16momtheorystudentguideWord文档下载推荐.docx
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16momtheorystudentguideWord文档下载推荐.docx
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Thefunctiong(x)isalinearapproximationoff(x)
Analternativedescriptionistohavef(x)bethesumoftwoshiftedtriangularfunctions.
Basisfunctionexpansion
Triangularbasisfunctionexpansion
Thetwotriangularfunctionscanbenormalizedtogobetween0and1withsomemultipliedcoefficientf0,f1,f2…
TheLinearapproximationtothefunctionf(x)canberepresentedasasumoftwoshiftedtriangularbasisfunctions.
fi:
functionvaluesatthenodes(I=0,1,2,3…),fi=f(x)
Ti:
basisfunctions(triangles)independentofthefunctionf(x)
CurrentModeling–StripandSlot
MomentumsolvesfortheunknowncurrentsJeandJm.
Magneticcurrent=electricfieldrotatedover90degrees
Momentumcurrentexpansion
ThecellcurrentdensityJisavectorinthehorizontal“x-y”plane.TheJvectorcanbedecomposedintoaJxandJyvector.Eachvectorismadefromtwohalfrooftops.
Thetotalcellcurrent=sumof4halfrooftops
Themethodofmomentsusesthefollowingapproachtoapproximatecellcurrent:
∙Createmeshofrectanglesandtriangles
∙Approximatecellcurrentsbylinearfunctions(rooftops)
∙Expandunknownsurfacecurrentintoasumofrooftopfunctionswithunknownamplitudes
Rooftops
Thecircuitismeshedwithsmallcellscomparedtothewavelength(typically30cellsperwavelength).
Theunknownsarethecurrentsthroughtheinneredgesofthemesh
Eachunknowniscombinedwithaunit-heightrooftopdefinedoverthecellsadjacenttotheedge.
MethodofMoments
ThegoalofMethodofMomentsistosetupandsolvethematrixfunctionZ(f)*I=V
∙Z(f)istheMomentuminteractionmatrixwhichisafunctionoffrequency
∙Iisthecurrentvectorwithunknowncurrents
Visthevoltagesourcevector
Divideandconquer…
Expandunknowncurrentintobasisfunctionswithunknownamplitudes.
Calculatecapacitiveandinductivecouplingbetweenbasisfunctions.
BuildanequivalentcircuitnetworkofmutuallycoupledLsandCsforeachsimulationfrequency.
SolveKirchoff’svoltageequations
MethodofMomentsmatrixequation
CapacitiveCoupling
Ifasinglemetalcellisisolatedyoucandescribeitintermsofhavingsomecapacitancetoground.
Iftwocellsareconsidered,theywillsomemutualcapacitancebetweenthem.
MethodofMomentsMatrixequation
InductiveCoupling
Ifyouconsiderthateachcellhaselectricallength,thenitactsasasmalltransmissionline.Eachcellcurrenthasaself-inductance.
Eachcellcurrentpairhasamutualinductance.
Self-Equivalentcircuit
Combinethecapacitorandinductorrepresentationstoformanequivalentcircuitforeachcell.Thisisthe“selfequivalentcircuit”.
MutualEquivalentCircuit
Replacetwocellsbyanequivalentcircuit.
Cellcurrentscouple:
mutualLs
Cellchargescouple:
mutualC
Allelementvaluesarefrequencydependent.
EquivalentMOMCircuit
Replaceallmetalcellsbyanequivalentcircuit.
Eachnodecorrespondswithacell;
eachbranchcorrespondswithanedge.
Foreachfrequencythereisadifferentnetwork,butwiththesametopology(fixedmesh).
Summary:
Thecircuitgeometryisbrokendownintorectangularandtriangularcells
AdaptiveFrequencySampling(AFS)
Smartsamplingandinterpolationalgorithmthat
∙Selectskeyfrequenciesautomaticallyinconsecutiveiterations
∙InterpolatesallS-parameterdatausingrationalfittingmodels
Keybenefits
∙Automaticsampleselection
∙NopriorknowledgeofdynamicsofS-parametersrequired(easeofuse)
∙Reducednumberofsamplesforagivenaccuracy
∙Noover-samplingorunder-sampling–majorspeedimprovement
AFSUseModel
INPUT:
∙Frequencyrange(startandstopfrequencies)
∙Maximumnumberoffrequenciestosample(astoppingcriteria)
∙Filereuse(toextendfrequencyrangeorincreasemax.numberofsamplingfrequencies)
OUTPUT:
∙S,S_50&
S_Z0:
S-parameterdata
∙PortZ:
referenceportimpedance
∙Z0&
Gamma:
portimpedanceandpropagationfactor
∙S_Error:
estimatedfittingerror
AFSFlowChart
1.Designplanarstructureandspecifyfrequencyrange.
