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SEPTEMBER2008 VOLUME56 NUMBER9 IETMAB (ISSN0018-9480) PAPERS LinearandNonlinearDeviceModeling CompactEmpiricalModelingofNonlinearDynamicThermalEffectsinElectronDevices ................................. ...................................................................... I.Melczarsky,J.A.Lonac,F.Filicori,andA.Santarelli 2017 SmartAntennas,PhasedArrays,andRadars APackagedMEMS-Based5-bit -BandHigh-Pass/Low-PassPhaseShifter........M.A.MortonandJ.Papapolymerou 2025 ActiveCircuits,SemiconductorDevices,andICs High-Power20–100-MHzLinearandEfficientPower-AmplifierDesign .... N.Sahan,M.E.Inal,S.Demir,andC.Toker 2032 A900-MHz29.5-dBm0.13- mCMOSHiVPPowerAmplifier .......................L.Wu,I.Dettmann,andM.Berroth 2040 A -Band Four-Element Phased-Array Front-End Receiver With Integrated Wilkinson Power Combiners in 0.18- m SiGeBiCMOSTechnology...................................................................... K.-J.KohandG.M.Rebeiz 2046 Millimeter-WaveandTerahertzTechnologies ApplyingMillimeter-WaveCorrelationRadiometrytotheDetectionofSelf-LuminousObjectsatCloseRange.......... ................................................................................................... J.A.NanzerandR.L.Rogers 2054 WirelessCommunicationSystems PerformanceAnalysisofSerialandParallelSix-PortModulators................................ B.LuoandM.Y.-W.Chia 2062 FieldAnalysisandGuidedWaves EfficientEvaluationofthe3-DPeriodicGreen’sFunctionThroughtheEwaldMethod ..................................... ....................................................................................... G.Lovat,P.Burghignoli,andR.Araneo 2069 CADAlgorithmsandNumericalTechniques Accurate Physical Modeling of Discretization Error in 1-D Perfectly Matched Layers Using Finite-Difference Time-DomainMethod .................................................................... J.M. López-VillegasandN.Vidal 2076 (ContentsContinuedonBackCover) (ContentsContinuedfromFrontCover) FiltersandMultiplexers ImprovedSynthesisfortheDesignofMicrowaveFiltersWithaMinimumInsertion-LossConfiguration ................. ................................................................................. A.Nasser,S.Bila,S.Verdeyme,andF.Seyfert 2086 Ultra-WidebandBandpassFilterUsingMultilayerLiquid-Crystal-PolymerTechnology ...... Z.-C.HaoandJ.-S.Hong 2095 MicrostripParallel-CoupledFiltersWithCascadeTrisectionandQuadrupletResponses .................................... .......................................................................................... J.-C.Lu,C.-K.Liao,andC.-Y.Chang 2101 Packaging,Interconnects,MCMs,Hybrids,andPassiveCircuitElements DevelopmentofThin-FilmLiquid-Crystal-PolymerSurface-MountPackagesfor -BandApplications ................ ......................................................................................... K.Aihara,M.J.Chen,andA.-V.Pham 2111 AnalyticalEvaluationofVia-PlateCapacitanceforMultilayerPrintedCircuitBoardsandPackages....................... ......................................................................Y.Zhang,J.Fan,G.Selli,M.Cocchini,andF.dePaulis 2118 InstrumentationandMeasurementTechniques AFastandAccurateAmplitude-OnlyTransmission-ReflectionMethodforComplexPermittivityDeterminationofLossy Materials ............................................................................................................ U.C.Hasar 2129 New Method for Noise-Parameter Measurement of a Mismatched Linear Two-Port Using Noise Power Wave Formalism................................................ D.Pasquet,E.Bourdel,S.Quintanel,T.Ravalet,andP.Houssin 2136 Microwave Photonics OpticalSpatialQuantizationforHigherPerformanceAnalog-to-DigitalConversion ........................................ .....................................................................M.Jarrahi,R.F.W.Pease,D.A.B.Miller,andT.H.Lee 2143 MEMSandAcousticWaveComponents AnAnalogRFMEMSSlotlineTrue-Time-DelayPhaseShifter ............. K.VanCaekenbergheandT.Vähä-Heikkilä 2151 Biological,Imaging,andMedicalApplications DevelopmentandLaboratoryTestingofaNoninvasiveIntracranialFocusedHyperthermiaSystem........................ .....................................................I.S.Karanasiou,K.T.Karathanasis,A.Garetsos,andN.K.Uzunoglu 2160 InformationforAuthors ............................................................................................................ 