Tài liệu High frequency transformer, design and modelling using finite element technique

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HIGH FREQUENCY TRANSFORMER, DESIGN AND MODELLING USING FINITE ELEMENT TECHNIQUE BY ADIL H. MUHAMMED,MPhil A THESIS IS SUBMITTED FOR THE DEGREE OF DOCTOR OF PHILOSOPHYIN THE FACULITY OF ENGINEERING NEWCASTLE UNIVERSITY ---------------------------201 29469 5 ---------------------------L- 7,: t 9, g LIBRARY THE UNIVERSITYOF NEWCASTLE UPON TYNE 2000 ABSTRACT in has density high field recent much attention The received power supplies of power frequency increase is to the The so as to switching the concern most area of years. in high frequency Such in power the concern size. power supply achievea reduction In (quasi, led has all to multi, and pseudo). many resonantstructures conversionunits is by load from the to the transfer the types, source controlled varying power resonant been frequencies. Every has to effort made to reduce the ratio of operating resonant the switchinglossesusing zero voltage and/or zero current techniques. In contrast, little attention has been given to the area of the design of the magnetic in is frequency It high that the usually accepted operation. weak point componentsat further high frequencypower supply design is in the magnetic devices( transformer into high ). No inductor taking the transformer the of accurate model account and frequency range has been performed yet. It is well known that as the frequency increasesso the transformermodel becomesmore complicated,due to the complexity frequency distribution, dependence the transformer of the and nature of element of kind directions, Indeed, this take these of can and the work many elements. some of attempt here is to introduce a number of mathematics,analytical, numerical, and high factors The directions the the transformer. to main model affecting practical frequencyperformanceare the eddy current losses,leakageflux and the effects due to the transformerelements,where the transformeris part of the resonantconverter. Two dimensionaltransformer finite element modelling 0 is used to examine different including open and short circuit conditions. The frequency dependencyof the cases, winding resistanceand leakageinductanceis fully explained.The practical design of the transformer and testing is used to valididate the simulation results. These results are supported by the results obtained from the mathematicalformulation. Special attention is given to reducingboth copper lossesand leakagein the windings. Three dimensionalmodelling of the high frequencytransformer and the solution using a program solving the full set of Maxwell's equationsis the original part of the present work. Frequencyresponsecharacteristicsare found and compared to that obtained from the test. Curves of these characteristics are used to predict a simplified transformerequivalentcircuit. This circuit is used with the simulation.of a full bridge I isolation, feedback, ( and switches, control, all units seriesresonantconverter, where transformer ) are representedby an equivalent circuit. The power supply operation frequency in behaviour its to the change of each of the transformer with respect and elementsare examined.Two casesare consideredthrough the simulation, when the it is below frequency. is The frequency the tank and resonant above when operating by building a practical power supply. are validated results simulated In addition, the numerical solution of modelling the transformer by an equivalent highest Q introduced. is (R, L, The are possible number and of elements network also found both FEM 2D the are using of magnetostatic all elements solution used,where is integration fields. This the trapezoidal solved using rule of network and electrostatic influences distribution The the the theory. of examination of and electric network frequency internal is the response characteristic carefully capacitanceson winding examined. The last work in the present researchis focussedon finding a general model of an frequency is The the to thesis transformer equivalent circuit cover wide range. exact completedwith a conclusion. 11 ACKNOWLEDGEMENT I would like to expressmy deepestthanks to Professor AG Jack, my supervisor for his support in all aspects of the work, and for being a constant source of encouragementthroughout the most difficult times. I would like to thank him for during final the stageof writing. painsundertaken I would also like to thank Dr. B. Mecrow for his contribution and help in the early stagesof the work. As with most projects of this kind, a great many academicand technical peopleare owed a great deal of thanks. I would like to expressmy specialthanks to all my colleaguesin the UG lab, Chris, Simon, Steve, Oystien, Ken, Phil, Jim, Hassan,Bernhard, Wander, and Christian, for their ffiendship and mutual encouragement. I am also thankful to my parents, brothers and sisters who have undergone all the in long absence. my pains Last but not least,my thanksto my wife Safiyawho has never failed to offer help and support at all times, and my son Ala!a, and daughterShaima'a. III SYMBOLS, PREFIXES, AND ABBREVIATION H magneticfield intensity Am-' J current density Ani-2 A B vector potential magnetic flux density wb m-' T D E electric flux density electric field intensity Cm-2 a area length ni! e C. Vnr' m F L capacitance inductance f frequency Hz R resistance impedance C1 Y admittance f2-1 I current A V voltage V z H Q Prefixes 9 ground M d magnetising distribution r resonant oc open secondarywinding sc short secondarywinding P primary winding s h secondarywinding history ( previoustime IV Abbreviation FEM finite elementmethod EC equivalentcircuit FFT fast fourier transformer HF high frequency V CONTENTS CHAPTER ONE: INTRODUCTION 1.1 : Introduction 1.2 : Literature Review 1.3 : ThesisStructure CHAPTER TWO: 1-1 ......................................................................... 