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NUMERICALANALYSISOFTHEVIBRATION
ANDACOUSTICCHARACTERISTICSOF
LARGEPOWERTRANSFORMERS
by
JustinJamesCase
BEng(Mech)(Hons),BBus(Finance)
Adissertationsubmittedinfulfilment
oftherequirementsforthedegreeof
MasterofAppliedScience(Research)
SchoolofChemistry,PhysicsandMechanicalEngineering
Science and Engineering Faculty
QueenslandUniversityofTechnology
Australia
2017
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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
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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.
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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
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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.
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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
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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
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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
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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
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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
Figure4-3 Comparisonof(a)theexperimentalmodeshapeat11Hzand(b)theexperimentalmodeshapeofthetransformerat25.87Hz.......................90
Figure4-4 Comparisonof(a)theexperimentalmodeshapeat11Hzand(b)theexperimentalmodeshapeofthetransformerat39.31Hz.......................91
Figure4-5 Comparisonofmeasuredandpredictedsoundpressurelevelsat16correspondingmeasurementpoints.......................................................................97
Figure4-6 Comparisonofpredictedsoundpressurelevelscalculatedwiththesimplifiednumericaltransformerassemblyandthecompletenumericaltransformermodel...........................................................................................99
Figure5-1 Comparisonofthetransformersacousticbehaviourasidentifiedbysoundpressurelevelsdeterminedat10Hzand20Hzintervals...............104
Figure5-2 Comparisonofpredictedsoundpressurelevelsat10Hzintervalscalculatedwiththesimplifiednumericaltransformerassemblyandthecompletenumericaltransformermodel..........................................................105
Figure5-3 Modalmassparticipationfactorofactivepartmodesintheverticalplane.............................................................................................................................................109
Figure5-4 Modeshapeexhibitingglobaldeformationofthetankcoveratapproximately400Hz.........................................................................................................111
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Figure6-1 Influenceofactivepartstiffnessontheacousticbehaviourofatransformer...............................................................................................................................116
Figure6-2 Updatedtanklayoutoftankwithadditionalstiffeningribs..........................118
Figure6-3 Influenceofstiffeningribsontheacousticbehaviourofatransformer...............................................................................................................................119
Figure6-4 Influenceofthedirecttransmissionpathbetweenthetankandtheactivepartofatransformer.....................................................................................120
Figure6-5 Influenceoftankdimensionsontheacousticbehaviourofatransformer...............................................................................................................................122
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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
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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
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[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
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
Chapter1:Introduction
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
Chapter1:Introduction
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.
Chapter1:Introduction
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.
Chapter1:Introduction
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.
Chapter1:Introduction
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.
Chapter1:Introduction
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
Chapter2:LiteratureReviewandTheoreticalFramework
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.
Chapter2:LiteratureReviewandTheoreticalFramework
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
Chapter2:LiteratureReviewandTheoreticalFramework
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].
Chapter2:LiteratureReviewandTheoreticalFramework
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.
Chapter2:LiteratureReviewandTheoreticalFramework
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.
Chapter2:LiteratureReviewandTheoreticalFramework
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
Chapter2:LiteratureReviewandTheoreticalFramework
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
Chapter2:LiteratureReviewandTheoreticalFramework
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
[ ] ⋅ { } + [ ] ⋅ { } + [ ] ⋅ { } = { ( )}
Chapter2:LiteratureReviewandTheoreticalFramework
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.
Chapter2:LiteratureReviewandTheoreticalFramework
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].
Chapter2:LiteratureReviewandTheoreticalFramework
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].
Chapter2:LiteratureReviewandTheoreticalFramework
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
Chapter2:LiteratureReviewandTheoreticalFramework
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
Chapter2:LiteratureReviewandTheoreticalFramework
23
EmployingEqn.2.12andEqn.2.13,thediscretisedelementaryweakformofEqn.2.11
maybedetermined.Assemblingofthesetofelementarycontributionsthenresultsin
thediscretisedglobalwaveequationmatrixformofEqn.2.11.Thisfinalglobalformis
presentedinEqn.2.14,wherethedependencyofallthetermsonthetestfunctionhas
beeneliminated.
Chapter2:LiteratureReviewandTheoreticalFramework
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.
Chapter2:LiteratureReviewandTheoreticalFramework
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.
Chapter2:LiteratureReviewandTheoreticalFramework
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].
Chapter2:LiteratureReviewandTheoreticalFramework
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.
Chapter2:LiteratureReviewandTheoreticalFramework
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.
Chapter2:LiteratureReviewandTheoreticalFramework
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.
Chapter2:LiteratureReviewandTheoreticalFramework
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.
Chapter2:LiteratureReviewandTheoreticalFramework
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
Chapter2:LiteratureReviewandTheoreticalFramework
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
Chapter2:LiteratureReviewandTheoreticalFramework
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
Chapter2:LiteratureReviewandTheoreticalFramework
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)
Chapter2:LiteratureReviewandTheoreticalFramework
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.
Chapter2:LiteratureReviewandTheoreticalFramework
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)
Chapter2:LiteratureReviewandTheoreticalFramework
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].
Chapter2:LiteratureReviewandTheoreticalFramework
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:
Chapter2:LiteratureReviewandTheoreticalFramework
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
Chapter2:LiteratureReviewandTheoreticalFramework
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.
Chapter2:LiteratureReviewandTheoreticalFramework
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)
Chapter2:LiteratureReviewandTheoreticalFramework
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
Chapter3:FiniteElementModeloftheTransformer
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.