2172 IEEEMICROWAVETHEORYANDTECHNIQUESSOCIETY TheMicrowaveTheoryandTechniquesSocietyisanorganization,withintheframeworkoftheIEEE,ofmemberswithprincipalprofessionalinterestsinthefieldofmicrowavetheory andtechniques.AllmembersoftheIEEEareeligibleformembershipintheSocietyuponpaymentoftheannualSocietymembershipfeeof$14.00,plusanannualsubscriptionfee of$22.00peryearforelectronicmediaonlyor$44.00peryearforelectronicandprintmedia.Forinformationonjoining,writetotheIEEEattheaddressbelow.Membercopiesof Transactions/Journalsareforpersonaluseonly. 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DigitalObjectIdentifier10.1109/TMTT.2008.2005057 This article has been accepted for inclusion in a future issue of this journal. Content is final as presented, with the exception of pagination. IEEETRANSACTIONSONMICROWAVETHEORYANDTECHNIQUES 1 Compact Empirical Modeling of Nonlinear Dynamic Thermal Effects in Electron Devices IlanMelczarsky, JulioAndrés Lonac, FabioFilicori,and Alberto Santarelli,Member,IEEE Abstract—Anoriginalempiricalapproachtodealwithnonlinear The relationship between the junction3 temperature and the dynamicthermaleffectsinelectrondevicesisproposed.Thenew dissipatedpowerinanelectronicdeviceis,ingeneral,nonlinear technology-independentapproachisverycompactandeasytoim- andtypicallyexhibitsbothshort,aswellaslongdynamics.Non- plementincomputer-aideddesigntools.Therefore,itcanbeeasily linearity stems from the temperature-dependent thermal con- coupledwithelectricaldevicemodelsinordertoobtainaccurate ductance of semiconductors (especially evident in III–V com- electrothermalmodelsthataresuitablefornonconstant-envelope RFapplications(e.g.,pulsedradar).Modelequationsandidentifi- poundsemiconductors),whereasmemoryeffectsareduetothe cationproceduresarederivedinthispaper.Validationresultsand thermalcapacitance.Moreover,thecharacteristicdimensionsof comparison with simplified models are also presented both for a thedifferentsectionsofanelectronicdevice,andhence,theas- simulatedfield-effecttransistordevice,aswellasforarealhetero- sociatedthermalconstants,candifferbyseveralordersofmag- junctionbipolartransistordevice. nitude;infact,heatistypicallygeneratedinsideasubmicrom- Index Terms—Behavioral modeling, electrothermal, elec- eterchannelvolume,andflowsthroughasubstratethatcanbe trothermal effects, heterojunction bipolar transistors (HBTs), several hundreds micrometers thick to reach a package and/or intermodulation distortion, microwave field-effect transistors athermalheatsink,whichcanhavedimensionsintheorderof (FETs), modeling, nonlinear, self-heating, semiconductor device several millimeters. This causes a very complex dynamic be- thermalfactors,thermal,thermalimpedance,Volterraseries. haviorwithtimeconstantsthatcanrangefromafewhertz,up tohundredsofmegahertz.Forconstant-amplitudeoperationat RFfrequencies,onlythedcvalueofthejunctiontemperatureis I. INTRODUCTION importantsince,inmostcases,theperiodicsteady-statejunction temperaturecontainsonlya dc value.Inthis case,the thermal T HERMALeffectsplayafundamentalroleinthedesign, part of the electrothermal device model simply reduces to the measurement,andmodelingofhigh-power1electronde- nonlinear static characteristic that relates the dissipated power vices and circuits. In fact, since carrier concentration and mo- tothejunctiontemperature(i.e.,anonlinearthermalresistance). bility in semiconductor materials are strongly temperature de- Ontheotherhand,forpulsedradarorfornonconstant-envelope pendent,boththestaticanddynamicelectricalcharacteristicsof telecommunications applications, where the dissipated power electrondevicesareaffectedbythedevicejunctionorchannel2 waveform contains baseband spectral components below the temperature.Moreover,thereliabilityandlifetimeofelectron devicethermalcutoff,thethermalmodelingproblembecomes devices have also been shown to be strongly correlated with more complex due to the interaction of nonlinearity and dy- the junction temperature. For bipolar devices, in addition to namiceffects.Theuseofanoversimplifiedmodelintheafore- the aforementioned interactions, there is also the problem of mentionedapplications(e.g.,anonlinearmemorylessmodelor thermalinstabilityduetocurrent-gaincollapseandthermalrun- alineardynamicmodel)canleadtoawrongestimationofde- away.Therefore,accurateelectrothermalmodelsareneededin viceperformancessuchasoutputpower,efficiency,intermodu- ordertobeabletoproperlydesign,measure,andmodeltheelec- lationdistortion,etc. tricalperformancesofpowerdevicesandcircuits. In this context, only a few approaches have been proposed in order to model the dynamics of the junction temperature [1]–[8].Notwithstandingthefactthattheyhavebeenprovento ManuscriptreceivedDecember5,2007;revisedMay8,2008.Thisworkwas workwellinspecificapplications,theseapproachessufferfrom supported in part by the Italian Ministry of Research and Higher Education (MIUR)andbytheInformationSocietyTechnologiesProgramoftheEuro- limitationsthatmayprecludetheiruseinsomecircumstances. peanUnionundertheframeworkoftheNetworkofExcellenceTopAmplifier In fact, [1], being based on Ritz vector reduction techniques, ResearchGroupsinaEuropeanTeam(TARGET). completelyneglectsnonlineareffects,andtherefore,isaccuracy I.MelczarskyiswiththeDipartimentodiIngegneriaElettronica(DEIS),Uni- versityofBologna,40136Bologna,Italy(e-mail:[email protected]). might be poor when dealing with high power densities and/or J.A.Lonac,F.Filicori,andA.SantarelliarewithMicrowaveElectronicsfor with strongly nonlinear thermal conductivities. An improved Communications(MEC)Srl,40123Bologna,Italy. version of [1] has been proposed in [2], which somewhat DigitalObjectIdentifier10.1109/TMTT.2008.2001956 overcomes this problem by using the Kirchoff transformation 1Strictlyspeaking,thermaleffectsarealsoimportantbothforlow-powerde- [3].Themethod,however,stillhasthelimitationthatitcanbe vices,wheneverself-heatingphenomenaareimportant(e.g.,duetopoorheat dissipation),aswellasfordevicesoperatinginawiderangeofcasetempera- 3We acknowledge that the term junction (or channel) temperature can be tureconditions. somewhatambiguoussincethejunctionisactuallyavolumeratherthanasingle 2Intheremainderofthispaper,thetermsjunctiontemperatureandchannel point.However,thisproblemisovercomeincompactbehavioralmodelingap- temperaturewillbeusedindistinguishablysincetheyarecompletelyequivalent proachessuchastheonebeingproposed,wherean“average”equivalenttem- forthesakeofthismodelingapproach. peraturecanbenormallyconsidered. 0018-9480/$25.00©2008IEEE This article has been accepted for inclusion in a future issue of this journal. Content is final as presented, with the exception of pagination. 2 IEEETRANSACTIONSONMICROWAVETHEORYANDTECHNIQUES rigorously applied only to homogeneous materials [3]. More- Thus,inordertoderivethethermalmodel,westartfromthe overthemodelsproposedin[1]and[2]canonlybeidentified dissipated power versus junction temperature characteristic, on the basis of 3-D thermal simulations and have relatively which can be generally described by means of a nonlinear complex computer-aided design (CAD) implementations. In functional4 [5], an analytic thermal model based on a combination of the Kirchoff transformation and a transformed time variable (1) [4] was proposed. Although this approach works well for moderate nonlinearities, it may not be accurate enough for Equation(1)definesthedeviceinstantaneousdissipatedpower very temperature-dependent materials. Moreover, [5] being in terms of the case or backside temperature and the an analytical model, it requires the knowledge of the device present and past values of , which stands for the instanta- structure, which is not always available to the user. More neous junction temperature increment with respect to . The recently, in [6]–[8], it was proposed to use Hammerstein- and parameter representsthemaximummemorydurationasso- Wiener-type [9] behavioral models to account for nonlinear ciatedwiththisdescription.Fromnowon,andinordertosim- dynamicthermaleffects.Whereastheresultingmodelsareeasy plify the notation, we will sometimes explicitly omit the time to extract either from simulations or from measurements and dependenceof and ,aswellasthedependenceon of the reported results are good, the general applicability of the allthefunctionsandfunctionals.Bydefininganauxiliaryvari- approachmightbequestionablesincenophysicalortheoretical able ,asthedynamictemperatureincre- insightisgivenin[6]–[8]inordertojustifythevalidityofthe mentdeviation,itispossibletoexpandthenonlinearfunctional Hammerstein/Wiener approximations. On the other hand, the in(1)intermsofamodifieddynamic-deviation-basedVolterra model we are proposing overcomes these limitations since, as series[10] willbeshowninthefollowingsections:1)althoughitisbased onanempiricalapproach,themodelingapproximationscanbe justified on physical grounds; 2) it is extremely compact and canbeeasilyimplementedinCADtools;3)itcanbeidentified not only on the basis of 3-D thermal simulations, but also from static and differential quiescent-temperature-dependent thermalimpedancemeasurements;and4)itispredictionshave proventobe inverygoodagreement