NUMERICAL ANALYSIS OF THE VIBRATION AND ACOUSTIC ... · NUMERICAL ANALYSIS OF THE VIBRATION AND ACOUSTIC CHARACTERISTICS OF LARGE POWER TRANSFORMERS by Justin James Case B Eng (Mech)

NUMERICALANALYSISOFTHEVIBRATION

ANDACOUSTICCHARACTERISTICSOF

LARGEPOWERTRANSFORMERS

by

JustinJamesCase

BEng(Mech)(Hons),BBus(Finance)

Adissertationsubmittedinfulfilment

oftherequirementsforthedegreeof

MasterofAppliedScience(Research)

SchoolofChemistry,PhysicsandMechanicalEngineering

Science and Engineering Faculty

QueenslandUniversityofTechnology

Australia

2017

ii

STATEMENTOFORIGINALAUTHORSHIP

The work contained in this thesis has not been previously submitted to meet

requirements for anaward at thisor anyotherhigher education institution.To the

bestofmyknowledgeandbelief,thethesiscontainsnomaterialpreviouslypublished

orwrittenbyanotherpersonexceptwhereduereferenceismade.

10February2017

QUT Verified Signature

iii

ABSTRACT

Theacousticemissionsoflargepowertransformers,beingasourceofenvironmental

noise pollution, are of concern to communities living in proximity to power

substations. Increasingly stringent noise emission limits are therefore placed on

transformers by both regulators and utility operators. However, the absence of

adequate modelling tools to predict a transformer’s noise levels during the design

phasemeansthatconsistentlymeetingacousticspecificationsisasignificantissuefor

manufacturers.

This thesis has presented a numerical methodology to identify the vibration

characteristicsandpredictthesoundpressurelevelsofalargepowertransformer.The

methodology involves developing the vibro-acoustic finite element model of a

transformer and exciting the model with forces that replicate the excitation

mechanisms present in a transformer’s active part. The characteristics of the

transformerassembly’soscillatorymotionresulting from theapplied forcesare then

mappedtoanexternalacousticbody,allowingsoundpressurelevelstobedetermined.

It isofnotethatallkeystructuraland fluidbodies influencingatransformer’svibro-

acoustic responsehave been included in the finite elementmodel presented in this

thesis. Structural components within the model therefore included the core, the

windings, and the tank of the transformer, as well as parts connecting the

aforementioned assemblies. Furthermore, the insulation oil within the transformer

andthebi-directional fluid-structure interactionbetweentheactivepart, theoiland

thetankhavebeenconsidered.Suchdetailedmodellinghasenabledthepredictionofa

transformer’svibrationandacousticresponsecharacteristicsundernominaloperating

conditions.

iv

Thevalidationofthevibro-acousticmodellingschemedetailedinthisthesishasbeen

conducted with appropriate experimental measurements. Specifically, it has been

shown thatexperimentallydeterminedmodalparametersandsoundpressure levels

are in favourable agreement with numerical results. Contrasting simulated and

measuredsoundpressurelevelsovera100Hzto400Hzspectrumhasalsoillustrated

that the modelling methodology identifies both the local and global trends in a

transformer’sacousticbehaviour.

The application of the vibro-acoustic modelling methodology with regard to the

computer-aided design of transformers has been presented through a sensitivity

analysis.Fromthisanalysisitisshownthattheacousticbehaviourofatransformeris

bestoptimisedbyincreasingthestiffnessoftheactivepart,resultinginamorelinear

acousticspectrumandlowersoundpressurelevels.Furthermore,evaluationmethods

presentedinthisthesis,includingthestudyofmodalmassparticipationfactors,have

allowedfortheidentificationofatransformer’scomplexvibrationcharacteristicsthat

resultinelevatednoiselevels.

The numerical prediction of a transformer’s vibration characteristics and acoustic

behaviourtogetherwithanunderstandingoftheparametersbestsuitedtooptimising

aunit’sacousticdesignwillbetterenable transformermanufacturerstoconsistently

producetransformersthatmeetacoustictargets.Manufactureswillthereforebeless

likely to incur the significant non-conformity costs thatmay result from exceeding

guaranteednoiselevels.Theacousticoptimisationoflargepowertransformerdesigns

willalsobenefitpersonsworkingorlivinginproximitytopowersubstations,reducing

theadversehealthimpactsassociatedwithextendedperiodsofnoiseexposure.

Keywords: TransformerNoise,FiniteElementMethod,Vibro-AcousticModelling,

Fluid-Structure Interaction, Resonance, Modal Mass Participation

Factor,ModalDamping,Curve-Fitting,ModalParameterExtraction

v

ACKNOWLEDGEMENTS

I ammostgrateful tomy supervisor,Dr.PietroBorghesani, for his encouragement,

interest,stimulusandguidance throughoutthisproject.His initiativeshaveelevated

the standard of this project and his enthusiasm has helped to see it through to

completion.

ThanksareduetoMrHelmutPregartner,theHeadofLargePowerTransformerNoise

andMechanicsR&D at SiemensAG, for hisvaluable advice and supportonvarious

issues.Iwouldalso liketothanktheothermembers iftheNoiseandMechanicsR&D

teamwhohaveprovidedinsightintothefieldofinterest.

IwishtoexpressmygratitudetoMrMartinStössl,MrGeraldLeberandagaintoMr

HelmutPregartnerwhohavegivenmetheopportunitytoconductmyresearchwithin

theLargePowerTransformerNoiseandMechanicsR&DteamatSiemensAG.Iwould

also like to extendmy thanks to theentireSiemensAGAustria,TransformersWeiz

R&Ddepartmentforprovidingasimulatingandenjoyableworkenvironment.

Specialthanksareduetomyparentsfortheirsupportduringthecourseofthiswork.

Withoutthemthisthesiswouldnothavebeencompleted.

vi

TABLEOFCONTENTS

StatementofOriginalAuthorship...................................................................................................................ii

Abstract.........................................................................................................................................................................iii

Acknowledgements..................................................................................................................................................v

TableofContents.....................................................................................................................................................vi

ListofFigures............................................................................................................................................................ix

ListofTables.............................................................................................................................................................xii

Nomenclature.........................................................................................................................................................xiii

Chapter1:Introduction......................................................................................................................1

1.1 BackgroundandMotivation.............................................................................................................2

1.2 ObjectivesandScopeofWork........................................................................................................4

1.3 SignificanceoftheStudy....................................................................................................................5

1.4 NoveltiesandAreasofSignificantContribution.................................................................5

1.5 AssumptionsandApproximations...............................................................................................7

1.6 LayoutoftheThesis..............................................................................................................................8

Chapter2:LiteratureReviewandTheorecticalFramework...........................................9

2.1 FundamentalsofTransformerNoiseandVibrations......................................................10

2.2 ApproachesforthePredictionofTransformerNoise.....................................................13

2.3 FiniteElementPredictionofTransformerNoise...............................................................14

2.3.1 TheoreticalBasis........................................................................................................................15

2.3.2 PastFiniteElementStudiesofTransformerNoise...................................................27

vii

2.4 ExperimentalModalAnalysis.......................................................................................................30

2.4.1 TheoreticalBasis........................................................................................................................31

2.4.2 ModalDampingExtraction...................................................................................................35

2.4.3 CorrelationofExperimentalandNumericalModalResults................................38

Chapter3:FiniteElementModeloftheTransformer.......................................................43

3.1 SpecificationsoftheTransformer..............................................................................................46

3.1.1 SimplificationoftheComputer-AidedDesignModel...............................................47

3.2 ModellingoftheTank........................................................................................................................48

3.3 ModellingoftheActivePart...........................................................................................................50

3.4 AcousticBodiesandFluid-StructureInteraction..............................................................53

3.5 ExcitationMechanisms.....................................................................................................................55

3.5.1 ExcitationMechanismsintheCore...................................................................................56

3.5.2 ElectromagneticForcesintheWindings.......................................................................59

3.5.3 SummaryoftheCharacteristicsoftheExcitationMechanisms.........................60

3.5.4 ModellingoftheForcesAppliedtotheNumericalModel......................................61

3.6 ModalDamping.....................................................................................................................................64

3.6.1 ModalDampingoftheActivePart....................................................................................66

3.6.2 ModalDampingoftheTankStructure...........................................................................69

3.6.3 ModalDampingintheFiniteElementModel..............................................................84

Chapter4:ValidationoftheNumericalModel.....................................................................87

4.1 ModalValidationoftheNumericalModel.............................................................................88

4.2 AcousticValidationoftheNumericalModel........................................................................95

viii

4.2.1 SoundPressureLevelMeasurements..............................................................................95

4.2.2 SoundPressureLevelPredictions......................................................................................99

4.3 Conclusions...........................................................................................................................................103

Chapter5:Discussion.....................................................................................................................105

5.1 IncreasedFrequencySamplingtoprovideDetailedAcoustic

Characteristics................................................................................................................................106

5.2 ExcitationMechanismsandActivePartVibrationCharacteristics......................109

5.3 ResonanceEffectsandExcessiveSoundRadiation.......................................................112

5.4 ComputingRequirements............................................................................................................114

Chapter6:SensitivityAnalysis..................................................................................................117

6.1 InfluenceofActivePartStiffness.............................................................................................118

6.2 InfluenceofStiffeningRibs..........................................................................................................120

6.3 InfluenceoftheDirectTransmissionPathbetweenActivePart

andTank.............................................................................................................................................123

6.4 InfluenceofTankDimensions...................................................................................................124

6.5 Conclusions...........................................................................................................................................126

Chapter7:ConculsionsandFutureWork.............................................................................127

7.1 ConcludingRemarks.......................................................................................................................128

7.2 FutureWork.........................................................................................................................................132

References...........................................................................................................................................135

ix

LISTOFFIGURES

Figure1-1 Illustrationofanoil-filledpowertransformer..........................................................2

Figure1-2 Developmentofthenoiseemissionsfromapowertransformer...................3

Figure2-1 Sourcesofnoiseemissionsfromapowertransformer.......................................11

Figure2-2 Threebroadcategoriesofstructuraldynamicssimulations............................17

Figure2-3 Envelopecomputationoftheimpulse-responsefunctionwithuseoftheHilberttransform.......................................................................................................37

Figure3-1 Stepsinthedevelopmentofthefiniteelementtooltopredicttransformernoise.....................................................................................................................45

Figure3-2 Geometryofthetransformerassembly.......................................................................46

Figure3-3 Finiteelementmodelof(a)thetransformertankand(b)thetransformertankwithauxiliaryequipment..............................................................49

Figure3-4 Finiteelementmodeloftheactivepart.......................................................................50

Figure3-5 Nodalvelocitiesvectorsmappedtotheacousticbodysurroundingthetransformertank...............................................................................................................54

Figure3-6 MagnetostrictiveeffectinSiFesheetsatdifferinginductionlevels.............57

Figure3-7 Typicalacousticspectrumproducedbya50Hzpowertransformer.........60

Figure3-8 Half-powerpointsonanFRFemployedtocalculatethedampingratiosoftheactivepartat(a)192Hzand(b)346Hz(c)andtheresultingRayleighdampingcurve...................................................................................67

Figure3-9 Estimateddampingincomparisontotheapplieddamping.............................74

Figure3-10 Errordistributionoftheapplieddampingmethodologywiththe0–400Hzanalysisspectrumdelineatedintofivebands..................................75

x

Figure3-11 SpectralleakagewhentakingtheIFFTof(a)acompleteFRFsignaland(b)adefinedafrequencybandofthecompleteFRFsignal................................................................................................................................................78

Figure3-12 Transmissibilityfunctionswithmodaldampingat4.95%,5.50%and6.05%....................................................................................................................................80

Figure3-13 Dampingestimatesin1/3octavebands.....................................................................82

Figure4-1 Numericalversesexperimentalnaturalfrequenciesofthecorrelatedmodeswithtrendlineindicatingperfectcorrelation..................88

Figure4-2 Comparisonof(a)theexperimentalmodeshapeat11Hzand(b)theexperimentalmodeshapeofthetransformerat10.24Hz.......................89



Figure4-5 Comparisonofmeasuredandpredictedsoundpressurelevelsat16correspondingmeasurementpoints.......................................................................97

Figure4-6 Comparisonofpredictedsoundpressurelevelscalculatedwiththesimplifiednumericaltransformerassemblyandthecompletenumericaltransformermodel...........................................................................................99

Figure5-1 Comparisonofthetransformersacousticbehaviourasidentifiedbysoundpressurelevelsdeterminedat10Hzand20Hzintervals...............104

Figure5-2 Comparisonofpredictedsoundpressurelevelsat10Hzintervalscalculatedwiththesimplifiednumericaltransformerassemblyandthecompletenumericaltransformermodel..........................................................105

Figure5-3 Modalmassparticipationfactorofactivepartmodesintheverticalplane.............................................................................................................................................109

Figure5-4 Modeshapeexhibitingglobaldeformationofthetankcoveratapproximately400Hz.........................................................................................................111

xi

Figure6-1 Influenceofactivepartstiffnessontheacousticbehaviourofatransformer...............................................................................................................................116

Figure6-2 Updatedtanklayoutoftankwithadditionalstiffeningribs..........................118

Figure6-3 Influenceofstiffeningribsontheacousticbehaviourofatransformer...............................................................................................................................119

Figure6-4 Influenceofthedirecttransmissionpathbetweenthetankandtheactivepartofatransformer.....................................................................................120

Figure6-5 Influenceoftankdimensionsontheacousticbehaviourofatransformer...............................................................................................................................122

xii

LISTOFTABLES

Table3-1 Geometricparametersofthetransformer.................................................................47

Table3-2 ModelSimplificationMethodology................................................................................48

Table3-3 VariablesemployedinthegenerationoftherandomFRF................................73

Table3-4 Effectoffrequencybandsontheaccuracyofdampingestimates................76

Table4-1 Naturalfrequenciesofthetransformertank............................................................87

Table4-2 Electricalexcitationfrequencyandcorrespondingsoundpressurelevelmeasurementpointsandvalues...........................................................................95

Table4-3 ModelValidationMethodsandCorrespondingErrorMargins..................101

Table5-1 Computingspecifications..................................................................................................112

xiii

NOMENCLATURE

BasicTerms,DimensionsandSubscripts

A Amplituderatio

a Acceleration

B Magneticfluxdensitythroughaunitarea

c Speedofsoundinafluidmedium

ex Experimentallydeterminedvalue

F Force

i √−1

J Electriccurrentdensity

Bulkmodulusofafluid

p Acousticpressure

m Mass

Numberofdegreesoffreedom

Naturalfrequency

nu Numericallycalculatedvalue

Q Masssourceappliedtoanacousticbody

r Numberofrigidbodymodes

Transmissibilityofasystem

t Instantaneoustimevalue

Virtualdisplacement

ϵ Errorpercentage

xiv

Fluiddensity

Dynamicviscosityofafluid

Phaseshiftofawaveform

Angularfrequency

Matrices,VectorsandScalars

[] Matrix

{} Columnvector

[] ;{} Transposeofamatrix;vector

[] Inverseofamatrix

[]∗ Complexconjugateofamatrix

[] Gerenalised/Pseudoinverseofamatrix

╲

╲ Diagonalmatrix

[ ] Identitymatrix

[] Matixrepresentingthepropertiesofafluid

[] Matixrepresentingthepropertiesofasolidstructure

SpatialandModellingProperties

[ ] Viscousdampingmatrix

[ ] Stiffnessmatrix

[ ] Massmatrix

[ ] Couplingmatrixatthefluid-structureinterface

[k ] Modalorgeneralisedstiffnessmatrix

xv

[m ] Modalorgeneralisedmassmatrix

{ } Elementshapefunction

{ } Outwardnormalvectoratthefluidboundary

{ } Nodalpressurevector

{q} Nodalmasssourcevector

{ } Displacementvector

{ } Velocityvector

{ } Accelerationvector

{ } Forcevector

Mass-proportionaldampingcoefficient

Stiffness-proportionaldampingcoefficient

Modaldampingratioofthe mode

Ω Acousticdomain

Acousticboundarydomain

FrequencyResponseProperties

[ ( )] Frequencyresponsematrix

[ ] Displacementofmassesmatrix

{ } Displacementtransformationvector

[Φ] Matrixofmass-normalisedeigenvectors

[ ] Generalisedformoftheeigenvectormatrix

xvi

( ) Fouriertransformofthetimesignalx(t)

( ) Impulsesignalinthetimedomain

Γ Modalparticipationfactoroftherthmode

The elementofthe eigenvector{ }

AbbreviationsandUnits

AC Alternatingcurrent

CAD Computer-aideddesign

dB Decibel

dB(A) A-weighteddecibel

DOF Degreeoffreedom

EMA Experimentalmodalanalysis

FE Finiteelement

FEM Finiteelementmethod

FFT FastFouriertransform

FRF Frequencyresponsefunction

FSI Fluid-structureinteraction

HEM HilbertEnvelopeMethod

IFFT InversefastFouriertransform

IFT InverseFouriertransform

IRF Impulseresponsefunction

LTP Largepowertransformer

MAC Modalassurancecriterion

xvii

MSF Modalscalefactor

MDOF Multipledegreeoffreedom

SDOF Singledegreeoffreedom

SPL Soundpressurelevel

Chapter1:Introduction

1

CHAPTER1

INTRODUCTION


2

1.1 BACKGROUNDANDMOTIVATION

Apowertransformerisadevicedesignedtotransferelectricalenergybetweentwoor

more circuits through electromagnetic induction. Serving as critical nodes in the

power grid, transformers have become essential in the effective and reliable

transmission and distribution of electricity. A diagram of a typical oil-filled

transformerisseeninFigure1-1.

Figure1-1 Illustrationofanoil-filledpowertransformer[1]

The essential components of an oil-filled transformer, whether it is designed for

transmissionordistributionpurposes,include:

· Theactivepart–comprisedofthecoreandthewindings;

· Insulationoil;

· High-voltageandlow-voltagebushings;and

· Thetank.

Largeroil-filledtransformersalsogenerallycarryauxiliarycoolingequipment,suchas

fans and radiators, and an oil conservator, which allows for the expansion of the

insulationoilwithinthetank.

Activepart(thecoreandthewindings)

Bushings

Coolingequipment(radiatorsandfans)

Oilconservator

Tank


3

Perhapsthebest-knowneffect linked totheoperationof apower transformer isthe

distinctivehum itproduces.This characteristic is causedbyvibrations in theactive

part,whichresult from electromagneticandmagnetostrictive forces in thecoreand

thewindings. The noise emissions from large power transformers have become a

serious environmental problem for communities living in proximity to power

substationsandthereforestringentacousticregulationsareincreasinglybeingplaced

on transformers.Largepowertransformers typicallygeneratesoundpressure levels

ranging from 60 dB(A) to 80 dB(A); levels which are widely recognised as

unacceptablefor24-hourexposureinaresidentialenvironment[2].

A significant issue facing transformer manufacturers is meeting the maximum

allowable sound levels specified by noise regulations and utility operators. It is

therefore critical that manufacturers are able to accurately identify the acoustic

characteristicsofapowertransformerbeforeproductioncommences.However,thisis

difficultdue to the complexphenomenarelated to thegenerationofa transformer’s

acousticemissions,asillustratedinFigure1-2.

Figure1-2 Developmentofthenoiseemissionsfromapowertransformer ( indicatestheprogressionofeventswhichresultintheacoustic

emissionsofapowertransformer)

Theinherentuncertaintyinthepredictionofpowertransformernoiselevelshasledto

largesafetymarginsintransformerdesigns.Thesesafetymarginsintroduceadditional

economicexpenseformanufacturersinboththedesignandmanufacturingprocesses.

However, failure to meet guaranteed noise levels can result in significant non-

conformitycosts.

Electromagnetic&

MagnetostrictiveExcitation

VibrationoftheActivePart

TransmissionofVibrations

VibrationoftheTransformer

Tank

AcousticSound

Emission


4

1.2 OBJECTIVESANDSCOPEOFWORK

Theaimofthisthesisistodevelopanumericalmethodologythataccuratelypredicts

thevibrationandacousticcharacteristicsofalargepowertransformer. Inparticular,

thisthesisisconcernedwithmodellingatransformer’ssoundradiationandidentifying

thephysicalpropertiesthatresultinelevatednoiselevels.Thefourmainobjectivesof

thisworkaresetoutasfollows:

i) Todefineasetofgenericproceduresfornumericallypredictingthevibration

characteristicsandnoiselevelsofalargepowertransformer;

ii) Tovalidatethenumericalmethodologybycomparingthesimulatedvibration

characteristicsandnoiseemissionsof a transformerwith thosedetermined

experimentally;

iii) To evaluate and subsequently optimise the vibration and acoustic

characteristics of a transformer employing the numerical methodology,

replicatingtheprocessthatmaybecarriedoutwithin thedesignphaseof a

transformer;and

iv) To determine the influence of key structural design parameters on the

acousticbehaviourofatransformer.

The scope of work covered by this thesis includes the numerical prediction of a

transformer’svibration and acousticbehaviourundernominaloperating conditions.

