Applied Science, Faculty of
Electrical and Computer Engineering, Department of
DSpace
UBCV
Galiano Zurbriggen, Ignacio
Doctor of Philosophy - PhD
University of British Columbia
Switch-mode power converters are a fundamental component of modern power systems; they are ubiquitous in renewable energy applications, electric vehicles, battery chargers, and power supplies. Controllers are an essential component in power converters as they improve the converter's dynamic behaviour during transients and in steady-state. For decades, the power electronics industry has preferred to utilize controllers based on converter small-signal analysis due to their low implementation requirements and in spite of their dynamic performance limitations and global stability issues. On the other hand, excellent dynamic response and global stability are achieved by non-linear boundary controllers based on state-plane analysis, which usually have much higher implementation requirements. This thesis focuses on the dynamic performance improvement of power converters by incorporating state-plane concepts while maintaining low implementation requirements to facilitate the large-scale adoption of the technology. By combining traditional averaging modelling tools with state-plane analysis, the unified Average Natural Trajectories (ANTs) are obtained to accurately model the large-signal dynamic behaviour of the fundamental topologies (buck, boost, and buck-boost). As a result, the proposed framework establishes the foundation for several dynamic performance improvement efforts introduced in this thesis. Employing the ANTs as a large-signal model, the theoretical limits of dynamic performance are defined and used to develop a powerful benchmarking tool, providing great value for design engineers. Furthermore, a unified controller based on the ANTs model is developed for the fundamental topologies in this work. This controller features a predictable large-signal response and outstanding dynamic performance while maintaining low implementation requirements. The ANTs modelling approach is also extended to photovoltaic applications to develop an extremely fast maximum power point tracking method for scenarios that include rapidly changing environmental conditions. Finally, the concept of dual-loop geometric control is introduced by combining state-plane analysis with an industry-standard dual-loop control structure, thereby bridging the gap between industrial applications and state-plane controllers. The concepts introduced in this thesis are supported by thorough mathematical analysis and validated by extensive simulation and experimental results. This thesis significantly contributes to the advancement of the field of modelling and control for power electronics.
Large-signal Transient Controlin Power Electronics: anAverage-Geometric FrameworkbyIgnacio Galiano ZurbriggenIng., Universidad Nacional de Co´rdoba, 2010M.A.Sc., The University of British Columbia, 2013A THESIS SUBMITTED IN PARTIAL FULFILLMENT OFTHE REQUIREMENTS FOR THE DEGREE OFDOCTOR OF PHILOSOPHYinThe Faculty of Graduate and Postdoctoral Studies(Electrical & Computer Engineering)THE UNIVERSITY OF BRITISH COLUMBIA(Vancouver)March 2020c© Ignacio Galiano Zurbriggen 2020The following individuals certify that they have read, and recommend to the Faculty ofGraduate and Postdoctoral Studies for acceptance, the dissertation entitled:Large-signal Transient Control in Power Electronics: an Average-Geometric Frameworksubmitted by Ignacio Galiano Zurbriggen in partial fulfilment of the requirements forthe degree of Doctor of Philosophyin Electrical and Computer EngineeringExamining Committee:Martin Ordonez, Professor, Electrical and Computer Engineering, UBCSupervisorJose´ Mart´ı, Professor, Electrical and Computer Engineering, UBCSupervisory Committee MemberJohn Madden, Professor, Electrical and Computer Engineering, UBCUniversity ExaminerRyozo Nagamune, Associate Professor, Mechanical Engineering, UBCUniversity ExaminerYan-Fei Liu, Professor, Electrical and Computer Engineering, Queens UniversityExternal ExaminerAdditional Supervisory Committee Members:William Dunford, Professor, Electrical and Computer Engineering, UBCSupervisory Committee MemberiiAbstractSwitch-mode power converters are a fundamental component of modern power systems; theyare ubiquitous in renewable energy applications, electric vehicles, battery chargers, and powersupplies. Controllers are an essential component in power converters as they improve the con-verter’s dynamic behaviour during transients and in steady-state. For decades, the powerelectronics industry has preferred to utilize controllers based on converter small-signal analy-sis due to their low implementation requirements and in spite of their dynamic performancelimitations and global stability issues. On the other hand, excellent dynamic response andglobal stability are achieved by non-linear boundary controllers based on state-plane analysis,which usually have much higher implementation requirements.This thesis focuses on the dynamic performance improvement of power converters byincorporating state-plane concepts while maintaining low implementation requirements tofacilitate the large-scale adoption of the technology. By combining traditional averagingmodelling tools with state-plane analysis, the unified Average Natural Trajectories (ANTs)are obtained to accurately model the large-signal dynamic behaviour of the fundamentaltopologies (buck, boost, and buck-boost). As a result, the proposed framework establishesthe foundation for several dynamic performance improvement efforts introduced in this thesis.Employing the ANTs as a large-signal model, the theoretical limits of dynamic perfor-mance are defined and used to develop a powerful benchmarking tool, providing great valuefor design engineers. Furthermore, a unified controller based on the ANTs model is developedfor the fundamental topologies in this work. This controller features a predictable large-signaliiAbstractresponse and outstanding dynamic performance while maintaining low implementation re-quirements. The ANTs modelling approach is also extended to photovoltaic applicationsto develop an extremely fast maximum power point tracking method for scenarios that in-clude rapidly changing environmental conditions. Finally, the concept of dual-loop geometriccontrol is introduced by combining state-plane analysis with an industry-standard dual-loopcontrol structure, thereby bridging the gap between industrial applications and state-planecontrollers.The concepts introduced in this thesis are supported by thorough mathematical analysisand validated by extensive simulation and experimental results. This thesis significantlycontributes to the advancement of the field of modelling and control for power electronics.iiiLay SummaryPower converters are an essential component in energy-related applications such as solarand wind power, electric vehicles, and battery chargers. Controllers are usually employed inpower converters to guarantee consistent behaviour under all operating conditions.Simple to implement small-signal controllers have been the preferred choice by the powerelectronics industry during decades despite their performance limitations. On the other hand,a much superior performance can be achieved by using non-linear controllers, which demandhigher implementation requirements.This thesis focuses on the dynamic performance improvement of power converters whilemaintaining controller implementation requirements low, facilitating the large-scale adoptionof the technology. By combining traditional averaging with geometric analysis, the large-signal behaviour of the converters is accurately modelled by geometric trajectories. Based onthese models, simple to implement high-performance controllers are proposed, implemented,and validated under different operating conditions, achieving outstanding dynamic perfor-mance in several applications.ivPrefaceThis work is based on research performed at the Electrical and Computer Engineering de-partment of the University of British Columbia by Ignacio Galiano Zurbriggen, under thesupervision of Dr. Martin Ordonez.Chapters 1 and 9 contain modified portions of text from all below-listed publications, aswell as some modified text from my master’s thesis [1], which included the basic conceptsof Average Natural Trajectories in buck and boost converters, as well as the time-optimalresponse of buck converters:• I. Galiano Zurbriggen, “Novel concepts for power electronics control: introducing thedynamic physical limits, the average natural trajectories, and the centric-based con-troller,” Master’s thesis, University of British Columbia, 2013. [Online]. Available:https://open.library.ubc.ca/collections/ubctheses/24/items/1.0103286Fragments of chapters 2 and 5 are published in IEEE Transactions on Industrial Electronicsand in the IEEE Applied Power Electronics Conference and Exposition (APEC) [2, 3]:• I. Galiano Zurbriggen, M. Ordonez, and M. Anun, “PWM-geometric modelling andcentric control of basic DC-DC topologies for sleek and reliable large-signal response,”IEEE Trans. Ind. Electron., vol. 62, no. 4, pp. 2297–2308, April 2015.• I. Galiano Zurbriggen and M. Ordonez, “Average natural trajectories (ANTs) for boostconverters: Centric-based control,” in Proc. IEEE Appl. Power Electron. Conf. andExpo., March 2014, pp. 1190–1197.vPrefaceParts of chapters 3 and 4 have been been published in IEEE Transactions on IndustrialElectronics and in the IEEE Energy Conversion Congress and Exposition (ECCE) [4, 5]:• I. Galiano Zurbriggen and M. Ordonez, “Benchmarking the performance of boost-derived converters under start-up and load transients,” IEEE Trans. on Ind. Electron.,vol. 63, no. 5, pp. 3125–3136, May 2016.• I. Galiano Zurbriggen, M. Anun, and M. Ordonez, “Dynamic physical limits of boostconverters: A benchmarking tool for transient performance,” in Proc. IEEE EnergyConv. Cong. and Expo., Sept 2014, pp. 4118–4124.Part of the content of this chapter 6 has been published in IEEE Transactions on IndustrialElectronics and in the IEEE Applied Power Electronics Conference and Exposition [6, 7]:• I. Galiano Zurbriggen and M. Ordonez, “PV energy harvesting under extremely fastchanging irradiance: State-plane direct MPPT,” IEEE Trans. Ind. Electron., vol. 66,no. 3, pp. 1852–1861, March 2019.• I. Galiano Zurbriggen, F. Paz, and M. Ordonez, “Direct MPPT control of PWM con-verters for extreme transient PV applications,” in Proc. 2016 IEEE Appl. PowerElectron. Conf. and Expo., March 2016, pp. 386–391.Parts of chapter 7 have been published in the IEEE Applied Power Electronics Conferenceand Exposition and in the IEEE Energy Conversion Congress and Exposition [8, 9]:• I. Galiano Zurbriggen, M. Anun, and M. Ordonez, “Dual-loop geometric-based controlof buck converters,” in Proc. 2015 IEEE Appl. Power Electron. Conf. and Expo.(APEC), March 2015, pp. 392–396.• I. Galiano Zurbriggen, M. Ordonez, and M. A. Bianchi, “Dual-loop geometric-basedcontrol of boost converters,” in Proc. 2015 IEEE Energy Conv. Cong. and Expo.(ECCE), Sep. 2015, pp. 2072–2077.viPrefaceA portion chapter 8 has been published in the IEEE International Symposium on PowerElectronics for Distributed Generation Systems (PEDG) [10]:• I. Galiano Zurbriggen, M. A. Bianchi, and M. Ordonez, “Fast-transient, low-THDgeometric control of boost-derived active rectifiers,” in Proc. 2016 IEEE Int. Symp.on Power Electron. for Distr. Gen. Syst. (PEDG), June 2016, pp. 1–6.As first author of the above-mentioned publications, the author of this thesis developedthe theoretical concepts and wrote the manuscripts, receiving advice and technical supportfrom Dr. Martin Ordonez, and developed simulation and experimental platforms, receivingcontributions from Dr. Ordonez’s research team. In particular, Matias Anun and MarcoAndres Bianchi contributed to developing experimental platforms and firmware for the pub-lications [2, 5, 8] and [9, 10] respectively. Furthermore Francisco Paz participated of insightfultechnical discussions about the existing literature and feasibility of the ideas presented in [7].viiTable of ContentsAbstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iiLay Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ivPreface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . vTable of Contents . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . viiiList of Tables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xivList of Figures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xviAcknowledgements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xxviDedication . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xxvii1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.1 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.2 Literature Review . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51.2.1 Modelling and Control of Fundamental DC-DC Topologies . . . . . . 51.2.2 Dynamic Performance Benchmarking . . . . . . . . . . . . . . . . . . 71.2.3 Maximum Power Point Tracking in Photovoltaic Applications . . . . 101.2.4 Modelling and Control of Single-Phase PFC . . . . . . . . . . . . . . 111.3 Contribution of the Work . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13viiiTable of Contents1.3.1 Large-signal State-plane Modelling of the Fundamental PWM BasedDC-DC Converters . . . . . . . . . . . . . . . . . . . . . . . . . . . . 131.3.2 Dynamic Performance Physical Limits and Benchmarking of Funda-mental Topologies . . . . . . . . . . . . . . . . . . . . . . . . . . . . 151.3.3 Centric Control: High Performance and Simple Implementation . . . 171.3.4 State-Plane MPPT for Photovoltaic Applications Under ExtremelyFast Irradiance Transients . . . . . . . . . . . . . . . . . . . . . . . . 191.3.5 Dual-Loop Geometric Control . . . . . . . . . . . . . . . . . . . . . . 211.4 Dissertation Outline . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 242 Unified Average Natural Trajectories of DC-DC Converters . . . . . . . 272.1 Unified Normalized Average Model Derivation . . . . . . . . . . . . . . . . . 272.1.1 Normalization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 292.1.2 Averaging . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 312.1.3 Unified Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 322.2 Steady State Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 332.3 Transient Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 352.4 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 383 Physical Limits of Dynamic Performance . . . . . . . . . . . . . . . . . . . 403.1 Saturation Trajectories . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 413.1.1 Buck Converter Saturation Trajectories . . . . . . . . . . . . . . . . 413.1.2 Boost Converter Saturation Trajectories . . . . . . . . . . . . . . . . 433.2 Theoretical Limits of Dynamic Performance in Buck Converters . . . . . . . 453.2.1 Start-Up Transient . . . . . . . . . . . . . . . . . . . . . . . . . . . . 463.2.2 Loading Transient . . . . . . . . . . . . . . . . . . . . . . . . . . . . 483.2.3 Unloading Transient . . . . . . . . . . . . . . . . . . . . . . . . . . . 51ixTable of Contents3.2.4 Experimental Validation . . . . . . . . . . . . . . . . . . . . . . . . . 553.3 Theoretical Limits of Dynamic Performance in Boost Converters . . . . . . 583.3.1 Start Up Transient . . . . . . . . . . . . . . . . . . . . . . . . . . . . 583.3.2 Loading Transient . . . . . . . . . . . . . . . . . . . . . . . . . . . . 603.3.3 Unloading Transient . . . . . . . . . . . . . . . . . . . . . . . . . . . 633.3.4 Input voltage step changes . . . . . . . . . . . . . . . . . . . . . . . 663.3.5 Effects of Losses on Theoretical Limits of Performance of Boost Con-verters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 673.3.6 Experimental Validation . . . . . . . . . . . . . . . . . . . . . . . . . 713.4 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 744 Benchmarking Dynamic Performance . . . . . . . . . . . . . . . . . . . . . 754.1 Direct Comparison Approach . . . . . . . . . . . . . . . . . . . . . . . . . . 754.2 Transient-Specific Approach . . . . . . . . . . . . . . . . . . . . . . . . . . . 784.3 Defining Performance Indices . . . . . . . . . . . . . . . . . . . . . . . . . . 794.4 Using the Dynamic Performance Indices . . . . . . . . . . . . . . . . . . . . 824.4.1 Buck Converter Dynamic Performance Benchmarking . . . . . . . . 824.4.2 Boost Converter Dynamic Performance Benchmarking . . . . . . . . 844.5 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 875 Unified Centric Control of DC-DC Converters . . . . . . . . . . . . . . . . 895.1 Target Approaching Technique . . . . . . . . . . . . . . . . . . . . . . . . . 895.2 Complete Control Law . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 945.3 Large Signal Stability Analysis . . . . . . . . . . . . . . . . . . . . . . . . . 975.4 Analysis of Non-Ideal Conditions . . . . . . . . . . . . . . . . . . . . . . . . 1005.4.1 Tolerances . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1005.4.2 Parasitic Resistances . . . . . . . . . . . . . . . . . . . . . . . . . . . 102xTable of Contents5.5 Experimental Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1065.6 Comparison with Dual-loop PI Controller . . . . . . . . . . . . . . . . . . . 1085.7 Centric Controller Benchmarking . . . . . . . . . . . . . . . . . . . . . . . . 1125.8 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1136 State-plane Direct MPPT Control for Solar PV . . . . . . . . . . . . . . . 1156.1 Irradiance Profiles in Typical Moving PV Applications . . . . . . . . . . . . 1156.2 Incremental Conductance MPPT Basis . . . . . . . . . . . . . . . . . . . . . 1166.2.1 Traditional IC MPPT with Direct Duty-cycle Control . . . . . . . . 1196.3 Converter Dynamic Behaviour . . . . . . . . . . . . . . . . . . . . . . . . . 1206.4 State-plane direct MPPT Control . . . . . . . . . . . . . . . . . . . . . . . . 1246.4.1 Main Control Law . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1246.4.2 Compensation of Potential Steady State Error . . . . . . . . . . . . 1266.4.3 Implementation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1276.5 Equilibrium Characterization . . . . . . . . . . . . . . . . . . . . . . . . . . 1276.6 Design Procedure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1326.7 Simulation Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1346.8 Experimental Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1406.9 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1437 Dual-loop Geometric Control of DC-DC converters . . . . . . . . . . . . 1457.1 Dual-loop Geometric Control Principle . . . . . . . . . . . . . . . . . . . . . 1457.2 Target Trajectories . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1467.3 Defining a Transient Response with Geometric Trajectories . . . . . . . . . 1487.3.1 Standard response: Straight Line . . . . . . . . . . . . . . . . . . . . 1487.3.2 Sleek Response: Circular Trajectories . . . . . . . . . . . . . . . . . 1527.3.3 Near Time-Optimal Response: Natural Trajectories . . . . . . . . . . 156xiTable of Contents7.4 Experimental Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1597.4.1 Straight Line Trajectories . . . . . . . . . . . . . . . . . . . . . . . . 1607.4.2 Circular Trajectories . . . . . . . . . . . . . . . . . . . . . . . . . . . 1617.4.3 Natural Trajectories . . . . . . . . . . . . . . . . . . . . . . . . . . . 1627.5 Benchmarking Dual-loop Geometric Controller . . . . . . . . . . . . . . . . 1647.6 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1668 Dual-loop Geometric Control of AC-DC converters . . . . . . . . . . . . . 1678.1 Dual-loop Geometric Control in AC/DC Converters . . . . . . . . . . . . . 1688.2 Steady State: Load Line & Target Operating Trajectory . . . . . . . . . . . 1708.3 Transient: Trajectory Convergence Conditions . . . . . . . . . . . . . . . . . 1728.4 Geometric Control Law . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1748.5 AC-DC Dual-loop Geometric Controller Results . . . . . . . . . . . . . . . . 1758.5.1 Simulation Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1758.5.2 Experimental Results . . . . . . . . . . . . . . . . . . . . . . . . . . 1788.6 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1809 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1829.1 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1829.1.1 Unified ANTs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1829.1.2 Physical Limits of Dynamic Performance . . . . . . . . . . . . . . . 1839.1.3 Dynamic Performance Benchmarking Tool . . . . . . . . . . . . . . . 1849.1.4 Centric Control for DC-DC Converters . . . . . . . . . . . . . . . . . 1849.1.5 State-plane Direct MPPT . . . . . . . . . . . . . . . . . . . . . . . . 1859.1.6 Dual-loop Geometric Control . . . . . . . . . . . . . . . . . . . . . . 1869.2 Specific Academic Contributions . . . . . . . . . . . . . . . . . . . . . . . . 1879.3 Future Work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 191xiiTable of Contents9.3.1 Average Natural Trajectories . . . . . . . . . . . . . . . . . . . . . . 1919.3.2 Dynamic Performance Benchmarking . . . . . . . . . . . . . . . . . . 1929.3.3 Dual-loop Geometric Control . . . . . . . . . . . . . . . . . . . . . . 192Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 194xiiiList of Tables1.1 Some theoretical limits . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82.1 Model Coefficients Definition . . . . . . . . . . . . . . . . . . . . . . . . . . . 332.2 Target Operating Points and Steady State Duty Cycle . . . . . . . . . . . . 353.1 Buck Converter Parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . 553.2 Boost Converter Parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . 713.3 Theoretical limits validation . . . . . . . . . . . . . . . . . . . . . . . . . . . 724.1 Dynamic Performance Benchmarks for Buck Converters . . . . . . . . . . . . 774.2 Buck Converters Dynamic Response Benchmarking . . . . . . . . . . . . . . 834.3 Benchmarked Boost Converters Parameters . . . . . . . . . . . . . . . . . . . 844.4 Boost Converters Benchmarking . . . . . . . . . . . . . . . . . . . . . . . . . 865.1 Global Stability Sufficient Conditions . . . . . . . . . . . . . . . . . . . . . . 995.2 Converter Parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1065.3 Centric Controller Benchmarking . . . . . . . . . . . . . . . . . . . . . . . . 1126.1 Simulation Parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1346.2 Dynamic MPPT Methods Comparison . . . . . . . . . . . . . . . . . . . . . 1397.1 Buck Converter Parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . 1607.2 Boost Converter Parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . 161xivList of Tables7.3 Dual-loop Geometric Control Benchmark . . . . . . . . . . . . . . . . . . . . 1648.1 Simulation Parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1758.2 Experimental Parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 178xvList of Figures1.1 Transient response to sudden load current changes for different current stepsin a boost converter. (a) Boost converter fundamental topology. (b) Unpre-dictable dynamic response obtained employing industry-standard controllers. 31.2 The concepts of average state-plane geometric modelling and control are intro-duced for the fundamental PWM topologies, and then extrapolated to furtherconverters and applications in order to illustrate the potential for expansionof the technology. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41.3 Buck converter Average Natural Trajectories (ANTs). (a) Converter topology.(b) Predictable large signal evolution of the state variables in PWM drivenconverter for different values of duty cycle, shown in time domain and stateplane. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 141.4 Dynamic performance benchmarking of two different boost converters. Thenormalized transients are compared to the ideal MRT and MVD responseto define six performance indices that characterize the dynamic recovery andregulation aspects of the converter’s response. . . . . . . . . . . . . . . . . . 161.5 Centric-based controller transient response in a boost converter compared withindustry-standard linear controller. . . . . . . . . . . . . . . . . . . . . . . . 181.6 Applications with fast irradiance changes (e.g. backpack or car mounted PV)require extreme dynamic performance in order to avoid wrong MPPT decisionsand maximize the energy extracted from the PV cells. . . . . . . . . . . . . . 20xviList of Figures1.7 Dual-loop trajectory-based control scheme. Traditional current loop + geo-metric voltage loop . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 221.8 Dual-loop geometric control scheme. Traditional current loop + geometricvoltage loop . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 232.1 Schematic topologies of the three fundamental DC-DC converters. . . . . . . 282.2 Normalized schematic topologies of the three fundamental DC-DC converters. 302.3 Ideal synchronous buck converter response shown in state plane and time do-mains for d = 0.5, Vccn = 2, and fswn = 10. Non-averaged trajectory and ANTare shown overlapped. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 322.4 State-plane representation of the load line for the three fundamental DC-DCconverters. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 342.5 Average Natural Trajectories in boost converters for several PWM duty cyclevalues (0, d1 = 1/3, d2 = 2/3, and 1) . . . . . . . . . . . . . . . . . . . . . . 362.6 Pseudo-ideal ANT forced in a real synchronous buck converter for d = 0.5,Vccn = 2, and fswn = 10. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 383.1 Buck converter ON and OFF switch states. . . . . . . . . . . . . . . . . . . . 413.2 Buck converter Natural Trajectories for ON (d = 1) and OFF (d = 0) statesof the switch, shown in geometric and time domains. . . . . . . . . . . . . . 423.3 Boost converter ON and OFF switch states. . . . . . . . . . . . . . . . . . . 433.4 Boost converter Natural Trajectories for ON (d = 1) and OFF (d = 0) statesof the switch, shown in geometric and time domains. . . . . . . . . . . . . . 443.5 Theoretical time-optimal start-up transient in a normalized buck converter.The transient is solved employing only ON and OFF saturation trajectories. 46xviiList of Figures3.6 Theoretical time-optimal load step-up transient in a normalized buck con-verter. The transient is solved employing only ON and OFF saturation tra-jectories. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 493.7 Load step-up transient in the normalized buck converter: optimal transientparameters as function of the load step for several normalized input voltages. 503.8 Theoretical time-optimal load step-down transient in a normalized buck con-verter. The transient is solved employing only ON and OFF saturation tra-jectories. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 523.9 Load step-down transient in the normalized buck converter: optimal transientparameters as function of the load step for several normalized input voltages. 533.10 Experimental validation of start-up transient in buck converter with low step-down conversion ratio (Vccn = 1). . . . . . . . . . . . . . . . . . . . . . . . . 563.11 Experimental validation of start-up transient in buck converter with high step-down conversion ratio (Vccn = 10). . . . . . . . . . . . . . . . . . . . . . . . . 563.12 Experimental results for start up, loading and unloading transients with Vccn = 2. 573.13 Ideal start-up transient shown in geometric and time domains. . . . . . . . . 593.14 Minimum normalized settling time as function of Vccn. . . . . . . . . . . . . . 603.15 Ideal loading transient shown in geometric and time domains. . . . . . . . . 613.16 Ideal unloading transient shown in geometric and time domains. . . . . . . . 643.17 Equivalence between step changes in input voltage and load current . . . . . 663.18 Normalized boost converter including parasitic resistances . . . . . . . . . . 683.19 Normalized boost converter trajectories and load line including the effect ofresistive losses for rLn = rCn = [0, 0.01, 0.025, 0.05, 0.1] . . . . . . . . . . . 68xviiiList of Figures3.20 Experimental validation of theoretical limits for start-up transient with (a)Vccn = 0.25, (b) Vccn = 0.5, (c) Vccn = 0.75, and (d) Vccn = 0.875. Prepro-grammed switching actions (open-loop) force the converter to follow minimum-time trajectories, validating the theoretical ideal response. . . . . . . . . . . 723.21 Experimental validation of theoretical limits for load step-up/step-down tran-sients with: (a) Vccn = 0.5, and (b)Vccn = 0.75. Preprogrammed switchingactions (open-loop) force the converter to follow minimum-time trajectories,validating the theoretical ideal response. . . . . . . . . . . . . . . . . . . . . 734.1 Dynamic performance benchmarking of a dual-loop linear controller: a) Start-up response, and b) Load step-up/Step-down. . . . . . . . . . . . . . . . . . 764.2 Benchmarking of the load step-up transient dynamic performance of severalbuck converters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 784.3 Performance indices ranges and categories. . . . . . . . . . . . . . . . . . . . 804.4 Buck converter benchmarking indices for near-time optimal response shown inFig. 3.12, and linear compensator response shown in Fig. 4.1. . . . . . . . . 834.5 Start-up transient response benchmarking example: Converter A . . . . . . . 844.6 Start-up transient response benchmarking example: Converter B . . . . . . . 854.7 Load step up/down transient benchmarking example: Converter A . . . . . . 854.8 Load step up/down transient benchmarking example: Converter B . . . . . . 864.9 Benchmarking indices for two different linear controllers. Converter B showsa balanced response while Converter A shows a rather fast response while thedynamic regulation is sacrificed. . . . . . . . . . . . . . . . . . . . . . . . . . 875.1 Average Natural Trajectories in buck converters for several duty cycle values. 905.2 Unified centric-based control law . . . . . . . . . . . . . . . . . . . . . . . . 91xixList of Figures5.3 Theoretical centric-based control trajectories for several initial conditions in aboost converter. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 925.4 Centric-based trajectories experimental validation for several initial conditionsin a boost converter . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 935.5 Closed-loop centric-based controller conceptual scheme . . . . . . . . . . . . 955.6 Complete centric control-law implementation. The same control process ap-plies to the three topologies under analysis. Topology-specific equations em-ployed for the case of a boost converter are detailed in blue in order to illustratethe low complexity of the implementation. . . . . . . . . . . . . . . . . . . . 965.7 Centric controller dynamic response simulation results for a) buck, b) boost,and c) buck-boost converters . . . . . . . . . . . . . . . . . . . . . . . . . . . 985.8 Centric-controlled buck-boost converter dynamic response simulation resultsfor a LC parameter variation of ±20% . . . . . . . . . . . . . . . . . . . . . 1025.9 Load factor and efficiency as function of load current for several parasitic seriesresistances in the three basic DC-DC converters . . . . . . . . . . . . . . . . 1035.10 Start-up and load current step up/down transient response in a buck converter 1065.11 Start-up and load current step up/down transient response in a boost converter1075.12 Load current step up/down transients in a boost converter compared to thetime-optimal response . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1075.13 Start-up, loading and unloading transient response in the normalized boostconverter with a) dual-loop linear controller, and b) centric controller. . . . . 1085.14 Buck converter transient performance and computation time comparison be-tween a) dual-loop linear controller, and b) centric controller. The duty cycle(C3) and computation flag (C4) signals are zoomed in the time axis in (Z3and Z4 respectively). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1105.15 Benchmarking indices for the centric controller in buck and boost converters. 112xxList of Figures6.1 Irradiance profiles measured in backpack, bike, and car mounted PV. Maxi-mum irradiance slopes identified as 1, 2 and 10 percent per millisecond. . . . 1166.2 Existing MPPT architectures comparison: Dual Stage Stand Alone MPPT vs.Single Stage Direct Duty Cycle MPPT . . . . . . . . . . . . . . . . . . . . . 1176.3 IC MPPT algorithm embedded into the boost converter control law. . . . . . 1186.4 PV voltage increment transient response of Traditional direct duty cycle MPPTin state plane and time domains. . . . . . . . . . . . . . . . . . . . . . . . . 1196.5 PV Norton equivalent circuit connected to the normalized inverted buck con-verter. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1216.6 Inverted buck converter dynamic state-plane behaviour: circular trajectorieswith centre at (d,iPV n), and angular speed ωon = 2pi. . . . . . . . . . . . . . 1236.7 PV voltage increment transient response of proposed direct duty cycle MPPTin state plane and time domains. . . . . . . . . . . . . . . . . . . . . . . . . 1256.8 Complete State-plane direct MPTT control scheme. . . . . . . . . . . . . . . 1286.9 State-plane MPPT controller equilibrium behaviour under no-damping condi-tions (worst case-scenario). . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1296.10 PV-voltage oscillation amplitude as function of the update angle α. . . . . . 1326.11 Simulation comparison of the State-plane direct (light blue - SPD), traditionaldirect duty-cycle (orange - Direct), sliding mode (purple - SMC), and lock-inamplifier (green - LIA) MPPTmethods under start-up and irradiance transientmeasured from car-mounted PV. Variables shown: PV voltage (vPV ), inductorcurrent (iL), PV power (pPV ), and PWM duty cycle (d). As predicted bythe theoretical analysis, the proposed method shows extremely fast dynamictracking performance maintaining MPP operation (dashed) during the entiresimulation time. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 135xxiList of Figures6.12 Comparison of simulated capacitor current for traditional direct duty-cycle(Direct), sliding Mode (SMC), lock-in amplifier (LIA) and State-plane direct(SPD) MPPT methods. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1366.13 Dynamic tracking efficiency (η) as function of irradiance slope (∆S∆t) for State-plane direct MPPT (blue), and traditional direct duty cycle MPPT (orange).Results of EN50530 10% to 50% irradiance change standard test with X1, X10,X100, X1000, and X10000 speed multipliers to reach irradiance slopes foundin moving applications such as backpack, bike, and car mounted PV. . . . . 1376.14 Dynamic tracking efficiency (η) as function of irradiance slope (∆S∆t) for State-plane direct MPPT (blue), and traditional direct duty cycle MPPT (orange).Results of EN50530 30% to 100% irradiance change standard test with X1,X10, X100, X1000, and X10000 speed multipliers to reach irradiance slopesfound in moving applications such as backpack, bike, and car mounted PV. . 1386.15 State-plane direct MPPT hardware implementation. . . . . . . . . . . . . . . 1396.16 Experimental results of the proposed scheme under start-up transient: Maxi-mum Power Point found in only 798µs . . . . . . . . . . . . . . . . . . . . . 