1-4 ......................................................................... 1-9 ......................................................................... PRACTICAL VALIDATION OF THE HF TRANSFORMER SIMULATION AND TEST 2-1 2.1 : Introduction ....................................................................................... 2-2 2.2 : Principles of resonance ....................................................................... .. 2.3 : Converterdesign 2-4 .................................................................................. 2-5 2.4 :,Power transformerdesign ................................................................... .. 2.4.1 Primary turns 2-8 ................................................................................... .. 2.4.2 Secondaryturns 2-8 ............................................................................... .. 2.4.3 Primary& secondarywiring specification 2-8 .......................................... 2.4.4 Temperaturerise 2-9 ................................................................................ . 2.5 : Control technique 2-10 ............................................................................... 2.6 : Principle of operation 2-11 ......................................................................... 2.7 : Practical considerationof the power supply 2-12 ......................................... 2.7.1 : Input stage 2-13 ....................................................................................... 2.7.2 : Power stage 2-14 ..................................................................................... 2.7.3 : Isolation stage 2-15 ................................................................................. . 2.8 : Converteroperationand test 2-16 ............................................................... 2.9 : Testing the HF transformer 2-16 ................................................................ . ' 2 10: Summary 2-18 . .......................................................................................... CHAPTER THREE: EFFECT OF EDDY CURRENT IN TRANSFORMER WINDING BY 2D FEM 3.1 Introduction 3-1 I ..................................................................... ...................... 3.2 2D finite elementmethod 3-2 ..................................................................... 3.2.1 2D FE transformermeshmodel 3-4 ......................................................... 3.2.2 Serieswinding representation 3-5 ............................................................ 3.3 : Eddy current lossesin transformerwinding 3-7 .......................................... andcircuit conductors 3.3.1 : Copperlossesin the circuit wiring vi 3-9 ..................................................... 3-9 3.3.1.1 : Skin effect .................................................................................... 3-11 33.3.1.2: Equivalent circuit model of the wire ............................................. . 3-13 3.3.1.3 : Proxin-dtyeffects .......................................................................... 3-14 3.4 : Short and open circuit analysis ............................................................ 3-14 3.4.1 : Short circuit analysis .........................................................***........... 3-16 3.4.2 : Open circuit analysis ....................................................................... 3-16 3.5 Core properties . .................................................................................. 3-18 3.6 Windings layers at HF applications .................................................... 3-22 3.6.1 : Layers topology .............................................................................. . 3-24 Analysis IMHz validity at 3.7: . ................................................................. . 3.8: Summary 3-25 ............................................................................................ . CHAPTER FOUR: THREE DIMENSIONAL TRANSFORMER MODEL BY FEM 4.1 : Introduction 4-1 ......................................................................................... 4.2 : Mesh generation 4-2 .................................................................................. 4.3 : Element shapesin 3D FEM 4-3 ................................................................. 4.4 : Field equations 4-5 .................................................................................... 4.5 : Guageand formulation 4-7 ........................................................................ 