withrespecttoboth3-D finite-element method (FEM) thermal simulations, as well as measurements. The remainder of this paper is organized as follows.SectionIIisdevotedtomodelformulationandidenti- (2) fication.SectionIIIpresentsvalidationresultsandcomparisons For sufficiently low frequencies of the controlling variable with a simplified model both for a simulated FET device, as (i.e., for ), (1) can be approximated through wellasforameasuredheterojunctionbipolartransistor(HBT) a“heat-controlled”quasi-static(QS)formulation(analogousto device.Finally,someconclusionsaregivenSectionIV. theclassiccharge-controlledformulationforelectrondevices) II. THERMALMODEL (3) Accurately modeling nonlinear dynamic thermal behavior can be difficult due to the combination of nonlinearity and memory effects. However, the difficulty of the modeling task where isthestatic,powerversusjunctiontemperature canvarydramaticallydependingonthechoiceofthevariables increment characteristic, represents the equivalent of a for the description. In fact, if for the behavioral description thermalcapacitance,and representstheerrorassociated the dissipated power is chosen as the forcing variable and the with the QS approximation, which vanishes for junction temperature as the response (e.g., as in the case of a 5Equation(3) can bewritten ina moreconvenient form,by thermal impedance), the interaction between nonlinearity and definingathermalequivalentofelectriccharge(i.e.,theamount memory can be difficult to modelaccurately since, ingeneral, ofheatstoredinacertainvolume)inthefollowingway: thedissipatedpowerhasharmoniccontentfromdcuptoatleast (4) twicethefrequencyoftheRFsignal.Ifinsteadajunction-temperature-controlled formulation is chosen (i.e., temperature is the independent variable and power is the response, as in a where , and thermaladmittanceformulation),theproblemisstillnonlinear with memory, but memory effects can be expected to be less represents the devia- importantandcan,therefore,bereasonablyapproximatedusing a simple model as the one we are proposing. This claim can be intuitively justified by taking into account that due to the 4Afunctionalisareal-valuedfunctiononavectorspaceoffunctions thermal low-pass filtering characteristics of the material, the 5Notethatananalogexpressioncanbewrittenfor(cid:18) (t)intermsofP(t). However,inthatcase,aQSapproximationsuchas(3)wouldbemuchworse junction temperature varies much more slowly than does the since,owingtothelow-passfilteringactionofthermalcapacitance,P(t)hasa RFdissipatedpower. muchbroaderspectrumthan(cid:18) (t). This article has been accepted for inclusion in a future issue of this journal. Content is final as presented, with the exception of pagination. MELCZARSKYetal.:COMPACTEMPIRICALMODELINGOFNONLINEARDYNAMICTHERMALEFFECTSINELECTRONDEVICES 3 tions from the QS approximation. Moreover, and similarly to what is done in [11] to model nonquasi-static (NQS) effects in electron devices, can be conveniently expressed bymeans ofan equivalentdynamic temperaturevariation , appliedtothenonlinearheatfunction,inthefollowingway: (5) where Fig.1. Equivalent-circuitschematicoftheproposedmodel. (6) capabilities for mildly nonlinear systems independently of the isstillanonlinearfunctionalofthepastvaluesofthetempera- memory length, the truncated modified series can be used for tureincrementuptotime .Theonlyassumptionthatwas highly nonlinear systems with short memory, as well as for madein(5)inordertoexpresstheheatvariations byvirtue mildlynonlinearsystemswithlongmemory.Aswillbeshown of a temperatureperturbation ,applied tothe heat function inthefollowingsections,therelationshipbetweenthedissipated ,isthattheheatversustemperaturecharacteristicisinvert- powerandthejunctiontemperatureincrementinadevice,when ible.As in(1), canalsobeexpandedaround interms thetemperatureincrementisconsideredtobethe independent of a modified Volterra series by defining a dynamic tempera- variable,asin(1),fitswellintothelattercategory.Althoughin tureincrementdeviation .Moreover, theorytheconvolutionkernel in(7)dependson , consideringthelimitedbandwidthof ,itcanbereasonably wehaveempiricallyfoundthatwithoutagreatlossofaccuracy, assumedthat,innormalconditions, willbesmallenough it can be assumed to be temperature independent. The expres- sothatthemodifiedVolterraseriescanbetruncatedafterthefirst sionfor thenbecomes convolution integral without a significant error. For instance, consideringasinusoidal oftheform ,the dynamicdeviation willbesmallinbothofthefollowing cases:1)when