Open Collections

UBC Theses and Dissertations

UBC Theses Logo

UBC Theses and Dissertations

Large- and small-signal average modeling of dual active bridge dc-dc converter considering power losses Zhang, Kai 2015

Your browser doesn't seem to have a PDF viewer, please download the PDF to view this item.

Notice for Google Chrome users:
If you are having trouble viewing or searching the PDF with Google Chrome, please download it here instead.

Item Metadata

Download

Media
24-ubc_2015_september_zhang_kai.pdf [ 2.28MB ]
Metadata
JSON: 24-1.0166507.json
JSON-LD: 24-1.0166507-ld.json
RDF/XML (Pretty): 24-1.0166507-rdf.xml
RDF/JSON: 24-1.0166507-rdf.json
Turtle: 24-1.0166507-turtle.txt
N-Triples: 24-1.0166507-rdf-ntriples.txt
Original Record: 24-1.0166507-source.json
Full Text
24-1.0166507-fulltext.txt
Citation
24-1.0166507.ris

Full Text

   LARGE- AND SMALL-SIGNAL AVERAGE MODELING OF DUAL ACTIVE BRIDGE DC-DC CONVERTER CONSIDERING POWER LOSSES by Kai Zhang  B.Eng., Huazhong University of Science and Technology, 2013  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)  August, 2015 © Kai Zhang, 2015 ii  Abstract  Detailed switching models are commonly used for analysis of power electronic converters, whereas the average-value modeling (AVM) provides an efficient way to study the power electronic systems in large and small signal sense. This thesis considers a dual-active-bridge (DAB) DCDC converter, as this topology is very common in applications that require bi-directional power flow and galvanic isolation between the input (primary) and output (secondary) sides. Although this type of converter is very common, the available state-of-the-art models often relay on many assumptions and neglect the losses, which make such models inaccurate for studies where the converter efficiency and small- and large-signal responses must be predicted with high fidelity in system-level studies.  We first present an improved detailed model of the DAB DCDC converter by including the conduction loss, switching loss and core loss, which are derived based on the conventional phase shift modulation approach while considering the energy conservation principle. According to the proposed methodology, the equivalent resistances representing switching loss and core loss have been appropriately derived and added to the final simplified circuit model. The proposed approach is simple to use for modeling DAB converters when considering non-ideal circuit components. The new detailed model increases the accuracy in efficiency predictions over wide range of converter operating conditions. Furthermore, this thesis presents a new reduced-order AVM that includes the parasitic resistance and input/output filters. Based on the large-signal AVM, the small-signal model and control-to-output transfer function are also derived. The proposed AVM is compared with full-order generalized average model and the detailed model in predicting large-signal transients in time domain and small-signal analysis in frequency domain. Experimental prototype of a 150W, 24/48 VDC DAB converter has been designed and built to validate the proposed modeling methodologies. The experimental results confirm that the proposed detailed and average-value models yield high accuracy in predicting the power losses and time-domain responses, which represents an improvement over the existing state-of-the-art models.   iii  Preface   The research results presented in this thesis have already been published in a conference paper. In the publication, I was responsible for deriving formulations, building models, doing simulations and hardware tests, analyzing results, as well as preparing the majority of the manuscripts. The co-author, Dr. Zhenyu Shan, designed the hardware prototype and provided a lot of valuable comments and advice. My supervisor, Dr Juri Jatskevich, gave supervisory corrections and feedback during the process of conducting my research and writing the manuscripts.  A version of Chapter 2 has been published as a conference paper. K. Zhang, Z. Shan, and J. Jatskevich, “Estimating Switching Loss And Core Loss in Dual Active Bridge DC-DC Converter”, In Proc. 2015 IEEE 15th Workshop on Control and Modeling for Power Electronics (COMPEL 2015), Vancouver, Canada, 12 – 15 July, 2015, 6 pages. Dr. Jatskevich and Dr. Shan provided useful discussion and modified the manuscript.   iv  Table of Contents  Abstract .......................................................................................................................................... ii Preface ........................................................................................................................................... iii Table of Contents ......................................................................................................................... iv List of Tables ................................................................................................................................ vi List of Figures .............................................................................................................................. vii List of Abbreviations ................................................................................................................... ix Acknowledgements ....................................................................................................................... x Dedication ..................................................................................................................................... xi Chapter 1 Introduction................................................................................................................. 1 1.1 Motivation ........................................................................................................................ 1 1.2 Literature Review ............................................................................................................. 2 1.2.1 Reduced-order Model ............................................................................................... 5 1.2.2 Full-order Model ....................................................................................................... 5 1.2.3 Losses in DAB .......................................................................................................... 6 1.3 Research Objectives ......................................................................................................... 6 Chapter 2 Estimation of Switching Loss and Core Loss in DAB Converter ........................... 9 2.1 Switching Loss in DAB.................................................................................................. 10 2.1.1 Conventional Switching Loss Analysis .................................................................. 10 2.1.2 Simplified Estimation of Switching Losses ............................................................ 14 2.2 Consideration of the Core Losses .................................................................................. 15 2.2.1 Introduction of Core Loss ....................................................................................... 15 2.2.2 Transformer Open-circuit Test ............................................................................... 16 2.3 Experimental Results of Switching Loss and Core Loss in the DAB Converter ........... 18 Chapter 3 Full-order Model of DAB ......................................................................................... 26 3.1 Full-order Model of DAB Converter ............................................................................. 26 3.2 Full-order Model Eigenvalue Analysis .......................................................................... 31  v  3.3 Small-signal AC Model based on Full-order Model ...................................................... 32 3.4 Full-order Model Case Studies ....................................................................................... 34 3.4.1 Steady State Prediction ........................................................................................... 35 3.4.2 Power Losses and Efficiency .................................................................................. 36 3.4.3 Open-loop Dynamic Response ............................................................................... 38 3.4.4 Frequency-domain Analysis ................................................................................... 39 Chapter 4 Reduced-order Model of DAB Converter .............................................................. 41 4.1 State Equations of DAB Converter in Steady State ....................................................... 41 4.1.1 Effect of Magnetizing Inductance and Core Losses ............................................... 44 4.1.2 Reduced-order Average Model of DAB ................................................................. 45 4.2 Reduced-order Model Eigenvalues Analysis ................................................................. 46 4.3 Small-signal ac Model based on Reduced-order Model ................................................ 47 4.3.1 Derivation of Small-signal Model .......................................................................... 47 4.3.2 Reduced-order Small-signal ac Model of DAB Converter ..................................... 50 4.4 Reduced-order Model Case Studies ............................................................................... 50 4.4.1 Steady State Predictions .......................................................................................... 50 4.4.2 Power Losses and Efficiency .................................................................................. 53 4.4.3 Open-loop Transient Study ..................................................................................... 56 4.4.4 Frequency-domain Analysis ................................................................................... 57 Chapter 5 Summary of Contributions and Future Work ....................................................... 59 5.1 Representation of Conduction, Switching, and Core Losses ......................................... 59 5.2 Detailed and Average-value Modeling of DAB ............................................................. 59 5.3 Small-signal Model of DAB Converter ......................................................................... 59 5.4 Future Work ................................................................................................................... 60 Bibliography ................................................................................................................................ 61 Appendices ................................................................................................................................... 66 Appendix A. DAB Converter Parameters ................................................................................. 66 Appendix B. DAB Converter PCB Design Layout ................................................................... 67   vi  List of Tables Table 2.1 Transformer open-circuit test core loss measurements under different frequencies. ... 17 Table 2.2 Accuracy precision of the detailed model with conduction losses only, and the improved detailed model with conduction, switch and core losses in predicting the total power losses and efficiency at 15kHz. ..................................................................................................... 19 Table 2.3 Accuracy precision of the detailed model with conduction losses only, and the improved detailed model with conduction, switch and core losses in predicting the total power losses and efficiency at 25kHz. ..................................................................................................... 21 Table 2.4 Accuracy precision of the detailed model with conduction losses only, and the improved detailed model with conduction, switch and core losses in predicting the total power losses and efficiency at 40kHz. ..................................................................................................... 23 Table 3.1 Eigenvalues of GAMs. .................................................................................................. 31 Table 3.2 Accuracy precision of 8th –and improved 10th –order GAMs in predicting the steady state input current, output current, and output voltage. ................................................................ 35 Table 3.3 Accuracy precision of 8th –and improved 10th –order GAMs in predicting power losses and efficiency. ............................................................................................................................... 36 Table 4.1 Eigenvalues of 6th –order RAVM and GAMs. ............................................................. 46 Table 4.2 Accuracy precision of improved 10th –order GAM, ideal reduced-order model, and the proposed new 6th –order RAVM in predicting the steady state variables. ................................... 52 Table 4.3 Accuracy precision of the improved 10th –order GAM, proposed new 6th –order RAVM in predicting the power losses and efficiency. ................................................................. 54   vii  List of Figures Figure 1.1 Topology of a general DAB converter depicting input bridge, a transformer with leakage inductances, and an output bridge. .................................................................................... 3 Figure 1.2 Simplified diagram of switching signals and transformer voltages. ............................. 4 Figure 2.1 Detailed model of a DAB converter interfacing input and output sources Vis and Vos.. 9 Figure 2.2 Equivalent detailed model of the DAB converter with Req.and Leq. ............................ 10 Figure 2.3 Simplified and magnified view of switch turn-on and turn-off transition waveforms. 