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Circuit averaging and numerical average value modeling of flyback converter in CCM and DCM including… Amini Akbarabadi, Soroush 2014

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CIRCUIT AVERAGING AND NUMERICAL AVERAGE VALUE MODELING OF FLYBACK CONVERTER IN CCM AND DCM INCLUDING PARASITICS AND SNUBBERS  by  Soroush Amini Akbarabadi  B.Sc., Amirkabir University of Technology, 2011   A THESIS SUBMITTED IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF  MASTER OF APPLIED SCIENCE  in  THE FACULTY OF GRADUATE AND POSTDOCTORAL STUDIES  (ELECTRICAL AND COMPUTER ENGINEERING)  THE UNIVERSITY OF BRITISH COLUMBIA  (Vancouver)     June 2014  © Soroush Amini Akbarabadi, 2014  ii Abstract  Modeling and analysis of basic DC-DC converters is essential for enabling power-electronic solutions for the future energy systems and applications. Average-value modeling (AVM) provides a time-efficient tool for studying power electronic systems, including DC/DC converters. Many AVM techniques including the analytical and numerical state-space averaging and circuit averaging have been developed over the years and available in the literature. Average-value modeling of ideal PWM converters neglects parasitics (losses) to simplify the derivations and modeling procedures, and the resulting models may not be sufficiently accurate for practical converters.   In this work, first we consider a second-order Flyback converter, which has transformer isolation and additional parasitics such as conduction losses that have not been accurately included in the prior literature. We propose three new AVMs using the analytical state-space averaging, circuit averaging, and parametric AVM approaches, respectively. By taking into account conduction losses, the accuracy of the proposed average-value models is significantly improved. The derived (corrected) models show noticeable improvement over the traditional (un-corrected) models.  Next, we consider the Flyback converter including the snubbers and leakage inductances in the full-order model. Snubbers reduce electromagnetic interfaces (EMI) during transients and protect switching devices from high voltage, and therefore are necessary in many practical converter circuits. Including snubbers into the model improves accuracy in predicting the circuit variables during the time-domain transients as well as predicting the converter efficiency. It is shown that conventional analytical/numerical methods of averaging do not result in accurate AVM for the full-order Flyback converter. A new formulation for the state-space averaging methodology is proposed that is functional for higher-order converters with parasitics and result in highly accurate AVM. The new approach is justified mathematically and verified experimentally using hardware prototype and measurements. The new model is demonstrated to achieve accurate results in large signal time-domain transients, and small-signal frequency-domain analysis in continuous conduction mode (CCM) and discontinuous conduction mode (DCM), which represents advancement to the state-of-the-art in this field.   iiiPreface  The research results presented in this thesis have been already published in several conference proceedings and/or ready for submission as a journal article. In all publications, I was responsible for developing the formulations, implementing models, doing simulations and hardware tests, compiling results, as well as preparing the majority of the manuscripts. My  research supervisor, Dr. Juri Jatskevich, has provided supervisory comments and corrections during the process of conducting research and writing the manuscripts. The other co-authors have also provided constructive comments and feedback.   A version of Chapter 2 has been published:  S. Amini Akbarabadi, H. Atighechi, and J. Jatskevich, “Circuit-Averaged and State-Space-Averaged-Value Modeling of Second-Order Flyback Converter in CCM and DCM Including Conduction Losses,” International Conference On Energy and Electrical Drives (POWERENG), May 2013, pp. 995-1000.  S. Amini Akbarabadi, M. Sucu, H. Atighechi, and J. Jatskevich, “Numerical average value modeling of second-order Flyback converter in both operational modes,” IEEE  Workshop on Control and Modeling for Power Electronics (COMPEL), June 2013, pp. 1-6.  S. Amini Akbarabadi, H. Atighechi,J. Jatskevich, “Corrected State-Space Averaged-Value Modeling of Second-Order FlybackConverter Including Conduction Losses,” IEEE Canadian Conference on Electrical and Computer Engineering (CCECE), 2013, Regina, Canada.    ivTable of Contents  Abstract .................................................................................................................................... ii Preface ..................................................................................................................................... iii Table of Contents ................................................................................................................... iv List of Tables .......................................................................................................................... vi List of Figures ........................................................................................................................ vii List of Abbreviations ............................................................................................................. xi Acknowledgements ............................................................................................................... xii Chapter 1 : Introduction ........................................................................................................ 1 1.1 Average-Value Modeling ...................................................................................................... 1 1.1.1 State-Space Averaging...................................................................................................... 2 1.1.2 Circuit Averaging ............................................................................................................. 2 1.1.3 Parametric Average Value Modeling ............................................................................... 3 1.2 Flyback Converters ............................................................................................................... 3 1.3 Motivations and Objectives .................................................................................................. 5 Chapter 2 : Second-order Flyback Converters .................................................................... 7 2.1 State-Space Averaging .......................................................................................................... 7 2.1.1 State-Space Averaging Case Studies .............................................................................. 13 2.1.1.1 Correction Matrix M .............................................................................................. 13 2.1.1.2 Effect of Including Parasitics on Predicted Efficiency of Converter ..................... 15 2.1.1.3 Closed-loop System ............................................................................................... 16 2.2 Circuit Averaging ................................................................................................................ 17 2.2.1 New Large-Signal Averaged Switching Cell .................................................................. 17 2.2.2 Ideal Averaged Cell ........................................................................................................ 18 2.2.3 Energy Conservation Principle ....................................................................................... 19 2.2.4 Circuit Averaging Case Studies ...................................................................................... 22 2.2.4.1 Time-Domain Transient ......................................................................................... 23 2.2.4.2 Frequency-Domain Analysis.................................................................................. 26 2.3 Numerical Average Value Modeling .................................................................................. 28  v2.3.1 Parametric Average Value Modeling ............................................................................. 28 2.3.2 Numerical Average Value Modeling Case Studies ........................................................ 32 2.3.2.1 Time-Domain Transient Studies ............................................................................ 32 2.3.2.2 Frequency-Domain Analysis.................................................................................. 37 Chapter 3 : Full-order Flyback Converter ......................................................................... 38 3.1 Effect of Input Snubber in SSA .......................................................................................... 38 3.2 Effect of Output Snubber in SSA-AVM ............................................................................. 41 3.2.1 Output Stage without Snubber (4th Order Flyback Converter) ....................................... 41 3.2.2 Output Stage with Snubber (5th Order Flyback Converter) ............................................ 43 3.3 Generalized Numerical SSA-AVM..................................................................................... 46 3.4 Construction and Implementation of the Generalized PAVM ............................................ 47 3.5 Eigenvalue Analysis ............................................................................................................ 55 3.6 Model Validation with Respect to Hardware ...................................................................... 56 3.7 Precision Evaluation in Steady State .................................................................................. 58 3.8 Case Studies ........................................................................................................................ 59 3.8.1 Performance of Proposed PAVM in Time-Domain Transients ...................................... 60 3.8.2 Performance of Proposed PAVM in Frequency-Domain ............................................... 65 Chapter 4 : Summary of Research and Future Work ....................................................... 66 4.1 Second-order Flyback Converter ........................................................................................ 66 4.2 Flyback Converter with Snubbers ....................................................................................... 66 4.3 Future Work ........................................................................................................................ 67 Bibliography .......................................................................................................................... 68 Appendices ............................................................................................................................. 73 Appendix A. Second-order Flyback Converter State-space Matrices ........................................... 73 Appendix B. Converters Circuit Parameters ................................................................................. 74 B.1 Second-order Flyback Converter Parameters with Basic Parasitics ............................... 74 B.2 Full-order Flyback Converter Parameters with all Parasitics and Snubbers ................... 74 Appendix C. Hardware Flyback Converter Prototype Circuit Diagram ....................................... 75   viList of Tables   Table  2.1 Output voltage and inductor current values and errors as predicted by different averaged models in DCM and CCM. .................................................... 24 Table  2.2 Eigenvalues of average-value models in DCM and CCM. ..................................... 25 Table  2.3 Efficiency comparison of average-value models. ................................................... 25 Table  2.4 Output voltage and inductor current as predicted by various models for the two steady state operating points. .................................................................. 34 Table  2.5 Converter efficiency as predicted by various models for the two steady state operating points. .......................................................................................... 34 Table  2.6 Simulation speed comparison of the detailed and average models in closed-loop transient study................................................................................... 36 Table  3.1 Eigenvalues of AVMs. ............................................................................................ 56 Table  3.2 Hardware prototype and detailed model comparison in terms of input current, output voltage and efficiency.................................................................. 57 Table  3.3 Accuracy precision of the proposed PAVM in predicting steady state variables. .............................................................................................................. 59 Table  3.4 Accuracy precision of the proposed PAVM in predicting converter efficiency. ............................................................................................................. 59   viiList of Figures  Figure  1.1 A detailed waveform containing switching ripple and the corresponding average-value ................................................................................. 1 Figure  1.2 Full-order Flyback converter circuit with snubbers on primary and secondary sides. ..................................................................................................... 