OF LOW PROFILE TRANSFORMERS FOR HIGH FREQUENCY...example, a high frequency, low volume,...

DESIGN OF LOW PROFILE TRANSFORMERS FOR

HIGH FREQUENCY OPERATION

by

Valentin Bolborici

A thesis submitted in confonnity with the requirements

for the degree of Master of Applied Science

Graduate Department o f Electricai and Cornputer Engineering

University o f Toronto

O Copyright by Valentin Bolborici 1 999

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Design of Low Profile Transfomers for High

Frequency Operation

Master of Applied Science, 1 999

Valentin Bo lborici

Graduate Department of Electricai and Computer Engineering

University of Toronto

Abstract

This thesis is concemed with the design of low profile transfomrs for high

fiequency operation that are used in dc/dc resonant converters.

The tirst objective of this work is to examine three different available design methods

for high fiequency transformers, and to make a comparison of the results for a specific

design A comparison with the Finite Eiement anaiysis results is also done in order to see

if these methods are accurate in terms of predicting losses.

The second objective of this work is to develop an algorithm that generates design

curves for a class of high fiequency transfomers that are height constrained. The key

starting point in developing this algorithm is that an optimal design can be achieved when

copper and core losses are in a certain ratio dictated by the constants that characterize the

specific loss curve of the magnetic material.

A set of design curves, developed using the proposeci algorithm for a specific power

level, is presented at the end of this thesis.

Design of Low Profile Transfwmers for High Frequency Operation University of Toronto

Ac kno wledgements .-. 111

Acknowledgements

The author wishes to express his sincere gratitude and appreciation to his supervisor,

Professor J.D. Lavers, for the opportunity to perfonn this work and for his guidance

thro ughout the preparation of this thesis.

1 would also like to thank to Professors P. Jain, J.D. Lavers and F.P.Dawson for

granting me financial support for this work

The author also extends gratitude to his fiend Philippe Blanchard for usefid discussions

during the preparation of this thesis.

Design of Low Rofile Transformas for High Frequency Operation University of Toronto

Table of Contents iv

Table of Contents

. . Abstract ...................,.,.......................................................................................................... il

... Ac knowledgements ...... ... .. .. ........................................................................................ 111

Table of Contents .................................. ............................. ........................................... iv

Nomenclature ........ .,. ...................................... ..... .............................................................. .vi

Chapter 1 Introduction ................................................................................................... 1

....................................................................................................... 1.1 Motivation 1

1.2 Thesis Focus ........................... .. ......................................................... 5

1 -3 Thesis Overview ....................................... ...,, ................................................. -7

Chapter 2 Identification of the Problem. ........................................................................... 9

2.2 Introduction .................... --. .... ,.. ....... .,. ...... 9

2.2 Area Produ ct ............ .,. .................................................................................. 9

2.3 The Choice of the Magnetic Material and Flux Density .................................. 12

.................................................... 2.4 The Choice of the Current Density 1 5

2.5 Summary ....................................................................................................... 17

Chapter 3 Revue of Available Design Methods ............................................................. 18

3.1 introduction .................... ... ... .. ............. .... ........... 18 3 -2 The Classicai Design Approach .................... ... ......................................... 19

3.3 Classical Design Approach with Improved Estirnate of Coil Loss .................. 27

3.4 Optimized Transformer Design Method ......................................................... 30

3.5 Cornparison of the Results Using the Design Methods Describeci in Sections

3.2, 3.3, and 3.4 ................................. .., ........................................................ 37

3.6 Conclusions ................................................ .... Chapter 4 Design Curves for High Frequency Transfoma ..................... .. ............... 43

4.1 Introduction ................................................................................................... 43

4.2 The Choice of the Core and its Geometrïcai Dimensions ................................ 44

Design of Low Profile Transfixmers for High Frquency Operation University of Toronto

Table of Contents v

Chapter 5

Appendix

Appendix

Appendix

Appendix

Appendix

Appendix

The Expression for Core Loss ........................................................................ 46

The Expression for Copper L o s ................... ... .......................................... 5 1

Calculation o f the Power Density and Current Density as a Function of

Transformer Height ....................................................................................... 54

Example of a Set of Design Curves ................................................................ 56

4.6.1 VariationofPowerDensitywithHeight ............ ........ .................... 57 ........................................... 4.6.2 Variation of Mount@ Area with Height -63

.......................................... 4.6.3 Variation of Current Density with Height .69

4.6.4 Validation of the Design Curves ......................................................... 75

Summary ..................... .. .......... .... ............................................................. 76

Conclusions and Fu- Work ........................................................................ 77 Conclusions .................................................................................................. 77 Future Work .................................................................................................. 79

Calculation of Window Utilizat ion

The value of coefficients Kj9 I(,. x

......... Factor Ku ...................................... 80

and y for dif5erent core co~gurat ions and

temperature bcreases ................... ... ......................................................... 82

Values of H and K factors .............................. .. ............................................ 83

Temperature rise AT versus -ce dissipation Y ........................................ 84

Mode1 Used in the F . E . Analysis of the Transformer ................... .. ........... -85

............................................. C haracteristics of K and F Magnetic Materials 86

......................................................................................................................... Re ferences 88

Design of Low Profile TfatlSformers for High Frequency Operation University of Toronto

Nomenclature vi

Nomenclature

Coefficients that descri'be the l o s curve for a ferrite.

Alternathg current.

Effèctive cross-section area of the center post.

Effective cross-section area of the selected cote-

Mounting area of a transformer-

Area produt of the core.

S&e area of a wound transformer.

Cross-section area of the single equivaient turn.

Cross section area of a wire filled with copper.

Necessary cross-section area of the wire used in the primary winding.

Necessary cross-section area of the wire used in the secondary winding.

Breadth of a square strand-

Magnetic flux density in the center pst.

Operating magnetic flux density in tesla unless specified.

Optimum magnetic flux density.

Magnetic flux density in the top and bottom plates at distance r.

Saturation magnetic flux density.

Widing width,

Fourier series coefficient at the fiequency considered.

Diameter of the finished cable over the strands in inches.

Diameter of individuai strands over the copper in inches.

Direct cment.

Power dissipation in dv.

Infinitesimal volume at distance r.

Operating kquency in Hz unless specified.

eddy-current b i s fkctor.

rms fàctor for the P harmonic (gi = 1 for i = 0, and gi = 2 for i 1).

Coefficient of heat transfer by convection.

Design of Low Profile Transfotmers fot High Frequency Operatiori University of Toronto

Nomenclature vii

Resistance ratio of individuai strarads when isoiated.

Thickness of a layer.

Height of transformer.

Height of window area.

Curtent density per unit of length in a iayer (Nm).

The nns value of the current in the primary winding.

The rms value of the current in the secondary winding.

Equivalent cunent in the single equivdent tuni,

The rms value of the current in a winding.

Current density.

Constant depending on the number of strands N.

Waveform fàctor.

Ratio between the height of window area h, and the height of transformer b. Current density coefficient.

Coefficients that depend on the con6iguration of the core

Coefficient that considers the skin effect.

Tunis ratio.

Window utilization factor.

Coefficient that considers the proximity effect.

Number of layers.

Mean-length tum.

Manganese-Zinc.

Number of strands in the cable,

Nic ke 1-2 inc.

Number of tunis in a layer.

Number of turns in the primary winding.

Number of tums in the secondary winding.

Total nwnber of turns made of square strands.

Number of tunis in a winding.

Power dissipation in the center pst.

Total power dissipation in a winding.

Design of Low ProfiIe Transformas for High Frequency Operation University of Taonto

Nomenclature *.* VUI

dc power dissipation in a winding.

Copper Ioss in the prhary windiag.

Copper bss in the secondary winding-

Power density,

Power dissipation in a core-

Specific power los of a magnetic materiai in m w/cm3.

Total power dissipation in a transformer.

Power dissipation in the top or bottom plate.

Pulse Wdîh Modulation,

Arbitrary distance fiom the center of the core.

Normalizing resistance.

Radius of the center pst.

Radius of one strand

Outer radius of the core.

ac resistance of the primary winding.

dc resistance per unit length for the Litz wire used in the primary-

dc resistance of the prirnary winding.

ac resistance of the secondary winding.

dc resistance of the secondary winding.

dc resistance per unit iength for the Litz wire used in the secondary.

Resistance of the single equivalent tum.

Inner radius of the window area.

Total apparent power of transformer.

Rated apparent power of transformer.

Apparent power in the prirnary.

Apparent power in the secondary.

Apparent power in the one equivalent tuni.

Maximum temperature.

Thickness of the top and bottom plates.

rms voltage induceci in a winding.

effective volume of the center p s t .

m i g n of Low Profile Transformers for High Frequency Operation University of Toronto

Nomenclature ix

Effective volume of the core.

rms voltage in the prïmary winding.

Volt-per-turn in a winding.

Volume of the wiadings.

Wmdow a m

Coefficients that depend on the configuration of the core.

Zero Voitage Switching.

Geornetnd dimension for the DS core.

Temperature coefficient of resistivity of copper at 20°C.

Skin depth at the hdamental kquency.

S b depth at the ih barmonic.

Ratio of the thickness of a layer of foil to the skin depth

Temperature rise.

Magnetic flux in the center post.

Magnetic flux in the top and bottom plates.

Conductor spacing factor.

Efficiency.

Surfixe dissipation.

Permeability of fiee space.

Resistivity of wpper at 20°C.

Electricai resistivity of a wlliding at Tm.

Conductivity of copper.

Angular fkquency for the ih harmonie.

Design of Low Profile Transformas fw High Freqwicy Operation University of Toronto

Chapter 1 : Introduction 1

Chapter 1

Introduction

1 .1 Motivation

In today's world, size rninimization and efficiency optixnization of electronic equipment

have becorne critidy important design issues. These design issues have naturaly appeared

£iom the fâct that in telecornmunications and computer systems, the present day trend is to

use distributed-power architecture topology. Also, as it is illustrated in two recently

published articles [l, 21, power supplies for the oew generatiom of ICs will require ratings

in the order of 160A at IV. The problem that appears here is the way these ICs will have to

be supplied and the fact that the power supply will have to be part of the IC in order to be 0

closer to the internai power grid.

Power supplies that are mainly used today in electronic equipment are switched-mode

converters. In order to achieve high power densities, there is a trend to increase the operating

fiequency. As the operating fiequency incfea~es, losses associated with the tum off and the

turn on of the electronic switches also increase. In switched-mode converters, these Iosses

becorne so significant at high fiequencies that the converter becornes impractical due to low

conversion efficiencies. In order to reduce switching losses, power supply designers

improved these converters by using the technique of sofl switching [3]. This technique

allows the electronic switches to tum on and off under the condition of zero voltage, zero

current or bath In this way, the power dissipation in the switch can be minirnized. This type

of converters is known as resonant converters,

A number of resouant converter topologies have been reporteci in the literature 141-[9].

These circuits exhîihit low switching losses and can be operated at high fiequency. For

example, a high frequency, low volume, point-of-load power supply for distributed power

systems is described in [4]. The key point in the operation of this resooant converter is that it

takes advantage of a very low transformer leakage inductance to achieve zen>-voltage

switching of ali its power semiconductor devices. Its resonant ringing fhquency is aiso

Design of Low Rofile Transformas for High Frequaicy Opedori University of Toronto

Chapter 1 : Introduction

independent of load current. A 50W prototype operating at 3.- and a discussion of the

cornponent rquirements wcessary to reach 10 MHz is also presented.

A 1kW 5ûûkHz fiont-end converter for a distriiuted power supply system is d e m i

in [5]. The converter's topology is a standard power MOSFET H-bridge that drives a

transformer. Its switching îhquency is 500- and I uses a phase-shifted PWM technique

to avoid prirnary side switching losses. The converter's efficiency at fidi ioad approaches

90%.

Miwa describes another type of resonant converter in [a. This paper describes a

prototype 50W 40V-SV dddc resonant wnverter operating at S M H z and wasb~cted with

chip and wire hybnd techniques on a ceramic subsaate with copper thick film conductors.

The efficiency of this converter was found close to 85%.

An improved füii-bridge zero-voltage-switching PWM converter using a saturable

inductor is presented by Hua in [A. A saturable inductor is employed in the fU-bridge to

improve converter's performance. The current and voltage stresses as weil as parasitic

oscillations are significantly reduced compared to those of the conventional full-bridge zero-

vo ltage-switchhg P WM converters.

Asymmetrical pulse-width-modulated resooant dcldc converter topologies that exhibit

near-zero switcbulg losses while operating at constant and very high fkquencies are

described in [8]. The converters iaclude a bridged chopper to convert the dc-input voltage to

a high-fiequency unidirectiod ac voltage, which in turn is fed to a high-frequency

transformer through a resonant circuit. Experimental results for a 48V-SV, 30W converter

show an efficiency of 88% at a constant operating fiequency of 1 MHr

Finally, in [9] two variations of a sofi-switched converter together, with a novel

asymmetrical PWM coatrol are proposed. The proposed convertedcontrol combination has

several advantages, such as simple aad effective w ntrol, low-device vo hage stress, effective

use of parasitic elements, ZVS, and high hll-load and partial-load efficiencies, which make

them suitable candidates for highsfficiency highaensity applications. A prototype dc/dc

converter has been built. It achieves 94% efficiency for SOOWsutput power a d 5 0 ~ 1 2

power density.

Al1 of the resonant converters mentioned above use lnagnetic components - inductors

and transformers. The size of a resonant converter is a stroag fùnction of the magnetic

Design of Low Profile Transformas for High Frequmcy Operation University of Tmmto


components. It is wefl known that the size of inductors and tramformers g n d y depends on

the operating frequency. The higher the fkquency, the smaller the inductors and

transformers are in physical size.

The design of hi@ frequency magnetic components becomes more complicated due to

several aspects that are not very important at low fiequency. The main issue for high

fiequency magnetic component design is tbat the winding loss imreases considerably due to

skin and proximity effects. Also, the magnetic flux density has to be reduced in order to

keep the core loss at acceptable levels.