Specifically, the work is concerned with the vibration and acoustic response of a

single-phase, special purpose large power transformer. Within this thesis the term

‘large power transformer’ (LPT) is used to describe a power transformer with a

maximumratedpowerof100MVAorhigher;however,it isofnotethatnoindustry

definition or criterion forwhat constitutes a LPT exists.The term ‘special purpose

transformer’ is utilised in thiswork to distinguish between generator-step up and

systemintertietransformersandallothertransformertypes.


5

1.3 SIGNIFICANCEOFTHESTUDY

Thesignificanceofthisstudyresidesinpresentinganumericalmethodologythatwill

enable transformer manufacturers to evaluate and optimize the acoustic

characteristics of LPTs in the design phase. This evaluation and optimisation

procedurewillcentreonsoundpressurelevelestimatesandthenumericalprediction

ofatransformer’svibrationcharacteristicsthatresultinelevatedacousticemissions.

Such adesign tool isof importance to transformermanufacturersas failure tomeet

acoustic targets specified by regulators and utilities operators can result in

considerablenon-conformityexpenses.

Togetherwithenablingtransformermanufacturerstomoreconsistentlymeetacoustic

targets, accuratesoundpressureestimateswillallow theacoustic safetymargins in

currentLPTdesignpracticestobereduced.Thesesafetymarginsintroduceadditional

costs inboththedesignandmanufacturingphases.Furthermore,theoptimizationof

LPT designs with respect to acoustic emissionswill limit the environmental noise

pollution introduced by these pieces of equipment. This would benefit persons

working or living in proximity to power substations, reducing the adverse health

impactsassociatedwithextendednoiseexposure.

1.4 NOVELTIESANDAREASOFSIGNIFICANTCONTRIBUTION

Withinthisthesisseveralareasofnovelresearcharepresented.Thesenoveltiesrelate

to the vibro-acoustic modelling of power transformers and to modal parameter

extractiontechniques.Thecurrentstateoftheart,asidentifiedinpastliterature,and

theareasofsignificantcontributionwithinthisthesisaredescribed in the following

paragraphs. A detailed literature review covering both theoretical topics and past

studiesispresentedinChapter2;hereonlyabriefstatementonthestateoftheartis

reportedtohighlightthenoveltyofthisthesis.


6

Thevibro-acousticmodellingofpowertransformersismostcomprehensivelydetailed

in the investigations published by Rausch, Kaltenbacher, Landes & Lerch [3] and

Kavasoglu[4].However,both theaforementionedstudiesomit the transformercore

from the respectivenumericalmodels and thereforepredict only coil-emittednoise

emissions.Theworkpresented in this thesis furtherstheworksofRauschetal.and

Kavasoglubydetailingavibro-acousticmodelthatincorporatesallkeystructuraland

fluid elements (i.e. the core, thewindings, the transformer tank and the insulation

fluid). This allows the model to predict a transformer’s acoustic behaviour under

nominaloperatingconditions,withacousticemissionsresultingfromthevibrationof

boththecoreandthewindings.

AsecondlimitationofthestudiesperformedbyRauschetal.andKavasogluisthatthe

prediction and subsequent validation of sound pressure levels is performed

exclusively at double the network frequency (being 100 Hz or 120 Hz). When

considering the acoustic behaviour of a large power transformer under nominal

operating conditions this simplification is not valid. Therefore, the study presented

withinthisthesis isnovel inthatitpredictsandvalidatesacousticemissionsoveran

extended frequency range. Explicitly, this thesis estimates and validates sound

pressure levels over a spectrumwhich includes the network frequency harmonics

considereddominantintheacousticspectrumofaLPT.

Inaddition toprovidingmoredetailedacousticpredictions, the extendedvalidation

spectrum of the vibro-acoustic model employed in this thesis allows for a

transformer’s key vibration characteristics to be identified and discussed. Such an

analysisispresentedinthisthesisforthefirsttime.Furthermore,numericalindicators

including modal mass participation factors are identified as key design-quality

parameters.Theseaforementionednumericalindicatorsmaybeusedtodeterminethe

modesthatsignificantlycontributetoexcessivenoisegeneration.


7

Finally, this thesis details the first modal damping analysis of a large power

transformer.Due to the limitationsof conventional curve-fitting techniques a novel

methodology to identify the modal damping behaviour of a complex fluid-filled

assemblyisalsosetoutandapartialnumericalvalidationofthemethodologyisgiven.

Additionally, the interpretation and application of the experimentally determined

modaldampingcharacteristicstothefiniteelementmodelofthestudiedtransformer

isconductedinawaythathasnotbeenpresentedinpastliterature.

1.5 ASSUMPTIONSANDAPPROXIMATIONS

While attempts have been made to make the numerical prediction of transformer

noisepresented inthisthesisasgenericaspossible,themodellingprocess inevitably

containsanumberofassumptions.Theseassumptionsmaytosomeextentexplainthe

limitations or discrepancies of the results shown in later chapters. The principal

assumptionsmadeinthisthesisareoutlinedasfollows:

· Thediscretisationofthemodelwasconductedbymeansoffiniteelements;

· Contact between singleparts or assemblieshave been considered bonded and

werethereforemodelledwithlinearbondedcontacts;

· Thedampingnatureofthetransformeranalysedinthisthesis isassumedtobe

thatofproportionalviscousdamping;and

· The electromagnetic andmagnetostrictive forces employed to excite the finite

element model have been determined analytically by Siemens AG Austria,

TransformersWeizandareassumedtobeaccurate.


8

1.6 LAYOUTOFTHETHESIS

Therearesevenchapters inthethesis.Chapter2providesareviewofpast literature

and the theoretical framework relevant to the work carried out in this thesis.

Chapter3details the finiteelementmethodologydeveloped topredict thevibration

and acoustic characteristics of a largepower transformer operating under nominal

conditions. The validation of the numerical methodology is detailed in Chapter 4.

Chapter5 identifiesanddiscusses thevibro-acousticcharacteristicsof theparticular

LPTanalysedinthisthesisanddetailsthecomputingrequirementsforthesimulations

performed. An evaluation of the key structural parameters affecting the acoustic

behaviourofatransformer ispresented inChapter6.Finally,Chapter 7outlines the

significant results obtained from this thesis and discusses the future works to be

conducted.

Chapter2:LiteratureReviewandTheoreticalFramework

9

CHAPTER2

LITERATUREREVIEWAND

THEORECTICALFRAMEWORK


10

The purpose of this chapter is to provide an overview of the key concepts and to

identifythestrengthsandweaknessesofpreviousstudiesrelatingtothepredictionof

transformernoise.Thechapterbeginsbyintroducingthefundamentalsoftransformer

noise including vibration sources, transmission paths and radiated sound. The

principal approaches thathavebeenemployed inprevious investigations topredict

transformernoisearethendetailed.Withinthischapterthereisaparticularfocuson

the finiteelementapproachtopredicttransformernoiseandthemethod’svalidation

throughexperimentalmodalanalysis(EMA).TheanalysisofEMAdatatodeterminean

oscillatingsystem’smodaldampingcharacteristics,therebyallowingthesystemtobe

moreaccuratelymodelled,isalsodiscussed.

2.1 FUNDAMENTALSOFTRANSFORMERNOISEANDVIBRATIONS

Undernominaloperatingconditionsatransformer’sacousticemissionsresultfromthe

vibrationsofthecoreandthewindings,andthenoisegeneratedbyauxiliarycooling

equipment. Estimation and the subsequent adjustment of the noise produced by

coolingequipment,suchas fansandpumps,cangenerallybemadethroughphysical

measurementsandanalyticalmodelling.However,predictingthenoiseresultingfrom

thevibrationsofthecoreandthewindingsissignificantlymorechallenging.

Thedifficulty inpredicting the acoustic emissionsgenerated by the vibrationsof a

transformer’s core and windings are the many complex phenomena involved.

Specifically,thefollowingaspectsmustbeaccountedfor:

· Theexcitationmechanismsinthecoreandthewindings;

· Thetransmissionpathsfromtheactiveparttothetransformertank;and

· Thestructuralresponsecharacteristicsoftheassembly.

Eachofthesefactorsisdiscussedinfurtherdetailwithinthischapter.


11

The vibrations of a transformer’s active part result from three distinct excitation

mechanisms: amagnetostrictive effect in the core;Maxwell forces in the core; and

Lorentz forces in the windings [5-7]. Figure 2-1 illustrates these excitation

mechanismswithinthecontextofpowertransformernoise.Thereismuchconjecture

about which excitation mechanism in a transformer’s active part is the dominant

sourceofacousticemissions(refertoRefs. [3,7-10] forconflictingarguments). It is

alsoplausiblethatthedominantsourceofnoise isdependentonanindividualunit’s

designandoperatingspecifications.ThisisviewsupportedbyRefs.[11,12].

Figure2-1 Sourcesofnoiseemissionsfromapowertransformer (---indicatesthenoisesourcesconsideredinthisthesis)

Although the dominant sourceof transformer noise is unclear, the three excitation

mechanisms that generate a transformer’s acoustic emissions have been studied

extensively.Keypointsofconsensusamongprecedinginvestigationsinclude:

· The Maxwell forces in the core and the Lorentz forces in the winding act

predominantlyatdoublethenetworkfrequency;and

· Themagnetostrictivestrainofthecore,duetonon-linearsaturationofthecore

material,producesvibrationsatevenharmonicsofthenetworkfrequency.

Furtherdetailoftheactivepart’sexcitationmechanismsandhowtheycharacterisethe

acousticspectrumofatransformerisprovidedinChapter3.6.

CoreVibrations

· Magnetostrictivestrain

· Maxwellforces

WindingVibrations

· Lorentzforces

AuxiliaryNoise · Coolingfansand

pumps

PowerTransformerNoise


12

The vibrations resulting from the excitation mechanisms in the active part of a

transformerareabletoreachthetankwallthroughtwotransmissionpaths.Refs.[13-

15]haveinvestigatedthesetwotransmissionmechanisms,being:

· Adirectcouplingthroughmechanicaljoints

Adirectcouplingoftheactivepartandthetankoccursthroughmechanicaljoints,

withtransmissionprimarilythroughthebaseofthetankonwhichtheactivepart

rests.

· Indirecttransmissionthroughtheinsulationfluid

An indirect coupling through the fluid-structure interaction (FSI) between the

insulationliquid,theactivepartandthetankallowspressurewavesgeneratedby

themovementoftheactiveparttotransmitthroughtheinsulationoiltothetank

structure.

In addition to acting as a transmission mechanism, the insulation fluid contained

withinatransformersignificantly influencestheassembly’sfrequencyresponse.This

is due to the fluid’smass andviscousdamping effects aswell as the bi-directional

couplingbetweenthefluidandthestructuralelementsofatransformer[16,17].

The transmission of vibrations through the insulation fluid and mechanical joints

eventuates inthetransformertankvibratingwithoscillatorymotion.However,there

have been few investigations that specifically examine how the vibration

characteristicsof a transformer tankaffect acoustic emissions.This absenceofpast

research is likely due to: the perception that the tank itself does not have any

significantinfluenceonthepropagationofnoise[12];andrelativelyrecentadvancesin

computing technology thathaveallowed for the numerical calculationof a complex

structure’svibrationcharacteristics[18].


13

In Ref. [19] the authors dismiss the notion that the tank has no effect on the

propagationof noise. Instead, the authorsof Ref. [19] suggest that excessive noise

emissionsresultifthenaturalfrequencyofthetankmatchestheoscillatingfrequency

oftheactivepart,resulting inaresonanceevent.Thisunderstanding thatasystem’s

externalhousingcanamplifynoiseemissionsisalso inaccordancewithgearboxand

automobilenoisestudies,suchasthosepresentedinRefs.[20-23].

2.2 APPROACHESFORTHEPREDICTIONOFTRANSFORMERNOISE

There have been a series of past attempts to model the acoustic characteristics of

powertransformers.Themajorityofthesepreviousstudieshaveattemptedtodevelop

parametric predictive models (i.e. statistical models formulated from experimental

data).Thesemodelsmaybroadlybeclassifiedintotwocategories:

· Correlationbetweentheprincipalparametersofatransformerandnoiselevels

A number of studies have usedmeasured data to develop acoustic prediction

models based on the principal parameters of a transformer, such as core

dimensions,tankdimensionsandpowerrating(seeRefs.[24-26]asexamples).

· Influenceofmaterialpropertiesontheacousticemissionsofatransformer

The mechanical properties of specific materials and components within a

transformer andhow theseaffectsound radiation levelshavebeen extensively

studied.Themajorityofresearchinthisareahasfocusedonthecharacteristicsof

themagneticcorematerial(seeRefs.[27-29]asexamples).

Theparametricmodelsdetailedinpastinvestigationsprovideanunderstandingofthe

factorsinfluencingthenoiseemissionsfrompowertransformers.However,asnotedin

Ref. [30], acousticmodels basedon experimental analyses areonly able toprovide

limitedinformationastheyfocusonafewvibrationpropertiesatcertainfrequencies.


14

Due to the limitations of statistical techniques, numerical methods have become

common in studying thebehaviour of transformers.Thesenumerical investigations

havedevelopedwiththeprogressofhigh-poweredcomputingandothertechnological

advances [31]. The numerical techniques that may be employed to study a

transformer’s behaviour include the finite element method, the finite difference

method, theboundaryelementmethod, and thepointmirroringmethod, toname a

selectedfew.

Althoughthethreephysicalfieldsinfluencingthesoundradiatedbyatransformerare

able to be numerically modelled – namely the electromagnetic, mechanical and

acoustic fields – no study hasyet to detail a comprehensive calculation scheme to

determinethenoiseemissionsofatransformerundernominaloperatingconditions.

However, numerical studies analysing the vibration characteristics of individual

componentsandkeystructuralassemblieswithinatransformerhavebeenconducted.

These numerical analyses have generally employed the finite element (FE)

methodology[31].

2.3 FINITEELEMENTPREDICTIONOFTRANSFORMERNOISE

ThecommercialsoftwarepackageANSYS®hasbeenused in this thesis to study the

vibro-acoustic behaviour of a transformer. Specifically, ANSYS® 16.2 Mechanical

software has been employed, which is a finite element analysis tool for linear,

nonlinearanddynamicstructuralanalyses.TheANSYS®Mechanicalsoftwarepackage

also supports coupled structural-acoustics simulations through the integration of

acoustic features and by accounting for fluid-structure interaction in the solution

process.


15

Thefirstofthefollowingsubchaptersdetailsthetheoreticalbasisofthefiniteelement

methodology as employed inANSYS®Mechanical. Specific topics relevant to theFE

model presented in this thesis, such as fluid-structure interaction (FSI), are also

discussed. The subsequent subchapter then provides a review of relevant past

literature that has employed the finite element method in an attempt to predict

transformernoise.

2.3.1 THEORETICALBASIS

The finite elementmethod (FEM) is anumerical technique for finding approximate

solutions to differential equations and is used in numerous fields of science and

engineering. The central notion behind FEM is to discretise a complex object into

componentsofsimplegeometrycalledfiniteelements.Assemblingtheequationsthat

describe each of the individual finite elements into a general system of equations

providesforadetailedmathematicalmodelofthestudiedobject.Thegeneralsystem

ofequationscanthenbesolvedthroughnumericalmethodstoobtainanapproximate

overallsolutiontothemathematicalmodel[32,33].

To predict the physical behaviour of systems and structures ANSYS® Mechanical

employs the d’Alembert principle [34]. The principle is an interpretation of the

classical lawsofmotionandpresentsanalternative formofNewton’ssecond lawof

motion.Theadvantageofutilisingd’Alembert’sprincipleisthataproblemindynamics

maybereducedtoaprobleminstatics,suchthat:

− = 0 (2.1)

Where: F istheforceactingonabody

m isthemassofthebody

a istheaccelerationofthebody


16

As seen in Eqn. 2.1, d’Alembert’s principle indicates that equilibrium may be

consideredtoresultundertheactionofarealforce andafictitiousforce− .More

specifically,thed’Alembertprinciplestatesthatthesumofthedifferencesbetweenthe

forcesactingonasystemofmassparticles(orbodies)andthetimederivativesofthe

momenta of the system itself along any virtual displacement consistent with the

constraintsofthesystemiszero[35]:

( − ) ∙ = 0 (2.2)

Where: i denotesaparticularparticle(orbody)inthesystem

isthevirtualdisplacementoftheithparticle(orbody)

Itistheequationgivenbyd’Alembert’sprincipletogetherwiththediscretisationofa

continuous structure thatANSYS®Mechanical employs to solve structuraldynamics

problems.Theresult isanequationofmotionaccounting for themass,dampingand

stiffnessofabody,whichisgivenby[34]:

[ ] ⋅ { } + [ ] ⋅ { } + [ ] ⋅ { } = { ( )} (2.3)

Where: [ ] isthestructuralmassmatrix

[ ] istheviscousdampingmatrix

[ ] isthestiffnessmatrix

{ } isthenodalaccelerationvector

{ } isthenodalvelocityvector

{ } isthenodaldisplacementvector

{ ( )} isthetimevaryingvectoroftheappliedforces


17

TakingdifferentrepresentationsofthetimevaryingforcevectorinEqn.2.3providesa

generalclassificationofthevarioustypesofstructuraldynamicsanalysesthatmaybe

conducted.Thisleadstothreebroadcategories,beingtransient,modalandharmonic

response,asillustratedinFigure2-2.

Figure2-2 Threebroadcategoriesofstructuraldynamicssimulations

Thedynamicanalysesemployedinthisthesistodeterminethevibrationandacoustic

characteristics of a power transformer are further described in the following

paragraphs.

· ModalAnalysis

Amodalanalysisisusedtodeterminethevibrationcharacteristicsofastructure

through identifying natural frequencies andmode shapes. Therefore, the time

varying forces applied to the structure are taken aszero andEqn.2.3may be

simplifiedto:

[ ] ⋅ { } + [ ] ⋅ { } + [ ] ⋅ { } = 0 (2.4)

Inamathematicalsensethecomputationofnaturalfrequenciesandmodeshapes

isequivalentwiththesolutionofaneigenvalueproblem.

ModalAnalysisTransientAnalysis HarmonicAnalysis

[ ] ⋅ { } + [ ] ⋅ { } + [ ] ⋅ { } = { ( )}


18

· HarmonicResponseAnalysis

Aharmonicresponseanalysisisemployedtosolvestructuraldynamicsproblems

inwhichappliedcyclical loadingproducesasustainedcyclicalresponse.Within

harmonic response analyses the applied loads and displacements vary

sinusoidallywith the sameknown frequencyalthoughnotnecessarily inphase.

Eqn.2.3maythereforebeexpressedas:

[ ] ⋅ { } + [ ] ⋅ { } + [ ] ⋅ { }

= ∙ ( )

(2.5)

Where: istheamplitudeoftheforce

istheangularfrequency

isthephaseshiftoftheforce

Althoughnotemployed inthisthesis, it isnotedthatatransientanalysisdiffers

fromaharmonicresponseanalysisinthatanyfunctionfortheloadvectormaybe

specified.

2.3.1.1Fluid-StructureInteractioninNumericalAnalyses

To accurately account for the influence of fluid bodies in numerical analyses the

couplingofstructuralandfluidfieldsisrequired.Thiscouplingallowspressurewaves

in the fluid to be generated by the vibration of a structure and/or structural

deformationtoresult fromappliedfluidpressure.Thecouplingofthestructuraland

fluid fields in ANSYS® Mechanical is accomplished by considering the structural

dynamicsequation,asgiven inEqn.2.3, togetherwith theNavier-Stokesequationof

fluidmomentumandtheflowcontinuityequation.


19

Within fluid-structure interaction (FSI) analyses ANSYS® Mechanical simplifies the

Navier-Stokes and the flow continuity equations. Assumptions made in this

simplificationincludethatthefluidiscompressibleandthatthereisnomeanflow.The

equation resulting from these assumptions, which accounts for the propagation of

pressurewaveswithinafluidbody,isgivenby[36]:

∇ ∙

1∇ −

1+ ∇ ∙

43 ∇

1

= − + ∇ ∙43 ∇

(2.6)

Where: c isthespeedofsoundinafluidmedium = ⁄

isthemeanfluiddensity

K isthebulkmodulusofthefluid

isthedynamicviscosity

istheacousticpressure[ = ( , , , )]

Q isthemasssourceinthecontinuityequation

t isthetime

As the viscous dissipation of energy has been taken into account using the Stokes

hypothesis1, thewave equation stated in Eqn. 2.6 is referred to as the lossy wave

equation for thepropagationof sound in fluids.Thediscretized structural equation

definedinEqn.2.3andthelossywaveequationmustbeconsideredsimultaneouslyin

FSI problems. The following paragraphs detail how the discretised fluid matrix is

derivedandpresentsthecoupledfluid-structurematrixutilisedinFSIanalyses.

1Stokeshypothesisstatesthattheaveragenormalstressesonanythreemutuallyperpendicular

normalforces inanelementofacompressiblefluidareequal tozero.Therefore, thesecondcoefficientofviscosityplustwo-thirdsofthedynamicviscosityisequaltozero[37].


