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Parametric average value modeling of flyback converters in ccm and dcm including parasitics and snubbers 2011

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PARAMETRIC AVERAGE VALUE MODELING OF FLYBACK CONVERTERS IN CCM AND DCM INCLUDING PARASITICS AND SNUBBERS  by  Mehmet Sucu  B.A.Sc., Marmara University, 2000 M.A.Sc., Marmara University, 2003  A THESIS SUBMITTED IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF  MASTER OF APPLIED SCIENCE  in  THE FACULTY OF GRADUATE STUDIES  (ELECTRICAL AND COMPUTER ENGINEERING)  THE UNIVERSITY OF BRITISH COLUMBIA  (Vancouver)     October 2011  © Mehmet Sucu, 2011  ii Abstract  Modeling of switched-mode DC-DC converters has been receiving significant interest due to their widespread applications. Averaged modeling is the most common approach (and tool) that has been used to analyze dynamic performance of converter circuits. Specifically, state- space averaged models are widely used because of their simplicity and generality. However, as has been shown in the literature, the challenges of directly applying this approach to predict the discontinuous variables (states) and include the parasitics and losses have limited application of this approach to a wider range of converter circuits. The recently introduced parametric average value models (PAVM) has a potential to overcome this problem.  In this Thesis, first of all a second-order flyback converter has been investigated. An analytical solution of state-apace averaging and small-signal analysis of the flyback converter in continuous conduction mode (CCM) and discontinuous conduction mode (DCM) is given without and with parasitics. The PAVM methodology has been applied to the second-order model to overcome the problem of discontinuous state during the DCM.  The snubber circuits in flyback converter have also been investigated. Appearance of snubbers in the model introduces a problem on the output voltage besides improving the efficiency prediction. It is shown that with the snubbers the conventional state-space averaging cannot predict the output voltage correctly in CCM and DCM. To solve this problem the model is partitioned into two different sub-circuits: i) switching sub-circuit circuit; and ii) non-switching sub-circuit. Thereafter it becomes possible apply the averaging on the switching sub-circuit only.  Finally, a full-order flyback converter with two RC snubber circuits and all the basic parasitics is considered. The PAVM methodology has been extended to this class of switching converter for the first time.  It is shown that including the snubbers and parasitics significantly improves the model accuracy in terms of predicting converter efficiency, which represents an appreciable improvement over all previously existing average models. The proposed model has been verified with detailed simulations and hardware measurements.  iii Table of Contents Abstract .................................................................................................................................... ii Table of Contents ................................................................................................................... iii List of Tables ........................................................................................................................... v List of Figures ......................................................................................................................... vi List of Abbreviations ............................................................................................................. ix Acknowledgements ................................................................................................................. x Chapter 1 : Introduction ........................................................................................................ 1 1.1 PWM DC-DC Converters ..................................................................................................... 1 1.2 Flyback Converters ............................................................................................................... 1 1.3 Average Value Modeling ...................................................................................................... 2 1.4 Parametric Average-Value Modeling ................................................................................... 4 1.5 Motivations and Objectives .................................................................................................. 4 Chapter 2 : Second Order Flyback Converters ................................................................... 5 2.1 Small-Signal AC Model and State-Space Averaging without Parasitics in CCM ................ 5 2.2 State-Space Averaging in DCM without Parasitics ............................................................ 15 2.3 Small-Signal AC Model and State-Space Averaging with Basic Parasitics in CCM ......... 19 2.4 State-Space Averaging with Parasitics in DCM ................................................................. 32 2.5 Parametric Average Value Modeling in CCM and DCM ................................................... 36 2.5.1 Correction Term ............................................................................................................. 37 2.5.2 Model Implementation .................................................................................................... 38 2.5.3 Case Studies .................................................................................................................... 43 2.5.3.1 Time domain .......................................................................................................... 43 2.5.3.2 Frequency domain .................................................................................................. 45 Chapter 3 : Analysis of Flyback Converter with Snubber Circuits ................................. 47 3.1 Fifth –order Flyback Converter with Snubbers ................................................................... 47 3.2 State-Space Averaging Phenomena with the Snubbers ....................................................... 49 Chapter 4 : Full-order Flyback Converter ......................................................................... 56 4.1 State-Space Averaging in CCM .......................................................................................... 56 4.2 State-Space Averaging in DCM .......................................................................................... 59  iv 4.3 Parametric Average Value Modeling in CCM and DCM ................................................... 62 4.3.1 Model Implementation .................................................................................................... 62 4.4 Case Studies ........................................................................................................................ 67 4.4.1 Time Domain .................................................................................................................. 67 4.4.2 Frequency Domain.......................................................................................................... 69 4.4.3 Efficiency Results ........................................................................................................... 70 Chapter 5 : Conclusion ......................................................................................................... 72 5.1 Future Work ........................................................................................................................ 72 Bibliography .......................................................................................................................... 74 Appendices ............................................................................................................................. 78 Appendix A. The Converters Circuit Parameters .......................................................................... 78 A.1 Second-order Flyback Converter Parameters without Parasitics in CCM ...................... 78 A.2 Second-order Flyback Converter Parameters without Parasitics in DCM ...................... 78 A.3 Second-order Flyback Converter Parameters with Parasitics in CCM ........................... 78 A.4 Second-order Flyback Converter Parameters with Parasitics in DCM ........................... 79 A.5 Fifth-order Flyback Converter Parameters in CCM ....................................................... 79 A.6 Full-order Flyback Converter Parameters in CCM ......................................................... 79 A.7 Full-order Flyback Converter Parameters in DCM ........................................................ 80 Appendix B. Flyback Converter Circuit Diagram ......................................................................... 81   v List of Tables  Table 4.1 Efficiency comparison of the average-value models .............................................. 71   vi List of Figures  Figure 2.1 (a) Assumed circuit for the second order Flyback converter without parasitics; (b) Circuit during subinterval 1; (c) Circuit during subinterval 2. .......................................................................................................... 6 Figure 2.2 The inductor current ................................................................................................ 6 Figure 2.3 Inductor current and capacitor voltage of second order Flyback converter without parasitics in CCM. .................................................................. 14 Figure 2.4 Capacitor voltage of second order Flyback converter without parasitics in CCM. ................................................................................................................ 14 Figure 2.5 Second-order Flyback converter without parasitics during third subinterval in DCM. ............................................................................................. 15 Figure 2.6 Magnetizing current in DCM for the load 2500R = Ω . ......................................... 16 Figure 2.7 Inductor current and capacitor voltage of second order Flyback converter without parasitics in DCM. .................................................................. 18 Figure 2.8(a) Second-order Flyback converter with parasitics; (b) Circuit during subinterval 1 (c) Circuit during subinterval 2. ..................................................... 19 Figure 2.9 Inductor current, capacitor voltage and output voltage of second order Flyback converter with parasitics in CCM. ......................................................... 31 Figure 2.10 Output voltage of second order Flyback converter with parasitics in CCM. .................................................................................................................... 31 Figure 2.11 Second-order Flyback converter with parasitics during third subinterval in DCM. ............................................................................................. 32 Figure 2.12 Inductor current, capacitor voltage and output voltage of second order Flyback converter with parasitics in DCM. ................................................ 35 Figure 2.13 Variable 3d  as a function` of duty-cycle ( )1d  and the load ( )R . ....................... 40 Figure 2.14 The correction term 1m  as a function of duty-cycle ( )1d  and the load resistance ( )R . ..................................................................................................... 40 Figure 2.15 The correction term 2m  as a function of duty-cycle ( )1d  and the load resistance ( )R . ..................................................................................................... 41  vii Figure 2.16 Implementation of the parametric average-value model. .................................... 42 Figure 2.17 Simulated inductor current, capacitor voltage and output voltage of the second order Flyback converter with parasitics in DCM. .............................. 44 Figure 2.18 Transients in inductor current, capacitor voltage and output voltage of the second order Flyback converter due to the step change in load. ............... 45 Figure 2.19 Control-to-output transfer function of the second-order Flyback converter evaluated at 717.05R = Ω  and 1 0.381d = . ............................................. 46   Figure 3.1 Fifth-order Flyback converter circuit. ................................................................... 47 Figure 3.2 Measured transformer secondary voltage: (a) without the diode snubber; and (b) with the diode snubber. ............................................................. 48 Figure 3.3 Simulated output filter capacitor voltage and the output voltage of the fifth-order Flyback converter with snubbers in CCM. ......................................... 50 Figure 3.4 The predicted secondary current and the diode snubber capacitor voltage of fifth-order Flyback converter in CCM. ............................................... 51 Figure 3.5 Forth-order Flyback converter without diode snubber. ......................................... 52 Figure 3.6 The simulated output filter capacitor and output voltage of the forth- order Flyback converter without the diode snubber in CCM. .............................. 53 Figure 3.7 Modified fifth-order Flyback converter circuit. .................................................... 54 Figure 3.8 Proposed state-space averaged model of the fifth-order Flyback converter using two sub-circuits and sub-models. ............................................... 54   Figure 4.1 Full-order Flyback converter circuit. ..................................................................... 56 Figure 4.2 Predicted state variables of full-order Flyback converter in CCM. ....................... 58 Figure 4.3 Simulated state variables of the full-order Flyback converter in DCM. ............... 60 Figure 4.4 Simulated transformer secondary current of the full-order Flyback converter in DCM. ............................................................................................... 61 Figure 4.5 The diode current waveform. ................................................................................ 63  viii Figure 4.6 Variable 3d  as a function of the duty-cycle ( )1d  and the load resistance ( )R . ..................................................................................................... 64 Figure 4.7 The correction term 2m  as a function of duty-cycle ( )1d  and the load resistance ( )R . ..................................................................................................... 65 Figure 4.8 The correction term 3m  as a function of the duty-cycle ( )1d  and the load resistance ( )R . ............................................................................................. 65 Figure 4.9 The correction term 4m  as a function of duty-cycle ( )1d  and the load resistance ( )R . ..................................................................................................... 