For fkequencies between 100 lrHz and 2 MHz, the most cornmon materiais used are

MnZn and NiZn ferrites. These are characterized by a high electric resistivity resulting in

low eddy curent losses. For such high hquencies, magnet wire is inefficient and Litz wire

and copper foil become attractive alternatives. Above 2 MHz, Litz wire and copper foi1 aiso

become inefficient and the designer has to support the resulhg losses.

One of the most important magnetic compownts of resonant converters is the high

fiequency transformer. This component plays an important role in detemiining the

efficiency and the size of the respective converter.

Different design methods for transformers that work at high fiequency have been

published over the last two decades. Some of these design methods, together with

investigations of different construction types of high fkquency transformers, are descriid

in reference [ 1 O] through [14]. For example, three types of high fiequency transformers with

dflerent magnetic and winding configurations used in resonant converters, are compared in

[IO] for a resonant converter appikation. The experimental results demonstrate the

characteristics of the self-inductance to voltage ratio versus fkquency. The magnetic flux

and eddy-current distriiutions of pot core, p h core and multi-element core tramformers

were calculated by a boundary element based CADKAE software package. Both

experimental and computer modehg resuits were used for optimum design of high

fiequency transformers and minimixing the copper losses in the transformer windings.

A transformer computer design aid is presented by Coonrod in [Il]. The design method

was developed for use at switching fiequencies above lOOktIz where the classical design

methods of using the saturation constraint results in an overheated transformer due to

excessive core loss. The operating flux density is selected instead by an optirnizatioa

- - - - --

Design of Low Profile T d o r m e r s for Hi& Ftequaicy Operation University of Tacmto


procedure that m h h h e s total transformer losses. Key equaîions used in the design

procedure are developed.

A highly efficient (99.5%) transformer and resonant inductor with very high power

density (1 S O O W / ~ ~ ~ ) aod low profle (height < 0.4 in), are d e s c n i in [12]. The design of

these components is based on a trade-off study that estabiishes the optimum operating

fiequency to be around lMH2. These magnetic components were developed for a 95%

efficient 1 kW resonant dc/dc converter.

Petkov presents a procedure for optimum design of a high-power high-fkquency

transformer in [13]. Tbe procedure is besed on both electncal and t h e r d processes in the

transformer and identifies the V A - r h g of ferrite cores in relation to the operatting

fiequency, the optimum flux density in the core, and the optimum current densities of the

windings providing maximum transformer efficiency.

In [14], a new plam integrated magnetic component with transformer and inductor

using multilayer printed wiring board is presented The transformer is located in an outer

ring, and the choke coi1 in an inwr ring. This magnetic component was designed to reduce

the loss and the height especiaiiy focusing on the relationship of winding resistance to the

thickness of copper foil The experimental resuits show that the component offers better

e fficiency and lower electromagnetic noise than conventional components.

A great deal of the present hi& fiequency magnetic compownt design involves the

development of micro-machined devices. Examples of designs of such magnetic

components can be found in [15] and [ l q .

It is known that relatively simple design methods exist for tramformers and inductors.

One of these design methods is presented by McLyman in [ l q . This method starts fiom the

simple approach that the critical dimensions of a core can be related to the design niting of

the transformer - the total apparent power, frequency, operathg flux density. Empirical

correction fkctors are used to account for skin and proximity effect Iosses, and for the fàct

that magnetic flux density is considered constant inside the core. When used for high

flequency design, the design based on this method can be improved by using Litz wire

instead of magnet wire. In order to consider losses due to skin effect, the fonnula that

calculates the correction k t o r for the AC resistance presenred in [18] can be used.

Design of Low Profile Transformas for High Frequeacy Operation University of Toronto


This method can be fbther improved considering both skin and proximïty eff'ects by

appiying the method to calculate copper l o s presented by Vandelac and Ziogas in [19]. The

key point in this method is that losses in any layer of the winding can be calculated if the

tangentid magnetic field on both sides of that iayer is known, This paper also describes the

effect of harmonies for the situation in which the current in the windings is not Sinusoidal,

This method was fiuther extendeci by Hurley [20J m developing an optimized approach. It

was shown in this paper that for any transformer core, it is possible to define a critical

fiequency. Above this critical hquency, the losses can be minimiled by selecting a flux

densÎty that is less than the saturation flux density. Below the critical fkquency, the

throughput of energy is resaicted by the Limitation that flux density canwt be gr- than

the saturation value for the core material in question.

As an altemative, in high fiequency transformer design there are many finite element-

modeling packages around These offer the possibility of representing the geometry and the

material properties more exactly- One of the disadvantages of the finite element approach is

that the results, whïie accurate, are for specific cases. It is thus dSfïcult to discem trends and

trade-O&.

Another objective in this field wouid be to devebp design curves for the range of

parameters that are of interest. It is known that a major constraint in the design of high

fkequency transformers that are used in rack mounted or printed circuit board converters, is

the height of the transformer. In order for the converter to M in narrow spaces, the height of

the transformer has to be limited by the width of the slot where the converter has to be

placed. An d y s i s regardhg the variation of the power density with the height of the

transformer has been shown by Ngo in [2 11. In this paper it has been developed an algorithm

to determine how does the height of the transformer infiuence the power density for a given

power, efficiency, fiequency, and volt per tuni.

1.2 Thesis Focus

As it has been mentioned in the 1s t seztion, in many advanced teiecommunication and

computer systems, distributeci architecture topology is becoming popular because of

Design of Low Profile Transformas for High Frequency Operation University of Toccmto


distn'buted themial profile, point+& use power regdation, and modular system design. Some

of the basic requirements on the point-&use power supplies emplo yed in this architecture

are 1) high power density, 2) high efficiency, 3) low EMI, and 4) constant operating

fiequency. One type of power supplies that meets these requirements is the resonant

converter topologies descri'bed by Jain in 181. These topologies use two magnetic

components: a high fkequency inductor and a high fkquency transformer, The losses in

these two çomponents will dictate the efficiency of the converter. As a result, the design of

these two components has to be done very carefùiiy. Another design issue for these

magnetic wmponents is thek height. The height of these wrnponents must not exceed a

certain value imposed by the overd size of the resonant converter.

The design of the high ikeqwncy transformer, used in these resonant topologies, rises

the question " how weli do the conventional methods work in the high fiequency region, and

what are the aade-offs when the optimized approach is used?" To answer this question, the

conventional methoch can be used for designs in the region of interest and the results

compared, for losses at least, with the finite element anaiysis approach. Amther question

that rises would be "is it possible to generate some design curves that show the dimensions

of the transformer as a fbction of height ?".

In view of the foregoing, this thesis examines the three available design methods for

high fiequency transformers descriid above, and makes a cornparison of the results for a

specific design, A cornparion with the results of a Finite Element analysis is also done in

order to see if these methods are accurate in tenn of predicting losses. Mer that, an

algorithm to generate design c w e s for high fkquency transformers is proposed. The key

point in this algorithm is that it takes into consideration the height of the transformer when

this is a constraint. The height of the transformer, output power, eficiency, working

fiequency, and volt-per-tuni are considerd as input for the design, The algorithm has as an

output the other dimensions of the transformer and the current density in the windings. This

algo rithm dBers nom the O ne d e s m i by Ngo in [2 11 as follo ws:

1). The algorithm uses a condition to get minimum loss in the transformer. It was shown

by Hurley in [20] that the minimum loss in a transformer happeos for a h e d operating

fiequency when the copper loss is equal to the core l o s multiplied by a coefficient d divided

by two, where d depends on the specific core loss and it wiil be explained in Chapter 2.

Design of Low Profile Transformas for High Frequaicy Operation University of Tamto


Having this condition in the design, one expects to obtain a high power density for a

specified efficiency.

2). The algorithm considers the windings made of Litz wire instead of copper foil,

3). The algorithm will yield the current density that the windings must support in order

to obtain the expected efficiency. Knowing the current demity, the çopper cross-section area

of the Litz wire that is going to be used in fabricating the windings can be specified.

4). The two algonthms d s e r in the way the caiculations are done. The algorithm

described by Ngo in [20] uses an iteration process. The caicuiations are repeated in two

loops with the increment of two parameters until the expected los is met. Ln the algorithm

proposed in this thesis, the condition to have minimum loss r d t s in a system of two

transcendental equations that can be solved numericaliy. The solution yields two of the

geornetrical dimensions of the transformer. Ail other parameters çan then be obtained fiom

specified constraints and conditions.

A short description of each chapter is presented in the foliowing subsection.

1.3 Thesis Overview

Chapter 2 identifies several specinc problems in the design of high fkquency

transfomers These include the choice of the core, magnetic ma te ra operating magnetic

flux density, and current density in the windings.

In Chapter 3 three of the available design methods for high fkequency transformer

design are briefly reviewed. The rnethods are compared for the case of a 500kHi1, 54VA

tramformer. The losses predicted by these three rnethods are compared to those predicted by

a 2-D Finite Element model.

In Chapter 4 an aigorithm to generate design curves for traasformers, having as an input

height, output power, efficiency, operating fkquency, and voit-per-turn, is presented. The

other dimensions of transformer and the current density in the windings are obtained as an

output of the algorithm. This chapter also presents a set of design curves generated by the

Design of Low Profile Transfomers for High Frequaicy Operation University of Tormto


algorithm, These design cuves were developed for a transformer with a specifïc range of

output power, Merent efficiencies, operating fkquencies and different magnetic maîenals.

A discussion of these resuits is provided.

Chapter 5 presents the conciusions and fhre work that should be done.

Design of Low Profile TCaflSformas fot High Frequency Operation University of Torooto

Chapter 2: Identification of the Problem 9

Chapter 2

Identification of the Problem

Introduction

In Chapter 1, it was mted tbat relatively simple design methods for transfomers exkt

in the technical iideralure. The purpose of the present chapter is to show how it is possiae to

link the size of a transformer core to the required VA rating, fiequency, operating flux

density, and current density. This link achially represents the basic concept for the standard

design methods.

When initiating the design of a high frequency transformer, the two parameters that are

not directly design parameters are the operating flux deasity and the current density in the

windings. These parameters must therefore be chosen. A second objective of this chapter is

to descni the factors that govern the choice of these parameters.

2.2 Area Product

Consider a vertical section in a transformer as shown in Figure 2.1. According to

Faraday's law, the rms voltage induced in a winding will be equal to:

where:

Kf is the wavefonn coefficient (&= 4.44 for sine wave and 4.0 for square wave).

Nw is the number of tum in the winding.

f is the operathg hquency in Hz.

B, is the maximum or peak value of the flux density in tesla,

A, is the effective cross-sectional area of the core in cm2 (see Figure 2.1).

Design of Low Profile Transformers for High Frequency Operation University of Toronto

Chapter 2: Identification of the Pmblem 10

Figure 2.1 Cross-sectionai area and window area of a core.

The current in a given winding will be quai to:

where:

J is the current density in Ncm2.

A, is the cross section area of the wire filled with copper in cm2.

W. is the available wimlow area in cm2 (see Figure 2.1).

Nw is the number of turns in a winding.

K,, is the window utilization tàctor and represents the amount of copper that appears

in the window area of the transformer. The calculation of K,, is shown in

Appendix A.

The totai apparent power of the transformer will be:

Design of Low Profile Transformers for High Frequaicy Operation University of Tamto


In (2.2.3) the product of the effective cross section area A, and the avaüable window

area W. of the core is kwwn in the technical iiterature as the Area Product 4 - (2.2.4).

These three parameters (A, WC Ap) are used by core suppliers to summarize

dimensionai properties in theu data sheets. They are available for larninations, C cores, pot

cores, powder cores, ferrite toroids, and toroidal tape-wound cores.

Using the notation of Area hduct, the expression of the apparent power will becorne:

From (2.2.5) one can fïnd the necessary area product the magnetic core needs to handle

the apparent power S for a selected operaîing tkquency, current density, magnetic flux

density and of course Kr and K,,.

Equation (2.2.6) is the key relation in standard design methods, and it ünks the size of

the core to the VA rat@, ûequency, operathg flux density, and current density in the

windings. In this expression, the parameters that are not directly design parameters are the

operaîing flux density B, and the current density in the windings J. These parameters m u t

therefore be chosen. Consequently, it is necessary to descrii the factors that govern the

choice of these parameters. The choices of the operating flux density and of the current

density in the windings are discussed in the next two sections.

Design of Low Profile Transformas for High Frequency Operaiion University of Tacmto

Chapter 2: 1denti.ficaîion of the Problem 12

The Choice of the Magnetic Material and Flux Density

For low fkquency desigas (50/60Hz), where the losses produced by eddy currents aod

hysteresis are low, the materials used are u s d y süicon-steel laminations. As the operathg

fiequency increases, tbe losses prduced by eddy currents and hysteresis increase

signif~cantly and silicon-steel laminations are no longer efficient.

For transformers h t operate at fkquencies between lOOkHz and 2MHz the most

cornmody used materials are MnZn and NiZn ferrites. These are cbaracterized by a high

electric resistivity that resuits in Iow to negligible eddy current losses. Further more, these

particular ferrites have low hysteresis losses in the specifïed frequency range.

The generic single valwd rnagnetiring curve for a magnetic material is shown in Figure

--

MAGNETlZlNG FORCE (H)

Figure 2.2 Magnetizing curve for a magnetic material

In Figure 2.2, four distinctive regions can be mted on the magnetizing c w e : (1) the

region between O and A where the rnagnetizing curve is wnlinear, (Ir) between A and B

- - - - -

Design of Low Profile Transformas for High Frequaicy Operation

--


C hapter 2: Identification of the Problern 13

where it is almost hear and bas a signifïcant slope, (ID) betweem B and C where the curve

again is nonlinear, and above C where it is linear and has a very small slope (equai to

the f i e space permeabiiity).

