant frequency of the LLC converter is 95.15kHz and 95.29kHz, under the lightand full loading conditions, respectively. This demonstrates that the series resonant frequency of theLLC converter slightly changes with the loading conditions and that the LLC converter has a resonantfrequency deviation of 4.85kHz and 4.71kHz under light and full loading conditions, respectively. Aspreviously mentioned, leakage and stray inductors, parasitic capacitors, loading condition, temperature,and switching frequency are the parameters creating the resonant frequency deviation. The input voltageof the LLC converter is 50V . Since the homopolarity cycle is 1 when the LLC converter operates atresonance, the output voltage should theoretically be 6.2V . The proposed resonant frequency trackingmethod enables synchronous rectification for the LLC converter. Therefore, as shown in Fig. 5.8 (a)and (b), the output voltage is very close to 6.2V under both light and full loading conditions.In Fig. 5.9 (a) and (b), the dynamic response of the LLC converter with the proposed resonantfrequency tracking method to a load step-up from half to full loading condition and a load step-downfrom full to half loading condition is illustrated. As shown in Fig. 5.9 (a) and (b), the response ofthe LLC converter to the transients is very fast as the settling time after the load step-up, and step-down is 40µs and 36µs, respectively. The dynamic response of the LLC converter with the proposedresonant frequency tracking method following a load step-up and step-down is compared with that with a121conventional method that minimizes the rectifier zero current time. This comparison is illustrated in Fig.5.10 (a) and (b). As shown in this figure, the settling time of the LLC converter with the conventionalmethod, following a load step-up from half to full loading condition is 1.5ms and following a loadstep-down from full to half loading condition is 1.82ms. These results show that the proposed methodis 33 and 45 times faster compared with the conventional method in response to the load step-up andstep-down, respectively.As discussed, the proposed resonant frequency tracking method (employed in the control layer 3)enables the use of synchronous rectifiers in the LLC converter. This significantly reduces the conductionlosses. As shown in Fig. 5.11, the average efficiency of the LLC converter with the proposed method, is96.04%. In addition, a flat efficiency curve is obtained when the proposed resonant frequency trackingmethod is used. In low-voltage-high-current applications, the use of synchronous rectifiers is crucial inorder to achieve high efficiencies.5.4 SummaryIn order to achieve higher efficiency, unregulated LLC converters, often used as DC transformers, needto operate at resonance. Variations in the temperature, frequency, load, and manufacturing tolerancescause the resonant frequency of the LLC converter to deviate. As a result of this deviation, the efficiencyand voltage gain of the unregulated LLC converter are negatively affected. In this chapter, a new res-onant frequency tracking method, based on the homopolarity cycle, was proposed in order to maintainthe LLC converter at resonance under all electrical and environmental conditions. As discussed before,the three-layer control strategy aims to address the following concerns: large and small-signal modelingand control, and degraded efficiency due to conduction losses and deviations in the resonant frequency.The proposed resonant frequency tracking method introduced in this chapter is employed in the controllayer 3 and addresses the deviations in the resonant frequency. The advantages of the proposed methodinclude low analysis complexity, fast convergence time, low-cost voltage-polarity-based sensing tech-nique, synchronous rectification, and high efficiency. These features were all enabled by developingthe theory of the homopolarity cycle in time-domain. The experimental and simulation results of alow-voltage-high-current unregulated LLC converter showed that an average efficiency of 96.04% andconvergence time of 45µs (following load steps) can be achieved when using the proposed resonantfrequency tracking method. In comparison with the conventional resonant frequency tracking method,the proposed method improved the convergence time by 39 times.122Chapter 6Conclusion6.1 Conclusions and ContributionsAlthough resonant converters feature low switching losses and electromagnetic interference (EMI), andhigh power density, conduction losses in their rectifier cause a significant drop in their efficiency, es-pecially in low-voltage-high-current applications. Moreover, resonant converters are such complicatedsystems to be analyzed because they process the electrical energy through a high-frequency resonanttank, causing excessive nonlinearity. The nonlinearity of resonant converters become even severe since,in practice, the resonant frequency of such converters deviates due to variations in the temperature, op-erating frequency, load, and manufacturing tolerances. As a result, large and small-signal modeling andcontrol, synchronous rectification, and resonant frequency tracking of resonant converters have been anactive research topic in the resonant power conversion. In this dissertation, a three-layer control strategy,based on the homopolarity cycle, was introduced to tackle the concerns listed above. The theoreticalanalysis was mainly focused on LLC converters; however, they could be developed for other types ofresonant converters. It