11 Figure 2.4 Waveforms of switching signals, transformer voltages, transformer current (it) and magnetizing inductance current (iM) during normal operation of DAB converter. ...................... 11 Figure 2.5 Assumed topological instances for DAB converter primary bridge: (a) circuit during interval from 0 to T; and (b) circuit during interval from T to 2T. ............................................... 13 Figure 2.6 Experimental results of transformer core loss and a corresponding fitting curve. ...... 17 Figure 2.7 Improved detailed model of the DAB converter considering switching loss and core loss. ............................................................................................................................................... 18 Figure 2.8 Photo of the 150W DAB converter experimental prototype. ...................................... 19 Figure 2.9 Power losses and efficiency as obtained from measurements at 15kHz and predicted by the considered detailed models. ............................................................................................... 20 Figure 2.10 Magnified plot of Figure 2.9. .................................................................................... 20 Figure 2.11 Power losses and efficiency as obtained from measurements at 25kHz and predicted by the considered detailed models. ............................................................................................... 22 Figure 2.12 Magnified plot of Figure 2.11. .................................................................................. 22 Figure 2.13 Power losses and efficiency as obtained from measurements at 40kHz and predicted by the considered detailed models. ............................................................................................... 24 Figure 2.14 Magnified plot of Figure 2.13. .................................................................................. 24 Figure 3.1 Power losses and efficiency as predicted by the considered models with respect to actual measurements. .................................................................................................................... 37 Figure 3.2 Output current and transformer current resulting from a phase-shift step change. ..... 39 Figure 3.3 Bode plots of control-to-output transfer function. ....................................................... 40 Figure 4.1 The transformer input and output currents iaci and n·iaco in half switching period. ..... 42 Figure 4.2 The DAB converter average model circuit .................................................................. 46  viii  Figure 4.3 Small-signal ac model of DAB.................................................................................... 50 Figure 4.4 Output current error trends as predicted by three models. .......................................... 53 Figure 4.5 Power losses and efficiency predictions compared to the actual measurements......... 55 Figure 4.6 Magnified plot of Figure 4.5. ...................................................................................... 56 Figure 4.7 Output current and transformer current transient due to a phase shift change. ........... 57 Figure 4.8 Bode plots of the control-to-output transfer functions. ............................................... 58    ix  List of Abbreviations  DAB   Dual active bridge AVM   Average-value model(ing) SSA   State-space averaging  CA   Circuit averaging  PAVM   Parametric averaging-value modeling PSM   Phase shift modulation  8th –order GAM 8th full-order generalized average model with leakage inductance and conduction loss, not including magnetizing inductance or core loss 10th –order GAM Improved 10th full-order generalized average model with leakage inductance, magnetizing inductance, conduction loss and core loss 6th –order RAVM Proposed new 6th reduced-order average value model with leakage inductance, magnetizing inductance, conduction loss and core loss SSMTF  Small-signal model transfer function  x  Acknowledgements    First of all, I would like to express my sincere gratitude to my research supervisor, Dr. Juri Jatskevich, for his expert, aspiring motivation and valuable encouragement during my master study at UBC. His patient guidance helped me a lot in my research and writing of this thesis. I could not have imagined having a better supervisor. I am also honored to have been a research and teaching assistant for Dr. Jatskevich. It was a great pleasure to work with such knowledgeable, dedicated and hardworking individual. In addition to that, I want to acknowledge financial support obtained from the Natural Science and Engineering Research (NSERC) of Canada, Strategic Project Grant entitled “Advanced Integrated AC-DC Systems for Energy Efficient Buildings and Communities in Canada” led by Dr. Juri Jatskevich.  Besides my supervisor, I would like to thank the rest of my thesis committee, Dr. Shahriar Mirabbasi and Dr. Christine Chen, who dedicated their time to provide me encouragement, insightful comments and constructive questions.  I would like to thank my friends and colleagues in Electric Power and Energy Systems research group and Power Lab at UBC for their help and support during my courses and research. I would like to especially thank to Dr. Zhenyu Shan and Mr. Hua Chang, who gave me much instruction and help in experimental part of this project. Also, I would like to give my thanks to Mr. Yajian Tong, who helped me a lot in preparation of defense slides and presentation.   Last but not the least, I would like to give my deepest thanks to my parents for their spiritual encouragement and support throughout my life.    xi   Dedication    To my parents and grandparents  1  Chapter 1 Introduction  1.1 Motivation It is envisioned that DC systems will be used more often in the future in combination with the traditional AC systems in order to reduce the number of conversion stages and energy loss, and achieve high power density. The standards for modern power distribution systems are now evolving by many industrial alliances to promote and integrate the DC technology and adopt the common DC voltage levels, e.g. 380VDC, 48VDC, 24VDC, etc. Many distributed energy resources including solar panels, battery and fuel cells, directly provide DC power. An increasing number of loads are also consuming DC power. In many of such applications, which include energy storage and DC loads, a backup battery with a high performance bi-directional converter for power management is needed to build a reliable and uninterruptable DC power system. Sometimes, the galvanic isolation may also be required to separate the primary (input) and secondary (output) sides. There are various topologies of isolated bidirectional DCDC converters. Most prominent topologies include the isolated bidirectional full-bridge converter [1], the bidirectional current doublers converter topology [2], and the dual-active-bridge (DAB) converter [3], [4], [5]. For the application of battery power management, a DAB converter is a good candidate, since it is capable of a bi-directional energy flow, inherent soft switching, high-power and high-power-density, and high-efficiency, and galvanic isolation [6], [7]. Bidirectional isolated DCDC DAB converters were initially proposed in [6] and [7] as candidates for high power density. Since that, this topology has been widely used in DC power systems for solid-state transformers [8], [9], smart grid [10] and electric/hybrid vehicles [11]. The comprehensive analyses of its design, operation, and control may be found in [12], [13]. Furthermore, recent achievements on DAB converters include detailed modeling for power loss evaluation [11], design optimization [13], advanced control strategies [14], [15], generalized average modeling [16], modeling for rms and average currents [17], and PWM control [18]. Conventional converter models focus on the operating principle and theory, thus often ignoring the existence of power losses. At the same time, for design and integration of DC systems with renewable energy source and storage, the  2  models capable of accurately predicting the input-output power characteristics and dynamics are highly desirable.  1.2 Literature Review The commonly used models of power electronic systems may be classified into two basic groups. The first group includes the detailed switching models, which may be readily developed in commercial transient simulators for the power electronic circuits. Such detailed models typically include all components (resistors, inductors, capacitors, diodes, transistors, etc.), which may also include some parasitic parameters (e.g. equivalent series resistances, on-conduction resistances of diodes and transistors, forward voltage drops, leakage inductances, etc.) for simulation accuracy. Due to existence of switching actions in detailed models, their time-domain simulations typically require small integration time steps and are computationally expensive. Also, due to switching, the detailed models are not well-suitable for system-level analysis in time- and frequency- domains [19]. Moreover, the instantaneous waveforms in detailed models will contain switching harmonics and ripples, which may not be needed for controller design and studying system-level interactions.  The second group includes the so called average-value models (AVMs) [20], which do not have switching and therefore are computationally more efficient and suitable for the system-level analysis, wherein the details of the switching ripples are not important and therefore may be neglected. By carefully averaging the corresponding state variables (e.g. currents and voltages), the switching is removed from the model and replaced by the respective equivalent average-value relationships connecting the circuits variables. Therefore, the AVMs are continuous and typically permit the use of much larger time steps in the transient simulations, which make a very efficient way in investigating the system-level dynamic behavior. Moreover, the AVMs may also be linearized about a desired operating point of interest for the subsequent small-signal analysis of the associated switching circuits [21], [22] and controller design [23], [24]. The AVMs are also very useful for investigation of system transients and stability [25]. Developing accurate AVMs for various power electronic topologies has been and remains to be an active area of research [20], [26]. The AVM may be derived using discrete-time and continuous-time formulations. In general, a continuous-time model is usually preferred because it provides more physical insight  3  and facilitates control design [16]. As seen from point of view of internal dynamics (and the corresponding number of state variables), the AVMs may also be classified into full-order and reduced-order models. On the one hand, a full-order model retains more information in high frequency and therefore can have fewer mismatches in frequency-domain analysis compared with that of reduced-order model. On the other hand, a full-order model is more complicated due to its higher order and generally harder to derive. If a system maintains most of its dominant poles and neglects some poles that are less pronounced, a simpler reduced-order model may be obtained that preserves the accuracy in frequency-domain and time-domain performance to some extent. The commonly used approaches to obtain the AVMs include the state-space averaging (SSA) [27], the circuit averaging (CA) [28], [29] and the parametric averaging-value modeling (PAVM) [30], [31]. Over the years, various methods have been proposed in the literature and attracted significant attention. Modeling and simulation of DAB converters can be extremely useful before the hardware implementation as it helps in component selection, controller design, and preliminary evaluation within the overall DC distribution system. The schematic of a general purpose DAB considered in this thesis is shown in Figure 1.1. The converter consists of a high-frequency transformer (with turn ration 1:n) and two active bridges, one on the input (primary) and one on output (secondary) side, respectively. The transformer provides galvanic isolation and energy delivery through its windings. The bridges produce square-wave voltages with 50% duty that are applied to the transformer windings. Figure 1.2 shows the MOSFET gate signals and transformer voltages, respectively.  Figure 1.1 Topology of a general DAB converter depicting input bridge, a transformer with leakage inductances, and an output bridge.   4   Figure 1.2 Simplified diagram of switching signals and transformer voltages.  To transfer the power between the input (primary) side and output (secondary) side, this thesis considers a simple phase shift modulation (PSM). According to this operation, the phase shift between the driving signals for the two bridges determines the power transfer. If an ideal transformer is assumed (no power loss is considered), the power delivered from the leading bridge to the lagging bridge may be calculated by   ddLnfVVPeqsosis  12,  (1.1) where n is the turns ratio of high-frequency transformer; Vis and Vos are the converter input and output voltages; d = φ / π is the phase shift ratio;  fs is switching frequency; and Leq = Ll1 + Ll2/n2 is the leakage inductance referred to the input side. In analysis of power electronic systems, such as DAB converter shown in Figure 1.1, the dynamic order basically equals to the number of energy storing elements, i.e. inductors and capacitors, since the current through inductance and the voltage across capacitor are usually chosen as the state variables. This notion directly carries for the DCDC converters. However, in power electronic systems containing AC stages, the transformers may be utilized for providing galvanic isolation and energy delivery through its windings [16]. The conventional state-space  5  averaging method [32], [33] may be used to derive the AVMs by eliminating the current ripple. However, the transformer current typically has a dominant AC component which is not negligible for the conventional state-space averaging method. This is also a big obstacle for applying the state-space averaging to the DAB converters. In previous literature, to the best of our knowledge, there have been two methods proposed to construct the average-value model for the DAB converter: 1) Neglect all conduction losses, i.e. assume ideal DAB converter circuit, for which a reduced-order model may then be derived; 2) Considering transformer leakage inductances and winding resistances, and use the generalized averaging method to build a full-order model for non-ideal DAB converter considering only the fundamental AC component. 