For low eequency designs (50/60Hz), wkre the losses produced by eddy currents and

hysteresis are low, the operating flux d e n s e chosen for a design is close to the knee point

region, i.e. below B (see Figure 2.2). As the opaating kquency increases, losses due to

eddy currents and hysteresis increase significantly and the flux density must be dropped to

the h e a r region between A and B sometimes very close to A (see Figure 2.2).

Manufhcturers of f d e s for high hpency applications typically provide performance

factor data as a meam of assisting in the choice of operathg flux density. The performance

factor generaily relates flux density, fkquency, and a r e loss. Figure 2.3 shows one such

factor. This represents the product of magnetic flux density and fkquency as a function of

~equency for a constant specific power los. The performance k t o r for different materials

is shown in Figure 2.3 for a specinc power loss of 300 mw/cm3 at 80°C.

Figure 2.3 - Performance factor for different magnetic materials '

Design of Low Rofile Transformas fa Hi& Frequency Operation University of Toronto


The perforniance fàctor can be used in the foilowing way to fInd the flux density thai

will be used in design. In Figure 2.3, assume that the K materiai will be used in a design at

500kHz operating fiequency and 80°C maximum temperature. In Figure 2.3, it can be seen

that at the SOOlrHz operating fiequency for the K material, the corresponding B x f product is

350 (kgauss x kHz), This yields a flux density o f 0.7 kgauss at 500kHz. The flux density

thus obtained provides a starting point for the design Depending on the cooling conditions

specific to the application, this flux density can be chosen higher or lower.

I B

Flux Density [kgauss]

Figure 2.4 Specitic core loss for specific fiequencies2

' ~ e ference 23, page 2.4

- - . . . - -- -- - - -

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In addition to the characteristics shown in Figure 2.3, it is also common for

manufacturers of magnetic materials to provide specific core loss curves developed îiom

empiricai data. An example of specinc loss curves for the K material can be seen in Figure

2.4. In Figure 2.4, the specific losses are shown at 80°C for specifïc fkquencies. For

example, assume that the K materiai will be used in a design at 5ûûkHz operaîing hquency,

80°C maximum temperature and a specinc power loss of 300 mw/cm3. In Figure 2.4 it can

be seen that for 300 mw/crn3 specüic power loss and 500 lrHz operating fiequency

corresponds a £lux density of 0.7 kgauss.

Functions that appruximate these speca?c l o s curves are aîso provided by ferrite

manufacturers. For the materials shown in Figure 2.4, the losses are approximated by the

fo llo wing expression:

d PL = a - f C - B , (2.3.1)

where:

PL is the specific power los in mw/cm3.

f is the fiequency in kHz.

&II is the magnetic flux density in kgauss.

a, c, d are coefficients specific to each type of magnetic material.

This section has shown the factors that govem the choice of the operating flux density

and the magnetic material. The objective of the next section is to show the factors that

govem the choice of the current density in the windings.

2.4 The Choice of the Current Density

In the last section it has k e n shown that for a nxed fkquency, the operating magnetic

flux density is the parameter that detemines the loss in the magnetic core. In a similar way,

the current density in the windings determines the copper 105s in the windings. The choice of

the current density has to be done in such way that the temperature of the hottest spot of the

transformer, due to core and copper losses, is lower than the insulation class temperature.

Design of Low Profile Transformets for High Frequency Operation University of Toronto


It bas k e n shown in [17] that there is a unique relationship between the area product A,

and the current density used in the windings, on the one han& and the resulting temperature

rise on the other band. This relationslip has the fom:

where:

J

Kj

A,

Y

is the current density in A./cm2.

is a constant related to core configuration and bas a positive value.

is area product m cm4.

is a constant related to core co~guration and has a negative value.

In Figure 2.5, the variation of current density as a finction of area product is shown for

a pot çore configuration and two temperature rises (2S°C and 50°C)-

0.0 1 O. 1 1 10 1 O0 0.0 1

1 -id A P

1 O00

Area Product [cmA4]

Figure 2.5 Current density vs. area pmduct for temperature increases of 25°C and

SO°C for pot cores 3.

Design of Low Profile Tramformers for High Frequency Operation University of Toronto

Chapter 2: Identification of the Pro blem 17

An important observation cegarding Figure 2.5 can be made by ooting tbaî for cores

with a s d area product, the curent density in the windings is high, and for cores with a

large area product, the current density in the wùidings is low. Specificaily, when the area

product increases ten times, the m e n t density has to decrease roughly one and a half t h e .

The teason this bappens is that the ratio between the total extemal area and the volume is

higher for small transformer than for iarge transformers. For a constant power dissipation

per volume, in the case of large transformers there wül be more heat to be evacuated pet unit

of external area than in the case of srnall transformers. As a remit, the power dissipation per

unit of vohune has to be bwered in order to obtain the same tenperahne rise. This WU lead

to the fact that the current density in the windings has to be lowered.

The objective of this chapter was to fàmiiiarize the reader with the following three things:

1) The possbility to link the size of a transformer core to the VA rating, fiequency,

operating flux density, and current density. This link actually represents the basic concept

for the standard design methods.

2) Describe the fàctors that govern the selection of the operating flux density. The

choice of the flux density is very important because it dictates the core loss.

3) Identify the factors that govem the choice of the current density in the windings.

Here an important observation bas k e n made: the current densÏty in tramformers has to

vary in an inverse relation to the area product in order to keep the maximum temperature

be low the insulation class temperature.

Design of Low Rofile Transformers for Hi& Frequency Operiitim University of Toronto

Chapter 3 : Revue of Available Design Methods 18

Revue of Available Design Methods

3.1 Introduction

In Chapter 2 of this thesis, the basic concept that the simple design methods rely on,

namely the possibility to lin. the size of a transformer core to the VA rating, fkquency,

operating flux density, and current density, has ken descriid. The hctors that govern the

cho ice of the operating flux density in the core and the current density in the windhgs were

also outlined.

Non-sinusoidal excitation at high fiequencies introduces new design issues: skin and

proximity effects in windings and increased eddy current and hysteresis losses in cores. The

starting point for an optimized design is the assumption that winding losses are

approximately equal to the core losses. However, m a high IÏequency transformer, the ratio

may be as high as 5:l. This is due to the hct that the flux density is limited by its saturation

value. At the high end of the fkequency scale, the transformer may be operating with a

maximum flux density that is weii below its saturation value in order to achieving an

optimum design.

The purpose of this chapter is to compare three available design methods that are

conventionally used for transformer design. The methods are compared using the specific

case of a 500kHz low profile transformer. An issue of particular interest is whether the

existing methods give reliable estimates of core losses. To answer this question, a 2-D Finite

Element analysis of the transformer is also undertaken

Al1 three methods use as input the total apparent power of the transformer S, the voltage

in the prLnary windhg V, the huns-ratio KT& the operathg fkquency E the temperature

rise AT, the waveform fàctor KG and the window utilization factor K,,.

The fist method is the classical design method descnbed by McLyman in [IV. This

method uses the basic concept, d e m i i d in Chapter 2, to Lùik the size of the transformer

Design of Low Profile Transfonners for High Frequency Operation University of Taonto

Chapter 3 : Revue of Available Design Methoàs 19

core (as measured by the area product) to its rated VA, fkquency, operating flux density and

current density. Copper losses due to skin effect are accounted for using an expression that

can be found in the iiterature [I8]. This method is shortly d e m i in section 3.2 of this

chapter.

The second design method was developed using the classical McLyman approach,

together with a winding loss calculation method which takes in consideration both skin and

proximity effects. The method was M y d e s c r i i in [19] and is briefly surnmarized in

section 3.3. A key feature of the method is that it incorporates an approximation of Litz

wire. This approximation is based on foi1 layers as suggested in [24].

The third design method is an optimized design method described by Hurley, WoIfle

and Breslin in [20]. The method incorporateci the approximation of Litz wire windings by

foi1 layers as suggested in [24]. Moreover, it was shown in [20] that for any transformer

core, it is possible to define a critical fiequency such tbat above this critical fkquency, the . . .

losses can be rrmmmed by seIecting a flux deusity which is less than the saniration flux

density of the core material being used, BeIow the critical fiequency, the throughput of

energy is restricted by the limitation that the flux density cannot be greater than the

saturation value for the core materiai in question.

The main point that diierentiates this design method fiom the one described by

McLyman is the fact that the optimal method minimizes losses in a transformer under the

condition of a constant operating kquency or constant operating flux density. The

optiniized design method is shortly described in section 3.4 of tbis chapter.

3.2 The Classical Design Approach

This method was developed using the classical design of transformers as described by

McLyman in [ln This method uses the basic concept, descriid in Cbapter 2, to Mc the

size of the transformer core (as measured by the Area Product) to its rated VA, fkquency,

operating flux density and current density. The current density is linked to the size of the

Design of Low Profile Transformas fot High Frequaicy Operation University of Torcmto

Chapter 3: Revue of AvailabIe Design Methods 20

transformer by a coefficient K,. This coefficient is dsectly dependent to the temperature rise

AT, and is specifiç for each core configuration

Core l o s is found using an effective volume of the core V, and considering the magnetic

flux density B, constant inside the core.

Copper losses due to sk i . effect are accounted for ushg an expression thai can be found

in the literature [18]. No effort is made to include proximitty effect losses.

This method checks the temperature rise AT by calculating a SUfface dissipation Y and

comparing the result with one found experirnentdly for the same temperature rise.

The method is presented shortiy m tbe following steps:

1. First, the input data for the design bas to be specified. The data used as input is the total

apparent power of the transformer S, the voitage in the primary winding V, the turns-

ratio Km the operating fkequency Ç the temperature rise AT, the waveform fmor Kt

and the window utilization factor &,.

2. Second, a magnetic materiai that is going to be used in the design must be selected. For

this material, the foilowing parameters have to be kwwn: the saturation flux demity B,,

and the parameters that describe the specific loss curve: a, c, and d. The selection of the

magnetic material depends on the operating fiequency and was discussed in Chapter 2 of

this îhesis.

3. For the magnetic material that will be used, the operatmg flux density Bm has to be

selected. The fàctors that govern the choice of the operating flux density have been

discwed in Chapter 2 of this thesis.

4. Based on the operating flux density selected at step 3, the area product A, necessary to

handle the total apparent power S must be determined The area product will be

determined using expression (3.2.1 ).

Design of Low Profile Transformers for High Frequency O p d m University of Tamto

Chapter 3: Revue of Available Design Methods 21

Note that (3.2.1) is similar to the area product expression (2.2.6) that was developed in

Chapter 2. However, the exponent x bas been introduced to account for the configuration

of the magnetic core. Typically, x is in the range of 1.14 to 1.20, depending on the core

configuration. In Appendix B is given the d u e of this coefficient for different core

configurations. In addition, the coefficient Kj has been introduced. This coefficient links

the current density in the windings to the size of the transformer. This coefficient is

specific for each core configuration. In Appendix B is given the value of this coefficient

for dif5erent core configurations and two temperature rises. The other parameters in

(3.2-1) are d e k d as fobws:

B, is the maximum value of the operating flux density.

f is the operating fkquency in Hz.

Kr is the waveform factor (TCf= 4.44 for sine wave and Kr = 4.0 for square wave).

K, is the window utilkation factor. The way this factor is calculated it is shown in

Appendix A.

5. Based on the area product determined at step 4, a core with an area product close to the

one calculated will be selected. For this core, the foilowing parameters must be known:

effective cross section area A, window area W,, the mean length tum MLT, and the

effective volume of the core V,- However, sornetimes, an exact or even nea. match is not

possible. This will have an impact on the power density and the temperature rise.

If the selected core has an area product that is much larger than the calculated one,

the resulted power density will be very low. This will yield a transformer that is too big

for the application where it has to be used.

If the selected core bas an area product that is much smaller than the calculated one,

the resulting temperature rise will be higher thaa the imposed one. This will make the

transformer t w hot and its winding isolation wili be stressed.

6. The next step is to determine the number of turns for the primary winding using

expression (3.2.2).

-- -

Design of Low Rofile Transformers foi. High Frequency Operation University of Toronto

Chapter 3 : Revue of Available Design Methods

The parameters in (3 -2.2) are de- as fokws:

V, is the rms value of the voltage in the primary winding.

4 is the effective cross section area of the selected core.

I f the result is not an integer, the closest integer number of turns for the primary

winding will have to be selected in such way that it will result in an integer number of

turns in the seconàary winding. Based on the selected number of turns in the primary

winding, the flux density will be recalculated later to be able to calculate the core loss.

The rms value 1, of the current in the primary winding m u t be determined. This curent

wiil be calculated using the rated power of the transformer S, and the rms vahie of the

voltage applied to the primary winding V,.

The current density J in the windings is detennined using:

J = K j . A p Y

were y is a coefficient that depends on the configuration of the core. The value of this

coefficient is given in Appendix B for different core configurations.

The necessary cross-sect ion area A, of the wire for the primary winding is given by :

10. Based on the result obtained at step 9, a Litz wire is selected such tbat it bas a cross-

section copper area close to the calculated one.

11. Determine the number of turns Ns in the secondary winding using the nwnber of tums

Np in the primary winding, that was calculated at step 6, and the given turns ratio KTx

Design of Low Profile Transformers foc High Frequaicy Operatiori University of Toronto

Chapter 3 : Revue of Avaiiable Design Methods 23

12. Determine I,, the effective value of the current in the secondary winding. The value of

this current will be determined ushg the rms value of the current in the primary 1, and

the tunis-ratio Km

13. Determine the necessary cross-section area of the wire for the secoadary Win- A,:

14. Based on the result obtaiwd at step 13, select a Litz wire for the secondary winding. The

selected wire should have a cross-section copper area close to the calculated one.

15. Check if the prirnary and the secondary windings fit in the bobbin. It has been noticed

that the turns-ratio of the transformer is a major drawkk in the design In order to have

an integer number of ninis in the primary and secondary windings, the number of turns

that is selected is usually larger than the one calculated with expression (3.2.2).

