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Novel concepts for power electronics control : introducing the dynamic physical limits, the average natural… Galiano Zurbriggen, Ignacio 2013

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Novel Concepts for PowerElectronics Control:Introducing the Dynamic Physical Limits, theAverage Natural Trajectories, and theCentric-Based ControllerbyIgnacio Galiano ZurbriggenIng., Universidad Nacional de Co?rdoba, 2010A THESIS SUBMITTED IN PARTIAL FULFILLMENT OFTHE REQUIREMENTS FOR THE DEGREE OFMASTER OF APPLIED SCIENCEinThe Faculty of Graduate and Postdoctoral Studies(Electrical & Computer Engineering)THE UNIVERSITY OF BRITISH COLUMBIA(Vancouver)September 2013c? Ignacio Galiano Zurbriggen 2013AbstractControllers are an essential component in power conversion systems that have a significantimpact on characteristic features such as performance, efficiency, size, and cost, among manyothers. During the last four decades, countless efforts have been made to find better con-trollers for power electronics systems in order to improve the converters steady state anddynamic behaviour, increase power densities and reduce losses in the system.Small-signal based linear controllers have been the preferred alternative during decades.This technique features fixed switching frequency and low computation/sensing requirements,while the dynamic response can be improved to only a limited extent and the global stabilitycannot be ensured. On the other hand, excellent dynamic performances and global stabilityare achieved by boundary controllers, in which the switching frequency is variable and fastersensors are required.The first part of this work presents a practical tool which allows to objectively quan-tize improvements made by the controllers to the performance of power converters. Thetheoretical optimal dynamic behaviour of buck converters is determined, analyzed, and char-acterized using closed-form mathematical expressions, setting a strong benchmark point forthe performance evaluation.Taking the physical limits of dynamic performance into account, and merging the ad-vantages of linear and boundary techniques, a novel control scheme is developed for buckconverters. The proposed controller is based on a large-signal model introduced here: theAverage Natural Trajectories (ANTs). Enhanced dynamic performance and global stabilityiiAbstractare achieved while low sensing and computational requirements are maintained, which makesthe technique very appealing for use in high-volume production applications.Due to the outstanding results in the basic buck converter, and in order to illustrate theapplication of the ideas introduced in this work for different topologies, the ANTs and thecentric-based controller are developed for boost converters. The obtained results confirmthe enhanced dynamic response and fixed frequency operation as natural advantages of theproposed control scheme.The theoretical findings are supported by detailed mathematical procedures and validatedby experimental results, which highlight the practical usefulness of the concepts introducedin this work.iiiPrefaceThis 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. Some experimental validation work was done in collab-oration with Matias Anun.Versions of chapter 2 and 3 have been published at the IEEE Applied Power ElectronicsConference and Exposition (APEC), 2013 [1, 2].An extended version of chapter 3 has been submitted to a power electronics journal [3].A version of chapter 4 has been submitted to a conference in power electronics [4].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 from the MASc. studentMatias Anun who developed some specific experimental tasks.ivTable of ContentsAbstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iiPreface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ivTable of Contents . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . vList of Tables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . viiiList of Figures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ixAcknowledgements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xiiDedication . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xiii1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.1 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.2 Literature Review . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31.2.1 Dynamic Physical Limits of Performance . . . . . . . . . . . . . . . 31.2.2 Control for Buck Converters . . . . . . . . . . . . . . . . . . . . . . 51.2.3 Control for Boost Converters . . . . . . . . . . . . . . . . . . . . . . 61.3 Contribution of the Work . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91.4 Thesis Outline . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11vTable of Contents2 Dynamic Physical Limits of Buck Converters: the T0/4 Transient Bench-mark Rule . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 132.1 Buck Transient Natural Trajectories and Response . . . . . . . . . . . . . . 142.2 Buck Loadability and Sudden Load Transients . . . . . . . . . . . . . . . . 182.2.1 Loading . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 202.2.2 Unloading . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 222.3 Benchmarking Procedure Example . . . . . . . . . . . . . . . . . . . . . . . 242.4 Experimental Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 262.5 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 283 Average Natural Trajectories (ANTs) for Buck Converters: Centric-BasedControl . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 303.1 Buck Converter Ideal ANTs Derivation . . . . . . . . . . . . . . . . . . . . 323.2 Approaching the Target . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 383.3 Closing the Loop with a Centric-Based Controller . . . . . . . . . . . . . . . 403.3.1 Periodic Center Calculation . . . . . . . . . . . . . . . . . . . . . . . 423.3.2 Domain Restrictions . . . . . . . . . . . . . . . . . . . . . . . . . . . 423.3.3 Steady State Error Correction . . . . . . . . . . . . . . . . . . . . . 423.4 Experimental Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 443.5 Comparison with Linear Controllers . . . . . . . . . . . . . . . . . . . . . . 453.6 Tolerances Sensitivity Analysis . . . . . . . . . . . . . . . . . . . . . . . . . 493.7 Buck Converter Non-Ideal ANTs Derivation . . . . . . . . . . . . . . . . . . 523.8 Closed-Loop Controller Including Estimation of Parasitics . . . . . . . . . . 563.9 Centric Controller Benchmark . . . . . . . . . . . . . . . . . . . . . . . . . . 593.10 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61viTable of Contents4 Average Natural Trajectories (ANTs) for Boost Converters: Centric-BasedControl . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 634.1 Normalized Boost Converter ANTs Derivation . . . . . . . . . . . . . . . . 654.2 Approaching the Target . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 694.3 Closing the Loop with a Centric-Based Controller . . . . . . . . . . . . . . . 714.4 Experimental Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 724.5 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 745 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 765.1 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 765.2 Future Work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78viiList of Tables1.1 Some theoretical limits . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42.1 Buck converter parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . . 243.1 Buck converter parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . . 453.2 Main parasitics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 453.3 Centric - linear controllers comparison . . . . . . . . . . . . . . . . . . . . . 473.4 Centric controller parameter deviations . . . . . . . . . . . . . . . . . . . . . 504.1 Boost converter parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . 744.2 Main parasitics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74viiiList of Figures1.1 Controls in power electronics . . . . . . . . . . . . . . . . . . . . . . . . . . . 21.2 Centric-based, sliding-mode and dual-loop linear controllers in buck convert-ers: a conceptual comparison. . . . . . . . . . . . . . . . . . . . . . . . . . . 81.3 Normalized buck converter . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101.4 Normalized boost converter . . . . . . . . . . . . . . . . . . . . . . . . . . . 112.1 Normalized buck converter . . . . . . . . . . . . . . . . . . . . . . . . . . . . 142.2 Start-up transient in geometrical and time domains. . . . . . . . . . . . . . . 152.3 Loading transient represented on geometrical and time domains. . . . . . . . 192.4 Normalized loading transient parameters as function of Vccn, using ?iloadnas parameter a) voltage drop ?von(min) b) minimum loading recovery timetrecn(min),Loading . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 212.5 Unloading transient represented on geometrical and time domains. . . . . . . 232.6 Comparison between physical limit operation and traditional compensationtechniques. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 252.7 Start-up experimental results for a) low step-down (Vccn = 1) and b) highstep-down Vccn = 10 transients. . . . . . . . . . . . . . . . . . . . . . . . . . 272.8 Start-up, loading and unloading transients for Vccn = 2. . . . . . . . . . . . . 283.1 a) Normalized PWM driven buck converter. b) Pseudo-ideal natural trajectoryforced in a real converter. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32ixList of Figures3.2 Ideal PWM natural trajectory with Vccn = 2, d = 0.5 and fsn = 10. . . . . . 333.3 Several PWM-driven buck converter ANTs departing from unique initial op-erating point. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 363.4 Ideal fixed duty cycle target approaching method for different initial conditions 393.5 Fixed duty cycle target approaching method forced in a real converter . . . . 403.6 Centric-based controller: a) scheme, and b) experimental results. . . . . . . . 413.7 Closed-loop controller concept . . . . . . . . . . . . . . . . . . . . . . . . . . 433.8 Comparison between dual-loop linear and centric-based controllers. Start-uptransient for a) linear, and b) centric-based. . . . . . . . . . . . . . . . . . . 463.9 Comparison between dual-loop linear and centric-based controllers. Load-ing/unloading transients with ?ion = 1 for a) linear, and b) centric-based. . 473.10 Comparison between dual-loop linear and centric-based controllers. Load-ing/unloading transients with ?ion = 0.5 for a) linear, and b) centric-based. 483.11 Centric-based controller performance under parameter deviations. Start-up,loading and unloading transients for a) Lreal = 0.8 L, Creal = 0.8 C; and b)Lreal = 0.8 L, Creal = 1.2 C; . . . . . . . . . . . . . . . . . . . . . . . . . . . 503.12 Centric-based controller performance under parameter deviations. Start-up,loading and unloading transients for a) Lreal = 1.2 L, Creal = 0.8 C; and b)Lreal = 1.2 L, Creal = 1.2 C. . . . . . . . . . . . . . . . . . . . . . . . . . . 513.13 a) Non-ideal normalized PWM driven buck converter. b) Real PWM naturaltrajectory. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 523.14 Non-ideal natural trajectory with Vccn = 2, d = 0.5 and fsn = 10. . . . . . . . 543.15 Centric-based controller with load current estimation: a) scheme, and b) ex-perimental results. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 573.16 Response of the centric-based controller with load current measurement forloading/unloading transients with ?ion = 1 (a), and ?ion = 0.5(b). . . . . . 59xList of Figures3.17 Performance comparison between the ideal centric-based controller and thedynamic physical limits. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 604.1 a) Normalized PWM driven boost converter. b) PWM natural trajectories ina real converter. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 644.2 Ideal natural trajectories with Vccn = 0.5, d = 0.5 and fsn = 10. . . . . . . . 664.3 Several PWM-driven boost converter ANTs departing from unique initial op-erating point. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 684.4 Fixed duty cycle target approaching method for different initial conditions . 704.5 Centric-based controller experimental results. a) Open-loop with duty cycleprecalculation. b) Closed-loop response for start-up, loading and unloadingtransients. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73xiAcknowledgementsI would like to sincerely thank my supervisor Dr. Martin Ordonez for accepting me as partof his research team. His support, extreme dedication and valuable technical advice duringmy Master?s program are greatly appreciated.I would also like to acknowledge my lab mates, the research team, for kindly sharinghis experience and knowledge and for making of these years an enjoyable experience beyondtechnical aspects.I must thank the professors of the courses I took as part of the program for the valuableknowledge transferred, and the staff of the Electrical and Computer Engineering departmentof the University of British Columbia for their assistance in academic matters.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.My English teachers deserve to be mentioned as well for having taught me the languagein a very short period of time and almost from scratch.Last but not least, I would like to thank you, the reader, for showing interest in this work.xiiTo those in my deep love and respectxiiiChapter 1Introduction1.1 MotivationSwitching power converters are present in almost every electrical/electronic device in today?sworld. The ever-increasing demand of higher power densities (reduced size for the samepower) and higher efficiencies, has turned power electronics ubiquitous in any apparatus. Theapplications range from mass-produced low power converters (mili Watts) used in portabledevices (ie: cell-phones, digital cameras, music/video players, etc.) to high power (MegaWatts) applications produced on a small scale for use in power distribution.Closed-loop controllers are usually implemented in power conversion systems in order toobtain a tight regulation of the output variables at the desired values. Controllers are the?brain? of the system, and the performance of the converters is strongly related to them.Higher efficiency, better performance, smaller size and reduced cost can be reached just byimproving the controllers. Among the main aspects that determine the performance of powerconverter control schemes, dynamic response and robustness are undoubtedly two of the mostimportant.The dynamic response of a power converter is characterized by two main parameters:recovery time and peak deviations in the state variables. The recovery time is given bythe period of time the converter takes to reach steady state after a change in the operatingconditions, for instance a step-up in the load current. The deviations in the converter statevariables during a transient determine how much the voltages and currents differ from the11.1. MotivationIMPLEMENTATIONCONTROL IN POWER ELECTRONICSPERFORMANCECentric-basedLinearNatural Switch.  SurfacesSliding surfacesCONTROLLERSSmall-signal modelNaturaltrajectoriesAverage NaturalTrajectoriesImposeddynamicsMODELSAverageGeometric analysisLinearizationSliding Mode existance& invarianceMODELING TOOLSDynamic physicallimits of performanceImplement. costs ($, size, time, ...)Lyapunov stability analysisEVALUATION TOOLS- Fixed  frequency- Global stability- Dynamic response- V & I peak deviations- Complexity- PWM modulation- Sensing/sampling req. - Amount of variablesConcepts introduced by this workFigure 1.1: Controls in power electronicsdesired steady state values, and the peak values reached can determine whether or not theconverter is suitable for a determined application. For instance, an electronic system thatrequires a 3.3 V power supply might require a recovery time of 1 ms and a maximum drop of0.3 V in order to function properly. Undesired saturation in magnetic elements due to largecurrent peaks, and system failures due to large and maintained voltage drops can be avoidedby implementing controllers featuring fast and predictable dynamic response.The robustness of a system is defined as its ability to deal with external and/or inter-nal disturbances. Tolerances in the converter components and inaccuracies of the modelsemployed are some of the factors that can cause unstable behaviours in power converters.Taking these effects into account when designing the controllers is essential to guarantee areliable operation during transients and in steady state.Other features related to the implementation, such as complexity, constant switching fre-quency operation, and sensing/sampling requirements, are very important as well, since theydefine the amount of resources required by the controller, and how well it can be implementedusing technologies currently available.21.2. Literature Review1.2 Literature ReviewCharacteristic features of the different control techniques are strongly related to the modelsemployed for their design, which inherently depend on the mathematical tools by means ofwhich they have been developed. An illustrative sketch including some of the most popularschemes and models is provided in Fig. 1.1.Linear compensators based on small-signal averaged models have been the preferred alter-native for controlling power converters during the last four decades. Although proven to beeffective by countless industrial applications, these control techniques present issues relatedto the model employed, which is based on a local linearization and therefore is only valid inthe neighborhood of a determined operating point.Nonlinear solutions of many different natures have been proposed, resulting in substantialimprovements to the robustness and dynamic response, boundary controllers being one ofthe most popular alternatives.However, the improvements cannot be properly quantized due to a lack of awarenessabout the limits of performance limits of the converters. The following literature review goesthrough the efforts that, during many years, have been made to find a controller performancebenchmark tool for buck converters and to improve controllers for basic Buck and Boost DC-DC topologies.1.2.1 Dynamic Physical Limits of PerformanceWith over 5,000 buck control related papers available at IEEE Xplore (2013), the engineeringand scientific community has been looking for ways to improve the dynamic performance ofbuck derived topologies for decades, either by developing new models and implementing newcontrol techniques [5?19] or by introducing modifications to the topology [20?24]. Manyother disciplines follow the same trend, such as electrochemistry in the search for fuel cell31.2. Literature ReviewTable 1.1: Some theoretical limitsDEVICE THEORETICAL LIMITWind Turbine 59.3% effic.(Betz law)Fuel Cell 1.23V(Nernst cell voltage)Solar Cell 33.7% effic.(Shockley-Queisser Limit)Lithium-Ion Battery Cell 4.1V100Ah/Kg capacityClass A Audio Amplifier 25% effic.Buck converter, To/4 [s]high step-down start-up transient(vin/vout ??)Buck converter, To/3 [s]low step-down start-up transient(vin/vout = 1)voltages closer to the theoretical limit of 1.23V (Nernst voltage), or the improvement ofsolar cells efficiency in the attempt to reach values closer to the theoretical maximum of33.7% (Shockley-Queisser limit); the pursuit of higher efficiencies in wind turbines in orderto approach the theoretical maximum of 59.3% (Betz law); and the improvement of steam andDiesel engines to obtain efficiency values closer to the theoretical maximums (given by Carnotand Diesel cycles respectively) in mechanical engineering (Table I). Unlike these disciplines,power electronic engineers/researchers do not use the theoretical limits of performance tobenchmark controllers. Unfortunately, the relative gains or improvements to the dynamicresponse of the converter cannot be fully determined if the results are not compared with thetheoretical transient time limits.Transient analysis of buck converters has been performed in the past using small-signalmodeling [25]. Due to constraints of small-signal analysis, the results do not provide any41.2. Literature Reviewinformation about performance limits. Critical parameters on the transient response and theireffects have been also analyzed [26?28]. However, the theoretical minimum transient times arenot specifically addressed. Based on small-signal analysis, an effort to establish a benchmarksystem using a standard 20W converter design was proposed [29], but the physical limits of thesystem are not discussed. Later interesting work used simulations and experimental resultsto explore the limits of buck performance [30?33], presenting advancements to the field. Thederivations and general expressions necessary to determine the theoretical physical limitsremained unaddressed and without a concrete solution. In fact, rather than being commonknowledge, the theoretical transient limits of buck converters remains unknown to technicalexperts, power designers, the power conversion community in general.1.2.2 Control for Buck ConvertersUsing linear control techniques in buck topologies, switching and sampling frequencies arekept constant and robust systems can be obtained by performing stability analysis and tuningthe compensator parameters [34?42]. However, the dynamic performance can be improvedto only a limited extent and, since the models of the converter are only valid for small-signalvariations, the controllers perform poorly under large load transients. Linear/non-linearalternatives based on output capacitor current of the buck converter have been proposedin [43?45], where non-linear solutions provide improvements in the dynamic response whileclassical linear controllers are used for steady-state regulation.Among the non-linear techniques, boundary controllers are a popular alternative for con-trolling buck converters. Typically, the main advantage of this type of controllers is its fastand predictable transient response, which in some cases reaches the physical limit of perfor-mance [1]. Conceptual boundary controlled systems are bounded stable by nature, as longas the sliding mode existence and invariance conditions are met [46, 47]. Chattering andvariable switching frequency are the major issues they present [48?50]. Several techniques51.2. Literature Reviewto obtain constant switching frequencies for first and second order switching surfaces havebeen successfully implemented [51?55]. Since the solution deviates from the ideal concept,unboundedness and steady state error become an issue. A different situation takes place inthe case of natural switching surfaces of buck converters, where the ideal behavior is boundedstable, the steady state switching frequency is fixed and large transients are solved in onlyone switching action [33, 56?58]. However, there are real factors that prevent the converterfrom exhibiting ideal behavior, and therefore the robustness of the system results highly de-pendent on factors like finite sampling frequency, reactive components value accuracies andlosses effect.1.2.3 Control for Boost ConvertersFixed frequency operation and a reliable steady state behaviour can be achieved in boosttopologies by using traditional linear controllers. However, due to the characteristic righthalf plane zero (RHPZ) of the small signal control-to-voltage transfer function, the robustcontrollers that can be designed present low bandwidth, which leads to a sluggish dynamicresponse [59]. Introducing a second loop to control the inductor current certainly allows anincrease in the controller bandwidth [41, 60, 61]. Nevertheless, the RHPZ is not eliminatedand therefore the dynamic response is still limited. Besides, since the employed modelsare only valid for small signal, the response under large transients cannot be predicted andlarge-signal stability cannot be ensured. Several approaches to solve this last issue in theimplementation of small-signal based controllers have been presented recently [38, 62?68].Although interesting results are obtained, the complexity of the implementation is greatlyincreased and the dynamic performance is still limited due to the nature of the modelsemployed.Non-linear geometric-based approaches present an appealing alternative for boost con-61.3. Contribution of the Workverters, where the effect of the characteristic non-minimum phase behaviour is eliminatedsince the control decisions are based purely on the system operating point. First orderswitching surface controllers are a popular alternative in which a predictable response is ob-tained once the sliding mode is reached [46?49]. Unpredictable current peaks before reachingsliding