In contrast, little attention has been given to the area of the design of the
magnetic ... exact transformer equivalent circuit to cover the wide frequency
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
THE UNIVERSITYOF NEWCASTLE UPON TYNE 2000
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
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.
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.
SYMBOLS, PREFIXES, AND ABBREVIATION
magneticfield intensity Am-'
vector potential magnetic flux density
wb m-' T
electric flux density electric field intensity
secondarywinding history ( previoustime
fast fourier transformer
1.1 : Introduction 1.2 : Literature Review 1.3 : ThesisStructure CHAPTER TWO:
1-1 ......................................................................... 1-4 ......................................................................... 1-9 .........................................................................
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
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
3-9 126.96.36.199 : Skin effect .................................................................................... 3-11 188.8.131.52: Equivalent circuit model of the wire ............................................. . 3-13 184.108.40.206 : 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 ........................................... ..................................
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 ...........................................................................................
APPENDIXES : A. I: Finite elementformulation in 2D analysis i ............................................................ A. 2 : -Usefulnessof the magneticvector potential iii ...................................................... B: Transienttransformerequivalentcircuit ...............................................................v
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
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
providing a clear overall picture of the various aspectsof the performancewhich are not easyto obtain by conventionalmethods.
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 , 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 in 1967.A two dimensional modelwasusedfor this purpose.A resistive-capacitive networkwasusedto avoidsolvingthe full field equations.Thiswasfollowedby anotherpublication, 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  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 , 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 ,
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  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
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.
1.3: 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
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.
CHAPTER TWO PRACTICAL
OF THE HF
The size of a switched mode power supply unit is reduced by increasing switching frequency or by improving efficiency. Increasing the switching frequency leads to a large ), filter ( in transformer while the and output magnetic components size reduction increasing efficiency leads to less overall losses and hence to a smaller heat sink in the power supply unit. The
transistor switching losses in the conventional pulse width
( for 500khz frequency limits ) ( PWM the the supply power working modulation lMz). In PWM, the current waveform is driven mainly square in shape which increases losses due both These losses to the overlap turn-on turn-off. are mainly and on switching failing falling turn-off, transistor and or rising current at current and rising voltage of dissipative due Using losses the to the turn-on, reverse rectifier recovery. plus voltage at 0
RC ) or nondissipative( LC ) snubbercircuits to reduce transistor switching lossesis not a practical solution at higher frequencies.This is becausethe switching lossesare not losses is is be happens if dissipative the that to that all snubbercircuit used, reduced a In from ( ) MOSFET that case the the the transistor to of circuit. resistance snubber shift a larger heat sink is required for both the resistanceand transistor. A nondissipative it is frequency, because hand high the troublesome acts as a snubbercircuit, on at other handle. that the the transistor transistor resonantcircuit around cannot and storesenergy Resonant converters offer the advantage of overcoming these problem and hence be losses. In the the these reducing converters, switching current waveform can sinusoidal (or quasi sinusoidal) instead of square which'minimises higher frequency harmonicsand reducesnoise. Also the transistor can turn-on and off at zero current or 2-1
dv/dt low di/dt losses, a with stresses or the give to and switching eliminate zerovoltage goodtransientresponse. different for been have Many resonantconverterstopologies proposed,which are used Most depend their these of the topologiesare mainly applications. on purposes,and few but density losses, areconcernedwith the transformer a etc., power about concerned in distributed The transformerelementvaluesare the capacitance. particular elements important, is Their higher frequencies megahertz. value approaching very sensitiveat first becausethe transformerelementsare resonantelementsand haveto be accounted for in the resonanttank, andsecondbecausewaysmustbe foundto reducethe effectof to get low EMI andreductionin stress. theseelements The topologyselectedin this chapteris aimedto introducea power supplythat canbe in later The toprovide the results chapters. which validate simulationcarriedout used simulationresultsare used to examinethe effect of eachof the transformerelements within the normaloperationof the powersupply.This topologyis a full bridgeclamped in manypapers[ 44,45 ]. Later modeseriesresonantconverter,that is well documented partsof this chapterconcerntestson the actualtransformer.
2.2: PRINCIPLES OF RESONANCE.
Resonancephenomenaare very important physical effects in any circuit having RLC elements.During resonance,the circuit is purely resistive and maximum power can be deliveredto the load if its value matchesthe load value. The circuit with RLC elements can have series,parallel, or a combinationof series-parallelresonance. In this section, a brief introduction to the principle of resonanceis given to establishterms but of course this is well known theory. Fig.( 2.1 ), shows a typical RLC circuit modelled in both the time and frequency domains,together with the magnitudeand phaseof the circuit impedance. The resonant condition occurs when the source voltage and current are in phase. This means, the 2-2
circuit impedance is a purely real number, which leads to the conclusion that the imaginary part of the circuit impedanceis equal to zero. Hence, during resonancethe circuit is equivalentto a single resistance.In order to satisfythis condition, the following applies; Z (j o) Z0
0j) .-. J( XL - XC
where f. ( Hz
XL - XC f) and,
0 => f- =I
-2; r --fL C is the resonancefrequency,and XL = coL, Xc =
From theserelations, three casescan be observed.First, when f)f.
1C 0) ( source frequency
larger than resonancefrequency ) i.e. above resonance,XL) XC2 0(
phase angle ) is positive. Hence the circuit is predominantly inductive. Second, when f(f.
i. e. below resonance, XL( XCI 0
capacitive. finally, when f=f,,
is negative. The circuit is predominantly
( at resonance), XL
XC 0 is zero, and the circuit is
purely resistive. The circuit impedancecurve showsthat when the sourcefrequencyis zero ( dc ), Xc is oo ( open circuit ), and the impedancemagnitudewill be oo . When the source frequency tends to oo XL is co and the impedancemagnitude will be at oo again. The only , , condition that the impedancemagnitude is minimum is where the source ( current or voltage ) frequencyis equal to the resonantfrequency.The opposite conclusion can be made for the case of a parallel resonant circuit. At resonant frequency, the circuit impedancewill reach its highest magnitude which equals the resistancevalue. If the resistanceis removed( circuit with L and C only ), the impedancemagnitudecould reach an infinite value ( open circuit ), but this caseis not practical since there is an internal resistanceaccompaniedwith the inductor. The resonantfrequencyhasthe samerelation in both seriesand parallel RLC circuit, but the behaviour in these two cases is completely different. However, both series and parallel resonancecould be presentin the samecircuit, as is the casewith filters or the high frequencytransformerdiscussedin the following chapters.
2.3: CONVERTER DESIGN.
Generally,the design of any type of converter is based on the numerical calculation of dc ]. [ The 46,47,48,49 dc voltage conversion the voltage conversionratio characteristics for different differential by modes of equations solving ratio curves are obtained for ON-OFF follow the These states the arrangement pulses switching modes operation. function The the as a of normalised conversion characteristicsare plotted of switches. frequency,which is the ratio of switching to resonantfrequenciesfor simplicity of the be All the the normalised currents and can results. voltages calculationsand generalityof _IC' impedance (VL, ), to the and the transformer characteristic resonant with respect turns ratio. However, the procedure of the full design is not illustrated in this chapter, the present have below frequency dc the already above and resonant characteristics power supply beengiven . In the converter shown in Fig.( 2.2 ), there are a few points of interest first is diodes ( The Ds2&Ds4). In deserve the series somecircuits attention. more which that require reversevoltage across the transistor, a blocking diode could be placed in drain ( ). In a resonant MOSFET the terminals the transistor the or of source serieswith circuit, the reverse voltage rises very quickly on cessation of forward current. The antiparallel diode within the MOSFET is not fast enough and a blocking diode is required. The blocking diode will prevent forward current from flowing through the transistor body diode, and the fast reverserecovery diode (D1 -D4) carries the required forward current. The diodes ( Ds2&Ds4 ) are then used to avoid conduction of the internal diode of the MOSFET. The load in a series resonant converter is resistive in the transformer secondary side reflected to the primary by the turns ratio squared,and in serieswith the resonatingLC elements.In the seriesresonantconverter,the output inductor is ornitted. This converter is usedfor high voltage applicationas it requiresno output inductor. An output inductor for high output voltage would have to support large voltage acrossitself and would be bulky. Hence this converter can be used for either high or low output dc voltages. The 24
series loaded circuit can safely survive an output short circuit as the reflected load impedanceis in serieswith the resonatingLr and Cr elements.This impedanceis small comparedto the characteristicimpedanceof the resonanttank elements( Lr and Cr).
2.4: POWER TRANSFORMER
Most switching power suppliestransformersuse Ferrite cores [33,46,50]. In general, Ferrites are ceramic ferromagnetic materialshaving a crystalline structure consisting of mixtures of iron oxide with either manganeseor zinc oxide. Each Ferrite manufacturer processesdifferent mixes:of oxidesfor specific purposes. Ferrite cores are manufactured in a wide variety of shapes,EC, ETD, EF, U, RKE, and many others shapes. With each of the shapes,there is a recommendationfrom the manufacturer to be used in an application which dependson the transformer power density. For instance, an E-core has been recommendedto be used for a power of 500 Watt and over by Philips . There are also many grade materialsforming these shapes,3C series,D series,S series, and many others. Each of thesegradematerialshas a different material permeabilityand resistivity to fit the required application [ 18,41 The performance of the power supply Ferrite core transformer is determined by the factors specifiedbelow: Geometry of the core and winding. The allowabletemperaturerise. The frequency. operating The input minimum and maximum voltage. The by the winding with respectto the window area. area occupied Peak flux density in the the core. value of The size of the transformer is determinedby the maximum volt-second product applied to the primary winding and the current flowing through the primary and secondary windings. The maximum volt-second product determines the maximum flux density 2-5
leading to core sectional area and primary turns selection. As the operating frequency increases,the volt-second product decreases,it is then possibleto reduce the number of the primary turns or select a core with smaller area. The minimum number of primary turns is limited by the required turns ratio, beyond this any decreasein volt secondsdue to frequency increaseis achieved by a reduction in the core area. There is a limited frequencyrange for any core, becauseas the frequency increasesthe core and copper lossesincreasewhich leads to a reduction in the transformer efficiency. Operating the transformerabovecertain frequency no longer results in a size reduction, since a larger core is requiredto handlethe losses. The first step of material selectionis the core loss definedby a curve of loss ( usually in milliwatts or kilowatts per cubic centimetre or meter ) versus peak flux density at different frequencieswhich is suppliedby the manufacturers.From Faraday'slaw, which is usedto calculatethe numberof primary turns, the larger the flux density,the lower the primary turns and the larger the allowable wire size, and the larger the output power. The limitation on higher flux density is the core loss, and hence increased core temperaturerise. In this casethe core loss is not the only limitation, the core may also go into saturation.The primary then cannot support the applied voltage, drawing large magnetisingcurrent and destroyingthe power transistor. The peak flux density has to be chosenso that the total core and copper lossesresult in an acceptablylow temperature rise over ambient(i. e. room temperature). In order to avoid this method of choosingthe peak flux density during the design,anotherway is desirable. The design for this work follows , and is built on selectingthe core temperature rise rather than peak flux densityat the operatingfrequency. This is a much safer way. The finite elementanalysis can be used for this purpose using magnetostaticand either two or three dimensional models.The finite elementresults can give the best selectionof the flux density at which the leakageis minimum. Theseresults can be comparedwith the practical results using a searchcoil aroundthe core [42,53]. An E-shapedcore with material grade 3173was used to designthe transformer at 1MHz operatingfrequency,with the core dimensionsas given in Fig.( 2.3 ). The choice is made 2-6
that the maximumtemperaturerise due to the total loss ( core and winding ) will 40 C*. It is also assumedas a starting point that half the temperaturerise occurs in the core due to iron loss and half in the winding due to J2R and eddy loss. The core thermal loss is In is 1.35 C*/watt. The 14.8 then transformer to watt. maximum resistance equal loss by [5 ], loss in 1 Philips the the this to curve given manufacturer's order use value of total loss hasto be given per unit volume, which is the effective core volume, and that is 3. / / 76.8 1.35 17.6 mWatt CM From the loss curve at IM]Hz, it is found that equalto = the maximumflux densityis 16 mtesla( peak ). Important relations to examinethe design limitation ( winding loss or hysteresisloss havebeenmentioned. Thesetwo relationsare: 11.1 (pi. ), 4 cm core size winding loss K0f p
core size where; Pi.
I 20-K 2 PY77 input
(Kh f+ Ke f2
K factor equalto 0.165 for full bridge f operatingfrequency K. hysteresiscoefficient = 4E -5 K, eddy current coefficient = 4E -10
According to the samereference,if the core size calculatedby relation (1) is smallerthan that using relation (2), the design is limited by winding loss and not iron loss. At an output power of 500watt, the peak to peak flux density is 32mtesla and at IMHz frequency,the first equation results in 2.8 cm4, and the second equation in 3.66 cm4These results indicate that according to the design specifications,the smallest core requiredýshould be in the range between 2.8 - 3.66 cm4. Therefore the core that is selectedis E42, where the area product is its window area ( Aw=1.78 cM2) times the effectivecrosssectionarea( A, ff.=1.82 cm' ), and that gives 3.2 cm4.
2.4-1: PRIMARY TURNS.
Primary turns are calculateddirectly using Faraday'slaw [52,53]. The minimum primary is: frequency IMHz input 150V turns to support a minimum of voltage at 104
N=i. = 14 turns P 2f AB A,, ff.
2.4-2: SECONDARY TURNS
In order to calculatethe number of secondaryturns to support 250V output voltage, the turns ratio hasto be calculatedand is given by : (Vp
* duty cycle VDIODE
are the voltage drops on the transistor and the output rectifier
is duty 0.9, then [50,54], the turns cycle of ratio given a primary voltage respectively 0.52, andthe secondarynumberof turns is 26.
2.4-3: PRIMARY- SECONDARY WIRING SPECIFICATION.
The selectionof the wire size is usually found from the current density. The allowable current density is nearly the same in all the winding. A large increase in any single winding current density over other windings in the transformerleadsto a hot spot in the coil. Wire tablesgive a current densityusually in Ampere per unit areawhich provides a very rough guide to heatinglimits basedupon typical ( conservativelyestimated)cooling. The cross section area of the wire required can then be estimated directly from the current density. The maximum primary current at minimum dc voltage of 150V and maximuminput power of 625 Watt is: 2-8
The current density is the ratio of the current to the cross sectional area of the wire. Using insulatedcopper wire of 450 A/ cni2 leadsto the size of the wire required, which is 0.009 cni2 The number of fine wires is selectedaccording to the skin depth. At . IM]Hz (0.0066cm) the number is 64 of wire gauge 35 with the following specification; insulator diameter 0.412-4 diameter 0.009 0.007cm, cni2 and copper area cm, copper , insulator area 0.7E-7 =2 The secondary current can be estimated directly from the turns ratio, giving 2.2 Ampere. Following the same procedure, the cross section area of the wire required to carry this current is 0.0048 cni2 . Using the same wire gauge the number of fine wires is 32.
It is importantto checkthe validity of the temperaturerise assumptionwhich is 40 C' as assumedearlier.The temperaturerise canbe found from the total losses( core and winding) andthe thermalresistance( 14.8 C' /W
of the selectedcore. The thermal
data for the core and is an experimentally resistancecomesfrom the manufacturers determined figure usingnormalwindingvolumeandfill factors.The core losshasbeen found earlier equalto 1.35W.The winding loss can be estimatedfrom the primary current(4.1 Amp.), meanlengthof the turn ( from the data sheet9.3 cm ), primary numberof turns ( 14 ), and the resistanceof the selectedwire ( wire specification 0.00032Ell cm whichgivesa windinglossof 1.22W( assumingthat eddylossesin the , winding*areinsignificant). The total lossis then 2.57W,andthe temperaturerise is 2.57 * 14.8) equal38C* . Sincethe estimatedvalue of the temperaturerise is less than that which has been the designis accepted.If this valueis biggerthan40C* largercoreneedsto assumed, ,a be selectedandthe samedesignprocedurerepeated.
2.5: CONTROL TECHNIQUE.
In resonantconverters,i. e. series,parallel, or series-parallel[45,48,55,56], the switching losses are reduced due to zero current or zero voltage techniqueswhich need to be has line load for The the control circuit current and voltage. maintained a wide range of duty of maintaining any change and reflecting that as a change in the switching pulse width. There are two types of control technique namely fixed and variable frequency. The variablefrequencycontrol technique, uses frequencymodulation to control the output voltage. The method usesthe slope of the circuit impedancecurve to control the output. The control circuit changesthe frequencyto move either toward or away from resonance,and thereby controls the energytransferredto the resonantcircuit and to the load accordingly. In general,this technique suffers from a number of limitations. First, MOSFETs with a small transition time are required to maintain control at high frequency,secondas the frequencyapproachesresonance,peak currents and voltages go higher stressingthe resonant componentswhich are designed to operate at a single frequency, and also the filter componentswhich are usually designed at the lowest frequency.Practically this is not an easytask with respectto the frequencyvariation of this control circuit. Therefore it is desirableto keep the switching frequency constant [45,55]. Constant frequency control uses the conventional pulse width modulation ( PWM principle [57,58], to changethe output. There are three kinds of constant frequency control strategy,direct duty cycle, voltage forward, and current mode. In all three types, the output voltage has to be comparedwith a fixed referenceto give an error voltage which is usedby the control circuit to modify the on-time. The total pulse frequencyon and off is kept constant. The constant frequency technique is the one currently used to control most power supplies.Dedicatedintegratedcircuits are availablefor this task for instanceUC3825A.
The clampedmode seriesresonantconverter as shown in Fig.( 2.2 ), is a seriesresonant duty frequency by be the cycle that controlling regulated at a constant can converter ) is held ( i. Clamped the tank the that the e. resonantcircuit voltage across ratio. means leg in MOSFET's for the the the opposite switching cycle until one of a period of at zero is triggered. The duty cycle can be used as the control parameterwhile the switching frequency is in is by four frequency MOSFETs Constant triggering the a operation realised constant. time sequenceso as to produce a quasi square voltage, which is applied across the bases ( ) Fig. 2.4 the theoretical tank. of the switching pulses, tank, shows resonant inductor 'resonant ( ), ( Ir). The Vcr the and resonant current voltage, capacitor'svoltage four switching pulses are arranged as shown in this figure, where each of the two switches( i. e. MOSFET's ) in one leg are delayed 180' with respect to the switches of the oppositeleg . At the start of the new switching cycle, T3 and T4 ( Fig. 2.2 ) are conducting, while TI and T2 are off. The resonant inductor (Lr) current starts to circulate and charge the resonantcapacitor(Cr). When T3 is turned off, the current still circulatesin the inductor through DI and the capacitorvoltage reachesa maximum.TI is now turned on, forcing the resonant inductor current to change its polarity, and causing the capacitor to be discharged.The voltage is clampedto zero until T4 is turned off. When T2 is on the resonantcapacitorchargesto negativepolarity and reachesmaximumvoltage when TI is turned off. The inductor current circulatesthrough D3 and T2 to dischargethe resonant capacitor,and the processis repeated. Two important points have to be considered.First, the resonantinductor current does not reverseuntil one of the switches in a leg is turned off. Considering the switching sequencesof the four MOSFETs during the period that TI is ON, T4 is ON for a short period shown in the Fig.( 2.4 ), and during this period the current circulatesthrough the tank only. When T4 is turned OFF, T2 will turn ON, therefore, the current polarity 2-11
OFF leg first leg in OFF, TI turn the two turns the the switches so remains sameuntil its before is diode loss. Second the conducting associatedanti parallel with no switching ON. force is ON transistor turn to turned switching at a zero voltage switch Further detail of the circuit operation can be found in publications on this type of converter [ 45,50].