Consequently, both the primary and the secondary windings WU have a larger number of

tunis that might not M in the bobbin. If the windings do not M, another core with a

bigger area product has to be selected, a d seps 6 to 14 have to be repeated. hother

way of dealing with this problem would be to select for the primary and secondary

windings Litz wire with a smaller cross-section copper area.

16. Detennine the dc resistance for the primary and secondary windings:

Rp-k =MLT*Np-$ - ,

Design of Low Profile Transformas for High Frequeucy Operation University of Tcnoato


The parameters in (3.2.9) and (3.2.10) are defined as foilows:

MLT is the mean length turn of the bobbin,

NP is the number of tunis in the primary winding.

NS is the number of tums in the secondary winding.

rp_dc is the dc resistance per unit of length for the Litz wïre used in the prirnary.

rs-dc is the dc resistaace per unit of lengîh for the Litz wire used in the

=ndary-

17. Determine the Ir, and k, factors (for the prbary and secoadary windings). These

factors account for the skin effect and are found using the foiiowing expression4:

and

2 Resistaace-to-Altm-kent = + *

k, = (y) -G (3.2- 11) Resistance-to-Direct-Current

The parameters in (3.2.1 1) and (3 -2.12) are defïned as follows:

G is the eddy-current basis factor.

H is the resistance ratio of individual strands w k n isolateci (see Appendix C,

Table 1).

f is the operaîing fiequency in Hz

N is the number of strands in the cable.

DI is the diameter of individual strands over the copper in inches.

Do is the diameter of the finished cable over tbe strands in inches.

K is a constant depending on the nurnber of strands N (see Appendix C, Table II).

Reference [I 81, page 37.

Design of Low Profile Transformers for High Frequaicy Operation University of Tamto


18. Using the fhctors k, and k , calculated at step 17, the ac resisiance of the p m and

secondary windings wiil be determined:

19. Based on the ac resistances determined at step 18, the loss in the windings will be

calcuiated,

Pa= = RP-= - 1: (3.2.15)

20. Next, the core los is determineci. First, the flux density mut be recalcdated she , at

step 6, an integral number of tums had to be selected.

Second, ushg the values of the operathg flux deasity and operathg frequency, the

specific core loss PL can be determined using the following expression:

Where a, c, and d are coefficients that describe the specinc loss curve for the magnetic

material used in design. Finaliy, multiplying the specific core los by the volume of the

core wiil yield the total core los.

Design of Low Profile Tramformers for High Frequency Operation University of Toronto

Chapter 3: Revue of Avaüable Design Methods 26 - --

2 1. Calculate the total power l o s ~ Pl, and the adàce dissipation Y in order to detennine

whether the temperature rise AT exceeds the Limit imposed by design for the specified

cooling conditions:

where & is the total e x t e d area of the transformer and is equal to:

In (3.2.22), K, is a coefficient that relates the area product of a core to the externai area

of a wound transformer that uses that core. 'Ilus coefficient depends on the codiguration of

the core. Values of K, are presented in Appendix B for different core configurations.

The s&e dissipation Y is shown in Appendix D as a fimction of the temperature rise

for ttiree different ambient temperatUres. It c m be seen fiom this graph that the surfafe

dissipation Y has a value 0.03 w/cm2 at 2S°C temperature rise, and a value of 0.07 w/cm2

at 50°C temperature rise for a 25°C ambient temperature.

The classical design rnethod presented above has the advantage tbat it is a

straightforward design method that wi be easily irnplemented. One limitation of thh

method is the k t that the flux density is assumed to be constant inside the core. Thus, the

power dissipation in the core xnay not be accurately known. The second limitation is that the

CO pper loss calcuiation does not include the calculation of losses due to pmximity effect.

Design of Low Profile Triansformers for High Frequency Operation University of Toronto


3.3 Classical Design Approach with Improved Estimate of Coi1 Loss

This design method was devebped using the classical approach that was outiined in the

previous section of this chapter, together with a winding loss calculation method which

takes in consideration both skin and proximity effects. The method was fkst descnbed in

[19] and is briefly summatized in this section. A key feature of the method is tbat it

incorporates an approximation of Litz wire. The approximation is based on foil layers as

suggested in 1241 and is iliustrated in Figure 3.1. The difference between this method and the

design method d d b e d in the previous section is the way that losses in the windings are

calculated- The method described in section 3.2 takes in consideration the skin effect only.

The present method considers both skin and proximity effeçts.

The basis of the copper loss calculation is that the loss in any equivalent layer can be

estimated provided the tangential magnetic field on both sides of the layer are known.

The copper loss calcuiation can be summarkd as follows:

1. An approximation of the wimlings made of Litz wire with foil layers is made. This

approximation was presented by Carsten in [24] and allows the calculation of loss in

Litz wire using the method presented by Vandelac in [19]. The steps in arriving at this

approximation c m be seen in Figure 3.1. First, the round Litz wire (Figure 3.la) is

approxirnated with a square matrix of square strands (Figure 3.lb). M e r thaf the

square matrix is approximated with a stack of foils (Figure 3. l c). Finally, al1 stacks of

foils are joined together resuiting layers of foils as it is sbown in Figure 3. Id.

2. Next, the current in each foil layer is detennined. It is assumed that each layer shares a

curent equai to the total cmrent divided by the number of layers.

3. Finally, the loss in the winding must be calculated. In order to do so, the mmf diagrams

of the windings must be developed. From the mmfdiagrams, the valw of the men t i a l

magnetic field at both sides of each layer can be inferreci*

Next, based on the values of the tangentiai magnetic field at each side of each layer, an

expression for specüïc power loss per square meter for each layer and for each

harmonic of the current can be developed ushg Maxwell's equations.

Design of Low Rofile Transformas for High Frequency Operation University of Taonto


Figure 3.1 Approximation of Litz wire with foil.

Design of Low Profile Transformas for High Frequency Operation University of Toronto


The expression for specific power los can then be integrated on the entire area of each

layer. This integration yields the total po wer dissipation in each layer-

F W y , d g the power dissipation in all layen will yield the fo ilowiag expression

for the total power loss in the winding:

where:

Design of Low Profile Transformas fot High Frequaicy Operation University of Tamto


The parameters in (3.3.1) to (3.3.9) are defïned as foliows:

is the breadth of a conductor-

is the winding width.

is the Fourier series coefficient for the ia harmonie.

is the rms factor for the i<h harmonie (gi = 1 for i = 0, aad gi = 2 for i >= 1).

is the thickness of a layer.

is the current density per unit of length in a layer (Mm).

is the mean-length-tuni,

is the number of layers.

is the total number of tums made of square strands.

is the number of turns made of square strands in a layer.

is the normalking resistance.

is the skin depth at the fidamental fiequency.

is the skin depth at the ih harmonic.

is the conductor spacing fàctor.

is the pemability of fiee space.

is the conductivity of copper.

is the angular fiequenc y for the im harmonic.

This method is more accurate in calcuiating the losses in the wùidings than the one used

in the classical design method. However, it has the disadvantage that is more compticated

due to the fàct that it has to approximate the windings with foil layers.

3.4 Optirnized Transformer Design Method

Hurley, Woifle, and Breslin described an optimized transformer design method in 1201.

The method incorporates the approximation of Litz wire windings by foil layers as

suggested in [24].

It was shown in [20] that for any transformer core, it is possible to d e k e a critical

fkequency such that above this critical fkequency, the losses can be minimized by selecting a

Design of Low Profile T d o n n a s f o ~ High Frequency Operation University of Toronto


flux density which is less tban the saturation flux densïty of the core material king used.

Below the critical kquency, the throughput of energy is restncted by the limitation that the

flux density cannot be greater than the saturation value for the core material in question

n i e method has as input the same parameters as the classical method demiibed by

McLymaa These parameters are: the total apparent power of the transformer S, the voltage

in the primary winding V, the turns ration K m the operating nPquency f; the temperature

rise AT, the waveform fàctor KG and the window utilization factor L. This design method was developed using the foiiowing five basic equatiom:

A). The vowe equafion for a wmding:

V=K,-f-Nw-B;A, (3 -4.1)

B). The power equotion for a transformer:

S=K,-Ku-J-f-B;A,

C). The copper loss equation in a winding:

P, =pw-V, , , -K , -~2

This equation is written under the assurnption that the windings are made of Litz

wire or foil and skin and proximity effects are not present. In this case wiii result that

the current density has a uniform distniution in the windings.

D). The core loss equation in a transformer core:

E). The thermai equation that relates heat flow to temperature rise AT, surfàce area &

and the coefficient of heat transfer h by:

Design of Low Profile Transformas for High Frequency Operation University of Tamto

Chapter 3 : Revue of Avaiiable Design Methods 32

Tt bas to be noticed that expressions (3.4. l), (3.4.2) and (3.4.4) were also used in the

classical design method dem'bed by McLyman

Besides (3.4.1) to (3.43, the foiiowing equations that reiate the physical quaotities V,

V , and & to the core size A, are also used:

where &, Kw, and K, are dimensionless coefficients and depend on the core

configuration.

It is shown in [20], using (3.4.2), (3.4.3), a d (3-4-4), that the minimum loss in a

transformer occurs:

a). for a fixed operathg fkequency when

L

b). for a fixed flux density when:

The key point in this design rnethod consisis in the minimi?ation of the total l o s in the

transformer. This is done at constant fiequency by using the expression (3.4.9). This is the

main ciifference between the optimal design method and the one describeci by McLyman.

Based on equation (3.4.1) to (3.4. IO), the optimized design method is shortly describecl

as follows:

1. First, the input data for the design has to be specified. The input data bas been presented

at the beginning of this section. It bas to be mticed that the input data is the sarne as the

input data for the classical design method descrt'bed by McLyrnan,

Design of Low Profile Transformas fm High Ftequency Operation University of Toronto

Chapter 3: Revue of Available Design Methods 33 - -

2. Second, a magnetic material tbat is going to be used in the design must be selected For

this materid, the following parameters have to be kwwn: the saturation flux density

B, and the parameters that descrii the specific loss curve: a, c, and d

3. Based on the data input specined at 1. and 2., an optimum flux density (Bo) will be

calculated.

where :

This is the flux density where the losses are minimum for a fixed operaihg fiequency f.

Expression (3.4.1 1) was found in the fo Uowing way:

a). From the condition (3 -4.9) to get minimum loss and the fgct that the total loss Pi, is

equal to the sum between the core loss and the copper loss results:

Elirninating the current density in (3.4.3) using (3.4.2) results:

Replacing (3.4.4), (3 .43 , and (3.4.15) in (3.4.14), an equation in A,, is yield. Solving this

equation for A, wili resuh:

- - --

Design of Low Profile Transformas foc High Frequaicy Operation University of Toronto


b). From (3.4.9), (3.4.3), and (3.4.9, an optimum vahe of the current density can be

found:

c). Substituthg the optimum value of J given by (3.4.17) and A, given by (3.4.16) into

the power equation (3-4.2) re& in expression (3.4.1 1) for the optimum flux density

Bo.

Calculate the neces- area product using the expression (3.4.16) in which B, is

subst ituted with Bo.

Based on the area product calculated at step 4, a core is seiected with an area product

that is close to the one calculated. For this core the following parameters have to be

known: effective cross section area A, window area W,, the mean length tum MLT,

and the effective voIume of the core V,. Sometimes, an exact or even near match is not

possible. This will have an impact on the minimum l o s condition imposeci at the

beginning of the design method.

Calculate the number of !uns for the prirnary and secondary windings. Select an integer

number of turns in the primary close to the one cdculated, so that an integer nurnber of

tums will result for the secondary winding, The nurnber of tums will be calculated

using the same expression as the one used in the classical design method.

Determine the current density in the windings J using expression (3 -4.17).

Design of Low Profile Transformas for Wigh Frequmcy Operation University of Toronto


8.

9.

1 O.

I I .

Calcuiate the nns value of the current in the primary winding 1, and the secondary

winding 1, using the same expressions as in the design rnethod described by McLyman

((3.2.3) and (3.2.7)).

Based on the values of the currents calculateci at step 8 and the current density

calculated at step 7, detemine the necessary cross-section area of the wire for the

primary winding A, and the secondary winding Am using the expressions (3.2-5) and

(3.2.8) fiom the design method descriid by McLyman.

Select Litz wires for the primary and the secondary windings.

Calculate the dc copper loss.

The parameters in (3 -4.1 9) are defined as fo llo ws:

N, - is the nurnber of tum in the winding.

p, - is the resistivity of copper at 20°C.

a 2 0 - is the temperature coefficient of resistivity at 20°C.

T, - is the maxllnum temperature.

12. Recalculate the flux density for the selected number of tums in the primary winding

using expression (3 -4- 1 a), and based on this value determine the core loss.

13. Calculate the factors k, and kr that take in consideration skin a d proximity effects.

First, the windings made of Litz wire must be approxïmated with foi1 layers as shown in

section 3.3. The k, fàctor, that takes in masideration the skin effect, is calculated with

the fo Do wing expressions:



where:

r, is the radius of one strand.

6 is the skin depth,

The prorcimity effect factor k. for m layers is given by Doweli in 1251.

where:

m isthe numberoflayers.

A is the ratio of the thichess of a layer of foi1 to the skin depth.

14. Determine the total copper loss, using the fàctors k, and k, calculated before.

Pm =k,-k;P,,

1 5. Calculate the total loss in the transformer PI, and the eficiency 0.

pi'x =Pau +Pr,

where S, is the rated power of the transformer.

One interesthg thhg can be noticed when compare this design method with the

previous ones. This design method considers the temperature rise AT a directly design

Design of Low Profile Transformas for High Frequaicy Operatian

- - - - - - - -


Chapter 3 : Revue of Available Design Methods 37 -

parameter. It can be seen that this parameter appears in the expression of the optimum flux

density, the expression of the area product, and the expression of the current density.