4 Figure  2.1 Simplified second-order Flyback converter with basic conduction parasitics................................................................................................................. 7 Figure  2.2 Idealized inductor current waveform assuming DCM. ........................................... 8 Figure  2.3 Assumed topological instances for the second-order Flyback converter without parasitics; (a) original circuit; (b) circuit during subinterval 1; (c) circuit during subinterval 2; and (d) circuit during subinterval 3. .................... 9 Figure  2.4 Effect of basic parasitics on inductor waveform and its peak in DCM. ................ 11 Figure  2.5 Simplified second-order Flyback converter with basic conduction parasitics represented as an equivalent resistor in the primary side. ................... 11 Figure  2.6 Steady state output voltage and inductor current in DCM as predicted by various models: (a) detailed model; (b) state-space averaged-value model (SS-AVM); and (c) corrected state-space averaged-value model (CSS-AVM.). ............................................................................................ 14 Figure  2.7 Efficiency prediction by various models as a function of the load resistance: (a) detailed model; (b) CSS-AVM without equivalent parasitic resistance; and (c) CSS-AVM with equivalent parasitic resistance. ............................................................................................................. 15 Figure  2.8 Efficiency prediction by various models as a function of duty cycle: (a) detailed model; (b) CSS-AVM without equivalent parasitic resistance; and (c) CSS-AVM with equivalent parasitic resistance. .................... 16 Figure  2.9 Output capacitor voltage and inductor current responses to the step change in load for a closed-loop system. ............................................................. 17 Figure  2.10 Averaged switching cell in second-order Flyback converter. ............................. 18 Figure  2.11 Resulting equivalent averaged cell for the second-order Flyback converter............................................................................................................... 19  viiiFigure  2.12 Typical inductor and switch waveform assuming CCM. .................................... 21 Figure  2.13 Output voltage and inductor current transients during load change as predicted by various models. ............................................................................... 23 Figure  2.14 Control-to-output transfer function magnitude and phase in DCM as predicted by various models. ............................................................................... 27 Figure  2.15 Control-to-output transfer function magnitude and phase in CCM as predicted by various models. ............................................................................... 28 Figure  2.16 Correction coefficient m1 for the example second-order Flyback converter............................................................................................................... 30 Figure  2.17 Correction coefficient m2 for the example second-order Flyback converter............................................................................................................... 30 Figure  2.18 Duty-ratio constraint d2 for the example second-order Flyback converter............................................................................................................... 31 Figure  2.19 Inverted matrix condition number for the example second-order Flyback converter. ................................................................................................ 31 Figure  2.20 Implementation of PAVM. .................................................................................. 32 Figure  2.21 Output capacitor voltage and inductor current transients due to change in the switch duty cycle as predicted by various models. ........................ 33 Figure  2.22 Closed-loop system of the considered second-order Flyback converter with PI controller to regulate the output voltage. ................................ 34 Figure  2.23 Closed-loop second-order Flyback converter with PI controller response to a load change. .................................................................................... 35 Figure  2.24 Control to output transfer function magnitude and phase. .................................. 37 Figure  3.1 Averaged detailed circuit of input stage of Flyback converter in steady state. ..................................................................................................................... 39 Figure  3.2 Input stage of Flyback converter depicting operation of snubber. ........................ 40 Figure  3.3 Averaged detailed circuit of the output stage Flyback converter without the snubber in steady state. ..................................................................... 41 Figure  3.4 Output stage topologies of Flyback converter without output snubber. ................ 42 Figure  3.5 Averaged detailed circuit of output stage of full-order Flyback converter in steady state. ...................................................................................... 43  ixFigure  3.6 Output side of full-order Flyback converter depicting the snubber operation............................................................................................................... 44 Figure  3.7 Inverted matrix condition number. ........................................................................ 49 Figure  3.8 Correction coefficient m1 for voltage of output capacitor Cv . .............................. 50 Figure  3.9 Correction coefficient m2 for voltage of primary snubber capacitor Cssv . ...................................................................................................................... 50 Figure  3.10 Correction coefficient m3 for primary side leakage inductor current Lpti . ....................................................................................................................... 51 Figure  3.11 coefficient m4 for secondary side leakage inductor current Lsti . ......................... 51 Figure  3.12 Calculated function of duty-ratio constraint d2. .................................................. 52 Figure  3.13 Correction coefficient m1 for voltage of output capacitor Cv . ............................ 52 Figure  3.14 Correction coefficient m2 for voltage of primary snubber capacitor Cssv . ...................................................................................................................... 52 Figure  3.15 Correction coefficient m3 for voltage of output snubber capacitor Cdsv . ...................................................................................................................... 53 Figure  3.16 Correction coefficient m4 for primary side leakage inductor current Lpti . ....................................................................................................................... 53 Figure  3.17 Correction coefficient m5 for secondary side leakage inductor current Lsti . ....................................................................................................................... 53 Figure  3.18 Block diagram depicting implementation of proposed extended PAVM. ................................................................................................................. 55 Figure  3.19 Measured and detailed model waveforms for the considered operating point. .................................................................................................... 57 Figure  3.20 Transients of state variables of fourth-order Flyback converter as predicted by uncorrected PAVM and the proposed corrected PAVM. ................ 61 Figure  3.21 Transients of state variables of full-order Flyback converter due to load change. ......................................................................................................... 62 Figure  3.22 Circuit state variables transients due to intense increase of duty cycle. .............. 64  xFigure  3.23 Control-to-output transfer function magnitude and phase in DCM as predicted by proposed PAVM and detailed model. ............................................. 65   xiList of Abbreviations  AVM Average-Value Modeling CA Circuit Averaged CCA Corrected Circuit Averaged CCM Continuous Conduction Mode  CSSA Corrected State-Space Averaged DCM Discontinuous Conduction Mode PAVM Parametric Average Value Model(ing) PWM Pulse Width Modulation SS State-Space SSA State-Space Averaged    xiiAcknowledgements  I would like to express my appreciation to my research supervisor, Dr. Juri Jatskevich, for his expert, sincere, valuable guidance and encouragement that he provided to me during all my years at UBC. The financial support for this research was made possible through the Natural Science and Engineering Research Council (NSERC) Discovery Grant entitled “Modeling and Analysis of Power Electronic and Energy Conversion Systems” and the Discovery Accelerator Supplement Grant entitled “Enabling Next Generation of Transient Simulation Programs” lead by Dr. Juri Jatskevich as a sole principal investigator.  I consider it an honor to have Dr. John Madden and Dr. Jose Marti as my committee members who dedicated their time to provide me constructive comments and take part in my M.A.Sc. examination.  I would like to thank the following friends and colleagues from the Electrical Energy Systems Lab at UBC for their continued support and helpful insight toward the problems that I faced in the course of my research project: Francis Therrien, Hamid Atighechi, Mehmet Sucu, and Mehrdad Chapariha.  Last but not the least, I would like to express my deepest gratitude to my parents for their patience while I was abroad, and for supporting me spiritually throughout my life.  1  Chapter 1 : Introduction  1.1 Average-Value Modeling Modeling and simulation enables engineers to design complicated power electronic converters, tune their parameters, and detect possible flaws before hardware implementation, which in turn increases the productivity and result in significant cost savings  [1]. Studying dynamic behavior of power electronic converters using their detailed models in commercial simulators may be time-consuming and not well-suitable for system-level analysis in frequency- and time-domains. However, in many applications such as design of controllers and system-level interactions, we are interested in knowing the average values of circuit variables rather than their instantaneous waveforms that might include the high-frequency switching ripples as depicted in Figure  1.1. Therefore, the so-called average-value models (AVMs) were introduced  [2]. Developing accurate AVMs for various power electronic converters has been an active area of research  [2],[3].  DetailedAveraged Figure  1.1 A detailed waveform containing switching ripple and the corresponding average-value  The average-value modeling of power electronic systems is receiving significant attention, as this approach results in improvement of system-level transient simulations as well as enables automatic linearization and small-signal analysis for design of controllers  [4]. The AVMs are also very useful for investigations of system transients and stability   [5]. The three common approaches to obtain the AVMs are the state-space averaging (SSA), the circuit averaging (CA), and parametric average-value modeling (PAVM). Over the years, each method has received a significant attention in the literature. 2   1.1.1 State-Space Averaging State-space averaging is a straightforward and general method of average-value modeling proposed first in  [6] and later extended in  [7]- [9], as well as many other publications. The SSA is used for large and small signal analysis of DC/DC converters. In state-space description of a system, the differential equations describing the converter circuit are written in canonical form. Extracting these equations manually is complicated for high-order converters with parasitics. Therefore, simulation and software tools that can automatically extract the state-space equation (matrices) become very useful. SSA is valid when natural frequencies (poles) of system are much lower than the switching frequency of the circuit  [10]. Due to a well-defined structure of this method, its equations can be easily transformed to the frequency-domain. A corrected full-order SSA model for basic converters has been introduced in  [9]. One of the main challenges in this method is calculating the length of each subinterval in a switching cycle. When parasitics are placed in the system, calculation of the duty ratio constraint becomes more complicated. However, a good AVM should include parasitics so as to approximate the hardware prototype with good accuracy  [11]. Including the parasitics in the state-space AVMs analytically has been considered in  [12] and  [13] for basic converters. A recently proposed numerical SSA approach is based on calculating the duty ratio constraint and correction terms numerically.  