mode, steady state error, chattering and variable switching frequency are some of theissues in the implementation of this technique. Although successful attempts to solve mostof these issues have been presented [55, 69, 70], the simplicity of the original concepts iscompromised. Using Second order switching surfaces, the dynamic response can be improvedwhile the achievement of constant switching frequencies can be maintained [52, 71]. Besidesthe increase in complexity needed to obtain constant frequency, overshoot and large currentspikes during large transients are some of the disadvantages the technique presents.The physical limit of dynamic performance of boost topologies is reached by boundarycontrollers using the Natural Switching Surfaces, which have been proven to be successful inseveral DC-DC topologies. Fixed frequency steady state operation and a predictable time-optimal transient solution performing only one switching action are the main advantages ofthis technique [1, 33, 57, 58, 72, 73]. On the other hand, large current peaks are needed toachieve the physical limit of dynamic performance; high sampling frequencies are requiredand the robustness of the system is tied to values accuracy and losses effect in the reactivecomponents. As a result, the technique presents an excellent alternative in high-power ap-plications where the high-end requirements for the controller implementation are justified bythe obtaining of time-optimal dynamic performance and the minimization of the number ofswitching actions.71.3. Contribution of the WorkCentric Start-up timeSliding-modeLinear dual-loopiCncontrolvariablevCntntntnCentricStart-up currentpeakSliding-modeLinear dual-loopiCnvCntarget0Figure 1.2: Centric-based, sliding-mode and dual-loop linear controllers in buck converters:a conceptual comparison.81.3. Contribution of the Work1.3 Contribution of the WorkThis work introduces valuable theoretical concepts to the field of controls for power electronicssystems as well as useful practical applications of the ideas developed:? The first concept comes after a clear need identified in the literature review: a bench-mark tool to objectively assess the dynamic performance of power electronics con-trollers. The dynamic physical limits of performance are introduced as a theoreticaloptimal response only achievable in ideal conditions, which cannot be exceeded.? Second, a novel way of modeling power converters that merges geometrical analysiswith traditional averaging techniques, is introduced: the Average Natural Trajectories(ANTs). The proposed model accurately describes the averaged large-signal behaviourof PWM-driven power converters.? Third, based on the obtained ANTs, the novel geometric-based control technique illus-trated in Fig. 1.2 referred to as centric-based control. The proposed technique featuresrepeatable and predictable dynamic response reaching values close to the theoreticallimit of performance, while the complexity of the implementation is kept low, fillingthe gap between small-signal-based linear controllers and geometric-based boundarytechniques. As shown in the conceptual figure, the proposed scheme features shortertransient times and lower peak current than two of the most popular techniques in thefield, linear and sliding mode controllers.It is worth mentioning that the ideas introduced in this work can be applied to otherswitching power converter topologies, and since the analysis is performed in a normalizedfashion, the results are valid for any combination of values of the reactive components.As part of the focus of this work, a geometrical description of the theoretical optimalresponse of the normalized buck converter illustrated in Fig. 1.3 is obtained. Analyzing91.3. Contribution of the WorkiloadnVccn vonLn  =12?Cn  =12?OFFON+-iLn T o= 2pi LCZo = L /Cvxn=vxvreftToixn=ixvrefZo tn =Normalization:Base quantities:Figure 1.3: Normalized buck converterthe optimal response, closed-form equations that determine the physical limits of dynamicperformance of buck converters are derived and the T0/4 start-up transient rule is derived,setting a strong benchmarking point.The Average Natural Trajectories are derived for buck converters and based on the modelobtained, the centric-based control is implemented. The enhanced dynamic response pre-dicted is validated by experimental results. In order to illustrate that the concepts introducedin this work can be developed for any other topology, and due to the outstanding perfor-mance obtained by the centric-based controller implementation, the analysis is extended toboost converters. The large-signal model is obtained by deriving the ANTs in the normalizedconverter shown in Fig. 1.4, and the controller is implemented in an experimental platform,obtaining excellent results. The controllers are implemented in low-cost DSPs, which makesthe technique suitable for implementation in large-scale production converters, highlightingthe important contribution of the work to the practical field.101.4. Thesis OutlineLn =12piCn =12piVccniLniloadnvon+- T o= 2pi LCZo = L /Cvxn=vxvreftToixn=ixvrefZo tn =Normalization:Base quantities:Figure 1.4: Normalized boost converter1.4 Thesis OutlineThis work is organized in the following manner:? In chapter 2, the buck converter physical limits of performance are developed. Anormalization procedure is applied to the buck converter state equations to obtainexpressions that are not dependent on the system parameters. These expressions aresolved in time domain first, and then merged to obtain a geometric representation ofthe converter behaviour for ON and OFF states, which defines the converter naturaltrajectories. Using the insight provided by the trajectories, the theoretical optimalresponse is found in a geometric domain, and characterized by mathematical expressionsfor three types of transient: start-up, loading and unloading. A benchmark procedureis introduced, in order to address the practical contribution of the concepts developedin this work. The findings are validated using simulations and experimental results inorder to illustrate the different transients analyzed.111.4. Thesis Outline? In chapter 3, the concepts of Average Natural Trajectories (ANTs) and centric-basedcontrols are developed for buck converters. The normalization of the ON and OFFstate equations of buck converters is followed by an average procedure to obtain anaveraged representation of the converter behaviour. The average state equations areobtained and solved in time domain to find expressions that describe the averaged evo-lution of PWM driven buck converters. The time-domain expressions are combined toobtain trajectories that model the natural evolution of the buck converter variables ina geometric domain: the Average Natural Trajectories (ANTs) are obtained for buckconverters. Based on the derived large-signal model, the centric-based controller is con-ceptually derived for buck converters. Simulation and experimental results are shownin order to validate the enhanced dynamic performance of the proposed technique.? In chapter 4, the concepts of Average Natural Trajectories (ANTs) and centric-basedcontrols are introduced for boost converters. The ON and OFF state equations of boostconverters are normalized and averaged over a switching period in order to obtaina set of duty-cycle dependent state equations that describe the behaviour of PWMdriven boost converters. The averaged equations are solved in time domain, obtaininga description of the natural evolution of the averaged variables. The averaged time-domain equations are combined to obtain the Average Natural Trajectories of boostconverters, which models the large-signal behaviour in a geometric domain.? Finally, in chapter 5 a summary and conclusions of this work are presented along withan account of work under development and ideas for future research.12Chapter 2Dynamic Physical Limits of BuckConverters: the T0/4 TransientBenchmark RuleAs identified in the literature review, and unlike researchers in other disciplines, the designerof power converter controls does not count with a tool that helps him/her to determine howimportant the improvements under development are. Furthermore, since the analysis is notusually performed in the normalized domain, there is no clear indication of how well thecontroller performs and how much better it could for the reactive components being used.It is the aim of this chapter to introduce the concepts of normalized analysis and physicallimits of performance, and to present the way in which these can be used as a tool to toenable the objective evaluation of the performance of different control techniques.This chapter presents the derivation and final equations of the theoretical performancelimits of buck converters and validates the findings with experimental results - a valuabledesign benchmark for power engineers and researchers. The theoretical optimal behavior isfully characterized in the normalized domain and the findings are valid and general for anycombination of specifications. As a result, the T0/4 rule (quarter of the filter natural period)is obtained, providing a remarkably useful benchmark equation. In addition, through thiswork, power designer and researchers are equipped with a set of transient response limits toaid filter design tasks and controller performance comparisons.132.1. Buck Transient Natural Trajectories and Responseiloadnvccn vonLn  =12?Cn =12?OFFON+-iLn T o= 2pi LCZo = L /Cvxn=vxvreftToixn=ixvrefZo tn =Figure 2.1: Normalized buck converterFig. 2.2 shows the conceptual start-up evolution of a buck converter in a normalized phase-plane. The figure illustrates the three critical points (normalized) of the ideal theoreticaltransient trajectory: turn on action ?, turn off action ?, and arrival to target point ?. Fig.2.7 shows the resulting experimental test to confirm the transient response limits for (a) lowstep down ratio Vccn = 1 and (b) high step down ratio Vccn = 10.2.1 Buck Transient Natural Trajectories andResponseIn order to add generality to the analysis, a normalized buck is employed for the derivationleading to the elimination of the inductor and capacitor filter values, output voltage, powerrating, input voltage, and time domain. The resulting normalized model is valid for anycombination of filter component values.The normalization:tn =tT0; Zxn =ZxZ0; vxn =vxvref; ixn =ixiref142.1. Buck Transient Natural Trajectories and Responsetn0.0.20.1 30von1von1123icn1tn0.30.20.10icn123tn0.30.20.10drivingsignal??2? 2?1 2 3?OFF?ONrONrOFFvccnvon1??icn11icnvon123Switch   on    action     Switch    off   action123 Target   operating   pointFigure 2.2: Start-up transient in geometrical and time domains.152.1. Buck Transient Natural Trajectories and Responseis performed using as base quantities the filter characteristic impedance and resonance periodand the reference voltage and current:Z0 =?L/C; T0 = 2pi?LC; vref ; iref =vrefZ0;The differential equations that characterize the response of buck converters can be ex-pressed as:Cdvodt= iL ? io (2.1)LdiLdt= u Vcc ? vo (2.2)where u takes the values 1 and 0 for ON and OFF states respectively.Performing the normalization, they become:12pidvondtn= iLn ? ion (2.3)12pidiLndtn= u Vccn ? von (2.4)Solving them, time domain expressions that determine the evolution of capacitor voltageand inductor current are found:von(tn) = [von(0)? u Vccn]cos(2pitn) + [iLn(0)? ion]sin(2pitn) + u Vccn (2.5)iLn(tn) = [iLn(0)? ion]cos(2pitn) + [von(0)? u Vccn]sin(2pitn) + ion (2.6)The capacitor current is derived from (2.6):iCn(tn) = [iLn(0)? ion]cos(2pitn)? [von(0)? u Vccn]sin(2pitn) (2.7)162.1. Buck Transient Natural Trajectories and ResponseBy combining (2.5) and (2.7), the time can be eliminated, obtaining the equivalent of thebuck in the geometrical domain defined by von and iCn:?ON : i2cn + (von ? Vccn)2 ? i2cn(0)? (von(0)? Vccn)2 = 0 (2.8)?OFF : i2cn + v2on ? i2cn(0)? v2on(0) = 0, (2.9)which are the natural circular trajectories moving at the angular speed ?on = 2pi.Since the aim of this work is to find the theoretical physical limits on the dynamic perfor-mance of buck converters, the paths that the operating point follows are determined by theON and OFF natural trajectories (?ON and ?OFF ). The normalized angular speed at whichthe operating point is moving is constant; therefore the transient times can be determinedby calculating the angles of the operating point circular paths required to reach the targetlocated at von = 1 and icn = 0, as depicted in Fig. 2.2.At the initial instant ? in Fig. 2.2, the switch is turned on and the operating pointfollows the ?ON trajectory covering the angle ?. Thereafter, at the instant ?, the ?OFF stateNatural Trajectory is followed (the switch is turned off), moving toward the target point andcovering the angle ?. Finally, the target operating point is reached at ?.By performing a geometrical analysis described above, the normalized start-up transientanalytical equation is obtained as follows:tstartn =? + ?2pi(2.10)where? = cos?1(1? 