variesslowly(i.e., ),irrespective of the amplitude or 2) when is small irrespective of .Withafewconsiderations,itiseasytoprovethattheformer reasoningcanbegeneralizedforanarbitrary .Therefore,it seemsreasonabletoadoptthesmall approximationand the resulting truncation of the series since, due to the intrinsic low-passnatureof ,forlowfrequencieswherethespectral contentof isimportant, willbemuchsmallerthan , whereasforhigherfrequencieswherethe approximation no longer holds, will still be small due to the smalleramplitudeoftheharmonicsof .Beyondanytheoret- ical considerations, the former assumption will be empirically (8) confirmedinSectionIII.Accordingly,weproposethefollowing expressionforthetemperatureincrementperturbation : where is the dc value of , and the symbol“ ”indicatestheconvolutionoperator.Thecancellation ofthesecondintegralin(8)occursbecause cannothave adcvaluesince,bydefinition,theNQScorrection must (7) bezeroindc.Thus,wehaveexpressedthetemperaturedevia- tionsin(8)bymeansofaconvolutionbetweenthetemperature increment and a linear impulse response , yet to be Equation (7) corresponds to the modified Volterra-series identified.Themodelequationsthenbecome expansion of the functional , truncated after the first convolution integral. The zeroth-order term has been omitted since obviously the NQS equivalent temperature correction must be identically zero under QS operating conditions. (9) It should also be noted that the dynamic-deviation Volterra serieshasverydifferentconvergencepropertieswithrespectto Fig. 1 shows an electrothermal equivalent circuit of the pro- theconventionalone[10]andthisaffectstheapproximationca- posed nonlinear dynamic thermal model, where as usual, the pabilities of the respective truncated version. In fact, whereas dissipatedpowerismodeledasacurrentandthejunctiontem- thetruncatedconventionalSeriesprovidesgoodapproximation peratureincrementandbaseplatetemperatureasvoltages. This article has been accepted for inclusion in a future issue of this journal. Content is final as presented, with the exception of pagination. 4 IEEETRANSACTIONSONMICROWAVETHEORYANDTECHNIQUES As can be seen from Fig. 1, for frequencies low enough to neglect any NQS memory effects, is negligible and the modelreducestoanonlinearQSmodelresultingfromthecom- bination of the dc nonlinear thermal resistance and the nonlinear thermal capacitance6 , as in (3) with (in dc just the static nonlinear characteristic remains). Moreover, nonlinear dynamic effects occurring at frequencies where the QS approximation is no longer valid are modeled through the voltage generator derivedin(6)and(7)andapproximatedby(8). A. ModelIdentificationandImplementation As can be seen in Fig. 1, the proposed model is completely Fig.2. Closeupofthedevicegeometryemployedforthe3-DFEMsimulations. defined by the two nonlinear static functions and , responsible for the QS behavior at low frequencies, where is the static incremental andbytheconvolutionkernel ,whichallowstoaccount thermal conductance. Equation (10) allows for the identifica- for NQS effects. The identification is, therefore, performed in tionof fromlow-frequencysmall-signalmeasurements twophases;thefirstphaseinvolvestheidentificationoftheQS or simulations of the -dependent thermal admittance of the partonthebasisofstaticandlow-frequencydifferentialthermal device. The NQS part of the model, , can also be deter- impedancemeasurementsandthesecondoneinvolvingthecal- minedfromthe -dependentthermaladmittancesprovidedthat culationof inordertoaccountforNQSbehavioroccurring thehigherfrequencies,wheretheNQSeffectsareimportant,are athigherfrequencies.Modelextractionbeginswiththeidenti- considered.Infact,byderivingthesmall-signalthermaladmit- fication of , which is simply the static versus tancefrom(9)at , characteristic for a given .7 Thenonlinear characteristiccan beobtainedeitherexperimentally,bymeasuringthesteady-state (11) junctiontemperatureeitherelectricallyorbyothermethods(see [13] and references therein), or by thermal simulations using where . From (11),the term can a FEM solver. If the geometry of the device and the thermal be identified on the basis of the small-signal high-frequency propertiesofthematerialsareknown,thelatternotonlyusually thermal admittance . The model depicted in Fig. 1 gives superior accuracy, but also allows to predict the thermal canbeeasilyimplementedinCADtools,aloneorasapartof properties of the device also including the entire heat sinking