was discussed that the complexity of the analysis is reduced when applying thedefinition of the homopolarity cycle to time-domain equations of the resonant converters. The sequenceof the control layers were arranged so as to, first, improve the large-signal dynamic response of res-onant converters by using the average geometric control (AGC), second, support the design of linearcompensators in the quiescent area by using a new small-signal modeling technique, and third improvethe efficiency through synchronous rectification or enable the optimum operating condition through theresonant frequency tracking. The control layers of the proposed three-layer control strategy can all orpartially be used when implementing a resonant converter.6.1.1 Large-Signal Modeling and Geometric ControlThe traditional technique to control power electronic converters for many years has been based on linearcontrollers which use small-signal modeling techniques. Since the small-signal models are just validnear the desired operating point, the converter response to large transients is usually poor; however,it provides a good behavior in steady-state. In Chapter 2, a nonlinear control method, called average123geometric control, was proposed to be used in the control layer 1. The AGC provides a robust and fastdynamic response for the LLC and series resonant converters with the availability of low-cost micro-processors. In order to provide an analytical tool for analyzing the converter large-signal behavior, anaverage large-signal model was introduced enabling the average circular trajectories of the LLC and se-ries resonant converters to obtain the control laws. On average, the experimental results of the proposedgeometric control method showed that the converter response to disturbances, including load and refer-ence voltage changes and also the start-up, is much faster than that of the designed linear proportionalintegral (PI) controller. Moreover, the overshoot after disturbances was virtually eliminated. The ex-perimental results also showed a significant improvement in the start-up dynamic response of the LLCconverter.6.1.2 Average Small-Signal Modeling of LLC ConvertersLLC converters are such complicated systems, and their small-signal modeling is traditionally per-formed using empirical/simulation methods or theory limited in the vicinity of the resonant frequency.Empirical/simulation methods yield limited insight into the dynamic behavior, and theory limited meth-ods suffer from low accuracy below and above resonance. In Chapter 3, a new average small-signalmodeling technique, based on the homopolarity cycle concept, was introduced for LLC converters.Through the analysis of the homopolarity cycle in time-domain, it was shown that the following out-comes are achieved while no simulation, numerical computing, or programming software packages areused: high accuracy, low complexity, and small-signal circuit representation from below to above reso-nant operations. Two small-signal circuit models were obtained, and they were used in the control layer2. These models can be used to design linear compensators to keep the steady-state error close to zero.The control layer 2 is enabled when the LLC converter operates in the quiescent region. In order to vali-date the theoretical analysis, a 650W LLC converter was practically implemented. Both the steady-stateand dynamic operations were justified through experimental and simulation results. The results showedthat the proposed circuit models predict the small-signal dynamic behavior of the LLC converter withhigh accuracy from below to above resonant operations.6.1.3 Synchronous Rectification of LLC Resonant ConvertersFurther improvement in the LLC converter performance requires the use of synchronous rectifiers (SRs).The main challenges in synchronous rectification for LLC converters is to analyze and detect the con-duction angles of the SRs. In Chapter 4, a synchronous rectification method, called homopolarity cyclemodulation (HCM), was proposed for LLC resonant converters. The proposed homopolarity cycle mod-ulation (HCM) was used in the control layer 3. The control layer 3 is enabled when the LLC converteroperates in the quiescent region. A new definition, called homopolarity cycle, was applied to time-domain equations of the LLC converter. The homopolarity cycle significantly reduced the complexityof the LLC converter analysis and related the conduction angles of the SRs to the gate driving signals ofthe inverter switches and polarity of the rectifier voltage. The proposed HCM method uses the advan-tage of a polarity-based sensing technique that reduces cost and the effects of noise. The homopolarity124plane, which can be used to graphically analyze the LLC converter and conduction angles of the SRs,was introduced. The theoretical analysis has resulted in a simple HCM algorithm that controls the SRsused in the LLC converter, and that has a low implementation cost. The proposed HCM method iscompared with a conventional synchronous rectification method and validated by experimental results.The results have shown that the proposed HCM synchronous rectification method has a near maximumsynchronous rectification coverage from light to full loading conditions.6.1.4 Resonant Frequency Tracking of LLC Converters Using Homopolarity CycleIn DC/DC applications where the voltage regulation is not mandatory, the overall cost can be reduced byusing unregulated LLC converters. In order to achieve higher efficiency, unregulated LLC