THE POWER SUPPLY.
Any power supply prototype consists of different stagessuch as, the input dc supply, in block isolation, the output circuit switching circuit, as shown control circuit, and diagram in Fig.( 2.5 ). Each of these stageshas its own problems. In addition to the specificconsiderationsthat will be explainedshortly, it is clear that in a high power, high switching speed converter electromagnetic interference i. e. noise is going to be a problem.The following points help to reduce noise.Practical experienceshowsthat they are necessary. I- The control circuit has to be well spaced from the power circuit to reduce the interference. 2- The resistorsthat are used as peripheral componentsaround the control circuit need to be of non-inductivetype. 3- Twisted fine wires needto be used to connect parts of control circuit and the supply switching pulsesshouldbe kept as short as possible. 4- The resistorsand capacitorshaveto be placedwith as short leadsas possible.
2.7-1: INPUT STAGE.
The dc input stage consistsof a bridge rectifier and filter which convert the main ac voltage to dc. The rectifier circuit consistsof an input transformer, full bridge diodes and an output capacitor.The input bridge diode circuits used in the presentpower supply are dual packageSKKD2608 diode modules. A capacitorinput filter is used to reduceripple and provide the requirqd dc supply. The for in be the design. If the value of this this to capacitor needs accounted value of capacitoris too small,the result is a large ripple and lower minimum input voltage. If the capacitor value is larger than necessary,the recharging current is narrow and large in amplitude which increasesEMI and gives higher losses.The energy required from the capacitorto provide the power supplyis given by: Pin Joules( Watt-second w= f Which is the input power over the line frequencyof 50 Hz. The input capacitorvalue is Cm
w = V2peak
The input capacitorusedin the power supply is an electrolytic of value 470PF ( 450V As a simpleapproach,the capacitorvalue can be found by using I gF for every volt, with a little extra as tolerance.
2.7-2: POWER STAGE.
Thepowerstageconsistsof four MOSFETsandthe resonanttank. The selectionof the MOSFET type dependson the rating of the input current and the maximuminput voltage.The current can be found from the design specificationof minimuminput voltage,efficiency,outputpowerandthe duty cycle,resultingin 4AAmp. following the
specificationgiven earlier. The selectionof at least a 5A MOSFET gives some margin. The MOSFETs currently used are IRF730 rated 400V and 5.5A. The size of the heat sink used for the MOSFET can be estimated from the drain to in in device ( ) due loss ON The MOSFET temperature the to the rise source resistance. this resistanceis one of the design specifications. This resistance is a function of temperature
( increasing as the temperature increases), and is supplied in the
manufacturer'sdata sheet.The value of this resistancefor IRF730 is I fl at 25C* and at the designvalue of 60C*
(i. e. a temperaturerise of 35 C' ) its value is 1.392 The . power dissipatedduring the switching ON time then is the product of current squared, resistanceand 0.45 ( the On time), giving 11.3 Watt. The thermal resistanceof the heat sink requiredat the specifiedtemperaturerise is (35/11.3) 3.1C* /W The heat sink used . is 5 C' /W slotted ABL with a dimensionof 33mm height and surfaceof 50mm*50mm. The presentpower supply is aimed to operatebelow resonantfrequency. The capacitor used in the resonant tank is I. InF (poly). The measuredleakage inductance of the transformeris 4.1ýffl and this could be used as the inductor of the resonanttank if the frequency is high enough. However at practical frequency this inductance is not large enoughto force zero voltage switching and to handlethe power of the tank, so a series inductor is required.The designprocedurefor the inductor is similar to that used for the transformer, where here a small toroid ferrite core is used. However, the design procedureusedto build this inductor is basedon the inductor design step given by Evans [59 ].
2-7-3: ISOLATION STAGE.
Isolation circuits are vital for any power supply.There are three circuits with their featuresandfunctionsasfollows: I- Isolation betweenthe power input and output lines. 2- To provide a virtual ground for the top MOSFETs of the converterbridge. 2-14
3- Isolation betweenthe output dc power and the control circuitry.
The first point relates to the IIF power transformer that was discussedearlier. The The MOSFET's to the top are ground referenced. whose sources not secondrelates input switching signal is formed with respect to the ground of the power circuit rather than the emitter of the switch, and hence a dc isolation is needed.Basically, there are three ways that can be used to provide dc isolation between the gates of the power devicesswitchesand the control circuit, a pulse transformer,DC-DC chopper circuit, or an optoisolator [50,52]. A pulse transformer can only sustain a limited duration pulse, inductance its leakage its from Furthermore, to and output also swings negative positive. has to be very small to achieve acceptablebandwidth. The chopper circuit also has bandwidthlimitations and is therefore only really suitablefor low frequencyapplications. The opto-couplerrequiresa separatedc supply for isolating the ground betweenits input and output ( specifically an NMA seriesdc-dc converter ). The optically coupled gate 6N137 used requires an additional stage at the output to regulate the switching current as shownin Fig.( 2.6 ). The third point relates to the isolation of the output voltage feedback to the error amplifier before entering the control circuit. The output voltage is first reducedusing a voltage divider, and the result is used with an adjustablevoltage regulator to drive the optoisolator.
2.8: CONVERTER OPERATION AND TEST.
The practical circuit is shownin Fig.( 2.7 ). The power supply currently used is operating below the resonantfrequency, where the ratio of switching to resonant frequenciesis designedto be 0.8. This ratio is used to achievebetter performance as recommended . The frequencyratio can be adjustedby varying the number of turns of the series 2-15
inductor. Of course this ratio is not expectedto be exactly 0.8, since that dependson how sensitivethe instrumentis that is usedto measurethe inductor. Fig.( 2.8a), shows measurementsof the resonant capacitor voltage and resonant tank from figure be It that zero voltage switching is clearly seen.The this voltage. can noticed capacitorvoltage and resonanttank voltage are rising from the point that both are almost zero. When the air gap of the core is changedor the frequency ratio goes higher ( for instance0.88), zero voltage switching is not guaranteed,and then external capacitors across the switches are required as shown in Fig.(2.8b). In this figure, zero voltage switching doesnot occur, and so the devicesare switchedon while the capacitor voltage is just starting to discharge.Therefore, external capacitors are required across each of the devicesto force zero voltage switching or resonatethis capacitor with an additional inductor in serieswith the transformer primary winding Fig.( 2.9 ) shows the resonant inductor current. The measuredwaveforms may be compared with the simulations which are shown in Fig. ( 2.10-2.11). Good agreementis clearly seen.The detail of the simulationprocedurefor this type of converteris given in chaptersix.
2.9: TESTING THE HF TRANSFORMER
The high frequencymeasurementsare not different in principle to that at low frequency. The methodsthat were used for low frequencycan be used with high frequencybut care needs to be exercised. There are many methods that are used to measure the BF transformerelements( capacitanceand inductance), including impedance,resonanceand a bridge. Using a bridge circuit such as an Owen bridge  can yield the measurements of the coil resistanceand inductance,but the impedancemethod is easy to use and it needs an ammeter and voltmeter only. The inductance here means any
magnetisingor self inductance,but the procedure of the test will decide which of these dominate the measurement.The leakage inductance could be approximately found for instanceby shorting the secondarywinding and measuringthe inductance, and primary resistance.It is not possible however to split the values between windings. A similar 2-16
impedance The for be using an circuit connection. magnetising open method can used ( inductances in be found different the transformer these can capacitances parts of using ). frequencies frequency ) low ( the and resonant series and parallel If measuredat it be directly by is than can rather calculation, measurementof capacitance required done using a capacitancebridge but the results are prone to error. Even given accurate its determination because the capacitance still exists, measurement, problem with distribution ( i.e. betweenturns and betweenturns and earth ) is important and there is no way to separatethe components. Transformer element measurement has been discussed in many publications [34,46,60,61,62].Researchershave usually used an indirect way to estimatethe element impedance functions ( transfer valuessuch as via short circuit and voltage gain )[63,64], together with fitting procedures. In the presentwork, open and short circuit impedancesare measuredand used to model the transformer. Using these impedances, all the resonances experienced by the transformercan be viewed. During the measurementsa variable frequencysignal from a function generatoris amplified and appliedto the primary winding of the transformer. It is more safeto apply the signal to the high voltage terminals of the transformer and then refer the impedanceto the winding required by the turns ratio of the transformer. The frequency responseof this impedanceis obtained from the ratio of input voltage to current or directly using an impedanceanalyser.A class AB amplifier, as detailed in ReQ54], is used to amplify the signal of the function generator. The parametersof the amplifier havebeenimproved by using high speed,high power transistorsand capacitors to handlea bandwidth of 3 MHz. The amplifier can drive a voltage of 80V and a current of 1.2 A at the highest frequency. Depending on which of the impedancesis to be measured,the secondarywinding is open or short circuited. The most critical point in the test is when the measurement approaches the series resonant frequency of the transformer.The circuit can deliver a maximum current enough to fuse the transistor if care is not taken. In order to avoid repetition, the practical results will postponedto a later chapterso asto comparewith thesefound using numericalmethods. 2-17
This chapter is concerned with two topics, the practical implementation of a high frequencypower supply, and transformer testing. The power supply is built to provide the ground for justifying the simulation results that will be carried out in a later chapter. The converterused is a clampedmode seriesresonantconverter, that clampsthe voltage is leg in turned off. This converter uses a the to zero until one of switches a converter duty keeping frequency the the technique, cycle controls while which control constant for frequency The selecting this type of resonant reason constant. switching pulse it is high (since is transformer testing a to the voltage a serve requirementof converter high voltage converter), as well as a clear insight into the interaction between this resonantconverterand its transformer. A ratio of 0.8 of the switching frequency to the resonantfrequency is used. When the frequency, high frequency the current resonant very switching more closely approaches flows through the tank causingunacceptablestressto the tank elementsand increasing losses accordingly. Simulation has been used to study the effect of the internal transformerelementson the converter performance. This kind of resonant converter can survive short circuit without damage, because during a short circuit, the load is reflected to the primary as a very low impedance connectingthe LC resonantelementsin series.Parallel resonantconverters on the other hand reflect the short circuit impedance across the resonant capacitor, destroying resonanceand sharplyincreasingswitching losses. It may seemadvantageousto use the resonantconverter to test the transformer, but this converterexperiencesdeficiencyduring open circuits. Here, the reflected load impedance betweenthe tank elementsis high and hencevery small current flow through the tank and primary winding ( i. e. no or smalltransformerinput current), and in addition :I- Many changesneed to be made at each frequency required. The changesinclude, control circuit, drive, resonant tank elements, and the peripheral components of the PWM chip. 2-18
2- The resonanttank elementsare in serieswith the transformer and could lead to an impedancemeasurementerror, becausethe tank elementsare part of the transformer internal ( parasitic)elements. For thesereasonsa linear power amplifier hasbeenusedto test the transformer.
j ý1\1ý ------
90 1------------------------------/ýIýnd Capacitive , Wo -90
Fig.( 2.1 ) Seriesresonancein an RLC circuit
TI-T4: IRF730 Ds2 & Ds4: RHRD660
Dol-Do4: RIIRP860 D2 & D4 I IDF4
Fig.( 2.2 ) Clampedmode seriesresonantconverter
43mm ;;z-1.9.5mm 12.2mm
Fig.( 2.3 ): E-42 core dimensions
Tl T2 T3 T4 Vtank
D3T2 D4T1 DlT4
The theoreticalbasesof switching pulses
AC I/P Rectifier & Filter
Power&l Resonantl Circuits
HFilter Rectifier &
Voltage Divider pr
Optodriv & Regulat ;u
Fig. ( 2.5 ): Block diagramof the power supply main stages
Fig.( 2.6 ) Opto-coupler drive circuit
Fig. (2.7) Photographs of practical power supply
C112 t IOVIDIV
-; .---! -..!-.
Fig.( 2.8 ratio,
m -3.5 -; Is
Resonantcapacitor(Ch2)and tank voltages(Chl)measured it 0.8 frequency a): Normal operation, (b): Core air gap changed
...... ...... P
... .... .. -i-. l. -ý-+-+.+--. + +44-i
.................................. Jr T..
Fig. ( 2.9
Resonantinductor current measuredat 0.8 resonantfrequency ( IA/Div, 0.5 ps )
Fig. ( 2.10 ): Simulatedwaveformsof resonantcapacitorand tank frequency (a) at ratio 0.8, (b at resonant voltages. ratio of 0.88.
2.50 2.00 1.50 1.00 0.50 0.00 -0.50 oo -i. -1.50 -2.00 -2.50
(b) Fig.( 2.11 ): Simulatedwaveforms:(a) ResonantInductor current at frequencyratio 0.8 (b) ResonantInductor current at frequencyratio 0.88
CHAPTER THREE EFFECT OF EDDY CURRENTS IN TRANSFORMER WINDINGS BY 2-DIMENSIONAL FINITE ELEMENET METHOD
Voltage is induced in a conductorwhen it links time varying magnetic field. The faster the changein flux, the more voltage is induced. If the conductor size is significant and / or the magnetic field is rapidly changing, voltage induced in various parts of the conductorscan causean internal current loop called eddy currents. The eddy currents representan advantagein some systemsand disadvantagein others. In application such as transformers, the existence of these currents affects the performanceof the transformer and causelosses.Away must found to minimise these effects. In some electroheatapplications, they representthe source of heat due to the energy dissipatedthrough the ohmic losses.
In our case,the existenceof thesecurrentsin transfonnerneedsto minimise for the following reasons: I- 'Mey causeundesiredohmic loss in the windingsboth due to skin and proximity effects 2- Theycauseextralossin the coreandreducedits effectivepermeability 3- Theycausestraylossin structuralandcontainmentcomponents. Thus,the study of the behaviourof thesecurrentsis very importantin the searchto reducelossesandtherebydesigna higherefficiencytransformer.
As the frequencyis increasedand the power transformer size is reduced, eddy currents become one of the major limiting factors. The winding ac resistance and leakage inductanceare strongly relatedto eddy current effects. In order to analysethe eddy currents, Maxwell's equations are used. These equations be form fields. The these can describecompletelythe magneticand electric of equations is in formulate best integral . The these differential to equations way or either terms of potentials. This chapter will consider eddy currents effects in some detail using one of the most is ). ( FEM This is finite the method elementmethod powerful numericalmethodswhich based on an approximate solution of
Maxwell's equations [15,16,32,65] using the
methodof weighted residuals. A closed form mathematicalapproach is available from the work done by Dowel [11, Greater [19,30]. is by and simple method quick as a which adopted some researchers is is this the7subject of the work and accuracy only possibleusing numerical methods describedhere.
3.2: 2-DIMENSION FINITE ELEMENT METHOD. By. its very nature a transformer is a three dimensionaldevice. Both magnetic and electric fields are involved and they act in a sensenormal to each other making the solution three dimensional and time varying. Three dimensional solutions  are expensiveand time consuming, and it would be very useful to reduce analysisto two dimensions. Sincethe magneticfield is the main source of eddy current generation,the solution can be approximately reduced from three to two dimensions. What is being ignored is the displacementcurrent which effectively in an ac sense"bleeds" charge betweenconductorsand to ground. At low frequencyignoring this current is acceptable.
At high frequency the approximation becomes troublesome. This two dimensional be but the to does include all calculate the not used currents, will of eddy effects analysis parametersof the equivalentcircuit model. The beauty of the FEM comesfrom its flexibility. It is not restricted in any way by the irregular geometryor inhomogenietyof the field areato be solved. The main procedures of the FF-Msolution is: I- Break up the geometryof the problem into a small elements. 2-Apply a different ( approximate) equation to each element.For instant, the magnetic
in is by linear is (A) the variation each vectorpotential unknown represented a which element. 3- Taken together the elemental equations form the overall system equations which becomea matrix of simultaneousequations.The best solution methods for the size of the problem consideredhere appear to be semi-iterative methods such as Incomplete CholeskýConjugateGradients( ICCG ) .
Thefinite elementformulationthat hasbeenadoptedusingthe magneticvectorpotential is given in appendix A. 1.
4- Post-processingof the results in order to obtain parametersof interest and to show the resultsin graphicalform. The.main equationto be solvedin the 2-dimensionalprogram is :
(PI-)= VxA x
(0 A, -a ýýO+vv,
During the solution, this equation is consideredto have a sinusoidal source variation, which meansa simpletransferenceof the eddy current term into the frequencydomain.
= jo). A,
In most power supplies,the current waveforms are not sinusoidal. Although, modelling this wave shapeis possible,it greatly addsto computation time. Therefore, attention has beenrestrictedto the fundamentalcomponentonly.
3.2.1: TWO DIMENSIONAL FE TRANSFORMER MESH MODEL. Apart from the assumptionswhich have already mentioned, further simplifications havebeenintroducedin order to makethe problem numericallytractable: I- The magneticvector potential( A ), and the sourcecurrent density (J) have values in the z-axis only, which are constantin that direction. 2- The problem is treated as linear, where a singlevalue of permeabilityand conductivity are applied. Hence, the hysteresis, saturation and temperature effects [41,43] are approximatedor neglected. 3- The displacementcurrents  are absent,i. e.: OD =0 .0t
Fig. (3.1a) shows the mesh model of the transformer. Valid solutions must also satisfy the condition that the length of each elementside has to be shorter than the skin depth. This condition is important to avoid excessiveerror. The well known skin depth relation is : F2 8= c0 adu
having area an with section cross in a square as The conductors the model are considered by been has . researchers This many considered approach to that wire. of a round equal [1,30] to makethe problem more manageablefrom a meshingviewpoint. The answersfor A and V( the net voltage at the end of the conductors) are effectively idea lead to This to obtain in direction length the turn. simple a can the of per unit of better accuracy, by noticing that the depth of the E-core viewed from the front is larger is for the In from path the that magnetic area available the sides. than that viewed case different in eachside. Therefore two models are used one for each side as shown in Fig. (3.1b). The total winding impedanceis the total of the impedanceof the two models. Z=
VV+j *ZVV) *y 2 I ,
Where 1, and 12are the length from the front and side section of the core, and N is the by 2% improve due the The two at these to result turns. models numberof calculation IKHz and by 8% at lNffh in comparisonwith that of using one model when compared with measurements.