1.2.1 Reduced-order Model The reduced-order model eliminates the switching frequency AC component of the transformer current and its dynamics which are changing much faster than the DC current transient. Due to this approximation, the current through the transformer leakage inductance is not state variable. The reduced-order models are usually simple and straightforward to implement. Such reduced-order models have been widely used for modeling DAB converters [34], [35], [36], wherein the average models are built based on the average power transmitted in one switching cycle. However, due to simplifying assumptions, these reduced-order models do not consider the transformer losses, i.e. copper losses and core losses. Therefore, the reduced-order models in [34], [35], and [36] are just an ideal case of AVMs that cannot be utilized in power losses and efficiency analysis of DAB converters. If the transformer leakage resistance and shunt resistance are considered in DAB circuit, the equations become nonlinear. Recent works on modeling DAB converters show that nonlinear effects is a great obstacle to development of accurate AVMs [5], [37]. 1.2.2 Full-order Model The full-order models consider the transient component of transformer current, which requires increasing the number of state variables. Basically, a full-order model is achieved by a so-called generalized averaging technique which uses more terms of Fourier series as the state variables. This approach has been widely used for modeling of DAB converters [16] and  6  resonant converters [38], [39]. In general, a full-order model can obtain higher accuracy than a reduced-order model [40]. However, in the continuous full-order model, the real and imaginary parts of transformer current are used as state variables [16]. Also, the full-order model derived by generalized averaging is based on the study of first harmonic, which means that the accuracy of steady state prediction may be poor if large harmonic distortions are present [38]. The steady state errors in generalized average modeling have been also noted in [39]. Moreover, the core losses and magnetizing inductance have not been included in the previous literature. 1.2.3 Losses in DAB The losses in DAB converter mainly consist of three parts, i.e. conduction losses, core losses, and switching losses. The conduction losses include the transformer windings resistance losses, the switching conduction losses, the wire and PCB conduction losses, etc. Basically, the conduction losses can be modelled by adding corresponding resistances in the DAB model. Also, these resistances can be treated as constants under low frequency range, e.g. under 50kHz [11]. The most common in previous literature method to model the core losses in transformer is to add a shunt resistance in the circuit. However, since switching frequency significantly affects the core losses, this resistance cannot be considered constant for accuracy. To model the switching losses based on [24], the switching power loss on one switch may be approximated as    swoffonswswsw fttIVP  21,  (1.2) where Vsw and Isw are the voltage and current of the switch at the instant of switching respectively; ton and toff  are the time intervals during ON and OFF which can be obtained from datasheet; fsw is the switching frequency. Therefore, Vsw, Isw and fsw should be the most important factors of switching losses as long as the type of switch has been chosen. Similar to the method simulating core losses, an equivalent resistance may also be added in the DAB model to represent the total switching losses. 1.3 Research Objectives When it comes to average-value modeling, different power converters have various challenges. In this thesis, the DAB converter (for which only the approximated/idealized average models are available in previous literature) is considered. Unlike the classical DC-DC converters, such as Buck, Boost, Cuk, etc., in DAB converter there is a galvanic isolation transformer, which  7  is the most challenging part for accurate modeling. If the conduction losses, core losses, and switching losses, are considered, the development of accurate AVM becomes very challenging. To the best of my knowledge, these challenges have not been addressed in the prior literature. Therefore, the purpose of this thesis is to develop accurate, simple-to-use, and straightforward-to-implement models for the DAB converter with power losses.  More specifically, this thesis has the following research objectives:   Objective 1: Modeling conduction losses, switching losses, and core losses in DAB converter, while including the effect of phase shift ratio and frequency. As a first step, we need to develop more accurate detailed model which includes the major power losses by considering the effect of frequency and phase shift ratio. To achieve this objective, we need to rigorously analyze the factors affecting the power losses in DAB circuit. The conduction losses are mainly caused by the winding resistances and the transistors. It is desirable to evaluate these power losses and derive equivalent operating point and frequency-dependent resistances that may be placed in a simplified detailed model, while preserving the energy conservation in the circuit. The resulting improved detailed model should demonstrate higher accuracy in predicting the energy losses of the DAB over a wide range of operating conditions.   Objective 2: Development of average-value model of DAB converter considering major power losses. Before the small-signal modelling and frequency-domain analysis, the accurate detailed and average-value models are needed. As a next step, it is desirable to develop both the reduced-order and full-order models that include the major conduction losses as well as transformer core losses. AVMs with combinations of all these losses have not been proposed in the prior literature. However, the resulting AVMs will be very accurate and desirable for predicting the input-output power and losses characteristics of the DAB converters in the system level studies.    8   Objective 3: Development of small-signal model of DAB based on the obtained AVM. For frequency-domain analysis and future controller design, it is necessary to develop small-signal model based on the obtained average-value models. We will consider both reduced-order and full-order models, and apply the Taylor Series to linearize the average models. It is desirable to construct a reduced-order small-signal model that has good accuracy up to the 1/3 of the switching frequency (basically the same accuracy as the full-order AVM), while being simpler and easy-to-use.  If Objectives 1 – 3 are achieved, a number of high-accuracy advanced models of DAB converters will become available. The large-signal AVMs will be very useful for the system-level transient studies, where the users will be able to use much larger simulation time steps and at the same time preserve all the energy losses and efficiency information that has not been previously possible using conventional models. Accurate small-signal models have always been desirable, which has been particularly difficult to obtain for the DAB converters. In the long run, such models which readily include the conduction and transformer losses may be included in transient simulation programs and design tools that are used for power electronic systems.     9  Chapter 2 Estimation of Switching Loss and Core Loss in DAB Converter  Compared with the general DAB shown in Figure 1.1, this chapter also considers the input/output filters as shown in Figure 2.1. These filters are designed as typical LC structures with RC damping branches to suppress the current and voltage ripples. Also, the magnetizing inductance and resistance (LM, RM), the leakage inductance (Ll), the winding resistance (Rl), and the turn-on resistance of all MOSFETs (Rs) are considered. The conduction losses and the core losses are represented by Rl, Rs and RM, respectively, while the dead zone effect is neglected for the purpose of this chapter. In order to simplify the derivation of state equations, the switches turn-on resistance (Rs) and the winding resistance (Rl) are lumped to the primary side as an equivalent resistance,  Figure 2.1 Detailed model of a DAB converter interfacing input and output sources Vis and Vos. The detailed circuit also includes the leakage inductance (Ll), the winding resistance (Rl), the magnetizing inductance (LM), the equivalent core loss (RM), and the MOSFET turn-on resistance (Rs), respectively.   22122 nRRRRR slsleq ,  (2.1)   221 nLLL lleq .  (2.2)   10    Figure 2.2 Equivalent detailed model of the DAB converter with Req.and Leq.  Taking into consideration (2.1) and (2.2), the equivalent detailed model is obtained with Req and Leq, as shown in Figure 2.2. In general case, the main loss in DAB is conduction loss, i.e. the loss caused by the equivalent resistance as shown in Figure 2.2. Some basic ideas on how to estimate the switching loss and conduction loss has been presented in [41]. In [42] and [43], improved core loss calculation methods are discussed. A comprehensive view of power losses in DAB converters has been proposed in [11]. However, a simple and accurate detailed model for switching loss and core loss estimation is still required. This chapter indicates the main factors that affect the switching loss and core loss, respectively. The equations to estimate the losses shown in Figure 2.2 will be modified to get more accurate results.  2.1 Switching Loss in DAB 2.1.1 Conventional Switching Loss Analysis  As shown in Figure 2.3 [44], the switching losses in one MOSFET can be calculated based on the following equation    swoffonswswsw fttIVP  21,  (2.3)  where Vsw and Isw are the voltage and current of one specific MOSFET at the time of switching respectively; ton and toff are the time intervals during ON and OFF which can be obtained from datasheet; and fsw is switching frequency. In order to estimate the switching losses, it is important  11  to know the instantaneous voltages and currents at the time of switching for the MOSFETs in both legs.  Figure 2.3 Simplified and magnified view of switch turn-on and turn-off transition waveforms.  Figure 2.4 Waveforms of switching signals, transformer voltages, transformer current (it) and magnetizing inductance current (iM) during normal operation of DAB converter.  12   According to the phase shift modulation is considered in this thesis, Figure 2.4 shows the switch gate signals and transformer voltage and current waveforms. Each bridge is driven by a 50% duty-ratio signal to generate square-wave voltage on the transformer windings with the phase shift ratio defined as d = φ/π. The period T denotes the half of the switching period. It should be noted in Figure 2.4, that presence of series equivalent resistance Req in the converter circuit would result in piecewise exponential waveforms of the transformer current it, which is shown as a piecewise linear waveform for simplicity. As shown in Figure 2.4, the absolute values of the current Isw for switches S1, S2, S3 and S4 on the primary bridge are all equal to It1, i.e.  1,,,, 4321 tSSSSsw II .  (2.4)  Also, due to the shunt current in the transformer magnetizing branch LM and RM, the values of Isw for switches S5, S6, S7 and S8 on the second bridge considering transformer turns ratio become     peakMMctSSSSsw iRnvInI ,32,,,,18765.  (2.5)  It should be noted that in ideal case, the secondary bridge voltage vc3 will have a step change at the switching instant of the secondary leg. Since the voltage across magnetizing inductance LM  may be assumed constant for the half of the switching period, the following holds   nvddiL cMM 3,  (2.6)  where τ is used to denote time. Considering half of the switching period, (2.6) becomes   nvTiiL cpeakMpeakMM 3,, .  (2.7)  Therefore,   McpeakM LTnvi  3, 21 .  (2.8)  Based on (2.4), (2.5) and (2.8), the current Isw for eight switches may be obtained.      13   Figure 2.5 Assumed topological instances for DAB converter primary bridge: (a) circuit during interval from 0 to T; and (b) circuit during interval from T to 2T.  To determine the switch Vsw for different switches, here we consider the primary side leg as an example, whereas the secondary side leg voltage may be determined similarly. As shown in Figure 2.5, when switches S1 and S2 are conducting, the voltages across S3 and S4 are both equal to vc1. When switches S3 and S4 are conducting, the voltages across S1 and S2 are still equal to vc1. Therefore,   1,,,, 4321 cSSSSsw vV .  (2.9)  Similarly, the voltage for the secondary side switches is determined to be  3,,,, 8765 cSSSSsw vV .  (2.10)  Substituting (2.4), (2.5), (2.9) and (2.10) into (2.3), the total switching loss in both converter legs are  14       swoffonSSSSswcSSSSswcswoffonSSSSswcswoffonSSSSswctotalswfttIvIvfttIvfttIvP8765432187654321,,,,3,,,,1,,,,3,,,,1,2421421.  (2.11)   2.1.2 Simplified Estimation of Switching Losses   To simplify the calculation of power losses, several assumptions are made. First, it is assumed that the shunt current in the transformer magnetizing branch LM and RM may be neglected since the impedance of LM and RM are much large than that of series impedance Leq and Req. Under this assumption, (2.5) can be simplified as   nII tSSSSsw 2,,,, 8765 ,  (2.12)  and (2.11) may be rewritten as    swoffontctctotalsw fttnIvIvP   2311, 2.  (2.13)  Second, since both the input and output sides of the DAB are connected to DC source or DC bus as shown in Figure 2.2 and Figure 2.5, and the voltage conversion ratio Vis/Vos is close to 1/n, it also follows that in steady state we have  nvvcc 131 .  (2.14)  Substituting (2.14) into (2.13), we obtain      swoffonttctotalsw fttIIvP  211, 2 .  (2.15)  The actual transformer current it will have a piecewise exponential waveform due to the existence of series equivalent resistance. In most practical transformers, however, the impedance of equivalent leakage inductance is much larger than the equivalent series resistance. Under this assumption, the waveform of it is really close to piecewise linear. If the transformer current it is assumed to be piecewise linear, based on Figure 2.4, during the interval from 0 to dT, we have   15      nvvdTIIL cctteq 3112,  (2.16)  which yields  eqcctt LdTnvvII   3121.  (2.17)  Noting that 2T = 1/fsw and substituting (2.17) into (2.15), a simple formula to estimate the switching power losses can be obtained as   dttnvvLvP offoncceqctotalsw   311,,  (2.18)  which simplifies to   dttLvP offoneqctotalsw 21,2 .  (2.19)  Equations (2.18) and (2.19) show that under some reasonable assumptions, the total switching loss on both converter legs is a function of the phase shift ratio d. Based on (2.19), it is possible to define an equivalent switch resistance in terms of the power loss as  dttLPvRoffoneqtotalswcsw )(2,21.  (2.20)  Equation (2.20) shows that an equivalent switch resistance Rsw can be added in parallel with capacitance C1 to represent the total switching losses on two inverter legs.   2.2 Consideration of the Core Losses 2.2.1 Introduction of Core Loss It is important to take into consideration the core losses when designing power electronic converters with inductive components and transformers. Most commonly, the core losses may be characterized by the so-called Steinmetz equation [42], [45],  BkfPv ˆ ,  (2.21)   16  where Pv is the time-average power loss per unit volume; Bˆ  is the peak induction of a sinusoidal excitation with frequency f ; and k, α and β are the coefficient parameters that may be found by curve fitting for a particular magnetic material [43]. These parameters can be assumed as constant for a limited range of frequencies and flux density.  If the ferrite core is not saturated and the magnetizing inductance is constant, then according to Figure 2.2, the value of magnetizing current can be calculated as  MswcMcM LfnvLnvi  233,  (2.22)  where fsw is switching frequency. Therefore,   Mr niB 0 ,  (2.23)  where μ0 is the magnetic permeability constant; μr is the material relative permeability; and n is number of turns per length. Substituting (2.22) and (2.23) into (2.21), and assuming that Bˆ  is proportional to B, the following relationship can be obtained   swv fP .  (2.24)  Equation (2.24) indicates that core loss is proportional to f α-β. Furthermore, the equivalent shunt resistance RM representing the core loss can be written as  swfVPnvRvcM1123 ,  (2.25)  where V is the ferrite core volume. Equation (2.25) shows that core loss resistance is a function of switching frequency. In (2.24) and (2.25), due to the unknown parameter k in Steinmetz equation and the exact proportional factor between Bˆ  and B, only proportional relation can be obtained. The actual proportionality constant and accurate model of RM as a function of switching frequency can be obtained through open-circuit test and curve fitting. 2.2.2 Transformer Open-circuit Test  The transformer used in our converter prototype is a ferrite-core high-frequency transformer with magnetic core part number B66335G0000X187, produced by TDK. The turns of the primary and secondary windings are 40 and 20, respectively. In the open-circuit test, a  17  +46V to -46V, 50% duty cycle square wave is applied on the transformer. The waveform imitates the transformer operation in the DAB converter. Table 2.1 shows the measured core loss under different switching frequencies. Based on the data in Table 2.1, a curve is fitted by Microsoft Excel solver as shown in Figure 2.6. The function of this curve is  627.03587.3  swcl fP .  (2.26)   Table 2.1 Transformer open-circuit test core loss measurements under different frequencies. Core loss/ W 1.1 0.9 0.6 0.4 0.4 0.3 0.3 Frequency/ kHz 5 10 20 25 30 40 50   Figure 2.6 Experimental results of transformer core loss and a corresponding fitting curve.  Based on (2.26), a core loss equivalent shunt resistance may be calculated as  627.023 6301 swclcM fPnvR   (2.27)   18  This shunt resistance is then added to the transformer circuit. Finally, a new detailed model for the DAB converter as shown in Figure 2.7 is proposed, wherein the switching loss and core loss are obtained according to (2.20) and (2.27).  Figure 2.7 Improved detailed model of the DAB converter considering switching loss and core loss.  2.3 Experimental Results of Switching Loss and Core Loss in the DAB Converter An experimental prototype has been designed and built to verify the modelling assumptions presented in this thesis. The final experimental prototype is shown in Figure 2.8, and its parameters are summarized in Appendix A. The detailed model of this DAB converter has been implemented in PLECS [46]. The power stage of the experimental DAB converter is composed of MOSFET switches (IPB144N12N3), a ferrite-core high-frequency transformer and input/output filters. The turns ratio of the primary to secondary winding is 40 to 20. A 47-H inductor is in series with the primary winding to achieve the anticipated leakage inductance for the proper operation of the DAB converter prototype. The control stage of the experimental DAB converter has been implemented using a 32-bit floating-point microprocessor TMS320F28335 from Texas instruments. To investigate the converter losses, various operating frequencies, i.e. 15kHz, 25kHz and 40kHz, have been considered. In addition to experiment results, two kinds of models are compared and discussed: 1) Detailed model that considers conduction loss only.  19  2) Improved detailed model with conduction loss, switching loss, and core loss, as shown in the converter circuit of Figure 2.7.  The comparisons are summarized in Table 2.2 to Table 2.4 and Figure 2.9 to Figure 2.14, respectively.  Figure 2.8 Photo of the 150W DAB converter experimental prototype.  Table 2.2 Accuracy precision of the detailed model with conduction losses only, and the improved detailed model with conduction, switch and core losses in predicting the total power losses and efficiency at 15kHz. Variable @ 15kHz Prototype/Model Phase shift ratio d 0.05 0.1 0.15 0.2 0.25 Power losses/ W Experiment 5.3511 13.9527 29.6121 46.64 69.8245 Detailed model with conduction loss only  4.55 13.07 28 48.8 74.57 Improved det. model with cond., switch and core losses  5.24 13.95 29.08 50 75.95 Efficiency Experiment 90.94% 87.31% 81.80% 78.21% 73.31% Detailed model with conduction loss only 92.7% 87.92% 81.9% 75.7% 69.5% Improved det. model with cond., switch and core losses 91.6% 87.15% 81.27% 75.15% 69.03%  20   Figure 2.9 Power losses and efficiency as obtained from measurements at 15kHz and predicted by the considered detailed models. The results from experimental hardware measurement are denoted with stars. The detailed model with conduction losses only is denoted by dash-dotted line. The improved detailed model with conduction losses, switch losses, and core losses is denoted by solid line.  Figure 2.10 Magnified plot of Figure 2.9. 21   Table 2.3 Accuracy precision of the detailed model with conduction losses only, and the improved detailed model with conduction, switch and core losses in predicting the total power losses and efficiency at 25kHz. Variable @ 25kHz Model Phase shift ratio d 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 Power losses/ W Experiment 2.2552 4.6755 9.28218 15.2021 22.0723 29.7918 37.8568 46.3488 54.3268 Detailed model with cond. loss only  1.34 3.515 7.28 12.5 19 26.59 35 43.97 53.2 Improved det. model with cond., switch and core losses  1.87 4.22 8.16 13.55 20.24 28 36.55 45.7 55.08 Efficiency Experiment 94.52% 93.09% 90.54% 88.01% 85.54% 83.00% 80.47% 77.75% 75.13% Detailed model with cond. loss only 96.30% 94.80% 92.50% 89.98% 87.30% 84.54% 81.60% 78.58% 75.33% Improved det. model with cond., switch and core losses  94.90% 93.75% 91.60% 89.17% 86.50% 83.80% 80.90% 77.87% 74.65%  22   Figure 2.11 Power losses and efficiency as obtained from measurements at 25kHz and predicted by the considered detailed models. The results from experimental hardware measurement are denoted with stars. The detailed model with conduction losses only is denoted by dash-dotted line. The improved detailed model with conduction losses, switch losses, and core losses is denoted by solid line.  Figure 2.12 Magnified plot of Figure 2.11. 23   Table 2.4 Accuracy precision of the detailed model with conduction losses only, and the improved detailed model with conduction, switch and core losses in predicting the total power losses and efficiency at 40kHz. Variable @ 40kHz Model Phase shift ratio d 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 Power losses/ W Experiment 1.3288 2.8045 5.3576 8.91251 13.0812 18.068 22.6561 28.239 33.8048 Detailed model with cond. loss only 0.853 2.12 4.24 7.14 10.7 14.83 19.4 24.27 29.32 Improved det. model with cond., switch and core losses 1.271 2.704 4.99 8.04 11.765 16.05 20.77 25.8 31 Efficiency Experiment 95.10% 93.04% 90.70% 87.94% 85.19% 82.24% 79.74% 76.56% 73.22% Detailed model with cond. loss only 96.12% 94.7% 92.55% 90.17% 87.65% 84.98% 82.2% 79.26% 76.15% Improved det. model with cond., switch and core losss 94.25% 93.3% 91.31% 89.01% 86.53% 83.9% 81.12% 78.18% 75.04%  24   Figure 2.13 Power losses and efficiency as obtained from measurements at 40kHz and predicted by the considered detailed models. The results from experimental hardware measurement are denoted with stars. The detailed model with conduction losses only is denoted by dash-dotted line. The improved detailed model with conduction losses, switch losses, and core losses is denoted by solid line.  Figure 2.14 Magnified plot of Figure 2.13. 25   Based on the results summarized in Tables 2.2 through Table 2.4, it can be seen that the new improved detailed model that considers the conduction, switching, and core losses has a better prediction of power losses and efficiency throughout a wider range of operating conditions. Specifically: 1) Under a given switching frequency, when the phase shift ratio d varies, the improved detailed model has a good accuracy in predicting the overall losses even when d is small. The reason is that under a light loading current, the switching loss and core loss are not negligible compared with conduction loss. When the phase shift ratio d increases, the prediction from the new model starts to deviate from experiment measurements, somewhat. This is may be possible due to small changes of the actual circuit parameters caused by the heat under heavy loading (which is not taken into account by our model). Also, due to high current, the switch turn-on and turn-off times might become longer than those specified on manufacture’s datasheet. The switch turn-on and turn-off times may actually become current-dependent (which is also not taken into account by our model). The conducted experimental studies also show that the conduction loss constitute a major part in total losses in practical DAB converters. It is important to consider this loss component accurately when the modeling goal is to predict the overall converter efficiency. 2) When the phase shift ratio d and switching frequency vary widely, it can be seen that frequency affects the precision of power loss and efficiency prediction. Basically, when the operating frequency become low, the impedance of leakage inductance is not high enough, and the assumption of piecewise-linear current waveforms in transformer used in (2.16) and (2.17) is no longer sufficient for calculation of switching losses. In fact, leakage inductance and switching frequency play significant roles in operation of the DAB converters, including their power losses and efficiency. For high efficiency operation at a given frequency, the leakage inductance should be much larger than the winding resistance so that the transformer current remains close to piecewise-linear.   26  Chapter 3 Full-order Model of DAB  A full-order model of DAB is proposed in this chapter based on generalized average modeling [16]. The full-order model presented in this chapter is derived based on the equivalent detailed model of Figure 2.2, i.e. variable resistances in Figure 2.7 will not be considered for conciseness during AVM derivation. However, in [16], the input filter, output the filter, the magnetizing inductance LM and the resistance RM representing the core losses are not considered. Therefore, new derivations are required to obtain a proper full-order model that includes the circuit elements of Figure 2.2. 3.1 Full-order Model of DAB Converter  The full-order model may be realized using the generalized averaging method. Under the same phase shift modulation, as seen in Figure 1.2 (if dead zone effect is not considered), the voltage at the transformer primary side, vp, can only have two states: 1) it is equal to vc1 when S1 and S2 are ON; and 2) it is equal to -vc1 when S3 and S4 are ON. Therefore,   11 cp vsv ,  (3.1) where vc1 is the voltage across the capacitor C1, and the switching function s1 at primary side is    TTTs 2,10,11 .  (3.2) Here, T denoted half of the switching period, τ is used to denote the time interval. Similarly, at the secondary side,   32 cs vsv  ,  (3.3) where vc3 is the voltage across the capacitor C3, and the switching function s2 at secondary side is    TdTTdTdTTdTs 2or0,1,12 .  (3.4) If voltage across capacitor and current through inductance are chosen as state variables, based on Figure 2.2, the topology can be described by the following set of differential equations,   27   111 cis vvddiL ,  (3.5)  oosc RivvddiL 2322 ,  (3.6)  12122 RvvddvC ccc ,  (3.7)  24344 RvvddvC ccc ,  (3.8)  1211111 RvvisiddvC cctc ,  (3.9)  243222333 RvvinsisnRviddvC ccMMctc  ,  (3.10)  eqtccteq RinvsvsddiL  3211,  (3.11)  nvsddiL cMM 32 ,  (3.12) where vc1, vc2, vc3 and vc4 are the voltages across capacitors C1, C2, C3 and C4; it is current through transformer leakage inductance; and iM is current through magnetizing inductance. Note that (3.10) can be further simplified as    2432232243222232243222333RvviRnvnsiiRvvisRnvnsiiRvvinsisnRviddvCccMcMtccMcMtccMMctc,  (3.13) since s22 = 1 is a constant function.   According to generalized average modeling [16], [47], [48], the derivative of the kth coefficient for variable x is  28         kskk xjkxddxdd ,  (3.14) where kxddrepresents the average of the differential of a state variable in (3.9), (3.11), (3.12) and (3.13). The kth coefficient of the product of two variables x and y is   i iikkyxxy,  (3.15) For simplicity, we only consider the 1st and -1st coefficient in Fourier series, which are the complex conjugates. The product of 0th coefficients is    IIRR yxyxyxxy 1111000 2  ,  (3.16) and the real and imaginary 1st coefficient terms are  01101 yxyxxy RRR ,  (3.17)  01101 yxyxxy III ,  (3.18) where the subscripts R and I denote the real and imaginary parts of a complex number, respectively.   Applying (3.15) - (3.18) into (3.9), (3.13), (3.11) and (3.12), the 0th and 1st coefficients of state variables vc1, vc3, it and iM are   ItIRtRtccc isisisRvvidvdC 1111110011020101011 22 ,  (3.19)   0111011111 121111111 tRRtIcsRcRcRRc isisvCRvvidvdC  ,  (3.20)   0111011111 121111111 tIItRcsIcIcIIc isisvCRvvidvdC  ,  (3.21)       IIMItRRMRtMtccMccsiisiisiinRvviRnvdvdC1211121102002040302203033221 ,  (3.22)  29       0211120013321413122131331siisiinvCRvviRnvdvdCRMRtRMtIcsRcRcRMRcRc  ,  (3.23)      0211120013321413122131331siisiinvCRvviRnvdvdCIMItIMtRcsIcIcIMIcIc  ,  (3.24)    IcIRcRcIcIRcRcteqteqvsvsvsnvsvsvsiRdidL1312131203021111111101010022122 ,  (3.25)    03121302011111011111cRRccRRcItseqRteqRteqvsvsnvsvsiLiRdidL  ,  (3.26)    03121302011111011111cIIccIIcRtseqIteqIteqvsvsnvsvsiLiRdidL  ,  (3.27)   IcIRcRcMM vsvsvsndidL 1312131203020 221 ,  (3.28)   0312130211 1 cRRcIMsMRMM vsvsniLdidL  ,  (3.29)   0312130211 1 cIIcRMsMIMM vsvsniLdidL  .  (3.30) It is assumed that capacitors C1 and C3 in Figure 2.2 are sufficiently large to suppress the ripple of input and output voltages. Therefore, the dynamic of the input voltage vc1 and output voltage vc3 is much slower than those of DAB converter. Therefore, <vc1>1R = <vc1>1I = 0, <vc1>0 = vc1, <vc3>1R = <vc3>1I = 0, <vc3>0 = vc3. It is also necessary to calculate the coefficients of the switching functions s1(τ) and s2(τ). Due to the assumption that the phase shift ratio is 50% as shown in Figure 1.2, the following is obtained  30   00201  ss,  (3.31)  011 Rs,  (3.32)  211 Is,  (3.33)  ds R sin212 ,  (3.34)  ds I cos212 .  (3.35) Substituting (3.31) - (3.35) into (3.19) - (3.30) and ignoring the angle brackets, the equations may be simplified into the following:   Itccc iRvviddvC 11211114,  (3.36)     IMItRMRtccMcc iindiindRvviRnvddvC 111124322333cos4sin4  ,  (3.37)  nvdiLiRddiL cItseqRteqRteq 3111 sin2  ,  (3.38)  nvdviLiRddiL ccRtseqIteqIteq 31111 cos22  ,  (3.39)  dnviLddiL cIMsMRMM sin2311 ,  (3.40)  dnviLddiL cRMsMIMM cos2311 .  (3.41) The generalized full-order average model of DAB is formed by (3.5) - (3.8) and (3.36) - (3.41). This model retains the dynamic of transformer including core losses and magnetizing inductance by using the 0th coefficients of input and output voltages (vc1 and vc3) and the 1st coefficients of the leakage current and magnetizing current (it1R, it1I, and iM1R, iM1I,) as the state variables. Counting the state variables, this is a tenth-order model. This model is referred as 10th order GAM (generalized average model) in this thesis.  31  Basically, the generalized average method is based on the Fourier series coefficients, and it significantly increases the model order. Therefore, more frequency domain information is maintained, and this model should offer good accuracy in frequency-domain analysis. In time domain analysis, the accuracy may suffer from the assumptions of steady state based considering only on the 1st fundamental harmonic. If large harmonic distortions are present in the transformer current, more Fourier series coefficients should be used to achieve higher accuracy. However, the resulting model would be too complex to provide insightful information for controller design [16]. A small-signal model will be derived based on this generalized full-order average model in Section 3.3.  3.2 Full-order Model Eigenvalue Analysis Based on the improved 10th –order GAM defined in Section 3.1, the eigenvalues of this 10th order GAM can be obtained. Assuming an operating point defined by d = 0.1, the eigenvalues of the full-order model with magnetizing inductance and core losses (improved 10th –order GAM) and the model without them (8th –order GAM [16]) are summarize in Table 3.1. Several conclusions can be made based on Table 3.1: Table 3.1 Eigenvalues of GAMs. Model Eigenvalues 8th-order GAM without magnetizing inductance or core losses [16] (-2.02±j2.0537)e4, (-1.5559±j3.0576)e4, -2.6011e3, -1.0399e4,  (-2.4256±j16.001)e4 Improved 10th-order GAM with magnetizing inductance and core losses (-2.0202±j2.0528)e4, (-1.5559±j3.0576)e4, -2.6021e3, -1.0399e4,  (-2.425±j16.001)e4, -14.161±j1.5716e5  1) It can be seen that due to the existence of magnetizing inductance, there are two more eigenvalues, i.e. -14.161±j1.5716e5, in the improved 10th –order GAM. The remaining eight eigenvalues are very similar.  2) Since the core loss resistance RM is much larger than the other resistances in the converter circuit, the absolute value of the real part of the two new eigenvalues are much smaller than that of the other eight eigenvalues. As a result, the system becomes numerically stiff  32  due to the presence of RM, which may require special considerations in MATLAB/Simulink simulations.  3) The two eigenvalues (-2.425±j16.001)e4 are caused by the leakage resistance and inductance. Note the values of imaginary part in the 8th –order and improved 10th –order GAMs. In fact, 16.001e4/2π = 25466 and 1.5716e5/2π = 25012, which are close to the switching frequency of 25kHz. These eigenvalues represent the dynamics of transformer. The values of 16.001e4 and 1.5716e5 are much larger than the imaginary part of others eigenvalues, which implies that the dynamics of transformer is much faster than that of the input/output filters. Based on this observation, a reduced-order model may be derived. 3.3 Small-signal AC Model based on Full-order Model Small-signal ac model can be derived based on the improved 10th –order GAM obtained in Section 3.1. The variables vis, vos, and d are assumed to be the input parameters, and they are set to quiescent values plus some superimposed small ac variations, i.e.,  isisis vVv ˆ ,  (3.42)  ososos vVv ˆ ,  (3.43)  dDd ˆ .  (3.44) In response to these inputs, in small-signal sense, the output averaged parameters and state variables (i1, i2, vc1, vc2, vc3, vc4, it1R, it1I, iM1R and iM1I) will also be equal to the corresponding quiescent values plus some superimposed small-signal ac variations [24].  111 iˆIi  ,  (3.45)  222 iˆIi  ,  (3.46)  111 ˆccc vVv  ,  (3.47)  222 ˆccc vVv  ,  (3.48)  333 ˆccc vVv  ,  (3.49)  444 ˆccc vVv  ,  (3.50)  RtRtRt iIi 111 ˆ ,  (3.51)  33   ItItIt iIi 111 ˆ ,  (3.52)  RMRMRM iIi 111 ˆ ,  (3.53)  IMIMIM iIi 111 ˆ .  (3.54) Substituting (3.42) - (3.54) into (3.5) - (3.8) and (3.36) - (3.41), eliminating the DC terms and neglecting second and higher order ac terms, the resulting small-signal ac model based on the improved 10th –order GAM considering magnetizing inductance and core losses can be obtained.    It should be noted that during derivations, the trigonometric functions needed to be linearized as well. According to Taylor series, when variable x is close enough to zero, the functions sin(x) and cos(x) can be expressed as an infinite convergent series,         1253 !121!51!31sin nnxnxxxx,  (3.55)         nnxnxxx242!21!41!211cos.  (3.56) If second and higher-order items are ignored, than (3.55) and (3.56) can be approximated as    xx sin ,  (3.57)    1cos x .  (3.58) Then, utilizing (3.57) and (3.58), take for example the term sin(dπ)·vc3, we obtain,                             DvDdVDVvVDdDvVdDdDvVdDvdccccccccccsinˆcosˆsinˆcosˆsinˆˆsincosˆcossinˆˆsinsin3333333333,  (3.59) Other nonlinear terms are approximated similarly.  Finally, the small-signal ac model based on the improved 10th –order GAM considering the magnetizing inductance and core losses is summarized in the following set of equations:  34   111 ˆˆˆ cis vvdidL ,  (3.60)  oosc RivvdidL 2322 ˆˆˆˆ ,  (3.61)  12122ˆˆˆRvvdvdC ccc ,  (3.62)  24344ˆˆˆRvvdvdC ccc ,  (3.63)  Itccc iRvvidvdC 1121111 ˆ4ˆˆˆˆ  ,  (3.64)      DiDdIDIDiDdIDInDiDdIDIDiDdIDInRvviRnvdvdCIMIMIMItItItRMRMRMRtRtRtccMcccosˆsinˆcoscosˆsinˆcos4sinˆcosˆsinsinˆcosˆsin4ˆˆˆˆˆ11111111111124322333,  (3.65)   DvDdVDVniLiRdidL cccItseqRteqRteq  sinˆcosˆsin2ˆˆˆ 333111 ,  (3.66)   DvDdVDVnviLiRdidL ccccRtseqIteqIteq  cosˆsinˆcos2ˆ2ˆˆˆ 3331111 , (3.67)   DvDdVDVniLdidL cccIMsMRMM  sinˆcosˆsin2ˆˆ 33311 ,  (3.68)   DvDdVDVniLdidL cccRMsMIMM  cosˆsinˆcos2ˆˆ 33311 ,  (3.69)  3.4 Full-order Model Case Studies  The same DAB converter experimental prototype shown in Figure 2.8 is used here to verify the accuracy of the full-order average models developed in this chapter. The operating frequency 25kHz is considered. Besides experimental results, two models are compared: 1) The 8th –order GAM [16] without magnetizing inductance or core losses.   35  2) The improved 10th –order GAM considering the magnetizing inductance and core losses.  3.4.1 Steady State Prediction  Table 3.2 summarizes the comparison of variables, including input current i1, output current i2, and output voltage vo, under different phase shift ratio d in steady state as measured experimentally and predicted by the two subject full-order models. It should be noted that the output voltage vo is different from output source voltage vos. In order to study the power loss in DAB converter, we define  ooso RiVv  2 .  (3.70) Table 3.2 Accuracy precision of 8th –and improved 10th –order GAMs in predicting the steady state input current, output current, and output voltage. All of errors in this table are normalized. Variable Model / prototype Phase shift ratio d 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 Input current /A Experiment  0.857 1.41 2.044 2.641 3.18 3.65 4.038 4.34 4.55 8th –order GAM 0.655 1.2386 1.837 2.43 2.996 3.5117 3.9572 4.314 4.568 Improved 10th –order GAM 0.655 1.238 1.836 2.4286 2.993 3.509 3.954 4.31 4.564 8th –order GAM error -23.57% -12.16% -10.13% -7.99% -5.79% -3.79% -2.00% -0.60% 0.40% Improved 10th –order GAM error -23.57% -12.20% -10.18% -8.04% -5.88% -3.86% -2.08% -0.69% 0.31% Output current /A Experiment  1.855 2.925 4.014 4.93 5.667 6.23 6.62 6.84 6.92 8th –order GAM 1.457 2.645 3.741 4.72 5.557 6.2316 6.727 7.031 7.136 Improved 10th –order GAM 1.417 2.603 3.699 4.677 5.513 6.187 6.6815 6.9852 7.09 8th –order GAM error -21.46% -9.57% -6.80% -4.26% -1.94% 0.03% 1.62% 2.79% 3.12% Improved 10th –order GAM error -23.61% -11.01% -7.85% -5.13% -2.72% -0.69% 0.93% 2.12% 2.46% Output voltage /A Experiment  20.96 21.54 22.13 22.63 23.04 23.34 23.56 23.68 23.71 8th –order GAM 20.73 21.3224 21.87 22.36 22.778 23.1158 23.364 23.5155 23.568 Improved 10th –order GAM 20.708 21.301 21.85 22.338 22.756 23.093 23.34 23.49 23.545 8th –order GAM error -1.10% -1.01% -1.17% -1.19% -1.14% -0.96% -0.83% -0.69% -0.60% Improved 10th –order GAM error -1.20% -1.11% -1.27% -1.29% -1.23% -1.06% -0.93% -0.80% -0.70%   Based on results presented in Table 3.2 it can be seen that the accuracy precision of 8th -and improved 10th –order GAMs in predicting steady state variables are more or less similar.  36  Overall, the accuracy is getting better when d becomes larger. This is because the generalized averaging method considered in this chapter is based on the first harmonic of Fourier series. When the phase shift ratio d increases, the transformer load increases and the total harmonic distortion (THD) of the current decreases (current becomes more sinusoidal). Therefore, the waveform of transformer current affects the accuracy of the generalized average models.  3.4.2 Power Losses and Efficiency  The accuracy of the 8th – and improved 10th –order GAMs in predicting the total power losses and efficiency is considered next. Based on the data summarized in Table 3.2, if the power losses in the input and output filters are ignored in steady state, the power losses and efficiency of DAB can be calculated   21 iviVP oinloss  ,  (3.71)  isin VV  ,  (3.72)  %10012  iVivEfficiencyino,  (3.73) where vo is defined in (3.70).  