12 V 2ccn), (2.11)and? = cos?1(12 Vccn)(2.12)172.2. Buck Loadability and Sudden Load TransientsAs can be seen in (2.10)-(2.12), the start-up physical limit transient time (normalized)depends only on the normalized input voltage (Vccn = Vcc/vref ) - no other parameter is in-volved. The resulting representation is remarkably simple and provides extraordinary insightinto the transient evolution. By using (2.10)-(2.12), the minimum theoretical normalizedstart-up transient time (tstartn) can be found for any input-to-output voltage ratio. For ex-ample, the absolute limit of operation for a buck converter occurs when the input voltage ismuch greater than the output voltage Vccn >> vo (or Vccn ? ?), yielding a total minimumtheoretical transient time tstartn = 1/4. By denormalizing, the rule of:tstart(min) = T0/4 =pi2?LC (2.13)is established, where T0 is the natural resonant period of the output filter. Equation (2.13)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 = vo (or Vccn = 1) the transienttime is given by tstartn = 1/3 (normalized) or T0/3 in the time domain. The experimentalcaptures depicted in Fig. 2.7 provide experimental validation for both cases, low step-downand high step-down, respectively.2.2 Buck Loadability and Sudden Load TransientsThese transients are produced once the system is in steady state, by adding or removing load.Steps up (loading) and down (unloading) in the load current, of a magnitude ?iloadn, are usedfor the analysis. Representations of the loading and unloading transients in both domains(geometrical and time) are shown theoretically in Figs. 2.3 and 2.5, and experimentally inFig 2.8. In both cases of analysis, the initial operating point is defined when the load stepoccurs.182.2. Buck Loadability and Sudden Load Transients?1tn0drivingsignal?2??22? 2?1 2 3icn1tn0icn123-?iloadnvonminvon112tn0von1 3?OFF?ON rONrOFFvccnvonmin?2?1?icn1-?iloadn1icnvon123von1Switch   on    action     Switch    off   action123 Target   operating   pointFigure 2.3: Loading transient represented on geometrical and time domains.192.2. Buck Loadability and Sudden Load Transients2.2.1 LoadingIn Fig. 2.3, the loading transient is shown in the geometrical and time domains, where theinitial operating point ? is located at (1;??iloadn). At the zero instant, the switch is turnedon and the operating point starts to follow the ?ON trajectory. It remains in that state untilthe ?OFF trajectory that contains the target operating point (for icn > 0) is reached at ?.The operating point first covers the angle ?1 to reach the zero current intersection, and thenit continues moving through ?2 to reach ?. The switch is turned off at ? and the operatingpoint covers the angle ? to reach the target ?. By performing a geometrical analysis of theloading transient (Fig. 2.3) the angles ?1, ?2 and ? are found:?1 = tan?1(?iloadnVccn ? 1)(2.14)?2 = tan?1(?iloadn?4 Vccn ??iloadn22 V 2ccn ? 2 Vccn + ?iloadn2)(2.15)? = tan?1(?iloadn?4 Vccn ??iloadn22 Vccn ??iloadn2)(2.16)And, since the angular speed is ?on = 2pi, the normalized theoretical minimum recoverytime is given by:trecn(min),Loading =?1 + ?2 + ?2pi(2.17)By using (2.17), the minimum recovery time for any loading transient can be calculatedto establish a benchmark recovery.Since the analysis is performed in the geometrical domain, the transient can be fullycharacterized, and particular aspects can be analyzed. One of the parameters of interest isthe theoretical minimum voltage drop (dynamic regulation), which given by:?von(min) = 1? Vccn +?(Vccn ? 1)2 + ?iloadn2 (2.18)202.2. Buck Loadability and Sudden Load Transients2 4 600. = .8(a)2 4 600.511.52vccnt recn(min),Loading. = .8(b)Figure 2.4: Normalized loading transient parameters as function of Vccn, using ?iloadn asparameter a) voltage drop ?von(min) b) minimum loading recovery time trecn(min),LoadingThe values of time recovery and voltage drop are general and valid for any combinationof parameter, and are synthesized in two normalized plots shown in Fig. 2.4 (a) and (b).These plots allow the power electronics designer to find the theoretical minimum recoverytime and voltage drop for any load current step-up in a fast manner. Normalizing the currentstep and determining the converter input to output voltage ratio are the only previous stepsneeded to read the optimal values from the plots. For instance, if a current step-up of anormalized magnitude ?iloadn = 1 happens in a normalized buck converter with Vccn = 2,the theoretical minimum recovery time and voltage drop are trecn(min),Loading = 0.3175 and?von(min) = 0.4142.212.2. Buck Loadability and Sudden Load Transients2.2.2 UnloadingThe unloading transient is shown in Fig. 2.5. The initial operating point ? is located at(1;?iloadn). At the zero instant, the switch is turned off and the operating point movesfollowing the trajectory ?OFF , covering the angle ?1 to reach the maximum voltage point(vonmax) and ?2 to reach the switching point ?. At that moment, the switch is turned on and?ON is followed during the angle ? to reach the target operating point ?.Performing geometrical analysis, the normalized minimum recovery time for unloadingtransients is found:trecn(min),Unloading =?1 + ?2 + ?2pi; (2.19)where:?1 = tan?1 (?iloadn) (2.20)?2 = tan?1(?iloadn?4 Vccn(Vccn ? 1)??iloadn22 Vccn + ?iloadn2)(2.21)? = tan?1(?iloadn?4 Vccn(Vccn ? 1)??iloadn22 V 2ccn ? 2 Vccn ??iloadn2)(2.22)The expression that defines the theoretical minimum overshoot voltage is found by ana-lyzing the evolution of the variables in the geometrical domain:vonmax =?1 + ?iloadn2 (2.23)This simple equation, indicates the normalized minimum value that the output voltagereaches during a load current step-down transient, as function of normalized step magnitudeonly. The effectiveness of the controller to avoid overshoot during this kind of transientscan be objectively assessed by comparing the obtained values with the theoretical minimumpresented.222.2. Buck Loadability and Sudden Load Transientstn0drivingsignal1 2 3?1 ??22? 2? 2?icn1 tn0icn123?iloadnvonmaxvon1 12tn0von1 3vccnicn1?iloadn1icnvon?OFF?ON rONrOFFvon1??1?21-1123vonmaxSwitch   off    action     Switch    on   action123 Target   operating   pointFigure 2.5: Unloading transient represented on geometrical and time domains.232.3. Benchmarking Procedure Example2.3 Benchmarking Procedure ExampleIn this section, the start-up and loading transients of the 30W buck converter detailed inTable 2.1 are studied and compared with the theoretical physical limits.Table 2.1: Buck converter parametersPARAMETER VALUE NORM.vref 12 V 1Vcc 24 V 2L 508 ?H 12piC 47.5 ?F 12piThe compensation is performed using small-signal modeling and implementing the tradi-tional lead technique. The closed loop response is shown in Fig. 2.6 for start-up and for acurrent step-up of 2.5A (no-load to full-load) transients.The normalization parameters are T0 = 985?s, Z0 = 3.25?, vref = 12V , and iref = 3.7A.The ideal normalized start-up transient can be calculated using (2.10), (2.11), and (2.12)and depends only on the value of the normalized input voltage. For this case, where Vccn = 2,the theoretical minimum settling time is tstartn = 0.2902. Denormalizing, the ideal start-uptransient for the specified converter is obtained:tstart = T0 tstartn = 285?s (2.24)The settling time of the controller is 1.1ms, which is 3.85 times the optimal. A clearopportunity for improvement is identified in the start-up transient.In order to benchmark the response of the compensator during a current step-up transient,it is necessary to normalize the converter parameters. The normalized current step-up is givenby:?iloadn =?iloadiref= 0.678 (2.25)242.3. Benchmarking Procedure Exampletnt [ms]0dutycycle0vo12V9.5V-2.5Aic0Theoretical minimumtransient times6429.5Vicvo12 VPhysicallimitConverter response-2.5A1.97 3.94 5.91Physical limitConverter responseFigure 2.6: Comparison between physical limit operation and traditional compensation tech-niques.252.4. Experimental ResultsUsing (2.17) and (2.18), the normalized theoretical minimum recovery time and voltagedrop for the transient under study are found as 0.235 and 0.208 respectively, denormalizing:trec(min),Loading = 230?s (2.26)and,?vo(min) = 2.5V (2.27)The characteristic parameters of the linear controller transient response are a recoverytime of 1.35ms and a maximum drop of 2.95V . The physical limit of operation is shown inFig. 2.6 and compared with the response of linear compensator. In this way, the opportunitiesto improve the classical control technique can be objectively assessed.e the classical controltechnique can be objectively assessed.2.4 Experimental ResultsExperimental results of the converter operating at the physical limit are shown in geometricaland time domains in Figs. 2.7 and 2.8, where the main parameters have been indicated andcompared with the theoretical values. The prototype filter values are L = 512?H andC = 48?F , which leads to a natural period T0 of 985?s.In Fig. 2.7 the start-up transients for two different normalized input voltages, Vccn = 1and Vccn = 10, are shown and compared with the ideal minimum times. For Vccn = 1 themeasured start up time is 340?s while the theoretical minimum is 328?s. For Vccn = 10the start up time is 262?s, which approximates the quarter of T0 (246?s) due to the highnormalized input voltage.Fig. 2.8 shows the three studied transient responses of the designed converter. Themeasured start up time is 290?s, while the theoretical one for Vccn = 2 can be obtainedfrom (2.10) and is 285.65?s. In the case of the loading transient, the design values for262.4. Experimental ResultsTurn on action     Turn off actionTarget operating point1 2 3123123340?s(physical limit: To/3 = 328?s)(a)Turn on action     Turn off actionTarget operating point1 2 3123123262?s(physical limit: To/4 = 246?s)(b)Figure 2.7: Start-up experimental results for a) low step-down (Vccn = 1) and b) high step-down Vccn = 10 transients.272.5. SummaryTurn on action     Turn off actionTarget operating point12 3123trec =242?s(Designed for: 230?s)?von =2.3V(Designed for: 2.5V)123Figure 2.8: Start-up, loading and unloading transients for Vccn = 2.recovery time and voltage drop are is 230?s and 2.5V , and the measured values are 242?sand 2.3V respectively. For the unloading transient, the theoretical values for recovery timeand maximum voltage, calculated using (2.19) and (2.23), are 230?s and 14.5V , while themeasured values are 244?s and 14.7V .2.5 SummaryA detailed derivation of the theoretical physical limits of buck converters has been performedin this chapter. The theoretical optimal response for start-up, loading and unloading tran-sients of buck converters have been fully characterized by general expressions obtained bygeometrical analysis of the normalized converter.Since the work is performed in the normalized domain, generality is gained and theexpressions are valid for any possible filter values combination. The absolute minimum startup time for buck converters has been found, and the T0/4 rule has been established.282.5. SummaryA benchmarking tool was introduced by determining the absolute limits of performance,providing a practical tool for evaluating the behavior and comparing the performance of con-trol strategies. A benchmarking procedure of a 30W, 24V/12V converter, based on transientresponse specifications, was performed and experimental results of the converter operatingat physical limits were shown to validate the obtained mathematical expressions.29Chapter 3Average Natural Trajectories (ANTs)for Buck Converters: Centric-BasedControlMany different control schemes have been developed for buck topologies in order to improveconverter performance, achieve higher efficiency and reduce size and cost. In the previouschapter, a useful tool