3.5 Cornparison of the Results Using the Design Methods

Described in Sections 3 .2,3 -3, and 3.4

In the last three sections of this thesis three available design methods for tramformers

that operate at high fiequency have been presented. The most widely used transformer

design method is the classical design togeiéer with the copper bss calculation desc~l'bed by

Vandelac and Ziogas in Cl91 since it is simple, but also complete. McLyman's method is

simpler, using empmcai factors to wrrect losses. On the other band, the optimal design

method appears to be attractive. It is therefore of interest to compare these methods for a

specific design case.

The purpose of this section is to m e r to the following questions:

1). How do the results for a specific design dif5er when these design methods are used?

2). How accurate are these methods in terms of predicting losses?

To answer these questions, the design resiitts for a 54VA transformer using the design

methods d e s c r i i in 3 -2, 3 -3, and 3 -4, together with the Finite Element analysis performed

on these designs are presented in Tables 3.1, 3.2, and 3.3. The core that was chosen is the

power pot core or the DS cors, as it is known in the techaicd literature. This core was

chosen for the following reasons: it bas a high power density, very good electromagnetic

shielding and the windings are very easy to manufacture. The magnetic material used for

these designs is K material manufactured by ~ a g w t i c s . ~ The reason this material was

selected is that it bas a low specific l o s at 500kHz operating hquency and it will yield a

transformer with a hi& power density- The Litz wire used for designs is round Litz wire

mmufactured by New England Electric Wiie corporation6

5 Reference [23j, page 2.4 Reference [26]

Design of Low Profile Transfotmers for High Frequency Operation University of Toronto


The ratings of the transformer are:

- operating fhquency 5ûûkHz

- voltage in the primary winding 18Vrms square wave.

- current in the primary winding 3Arms Wamental plus 20% third harmonie. - tum ratio 2.5 step-down with center tap on the secondary winding.

- 50°C operating temperature.

The Finite Element software package that was used is PC-OPERA issued by Vector

Fields. PC-OPERA is a suite of programs for 2-dimensionai electromagnetic field d y s i s .

The programs use finite element method to solve the partial dzerential equations that

describe the behavior of fields. The solution of these equations is an essential part of design

in rnagnetostatics, electrostatics and e l e c t r o ~ t i c s . S ince much information is required

before the analysis may be @orme& data entry is carried out using a powerfûl interactive

preprocessor.

Using the graphical interaction within the preprocessor, the mode1 space was divided

into a contiguous set of triangular elements. The physicaI model may be describeci in

cartesian or cylindrical coordinates. Once the model bas been prepared, the solution is

calculated using a suitable analysis module. The analysis program iteratively determines the

correct solution induding non-linear effects if these are modeied. The result may then be

examined using a versatile interactive postprocessor.

The power pot core was modeled in this software in cylindrical coordinates. The

element type that was used to generate the mesh was the triangular linear element. The

rnagnet ic material was considered nonlinear. A separate analysis was done with Litz wire to

see how much the resistance increases due to skin and proximity effects. It was noticed an

increase of the ac resistance of 1 -05 times the dc resistance for the Litz wire used in the

design In Appendix E, a Finite Element adysis mode1 that was used in the calculation of

the core l o s for the transformer that was designeci using the optimized design method is

shown.

The following observations can be made regarding tbe data fiom Tabies 3.1 to 3.3:

1) When using the design method presented in section 3.4, a bigger wre wiii be yield than

if the methods presented in 3.2 and 3.3 are used. Consequentiy, the transformer

--

Design of Low Profile Transf'mers for Hi@ Frquency Operation University of Toronto


designed using the optimized design method has a power density that is almost half the

power density obtained when using the design methods descriid in 3.2 and 3 -3. This

cornes fiom the fàct that the optimized design rnethod minimizes the 105s in transformer

and a bigger core d l have to be used-

2) The dominant losses for ail designs are the copper losses, Looking at tables 3.2 and 3.3

it can be seen that the copper losses are roughly seven times the core losses. This

happens because of the drawback caused by the tu=-ratio. The condition to have an

integer number of tum in the primary and secondary windings leaâs in selecting a

larger number of tum in the primary and secondary windings. Consequently, the

operating flux density will decrease and the core loss will decrease as weil. On the other

band., the copper loss will increase as a result of increasing the length of the winding.

3) When using the design method presented in 3.4 the total loss is lower than when using

the design methods presented in 3 -2 and 3.3- Lookmg at Tables 3.2 aud 3.3 it is noticed

that the total l o s for the design using the method presented in 3.4 is about 1.4 times

1ower than when using the methods presented in 3.3 and 3.4. Hence the efficiency of the

transformer increases h m 99.2% when using the method presented in 3.2 and 99.3%

when using the method presented in 3-3 to 99.5% when using the method presented in

3.4. These high efficiencies are validated by the fact that there were magnetic

components built before that bave similar efficiencies. For example, in [12] are

presented the experimental results for a low profile inductor and transformer. It is

noticed that the efficiency of the transformer is 99.5%, which is the eficiency that was

obtained for the optimïzed design.

4) If Finite Element analysis is pedormed on the designs yield with the three methods

describeci above, the total loss for the design using the rnethod described in 3.4 is lower

than when using the design methods presented in 3.2 and 3.3. Looking at the results

presented in Tables 3.2 and 3.3, it can be observed that the total l o s for the Finite

Element analysis perfiiormed on the design using the method presented in 3.4 is about

two times lower than for the Finite Element analysis performed on the designs using the

Design of Low Profile T r a n s f i a s foc High Frequaicy Operation University of Tacmto


rnethods presented in 3.3 and 3.4. Hence the efficiency of the transformer incteases

from 99.3% when using the designs performed with the methods presented in 3 3 and

3.3 to 99.7% when using the design performed with the method presented in 3.4. This

shows that the resuits of the Finite Element analysis follow the theoretical resuits of the

h e analytical methods.

5) The total losses given by the Finite Element anaiysis are lower than the losses predicted

by the anaiytic methods. The Finite Element analysis is more accurate than the

adytical methods because it calculates the magnetic flux density in almost each point

inside the core. Hence, it results that the design methods overestimate the losses. In

order to see if this is valid at higher muencies, a tramformer operaihg at 1 MHz was

designed using the design method d e s c r i i in section 3.2. The calculateci core loss was

8.8 mW- When the Finite Element analysis was performed on the design, the core 1 0 s

was 5.3 mW, which confirmed that the analytical method overestimates the loss.

Design of Low Profile Tmnsfimners for High Frequency Operation University of Tamto


Table 3.1 Design Results

Mount Area [cm2 ]

4.10

Volume [cm3 1

Power Density

~A./crn3 ]

Design Method

Core Type

McLyman (section 3.2)

McLyman + Vandelac

(section 3.3) Optimized

Tram former Design

(section 3.4)

Table 3.2 Losses and Efficiencies for McLyman and McLyman +Vandelac Design Methods

Design Method [tes la] F-18

McLyman (section 3 2 )

McLyman + Vandelac

(section 3.3)

Finite Element Analys is

Table 3 -3 Losses and Efficiencies for Optimized Transformer Design Method

Optimized Transformer

Design

E fficiency I%I

Design Method

ffe rw

Element Analysis

Ptot Lw3

- - - - -

Design of Low Profiie Transformas fof Hi& Frequeacy Operatioa

Core Type


Bave [tes la]

PCU ['KI


3.6 Conclusions

The purpose of this chapter was to present three available design methods that are bIised

on the materiai presented in Chapter 2. The results for a specific design using these design

methods together with a cornparison with Finite Element analysis resuits were also

presented here in order to see if these methods are accurate in terms of predicting losses.

Finally, the fo llowing conclusions can be drawn:

1). Using the optunized design method a bigger core wiii be r e q d h m the design

but the efficiency of the transformer will be higher than if the rnethods presented in 3.2 and

3.3 are used. The increase in efficiency cornes nom the fact that the optimal design metliod

minimïzes the total loss in transformer. That means the price that has to be paid to have a

higher efficiency is to use a bigger core.

2). From the results of the designs it can be noticed that the copper bsses are much

higher than the core losses. This is a result of the drawback caused by the tums-ratio. From

here it follows that there is no big gain in using a magnetic materiai with a very low specific

loss at this fiequency since the operating flux density is weii beiiow saturation.

3). The results of the Finite Element analysis follow the resuits of the analytical

calculations. This means the design methods have a good prediction of losses when are used

at high fiequency operation.

Design of Low Rofile Transfocmers for High Frequency Operation University of Toronto

Chapter 4: Design Curves for High Frequency T d o r m e r s 43

Chapter 4

Design Curves for High Frequency Transformers

4.1 Introduction

The design rnethods presented in Chipter 3 of this thais do not take the height of the

transformer into consideration. In certain applications, this parameter represents a significant

design constraint. For example, if a resonant converter bas to be designed, and the converter

has to fit in a Limited space, the height of the transformer becornes an issue of the design.

The purpose of this chapter is to develop an algorithm that generates design curves for a

class of high fiequency tninsfbrmers that are kight constrained. In order to develop this

algonthm, it is nrst necessary to identify the parameters tbat cm be used as input. These

parameters are: the height of the transformer 4, rated apparent power S, efficiency q,

operating fkquency f, and voit-per-tum VT. AS an output, the other physical dimensions of

the transformer, together with the current density in the windings, will be obtained.

The key starting point in developing this algorithm is that an optimal design can be

achieved when copper and core losses are in a certain ratio dictated by the constants that

characterize the qecific l o s curve of the magnetic material. Using this condition, together

with the fact that the total loss in a transformer is equal to the sum between the core and

copper losses, a system of two equations will be obtained. Solving this system of equations

and using the conditions set for the geometrical dimensions of the core, the other dimensions

of the transformer together with the current demity in the windings will be obtained.

A set of design curws generated with the proposed aigorithm is presented at the end of

this c hapter. A dimission regardhg the trade-off s when using different magnetic m a t e s ,

different operaîing fiequencies and different efficiencies is also presented here.

Design of Low Profile Transformas for High Frquency Operation University of Tamto

Chapter 4: Design Curves for High Frequency Tramdiormers 44

4.2 The Choice of the Core and its Geomeîrical Dimensions

The algorithm will be developed for a specific type of core. The core that was chosen is

the power pot core or the DS core, as it is known in the technical literatute. This core was

chosen for the foiiowing reasons: it bas a high power density, very good electromagnetic

shielding and the windings are very easy to manufacture.

The top view of one halfof a DS core can be seen in Figure 4.1. Usually, for this type of

core, the average value of the angle a is 42 degrees (see [21]). The magnetic flux hes will

cross the outer walls through the area A2. This magnetic £lux wïii cross the center post

through the area Ai. It c m be noticed that the area Ai is d e r than the total cross-section

of the center post This happens because the magnetic flux lines tend to follow the path with

the lowest reluctance. Of course there wiil be flux lines that WU cross the remainder of the

center post d c e as weil, but the concentration of these lines will be very low. Hence, one

can consider, with a good approximation, that the magnetic flux lines are concentrated only

in area Al.

A cross section of the whole core is presented in Figure 4.2. In order to have the core

weli designed fiom a magnetic point of view, the foilowing conditions should be imposed:

1. The height of the window area hW will be proportional to the height of the transformer h(

by a coefficient &.

Consequently, the thickness of the top and bottom plates will be:

2. In order to have a magnetic flux density in the top and bottom plates equal to or smaller

than the magnetic flux density in the center pst, the following condition must be

irnposed:

A, = A3 (4.2.3)

Design of Low Profile Transformas foc Hi& Frequaicy Operation University of Toronto

Chapter 4: Design Curves for High Frequency Transfomers 45

Figure 4.1 DS core - top view

Figure 4.2 DS core - vertical cross section


Chapter 4: Design Cumes for High Frequency Transformers 46

where A3 is the surfàce area of a cyiinder that has the radius of the anter post R, and the height equal to the thickness of the top or bottom plate t, as it is shown in Figure

4.2. Thus,

3. Ln order to have the same mgnetic flux density in the center pst and in the outer wall, it

is necessary that:

Therefore, the effective area of the center p s t where the magnetic flux circulates is:

Based on these expressions, an expression for the core loss wiii be developed in the next

section-

4.3 The Expression for Core Loss

The objective of this section is to generate an expression for the core loss of a

transformer that uses a DS çore and has an arbitrary volts-pet-tum Vp/Np in the primary

winding. This expression will be firrther used in the algorithm to calculate the other

dimensions of the transformer and the current density in the windings.

The magnetic flux density in the effective area of the center pst can be considered

constant and equal to:

Design of Low Profite Transformas for High Frequency O p d o r i University of Toronto

Chapter 4: Design Curves for High Frequency Tramformers 47

where:

fis the operathg fhquency in lrHz

kptr is the effective cross-section area of the center p s t in cm2.

Vp is the rms voltage in the primary winding in volts.

The expression (4.3.1 ) can be written as:

where VT is the voit-per-tum of the winding.

If R, is substituted with expression (4.2.4), one obtains:

- 10-V, [tesla] Bq - 4 - f -0.466-R-h: - ( I - K , ) ~

Magnetic materiais rnanufacturers generally give the specific power loss for a f d e . For

examp le, one specific manufacturer7 has develo ped the fo llowing expression:

where:

fis the fiequency in kHz.

B, is the magnetic flux density in kgauss.

a, c, and d are coefficients that descnk the specific loss curve of the magnetic

material.

7 Reference [23], page 2.1 1

Design of Low Profile Transformas for Hi@ Frequaicy Operation University of Toronto

Chapter 4: Design Curves for High Frequency Trdormers 48

SpeciQing the flux density B, in tesia, (4.3.4) becomes:

Expressions for the power dissipation in the center pst, the outer waiis, and the top and

bottom plates can be developed ushg (4.3.5) and the expressions for the core dimensions.

These losses can then be summed in order to detennine the total core 105s.

The power dissipation in the center pst will be

center p s t muftiplieci by the specific power loss of the

Substituting (4.2.7) and (4.3.5) in (4.3.6), one obtains:

equal to the effective volume of the

magnetic material.