4.6: Programvalidity 4-10 ................................................................................ 4.7 : Open and short circuit impedancescalculations 4-11 ................................. 4.8 : Commenton the results 4-14 ..................................................................... 4.9 : Transformerequivalentcircuit 4-16 ........................................................... 4.10: Summary 4-19 ......................................................................................... CHAPTER FIVE: PARAMETERS ESTIMATION AND WINDING NETWORK ANALYSIS 5.1 : Introduction 5-1 ......................................................................................... 5.2 : Network parameters 5-2 ............................................................................. 5.2.1 : Capacitance 5-3 ...................................................................................... 5.2.2 : Inductance 5-5 ............. ........................................ ................................... 5.2.3 : Resistance 5-6 ..................................................... ................................... 5.3 : Equivalent network of the transformerwinding 5-8 ................................... 5.4 : Trapezoidalintegration 5-9 ........................................................................ 5.4.1 : Trapezoidal rule for inductance 5-10 ...................................................... 5.4.2 : Trapezoidal rule for capacitance 5-10 ................... .................................. 5.5 : The numerical solution 5-12 .................................... .................................. 5.5.1 : LU Factorisation 5-13 ........................................... .................................. Vil 5-15 5.5.2 : Analysis of linear network ............................................................... 5-16 5.6 Programdescription ........................................................................... 5-18 5.7 Programaccuracy .............................................................................. 5-20 5.8 Network transientresults ................................................................... 5-24 5.9 Modelling the HF transformerby network representation .................. 5-25 5.10: Summary ......................................................................................... CHAPTER SIX: TRANSFORMER ELEMENTS SIMULATION USING SPICE CONVERTER MODEL 6.1 : Introduction 6-1 .......................................................................................... 6.2 : Simulation advantage 6-3 ........................................................................... 6.3 : Converter model 6-4 .................................................................................. 6.3.1 : Input stage 6-5 ........................................................................................ 6.3.2 : Power, isolation and control stages 6-6 ................................................... 6.3.3 : Transformer model 6-7 ........................................................................... 6.4 Principle of operation 6-8 .......................................................................... 6.5 Transformer elements and power supply performance 6-9 ......................... 6.6 Transformer elements effect on the output 6-12 ......................................... 6.7 Summary 6-13 ........................................................................................... CHAPTER SEVEN: EXACT TRANSFORMER EQUIVALENT CIRCUIT 7.1 : Introduction 7-1 ......................................................................................... 7.2: Physical meaning of elements 7-3 .............................................................. 7.3 : EC elements calculation 7-4 ...................................................................... 7.4 : Open & short circuit impedances calculation 7-8 ....................................... 7.5 : Transient response 7-9 ............................................................................... 7.6 : Summary 7-11 ........................................................................................... CHAPTER EIGHT: CONCLUSION APPENDIXES : A. I: Finite elementformulation in 2D analysis i ............................................................ A. 2 : -Usefulnessof the magneticvector potential iii ...................................................... B: Transienttransformerequivalentcircuit ...............................................................v REFERENCES: Ni .................................................................................................. Vill CHAPTER ONE 1.1: INTRODUCTION have in 1970's, MOSFET first they introduction the Since the practical power of the in improvements, and are now widely acceptedand used undergonemajor performance for features They them a that suitable make combine many power electronic equipment. losses, low high including, light and switching speed, weight, wide range of applications, high power density. One of the areaswhere they are used is in power supply units. There Each implement basic to topologies supply. a power switching commonly used are many topology hasunique propertieswhich make it best suited for a particular application, such is