1406.17 Experimental results of the proposed scheme operating in equilibrium: reliableoperation and 97.5% tracking efficiency. . . . . . . . . . . . . . . . . . . . . . 1416.18 Experimental results of the proposed scheme under step-up and step-downirradiance transients . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1416.19 Experimental results of the proposed scheme under step-up and step-downirradiance transients: Step-up zoom showing MPP. . . . . . . . . . . . . . . 1426.20 Experimental results of the proposed scheme under step-up and step-downirradiance transients: Step-down zoom showing MPP. . . . . . . . . . . . . . 142xxiiList of Figures7.1 Dual-loop trajectory-based control scheme. Traditional inner current loop plusgeometric outer voltage loop. . . . . . . . . . . . . . . . . . . . . . . . . . . . 1467.2 Geometric controller trajectories: standard (λxLine), sleek (λxCircle), and nearTime-Optimal (λxON,xOFF ) responses. . . . . . . . . . . . . . . . . . . . . . . 1477.3 Buck Converter state plane and time domain response using a dual-loop geometric-based controller with a straight line trajectory as voltage compensator. . . . 1497.4 Boost Converter state plane and time domain response using a dual-loopgeometric-based controller with a straight line trajectory as voltage compensator.1517.5 Buck Converter state plane and time domain response using a dual-loop geometric-based controller with circular trajectories as voltage compensator. . . . . . . 1537.6 Boost Converter state plane and time domain response using a dual-loopgeometric-based controller with circular trajectories as voltage compensator. 1557.7 Buck Converter state plane and time domain response using a dual-loop geometric-based controller with Natural trajectories as voltage compensator. . . . . . . 1577.8 Boost Converter state plane and time domain response using a dual-loopgeometric-based controller with Natural trajectories as voltage compensator. 1597.9 Dual-loop geometric controller start-up, load step-up and load step-down tran-sients experimental results in a buck converter using a straight line as targettrajectory. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1617.10 Dual-loop geometric controller start-up, load step-up and load step-down tran-sients experimental results in a boost converter using a straight line as targettrajectory. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1627.11 Dual-loop geometric controller start-up, load step-up and load step-down tran-sients experimental results in a buck converter using circular target trajectories.1627.12 Dual-loop geometric controller start-up, load step-up and load step-down tran-sients experimental results in a boost converter using circular target trajectories.163xxiiiList of Figures7.13 Dual-loop geometric controller start-up, load step-up and load step-down tran-sients experimental results in a buck converter using ON/OFF saturation (nat-ural) target trajectories. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1637.14 Dual-loop geometric controller start-up, load step-up and load step-down tran-sients experimental results in a boost converter using ON/OFF saturation(natural) target trajectories. . . . . . . . . . . . . . . . . . . . . . . . . . . . 1647.15 Dual-loop geometric controller dynamic performance benchmark for straightline, circular and natural trajectories in buck and boost converters. . . . . . 1658.1 Dual-loop geometric control scheme. Traditional current loop + geometricvoltage loop . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1688.2 Steady state trajectories for boost-derived PFC shown in both time and geo-metric domains. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1698.3 Convergence conditions in a boost-derived PFC shown in state plane and timedomains. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1728.4 Three different Geometric Compensators that satisfy the convergence condi-tions: λxLine, λxQuadratic, and λxCircle. . . . . . . . . . . . . . . . . . . . . . . 1738.5 Implementation block diagram for the straight line geometric compensatorλxLine. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1758.6 Simulation results of the straight-line geometric compensator performance dur-ing start-up transient. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1768.7 Simulation results of the straight-line geometric compensator performance dur-ing an input voltage amplitude step-down transient. . . . . . . . . . . . . . . 1768.8 Simulation results of the straight-line geometric compensator performance dur-ing a load current step-up transient. . . . . . . . . . . . . . . . . . . . . . . . 177xxivList of Figures8.9 Simulation results of the straight-line geometric compensator performance dur-ing a load current step-down transient . . . . . . . . . . . . . . . . . . . . . 1778.10 Experimental results of the straight-line compensator under load current step-up transient (0.8A to 1.8A) at the input voltage peak. . . . . . . . . . . . . . 1788.11 Experimental results of the straight-line compensator under load current step-up transient (0.8A to 1.8A) at the zero crossing level. . . . . . . . . . . . . . 1798.12 Experimental results of the straight-line compensator under load current step-down transient (1.8A to 0.8A) at the input voltage peak. . . . . . . . . . . . 1798.13 Experimental results of the straight-line compensator under load current step-down transient (1.8A to 0.8A) at the zero crossing level. . . . . . . . . . . . 180xxvAcknowledgementsI would like to sincerely thank my research supervisor, Dr. Martin Ordonez, for his continuedsupport, encouragement and valuable advice.I would also like to acknowledge my lab colleagues for kindly sharing their experience andknowledge with me and for making of these years an enjoyable experience.The members of my PhD committee Dr. Jose´ Mart´ı and Dr. William Dunford deserverecognition for their continued encouragement and insightful suggestions on how to improveand extend the work.I feel the need to express my deepest gratitude to my girlfriend Silvita, to my parents Anaand Ernesto, to my brothers Alejandro, Federico, and Santiago, and to my closest friends,for their constant encouragement and moral support without which this work would not havebeen possible.Last but not least, I would like to thank you, the reader, for showing interest in this work.xxviTo those who I will always admirexxviiChapter 1Introduction1.1 MotivationThe ever-increasing demand of higher power densities and higher efficiencies, has turnedpower electronics devices into a key component in modern power conversion systems. Switch-ing power converters are vastly utilized across the entire spectrum of electrical/electronicdevices to make power comply with the desired specifications. These applications range frommass-produced low power converters (milli-Watts) used in portable electronic devices, to veryhigh power (Mega-Watts) applications produced on a small scale for use in distributed powergeneration and distribution. Furthermore, the increasing popularity of renewable energies,electric vehicles and battery chargers will continue to broaden the spectrum of applicationsfor switch mode power converters. In fact, the global power electronics market was valuedat USD 83.46 Billion in 2018 and it is projected to reach USD 140.79 Billion by 2026 [11].Closed-loop controllers play a fundamental role in power conversion systems by ensuringproper regulation of the voltages and currents to the desired values. Controllers guaranteereliable operation under diverse operating scenarios, which may or may not have been takeninto account when the power conversion system was designed. Controllers are the ‘brain’ ofthe system, and the performance of the converters is strongly related to them. Improvementsin the controller can translate into higher efficiency, smaller size, and reduced cost for theentire power conversion system.The dynamic performance is undoubtedly one of the most important aspects when evalu-11.1. Motivationating the performance of power converter control schemes. This aspect becomes of especialinterest when operating conditions change suddenly and a sharp transient is produced. Thedynamic performance of a power converter is characterized by two main parameters: recov-ery time and peak deviations in the state variables (voltage in capacitors and current ininductors). The recovery time is given by the period of time the converter takes to reachsteady state after a change in the operating conditions, for instance a step-up in the load cur-rent. The deviations in the converter state variables during a transient determine how muchthe voltages and currents deviate from the desired steady state values, and the peak valuesreached can determine whether or not the converter is suitable for a given application. Forinstance, an electronic system that requires a 3.3 V power supply might require a recoverytime of 1 ms and a maximum drop of 0.3 V in order to function properly. System failuresdue to large and maintained voltage drops, and undesired saturation in magnetic elementsdue to large current peaks can be avoided by implementing controllers featuring fast andpredictable dynamic response.Implementation-related features, such as robustness, complexity, constant switching fre-quency operation, and sensing/sampling requirements, are very important as well. Thesecharacteristics are critical to determine the controller-application compatibility, and to de-fine the resources required for the controller to be implemented. Industry-standard controllersemployed in large volume applications usually prioritize low implementation complexity overperformance in order to achieve a cost-effective solution. As a result, the dynamic responsemight become unpredictable leading to potential performance-related issues as illustrated inFig. 1.1 for load step transients in a boost converter.This thesis explores methods to improve the dynamic performance of power electronic con-verters while maintaining low implementation requirements by combining industry-standardpractices with geometric modelling and analysis in the state-plane. State-plane analysis hasbeen employed for controlling power converters in very specific high-end applications. Nat-21.1. Motivation12 V 24 V 36 V-5 A05 A10 AvCi L12 V24 V36 VvC-5 A05 A10 Ai L0 4 8 1200.51td4 A3 A2 A∆Io=1 A(b)[ms]dIoVccPWMCiLLvC(a)Boost converterFigure 1.1: Transient response to sudden load current changes for different current steps ina boost converter. (a) Boost converter fundamental topology. (b) Unpredictable dynamicresponse obtained employing industry-standard controllers.urally, in this type of applications the higher controller implementation requirements arejustified by the dynamic performance improvements achieved. The valuable insight providedby geometric state-plane analysis is currently not being employed to benefit most industrialapplications.The main goal of this work is to bridge the gap between high-performance state-plane con-trollers and the low implementation complexity required by large-volume industrial applica-tions. This work aims to generate theoretical insight, models, methodologies and techniquesthat can directly benefit industrial power electronics applications. The advantages of state-31.1. MotivationCORE TECHNOLOGY:FUNDAMENTALTOPOLOGIESDERIVED TOPOLOGIESAPPLICATIONSSTATE-PLANEGEOMETRICMODELLING& CONTROLBuck, Boost,Buck-BoostInverted Buck,PFC,...RenewableEnergies,Grid-tie, ...CORE TECHNOLOGY EXPANSIONFigure 1.2: The concepts of average state-plane geometric modelling and control are intro-duced for the fundamental PWM topologies, and then extrapolated to further converters andapplications in order to illustrate the potential for expansion of the technology.plane analysis, such as large signal validity and fast recovery controllers, are integrated intoestablished technologies and techniques in this work. The objective of this thesis is to intro-duce the basis of state-plane geometric analysis for the fundamental topologies, facilitatingthe later extrapolation to the ample spectrum of potential power conversion applications.This thesis provides an insightful large-signal model of the fundamental DC-DC topolo-gies, which is employed to develop a powerful dynamic performance benchmarking methodand high-performance, simple implementation control alternatives for large-volume applica-tions. As illustrated in Fig. 1.2, the technology is developed first for the three fundamentaltopologies in power electronics: buck, boost, and buck-boost converters. Based on the coretechnology developed for the fundamental topologies, solutions for inverted buck and boost-derived PFC converters to be used in photovoltaic and grid-tie applications are successfullyexplored in this work. These solutions highlight the potential of the proposed state-planebased methodology to benefit multiple advanced applications in power electronics.41.2. Literature Review1.2 Literature ReviewModelling, and control of power converters has been one of the most popular research top-ics since the origins of the power electronics field. Controllers with extreme performances,achieving very reliable behaviour and extremely fast transient recovery have been exploredfor multiple topologies, as detailed in the following sub-sections. However, to date mostcost-sensitive industrial applications employ controllers whose design is based on averagedsmall-signal linear equivalent models, which were introduced in [12] for the fundamentalpower conversion topologies. Small-signal linear control approaches have been extensivelyproven to be useful in the power electronics field and this work introduces several conceptsthat aim to complement the large-signal understanding of the dynamic behaviour of PWMbased converters. The following subsections provide an overview of some of the existing lit-erature in the areas of: modelling and control of fundamental DC-DC topologies, dynamicperformance benchmarking, maximum power point tracking for PV applications, and controlof boost derived power factor correctors.1.2.1 Modelling and Control of Fundamental DC-DC TopologiesLinear techniques are the preferred alternative to control the fundamental DC-DC powerelectronic converters due to their flexible and simple implementation. Fixed frequency PWM-based operation and robust behaviour can be achieved by properly tuning single-loop ordual-loop compensators [13–21]. The averaged nature of the model enables implementationusing low bandwidth sensors and low cost analog or digital controllers. On the other hand,the dynamic performance can be improved only to a limited extent and a high switchingto resonance frequency ratio is required in order to achieve stable operation. Furthermore,since the models are only valid for small signal variations, the large transient performance isusually poor.51.2. Literature ReviewSeveral linear and linear/non-linear mixed alternatives have been proposed to enhancethe dynamic response of basic DC-DC topologies while ensuring a robust behaviour [22–27].Although very interesting results have been obtained, the complexity in the implementationis often increased, and the solutions may not be suitable for cost-sensitive applications.Interesting transient enhancements can be achieved by including the load or capacitorcurrent as an extra input for the controller [28–32]. Nevertheless, since frequency responseis the main design criteria, the time-domain response remains outside of the design scope.Therefore, the state variables peak values can reach undesired values during large transients,leading to failures in the system due to the saturation of magnetic components or largevoltage oscillations.Boundary controllers are an appealing alternative for controlling the fundamental DC-DCconverters based on state plane analysis, in which the control decisions are based purely onthe system operating point. Conceptual boundary controlled systems are bounded stableby nature, as long as the sliding mode existence and invariance conditions are met [33, 34].First order switching surface controllers have been reported, which provide a reliable responseonce the sliding mode is reached. Some of the issues involved in the implementation ofthis technique are: large current spikes before sliding mode is reached, steady state error,chattering and variable switching frequency [35–37]. Successful attempts to solve most ofthese issues have been presented in [38–45]. Since the solution deviates from the ideal concept,unboundedness and steady state error become an issue, while the simplicity of the originalsolution is compromised.The dynamic response can be improved even further by employing Second Order Switch-ing Surfaces (SOSS), while constant switching frequencies can be maintained [46]. As well,the physical limit of dynamic performance is reached when the Natural Switching Surfaces(NSS) are employed in boundary controllers [47–50]. The main advantages of SOSS and NSSare predictable near time-optimal transient solution performing only one switching action61.2. Literature Reviewand fixed frequency steady state operation. However, the state variables must be sampledand processed in a very fast manner in order to ensure constant switching frequency. Further-more, large inductor current peaks as well as capacitor voltage drops are needed to achievenear-optimal dynamic performance in the non-minimum phase converters, and the overallrobustness of the system is conditioned by the reactive components’ tolerances and para-sitic elements. These techniques are an excellent alternative for high-end applications wherethe extra cost for sensors and larger inductor peak currents can be justified by the neartime-optimal response achieved. However, they are not suitable for high volume productionapplications, where reducing the cost of sensors, processors, and inductors is a must. In thisway, existing control methods employed in DC-DC power converters nowadays are unableto provide fast and reliable large-signal transient response (disregarding point of operation)while maintaining complexity and implementation requirements low.This thesis draws into the advantages of combining geometric analysis with PWM-basedimplementation, in the search of achieving high-performance controllers suitable for imple-mentation in low-end microcontrollers and low bandwidth sensing stages.1.2.2 Dynamic Performance BenchmarkingThe search for improvements in performance is present in most of the diverse disciplinesin engineering. In many of them, the theoretical limits are well known and provide a veryuseful reference for the performance benchmarking. Clear examples of these limits are thetheoretical maximum voltage of 1.23V for a single fuel cell (Nernst voltage) [51], the Shockley-Queisser efficiency limit of 33.7% for solar panels [52], the maximum efficiency of 59.3% forwind turbines (Betz law) [53], the Carnot and Diesel cycles as limits of efficiency in steamand Diesel engines [54], and the theoretical maximum voltage and capacity of a Lithium-Ion battery of 4.1V and 100Ah/kg respectively [55], among many others (Table 1.1). Theselimits are essential for the design and use of the mentioned devices because they provide an71.2. Literature ReviewTable 1.1: Some theoretical limitsDEVICE THEORETICAL LIMITWind Turbine 59.3% effic. (Benz law)Fuel Cell 1.23V (Nernst cell voltage)Solar Cell 33.7% efficiency(Shockley-Queisser Limit)Lithium-Ion Battery Cell 4.1V100Ah/kg capacityClass A Audio Amplifier 25% efficiencyBuck converter To/4 [s]high step-down start-up transient(vin/vout →∞)Boost converter To/4 [s]low step-up (Vcc → vref ) start-up transientabsolute performance reference. For instance, the designer of a wind turbine would neverattempt to reach performances higher than the limit given by the Betz law. As opposed to thepractices followed in these disciplines, the theoretical limits are not employed for the dynamicperformance evaluation in power converters. As a result, the improvements introduced bynovel control ideas and hardware modifications to the basic topologies cannot be objectivelyquantified. Furthermore, since power electronic designers do not count with this theoreticalideal reference point, their work can derive in a blind search for improvements aiming forperformances that might not be possible to reach. The limits of dynamic performance forbuck and boost converters are explored in this thesis. The absolute minimum settling timesfor both analysed topologies are detailed in Table 1.1 in order to illustrate the strong referencepoints that will be introduced.81.2. Literature ReviewBuck ConvertersThe dynamic performance improvement of buck converters and their derived topologies,either by implementing new control techniques [32, 40, 46, 56–63] or by introducing modifi-cations to the topology [64–69] has motivated extensive research.Transient analysis of buck converters has been performed in the past using small-signalmodelling [70]. Due to constraints of small-signal analysis, the results do not provide anyinformation about performance limits. Critical parameters on the transient response andtheir effects have been also analysed [71, 72]. However, the theoretical minimum transienttimes are not specifically addressed. Based on small-signal analysis, an effort to establish abenchmark system using a standard 20W converter design was proposed [73], but the physi-cal limits of the system are not discussed. Later interesting work explored with simulationsand experimental results the limits of dynamic performance [49, 50, 74–76] in buck-derivedtopologies, presenting advancements to the field. However, the derivations and general ex-pressions to determine the theoretical physical limits remained unaddressed and without aconcrete solution. In fact, rather than being common knowledge, the theoretical transientlimits of buck converters remain unknown to technical experts, power designers, the powerconversion community in general.Boost ConvertersDue to its non-minimum phase behaviour, boost-derived topologies present a special challengefor the designers of controllers for power electronics. Interesting control ideas have beenimplemented in order to improve the transient response of boost topologies [21, 24, 38, 77–83]. Other approaches focused on introducing hardware modifications to the basic boosttopology in order to eliminate the Right Half Plane Zero (RHPZ), facilitating the design ofcontrollers [84–86]. Although the dynamic behaviour has been successfully improved, thelack of a standard reference makes the improvements impossible to quantify.91.2. Literature ReviewThe performance limits of boost-derived power factor correctors were studied in [87]. How-ever, the efforts were only focused on the converter capability to track a sine-wave currentreference. A suite of examples of good traditional designs for 20W converters was presentedin [73], but the limits of performance were not included as part of the proposed benchmarkprocedure. The dynamic performance of DC-DC boost converters was studied in [88]. Dueto limitations of small-signal based analysis, the theoretical limits of dynamic response re-mained unaddressed. The minimum recovery time controller was conceptually introducedin [89]. Nevertheless, the practical implementation of the controller was limited by theprocessing power available at the time. More recent works introduced controllers showingtime-optimal [48, 76], and minimum voltage deviation [57] transient responses. Although thelimits of dynamic performance were explored, expressions that describe them as functions ofthe system parameters have not been presented.1.2.3 Maximum Power Point Tracking in PhotovoltaicApplicationsPhotovoltaic (PV) cells are one of the most popular renewable energy sources with a Com-pound Annual Growth Rate (CAGR) of 24% from 2000 to 2017 [90]. Growth is expectedto continue, resulting in a cumulative global installed capacity of 1,026 GW by 2022 [91].Battery chargers and grid-tied applications are amongst the most common targets for PVapplications. However, the popularity of alternative applications, such as manned [92, 93]and unmanned vehicles [94–97], and wearable technology [98, 99], is increasing. MaximumPower Point Tracking (MPPT) algorithms must be employed in order to maximize the powertransfer from the solar panel to the load. The MPPT algorithm must be implemented atthe first stage of the energy conversion chain, where DC-DC converters are undoubtedly themost popular choice.101.2. Literature ReviewMPPT techniques with degrees of accuracy and complexity have been proposed [100, 101].Hill-climbing techniques, such as Incremental Conductance (IC) and Perturb and Observe(P&O), are a very popular alternative due to their flexibility and balance between complexityand accuracy. Very high tracking efficiency (fraction of electrical energy absorbed as percent-age of the total available) can be achieved by employing these algorithms in static applicationswhere irradiance and temperature change slowly. However, the traditional implementationof these algorithms under rapidly changing environmental conditions is especially prone toerrors, leading to lower extraction efficiencies [102, 103].Hill-climbing methods are commonly implemented using direct duty cycle control or multi-loop alternatives. Direct duty cycle MPPT algorithms show simple implementation while theupdating frequency is limited by lengthy oscillations occurring before reaching the new steadystate [102–106]. Lower settling times can be achieved with properly designed linear or non-linear (e.g. sliding-mode) controllers in a multi-loop approach [107–112], enabling higherdynamic tracking performance only to a limited extent, while increasing the implementationcomplexity. In both cases, the MPPT algorithm must wait for the converter to reach thesteady state before taking a new action, leading to limited dynamic tracking performance evenwhen the algorithms are optimized. A single-loop, non-perturbative lock-in-amplifier (LIA)based solution was proposed in [113], showing significant dynamic performance improvementsat the expense of higher implementation complexity. Hill-climbing methods have also beencomplemented with methods to solve well-known issues for PV panels such as partial shading[114, 115] and thermal stress [116].1.2.4 Modelling and Control of Single-Phase PFCActive rectifiers with Power Factor Correction (PFCs) are an essential element in today’spower electronics scene. The ever increasing demand of battery chargers for portable elec-tronics, electric vehicles, and power back-up systems, the growing popularity of distributed111.2. Literature Reviewsources of renewable energy producing electrical energy of AC and DC types, as well as theemergence of DC microgrids into the mainstream of power systems, require of reliable waysof interacting with the existing AC distribution systems. Efficient and reliable AC/DC con-verters with high power quality and good dynamic performance are a key element for theproper incorporation of these new elements into the currently existent power systems layout.The control of PFCs shows special challenges and it has been the object of extensiveresearch throughout recent decades. Two main control objectives can be identified in theseAC/DC conversion systems: to achieve high quality sinusoidal input current, and to obtain areliable DC voltage at the output. These two objectives are usually achieved by implementinga dual-loop scheme in which the inner-loop controls the sinusoidal input current and theouter-loop ensures the DC output voltage is maintained within the desired operating range.Due to the sinusoidal nature of the input voltage and current, the power absorbed by activerectifiers from the grid fluctuates during the sine wave cycle. As a consequence of this powervariability, the energy stored in the output capacitors changes, which implies output voltagevariations of twice the line frequency. When traditional dual-loop linear control techniquesare employed, the control of the output voltage and its ripple can cause distortion in theinput current. The two standard approaches taken to avoid this issue are to increase theoutput capacitance (lowering the ripple), and/or lowering the bandwidth of the voltage-loopcontroller (reducing the output voltage quality). Traditional solutions generate a trade-offbetween capacitor volume/cost and voltage loop performance. These trade-off conditionshave been thoroughly analysed in [117–120]. As a result, the voltage control loop bandwidthis usually set to low values resulting in sluggish dynamics.Several interesting alternatives that aim to improve the dynamic response employing lin-ear control techniques have been reported in [121–124]. These solutions mainly focus onminimizing the effect of the capacitor voltage ripple by either filtering it or cancelling it withestimated values. These approaches provide significant improvements to the voltage loop121.3. Contribution of the Workdynamics. However, it takes several line-frequency cycles for the output voltage to reach thedesired value after sudden load transients.Non-linear boundary controllers can achieve extraordinary transient performances [125–129], which approach the physical limits of performance. However, since the switching actionsare based on the status of the state variables (no PWM), very high sample frequencies arerequired and achieving constant switching frequency is not straightforward.1.3 Contribution of the WorkThis work makes significant contributions to the areas of modelling, characterization andcontrol in the power electronics field. The main contribution of this thesis fulfils the objectiveof bringing the understanding and implementation of high-performance state plane controllerscloser to the requirements posed by large-scale industrial applications, resulting in significantdynamic performance improvements. The specific aspects in which this work helps advancingthe state of the art of the technology are detailed in the following subsections.1.3.1 Large-signal State-plane Modelling of the FundamentalPWM Based DC-DC ConvertersOne of the most important contributions of this work is an alternative framework to un-derstand the behaviour of the fundamental power electronics converters. As opposed to thetraditional frequency response based methods employed to describe the small-signal dynamicresponse of the converters, the large-signal state-plane based model proposed in this workaccurately describes the behaviour of buck, boost and buck-boost converters for any valuethe state variables and PWM duty cycle may take, as illustrated in Fig. 1.3.The unified Average Natural Trajectories (ANTs) introduced in this thesis model thenatural evolution of the averaged state variables in a PWM converter on the state-plane131.3. Contribution of the WorkiL initial conditionsvCIo¸ON¸d1 ¸d2¸OFFvC (0)iL (0)t10.50vCvC (0)ToiLiL (0)t10.5 ToIodONd2d1 OFF t10.5 To(b)iLIoVccd PWMCLvCBuck converter(a)ONOFFFigure 1.3: Buck converter Average Natural Trajectories (ANTs). (a) Converter topology.