1.1.2 Circuit Averaging Circuit averaging is a well-known technique of deriving equivalent circuit for converters, wherein the manipulations are carried out based on a circuit diagram  [14]. Generally, CA methods replace the PWM switching cell of the converter with averaged circuit components, which gives physical insight into the converter operation  [4]. Including the parasitics and correct value of losses improves dynamic performance, robustness, and accuracy of models. This has been generally shown for boost and buck converters in  [15] and  [16]. For the Flyback converter, a small-signal ac equivalent circuit is developed for CCM in  [14]. In this work, the averaged circuit is derived based on the method presented in  [17] for obtaining the average current and average voltage drop across semiconductor devices. Therein, the conduction losses of Flyback converter are incorporated into the averaged 3  circuit, which improves the model accuracy. In this thesis, a corrected CA model is derived for the Flyback converter circuit that takes into account the basic conduction losses and properly respects the energy conservation principle  [17]- [19].   1.1.3 Parametric Average Value Modeling Analytical derivation of corrected SSA method could be done simply for ideal and low-order converters such as boost, buck, and buck-boost converters  [8],  [9],  [20]. But for high-order power electronic circuits that have parasitics and non-idealities, such as full-order Flyback converter, the analytical SSA becomes very challenging due to complexity of the circuit, non-linearity of the currents/voltages waveforms, and large ripple in DCM. Therefore, a computer-aided version of the methodology, known as parametric average-value modeling (PAVM), has been proposed and developed in  [21]- [24].  The PAVM is based on corrected full-order version of SSA, and it generates the AVM of system on the basis of numerically-constructed functions.  In this thesis, the deficiencies of previously developed numerical corrected SSA model in dealing with non-ideal circuits are presented. The PAVM methodology is further explored mathematically, and a new set of equations is shown to work very accurately for the considered Flyback converter. It is envisioned that the new and extended PAVM methodology can be readily applied to other types of PWM converters with parasitics and transformers.  1.2 Flyback Converters The PWM Flyback converters is widely used in computers, power supplies, and electronics  [25]- [26]. Such converters are often used in power electronic applications, wherein new topologies and control schemes are proposed to improve their performance  [27]. Various design considerations and control approaches are applied to Flyback converter to achieve higher efficiency and improve output voltage regulation  [28]- [29]. Improvements are made by introducing new snubbers to the circuit [30]- [33], replacement of passive semiconductor elements by active and controlled switches  [28],  [34]- [35], changing switching and modulation schemes  [28],  [36], and increasing switching frequency in some applications  [36]. The presence of a transformer in the circuit provides a galvanic isolation 4  between the input and output. Galvanic isolation separates grounds with different potentials and prevents unwanted current flow, e.g. in the case of output short circuit. For miniaturizing transformer size, converter circuit should operate at high frequency  [25], which in turn results in increase of leakage inductances. Therefore, it becomes necessary to use snubbers in the circuit in order to alleviate the hard-switching or non-zero current/voltage switching problem that would happen otherwise. A snubber is not a fundamental part of a power electronic converter circuit, and adding it to a semiconductor device reduces the stresses to a level that is tolerable according to that device electrical ratings  [37]. The considered full-order Flyback converter is illustrated in Figure  1.2. This circuit has both primary and secondary snubbers. The RC snubbers  [39] in the circuit help MOSFET and diode operate under hard-switching conditions. As the converter works with a relatively high switching frequency (e.g., 250 KHz for the hardware prototype considered in this work), the absence of snubbers in the circuit would cause failure of MOSFET and generate broadband noise that can cause problem for data transmission equipment  [25]. These dissipative RC snubbers used in the circuit reduce the stress on the switches, reduce dtdv and  dtdi, etc.  [40]- [41].   v(t)RloadCRcnLmRswvg(t)LstLptCssRptRstRCssCdsRCds Figure  1.2 Full-order Flyback converter circuit with snubbers on primary and secondary sides.     5    1.3 Motivations and Objectives Different power converters have various challenges when it comes to average-value modeling. In this thesis, I am focusing on the Flyback converter for which only very simplified average models have been developed in the prior literature. Unlike the classical DC/DC converters i.e. Buck, Boost, Buck/Boost, Cuk, etc., for which there have been extensive prior research with advanced results, the Flyback converter includes a transformer and two RC snubbers in its circuitry, which represents additional challenges for developing the accurate dynamic models. To the best of our knowledge, these challenges have not been addressed in the published literature. Thus, the purpose of this work is to develop accurate and straightforward-to-use AVMs for Flyback converter for both widely used approaches, CA and SSA. Specifically, the objectives of this research project are:  • Objective 1: Development of CA-AVM for Flyback converter Before considering the full converter circuit with all parasitics and snubbers, we want to develop the CA-AVM for Flyback converter with basic parasitics that represent the conduction losses. In working on this objective, we will need to include the energy conservation principle for both modes of operation, i.e. CCM and DCM, which has not been achieved before.   • Objective 2: Development of SSA-AVM for Flyback converter  To develop SSA-AVM for Flyback converter with basic parasitics that represent the conduction losses. For simplified (second-order) Flyback converter with ideal components are piece-wise linear. The objective is to derive a more accurate analytical SSA model and numerically-constructed AVMs by appropriately correcting the system state-space matrices and the equations for the duty ratio constraint.   • Objective 3: Include snubbers in PAVM Extend the prior analysis and AVM methodology by including the effects of snubbers. Flyback converter has one input snubber (for protecting switch) and one output 6  snubber (for protecting diode). The presence of these snubber circuits has been omitted in most, if not all, prior work on AVM, due to analytical complexity that they represent. It is very desirable to understand the effects of snubbers for the analytical and PAVM methodologies, and to develop a generalized methodology that is capable of automatically including both the conduction parasitics as well as the energy losses due to all possible snubber circuits.                           7  Chapter 2 : Second-order Flyback Converters  As shown in Figure  1.2, the complete full-order Flyback converter circuit includes the snubbers, the leakage inductors of the transformer, and the conduction losses of the two switches. Oscillation of electrical energy between inductive and capacitive elements in this circuit in the form of ringing at switching instants makes the process of deriving the AVMs particularly challenging. Therefore, in the first step, we consider just very basic conduction parasitics as depicted in Figure  2.1. The resulting circuit will have only two energy storing elements, and correspondingly will have a second-order. Such simplified Flyback converter circuit has been considered in many classical and contemporary research literature sources.       RloadCRcnLmiL(t)Rswvg(t)vfdMosfetRptRst Figure  2.1 Simplified second-order Flyback converter with basic conduction parasitics.  2.1 State-Space Averaging In state-space average representation, the derivatives of the averaged state variables are calculated as a function of inputs and state variables. Among different versions of this method, the corrected full-order state-space averaging  [41] is the most accurate one for both low and high frequencies. In all versions of SSA methods, the state equations for each topology are weighted relative to the time duration spent in each switching subinterval and added together in a canonical form. For DCM, each switching cycle can be divided into three subintervals denoted by 1d , 2d , and 3d , respectively. A typical inductor current waveform for the DCM is shown in Figure  2.2, wherein the converter is assumed to be without any parasitics and the ideal current waveform is triangular. In the first subinterval )1( =k , the 8  switch (transistor) is ON and the diode is OFF.  In the second subinterval )2( =k , the switch is OFF and the diode is ON. In third subinterval )3( =k , both switch and diode are OFF and not conducting. The resulting converter topology in each subinterval )3,2,1( =k  is depicted in Figure  2.3 with circuits labeled by (b), (c), and (d) respectively.  i L, A 0Tsd1Tsipkd2Ts d3TsiLtTimek=1 k=2 k=3 Figure  2.2 Idealized inductor current waveform assuming DCM.  9  dloadloadloadloadiL(t)iL(t)iL(t)iL(t)  Figure  2.3 Assumed topological instances for the second-order Flyback converter without parasitics; (a) original circuit; (b) circuit during subinterval 1; (c) circuit during subinterval 2; and (d) circuit during subinterval 3.   10  Herein, in-order to accurately represent the dynamics of the converter circuit, a corrected state-space average-value model (CSS AVM) is considered, wherein the correction matrix, ( )[ ]1,121 −+= dddiagM , is defined based on  [9] to compensate for the errors in DCM. The corrected full-order model is defined as,  ,3131uBxMAx += ∑∑ ====kkkkkkkk dddtd ( 2.1) The state-space matrices kA  and kB  are generated for the converter circuit in three switching subintervals based on the topological states in DCM. These matrices are summarized in  Appendix A. The weighting factors for each subinterval are defined as 1d , 2d , and 213 1 ddd −−= . As shown in Figure  2.2, for an ideal Flyback converter without parasitics, the inductor current is triangular and its average value is calculated as   ,)(2 21ddii pkL +=  ( 2.2) Since in the first subinterval the switch is closed, the input voltage is directly connected to the inductor and voltage drop across the inductor is onv . Therefore, the peak of the inductor current is   .1 sonp TdLvi =  ( 2.3) Based on ( 2.2) and ( 2.3), the duty ratio constraint is calculated as   .2112 dvTdiLdonsL−=  ( 2.4) Since ( 2.4) is obtained for DCM, to include the CCM, a general expression for the duty ratio is defined as   .1,2min 1112 −−= ddvTdiLdonsL ( 2.5) In-order to enhance the accuracy of the modeling approach, the basic parasitic losses should be considered in the AVM. Although adding resistors to model complicates the procedure, it makes the AVM more accurate and closer to the hardware prototype. As shown in Figure  2.4, the converter is operating in DCM, and when the circuit parasitics such as 11  resistors and diode voltage drop are included, the value of the current peak would drop from pi to pi′ .    iL , A 0d1Tsd2Tsd3TsTsipi'p Figure  2.4 Effect of basic parasitics on inductor waveform and its peak in DCM.  Therefore, to include the effect of basic parasitics in the model, the MOSFET ON-resistance swR , the transformer primary and secondary side (referred by the turns-ratio a) resistances ptR and stR , are included using an equivalent resistor depicted the second-order Flyback converter circuit illustrated in Figure  2.5. Herein, the equivalent resistor is calculated as  .2stptsweq RaRRR ++=  ( 2.6)   RCRcnLmiL(t)Reqvg(t)p'vfdPWMPI Controllervrefload Figure  2.5 Simplified second-order Flyback converter with basic conduction parasitics represented as an equivalent resistor in the primary side.  12  With presence of parasitic resistors in the circuit, the voltage drop across the magnetizing inductance is no longer equal to onv and an equation for the loop on the primary side becomes   .onLeqL viRdtdiL =+  ( 2.7) Calculating the local average of ( 2.7) over the first subinterval, results in the following   ,)(1)(111000∫∫∫ =+sss TdonsTdLseqTdLsdvTdiTRddtdiTLτττττ  ( 2.8) which is then simplified to   .)( 101onTdLseqpsvddiTRiTL s=+ ∫ ττ  ( 2.9) As the waveforms in Figure  2.2 are assumed to be piece-wise linear, ( 2.9) is further simplified to    .2 11onpeqpsvdiRdiTL=+  ( 2.10) Based on ( 2.10), the new corrected peak of the inductor current shown in Figure  2.2 is calculated as   .211eqsonp RdTLvdi+=′ ( 2.11) Substituting ( 2.11) in ( 2.2) and solving for 2d  gives the following result   .)2(112 dviRTdLdonLeqs−+=  ( 2.12) Similar to ( 2.5) for ideal converter, a general expression of the duty ratio constraint for both DCM and CCM is obtained as   .1,)2(min 1112−−+= ddviRTdLdonLeqs ( 2.13)  13  2.1.1 State-Space Averaging Case Studies  The detailed model of the Flyback converter shown in Figure  2.5 with parameters summarized in Appendix in  B.1. is used as reference for validating the proposed average value model. Both detailed model and average-value model have been implemented in Matlab toolboxes, specifically PLECS  [42] and Simulink  [43] for each model respectively. Improvements of state-space average-value model are demonstrated in the following sequential steps.  In the first step, the necessity of using the correction matrix M is shown. After that, the importance of including parasitics and dissipative elements is explained by efficiency studies over a wide range of circuit operating conditions. Finally, a proportional-plus-integral (PI) controller is added to the models, and closed-loop system's ability to regulate the output voltage is examined. 