Essentially, low high the the unit size supply of power output power, voltage etc. or as inverselyproportional to the switching frequency,therefore most of the researchnowadays is concernedwith the high frequencyrange and Megahertz in particular. Higher switching frequenciesmade possible by power MOSFET transistors, new topologies and PWM ) integrated circuits (which pack more control and supervisory pulse width modulation featuresin a small volume), have contributed to making modern power supplies smaller. Power supplieshave a very wide rangeof applicationincluding TV, PC, power system,Xin ray, etc., and a reduction power supply size has a significant effect on the cost of the overall system. The developmentof high frequency power suppliesencompassescircuit analysis,control theory and magneticcircuit design. It is generallyacceptedthat the weak point in further increasesin switching frequency comesmainly from the magneticdevices.Unfortunately, thesemagneticdevices( transformersand inductors ) are unavoidable. Transformersare presentin most circuits servingmany purposes,such as isolation, step up and down. Practically,there is no upper limit to their power handling capability, if proper designis achieved,but it is also one of thernost difficult devicesto model accuratelyas the operating frequency goes higher. An accurate model of the transformer with proper account of the effects of high frequencyis essentialto further designprogress.When such a model is achieved,it should be possibleto establishthe important.characteristicsof the 1-1 be higher frequency. for The in drive must model magneticstructure and thereby assist the into in taking transformer account parts all the currents eddy of effect predicting of capable include loss leakage dependence frequency capacitive effects and elements, and the of betweenturns and betweenwindings and earth. There are many types of transformer categorisedaccording to their operating frequency band, The pulse, etc. power transformerusually operates power, wide such as and power, for frequency i. line ). band ( A frequency typical the power application e. over a narrow transformers is in distribution systems for electric utilities. Power transformers are designedmostly to operate near the maximum allowable flux density under steady state density highest highest turn the the current and at possiblevoltage per conditions, near be limits Transformer these the will cooling mechanism. operation above consistentwith destructive. A pulsetransformeris a transformerthat operatesin high frequencycircuits providing isolationandlow power signalamplification.Typicalapplicationsincludecommunication andfeedbackcircuits. equipments Sincethe input to a power supply transformersis high frequency,high power, these transformerscombinethe featuresof power andpulsetransformersand are termedwide bandpowertransformers. in wide bandtransformerdesignis the efficiency.The high Amongthe first considerations in be those efficiencythat is achievedin low frequencytransformers cannot comparedwith a highfrequencytransformer( wide band). The transformerlossesarestrongly relatedto frequency.Theselossescontributeto the economicsof the systemin which they operate. Theheatdueto transformerlossesneedto be considered aspart of the equipmentin which limit, sinceall they areinstalled. It is crucialthento reducetheselossesto an acceptable switches,thetransformer,andthe controlcircuitsarevery closeto eachotherin the power supplyunit. Reducingthe lossesso asto reachhigherefficiencyis currentlythe subjectof muchattention.Thereare two lossesmainlycontributingto the total transformerlosses, the coreloss( whichrepresents the no load loss),andthe windingor copperloss( which the load loss).The core lossi.e. the power dissipatedin the core consistsof represents 1-2 in loss is losses. hysteresis Hysteresis the continuous reversal consumed current and eddy is loss This direction field due the to the changing of magnetisingcurrent. of the magnetic is loss loss. Eddy design the than current eddy current stage easierto control through the due is in body This to the by the the core. of current produced caused circulating currents inducedvoltage when the magneticflux is changing.In principle, the induced voltage per is direction in is The in this the current of secondarywinding. turn the core the sameas at it. low frequency, flux direction At that the eddy currents to the produced magnetic normal induced frequency in direction by laminating As be the the the of core voltage. can reduced impracticable, is laminations become the and resort made to Ferrites which required rises is by loss, have low The the granular structure. core which eddy currents virtue of naturally determined by the core materials and the design is a function of the amplitude and frequency of the applied voltage. Core manufacturers have gradually improved core ideal including The Ferrites the which are widely used at -present. material properties, transformercore materialwould have an infinite magnetisingpermeabilityand zero loss. In infinite have the core material would addition, saturationflux density,unfortunately current ideal. for fall this short of materials Winding loss and leakage component calculations still