that enables the objective assessment of dynamic performance was in-troduced and, following a similar geometrical approach, a novel type of controller is developedhere.Linear techniques have been the preferred alternative since the 1970s [5, 6], while bound-ary techniques started gaining popularity in the late 1990s [47, 49]. These popular controlschemes present advantages and disadvantages, inherent to the different natures of the ap-proaches. For instance, a fixed frequency behaviour is obtained in linear techniques as aresult of the averaged model employed, and a naturally bounded stable response is the resultof the geometric-analysis performed in boundary techniques.Combining the advantages of linear and boundary approaches, a new type of geometric-based controller is introduced in this chapter. The proposed scheme features fast and pre-dictable behaviour as well as reliable operation during transients and in steady state. Thecornerstone of the proposed controller is provided by a large-signal geometric-based modelof the natural evolution of the averaged state variables, the Average Natural Trajectories30Chapter 3. Average Natural Trajectories (ANTs) for Buck Converters: Centric-Based Control(ANTs). The ANTs of PWM driven buck converters are derived by merging traditionalaveraging techniques [5] and geometrical analysis. These trajectories are simple circles thatrepresent the averaged paths the converter operating point follows for a given PWM dutycycle, providing an accurate model of the converter?s large-signal behaviour.Using the insight given by this model, the duty cycle necessary to achieve the targetoperating point can be calculated departing from any arbitrary initial condition. In contrastwith boundary control techniques, where the control actions are ON/OFF , in the centric-based controller they are based on the definition of a PWM duty cycle. As a result, a newtype of controller that narrows the gap between linear and boundary techniques is developedand referred to as centric-based control. Conceptual illustrations of the behaviour duringstart up of the centric-based, linear and sliding-mode controllers are detailed in Fig. 1.2. Asindicated in the figure, the centric-based controller features shorter settling time than thelinear and sliding-mode solutions, while the transient current peak is maintained at a lowervalue, which highlights the enhanced dynamic behaviour of the technique developed in thiswork. Fixed frequency operation (using PWM) and single duty-cycle transient solution areother characteristic features of the proposed controller depicted in the conceptual figure.The defined trajectories are averaged over one PWM switching period, and therefore, fixedswitching frequencies and low bandwidth requirements for sensing and signal conditioningsystems are among the advantages of the proposed centric-based control. The computationalburden of the proposed control technique is very low since, in an ideal case, calculations areperformed just once to direct the operating point from an initial condition to the target op-erating point. The proposed normalized geometrical design framework provides the designerwith not only generality, but also with an intuitive graphical representation of the behaviorof the converter during transients. A detailed theoretical derivation to obtain the ANTsfor buck converters is included in this work as well as the derivation of the control laws toimplement the proposed technique. Experimental results of the target approaching principle313.1. Buck Converter Ideal ANTs DerivationLn =12piCn =12pi+-VccniLnvCnion(a)iCnvCndrivingsignal d = 0.53 d = 0.47(b)Figure 3.1: a) Normalized PWM driven buck converter. b) Pseudo-ideal natural trajectoryforced in a real converter.are shown in order to validate the obtained ANTs. Closed-loop results under different tran-sients validate the enhanced performance of the proposed control scheme and highlight thepractical usefulness of the concepts introduced in this chapter.3.1 Buck Converter Ideal ANTs DerivationThe procedure to obtain the ANTs of the ideal PWM driven normalized buck convertershown in Fig. 3.2 is presented in this section. The effects of parasitic resistances are includedlater in this chapter.323.1. Buck Converter Ideal ANTs Derivation     ?2??2?iCn0tn1tn0drivingsignal1tn0vCn11???1ON?1OFF?2ON?2OFFVccnVccniCnvCn-110 1?AVGFigure 3.2: Ideal PWM natural trajectory with Vccn = 2, d = 0.5 and fsn = 10.333.1. Buck Converter Ideal ANTs DerivationThe differential equations that rule the behavior of buck converters are, for ON state:CdvCdt= iL ? io (3.1)LdiLdt= Vcc ? vC (3.2)and for OFF state:CdvCdt= iL ? io (3.3)LdiLdt= ?vC (3.4)In order to gain generality, the analysis is taken to the normalized domain, where:tn =tT0; Zxn =ZxZ0; vxn =vxvref; ixn =ixirefand the employed base quantities are the filter characteristic impedance and resonance pe-riod, as well as the reference voltage and current:Z0 =?L/C ; T0 = 2pi?LC; vref ; iref =vrefZ0As a result, the normalized differential equations of the buck converter are found for ON,12pidvCndtn= iLn ? ion (3.5)12pidiLndtn= Vccn ? vCn (3.6)and OFF states.12pidvCndtn= iLn ? ion (3.7)12pidiLndtn= ?vCn (3.8)343.1. Buck Converter Ideal ANTs DerivationThe averaged differential equations describe the behavior of the converter when the nor-malized switching frequency (fsn = fs T0) is much higher than the unity. Traditional averag-ing techniques are implemented to obtain the averaged differential equations in normalizedform:12pidvCndtn= iLn ? ion (3.9)12pidiLndtn= d Vccn ? vCn (3.10)where d is the PWM duty cycle determined by the ratio between the ON time (TON) andthe PWM period (Ts).Solving the averaged differential equations the expressions that describe the time evolutionof the averaged variables (capacitor voltage and inductor current) are found:vCn(tn) = [vCn(0)? d Vccn]cos(2pitn) + [iLn(0)? ion]sin(2pitn) + d Vccn (3.11)iLn(tn) = [iLn(0)? ion]cos(2pitn) + [vCn(0)? d Vccn]sin(2pitn) + ion (3.12)By performing nodal analysis, the time domain expression of the capacitor current isdetermined:iCn(tn) = [iLn(0)? ion]cos(2pitn)? [vCn(0)? d Vccn]sin(2pitn) (3.13)Combining (3.11) and (3.13) the normalized time tn is eliminated, yielding to the para-metric representation of the Averaged Natural Trajectories (ANTs) for ideal buck converters:?AV G : i2Cn + (vCn ? d Vccn)2 = i2Cn(0) + (vCn(0)? d Vccn)2 (3.14)353.1. Buck Converter Ideal ANTs Derivation0 vccniCn?ON!0nrONcON?OFF!0nrOFFcOFF?AVG d1!0nrd1cd1?AVGd2!0nrd2cd2[vCn(0); iCn(0)]vCntn10.50vCnVccnvCn(0)0iCniCn(0)tn10.5d 1d2 d1  0tn10.5Figure 3.3: Several PWM-driven buck converter ANTs departing from unique initial operat-ing point.363.1. Buck Converter Ideal ANTs DerivationThis equation represents the averaged trajectory that the operating point follows for anygiven duty cycle. It can be represented in the plane formed by vCn and iCn by a circle withcenter at (d Vccn; 0), and radius determined by the initial conditions [vCn(0); iCn(0)]. Fig. 3.3shows the natural evolution of the averaged variables in both geometrical and time domainsfor several duty cycle values (centers).PWM duty cycle values of dOFF = 0, d1 = 0.33, d2 = 0.67, and dON = 1, illustrate thewhole range of possible ANTs corresponding to an initial operating point [vCn(0); iCn(0)] .The direct relationship between the circular trajectory center and the PWM duty cycle,given by c = d Vccn, must be highlighted, due to the importance it represents for the controltechnique proposed in the following sections.From (3.11) and (3.13), the normalized angular speed at which the averaged operatingpoint moves across the circular path is constant and is given by ?on = 2pi, as depicted inFigs. 3.1(b) (experimental) and 3.2 (simulation).When the duty cycle saturates at either of its extremes given by d = 1 and d = 0, theconverter follows the ?ON and ?OFF trajectories (shown in Fig. 3.3), which are referred toas natural trajectories rather than ANTs, since the switch is not in PWM mode for thosecases. Equations (3.11), (3.13), and (3.14) are still valid for those cases, and since there areno switching actions involved, there is no average and the trajectories determine the actualnatural trajectories of the converter.It is also possible to find the real (non-averaged) paths that the operating point follows,shown in Fig. 3.2. The ?kON and ?kOFF trajectories are followed for (k ? 1)TSn ? Tn ?(k?1+D)TSn and (k?1+D)TSn ? Tn ? kTSn respectively, where k is the number of switchingperiod, and the initial conditions of each subinterval are given by the final conditions of theprevious state. Although the results would describe exactly the path followed, the advantagesof working with the simplified averaged model would be lost and the complexity increasedsignificantly.373.2. Approaching the Target3.2 Approaching the TargetAs can be seen in Fig. 3.3, for any given initial conditions, there is a theoretically infinitenumber of averaged circular trajectories (in practice, the number is as high as the PWMresolution). One of those trajectories, and only one, is a circle that contains both initial andtarget points. For the ideal converter, the center of that circle can be found by performing ageometrical analysis from Fig. 3.4:cx =vCn(0)2 + iCn(0)2 ? 12 (vCn(0)? 1)(3.15)Since c = d Vccn, then:dx =vCn(0)2 + iCn(0)2 ? 12 Vccn (vCn(0)? 1)(3.16)Using this equation, the duty cycle needed to reach the target operating point followingjust one averaged circular path can be determined.The time that it takes for the operating point to reach the target can be determinedfrom Fig. 3.4 by finding the angle ?1 and dividing it by ?on = 2pi. Using trigonometricaldefinitions, the transient time is found as,tD =12picos?1(vCn(0)? c1? c)(3.17)Once the target is reached, the center must be moved to the steady state operating pointby setting the duty cycle to 1Vccn , in order to obtain an averaged circular trajectory with nullradius.In theory, and for infinite switching frequency, the operating point remains at the targetwithout moving. However, in practice and for limited switching frequencies, the operatingpoint becomes an operating path determined by ?ON and ?OFF trajectories, which definevoltage and current ripples.383.2. Approaching the TargetiCn1(0)iCn2(0)iCn3(0)iCn4(0)0tniCnvCn1(0)vCn2(0)vCn3(0)vCn4(0)tn0vCn1?1,4tD 1,4 tD 2,32??2,32?d41Vccn0.5 tn0dd1d3d2d0VccniCnvCn10 1[vCn1(0);iCn1(0)][vCn3(0);iCn3(0)][vCn4(0);iCn4(0)]c2c3r1r0r3r4c1 c0c4?2?3r2[vCn2(0);iCn2(0)]?1?4?OFF?ONFigure 3.4: Ideal fixed duty cycle target approaching method for different initial conditions393.3. Closing the Loop with a Centric-Based ControllerTargetInitial conditionsFigure 3.5: Fixed duty cycle target approaching method forced in a real converterThe result of the analysis reveal a simple and powerful concept: the target operatingpoint can be reached following only one ANT by setting a duty cycle corresponding to adetermined center location which depends only on the initial operating point location.3.3 Closing the Loop with a Centric-Based ControllerA new geometric-based, fixed-frequency control technique based on (3.15) is presented inthis section. Fig. 3.6 (a) shows a block diagram of the proposed controller, and Fig. 3.6 (b)shows the results of the implementation in an experimental setup.The control scheme is composed of three main principles. A periodic center recalculationusing (3.15) defines the main control law, which is complemented with extra restrictions tothe function domain to improve dynamic response and a linear term to eliminate possiblesteady state error. The three principles are described in the following subsections.403.3. Closing the Loop with a Centric-Based ControllerLinearCompensatorSaturationiCnvonLinearON OFF CentricCentriccLnCnrCnrLniCnVccn vonion(a)iCndvonStart-upduty cycle saturationLoadingUnloadingion10?ion=-1?ion=1(b)Figure 3.6: Centric-based controller: a) scheme, and b) experimental results.413.3. Closing the Loop with a Centric-Based Controller3.3.1 Periodic Center CalculationUsing (3.15), the center of the circle that directs the operating point to the target is calculatedbased only on the location of the averaged operating point. Tolerances in the values of reactivecomponents, as well as losses present in the system, can cause the converter?s averagedbehaviour to differ from the ideal circular trajectories derived. Performing a periodical