converters,often used as DC transformers, need to operate at resonance. Variations in the temperature, frequency,load, and manufacturing tolerances cause the resonant frequency of the LLC converter to deviate. As aresult of this deviation, the efficiency and voltage gain of the unregulated LLC converter are negativelyaffected. In chapter 5, a new resonant frequency tracking method, based on the homopolarity cycle, wasproposed in order to maintain the LLC converter at resonance under all electrical and environmentalconditions. The proposed resonant frequency tracking method is used in the control layer 3. The controllayer 3 is enabled when the LLC converter operates in the quiescent region. The advantages of theproposed method include low analysis complexity, fast convergence time, low-cost voltage-polarity-based sensing technique, synchronous rectification, and high efficiency. These features were all enabledby developing the theory of the homopolarity cycle in time-domain. The experimental and simulationresults of a low-voltage-high-current unregulated LLC converter showed that an average efficiency of96.04% and convergence time of 45µs (following load steps) can be achieved when using the proposedresonant frequency tracking method.6.1.5 Specific Academic ContributionsThe work, presented in this dissertation, is reported in a patent (under preparation), 4 journal (3 pub-lished and 1 submitted) and 5 published international conference papers. The work encompasses anal-ysis, modeling, control and modulation of series resonant converters (SRCs) and LLC converters, andproposes a three-layer control strategy to cover large and small-signal modeling and control, and con-duction loss reduction through synchronous rectification and resonant frequency tracking.The following publications address the large-signal transient response of the LLC and series resonantconverters in DC-DC applications:• M. Mohammadi, M. Ordonez, “Fast Transient Response of Series Resonant Converters UsingAverage Geometric Control,” IEEE Transactions on Power Electronics, vol. 31, no. 9, pp. 6738-6755, Sep. 2016.• M. Mohammadi, M. Ordonez, “Inrush Current Limit or Extreme Start-Up Response for LLCConverters Using Average Geometric Control,” IEEE Transactions on Power Electronics, vol.33, no. 1, pp. 777-792, Jan. 2018.125• M. Mohammadi, M. Ordonez, “Fast Transient Response of Series Resonant Converter Using anAverage Large Signal Model,” IEEE Energy Conversion Congress and Exposition (ECCE), pp.187-192, Montreal, Canada, Sep. 2015.• M. Mohammadi, and M. Ordonez, “Extreme start-up response of LLC converters using averagegeometric control,” IEEE Energy Conversion Congress and Exposition (ECCE), pp. 1-7, Milwau-kee, WI, USA, 2016.The following paper addresses the small-signal modeling of LLC converters using homopolaritycycle:• M. Mohammadi, F. Degioanni, M. Mahdavi, and M. Ordonez, “Small-Signal Modeling of LLCConverters Using Homopolarity Cycle,” IEEE Transactions on Power Electronics, under review,submitted in Dec. 2018.The following patent and publications address the conduction losses of LLC resonant converters inDC-DC application:• M. Mohammadi, and M. Ordonez, “Synchronous Rectification of LLC Resonant ConvertersUsing Homopolarity Cycle Modulation,” IEEE Transactions on Industrial Electronics, vol. 66,no. 3, pp. 1781-1790, Mar. 2019.• M. Mohammadi, M. Ordonez, P. Ksiazek, R. Khandekar, and N. Shafiei, “Synchronous Recti-fication of LLC Converters Based on The Homopolarity Cycle and Related Methods and Algo-rithms of Conduction Loss Reduction,” US Patent application, Under preparation.• M. Mohammadi, and M. Ordonez, “LLC synchronous rectification using homopolarity cyclemodulation,” IEEE Energy Conversion Congress and Exposition (ECCE), Cincinnati, OH, USA,pp. 3776-3780, 2017.• M. Mohammadi, N. Shafiei, M. Ordonez, “LLC Synchronous Rectification Using CoordinateModulation,” IEEE Applied Power Electronics Conference & Exposition (APEC), pp. 848-853,Long Beach, USA, Mar. 2016.The following paper addresses the resonant frequency tracking of LLC converters in unregulatedDC/DC applications:• M. Mohammadi and M. Ordonez, “Resonant LLC bus conversion using homopolarity widthcontrol,” IEEE Energy Conversion Congress and Exposition (ECCE), Cincinnati, OH, USA, pp.225-229, 2017.1266.2 Future WorkThis dissertation proposed a three-layer control strategy for resonant converters in order to overcomethe following issues:• Uncertain large-signal transient behavior and sluggish dynamic/recovery response• Limited insight and low accuracy in small-signal modeling• Reduced efficiency due to conduction losses in the rectifier• Reduced efficiency due to the deviation in the series resonant frequencyDue to the high demand for more robust and reliable resonant power converters, it is important toperform more research and study on modeling, control, modulation and reliability of resonant powerconverters. Although this work has tackled the above issues by developing novel theoretical and prac-tical approaches for LLC converters, it has opened many full areas of research for Masters and Ph.D.students. As a starting point for future work, some research has been done by the author on the dy-namic performance of synchronous rectification in resonant converters and the modular and multi-phaseresonant power conversion technology. Future research topics are listed below:• Reliability and robustness of synchronous rectification in transients: Although synchronous rec-tification of resonant converters in the quiescent area has been achieved, the operation of SRs intransient is still questionable. In order to increase the synchronous rectification coverage, thisdissertation proposed the use of the homopolarity cycle concept in time-domain equations. Thehomopolarity cycle concept mathematically explains the volt-amp-second balance condition inresonant converters and is valid only if the converter operates within the quiescent area. Out-side the quiescent area, the behavior of the resonant converter cannot be anticipated for syn-chronous rectification. Therefore, it is highly important to study the reliability and robustness ofsynchronous rectification in transients.