20

The finite element formulation of the definition of the wave equation is obtained

applyingtheGalerkinproceduretoEqn.2.6.ThewaveEqn.2.6isinitiallymultipliedby

theweighting functionψ(alsoreferredtoasatest function)and integratedoverthe

volumeoftheacousticdomaintoyieldthefollowing[36]:

1ψ + ψ ∙

43

+ ψ ∙1

− ψ1

+4

3∙

+ ψ43 ∙

= ψ1

+ ψ ∙43

(2.7)

Where: isthevolumedifferentialofacousticdomainΩ

isthesurfacedifferentialofacousticboundarydomain

istheoutwardnormalunitvectortotheboundaryΓ

Multiplyingasystemofpartialdifferentialequations (PDEs)byaweighting function

andthenexpressingthePDEs inan integral formulation,asconducted inEqn.2.7,is

referred to as theweighted residual method. The Galerkin method, as utilised by

ANSYStodiscreditsthewaveEqn.2.6,employsaspecificformulationoftheweighting

function.AdetaileddescriptionoftheGalerkinfiniteelementmethodmaybefoundin

Ref.[38].


21

Fromtheequationofmomentumconservation,thenormalvelocityontheboundaryof

anacousticelementisgivenby:

, = ∙= −

1+

43 ∙ ∇p +

43 ∙ ∇Q

(2.8)

SubstitutingthedefinitionprovidedinEqn.2.8intoEqn.2.7givesthe“weak”formof

acousticwaveequation,notatedas[36]:

1ψ + ∇ψ ∙

43 ∇

+ ∇ψ ∙1∇ + ψ , ds

Γ

= ψ1

+ ∇ψ ∙4

3∇Q

(2.9)

Consideringthatthenormalaccelerationofafluidparticlecanbepresentedusingthe

normaldisplacementofthefluidparticle,suchthat,

, = ∙ (2.10)

Where: isthedisplacementofthefluidparticle


22

theweakformoftheacousticwaveequationinEqn.2.9maythenbeexpressedas:

1ψ + ψ ∙

43

+ ψ ∙1

+ ψ

= ψ1

+ ψ ∙4

3

(2.11)

Thedefinitiongiven inEqn.2.11,which contains the fluidpressure and the fluid

displacement asthedependentvariables,canbediscretisedandnotatedinmatrix

form.Thegenericdiscretisationprocessofaweak form integral toreachthematrix

formulation,aspresentedinRef.[38],involvesthefollowingprocess:

i) Representationof thedomainof calculation by a setof finite elements

( = 1 … );

ii) Approximationofthesolutionfunctionandthetestfunctiononeachelement

asacombinationofthecorrespondingvaluesattheelementnodes,weighted

byashapefunction ( ):

( , ) = ( ) ( )

(2.12)

ψ ( ) = ( )ψ

(2.13)

Where ( ) arenodalvariablesofthesolutionfunction

ψ arenodalvariablesofthetestfunction

is the interpolation (or shape) function linking thesolution at coordinates with the solution the innode


23

EmployingEqn.2.12andEqn.2.13,thediscretisedelementaryweakformofEqn.2.11

maybedetermined.Assemblingofthesetofelementarycontributionsthenresultsin

thediscretisedglobalwaveequationmatrixformofEqn.2.11.Thisfinalglobalformis

presentedinEqn.2.14,wherethedependencyofallthetermsonthetestfunctionhas

beeneliminated.


24

{ } + { } + { } + [ ] , = (2.14)

Where: = ∭ { }{ } istheacousticfluidmass

matrix

= ∭ [∇N] [∇N] istheacousticfluid

dampingmatrix

= ∭ [∇N] [∇N] istheacousticfluid

stiffnessmatrix

[ ] = ∯ { }Γ

{ } { ′} istheacousticfluid

boundarymatrix

. = ∭ { }{ } {q}

+ ∭ [∇N] [∇N] {q} istheacousticfluidload

vector

{ } isthenodalpressurevector

istheacousticmassfluidconstant

{ } istheelementshapefunction2forpressure

{ ′} istheelementshapefunction2fordisplacements

{ } istheoutwardnormalvectoratthefluidboundary

{q} isthenodalmasssourcevector

2 The element shape function is the function that interpolates the solution betweendiscrete

values obtained at themesh nodes.Low order polynomials are typically chosen as shapefunctions.Thedegreeofthepolynomialrepresentingtheshapefunctionisdependentonthenumberofnodesoftheelementemployed.


25

Simulationsthatincorporatebothstructuralandfluidfields,andthecouplingbetween

them, requiremass, damping and stiffness matrices that describe both fields. This

culminatesinacoupledfluid-structurematrixgivenby[39]:

0 +

00

+

−0 =

(2.15)

Where: s denotesthepropertiesofasolidstructure

f denotesthepropertiesofafluid

Theextradiagonalterm– ofthestiffnessmatrix inEqn.2.15transferstheaffectof

thefluidpressuretothecoupledsolidstructurewhiletheextradiagonalterm of

themassmatrixrepresents theaffectof thestructure’s inertial forceson thesecond

derivativeofthefluidpressure.Thebi-directionalcouplinginFSIanalyses,asgivenby

Eqn.2.15,thereforerequiresasinglelayeroffiniteelementsattheinterfacebetween

the fluid and structural fields thathave bothpressure anddisplacementdegreesof

freedom.

WithregardtoEqn.2.15itisofnotethattheglobalmassandglobalstiffnessmatrices

of thecoupled fluid-structurematrixareasymmetric (i.e. theoff-diagonal termsare

nottransposesofeachother).Solvingforthenodalpressureanddisplacements,which

requires the inversion of the asymmetric matrix, therefore requires a significant

amountofcomputingresources.


26

To reducecomputational requirements,under the assumptionsof a linearharmonic

analysisasymmetricmatrixformulationforFSIproblemsmaybeachieved.Forlinear

harmonicanalysesEqn.2.15canbereducedtoanexpressionwithoutdifferentialsas,

− + + −− − + + =

(2.16)

A symmetrical formulation of Eqn. 2.16 may then be accomplished by defining a

transformationvariableforthenodalpressuressuchthat,

= = (2.17)

andsubstitutingEqn.2.17intoEqn.2.16sothatthesystemofequationsbecomes:

− + + −

− + +=

(2.18)

Eqn.2.18presentsasymmetricfluid-structurematrixformulationthatcanbeinverted

and solved for the vectors of the structural nodal displacements and the

transformation variable for nodal pressures , faster than the unsymmetric

formulationinEqn.2.15.Thenodalpressures canthenbecalculatedusingEqn.2.17.

As previously noted, the fluid-structure interaction detailed by the formulations in

Eqn.2.15andEqn.2.18account forthetwo-waycouplingbetweenstructuralbodies

andfluids.Thismechanismissignificantifastructureisradiatingintoaheavier-than-

airmediumsuchaswater,orifthestructureislightweight.Insuchcasestheacoustic

field is dissipated by the inducedvibrationof the structure,whichhas the effectof

dampingthevibrationandacousticresponseofthesystem[39].


27

2.3.1.2NumericalErrorAssociatedwiththeFiniteElementMethod

As a numerical technique, the finite elementmethodprovides only an approximate

solutionratherthananexactanswer.Thetwoprimarlytypesoferrorsassociatedwith

finiteelementanalysesaremodellinganddiscretisationerrors.Theformerarisesfrom

anincorrectdecriptionoftheboundaryvalueproblem–beinggeometricandmaterial

properties, loading, applied boundary conditions and the analysis type. The latter

arises from the creationof themesh.Numerical errors,which arise from theuseof

numericalmethods,mayalsobesignificantifthemodelisill-conditionedorifthereis

alackofcomputingpower.

2.3.2 PASTFINITEELEMENTSTUDIESOFTRANSFORMERNOISE

Therehavebeenaseriesofpastinvestigationsemployingthefiniteelementmethodto

studying the vibration characteristics of individual transformer components. These

studieshaveconsideredthestructuralresponseofthewindings[40-42],thecore[42-

44], the active part (i.e. the core and the windings) [30, 45], and the tank [46].

However,thelimitedscopeoftheaforementionedanalysesandtheabsenceofacoustic

elements limit their ability to qualitatively or quantitatively predict the noise

emissionsofatransformer.

To overcome the shortcomings of empirical prediction formulas and numerical

structural vibration analyses in the prediction transformer noise, two past

investigationshavepresentedmulti-physics finiteelementmodels.Thesestudiesare

themostcomparabletothatwhichispresented inthisthesisandtheyaretherefore

discussedinfurtherdetail.


28

· M.Rausch,M.Kaltenbacher,H.Landes,andR.Lerch,Numericalcomputationof

theemittednoiseofpowertransformers,2001(fullcitationinRef.[3])

Ref. [3] is the earliest studywhichhas employed avibro-acousticFEmodel to

determineapowertransformer’snoiseemissions.Muchofthispaperfocuseson

thetheoreticalbasisandgoverningequationsbehind thephysical fields. Inthis

sensetheworkpresentedinRef.[3]isanapplicationoftheprecedingtheoretical

paperspublishedbythecontributingauthors(seeRefs.[47,48]).

Theprincipalshortcomingofthevibro-acousticmodelpresentedinRef.[3]isthe

omissionof the transformercore.Thishas led tonumerous simplificationsand

assumptions,whichpreventthemodel frompredicting atransformer’sacoustic

emissionsundernominaloperatingconditions.Limitationsofthevibro-acoustic

modelpresentedinRef.[3]include:

- Thevibrationcharacteristicsofthetransformercore,whicharenotedin

Refs.[7,28,44,49-51]tosignificantlyaffecttheoverallsoundpressure

levelofatransformer,areoverlooked;

- Onlytheacousticemissionsatdoublethelinefrequencyareconsidered,

being those at 100 Hz in the study conducted in Ref. [3]. This is a

simplification that does not hold true in reality due to the significant

sound emissions from a transformer at even harmonics of the line

frequency[10,15,52];and

- Theomissionofthetransformercorehasnecessitatedtheuseofspring

elements to support thewinding blocks within the numerical model

presented in Ref. [3]. However, the values assigned to these spring

elementshavenotbeenexperimentallyvalidated.


29

Followingtheirinitialpaper,theauthorsofRef.[3]presentedthevibro-acoustic

computation scheme again inRef. [53].This secondpaper furtherdetailed the

applicationofavibro-acousticsimulation,describinghowitmaybeemployedto

understand the influence of various components on a transformer’s sound

radiationlevels.However,thenumericalmodelinRef.[53]isidenticaltothatin

Ref.[3]andassuchhasthesameshortcomings.

· M.Kavasoglu,Loadcontrollednoiseofpowertransformers:3Dmodellingofinterior

andexteriorsoundpressurefield,2010(fullcitationinRef.[4])

AsimilarinvestigationtothatpresentedinRefs.[3,53]wascarriedoutinRef.[4],

athesispublishedbytheKTHRoyalInstituteofTechnology.Theinvestigationin

Ref.[4]appearstobecarriedoutonthesameFEmodelpresentedinRefs.[3,53].

As such, theprincipal limitationof the study inRef. [4] is theomissionof the

transformer core. This leads to identical shortcomings noted of the preceding

works, including assumptions that have not been validated and only the

predictionof the coil emittednoiseat a singlepointon theacoustic frequency

spectrum.ThestudyinRef.[4]isalsopublishedinRef.[54],

InadditiontothelimitationspreviouslynotedofRefs.[3,4,53,54],theinvestigations

did not conduct a detailed experimental modal analysis of the assembly. Such an

analysiswould have allowed forgreatervalidation of themodelling procedure and

subsequentvibrationandacousticestimates.Asnoted inRef.[12], thesimulationof

any structural system that results in an acoustic response should begin with a

numericalmodalanalysisvalidatedbyexperimentalresults.


30

2.4 EXPERIMENTALMODALANALYSIS

ExperimentalModalAnalysis (EMA) is theprocessof experimentally extracting the

vibration characteristics of a structure, being the natural frequencies,mode shapes

andmodaldamping.Thepurposeof theEMA resultspresented in this thesisare to

correlate a transformer’s measured vibration response to that simulated through

numericalmethods.Thevalidatednumericalmodalmaythenbeusedto:

· Predicttheresponseofthestructurewithcomplexexcitationforcesapplied;

· Provideresponse informationatpointswherephysicalmeasurementswerenot

taken;and

· Analysehow thevibro-acoustic characteristicsof a transformer are affectedby

modificationstokeystructuralelements.

WithinthisthesisthefastFouriertransforms(FFTs)oftheimpulseresponsedata(i.e.

frequency response functions),which are thebasisofEMA,havebeenprovidedby

SiemensAGAustria,TransformersWeizandwereacceptedasastartingpointforthis

investigation.Thedesignoftheexperimentsandthemeasurementchoicestherefore

didnotformpartofthisthesis.Furthermore,SiemensAGAustria,TransformersWeiz

had previously analysed theEMAdata todetermine themode shapes and natural

frequenciesof the transformer studied in this thesis. As such, only the theoretical

basisof EMA andmodal damping extraction are further discussed in this chapter.

MethodstocorrelateEMAresultstonumericalmodalpredictionsarealsopresented.

TwocommercialsoftwarepackageswereusedintheEMAconductedbySiemensAG

Austria,TransformersWeiz.BrüelandKjærPulsesoftwarewasutilisedforacquiring

rawdataandperforming thespectralanalysis.TheMe’Scopesoftwarepackagewas

employed for extracting the experimentalmodal parameters, including the natural

frequenciesandthemodeshapesofthestudiedtransformer.


31

2.4.1 THEORETICALBASIS

Thebasis forexperimentallydetermining thevibrationcharacteristicsof asystem is

the frequency response function (FRF). The function, defined as the ratio of the

Fourier-transformedresponsetotheFourier-transformedinputforce,allowsthebias

of the force spectral distribution to be removed from the response measurement;

thereforedescribingtheunbiasedstructuralresponseofasystem[55].To introduce

modalparameter extractionmethods and numericalmodel validation techniques, a

derivationoftheFRFassumingviscousdamping isprovided.Thederivationisbased

onthatprovidedinRef.[56].

Theequationofmotion for adiscrete linearsystem (or asystem inwhich the finite

elementapproachhasbeenappliedsuchthattheresponseisconsideredataseriesof

discrete locations) has been notated in Eqn. 2.3. Considering the case where the

structureisexcitedsinusoidallybyasetofforcesallatthesamefrequency,ω,butwith

variousamplitudesandphases,suchthat:

{ ( )} = {F} (2.19)

andassumingthatasolutionexitsintheform:

{ ( )} = {U} (2.20)

Where: {F} istheforceappliedateachdegreeoffreedominthesystem

{U} istheresultantdisplacementateachdegreeoffreedominthe system

{F},{U} are × 1vectorsofthetime-dependentcomplex amplitudes

isthenumberofdegreesoffreedom


32

theequationofmotionexpressedinEqn.2.3becomes:

([K] + [C] − [ ]){ } = { } (2.21)

or,rearrangingtosolvefortheunknownresponses,

{ } = ([K] + [C] − [ ]) {F} (2.22)

whichmaybeexpressedas,

{ } = [H( )]{F} (2.23)

Where: [H( )] isthe × frequencyresponsematrixforthesystem andconstitutesitsmodalresponse

The general element in the frequency response matrix h ( ), defined as the

displacement at coordinate due to a single harmonic excitation force applied at

coordinate withafrequencyof ,isnotatedasfollows:

h ( ) =

{U}{F}

(2.24)

BeforeprecedingwiththedeviationoftheFRFandrelatingthemodalparametersto

thesystemmatricesitisimportanttonotethatthemodalmodelpossessorthogonality

properties.Thesepropertiesmaybestatedas,

[ ] [M][ ] = m╲

╲ (2.25)

and, [ ] [K][ ] = k╲

╲ (2.26)

Where: [ ] isthegeneralisedeigenvectormatrix

m╲

╲ isthemodalorgeneralisedmassmatrix

k╲

╲ isthemodalorgeneralisedstiffnessmatrix


33

Aproofoftheorthogonalitypropertiesofthegeneralisedmassandstiffnessmatrices

ispresentedinRef.[56].

Iftheeigenvectormatrix[ ]isnormalisedwithrespecttothemassmatrixthenthe

mass-normalisedeigenvectors[Φ]havetheparticularpropertythat,

[Φ] [ ][Φ] = [I] (2.27)

andtherefore, [Φ] [ ][Φ] =╲

╲ (2.28)

Where: [I] istheidentitymatrix

╲

╲ iscomprisedofeigenvalues to .

Therelationshipbetween themass-normalisedmodeshape formode r{ } and the

generalizedform{ } isgivenby{ } =√

{ } .

ReturningtoEqn.2.23itmaybestatedthat,

[H( )] = ([K] + [C] − [ ]) (2.29)

Premultiplyingtheabovedefinitionby[Φ] andpostmultiplyingbothsidesby[Φ],

[Φ] [H( )] [Φ] = [Φ] ([K] + [C] − [ ])[Φ] (2.30)

and assuming proportional damping in the form [C] = α[M] + β[K] (i.e. Rayleigh

damping),Eqn.2.29maybenotatedas:

[Φ] [H( )] [Φ] = [ ] + α[I] + β[ ] − (2.31)

Where: isthemass-proportionaldampingcoefficient

isthestiffness-proportionaldampingcoefficient


34

Notingthat[C]isalinearcombinationof[M]and[K]andtherefore[Φ] [C][Φ]isalso

orthogonal,theleft-handsideofEqn.2.31isorthogonalandthe itemis,

([Φ] [H( )] [Φ]) = + 2 − (2.32)

Where: isthemodaldampingratioofthe mode,definedas,

=12 + β

(2.33)

As the mass-normalised eigenvector matrix [Φ] has orthogonality properties with

respect to the mass, stiffness and viscous damping matrices, it can be noted that

[Φ] = [Φ] .Therefore,thefrequencyresponsematrix[H( )]canberelatedtothe

modalfrequency,modeshapesanddampingas,

[H( )] = [Φ]

⋱1

+ 2 −⋱

[Φ]

(2.34)

ResultingfromtheorthogonalityofthematricesinEqn.2.34,eachrowoftheequation

inisindependentandrepresentsasingledegreeoffreedomsystem.TheFRFh ( )is

thereforegivenby,

ℎ ( ) =

{U}{F} =

+ 2 −

(2.35)

Where: isthe elementofthe einvector{ } (i.e.therelativedisplacementatthatpointduring vibrationinthe mode)

N isthenumberofdegreesoffreedom(ormodes)


35

The expression in Eqn. 2.35 has defined the receptance FRF ℎ ( ) as the ratio

betweentheharmonicdisplacementresponseand theharmonic force.Similarly, the

acceleration ( ) = ( ) can be used as the response parameter, noting that it is

customarytomeasureaccelerationduringexperimentaltesting.Inthis formtheFRF

h ( )wouldbegivenas,

ℎ ( ) =

U

{F} = −

+ 2 − (2.36)

WithrespecttoEqn.2.35andEqn.2.36itcanbeseenthat:

· TheFRFrelatesthemodalparameterstothesystemmatrices;and

· Thesystem’svibrationresponseisasummationofthecontributionofallthe

system’smodes.

TheprocessofidentifyingthemodalparametersfromFRFdataiscommonlyreferred

toascurve-fittingormodalparameterextraction.

2.4.2 MODALDAMPINGEXTRACTION

Modal damping is a fundamental feature required to fully describe an oscillating

system, noting that damping exists in all vibratory systems that experience energy

dissipation. Ref. [57] provides a comparison of several curve-fitting methods for

extractionof themodalparameters fromFRFdata,withaparticular focusonmodal

dampingestimates.Forthepurposeoftheworkconductedinthisthesisanoverview

oftheHilbertEnvelopMethod(HEM)isprovided.


36

· HilbertEnvelopMethod(HEM)

The smooth curve outlining the amplitude of a rapidly oscillating signal that

variesgraduallywithtimeisreferredtoasthesignal’senvelope.Theenvelopeis

animportantcharacteristicofatime-domainsignalasitallowstheinstantaneous

attributesoftheseriestobedetermined.ByutilizingtheHilberttransform,which

providesananalytic signal from arealdatasequence, therapidoscillationof a

signalcanberemoved.Arepresentationofthedatasequence’senvelope isthus

constructed.

TheHilberttransformisemployedtocomputeatimesignalℎ( )fromtheoriginal

timesignalℎ( ),whereℎ( )retainstheamplitudesofspectralcomponentsbut

thephasesarealteredby /2suchthat:

ℎ( ) = sin (2.37)

ℎ( ) = − cos (2.38)

Where: istheamplitudeofthecomplexexponential

isthedampingfactor

istheangularfrequency

istheinstantaneoustimevalue

Themagnitudeoftheanalyticsignalℎ( ),whichdescribestheHilbertenvelope,

canbedirectlycalculatedfromℎ( )andℎ( )asfollows:

ℎ( ) = ℎ( ) + ℎ( )

(2.39)

Where: ℎ( ) = (2.40)


37

FromEqn.2.40 itcanbedeterminedthatthe logarithmof theHilbertenvelope

gives a first order polynomial whose slope is the negative of the damping

coefficient.Thismaybeexpressedas:

= − log ℎ( ) ⁄ ⁄ (2.41)

Figure2-3showstheimpulseresponseofanSDOFsystemandhowitsenvelope

canbeobtainedbymeansofaHilberttransform.