2.1.1.1 Correction Matrix M Herein, a study is performed to indicate how the correction matrix M  helps the average value model. As matrix M  role becomes evident in DCM, an operating point defined by load resistance Ω= 3100R  and duty cycle 5.01 =d  is chosen, and the models are run until reaching the steady state condition. The corresponding results are shown in Figure  2.6.  14  -155-150-145-140v C , VDetailedSSACSSA-150.99-150.98-150.98-150.97-150.97-150.96v C , V0.5 0.5 0.5 0.5 0.5 0.5 0.5-0.500.511.52Time, si L , A0.5See lower plot Figure  2.6 Steady state output voltage and inductor current in DCM as predicted by various models: (a) detailed model; (b) state-space averaged-value model (SS-AVM); and (c) corrected state-space averaged-value model (CSS-AVM.).  As Figure  2.6 shows, the corrected state-space averaged-value model has a much better accuracy in computing the average-value of the circuit state variables compared with the uncorrected version of the model. According to Figure  2.6, the average inductor current and output voltage predicted by the CSSA-AVM pass through the ripples of the detailed model as opposed to uncorrected SSA-AVM. Therefore, in the future investigations in this thesis, we consider the CSSA-AVM as a basis. 15   2.1.1.2 Effect of Including Parasitics on Predicted Efficiency of Converter Since calculation of efficiency requires several circuit variables, the predicted efficiency can be considered for evaluating accuracy of different AVMs. In this Subsection, we summarize the results obtained from many time domain studies for calculating the efficiency as predicted by each model. Figure  2.7 demonstrates the results obtained by the CSSA-AVMs with and without considering the transformer and switch equivalent resistances. In Figure  2.7, the duty cycle is kept constant at 0.6 and the load is changed from Ω500  to Ω2300  in steps of Ω200 . In Figure  2.8, the load is kept constant at Ω700  and the duty cycle is changed from 1.0 to 9.0 . As can be seen in these figures, the CSSA-AVM that includes the equivalent parasitics resistance predicts the same efficiency as the corresponding detailed model. At the same time, the CSSA-AVM that does not include the equivalent parasitics resistance predicts unrealistically high efficiency.   600 800 1000 1200 1400 1600 1800 2000 220080859095100Rload, OhmEfficiency (a),(c)(b)CSS Avg w/o ReqCSS Avg with ReqDetailed(a)(b)(c) Figure  2.7 Efficiency prediction by various models as a function of the load resistance: (a) detailed model; (b) CSS-AVM without equivalent parasitic resistance; and (c) CSS-AVM with equivalent parasitic resistance.  16  0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9406080100d1Efficiency (a),(c)(b)CSS Avg w/o ReqCSS Avg with ReqDetailed(a)(b)(c) Figure  2.8 Efficiency prediction by various models as a function of duty cycle: (a) detailed model; (b) CSS-AVM without equivalent parasitic resistance; and (c) CSS-AVM with equivalent parasitic resistance.  2.1.1.3  Closed-loop System Next, the accuracy of AVM in predicting the large-signal time-domain transient behavior is verified in closed loop with PI controller ( 07.0=iK and .180=pK ) The controller is set to regulate output the voltage at V70− . Initially, the converter is assumed to operate in DCM loaded by Ω2200  resistor. At sec35.0=t , the load is changed by connecting another resistor of Ω150  in parallel to the original load.  As it can be observed in Figure  2.10, the inductor current significantly increases and the converter mode changes from DCM to CCM.  Both current and voltage undergo transient oscillations and settle at a new steady state in CCM. As it can be seen, the considered CSSA-AVM that includes the same equivalent parasitic resistance as the detailed model predicts the entire transient response of the detailed model with good accuracy.   17  -71-70-69-68-67v C , VDetailedCSSA0.35 0.351 0.352 0.353 0.354 0.355 0.356 0.3570246810Time, si L , A Figure  2.9 Output capacitor voltage and inductor current responses to the step change in load for a closed-loop system.   2.2 Circuit Averaging The circuit averaging methodology is considered next. In CA, the switching devices are replaced by dependent sources to derive an equivalent continuous circuit with nonlinear elements and dependent sources. A further improvement that could be done in CA method is the correction of power dissipation on the circuit elements according to the energy conservation principle  [18]- [19]. The difference arises due to the fact that the dissipated power is related to the rms current, whereas the conventional circuit averaging method gives the average currents  [17].  2.2.1 New Large-Signal Averaged Switching Cell According to the methodology used in  [5],  [41] for obtaining the averaged circuits of basic DC-DC converters, a three terminal switching cell containing a diode and a transistor (MOSFET) and an inductor is modeled by a non-switching cell wherein the switching 18  devices are substituted by dependent voltage and/or current sources. However, in the Flyback converter shown in Figure  2.10 the switches are separated from each other by the isolation transformer. Therefore, in this thesis, we introduce a more general switching cell with four terminal-ports, which in addition to other elements also contains the transformer inside the cell as depicted in Figure  2.10. Here, the switching cell ports are labeled by a, b, c, and p, respectively. In-order to construct the averaged circuit, it is required to find a dynamically accurate equivalent cell for the four-port network. In the second-order Flyback converter circuit shown in Figure  2.10, the basic parasitics are incorporated. Specifically, the power dissipation components including the MOSFET ON-resistance swR , the transformer primary and secondary side (referred by the corresponding turns-ratio) resistances ptR and stR , are added up together and represented by an equivalent resistor defined in ( 2.6). For convenience of analysis, the element eqR is brought outside the switching cell as shown in Figure  2.10.  capbv(t)CRcnLmiL(t)Reqvg(t)p'a'vfdRload Figure  2.10 Averaged switching cell in second-order Flyback converter.  2.2.2 Ideal Averaged Cell Based on Figure  2.2 and Figure  2.10, and considering each subinterval, the following expressions for the current through switch and the voltage across diode can be obtained for the switching cycle,   19    [ ][ ][ ],,)(0)(,0,0212111+∈+∈∈=sssssLswTTddtTddTdtTdtii  ( 2.14)     .],)[(])(,[0],0[212111+∈+∈∈−=sspbssscapbdiodeTTddtvTddTdtTdtnvvv  ( 2.15) Averaging ( 2.14) and ( 2.15) over the switching cycle sT  yields   ,2 2111ddididi Lpksw +==  ( 2.16)   .)1( 12 capbdiode vndvdv −−=  ( 2.17) Finally, to get the equivalent average circuit cell, the transistor and diode are replaced by dependent current and voltage sources as shown in Figure  2.11. The values of these sources are defined by ( 2.16) and ( 2.17) respectively. To get the final AVM, the derived averaged cell of Figure  2.11 is inserted back into the original converter circuit shown in Figure  2.10. Transformers are designed to transmit AC voltages but no the DC voltages. In the following figure, the transformer represents a dependent voltage source wherein the conversion ratio is equal to n ( i.e. the output voltage is equal to n times input voltage).    capbvdiodeiL(t)Lmiswn Figure  2.11 Resulting equivalent averaged cell for the second-order Flyback converter.  2.2.3 Energy Conservation Principle For a second-order Flyback converter considered in this Chapter, the conduction losses are represented by the equivalent series resistance of the output capacitor, the switch on resistance, and the transformer primary and secondary side resistances, respectively. 20  Neglecting these basic parasitics will lead to obvious discrepancies between the AVM and the detailed model. According to the energy conservation principle  [17],  [18], and  [19] the energy loss in the resistance is characterized by the RMS value of the current rather than the average value of the current. Thus, assuming the DCM or boundary mode as in  [17],  [18], and  [19], and the current waveform depicted in Figure  2.2, the average and RMS values can be calculated as follows   ,2)(1)( 1∫−==tTtpkLsaveswsiddttiTi  ( 2.18)   .3)(1)( 12 pktTtLsrmssw iddttiTis== ∫− ( 2.19) Therefore, in-order to have the same energy loss in eqR  as in the detailed model, an equivalent resistance should be appropriately modified (corrected). On the basis of work presented in  [17], the dissipated energy on resistance in each switching cycle is calculated, and the equivalent resistance value is modified accordingly to match the required energy dissipation. Based on ( 2.18) and ( 2.19), the energy dissipated on eqR  is equal to seqrmssw TRi 2)( . The energy conservation principle implies that, in each switching cycle, the energy dissipated in the averaged circuit has to be equal to its actual value as it would be predicted by the detailed switching circuit. We therefore introduce a modified equivalent resistance, eqR′ , for which the energy conservation principle holds. Thus, considering ( 2.18) and ( 2.19), the following energy balance should hold:   ( ) ( ) seqrmsswseqavesw TRiTRi 22 =′  ( 2.20) which yields the following formula for calculating the modified equivalent resistance,   .341dRR eqeq =′  ( 2.21) Finally, since a new resistance is introduced to the converter circuit, the equation for the duty ratio constraint 2d  needs to be modified as well. In particular, when the switch is closed, the voltage across the inductor is   .3421 ddiRvv Leqgon +−=  ( 2.22) 21  Substituting ( 2.22) into ( 2.4), we obtain    .)34(212112 dddiRvTdiLdLeqgsL−+−=  ( 2.23) Finally, isolating 2d  on one side, an explicit expression for the new duty ratio constraint is derived,   .)342(112 dviRTdLdgLeqs−+=  ( 2.24) Equation ( 2.21) is derived on the basis of DCM. As shown in Figure  2.12, in CCM, the inductor current never goes to zero and the switching interval is divided into two subintervals. Therefore, ( 2.18) and ( 2.19) are no longer valid for CCM, and new relationships need to be re-derived.      0d1Tsd2TsTsiLave iL , A 0d1Tsd2TsTsiLave ipk  isw , A  Figure  2.12 Typical inductor and switch waveform assuming CCM.    According to Figure  2.12, the average current of the switch is    .)()( 1 aveLavesw idi =  ( 2.25)  22  The RMS value of the switch current is    ,)(2)(22/2/ 111dtitdiiddaveLpkrmssw ∫−+=  ( 2.26) which after taking the integration and simplifying manipulations gives   .12)()(221 += pkaveLrmsswiidi  ( 2.27) In (2.27), pki  denotes the current ripple that is shown in Figure  2.12, and is calculated based on the following formula   .)(1  −=LiRvTdi aveLeqgspk  ( 2.28) To observe the energy balance in CCM in the averaged circuit, ( 2.25) and ( 2.27) are inserted into ( 2.20) and the resulting equation is solved for eqR′ , which yields   .)(121 221+=′aveLpkeqeq iidRR  ( 2.29) Depending on the converter mode of operation, either ( 2.21) or ( 2.29) will be the correct resistor value such that the energy conservation principle accurately predicts/captures the energy dissipation in both DCM and CCM, respectively.  2.2.4 Circuit Averaging Case Studies The parameters of the Flyback converter considered in this Section, including parasitics, are summarized in Appendix  B.1. The detailed model has been implemented in Matlab/Simulink using PLECS toolbox, and it is used as a reference for evaluating the accuracy of averaged models derived in previous Subsections. To demonstrate the benefits of the new AVMs, we have implemented the following models: circuit-averaged (CA); the corrected circuit-averaged (CCA); conventional state-space-averaged (SSA); and compensated state-space-averaged (CSSA) models, respectively. The large-signal time-domain and small-signal frequency-domain studies are performed on all models, and the results are compared with reference solutions produced by the detailed switching model.   23  2.2.4.1 Time-Domain Transient An informative time-domain study for analysis of PWM converters should span different conduction modes. In the transient study considered in this Subsection, the duty cycle is kept constant at 381.01 =d . Initially, the converter is assumed to operate in a steady state in DCM under load 2500=loadR . Then, at st 3.0= , a parallel resistor  200=pR  is added to the load of the system. The converter undergoes a large transient and the mode of operation changes to CCM. The corresponding transient responses produced by the considered models are superimposed in Figure  2.13. As can be seen in Figure  2.13, the decrease in load resistance results in decrease of the output voltage. The actual values of the output voltage and inductor current in steady state in DCM (before transient) and in CCM (after transient) as predicted by the considered models are summarized in Table  2.1.  0.3 0.301 0.302 0.303 0.304 0.305012345Time, si L , A-110-100-90-80-70-60v out , VDetailedC AvgCC AvgSS AvgCSS Avg(a)(b)(e)(d)(c)(a),(c),(e)(d)(b)(a)(d)(b)(c),(e) Figure  2.13 Output voltage and inductor current transients during load change as predicted by various models.  