2.PerformEManalysisofnewsamplepoints
3.Builddifferentphysicallypossiblerationalfittingmodels.Checkerrorinsamplepointsandcomparedifferentfittingmodels
4.Addnewsamplepointsiftheerroristoolarge.Repeat2-4untilconvergence
5.Output
AFSConvergenceIllustration
STEP1:
AFSstartswithaminimumoffoursimulationdatapoints(thetwoendpointsplus
twoadditionalfrequencies:
Afittingmodelisgeneratedforthesefourdatapoints,andanerrorfunctionisestimated(complexdatafit).Iftheerrorisbelowthegoal,thenithasconverged.Ifnot,itselectsanadditionalpointtosimulate.
STEP2:
Anewsampleisselectednearthelargestfittingerror.
AFSConvergenceIllustration(cont’d)
STEP3and4:
Additionaldatapointsareselected,andtheerrorisestimated.
AFSConvergenceIllustration(cont’d#2)
STEP5:
Eventually,AFScalculatesafittingfunctionthatreducedtheerrorbelowthegoal.Atthispoint,thedatahasconverged.
AFSRuleofthumb–ForeachloopoftheS-parametersintheSmithchart,AFSneedsabout6frequencysamplesinordertomodelthephasevariationaccurately.
DataReuse–there-useofdataisonlypossibleifthefollowingareallthesame:
∙Thesubstrate
∙Thelayout
∙Themesh
∙Theversionofthesimulator
∙Thetypeofcomputer
CalibrationProblem
Highfrequencyeffectscauseaproblemwhenasourceisconnecteddirectlytothetestcircuit.
Twodifferentwavesareexcited:
∙Themodepropagatedalongthemicrostrip
∙Cylindricalsurfacewaves
Calibrationwasdesignedforremovinglocalparasiticeffectsfromsources,notforseparatingtwodifferentwaves.
Calibrationislessaccurateatveryhighfrequencies.
HighFrequencyCalibration
Lengthofcalibrationline=½
wavelengthateveryfrequencysimulated.
Theeffectoftwocalibrationlinesplacedbacktobackisdetermined.TheequivalenttoaTRLcalibrationisperformed.ThentheeffectofthecalibrationlinesisremovedfromtheMomentumresults.
Forlowfrequencies,½
wavelengthisverylong.Forlowfrequencies,staticshieldingeffectsaremoreimportantthanwaveeffects.
∙Calibrationlinelengthcanbelimitedtoafewtimesthesubstrateheight
∙Afewcellsareusedinsteadof½
wavelength.
Example:
for100umGaAs,fourcellsareusedbelow5GHz.
LowfrequencyCalibration
Atacertainlowfrequency,theequivalentcircuitforthetransmissionlineisdeterminedusinganRLGCmodel.
Belowthatfrequency,Z0isfoundbyextrapolatingtheZ0formulaforRLGC.
InternalPorts–interiorandedge
directexcitationpointfeed
directexcitation
linefeed
-1.i
DifferentialPort
EdgeMesh
Efficiency=accuracy/resources
Simulationaccuracyisafunctionofthemesh.
Generalrule:
densermeshesgivemoreaccurateresults
Resources:
memoryisproportionaltoN^2(Nisnumberofcellsinthemesh)
ThegoalofMomentumistogeneratethemostaccuratesolutionwiththefewestnumberofcells(theshortestsolutiontime).
Settingthenumberofcellsperwavelengthcontrolsthemesh.Inaddition,Momentumcanlookforstructuresthatappeartobetransmissionlines,andmeshthemwithauserdefinednumberofcellsperwidth.ItisrecommendedthattheTMLbeusedonlyfortransmissionlinegeometries.Itmaydeterioratethesimulationefficiencyforarbitrarycircuits.
Inanedgemesh,thesmallcellsalongtheedgeofthegeometryallowamoreaccuratemodelingofthesingularbehaviorofthecurrentneartheedges,hencemoreaccurateresults.
EdgeMeshAccuracy
Astudywasdoneofimpedanceaccuracyversuspercentofedgemesh.0%edgemeshisthesameasonecellacrossthewidthofthetransmissionline.33%edgemeshproduces3uniformsizecellsacrossthewidth.Thehighestaccuracyisachievedwithedgemeshbetween2%and12%.Anominalvalueof5%ischosenforMomentum.
EdgeMesh–Discretizationerrors
Thisillustrationshowsthephysicalcross-sectionalvariationofthecurrent.Theedgemeshistryingtomoreaccuratelyapproximatethecurrentcrowdingattheedges.
Longitudinalvariationofthecurrent,andit’sapproximationwithdiscretecells.
MiscellaneousTopics
Green’sfunctionsubstratecalculationtime
TheGreen’sfunctionsubstratecalculationcanbetimeconsumingwhentherearemanymetallizationlayersandvialayers.Forthecaseillustratedhere,9conductorlayersplus9viasproduce18*18(324)differenttypesofcoupling.Thesemustallbecomputedandstored.Thiscanbe324timesslowerthanthecalculationforasinglemicrostripmetalwithoutavia.
MoMvs.FEM
Analternativetothemethodofmomentstechnique(MoM)isthefiniteelementtechnique(FEM).
WithFEM,the3Dspaceisbrokeninto3Dtrianglescalledtetrahedra.Byapplyingtheboundaryconditions,theEfieldisdeterminedforallofthetetrahedraandtheS-parametersarecalculated.
Themethodofmomentsprodu
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