(b) Predictable large signal evolution of the state variables in PWM driven converter fordifferent values of duty cycle, shown in time domain and state plane.formed by capacitor voltage and inductor current. These trajectories are circles that representthe paths the converter operating point follows in the state-plane for a given PWM dutycycle, providing an accurate model of the converters large-signal behaviour. The analysis141.3. Contribution of the Workis performed in the normalized domain, and therefore the model obtained is valid for anycombination of L-C parameters as well as input/output voltage levels.Furthermore, the unified ANTs for the fundamental topologies set the basis to develop themodelling tool across the diverse existing and future topologies in the power electronics field.In this way, the unified ANTs, and the methodology followed to determine them, show thepotential of complementing the way in which all industrial power electronics control systemsare designed and implemented.1.3.2 Dynamic Performance Physical Limits and Benchmarkingof Fundamental TopologiesThis thesis includes significant contributions to dynamic performance characterization in thefield of power electronics. The physical limits of dynamic performance of buck and boostconverters are explored and close-form expressions that characterize the ideal theoreticaltransient response are derived. Furthermore, a practical tool to use these physical limits ofperformance as a controller benchmarking tool is developed in this thesis. This powerfultool finds its cornerstone in the state-plane geometric analysis of the converters’ ANTs forsaturations of the PWM duty cycle (or simply ON and OFF natural trajectories).Six performance indices are defined based on the theoretical ideal responses, providingan absolute dynamic performance evaluation. These indices provide deep insight into theconverters’ dynamic behaviour, enabling thorough and quantitative performance character-ization. Since the theoretical ideal performance cannot be overtaken, the proposed index-based benchmarking method provides the unique feature of clearly indicating if, and to whichextent, the dynamic behaviour of the converters can be improved. The proposed indices char-acterize the dynamic recovery time regulation under three different transients: start-up, loadstep-up and load step-down. The performance benchmarking indices are defined in a scale151.3. Contribution of the WorkConverterBConverterAVcc Avref AvCALACAIstepAController ASensedvariablesONOFFVcc Bvref BvCBLBCBIstepBController BSensedvariablesONOFFTheoretical Limits of Dynamic PerformanceNormalizationvref A,Z0A,!0Avref B,Z0B,!0BvCAntn0.250.010.86vCBntn0.173.6vCntn0.140.34Minimum Recovery Time (MRT)Minimum Voltage Deviation (MVD)Normalized response analysis Dynamic Performance BenchmarkingRTiLSTi SOiDRiLRTiU DRiURTiLSTi SOiDRiLRTiU DRiUConverter A Converter BTimeIndicesRegulationIndicesTimeIndicesRegulationIndicesFigure 1.4: Dynamic performance benchmarking of two different boost converters. Thenormalized transients are compared to the ideal MRT and MVD response to define sixperformance indices that characterize the dynamic recovery and regulation aspects of theconverter’s response.that ranges from zero (for extremely poor performance) to one (for the theoretical perfor-mance limit). The analysis is performed in a normalized domain, and therefore the resultsare valid for any combination of reactive components and voltage levels. In this way, the161.3. Contribution of the Workdynamic performance of the converters can be quantitatively evaluated in terms of severalcritical parameters.The proposed benchmarking procedure is illustrated in Fig. 1.4. The load step-up tran-sient responses provided by converters A and B are normalized and compared against the twotheoretical ideal responses: Minimum Recovery Time (MRT) and Minimum Voltage Devia-tion (MVD). As a result of the comparison, two out of the six dynamic performance indicesdefined in this thesis can be calculated.In this way, the theoretical limit of performance reference point and the benchmarkingprocedure included in this thesis enable an objective quantification of the performance im-provements introduced by modifications to the controllers. The proposed method is usedalong this thesis to verify the outstanding performance of the proposed controllers and tocompare them with industry-standard solutions.1.3.3 Centric Control: High Performance and SimpleImplementationAnother contribution of this work is the centric-based controller: a novel and unified geometric-PWM strategy to control buck, boost and buck-boost topologies. The technique combinesthe implementation benefits of fixed-frequency PWM and the fast and reliable dynamics ofgeometrical controllers. As in linear controllers, the switching actions are driven by a PWMmodule, but in this case the circular trajectories in the state-plane dictate the duty cyclevalues.The first objective of the proposed control technique is to achieve a reliable and consis-tent circular large signal transient response, therefore the control decisions are based on thelocation of the variables in the state plane shown in Fig. 1.5. It is worthwhile highlightingthat all transients are resolved following predictable circular patterns. The second objective171.3. Contribution of the WorkiLtvCiLtdtvCTargetoperatingpointsEnhanced dynamic performanceIndustry-standard linear controllerProposed centric controllerPredictabletransientresponseFigure 1.5: Centric-based controller transient response in a boost converter compared withindustry-standard linear controller.is to avoid large current and voltage spikes in order to improve the dynamic regulation andto obtain fast and sleek trajectories. As shown in Fig. 1.5, the proposed controller exhibitsa smooth response with reduced current and voltage peak deviations from the target values.The third objective of the technique is to maintain the simple implementation characteristicof linear techniques. As illustrated in Fig. 1.5, the proposed control strategy is based inthe ANTs, which enables the implementation using fixed-frequency PWM operation, lowbandwidth sensors and low-cost digital processors. Based on the ANTs model, the controlactions are performed setting the PWM duty cycle to the value that directs the operating181.3. Contribution of the Workpoint to the target following a consistent and reliable circular path. Saturations on the dutycycle are performed only to improve the dynamic regulation during load transients and theyare rapidly removed, avoiding large spikes in the inductor current and capacitor voltage.The derivation of the control laws to implement the centric-based controller is thoroughlydescribed in this thesis. Experimental results of the large-transient target approaching prin-ciple are shown in order to validate the large signal model presented. Closed-loop results ofthe response under different transients validate the fast, sleek, and reliable performance of theproposed control scheme and highlight the practical usefulness of the introduced concepts.Furthermore, the six-index benchmarking method introduced in this work is employed toillustrate the superior performance of the centric controller when compared with industry-standard methods.1.3.4 State-Plane MPPT for Photovoltaic Applications UnderExtremely Fast Irradiance TransientsA state-plane based MPPT method for PV applications is developed in this thesis to highlightthe potential of geometric analysis beyond the fundamental DC-DC topologies. The ANTsare developed for an inverted buck converter, which is a very common topology employed inDC-DC PV applications. Based on this large-signal model, a fast and reliable MPPT methodis developed for applications that demand extremely fast dynamic performance.Examples of rapidly changing environmental conditions are shown in Fig. 1.6. The fastirradiance transients caused by the sunshine-to-shade sudden transient found in moving PVapplications lead to wrong decisions in the traditional hill-climbing method (in red), takingthe operating point away from the MPP. In contrast, the state-plane direct MPPT (in blue)shows a much superior dynamic tracking performance enabling significant energy gains duringirradiance transients.191.3. Contribution of the WorkAvailable PowerTraditionaldirect MPPTEnergy gainState-planedirect MPPTPWMdutycycleIrradiance100%50%Traditional direct duty cycle MPPT wrong decisionsVery sharpirradiance changesFigure 1.6: Applications with fast irradiance changes (e.g. backpack or car mounted PV)require extreme dynamic performance in order to avoid wrong MPPT decisions and maximizethe energy extracted from the PV cells.This thesis proposes a novel theoretical framework based on state-plane modelling of thepower converter. The analysis results in a new method of operating direct MPPT algorithmsto enable extreme dynamic tracking improvements while maintaining simple implementationwith no requirement of temperature or irradiance sensing. By employing state-plane analysisof the dynamic behaviour of the converter, the necessity of reaching steady-state beforeperforming a new MPPT iteration is eliminated. In this way, no fixed set-point is requiredand the control law continuously leads the operating point to the MPP. A dynamic solutionis proposed to solve the dynamic MPPT problem.The analysis included in this work focuses on single PV cell type of I-V curve. However,201.3. Contribution of the Worksince the method being proposed is based on hill-climbing principles, it is possible to combinethe algorithm with currently existing solutions to solve issues like partial shading and thermalstress relief. In this way, the state-plane analysis applied to MPPT algorithms opens awhole new set of research opportunities. The theoretical findings are supported by detailedmathematical procedures and validated by simulations and experimental results.This thesis includes detailed theoretical analysis including the effect of losses and PVchanges in incremental conductance, a step-by-step design procedure, enhanced implemen-tation descriptions, data of irradiance profiles, comparison with traditional direct duty cycleMPPT, dynamic testing following the EN50530 standard, comparison with popular and highperformance methods found in the literature, simulation and experimental results.The application of state-plane analysis to MPPT for PV applications is employed in thisthesis to illustrate how the methodology introduced here can be extended to renewable energyapplications yielding outstanding dynamic performance improvements while maintaining lowimplementation requirements.1.3.5 Dual-Loop Geometric ControlDual-loop controllers are very common in industrial applications due to their improved dy-namic characteristics and higher stability margins. Linear controllers are traditionally em-ployed for both current control (inner-loop), and voltage control (outer-loop). This thesisintroduces the concept of dual-loop geometric control in which the benefits of dual-loop con-trol are combined with the fast and predictable response of controllers based on state-planeanalysis. Non-linear solutions such as peak mode, or hysteresis mode are commonly used inindustry to control current, but the outer voltage loop is usually solved through linear meth-ods. The dual-loop geometric controller proposes employing geometric analysis to design ageometric-based voltage control loop, while maintaining PWM-based current control in theinner loop. As a result, very fast, reliable and well defined dynamic responses are obtained211.3. Contribution of the WorkLnCnVccnvCnionCiLn PWMiLn1iLn*CurrentLoopGeometricVoltageControllerFigure 1.7: Dual-loop trajectory-based control scheme. Traditional current loop + geometricvoltage loopeven for large signal transients (similarly to boundary controllers). At the same time, theaverage nature of the proposed technique makes its implementation straightforward, placinglow demands on sensing stages and digital processors. Alternatively, the geometric voltagecontrol loop can be combined with any of the existing current control methods (i.e. peakmode, hysteresis mode, average PI, average PR, etc.), at the expense of loosing the fixed-frequency PWM operation.The solution is developed for buck and boost DC-DC converters, and for boost-derived AC-DC converters. Even for the non-minimum phase systems, the dual-loop geometric controllermaintains its very high performance. Since the analysis and design are performed in thenormalized domain, the results are valid for any combination of L-C parameters as well asinput and output voltages.The dual-loop geometric control scheme for buck converters is shown in Fig. 1.7, a similarstructure is employed for the case of boost converters. Based on the output voltage error andload current, the geometric voltage controller defines the reference current the inner currentloop must track in order to reach the target operating point following the desired path. Inthis way, not only a target steady state operating point is defined, but also target trajecto-ries are defined to solve transients following a reliable path. Since the voltage loop design221.3. Contribution of the WorkLnCnPWMMAFPRiLnvgnPR Current LoopBoost-Derived PFCionGeometricVoltageCompensatoriLn*vCnvCntFigure 1.8: Dual-loop geometric control scheme. Traditional current loop + geometric voltageloopis based on geometric analysis instead of the traditional frequency response, non-minimumphase behaviours no longer represent an issue for the voltage control design. Several trajec-tories and their time-domain response are analysed in order to illustrate the advantages andimplementation requirements. It is shown in this work that straight line reference trajectoriescan provide a reliable solution for applications with limited bandwidth in the current loop,while circular trajectories can provide a much faster response, reaching near time-optimalrecovery times. The time-domain dynamic response obtained with each one of the proposedtarget trajectories are described by mathematical expressions, and validated by experimentalresults carried out in experimental prototypes for buck and boost converters.In the case of PFCs, the analysis changes since in this case the state variables are desired tochange in a specific way defining a target operating trajectory in the state-plane. The PWM-based dual-loop geometric controller for PFCs is shown in Fig. 1.8, where the sinusoidalcurrent reference is generated based on a desired target trajectory. The proposed methodis validated by simulation and experimental results confirming the predicted outstandingdynamic performance of the dual-loop geometric compensator for boost-derived PFCs.In this way, the concept of dual-loop geometric control brings the benefits of state-plane231.4. Dissertation Outlineanalysis closer to solutions currently being employed in the power electronics industry. Fur-thermore, the application of the dual-loop geometric controller is extended to boost-derivedPFCs in order to illustrate how the concept can be extrapolated to other topologies, regard-less of their AC or DC nature, in a simple and straightforward manner.1.4 Dissertation OutlineThe organization of this thesis is described below:• In Chapter 2, the unified Average Natural Trajectories (ANTs) are introduced for thefundamental DC-DC topologies (buck, boost and buck-boost) as a large signal modeldeveloped in the state-plane formed by the normalized capacitor voltage and normalizedinductor current. The methodology followed to obtain the ANTs comprises: normal-ization, averaging, unification, steady state analysis, and transient analysis to obtain alarge signal model given by circular shapes that describe the converters’ behaviour .• In Chapter 3 the theoretical limits of dynamic performance of buck and boost con-verters are explored for start-up, load step-up and load step-down transients. Thenatural trajectories are obtained by evaluating the ANTs for saturations of the PWMduty cycle. These ON and OFF saturation trajectories are employed to determine theMinimum Recovery Time (MRT) and Minimum Voltage Deviation (MVD) responsesto solve the transients optimally, setting a strong reference point.• In Chapter 4 a dynamic performance benchmarking method is introduced using theMRT and MVD responses as a reference point. Several alternatives to use the theoreti-cal limits as a reference point for the dynamic performance benchmarking are exploredincluding a direct comparison and a transient-specific methods. Finally, the six-index241.4. Dissertation Outlinemethod is defined to enable the objective and visual performance comparison of multi-ple controllers.• Chapter 5 introduces the unified centric control for the fundamental DC-DC convertersas an application of the unified ANTs. A closed-form equation that determines theduty PWM cycle required to place the centre of the circular trajectories in the locationthat leads the operating point to the target is derived to be used as target approach-ing technique. This target approaching technique is complemented with saturationregions and steady state regions that compose the unified centric control technique.The proposed control method is tested in simulations and experimental platforms un-der multiple challenging conditions and scenarios, compared with industry-standardtechniques, and assessed employing the dynamic performance benchmarking methodintroduced in this work yielding outstanding results.• The state-plane direct MPPT method for photovoltaic applications is introduced inChapter 6 to obtain extreme improvements in maximum power point tracking speed.Irradiance profiles of moving PV applications are detailed, and the need of very fastMPPT tracking is identified. The ANTs for a PV connected inverted buck topologyare derived, and the MPPT-based geometric control law is defined. A detailed designprocedure example is provided, and the proposed MPPT method is tested under sev-eral operating scenarios in simulations and experimental setup validating the fast andreliable performance.• Chapter 7 introduces the concept of dual-loop geometric control by merging PWM-based current control with a voltage compensator based on state-plane analysis. Targettrajectories are defined and analysed to solve the transient in a predictable fashion. Suf-ficient convergence conditions are found, and several trajectories that satisfy them areanalysed. The dynamic characteristics of straight-line, circular and near time optimal251.4. Dissertation Outlinetrajectories are obtained and validated through simulation and experimental results forbuck and boost converters. Finally, the dynamic performance obtained is characterizedby using the six-index method confirming the outstanding performance obtained.• The dual-loop controller is extended to boost-derived AC-DC converters in Chapter 8.A moving current setpoint is generated by the geometric voltage compensator accordingto a defined steady-state target operating trajectory. The convergence conditions for thetransient trajectories are defined and the dynamics defined by a straight line trajectoryare analysed. The high performance of the controller is validated by simulation andexperimental results under start-up and several load transients.26Chapter 2Unified Average Natural Trajectoriesof DC-DC ConvertersA unified geometric model that describes the large signal normalized averaged behaviour ofthe three basic DC-DC topologies (buck, boost and buck-boost) is developed in this chapter.The model being introduced and the methodology employed to obtain it set the basis fornumerous applications, some of which are explored throughout this thesis.Some of the content of this chapter is included in a journal paper published in IEEETransactions on Industrial Electronics [2].2.1 Unified Normalized Average Model DerivationThe behaviour of the three basic topologies shown in Fig. 2.1 is ruled by the followingdifferential equations for ON and OFF states of the switch:BuckONLdiLdt= Vcc − vC (2.1)CdvCdt= iL − Io (2.2)OFFLdiLdt= −vC (2.3)CdvCdt= iL − Io (2.4)BoostLdiLdt= Vcc (2.5)CdvCdt= −Io (2.6)LdiLdt= Vcc − vC (2.7)CdvCdt= iL − Io (2.8)272.1. Unified Normalized Average Model DerivationdIoVccPWMCiLLvCON OFFiLIoVccdPWMCLvCONOFFdIoVccPWMCiLLvCONOFFBuckBoostBuck-BoostFigure 2.1: Schematic topologies of the three fundamental DC-DC converters.Buck-boostONLdiLdt= Vcc (2.9)CdvCdt= Io (2.10)OFFLdiLdt= vC (2.11)CdvCdt= Io − iL (2.12)As shown by the three sets of differential equations, the dynamic behaviour of the basicDC-DC converters shown in Fig. 2.1 is highly dependent on the values of the reactive elementsL and C as well as the input and output voltage levels. As a consequence, controllers designedfollowing traditional practices are specific to the converter parameters.282.1. Unified Normalized Average Model Derivation2.1.1 NormalizationIn order to gain generality and to eliminate the dependence of the controller on the converter’sreactive components and voltage levels, the differential equations that characterize the be-haviour of the power converters are taken to a normalized domain where time, impedances,voltages and currents are given by:tn =tT0; Zxn =ZxZ0; vxn =vx| vref | ; ixn =ixirefand the employed base quantities are the filter characteristic impedance (Z0) and resonanceperiod (T0), as well as the desired output voltage (vref ) and the reference current (iref ):Z0 =√L/C ; T0 = 2pi√LC; vref ; iref =| vref |Z0The differential equations obtained as a result of the normalization procedure characterizethe behaviour of the normalized converters in ON and OFF states:BuckON12pidiLndtn= Vccn − vCn (2.13)12pidvCndtn= iLn − Ion (2.14)OFF12pidiLndtn= −vCn (2.15)12pidvCndtn= iLn − Ion (2.16)Boost12pidiLndtn= Vccn (2.17)12pidvCndtn= −Ion (2.18)12pidiLndtn= Vccn − vCn (2.19)12pidvCndtn= iLn − Ion (2.20)Buck-boost12pidiLndtn= Vccn (2.21)12pidvCndtn= Ion (2.22)12pidiLndtn= vCn (2.23)12pidvCndtn= Ion − iLn (2.24)292.1. Unified Normalized Average Model DerivationiLnIonVccndBuckPWMvCn1/(2pi)1/(2pi)dIonVccnBoostPWMiLn1/(2pi)vCn1/(2pi)dIonVccnBuck-BoostPWMiLn1/(2pi)vCn1/(2pi)Figure 2.2: Normalized schematic topologies of the three fundamental DC-DC converters.The resulting normalized differential equations describe the behaviour of the convertersshown in Fig. 2.2, which are independent from the L-C parameters and input-output voltagelevels. Instead, the normalized reactive elements become 12pi, and the voltage levels becomerelative to the desired reference.The practical implementation of this normalization procedure demands only extra gainson the sampled voltages and currents and a proper normalized switching frequency. Thisnormalization procedure has been successfully employed in several scenarios including theimplementation of universal linear controllers for DC-DC converters [130].302.1. Unified Normalized Average Model Derivation2.1.2 AveragingThe normalized differential equations obtained in the previous section detail the behaviour ofthe converters in ON and OFF states. Since the model being proposed targets PWM-basedapplications, the two states can be averaged based on the converter’s duty cycle. Employingtraditional averaging techniques [12], the differential equations that represent the averagedbehaviour of the three normalized PWM-driven converters can be obtained:BuckAVG12pidiLndtn= d Vccn − vCn (2.25)12pidvCndtn= iLn − Ion (2.26)Boost12pidiLndtn= Vccn − d′ vCn (2.27)12pidvCndtn= d′ iLn − Ion (2.28)Buck-boost12pidiLndtn= d Vccn + d′ vCn (2.29)12pidvCndtn= −d′ iLn + Ion (2.30)The averaging procedure is illustrated in Fig. 2.3, where the non-averaged path the statevariables describe in a fixed-duty-cycle buck converter is shown along with the Average Natu-ral Trajectory (λAV G). As illustrated in the figure, the ON (λ1ON,2ON) and OFF (λ1OFF,2OFF )trajectories describe constant angles (α and β) corresponding to the ON and OFF states ofthe switch.312.1. Unified Normalized Average Model Derivation ¯2¼®2¼iLn0tn1tn0drivingsignal1tn0vCn11Vccn®¯¸1ON¸1OFF¸2ON¸2OFFVccniLnvCn-110 1¸AVGFigure 2.3: Ideal synchronous buck converter response shown in state plane and time do-mains for d = 0.5, Vccn = 2, and fswn = 10. Non-averaged trajectory and ANT are shownoverlapped.2.1.3 Unified ModelAs it can be observed in (2.25)-(2.30), the averaged differential equations of the three con-verters have a similar structure. By defining the auxiliary coefficients m1, m2, and kw, asdetailed in Table 2.1, a unified system of equations valid for the three topologies is obtained:322.2. Steady State AnalysisTable 2.1: Model Coefficients DefinitionTopology m1 m2 kωBuck d 1 1Boost1d′1d′d′Buck-boost − dd′1d′−d′12pidiLndtn= kω (m1 Vccn − vCn) (2.31)12pidvCndtn= kω (iLn −m2 Ion) (2.32)As a result of normalization, averaging and unification, the system of equations deter-mined by (2.32) and (2.31) provides a unified characterization of the three basic DC-DCtopologies. The following sections analyse the steady state and transient behaviour of theunified equations in order to find a large signal state-plane model.2.2 Steady State AnalysisThe possible steady state points the converter may take define the load line, which is para-metrically described by:0 = m1 Vccn − vCnSS (2.33)0 = iLnSS −m2 Ion (2.34)Combining these conditions with the definition of m1 and m2 shown in Table 2.1, andsolving to eliminate d, the load line equations are found:λLLBuck : iLnSS − Ion = 0 (2.35)332.2. Steady State AnalysisVccniLnvCnIon1iLnvCnIonVccn-1iLnvCnIonTargetVccn1¸LLBuck¸LLBoost ¸LLBuck−BoostBuck Boost Buck-BoostTargetTargetIonVccnIon 1 +1VccnFigure 2.4: State-plane representation of the load line for the three fundamental DC-DCconverters.λLLBoost : iLnSS −IonVccnvCnSS = 0 (2.36)λLLBuck−boost : iLnSS − Ion(1− vCnSSVccn)= 0 (2.37)The load lines for the three normalized DC-DC converters are depicted in Fig. 2.4.The desired combination of normalized voltage and current is defined as the target oper-ating point (vCn = vCnt; iLn = iLnt). As the target operating point must be stable, it must becontained in the load line. Plugging the target operating point into (2.33) and (2.34) yields:vCnt =M1 Vccn (2.38)iLnt =M2 Ion (2.39)where M1 and M2 correspond to the steady state duty cycle.Since the normalizing voltage is the absolute value of the desired output, the target voltageis 1 for the cases of buck and boost converters, and −1 for the buck-boost converter. Thecoefficient M1 determines the steady state input output voltage ratio, and M2 determinesthe steady state ratio between output and inductor current. The target inductor current342.3. Transient AnalysisTable 2.2: Target Operating Points and Steady State Duty CycleParameter Buck Boost Buck-boostvCnt 1 1 −1iLnt IonIonVccnIon(1 +1Vccn)D1Vccn1− Vccn 11 + Vccnis expressed as a function of the load current and input voltage for each topology in Table2.2, and graphically represented in the state-plane in Fig. 2.4. In order to complete thesteady state parameter values, the steady state duty cycle D can be determined using (2.38),and replacing vCnt and M1 by their equivalent expressions for each converter, leading to theexpressions detailed in Table 2.2.2.3 Transient AnalysisGeneric expressions that describe the time domain evolution of the averaged state variablesin the three fundamental topologies can be found by solving the unified differential equations(2.31) and (2.32), yielding:vCn(tn) = [vCn(0)−m1 Vccn] cos(kω 2pi tn) + [iLn(0)−m2 Ion] sin(kω 2pi tn) +m1 Vccn (2.40)iLn(tn) = [iLn(0) −m2 Ion] cos(kω 2pi tn) + [vCn(0) −m1 Vccn] sin(kω 2pi tn) +m2 Ion (2.41)which indicate both state variables describe sinusoidal waveforms with angular frequency:ωn = kω 2pi (2.42)giving a more meaningful definition to the constant kω, as resonance frequency constant.352.3. Transient Analysis¸ON¸OFFrOFFcOFF¸AVG d1rd1cd1¸AVGd2rd2cd2iLnvCnVccnIon0!0n!d1n!d2nVccnion 1d2d1 0tn31 1.5dvCn(0)tn0vCn 2¼!0n 2¼!d1n 2¼!d2niLn(0)tniLn0VccnLnionCnFigure 2.5: Average Natural Trajectories in boost converters for several PWM duty cyclevalues (0, d1 = 1/3, d2 = 2/3, and 1).As shown by (2.40) and (2.41), the parameters Vccn, Ion, along with the PWM duty cycleand the initial conditions are sufficient to determine the converters’ time-domain behaviour.Combining (2.40) and (2.41) the normalized time variable tn is eliminated yielding aparametric equation that describes the averaged path the state variables follow, the Average362.3. Transient AnalysisNatural Trajectories (ANTs):λAV G : (vCn −m1 Vccn)2 + (iLn −m2 Ion)2 = (vCn(0) −m1 Vccn)2 + (iLn(0) −m2 Ion)2 (2.43)The equation indicates that, for constant input voltage and load current, the converter’soperating point follows circular average paths, which centres depend on the duty cycle, andare located on the load line at (m1 Vccn;m2 Ion). The radius of the circle is determined bythe initial conditions (vCn(0); iLn(0)), and the speed at which the operating point moves acrossthe circular path is constant and given by (2.42). It is worthwhile noticing that even if theinput voltage, load current and duty cycle are not constant, but known, (2.43) still describesthe behaviour of the converters deriving in time-dependant geometric trajectories. The loadcurrent Ion can either be measured or estimated through an observer, avoiding the need ofa physical current sensor. Since the latter approach is based on the derivative of the outputvoltage, it is more sensitive to measurement noise.Average Natural Trajectories for a normalized boost converter are shown in Fig. 2.5.The trajectories shown correspond to the same initial operating point and several duty cyclevalues (0, d1 = 1/3, d2 = 2/3, and 1). Resonance periods corresponding to the different dutycycles, as well as the ON trajectory slope are indicated in both geometric and time-domains.It is worthwhile highlighting that (2.43) is valid even in large ripple scenarios, for the entirerange of possible operating points and duty cycle values including the saturation extremes.Therefore, the ANTs accurately model the average large-signal behaviour of the three basictopologies in a geometrical domain. This powerful concept will be employed in the followingsections to determine the physical limits of performance of buck and boost converters, as wellas developing state-plane based controllers.The effect of dissipative components is not included in the analysis in order to gain sim-plicity in the obtained models. Since the completely undamped scenario represents the most372.4. SummaryiLnvCndrivingsignal d = 0.53 d = 0.471msFigure 2.6: Pseudo-ideal ANT forced in a real synchronous buck converter for d = 0.5,Vccn = 2, and fswn = 10.challenging condition from a control point of view, the ANTs large signal model can be em-ployed to design rugged controllers. Pseudo-ideal ANTs forced in a buck converter are shownin Fig. 2.6, which illustrates how the ideal behaviour can be obtained in a real converter byadjusting the duty cycle. Very large excursions in the state variables are shown in the ex-perimental capture (reaching 24V and 4A), validating the large-signal model proposed. Theresonance period obtained is 1ms, while the expected value for the parameters employed(L = 512µH, C = 48µF ) is T0 = 985µs, showing a difference of only 1.5%.2.4 SummaryThe unified Average Natural Trajectories for the three basic DC-DC topologies were intro-duced in this chapter as a model that can accurately describe the large signal behaviour ofany buck, boost, or buck-boost converter regardless the input and output voltage levels, aswell as L and C reactive component values.The differential equations that represent the behaviour of the three converters were nor-malized first in order to gain independence from voltage levels and components value. The382.4. Summaryobtained normalized differential equations of the three topologies were averaged using tradi-tional averaging methods. The normalized average differential equations were then mergedinto a unified single set of differential equations including parameters that take different val-ues for each individual topology. This unified set of differential equations establishes thebasis to obtain the unified ANTs by analysing its steady state and dynamic solutions. Asa result of the analysis the load line equations were obtained as the set of possible equilib-rium points the converters may take. The Average Natural Trajectories were obtained fromthe dynamic analysis in the state plane, and represented by parametric circles to accuratelydescribe the large signal transient response of the converters, where the circle centres are onthe load line and their location is determined by the PWM duty cycle, while the circle radiusis determined by the distance between the centre and the converter’s initial conditions.The unified ANTs, the load lines, and the procedure followed to obtain them, are be usedin the rest of this thesis to propose interesting solutions to several issues identified in thepower electronics field.39Chapter 3Physical Limits of DynamicPerformanceThe transient performance of DC-DC power converters is defined by the output voltage re-sponse to external perturbations, such as load change or start-up operation. The performance-determining parameters of the voltage waveform during start-up transient are the settlingtime and overshoot. For load step changes the critical parameters are the recovery time,peak voltage deviation and voltage over/under-shoot. In this work, the converter’s transientperformance is assessed by comparison with the theoretical limits of performance. Two typesof theoretical limits of performance are identified and defined as Minimum Recovery Time(MRT) and Minimum Voltage Deviation (MVD) transients.In order to achieve time-optimality in a system with limited ability to absorb (release)energy from (to) a source, the maximum amount of power should be taken (released) at alltimes until the desired steady state is reached. This somewhat intuitive concept is formalizedby Theorem 2 in [131], which states that: if there is an optimal control function, then thereis always a ’bang-bang’ control function that is optimal. Or the analogue statement, if thereis a limit of dynamic performance, then it can always be achieved by a saturating controlfunction. Saturation duty cycle ANTs for ON (d = 1) and OFF (d = 0) states of the switchesare employed to characterize the time-optimal transient responses in this chapter.Part of the findings reported in this chapter are included in a journal paper published onIEEE Transactions on Industrial Electronics [4].403.1. Saturation TrajectoriesiLnIonVccn vCn1/(2pi)1/(2pi)ON d=1d=0OFFBuckFigure 3.1: Buck converter ON and OFF switch states.3.1 Saturation TrajectoriesIn order to determine the ideal MRT and MVD transient responses, the saturation trajectoriesof the converters must be analysed. The natural evolution of the state variables of the buckand boost converters, and the shapes they describe on the state plane for ON and OFFsaturation extremes of the PWM duty cycle are described in this section. The results of thisanalysis will be employed in the next chapter to define theoretical indices and benchmarkthe dynamic performance obtained through different control methods for buck and boostconverters.3.1.1 Buck Converter Saturation TrajectoriesThe time domain expressions that represent the evolution of the state variables in a buckconverter can be found by replacing the coefficients detailed in Table 2.1 in (2.40) and (2.41).Evaluating the equations for d = 1 and d = 0, the saturation trajectories are found for ONstate:vCn(tn)ON = [vCn(0)− Vccn] cos(2pitn) + [iLn(0)− Ion] sin(2pitn) + Vccn (3.1)iLn(tn)ON = [iLn(0)− Ion] cos(2pitn) + [vCn(0)− Vccn] sin(2pitn) + Ion, (3.2)413.1. Saturation TrajectoriesiLniLn(0)Iontn21tn2100vCnVccnvCn(0)¸OFF¸ONVccn!0n!0niLnIonvCnvCn(0)iLn(0)rONrOFF0Figure 3.2: Buck converter Natural Trajectories for ON (d = 1) and OFF (d = 0) states ofthe switch, shown in geometric and time domains.and for OFF state of the switch:vCn(tn)OFF = vCn(0) cos(2pitn) + [iLn(0)− Ion] sin(2pitn) (3.3)iLn(tn)OFF = [iLn(0)− Ion] cos(2pitn) + vCn(0) sin(2pitn) + Ion (3.4)423.1. Saturation TrajectoriesIonVccnBoostiLn1/(2pi)vCn1/(2pi)ONOFF d=1d=0Figure 3.3: Boost converter ON and OFF switch states.As detailed in (3.1)-(3.4), and illustrated in Fig. 3.2, for each saturation state of the switch,the normalized voltage and current show sinusoidal waveforms of the same amplitude andfrequency, and with ninety degree phase shift between them. The normalized time variabletn can be eliminated by combining (3.1) with (3.2), and (3.3) with (3.4):λON : [iLn − Ion]2 + [vCn − Vccn]2 − [iLn(0)− Ion]2 − [vCn(0)− Vccn]2 = 0 (3.5)λOFF : [iLn − Ion]2 + [vCn]2 − [iLn(0)− Ion]2 − [vCn(0)]2 = 0, (3.6)These expressions represent the natural trajectories the converter operating point followsfor ON and OFF states as shown in Fig. 3.2. The normalized angular speed at which theoperating point moves across the circular trajectories is given by (2.40) and (2.41), and itsvalue is constant for both states of the switch ωon = 2pi. These large-signal ideal trajectories(3.5) and (3.6) will be employed to obtain the physical response limit of the buck converter.3.1.2 Boost Converter Saturation TrajectoriesTime domain expressions that describe behaviour of boost converters can be found by replac-ing the coefficients detailed in Table 2.1 in (2.40) and (2.41). Evaluating the time-domain433.1. Saturation Trajectoriestn0vCnvCn(0)0.750.50.25 1tn0iLn0.750.50.25 1iLn(0)¸OFF¸ONVccn 1iLnvCniLntIonVccnIon¸LLvCn(0),iLn(0)Figure 3.4: Boost converter Natural Trajectories for ON (d = 1) and OFF (d = 0) states ofthe switch, shown in geometric and time domains.solutions for duty cycle extremes, saturation trajectories are found for ON state:vCn(tn)ON = vCn(0)− 2pi Ion tn (3.7)iLn(tn)ON = iLn(0) + 2pi Vccn tn (3.8)and for OFF state:vCn(tn)OFF = [vCn(0)− Vccn] cos(2pitn) + [iLn(0)− Ion] sin(2pitn) + Vccn (3.9)iLn(tn)OFF = [iLn(0)− Ion] cos(2pitn)− [vCn(0)− Vccn] sin(2pitn) + Ion (3.10)443.2. Theoretical Limits of Dynamic Performance in Buck ConvertersCombining the equations in order to eliminate the time variable, both ON and OFF statesof the boost converter can be graphically represented in the state plane:λON : Ion [iLn − iLn(0)] + Vccn [vCn − vCn(0)] = 0 (3.11)λOFF : [iLn − Ion]2 + [vCn − Vccn]2 − [iLn(0)− Ion]2 − [vCn(0)− Vccn]2 = 0 (3.12)which represent the natural trajectories of boost converters.The response of the converter in ON and OFF states is shown in both geometric andtime domains in Fig. 3.4. The ON natural trajectory is a straight line perpendicular to theload line. The OFF natural trajectory is a circle with centre located at (Vccn; Ion) and radiusdefined by the initial conditions vCn(0) and iLn(0). As shown in Fig. 3.4, a unique startingpoint has been employed for both ON and OFF trajectories. Although these points are notnecessarily coincident, this selection was made in order to illustrate how, starting from thesame conditions, the operating point can move following different paths determined by theON/OFF status of the switch. Based on these findings, the theoretical ideal responses andcritical parameters will be characterized and employed to define the performance indices.3.2 Theoretical Limits of Dynamic Performance inBuck ConvertersThe objective of this section is to characterize the optimal transient responses in buck con-verters. Both Minimum Recovery Time (MRT) and Minimum Voltage Deviation (MVD)transients can be achieved by following two natural trajectories corresponding to one ON(λON) and one OFF state (λOFF ). Since the normalized angular speed at which the operat-ing point moves is constant, the transient times can be determined by calculating the anglesthe operating point covers in order to reach the target located at vCnt = 1 and iLnt = Ion.453.2. Theoretical Limits of Dynamic Performance in Buck Converterstn0.0.20.1 30vCn1vCn1123iLn1tn0.30.20.10iLn123tn0.30.20.10drivingsignal¯®2¼ 2¼1 2 3¸OFF¸ONrONrOFFVccnvCn1®¯iLn11iLnvCn123Switch on action Switch off action123 Target operating pointFigure 3.5: Theoretical time-optimal start-up transient in a normalized buck converter. Thetransient is solved employing only ON and OFF saturation trajectories.3.2.1 Start-Up TransientThe time-optimal start-up transient is shown in state-plane and time domain in Fig. 3.5.The analysis in this work focuses on the no-load starting condition to find the theoreticalminimum start-up time. Lightly loaded scenarios, which are less common, would derive in ahigher minimum start-up time.At the initial instant ➊, the switch is turned on and the operating point follows the λON463.2. Theoretical Limits of Dynamic Performance in Buck Converterstrajectory covering the angle α. When point ➋ is reached, the switch is turned off, and theoperating point follows the trajectory λOFF to move towards the target, covering the angleβ. The target operating point is reached at ➌, and the converter continues to operate insteady state as indicated by the grey part of the waveforms. By performing the geometricalanalysis described above, the coordinates of the switch-off point ➋ are found as:vCn1 =12Vccn(3.13)iLn1 =√1− 14V 2ccn(3.14)and the theoretical minimum normalized start-up time is found as:tMSn =α + β2pi(3.15)where the angles α and β are given by:α = cos−1(1− 12 V 2ccn)(3.16)β = cos−1(12 Vccn)(3.17)As shown by (3.15)-(3.17) the theoretical minimum start-up time depends only on thenormalized input voltage (Vccn = Vcc/vref ). By using (3.15)-(3.17), the theoretical mini-mum settling time (tMSn) can be found for any input-to-output voltage ratio. As shown inFig. 