3.2.2: SERIES WINDINGS REPRE SENTATION. In the solution of the FE equations,the sourcethat is used to drive equation ( 3.1 is either vOýltage'orcurrent. The term VV representsthe voltage appearingat the ends for is length If the of conductor expressedper unit ofthe conductor. voltage not given, instancewhen the winding is connectedto an impedance,an extra equation is required to solve the extended circuit. For example, when the impedanceis a short circuit, the summationof VV is equalto zero, but its value for eachindividual conductor is not. The extra equationrequiredis : V=I*
j: VV N
is is is VV V N the the the turns, the voltage across winding, and where number of is between formed from two separatesections. turn turns each voltage sections,where This equationgives the flexibility required to link extendedcircuit with the FE analysis. A simple caseis a capacitor connectedacross the terminals as carried out in reference , or by adding an external circuit as seen in the Fig. (3.2). When this figure is considered, equation( 3.6 ) can be rewritten as : V+IZ=
The equationthat representsthe inducedcurrent in the windipg iis I i-ff Jds= -fficocrAdcdy-ff 3
Whereds is the conductorcrosssectionarea.This equationcanbe rearrangedIn order to include the serieswinding equationas : -ffc; -VVdxdy=l+jo)
YffaAdxdy I, conductors
Equations (3.1), (3.6), and (3.9) can be formed into a matrix that is used to solve for A
and VV. The useof the magneticvector potentialto calculatethe winding inductance and resistanceis given in appendixA. 2. In order to help to validate theseequations,the transformer describedin chaptertwo has beenmodelledbut with very simpleprimary and secondarywindings of only 4 turns each ( eachturn cross sectionareais the sameas the true transformer) The table below showsthe results for the short circuit condition. Two conclusionscan be drawn from the table. First, that the input voltage is equal to the summationof the voltage at the end of conductorsin the primary winding, while their net value is equal to zero in the secondaryas required. Secondly, the voltages across all conductors are relatively the samedue to the absenceof any capacitive effect. The small changein the
flux leakage due in be difference to the the value of explained secondaryvoltage can linked by eachturn. The high frequency losses in the winding will depend on the depth of the conductor relative to its skin depth and also on the position of the conductor relative to its neighbours ( proximity effect ). This very slight differencein inducedvoltage per turn at this low frequency( IKHz ) will becomelarger as the frequencyis raised.
SF 11 22 33 44
voltage / meter
primary winding I
-1.18635E+01 +j -8.11604E-01
5E+O1 +j -8.11605E-0 1 8.11426E-01 +j -1.18635E+01
secondarywinding 3.66716E-01 +j 2.23612E-01 2
+j -3.67713E-01 -2.25624E-01
+j -3.65717E-01 -2.24668E-01 3.66714E-01 +j 2.26679E-01
3.3: EDDY CURRENT LOSSES IN TRANSFORMER WINDINGS AND CIRCUIT CONDUCTORS. As switching frequency is increased, optin-ýisation of the power transformer
becomes a major concern. Eddy current losses and variation of inductance with frequencycan greatly affect the transformerperformance. The changein inductancewith frequencybecomes significantin sizeas comparedto the total circuit inductance. It will be useful to define some general principals and resulting phenomenon at the outset. It will also help to relate the effects to the terms often used to describe them. If an inductor is suitably manufactured from not vary with frequency.
fine wire, its resistance and inductance will
In, practice, there will always be a frequency at which these
eddy current effects start to become significant, and the current is forced into a surface layer by its own magnetic field. The resistance will clearly rise and hence the term " ac resistance " is coined.
At the same time, as the resistance rises, the inductance falls
because the area of the flux path is reduced by the width of the conductor. If the inductor has an iron core, the laminated or sintered core will also be affected by eddy currents, and its reluctance will rise which will also tend to reduce the coil's self and leakage inductance.
Yet a further effect is that any conducting componentsnearby will increasingly have opposing eddy currents induced in them, which will again appear as reduced coil inductanceand increasedcoil resistance. The term ac inductanceis often used to point to the fact that the effective inductance( self and mutual ) reducesas the frequencyrises. More confusinglythe term " parasitic inductance" is also used. The implication is that
as frequencyrisesthe " parasiticinductance" increases,and that thereforeone could view the parasiticinductanceasbeing negativein character. Proximity is the effect whereby current is forced to flow in the edge of a conductor by the proximity to current in nearby conductors. It will have the same effect as
larger be In transformer a windings, proxin-dty will many straightforward skin effect. effect than simpleskin effect [2,31,52,68]. A numericaltechniquethat includesall of the important effects and is in good agreement with the practical resultsis thereforeessential. In this section, different physicalaspectsof the winding arrangementare discussed.The aim is to understandthe underlying principle and thereby lower leakagecomponentsand eddy current losses.
3.3.1: COPPER LOSSES IN THE CIRCUIT WIRING.
It is well known in eddy current analysisthat extremecare must be taken in modelling correctly the rapid decay in field that occurs at conductor boundaries [28,52,69]. This problemwill be troublesomeas the frequencygets higher and lossesare increasedaccordingly. The main two effectsthat dominatethe copper lossesare the skin and proximity effects. The procedureof numericalsolutions will start by consideringone conductor only, and the principlesdeducedcan be extendedto include all the turns in the winding.
220.127.116.11: SKIN EFFECT. I
Fig. (3.4) showsa copperconductorconsistingof 64 ( parallel untwisted) fine , wires forming the " go " half of a turn of the winding. The return half is not modelled due to the symmetryof the turn aroundthe core.
In the top flux plot in Fig.(3.4), the conductor is carrying current at I KHz.
frequencythe skin depth is 2mm. This is much greater than the depth of bundle of wires differential field is The 0.48 the conductor causing varies across mm. magnetic which insufficient in but to redistribute the the result are currents which eddy voltage eachwire, field. When the frequencyis increasedto the level where the skin depth is much smaller than the wire thickness,the current inside the conductor will be non uniformly distributed as seenin Fig. (3.5). The bottom flux distribution of Fig.(3.4) is with a frequencyof IMHz. The skin depth is now 0.063mm,and the eddy currentsredistribute the flux, so that every turn links the sameflux. Thesecurrents are in the samedirection as the main current at the conductor surface and oppose it further in.
Hence, the current density falls
exponentiallywithin the conductor. The actual area of the conductor that carries current is reduced as the frequency increases,and so the ac resistancewill be greater than its value at dc or low frequencyand the loss increasesaccordingly. In order to reducethis skin effect, the wire has to be divided into many fine wires and twisted, so that eachof the fine wires occupiesequally every position in the bundle. The main issuehere is to seehow fine the wire has to be. Many researchers[30,33,59] have usedthe commonapproximationthat the current is evenly distributed acrossa skin depth (see sketchbelow). In this casethe actual curve of the current density is replace by an equivalent rectangular distribution as shown in the diagram, which of course is much easierto apply. In fact this approximationis only valid provided that the conductor is far greater than the skin depth. Extending the argumentto suggestthat conductors should be no more than one skin depth deepin fact doesnot makebest use of material.
Equivelant (One Skin Depth)
The finite elementtechniquehas beenused to estimatean "optimum" value of wire size providing the best compromise between number of wires ( and hence cost ) and ac resistance.Different wires sizeswith respectto the penetrationdepth relation have been modelled. The results show that a good ac;to dc resistanceratio is achievedwhen the wire thicknessis 1.2 times the skin depth. In order to give additional support for this value, a method is derived for calculating the variation of the conductor resistancefor a single square conductor against frequency. The resulting resistancevalue for a sinusoidalvariation, for a single conductor is given by : R" Re = Rdý:
FpL (-E 9,u
Where the dc resistanceis ýE-t and b is the side length. A full derivation of this b' , equation is given in ref  and discussedin ref.
The resistanceratio is shown
graphically in Fig.(3.17)as comparedwith that in ref , which shows how its value is increasedas the thicknessto skin depth ratio gets higher. As can be seen,the ratio takes 118 for the curve around a sharprise = 1.2. The accuracyof the finite elementresult providesvalidation for the method and helpsjustify its use for more complex problems.
18.104.22.168: EQUIVALENT CIRCUIT MODEL OF THE WIRE. It is often useful to form an equivalentcircuit model. This approach is probably deal designer for to with. easier a circuit In the finite element modelling, if the wire is placed in free space it will set up an infinitely extending magnetic field. To restrict computation the wire is encasedin a boundedareato enclosethe magneticfield at a reasonabledistance.However, its size is still arbitrary, but it is a matter of numerical convenienceto satisfy the requirementsof the finite element solution. In practice there will always be a return wire(s) and the boundaryconditions imply their existence. Returning once more to the 64 strandedwires shown in Fig.(3.4), an equivalent circuit can be producedfor thesefine wires as shown in Fig. (3.3a). If thesewires are modelled in one equivalentcircuit, three elementscan be clearly recognised.One of theseelements is the resistanceassociatedwith the loss in eachof the wires. The secondelementis the assemblyof inductancesassociatedwith the operation of all wires acting together, this is termed the "external inductance" [3,46] and is given in the symbol Lext in Fig.(3.3a). The other inductanceis associatedwith flux which lies between each parallel wire and reflects the fact that there is flux produced by each wire, which does not couple with other wires. It would be logical to describethe fine wire elementsby an inductance matrix made of self and mutual terms, and the effect of proxin-dty would demand the samefor a resistancematrix . However, The size of thesematriceswould be 64*64, which by any standardsis large and cumbersome.In order to keep the computations as simpleas possible,a singlefine wire is considered. In this casethe equivalentcircuit can be reduced to the model shown in Fig.
(3.3b). It should be noted that point "V
representsthe outer surface of the conductor, while "T' is at the centre. In order to calculate the terms ( elements) in the model, the impedanceof the wire is calculated using finite elementsat two different frequenciesand the results are as given below.
R/m ( 921m)
Urn ( P"Im
0.06 +j 0.0021
The inductancevalues above describethe total of two components( self inductance). One comesfrom the energystoragein the externalmagneticfield, and the other accounts for the magnetic energy distributed throughout the wire. At very high frequencies,the flux is effectively excludedfrom the wire and the effective inductancefalls towards the external'inductance.From the energy calculations(ILV), 2
the internal inductancesat
these two frequenciesare 34 nWm and 13.7 nIVm while the external inductancesare 0.316 ýtWrn and 0.31 gWm respectively.Thesethree components:- internal inductance,
externalinductanceandresistance arethe elementsof the equivalentcircuit of onewire. The relevanceof the skin depth can be seenclearly by applying a sinewave signal to the circuit of Fig.(3.3a). The values of the elementsare the sameas those calculatedabove. When the frequencyof the voltage wave form is 1 KHz the current distribution will be uniform as seenin Fig.(3.6), becausethe internal reactanceis relatively small compared to R( the resistive component). In the high frequency case, the current flow will be greatestat the surfaceand fall exponentially toward the centre of the conductor, due to the high value of reactancein comparisionwith resistivecomponent.
Sofar only oneconductorhasbeenconsidered, wherethe selfinducededdycurrentsare called the skin effect eddy currents. Another important eddy current effect is the proximity effect [31,33,52]. This effect exist due to the flux linking surrounding conductors,whichcausea circulatingcurrentin adjacentconductorsasshownin the Fig. I
(3.7). Flux is not going to be radially uniform when two adjacent conductors carrying the samecurrent are considered.As the frequency is increased,current will concentrate on the outer surfacesof the conductors. The field will be additive on the outside and cancelin the middle. If the bundle of fine wires are in parallel and untwisted, the current will end up flowing in the outer wires, and the situation would be identical to having a solid conductor. For this case the current distributions in different wires within the conductor bundle including face, middle and far end at low and high frequencies respectivelyare given in Fig.(3.8 a,b). The effect of the eddy and proximity effects in the middle area which are opposing the main current in the surface is clearly shown in the figure. As the spacebetween the conductors is reduced, more field cancellation occurs. The losseswill then be higher and the leakageinductancereduced. If eachwire was very far from its neighbour,the proximity effect would be negligible but the leakage inductance would be high. This is clearly not a reasonablesolution, the obvious solution is to twist the conductor bundle. This has the effect of forcing every wire to carry the same current, and hencereducinglossesbut also to minimiseleakageinductance.
3.4: SHORT & OPEN CIRCUIT ANALYSIS
The aim of the analysisis to make an attempt to minimise the copper  and eddy losses[28,70]. Theselossesdependon the total current carried by transformer windings. There are three current componentsforming this current, they are, load, magnetising, and capacitive currents. This analysis neglects the capacitive effects for the reasons already stated, and they are in general small in comparisonto the leakage and copper loss.But their effectscan not neglectedfor frequencies greater then IMHz .
The load and magnetisingcurrents can almost be separatedby open and short circuit analysis.Sincea Ferrite core of high permeabilityis used, the magnetisingcurrent will be small in comparisonto the load current. It may be quite appreciableif the core has an air gap, and transformersused in power suppliesneed an air gap for two reasons,first to provide the required magnetisingcurrent to maintain zero voltage switching at light load , and secondto avoid saturationcausedby dc flux components in the transformer. Finite Element analysis is used to consider the effects due to the two current components,and the transformeranalysedis as before.
This analysis enables the determination of the copper loss and leakage flux
componentscausedby the load current. In this analysisa rated current of 5Ampere is applied to the primary side, while the secondary terminals are short circuited. The equivalentcircuit for this two dimensionalmodel is shown in Fig.(3.9a). The values of the total ac resistanceand leakage reactance for both windings are calculated and referredto the side on which the voltage is supplied. 2
NJ X., NP)
On shortcircuit,the flux is low in the coreandthe magnetising currentis negligible.The valuesof the circuit elementscanbe difficult to determineseparatelyand accuratelyby test becauseof the dominationof the inductorreactance.In the finite elementanalysis thevaluescanbe determined from the lossandthe storedenergydirectly.
The field distributions during a short circuit in the secondary winding at different frequenciesare shown in Fig.(3.10). As the frequency is increased,the flux tends to field distribution in between This the the reflects of change concentrate space windings. losses due to the proximity effect and also changes in the leakage inductance. The calculated impedanceis the sum of the two finite element models
front and side
elevations) as shown in Fig.(3.11) (Pleaserefer to fig. (2.3) and ref [5 1 for transformer overall dimensions).The total short circuit impedancein the primary side is given in Fig.(3.12) along with measureddata and it can be seemthat the agreementis reasonable. The curve shows that the calculation is viable up to I M]Hz, but above that capacitive effects cannot be neglected, the calculated impedancetherefore only rises whilst the measuredimpedanceturns over as capacitancestarts to dominate. The total winding resistanceand leakageinductanceare given in Fig. (3.13a and b) respectively. The most important factor that effects the leakage inductance is the gap between the windings, i. e. inductanceincreasesasthe gap increases.The effect of the gap on the loss is the opposite,i. e. falling with increasinggap as proximity effects reduce. The effect of winding separationis clearly seenin Fig.(3.14).
3.4.2: OPEN CIRCUIT
Traditionally, this test will establishmagnetisingcharacteristics,core loss and winding coupling ( i. e. effective turns ratio ). The assumptionhas been made for the core test that the permeability will be a single value, this meansthat the elementsof the open circuit test to do with saturation, and harmonic production [6,43,72] are beyond the scope of the study. It is the case however that proximity effects are changedby the presenceof the main field and effective winding resistanceswill not be the sameon open circuit as they are on short circuit. An example of this effect is that even though the
it be is which in there within currents eddy can still the zero, winding secondary current
sumto zero. is bulk ferrite be It should noted that whilst conductivity which around cores exhibit a 10' smallerthan laminationsteel,it still becomesimportant at high frequency.This effect both The it is be of that results not a negligible effect. can rnodelledand the results show in in (3.15) this inductance Fig. and respectively, are shown primary resistanceand self casethe magnetisinginductancefalls as core eddy currents occur at about 1MHz.
3.5: CORE PROPERTIES The core used in this analysis is Ferrite E42 [5 1], the dimensions and all relevant data are
given in chapter two. Fixed values of relative permeability and conductivity are used during the finite elementmodelling. These values are p= 2200 and a= 2(n. m)-', for
Ferrite be best ( the to to the taken match permeabilityand conductivityrespectively manufacturersdata ). In general,there are two types of lossesin the core, eddy current and hysteresislosses. The finite elementmodelling showedthat the largest effect was eddy current rather than hysteresisloss. The core properties have virtually no effect on the transformer parametersduring the short circuit, becauseflux in the core is small and hencethe eddy currents in the core are negligible as comparedto those in the winding. The short circuit test's main use is to examine'theeddy current loss in the winding and the subsequenteffect on the leakage inductanceand winding resistancerather than effects in the core. The opposite state can be expectedduring the open circuit case,where the eddy current in and leakage of the winding are very small and can be neglectedin comparisonto the propertiesof the core.
Essentially,the open circuit and short circuit modelling ( with conditions chosento avoid saturation ) give relatively good results as shown previously in Fig.(3.13a,b) and Fig.(3.15).
The small difference between calculation and measurementvalues can
for instance be by the single real value of the employed, probably explained assumptions the permeability. In order to examinethe effect of the core properties on the transformer parameters,high and low values of the permeability and conductivity are considered as shown in Fig.(3.16). This figure indicates ( not surprisingly ) that the self inductance is more sensitiveto the value of permeability,while any changein the core conductivity results in a greater changein the winding resistancedue to the loss effects of the eddy currents induced in the core. As the permeability is reduced, the frequency at which the self inductancecurve falls is reduced. This frequencyrepresentsthe maximum frequencythat the core could realisticallybe used with such a value of permeability. This is in addition to the more obvious reduction in inductancewith reducing permeability. The frequency at which the self inductancefalls away shows the point at which skin effect starts to dominate. Laminated steels require unpractically thin laminations to operate at very high frequencies. Ferrite have a very low bulk conductivity which allows high frequency operation, but it is easy to see from the results that skin depth eventually sets in the region of 1MHz. Extreme operating frequencywill demanda reduction in conductivity, or acceptanceof a reduction in permeability. It would also be possibleto laminate the Ferrite.