Cm Wl

However, R, and B, are given by (4.2.4) and (4.33, respectively. Making these

substitutions in (4.3.7) and sirnpli-g the resulting expression, one obtains:

The power dissipated in the outer wali will be e q d to the power dissipated in the

center post from the condition to have the same flux density in the center pst and the oiiter

wall (see condition (4.2.5)).

in order to calculate the power loss in the top and bottom plates, it is necessary to kaow

the value of the flux density in the top aml bottom plates. The basic assumption is that the

magnetic flux has to remain the same in ail-msgnetic path; there is no leakage f?om the core.

This is a reasonable assumption since the core operates well below saturation.

The magnetic flux in the center pst will be:

Design of Low Profile Transformas fot High Frequency Operation University of Tamto

Chapter 4: Design Curves for High Frequeocy Transformers 49

In the top a d bottom plates the magnetic flux at a distance r &om the center will be

equal to :

where Br is the flux density at distance r from the center as it is shown in Figure 4.3-

Figure 4.3 Flux deasity Br at distance r from the center.

The condition for the two fluxes to be equal can be written as follows:

q,, =O, (4.3.1 1)

Using (4.3.9) and (4.3.10) to equate 0, and 0- the foUowing expression is obtained:

Using (4.2.2) and (4.2.4) to substitute for t,, and R, an expression for the flux density in the

top and bottom plates as a function of the flux density in the center post is obtained:

Design of Low Profile Transfomers for High Frequency Operation University of Toronto

Chapter 4: Design C m e s for High Frequency T d o r m e r s 50

In order to calculate the loss in the top and bottom plates, it is first necessary to calculate the

power Ioss in an infinitesimal volume dv that is situated in the magnetic path. Let dv be an

in£initesimal volume at distance r fiom the center.

The power loss dpl in dv wiIl be eclual to the specific power loss multiplied by the volume

dv. It can be assumed that for a s m a l l dr, the flux density Br is constant inside the volume

dv-

dp, = PL -dv (4.3.15)

Using (4.3.5) and (4.3-14) to substitute for the specific power loss and volume dv, one

obtains:

a dp, = a - f c - ( l ~ - ~ r ) d .4---n-t , - r -dr

180"

Substituting (4.2.2) and (4.3.13) in (4.3.16) the foiiowing expression is obtained:

a 1-K, n-h, --- r-dr

r r

Finally, substituting (4.3 -3) in (4.3.1 7) and simpliQing the resulting expression, one obtains:

---- - --

Design of Low Rofile Transformas for High Frequency Operatiori University of Toronto

Chapter 4: Design Curves for High Frequency Transformers 51

The power dissipation in the top or botîorn pkte can be found integrating the power

dissipation in the volume dv over the entire pkte.

Substituting (4.3.1 8) in (4.3.19) and integrating between R, a d Rw the total 1 0 s in the top

or bottom plate is obtained-

The total power dissipation in the core wiU be equal to:

The total power dissipation in the core will be f i d e r used in the algorithm to be able to

calculate the other dimensions of the core and the current density. If the purpose of this

section was to yield an expression for the core loss, in the next section an expression for the

copper loss will be developed.

4.4 The Expression for Copper Loss

It can be demonstrated that the design of a winding with an arbiîrary voltage d

number of tum is quivalent to the design of a one-- winding having the same volt-per-

tum and power rating. For example, consider the design of a winding with N tums h g a

resistance R, voltage V, and power P. The rated current for this winding will be 1 = PN. The

winding loss will be RI^ anà its core loss will be deterrnined by the volt-per-him VIN. Ushg

the same type of core, consider the problem of designing a wiading that bas only one tum,

the same volt-per-han as the wioding with N tums, and the same power P. The two

windings (singie turn and N turns) shouId also have the same current density. The resistance

Design of Low Profile Transfmers for Hi@ Frequency Operatiori University of Tamto

Chapter 4: Design Cumes for High Frequency Transfomers 52 - --

of the one-tiim wiading WU be & because the single tum has 1/N times the length a . N

times the area of the N-turn winding. Since both windings have the same volt per tu- the

core iosses will be the same in both cases. If the power ratings of the two windings are the

same, the current in the one-tum winding would be NI. Therefore, the winding 10% in the

one-tum winding is (R/N2)O2, or lU2 , which is the same as the wiading loss in the N-han

winding. Since an N-tum aad one-tum design, each with the same volt-per-tum and power

rating and using the same fore, have the sarne wioding a d core losses, they are equident.

Having considered this, an expression for the copper loss, considering one equivalent tiim in

the transformer, will be developed It will be assumed that the modings are made of Litz

wire and the skin and prolomity effects are not present. In this case, the AC copper l o s will

be equal to the DC copper loss.

The total apparent power m the one equivalent tum bas to be equal to the apparent

power in the prirnary plus the apparent power in the secondary of the tlansfonner. This can

be written as:

where:

S, is the rated apparent power of the transformer.

q is the efficiency of the transformer.

The apparent power of the one equivalent turn will be also equal to:

VT is the voit-per-hum

ITq is the equivalent current tbat WU produce the total power.

Design of Low Profile Transformas for High Frequaicy Operation University of Taonto


From (4.4.1) and (4.42), it is possible to determine the expression for the equivaient

current as a function of the power delivered by the secondary whding, the volt-per-han, and

the effic ienc y:

The resistance of the single equivalent tum wiü be:

where:

& is the radius of center pst (see Figure 4.2).

Rw is the radius of window area (see Figure 4.2).

p, is the resistivity of copper.

& is the cross-section area of the one equivaient turn and is given by :

where :

hW is the height of window area (see Figure 4.2).

K, is the window utilization factor (see Appendix A for calculation of Ku).

The power loss dissipation in the O ne equivalent tum will therefore be:

The expression for the copper loss developed in tbis section together with the

expression for the core loss developed in 4.3 will be used in the next section to calculate the

other dimensions of the transformer and the current density in the windings.

Design of Low Profile 'irausformers for High Frequaicy Operatim University of Toranto

Chapter 4: Design Curves for High Frequency Tdormers 54

4.4 Calculation of the Power Density and Current Density as a

Function of Transformer Height

It has been demonstrated in 1191 that the minimum fosses in a high fkquency

transformer occur when:

for a fixed fkqueucy f where d is a coefficient in (4.3.5) which govems the core loss

density.

The total power loss in a transformer is equal to the core loss plus copper los.

From (4.5.1) and (4-5.2) it foliows that:

P,- = (d + 2) - P, and

Using (4.3.8) and (4.3.20) to substitute for Pq and Ppi, in (4.3.21), a d after that

subst ituting the r d i n g expression in (4.5.3), the fo Mowing expression d l be obtained:

Substituting (4.4.6) in (4.5.4) d l resuit in:

Design of Low Rofile Transformas fot High Frequaicy Operation University of Toronto


Equations (4.5.5) and (4.5.6) represent a system of two eqdons with two unlmown

geometricai factors & a d Rw To solve this system, Rw will be expressed as a function o f

&,, using expression (4.5.6) and resulting in expression (4.5.7).

Using (4.5.7) to suMitute for Rw in (4.5.5), the foliowing transcendental equation with one

4 - n - a - a I-d -fs-d - ( I - K h ~ - d P,, =(d+2)- +

360 - (2 - d): ht

Equation (4.5.8) has one uakoown I(b. This equation can be solved numeridy for a

range of values of the height of the transformer ht aod h d out how does &, Vary with the

height 4.

Design of Low Profite T d m e r s for High Fre~uarcy Operation University of Toronto

Chapter 4: Design C w e s for High Frequency Transformefi 56

M e r kt, the red thg values of &, can be substinited in expression (4.5.7) for the

same range of values of the height of the transformer. T'us, it is possible to determine how

the inner radius of the window area R, varies with the height of the transformer.

The next step would be to calculate to calculate hW, R, and &, for the same range of

values of the height of the transformer & using expressions (4.2.1 ), (4.2.4), and (4.2.6).

The mounting area, the power density, and the current density wiii be calcdated using

the following expressions:

The expressions developed in this section will be used to generate a set of design curves

for a specific application. These curves, together with some observations, are presented in

the next section

4.6 Example of a Set of Design Cuwes

In the previous sections, an algorithtn was developed to generate design curves for

transfomers that operate at high fkquency. The power rating, the efficiency, the operating

fiequency, the volt-pet-turn, and the height of the transformer were used as input for the

algorithm. The power density, mounting area, and the current density in the windings were

obtained as output.

This section will present a set of design curves for a specific power leveL These design

curves represent the variation of pwer density, mountiog area and current density, as a

function of the height of transformer. The design curves have been generated for two

Design of Low Profile Transformas fot High Frequeacy Operation University of Toronto


efficiencies (99.0% and 99.5%), two kquencies (500kHi anci IMHz), two materials (an

expensive material - K material and a cheap material - F material)', and for four values of

volt-per-turn (2,3,4, and 5 volt-per-tum).

4.6.1 Variation of Power Density with Height

The variation of power density with height for a high fiequency transformer that uses a

DS core can be seen in Figures 4.4 to 4.1 1. Clearly, there is an optimal core height at which

the transformer power demity reaches a maximum. In Table 4.1, peak vahes of power

density, together with the heights where these peaks occur, are Summarized. The variation of

the height was taken between zero and 3 cm. The upper limit for the height is set t 3 cm

because the primary interest is in low profile. The following observations c m be made:

1) As noted above, for any of the situations presented in Figures 4.4 to 4.1 1, there is a

criticai height where the power density has a peak. Above this critical height, the power

density decreases slowly but below it, degrades drastically.

2) It can be noticed that for the designs with lower efficiency (99.0%) the peak of the

curves is sharper than the peak for the designs with higher efficiency (99.5%). This means

the design with a higher efficiency bas a wider range for height to yield a transformer with a

power density close to the maximum,

3) From Table 4.1 it can be noticed that the peak occurs at a lower height when a lower

efficiency is wanted (99.0%) and at a higher height when a higher efficiency is wanted

(99.5%).

4) For a range of 2 to 5 volt-per-tuni, the peak of the power density occurs for the most

cases at 3 volt-per-tum.

5 ) When using the same operatmg ikquency and the same magnetic materiai, the peak

of the power density is lower for the design with higher efficiency, and higher for the design

with lower efficiency. For example, in Table 4.1 it can be noticed that when the K material

is used in the design, the peak of the power density is approximately thtee times higher for

the design with 99.W efficiency than the design with 99.5% efficiency. In the same table it

*K and F materiah are rnanufictmed by Magnetics The charaderistics of these materials are shown in Appendix F.

Design of Low Profile Transformas for High Frequency Operation University of Toccmto


c m be wticed that when the F rnateriai is used in the design, the peak of the power density

is approximately five times higher for the design with 99.0% efficiency than the design with

99.5% efficiency. With other words, it can be said that if the total loss is doubied, the peak

of the power density increases three times when the K maîerial is used and five tirnes when

the F material is used.

6) For the same materiai and the same efficiency it is mticed an increase in power

density when the operating kquency is increased. In Table 4-1 it is noticed an increase in

power density of roughly 1.4 times when the operathg fkquency increases fiom 500kHi to

lMHz

7) For the same design conditions (same efficiency and operatiog kquency) but using

different magnetic materials, it has been aoticed an increase of the power densïty when an

expensive magnetic material is used than when a cheap -tic material is used. This

cornes fiom the k t that expensive ferrites bave a lower specific l o s at high Gequency

operation than cheap ferrÏtes. For example, it can be noticed in Table 4.1 that for a 99.0%

efficiency and 500kHz or lMHz operathg fkquency, the peak of the power density when

using the K material is two times higher than when using the F materiai. Also, for a 99.5%

efficiency, the peak of the power density increases three times when the K materiai is used

than when the F materiai is used,

Table 4.1 The peak of the power density for ciifferent materials, efficiencies and Eequencies

Mag net ic Material

Efficiency [%]

Design of Low Profile Trausformas fot High Frquency Operatioa University of Toronto

operating Frequency ml

Peak of the po wer density

~ A / c m 3 ] Optimal Height

km]

p.

K

500

37

0.95

F

99

1000

51

0.87

99 99.5 99.5


Power Density, 54VA, K mat, 500kHz, 99% 40 1

O 0 2 0-4 0.6 0.8 1 12 1.4 1.6 1.8 2 2 2 2.4 2.6 2.8 3 3 2 O t0.k I * h t ~ . k ~ ' ~ W . k ~ 3 3

height [cm]

Figure 4.4 Variation of power density with height at 500kHz, 99% eff., and K material

14 Power Density, S4VA, Kmat, SOOk.Hz, 99.5%

O 03 0.4 0.6 0.8 1 1 2 1.4 1.6 1.8 2 2.2 2.4 2.6 2.8 3 3 3 O t0.k I'h t ~ . k ~ ~ ~ tO,kjvh f0,k4 3 3

height [cm]

Figure 4.5 Variation of power density with height at SOOkHz, 99.5% eE, and K material

Design of Low Profile Transfiers for High Frequaicy Operation University of Toronto

Chapter 4: Design &es for High Frequency Transfomiers 60

Go 60 Power Density, 54V4 K mat, lMHz, 99%

O 0.2 0.4 0.6 0.8 1 1 2 1.4 1.6 1.8 2 2.2 2.4 2.6 2.8 3 3 3 O h t ~ , k I*ht0,k2'ht~.k3'ht0,k4 3 3

height [cm]

Figure 4.6 Variation o f power density with height at 1 MHz, 99% eE, and K material

Power Density, 54VA K mat, IMHz, 99.5%

O O 2 0.4 0.6 0.8 1 1 2 1.4 1.6 1.8 2 2.2 2.4 2.6 2.8 3 3 3 O t0.k I*h t ~ , k ~ * ~ t ~ . k ~ ' ~ t0.k4 3 3

tieight [cm]

Figure 4.7 Variation o f power density with height at lMHz, 99.5% eff., and K matenal

Design of Low Profile Transfotmers for High Frequency Operation University of Toronto

Chapter 4: Design Curves for High Frequency Transfomrs 61

>-- - 3 dens-4V0, k .- Cr.