66 Figure 4.10 The correction term 5m  as a function of duty-cycle ( )1d  and the load resistance ( )R . ..................................................................................................... 67 Figure 4.11 Measured and simulated output voltage, primary and secondary current in CCM at constant duty-cycle. ............................................................... 68 Figure 4.12 Simulated output voltage, primary and secondary current during the transient from DCM to CCM due to the step change in load. ............................. 69 Figure 4.13 Control-to-output transfer function of the full-order Flyback converter evaluated at 717.05R = Ω  and 1 0.381d = ............................................... 70   ix List of Abbreviations  CCM Continuous Conduction Mode DCM Discontinuous Conduction Mode PAVM Parametric Average Value Modeling PWM Pulse Width Modulation    x Acknowledgements  I would like to express my appreciation to my supervisor, Dr. Juri Jatskevich, whose strong academic support have been the most precious assets to my studies and research. I am also very grateful for the partial financial support that has been made available to me through the NSERC  under the Discovery Grant.  I also like to thank Dr. William Dunford and Dr. Shahriar Mirabbasi who have accepted to be the committee members and dedicated their time and effort for reading this thesis and providing their constructive and valuable comments.  My special thanks go to previous and current members of the Power and Energy Systems Group at UBC who have always supported me and gave their valuable insights into my research.   1  Chapter 1 : Introduction  1.1 PWM DC-DC Converters Switch mode dc-dc converters have become an essential element of many commercial and military applications. Due to their high efficiency, light weight and relatively low cost, the switching dc-dc converters have generated a significant research interest in the area of their modeling, analysis, and control. Among various types of dc-dc converters, the Pulse-Width Modulated (PWM) converters constitute by far the largest group. They have displaced conventional linear power supplies even at low power levels. Switch-mode dc-dc converters can be categorized as non-linear periodic time-variant systems due to their inherent switching operation. The topology depends on instantaneous states of the power switches. This is what makes their modeling a complex task. Nevertheless, accurate analytical models of PWM dc- dc converters are essential for the analysis and design in many applications e.g., automobiles, aeronautics, aerospace, telecommunications, submarines, naval ships, mainframe computers and medical equipments. Many efforts have been made in the past few decades to model dc- dc converters and several new models have been proposed. These models are widely used to study the static and dynamic characteristics of the converters as well as to design their control systems to achieve specific regulation characteristics [1-8].  1.2 Flyback Converters A flyback converter is a switching power supply topology widely used in low power applications such as chargers and PC power supplies. It is basically an implementation of buck-boost converter and has transformer isolation. The most important advantage is that it becomes possible to have multiple outputs with a simple modification on the transformer (adding another secondary winding) and adding few extra components (a diode and a filter capacitor). Another important advantage is that it has natural isolation between input and output, which is required by many standards for design of power supplies [9-13].  A detailed model of a flyback converter can be easily implemented using widely available simulation packages (e.g. Matlab/Simulink, ASMG, PLECS, etc.) [14-16]. Detailed models are often used during the design process as such models have all the required information to 2  calculate the exact switching transients and component stresses and characteristics. But the large computation time required for such detailed switching models makes them less applicable for system-level studies. Instead, the average-value modeling has been used very effectively for the system-level analysis and studies, wherein the effects of fast switching are neglected or averaged with respect to the switching interval. A classical state-space averaged model of a flyback converter [10] considers only the switch losses without any snubber circuits and has the simplest first-order transformer approximation. A simplified linear circuit model for obtaining dc and small-signal circuit model is given in [17], which has the basic parasitics but does not include any snubber circuits and transformer primary and secondary copper losses. A dc and small-signal circuit model models for a flyback converter operating in CCM can be found in [12], which has the basic parasitics but again does not include any snubber circuits and has only a the simplest transformer model without the primary and secondary losses.  1.3 Average Value Modeling The averaged-value modeling, wherein the effects of fast switching are “averaged” over a switching interval, is most frequently applied when investigating power-electronics-based systems. Continuous large-signal models are typically non-linear and can be linearized around a desired operating point. Averaged models of dc-dc converters offer several advantages over the switching models. These advantages are: i) straightforward approach in determining local transfer-functions; ii) faster simulation of large-signal system-level transients; and iii) use of general-purpose simulators to linearize converters for designing the feedback controllers.  A typical switched-inductor dc-dc converter can operate in two modes. One is the Continuous Conduction Mode (CCM) in which inductor current never falls to zero, and the second mode is Discontinuous Conduction Mode (DCM) allowing inductor current to become zero for a portion of switching period. The DCM typically occurs at light loads and differs from CCM since this mode results into three different switched networks over one switching cycle (as opposed to two switched networks in the case of CCM operation). Models for PWM converters operating in CCM based on well-known state-space averaging 3  technique were first introduced in 1970’s [18]. Since then, several circuit-oriented averaging approaches have also been proposed [3, 19]. Numerous method have been developed for the average value modeling of PWM dc-dc converters in DCM such as reduced-order state-space averaging [20], reduced-order averaged-switch modeling [7], equivalent duty ratio models [3], loss-free resistor model [10], full-order averaged-switch modeling [21], and full-order state-space averaging [4].  Average value models may be categorized as resulting system of equations (reduced-order vs. full order); or by derivation methodology (sampled data modeling, circuit averaging, state-space averaging). The full-order as well as reduced-order models can be obtained by averaging approaches including sampled data modeling, circuit averaging or state space averaging. The conventional reduced order models treat the discontinuous variable as a dependent variable and eliminate its dynamic from the state equations. The elimination of fast/discontinuous variable is undesirable for application in which this variable is used for control purposes, which limits the range of applications of such reduced-order models.  State-space averaging is based on the classical averaging theory and involves manipulation of state-space equations of a converter system. First, a state-space representation of converter is obtained for each topology and subinterval. Then, the obtained piece-wise linear equations are weighted by the corresponding time subinterval length and added together. State-space averaging has been demonstrated to be an effective method to analyze PWM converters. Analytical averaging, however, is based on so-called small-ripple approximation. Most of the previous works on averaging methods were derived for a specific ideal topology. In addition, derivation of state-space average-value model, the equivalent series resistance (ESR) of circuit components are often neglected and the state variables are considered as linear segments. Such assumptions result in inaccuracy of the corresponding time constants as well as the waveforms. If the losses due to the switch and/or active elements are taken into account, whereby the linear shape of the current waveform would change into exponential form, the analytically derived models would become significantly more complicated and challenging. The analytical derivation also becomes more complicated when the number of energy storage elements (inductors and capacitors) is high. 4  1.4 Parametric Average-Value Modeling Parametric average-value modeling methodology has been set forth by the UBC researchers. This methodology has been successfully demonstrated for synchronous machine-converter systems in [22, 23]. The major point of this approach is to use the detailed simulation for numerically calculating the key relationships needed for constructing the average-value model of a certain well-defined form. In doing so, the effect of parasitics included in the detailed model becomes automatically included in the numerically constructed parametric functions, which are then used for the state-variable-based average-value models. This approach also reduces the effort of the model developer and avoids many complicated analytical derivations. This method has been extended to the PWM dc-dc converters in [24- 27] based on corrected full-order averaged models proposed for circuit averaging [28] and state-space averaging [4] that very accurately capture the high-frequency dynamics of fast state variables.  1.5 Motivations and Objectives The detailed models of PWM dc-dc converters are widely used for design purposes but they are not desirable for system level studies due to very high computational times, wherein it has been always required to have more efficient average models. Although there are various averaged models of the flyback PWM converter available, none of the previously established models have full order and include all realistic parasitics. Most of the models use the simplest transformer representation and none of them include the snubber circuits. At the same time, the snubber circuits are very important components of the flyback PWM converters and have significant effect on the converter dynamics and efficiency.  This Thesis makes an original contribution and extends the parametric average-value modeling the flyback converters. The considered converter model includes all the basic parasitics and high order transformer model with primary and secondary resistances and leakage inductances. The propose model also includes two RC (resistance and capacitance) snubber circuits to protect the switch and the diode during the on-off operation. To the best of our knowledge, this has not been done in any published research on this subject.  5  Chapter 2 : Second Order Flyback Converters   In this Chapter, we consider and approximate (simplified) circuit of the flyback converter, wherein only two energy storage elements are considered, hence second order converter. Such approximate converter circuit has been used in the literature for carrying out basic analysis and averaging methods. A number of modeling techniques have appeared in the literature, including the current injected approach [19], circuit averaging [7, 21, 29], and state-space averaging [18] method.  2.1 Small-Signal AC Model and State-Space Averaging without Parasitics in CCM The state-space description of dynamical systems is a basis of modern control theory. The state-space averaging method makes use of this description to derive the small-signal averaged equations of the PWM switching converters. The state-space averaging method is otherwise identical to the procedure of deriving the small-signal ac model. A benefit of the state-space averaging procedure is its results: a small-signal averaged model that can always be obtained, provided that the state equations of the original converter can be written.  Obtaining a small-signal ac model of a basic switched converter circuit, such as buck, boost without parasitics, can be readily achieved using analytical derivations. But when the converter circuit has parasitics, it becomes almost impossible and impractical to derive the higher order state equations. In this case, the state-space equations can be obtained from the detailed model by using commercially available simulation packages [16, 30], and then used the state-space description (matrices) to establish the small-signal model [10] (see Section7.3.2).  In this Section, a small-signal ac model will be derived for a second-order flyback converters without parasitics. Based on that, the state-space equations will be derived.  A second order flyback converter without parasitics is shown in Figure 2.1(a). Here, n  is the turn ratio of the transformer ( )1 2N N . 6  + mL n ( ) i t ( )gi t ( )ci t ( ) v t ( )Lv t C R D Mosfet + mL n ( ) i t ( )gi t ( )ci t ( ) v t ( )Lv t C R + mL n ( ) i t ( )gi t ( )ci t ( ) v t ( )Lv t C R ( ) a ( ) b ( ) c ( )gv t ( )gv t ( )gv t  Figure 2.1 (a) Assumed circuit for the second order Flyback converter without parasitics; (b) Circuit during subinterval 1; (c) Circuit during subinterval 2.  The inductor current ( )i t  in CCM is shown in Figure 2.2.  During the first interval, Figure 2.1(b), the inductor stores some energy and transfers this energy to the secondary side during the second interval, Figure 2.1(c). 0.4 0.6 0.8 1 1.2 1.4 1.6 I (A m p .) 