3-6: WINDINGS LAYERS AT HIGH FREQUENCY In this section, the effect that occurs due to increasesin the number of layers is considered(a layer here is taken to meanin its conventionaltransformer parlance,i. e. a
). Again, the helically layers from is coils wound of successive made up winding The depth. 1.2 the main is times thickness skin of simulation carried out with a wire inductance. find the and the resistance the value of ac; component of concern is to Firstly, this gives an immediatefeeling as to the limiting factors that restrict the design, inductance, leakage in losses lead the the to the and a reduction study can and secondly . interference due in factors to the electromagnetic minimising noise main which are As the frequencyis increased,the field in the gap between the windings is not effected, density inside it depends dimensions. But the the the current conductor, since only on falls exponentiallywith increasingdepth into the conductor. When the flux within the is is is leakage inductance But this the not a good way also reduced. conductor reduced, to reducethe leakageinductance,sincethe penaltywill be increased eddy current loss. The problem of reducing the leakage inductance and keeping the eddy losses within [1,19,30,52]. from limits has many researchers reasonable received great attention Dowell solved the problem mathematicallyand the equationshe used are fully detailed . The principle works well but the main drawbacks are that the geometrical distribution of leakagefield is assumedto be purely acrossthe transformer window, and eachlayer of the winding is coalescedinto a current sheetwith simple assumptionsmade on current distribution within the sheet. In this work, the calculations are done using FE analysis,where all the turns of the windings are included, and the core is fully modelled.The elementsare chosento be no larger than one third of a skin depth at the highestfrequency.The primary and secondary windings are arranged in layers in the same way as the true transformer. The model allows variation in the numberof layersto be considered,and the FE results from this are presented. The dc componentscan be found either by solving the problem at zero frequency,where the equationto be solved is the magnetostaticform of Maxwell's equation ( i. e. the eddy
form ), ) in ( 3.1 the to term equations closed madeequal zero or using current equation discussedin chapterfive and given by Dowel [I]. There are two ways of calculating the ac components of the winding resistanceand leakageinductance.The first is the simplestand involves a short circuit calculation, this gives a good approximationof the combinationof the leakagein the primary winding and the leakagein the secondaryreflected to the primary by the turns ratio. During a short circuit, the shunt inductanceand resistancedue to the magnetisingflux and loss in the core are small and can be neglected.That means,during the short circuit, it appearsthat the impedanceseen at the terminal is the primary and secondary leakage branchesin series.The secondmethod involves exciting a section of the winding, in this case each layer in turn and calculating the resulting flux linkage in the excited and all the other nonexcited section (layers). This procedure has to be repeated for each layer so as to build a matrix ( detailedin chapter five ). This matrix can then be used to determinethe effectiveimpedanceresulting from any connection. In the Dowell analytical solution, the leakageimpedancecalculation is done by dividing the primary winding into layeredportions from the inside of the winding outwards. Each of the portions consistsof two layers. There are two principles clearly explained in his work: I- The leakageimpedanceof a particular layer needsto consider the effect of all of the other layersin the winding on the consideredlayer. 2- The leakage flux in each layer dependson the current in that layer and the total
currentbetweenthe layerandanadjacentpoint of zeronmE Hence,eachportion of the winding is consideredseparately,and the primary leakage impedance is the sumof the leakageimpedance of the portions.The analyticalresultsare given in Dowel'swork as a graphshowingthe differentlayersin the primarywinding portions.It is clearly recognisedthat to includethe secondarywinding, the leakage
impedanceof the portion forming part of the secondaryhas to be divided by the turns has it two principal problems: to to the primary, which squared refer I- In the Dowell model, the flux in the gaps between the conductors is considered independentof frequency.In fact the leakageflux as it is expelled from the conductors tends to fringe around the conductors and make the leakage field non-uniform. This fringing involvesthe spacearoundthe windings. 2- There is a possibility of proximity effect between the primary and secondarylayers, and consequentlyfor a good solution both primary and secondaryneed to be modelled together. Here, the ac to dc ratios of the leakageinductanceand resistanceare found using FE with the layersof the primary consideredfirst. Each layer consistsof 14 turns, therefore; two layers means28 turns and so on, a half layer means7 turns placed in the middle of the window. In considerationof primary layers only, it is easy to compare the results with Dowell's analytical solution. Fig.(3.17) shows, the comparison between analytical and FE, where a reasonablyclose agreementis seen.The vertical scalesare the ac to dc inductanceand resistanceratios respectively.The horizontal scaleis effectively frequency ( or rather the root of the frequency ), but it is presentedin terms of the ratio of the conductor sizeto the skin depth. The figure showsthat an increasein the numberof layer apparentlyshows an increasein the ac resistanceas comparedto the dc, and a reduction in the inductanceper turn for ac as compared to dc. It should be rememberedthat these results are taken against conductor thickness as a ratio of skin depth. Taking results for a fixed dimension of conductor and simply upping the layers actually puts more copper into the transformer window. When the fields become skin limited there is less spaceavailable for leakage flux and the inductance per turn falls. The conductors are close together and hence proximity is increasedcausing an increasein the ac resistance.Normally a choice for more layers implies a thinner layer and hence design changesin layers are not simple 3-21
be design is however to The that the these graphs allow any verticals on graphs. point analysedwithin the transformerwindow becauseof the way they have beennormalised.
If the secondaryis now addedit is possibleto use Dowell's work by reflectingthe leakageimpedance valuesobtainedacrossthe turnsratio. The finite elementresultsare obtaineddirectlyusingstoredenergyandlossasbefore.Thetwo resultsarecomparedin Fig.(3.18), and the resultsshow that the inductanceper turn is reduced,and the loss resistanceis increasedby almost30% as the conductorthicknessto skin depth ratio variesfrom I to 10. The existenceof the secondarylayersadd more loss which is reflectedto the primaryby the turns ratio. The caseis not the samewith the leakage The leakageflux dependson the spacebetweenlayersandthat of the rest components. of the window, and since the window spaceis part of the solution, the leakage do not showmucheffectby alteringthe secondary layers. components
3.6.1: LAYERS TOPOLOGY In order to illustrate the effect of increasingthe number of layers on the leakageand losses, the diagrambelow may be considered[30,52]. 4 -3 91
3 -2 2 -1 U 11
Assumea current of one ampereflows through four layers, eachwith two turns. The skin depth is 1.2 times the conductor thickness. The current in layer one causes a
the layer The between in field two. of the surface to opposite and one gap exist magnetic flux. The in it by induced layer has this net a nearly equal and opposite current next been has in has layer be two to assumed, and so a2 amp current as one ampere current flows in the far surface of layer two, and so on for the rest of layers as shown in the above diagram. The mmf curve rises with layer responding to net current. The current density squared determines the loss density, the two surface currents in layer two alone (2)2 +(_1)2 be in layer losses The times that can applied to the one. same cause of =5
four in layer layers, in layer losses 13 25 the the times three times rest of where are and with respectto layer one. The averageresistancein all four layers is 25 +13 +5+I/4= 11 times that in layer one. Since the ac resistancein layer one is ( for generality the whole areais 100%, and the areaconsideredis 12%) 100 / 12 = 8.33 with respectto the dc acrossthe whole conductor area,and it hastwo surfaces,the ac resistanceis therefore equalto 4.167. That meansthe ac to dc ratio would be 4.167:1( the ratio of averageac to dc resistanceis 11 * 4.167 = 45.8 ). In order to confirm the obtained result, the graph in Fig.Q. 18) showsthat four layerswith depth ratio of 1.2 gives an ac to dc resistanceof about 4( averageac to dc ratio is II*
4= 44 ). If the penetration is larger than the
conductor thickness,the mmf will not be zero at the middle of the conductor and the current cancellationwill be complete.In this casethe value of ac would be the sameas the dc, and the current is distributed uniformly within the area of the conductor. The ratio of the ac to dc resistanceis then equalto one . Another considerationin the value of ac to dc ratios of leakageinductanceand resistance can be observed.One of the main concernsusing the graph is the number of fine wires that are used for a certain operational frequency. Consider for example, a single wire which f6rms one turn of a ten turns layer, with a wire gauge of area 0.033 CM2, and a transformer designedto operateat IMHz, (where the depth of penetrationis 0.066mm). This meansthat the conductor thicknessis much greater than the depth of penetration. The ratio of conductor thicknessto skin depth ( sayK) is 27.5. Referring this value of K to a single layer on the graph of fig. (3.18), it shows that there is a large difference
between ac and dc resistanceas expected. In this casethe systemwill not be able to handlethe high losses. If this conductor is divided into four conductors each with 1/4 the original diameter,the numberof conductorsis then 40 with two layers, eachwith 20 dc be 6.75 The K to to to the ratio ac conductors, will original wire. equal with respect is still high and needs to be reduced. Again by dividing each of the four section of conductor into six, there are 60 wires placedin 2.5 layers, 25 wires per layer, eachwith diameter0.075mm, and K is less and so on. Therefore, dividing the original conductor into many fine conductors is a very important way to reduce losses.This information can be obtained by deriving a graph of the type shown in Fig.(3.18) at the design frequency and number of primary and secondarylayers. Further detail can be obtained from publications[1,30,52].
3.7: ANALYSIS VALIDITY AT I MHz In order to give the required validity to the FE analysis at I MHz, example 6.2 in reference [ 31 ] is considered.In this example an inductance needs to be built at a requiredvalue of 28 uH at IN411z.The inner diameterof the toroidal core is 28mm, and the outer diameter is 40mm. The core material is Nickel-Zinc Ferrite with resistivity equal to 5x 10' Q. m. The procedure that the author used to design this inductor is the sameas that used by Dowell , where the conductor is consideredto have a square section area to make the mathematical calculations more manageable. Using the referenceequations,somecalculationsneedto be prepared.The height of the core ( H, equal 5. lmm, the inner and outer diameters are D, = 28mm D. = 40mm, and the numberof turns and strandsare 77 and 16 respectively(as shown in the diagrambelow). The author predicted from the calculationthat the optimum ac resistancefor the winding length of 1.9mis 0.41 fl, and the measuredvalue is 0.48 fl. Finite Element analysis is used for the above specifications. The field distribution is shownin Fig.(3.19), at I MHz. From the loss relation the ac resistanceis 0.456 fl. The 3-24
finite elementanalysishas resulted in a value close to the practical result. The reasons behindthe differencebetweenthe measuredand calculatedvalues may possiblybe due to derived is the mainly the neglect of conductor curvature, and secondly, equation it is for Using FE, infinite the possible core. assuming permeabilityand zero conductivity to model circular wires and the core is more exactly and accurately modelled. This exampleshowsthe capability of the finite elementanalysisto solve any problem, and to be the first choicein comparisonwith other numericalmethods. O. lmm
13 E3 E3 E3 E3 1:1 E3 1:1 E3 E3 E3 E3 E3 E3 E3 E3
ý 0.51 mm
: 5.1 mm
The ac component of magnetic field that exists between the primary and secondary windings is the main reasonfor proximity effects. This field is shown to be concentrated in the gap between the windings as the frequency goes up. This causes an induced current that is circulating within eachconductor, adding to the conductor current in one surfaceof the conductor and opposing or cancelling current in the other. Effects like thesetogether with the effects due to the self induced currents (i. e. skin effects ) can be removed or at least reduced to acceptablelevel by using either twisted fine wires, or reducing the operating magnetic field strength. The field strength can be reduced by using a wider window areato accommodatethe number of turns in less layers. Reducing
less is field has to the transformer that make work with this another advantageas well, saturation. Another way to reduce the effective number of layers is by splitting the winding into faces by interleaving. In the this one section of caseeach primary winding small sections full balance is in layer, If the ampere-turn achieved each secondarywinding. section of in field field Because for the the the conductors. of outside reduced cancellationoccurs suchan arrangement,the leakageinductanceis reducedas required. Also, the study in this chapterhas shown that the winding resistanceis a function of core conductivity, while the self inductanceis a function of core permeability. So, by choosing a core material with a very high permeability and very low conductivity, the lossesand leakagecan be reduced greatly.
N \ \NIN
Principle of Finite Element Analysis with seFIeSWIndIng .
Center FIG. (3.3) ý Single conductor equivalent circuit
: Flux distribution for sin-fe conductor at I kHz and I MHz
Current Density Modulus Against Position
30 ,I 25 10 20 15
10 5 LL 0
2345 distancealong line (m) xlO**-4
234 distancealong line (m) xlO**-, 0
Modulus current density along the conductor at I kHz and 1 MIR.
0.8 0.61 , 0,4 0.21 01 ',0
: -0.20 1 -G4 -ü6!
brre ( us)
Current Oscillation Behaviour due to the skin efrect
main current eddy current
): Proximity Effects Principal
Instantaneous Current Density Against Position
2345 ýistancealong line (m) x 10**-4
InstantaneousCurrentDensity Against Position 11) 25
forward solution on first right I on eqn. 5.32, U-matrix
Fig.( 5.8 ): Program flowchart
0.018, 0.016-0.014-0.012-0.01 0.008 Ln 0.006-0.004-0.002-0--l 6qq
M cm R co
0i C: )
Fig.( 5.9 ) Numerical error as a function of time step
150 100 .. I-, 50 0
5 -50 -100 -150 Time (us) Node I
80.00 60.00 40.00 20.00 0.00 0 -zu. uu -40.00 -60.00 -80.00
Time (us) Node6
40.00 30.00 20.00 10.00
ca C -1 U.UU -20.00 -30.00 -40.00 Time (us) Node II
Fig.( 5.10 ): Voltage of the selectednodes
1.50 1.00 0.50 0.00 u
-0.50 -1.00 -1.50
0.06 0.04 0.02 0.00
1 -0.02 -0.04 -0.06 Time (us)
0.30 0.20 0.10 0.00 U
0.10 -0.20 -0.30 Time (us)
Branchescurrent, (a): Load, ( b): branch 1-7 (c): branch 1-14
1.OOE+05 I. OOE+04 1.OOE+03 1.OOE+02 CL
OOE+OOL4mi66m 0.1 0.01
( MHz )
Fig. ( 5.12 ) Winding input Impedance
-, E = 1.OOE+04 0 1.OOE+03
I. OOE+02 1. OOE+01 4) CL 1. OOE+00 0.01
Fig. ( 5.13 ) Winding input impedanceWhen C between Adjacent conductorsis increased( 0.152nF )
1. OOE+06 I. OOE+05 1. OOE+04 I. OOE+03 1. OOE+02 1. OOE+01 CL
E 1.OOE+00 0.01
Fig. ( 5.14 ) Winding input impedancewhen C between is adjacentconductors decreased(6.1pF)
, E = I. OOE+04 0 -V 1.OOE+03 1.OOE+02 M 1.OOE+01 10 CL E
1. OOE+00 0.01
Fig. ( 5.15 ) Winding input impedancewhen ground capacitances decreased( 8.146pF)
I. OOE+03 'r-, 0 1.OOE+02 "0 1.OOE+01 d) 'm CL. I. OOE+00 0.01
Fig. ( 5.16 ) Winding input impedancewhen ground capacitances increased( 0.203nF)
TRANSFORMER ELEMENT SIMULATION USING SPICE CONVERTER MODEL.
The Spice simulation program is a useful and important tool for power supply design. This program can be consideredas an intermediate step between analytical design and into before insight design It their testing. problems and solution practical provides more for forms basis it is implemented. Hence time, cost and a system saves any practicalwork improvement. Spice can also be used as a tool for evaluation of potential problems, which could destroy the semiconductordevicesif a practical circuit was built without a imbalance instance, For that can the prior understandingof volt-second and problem. pushthe transformerto work in the saturationarea. The Spice program starts by placing the required circuit elementsbetween nodes. The program reads all the circuit information through a file called the "net list" file. In this file, the program translates the original circuit drawing, their elements, and initial conditions. A circuit model can either be built from discrete componentsor using the componentsthat already exist in its library. The computer run time tends to be long due to the numberof iterations attempted. Of coursethis is aided through the use of a very fast computer but Spice none the less can be very cumbersome.This problem will be exacerbatedif many sub-circuits have'been selected and linked together to form the whole circuit as is the case here, where a full bridge converter needsto be simulated. Finally, the associatedanalysisspecifiedin the input net-list file will then be executedby entering the "spice" command. The Spice program, stores the simulation results requestedin either an output listing file, or as a graph data file. It has a complete set of prints and graphscalled "probe" for viewing the analysisresults. 6-1
The main concernof this chapteris to examinethe transformer element's behaviour and their interaction with the resonant tank elements during normal converter operation. Generalinformation is given about this converterincluding its principles of operation, the Nowadays, in it, for interest the transformer. the problems associated with and reasons there is much interest in resonantconverters, series,parallel or a combination of seriesdc-dc is depends being The the on whether power conversion classification parallel. is A the through the or resonant current voltage. called converter resonant achieved seriesloaded when the load is in serieswith the resonantcircuit elements( inductor and in capacitor series).The operatingcharacteristicof this type of converter tends towards a current sourcewith a high output impedance( high output voltage ). A parallel loading converter has the opposite characteristics. The series-parallelconverter combines the operatingcharacteristicsof both types. The magnetic components( transformer and inductor ) that exist in any power supply unit reducethe opportunity for improvement.Therefore, a better understanding through the simulationof thesecomponentscan turn theseundesiredeffectsto an advantage. The work in this chapter is an extensionof previous researchin this area . In this reference,simulationwas carried out to examinethe transformer elementsfor a 20KHz half bridge converter.'The ratio of the switching frequencyto the resonantfrequencywas 0.4. In the presentchapter,the work is extendedto simulatea full bridge seriesresonant converter. The transformer equivalent circuit currently used is a high frequency transformerand it obviously containsmore elementsthan that at 20KHz. In addition, the simulation is carried out above and below resonant frequency and at a different frequenciesratios. The simulation program used in this chapter is not the same as that usedby .
6.2: SIMULATION ADVANTAGE.