24  Table  2.1 Output voltage and inductor current values and errors as predicted by different averaged models in DCM and CCM. Model Output voltage Inductor current  DCM CCM DCM CCM Detailed Model Value -101.87 -66.68 0.46 3.49 CA-AVM Value -102.61 -70.26 0.46 3.68 Error 0.81% 5.36% 0.62% 5.34% CCA-AVM Value -101.35 -66.71 0.45 3.5 Error 0.42% 0.03% 0.61% 0.01% SSA-AVM Value -94.17 -66.69 0.48 3.49 Error 7.48% 0.01% 4.19% 0.02% CSSA-AVM Value -101.79 -66.69 0.46 3.49 Error 0.01% 0.01% 0.18% 0.02%  As it can be seen in Table  2.1, the accuracy of the CCA-AVM is significantly improved as compared to the CA-AVM, which is achieved by modifying the equivalent conduction loss resistance eqR  according to the energy conservation principle. Figure  2.13 also indicates that when value of eqR  is not corrected (CA-AVM), the inductor current takes a longer time to settle down to its steady state value and it oscillates with large amplitude during the transient. But in corrected averaged circuit (CCA-AVM), the value of eqR  is larger (the energy is dissipated faster), and no oscillations are observed. It can also be noted that this model response is somewhat slower (more dissipative) than what is predicted by the detailed model.  The results from conventional state-space averaging (SSA-AVM) method, wherein no correction matrix is added, and that of the corrected state-space averaging (CSSA-AVM), are also shown in Figure  2.13 and Table  2.1. As indicated in Table  2.1, the absence of correction matrix in conventional state-space average-value model causes an obvious error in DCM in predicting state variables average value even in steady state. This source of error is significantly reduced when the state-space model is corrected (CSSA-AVM).  To get more insight into the difference between the corrected and un-corrected circuit-averaged models, their eigenvalues have been extracted in DCM and CCM and are summarized in Table  2.2. As it can be seen in Table  2.2, both models have very similar real eigenvalues in DCM, which implies that the system will not have oscillatory behavior, as is verified in Figure  2.13. However, when the converter is in transient going into CCM, the CA-25  AVM shows some oscillations. As can be seen in Table  2.2, the eigenvalues in CCM are complex, and the imaginary part of complex eigenvalues for the CA-AVM is dominant, which explains the oscillations. At the same time, the CCA-AVM in CCM has very large negative real part of the eigenvalues, which explains its significantly more damped response.  Table  2.2 Eigenvalues of average-value models in DCM and CCM. Models DCM CCM Circuit averaging (CA-AVM) -1.14092e6 -36.143 -1377.08 ±j 4076.81 Corrected circuit avg.( CCA-AVM) -1.14101e6 -36.140 -3386.10 ± j 2835.13 State-space averaging (SSA-AVM) -1.06311e6 -45.90 -3397.00 ± j 2823.0 Corrected state-space avg. (CSSA-AVM) -1.14097e6 -36.141 -3396.99 ± j 2823.0  The converter efficiency is considered as an important parameter, and it is very desirable that the averaged models predict its value with high accuracy. To compare the considered models, it is assumed that the converter operates in steady state and in DCM ( 381.01 =d , Ω= 2500loadR ) and CCM ( 381.01 =d , Ω= 185loadR ). The predicted efficiency as calculated by different models is summarized in Table  2.3. As can be seen in Table  2.3, the circuit averaging (CA-AVM) model over-estimates the efficiency in both operating modes. At the same time, the corrected circuit averaging (CCA-AVM) model predicts the results that are much closer to the reference, especially in DCM. Both state-space averaging models predict the steady state efficiency in CCM quite well. However, in DCM, the conventional state-space averaging model is a bit off.  Table  2.3 Efficiency comparison of average-value models. Efficiency DCM CCM Detailed model (reference) 97.11% 90.20% Circuit averaging (CA-AVM) 98.31% 95.08% Corrected circuit avg.( CCA-AVM) 97.10% 90.27% State-space averaging (SSA-AVM) 97.60% 90.25% Corrected state-space avg. (CSSA-AVM). 97.52% 90.25% 26   It can also be noted in Table  2.3 that corrected circuit averaging (CCA-AVM) model over-estimates the losses in CCM. This model also has higher damping in CCM as compared to the other models based on the transient study in Figure  2.13 and the eigenvalues in Table  2.2. This phenomenon is attributed to the fact that the energy conservation correction  assumed the DCM, and the value of eqR′  becomes over-compensated (too high) when the converter operates in CCM.  2.2.4.2 Frequency-Domain Analysis To verify the small-signal behavior of the considered models, the control-to-output transfer function has been extracted and compared with that of the detailed model. This transfer function can be used for design and implementation of various controllers. There are different methods of obtaining frequency response of a system such as AC sweep, impulse response, small-signal injection, and linearization around an operating point. Figure  2.14 shows extracted transfer function of the averaged models and the detailed system for the operating point defined by 381.01 =d and Ω= 2500loadR using the Matlab Simulink and PLECS toolboxes. As can be seen in Figure  2.14, the averaged models predict the small-signal control-to-output transfer function in DCM with good agreement with the detailed switching model. For higher frequencies, the averaged models become somewhat less accurate because at these frequencies the basic assumptions of averaging are no longer valid. A more detailed view of Figure  2.14 reveals that proposed corrected state-space averaged model and corrected averaged circuit model predict the magnitude and phase with a very good match to the detailed model.  27  -50050100Magnitude, dBDetailedC AvgCC AvgSS AvgCSS Avg100 101 102 103 104 10550100150200Frequency, HzPhase, deg Figure  2.14 Control-to-output transfer function magnitude and phase in DCM as predicted by various models.  To investigate the difference between the circuit-averaged models further, the control-to-output transfer function has been extracted in CCM, and the results are superimposed in Figure  2.15. As can be seen in Figure  2.15, a slight bump in the magnitude curve predicted by the circuit-averaged (CA-AVM) model corresponds to its complex eigenvalue in CCM around 700 Hz frequency, which is consistent with the results shown in Table  2.2 and the transient observed in Figure  2.13. At the same time, the corrected circuit-averaged (CCA-AVM) model is more dissipative which also improves its accuracy. 28  100 101 102 103 104 105-1000100200Frequency, HzPhase, deg-50050100Magnitude, dBDetailedCC AvgC Avg Figure  2.15 Control-to-output transfer function magnitude and phase in CCM as predicted by various models.  2.3 Numerical Average Value Modeling Analytical derivation of an average model that is valid in different operation modes is difficult, especially when parasitics and non-idealities are incorporated into the converter circuit. Recently, a numerical computer-aided method of obtaining the average-value model of power electronic systems has been proposed based on corrected state-space average-value modeling, wherein several parametric functions are obtained using the detailed simulation.  2.3.1 Parametric Average Value Modeling The parametric average-value modeling approach proposed in  [23] and  [24] is based on corrected full-order state-space averaged model  ,3131uBxMAx += ∑∑ ==∆==kkkkkkkk dddtd ( 2.30) 29  Unlike (2.1) where the correction matrix is derived analytically, here ∆M  is a diagonal correction matrix which is computed numerically; kd  is the relative length of subintervals for each topology within the switching interval; kA and kB  are state-space matrices; u  is the input vector; and x  is the vector of averaged state variables  [44]. In CCM, there is no third subinterval and 2,1=k . When the state variables are in steady state condition, their derivatives are equal to zero. Thus, under steady state condition, ( 2.30) becomes equal to zero, yielding   .03131uBxMA += ∑∑ ==∆==kkkkkkkk dd  ( 2.31) Solving ( 2.31) for xM∆  gives    .31131uBAxMp −== ∑∑ ==−==∆kkkkkkkk dd  ( 2.32) Using ( 2.32), the following expression can be utilized to calculate ),( 1 Lid∆M  for each operating point   .)()(),( jjjjxpM =∆  ( 2.33) Expression ( 2.33) gives the diagonal elements of correction matrix ),( 1 Lid∆M , which makes the correction terms of state variables independent of each other. For the considered Flyback converter, the parametric functions have been calculated using the approach described in  [22]. Figure  2.16 and Figure  2.17 show the correction term for inductor current, ( ) 11,1 m=∆M , and capacitor voltage, ( ) 22,2 m=∆M , respectively.  As Figure  2.17 shows, when the converter enters DCM, the correction coefficient 1m  starts increasing to compensate for the deficiency in conventional state-space averaging in predicting the average value of the discontinuous state variables. The value of 2m  shown in Figure  2.17 is always close to one, because the voltage of output capacitor is a continuous state variable and does not require much (if any) correction.  Figure  2.18 illustrates the duty ratio constraint over an extended range of operating conditions. Since calculating vector p is accompanied by inversion of matrix∑ ==31kk kkd A , numerical robustness of the procedure depends on the condition number of this matrix. The 30  condition number of this matrix has been calculated and is depicted in Figure  2.19.  As can be seen in Figure  2.19, the condition number remains relatively small over the range of considered operating conditions, suggesting that the error resulting from the matrix inversion is negligible.  0 0.2 0.4 0.6 0.8102000400011.52d1R, Ohmsm1 CCMDCM Figure  2.16 Correction coefficient m1 for the example second-order Flyback converter.  00.510200040000.9511.051.1d1R, Ohmsm2 Figure  2.17 Correction coefficient m2 for the example second-order Flyback converter.   31  00.51 0 10002000 3000400000.51R, Ohmsd1d 2 Figure  2.18 Duty-ratio constraint d2 for the example second-order Flyback converter.   00.510200040000200400d1R, OhmsCondition number Figure  2.19 Inverted matrix condition number for the example second-order Flyback converter.   The output variables in PAVM are calculated on the basis of the following equation   .3131uDxMCy += ∑∑ ==∆==kkkkkkkk dd  ( 2.34)  After all parametric functions have been calculated numerically, the AVM is implemented according to the block diagram depicted in Figure  2.20. 32   Figure  2.20 Implementation of PAVM.  2.3.2 Numerical Average Value Modeling Case Studies The same second-order Flyback converter with parameters summarized in Appendix  B.1 is considered here. For convenience, the switch ON resistance and transformer primary and secondary resistances are added together and indicated by eqR  in the AVMs. The detailed model of the considered Flyback converter has been implemented in PLECS, and the AVMs, including PAVM and CA-AVM (Examined in section  2.2), have been implemented in Matlab/Simulink.  2.3.2.1 Time-Domain Transient Studies For model verification, a fast change in the MOSFET’s duty cycle is considered in this study. Initially, the converter operates in steady state in DCM defined by 3.01 =d  and Ω= 2500loadR . Then, the duty cycle is increased from 3.0 to 8.0  over a period of 05.0 second. After the transition, the converter operates in CCM with the new duty cycle. Figure  2.21 shows the state variables of the detailed and average models during this transition. In-order to compare the precision of average models, their predicted inductor current and output voltage for the DCM and CCM are summarized in Table  2.4. In addition, the converter efficiency predicted by each model is calculated and summarized in Table  2.5. As can be observed in Figure  2.21, the results of the detailed model imply that when converter duty cycle is changed, the magnitude of both output capacitor voltage and inductor current increases after undergoing some transient. According to plots in Figure  2.21, the state 33  variables predicted by the PAVM is very accurate and completely overlaps with average values from the detailed model. However, voltage predicted by the circuit averaging model (CA-AVM) clearly deviates from the detailed model during transients and in steady state at the end of simulation. Similar discrepancies are observed for the inductor current. In fact, these deficiencies in the circuit averaging model have two reasons. First, the derived averaged circuit in  [27] is valid for DCM and light CCM operating conditions, while in our case study after the increase in duty cycle the converter is working in heavy CCM. Second, the energy conservation principle is not used in this circuit averaging model, and the CA-AVM dissipates less power than that of detailed model.   0.3 0.32 0.34 0.36 0.380246810Time, si L , A-400-300-200-1000 v C , VDet. Sim.C AvgPAVM Figure  2.21 Output capacitor voltage and inductor current transients due to change in the switch duty cycle as predicted by various models.   34  Table  2.4 Output voltage and inductor current as predicted by various models for the two steady state operating points.        Table  2.5 Converter efficiency as predicted by various models for the two steady state operating points. Efficiency DCM CCM Det. Model 97.14% 92.04% CA-AVM 98.17% 94.05% PAVM 97.46% 92.31%   In the next study, we test the PAVM functionality in a closed-loop system with commonly-used PI controller  [45].  Here, a PI controller with parameters 05.0=iK  and 110=pK  is chosen. In-order to prevent error accumulation during the rise time, the output of controller integrator is limited by a saturation block. The controller diagram together with the converter plant is shown in Figure  2.23.  Converter ModelkiskpRef.Output Figure  2.22 Closed-loop system of the considered second-order Flyback converter with PI controller to regulate the output voltage.  