0.999991 0.999993 0.999995 0.999997 Time (s) 1 sd T 2 sd T sT  Figure 2.2 The inductor current 7  During the first subinterval, when the MOSFET conducts and the diode is off, the circuit reduces to Figure 2.1(b). The inductor voltage ( )Lv t , capacitor current ( )ci t , and converter input current ( )gi t  can be expressed as follows:  ( ) ( )L gv t v t=  (2.1)  ( ) ( )c v ti t R= −  (2.2)  ( ) ( )gi t i t=  (2.3) Applying the small ripple approximation [10] and replacing the voltages and currents with their respective average values, we obtain  ( ) ( ) s L g T v t v t=  (2.4)  ( ) ( ) sTc v t i t R = −  (2.5)  ( ) ( ) s g T i t i t=  (2.6) During the second subinterval, MOSFET is off and the diode conducts, which results in circuit of Figure 2.1(c). The inductor voltage ( )Lv t , capacitor current ( )ci t , and converter input current ( )gi t  are given by  ( ) ( )Lv t v t n=  (2.7)  ( ) ( ) ( )c v ti t i t n R   = − +     (2.8)  ( ) 0gi t =  (2.9) The small ripple approximation leads to  ( ) ( ) s L T v t v t n=  (2.10)  ( ) ( ) ( ) s s T c T v t i t i t n R = − −  (2.11)  ( ) 0gi t =  (2.12) The average inductor voltage now can be found by averaging the subintervals over one complete switching period. The result is 8   ( ) ( ) ( ) ( ) ( )1 2 s ss L gT TT v t v t d t v t nd t= +  (2.13) This leads to the following state equation for the average inductor current  ( ) ( ) ( ) ( ) ( )1 2s ss T m g TT d i t L v t d t v t nd t dt = +  (2.14) The average capacitor current now can be found by averaging the subintervals over one switching period resulting in the following:  ( ) ( ) ( ) ( ) ( ) ( ) ( )1 2 2s s s s T T c T T v t v t i t d t i t nd t d t R R = − − −  (2.15) After collecting terms, the average capacitor current becomes  ( ) ( ) ( ) ( )( ) ( ) ( )1 2 2 1 s s s T c T T v t i t d t d t i t nd t R = − + − 1442443  (2.16) This leads to the following state equation for the average capacitor voltage  ( ) ( ) ( ) ( )2s s s T T T d v t v t C i t nd t dt R = − −  (2.17) The converter input current now can be found by averaging the subintervals over one switching period. The result is  ( ) ( ) ( )1 0 ss g TT i t i t d t= +  (2.18)  The equation (2.14), (2.17), and (2.18) are nonlinear set of differential equations. In order to construct the converter small-signal ac model, the next step is to perturb and linearize these equations. Here, we assume that the converter input voltage ( )gv t  and duty cycle ( )1d t  can be expressed as quiescent values plus small ac variations, as follows  ( ) ( )ˆ s g g gT v t V v t= +  (2.19)  ( ) ( )1 1 1ˆd t D d t= +   (2.20) In response to these inputs, and after all transients have decayed, the average converter variables can also be expressed as quiescent values plus small ac variations,  ( ) ( )ˆ sT i t I i t= +  (2.21)  ( ) ( )ˆ sT v t V v t= +  (2.22) 9   ( ) ( )ˆ s g g gT i t I i t= +  (2.23) With these substitutions, the large-signal averaged inductor equation (2.14) becomes  ( )( ) ( )( ) ( )( ) ( )( ) ( )( )1 1 2 1ˆ ˆ ˆˆ ˆm g gd I i tL V v t D d t V v t n D d tdt + = + + + + −  (2.24) Upon multiplying this expression out and collecting terms, we obtain  ( ) ( ) ( ) ( ) ( ) ( )( ) ( ) ( ) ( ) ( )( ) 1 2 1 1 1 2 1 ( ) 1 1 2 ( ) ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ m g g g Dcterms st order ac terms linear g nd order ac terms nonlinear di tdIL V D VnD V d t v t D Vnd t v t nD dt dt v t d t v t nd t   + = + + + − +     + − 1442443 14444444244444443 14444244443  (2.25)  As usual, this equation contains three types of terms. The dc term contain no time-varying quantities. The first order ac terms are linear functions of the ac variations in the circuit, while the second order ac terms are functions of the products of the ac variations. At this point, we make an assumption that the ac variations are small in magnitude compared to the dc quiescent values,  ( ) ( ) ( ) ( ) ( ) 1 1 ˆ ˆ ˆ ˆ ˆ g g g g v t V d t D i t I v t V i t I <<  <<   <<   <<   <<     (2.26) If the small signal assumptions (2.26) are satisfied, then the second-order terms are much smaller in magnitude than the first-order terms and hence can be neglected. The dc terms must satisfy  1 20 gV D VnD= +  (2.27) The first order ac terms must satisfy  ( ) ( ) ( ) ( ) ( )1 1 1 2 ˆ ˆ ˆ ˆ ˆ m g g di t L V d t v t D Vnd t v t nD dt = + − +  (2.28) This result is the linearized equation that describes ac variations in the inductor current.  10  Upon substation of (2.19), (2.20), (2.21), (2.22), and (2.23) into the averaged capacitor voltage state equation (2.17), we obtain the following  ( )( ) ( )( ) ( )( ) ( )( )2 1ˆ ˆ ˆˆd V v t V v tC I i t n D d tdt R + + = − − + −  (2.29) Upon multiplying this expression out and collecting terms, we obtain  ( ) ( ) ( ) ( ) ( ) ( )( ) 2 1 2 1 ( ) 1 2 ( ) ˆ ˆ ˆ ˆ ˆˆ Dcterms st order ac terms linear nd order ac terms nonlinear dv t v tdV VC InD Ind t i t nD dt dt R R i t nd t      + = − − + − + −          + 1442443 1444442444443 142443  (2.30) Here, we neglect the second-order terms. The dc terms of equation (2.30) must satisfy  20 V InD R = − −  (2.31) The first-order ac terms of (2.30) lead to the small-signal ac state equation for capacitor voltage  ( ) ( ) ( ) ( )1 2ˆ ˆ ˆ ˆdv t v tC Ind t i t nDdt R= − + −  (2.32) Substation of (2.19), (2.20), (2.21), (2.22), and (2.23) into (2.18) results in the following:  ( ) ( )( ) ( )( )1 1ˆˆ ˆg gI i t I i t D d t+ = + +  (2.33) Upon multiplying this expression out and collecting terms, we obtain  ( ) ( ) { ( ) ( )( ) ( ) ( )( )1 1 1 1 1 ( ) 2 ( ) ˆ ˆˆ ˆ ˆ g g Dcterms st order ac terms linear nd order ac terms nonlinear I i t ID Id t i t D i t d t+ = + + + 1442443 14243  (2.34) The dc term must satisfy  1gI ID=  (2.35) We neglect the second-order terms in (2.34), leaving the following linearized ac equation  ( ) ( ) ( )1 1ˆˆ ˆgi t Id t i t D= +  (2.36) This result represents the low-frequency ac variations in the converter input current.  The equation of the quiescent values, (2.27), (2.31), and (2.35) are collected below, 11   1 2 2 1 0 0 g g V D VnD V InD R I ID = +    = − −   =   (2.37) For given quiescent values of the input voltage gV , and duty cycle 1D , the system of equations (2.37) can be evaluated to find the quiescent output voltage V , inductor current I , and input current dc component gI . The results are then inserted into the small-signal ac model. The final small signal ac model is summarized below,  ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) 1 1 1 2 1 2 1 1 ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆˆ ˆ m g g g di t L V d t v t D Vnd t v t nD dt dv t v t C Ind t i t nD dt R i t Id t i t D  = + − +    = − + −   = +    (2.38) The final step is to construct an equivalent circuit of (2.37) and (2.38) using commercially available simulation tools.  Let us now apply the state-space averaging method to the second order flyback converter circuit depicted in Figure 2.1(a). The independent state variables of the converter are the inductor current ( )i t  and the capacitor voltage ( )v t , which form the state vector  ( ) ( )( ) i t x t v t   =      (2.39) The input voltage ( )gv t , is an independent source which should be placed in the input vector,  ( ) ( )gu t v t =    (2.40) To model the converter as a system with input and output, we need to find the converter input current ( )gi t . To calculate this dependent current, it should be included in the output vector ( )y t . Therefore,  ( ) ( )gy t i t =    (2.41) Note that it’s not necessary to include the output voltage ( )v t  in the output vector since the voltage is already included in the state vector ( )x t . 12  Next, let us write the state equations for each subinterval. When the switch is on and the diode is off, the converter circuit of Figure 2.1(b) is obtained. The inductor voltage, capacitor current, and converter input current are  ( ) ( )m gdi tL v tdt =  (2.42)  ( ) ( )dv t v tC dt R = −  (2.43)  ( ) ( )gi t i t=  (2.44) Similar to (2.1), (2.2), and (2.3), after organizing (2.42), (2.43), and (2.44), these equations can be written in the following standard state-space form:  ( ) ( ) ( ) ( ) ( ) ( ) { ( ) ( ) 1 1 10 0 10 0 m g u t x t A B dx t dt di t i tdt L v t v tdv t RC dt              = +      −            14243 123 14243 14243  (2.45)  ( ) ( ) [ ] { ( ) ( ) ( ) [ ] { ( ) ( )1 1 1 0 0g g C Ey t u t x t i t i t v t v t      = +      123 14243123  (2.46) So, (2.45) and (2.46) define the state-space equation for the first subinterval.  When the MOSFET is off and the diode is on, the converter circuit of Figure 2.1(c) is obtained. For the second subinterval, the inductor voltage, capacitor current, and converter input current are given by  ( ) ( )m di tL v t ndt =  (2.47)  ( ) ( ) ( )dv t v tC i t n dt R = − −  (2.48)  ( ) 0gi t =  (2.49) After organizing terms, the following state-space equation for the second interval is obtained: 13   ( ) ( ) ( ) ( ) ( ) ( ) { ( ) ( ) 2 2 0 0 01 m g u t Bx t Adx t dt ndi t i tLdt v t v tdv t n C RCdt                  = +            − −      14243 123 144244314243  (2.50)  ( ) ( ) [ ] ( )( ) ( ) [ ] { ( ) ( )22 0 0 0g g ECy t u t x t i t i t v t v t      = +      123123 14243123  (2.51) The next step is to evaluate the averaged state-space model, which is achieved as follows:   2 1 1 2 2 1 2 2 0 00 0 10 1 1 m m nDn L LA A D A D D D n nD RC C RC C RC             = + = + =   −    − − − −         (2.52)  1 1 1 2 2 1 2 1 0 0 0 0 m m D L LB B D B D D D         = + = + =              (2.53)  [ ] [ ] [ ]1 1 2 2 1 2 11 0 0 0 0C C D C D D D D= + = + =  (2.54)  [ ] [ ] [ ]1 1 2 2 1 20 0 0E E D E D D D= + = + =  (2.55)  So the final averaged state-space model of the second order Flyback converter without parasitics in CCM is   ( ) ( ) ( ) ( ) ( ) 2 1 2 0 1 0 m m g nDdi t D i tLdt L v t v tdv t nD C RCdt                  = +            − −         (2.56)  ( ) [ ] ( )( ) [ ] ( )1 0 0g g i t i t D v t v t      = +        (2.57)  To validate the analytically derived averaged state-space model in CCM, a detailed model of Figure 2.1(a) has been constructed in PLECS [16]. The averaged state-space equations have been constructed in Matlab/Simulink [30]. A hardware prototype of the subject converter has 14  been built to validate the results. The parameters of the second-order flyback converter without parasitics in CCM are given in Appendix A.1. The predicted and measured inductor current and capacitor voltage are shown in Figure 2.3. 0.4 0.6 0.8 1 1.2 1.4 1.6 I (A m p .) 0.999991 0.999993 0.999995 0.999997 Time (s) Detailed Model Actual Average Analytical State-Space Hardware Prototype Detaied Model Actual Average Analytical State-Space Actual Average and Analytical State-Space See Figure 2.4 -74 -73 -72 -71 V  ( V o lt )  Figure 2.3 Inductor current and capacitor voltage of second order Flyback converter without parasitics in CCM.  0.999991 0.999993 0.999995 0.999997 -73.896 -73.894 -73.892 -73.89 Time (s) V  ( V o lt ) Detailed Model Actual Average Analytical State-Space  Figure 2.4 Capacitor voltage of second order Flyback converter without parasitics in CCM.  As can be seen in Figure 2.3, the analytically derived state-space averaged model predicts the averaged inductor current very well. As can be seen in Figure 2.3, there is a difference between the capacitor voltage waveforms predicted by the detailed model and the measurements from the hardware prototype, which comes from the simplified detailed model 15  (second-order without parasitics). As can be seen in Figure 2.4, the analytically derived state- space averaged model predicts the average capacitor voltage very well.  2.2 State-Space Averaging in DCM without Parasitics When designing a flyback converter, one of the very first challenges is the decision on the mode of operation. It is known that performance of the flyback converters in CCM and DCM differs significantly in terms of components stress, output voltage regulation, transient response, and efficiency. Interested reader can find a comparison of CCM and DCM for the flyback converters in [31]. If the converter is designed to operate in DCM, it will operate in DCM for almost all specified loads. If the converter is designed to operate in CCM, it will operate in CCM at nominal load up to a boundary between CCM and DCM. The value of magnetizing inductance at the boundary between CCM and DCM is given as [32],  ( ) 221 1 2 1 2 L m D R NL f N −   =      (2.58) Here, 1D  is the duty cycle, LR  is the load resistance, f  is the switching frequency, and 1 2N N  is the turn ratio of the transformer.  In addition to 2 subintervals that occur in CCM, in DCM there is another subinterval that occurs at light loads. The third interval results in the topological instance of the flyback converter circuit shown in Figure 2.5.  + mL n ( ) i t ( )gi t ( )ci t ( ) v t ( )Lv t C R ( )gv t  Figure 2.5 Second-order Flyback converter without parasitics during third subinterval in DCM.  During the first subinterval [see circuit of Figure 2.1(b)], the MOSFET is on and the diode is off, and the magnetizing inductance stores the energy. During the second subinterval [see Figure 2.1(c)], the MOSFET is off and the diode is on, and the energy stored in the 16  transformer field is transferred to the secondary side. This energy then flows through the filter capacitor and is supplied to the load resistor. If the magnetizing current during the second subinterval goes to zero before the end of the second subinterval, the converter goes into another stage (DCM, subinterval 3) before it goes back to the first interval again. The magnetizing current predicted by the detailed model in DCM  is shown in Figure 2.6 for the load 2500R = Ω . 0.123543 0.123546 0.123549 0 0.4 0.8 1.2 Time (s) I (A m p .) 1 sd T 2 sd T 3 sd T sT  Figure 2.6 Magnetizing current in DCM for the load 2500R = Ω .  For the flyback converter without parasitics, the first and second subintervals in DCM are the same as in CCM, as described in Section 2.1. The third subinterval comes into account when the MOSFET and the diode are both off as shown in Figure 2.5. For this topology, the inductor voltage ( )Lv t , capacitor current ( )ci t , and converter input current  ( )gi t  are  ( ) 0Lv t =  (2.59)  ( ) ( )c v ti t R= −  (2.60)  ( ) 0gi t =  (2.61)  Let us now apply the state-space averaging method to third sub interval. The state vector ( )x t , the input vector ( )u t , and the output vector ( )y t  are as define in Section 2.1 in (2.39), (2.40), and (2.41), respectively. Next, let us write the state equation for third subinterval. When the switch and the diode are off, the converter circuit of Figure 2.5 is obtained. The inductor voltage, capacitor current, and converter input current are 17   ( ) 0m di tL dt =  (2.62)  ( ) ( )dv t v tC dt R = −  (2.63)  ( ) 0gi t =  (2.64) Similar to (2.59), (2.60), and (2.61), after organizing the terms in (2.62), (2.63), and(2.64), the following state-space for the third subinterval is formed   ( ) ( ) ( ) ( ) ( ) ( ) { ( ) ( ) 3 3 0 0 0 1 00 g u t Bx t A dx t dt di t i tdt v t v tdv t RC dt             = +       −         14243 123 14243 14243  (2.65)  ( ) ( ) [ ] ( )( ) ( ) [ ] { ( ) ( )33 0 0 0g g ECy t u t x t i t i t v t v t      = +      123123 14243123  (2.66)  So the next step is to evaluate the state-space averaged model, which goes as following:   1 1 2 2 3 3 2 1 2 3 2 0 00 0 0 0 1 10 01 1 m m A A D A D A D nDn L LD D D n nD RC RC C RC C RC = + +                 = + + =     − −    − − − −           (2.67)  1 1 1 2 2 3 3 1 2 3 1 0 0 0 0 0 0 m m D L LB B D B D B D D D D           = + + = + + =                  (2.68)  [ ] [ ] [ ] [ ]1 1 2 2 3 3 1 2 3 11 0 0 0 0 0 0C C D C D C D D D D D= + + = + + =  (2.69)  [ ] [ ] [ ] [ ]1 1 2 2 3 3 1 2 30 0 0 0E E D E D E D D D D= + + = + + =  (2.70)  So the final state-space averaged equation of the second order flyback converter without parasitics in DCM becomes: 18   ( ) ( ) ( ) ( ) ( ) 2 1 2 0 1 0 m m g nDdi t D i tLdt L v t v tdv t nD C RCdt                  = +            − −         (2.71)  ( ) [ ] ( )( ) [ ] ( )1 0 0g g i t i t D v t v t      = +        (2.72) Equations (2.71) and (2.72) are the state-space averaged model for the DCM, which is the same as (2.56) and (2.57). It only happens when there are no parasitics.  