'Of Power Density, S4VA F mat, SOOZcHz, 99%

O 0.2 0.4 0.6 0.8 1 12 1.4 1.6 1.8 2 2 2 2.4 2.6 2.8 3 3.2 O 4),k l w h t0 ,k2'~ t 0 . t ~ ~ ~ t 0 . k ~ 3 2

height [cm]

Figure 4.8 Variation of power density with height at 500- 99% eK, and F material

Power Density, 54VA, Fmat, 500kHiq 99.5%

O 0.2 0.4 0.6 0.8 1 12 1.4 1.6 1.8 2 2 2 2.4 2.6 2.8 3 3 2 O 10,k l * h t 0 . I ~ ~ ' ~ t ~ , k ~ * ~ tO,kq 3 2

height [cm]

Figure 4.9 Variation of power density with height at SOOkHi, 99.5% eE, and F material

Design of Low Profile Transformas f i High Frequmcy Operation University of Toronto


Power Densitvnsitv 54VA F mat, lMHz 99??

O 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 2 2 2.4 2.6 2.8 3 3 3 O r ~ , k l'h t0,kZvh t ~ . k ~ ' ~ t0,k4 3 2

height [cm]

Figure 4.10 Variation of power density with height at IMHz, 99% eE, and F material

Power Density, 54V4 F mat, IMHt, 99.5%

O 0 2 0.4 0.6 0.8 1 1 2 1.4 1.6 1.8 2 2 2 2.4 2.6 2.8 3 3.2 O htO,k I ' h t ~ , k 2 ' h t ~ . k j ' h t ~ . k 4 3 3

height [cm]

Figure 4.1 1 Variation of power density with height at 1 MHz, 99.5% eE, and F materiai

Design of Low Rofile Transformas for High Frequaicy Operation University of Tamto


4.6.2 Variation of Mounting Area with Height

Besides the variation of the power density with the height of the transformer, another

important thing is to see how the mounting area varies with the heigk Figures 4.12 to 4.19

show the variation of the mounting area with height for a transformer with a specific power

level, for dinerent opaeting muencies, diffèrent efficiencies and different rnagnetic

materials. It is a h interesthg to see how the mounting area that corresponds to the peak of

the power density varies when dinerent materials or ditferent operating fiequedes are used

or different efficiencies are wanted. In order to notice this variations, Table 4-2 shows the

values of the mounting area for the heights where the peaks of the power d e n s e occur.

Looking at Figures 4.12 to 4.19 and Table 4.2, the fo llowing observations can be made:

1) The design curves presented in Figures 4.12 to 4.19 have similar shapes. At low

heights the mounting area is very big. As mon as the height increases, the mounting area

decreases rapidly until reaches a minimm If the height fùrthet increases, the mounting area

increases slowly remaining almost constant especially for low volt-per-tum. It should also

be noticed that the minimum mountiog area does not occur at the peak of the power density.

In fact, it occurs at a bigger height-

2) It can be noticed that above the height where the minimum occurs, the mounting area

is smaller for the design with a low volt-per turn, and bigger for the design with a high volt-

per-tum.

3) When using the same operating fkquency and the same rnagnetic material, the

mounting area is s d l e r for the design with lower efficiency, and bigger for the design with

higher efficiency. This happens due to the fact that at a lower efficiency the power density is

higher than at a higher efficiency. Hence, the volume of the transformer is lower and the

mounting area will be smaller. As it is wticed in Table 4.2, the mounting area increases

about 2.5 times when K material is used and 2.8 times when F material is used when the

efficiency increases ffom 99% to 99.5%.

4) For the same design conditions (same efficiency and operating fiequency) but using

different magnetic materials, it has been wticed an imxpase in mounthg area when a cheap

magnetic materiai is used instead of an expensive material. It is mticed, when looking at the

data presented in Table 4.2, tbat for the same design conditions when the F material is used

- - --- - -- - - -

Design of Low Profile Ttansformers for High Frequency Operatiori University of Toronto

Chapter 4 Design Curves for High Frequency Transformers 64

in the design, the mounting area increases roughly tmce compareci to the one obtained when

the K material is used.

5) For the same magnetic matenal and the same efficiency it is noticed that the

mounting area slowly decreases when the operating fkquency is increased In Figures 4.12

to 4.1 9 it can be noticed a decrease in mounting area of roughly 1.25 when the operating

frequency is increased h m 500kHz to 1 MHz

Table 4.2 The values of the mounting area that correspond to the peak of the power densities

Efficiency [%]

operating Frequency

m l Optimal Height

km1

Mounting area Lr

Design of Low Profile Transformas fa High Frquency Operaîion University of Tacmto


O 0 2 0.4 0.6 0.8 1 12 1.4 1.6 1.8 2 2 2 2-4 2.6 2.8 3 3 2 O t0,k l'h t ~ . k ~ ' ~ t ~ . k ~ ' ~ t ~ , k ~ 3 2

height [cm]

6 Mounting Area, 54V& K mat, SOOkHz, 99%

Figure 4- 12 Variation of mounting area with height at SOOkHq 99% eE, and K material

5

4

3

2

1

Mounting Are% 54Vk Kmat, SOOkHz, 99.5% & \ Z

I: \ " \ s ..

_*__________*--*----_.______________________-_._______________________.______________________ ___-------- - * - - - - - - - - - - - O - - - -

- - - -

O O 2 0.4 0.6 0.8 1 13 1.4 1.6 1.8 2 22 2.4 2.6 2.8 3 3 2 O h t ~ , k I ' h ~ ~ , k 7 ' h t 0 , k 3 ' h W,k4 3 2 -

height [cm]

Figure 4.1 3 Variation of mounting area with height at 500- 99.5% eE, and K materiai

Design of Low Profile Transformes for High Frequency Operation University of Tamto


O O 2 0.4 0.6 0.8 1 12 1.4 1.6 1.8 2 2 2 2.4 2.6 2.8 3 3.2

O htO,k I 'h t~ ,k7 'h t~ .k3vh t0,k4 3 3 - height [cm]

Figure 4.14 Variation of mounting area with height at IMHz, 99% eff., and K material

Figure

Mounting Area, S4VA K mat, IMHz, 99.5%

__-- - - - - __---*--

--

htO,k 1 ' h t ~ T k 2 * h t ~ . k 3 T h t0.k4

height [cm]

4.15 Variation of mounting area with height at IMHz, 99.5%

3 3

e E , and K material

Design of Low Profile Transfmers for Hi& Frquency Operation University of Toronto

Chapter 4: Design Curves for High Frequency Transfomrs 67

O 0 2 0.4 0.6 0.8 1 12 1.4 1.6 1.8 2 2 2 2.4 2.6 2.8 3 3 2 O h t ~ , k I * h ~ , k 2 p h t 0 , k 3 ' h t0,k4 3 3

height [cm]

Figure 4.16 Variation of mounting area with height at SOOicNz, 99% eE, and F material

O 0.2 0.4 0.6 0.8 1 1.2 1-4 1.6 1.8 2 2 2 2.4 2.6 2.8 3 3 2 O h ~ ~ , k I*ht~,k2*ht09kj*h 1 0 . k ~ 3 3

height [cm]

Figure 4.17 Variation o f mounting area with height at 500- 99.5% eE, and F rnaterial

Design of Low Profile TfatlSformm fa High Frequency Operation University of Toconto


O 0.2 0-4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 2 2 2.4 2.6 2.8 3 3 2 O t0.k l * h t0,k27h t ~ , k ~ ' ~ t0,k4 3 3

height [cm]

Figure 4.1 8 Variation of mouoting area with height at 1 MHz, 99% eE, and F material

O 0 2 0.4 0.6 0.8 1 12 1.4 1.6 1.8 2 2 2 2.4 2.6 2.8 3 3-2 O h t ~ , k l * h ~ 0 , k 2 * h t ~ , k 3 ' h t ~ , k 4 3 3

height [cm]

Figure 4.19 Variation of rnounting area with height at 1 MHZ, 99.5% eE, and F material

Design of Low Profile Transfmers fot High Frequaicy Operation University of Taonto


4.6.3 Variation of Current Density with Height

The algorithm presented in this chapter bas as an output, besides the power density and

mounting area, the current density that bas to be in the windings in order to obtain the

desired efficiency. The variation of the current densÏty with height for a transformer with a

specific power level is presented in Figures 4.20 to 4.27. Tt is also interesthg to see how

does the current density that corresponds to the peak of the power density vary when

different materials or different operating fiequemies are used or different efficiencies are

wanted- In order to notice this variations, Table 4.3 shows the values of the current density

for the heights where the peaks of the power density occur. Lookiog at Figures 4.20 to 4.27

and Table 4.3, the following observations can be made:

1) The design curves presented in Figures 4.20 to 4.27 have similar shapes. At low

heights the current density is low. As soon as the height inmeases, the current density

increases rapidly untii reaches a maximum If the height increases, the current

density decreases slowly remaining almost constant.

2) It c m be noticed that the current density is higher for a high volt-per-tuni and lower

for a low volt-per-tum. The reawn is that at high volt-per-tum, the length of the winding is

sbrter than the length of the winding at low volt-per-tum for two windings that have the

sarne induced voltage. Ln order to have the sarne power dissipation in the longer wuidhg as

in the shorter wioding, the current density in the windhg wÏth low volt-per-tum bas to be

decreased.

3) When using the same operating frequency and the same magnetic material, the

current density is lower for the design with higher efficiency, and higher for the design with

lower efficiency. In Table 4.3 it can be noticed that the current density is roughly 2.4 times

higher when the K material is used and 3.4 times higher when F material is used when the

efficiency decreases fiom 99.5% to 99%.

4) For the same design conditions (same efficiency and operating hquency) but using

dinerent magnetic materials, it has been noticed an increase in current density when an

expensive magnetic material is used instead of a cheap one. This wmes h m the k t that

when an expensive ferrite is used, the power density is higher than when a cheap f d e is

used. That means the dimensions of a core made of an expensive ferrite are smailer tban the

Design of Low Profile T d o n i i a s for High Frequency Operation University of Toronto


dimensions of a core made of a cheap ferrite. Hence, the length of the windings for a mre

made of an expensive ferrite is shorter tban the length of the windings for a core made of a

cheap ferrite. In order to obtain the same winding losses, the current deasity for the design

that uses a cheap material has to be decreaSed. It can be noticed in figures 4.20 to 4.27 that

for the core d e of F material the current density is almost haif of the one used in the

design that uses a core made of K material.

5) For the same magnetic material and the same efficiency it can be observeci thaî the

current density cemains almost constant when the operating kquency is increased. In Table

4.3 it is noticed an increase m -nt den- of roughty 1.1 times when the operating

fiequency inmeases fiom 5ûûlrHz to 1 MHz

Table 4.3 The values of the current density that correspond to the peak of the power

- - - - - - -

Design of Low Profile TratlSformers for High Frequency Operation

densities

University of Tamto

Magnet ic Material

Efficiency [%]

Operat ing Frequency WI

Optimal Height km]

Current Density

F K

99

500

1.1

12

99 99.5

1000

1

13.5

500

99.5

500

1.95

3.5

1000 500

1.25

1OOO

1.8

4

1000

1.18 0.95 0.87

7.5 8 17.5 20


O 0 2 0-4 0.6 0.8 1 1 3 1.4 1.6 1.8 2 2 2 2.4 2.6 2.8 3 3.2 O h t ~ , k 1 * h t ~ . k 2 ' h t ~ . k 3 ' h to.k4 3 2

height [cm]

Figure 4.20 Variation of current density witb height at SOOkFh, 99% eff., and K material

Figure

O 0 2 0.4 0.6 0.8 1 1.2 1.4 1-6 1.8 2 2 2 2.4 2.6 2.8 3 3 2 O t0.k l'b t ~ , k ~ * ~ f o , k ~ ' ~ Q,k4 3 2

height [cm]

4.2 1 Variation of current density with height at SOOkHz, 99.5% eff., and K material

Design of Low Rofile Transfiers for tligh Frequency Operation University of Toronto

Chapter 4: Design Curves for High Frequency Transformers

Current Density, 54V4 K mat, 1 MHz, 99%

O 0 2 0.4 0.6 0.8 1 1 2 1.4 1.6 1.8 2 2 2 2.4 2.6 2.8 3 3.2 O t0.k 1 ' h t ~ , k 2 * h t ~ . k ~ ' ~ t O . k ~ 3 2

height [cm]

Figure 4.22 Variation o f current density with height at lMHz, 99% eE, and K materiai

Current Deusity, 54V4 Kmat, IMHz, 99.5%

O 02 0.4 0.6 0.8 1 1 2 1.4 1.6 1.8 2 2 2 2.4 2.6 2.8 3 3 2 O t0.k t ~ , k ~ ' ~ t ~ . k ~ ' ~ t0,k4 3 2

height [cm]

Figure 4.23 Variation of current density with height at 1 Ml& 99.5% eff., and K material

Design of Low Profile Transformers fa High Ftequency Openiticin University of Toronto


Current Dens, 54V4 F mat, 500kf4 99%

O 0 2 0-4 0.6 0.8 1 12 1.4 1.6 1.8 2 2 2 2.4 2.6 2.8 3 3.2 O t0.k I'h f ~ . k ~ * ~ f ~ , k ~ * ~ t0,k4 3 2

height [cm]

Figure 4.24 Variation of current density with height at 500kH.q 99% e&, and F materiai

Figure

Current Dens, 54V4 F mat, 500- 99.5%

4.25 Variation of current

0.8 1 1 2 1.4 1.6 1.8 2 2 2 2-4 2.6 2.8 3 3 2

t0.k I*h t ~ . k ~ * ~ t 0 , k ~ ' ~ t 0 , k ~ 3 3

height [cm]

density with height at 500kHk, 99.5% efE, and F material

Design of Low Profile Transfmers for High Frequaicy Operation University of Taonto


20 Current Density, 54V4 F mat, IMHz, 99%

O 0 2 0.4 0.6 0.8 1 12 1.4 1.6 1.8 2 2.2 2.4 2.6 2.8 3 3 2 O t0.k I 'ht~.k2*ht0,k3'ht0,k4 3 2

height [cm]

Figure 4.26 Variation of current density with height at 1 MHz, 99% eff., and F material

Current Density, 54Vq Frnat, 1MHq 99.5% 6

O O 2 0.4 0.6 0.8 1 1.2 1-4 1.6 1.8 2 2 2 2.4 2.6 2.8 3 3 3 O t ~ . k ~ ' ~ 1 0 . k ~ ' ~ t0.k4 3 3

height [cm]

Figure 4.27 Variation of current density with height at MHz, 99.5% eE, and F material

Design of Low Profile Transformas for High Frequency Operaîion University of Toronto


4.6.4 Vaiidation of the Design Curves

The design curves expiicitly point to optimal core dimensions. However, sometimes, an

exact or even near match with available cores is not possible. This will have an impact on

the transformer optimization by decreasing the power density or increasing the temperature

rise.