In the following closed-loop study, load is initially set to Ω= 220loadR  and converter is assumed to operate in CCM. Then, at 2.0=t sec, the load loadR  is increased to Ω1800 . As Figure  2.23 illustrates, the controller is able to regulate the output voltage at 70− V within a Model Output voltage Inductor current  DCM CCM DCM CCM Det. Model Value -80.23 -332.38 0.33 3.2 CA-AVM Value -80.81 -338.72 0.33 3.25 Error 0.73% 1.91% 0.59% 1.7% PAVM Value -80.23 -332.38 0.33 3.2 Error 0% 0% 0% 0% 35  short settling time and relatively small overshoot. Also, as can be seen in Figure  2.23, the results predicted by the PAVM are in very good agreement with the detailed model.   -75-70-65v C , VDet. Sim.PAVM01234i L , A0.19 0.195 0.2 0.205 0.21 0.215 0.22Time, s0.20.250.30.350.4Control signal Figure  2.23 Closed-loop second-order Flyback converter with PI controller response to a load change.   To give the reader an idea how the simulation speed of the AVM may differ from that of the original detailed switching model, Table  2.6 compares the simulation speed of models in the last study. In order to make the comparison reasonable, the same solver Ode23s with relative/absolute tolerances set to 1e-3 were used for each model. As can be seen in Table 36   2.6, the detailed model handles all switching events and requires very large number of time steps (90,8313) and takes appropriately long time (70.43s) to complete the study. At the same time, the PAVM is not switching, which allows it to use larger time steps (taking only 2,397 time steps) and complete the same study much faster (0.35s). This very significant increase in simulation speed while preserving the slower dynamics at the control and input-output terminals of the converter suggest that such AVMs can be used very effectively for the system-level studies where the focus is on the input-output interactions and controller design, and where the details of the switching waveforms can therefore be neglected.   Table  2.6 Simulation speed comparison of the detailed and average models in closed-loop transient study. Model # of time steps Elapsed time, s Det. Sim. 908,313 70.43 PAVM 2,397 0.35 37  2.3.2.2 Frequency-Domain Analysis To demonstrate the performance of the AVMs in frequency-domain, we consider the control-to-output voltage transfer function. The considered operating point of interest in defined by the steady state operating condition with 381.01 =d  and 2500=loadR , with corresponds to the DCM.  The corresponding transfer functions have been extracted with Matlab/Simulink and PLECS toolboxes, and the results are shown in Figure  2.24. As Figure  2.24 shows, the derived CA-AVM and PAVM are in a good agreement with the detailed model at frequencies up to around 100 KHz. At frequencies above 100 KHz,  the AVMs start deviating from the detailed model a little bit. This is because the primary assumptions used for deriving the average models are valid below the switching frequency.  -50050100Magnitude, dB DetailedC AvgPAVM100 101 102 103 104 105050100150200Frequency, HzPhase, deg Figure  2.24 Control to output transfer function magnitude and phase.     38  Chapter 3 : Full-order Flyback Converter  Snubbers bring many benefits to the hardware circuit (Figure  1.2) as they enable semiconductor components to switch in a softer mode and reduce the voltage stresses. At the same time, snubbers also increase power loss of the circuit and make modeling of power electronic systems more complicated. The full-order Flyback converter would typically include two snubbers: one snubber is used on the input side to protect MOSFET; and another one is used on the output side of converter to absorb spikes resulting from the secondary side leakage inductance.  In this work, it is shown that due to the presence of RC snubbers of in Flyback converter, the previously established SSA-AVM method is not capable of presenting an accurate AVM. Therefore, a new PAVM is developed which corrects the average values of state variables in a more general different way. The methodology is based on extending the SSA formulation to include the corrections are of the state variables as well as the state-space matrices (their specific entries corresponding to snubbers).      3.1 Effect of Input Snubber in SSA The input RC snubber absorbs energy stored in primary side leakage inductance when the MOSFET opens. When the switch opens at the end of the ON state, the energy stored in the primary leakage inductance results in a voltage surge across the switch. The RC snubber protects semiconductor device by suppressing voltage transients across it. The leakage inductor energy is absorbed and then released back by capacitor of the snubber, which causes resonance and oscillation in the waveforms of the circuit at switching moments. This energy is gradually dissipated in the large resistor of the RC snubber. As shown in Figure  3.2, based on volt-second balance (for inductors) and charge balance (for capacitors) average value of Cssv in steady state can be calculated as follows    LptptgCss iRvv −= . ( 3.1) 39  Rswvg(t)RptRCssInputOutputvCssiLptiLmvsw Figure  3.1 Averaged detailed circuit of input stage of Flyback converter in steady state.   According to Figure  3.2, in the first topology )1( =k , when the switch is closed, the input voltage source transfers energy to the magnetizing and leakage inductors as the current increases. During this subinterval, the snubber does not play a significant role, and the capacitor energy is dissipated on the resistors of snubber and switch. During this subinterval )1( =k , the capacitor current is      .1LptssswswCssssswCssssCss iRRRvRRdtvdCi+++−== ( 3.2) When the switch opens, second topology )2( =k , the energy in the magnetizing branch is transmitted into the secondary side to feed the load. In this subinterval, the snubber capacitor becomes active and is charged with energy from the current in leakage inductor. Finally when the current of magnetizing branch goes to zero, the diode stops conducting, and converter enters the third topology )3( =k . According to Figure  3.2, during these subintervals )3,2( =k , the capacitor current is    .LptCssssCss idtvdCi == ( 3.3)  40  LmRswvg(t)LptCssRptRCssSwitch ON  (K = 1)Switch OFF  (K = 2, 3)InputOutputLmRswvg(t)LptCssRptRCssInputOutputiLpt(t)iLpt(t)iCss(t)iCss(t) Figure  3.2 Input stage of Flyback converter depicting operation of snubber.  Based on conventional SSA-AVM, averaging the state-space equations (3.2) and (3.3) over the entire switching cycle gives    ( ) LptLptssswswCssssswCssssCss idiRRRvRRddtvdCi 11 11−++++−== ( 3.4) In steady state, the derivative term in (3.4) goes to zero, and Cssv  is obtained as    LptssswCss idRdRv11)1( −+= ( 3.5) Equation (3.5) shows that if Lpti is calculated correctly by the SSA-AVM, then Cssv would be calculated correctly too by the AVM and does not need any additional correction. Therefore, the correction term for Cssv  should be close to one. This will be proven with numerical graphs in the following sections.   41  3.2 Effect of Output Snubber in SSA-AVM The output snubber protects the diode from voltage and current spikes. To investigate the effect of output snubber in SSA-AVM, we examine and compare the output stage of the Flyback converter with and without the snubber branch.  3.2.1 Output Stage without Snubber (4th Order Flyback Converter) When the converter circuit is operating in steady state, the average current going into the output capacitor during a switching cycle is zero, and the average voltage across the inductors is zero too. Therefore, the averaged detailed circuit of output stage of Flyback converter could be presented as shown in Figure 3.3.  RloadRcRstInputOutputvCiLstvd Figure  3.3 Averaged detailed circuit of the output stage Flyback converter without the snubber in steady state.  According to Figure  3.3, the average value of capacitor voltage in steady state could be calculated as  .LstloadC iRv =  ( 3.6) To calculate the predicted average value of the output capacitor voltage by conventional SSA-AVM, all topologies occurring during a switching cycle have to be considered.  42  RloadCRcLstRstDiode ON    (K = 2)Diode OFF   (K=1, 3)InputOutputiLst(t)iC(t)RloadCRcLstRstInputOutputiLst(t)iC(t) Figure  3.4 Output stage topologies of Flyback converter without output snubber.  According to Figure  3.4, when the diode is in ON )2( =k , the charging current of output capacitor is  .1LstCCloadCC ivRRdtvdCi ++−==  ( 3.7) In the first and third topology )3,1( =k , the current of secondary side leakage inductor becomes zero )3( =k  and increases to a positive value )1( =k . During these subintervals, the diode stops conducting. During this subinterval, the output capacitor discharges and supplies the load with current  .1CcloadCC vRRdtvdCi+−==  ( 3.8) Based on SSA-AVM, using (3.7) and (3.8), we take the average of the output capacitor current over the entire switching cycle   .12 LstCCloadCC idvRRdtvdCi ++−==  ( 3.9) The derivative term in (3.8) goes to zero in steady state. Solving the resulting equation for Cv  in steady state gives,  ( ) .2 LstCloadC iRRdv +−=  ( 3.10) 43  Equations (3.6) and (3.10) can be used to calculate the correction term for Cv  analytically. In particular, (3.10) shows that for fourth-order Flyback converter, if Lsti  is calculated correctly, then Cv  is accurate too, and its correspondent correction term will be close to one. This is observation will be proven with numerical graphs in the later sections (Section  3.4).  3.2.2 Output Stage with Snubber (5th Order Flyback Converter) Figure  3.5 shows the averaged circuit of the output stage of full-order Flyback converter in steady state. According to Figure  3.5, and on the basis of charge balance for the output capacitor, the average output capacitor voltage is  .LstloadC iRv =  ( 3.11) Similarly, the average value of the snubber capacitor voltage is  .LststCds iRv −=  ( 3.12) RloadRciLstRstRCdsInputOutputvCvCdsvd Figure  3.5 Averaged detailed circuit of output stage of full-order Flyback converter in steady state.  According to Figure  3.6, when the diode is ON )2( =k , the output snubber capacitor charges with current coming from transformer. In this subinterval, the snubber branch is parallel with the main output capacitor and it absorbs part of the transmitted energy to the secondary side. The output capacitor current in this subinterval is  [ ] ( )[ ] .)||()||(||)||(1LstCloadCdsloadCdsCdsloadCCdsCloadloadCloadCdsCCCiRRRRRvRRRRRRvRRRdtvdCi++++++−== ( 3.13) Correspondingly, the snubber capacitor current is 44   [ ] ( )[ ].)||()||()||(1||LstCdsloadCloadCCdsloadCCdsCloadCdsCCdsloadloadCdsdsCdsiRRRRRvRRRvRRRRRRdtvdCi+++−++== ( 3.14)  The diode turns OFF in the first and third subintervals )3,1( =k . In the third subinterval, the current stored in secondary side leakage inductor results into a voltage spike that is partly absorbed by the output snubber capacitor. In this subinterval, the output capacitor current is  ,||1CloadcCC vRRdtvdCi −== ( 3.15) and the snubber capacitor current is  .LstCdsdsCds idtvdCi == ( 3.16)  RloadCRcLstRstCdsRCdsDiode ON   (K = 2)Diode OFF   (K=1, 3)InputOutputiLst(t)iC(t)iCds(t)RloadCRcLstRstCdsRCdsInputOutputiLst(t)iC(t)iCds(t) Figure  3.6 Output side of full-order Flyback converter depicting the snubber operation.  Based on SSA-AVM, we take the averages of the output capacitor current and the snubber capacitor voltage over the entire switching cycle, respectively, which yields 45   [ ] ( )[ ] ,)||()||(||)||(||)1(2222LstCloadCdsloadCdsCdsloadCCdsCloadloadcloadCdsCloadCCCiRRRRRdvRRRRRRdvRRRdRRddtvdCi+++++++−−== ( 3.17) and  [ ] ( )[ ].)1()||()||()||(||2222LstCdsloadCloadCCdsloadCCdsCloadCdsCCdsloadloadCdsdsCdsidRRRRRdvRRRdvRRRRRRddtvdCi−++++−++== ( 3.18) Note that in steady state, (3.17) and (3.18) are equal to zero. Solving (3.15) for Cv  and (3.16) for Cdsv  gives the respective average values as predicted by SSA-AVM in steady state  [ ] ( )[ ],)||(||)1()||()||(||12222−++−++++=loadCdsCloadCLstCloadCdsloadCdsCdsloadcCdsCloadloadCRRRdRRdiRRRRRdvRRRRRRdv ( 3.19) and  ( )[ ] ( )[ ]( ).)1()||()||()||(||)||(22222LstCdsloadCloadCloadCCdsCloadCdsCCdsloadloadloadCCdsCdsidRRRRRddRRRvRRRRRRddRRRv−+++−+++−= ( 3.20) As shown in (3.19) and (3.20), the Cv  and Cdsv depend on two state variables. Specifically, the variable Cv  is a function of Lsti  and Cdsv ; and Cdsv is a function of Lsti  and Cv . Moreover, Cv  and Cdsv  are dependent on each other. This means that for accurate AVM, both of these state variables will have to be corrected by appropriate functions. Therefore, in the case of full-order Flyback converter, the correction terms for Cv  and Cdsv are not expected to be close to one. This will be proven numerically in Section  3.4.     46  3.3 Generalized Numerical SSA-AVM In conventional SSA-AVM  [6], state equation has the following form  ,3131uBxAx += ∑∑ ====kkkkkkkk dddtd ( 3.21) which is a weighted sum of state equations corresponding to the topological instances of the detailed converter circuit. In  [9], a correction matrix M  is added to ( 3.21) to correct for the errors in DCM. Therefore, a new formula was proposed which is given in ( 2.1). However, for a full-order Flyback converter, even the continuous state variables such as output capacitor voltage, have to be corrected due to the presence of snubbers. In the extended methodology, we consider a possibility for correction of all state variables in the circuit. A vector of state variables of the circuit has the following general form    .]