anelectrothermalmodel,usingsuitablelookuptablesand/oran- systemevenbeforeitisactuallyfabricated.Duetothesuperior alyticapproximatingfunctionsfor and . accuracy,wehavechosenthisapproachinordertoidentifyand preliminarilyvalidatethemodelinSectionIII-A.However,we III. MODELVALIDATION also performed model identification from and validation with A. SimulatedField-EffectTransistor(FET)Device realmeasurements,asdescribedinSectionIII-B. Inordertoidentifythethermalmodel,3-DFEMsimulations Once hasbeen identified, can be deter- wereperformedusingacommercialsolver.8Thedeviceconsid- mined on the basis of low-frequency small-signal thermal ad- eredwasa0.25- mGaAsFEThavingfourfingersand480 m mittance measurements or simulations performed at different ofgateperiphery.Thetopsideofeachfingeristhermallycon- quiescent conditions.Infact,forfrequencieslowenoughfor nected to the others through 4- m-thick gold air bridges. Un- thecharge-controlledQSapproximationgivenby(3)tobevalid, derneath the device there is a 100- m GaAs substrate on top and by considering a small-signal sinusoidal superimposed of a 25.4- m AuSn solder layer. Fig. 2 displays a closeup of onthequiescentvalueofthejunctiontemperatureincrement , thedevicegeometryunderneathonegatefinger.Typicalvalues the temperature-dependent low-frequency thermal admittance wereassignedtoeachmetallayer’snormalandlateralthermal canbewrittenas conductivities, whereas lookup-table-based nonlinear thermal (10) conductivitywasusedfortheGaAssubstrate.Typicalspecific heatanddensityvalueswerealsoassignedtoeachmaterial.The “case” or “backside” temperature was defined as a boundary 6Althoughanonlinearthermalcapacitancemightseemstrangefromaphys- icalpointofview,itspresenceintheproposedmodelcanbetheoreticallyjus- condition on the lower plate of the solder layer and was fixed tified.Infact,atlowfrequencies,themodelreducestoasingleRCcell,which to25 Cinallthesimulations. maynotinpracticebeassociatedwiththethermalconductivityandspecificheat Due to the geometric symmetry of the problem, only one- ofaparticularmaterialorsectionofthedevice(i.e.,itcancondensedifferent materialsandtimeconstants),andthus,C canturnouttobetemperaturede- quarter of the entire device was simulated. The geometry was pendent.Nevertheless,ourexperienceshowsthatbothforsimulated,aswellas meshedusinganonuniformgridofapproximately10000nodes forrealdevices,thevariationofthiscapacitancewiththejunctiontemperature givingapproximately15000brickelements.Fig.3showsa3-D is,inmostcases,modest. viewofthestructureincludingthemesh.Theheatsource,ascan 7Wehavemadetheassumptionthatthecasetemperatureisafixedboundary condition,andtherefore,actsasaparameterofthemodel.Ifneeded,itisclearly beseenfromFig.2,wasplacedinsidea0.25 0.06 60 m possibletoidentifyafamilyofthermalmodelseachforadifferentcasetemper- atureintherangeofoperation. 8ThermalAnalysisSystem,HarvardThermal,Harvard,MA. This article has been accepted for inclusion in a future issue of this journal. Content is final as presented, with the exception of pagination. MELCZARSKYetal.:COMPACTEMPIRICALMODELINGOFNONLINEARDYNAMICTHERMALEFFECTSINELECTRONDEVICES 5 Fig.4. Measured(o)andmodeled(—)junctiontemperatureincrementfora periodicpulseddissipatedpowerexcitation(T:12ms,dutycycle:1/3).For comparison the results obtained using a simple linear model (---) are also Fig.3. 3-DmeshemployedfortheFEMsimulations. displayed. cuboid located 0.1 m underneatheach gate and,accordingly, channeltemperaturewasdefinedastheaveragetemperaturein- sidethatvolume. Steady-state simulations were performed in order to obtain the static power versus junction temperature characteristic, whereas transient simulations were performed to derive the differentialthermaladmittancesatseveraljunctiontemperature quiescent conditions in the range 26 C to 190 C. In that range, due to the variation in the GaAs thermal conductivity, the incremental static thermal resistance varies from 138 to 220 K/W. The proposed model was then identified using (10) and (11) and implemented in a circuit simulation SW.9 The nonlinear resistor, nonlinear capacitor, and voltage-controlled Fig.5. Modelingerror(inabsolutevalue)betweenthe3-DFEMsimulated linear voltage generator in Fig. 1 were implemented using a junctiontemperatureincrementandboththeproposedmodel(—)andasimple symbolicallydefineddevice(SDD)andlookuptables. linearmodel(---)(T:5ms,dutycycle:20%). Large-signal transient simulations were then performed in order to validate the nonlinear dynamic thermal model. In all modelandasimplifiedlinearmodelthatareplottedinFig.4.As