Megahertz frequencypower supplieshave receiveda high amount of researchsattention recently [33,48]. In this zone of operation, the resonant converter types are the only frequency, designed ( The two) to this transformer chapter at was work choice available. finite The it the elements program. effects of the transformer simulated using and was elementsare the main concern of the presentwork. The transformer was designedas a has be ( high the to type transformer, therefore converter series of and voltage step up converter ). The practical circuit of the presentpower supply ( chapter two) was given frequency . below 500KHz The same circuit the of above and resonant previously has been improved and built here again at I MHz frequency operation. The results in chapter two provide the basis of the simulation validity for the case currently studied. Bearing in mind that the aim here is to examine the behaviour of the transformer elements within the converter, the reasons behind not examining these elements practically is as follows: I- The principlesof the transformer equivalentcircuit is well known which simplifies the analysis.It is difficult to implementpractically, unlessfive or more different transformers are designed.Each of these transformerswill provide an examinationof a combination such as high / low inductance( leakageand magnetising),and capacitance( ground and distribution). This is rather difficult practically. Even assuming that all the above is possible,difficulties still exist due to the following: a- The difficulty of controlling the values of these elements,where examining these elementspractically meansthere are some degreesof flexibility to change their values within a specificrange. b- Many converters have to be designed for different duties, so a different control techniqueis required in each case.It is well known that each of these elementshas a great influenceon the power supply particularly in resonantconverters. 2- The optimum power supply solution and performance is found when the supply frequencymatchesthe transformerresonantfrequency.From the open circuit impedance 6-3
is ) ( OOKHz I the exists either about which not curve, voltage resonant parallel resonant is 2.4MEz, the one currently the approximately which under present concern or at investigated in this chapter. Care must be taken at this frequency where the if destroyed be devices could semiconductor
resonant frequenq is approached too
closely. In any case,the simulationcan provide a clear insight in the behaviour of the transformer interaction in Megahertz their the with the re',onant tank elements. and zone elements The existence of inductances ( leakage, magnetising ), and capacitances( ground, distributed ) can turn to an advantage.Theseelementscan be arraiged to form a series or parallel resonanttank, with no additional discretecomponentsmIquired.For instance, many researchershave alreadyused the leakageinductance as a resonant tank element . The magnetisinginductancehas also been used as resonant alements[47,48], but under specialtreatment.The treatmentwas basedon the fact that the primary winding is shorted during the resonant stage due to the simultaneousconduction of the output rectifier ( when a full bridge output rectifier is used).
The model used to simulate the converter above and below resonant frequency is not significantly different to the practical model. The simulated model follows the same stagesthat were discussedin chaptertwo. The simulation difficultias involved with each stagewill be explained.Every effort hasbeenmadeto reducethe differencesbetweenthe practiceand the simulation.In both resonantcases( aboveand below ) the resonanttank inductor consistsof the reflected leakageinductanceof the transfojmer in the primary in serieswith an external inductor. The transformer has been placed betweenthe resonant inductor and the capacitor.Practically, this arrangementhas an ad-vantage,where in the 6-4
full bridge converter, an ON pair in one half cycle may have a different storagetime than the secondhalf. This means,the on-voltage drop in both MOSFET legs may be unequal, is This to the transformer the primary unequal as well. applied, and volt-second product loop hysteresis to changewhich saturatesthe core and this the the centre of can cause flux imbalance is damage device. In this to the avoid a capacitor order usually placed can in serieswith the transformer. It is obvious, that the switching signal of the present converter is arranged to give a both below frequency. tank the above and across resonant quasi-squarewave voltage The inductor ( and/or transformer leakage inductance) and the capacitor forming together the tank of the seriesresonantcircuit. Another advantageof the inductor is to charge and dischargethe capacitanceof the devicesto achieve zero voltage switching and hencereducethe switching losses. In general,when any of the devices(MOSFET) is turned off, its capacitanceis charged by the main supply dc voltage. This capacitancehas-tobe dischargedprior to turning the device on to achievezero voltage switching, and this reducesthe switching losses.there are two ways of doing this, either supplying a negative current to discharge the capacitance,or to causeit to oscillatewith an externalinductor. The sametechniquecan be employedto achievezero current switching, but'.instead of a capacitor an inductor is used, and a negativevoltage applied to dischargethe inductor energy or to cause it to oscillate with a capacitor. Thesetwo techniques( zero-current, zero-voltage ) are very important for any converter operating in the Megahertz zone to damp the switching losses.
6.3.1: INPUT STAGE.
'Theinput stagehasalreadybeendiscussedin chaptertwo and consistsof two parts, a rectifier and a filter circuit. The simulatedinput dc voltage can be taken from a full bridgerectifiermodelof a sinewavesourceinput.Thevoltagedrop in the input rectifier diodescanalsobe takeninto account.For the bestaccuracy,the wholeconverterhasto 6-5
be simulatedin lessthan onenanosecond time steps.Oneproblernthat arisesis that of ( ) input time the step msec. with the converter time step of matching rectifier faster Hence, than the whole converter. the can run circuit alone rectifier nanoseconds. This problemmaybe overcomeby runningthe program to msec.,which allowsthe dc is it large build However, this to a quite expensive way as requires'a amount up. voltage is dc input An time to time. run of saving E. pure source alternative way use of computing that containsa completelyripplefreeoutput.
6.3.2: POWER, ISOLATING, AND CONTROL STAGES.
The equivalentn-channelMOSFET circuit model that is used for the transient analysisis shown in Fig.( 6.1 ), in which all the data can be obtained from the manufacturersdata sheet. This model was given previously by Ram et al . 'rhe drain to source capacitance(Cds) is introducedin the presentwork for high frequencyvalidation . In this model, three capacitancesare typically used, and their values are given in the manufacturersdata sheet.First is the gate to source capacitance,usually referred to in the manufacturerdata sheetas the input capacitance( Ciss ). Secolid, is the gate to drain capacitancereferred to as the reversetransfer capacitance( Crss ). Third, is the drain to source capacitanceusually referred to as the output capacitancei. Coss ). The gate to drain capacitance(Cgd) is equalto Crss at high Vds, Cgs is equal to Ciss minus Crss at high Vds, and Cds is Coss minus Crss at zero Vds. A non-linear current source (J)
used to model the forward drain source current and the static body diode. The step by stepdetail of calculatingthe elementsof the MOSFET equivalentc rcuit is given in many references[85,86]. The clamped shotcky diode is available in the spice program lit rary. The rest of the diodes are modelled by the diode model given in Fig.( 6.2 ) [8 4]. In this model the parametersconsist of resistance,two non-linear current sources-:o model the forward 6-6
detail The layer depletion inductance, leakage capacitance. and and backward currents, in [85,86]. is of the way to calculatetheseparameters explained in bridge looks MOSFETs the isolating easier converter the two top The problem of of be the in There this using two either achi. can -ved, ways than are practice. simulation Spicelibrary optoisolator model or assigningthe circuit to a different ground. involving by is a type three control circuit The switching signal provided a very simple (6.4). in The-e Fig. six unknown are shown as comparator and amplifier error be By two the in of resistances to choosing any selected. the amplifier error components design be found four the ) R3 Rl, the at can readily remaining arbitrarily, and say frequency[50,85]. The four equationsrequired are two zero's frec.uenciesat C2 R3 and RIR2CI, and two pole frequencies R3C2C3, and CIR2. The output of the error be 5V is is 3V to sawtooth a compared with amplitude within a amplifier -which four hen,. MOSFETs the During and e rapidly, changes the simulationnothing waveform. i. the from different be control c. signals, without switching as sources supplied may low is has Nevertheless, this proved to give an error which 1elatively small at circuit. frequencybut grows asthe frequencygoeshigher.
The simplifiedtransformerequivalentcircuit model derived in chapterfour is used in this hence losses In this the no resistances are neglected,and core and copper analysis. model heavily in known has However, the was studied effect a well which exist. resistance chapterthree. The model consists of leakageinductancesand diaribution capacitances for both primary and secondary windings, magnetising inductance, and ground by the turns ratio All to the the side primary capacitance. secondaryelementsare referred of 14/26as explainedpreviously. Surprisingly,researchersin this a-ea [36,44], have been in their transformer value the audio concernedabout elementsand ways of estimating frequencyrange ( 20KHz ), but no further study, very few have been concernedabout 6-7
theseelementsfor higher frequenciessuch as those in the megaher-: z zone. It is logical to think that the most visible effect of these elementsis present at this higher frequency. Again, all the power suppliesoperating in the megahertzzone art: of the resonanttype, be This the tanlic these the could resonant elements. can added with elements and value of lead to a seriousconsequenceas it may push the power supply to work out of its design frequency which can cause failure. A further study of these el4%, ments can turn their inductors be for into there then or and will no need extra an advantage, undesiredeffect capacitors. The transformer simplified equivalent circuit replaces the origirial transformer of the power supply circuit. However, the equivalentcircuit can be arrangedin a different way if required using a, dependentcurrent and voltage source . But in this case the is does be-muse this transformer the elements not purpose of examining possible, circuit not containtheseelements.
6.4: PRINCIPLE OF OPERATION.
Fig.( 6.3-6.4 ) show a seriesresonantconverter and its gating ignals above resonant frequency . The converter circuit, gate signals, and principle of operation below resonant frequency has already been discussedin chapter two. Hence, the operation principle of this section is concernedwith the case where the switching frequency is higher than the resonant tank frequency. The ratio of the switcling frequency to the resonantfrequencyis consideredto be 1.2, in a similar mannerto tl le value of 0.8 for the casebelow resonantfrequency. The way of selectingL and C of the resonanttank is not arbitrary. During the resonant frequencycalculation, the selection of any L or C will be at the expenseof the other. Usually for optimum operation, the characteristicimpedanceof the resonanttank (Z) has to be matched with the load impedance( Zload ), which I -.ads to the following relationships: 6-8
J Lý Zload, 2; rf coo = -,
C= and , 0)0 cooZ
The switchingfrequencycanbe selectedthen from eitherof 1.2 or 0.8 at the resonant frequency(w,, Although,the switchingsignalsin Fig.( 6.3 ) haveshownthe sequences of operationof be further here. beginning diodes, At. MOSFETs the point will mentioned a of a and all new cycle, TI and T2 are conducting,and so the current is circulatingthrough the diodes ( DI and D2 ), are conductinguntil the inductor. The two antiparallel resonant its inductor resonant currentchanges polarity. At this momentthe resonantcapacitor chargesto a maximumvoltage.WhenDI andD2 areturnedoff, the currentdischarges the capacitorvoltage,whereit reacheszerowhenTI is turnedoff. The inductorcurrent startsto flow throughT2 and D3 chargingthe capacitor.The negativecycle of the resonanttank followsthe sameprocedure,andthe processrepeatsitself. It clearlyfollowsthat the antiparalleldiodesof anytwo MOSFETsare conductingfor a shortperioduntil the resonantcurrentchangesits polarity.Therefore,the principleis to turn the devicesoff at zero voltage.This proceduremay not help to reducethe losses Megahertzunlessa capacitoris placedacrosseach practicallyat frequencies approaching of the MOSFETs.This capacitorcanhelpto force zero voltageswitchingto reducethe turn on lossesof the device.Anothersolutionis to introducea short deadgap between eachof the two legs of the MOSFETs,where the diode is conductingprior to its MOSFET.ThezerovoltageswitchingON canthenbe forcedto work. associated
6.5: TRANSFORMER ELEMENTS AND POWER SUPPLY PERFORMANCE.
In the simplified transformer equivalentcircuit given in chapter four, some points are to
beobserved.Thelow cut off frequency( first parallelresonantfrequency) is determined 6-9
by the magnetisinginductanceand ground capacitance. The high cut off frequency ( second parallel resonant frequency ) is determined by the leakage inductance and distribution capacitance. The arrangement of all the elements in the transformer equivalent circuit determinesthe seriesresonantfrequency. Therefore, any changein the values of theseelementswill reflect a changein the appropriate resonantfrequency. The clear insight to the influenceof theseelementsthen is by examiningeach of the elements individually. Fig. ( 6.5 ) and Fig. ( 6.6 ) show the results of the power supply simulation. Since all of the transformer elementsare referred to the primary side, the resonant voltages and by the turns ratio. These figures currents are referred to the transformer primary side show the resonant tank and resonant capacitor voltages. In addition, the figures also show the resonantinductor
current ( primary current ), and the secondarycurrent of
the transformer.The results of thesetwo figures are calculatedfor both casesi.e. where the switching frequency is above and below resonant frequency. The ratio of the switching frequencyto the resonanttank frequency is consideredas 0.8 and 1.2 below and above the resonantrespectively.The output load is assumedto match the resonant _IC, impedance tank characteristic (Z=J, ). In order to show the effect of the transformer elements,waveforms under the normal operation of the power supply are consideredfirst. The first element to be examinedis the magnetisinginductance. When the magnetising inductance is removed ( i. e. open circuited ) as shown in (b)
of both figures, the gFoundcapacitancenow is part of the
resonanttank capacitor. The voltage across-the resonant tank capacitor is reduced by 0.72 and increasedby 1.2 of the normal operation for below and above resonance respectively. The reason is that as the resonant capacitor is increased, the resonant frequencyis reduced,and hencethe resonantfrequencyis away from the seriesresonant frequencyof the transformer when working below resonant frequency, and close to the seriesresonantfrequencyin the caseof working above resonantfrequency.That means referring to the impedancecurve in chapterfour), the voltage is reduced below resonant and increased above resonant frequency. It should be noticed, that the switching 6-10
I frequency for is be frequency the the transformer to a series resonant of near selected better performance.
The observation of the currents in both windings indicates an
is first due This in to the the cut off shifting oscillation secondary current. oscillation frequency to the value that it very close to the dc, by removing the magnetising inductance. The switching frequency is far away from the series resonant frequency, and the frequency ratio may be less than 0.05. The oscillation no longer exists in the above indicates inductance has fact for This that the the magnetising resonant case same reason. the major effect in the below resonant case but not in the above resonant case.
The samesituation is shown in the casewhen the ground capacitanceis removed (c For the samereasonabove,the voltage acrossthe resonanttank is increasedby 1.1 and is frequency by The below 0.7 oscillation respectively. reduced and above resonant reducedby 50% as comparedto that when the magnetisinginductanceis removed. This oscillation no longer exists in the casewhen the switching frequency is higher than the resonanttank frequency. Neglecting the distribution capacitancebetweenthe winding turns has not introduced any resonant tank voltage effect in both casesunder study ( above and below ). This is becauseit has nothing to do with the resonanttank frequency, but this effect becomes greater if the converter is parallel loaded rather.than series.Therefore, it can be used as resonanttank element in the parallel resonant converter but not in a series converter. However, in the case under study the waveforms given in (d)
of both figures have
shown a large reduction in the secondarycurrent oscillation when this capacitanceis removed. Practically, this reduction in the capacitancewill be at the expense of an increasedleakageinductance.
Throughthe examinationof the aboveandbelow resonantcases,it seemsthat the best caseis the aboveresonantcase.Thetransformerelementshaveshowna limited effecton the power supplyin comparisonwith the below resonantcase.In the below resonant inductanceandthe groundcapacitance casewhenboth the magnetising areremoved,the result is shownin (e). Removingthe magnetisinginductancemeansno magnetising first Here, the currenti.e. aninfinitelypermeable cut off frequencyno longerexists. core 6-11
The seriesresonantfrequencyof the transformer is dominatedby the leakageinductance both distribution the primary and secondarywindings. It should be and capacitanceof is An transformer. to this that core an air air core will equivalent noticed case not increase the magnetising current i.e. make the magnetising inductance smaller. The ground capacitancecannotbe removedby an air core due to the existenceof the winding ground, but this is lessthan using a core. Usually the closest ground is the core, and the problem of increasingthe leakagewill be much higher in the air core than when using a core. It would be better if the reduction of the ground capacitancewas of the same magnitudeas the reduction in the distribution capacitanceas shown in (f)
of the same
figures. This can be consideredasthe best caseof working below resonantfrequency. The ratio of the switching frequency to the resonant frequency has also been under attention. The simulationhas shown that when the ratio is reducedto 0.4 insteadof 0.8, a change is introduced to the resonant tank voltage. Within the normal tank voltage waveform there is a partial chargeand discharge.In addition the oscillation gets worse as the ratio is reducedlower than 0.8. When the ratio goes higher than 1.2 above resonant, the low frequencyoscillation (a little bit is shown in the secondarycurrent in Fig. ( 6.6 )) is getting worse. In this case,the resonant tank capacitor has to be split between the primary and secondaryto reducethe low frequencyoscillation .
6.6: EFFECTS OF TRANSFORMER ELEMENTS ON THE OUTPUT.
In this section, the investigation is continued to study the effect of the elementson the output voltage of the power supply. The examinationtook place by changingthe values of each of the elementsindividually. The calculated results.are given in normalised values.The normalisedrelation of ( Vo/nVin ), where n is the turns ratio is used for this 6.12
in 0.4,0.8, 1.2 frequency Three are considered as shown of and ratios casesat purpose. Fig. (6.7abc). In Fig.( 6.7 a ), the value of the distribution capacitanceis changedto higher and lower by is different 1.65 its The to at a value than reduced output original value. -which frequency ratio of 0.8, compared with a frequency ratio of 0.4 at below resonant frequency. At one value of the distribution capacitancethe output reachesthe minimum is level the indicates to the The that the maximum minimum of magnitude graph value. best at 0.8 rather than that at 0.4. As previously discussed,the distribution capacitance determines the second cut off frequency, where as its value increases the cut off frequencyis reduced.Therefore, the minimum value of the curve representsthe point at The frequency the transformer. the of resonance switching matches with parallel which frequency 1.2 in happens the the above of ratio of case same and opposite situation both inductance frequency. If are the magnetising and ground capacitance resonant is 1.8% drop 4.2%, 2.3%, is The drop at and the small. removed, output voltage frequencyratio of 0.4,0.8 and 1.2 respectively. Fig.( 6.7 b) shows the case of changing the value of the ground capacitance.In both below resonant cases ( 0.4, and 0.8 ), the output is reduced as the capacitance is increased.The reduction is steeperat a frequencyratio of 0.4 but less of a reduction at a' frequencyratio of 0.8. The effect of changingthe capacitanceis less at the ratio of 1.2, and the output increasesas the ratio is increasedabove the resonant frequency. If the magnetisinginductance is removed, the voltage drop on the output is greater. At a ground capacitanceof 12nF, the drop is 17%, 11.3%, and 7.5% at frequency ratios of 0.4,0.8, and 1.2 respectively. In Fig.( 6.7 c), the magnetisinginductancevalue is plotted againstthe output voltage. As is well known this inductanceis a function of the core current. As -the core current is increased,the value of this inductanceis decreaseduntil saturationwhen its value is zero. Practically this inductancevalue can be changedwithout affecting the other transformer elementsby changingthe core air gap. In both below resonantcases,the output reaches a maximumat one value of the magnetisinginductance,this is called the optimum value 6-13
. The magnetisinginductanceat which this maximum value occurs is different at 0.8 than at 0.4. This meansthat less air gap is required as the ratio is increased.The same effect can be seen in the ratio of 1.2, but the output voltage is increased as the for is inductance increases. The this that as reason maximum occurring magnetising value the magnetisinginductancevalue is reduced from that originally used by the equivalent increases frequency. first frequency to that the the match circuit, cut off of switching This meansthat the switching power supply is working in the area of resonance.If the ground capacitanceis removed,the effect of the magnetisinginductanceon the output is negligible. the voltage drop with respect to the normal operation is 2.2%, 4.1%, and 1.8% at the frequencyratios of 0.4,0.8 below, and 1.2 aboveresonantrespectively.