...[ 21 Tnxxx=x  ( 3.22) In the case of full-order Flyback converter, this vector is   [ ] .TLstLptCdsCssC iivvv=x  ( 3.23) For second/fourth-order Flyback converters, fewer state variables will be included as appropriate for each case. The generalized diagonal correction matrix has the following form   ( )....,, 21 nmmmdiag=M  ( 3.24) Next, we define the state vectors of the conventional SSA-AVM as SSAx , and that of the generalized corrected SSA-AVM as SSAC−x , respectively. Thereafter, the correction matrix is used to relate the state vectors of the conventional and the generalized corrected SSA as   .SSASSAC xMx =−  ( 3.25) The state model ( 3.21) is corrected by inserting SSAC−x from ( 3.25), which yields  ,3131uBxMAxM += ∑∑ ==−==−kkkkSSACkkkkSSAC dddtd ( 3.26) which after multiplying both sides by 1−M  gives   .3131uBMxMAMx 11 += ∑∑ ==−−==−−kkkkSSACkkkkSSAC dddtd ( 3.27) 47  Therefore, ( 3.27) would provide the correct average values of all state variables if matrix M  is computed appropriately. It is also noted that the AVMs based on ( 2.1) and ( 3.27) are different as these models will have different equivalent state space matrices.   3.4 Construction and Implementation of the Generalized PAVM To make detailed models more representative of hardware, the parasitics and non-idealities are included as part of model. Here, for the fourth and full-order Flyback converter, the switch and capacitor parasitics, the non-ideal transformer, and snubbers are added to the converter circuit. ASMG  [47] and PLECS are two software tools that could be used for building the detailed model of the Flyback converter. For implementation of the proposed AVM, the system matrices ,,, kkk CBA and kD  relating to different topologies of the Flyback converter are extracted from detailed model. This could be done readily using software facilities such as state-space functions of PLECS.  The detailed model of the converter is ran in a wide range of operating conditions to numerically calculate the parametric functions. By changing duty cycle and load resistor of the converter, the model is run under different operating conditions until the circuit reaches steady state. In steady state, the dynamic average value of state variables is obtained as  .1 ∫+=sTttsdtTxx ( 3.28) The duty ratio constraint 2d  is calculated on the basis of the period that the diode is conducting. The averaged state variables and length of switching subintervals are calculated and saved for construction of the appropriate correction terms.    The correction terms are then calculated using the following procedure. When the model is in steady state, from ( 3.27) we have    ,03131uBMxMAM 11 += ∑∑ ==−−==−kkkkSSACkkkk dd  ( 3.29) Multiplying both sides of ( 3.29) byM gives   .03131uBxMA += ∑∑ ==−==kkkkSSACkkkk dd  ( 3.30) 48  Matrix M and its calculation are also described in  [22]. For consistency with the prior work and for computational convenience, a diagonal correction matrix, namely ∆M is assumed here. This is done by solving ( 3.30) for xM and defining vector p  as follows   .31131∆ uBAxMp −== ∑∑ ==−==kkkkkkkk dd  ( 3.31) Thereafter, the entries of the diagonal of correction matrix ∆M are calculated as following:   .)()(),(∆ jjjjxpM =  ( 3.32) It can be seen from ( 3.31) that computing vector p  requires matrix inversion. To verify that the procedure is not ill-conditioned, it is possible to calculate the condition number of the matrix ∑==31kkkkd A  as has been done in  [46]. The result is plotted in Figure  3.7. As illustrated in Figure  3.7, the condition number of this matrix increases for some operating conditions but remains sufficiently low to cause a numerical problem for the conventional double-precision arithmetic in Matlab. To show that the resulting errors will be sufficiently small, a residual equation is defined as   ,ˆˆ3131uBpAr += ∑∑ ====kkkkkkkk dd  ( 3.33) wherein pˆ  is an approximate solution obtained by numerical methods. Due to the round-off errors, rˆ  is always nonzero. If p  is an exact solution, then   ,3131uBpA −= ∑∑ ====kkkkkkkk dd  ( 3.34) and   .ˆˆ131rApp−===− ∑kkkkd  ( 3.35) Choosing a vector norm, an upper bound can be obtained for the absolute error in pˆ .   .ˆˆˆ131131rArApp ∗≤=−−==−==∑∑ kkkkkkkk dd  ( 3.36) 49  It is understood from ( 3.34) that   ,3131uBpA ≥∗ ∑∑ ====kkkkkkkk dd  ( 3.37) In other words,   .13131uBAp ≤∑∑====kkkkkkkkdd ( 3.38) Combining ( 3.36) and ( 3.38), gives a bound for the relative error in pˆ  as   .ˆˆ3131131uBArAppp∗∗≤−∑∑∑====−==kkkkkkkkkkkkddd ( 3.39) As it is seen in ( 3.39), part of the inequality on right hand side is    .31131∗= ∑∑ ==−==kkkkkkkk dd AAκ  ( 3.40) which is the condition number plotted in Figure  3.7.  To see how large the relative error could be, ( 3.39) is calculated for an example operating point defined by 2900=loadR  and 8.01 =d . This operating point is close the worst case, and the calculated maximum relative error for this case is 1.64e-5. Such errors are considered to be sufficiently small and acceptably for the purpose of this thesis.      0.2 0.40.6 0.810100020003000051015x 107d1condition numberRload, Ohm Figure  3.7 Inverted matrix condition number. 50   The correction functions for the fourth-order Flyback converter (without the output snubber) have also been calculated using the same general procedure. The functions 321 ,, mmm  and 4m  corresponding to the voltage of output capacitor, the voltage of input (switch) snubber capacitor, the current of primary side leakage inductor, and current of secondary side leakage inductor, respectively, are plotted in Figure  3.8 through Figure  3.11.  00.5101000200030000.511.5d1m1Rload, Ohm Figure  3.8 Correction coefficient m1 for voltage of output capacitor Cv .  00.51500100015002000250030000.511.5d1m2Rload, Ohm Figure  3.9 Correction coefficient m2 for voltage of primary snubber capacitor Cssv .   51  0.20.40.60.8101000200030000510d1m3Rload, Ohm Figure  3.10 Correction coefficient m3 for primary side leakage inductor current Lpti .  0.2 0.40.6 0.8101000200030000510d1Rload, Ohmm4 Figure  3.11 coefficient m4 for secondary side leakage inductor current Lsti .  The calculated duty ratio constraint for full-order Flyback converter is plotted in Figure  3.12 with respect to 1d  and loadR . The diagonal entries of ∆M are calculated according ( 3.32) using the element-wise operations. The resulting correction functions 4321 ,,, mmmm  and 5m  corresponding to the voltage of output capacitor, the voltage of first (transistor) capacitor snubber, the voltage of second (diode) capacitor snubber, the current of primary side leakage inductor, and the current of secondary side leakage inductor, respectively. These numerically calculated parametric functions are plotted in Figure 3.12 through Figure  3.13.  52  00.51500 1000 1500 2000 2500 300000.20.40.60.8d1d 2Rload, Ohm Figure  3.12 Calculated function of duty-ratio constraint d2.  0.2 0.40.6 0.8101000200030000.040.020d1R, Ohmm 1 Figure  3.13 Correction coefficient m1 for voltage of output capacitor Cv .   0.2 0.40.6 0.8101000200030000.511.5d1m2Rload, Ohm Figure  3.14 Correction coefficient m2 for voltage of primary snubber capacitor Cssv .   53   0.2 0.40.6 0.8101000200030000.040.020d1m3Rload, Ohm Figure  3.15 Correction coefficient m3 for voltage of output snubber capacitor Cdsv .  0.2 0.40.6 0.8101000200030000510d1m4Rload, Ohm Figure  3.16 Correction coefficient m4 for primary side leakage inductor current Lpti .   0.2 0.40.6 0.8101000200030000.040.020d1m5Rload, Ohm Figure  3.17 Correction coefficient m5 for secondary side leakage inductor current Lsti .  54   The numerically calculated parametric functions, including correction terms and duty ratio constraint, are stored in lookup tables and is used in the AVM in the form of non-linear functions, wherein interpolation/extrapolation is automatically implemented. The output of these functions depend on the input voltage gv , the duty cycle 1d , and the average value of state variables in vector x . Instead of using complex, multi-input and multi-dimensional lookup tables, it is preferred to define a dynamic impedance of the converter switching cell and use it for specifying the converter operating condition. For the work presented in this thesis, two dynamic impedances are defined as follows:   ,LstCdsCdiode ivvz−= ( 3.41)  .LptCmid ivz = ( 3.42) The impedance diodez  defined in ( 3.41) combines three state variables (two of which are fast state variables). This impedance is used as input argument for the 1m  lookup table that performs the output capacitor voltage correction. All other correction functions are implemented using the input argument impedance midz  defined in ( 3.41)-( 3.42). Another advantage of using the dynamic impedances - ( 3.42) is that they define the converter operating point independently of the input voltage. For example, when the input voltage is increased while 1d  and loadR  are kept constant, all the state variables will increase proportionally and the value of dynamic impedance that specifies the operating point of the converter will not change.  Once the state-space matrices and parametric functions are available, the final extended PAVM is implemented according to the block diagram shown in Figure  3.18. In each time step,  the state-variable-dependent matrix M  (and its inverse) containing correction coefficients is calculated and appropriately multiplied with the previously extracted system matrices ,,, kkk CBA and kD . The resulting state space model is nonlinear but continuous (has no switching), and can be readily implemented in any state-variable-based simulation environment.  55  M (d1 , zdiode , zmid )d2 (d1 , zdiode , zmid )d3=1- d1 - d2Ak , Bk , Ck , Dkx = M-1AT M x + M-1BT uy = CT  x + DT  uAT  = Σ (dk  Ak), k = 13CT  = Σ (dk  Ck),k = 13BT  = Σ (dk  Bk)k = 13DT  = Σ (dk  Dk)k = 13zdiode (d1 , x )zmid (d1 , x )d2 , d3d1xyuMz Figure  3.18 Block diagram depicting implementation of proposed extended PAVM.  3.5 Eigenvalue Analysis Based on ( 3.27), it may appear that the new formulation uses MAM T1− , which is MAM T1− is a similarity transformation  [46] that does not change the eigenvalues of the matrix = ∑==31kkkkT d AA , which is the same matrix used in conventional (uncorrected SSA) in ( 2.1). To verify the effect of correction in ( 3.27), we assume a CCM operating point defined by 381.01 =d  and Ω= 717loadR . The resulting correction coefficients (the diagonal entries of M) and the eigenvalues of various matrices are summarized in Table 3.1. As is shown in Table 3.1, the matrix product MAT  contains some eigenvalues with positive real part (which is a result of negative entries present in M). Therefore, if the PAVM is formed using conventional approach, the resulting model will also contain positive eigenvalues and be unstable. At the same time, if the PAVM is formed using the proposed generalized approach ( 3.27), the matrix product MAM T1−  as well as the final resulting PAVM will have eigenvalues with negative real part. It is also important to point out that the eigenvalues of MAM T1− are different from those computed by numerical linearization of the corresponding PAVM. This is due to the fact that the matrix M is state-variable-dependent, which affects the model’s eigenvalues and makes that different from those of MAM T1− and = ∑==31kkkkT d AA  (which would obviously be incorrect due to lack of correction in DCM).   56   Table  3.1 Eigenvalues of AVMs. Elements of correction matrix M  -0.01; 0.99; -0.01; 2.89; -0.01 Eigenvalues of MAT  matrix (-8.15±j1.03)e8; 4.53e5; 1.88e3; 6.3 Eigenvalues of PAVM -7.16e7; -1.36e7; 4.71e5; 14.86; 1.02e3 Eigenvalues of MAM T1− matrix (-5.86±j5.8)e7; -3.31e7; -4.29e2; -1.06e5 Eigenvalues of corrected PAVM -7.8e7; -2.63e7; -1.65e7; -7.54e4; -310.56    3.6 Model Validation with Respect to Hardware For hardware validation, the prototype converter circuit whose parameters as summarized  Appendix C, is considered. The converter is assumed to operate in nominal operating point when supplied from 20 Volts dc input, with 0.381 duty cycle, and a resistive load of 717 Ohm on the output side. The voltages at various points of the converter circuit were measured and recorded using a 500 MHz digital oscilloscope. The measured and simulated results are shown in Figure  3.19, which includes the converter output voltage, the transformer secondary side voltage, and the voltage across primary side snubber branch. As can be seen in Figure  3.19, the waveforms from hardware are somewhat noisy and include the switching spikes and high frequency ringing of the voltages. The simulated results appear to be in very good agreement with the measurements, although some difference in the measured and predicted spikes can be attributed to the absence of the diode the reverse recovery current in the detailed simulation. The values of the input current, output voltage, and the resulting converter efficiency are also compared in Table 3.2, which demonstrates that the detailed model captures the necessary major sources of losses and predicts the terminal characteristics of the actual experimental converter circuit with very good accuracy. Therefore, the detailed model is assumed to be sufficiently accurate and it is used in the following sections for evaluating performance of the subject AVMs.     57  Table  3.2 Hardware prototype and detailed model comparison in terms of input current, output voltage and efficiency.  