representa great challenge for a high frequencytransformer designer.They are related to each other in the sensethat any reduction in one tends to be at the expenseof the second.Reducing the leakagemagnetic fields is vital to avoid interference with other circuits within the power supply unit. Unfortunately, reducing the leakagealso results in an increasein the distributed winding capacitance. Many computationaltechniqueshave been applied to improve the analysisof transformer performancewhich havebeenaidedby rapid developmentsin digital computers.One of the techniqueswhich has been applied successfullyto predict transformer performanceat low and audio frequenciesis the finite element method. This method provides a numerical solution of the electromagneticfield in eachpart of the transformer.The calculation results -canbe used to predict performanceat the design stage. They also have a tutorial value in 1-3 providing a clear overall picture of the various aspectsof the performancewhich are not easyto obtain by conventionalmethods. 1.2: LITERATURE REVIEW The major thrust of this thesis is to design, model, and analyse high frequency power transformers.To achieveeconomy in the design stage, and high operational performance of the transformer, it is vital to model the transformer accurately.The main factors that disturb transformer operation at high frequency,are winding eddy current losses,leakage, and the increasinginfluenceof capacitance.The combinationof distributed inductanceand capacitanceproducesmany natural frequencieswhich are troublesomeif the transformer is usedin a resonantconverter. When the frequency of the excited.waveform is increased, current is not distributed uniformly through the conductor but in a skin around its periphery. This gives rise to the eddy current lossesin the winding, i.e. as skin effect in a particular conductor, and as proximity effect with respectto currents in all other conductors. The proximity lossesare minimisedif the sub conductors are far apart, but the down side is an increasein leakage flux. The problem of reducingthe leakageflux while keepingthe winding losseswithin acceptablelimits has receivedconsiderableattention.In 1966,Dowell [1], derived an analyticalformulationto predict the frequencydependence of winding resistanceand leakageinductance. The relationwaslimitedto the window areaof the core ( i.e. the core is not included). It only includesthe effect of high frequencyon the inductanceand resistance,capacitanceeffects are not included.Different winding arrangements were from a physicspoint of view. Anothermathematical formulation calculatedanddiscussed to solvethe skineffectproblemin a conductorwasgivenby Silvester[2] in 1967.A two dimensional modelwasusedfor this purpose.A resistive-capacitive networkwasusedto avoidsolvingthe full field equations.Thiswasfollowedby anotherpublication[3], using a resistive-inductive networkto solveboth skinandproximityeffectlosses.The procedure 14 individual into i. by sub the the conductor subdivision of same e. of solution was branch. The by methods were a network represented conductors, each of which was has hopelessly it becomes but the system complex when applied to the single conductor, many conductors. Transformer models using equivalent RL and C network have long been a common different for fast [4] Fergestad type transients. a presented method of analysisparticularly based but distributions, to the transient on still calculate voltage of numerical method be into divide The to to the solved a number of sections approachwas winding networks. numerically.Kasturi [5], introduced a method of solving the winding equivalentnetwork through a companionnetwork. Each elementin the network was replacedby its equivalent is is derived integration. As trapezoidal the network which using a rule of a result, transfeged at eachtime step to an entirely resistive systemthat can be easily solved. Many been instance have for to the transient published other papers model characteristics [ 6,7,8 ] are a selection. Once capacitance effects come into operation at high frequency, the prediction of transformerfrequencyresponseis far simplerthan its time (i. e. transient) response.This is because a transient excites all frequency modes. Therefore, the determination of transformer frequency response characteristics have received attention as the only reasonablemethod to model the high frequencytransformer.In 1977, Degeneff [9], used the familiar equivalentwinding network to calculateterminal and internal impedancesand to predict the resonantand anti resonantfrequencies.Many papershave also paid the same attention to the transformer frequencyresponse[9,10,11,12]. Certainly, measurementcan be usedbut it is difficult to predict the internal winding response,even with a capacitively coupledprobe, damageto the winding insulationis unavoidable. The enormous difficulty of modelling a high frequency power transformer is well recognised.Continuedgrowth in computer scienceand technology has made it possibleto contemplateincreasinglyaccuratetransformermodels. The rapid developmentof the finite element method since the 1970's has provided an improved method for the solution of transformerelectromagneticfields. The