To validate the analytically derived state-space averaged model in DCM, a detailed model of the converter depicted in Figure 2.1(a) has been constructed in PLECS [16]. The state-space averaged model has been constructed in Matlab/Simulink [30]. The interval 2d  has been calculated as 0.4409  using (2.58) and 2500LR R= = Ω . The parameters of the second-order flyback converter without parasitics in CCM are given in Appendix A.2. The resulting waveforms of inductor current and capacitor voltage are shown in Figure 2.7.  0 0.4 0.8 1.2 I (A m p .) 1.699991 1.699993 1.699995 1.699997 1.699999 1.7 -102.547 -102.545 -102.543 -102.541 Time (s) V  ( V o lt ) Detailed Model Actual Average Analytical State-Space  Figure 2.7 Inductor current and capacitor voltage of second order Flyback converter without parasitics in DCM.  19  As can be seen in Figure 2.7, the analytically derived state-space averaged model predicts the magnetizing current with a large error of 18.25% . This error is because the actual inductor current is discontinuous, which is not properly accounted by the classical state-space averaging as will be explained in Section 2.5. At the same time, the analytically derived state-space averaged model predicts the capacitor voltage and the output voltage with a very small error because these state variables are continuous. The small error in this case comes from the error in representing the magnetizing current according to (2.71).  2.3 Small-Signal AC Model and State-Space Averaging with Basic Parasitics in CCM A second order flyback converter with basic parasitics is shown in Figure 2.8(a). Here, n  is the turn ratio of the transformer ( )1 2N N . + mL n ( ) i t ( )gi t ( )ci t ( ) v t ( )Lv t C R D Mosfet ( ) a ( ) b ( ) c ( )gv t + cR swR dV ( )cv t + mL n ( ) i t ( )gi t ( )ci t ( ) v t ( )Lv t C R ( )gv t cR swR ( )cv t + mL n ( ) i t ( )gi t ( )ci t ( ) v t ( )Lv t C R ( )gv t + cR dV ( )cv t  Figure 2.8(a) Second-order Flyback converter with parasitics; (b) Circuit during subinterval 1 (c) Circuit during subinterval 2.  20  During the first subinterval, when the MOSFET conducts and the diode is off, the circuit reduces to Figure 2.8(b). For this interval, the inductor voltage ( )Lv t , capacitor current ( )ci t , converter output voltage ( )v t , and converter input current ( )gi t  are  ( ) ( ) ( )L g sw gv t v t R i t= −  (2.73)  ( ) ( )cc c v t i t R R = − +  (2.74)  ( ) ( )c c v t R v t R R = +  (2.75)  ( ) ( )gi t i t=  (2.76)  We next apply the small ripple approximation and replace the voltages and currents with their respective average values to obtain the following:  ( ) ( ) ( ) s s L g sw gT T v t v t R i t= −  (2.77)  ( ) ( ) sc Tc c v t i t R R = − +  (2.78)  ( ) ( ) sc T c v t R v t R R = +  (2.79)  ( ) ( ) s g T i t i t=  (2.80)  During the second subinterval, the MOSFET is off and diode conducts, which results in the circuit of Figure 2.8(c). For this interval, the inductor voltage ( )Lv t , capacitor current ( )ci t , converter output voltage ( )v t , and converter input current ( )gi t  are given by  ( ) ( ) ( )( )L c c c dv t v t i t R V n= − −  (2.81)  ( ) ( ) ( )c v ti t i t n R   = − +     (2.82)  ( ) ( ) ( )c c cv t v t i t R= −  (2.83)  ( ) 0gi t =  (2.84) Applying the small ripple approximation leads to the following: 21   ( ) ( ) ( ) s s L c c c dT T v t v t n i t R n V n= − −  (2.85)  ( ) ( ) ( ) s s T c T v t i t i t n R = − −  (2.86)  ( ) ( ) ( ) s s c c cT T v t v t i t R= −  (2.87)  ( ) 0gi t =  (2.88)  The average inductor voltage now can be found by averaging the subintervals over one complete switching period. The result is  ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) 1 1 2 2 2 s s s s s L g sw gT T T c c c dT T v t v t d t R i t d t v t nd t i t R nd t V nd t = − + − −  (2.89) This leads to the following equation for the average inductor current  ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) 1 1 2 2 2 s s s s s T m g sw gT T c c c dT T d i t L v t d t R i t d t dt v t nd t i t R nd t V nd t = − + − −  (2.90)  The average capacitor current now can be found by averaging the subintervals over one switching period, which results in the following:  ( ) ( ) ( ) ( ) ( ) ( ) ( )1 2 2s s s s c T T c T T c v t v t i t d t i t nd t d t R R R = − − − +  (2.91) This leads to the following equation for the average capacitor voltage  ( ) ( ) ( ) ( ) ( ) ( ) ( )1 2 2s s s s c cT T T T c d v t v t v t C d t i t nd t d t dt R R R = − − − +  (2.92)  The converter output voltage now can be found by averaging the subintervals over one switching period, which results in  ( ) ( ) ( ) ( ) ( ) ( ) ( )1 2 2s s s s c T c c cT T T c v t R v t d t v t d t i t R d t R R = + − +  (2.93)  22  The converter input current can now be found by averaging the subintervals over one switching period, resulting in the following:  ( ) ( ) ( )1 0 ss g TT i t i t d t= +  (2.94)  Equations (2.90), (2.92), (2.93), and (2.94) are nonlinear differential equations. Hence, to construct the converter small-signal ac model, the next step is to perturb and linearize them. We assume that the converter input voltage ( )gv t  and duty cycle ( )1d t  can be expressed as quiescent values plus small ac variations, as follows  ( ) ( )ˆ s g g gT v t V v t= +  (2.95)  ( ) ( )1 1 1ˆd t D d t= +   (2.96) In response to these inputs, and after all transients have decayed, the averaged converter waveforms can be expressed as quiescent values plus small ac variations as  ( ) ( )ˆ s g g gT i t I i t= +  (2.97)  ( ) ( )ˆ sT i t I i t= +  (2.98)  ( ) ( )ˆ s c c cT v t V v t= +  (2.99)  ( ) ( )ˆ s c c cT i t I i t= +  (2.100)  ( ) ( )ˆ sT v t V v t= +  (2.101) With these substitutions, the large-signal averaged inductor, (2.90), becomes   ( )( ) ( )( ) ( )( ) ( )( ) ( )( ) ( )( ) ( )( ) ( )( ) ( )( ) ( )( ) 1 1 1 1 2 1 2 1 2 1 ˆ ˆ ˆˆ ˆ ˆ ˆˆ ˆ ˆ m g g sw g g c c c c c d d I i t L V v t D d t R I i t D d t dt V v t n D d t I i t R n D d t V n D d t + = + + − + + + + − − + − − −  (2.102)  Upon multiplying this expression out and collecting terms, we obtain 23   ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) 1 1 2 2 2 1 1 1 1 1 1 2 1 2 1 1 ( ) ˆ ˆ ˆ ˆˆ ˆ ˆ ˆˆ ˆ m g sw g c c c d Dcterms g g sw g sw g c c c c c c d st order ac terms linear di tdIL V D R I D V nD I R nD V nD dt dt V d t v t D R I d t R i t D V nd t v t nD I R nd t i t R nD V nd t   + = − + − −      + − − −  +  + + − +  1444444442444444443 144444444 2 ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) 1 1 1 1 1 1 2 ( ) ˆ ˆ ˆˆ ˆ ˆ ˆ ˆˆ g sw g c c c d nd order acterms nonlinear v t d t R i t d t v t nd t i t R nd t V nd t   − −  +  + +  44 444444444443 1444444442444444443  (2.103)  As usual, this equation contains three types of terms. The dc term contains no time-varying quantities. The first order ac terms are linear functions of the ac variations in the circuit, while the second order ac terms are functions of the products of the ac variations. At this point, we make an assumption that the ac variations are small in magnitude compared to the dc quiescent values,  ( ) ( ) ( ) ( ) ( ) ( ) ( ) 1 1 ˆ ˆ ˆ ˆ ˆ ˆ ˆ g g g g c c c c v t V d t D i t I i t I v t V i t I v t V <<  <<  <<   <<   <<   <<   <<    (2.104) If the small signal assumptions (2.104) are satisfied, then the second-order terms are much smaller in magnitude than the first-order terms and hence be neglected. The dc terms must satisfy  1 1 2 2 20 g sw g c c c dV D R I D V nD I R nD V nD= − + − −  (2.105) The first order ac terms must satisfy  ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) 1 1 1 1 1 1 2 1 2 1 ˆ ˆ ˆ ˆˆ ˆ ˆ ˆˆ ˆ m g g sw g sw g c c c c c c d di t L V d t v t D R I d t R i t D V nd t dt v t nD I R nd t i t R nD V nd t = + − − − + + − +  (2.106) This is the linearized equation that describes ac variations in the inductor current. 24  Upon substation of (2.95)-(2.101) into (2.92), we obtain  ( )( ) ( )( ) ( )( ) ( )( ) ( )( ) ( )( ) ( )( ) 1 1 2 1 2 1 ˆ ˆ ˆ ˆˆ ˆ ˆ c c c c c d V v t V v t C D d t I i t n D d t dt R R V v t D d t R + + = − + − + − + + − −  (2.107)  Upon multiplying this expression out and collecting terms, we obtain   ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) 1 2 2 1 1 1 2 1 2 1 ( ) 1 1 ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆˆ cc c c Dcterms c c c c st order ac terms linear c c dv tdV V D VDC InD dt dt R R R V d t v t D Vd t v t D Ind t i t nD R R R R R R v t d t v t i t nd t R R     + = − − −    +     + − − + − + −   + +  + − + + + 14444244443 14444444444424444444444443 ( )1 2 ( ) ˆ nd order ac terms nonlinear d t R        14444444244444443  (2.108)  Here again we neglect the second-order terms. The dc terms of (2.108) must satisfy  1 2 20 c c V D VDInD R R R = − − − +  (2.109) The first-order ac terms of (2.108) lead to the following small-signal for the ac capacitor voltage  ( ) ( ) ( ) ( ) ( ) ( ) ( )1 1 1 21 2 ˆ ˆ ˆ ˆ ˆ ˆ ˆ c c c c c dv t V d t v t D Vd t v t D C Ind t i t nD dt R R R R R R = − − + − + − + +  (2.110)  Substation of (2.95)-(2.101) into (2.93) leads to  ( )( ) ( )( ) ( )( ) ( )( ) ( )( ) ( )( ) ( )( ) 1 1 2 1 2 1 ˆ ˆ ˆ ˆ ˆ ˆˆ c c c c c c c c V v t V v t R D d t V v t D d t R R I i t R D d t + + = + + + − + − + −  (2.111) Upon multiplying this expression out and collecting terms, we obtain 25   ( )( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) 1 2 2 1 1 1 2 1 2 1 ( ) 1 1 1 ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆˆ ˆ c c c c c Dcterms c c c c c c c c c c st order acterms linear c c c c c V D RV v t V D I R D R R V Rd t v t RD V d t v t D R R R R I R d t i t R D v t Rd t v t d t i t R d t R R   + = + −  +    + − +  + ++     + −    + − +  + 14444244443 1444444442444444443 2 ( )nd order ac terms nonlinear   144444442444444443  (2.112)  The dc term must satisfy  1 2 2 c c c c c V D RV V D I R D R R = + − +  (2.113) We neglect the second-order terms in (2.112), leaving the following linearized ac equation  ( ) ( ) ( ) ( ) ( ) ( ) ( )1 1 1 2 1 2 ˆ ˆ ˆ ˆ ˆ ˆ ˆ c c c c c c c c c c V Rd t v t RD v t V d t v t D I R d t i t R D R R R R = + − + + − + +  (2.114)  Substation of (2.95)-(2.101) into (2.94) leads to  ( ) ( )( ) ( )( )1 1ˆˆ ˆg gI i t I i t D d t+ = + +  (2.115) Upon multiplying this expression out and collecting terms, we obtain  ( ) ( ) { ( ) ( )( ) ( ) ( )( )1 1 1 1 1 ( ) 2 ( ) ˆ ˆˆ ˆ ˆ g g Dcterms st order ac terms linear nd order ac terms nonlinear I i t ID Id t i t D i t d t+ = + + + 1442443 14243  (2.116) The dc term must satisfy  1gI ID=  (2.117) We neglect the second-order terms in (2.116), leaving the following linearized ac equation  ( ) ( ) ( )1 1ˆˆ ˆgi t Id t i t D= +  (2.118) This result represents the low-frequency ac variations in the converter input current.  The equations of the quiescent values, (2.105), (2.109), (2.113), and (2.117) are collected below as  26   1 1 2 2 2 1 2 2 1 2 2 1 0 0 g sw g c c c d c c c c c c c g V D R I D V nD I R nD V nD V D VDInD R R R V D RV V D I R D R R I ID = − + − −    = − − − +    = + − +  =   (2.119)  For given quiescent values of the input voltage gV , the diode voltage drop dV , and the duty cycle 1D , the system (2.119) can be evaluated to find the quiescent output voltage V , inductor current I , input current gI , capacitor voltage cV , and capacitor current cI . However, in this problem there are 5 variables but there are only 4 equations. The fifth equation can be the following  c c cV V I R= −  (2.120) The results are then inserted into the small-signal ac model.  The small signal ac model, (2.106), (2.110), (2.114), and (2.118), is summarized below   ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) 1 1 1 1 1 1 2 1 2 1 1 1 1 2 1 2 1 1 1 2 1 ˆ ˆ ˆ ˆˆ ˆ ˆ ˆˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ m g g sw g sw g c c c c c c d c c c c c c c c c c c c c c di t L V d t v t D R I d t R i t D V nd t dt v t nD I R nd t i t R nD V nd t dv t V d t v t D Vd t v t D C Ind t i t nD dt R R R R R R V Rd t v t RD v t V d t v t D I R d t i R R R R = + − − − + + − + = − − + − + − + + = + − + + − + + ( ) ( ) ( ) ( ) 2 1 1 ˆˆ ˆ c g t R D i t Id t i t D              = +   (2.121)  The final step is to construct an equivalent circuit of (2.119) and(2.121) using the commercially available simulation tools.  27  Let us now apply the state-space averaging method to the second order flyback converter of Figure 2.8(a). The independent state variables as usual are the inductor current ( )i t  and the capacitor voltage ( )cv t , which form the state vector  ( ) ( )( )c i t x t v t   =      (2.122) The input voltage ( )gv t , and the diode voltage drop is an independent source, which should be placed in the input vector as  ( ) ( )g d v t u t V   =      (2.123) To model the converter input port and output port, we need to find the converter input current ( )gi t  and output voltage ( )v t . To calculate this dependent current and voltage, it should be included in the output vector ( )y t  as  ( ) ( )( ) gi ty t v t   =      (2.124)  Next, let us write the state equations for each subinterval. When the switch is on and the diode is off, the converter circuit of Figure 2.8(b) is obtained. The inductor voltage, capacitor current, output voltage, and converter input current are  ( ) ( ) ( )m g swdi tL v t R i tdt = −  (2.125)  ( ) ( )c c c dv t v t C dt R R = − +  (2.126)  ( ) ( )c c v t R v t R R = +  (2.127)  ( ) ( )gi t i t=  (2.128) Similar to (2.73)-(2.76), after organizing the terms in (2.125)-(2.128), the result can be written in the following state-space form 28   ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) 1 1 0 1 0 10 0 0 sw m g m c dc u tx tc B dx t A dt Rdi t L i t v tdt L v t Vdv t RC R Cdt −              = +          −       +    14243123 14243 14243 144424443  (2.129)  ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )1 1 1 0 0 0 0 0 0 g g c d c E u ty t x t C i t i t v t R v t v t V R R           = +              +   12314243123 123142443  (2.130)  So the state-space equations for the first subinterval have been identified.  In the second subinterval, when the MOSFET is off and the diode is on, the converter circuit of Figure 2.8(c) is obtained. For the second subinterval, the inductor voltage, capacitor current, output voltage, and converter input current are given by   ( ) ( ) ( )2 c c m d c c di t i t n R R v t nR L V n dt R R R R = + − − −  (2.131)  ( ) ( ) ( )c c c c dv t i t nR v t C dt R R R R = − − − −  (2.132)  ( ) ( ) ( )c c c c i t nR R v t R v t R R R R = + − −  (2.133)  ( ) 0gi t =  (2.134)  After organizing them and writing in state-space form, we get   ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) 2 2 2 0 1 0 0 c gm c m m c m m c dc u tx tc c B dx t A dt n R R nRdi t n i t v tRL R L RL R Ldt L v t VnRdv t RC R C RC R Cdt        −   − −     = +          − −       − −    14243123 14243 14243 1444442444443  (2.135) 29   ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )2 2 0 0 0 0 0 0 g g c c d c c E u ty t x t C i t i t v t nR R R v t v t V R R R R           = +               − −   12314243123 123144424443  (2.136) So the space-space equation of the second interval has also been identified.  