centerrecalculation using updated values of the averaged capacitor current and voltage as initialconditions, these deviations are limited to the magnitude they reach over one recalculationperiod.3.3.2 Domain RestrictionsThe domain of the control law equation (3.15) is determined by the saturation trajectories?ON and ?OFF shown in Fig. 3.4. Otherwise, the converter?s operating point must betaken back into the function domain by setting the duty cycle to one of its extremes beforeattempting to perform the center calculation. Extra restrictions are applied to the functiondomain in order to improve the converter dynamic behaviour as shown in Fig. 3.6(a). Therestricted domain is defined as the positive capacitor current half of ?OFF and negativecapacitor current half of ?ON . In this way, since saturation trajectories are followed rightafter load transients, the magnitude of drops and spikes in the output voltage are minimized,reaching near-optimal values.3.3.3 Steady State Error CorrectionEffects not being considered in the model can also affect the steady state behaviour, leadingto undesired steady state error. For instance, the duty cycle needed in a buck converterwith Vccn = 2 working at full load might be 0.51 instead of 1Vccn = 0.5. Even when thiseffect might be minimal in high-efficiency converters, it is worth addressing it in order to423.3. Closing the Loop with a Centric-Based Controllerideal Center recalculationLinear compensationlinearregion0tniCn tn0von110.5 tn0d1Vccn1fsn=0.5??2?iCn?c von0.50 1??Figure 3.7: Closed-loop controller concept433.4. Experimental Resultspresent a complete solution. The approach proposed here is based on the implementation ofa linear controller that is engaged once the operating point reaches the target neighborhood,as indicated in the scheme of Fig. 3.6 (a).In this way, when the operating point is close enough to the target, the duty cycle is setto the ideal value ( 1Vccn ) plus the output of a linear term. Implementing a dual-loop com-pensator using output voltage and capacitor current becomes an appealing option for thispurpose since the variables are already being measured.A conceptual illustration of the behaviour of the complete controller is provided in Fig.3.7 for a start-up transient. The differences between the ideal trajectories and the obtainedones, as well as the effect of the center recalculation, can be observed during the initial partof the transient. When the indicated target neighborhood area is reached, the linear termis engaged, and the steady state error is eliminated. It is also worthwhile to observe the 1fsndelay introduced to the control action, which is produced by the fixed-frequency nature ofthe approach.3.4 Experimental ResultsExperimental results that validate the theoretical concepts and the presented control tech-nique have been obtained using the 44W buck converter detailed in Tables 3.1 and 3.2.The experimental capture of 3.1(b) illustrates pseudo-ideal ANTs obtained from a realconverter operating at a normalized switching frequency fsn = 10. In Fig. 3.5, the fixed dutycycle target approach method is shown for four different initial conditions with fsn = 20.Experimental results of the proposed closed-loop method are shown in Fig. 3.6(b), show-ing a fast and predictable response during start-up and extreme loading/unloading transients(?ion = 1). Small oscillations are observed when the operating point enters the target neigh-443.5. Comparison with Linear ControllersTable 3.1: Buck converter parametersPARAMETER VALUE NORM.vref 12 V 1Vcc 24 V 2L 508 ?H 12piC 47.5 ?F 12piT0 985 ?s 1Z0 3.25 ? 1Table 3.2: Main parasiticsPARAMETER VALUE NORM.rL 180 m? 33.4mrC 71 m? 21.8mrswitch1 20 m? 6.15mrswitch2 20 m? 6.15mborhood area. Due to the effect of losses, when the step in the duty cycle is produced, thecenter is not located exactly at the target, causing a non-null averaged trajectory whichtranslates into small oscillations. The linear term is engaged at the same moment and leadsthe operating point to finally reach the target.Further experimental results of the proposed centric-based controller are shown in thefollowing sections, which compare the technique with classical linear schemes and analyze thesensitivity to changes of the value of reactive components. The saturation of the duty cyclefor operating points not contained in the restricted domain, as well as the centric controlledtrajectories, can be noticed in all the centric-based controller experimental captures.3.5 Comparison with Linear ControllersThe control technique presented here shows a fast and predictable response for large transientssuch as start-up and high step loading/unloading. In this section, experimental results of acentric-controlled buck converter are compared with the ones obtained using a classical dual-loop linear compensator in order to benchmark the performance of the proposed scheme.The performances of both controllers under the same transient conditions are presentedin Figs. 3.8, 3.9, and 3.10. The main normalized transient characteristics (normalized tran-sient time, voltage and current peak deviations) have been indicated in the figure and are453.5. Comparison with Linear Controllers10iCndvonipeak  = 0.95ttrn  = 5.5ipeak?von = 0.12?von (a)10iCndvonipeak  = 0.44ttrn  = 0.56ipeak ?von  = 0.02(b)Figure 3.8: Comparison between dual-loop linear and centric-based controllers. Start-uptransient for a) linear, and b) centric-based.summarized in Table 3.3 in order to facilitate an objective assessment.The steady state region employed to measure transient recovery times is defined as ?2%of the reference voltage.Start-up transients are shown in Fig. 3.8 for linear and centric-based controllers respec-tively. In the case of the linear controller, steady state is reached after 5.5 resonance periodsT0 (5.4ms) with a voltage overshoot of 12% (1.44V ) and a normalized current peak of 0.95(3.5A). The proposed controller exhibits a settling time almost ten times faster (0.56 T0 or0.55ms) with a normalized current peak of 0.44 (1.63A) and almost no overshoot (0.02 vref ).463.5. Comparison with Linear Controllers10iCndvonion0.480.330.730.752.652.12(a)10iCndvonion0. 3.9: Comparison between dual-loop linear and centric-based controllers. Load-ing/unloading transients with ?ion = 1 for a) linear, and b) centric-based.Table 3.3: Centric - linear controllers comparisonTRANSIENT ttrn ?von ipeak?ion Lin. Cen. Lin. Cen. Lin. Cen.Start-up 5.5 0.56 0.12 0.02 0.95 0.44Loading 1 2.65 0.72 0.73 0.5 0.48 0.270.5 1.88 1 0.35 0.19 0.25 0.12Unloading 1 2.12 1.11 0.75 0.48 0.33 0.250.5 1.78 1.1 0.36 0.18 0.22 0.1473.5. Comparison with Linear Controllers10iCndvonion0.250.220.350.361.881.78(a)10iCndvonion0. 3.10: Comparison between dual-loop linear and centric-based controllers. Load-ing/unloading transients with ?ion = 0.5 for a) linear, and b) centric-based.Regarding the geometrical-domain plots, it is worth highlighting the predictable behaviorfeatured by the centric controller.The responses of the controllers to unity normalized magnitude current step-up and step-down transients (3.7A - extremely large transient) are shown in Fig. 3.9. The normalizedrecovery times presented by the linear-controlled converter are 2.65 (2.61ms) for step-uptransient and 2.12 (2.08ms) for step-down transient. In the centric-based controller imple-mentation, these times are reduced to 0.72 (0.71ms) and 1.11 (1.09ms) respectively.483.6. Tolerances Sensitivity AnalysisThe normalized voltage drops and peaks during the transients are also smaller in thecentric-based case, featuring 0.5 (6V ) and 0.48 (5.76V ) against the 0.73 (8.76V ) and 0.75(9V ) presented in the linear case. Following the same trend, the current peaks are reducedfrom the values 0.48 (1.77A) and 0.33 (1.22A) they present in the linear case, to 0.27 (1A) and0.25 (0.93A) for the centric controller. In the geometrical domain, the large transient responseof the centric-based controller is, once more, proven to be very predictable. Small unpredictedbehaviours are shown in the target neighborhood due to the center misplacement issue, whichis addressed in previous sections and solved by a complementary linear compensator.One more performance comparison between the controllers is shown in Fig. 3.10; in thiscase the current step-up and down magnitude is one half of the reference current (1.85A).For this transient, half of the centric-based controller recovery time is determined by thecomplementary linear term. For this reason, the difference between the controllers perfor-mance is reduced, as detailed in Table 3.3. Even in this case, the performance of the centriccontroller is still 60% faster than the traditional dual-loop linear controller (at worst), whilethe voltage and current deviation peaks are still 50% lower.3.6 Tolerances Sensitivity AnalysisThe converter ANTs are strongly related to the values of the main reactive components in thecircuit and their tolerances. Therefore, and in order to illustrate how well the proposed controltechnique behaves under variations of these parameters, experimental results including ?20%deviations in the component values are shown in Figs. 3.11 and 3.12. The deviations in thecomponent values cause the system characteristic impedance Z0 and natural resonance timeT0 to differ from the values used in the normalization, producing distortions in the expectedbehaviour. The resulting characteristic parameters and responses are shown in Table 3.4 foreach one of the cases under analysis.493.6. Tolerances Sensitivity AnalysisiCndvonion10(a)iCndvonion10(b)Figure 3.11: Centric-based controller performance under parameter deviations. Start-up,loading and unloading transients for a) Lreal = 0.8 L, Creal = 0.8 C; and b) Lreal = 0.8 L,Creal = 1.2 C;Table 3.4: Centric controller parameter deviationsParameter deviation Loading transient responseFig. L C Z0 T0 trecn ?von ipeak3.11 (a) ?20% ?20% 0 ?20% +19% +4% ?11%3.11 (b) ?20% +20% ?18% ?2% ?19% ?36% +30%3.12 (a) +20% ?20% +22% ?2% +33% +24% ?15%3.12 (b) +20% +20% 0 +20% +22% ?3% ?20%503.6. Tolerances Sensitivity AnalysisiCndvonion10(a)iCndvonion10(b)Figure 3.12: Centric-based controller performance under parameter deviations. Start-up,loading and unloading transients for a) Lreal = 1.2 L, Creal = 0.8 C; and b) Lreal = 1.2 L,Creal = 1.2 C.The deviations in resonance period and characteristic impedance can be observed in bothtime and geometrical domain plots. Comparing the time domain plots of Figs. 3.11 (a) and3.12 (b), the variations in T0 become evident when it is observed how the recovery time inthe first one is shorter than in the second one. The effects of changes in Z0 are evident in thegeometrical domain plots of Fig. 3.11 (b) and 3.12 (a), in which the circular trajectories aredistorted. As proven by the experimental results in shown Figs. 3.11 (a) and 3.12 (b), theproposed control scheme behaves reliably under large variations in the system parameters.513.7. Buck Converter Non-Ideal ANTs DerivationLnCnrCnrLnVccniLnvonvCnion(a)iCnvondrivingsignal?rd = 0.5(b)Figure 3.13: a) Non-ideal normalized PWM driven buck converter. b) Real PWM naturaltrajectory.Fast responses are obtained and the predictability of the trajectories is kept within acceptablemargins even for large parameter deviations.3.7 Buck Converter Non-Ideal ANTs DerivationThe ANTs of the non-ideal normalized buck converter shown in Fig. 3.13, including nor-malized parasitic resistances in the inductor and capacitor (rLn and rCn respectively) arederived in this section. The insight provided here features important conceptual value andenables the derivation of a closed-loop centric-based controller without the need of a linearcomponent to solve steady state issues.523.7. Buck Converter Non-Ideal ANTs DerivationThe differential equations that represent the normalized converter are:12pidvCndtn= iLn ? ion (3.18)12pidiLndtn= d Vccn ? iLn rLn ? iCn rCn ? vCn (3.19)Rearranging terms, (3.19) can be expressed as:12pidiLndtn= d Vccn ? ion rLn ? iCn (rCn + rLn)? vCn (3.20)By solving the differential equations system formed by (3.18) and (3.20) time domainexpressions that describe the time evolution of the state variables are identified as:vCn(tn) = Att(tn){(vCn(0)? c?) cos(?dntn)+[(vCn(0)? c?)rCn + rLn2+ (iLn(0)? ion)] ?on?dnsin(?dntn)}+ c? (3.21)iLn(tn) = Att(tn){(iLn(0)? ion) cos(?dntn)?[(iLn(0)? ion)rCn + rLn2+ (vCn(0)? c?)] ?on?dnsin(?dntn)}+ ion (3.22)with:?on = 2pi; (3.23) ?dn = ?on?1?