Table 3.3 Accuracy precision of 8th –and improved 10th –order GAMs in predicting power losses and efficiency. Power losses errors are normalized. Efficiency errors are absolute. Variable Model Phase shift ratio d 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 Power losses /W Experiment  2.2552 4.6755 9.2821 15.202 22.072 29.791 37.856 46.348 54.326 8th –order GAM 1.228 3.064 6.364 11.11 17.2 24.51 32.78 41.74 51.06 Improved 10th –order GAM 2.092 3.975 7.32 12.1 18.24 25.56 33.84 42.8 52.127 8th –order GAM error -45.5% -34.4% -31.4% -26.9% -22.1% -17.7% -13.4% -9.9% -6.0% Improved 10th –order GAM error -7.2% -14.9% -21.1% -20.4% -17.4% -14.2% -10.6% -7.6% -4.0% Efficiency Experiment  94.52% 93.09% 90.54% 88.01% 85.54% 83.00% 80.47% 77.75% 75.13% 8th –order GAM 96.10% 94.84% 92.78% 90.47% 88.03% 85.45% 82.74% 79.84% 76.71% Improved 10th –order GAM 93.35% 93.31% 91.70% 89.62% 87.31% 84.82% 82.17% 79.31% 76.21% 8th –order GAM error 1.58% 1.75% 2.24% 2.46% 2.49% 2.45% 2.27% 2.09% 1.59% Improved 10th –order GAM error -1.17% 0.22% 1.16% 1.61% 1.77% 1.82% 1.70% 1.56% 1.09%  37   It can be seen in Table 3.3 that improved 10th –order GAM has better accuracy in prediction of power losses and efficiency compared with 8th –order GAM, especially when d is small. This is because the core losses of transformer are not negligible when d is small. Al low power level, the current is low as well, and the core losses become significant in proportion to the total power conversion. However, due to the assumption of the 1st harmonic, the error of power losses always exists in such generalized average models. The experimentally measured power losses are always a bit higher than what is predicted by the models.    Figure 3.1 Power losses and efficiency as predicted by the considered models with respect to actual measurements. The results from experiment measurement are denoted by stars. The results of detailed model with RM are denoted by crosses. The detailed model without RM, i.e. without core losses is denoted by points. Finally the 8th –order GAM is denoted by solid line, and the improved 10th –order GAM is shown by dash-dotted line. 38    Figure 3.1 shows the power losses and efficiency comparison between the experiment results, the detailed model with core losses, the detailed model without core losses, and the 8th –order and improved 10th –order GAMs, respectively. It can be seen in Figure 3.1 that the improved 10th –order GAM has a better accuracy in predicting the power loss and efficiency when phase shift ratio d is small. The 8th –order GAM matches the detailed model without core losses well, since both of these models have similar assumptions. Figure 3.1 clearly shows that core loss become significant part when d is small. Also, Figure 3.1 indicates that the results from the detailed model match the experiment values very well. However, the averaging method used to obtain the 8th –order and improved 10th –order GAMs are based on considering only the fundamental 1st harmonic of transformer current, which may be not accurate when the actual transformer current is distorted. Therefore, it is necessary to develop another average model based on the equivalent detailed model to further improve the steady state accuracy. 3.4.3 Open-loop Dynamic Response The full-order generalized averaged models presented in this chapter can also be used in predict the dynamic response of the converter circuit. In the following study, the converter is assumed to initially operate in steady state defined by d = 0.1. At time instant of 0.05s, the control variable d is increased from 0.1 to 0.3. The output current i2 and the transformer current itr during the transition period are shown in Figure 3.2. The output current i2 and transformer current itr rising is caused by the increase of phase shift ratio. As can be observed in Figure 3.2, the trend predicted by the 8th –order and improved 10th –order GAMs match the experiment result, although the steady state error before the time instant 0.05s is also visible. However, the detailed model matches the hardware measurements to a very great accuracy.    39   Figure 3.2 Output current and transformer current resulting from a phase-shift step change. The experiment measured results are denoted by dashed line. The detailed model is shown by the solid line. The 8th –order and improved 10th –order GAMs are denoted by the dotted and dashed-dot lines, respectively.   3.4.4 Frequency-domain Analysis One important feature of the average models is that they are also used for the small-signal analysis in frequency-domain. In this section, we consider the control-to-output transfer function of the DAB converter, which is often used for controller design. Here, the phase shift ratio d is considered a control variable, and current i2 is the output variable. In this study, the quiescent operating point defined by d = 0.1. The control-to-output transfer function in small-signal sense is extracted from the proposed equivalent detailed model using the PLECS toolbox [46]. The same transfer function is then extracted from the 8th –order and improved 10th –order GAMs using MATLAB/Simulink [49]. The comparison of corresponding bode diagrams is shown in Figure 3.3. As it can be seen in Figure 3.3, the 8th –order and improved 10th –order GAMs are in good agreement with the equivalent detailed model in a wide range of frequency (even close to the switching frequency). This is the advantage of generalized average models which consider the dynamics of the fundamental component of the transformer current (using real and imaginary  40  parts of Fourier series coefficients). It also seems that the 10th –order GAM has no significant advantage over the 8th –order GAM in frequency domain analysis. This is because the magnetizing inductance LM and core loss resistance RM are sufficiently large and do not have pronounced effect in frequency domain.   Figure 3.3 Bode plots of control-to-output transfer function. The results from the detailed model are based on using the PLECS toolbox (dashed line). The results of 8th –order (dashed-dot line) and improved 10th –order (dotted line) GAMs also superimposed.      41  Chapter 4 Reduced-order Model of DAB Converter   Since in DAB converter the transformer current is AC, the conventional state-space averaging method is not easily applicable in the same way it is applied to other DC-DC converters. The approaches to extend the classical state-space averaging method may include increasing the state-space order and extract the transient component of transformer current to form full-order model, either in discrete time [22], [40] or continuous time [16]. In Chapter 3, a new generalized full-order average model with core loss and magnetizing current is obtained using continuous time formulation. Another approach is to eliminate the switching frequency AC component of the transformer current, and then reduce the model order by removing the state variables which are changing much faster than the DC currents. In [34], [35] and [36], the reduced-order models without losses have been proposed. In this chapter, a new reduced-order model of DAB converter considering the power losses, including switch conduction losses, the transformer copper and core losses is proposed. The proposed new reduced-order model is derived based on the equivalent detailed circuit shown in Figure 2.2.  4.1 State Equations of DAB Converter in Steady State  As shown in Figure 2.2, the converter circuit can be divided into three sub-circuits: Part I, Part II, and Part III. The sub-circuits corresponding to Parts I and III can be described by the following equations,   111 cis vvddiL ,  (4.1)   oosc RivvddiL 2322 ,  (4.2)   12122 RvvddvC ccc ,  (4.3)   24344 RvvddvC ccc ,  (4.4)   42   121111 RvviiddvC ccacic ,  (4.5)   243233 RvviiddvC ccacoc ,  (4.6)  where vc1, vc2, vc3 and vc4 are voltages across capacitors C1, C2, C3 and C4, respectively; iaci and iaco are the average values of input and output current corresponding to Part II; and τ is used to denote time.   Figure 4.1 The transformer input and output currents iaci and n·iaco in half switching period. The values of transformer current peaks are denoted by It1 and It2.   The input/output currents (iaci and iaco) include substantial AC component as shown in Figure 4.1. It is noted that period T in Figure 4.1 is half of the switching period. Due to the existence of Req, the actual curves of iaci and niaco waveforms are piecewise exponentials (not piecewise linear). Assuming that the currents of the magnetizing inductance and resistance (LM and RM) are negligible, iaci can be obtained using the following equation       TdTeRvvIRvvidTeRvvIRvviidTLReqccteqccLReqccteqccacieqeqeqeqfor;0for;3123131131,  (4.7)   43  where v’c3 = vc3/n, It1 and It2 are the transformer peak currents. It is further assumed that the ripple on vc1 and vc3 are sufficiently small, and then vc1 and vc3 may be assumed constant inside interval T.  Further analyzing Figure 4.1, the values of inductor current at time instants of 0, dT and T are respectively denoted as –It1, It2 and It1, i.e.,    10 tIi  ,  (4.8)     2tIdTi  ,  (4.9)     1tITi  .  (4.10)  Referring (4.7) to (4.10), the currents It1 and It2 can be derived as   TLRTLReqccdTTLReqceqccteqeqeqeqeqeqeeRvveRvRvvI12 313311 ,  (4.11)   TLRTLReqccdTLReqceqccteqeqeqeqeqeqeeRvveRvRvvI12 311312 .  (4.12)  Note that the waveforms of iaci and n·iaco from 0 to dT are symmetrical about the horizontal axis. Therefore, the average value of iaci and n·iaco for the period T has the following form     dT TdTaci didiTi 01  ,  (4.13)  and       dT TdTaco didiTin 01  ,  (4.14)  respectively. Substituting (4.7) into (4.13) and (4.14), the value of iaci and n·iaco are expressed as  44      1112313131131dTTLRteqcceqeqeqccdTLReqccteqeqeqccacieqeqeqeqeIRvvRLTdRvveRvvIRLdTRvviT,  (4.15)      1112313131131dTTLRteqcceqeqeqccdTLReqccteqeqeqccacoeqeqeqeqeIRvvRLTdRvveRvvIRLdTRvvinT,  (4.16)  Therefore, (4.1) - (4.6), (4.11), (4.12), (4.15) and (4.16) constitute the reduced-order average model without core losses and magnetizing inductance. 4.1.1 Effect of Magnetizing Inductance and Core Losses In the model defined by (4.1) - (4.6), (4.11), (4.12), (4.15) and (4.16), the magnetizing current through LM and the current through RM have not been considered. Actually, the elements LM and RM will draw some current from the power source. Considering this effect, (4.14) is modified as   McTMdT TdTaco RnvdiTdidiTin     300 11  ,  (4.17)  where iM is the current in LM.  It should be noted that for the magnetizing inductance, the period of its charging and discharging depends on voltage vc3 during the interval 2T. Therefore, the averaging of iaco should be taken in 2T and (4.17) is further modified as   McT MdT TdTaco RnvdiTdidiTin     3200 212221  .  (4.18)  Moreover, the one-period integral of iM in steady state should be zero, i.e.   T M di20 0 .  (4.19)   45  Therefore, the existence of LM will not affect the modeling of DAB converter, and (4.18) may be further simplified as   McdT TdTaco RnvdidiTin     301  ,  (4.20)  The reduced-order average model with core losses and magnetizing inductance is defined by (4.1) to (4.6), (4.11), (4.12), (4.15), (4.16) and (4.20), which is a 6th –order model. This model is referred as 6th –order RAVM (reduced-order average value model) in this thesis. Based on available average values of the input and output currents, iaci and iaco, it is possible to express the average input and output power transferred by DAB converter considering the copper loss and core loss as  acicin ivP  1 ,  (4.21)   acocout ivP  3 .  (4.22)  The total power loss and efficiency are  outinloss PPP  .  (4.23)   %100inoutPPEfficiency.  (4.24)  Equation (1.1), which is based on an ideal DAB converter without power losses, can be replaced by (4.21), (4.22) and (4.23) to analyze the DAB converter loss. Note that (3.71), (3.72) and (3.73) also calculate the DAB converter power losses and efficiency, but ignore the power losses from the filters. However, (4.21) - (4.23) directly calculate the power transferred by the DAB converter.  4.1.2 Reduced-order Average Model of DAB The proposed 6th –order RAVM is shown in Figure 4.2, where the variables are referring to (4.1) - (4.6), (4.11), (4.12), (4.15) and (4.20). Here, the currents iaci and n·iaco are represented as dependent current source, and the transformer is treated as ideal DC current transformer (to scale variables). Since there are no switching components in the proposed new 6th –order RAVM, its implementation and simulation may be much faster than that of the detailed model. Also, the  46  small-signal model and frequency-domain analysis can be obtained by numerical linearization of this new 6th –order RAVM, which are analytically derived in the following Section 4.3.   Figure 4.2 The DAB converter average model circuit, where the currents iaci and niaco are dependent current source which are functions of vc1, vc3 and d.  4.2 Reduced-order Model Eigenvalues Analysis Based on the reduced-order model defined in Section 4.1, the eigenvalues of this sixth new 6th –order RAVM can be obtained and analyzed. Assuming an operating point defined by d = 0.1, the eigenvalues have been calculated and are shown in Table 4.1. For comparison, the eigenvalues of 8th –order and improved 10th –order GAMs are also provided. As it can be observed in Table 4.1, the proposed new 6th –order RAVM neglects the eigenvalues corresponding to the much faster dynamics of the transformer. This model is also non-stiff, which may result in faster simulations. In fact, the eigenvalues corresponding to the input and output filters present in all models are very similar.  Table 4.1 Eigenvalues of 6th –order RAVM and GAMs. Eigenvalues of proposed new 6th –order RAVM (-2.0394±j2.0816)e4, (-1.5766±j3.0707)e4, -2.6055e3, -1.0402e4 Eigenvalues of 8th –order GAM (-2.02±j2.0537)e4, (-1.5559±j3.0576)e4, -2.6011e3, -1.0399e4,  (-2.4256±j16.001)e4 Eigenvalues of improved 10th –order GAM (-2.0202 ± j2.0528)e4, (-1.5559 ±j3.0576)e4, -2.6021e3, -1.0399e4,  (-2.425±j16.001)e4, -14.161±j1.5716e5   47  4.3 Small-signal ac Model based on Reduced-order Model 4.3.1 Derivation of Small-signal Model To construct a small-signal ac model at a DC operating point, the variables vis, vos, and d are assumed to equal to their quiescent values, plus some superimposed small ac perturbations, i.e.,  isisis vVv ˆ ,  (4.25)   ososos vVv ˆ ,  (4.26)   dDd ˆ .  (4.27)  The output averaged variables i1, i2, vc1, vc2, vc3 and vc4 will also be equal to their corresponding quiescent values, plus some small superimposed ac variations [24].   