The simulationwas carriedout to studythe effect of the transformerelementson the Many caseshavebeenconsideredby examiningeachof the powersupplyperformance. transformerelementsindividually. The full circuit was simulatedin a step by step procedureusingthewell knownSpiceprogram.Thetwo casesat whichthe simulationis performedarewherethe switchingfrequencyof the powersupplyis higherandwhereit is lowerthanresonanttank frequency. It was found that the best performanceis achievedat frequencyratio of 0.8 below resonantand 1.2 aboveresonant.As the switchingfrequencyis increasedto megahertz for instance,the best choiceavailableis the caseof 1.2 aboveresonant.This casehas shown a stablepower supply performanceand a very limited deteriorationof the transformerelements.If the below resonantpower supplyrequiredthen the frequency ratio of 0.8 is advisable.Whenthis is requiredat a higherfrequencyoperation,more A circuit requiring effort is requiredto reduceboth groundanddistributioncapacitances. low capacitance to groundpresentsa specialchallengeto the transformerdesigner.In orderto reducethis capacitance a corewith a largewindowis required.In this casethere is no point in increasingthe frequencyto achievelower sizedmagneticcomponents. The 6-14
results also show that reducing this capacitancecan causean undesiredoscillation in the distribution Reducing the capacitanceis a better choice than reducing secondarycurrent. the ground capacitance.The distribution capacitancecan be reduced by increasingthe by dielectric dielectric between lower to and changing a space conductors material with constant. There is an unavoidable increase in the leakage inductance when the is between decreased. turns capacitance winding In order to avoid all of these difficulties, the above resonantfrequencyratio of 1.2 is the best solution at higher frequencies( above500KHz ).
Rd I Cgd
Jm Cd'ý Cgs
Fig.( 6.1 ): High frequencyMOSFET model
Fig.( 6.2 ): Diode model
Tl T2 E- T3 : T4
1 TI T2 D3 T41T4 _4
3 D3 4 D, 2'ý: Vcr
Fig. ( 6.3 ): Switching signals
RI IK KCI R3 c- R2
Fig.( 6.4 ): Spicepower supply model
3300fl 03uF . . + -ý Vreference
Below resonant, simulated waveforms: (a) Normal operation, (b) no Lm, (c) no Cg, (d) no Cd, (e) no Lm, and no Cg, (f) no C9, and no Cd
............................... 2.. V(a2) -V(,. 1 . V2(Cr)
op, r, -. L,. -
- ýl (Ct)
-----------------........................... V2 (C.. ) -1 ic')
d.. y ji
V(.2) -J, J) VZ(Cr)- VI(C,j 06U. TILII - ZILIP
(d) 2 OOV
W ......... -2.01
2.. . VZ (Cr)
y C. 'r-t
", t tV-
- VZ(C, ) -VI(C,
v lo, .
- 11 C9,
Fig,.(6-6)ý Above resonant, simulated waveforms.. (a) Normal operation, (b) no Lm, (c) no Cg, (d) no Cd, (e) no Lm, and no Cg, (0 no Cg, and no Cd I
. V2(C. ) -
................... 2,2ý,. V I-.
v"LL. LJ Ci
------------------------------------------------------------", 11 -----------V(62)
- . 11.1 Ti..
!C -I -
------------------------------------------------------------2ý . V2(C, ) - vl(Cý)
J»\ }' iK,, ; : I'
. V;. 2)
........................ ýl (C.
(L ýI Ti-
Vo /n Vin
1 0.9 0.8
0.7 0.6 1-fo/fs=0.8
L, = 3.36pH, 4=0.8 6pH, L. = 187pH, Cs = 12.3nF (a) Vo/nVin
1.2 1 0.8 0.6 0.4 0.2 0
Cg (nF) 0.86pH, L. = 187,uH, Cd = C, 1.3nF& C, 5.1InF (b)
1 0.8 0.6 0.4 0.2 -fo/fs=0.8
-*-fo/fs=0.4 -n-fo/fs=1.2 300
4=3.36pH, 4 =0.86pH, Cg = 12.3nF, Cd =C, 1.3nF&C25.llnF (C) Fig.( 6.7 Output voltage againstthe transformerelements
CHAPTERSEVEN "EXACT" TRANSFORMER EQUIVALENT CIRCUIT
7.1: INTRODUCTION. In any circuit, the behaviour of the transformer can be found from its dimensionsand the material properties, by meansof solving the electromagneticfield equations as previously discussed.An alternativeway of predicting such behaviour is by using an equivalent circuit representation. There are many advantages of treating the transformer by meansof an equivalent circuit. The simultaneouspartial differential equationsthat describethe electromagneticfield can be avoided and the equivalent circuit gives an alternative method of solving these equations. In addition, the imagination is than differential' to the equivalent circuit scale of practical closer equations,aswell as easyto use and remember. A given transformer may be representedby many equivalent circuits depending on the required purpose. Usually the approximation is considered to avoid analysis complexity. For instance, in an actual transformer, the existence of capacitances betweenwindings and to earth are not important at low frequencybut are significant at high frequency.Nevertheless,the "exact" equivalentcircuit at very high frequency ( Megahertz ) could never be achieved as it would be too complicated. In this chapter, the exact does not means an exact practical representationbut an exact physicaldefinition of the transformer elements.In fact, the presenceof an elementis a sign of comparison witý an ideal transformer. The ideal transformer is not a practical state but only a convenientway of comparing the actual transformer with the ideal. The ideal transformer could possibly be realised at low frequency by the use of a very high permeability core material and an arrangement'allowing both primary and secondarywindings to be very close to each other. The samemagnetic field is then led through the windings, and the capacitivereactanceis so high as to be 7-1
frequency, in high is The the that as as at any reduction neglected: case not as simple leakageflux will be at the.expenseof increasedcapacitance. Electrostatic coupling is the phenomenabehind the appearanceof capacitance.It resides between the componentswithin the actual transformer such as the core, desirable it is Theoretically, to treat the transformer turns, shields etc. winding basis for but the of on separation simplicity, aspects electromagneticand electrostatic in practice,the caseis far away from this simpleassumption. In the present chapter, care is taken to derive the circuit in such a way that its elementshave a physical meaning. A three dimensional FE model could be used directly to build a highly accurate equivalent circuit, although, there are still many difficulties with calculating the capacitanceof the different parts. This could be solved by using the electrostatic energy calculation by means of treating the transformer as a three port network . This technique attempts to model the whole transformerwhich is very time consumingand expensiveas it requires a huge computer memory to model the transformer FE mesh, and much preparation and computationtime. Therefore, although the later techniqueis just about practicable,it is not used as researchersare usually looking for a simple and general method that can be used to derive the circuit. A model derived from calculation has a great meaning practically since a computer can be used instead of building prototypes. Nevertheless, a 3D finite element program is not generally available in many institutions and companies. One of the well establishedmethods to derive equivalent circuits is by fitting elementsto match the performance measured(or calculated) in short circuit and open circuit tests (or calculations).In analysisterms this approach has the merit of simplifying the field model that must be solved from a 3D transient load model to the open and short circuit results at fixed frequency.The equivalentcircuit is also easier to solve for transientanalysisthan trying to time step a 3D finite elementmodel. This chapter persuesthis approachby using the open and short circuit input impedance
measured(or calculated) across a range of frequencies to obtain parametersin a selectedequivalentcircuit topology.
7.2: PHYSICAL MEANING OF ELEMENTS
Fig.(7.1) shows the proposed equivalent circuit of the high frequency transformer. The circuit elementsshown are referred to the primary winding using the transformer turns ratio. The circuit elementswill be explainedfrom the physicalpoint of view. The primary and secondarywindings are madefrom conductors, and lossesexist due to the finite copper conductivity ( 5.7 x 107 mho.m-1). The windings losses can be modelledusing two resistancesone eachto representthe two windings. The primary winding resistance( Rp) carries both load and no load ( magnetising) current, and hence it has to be placed prior to the magnetisingbranch. The secondarywinding it is in ( load ) Rs the so and a serieswith the load. In resistance current, carriesonly addition to the winding losses,there are anothertwo kinds of lossesthat exist in the core, eddy and hysteresislosses.The combinationof both eddy and hysteresislosses are called the transformercore loss. The core loss can be modelled by a resistancein parallel with the magnetisingbranch. In simplistic terms, it could be assumedthat the flux generated by the primary winding links all the turns in the secondary.Practically, there is an amount of flux that does not link the secondary(and vice versa), and so it is consideredas a leakage flux. In order to model the leakageflux two inductances(Lp & Ls
are placed in
serieswith the two windings resistances( Rp & Rs ) respectively. If these inductancesdid not exist the input / output voltage ratio would equal the transformerturns ratio (ideal transformer ). In the sameway, if the no-load current did not pxist or it is not taken into account, the input / output current ratio would equal the reciprocal of the turns ratio (ideal). Practically both load and no load currents are presentand they flow in the primary winding. If the core is made from a very high permeabilitymaterial, then the no load current is negligible in comparison7-3
with the load current, but it will be quite appreciableif the core has an air gap. This no load ( magnetising) current can be modelledby an inductance( Lm ). The presenceof conductors within the actual transformer separatedwith respect to eachother and to the core makescapacitancebetweenthem. In this equivalentcircuit representation,there are two types of capacitance namely, the distribution and ground capacitances.The distribution capacitancesare either those belonging to successiveturns of the samewinding (CI&
C2 ), or that betweenwindings ( C12 ).
The ground capacitance( Cg) is the capacitanceappearingbetweenthe total winding turns and earth. The core representsthe closest ground for both windings, and so capacitancemay be placed in parallel with the magnetisingbranch. Nevertheless,the modelling of these capacitancesin the equivalent circuit does not in general correspondexactly to any actual transformer capacitance,they are only a convenient way of approximation.All these elementscan be justified from the physical point of view, but the reality is very complicated.It should be rememberedthat the main issue here is to generalisethe prediction of the equivalent circuit. Since, the impedance curves are the only data being considered,the equivalent circuit can be built for any transformer regardlessof differencesin design.
7.3: EQUIVALENT CIRCUIT ELEMENTS CALCULATIONS
The equivalent circuit can provide a reasonabletool for finding the magnetic field characteristics( flux density, field intensity, etc.), as well as currents 'andvoltages. In the first instance,the main two questionsthat may be raised and needto be explained are as follows: First, supposethe flux in the core is calculated using the inductive elementsof the equivalent circuit. The single value obtained is assumedto be the sameat all points within the core. Second,in an actual transformer, the magnetising inductance will fall its is linear within the equivalent saturation while value -with 7-4
circuit. At high frequency,the capacitancehas the effect of shunting the current to ground and transferring current between primary and secondary. Clearly the distribution of the magneticfield is affected and the results implied by the equivalent circuit can only provide answersfor local field distribution within the limits of the topology chosenfor the circuit. It is in thesedetails that finite elementmodelling can provide further insight. Fortunately, due to the high permeabilityFerrite core material used, the magnetising current is small, and hencethe assumptionof linearity does not create much error in the calculations.The previous 3D finite element impedancecalculations provide a basis to validatethe linear model. The problem in any proposed equivalent circuit parameter derivation is the complexity of deriving the equations that represent both impedances( open and short) to specify adequatelythe poles and zeros. These points are vital to estimate the resonant and anti-resonant frequencies. Once these points are found, the equivalent circuit elementsestimation follows a straight forward procedure. The proceduregiven here is in easy stepsthat avoids such complexity. The assumptions madefrom the point of view of Fig.(7.1) is that the current in the capacitanceof the samewinding is negligible comparedwith that flowing in the capacitancebetween windings during the short circuit. In the sameway, the opposite assumptioncan be madeduring the open circuit. During the open circuit, the secondarywinding current is zero and so the capacitancebetween windings is part of the capacitanceof the samewinding and it leaks current through the ground capacitance. During the short circuit, the equivalent circuit can be reduced to the one shown in Fig.(7.2 a&b)
. The winding to winding capacitancecan be referred to the
primary winding using the turns ratio as shown in the Fig.(7.2 b ). Since the flux required to induce a certain voltage is inversely proportional to the frequency, as the frequencygoes higher, the assumptionof neglecting the magnetisingbranch during the short circuit is increasingly valid . In general, the magnetising current is negligible in comparison with the load current during the short circuit. Using 7-5
Fig.(7.2b), where both winding resistanceand inductance are in series, the short circuit impedanceis given by: ZSC
W C112 2PS
2 Lp. 0)
The parallel resonant frequency can be found by setting the imaginary part of is (7.1 ) the to equation result zero, and Rp
C12/ PS Is
= Req =
R'P. + (t)
Where the magnitude of the short circuit impedance at the parallel resonant frequencyis equalto 12M. The total of the primary and secondary windings resistance (Rps) can now be deduced directly from the short circuit impedance curve at low frequency. This is if impedance is the to the curve found resistance automatically referred primary 2 (ýN, ) (N2 ZL fl. Rs 0.2 Rp from the primary winding side, and hence Z + = sc = j From equations(7.1, and 7.2 ), the elementsof the circuit shown in Fig.(7.2b) are calculateddirectly at a resonantfrequencyof 2.408 MHz, and are given by : Lps = 3.24 pH, and C, = 1.34 nF 21 It is well known ( as previously discussedin chapter four ) that the highest cut off frequency obtained is due to the leakage inductanceand distribution capacitanceof the winding. In the same way the lowest cut off frequency is due 'to the ground capacitance,becausethe ground capacitanceis usually bigger than the distribution capacitance.Hence, the parallel resonantfrequency of 2.408 N1Hzis used to predict the leakageinductanceand distribution capacitance. During the open circuit., it is assumedthat the current leaking to the secondary through C12 is small in comparison with the current through Cl, and hence the
circuit shown in Fig.(7.3) is the equivalent circuit during an open circuit. Following the sameprocedure as the short circuit, the open circuit impedanceis the total of primary and magnetisingimpedancesin a seriesconnection as shown in the figure. The resonantfrequencyrelationshipis given by : Rc = OCI.PI
a) PI =IIZ--C-9 L (ETP (R'p 0)
Where the magnitudeof the open circuit impedanceat the first and second parallel resonantfrequenciesare about 32 KIQ and 22 KfI respectively.The seriesresonant frequencyis equalto : Lm
= 40 SELC + m
eq A- Ct) 2 Vp
2 Lp o)
r x 751 KHz
The value of the open circuit impedancecurve at low frequency gives Rp only. Because Rc is compensatedby Lm and Cg , its value only appears at the resonantfrequencyof Lm and Cg, and so Rp is equal to 0.09 C1at low frequency. Using equations(7-3) and (7.4) leadsdirectly to the following :
Rc = 32 M
C1 = 1.49 nF
Rs = Rps - Rp = 0.11 f2
Lp = 2.9 ýffl
Lm = 161 ýffl
Cg = 14.3 nF
and Ls = Lps - Lp = 0.34 ýffl.
C2 can be estimatedfrom the resonantfrequency,when Rs and Ls are known, or by referring the open circuit impedancecurve to the secondaryusing the turns ratio. The sameprocedure that is used to estimate Cl can then be applied. If the later is used, its value has to be referred again to the primary, and then C2 is equal to 12.9 nF.
7.4: OPEN AND SHORT CIRCUIT IMPEDANCE CALCULATIONS
The proposed equivalent circuit together with it estimated elementsis used to reit is is impedances. Because the simple, circuit predict the open and short circuit frequencies by the impedances both determine within at all point to point easy ( ) ) ( 7.5 7.4 Fig. the frequency open and short circuit show and range. required impedancecurvesrespectively,given in phaseand magnitudein comparisonwith the in both figures is The finite 3D seen clearly agreement elementmodel. results of the be follows: in both noted as curves,a numberof observationscan and I- If the windings resistancesare neglectedand keeping the elementvalues as they is MHz 2.408 frequency the which are, parallel resonant PLIC2.413 PI
M]Hz, with a difference of 5.1 KHz. In the megahertz
In is difference the circuit previously this unrecognisable. clearly as zone such a have doesnot appearwherethe resistances derivedin chapterfour, sucha difTerence been neglected.This is because,the values of the windings resistancesare distribution inductance leakage by and capacitance at the resonant the compensated frequencies[87). Comparingthe elementvaluesderivedin this chapterwith that in how four thesevalues changewhen the winding of chapter gives a clear view is considered. resistance into account,both impedance 2- By takingthe windingresistance curveshavea finite infinity do ( parallelresonance) or frequency to not go and valueat eachresonant ) asis the casederivedin chapterfour. zero( seriesresonance 3- At very low frequencythe impedancecurves are constantwhich give the resistancevalue. At a certainfrequency,where the curves start to increase,the inductance beginsto appear,andsothe impedance valueat this frequencyis the total it into Whenthe capacitance of resistance with andinductance. comes play, resonates the inductanceat a certain frequency,which dependson which of them is
inductive be for to the or capacitive and then the resonant predominant circuit frequencyis either seriesor parallel. 4- If the lossesare not considered,the real part of the impedancesis equal to zero, and so the phaseis a straight line alwaysequal to 900 .
7.5: TRANSIENT RESPONSE
Transformers are subject to many transient phenomenaduring their working life. Theseinclude surges,inrush current, suddenshort circuit, etc. During the transient a high frequencyoscillation occurs and thesecan result in an over voltage stresson the windings. Such stress can be enough to destroy the insulation of the windings. A large amount of work has beendirected towards modelling the transient responseof transformers[9,39,63,89]. These efforts were mainly devoted to modelling the over voltage transient response in the windings ( as discussed in chapter five ). Measurementof the transformertransientresponseis the most reliable method that is difficulties involved in There this measurement the time. used at are many present . These difficulties are increased as the frequency rises, for instance the high sampling rates of the measuring instruments required, therefore an adequate transformer model is unquestionablyrequired to determine the transient response numericallyand analytically. Current attention is being concentrated on a high frequency transformer model suitablefor transientcalculation.The determinationof the equivalentcircuit elements from the transient response at 100 KHz was found previously . At high frequencies, the capacitancebecomes predominant, and it is harder to use the transient responseto obtain these elements.Therefore, researchershave paid more attention to modelling the frequency response rather than the transient response [9,35,87,90]. Using the frequency response,the transformer can be modelled for a 7-9
certain frequency range. The transient waveforms ( currents or voltages) in the transformer compriseall the frequency components,and there will be no frequency range specified. All the frequency components are excited at once during the transientresponse.The-difficulty in the time responsesis then clear and the question it frequencies is left is how the to transformer the to covering all which model which is subject. In order to avoid such a difficulty, the transformer is modelled first in the frequency domain. The frequency response characteristics have been calculated for the proposed equivalent circuit. These characteristicsfor the open and short circuit impedancesare given as follows: Vin
Two methodsare availableto determinethe transient response.First by transferring the frequency responsecharacteristicsinto the time domain using a well known method, such as Fourier transformer (83], state space[l 1,77] etc. The input voltage or current can usually be consideredas a step changefunction. Mathematically, the step changein the frequency domain gives an impulse function in the time domain and vice versa. These relationshipsbetween step change and impulse functions are given by Gupta . The secondmethod is that the equivalent circuit can be time stepped directly (as detailed in chapter five). The present transient simulation is based on using the trapezoidal rule of integration. Each of the equivalent circuit elements is transferred to an equivalent of a current source and a resistance. Appendix B shows the way of applying this method to the proposed equivalent circuit. The I-IP348I OA Bench link software can provides a communicationlink betweenthe PC andthe IIP54500 Oscilloscope.The results can be stored as a table of data which is easy to use and compare with the calculated results. Fig.(7.6)
comparisonof the actual and calculated transient responsesof the primary voltage during the short circuit at high frequencyexcitation. In order to simplify the analysis, it is assumedthat the step occurred at t=O. This can be clearly recognisedfrom the is instant fault if the However, this cycle not required another the of ac at curves. instant can be analysed. The small differences shown could be caused by the insufficient in is the transient devices the sampling rate or sampling rate of the is fairly Nevertheless, the acceptable. agreement simulationresults.