3.5, the minimum start-up overshoot for any combination of reactive components andinput/output voltages is zero. These values will employed later in this work to define theSettling Time Index (STi), and the Start-Up Overshoot Index (SOi) for buck converters.The absolute minimum settling time occurs when the input voltage is much greater thanthe output voltage Vcc >> vref (or Vccn →∞), yielding a total minimum theoretical settling473.2. Theoretical Limits of Dynamic Performance in Buck Converterstime tMSn = 1/4. By denormalizing, the rule of:tMSmin = T0/4 =pi2√LC (3.18)is established, where T0 is the natural resonant period of the LC resonant tank. Equation(3.18) is referred to as the quarter of the filter natural period rule and represents the absolutephysical limit of the system. On the other extreme, when Vcc = vref (or Vccn = 1) the transienttime is given by tMSn = 1/3 (normalized) or T0/3 in the time domain. The experimentalcaptures depicted in Fig. 3.10 and 3.11 provide experimental validation for both cases, lowand high step-down, respectively.3.2.2 Loading TransientLoading transients are produced with the converter operating in steady state by suddenlyincreasing the load current. A load current step-up of a magnitude ∆Ion, is used in thissection. The loading transient is shown in Fig. 3.6 in geometrical and time domains, wherethe initial operating point ➊ is located at (1; Ion(0)), and the target operating point is located,at a distance ∆Ion, at (1; Ion). At the zero instant, the switch is turned on and the operatingpoint starts to follow the λON trajectory. It remains in that state until the λOFF trajectorythat contains the target operating point (for iLn > Ion) is reached at ➋. The operatingpoint covers first the angle α1 to reach the zero current intersection, and then it continuesmoving through α2 to reach ➋. The switch is turned off at ➋ and the operating point coversthe angle β to reach the steady target operating point➌ (indicated by grey waveforms).The transient represented by the points ➊, ➋, ➌ is controlled by saturation control actions,and it exhibits Minimum Recovery Time (MRT) and Minimum Voltage Deviation (MVD)behaviour. Parametric expressions that determine the MRT and MVD optimum values canbe derived by analysing this transient response geometrically.483.2. Theoretical Limits of Dynamic Performance in Buck Converters®1tn0drivingsignal¯2¼®22¼ 2¼1 2 3iLn1tniLn123Ion(0)IonvCn(min)vCn1 1 2tn0vCn1 3¸OFF¸ON rONrOFFvCn(min)®2®1¯iLn1IonIon(0)1iLnvCn123vCn1Switch on action Switch off action123 Target operating pointVccnDIonFigure 3.6: Theoretical time-optimal load step-up transient in a normalized buck converter.The transient is solved employing only ON and OFF saturation trajectories.Minimum Recovery Time (MRT):By geometrically analysing the loading transient, the angles α1, α2 and β are found:α1 = tan−1(∆IonVccn − 1)(3.19)493.2. Theoretical Limits of Dynamic Performance in Buck Converters00.10.20.30.40.50.60 0.2 0.4 0.6 0.8 1 1.200.10.20.30.40.50.60.70.8∆ IonVccn = 1Vccn = 2 Vccn = 4Vccn = 8Vccn = 1Vccn = 2 Vccn = 4Vccn = 8∆vMDLntMRLnFigure 3.7: Load step-up transient in the normalized buck converter: optimal transientparameters as function of the load step for several normalized input voltages.α2 = tan−1(∆Ion√4Vccn −∆Ion22V 2ccn − 2Vccn +∆Ion2)(3.20)β = tan−1(∆Ion√4Vccn −∆Ion22Vccn −∆Ion2)(3.21)And, since the angular speed is ωon = 2pi, the normalized theoretical minimum recoverytime is given by:tMRLn =α1 + α2 + β2pi(3.22)503.2. Theoretical Limits of Dynamic Performance in Buck ConvertersBy using (3.22), the minimum recovery time for any loading transient can be calculatedto establish a strong benchmark point that will be employed later in this thesis to determinethe Loading transient Recovery Time Index (RTiL).Minimum Voltage Deviation (MVD):Since the analysis is performed in the geometrical domain, the transient can be fully char-acterized, and particular aspects can be analysed. One of the parameters of interest is thetheoretical minimum voltage drop (dynamic regulation), which is given by:∆vMDLn = 1− vCn(min) = 1− Vccn +√(Vccn − 1)2 +∆Ion2 (3.23)This closed-form expression will be employed to determine the Loading transient DynamicRegulation Index (DRiL).3.2.3 Unloading TransientThe unloading transient, illustrated in Fig. 3.8, also occurs with the converter operating insteady state when the load current is decreased suddenly. The initial operating point ➊ islocated at (1; Ion(0)), and the target is defined at (1; Ion). At the zero instant, the switch isturned off and the operating point moves following the trajectory λOFF , covering the angleβ1 to reach the maximum voltage point (vCn(max)) and β2 to reach the switching point ➋. Atthat moment, the switch is turned on and λON is followed during the angle α to reach thetarget operating point ➌, resuming steady state operation (grey waveforms). As in the caseof the loading transient, the analysed unloading transient response includes MRT and MVDcharacteristics.513.2. Theoretical Limits of Dynamic Performance in Buck Converterstn0drivingsignal1 2 3¯1 ®¯22¼ 2¼ 2¼iLn1tniLn123Ion(0)IonvCn(max)vCn1 12tn0vCn1 3iLn1Ion(0)Ion1iLn¸OFF¸ON rONrOFFvCn1®¯1¯2123Switch off action Switch on action123 Target operating pointVccnvCnvCn(max)DIonFigure 3.8: Theoretical time-optimal load step-down transient in a normalized buck converter.The transient is solved employing only ON and OFF saturation trajectories.Minimum Recovery Time (MRT):Performing geometrical analysis, the normalized minimum recovery time for unloading tran-sients is found:tMRUn =β1 + β2 + α2pi(3.24)523.2. Theoretical Limits of Dynamic Performance in Buck Converters00.10.20.30.40.50.60 0.2 0.4 0.6 0.8 1 1.200.050.10.150.20.250.30.350.4∆ IonVccn = 2 Vccn = 1 ~ ∞ Vccn = 4Vccn = 8∆vMDUntMRUnFigure 3.9: Load step-down transient in the normalized buck converter: optimal transientparameters as function of the load step for several normalized input voltages.where:β1 = tan−1 (∆Ion) (3.25)β2 = tan−1(∆Ion√4Vccn(Vccn − 1)−∆Ion22Vccn +∆Ion2)(3.26)α = tan−1(∆Ion√4Vccn(Vccn − 1)−∆Ion22V 2ccn − 2Vccn −∆Ion2)(3.27)The Unloading transient Minimum Recovery time defined by (3.24) will be employed to533.2. Theoretical Limits of Dynamic Performance in Buck Convertersdetermine the Unloading transient Recovery Time Index (RTiU).Minimum Voltage Deviation (MVD):Analysing the evolution of the variables in the geometrical domain, the minimum transientvoltage overshoot is found as:∆vMDUn = vCn(max) − 1 =√1 + ∆Ion2 − 1 (3.28)This simple equation indicates the normalized minimum peak deviation value that theoutput voltage reaches during a load current step-down transient, as function of normalizedstep magnitude only. The effectiveness of the controller to avoid overshoot during this kind oftransients can be objectively assessed by comparing the obtained values with the theoreticalminimum given by (3.28), and calculating of the Unloading transient Dynamic RegulationIndex (DRiU), which will be introduced later in this work.The above-detailed transient analysis considers the angle α to be within the range [0−pi].Therefore, the domain of analysis for both MRT and MVD cases must be restricted to:Vccn ≥ 1 + ∆vMDUn2=1 +√1 + ∆Ion22(3.29)Which indicates that the input voltage must be larger than the output by at least half ofthe minimum unloading transient peak voltage deviation. Analysing the response shown inFig. 3.8, it can be observed that the required condition naturally holds true for most buck-derived topologies even for extremely large transients. For instance, for a normalized currentstep-down as large as ∆Ion = 1.2, an input voltage 30% larger than the output would beenough for (3.24) to hold true. The restriction only applies for the load step-down recoverytime, no voltage ratio restrictions are required for the rest of the analysed parameters.The values of recovery time and voltage drop are general and valid for any combination543.2. Theoretical Limits of Dynamic Performance in Buck ConvertersTable 3.1: Buck Converter ParametersPARAMETER VALUEVcc 12, 24, 120 Vvref 12 VL 495 µHC 46 µFZ0 3.28 ΩT0 948 µsof L-C parameters, and are shown in the plots included in Fig. 3.7 and 3.9. These plotsenable power electronics designers to rapidly find the theoretical minimum recovery timeand voltage drop for any load current step-up/step-down. Normalizing the current stepusing iref =vrefZ0and determining the converter input to output voltage ratio are the onlyprevious steps needed to read the optimal values from the plots. For instance, if a currentstep-up of a normalized magnitude ∆Ion = 1 happens in a normalized buck converter withVccn = 2, the theoretical minimum recovery time and voltage drop are tMRLn = 0.3175 and∆vMDLn = 0.4142. It must be noted that there is an infinite combination of converters andtransients that fulfil the conditions of this example Vccn = 2 and ∆Ion = 1 depending on inputvoltages as well as L and C parameters. All these combinations share the same normalizedtheoretical minimum values. Changes in the output voltage vref or in the characteristicimpedance Z0 would require adjusting the input voltage and load current step to maintainthe conditions of this example Vccn = 2 and ∆Ion = 1 to enable a fair comparison in thenormalized domain.3.2.4 Experimental ValidationNear time-optimal experimental results were obtained employing the prototype detailed inTable 3.1. These experimental results validate the theoretical limits of performance for buckconverters derived in this section. The experimental captures shown in Fig. 3.10 and Fig.553.2. Theoretical Limits of Dynamic Performance in Buck ConvertersTurn on action Turn off actionTarget operating point123123Vccn=1iLndvCn317¹s(Physical limit: To/3=316¹s)1 2 3Figure 3.10: Experimental validation of start-up transient in buck converter with low step-down conversion ratio (Vccn = 1).Turn on action Turn off actionTarget operating point123123Vccn=10iLndvCn247¹s(Physical limit: To/4=237¹s)123Figure 3.11: Experimental validation of start-up transient in buck converter with high step-down conversion ratio (Vccn = 10).3.11 exhibit near-optimal start-up transients for extreme normalized input voltage values of1 and 10 respectively. In the case of the extremely low input voltage (Vcc = vref ) the start-up563.2. Theoretical Limits of Dynamic Performance in Buck ConvertersTurn ON Turn OFF11223435465 6123 64 5Vccn=2iLndvCnIon228¹s(0.240)219¹s(0.231)286¹s(0.301)2.43V(0.203)2.16V(0.180)Figure 3.12: Experimental results for start up, loading and unloading transients with Vccn = 2.transient is solved in 317µs, while the theoretical optimum is given by T03= 316µs. Whenthe input voltage takes an extremely high normalized value (Vcc = 10 vref ), the settling timeobtained is 247µs, which is only 4.2% higher than the absolute theoretical limit given byT04= 237µs.A full set of transients (start-up, loading and unloading) of the converter detailed in Table3.1 operating with Vccn = 2 (Vcc = 24 V ) is shown in Fig. 3.12. A settling time of 286µs isobtained, while the theoretical minimum calculated using (3.15) is 0.29T0 = 278µs (less than3% lower). The measured recovery times and voltage deviations are 228µs and 2.43V forthe load step-up transient, and 219µs and 2.16V for the load step-down, while the optimalparameters for both transients, obtained by evaluating (3.22), (3.23), (3.24) and (3.28), are217µs and 2.35V .The small differences found in the measured parameters are caused by changes in thereactive component values for different operating points, as well as parasitic elements nottaken into account in the analysis. Most of the measured values are within a 5% differencerespect the physical limits of performance validating the theoretical findings. The most573.3. Theoretical Limits of Dynamic Performance in Boost Converterssignificant losses effect are seen in the unloading transient peak voltage deviation, whichreaches a value of 8%.3.3 Theoretical Limits of Dynamic Performance inBoost ConvertersThe ideal transient responses of boost converters under start-up and loading transients aredefined and characterized in this section. The Minimum Recovery Time (MRT) response isachieved by following one ON and one OFF natural trajectories to reach the target operatingpoint, therefore only one switching action is required to solve the transient. In contrast,the Minimum Voltage Deviation (MVD) response can be theoretically achieved performingswitching actions as soon as the operating point crosses the load line.As described by (3.7) and (3.8), the time evolution of the state variables is linear duringthe ON time, which implies time intervals can be determined by analysing the increment ofthe state variables in the geometrical domain. In an similar way, (3.9) and (3.10) indicate theangular speed at which the operating point covers the circular OFF trajectory is constantand equal to ω0n = 2pi, therefore time intervals can be read as angles in the geometricaldomain.3.3.1 Start Up TransientThe evolution of the state variables during the unloaded start-up transient is analysed in thissection to find the theoretical minimum settling time. As indicated in Fig. 3.13, the initialoperating point of the converter is ➊, with the capacitor charged to the input voltage Vccnand no load at the output (Ion = 0). It is worthwhile highlighting that the inrush transientthat charges the output capacitor from 0 V to Vccn is not included in this analysis. As shown583.3. Theoretical Limits of Dynamic Performance in Boost ConverterstnvCn1123iLn1tniLn1230.3 0.40.20.1 tn0drivingsignal1 2 3vCn¸OFF¸ONVccnVccn¯1iLn1iLn123Switch ON action Switch OFF action123 Target operating point1Figure 3.13: Ideal start-up transient shown in geometric and time domains.in the figure, the switch is turned ON at ➊ and the operating point follows the path λONuntil the current reaches iLn1 at ➋. Then the switch is turned OFF and the operating pointfollows the trajectory λOFF covering the angle β1 to reach the target at ➌. Once at this point,the fast switching actions on the driving signal represented by the solid grey block in Fig.3.13 maintain the converter operating in steady state. Performing a geometric analysis, thepeak current iLn1 and the angle β1 are found:iLn1 = 1− Vccn (3.30)593.3. Theoretical Limits of Dynamic Performance in Boost Converters0 0.2 0.4 0.6 0.8 10.2512tMSnVccnTo4Figure 3.14: Minimum normalized settling time as function of Vccn.β1 =pi2(3.31)Using (3.30) and (3.31), and the insight provided by (3.8)-(3.10) as described above, theminimum start-up time is found:tMSn =iLn12pi Vccn+β12pi=12pi(1Vccn− 1)+14(3.32)As described by (3.32) and illustrated by the plot in Fig. 3.14, the minimum settlingtime in a boost converter is given by the input-output voltage ratio. In the limit case whenVcc → vref , the minimum start-up time limit is found as tMSn → T04 . Since there is no boostoperation at Vcc = vref , this limit cannot be reached.The theoretical minimum settling time given by (3.32) will be employed to define theSettling Time Index (STi). As shown in Fig. 3.13, the minimum start-up overshoot for anycombination of reactive components and input/output voltages is zero. This value will betaken into account to define the Start-Up Overshoot Index (SOi) later in this work.3.3.2 Loading TransientThe loading transient is illustrated in Fig. 3.15. At the initial instant the operating point islocated at ➊, and the target has just moved to ➌ due to a step-up in the load current. In this603.3. Theoretical Limits of Dynamic Performance in Boost ConverterstnvCn11423von2iLn2tniLn123iLnt4iLn(0)tn0.60.40.20drivingsignal2 31¸OFF¸ONVccn vCn2¯2®1iLnvCn1234iLn2iLntIonDvMDLniLn(0)Switch ON action Switch OFF action123 Target operating point4 MVD pointtMRLnFigure 3.15: Ideal loading transient shown in geometric and time domains.case the two different conditions of optimality (MRT and MVD) imply different transients,therefore they must be analysed separately.Minimum Voltage Deviation (MVD):For the MVD response during loading transient, the switch is turned ON at the initial instant➊. In this case the switching actions should start when the trajectories λON and λOFF showthe same slope, once the load line is reached at the point ➍. The minimum voltage deviation613.3. Theoretical Limits of Dynamic Performance in Boost Convertersfor loading transient (∆vMDL) is found by geometrical analysis:∆vMDLn =(iLnt − iLn(0)) iLnti2Lnt + 1(3.33)As shown in Fig. 3.15, the theoretical minimum overshoot for this type of transients iszero. This value, along with (3.33), will be used as a benchmark point to define the Loadingtransient Dynamic Regulation Index (DRiL).Minimum Recovery Time (MRT):In order to achieve the MRT response, the switch is turned ON at the initial instant and theoperating point follows the trajectory λON until it reaches the point ➋. Then, the switch isturned OFF and the operating point directs towards the target ➌ covering the angle β2. Thecoordinates of the point ➋ are given by:vCn2 =1 + iLnt iLn(0)− i2Lnt k2i2Lnt + 1(3.34)iLn2 = iLn(0) +iLnt (1 + k2)− iLn(0)i2Lnt + 1(3.35)with:k2 = (iLn(0)− iLnt)√2 (Vccn − 1) (i2Lnt + 1)(iLn(0)− iLnt) iLnt + 1 (3.36)The minimum recovery time for loading transient (tMRL) is found by geometric analysis:tMRLn =iLn2 − iLn(0)2pi Vccn+β2 − α2pi(3.37)where:α = tan−1(IonVccn)(3.38)623.3. Theoretical Limits of Dynamic Performance in Boost Convertersβ2 = tan−1(iLn2 − IonvCn2 − Vccn)(3.39)The theoretical limit given by (3.37) provides a valuable benchmarking point that will beused to define the Loading transient Recovery Time Index (RTiL).The extra peak voltage deviation required to achieve the MRT response can be found bysubtracting the MVD voltage deviation from the MRT peak deviation (1− vCn2):∆vMRTLn = 1− vCn2 − vMDLn =k2 iLnti2Lnt + 1(3.40)3.3.3 Unloading TransientThe unloading transient is illustrated in Fig. 3.16. The initial operating point is located at➊, while the target operating point has moved to ➌ due to a load current step-down.Minimum Voltage Deviation (MVD):In order to obtain the MVD in a load step-down transient, the switch must be turned OFFat the point ➊, and remain in that position until the load line is reached at ➍.Then, the minimum voltage deviation for unloading transient is given by:∆vMDUn = Vccn − 1 +√(iLn(0)− Ion)2 + (1− Vccn)2i2Lnt + 1(3.41)This expression characterizes a theoretical minimum that will be employed to define theUnloading transient Dynamic Regulation Index (DRiU).Minimum Recovery Time (MRT):In order to achieve MRT response for the load step-down transient, the switch must be turnedOFF instantaneously when the load step-down occurs at the initial point ➊. The switch is633.3. Theoretical Limits of Dynamic Performance in Boost Converters¸OFF¸ONrOFFVccn¯0¯3®1iLnvCniLn3IoniLn(0)iLnt1234DvMDUnvCn3iLn3 tniLniLntiLn(0)1234tn0.60.40.20drivingsignal1 2 3tnvCn11234Switch OFF actionSwitch ON action123 Target operating point4 MVD pointtMRUnFigure 3.16: Ideal unloading transient shown in geometric and time domains.maintained in OFF position and the operating point starts following the curve λOFF coveringthe angles β0 and β3 to reach the point ➋. At ➋ the switch is turned ON and the operatingpoint follows the trajectory λON to reach the target at ➌.The coordinates of the point ➋ are given by the intersection of the OFF trajectory (λOFF )that contains the initial operating point (1, iLn(0)) and the ON trajectory (λON) that containsthe target (1, iLnt):643.3. Theoretical Limits of Dynamic Performance in Boost Convertersk3 =√√√√( iLn(0)iLnt − Vccn)2 − (1− Vccn)21 + i2Lnt(3.42)vCn3 = 1 + i2Lnt k3 (3.43)iLn3 = iLnt (1− k3) (3.44)Then, the minimum unloading transients recovery time is:tMRUn =iLnt − iLn32pi Vccn+β0 + β32pi(3.45)where:β0 = tan−1(iLn(0)− Ion1− Vccn)(3.46)β3 = tan−1(Ion − iLn3vCn3 − Vccn)(3.47)The theoretical parameter (3.45) is critical in the definition of the Unloading transientRecovery Time Index (RTiU).The peak voltage deviation during the MRT unloading transient is given by:vCn(pk)U = Vccn +√(iLn(0)− Ion)2 + (1− Vccn)2 (3.48)Then, the extra voltage drop required to achieve the MRT unloading transient is found:∆vMRTUn =√(iLn(0)− Ion)2 + (1− Vccn)2(1−√1i2Lnt + 1)(3.49)653.3. Theoretical Limits of Dynamic Performance in Boost ConvertersiLnvCnIoniLtn0iLtnIon0Vccn 1Vccn012DiLntDIonDVccn¸LL¸LL0¸OFF¸ONFigure 3.17: Equivalence between step changes in input voltage and load current3.3.4 Input voltage step changesThis section provides a detailed explanation on how to analyse the dynamic response understep-changes on the normalized input voltage. As shown in Fig. 3.17, a step change in thenormalized input voltage, from Vccn0 to Vccn, modifies the converter load line slope and causesa displacement in the target current from iLnt0 at ➊ to iLnt at ➋. The load line correspondingto the operating point before the input voltage step-down λLL0 can be defined as:λLL0 :iLnSSvCnSS=IonVccn0=Ion0Vccn(3.50)which indicates there is a load current step-up ∆Ion that, for a constant input voltage Vccn,causes the same change on the target current ∆iLnt as the voltage step-down ∆Vccn withconstant current Ion. Therefore, the MRT and MVD responses for input voltage step changescan be considered equivalent to the load step change responses, which have been analysed inthe previous sub-sections. While the load current and input voltage employed for the analysismust be maintained at the same values (Vccn, Ion), the initial voltage Vccn0 must be mappedto the equivalent initial current:Ion0 = VccnIonVccn0(3.51)663.3. Theoretical Limits of Dynamic Performance in Boost ConvertersSince the MRT and MVD responses are maintained, the same indices can be used for thetransient performance evaluation under load current and input voltage step changes.3.3.5 Effects of Losses on Theoretical Limits of Performance ofBoost ConvertersParasitic resistances on the reactive components and switches cause losses that make theconverter’s behaviour differ from the ideal case. The effects of these losses on the timeresponse of the converter are analysed in this section.The topology including the parasitic elements is shown in Fig. 3.18. The parasitic resis-tances of inductor and switches are lumped together because they are connected in series.The differential equations that describe the behaviour of the system including the resistiveelements are:12pidvCndtn= u′ iLn − Ion (3.52)12pidiLndtn= Vccn − iLnrLn − u′ v′Cn (3.53)v′Cn = vCn + iCn rCn (3.54)where v′Cn is the voltage at the terminals of the capacitor.The load line including the effect of losses is shown in Fig. 3.19. The mathematicalexpression that describes it can be found by making (3.52) and (3.53) equal to zero to findthe set of possible steady state capacitor voltage vCnSS and inductor current iLnSS:λLL : iLnSS =12rLn√V 2ccn − 4rLnIonvCnSS +Vccn2rLn(3.55)As shown by (3.55) the load line is not a straight line as in the ideal case. The targetinductor current i′Lnt is found when vCnSS = 1 in (3.55), as shown in Fig. 3.19.673.3. Theoretical Limits of Dynamic Performance in Boost ConvertersVcc nIonvCnvCnLn =12πCn =12πONOFFiLn rLnrCn'Figure 3.18: Normalized boost converter including parasitic resistancesAtt(tn)¸OFF ¸ON1iLnvCniLntIonVccnIon¸LLVccn' 'vCn(0),iLn(0)tn0vCn'0.750.50.25 1vCn(0)'tn0iLn0.750.50.25 1iLn(0)Figure 3.19: Normalized boost converter trajectories and load line including the effect ofresistive losses for rLn = rCn = [0, 0.01, 0.025, 0.05, 0.1]Time-domain expressions that describe the inductor current and output voltage dynamicresponses including the effects of losses are shown in Fig. 3.19 in both geometric and timedomains, for normalized parasitic resistances rLn and rCn of 0, 0.01, 0.025, 0.05, and 0.1.683.3. Theoretical Limits of Dynamic Performance in Boost ConvertersThe mathematical expressions that describe these shapes are found by solving (3.52)-(3.53)and replacing by (3.54). As a result, time-domain expressions are obtained for ON:v′Cn(tn)ON = v′Cn(0)− 2pi Ion tn (3.56)iLn(tn)ON =(iLn(0)− VccnrLn)e−2pirLntn + VccnrLn(3.57)and OFF states:v′Cn(tn)OFF = V′ccn + Att(tn){(v′Cn(0)− V ′ccn) cos(ωdntn)+[(v′Cn(0)− V ′ccn) rLn−rCn2 + (iLn(0)− Ion)(1− rCnrLn)]2piωdnsin(ωdntn)}(3.58)iLn(tn)OFF = Ion + Att(tn){(iLn(0)− Ion) cos(ωdntn)− [(iLn(0)− Ion) rLn−rCn2 + (v′Cn(0)− V ′ccn)] 2piωdn sin(ωdntn)} (3.59)Where:Att(tn) = e−pitn(rCn+rLn) (3.60)ωdn = 2pi√1− ( rCn+rLn2)2(3.61)V ′ccn = Vccn − Ion rLn (3.62)The time variable tn is eliminated by combining (3.56) and (3.57) to obtain the ONgeometric trajectory:λON : iLn =(iLn(0)− VccnrLn)erLnIon[vCn−vCn(0)] + VccnrLn(3.63)693.3. Theoretical Limits of Dynamic Performance in Boost ConvertersIn a similar way, (3.58) and (3.59) are combined to obtain the OFF state trajectory:λOFF : (v′Cn − V ′ccn)2 + (1− rLn rCn) (iLn − Ion)2+ (rLn − rCn)(v′Cn − V ′ccn)(iLn − Ion) = r(0)Att(tn) (3.64)with:r(0) = (v′Cn(0)− V ′ccn)2 + (1− rLn rCn) (iLn(0)− Ion)2+ (rLn − rCn) (v′Cn(0)− V ′ccn) (iLn(0)− Ion) (3.65)The ON and OFF geometric trajectories given by (3.63) and (3.64) describe exponentialand spiral shapes respectively.As shown in Fig. 3.19, in the neighbourhood of the starting point v′Cn(0), iLn(0), theexponential ON trajectory shows the same slope as in the ideal case. This slope decreasesas the operating point moves forward due to the exponential shape of the current given by(3.57).In case of the OFF trajectory, four differences respect to the ideal case are identified.One of the main differences found is an attenuation on the circle’s radius Att(tn) given by(3.60). Another important difference is the change on the angular frequency ωdn definedin (3.61). The two least important differences are a small deviation of the circle centre toV ′ccn defined in (3.62), and a slight deformation of the circle, which becomes an ellipse givenby the (1 − rLnrCn) coefficient and the non-quadratic term in (3.64). The most significantdifferences will be used in the next section in order to validate the obtained mathematicalexpressions of the physical limits of performance with experimental results.703.3. Theoretical Limits of Dynamic Performance in Boost ConvertersTable 3.2: Boost Converter ParametersPARAMETER REAL VALUE NORM. VALUEvref 24 V 1L 238 µH T0 = 649 µsC 44.9 µF Z0 = 2.3 ΩrL 111 mΩ rLn = 48.2 mrS 20 mΩ rSn = 8.7 mrC 14 mΩ rCn = 6.1 mfsw 150 kHz fswn = 97.43.3.6 Experimental ValidationThe theoretical limits of performance are validated by experimental results in this section.The results obtained in section 3.3 are complemented with the effect of losses describedin section 3.3.5 and compared with experimental results. The boost converter detailed inTable 3.2, is operated by pre-programmed switching actions (open loop). Two MOSFETs(IRFB4127) are driven in complementary mode, and their ON resistances (rS) are lumpedtogether with the inductor’s.Four start-up transients corresponding to 6, 12, 18 and 21V are shown in Fig. 3.20. Loadtransients for input voltages of 12 and 18V are shown in Fig. 3.21 (a) and (b) respectively,where The driving signal is labelled as sw. The transients shown in Fig. 3.21 correspond toload current step-up/step-down of 2.6/1.3A for (a), and 4.16/2.08A for (b). The measuredrecovery times are indicated in the figures and summarized in Table 3.3.The theoretical recovery times are compared with the experimental results in Table 3.3.The following corrections were applied to the recovery times calculation in order to accountfor the effect of losses: a) from (3.61), a factor of 2piωdnmust applied to the OFF times; b)The ON time must be calculated by employing (3.57); c) an attenuation factor of Att(tOFF )defined in (3.60) must be applied to the OFF trajectory radii; and d) the target inductor713.3. Theoretical Limits of Dynamic Performance in Boost Converters12312von(a)(a)(a)(b)(b)(b)(c)(c)(c)(d)(d)(d)iLnTarget Switch ON 535 μs276 μs200 μs179 μs3Switch OFF Figure 3.20: Experimental validation of theoretical limits for start-up transient with (a)Vccn = 0.25, (b) Vccn = 0.5, (c) Vccn = 0.75, and (d) Vccn = 0.875. Preprogrammed switchingactions (open-loop) force the converter to follow minimum-time trajectories, validating thetheoretical ideal response.Table 3.3: Theoretical limits validationTransient Vcc Calculated Measured ErrorStart-up 6 V 521 µs 535 µs 2.6 %12 V 274 µs 276 µs 0.61 %18 V 199 µs 200 µs 0.53 %21 V 178 µs 179 µs 0.58 %Loading 12 V 284 µs 293 µs 2.9 %18 V 252 µs 253 µs 0.58 %Unloading 12 V 187 µs 193 µs 3.0 %18 V 213 µs 216 µs 1.3 %current must be calculated by employing (3.55). Therefore, the adjusted ideal minimumstart-up time (3.32) is:t′MSn = t′OFFnMS+ t′ONnMS (3.66)723.3. Theoretical Limits of Dynamic Performance in Boost Converters1 2 3 4 5LoadingUnloading12345vCniLnIon293μs5.72A2.81A193μssw(a)1 2 3 4 5LoadingUnloading12345vCniLnIon253μs5.9A2.9A216μssw(b)Figure 3.21: Experimental validation of theoretical limits for load step-up/step-down tran-sients with: (a) Vccn = 0.5, and (b)Vccn = 0.75. Preprogrammed switching actions (open-loop) force the converter to follow minimum-time trajectories, validating the theoretical idealresponse.witht′OFFnMS =12√4− (rCn + rLn)2(3.67)t′ONnMS =−12pirLnln[1−(1Vccn− 1) rLnAtt(t′OFFnMS)](3.68)For simplicity c) was omitted from the load transient calculations due to the small angles(lower than pi2) covered by the operating point. As a result, the minimum recovery time for733.4. Summaryloading transient (3.37) becomes:t′MRLn =−12pirLnln(iLn2 rLn − VccniLn(0) rLn − Vccn)+β2 − αωdn(3.69)In a similar way, the unloading transient analysis including the effect of losses yields:t′MRUn =−12pirLnln(i′Lnt rLn − VccniLn3 rLn − Vccn)+β0 + β3ωdn(3.70)The minimum recovery times including the effect of losses were calculated by employing(3.66), (3.69), and (3.70). As shown in Table 3.3, the comparison between measured andcalculated values yields errors of 3% or lower for the whole set of experiments performed.The limits of performance developed in this chapter are theoretical and therefore they cannotbe reached. However, a very good fit with experimental results for boost converters wasachieved by including the effects of parasitic elements into the analysis.3.4 SummaryIn this chapter, the Natural ON and OFF trajectories of buck and boost converters were ob-tained based on the unified ANTs model derived in chapter 2. The obtained PWM duty cyclesaturation trajectories were employed to find the theoretical limits of dynamic performancefor buck and boost converters under start-up and load current step transients. Two differenttypes of ideal responses were identified as Minimum Recovery Time (MRT) and MinimumVoltage Deviation (MVD). The geometric analysis is supported by detailed mathematicalprocedures and validated through experimental results obtained by pre-loading duty cyclesaturation actions in buck and boost experimental platforms. The obtained physical limitsof performance will be employed in next chapter as a powerful benchmark point to assess thedynamic performance of controllers for buck and boost converters.74Chapter 4Benchmarking Dynamic PerformanceThis chapter includes the analysis of different alternatives to employ the physical limits ofdynamic response to enable an objective assessment of controller’s dynamic performance inbuck and boost derived topologies. The analysed methods include direct dynamic responsecomparison, and a parameter-focused approach which are useful to gain insight but fail toprovide a clear, quantifiable benchmark. An index-based approach is proposed in order toenable a fast, intuitive yet thorough dynamic performance benchmarking method. Part ofthe findings reported in this section are included in a journal paper published on IEEETransactions on Industrial Electronics [4].4.1 Direct Comparison ApproachExperimental results of a dual-loop controlled buck converter are shown and compared withthe time-optimal response for start-up, load step-up and step-down transients in Figs. 4.1(a)and (b). As shown in Fig. 4.1(a), the implemented linear controller exhibits a settling timeof 3.75ms, which is 13 times higher than the limit of performance of 286µs. The start-uptransient response shows an overshoot of 9%, while the ideal response shows no overshoot.A load step-up/step-down transient of 1A − 3.8A − 1A is shown in Fig. 4.1(b) for thesame compensator. The measured recovery times and voltage peak deviations are 920µs and5.7V for the load step-up, and 939µs and 5.7V for the load step-down transient, while thetheoretical limits for both transients are 255µs and 3V . The recovery time of the compensator754.1. Direct Comparison ApproachiLnvCn3.75ms (3.9)1.09V(0.09)265¹s (0.28)(a)Vccn=2iLnvCnIon 2.8A (0.75)5.7V (0.47)934¹s (0.98)920¹s (0.97)242¹s (0.255)242¹s (0.255)5.7V(0.47)3V(0.25)3V(0.25)(b)Figure 4.1: Dynamic performance benchmarking of a dual-loop linear controller: a) Start-upresponse, and b) Load step-up/Step-down.under analysis is only 3.7 times the optimal, the voltage peak deviation reaches 1.9 times theminimum, and no overshoot/undershoot are found during the recovery. The analysis shows764.1. Direct Comparison ApproachTable 4.1: Dynamic Performance Benchmarks for Buck ConvertersREF Vccn ∆Ion trecn trecn(min) ∆vCn ∆vCn(min)1 Fig. 3.12 2 0.657 0.237 0.229 0.242 0.1962 Fig. 4.1(b) 2 0.766 0.939 0.259 0.475 0.2603 [56] 3 0.227 1.32 0.0493 0.0702 0.01294 [40] 2 0.101 1.05 0.0387 0.0222 0.005085 [57] 6.67 0.571 0.0929 0.0574 0.0556 0.02876 [58] 2 0.490 0.711 0.177 0.125 0.1147 [59] 2 0.380 0.266 0.141 0.178 0.06998 [32] 2 0.512 0.687 0.184 0.224 0.1239 [60] 3.33 0.249 0.213 0.0480 0.0333 0.013310 [61] 1.5 0.152 0.348 0.105 0.111 0.022711 [62] 3 1.16 0.360 0.236 0.400 0.31212 [46] 2 0.242 0.130 0.0915 0.0667 0.028913 [63] 2 0.131 0.177 0.0499 0.0328 0.0084714 [50] 2 0.950 0.310 0.306 0.416 0.379good recovery time and peak voltage deviations for a simple linear dual-loop controller (sameorder of magnitude as the optimal values). On the other hand, the settling time is muchhigher than the theoretical minimum, which indicates a large room for improvements in thisparticular transient.The transient comparison versus the time-optimal response is an insightful alternativethat enables thorough analysis of the transient response obtained. On the other hand, theresulting waveforms and ratios require of a certain level of expertise for the interpretationto be meaningful. Furthermore, the results are specific to a particular normalized transient,which makes the comparison among different existing responses more difficult.774.2. Transient-Specific Approach2468 00.5100.20.40.6∆ IonDynamic Performance BenchmarkingDynamicperformancelimit surfacesVccn∆vCn(min)2468 00.5100.51∆ IonVccn11145218679312131041114521867931213104t recn(min)Figure 4.2: Benchmarking of the load step-up transient dynamic performance of several buckconverters4.2 Transient-Specific ApproachThe load step-up transient theoretical minimum voltage drop and recovery times of severalbuck converters were normalized and are compared with the theoretical limits of performancein Table 4.1 and Fig. 4.2.784.3. Defining Performance IndicesThe first column of Table 4.1 shows identifier numbers, which correspond to the differentpoints plotted in Fig. 4.2. References to the respective figures and publications are shown inthe second column. The third and forth columns in Table 4.1 detail the normalized conditionsof the current step-up transient, namely the normalized input voltage (Vccn), and current stepmagnitude (∆Ion) respectively. The four remaining columns detail the normalized measuredand theoretical minimum parameters. The theoretical minimum load transient parametersare represented graphically by surfaces in Fig. 4.2. The data points detailed in Table 4.1 areincluded in the same three dimensional plots in order to illustrate the proposed benchmarkingprocedure. In this way the dynamic performance of any closed-loop controlled buck-derivedconverter can be objectively assessed, providing a powerful and practical tool for controldesigners in power electronics.This benchmarking method facilitates the transient-specific comparison of multiple con-trollers and their dynamic response. However, the meaningful interpretation of the resultsrequires of a certain level of expertise and analysis. Based on this, and in order to facilitatethe performance benchmarking, performance indices are defined in the next section.4.3 Defining Performance IndicesIn order to quantitatively evaluate start-up, load step-up and load step-down transients,performance indices related to each one of them are defined as: Settling Time Index (STi),Start-up Overshoot Index (SOi), Load/Unload Recovery Time Index (RTiL/RTiU), andLoad/Unload Dynamic Regulation Index (DRiL/DRiU). The indices values range from 0 to1 and they are based on the theoretical ideal responses of the converters MRT and MRV.The performance indices are defined in order to enable the objective evaluation of differentcontrollers based on two main aspects of the output voltage transient waveform: time elapsedto reach steady state and peak voltage deviations. In this way, two performance indices are794.3. Defining Performance IndicesRTiLSTiSOiDRiLRTiUDRiURTiLSTi SOiDRiLRTiU DRiUPerformance Index value00.50.70.85100.50.70.851PoorRegularGoodExcellentFigure 4.3: Performance indices ranges and categories.defined for each transient included in the analysis. It is worthwhile mentioning that theseindices can be calculated in normalized or de-normalized domains, by either normalizing theconverter response, or by de-normalizing the theoretical ideal parameters.The formulae employed to calculate the indices are defined in order to provide a meaningfuland straightforward quality indication, resembling the performance ranges detailed in Fig.4.3. Excellent performances are indicated by indices of 0.85 or higher, the good performancerange is delimited by indices from 0.7 to 0.85, regular performances are indicated by indicesbetween 0.5 and 0.7, and poor performances correspond to indices lower than 0.5.The performance ranges for time-dependent indices are defined as: excellent for recov-ery/settling times up to twice the theoretical minimum, good for times between two and fourtimes the ideal value, regular for values ranging from four to ten times the ideal time, andpoor for recovery/settling time periods larger than ten times the physical limit value. Inorder to satisfy these ranges, the time-dependent indices are defined as Settling Time Index :STi = 1− 0.5 log(tstarttMS)(4.1)804.3. Defining Performance Indicesand Recovery Time Index :RTiL/U = 1− 0.5 log(trecL/UtMRL/U)(4.2)where tMS and tMRL/U are the ideal parameters, and tstart and trecL/U the measured values.Several formulae were analysed to calculate the time-related performance indices, and (4.1)-(4.2) showed the best matching with the desired performance index ranges.In case of the start-up voltage overshoot, the reference parameter employed is the differencebetween the desired operating voltage and the initial voltage (vref − vc(0)). The ranges aredefined as: excellent for overshoots of 8% of the reference or lower, good for values in therange 8− 16%, regular for 16− 50%, and poor for overshoots higher than 50%. The Start-upOvershoot Index is defined accordingly in order to match these ranges:SOi =vref − vc(0)vref − vc(0) + 2 ∆vOS (4.3)where ∆vOS is the start-up voltage overshoot.The Dynamic Regulation Index is defined by comparing the peak-to-peak voltage excursionduring load transients with the theoretical minimum voltage deviation:DRiL/U =∆vMDL/U∆vpkL/U +∆vOL/U(4.4)where ∆vpkL/U is the measured peak voltage deviation and ∆vOL/U the overshoot for the anal-ysed transient. The dynamic regulation is defined as: excellent for peak-to-peak deviationsup to 18% larger than the minimum deviation, good for values between 18− 33%, regular for33− 100%, and poor for values larger than twice the theoretical minimum.These indices provide an objective evaluation of the converter’s transient response withrespect to its ideal capabilities. An absolute reference frame for the performance evaluation is814.4. Using the Dynamic Performance Indicesestablished, i.e. the indices get closer to the unity for performances approaching the theoret-ical limit. Applications with specific requirements can be mapped to a set of required indicesin a given converter, which define the performance requirements placed in the controllers.It is worthwhile highlighting that all the indices defined in this section are the result of arelative comparison of the measured parameters against a reference parameter of the sameunit. The minimum transient times (tMS and tMRL/U ), as well as minimum voltage deviations(vMDL/U ) are calculated based on the MRT and MVD optimum transient responses obtainedusing the converters’ ON and OFF Natural Trajectories.4.4 Using the Dynamic Performance IndicesThe performance indices are used in this section to benchmark the dynamic response ofseveral buck and boost converters. These examples provide a clear indication on how theproposed indices can facilitate the direct performance comparison of diverse linear, non-linear,and state-plane controllers, employed in converters that might have different L-C parametersas well as input-output voltages.4.4.1 Buck Converter Dynamic Performance BenchmarkingThe normalized dynamic performance parameters that characterize the near-optimal tran-sient response shown in Figs. 3.10, 3.11 and 3.12, as well as the linear controller responseshown in Fig. 4.1 are summarized in Table 4.2. The parameters are employed in this sectionto illustrate the application of the proposed dynamic performance benchmarking methodto buck converters. The six performance indices are calculated using (4.1)-(4.4). The re-sulting values are detailed in Table 4.4, and graphically shown in Fig. 4.4. As expected,the near-optimal response yields values close to 1 in all aspects. This is due to the fact thatboth MRT and MVD optimal responses can be achieved at the same time for all transients in824.4. Using the Dynamic Performance IndicesTable 4.2: Buck Converters Dynamic Response BenchmarkingNormalized Value Index ValueTIME Near-Optimal Linear Near-Optimal LinearStart-up: STi 0.30 3.9 0.99 0.44Loading Recovery: RTiL 0.24 0.97 0.99 0.72Unloading Recovery: RTiU 0.23 0.98 0.98 0.7Normalized Value Index ValueVOLTAGE Near-Optimal Linear Near-Optimal LinearStart-up Overshoot: SOi 0 0.09 1 0.85Loading Deviation: DRiL 0.20 0.47 0.97 0.55Unloading Deviation: DRiU 0.20 0.47 1 0.55RTiLSTi SOiDRiLRTiU DRiURTiLSTi SOiDRiLRTiU DRiUNear time-optimal response Linear Compensator responseTimeIndicesRegulationIndicesTimeIndicesRegulationIndicesFigure 4.4: Buck converter benchmarking indices for near-time optimal response shown inFig. 3.12, and linear compensator response shown in Fig. 4.1.buck converters. On the other hand, the dynamic response achieved with the linear controlleryields indices that range from 0.44 to 0.85, covering all performance categories (poor to ex-cellent). This unbalanced performance can be quickly identified by observing the right-handside performance hexagon shown in Fig. 4.4. The figure indicates that the controller underanalysis prioritizes overshoot minimization over settling time during the start-up transient,while showing a balanced dynamic regulation to recovery speed ratio during load transients.834.4. Using the Dynamic Performance IndicesTable 4.3: Benchmarked Boost Converters ParametersPARAMETER A BVcc 12 V 18 Vvref 24 V 36 VL 240 µH 490 µHC 112 µF 112 µFT0 1.03 ms 1.47 msZ0 1.46 Ω 2.09 Ωfsw 46.1 kH 32.6 kHzPout 69 W 108 WvC2.38ms (2.31)8.53V (0.35)12V (0.5)Figure 4.5: Start-up transient response benchmarking example: Converter A4.4.2 Boost Converter Dynamic Performance BenchmarkingThe performance of the two boost converters detailed in Table 4.3 assessed by employingthe proposed indices in this section. Both converters are controlled employing dual-loop PIschemes. As it can be observed in the table, the two converters show different input/outputvoltages and L-C values. This illustrates how, by employing the proposed performanceindices, the performance of different boost converters can be compared.The normalized load current transients selected for the analysis are 0.05-0.35 (step-up),844.4. Using the Dynamic Performance IndicesvC4.24ms (2.87)3.72V (0.1)18V (0.5)Figure 4.6: Start-up transient response benchmarking example: Converter BvC2.46A (0.3)0.41A (0.05)7.07V (0.29)1.72ms (1.66)1.96ms (1.9)6.55V (0.27)IoFigure 4.7: Load step up/down transient benchmarking example: Converter Aand 0.35-0.05 (step-down). The theoretical ideal values for the MRT and MVD responsescan be calculated employing (3.32), (3.37), (3.33), (3.45), and (3.41).The start-up and load/unload transients are detailed in the experimental captures shownin Figs. 4.5-4.8. The critical transient parameters have been indicated in the figures in realand normalized values (in brackets). These parameters and the indices calculated employing(4.1)-(4.4) are summarized in Table 4.4. The six performance indices are shown in Fig. 4.9forming a dynamic performance hexagon. This geometric representation enables a fasterinterpretation of the results. The time-related indices (STi, RTiL , and RTiU ) are shown on854.4. Using the Dynamic Performance IndicesvC2.58A (0.3)0.43A (0.05)3.48ms (2.36)3ms (2.04)7.28V (0.2)6.39V (0.18)IoFigure 4.8: Load step up/down transient benchmarking example: Converter BTable 4.4: Boost Converters BenchmarkingNormalized Value Index ValueTIME MRT A B A BStart-up: STi 0.41 2.31 2.88 0.62 0.57Loading Recovery: RTiL 0.28 1.66 2.36 0.61 0.53Unloading Recovery: RTiU 0.28 1.9 2.04 0.58 0.57Normalized Value Index ValueVOLTAGE MVD A B A BStart-up Overshoot: SOi 0 0.35 0.1 0.41 0.71Loading Deviation: DRiL 0.09 0.29 0.2 0.32 0.46Unloading Deviation: DRiU 0.09 0.27 0.18 0.35 0.54the left side of the hexagons, and the dynamic regulation indices on the right. As it can beobserved in the figure a balanced response is achieved in the Converter B, with most of theindices in the regular performance range. On the other hand, the performance hexagon isslightly shifted to the left in Converter A, indicating a response that prioritizes speed ratherthan dynamic regulation. In this way, the dynamic response obtained in different convert-ers can be quantitatively compared even for platforms with different L-C components andinput/output voltages. The proposed benchmarking procedure not only enables the relative864.5. SummaryRTiLSTi SOiDRiLRTiU DRiURTiLSTi SOiDRiLRTiU DRiUConverter A Converter BTimeIndicesRegulationIndicesTimeIndicesRegulationIndicesFigure 4.9: Benchmarking indices for two different linear controllers. Converter B showsa balanced response while Converter A shows a rather fast response while the dynamicregulation is sacrificed.comparison between different techniques, but also provides a theoretical ideal reference foran absolute dynamic performance evaluation.4.5 SummaryBased on the theoretical limits of performance derived in chapter 3, alternatives to evaluatecontroller’s dynamic performance were presented and analysed in this chapter. The directcomparison method analysed enables a detailed assessment of the controller’s performanceover time, while it requires both time domain response waveforms to be overlapped, makingthe benchmarking less straightforward and more subjective. The second alternative analysed,the transient-specific approach focuses the comparison on the values of recovery time andpeak voltage drop measured for a specific load current step. Since the method is basedin comparing numerical values, the benchmarking becomes more straightforward, but theprocess remains subjective to the user interpretation.In order to eliminate subjectivity from the benchmarking procedure, performance indiceswith clearly defined numeric categories, ranging from poor to excellent, were defined. Based874.5. Summaryon six numerical values measured from the converter’s dynamic response, the benchmarkingprocedure proposed provides an objective metric based on six numerical indices that canrange from 0 to 1. The proposed benchmarking method was employed to characterize thedynamic response of several sample responses found in buck and boost converters in orderto showcase its simplicity and applied value. The calculated indices and their graphicalrepresentation in a performance hexagon enable a thorough, direct and accurate assessmentof the controllers under evaluation on relative and absolute basis.88Chapter 5Unified Centric Control of DC-DCConvertersThis chapter introduces a PWM-geometric control method for DC-DC converters that canprovide reliable, smooth and fast transient response. The controller being introduced is basedon the insight provided by the ANTs derived in chapter 2. The performance indices intro-duced in chapter 4 are employed to benchmark the centric controller transient performance,yielding significant performance improvements respect to similarly complex techniques suchas dual-loop linear controllers. Some of the topics included in this section are part of thejournal paper published in IEEE Transactions on Industrial Electronics [2].5.1 Target Approaching TechniqueAs illustrated in Fig. 5.1 for the case of a buck converter, for any given initial conditions,a theoretically infinite number of averaged circular trajectories (in practice, as many as thePWM resolution enables) could be followed. One of those trajectories, and only one, is acircle that contains both initial and target operating points. The main control law to beemployed in the controller being proposed is represented precisely by this circle. In this way,as shown in Fig 5.2 for the unified model, and in Fig. 5.3 for boost converters, the targetoperating point can be reached by following a predictable and reliable circular trajectory.The desired circle can be obtained by evaluating the state variables at the target operating895.1. Target Approaching Technique0 vccniLn¸ON!0nrONcON¸AVG d1!0nrd1cd1¸AVGd2!0nrd2cd2[vCn(0); iLn(0)]vCnIon¸OFF!0nrOFFcOFFvCn(0) Vccntn10.50vCnVccnvCn(0)0iLniLn(0)tn10.5d 1d2 d1 0tn10.5IonFigure 5.1: Average Natural Trajectories in buck converters for several duty cycle values.point (vCn = vCnt ; iLn = iLnt) in (2.43):(vCnt −m1x Vccn)2 + (iLnt −m2x Ion)2 = (vCn(0) −m1x Vccn)2 + (iLn(0) −m2x Ion)2 (5.1)where m1x and m2x correspond to the control action necessary to direct the operating pointto the target.905.1. Target Approaching TechniqueiLntLoad line Initial conditionsTarget pointControlled centreiLniLn(0)vCn(0) vCntvCnm1xVccnm2xIon¯!xn = k!x 2¼Figure 5.2: Unified centric-based control lawRearranging terms, the circular trajectory to be followed can be represented as:λx : m1x 2Vccn(vCn(0) − vCnt)︸ ︷︷ ︸a+m2x 2Ion(iLn(0) − iLnt)︸ ︷︷ ︸b− (v2Cn(0) − v2Cnt + i2Ln(0) − i2Lnt)︸ ︷︷ ︸c= 0(5.2)where the auxiliary variables a, b, and c are defined such that: λx : m1x a+m2x b− c = 0.By replacing m1x and m2x with the functions of duty cycle they represent for each topol-ogy (detailed in Table 2.1) and rearranging terms, the control laws are obtained:Buck:dx =c− ba(5.3)Boost:dx = 1− a+ bc(5.4)Buck-boost:dx =c− bc− a (5.5)The obtained expressions determine the PWM duty cycle required to direct the converter’soperating point to the target. Since the duty cycle values range from 0 to 1, the determinedpaths to be followed (λx) are always contained inside the trajectories corresponding to dutycycle saturation values:λx ⊂ [(λxON > 0) ∪ (λxOFF > 0)] (5.6)915.1. Target Approaching Technique¯2!n2¯1!n1¯4!n4¯3!n3iLn1(0)iLn2(0)iLn3(0)iLn4(0)tniLniLnt1-VccnvCn1(0)vCn2(0)vCn3(0)vCn4(0)tn0vCn1d4tn02dd3d2d1[vCn3(0);iLn3(0)][vCn4(0);iLn4(0)]r1c1¯2¯1c3r3¯3c2r2[vCn1(0);iCn1(0)][vCn2(0);iLn2(0)]r4c4¯4iLnvCnVccn 1ioniLnt0¸xOFF¸x1¸x2¸x3¸x4¸xON!n1!n2!n3!n4Figure 5.3: Theoretical centric-based control trajectories for several initial conditions in aboost converter.where:λxON = m1ON a+m2ON b− c (5.7)λxOFF = m1OFF a+m2OFF b− c (5.8)Since the operating point angular speed is constant, the time in which the target is reached925.1. Target Approaching TechniqueTargetInitial conditionsLoad lineFigure 5.4: Centric-based trajectories experimental validation for several initial conditionsin a boost converteris defined by:tx =βkω 2pi(5.9)where the angle β is graphically found as:β = tan−1(iLn(0) −m2x IonvCn(0) −m1x Vccn)− tan−1(iLnt −m2x IonvCnt −m1x Vccn)(5.10)Once the target is reached after the time tx, the centre must be moved to the desiredsteady state operating point in order to determine a circular average trajectory with nullradius. The steady state duty cycle is determined using (2.38), and replacing vCnt and M1by their equivalent expressions for each converter:935.2. Complete Control LawBuck:D =1Vccn(5.11)Boost:D = 1− Vccn (5.12)Buck-boost:D =11 + Vccn(5.13)The analysis reveals a simple and powerful concept applicable to the three basic topologies:for a known input voltage and load current, the target operating point can be reached in awell determined time period given by (5.9), by following a reliable circular trajectory. Theonly required control action to solve the transient is to set the PWM duty cycle to a fixedvalue determined by (5.3), (5.4) or (5.5) according to the topology.The conceptual response of the control law applied to a boost converter is illustrated inFig. 5.3 in both geometric and time domains for several initial conditions. The figures showthe domain to which the control law must be restricted given by λxON and λxOFF . For anyinitial conditions located inside of the restricted domain, there is a constant duty cycle thatleads the operating point to the target following a circular trajectory λxn. The time takento reach the target is given by the angles covered βxn and the resonance frequency ωxn. Thecentric-based control concept is validated by experimental results shown in Fig. 5.4 wherepre-calculated duty cycle values are set to lead the operating point to the target.5.2 Complete Control LawBased on the insight provided by the ANTs, and employing the target approaching techniquedeveloped in the previous section, the complete centric-based controller is presented in thissection. Duty cycle saturations based on the load lines (2.35)-(2.37) and duty-cycle saturationtrajectories (5.7)-(5.8) are included in order to improve dynamic regulation and transientrecovery times.As a result, the controller being proposed, graphically described in Fig. 5.5, shows fivesectors: one corresponding to the target operating point neighbourhood, two corresponding945.2. Complete Control LawiLndVccnvCnd=dxd=dxIonIon¸xOFF¸xONDC-DC ConverteriLnvCnvCntd=1d=0d=D¸LL=0iLntFigure 5.5: Closed-loop centric-based controller conceptual schemeto ON and OFF duty cycle saturations, and two centric-based regions in which the dutycycle is ruled by (5.3)-(5.5). The complete control law is mathematically given by:d =D, for: (vCn(0) − vCnt)2 + (iLn(0) − iLnt)2 < δ2dx, for: λLL > 0, & λxOFF > 00, for: λLL > 0, & λxOFF ≤ 0dx, for: λLL < 0, & λxON > 01, for: λLL < 0, & λxON ≤ 0 (5.14)where δ defines a small neighbourhood of the target point.The duty cycle is periodically recalculated using the updated capacitor voltage and in-ductor current values as initial conditions (vCn0, iLn0), providing a rugged dynamic response.Ideally, the value dx should be calculated only once and a unique averaged circular trajec-tory would be followed. Nevertheless, a periodic recalculation of dx is performed in orderto compensate for deviations in the reactive component values, losses, and any extra effectnot being considered in the derivation of the ANTs that could take the trajectory away from955.2. Complete Control LawNormalizationcalculate: i Lnt, v Cnt v CVcc i L I oVccn , i Ln , v Cn , I on¸LL > 0¸xON > 0¸xOFF ≤ 0a = 2 Vccn [vCn − v Cnt]b = 2 I on [i Ln − i Lnt]c = vCn 2 − vCn t2 + i Ln 2 − i Lnt2ε2 = [i Ln − i Lnt]2 + [vCn − vCn t]2d = d xd = 0 d = 1 d = Dcalculate: a, b, cYesYesNoε2 > δ2Yes NoNo Yes Noi Lnt ,Boost = I on /Vccn , v Cnt ,Boost = 1 ¸xOFF,Boost = a + b − c ¸xON ,Boost = a + b ¸LL ,Boost = i Ln Vccn − I on vcnd x,Boost = 1 − a + bcD Boost = 1 − VccnFigure 5.6: Complete centric control-law implementation. The same control process appliesto the three topologies under analysis. Topology-specific equations employed for the caseof a boost converter are detailed in blue in order to illustrate the low complexity of theimplementation.ideal circles. In this way, the distortion in the circular trajectories is limited to the valueit reaches during one recalculation period. Further analysis regarding this particular aspectof the centric controller implementation is included in section 5.4. Since the state variablesare periodically sensed and compared to target values, the proposed control scheme can beunderstood as a particular, highly non-linear discrete-time state feedback controller.A flowchart of the complete controller implementation is shown in Fig. 5.6, where thegeneral case for the three topologies is shown in black, and the case-specific equations for965.3. Large Signal Stability Analysisa boost converter are included in blue in order to illustrate the simplicity of the equationsinvolved. A slow linear estimator of the inductor series resistance can also be implementedin order to account for potential steady state errors due to drops in this parasitic elements.Even though in high efficiency converters the DC resistances in the inductor and switchesare minimal, a small steady state error might be present, but is eliminated using the loadcurrent and the estimated value for the series resistance. It is worthwhile mentioning thatthe load current can be directly measured or estimated using the information on the outputvoltage derivative. A direct measurement was employed in this work to avoid potential noiseissues caused by the derivative term.Simulation results of the centric controller performance are shown in Fig. 5.7 for start-upand loading/unloading transients in the three topologies. As it can be observed, the centric-based control provides fast and sleek trajectories even for extremely large load transients,providing low peaks in inductor current and capacitor voltage and maintaining recovery timesin the same order of magnitude as the LC resonance period (T0n = 1).5.3 Large Signal Stability AnalysisThis section includes analysis to confirm the large-signal stability analysis of the controllaw being proposed. Given the geometric and highly non-linear nature of the centric-basedcontroller, the stability is analysed by employing Lyapunov’s method.Let the function f : ℜ2 → ℜf = (iLn − iLnt)2 + (vCn − vCnt)2 (5.15)be a proposed Lyapunov function, whose derivative must be a negative semi-definite function975.3. Large Signal Stability Analysis0 0.5 1 1.5−0.500.511.5vCni Ln01vCn01i Ln0 2 4 600.51tnd1 231 2 23Duty cycle saturationCentric controlled trajectoriesLoad line0.5 0.75 1 1.25−0.2500.250.50.75vCni Ln1vCn00.5i Ln0 2 4 600.51tnd1 231 2 23Duty cycle saturationCentric controlled trajectoriesLoad line−1.5 −1 −0.5 0−0.500.511.5vCni Ln−10vCn01i Ln0 2 4 600.51tnd1231 2 23Duty cycle saturationCentric controlled trajectoriesLoad line(a)(b)(c)Figure 5.7: Centric controller dynamic response simulation results for a) buck, b) boost, andc) buck-boost converters985.3. Large Signal Stability AnalysisTable 5.1: Global Stability Sufficient ConditionsTopology Sufficient ConditionBuck (d−D) (iLn − Ion) ≤ 0Boost (d−D) (iLnVccn − vcnIon) ≤ 0Buck-boost (d−D) [iLnVccn + Ion(vcn − Vccn)] ≤ 0in order to ensure global stability:dfdtn= 2(iLn − iLnt)diLndtn+ 2(vCn − vCnt)dvCndtn≤ 0 (5.16)Employing the converters characteristic equations (2.31)-(2.32), as well as the definitionof the target operating point (2.38)-(2.39), and rearranging terms the condition becomes:4pikω[iLn Vccn (m1 −M1) + vCn Ion (M2 −m2) + Vccn Ion (M1m2 −m1M2)] ≤ 0 (5.17)Replacing the coefficients by their respective values for each topology, the sufficient globalstability conditions detailed in Table 5.1 are found.The obtained sufficient conditions are composed by two multiplying factors, which musthave opposite sign in order to ensure global stability. Since the sign of the second factor isequivalent to the sign of λLL, the sufficient condition for global stability can be expressed as:(d−D) λLL ≤ 0 (5.18)Analysing the proposed controller illustrated in Fig. 5.5 it can be observed that:∀ λx : (m1x a+m2x b− c = 0) ⊂ (λxOFF > 0),m1x ∈ (m1OFF ,M1), and m2x ∈ (m2OFF ,M2)⇒ dx ∈ (0, D) (5.19)and, in an analogue way:995.4. Analysis of Non-Ideal Conditions∀ λx : (m1x a+m2x b− c = 0) ⊂ (λxON > 0),m1x ∈ (M1,m1ON ), and m2x ∈ (M2,m2ON )⇒ dx ∈ (D, 1) (5.20)Therefore, three stability regions are found in the proposed control law:Sector I: f < δ2, d = D⇒ (d−D) λLL = 0⇒ Bounded StabilitySector II: f > δ2, λLL > 0, dx ∈ [0, D)⇒ (d−D) λLL < 0⇒ Asymptotic StabilitySector III: f > δ2, λLL < 0, dx ∈ (D, 1]⇒ (d−D) λLL < 0⇒ Asymptotic StabilityThen, according to the Lyapunov stability criterion, the only potential oscillations will bebounded to δ inside of the steady state region indicated as sector I, which proves the stabilityof the proposed control law.5.4 Analysis of Non-Ideal ConditionsThe effects of non-ideal conditions such as deviations in the values of the reactive componentsand parasitic dissipative elements are discussed in this section.5.4.1 TolerancesTolerances in the values of reactive components affect the normalization base quantities andcause the converter’s behaviour to differ from the ideal case.Including this phenomena in the analysis, the normalization results:tn =tT0 k∆T0(5.21)1005.4. Analysis of Non-Ideal Conditionsvxn =vxvref(5.22)ixn =ix Z0 k∆Z0vref(5.23)where it is worthwhile highlighting the voltage normalization remains unaltered. The varia-tion coefficients are given by the components deviation:k∆T0 =√(1 + ∆L)(1 + ∆C) (5.24)k∆Z0 =√(1 + ∆L)/(1 + ∆C) (5.25)Since the control decisions are geometric-based, mismatches in the normalizing time wouldnot affect the dynamic behaviour as long as the normalized switching frequency remains muchhigher than unity. Deviations in the characteristic impedance make the trajectories differfrom ideal circles, compressing or expanding them in the direction of the iLn axis.Transient response simulations of the centric controller considering uncertainties of ±20%in L and C are shown in Fig. 5.8, for the normalized buck-boost converter. As it canbe observed, the trajectories that differ the most from the ideal case are those in whichthe deviations in the parameters present opposite sign maximizing the variations in thecharacteristic impedance.The effect of tolerances on the steady state operation can be taken into account byanalysing the steady state equations (2.33)-(2.34). Taking tolerances into account, the volt-ages remain unaltered and the currents are affected by the constant k∆Z0 :0 = m1 Vccn − vCnSS (5.26)0 = k∆Z0(iLnSS −m2 Ion) (5.27)since k∆Z0 6= 0, these equations are equivalent to (2.33)-(2.34). Therefore, the load line and1015.4. Analysis of Non-Ideal Conditions−1.5 −1 −0.5 0−0.500.511.5vCni Ln−10vCn01i Ln0 2 4 600.51tnd-20% +20%L C+20% -20%+20% +20%-20% -20%Figure 5.8: Centric-controlled buck-boost converter dynamic response simulation results fora LC parameter variation of ±20%the steady state behaviour remain unaltered under variations on the reactive componentvalues. The analysis highlights the rugged nature of the proposed centric-based control,which is able to successfully handle large parameter variations while still providing a reliableand predictable response with minor dynamic response deviations from the ideal case.5.4.2 Parasitic ResistancesDissipative losses in the system can be electrically synthesized as resistors. The losses onswitching devices are lumped together with the inductor series resistor because these elementscarry the same current while each of the switches conduct.Steady State EffectsParasitic losses modify the load line and can cause steady state error in the converter. Theload line equations including the parasitic resistances result:0 = m1 Vccn − vCnSS −iLnSS rLnkω(5.28)1025.4. Analysis of Non-Ideal Conditions0 0.5 111.051.11.151.2kiLIon0.750.80.850.90.951ηVccn = 20 0.2 0.411.051.11.151.2kiLIon0.750.80.850.90.951ηVccn = 10 0.2 0.411.11.21.31.4kiLIon0.750.80.850.90.951ηVccn = 0.5Buck-BoostBoostBuckrLn= 0 rLn= 00.080.0250.02500 ~ ∞0.050.050.0750.0750.10.1rLn= 00.030.0300.060.060.090.090.120.120.160.240.32Figure 5.9: Load factor and efficiency as function of load current for several parasitic seriesresistances in the three basic DC-DC converters0 = iLnSS −m2 Ion (5.29)In order to avoid steady state error, this effect must be taken into account to calculatethe target inductor current as well as steady state duty cycle. The target inductor currentis determined solving (5.28) and (5.29) at the target operating point. As a result, the factorkiL is defined such that:iLnt = kiL iLnt(ideal) (5.30)This factor is shown in Fig. 5.9 for the three topologies along with the converter’s efficiency,as a function of the load current using the series parasitic resistance as parameter. Theparasitic series resistance can be determined from the curve for a given efficiency and loadvalue, then the corresponding kiL curve is selected. The curve is then loaded into a look-uptable that determines the target current multiplying factor for a given load current.Solving (5.28) and (5.29), the non-ideal steady duty cycles are found for buck, boost andbuck-boost converters:DBuck =1 + iLnt rLnVccn, (5.31)1035.4. Analysis of Non-Ideal ConditionsDBoost = 1 + iLnt rLn − Vccn, (5.32)DBuck−boost =1 + iLnt rLn1 + Vccn(5.33)These equations provide the foundation to achieve excellent and predictable dynamicresponse, even for low efficiency converters with noticeable resistive losses.Dynamic EffectsParasitic series resistances produce damping in the system, making the circular trajectoriesbecome spirals (radius attenuation along time). Since the duty cycle is periodically recalcu-lated, the effect of parasitic impedances during transient is limited to the values it reachesduring one recalculation period. Due to the low computation time required, the recalculationcan be performed in a fast manner, compensating for the dynamic effect of losses.The analysis including parasitic elements for buck converters is performed below, as anexample aimed to mathematically illustrate the validity of the stated assumption. The dif-ferential equations that represent the normalized buck converter including losses are:12pidiLndtn= d Vccn − iLn rLn − iCn rCn − vCn (5.34)12pidvCndtn= iLn − Ion (5.35)where rLn represents the series resistance in the inductor and switches, and rCn indicatesthe capacitor series resistance. Solving the differential equations system (5.35)-(5.34) timedomain expressions that describe the time evolution of the state variables are found as:vCn(tn) = Att(tn){(vCn(0)− c′) cos(ωdntn)+[(vCn(0)− c′) rCn + rLn2+ (iLn(0)− Ion)] ωonωdnsin(ωdntn)}+ c′ (5.36)1045.4. Analysis of Non-Ideal ConditionsiLn(tn) = Att(tn){(iLn(0)− Ion) cos(ωdntn)−[(iLn(0)− Ion) rCn + rLn2+ (vCn(0)− c′)] ωonωdnsin(ωdntn)}+ Ion (5.37)with:ωon = 2pi; (5.38)ωdn = ωon√1−(rCn + rLn2)2(5.39)Att(tn) = e−ωontnrCn+rLn2 (5.40)c′ = d Vccn − Ion rLn (5.41)where ωon and ωdn are the ideal and damped normalized resonance frequencies respectively.The time-dependent function Att(tn) describes the attenuation of the radius along the nor-malized time, and c′ gives the location of the new centre.The difference given by the radius attenuation over one recalculation period given by:∆r = 1− Att(1frecn)= 1− e−ωonrCn+rLn2 frecn (5.42)is negligible if:rCn + rLn2 frecn<< 1 (5.43)Which holds true for modern power electronics applications since rCn and rLn are small(order of magnitude as high as 0.1 Ω), and the recalculation frequency frec can be set torelatively large values (order of magnitude as low as kHz) due to the low computation timedemanded by the technique.1055.5. Experimental ResultsTable 5.2: Converter ParametersPARAMETER BUCK BOOSTvref 12V 24VVcc 24V (2) 12V (0.5)L 82µH 240µHC 58µF 45µFT0 433µs 654µsZ0 1.2Ω 2.3Ωfsw, frec 80kHz (35) 68.8kHz (45)iLndriving signalvCnStart-upCentric-controlled trajectoriesduty cycle saturationLoadingUnloadingIon245¹s (0.57)259¹s (0.6)7.5A (0.75)295¹s (0.68)1231 2 3 2Figure 5.10: Start-up and load current step up/down transient response in a buck converter5.5 Experimental ResultsExperimental results of the centric-controlled buck and boost converters are shown in Figs.5.10 and 5.11. The parameters of each converter are detailed in Table 5.2, where the nor-malized parameters are indicated between brackets. Start-up and large current step-up/step-down transients are included in the experimental captures. As shown in the experimentalcaptures the converters’ operating points follow sleek circular trajectories with small currentpeaks and voltage drops, reaching steady state in a fast (same order of magnitude as T0) andreliable manner even for extremely large load transients.1065.5. Experimental ResultsiLndvCnStart-upCentric-controlled trajectoriesduty cycle saturationLoadingUnloadingIon0.64ms (0.98)1ms (1.53)1.3ms (1.98)6.8V (0.57)6.6V (0.55)123101 2 3 22.6A (0.25)Figure 5.11: Start-up and load current step up/down transient response in a boost converterLoadingTime -optimal responseUnloadingiLndvCnIon2A (0.2)3A (0.29)8A (0.78)Figure 5.12: Load current step up/down transients in a boost converter compared