222 iˆIi  ,  (4.28)   111 iˆIi  ,  (4.29)   111 ˆccc vVv  ,  (4.30)   222 ˆccc vVv  ,  (4.31)   333 ˆccc vVv  ,  (4.32)   444 ˆccc vVv  .  (4.33)  The small-signal model can be derived by substituting (4.25) to (4.33) into the previous large-signal average model. In this process, the DC terms will be eliminated, and the second or higher order ac terms will be neglected. Noting that during the derivation process, there will be equations with terms dae ˆ , where a is a constant coefficient. According to Taylor Series, when dˆ  is close enough to zero, the term dae ˆ  can be expressed as an infinite convergent series,   022ˆ ˆ!ˆ!2ˆ!11!ˆnnnnda dnadadandae .  (4.34)  If the second and higher order items are ignored, dae ˆ  will be approximated as  48   dae da ˆ1ˆ  .  (4.35)  When (4.35) is utilized to linearize the average model, the resulting small-signal model can be obtained as follows,  111 ˆˆˆcis vvdidL ,  (4.36)   oosc RivvdidL 2322 ˆˆˆˆ ,  (4.37)   12122ˆˆˆRvvdvdC ccc ,  (4.38)   24344ˆˆˆRvvdvdC ccc ,  (4.39)   121111ˆˆˆˆˆ RvviidvdC ccacic ,  (4.40)   243233ˆˆˆˆˆ RvviidvdC ccacoc ,  (4.41)  where   31 ˆˆˆˆ ccaci vrvqdpi  ,  (4.42)   31 ˆˆˆˆ ccaco vzvydxin  .  (4.43)  Here, the variables p, q, r, and x, y, z are defined as    nVeRenRVp cTLReqTDLReqceqeqeqeq313142,  (4.44)   49   11211TLReqTLReqeqeqeqeqeqeqeReRLTRq,  (4.45)    112211211TLReqTDLRTLReqeqeqeqeqeqeqeqeqeReeRLTRDnr,  (4.46)   11142cTLReqDTLReqcVeReRVxeqeqeqeq,  (4.47)   1122121TLReqDTLRTLReqeqeqeqeqeqeqeqeqeReeRLTRDy,  (4.48)   TLReqTLReqeqeqeqeqeqeqeReRLTRnz112111.  (4.49)  If Laplace Transform is applied to (4.36) - (4.43), the control-to-output transfer function can be defined as           0ˆ,0ˆ2ˆˆsvsvidosissdsisG.  (4.50)   50  4.3.2 Reduced-order Small-signal ac Model of DAB Converter Figure 4.3 shows the small-signal ac equivalent circuit of the DAB converter. Compared with Figure 4.2, it can be seen that the construction of small-signal ac model is the same as for the proposed new 6th –order RAVM, except the coefficients of dependent current source are somewhat different.   Figure 4.3 Small-signal ac model of DAB, where aciiˆ  and acoin ˆ  are dependent current source which are functions of vc1, vc3 and d.  4.4 Reduced-order Model Case Studies  In this section, the proposed new 6th –order RAVM is studied to verify and compared to the previous models and the experimental prototype. The converter prototype has been already shown in Figure 2.8. In this section, in addition to experiment results, the following three models are considered: 1) Full-order model considering magnetizing inductance and core losses (improved 10th –order GAM). 2) Reduced-order model with ideal transformer, i.e. no winding resistance, no magnetizing inductance and no core losses, as defined in [34], [35], and [36]. 3) The new 6th –order RAVM model considering winding resistance, switch conduction resistance, magnetizing inductance, and core losses (6th –order RAVM), as proposed in this thesis.  4.4.1 Steady State Predictions Table 4.2 shows the comparison of the input current i1, output current i2, and output voltage vo, under different phase shift ratio d in steady state. As it can be seen in Table 4.2 and  51  Figure 4.4 (take the output current as an example), the three considered models have the following characters: 1) The accuracy of the improved 10th –order GAM depends on phase shift ratio d. As it was observed in Section 3.4.1, the full-order generalized averaging method is based on the fundamental waveform in Fourier series, and its accuracy becomes better when d increases and THD decreases. 2) For the ideal reduced-order model, however, it happens to be the opposite case - smaller d results in higher accuracy. The reason is that when d is small, the current is at a low level and the DAB converter can be treated as lossless, i.e. ideal converter. However, as d becomes larger, the error of ideal reduced-order model grows.  3) The proposed new 6th –order RAVM with losses has good accuracy in prediction of all variables in wide range of d. Moreover, this is model is not derived based on the fundamental component (as the 8th –order and improved 10th –order GAMs are), and its overall errors remain very low. Note that all three models cannot predict the performance very well when d is very small, i.e. d = 0.05, the third column in Table 4.2 and Figure 4.4.   52  Table 4.2 Accuracy precision of improved 10th –order GAM, ideal reduced-order model, and the proposed new 6th –order RAVM in predicting the steady state variables.All errors are normalized. Variable Model Phase shift ratio d 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45  Input current /A  Experiment  0.857 1.41 2.044 2.641 3.18 3.65 4.038 4.34 4.55 Improved 10th –order GAM with losses 0.655 1.238 1.836 2.4286 2.993 3.509 3.954 4.31 4.564 Ideal reduced-order model 0.677 1.3265 1.933 2.4848 2.97 3.38 3.7065 3.944 4.088 Proposed new 6th –order RAVM with losses 0.76 1.4 2.01 2.58 3.095 3.545 3.925 4.224 4.438 Improved 10th –order GAM error -23.57% -12.20% -10.18% -8.04% -5.88% -3.86% -2.08% -0.69% 0.31% Error of ideal reduced-order model -21.00% -5.92% -5.43% -5.91% -6.60% -7.40% -8.21% -9.12% -10.15% Error of proposed new 6th –order RAVM -11.32% -0.71% -1.66% -2.31% -2.67% -2.88% -2.80% -2.67% -2.46% Output current /A Experiment  1.855 2.925 4.014 4.93 5.667 6.23 6.62 6.84 6.92 Improved 10th –order GAM with losses 1.417 2.603 3.699 4.677 5.513 6.187 6.6815 6.9852 7.09 Ideal reduced-order model 1.564 2.964 4.199 5.269 6.175 6.916 7.492 7.904 8.151 Proposed new 6th –order RAVM with losses 1.645 2.924 4.013 5.06 5.645 6.194 6.573 6.784 6.832 Improved 10th –order GAM error -23.61% -11.01% -7.85% -5.13% -2.72% -0.69% 0.93% 2.12% 2.46% Error of ideal reduced-order model -15.69% 1.33% 4.61% 6.88% 8.96% 11.01% 13.17% 15.56% 17.79% Error of proposed new 6th –order RAVM -11.32% -0.03% -0.02% 2.64% -0.39% -0.58% -0.71% -0.82% -1.27% Output voltage /A Experiment  20.96 21.54 22.13 22.63 23.04 23.34 23.56 23.68 23.71 Improved 10th –order GAM with losses 20.708 21.301 21.85 22.338 22.756 23.093 23.34 23.49 23.545 Ideal reduced-order model 20.782 21.482 22.1 22.635 23.087 23.458 23.746 23.952 24.075 Proposed new 6th –order RAVM with losses 20.822 21.462 22.01 22.46 22.822 23.097 23.29 23.39 23.416 Improved 10th –order GAM error -1.20% -1.11% -1.27% -1.29% -1.23% -1.06% -0.93% -0.80% -0.70% Error of ideal reduced-order model -0.85% -0.27% -0.14% 0.02% 0.20% 0.51% 0.79% 1.15% 1.54% Error of proposed new 6th –order RAVM -0.66% -0.36% -0.54% -0.75% -0.95% -1.04% -1.15% -1.22% -1.24%   53   Figure 4.4 Output current error trends as predicted by three models.  4.4.2 Power Losses and Efficiency  In this section, only improved 10th –order GAM and the proposed new 6th –order RAVM will be discussed. For comparison, detailed model will also be included. The results are depicted Figure 4.5 and Figure 4.6 for better comparison.  As it was determined in Chapter 3, the improved 10th –order GAM has a better accuracy in prediction of power loss and efficiency than 8th –order GAM. As it can be observed in Figure 4.5, the prediction error from the proposed new 6th –order RAVM is very little with respect to the experimental results. The new 6th –order RAVM includes the majority of power losses in the circuit. However, due to the existence of switching losses and losses in wire and PCB, the experimental power losses are always a bit higher than in any of the models. Additionally, it is observed that the proposed new 6th –order RAVM order model is more accurate for different steady state operating points than the improved 10th –order GAM.   54   Table 4.3 Accuracy precision of the improved 10th –order GAM, proposed new 6th –order RAVM in predicting the power losses and efficiency. Power losses errors are normalized. Efficiency errors are absolute. Variable Model Phase shift ratio d 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 Power losses /W Experiment  2.2552 4.6755 9.28218 15.2021 22.0723 29.7918 37.8568 46.348 54.326 Improved 10th –order GAM with losses 2.092 3.975 7.32 12.1 18.24 25.56 33.84 42.8 52.127 Proposed new 6th –order RAVM with losses 2.205 4.412 8.17 13.34 19.732 27.145 35.34 44.06 53.04 Improved 10th –order GAM error -7.24% -14.98% -21.14% -20.41% -17.36% -14.20% -10.61% -7.66% -4.05% Error of proposed new 6th –order RAVM  -2.23% -5.64% -11.98% -12.25% -10.60% -8.88% -6.65% -4.94% -2.37% Efficiency Experiment  94.52% 93.09% 90.54% 88.01% 85.54% 83.00% 80.47% 77.75% 75.13% Improved 10th –order GAM with losses 93.35% 93.31% 91.70% 89.62% 87.31% 84.82% 82.17% 79.31% 76.21% Proposed new 6th –order RAVM with losses 93.95% 93.43% 91.50% 89.20% 86.71% 84.05% 81.24% 78.27% 75.10% Improved 10th –order GAM error -1.17% 0.22% 1.16% 1.61% 1.77% 1.82% 1.70% 1.56% 1.09% Error of proposed new 6th –order RAVM -0.57% 0.34% 0.96% 1.19% 1.17% 1.05% 0.77% 0.52% -0.02%   55   Figure 4.5 Power losses and efficiency predictions compared to the actual measurements. The results from experiment measurements are denoted by stars. The detailed model is denoted by crosses. The improved 10th –order model is denoted by dash-dotted line. The proposed new 6th –order RAVM is denoted by solid line.     56   Figure 4.6 Magnified plot of Figure 4.5.  4.4.3 Open-loop Transient Study  The proposed new 6th –order RAVM should be able to accurately predict the dynamic response of an actual circuit. In this study, similar to the Section 3.4.3, the converter is assumed to initially operate under d = 0.1. At t = 0.05s, the control variable d is increased from 0.1 to 0.3. The output current i2 and the transformer current itr during this transition period are shown in Figure 4.7, as predicted by experiment and various models. As observed in Figure 4.7, the new 6th –order RAVM and detailed model match the experimental results extremely well. A small mismatch between these models and the experimental results are mainly caused by the additional higher-order parasitics in the actual circuit (such as series resistance from skin effects) which are not considered in the modeling. The errors in the output current predicted by the improved 10th –order model are larger when d = 0.1. The reason for this is that the transformer current contains more harmonics under this loading condition (as compared to when d = 0.3), and the accuracy captured only by the fundamental component is not sufficient.  57   Figure 4.7 Output current and transformer current transient due to a phase shift change. The results from experiment hardware measurement are denoted by dashed line. The results from the detailed model are denoted by solid line. The results of the improved 10th –order GAM are shown by dotted line. The results of proposed new 6th –order RAVM is denoted by the dashed-dot line.  4.4.4 Frequency-domain Analysis  Similar to Section 3.4.4, the quiescent operating point with d = 0.1 is considered for the frequency-domain analysis in this section. A control-to-output transfer function in small-signal sense is extracted from the detailed model using PLECS toolbox [46]. The same transfer function is then obtained for the three average models using MATLAB/Simulink [49]. Additionally, one more result is derived from the proposed new 6th –order RAVM using the analytical expression (4.50) for the small-signal model transfer function (SSMTF). The comparison of bode plots are shown in Figure 4.8. The small-signal ac model derived from the proposed new 6th –order RAVM is in good agreement with that extracted from the detailed model, particularly at the low-frequency. There is little mismatch between the proposed new 6th –order RAVM, the improved 10th-order GAM and the detailed model results at low frequencies. In high frequency range (around 10kHz), the curve from the proposed 6th –order RAVM starts deviating from the detailed model result. This is because the analysis is only applicable when the frequency is lower than the  58  converter switching frequency. The result from improved 10th –order GAM matches the detailed model in high frequency very well (which is due to its higher order). As mentioned in [16], [40], the accuracy of ideal reduced-order model is worse than that of the improved 10th –order GAM, especially in magnitude. However, the proposed new 6th –order RAVM does not have this problem, and it offers great accuracy and simplicity.  Figure 4.8 Bode plots of the control-to-output transfer functions. The results from detailed model based on PLECS are denoted by dashed line. The results of the improved 10th –order GAM are shown by dashed-dot line. The ideal reduced-order model is shown by dotted line. The proposed new 6th –order RAVM is denoted by dotted line. The analytical result predicted based on the proposed new 6th –order RAVM SSMTF is denoted by solid line.  59  Chapter 5 Summary of Contributions and Future Work 5.1 Representation of Conduction, Switching, and Core Losses This thesis explores the possibility to add variable components in the detailed model to pursue higher accuracy, which has not been done in the previous literature. Specifically, variable resistances representing core losses and switching losses, respectively, are added into the improved detailed model of DAB converter, which achieves Objective 1. Simple formulas are derived to estimate switching losses, in terms of phase shift ratio, and core losses in terms of switching frequency. 5.2 Detailed and Average-value Modeling of DAB This thesis presents DAB circuit analysis, detailed modeling, equivalent detailed modeling, improved 10th –order GAM and proposed new 6th –order RAVM, which achieves Objective 2. We present the results for steady state performance in wide range of operating conditions and open-loop dynamics for the considered models. Numerous simulations are shown to verify proposed advanced models with respect to experimental converter prototype.  