The next step is to combine the result and obtain the state-space averaged model as   2 1 1 2 2 1 2 2 1 2 2 2 1 2 0 1 10 sw c m m c m m c m c c c sw c m m c m m c m c c c R n R R nR L RL R L RL R L A A D A D D D nR RC R C RC R C RC R C R D n R RD nRD L RL R L RL R L nRD D D RC R C RC R C RC R C −       − −  = + = +    − − −  + − −     − +  − −  =   − − −  − + −   (2.137)  1 2 1 1 2 2 1 2 1 0 0 0 0 0 0 0 0 m m m m D nDn L L L LB B D B D D D      − −     = + = + =                 (2.138)  1 1 2 2 1 2 1 2 1 2 0 01 0 0 0 c c c c c c c c C C D C D D DnR RR R R R R R R R D nR RD RD RD R R R R R R       = + = +    +  − −          =  +  − + −    (2.139)  1 1 2 2 1 2 0 0 0 0 0 0 0 0 0 0 0 0 E E D E D D D       = + = + =             (2.140)  Therefore, the final state-space averaged model of the second order flyback converter with parasitics in CCM becomes 30   ( ) ( ) ( ) ( ) ( ) 2 1 2 2 2 1 2 1 2 0 0 sw c m m c m m c m cc c c c g m m d R D n R RD nRDdi t i tL RL R L RL R Ldt v tnRD D Ddv t RC R C RC R C RC R Cdt D nD v t L L V  −  +    − −   =       − − −   − + −      −   +         (2.141)  ( ) ( ) ( ) ( ) ( )1 2 1 2 0 0 0 0 0 g g c c d c c c D i t i t v t nR RD RD RD v t v t V R R R R R R           = +       +        − + −    (2.142)  To validate the analytically derived state-space averaged model in CCM, a detailed model of Figure 2.8(a) has been constructed in PLECS [16]. The state-space averaged model has been constructed in Matlab/Simulink [30]. The same hardware prototype has been used here to validate the results. The parameters of the second-order flyback converter with basic parasitics in CCM are given in Appendix A.3. The predicted and measured inductor current, capacitor voltage, and output voltage are shown in Figure 2.9.  As can be seen in Figure 2.9, the analytically derived state-space averaged model predicts the averaged inductor current and capacitor voltage very well. There is a difference between the detailed model and hardware prototype in terms of the output voltage (see Figure 2.9), which comes from the simplified detailed model (second-order with parasitics). As seen in Figure 2.10, the analytically derived state-space averaged model predicts the average output voltage very well. 31  0.5 1 1.5 I (A m p .) -72.496 -72.492 -72.488 V c  ( V o lt ) 0.999991 0.999993 0.999995 0.999997 -72.6 -72.2 -71.8 -71.4 Time (s) V  ( V o lt ) Actual Average and Analytical State-Space See Figure 2.10 Detailed Model Actual Average Analytical State-Space Detailed Model Actual Average Analytical State-Space Hardware Prototype  Figure 2.9 Inductor current, capacitor voltage and output voltage of second order Flyback converter with parasitics in CCM.  0.999991 0.999993 0.999995 0.999997 -72.5 -72.495 -72.49 -72.485 -72.48 Time (s) V  ( V o lt ) Detailed Model Actual Average Analytical State-Space  Figure 2.10 Output voltage of second order Flyback converter with parasitics in CCM.       32  2.4 State-Space Averaging with Parasitics in DCM In addition to the two subintervals that occur in CCM, here there is another subinterval that occurs at light loads. The topology of the flyback converter in this subinterval is shown in Figure 2.11.  + mL n ( ) i t ( )gi t ( )ci t ( ) v t ( )Lv t C R ( )gv t cR ( )cv t  Figure 2.11 Second-order Flyback converter with parasitics during third subinterval in DCM.  During the first subinterval depicted in Figure 2.8(b), the MOSFET is on and the diode is off, and the magnetizing inductance stores some energy. During the second subinterval depicted in Figure 2.8(c), the MOSFET is off and the diode is on. During this interval, the stored energy is transferred to secondary side and it flows through the filter capacitor to the load resistor. If the magnetizing current during the second interval goes to zero before the end of the second interval, the converter goes to another stage resulting in third subinterval and DCM, before it goes back to the first interval in the next cycle.  For the flyback converter without parasitics in DCM, the first and second sub intervals remain of the same as in CCM as described in Section 2.3. The third interval comes into account when the MOSFET and the diode are off as in Figure 2.11. For this case, the inductor voltage ( )Lv t , capacitor current ( )ci t , output voltage ( )v t , and converter input current  ( )gi t  are  ( ) 0Lv t =  (2.143)  ( ) ( )cc c v t i t R R = − +  (2.144)  ( ) ( )c c v t R v t R R = +  (2.145) 33   ( ) 0gi t =  (2.146)  Let us now apply the state-space averaging method to the third interval of Figure 2.11. The inductor voltage, capacitor current, output voltage, and converter input current can be written as  ( ) 0m di tL dt =  (2.147)  ( ) ( )c c c dv t v t C dt R R = − +  (2.148)  ( ) ( )c c v t R v t R R = +  (2.149)  ( ) 0gi t =  (2.150) Similar to (2.143)-(2.146), after organizing terms in (2.147)-(2.150), these equations can be written as   ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )3 3 0 0 0 0 10 0 0 g c dc c B u tx t A dx t dt di t i t v tdt v t Vdv t RC R C dt             = +      −      +      12314243123 14424443 14243  (2.151)  ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )3 3 0 0 0 0 0 0 0 g g c d c E u ty t x t C i t i t v t R v t v t V R R           = +              +   12314243123 123142443  (2.152)  which defines the state-space equation for the third subinterval. Hence, the next step is to evaluate the state-space averaged equations, which goes as following:  34   1 1 2 2 3 3 2 1 2 3 2 1 2 2 1 32 2 0 0 0 101 10 sw c m m c m m c m c c c c sw c m m c m m c m c c c A A D A D A D R n R R nR L RL R L RL R L D D D nR RC R C RC R C RC R C RC R C R D n R RD nRD L RL R L RL R L D DnRD D RC R C RC R C RC R C = + + −        − −    = + +   −   − +− −      + − −    − + − − =  + − − − − + −        (2.153)  1 2 1 1 2 2 3 3 1 2 3 1 0 0 0 0 0 0 0 0 0 0 0 0 m m m m D nDn L L L LB B D B D B D D D D      − −       = + + = + + =                    (2.154)  ( ) 1 1 2 2 3 3 1 2 3 1 1 32 2 0 01 0 0 0 0 0 0 c c cc c c c c c C C D C D C D D D DnR RR RR R R R RR R R R D R D DnR RD RD R R R R R R           = + + = + +     + + − −             = + +   − + −   (2.155)  1 1 2 2 3 3 1 2 3 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 E E D E D E D D D D         = + + = + + =                 (2.156)  The final state-space averaged model for the second order flyback converter with parasitics in DCM becomes:   ( ) ( ) ( ) ( ) ( ) 2 1 2 2 1 32 2 1 2 0 0 sw c m m c m m c m cc c c c g m m d R D n R RD nRDdi t i tL RL R L RL R Ldt v tD DnRD Ddv t RC R C RC R C RC R Cdt D nD v t L L V  −  +    − −   =   +    − − −   − + −      −   +         (2.157)  ( ) ( ) ( ) ( ) ( ) ( )1 1 32 2 0 0 0 0 0 g g c c d c c c D i t i t v t R D DnR RD RD v t v t V R R R R R R           = ++       +         − + −   (2.158)  35  To validate the analytically derived state-space averaged model in DCM, a detailed model of the converter circuit depicted Figure 2.8(a) has been constructed in PLECS [16]. The state- space averaged model has been constructed in Matlab/Simulink [30].The variable 2d  has been calculated using (2.58) and set to 0.4409 . The load was assumed as 2500LR R= = Ω . The parameters of the second-order flyback converter without parasitics in CCM are given in Appendix A.4. The predicted inductor current and capacitor voltage are shown in Figure 2.12.  0 0.5 1 I (A m p .) -102 -101.5 -101 V c  ( V o lt ) 0.999991 0.999993 0.999995 0.999997 -102 -101.5 -101 Time (s) V  ( V o lt ) Detailed Model Actual Average Analytical State-Space Detailed Model and Actual Average Detailed Model and Actual Average  Figure 2.12 Inductor current, capacitor voltage and output voltage of second order Flyback converter with parasitics in DCM.  As can be seen in Figure 2.12, the analytically derived state-space averaged model predicts the magnetizing current with a large error of 17.5% . This error is because the current is discontinuous, while the conventional state-space averaging fails to correctly take this into 36  account. At the same time, the analytically derived state-space averaged model predicts the capacitor voltage with a very small error because the capacitor voltage is a continuous state variable. The small error comes from the error in the magnetizing current defined by (2.157).  2.5 Parametric Average Value Modeling in CCM and DCM To see the challenges with the conventional state-space averaging in DCM and representation of parasitics, we take another detailed look at this approach. The state-space averaging is a well defined approach [18] that has been presented previously in numerous publications, e.g. [8, 10, 33]. In CCM, the state-space equation is  ( ) ( ) ( )( )( ) ( ) ( ) ( )( )( ) ( )1 2 1 21 1x t q t A q t A x t q t B q t B u t• = + − + + −  (2.159) where ( )q t is the switching function, ,k kA B are the system matrices, and ( )u t is the input vector. By definition, the so-called fast average of a state variable ( )x t  over a switching interval is  ( ) ( )1 st T s t x t x t dt T + = ∫  (2.160)  where ( )x t  is the actual or the true average of ( )x t . Since the averaging is commutative with respect to differentiation, taking the fast average of (2.159) over a switching interval yields  ( ) ( ) ( )( )( ) ( ) ( ) ( )( )( ) ( )1 2 1 21 1x t d t A d t A x t d t B d t B u t• = + − + + −  (2.161) This result follows from the fact that the fast average of the switching function over a switching interval is the duty cycle function  ( ) ( )1 st T s t d t q t dt T + = ∫  (2.162) It is also assumed that the average of the product is equal to the product of the averages, especially  Ax A x= ⋅  (2.163)  Bu B u= ⋅  (2.164) Assumption (2.163) is acceptable if the original switching variables do not deviate significantly from their average values and also the system matrices 1A  and 2A  are 37  commutative [18, 34]. But in general, the matrices 1A  and 2A  are not commutative [35, 36]. So, if we make assumption (2.163), the equation (2.161) is not an exact solution as mentioned in [18] and Appendix A, but can only be an approximation. Assumption (2.164) is usually accepted when the source ripple is neglected.  The DCM operation of PWM converters differs from CCM operation by an additional time interval in each switching cycle during which the inductor current or capacitor voltage is clamped to zero (or a constant when there are multiple energy storage elements).  Conventional state-space averaging for converters working in DCM has been summarized in [4, 10, 18]. For this mode, the direct extension of (2.159)-(2.161) results in  ( ) ( ) ( ) ( ) ( )1 1 2 2 3 3 1 1 2 2 3 3x t d A d A d A x t d B d B d B u t • = + + + + +  (2.165) which is no longer accurate. In particular, the local average of the magnetizing current in the third interval is zero, whereas the state-space averaging implies that this value should be 3d i . Since 3d  and i  are not zero, the result of the state-space averaging is not zero. That is why the discontinuous averaged variable in DCM is higher than the actual averaged value that can be seen in Figure 2.7 and Figure 2.12.  As shown in Sections 2.2 and 2.4, the conventional state-space averaging is no longer accurate in DCM. The parametric average-value modeling has been proposed to solve this problem as documented in numerous publications [22, 24-27].  2.5.1 Correction Term The state-space averaging involves the weighted sum of the state-space equations corresponding to different topological instances within a switching interval. In DCM, a prototypical switching interval is divided into three subintervals as seen in Figure 2.6. To accurately represent the dynamics of the underlying converter circuit, a corrected full order state-space averaged model [4] has been proposed for an ideal converter circuit as  ( )3 3 1 1 k k k k k k x d A Mx d B u • = =     = +        ∑ ∑  (2.166) 38  To make state-space averaging work properly in DCM, the so-called correction matrix M  is added in (2.166). The analytical derivation of the correction matrix for an ideal topology (without parasitics) is given in [4]. Since the circuit of Figure 2.8(a) with parasitics has been considered in this study, it would be very complicated to extend that analytical solution. Hence, a numerical solution has been adopted in this Thesis.  The correction term is a diagonal matrix, wherein each state variable has its own correction coefficient. In this implementation presented in this Thesis, instead of a diagonal matrix, a column correction vector (each state variable has its own correction term as an element of column matrix) is proposed as  ( ) ( )3 3 1 1 .k k k k k k x d A x M d B u • = =     = ∗ +        ∑ ∑  (2.167) where ( ).x M∗  denotes the element wise multiplication. This operation uses less computational time compare to direct implementation of (2.166).  2.5.2 Model Implementation A detailed model of the converter depicted in Figure 2.8(a) has been constructed using the PLECS toolbox. The system matrices ( ), , ,k k k kA B C D  in each subinterval can be extracted numerically using PLECS and Simulink built-in feature for numerical model linearization and analysis. Since the detailed model includes all the parasitics, the extracted system matrices will have all the necessary information in them by construction and without any extra analytical derivations. The elements of correction vector ( )M  and ( )3d  are obtained as functions of the duty-cycle ( )1d  and the average value of the state variables ( )x . To obtain the values of ( )3 1,d d x  and ( )1,M d x , the detailed model is run in the operation region of interest (for example; duty-cycle changes between 0.1 and 0.9, and the load ( )R  changes from very low load to very high load) whereas the state variables are averaged numerically over the prototyping switching interval and saved for the future use. In particular, the average-value of the state vector ( )x  is computed in a steady-state corresponding to a given operation point. Specifically, in steady-state we have 39   ( ) ( )3 3 1 1 0 .k k k k k k d A x M d B u = =     = ∗ +        ∑ ∑  (2.168) From which an intermediate variable vector p  is computed as  ( ) ( ) 13 3 1 1 . k k k k k k p x M d A d B u − = =     = ∗ = − ⋅        ∑ ∑  (2.169) Thereafter the elements of M  can be found using the following  j j jM p x= ⋅ ÷  (2.170) where j  denotes number of state variables and ( )j jp x⋅ ÷  denotes element wise division.  To obtain the functions  ( )3 1,d d x  and ( )1,M d x  for the desired operation range, the detailed simulation is run with different values of control variable ( )1d  as well as the load ( )R . The variables resulting from this procedures are 1 2 3, , , , ,cd d d R v i  and the correction vector ( )M  is computed using (2.169) and (2.170). These variables are stored for future use in lookup tables, wherein an interpolation/extrapolation may be used as necessary. The real challenge here is to calculate the 2d  or 3d  at any given operation point. If one can calculate either one, thereafter it becomes easy to calculate the other one since 1d  is the control variable which can be calculated from the magnetizing current. As can be seen in Figure 2.12, the magnetizing current is zero during the third subinterval whereas the variable 3d  can be calculated.   