Figure2-3 Envelope computation of the impulse-response functionwithuseoftheHilberttransform[58]

WithregardtotheHEM,itshouldbenotedthatifaone-sidedspectrumisinverse

transformed tothetimedomain, theresulting timesignal isanalytic.Therefore

the imaginary component of the computed signal is automatically the Hilbert

transformoftherealpartanditsmodulusistheenvelope[57].SincetheHilbert

transform corresponds to a convolutionwith a hyperbolic function, it cannot

produceasuddenstepas illustrated inFig.2-3buta littleawayfromthe initial

partoftheresponsetheestimatedenvelopeisaccurate[59].


38

ThedescriptionoftheHEMprovidedso far inthischapterappliesonlytoSDOF

systems.WhenapplyingtheHEM toMDOFsystems, individualresonancepeaks

arebandpass filtered inasignal’s frequencydomainandinversetransformedto

give the equivalent SDOF impulse response function [57]. The HEM is then

appliedaspreviouslydiscussed.OfnoteisthattheHEMmayprovideinaccurate

modaldampingratio estimateswhen structureshavehighmodaldensitiesand

thereforeexhibitheavilycoupledmodes.

2.4.3 CORRELATIONOFEXPERIMENTALANDNUMERICALMODALRESULTS

TheprimaryapplicationofEMAresults in this thesis istovalidate theemployedFE

computation methodology. In practice, the accuracy of a FE model is limited by

uncertaintiesrelatingtogeometricandmaterialproperties,boundaryconditionsand

loadingappliedtothemodel.Reliablevibrationandacousticpredictionscantherefore

onlybemadeifthenumericalmodelisvalidatedwithexperimentalresults.

Thecorrelationoffiniteelementandexperimentalresultsisusuallyconductedwithin

thecontextoffiniteelementmodelupdating.However,asthisthesisisconcernedwith

developing anumericalmethodology tobeutilisedwith thedesignphase,nomodel

updatingwillbeconducted.Thecorrelationofnumericalandexperimentalresultsis

therefore used as a tool to validate themodellingmethodology employed. Several

correlationmethodsaredetailedasfollows:


39

· Naturalfrequencycorrelation

Thesimplestandmostfrequentlyemployedapproachtocorrelatetwomodal

models is through the direct comparison of natural frequencies. If the

experimental values and thenumerical results lie on a straight linewith a

gradient of one when plotted against each other the data is perfectly

correlated.Apercentagedifferencebetweentheexperimentalandnumerical

modescanalsobedefinedas:

ϵ = −

× 100

(2.42)

Where: ϵ is the error percentage between experimental and numericalnaturalfrequenciesatmode

isthenumericalnaturalfrequencyatmode

istheexperimentalnaturalfrequencyatmode


40

· Directmodeshapecorrelation

In addition to comparing natural frequencies, mode shapes may also be

directly compared by plotting numerical modes against those determined

experimentally.Themodalscale factor(MSF)allows formodalvectorstobe

compared and contrasted by normalising all estimates of the same modal

vector, taking into account magnitude and phase differences. Once two

differentmodalvectorestimatesarescaledsimilarly,elementsofeachvector

canbeaveraged,differentiated,orsortedtoprovideanindicationoftheerror

vector superimposed on the modal vector [60]. The modal scale factor is

definedas:

( , ) =

{ } { }∗

{ } { }∗

(2.43)

Where: isthenumericallydeterminedmodeshape

istheexperimentallydeterminedmodeshape

{}∗ denotesancomplexconjugatecolumnvector

If a numerical and an experimental vector are considered to represent the

samemode,thenumericallycomputedvectorandtheMSFmayalsobeusedto

complete theexperimentalmodalvector[60]. It isofnote that inalmostall

cases theexperimentaldataset is incompleteasmeasurementsaretakenat

selectedlocationsandinparticulardirections.


41

· TheModalAssuranceCriterion

ThefunctionoftheModalAssuranceCriterion(MAC)istoprovideameasure

of consistency (i.e. the degree of linearity between estimates of a modal

vector) [61]. It is one of the most popular tools for the quantitative

comparisonofmodalvectors.TheMAC is calculatedas anormalised scalar

productofthetwosetsofvectors{ }and{ }asfollows:

=

{ } { }∗

{ } { }∗ ({ } { }∗ )

(2.44)

AMAC value of one indicates that the two modes are perfectly correlated

whileavalueofzeroindicatesthatthereisnocorrelationatall.

· Frequencyresponsefunctioncorrelation

Inadditiontocontrastingmodalparameters,adirectcomparisonofFRFsmay

be conducted to determine the correlation between numerically and

experimentally determined data. To compare numerical and experimental

transfer functions a visual inspection is often sufficient to determine

agreement. Additionally, employing theEuclidean norm of theFRF vectors

measured at discrete frequencies allows for an error indication to be

computedas[62]:

ϵ =

( ) − ( )( )

(2.45)


42

Based on the MAC technique and on the concept of frequency shifting,

Ref.[63] proposed to measure the correlation between experimental and

numerically determined FRFs by using the Frequency Domain Assurance

Criterion(FDAC)as,

( , , )

= { ( )} { ( )}

{ ( )} { ( )} { ( )} { ( )}

(2.46)

Where: correspondstothemeasuredcolumnof[ ]

isthefrequencyatwhich{ }iscalculated

isthefrequencyatwhichtheFRFwas measuredexperimentally

TheFDACmeasurecanbeviewedasanequivalenttoMAC in the frequency

domain.Aswith theMAC criterion, avalueof zero indicatesno correlation

between the FRFs considered while a value of one indicates perfect

correlation.

Chapter3:FiniteElementModeloftheTransformer

43

CHAPTER3

FINITEELEMENTMODEL

OFTHETRANSFORMER


44

This chapter presents the finite element modelling methodology employed in this

thesis to determine the vibration and acoustic characteristics of a large power

transformer. The boundary conditions and electromagnetic and magnetostrictive

forcesappliedtothefiniteelementmodelarealsodiscussedindetail.Thestepstaken

inthedevelopmentoftheFEpredictiontoolareshowninFigure3-1.Forareviewof

thetheoreticalconstructsofthefiniteelementmethodologyrefertoChapter2.4.1.

Thefiniteelementcomputationschemepresented inthischaptercanbedividedinto

thefollowingfoursteps:

· Modellingtheactivepart,namelythecoreandthewindings;

· Modellingthetankandauxiliarycomponents;

· Definingacousticbodiesandfluid-structureinteractionintheFEmodel;and

· ApplicationofexcitationforcestotheFEmodel.

The abovemodelling steps are furtherdiscussed in the following subchapters.The

finalpartofthischapterdetailsthedampingcharacteristicsappliedtotheFEmodel,

as determined from the FRF data provided by Siemens AGAustria, Transformers

Weiz.


45

ParasolidModel

ConversiontoANSYSModel

DiscretisationandMeshing

SetMaterialProperties

SetBoundaryConditions

SimplificationofGeometry

NumericalPath

NumericalModalAnalysis

Experimental/AnalyticalPath

ModalTest

ExperimentalModalValues

PerformModalCorrelation

ModelValidation

ApplyLoadingtoModel

DetermineExcitationForces

ExperimentalSoundPressureMeasurements

AcousticFrequencyResponseAnalysis

PerformAcousticCorrelation

ModelValidation

AcousticTesting

Figure3-1 Stepsinthedevelopmentofthefiniteelementtooltopredictthevibrationandacousticcharacteristicsofapowertransformer


46

3.1 SPECIFICATIONSOFTHETRANSFORMER

The transformer analysed within this thesis is a single-phase, special purpose

transformer. A single-phase unit has been studied to simplify the electro-magnetic

loadingconditionsappliedtothemodel,wherebyreducingapotentialerrorsourcein

the numerical model. It is anticipated that simulating a three-phase transformer,

provided that the loadingconditionsof themodelareaccurately calculated,willnot

affectthevalidityofthevibro-acousticmodellingmethodologypresentedinthisthesis.

Thesingle-phase transformer studied in this thesis features a two-limb coredesign,

which decreases the magnitude of potential short-circuit forces by reducing the

ampere-turnsperheight.Thetwo-limbcoredesignhasbeenadoptedduetothehigh

rating of the unit. Siemens AG Austria, Transformers Weiz have provided the

computer-aided design (CAD)model of the transformer. An illustration of the unit

fromtheCADmodelisshowninFigure3-2.

Figure3-2 Geometryofthetransformerassembly


47

Duetothesensitivecommercialnatureofthedatapresented inthisthesis, including

vibration characteristics and sound pressure measurements, the exact technical

specifications of the transformer will not be stated. However, the basic geometric

parametersofthetransformeraregiveninTable3-1.

Table3-1 Geometricparametersofthetransformer

PartApproximateDimensions

(m)

ApproximateMass

(kg)

Transformertank · 4.0(l)x2.0(w)x3.5(h) 19,000

Core(single-phase,two-limbdesign)

· 3.3(l)x2.7(h)· Ø =0.3

19,000

Windings

· 1.6(h)· Ø =1.6

28,000

Radiators - 5,500

Conservator - 1,500

Bushings - 1,400

Insulationfluid - 18,000

3.1.1 SIMPLIFICATIONOFTHECOMPUTER-AIDEDDESIGNMODELThe illustration of the transformer shown in Figure3-2 is a simplified copy of the

originalcomputer-aideddesignmodel.Simplificationofthemodel,whichsuppresses

parts and featuresnot likely to influence the accuracyof analysis results, has been

carried out to increase computational performance both in terms ofmesh sizeand

meshquality.Thesimplificationmethodsthathavebeenemployedwithinthisthesis

aredetailedinTable3-2.


48

Table3-2 ModelSimplificationMethodology

FeatureType SimplificationMethodology

Fasteners Thefastenersconnectingbodies,i.e.bolts,nutsandscrews,havebeensuppressedwhensimplifyingtheCADmodel.Theconnectionbetweenbodiesandcomponentsconnectedwithfastenershasbeenconsideredasabondedcontact.

Surfacesheldtogetherwithadhesivesorsurfacesweldedtogetherhavealsobeenconsideredasexhibitingthecharacteristicsofabondedcontact.

Holes Holes,suchasthosecreatedfortheboltedconnections,havebeenfilled.Holesthatwerefilledhaveadiameteroflessthan20mmandarenotinproximitytoboundaryconditionsorappliedloads.

Chamfers Chamfersnotinproximitytoboundaryconditionsorappliedloadsandwithanedgelengthoflessthan20mmhavebeensuppressedinthemodel.

Rounds Roundsnotinproximitytoboundaryconditionsorappliedloadsandwitharadiusoflessthan5mmhavebeensuppressedinthemodel.

3.2 MODELLINGOFTHETANK

There are several reasons for modelling the tank of the transformer. The most

significantoftheseistheneedtoidentifythesituationswherevibrationspropagating

fromtheactivepartexciteanaturalfrequencyofthetankstructure.Attheseinherent

natural frequencies the tankcanamplify thevibrationspropagating from theactive

part.Thisphenomenon,referredtoasaresonance,resultsinelevatedsoundradiation

levels.

Todiscretisethetransformertankthree-dimensionalsolidelements,namelySolid185

andSolid186, have been employed.Thequadratic displacementpropertiesof these

elementslendthemtomodellingcomplexgeometry.Thediscretisedtankstructureis

illustratedinFigure3-3.


49

(a)

(b)

Figure3-3 Finiteelementmodelof(a)thetransformertankand(b)the transformertankwithauxiliaryequipment


50

Within the finiteelementmodeltheconnectionsbetween the tank,thebushings,the

conservatorandtheauxiliaryequipmenthavebeenmodelledasbondedcontacts.The

bondedcontactsreplicatethefastenersandweldedjointsthatconnectthesestructural

elements.To replicate the test setupof the transformeranelasticsupporthasbeen

modelledon thebaseof the transformer tank.Thevalueof theelastic supportwas

calculated from the mechanical properties and dimensions of the rubber mat

positionedunderthetransformerduringtesting.

3.3 MODELLINGOFTHEACTIVEPART

Thecoreandthewindingsarethetwoprincipalcomponentsthatcomprisetheactive

partofatransformer.Additionalcomponentsintheactivepartassemblyincludepress

plates, press rings, support blocks and tie-rods, which act primarily to maintain

clamping pressure on the core and thewindings.Aswith the tank structure, solid

elements Solid185 and Solid186 have been employed to discretise the active part

assembly.ThefiniteelementmodeloftheactivepartisillustratedinFigure3-4.

Figure3-4 Finiteelementmodeloftheactivepart


51

AsseeninFigure3-4,asignificantfocusofthemodellingprocedureemployedinthis

thesishasbeentoincorporateallkeystructuralelementsintothefiniteelementmodel

of the active part. Creating such a detailed model, whereby all components that

comprise the activepartaremechanically coupled,hasnegated theuseof fictitious

springelementsorother constraints3.Althoughnot illustrated inFigure3-4, it isof

notethattheactivepartisalsodirectlycoupledtothebaseofthetank.

Withinthenumericalmodeloftheactivepartfastenershavebeensuppressedandthe

connectionsbetweencomponentsheldtogetherwithfastenersoradhesiveshavebeen

modelledasbondedcontacts.Furthermore,bondedcontactshavebeenusedbetween

thewindings,thepressringsandthepressplates.Thebondedcontactbetweenthese

aforementionedelementsreplicatestherigidclampingsystemthatmaintainspressure

onthewindingassemblyoverthelifeofthetransformer4.

Severalmaterialsandcomponentswithintheactivepartassemblydisplayanisotropic

mechanical behaviour (i.e. they exhibit directionally dependent mechanical

properties).To accuratelyaccount for the structural responseof theactivepart the

anisotropic material properties of these parts have been accounted for in the FE

model.Theanisotropicmaterialsand componentswithin theactivepartare further

describedinthefollowingparagraphs.

3Fictitiousspringelementshavebeenrequiredinpreviousfiniteelementstudiesinvestigating

the acousticemissionsofpower transformer.These springelementshavebeen required toconstrain theactivepartwithin the tankasnomechanicalcouplingbetween theactivepartandthetankhasbeenmodelled(seeRef.[3]andRef.[4]).

4 In order towithstand themechanical stresses during lifting, transportation, operation andshortcircuitsevents thewindingassemblyofa transformermustbeappropriatelyclamped.Clampingisgenerallyconductedbyapplyingaforcetothepressringandthenfillingthegapcreatedbetweenthepressringandtheclampingplatewithinsulationmaterial.


52

· Windingblockassembly

Transformerwindingsaremanufacturedfrompaperinsulatedcopperconductors

and pressboard spacers. Depending on the specifications of the unit and

manufacturerpreferences,theseconductorsmaybeintheformofindividualstrip

conductors, bunched conductors or continuously transposed cable conductors.

Thecomplexityofthewindingsmakesaccurategeometricmodellingimpractical.

Therefore, within this study the winding block has been considered as a

homogeneousbody.

The mechanical properties of the homogeneous winding block have been

determined from measurements relating the deflection of the winding to the

forcesappliedduringtheassemblyprocess.Thesemeasurementswereprovided

bySiemensAGAustria,TransformersWeiz.Fromthisdatatheelasticmodulusof

thewindings inthepressingplanehasbeendetermined.Intheradialplanethe

windingswereassumedtohaveanelasticmodulusequaltothatofcopper.

· Electricalsteel

Thepurpose of a transformer core is to provide a low-reluctancepath for the

magnetic flux that links theprimary and secondarywindings.Themajorityof

modern transformer cores are therefore produced from grain-orientated steel,

which reduces hysteresis losses, increases permeability and also increases

resistivity [52].However,asaresultof thegrain-orientation,coresteelsexhibit

anisotropic properties. The propertiesof the corematerial utilisedwithin this

thesishavebeentakenfromvaluespublishedbyelectricalsteelproducers.


53

· Clampingrings

Clampingringsareanintegralcomponentofatransformer’sactivepart,designed

topreventthewindingsfromsustainingpermanentdistortionordamageduring

short-circuitevents.Theclampingringsoftheunitanalysed inthis thesiswere

produced from densified cellulous pressboard layers and adhesive. This

construction results in weaker mechanical properties in the radial plane. The

anisotropicmaterialpropertiesofthisproductionmethodhavebeendetermined

bySiemensAGAustria,TransformersWeizthroughexperimentalmeasurements.

3.4 ACOUSTICBODIESANDFLUID-STRUCTUREINTERACTION

Theoilwithinatransformer isvital to itseffectiveoperation,especiallyso in larger

units.Thisisduetothedualrolesthatthefluidperforms,actingasacoolantandasan

insulator.Theenergy lossesofatransformermanifest themselvesprimarilyasheat,

and efficient cooling is therefore required inorder to limit themaximumoperating

temperature. Furthermore, theoil helps to insulate components of the transformer

assembly that are at different electrical potentials from one another. Through the

impregnationofvariousporouscomponentstheoilalsoaddstotheefficiencyofsolid

insulation.

In this study the insulationoilhasbeen consideredasan acousticdomain.Thishas

allowed for the bi-directional couplingof fluid and structural elements, creating an

indirect transmission path between the active part and the tank structure. Fluid

elements Fluid220 and Fluid221 have been employed to model the acoustic body

representing the insulation oil. These three-dimensional solid elements exhibit

quadraticpressurebehaviourandarethereforesuitedtomodellingthefluidmedium

andthefluid-structureinterface.


54

Inaddition totheacousticdomainrepresenting the insulationoil, the finiteelement

predictionofsoundradiationhasrequiredtheoscillationsofthetransformertankto

be applied to an external acoustic body. Specifically, the nodal velocities from the

transformer’sexternalsurfaces,whichresulted fromexcitationof the finiteelement

model,havebeeninterpolatedandmappedtoanacousticmesh.Figure3-5illustrates

thenodalvelocityvectorsofthetransformertank.TheforcesappliedtoexcitetheFE

model,replicatingtheelectromagneticandmagnetostrictive forcesintheactivepart,

arefurtherdescribedinChapter3.7.

Figure3-5 Nodalvelocityvectorsmappedtotheacousticbodysurrounding thetransformertank

As opposed to the bi-directional fluid-structure interaction between the insulation

fluid, the active part and the tank, a single direction coupling has been modelled

between the tank and the external acoustic body. This single direction coupling is

assumed tobevalid,as theairsurrounding the transformer tankdoesnotaffectthe

amplitudeorothervibrationcharacteristicsofthetank.Furthermore,neglectingthe

acousticeffecton thestructureand thereforedefining aone-way coupling from the

structuraltotheacousticfieldhasprovidedamorecomputationallyefficientanalysis.


55

Theacousticfiniteelementdomainsurroundingthetransformerhasbeenmodelledto

geometrically and physically replicate the transformer’s environs. Specifically, the

acousticbodyaimstoreproducethetestlaboratory inwhichtheexperimentalnoise

measurementsoftheunitwereobtained.Fluid221elementswereusedtomodelthe

externalacousticbodysurroundingthetransformer.

3.5 EXCITATIONMECHANISMS

Theelectromagneticandmagnetostrictiveforcesthatactofatransformer’sactivepart

have been extensively studied. Detailed mathematical descriptions and calculation

proceduresoftheseforcesarefoundinmanytextbooks,withRefs.[7,64,65]beinga

selectedfew.Therefore,comprehensivemathematicalformulationsandcalculationsof

theelectromagnetic andmagnetostrictive forceswillnotbepresented in this thesis.

Insteadthischapterwillfocusonthecharacteristicsoftheexcitationmechanismsina

transformer’s active part and how these characteristics influence the acoustic

behaviourofatransformer.

TheelectromagneticandmagnetostrictiveforcesappliedtoexcitetheFEmodelinthis

thesis have been calculated with Siemens AG’s internal noise prediction tool. The

calculationtoolaccountsforboththephysicaldescriptionoftheforcesandthespecific

designcharacteristicsoftheactivepart,suchasdimensionsandmaterialgrades.The

three excitation mechanisms that generate the vibrations of the active part are

Maxwell andmagnetostrictive forces, acting on the core, and Lorentz forces in the

windings.