The improved 10th –order GAM includes core losses and magnetizing inductance, which is an improvement compared to the best known 8th –order GAM available in the previous literature. The new 6th –order RAVM overcomes the drawback of the previous full-order models and offers greater accuracy in predicting the total power losses and efficiency in steady state. The new model also shows very good performance for the open-loop dynamic response compared with accurate detailed model and the experiment results. The proposed new 6th –order RAVM includes the leakage resistance and shunt resistance in transformer, the switches turn-on resistance, which have not been considered in previous reduced-order models [35], [36]. The results from hardware measurements verify that the new proposed model has a good accuracy in predicting the power losses and efficiency of DAB converter circuit. 5.3 Small-signal Model of DAB Converter  Based on the improved 10th –order GAM and the proposed new 6th –order RAVM, the new small-signal models have been derived to realize the Objective 3. The proposed new 6th –order RAVM neglects the non-dominant poles of system to simplify small-signal model. The 8th  60  –order and/ or improved 10th –order GAMs generally have a better performance in frequency-domain than the proposed new 6th –order RAVM. However, the new 6th –order RAVM is simple and easy to use without losing much accuracy. The small-signal frequency-domain response from the new 6th –order RAVM agrees well with the equivalent detailed model built in PLECS [46]. This model has no mismatch for magnitude at low frequency as mentioned in [16] and [40]. The proposed model also shows that Taylor series method may be used very effectively for analytical linearization of the AVMs to derive appropriate small-signal models. 5.4 Future Work  It is envisioned that the modeling methodology developed in this thesis for including the effect of losses can also be used in other converter topologies containing transformers and non-ideal components. Since the leakage inductances, winding resistances, magnetizing inductance, and core losses are considered with good results, the proposed approach may be potentially integrated in commonly-used simulation programs used for analysis of power electronic systems. Various controller strategies can be designed based on the obtained average-value models and small-signal models. Some of these topics are presently considered by other graduate students in the UBC’s Electrical Power and Energy Systems research group.  61  Bibliography  [1] L. Zhu, “A novel soft-commutating isolated boost full-bridge ZVS-PWM DC–DC converter for bidirectional high power applications,” IEEE Trans. Power Electron., vol. 21, no. 2, pp. 422–429, Mar. 2006. [2] H. Xiao and S. Xie, “A ZVS bidirectional DC–DC converter with phaseshift plus PWM control scheme,” IEEE Trans. Power Electron., vol. 23, no. 2, pp. 813–823, Mar. 2008. [3] M. H. Kheraluwala, R. W. Gascoigne, D. M. Divan, and E. D. Baumann, “Performance characterization of a high-power dual active bridge dc-todc converter,” IEEE Trans. Ind. Appl., vol. 28, no. 6, pp. 1294–1301, Nov./Dec. 1992. [4] H. Bai and C. Mi, “Eliminate reactive power and increase system efficiency of isolated bidirectional dual-active-bridge DC–DC converters using novel dual-phase-shift control,” IEEE Trans. Power Electron., vol. 23, no. 6, pp. 2905–2914, Nov. 2008. [5] B. Hai, C. Mi, and S. Gargies, “The short-time-scale transient processes in high-voltage and high-power isolated bidirectional dc-dc converters,” IEEE Trans. Power Electron., vol. 23, no. 6, pp. 2648–2656, 2008. [6] R. De Doncker, D. Divan, and M. Kheraluwala, “A three-phase soft-switched high-power-density dc/dc converter for high-power applications,” IEEE Transactions on Industry Application, vol. 27, pp. 63–73, Jan./Feb. 1991. [7] M. N. Kheraluwala, R. W. Gascoigne, D. M. Divan, and E. D. Baumann, “Performance characterization of a high-power dual active bridge,” IEEE Transactions on Industry Application, vol. 28, pp. 1294–1301, Jun. 1992. [8] S. Bhattacharya, T. Zhao, G. Wang, S. Dutta, S. Baek, Y. Du, B. Parkhideh, X. Zhou, and A. Q. Huang, “Design and development of generation-isilicon based solid state transformer,” in Proc. 25th Annu. IEEE Appl. Power Electron. Conf. Expo., Palm Springs, CA, 2010, pp. 1666-1673. [9] J. Shi, W. Gou, H. Yuan, T. Zhao, and A. Huang, “Research on voltage and power balance control for cascaded modular solid-state transformer,” IEEE Trans. Power Electron., vol. 26, no. 4, pp. 1154–1166, Apr. 2011.   62   [10] S. Inoue and H. Akagi, “A bidirectional isolated dc-dc converter as a core circuit of the next-generation medium-voltage power conversion system,” IEEE Trans Power Electron., vol. 22, no. 2, pp. 535–542, 2007. [11] F. Krismer and J. Kolar, “Accurate power loss model derivation of a high-current dual active bridge converter for an automotive application,” IEEE Trans Ind. Electron., vol. 57, no. 3, pp. 881–891, Mar. 2010. [12] A. Alonso, J. Sebastian, D. Lamar, M. Hernando, and A. Vazquez, “An overall study of a dual active bridge for bidirectional dc/dc conversion,” in Energy Conversion Congress and Exposition (ECCE), 2010 IEEE, 2010, pp. 1129 –1135. [13] C. Mi, H. Bai, C. Wang, and S. Gargies, “Operation, design and control of dual H-bridge-based isolated bidirectional dc-dc converter,” IET Power Electron., vol. 1, no. 4, pp. 507–517, Apr. 2008. [14] S. Dutta and S. Bhattacharya, “Predictive current mode control of single phase dual active bridge DC to DC converter,” in Energy Conversion Congress and Exposition (ECCE), 2013 IEEE, 2013, pp. 5526-5533. [15] H. Wen and W. Xiao, “Bidirectional dual-active-bridge DC-DC converter with triple-phase-shift control,” in Applied Power Electronics Conference and Exposition (APEC), 2013 Twenty-Eighth Annual IEEE, 2013, pp. 1972-1978. [16] H. Qin and J. W. Kimball, “Generalized average modeling of dual active bridge DC-DC converter,” IEEE Trans. Power Electron., vol. 27, no. 4, pp. 2078–2084, Apr. 2012. [17] R. T. Naayagi, A. J. Forsyth, and R. Shuttleworth, “High-power DC-DC converter for aerospace applications,” IEEE Trans. Power Electron., vol. 27, no. 11, pp. 4366–4379, Nov. 2012. [18] A. Jain and R. Ayyanar, “PWM control of dual active bridge: Comprehensive analysis and experimental verification,” IEEE Trans. Power Electron., vol. 26, no. 4, pp. 1215–1227, Apr. 2011. [19] S. A. Akbarabadi, M. Sucu, H. Atighechi, J. Jatskevich, “Numerical average value modeling of sceond order flyback converter in both operational modes,” in Proc. IEEE 14th Workshop COMPEL, 2013, pp. 1–6.   63   [20] M. Plesnik, “Use of the State-Space Averaging Technique in Fast Steady-State Simulation Algorithms for Switching Power Converters,” Canadian Conference on Electrical and Computer Engineering(CCECE), May 2006, pp. 2224-2227. [21] G. W. Wester and R. D. Middlebrook, “Low-frequency characterization of switched dc–dc converters,” IEEE Trans. Aerosp. Electron. Syst., vol. AES-9, no. 3, pp. 376–385, May 1973. [22] F. Krismer and J. W. Kolar, “Accurate small-signal model for the digital control of an automotive bidirectional dual active bridge,” IEEE Trans. Power Electron., vol. 24, no. 12, pp. 2756–2768, 2009. [23] G. Nirgude, R. Tirumala, and N. Mohan, “A new, large-signal average model for single-switch DC-DC converters operating in both CCM and DCM,” Power Electronics Specialists Conference (PESC), 2001, vol. 3, pp. 1736-1741. [24] R. W. Erickson and D. Maksimovic, Fundamentals of Power Electronics, 2nd ed. New York: Springer-Verlag, 2001 [25] A. Davoudi, J. Jatskevich, and P.L. Chapman, “Averaged modelling of switchedinductor cells considering conduction losses in discontinuous mode,” IET, May 2007, vol. 1, no.3, pp. 402-406. [26] D. Maksimovic, A.M. Stankovic, V.J. Thottuvelil, and G.C. Verghese, “Modeling and simulation of power electronic converters,” Proceedings of the IEEE, Jun 2001, vol. 89, no. 6, pp. 898–912. [27] R. D. Middlebrook and S. Cuk, “A General Unified Approach to Modelling Switching Converter Power Stage,” IEEE PESC Record, 1976, pp. 18-34. [28] A. Reatti, L. Pellegrini, and M. K. Kazimierczuk, “Impact of boost converter parameters on open-loop dynamic performance for DCM,” ISCAS, 2002, vol. 5, pp. 513-516. [29] L.R. Pujara, M. K. Kazimierczuk, and A. N. I. Shaheen, “Robust stability of PWM buck DC-DC converter,” Proceedings of the IEEE International Conference, Sep 1996, pp. 632-637. [30] J. Chen and K. D. T. Ngo, “Alternate forms of the PWM switch model in discontinuous conduction mode”, IEEE Trans. Aerosp. Electron. Syst., 2001, vol. 37, no. 2, pp. 754-758.   64   [31] J. Sun, D. M. Mitchell, M. F. Greuel, P. T. Krein, and R.M. Bass, “Averaged modeling of PWM converters operating in discontinuous conduction mode,” IEEE Trans. On Power Electronics, Jul 2001, vol. 16, no. 4, pp. 482-492. [32] R. D. Middlebrook and S. Ćuk, “A general unified approach to modelling switching-converter power stages,” in Proc. IEEE PESC, Cleveland, OH, Jun. 8–10, 1976, pp. 18–34. [33] S. Bhattacharya, T. Zhao, G. Wang, S. Dutta, S. Baek, Y. Du, B. Parkhideh, X. Zhou, and A. Q. Huang, “Design and development of generation-isilicon based solid state transformer,” in Proc. 25th Annu. IEEE Appl. Power Electron. Conf. Expo., Palm Springs, CA, 2010, pp. 1666-1673. [34] H. K. Krishnamurthy and R. Ayyanar, “Building block converter module for universal (ac–dc, dc–ac, dc–dc) fully modular power conversion architecture,” in Proc. IEEE Power Electron. Spec. Conf., 2007, pp. 483–489. [35] H. Bai, M. Chunting, W. Chongwu, and S. Gargies, “The dynamic model and hybrid phase-shift control of a dual-active-bridge converter,” in Proc. 34th Annu. Conf. IEEE Ind. Electron. (IECON), 2008, pp. 2840–2845. [36] H. Bai, Z. Nie, and C. C. Mi, “Experimental comparison of traditional phase-shift, dual-phase-shift, and model-based control of isolated bidirectional dc-dc converters,” IEEE Trans. Power Electron., vol. 25, no. 6, pp. 1444–1449, 2010. [37] Y. Xie, J. Sun, and J. S. Freudenberg, “Power flow characterization of a bidirectional galvanically isolated high-power dc/dc converter over a wide operating range,” IEEE Trans. Power Electron., vol. 25, no. 1, pp. 54–66, 2010. [38] Jean-Romain Sibue, Jean-Paul Ferrieux, G´erard Meunier, Robert P´eriot, Edith Clavel, “Generalized Average Model of Series - Parallel Resonant Converter with Capacitive Output Filter for High Power Application,” 2010, <hal-00521993>. [39] J. M. Ramos, J. Diaz, A. M. Pernía, F. Nuño, and J. M. Lopera, “Dynamic and steady-state models for the PRC-LCC resonant topology with a capacitor as output filter.,” IEEE Trans. Ind. Electron., vol. 54, no. 4, pp. 2262–2275, Aug. 2007. [40] C. Zhao, S. D. Round, and J. W. Kolar, “Full-order averaging modelling of zero-voltage-switching phase-shift bidirectional dc-dc converters,” IET Power Electron., vol. 3, no. 3, pp. 400–410, 2010.   65   [41] Y. H. Abraham, W. Xiao, W. Huiqing, and V. Khadkikar, "Estimating power losses in Dual Active Bridge DC-DC converter," in proc. 2nd International Conference on Electric Power and Energy Conversion Systems (EPECS), 2011. pp, 1-5. [42] J. Muhlethaler, J. Biela, J. W. Kolar, and A. Ecklebe, “Improved core-loss calculation for magnetic components employed in power electronic systems,” IEEE Trans. Power Electron., vol. 27, no. 2, pp. 964–973, Feb. 2012. [43] K. Venkatachalam, C. R. Sullivan, T. Abdallah, and H. Tacca, “Accurate prediction of ferrite core loss with non-sinusoidal waveforms using only steinmetz parameters,” in Proc. 8th IEEE Workshop Comput. Power Electron., Compel 2002, Jun., pp. 36–41. [44] A D. Rajapakse, A M. Gole, and P. L. Wilson, "Electromagnetic Transients Simulation Models for Accurate Representation of Switching Losses and Thermal Performance in Power Electronic Systems," IEEE Transactions On Power Delivery, vo1.20, no. 1, Jan. 2005. [ 45 ] E. C. Snelling, Soft Ferrites, Properties and Applications, 2nd ed. London, U.K. Butterworths, 1988. [46] Piece-wise Linear Electrical Circuit Simulation for Simulink (PLECS), Plexim GmbH, 2013. [Online]. Available: www.plexim.com [47] S. R. Sanders, J. M. Noworolski, X. Z. Liu, and G. C. Verghese, “Generalized averaging method for power conversion circuits,” IEEE Trans. Power Electron., vol. 6, no. 2, pp. 251–259, 1991. [48] V. A. Caliskan, O. C. Verghese, and A. M. Stankovic, “Multifrequency averaging of dc/dc converters,” IEEE Trans. Power Electron., vol. 14, no. 1, pp. 124–133, 1999. [49] Simulink: Dynamic System Simulation for Matlab, Using Simulink, Version R2012b, The MathWorks Inc., 2012.  66  Appendices Appendix A. DAB Converter Parameters C1 = 44 μF, C2 = 180μF, C3 = 94μF, C4 = 330μF L1 = 15μH, L2 = 22μH, Ll1 = 52.65μH, Ll2 = 1.41μH, LM = 1.4mH R1 = 0.68Ω, R2 = 0.68Ω, Ro = 0.5Ω,  Rl1 = 0.64Ω, Rl2 = 0.16Ω, Rs = 0.0147Ω, RM = 4740Ω (at fsw = 25kHz) Vis = 48V, Vos = 20V, fs = 25kHz, n = 0.5  67  Appendix B. DAB Converter PCB Design Layout   68     

Cite

Citation Scheme:

        

Citations by CSL (citeproc-js)

Usage Statistics

Share

Embed

Customize your widget with the following options, then copy and paste the code below into the HTML of your page to embed this item in your website.
                        
                            <div id="ubcOpenCollectionsWidgetDisplay">
                            <script id="ubcOpenCollectionsWidget"
                            src="{[{embed.src}]}"
                            data-item="{[{embed.item}]}"
                            data-collection="{[{embed.collection}]}"
                            data-metadata="{[{embed.showMetadata}]}"
                            data-width="{[{embed.width}]}"
                            data-media="{[{embed.selectedMedia}]}"
                            async >
                            </script>
                            </div>
                        
                    
IIIF logo Our image viewer uses the IIIF 2.0 standard. To load this item in other compatible viewers, use this url:
https://iiif.library.ubc.ca/presentation/dsp.24.1-0166507/manifest

Comment

Related Items