• Homopolarity cycle concept for other types of resonant converters: As discussed in detail in thisdissertation, to mathematically describe the volt-amp-second balance principle in LLC converters,the homopolarity cycle definition was applied to the converter time-domain equations. By apply-ing the homopolarity cycle definition to time-domain equations, the complexity of the analysisfor LLC converters significantly reduced and based on that large and small-signal modeling andcontrol, synchronous rectification and resonant frequency tracking methods were developed. Al-though the homopolarity cycle was mainly used to analyze LLC converters, theoretically, it can beused to analyze any type of resonant converters. Therefore, there is an open research opportunityto study homopolarity cycle concept in other resonant converters.• Modular resonant converters: In order to provide redundancy and availability in high powerDC/DC applications, modular resonant power conversion is required. The primary challenge127in modular structures is providing current/power sharing. This dissertation developed the theoryof the homopolarity cycle for LLC converters, and there is a potential use of it in modular LLCconverter structures. Through the analysis of the homopolarity cycle, the current sharing can beformulated, and consequently, a new control strategy can be developed.128Bibliography[1] M. Mohammadi, M. Ordonez, “Fast Transient Response of Series Resonant Converters Using Av-erage Geometric Control,” IEEE Transactions on Power Electronics, vol. 31, no. 9, pp. 6738-6755,Sep. 2016. → pages v, vii, 1, 4, 6, 13, 60[2] M. Mohammadi, M. 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Three-layer control strategy for LLC converters : large-signal, small-signal, and steady-state operation Mohammadi, Mehdi 2019
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Title | Three-layer control strategy for LLC converters : large-signal, small-signal, and steady-state operation |
Creator |
Mohammadi, Mehdi |
Publisher | University of British Columbia |
Date Issued | 2019 |
Description | Resonant converters, particularly LLC converters, feature low switching losses and electromagnetic interference (EMI), and high power density and efficiency. As a result, they have been widely used in DC/DC applications. Although LLC converters naturally provide soft switching conditions and therefore, produce relatively less switching losses, conduction losses in their rectifier have remained a barrier to achieving higher efficiencies. Moreover, the analysis of LLC converters is complicated since they process the electrical energy through a high-frequency resonant tank that causes excessive nonlinearity. The issue of this complexity becomes even worse since, in reality, the resonant frequency of such converters deviates due to variations in the temperature, operating frequency, load, and manufacturing tolerances. This complexity has caused: a) limited research on large-signal modeling and control of LLC converters to be performed (this leads to uncertain large-signal transient behavior and sluggish dynamic/recovery response), b) limited insight into small-signal modeling of LLC converters (this often leads to low accuracy), c) unregulated LLC converters not to operate in their optimum operating point (this leads to degraded efficiency and gain), d) conduction losses in the LLC rectifier to remain the main challenge to achieve higher efficiency. To address the above concerns, in this dissertation, a three-layer control strategy is introduced. Based on the need, all the three layers or just one of them can be used when implementing the LLC converter. The three-layer control strategy produces accurate and fast dynamics during start-up, sudden load or reference changes with near zero voltage overshoot in the start-up, obtains a near zero steady-state error by employing a second-order average small-signal model valid below, at, and above resonance, improves efficiency by a new synchronous rectification technique, and also tracks the series resonant frequency in unregulated DC/DC applications. |
Genre |
Thesis/Dissertation |
Type |
Text |
Language | eng |
Date Available | 2019-07-18 |
Provider | Vancouver : University of British Columbia Library |
Rights | Attribution-NonCommercial-NoDerivatives 4.0 International |
DOI | 10.14288/1.0379920 |
URI | http://hdl.handle.net/2429/71048 |
Degree |
Doctor of Philosophy - PhD |
Program |
Electrical and Computer Engineering |
Affiliation |
Applied Science, Faculty of Electrical and Computer Engineering, Department of |
Degree Grantor | University of British Columbia |
GraduationDate | 2019-09 |
Campus |
UBCV |
Scholarly Level | Graduate |
Rights URI | http://creativecommons.org/licenses/by-nc-nd/4.0/ |
AggregatedSourceRepository | DSpace |
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