power of the 1-5 field its to is finite element method well recognised, and electromagnetic application drawn first it to the attention of dramatically has was since expanded problems has been ]. [ The 13 by Zienkiewicz applied with success method electromagneticanalysts ] [ 18,19 frequency ], [ frequency 14,15,16,17 low to 2D and medium eddy currents discussed been have dimensional Three extensively also applications problems. [13,20,21,22,23,24,25]. The finite elementmethod can be used within an equivalent winding network model to field At the time, the the attempts using electromagnetic same predict network elements. equationsdirectly has grown dramatically.In 1979, Perry [ 26 ] examinedthe variation of based in dimensional density The on was a multilayer coil. work a one reduction of current finding a relation of power dissipationwith respectto the thicknessand number of layers. On the samegrounds,Beland [ 27 ] computedthe eddy current lossesin different shapes induced flow that the the currents on paths parallel to the assumption under of conductors FEM 2D impedance in frequency The the responseof multiconductor systemsusing sides. by impedance from introduced [ ]. Weiss 28 The the calculation of was predicted was alsq loss densityand stored magneticenergy.Many numericalmethodshavebeenused for eddy current field computations and magnetic device modelling. Konrad [ 17 ] presented a surveyof suchmethods. Some have given direct attention to the capacitancedistribution through the winding. Chowdhuri [ 29 ] presenteda method in 1987 to calculate the equivalent of the winding seriescapacitances.Laplacetransformswere usedto predict the input impedanceresponse. In 1988, Vaessen[ 10 ], presenteda method to model a high frequencytransformer using the principle of two port networks aided with the measurementof admittanceand transfer functions. Vandelac [ 30 ], presentededdy current losses( skin and proximity ) using a field approach. The work provides an insight into minimising copper losses at high frequency including interleaving the winding. Proximity effects have also been studied widely by many [ 2,31,32 ]. An approachto calculatehigh frequencyconductor resistance was presentedby Goldberg [ 33 ] in 1989. In this work a layer of winding composedof discretewires is convertedinto an equivalent continuous sheet.Further simplification was 1-6 length, infinite the has finite layer by so thickness and that and a each assuming achieved includes detailed A dimension. is transformer to model which one problem reduced from derived in ] the The by [ 1989. losses 34 Wilcox model was winding was given from derived by impedances A test. classical model aided calculation of self and mutual transformertheory was also presentedin 1991by Wilcox [ 35 ]. The procedureconsistsof decomposingthe transformer into sections,and each section is representedby a network. The voltages and currents of eachsection are arrangedin matrix form. The model can not be used to design a transformer, but it allows better a understanding of the transient phenomena. The first confirmation of interaction between a resonantconverter and a transformer was given by Wint [ 36 ] in 1991. A low frequency oscillation of the input and output of the power supply transformer was noticed and solved mathematicallyusing an equivalent model. In 1993,Woivre [ 37 ] presentedanalyticaland numericalmodels( FEM ) to model the transformerand to calculatethe frequencyresponsecharacteristics.Fourier transforms were usedto computethe over voltage transient effect. Morched [38] introduced a model to simulate the behaviour of a multiwinding transformer, over a wide frequency range. Another numerical solution to the winding of the high frequency transformer was presentedby Leon et. al [ 39 ]. Ahmad et al [ 40 ] provided in 1994 a generaltransformer model to predict the high frequency behaviour. The capacitanceswithin the model were computed using 2D electrostatic FEM, and magnetostatics to predict the winding resistance and leakage inductance. Frequency response characteristics were studied covering a frequency range of 0 to IMHz. The core eddy current losses and hysteresis havereceivedequal attention. Ahmed [ 41 ] and Basak [ 42 ] solvedthe core loss using 2D FEM in 1994. The confirmation of the need to use the core conductivity as a function of frequency was recognised in their work. Hysteresis losses in high frequency power transformerswere also under attention, seefor instance,Leon [ 43 ] in 1995. Lotfi [ 19 presenteda method aswell to calculatethe frequencydependentresistancein a rectangular conductor. The solution is basedon the ellipse formulation from the fact that the shapeof 1-7 current in the conductor cross section is equivalent to an ellipse. The method was comparedto the FEM result and appliedat the power supply switching frequency. All the methodsdiscussedso far have limitations as well as lack of generalisationand so cannot used to model high frequencytransformers.The attempt toward further frequency range with a fbll transformer model is hardly found. Therefore, the attention is given currently to model the whole transformer regardless of the design differences. High frequencyeffects are taken into account correctly. The model is solved using the full field equationsby finite elementtechnique. 