cases,apulsedlarge-amplitudedissipatedpowerwasemployed canbeseenfrombothofthesefigures,theproposednonlinear asaperiodicexcitationfortheFEMtime-domainsimulations. dynamicmodelpredictsverywellthejunctiontemperaturein- The type of excitation was chosen considering a hypothetical crementduringthefullperiodofthedissipatedpowerpulsewith radarapplication,inwhichthepoweramplifieroperatesinRF amaximumerrorofapproximately2.2K.Ontheotherhand,the pulsed mode of fixed or variable duration and duty cycle, de- useofasimplifiedlinearmodel,asshownbythesefigurescan pendingontheapplication.Comparisonswerethenperformed giveinaccurateresultswhenalarge-signalexcitationisconsid- between the junction temperature increment obtained with the ered; in this case, the maximum error in the prediction of the FEMsolverandtheonepredictedusingthenewthermalmodel junctiontemperatureisapproximately18K. for different types of pulsed excitations. As a typical example Wehavealsoperformedothertestsusingpulsedexcitations ofthemodelcapabilities,Fig.4showsthecomparisonbetween ofdifferentdutycyclesandfrequencies.Duetolackofspace, thejunctiontemperatureincrement obtainedwiththeFEM however,weonlyincludeinTableIasummaryofthemaximum, solverandtheoneobtainedusingourcompactthermalmodel. average,andrmsvalues( , ,and )oftheabsolute The examplecorresponds to a pulsed dissipatedpower excita- value of the error between the FEM simulation and both the tionhavingaperiodof5msandadutycycleof20%;thedis- proposedmodelandasimplifiedlinearmodel.Ascanbeseen sipated power during the “on” part of the cycle is 1.08 W and from Table I, the proposed nonlinear dynamic thermal model is7.2mWduringtherestoftheperiod.InFig.4,wehavealso givesaverygoodpredictionofthejunctiontemperatureovera includedtheresultsthatwouldhavebeenobtainedifaconven- broadrangeofoperatingconditions,whereasasimplifiedlinear tionallinearmodelhadbeenused. modelfails,inmostconditions,togiveanacceptableprediction Fig.5showstheabsolutevalueofthedifferencebetweenthe ofthejunctiontemperature. junction temperature increment obtained with the FEM solver As a further test, the proposed model was used to predict andtheonespredictedbyboththeproposednonlineardynamic thesmall-signalquiescent-dissipated-power-dependentthermal 9AdvancedDesignSystem(ADS),AgilentEEsof,PaloAlto,CA. impedance. As can be seen in Fig. 6, small-signal simulations This article has been accepted for inclusion in a future issue of this journal. Content is final as presented, with the exception of pagination. 6 IEEETRANSACTIONSONMICROWAVETHEORYANDTECHNIQUES TABLEI VALIDATIONRESULTSFOROTHERPULSEDEXCITATIONS Fig.7. Measured(symbol)andmodeled(—)magnitudeofthesmall-signal thermalimpedanceatdifferentdissipatedpowerquiescentconditions. ( :150,175,and200mW)andinafrequencyrangefromdc upto71kHzusingthemethodproposedin[12].Thesymbols inFig.7indicatethemeasuredthermalimpedances. The capacitor in Fig. 1 was then identified from the imaginarypartofthemeasuredthermaladmittanceatlowfre- quencies (from dc to 50 Hz) where a QS model composed of a nonlinear resistance and a nonlinear capacitor could be assumed.TheFouriertransform ofthevoltage-controlled voltage source inFig. 1was then identifiedfrom the high-fre- quencypartsofeachthermaladmittanceusing(11).Asassumed during the derivation of the model in Section II, it was found that wasonlymoderatelydependentonthequiescent junction temperature , and therefore, the value corre- Fig.6. Measured(symbol)andmodeled(—)magnitudeofthesmall-signal thermalimpedanceatdifferentdissipatedpowerquiescentconditions. spondingto C ( mW)wasadopted forthemodel. ThefullthermalmodelinFig.1wasthenimplementedinthe performedwiththeproposedmodelareverygoodapproxima- frameworkofaCADtoolforcircuitsimulationbymeansofan tions of the thermal impedance obtained with the 3-D FEM analyticapproximatingfunctionforthenonlinearconductance thermalsolver,overaverywiderangeofoperatingconditions. ,alookup-table-based nonlinear capacitance, and a linear frequency-domain-defined B. MeasuredHBTDevice voltage-controlledvoltagesource. Asafurthervalidationofthemodel,weemployedittoobtain The identified thermal model was then used to obtain the thelarge-signaldynamicjunctiontemperatureandthedifferen- values of the measured thermal admittances, as shown in tial quiescent-dissipated-power-dependent thermal impedance Fig. 7. As can be seen from Fig. 7, the proposed model is of a real device. The device was an InGaP/GaAs power HBT able to model the thermal admittances for different quiescent consisting of four emitter fingers