to thetime-optimal responseThe transient response of the centric-controlled boost converter for load-step transientsis compared with the time-optimal solution in Fig. 5.12. The recovery times achieved withthe centric controller are in the same order of magnitude (3 and 2.5 times) as the absolutetheoretical limit, which highlights the fast response of the proposed technique. Furthermore,the dynamic regulation is largely improved: the voltage drop and current overshoot duringthe loading transient are respectively reduced to half and one third of the time-optimalresponse, while during the unloading transient the voltage peak reaches 1.1 times the time-1075.6. Comparison with Dual-loop PI Controller0.5 1 1.5−0.500.51vCni Ln0.511.5vCn−0.500.51i Ln0 6 12 1800.51tnd0.40.30.2∆Ion=0.1(a)(b)0.5 1 1.5−0.500.51vCni Ln0.511.5vCn−0.500.51i Ln0 6 12 1800.51tnd0.40.30.2∆Ion=0.1Figure 5.13: Start-up, loading and unloading transient response in the normalized boostconverter with a) dual-loop linear controller, and b) centric controller.optimal response and the current undershoot is reduced to one quarter.5.6 Comparison with Dual-loop PI ControllerThe proposed control technique is compared with industry standard PI controllers in thissection. Transient response plots of the normalized boost converter are presented in Fig. 5.13for linear (a) and centric (b) controllers. In both cases the start-up transient is performedwith no load, the normalized switching frequency is 35 (same as in the experimental results1085.6. Comparison with Dual-loop PI Controllerof Fig. 5.11) ,the normalized input voltage is Vccn = 0.5, the normalized inductor parasiticresistance is rLn = 0.1 and the load steps (up and down) are ∆Ion = [0.1, 0.2, 0.3, 0.4].The implemented linear compensator is a dual-loop (inner current, outer voltage loops),and it features two poles and two zeroes in each loop. The compensator was tuned based onthe loading transient of ∆Ion = 0.2, in order to obtain a smooth yet not- sluggish response.As it can be observed in Fig. 5.13 (a) for the dual-loop linear controller, the inductor currentshows an overshoot of 18% for the transient the design was based on (∆Ion = 0.2). The peakvoltage drop reaches 30%, and the normalized transient recovery time is 1.6. For the rest ofthe load step-up transients shown in the figure, the response of the dual-loop linear controlleris either sluggish (for smaller load steps) or too aggressive (for larger load steps), producingoscillations in the inductor current and capacitor voltage. For all the start-up and unloadingtransients analysed, the dual-loop linear controller shows overshoot in the capacitor voltageand undershoot in the inductor current. Some signs of reaching instability (large oscillationsin the duty cycle) can be observed in the larger load step-down transients.On the other hand, sleek, fast and reliable responses are provided by the centric controllershown in Fig. 5.13 (b). For instance, for the load step-up transient of ∆Ion = 0.2 (themost favourable for the dual-loop linear), an inductor current overshoot of 18% is observed(same as in the dual-loop controller), while the peak voltage drop reaches 16% (almost halfof the observed in the linear case), and the normalized recovery time is 1 (60% lower than inthe linear counterpart). As it can be observed in the comparison the centric scheme showssuperior performance for all the transients analysed.A digital platform (microcontroller, DSP, or FPGA) is required to implement the centriccontroller, and besides the two state variables (inductor current and capacitor voltage), theinput voltage and load current are part of the control law.Experimental results of load step-up transients in linear and centric controllers for thebuck converter detailed in Table 5.2 are shown in Fig. 5.14. The load step-up transient is in1095.6. Comparison with Dual-loop PI Controller7A (0.7)546us (1.26)840nssamples & averageprocessingtimeInductor saturation4.75A(0.475)5.6V (0.47)d (zoom)diLnvCnVccn=2∆Ion = 0.7(a)1120nssamples & averageprocessingtime7A (0.7)1.75A(0.175)315us (0.73)3.95V (0.33)d (zoom)diLnvCnVccn=2∆Ion = 0.7(b)Figure 5.14: Buck converter transient performance and computation time comparison be-tween a) dual-loop linear controller, and b) centric controller. The duty cycle (C3) andcomputation flag (C4) signals are zoomed in the time axis in (Z3 and Z4 respectively).both cases 0−to−7A, and the switching frequency 80kHz.As shown in Fig. 5.14 (a), hinted by the inductor current reaching saturation levels,the linear controller was tuned to provide an aggressive response, aiming to minimize the1105.6. Comparison with Dual-loop PI Controllercapacitor voltage drop and recovery time. The current spikes reach 4.75A (0.475 normalized),while the peak voltage drop is 5.6V (0.47) and the recovery time is 546µs (1.26). The centriccontroller response for the same transient is shown in 5.14 (b). The current spikes reach1.75A (0.175 normalized), almost one third of the value obtained with the linear controller.The peak voltage drop is 3.95V (0.33 normalized), which is 40% lower than in the linearcontroller, and the recovery time is 315µs (0.73 normalized), showing a 73% improvement.The oscilloscope captures shown in Fig. 5.14 also provides details regarding the compu-tation burden of each control technique, which were implemented in an industry-standardTI C2000 DSP running at 80 MHz clock. The signals shown in channels C3 and C4 indi-cate the state of the switch and a computation flag respectively, and are zoomed for betterappreciation in the waveforms Z3 and Z4. As it can be observed for both controllers, thecomputation flag (Z4) is raised and lowered five times per PWM cycle. The first four timesare evenly distributed over the switching cycle, and they correspond to four ADC samplesand their average in the last sampling (which takes longer than the previous three). Once theaverage values are ready, the averaged signals are processed by the controllers. The compu-tation time is given by the fifth raise and fall of the computation flag. The linear controlleris dual-loop, with two-poles and two-zeroes and anti wind-up implemented in each loop, andit takes 0.84µs to be computed. The centric controller computation time obtained is 1.12µs,marking a 33% increase respect to the linear controller. The low computation time for theproposed controller, comparable to a dual-loop 2P2Z controller reflects the typical processingpower of industry-standard microcontrollers/DSPs and their associated rapid multiplicationtimes.1115.7. Centric Controller BenchmarkingTable 5.3: Centric Controller BenchmarkingNormalized Value Index ValueTIME Buck Boost Buck BoostStart-up: STi 0.57 0.98 0.86 0.81Loading Recovery: RTiL 0.56 1.53 0.81 0.71Unloading Recovery: RTiU 0.68 1.99 0.77 0.66Normalized Value Index ValueVOLTAGE Buck Boost Buck BoostStart-up Overshoot: SOi 0.03 0 0.95 1Loading Deviation: DRiL 0.38 0.25 0.66 0.8Unloading Deviation: DRiU 0.32 0.28 0.78 0.75RTiLSTi SOiDRiLRTiU DRiURTiLSTi SOiDRiLRTiU DRiUTimeIndicesRegulationIndicesTimeIndicesRegulationIndicesCentric BuckLinear BuckCentric BoostLinear BoostFigure 5.15: Benchmarking indices for the centric controller in buck and boost converters.5.7 Centric Controller BenchmarkingThe index-based benchmarking method introduced in Chapter 4 is employed in this sectionto assess the dynamic performance of the centric-based controller. The transient times andvoltage deviations found in the experimental results shown in Figs. 5.10 and 5.11 are usedas inputs for the index calculation through (4.1)-(4.4).The normalized measured start-up time, loading/unloading recovery times, start-up over-1125.8. Summaryshoot, and loading/unloading voltage deviations are detailed in Table 5.3 along with theresulting indices. As shown in the table, the index values obtained for the centric controllerrange from 0.66 to 0.95 in buck converters, and 0.66 to 1 in the case of boost converters.The indices values obtained employing industry-standard controllers (detailed in Tables 4.2and 4.4), range from 0.44 to 0.85 for buck, and from 0.46 to 0.71 for boost converters. Thesuperior performance of the centric-based controller when compared to similarly complexindustry standard linear controllers is illustrated by the overall higher indices obtained. Theindices for both centric and linear controllers in buck and boost converters are graphicallyshown in the performance hexagons detailed in Fig. 5.15. The figure provides a more clear vi-sualization of the superior performance achieved by the centric controller, while highlightingthe practical value of the benchmarking method proposed.5.8 SummaryA large signal, high performance control scheme for the three basic DC-DC topologies wasintroduced in this chapter based on the ANTs introduced in chapter 2.A target approaching technique, based on the simple principle of following a unique circulartrajectory of constant duty cycle to reach the desired operating point, was introduced. Thismethod was complemented with extra duty cycle saturation rules aimed to enhance thedynamic response to constitute the centric controller. The large-signal stability was analysedthrough Lyapunov analysis, proving that the proposed control method is stable over the entirestate plane providing reliable and predictable responses even for extremely large transients.Non-ideal conditions such as component tolerances and the static and dynamic effects ofparasitic resistances over the converter’s transient response are analysed to guarantee stableoperation.The centric controller was implemented in buck and boost converter experimental plat-1135.8. Summaryforms through an industry standard microcontroller in order to validate the fast and pre-dictable dynamic response expected. The centric controller transient performance was com-pared with the one obtained through similarly complex dual-loop controllers, where the fasterand more predictable performance of the proposed method can be clearly observed. Thebenchmarking procedure introduced in chapter 4 was employed to support the performanceassessment of the centric controller.114Chapter 6State-plane Direct MPPT Control forSolar PVThis chapter introduces a new Maximum Power Point Tracking (MPPT) method for Photo-Voltaic (PV) applications based on the Average Natural Trajectories. The large-signal ANTsof a fixed output voltage boost (or inverted buck) topology are derived including the ef-fects of the PV cells connected to the input. The insights provided by this accurate modelare combined with the traditional Incremental Conductance (IC) method to create a high-performance MPPT. The proposed method can find the MPP in a period of time up to twoorders of magnitude shorter than traditional IC implementations, which makes the techniquevery appealing for scenarios with extremely fast changing environmental conditions such assolar backpacks, bikes or even car rooftop mounted PV applications.The content of this section is partially included in the journal paper published in IEEETransactions on Industrial Electronics [6].6.1 Irradiance Profiles in Typical Moving PVApplicationsThe changes in irradiance due to sun-to-shade displacement of PV cells was measured forthree different applications in order to define the dynamic requirements placed on the MPPTalgorithm. The plots in Fig. 6.1 show fast irradiance changes measured on a 10W PV panel1156.2. Incremental Conductance MPPT BasisBackpack20s timespan 0.5s timespan0%50%100%Bike0%50%100%t [s]0 5 10 15Car0%50%100%t [ms]0 250Irradiance Profiles Measured during Typical Operationas percentage of 1,000 =1 %ms=2 %ms=10 %ms50ms50%25ms50%5ms50%Figure 6.1: Irradiance profiles measured in backpack, bike, and car mounted PV. Maximumirradiance slopes identified as 1, 2 and 10 percent per millisecond.placed on a backpack, bike and car wind shield during typical operation. The irradiance wasmeasured by recording the short-circuit current at a 1ms sample time and it is shown aspercentage of the nominal irradiance of 1,000 Wm2. Irradiance change rates as high as 1%, 2%,and 10% per millisecond are found in the backpack, bike, and car scenarios respectively.6.2 Incremental Conductance MPPT BasisHill-climbing methods, such as IC, provide a mean to find the maximum power point andthey can be implemented following dual-loop, or direct duty cycle control approaches, asshown in Fig. 6.2. This section reviews the fundamental principles IC MPPT as well as anexample of the traditional implementation using direct duty-cycle control.1166.2. Incremental Conductance MPPT BasisDual Stage IntegratedStand AloneMPPTDual LoopPI ControlvCiLdPWMDirectDuty CycleMPPTControldPWMFigure 6.2: Existing MPPT architectures comparison: Dual Stage Stand Alone MPPT vs.Single Stage Direct Duty Cycle MPPTThe power delivered by a PV cell is given by:PPV = iPV vPV (6.1)Therefore, at the maximum power point:dPPVdv(MPP ) = vMPPdiPVdvPV(MPP ) + iMPP = 0 (6.2)Rearranging terms to the form of conductances, and taking only positive values:gMPP =|diPV ||dvPV |(MPP ) =iMPPvMPP(6.3)The single-cell-type I-V curve of a PV panel is shown in Fig. 6.3 and mathematicallydefined by:iPV = Iph − I0(evPVVT − 1) (6.4)The PV absolute conductance gDC =iPVvPVand the PV incremental conductance gˆac =|diPV ||dvPV |are determined by the PV cell curve. As given by (6.4) and shown in Fig. 6.3, gDCcontinuously decreases with the increase of the PV cell voltage vPV , while gˆac continuously1176.2. Incremental Conductance MPPT BasisiPVInState-plane MPPT ControlOutgDCgacgac<gDC gac>gDCPPVvMPPiMPPgMPPMPvPVVDCdvPViPV LPWMCFigure 6.3: IC MPPT algorithm embedded into the boost converter control law.increases along the entire range. Therefore, (6.3) is only satisfied at the MPP where vPV =vMPP , yielding:vPV < vMPP ⇒ gˆac = |diPV ||dvPV | <iPVvPV= gDC ; andvPV > vMPP ⇒ gˆac = |diPV ||dvPV | >iPVvPV= gDC (6.5)As illustrated in Fig. 6.3, (6.5) provides sufficient information to determine the directionin which the MPP is located. The main IC control law is found from (6.5):if: |∆iPV | vPV ≤ |∆vPV | iPV ⇒ dvPVdt > 0else if: |∆iPV | vPV > |∆vPV | iPV ⇒ dvPVdt < 0(6.6)These expressions can be employed to update the duty-cycle according to the desiredchanges in the PV voltage.1186.2. Incremental Conductance MPPT BasisiniLn’δ δ δ δvPVn’iPVniLn’vPVn’pPVn’pPVmaxtU’iPVn’d’tU’ tU’ tU’δδδδttttvipuFigure 6.4: PV voltage increment transient response of Traditional direct duty cycle MPPTin state plane and time domains.6.2.1 Traditional IC MPPT with Direct Duty-cycle ControlIn a traditional direct MPPT algorithm, (6.6) is employed to modify the duty cycle d everyupdating interval t′U taking the converter to a new steady state condition. Since the convertermust reach steady state before a new MPPT iteration can be performed, then:t′U >> T0 = 2pi√LC (6.7)1196.3. Converter Dynamic Behaviourwhere T0 is the LC resonance period. Clear guidelines to define t′U in traditional applicationsare provided in [104, 105]. Typical values of the updating period t′U range between 20 to 30times the resonance period T0, as illustrated in Fig. 6.4. As shown in Fig. 6.4 in XY andtime domains, the state variables (inductor current - i′Ln, and PV voltage - vPV n) show adamped oscillatory behaviour and converge to steady state within the update time t′U .As opposed to traditional methods, the State-plane direct MPPT avoids the necessityof reaching steady state before making a new decision, achieving extreme dynamic trackingimprovements. The method being introduced provides a dynamic solution for the dynamicproblem of MPPT, which is enabled by a dynamic State-plane analysis of the converter’sbehaviour as opposed to traditional steady-state based approaches.6.3 Converter Dynamic BehaviourThe inverted-buck topology shown in Fig. 6.3, is a very common choice for the front-endDC-DC stage in PV systems. The average behaviour of the converter is described by:LdiLdt= vPV − d VDC (6.8)CdvPVdt= iPV − iL (6.9)where d is the PWM duty-cycle, and the PV current iPV is given by (6.4) as a function ofthe PV voltage vPV .In order to gain generality, and obtain the Average Natural Trajectories that describe thebehaviour of the topology, the equations are normalized as: tn =tT0, Zxn =ZxZ0, vxn =vxVDC,and ixn = ixZ0VDC. Using the LC characteristic impedance Z0 =√L/C, and resonance periodT0 = 2pi√LC as base-quantities.The resulting normalized expressions are independent from the LC parameters and input-1206.3. Converter Dynamic BehaviourVDCn=1dvPVniPVniLnPWMPV Norton Equivalent Ln =12πCn =12πiPVnĝacnFigure 6.5: PV Norton equivalent circuit connected to the normalized inverted buck con-verter.output voltage levels:12pidiLndtn= vPV n − d (6.10)12pidvPV ndtn= iPV n − iLn (6.11)The normalized inverted buck converter with a Norton equivalent of a PV cell connectedat the input is illustrated in Fig. 6.5. The equilibrium conditions are defined by nil statevariables derivatives, i.e.:vPV n = d (6.12)iLn = iPV n(d) (6.13)The AC conductance gˆacn is defined by the slope of the PV cell I-V curve (6.4):gˆacn =|diPV n||dvPV n| (6.14)The PV current can be linearised around a given operating point (d, iPV n(d)) as:iPV n = iPV n(d) − gˆacn(vPV n − d) (6.15)1216.3. Converter Dynamic Behaviourwhich can be replaced in (6.11), yielding:12pidvPV ndtn= iPV n(d) − gˆacn(vPV n − d)− iLn (6.16)Time-domain expressions that describe the dynamic behaviour of the system are obtainedby solving (6.10) and (6.16):vPV n(tn) = d+ Att(tn){∆v(0) cos(ω′0ntn)−[∆v(0)gˆacn2+ ∆i(0)]1k1sin(ω′0ntn)}(6.17)iLn(tn) = iPV n(d) + Att(tn){∆i(0) cos(ω′0ntn) +[∆v(0) +∆i(0)gˆacn2]1k1sin(ω′0ntn)}(6.18)where ∆v(0) and ∆i(0) are the initial voltage and current distances to the equilibrium point:∆v(0) = vPV n(0) − d (6.19)∆i(0) = iLn(0) − iPV n(d), (6.20)the function Att(tn) defines the damping of the oscillations:Att(tn) = e−pigˆacntn , (6.21)the damped resonance frequency ω′0n is:ω′0n = k12pi, (6.22)and the coefficient k1 is given by:k1 =√1−(gˆacn2)2(6.23)1226.3. Converter Dynamic BehaviouriPVniPVn(u)iLnvPVndĝacnω0n=2π k1ʹvPVn(0)iLntniLn(0)iLn(0)vPVn(0)vPVntniLn(0)Figure 6.6: Inverted buck converter dynamic state-plane behaviour: circular trajectories withcentre at (d,iPV n), and angular speed ωon = 2pi.The waveforms (6.17) and (6.18) are illustrated in Fig. 6.6 in time domain, and in theXY state-plane formed by vPV n and iLn. Since the converter dynamics are determined bythe PV incremental conductance (gˆacn), the response changes along the I-V curve. Responsescorresponding to gˆacn = 0 (blue) and gˆacn > 0 (yellow) are shown in Fig. 6.6. The spiraltrajectory described in the state plane is defined by the equilibrium conditions (d,iPV n(d)),initial conditions (vPV n(0),iLn(0)), and PV incremental conductance (gˆacn).In order to simplify the analysis while ensuring predictable behaviour, the control worst-case scenario (i.e. no damping, gˆacn = 0) is adopted. Plugging this condition into (6.21)yields:vPV n(tn) = d+ [vPV n(0) − d] cos(2pi tn)− [iLn(0) − iPV n(d)] sin(2pi tn) (6.24)iLn(tn) = iPV n(d) + [iLn(0) − iPV n(d)] cos(2pi tn) + [vPV n(0) − d] sin(2pi tn) (6.25)Merging (6.24) and (6.25) by eliminating the time tn, the geometric shape described bythe state variables is found:λd : (vPV n − d)2 + (iLn − iPV n(d))2 = (vCn(0) − d)2 + (iLn(0) − iPV n(d))2 (6.26)1236.4. State-plane direct MPPT ControlAs shown in (6.26) the state variables describe a circular trajectory with its centre at(d,iPV n(d)), and its radius defined by the initial conditions, as illustrated in Fig. 6.6 forgˆacn = 0. This circular trajectory describes the most challenging condition from a controlstandpoint, including all possible spirals inside of it, and ensuring reliable and predictablebehaviour.6.4 State-plane direct MPPT ControlThe control scheme is developed in this section by combining the basis for MPPT set insection 6.2 with the large-signal, state-plane model of the converter developed in section 6.3.In this way, the proposed controller ensures large-signal reliable operation while ensuringmaximum power extraction.6.4.1 Main Control LawAs opposed to the traditional steady-state-based direct duty-cycle MPPT, the State-planedirect MPPT finds its cornerstone in the simple idea of continuously operating the powerconverter slightly away from its equilibrium conditions given by (6.12) and (6.13). Thedynamic behaviour of the proposed method is shown in Fig. 6.7. As shown in the figure, thecircle centre is periodically set to a distance δ from the measured PV voltage (i.e.:d[z+1] =vPV n[z] + δ). In this way, the equilibrium conditions are never met, and the operating pointcontinuously moves in search of the maximum power point. The centre-updating mechanismis shown in Fig. 6.7: the PV voltage vPV n is sampled at the instant indicated by , and thecentre (given by d) is then placed at a distance δ to the right, remaining at that locationuntil a new update is performed. As shown in the figure, the inductor current (iLn, blue)describes fractions of spirals during the time the centre remains at a given location. The angle(α) covered by the operating point during an update period (tU) is defined as a fraction of a1246.4. State-plane direct MPPT ControlvPVnλuiPVnδ δ δ δ δiniLn’iLnαiLn’vPVn’pPVn’d’ttttiLnvPVnpPVnvPVnpPVmaxtUiPVndtU tU tUδδδδvipuFigure 6.7: PV voltage increment transient response of proposed direct duty cycle MPPT instate plane and time domains.turn given by the ratio between update and resonance periods:α = 2pitUT0< pi (6.27)The sign of the displacement is defined according to the desired PV voltage derivativedefined by (6.6). The embedded MPPT control rules can be found by plugging the above-1256.4. State-plane direct MPPT Controldescribed control actions into (6.6):if:∣∣∆iPV [z]∣∣ vPV [z] ≤ ∣∣∆vPV [z]∣∣ iPV [z]then: d[z+1] = vPV n[z] + δelse: d[z+1] = vPV n[z] − δ (6.28)where ∆x[z] = x[z] − x[z−1], vPV n[z] is the normalized PV voltage at the update moment ( ),and δ << 1 is the displacement from the converter’s equilibrium condition. In this way,the operating point is continuously moving in search of the maximum power point, and theproposed technique shows a dynamic solution for the dynamic problem of MPPT.The orange, dotted waveforms shown in Fig. 6.7 depict the trajectories described in thetraditional direct duty cycle MPPT shown in Fig. 6.4, and highlight the dynamic improve-ments introduced by the State-plane direct MPPT. The circular areas λd given by (6.26) areshown in solid grey and they contain all possible trajectories (no damping).6.4.2 Compensation of Potential Steady State ErrorThe equilibrium conditions (6.12) and (6.13) are valid only for a lossless scenario. Resistivelosses in the inductor, PCB and switches can be lumped together as rLn, turning (6.12) into:vPV n = u+ iPV n(d)rLn (6.29)Solar PV converters usually show high efficiency leading to d >> iPV n(u)rLn, and (6.29)approximates to (6.12). However, these losses are taken into account in the design procedureshown in section 6.6 to determine the minimum magnitude of the displacement δ.In an ideal State-plane direct MPPT controlled converter, the steady-state waveform ofthe displacement δ is symmetric with nil cycle mean value, as proven in section 6.5. Lossesand other non-ideal conditions affect the centre placement resulting in a waveform withnon-zero mean value. Therefore, a very slow integration of the displacement δ waveform1266.5. Equilibrium Characterizationcan be implemented in order to compensate for potential steady state errors. Multiplyingthe result of the integration by the PV current iPV n ensures the term compensates for theseries resistance rLn, which changes very slowly with temperature variations and componentsageing. The resulting control law can therefore be expressed as:d[z+1] = vPV n[z] ± δ − iPV n[z] Kint∑(δ)︸ ︷︷ ︸≈rLn(6.30)with Kint << tU .6.4.3 ImplementationA block diagram of the State-plane direct MPPT is shown in Fig. 6.8. As in traditionalaverage controllers, the PV voltage and current are averaged over an entire switching period(tsw) in order to remove ripple components. As shown in the figure, the implementation ofthe State-plane direct MPPT maintains the simplicity of the traditional direct duty cycleMPPT, making the algorithm very suitable for implementation in low-cost microcontrollers.The main difference versus traditional direct duty-cycle MPPT is given by the much higherupdate frequency found in the proposed method (calculated in the design procedure providedin section 6.6), which is obtained by employing state-plane analysis and enables extremeimprovements in the dynamic tracking efficiency.6.5 Equilibrium CharacterizationThe equilibrium behaviour of the State-plane MPPT controlled converter is analysed in thissection. Assuming worst-case scenario conditions, the PV current is considered constant andindependent from the PV voltage (no damping), as shown in Fig. 6.9. In order to simplifythe notation, the variables: vk = vPV n[k] − vMPP , and ik = iLn[k] − iPV n[k] are defined, where1276.5. Equilibrium Characterizationv P Vv DC-1v P V [z ]d [z +1]AVGPWMt swt swZOHδ sgn(x)z-1|x|K int (x) ≈ r Lnt UAVG t swi P Vi P V [z ]|∆i P V [z ]| v P V [z ]|∆vP V [z ]| i P V [z ]ZOHz-1|x|t UFigure 6.8: Complete State-plane direct MPTT control scheme.k indicates the kth sample, taken every update time interval tU .Given the predictable circular path the state variables follow, and the fixed update rate,the location of future samples (vk+1, ik+1) can be calculated at any given sampling instant k.vk+1 = vk + δ sgn(vk)[cos(α)− 1]− ik sin(α) (6.31)ik+1 = δ sgn(vk) sin(α) + ik cos(α) (6.32)1286.5. Equilibrium CharacterizationαEquilibrium trajectoryTransient trajectorySamplingEquilibriumttvivqiqq= 0 1 2 3 m-1 mq=0123m-1mα2πT0ΔvqviFigure 6.9: State-plane MPPT controller equilibrium behaviour under no-damping conditions(worst case-scenario).The increment of the variables during the time between samples (tU) is then given by:∆vk = δ sgn(vk)[cos(α)− 1]− ik sin(α) (6.33)∆ik = δ sgn(vk) sin(α) + ik[cos(α)− 1] (6.34)1296.5. Equilibrium CharacterizationThis process can be re-iterated to find the operating point location at any future samplinginstant. The PV voltage and inductor current values oscillate around the maximum powerpoint, as shown in Fig. 6.9. These oscillations converge to an equilibrium condition where theoperating point repeats its locations (vq, iq) after m iterations. The equilibrium conditionsare given by:m−1∑q=0∆vq = 0 (6.35)m−1∑q=0∆iq = 0 (6.36)where the index q < m indicates the different steady state points, as shown in Fig. 6.9.Plugging (6.33) and (6.34) into (6.35) and (6.36):δ [cos(α)− 1]m−1∑q=0sgn(vq)− sin(α)m−1∑q=0iq = 0 (6.37)δ sin(α)m−1∑q=0sgn(vq) + [cos(α)− 1]m−1∑q=0iq = 0 (6.38)Combining (6.37) and (6.38):2δ [cos(α)− 1]m−1∑q=0sgn(vq) = 0 (6.39)Since δ 6= 0 and α 6= 0, (6.39) yields:m−1∑q=0sgn(vq) =m−1∑q=0δ = 0 (6.40)which indicates an equal amount of positive and negative displacements δ per cycle. Thiscondition is employed as redundancy to ensure any potential steady state error is eliminated,as detailed in section 6.4.2.1306.5. Equilibrium CharacterizationPlugging (6.40) into (6.37) or (6.38) yields:m−1∑q=0iq = 0 (6.41)From these equilibrium conditions it can be determined that stable periodic behavioursrequire even values of m, with 2 < m < 3piα. Therefore, the possible values of m for a givenα are:m ={4, 6, 8, ..., 2floor(3pi2α)}(6.42)where floor(x) is the flooring function, which rounds numbers down to the previous integer.Starting from (v0, i0), the mth operating point location can be found by employing (6.31)and (6.32) re-iteratively m times:vm = v0 + δ [cos(α)− 1]m−1∑k=0sgn(vk)− sin(α)m−1∑k=0ik (6.43)im = i0 cos(α)m + δ sin(α)m−1∑k=0sgn(vk) cos(α)m−1−k (6.44)Inserting (6.40) and (6.41) into (6.43) yields:vq = vm = v0 (6.45)This indicates there are no extra constrains for the equilibrium voltage values. As a result,the equilibrium voltage values vq can change as long as sign(vq) remains constant.In case of the current, equilibrium implies: iq = i0 = im, substituting into (6.44) yields:iq =δ sin(α)1− cos(α)mm−1∑k=0(−1)ceil(2 q+km ) cos(α)m−1−k (6.46)1316.6. Design Procedure15 30 45 60 75 90 105 120544.55.56 [deg]4812162024283236Figure 6.10: PV-voltage oscillation amplitude as function of the update angle α.where ceil(x) is the ceiling function, which rounds up to the next integer, and the initialsample q = 0 is defined such that vq=0 > 0 and vq=1 < 0 as shown in Fig. 6.9.The maximum excursion values of vq were numerically computed and are shown in Fig.6.10 as a function of the update angle α. As a result of this equilibrium characterization, thevoltage oscillations are found to be directly proportional to the output voltage VDC , the dutycycle increment δ and a function of the update angle α given by the plot shown in Fig.6.10.The voltage oscillations relate directly to the amplitude of the power oscillations around theMPP which determine the tracking efficiency.The analysis performed in this section reveals a high-efficiency, reliable equilibrium be-haviour can be obtained with the State-plane MPPT controller.6.6 Design ProcedureA design example based on a 65W panel (BP365) is shown in this section. A switchingfrequency of fsw = 100kHz is selected as a good trade-off between size of components andefficiency for the 65W converter. In order to enable enough ripple attenuation, the LCresonance frequency is determined as F0 =fsw25= 4kHz. An angle α = pi4is selected to1326.6. Design Procedurecomply with (6.27), which results in an MPPT duty cycle update time of:tU = T0α2pi= 0.031ms (6.47)Since the switching period is tsw = 0.01ms the update time is slightly modified to tU =0.03ms in order to synchronize the MPPT algorithm with the PWM operation. From thecommercially available LC values, the combination of L = 68µH, and C = 24µF results in aresonance frequency F0 = 3.94kHz. Therefore, the final value of the update angle is α = 43◦,which still complies with (6.27).The minimum duty cycle increment δ is found by taking into account the centre shift dueto the series resistance rL.δ > rLiPV scVDC= 0.1Ω3.99A25V= 1.6% (6.48)A value of δ = 3% is selected to provide enough margin.The maximum irradiance percentage slope of 10% per ms measured in the car setup andshown in Fig. 6.1 is employed as a design reference to verify the algorithm is able to achieveMPPT rapidly enough. Since the PV maximum power is 65W, then the maximum powerslope due to irradiance is found as dSdt= 6.5Wms(or 6,500 W/s). Through geometric analysisof Fig. 6.7, the minimum power change due to the duty-cycle change δ is found as:∆Pδ =(2 + sin(α)δVDCVMPP)sin(α)δVDCIMPP (6.49)which indicates a 3.8W power change during the selected update time. The irradiance powerstep ∆PS =dSdttU must be lower than the incremental step power ∆Pδ to ensure proper1336.7. Simulation ResultsTable 6.1: Simulation ParametersSPD Direct LIA SMCPV PANEL BP365UVDC 25 VfSamp 100 kHz 800 kHzfSw 100 kHz 20 kHzf0 4 kHz 0.8 kHzZ0 1.68 ΩL 68 µH 334 µHC 24 µF 118 µFfU 33 kHz 0.2 kHz 100 kHz 0.66 kHzδ 3% 5% N/A 5%maximum power point tracking. Then the MPPT update period must comply with:tU < ∆PδdSdt−1=3.8W6.5W/ms= 0.58ms (6.50)which is largely satisfied by the selected tU = 0.03ms. It is worthwhile highlighting that(6.50) would not be possibly satisfied employing traditional methods, which would require aminimum update time of t′U = 20 T0 ≈ 5ms.6.7 Simulation ResultsSimulation results of the State-plane direct MPPT (SPD) are provided in this section andcompared with traditional direct duty-cycle MPPT (Direct), as well as more complex methodssuch as sliding mode MPPT (SMC) and lock-in amplifier MPPT (LIA). The parameterscalculated in the design procedure are employed in the State-plane direct MPPT. As shownin Table 6.1, most parameters are maintained constant across all methods with the exceptionof those that require a different value to enable proper implementation. All parametersare selected prioritizing dynamic tracking performance and result in a fair comparison to1346.7. Simulation Results5101520024020406000.50 20 40 60 80MPP LIASMCDirect SPDFigure 6.11: Simulation comparison of the State-plane direct (light blue - SPD), traditionaldirect duty-cycle (orange - Direct), sliding mode (purple - SMC), and lock-in amplifier (green- LIA) MPPT methods under start-up and irradiance transient measured from car-mountedPV. Variables shown: PV voltage (vPV ), inductor current (iL), PV power (pPV ), and PWMduty cycle (d). As predicted by the theoretical analysis, the proposed method shows ex-tremely fast dynamic tracking performance maintaining MPP operation (dashed) during theentire simulation time.evaluate the performance of the State-plane direct MPPT. The sampling frequency waslimited to 800kHz to emulate implementation in an industry-standard DSP. The sampling-to-switching frequency ratio is set to 1 for the SPD and Direct methods, while values of8 and 40 are required for the LIA and SMC cases respectively. The switching-to-resonantfrequency ratio is set to 25 for all techniques. Compensator gains of kp = 0.001 and ki = 0.5are employed in the lock-in amplifier MPPT.1356.7. Simulation Results-1190 91 92 93 94-11-11-11LIASMCDirect SPDFigure 6.12: Comparison of simulated capacitor current for traditional direct duty-cycle(Direct), sliding Mode (SMC), lock-in amplifier (LIA) and State-plane direct (SPD) MPPTmethods.Simulation results under start-up and irradiance transients measured on the car-mountedPV scenario are shown in Fig. 6.11 for the four methods being compared. As shown in thefigure, the State-plane direct MPPT solves the start-up transient in only 0.45 ms, maintainingthe tracking efficiency above 95% to achieve an overall value of 99%. In contrast, the directduty cycle MPPT, which shows a similar level of implementation complexity as the proposedmethod, shows a much slower response. It reaches the MPP in 35 ms and looses MPPtracking during the irradiance transient taking 20 ms to recover, which derives in an overallefficiency of 84%. In case of the sliding mode MPPT, the maximum power point is reached in3.6 ms, MPP tracking is lost during the irradiance transient and reached back in 20 ms. Theoverall efficiency achieved by the SMC MPPT is 92.4%. The lock-in amplifier MPPT reachesthe MPP in 15 ms and slightly deviates from it during the irradiance transient, maintainingthe tracking efficiency above 95%, achieving an overall value of 97.5%. In this way, thecomparison highlights how the proposed MPPT method shows excellent dynamic tracking1366.7. Simulation Results1001021040.20.40.60.81x1 x100x10 x1,000x10,000100Dt=10st5001 2 n...0.5122n2 33 45 67 81014101020 1030 1050 10SpeedmultiplierIrradiancechange speedSlower (Static Applications)Faster(Moving Applications)10 - StaticTrad. 98.1%Prop. 98.7% - BackpackTrad. 61.5%Prop. 98.7%104 - BikeTrad. 45.1%Prop. 98.7%102x4 - CarTrad. 