56

3.5.1 EXCITATIONMECHANISMSINTHECOREThevibrationsof the core,whichproduceacousticemissionsreferred toasno-load

transformernoise,resultfromalternating flux intheactivepart.Thealternatingflux

causesmagnetostrictiveforcesintheferromagneticcoresheetsandMaxwellforcesat

corner joints and air-gaps. The amplitudes of thesemagnetic andmagnetostrictive

forces are dependent on both the flux density in the core sheets and thematerial

propertiesofthemagneticcoresteel.Furtherdescriptionsofthesetwo forcesacting

onthetransformercorearegivenasfollows:

· Maxwellforces

Maxwellforcesactontheboundariesbetweentwomagneticmediawithdifferent

permeabilities. In transformer cores these forces are therefore typically

associatedwithair-gapregionsand joints.Theamplitudeof theMaxwell forces

canbecalculatedbythesurfaceintegrationofthelocalforcedensitybasedonthe

Maxwellstresstensoras[5,66]:

=

1 ( )−

12

2 (3.1)

Where: ismagneticconstantpermeabilityoffreespace

isthemagneticfluxdensityflowingthroughthe closedsurfaceareaA

isaunitvectornormaltosurfaceA

istheareadifferentialoftheclosedsurfaceareaA aroundthedomainofinterest


57

FromEqn.3.1itcanbedeterminedthattheMaxwellforcesactpredominantlyat

twice thepower frequency.This is due to the forces beingproportional to the

squaredmagnetic flux density in the core sheets.Therefore,assuming that the

structuralresponseofatransformerislinear,thesoundradiationresultingfrom

Maxwellforcesinthecoreisdominantattwicethenetworkfrequency.

· Magnetostrictivestrain

Magnetostriction refers to thechange in thedimensionsof amagneticmaterial

whenitissubjectedtoamagneticfield.Thedimensionalchangeresultsfromthe

rotationofuniformmagneticdomains–regionswithinamagneticmaterialthat

have uniform magnetisation – under an applied magnetic field. Within a

transformercorethemagnetostrictivestrainreachessaturationtwicepercycleof

thenetworkfrequency[6].

Magnetostrictionexhibitshystereticbehaviour(i.e.timebaseddependence)with

respect to the applied magnetic field [67]. This complex hysteretic behaviour

reflectsthenonlinearchangeofthemagneticdomainswithinamagneticmaterial

during AC magnetisation. Magnetostrictive strain curves in the time domain

illustratingthisphenomenonareshowninFigure3-6atdifferinginductionlevels.

The measurements provided in Figure 3-6 were recorded during quality

assurancemeasurementsperformedongenerictransformercorematerial.


58

Figure3-6 MagnetostrictiveeffectinSiFesheetsatdifferinginductionlevels (Source:SiemensAGAustria,TransformersWeiz)

Thenonlinearcharacteristicsof themagnetostrictivestrainoccurring in transformer

core steel result in the sound radiation from magnetostriction containing even

harmonicsof thenetwork frequency.Thesignificanceofeachof theseharmonics is

dependentontheinductionlevelinthecore.


59

3.5.2 ELECTROMAGNETICFORCESINTHEWINDINGSTheelectromagnetic excitationof a transformer’swindings,whichproduceacoustic

emissionsreferredtoas loadnoise,resultsfromthe interactionbetweenthecurrent

carrying winding loops and the magnetic stray flux. These forces, referred to as

Lorentz forces, are determined by the current amplitude and themagnitude of the

magneticstrayfluxatthewindinglocationsuchthat:

= × (3.2)

Where: istheLorentzforce

istheelectriccurrentdensity

isthemagneticfluxdensity

AswithMaxwellforces,theLorentzforcesaredominantattwicethepowerfrequency.

ThisisduetotheLorentzforcesbeingproportionaltothesquaredwindingcurrent[6,

52,68].Thesound radiationresulting fromLorentz forces is thereforeprevalentat

twice the network frequency, assuming that a transformer’s structural response is

linear.


60

3.5.3 SUMMARYOFTHECHARACTERISTICSOFTHEEXCITATION

MECHANISMSFrom the preceding descriptions of the excitation mechanisms in a transformer’s

activepartitcanbesummarisedthat:

· The Maxwell forces in the core and the Lorentz forces in the windings act

predominantlyattwicethelinefrequency;and

· Thenon-linear characteristicsof themagnetostrictive strain in a transformer’s

core results in significantmechanical forcesat evenharmonics of thenetwork

frequency.

Higherharmonicsintheacousticspectrumofatransformermayalsoresultfromthe

presence of nonlinear magnetic stray flux in the active part and the nonlinear

behaviourofthewindingsandothercomponents.

Pastliteratureonthetransformernoisecharacteristics[67,69,70]identifiesthefirst

threeevenharmonicsofthelinefrequencyasbeingdominantinthesoundradiatedby

transformers (i.e. assuming anetwork frequencyof 50Hz thenoise emissionsof a

transformermaybeconsidereddominantat100Hz,200Hzand300Hz).Thisnotion

is furthersupportedby the typicalacoustic frequencyspectrumof a transformer,as

illustratedinFigure3-7. Itcanthereforebeconcludedthatanoisepredictiontoolto

estimateacousticemissionsofatransformeroperatingundernominalconditionsmust

incorporatethefirstthreeevenharmonicsofthenetworkfrequency.


61

Figure3-7 Typicalacousticspectrumproducedbya50Hzpowertransformer (Source:SiemensAGAustria,TransformersWeiz)

3.5.4 MODELLINGOFTHEFORCESAPPLIEDTOTHENUMERICAL

MODELWithin this study multiple mono-harmonic linear simulations (i.e. simulations at

discrete frequencies in which the FE model is excited with sinusoidal forces that

produce aharmonicresponse)across a100Hz to400Hz frequency spectrumhave

beenperformedtodeterminethevibrationandacousticresponseofthenumerically

modelled transformer. This approach has been adopted as it is able to identify the

distanceoftheassembly’sresonancepeaksfromthemechanicalexcitationfrequencies

intheactivepart,being100Hzor120Hzandhigherharmonicsthereof.Aprediction

ofthisdistancebetween theresonance frequenciesand theexcitation frequencies is

criticalinunderstandingtheriskofhighnoiseemissionsfromatransformer.


62

Themultiplemono-harmonicsimulationapproachemployedinthisthesisaccountsfor

variation in the natural frequencies’ of a transformer arising from both model

uncertainty andmanufacturing variability. A simulation approachwith fixedmulti-

harmonicexcitationwouldonlyhaveprovidedthreenoisepredictionpointsoverthe

100Hzto400Hzrangeofinterest(beingatthedominantharmonicsoftheexcitation

mechanisms). Furthermore, a fixed-multi-harmonic simulation would not have

accounted for any variability between numerical and experimental natural

frequencies.

The forcesemployed in this thesis toexcite theFEmodelateachdiscreteexcitation

frequency replicate the fundamental frequency component of theMaxwell, Lorentz

and magnetostrictive forces present in the transformer’s active part. These

aforementioned forceshavebeenappliedto thesurfacesoftheactivepartgeometry

onwhichtheyprimarilyact.Assuchandwherevalid,Lorentzforceshavebeenapplied

to the windings, Maxwell forces have been applied to the core joints, and

Magnetostrictive strain has been applied to the electrical core sheets. Further

discussion on the characteristics of these forces has been provided previously in

Chapter3.5.3.


63

Although not considered in this thesis, the multiple mono-harmonic simulation

approachmayalsobeemployedtoestimate a transformer’s totalnoise levelsunder

nominal operating conditions. This total noise prediction would require the non-

linearity of the excitation mechanisms to be accounted for, which has not been

considered in this thesis.Specifically, thediscrete amplitudesof theelectromagnetic

andmagnetostrictiveforcesatthefirstthreeevenharmonicsofthenetworkfrequency

wouldneedtobedeterminedandappliedwithinthenumericalmodel.

Employing the multiple mono-harmonic simulation approach, the prediction of a

transformer’s total operational acoustic emissions would be made through the

summation the sound pressure levels determined at each simulation frequency.

Therefore, the total soundpressureof transformer accounting formeasurementsat

thenetworkfrequency’sfirstthreeevenharmonicswouldbedeterminedby:

∑ = 10 log 10 + 10 + 10 (3.3)

Where: indicatesindividualthesoundpressurelevelsat eachofthedominantharmonics

It is of note that employing Eqn. 3.3 is similar to conducting a multi-harmonic

simulationtocomputeasingletotalsoundpressurelevelestimate.

Withinthisthesistheoperationalacousticcharacteristicsoftheanalysedtransformer

are not stateddue to the sensitive commercialnatureof thedata.However, adhoc

acousticmeasurementsforthevalidationofthefiniteelementmodeloverarangethat

incorporatesthefirstthreeevenharmonicsofthenetworkfrequencyarepresentedin

Chapter4.3.Thevalidation of the numerical vibro-acousticmodel represents akey

noveltyofthisthesis.


64

3.6 MODALDAMPING

Dampingexistsinalloscillatorysystemsthatexperienceenergydissipation,havingthe

effect of diminishing the system’s vibration in the neighbourhood of a resonance.

Dampingalsoaffectsthetransmissibilityofvibrationsthroughasystem,i.e.theratio

betweenthevibrationoutputandimposedvibrationinput.Accuratemodellingofthe

damping loss factor in vibro-acoustic predictions is therefore critical in order to

accountforthecorrecttransmissibilityofvibrationsandsound.

Asopposedtothemassandstiffnessmatrices in finiteelementmodels,thedamping

matrixofanassemblycannotbepracticallyevaluated fromstructuralmaterialsand

member sizes. Instead modal damping is determined from dynamic tests and

experimental results. To fully describe the vibration response of the transformer

analysedwithinthisthesis,themodaldampingcharacteristicsofboththeactivepart

and the oil-filled tank assembly have been determined through the evaluation of

experimentalmodalanalysis(EMA)results.

TheEMAsoftheactivepartandtheoil-filledtankassemblywereconductedbymeans

ofrovingimpacthammertests.Adetailedtheoreticaldescriptionofthismethodcanbe

foundinRefs.[71,72].SiemensAGAustria,TransformersWeizperformedtheEMAsof

theactivepartand thetankassemblyandprovided therelevantdatautilisedwithin

this thesis.Of note is thatnoEMAwas conducted on the activepartof the specific

transformermodelledandanalysedwithin this thesis.Therefore,data fromanEMA

conductedontheactivepartofasimilarunithasbeenutilised.


65

FollowingtheevaluationofdampingfromEMAresults,thischapterwilldetailhowthe

modaldampingcharacteristicsofthetransformerhavebeenaccounted for in theFE

model. The most common formulation to numerically model modal damping in

dynamic FE simulations is that of Rayleigh damping. Expressed in matrix form,

Rayleighdamping is a linear combination of themass and stiffnessmatricesof the

structure:

[ ] = [ ] + [ ] (3.4)

Where: [ ] istheviscousRayleighdampingmatrix

[ ] isthemassmatrix

[ ] isthestiffnessmatrix

isthemass-proportionaldampingcoefficient

isthestiffness-proportionaldampingcoefficient

In this thesis ithasbeenassumed thatdamping is in the formofRayleighdamping

(i.e.mass and stiffness proportional damping). Therefore the modal damping

investigationsdetailedinthefollowingsubchaptersareconductedwiththeobjective

ofdeterminingtheRayleighdampingcoefficientsalphaandbeta.


66

3.6.1 MODALDAMPINGOFTHEACTIVEPART

Themostcommonapproachtodescribingthedampingcharacteristicsofasystemis

throughthemodaldampingratio.Thisparameter,usuallydenotedbyzeta,ζ,provides

amathematicalmeansofexpressing the levelofdamping in asystemrelativeto the

system’scriticaldamping5.Inthisinvestigationthehalf-powerbandwidthmethodhas

been employed to determine the damping of the active part. The aforementioned

methodology is themostwidelyused frequencydomainapproach fordetermining a

system’sdampingbehaviour.

Thehalf-powerbandwidthmethodutilisesthefrequencyresponsecurvefromaforced

vibration test to determine of themodal damping ratio.Employing the half-power

bandwidthmethodologythemodaldampingratioiscomputedfrom:

ζ =−

/2 (3.5)

Where: ζ isthemodaldampingratio

isthenaturalfrequency

, arepointsonthefrequencyresponsecurvethatthat relatetothemaximumresponsemultipliedby1/√2

5Asystem’scriticaldampingreferstothedampingrequiredtoreturnaperturbedsystemtoits

equilibriumpositionintheshortestpossibletimewithoutanyoscillation


67

InthisstudytheFRFdatadescribingthevibrationcharacteristicsoftheactiveparthad

qualitieswellsuited totheuseof thehalf-powerbandwidthmethod.Thesequalities

included:

· Clearlydefinedandisolatedmodeswithinthefrequencyresponsefunction,which

allowedfortheaccurateidentificationofthehalf-powerpoints;and

· Ahighresolutionofthe frequencyresponse function,withmeasurements taken

at 0.25Hz intervals.This allowed for accurate identificationof thenormalised

accelerationmagnitudeoftheFRFfunctionsatresonancepeaks.

Applying thehalf-powerbandwidthmethod atresonancepeaksalong themeasured

frequencyspectrumhasallowedfortothemodaldampingcharacteristicsoftheactive

parttobedetermined.Forthepurposesofthisstudythedampingcharacteristicshave

beendeterminedbetween100Hzand for400Hz,matching the frequencyspectrum

overwhichnumericalvibrationand acousticanalysesof the transformerhavebeen

performed.

Illustrationsof thehalf-powerbandwidthmethodology as applied to the resonance

peaksmeasuredat192Hzand346Hzarepresented inFigure3-8(a)andFigure3-

8(b),respectively.Figure3-8(c)showstheRayleighdampingcurvedetermined from

themodaldamping ratios calculated at 192Hz and 346Hz.TheRayleighdamping

curveinFigure3-8(c)hasbeencomputedfromthedefinitionprovidedinEqn.3.16.


68

(a)

(b)

(c)

Figure3-8 Half-powerpointsonanFRFemployedtocalculatethedampingratiosof theactivepartat (a)192Hzand (b)346Hz (c)and the resultingRayleighdampingcurve


69

3.6.2 MODALDAMPINGOFTHETANKSTRUCTURE

Conventionalcurve-fittingmethodssuchasthehalf-powerbandwidthmethodandthe

Hilbert EnvelopeMethod (HEM) determine themodaldamping characteristics of a

system at isolated resonance peaks. However, the complexity of the transformer

assembly studied in this thesis, with a high modal density and therefore heavily

coupledmodes,hasnullifiedconventionalcurve-fittingapproaches.Anoveladaptation

of the HEM, herein referred to as the adapted damping estimation technique, is

thereforeemployedinthisthesis6.

To determine the studied transformer’s modal damping characteristics, the novel

dampingtechniqueestimatesmodaldampingwithindefinedbandwidthsasopposed

toat isolated resonancepeaks.The adapteddampingestimation techniqueassumes

thatasingleprincipalresonance,whichoccursatanunknown frequency,dominates

within each defined bandwidth. All other resonance peaks that occur within the

bandwidth are therefore considered as noise.Under this assumption theprevailing

resonanceineachbanddominatesthedecayofthetimedomainimpulseresponse.The

unknown frequency of the dominant resonance is approximated by the geometric

meanvalueoftheband(thecentrefrequency).

The adapted damping estimation technique is further detailed in the following

paragraphswithamathematicaldescription followedbyanumerical justification.An

overviewofthemethodologyasappliedtotheEMAdataoftheoil-filledtransformer

assemblyanalysedwithinthisthesisisalsopresented.

6TheHilbertEnvelopeMethod isusedas the theoretical foundationof the adapteddamping

estimationtechniqueduetoitsrelativesimplicityanditshighlevelofaccuracyincomparisontoothermethods(refertoRef.[73]foracomparisonofcurve-fittingtechniques).


70

· MathematicalDescription

Withinthisthesisthedampingcharacteristicsofthetransformerassemblywere

calculatedwithinone-thirdoctavebands.However,otherbandwidths,depending

on the vibration characteristics of the structure analysed, may have been

employed. The one-third-octave bandwidth delineation of the FRF signal is

notatedby:

Δ = ℎ /2 ,ℎ 2 ; ℎ = ℎ 2 (3.6)

Where: Δ isthefrequencyband

ℎ isthegeometricmeanofthefrequencyband

AsanadaptionoftheHilbertEnvelopMethod,themodaldampingestimationwas

conducted in the timedomain.Thisrequired the inverse fastFouriertransform

(IFFT)oftheFRFtobetaken,suchthat:

( ) = ( ( ) ⋅ ( ;Δ )) (3.7)

Where: ( ) istheimpulsesignalinthetimedomain

( ) istheFouriertransformofthecomplexseriesx(t)

Theterm ( ; Δ )inEqn.3.7actsasanindicatorfunction(idealfilter),allowing

only the frequencydatapointswithin thegivenone-third-octavebandwidth to

pass through the IFFT algorithm. The remaining FRF data points, which fall

outsidetheband,returnazerovalue,suchthat:

( ;Δ ) = 1 ∈ Δ0 ℎ (3.8)

The time domain vector resulting fromEqn. 3.7 therefore retains the original

lengthoftheFRFvector.


71

SincetheFRFfunction ( )inversetransformedtothetimedomaininEqn.3.7is

one-sided(nonegativefrequencycomponent)theresultingtimesignalisanalytic.

Furthermore, itcanbeassumed that theanalytic signalmaybemathematically

approximatedasasimpleharmonicoscillationintheform,

( ) = sin( ) (3.9)

Where: istheanalyticsignal

isthedampingfactor

istheangularfrequency( = 2 )

Takingtheabsolutevalueoftheanalyticsignalresultsinanexponentialfunction

thatdescribesthesignal’senvelope.Thismaybenotatedas:

| ( )| = (3.10)

Where: isaconstant

ThenaturallogarithmofEqn.3.10representstheenvelopeoftheanalyticsignal

asalinearfunction:

log(| ( )|) = log( )− (3.11)

The function inEqn.3.11 isa firstorderpolynomial.Therefore, identifying the

coefficients of the first order polynomial that best fit the function, in a least-

squaressense,providesanestimateof− .


72

The calculation of the modal damping ratio is computed assuming that the

frequencyof thedominant resonancewithin eachdefinedband isequal to the

geometricmeanvalueofthebandofinterest.Althoughthisassumptionmaylead

tohighlyapproximatemodaldampingratiosforeachband, it isconsideredthat

fitting the Rayleigh damping coefficients to a number of damping estimates

calculated over the frequency spectrum provides a reasonably accurate

approximationoftheproportionaldampingbehaviourofthestudiedsystem.The

fitting of the Rayleigh damping coefficients to the calculated modal damping

ratiosisfurtherdetailedinChapter3.6.3.

· NumericalJustification

In the absence of an empirical validation, a numerical study of the adapted

dampingestimationtechniquehasbeenconducted.Ananalyticalstudyandafull

empiricalvalidationwillformpartofthefutureworkstobeconductedfollowing

this thesis. Further detail of these futureworks is provided in Chapter 7. The

numericalstudy,whichconsidersonlymass-proportionaldamping,providesan

indicationoftheerrordistributionassociatedwiththemethodology.Theeffectof

the frequencyresponse function’sbandwidthdelineationon theaccuracyof the

methodologyisalsoconsidered.


73

The partial numerical validation of the adapteddamping estimation technique

wasconductedbycreatingarandomFRFtowhichthetechniquewasapplied.The

randomFRF featureda fixednumberofresonancepeaks withthepositionin

the frequencydomain andtheamplitudeandphaseoftheresonancesbeing

randomlyassigned.Thedampingateachofthe resonancepeakswasimposed

inthe form: = + β .Specific featuresoftherandomFRFusedwithin

thisthesisincluded:

- 400randomresonancepeaksweregeneratedwithuniformdistribution

alongafrequencyspectrumfrom0Hzto1600Hz;

- Thedampingfactorateachresonancepeakwasobtainedwithregardto

mass-proportionalRayleighdamping,suchthat

( ) = 1

2 (3.12)

( :βissettozeroinEqn. 3.12)

- A secondrandomvectorwithuniformdistribution [0,1]determined

therelativesignificanceofeachresonance.

Considering the above three points, the random generationof the FRF can be

notatedmathematicallyas:

FRF( ) =

g

1 − + 2 ζ (3.13)

Where: g isthemagnitudeofresonancen

/ is the angular frequency vector of the FRFnormalised to the angular frequency at which theresonancenoccurs

ζ isthedampingfactoratresonancepointn


74

Variables within the generation of the random FRF included the number of

resonancepeaks,thesamplingrate,thenumberofsamplestakenandthemass-

proportional damping factor. These variables were assigned values of, or

considered representative of, those in the FRF of the oil-filled transformer

assembly.Thevaluesof thevariablesemployed in the random FRFgeneration

andfurtherexplanationofthesevaluesisprovidedinTable3-3.

Table3-3 VariablesemployedinthegenerationoftherandomFRF

Parameter Value Relationtotheexperimentalcase

Samplingrate 4096 Thevalueutilisedduringtheexperimentalmodalanalysisofthetransformerassembly

Numberofsamplestaken

4096 Thevalueutilisedduringtheexperimentalmodalanalysisofthetransformerassembly

Mass-proportionaldamping

50 Consideredrepresentativeofthemass-dampingvalueofthetransformer,withtheoilcontainedwithinthetransformertankhavingasignificantmass-dampingeffect

Naturalfrequenciesgenerated

400 Consideredrepresentativeofthetransformer’shighmodaldensity

The error estimate associatedwith the adapteddamping estimation technique

wasdeterminedbycomparing:

- theknowndampingwithintherandomlygeneratedFRF;and

- the damping value computed when applying the adapted damping

estimationtechniquetotherandomlygeneratedFRF.