The final numerical function for 3d  is plotted in Figure 2.13, which shows that this function has a flat surface corresponding to CCM, and varies linearly along the 1d . In DCM, the surface of 3d  becomes non-linear and increases.   40  0 1000 2000 3000 4000 5000 0.10.20.30.4 0.50.60.70.8 0.91 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 R (ohm) d 1 d 3 CCM DCM  Figure 2.13 Variable 3d  as a function` of duty-cycle ( )1d  and the load ( )R .  The correction term ( )1m  of the magnetizing current ( )I  is plotted in Figure 2.14 as a function of duty-cycle ( )1d  and the load resistance ( )R . As can be seen in Figure 2.14, on the one hand, the correction term ( )1m  has a flat surface and value of 1 in CCM, which implies that the state-space averaging captures the correct value of this current in CCM and no correction is required. On the other hand, it has values higher than 1 corresponding to the DCM because the conventional state-space averaging does not capture the correct average values in this region of operation as explained in Section 2.5, and therefore the correction effort is required.  0 1000 2000 3000 4000 5000 0.10.20.30.4 0.50.60.70.80.91 0.8 1 1.2 1.4 1.6 1.8 2 2.2 2.4 2.6 R (ohm) d 1 m 1 CCM DCM  Figure 2.14 The correction term 1m  as a function of duty-cycle ( )1d  and the load resistance ( )R . 41  The correction term ( )2m  of filter capacitor voltage ( )cv  is plotted in Figure 2.15 as a function of the duty-cycle ( )1d  and the load resistance ( )R . As can be seen in Figure 2.15, the calculated value for this correction term is 1 for either CCM or DCM. This value is consistent with the fact that the capacitor voltage is a continuous state variable with relatively small ripple, so the conventional state-space averaging predicts the correct average value for this variable for both modes.  0 1000 2000 3000 4000 5000 0.10.20.30.40.50.6 0.70.80.91 0.98 0.99 1 1.01 1.02 R (ohm) d1 m 2 CCM and DCM  Figure 2.15 The correction term 2m  as a function of duty-cycle ( )1d  and the load resistance ( )R .  Once the parametric functions ( )3 1,d d x  and ( )1,M d x  have been calculated and stored, these functions become available for the model implementation. Finally, the parametric average- value model is implemented according to the block diagram shown in Figure 2.16. The system matrices ( ), , ,k k k kA B C D  for each subinterval are calculated numerically. For a given value of control variable 1d  and state vector x , the values of 3d  and correction vector ( )M are acquired through the lookup tables. Total average system matrices ( ), , ,T T T TA B C D  are computed using the system matrices ( ), , ,k k k kA B C D  and the variables 1 2,d d  and 3d . These matrices are then used to build the new continuous non-linear state-space average-value model that replaces the discontinuous detailed model. Thereafter, this parametric average- value model can be used for large-signal transient studies as well as for numerical linearization and subsequent small-signal frequency-domain analysis.  42  3 d M x 1 d u y ( ) ( ) ( ) ( ) 2 1 3 3 1 3 1 3 1 3 1 1 T k k k T k k k T k k k T k k k d d d A d A B d B C d C D d D = = = = = - - = = = = å å å å ( )T T T T x A x M B u y C x D u · = ×* + = + , , ,k k k kA B C D ( ) ( ) 3 1 1 , , d d x M d x  Figure 2.16 Implementation of the parametric average-value model.  The previously proposed parametric average-value models [24-27] had problem with the correcting the output equation. In particular, in the previous formulation the output equation was defined as  ( )3 3 1 1 k k k k k k y d C Mx d D u = =     = +        ∑ ∑  (2.171)  However, my simulation results (which are not shown in this study) have shown that if there is a correction term ( )M  in the output equation, in this case no correction occurs on the output voltage ( )V . The reason for this is that the state vector ( )x  has already been corrected in the state equation (2.167), and therefore no additional correction is required in the output equation. The analytical proof of this argument goes as follow: During the steady-state, we have  1 10 T T T TA Mx B u x M A B u − − = + ⇒ = −  (2.172) When (2.172) is inserted into (2.171), the output equation becomes  ( )1 1T T T Ty C M M A B u D u− −= − +  (2.173) After required calculations in (2.173), we get  1 T T T Ty C A B u D u − = − +  (2.174) If there is a correction term in (2.171), there is no correction occurs in (2.174). Therefore, the correct output equation is 43   ( )3 3 1 1 k k k k k k y d C x d D u = =     = +        ∑ ∑  (2.175)  2.5.3 Case Studies The flyback converter shown in Figure 2.8(a) is used here for the case studies. The detailed model has been implemented using PLECS. The converters parameters are summarized in the Appendices A.3 and A.4 for CCM and DCM, respectively. The proposed parametric average-value model has been implemented and compared to the hardware prototype, the detailed model and the conventional state-space averaging model in both time and frequency domains.  2.5.3.1 Time domain To demonstrate the parametric average-value model (PAVM) in DCM, the system is operates in steady-state defined by 1 0.381d =  and 2500R = Ω . The resulting simulated waveforms of the magnetizing current, filter capacitor voltage and the output voltage are plotted in Figure 2.17. As can be seen in Figure 2.17, the PAVM predicts the average-value of the steady state waveforms in DCM very well compared to the conventional state-space averaging. Actual average of the inductor current is 0.46Amp , and the PAVM can predict this value very well. On the other hand, the analytical state-space averaging can predict this value as 0.55 .Amp which is equal to 18.18%  error. Actual average of the capacitor voltage is 101.56V− , and the PAVM can predict this value very well. The analytical state-space averaging can predict this value as 101.2V− .  44  0 0.5 1 I (A m p .) -102 -101.5 -101 V c  ( V o lt ) 0.999991 0.999993 0.999995 0.999997 -102 -101.5 -101 Time (s) V  ( V o lt ) Detailed Model, Actual Average and PAVM Detailed Model Actual Average Analytical State-Space PAVM Actual Average and PAVM Detailed Model, Actual Average and PAVM  Figure 2.17 Simulated inductor current, capacitor voltage and output voltage of the second order Flyback converter with parasitics in DCM.  The accuracy of the PAVM in predicting the large-signal behaviour in time-domain has also been verified by studying the effect of sudden change in load. In the following study, the output of the flyback converter was regulated using a PI controller. The PI controller was designed to regulate the output voltage at 72V−  by adjusting the duty-cycle ( )1d . The controller parameters are 0.7pK =  and 25iK = . In this study, the converter run with a resistive load of 2000Ω  (which results in DCM) until it reaches a steady-state. At 0.2t s= , a parallel load of 500Ω  is added (which results in CCM). The resulting time-domain transients are shown in Figure 2.18.  45  0 1 2 3 I (A m p .) -72 -71.8 -71.6 V c (V o lt ) 0.1996 0.1998 0.2 0.2002 0.2004 0.2006 0.2008 0.201 -72 -71.8 -71.6 Time (s) V  ( V o lt ) PAVM Detailed Model Detailed Model and PAVM Detailed Model and PAVM  Figure 2.18 Transients in inductor current, capacitor voltage and output voltage of the second order Flyback converter due to the step change in load.  As it can be observed in Figure 2.17, after the load change at 0.2t s= , the converter switches operation from DCM to CCM. After load changes the control action brings the output voltage to about the desired 72V−  after about 0.15s  which wasn't shown in Figure 2.18 to demonstrate the response of the PAVM to load change.  At the same time, as it can be seen in Figure 2.17, through all transients the large signal behaviour of the detailed model is accurately predicted by proposed PAVM. 2.5.3.2 Frequency domain The control-to-output transfer function is often considered in literature for verifying the small-signal behaviour of the converter models. The small-signal injection and subsequent frequency sweep method has been implemented to extract the small-signal transfer function 46  from the detailed simulation and the PAVM corresponding to full load operation condition defined by 717.05R = Ω  and 1 0.381d = . The magnitude and phase of the corresponding control-to-output transfer function are plotted in Figure 2.19.  The transfer function is evaluated up to150kHz , which is more than one-half of the switching frequency ( )250kHz . Closer to the switching frequency the results become distorted due to the interaction between the injected perturbations and the converter switching. In general, considering the frequencies closer to and above the switching frequency has limited use for the average-value model since the basic assumptions of the averaging are no longer valid.  -20 -10 0 10 20 30 A m p lit u d e  ( d B ) 10 2 10 3 10 4 10 5 -90 -80 -70 -60 -50 -40 -30 -20 -10 0 P h a s e  ( ° ) Frequency (Hz) PAVM Detailed PAVM Detailed  Figure 2.19 Control-to-output transfer function of the second-order Flyback converter evaluated at 717.05R = Ω  and 1 0.381d = .       47  Chapter 3 : Analysis of Flyback Converter with Snubber Circuits   Real semiconductor devices experience voltage and current stress during the turn-on and turn-off transitions which can easily damage the circuit elements. If a power electronic converter stresses a power semiconductor device beyond its ratings, there are two basic ways of relieving the problem. Either the device can be replaced with one whose ratings exceed the stresses, or a snubber circuits can be added to the converter in order to reduce the stress to safe levels. The final choice always involves a trade-off between the cost and availability of semiconductor devices with required electrical ratings compared to the cost and additional complexity of using the snubber circuits [9, 37-40]. In some topologies, such as a transformer isolated Flyback converter, the snubber circuit may be required to  protect the switching transistor due to non-zero transformer leakage inductance.  3.1 Fifth –order Flyback Converter with Snubbers The circuit with snubbers considered in this Chapter is shown in Figure 3.1. Also, a hardware prototype that has been built includes the snubbers. The parameters and the detailed circuit diagram are given in Appendices A.5 and Appendix B, respectively. The circuit is has fifth- order because it has a high order transformer model with primary and secondary leakage inductances, and two RC snubber circuits ( ),ss ss ds dsC R C R− −  to protect the MOSFET and diode during turn-on and turn-off switching transitions.  + + sL mL R cR ssR dsR swR C dsC ssC gV Mosfet V dVD n pL  Figure 3.1 Fifth-order Flyback converter circuit. 48   To demonstrate the role and effect of the snubber circuit, the measured transformer secondary voltage waveforms with and without the diode snubber is shown in Figure 3.2.  -100 0 100 200 300 V s (V o lt ) 5.983 5.984 5.985 5.986 5.987 5.988 5.989 x10 -3Time (s) -100 0 100 200 300 V s (V o lt ) ( ) a ( ) b  Figure 3.2 Measured transformer secondary voltage: (a) without the diode snubber; and (b) with the diode snubber.  As can be seen in Figure 3.2(a), when the snubber is not used on the secondary side, the voltage spikes up to 310V  and the oscillations die pretty much at the end of the switching interval. This phenomena causes two problems. First, the spike voltage can cause breakdowns on the switching device. Secondly, the ringing energy will be radiated creating noise and electro-magnetic interference (EMI) issues with potential for logic and control errors. This type of ringing is not acceptable and it is necessary to add the snubber circuit elements to damp the ringing, or clamp the voltage (with RCD clamps), or active snubbers [40]. In the hardware prototype, a conventional RC snubber is used because of its simplicity and low part count. Calculation of the RC elements is not the interest of this Thesis, whereas the reader can find more information in [9, 37-39].  49  It must be emphasized that the snubbers are not a fundamental part of a power electronic converter circuit. However, when it comes to averaging of PWM converters, the snubbers significantly add to the complexity of the problem, and to the best of our knowledge this has not been considered in the prior literature. At the same time, since the snubber circuits have an important role in overall system dynamics, it is very desirable to take them into account in the average-value modeling. On the one hand, the benefit of considering the snubbers will be the improved accuracy of the average model in terms of capturing and predicting the converter losses and efficiency, which will be used in Section 4.4.3. On the other hand, their presence in the circuit has introduced more complexity and theoretical problems which are almost impossible or impractical to resolve using conventional analytical derivations of the average-value models as will be explained in more detail in Section 3.2.  3.2 State-Space Averaging Phenomena with the Snubbers The circuit considered in this Section is shown in Figure 3.1. This circuit is a fifth-order system which makes it more difficult to derive state equations analytically. A more practical method to calculate the state-space equation is to built a detailed switching model using any commercially available software programs and extract the state-space matrices numerically from the detailed model [16, 30] for each subinterval. These matrices can then be used for the state-space averaging.  A detailed analysis of the fifth-order Flyback converter will be given in Chapter 4. In this Section, we will focus on only the problem caused by the snubbers, not the whole converter circuit. The problem occurs on the output filter capacitor voltage, which also translates to the output voltage which has been defined as an output in state-space output equation. To demonstrate this problem caused by the snubbers, a detailed model has been built using PLECS [16]. The corresponding converter parameters are summarized in Appendix A.5. The state matrices , , ,k k k kA B C D  have been extracted for each subinterval in CCM. Finally, the state-space averaged model has been calculated using these matrices and given duty ratio. The simulated output filter capacitor voltage and the output voltage are shown Figure 3.3.   50  0.0499 0.04991 0.04992 0.04993 -80 -60 -40 -20 0 Time (s) V  ( V o lt ) Detailed Model Actual Average State-Space Averaging c -80 -60 -40 -20 0 V  ( V o lt ) Detailed Model and Actual Average Detailed Model and Actual Average  Figure 3.3 Simulated output filter capacitor voltage and the output voltage of the fifth-order Flyback converter with snubbers in CCM.  