each having 2 40 m of dissipatedpowers,confirmingthevalidityoftheassumptionof activearea.The100- m-thickdiewasbondedtoabrasscarrier atemperature-independent . (thickness mm) using 20 m of thermally and elec- As a last validation test, the model was used to predict the trically conductive epoxy. In order to identify the model, the junctiontemperatureofthedeviceinlarge-signaloperatingcon- static characteristic was measured using two dif- ditions. To this aim, a simple RF power amplifier operating at ferentmethods[13],[14]andfittingalinearmodelforthede- 1GHzwasimplemented,whichconsistedofthemodeledHBT pendenceofthethermalresistanceonthejunctiontemperature devicebiasedinaclass-Aoperatingcondition( mA, (i.e., ).Theobtainedthermal V). The source and load RF terminations were set resistancevariedfrom539to581K/Wfor intherange to 50 , and a significant amount of pulsed RF power was 117 C–154 C.Thefrequency-dependentdifferentialthermal applied in order to have a significant amount of self-heating impedanceofthedevicewasthenmeasuredforafixedbackside during the “off”part of the RF pulses temperature( C)atthree differentdissipatedpowers mW ms .Theevolutionofthejunction This article has been accepted for inclusion in a future issue of this journal. Content is final as presented, with the exception of pagination. MELCZARSKYetal.:COMPACTEMPIRICALMODELINGOFNONLINEARDYNAMICTHERMALEFFECTSINELECTRONDEVICES 7 tool. Whenever nonconstant envelope RF applications need to be simulated, the proposed model could be easily employed to accurately estimate the transient behavior of the junction temperatures, and thus to predict important device perfor- mancessuchaspeakjunctiontemperatureinpulsedoperation, intermodulationdistortion,gain,outputpower,etc. ACKNOWLEDGMENT The authors would like to thank Dr. R. Cignani, University ofBologna,Bologna,Italy,andDr.A.Musio,MicrowaveElec- tronicsforCommunications(MEC)Srl,Bologna,Italy,fortheir assistanceduringthemountingofthecharacterizeddevice. REFERENCES [1] R.Sommet,D.Lopez,andR.Quéré,“From3Dthermalsimulationof HBTdevicestotheirthermalmodelintegrationintocircuitsimulators viaRitzvectorreductiontechnique,”inProc.8thIntersoc.Thermal Thermomech.PhenomenaElectron.Syst.Conf.,SanDiego,CA,2002, Fig.8. 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As can and G.Ghione,“Considerations on theimpact of power devicedy- namicelectrothermalphenomenaonpoweramplifiernon-linearities,” be seen from Fig. 8, the proposed model very accurately pre- presentedattheTARGETWorkshop5—CircuitLevelLinearisation dictsthedynamicjunctiontemperaturewithamaximumerror Tech.,Manchester,U.K.,Sep.10,2006. of approximately 3.5 K, even for relatively large temperature [7] A.Angelini,F.Bonani,V.Camarchia,F.Cappelluti,S.D.Guerrieri, G.Ghione,andM.Pirola,“DynamicelectrothermalmodellingofGaN excursionswherenonlineardynamiceffectsareimportant.For HEMTsastoolfortechnologyassessment,”inProc.TARGETDays comparativepurposes,alsothepredictionsofaclassicallinear 2006,Frascati,Italy,2006,pp.59–62. model identified around 120 C of junction temperature have [8] V.Camarchia,F.Cappelluti,M.Pirola,S.D.Guerrieri,andG.Ghione, “Self-consistentelectrothermalmodelingofclassA,AB,andBpower beenincluded.Itcanbenoticedthatthestandardlinearmodel GaNHEMTsundermodulatedRFexcitation,”IEEETrans.Microw. givesratherinaccurateresultsfortheevolutionofthejunction TheoryTech.,vol.55,no.9,pp.1824–1831,Sep.2007. temperaturewithamaximumerrorof 10K. 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The latter was possible due to a judicious tanceofHBTs,”inProc.GAAS’05Conf.,Paris,France,Oct.2005,pp. choice of the modeling variables: in fact, due to the low-pass 197–200. filtering nature of the materials, a power versus temperature [13] I.Melczarsky,J.A.Lonac,andF.Filicori,“Electricalmeasurement of the junction temperature and thermal resistance of HBTs,” IEEE description yields less important NQS effects than the classic Microw.WirelessCompon.Lett.,vol.16,no.2,pp.78–80,Feb.2006. temperatureversuspowerdescriptiondoes.Theresultingmodel [14] S.P.Marsh,“Directextractiontechniquetoderivethejunctiontemper- is extremely compact, easy to identify on the basis of static atureofHBT’sunderhighself-heatingbiasconditions,”IEEETrans. Electron.Devices,vol.47,no.2,pp.288–291,Feb.2000. and small-signal thermal measurements or simulations, and [15] I.MelczarskyandJ.A.Lonac,“Characterizationofnonlineardynamic very easy to implement in any CAD circuit/system simulation self-heatinginHBTsthroughelectricalmeans,”,unpublished.