This techniquehastwo advantages.It is a generalsolution so that it could be applied to any circuit regardlessof the design differences.The accuracy of the simulation dependson the accuracyof the proposed equivalentcircuit and its elementsand the in directly found be is Second a numerical or the either that can results samplingrate. by an analyticalsolution.
This chapter presentsa method to calculate the frequency responsecharacteristics for a lumped parameter transformer model. Previous characteristics of open and The impedances these the to of values elements. existenceof short are used predict theseelementsin the proposedequivalentcircuit is justified at high frequency from a physical point of view. The circuit is reused to. compare the predicted frequency responseswith the actual. This chapter has also paid attention to the transient response for the proposed equivalent circuit. During the transient responsea time stepping technique is used which transformsthe differential equationsinto equationsin which eachof the circuit elementsis transferred into an equivalentcurrent source and constant resistance(as discussedin chapterfive). The circuit is used to predict the transient primary voltage due to a step changefunction in the input of the primary.
Rp .41% lip
Fig.(7.1): The proposedIHFtransformer model
C12 nl: n2
Ip / :)
Fig.(7.2) : Equivalentcircuit during shortcircuit of the secondary winding
Rcý Lm: ý Cg
Equivalent circuit during open circuit of the secondarywinding
0.1 0.01 1
EC -a-Phase,FE -*-P:h7a7s7eý, EC
Fig.( 7.4 ): Calculatedopen circuit impedance
1 0.1 10
I-Mag. FE FE -4-*-Mag. EC -m-Phase,
Fig.( 7.5 ): Calculatedshort circuit impedance
80 1 60 40 20 >o ý% (D -20 E r. -40
-60 -80 Tirm us
Fig. ( 7.6
Primaryvoltageduringthe shortcircuit dueto a unit stepprimarycurrent (a): Calculated (b) : Measured
CHAPTER EIGHT CONCLUSION
The transformer is a device that is present in all power supplies serving many down isolation, up and step power etc. The size of the power purposes,such as supply is reduced with an increase in the switching frequency. Increasing the switching frequencycan lead to a large reduction in the magnetic componentssizes and hence a smaller power supply unit. Nevertheless, high frequency operation meanshigher lossesin the power supply both from device switching lossesand in the transformer from eddy currents. The switching losseshave been reduced by using a resonant type converter and other techniques such as zero voltage and / or zero current switching. As the power transformer size is reduced by increasing the frequency,eddy currentsbecomeone of the major limiting factors. Eddy currents in the transformer'core can be reducedto an acceptablelevel by using high quality core materials.The windings eddy currents are the most complex to analyse. The winding ac resistanceand leakage inductanceare strongly related to the eddy current effects. The term ac inductanceis used to point to the fact that effective inductance( self and mutual ) is reducedas the frequencyrises.Practically, there will alwaysbe a frequencyat which eddy currents becomesignificant, and the conductor current is forced by its own magneticfield towards the surf ace.When the inductance falls becausethe areaof the flux path is reduced,the resistanceincreases. The two main effects that dominate winding eddy currents losses are skin and proximity effects. The self induced eddy currents in one conductor is called the skin effect eddy current. The proximity effect is due to the flux linking surrounding conductors which causesa circulating current in adjacent conductors. In addition, since the power supply topology at high frequency is resonant, the transformer parasitic elementsform the secondmajor design challenge.These elementsare also resonantin nature and their minimisationis vital to reducenoise,EMI, and stress.
Attempts to obtain an accuratefrequencyresponseby detailed transformer modelling has attracted the attention of researchers.In parallel to experiment developments level high is of a with various test methods, numerical computation also attracting field is One computation using of the computing methods electromagnetic interest.. the finite elementmethod. Returning to the main purpose of this work which is to study the winding eddy The to their effects. transformer nihmise so as elements parasitic currents and high frequency implementation power supply and transformer testing a of practical built in A two. mode series resonant converter was clamped was carried out chapter to justify the simulation results. Full design of transformer and power supply was given in this chapter as well as a linear power amplifier which was used to test the high frequencytransformerpractically. The magnetic field is the main source of eddy current generation and so two dimensional finite element analysis was used (chapter three). Two dimensional analysiscan analysethe effects of eddy currents, but for a limited bands only, or more specificallyuntil the appearanceof the capacitive effect. The conductors in the model were consideredas having a squarecross sectionalarea equal to that of round from to the manageable a meshingview point. In this wire so as make problem more analysis care had be taken to model correctly the rapid decay in the field at the conductors boundaries.These boundarieshave to be modelled with as many small elementsas possible.Again, the depth of the modelledE-core viewed from the fr6nt is larger than that viewed from the sides. That means the area available for the magneticpath is different. Therefore, two modelswere used within the finite element computation. This has lead to the numericalresults to be improved by 2% at 1KHz, and 8% at IMHz in comparisonwith using one model only. The transformer elements( resistanceand inductance) can be computed both from short and open secondarywinding computations. The finite element derived short circuit flux distribution (chapter three) shows that as the frequency is increased,the flux is concentratedin the spacebetween the windings ( primary and secondary). 8-2
This reflects losses due to the proximity effect and also changes in the leakage inductance.Two dimensionalanalysiscan be used within the frequency range of dcMHz. At higher frequencies,the capacitive effect cannot be neglected and so two dimensionalanalysisis no longer valid. The results also show that when the space between windings is half of a certain limit, the losses will be three times more (chapter three). When the spaceis doubled, the proximity effect is reduced but the leakageinductanceis increasedby almost one and half times its original value. The investigationcontinuesto suggestthat the best spacebetweenboth windings so as to optimisethe conditions and reduceto a minimum the lossesand leakage,is when the space (between windings) is equal to or less than the thickness of the windings conductors.The investigation also includes the skin depth effect in the conductor ( wire ). The results show that when this depth is much greater than the depth of the wire, the magnetic field varies across the conductor. This field causesdifferential voltage in the wire and the resultant eddy currents are insufficient to redistribute the field. When the frequencyincreasesto the level that the skin depth is much smaller than the wire thickness,the current inside the wire will be non uniformly distributed. Hence, the current density falls exponentially within the wire. In addition, as the frequencyis increased,the actual area of the wire that carries current is reduced and then the ac resistanceis far higher than dc. The finite elementtechniquewas used to estimatean optimum value of wire size that provides the best compromisebetween the numberof wires and ac resistance.The investigationhas lead to the point that the best ac to dc resistanceratio is achievedwhen the wire thicknessis 1.2 times the skin depth. This result is supportedanalytically.It is vital to subdividethe wire into many fine wires so as to keep the eddy currentsto a minimum, and then eachof thesewires is to occupy equally every position in the bundle. The problem of keeping both the leakageinductanceand ac resistanceto reasonable limits at a certain frequency was also investigated. During the examination of this effect due to the increasingthe numberof layers, curves were computed to show the ratio of ac to dc components of leakage inductance and resistance.These values 8-3
in frequency, term of conductor size as a ratio to presented and were given against the skin depth. Using thesecurvesis vital for the pre-designstage of the transformer. From these curves the required number of layers and how many fine wires at a certain frequencycan found directly. The results show that an increasein the number inductance in in layers increase the the ac resistance and a reduction of can give an ac; per turn as comparedto dc values.It should be noticed that increasingthe number of layers actually increasesthe copper content with respect to the air gap in the transformer window. When the field becomes skin limited there is less space available for the leakageflux and so it gives a reduction in the inductance. In the sameway, the layerswill lie close to eachother and so the ac resistancewill increase due to the proximity effects. The core properties were also investigated with the secondarywinding was open circuited. The results (not surprisingly) show that the self inductance is sensitiveto the core permeability, while the resistanceis sensitiveto the core conductivity. As the permeabilitywas reduced,the frequencyat which the gelf inductancecurve falls is reduced. This frequency representsthe maximum frequency at which the core could realisticallybe used, and also showsthat skin effect starts to dominate.For the best materialsavailable,the megahertzzone representsthe areawhere the skin effect eventuallysetsin. At a certain operating frequency(say I M11z),two dimensionalanalysisis no longer useful. Becausethe transformeris a three dimensionaldevice in nature where there is almost no axis of symmetry,both magneticand electric fields are required to model the transformer, and they are working in a normal senseto each other. A three dimensionaltransformer model was built and solved using the full set Maxwell's of equation,this was describedin chapter four. It is believedthat this model is original. The transformer parasitic elementsand their effects was the next concern of this work, where this three dimensional model was used to compute the frequency responsecharacteristicsof the transformer. These characteristicsinclude the open and short circuit impedancecomputation.The computed results were comparedwith 8-4
is it This that model proves the actual results and good agreementwas achieved. in be investigation tool future to a good and promises capable of handling any designinghigher frequencytransformers. The modellingof the transformertransient characteristicwas given as much attention frequency high frequencies, five). At (chapter frequency response as was given to the due the is transient, to the than of appearance more effective responsecomputation capacitive effects. During the frequency response, the transformer was modelled during be frequency This the transientwhere all rangecannot specified across range. the transformernatural frequenciesare excited at once. Therefore a very high rate of instruments limits is the the this analysis as well as practical sampling required and introducing frequency As time the without and steps available. an example, maximum 150Hz frequency, 0.008psec low 3Hz 2psec at and and aliasing error was at and high frequency. If this figure is considered, the number of samples required for transientcalculationis about 500000 ( this figure is approximatedwhere the number is high by ), be two any means. very of samples should which a power of Nevertheless,the transient computation method used currently can easily handle a for This the equivalentnetwork solution time nanosecond method step computation. usesthe trapezoidalrule of integration. Each of the !nductancesor capacitancesthat exist in the network is transferred to an equivalent circuit of current source and resistance.The whole network looks like a resistive network that can be solved by well known methods. The winding is divided into a number of sections, each is representedby self and mutual inductances,distribution and ground capacitances, and resistancesto take the lossesinto account. Theseelementswere calculatedusing the finite element method. The transformer winding was transferred into an equivalentnetwork. The first transient effect to be examinedin this network was the over voltage stress. This stress is well known in industry and is subjected to an impulse signal to examining the winding insulation. The results showed that the voltage in the mid point of winding goes higher than the terminal voltage. It was believed that this resonant phengmenais related to the relative location of the 8-5
to Reducing stress the voltage allows over can capacitance winding poles and zeros. be reduced. The ground capacitancecannot be reduced much because it would involve using a bigger core, and,so it is meaninglessto increasethe frequency for frequency terminal the The of computation response transformers. size small impedanceconfirms the need to reduce both ground and distribution capacitances. This response calculation also shows that the network response at which the Nevertheless, from is different the accuracy of the actual. resonanceoccurs always the calculationis believedto be limited by the way of representingthe winding by an equivalent circuit and the number of elementsinvolved. The disadvantageof this include huge is it to the that capacity admittance method requires more computer it but is The transformer, to the whole matrix. method expensivewhen used model can be usedto study the transientphenomenain the winding. The frequencyresponsecharacteristicswere used to derive an equivalent circuit of the transformer. The elementsof this circuit were found from the poles and zeros that the impedancecurves have shown. The simplified equivalentcircuit was derived (chapter four) first and under the sameprinciple, a wide frequencyband transformer equivalentcircuit was derived (chapter seven).The simplified equivalent circuit was used to examinethe effect of transfqrmer parasitic elementson the power. supply performance(chapter six). The Spice program was used to simulate the converter model and to examinethe transformer element'sbehaviour and their interaction with the resonant tank. Since the transformer elements are resonant elements, any reduction of any of them will be at the expenseof the others, and so the elements need to be examinedindividually during the simulation. The transformer equivalent circuit was placedin series,and betweenthe resonanttank elementsof the converter. The normal operatin'gwaveformsof the converter during the simulation is justified as it correspondedwith that actually obtained using the practical power supply. The converterwas simulatedaboveand below resonantfrequencyoperation When the magnetising inductance ( Lm ) was removed, the voltage across the resonanttank was reducedto 72% and increasedto 120% of the normal operation at 8-6
below and above resonant respectively. When Lm is removed, the ground capacitance( Cg ) becomespart of resonanttank capacitor (Cr) and so the resonant frequencyis reduceddue to the increasein the capacitancevalue. This frequency is then further away from the series resonant frequency for the below resonant case, in This it for to the results the voltage decreasingand and close aboveresonantcase. increasingfor the below and above casesrespectively.Removing Lm also shows that an oscillation exists in the winding current, where the resonantfrequency is near to dc, and so the operating frequency is much lower than resonant. The same and opposite situation happens when Cg is removed. The resonant tank voltage is increased to 110% and reduced to 70% below and above resonant frequency respectively.Removing Cg has shown a reduction in the winding current oscillation and it no longer existsabovethe resonantfrequency. The simulation waveforms of input/ output currents and voltages have shown less transformer elementseffect on the power supply performance above resonancein comparisonwith below resonance.The ratio of switching to resonantfrequency has also beenconsidered,where the current researchsupports using a ratio of 0.8 below resonantand 1.2 above resonant.When the ratio is reduced further than 0.8, partial charge and discharge (which can be seen clearly in the resonant.tank voltage waveform) will increase.In the same way, as the ratio goes higher than 1.2, the oscillation in the winding current will increase. The effects of eachof the elementson the output voltage have been investigatedand given as graphs. These graphs have shown that any increasesin Cg will result in a reduction in the output voltage. This reduction is greater as the frequencyratio goes lower than 0.8, but above resonanceit is much lower than below resonance.When Lm is changingat a resonantratio of 0.8 and 0.4, both caseshave shown a point at which the output reachesa maximum.This point is different at the resonantratio of 0.8 than it is at 0.4 and this meansless air gap in the core is required as the ratio increases.This does not occur in the case where the converter is operating above resonant frequency. There is no significant effect on Lm if Cg is removed. The 8-7
simulation results confirm that for the best performance the reduction in the distribution capacitancehasto be linked with a correspondingreduction in Cg. The frequencyresponseimpedancescan give all the information required to derive an equivalent circuit of the wide frequency band transformer (chapter seven). The equivalentcircuit is used to reproduce the frequency responseand good agreement was clearly obtainedwith the measuredand finite elementresults.
A. l: TWO DIMENSIONAL FINITE ELEMENT PROGRAM
is in finite The magneticvector potential (A) w1-: the tool the main tich elementprogram can be expressedin two equations.One from the divergenceof A which is always equal to zero everywhere,and secondas the curl of anothervector function which is the flux density. VxA=B
By taking the curl of equation ( 1) and consideringthe relation between flux density B and flux intensityH as B=pH(
is hystereses ), the result neglect
The equationto be solvedthen is VxI
J is the current density and an important equationcan be derived to representits value. The ,0B which differential form of Maxwell's equation describingthe electric field is VxE= -, t 10 states that the curl of E is not zero, therefore E= -V V is not enough as the case of Electrostaticfield. So this equationcan be arrangedto 0 A) VxE= -V x0A E+ =0 =>Vx ,9tat The curl of the terms in the parenthesesequal to zero, therefore, it equalsto the gradient of a scalarfunction. E+0A= 0t0t
E0A_VV V => -V
In the conductor,the relationbetweenthe electricfield intensityandcurrentdensitycanbe definedby ohm'slaw (J= A *? c j=_a UVV
aE ). Equation 6 can be written as: (7)
The final equationto be solvedthen is
In orderto solvethis equation,the methodof Weightedresidualwasusedin the program. Togetherwith the integrationoverthe volumeof the problemwhichis givenas fff WVx (9) dV=O VxA-J 9 have. J A dimensions equation In the two only z-direction components, and where length to : equivalentper unit ffw Vx VxA. -J, dxdy=O VxVxA-
6 (. 0 Az 19xy0 '0
(10) (46 A,
By integratingthe aboveequations( 10 &II) by parts ( Green!s Theorem ), the result is : I ffw (12) dt =0 w10A' Vx VxA, -J. dxdy+f On Pp a boundary Where the to the surfaceof the xy plane, so represent normal component On I OA, I (13) Ht -, On = -B,, = p P Within equation(12), the componentof the current density can be simplified using the third is: Maxwell, equationof which 0B (V x A) VxE=0tt
The integration of this equationfor a singlefrequencyvariation is : E=-j o)A +VV Where V is the integration constant. This equation shows that the electrical field comes from two components,one induced in a conductor by changein the magneticfield which is the first part of equation(15), and the secondarisefrom immersionin the electrostaticfield. The basic step of the finite elementis to break up the area of interest into small elements,
is the within eachof which vector potential definedat the vertices( nodes). In the case I the shapeof elementsare triangular, the vector potential is: whereZNkAk A, = (16) k-1.3
by the to consideringthe weighted term Galerkin problem The techniquewas used simplify formulation The the of whole the considered. element of
W as the shapefunction (N)
is region that containsm numberof elements : [ff I ONj P
ff Njaj 1: co
I] k-1,3 19Y 1: ff Nia VV dx dy
dx dy +
Ni J,. dx dy -
Ni H, dC -=0
A. 2: THE USEFULNESS OF THE MAGNETIC VECTOR POTENTIAL. As explainedearlier in chapterthree, the main parameterto be solved for by FE analysisis the magnetic vector potential (A),
and all of the calculated quantities required can be
found directly from it. The singlevalue of A( in z direction ) can be calculatedat each node is induced from flux The then the the current resulting easily obtained and within model. definition of A. The magneticflux can then be usedto predict voltage and inductance. The relationshipbetweenthe magneticflux and A can be given as: f T" =fB. ds= VxA
This surface integral can be replaced by a line integral enclosing the surface theorem), T, =fA. dt
In the caseof steadystate two-dimensional analysiswith no capacitive effect, the complex ( Faraday's Law, be together with equation which value of the voltage can calculatedusing 2) gives: A. df V= -N 40ýv '= -N 01 01 . .
whereN is the numberof turns. The inductance can be found from the definition of the flux linkage due to the current flowing in the conductor. L=AII=Nlýll
If bothcurrentandflux in equation(4) belongto the samecoil, the inductanceterm is the 0 self inductance.If the flux in the coil is due to the anothercoil current,it is the mutual inductance. The resistancecomponentsof either dc or ac can be found from the value of A. The dc component does,not need to be found by FEA, since it is given in the wire specification data. However, it can be calculatedby consideringthe frequencyto be zero ( i.e. static ) for a uniform conductor cross section area. In the case where the conductor area is not uniform, then a steadycurrent solution is required. The ac component due to
frequency increases,can be found from the losses in the
conductor,wherethe conductorcurrentdensity(J) is: PotOA_VV
Thepowerlossperunit metrein the two dimensional is analysis. then: ' P=lj2
The figure shows a simplified equivalentcircuit suitable for transient analysis.The winding to winding capacitanceis not included for simplicity. During the simulation of any circuit, into is transferred the an equivalent current source and constant each of circuit elements in detail in it is here for five, This technique explained chapter resistance. was and repeated completeness. il
44ZD12--li12-1ý1-ZDI3 "' vi Rl
The circuit can be analysedusing the input output relations as follow: ijR, + E, = V, +IIRI '2R2+ n, i,
n2E, - n,
Theserelationscan be arrangedin a matrix form as follow [II 10 [1 1] [RI R, 0]I I] 0 R2 [i(t)] R2 [0 =I10 [0n, n2] 01 [0 0 E(t) 0
[1 V(t) (t - At)]
If the input voltage is given, the matrix can be used to solve for input and output current. The matrix can be simplified in final form and given as NO]=
Where Y is a matrix contain the known resistancesand turn ratios. Further details of this method can be found in many publicationsas for instanceChimklai, et al .