Figure3-9 provides an illustration of this comparison between the estimated

dampingandthedampingimposedontheFRF.


75

Figure3-9 Estimateddampingincomparisontoapplieddamping –0Hzto400Hz

Forthepurposesofthisinvestigation,theerrorestimateoftheadapteddamping

estimation techniquewas conducted over a range from 0Hz to 400 Hz. This

encompassestherangeconsidereddominant inatransformer’snoisespectrum.

Furthermore,with the lengthof the randomFRFexceeding thatof the analysis

range, the effects of resonance peaks partially within thewithin the analysis

spectrumwereaccounted for.Theoccurrenceof thesepartialpeaks is realistic

and likely has the effect of increasing the error associated with the adapted

dampingestimationtechnique.


76

Asabasecaseforstudyingtheadapteddampingestimationtechniquethe0Hzto

400Hzanalysisspectrumof therandomlygeneratedFRFsignalwasdelineated

into five constant-size frequency bands.Withineach frequencyband the initial

500samplesofthetime-domainsignalhavethenbeenconsideredinestimating

themodaldamping factor.Toprovideanerrordistributionof themethodology

the random FRF generation and the subsequent application of the adapted

dampingestimationtechniquewasiterated100times.Figure3-10illustratesthe

resultingerrordistribution.

Figure3-10 Error distribution of the applied dampingmethodologywith the0–400Hzanalysisspectrumdelineatedintofivebands (basecase)

As seen inFigure3-10, considering the base caseparametervalues, the curve

fitting methodology provides a good estimate of the random FRF’s damping

characteristics.Specifically,theaverageerrorassociatedwiththe100iterations

ofthedampingestimationpresentedinFigure3-10wasdeterminedtobe1.64%.


77

Inadditiontothebasecase,theaverageerroroftheadapteddampingestimation

technique with differing bandwidth delineations of the FRF signal has been

considered. Specifically, the average error associatedwith the techniquewhen

delineatingtheFRFsignalintofour,fiveandsixfrequencybandsoverthe0Hzto

400Hz analysis spectrumhas been analysed.The results from these analyses,

with each analysis considering 100 iterations to provide an average error, are

presentedinTable3-4.

Table3-4 Effectoffrequencybandsontheaccuracyofdampingestimates

NumberofFrequencyBands AverageErrora

4 1.23%

5(basecase) 1.64%

6 0.20%

Note: aDuetothesignalgenerationbeingarandomprocesstheresultspresented.arenotreproducible;however,theyprovideanunderstandingofkey.trends

FromtheresultspresentedinTable3-4itisapparentthatincreasingthenumber

of bands intowhich the analysis spectrum is divided significantly reduces the

averageerror.This is likelyaresultof lessspectralnoisebeing introduced into

the impulse response functions when taking the IFFTs within each band. By

delineating the analysis spectrum into smaller frequency bands the adapted

dampingestimationtechniquealsomorecloselyreplicates conventionalcurve-

fittingmethods,whichdeterminedampingestimatesatisolatedresonancepeaks.


78

Atthispoint it isworthnotingthatthenumericalstudyoftheapplieddamping

estimationtechniquehasidentifiedthattakingtheIFFTofthefrequencydomain

FRF signal likely introducesan error factor.This sourceof errorhas alsobeen

noted in previous studies. The error arises from the ‘spectral leakage’ of the

transformedsignalfromoneperiodintothefollowingperiod.Thisiscausedby:

- TheIFFTnotrecreatingtheimpulseresponseoftheoriginaltimesignal

butratherarepeatedpulsetrainresponse;and

- Thenon-casualspreadingofIFFTsignal–withthepulsearrivingbefore

theimpulseandcontinuingafterit.

Ofsignificanceinthisstudyisthatthespectralleakageappearstobeexaggerated

when taking an IFFT across a defined frequency band of a FRF as opposed to

across thecompleteFRFsignal.Thisaforementionedphenomenon is illustrated

inFigure3-11.


79

(a)

(b)

Figure3-11 Spectral leakagewhen taking the IFFTof (a) acompleteFRFsignaland(b)adefinedafrequencybandofthecompleteFRFsignal


80

Togainaninsightintothepracticalapplicationofthenoveldampingestimation

technique detailed in this chapter, the error associated with the estimated

damping factors should be observed within the context of a system’s

transmissibility. This parameter relates the amplitude of a system’s vibration

responsetotheamplitudeoftheexcitationvibrationapplied.Thetransmissibility

factor,beingdependentonthedampingratio,thefrequencyofvibrationandthe

naturalfrequenciesofthesystem,ismathematicallynotatedas:

= =1 + 2

1− + 2 (3.14)

Where: isthetransmissibilityfactor

istheamplitudeofthevibrationresponse

istheamplitudeofthevibrationinput

isthefrequencyoftheexcitationvibration

isthenaturalfrequencyofthesystem

FromEqn.3.14itisobservedthatwhentheexcitationfrequencyisequaltothat

ofasystem’snaturalfrequencythetransmissibilityfactorisatamaximum.Itcan

also be shown that, for small values of , when the excitation and natural

frequenciesofasystemcoincidethetransmissibilitycanbeapproximatedas[74]:

( = ) =

1 + 42 ≈

12 (3.15)


81

Toillustratehowthedampingfactoraffectsthetransmissibilityofasystemthree

modaldampingratioshavebeenselected,being4.95%,5.50%and6.05%.An

illustrationof a system’s transmissibilitywith thesedamping factors applied is

showninFigure3-12.The10%differencebetweenthemodaldampingratiosis

representativeofthemaximumerrorconsideredtobepresentinthenovelcurve-

fittingmethodology(refertoFigure3-10foraplotoftheerrordistribution).

Figure3-12 Transmissibilityfunctionswithmodaldampingat4.95%,5.50%and6.05%

Eqn.3.15andFigure3-12illustratethatwhenaforcingfrequencyisinproximity

toasystem’snaturalfrequencythestructure’sdampingpropertiesinfluencethe

transmission of vibrations through the system. However, a slight difference

betweentheexcitationfrequencyandthesystem’snaturalfrequencysignificantly

reducesthedependenceofasystem’stransmissibilityonthedampingproperties.


82

Consideringthischaracteristicofthetransmissibilityfunction,togetherwiththe

absence of conventional curve fitting methodologies and largely accurate

damping estimates provided by the novel methodology detailed, it may be

concluded that the adapted damping estimation technique is appropriate to

provideanapproximationofthetransformerassembly’smodaldampingoverthe

frequencyrangeofinterest.

· ApplicationoftheDampingMethodology

Anillustrationoftheadapteddampingestimationtechniqueasappliedtoasingle

FRFsignalgeneratedduringtheexperimentalmodalanalysisofthetransformer

analysedwith this study is provided in Figure 3-13. The figure illustrates the

linear best-fit trend line,which allows the calculationof the damping factor

fromtheexpressiongiveninEqn.3.11.


83

Figure3-13 Transformerassemblydampingestimatesin1/3octavebands


84

3.6.3 MODALDAMPINGINTHEFINITEELEMENTMODEL

Themodal damping characteristics of the transformermodelled in this thesis have

beenaccountedforintheFEmodelemployingtheRayleighdampingformation(refer

toEqn.3.3 foramathematicaldescriptionofRayleighdamping).TheRayleighmass

and stiffness proportional damping coefficients, α and β, have conventionally been

evaluated from damping ratios and natural frequencies corresponding to two

referencemodesofvibrationas:

=12

⎣⎢⎢⎢⎡

1

1⎦⎥⎥⎥⎤αβ (3.16)

Where: , denotethetworeferencemodesofvibration

It is well known that calculating the Rayleigh damping coefficients using the

expressioninEqn.3.16resultsinerroneousmodaldampingratiosexceptforthetwo

reference modes of vibration considered. This error is a significant source of

inaccuracy in estimation of system’s structural response [75]. Therefore, in this

investigation a Moore-Penrose pseudoinverse matrix has employed, allowing the

Rayleighdamping coefficients tobe fitted toanumberof referencemodesover the

frequencyspectrum.


85

Theuseofthepseudoinversematrixallowsforthecomputationofabest-fitsolution,

in a least-squares sense, to a systemof linearequations that lackaunique solution.

Therefore,theuseoftheinversematrixhasallowedforbetterestimateoftheRayleigh

damping coefficients over the frequency spectrum under consideration. The

estimation of Rayleigh damping coefficients employing the Moore-Penrose

pseudoinversematrixisnotatedas:

αβ = 2

⎣⎢⎢⎢⎡

1

⋮ ⋮1

⎦⎥⎥⎥⎤

⋮ (3.17)

Where: [] denotesaMoore-Penrosepseudoinversematrix

UtilizingEqn.3.17theRayleighdampingcoefficientsweredeterminedindependently

fortheoil-filledtankassemblyandtheactivepart.Themodaldampingratios … ,

to which the Rayleigh damping factors have been fitted, were determined from

experimentalmodalanalysesresultsasdiscussedinChapter3.6.1andChapter3.6.2.

The computedRayleighdamping coefficientsof the active partand the transformer

tankwereappliedtotherespectivestructuralassembliesforwhich and hadbeen

determined.Ofnoteisthatthedampingcharacteristicsofthefluidwerenotaccounted

forbymanualinputof and coefficients.InsteadANSYSaccountedforthedamping

characteristics of the fluid employing the acoustic properties of the fluid body, as

notatedinEqn.2.15.


86

Differences between the damping matrix formulation in FSI simulations and the

experimental setup employed to determine the damping characteristics of the

transformermayhave leadtoslightdifferencesbetweenthedampingcharacteristics

determined experimentally and those exhibited by the numerical model. However,

thesedifferenceshavebeenconsideredofnegligibleinfluence.

Chapter4:ValidationoftheNumericalModel

87

CHAPTER4

VALIDATIONOFTHE

NUMERICALMODEL


88

As discussed in Chapter 2, the accuracy of finite element models is limited by

uncertaintiesrelatingtogeometricandmaterialproperties,boundaryconditionsand

loading applied to themodel. The validation of finite elementmodels through the

comparisonofexperimentalandnumericalresultsisthereforenecessary.Withinthis

thesis experimental modal analysis results have allowed for the validation of the

structural finite element model while measured sound pressure levels have been

employedtovalidatetheacousticnoisepredictions.Thefollowingsubchaptersdiscuss

thestructuralandacousticvalidationofthevibro-acousticmodelinfurtherdetail.

4.1 MODALVALIDATIONOFTHEASSEMBLY

Thestructuraldynamicscharacteristicsof thenumerical transformermodeldetailed

in Chapter 3 have been validated by comparing numerically computed modal

parameterstothosedeterminedexperimentally.Theexperimentalmodalparameters

employed in this validation procedure were extracted from impact hammer tests

conductedwithtri-axialaccelerometerspositionedat197pointsonthetransformer.

Therefore, the experimentalmodal model consisted of 591 degrees of freedom. In

comparison, the numerical model comprises in excess of 9.5 million degrees of

freedom.

The significant difference in thenumberof degrees of freedom between themodal

modelshasresultedinthenumericalmodelcomputingaconsiderablygreaternumber

of modes than were determined experimentally. Validating the numerical model

throughmethodssuchasthemodalassurancecriterion(MAC)isthereforeimpractical.

Inparticular,MACmaymistakenlyindicateunitywhentheexperimentalmodalvector

is incompletelymeasured (i.e. too few response stationshave been included in the

experimentaldeterminationofthemodalvector).Insuchcasestheincompletenessof

theexperimentalmodalvectorresultsinthecomparisonoftwouncorrelatedmodes.


89

Inthisthesisthestructuralvalidationofthefiniteelementmodelhasbeenconducted

bycomparingexperimentalandnumericalnatural frequenciesatmodesshapesthat

exhibitvisualcorrelation.Thissimplisticcomparisonmethodhasbeenadopteddueto

limitationsassociatedwiththeincompleteexperimentalmodalmodel,asdiscussedin

the preceding paragraph. Also of note is that because experimental modal

measurementsweretakenontheexternalsurfacesofthetank,theshapeofthetankat

naturalfrequencypointshasthereforeformedthebasisofvisualcomparisonsmadein

thisthesis.

Duetothecomplexityofmodeshapesoccurringathigherfrequenciesnumericaland

experimentalmodeshavebeencomparedatthelowerendofthefrequencyspectrum

duringthisvalidationprocedure.Thecomparisonofmodesinlowerfrequencyranges,

wheremodesarecharacterisedbygreaterglobaldeformation,hasalsodecreasedthe

probability of comparing uncorrelatedmode shapes.Thenatural frequenciesof the

firstsixcorrelatedexperimentalandnumericalmodesarepresentedinTable4-1.

Table4-1 Naturalfrequenciesofthetransformertank

ModeNo. ExperimentalAnalysis NumericalAnalysis.

1 11Hz 10.24Hz

2 17Hz 20.03Hz

3 25Hz 25.87Hz

4 29Hz 32.33Hz

5 39Hz 39.31Hz

6 46Hz 42.02Hz


90

Theaverageddifferencebetweennatural frequenciesof the six correlatedmodesas

identifiedinTable4-1is2.9%.Notingthatnomodelupdatinghasbeenconducted–as

thecomputationschemepresented inthisthesisisintendedtobeused inthedesign

phaseofatransformer–thisrepresentsaveryhighdegreeofcorrelation.Figure4-1

illustratesthedistributionofthenumericalversesexperimentalnaturalfrequencies.

Figure4-1 Numericalversesexperimentalnaturalfrequenciesofthecorrelated modeswithtrendlineindicatingperfectcorrelation

Thecorrelatednumericalandexperimentalmodeshapesofthetransformerat~11Hz,

~25 Hz and ~39 Hz are illustrated in Figure 4-2, Figure 4-3 and Figure 4-4,

respectively. For illustrative purposes, only the deformation of the tank has been

presented; however, for both the numerical and experimental modal analyses the

transformerwasfullyassembled,aspresentedinFigure3-3(b)andFigure3-4.


91

Inaddition tomodalparameter correlation, the comparisonofFRFsmayhavebeen

utilised to validate the structural dynamics characteristics of the numerical model.

However, thiscorrelationmethodhasbeendismissedas aresultoftwo factors.The

primaryreason thatFRFshavenotbeencompared is thecomplexityoftransformer

assembly, with a high modal density and heavily coupled modes, making the

correlation of FRFs a difficult task. Furthermore, the computing requirements to

calculate a numerical FRF with a sampling frequency matching that of the

experimentalFRFwouldbeexcessive.Thecomputingrequirementsofthenumerical

analysesemployedwithinthisthesisarefurtherdiscussedinChapter5.4.


92

(a)

(b)

Figure4-2 Comparisonof(a)theexperimentalmodeshapeat11Hzand (b)theexperimentalmodeshapeofthetransformerat10.24Hz


93

(a)

(b)



94

(a)

(b)



95

4.2 ACOUSTICVALIDATIONOFTHENUMERICALMODEL

Theacousticvalidationofthenumericalsimulationhasactedtovalidateallpreceding

steps in the modelling of the transformer. The validation has been conducted by

directlycomparingexperimentallyandnumericallydeterminedsoundpressurelevels

over aspectrum from100Hzto400Hz.Thisvalidationapproach isopposed those

employed in comparable preceding studies, which validated acoustic results at a

singlefrequencypointofinterest.

Validationofthenumericalmodelfrom100Hzto400Hz issignificant,ashasbeen

discussedindetailinChapter3.6.Thisisbecausethespectralrangeisinclusiveofthe

networkfrequency’sfirstthreeevenharmonics,whichareconsidereddominantina

LPTsacousticresponse.Furthermore,validationovertheaforementionedfrequency

spectrumwillallow for futureworks toestimate the totalaudiblenoise levelsof a

transformer under nominal operating conditions employing the computational

schemepresentedinthisthesis.

4.2.1 SOUNDPRESSURELEVELMEASUREMENTS

The sound pressure levelmeasurementsof the transformer analysed in this thesis

wereprovidedbySiemensAGAustria,TransformersWeiz.Themeasurements,which

have been carried out in the Transformers Weiz test laboratory, were taken in

accordancewith:

IEC60076-10PowerTransformers–Part10:DeterminationofSoundLevels

Inagreementwith IEC60076-10[76]soundmeasurementsweretakenatone-third

andtwo-third-heightsofthetransformer.Furthermore,asnocoolingequipmentwas

inoperationduring themeasurements, soundpressure levelsweredeterminedat a

distanceof0.3metresfromtheradiatingsurfaces.


96

Withrespecttothesoundpressurelevelmeasurementsofthestudiedtransformer it

should be noted that a narrow-band measurement method has been employed.

Specifically, the fast Fourier transformation (FFT) frequency analysis method, as

described inIEC60067-10,hasbeenused.Thismethodprovidesconstantbandwidth

acoustic measurements across a defined spectrum. When performing the acoustic

measurementsofthetransformerstudiedwithinthisthesisabandwidthresolutionof

2Hzwasemployed.

The FFT narrow-band method uses an FFT algorithm to transform the measured

digital sound pressureversus time spectrum to a soundpressureversus frequency

spectrum.Thebandwidthofthespectrumthereforedependsonthesamplingrateof

the analogue-to-digital converter. As noted in IEC60067-10, when employing a

narrow-bandmeasurementmethod the summation of sound levels for the first 10

bandscontainingfrequenciesequaltotwicetheratedfrequencyandmultiplesthereof

isadequatetoapproximatetotalsoundlevelsformosttransformers.Thesummation

ofsoundpressurelevelscanbecalculatedasfollows:

= 10 10 , (4.1)

Where: isthesoundpressurelevelatratedvoltageandrated frequency

isthesoundpressurelevel(orsoundintensitylevel) measuredoverthechosenbandwidth, ,centredona frequencyequalto2 ,atratedvoltageandratedfrequency

istheratedfrequency

isthesequencenumber(1,2,3,etc.)ofmultiplesofeven harmonicsoftheratedfrequency;

= 10


97

Duringtheacousticmeasurementsconductedonthetransformerstudiedinthisthesis,

theunitwasexcitedunderloadconditionsataseriesofdiscreteelectricalfrequencies

rangingfrom50Hzto200Hz.Theexcitationoftheunitresultedinmechanicalforces

actingontheactivepartattwicetheappliedelectricalfrequencyandhigherharmonics

thereof.Adetailedreviewoftheexcitationforcesinatransformer’sactivepartcanbe

foundinChapter3.6.

For comparison of the experimental sound pressure levels and the numerical

predictions, the experimental sound levelswere evaluated at a single point on the

acoustic frequency response spectrum. Specifically, the sound pressure levelswere

evaluatedatthesecondharmonicoftheelectricalexcitation frequency,whichisalso

the fundamental frequency of theMaxwell, Lorentz andmagnetostrictive excitation

forces that act on the core and the windings. The electrical excitation frequencies,

fundamental frequency of the excitation forces, the frequency at which the sound

pressure levels were determined and the measured sound pressure levels are

presentedinTable4-2.


98

Table4-2 Electricalexcitation frequencyandcorrespondingsoundpressure level measurementpointsandvalues

MeasurementPoint

ElectricalExcitationFrequency

(Hz)

FundamentalFrequencyoftheExcitationForces

(Hz)

SoundPressureLevelEvaluation

Frequency

(Hz)

SoundPressure

Level

(dB)

1 50 100 100 42.44

2 60 120 120 44.74

3 70 140 140 48.18

4 80 160 160 51.68

5 90 180 180 41.38

6 100 200 200 40.06

7 110 220 220 38.11

8 120 240 240 38.87

9 130 260 260 39.88

10 140 280 280 36.86

11 150 300 300 37.07

12 160 320 320 37.79

13 170 340 340 37.37

14 180 360 360 37.33

15 190 380 380 46.11

16 200 400 400 50.48

WithregardtotheacousticmeasurementspresentedinTable4-2,itshouldbenoted

that although the unit was excited under load conditions, the results are not

representative of load noise acoustic measurements. This is because the acoustic

measurements have been evaluated at a single point on the acoustic frequency

responsespectrumi.e.thesecondharmonicoftheelectricalexcitationfrequency.


99

Theindustrydefinitionsofloadandno-loadnoisegenerallyrelatetototalnoiselevels

instead of noise levels at an individual frequency component. Sound emissions at

higher harmonics of the excitation frequency are therefore considered in the

calculationof loadandno-loadnoise. It isalso importanttonotethat,aspreviously

discussed in Chapter 2.1 and Chapter 3.5, load and no-load noise levels may be

dominatedbydifferentharmonicsoftheexcitationfrequency.

As sound levels at only a single harmonic of the excitation frequency have been

evaluatedinthisthesisithasbeenconsideredinappropriatetocontrastresultsunder

loadandno-loadconditions.Suchresultswouldhavenocontextwithinthe industry

interpretation of load and no-load transformer noise. The viewpoint taken to not

contrast loadandno-loadnoisemeasurementshasalsobeencarried forwardto the

followingsectionsofthisthesis,whichpresentnumericallypredictednoiselevels.