28.2%Prop. 96.7%105Figure 6.13: Dynamic tracking efficiency (η) as function of irradiance slope (∆S∆t) for State-plane direct MPPT (blue), and traditional direct duty cycle MPPT (orange). Results ofEN50530 10% to 50% irradiance change standard test with X1, X10, X100, X1000, andX10000 speed multipliers to reach irradiance slopes found in moving applications such asbackpack, bike, and car mounted PV.performance, superior, in this case, to the one obtained through methods with much morecomplex implementation.The capacitor currents found in each one of the analysed methods are shown in Fig. 6.12.The waveform RMS values are 0.30 A for the direct duty-cycle MPPT, 0.31 A for the slidingmode MPPT, 0.24 A for the lock-in amplifier MPPT, and 0.41 A for the State-plane directMPPT. Due to the dynamic nature of the proposed method, the operating point continu-ously oscillates around the MPP causing the capacitor current to oscillate continuously. Theamplitude of these oscillations can be adjusted by selecting adequate values for the updatingfrequency fU , and displacement δ to fit the design requirements. This point must be takeninto account in order to avoid artificial ageing of capacitors caused by excessive currents.1376.7. Simulation Results1021040.20.40.60.81x1 x10 x100 x1,000 x10,000300Dt=10st1,0001 2 n...10141010n20 1030 1050 10100 10Speedmultiplier10 - StaticTrad. 97.9%Prop. 99.1% - BackpackTrad. 97.0%Prop. 99.1%104 - BikeTrad. 69.7%Prop. 99.1%102x4 - CarTrad. 50.8%Prop. 99.0%105Irradiancechange speedSlower (Static Applications)Faster(Moving Applications)Figure 6.14: Dynamic tracking efficiency (η) as function of irradiance slope (∆S∆t) for State-plane direct MPPT (blue), and traditional direct duty cycle MPPT (orange). Results ofEN50530 30% to 100% irradiance change standard test with X1, X10, X100, X1000, andX10000 speed multipliers to reach irradiance slopes found in moving applications such asbackpack, bike, and car mounted PV.The tracking efficiencies of the state-plane direct MPPT and the traditional direct dutycycle MPPT were obtained through the EN50530 dynamic test standard and are plotted inFig. 6.14 in blue and orange respectively. Two different irradiance transient tests detailedby the EN50530 standard (10% to 50%, and 30% to 100%) have been performed and areshown in the sub-figures (a) and (b). The plots show the dynamic tracking performance(η [%]) obtained for different irradiance slopes (∆S∆t[ Wm2s]). Since the EN50530 standard testwas designed for steady PV applications, the original irradiance slopes do not include thetypical values found in moving applications. A speed multiplier with values of 100, 101,102, 103, 104, and 105 has been introduced in order to evaluate the performance of theMPPT methods for the applications under analysis. Both MPPT solutions show very high1386.8. Experimental ResultsTable 6.2: Dynamic MPPT Methods ComparisonMethod ∆t1∗ ∆t2∗∗ Complexity Sensors Req. BW Pert. Implement.SPD 0.7 1.2 Very Low vPV , iPV Low Fixed Digital ($)Direct 62.0 248.2 Very Low vPV , iPV Very Low Fixed Digital ($)Var. δ 121.5 194.4 Low vPV , iPV Very Low Variable Digital ($)SMC 79.6 19.9 Moderate vPV , iPV , Very High Fixed Digital ($$$), oriL Digital + Analog ($)LIA 34.6 17.3 Moderate vPV , iPV Moderate None Digital ($$)∗ Tracking time under 50-to-100% Irradiance step transient∗∗ Tracking time under Start-up transientSignal cond.dRshuntMicrocontrollerCPUv P V i P VADC PWMControl & MPPT :<20 simple math.operations perMPPT stepLow update rateStandardaverage-modesamplingLow BW req.Low-cost sensingFixed-frequencystandard peripheralPowerConverterLow fsw /f 0 ratioFigure 6.15: State-plane direct MPPT hardware implementation.tracking efficiencies at low irradiance change rates. However, the response of the traditionalimplementation deteriorates very rapidly as the irradiance change rate reaches the valuesfound in the proposed applications. In contrast, the state-plane MPPT maintains very hightracking efficiency throughout the whole range of irradiance change rates. As shown in thefigure, the State-plane direct MPPT shows extreme improvements when compared with thetraditional implementation in the backpack, bike, and car scenarios.1396.8. Experimental ResultsvPVvPViPVpPVδ798 μspPVMPPFigure 6.16: Experimental results of the proposed scheme under start-up transient: MaximumPower Point found in only 798µs6.8 Experimental ResultsThe hardware platform detailed in Table 6.1 was employed to validate the State-plane directMPPT controller. A hardware implementation diagram is shown in Fig. 6.15. The PVpanel voltage and current are respectively measured through a resistor divider and a 10mΩshunt. The voltage levels of the measured signals were adapted to the 0-3V range required bythe analog-to-digital converter (ADC) by using operational amplifiers. An industry-standardC2000 TI microcontroller was employed to perform the analog to digital conversion, digitalprocessing and PWM generation.Since the State-plane direct MPPT is based on the converter’s average behaviour, stan-dard average-mode sampling methods can be employed, placing low demands on the ADCconversion speed and sensing/signal-conditioning stages bandwidth. The average currentand voltage digital values are read by the CPU every MPPT update period tU . The CPUprocesses the complete algorithm shown in Fig. 6.8, which demands less than 20 simple math-ematical operations. A new duty cycle value is sent from the CPU to the fixed-frequencyPWM peripheral every update period tU .Experimental results are shown in Figs. 6.16 to 6.20, in time and XY domains. The start-1406.8. Experimental ResultsMPPη=97.5%vPViPVpPVδ vPVpPVFigure 6.17: Experimental results of the proposed scheme operating in equilibrium: reliableoperation and 97.5% tracking efficiency.vPViPVpPVvPVpPVMPP = f(G)Figure 6.18: Experimental results of the proposed scheme under step-up and step-downirradiance transientsup transient detailed in Fig. 6.16 shows the MPP is reached in 798µs, highlighting the ex-tremely fast transient response achieved with the proposed method. The XY plot (PV powervs. voltage) shows how the operating point reaches the MPP following a straightforwardpath, remaining oscillating around it. Equilibrium operation around the MPP is depicted inFig. 6.17 showing very stable behaviour, with a tracking efficiency of 97.5%. The XY plotshows a zoomed view of the oscillatory behaviour around the MPP. Due to the extremely fastdynamics, the effect of the PV panel parasitic output capacitance is manifested as two dif-1416.8. Experimental ResultsvPViPVpPVvPVpPVMPPFigure 6.19: Experimental results of the proposed scheme under step-up and step-downirradiance transients: Step-up zoom showing MPP.vPViPVpPVvPVpPV MPPFigure 6.20: Experimental results of the proposed scheme under step-up and step-downirradiance transients: Step-down zoom showing MPP.ferent trajectories when the voltage slope changes directions. The capture in Fig. 6.18 showsthe response of the proposed technique under changes of irradiance. The irradiance slope islimited to 0.53%/ms due to thermal inertia of the lamps employed. The proposed algorithmshows reliable behaviour over the whole range of irradiance values. Zoomed views of the PVvoltage, current and power are shown during step-up and step-down transients in Figs. 6.19and 6.20 respectively. The XY plots show the operating point continues to oscillate around1426.9. Summarythe MPP even during fast transients, achieving tracking efficiencies above 90% at all times.The experimental results provided in this section highlight the contribution of this workto the applied field. An extreme dynamic performance is achieved while maintaining simpleimplementation.A comparison against the results obtained by employing some of the most popular andhighest performance dynamic MPPT methods currently available is shown in Table 6.2. Twodifferent tracking times are included in the comparison: Start-up and irradiance step from50% to 100%. Due to the diverse power and voltage levels employed in the different so-lutions, the tracking times are shown as a fraction of the converter’s LC resonant periodT0 = 2pi√LC. Furthermore, the comparison includes implementation complexity, measuredvariables required for the implementation, required sensing and signal conditioning band-width (BW), and type of perturbation employed.As shown in the table, the State-plane Direct MPPT shows the highest tracking speed ofall compared techniques, while maintaining very low implementation requirements.6.9 SummaryCharacteristic irradiance profiles found in moving PV applications were measured and anal-ysed in this chapter in order to determine MPPT dynamic requirements for this type ofapplications. Irradiance slopes up to 1%/ms, 2%/ms, and 10%/ms were found in backpack,bike and car mounted PV scenarios respectively. The findings highlight the need of fasterMPPT methods for moving PV applications, since priorly existing solutions are not able toproperly track the maximum power point during such fast transients.Incremental Conductance principles were reviewed to set the basis for the MPPT methodbeing introduced, and the performance of the traditional direct duty cycle IC solution wasanalysed in order to set a benchmarking point using a method of similar complexity to the1436.9. Summaryone State-plane Direct MPPT.The Average Natural Trajectories of the PV connected inverted buck converter were de-rived following a similar procedure to the one employed in chapter 2 for the three basictopologies. Based on the obtained large signal model, the State-plane Direct MPPT con-troller was derived as a dynamic method to solve the dynamic problem of maximum powerpoint tracking avoiding the delays required by traditional solutions. As a result, the proposedMPPT method is proven to achieve equilibrium only around the maximum power point.The dynamic response of the proposed method was thoroughly evaluated employing sim-ulation and experimental platforms under diverse scenarios. In this way, the extremely fasttracking performance was validated through various methods including an accelerated versionof the standard PV test EN50530 (up to 10,000 times), and a start-up transient in which theMPP was found in less than 800 µs.144Chapter 7Dual-loop Geometric Control ofDC-DC convertersThe dual-loop Geometric Control concept is introduced for buck and boost converters inthis chapter. A combination of traditional dual-loop control architectures with state-planeanalysis is explored to achieve a fast and reliable transient response while eliminating steadystate error maintaining an industry-standard control architecture. As a result, the dual-loop Geometric controller matches the performance obtained with more complex alternativessuch as first-order sliding mode, Centric, and Natural Switching Surfaces controllers. Thedynamic performance obtained through three different geometric trajectories is characterizedemploying the benchmarking procedure introduced in chapter 4. Some of the content includedin this chapter has been published in the IEEE APEC and ECCE conference papers [8, 9].7.1 Dual-loop Geometric Control PrincipleThe dual-loop Geometric controller is, as its name indicates, based on a traditional dual-loopscheme, in which the voltage loop compensator is determined by a geometric equation (e.g.:straight line, circle, etc.). The concept is based on state-plane analysis of the normalizedconverters response, the ANTs introduced in chapter 2. As illustrated in Fig. 7.1 for aboost converter, a geometric-based algorithm is implemented in the outer voltage loop, whichprovides the reference to the inner inductor current loop. The inductor current control can1457.2. Target TrajectoriesLnCnVccnPWMPIiLnPI Current LoopBoost ConverterionGeometricVoltageCompensatorvCnvCntiLn*Figure 7.1: Dual-loop trajectory-based control scheme. Traditional inner current loop plusgeometric outer voltage loop.be carried out by using any of the several existent industry-standard techniques (e.g. peakcurrent mode, hysteresis mode control). An average current mode scheme is used in thiswork.7.2 Target TrajectoriesAs illustrated in Fig. 7.2, the geometric voltage compensator is defined by target trajectories(λx), which define the desired path for the operating point to describe in order to reach thetarget value. It is worthwhile highlighting that the target operating point must be the onlyequilibrium operating point contained in the target trajectory. Hence, the target trajectoriesλx must intersect the load line λLL only at the target operating point, as shown in Fig. 7.2.Furthermore, the domain of the target trajectories must be restricted in order to guaranteeconvergence towards the target operating point. Mathematically, the convergence conditionsare given by:dvCndtn> 0, for vCn < vCnt, and (7.1)dvCndtn< 0, for vCn > vCnt (7.2)1467.2. Target TrajectoriesiLniLntvCnvCntVccnionTarget Operating PointTransient paths:Sleek (¸xCircle )Standard (¸xLine )Load line(¸LL ) TrajectoriesDomainTime-optimal (¸xON, xOFF ) Figure 7.2: Geometric controller trajectories: standard (λxLine), sleek (λxCircle), and nearTime-Optimal (λxON,xOFF ) responses.Plugging 7.1 and 7.2 into the normalized unified DC-DC converter model equations, (2.31)-(2.32):iLn > m2Ion, for vCn < vCnt, and (7.3)iLn < m2Ion, for vCn > vCnt (7.4)Combining these conditions with the load line definition (2.33)-(2.34), the equilibrium con-ditions become:iLn = iLnss for: vCn = vCnt,iLn > iLnss for: vCn < vCnt, andiLn < iLnss for: vCn > vCnt, (7.5)which can be graphically interpreted as the areas above the load line for voltages lower thanthe target, and below the load line for voltages above the target, as shown in Fig. 7.2.Due to the limited range [0, 1] on the duty cycle value, extra restrictions must be includedin order to define a domain of realizable trajectories. The target trajectories must be insideof the saturation trajectories, which results in a restricted domain illustrated in Fig. 7.2,along with several proposed geometric trajectories.1477.3. Defining a Transient Response with Geometric TrajectoriesThe analysis shows that there are a theoretically unlimited amount of target trajectoriesthat lead the operating point to the target. Each one of the possible trajectories shows aunique set of transient qualities that might turn it into the most suitable solution for specificapplications according to the given requirements and restrictions.7.3 Defining a Transient Response with GeometricTrajectoriesSeveral target trajectories (Straight-line, Circular and Natural) employed to defined the ge-ometric compensator are analysed in this section in order to illustrate their advantages andrequirements. As illustrated in Figs. 7.3-7.8, during the first part of the transients (start-up,load step-up/step-down) the operating point moves towards the target trajectory. Once thetarget trajectory is reached, the converter operating point follows the established path toreach the desired steady state operating point in a reliable and predictable manner.7.3.1 Standard response: Straight LineEmploying a straight line as a compensator allows to tune the voltage loop for robust or fastresponse by only adjusting the line slope. It must be noted that, although a straight line isfollowed, the proposed geometric compensator is not a sliding-mode (boundary) scheme, buta PWM-based dual-loop controller. A straight line that contains the target operating pointis mathematically defined by:λxLine : (iLn − iLnt) + k (vCn − vCnt) = 0 (7.6)where k > 0 defines the line slope.1487.3. Defining a Transient Response with Geometric TrajectoriesiLn10ion1ion2TargetOperating PointsReferenceTrajectoriesControlledTrajectoriesvCntniLntnvCntn2 4 6d10100ion1ion2ExponentialDuty cycleReferenceCurrentFigure 7.3: Buck Converter state plane and time domain response using a dual-loopgeometric-based controller with a straight line trajectory as voltage compensator.BuckFor buck converters, since the load line is horizontal, the straight-line target trajectory canbe maintained within the desired domain by making: k > 0. Time-domain expressions thatdescribe the dynamics found in a normalized buck converter once the target line is reached1497.3. Defining a Transient Response with Geometric Trajectoriescan be obtained by plugging (7.6) into the normalized buck converter equations (2.25)-(2.26),and solving for vcn(tn) and iLn(tn):vCnLine(tn) = 1 +(vCn(0) − 1)e−2pik tn (7.7)iLnLine(tn) = ion +(iLn(0) − ion)e−2pik tn (7.8)where the coordinates (vCn(0); iLn(0)) indicate the point where the target line is reached.The time-domain duty cycle function required to achieve the linear trajectory is given by:dLine(tn) =(vCn(0) − 1)(k2 + 1)e−2pik tn + 1Vccn(7.9)As shown by (7.9) in order to obtain a straight line trajectory, the duty cycle must evolvefollowing an exponential function which time constant is defined by the line slope k.BoostFor boost converters, the straight line target trajectory remains within the convergence do-main when its slope complies with: k > −iLnt. Time-domain expressions that describe thenormalized boost converter response once the target line is reached are obtained by plugging(7.6) into the boost converter model(2.27)-(2.28), and solving for vcn(tn) and iLn(tn).vCnLine(tn) = 1 +W(z(tn)) 1k1(7.10)iLnLine(tn) = 1 + iLnt −W(z(tn)) kk1, (7.11)where W (z) is an implicit function known as the Lambert function defined as the solution to1507.3. Defining a Transient Response with Geometric TrajectoriestniLn tnvCntn3 6 9dVccn1100ReferenceCurrentiLnt1iLnt2iLnvCnVccn 10ReferenceTrajectoriesiLnt1iLnt2TargetOperating PointsFigure 7.4: Boost Converter state plane and time domain response using a dual-loopgeometric-based controller with a straight line trajectory as voltage compensator.W (x ex) = x, the argument z(tn) is given by:z(tn) = k1(vCn(0) − 1)e−k2tn−k1(1−vCn(0)), (7.12)vCn(0) is the voltage at which the target line is reached, and the constants k1 and k2 are:1517.3. Defining a Transient Response with Geometric Trajectoriesk1 =(k2 + 1)1− k iLnt , (7.13) k2 =2piVccn(k + iLnt)1− k iLnt . (7.14)The complementary duty cycle evolution required to track the straight line is given by:d′ =VccnW (z(tn)) + 1(7.15)The function W (z(tn)) is implicit can be accurately approximated for −1e < z(tn) < 1, as:W (z(tn)) = ln (1 + z(tn))1 + 12340z(tn) +2110z2(tn)1 + 14340z(tn) +71310z2(tn)(7.16)As shown in Fig. 7.4 the resulting duty cycle waveform, described by (7.15), shows apseudo-exponential behaviour which time constant is defined by the line slope k.In both cases, for buck and boost converters, the inner current loop must be fast enoughto generate the time-domain functions (7.9) and (7.15). Therefore, the maximum slope kthat can be achieved is limited by the inductor current control loop bandwidth. For higher-bandwidth current controllers, the slope can be set to steeper values producing a fastertransient response. If much faster transient responses are required, circular trajectories canbe employed as discussed in the next sub-sections.7.3.2 Sleek Response: Circular TrajectoriesSince the three basic DC-DC converters describe circular average natural trajectories whenthe duty cycle is maintained constant, employing circular trajectories for the geometric com-pensator is a direct alternative in which the duty cycle is maintained at a constant value whilethe target trajectory is followed. In this way, the circular trajectories provide a predictablesleek (fast and smooth) response.The generic equation that describes a circle containing the target operating point is given1527.3. Defining a Transient Response with Geometric TrajectoriestniLntnvCntn2 4 6d10100ion1ion2ReferenceCurrentConstantDuty cycleiLnµ0µ0µ0vCnion1c1ion21ControlledTrajectoriesc2ReferenceTrajectoriesTargetOperating PointsFigure 7.5: Buck Converter state plane and time domain response using a dual-loopgeometric-based controller with circular trajectories as voltage compensator.by:λxCircle : (iLn − ci1,2)2 + (vCn − cv1,2)2 = (iLnt − ci1,2)2 + (vCnt − cv1,2)2 (7.17)where two circles with centres at (cv1; ci1) and (cv2; ci2) must satisfy the convergence conditions(7.5), and the centres voltage and current coordinates are located on the load line. Taking1537.3. Defining a Transient Response with Geometric Trajectoriesinto account the specific load line for each topology, the circle equation can be simplified foreach particular case.BuckAs illustrated in Fig. 7.5, the circular target trajectories employed to determine the geometricvoltage compensator in a buck converter are defined by semi-circles at each side of the targetvoltage line:λxCircle : (iLn − ion)2 + (vCn − c1,2)2 − (1− c1,2)2 = 0 (7.18)where c1 < 1 defines the centre of the upper semi-circle at the left side semi-plane of thetarget voltage, and c2 > 1 the centre of the lower semi-circle at the right side, as indicatedin Fig. 7.5. Time domain expressions that describe the behaviour of the converter once thecircular target trajectory is reached can be obtained plugging (7.18) into the normalized buckconverter equations (2.25)-(2.26), and solving for vcn(tn) and iLn(tn):vCnCircle(tn) = (1− c1,2)cos(2pitn − θ0) + c1,2 (7.19)iLnCircle(tn) = (1− c1,2)sin(2pitn − θ0) + ion (7.20)where θ0 is the angle at which the operating point reaches the target trajectory as indicatedin Fig. 7.5. The duty cycle required to track the circular trajectory is given by:dCircle(tn) =c1,2Vccn(7.21)which indicates the duty cycle must be constant once the circular trajectory is reached.1547.3. Defining a Transient Response with Geometric TrajectoriestniLn tnvCntn1 2 3d1100ConstantDuty cycleVccniLnt1iLnt2ReferenceCurrentµ0c1 c2iLnvCnVccn 10iLnt1c1iLnt1c2iLnt2iLnt2µ0µ0TargetOperatingPointsFigure 7.6: Boost Converter state plane and time domain response using a dual-loopgeometric-based controller with circular trajectories as voltage compensator.BoostAs illustrated in Fig. 7.6, circular target trajectories with centre on the load line are definedfor the boost converter by semi-circles at each side of the target voltage:λxCircle : (iLn − c1,2 iLnt)2 + (vCn − c1,2)2 − (iLnt − c1,2 iLnt)2 − (vCnt − c1,2)2 = 0 (7.22)1557.3. Defining a Transient Response with Geometric Trajectorieswhere c1,2(1; iLnt) defines the centres of the upper and lower semi-circles at the left and rightsides of the target voltage, as shown in Fig. 7.6.The time domain expressions that describe the behaviour of the converter once the circulartarget trajectory is reached can be obtained combining (7.22) with the normalized boostconverter model (2.27)-(2.28), and solving for vcn(tn) and iLn(tn):vCnCircle(tn) = (1− c1,2) cos(2pic1,2tn − θ0)+ iLnt (1− c1,2) cos(2pic1,2tn − θ0)+ c1,2 (7.23)iLnCircle(tn) = iLnt (1−c1,2) cos(2pic1,2tn − θ0)+(1−c1,2) sin(2pic1,2tn − θ0)+c1,2 iLnt (7.24)where θ0 is the angle at which the operating point reaches the target trajectory as indicatedin Fig. 7.6. The duty cycle required to track the circular trajectory is given by:dCircle(tn) = 1− Vccnc1,2(7.25)which shows the duty cycle must be kept constant once the circular trajectory is reached.Although for both buck and boost converters the transients are solved employing constantduty cycle values, the necessary rapid transition between these different values places a high-bandwidth requirement on the current control loop. In both cases, the circular trajectorycan be complemented with a straight line (given by (7.6)) in the neighbourhood of the targetoperating point to enable smoother duty cycle transitions that can be fulfilled by the availablecurrent controllers.7.3.3 Near Time-Optimal Response: Natural TrajectoriesThe ON and OFF saturation trajectories are a particular case of circular trajectories that isconsidered of interest since using them as a voltage loop compensator produces near time-optimal dynamic response.1567.3. Defining a Transient Response with Geometric TrajectoriestniLntnvCntn2 4 6d10100ion1ion2ReferenceCurrentiLnvCnion1ion2ControlledTrajectories10ReferenceTrajectoriesTargetOperating Pointsµ0µ0µ0VccnFigure 7.7: Buck Converter state plane and time domain response using a dual-loopgeometric-based controller with Natural trajectories as voltage compensator.BuckAs shown in Fig. 7.7, near time-optimal dynamics can be achieved in buck converters byemploying the saturation trajectories λxON and λxOFF to define the voltage compensator:λxON,xOFF : (vCn − u Vccn)2 + (iLn − ion)2 − (vCnt − u Vccn)2 = 0 (7.26)1577.3. Defining a Transient Response with Geometric Trajectorieswhere u = 1 for λxON , and u = 0 for λxOFF . This simple equation form the basis for thegeometrical natural compensator.The converter dynamics once the target trajectories are reached can be described byemploying (7.19) and (7.20), with c1 = 0 and c2 = Vccn.BoostFor the case of boost converters, illustrated in Fig. 7.8, the natural trajectories λxON andλxOFF to be used as voltage compensator are given by:λxON,xOFF :(vCn − Vccnu′)2+(iLn − ionu′)2−(vCnt − Vccnu′)2−(iLnt − ionu′)2= 0 (7.27)where u′ = 0 for λxON , and u′ = 1 for λxOFF .Once the target trajectories are reached, the converter dynamics are described by (7.23)and (7.24), with c1 = Vccn and c2 →∞.As illustrated in Figs. 7.7, and 7.8, both buck and boost converters operate at their limitof performance since the transients are solved employing only saturation trajectories. Thefast variations in the duty cycle required to achieve the time-optimal response indicate thata very high bandwidth current-loop is required. In case a high bandwidth current controlleris not available, the curve can be replaced by a straight line with lower speed requirementsin the target neighbourhood, as in the circular case. The response, in that case would differfrom the time-optimal, but near time-optimal results can be reached.As it can be observed in Figs. 7.7, and 7.8, a large current peak is required to achieve atime-optimal response. According to the application, maximum current limits can be imposedon the target trajectory in order to avoid the magnetic saturation of inductors.1587.4. Experimental ResultstniLn tnvCntn1 2 3d1100VccniLnt1iLnt2ReferenceCurrentSaturatingDuty cycleiLnvCnVccn0iLnt2TargetOperatingPointsµ0µ01iLnt1ion1ion2Figure 7.8: Boost Converter state plane and time domain response using a dual-loopgeometric-based controller with Natural trajectories as voltage compensator.7.4 Experimental ResultsExperimental results of the dual-loop geometric controller implemented in buck and boostconverters are analysed in this section. The dual-loop digital controllers were implementedusing an industry-standard TIC2000 DSP, the inner current loop is implemented with tradi-1597.4. Experimental ResultsTable 7.1: Buck Converter ParametersPARAMETER VALUEVcc 24 Vvref 12 VL 495 µHC 46 µFZ0 3.28 ΩT0 948 µstional PI controllers, and the voltage loop geometric compensator equations are solved in realtime. In applications where an implementation in lower processing power micro-controllersis required, look-up tables containing the geometric trajectories can be employed.Start-up and load current step-up/step-down transients are shown for each one of the threeanalysed trajectories. The settling and recovery times have been indicated in the figures inseconds and normalized values (between brackets).The buck and boost converters employed to validate the dual-loop geometric controllerare detailed in Table 7.1 and 7.2. The dynamic response obtained in both converters forstraight line, circular and natural target trajectories are shown in the experimental capturesin Figs.7.9-7.14. The buck converter experimental transient responses shown Figs. 7.9, 7.11,and 7.13 include the output voltage (vCn), inductor current (iLn), duty cycle (d), and loadcurrent (Ion) waveforms. In case of the boost converter oscilloscope captures shown in Figs.7.10, 7.12, and 7.14 the reference current waveform (iLn∗) is included instead of the dutycycle, in order to highlight the importance of a fast inner current control loop.7.4.1 Straight Line TrajectoriesThe transient response of the straight-line implementation of the dual-loop geometric con-troller is shown in Figs. 7.9 and 7.10 for buck and boost converters. Slopes of k = 0.4and k = 0.25 were employed for the buck and boost converters respectively. As it can be1607.4. Experimental ResultsTable 7.2: Boost Converter ParametersPARAMETER VALUEVcc 12 Vvref 24 VL 490 µHC 224 µFZ0 1.48 ΩT0 2.08 msStart-upUnlo ading2.15ms(2.27)1.6ms (1.69)2ms (2.1)1 23101 2 3 22.75A (0.75)TargetTrajectoriesLoadingiLndvCnIonFigure 7.9: Dual-loop geometric controller start-up, load step-up and load step-down tran-sients experimental results in a buck converter using a straight line as target trajectory.observed in the figures, the linear target trajectory is properly tracked by the controller inthe state plane, providing the expected exponential-like shapes on the time domain transientresponses in voltage and current variables. In this way, the straight line trajectory providesa dynamic performance that resembles the one obtained with sliding-mode controllers, whilemaintaining fixed frequency PWM operation, and a traditional dual-loop architecture.7.4.2 Circular TrajectoriesEmploying circular trajectories with centres at c1 = 0.5, c2 = 1.5, much faster transientresponses than in the straight-line case were obtained and are shown in Figs. 7.11 and 7.121617.4. Experimental ResultsTargetTrajectoriesUnlo ading4.1A (0.25)LoadingiLnvCnIon1 22 3Start-up1 2311.65ms(5.6)3.02ms(1.45)13.32ms(6.4)iLn*Figure 7.10: Dual-loop geometric controller start-up, load step-up and load step-down tran-sients experimental results in a boost converter using a straight line as target trajectory.iLndvCnStart-upLoadingUnlo adingIon0.4ms(0.42)0.47ms(0.49)0.31ms (0.33)123101 2 3 22.75A (0.75)ReferenceTrajectoryFigure 7.11: Dual-loop geometric controller start-up, load step-up and load step-down tran-sients experimental results in a buck converter using circular target trajectories.for buck and boost converters respectively. As shown in the figures, the operating pointfollows the desired circular path providing a sleek, fast and reliable response. The constantduty cycle value required to maintain the circular path trajectory is shown in Fig. 7.11.7.4.3 Natural TrajectoriesNear-optimal transient recovery times are shown in Figs. 7.13 and 7.14 by employing theconverter saturation trajectories as geometric compensator. Since the requirements placed1627.5. Benchmarking Dual-loop Geometric ControllerUnlo ading4.1A (0.25)LoadingiLnvCnIon1 22 3TargetTrajectoriesStart-up1 231.33ms(0.64)1.57ms(0.76)1.39ms(0.67)iLn*Figure 7.12: Dual-loop geometric controller start-up, load step-up and load step-down tran-sients experimental results in a boost converter using circular target trajectories.iLndvCnStart-upLoadingUnlo adingIon0.29ms(0.31)0.28ms(0.29)0.28ms(0.29)1 23101 2 3 22.75A (0.75)ReferenceTrajectoriesFigure 7.13: Dual-loop geometric controller start-up, load step-up and load step-down tran-sients experimental results in a buck converter using ON/OFF saturation (natural) targettrajectories.on the inner-loop current controller are higher, minor deviations from the desired trajectorycan be observed during the transients.1637.5. Benchmarking Dual-loop Geometric ControllerUnlo ading4.1A (0.25)LoadingiLnvCnIoniLn*1 22 3 TargetTrajectoriesStart-up1 230.98ms(0.47)1.23ms(0.59)1.04ms(0.5)Figure 7.14: Dual-loop geometric controller start-up, load step-up and load step-down tran-sients experimental results in a boost converter using ON/OFF saturation (natural) targettrajectories.Table 7.3: Dual-loop Geometric Control BenchmarkBuck BoostTIME Line Circular Natural Line Circular NaturalStart-up: STi 0.55 0.92 0.99 0.43 0.9 0.97Loading Recovery: RTiL 0.52 0.92 0.94 0.72 0.87 0.92Unloading Recovery: RTiU 0.5 0.82 0.93 0.4 0.9 0.96Buck BoostVOLTAGE Line Circular Natural Line Circular NaturalStart-up Overshoot: SOi 1 1 0.91 1 1 1Loading Deviation: DRiL 0.52 0.69 0.69 0.7 0.61 0.51Unloading Deviation: DRiU 0.62 0.69 0.71 0.84 0.87 0.927.5 Benchmarking Dual-loop Geometric ControllerThe dynamic performance of the dual-loop Geometric controller responses shown in Figs.7.9-7.14 is assessed in this section using the benchmarking procedure introduced in chapter 4.The calculated indices are shown numerically in Table 7.3, and graphically in Fig. 7.15 forstraight-line, circular and natural trajectories in both buck and boost converters.1647.5. Benchmarking Dual-loop Geometric ControllerRTiLSTi SOiDRiLRTiU DRiURTiLSTi SOiDRiLRTiU DRiUTimeIndicesRegulationIndicesTimeIndicesRegulationIndicesStraight Line TrajectoryCircular TrajectoryNatural TrajectoryStraight Line TrajectoryCircular TrajectoryNatural TrajectoryDual-loop Geometric Controller BenchmarkingBuck Converter Boost ConverterFigure 7.15: Dual-loop geometric controller dynamic performance benchmark for straightline, circular and natural trajectories in buck and boost converters.As indicated by the indices, the transient recovery speed obtained using straight-linetrajectories is comparable with the resulting from traditional dual-loop control schemes.However, the straight-line dual-loop geometric controller guarantees a large-signal predictableresponse for all possible transient, whereas traditional small-signal based controllers canbecome unstable under large transient perturbations.The superior dynamic performance obtained by employing circular and natural trajectoriesas target is clearly indicated by the indices obtained. As the dynamic performance approachesthe theoretical limits of performance, the limitations imposed by the inner current loopcontroller become more evident. For this reason, the dual-loop geometric controller thatuses the Natural trajectories of the converter as voltage compensator is affected the mostby the current loop limitations, and the one in which the obtained indices deviate the mostfrom their theoretical values. Faster current control schemes, such as peak current mode orhysteresis current control, could be employed to further improve the dynamic performance.1657.6. Summary7.6 SummaryThe concept of dual-loop geometric control in power electronics was introduced in this chapteras an alternative that merges the benefits of industry-standard dual-loop controllers with theadvantages of state plane analysis. The aim of this compensator is to get state-plane basedcontrollers closer to regular industrial applications.The dual-loop geometric controller introduced includes an inner current control loop,and an outer geometric compensator based on geometric trajectories in the state plane.Domain restrictions for the state-plane based trajectories were identified in order to guaranteeconvergence towards the desired target operating point, and three potential trajectories wereproposed. Mathematical expressions that determine the dynamic responses of buck andboost converters obtained through voltage loop controllers represented by linear, circularand natural trajectories were derived and analysed.Experimental results for the three different trajectories were obtained in buck and boostconverters, validating the high performance predicted by the mathematical analysis. Thesix performance indices were calculated for each of the geometric voltage compensators andrepresented in two performance hexagons corresponding to both buck and boost converters.The indices obtained for the linear trajectory resemble the performance of a traditionalcompensator while the circular and natural compensators achieve indices in the excellentand good ranges.166Chapter 8Dual-loop Geometric Control ofAC-DC convertersThe control of active rectifiers, or Power Factor Correctors (PFCs), necessarily implies a dual-loop scheme due to the sinusoidal inductor current and static capacitor voltage waveformsdesired. The dual-loop geometric Control concept introduced in chapter 7 is extended in thischapter to the case of boost-derived active rectifiers by proposing a voltage loop compensatordetermined by a geometric equation (e.g.: straight line, quadratic, circle, etc.) in combinationwith a linear current controller. The concept is based on state-plane analysis of the normalizedboost-derived PFC behaviour. As illustrated in Fig. 8.1, a geometric-based algorithm