REFERENCES I-P. L. Dowell, " Effect of eddy currentsin transformerwinding.", IEE Proc., Vol. 113, No. 8, Aug. 1966,PP. 1387-1394. 2- P.Silvester," Network analogsolution of skin and proximity effect problem.", IEEE Trans. on PAS, Vol. 86, No. 2, Feb.1967,PP.241-247. 3- P.Silvester," Ac resistanceand reactanceof isolated rectangularconductor.", IEEE Trans. on PAS, Vol. 86, No. 6, June 1967,PP. 770-775. 4- P.I. Fergested, and T.Henriksen," Transient Oscillation in mutiwinding transformers.", IEEE Trans. on PAS, Vol. 93, March 1974,PP. 500-509. 5- R.Kasturi, G.R.K. Murty, " Computation of impulse voltage stressesin transformer windings.", IEE Proc. Vol. 126,No. 5, May 1979,PP. 397-400. 6- C.E.Lin, C.L. Cheng, et al." Investigation of magnetizing inrush current in transformers.", IEEE Trans on PD, Vol. 8, No. 1, Jan.1993,PP. 246-253. 7- W.J.McNutt, T.J.Blalock, et al.," Response of transformer winding to system transientvoltage.", IEEE Trans. on PAS, Vol. 93, March 1993,PP. 457-467. 8- Adielson T., Carlson A., et al, " Resonant over-voltage in EHV` transformers, modelling and application''JEEE Trans., Vol. PAS-100,1981, PP. 3563-3572. 9- R.C.Degeneff," A general method for determining resonancesin transformer windings.", IEEE Trans. on PAS, Vol. 96, No. 2, April 1977,PP. 423-430. 10- P.T.M. Vaessen," Transformer model for high frequencies.", IEEE Trans. on PD, Vol. 3, No. 4, Oct. 1988,PP. 1761-1768. * I I- Keyhani,A., Chua S.W., et al, " Maximum likelihood estimation of transformer high frequencyparametersfrom test data ", IEEE Trans. on PD, Vol. 6, No. 2, April 1991,PP. 858-865. 12- Chimklai S., Marti J.R., Simplified three phase transformer model for electromagnetictransient studies IEEE Trans. on PD., Vol. 10, No. 3, July 1995,PP 1316-1325. 13- O.C.Zienkiewicz, " The finite elementmethod.",3rd edition, McGraw-Hill, Newyork, 1977. 14- R.L. Stoll," The analysisof eddy currents.", Oxford, 1974 15- O.W.Anderson," Transformer leakageflux programn basedon the FEM. ", IEEE Trans. on PAS, Vol. 92, July.1973,PP.682-689. 16- M. V. Iý.Chari," FE solution of eddy current problem in magneticstructures.", IEEE Trans. on PAS, Vol. 93,1974, PP. 62-72. 17- A. Konrad," Eddy currents and modelling.", IEEE Trans. on Mag., Vol. 21, No. 5, Sep.1985,PP. 1805-1810. 18- S.Yamada, and E.Otsuki," Analysis of eddy current loss in Mn-Zn Ferrites for power supplies.", IEEE Trans. on Mag., Vol. 3 1, No. 6, Nov. 1995,PP. 4062-4064. 19- A. W.Lotfi, F.C.Lee, et al." Two dimensionalskin effect in power foils for high frequencyapplication.", IEEE Trans. on Mag., Vol. 31, No. 2, March1995, PP. 10031006. 20- J.H.McWhirter, " Computation of three dimensional eddy currents in thin conductors.", IEEE Trans. on Mag., Vol. 18, No. 2, March 1982,PP. 456-460. 21- M. J.Balchin, J.A. M. Davidson," Numerical method for calculating magnetic flux and eddy current distribution in three dimensions.", IEE Proc., Vol. 127, Part-A, No. 1, Jan. 1980,PP. 46-53.
22- Y. Ohdachi, Y. Kawase, et al.," Load characteristics analysis of coupling transformer using 3D finite element method with edge elements.", IEEE Trans. on Mag., Vol. 30, No. 5, Sep.1994,PP.3721-3724. 23- Tutorial by Kriezis, Tsiboukis, et al.," Eddy currents, theory and applications.", IEEE Proc., Vol. 80, No. 10,Oct. 1992,PP. 1556-1588. 24- Y. Tanizume, H. Yamashita, et al.," Tetrahedral elements generation using topological mapping and spacedividing for 3D magneticfield FEM. ", IEEE Trans. on Mag., Vo126,No. 2, March 1990,PP. 775-778. 25- O.Biro, K. Pries," Finite element analysisof 3D eddy currents.", IEEE Trans. on Mag., Vol. 26,1990, PP.418-423. 26- M. P.Perry," Multiple layer seriesconnectedwinding designfor minimum losses. IEEE Trans. on PAS, Vol. 98, No. 1, Jan.1979,PP. 116-123. 27- B.Beland," Eddy current in circular, squareand rectangularrods.", IEE Proc., Vol. 130,No. 3, Part-A, mayl983, PP. 112-121. 28- J.Wiess, V. K. Garge,et al." Eddy current loss calculation in multiconductor system.", IEEE Trans on Mag., Vol. 19,No. 5, Sep.1983,PP. 2207-2209. 29- P.Chowdhuri," Calculation of series capacitance for transient analysis of windings.", IEEE Trans. on PD, Vol. PWRD2, No. 1, Jan.1987,PP. 133-138. 30- J.Vandelac, and P.D. Ziogas," A novel approach for minimizing high frequency transformercopper losses.", IEEE Trans. on PE, Vol. 3, No. 3, July 1988,PP. 266-277. 31- I. R.Ciric," Proximity effect between a wire carrying current and a rotating cylindericýdconductor.", IEE Proc., Vol. 130, Part-A, No. 3, may 1983,PP. 122-128. 32- S.Isaka, T. Tokumasu, et al.," Finite element analysis of eddy currents in transformer parallel conductors.", IEEE Trans. On PAS,.Vol. 104, No. 10, Oct. 1985, PP. 2731-2737. 33- A. F.Goldberg, T. G. Kassakian, et al., " Issues related to I-l0MHz transformer design.",IEEE Trans. on PE, Vol. 4, No. l, jan. 1989,PP. 113-124. 34- D.J.Wilcox, W.G.Hurley, et al.," Calculation of self and mutual impedance between sections of transformer windings.", IEE Proc. Part-C, Vol. 136, No. 5, Sep.1989,PP. 308-314. 35- D. J.Wilcox, " Theory of transformer modelling using modal analysis.", IEE Proc. Part-C, Vol. 138, No. 2, march 1991,121-128. 36- K. van der Wint, " Interaction between a series resonant converter and a transformer.". EPE Conference,Firenze 1991,PP. 2.1-2.7. 37- V. Woivre, J.P.Arthaud, et al." Transient over voltage study and model for shell , type power transformer.", IEEE Trans. on PD, Vol. 8, No. 1, Jan.1993, PP. 212-219. 38- A. Morched, L. Marti, et al." A high frequencytransformer model for the EMTP. ", IEEE Trans. on PD, Vol. 8, No. 3, July 1993,PP. 1615-1624. 39- F.De leon, and A. Semlyen," Detailed modelling of eddy current effects for transformertransient.", IEEE Trans. on PD, Vol. 9, No. 2, April 1994, PP. 1140-1150. 40- A. Ahmad,P.H. Auriol, et al.," Shell form power transformer modelling at high frequency.", IEEE Trans. on Mag., Vol. 30, No. 5, Sep.1994,PP.3729-3732. 41- B. Ahmed, J.Ahmad,et al.," Computing Ferrite core losses at high frequency by finite elementsmethodincluding tempratureinfluence.", IEEE Trans. on Mag., Vol. 30, No. 5, Sep.1994,PP. 3733-3736. 42- A. Basak, Y. U. Chi-Hang, et al." Effecient transformer design by computing core loss using a novel approach.". IEEE Trans. on Mag., Vol. 30, No. 5, Sep.1994, PP. 3725-3728.
in hysteresis losses dynamic " Asimple Semlyen, A. leon, representationof 43- F.De and PP. 315-321. No. Jan. 1995, Vol. 10, 1, PD, Trans. ", IEEE on power transformers. high " A the the to X. Tian, of B. Tala-Lghil, effects 44new method eliminate et al., Conference, in ", EPE supply. voltage transformer parasitic elements a static resonant Firenze 1991,PP. 2.468-2.473. 45- J.A. Sabate, and F.C.Lee," Off-line application of the fixed frequency clamped PP. 396, No. Jan. 1991, Vol. 1, PE, Trans. ", IEEE on mode seriesresonantconverter. 46. 46- D.T.Khai, t al. " Modelling of magnetizinginductanceand leakageinductancein a April 1993, PP. 200-207. No. 2, PE, Vol. 8, Trans. ", IEEE transformer. on matrix 47- R.Farrington, M. M. Jovanovic,et al." A new family of isolated convertersthat uses the magnetizinginductance of the transformer to achieve zero voltage switching IEEE Trans. on PE, Vol. 8, No. 4, Oct. 1993,PP. 535-545. 48- M-M-Jovanovic, A. Wojciech, et al." High frequency off line power conversion using zero voltage switching quasiresonant,and multiresonant techniques.", IEEE Trans. on PE, Vol. 4, No. 4, Oct. 1989,PP. 459-469. 49- K. Chen, and T. A Stuart, "A1.6KW, I IOKHz, dc-dc converter optimized for IGBT's.", IEEE Trans. on PE, Vol. 8, No. 1,1993, PP. 18-25. 50- AbrahamI. Pressman! 'Switching Power Supply Design'', McGraw-Hill, 1991. 51-" Soft Ferrites, data handbook", Philips component,MAO 1,199 1. 52- " Unitrod switching regulatedpower supply design ", Seminarmanual,USA, 1993. 53- S.M. Miri, T.U.Feng, et al., " Design of low leakagepower supply transformer for high precision electronic instruments.", IEEE Trans. on IM., Vol. 42, No. 4, Aug. 1993, PP. 854-859. 54- IR, " Product selectorguide, power MOSFET ", HEXFET data book, 1985. 55- Y. Chin, and F.C.Lee," Constant frequency parallel resonant converter."JEEE Trans. on IA, Vol. 25, No. 1, Jan. 1989,PP. 133-142. 56- Bhat,A.K., " Analysis and design of series-parallelresonant converter", IEEE Trans. on PE, Vol. 8,No. Ijan. 1993,PP 1-7. 57- P.H. Swanepoel, J.D. Vanwyk,et al.," transformer coupled direct base drive technology for high power-high voltage bipolar transistor PWM converter,", IEEE Trans. on IA, Vol. 25, No. 6, Nov. 1989,PP.1158-1166. 58- F.U. Cheng-Tsai," Small signal and transient analysisof a zero voltage switched, phase-controlledPWM converter using averaged switched model.", IEEE Trans. on IA, Vol. 29, No. 3, May 1993,PP. 493-499. 59- K. W. Cheng, and P.D.Evans," Calculation of winding losses in high frequency toroidal inductors using multi-strand conductors,", Birmingham University, EPA1771, Sep.1993. 60- A.Hund, " High Frequencymeasurements.", 2-nd edition, McGraw-Hill, London, 1951. 61- V. J.Thottuvelil,T. G.Wilson, et al., " High frequency measurmenttechniques for magneticCores.", IEEE Trans. on PE, Vol. 5, No. 1, Jan.1990,PP. 41-48. 62- Hurtley W.G., et al, " Calculation of leakageinductancein transformer windings IEEE Trans. on PE, Vol. 9, No. 1, Jan. 1994,PP 121-126. 63- Q.Su, R.E.James,et al. "A Z-transform model of transformer for the study of electromagnetictransients in power systems.", IEEE Trans on PS, Vol. 5, No. 1, Feb.1990,PP. 27-33.
by " Diagnosis transformer the A. Arri, windings 64- E. state of power of carta, et al., April No. Trans. 2, Vol. ", IEEE IM, 42, on on-line measurementof stray reactance. 1993,PP. 372-378. 65- Kraus, J.D. " Electromagnetics ", Fourth edition, McGraw-Hill International edition, 1991. 66- D. S.Kershaw," The Incomplete CholeskyConjugateGradient Method for iterative 43-65. Vol. PP. Phys., 26,1978, ", Journal linear comp. equations. solution of systemof 67- O.H.Zinke and W.F.Schmidt," Linear ac magneticcircuit theory.", IEEE Trans. on Mag., Vol. 29, No. 5, Sep.1993,PP. 2207-2212. 68- A.Konarad, M. V. K. Chari,et al.," New finite element techniques for skin effect PP. No. March1982, 450-455. Vol. 2, Mag., 18, ", IEEE Trans. problems. on 69- A. F.Goldberg, T. G.Kassakian,et al." Finite element analysisof copper loss in IIOMHz transformers.", IEEE Trans.on PE, Vol. 4, No. 2, April 1989,PP. 157-167. 70- C.F.Bryant, B.Dillon, et al.," Solving high frequency problems using the magnetic vector potential with Lorentz gauge.", IEEE Trans. on Mag., Vol. 28, No. 2, March 1992,PP. 1182-1185. 71- W.G.Hurley, and M. C.Duffy, " Calculation of self and mutual impedancein planar magneticstructures.", IEEE Trans. on Mag., Vol. 3 1, No. 4, July 1995,PP. 2416-2423. 72- Masourn M. A. S., et al, " Transformer magnetisingcurrent and iron core lossesin harmonicpower flow", IEEE Trans. on PD, Vol. 9, No. 1, Jan.1994,PP 10-20. 73- P.Fernandes,P. Girdinio, et al., " Local error estimates for adaptive mesh refinement.", IEEE Trans. on Mag. vol.24, No. 1,198 8, PP. 299-302. 74- C.F.Bryant, C.R.I. Emson, et al.," A comparisionof Lorentz gaugeformulations in eddy current computations.", IEEE Trans. on Mag., Vol. 26, No. 2 Marchl990, PP. 430-433. 75- M. Ehsani,O.H. Stielau, et al.," Integrated reactive componentsin power electronic circuits.", IEEE Trans on PE, Vol. 8, No. 2, April 1993,PP. 208-215. 76- T. fawzi, P.E.Burke," The accuratecomputation of self and mutual inductanceof circular coil.", IEEE Trans. on PAS, Vol. 97, No. 2,1978. 77- D. A. Calahan," Computer aided network design.", Revised edition, McGraw-Hill, 1972. 78- L. O.Chua, and P.M. Lin, " Computer aided analysis of electronic circuits.",PrenHall, 1975. 79- F.H.Branin," Computer methods of network analysis.% IEEE Proc., Vol. 55, No. 11, Nov. 1967,PP.1787-1800. 80- W.F.Tinney, and J.W.Walker," Direct solution of sparse network equations by optimally ordered triangular factorization.", IEEE Proc., Vol. 55, No. 11, Nov. 1967, PP. 1802-1807. 81- W.D.Hermann," Digital computer solution of electromagnetictransient in single and multiphasenetwork.", IEE Trans. on PAS, Vol. 88,1969, PP.388-393. 82- Hosseinian S.H., " Improvements in simulation methods for power system harmonicand transientstudies",Ph.D. Thesis, 1996,NewcastleUniversity,UK. 83- Muhammed A.H., " Calculation of transient fields in turbogeneratorsusing finite elementsand Fourier transfornf', M. Phil Thesis, 1993,NewcastleUniversity, UK. 84- R.Kelkar, R.Wunderlich, et al.," Software breadboards for power electronic circuits.", IEEE Trans. on PE, Vol. 6, No. 2, April 1991,PP. 170-177. 85- "Mospower application.", Siliconix Incorporated, data book, USA 1985. 86- K. Shenai," A circuit simulation model for high frequency power mosfets.11,IEEE Trans. on PE, Vol. 6, No. 3, July 1991,PP.539-547. iv
87- B. Cogitore, J.P.Keradec,et al.," The two winding transformer, an experimental On IM, frequency ", Trans. IEEE to circuit. range equivalent method obtain a wide Vol. 43, No. 2, April 1994,PP. 364-371. 88- S.Chimklai, and J.R.Marti, " Simplified three phase transformer model for ", PP. IEEE Trans. PD, Vol. No. July 1995, 10, 3, transient studies. on electromagnetic 1316-1325. 89- de Leon,F.,and SemlyenA., " Reduced order model for transformer transiente', IEEE on PD,Vol. 7, No. 1, PP. 361-369, Jan. 1992. 90- A. Keyhani, S.W. Chua, et al.," Maximum liklihood estimation of transformer high frequency parametersfrom test data.", IEEE Trans.on PD, Vol. 6, No. 2, April 1991, PP. 858-865. 91- Gupta S.C.," Delta function ", Math. of Comput., Vol. 24,1964, PP 16. 92- I. Marinova, Y. Midorikawa, et al. " Thin film transformer and its analysis by integral equationmethod.", Trans. on Mag., Vol. 3 1, No.4, Julyl.995, PP. 2432-2437.