(rCn + rLn2)2(3.24)Att(tn) = e??ontn?1?(?dn?on )2(3.25)c? = d Vccn ? ion rLn (3.26)Where ?on and ?dn are the ideal and damped normalized resonance frequencies respec-tively. The time-dependent function Att(tn) describes the attenuation of the radius along533.7. Buck Converter Non-Ideal ANTs DerivationiCn0tn1tn0drivingsignal1tn0von11VccnVccniCnvonc' c?c-110?AVG?AVG (non-ideal)?AVG?AVG?rFigure 3.14: Non-ideal natural trajectory with Vccn = 2, d = 0.5 and fsn = 10.the normalized time, and c? gives the location of the new center.543.7. Buck Converter Non-Ideal ANTs DerivationAs in the lossless case, the capacitor current is given by:iCn = iLn ? ion (3.27)Then, the time domain expression that describes it can be derived from (3.22) as:iCn(tn) = Att(tn){iCn(0) cos(?dntn)?[iCn(0)rCn ? rLn2+ (von(0)? c?)] ?on?dnsin(?dntn)}(3.28)Since the capacitor equivalent series resistance is different than zero, the output voltagevon is now determined by the addition of capacitor voltage and the drop on its ESR:von = vCn + iCn rCn (3.29)Replacing (3.21) and (3.28) in (3.29), the time domain describing equation of the outputvoltage is found:von(tn) = Att(tn){(von(0)? c?) cos(?dntn)+[(von(0)? c?) + iCn(0) (1? rLn rCn)] ?on?dnsin(?dntn)}+ c? (3.30)Combining (3.30) and (3.28) an implicit expression that describes the ANTs in non-idealconverters is found:?AV G(non?ideal) : (von?c?)2+(1?rLn rCn) i2Cn+(rLn?rCn)(von?c?)iCn = r(0) Att(tn) (3.31)553.8. Closed-Loop Controller Including Estimation of Parasiticswhere:r(0) = (von(0)? c?)2 + (1? rLn rCn) iCn(0)2 ? (rLn ? rCn) (von(0)? c?) iCn(0) (3.32)Which represents a spiral given by an ellipse with initial radius r(0) which exponentiallydecreases with the normalized time following the function Att(tn).Fig. 3.14 shows the non-ideal ANTs for a normalized buck converter with normalizedinput voltage Vccn = 2, switching frequency fsn = 10, duty cycle d = 0.5, and series parasiticresistances rCn = rLn = 0.1.The analysis performed in this section reveals differences between the particular case,when rCn = rLn = 0, and the general one; these differences can be summarized in three mainpoints:? The circle becomes a rotated ellipse? The radii are attenuated by the function Att(tn)? The center is shifted to the left by ?c = ion rLnTaking these differences into account, a purely-geometrical alternative is proposed in thefollowing section.3.8 Closed-Loop Controller Including Estimation ofParasiticsAs found in the previous section and shown in Fig. 3.14, the ANTs centers are shifted tothe left in non-ideal converters. The insight provided is used in this section to develop apurely geometrical alternative with no steady state error and with an absolutely predictabletransient response.563.8. Closed-Loop Controller Including Estimation of ParasiticsLnCnrCnrLniCnVccn vonSaturationiCnvonON OFF CentricCentriccionrLnestimator(a)iCndvonStart-upduty cycle saturationLoadingUnloadingion10?ion=-1?ion=1(b)Figure 3.15: Centric-based controller with load current estimation: a) scheme, and b) exper-imental results.573.8. Closed-Loop Controller Including Estimation of ParasiticsFrom (3.26), the relation between duty cycle and ANTs center is given by:d = c? + ion rLnVccn= c? + ?cVccn(3.33)which combined with (3.15) leads to:dx =vCn(0)2 + iCn(0)2 ? 12 Vccn (vCn(0)? 1)+ ion rLnVccn(3.34)In this case, the duty cycle needed to lead the operating point to the target depends alsoon the load current and inductor normalized parasitic resistance.The inductor series resistance varies in response to environmental parameters (eg: tem-perature); therefore a linear estimation is performed to approximate its value. The valueof rLn changes in a very slow manner (order of seconds) which leads to very low dynamicrequirements for the linear estimator.The proposed controller is described by the block diagram of Fig. 3.15(a). Experimentalresults are shown in Fig. 3.15(b) and 3.16, validating the described behaviour. Since in thisapproach more information is required for the implementation of the controller, it has beenexcluded from the comparative analysis of section VI in order to perform a fair one-to-onecomparison. Nevertheless, experimental results of this last approach for the same transientsare presented in Fig. 3.16 (a) and (b) to illustrate the enhanced performance obtained. Asexpected, the voltage and current peak deviations during the transients show similar values tothose in the previous centric-based case. The main differences are found in the neighborhoodof the target operating point, where the small oscillations are avoided and the steady stateis .reached merely by recalculating the center and setting the right duty cycle. It is alsoworthwhile noting that in this last case the duty cycle value changes by fewer larger steps,remaining steady most of the time. According to the requirements of the application, theextra variable needed to implement this version of the controller might be justified by theimprovement in the dynamic response and the 100% predictable response obtained.583.9. Centric Controller Benchmark10iCndvonion0.290.260.510.510.590.61(a)10iCndvonion0. 3.16: Response of the centric-based controller with load current measurement forloading/unloading transients with ?ion = 1 (a), and ?ion = 0.5(b).3.9 Centric Controller BenchmarkIn this section the centric-based controller is compared with the physical limits of performancederived in the previous chapter. In order to allow an objective assessment, the start-up,loading and unloading transient responses of the ideal centric-based controller are shown inFig. 3.17, along with the physical limits of performance in both time and geometric domains.593.9. Centric Controller Benchmarktn0dutycycle0von10Theoretical minimumtransient times321vonicnPhysicallimitoperationCentriccontrollerresponse-1112 V1icnFigure 3.17: Performance comparison between the ideal centric-based controller and thedynamic physical limits.603.10. SummaryAs mentioned in the previous chapter, the ideal start-up transient only depends on thenormalized input voltage and can take values from T0/4 to T0/3. In the case of the centriccontroller, the normalized settling time is constant. It is simple to observe that the start-uptransient describes a half-circle for any input-output voltage ratio. Since the normalizedangular speed is constant, the start-up is found to be:tstart?up(centric) =pi2pi= 0.5 (3.35)which, as expected, is always higher than the theoretical limit of performance. The start-upcurrent peak of the novel technique is also constant, and presents a normalized value of 0.5.Since the response shown in Fig. 3.17 corresponds to the ideal behaviour of the centriccontroller, and due to the duty cycle saturation algorithm implemented, the first portion ofthe loading/unloading transients coincide with the theoretical limits of performance. Oncethe capacitor current reaches zero, the transient response differs from the ideal and describesa half-circle, taking a time of T0/2 to reach steady state. In this way, the theoretical voltagedrops/peaks reached present values close to the ideal, while the recovery current peaks aremaintained at lower values than in the ideal case.3.10 SummaryThis chapter presented the Averaged Natural Trajectories (ANTs) for buck converters, whichwere derived by using traditional averaging techniques on the converter state equations andrepresenting them in the geometrical domain. Mathematical expressions to direct the op-erating point towards the target by performing calculations only at the initial instant werefound and verified by experimental results.A new type of geometric-based controller for buck converters, based on the large-signalmodel obtained in this chapter, was developed. The control laws to implement the novel613.10. Summaryscheme were derived and validated by experimental results in a 44W prototype, showingfixed switching frequency and excellent transient performance.The performance of centric-based and traditional dual-loop controllers was compared us-ing experimental results corresponding to different transients. As a result of the comparison,the enhanced performance of the centric-based controller was verified by a faster recover,lower peak deviations in voltage and current, and a predictable evolution of the variables.The proposed control scheme was also tested under parameter deviations to ensure its reli-able operation and suitability to be implemented in high volume applications. The controllerperformance was compared against the theoretical performance limits using the benchmarktool introduced in the previous chapter.The work provides valuable conceptual and practical contributions to the field: the Av-erage Natural Trajectories which accurately model the PWM driven converter large-signalbehavior, and the high-performance control strategy that results from a centric-based PWMmanipulation.62Chapter 4Average Natural Trajectories (ANTs)for Boost Converters: Centric-BasedControlTwo main novel concepts were introduced in the previous chapter for buck converters: theANTs and the centric-based controller. The ideas are new in the field of controls for powerelectronics, and can be applied in any power conversion topology. In order to illustrate thisfact, and due to the outstanding results obtained in buck converters, this chapter extendsthe theory and implementation to the basic boost DC-DC topology. A particular extra lim-itation is imposed by the characteristic non-minimum phase behavior of the boost topology,which gives traditional linear controllers a sluggish response. As in the previous case, thenovel geometric-based controller is developed to combine the advantages of averaged andgeometrical analysis. As a result, a fast and predictable dynamic performance is obtainedwhile keeping fixed PWM frequencies and low-bandwidth sensing/sampling systems.The natural response of the PWM driven boost converter is modeled by Averaged NaturalTrajectories (ANTs), which are obtained by merging geometrical analysis and traditionalaverage techniques. As a result of the normalized analysis, circular averaged trajectoriesvalid for any combination of LC parameters are obtained as illustrated in Fig. 4.2. Theinsight provided by the obtained ANTs is used to develop a reliable fixed-frequency controlscheme featuring fast and predictable dynamic performance and low bandwidth requirements63Chapter 4. Average Natural Trajectories (ANTs) for Boost Converters: Centric-Based ControlLn =12piCn =12pi+-VccniLnvCnion(a)iLnvoniondrivingsignal?r1.31 ms(b)Figure 4.1: a) Normalized PWM driven boost converter. b) PWM natural trajectories in areal converter.for sensing and signal conditioning stages. In contrast with boundary controllers wherethe control is performed by ON-OFF actions, in the centric-based controller it is done bysetting a PWM duty cycle based on the current location and averaged behaviour of theconverter. In the ideal case, the center is calculated only once in order to direct the operatingpoint to the target, requiring a very low processing cost. The derivation of the controllaws is presented along with the normalization procedure and the ANTs derivation for boostconverters. Experimental results in open and closed loop validate the ANTs and the enhanceddynamic response of the centric controller, highlighting the strong contribution this workmakes to both theoretical and applied fields.644.1. Normalized Boost Converter ANTs Derivation4.1 Normalized Boost Converter ANTs DerivationThe Averaged Natural Trajectories of the ideal PWM driven normalized boost convertershown in Fig. 4.2 are developed in this section.The differential equations that rule the behavior of boost converters are, for ON state:CdvCdt= ?io (4.1)LdiLdt= Vcc (4.2)and for OFF state:CdvCdt= iL ? io (4.3)LdiLdt= Vcc ? vC (4.4)In order to gain generality, the analysis is taken to the normalized domain, where:tn =tT0; Zxn =ZxZ0; vxn =vxvref; ixn =ixirefThe base quantities employed are the filter characteristic impedance and resonance pe-riod, and the reference voltage and current:Z0 =?L/C; T0 = 2pi?LC; vref ; iref =vrefZ0As a result, the differential equations that describe the behaviour of the normalized boostconverter are found for ON,12pidvCndtn= ?ion (4.5)12pidiLndtn= Vccn (4.6)654.1. Normalized Boost Converter ANTs DerivationVccnion?AVGiLnvCn?vCn?iLn0?kON?kOFF     Vccnd?     