In order to see if the design curves give diable data, a comparison of the design resuits

f?om Chapter 3, when the optimized design method was used, with the data 60m the design

curves generated in this chapter can be done. Looking at the d t s presented in Tables 3.1

and 3.3 it can be noticed that for the optimized design the height of the transformer is 1 -6 cm

and the power density is 6.6vNcm3. The resuhed volt-per-tum is 3.6 and the efficiency of

the transformer is 99.5%. Looking at Figure 4.5 it can be noticed that for a height of 1.6 cm

and 4 volt-per-turn it results a power density of 1 1 ~ ~ c m ~ . The reason the power density of

the optitnized design is lower is because a core with an area product double than the one

calculated was selected for the design. The selection of the core was irnposed by the fact that

both the secondary and the primary windings must have an integer nurnber of tums and the

windings must fit in the bobbin.

Assume w w that fiactional numbers of tunis cm be used in both windings. As a result,

a core with an area product equal to the calculated one can be used. This core will have a

height of 1.3 cm and an outer diameter of 2.2 cm. The resulted volt-per-tuni will be equal to

5.4. The resulted power density will be I 1 ~ ~ 1 c r n ~ . Looking at Figure 4.5 it can be noticed

that for a height of 1.3 cm and 5 volt-per-turn it results a power density of ~o . svA/c~~ , which is very close to the one found using the optimal design method. From here it results

that the design curves give reiiable data.

Design of Low Profile Transfonners for High Frquency Operation University of Taonto


The purpose of this cbapter bas been to develop an algorithm that generates design

c w e s for a class of high tiequency transfomers that are height constrained-

First, the pararrieters that can be used as input were identifïed as the height of the

transformer output power S, efficiency q, operating fkquency f; and volt-per-turn VT.

The type of the core that has been chosen is the power pot core. For this type of core, a set of

conditions for the geometricai dimensions of the core have been imposed in order to have

the core well designed.

Second, an expression for the core l o s and one for the copper loss have been

developed. These expressions are based on the conditions set for the geometrical dimensions

of the core.

Third, based on the condition for optimum design and the hct that the total l o s in a

transformer is quai to the sum between the copper l o s and the core los, a system of two

equations with two unknowns bas been obtained- Solving this system of equaîions and using

the conditions set for the geometrid dimensions of the core, the other dimensions of the

core together with the current density have been obtained.

Finally, a set of design curves generated with the proposeci algorithm for a specific

power level transformer has ken presented. A discussion regarding the trade-offs when

ushg diffierent magnetic matkials, différent operating fkquencies and diffetent efficiencies

has been also presented here.

Design of Low Profile Transformers for High Fquaicy Operation University of Toronto

Chapter 5: Conclusions and Future Work 77

Chapter5

Conclusions and Future Work

The present work has two main purposes in the field of the design of transformers for

high fkequency operation used in resonant converters,

The first purpose of this work was to investigate three available design rnethods for

transformers that operate ai high fkequency. The results of a design for a specific power

level using these methods together with a cornparison to Finite Element anaiysis resuhs

performed on the design d t s bas been done in order to see if these methods are accurate

in terms of predicting losses. Analyzing the design resuits the following conclusions were

O btained:

1). It bas k e n noticed tbat using the optimized design method a bigger core will be

yield f?om the design but the efficiency of the transformer will be higher than if the methods

presented in 3.2 and 3.3 are used This means that the pnce that has to be paid to have a

higher efficiency transformer is to use a bigger core-

2). From the resufts of the designs it bas been noticed that the copper losses are much

higher than the core losses. This is a result of the drawback caused by the tunis-ratio. From

here it follows that there is no big gain in using a magnetic material with a very low specific

loss at this fiequency since the operating flux density is well bellow saturation.

3). The results of the Finite Element analysis follow the results of the analytical

calculations. This means the design methods have a gmd preâiction of losses when are used

at high fiequency operation.

The second purpose of this tbesis was to develop an algorithm that generates design

curves for a class of high fkquency transfomers that are height constrained. The key

starting point in developing this algorithm was that an optimal design can be achieved when

Design of Low Profile Transformers for Hi& Frequaicy Operation University of Taonto


copper and core iosses are in a certain ratio dictated by the constants that characteriz the

specific loss cuve of the magnetic materiaL Using this condition, together with the fact tbat

the total loss in a transformer is equal to the sum between the core and copper losses, a

system of two equations will be obtaiwd. Solving this system crf equations and using the

conditions set for the geometrid dimensions of the core, the other dimensions of the

transformer together with the current density in the windings were obtained.

A set of design curves developed using the proposeci algonthm for a specifïc power I )

level bas been presented at the end of this work. These design cmves represent the variation

of power density, rnounting area and current density, as a fünction of the height of

transformer. The design curves have k e n generated for two efficiencies (99.0% and 99.5%),

two fiequencies (5001rHz and IMHi), two materials (an expensive materiai - K material and

a cheap material - F material), and for four values of volt-per-turn (2, 3, 4, and 5 voit-per-

turn). Looking at these design curves the foiiowing generai conclusions couid be d r a m

1). The design curves can be used for the design of high kquency transformers where

the height is a constraint.

2). For each design curve there is an optimal height where the power density has a

maximum, That means the design curves can be used for minimizing the size of a

transformer for a given set of design parameters.

3). Far a given range of the voit-per-tuni, there is a certain value of the voit-per-turn

where the power density bas a maximum. With other words, using the design curves, a

global minimization can be obtaiaed choosing a specific volt-per-turn and a specific height

that correspond to the peak of the power density.

4). Using a cheaper material in the design wiii result in a bigger transformer than using

an expensive material with a low specific loss. That means fiom the design curves it can be

seen what are the trade-off s when using dBerent magnetic materiais in the design.

5). For the same design conditions but using different opemathg hquencies, it bas been

noticed an increase in power density when the operating fkquency is increased. This rneans

fiom the design curves it can be seen what are the trade-off s wkn using dserent operating

fiequencies.

-- -

Design of Low Profile Tmsformers for High Fre~uency Operation University of Toronto


6). When a higher efficiency is wanted, a transformer with a lower power density will

result ftom design. Wth other words, the design curves show the trade-offs when difXerent

efficiencies of the transformer are wanted-

7). The design curves explicitly point to optimal core dimensions. However, sometimes,

an exact or even war match with avaihble cores is not possible. This wiil bave an impact ou

the transformer optmiization by deaeasing the power density or increasing the temperature

rise.

5.2 Future Work

There are several possible extensions of the work presented in this thesis:

1 ). In the proposed algorithm it was assumeci that Litz wire is used in the design and the

ac resistance is equal to the dc resistance. This is tnie for Litz wire until it reaches a certain

diameter. Mer that, the ac resistance increases more and the assumption is w t valid

anyrnore. It results that in the design cuves it shouid be taken in consideration the iacrPase

of the ac resistance d e r the diameter of Litz wire exceeds a certain value.

2). The algorithm was developed for a specific core configuration (the power pot core).

A similar algorithm can be developed for other core configurations.

3). The algorithm was developed using the efficiency of the transformer as an input

parameter. A similar algorithm can be developed using the temperature rise as an input

parameter instead of efficiency.

4). The biggest question relates to transformer design when the output voltage is low

and the current high The next generation of ICs wiil require power supplies at IVy 30A and

operating fiequemies of 1-2MH.z. In this situation, copper foils coawcted in paralle1

become an attractive alternative. A s i d a r algorithm can be developed using copper foils

instead of Litz wire, and considering a low volt-per-tum in the design.

Design of Low Profile Transformers fm High Frequaicy Operation University of Tamto

Appendixes 80

Appendix A

Calculation of Window Utilkation Factor &9

The window utilization fàctor K,, is a measure of the amount of copper thai appears in

the area of the transformer. The window u t w t i o n factor is influenced by four factors:

1. Wire insuiation

2. Wie lay (fil1 factor)

3. Bobbin area (or, when using a toroid, the clearance hole for passage of the shuttle)

4. Insulation required for multilayer windings or between windings

The fiaction K,, of the available core window space that wilï be occupied by the winding

(copper) is calculated fiom areas Si, St, S3, and Sq:

Ku =S, 4 , - S , - S , where:

conductor area S, =

wire area

- womd area ' 2

- usable window area

usable window area S, =

window area

usable window area S, =

usable window area + iasulation area ( A 3

in which:

9 Reference [la, page 25 1.

Design of Low Profile Tramformers for Hi& Frequency Operatiori University of Tamto

Conductor area = copper area

Wire area = copper area + iosulation area

Wound area = nwnber of tunis x wire area of one turn

Usabie window area = available d o w area - residuai area that resuits h m the particular

winding technique used

Window area = available window area

Insulation area = area usable for winding insulation

For Litz wÙe:

SI = 0.449

S2 = 0.60

S3 = 0.55 for pot core

S4 = 0.9 for one secondary

It wiil resuit:

K,,= 0.13

Design of Low Profile Transformers for Mgh Frequaicy Operation University of Toronto

Appendixes 82

Appendix B

The value of coefficients Kj, x, y, and K, for different core configurations and temperature increases of 25°C and 50°C 'O

1 1 I I I

Powder core 1 403 1 590 1 1.14 1 -0.12 / 32.5

Pot cote 632 433

Lamination

C core

1.20

366

Single-coi.1 C core

10 Reference [ 17, pages 79 and 106.

322

Tape-wound core

Design of Low Profile Transfc~mers for High Frequency Operation University of Tamto

-0.17

534

395

1

33.8

468

250

1.14

569

1.16

365

-0.12

1.16

41.3

-0.14

1.15

39.2

-0.14 44.5

-0.13 50.9

Appendices 83

Appendix C

Values of H and K factors used in (3.2.1 1)

FREQUENCY

850 kWz to 1.4 MHz

GAUGE COPPER

DC RES. OHMSIM' O

SINGLE STRAND Rac/Rdc

b'H"

1,0000

Table II

Design of Low Profile Transformers fot High Frequaicy Operation University of Toronto

Appendixes 84

Appendix D

1 ernperanue nse n 1 versus surrace aissipanon Y

ambient temperatures 0,25 and 50°C

' ' Reference [ 171, page 278.

10 1 O0 X

Temperature Rise [Cl

Design of Low Profile Transformers for High F q u a i c y Operation University of T m t o

Appendixes 85

Appendix E

Model used in the Finite Element Analysis of the transformer

UNKS Le?#' : ni Flux demity : T Field strength :A nr Poîential :Wb m- Canduuhity : S m-. Saune density,A w Power :W Force : N en erg^ : J Mass :kg

F I L E S

PRORLEM DATA itnf comlossl0l Kop2 Lin& elements

mmetry ~ tmL m~ pol. Magnetic fields No rnesh

7 regions

l

In the figure above a Finite Element analysis mode1 that was used in the calculation

of the core loss for the transformer that was designed using the optimized design method is

shown. The mode1 is described in cylindrical coordinates. In this figure region 1 represents

the core of the transformer, and regions 2-6 represent the wiadings in the transformer. The

background is represented by region 7. The m w t i c material in the core was considered

no nlinear.

I

Design of Low Profile Transfanners for High Frequency Operation University of Toronto

Appendix F

The characteristics of K and F magnetic materials12

1 . Material Characteristics

Initial Permeabiiity I I

Maximum Usable Frequency

I

Saturation Flux Density 1

1

Density

2. Core Loss Information

For cornputer p r o g r d g purposes, the core loss curves c m be represented by the

equation below. The factors indicated in the table are split into dimete fkequeacy ranges, so

that the equation offers a close approximation to the experimental core loss curves.

where: - B is the flux density in kgauss. - fis the operating hquency in kHz.

'* Reference [23], pages 22 and 2.2 1.

- --- - -- - - -

Design of Low Profile Transformas for High Frquency Operation University of Toronto

Appendixes 87

Design of Low Profile Transformers foc High Frequency ûperation University of Toronto

Re ferences 88

References

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Applications, and Design (second edition)", John W~ley & Sons, Inc., 1 995.

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power supply for distriiuted power systems", IEEE Transactions on Power

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- - .

Design of Low Profile Transfîmers for High Frequency Operation University of Toronto

References 89

Ramesh Oniganti, Phua Chee He*, Jeffrey Tan Kian Gu- and Liew Ah Choy,

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Electronics, Vol. 2 1, No. 2, March 1996, pp. 228-238.

Design of Low Profile TtatlSfmers f w High Frequency Operatian University of Toroato

References 90

[16] Srinivasan Iyengar, Trifon M. Liaicopoulos, and Chong H. Ahn, "A dcldc boost

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[ 1 71 W. T. McL yman, "Transformer and inductor design handbook (second edition)",

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[ 1 8f Frederick Emmons Terman, "Radio Engineer's Handbook", McGraw-Hili Book

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Design of Low Profile Transformas for High Frequaicy Operation University of Toronto

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P. L. Dowei, "Effects of eddy currents in tdormer windings", Proc. IEE, voL113,

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1997.

Design of Low Profile Tdormers for High Frquaicy Operation University of Toronto

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