As can be seen in Figure 3.3, the output voltage and capacitor voltage is about 72V− , which corresponds to the considered operating point in CCM. However, the state-space averaged predicts the very low value of about 1.25V− . Since the output voltage and the capacitor voltage are continuous and slow variables, the conventional state-space averaging should predict the average value correctly but in this can it does not! The reason why this is happening can be understood by examining the Figure 3.1. The output filter capacitor voltage is created by the charging current that flows through the diode. On the one hand, if the average current is predicted correctly, the capacitor voltage will also be correct. But on the other hand, if we look at the final state-space averaged equation (3.1), the extracted from the detailed model diode current is not a state variable. As a result, the output capacitor voltage does not depend on the diode current in the averaged state-space model. Instead, the output filter capacitor voltage depends on the secondary side current ( )sI , the diode snubber capacitor voltage ( )dsCV , and the diode voltage drop ( )dV . The diode voltage drop is a small constant that is not significant in this discussion. Instead, let us look at the averaged secondary current ( )sI , and the diode snubber capacitor voltage ( )dsCV  shown in Figure 3.4. 51   7 9 5 6 7 7 5 5 6 6 7 9 7 205.33 0 0 28393 141.96 0 7.946 10 1.333 10 0 0 2.593 10 1.572 10 1.572 10 3.115 10 1.557 10 43538 2.6 10 2.6 10 5.231 10 26157 3.123 10 0 0 3.752 10 3.123 10 ss ds c C p s C v v i i v • • • • •     −    − × ×   = × − × − × × ×    − × × − × −   × × − ×       6 5 5 7 0 141.96 0 0 2.509 10 2.593 10 4.151 10 43543 0 3.123 10 ss ds c C p s C g d v v i i v V V                        −         + × ×       − × −   ×   (3.1)  -0.4 -0.2 0 0.2 0.4 I (A m p .) s 0.01232 0.012321 0.012322 0.012323 0.012324 0.012325 0.012326 -100 0 100 200 Time (s) V C d s (V o lt ) Detailed Model Actual Average State-Space Averaging Actual Average and State- Space Averaging  Figure 3.4 The predicted secondary current and the diode snubber capacitor voltage of fifth-order Flyback converter in CCM.  As can be seen in Figure 3.4, the averaged state-space model predicts the secondary current very low at 0.01− , but its actual value must be about 0.1− . The reason being is that the secondary current is a discontinuous variable even though the converter operates in CCM. Also, the diode snubber capacitor voltage represents another problem. Here, he averaged state-space model predicts this value as zero which is also the actual averaged because of the 52  characteristic of the snubber as it releases the energy stored during the ringing. As a result, the output filter capacitor voltage has a very large error which translates into the large error in predicted output voltage.  For the purpose of further investigation, let us look at a circuit without a diode snubber, shown in Figure 3.5. This circuit is a forth-order system.  + + sL mL R cR ssR swR C ssC gV Mosfet V dVD n pL  Figure 3.5 Forth-order Flyback converter without diode snubber.  A detailed model of this simplifies fourth-order converter has been built in PLECS [16]. The corresponding parameters are summarized in Appendix A.5. The state-space matrices , , ,k k k kA B C D  have been extracted for the CCM in each subinterval. Finally, the state-space averaged model has been implemented using these matrices and the given duty ratio. The corresponding simulated output filter capacitor voltage and the output voltage are shown in Figure 3.6. As can be seen in Figure 3.6, the state-space averaged model predicts the output filter capacitor cV  and output voltage V  with a very small error. The reason for this small error is the discontinuous secondary current, which the state-space averaging cannot predict this current correctly.   53  -72.256 -72.252 -72.248 -72.244 -72.24 V c  ( V o lt ) 0.199991 0.199993 0.199995 0.199997 0.199999 -72.28 -72.27 -72.26 -72.25 -72.24 -72.23 Time (s) V  ( V o lt ) Detailed Model Actual Average State-Space Averaging  Figure 3.6 The simulated output filter capacitor and output voltage of the forth-order Flyback converter without the diode snubber in CCM.  Let us now look at the averaged state-space equation (3.2) of the fort-order flyback converter. The output filter capacitor voltage depends on only the secondary current. Since the state- space averaging predicts the secondary current with an error, because of the discontinuity, the output filter capacitor has the error but much less than the fifth-order model.  7 9 5 6 7 5 6 6 5 5 63.383 0 0 28042 0 8.116 10 1.316 10 0 2.561 10 1.548 10 1.548 10 23049 43000 2.561 10 2.561 10 3870 0 0 0 0 1.562 10 2.561 10 2.561 10 ss ss c c C C p p s s v v v v ii i i • • • •      −       − × ×     =    × − × − ×       − × × −          + × × − × 43005 g d V V               −   (3.2)  As it is shown in this Section, the state-space averaging has a problem when the diode snubber is present. The way to fix this problem is to introduce the diode current as a variable 54  in the averaged state-space model. To achieve that, the circuit has to be separated into two circuits as shown in Figure 3.7.  + + sL mL R cR ssR dsR swR C dsC ssC gV Mosfet V dVD n pL 1Circuit 2Circuit  Figure 3.7 Modified fifth-order Flyback converter circuit.  The Circuit 1 is a switched circuit, and on the other Circuit 2 is non switched and can be removed from the state-space averaging. After such partitioning, it is easy to extract the state matrices of the Circuit 1 in CCM or DCM for each subinterval using detailed model. Here, the diode current must be defined as an output in the state-space model, which will be the input of the Circuit 2. Then, the state-space equations of the Circuit 2 can either be extracted from the detailed model or calculated analytically. After extracting state-space model for these circuits, the overall system can be modelled in CCM as shown in Figure 3.8.  x Ax Bu y Cx Eu · = + = + ( ) ( ) ( ) ( ) 1 1 2 2 1 1 2 2 1 1 2 2 1 1 2 2 x A d A d x B d B d u y C d C d x E d E d u · = + + + = + + + State Variables gV dV di cv v ssC v dsC v pi si u u State Variable 1Circuit 2Circuit  Figure 3.8 Proposed state-space averaged model of the fifth-order Flyback converter using two sub- circuits and sub-models.  If the proposed state-space averaged model is implemented, it will be seen that the predicted output filter capacitor voltage cv  and output voltage v  is same as Figure 3.3, which is still 55  incorrect. The reason is that the diode current is a discontinuous variable, and so the state- space averaging of the Circuit 1 does not predict the diode current correctly, which translates into the corresponding error in the predicted output voltage.                             56  Chapter 4 : Full-order Flyback Converter   4.1 State-Space Averaging in CCM The full circuit considered in this Chapter is shown in Figure 4.1. This circuit corresponds to the full-order Flyback converter with parasitics and two snubber circuits (one snubber in the MOSFET and another snubber is on the secondary side with the diode). As mentioned earlier, when the circuit gets complicated, it is no longer practical to derive the state equations analytically. There are well defined algorithms [41] and software tools [14-16] that automatically generate and dynamically update the state-space model for each new topological state of the system being used. Regardless of the approach or tool used, it is assumed that inside each subinterval ( )k  the system state model may be expressed by the system matrices , , ,k k k kA B C D .  + + sL mL R cR ssR dsR swR C dsC ssC gV Mosfet V dVD n pLpR sR  Figure 4.1 Full-order Flyback converter circuit.  The detailed model has been implemented using PLECS [16]. The converter parameters are summarized in Appendix A.6. The state matrices , , ,k k k kA B C D  have been extracted for each subinterval in CCM. Finally, the state-space model has been implemented using these matrices and the given duty ratio. The simulated state variables are shown in Figure 4.2.  57  As can be seen in Figure 4.2, the state-space averaging predicts the switch snubber capacitor voltage ( ) ssC V , and the diode snubber capacitor voltage ( )dsCV  correctly as expected because these are continuous state variables. However, the state-space averaging predicts the output filter capacitor voltage ( )cV  with a large error. Moreover, the transformers primary ( )pI  and secondary current ( )sI , are also predicted with an error. First, the currents deviate from their average values because of the oscillations seen during the topology changes and as a result the assumption (2.163) is no longer valid as explained in Section 2.5. Second, both currents are discontinuous variables although the converter operates in CCM. Regardless of the mode of operation, CCM or DCM, these currents as individual state variables are discontinuous and (2.161) is no longer accurate. In particular, the local average of the primary current in the second interval ( )2d T  is zero (see Figure 4.2), whereas the state-space averaging implies that this value should be 2 pd i . Since 2d  and  pi  are not zero, the result of the state-space averaging is not zero. Similar scenario is applied to secondary current.  As a result, when it comes to state-space averaging of Flyback converters (if the transformer model has leakage inductances, which means primary and secondary currents are state variables), it needs special consideration in CCM as well as in DCM which will be explained in Section4.2.  58  -80 -60 -40 -20 0 V c (V o lt ) 0 10 20 30 40 V C s s (V o lt ) -1 0 1 2 3 I p (V o lt ) -0.4 -0.2 0 0.2 0.4 I s (V o lt ) 0.071399 0.071401 0.071403 0.071405 0.071407 -100 0 100 200 Time (s) V C d s (V o lt ) Detailed Model Actual Average State-Space Averaging Detailed Model Actual Average Actual Average and State-Space Averaging Actual Average and State-Space Averaging  Figure 4.2 Predicted state variables of full-order Flyback converter in CCM.    59  4.2 State-Space Averaging in DCM The DCM operation of PWM converters differs from CCM operation by an additional time interval in each switching cycle during which the inductor current or capacitor voltage is clamped to zero. In Flyback converters, there is an exception in this case if there is a diode snubber in the circuit. To demonstrate this point, a detailed model of the converter circuit depicted in Figure 4.1 has been implemented using PLECS [16]. The corresponding parameters are summarized in Appendix A.7. The state matrices , , ,k k k kA B C D  have been extracted for each subinterval in DCM. Finally, the state-space averaged model has been implemented using these matrices and the given duty ratio. The resulting simulated state variables are shown in Figure 4.3.    60  -100 -50 0 V c (V o lt ) 0 20 40 V C s s (V o lt ) -0.2 0 0.2 I s (A m p .) 0.099959 0.099961 0.099963 0.099965 0.099967 -100 0 100 200 Time (s) V C d s (V o lt ) -0.5 0 0.5 1 1.5 I p (A m p .) Detailed Model Actual Average State-Space Averaging Detailed Model Actual Average Actual Average and State-Space Averaging Actual Average and State-Space Averaging 1 sd T 2 sd T 3 sd T sT see Fig 4.4  Figure 4.3 Simulated state variables of the full-order Flyback converter in DCM.  61  0.099963 0.099964 0.099965 -0.04 0 0.04 0.08 Time (s) I s (A m p .) 3 sd T  Figure 4.4 Simulated transformer secondary current of the full-order Flyback converter in DCM.  During the first interval ( )1 sd T , when the MOSFET is on and the diode is off, the primary side of the transformer stores the energy in the field. Right after the switch change its state, the stored energy on primary side is transferred to the secondary side, and this energy is spent during the second interval ( )2 sd T . In CCM, the switches go back to their original position before this energy being spent. But in DCM this energy is spent before the switches change their positions. That is where there is an exception. If there is no snubber circuit, after the energy is spent, the secondary current will stay at zero until switches go back to their original state. If there is a snubber circuit, after the energy is spent, the energy stored in the diode snubber capacitor will flow in the other direction until it reaches zero or switches go back their original state. This special case makes it difficult to identify 2d  and 3d  in the model with the presence of parasitics, which will be explained is Section 4.3.1.  As can be seen in Figure 4.3, the state-space averaging predicts the switch snubber capacitor voltage ( ) ssC V , and the diode snubber capacitor voltage ( )dsCV  correctly as expected because these are continuous state variables. However, the state-space averaging predicts the output filter capacitor voltage ( )cV  with a large error. The reason has already been explained in Section 3.2. Also the transformer primary ( )pI  and secondary current ( )sI  are predicted with an error. As explained in Section 4.1, these currents are discontinuous in either CCM or DCM. As a result, the conventional state-space averaging method also does not produce the correct results in DCM. 62  4.3 Parametric Average Value Modeling in CCM and DCM In this Section, the parametric average-value modeling is extended to the full-order converter operation in CCM and DCM, which to the best of our knowledge has not been done for the transformer isolated topologies.  4.3.1 Model Implementation A detailed model of the converter depicted in Figure 4.1 has been implemented using PLECS. The system matrices ( ), , ,k k k kA B C D  for each subinterval have been extracted numerically using PLECS and Simulink. Since the detailed model has all the parasitics, the extracted system matrices have all information required. The element of correction vector ( )M  and ( )3d  are obtained as parametric functions of the duty-cycle ( )1d , the diode current ( )di  and the average value of the state variables ( )x . To obtain the values of ( )3 1,d d x  and ( )1,M d x , the detailed model has been run in the operation region of interest (for example; duty-cycle changes between 0.1 and 0.9, and the load ( )R  changes from very low load to very high load) whereas the state variables are averaged numerically over the prototyping switching interval. In particular, the average-value of the state vector ( )x  is computed in a steady-state corresponding to given operation point. Specifically, in the steady-state from equation(2.168) an intermediate vector p  is computed using (2.169). Thereafter, the elements of M  are found using (2.170).  To obtain the functions  ( )3 1,d d x  and ( )1,M d x  for the desired operation range, the detailed simulation is run with different values of control variable ( )1d  as well as the load resistance ( )R . The variables resulting from this procedure are 1 2 3 ,, , , , , , , ,ss dsd c C p s Cd d d R i v v i i v . Then, the correction vector ( )M  is computed using (2.169) and (2.170). These variables are stored for future use in lookup tables. The real challenge here is to calculate the 2d  or 3d  at any given operation point. As can be seen in Figure 4.3 and Figure 4.4, there is no easy way to read the intervals from the state variables' waveforms because of the ringing waveforms caused by snubbers. To achieve that, the diode current is used, as seen in Figure 4.5. As can be seen in 63  Figure 4.5 , the diode current is zero during the first and third subinterval. Since we know the first interval, it becomes easy to calculate the third and then second subintervals.  