iond??!nLoad LineiLn0 tntn0drivingsignal1tn0vCn2     Vccnd?     iond?     Vccn     ion2?!n?!n?vCn?iLnFigure 4.2: Ideal natural trajectories with Vccn = 0.5, d = 0.5 and fsn = 10.664.1. Normalized Boost Converter ANTs Derivationand OFF states.12pidvCndtn= iLn ? ion (4.7)12pidiLndtn= Vccn ? vCn (4.8)Employing traditional averaging techniques [5], the averaged differential equations innormalized form are obtained:12pidvCndtn= d? iLn ? ion (4.9)12pidiLndtn= Vccn ? d? vCn (4.10)where: d? = 1? d, is the complementary PWM duty cycle determined by the ratio betweenthe OFF time (TOFF ) and the PWM period (Ts).Time-domain expressions that describe the averaged evolution of the state variables (ca-pacitor voltage and inductor current) can be found by solving the differential equations (4.9)and (4.10):vCn(tn) = [vCn(0)?Vccnd?]cos(d? 2pitn) + [iLn(0)?iond?]sin(d? 2pitn) +Vccnd?(4.11)iLn(tn) = [iLn(0)?iond?]cos(d? 2pitn) + [vCn(0)?Vccnd?]sin(d? 2pitn) +iond?(4.12)The expression that describes the Averaged Natural Trajectories (ANTs) that the op-erating point follows for the ideal PWM driven normalized boost converter is obtained bycombining (4.11) and (4.12) and eliminating the normalized time variable tn:?AV G :[iLn ?iond?]2+[von ?Vccnd?]2?[iLn(0)?iond?]2?[von(0)?Vccnd?]2= 0 (4.13)The parametric equation obtained represents a circle with center located at (Vccnd? ;iond? ),and radius determined by the initial conditions [vCn(0); iLn(0)].674.1. Normalized Boost Converter ANTs Derivation?ON?OFFrOFFcOFF?AVG d1rd1cd1?AVGd2rd2cd2iLnvCnVccnion0!0n!d1n!d2nVccnion 1d2d1 0tn31 1.5dvCn(0)tn0vCn 2?!0n 2?!d1n 2?!d2niLn(0)tniLn0VccnLnionCnFigure 4.3: Several PWM-driven boost converter ANTs departing from unique initial oper-ating point.684.2. Approaching the TargetSeveral circular trajectories corresponding to the same initial conditions but differentcenters are shown in Fig. 4.3. It is worth noting that the center is always located on the loadline, at a distance from the origin determined by the scalar c = 1d? multiplied by the basevector (Vccn; ion).Taking (4.11) and (4.12) into account, the normalized angular speed at which the op-erating point moves across the circular trajectory is found to be constant and defined by?n = d? 2pi, as illustrated by the time-domain plots in Fig. 4.3.From a practical point of view, the natural resonance of the L? C system is defined bythe fraction of time the reactive components are connected together. For instance, when thesystem remains in OFF state, the inductance L and the capacitance C are connected all thetime, which corresponds to the condition d = 0 and therefore ?n = 2pi (? = 1?LC ), whichrepresents the natural frequency of an LC resonant tank. It is also worth pointing out thatwhen the system remains in ON state (d = 1), there is no interaction between the reactivecomponents and therefore ?n = 0 and the circle becomes a straight line that is perpendicularto the load line as the center location tends towards infinity in the load line direction.As indicated, the derived expressions are valid for the whole range of possible PWM dutycycles, including the saturation extremes d = 0 and d = 1. Therefore (4.13) represents acomplete ideal averaged model of the normalized boost converter in the geometrical domain.4.2 Approaching the TargetAs determined in the previous section, for any given initial conditions the operating pointfollows an ANT determined by the PWM duty cycle. There is an ANT that leads theoperating point to the target located at (1; iLnt), where the normalized target inductor currentis: iLnt = ionVccn .The center of the circle that contains both initial and target operating points can be694.2. Approaching the Target?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?OFF ?ON!n1!n2!n3!n4Restricted domainFigure 4.4: Fixed duty cycle target approaching method for different initial conditions704.3. Closing the Loop with a Centric-Based Controllerfound by performing a geometrical analysis from Fig. 4.4:c = 0.5 vCn(0)2 + iLn(0)2 ? i2Lnt ? 1Vccn(vCn(0)? 1) + ion(iLn(0)? iLnt)(4.14)Taking into account that c = 11?d , the duty cycle to obtain the desired circle is found:d = 1? 2 Vccn(vCn(0)? 1) + ion(iLn(0)? iLnt)vCn(0)2 + iLn(0)2 ? i2Lnt ? 1(4.15)This expression is valid for any initial point located inside of the domain limited by theON and OFF trajectories as indicated in Fig. 4.4, and determines the duty cycle neededto reach the target operating point following just one averaged circular path. The time theoperating point takes to reach the target can be determined from Fig. 4.4 by finding theangle ?k and dividing it by ?kn = d? 2pi:tD =1d? 2pi(tan?1(ionVccn)? tan?1(d? iLn(0)? iond? vcn(0)? vcn))(4.16)Once the target is reached, the center must be moved to the steady state operatingpoint (1; iLnt) by setting the duty cycle to 1 ? Vccn, in order to obtain an averaged circulartrajectory with null radius. A simple and powerful principle is found: the desired steadystate operating condition can be reached in a known time period and following a predictablecircular trajectory. The only control action required to achieve this is to set the PWM dutycycle to a fixed value determined by the location of the initial and target operating points.4.3 Closing the Loop with a Centric-Based ControllerBased on the target approaching method presented in the previous section, a novel fixed-frequency geometrical control scheme is proposed here. A periodical recalculation of the714.4. Experimental Resultscenter location using (4.15), allows the controller to compensate for deviations in the reactivecomponents values, losses and any extra effect not being considered in the ANTs derivationthat could move the trajectory beyond ideal circles. In this way, the distortion in the circulartrajectories is limited to the value it reaches during one recalculation period.As mentioned above, (4.15) is valid for any initial point located inside the ON and OFFnatural trajectories. When the operating point is located outside of this domain, it is takenback in by saturating the duty cycle at its corresponding extreme. In order to improve thedynamic response, the function domain is restricted to the half of the ?OFF circle locatedabove the load line, and to the portion at the right of ?ON below the load line as indicated inFig. 4.4. In this way, the drops and peaks in the output voltage can be kept low during loadtransients, while large current spikes at the inductor are avoided, and a fast, predictable andreliable transient response is obtained.A slow estimator of the series resistance is also implemented in order to account for seriesvoltage drops. Even though in high efficiency converters the DC resistance in the inductorand switches are minimal, a small steady state error might be present but can be eliminatedby using the load current and the series resistance estimation.4.4 Experimental ResultsThe theoretical and applied concepts introduced in this chapter have been validated in the65W boost converter detailed in Tables 4.1 and 4.2.The Averaged Natural Trajectories are validated by experimental results in Fig. 4.1 (b),where the normalized switching frequency has been set to fsn = 10 and the duty cycle to50%. Observing the time-domain waveforms, the normalized resonance period of the systemis found to be 2 (1.31 ms), as is to be expected for a d? = 0.5.724.4. Experimental ResultsTargetInitial conditionsLoad line(a)iLndvonStart-upCentric-controlled trajectoriesduty cycle saturationLoadingUnloadingion0.981.531.981 2 3 4 5 21234510(b)Figure 4.5: Centric-based controller experimental results. a) Open-loop with duty cycleprecalculation. b) Closed-loop response for start-up, loading and unloading transients.734.5. SummaryTable 4.1: Boost converter parametersPARAMETER VALUE NORM.vref 24 V 1Vcc 12 V 0.5L 240 ?H 12piC 45 ?F 12piT0 654 ?s 1Z0 2.3 ? 1Table 4.2: Main parasiticsPARAMETER VALUE NORM.rL 150 m? 65.2mrC 43 m? 18.7mrswitch1 20 m? 6.15mrswitch2 20 m? 6.15mAs shown in Fig. 4.1 (b), the real trajectories present differences from those derived forthe ideal case; these differences are due to the effects of losses in the system.Experimental results of the system working in open-loop with pre-calculated duty cyclesthat lead the operating point to the target for different initial conditions are shown in Fig.4.5 (a), where fsn = 10.The enhanced dynamic performance of the closed-loop controller is validated by the fastand predictable recoveries from extremely large transients shown in Fig. 4.5 (b). Even thoughthe normalized current step-up and step-down transients are extremely large, steady stateis reached back in periods of time of the same order of magnitude (1.53 to 1.98) than theLC resonance period. Since the duty cycle saturation is implemented, the voltage drops andpeaks are close to optimal, and due to the center placement technique, the inductor currentpeaks are kept low.4.5 SummaryThe Averaged Natural Trajectories (ANTs) for DC-DC boost converters were derived andpresented in this chapter. The proposed large-signal model was developed combining tra-ditional averaging techniques with a geometrical domain analysis of the normalized boostconverter state variables.744.5. SummaryA closed form expression to calculate the duty cycle necessary to direct the operating pointtowards the target following one circular ANT was found and verified by experimental results.A new type of closed-loop control technique for boost converters, featuring enhanced dynamicperformance, fixed switching frequency and low bandwidth requirements, was developed andvalidated employing a 65 W prototype.75Chapter 5Conclusions5.1 SummaryThis thesis introduced three novel concepts to the field of controls for power electronics: thephysical limits of performance, the Average Natural Trajectories and the centric-based con-troller. The theoretical optimal response of the normalized buck converter to start-up, loadingand unloading transients was found, analyzed and characterized by closed-form mathemati-cal expressions. A powerful benchmark tool to analyze transient response in buck converterswas introduced, and illustrated by an example of benchmarking procedure.A novel model was developed for buck and boost converters by combining average, nor-malization and geometrical analysis. The Averaged Natural Trajectories provide valuableinsight into the behaviour of power converters and accurately model the large-signal dynam-ics of the system. Due to the normalized approach, the models are valid for any combinationof reactive components. The significant theoretical contribution to the field of modeling andcontrols must be highlighted.Based on the model introduced in this work, a new type of geometric-based controller,suitable for high-volume applications, was developed for buck and boost converters. Due tothe nature of the model employed, the controller features global stability, fixed switchingfrequency and an excellent dynamic performance. Furthermore, the control laws developedfor both topologies can be implemented in converters with any combination of L?C values,providing a universal solution and highly simplifying the design.765.2. Future WorkPerformance comparisons with traditional linear techniques and a benchmark procedureusing the physical limits of performance, were carried out for buck converters, confirming theenhanced dynamics and predictable behaviour of the proposed control scheme. The centric-based controlled buck converter was tested under deviations in the reactive components valuesin order to illustrate the robust characteristic of the proposed control technique.The theoretical concepts introduced were supported by detailed mathematical procedures,and the control applications validated by simulation and experimental results.The contribution of the work to the power conversion community is proven by the publi-cation of [1?4]5.2 Future WorkThe concepts introduced in this work are original and were not known in the past, whichhighlights the importance of the contribution. A research paper about the physical limitsof performance boost converters is being developed. A clear need to develop the theoreticaloptimal performance of other topologies and to work on the characterization of the controllersthat are already on the field is identified.The proposed large-signal model and control technique can be extended to other powerconverter topologies. 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