isimplemented in the outer voltage loop, which provides the reference to the inner inductorcurrent loop. The inductor current control can be carried out by employing any of theseveral existent techniques (e.g. peak current mode, hysteresis mode control). An averageProportional + Resonant current controller is employed in this work.The content included in this chapter has been partially published in the IEEE PEDGconference paper [10].1678.1. Dual-loop Geometric Control in AC/DC ConvertersLnCnPWMMAFPRiLnvgnPR Current LoopBoost-Derived PFCionGeometricVoltageCompensatoriLn*vCnvCntFigure 8.1: Dual-loop geometric control scheme. Traditional current loop + geometric voltageloop8.1 Dual-loop Geometric Control in AC/DCConvertersThe behaviour of the boost-derived PFC topology shown in Fig. 8.1 can be modelled inthe normalized domain in a similar way as the DC-DC boost converter is represented by(2.27) and (2.28). The differential equations that rule the behaviour of the normalized,boost-derived PFC are:12pidvCndtn= d′ iLn − ion (8.1)12pidiLndtn= vgn − d′ vCn (8.2)where d′ ∈ [0, 1] is the complementary PWM duty cycle, and vgn = Vgn |sin(ωgntn)| is therectified grid voltage after amplitude and time normalization. Since unity power factor isdesired, the normalized inductor current must follow the waveform shown in Fig. 8.2, whichis described by the equation:iLn = ILn |sin(ωgntn)| (8.3)1688.1. Dual-loop Geometric Control in AC/DC ConvertersiLnILntvCnvCniLnVgn2 ionLoad line (¸LL ) vCntvCnSS1ILnSS1ILnSS2vCnSS2vCntvCnSS1vCnSS2ILnSS2ILnSS1ILntTarget OperatingTrajectory (¸target ) EquilibriumTrajectories (¸SS ) tntn2fgn14fgn14fgn3fgn12fgn1fgn1,4fgn14fgn3,Figure 8.2: Steady state trajectories for boost-derived PFC shown in both time and geometricdomains.This waveform requirement poses a restriction to the boost topology describing equations(8.1) and (8.2). Solving for dvCndtn:12pidvCndtn=Vgn ILn2 vCn[1−R cos(2 ωgntn − δ)]− ion (8.4)where:R =√1 +(ωgnILn2pi Vgn)2≈ 1 ; (8.5)δ = tan−1(ωgnILn2pi Vgn)≈ 0 (8.6)1698.2. Steady State: Load Line & Target Operating Trajectorywhere the approximations hold true for most PFC applications since the grid frequency ismuch lower than the LC resonance (ωgn << 2pi) and the normalized peak grid voltage ismuch larger than the normalized peak inductor current (Vgn >> ILn). The solution of(8.4) describes the evolution of the normalized capacitor voltage vCn when the inductorcurrent is sinusoidal and in phase with the input voltage, as shown in Fig. 8.2. Sincethe differential equation is implicit, its direct solution exceeds the complexity desired forthis work. An approximate solution is achieved by separating the capacitor voltage in twoparts corresponding to its DC and AC components: vCn = v¯Cn + ∆vCn, and assumingv¯Cn >> ∆vCn. As a result, the approximate solution is given by:vCn = v¯Cn(0) +(Vgn ILn2 v¯Cn− ion)2pi tn − pi Vgn ILn2 v¯Cnωgnsin(2 ωgntn) (8.7)8.2 Steady State: Load Line & Target OperatingTrajectorySince the desired current waveform is sinusoidal, the ripple component of twice the linefrequency is part of the desired steady state capacitor voltage. Therefore, the equilibriumcondition to maintain a constant cycle mean capacitor voltage can be obtained by eliminatingthe ripple from (8.4), and making the derivative equal to zero:λLL :Vgn ILnSS2 v¯CnSS− ion = 0 (8.8)which contains all possible combinations of equilibrium inductor peak current (ILnSS) andcapacitor DC voltage (v¯CnSS), and defines the load line λLL shown in Fig. 8.2. Includingthis condition into (8.3) and (8.7), the normalized equilibrium inductor current and capacitor1708.2. Steady State: Load Line & Target Operating Trajectoryvoltage are obtained:iLnSS = 2 ionv¯CnSSVgn|sin(ωgntn)| (8.9)vCnSS = v¯CnSS − pi ionωgnsin(2 ωgntn) (8.10)These normalized equilibrium current and voltage waveforms can be combined, eliminatingthe time variable, in order to describe the equilibrium geometric trajectories shown in Fig.8.2, which are mathematically described by:λSS : 4y2(1− y2)− x2 = 0 (8.11)with:x = (v¯CnSS − vCnSS) ωgnpi ion(8.12)y =iLnSS Vgn2 ionv¯CnSS(8.13)The target operating trajectory λtarget, formed by the desired inductor current (iLnt) andcapacitor voltage (vCnt) waveforms, is found when the DC component of the capacitor voltageequals the target (v¯CnSS = v¯Cnt):λtarget :(iLnt Vgnion v¯Cnt)2 [1−(iLnt Vgn2 ion v¯Cnt)2]−[(v¯Cnt − vCnt) ωgnpi ion]2= 0 (8.14)The time-domain expressions of the target waveforms are:iLnt =2 ion v¯CntVgn|sin(ωgntn)| (8.15)vCnt = v¯Cnt − pi ionωgnsin(2 ωgntn) (8.16)1718.3. Transient: Trajectory Convergence ConditionsiLnILntvCnvCniLn¸LLvCntvCntvCn(0)1vCn(0)1vCn(0)2vCn(0)2ILn2ILn1ILn2ILn1ILnt¸targettntn2fgn14fgn14fgn3fgn1ConvergenceRegionsFigure 8.3: Convergence conditions in a boost-derived PFC shown in state plane and timedomains.8.3 Transient: Trajectory Convergence ConditionsWhen the equilibrium conditions are not satisfied, the cycle mean capacitor voltage, andtherefore the operating trajectory, are not constant (as shown in Fig. 8.3). The geometriccontrollers being proposed in this work find their foundational basis in geometric trajectoriesλx, which define the desired peak reference current (ILn) as a function of the cycle meancapacitor voltage (v¯Cn). These geometric compensators must direct the operating trajectoryto the defined target λtarget. This sub-section shows the analysis of convergence conditionsthe trajectories λx must satisfy.1728.3. Transient: Trajectory Convergence ConditionsiLnILntvCn¸LLvCnt¸xCircle ¸xLine¸xQuadraticTrajectoriesDomain¸targetFigure 8.4: Three different Geometric Compensators that satisfy the convergence conditions:λxLine, λxQuadratic, and λxCircle.The normalized capacitor voltage evolution over time is ruled by (8.4). Eliminating theripple component, which is part of λtarget, the dynamics of v¯Cn are found:12pidv¯Cndtn=Vgn ILn2 v¯Cn− ion (8.17)which, for the equilibrium condition dv¯Cndtn= 0, defines the load line λLL (8.8). Combining(8.8) and (8.17), the convergence conditions that make the capacitor average voltage v¯Cnmove towards the desired steady state value v¯Cnt are found:λLL = 0, for: v¯Cn = v¯Cnt;λLL > 0, for: v¯Cn < v¯Cnt; andλLL < 0, for: v¯Cn > v¯Cnt. (8.18)which define the state plane area inside of which the geometric compensator λx must becontained, as shown in Fig. 8.4.1738.4. Geometric Control Law8.4 Geometric Control LawThe transient performance of the voltage loop is defined by the geometric trajectory λxemployed as the voltage controller. In this way, different trajectories that provide differentdynamic responses could be implemented as illustrated in Fig. 8.4. Any geometric equationλx that satisfies (8.18) would be a valid geometric compensator. This work focuses on theanalysis for straight line voltage controllers λxLine. The equation of a line that includes thepeak target inductor current ILnt =2ionv¯CntVgnand the mean target capacitor voltage (v¯Cnt) is:λxLine : I∗Ln = ILnt − k (v¯Cn − v¯Cnt) (8.19)where I∗Ln is the peak reference current and k defines the line slope. In order to maintain thetrajectory inside of the domain defined in section 8.3, the coefficient must satisfy: k > −ILnt.Plugging (8.19) into (8.17), and assuming perfect current reference tracking , the dynamicsof the straight line geometric compensator are defined by the non-linear differential equation:12pidv¯Cndtn=(ion +k Vgn2)(v¯Cntv¯Cn− 1)(8.20)which solution is implicit and it resembles the one of the DC-DC boost converter (7.10), wherethe slope k is directly related to the geometric compensator’s dynamic response. Higher valuesof k produce faster dynamics at the output-voltage control loop, and vice-versa.The complete implementation of the geometric controller is shown in Fig. 8.5, in whicheach term of the geometric compensator can be identified. An extra integrator term (slowdynamics) is added in order to eliminate any steady state error caused by losses and/orapproximations performed. Since the capacitor voltage ripple is part of the desired trajectory,the feedback signal employed by the geometric controller is the capacitor voltage cycle meanvalue v¯Cn, which, as shown in (8.19), is obtained through a Moving Average Filter (MAF).1748.5. AC-DC Dual-loop Geometric Controller ResultsMAFPk21/x∫vgnVgnkionStraight Line Geometric Compensator: ¸xLineiLn*vCnvCntvCnFigure 8.5: Implementation block diagram for the straight line geometric compensator λxLine.Table 8.1: Simulation ParametersPARAMETER VALUEv¯cnt 1Vgn 0.7fgn 0.2Ln 1/(2pi)rLn 0.05Cn 1/(2pi)fswn 2008.5 AC-DC Dual-loop Geometric Controller ResultsThe dual-loop straight line compensator was implemented in simulation and experimentalplatforms following the architecture illustrated in Fig. 8.5.8.5.1 Simulation ResultsSimulation results performed using the normalized boost-derived PFC detailed in Table 8.1are shown in Figs. 8.6 to 8.9. The normalized inductor parasitic resistance rLn was includedin order to represent more realistic conditions, and the normalized switching frequency fswnwas set to a high value in order to reduce the current ripple.1758.5. AC-DC Dual-loop Geometric Controller Results0.8 100.040.08vCni Ln0.81vCn00.1i Ln0 5 10 15 2000.5tn|vgn|Figure 8.6: Simulation results of the straight-line geometric compensator performance duringstart-up transient.0.8 0.9 1 1.100.050.10.150.2vCni Ln0.81vCn00.10.2i Ln0 10 20 3000.5tn|vgn|Figure 8.7: Simulation results of the straight-line geometric compensator performance duringan input voltage amplitude step-down transient.In the start-up transient shown in Fig. 8.6 the initial capacitor voltage equals the nor-malized peak input voltage (0.7). It must be remarked that no soft-start routine is includedin the algorithm. As shown by the simulation capture, the geometric controller provides fastresponse reaching the target voltage in less than one line cycle with under 10% overshoot.The grid voltage step down transient shown in Fig. 8.7 corresponds to a sudden change inthe normalized input voltage amplitude from 0.7 to 0.5. As it can be observed, the geometric1768.5. AC-DC Dual-loop Geometric Controller Results0.9 1 1.100.050.1vCni Ln0.911.1vCn00.1i Ln0 5 10 1500.04tni onFigure 8.8: Simulation results of the straight-line geometric compensator performance duringa load current step-up transient.0.9 1 1.100.050.10.15vCni Ln0.911.1vCn00.1i Ln0 5 10 1500.04tni onFigure 8.9: Simulation results of the straight-line geometric compensator performance duringa load current step-down transientcontroller recovers steady state operation in less than a line cycle, showing an outstandingresponse even for extremely large and sudden condition changes.Simulation results of load current step-up and step-down transients are shown in Figs. 8.8and 8.9 respectively. As shown in the simulation captures, the geometric voltage compensatorreacts in a fast manner, changing the peak current setpoint almost instantaneously andenabling excellent dynamic regulation.1778.5. AC-DC Dual-loop Geometric Controller ResultsTable 8.2: Experimental ParametersPARAMETER VALUEv¯ct 200 VVg 155 Vfg 60 HzL 550 µHrL 0.056 ΩC 560 µFf0 287 Hzfsw 60 kHziLvCiovgFigure 8.10: Experimental results of the straight-line compensator under load current step-uptransient (0.8A to 1.8A) at the input voltage peak.8.5.2 Experimental ResultsExperimental results obtained employing the converter detailed in Table 8.2 are shown inFigs. 8.10 to 8.13. As shown in the table, the grid frequency is approximately five times lowerthan the L-C resonance frequency (fgn = 0.21), while the normalized switching frequency is:fswn = 209, and the normalized peak grid voltage is: Vgn = 0.775.Load current step-up transients from io = 0.8A to io = 1.8A occurring when the inputvoltage is at its peak value and when it crosses zero are shown in Figs. 8.10 and 8.11respectively. Load current step-down transients from io = 1.8A to io = 0.8A occurring at the1788.5. AC-DC Dual-loop Geometric Controller ResultsiLvCiovgFigure 8.11: Experimental results of the straight-line compensator under load current step-uptransient (0.8A to 1.8A) at the zero crossing level.iLvCiovgFigure 8.12: Experimental results of the straight-line compensator under load current step-down transient (1.8A to 0.8A) at the input voltage peak.peak and zero levels of the input voltage are shown in Figs. 8.12 and 8.13.These results highlight the extremely fast and predictable response of the proposed con-troller, solving large constant-current step transients in small fractions of a line cycle in apredictable manner regardless the instant when the transients occur. The physical limitsof performance of this boost-derived application follow the same principles as in a DC-DCconverter, in which the normalized current steps plays a fundamental role. In this case,and due to the typically higher voltages of AC-DC applications, the normalizing current1798.6. SummaryiLvCiovgFigure 8.13: Experimental results of the straight-line compensator under load current step-down transient (1.8A to 0.8A) at the zero crossing level.is iref =vrefZ0= 201.8 A. For this reason the normalized current steps analysed here are∆ion = 0.005. This very low value is typical in low power AC-DC converters, and enablesthe extremely fast transient recovery shown in the experimental captures.The results obtained illustrate how the benefits of fast and reliable dynamic response foundin geometric controllers can be maintained in applications other than the three basic DC-DCconverters. This validation opens the door to a diverse range of potential applications.8.6 SummaryA dual-loop geometric control scheme for AC/DC converters was explored in this chapteras an extension of the concepts introduced in chapter 7. The differential equations that de-scribe the behaviour of boost-derived PFCs were normalized and solved in the time domain.Evaluating for equilibrium conditions, the load line and the steady state target operatingtrajectory were obtained. The target trajectories were defined as the path to be followedby the average capacitor voltage and the cycle peak inductor current. Convergence con-ditions that guarantee the converter operating trajectory is continuously directed towardsthe target operating trajectory were analysed, and the corresponding necessary restrictions1808.6. Summaryto the state-plane domain were defined. The geometric voltage-loop controller that definesthe current loop reference was mathematically derived and its practical implementation wasclearly described including the geometric trajectory and a steady state error correction term.Simulation and experimental results under different steady state and transient conditionswere provided in order to validate the extremely fast performance achieved by the dual-loopgeometric controller.181Chapter 9Conclusions9.1 ConclusionsThis dissertation explored the geometric behaviour of power electronic converters in the stateplane domain with the objective of enabling significant dynamic performance improvements.The work includes: a unified large signal model that describes the behaviour of the threefundamental topologies; the derivation of the dynamic physical limits of performance forbuck and boost converters; a benchmarking tool to characterize the dynamic response ofbuck and boost converters; two entirely geometric-based, non-linear control methods forclassical DC-DC applications and Photovoltaic Maximum Power Point Tracking; and twohybrid linear-geometric controllers for DC-DC and AC-DC converters.9.1.1 Unified ANTsThe Unified Average Natural Trajectories were developed in Chapter 2, introducing an alter-native way of modelling DC-DC converters to the traditional small-signal models. The ANTsaccurately represent the large signal averaged behaviour of PWM based DC-DC convertersin the entire state-plane. The normalized approach followed makes the model representativeof any converter independently of its L-C values and voltage levels at the input and output.In contrast with other non-averaged geometric models, the average nature of the unifiedANTs enables employing them in multiple PWM-based applications, as shown throughoutthis thesis. Furthermore, the ANTs concept is simple and intuitive, as they are represented1829.1. Conclusionsby circles in the state-plane domain. In this way, the model provides thorough insight on thedynamic behaviour of the physical variables interacting within the power converters, whichcan be leveraged in any type of applications including traditional linear compensators.A detailed step-by-step mathematical procedure to obtain the ANTs in DC-DC convertersis included, setting the modelling tool basis for other applications such as the inverted buckfor PV applications detailed in chapter 6. The large-signal accuracy of the model beingproposed was validated by experimental results obtained in several DC-DC converters.9.1.2 Physical Limits of Dynamic PerformanceThe physical limits of dynamic performance of buck and boost converters were analysed inchapter 3. The natural evolution of the converters’ state variables for ON and OFF positionsof the switch was modelled by mathematical equations and geometric shapes in both timeand state plane domains. The normalized domain employed for the analysis enables validsolutions for any combination of L-C parameters as well as input/output voltages.Employing these ON and OFF natural responses, the physical limits of performance of theconverter dynamic behaviour under load current and input voltage transients were found forMinimum Recovery Time and Minimum Voltage Deviation responses. The critical parametersin the ideal responses were characterized by closed-form expressions providing an absolutebenchmarking point for the dynamic performance evaluation. The absolute minimum start-up time for buck and boost converters was found, establishing the T0/4 start-up transientbenchmark rule.The work included in this chapter shows strong contributions to the theoretical fieldby providing the analysis of the buck and boost converters’ natural trajectories (with andwithout losses), and the theoretical ideal MRT and MVD responses.The theoretical findings were validated by near-optimal experimental results for the threetransients studied and different input/output voltage ratios.1839.1. Conclusions9.1.3 Dynamic Performance Benchmarking ToolA powerful benchmarking tool to objectively characterize the dynamic performance of buckand boost derived converters was introduced in chapter 4.The work enables the quantitative evaluation of the converters’ dynamic performancebased on the ideal MRT and MVD responses derived in chapter 3. Six performance indiceswere defined in order to characterize each one of the two types of performance parametersof analysis (speed and dynamic regulation) under three different types of transient (start-up,load step-up and load step-down).The idealized analysis of the evolution in the state-variables was extended for the inclu-sion of parasitic resistances in the inductor and capacitor. The effect of these losses werepresented in time and geometrical domains. Experimental results of the theoretical limitsof performance were provided. The results obtained validate (within 3% error) the valuesobtained theoretically including the effect of losses.The six indices obtained as a result of the analysis were summarized in a graphical repre-sentation to allow the fast interpretation of results. The benchmarking method introduced inthis work was illustrated by evaluating the performance of several buck and boost convertersthroughout this thesis.The six-indices method proposed in this chapter shows direct impact to the applied field,providing practical engineers with an absolute reference frame, and a benchmarking methodfor the dynamic performance evaluation.9.1.4 Centric Control for DC-DC ConvertersA novel PWM-based geometric control scheme for the three basic DC-DC topologies wasintroduced in chapter 5. The three main objectives stated for the proposed control tech-nique are: to obtain a reliable and consistent transient response, with an enhanced dynamic1849.1. Conclusionsregulation, and a simple implementation. These objectives were accomplished by developinga high-performance low-complexity control scheme based on the ANTs, which describe theconverters’ large-signal behaviour in an intuitive geometric domain.The control decisions are based in the state variables location in the geometric plane,which allows a sharp control of trajectories described by the operating point ensuring reliablebehaviour. Improvements to the dynamic regulation are obtained by allowing duty cyclesaturations only for a well-determined period of time, avoiding large spikes in the inductorcurrent and capacitor voltage. Due to its averaged nature, the controller is suitable forimplementation in low-cost DSPs using fixed-frequency PWM, and the bandwidth requiredfor sensing stages is maintained low. Furthermore, due to the normalized approach adopted,the controller can be implemented for any combination of LC parameters and input/outputvoltages maintaining the fast, sleek and reliable dynamic response.As a result, the maximum ratings of the reactive components are reduced, as well asthe implementation requirements for the controller, which turns the technique into a veryappealing alternative for high-volume applications.Detailed mathematical procedures the centric-based controller are included in the analysis.The enhanced dynamic response of the controller proposed were validated by experimentalresults for several DC-DC converters.9.1.5 State-plane Direct MPPTChapter 6 introduced a novel approach for performing direct duty-cycle MPPT in photo-voltaic energy harvesting applications: the State-plane direct MPPT.By employing state-plane analysis, the large-signal dynamic behaviour of the PV-connectedpower converter was modelled in a straightforward geometric manner. Using this geomet-ric model, an increase in the direct MPPT algorithm updating frequency of two orders ofmagnitude was enabled (e.g. from 20 T0 toT08in the analysed example). The much higher1859.1. Conclusionsupdating frequency resulted in extreme enhancements in the MPPT dynamic tracking per-formance compared to traditional solutions.Detailed mathematical procedures were provided to support the proposed methodology.The extreme improvements were confirmed through simulation results and under the dynamictest standard EN50530 including a speed multiplier to reach the typical irradiance changerates found in moving PV applications (i.e. backpack, bike, car).A step-by-step design procedure, an example application, and a detailed hardware im-plementation diagram were provided, highlighting the relevance of the contribution of thework to the applied field. The theoretical framework and design procedure were validated bysimulation and experimental results in 65W and 50W platforms, respectively.9.1.6 Dual-loop Geometric ControlThe concept of dual-loop geometric-based control was introduced in chapter 7 and 8 forDC-DC and PFC converters respectively by combining traditional dual-loop schemes with ageometric-based voltage loop design.The trajectory-based approach adopted for the voltage compensator design enables fastand reliable performance, eliminating the issues related to the inherent RHPZ of boost-derived applications. The proposed technique combines the simple implementation charac-teristic of traditional control schemes with the enhanced dynamic response characteristic ofstate-plane based controllers. The time-domain evolution of the state variables can be con-trolled by employing the proposed scheme, providing a reliable transient response by trackingtarget trajectories in the state plane to reach the target operating point.The theoretical analysis was performed in the normalized geometric domain providinggenerality and valuable insight into the topologies dynamic behaviour. Theoretical analysisof the geometric control principles were provided including target operating trajectories andconditions for convergence of the controllers. Several target trajectories were tested as volt-1869.2. Specific Academic Contributionsage controller for buck and boost converters, and their characteristic features and dynamicbehaviour were analysed. Full details on the controllers implementation were provided forthe three converters under analysis.The expected fast and predictable dynamic response of the proposed geometric controllerswas validated by simulation and experimental results employing 50W, 100W, and 360Wplatforms for buck, boost and PFC converters respectively. The desired trajectories wereproperly followed by the converters’ operating point, highlighting the strong contribution ofthe work to both the theoretical and the applied field.9.2 Specific Academic ContributionsThe derivation of the unified ANTs for DC-DC converters, along with the Unified CentricControl of DC-DC Converters culminated in the following publications:• I. Galiano Zurbriggen, M. Ordonez, and M. Anun, “PWM-geometric modelling andcentric control of basic DC-DC topologies for sleek and reliable large-signal response,”IEEE Trans. Ind. Electron., vol. 62, no. 4, pp. 2297–2308, April 2015.• I. Galiano Zurbriggen and M. Ordonez, “Average natural trajectories (ANTs) for boostconverters: Centric-based control,” in Proc. IEEE Appl. Power Electron. Conf. andExpo., March 2014, pp. 1190–1197.These works introduce the concept of state-plane large signal modelling for the threefoundational PWM based DC-DC topologies, and highlight the applied contribution of theunified ANTs by proposing an ANTs-based high performance control method.The following publications explore the physical limits of dynamic performance of boost-derived topologies, providing close form expressions for the theoretical ideal responses MRT1879.2. Specific Academic Contributionsand MVD. These strong reference points are employed to develop an objective dynamicperformance benchmarking method based on six performance indices:• I. Galiano Zurbriggen and M. Ordonez, “Benchmarking the performance of boost-derived converters under start-up and load transients,” IEEE Trans. on Ind. Electron.,vol. 63, no. 5, pp. 3125–3136, May 2016.• I. Galiano Zurbriggen, M. Anun, and M. Ordonez, “Dynamic physical limits of boostconverters: A benchmarking tool for transient performance,” in Proc. IEEE EnergyConv. Cong. and Expo., Sept 2014, pp. 4118–4124.The principles of state-plane MPPT control for PV applications under extremely fastchanging irradiance were included in the following publications:• I. Galiano Zurbriggen and M. Ordonez, “PV energy harvesting under extremely fastchanging irradiance: State-plane direct MPPT,” IEEE Trans. Ind. Electron., vol. 66,no. 3, pp. 1852–1861, March 2019.• I. Galiano Zurbriggen, F. Paz, and M. Ordonez, “Direct MPPT control of PWM con-verters for extreme transient PV applications,” in Proc. 2016 IEEE Appl. PowerElectron. Conf. and Expo., March 2016, pp. 386–391.These papers include the theoretical principles that set the foundations of the MPPTmethod, detailed design and implementation procedures, and exhaustive testing throughsimulations and experiments under several scenarios.The concept of dual-loop geometric control was introduced in the following publicationsfor buck, boost, and PFC converters:• I. Galiano Zurbriggen, M. Anun, and M. Ordonez, “Dual-loop geometric-based controlof buck converters,” in Proc. 2015 IEEE Appl. Power Electron. Conf. and Expo.(APEC), March 2015, pp. 392–396.1889.2. Specific Academic Contributions• I. Galiano Zurbriggen, M. Ordonez, and M. A. Bianchi, “Dual-loop geometric-basedcontrol of boost converters,” in Proc. 2015 IEEE Energy Conv. Cong. and Expo.(ECCE), Sep. 2015, pp. 2072–2077.• I. Galiano Zurbriggen, M. A. Bianchi, and M. Ordonez, “Fast-transient, low-THDgeometric control of boost-derived active rectifiers,” in Proc. 2016 IEEE Int. Symp.on Power Electron. for Distr. Gen. Syst. (PEDG), June 2016, pp. 1–6.The publications listed below were the result of academic collaborations.Geometric controllers aimed to overcome characteristic issues found in Constant PowerLoad applications were explored in the publications listed below:• M. A. Bianchi, I. Galiano Zurbriggen, F. Paz and M. Ordonez, ”Improving DC Micro-grids Dynamic Performance Using a Fast State-Plane-Based Source-End Controller,”IEEE Transactions on Power Electronics, Early Access, 2019.• M. Anun, M. Ordonez, I. Galiano Zurbriggen, and G. Oggier, Circular Switching Sur-face Technique: High Performance Constant Power Load Stabilization for Electric Ve-hicle Systems,” in IEEE Transactions on Power Electronics, vol.30, no.8, pp.4560-4572,Aug. 2015.• M. A. Bianchi, I. Galiano Zurbriggen, M. Ordonez, and M. Anun Dynamic performanceimprovement of DC microgrids during Constant-Power Load transients, in Proc. IEEEInternational Symposium on Power Electronics for Distributed Generation Systems(PEDG 2016), Vancouver, (BC, Canada),. Jun 2016.A dual-loop geometric controller for full bridge inverters was introduced in the followingpublication:• M. A. Bianchi, I. Galiano Zurbriggen, and M. Ordonez, Dual-loop geometric-basedcontrol of full-bridge inverters for stand-alone distributed generation systems, in Proc.1899.2. Specific Academic ContributionsIEEE International Symposium on Power Electronics for Distributed Generation Sys-tems (PEDG 2016), Vancouver (BC, Canada), Jun 2016.A unified approach to design linear controllers for the fundamental DC-DC topologies wasintroduced in:• F. Degioanni, I. Galiano Zurbriggen and M. Ordonez, ”Universal Controllers for PWMConverters: a Normalized Approach,” Accepted for 2019 IEEE Applied Power Elec-tronics Conference and Exposition (APEC 2019), Anaheim (U.S.A.), March 2019.A novel modelling approach for LLC converters was introduced in the papers listed belowalong with a dual-loop control scheme:• F. Degioanni, I. Galiano Zurbriggen, and M. Ordonez, Dual-Loop Controller for LLCResonant Converters Using an Average Equivalent Model,” in IEEE Transactions onPower Electronics, vol. 33, no. 11, pp. 9875-9889, Nov. 2018.• F. Degioanni, I. Galiano Zurbriggen, and M. Ordonez, Dual-Loop Controller for LLCResonant Converters Using an Average Equivalent Circuit, in Proc. IEEE EnergyConversion Congress and Expo (ECCE17), Cincinnati (OH, USA), Oct. 2017.An optimization algorithm to dimension components for PV installations based on anenhanced model was introduced in the paper:• G. Bogado, F. Paz, I. Galiano Zurbriggen and M. Ordonez, ”Optimal Sizing of a PVand Battery Storage a Using Detailed Model of the Microgrid for Stand-Alone Applica-tions,” Accepted for 2019 IEEE Applied Power Electronics Conference and Exposition(APEC 2019), Anaheim (U.S.A.), March 2019.A controller design procedure for inverted buck converters employed in PV applicationswas presented in the publication:1909.3. Future Work• P.F. Ksiazek, I. Galiano Zurbriggen, F. Paz, J.M. Galvez, and M. Ordonez, ”Reversecontrol of solar power converters for modular telecommunications UPS,” in Proc. IEEEInternational Telecommunications Energy Conference (INTELEC14), Vancouver, (BC,Canada), Sep. 20149.3 Future WorkThe outcomes presented throughout this thesis are the result of specifically selected scenariosand applications that aim to contribute to setting the basis of PWM-geometric control inpower electronics. Numerous applications within power electronics that could benefit fromemploying similar methodology to the one employed in this work have been identified andare currently being explored or in the process of being developed.Furthermore, multiple fields outside of power electronics could largely benefit from em-ploying a similar framework to accurately model the large signal behaviour of the systems.Since, the methodologies introduced in this work are based on state-plane analysis of the timedomain solution of a set of normalized differential equations, extrapolating the solutions toother fields is feasible. Higher order equation systems might require of higher dimensionalorder state-plane analysis, making the visualization of the solutions an interesting challengefor future research.9.3.1 Average Natural TrajectoriesAs demonstrated throughout this thesis, there are multiple advantages in combining PWMoperation with geometric state-plane analysis for the modelling and control of switch-modepower converters.Developing the ANTs for three-phase converters in a Rotary Reference Frame (RRF) is avery appealing alternative due to the large number of potential applications which include:1919.3. Future Workboost-derived PFCs, Grid-tie and Stand-alone Inverters, Motor Drivers for EV powertrain.Employing the ANTs developed in this work, multiple DC-DC applications can be ex-plored. MPPT algorithms for wind turbines, Thermo-Electric Generators (TEG), and oceanturbines are natural courses of action due to the similarities with the PV solution includedin this thesis.9.3.2 Dynamic Performance BenchmarkingThe benchmarking procedure introduced in this work for buck and boost converters canbe extended to other buck and boost derived topologies. Although the physical limits ofperformance remain the same for all topologies based on the fundamental converters, thebenchmarking indices ranges should be re-defined to better represent the excellent, good,regular, and poor performances in each topology. The benchmarking procedure introducedin this thesis can also be extended to other DC-DC topologies, such as multi-level, resonanttopologies, and Dual Active Bridge, among many others.Furthermore, including response predictability and implementation complexity aspectsinto a benchmarking procedure would provide interesting value to design engineers and enablea comprehensive controller assessment tool.9.3.3 Dual-loop Geometric ControlThe dual-loop geometric control is a powerful concept, and there are numerous applicationsthat could benefit from its application. The geometric voltage loop is especially effective insolving the challenges posed by right half plane zeroes, which are present in all non-minimumphase systems such as boost and buck-boost topologies and all converters derived from them.A unified solution for the three fundamental DC-DC topologies would be a very interestingcomplement to the ideas introduced in chapter 7. Extending the solution introduced in chap-1929.3. Future Workter 8 to bridgeless PFC converters, and to single phase grid-tie inverters would extensivelybroaden the range of applications of the controller due to the popularity of the topologies.Three-phase applications are of particular interest for the implementation of the dual-loopgeometric controller. Employing a RRF the these applications can be represented by DC-DCtopologies for Direct, Quadrature and Zero (DQ0) components.Besides of extending the range of applications, interesting results could be obtained byanalysing the combination of the dual-loop geometric controller with different current con-trol solutions. Employing the normalized linear controller introduced in [130] would pro-vide a unique design for all possible voltage ranges and L-C parameters. Combining withpeak-current mode, hysteresis, and other non-linear current control solutions may yield veryinteresting dynamic performance improvements.193Bibliography[1] I. 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Thesis/Dissertation
eng
Electrical and Computer Engineering
Vancouver : University of British Columbia Library
Attribution-NonCommercial-NoDerivatives 4.0 International
Graduate
Large-signal transient control in power electronics : an average-geometric framework
Text