0.105993 0.105995 0.105997 0.105999 -0.16 -0.12 -0.08 -0.04 0 Time (s) I d (A m p .) 2 sd T ( )1 3 sd d T+ sT  Figure 4.5 The diode current waveform.  The final numerical function for 3d  is plotted in Figure 4.6. The variable 3d  has a flat surface corresponding to CCM and varies linearly along the 1d . In DCM, the surface of 3d  becomes non-linear and increases.  As explained in Section 3.2, the filter capacitor voltage ( )cV  has a large error even though it is a continuous voltage. To fix this problem a model proposed in Section 3.2 (see Figure 3.8) has been used here. If the secondary current is corrected, the diode current will be corrected as well, and as a result the output capacitor voltage and the output voltage will be fixed without a correction term. In the PAVM, the correction term 1m  of filter capacitor voltage ( )cV  can be set to 1. 64  0 1000 2000 3000 0.10.20.30.40.50.60.70.80.91 0 0.1 0.2 0.3 0.4 0.5 R (ohm) d 1 d 3 CCM DCM  Figure 4.6 Variable 3d  as a function of the duty-cycle ( )1d  and the load resistance ( )R .  The correction term ( )2m  of the primary side snubber capacitor voltage ( )ssCV  is plotted in Figure 4.7 as a function of the duty-cycle ( )1d  and the load resistance ( )R . The correction term ( )3m  of the primary current ( )pI  is plotted in Figure 4.8 as a function of the duty-cycle ( )1d  and the load resistance ( )R . The correction term of the primary current has no flat surface because some correction is needed in CCM and DCM as explained in Section 4.1. The correction term ( )4m  of the secondary current ( )sI  is plotted in Figure 4.9 as a function of the duty-cycle ( )1d  and the load resistance ( )R . Similar to correction term the secondary current ( )3m , the current ( )sI  is always discontinuous and needs correction in CCM and DCM.   65  0 1000 2000 3000 0.10.20.30.40.50.60.70.8 0.96 0.98 1 1.02 1.04 R (ohm) d 1 m 2 CCM and DCM  Figure 4.7 The correction term 2m  as a function of duty-cycle ( )1d  and the load resistance ( )R .     0 1000 2000 3000 0.10.20.30.40.50.60.70.8 0 2 4 6 8 10 12 14 R (ohm) d1 m 3 CCM and DCM  Figure 4.8 The correction term 3m  as a function of the duty-cycle ( )1d  and the load resistance ( )R .    66  0 1000 2000 3000 0.10.20.30.40.50.60.70.80.91 0 0.2 0.4 0.6 0.8 1 1.2 1.4 R (ohm) d1 m 4 CCM and DCM  Figure 4.9 The correction term 4m  as a function of duty-cycle ( )1d  and the load resistance ( )R .  The correction term ( )5m  of the diode’s snubber capacitor voltage ( )dsCV  is plotted in Figure 4.10. This correction term has a problem. Since the diode’s snubber capacitor voltage ( )dsCV is a continuous state variable, the correction term must be equal to 1. In theory, the average value of the diode snubber capacitor voltage is zero because it stores energy during one interval and discharge this energy during another interval. As a result, the average value is should be equal to zero. But when the value of p  calculated using (2.170), it doesn’t give exactly zero but gives very small numbers such as 132 10−⋅  due to the numerical precision of calculations. Then this is used to calculate the correction term ( )5m . In theory, dsCv  must be equal to zero but in the detailed simulation again we get very small numbers because of the ringing waveforms. Since the capacitor voltage is a continuous state variable, in the model implementation the correction term ( )5m  is set to 1. 67  0 1000 2000 3000 0.10.20.30.40.50.60.70.8 -2 -1.5 -1 -0.5 0 0.5 1 1.5 2 x10 -5 R (ohm) d1 m 5 CCM and DCM  Figure 4.10 The correction term 5m  as a function of duty-cycle ( )1d  and the load resistance ( )R .  Once the functions  ( )3 1,d d x  and ( )1,M d x  are available and stored, the parametric average- value model can be implemented as explained in section 2.5.2.  4.4 Case Studies The flyback converter shown in Figure 4.1 is used for case studies. The detailed model is implemented in PLECS [16]. The converters parameters are summarized in the AppendixA.6 and A.7 in CCM and DCM, respectively, correspond to the hardware prototype that has been built, Appendix B. The proposed model, Figure 2.16, has been implemented and compared to the hardware prototype, the detailed model and conventional state-space averaging model in both time and frequency domains.  4.4.1 Time Domain To demonstrate the parametric average-value model in CCM, the following study has been implemented in the detailed model, the hardware prototype, the conventional stat-space averaged model and the proposed PAVM. The system is assumed to initially operate in a steady state defined by 1 0.381d =  and 717.05R = Ω . The resulting measured and simulated output voltage, the primary current and the secondary current shown in Figure 4.11.  As can be seen in Figure 4.11, the developed PAVM predicts the average-value of the waveforms well in CCM as compared to conventional state-space averaging which does not. 68  There is a difference between measured output voltage and detailed model’s output voltage. The reason is that the detailed model doesn’t have all the parameters the hardware model has such as stray capacitances.  -71.7 -71.6 -71.5 -71.4 V  ( V o lt ) 0 1 2 I p (A m p .) 0.009992 0.009993 0.009994 0.009995 0.009996 -0.4 -0.2 0 0.2 0.4 Time (s) I s (A m p .) Detailed Model Hardware Prototype Actual Average PAVM Detailed Model State-Space Averaging Actual Average PAVM Detailed Model Actual Average and PAVM Actual Average and PAVM Actual Average and PAVM  Figure 4.11 Measured and simulated output voltage, primary and secondary current in CCM at constant duty-cycle.  The accuracy of the PAVM in predicting the large-signal behaviour in time-domain has also been verified by studying the effect of sudden change in load. In the following study, the output of the flyback converter was regulated using the same PI controller designed to regulate the output voltage at 72V− . In the study being considered, the converter initially operate in DCM with a 2000Ω  load. At 0.01t s= , a parallel load of 500Ω  is added which changes the mode to CCM. The resulting time-domain transients are shown in Figure 4.12. After load changes the control action brings the output voltage to the desired 72V− . Through 69  all transients, the large signal behaviour of the detailed model is accurately predicted by developed PAVM.  -72 -71.9 -71.8 -71.7 V  ( V o lt ) -2 -1 0 1 2 3 4 I p (A m p .) 0.0098 0.0102 0.0106 0.011 0.0114 -1 -0.5 0 0.5 Time (s) I s (A m p .) Detailed Model PAVM Detailed Model PAVM Detailed Model PAVM  Figure 4.12 Simulated output voltage, primary and secondary current during the transient from DCM to CCM due to the step change in load.  4.4.2 Frequency Domain The control-to-output transfer function of the full-order converter predicted by the detailed and the developed PAVM models have been extracted again for the full load operating point defined by 717.05R = Ω  and 1 0.381d = . The magnitude and phase of the corresponding function are plotted in Figure 4.13. As can be seen in this figure, the developed PAVM predicts the small-signal characteristic with good agreement with the detailed switching model. 70  -40 -20 0 20 40 60 A m p lit u d e  ( d B ) 10 2 10 3 10 4 10 5 0 20 40 60 80 100 120 140 160 P h a s e  ( ° ) Frequency (Hz) PAVM Detailed Model Detailed Model PAVM  Figure 4.13 Control-to-output transfer function of the full-order Flyback converter evaluated at 717.05R = Ω  and 1 0.381d = .  4.4.3 Efficiency Results One of the most common use of averaged models is system level modeling, wherein the models only appear as a black box in the system level modeling with input and output ports. It is therefore very important that the developed model accurately predicts the terminal characteristics of the converter module. Among such terminal characteristics the total converter efficiency is of significant importance. The comparisons of efficiencies predicted by various averaged models are shown in Table 4.1 assuming the converter operates at full load in CCM, ( )1717.05 , 0.381R d= Ω = .        71  Table 4.1 Efficiency comparison of the average-value models Model Type Efficiency (%) 2nd Order Model without Parasitics Detailed Model  99.88 State-space Averaging  99.77 2nd Order Model with Parasitics Detailed Model  98.12 State-space Averaging 98.14 PAVM  98.11 5nd Full-order Detailed Model  85.33 State-space Averaging  cannot predict PAVM  85.41 Measured from the base hardware prototype 83.45  As can be seen in Table 4.1, the ability of the model to account for the losses and predict the efficiency improves with increasing the model order from 2 to 5, wherein the traditional second-order models are significantly off and over estimate the efficiency to the 98%. Moreover, the classical state-space averaging cannot be simply extended to the 5th order model and therefore cannot be used for calculating the efficiency correctly. At the same time, the proposed PAVM of full-order model predicts the efficiency with only a small error compared to the hardware prototype and its detailed switching model from which it was established.        72  Chapter 5 : Conclusion   In this Thesis, the recently established parametric average-value modeling methodology has been extended to the transformer-isolated Flyback converter topology which includes parasitics and snubbers.  It is shown that the developed model captures includes the effect parasitics and losses, and is therefore capable of accurately predicting the terminal characteristics of the converter such as efficiency with the accuracy that has not been attainable by any previously established average-value models. The developed method overcomes the complexity and challenges common to many previously developed models when the parasitics of the circuit elements and snubbers are considered. The numerically constructed model can function in both CCM and DCM. It has been shown that obtaining an accurate full order average-value model requires extracting the duty-ratio constraint and the correction term. The functions of the duty-ratio constraint and correction terms were obtained numerically by running the detailed simulation at desired operation range. Once the model established, the resulting model is continuous and valid for large-signal time-domain transient studies as well as for linearization and subsequent small-signal characterization of the overall system over a wide range.  It has also been shown that direct application of conventional state-space averaging method is no longer accurate for flyback converters in CCM when the transformer leakage inductances are taken into account. A detailed analysis of flyback converters working in CCM has been done to present the problems in CCM. Regardless of the converter operating mode (CCM or DCM) the primary and secondary currents are discontinuous. As a result, the conventional assumptions used for DC-DC converters are no longer acceptable for flyback converters.  5.1 Future Work As a next step, the parametric average-value modeling can be extended to other DC-DC topologies such as forward converters (H Bridge) and Push Pull converters for high voltage applications to have a full set of models for DC-DC converters. Another potential topic for 73  future research could be to investigate the converters with source ripple and input filters using PAVM. Since it is very easy to include parasitics in PAVM, one could also include in the model stray capacitances and core losses in the magnetic components.  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The Converters Circuit Parameters A.1 Second-order Flyback Converter Parameters without Parasitics in CCM 1 20.009 :3.8 ,55 , , Re 2705 250 0.381 : 0635 27 1 / 6 :1 ,400 , cov , 1 04 2 g s m v V Mosfet A V N chanel International ctifier IRLL f kHz d Transformer ICE Components ICA L H n Diode A V ultra fast re ery diode Central Semiconductor CorpCMR U C µ = − = = − = = − − = 2 ,100 , min , 100 22 717.05 F V alu umelectrolyticcapacitor Sanyo MV AX R µ = Ω  A.2 Second-order Flyback Converter Parameters without Parasitics in DCM 1 220.009 , 250 , 0.381, 0.4409 27 , 1 / 6, 22 , 2500 g s m v V f kHz d d L H n C F Rµ µ = = = = = = = = Ω  A.3 Second-order Flyback Converter Parameters with Parasitics in CCM 1 20.009 :3.8 ,55 , , Re 2705 0.04 250 0.381 : 0635 27 1 / 6 :1 ,400 , cov , g sw s m v V Mosfet A V N chanel International ctifier IRLL R f kHz d Transformer ICE Components ICA L H n Diode A V ultra fast re ery diode Central Semiconductor CorpCM µ = − = Ω = = − = = − 1 04 1.25 22 ,100 , min , 100 22 0.09 717.05 d c R U V V C F V alu umelectrolyticcapacitor Sanyo MV AX R R µ − = = = Ω = Ω   79  A.4 Second-order Flyback Converter Parameters with Parasitics in DCM 1 220.009 , 250 , 0.381, 0.4409 27 , 1 / 6, 22 , 2500 0.04 , 1.25 , 0.09 g s m sw d c v V f kHz d d L H n C F R R V V R µ µ = = = = = = = = Ω = Ω = = Ω  A.5 Fifth-order Flyback Converter Parameters in CCM 1 20.009 :3.8 ,55 , , Re 2705 0.04 , 250 0.381 470 , 10 : 0635 27 , 0.2 , 0.8 1 / 6 :1 ,400 , co g sw s ss ss m p s v V Mosfet A V N chanel International ctifier IRLL R f kHz d C pF R Transformer ICE Components ICA L H L H L H n Diode A V ultra fast re µ µ µ = − = Ω = = = = Ω − = = = = − v , 1 04 1.25 100 , 200 22 ,100 , min , 100 22 , 0.09 717.05 d ds ds c ery diode Central Semiconductor CorpCMR U V V C pF R C F V alu umelectrolyticcapacitor Sanyo MV AX R R µ − = = = Ω = = Ω = Ω  A.6 Full-order Flyback Converter Parameters in CCM 1 20.009 :3.8 ,55 , , Re 2705 0.04 , 250 0.381 470 , 10 : 0635 27 , 0.2 , 0.8 210 , 1.35 1 / 6 :1 ,40 g sw s ss ss m p s p s v V Mosfet A V N chanel International ctifier IRLL R f kHz d C pF R Transformer ICE Components ICA L H L H L H R m R n Diode A µ µ µ = − = Ω = = = = Ω − = = = = Ω = Ω = 0 , cov , 1 04 1.25 100 , 200 22 ,100 , min , 100 22 , 0.09 717.05 d ds ds c V ultra fast re ery diode Central Semiconductor CorpCMR U V V C pF R C F V alu umelectrolyticcapacitor Sanyo MV AX R R µ − − = = = Ω = = Ω = Ω   80  A.7 Full-order Flyback Converter Parameters in DCM 1 20.009 :3.8 ,55 , , Re 2705 0.04 , 250 0.381 470 , 10 : 0635 27 , 0.2 , 0.8 210 , 1.35 1 / 6 :1 ,40 g sw s ss ss m p s p s v V Mosfet A V N chanel International ctifier IRLL R f kHz d C pF R Transformer ICE Components ICA L H L H L H R m R n Diode A µ µ µ = − = Ω = = = = Ω − = = = = Ω = Ω = 0 , cov , 1 04 1.25 100 , 200 22 ,100 , min , 100 22 , 0.09 2500 d ds ds c V ultra fast re ery diode Central Semiconductor CorpCMR U V V C pF R C F V alu umelectrolyticcapacitor Sanyo MV AX R R µ − − = = = Ω = = Ω = Ω 81  Appendix B. Flyback Converter Circuit Diagram 

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