4.2.2 SOUNDPRESSURELEVELPREDICTIONS

Thesoundpressure levelpredictionsofthetransformeranalysedinthisthesiswere

determined employing the computation scheme detailed in Chapter 3. For the

validation of the numerical model simulations at 16 discrete frequencies across a

100Hz to 400 Hz spectrum were conducted. These simulations allowed for the

vibrationandacousticresponseofthenumericalmodeltobedetermined.Theforcing

frequenciesofthese16simulationsmatchedthesecondharmonicofthe16electrical

excitation frequencies applied to the transformer during experimental testing.

Therefore,notingthattheexperimentalresultswereevaluatedatthesecondharmonic

of theelectrical excitation frequency,an acoustic frequencyresponsespectrumwith

16correspondingexperimentalandnumericalmeasurementpointshasbeencreated.


100

Toensuresimilaritybetweenexperimentalandsimulatedresultsthenumericalsound

pressurelevelswerecalculatedonacousticsurfacessurroundingthetransformerthat

approximatedtheexperimentalmeasurementarea.Itisalsoofnotethatvalidationof

thenumericalmodel though anumerical frequencypoint toexperimental frequency

point comparison allowed the non-linearity of the excitation forces and structural

elements to be disregarded. A comparison of experimental and predicted sound

pressurelevelsisillustratedinFigure4-5.

Figure4-5 Comparisonofmeasuredandpredictedsoundpressurelevelsat16correspondingmeasurementpoints

From Figure 4-5 it appears that there is a high degree of correlation between the

numerical and experimental sound pressure levels. At double the 50 Hz (60 Hz)

network frequency, being the fundamental frequency of the Maxwell, Lorentz and

magnetostrictive forces in the active part, the error between experimental and

numericalsoundpressure levels is1.91dB (3.60dB).Theaverageerrorat the first

threeevenharmonicsofthenetworkfrequency,whicharethefrequenciesconsidered

dominantintheacousticresponsespectrumofaLPT,is1.96dB(5.72dB).


101

Thenumericalandexperimentalacousticresultsofthestudiedtransformerremainin

favourableagreementacrossallmeasurementpointswithinthe100Hzto300Hz(100

Hzto360Hz),withanaverageerrorof2.48dB(4.86dB)overthisspectrum.However,

abovethisfrequencyrangeofinteresttheerrorbetweennumericalandexperimental

results increases. The total error across the 100 Hz to 400Hz analysis spectrum

consideredinthisthesisis5.67dB.

The sound pressure level results presented in Figure 4-5 are comparable to those

presented in preceding studies. Specifically, the results are comparable to those

presented in Ref. [3], which contained a 1.5 dB difference between simulated and

experimental values (at the single 100 Hz validation point considered) and those

presented in Ref. [4], which included a 6.61 dB error (again at a single frequency

point). The numerical sound pressure levels presented in this thesis are therefore

considered accurate, as the aforementioned preceding studies have predicted only

coil-emittednoiseatasinglefrequencypoint.

FurtherexaminationofFigure4-5illustratesthatthepredictedsoundpressurelevels

exhibitthelocalandglobaltrendsofthetransformer’sacousticbehaviour.Theglobal

behaviourischaracterisedwiththesoundpressurelevelsincreasingbetween100Hz

and160Hzandagainbetween360Hzand400Hz.Additionally, thepeaks insound

pressure,whichmay be attributed to natural frequencies of the transformer being

excited,illustratethefrequency-localisedcharacteristicsoftheunit.

ThefinalpointofnoteregardingFigure4-5isthatthenumericalresultsshowhigher

correlation tomeasuredsoundpressure levelsat the lower frequencypointswithin

the100Hzto400Hzanalysisspectrum.Thisisofsignificanceaslowerfrequenciesin

the spectrum consideredaregenerallymoredominant in theacoustic responseof a

largepowertransformer.RefertoFigure3-7foranillustrationofthischaracteristic.


102

As thenumericalvibro-acousticmethodologyproposed in thispaper is intended for

use in the designprocess, computing time is a significant consideration.As such, a

simplified acoustic prediction simulation was run, with the transformer assembly

comprising of the active part, the tank and the insulation fluidwhile excluding the

conservator, radiators, bushings and other auxiliary equipment. The results of this

simulation,with a comparison to the results from the fully assembled transformer

model,arepresentedinFigure4-6.

Figure4-6 Comparisonofpredictedsoundpressurelevelscalculatedwiththesimplifiednumericaltransformerassemblyandthecompletenumericaltransformermodel

TheresultsinFigure4-6illustratethatthesimplifiednumericalmodelprovidessound

pressure level estimates in favourable agreement with those from the numerical

analysisof thecompletetransformerassembly.Furthercomparisonofthesimplified

and complete numerical models is provided in Chapter 5.1. A discussion of the

computing requirements for the simplified and complete simulations is given in

Chapter5.4.


103

4.3 CONCLUSIONS

Thischapterhasdetailedthevalidationofthenumericaltransformermodelpresented

inChapter3.Experimentalmodal analysis resultshaveallowed for thevalidationof

the structural finite element model by direct comparison of numerical and

experimental mode shapes and corresponding natural frequencies. The numerical

results have shown favourable agreement with experimental measurements, the

averagedifferencebetweenthemodalfrequenciesofthefirstsixcorrelatednumerical

andexperimentalmodesbeing2.9%.

Followingthevalidationofthestructuralmodel,measuredsoundpressurelevelswere

employed tovalidate theacousticnoisepredictions.Specifically,16 frequencypoints

acrossa300Hzspectrum from100Hzto400Hzhavebeenanalysedemployingthe

numericalmodel, corresponding to the16 frequenciesatwhich experimentalsound

pressurelevelswereevaluated.Validationofthenumericalmodelthoughanumerical

frequencypoint to experimental frequency point comparison has allowed the non-

linearityoftheexcitationforcesandstructuralelementstobedisregarded.

Comparisonofthenumericalandexperimentalsoundpressure levelresultsoverthe

analysis spectrum showed a high degree of correlation. At the first three even

harmonicsof the50Hz (60Hz)network frequency,being theharmonicswhichare

considered dominant in the sound radiationof a transformer, comparison between

numerical andexperimentalmeasurements resulted in an average errorof1.96dB

(5.72dB).Theaccuracyoftheseresults is in linewithcomparableprecedingstudies.

Furthermore, comparison of the acoustic prediction andmeasured sound pressure

levelshasillustratedthattheappliedmethodologyidentifiesboththeglobalandlocal

acousticbehaviourofthetransformerovertheanalysisspectrum.


104

The error margins relating to the respective numerical models presented in this

chapter have beenprovided in Table 4-3. The errorvalues stated inTable 4-3 are

average error values accounting for all corresponding experimental and numerical

datapointsconsideredinthemodalandacousticmodels.

Table4-3 ModelValidationMethodsandCorrespondingErrorMargins

ModelValidationMethod ModelError

ModalValidation 2.9%

AcousticValidation 5.67dB

AcousticValidation(simplifiedtransformer)

4.95dB

Chapter5:Discussion

105

CHAPTER5

DISCUSSION

Chapter5:Discussion

106

Throughthecomparisonofexperimentalandnumericalresultsithasbeenshownthat

the computation scheme described in Chapter 3 is able to predict a transformer’s

acoustic emissions with a level of accuracy in line with comparable preceding

investigations. Additionally, the inclusion of all key structural and fluid elements

togetherwithvalidationovera100Hzto400Hzspectrumhasallowed forthe first

timeadetailedinsightintothevibrationandacousticcharacteristicsofatransformer.

This chapter identifies these vibration and acoustic characteristics and provides

methodsforidentifyingthemduringthedesignprocess.Thecomputingrequirements

forthenumericalsimulationsemployedwithinthisthesisarealsodiscussed.

5.1 INCREASEDFREQUENCYSAMPLINGTOPROVIDE

DETAILEDACOUSTICCHARACTERISTICS

Following the acoustic validation of the finite elementmodel, soundpressure level

predictionsweremade at frequency pointswhere physical measurements had not

been not been taken. This has allowed for a more detailed image of the studied

transformer’sacousticbehaviourtobeconstructed.Specifically,soundpressure level

predictionshavebeenmadeat10Hz intervalsacrossthe100Hzto400Hzanalysis

spectrum.Thisisopposedtothe20Hzintervalsatwhichacousticmeasurementswere

performedtovalidatethenumericalmodel,aspresentedinChapter4.2.1.Therefined

acousticspectrumispresentedinFigure5-1.

Chapter5:Discussion

107

Figure5-1 Comparisonofthetransformer’sacousticbehaviourasidentifiedbysoundpressurelevelsdeterminedat10Hzand20Hzintervals

Figure 5-1 illustrates that increasing the number of frequency sampling points

provides ahigherresolution frequency response spectrum.This isof importance as

predicting the distance of resonance peaks from 100 Hz (120 Hz) and higher

harmonicsthereofiscriticalinunderstandingtheriskofhighnoiseemissionsfroma

transformer. The aforementioned frequencies and their higher harmonics are of

significanceastheyarethefrequenciesatwhichtheexcitationforcesintheactivepart

occur.

Whendeterminingifanumericallypredictedresonancepeakislikelytoproducehigh

noise emissions from amanufactured transformer a confidence interval should be

accounted for.Thisconfidence intervalallows fordifferencesbetween thenumerical

and experimental resonance frequencies, noting that the numerical model is an

idealizedrepresentationofthephysicalunit.Inparticular,thenumericalmodeldoes

not account for variables including manufacturing tolerances, assembly process

variancesandnon-linearmaterialproperties.

Chapter5:Discussion

108

ThesignificanceofahighresolutionacousticspectrumisclearlydepictedinFigure5-

1.Asshown,therefinedacousticspectrumidentifiesaresonancepeakat150Hz,made

visible through elevated sound pressure levels (being approximately 10 dB above

othersoundlevelpredictionsmadeinproximity).Thiselevatednoiselevelestimation

isinfavourableagreementwiththeelevatednoiseleveldeterminedexperimentallyat

160Hz.Thedifferencebetweentheexperimentalandnumericalresonancepeakmay

beaccountedforbythevariablesidentifiedintheprecedingparagraph.

In addition a more providing a more detailed acoustic frequency response of the

studied transformer, the increased sampling of sound pressure levels has been

employedtofurthervalidatethesimplifiedmodelpresentedinChapter4.3.Figure5-2

providesacomparisonofsoundpressurelevelscalculatedat10Hzintervalswiththe

simplified and the complete numerical models. The simplified model included the

activepart, the tankand the insulation fluidbutexcluded theconservator,radiators,

bushingsandotherauxiliaryequipment.

Figure5-2 Comparisonofpredictedsoundpressurelevelsat10Hzintervalscalculatedwiththesimplifiednumericaltransformerassemblyandthecompletenumericaltransformermodel

Chapter5:Discussion

109

Thecalculatedsoundpressure levelspresentedinFigure5-2again illustratethatthe

simplifiednumericalmodelprovidesacousticestimatesinfavourableagreementwith

thoseof thecomplete transformerassembly.From these results it is concluded that

simplified numericalmodels can be employedwithin the design process to reduce

modelling, discretisation and simulation time. A discussion of the computing

requirementsforthecompleteandsimplifiedsimulationsispresentedinChapter5.4.

5.2 EXCITATIONMECHANISMSANDACTIVEPARTVIBRATION

CHARACTERISTICS

Past literature[7,28,44,49-51]has identified thatthenoise levelsof atransformer

are significantly influencedby thevibration characteristicsof theactivepart and in

particular the core. This connection between vibration characteristics and sound

generation is a consequence of the active part’s modes being excited by

electromagnetic and magnetostrictive excitation forces. The excitation of the

assembly’smodes results inanamplificationofvibrations, leading toelevatednoise

levels. Despite being identified as a source of excessive noise generation in past

literature,thisthesispresentsforthefirsttimearigorousvalidationofthiscorrelation.

The likelihood of an active part’s mode being excited by electromagnetic and

magnetostrictive forces is analyzed in this chapter through determination of the

assembly’smodalmassparticipationfactors.Themodalmassparticipationfactorisa

measureof themassdisplacedby an individualmodeshape; therefore,modeswith

high modal mass participation factors are expected to involve large areas of the

transformerandarelikelytobeexcitedbyinternalforces.Amathematicaldescription

ofthemodalparticipationfactor,aspresentedinRef.[77],isgivenbelow.

Chapter5:Discussion

110

Notingthatasystem’sgeneralisedmassmatrix[m ]isgivenby,

[m ] = [ ] [M][ ] (5.1)

Where: [ ] istheeigenvectormatrix

[M] isthemassmatrix

and a matrix describing the displacement of the masses resulting from a unit

displacementspectrumineachoftheCartesiandirectionsmaybedefinedas,

[L] = [ ] [M]{s} (5.2)

Where: {s} isadisplacementtransformationvectorthatexpressesthe displacementofeachstructuralDOFduetostaticapplication ofaunitmovementineachoftheCartesiandirections

Themodalparticipationfactorforthe modemaythenbedefinedas:

Γ =

[L][m ] (5.3)

Where: Γ isthemodalparticipationfactoroftherthmode

AsdiscussedinChapter2.4,iftheeigenvectormatrix[ ]ismass-normalised,withthe

resultingmatrixdenotedas[Φ],then[m ] = [I].

With regard toEqn.5.3 it isof note that thatmodalmassparticipation factors are

analysed within a translational or rotational direction of motion. In ANSYS®

Mechanical these factors are calculated in the global x-, y-, and z-axes. Due to the

movement of magnetic flux through the core, electromagnetic forces acting in the

verticalplanewere consideredmost likely to excite the two-limbcoredesignof the

transformeranalysedinthisinvestigation.

Chapter5:Discussion

111

Thenumericalmodelemployedtodeterminethemodalmassparticipation factorsof

the activepart in this thesishasnotdirectlyaccount for fluid-structure interaction.

Thefluid-structurecouplinghasbeenomittedduetotheexcessivecomputingpowerit

required.A20% reduction in the calculatednatural frequencieshas thereforebeen

introduced. This factor accounts for the added-mass and viscous damping effects

associatedwiththesubmersionoftheactivepartinmineraloil.

The factor used to relate the numerically calculated natural frequencies to the

resonances of the submerged active part has been determined from experimental

measurements. The measurements, which were provided by Siemens AG Austria,

TransformersWeiz,contrastthevibrationresponseofa transformercore inairand

submerged inmineral oil. The factor employed in this thesis is also in favourable

agreementwithanalyticalapproximations,determinedfromtheexpressionsfoundin

Ref. [78] and Ref. [79]. Figure5-3presents the updated natural frequenciesof the

model and the correspondingmodal mass participation factors, normalised to the

largestwithintheset.

Chapter5:Discussion

112

Figure5-3 Modalmassparticipationfactorofactivepartmodesintheverticalplane(○indicatesthenaturalfrequencyandmassparticipationfactorofanumericallycalculatedmode)

From Figure 5-3 it may be predicted that the sound radiation levels significantly

increase when the transformer is excited at 160 Hz. This agrees favourably with

measured sound pressure levels, which have also been illustrated in Figure 5-3.

Furthermore, contrasting the measured sound pressure levels and the mass

participationfactor’softheactivepartillustratesthatthenumericalparameterisable

toprovideaqualitativeacousticestimateovertheentirefrequencyspectrum.

5.3 RESONANCEEFFECTSANDEXCESSIVESOUNDRADIATION

Inadditiontomodalexcitationoftheactivepart,resonancesofthetankmayalsolead

to elevated sound levels. However, due to the complexity and size of large power

transformers, predicting such resonances is challenging.This is particularly true at

higher frequencies,with the occurrenceof closely groupedmodes characterised by

multiple local deformations on the transformer tank. Suchmodesdonot generally

contributesignificantlytotheoverallnoiselevelofatransformer.

Chapter5:Discussion

113

Togain insight into thepresenceofglobalmodesshapeson the transformer tankat

higherfrequencyrangesaseriesofforcedexcitationsimulationshavebeenconducted

employing the numerical model detailed in Chapter 3. These forced excitation

simulations followed an identical procedure to that used to predict the acoustic

characteristics of the transformer. The forced excitation simulations conducted in

proximity to 400 Hz have identified a mode shape characterised by the global

deformationofthetankcover.

Inaddition to theglobalmodeshapeof the tankoccurringatapproximately400Hz,

the frequency spectrum illustrated inFigure5-3 indicates that the activeparthas a

high modal density between 360 Hz and 400Hz. A number of active part modes

(representedbycirclesinFigure5-3)maythereforebeexcitedbyelectromagneticand

magnetostrictive forces within this frequency range. The abovementioned

observationsindicatethatthehighnoiseemissionsofthetransformerstudiedinthis

thesisbetween350to400Hzlikelyresultedfromthecoincidenceof:

i) A mode shape of the active part being excited by electromagnetic and

magnetostrictiveforces;and

ii) Aglobalmodeshapeofthetankbeingexcitedbyvibrationspropagatingfrom

theactivepart.

The shape of the tank at approximately 400 Hz, characterised by the global

deformationofthetankcover,isillustratedinFigure5-4.

Chapter5:Discussion

114

Figure5-4 Modeshapeexhibitingglobaldeformationofthetankcoveratapproximately400Hz

5.4 COMPUTINGREQUIREMENTS

Noting that thecomputation schemepresented in this thesishas focusedon itsuse

within the design phase of a transformer, computing requirements and simulation

timearesignificant factors.Fromadesignperspective,thenumericalprediction tool

shouldnotrequireexcessivesimulationtime,allowingforthemodeltobe iteratively

updatedandanoptimisedsolutionachieved.Thecomputingspecificationsusedforthe

variousnumericalsimulationswithinthisinvestigationwereasfollows:

· Processors: [email protected]

· AvailableMemory: 256GB

· DriveType: SolidState

With respect to the drive type, for technical computing loads solid states drives

provideupto7.3timesincreasedperformancewhencomparedtoHDDdrives[80].

Chapter5:Discussion

115

Modal and harmonic response analyses are the two dynamic structural response

simulations thathave been conductedwithin this thesis.The computation time for

eachfortheseanalysesmaybenormalisedasfollows:

· Modal analysis – minutes per calculation of a single mode within a defined

frequencyspectrum;and

· Harmonic response analysis – minutes per simulation of the acoustic sound

pressurelevelatasinglefrequencypoint.

Basedonthesenormalisation factors,Table5-1presentstheapproximatecomputing

timeforthedynamicstructuralresponsesimulationsperformed.

Table5-1 Computingspecifications

Analysis ReferenceApproximate

computingtime

Modalanalysis–completetransformerassembly Chapter4.2 300minutespermode

Modalanalysis–activepartwithfluid-structureinteraction Chapter5.1

Greaterthan100minutespermodea

Modalanalysis–activepartwithoutfluid-structureinteraction Chapter5.1 0.37minutespermode

Harmonicresponseanalysis–completetransformerassembly Chapter4.3

120minutesperfrequencypoint

Harmonicresponseanalysis–simplifiedtransformerassembly Chapter4.3

80minutesperfrequencypoint

Note: athesimulationwasabortedbeforecompletionduetoexcessivecomputingtime

Chapter5:Discussion

116

FromthecomputingtimespresentedinTable5-1itisapparentthatperformingafully

coupled fluid-structure interaction modal analysis not likely to be feasible in the

designphase.Thisisduetothesignificantcomputingrequirementsoftheasymmetric

matrix formulation in such simulations. However, performing a structural modal

analysisoftheactivepartandanalyticallyaccountingfortheadded-masseffectofthe

fluidwithinthedesignprocessisrealistic.AspresentedinChapter5.3,thisanalytical

correction allows for an accurate quantitative estimate of noise levels and the

identificationofmodesthatmaysignificantlyinfluencetheacousticcharacteristicsofa

transformer.

Inadditiontoamodalanalysisoftheactivepart,thecomputingrequirementsdetailed

inTable5-1illustratethatemployingforcedexcitationharmonicresponseanalysesto

aid in transformer design is feasible. It has also been shown in Chapter 5.2 that

simplification of the transformer assembly – whereby removing all auxiliary

equipment, including the conservator, the radiators and the bushings – reduces the

computing time by approximately 33% while not significantly influencing the

accuracy of the results. Furthermore, the forced excitation harmonic response

simulationspresentedinthisthesishavealsobeenshowntoidentifytheglobalmodes

ofthetankstructure.

Chapter6:SensitivityAnalysis

117

CHAPTER6

SENSITIVITYANALYSIS


118

This chapter presents a sensitivity analysis to better understand the influence of

selectedparametersonatransformer’svibro-acousticcharacteristics.Itisanticipated

thattheunderstandinggainedfromthisanalysismayaidintheacousticoptimisation

ofatransformerduringthedesignphase.Theparameterswithinthisstudyhavebeen

selected on the basis that they do not affect a transformer’s electrical behaviour.

Parametersthathavebeenanalysedincludethestiffnessoftheactivepart,thenumber

ofstiffeningribsonthetankwalls,thetransmissibilityofvibrationsbetweentheactive

partandthetankandthedimensionsofthetank.

Thesensitivityanalysishasbeenconductedbychangingselectedparameterswithin

thenumericaltransformermodelpresentedandvalidated intheprecedingchapters.