1-8 1.3: THESIS STRUCTURE In the presentwork, an attempt is madeto introduce a numberof numerical-andanalytical by is The to examine the supported practical and mathematical processes. aim methods frequency in factors high that the transformers affect of power performance main used supply units. These factors are, winding losses,leakage,winding capacitances,core loss and their variation with transformer design, particularly with referenceto use in resonant power supplies. A practical transformerhasbeendesignedand built as detailedin Chapter two, and a power amplifier is used to test the transformer impedancecharacteristics. Another concernof chapter two is the operation of the transformer in a full bridge series resonant converter. Particular attention is paid to consideration of an adequateway to simulate the transformer within the power supply unit. Practical aspectsof building the power supply are also discussedin detail. The current and voltage waveform at the input and output transformerterminalsprovide a basisfor justification of the simulation. In fact, the actual power supply can be used for the test purposes, since the series resonant converter can safely handle a short circuit, but the resonant tank elements could be coincident with the transformer elementswhich results in curves which differ from the transformeralone. The designprocedurewas used to examinea large number of casesto examineboth the leakageand lossesin the winding using a two dimensionalfinite element model as detailed in Chapter three. In this chapter, attention is paid to the frequency dependenceof the winding resistanceand leakage.inductance. Since a two dimensional model is used for the magnetic field, capacitive effects cannot be representedplacing a limit on the calculation below the frequency at which such effects appear. Winding arrangementssuch as the number of layers and turns are examined. The frequency dependenceof these elementsare also introduced analytically and compared with finite elementand practical results. The electromagneticequationsgiven by Maxwell are usually approximatedat power line frequencyby neglectingthe displacementeffects. This approximation be valid as may not the frequency goes into the Megahertz zone, where capacitance effects become 1-9 increasinglyimportant. The magneticand electric fields are coupled and three dimensional formulation four, in Chapter is is That a numerical where covered analysis required. involving the full set of Maxwell's equations is derived and then solved using a three dimensionalfinite element model. A simple model is used first for validating the results ) i. ( just transformer and core e. an air core primary and secondarywindings and no with is The transformer then used to predict the model results. whole practical with compared impedances. in is Later Chapter, this short circuits attention given to the actual open and derivation of a simplified transformer equivalent circuit. The elementsof this circuit are found directly from the resonantfrequenciesof both impedancescurves. Indeed, most of theseelementscan be predicted directly from the finite elementmodel calculations.There is however considerabledifficulty in predicting the capacitancedue to its complicated distribution in the actual transformer. Usually the capacitanceeffects are consideredas frequencyindependentand can be solved electrostatically.This procedure can reduce the model to two dimensionsbut will never model all of the transformer elementsaccurately, becausemagneticand electric fields are coupledto eachother. Chapter five, pays particular attention to capacitance effects within the transformer winding. The transformer winding is modelled by an "accurate" equivalent circuit which has series/ parallel branchesto representthe distributed nature of the field. The elements of this circuit ( R,L, and C) are found by solving the 2DFE model magnetostaticalyand electrostaticalyindividually. The full circuit is solved for transient problems numerically using the trapezoidal rule of integration. The combined effects of capacitance and inductanceare viewed through the frequencyresponseof the input impedance. In Chapter six, fiill details of the power supply simulation is given, and the aim is to examinethe effects of the transformer elementson the performanceof the power supply. Each part of the power supply including switches( MOSFET ), control, feedbackisolation, and transformeris implementedusing its equivalentrepresentation,taking into account the high frequencyeffect of the switches.Two caseshave been consideredone above and the other below the resonantfrequency.The simulated results are validated practically using the measurements'described in Chaptertwo. 1-10 The high frequency responseof wide band transformers is usually analysedby meansof is Although circuit any not an exact representation of an actual equivalent circuits. transformer, it can be a convenient way of approximation. The representation of a transformerby an improved equivalentcircuit is the subjectof Chapterseven.The thesisis completedwith conclusions. 1-11
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