Input current Output voltage Efficiency Hardware 0.432 -72.67 85.33% Detailed Model 0.42 -72.63 87.52%   -72.9-72.8-72.7-72.6-72.5-72.4v out , VHardware measurementDetailed model-1000100200v  Trans.s , V0 0.5 1 1.5x 105-10010203040Time, sv Snub.p , V300Output voltageTransformer secondary side voltageSwitch snubber voltage Figure  3.19 Measured and detailed model waveforms for the considered operating point.  58   3.7 Precision Evaluation in Steady State To evaluate precision of the new generalized PAVM, this model is compared against the detailed model in DCM )2500,381.0( 1 Ω== loadRd  and in CCM )717,381.0( 1 Ω== loadRd  operating points. The average-values of all state variables, including the voltage of snubber capacitors, obtained from the corrected PAVM and the detailed model is summarized in Table  3.3. As Table  3.3 demonstrates, the new PAVM is capable of predicting average-value of circuit variables very accurately, with a very small relative error.  Next, the precision of the new generalized PAVM in terms of predicting the converter efficiency is considered. For consistency, the same DCM and CCM operating points are considered here. The calculated results are summarized in Table  3.4. As can be observed in Table  3.4, the new model works very well in both modes DCM and CCM. Comparing the results in Table  3.4 corresponding to the 4th and full-order converter circuit, one can also observe that the converter efficiency reduces when the output snubber is used (due to additional energy dissipation).                59  Table  3.3 Accuracy precision of the proposed PAVM in predicting steady state variables. Variable 4th-order Flyback converter Full-order Flyback converter DCM CCM DCM CCM Output capacitor voltage Cv (V) Correct Value -102 -70.87 -85.78 -72.63 Predicted Value -102.1 -70.87 -85.78 -72.66 Error 0.10% 0% 0% 0.04% Input snubber capacitor voltage Cssv (V) Correct Value 19.96 19.93 19.97 19.92 Predicted Value 19.96 19.93 19.97 19.92 Error 0% 0% 0% 0% Output snubber capacitor voltage Cdsv (mV) Correct Value --- --- 47.1 136.8 Predicted Value 47.1 137.2 Error 0.08% 0.29% Transformer primary side current Lpti (mA) Correct Value 220.6 370.8 167.4 420.1 Predicted Value 220.8 370.9 167.4 417.9 Error 0.09% 0.03% 0% 0.53% Transformer secondary side current Lsti (mA) Correct Value -40.8 -98.8 -34.31 -101.3 Predicted Value -40.82 -98.8 -34.32 -101.3 Error 0.05% 0% 0.03% 0%   Table  3.4 Accuracy precision of the proposed PAVM in predicting converter efficiency. Model 4th-order Flyback converter Full-order Flyback converter DCM CCM DCM CCM Detailed model 94.31% 94.41% 87.88% 87.52% Generalized/Corrected PAVM 94.31% 94.41% 87.89% 88.05%   3.8 Case Studies The detailed model of full-order Flyback converter including all major parasitics is depicted in Figure  1.2. The hardware prototype utilized for validating the detailed model, its parameters and circuit diagram are shown in  Appendix B and  Appendix C, respectively. First, in Section  3.6 we demonstrated that the results of the base line detailed model is in very 60  good agreement with the corresponding experimental measurements from the laboratory converter prototype. In the following sections, the proposed PAVM is further evaluated in time-domain transients as well as in frequency-domain.  3.8.1 Performance of Proposed PAVM in Time-Domain Transients The new PAVM has been implemented according to the diagram shown in Figure  3.18 in Matlab/Simulink environment using toolboxes such as ASMG and PLECS. In order to investigate the difference between the previously established PAVM and the corrected/generalized PAVM, the fourth-order Flyback converter is considered first. The converted is assumed to initially operate in steady state DCM under load Ω= 2500loadR  and duty cycle 381.01 =d . While the converter duty cycle is kept constant, another Ω2500  resistor is added in parallel to the load. The converter transient response as predicted by the PAVMs is shown in Figure  3.20.  As it is seen in Figure  3.20, the predicted steady state values (before and after the transient) are same for both models, but the transient is predicted somewhat differently with the uncorrected PAVM showing a delayed response. This is because formulas that the AVMs are based on (( 2.1) for uncorrected PAVM and ( 3.27) for corrected PAVM) are equivalent with ( 3.30) in steady state, and therefore both AVMs predict same averaged-values.  Next, the same study is performed using the full-order Flyback converter model. As discussed in Section  3.5, the uncorrected PAVM is not able to operate due to positive eigenvalues, and is therefore not used in this study. The simulations result obtained by the generalized/corrected PAVM are shown in Figure  3.21. As it is observed in Figure  3.21, the new PAVM very accurately predicts the steady state values as well as the entire transient response of the system.   61  -110-100-90-80-70v C , VDetailedCorrected PAVM0.3 0.31 0.32 0.33 0.34 0.35-0.25-0.2-0.15-0.1-0.0500.05Time, si Lst , A-0.500.511.5i Lpt , A010203040v Css , VPAVM Figure  3.20 Transients of state variables of fourth-order Flyback converter as predicted by uncorrected PAVM and the proposed corrected PAVM.   62  -90-85-80-75-70v C , VDetailedCorrected PAVM010203040v Css , V-100-50050100150v Cds , V-0.500.511.5i Lpt , A0.29 0.295 0.3 0.305 0.31 0.315 0.32 0.325 0.33-0.2-0.15-0.1-0.0500.05Time, si Lst , A Figure  3.21 Transients of state variables of full-order Flyback converter due to load change. 63   Another time-domain study is considered using the full-order Flyback converter model. In the study presented here, the converter undergoes an increase in the duty cycle which causes a change in the operating mode. The converter is assumed to initially operate in steady state under load Ω= 717loadR  and duty cycle 5.01 =d . Then, at 3.0=t s, the control duty cycle is decreased from 0.5 to 0.2 over a period of 0.05s. The resulting transient responses are shown in Figure  3.22. As it can be observed in Figure  3.22, the converter initially operates in CCM, but after the transients and duty cycle change, the operating mode changes to DCM. With the decrease in duty cycle, the transistor on-state becomes shorter in each switching cycle, and the output voltage amplitude decreases as well. Therefore, the primary side leakage inductor current (input current) has smaller oscillation amplitude, which in turn results in decrease of the average value of the input current. In consequence, less voltage is delivered to the load. This is verified by the output capacitor voltage in Figure  3.22.          64  -120-100-80-60-40-20v C , VDetailedCorrected PAVM-200-1000100200v Cds , V-200204060v Css , V-20246i Lpt , A0.3 0.32 0.34 0.36 0.38 0.4-1-0.500.51Time, si Lst , A0.28 Figure  3.22 Circuit state variables transients due to intense increase of duty cycle. 65   3.8.2 Performance of Proposed PAVM in Frequency-Domain In the following study, the converter is assumed to operate in DCM operating point defined by 381.01 =d  and Ω= 2500loadR . The control-to-output transfer function is considered, and the frequency response is calculated by injecting a small amplitude sinusoidal signals to the control duty cycle of detailed model and corrected PAVM. The result of this control-to-output transfer function magnitude and phase are shown in Figure  3.23. As it can be seen in Figure  3.23, the proposed PAVM predicts the small-signal transfer function with a good agreement with the detailed model. At higher frequencies, the results become less accurate due to interaction of the injected sinusoidal wave with switching  [9],  [49].  -50050Magnitude , dB DetailedCorrected PAVM101 102 103 104 105-50050100150Frequency, HzPhase , deg Figure  3.23 Control-to-output transfer function magnitude and phase in DCM as predicted by proposed PAVM and detailed model. 66   Chapter 4 : Summary of Research and Future Work  4.1 Second-order Flyback Converter This thesis presented three new average-value models (CA-AVM, SSA-AVM, and PAVM) for second-order Flyback converter, which achieves Objective 1 and Objective 2. The conventional state-space averaging method was applied to the same Flyback converter to set the stage for the new models. A corrected state-space averaged model is proposed, where the basic losses are included into the state matrices as well as the new expression for the duty ratio constrain. Next, a new circuit-averaged model is proposed, where the equivalent conduction loss resistance is modified to observe the energy conservation principle in both DCM and CCM. Then, an improved numerical average-value model of second-order Flyback converter with basic conduction losses and parasitics is presented. The presented numerical SSA method is straightforward and does not have usual complexity of analytical methods. All new average-value models have been demonstrated and compared with the detailed simulation and alternative/existing state-of-the-art models in DCM and CCM. The proposed AVMs are shown to be more accurate with respect to the detailed model in both time- and frequency-domains.   4.2 Flyback Converter with Snubbers In Chapter 3, which addresses Objective 3 of this thesis, the previously established SSA approach is investigated and shown not able to result in an accurate AVM for full-order Flyback converter with both input and output snubbers. Therefore, the SSA formulation is re-derived assuming a more general correction of state variables and the state-space matrices using one diagonal correction matrix. Based on the generalized corrected SSA formulation, the new PAVM is constructed using similar numerical procedures and detailed simulations as was used for the previously established PAVM approach. The proposed PAVM has also been verified against the detailed model and the experimental converter prototype.  The proposed PAVM is easy to implement in many simulation programs such as Matlab/Simulink and related toolboxes such as PLECS, ASMG, and SimPowerSystems. The new PAVM for the Flyback converter is shown to be very accurate in large-signal transients covering both DCM 67  and CCM with transformer isolation, basic parasitics, and snubbers, which to the best of our knowledge has not been achieved in the prior literature.   4.3 Future Work It is envisioned that the extended/generalized PAVM methodology can also be applied to other converters with more complicated topologies. In particular, the full-bridge bidirectional DC/DC converter is another example of a converter that is very commonly used in power supply applications, but for which there are only approximate/idealized average models available in the literature due to the overwhelming complications that arise in conventional analytical approaches when considering parasitics such as conduction losses and transformer leakage inductances. Converters with resonant switching may also be very interesting to consider for the PAVM since the operation of resonant circuits may have a similar effect of the modeling as the snubber circuits. Also, more research could be conducted to develop the proposed methodology for the multiple-input multiple-output converters. Another very hot research direction is to apply the PAVM to the multi-level and modular converters that are becoming particularly attractive for high power utility and HVDC applications. Presently, some of these topics are under investigation other graduate students of the UBC’s Electrical Power and Energy Systems research group.   68   Bibliography  [1] P. Chrin and C. Bunlaksananusorn, “Large-Signal Average Modeling and Simulation of DC-DC Converters with SIMULINK,” Power Conversion Conference, Nagoya, April 2007, pp. 27–32. [2] M. 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Hagglund, “New tuning methods for PID controllers”, Proceedings of the 3rd European Control Conference, 1995. [46] U. M. Ascher and C. Greif, A first course in numerical methods, SIAM, Philadelphia, 2011. [47] Automated State Model Generator (ASMG), PC Krause and Associates Inc., 2002. [Online]. Available: www.pcka.com. [48] W. Gautchi, Numerical Analysis: An Introduction, Birkhauser Publisher, Boston, 1997. [49] J. Sun, D. M. Mitchell, and D. E. Jenkins “Delay effects in averaged modeling of PWM converters,” Power Electronics Specialists Conference(PESC), 1999, vol. 2, pp. 1210-1215. [50] M. Sucu, “Parametric Average Value Modeling of Flyback Converters in CCM and DCM Including Parasitics and Snubbers,” M.Sc. Thesis, University of British Columbia, Vancouver, BC, Canada, 2011.      73  Appendices Appendix A. Second-order Flyback Converter State-space Matrices =+−−=0001,)(10011 mcloadmeqLRRCLRBA   −=+−+−++−=0010,)(1)()()(222 mcloadcloadloadcloadmloadcmcloadnLRRCRRCnRRRnLRRRLnRRBA =+−=0000,)(100033 BAcload RRC                  74  Appendix B. Converters Circuit Parameters  B.1 Second-order Flyback Converter Parameters with Basic Parasitics Ω===−−==Ω=′=−==Ω=−=mRAXMVSanyocapacitoricelectrolytumAluVFCVvUCorpCMRtorSemiconducCentraldiodeeryrefastultraVADiodenHLmRRICAComponentsICErTransformedkHzfRIRRLctifiernalInternatiochanelNVAMosfetVvcfdmstptsswg9022100,min,100,2225.1041,cov,400,1:6/1272100635:381.025004.02705Re,,55,8.3:201µµ B.2 Full-order Flyback Converter Parameters with all Parasitics and Snubbers Ω==Ω===−−====Ω=′=−Ω====Ω=−=mRAXMVSanyocapacitoricelectrolytumAluVFCRpFCVvUCorpCMRtorSemiconducCentraldiodeeryrefastultraVADiodenHLHLHLmRRICAComponentsICErTransformeRpFCdkHzfRIRRLctifiernalInternatiochanelNVAMosfetVvcdsdsfdspmstptsssssswg9022100,min,100,22200,10025.1041,cov,400,1:6/18.0,2.0,272100635:10,470381.025004.02705Re,,55,8.3:201µµµµ  75  Appendix C. Hardware Flyback Converter Prototype Circuit Diagram  

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