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