Ofnoteisthatthesimplifiednumericalmodel–consistingoftheactivepart,thetank

andtheinsulationfluidbutexcludingtheconservator,theradiators,thebushingsand

otherauxiliaryequipment–hasbeenemployed in theseanalyses.This isdueto the

reduced simulation time associated with the simplified model. All acoustic results

presented in this chapter have been calculated at 10Hz intervalsover an analysis

spectrumfrom100Hzto400Hz.

6.1 INFLUENCEOFACTIVEPARTSTIFFNESS

The stiffness of the active part is an important parameter as it influences the

assembly’s structural dynamics characteristics,with these characteristics known to

significantly affect the acoustic behaviour of a transformer. As noted in Ref. [44],

decreasing in the stiffness of the core results in closer spacing between the active

part’snaturalfrequencies.Thehighermodaldensitythereforeincreasesthelikelihood

ofanactivepartmodebeingexcited,eventuatinginhighernoiselevels.


119

Withinthisparametricstudythestiffnessoftheactiveparthasbeenincreasedinan

efforttolowertheassembly’smodaldensitywithinthefrequencyspectrumofinterest.

Specifically,theYoung’smoduliofboththecorematerialandthewindingshavebeen

increasedby15%.The resulting acoustic characteristics of the transformer,with a

comparison to results calculated with original numerical model, are presented in

Figure6-1.

Figure6-1 Influenceofactivepartstiffnessontheacousticbehaviourofa transformer

ItisshowninFigure6-1thatthetransformer’sacousticcharacteristicsareinfluenced

bythestiffnessoftheactivepart.Inparticular,aspredictedbythestudyinRef.[44],

increasing the stiffness of the active part has resulted in a more linear acoustic

spectrum. This indicates a lowermodal density of the assembly and subsequently

fewernaturalfrequenciesbeingexcitedovertheanalysedfrequencyspectrum.Thisis

particularlytrueatthehigherfrequencyrangesofthespectrumconsidered.


120

ThesecondtrendillustratedinFigure6-1isthenotablylowernoiselevelscalculated

whenthestiffnessoftheactivepartisincreased.Thissuggeststhat,inadditionalower

modaldensity, increasing the active part’s stiffness has had the effect reducing the

amplitudeoftheassembly’svibrations.Theapplicationofthisresulthasalreadybeen

implemented in shunt reactor cores,which are designed as very stiff structures to

eliminateexcessivevibrations[7].

Increasing the stiffness of the activepart tooptimise a transformer’svibration and

acoustic characteristics may be carried out in numerous ways. As identified in

Ref.[44],thelaminationsofthecoresignificantlyaffectthestructure’sflexibility(with

anincreaseinlaminationmaterialdecreasingthestiffnessofthecore).Therefore,the

degreeof laminationappliedto thecoresheetsand thepropertiesofthe laminating

materialmaybeoptimisedtoprovide astiffercore.Additionalmethodsthatmaybe

employedtoincreasethestiffnessoftheactivepart includefurthertensioningofthe

tierodsandincreasingtheclampingpressureappliedtothewindings.

6.2 INFLUENCEOFSTIFFENINGRIBS

The second parameter analysed within this chapter with respect to the acoustic

emissionsofapowertransformer isthenumberofstiffeningribsplacedonthetank

walls.Theadditionof stiffening ribs is a technique frequentlyemployed tostiffen a

structure, thereby reducing the structure’s vibration levels and subsequent noise

radiationlevels.However,asnotedinRefs.[81,82],increasingthenumberofribsona

structuredoesnotassureasystematicreductioninradiatedsoundpower.Thisisdue

tovariationsinthesoundradiationefficiencyresultingfromstiffenedstructures.


121

Within this parametric study additional stiffening ribs have been added to those

alreadypresenton the tankwalls. TheU-shaped stiffeners included in the original

design fulfil theduel purposes of increasing the tank’s vacuumpressurewithstand

capabilities(necessaryasoil-fillingofatransformeriscarriedoutundervacuum)and

reducing the assembly’s acoustic emissions. Within this parametric study, an

additional two stiffening ribswere placed on each of the tankwalls. The updated

layoutofthetankisillustratedinFigure6-2.

Figure6-2 Updatedtanklayoutoftankwithadditionalstiffeningribs

Thesimulationwithadditionalstiffeningribsofthetankwallhasbeenconductedonly

to understand the degree of influence these elements have on the transformer’s

acoustic response. Therefore, the ideal placement of the additional ribs to reduce

vibrationshasnotbeenconsidered.Such calculation schemesarepresented inRefs.

[81,83].Theacousticcharacteristicsofthetransformerwithadditionalstiffeningribs

andacomparisontoresultscalculatedwiththeoriginalnumericalmodelispresented

inFigure6-3.


122

Figure6-3 Influenceofstiffeningribsontheacousticbehaviourofa transformer

Withadditionalstiffeningribson the tanktheacousticbehaviourof the transformer

hasbeensignificantlyaltered,asshowninFigure6-3.Inparticular,theresultsappear

to show that some natural frequencies of the assembly have been amplifiedwhile

othersdampedbythealterationsmade tothetank layout.Thisoccurrence,which is

particularlyvisibleatlowerfrequencies,maybeattributedto ‘stopping’and ‘passing’

bands introduced by the additional stiffeners. Further reading on the formation of

thesebandsmaybefoundinRefs.[20,82,84].

Reducingtheacousticemissionsfromapowertransformerbyoptimisingthenumber

and the positioning of stiffening ribs on the tank walls would require a detailed

algorithm.Suchastudy,asappliedtoaconstantspeedgearbox,hasbeenconductedin

Ref.[20].Thereductionofatransformer’snoiseemissionswouldhoweverrequirea

significantlymoredetailedalgorithmthanthatemployedinRef.[20].Thisisduetothe

complexityandindividualityoftransformertankdesigns.


123

6.3 INFLUENCEOFTHEDIRECTTRANSMISSIONPATHBETWEEN

ACTIVEPARTANDTANK

Aswithstiffeningribs,vibration-dampingpadsareoftenusedtominimisestructure

bornevibrations. In transformers, such pads are generally employed to isolate the

activepart,beingthesourceofvibrations,fromthebaseofthetank.However,inoil-

filled transformers isolating the active part from the tank base only limits one

transmissionpart,as thevibrationsare stillable to travel indirectly from theactive

parttothetankstructurethroughthefluidmedium.

Toquantifytheacousticeffectsofisolatingtheactivepartfromthebaseofthetankin

an oil-filled transformer a simulationwith additional damping applied to material

elements connecting theactivepartand the tankhasbeen conducted.Specifically, a

constantdampingfactorof0.3hasbeenappliedtotherubbercompoundthatconnects

theactivepart to thebaseof the transformer tank.Therubbercompoundmeasures

27mminheight.TheresultsofthesimulationareillustratedinFigure6-4.

Figure6-4 Influenceofthedirectvibrationtransmissionpathbetweenthetank andtheactivepartofatransformer


124

AsseeninFigure6-4additionaldampingappliedtotheconnectionbetweentheactive

partandthetankstructurehashadalimitedinfluenceontheacousticcharacteristics

of the studied transformer. The average decrease in the acoustic emissions of the

transformerwith additionaldampingmeasures0.1dBacross the31 corresponding

simulationpoints9.Therefore,itappearsthattheprimarytransmissionpathfromthe

activeparttothetankofthetransformeristhroughthefluidmedium.Itshouldalsobe

noted thatno assertion ismade in this thesis as to theprobabilityof being able to

achieveadditionaldampingof0.3betweentheactivepartandtankofatransformer.

6.4 INFLUENCEOFTANKDIMENSIONS

Thefinalparameterstudiedwithinthischapteristheinfluenceoftankdimensionson

a transformer’s acoustic response. Tank dimensions are generally calculated with

regardtothedimensionsoftheactivepart,theeddycurrent lossesinthetankwalls

(which result from magnetic field leakage) and material costs. These factors make

alteringthedimensionsofthetankmorecomplicatedthantheparameterspreviously

discussed.

To identify how tank dimensions influence a transformer’s acoustic response the

lengthofthetankinthenumericalmodelhasbeenincreasedbyapproximately10%.

It isnotedthatdue totheelongationof the tanktheplacementof thestiffeningribs

relative to the four corners of the tank has also been altered. The acoustic

characteristicsof the transformerwithanelongated tankandacomparisonto those

soundpressure levelscalculatedwith theoriginalnumericalmodelarepresented in

Figure6-5.

9The greatest vibrationdamping effecthasbeennoted in the lower frequency rangesof the

numericalsimulation.Specifically,between100Hzand150Hzanaveragedecreaseof0.3dBis calculated across the six simulation points. Above 150 Hz it is likely that the stiffnessdampingcoefficientappliedgloballytothenumericalmodel(refertoChapter3.6.3)becomesdominantandthereforethelocalisedconstantdampingappliedtotheelementsconnectingtheactivepartofandthebaseofthetankhasminimaleffect.


125

Figure6-5 Influenceoftankdimensionsontheacousticbehaviourofa transformer

From Figure 6-5 is it apparent that the elongation of the transformer tank has

influenced the acoustic response of theunit.However, there is no clear correlation

between acoustic response of the original design and the updated assembly.

Furthermore,fromtheresultspresentedinFigure6-5itcannotbeconcludedwhether

thechange inacousticcharacteristics isdueto the increasedoilvolumewithintank,

thetank’srevisedstructuraldesignoracombinationoftheseaforementionedfactors.


126

6.5 CONCLUSIONS

This chapter has detailed a sensitivity analysis to better understand influence of

selectedparameterson the soundpressure levels andvibration characteristicsof a

large power transformer. Four parameters selected on the basis that they do not

influencetheelectricalbehaviouroftheunithavebeenconsidered,being:thestiffness

of the active part; the number of stiffening ribs placed on the tank walls; the

transmissibilityofvibrationsbetweentheactivepartandthetank;andthedimensions

ofthetank.Theanalysiswasconductedbychangingtheselectedparameterswithin

thenumericalmodelpresentedandvalidatedintheprecedingchaptersofthisthesis.

Fromtheanalysespresentedwithinthischapterithasbeenseenthatthestiffnessof

the activepart is theparameterbest suited tooptimise the acousticbehaviourof a

transformer.Specifically,increasingtheactivepart’sstiffnesshasbeenshowntoresult

in amore linear acoustic spectrumwith lower soundpressure levels.These results

indicate that the increased stiffness has had the effect of reducing the assembly’s

modaldensitywhilealsoeliminatingexcessivevibrations.

Chapter7:ConclusionsandFutureWork

127

CHAPTER7

CONCLUSIONSANDFUTUREWORK


128

7.1 CONCLUDINGREMARKS

This thesis has presented the vibro-acoustic model of a large power transformer,

whichallows thevibrationandacousticcharacteristicsoftheunit tobedetermined.

The model has been discretised and solved using the finite element method and

accountsforthebi-directionalfluid-structureinteractionbetweentheactivepart,the

insulation fluid and the tank. Although a series of previous investigations have

employed the finite element method to predict the vibration characteristics of

individualtransformercomponents,thenarrowscopeofthesepreviousanalysesand

the absenceof acoustic elements limit their ability toqualitatively orquantitatively

predictthecomplexacousticcharacteristicsofapowertransformer.

Thetwomostcomparablestudiestothatpresentedinthisthesis,beingthosedetailed

in Ref. [3] and Ref. [4], are limited by the omission of the transformer core. This

omission has led to simplifications and assumptions that do not allow for the

predictionofatransformer’sacousticbehaviourundernominaloperatingconditions.

TheshortcomingsofthemodelspresentedinRefs.[3,4]include:

· The vibration characteristics of the transformer core,which may significantly

affectatransformer’soverallsoundpressurelevels,areoverlooked;

· Only the acoustic emissionsatdouble the line frequency are considered, being

thoseat100Hz.This isasimplificationthatdoesnotholdtrueinrealitydueto

thesignificantsoundemissionsfromatransformeratevenharmonicsoftheline

frequency;and

· Theomissionofthecorestructurehasnecessitatedtheuseofspringelementsto

supportthewindingblockswithinthenumericalmodelspresentedinRefs.[3,4].

However, the values assigned to these spring elements have not been

experimentallyvalidated.


129

The vibro-acousticmodellingmethodology adopted in this thesis has attempted to

address the limitations in precedingworks. Specifically, all key structural elements

(i.e.the core, the windings, and the tank, as well as the significant components

connectingtheseassemblies)andtheinsulationfluidhavebeenincorporatedintothe

numerical model. Furthermore, validation of the acoustic predictions has been

conductedat16frequencypointsacrossa100Hzto400Hzanalysisspectrum.Ofnote

is that this spectrum is considered asdominant in the acoustic responseof a large

powertransformer.

Theproceduresemployed inthisthesishavebeendiscussed indetail intheprevious

chapters. A summaryof themethodologyemployed forpredicting thevibrationand

acousticcharacteristicsofatransformerduringthedesignprocessisgivenasfollows:

i) Developavibro-acousticnumericalmodelofthestudiedpowertransformer

employing the finiteelementmethodwith the activepart, the insulationoil

andthetransformertankincludedinthemodel;

ii) Determinethevibrationcharacteristicsoftheactivepartthroughastructural

modalanalysis,notingthattheadded-massandviscousdampingeffectsofthe

insulation fluid may be accounted for through analytical correction of the

assembly’snaturalfrequencies;

iii) From the modal analysis conducted in step ii analyse the modal mass

participationfactorsoftheactiveparttogainaqualitativepredictionofnoise

levelsoverthestudiedfrequencyspectrum;

iv) Applyexcitation forces to thenumericalmodelof the complete transformer

(which incorporates the active part, the insulation oil and the transformer

tank), with the applied forces simulating the electromagnetic and

magnetostrictiveforcesactingontheactivepart;


130

v) Mapthevibrationvelocityofthetankstructure,ascalculatedinstepiv,toan

externalacousticbodythatreplicates thetestoroperatingsurroundsof the

transformerandhencecalculatetheradiatedsoundpressurelevels;

vi) Repeatstepsivandvatfrequencypointsspacedoverthefrequencyspectrum

ofinterest;and

vii) Calculate of the operational acoustic emissions through the combinationof

soundpressurelevelsdeterminedateachofthethreedominantharmonicsof

theelectromagneticandmagnetostrictive forces (notingthatstepviihasnot

beenperformedinthisthesis).

Theproceduresdetailedabovearenonspecificandmaybeappliedtoanytransformer.

In addition to thesegeneric procedures this thesishas: validated thevibro-acoustic

numericalmethodology;identifiedthekeyvibrationcharacteristicsofthetransformer

studied;conducted a sensitivity analysisof thekeydesignparameters influencing a

transformer’s acoustic behaviour; and detailed a novel damping estimation

methodology.Concludingremarksapplicabletothesestudiesareasfollows:

i) Validation of the vibro-acoustic methodology was conducted through the

comparisonofnumericalandexperimentalresults.Both a structuraland an

acousticvalidationofthemodelwasperformed,noting:

- Numerical and experimental modal results showed a high level of

correlationwiththeaveragedifferencebetweenthenaturalfrequencies

ofthefirstsixcomparablemodesfoundtobe2.9%;

- Theacousticvalidationof thenumericalmodelwas conductedover a

100 Hz to 400 Hz spectrum and illustrated that the applied

methodologyidentifiedboththetransformer’sglobalandlocalacoustic

behaviour.The accuracyof the soundpressure levelpredictionswas

showntobeinlinewithcomparableprecedingstudies.


131

ii) Analysesconductedwiththevalidatedvibro-acousticmodelshowedthatthe

vibration characteristicsof theactivepart significantly influence theoverall

acousticresponseofatransformer.Thenumericalmodelalsoillustratedhow

the frequencyoverlapofactivepartand tankmodescanresult in increased

noiselevels.

iii) Thesensitivityanalysisconductedinthisthesisconcludedthatthestiffnessof

the active part is the parameter best suited to optimise a transformer’s

acousticbehaviour.Specifically,increasingtheactivepart’sstiffnessisshown

toproduceamorelinearacousticspectrumwithlowersoundpressurelevels.

iv) Anoveldampingestimationtechniquewaspresentedtodeterminethemodal

dampingofthecomplexoil-filledtransformerassembly,noting:

- The damping estimation technique methodology estimates modal

damping within defined bandwidths as opposed to at natural

frequencypoints,allowingfordampingestimatestobecomputedof

complexstructureswithhighmodaldensities;and

- The partial numerical validation conducted suggests that the

methodology is able to provide a good estimate of a complex

system’sdampingcharacteristicsoverawidefrequencyspectrum.

Insummary,thenumericalmethodologypresentedinthisthesis isshownto identify

the acoustic behaviour of a transformer through numerically calculated sound

pressurelevels.Theseacousticemissionsresultfromexcitationforcesappliedtoboth

the core and the windings. Furthermore, the vibro-acoustic modelling scheme and

analysesmethods, including amodalmassparticipation factoranalysisof theactive

part,allowforthecomplexvibrationcharacteristicsofatransformertobeidentified.

Anunderstandingofthesevibrationcharacteristicstogetherwithsoundpressurelevel

predictionswillbetterenabletransformermanufacturerstoconsistentlymeetacoustic

targetswhilealsoreducingthecostlysafetymarginsincurrentdesignpractices.


132

7.2 FUTUREWORK

As presented in the preceding chapters, this thesis has detailed and validated a

numericalmethodology able to predict the noise and vibration characteristics of a

largepowertransformer.However,areasoffurtherresearchhavealsobeenidentified.

These future works focus on the assumptions and limitations of the modelling

methodology, which are potential sources of error and may account for the

discrepanciesbetweenpredictedandmeasuredvaluesinthisthesis.Thelimitedscope

oftheworkpresentedinthisthesisisalsotobeaddressedinfutureinvestigations.

It is proposed that the first study to follow this thesis is the application of the

modellingmethodologytoathree-phaselargepowertransformer.Thisworkisseenas

significantas three-phase largepower transformersare themostcommonlyutilised

class of transformer in transmission networks. The principal challenge of this

proposed investigation is the determination and application to the finite element

modeloftheelectromagneticandmagnetostrictive forcesresulting fromthree-phase

excitation.Inthisthesisasingle-phaseunithasbeenconsideredwhich,incomparison

toathree-phasetransformer,greatlysimplifiedthecalculationandapplicationofthe

excitationmechanisms.

Havingvalidated thenumericalmodellingmethodology in this thesis, it isproposed

thatafuturestudydeterminesthetotalsoundpressurelevelsofatransformerunder

nominal operating conditions employing the computational scheme presented.This

would require the non-linearity of the excitation forces to be accounted for.

Specifically,themagnitudeoftheexcitationforcesandthefirstthreeevenharmonics

of the networkwould need to be determined and applied to the numericalmodel.

Estimationofatransformer’sacousticemissionsundernominaloperatingconditions

wouldthenbemadethroughthecombinationofthesoundpressurelevelsdetermined

attheaforementionedharmonics.


133

Aspreviouslydiscussed inthisthesis, thevibrationcharacteristicsof theactivepart

areasignificantfactorintheacousticbehaviourofatransformer.However,noexplicit

experimental modal analysis of the active part was conducted within this thesis.

Conducting adetailedEMA tovalidate thenumericalmodellingof the activepart in

future works is therefore of importance, with a focus on experimental modal

measurementsof:

· Thewindingsblocks– tovalidate thehomogeneousbodyemployed tosimulate

thewindingswithinthefiniteelementmodel;and

· The core – to validate modal mass participation factors and mode shapes

resultingfromthenumericalmodalanalysisoftheactivepart.

Within this thesis all thermal effects on the vibration and subsequent acoustic

characteristicsof a transformerhavebeendisregarded.However,assuming that the

thermalenvironsandoperating temperaturesdonotaffect thesoundemissionsof a

transformer isnotvalid.Futureworksmay therefore considerhow thermal factors

(i)affect and (ii) may be accounted for within the numerical modelling of power

transformernoise.

Inadditiontofiniteelementmodellingprocedures,thedampingestimationtechnique

presented in this thesis to determine the modal damping characteristics of the

transformer tankmaybe improvedupon and furthervalidated. Inparticular, ithas

been noted previously that the primary source of error associated with the

methodologyisconsideredtoresultfromtakingtheIFFToftheFRFsignal.However,

within this thesis no steps have been made to correct this purported error.

Furthermore, thedampingestimation techniquemaybe furthervalidated through a

detailedanalyticalstudyaimedatdescribingthestatisticalpropertiesoftheproposed

estimators.


134

As noted in the assumptions detailed at the beginning of thiswork, the excitation

forcesemployedinthisthesisweredeterminedanalyticallyandprovidedbySiemens

AG.Although itmaynot improve theaccuracyof thevibro-acousticpredictions, the

numericalmethodology presented in this thesismay be extended to incorporate a

coupled electromagnetic finite element calculation. The primary advantage of the

coupled electromagnetic simulationwould be the detailed force distribution of the

excitation mechanisms to the structural finite element model. Accounting for all

excitation mechanisms within a numerical multi-physics simulation to predict

transformernoisewouldamounttoasignificantbodyofresearch.

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135

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