Chapter3:FiniteElementModeloftheTransformer
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
Chapter3:FiniteElementModeloftheTransformer
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
Chapter3:FiniteElementModeloftheTransformer
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.
Chapter3:FiniteElementModeloftheTransformer
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.
Chapter3:FiniteElementModeloftheTransformer
49
(a)
(b)
Figure3-3 Finiteelementmodelof(a)thetransformertankand(b)the transformertankwithauxiliaryequipment
Chapter3:FiniteElementModeloftheTransformer
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
Chapter3:FiniteElementModeloftheTransformer
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.
Chapter3:FiniteElementModeloftheTransformer
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.
Chapter3:FiniteElementModeloftheTransformer
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.
Chapter3:FiniteElementModeloftheTransformer
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.
Chapter3:FiniteElementModeloftheTransformer
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.
Chapter3:FiniteElementModeloftheTransformer
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
Chapter3:FiniteElementModeloftheTransformer
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.
Chapter3:FiniteElementModeloftheTransformer
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.
Chapter3:FiniteElementModeloftheTransformer
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.
Chapter3:FiniteElementModeloftheTransformer
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.
Chapter3:FiniteElementModeloftheTransformer
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.
Chapter3:FiniteElementModeloftheTransformer
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.
Chapter3:FiniteElementModeloftheTransformer
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.
Chapter3:FiniteElementModeloftheTransformer
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.
Chapter3:FiniteElementModeloftheTransformer
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.
Chapter3:FiniteElementModeloftheTransformer
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
Chapter3:FiniteElementModeloftheTransformer
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.
Chapter3:FiniteElementModeloftheTransformer
68
(a)
(b)
(c)
Figure3-8 Half-powerpointsonanFRFemployedtocalculatethedampingratiosof theactivepartat (a)192Hzand (b)346Hz (c)and the resultingRayleighdampingcurve
Chapter3:FiniteElementModeloftheTransformer
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).
Chapter3:FiniteElementModeloftheTransformer
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.
Chapter3:FiniteElementModeloftheTransformer
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− .
Chapter3:FiniteElementModeloftheTransformer
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.
Chapter3:FiniteElementModeloftheTransformer
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
Chapter3:FiniteElementModeloftheTransformer
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.
Chapter3:FiniteElementModeloftheTransformer
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.
Chapter3:FiniteElementModeloftheTransformer
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%.
Chapter3:FiniteElementModeloftheTransformer
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.
Chapter3:FiniteElementModeloftheTransformer
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.
Chapter3:FiniteElementModeloftheTransformer
79
(a)
(b)
Figure3-11 Spectral leakagewhen taking the IFFTof (a) acompleteFRFsignaland(b)adefinedafrequencybandofthecompleteFRFsignal
Chapter3:FiniteElementModeloftheTransformer
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)
Chapter3:FiniteElementModeloftheTransformer
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.
Chapter3:FiniteElementModeloftheTransformer
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.
Chapter3:FiniteElementModeloftheTransformer
83
Figure3-13 Transformerassemblydampingestimatesin1/3octavebands
Chapter3:FiniteElementModeloftheTransformer
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.
Chapter3:FiniteElementModeloftheTransformer
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.
Chapter3:FiniteElementModeloftheTransformer
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
Chapter4:ValidationoftheNumericalModel
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.
Chapter4:ValidationoftheNumericalModel
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
Chapter4:ValidationoftheNumericalModel
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.
Chapter4:ValidationoftheNumericalModel
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.
Chapter4:ValidationoftheNumericalModel
92
(a)
(b)
Figure4-2 Comparisonof(a)theexperimentalmodeshapeat11Hzand (b)theexperimentalmodeshapeofthetransformerat10.24Hz
Chapter4:ValidationoftheNumericalModel
93
(a)
(b)
Figure4-3 Comparisonof(a)theexperimentalmodeshapeat25Hzand (b)theexperimentalmodeshapeofthetransformerat25.87Hz
Chapter4:ValidationoftheNumericalModel
94
(a)
(b)
Figure4-4 Comparisonof(a)theexperimentalmodeshapeat39Hzand (b)theexperimentalmodeshapeofthetransformerat39.31Hz
Chapter4:ValidationoftheNumericalModel
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.
Chapter4:ValidationoftheNumericalModel
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
Chapter4:ValidationoftheNumericalModel
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.
Chapter4:ValidationoftheNumericalModel
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.
Chapter4:ValidationoftheNumericalModel
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.
Chapter4:ValidationoftheNumericalModel
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).
Chapter4:ValidationoftheNumericalModel
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.
Chapter4:ValidationoftheNumericalModel
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.
Chapter4:ValidationoftheNumericalModel
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.
Chapter4:ValidationoftheNumericalModel
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
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.
Chapter6:SensitivityAnalysis
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.
Chapter6:SensitivityAnalysis
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.
Chapter6:SensitivityAnalysis
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.
Chapter6:SensitivityAnalysis
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.
Chapter6:SensitivityAnalysis
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
Chapter6:SensitivityAnalysis
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.
Chapter6:SensitivityAnalysis
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.
Chapter6:SensitivityAnalysis
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
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.
Chapter7:ConclusionsandFutureWork
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;
Chapter7:ConclusionsandFutureWork
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.
Chapter7:ConclusionsandFutureWork
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.
Chapter7:ConclusionsandFutureWork
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.
Chapter7:ConclusionsandFutureWork
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.
Chapter7:ConclusionsandFutureWork
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.
References
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